Introduction to the Standard Model and Electroweak Physics

0.84 MB·English

by  Paul Langacker

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INTRODUCTION TO THE STANDARD MODEL AND
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2 ELECTROWEAK PHYSICS

n
a PAULLANGACKER
J
School of Natural Sciences, Institute for Advanced Study

2 Princeton, NJ 08540, USA
E-mail: [email protected]

]
h
p Aconciseintroductionisgiventothestandardmodel,includingthestructure
- of the QCD and electroweak Lagrangians, spontaneous symmetry breaking,
p
experimentaltests,andproblems.
e
h Keywords:Standardmodel,Electroweakphysics
[

1
1. The Standard Model Lagrangian
v
1 1.1. QCD
4
2 The standard model (SM) is a gauge theory1,2 of the microscopic interac-
0 tions.Thestronginteractionpart,quantumchromodynamics(QCD)a isan
.
1 SU(3) gauge theory described by the Lagrangian density
0
1 (cid:88)
9 L =− Fi Fiµν + q¯ iD(cid:54) αqβ, (1)
0 SU(3) 4 µν rα β r
r
:
v where g is the QCD gauge coupling constant,
s
i
X Fi =∂ Gi −∂ Gi −g f Gj Gk (2)
µν µ ν ν µ s ijk µ ν
r
a is the field strength tensor for the gluon fields Gi, i = 1,··· ,8, and the
µ
structure constants f (i,j,k =1,··· ,8) are defined by
ijk
[λi,λj]=2if λk, (3)
ijk
where the SU(3) λ matrices are defined in Table 1. The λ’s are normalized
by Trλiλj =2δij, so that Tr[λi,λj]λk =4if .
ijk
TheF2 termleadstothreeandfour-pointgluonself-interactions,shown
schematicallyinFigure1.ThesecondterminL isthegaugecovariant
SU(3)
aSee3forahistoricaloverview.Somerecentreviewsinclude4andtheQCDreviewin5.
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derivative for the quarks: q is the rth quark flavor, α,β =1,2,3 are color
r
indices, and
Dα =(D ) =∂ δ +ig Gi Li , (4)
µβ µ αβ µ αβ s µ αβ
wherethequarkstransformaccordingtothetripletrepresentationmatrices
Li =λi/2. The color interactions are diagonal in the flavor indices, but in
generalchangethequarkcolors.Theyarepurelyvector(parityconserving).
Therearenobaremasstermsforthequarksin(1).Thesewouldbeallowed
byQCDalone,butareforbiddenbythechiralsymmetryoftheelectroweak
partofthetheory.Thequarkmasseswillbegeneratedlaterbyspontaneous
symmetry breaking. There are in addition effective ghost and gauge-fixing
termswhichenterintothequantizationofboththeSU(3)andelectroweak
Lagrangians, and there is the possibility of adding an (unwanted) term
which violates CP invariance.
Table1. TheSU(3)matrices.
„τi 0«
λi= , i=1,2,3
0 0
00011 000−i1
λ4=@000A λ5=@00 0A
100 i 0 0
00001 000 01
λ6=@001A λ7=@00−iA
010 0 i 0
010 01
λ8= √1 @01 0A
3
00−2
u d
gs gs gs g2
s
G
u d
Fig.1. InteractionsinQCD.
QCD has the property of asymptotic freedom,6,7 i.e., the running cou-
pling becomes weak at high energies or short distances. It has been exten-
sively tested in this regime, as is illustrated in Figure 2. At low energies
–TypesetbyFoilTEX– 1
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or large distances it becomes strongly coupled (infrared slavery),8 presum-
ably leading to the confinement of quarks and gluons. QCD incorporates
the observed global symmetries of the strong interactions, especially the
spontaneously broken global SU(3)×SU(3) (see, e.g., 9).
Average
Hadronic Jets
e+e- rates
Photo-production 0.3
Fragmentation
Z width
)
ep event shapes m(s0.2
Polarized DIS a
Deep Inelastic Scattering (DIS)
t decays
0.1
Spectroscopy (Lattice)
U decay
0.1 0.12 0.14 0 2
a s(MZ) 1 1m0 GeV 10
Fig. 2. Running of the QCD coupling αs(µ) = gs(µ)2/4π. Left: various experimental
determinationsextrapolatedtoµ=MZ usingQCD.Right:experimentalvaluesplotted
at the µ at which they are measured. The band is the best fit QCD prediction. Plot
courtesyoftheParticleDataGroup,5 http://pdg.lbl.gov/.
1.2. The Electroweak Theory
The electroweak theory10–12 is based on the SU(2)×U(1) Lagrangianb
L =L +L +L +L . (5)
SU(2)×U(1) gauge φ f Yuk
The gauge part is
1 1
L =− Wi Wµνi− B Bµν, (6)
gauge 4 µν 4 µν
where Wi, i=1, 2, 3 and B are respectively the SU(2) and U(1) gauge
µ µ
fields, with field strength tensors
B =∂ B −∂ B
µν µ ν ν µ
Wi =∂ Wi−∂ Wi −g(cid:15) WjWk, (7)
µν µ ν ν µ ijk µ ν
bForarecentdiscussion,seetheelectroweakreviewin5.
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where g(g(cid:48)) is the SU(2) (U(1)) gauge coupling and (cid:15) is the totally
ijk
antisymmetric symbol. The SU(2) fields have three and four-point self-
interactions. B is a U(1) field associated with the weak hypercharge Y =
Q−T3, where Q and T3 are respectively the electric charge operator and
the third component of weak SU(2). (Their eigenvalues will be denoted by
y, q, and t3, respectively.) It has no self-interactions. The B and W fields
3
will eventually mix to form the photon and Z boson.
The scalar part of the Lagrangian is
L =(Dµφ)†D φ−V(φ), (8)
φ µ
(cid:18)φ+(cid:19)
where φ = is a complex Higgs scalar, which is a doublet under
φ0
SU(2) with U(1) charge y =+1. The gauge covariant derivative is
φ 2
(cid:18) τi ig(cid:48) (cid:19)
D φ= ∂ +ig Wi + B φ, (9)
µ µ 2 µ 2 µ
where the τi are the Pauli matrices. The square of the covariant derivative
leads to three and four-point interactions between the gauge and scalar
fields.
V(φ)istheHiggspotential.ThecombinationofSU(2)×U(1)invariance
and renormalizability restricts V to the form
V(φ)=+µ2φ†φ+λ(φ†φ)2. (10)
For µ2 < 0 there will be spontaneous symmetry breaking. The λ term de-
scribesaquarticself-interactionbetweenthescalarfields.Vacuumstability
requires λ>0.
The fermion term is
F
L = (cid:88) (cid:0)q¯0 iD(cid:54) q0 +¯l0 iD(cid:54) l0 +u¯0 iD(cid:54) u0
f mL mL mL mL mR mR
(11)
m=1
+ d¯0 iD(cid:54) d0 +e¯0 iD(cid:54) e0 +ν¯0 iD(cid:54) ν0 (cid:1).
mR mR mR mR mR mR
In (11) m is the family index, F ≥ 3 is the number of families, and L(R)
refer to the left (right) chiral projections ψ ≡ (1∓γ )ψ/2. The left-
L(R) 5
handed quarks and leptons
(cid:18)u0 (cid:19) (cid:18)ν0 (cid:19)
q0 = m l0 = m (12)
mL d0 mL e−0
m L m L
transformasSU(2)doublets,whiletheright-handedfieldsu0 , d0 ,e−0 ,
mR mR mR
andν0 aresinglets.TheirU(1)chargesarey = 1, y =−1, y =q .
mR qL 6 lL 2 ψR ψ
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Thesuperscript0referstotheweakeigenstates,i.e.,fieldstransformingac-
cording to definite SU(2) representations. They may be mixtures of mass
eigenstates (flavors). The quark color indices α = r, g, b have been sup-
pressed. The gauge covariant derivatives are
(cid:16) (cid:17) (cid:16) (cid:17)
D q0 = ∂ + ig(cid:126)τ ·W(cid:126) + ig(cid:48)B q0 D u0 = ∂ + 2ig(cid:48)B u0
µ mL µ 2 µ 6 µ mL µ mR µ 3 µ mR
(cid:16) (cid:17) (cid:16) (cid:17)
D l0 = ∂ + ig(cid:126)τ ·W(cid:126) − ig(cid:48)B l0 D d0 = ∂ − ig(cid:48)B d0
µ mL µ 2 µ 2 µ mL µ mR µ 3 µ mR
D e0 =(∂ −ig(cid:48)B )e0
µ mR µ µ mR
D ν0 =∂ ν0 ,
µ mR µ mR
(13)
from which one can read off the gauge interactions between the W and B
and the fermion fields. The different transformations of the L and R fields
(i.e., the symmetry is chiral) is the origin of parity violation in the elec-
troweak sector. The chiral symmetry also forbids any bare mass terms for
thefermions.WehavetentativelyincludedSU(2)-singletright-handedneu-
trinos ν0 in (11), because they are required in many models for neutrino
mR
mass. However, they are not necessary for the consistency of the theory or
for some models of neutrino mass, and it is not certain whether they exist
or are part of the low-energy theory.
The standard model is anomaly free for the assumed fermion content.
There are no SU(3)3 anomalies because the quark assignment is non-
chiral, and no SU(2)3 anomalies because the representations are real. The
SU(2)2Y and Y3 anomalies cancel between the quarks and leptons in each
family, by what appears to be an accident. The SU(3)2Y and Y anoma-
lies cancel between the L and R fields, ultimately because the hypercharge
assignments are made in such a way that U(1) will be non-chiral.
Q
The last term in (5) is
F
(cid:88) (cid:104)
L =− Γu q¯0 φ˜u0 +Γd q¯0 φd0
Yuk mn mL nR mn mL nR
m,n=1 (14)
(cid:105)
+ Γe ¯l0 φe0 +Γν ¯l0 φ˜ν0 +h.c.,
mn mn nR mn mL nR
where the matrices Γ describe the Yukawa couplings between the single
mn
Higgs doublet, φ, and the various flavors m and n of quarks and leptons.
One needs representations of Higgs fields with y = +1 and −1 to give
2 2
masses to the down quarks and electrons (+1), and to the up quarks and
2
neutrinos (−1). The representation φ† has y = −1, but transforms as the
2 2
2∗ rather than the 2. However, in SU(2) the 2∗ representation is related to
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(cid:32) (cid:33)
φ0†
the2byasimilaritytransformation,andφ˜≡iτ2φ† = transforms
−φ−
as a 2 with y = −1. All of the masses can therefore be generated with a
φ˜ 2
single Higgs doublet if one makes use of both φ and φ˜. The fact that the
fundamental and its conjugate are equivalent does not generalize to higher
unitarygroups.Furthermore,insupersymmetricextensionsofthestandard
model the supersymmetry forbids the use of a single Higgs doublet in both
ways in the Lagrangian, and one must add a second Higgs doublet. Similar
statements apply to most theories with an additional U(1)(cid:48) gauge factor,
i.e., a heavy Z(cid:48) boson.
2. Spontaneous Symmetry Breaking
Gauge invariance (and therefore renormalizability) does not allow mass
terms in the Lagrangian for the gauge bosons or for chiral fermions. Mass-
less gauge bosons are not acceptable for the weak interactions, which are
known to be short-ranged. Hence, the gauge invariance must be broken
spontaneously,13–18 which preserves the renormalizability.19–22 The idea is
thatthelowestenergy(vacuum)statedoesnotrespectthegaugesymmetry
and induces effective masses for particles propagating through it.
Let us introduce the complex vector
v =(cid:104)0|φ|0(cid:105)= constant, (15)
whichhascomponentsthatarethevacuumexpectationvaluesofthevarious
complex scalar fields. v is determined by rewriting the Higgs potential as a
functionofv,V(φ)→V(v),andchoosingvsuchthatV isminimized.That
is, we interpret v as the lowest energy solution of the classical equation
of motionc. The quantum theory is obtained by considering fluctuations
around this classical minimum, φ=v+φ(cid:48).
The single complex Higgs doublet in the standard model can be rewrit-
ten in a Hermitian basis as
(cid:18)φ+(cid:19) (cid:32)√1 (φ1−iφ2)(cid:33)
φ= = 2 , (16)
φ0 √1 (φ3−iφ4
2
cIt suffices to consider constant v because any space or time dependence ∂µv would
increase the energy of the solution. Also, one can take (cid:104)0|Aµ|0(cid:105)=0, because any non-
zero vacuum value for a higher-spin field would violate Lorentz invariance. However,
these extensions are involved in higher energy classical solutions (topological defects),
suchasmonopoles,strings,domainwalls,andtextures.23,24
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where φ =φ† represent four Hermitian fields. In this new basis the Higgs
i i
potential becomes
(cid:32) 4 (cid:33) (cid:32) 4 (cid:33)2
1 (cid:88) 1 (cid:88)
V(φ)= µ2 φ2 + λ φ2 , (17)
2 i 4 i
i=1 i=1
which is clearly O(4) invariant. Without loss of generality we can choose
the axis in this four-dimensional space so that (cid:104)0|φ |0(cid:105)=0, i=1,2,4 and
i
(cid:104)0|φ |0(cid:105)=ν. Thus,
3
1 1
V(φ)→V(v)= µ2ν2+ λν4, (18)
2 4
which must be minimized withrespect to ν. Two important cases are illus-
trated in Figure 3. For µ2 > 0 the minimum occurs at ν = 0. That is, the
vacuumisemptyspaceandSU(2)×U(1)isunbrokenattheminimum.On
the other hand, for µ2 <0 the ν =0 symmetric point is unstable, and the
minimumoccursatsomenonzerovalueofν whichbreakstheSU(2)×U(1)
symmetry. The point is found by requiring
V(cid:48)(ν)=ν(µ2+λν2)=0, (19)
which has the solution ν =(cid:0)−µ2/λ(cid:1)1/2 at the minimum. (The solution for
−ν canalsobetransformedintothisstandardformbyanappropriateO(4)
transformation.)Thedividingpointµ2 =0cannotbetreatedclassically.It
is necessary to consider the one loop corrections to the potential, in which
case it is found that the symmetry is again spontaneously broken.25
V (φ)
ν ν
−
φ
Fig.3. TheHiggspotentialV(φ)forµ2>0(dashedline)andµ2<0(solidline).
We are interested in the case µ2 <0, for which the Higgs doublet is re-
(cid:18) (cid:19)
0
placed,infirstapproximation,byitsclassicalvalueφ→ √1 ≡v.The
2 ν
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generatorsL1,L2,andL3−Y arespontaneouslybroken(e.g.,L1v (cid:54)=0).On
theotherhand,thevacuumcarriesnoelectriccharge(Qv =(L3+Y)v =0),
so the U(1) of electromagnetism is not broken. Thus, the electroweak
Q
SU(2) × U(1) group is spontaneously broken to the U(1) subgroup,
Q
SU(2)×U(1) →U(1) .
Y Q
Toquantizearoundtheclassicalvacuum,writeφ=v+φ(cid:48),whereφ(cid:48) are
quantumfieldswithzerovacuumexpectationvalue.Todisplaythephysical
particle content it is useful to rewrite the four Hermitian components of φ(cid:48)
in terms of a new set of variables using the Kibble transformation:26
(cid:18) (cid:19)
φ= √1 eiPξiLi 0 . (20)
2 ν+H
H isaHermitianfieldwhichwillturnouttobethephysicalHiggsscalar.If
wehadbeendealingwithaspontaneouslybrokenglobalsymmetrythethree
Hermitian fields ξi would be the massless pseudoscalar Nambu-Goldstone
bosons27–30 that are necessarily associated with broken symmetry genera-
tors.However,inagaugetheorytheydisappearfromthephysicalspectrum.
To see this it is useful to go to the unitary gauge
(cid:18) (cid:19)
φ→φ(cid:48) =e−iPξiLiφ= √1 0 , (21)
2 ν+H
inwhichtheGoldstonebosonsdisappear.Inthisgauge,thescalarcovariant
kinetic energy term takes the simple form
1 (cid:20)g g(cid:48) (cid:21)2(cid:18)0(cid:19)
(D φ)†Dµφ = (0ν) τiWi + B +H terms
µ 2 2 µ 2 µ ν
M2
→M2 W+µW−+ ZZµZ +H terms, (22)
W µ 2 µ
wherethekineticenergyandgaugeinteractiontermsofthephysicalH par-
ticle have been omitted. Thus, spontaneous symmetry breaking generates
mass terms for the W and Z gauge bosons
1
W± = √ (W1∓iW2)
2
Z =−sinθ B+cosθ W3. (23)
W W
The photon field
A=cosθ B+sinθ W3 (24)
W W
remains massless. The masses are
gν
M = (25)
W
2
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and
M =(cid:112)g2+g(cid:48)2ν = MW , (26)
Z 2 cosθ
W
where the weak angle is defined by
g(cid:48) M2
tanθ ≡ ⇒ sin2θ =1− W. (27)
W g W M2
Z
One can think of the generation of masses as due to the fact that the W
andZ interactconstantlywiththecondensateofscalarfieldsandtherefore
acquire masses, in analogy with a photon propagating through a plasma.
The Goldstone boson has disappeared from the theory but has reemerged
as the longitudinal degree of freedom of a massive vector particle.
√
ItwillbeseenbelowthatG / 2∼g2/8M2 ,whereG =1.16637(5)×
F W F
10−5 GeV−2 is the Fermi constant determined by the muon lifetime. The
weak scale ν is therefore
√
ν =2M /g (cid:39)( 2G )−1/2 (cid:39)246 GeV. (28)
W F
Similarly, g = e/sinθ , where e is the electric charge of the positron.
W
Hence, to lowest order
√
(πα/ 2G )1/2
M =M cosθ ∼ F , (29)
W Z W sinθ
W
where α ∼ 1/137.036 is the fine structure constant. Using sin2θ ∼ 0.23
W
from neutral current scattering, one expects M ∼78 GeV, and M ∼89
W Z
GeV. (These predictions are increased by ∼ (2−3) GeV by loop correc-
tions.) The W and Z were discovered at CERN by the UA131 and UA232
groupsin1983.Subsequentmeasurementsoftheirmassesandotherproper-
tieshavebeeninexcellentagreementwiththestandardmodelexpectations
(including the higher-order corrections).5 The current values are
M =80.398±0.025 GeV, M =91.1876±0.0021 GeV. (30)
W Z
3. The Higgs and Yukawa Interactions
The full Higgs part of L is
L =(Dµφ)†D φ−V(φ)
φ µ
(cid:18) H(cid:19)2 1 (cid:18) H(cid:19)2
=MW2 Wµ+Wµ− 1+ ν + 2MZ2ZµZµ 1+ ν (31)
1
+ (∂ H)2−V(φ).
µ
2
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The second line includes the W and Z mass terms and also the ZZH2,
W+W−H2 and the induced ZZH and W+W−H interactions, as shown
in Table 2 and Figure 4. The last line includes the canonical Higgs kinetic
energy term and the potential.
Table 2. Feynman rules for the gauge and Higgs interactions after SSB, taking
combinatoricfactorsintoaccount.Themomentaandquantumnumbersflowinto
thevertex.NotethedependenceonM/ν orM2/ν.
Wµ+Wν−H: 12igµνg2ν=2igµνMνW2 Wµ+Wν−H2: 21igµνg2=2igµνMνW22
ZµZνH: 2igcoµsν2gθ2Wν =2igµνMνZ2 ZµZνH2: 2icgoµs2νgθ2W =2igµνMν2Z2
H3: −6iλν=−3iMH2 H4: −6iλ=−3iMH2
ν ν2
Hf¯f: −ih =−imf
f ν
Wµ+(p)γν(q)Wσ−(r) ieCµνσ(p,q,r)
Wµ+(p)Zν(q)Wσ−(r) itaneθWCµνσ(p,q,r)
Wµ+Wν+Wσ−Wρ− isine22θWQµνρσ
Wµ+ZνγσWρ− −itane2θWQµρνσ
Wµ+ZνZσWρ− −itane22θWQµρνσ
Wµ+γνγσWρ− −ie2Qµρνσ
Cµνσ(p,q,r)≡gµν(q−p)σ+gµσ(p−r)ν+gνσ(r−q)µ
Qµνρσ ≡2gµνgρσ−gµρgνσ−gµσgνρ
After symmetry breaking the Higgs potential in unitary gauge becomes
µ4 λ
V(φ)=− −µ2H2+λνH3+ H4. (32)
4λ 4
ThefirsttermintheHiggspotentialV isaconstant,(cid:104)0|V(ν)|0(cid:105)=−µ4/4λ.
It reflects the fact that V was defined so that V(0) = 0, and therefore
V < 0 at the minimum. Such a constant term is irrelevant to physics in
the absence of gravity, but will be seen in Section 5 to be one of the most
seriousproblemsoftheSMwhengravityisincorporatedbecauseitactslike
a cosmological constant much larger (and of opposite sign) than is allowed
by observations. The third and fourth terms in V represent the induced
cubic and quartic interactions of the Higgs scalar, shown in Table 2 and
Figure 4.
The second term in V represents a (tree-level) mass
√
(cid:112)
M = −2µ2 = 2λν, (33)
H