THETHEORY OF FUNDAMENTAL PROCESSES
ADVANCED BOOKCLASSICS David Pines, Series Editor Anderson, P.W., Basic Notions of Condensed Matter Physics Bethe H.and jackicv, K., Inte diate Quanwsn Mechnics, Third ErEitian Feynman, R., Photon- Wdron Interactions Feynman, R., Quantum Electrodynamics Feynman, R., Statistical Mechanics Feynman, R., The Theory of Fundamental Processes Nozikres, I?., Thew of frrterctaig-rgFemi Systenzs Pines, D,, The Many-BodyProblem Quigg, C., Gauge Theories of the Strong, Weak, and Electromngnetic Interactions
HEORY OF
RICHARD P. FEYNMAN late, California Institute of Technology
i
. I . /
A hifernher of the krseus Books Group
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Copyright @ 1961, 1998 by Wegviecv Press
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Westview Press is a mernher afthe Perseus Books Group Cover design by Suzmne Heiser
Editor's Foreword
Addison-Wesley's Frontiers in Physics series has, since 1961, made it ~ossiblefor leading physicis& to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-w ithout having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these informal accounts would be replaced by more formal counterparts-textbooks or monographs-as the cutting-edge topics they treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has not proven to be the case for a number of the volumes in the series: Many works have remained in print on an on-demand basis, while others have such intrinsic value that the physics community has urged us to extend their life span. The Advanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in Frontiers in Physics or its sister series, Lecture Notes and Suppkments in Physics, that continue to provide a unique account of a topic of lasting interest. And through a sizable printing, these classics will be made available at a comparatively modest cost to the reader. These notes on Richard Feynman's lectures at Cornelt on the Theory of Fundamental Processes were first published in 1961 as part of the first group of lecture note volumes to be included in the Frontiers in Physics series. As is the case with all of the Feynman lecture note volumes, the presentation in this work reflects his deep physical insight, the freshness and originality of his
vi
EDITOR'S FOREWORD
approach to understanding high energy physics, and the overall pedagogical wizardry of Richard Feynman. The notes provide both beginning students and experienced researchers with an invaluable introduction to fundamental processes in particle physics, and to Feynman's highly original approach to the topic, David Pines Urbana, Illinois December f 991
Preface
These are notes on a special series of lectures given during a visit to Come11 University in 1958. When lecturing to a student body different from the one at your own institution there is an irresistible temptation to cut comers, omit difficult details, and experiment with teaching methods. Any wounds to the students' development caused by the peculiar point of view will be left behind as someone else's responsibility to heal. That part of physics that we do understand today (electrodynamics, /3 decay, isotopic spin rules, strangeness) has a kind of simplicity which is often lost in the complex formulations believed to be necessary to ultimately understand the dynamics of strong interactions. To prepare oneself to be the theoretical physicist who will some day find the key to these strong interactions, it might be thought that a fbll knowledge of all these complicated formulations would be necessary. That may be so, but the exact opposite may also be so; it may be necessary to stay away from the comers where everyone else has already worked unsuccessfully. In any event, it is always a good idea to try to see how much or how little of our theoretical knowledge actually goes into the anatysis of those situations which have been experimentally checked. This is necessary to get a clearer idea of what is essential in our present knowledge and what can be changed without serious conflict with experiments. The theory of all those phenomena for which a more or less complete quantitative theov exists is described. There is one exception; the partial successes of dispersion theory in analyzing pion-nucleon scattering are omitted. This is mainly due to a lack of time; the course was given in 1959-1 960 at Cal Tech, for which these notes were used as a partial reference. There, dispersion theory
viii
PREFACE
and the estimation of cross sections by dominant poles were additional topics for which, unfortunately, no notes were made. These notes were made directly from the lectures at Cornell university by P. A. Carruthers and M. Nauenberg. Lectures 6 to 14 were originally written as a report for the Seond Conference on Peaceful Uses of Atomic Energy, Geneva, 1958. They have been edited and corrected by H.T. Yura.
R, I;: Feynman Pasadena, CaIifornia November 1961
Contents
t 2
3
4 5 6 7 8 9 10 11 12
13 14 15
16 17 18 19 20 21 22 23 24
Editor's Foreword Preface Review af the Principles of Quantum Mechanics Spin and Statistics Rotations and Angular Momentum Rules of Composition of AnmIar Momentum Relativity Electromagnetic and Fermi Couplings Fermi Couplings and the Failure of Parity Pion-Nucleon Coupling Strange Particles Some Consequences of Strangeness Strong Coupling Schemes Decay of Strange Particles The Question of a Unbersal Coupling Coe%cient Rules for Strangeness Changing Decays: Experinnen& Fundamental laws of Electrornagnetics and @+DecayCoupling Density of Final States The Propagator for Scalar Particles The Propagator in Configuration Space Particles of Spin 1 Virtual and Real Photons Problems Spin- 112 Particles Extension af Finite Mass Properties of the Four-Component Spinor
V
vii
1
1 11
19 23 29
33 38
43 48 51 55 661 64
67 73 18 83 88 95 102
206 112 118
CONTENTS
The Cornpton Egect Direct Pair Production by Muons Higher-Order Processes Seti-Energy of the Electron Quantum Etecrrudynamics Meson Theory Theory sf p Decay Properties of the PDecay Coupling Summary of the Course References Tabte of the Frndamcntat. Particles
T h e Theory of Funda ental Processes
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Revie of the Principles of Quantu Mechanics These lectures will cover all of physics, Since we believe that the b b v i o r of systems of many particles can trc? understood in Lems of the interactions of a small n u m b r of particles, we shall k concerned primarily with the l a t b r . Beariw in mind that the present theories need mdifications o r revision to account for observed phenomena, we shall want t;o consider the fom&tfon of quantum mechanics in their meet general form* This i s s o we eztn get some idea of the minimum assumptions (and a e f r chrzmckr) which we use to formulate those parts of the theory we use in dealing wi* the new phenomena of the strmge particles, A rough outline of the book fallows: First, we discuss the ideas of quantum mecbnics, mainly the concept of amplitudes, emphsizing that other thiws such a s the com'bimtion laws of a q u l a r momenh a r e largely consequences of this concept. Next, briefly, relativity and the idea of antipartleles, Following tMs, we give a camplek qlualihtiv@deseriptiorr of all, the h o w n particles and all that i s k n o w a b u t the couplings h t w e n them. A f k r t h t , we return to a delailed qluwGlative stndy of the two couplings far wMch calculations can k carried out today; namely, the B-decay coupling and the electrowgnetic coupliq. The stucfy. of the latter i s called quantum electrodynamics, and we s b l l spend most of our time with it, Accordingly, we b g i n with a review of the primiplea of quantum meehnfcs, T t has k e n found t k t all proceases ao f a r observed can be understood in, b r m s of the fallowiw prescription: To every Process there corresponds an anzfililudet; with. proper normalization the probability of the process is equal to the absolute square of thia amplitude. The precise meaning of terms will become more clear frorn the emmples t b t follow. Later we s h l 1 find rules for calculating amplitudes. First, we consider in deLail t b double-slit eweriment for electrons, A uniform beam of electrons of momentum p i ~ incident l on the double sfit. To be more precise, we consider successive electrow, randomly distributed in the vertical direction (we prepare each electron with p = p,, = p, = 0). { F g y ~ m n :They should come frorn a hole, at deffnik anerm.)
t~ complex aumber.
2
THE THEORY O F FUNDAMENTAL PROCESSES
When the electron hits the screen we record the position of the fait. The process eonsidered i s thecs: An electron with well-defined momentum somehow goes through the slit ~y~dfEm a d make6 its way t-0 the screen, (Fig, 1-2). Now we a r e not allowed to ask which slit the electron went through unless electrons
# light
souree
slit
sereen
we actually set up a device to dekrmine w b t h e r o r not it did, Bgt. Elzm we ulcruld be emsz'deriw a different P ~ O G E ? IS SHowever we can, relate, the am plitude of the considered process Lo the? sc?parate
[email protected]%rdesf o r the electron to h v e gone through sfit (I), (ai), and through slit (21, [a2).[For example, when slit (%) i s closed the amplitude for the electron ts hit the sereen i s a t (prob. \all2)etc.1 Nature gives the following simple rule: a = at + al. This i s a s p e i a l case of the principle of supernosition in qumturn mechmics (cf, reference I), Thus the probabilily of an electron reaching the screen i s Pa = la/' = lal + a2/2.Clearly, in general we have P, @ Pa$+P,,(PaI = /allr,
Pap= larj2), a s distinguished from the classical case. We speak of "interfeience" between the probabilities (see reference 2). The actual form of P, i s familiar from optics, Mow suppose we place a lighL souree between slits 1 and 2 (see Fig, 1-1) to find out which slit tXle electron u'really'p did ggo tfirowh (we observe the scattered photon). In this case the interference pattern becomes identical to that of the two slits considered independently. h e way of intewreting this situation is to say that the act of measurement, of the position of the electron imparts an uncertainty in the momentum (APY), a t the same time changing the phase of the amplitude in an uncontrollable way, s o that the average over many electrons yields zero for the "interference" terms, owing to the randomness of the uncontrollable phases (see ohm' for details of this view). However, WC? p r e k r the following viewoint: By looking a t thc3 electrons we have actually c h a q e d the process under consideration. Now we must eonsider the photon and its interaction with the electron. So we consider the following amplitudes: ail = amplitude that electron came through slit 1 and the photon was scatbred bebind slit 1.
azl = amplitude that electron came through slit 2 and the photon was s c a t b r e d &hind slit I afz = amplitude that electron came through slit I and the photon was scattered beMnd slit 2 a2, = amplitude that electron came through slit 2 and the photon waa scattered behind slit 2 The amplitude tlzat an electron seen at slit 1 arrives a t the screen i s therefore a' = ali + azi; for an electron seen a t slit 2, a" = aiz + azp Eviso that a t l at, dently f o r a properly d e s i ~ e dexperiment aiz cs 0 az2 a az of the previous experiment. Now the amplitudes a' and a" correspond to different processes, s o the probability of an electron arriving at the screen i s P: = 1%' j2 + /a"i2 = [%I2 + 1a2I2. Another emmple is neutron scattering from crystals. (1)Ignore spin: At the observation point the total amplitude eqilials the sm of the amplitudes for ~ c a t t e r l efrorn each atom, 0 ~ gets e the usual Bragg pattern. (2) Spin effects: Suppose all atoms have spin up, the neutrons spin down (arssme the atom spins a r e localized): (a) no spin. flip-as before, (b) spin flip-na diffraction. pattern shorn even tboxfi the energy and wavelexzgths orE the scattered waves a r e the 883118 a8 in case a, The reason for this is simply that the atom wMeh did the scatteriw h s its spin flipped down; in grineiple we can d i s t i x a s h t it from the other atoms, In this ease the scatteriw from atom i is a different process frorn the scattering by atom j s i. If instead of flocalized) spin flip of the atom we excite (unlocalized) spin waves with vvaverruber k = kkc ksCatt, we can agdm exwct some part;ial diffraction effects, Conaider scatteriw a t 90" in the c.m, system [see Fig. f -2: (a to d)): (a) Two identical spinless particles: There a r e two indistingdshable ways f o r scatter to occur. Here, total amplitude = 2a and P = 4 /&I2,which i s twice what we expected classica2ly.
--
amp, a
amp, a
(b) Two dietinguj;skble s p i d e s s particles. Here these processes a r e dietinguishable, so that P = la/' + /aI2= 2 (c) Two electrons with epin. Mere these processes a r e distiwuishable, s o that P = /a12+ = 2 /a/'.
T H E THEORY" O F FUPJDAMIENTAL, P R O C E S S E S
amp. a
n a,
amp. a
amp. a
amp. -.a
amp.
(d) But if both the incident electrons have spin up, the processes a r e indistiwuishble, The total. amplitude = a a =: 0, 8s here we h v e a new feature. We diseuas tMs further in the mxt lecture*
-
P~oblem2 - 1 : Suppose we have two sources of r d i o w w e s (e.g.t radio stars) and need to h a w how f a r aparl; they are, We m w s u r e this intensity in two receiver@at th@same time m d r e c o d the product of the intensities as a, f w c t i m of their rehtive position, This measurement of the correlation permits the required distanae to be computed, With one receiver there is no pattern on the average, because the relative phase of A, and B sources is rartdom and ffuctuatiw. For example, in Fig. 1-3 we have put the recefvers a t a sepa-
REVIEW OF PRINCIPLES
r m o n correspading to that of Wo maxiTna of the gaeterzl ff the relative. phase is O mble 1-1). E L aad R arc3 at separation between a m & m w and a mirzirnu w e have Table 1-2, Thus find the probability of reception of photon coincidence in tbe eomters. Examine the eEfizct of thawing tbe separation btween the reeaivers, Consider the woeess from the point of view of qwtatum mecbrries.
Relative phases of sources
L
ft
(common)
(ma)
Belative phase@of aources
L (common)
(mu)
Produet
R
Product
6
THE T H E O R Y O F FUPJDAMENTAL PROCESSES
Discwsion of Problem 1-1, There a r e four ways in wMch we can b v e photon coincidences: (I) Both photons come from A: amp, a$. (2) Both photons come from B: amp. a2. (3) Receiver L receives photon from A, R from B: amp, as. (4) Receiver L receives photon frorn B, R frorn A: amp. a,. Processes (1) and (2) are d i s t i w u i s b b l e from each other and from (3) and (4). However, (3) aPrd (4) a r e indistingui~hable. [For instance, we could, in principle, measure the energy content of the emitters to find which bad emitted the photon in case (I) and (23.1 Thus, P = la1j2 + ja212+ [a3+ a,]? The term /a3 + &, I 2 contains the interference effects. Note that if we were examining; electrons instead of photons the latter term would be la3 a,I2.
-
W e shodd learn to think dilrsctly in terms of q u a n t m mecfiad~s. The only thing mysterioue is why we must add the amplitudes, and the rule that P = /total amp.12 for a specific process. We return to consider the rules f o r adding amplitudes when the two alternative processes involve exchange of the two pafiieles* Conaider a process P (amp. a) and the exchnge proeess Pe, (aknp* )a,, findistiwishable f ~ $0. m We find the h l l o w l ~rernarbble r d e in nature: For one elass of particles (calbd bsonsf the total amplitude is a + a,,; for another class ffermiona) the total amplitude is a -aa, It t w n s out that partia r e fermions, and pasticles with spin Q, 1, 2, cles with spin 1/22, 3b, a r e bosons, This is deducible from qmntum meehadcs p l w relativity plus something else. This ie discussed in the literature by pauli4 and, more recently, by LUders and ~umino." It is important to notice that, for this sefneme to work, we must b o w all the? state8 of wMch the p a ~ i c l e(or system) is eapble?, For example, if we did not h o w a b o a polarization we would not d e r s t a a d the lack of intesference f o r dgferent polarizatians, If we discovered a failure; of any of our laws fe'g., f o r some new p a ~ i c l e we ) would look for ~ o m new e degree of freedom to completely ~ p ~ i the? f y r~hk. Degenera~y.Consider a beam of light polarized in a. given direction, Suppost?we put the axis of an malyzer (e,g., polaraid, nical priam) successively in two perpndiclxfar directions, x and y, to measure the n m b e r of photons of corresponding pokrizrrLion in the beam (x and y a r e of course perpendicular to the direction of the beam). Calf,the amplituxfe for the a r rival of a photon with ~ l a r i z a t i o nin the x direction a,, in the y direction a,. Now, if we rotate the analyzer 45", what is the amplitude a,,. for a r + ay); for rival of a photon in that direction? W e find that ar6o = (I/ a general angle 8 (from the x =is) we have a(8) = cog 8a, +- sin 8 a, The point is that only two numbers mere a, and ay) a r e required to swcify the amplitude for any polarization. etate. We ahall find thia r e s d t to be connected in;timately with the fact that any other choice of u e s is equally valid for the dc?eeription of the photon,
.,.
.,,
a)(%, .
T H E TIEXEORY O F IFUHDAMENTAL, P R O C E S S E S
LI x" FIG,2-1 For example (Fig, 2-1) consider the aystem of' coordinates x', y' rotawd -45" with reswot to (x@y),An obsemer using tMs reference frame has a'%? = (a,
-
= (a, +
%)/a
=
a, (as f t shodd be!)
We could represent the a t e of the Moton by a vector e = a,i -t. ay) in some two-dirnensiomf space, Then the amplitude for f&@ photon to bo fowd with polarization in direction v = icos 4 + J sin B is e e v . The hypothesis that the behavior of a system eamot dep111.don the orientation in spctce inapses great reslriction~on the propertiees: of the possible states, Consider (Fig, 2-24) a. xlucleus o r an atom wMeh emit8 a y ray preferably along the z W s . Now rotate e v e q t u ~ nucleus , plus detectiw appamtua, We ~ h o d dexpect that the phaton is emitted in the eorrespondixxg direction. Xf the nuclew could be characterized by a single amplitttdcs, Bay, its energy, then the y ray wodd lave to be emilted with equal likelibmd i n all dfrections. m y 3 Because othertvise ws could set tMngs up sa that the y ray comes o-a%in the x direction (for we em always rotate thf?;apparatus, the worMw system; and the laws of physics do not d e p n d on the &reation of the axis), Th18 i ~ &3 differ@&condition because tbs slxb~equentphenomsnon ( y enai~sion)ia predicted dgfercilrnlly, me amplitude far our state cannot yield two pred;fctlons. The system must be d e s c ~ b e dby more ampli-
very ssbrp we need a large n m b e r of ampltudes to c h s r a e t e ~ z the e state of the nucfeua. Suppocse &ere a r e emctly n amplitudea which; deseribe a syatem
t d e s . If the a w u l a r distribution is
How the problem: S u p p s e we h o w it is in the st&@ai = I, a2 a, = 0, N t e r rotation what a r e ths amplltudelsir chracterizing the system in the new coodimtes ? We define them aa a = * =
Similarly if it starts in the a h t e
= 1, as =
= * * * = a, = O, we have
Therefore we need an entire matrFx Dfj(R). A, more 60mpUcated C ~ B @ OCCtltrs If idt;ially the system i s in a state
THE THEORY OF FUNDAMENTAL PROCESSES
After the rotation the new state ia
whereas a1 = f: D t j (R)ap Think about; why Ws is so. J
3
Rotations and Angular
3Cn the last lecture we spoke about an apparatus t h t p r d u c e d an object in condition a:
This requires f u d h e r mpfanation; since we have introduced so f a r only the ooneept of an a m p x t d e for the complete event: the production md detection of the object, This amplitude ean be obtained a s follows: We assume that we have an amplitude bi t h t the object prociuced i s in. same condition c h r a c t e r i z e d by the index i,If it is in this condition, i, let a i be the amplitude that i t will activate some dekctor, Then the amplitude for the complete event (produetion and detc3ctimf i s a l b3, summed over the possible intermediate emditions i. Consider again the expedmenl of an electron paa3sing~through two glits
(Fig. 3-1). E z+3 ie the ampEtude for an electron to go thsowh one slit and the amplitude for an electron a t this slit t~ reaah ths screen a t 2, then the amafitde for the complete event; i s the product a*,, x aae2. N m rotate the apparatus t3_zro~hR()RJ= angle of rotation, WIRI = axis
12
T H E THEORY OF FUNDAMENTAL PROCESSES
of robtion) s o that the object fixed detector
18
prdueed in condi_Llion a' with respect to the
\
We h v e p i n t & out that We3 must be rebted to the a by an. eqwtion of the form d = B(IR;)a,where the matrix Dm) doea not depend on the particular piece of app'aratw, Xlr another exwriment (Fig, 3-23) we could have the same abject prduced in some other conditions b and b', Then b' = I)(Et;)b, and
FIG, 3-2
the same D(B) is expcted, m y mu& W s relation b linear? %cause &j e ~ t acan be made to i n t e ~ e r e ,Suppose we have Wo pieces of agparatura, one produciw an object in conditian a, the other p r a d u ~ i wthe same abject in condition k, and together produciw it in condl.t-ion a + b, After rotation we would have a', W , a d also at + If, and alao a' + b', in. a d e r that the interference phrtmamena a p p a r the came way In the rotated system, Then we have but (a+ b)@= at + b", &erefore B(R)(a+ b) = B(RL)a tQ(R)b. What else can ws deduce 3 Buppase we consider the apwratus that we rotated through R as a new appratus, which p r d u c e s the abject fn eonation a', Mow wa rotate it LBroug-h 8, a s shown in. Fig, 3-3, Aacczrding to our rule, the object ia now produced in a condition a@,where an = D(S) at. Since a' = R(R)a, we have a@= r>(@D(R)a,whiah means D(SR) = D(S)D(R),P Rotations form a group, and the f)% ere m a t r k reprefsentations of this group. X t is by no means sex-evident how to find them. , e a t pmve that the amplitudes afLer robtion tStrfctly ~ p e a k h g we must be the eame in b t h cases; only the equarea must be the same, The b w e v e r , VViwer k s ~hclwnWt amplitudes could the D%. it could alwayes b
ROTATIONS AND ANGULAR M O M E N T U M
Ex~mptes: (I) An object represented by a r~hgiecomplex n u m k r . The 3)'s arc? I x 1 matrices, lee,, a complex number can be chosen do be 1.
(2) An object represenkd by a veclor, henee by three amplitudes, the x, y,z components of the veehr. The D% sare the familiar
matrices relating rodabd coodinaks, Let us now go ta the general analysis, Suppose we h o w a matrix for an idinitesfrrral rotation. %y, Lhs r a t i o n of 11" about the z ass. Then tfie rotation n" about f%ez =is is repretsented by
More generafly, if we h o w D(G"arowd z), then
D ( B around z) = [D( E & r o d z)] @/. Haw, if we r o a t e just; a litcGfe we have approAmately the identity, so to first order in E , D(E around 2;) == 1 + if;M,, Also,
D(E around y) = X + i E My
-
Now, we have D(B around z) = (1+ IEM,)@/@and using the binomial expamion, one obhins, when X, 0,
whloh ia often written eieMz. The binomial expansion works, since M, behaves like orelinary numb@~& m d e r addition atad multipfication, E we want to rotate E about an a ~ slow s the wit vector v, we find
THE THEORY O F F U N D A M E N T A L PROCESSES
and for a f i n i h 19 about v,
D(B around v) = exp [iB(v,M, + yYMY+
V,
M,)]
But now we must be careful about tbe relative order of M,, N&y, alld M, in the matrix products that appear in the series; these matrices do not commute This follows from the fact elrat f i n i b rotations do not eornmutt3, Consider the r o b t i o n of an e r a s e r , Fig, 3-4 (a and b). (1)Rota& it 90" about the z =is and then 90" about Lhe x =is (Fig, $-.la); (2) rotaate i t 90" about the x axis, and then 90" about the z axis (Fig. 3-4b); axld we get two entirely different results,
FIG. 3-4b
Let us discover the commutation relations b t w e e n Mx and ilk$, We eonaider a rotation E about the x s i s , hllowed by q about the y axis, then ---L: about the x axis and -q a b u t the y =is as In Fig. 3-45. We fallow the motion of a point s h r t i w on the y axis, Clearly the result i s a second-order effect. It ends up Just displaced by about EV tom& the x
ROTATIONS AND ANGULAR M O M E N T U M
axis. We no& a l s o that a paint which sLitrti3 on the z axis returns to the origin, and t b r e f o r t ; the net displacement of the point on tbe sphere is just a roLation by an angle ~q a b u t the z axis. Keeping k r m s up to tbe second order,
we h v f ?
Collecting coefff'fieien& of G q we find
Similarly,
.
These a r e the rules of eommztation f o r the matrices M,, My, and M, Every thing e l s e can be derived from these rules. Haw this i s done is given in d e b i l in. many books (e,g,, SehiE), We give only a bare outline here. F i r s t we grove that M: + &$ + M: = M' commutes with all M8s. Then we can choose our a's s o that they satisfy M2a = ka, where k is some number. Construct
and no%
THE T H E O R Y OF F U N D A M E N T A L P R O C E S S E S
16
MOW,
suppose
am)
satisfies
where m 1s another number; then
=
(m
- 1)b
Therefore,
We normalize atm3to unity; lee..
where M, = M, +
iq.Now
and ~ 2 ~ [ . ( "= , ka"
""
Therefore c = lk
- m(m - l)llh
Let rn = --j be Lihs "hast'bstak. How can we fail. to get mother if we aperate by 24-7 Only if ~ - a ' - j ) = 0 o r c = O for m = -1, 80 k = -j(-j 1) = j(j 3.1).
-
ROTATIONS AND ANGULAR MOMENTUM
1'7
The same kind of s k p s (using M, , wbich rafseaa m, by one, just like &Ilowars it) prove dkt ff the largeet value of rn is + j ",then k = j e l l p + +), s~ that j = 1'. Henee 21' i s an fnlger. The total n u m b r of s h h s i s 2j t 1.
Bzamples: (1) 1 sbb: j = 0 (2) 3 stabs: f = f.
(3) 2 s h b s : j = 1/2, This ia a very in;%restingcase. Let
Uairxg- our general reeulb we obhin
T H E T H E O R Y Of" F U N D A M E N T A L P R O C E S S E S
Similarly we can show that
s o that
tve
can write
The above expressions a l s o serve as the definition of the three i m p o r k n t 2 x 2 matrices, the Pauli matrices ux, cY, cz. Check also that t~: = o; = o$ = 1, D ~ % = -% txX = icZ The main point of this is, that i t all came out of nothing: t h t m t u r e b s no preferred axis and the nature of the principle of superposition were the only assumptions invoked. However, we have made a very. imporbfit h s o t h e s i s : We have assumed that the processers- of production and deection a r e well separated md that in beween one can taf k of an m p l i l u d e that characterizes the object. This hypot;hesis bas always h e n made (particularly in field theory) no nnat%r how small the d i s t m c e &tureen the apparatus and the d e b c t o r . I t may turn out that it i s not v d i d if these a r e too close together. Another i m p o r b n t assumption was to disregard any dynamic i n b r f e r ence: There am no forces b t w e e n our producing and measuring apparatus ad least that a r e not d e s c r i h b l e by transfer of our object between them, An amplitude f o r two independent events 1s then a l s o the prtlduct of the amplitude f o r each s e p a r a k event, Laok a t the example of the turo a k r s A, B and the counters X, U (Fig. 3-6). If ag+x i s the ampliLude f o r the photon emitted at B to maeh counter X and anty is the corresponding amplitude f o r the photon emitted a t A to reach counter Y, then a = a~ +, x ahty is the amplitude f o r occurrence of both events,
.
FIG. 3-6
4
Rules of position of Angular
A spin 21;! st;tb i s characterized by two amplitudes, In general a = a,(1/2)
+ a-(-1/21 where (1/2) s h n d s for For i n s m c e , the solution of
corresponding to spin up along the x axis i s
*
Also, down in x, ( 1 / 6 )(1/2) - (1/fi)(-1/2): up in J', (l/fi)(1/2) filG) shown down in y, ( 1 / 6 ) ( 1 / 2 ) (i/a)(-1/21. In fact, it can t k t every s k t e represenl.s spin in some direction. Any system that has two complex n u m b r s &is an analogy In spin X/2. F o r i n s a c s let us consider the polarizatfon of fight, Let x, polarization b spin up md y palarfzation be spin dam along an axis El in a "crazy" threedimenfiifonal space, The ather two axes we l a k l 5 and 77. Then spin up along lj = 45" polarization; d a m , 6 = -45" pohrization; up, q = RHC: (righthand circular polarization); down, q = LHG (left-hand circular pslarization), If we draw a unit sphere cenbred a t the origin of this space (Fig, 8-11,every state of polarization i s represenkd by a point on it. A general direction corresponds to elliptical polarization, Passingl light through a 114-wve plate i s a ce&in roktion, The cornsetion &tween the polarization of light and direc-t;lon in a three-dimensional space was exprocesses, ploited long ago by SLokes, Xt i s very useful -Cr, undersand ce*in for emmple, masers, (The m s e r ia a device using a sys%m, the ammonia molecule, makhg transitions &tween two s b t e s under the idluence of elect r i c fieMs. Ite analysis can be more easily understood by representing tha @Lateof the ammonia molecule a t m y Lime as a direction in same threedimexlaioml space, analogous to the ordimry space for a spin-2/2electron.)
x (-1/2);
-
THE T H E O R Y Q F FUNDAMErJTAI, PROCESSES
Rules of Comporit;lon ol" p r d u e e s two particles which we label A m d If. Suppose padlcle A t . 1 spin ~ 1and exists in three s k k s with m = +1, 0, --X; grid that pardicle B has spin 112 and exisLs in two a b b a with m = +1/2 and -112. F o r each of A's .three g a t e s , B em k v e two, s o there are six po~slbI88etes of the two pa&icles bgether. We may be thinung of an electron revolving around a nucleus. How do we cbracktxl.ize t b combined aystem l We b v s matrices M A md Mg wbich operaw on the s l a b s +A and Jig. Then
M,
Mz, The sht;sa of the combined s y s k m a r e given in Table 4-1, There are eix states d one codd jurnp to tb conclusion that j = 512. Wowever, tbre is no value of rn = * 512 and a h a rn = k 112 appears twice, + M B)' h88 ~ W OV B ~ U ~for S 1: Actually M' = =
M Z A
+
t More precisely M, = MZAle + M I A md B. matrices for s b b ~
IA where IA and IB a r e the identity
RULES O F COMPOSITION O F ANGULAR MOMENTUM
21
Clearly the st&& j = 312, rn = 3/2 is (+ 1)(1/2). But which aLak corresponds to j = 312, rn = 1/2? Recall
Then
and
The e h h (1/2, 1/21 is obkined by formisg the l i m a r combination of {0)(1/2) and (I)(-1/21], *ich ie o f i ~ g o m lto (312, 1/22), We ohkin t3343 msulls given in Table 4-2.
312 1/2
(1)(1/2) (2/fi)(0)(1/2) + t1/&)(1)(-1/21 (2/G][@)(-1/2) + t-1)(1/2) (-I)(-1/21
-1/2 -3/2
-
t l / G l ( @ l ( l / 2 ) (2/5)t1)(-1/2) -(1/6j(0)(-1/2)
+ 42/6)(-1)(1/2)
More examples: Add two spin = 112 states {Table 4-3) under exchnge of spins. How add two 8pin = 1 states (Table 4-4). For the additim of t;urs equal TABLE 4-3
rn
j =1
j = O
(symmetrical)
(antisymmetrical)
-
TABLE 4-4 1x2
IM
(symmetrical)
I>(+ 1) (1/fil[(+l)(O) + (O)(+ 1)l O ~l/G)[(+%)(-1) + (-1){+1) + 2(0)(0)1 -1 [l/K)!(-l)(0) + (@)(-111 2 1
-2
rn
j =1 (antisymmetricat)
(I-
(1/Kli (+ I)(@)- to)(+ 111 (1/fi) i(+ I)(-1) - (-I)(+ 1) (l/Gl (O}(-I)
- [-l)(O)l
(-l)(-l) j=0
(symmetrical)
2
angular momentum the biggest stab ia symmetric, the next antisymmelrie, and so on,
mobtern 4-2: Consider the adation of three spin = 1 angular mot total angular n-mmen&. Find the compteLeEy symmetric stabs. men& occur?
Relativity You a r e all hnnifiar Mth the b r e n t z transformation. For motion along tion eqwtions beween two Lorent%fmmeatl are the z axis the transfo
tt = ( t - v z ) / ( ~ - v l ~ =/ ~t cosh u-z sinh u
where we have put e = 1 and introduced the quantiv u (called "rapidity" by &he experts l )
Note the eqdvatence of the second farm of,the transformation e w t i o w to a rotation throu@ aome lma@mry angle. For tran~formationsIn f i r s same areetion rapi&tiea are addiave, i.e,, if Lhe rapi&ty &Ween syatema f, and 2 is u, btween 2 and 3 v, then the trmsformalion. from system 1 to 3 i s charactefized by the ~rapi&lyw = ct + v. Transformations for &ffereIlt d2reclions do not commute. The set of all brentz, tranf~formationa(including rotations) form a group. Problem 5-1: Suppose them exists an object with spin 1/23 in three dlmenoiona; o r conaider a mare-generaf stah d e s c r i b d by
t happns to a mdier h r e n t z tranaformtion~? t h a p w n ~mder r&aaon is the same ae b f o r s . If you have time, a180 cowider the probkm of normrtllzation,
- -
Recall that the quantity t2 -2 y2 $ is invariant. We introduce the following notation: x,, is the vector with components x4= t, xg = x, xg = y, xa = z.
THE THEORY OF F U N D A M E N T A L PROCESSES
24
If the vector ap = (ac,at, a2, aS) transforme like x, then we call a u a fourvector, For example p, = (E,p) is a four-vector, with
-
E' = (E VP,)/(X
- S)'12,etc.
The four-dimensional invariant scalar product of two four-vectors aP,
b p sis
I n t r d u c e the quantity
Note that 6p, ay = a p eAlso we define
A useful invariant is p * p= E~ - p e p = m2.The skillful use of invariants in calculations ofkn avoids maWng a Loren& trmsformstion. As a simple example, we c o n ~ i d e rp-p fscattering: t i s the minimum energy necessary to produce a proton-antiproton pair?
(EtP)
P
P*
"P
-
(M,O) *
P
hfore (in the laboratov frame) d b r (in the cen&r-of-mass frame)
-
We hsve g2 pa = ( 4 ~ ) ' . Hence (E+ MI' pZ = 16 M', giving E = 7M. Thus the necessary Irinetie energy is 6 M w 5.6 b v , Wer,vee1 Ure know that a pzkrtiefe with energy-momentum pB lzas asaociated a wave p = u exp (-ipl,x(,) = u exp [-i(Et - p a x $ ] . It is at once apparent that the pham of the wave is invariant under Lorentz transformation. This was, in fact, bow b B r o g l i e fomd the relation h w e e n energy momentum and wavelength frequency. Probllexn 5-1 w8 to find how u transforme. Hate that
R E LATIVITU
25
Poeltlve 8nd Negative Energies. The equation E~= m2+ p2 b s the two solutians E=
* (m2+ p2)1/'
I t i s a remarkable fact that we must take both solutions seriously. We find there a r e particles described by both the positive and negative frequency solutions. F o r E 3 0, 9 exp (-iEt); for E = -We W 3 0, cp exp (iWt). These two c a s e s correspond to particles and antiparticles, respectively. Represent the scattering of a classical particle by a space-time diagram. Fig. 5-1. (The shaded area represents the presence of an external potential that s c a t k r s the? particle.)
-
-
How suppose f o r a moment what would fiapmn if the trajeclories (or rat;her, in quantum mechanics, the waves) could ga bachard in Nme l Such a zsituati~nis sfiavrrn in Fig. 5-2. Conventionally, this process appears as fol-
26
THE THEORY O F FUNDAMENTAL PROCESSES
lows (see Fig, 5-3): F o r t we have only the scattared electron, fns&ad we prefer to generalize the idea of s e a t k r i n g , s o t b t the electron is eonsidered .to be s c a t b r e d h c k w a r d in time from t2 to tt. Then the conventional positron b c o r n e s a n electron g o i w b a c k a r d in time, T b two 66"duble-seatkring" pprocesses then differ only in the time o r d e r of the successive s c a t b r i n g s , a s we follow the electron d o n g its world liae, (One might think that this interprektion would imply that one could g e t idormation from the future; however, the full a m l y s i s shows that causality is not viotahd.)
Mow we shall s e e how this vfewoint st might en^ out the difficulty of negative energies, We shall speak of initlaf (past) and final (future) slabs , WB i n l r d u c e the notion of erztry and e k t s k & s (which have nothing to do with i s the entry s h b and x the edE time), In the m a t r k element x *Mtg, s h b , To de&rmine edvy and exit s h t e s we follow the world line of the garticte even if we have to go backward in time. Considering the positron in Fig. 5-3, we s e e that f o r the exit state f a r s e a t k r i n g at tz we must use the initial, s&Le of the positron. F o r example, the matrix element for electron scadbring i s
while t b t for p a ~ i t r o nscattering b c o m e s
Tire compleb rule i s thus: Elscttmns: e n t m s b W in the matrix element is the initial s h t e , and exit s b t e in the matrix element i s the final atate;
Positrons: e n t q state in the m a t r k element is the final s k k , and exit state in the matrix element i s the initial a h t e . As a @sting example: Suppose an electron gives up Borne energy to the matrix machinery, Ei = Ef + o 2 Ef , Then the m t r i x element has the time dependence exp (iEl t) exp (iut) exp f-iEi t) lso we have shorn t b t M varies like exp fiwt) if it extracts energy]. How consider the positron ease: exp (-iEt) = exp (iWt). P-E the positron is treated in, the old way (wrowf ) the time dependence is
o r Wf = 4+ w, tee., the process creates energy 1 But aecodi% ta our correct prescripUon we should have
s o that W, = Wf + o and the energy extracbd by the IM machinen shows a s a loss Lo the positron, a s we ahould like, We a m take more complicated cases: For example, the amplitude f'or annihilation of a pair is
while the amplitude for pair creation is
It i s easy to see that if M is e n e q y abssrlbing, in (1) it soak up all the energy while (2) gives zero, etc, The first inQrprebtion of the negative energy s a t e s was giwn by Dlrae,
FIE. 5-4
28
T H E THEORY Q F FUNDAMEPJTAL P R O C E S S E S
who invoked the exclusion principle to prevent electrons from falling d a m to the negative energy s b t e s (Fig, 5-41, Aecoding to his picture, all negative energy states a r e filled up to -mc2. This infinite s e a i s Ulen unobservable, But we can see bubbles in the sea, i ,e., the absence of an electron from a negative e n e q y s t a b , This bubble is then. a positron, For emmple, consider positrxon seatwring: How c a d d a posftron get to a different; s b t e ? h electron fills the initial, hofs, leavtw a hole (positron) in the sL;tLef from wMeh the electron jumwd (Fig. 5-5), The matrix element for this process i s
which is the same result t h t the time-reversal argument gave,
initial
There i s a. slight advan-e in our formulatian in that you do not b v e Lo deal with an infrinib s e a of slect;rons, But for base particles, you eoufd not fill the s e a in a million years. Xt b o k eight y e a r s d k r the discovery of quantum mechanics bfom the Klein-Gordon equation was proper-ly h a d by Pauli, and Weisskopf. Their prescription for inteqreting the negative energy states of bosons was completely different from Diracps theory and involves ideas of second quantization (Pauli and ~ e i s s k o ~ f B But ) . their intf?rpretation is again only equivalent to our rule t h t antiparticles simply reverse the role of entry and exit stabs. (Lectures 6 to 14 a r e essenUalXy the calftents of aa unpublished review article on strange particles by R, P, Feynmm and M. Cell-Mann.)
?But, sf course, b v i n g e n e r m -E = W md momentum -p,
6
Electro agnetic and i Couplings
We w m t to describe haw we s h n d today in the age-old atkrnpt to emlain all of nature in k r m s of the working sf a few elements in inftnik variety of combinations; in particular, w h t a r e the elements 'li The many unewected results of high-energy e w r i n n e n t s have revealed the incompleteness of our bowledge of these elements. We shlSL d e a c r i b those theoretical ideas which b v e k e n most useful. in partially sorting out this material. We shall concentraM on the ideas themselves and shall not have time to discuss carefully their origin o r the history of their development. Furthermore, we can only d e s c r i b how things seem a t present, Every sentence might be prefaced by: Of course it might turn out ta look quite different, but ... We a r e well aware sf the fragility and incompleteness of our present bowkedge and of the m a i f o l d of speculative psesiWlities, but the eqosition b c o r n e s awkw a d if we must continually refer to them, This ia a survey of work from all over the world and not a report on any new contribution by the authors, AIX the lnultituds of form@and the varieties of b h a v i o r of m a t h r seem to be describable in k r m s of a limited n u m b r of fundamenttzl padieles int e r a c t $ in ~ definite ways. They obey the general principles of quantum mechmies md the principle of relativity. According to these principles, thsise of quantum field theory, there i s nothing else b s i d e s particles, They b v e the intrinsic properzt;ies of r e s t mass and spin and a relation among one another d e s c r i b d as eougllw. The EXsctrona&pe#cCouplhg. Light, aa m emmple, i s regresened by a particle, the phoLon of mst mass O, spin I. The? emission of light by m excikd atom i s represenkd as the result sf a fundannenGaX couplliw, ar vobess e e,y (e for electron, y for phoLon) mewing t b t there is a poe stbifity (deecribd more precisely by a mathematical qwntity, ;urr amplitude) that am electron may ubecornepya photon md an electron; the precioe law of this coupling @ow the amplitude d e p n d s an the directions of motion and spin of the particles concerned) is h o r n v e q accur%hly (at least for energies less thm f gev). m e n an electron in m atom does this, light i s ennithd by the atom. Each process implies its reverse a t a corresponding amplitude; the arrows ~sbould double-ended. The reverse occurs in light absowtion,
.
-
-
T H E T H E O R Y O F FUNDAMENTAL PROCESSES
These relations a r e represenkd either by a double arrow, o r by a diagram, the lines represeding the particles coming in o r out. The diagram is written ta the right of Eq. (6-1). A single free electron cannot emit one photon b c a u s e of conservation of e n e r a and momentum, but if two electrons a r e near one mother, one may emit a photon which the o a e r immedlahly absorlr>s. Quantum. mechnics ~ r m i t the s temporary exfshnee of s l a b s , whieh, if m a b i n e d , could not conserve e n e r a . The penetration of a 'tzarrier In a decay of radioactive @liernents is a well-knom example, The effect of this photon exehmge we rec ognize in an inbraction bt-uveen the electrons, that is, as the electrical inverse-~qrrarerepulsive forces, Thus all electric and mametic forces among electrons, as well as the emission, seaEering, and absorption of radio waves, fight, and X-rays by efeetrom are described precisely and in, d e k i l by the sfmple law [6-1). The aulallysis of all this i s called quantum electrodpamies , A proeess only occurriw by means of a tamporary violation of energy conservation f s called a virtual process. The diagram for electron inhraction. via virtual photon excknge i s
-
e,e
-
e,e indirect
Only t b real padicles a r e representcld by open-ended lines (tvvo electrons in and out), the virtuall phoLon has b t h its ends tied by the f w b m e n t a l couplrxlg (8-1). Actually there i s mother principle, relatcitd to rewrsibllity in time-tbt there a r e antiparCicIes to all parl;icles. (For some neutral particles, like the photon, the antipaAicle i s the same a s the particle,) The laws for this can be got by putting n particle on the other side of an equation and changing it to an antiparticle, The antielectron i s the positron, s o (6-1) implies
-
fhflation and creation art? also complet~ly and in. fact the laws of pair specified by (6-1). Putting the photon on the other side of (6- 11, e,? e only represents the same equstion (6-1) again, for in fact the photon has no antiparticle, o r more preoisely the antiparticle to a photon i s again a photon.
Qther fuxlctamental particles also couple with a photon; for example, if p represents a pmtorr wcshave
All particles t b t do so a m called '"charged. " There a r e two rexnarbble l a w about the numerical value of the charge e , neither of which i s we11 und e r s t o d , %e i s t b t all fundamental particles carrry the same amount of e b ~ t ; (but e i t may be plus o r minus), The other i s that all other couplings are such t b t the t o k l c b r g e of all the particles in my reaction can never c h w e . Finally the value of the cfrrtrge e i s measured in dimensionless form by the ratio e2/ric = 1/137.039. The value and origin of this number, which measures what we call the strength of the inbraction (13-1) of electron witkt photon, a r e also myskrious. Its size b s been debrmined empiriealfy, &cause 11231 i s a small n u m b r we say elr?ctrodyaamics i s a fairly weak inkraction. F a m i CToupllage. Beside electrodynamics &ern i s mother even weaker coupling, Permi c~upiing,required to account for the decay of nuclei. For example, the decay of the neutron n into prohn, electron, a& antineutrino n p,e,?i i s taken to be the direct result of a coupling
-
which we s h d l eaII iz Ferml. coupling, It i s also ofbn called the weak inwraction. h o t h e r errample of Fermi caupllnll;~is the decay of the mum g , a charged parLieXe just llib iut electron, but h v i n g mass 208.8 a f times a s much,l,fiee +- v + ii,
A third Fermi coupling is responsible for muon capture by nuclei p + p
n+
Y,
-
THE: T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
Others are mdoubtebly involved in the slow decay of the str;ange padicfes that we skll diseuss later,
7
i Couplings and the Failure of Parity
Remarbbly, the strength of the couplixlg for each of the three examples alone seems to be equal. I t i s measured by a constant O satisfying G M ~ / R ~ = 1.0 1 a 0.01 x 10- 5, where M i s the mass of the proton, included to make a dimensionless ratio, Xt i s seen to be v e v small; the coupling very weak, The only couplings b o w for the neutrino {rest mass O, spin 112) a r e Fermi couplings, s o the inbraction with matter of this particle is very small and its direct detection extremely difficult. The deLttiXed form of this Ferrni e o u p l w was estaltalisvlzerl in 1957, It i s r e m a r h b l e in being the only one t h t dolates the principle of refleetien symmetry of physical laws (also b o w aa the law of conservation of pari*) I t was believed for a long time that for each physical process there existed (or could, in principle, exist) the mirror-image process, Con~eqluently the distinction betureen right and left handedness was thought to h relative; neither could bEt defined in an absolub sense. Of course, we must leave out hi~torical,eEects (for insbnce, the sense of rohtion of our planet), because these correspond to a particular choice of initial conditions. If we mdioed to an inlaabibnt of a different g a l w the most detailed instructions to construct a cerhixl apparatus, he might end up buildiw the mirror-image apparatus, since we could not c o x u x m i ~ i e ato~ him what our convention for right and left is, Afl experiments involving electromagnetic o r nuclear lForces eonnpiebly supporkd this view, This idea had in quantum mechanics the consequence of a property, called parity, of a. sak, Suppo~et h t m apparatus p r d u e e s a~ object in the s b & y, while the mirror-image apparatus p s d u c e s the mirror object in st;ztk p' The principle of superposition requires that
.
.
where P is a Einsar operabr. But
within the phase factor. Therefore saks:
= 1; and there a r e only two possible
T H E THEORY O F FIJNRAMEHTAL P R O C E S S E S
+
even parity
-
odd p r i t y
The principle of reflection, symmetry requires that a s y s b m remain in a s t a h of a given pariQ for all times, In early 1957, however, a t the suggestion of Vmg and Lee, a series of expriments in ,6 deeay were psrf;3rmed which. violakd tbis principle, Consider the co6' experiment shown in Fig. 7-1. The toe' spin was aligned at
mirror FIG, "73,
very low temperatures in a mametie field and the angular distribution of the ennitkd electrons observed. In turned out that the electrons came preferentially backward with respect to the ~ o " spin. The mirror-image experiment indicrtks, however, that the electrons a r e emitted preferentially f o r w a d with respect to ~ o "spin. It is therefore a physical process that cannot exist in our world, It seemed a t last t h t we could radio our man in o u k s space how to distinguish right and left. We tell him to align some coas and define the direction of the mametie field so that the electrons eomr! out preferentially b e k m r d , But w h t if our fellow i~ m d e of mtixnamr, uses anti-&", and observes positrons 3 At present our b l i e f is t b t he will come up with the left-hmd rule, That is, we think the mirror image i s a possible world provided we also e b n g e m a w r intQ antinaatkr, Then, the positrons of anti-~o" would be emitted parallel to the magnetic field. Gmvibtion, In addition ta these couplings there i s one other, evenweaker, gravitation, The laws of gravitatfan, very satisfactorily h o r n in the classical limit, b v e not yet 'been completely satisfaetsrfly fitkd into the Ideas of quaturn, field theory, but pre~umablyif it i s done, them will b a padicIe (gravitsn, O rest mass, spin 2) coupled universally to e v e v particle
PERM1 CQUPLINGS ARB THE FAILURE O F P A R I T Y
with a coupling conslant s o small that the gravihtioml force btween electrons i s lo4$ times the electrical force, Nuclear For~es.Be sides these weak couplings, gravibtion, Ferrni coupling, and electrodynamics, there must be some v e q much stronger, The farces binding neutrons together in the nucleus a r e much too s t r o w to be explained othemise , I t i s these str0n-g couplings and the particles exhibiting them, wMch we wish now to discuss. There is no strong coupling to electron, muon, and neutrino (collectively called leptons o r weakly interacting partfcfes) nor to the photon o r graviton. The paracles that eAibit strong couconsist of t;lse hyperplings a r e called "strongly inkracting particlesMnd'a ons (amow which a r e the neutron and proton) and the mesons. (The muon is &chically not comidered to be a meson.) The foree t b t holds the electrons around the nucleus is, of course, simply the electric attraction resisultiw from the virtwl photon exchmge between proton and electron implied by combining (6-1) and (6-3). But the nucleus Is composed of neutrons and protons held together by a s t r o w attraeLive foree between them, Tbis nuelear force has k e n studied very carefully by studying nuclei and by scatkring neutrons and prohns by protons, Xt turns out to be not only much stronger than electrical force but also very much more camplieaQd, Xn fact, except for one Xittls unewecbd thing, it turns out to b almost as complicabd a s i t can be, Instead of the inverse-square force, it i s a very s t r o w repulsion at short distances, an attraction a t somewht larger distances falling rapidly to zero beyond 10-'' cm. The force depends on the relative spin directions of the p and n, 6nd on the relation of these spin directions to the line joining the two particles, I t even depends on the velocity of the parLicles and its relatian to the spin direction (spin-orbit inbraction), But, the one ligXe unewecQd thing: The force between g and p, that ktween p and n, and &at between n and n all seem t~ be practically equal, Of course there a r e also electrical forces btween p and p that do not b v e a c o u n e ~ a r in t p and n, but when these a r e allowed far, by saying the tatal force i s nuclear plus electrical, a s nearly a s we can tell the nuclear past of tfie p,p force, the p,n force, and the n,n force a r e all equal, w h ~ nthe particles a r e in corresponding s b k s , I @ ~ b g8pinf ~ Thus the origin of the strong nuclear farces has some kind of symmetry (called isotopic spin symmetry) of such a kind that it i s irrelevant wbetber a pa&icle is a praton o r neutron. Even the small rest mass difference btween neutmn and prohn most likely i s the mass associakd wiLh the electromagnetic field surroundfm these pa&iclee. We learn our first lesson. SLroqly interactiw particles come in sets, The nucleons a r e a set of t w , proton and neutron. We say the single nucleon has two s h b s , the proton and the neutron. These sbks b v e the same energy, It i s malogous to the two spin sahs of an electron spin "up" and "down'klong some axis, which s k b s have tbe same energy if no msmetic field i s present,
36
T H E T H E O R Y O F FUNDAMENTAL PROCESSES
In fact k c a u s e the qwntum nnechmics of a two-skte system i s s o thoroughly b o w to theoretical physicists fmrn a study of a spin-112, s y s b m , they like to make full use of the analow and to say the two s t a h s of the nucleon represent the "up" and '"Lawn" s a t e s of an otsject of spin 1/23 in an imwimxy three-dimensional space. W s space is called isobpic apin space, We say the nucleon has isobpic spin 112, The equality of the forces results from the hypothesis that the direction of the axis can b taken to be in m y dimetion in isotopic spin space, In o r & m q space this poersibilily of choosing the axis arbitrarily has a eonsequence the conservation of mgular momentum, The s t r o w couplings &en satis@ a corresponding law, the law of conservation of total isotopic spin. a particle of spin O,1/2, 1,3/2, has 1,2,3,4, states, r e s p c tively, when we find that say three particles form a set, we say it has isotopic spin 1, eke. Then we can use the b o r n laws of combining states of different angular momenta, to determine how these particles may be coupled to each other to preserve the symmetry between pmton and neutron, or more pnerally, a s we say, to p r e s e r w isotopic spin symmetry. This applies only to the strong couplings; the isotopic spin conservation is destroyed by weak inkraetfons like electroctynamics, A, proton and a neutron have, of course, connpleteEy different couplings to a photon. The Pian-Nuclean Coupling. To give an emmple of these ideas, suppose, a s Yukwa suggesbd, that the nuclear h r c e s a r e the result of a process analogous to (6-2) but insbad of the electron there s a n d s a nueleon and in piace of the photon there i s anather parlfcle. Let us try to do t h i ~ .Supwee there i s a f u d a m e n h l process amlogous to (6-I), such a s
...
...
where r +s-tands for a positive pion, a m w particle, carwing positive charge to k ~ e p the law of conservation of char@ inbet, Now theme would b a force between a proton md neutron by the virtwl transfer of a a* :
(Incideablly the p and n get e x c h g e d , s o it i s called an exe Beween two protons Eq. 47-2) can also make a force by exchanging two
F E R M Z COZTPLIPITCS A N D T H E F A I L U R E O F P A R I T Y
37
pions, But the fome from the exchange of two cannot give the s&me answer as the force resulting f m m the exehawe of one. So there rnust be some way f a r two protons to exchange one pion, There rnust be a neutral pion and a process like?
and also f o r neutrons,
The new possibilities c h n g e the n,p force, too, giving a nonexcknge pa&
After some trials it i s found that if (7-1) i s true, (7-2), (7-31, and (7-4) a r e all necessary, but; the amplitude (the anttlogue of electric charge, but for pion couplings) f o r (7- 1) and (7-4) must both equal fi times (7 -2) and that of (7-3) eqttdls ( 7 - 2 ) but of oppasik sign. Then one can show that the symm e t m of nuclear farces pp = pn = nn will remain true in all circumstances and no matter how many pions a r e exchmged:
8
Pion-Nucleon Coupling
VVe assume the three fundament;aI interactions
where a , lo, and c a r e the amplitudes f o r processes (I), (21, and (31, respectively, and we want to determine the coupling constants a, b, and G which give r i s e to the symmetry in nuclear force (p,p) = (n,n) = (p,n) when they a r e in corresponding stales. Xn lowest order we have the: processes of Fig. 8-1. Wc? must therefore have bc + a2 = b" ce",If b = c a = O there i s no interaction btween. the e b r g e d n and nucleons contrary to experimenbl fact, Consequently,
The choice a = -(2b)'12 corresponds to a different definition of the n-meson pthassse, which is arbitrary anyway. But this result i s also easily gained another way. If we have a triplet pion fn", re,~'"'1i t i s isotopic spin I, In tbe reaction N -Pif -t- n (w represent a nucleon, neutron, or proton by N) the left side, a sfwle nucleon, has isotopic spin 112. The right side has six st%t;es (pn+)(p~')(pn"")(nn~ )(nzO)(nk-) but these may b analyzed into a doublet and quartet, for we combine an fsotopic spin 112 nucleon and an isotopic spin I pion and can make up isobpic spin 1/2 and 312, If isohpic spin is conserved in strong couplings, the: state on the right must be isotopic spin 112, too, and by malorn to laws coupliw angular momentum we deduce
PION-NUCLEON
CQTJPLENCS
Scattemd amplitude:
T T b t we must add the amplitudes in the firat process i~ s o m e w b t tricky. To compare the (p,n) with the (p,p) and the (n,p) s y s h m s we should have considered a (p,n) s a t e t b t i s symmetric under the exchn$e of l a b l s , We h v e
which, carrespond tx, transitions beween. the same initial md final s t a t e s In both. eases, and according to our rule the amplitudes muet b added.
40
THE: " T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
where the coefficients give the amplitude of each state. Probabilities a r e the square--so if a nucleon and pion a r e in a state of isotxlpic spin 112 of plus charge, the odds a r e 2 : 1 to be n and n+ rather than p and rr , These n mesons, o r pions, have indeed been found, They have O spin. The T* and n- have q u a 1 masses 276 times the electron, while the no dif f e r s only slightly, 268 times the electron, The difference is probably simply the aciditioml electrical energy of the c b r g e d pions. At1 the implications of isobpie spin symmetry regarding the coupling coefficients have k e n verzied, if corrections for electrodynamics a r e included, X s the nuclear foree correctly given now from the exchange of pions btween nucleons? This brings us to a new w r i o u s matter. We a r e unable to calculate, with any precision, the consequences of a strong coupling! So we cannot ealcuXaLt3i the nuclear force directly and see if it agrees with the hypothesis of the couplings (8-1) and (8-2). We saw that there were forces resulting from the e x c h w e of one, o r of two, and of course of three o r more pions, I t i s easy to calculab the foree from one exclrstnge, a bit harder f o r two, etc., but we do not h o w how to do the sum a t all well. f n efeclrodynamfcs there is also a force from the exchange of one, two, etc ., photons, but the amplitude contributed by a diagram containing an extra photon is ez/tic o r 11137 times a s large, Thus the dominant contribution i s one photon exchange, with only a few per cent correction for two photon e x e h n g e , eta . We thus work out a s e r i e s of rapidly decreasing t e r m s (called a perturbation eqansion), But f o r the mesons the eoupliw consknt g correspondlq to e of photon coupling satisfies g2/tic = 15. This is very large, justifying the te mn ''strong" coupling, but also forbidding the pe rturb;alion expansion, A great &a1 Irm h e n done by invoking over-all theorems, isotopic spin symmetries, and dispersion relations (relations connected tcr the principle t h t s i p l a cannot travel faster t h light, which, we cannot comider fiere), Suffice t o say that a t present it i s not possible to calculate most t&ngs involving strong coupling, A serious problem i s holding up the a m l y s i s of these couplings, There a r e even serious douMs that a strong coupling i s a logically consisbnt possibility in qumtum field t h e o q . Xndirset Xntsractiana, To k k e an example of the kind of problem involved, connsider the inbraction of neutrons and photons. Experimentally there f s one; the neutron has a magnetic moment, h o r n to a few p a d s per million. But we may still assume there is no direct neutron-photon coupling. For, by (7-41,the neutron can transform to c b r g e d particles in a virtual s t a b , and thus indirectly interact with photons. One possibility is represented by Fig. 8-2 but; there a r e maxry more di~lgramsinvolving mare v i r t u 1 mesons, @
We a r e unable to caleulab the moment and thus c a m o t use the kautifully precise measurement as a test of our theories. All we can do i s explain qualitatively the electrical and decay properties of pions and nucleons by such intermediate processes, The! chal-ged and neulral pion differs markedly in their disinbgration disintegrates very rapidly (< 10-15 sec) into two photons, properties. The
We eaanot argue that isotopic spin symmetry implies an analogous reaction f o r the T+. In fact, i t i s impossible &cause of conservation of charge. But (8-3) can be an electromawetic interaction and thus isotopic spin symmetry is lost for this, We maJyhope to explain (8-3) as a result of the passage through a vidual. pair of proton and antiproton:
The f i r s t coupling is s t r o w , a consequence of (7-3), next, one p emits a photon, marked y ' , by (6-31, then the nucleus annihilates, emitting the second y" via (6-3) again, (Two photorta a r e necessary, o r momentam md ermereTy c a m o t be conserved a t the end. At3 USUS~X e n e r m naed not be conserved in the transient intermediab @-Cages,) Unfortunably we c oL cornguk this rate efther, because tbe f i r s t step involves strong interaction. The 7i decays much more slowly. in 2.6 x lom8 s e e mean ltfe it disintegraks into a yf and v,
(As expected, the antiparticle n" has the same lifetime for disintegration into p and v,) This could be m Indirect result through virLuaX sf;inbs:
We should also e w e c t the disintegration
T H E THEORY O F FUNDAMENTAL PRBCESSE8
Again, b e a u s @a strong coupling is irnvolved, we cannot eompuk the rates directly, but we am able to cornpub the ratio of the rah of (8-4)and (8-5). We e q e c t do find 3 v Enshad of pv as the producb of pion disinbgratian in one case out of 7400. Thia has k e n recently eoxlfirnoed experimenhlliy (aeeumcy about f 15 per cent).
9
Strange Particles
There a r e other particles strongly coupled to pions md nucleons. About
7 o r 8 gears ago some new pafiicles w r e discovered in cosmic rays. For
example, there was (now called A) a neutral particle which disintegrated into a p and n-:
f t has, accortling to recent measurements a mass 2182 electrons and lifetime 2.6 & 0.2 x 1 0 - ' ~see. This i s very slow compared to the times natural
see, the time for light to go between adjacent for strong reactions nucleons in the nucleus). Therefore, this decay i s a weak reaction, p r a k b l y rehted to /3 decay. ff we limit ourselves ta strong couplings, then we h o w there i s not such a coupling as
It is forbidden a s a strong inbraction f o t h e r a s e (9-1)would go very fast]. But then how a r e the A produced? The cosmic rays consist of fast protons hitting nuclei that conain p r o b a s and neutrons and virtual pions via the strong coupling (6-4). The production of A is s o copious e ~ e r i m e n h l l ythat i t must be via a s t r o q coupling, Ld cixnnot be p +no It, for this is not A,+-p, for this strong, a s we have seen. Nor can it bc; anything like p -+ rt -would imply A - p + n + P Is strong, and since $5 + n- n"- is strong by (7-4), reaction (9-1) would b strong, Nor i s a reacUoa such as +
possible. Because, even though (9-1)i s weak, it does exist, md we would then h v e ! the possibility of three neutrons in a nucleus turning to one via the vi &ual reaetion
44
THE THEORY O F FUNDAMENTAL PROCESSES
This would release a large enerm, the rest mass of two neutrons, m d no nucleus but Qdrogen would be s a b l e . The shbifity of matter like a piece of carbon is very strikiw; such delicak experiments to detect disinagrations have failed that we h o w w h t a r e commonly called s k b l e nuclei have a lifetime of at least 10" years. This leads us to another principle which we use in choosing couplings. No couplings must exist weak o r strong, s o t k t taken all together, nucleons can disapjpctar o r dlsintegrah into something lighbr, Thus, since one p r o b n is produced on disintegration of the A, just one nucleon must be consumed in its production, The easiest way to keep track of this is to give for each particle the number of nucleons ""fidden" in it; more precisely, the net number of protons t k t a p w a r in its ultirnab decay products (antiprotons counkd as minus). This number still h s no generally accepted name; nucleonic charge has been proposed, Thus the nucleonic charge of the II, i s 1, as i s that of p and n, Electrons o r .rr mesons have O nucleonic c h r g e , antiprotons k v e a nucleonic c b r g e of minus 1, No fundstxnenkl particles a r e h o w t b t b v e nucleonic c k r g e greater than 1, Then we have the principle that all couplings must satis@ the rule: If%cleonic chatpgg f s always conserued. Aaaociated Productfont, K: Mesons, Following arguments of this kind it became clear that In the strong production more thxl one strange particle A i- hj. Actually other parmust km produced a t once (for example, n + n ticles bad been discovered in cosmic r w s , for emnnple, a neutral pa&icle, now called Ka meson o r n e u t r ~ bl a n , wMch d i s i n k g r a b s intx, two piom
-
and has a lifetime of 1 0 ~ 'sec. ~ It has a mass 966 and, clearly. O nucleonic c b r g e , b a i n we e ~ c ~ u n a,b rlow decay, presenting the same problem a s the A with mgard to its production. Pais m d Oell-Mam sugge8lrt.d that they must be produced together and that the true pmduction reaction was the result of a strong coupling
It has since been verified directly that theae particles a r e produced a t the time when nucleons collide, But a strong couplixlg like (9-4) for the p r o h n would &feet the nuclear forces (see Fig, 9-11. The balance of n,n and p,p forces then could not be maintained m l e s s there was m malogous cotl-pling for the proQn. There does not appear to be alny c h a r e d particle analogous ( t b t i s of nearly the same mass) to the R s o we a r e fed to expect a coupling
S T R A N G E PARTZCLES
and there is indeed a charged counterpart to the Ke found first in cosmic rays; a K1 of mass 967. To put it in the language of isotopic spin (8-1.8-21, t-he nuclteon on the left side is isotopic spin l/2, 80 if fsobpic spin i s conserved the total on the right side must be 1/25, The A i s a singlet and s o i s an isotopic spin zero, s o the K must be of isotopic spin 1/2, a doublet, coming in two varieties, Although its stmng coupling (9-5)i s the same a s the Ke in (9-4) and the decay [amlogous to (9-3)f
does occur, the lifetime of the ' K i s 1 . 2 ~10-' sec, very much longer than the KQ, a reminder that the weak decays do not mainbin isotopic spin s m metry. There were still other particles discovered in cssmic rays in a, codusing profusion. The clue to organizing this mkrtal was to formulak this idea, t;fiat the strong couplhgs involve ~ u c hparticles in pairs, in b r m s of a new principle. ess. Let us suppoae t b t the Ka i s considered to carry. some c b r g e , which nucleoxla and pions lack, and supmse this c b r g e a k d ar destmyed in strong couplings. Then a s i q l e i(; c be created by strong interactions, but if the A i s considered to carry. a negative unit of this charge, its creation slow with the K Is allowed, Tltis charge has k e n called "strangeness,?TThe K@ and K* have strangeness 1, the A has strmgeness minus 1, nucIeona and mesons s t r q e n e s s 0. T o k l straweness on both sides of a strong coupling must balance, Strangeness may c b ~ via e a weak coupling evidenced by t b r?i*sbnce af (9-3)f, Following these ideas Cell-M and PJishijima were led, independently, to propose a scheme f o r organiziw our bowledge of sttaage padicles and to predict many relations among them. 70stlfw %is scheme, amlogous ts the periodic talble of &e chemical elements, we shall drrscrib the strongly inbracting pa&icfe~believed Lo exist M a y , &vans. First we take the particles at' nucleonic c h a a e 1. These a s a
THE THEORY O F FUNDAMENTAL PROCESSES
46
group have k e n called hyperons. They a r e illustraded in Table 9-1, The ext wa9 predicbd t h t it i s b n c e of the x* was predickd by this scheme, X would decay in an extremely short time to the A by photon emission, I t was sub~equentlydiscovered a d did Just tbt, Mass near 2684, a dsublet (T = 1/21 Ee, Ea , s t r a w e n e s s -2. The o r c a s e d e particle was already h o r n
Isotopic spin I :
In o d e r of their masses we fixld: Mass near 1836 la doublet (T= I/2), the nucleons n,p
0
Mass 2182 a singlet (T -. 01, the A, neutral
-1.
Mass near 2330 a triplet (T = Z), X"ZO2'
-1
from cosmic-ray experiments. (ft has recently also b e n prsdtu~edin the laboratory by high-energy machines .) It decays slowly (about 3x 10-" sec) into a R and n", This could be explained easily only if i d bad strangeness -2, In fact, two K mesons have been observed to be c r e a k d with it, That i t should be a doublet i s the consequence of a relation af strangeness charge and isobpie spin suggeshd by Cell-Mann and NishiJirna: The stl-aweness S i s twice rthe average electrz"~charge q of each mattiptet, mz"~as the meteonic charge N, F o r h y p r o n s then with nucleorr;ic charge 1 i t i s S = 2q 1. So the average charge of the E multiplet must be -1/2 since S = -2, and thus i t must be a doublet. The prediebd $@particle fias recently h e n fo w d Anttbaryons. With each of these hyperons there should be a corresponding antiparticle of nucleonic e b r g e -1. Thus the chart for m t i h y p r o n s i s exactly the same, with padicles of tht3 same mass but of opposite electrical charge and opposite straweness. Of these, the axltineutron 8 , the antiproton 6, and, very recently, the antilambda 11 have been produced artificially in the lstbratary. Marsons. Next come the s t r o w l y coupled particles of nucleonic c k r g e 0 (generically called mesons). They a r e given in Table 9-2. The nt and na r e antiparticles and the s o is its own antiparticle. But since the kaons , ' K K0 have strangeness + 1, their antiparticles must have strangeness -1, and
-
.
STRAHCE PARTICLES in particular there must be two neutral b o a s , of strangeness 1 and -1, respectively, These a r e a 1 Ihe particles generally accepkd to exist a t presenls, There a r e a very few cosmic-ray events whose interpretation remains puzzling and wMch may be evidence of still other particles. Furthermore, cosmicray evidence for the existence of particles of mass near 500 electrons has been found by one laboratory but other attempts to find these particles have failed so f a r , m e existence of all but some of the antihyperons i s well esbblisbed e q e r i r n e n b l t y
.
Problem 9-1: An e ~ e r i m e nhas t h e n done on K capture in deuterium: K .t D -- hyperon t- pion + nucfeon. The data given in Table 9-3 a r e available. At present one cannot distinguish the he from the C5 Can you & s t the principle of conservation of isotapic spin? M&e as many predictfans as you can, fn particular for the results if A" and 22" could be distinguished,
Isotopic spin T
-
Charge 0
4-
In order of mass we find: Mass near 276 a triplet (T = I) the pions
K"
f K'
Mass near 965 a daublet (T = 1/21 the k o n s KO, K" and their antiparticles ('=I' 1/21 the k o n s
p,K M
TABLE 9-3 Pr~duct;LS
No, of cases observed
Strangeness
0
Consequences of Strangeness
The concept of strangeness and its conservation in strong inbractions has led to a l a q e n u m k r of predictions, none of which l ~ &en ~ violated e by e w r i e n c e . It has servad very fitiWu1ly to help orgmize the e q e r i m e n tal material, These predictions are, for example, sat when a A o r Z is produced in nuclear collisions, Ke ar K" rnust b also, Again, a prodluction reaction such a s n -t- n; A -t A is impossible for the t o b l strangeneas
--.
of the two A% i ~ s-2, and of the neutrons 0. AS another example, K- particles colliding with nuclei in night may produce A, but TC" cmnot do so, The Decrty af The Hautra1: Ifken. &e of the most stri&ngfy brilliant predictions of this theory of ~tr&n;gen@ss was mads by Pais and Gell-Maw, It is mlated to the prediction that there must be two ~ u t r a Kl particles, having opposite stmrmgeness, Ke, and its antiparticle Ka . Now t;hs K* appears to d e e w into two pions, for example, the disinbgration
i s observed (lifetime about 10-gosec). This violates strangeness, of course, as weak interactions do. The mtiparLict1e should decay wi& the same probability into the correapnding antiparticles
The prOdu~tEIin (9-3) and (10-1) a r e of course identical, 'This k s , a s a consequence, a very i n b r e s l i w qumtum meebnieal inbrference effect, 331s exisbnce of (9-3) means t k t , even though it mag be via complex virtwl processes, &em ia aome amplitude, x say, for a KWto become n",nW: KW + -
-
. %
T*,T-
amplitude x
(10-2)
Also, from, the relation of particle and antiparticle, the amplitude for the antiparticle rnust the same,
SOME CONSEQUENCES O F STRANGENESS
-
K0- 7 r * , n " -
amplitude x
(10-3)
(Btrictly it might have opposite sign, but either possibility leads to the same conclusi6n,) but Now suppose we had gotten the wrticle in a state neigfer K* nor ra&er with equal but opposite amplitudes to be Ka and IC;a ; call the s t a h IQ :
(The amplitude must be 1 / 6 because the probability (amplitude squared) is 1/23 to be Ka and 1/2 to be @.) Then the particle in this state K could not decay into ' s and a*. for the amplitude for it to do s o is ((I/ 2 )x ( l / a ) x j = 0, from (10-2) and (10-311 The state
P
could, of course, decay with amplitude f i x . Thus tbe proper s t a b s to US@ to d e s c r i b disin&gration a r e Kj and Kg, for the first may, and the second may not, decay into two pisnn. m e s e two shbs will b v e very digerent lifetimes and disintegration prducts. (It turns out the K3 can disinbgrak, far emmple, into three particles, and b s a lgetime a t least LOO times a s >rr:~ga s the Ki going to two pions.) But when a b a n is produced, say along w:;h a A, it haa a definib slrmgeness -1; it i s Ka, and is neither K4 nor Kg:
an eqmtion immediably deduced by adding (10-4) and (10-5). The probabiliaes a r e thus 1/2 to be Kf and 112 to b K;, the pmtlpes ysis of disinkgration. Thus when the newly prduced k o n s disinkgrate, o d y half of %ern should e a i b i t %heshort lifetime meiQsee and decay into Lvvo pions. The remainder should have a much longer lifetime, la-@ see, and decay into three particles. That is, the remarkable predietion was made that the neutral k o n should e h i b i t two d@farent lifetimes, witan two s e b of decay products, mis has now &en verified. A fur&er asm e t h e also been verified. Since tbe Ma has slmweness + 1, it cannot produce It, in collieion wt% nuclei. However, if we move Ear enough away from the source of a K* beam sb taat Kj would nearly certainly have disintegrated (but not so far t b t a w u l d also disintegrab appreciably) the barn mst bstcome nearfy purely in tke state K;. But by (10-4)the etrtlawness i s not definite now; there is an amplitude - 1 / f i , and thus a probability 1/21to be of strangeness -1. Then 11% can be produced by the kaon beam hitting nuclei. This A production has n w been demonstrakd experimentally. 1%may be asked, in an over-all way: How did the strangeness c b n g e from -+ 1 on production to -1 further d o n in tihe b a r n ? ms answer is via
50
TEE THEQRU O F FUNDAMENTAL PROCESSES
the virtual process (10-3) followed by (10-3) in reverse, The strangeness violation is-via a very weak coupling x2, it i s true, but the equality of mass of Ke and Ke makes a resonance possible, s o t b t even the small amplitude for the process gradwlly builds up a large effect, This is one of the greatest achievements of theoretical physics, It is not based on an elegant ~ L h e m a t i c ahocus-pocus l such i3ts the general theosy of relativiQ yet the predictions are just a s imgomnt as, say, the predict;rion of positrons. EsmeiaHy inkresting is the faat that we have &ken the principle of supewosition to iL,s ultimatt;ly logleal conclusion. Bohm and co-workers thought that the principles of quantum mecbnics were only temporary and would eventwlfy fail to e q l a i n new phenomem, But it works. It d m s not prove i t right, but for my money, the principle of superposition i s here k, stay 1
Strong Coupling Given now the pardicles, hyperons, and mesons, and bowing tftat a e y a r e strongly coupled, the next problem is to find which a r e strongly coupled to which and in w h t mmner, These coupliws must satisfy con~ervationof nucleon number, charge, isobpic spin, and straweness, but that i s f a r from dewmining them complekly, For example, w h t coupling does the R have besides &e nucleon coupliog (8-11,(8-2)7 Is there a coupling A-- Z,T and of w b t stren* and kind, etc,? Again w b t i s the law of coupling of the K p s in (9-4) and in raactiorms s u c h a s PIT22,K o r f=-$,K? Even thc: spine of the particles a m uncer&in. Studies of the decay K* n* + nC + r' indic a k strongly that no angular momentum is carried out by the pions, s o the &on spin i s probably 0. But the emerinnenkl evidence it3 incomplete on A, Zf , and El, atthough spin 1/23 i s fndlcakd for the A and 2= Probably the hyperons all have spin 112, the mesons spin 0, Farmi and Y w MdeX, A ~ an P example of a host of models proposed to understmd the s t r o w interactions, I s b l l discuss an idea of Fermi and Yaw. logow to efectrical Suppose a neutron md a proton b v e a c b r g e charge but of the same aim, which coupl-hjsthem to a vector meson af vesy large mass. Then the nfi eyabm would feel a short-raw@ attractive force andagous to the longer-range electrical attraction eG, The n and fi have bgether a mss of 2 x 938 Mev and we f magine t h t they a r e b u n d bgether very. strongly, sa;y, by about 1600 Mev, If the forces a r e rigkt, the b1%l angular momenlam would tw 0, the pariQ -1, and we obkln the no: [nfil = .rr-, A1so, [pfi] = n" T n i c a l diagrams a r e of the form shorn in Fig. 11-1, *ere the wiggly l i resents vector-meson exchmge , We af s o e w e e t a p$ grid En system, of these i s the 7t ? We n o k &at the pe arrd n5 rzygbms b v e addi-e;ioml diagrams (Fig, 12-2) wMch do not occur in the nfi and the pii cases. Z t turns out that the sysbm?
-
.
.
T&cause the fip and fin 'both b v e the same amplitude to amihillak, s o the state ( 1 / f i )(@p rin) cannot annihilate. In judging whether it is an isotopic spin triplet, one must r e m e m b r to count 1? as -(1/2),
-
THE THEORY OF FUNDAMENTAL PROCESSES
which has the same e n e w as a (1/fi)([pp
T
" and a n - and does not a m i M h b i s
- iinl) = s o
These form an isotopic spin triplet. The other a b b ,
might not be bound o r might have a different e n e r a , p r b s ~ p sbigher, and tm a new meson of total isotopic @pin T = 0, which. b s not yet h e n observed, To get the remairning particles id igs necessary to introduce at least one
STRONG-COUPLING SCHEMES
"'fundamenta1'"particle
which carries strangeness, Choose the A. Then
Strangeness i s just the n u m b r of A's1 So you see, i t is possible t;a imagine all the s t m w l y inbracling particles a s composiks af n, p, and R and deduce isotopic spin and stra,~3;enessconservation, f s h l 1 tell you my secret hypothesis: You cannot tell whether a pamclicle i s "elemenhry" o r w b t h e r it irir a composib of '%elemenlaw" "rticles, In other w a d s , a 1 the theories of eomposi% particles would give equivalent results (if we could calculate them) and there would b no way to distinguish among these, ff you have a system composed of parGicles of mass large compared to the total binding e m r w (nuclei, atoms) then i t makes sense to speak of a cornpasite s y s k m , But when the binding e n e r m i s a good fraction af the rnass of the f r e e padicles i t is incorrect to make a distinction b t w e e n composite and elementary particles, How ean this idea be more clearly s b k d and how can practical use be made of i t ? 1 don%t o w , The theoretical proposals for the debiled scheme of strong couplings a r e a11 nearly entimly speculative. We shall give two others; one i s the GeX1~ a n n suggestion, ' called global symmetry. It proposes that all the hyperons would have the same rnass and be various sbbs of an eightfold rnultlplet if i t were not for the K eauplings. Two new slates a r e formed by linear cam~ Z') and Z = (%/&)(A +- ZiO). binations of A and Z" ; namely, Y = ( 1 / a ) ( Then it i s assumed that the form and size of the pion couplings a r e unehmged if, in the coupling, (8-11, (8-2) of the pions with n and p, the n,p a r e replaced by SLT, El;", o r a r e replaced by Z", 25, o r a r e replaced by E-, $", The kaon eauplings that destroy this symmetry r e m i n undetermined, The other swgestlon i s that the pions a r e coupled directly to the total isotopic spin veclor, and this coupling is responsible f"or the 11, f: splitting, The h o w couple be*eexx n, p and the A, Z: quartet, and fietween 23" , ED and the A, 2 q w r k t in such a way a s not Lo split the A, Z qutarkt (but the coupling to n, p and Lo 22-, ED a r e with different coefficients). (In the no, = 1/23; g = g,, , g,, = g,, ; tation of ell- am, r we take g,, = 0, g , and h = hnK,) The pattern of mass values e q e c t e d in this scheme fits v e q well to that o b s e m d . U&ortunatc31y critical teats of this hmotkeais have? not yet h e n fomd, There i s a large e ~ e r i m e n t a lprogram , on to determine production of kaons by nuclear collisions and by photons, s c a t b r i n g and inbraction of these mesans with nucLei, eGC. But just b b e e n us theoretical physicists: t da we do wiWa all these data? We can't do a n f i ~ r r g ,We a r e f a c i w a
,,
,,
54
THE THEORY 0 F FUNDAMEMTAL PROCESSES
very serious problem. and we need a revolutiowry idea; sometMq like Einstein" t h e o v , Perhaps the resulta of all exprinnents will produce some idiotic s u v r i s e s , and someone will be able to caleuitab everything from some simple rule, m a t we a r e doing can h compared Mtb those complic a b d models invented to exptlstixl the hydrogen spectra which turned out to satisfy very simple regularities. One more thing a b u t the question of strong couplings, n e r e i s also direct evidence that the R interacts strongly wi& nucleons. merc3 exist bypefirwments (a b t t e r =me wuXd be h;ypc;rnuclei), in which a A i s bound to a numher of nucleons. For example, the hypernucleus , ~ e ' has been found a s a f r a p e n t ; resulting from the capture of a KT by a nucleus, This hsernucleus consists of two protons, a neutron, and a A bound together, The R is b o u d by a few Mev. The system i s unshble of course, for the weak decay (9-1) of the R provides a mechnism for disappearance of khe h wit31 release of a pion and 37 Mev (or the pion may be virtual o r recaptured and its rest energy a p p a r s as Urntic enerm of nueleom in a s k r ) . From a study of a e s e hypernuclei we may eventually get information an the A-nucleon interaction force. At m y rate, it is nearly as strong a s the nueleon-nucleon interaction. For further details see the review article by ~ a l i t z . ~ ~ d d i t i o n a evidence l on these strong couplings should eventually come from a study of their relatian to weak coupling~i, For example, magnetic moments and electromapetie mass differences, as well ils the relative rabs of valrious weak decay processes must tell us someaing about the structure of the strongly inkractlng particles, Yet the tbreoretical malysis of all, this evidence r e l a t d to strong couplings i s severely cri~tpledby the inability to make quantibtive calculations with such esuplings.
2
Decay of Strange Particles
We next turn to the evidence on the weak decay of those particles. The exwrfmenbl informatian on masses and decay properties of all the particles i s given in Table 12-2, We are concerned here only Mth the decays of h y p r o n s and mesons, "Two of the decays clearly are the result of eleelromawetie couplings (in association wi-efi virtual s b t e s implied by strong cou-
TABLE 12-1 Produets
pliw), the a"
+
y + y and the Z0
etio,
-
A +y
55
Lifetime, sec
. m e s e a m alao the only o ~ e s
56
T'RE TI-IEORY O F F U N D A M E N T A L P R O C E S S E S
allowed by conservatim of charge and the principle t b t electromwnetic couplings cmnot alter stranrnness. The remaining decays all have lifetimes of the same general order of magnitude, It is k l i e v e d t b t they a r e all the result of coupling of the F s r m i t m e (in association vidual stabs implied by strong couplings, a s always). 'This hypothesis would already accomt for several features including the general order of magnitude of lifetimes, C e m i n l y lepbn emlissian~ would not be unexpected, But even when na leptons a r e involved a lack of conservation of pariw fan asymmetry under reflection of left f a r sight) speaks for a Fermi coupliw. The fact t b t the Sam*@particle K could decay into two rand into a r e e pions (of 0 total angular momentum) was in fact the f i r s t suggestion t h t conservatiom of pariQ may b violated in physical law, More recently an a s p m e t q demonstrating a failure of reflectian s p m e t r y has been found in the direction of the products in the decay of the A to p + 71.". We have already written three Fernni couplings (6-41, (6-51, and (6-81, but in. each case, counting the leptons a s of O strangeness, no strangeness change is implied, Thus with these three, we can only hope to explain decays for which AS Q where S i s t o h l strangeness. They a r e only the neutron decay n p -i- e -t E and the pion decay nf -+p' + v , which we have a1react;y discussed as an indirect consequence of ( 6 - 6 ) via virtual skdes, The remaining decays involve a cbngc: of strangeness of one unit, A53 = zt 1. The fwzctamenbl couplings that produce them kztve not been identified, although they a r e almost certainly Fermi couplings. The problem of their identification i s interesting, s o we shall go into i t in some d e h i l . Fermi, Couplfng Sohemea. We shall f i r s t count the minimum number of new couplings meded. F i r s t of all, the exishnee of a decay like IC' e+ + u + n o implies that the Le a r e coupled to a strangeness changing pair. It need not be K'nO directly, a s the strong couplings would allow such a decay if any other pair, such a s GA were_ coupled to C e. For example (Fig. decay to pEv 12-11, the Kt could become virtually h and p, and the =;
+
+
FIG. 12-1
with the p and coupling
.
annihilating to n o Let us take, a s an smmple, a Ferrnl
DECAY O F STRANGE PARTICLES
57
analogous to (5-4). But muons a r e emitted also in AS = -1 decays (e.g., K* --+ p+ + v o r l3?' p' + v -+ r a), eo w must have an sldditional coupling +
Finally there a r e decays involving no neutrinos a t all. These could come from a Fermi coupling of the @pe
of any equivalent via strong couplings. F o r example, the A would be via the virtual process, such a s
+
p
-I-
n" decay
plus more connplicabd diagrams, With these three couplings all h o w decays would b qualitatively explainable, The proposal of six independent F e m i couplings might appear complicahd, but the simultaneous appearance of the a r e s new ones will appear natural from one assumption, That Ss that the Fernni couplings are of the nature of the inbraction of a kind of current with itself:
and the problem i s to find the composition of the current J, the sum of seve r a l parb, The coupliws (6-41,(6-5), m d (6-6) d e s c r i b d previously result if J i s writkn
Experimenblly the eosfficieat of all f i r s t three k r m s a r e equal, All our three new couplings will result if we add to J just axle b r m , say X, which changes strangeness, Above we h v e suggested solely as an example what X might be but we ahall now k v e to consider more seriously what properties the tern X might have, An immediate consequence of this idea is that the coe1Eficienls of X to each of the three currents ( Fe), ( 9p. ), and (fin) a r e equal, That is, the cauplings (12-l), (12-21, and (12-3) must all have the same coefficient falthough it need not equal the coefficient of (6-42, (6-51, and (6-Ei)] If the couplings. of (Fe) and ( Lo X a r e equal, it can be cafeulakd that
.
THE THEORY O F FUNDAMENTAL PEOCESSES
58
a b u t aa mmy K* decays go Lo e" + v -t no a s go to @* I= v + no. This is s u p p o h d by the &b to an accuracy of 30 par cent, The absence f o d s o f a r af K' e" -t v i s also consisknt if the K* b s spin 0, far &en the r a b af K" e+ + Y should be v e q much less t h n the r a b KC y* -+ v, Can we check t b pmdiction t h t the couplings of (pn) and of (Zp) to X a r e equal? Unfoi%unakly, because of our inabilitjr to anctlgze strong couplings, we have found no way to do i t yet, ]&rapahladS ~ m e t Raaer q af the 8 nasa C h g t w Dsotrys, we say a b u t the eurmnt X? We try to make as restrictive h m o h s e s a s possible; in that way we make the maimurn of predictions, although some of &em m y be proved wrong in the future, First we nokice %at the presence of K" -+ e" -t- v -+ n" requires t b t , if strangemaa decreases, positrons a r e smitbd, So X must have at least one k r m like (FA), comisting of a pair of parLicles of total strangeness--1 (and in any case of negative charge). Can there br? also a term in X like ( 2"n) of stmngeness + 1 2 m e r e i s no evidence that these must be, s o we shall assume: 1. X contains only terms of strarlzgewss -2. This ha6 88 a eonsequence that a decay vvith AS -- 2 i s forbidden. Thus Z;lre E" could not decay into n + n- and so far it k s not k e n seen to d s SO, Furthemom, although a P2 could disinkgratc: equally into 8 -t K- t. v and e I= w" + 5,only the first ia allowed for K a and the second for K O . So, if the rare teptan decays of a K 0 barn be observed close to the course (before the Ic! mode has decayed away), the charged leptons (e o r p ) should predominantly pasitive, Eweriments to verify a i s have not yet b e n perhrmd, Me*, the current X, if (FA) would be of iaotopic spin 112; o a e r combi~ ) would b r i q in some isobpie spin 312, If it nat;ions, such a s ( 6 ~alone, were purely isotopic spin 312, %ere would be a rule, AT = 3/ 2, in a leptic iu+ + ;v of &e K' of isotopic spin 1/2 would then decay. But the decay K* he impossible (forhere AT -- 1/23), So X must eonkin a t least some eornponent of. isotopie spin 2/2. U"e shall assume it i s pure isotopic spin 1/23 [SOif (52 * ) appears it must be in the combination -(? 2' ) + f l ( z Z-)I m e n we would kave the rule: 2, In a Eeptie decay the isoto@c spin cha~gecaPz only be f / 2 . We can teat this rule by eompasiq the decays
-
+
+
-
.
and
What the rule says i s that if the-K's a r e brought to the other side to make antix's the pairs za)and (Ken-) must have amplitudes in the propor~to" make up a sL;aLe of isotopic spin I f 2 [malotions -(1K" no)+ 6 ( w")
(z"
DECAY O F STRANGE PARTICLES
gous b emreasion (8-211. Thus the amplttrtde for the second reaction (12-7) is times the first (12-6), s o the rate of the K D decay i s twice that of the ES;" The decay of KO must bet
-a .
the antipaxrdicle reaction corresponding to (12-7) and it must occur at the sarne rate as (12-7) [i.e,, twice the rate of (12-8)]. The K b n a t i e l e , being of amplitude 1 / f i to bet K0 and - 1 / a to be will therefore decay into n- + e" -+ v at the rate of (12-8) and into n" + e" + v a t the sarne r a k e A, esrms;po&lng relation applies to the decays with a muon replacing the elieetmn. These predictions a r e confirmed by e w r i m e n t (so far),
The Question of a U n i v e r s a l Couplling Coefficient Nonleptic decays occur from the combimtion of X ~ t the h h r m ( Fn) in the current J. This Itetrm (Pn) i s of isotopic spin I, s o if combined with an )I assumed t;o be pure isotopic spin 1/2 we can form only isotopic spin 112 and 3/2, and suggest the third rule pverning strangemss changing decays: 3. IE nonlefitie decays f i e ~Izanepeof isotopic spin caliz be only
This does not appear very restrictive, yet id does have consequences that we can check. First, we can predict c b r g e ratios for the three pion decays of the b a n s , The three pions in the decay
seem from their morne%llctmdistribution to be in a stak of zero mgular momentum, and therefore th@wave fmclion of t h ~ pions ? i s completely s m m e t rical. Xt can be s h o w t b t the only s p m e t r i c a l isotopic spin s b t e s available to three particles, each of isotopic spin 1, a r e T = 2 and "r = 3 , If rule 3 is correct, the s k k T = 3 eamot be generahd from the original T 1/2 of the &on, for a c b n g e of a t least 512 would b-F? involved. Tkus the fiml sLab must be T = 3, and from the rules for combining sbbs it i s easy to show that the rate of the decay K " -+ s" 4 a O-t. n0 should be 112 of the r a k of f 23-1)fexcept for a rate increase of aborill 9 per cent for each n" resulting from the small mass difference of " and n e), Experimentally the ratio i s 0.30 & 0.06, wMch is consistent with the predicted 0.25(1.2) =I
= 0.30,
By exactly the same argument, the three pion decagps of the K ; ~
A UNIVERSAL COUPLING COEFFICIENT
should occur in ratio 213 [or correckd for no mass difference, ratis 2(1.1)/3(1.3) = 0.561 if the final state i s T = 1. Measurements on the K; a r e just beginning to be made, and a r e so far in agreement with this prediction. There is also a consequence for the two-pim disinlegration of the &on. The d a b a r e K;
IC"
+
T*
n"
+1~-
-t-
n
78
* 6 per cent
0.002 times K: decay rate
A. rennarbble feature i s t b t the
K" decily is s o much less rapid than the
K: decay.
For two piom in a s p m e t r i c a l s b t e , the teal isotopie spin i s T = O o r T = 2. Only the T = 2 state i s available for the case K * of one n* and one r e . Now this couid be reached in general from the ban T = 1/23 either by A T = 3/2, or by A T = 5/2 (we add isotopic spin as vecbrs). According to hypoLhesis 3, however, m l y the AT = 3/2 operabs, That means that the r a b of K " d e a y gives us the relative amplitude of T = 2 in the decay of K Q, Actually it gives us only the square, but; we h o w that the amplitude is 0.052 Limes some complex phase for the tws pions of KT to be in s b t e T = 2. This amplitude is so small that the KT should decay almost purely into T = O. IT it did so, t&e ref ativci: proporti ons of n+ +- n" to r e t- n@ should be 2:X o r 67 per cent should k charged. If the amplttttde for T = 2 bas its p h s e for maimurn sr for minimum inbderence, the percenkge prdictions a r e 12 and 62 p r cent, respectively, Theoretically then, the proporlion of n* + a"' in the K; must lie somewhere between 62 and 72 per cent if hypothesis 3 is valid, We must wait for more precise data .to see if Wa i s true; the pmsent resulls art3 just consistent wi& it, As far a s we b o w , then, the X current can be restrickd Lo one wkose strangeness is -1 m d isotopic spin is 112, In k r m s of the h o r n particles then, it would have the f o m
wi.ttx the coeffieienta a , 6, y , 8 , and .k: to b determined. m i s i s a s far a s we have k e n able da proceed, The difficulties In proceeding further wit1 now be pointed out, The Queatlcnn of a U~iversalCoapltng GoaEioiant, In view of the apparent equality. of the coefficients in 8 of such different terms a s ( Fe), ( FP), and f i n ) , it is mtural. to suggest a kind of univerealiQ and propose that the cou-
62
T H E T H E O R Y O F F U N D A M E N T A L PROCESSES
pling coefficients of all particles are equal (udversal), s o that all the coefficients ol to c a r e equal and equal to unity. [Or at least if factors fi a r e differently distributed, some of them may be 1and others l/ato provide some special symmetry. For example, if a = 1 / f i= - @ the first two terms become ($2;) -t- (EZ"), a. combimtion erapciaIly simple fram the vie global s p m e t r y , l A s a fudher emmple in the second s t r o w coupling; scheme suggesled above, a particularly simple hnothesis would be t b t (1)the nonstrangeness chmging Fermi current i s eou;pIed t;o the same combimtion of parlisles a s i s the n * , namely, a; component of isotopic spin, and &so (2) the strangeness c h a w i w F e m i current is coupled to the same c~mbimtionof particles, a s i s the K * , This would mean a! = --@ , 6 = E , and possibly y = 0, But there i s direct evidence that this is not the case. If y = I, d i s m g a d ing the o a e r terms, the decay K* n o -t e* + v could h a direct process, and its rate calculated, It comes out 170 times too fast I m e r e may b some mdificaaon from other diagrams but i t surely camot be s o d m reduced. We deduce that either y is 0 o r of the order of 0.08 (i,e., 11 for the rate goes a s y 2). If y were O the process would have to be an rect one, wNcb we c ot calculab (althougk a t ffrst guess i t is not easy to x s o slow even if y were O and the o a e r consan& were of see how it could t order unity-yet no firm conelusion can b d r a m this way about the o&er c o m a l s ) , Again, if a = 1, we can c&cLzllab a rate for the pmcess A p + e -t i , mis pmcesrs and the one Mth p replacing e have not h e n Been, yet we predict it should appear in I6 p r cent of the h disinbgmtions, Experimentally i t occurs at least less m n one-bnth a s often as Ws. It i s mlikely tkat this is an effect of inkrference from o a e r diagrams, s o a i s probably less b n 0.3, In addition no lepton decay of the ;r;i'" k s h e n seen 10 per cent of that expected with /3 = 1/3/"%", s o must also be less &m 0.3. It fierefore does not look a s if the X i s coupled to leptons with the full coefficient exwcted from the universal coupling; in fact, a ccaeEicient of order 0.1 seems more likely, (It is not possible to disprove this fmrn the mpidity of the K" '1.1 + v decay, again because of m c e a i n ties in all such qwntitative calculations,) We can summarize these poinb by the ~ k e r v a t i o nWt, although theoretically unexpecbd, the data may indicate that 4. Leptic decays with chaw@ of strangeness are relatively much slower Ghan those without. chnge of stm~gelzoss(although the K" I.lf + v is a possibte violation), But if the eoegfcienls in X a r e of the order of 0.1 for lepbn coupling, we should eweet &em to be emet1y the game for the f Fn) coupling, This fs uncomfortable b e a u ~ ethe nonleptlc decays seem too fast f a r this, They seem to require coegicierrts of order unity, but we camot be sure, for we eamat really eaXculab Lhese processes b c a u s e of the ~ r t wsbWs l of strongly interactiw particles &at a r e involved.
--
+
@/a
-
-
Xn addition %ere is a fu&her approxfnnak symmetry rule suggesbd Xry the data far wMeh we h v e no theoretical expl 5. N~rzlept$cs l r a w e ~ e s sthawing decays with AT' = 312 are relatively much slower than those with A T = 1/23, i.e,, weaker.
4
Rules of Strangeness Changing Decays: Experi
We have already noticed this f o r the K- n + n" decay^, in which the neut r a l kaan (AT =. 1/23) decayed 500 times f a s t e r than the e b r g e d b a n (AT = 312). The amplitude f o r A T 312 here was only 0,052 of that f o r A T = 112" Let us a s k kf a similar predominance exists elsewhere. f t i s best studied by seeing to whit extc3nt the other data could satisfy the rule that the nonleptic deeays were entirely AT' = 112, F i r s t , in the 11. decay coming from T = 0, the final s a t e m u l d have tu be T = 112, and the mtfo of p + n- cases to a + would have to be 221, or charged p r d u c t s in 67 p e r cent of the decays. The data shows 63 A 3 per cent, a discrepancy that may either be e m e r i m e n h l e r r o r o r the result of a very s m d l interference with T" = 3/23, Second, the= a r e same predfctions about the 2-decay asymmetries, but f o r the present incomplete d a k they only represent inequalities which a r e indeed satisfied by the data. Third, we can now d e b m i n e the rate of K; decays to three pions, (13-2) and (13-31,relative to that of the K + (13-I), f o r we must reach the T = 1 s t a k in a unique way if A T = I,&!. The prediction is that the total rate of K l to &ree pions equals the kohl rate of IC' to three pions (if the corrections of 9 p e r cent f o r each. n a a r e allowed far). The preliminary measurements on the ff; a r e not in disagreement with this, We would predict that the Ee d + r e r a k should be half a s fast as the wA + n rate, but d a k a r e not available on the Ee The origin of this rule i s unhown, f o r if the decays a r e via some X eoupled to ( Po), there i s no apparent reason why the A T = 1/2 and A T = 3 / 2 amplitudes should not be of the same general o r d e r of m a e i t u d e . We a r e left also with the m y s t e w of the origin of the small coefficients in X. Qne speufation made by the authors (unpublished d a b ) i s that one particular diagmm i s very much more i m p o r a n t than the others and about ten times a e big a s one m u l d estimate, Let us ilEustmLe this idea f o r the
[email protected] the fj can be ple that 2% is simply 0.1 (FA), Then in the esupliw gn eliminated, gidng a direct amplitude f o r transformation of n A, The =I
T O
C1
-
-
-
.
- -
STXEANGENESS C H A M E I P J G D E C A Y S
mechsurism of such an elimination i s the diagram in Fig, 14-1, with the proton in a closed loop. We imagine this diagram with the loop undisturbed i s much larger than one would expact, and larger than all other diagrams, The large size compensabs the m a l l coefficient 0.1 in I(, and makes nonleptic decays appear a t a normal rate. F u ~ h e since r Lbe dominant coupling is now effectively n A, a transformation for which A T is restricted to + 1/2, this restriction appears a s rule 5, The A T = 3/2 can came only from the more complicahd usual diagrams, f o r which the small cmfficient 0.1 i s not compensated, But we e q l a i n the two m y s h r i e s (4 and 5) by two ad hoc assumptions (that the X coefficient i s small and that a c e r h i n diagram i s large), s o it i s not clear tbat we are getting anywhere, It i s true t b t all the A and I: d e c w details come out q u i k closely if we kke the coupling to be equally n fl and p 22" using the global symmetry h s o t h e s i s and a perturbation caleulation, but the pe rturbatian. approximation camot be justified. $urnmarye At this paint i t would be well to summarize the salient features of the problem we face. A c c o d i w to the principles of qumtunn. Beld theory there exist in naturct only particles k v i n g rnaes and spin. (even o r odd blf-inhgral), which hztve a relation a m o w each other called coupling, These particles fall irtto two groups: the weakly and the strongly interacting particles, The weakly interacting pa&icles include: the photon, the graviton (which everyone Ignores), and the leptons (a, p , v). The strongly inkraeting particles are the mesons and the baryons (see Tables 9-1 and 9-2). You h o w the garl;icles and the couplings and s o you h o w eveqtlning. P ~ Y S ~in C aB nutshell l Couplings other than gravitation (which conserves every.t;hing o r nothing b e p n d i w on haw you look at it) have the properties I i s b d in Table 14-1.
-
-
-
TABLE 24-1 Conservation laws satisfied Straweness Parity
Relative stren@h
f spin
Ferml:
10""
Na
No
No
Electrodynamics
lom2
Ho
Yes
Yes
Strong
10
Y ~ B
Yes
Yes
Coupling
66
THE T H E O R Y O F FUNDAMENTAL PROCESSES The conservation laws satisfied by all couplings a r e of two types:
Geometricat taws: Angula r momentum (roLEttion) Enerm, momentum. (translation) Parity x ebarge eonjugation Time reversal Mamber laws: Nurnkr of leptons E lectric charge N u m b r of baryons (nucleonic charge conservation) We see t b t there a r e 31 particles; fince, according to sbndard field theory, we need 31 fields to describe these partielea f However by k i n g elever we can try to reduce the number of fields we need, for some of the particles m q be compounds, m a t 18 the midmum number needed? First we need a baryon, Also we need aornething with i spin 112 (two slates), a d we need the qrlafiw of strangeness. So, for example, we could get by with three baryons: n, p, h. But this tells u s nothing about leptons, We need t h ~ n v, e, y and probably the graviton, No one is quite sure what to do with the p . then kave to e q l a i n a t least eight fields a ~ four d If we also include it couplings, This i s a s far as one bas been able to proceed, s o far, I t is clear that veS)r considerable progress is made by noticing symmetries, but that no eafculational work has otherwise yielded much, information, We are sorely In need of reliable methods for a quantiktive amlysis of these problems. But even more profound and exciting is the problem that i s acitfy implied I; i s the slpificance throughout these lectures but never clearly sbted. o r the pattern b M n d all these inhrrelated symmetries, partial symmetries, and asymmetries l
agnetic and 6-Decay Coupling UfSf s k 1 1 now see how tO make quantihtive calculations for those processes that we crul caleulak 1 I shall just k l 1 you the results and give heuristic arguments for their correctness, I don't feel t b t it is necessary to start from field theory since it is, in fact, not internally consis@nt. Anyway, X want to leave room for new ideas, You may have great pedgogical d i f iculty in learning the physics presented in this way. I t would b easier to learn it more historically, by moving from the SchrMinger equation 4x1 the Dirac equation, and from the quilntization of b r m o n i e oscillators to the creation and annihilation operators, and finally to the resulting amplitude for various processes, Instead, we s h l l just give rules for finding the resulting amplitude-bcause t b rules a r e s o much simpler than the steps leading to thtem. Furthermore, the things we would o a e m i s e s h r t with (e,g,, the SchrGdinger equation) a r e approximations to the end result, useful only mder certain conditions. m a t is ultimately desired for a true physical underslanding i s bow the SchriJding e r equation is a consequence of the more fundamenlal laws, I t is true that for historic& or pdagogical undershnding it would be? bttsr to start from SchrSdiwer and go the o%er way--although of course no real deduction can be made that way; new sings like Dfrac matrices, ete ,, must km added from time to Lime. But it is a long, b r d climb do the frontier: of physics that way, Let" s a k e an effort to try an experiment in learning, Let us see if we can put you right d o m on t h frontier ~ s o you can do two things, First, you can look forward into the unknom a d see the problems and the progress k i n g rnacde and perhaps 'fialp to solve some of them, Second, you can. look back and try ta see how f&e variou8 things you have learned, from BeMan% saws to Muwell% equations and to Sebr&dinger%equations, a r e aXl consequences of what. you a r e learning now. The latter will no2 be obvious, and will make it hard to accept the apparently ad 3106 rules we discuss now. But seally, that i s bow nature does it; she ""undersknds" the SchrGdfqer equation as an approxima@ quation describing large n u m b r s of interactions among many pa&icles n n o v i ~slowly. The fundamental elements a r e the keg inkractions among small n u m b r s af particles moving at arbitrarqy s p e d , To these we address our atknlian,
THE THEORY QF F U N D A M E N T A L PROCESSES The only inkractioioxrs for which a reasonably satisfactory quantiktive quantum mechanical description can be given -today a r e the electromagnetic coupling and the Permi o r f3 -decay coupliw, We consider processes that involve a small n u m b r of particles wMch may interact, c r e a k other particles, decay, etc, For each such process we have an amplitde; the s q m r e of -the amplitude gives the probability for the process to oeeur. We start with the case where there a r e no virl;uatl parl;ieles. This is easier than the processes involving virtml pa&iclcss, .uvhich we shall treat later. Also \;ve b g i n with spin 8 (scalar) particles, in order not to confuse things by introducing spin and relativity a t the same time, The wave function of the scalar particle has only one component, Under the transformation x p x; (rotation o r Lorentz transformation),
-
How about space reflection ?
Z f p(x,t) = p(-x,t), we speak of a "scalar" particle; cp (x,t) = -p(-x,t) i s a ""peudosealar..'Wf course yl migbt not obey either of the above eqwtions, We shall assume at a free particle is represen&$ by the plme wave
where p x p,x, = Et - p o x . This is DeBroglie2s assumption; u does not change under coorr;linade transformations Next we seek an e q r e s s i a n for $be probability, This has to be the fourth component of a four-vector (SF), since the probability of finding the particle somewhere: a
.
must be invariant. Here S4 = the probability of finding the particle p e r cm3 (also the probability of passing from past to future), and S = the prokbtlity of passing through a surface normal to S per cm2 sec. (e*p i s a scalar, and henee not a satisfactoq S4. The space inbgral of S4 i s really a s u d a c e integral in four dimensions (see Fig, 15-1). The generalf zation i s evidently Probability to pass through surface =
Js, HI,dS surface
where N p i s the unit normal to the surface, N pNp = -1.
ELECTROMAGNETIC AND @-DECAY COUPLING
path of integration
Suppose the padiele i s locahd in some ffni* region of space (so that u i s not a plane mve), Can aoxnebsdy else in another Lorentz frame perform the probability integral in his frame a d get the same result we do4 b c a l l ing t b t in the space-time diagrams the C%&vfngM ssyskm i s robed, we draw the schematic picture of Fig. 15-2.
Since the partiole is localized as shown, S p i s zero a t some distance, s o t b t we cizn close the path of inkgration indicated by the dotkd lines in the figure. Then both o b e r v e r s get the same answer if
N~ dS surface = o Using Gauss9haorem, this is assured if
10
T H E T H E O R Y QF F U N D A M E N T A L P R O C E S S E S
TNrs have seen that for a plane wave, 6u i s unsatisfacbry for a prohbility density. Since pl, i s the only available four-vector,
(The factor 2 is just convenUon,) m i s gives
Does this make sense? Ha& a a t the density in a moving s y s b m appears greater by just the same factor the St: has, Thus keep iiu = 1 in e v e q sgstem; the relativistic normalization i s then 2E per em3. This i s a crazy normalization but very useful in practice, We s b l t always normalize in tbis way, How about a more general expression for SI, 7 For the plane wave
TR the general ease we must take the more symmetrical forms
NOWthat we have e v s r y a i w defined, we sbll see how to c a l c d a b , Becall the famous formula for &@ transition probability W r secod:
Prob./sec = 2%IM ri l2
(unit energy inkrval)
evaluated for E = E , This form i s inconvenient for our puwoses, I shall r e w r i b it 90 L h t you carnot reeogxlize it. First, in order to use our narrnalization we must divide by 2E for each part;iclle enQring in the process. Since we shall a l w a y ~be workiw in a continuum of skks we have
Note that the expression for the density of states lacks d3w/(2n)3 because of momentum conservation, To wtake the formula s p m e t r i c a l among all 421s p a ~ i c l e s ,we add the factor
Also we replace IldE by ti f r: E in
-.
C E). In fact we could ht.e s k d d
out
ELECTROMAGNETIC AND B-DECAY
COUPLING
with the f o r m u l ~in thfsform: Between any two s b b s ,
Further recall that
This allows us to remove the Eack of symmetry- &tureen p and I3 in our formula, since
where pi = (Ei,pi). Collecting all aese factors wa finally k v e
The factor b 4 @ ( ? p m
- wtI: p) corresponds to over-all
conservation of energy
and momentum, We s'hiall always be going back and forth from codimration space to momentum space. Our convention is &a follows:
W I a this convention dp i s a l w a y ~accompanied by a factor 1 1 2 ~and the d function by 2%. There is one more imporhnt remark to be made. The great utility of thla method i s t b t &! t u r ~ out s to be invariant, So we c w choose the ~ y 8 tern in wMch we evalu;it;es M a t our convenience, ple, consider the dlsinbgratiomr of a particle, The probability of decay when it i s moving i s less by a factor M/E, which ia of course the relatidstic time dilation. I s u ~ e s that t you try the fallowiw example, which, we shall do in detail Eater. Consider the decay of a ken into two ~'8s:
THE: T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
(Forget about charge; just rtow.) The K and n have spin O and amplitude uk, use ASSUIZ?~ that the amplitude for the process i s
The factor ( 4 ~ ) ' i/s ~conventional (rationalized units), f measures the stren@h of the inkraction, and Mk is inserkd Lo make f dimenaianless. The u's a r e just Lo remind u s of the process (the u* % a r e the; mrticXes created and the ups the p a d i e l e ~destroyed) but they are s e t equal to unity in the calculatrians, Find the value of f that give^ the experimenhl lifetime,
6
Density of Final States
Generally we shall need to consider only two eases for the ineomixlg s t a e : (1) decay of a particle (prob, transelsee = Z/mean life), alld ( 2 ) tvvo padieles in collision (prob. trarls./sec = rrv, where cr i s the cross section and v i s the mla-d-ive velociQ), We write the transition probability per second. In the form Prob. trans ./set = 2n ( M ~ /I3 2E II 2E) D in
out
is the density of stale per unit range, There are other useful. expressions for I): (1) two outgoing partlcies :
when m2
-
=, D = [ ~ ~ ~ ~ /dQl ( 2 In r the ) ~ c.m. ] system,
(2) three outgoing particles:
THE THEORY O F F U H D A M E N T A L PROCESSES
--
Consider now ths b o n decay into t;wo T%, whj~h.f suggeskd in the last n,n. just l;o illustrab the lecture. X hsve assumed a dimct coupliw K h c h i q u e s we 8faaZ.l b using; wc: do not belie= t h t this is actually a fundamental c a u p l i q in nature, A word a b u t units: We h v e chosen E = c = 1, Then
rn = mass = enerI;l;y = f /length = I/time
F a r t b electsort, me = 9.1 x 10-" g
= 0.511
Mev = 1/3.86~10-" crn
These n u m b r s should be memorized, F o r the proton all q t;;iplied by 1836, and similarly, for other pafiicles of mass m by mime, At the end of a eailculation it will alvvaye be clear wEch units rn mpmsen&. We can still check dimensions, e,g,, a lifetime must b p r o p o ~ i o m f . to m. Keeping track of ti and c is a compleb wash of time I The firat time we tshll do the calculation the hard way. We have
-
-
Sening pz = m( pi we can cross out ( 2 ~ ) d4(rn( ' - pi R)and d$2/{2r)4* = (E ,g,),m = (AXK, @(in the rest fmme sf the XK). men Let
Decay rate = 117 = ( 2 l r ) P e (2%)6 (E'
- pe - m:)ilrr&(~k-2MKE)
D E N S I T Y O F FXNAL S T A T E S
76
The whale p u m s e of physics i s to find a n u m b r , with decimal points, etc, I @herwise you haven't done anything, Find
Experimentally the lifetime of the K = 0.99 of the decays go into 2a. Therefore, l/rexp= 0.78/0.99 x
X
10-" sec. Only 78 per cent
sec
and
This is a very small dimensionfess constant so we a r e eonfronted with a weak eoupliw, I want to impress on you that this i s just an example that we have made up; we do not bfieve that the true mechanism of this decay ia this fmdamental coupling, but rather t b t the decay occurs as m indirect effeet af some other couplings. However, let's go along with it.
P ~ ~ b L e n116-1: z The K also decays into 3 a's. I am going to assume a coupling
Qbkizz the spectral distribution, check it with the experimenbl results, and dekrmine g
.
Consider aaw a slightly more difficult problem, namely n - K scattering. Forget the c h r g e of the K and the n, a possible way in which this could occur i s iflustrabd in Fig. 26-3. This i s an indirect process involving a virtual a, I shall now give you the rules for finding the amplitude for such, a process ( l a k r on we shall m a k them more plausible), Follow slow tlfe pa&iele lines and w r i h (from right to left): (1) f o r each vertex an amplitude (4s)"' f~~ (2) for the propagation of the nr b t w e e n two v e ~ i e e an s amplitude
T H E THEORY" O F F U N D A M E N T A L P R O C E S S E S
FIE. 16-1 I / @ ~- mi), where p is the 4-momentum and m, is the mass of tbe n. This the eqmtion of motion: 43) enerQ momentum must be conserved a t each vertex The product of these amplitudes is then the amplitude M for the process. For Ffg. 16-1 we get i8
However there is another way in which the same transition can occur, s h o w in Fig. 16-2, *icb, is topalogically different from the first diagram (the vertices emnot be arranged in space time to get the first d i a r a m ) . The amplitude for this process i s
DENSITY O F FINAL STATES
-
(4a1'/~f ~ ,{ I/ (pl q112- mF2]')(4%)'" ~ M K and the l o b 1 amplitude
M
X?[
for the transition is obtained by adding these two:
= 4 r f 2M K f{l/[(pl ~ + q112
- m i 1) +
PI -92)'' -mill
)
7
The Propagator for Scalar Particles
I s k l l try to make the rule for the propagabr less artificial by connecting it with something you already b o w . Consider the - K s c a t k r i w example (Fig, TI-1).
'VVe have h e n saying that the amplitude has the form
Now consider the lowest-oder term In ordinary perturbation theory:
The sum is ta be taken over the i n b r m e d i a b s4.ea.b~ n. The contribution of the diagram in Fig, 17-1 i s
THE PROPAGATOR FOR SCALAR PARTICLES
But we must r e m e m b r t b t in ordfnstq perturbation t h e o v the process with opposite time order is cohsidered separately as pair production followed by annihilation of the positron with the incoming electron. The energy in the intermediate state i s thus 2Eg + E p (Fig. 17-2). Recalling our pre-
vious rule for @ d r yand exit s b b s we see that T(i) ia the e n t q (exit) @kt@. So we have
The sum of the two relevant m a t r k elements i s
and the factor 2 E P i s just the normalization constant. So the backward-intime idea simplifies the reault, Each of the above k r m s (cowldered aep-
THE THEORY O F F U N D A M E N T A L PROCESSES
80
arately) is not invariant. By combining them we obbin an obvtously invariant expression, This demonstration i s not inkaded do br? a proof of our propagator rule, but doe8 show the phycsical notions involved. There i s another way to get our propagator rule, For a free particle a special solution is
We ern form any free-particle solution by the sum
Noh that
But tMs is the same a s
where
Suppose &em i s a source S(x,t) for the particles. We then postulak that
We solve this equation by introducing the Fourier transforms
(Note that p4 * (p2+ m2)Ii2 since we are no longer dealing with a free particle,) The transformed equation Is
THE PROPAGATOR FOR SCALAR PARTICLES
So if we b o w the source, then rp (p) = [L4p2- m2)1B(P)
showing the origin of the propagator. Solving far tp (x,t) = Sexp (-ips x ) fl/(p2 - mZ)l
yt (x,t),
Sexp 1-ip
*
x8)S(x') dPx'
x [d"//(2a)4~
where
is the propagator in space and time.
We notice that in o d e r to give a meaning to D, i t its necessar;yl lo define the pole in the inbgrand, To do this we add an inffnihslmal negalive imaginary part; to the m a s s (an invariant) and i n k g r a b f i r s t over d w = dp4:
-
The limit E O is mdersboct. to be Laken d k r w a r d . This prescription displaces the poles from w = + E p = + ( p2 + m2)Ii2 to w = k E pT ie and i s equivalent to the eonlour in Fig. 17-3. For t 3 t' we
complete the contour in the lower b i l f -plane, m e r e f o r e , D+(t > t ' ) = -2ai Res(EP) = i [ e x p ( i ~ ~ ) ( t - t ' ) / ~exp[ip*(x-*)I d
We ~ o t %at e for t t h n l y positive e n e a i e s cantribub. F o r t < t f we must close the contour in the u p p r b l f -plaae. We get
T H E T H E O R Y QIF
-
F U N D A M E N T A L PROCESSES
s h o w i w b t only negative energies contrIbuM for t < t'. In this way we see how the rn m - i~ rule summarizes our prescriptions for going k c h a r d and farward in. time. Finally, the c o r r e c t formula For the propagator of a spin-0 pavlticle I s
8
The Propagator in Configuration Space
-
We have seen in the 1~ K scatbring emmpIe Wt in o d i n a r y seeondorder perturbation theory the separate contributions to the amplitude were not relativi~ticallyinwriant, but t k t their sum was invariant. Two Lorentz frames A and B are related by a robt;ion in space time, For example, the time order of the two successive iateractians would be differen$ in A and B in the situation in Fig. 18-2, mis i s true when the seeond vertex does not
lie within the light cane of the firat (otherwise the time order could not be reversed by a b r e n b transformation). Che might think t h r e is 0 arnplikde if %e two events a r e separabd in apace but not time, This i s not the case. AZI pasitions contribuk, Ta see that tbis m k e sense ~ we exsmine some of the properties of the propagator in eodfguration space:
How does I), ( x ) blzave with respect to the ligfit cone PI In configuration
84
T E E THEORY O F FUNDAMENTAL PROCESSES
space the propagator i s much more eomplicakd than in momernturn space. Explicitly,
where
H ' ~ ' i s a Hankel function of the second kind.' For large s,
Noh m t for low vctloeiWes using s r t
-.
x2/2t,
where $S solves the Schradiager equation. Qutsfde the light Gone for x2 > t2, D+ dies off exponentially:
-
exp (-mr)
far t2
As a physical, illustration s u p p s e that we measure the pasition of an electron, with, a shutkr, for emmple, At the same time, but a t a different position, we m&e a measurement to see if we can find the electron there (Fig. 18-2). The? probMlil;y- is not zero, kcause In the aet of measurement a pair could be creabd, the positron then amihilating the original electron. Pauff invenbcl this thought e m e r i m ~ n after t be Zlztd thought the idea was wrong.
THE PROPAGATOR I N CQhTFIGtJRATI6N S P A C E
85
NOWe o n ~ i d e ra particle of high velocity; see Fig. 18-3. Consider the b b v i o r of the amplitude (Fig. 18-4) as we go a c r o s s the light cone along AP, How,does the wavelength at the point A (for exrtmple) eorrespod to the cla;ssicaiS velocity x/t? Let u s examine the phase of
When x chmges by X the phaae must change by 2%: m[t2 - (x +
- m(t2 - x2)'/2 = 2 1 ~
THE T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
Therefore,
-
c) we approach a t3 singularkey on the light We note that as x t (v cam. A possible physical reason for this is that all momenk contribute, but f o r most momenta v is nearly c, s o there i s a great accumulation of amplitude near the light cone, We have written the eqwtion of motion for spin-0 boson fields a s
Now we must consider wbat S f s; i,e ,, w b t can c r e a k the particles rp 3 We consider the T - E(: emmple again with the coupling
Sening f' = ( 4 ~ ) ' / ~ the f , equations for 9, and
CQK a r e
Xd i s possible to obtain these results from a principle of least action. Conslider the action
Variation of cp, inbgsatIon by parts (omitting the sudaee term), and setting 683 = 0 gives the equation
"Phis is the eqwtion for the free pion, For the R IC example we add at sfmilar b r m for the free K and an fntemction t-erm:
-
THE PROPAGATOR I N CONFIGURATION SPACE
87
Variation with respect to psr,phc: yields the equations of motion given above. (Here we have bcitly assumed a real field describing neutral particles-tlze generalization i s easy .) More generally we have
when the Lagrmgian density f: must be r e l a l i ~ s t i c a l l yinvariant. This requirement greatly restrieta the n u m k r of admissible Lagrangian densities. Nofe the relation to the usual classical form,
We regard the action to be the more fundarnenla1 quantiQ, From it we can immediately read off the rules far the propagators, the eoupXfng, and the eqwtions of motion, But we still do not h o w the reason for the rules for the diagmms, or why we can get the propagadors out of S.
9
Particles of Spin 1
In general we want to find amplitudes wMch transfarm under h r e n B transformations linearly:
u ' = $(L)u
(L = Lorentz transformation)
where
One solution i s scalar. We can easily find another; 4-vectors transform linearly-so surely a $-vector is a possible answer. We note the same f a r rotations-a 3-vector representing angular momentum 1 was permissible. Thus a particle e m be represented by an m p l i t u d e which is a 4-veetor. T;Ve e x p c t i t to have spin 2 . There is just one complication, however: Under r o h t i o n s the space componenh transform like a vector, bud the time camponent t r a n s f o m s like a s c a l a r , So i t looks as if we a r e representing two p a ~ i c l e s .We can get around this difficulty by requiring that
Then in the r e s t frame of the particle (p = 0)
Photons. The photon i s the only known elementary particle of spin 1. I t h a s O mass. If we b o w the laws of propagation of photons and their coupliw to other particles, tken we h o w all about electrodymnnics. A very useful guide in formulating these Xawa i s the requirement that in the classical limit the t h e o q correspond to M a w e l l % equations , The amplitude f o r finding F>&otonsin quantum electrodynamics i s -taken to be the 4-vector potential A p (x,y ,z,t), which in the absence of sources satisfies the equation
PARTICLES O F SPIN I
A free photon i s represented by a plane wave
where izl,i s called the polarization vector. Substituting in Eq. (19-1) we find that ICZ = 0, o r rn = 0 , and from Eq. (19-2) we find KpE , = 0; the polarizat;ion i s perpendicular to Kp . The W o w must also be gauge invariant: If someone solves a problem with Al, and somebody else with A$ = Ap + Vpx where El2x = 0, both should get the same physical result, For plane mves this tell6 US SometPling like this: Let A = c , exp (-iKx) represent a photon of momentum K and polarization E: and
Then
Therefare, if two polarization vectors differ only by a multiple af the 4momentum, they must represent the same photsn. By a suihble gauge # 0. Let transfornation we can always choose c 4 = 0. Suppose
Then
and
A free photon is therefore mpresented by only t w sbks af polarization, We can choose for these any two directions perpendicular Lo its momentum, o r resolve &em into right- and left-bnded circular polarization (see Zecturn 2). M C (LHC) correspond to spin 1 a l o w (opposite) the momentum of the photon, This can be seen easily as foflows:
THE T H E O R Y OF FUNDAMENTAL PROCESSES
90
where E ,and c are two unit vectors nornral to the direelion of propagation. Rohting 8 degmes about the z axis,
C:
= E x cosB
- E y sin B
Substituting in ukHc we get
ukHc = exp ti@)U R W Q . In the same way,
We proceed now to find the laws of coupling and propagation sf phowns, Prlnaipla sf MEidmal Elec;tram&@@tio Coupling, There is a very in.l;erest;ing principle by means of which, we can obtirin the coupling of photons with a charged wrticle whenever the equation of motion of t b t particle is b o w . Take, for i n s b c e , the equation for a free scalar particle,
-
e iCj, to ivp @Ap. This gives an equation which Then the rules is to c contains the effecb of the e l e c t r o r n a ~ e t i cfield:
t to no& that this principle keeps the equations gaugeinvariant, Let
Then tp' satisfies the same equation as 9 with Ap replaced by Ap + Vpx. But tp and q q i f f e r only by a p k s e factor (vvhich, however, may d e ~ on d space time), and consequently they represent the same physical s k t e . Ule can write the quation for cp in the form
PARTICLES O F SPIN 1
91
The r i g h t - h d aide i s the source of the scalar field. We can ohkin the rules f o r the amplitude of the funhmenl-ill processes as follows: l%e amplitude far a particle +dh momentum pi [ql= exp (-ipix)] to emit a photon of momentum q and polarization E [Ap = Cp exp (iqx)] and continue with momentum p2 f p 2 = exp (ips)l i s proportional to
The last factor exgresses the conservation of s n e m and momentum a t Ute ve&x: pg + q = pt. If the photon i s absorbed, replace q by -q. Xn either ease the amplitude is given by
Amp. = -i(4*)'
e(pz + PI) ' E
The factor (4n)Ih i s introduced so that e is the unrationalized coupling eonstant e" =/I37 in wits whem A = e = I. The factor -I i s essenlial in order t a keep the correct p b s e relationship when indi~tingui~lkttble processes wEch am of higher o d e s in the coupling cozrshnt are included, but othervvise i d can b left out, The tf?rm qwdmtic in e gives the amplitude for the simul-eous emission (&sawtion) of 4x0 photon^, The amplitude i s groporlfoml to
-
The factor E , Cb appears twice, since either of the two Ap's could have s consere m i t b d pholwn a o r phobn b, Again, the fast factor e q r e ~ s e the vation of 4-mornenturn: p2 -I-qa -t- q b = pi, The amplitude i s now
THE THEORY OF FUNDAMEPJTAf, PROCESSES
Amp. = -4ne2
(8,
Let u s e m p h s i z e again that the connection between the rules f o r amp1itudes and the classical equations of motition i s only heuristic. I t i s clearly impossible to 'bderive'9quantum eleetrobynamies from Maxwell" eequatians; these can only serve as a guide, A l h r m t i v e l y we could have skr.t;ed with the Lagrangian density af the free s c a l a r fie'id y,
Changing iVp
-
iVp
- eA
we get
Expanding we clan w r i k
i s the contribution due to the coupling between particles and photons. The read from rule8 f o r tha amplitudes of the f u n b m e n b l processes can also ce.
The coafficient of e LeIIs u s for insknce, that there i s a process in. which a particle e t b momentum pi I p = exp f-ipix)l e m i t s a real o r virtual. phot;on with momentum q and polarization c [AI, = gp exp (iqx)] and goes on with momentum p2 [p = exp f -ip2x)], Sub~tidutingin 6, we get
e J C~ exp (iqx)(pzpexp (ip2x) exp (-iplx)
+
exp (ip2~1 pip ~ X PPIX)]
x d4x
The last factor tells us that p2 + ql = pl, The amplitude i s then
Amp. = -i(4n)lh e(pz + pI) E
PARTICLES O F SPIN 1
93
The inclusion of the factor (4n)lh and -i was discussed previously. The c w f f t c i ~ n of t e2 corresponds to the simultaneous emission of two photons: One of the A&'s i s
and correspondingly the other .AI, is
The amplitude f o r the process i s therefore
Amp. = -4m2 2 C a eEb
The Phokn Prwambr, The photon propagabr can also be obtained from the equations of motion. This amplitude AP (x,y,z,t) for a photon satisfies M a w e l l equation
"
VpVpA, = j,
j, i s the source of photons.
Since %Ay = Ox it follows t k t \j, = 0; we shall say more a b u t this later. Folfowiw the procedure fiescribd in Lecture 17 let A, @) =
.f c, (k) exp (-ik)ld%/(2a)41
Substituting in the diEerentia1 equation we get
consequently the propagator for virtual photons i s
The factor b p, serves to remind us what kind of sources produce what polarization, the factor i i s included because of tlre factor -i in the coupling, A s an example, cons ider the n - K scattering via photons (Fig, 19- 1). (Forget the direct K r n inkraction which we imagined previously,)
T H E T H E O R Y O F FWNDAMEPJTAL P R O C E S S E S
The t o b l amplitude; M is m d e up of ~ r e factors: e (1) amplitude for the K meson of momentum pi ta emit a virtual phobn of momentum pi pz, polarization 8: -.
(2) amplitude for the photon propagation:
(3) amplitude for the n meson w i a momentum g3 to absorb the virtual photon:
On summing over all four directions of palasfzation of the virLual phot;ons,
Later on we will discuss why we must consider only two polarizations for real photons,
Problems: 19-1, Obtain the n"" K- scatbring m ~ t r i xtn the earn, system. 19-2. Do the? same for the R" a*. 19-3, Dekrmine the eomptoa effect for n ' in the frame of reference where the initial n * i s a t rest. 19-4. Caleulab the n" " - r'" pair annihilation from flight with n at rest.
-
-
20
Virtual and Real Photons
Let us discuss the relation between virtual and real emission of photons. Why, for instance, for a real photon do we need to consider only two trans-
verse states of polarization, while for a virtual process we have summed over all four possible s k & s ;2 Suppose we send a photon to the moon. A f b r the process i s over we could describe It by a diagram, Fig, 20-f,
moon
ear& FIG, 263-3
In a sense every real phodon is actually vifiual if one l o o k over sufficiently long time scales, It is always a b o r b d somewbere in the universe, t characterizes a real photon is that k2 O (since it i s not real a t all times, by the uncertain@ principle, k? i s not identically = 0) and therefore the propagator l/k2-- w . Before we proceed further with this discussion we must eorr~tiderthe Iaw of conservation of c h r g e . 6anm1~"mUan of 6b-e, The aedion S for the particle plus the photon field is given by the hflothesis of minimal eleetrornagnetic interaction:
-
T H E T H E O R Y OF F U N D A M E N T A L P R O C E S S E S
96
Requirixlg that tke change in the action vanish f o r f i r s t - o d e r variation in the particle and photon fields we obbin the equation of motion for parLi~Xe
and the photan
(We have used the condition VpAg = 0.) The kharge-current vector is therefore
, Tbe first-order c b n g e of action f o r any e b n g e in A must vanish. F o r the special change &Ap= V px for arbitrary x the fields do not change, so the only change in action is that of the coupling k r m , o r
Since
x
i s arbitram, and this c k w e of action vanishes, we must have
This i s the law of charge-current conservation. f t i s implied by the principle 0% gauge invarhnce and holds even if the h n o t h e s l s of minimal electromagnetic inkractions does not. We return now to the relation b t w e e n virtual and real photons. Consider the scattering of two particles, a and b, which give r i s e to a current j; (x) and j; (x), respectively. The amplitude f o r emitting a pbatan of momentum q and polarization L: is j, ( 9 ), ~where ~ j (q) is the Fourier transform of j (x). The contribution to &e s c a t b r i n g amplitude, due to exclkange of one photon of momentum g = (o,Q), pol E is, aceording to a u r rttle8r given biy
VIRTUAL AND REAL PHOTONS
97
F o r the four possible dimctims of polarization of the phaLon we &kc? the space-time axis, with &e? 3 - a i ~oriented along the direction of propagation of the photon. On summing over polarization,
The last two trtrxrrs are? the e q e c t e d cantributtons of the two transversely polarized photons, i s then the meaning of the f i r s t two terms 7 Tb@ conservation of c b r g e current requires
or, since the 3-axis i s along Q,
Substituting j3 = (w/Q)j4 in M we find
If the photon transferred i s real, o r" Q , Then the contribution of Xangitudind plus timelike photons to M (first t e r n ) vanishes, compared tm that of transverse phobns, Hawevar, in general, the virtual longitudinal and timelike photons cannot be neglected and, in fact, play a very innporbnt role, To see w h t this role is, we express the contribution of the first term in IVI f a r all momenh & and frequency o in coodinale space. Substituting,
we get
The integral over w gives 2s8(tl - tz),and that an (a- gives 4a/ 1x1- 31 (since exp [-iQ R ( ~ ~ Q / Q= ~4n/R1 ) so we get
This i s the instantaneous Coulomb inbraction between two c b r g e d p a ~ i c l e s .
98
THE THEORY O F F U N D A M E N T A L PROCESSES
The total interaction, wMch includes the i n k s c b g e of transverse photons, then gives rise to the rehrrled inQractian. Bremsatrcafitlung, Suppose that a n meson scatters from a heavy. particle of spin 0, for irrslance a K meson, Then it i s possibZe f a r i t b emit light, ( L a k r we shall work out the more pmctical case of spin 1/2,) There a r e ~ e v e mdiagrams l i~ lowest order (Fig. 20-2) and similar diagrams, where
the photcn i s @ m i a dby the K, However, we are i n k r e s k d in iz very heavy K, in which ease one can show that a e s e other diagrams can be xregleckd, The amplitudes for processes (a), (k), and (c) are?
Some simpliifications a r e innmediably apparent:
We ~ b 1 1 c, o ~ i d e the r case in which the K meson i s initially at seat md its mass Mr(:--ao The conservation of enerm at the photon --Kmeson verk x . then require6 that the virtual photon e n e r a
.
VIRTUAL AND R E A L PHOTONS Fu&her K1 and K2 have practically only time components M. We o b a i n
The h e a v K meson md the a meson only exchange timelike virtual photons of z e r o energy. The photon propagator l/$ i s then equal to I / Q ~ ,oorrespanding -to a s b t i e Coulomb inhractfon, Show t b t the sum a +- b + c i s gauge-invariant by sltoMng that i t vmishes if e iss in the direcaon k, x -. ak, ff we choose E s p a c e l i b , diagram c vanishes. T b differential c r a s s section f o r the s c a t b r i n g of the R meson ints a solid angle dSZ2 with emission of a, pfioblr with energy u into dQ, i s (choosing E spacelike)
where I3 its the density of final states (see Lecture X6):
Substituting o u r e w r e s s i o n f a r a , b, and I3 in do we get
The conservaUon of total e n e r o and momentum requires that
/
/
FIG, 20-3
100
THE THEORY O F FUNDAMENTAL PROCESSES
(See Fig, 20-3,) Summing over the polarization of the emitMd photons we ab-tain
- 2EiE2 vlvz sin e1 s i n B2 clas cp (1 - ~l eos @l)(f- ~2 cas e2) This i s the equivalent, for particles of spin 0, of the famous Bethe-Heitler formula far particles of spin 212.
2
Proble
P~obEem22-1: n " grams:
-- a -
scattering the e,m, syst-em, There are two dia-
Amp. = [(4n)jhe~2(pl+ P ~ I( ~ +3 PS(~/Q') Pi" Ps = Pz
+
P4
and the "exehmge" diagram pz
In the c.m. system,
g = Pl -Pz
-
pd
PI= -Pa= P; P2= -P4= Q; p2= Q'
where 6 fs the angle b-eween P and &, a s in Fig, 21-1, and v
E
PLE,
102
THE THEORY OF FUNDAMENTAL PROCESSES
Similarly,
Adding, we get
P~ob1em21-2: s * - n" seatkdng (a very irtbrestfag case), We have a diagram (Fig, 21-2). A s we discussed In Lecture 5, a n* (antipaI-Liele of n - )
FIG. 21-2
of energy moxnenhxn P i s represenbd by a, n- of 4-momentum p = -I" moving k c b a r d in time, The axrtplihde for the process i s
which shows that the n' has apposih electric charge of the n-, This i s always true of c h r g e d p a ~ i c l e sand their anlipa&icles, The amplitude f o r -Lhls praeess i s therefore
PROBLEMS
Since the n" and the 7 ~ - a r e distinct them is, of course, no excknge diagram. However, there i s an analogue to this diagram. Look at the diagram we get by changing connections s o that pi goes to p~ instead of p2,
in which the n- and the n+ amihilab, and the virtual photon r e c r e a k s the pair in the f i d stab, Vtre find for the seatbring matrix In the c,m. system,
-
PraltEeln 22 -3: Compton effect for K - , We consider the process y + A" + y . Them a r e three ways this can happen:
A-
since q s E = O and (p+ q)Z- rn2 = 2 p * q .
T H E T H E O R Y O F FZIPJDAMENTAL P R O C E S S E S
104
Consider the fmme of reference with tbe Initial pion a t rest, where Pi= 0. Take E~ = 0, Then
and the only contribution comes from diagram c above, This result comes from our particular choice of gauge Ea = 0. Note that the amplitude for each diagram is not gauge-invariant . The lilerature is full of false remarks a s to the relative masitude of varf otls diagrams, bXy the sum is gaugeinvariant, Show that the result is gauge-invariant, by shoMw t k t subs#tuting = E p + o q p produces no change in cross section; that is, substituting = aiql o r Q = as2give8 zero, Consider the frame with the initial pion at reat, Pi = 0:
EL
If the photon i s observed, w d not the pion pz , we can get a convenient, formula by eliminating & from th@equations by substituting pz = pl + kt k2. Squaring this gives mZ= rnZ + 2pg* kt 2pz* k2 2kl kz, or in our system,
-
-
-
a
o r the famoues Campton formula, for the change of kequency of ligM scattering from a free parGicle a t rest,
-
Pyoblem 21-4: rr " --7t. pair mnihifation in flight, Thia i s exactly analogous to the Compton effect, except that one of the n's i s now going backward in time.
A s before, we consider the frame in which PI= 0. Show by s q u a r i ~ gl = pS1 - 1%- Itz &at nn + Ez = w2(m + E2 - Pz cas 0). Then
We 8.658 that the total, cross section i s proportional to 2/v and becomes infinib a s v -+ 0, M a t does this mean? Suppost: w had a gas composed of n n's per unit volume, The probability per unit time that a n + moving through this gas with velocit;y v annihilahs 1s X / T = ngv, which i s a finik qumtity * F o r a baud R" - n' s y a b m (amlogous t;o pasitromium) n would bc? equal the squam of the wave funcUon at the origin and T i s the lifetime of the sysbm,
22
Spin-I Particles
Recall the two-component spinor, whose b h a v i o r under space robtions of angle B about unit vector a was d e s c r i b d by the o w r a t o r exp (lBn * M), where M = (1/2)o. In Problem 21-3 you were concerned with the b h a v i o r of this spinor under Larentz, transformation, As was the ease in space rotations i t is sufficiexlt to consider idinitesirnal transformations, We w r i h the corresponding operator as
where v is an infinikefmd velocity; c =. 1, Proceeding ae &fore, wle have for a fini* veXoci9 v in the z direction the o p r a t o r e q (iwN, ) with tanh w = v/e. Thus we need six o p e r a b r s to represent a general Larentz transformation:
corresponding Lo the six roktions in four-dimensional space. These quantiLies form an antisymmetrie LEtlnsor with camponen&
Either by algebra (studying successive b m n t z transhrmations) or by drawing figures we find the! earnmutation relations
All others commute; i.e.,
Tl.ree;e rules a r e a11 summarized by
Now we find the representation of the o w r a l o r N that acts on the twocomponent spinor u, F i r s t of all, N must be a 2 x 2 matrix,
We could put in some unknowns for the question marks and grind out the solution, using the commu.tation relations and M = (1/2) Q. I t i~ easier, however, if we notice that any 2 x % matrix e m be formed from a linear combination of the four matrices 1, a,, ~ j r qz. , SO we w r i k
We notice that N, and o, comnnutrt. Therefore g = h = 0. We find
We put these in t b commut;ation relation
Theref ore,
By c y ~i cf permulatian,
THE THEORY O F FUNDAMENTAL PROCESSES
Therefore,
To determine a, we substitute N = a@in N,NY
- NyN,
= iM,:
We can choose either sign far a. Suppose we choose the + sign, Then
-
However, consider the transformation proparties of the mirror-image spi--N,since v -v and N v ts a scalar; &M. I-+ nor. Under reflection N Thereflore a two-component spinor and its rntrrar image do not tmnsform in the same way under Lorentz transformation. In order: to have reflection invariance we need a four-component spinor. By writing 0; = a*v/lvl the o p e r a b r transforming u u d e r Lorentz transformation i s +
For instance consider the plane-wave s b t e u exp (-ip * x), For a. Lorentz transformation along the z axis, a, = rr, and u ' = expf -o, w/2) u, We can construct the general ease from the transformation of u
=
Since IV i s not Iferrniltian u* u ia not a scalar. Consider the transformation of u* u:
Now,
S P I N - 1/2 P A R T I C L E S
exp (--o; w) = cash w
- ui
u r * u ' = ccosh w (u* u)
~ i n MI h
- sinh w(u*uZu)
Also,
lxie notice immediately that u* u and u* a,u transform exactly like t,z, under Lorentz transformation:
Before we can conclude that u*u, u*cu form a 4-vector, we have to cheek. the analogue of x b x, y ' = y:
= u* exp f-u2w/2)
exp (+ a,w/2) o,u
Therefore we have discovered a new 4-vector, which we give a symbol Sp:
Now it appears that SI, might be satisfactory for the probability current. A s before, normalize s o that u*u = 2E, Then we have
Suppose we have a particle with spin up in the z direcuon:
110
TEIE T H E O R E " O F F U N D A M E N T A L P R O C E S S E S
There is trouble, b e a u s @the probabiliQ current u*czu is always rushing off in the z direction. This means that this probability current cannot represent a parlicle a t rest. Note that for this special case (u* u ) =~ (uf m)'. This is invariant under rotations, s o t h t , since any spinor must m p r ~ s s n at particle spinning in some direction, calling this z, we deduce tlre above is true in general, T b t is,
always, o r if we have SP = 2pp, we would have to have
Hence the prcjsent development is only valid for particles sf 0 mass ( a d spin I/2). We b o w of only one such particle--the neutrino. I t is posst'Pite to prove in general that Sl, opu = 0. (Prove it by first taking the case
and then arguing that it must then be true for any u . ) If we take Sp = 2pp we must have
We h k e %is a s the taw describing the neutrino, I t is true for each momentum plane wave, and hence for any superposition of such m @ ~ ,
where c p is any function of momentum. We can also convert this to an eqmtion in coordimh space, This quation ta simply
with
S P I H - % / ~P A R T I C L E S
Writkn out in full, the general equation is
Define crp 'cr*p/lpl. Since p = E, the equation (E - p o)u = O is equivalent to
This m e m s that the particle always spins clockwise in the direction of motion. Actually from e x p r i r n e n t we know t b t the neutrino p e s counterclockMse, However, r e m e m b r the other possibiliv for the slim of MI F o r N = i ~ [ 2 we find that the quantity which transforms like a 4-vecbr is
Xrt:
tMs case we get the equation
The particle described by v spins com&retackwise:
Xt i s essential Lo note that u and v transsform differently:
We say t k t u and v are, respectively, cospfnors and contmspinors, The corresponding t r m ~ f o m a t k o n sa r e called covariant and contravariant,
23
Extension of Finite M a s s
In Lecture 22 we saw that SI, = (u*opu) transforms like a 4-vector. This means that, f o r arbitrary B,,
i s an invariant, From this we can see that
b e h v s s difkrently from u under Lorentz transformations. Since uf = u* exp (-cr,w/2), (B u pu)' = exp (+ a,w/2) (BI, upU) Thus BP cr, u transforms like a contravariant spinor (u being a cospinor); if v i s a contraspinor then Bpopv i s a eavariant spinor. Exbnsion $0 Bfni* msm, We have found that f o r spin-l/2 parLlcles of mass 0, the equlations of mation a r e
(E - ~ * Q )=u O
right-handed
(Et-p . q v = O
left-handed
We notice Ghat such equations a m not invariant under space inversion, p being a polar v e c b r , o an m i a l vectoro (A few years ag;o this would have been sufficient reason for dmpping these equation%-as was done 25 years ago by PauXi in C%ffandbuettd e r Pkysik," p. 2226--but now we h o w that parity i s not conserved anyway, s o we shfl tick with our results.) Writing the first equation In the form
E X T E N S X O H O F FI-NIITIE: M A S S
113
we observe %at on the left-hand aide we have a contravariant quantit;y. Therefom if wf: want to add some b r m that deseribs the m a s s o r inte-raction of the particle, we k v to~b careful that it hzm the same t m n s f o m a -
tion prow*, F o r emmple, mu would b wrong, since u. i s a eovariant spinor. A f k r this, the simplest eligible t e r m for a source (linear in u) is of the fo mm
F o r example, the coupling for fi decay re~ent1Iydiscovered i s of this form; the interaction i s
For the p-decay, ul represents the neutrino, uz the p meson, ug the electron, and u4 the antineutrino, To find the t e r m in the eqmtlon of motion of ut we vary with respect to u: ; noting that u$oP U& is a w c t o r ( A p ) we s e e that we get the proposed form Al, ol,u2. Now consider mass %&in. The equation
behaves correctly under transformation, but since u Xzas M o components, this describes two indepndent particles of spin 0. The real difficulty shows up when we include electromagnetic interaction. Then the above eqmlion bcomes
The t e r n oeUI c k r a c h r i s t j c of spin-112 particles dws not follow from this equation. 1;Ve notice further that in the absence of in%raction there is no way to distinguish b t w e e n
and (E'
- p2]u = m 2 u
However in the presence of interaction, the substituMons E-E-y,
P-P-A
cla give different results for the two equations,
114
THE THEORY 0 F FUNDAMENTAX, PROCESSES
-
Nob t h t u i s covariant; (E p * 43 makes it contravariant and (E: + p ol) makes it cowriant &gain, Thus we introduce the contmvarjimt spinor v by
These coupled eqwtions t r a n s f a m correelly; when rn = O they give the previous results (but a r e no lower coupled), Together they a r e equivalent to the equation
We can combine (23-1) and (23--2) in f our-component spinor
a,
single equation by introducing the
Define the matrices
Similarly we have
Equations (23-1)m d (23-2) a r e then eummarized by
E X T E N S I O N O F FXNXTE M A S S
m9= (Eyt
- p*~)'lj[l
Equation (23-4)[or Eqs. (23-1) and (23-2)] i s known as the Dirac equation. It conbins mass and has the ~0rrf3-cttransformatian p r o p e ~ i e s .Qne can think of y , a s behaving like a 4-vector. [The Dirac equation is sometimes w r i t b n in the form
This is equivalent to (23-4) Mth the relations
It is useful to bow. the properties of the y matrices. We see eaaily that
The complete rule i s
In most problems one need not use an explicit represenbtion f o r the 7% but can derive everylthing from the commutation relations (23-51, The Current, By constructing a mixture of the sdates u and v we can find a probability current that can also represent parr*diclee a t rest. Recall that the quantities
a r e $-vectors. Suppoxse t b t we have a. particle with spin up in the r e s t fmme:
116
T'I1;E T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
We n a b t h t we can cancel the space parts by defining a new 4-vecbr that I s just the sum of the above vectors:
The new current has the exemplary p r o p d y that its s w c e component is O in the r e s t frame of the parliele, F u d h e r simplification may be made by writing SF in terms of 9. I t is easily seen that
where 9" i s the Hermitian conjugate: matrix to convenient form, we define
Jj,
To put this in a more
Then SI,= (Gy,@,Gy+) assumes the form
11;i s easy Lo s e e that (23-6) satisfies the continuiQ equation,
For, consider the Dfrac eqwtion and its coxljugate3,
Multiplying these equations by tively, and adding, we o b a i n
@ on the left and
9 on the right, respec-
which is Eq. (23-7). However u o r v by themselves cmnat f o m a conserved current. F o r example, Vp(u* cr, u) = 2m Im(u* v) * 0. [This follows from Eq. (23-1) for u.1 Finally we no@ that Eqs. (23- 1) and (23-2) are chaw@&i n b each oLher by the transformation
Sp is unchanged by this transformation. Thus the equations are invariant under reflection (but the P-coupEiw krn is not),
EXTENSION OF
FINITE MASS
117
AeHm Priaaiple. The D i m c equauon (23-4) [and hence (23-1) and (23-211 n n q be derived from the action
Introducing the useful notation (ap i s a 4-vector),
iI apYp we write the action for a particle of spin 1/2 In an electromagnetic field:
Varying 8' with respect to
gives the equation of motion f o r the particle
From this equation we s b l l see that the propagator f o r a particle of spin 1/2 is I/($ rn) F o r making calculations, since (I# m)($ + m) = p2 - rn2 we shall often w e the relation
-
-
From the coupling term e' %&@ in the Lagrangian we obtain the fundamental amplitude f o r the inbractfan of spinors and photons:
Properties of the Four-
We shall consider now the properties of the four-component spinor
which satisfies the Dirac equation
or in two-component form,
First of all, there a r e only two linearly independent solutions sf this eqwtion, Consequently it represents a pa&iele of spin 2/2, Wow does U transform P WB have Been tfrat under a Loren& transformation along the z =is,
Therefom, exp (-cr, w12)u
U"
We can w r i b this transfomatkon in more cornpael b r m usiw the 4 x 4 matrices
P R O P E R T I E S QF THE FOUR-COMPONENT
SPINOjR
219
introduced earlier. We have
and
Since this transformati on eorresgonds to a rotation in the tz plane, we expect similarly that M, = (i/2) y,yy f o r a rotation in the xy plane. Let us cheek, Substituting our representation f o r the y matrices we find
and
Let u s return to the problem of describing the spin staks. I f the particle i s a t r e s t the Dirac equation i s just
This shows that there a r e only two solutions, which we can k k a as spin up and down along some axis. F o r ins.lance, f o r spin up along the z axis, we have
o,U=
u
However, if the particle is moving, u
* v (since
u and v b h a v e differ-
f 20
T H E T H E O R Y 03F F U N D A M E N T A L P R O C E S S E S
ently, under Lorentz tramformations). We must b more careful in deseribin%t;he direction of' spin of a moviw pare-iele, If we Cake cr along the dire@tion of motion, it is possible to describe the solutions a s spin up (&$fit helicity) o r d o w (left helicity):
But o along the directfan of motion is not a Lorentz-invariant idea. If q i s in an arbitrary direction, we cannot find a solution of the Dirac equaMon which i s also an efgenfunctlon of rt (Q and $ do not commute). Let us t r y to find another way of describiw the spin shbs. Returniw t o the r e s t frame we have
Then also
Now, we i n t r d u c e the matrix
whie h i s Losentz -invariant, and w r i b
Also, let W be a 4-vector satisfying WppP = 0, W W = -1. In the r e s t p. lJ frame, W, = 0 and W i s a unit vector in any direct~on.In particular if W is along the z axis we h v e C F ~ Y=, iWyS. Therefore tT satisfies i w 6 U = tT, We started in the rest frame, but now the equation ia tarentz-invariantt i.e,, valid in any frame, Henee for a moving particle, the 2-spin shztes a r e = -1. Physically they represent eigenstates of iWy5, where ( W e p) = 0, spin up o r down along some =is in the reat frame of the padiele. m e n we do a problem, we shlt find in general that the amplitude is of l , M is a combination of y matrices and U,, U2 the form m = g 2 ~ uwhere a r e the inktiat md Bnal rapin sCabs, respectively. The task i s to conrxpuk the pr-ohbilitcy, which i s proportiom2 to
where M is M with the order of all y's reversed and each explieit i-- -i.
P R O P E R T I E S O F T H E F O U R - C O M P O N E N T SPIXIJOR
121
[From the definition 6= U+ y, we s e e .% = y, ( y , MI*, where * means Hermitian adjoint. This rule f o r % does not show the invariance clearly. The rule gitren above i s simpler, Check that they %reel f o r yourself, F o r insbnce,
which i s very useful.] There a r e two ways of calculating this. The f i r s t i s the obvious way. Sol% the pair of equations
f o r U1 and U2 and then cornpub
A much b t k r way, which is ordinarily ussed in practice, i s the fofloMng trick. Suppose that we a r e not i n b r e s t e d in the final spin states, Then what we want i s
22
2 spin s h t m of lfs
(~IMu~)(~~Mu,)
This can be writ&n in the form
where X = 2, spinsUZU2i s a 4 X 4 matrix (note the "wrong" o r d e r of U and U). m a t is this matrix? Let u s take a co-ardinab system in which the particle is a t r e s t 6 = my,. Solutions a r e (normalizing 6~to 2rn and dropping the subscript 2):
Then
T H E T H E O R Y OF F U N D A M E N T A L P R O C E S S E S
122
and
o r , in invariant form, X = pl" +- m, which fs then valid in all frames of reference, Incidentally, another way this can be ~ n d e r s t o s di s to note t b t the law of matrix multiplication implies that
r;
(E,Au)(GBu,)
=
d I 4 stat= of If
zm(G,~su,)
The four shks U a r e not only the two belonging to the eigenvalue + m of ftj, $U = mZT which we want, but also two other stzltes blonging t o the other elgenvalue -m, $U' = -mU' But if we write A = M ( $ + m), we sball get zero f o r AU" 0 f o r the mwanted states and AU =; %U 2m f o r the wanted states. Therefore,
.
If in addition the incident s k t e is rxnpolarized we must average over the two spinors U1, If we now use the fact that 4skW
(tiiA U ~ = 2m spur A
we s e e that
I: ( G l ~ ~ g ) ( i j 2 =~ spur[M($g ~i)
spin 1, spin 2
+
m)($i
+
m)I
L a k r on we shall dfscusss what to do when wt? are i n h r e s t e d in, the spin sbtes, Our whole proMem has been reduced to the calculation of the spur of a combination of y matrices. How do we calculate these s p u r s ? We note (look a t the sum of the diagonal elements of the matrices y, ,y, , given g r e viously)
...
PROPERTIES O F THE FOUR-COMPONENT
SPXNOR
123
SPY, = QZ
F o r any two matrices A, B,
sp(crA + /3 B) = cr s p A, + P sp B
a,@ complex n u m b r s
Using this rule we find
but
Only one spur is not zero-the spur of t b unit matrk: spur 1, = 4, This is a tremendous simplification; to find the spur of any complex praduct of y matrices we need only find the component along the unit matrix, (There a m sixleen linearly independent products of y matrices and any 4 x 4 m a t r h e m be reduced as a linear combination of these, just a s any 2 x 2 m a t r h can be? wri&en as a linear corvlbinatioa of the three Pauli spinors B and the identity,) The swr af any produet of an A d nuxnbr of y matrices must vmish, To reduce a produet of an even n u m b r of y m a t ~ c e we s proceed as fallows:
but
T H E T H E O R Y 01;;F"U N D A M E N T A L P R O C E S S E S
sp (&tB) = 4E(a * b)(c d)
- (a c)(b d) *
*
I- (a, * d)(b
- el]
The idea i s to push the first finear eombimtion of y matrices to the right, a t each s k p using the identi@
Wen ;k maehes the other side we get back; t k spur we sLarkd, but d t h opposik s i p , since we have an odd n u r n b r af t r a n ~ p s i l i o n s ,The remaining spurs will now c o n ~ i s of t a product of two less y matrices, and the whole procedure ia repeakd until tve get to the unit matrix,
2
The Effect
To get some familiarity with the spur techniqwe we shall ealcul& in detail the Compton effect, the scatksing of a photon from a free electron. Two diagrams ~ o n t d b u Mto this process:
'/'
Amp. = E2(&a)
@
dz
x Il/(plf + 4'
- w1
For complex polarization El,%; wave going out couples with C: (like an outgoing wave function). From left to right we have: (4~)'' ' e amplitude to absorb the incident photon, I/($i + (Ift - m) = amplitude for propagation af the virtual electron, ( 4 ~ ) ' / ~g1 e2 = amplitude for emission of the photon in the f i m l st;ak* At each v e d x energy and momentum must bc? conserved, The totall annplj;tude ia the eomnponrtti~t&Ween the initial, and f i n d s k k oE the electron. Adding the d i a g r a ~with the order of absorption and emission reversed we find then m = 4rre2UzMU,, where
el,
This is all the physic8 in the problem; the rest i s pure algebra, First, we rationalize denomrimtf)r~,
THE THEORY OF FUNDAMENTAL PROCESSES
126
Also, (pi
+ q1)2
- mZ= pi
+ 2pl *ql+ q:
- m Z= 2pl *ql
- 4 1 j 2 - m2= - 2 ~ 2.q2 and, therefore,
Note that M is taken between
e2 and U1, since
rllTIf2 =
We obtain further simplification by moving
$it
to the right, Note that
Fimlly if we choose the frame of referenee in which the electron is initially at rest, we have
since
E1
and E 2 are spacelike a&
To cafcufate the scattering cross section w need 1/2
z
x
spin 1 spin 2
= 1/2 spur
(see Lecture 24).
(ijlM~Z)(c2~~I)
[M($2 + m) ~ ( f m)] i ~ +
THE CQMPTON E F F E C T
Let
Then
Consider A. Since dl & = -1, we shall try to get the two Bob that 4% = -&(1 6%.
Now we use
to get
3E we inhrehaqe X 1 =8
For pi ;= (m,O)
-
I~(R
E; 2,
qr
-
E~)(qz* El)
qz
+
WE? get
fC& * ~ z ) (92 l ' PI)
together.
THE THEORY O F F U N D A M E N T A L PROCESSES
128
and
Next consider 13. WE: begin
by moving
$1
to the right:
B = sp tdt d1@2(&+ m)B(sd2BI: ($1 mil +
=
z(c ,*qg){ ~p[gl;Ld1#2($2
-2
+
Pushing
&'2
+
m)l = a)
( ~* t f I{ sp [d:PIrdg (62 + mI4z ($2
+{BP [drdz(Ilz + m)dz $:($I
we get
m)s':~t
+ mII =
PI
- mil = 71
do the right in cr a d y and substituting g& = 161 + dl
- dz in@
129
T H E COMPTON E F F E C T
The last bmm can b simplified by substituting
Finally,
-
B = 8 { 1 ~ ~ . ~ 1 ~\ qml w * ~- 2 1 2+mrnwlmwZ ~2 [ 2 / ~ 1 * ~ $ 11 1~ + [2(cTeE ~ ) ( E $-q1)(E1 *f&)
- 2(61
*
cf)(c2 * 4 t ) ( ~ . ?42)1 m w l l
A similar cdculatian fox: G gives B* = C (so tbat the last two brms in B cancel out In the sum B -+ C), Note that this result earnot be ~btSLinedby just ~ q l - % in the fimf expression for B, beauae inhrcbangiq c ~ - & and we obtained it in a swelal frame of referenee; mmely, that one for wMch pi = (m,O), (One gets it by reversing the o d e r of all f a c b r s in. B,)Gollecting our results we get
1/2 spur [M($2
+
m)(M(dr+
mil
The scathring eraas section i s given by ( ~ e ebetare 163)
D i s the den~i"cyof outgoiw s a t e s per unit range
130
TIfE T H E O R Y O F F U N D A M E N T A L P R O C E S S E S
8 i s the angle b t w e e n the incident and outgoing photon, Substituting in dcr we ob,t;ain finally
In the nonrelativistic limit (wt > m) we have wz cept near 8 = 01, and
rn
