Theoretical Advanced Study Institute in Elementary Particle Physics
FLAVOR PH SICS FOR THE MILLENNIUM TASI OOO
H
Theoretical Advanced Study Institute in Elementary Particle Physics
FLAVOR PHYSICS FOR THE MILLENNIUM TASI2000
This page is intentionally left blank
Theoretical Advanced Study Institute in Elementary Particle Physics
FLAVOR PHYSICS FOR THE MILLENNIUM TASI2000 Boulder, Colorado, US
4 - 3 0 June 2000
Editor
Jonathan L. Rosner University of Chicago
fe World Scientific m
New Jersey Singapore • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
TASI2000 Flavor Physics for the Millennium Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4562-9
Printed in Singapore by Uto-Print
FOREWORD The Theoretical Advanced Study Institute (TASI) in Elementary Particle Physics has been held every summer since 1984, and since 1989 at the University of Colorado in Boulder. It is aimed at graduate students desiring training in areas not typically covered in sufncent breadth and depth at their home institutions. No other North American summer school in elementary particle theory covers such a variety of topics with such pedagogic care. Topics of the four-week summer school have been chosen to alternate among several areas, including string theory, phenomenological particle theory, and particle astrophysics. The lecture note series have been in great demand for many years after their publication, as a reference for students, researchers, and teachers. The TASI 2000 Summer School was devoted to the broad subject of flavor physics, embracing the question of what distinguishes one type of elementary particle from another. Lectures ranged from the forefront of formal theory (treating the physics of extra dimensions) to details of particle detectors. Although special emphasis was placed on the physics of kaons, charmed and beauty particles, top quarks, and neutrinos, lectures were also devoted to electroweak physics, quantum chromodynamics, supersymmetry, and dynamical electroweak symmetry breaking. Violations of fundamental symmetries such as CP and time-reversal invariance were discussed in several contexts. The physics of the Cabibbo-Kobayashi-Maskawa matrix and of quark masses was described, both from the standpoint of present and future experimental knowledge and from the viewpoint of an eventual underlying theory. All the lectures of the Summer School are reproduced here, with the following exceptions. Carl Wieman felt that his lectures dealt with already-published material and has kindly supplied a bibliography. Because of other pressures, Scott Thomas and John Wilkerson were unable to supply manuscripts. I am grateful to the lecturers for their participation in the Summer School, to Hitoshi Murayama for helping me organize the scientific program, to Tom DeGrand for helping participants share the beauties of Colorado during the weekends, to Mu-Chun Chen for her assistance in the daily operation of the Institute, and especially to K. T. Mahanthappa and Kathy Oliver for their careful attention to all details of the school. Partial support for the TASI program was provided by the United States National Science Foundation, the United States Department of Energy, and the University of Colorado. Jonathan L. Rosner Chicago, Illinois May 3, 2001 v
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CONTENTS
Foreword Jonathan L. Rosner
v
The Electroweak Theory Chris Quigg
3
CP Violation Lincoln Wolfenstein
71
Precision Electroweak Physics Young-Kee Kim
87
Kaon and Charm Physics: Theory G. Buchalla
143
Kaon Physics: Experiments A. R. Barker
209
The Status of Mixing in the Charm Sector J. P. Cumalat
243
Basics of QCD Perturbation Theory Davison E. Soper
267
Lattice QCD and the CKM Matrix Thomas De Grand
319
The Strong CP Problem Michael Dine
349
A Bibliography of Atomic Parity Violation and Electric Dipole Moment Experiments Carl E. Wieman
373
The CKM Matrix and the Heavy Quark Expansion Adam F. Falk
379
CP Violation in B Decays Jonathan L. Rosner
431
VIII
Lectures on the Theory of Nonleptonic B Decays Matthias Neubert
483
Asymmetric e+e~ Colliders Aaron Roodman
539
Pathological Science Sheldon Stone
557
Top Physics Elizabeth H. Simmons
579
Neutrino Mass, Mixing, and Oscillation B. Kayser
625
Flavor in Supersymmetry Hitoshi Murayama
653
Technicolor and Compositeness R. Sekhar Chivukula
731
Models of Fermion Masses Graham G. Ross
775
Physics of Extra Dimensions Joseph D. Lykken
827
liilil
Chris Quigg
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T H E E L E C T R O W E A K THEORY CHRIS QUIGG Theoretical Physics Department Fermi National Accelerator Laboratory Batavia, IL 60510 USA E-mail:
[email protected] After a short essay on the current state of particle physics, I review the antecedents of the modern picture of the weak and electromagnetic interactions and then undertake a brief survey of the SU(2)L ® U(l)y electroweak theory. I review the features of electroweak phenomenology at tree level and beyond, present an introduction to the Higgs boson and the 1-TeV scale, and examine arguments for enlarging the electroweak theory. I conclude with a brief look at low-scale gravity.
1 1.1
Introduction Our picture of matter
At the turn of the third millennium, we base our understanding of physical phenomena on the identification of a few constituents that seem elementary at the current limits of resolution of about 10~ 18 m, and a few fundamental forces. The constituents are the pointlike quarks
CO. C). (0,and leptons
(A (A (A-
(2)
with strong, weak, and electromagnetic interactions specified by SU(3)C SU(2)L ® U(l)y gauge symmetries. This concise statement of the standard model invites us to consider the agenda of particle physics today under four themes. Elementarity. Are the quarks and leptons structureless, or will we find that they are composite particles with internal structures that help us understand the properties of the individual quarks and leptons? Symmetry. One of the most powerful lessons of the modern synthesis of particle physics is that symmetries prescribe interactions. Our investigation of symmetry must address the question of which gauge symmetries exist (and, eventually, why). We must also under3
4
stand how the electroweak symmetry" is hidden. The most urgent problem in particle physics is to complete our understanding of electroweak symmetry breaking by exploring the 1-TeV scale. Unity. We have the fascinating possibility of gauge coupling unification, the idea that all the interactions we encounter have a common origin—and thus a common strength—at suitably high energy. Next comes the imperative of anomaly freedom in the electroweak theory, which urges us to treat quarks and leptons together, not as completely independent species. Both ideas are embodied in unified theories of the strong, weak, and electromagnetic interactions, which imply the existence of still other forces—to complete the grander gauge group of the unified theory—including interactions that change quarks into leptons. Supersymmetry and the self-interacting quanta of non-Abelian theories both hint that the traditional distinction between force particles and constituents might give way to a unified understanding of all the particles. Identity. We do not understand the physics that sets quark masses and mixings. Although experiments are testing the idea that the phase in the quark-mixing matrix lies behind the observed CP violation, we do not know what determines that phase. The accumulating evidence for neutrino oscillations presents us with a new embodiment of these puzzles in the lepton sector. At bottom, the question of identity is very simple to state: What makes an electron an electron, a neutrino a neutrino, and a top quark a top quark? 1.2
QCD is part of the standard model
The quark model of hadron structure and the parton model of hard-scattering processes have such pervasive influence on the way we conceptualize particle physics that quantum chromodynamics, the theory of strong interactions that underlies both, often fades into the background when the standard model is discussed. I want to begin these lectures on the electroweak theory with a clear statement that QCD is indeed part of the standard model, and with the belief that understanding QCD may be indispensable for deepening our understanding of the electroweak theory. Other lecturers will explore the application of QCD to flavor physics. Quantum chromodynamics is a remarkably simple, successful, and rich theory of the strong interactions. 6 The perturbative regime of QCD exists, a a n d , no doubt, others—including the symmetry that brings together the strong, weak, and electromagnetic interactions. ''For a passionate elaboration of this statement, see Frank Wilczek's keynote address at PANIC '99, Ref. ' . An authoritative portrait of QCD and its many applications appears in the monograph by Ellis, Stirling, and Webber, Ref. 2 .
5 1
10
100
1000
Figure 1. The structure function F2 measured in uN interactions, from Ref.
3
.
thanks to the crucial property of asymptotic freedom, and describes many phenomena in quantitative detail. The strong-coupling regime controls hadron structure and gives us our best information about quark masses. The classic test of perturbative QCD is the prediction of subtle violations of Bjorken scaling in deeply inelastic lepton scattering. As an illustration of the current state of the comparison between theory and experiment, I show in Figure 1 the singlet structure function ^ ( z , Q2) measured in l/N chargedcurrent interactions by the CCFR Collaboration at Fermilab. The solid lines for Q2 ^ ( 5 GeV/c)2 represent QCD fits; the dashed lines extrapolate to smaller values of Q2. As we see in this example, modern data are so precise that one can search for small departures from the QCD expectation.
6
CDF Preliminary
50
100
150
200
250
300
350
400 450 ET (GeV)
Figure 2. Cross sections measured at y/s = 1.8 TeV by the CDF Collaboration for central jets (defined by 0.1 < \r)i | < 0.7), with the second jet confined to specified intervals in the pseudorapidity 772.4 The curves show next-to-leading-order QCD predictions based on the CTEQ4M (solid line), CTEQ4HJ (dashed line), and MRST (dotted line) parton distributions.
Perturbative QCD also makes spectacularly successful predictions for hadronic processes. I show in Figure 2 that pQCD, evaluated at next-toleading order using the program JETRAD, accounts for the transverse-energy spectrum of central jets produced in the reaction pp —> jet j + jet 2 + anything
(3)
over at least six orders of magnitude, at -y/i = 1 . 8 TeV. c c
F o r a systematic review of high-Ex
jet production, see Blazey and Flaugher, Ref.
.
7
CDF Preliminary ots(Mz) = 0.1106 +/- 0.0001 (exp.stat.) weighted average over E- region 40 - 250 GeV
0.12 0 W 0.10 *
0.08
-B(ET) as function of E-
0.06 0.04 0.02 0
M R S A ' parton distributions, n=E-/2 Input a s ( M z ) = 0.110
50
100
150
200
250
300
350
400
Transverse Energy [GeV] Figure 3. Determinations of cts inferred from the comparison of measured inclusive jet cross sections with the JETRAD NLO Monte-Carlo program. Source of this figure is http://www-cdf.fnal.gov/physics/new/qcd/qcd9943ub_blessed.html.
The (Revolution of the strong coupling constant predicted by QCD, which in lowest order is l/as(Q2)
= l/as(n2)
+
33
-In 12TT
log(Q2/y
(4)
where nj is the number of active quark flavors, has been observed within individual experiments 6,7 and by comparing determinations made in different experiments at different scales/ Figure 3, from the CDF Collaboration, shows the values oi as{Ex) inferred from jet production cross sections in 1.8-TeV pp collisions. The curve shows the expected running of the strong coupling constant. For a review, see Hinchliffe and Manohar, Ref.
10 Q [GeV] Figure 4. Determinations of 1/cts, plotted at the scale \x at which the measurements were made. The line shows the expected evolution (4).
1
'
1 '
1
1
1 ,T
, e+e~ rates
1
'
1
J
T„+c,
*"
e+e~ event shapes
• fi + X, but found no evidence for the production of electrons. Their study established that the muon produced in pion decay is a distinct particle, v,,, that is different from either ve or ve. This observation suggests that the weak (charged-current) interactions of the leptons display a family structure, (30)
We are led to generalize the effective Lagiangian (24) to include the terms
£v-\ = "rTf"""7^1 ~ 75 ^ ^"t 1 ~ 75^e + hx- >
(31)
in the familiar current-current form. With this interaction, we easily compute the muon decay rate as T(n -» eveVp) =
1927T3
(32)
22 With the value of the Fermi constant inferred from /? decay, (32) accounts for the 2.2-/US lifetime of the muon. The resulting cross section for inverse muon decay, o-(^e -> fj,ve) = (TV-A{vee -> vee) 1
{ml - ml) 2meEl/
2
(33)
is in good agreement with high-energy data in measurements up to E„ as 600 GeV. However, partial-wave unitarity constrains the modulus of an inelastic amplitude to be \Mj\ < 1. According to the V — A theory, the J = 0 partial-wave amplitude is M0
GF • 1meEv TIV2
{ml-ml) 2meE„
(34)
which satisfies the unitarity constraint for Ev < ir/GFme\/2 « 3.7 x 108 GeV. These conditions aren't threatened anytime soon at an accelerator laboratory (though they do occur in interactions of cosmic neutrinos). Nevertheless, we encounter here an important point of principle: although the V — A theory may be a reliable guide over a broad range of energies, the theory cannot be complete: physics must change before we reach a c m . energy y/s fa 600 GeV. A few weeks after my TASI00 lectures, members of the DONUT (Direct Observation of NU Tau) experiment at Fermilab announced the first observation of charged-current interactions of the tau neutrino in a hybrid-emulsion detector situated in a "prompt" neutrino beam. 39 The vT beam was created in the production and decay of the charmed-strange meson D+^ vT + anything.
(35)
Their "three-neutrino" experiment was modeled on the two-neutrino classic: a beam of neutral, penetrating particles (the tau neutrinos) interacted in the hybrid target to produce tau leptons through the reaction vTN —> T + anything.
(36)
Although extensive studies of r decays had given us a rather complete portrait of the interactions of vT, the observation of the last of the known standardmodel fermions gives a nice sense of closure, as well as a very impressive demonstration of the experimenter's art. We have a great deal of precise information about the properties of the leptons, because the leptons are free particles readily studied in isolation. All of them are spin-4, pointlike particles—down to a resolution of a fewx 10""l cm.
23 Table 1. Some properties of the leptons.
Lepton
Mass
Lifetime
e" ve H~ j/„
0.510 999 07 ± 0.000 000 15 MeV/c < 10 - 15 eV/c2 105.658 389 ±0.000 034 MeV/c2 < 0.19 MeV/c2 (90% CL)
T~ i/T
1777.06to 26 MeV/c2 < 18.2 MeV/c2 (95% CL)
2
> 4.3 x 10 23 y (68% CL) 2.197 03 ± 0.000 04 x 10~ 6 s 290.2 ± 1.2 x 10~ 15 s
The kinematically determined neutrino masses are all consistent with zero, though the evidence for neutrino oscillations argues that the neutrinos must have nonzero masses. A brief digest of lepton properties is given in Table 1. An important characteristic of the charged-current weak interactions is their universal strength, which has been established in great detail. We'll content ourselves here with the most obvious check for the lepton sector. Using the generic formula (32) for muon decay, we can use the measured lifetime of the muon to estimate the Fermi constant determined in muon decay as l
G, = fl^lY V Ttirnfi
= L
i638
x
io-5
GeV-2
(37)
J
Similarly, we can evaluate the Fermi constant from the tau lifetime, taking into account the measured branching fraction for the leptonic decay. We find l
3 (T(T^evevT) 1927r „3t /i\2(38) °r = - b TTT1 r= 1-1642 x 10~ 5 GeV~ 2 . V T(T -> all) rrm\ ) Both are in excellent agreement with the best value of the Fermi constant determined from nuclear (3 decay,0
Gp = 1.16639(2) x 10- 5 G e V - 2 .
(39)
The overall conclusion is that the charged currents acting in the leptonic and semileptonic interactions are of universal strength; we take this to imply a universality of the current-current form, or whatever lies behind it. °In this discussion, but not in the number quoted, I'm glossing over the complication that the strangeness-preserving transition is not quite full (universal) strength. We'll encounter "Cabibbo universality" in §3.5.
24
3
The SU(2) L ® U ( 1 ) Y Electroweak Theory
Let us review the essential elements of the SU(2)L ® U(l)y electroweak theory. 37 ' 40,41 The electroweak theory takes three crucial clues from experiment: • The existence of left-handed weak-isospin doublets, ve e
and
;,
;.
(i
The universal strength of the (charged-current) weak interactions; • The idealization that neutrinos are massless. 3.1
A theory of leptons
To save writing, we shall construct the electroweak theory as it applies to a single generation of leptons. In this form, it is neither complete nor consistent: anomaly cancellation requires that a doublet of color-triplet quarks accompany each doublet of color-singlet leptons. However, the needed generalizations are simple enough to make that we need not write them out. To incorporate electromagnetism into a theory of the weak interactions, we add to the SU{2)L family symmetry suggested by the first two experimental clues a U{\)Y weak-hypercharge phase symmetry/ We begin by specifying the fermions: a left-handed weak isospin doublet L = ( - )
L
m
with weak hypercharge YL — — 1, and a right-handed weak isospin singlet R = eR
(41)
with weak hypercharge YR = — 2. The electroweak gauge group, SU{2)L ® U(l)y, implies two sets of gauge fields: a weak isovector 6^, with coupling constant g, and a weak isoscalar p
W e define the weak hypercharge Y through the Gell-Mann-Nishijima connection, Q I3 + hY, to electric charge and (weak) isospin.
25
Ap, with coupling constant g'. Corresponding to these gauge fields are the field-strength tensors Flu = dX, - dX
+ gejkifyl
,
(42)
for the weak-isospin symmetry, and
for the weak-hypercharge symmetry. We may summarize the interactions by the Lagrangian L — -Cgauge "T AWeptons t
V
/
with •£>gauge =
—
~ir p^r
— -ijpvj
,
(45)
and Aeptons = R il" (d, + ijA^Y) + L i 7 " (d,, + i^A^Y
R
(46)
+ i9-f • b^j L.
The SU(2)L ® U(l)y gauge symmetry forbids a mass term for the electron in the matter piece (46). Moreover, the theory we have described contains four massless electroweak gauge bosons, namely _4M, 6* 62 and 6^, whereas Nature has but one: the photon. To give masses to the gauge bosons and constituent fermions, we must hide the electroweak symmetry. The most apt analogy for the hiding of the electroweak gauge symmetry is found in superconductivity. In the Ginzburg-Landau description 42 of the superconducting phase transition, a superconducting material is regarded as a collection of two kinds of charge carriers: normal, resistive carriers, and superconducting, resistanceless carriers. In the absence of a magnetic field, the free energy of the superconductor is related to the free energy in the normal state through ( ^ s u p e r (UJ — ^ n o r m a l
(0) + a H 2 + / ? k / > | 4
,
(47)
where a and /? are phenomenological parameters and \ip\ is an order parameter that measures the density of superconducting charge carriers. The parameter /? is non-negative, so that the free energy is bounded from below. Above the critical temperature for the onset of superconductivity, the parameter a is positive and the free energy of the substance is supposed
26
Order Parameter \\t
Order Parameter \|/
Figure 11. Ginzburg-Landau description of the superconducting phase transition.
to be an increasing function of the density of superconducting carriers, as shown in Figure 11(a). The state of minimum energy, the vacuum state, then corresponds to a purely resistive flow, with no superconducting carriers active. Below the critical temperature, the parameter a becomes negative and the free energy is minimized when rj> = rpo = yj—a/p ^ 0, as illustrated in Figure 11(b). This is a nice cartoon description of the superconducting phase transition, but there is more. In an applied magnetic field H, the free energy is Gsuper(-n
) — Gsuper(O) +
H2
1
+ ^ ^ I
ihVip -
(e*/c)Aip\2
(48)
where e* and m* are the charge (—2 units) and effective mass of the superconducting carriers. In a weak, slowly varying field H m 0, when we can approximate ip « ipo and V ^ « 0, the usual variational analysis leads to the equation of motion,
V2I-i?^o|2I=0,
(49)
the wave equation of a massive photon. In other words, the photon acquires a mass within the superconductor. This is the origin of the Meissner effect, the exclusion of a magnetic field from a superconductor. More to the point for our purposes, it shows how a symmetry-hiding phase transition can lead to a massive gauge boson. To give masses to the intermediate bosons of the weak interaction, we take advantage of a relativistic generalization of the Ginzburg-Landau phase
27
transition known as the Higgs mechanism. 43 We introduce a complex doublet of scalar fields
with weak hypercharge Y$ — + 1 . Next, we add to the Lagrangian new (gaugeinvariant) terms for the interaction and propagation of the scalars, ^scalar = {V)\V^
- Vtfj),
(51)
+ i9-r • b^ ,
(52)
V(^)2-
(53)
where the gauge-covariant derivative is Vll = dll+ i^AMY and the potential interaction has the form
We are also free to add a Yukawa interaction between the scalar fields and the leptons, £ Yukawa = -Ce [R(^L) + (L»R] .
(54)
We then arrange their self-interactions so that the vacuum state corresponds to a broken-symmetry solution. The electroweak symmetry is spontaneously broken if the parameter /i 2 < 0. The minimum energy, or vacuum state, may then be chosen to correspond to the vacuum expectation value
Wo = (
^ ,
(55)
where v = \J—n2/ |A|. Let us verify that the vacuum (55) indeed breaks the gauge symmetry. The vacuum state (^)o is invariant under a symmetry operation exp (iaQ) corresponding to the generator Q provided that exp(iaQ)()o, is., if G{)o = 0. We easily compute that ^ ) o = ( : : ) ( ^ ) = ( ^ ) ^
—
^=(!7)U)-(-^)^broken! M*)o = (1°)
(JK) 0-lJ\v/y/2j
=L,/./o^0
\-v/y/2/
Y()0 = Y^)0 = +l(0)o = (J^\
#0
broken! broken!
(56)
28
However, if we examine the effect of the electric charge operator Q on the (electrically neutral) vacuum state, we find that
QWo
= i ( r 3 + Y)Wo
=
(oo) (v/U)
= ! ( \
=
{I)
+ 1
y
0
° _ i ) w °
unbroken!
(57)
The original four generators are all broken, but electric charge is not. It appears that we have accomplished our goal of breaking SU(2)L ® U{\)y —> C^(l)em- We expect the photon to remain massless, and expect the gauge bosons that correspond to the generators T\, T^, and K = \{T3 — Y) to acquire masses. As a result of spontaneous symmetry breaking, the weak bosons acquire masses, as auxiliary scalars assume the role of the third (longitudinal) degrees of freedom of what had been massless gauge bosons. Specifically, the mediator of the charged-current weak interaction, W± = (b\ =Fi&2)/v2, acquires a mass characterized by MW
= Y
•
( 58 )
With the definition g' = g tan Qw, where 8\y is the weak mixing angle, the mediator of the neutral-current weak interaction, Z — 63 cos 6\y — A sin 9\y, acquires a mass characterized by Mf = M^, / cos2 8\y • After spontaneous symmetry breaking, there remains an unbroken U(l)em phase symmetry, so that electromagnetism, a vector interaction, is mediated by a massless photon, A = A cos 9w + b3 sin 6\y, coupled to the electric charge e = gg'/\/g2 + g'2. As a vestige of the spontaneous breaking of the symmetry, there remains a massive, spin-zero particle, the Higgs boson. The mass of the Higgs scalar is given symbolically as Mjj = — 2fi2 > 0, but we have no prediction for its value. Though what we take to be the work of the Higgs boson is all around us, the Higgs particle itself has not yet been observed. The fermions (the electron in our abbreviated treatment) acquire masses as well; these are determined not only by the scale of electroweak symmetry breaking, v, but also by their Yukawa interactions with the scalars. The mass of the electron is set by the dimensionless coupling constant £e = mey/2/v, which is—so far as we now know—arbitrary.
29 3.2
The W boson
The interactions of the W-boson with the leptons are given by Av-iep = ^ | [*e7"(l -7s)eW+
+ e 7 " ( l -fs)veW-]
.etc.,
(59)
so the Feynman rule for the veeW vertex is
2y/2
The W-boson propagator is ^ \ / v / \ / \ .
7A(1-75J
' ^
=
k
^lMw
Let us compute the cross section for inverse muon decay in the new theory. We find ff4me£„ e+e-) = T(Z -»• i/P) [L + i? ] . The neutral weak current mediates a reaction that did not arise in the V — A theory, i/^e —>• i/^e, which proceeds entirely by Z-boson exchange:
e
This was, in fact, the reaction in which the first evidence for the weak neutral current was seen by the Gargamelle collaboration in 1973. 44 It's an easy exercise to compute all the cross sections for neutrino-electron elastic scattering. We find
,( v -, v ) = ^
a(vee -» vee) =
[
(
3^L
9ee) = Gr™°E"
^
+
^/3] ,
[{Le + 2f
+ i^/3] ,
[(Le + 2) 2 /3 + R2e] •
(77)
By measuring all the cross sections, one may undertake a "modelindependent" determination'' of the chiral couplings Le and Re, or the traditional vector and axial-vector couplings v and a, which are related through a = \{Le-Re) Le = v + a
v=
\{Le-Re)
Re = v — a
By inspecting (77), you can see that even after measuring all four cross sections, there remains a two-fold ambiguity: the same cross sections result ' I t is model-independent within the framework of vector and axial-vector couplings only, so in the context of gauge theories.
33
if we interchange Re • — Re, or, equivalently, v a. The ambiguity is resolved by measuring the forward-backward asymmetry in a reaction like e+e~ —> n+n~ at energies well below the Z° mass. The asymmetry is proportional to (Le — i? e )(I/ /J — Rfi), or to aea^, and so resolves the sign ambiguity for R or the v-a ambiguity. 3.4
Electroweak interactions of quarks
To extend our theory to include the electroweak interactions of quarks, we observe that each generation consists of a left-handed doublet
h
Q Y =
2(Q-h) (79)
+ and two right-handed singlets, h
Q Y =
Ru = uR
0 +|
Rd = dR
0 —g
2{Q~I3) +|
(80)
—3
Proceeding as before, we find the Lagrangian terms for the VF-quark chargedcurrent interaction, iV-quark = ^
[ « e 7 " ( l ~ fs)dW+
+ d7"(l -
l5)u
W~]
,
(81)
which is identical in form to the leptonic charged-current interaction (59). Universality is ensured by the fact that the charged-current interaction is determined by the weak isospin of the fermions, and that both quarks and leptons come in doublets. The neutral-current interaction is also equivalent in form to its leptonic counterpart, (73) and (74). We may write it compactly as £z-quark =
. ~ \ 4 COS fjy
Y— ) qa" ^- ' i=u,d
[Li{l
- 75) +
fl,(l
+ 75)] Qi Z„ ,
(82)
where the chiral couplings are Li = r 3 - 2Qi sin 2 9\y , Ri = -2Qi sin2 9W .
(83)
Again we find a quark-lepton universality in the form—but not the values—of the chiral couplings.
34
3.5
Trouble in Paradise
Until now, we have based our construction on the idealization that the u { « 7 M [ - M * ~ 75) + - f t u ( l + J5)] U
4 COS U\v
+ J 7 " [Ld(l - 75) + Rd(l + 75)] d cos2 9C +sf" [Ld(l - 75) + Rd{l + 75)] s sin2 9C +d^ [Ld(l - 75) + Rd{l + 75)] * sin 9C cos 9C + s1'i[Ld(l-l5) + Rd(l+-f5)]d sin 9C cos9 C } , (85) Until the discovery and systematic study of the weak neutral current, culminating in the heroic measurements made at LEP and the SLC, there was not enough knowledge to challenge the first three terms. The last two strangenesschanging terms were known to be poisonous, because many of the early experimental searches for neutral currents were fruitless searches for precisely this sort of interaction. Strangeness-changing neutral-current interactions are not seen at an impressively low level.* Only recently has Brookhaven Experiment 787 47 detected a single candidate for the decay K+ —> ir+i/i>, r T h e arbitrary Yukawa couplings that give masses to the quarks can easily be chosen to yield this result. s F o r more on rare kaon decays, see the TASI 2000 lectures by Tony Barker 4 and Gerhard Buchalla. 4 6
35
and inferred a branching ratio B{K+ -» TY+I/I>) = 1.5;f;| x 1 0 - 1 0 . The good agreement between the standard-model prediction, B(KL —> ^+^~) — 0.8±0.3 x 1 0 - 1 0 (through the process Ki —> 77 —>• p + / i _ ) , and experiment 48 leaves little room for a strangeness-changing neutral-current contribution:
that is easily normalized to the normal charged-current leptonic decay of the K+: .; > 113.5 GeV/c2, excluding much of the favored region, either the Higgs boson is just around the corner, or the standard-model analysis is misleading. Things will soon be popping!
53
OU.O"
i
i
i
i
i
i
i
|
i
i
'
I
'
'
' A
— LEP1.SLD vN Data LEP2, ppDatd 68% CL 80.5-
>
/
CD
£ 80.4•'
E /
80.3-
/ •
on o
m H [GeV| 113, 300 1000
1CJO
150
170
'
Preliminary 190
21
mt [GeV]
Figure 20. Comparison of the indirect measurements of Mw and mt (LEP I+SLD+i/AT data, solid contour) and the direct measurements (Tevatron and LEP II data, dashed contour). Also shown is the standard-model relationship for the masses as a function of the Higgs mass. (From the LEP Electroweak Working Group, Ref. 5 4 .)
6
The electroweak scale and beyond
We h ave seen that the scale of electroweak symmetry breaking, v = ( G F \ / 2 ) ~ 2 « 246 GeV, sets the values of the W- and 2-boson masses. But the electroweak scale is not the only scale of physical interest. It seems certain that we must also consider the Planck scale, derived from the strength of Newton's constant, and it is also probable that we must take account of the S77(3)c SU(2)L U{1)Y unification scale around 1 0 1 5 - 1 6 GeV. There may well be a distinct flavor scale. The existence of other significant energy scales is behind the famous problem of the Higgs scalar mass: how to keep the distant scales from mixing in the face of quantum corrections, or how to stabilize the mass of the Higgs boson on the electroweak scale.
54
Cithco:1! j ' -:rtair • 0.02804I0.00065J 0.0275510.00046
CvJ
If
Excluded
Preliminary
10
10
10
m H [GeV]
from a global fit to precision d a t a vs. the Higgs-boson mass, Figure 21. A x 2 MJJ. The solid line is the result of the fit; the band represents an estimate of the theoretical error due to missing higher order corrections. T h e vertical band shows the 95% CL exclusion limit on MJJ from the direct search at LEP. The dashed curve shows the sensitivity to a change in the evaluation of ct(MV). (From the LEP Electroweak Working Group, Ref. 5 4 .)
6.1
Why is the electroweak scale small?
To this point, we have outlined the electroweak theory, emphasized that the need for a Higgs boson (or substitute) is quite general, and reviewed the properties of the standard-model Higgs boson. By considering a thought experiment, gauge-boson scattering at very high energies, we found a first signal for the importance of the 1-TeV scale. Now, let us explore another path to the 1-TeV scale. The SU(2)i U(1)Y electroweak theory does not explain how the scale of electroweak symmetry breaking is maintained in the presence of quantum corrections. The problem of the scalar sector can be summarized neatly as
55
follows.73 The Higgs potential is
v{4>H) = ii2{H) + \\\(H)2.
(123)
With fi2 chosen to be less than zero, the electroweak symmetry is spontaneously broken down to the U(l) of electromagnetism, as the scalar field acquires a vacuum expectation value that is fixed by the low-energy phenomenology, {4>)o = ^ - / i 2 / 2 | A | = {GFV$y1/2
» 175 GeV .
(124)
Beyond the classical approximation, scalar mass parameters receive quantum corrections from loops that contain particles of spins J = 1,1/2, and 0:
m'ip^m2^
^ J=1
+
O
+
J=1/2
Q J=0
(125) The loop integrals are potentially divergent. Symbolically, we may summarize the content of (125) as m2{p2) = m 2 (A 2 ) + Cg2 /
dk2 + • • • ,
(126)
where A defines a reference scale at which the value of m2 is known, g is the coupling constant of the theory, and the coefficient C is calculable in any particular theory. Instead of dealing with the relationship between observables and parameters of the Lagrangian, we choose to describe the variation of an observable with the momentum scale. In order for the mass shifts induced by radiative corrections to remain under control (i.e., not to greatly exceed the value measured on the laboratory scale), either A must be small, so the range of integration is not enormous, or new physics must intervene to cut off the integral. If the fundamental interactions are described by an SU(i)c ® SU(2)L (§) U(1)Y gauge symmetry, i.e., by quantum chromodynamics and the electroweak theory, then the natural reference scale is the Planck mass, y ^It is because Mpi a n c i, is so large (or because Gjjewton ' s s o small) that we normally consider gravitation irrelevant for particle physics. The graviton-quark-antiquark coupling is generically ~ E/Mp^nck, so it is easy to make a dimensional estimate of the branching fraction for a gravitationally mediated rare kaon decay: B(K]J —• ir°G) ~ (Aff 4 -/Mpi anc i;) 2 ~ 1 0 ~ 3 8 , which is truly negligible!
56
( h \ A ~ Mpi anck = ( - — ^ — )
1/ 2
'
w 1.22 x 1019 GeV .
(127)
V ^Newton/
In a unified theory of the strong, weak, and electromagnetic interactions, the natural scale is the unification scale, A ~ Mu « 10 15 -10 16 GeV .
(128)
Both estimates are very large compared to the scale of electroweak symmetry breaking (124). We are therefore assured that new physics must intervene at an energy of approximately 1 TeV, in order that the shifts in m2 not be much larger than (124). Only a few distinct scenarios for controlling the contribution of the integral in (126) can be envisaged. The supersymmetric solution is especially elegant. 74,75,76,77 Exploiting the fact that fermion loops contribute with an overall minus sign (because of Fermi statistics), supersymmetry balances the contributions of fermion and boson loops. In the limit of unbroken supersymmetry, in which the masses of bosons are degenerate with those of their fermion counterparts, the cancellation is exact:
J2 •
Cifdk2 = 0.
(129)
fermions -[-bosons
If the supersymmetry is broken (as it must be in our world), the contribution of the integrals may still be acceptably small if the fermion-boson mass splittings A M are not too large. The condition that g2AM2 be "small enough" leads to the requirement that superpartner masses be less than about 1 TeV/c2. A second solution to the problem of the enormous range of integration in (126) is offered by theories of dynamical symmetry breaking such as technicolor. 78,79,80 In technicolor models, the Higgs boson is composite, and new physics arises on the scale of its binding, ATC — 0 ( 1 TeV). Thus the effective range of integration is cut off, and mass shifts are under control. A third possibility is that the gauge sector becomes strongly interacting. This would give rise to WW resonances, multiple production of gauge bosons, and other new phenomena at energies of 1 TeV or so. It is likely that a scalar bound state—a quasi-Higgs boson—would emerge with a mass less than about 1 TeV/c 2 . 81 We cannot avoid the conclusion that some new physics must occur on the 1-TeV scale/ ^Since the superconducting phase transition informs our understanding of the Higgs mech-
57
6.2
Why is the Planck scale so large?
The conventional approach to new physics has been to extend the standard model to understand why the electroweak scale (and the mass of the Higgs boson) is so much smaller than the Planck scale. A novel approach that has been developed over the past two years is instead to change gravity to understand why the Planck scale is so much greater than the electroweak scale. 86 Now, experiment tells us that gravitation closely follows the Newtonian force law down to distances on the order of 1 mm. Let us parameterize deviations from a 1/r gravitational potential in terms of a relative strength £G and a range AQ, so that V(r) = -JdnJ
G rfr2
NewtonPMp(r2)
{1 + £G e x p ( _ r i 2 / A Q ) ]
>
(130)
where p(r,) is the mass density of object i and T\I is the separation between bodies 1 and 2. Elegant experiments that study details of Casimir and Van der Waals forces imply bounds on anomalous gravitational interactions, as shown in Figure 22. Below about a millimeter, the constraints on deviations from Newton's inverse-square force law deteriorate rapidly, so nothing prevents us from considering changes to gravity even on a small but macroscopic scale. For its internal consistency, string theory requires an additional six or seven space dimensions, beyond the 3 + 1 dimensions of everyday experience. 93 Until recently it has been presumed that the extra dimensions must be compactified on the Planck scale, with a compactification radius /^unobserved * 1/Mpianck « 1 . 6 x l 0 ~ 3 5 m . The new wrinkle is to consider that the SU(3)C ® SU(2)L U(\)Y standard-model gauge fields, plus needed extensions, reside on 3 + 1-dimensional branes, not in the extra dimensions, but that gravity can propagate into the extra dimensions. How does this hypothesis change the picture? The dimensional analysis (Gauss's law, if you like) that relates Newton's constant to the Planck scale changes. If gravity propagates in n extra dimensions with radius R, then GNewton ~ M p , ^ ~ M^^^R"1
,
(131)
where M* is gravity's true scale. Notice that if we boldly take M* to be as small as 1 TeV/c2, then the radius of the extra dimensions is required to be anism for electroweak symmetry breaking, it may be useful to look to other collective phenomena for inspiration. Although the implied gauge-boson masses are unrealistically small, chiral symmetry breaking in QCD can induce a dynamical breaking of electroweak symmetry. 8 2 (This is the prototype for technicolor models.) Is it possible t h a t other interesting phases of QCD—color superconductivity, 8 3 ' 8 4 , 8 5 for example—might hold lessons for electroweak symmetry breaking under normal or unusual conditions?
58 1
e
10
1
o
w
sz D) •1—•
c = ^
(\K°) - | * ° » ,
\K°) = CP \K°). The observed eigenstates were Kg (lifetime rg = 0.9 x 10~ 10 sec) and KL (TL = 5.2 x 10~ 8 sec) with TUL - ms — 0.48 Ts ~ 10~5eV. The primary decays were Ks ->• TT+ -K~~ and KL -> 3TT,
TT°
7r°,
consistent with the CP assignment Ks = ^ i and KL = ^ 2 - The discovery in 1964 was that KL also decayed into w+ n~ with a small branching ratio. There appeared then two alternatives: 1. Modify the AS — 1 interaction in some small way from the standard (V - A) form. 2. Assume there exists a much weaker AS = 2 interaction that violates CP. This would be described by an effective Hamiltonian (in terms of quarks) naw=GswsOdsOd+h.c,
(2)
where O is some Dirac operator. It is sufficient that Gsw ~ 1 0 - 1 0 to l O - 1 1 ^ .
(3)
The superweak idea is that CP violation is confined to K° — K° mixing, which is a AS = 2 process, second order for the standard theory. In the Ki — K2 representation M
l
£ _ / Ml 2 ~ \-im'/2
M2
im'/2\_i(T1 J 2\ T2
73
where m' is the superweak term and the i is required by CPT invariance. Then \Ks) = (\K1)+e\Ki))Hl \KL) = (\K2)+e\K2))/(l
+ \§\*), + \i\2),
(Mi-M 2 )-i(ri-r 2 )/2' Mi - M2 « Ms - ML = -AMjf,
( ) [
'
v
'
ri - r 2 « r s - rL « rs. The observations give e ~ 2 x 1 0 - 3 which then determines m' from which Eq. (1.3) follows. The superweak theory made three predictions: 1. The CP violation is completely described by e; in particular the observables V+- = %o = £> where T)+- = A(KL ->• 7r+ ir~)/A(Ks
-)• TT+ TT~),
IJdO = ^ ( i ^ L -»• 7T° 7 r ° ) / A ( ^ S -> 7T° 7T°).
2. The phases of T]^ and 7700 are equal and determined by Eq. (1.5) to be about 43.5° using the empirical values of AMK and Ts3. The CP violation is so small that it will not be seen anywhere else. These predictions proved all too true for more than 25 years. The alternative to superweak says that CP is violated in the decay amplitude AQ el6° and A2 el52 corresponding to final 1 = 0 and I = 2 irir states, where 6j is the strong phase shift. Prom CPT and unitarity ImAj give the amplitudes for the CP-violating transitions K2 —> TTTT. Since there is still in any model the contribution m' there are now three CP-violating quantities 2 m', ImAo, ImA 2 . However there is a choice of phase convention using the £7(1) transformation s —> se~ia under which the strong and electromagnetic interactions are invariant. For the infinitesimal f/(l) as an example s —> s(l — ia), ImAi -> ImAi - a Re Ai, m' -+ m' + a(Mi - M 2 ).
74
Thus there are only two independent quantities which may be chosen as e' oc ImAo/ReAo - Im A 2 /Re A 2 , e ~ £ + i(ImA 0 /ReAo).
,„> {)
Then e'
1+-=£ +
l+u,Jrf' 2£
(7)
1 + V2w where u = Re A 2 /Re A0 ~ 0.045 and the last numerical result is empirical and a sign of the AI = 1/2 rule. The quantity e' is a measure of CP violation in the decay amplitudes and so not due to a superweak interaction. 2
The Standard Model vs Superweak
For 35 years experiments have sought to determine e' by finding the difference between TJ^ and 7700 and thus detecting CP violation in the decay amplitude. The experiments actually measure V+-
2
~ l + 6Re(e'/e).
(8)
?7oo
By chance the phase of e given to a good approximation by Eq. (1.5) and the phase of e' given by (7r/2 — K -K 6 suggest P/T as high as 0.4. Assuming the strong phase A is small, then \ p+ TT~ and p~ n+. An alternative would be to consider decays that are dominated by the b —> d penguin. In this case • dss yielding B° -»• Ks Ks, B° -» p° 77, etc. Note that even if the decay were not pure penguin one would expect an asymmetry very different from that for 5 ° -» * Ks. Another possibility is a CP-violating effect in which mixing is not involved. One can look for a difference between the rates of B+ —>• / and B~ —> / or in the case of B° looking at time t = 0 before mixing for the difference between B° -> f and B° -» / . In practice this means looking for the cos ATUB t term 5 rather than the sin A m ^ t. As an example, consider B°(B°) —• 7r+ IT~; from Eq. (2.5) _ r ( £ ° -> 7T+ 7T~) - T(B° -> 7T+ 71-) ^ ~ T(B° ->• 7T+ 7T-) + r(J5° -4 7T+ 7T-) '
78
2TPsin(/? + 7)sinA T 2 + P 2 + 2TPcos(/3 + 7 ) c o s A ' For 0 + 7 ~ 90° this gives (with P/T = r) 1r sin A. 1 + rz Thus a very large CP-violating asymmetry is possible if r is greater than 0.3, but it all depends on the strong phase A. It is very difficult to make definite statements about the strong phases in B decays in contrast to K decays. For K -» TTTT the final state interaction can be thought of as elastic scattering with phase shifts S2 and So corresponding to 7T7T states with 1 = 2 and I = 0. However, KIT s-wave scattering at 5 GeV is highly inelastic involving many channels. The phase A arises from the absorptive parts of diagrams corresponding to the strong scattering from other final states into the TTTT state. For any weak interaction operator Oi we can define the real decay amplitude in lowest order An = -
Mti = M% =
(f\Oi\B).
If / were an eigenstate one would then multiply this by el Sf. Going from the eigenstate basis to the states of interest Mfi = £ < / | S 1 / 2 | / ' > < m i - B > , (14) /' where S is the strong-interaction S matrix. The sum in Eq. (2.7) is over a large number (almost uncountable) of states. One can only make some general comments about it: 1. The strong phase depends on the operator 0 , that affects the relative importance of different states / ' . The phase A in Eqs. (2.5) and (2.6) is the difference between the strong phase of the "tree" operator and that of the "penguin". 2. Since the strong scattering is expected to be very inelastic the diagonal element (/IS 11 / 2 !/) has as its major effect the reduction of M / J ; this is a kind of absorption effect. Thus we could write Mfi = M°fiai + i Ri =
\Mfi\ei5i,
where a < 1 is the reduction due to absorption. For a "typical" state, by unitarity, the scattering "in" due to R compensates for the scattering "out" so
Ri = \Jl-a] =sm5i.
(15)
79
3. An estimate can be based on a crude statistical argument 7 in which case one can reduce the multichannel problem to an equivalent 2-channel problem „ _ / cos2
81
in the decay K -> it + fi + v. This does not really violate T except in the Born approximation when final state interactions (FSI) can be avoided. As a simple didactic example, consider the scattering from a potential Vo +
Via-L
which certainly is T-invariant. The resulting amplitude is A = fo +
ifia-h
where n is the normal to the scattering plane. In the Born approximation / 0 and / i are real and so < a • h > = 0, but beyond the Born approximation / 0 and / i are complex; for example, if s and p waves dominate / 0 would have a phase el s° and / i the phase el ^. For the case of K° -> ir+ +/x~ + PM there is a Coulomb FSI so that without T violation, P ~ 10~ 3 ; for the case of K+ ->• TT° + ju+ + v^ the FSI involves 2-y exchange so that P ~ 10~ 6 . In the Standard Model the real TRV is expected to vanish in semi-leptonic decays. One would expect a real TRV in non-leptonic decays such as A —> pn~ where there is a defined parameter (3 oc< aA • a* x k > . However, the FSI effect is much larger being proportional to sin(J p — Ss) where 6P, 6S are the n-p phase shifts. If the experiment is also done with A, then /3 + /? is a clear CP-violating effect and is associated with true TRV, but this is hardly "direct evidence" of TRV. A large "T-odd observable" has been found in the decay KL —> n+ ir~ e+ e~ C = < ne x n^ • z > < he • nn >, where n^h^) are the normals to the e+ e~(ir+ ir~) planes and z is the unit vector between the pairs. This was predicted as a result of K - K mixing as an interference between an M l virtual 7 from if2 -> TTTTJ and an Ei virtual bremsstrahlung from K\ —> n tt 7. The theoretical result 9 is C = 0.15 sin(v?e + A), where A ~ 30° comes from mr phase shifts. The experimental result 10 verifies this; the result is so large because for the e + e~ energy considered the El is much larger than M l which compensates for the small admixture |e| of K\. Since A is involved this is not again obvious TRV. It is of didactic interest to consider the limit A —)• 0. In this case C is proportional to sin£. We know
82
that / i + / i candidate (Top) recorded at the OPAL experiment and candidate (Bottom) recorded at the L3 experiment at LEP-1.
92
02/03/2001
LEP
Preliminary *
*4-T«u
Racoon WW / YFSWW 1.14 no ZWW vertex (Gentle 2.1) onlyv exchange (Gentle 2.1)
160
170
180
190
200
210
ERm [GeV]
Figure 4: LEP-2 average W pair production cross-section, corrected to correspond to the three doubly-resonant W production diagrams.
93
' D E L P H I R I P ' *&£, l.im; I * J . 4 CiV "TH^I DASi M - A p r - I M !
Ijgr^
| | : 43; it
109372 E M . 5483 P r • c ;1 » -Api -1 1uI S e i . • 1P•A pr - ! 0) 8
n • J
n X2oo J
c I
o I
Figure 5: An e + e —> W+W candidate recorded at the DELPHI experiment at LEP-2, where W+ and W~ decays each into a pair of quarks giving a four jet topology.
100 Vs [GeV]
200
Figure 6: Cross sections in e+e
500
annihilation.
94
Z and W production at the Tevatron The primary production of Z and W bosons at the Tevatron arises from the reactions: uu -> Z, dd -» Z, ud -> W+, and ud -t W~, where the up and down quarks (and antiquarks) can be Valence or sea quarks in the proton c :
E xiE u
— - — « -
E X2E •*-=•—
7 d
Although both Z and W± are produced through quark-antiquark annihilation, the dominant contribution is not from the valence-valence collisions but from valence-sea collisions. The typical qq center-of-mass energy %/I for W, Z production is the mass of the boson, %/I ~ Mz,w- Since these boson masses are around 100 GeV, about 1/20 of the pp center-of-mass energy, both valence and sea quarks have a good probability for carrying a sufficient fraction of the proton's energy to produce a gauge boson (see Figure 1). The valence-sea production mechanism is about 4 times larger than the valence-valence and sea-sea production mechanisms. It is coincidental that the valence-valence and sea-sea mechanisms are about equal at this energy. At higher energies such as LHC, the sea-sea mechanism dominates; at lower energies such as SppS where the center-of-mass energy was 560 GeV, the valence-valence mechanism dominates.
Top-quark production at the Tevatron The Tevatron has been (and will be until LHC turns on) the unique place to produce top quarks. The dominant top-quark production mechanism at the Tevatron is the annihilation of the valence quark and the valence antiquark into a gluon, which then decays into a tt pair. One of the tt cadidates recorded at the Tevatron is shown in Figure 7. The dominant top-quark production mechanism at the LHC (proton-proton collider) will be the annihilation of a gluon and a gluon into a gluon, which in turn decays into a tt pair. c T h e proton consists of three valence quarks (uud) which carry its electric charge and baryon quantum numbers, and an infinite sea of light qq pairs.
95
CDF Top Event M
= 79 GeV/c2
Jet 2 Jet 3
Figure 7: A tt candidate recorded at the CDF experiment, where t —> W+b t —• W~b —y qqb.
—• e+ffe and
Summary Table 2 summarizes the number of W, Z and tt events identified at LEP-1, LEP-2, SLC and the Tevatron. 2.4
Survey ofW,
Z, and Top quark Properties
The W, Z, and top particles (and the other elementary particles in the Standard Model) are structureless upto our current resolution, which is about 10~ 16 - 10~ 17 cm; Note that the top-quark mass is enormous, as heavy as a gold atom which consists of 79 protons and 118 neutrons and yet it is structureless. Coincidentally the top-quark mass is about the sum of the Z boson mass and the W boson mass. The top quark and W and Z bosons decay immediately after they are produced; their lifetimes, provided by their width measurements, are
96 Table 2: The number of W's, Z's and top quarks.
Heavy Particle Z
W top
Accelerator LEP-1 SLC Tevatron LEP-2 Tevatron Tevatron
C M . Energy ~91 GeV ~91 GeV 1.8 TeV 132 ~ 208 GeV 1.8 TeV 1.8 TeV
# of events 17,221,000 557,000 9,000 10,000 180,000 100
around 10~ 25 sec. With current date techniques, we can not trace their decays in space. Z's decay into a fermion (a quark or a lepton) and its anti-fermion, and W's decay into a fermion and its weak interaction partner: W+ —> e+ve, n+Vn, T+VT, ud, and cs. Top quarks decay about 100% of time into a W and a b quark, the weak interaction partner of the top quark (t —> W+b). B hadrons (hadrons containing b quarks) live long, 1.5 x 10~ 12 s, at the elementary particle scale. For instance, b quarks from the top decay which have a momentum of ~50 GeV travel about 4 mm before decaying, which is large enough to trace down in space. This is demonstrated in Figure 7 by the displaced vertices (seconary vertices). 2.5
Detectors
In order to identify various types of particles such as electrons, muons, taus, neutrinos, and 6-quarks, which are the products of Z, W, and top decays, a typical experimental detector consists of layers of devices as shown in Figure 8. Imagine you are riding on a particle that was just produced by the collision of a proton and an anti-proton or an electron and a positron. It encounters a thin beam pipe. It then zips through a silicon device, capable of resolving very tiny distances (~100 /xm), thus identifing fo-quarks in the event as shown in Figure 7. After the silicon device, it zips through a gas containing an immense number of very thin gold wires. The particle passes ~100 of these wires. If the particle is charged, each nearby wire records its passage, and the particle's path is determined (see Figures 3,5,7). A measurement of the curvature (this device and the silicon device are typically inside of magnetic field B produced by a solenoid) gives us the momentum of the particle: qBr
97
Figure 8: The beam's view of a typical detector.
where q is the charge of the particle, r is the inverse curvature, and c is the speed of light. If the particle is neutral, there will be no signal on the wire. Next the particle passes through coil of solenoid magnets. It then passes into a calorimeter section, which measures particle energy except muons and neutrinos. Different particles interact differently with matter, thus with calorimeter. If an electron, it fragments on a series of closely spaced thin lead plates and scintillator between the plates, giving up its entire energy in 3 or 4 inches. If a hadron, it penetrates 10 to 20 inches of calorimeter material before exchausting all of its energy through nuclear collisions. If it is a muon, it zips through the calorimeter sections and leaves hits in a gas containing thin gold wires which is located outside of the calorimeter. Neutrinos leave the detector entirely, leaving behind not even a hint of their fragrance. The system stores the data with about one million bits of information for each event.
98
3 3.1
Precision Electroweak Measurements Tree-level formulation
QED can be described by one parameter e or a = e2/(47r) where l/a — 137.03599959(38)(13), most precisely derived from g - 2 measurements. 3 In the electroweak theory, the strengths are specified by three paramters. They are two gauge coupling constants g and g', and v, the vacuum expectation value of the Higgs field (see Chris Quigg's lectures in this proceedings). These three parameters as inputs predict charges, Mw, Mz, and Gp via the following relationships:
sin2 w = 1
°
~M
=
n™ V2GFM^ e2
(1) (2) V ;
(3)
9
2
e
The theory is democratic so that any three variables in the above equations may be chosen as inputs. It makes much more sense to choose the three parameters most accurately measured so that the predictions for any other measurements become as precise as possible, allowing sensitive tests of the theory. In the early 1980s, those were e (or a), GF d and the electroweakmixing parameter s'm29w- With the measured values of these parameters, the masses of the W and Z bosons were prediced to be around 80 GeV and 90 GeV, respectively. This prediction lead to the discovery of the W and Z boson at SppS at CERN. 1 By the end of August 1989, the 4 LEP experiments measured Mz to within 160 MeV. Since then, a, GF and Mz were adopted as the basic input parameters. The current Mz accuracy by LEP is 2 MeV (0.002%). 4 Most likely, this measurement won't be improved in the near future. 3.2
Higher Order Corrections
Once higher order corrections are included, all these equalities are no longer exactly true. In gauge theories, higher order corrections such as the loop diagrams lead to infinities which require renormalization. A general consequence d
GF = (1.16637 ± 0.00001) x 10~ 5 GeV~ 2 is derived from the muon lifetime measurements using the radiative corrections of the V — A Fermi theory.
99
of this is the introduction of Q2 dependent corrections to the parameters of the theory, therefore corrections to the relationships among the parameters. For instance, the Born level relationship GF
- y/MPy, Ml - M ^
W
will be modified by M r - l ™ z ^"l-Ar^M^Ml-M^
(to W
in higher orders. Here Ar ~ Aro — pt/tan#iy Ar 0 ~ 1 - a{0)/a(Mz) ^ 0.06
(7) (8)
where a(Mz) is the electromagnetic coupling constant at the scale of the Z mass and the top quark contributes pt- As shown in Aro, the bulk of the corrections can be observed in "running" (Q 2 -dependent) coupling. The uncertainty in a(Mz) is dominated by the contribution of the light quarks (it through 6), A a ^ ( M f ) 1 2 :
This is evaluated using dispersion relations and a{e+ e~ —> hadrons) measurements at low \ / s , and perturbative QCD calculations at large yfs. The relationship can also be written as Mw MzcosOw
l2
= 1
(10)
in the lowest order, and MW Mzcos0w
l
ia g 1 +
3GFMJ 8\/27r2
~ 1 + 0.0096
( n )
Mt 175GeV
100
with higher order corrections. The second term in this equation (or pt in Eq. 9) is contributed by the top quark in loop diagrams: t t W+-
W+
With Mtop ~ 175 GeV, the correction is about 1% level. Thus precision measurements of My/, Mz, and cos9y/ with much better than 1% accuracy can predict the top-quark mass. The predicted top mass from electroweak measurements is Mtop = 172+Jj GeV. 4 The top mass measured by the CDF and D 0 Tevatron experiments is Mtop — 174.3 ± 5.1 GeV, 5 which agrees with the prediction very well. This is a good example of the successful interplay between theory and experiments. Any inconsistency between the predicted and measured values would have hinted at new physics. Secondary contributions to electroweak observables (or to Eq. 11) are from the Higgs boson in loop diagrams H H W+-
W+
w+ which are proportional to HM2H/M2W) where the Higgs mass M # is unknown in the Standard Model. New particles beyond the Standard Model could also contribute to electroweak observables through loop diagrams. Difficulties arise due to these unknown masses, Mjj and M n e w particle- This could be bad news since the predictions are uncertain and it is hard to test the theory. Or this could be good news since the predictions depend on the unknown masses. Indeed precision measurements provide information about these unknown masses of particles which are too heavy to be produced directly, or whose production cross section is too small to be observed. For example, with ~30 MeV uncertainty in Mw and ~ 2 GeV uncertainty in Mtop, we can predict the Higgs boson mass within ~30%. When precision measurements are inconsistent each other (for example, different sets of precision measurements predict different Higgs masses), this may signal the presence of new physics beyond the Standard Model.
101
3.3
The Experimental Inputs
The last decade has been a remarkable improvement in our knowledge of various electroweak parameters. Much of the improvement is due to the study of the Z resonance at LEP-1 and the SLC, and the study of W and top quark at LEP-2 and the Tevatron. Z boson parameters The precise determination of the Z boson parameters from the measurements at the Z resonance by the four collaborations ALEPH, DELPHI, L3 and OPAL in e + e~ collisions at LEP and by the SLD collaboration in e+e~ collisions at the SLC is a landmarkforprecision tests of the electroweak theory. Cross section at the Z peak The coupling of the Z to a fermion (/) and an anti-fermion (/) pair is described by the following Lagrangian density, L = (^f)1/2*n»(vf
- an5)$fZ»
(12)
= (^M)i/2*/7/i[ff/(i_75)+5/(i+75)]$/^
(13)
where vj and a/ are vector and axial vector coupling constants, and g*L = (vf + a,f)/2 and g^ = (vf — a/)/2 are left-handed and right-handed combinations. The vector and axial vector couplings are related to the quantum numbers of the fermion as follows vf = y/pj(2l(
- 4Q / sin 2 0 / )
(14)
af = y/pj(2l()
(15)
where I[ is the third component of weak isospin, Qf is the electric charge, and the parameters pf ~ 1 and sin 2 0/ ~ 0.23 incorporate electroweak radiative corrections. The cross section for the process e+e~(Pe) -» Z —> ff is described in the center-of-mass frame by the following expression, daf/ dtt
=
9 sTeeTff/Ml 4 (s - M | ) 2 + s 2 r | / M | [(1 + cos26>)(l - PeAe) + 2cos6Af{-Pe
l
+ Ae)]
j
102
where Pe is the polarization of the electron beam, s is the square of the centerof-mass energy, Tz is the total width of the Z, 9 is the angle between the incident electron and the outgoing fermion, Fff is the partial width for Z —• / / , and Af is the left-right coupling constant asymmetry. The partial widths and coupling constant asymmetries are related to the couplings defined in the Lagrangian,
2vIaL_
=
v
=
a
jgtf
- (g/)» 2
(^) + (
