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Thisoneappearslogarithmicallydivergent:
M
µνρσ
=clogΛ
2
µν
η
ρσ
µρ
η
νσ
µσ
η
νρ
)+finite
(22.9)
For someconstantc. . The e η
s havebeensymmetrized. . However,weknowbytheWardidentitythatthis
mustvanishwhenoneofthephotonsisreplacedbyit’smomentum. SayA
µ
hasmomentumq
µ
. Then
0=q
µ
M
µνρσ
=clogΛ
2
(q
ν
η
ρσ
+q
ρ
η
νσ
+q
σ
η
νρ
)+q
µ
·finite
(22.10)
Thismustholdforallq
µ
,whichisimpossibleunless c=0. . Thusgaugeinvariancemakes s this amplitude
finite.
22.2.2 5,6,... . pointfunctions
For1-loopcontributionsto1PIGreen’sfunctionswithmorethan4legs,wegetthingslikepentagondia-
grams
0|T{ψ
¯
ψA
ν
A
ρ
A
σ
|0
d
4
k
(2π)4
1
k
1
k
1
k
1
k
1
k2
(22.11)
This will all l have at least 5 5 propagators, , with h 5 5 factors s of loop momentum, so they will l be finite. . It
doesn’t matter if the e propagators s are fermions or photons anymore, we e will l always s have more than 4
powersof kinthedenominator.
So,the4countertermsδ
1
2
3
andδ
m
sufficetocancelallthedivergencesinanyGreen’sfunctionof
QEDat1-loop. Thegeneraldefinitionofrenormalizableis:
Renormalizable:allUVdivergencesarecanceledwithafinitenumberofcounterterms
Thus,wehaveshownthatatQEDisrenormalizable at1-loop.
22.2.3 renormalizabilitytoallorders
Whatabout2-loopandhigherloop1PIgraphs? Wecanshowtheyarefinitebyinduction.
Whenwe addaloop,wecanaddeither aphoton n propagator r ora a fermion propagator. . If f weaddan
internalphotonpropagator,
(22.12)
it must split two o fermionlines,so the new loophas s two o additionalfermionpropagators s as well. . By y the
definition of f 1PI, loop p momenta go through all internal lines. . So o for k k ≫ p
i
, where e p
i
is any external
momentum,eachinternallinewillget
1
k
or
1
k2
.Sothematrixelementgetsmodifiedto
M→
d
4
k
(2π)4
1
k2
1
k
1
k
M
(22.13)
Ifweaddafermionloop,weneedtocutaphotonline,
(22.14)
22.2 Renormalizabilityof f QED
231
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giving
1
k2
andby Lorentz z invariance, we need d to add at least 2 fermion propagators
1
k
1
k
, so afermion
insertionalsodoesn’tchangethepowercounting.
Thisassumesthatdominantdivergencecomesfromwhenalltheloopmomentagotoinfinitytogether.
Clearlyifthe graphisnotdivergentwhenallthemomentaareblowingup,itwillnot bedivergentwhen
onlysomeofthemomentaareblowingup.
There is another r special case,when the e two loop p momenta come e inas s k
1
− k
2
in the denominator.
Thenthedegreeofdivergencedependsonpreciselyhowwetakethemomentatoinfinitytogether. Butif
we hold d p
µ
= k
1
µ
− k
2
µ
fixed in the limit, then we e are e effectively integrating over only y one fewer r loop
momenta. Thusthisisequivalenttoagraphofonefewerloops. Theactualproofthatthesespecialcases
do not t screw w things s upis s fairly y involved. For QED it involves gaugeinvariance in n an essentialway(for
example,aswesawinthelight-by-lightscattering4-pointfunctionabove).
TheresultthatthereareonlyafinitenumberofdivergentloopsisknownastheBPHZtheorem,after
Bogoliubov, Parasiuk, Hepp, and Zimmermann. . We e have only sketched d the e proof here. . Since e we e can
cancel all l the 1-loop divergences s already, we can now cancel all the 2 and higher r loop p divergences by
addingpiecestothesamecountertermstohigherorderinα.
Wecanconcludethat
QEDisrenormalizable
Thismeans that that alltheUV divergencesinQED arecanceledbythesame4 4 counterterms s we’ve
introducesat1-loop. Thesearefit by2numbers:thephysicalvalueoftheelectricchargee
R
measuredin
Coulomb’slawat p=0andthephysicalvalueoftheelectronmass. Theothertwocountertermsarefixed
by canonicallynormalizingthe electron n andphotonfields. . (The e physical observable for theseis that we
see1-particleelectronandphotonstates,whichsoundslikearathercoarseobservable,butitisanobserv-
ablenonetheless.)
Thisais prettyamazingconclusionactually. . Withjust2freeparameters s fittodata,wecanmakean
infinitenumberofpredictionsinQED.
Renormalizability playedavery important role in n the e historical development of quantum fieldtheory
andgaugetheories. Nowadays,wehavealargercontexttoplacerenormalizabletheories. Asitturns s out,
renormalizabilityis not obviously aqualityweneed d ina theory, andit t mayevenhurtus. . For r example,
QEDhasabigproblem:ithasaLandaupole. Sowe cannotpredictevery observablejust because
we can n cancel l all the UV V divergences. Forexample,Coulombscattering aboveE=10
300
GeVis a
completelymysteryinQED. Moreover,preciselybecauseQEDisrenormalizable,wehavenosensitivityat
lowenergyto the UVstructure of the theory, sowe have noway of probingthe mysterious s high-energy
regime without building g a a 10
300
GeV collider. . Other r renormalizable theories are unpredictive in n much
morerelevantregimes:thestandardelectroweakmodel(withouttheHiggs) isnon-perturbative at t ∼10
3
GeV,aswewilldiscussinamoment,andQCD doesnotmake perturbativepredictionsbelow w ∼1GeV.
Bothofthesetheoriesarerenormalizable.
Wecanunderstandtheseissuesinmoredetailbyaskingaverysimplequestion:whattonon-renormal-
izabletheorieslooklike?
22.3 Non-renormalizablefieldtheories
WesawthatinQED,therearefinitenumberofdivergentone-particleirreduciblecontributionstoGreen’s
functions. The superficialdegree of divergence ofaGreen’s s functionis theoverallnumber of powersof k
inthe loopmomenta,assumingallk≫ p
i
for any externalmomentum m p
i
. Callthe e superficialdegree of
divergenceofaGreen’sfunctionwithnexternalfermionsandmexternalphotonsD
n,m
. Thenwefound
0|T{AA}|0∼
d
4
k
(2π)4
1
k
1
k
D
2,0
=2
(22.15)
0|T{ψ
¯
ψ
|0
d
4
k
(2π)4
1
k2
1
k
D
2,0
=1
(22.16)
232
RenormalizabilityofQED
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0|T{ψ
¯
ψA
|0
d
4
k
(2π)4
1
k2
1
k
1
k
D
2,0
=0
(22.17)
0|T{A
µ
A
ν
A
ρ
A
σ
}|0∼
d
4
k
(2π)4
1
k
1
k
1
k
1
k
D
0,4
=0
(22.18)
0|T{ψ
¯
ψA
µ
A
ν
|0
d
4
k
(2π)4
1
k2
1
k
1
k
1
k
D
2,2
=−1
(22.19)
0|T{ψ
¯
ψψ
¯
ψ
|0
d4k
(2π)4
1
k2
1
k2
1
k
1
k
D
4,0
=−2
(22.20)
Andthatanythingwithmorethan4fieldshasD<0. Itisnothardtoworkoutthat
D
n,m
=4−
3
2
n−m
(22.21)
ThescalingcomessimplyfromthedimensionoftheGreen’sfunction:fermionshavedimension
3
2
andpho-
tonsdimension1.Withscalarexternalstates,thiswouldbe
D
n,m,s
=4−
3
2
n−m−s
(22.22)
wheresisthenumberofscalars. OnlytheGreen’sfunctionswithD>0canpossiblybedivergent.
QEDisaspecialtheorybecauseitonlyhasasingleinteractionvertex
L
QED
=L
kin
+eψ
¯
A
µ
γ
µ
ψ
(22.23)
the coefficient of this interactionis the dimensionless charge e. . More e generally, we might t have e atheory
withcouplingsofarbitrarydimension. Forexample,
L=−
1
2
φ(+m2)φ+g
1
φ3+g
2
φ2φ3+
(22.24)
Thisscrewsupthepowercounting.
Call the dimension of f the e coefficient of f the e ith h interaction ∆
i
. For r example, the g
1
term above e has
1
=1andg
2
has∆
2
=−5. Nowconsider r aloopcontributiontoaGreen’sfunctionwithn
i
insertionsof
theverticeswithdimension∆
i
. Fork≫p
i
,theonlyscalesthatcanappeararek
sand∆
s.Sobydimen-
sionalanalysis,thesuperficialdegreeofdivergenceofthesameintegralchangesas
k
D
→g
i
n
i
k
D−n
i
i
(22.25)
Thus
D
n,m
=4−
3
2
n−m−s−n
i
i
(22.26)
Soifthereareinteractionswith∆
i
<0thentherecanbeaninfinitenumbervaluesof nandm,andthere-
foreaninfinitenumberof1PIloopswithD>0. Thuswewillneedaninfinitenumberofcountertermsto
cancelalltheinfinities. Suchtheoriesarecallednon-renormalizable.
Non-renormalizable is s equivalent to o there being g interactions with mass dimensions ∆
i
< 0. . On n the
otherhandifalltheinteractionshavemass dimension∆
i
>0,thenthetheory iscalledsuper-renormaliz-
able.φ
3
theoryisasuper-renormalizabletheory.
Wegeneralizedthisterminologyalsotodescribeindividualinteractions
• renormalizableinteraction:couplingconstanthasdimension n 0
• non-renormalizableinteraction:couplingconstanthasdimension n <0
22.3 Non-renormalizable e field theories
233
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Alsopeoplesometimes describe interactions of dimension0as marginal,dimension> 0 0 as relevant,and
dimension<0asirrelevant.Theselabelscomefromtherenormalizationgroup.
22.3.1 non-renormalizabletheoriesarefinitetoo
Non-renormalizabletheorieshaveaninfinitenumberofsuperficiallydivergentintegrals.Thisisduetothe
presenceofcouplings g
i
withnegativemassdimension.Forexample,g=
1
M
forsomescaleM.
Thedivergentintegralscanalwaysbewrittenassumsoftermsoftheform
I
div
=(p
µ
p
ν
)g
1
g
n
dk
k
j
(22.27)
for some e number r m m of the various s external momenta a p
µ
. This s is s in n the e region n of f loop p momentum for
whichk≫pforallexternalmomentap.Theseintegralscanproducelogarithmsoftheregularizationscale
Λ,orpowersofΛ
I
div
=
g
n
(p
µ
p
ν
)[c
0
logΛ+c
1
Λ+c
2
Λ
2
+c
3
Λ
3
+
]
(22.28)
Itis veryimportantthattherecanneverbetermslikelogp
2
inthedivergent partoftheintegral,thatis,
nothinglikeΛ
2
logp
2
canappear. Thisissimplybecausewedidn’thaveanlogp
2
termstobeginwithand
we can n go tothedivergent t regionofthe integral bytaking g k≫ ≫ pbeforeintegrating g over r anythingthat
might givealog. ItisnotjustΛ
2
logp
2
termsthatwillnever appear,butanythingthatisnotanalyticin
theexternalmomentaareforbiddenaswell.
Anotherwaytosee that thedivergences areapolynomialintheexternal momentacomes fromusing
the derivative e trick k to evaluate e the integrals. . Weinberg g takes this approach in section 12.2. . A general
divergentintegralwillhavevariousmomentafactorsinit,suchas
I(p)=
0
kdk
k+p
(22.29)
There will l always s be e at least one e denominator r with h a factor of f p. If f we e differentiate the integral with
respecttopenoughtimes,theintegralbecomesconvergent
I
′′
(p)=
0
2kdk
(k+p)3
=
1
p
(22.30)
Thuswecanthenintegrateoverptoproduceapolynomial,uptoconstantsofintegration
I(p)=plog
p
Λ
1
−p+Λ
2
=plogp−p(logΛ
1
+1)+Λ
2
(22.31)
So we do get t non-analyticterms inthe integral, but only polynomials in n p can ever r multiply the diver-
gences.
Thepointisthatpolynomialsinexternalmomentaareexactlywhatwegetattreelevelfromterms in
theLagrangian. Thatis,wecanhavecountertermsforthem. Forexample,supposethisintegralI(p)were
contributiontosomethinglikeaφ
6
correlationfunction:
0|T{φ(p
1
)
φ(p)
}
|0
=−plogΛ+Λ+finite
(22.32)
Thenwecouldaddterms
L
new
=Z
5
(∂φ)φ
5
+Z
6
φ
6
=(∂φ)φ
5
6
5
(∂φ)φ
5
6
φ
6
(22.33)
Thesenewtermscontributetothe6pointfunctionas
0|T{φ(p
1
)
φ(p
6
)}|0=p+1+pδ
5
6
(22.34)
Thecounterterms cannowbe chosento o cancel the infinities:δ
5
=logΛandδ
6
=−Λ. Thenthe e Green’s
functionis madefinite. . Thiscanbedoneorder r byorderinperturbationtheorytocancelalloftheinfini-
tiesinthetheory.
234
RenormalizabilityofQED
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Inordertohaveacounterterm,weneedthecorrespondingtermtoactuallybeinour Lagrangian. . So
theeasiestthingtodoisjusttoaddeverypossibletermwecanthinkof. Butdoweneedtermslike∂
x
φ
4
,
which break k Lorentz z invariance? ? Obviously y not, as s we e will never get Lorentz z violatingdivergences in a
Lorentzinvarianttheory. Clearly,weonly y needtoaddterms thatpreservethesymmetriesofourtheory.
But, every possible term must beincluded, otherwise we cannot guarantee that all theinfinities s canbe
canceled. Thiscanbesubtleas s symmetries aresometimes violatedinthequantumtheoryorinthepath
integralmeasure,eventhoughthey are symmetries of theclassicalLagrangian. Suchviolationsarecalled
anomalies,andwewillgettothemeventually.
Insummary,
• non-renormalizabletheoriesmustincludeeverytermnotforbiddenbysymmetries
• non-renormalizabletheoriesneedaninfinitenumberofcounterterms
• non-renormalizabletheoriesarejustasfiniteasrenormalizabletheories
• non-renormalizabletheories s canberenormalized. . Their r physicalpredictions arejustasfiniteas in
renormalizabletheories.
The difference is s that the infinities in renormalizable e theories s can be e canceled d with a finite e number r of
counterterms,whileaninfinite numberof counterterms is oftennecessaryinnon-renormalizable theories.
However,non-renormalizabletheoriesarestillverypredictive,aswewillnowsee.
22.3 Non-renormalizable e field theories
235
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Chapter23
Non-Renormalizableand
Super-RenormalizableFieldTheories
23.1 Introduction
Almosteveryquantumfieldtheoryweever lookatis non-renormalizable. . Thisissimplybecauseeffective
fieldtheoriesarenon-renormalizable,andevery theoryweknowof iseither aneffective theoryor canbe
treated as an effective e theory y for practical l purposes. . So o youalready know that these e theories s are won-
derful for doing g physics. . We e will look at 4 examples, , corresponding g to o the e 4 forces of f nature: : the
Schrodingerequation(electromagnetism),the4-Fermitheory(theweakforce),thetheoryofmesons (the
strongforce),andquantumgravity. Ineachcasewewillseehowthenon-renormalizabletheoryis s predic-
tivedespitetheneedforaninfinitenumberofcounterterms.
After studyingsome examples ofnon-renormalizable theories,we willlook at the superrenormalizable
theories. These e theories all l have couplings withpositive mass dimension and have serious problems in
makingphysicalpredictions..
23.2 SchrodingerEquation
ConsidertheSchrodingerequation
i∂
t
ψ=Hψ=
p
2
2m
+V(r)
ψ
(23.1)
Thisisanon-renormalizableeffectivefieldtheory. Theparameterwithnegativemass-dimensionissimply
thecoefficientof p
2
:
1
m
.
Whyaretherenotmoretermsofhigherorderinpinthisequation? Well,thereare,asweknowfrom
takingthenon-relativisticlimitoftheDiracequation
H=
p
2
2m
1+a
1
p
2
m2
+a
2
p
4
m4
+
+V(r)
(23.2)
Allwearedoinghereis writingdownthemostgeneralHamiltonianconsistentwithrotationalsymmetry.
Thefactors of marejustdimensionalanalysis. . Itturnsoutthata
1
=−
1
4
fortheDiracequation,andthe
othersasarealsocalculable.
TheSchrodingerequationisusefulevenifwedon’tknowabouttheDiracequationorthata
1
=−
1
4
.In
thenon-relativisticlimit p≪m,thehigherordertermshaveaverysmalleffect. Buttheyaremeasurable!
a
1
contributes tothefinestructureof theHydrogenatom,anda
2
contributes tothehyperfinestructure.
It’snicethat we cancalculatea
1
anda
2
from theDiracequation. . But t evenif wedidn’t have the Dirac
equationtocalculate a
1
anda
2
,we couldhave measured d them m form they Hydrogenatomand d predicted
things about Helium,or H
2
or lots of other things. . So o this s non-renormalizable theory is very predictive
indeed.
237
Alsonotethat theSchrodinger equationis not predictive for energies s pm. . Thenall l of thehigher
ordertermsareequallyimportant. So,theSchrodingerequationispredictiveatlowenergy,butalsoindi-
cates the scale at whichperturbationtheory breaks down. . Ifwecanfindatheorywhichreduces s tothe
Schrodinger equationat t low energy, , but for r whichperturbationtheory still l works at t highenergy,it is
called UV completion. Thus, , the Dirac equationis a UV completion of the Schrodinger r equation. . The
Diracequation(andQED) arepredictivetomuchhigher energies(butnotatallenergies,becauseofthe
Landaupole,nottomentiontherestofthestandardmodelandgravity). TheKlein-Gordonequationis
adifferentUVcompletionoftheSchrodingerequation.
I want toemphasize that the Schrodinger equationis an extremelypredictive quantum theory, inde-
pendent of theDirac equation. . It t made quantum predictions many years before the Dirac equationwas
discovered. So o you already haveseenhow predictive anon-renormalizabletheory can be,as s longas we
stayatlowenoughenergy.
Letusnowlookatsomeothernon-renormalizablequantumfieldtheoriesandtheirUVcompletions.
23.3 The4-Fermitheory
Weak decays were first modeledby EnricoFermi. . He e observedthat a protoncandecay to aneutron, a
positronandaneutrino,soheaddedaninteractiontothefreeLagrangianforthesefieldsoftheform
L
fermi
=G
F
ψ
¯
p
ψ
n
ψ
¯
e
ψ
ν
(23.3)
withmaybesomeγ
µ
sorγ
5
’sthrownin. This s isknownasa4-Fermiinteraction,bothbecausethereare4
fermionsinitandalsobecauseFermiusedthistermasaverysuccessfulmodelofradioactivedecay
Onepredictionfromthisinteractionisthattheratefor β-decay p+→ne+νwillberelatedtotherate
forn→p
+
e
ν¯. Bothofthesereactionsoccur r innuclei. . It t alsomakesapredictionfor theangular depen-
denceandenergydistributionofthedecayproducts. Inaddition,the4-Fermitheorycanalsobe usedto
study parity violation, say,by comparingthepredictions ofthis interactiontothatofaninteractionlike
ψ
¯
γ
5
γµψψ
¯
γ
5
γµψ.
TheFermiconstantis
G
F
=1.166×10−5GeV−2
1
300GeV
2
(23.4)
SinceG
F
hasmassdimension −2,thisisanon-renormalizableinteraction.
Sinceitisnon-renormalizable,weshouldaddalltermsconsistentwithsymmetry
L=G
F
ψ
¯
ψψ
¯
ψ+a
1
G
F
2
ψ
¯
ψψ
¯
ψ+a
2
G
F
3
ψ
¯
ψ
2
ψ
¯
ψ+
(23.5)
We knowthe firsttermhas coefficient G
F
. Keepinmind,thisis s therenormalizedvalueofthecoupling,
becauseit wasextractedfromaphysicalexperiment. . Thea
i
s arearbitrary,andtheG
F
s thataccompany
themhavebeenaddedbydimensionalanalysis.
Despitethese additionalterms,the4-Fermitheoryis very predictive. Forexample,these higherorder
terms willeffecttheβ-decayratebyfactorsof(G
F
E
2
)
j
whereE issomeenergyintheprocess. Sincethe
masses of f the particles s and energies involved in n β-decay y are much less than G
F
−1/2
, these higher order
termswilldopracticallynothing
Or,consider alow energy ψψ→ ψψ scattering. . Thematrix x element will get tree-level contributions
fromalloftheseterms,andwewouldfind,intheschannel
M(s)=G
F
+a
1
G
F
s+a
2
G
F
s
2
+
(23.6)
238
Non-Renormalizableand
Super-Renormalizable FieldTheories
Nomatterwhata
1
is,fors≪(a
1
G
F
)−1 thissecondtermwillbenegligible. Ifitturnsoutthata
1
∼a
2
∼1
then for r s
≪ 300GeV,the entiredistributionis givenpractically by thefirst term,which h is s from the
original4-Fermicoupling(that’swhathappensinnature). Ontheotherhand,ifwegomeasurethisscat-
tingcross sectionasafunctionof s,andwefindthatitdoesdependons,we canfit a
1
anda
2
todata.
Thenwehave amoreadvancedtheory, whichis stillvery predictive. Weonlyneedtomeasurethecross
sectionat3energiestofit G
F
,a
1
anda
2
,thenwecansayalotaboutthedetailedenergydependence,or
othercrossed(e.g.t-channel)scatterings,ordecays. Itistruewewouldneedaninfinitenumberofexperi-
ments to fit all the e a
j
s, but t from m a a practical point of view, at low energies, this higher order a
j
s are
totally irrelevant, , just t as s the p
10
term in n the e Schrodinger equation is irrelevant. . All l our discussion of
renormalizabilityissecondarytothefactthatnon-renormalizabletheoriesareextremelypredictiveattree
level,thatisasclassicalfieldtheories.
Butwhat happens if s
1
G
F
? Thenall orders s inperturbationtheorywillcontribute. . Thusperturba-
tiontheorybreaksdown. SoweneedaUVcompletiontocalculatescatteringfors>
1
G
F
.
23.3.0.1 quantumpredictionsoftheFermi i theory
Perhapscounter-intuitively,the4-Fermitheoryispredictiveevenatthequantumlevel. Manypeopleoften
forgetthisfact!
Considertheloopcorrectiontothe4-pointfunction:
iM
2
(s)=
p
2
p
1
p
4
p
3
k
1
k
2
∼G
F
2
d
4
k
(2π)4
1
k
1
k
∼G
F
2
(b
1
s+b
2
slog
Λ
2
s
+b
3
Λ
2
)
(23.7)
Thisgraphisactuallyverysimilartothevacuumpolarizationgraph. Wehavejustparametrizedthepos-
sible form the result t could take e with h 3 constants s b
1
, b
2
, and b
3
. Without any y symmetry argument, like
gaugeinvariance,thereisnoreasontoexpectthatanyoftheseconstantsshouldvanish.
The quadratically divergent part b
3
Λis an(infinite) number, , which gives s an(infinite) correction n to
G
F
.Wecanrenormalizebywriting
L=Z
F
ψ
¯
ψψ
¯
ψ
(23.8)
=G
F
ψ
¯
ψψ
¯
ψ+δ
F
ψ
¯
ψψ
¯
ψ
(23.9)
andusingthenewcountertermδ
F
toabsorbthequadraticallydivergentcorrection.
δ
F
=−G
F
2
b
3
Λ
2
(23.10)
Inthesamewaytheb
2
logΛ
2
termwouldcomefromtherenormalizationofthea
1
termintheLagrangian
L=Z
a
1
ψ
¯
ψψ
¯
ψ
(23.11)
ThuswewantZ
a
1
=a
1
−b
2
G
F
2
log
Λ2
µ2
+
tocanceltheb
2
G
F
2
slogΛ
2
termfromtheloop. Thenweget t for
the4-pointfunction
M
tot
(s)=Z
F
+Z
a
1
s+M
2
(s)=G
F
+G
F
2
b
1
s+a
1
s+b
2
slog
µ
2
s
(23.12)
Sowehaveabsorbedtheinfinitiesintotherenormalizedvaluesof G
F
anda
1
. Notethatweneededthea
1
term in the e Lagrangian n to do this, which agrees withour general argument that t all operators s not t for-
biddenbysymmetrymustbeincluded.
The quantum prediction n comes s from the logterm. . This s term is non-analytic ins, , sonoterm inthe
Lagrangiancouldgivea tree-levelcontributionof this s form. . Thecoefficient t of the log,b
2
is a a calculable
number. Thus,forexample,wecanpredictthat
M(s
1
)−G
F
G
F
2
s
1
M(s
2
)−G
F
G
F
2
s
2
=b
2
log
s
1
s
2
(23.13)
23.3 The e 4-Fermi theory
239
Theleft-handsideofthisis measurable,sotherighthandsideisagenuine measurablepredictionofthe
quantum theory. . It is s not sensitive e to o higher-order r terms in n the Lagrangian, since e they would d be e sup-
pressedparametricallybyadditionalfactorsofG
F
s.
23.3.0.2 UV-completingtheFermi i theory
Today, we know that the 4-Fermi i interaction n comes from the the low-energy limit of f a renormalizable
theorywithmassivevectorbosons. Theactual4-Fermiinteractionhasγ
µ
sinitandlookslike
L=G
F
ψ
¯
γ
µ
ψψ
¯
γ
µ
ψ+a
1
G
F
2
ψ
¯
γ
µ
ψψ
¯
γ
µ
ψ+a
2
G
F
3
ψ
¯
γ
µ
ψ
2
ψ
¯
γ
µ
ψ+
(23.14)
There areadditionaltermswith h γ
5
s,whichare phenomenologicallyimportant,but irrelevant for the fol-
lowingdiscussion.
ConsidertheLagrangianforafermioninteractingwithamassivevector
L
EW
=−
1
4
F
µν
2
+
1
2
M
2
A
µ
2
¯
(i∂
+g
w
A
µ
(23.15)
The vector is s the e Z Z or r W W boson, , and the e fermions s are the various s quarks and leptons. . The e tree e level
ψψ→ψψmatrixelementintheschannelisgivenby
iM=
ψ1
ψ2
ψ1
ψ2
=(ig
w
)
2
1
γ
µ
u
1
−i(gµν+
pµpν
M2
)
s−M2
2
γ
µ
v
2
(23.16)
Atlowenergy,s≪M,thisiswellapproximatedby
M=
g
w
2
M2
1
γ
µ
u
1
2
γ
µ
v
2
=
ψ1
ψ
2
ψ1
ψ2
(23.17)
This is the samematrix element as wewouldget from the 4-Fermiinteractionif f G
F
=
g
w
2
M2
. The e actual
expressionfortheFermiconstantintermsoftheweakcouplingconstantandtheW massis
G
F
=
2
8
g
w
2
m
W
2
(23.18)
wherem
W
=80.4GeVandg
w
=0.65.
Anotherway toderive the 4-Fermiinteractionfrom theelectroweak Lagrangianis byintegratingout
theheavyvector. Atenergies s muchlessthanm
W
,pair creationof W W bosonsisaverysmalleffect. Then
only tree-level l diagrams s involving the e W W are e relevant and the W W can n be treated classically. . The e W
s
equationofmotionis
A
µ
=
g
w
+M2
ψ
¯
γ
µ
ψ=
g
w
M2
1−
M2
+
ψ
¯
γ
µ
ψ
(23.19)
PluggingbackintotheLagrangianwefind
L
EW
=−
1
2
A
µ
(+M
2
)A
µ
+g
w
A
µ
ψ
¯
γ
µ
ψ
(23.20)
=
1
2
ψ
¯
γ
µ
ψ
g
w
2
+M2
ψ
¯
γ
µ
ψ
(23.21)
=
g
w
2
M2
ψ
¯
γµψψ
¯
γµψ−
g
w
M2
ψ
¯
γµψ
M2
ψ
¯
γµψ+
(23.22)
240
Non-Renormalizableand
Super-Renormalizable FieldTheories
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