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whichincludes theFermiinteraction. . Italsopredictsthatthea
1
term aboveis just1/g
w
2
. Sothetheory
withamassivevectorbosonisaUVcompletionofthe4-Fermitheory. Sincewealreadyobservedthatwe
canmeasurea
1
,wemighthaveusedthistoguesstheUVcompletionwereitnotalreadyknown.
Inconclusion,fornon-renormalizablefieldtheories
• DependenceonpowersofexternalmomentacanbefittodataandgivehintsabouttheUVcomple-
tion.
• Non-analyticcorrectionsaregenuinepredictionsofthequantumtheory
• The e dimensionfulcoupling indicates s abreak k down n of f perturbationtheory at thescaleof thecou-
pling
By theway,thetheory withthe massivevectorbosonis renormalizable,butithas itsownproblems.
Duetothemassivevectorpropagator
M∼p
ν
i(g
µν
pµpν
M2
)
s−M2
p
µ
∼g
2
s−
s2
M2
s−M2
(23.23)
For large s≫M
2
, this s goeslike e g
2
s/M
2
(this is duetoexchange ofthe longitudinalpolarizationof the
massivevectorε
µ
1
M
k
µ
). Loopcorrectionsmaygolikeg
4
s2
M4
andsoon. Thus,eventhoughthetheoryis
renormalizable,perturbationtheorybreaksdownatenergies s
M
g
. FortheW W boson,this s scaleis s ∼1
TeV. Wehavenoideawhatwillhappenabovethatscale,andsowearebuildingtheLHC.
23.4 TheoryofMesons
Thefirst fieldtheoreticmodel of cuclear structure wasconceivedby HidekiYukawain1934(Nobelprize
1949). Henotedthat t nuclear interactions seem tobe confinedwithinthenucleus,sothe are very short
range. Keep p inmind, he e was s trying to explain why y neutrons andprotons stuck together, not t anything
have todowiththe structureof the neutronorprotonthemselves,whichwere stillthoughttobepoint
particles. Theconfusioninthe30swaswhetherthethingbindingtheneutronsandprotonshadanything
todowiththethingthatcausedradioactivedecay(theweakforce).Yukawawasthefirstpersontospecu-
latethattheyaredifferent. Actually,themoreprofoundandlastinginsightthathemadewastheconnec-
tionbetweenforcesandvirtualparticleexchange. In1934peoplewerestillusingold-fashionedperturba-
tiontheory,andnobodythoughtofvirtualparticlesasactuallyexisting.
We alreadyknowthat theexchange of a massive particleleads not to aCoulombpotential but toa
Yukawapotential
V(r)=
1
r
e−mr
(23.24)
Yukawasawatm∼100MeVwas the appropriatescalefornuclearinteractions,andthereforepostulated
thatthereshouldbeparticles ofmass intermediatebetweenthenucleons(∼1GeV)andtheelectron(∼
1MeV) andhe calledthem mesons. We e know definemesons as s bosonic c quark anti-quark boundstates.
Themesonsresponsibleforthenuclearinteractionsarecalledpions.
Incidentally, the first t meson n was disovered in 1936 by Carl Anderson (Nobel l 1936 6 for r positron n dis-
covery in n 1932). . It t hadamass s of f 100 MeV,very nearlywhat Yukawa predicted. . However,this s was the
muonandnot thepion. . It t tookanother10years,until1947for thepiontobediscovered(CecilPowell,
Nobel1950). Pions s arestrongly interactingandshorter livedthanmuonssothey are harder tosee. . The
confusionoftherelationshipbetweenthecosmicraythatAndersonfoundandYukawastheoreticalpredic-
tionledtotherapidadvancement of quantum fieldtheory inthe30s andhelpedpeople to o start t taking
virtualparticlesseriously.
23.4 TheoryofMesons
241
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Thepionsprovideaneffectivedescriptionofnuclearforces. Thesedays,wedon’tthinkpionsarereally
responsible for nuclear r forces. . QCD D is the real theory of the strongforce. . But t it is very difficult touse
QCDtostudynuclearphysics. Eventhesimpleexplanationofwhythestrongforceisshort rangehadto
wait until asymptotic c freedom m was understood d inthe e 1970’s, 40 years after Yukawa’s s phenomenological
explanation. Thelowenergy theoryofpions s is knownasthechiralLagrangian,andit is avery powerful
non-renormalizablefieldtheorystillbeingactivelyexploredtoday.
Thenucleons,moregenerallyknownashadrons,includetheprotonandneutron,as wellasthemeson.
Therearethreeπmesonslabelledbytheirelectriccharges:π
0
+
andπ
. WecanguessthatLagrangian
forthepionshasnormalkinetictermsandphotoninteractionsfromscalarQED:
L
kin
=−
1
2
π
0
(+m
0
2
0
+(D
µ
π
+
)
2
1
2
m
+
2
+
)
2
+(D
µ
π
)
2
1
2
m
2
)
2
(23.25)
Themassesareoforder ∼140MeV.Thepionsshouldalsointeractwitheachother,withtermslike
L
int
=
1
f
π
2
π0π0
µ
π+
µ
π+
1
f
π
4
π+)2
µ
π0
µ
π0+
(23.26)
wheref
π
issomeconstantwithdimensionsofmass,makingtheseinteractionsnon-renormalizable. Thisis
asfaraswecangetjustusingLorentzinvarianceandgaugeinvariancetoconstraintheLagrangian.
It turns s out t the interactions s are e strongly constrained d by y an n additional symmetry, calledchiral sym-
metry,whichisrelatedtotheisospinsymmetrywhichrelatestheneutronandtheproton. Wenowknow
that this symmetry is s ultimately due to o the e fact that all l these particles s are e made e upof up p and down
quarks uandd,whicharealmost massless,andthesymmetry is anSU(2) symmetrywhichrotatesthem
intoeachother. (There e areactually 2SU(2) symmetries, onefortheleft handedquarksandone for the
right-handed quarks, and the chiral symmetry y is s one e linear r combination. . But t lets s not t worry y about the
detailsnow.)
The point t of f this s symmetry y on n the e pions s is s that all l the interactioncan be expressedin terms of f a
singlematrix
U=exp
i
f
π
π
0
π
+
+iπ
π
+
−iπ
−π
0

=exp
i
f
π
τ
a
π
a
(23.27)
whereτ
a
arethePaulimatrices.ThenthesimplesttermwecanwritedowninvolvingU is
L=
f
π
2
4
Tr
(
D
µ
U
)
D
µ
U

+
(23.28)
ThisisknownastheChiralLagrangian.
Thisleadingorderterm containsinteractions,suchas the onesinL
int
above. As s wellasbeingSU(2)
invariant, allthe interactions s comingfrom this termmusthave2 2 derivatives. . Sotheinteractions s have a
very special form. . That t form has beencheckedto o great t accuracy by lowenergy pionscattering g experi-
ments.
Since theory is non-renormalizable, , we should also add more e terms. . These e terms s must always s have
derivativesactingontheU
s,sinceU
U=1. Infact,thereareonly3termsyoucanwritedownwith4-
derivatives
L
4
=L
1
Tr[(D
µ
U)(D
µ
U
)]
2
+L
2
Tr[(D
µ
U)(D
ν
U
)]
2
+L
3
Tr
(D
µ
U)(D
µ
U
)(D
ν
U)(D
ν
U
)
Thecoefficientsofthesetermscanbefit fromlowenergy pionscatteringexperiments. . Ithasbeenfound
thatL
1
=0.65,L
2
=1.89andL
3
=−3.06. FromtheLagrangianwiththeseinteractionswecansaypracti-
callyeverythingwewouldwantaboutpionscattering. Additionalinteractionsaresuppressedbypowersof
momentum dividedbytheparameter f
π
∼92. Actuallyifyouarecarefulaboutfactorsof f πyouwillfind
thatthisisapredictivetheoryforE<4πf
π
∼1200 GeV. Sowearedoinganexpansionin
E
4πf
π
.
Moreover,thequantumeffects arecalculableandmeasurable as well. . Wearetalkingabout the non-
analytic logarithmiccorrections. These havebeenworkedout ingreat dealforthechiralLagrangianand
agreewonderfullywithexperiment.
242
Non-Renormalizableand
Super-Renormalizable FieldTheories
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Aswithanynon-renormalizabletheory,thechiralLagrangianindicatesthatit becomesnon-perturba-
tiveatascaleΛ∼4πf
π
.Abovethisscale,allthehigherorderinteractionsbecomerelevantandthetheory
isnot predictive. . TheUV V completionof thechiral Lagrangianis QuantumChromodynamics,thetheory
ofquarksandgluons. This s is acompletelydifferent typeof UVcompletionthantheelectroweaktheory
thatUVcompletedtheFermitheoryortheDiracequationcompletingtheSchrodingerequation. Forboth
ofthesetheories,thefermions inthelowenergytheorywerepresentintheUVcompletion,butwithdif-
ferent interactions. . Thetheory y of QCDdoes nothavepionsinit at all! Thus onecannotask aboutpion
scattering at t highenergyinQCD. . Instead,onemust t try tomatchthetwotheories more indirectly, for
example,throughcorrelationfunctionsofexternalcurrents. Experimentally,wedothisbyscatteringpho-
tonsorelectronsoffpions.Theoretically,weneedanon-perturbativedescriptionofQCD,suchasthelat-
tice. Lattice e QCD D can show the e existence e of pions s as s poles s in n correlation functions. . It t canalso give e a
prettygoodestimateof f
π
andotherparametersofthechiralLagrangian.
QCD is renormalizable. . It t is s aUV V completion n of f thechiral Lagrangian n in the e sense that it is well-
definedandperturbativeuptoarbitrarilyhighenergies.However,itcannotanswerperturbativetheques-
tions that the chiral l Lagrangiancouldn’t answer:whatdoes s ππ π scattering look likefor r s≫ f
π
2
? Infact,
QCDdoesn’tevenhaveasymptoticstates. WecannottalkaboutanS-matrixofquarksandgluons,since
quarks andgluonsareconfined. . Thuswehavetorephrasethequestions s weask. . Moreover,ifweactually
wanttoscatter pions,thechiralLagrangianis muchmoreusefulthanQCD:wecancalculatethingsper-
turbatively.
23.5 QuantumGravity
The final non-renormalizable field d theory y I want to discuss is Quantum Gravity. . This s is the effective
descriptionofamasslessspin-2particle. AswesawwithWeinberg’ssoftgravitontheorems,aninteracting
theory of a a massless s spin-2 2 fieldmust t have anassociated d conservationlaw. . This s is s the conservation n of
momentum. Thesymmetryitcorrespondstoissymmetryunderspacetimetranslations
x
α
→x
α
α
(x)
(23.29)
Thisgeneratesthegroupof generalcoordinatetransformations. TheNoethercurrentforthissymmetryis
theenergy-momentumtensorT
µ
α
.
The spin n two fieldh
µν
is a a gauge field for this s symmetry. . For r infinitesimaltransformation, it trans-
formsas
h
µν
→h
µν
+∂
µ
ξ
ν
+∂
ν
ξ
µ
+(∂
µ
ξ
α
)h
αν
+(∂
ν
ξ
α
)h
µα
α
α
h
µν
(23.30)
Thefirsttwotermsarethegaugepart,theyaretheanalogof
A
µ
→A
µ
+∂
µ
α
(23.31)
butthereare4α
snow,calledξ
α
. Thelastthreetermsarejustthetransformationpropertiesofatensor
representationofthePoincaregroupundertranslations.Forexample,ascalartransformsas
φ(x)→φ(x+ξ)=φ(x)+ξα
α
φ(x)+O(ξ2)
(23.32)
VectorsandtensorsgetadditionalpiecesbecauseoftheirLorentzproperties. Requiringthat∂
µ
V
µ
begen-
eralcoordinateinvariantimpliesthat
V
µ
(x)→∂
µ
(x
α
α
)V
α
(x+ξ)=V
µ
(x)+ξ
α
α
V
µ
(x)+(∂
µ
ξ
α
)V
α
(x)
(23.33)
andsoon. Notethattransformationonhisnon-linear:ithastermsproportionaltohitself.
Iamparticularlyfondofthislanguage. ByTaylorexpandingthetransformationweneverhavetotalk
about xchanging,justfieldschanging. xalwaysrefers toaspace-timepoint,whichisindependentofany
coordinaterepresentationwemighthaveforit.
23.5 Quantum m Gravity
243
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InthesamewaythatF
µν
2
withF
µν
=∂
µ
A
ν
−∂
ν
A
µ
is the uniquekineticterm wecanwritedownfor
A
µ
,thereisauniquekinetictermwecanwritedownforh
µν
.Itis
L
kin
=
1
4
(∂
α
h
µν
)(∂
α
h
µν
)−
1
2
(∂
ν
h
βν
)(∂
µ
h
βµ
)−
1
2
(∂
ν
h
µν
)(∂
µ
h
ββ
)+
1
2
(∂
µ
h
αα
)(∂
µ
h
ββ
)
(23.34)
Youcancheckthatthisisinvariantunderthegaugesymmetry,uptotermscubicinh. Thosecubicterms
canthenbedeterminedbydemandinginvarianceunderthefulltranslation. Wegetabunchoftermslike
L
int
=a
1
1
M
P
(∂
µ
h
αβ
)(∂
ν
h
βα
)h
µν
+a
2
1
M
P
(∂
α
h
µν
)(∂
β
h
µα
)h
νβ
+
(23.35)
Where we have addeda scaleM
P
by dimensional analysis. . Youcancontinue,andwork k outwhat those
cubic terms are,andthenthe quarticterms,andso on. This approachis done beautifully inFeynman’s
LecturesonGravitation.
Itturnsoutyoucansumthewholeseriesandwriteitinacompactform
L
kin
+L
int
=M
P
2
det(g)
R
µµ
(23.36)
whereg
µν
µν
+
1
M
P
h
µν
,withη=diag(1,−1,−1,−1)theMinkowskimetric. R
µν
istheRiccicurvature
tensorforaRiemannianmanifoldbasedonthemetricg
µν
. ThisistheEinsteinLagrangian.
23.5.0.3 Predictionsofgravityasanon-renormalizablefieldtheory
InsomesensetheRiemanncurvaturetensorlookslike
R
µναβ
∼∂
µ
ν
exp
1
M
P
h
αβ
, R
µν
∼R
µναα
(23.37)
Thisisvery heuristic,butshowsthatallthetermsintheexpansionofthecurvaturehavetwoderivatives
and lots of h
s. So o then R
µµ
= Tr[R
µν
] becomes s very y similar r to o the form m of the chiral l Lagrangian
Tr[D
µ
UD
µ
U
]. Just t like the chiral Lagrangian, each h term m has two derivatives, and d the e interactions s are
stronglyconstrainedbysymmetries.Wecouldaddhigherorderterms
L=M
P
2
R
µµ
+L
1
R
αα
R
ββ
+L
2
R
αβ
R
αβ
+L
3
R
αβγδ
R
αβγδ
(23.38)
Inthiscase,thereare3terms,justlikeinthechiralLagrangian. (Actuallyoneisatotalderivative,called
theGauss-Bonnetterm,sotherearereallyonly2). SinceRhas2derivatives,theseallhave4derivatives,
sotheleadinginteractionsgolike
L=
+L
i
1
M
P
3
2
h
3
(23.39)
ThereasongravityispredictiveisbecauseM
P
=10
19
GeV,soE≪M
P
foranyreasonableE. Infact,itis
verydifficulteventotestthetermsintheLagrangiancubicinhwith2derivatives. Thatistermslike e ∼
a
1
M
P
h
2
hcomingfrom
g
R. Forthese,wedon’t get additionalE/M
P
suppression,butsincetheinterac-
tionhasanextrafactorof h/M
P
,wouldneedlargefieldvalues hM
P
. Thishappens,forexample,from
thegravitationalfieldaroundthesun.There,
h∼φ
newton
M
sun
M
P
1
r
∼1038
1
r
(23.40)
Bytheway,M
P
∼G
N
−1/2
,soh∼M
P
forr∼G
N
M
sun
∼100m,whichisstillmuchlessthantheradiusof
thesunr
sun
∼10km. Nevertheless,thesecubictermshavebeenmeasuredintheperihelionshiftofMer-
cury. Mercuryis s about10kmfromthesun,sotheeffectis onlyonepartin108. Stillithas s beencalcu-
latedpreciselyandmeasuredandtheagreementisperfect.
Toseetheeffectofthehigherorder terms,likeL
1,2,3
,wewouldneedtouseinteractions like
L
1
M
P
3
2
h
3
.
These have additionalfactors of
E
M
P
1
M
P
r
suppression. For r mercury,this factor is 10
−45
,sowewould
needtomeasuretheperihelionshiftto1partin10
−98
tosee these terms. . Needless s tosay,thatisnever
goingtohappen. ThusthehigherordertermsintheLagrangianaretotallyirrelevantforastrophysics.
244
Non-Renormalizableand
Super-Renormalizable FieldTheories
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So we can’t t measure the higher order r terms. . But t there are stilladditional predictions s we e canmake
fromthequantumtheory. Wecancalculatealoopcorrectiontogravitonexchange. Thisisacorrectionto
thegravitonpropagator,whichhasthesamegeneralformasthevacuumpolarizationgraph
+
p
p
k
p−k
1
p2
+
1
p2
p
4
M
P
2
c
1
+c
2
log
Λ
2
p2
+c
3
Λ
2
p
2
M
P
2
1
p2
(23.41)
wherethep4/M
P
4
comesfromthetwovertices whichhavefactors of
p2
M
P
inthem. Therest is s dimensional
analysis.Tobeconcrete
c
2
=
141
30π2
(23.42)
YoumighthavealsoexpectedaterminbracketsgoinglikeΛ
4
,but thatwouldproduce adoublepoleat
p=0,andwhenthe1PIgraphsareresummeditwouldshiftthegravitonmass. Themassisprotectedby
gaugeinvariance,sothiscan’thappen,justlikeforthephoton.So,
Π
grav
=
1
p2
1+
p2
M
P
2
(c
1
+c
2
log
Λ2
p2
+c
3
Λ2
M
P
2
(23.43)
This is exactly thesame formthatwegot from the radiativecorrectionintheFermi i theory above. . The
quadratic divergencec
3
is absorbedintotherenormalizationof f M
P
,The c
1
term is s absorbedinto coun-
tertermsforL
1,2,3
. Thatleavesthelogarithmwithit’scalculablecoefficient.
IfweFouriertransformbackintopositionspace,andputitalltogether wefindtheNewtonpotential
becomes
h(r)=
M
sun
M
P
1
r
1+
c
2
M
P
2
r2
+
L
1
M
P
3
r3
+
M
sun
M
P
a
1
M
P
r
+
L
1
M
P
3
r3

(23.44)
Thefirstthreetermscomefromhh,thelogpquantumcorrection,andtheL
1
h2hterminEq.(23.38).
Thefinaltwoarethehtermfrom
g
RandtheL
1
2htermfromEq. (23.38).L
1
isafreeparameter,
butbothc
2
anda
1
areknown. Thus s theradiativecorrection,thec
2
term,givesacalculableeffectthis is
parametricallymoreimportantthantheL
1
term. Thistermhasnotyetbeenobserved,butitisagenuine
predictionofquantumgravity.Infact,itistheonlypredictionofquantumgravityIknowof.Don’t
forget:thispredictionisentirelyindependentontheUVcompletionoftheEinsteinLagrangian.
Inanycase,thisdiscussionshouldmakeclearthat
Gravityandquantummechanicsarecompletely compatible
There is nothinginconsistentabout general relativity andquantum mechanics. . Gravity y is theonlycon-
sistent theoryofaninteractingspin2particle. Itisaquantumtheory,justassolidandcalculableasthe
4-Fermitheory. It t is justnon-renormalizable,andthereforenon-perturbativeforenergiesEM
P
,but it
isnotinconsistent. Atdistancesr∼
1
M
P
∼10−33 cm,allofthequantumcorrections,andallof thehigher
ordertermsintheLagrangianbecomeimportant. Soifwewanttousegravityatveryshortdistanceswe
needaUVcompletion.
Stringtheory isone suchtheory. . It t is capableof calculatingtheL
i
terms inEq (23.38). . It does s this
veryindirectly. IfwecouldmeasuretheL
i
,thenwecouldteststringtheory,butaswenotedabove,we
wouldneedtoimproveourmeasurementsby90orders ofmagnitude. . InthesamewaythatQCDisnota
perturbativedescriptionofpions,stringtheoryisnotaperturbativetheoryofgravitons. However,wecan
prove instringtheorythatthereis amassless spin-2excitation,as we can(almost) prove that there are
boundstatesinQCD. Andsincetheonlyconsistenttheory ofspin-2particles s is theEinsteinLagrangian
plus higher order terms, since e string theory is s consistent, it must match on to Einstein gravity y at t low
energy. However, , unlike QCD, , string theory does not t (yet) have e a non-perturbative formulation n so we
don’tknowhowtodefineit,orevenifitexists!
23.5 Quantum m Gravity
245
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23.6 Summaryofnon-renormalizabletheories
Wehavelookedat4importantnon-renormalizabletheories.
• TheSchrodingerequationisperturbativeforE<m
e
. It’sUVcompletionistheDiracequationand
QED,whichisperturbativeuptoitsLandaupole,E∼10
100
GeV.
• TheFermitheoryofweakinteractionsisperturbativeforE<G
F
−1/2
∼300GeV. It’sUVcompletion
is the electroweak theory withmassive vector bosons W W andZ. The e electroweak theory is renor-
malizable,butonlyperturbativeuptoE∼1TeV. OnepossibleUVcompletionoftheelectroweak
theoryistoaddaHiggsboson.
• ThechiralLagrangianis s the lowenergytheoryofpions. . Itis s perturbativeandverypredictive for
E<4πf
π
∼1200MeV. It’sUVcompletionisQCD. QCDispredictiveathighenergies,butithas
no asymptoticstates. Atlowenergy,QCD D must bestudiednon-perturbativelyonthe lattice. . So,
atlowenergythechiralLagrangianwhichisperturbative,ismuchmoreuseful.
• Einsteinquantumgravityisthelowenergytheoryofgravity. . ItisperturbativeforE<M
P
∼10
19
GeV. Itisextremelypredictiveatlowenergies,includingpredictivequantumcorrections. Onepos-
sibleUVcompletionisstringtheory.
Incidentally, these 4examples s correspond d to the e 4 forces of nature:the electromagnetic force,the weak
force,thestrongforce,andgravity. NoticethattheUVcompletionsareallqualitativelyverydifferent.In
some cases, certainly for many physical applications, the non-renormalizable theory is more useful than
the renormalizableone. Renormalizablejustmeansthereareafinitenumberofcounterterms,itdoes not
meanthatyoucancalculateeveryobservableperturbatively.
Iwanttoemphasizeagain:non-renormalizabletheoriesareverypredictive,notjustattree-level(clas-
sically) but t also o through quantum m effects. The e quantum m effects are calculable and non-analytic in
momentum space. . Infact,non-renormalizabletheories s aresopredictive that it isoftenbetter tomakea
non-renormalizable approximation n toa renormalizable e theory thanto use the full renormalizable theory
itself. ExamplesincludeEinsteingravityinsteadofstringtheory,HeavyQuarkEffectiveTheory(HQET)
insteadofQCD,Ginsburg-Landautheoriesincondensedmatter,theSchrodingerequationinsteadofthe
Diracequation,andchemistryinsteadofphysics.
23.7 Quantumpredictionswhicharenotlogs
We have seenthat non-renormalizable field d theories s are predictive at the classical l level, but also at the
quantumlevelthroughtermslike logp
2
. Isit t possible that wecouldalsocomputecorrections whichare
analyticinp
2
?
We already knowof oneexample. RecallthatinQEDwe wereabletocalculateafinite correctionto
themagneticmomentoftheelectron:
µ
e
=
e
2m
e
2+
α
π
+O(α
2
)
(23.45)
WhathappenstothispredictionifweembedQEDinalargertheory,suchasquantumgravity? Thecon-
tributionofgravitytotheelectronmagneticmomentisdivergent:
m
e
M
P
2
log
Λ
2
m
e
2
+b
σ
m
e
M
P
2
(23.46)
whereb
σ
isfiniteandΛistheUV cutoff. . (Youdon’tgetalinearlydivergentpiecebecause
kµ
k4
d
4
k=0.
Also,fora magnetic moment,you alwaysa mass s insertionbecause ψ
¯
σµνψ couplesleft t andrighthanded
spinors.)
SoweneedanothertermintheLagrangiantoabsorbtheinfinity:
L
σ
=Z
σ
ψ
¯
σµνψF
µν
(23.47)
246
Non-Renormalizableand
Super-Renormalizable FieldTheories
Asusual,wesplitZ
σ
intoafinitepartc
σ
andcounterterm:Z
σ
=c
σ
σ
withδ
σ
=−
m
e
M
P
2
log
Λ
m
e
. Then
µ
e
=
e
2m
e
2+
α
π
+c
σ
+b
σ
m
e
M
P
2
(23.48)
Nowweneedarenormalizationconditiontofixc
σ
. Ifwedemandthatthemeasuredvalueofthemagnetic
momentisµ
e
experiment
,then
c
σ
e
experiment
e
2m
2+
α
π
−b
σ
m
e
M
P
2
(23.49)
⇒µ
e
e
experiment
(23.50)
Butthisimpliesthatthe
α
π
partwestartedwithistotallyunobservable. Butthatdoesn’tmakeanysense,
sincewehaveobservedit!Soit’saparadox!
23.7.1 fromphysicstophilosophy
Theresolutionofthisparadoxliesoutsidetherealm of physics. . I’mgoingtostart t italicizingwords that
aremore relatedtophilosophy thenexperiment,andI willtryto be veryclearabout distinguishingthe
two.
Onewaytothinkabout itistosupposethattheultimatetheoryof nature werefinite. . Thenwedon’t
needthecounterterm,andcanjustcomputetheloopingravity. Wewouldfind
µ
e
=
e
2m
e
2+
α
π
+
m
e
M
P
2
log
Λ
2
m
e
2
+b
σ
m
e
M
P
2
(23.51)
wherenowΛissomephysicalcutoffscale. WeneedatleastΛ>M
P
. Forexample,instringtheory,Λ∼
1
g
string
M
P
. b
σ
iscalculable,andprobablynotmuchlargerthan10. Thusaslongas
Λ<m
e
exp(
M
P
2
m
e
2
)∼1010
44
eV
(23.52)
theeffectofgravityong-2iscompletelynegligible.
What if the ultimate theory y of f nature is s simply y renormalizable? ? For r example, , imagine that the
gravitonisaboundstateinsomeexotictheory(likestringtheory),asthepionsareboundstatesofQCD.
Thentherearenocoefficientsofpositivemass dimensionand d g-2willcomeoutfinite. Thus s the effective
value of Λ Λ would bearoundthe e compositeness scale of thegraviton,which h wouldlikely be e aroundM
P
.
ForΛ∼M
P
,thecontributiontog-2ofthegravitonisstillexponentiallysmall.
Anotherway tothinkabouttheparadoxistosupposethatweactuallyhadmeasuredsomedeviation
of g-2from m it’s value inQED (or in n the e renormalizable standard d model). . For r example, there e is s alittle
roomfor this at order α
3
andabettermeasurementcouldconceivablerevealadeviation. If f youarereal
positivistyoucanthenjustaddL
σ
andtunetherenormalizedvalueof g-2tomatchexperiment. . Froma
practicalpointofviewthatwouldmeanthatc
σ
∼α
3
1
m
e
1
1300GeV
. Sowe’veintroducedanewscaleinour
theory, andmade e it non-renormalizable. . It t is s unlikely, , but not impossible, , that t this effect t comes s from
quantumgravity. Forthattohappen,wewouldhaveneededΛ∼10
1044
eV. Moreover,ifΛ∼10
1045
eVthe
effectwouldhavebeensoenormousthatthewholeuniversewouldhavebeenparamagnetic.
Solet’s addthisoperator toaccountforthe e ∼α
3
deviation. Thatworksfinefor r g-2,butitscrewsup
everything else. . That t is, , this s new w operator would contribute to o other r observables, and we would need
more higher r dimension operators, etc. . Now w we cankeep choosing the e coefficients of f those operators s to
cancelthecontributionfromthisoneandagreewiththepredictionsofthestandardmodelforeverything.
Sonothingis guaranteedtohappenfromthis term. . (Actually,therewillbecalculablequantumeffectsof
thisterm. Butsincethesemust showup,theydon’t tellusanythingnew.)Itmakesmoresensetoexpect
deviationstoshowupelsewhere. Thesedeviationsaremotivated bytheg-2observation,buttheyarenot
guaranteedtoshowup. Bylookingfor r thosedeviations,wecanfitmorehigher-dimensionoperators,and
thismayhelpusguesstheUVcompletion.
23.7 Quantum m predictions whichare not logs
247
Insummary,lookingfordeviationsinfinitepredictionsofthestandardmodelisacluetonewphysics.
Findingsuchasuchadeviationprobably indicatesanewscalewherewemightfindotherneweffects,but
nothingis guaranteed. . Inthe e context of a non-renormalizable theory,there areinfinite contributions to
the same quantity. . That t formally makes thetheory unpredictivefor this s quantity. . However r inpractice,
withanyreasonablefinitecutoff,or throughanembeddingintoarenormalizabletheory,theinfinitecor-
rection from m the non-renormalizable e terms s becomes s calculable e and small. . The e modern viewpoint is s to
thinkof the higher dimensionoperators inthe non-renormalizable theory asencoding informationabout
theUVcompletion.
23.8 Super-renormalizablefieldtheories
Nowlet us s turn n to theories with coupling constants of positive e mass dimension. . In n 4 dimensions there
aren’tmanyoptions.
23.8.1 tadpoles
Onepossibilityisthatwecouldhaveatermlinearinascalarfield.Forexample,
L=−
1
2
φ(+m
2
)φ+Λ
3
φ
(23.53)
forsomenumberΛ
3
withmass-dimension3,this wouldgenerate0|φ|0=Λ
3
0. But alsocontributeto
things like
0|T{φ
4
}|0
,andeverythingelse. Thewayto o dealwiththis s term istoredefine φ→φ+ φ
0
,
whereφ
0
=
Λ3
m2
. Thenweremovethetadpole. SotheexistenceofatermlikeΛ
3
φintheLagrangianmeans
wehavebeenworkingwiththewrongdegreesoffreedom.That’saseriousproblem.
Theonlypossibletermofmassdimension4aconstant
L=
(23.54)
Thisconstanthasaname:ρisthecosmologicalconstant. Byitself,thisterm doesnothing. Itcouples s to
nothingandinfactjustbepulledoutofthepathintegral. Thereasonitisdangerousisbecausewhenone
couplestogravity,itbecomes
ρ→
g
ρ=
1
2
h
µµ
ρ+h
µν
2
ρ+
(23.55)
Thefirsttermgenerates atadpolediagram,so0|h
µµ
|0=
1
2
ρ
0. Thisindicates s thatweareexpanding
aroundthewrongvacuum. Byredefiningh
µν
→h
µν
+h
µν
0
forsomeconstantbutx-dependentfieldh
µν
0
(x),
we can n remove e this term. . We e know it t has s to o be e x-dependent because e all l the terms s in the e Einstein
Lagrangianhavederivatives,sotheywillkillanyspacetime-independenth
µν
0
.Thathashugeconsequences.
For example, , it violatesLorentzinvariance. Inthe end,it preserves s someothersymmetrygroup,the de-
Sittergrouporantide-Sittergroup,dependingonthesignof ρ.Still,ifthistermisaround,wehavetogo
backandrethinkalltheresultswehadinquantumfieldtheorythatdependonLorentzinvariance.
So far, , these e are pretty awful l things. . Other r dimensionful couplings include e mass s terms, like m2φ2,
m
2
A
µ
2
ormψ
¯
ψ,orperhapsacubiccouplinginascalarfieldtheory,gφ
3
. Thatactuallyexhauststhepossi-
bilitiesin4dimensions. Thereareonlyahandfulofsuper-renormalizableterms.
23.8.2 relevantinteractions
Let’s look at φtheory first, then the mass terms. . We e are e interested d in radiative corrections, so o let’s
couple φ φ to something else, , say y a fermion n ψ. . We e will set the mass of f φ φ to o zero o for r simplicity. . The
Lagrangianis
L=−
1
2
φφ+
g
3!
φ3+λφψ
¯
ψ+ψ
¯
(i∂
−M)ψ
(23.56)
248
Non-Renormalizableand
Super-Renormalizable FieldTheories
Nowconsidertherenormalizationof g. Wecancomputethisbyevaluatingthe3-pointfunction
0|T{φ(p)φ(q
1
)φ(q
2
)}|0=g+
(23.57)
Thereisaradiativecorrectionfromtheloopof φ. ThisisUVfinite,proportionaltog
3
andnot particu-
larlyinteresting.Amoreimportantradiativecorrectioncomesfromtheloopofthefermion.
M
2
=
p
q
2
k
q
1
k−q
1
k+q
2
3
d4k
(2π)4
Tr[
1
k
−M
1
k
−q
1
−M
1
k
+q
2
−M
]
(23.58)
=
λ
3
2
M
0
1
dz
0
1−z
dy
M
2
+p
2
(y+z−3yz−
1
2
)
M2−p2yz
+3log
Λ
2
M2−p2yz
(23.59)
Takingp
2
=0,thisgives
0|T{φ(p)φ(q
1
)φ(q
2
)}|0=g+
λ
3
16π2
M
1+3log
Λ
2
M2
(23.60)
Sowefinda(divergent)shifting proportionaltothemassofthefermionM. . Thisisfineif M∼g,butif
parametrically it’s really weird. . As s M M gets s larger,the correction n grows. . That t means that the theory is
sensitivetoscalesmuchlargerthanthescaleassociatedwithg.So,
• Super-renormalizabletheoriesaresensitivetoUVphysics
Isthisaproblem? Notinthetechnicalsense,becausethecorrectioncanbeabsorbedinarenormalization
of g.Wejustaddacountertermbywriting
g=g
R
λ
3
16π2
M
1+3log
Λ
2
M2
(23.61)
Thiswillcanceltheradiativecorrectionexactly(atp=0).Again,thisistotallyreasonableif g
R
∼M,and
theoreticallyandexperimentallyconsistentforallg,butif g
R
≪M itlooksstrange.
Willtherebeproblemsduetothequantumcorrections? WecanlookattheGreen’sfunctionatadif-
ferentvalueof p
2
. ThiswouldcontributetosomethinglikethescalarCoulombpotential. Wefindthatof,
course,thedivergencecancels,andtheresultatsmallpissomethinglike
0|T{φ(p)φ(q
1
)φ(q
2
)}|0
≈g
R
+
λ
3
16π2
p
2
M
+3Mlog
1−
p
2
M2
+
(23.62)
≈g
R
+
λ
3
16π2
p
2
M
(23.63)
Sofor p≪M thesecorrectionsaretinyanddoinfactdecoupleasM→∞.
As for thenon-renormalizable case, , we enjoy speculating that theultimate theory y of f nature is s finite.
ThenΛ isphysical. . Ifwesuppose e that thereis somebare g which h is s fixed,thenwe needavery special
valueofΛtogetthemeasuredcouplingtobeg
R
≪M.Ofcourse,thereissomesolutionforΛ,forany g
0
andg
R
. ButifwechangeΛbyjustalittlebit,thenweget
g
R
→g
R
+
λ
3
16π2
Mlog
Λ
old
2
Λ
new
2
(23.64)
whichisahugecorrectionif M≫g
R
.
Thismeansthatgenericallythetheorywantstohaveg
R
∼M. Itisunnatural tohavearelevantcou-
plinginaquantumfieldtheory.
23.8 Super-renormalizablefieldtheories
249
23.8.3 scalarmasses
The only other possible terms of positive mass dimensionina Lagrangianare masses. . First,consider r a
scalarmass. Let’sgobacktoourYukawatheoryanddropthatridiculous s φ
3
term.
L=−
1
2
φ
+m
2
φ+λφψ
¯
ψ+ψ
¯
(i∂
−M)ψ
(23.65)
Nowconsidertheradiativecorrectionstom.
Weneedtocomputetheself-energygraphforthescalar(seeP&S10.33)
Σ
2
(p2)=
=4
λ2(d−1)
(4π)
d/2
Γ(1−
d
2
4−ε
0
1
dx(M2−x(1−x)p2)
d/2−1
(23.66)
Thisisquadraticallydivergent,asevidencedbythepoleatd=2.Ifweexpandforsmallp,wefind
Σ
2
(p
2
)=
λ
2
2
p
2
−6M
2
ε
1
2
(p
2
−6M
2
)log
M
2
µ2
+p
2
−5M
2
+
p
4
20M2
+
(23.67)
ItissomewhatnicertousearegulatorwhichmakestheUVdivergencesexplicit.Thenwewouldhave
Σ
2
(p
2
)=
λ
2
2
Λ
2
+(2p
2
+3M
2
)log
Λ
2
M2
+4p
2
+5M
2
+6
p
4
M2
+
(23.68)
whereIhavejustmadeupthecoefficientsoftheseterms.
The divergences are canceled d with h counterterms from the field strength and mass renormalizations
giving
=−p
2
δ
φ
−δ
m
(23.69)
Wecannowusephysicalrenormalizationconditionstosetthepoleofthepropagator attherenormalized
mass,withresidue1.Then,
Σ
2
(m
2
)=m
2
δ
φ
m
(23.70)
Σ
2
(m2)=δ
φ
(23.71)
Explicitly,
δ
φ
=
λ
2
2
(2m
2
)log
Λ
2
M2
+4+12
m
2
M2
+
(23.72)
δ
m
=
λ
2
2
Λ
2
+3M
2
log
Λ
2
M2
+5M
2
+6
m
4
−2m
2
M
2
M2
+
(23.73)
⇒Σ(p
2
)=Σ
2
(p
2
)+m
2
δ
φ
m
=
λ
2
2
1
20
(p
2
−m
2
)
2
M2
(23.74)
Notethatδ
m
istheshiftinthemass. Itisgettingnotonlyaquadraticallydivergentcorrection,butalsoa
correctionproportionaltoM
2
,whichcanbehugeifM≫m.
Supposeweusedminimalsubtractioninstead. Calltherenormalizedmassm
R
,todistinguishfromthe
polemassmabove.
δ
φ
=
λ
2
2
2log
Λ
2
M2
(23.75)
δ
m
=
λ
2
2
Λ
2
+3M
2
log
Λ
2
M2
(23.76)
⇒Σ(p
2
)=Σ
2
(p
2
)+m
R
2
δ
φ
m
=
λ
2
2
4p
2
+5M
2
+6
p
4
M2
+
(23.77)
250
Non-Renormalizableand
Super-Renormalizable FieldTheories
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