The Standard Model of Electroweak Interactions

1.2 MB·English

by  Antonio Pich

The Standard Model of Electroweak Interactions
A.Pich
IFIC,UniversityofValència–CSIC,
València,Spain
Abstract
Gauge invariance isapowerful tooltodetermine thedynamical forces among
the fundamental constituents of matter. The particle content, structure and
symmetries of the Standard Model Lagrangian are discussed. Special empha-
sis is given to the many phenomenological tests which have established this
theoretical frameworkastheStandardTheoryoftheelectroweak interactions:
2
electroweak precision tests, Higgs searches, quark mixing, neutrino oscilla-
1
0 tions. Thepresent experimental statusissummarized.
2

n
1 Introduction
a
J
TheStandardModel(SM)isagaugetheory,basedonthesymmetrygroupSU(3) SU(2) U(1) ,
C L Y
2 ⊗ ⊗
which describes strong, weak and electromagnetic interactions, via the exchange of the corresponding

] spin-1gaugefields: eightmasslessgluonsandonemasslessphoton,respectively, forthestrongandelec-
h
tromagnetic interactions, andthreemassivebosons, W andZ,fortheweakinteraction. Thefermionic
p ±
- matter content is given by the known leptons and quarks, which are organized in a three-fold family
p
structure:
e
ν u ν c ν t
h e , µ , τ , (1)
[ e d µ s τ b
(cid:20) − ′ (cid:21) (cid:20) − ′ (cid:21) (cid:20) − ′ (cid:21)

1 where(eachquarkappearsinthreedifferent colours)
v
7
ν q ν q
3 l u l , u , l , q , q , (2)
05 (cid:20) l− qd (cid:21) ≡ (cid:18) l− (cid:19)L (cid:18) qd (cid:19)L R− uR dR
.
1 plusthecorresponding antiparticles. Thus,theleft-handed fieldsareSU(2) doublets,whiletheirright-
L
0 handed partners transform as SU(2) singlets. The three fermionic families in Eq. (1) appear to have
2 L
identicalproperties(gaugeinteractions);theydifferonlybytheirmassandtheirflavourquantumnumber.
1
v: Thegaugesymmetryisbrokenbythevacuum,whichtriggers theSpontaneous SymmetryBreak-
i ing(SSB)oftheelectroweak grouptotheelectromagnetic subgroup:
X
r
a SU(3) SU(2) U(1) SSB SU(3) U(1) . (3)
C L Y C QED
⊗ ⊗ −→ ⊗
The SSB mechanism generates the masses of the weak gauge bosons, and gives rise to the appearance
ofaphysical scalarparticleinthemodel,theso-called Higgs. Thefermionmassesandmixingsarealso
generated through theSSB.
The SM constitutes one of the most successful achievements in modern physics. It provides a
very elegant theoretical framework, which is able to describe the known experimental facts in particle
physics with high precision. These lectures [1]provide an introduction to the SM, focussing mostly on
its electroweak sector, i.e., the SU(2) U(1) part [2–5]. The strong SU(3) piece is discussed in
L Y C
⊗
more detail in Refs. [6,7]. The power of the gauge principle is shown in Section 2, where the simpler
Lagrangians of quantum electrodynamics and quantum chromodynamics are derived. The electroweak
theoretical framework is presented in Sections 3 and 4, which discuss, respectively, the gauge structure
and the SSB mechanism. Section 5 summarizes the present phenomenological status; it describes the
main precision tests performed at the Z peak and the tight constraints on the Higgs mass from direct
searches. The flavour structure is discussed in Section 6, where knowledge of the quark mixing angles
and neutrino oscillation parameters is briefly reviewed and the importance of violation tests is em-
CP
phasized. Finally, a few comments on open questions, to be investigated at future facilities, are given
in the summary. Some useful but more technical information has been collected in several appendices:
aminimal amount of quantum field theory concepts are given in Appendix A;Appendix Bsummarizes
themostimportant algebraic properties ofSU(N)matrices; andashortdiscussion ongauge anomalies
ispresented inAppendixC.
2 GaugeInvariance
2.1 Quantumelectrodynamics
Letusconsider theLagrangian describing afreeDiracfermion:
= iψ(x)γµ∂ ψ(x) mψ(x)ψ(x). (4)
0 µ
L −
isinvariant underglobalU(1)transformations
0
L
U(1)
ψ(x) ψ (x) exp iQθ ψ(x), (5)
′
−→ ≡ { }
whereQθ isanarbitrary realconstant. Thephase ofψ(x)isthenapure convention-dependent quantity
without physical meaning. However, the free Lagrangian is no longer invariant if one allows the phase
transformation to depend on the space-time coordinate, i.e., under local phase redefinitions θ = θ(x),
because
U(1)
∂ ψ(x) exp iQθ (∂ +iQ∂ θ) ψ(x). (6)
µ µ µ
−→ { }
Thus, once a given phase convention has been adopted at one reference point x , the same convention
0
mustbetakenatallspace-timepoints. Thislooksveryunnatural.
The‘gauge principle’ istherequirement thattheU(1)phase invariance should hold locally. This
is only possible if one adds an extra piece to the Lagrangian, transforming in such a way as to cancel
the ∂ θ term in Eq. (6). The needed modification is completely fixed by the transformation (6): one
µ
introduces anewspin-1(since∂ θ hasaLorentzindex)fieldA (x),transforming as
µ µ
U(1) 1
A (x) A (x) A (x) ∂ θ, (7)
µ −→ ′µ ≡ µ − e µ
anddefinesthecovariantderivative
D ψ(x) [∂ +ieQA (x)] ψ(x), (8)
µ µ µ
≡
whichhastherequiredproperty oftransforming likethefielditself:
U(1)
Dµψ(x) (Dµψ)′(x) exp iQθ Dµψ(x). (9)
−→ ≡ { }
TheLagrangian
iψ(x)γµD ψ(x) mψ(x)ψ(x) = eQA (x)ψ(x)γµψ(x) (10)
µ 0 µ
L ≡ − L −
istheninvariant underlocalU(1)transformations.
The gauge principle has generated an interaction between the Dirac fermion and the gauge field
A , which is nothing else than the familiar vertex of Quantum Electrodynamics (QED). Note that the
µ
correspondingelectromagneticchargeQiscompletelyarbitrary. IfonewantsA tobeatruepropagating
µ
field,oneneedstoaddagauge-invariant kineticterm
1
F (x)Fµν(x), (11)
Kin µν
L ≡ −4
2
where F ∂ A ∂ A is the usual electromagnetic field strength which remains invariant under
µν µ ν ν µ
≡ −
the transformation (7). A mass term for the gauge field, = 1m2AµA , is forbidden because it
Lm 2 µ
would violate the local U(1) gauge invariance; therefore, the photon field is predicted to be massless.
Experimentally, weknowthatm < 1 10 18 eV[8,9].
γ −
·
ThetotalLagrangianinEqs. (10)and(11)givesrisetothewell-knownMaxwellequations:
∂ Fµν = eJν eQψγνψ, (12)
µ
≡
whereJν isthefermionelectromagnetic current. Fromasimplegauge-symmetry requirement, wehave
deduced therightQEDLagrangian, whichleadstoaverysuccessful quantum fieldtheory.
2.1.1 Leptonanomalousmagneticmoments
W(cid:13) W(cid:13) f(cid:13)
f(cid:13) (cid:13)g
g , Z(cid:13) (cid:13)n
(a)(cid:13) (b)(cid:13) (c)(cid:13) (d)(cid:13)
Fig.1: Feynmandiagramscontributingtotheleptonanomalousmagneticmoment.
ThemoststringentQEDtestcomesfromthehigh-precisionmeasurementsofthee[10]andµ[11]
anomalous magneticmoments a (gγ 2)/2, where ~µ gγ(e/2m )S~ :
l ≡ l − l ≡ l l l
a = (1 159 652 180.73 0.28) 10 12, a = (11 659 208.9 6.3) 10 10. (13)
e − µ −
± · ± ·
Toameasurablelevel,a arisesentirelyfromvirtualelectronsandphotons;thesecontributionsare
e
fullyknowntoO(α4)andmanyO(α5)correctionshavebeenalreadycomputed[12–14]. Theimpressive
agreement achieved between theory and experiment has promoted QED to the level of the best theory
everbuilt todescribe Nature. Thetheoretical error isdominated by theuncertainty inthe input value of
theQEDcoupling α e2/(4π). Turningthingsaround, a provides themostaccurate determination of
e
≡
thefinestructure constant[10,15]:
α 1 = 137.035 999 084 0.000 000 051. (14)
−
±
Theanomalousmagnetic momentofthemuonissensitive tosmallcorrections fromvirtualheav-
ier states; compared to a , they scale with the mass ratio m2/m2. Electroweak effects from virtual
e µ e
W and Z bosons amount to a contribution of (15.4 0.2) 10 10 [16–18], which is larger than the
± −
± ·
present experimental precision. Thus a allows one to test the entire SM. The main theoretical uncer-
µ
tainty comes from strong interactions. Since quarks have electric charge, virtual quark-antiquark pairs
inducehadronic vacuumpolarization corrections tothephotonpropagator (Fig.1.c). Owingtothenon-
perturbative character of the strong interaction at low energies, the light-quark contribution cannot be
reliably calculated at present. This effect can be extracted from the measurement of the cross-section
σ(e+e hadrons) and from the invariant-mass distribution of the final hadrons in τ decays, which
−
→
unfortunately provideslightlydifferent results[19]:
(11659180.2 4.9) 10 10 (e+e data),
ath = ± · − − (15)
µ (11659189.4 5.4) 10 10 (τ data).
−
(cid:26) ± ·
The quoted uncertainties include also the smaller light-by-light scattering contributions (Fig. 1.d) [20].
Thedifference between theSMprediction andtheexperimental value (13)corresponds to3.6σ (e+e )
−
or 2.4σ (τ). Newprecisee+e andτ datasetsareneededtosettlethetruevalueofath.
− µ
3
2.2 Quantumchromodynamics
2.2.1 Quarksandcolour
e– q
g , Z
e+ q
Fig.2: Tree-levelFeynmandiagramforthe e+e− annihilationintohadrons.
Thelarge number ofknownmesonic and baryonic states clearly signals the existence ofadeeper
level of elementary constituents of matter: quarks. Assuming that mesons are M qq¯ states, while
≡
baryons have three quark constituents, B qqq, one can nicely classify the entire hadronic spectrum.
≡
However, in order to satisfy the Fermi–Dirac statistics one needs to assume the existence of a new
quantum number, colour, such that each species of quark may have N = 3 different colours: qα,
C
α = 1,2,3(red,green,blue). Baryonsandmesonsarethendescribedbythecolour-singletcombinations
1 1
B = ǫαβγ q q q , M = δαβ q q¯ . (16)
α β γ α β
√6 | i √3 | i
In order to avoid the existence of non-observed extra states with non-zero colour, one needs to further
postulate that all asymptotic states are colourless, i.e., singlets under rotations in colour space. This
assumption is known as the confinement hypothesis, because it implies the non-observability of free
quarks: sincequarkscarrycolourtheyareconfinedwithincolour-singlet boundstates.
Adirecttestofthecolourquantumnumbercanbeobtained fromtheratio
σ(e+e hadrons)
−
Re+e− ≡ σ(e+e → µ+µ ) . (17)
− −
→
The hadronic production occurs through e+e γ ,Z qq¯ hadrons (Fig. 2). Since quarks are
− ∗ ∗
→ → →
assumed to be confined, the probability to hadronize is just one; therefore, summing over all possible
quarks in the final state, we can estimate the inclusive cross-section into hadrons. The electroweak
production factors which are common with the e+e γ ,Z µ+µ process cancel in the ratio
− ∗ ∗ −
→ →
(17). At energies well below the Z peak, the cross-section is dominated by the γ-exchange amplitude;
theratioRe+e− isthengivenbythesumofthequarkelectriccharges squared:
2N = 2, (N = 3 : u,d,s)
Nf 3 C f
Re+e− ≈ NC Q2f =  190 NC = 130 , (Nf = 4 : u,d,s,c) . (18)
Xf=1  11 N = 11 , (N = 5 : u,d,s,c,b)
9 C 3 f
This result involves an explicit sum overthe N quark flavours which are kinematically accessible
f
[4m2q < s ≡ (pe− + pe+)2], weighted by the number of different colour possibilities. The measured
ratio is shown in Fig. 3. Although the simple formula (18) cannot explain the complicated structure
around the different quark thresholds, it gives the right average value of the cross-section (away from
thresholds), providedthatN istakentobethree. Theagreementisbetteratlargerenergies. Noticethat
C
stronginteractions havenotbeentakenintoaccount; onlytheconfinementhypothesis hasbeenused.
Electromagnetic interactions are associated with the fermion electric charges, while the quark
flavours (up, down, strange, charm, bottom, top) are related to electroweak phenomena. The strong
forcesareflavourconserving andflavourindependent. Ontheotherside, thecarriers oftheelectroweak
interaction (γ, Z, W ) do not couple to the quark colour. Thus it seems natural to take colour as the
±
chargeassociated withthestrongforcesandtrytobuildaquantum fieldtheorybasedonit[21,22].
4
Υ
3
10 J/ψ ψ(2S)
Z
2
10
φ
R ω
10
ρ′
1
ρ
-1
10
2
1 10 10
√s [GeV]
Fig. 3: World data on the ratio Re+e− [9]. The broken lines show the naive quark model approximation with
N =3. Thesolidcurveisthe3-loopperturbativeQCDprediction.
C
2.2.2 Non-Abeliangaugesymmetry
Letusdenoteqα aquarkfieldofcolourαandflavourf. Tosimplifytheequations, letusadoptavector
f
notation incolourspace: qT (q1, q2, q3). ThefreeLagrangian
f ≡ f f f
= q¯ (iγµ∂ m )q (19)
0 f µ f f
L −
f
X
isinvariant underarbitrary globalSU(3) transformations incolourspace,
C
qα (qα) = Uα qβ , U U = U U = 1 , detU = 1 . (20)
f −→ f ′ β f † †
TheSU(3) matricescanbewrittenintheform
C
λa
U = exp i θ , (21)
a
2
(cid:26) (cid:27)
where 1λa (a = 1,2,...,8) denote the generators of the fundamental representation of the SU(3)
2 C
algebra, θ arearbitrary parameters andasum overrepeated colour indices isunderstood. Thematrices
a
λa aretraceless andsatisfythecommutationrelations
λa λb λc
, = ifabc , (22)
2 2 2
(cid:20) (cid:21)
with fabc the SU(3) structure constants, which are real and totally antisymmetric. Someuseful prop-
C
ertiesofSU(N)matricesarecollected inAppendix B.
AsintheQEDcase, wecannow require theLagrangian tobealsoinvariant under local SU(3)
C
transformations, θ = θ (x). To satisfy this requirement, we need to change the quark derivatives by
a a
covariant objects. Sincewehavenoweightindependent gaugeparameters, eightdifferent gaugebosons
Gµ(x),theso-called gluons, areneeded:
a
λa
Dµq ∂µ+ig Gµ(x) q [∂µ+ig Gµ(x)] q . (23)
f ≡ s 2 a f ≡ s f
(cid:20) (cid:21)
5
Gma Gma Gnb Gmb Gsd
q q
a b
G c G c G e
g l aab g gs fabc s n g2 f f r
s 2 m s abc ade
Fig.4:InteractionverticesoftheQCDLagrangian.
Noticethatwehaveintroduced thecompactmatrixnotation
λa
[Gµ(x)] Gµ(x) (24)
αβ ≡ 2 a
(cid:18) (cid:19)αβ
andacolour identitymatrixisimplicitinthederivativeterm. WewantDµq totransform inexactlythe
f
samewayasthecolour-vector q ;thisfixesthetransformation properties ofthegaugefields:
f
i
Dµ (Dµ) = U DµU , Gµ (Gµ) = U GµU + (∂µU)U . (25)
′ † ′ † †
−→ −→ g
s
Underaninfinitesimal SU(3) transformation,
C
λa
qα (qα) = qα + i δθ qβ,
f −→ f ′ f 2 a f
(cid:18) (cid:19)αβ
1
Gµ (Gµ) = Gµ ∂µ(δθ ) fabcδθ Gµ. (26)
a −→ a ′ a − g a − b c
s
Thegauge transformation ofthe gluon fields ismorecomplicated than the one obtained in QEDforthe
photon. The non-commutativity of the SU(3) matrices gives rise to an additional term involving the
C
gluon fields themselves. For constant δθ , the transformation rule for the gauge fields is expressed in
a
terms of the structure constants fabc; thus, the gluon fields belong to the adjoint representation of the
colour group (see Appendix B). Note also that there is a unique SU(3) coupling g . In QED it was
C s
possible to assign arbitrary electromagnetic charges to the different fermions. Since the commutation
relation (22) is non-linear, this freedom does not exist for SU(3) . All colour-triplet quark flavours
C
coupletothegluonfieldswithexactlythesameinteraction strength.
To build a gauge-invariant kinetic term for the gluon fields, weintroduce the corresponding field
strengths:
i λa
Gµν(x) [Dµ,Dν] = ∂µGν ∂νGµ+ig [Gµ,Gν] Gµν(x),
≡ −g − s ≡ 2 a
s
Gµν(x) = ∂µGν ∂νGµ g fabcGµGν. (27)
a a − a − s b c
UnderaSU(3) gaugetransformation,
C
Gµν (Gµν) = U GµνU (28)
′ †
−→
andthecolourtrace Tr(GµνG ) = 1 GµνGa remains invariant. Takingtheproper normalization for
µν 2 a µν
thegluonkinetic term,wefinallyhavetheSU(3) invariant Lagrangian ofQuantum Chromodynamics
C
(QCD):
1
GµνGa + q¯ (iγµD m ) q . (29)
LQCD ≡ −4 a µν f µ− f f
f
X
6
Fig.5: Two-andthree-jeteventsfromthehadronicZ bosondecays Z qq¯ and Z qq¯G (ALEPH)[23].
→ →
TheSU(3) gauge symmetryforbids toaddamasstermforthegluon fields, 1m2GµGa,because itis
C 2 G a µ
notinvariant underthetransformation (25). Thegaugebosons are,therefore, masslessspin-1particles.
Itisworthwhiletodecompose theLagrangian intoitsdifferentpieces:
1
= (∂µGν ∂νGµ)(∂ Ga ∂ Ga) + q¯α (iγµ∂ m ) qα
LQCD −4 a − a µ ν − ν µ f µ− f f
f
X
λa
g Gµ q¯αγ qβ (30)
− s a f µ 2 f
f (cid:18) (cid:19)αβ
X
g g2
+ s fabc(∂µGν ∂νGµ)Gb Gc s fabcf GµGνGd Ge .
2 a − a µ ν − 4 ade b c µ ν
Thefirstlinecontains thecorrect (quadratic) kinetic termsforthedifferent fields, whichgiverise tothe
corresponding propagators. The colour interaction between quarks and gluons is given by the second
line; it involves the SU(3) matrices λa. Finally, owing to the non-Abelian character of the colour
C
group, the GµνGa term generates the cubic and quartic gluon self-interactions shown in the last line;
a µν
thestrengthoftheseinteractions(Fig.4)isgivenbythesamecouplingg whichappearsinthefermionic
s
pieceoftheLagrangian.
In spite of the rich physics contained in it, the Lagrangian (29) looks very simple because of its
colour symmetry properties. Allinteractions aregivenintermsofasingle universal coupling g ,which
s
iscalledthestrongcouplingconstant. Theexistenceofself-interactions amongthegaugefieldsisanew
featurethatwasnotpresentinQED;itseemsthenreasonabletoexpectthatthesegaugeself-interactions
couldexplainproperties likeasymptotic freedom (stronginteractions becomeweakeratshortdistances)
andconfinement(thestrongforcesincrease atlargedistances), whichdonotappearinQED[6].
Without any detailed calculation, one can already extract qualitative physical consequences from
. Quarks can emit gluons. At lowest order in g , the dominant process will be the emission of a
QCD s
L
singlegaugeboson;thus,thehadronicdecayoftheZ shouldresultinsomeZ qq¯Gevents,inaddition
→
tothe dominant Z qq¯decays. Figure 5clearly showsthat 3-jet events, withtherequired kinematics,
→
indeedappearintheLEPdata. Similareventsshowupine+e annihilationintohadrons,awayfromthe
−
Z peak. Theratiobetween3-jetand2-jeteventsprovidesasimpleestimateofthestrength ofthestrong
interaction atLEPenergies (s = M2): α g2/(4π) 0.12.
Z s ≡ s ∼
7
3 Electroweak Unification
3.1 Experimentalfacts
Low-energy experiments have provided a large amount of information about the dynamics underlying
flavour-changing processes. The detailed analysis of the energy and angular distributions in β decays,
such as µ e ν¯ ν or n pe ν¯ , made clear that only the left-handed (right-handed) fermion
− − e µ − e
→ →
(antifermion) chiralities participate in those weak transitions; moreover, the strength of the interaction
appears to be universal. This is further corroborated through the study of other processes like π
−
→
e ν¯ or π µ ν¯ , which show that neutrinos have left-handed chiralities while anti-neutrinos are
− e − − µ
→
right-handed.
Fromneutrinoscatteringdata,welearnttheexistenceofdifferentneutrinotypes(ν = ν )andthat
e µ
6
thereareseparately conserved leptonquantum numberswhichdistinguish neutrinos fromantineutrinos;
thus weobserve thetransitions ν¯ p e+n, ν n e p, ν¯ p µ+n or ν n µ p,but wedo
e e − µ µ −
→ → → →
notseeprocesses like ν p e+n, ν¯ n e p, ν¯ p e+n or ν n e p.
e e − µ µ −
6→ 6→ 6→ 6→
Togetherwiththeoretical considerations related tounitarity (aproper high-energy behaviour) and
the absence of flavour-changing neutral-current transitions (µ e e e+, s dℓ+ℓ ), the low-
− − − −
6→ 6→
energy information wasgood enough todetermine thestructure ofthe modern electroweak theory [24].
The intermediate vector bosons W and Z were theoretically introduced and their masses correctly
±
estimated, before their experimental discovery. Nowadays, wehave accumulated huge numbers of W
±
andZ decayevents,whichbringmuchdirectexperimental evidence oftheirdynamical properties.
3.1.1 Chargedcurrents
n(cid:13) m(cid:13) -(cid:13)
n(cid:13) m(cid:13)
m(cid:13)
m(cid:13) -(cid:13)
+(cid:13)
-(cid:13) W(cid:13)
-(cid:13) e(cid:13)
W(cid:13)
n(cid:13) -(cid:13) n(cid:13)
e(cid:13) e(cid:13) e(cid:13)
Fig.6:Tree-levelFeynmandiagramsfor µ− e−ν¯ ν and ν e− µ−ν .
e µ µ e
→ →
Theinteraction ofquarksandleptons withtheW bosons(Fig.6)exhibitsthefollowingfeatures:
±
– Only left-handed fermions and right-handed antifermions couple to the W . Therefore, there is
±
a 100% breaking of parity ( : left right) and charge conjugation ( : particle antiparticle).
P ↔ C ↔
However,thecombinedtransformation isstillagoodsymmetry.
CP
– TheW bosons coupletothefermionicdoublets inEq.(2),wheretheelectric charges ofthetwo
±
fermionpartnersdifferinoneunit. Thedecaychannels oftheW arethen:
−
W e ν¯ , µ ν¯ , τ ν¯ , d u¯, s c¯. (31)
− − e − µ − τ ′ ′
→
Owing to the very high mass of the top quark [25], m = 173 GeV > M = 80.4 GeV, its
t W
on-shell production through W b t¯ iskinematically forbidden.
− ′
→
– Allfermiondoublets coupletotheW bosonswiththesameuniversalstrength.
±
– The doublet partners of the up, charm and top quarks appear to be mixtures of the three quarks
withcharge 1:
−3
d d
′
s = V s , VV = V V = 1. (32)
′ † †
   
b b
′
   
8
e– m – e– n
g , Z Z
e+ m + e+ n
Fig.7: Tree-levelFeynmandiagramsfor e+e− µ+µ− and e+e− νν¯.
→ →
Thus, the weak eigenstates d , s , b are different than the mass eigenstates d, s, b. They are
′ ′ ′
relatedthrough the3 3unitary matrixV,whichcharacterizes flavour-mixing phenomena.
×
– The experimental evidence of neutrino oscillations shows that ν , ν and ν are also mixtures
e µ τ
of mass eigenstates. However, the neutrino masses are tiny: m2 m2 2.4 10 3eV2,
ν3 − ν2 ∼ · −
m2 m2 7.6 10 5eV2 [9].
ν2 − ν1 ∼ · − (cid:12) (cid:12)
(cid:12) (cid:12)
3.1.2 Neutralcurrents
Theneutralcarriersoftheelectromagnetic andweakinteractions havefermioniccouplings(Fig.7)with
thefollowingproperties:
– All interacting vertices are flavour conserving. Both the γ and the Z couple to a fermion and its
own antifermion, i.e., γff¯ and Zff¯. Transitions of the type µ eγ or Z e µ have
± ∓
6→ 6→
neverbeenobserved.
– The interactions depend on the fermion electric charge Q . Fermions with the same Q have
f f
exactly the same universal couplings. Neutrinos do not have electromagnetic interactions (Q =
ν
0),buttheyhaveanon-zero couplingtotheZ boson.
– Photonshavethesameinteractionforbothfermionchiralities,buttheZ couplingsaredifferentfor
left-handed and right-handed fermions. Theneutrino coupling to the Z involves only left-handed
chiralities.
– Therearethreedifferent lightneutrino species.
3.2 The SU(2)L ⊗U(1)Y theory
Using gauge invariance, we have been able to determine the right QED and QCD Lagrangians. To
describe weak interactions, we need a more elaborated structure, with several fermionic flavours and
different properties for left- and right-handed fields; moreover, the left-handed fermions should appear
in doublets, and we would like to have massive gauge bosons W and Z in addition to the photon.
±
Thesimplestgroup withdoublet representations isSU(2). Wewanttoinclude alsotheelectromagnetic
interactions; thusweneedanadditional U(1)group. Theobvious symmetrygrouptoconsider isthen
G SU(2) U(1) , (33)
L Y
≡ ⊗
whereLreferstoleft-handed fields. Wedonotspecify, forthemoment,themeaningofthesubindex Y
since,aswewillsee,thenaiveidentification withelectromagnetism doesnotwork.
Forsimplicity, letusconsider asinglefamilyofquarks,andintroduce thenotation
u
ψ (x) = , ψ (x) = u , ψ (x) = d . (34)
1 d 2 R 3 R
(cid:18) (cid:19)L
Ourdiscussion willalsobevalidfortheleptonsector, withtheidentification
ν
ψ (x) = e , ψ (x) = ν , ψ (x) = e . (35)
1 e 2 eR 3 −R
(cid:18) − (cid:19)L
9
AsintheQEDandQCDcases, letusconsider thefreeLagrangian
3
= iu¯(x)γµ∂ u(x) + id¯(x)γµ∂ d(x) = iψ (x)γµ∂ ψ (x). (36)
L0 µ µ j µ j
j=1
X
isinvariant underglobalGtransformations inflavourspace:
0
L
G
ψ (x) ψ (x) exp iy β U ψ (x),
1 −→ 1′ ≡ { 1 } L 1
G
ψ (x) ψ (x) exp iy β ψ (x), (37)
2 −→ 2′ ≡ { 2 } 2
G
ψ (x) ψ (x) exp iy β ψ (x),
3 −→ 3′ ≡ { 3 } 3
wheretheSU(2) transformation [σ arethePaulimatrices(B.3)]
L i
σ
U exp i i αi (i = 1,2,3) (38)
L ≡ 2
n o
only acts on the doublet field ψ . The parameters y are called hypercharges, since the U(1) phase
1 i Y
transformation is analogous to the QEDone. Thematrix transformation U isnon-Abelian as in QCD.
L
NoticethatwehavenotincludedamassterminEq.(36)becauseitwouldmixtheleft-andright-handed
fields[seeEq.(A.17)],therefore spoilingoursymmetryconsiderations.
We can now require the Lagrangian to be also invariant under local SU(2) U(1) gauge
L Y
⊗
transformations, i.e.,withαi = αi(x)andβ = β(x). Inordertosatisfy thissymmetry requirement, we
needtochange thefermion derivatives bycovariant objects. Sincewehavenowfour gaugeparameters,
αi(x)andβ(x),fourdifferentgaugebosons areneeded:
D ψ (x) ∂ +igW (x)+ig y B (x) ψ (x),
µ 1 ≡ µ µ ′ 1 µ 1
D ψ (x) [h∂ +ig y B (x)] ψ (x), i (39)
µ 2 ≡ µ ′f2 µ 2
D ψ (x) [∂ +ig y B (x)] ψ (x),
µ 3 ≡ µ ′ 3 µ 3
where
σ
W (x) i Wi(x) (40)
µ ≡ 2 µ
denotes aSU(2) matrixfield. Thuswehavethecorrect numberofgaugefieldstodescribe theW ,Z
L f ±
andγ.
We want D ψ (x) to transform in exactly the same way as the ψ (x) fields; this fixes the trans-
µ j j
formationproperties ofthegaugefields:
1
G
B (x) B (x) B (x) ∂ β(x), (41)
µ −→ µ′ ≡ µ − g µ
′
i
G
Wµ −→ Wµ′ ≡ UL(x)WµUL†(x)+ g ∂µUL(x)UL†(x), (42)
where U (x) exp ifσi αi(x) . Tfhetransformatfion ofB isidenticaltotheoneobtainedinQEDfor
L ≡ 2 µ
thephoton, whiletheSU(2) Wi fieldstransform inawayanalogous tothegluonfieldsofQCD.Note
(cid:8) L (cid:9)µ
that the ψ couplings to B are completely free as in QED, i.e., the hypercharges y can be arbitrary
j µ j
parameters. Since the SU(2) commutation relation is non-linear, this freedom does not exist for the
L
Wi: thereisonlyauniqueSU(2) coupling g.
µ L
TheLagrangian
3
= iψ (x)γµD ψ (x) (43)
L j µ j
j=1
X
10