c# display pdf in window : Add contents page to pdf application SDK tool html winforms asp.net online QFT-Schwartz19-part450

SonowtheO(e2)electronpropagatoris
Π(q
)=
+
q
q−k
k
q
=
i
q
+
i
q
−iΣ
2
(q
)
i
q
(19.20)
=
i
q
1−
α
log
Λ
2
m
γ
2
(19.21)
Asexpectedwehavefoundaninfinitecorrectiontothetwopointfunction
G
2
(q)=
0|T{ψ
¯
ψ
|0
=
i
q
1−
α
log
Λ
2
m
γ
2
(19.22)
Whatarewetodoaboutthisinfinity? FortheCoulombpotential,wewereabletoabsorbtheinfinity
intothe electriccharge. Buthere,the leadingorderexpressiondoesn’tseem tohaveanyparameters into
whichwecanabsorbtheinfinity.
Infact,thereisaparameter:thenormalizationofthefermionwavefunction.Nobodyevertolduswhat
thewavefunctionsweresupposedtobe!Saywehaddefined
ψ(x)=
1
Z
2
d
3
p
(2π)3
1
p
a
p
e
ipx
+a
p
e
−ipx
(19.23)
for somenumber Z
2
. This s isthe eponymous renormalization: ψ
R
=
1
Z
2
ψ
0
. Wesay y Rfor renormalized,
physicalquantities,and0forinfinitebarequantities.
StickingabunchoftheseZ factorsintheLagrangian,itbecomes
L=−
1
4
Z
3
F
µν
2
+iZ
2
ψ
¯
R
ψ
R
+Z
2
m
0
ψ
¯
R
ψ
R
+e
0
Z
2
Z
3
ψ
¯
R
A
R
ψ
R
(19.24)
Whatkindofobservables wouldthischange? ? Well,weareintheprocessofcatalogingallthe e observ-
ables,soweareboundtofindout. Thetadpolesvanishnomatterwhat,andweareworkingonthetwo
pointfunction.
Forthetwo-pointfunction,thetreelevelpiecegetsa
1
Z
2
. Fortheloop:
i
q
−m
e
2
1
q
−k−m
1
k2
i
q
−m
i/Z
2
q
−m
(eZ
2
Z
3
)
2
1/Z
2
q
−k−m
1/Z
3
k2
i/Z
2
q
−m
(19.25)
=
1
Z
2
i
q
−m
e2
1
q
−k−m
1
k2
i
q
−m
(19.26)
So,
G
2
(q)=
1
Z
2
i
q
1−
α
log
Λ
2
m
γ
2
(19.27)
WeknowalreadythatZ
2
=1toleadingorder. IfwetakeZ
2
=1+δ
2
with
δ
2
=−
α
log
Λ
2
m
γ
2
(19.28)
then
G
2
(q)=
i
q
1+O(α2)
(19.29)
whichrenormalizesthepropagatorbacktowhatwestartedwith.
There is s no physical prediction in this calculation, because, as we said, it is s hardto o come up p with
observables.TheimportantpointisthatwehavegottenridoftheUVdivergenceinthe2-pointfunction.
19.3 Electronself-energy
191
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Thefactthatonlyonefactorof Z
2
comesthroughtheexpressionis becausetheonlythingthatactu-
allydependsonZaretheexternalstatesintheGreen’sfunctionG
2
=
ψ
¯
ψ
.
19.3.3 wave-functionrenormalizationwithamass
Nowlet’sputtheelectronmassbackin. Thenthetwopointfunctionbecomes
G
2
(q
)=
+
q
q−k
k
q
=
i
q
−m
e
+
i
q
−m
e
[−iΣ
2
(q
)]
i
q
−m
e
(19.30)
Form
e
≫m
γ
,wecansetm
γ
=0.Then,
Σ
2
(q
)=
α
0
1
dx(4m
e
−2xq
)log
2
(1−x)m
e
2
−x(1−x)q2
(19.31)
We expectthatthis tobezerowhenq
=m
e
,tocanceloneofthepolesintheexpansionof G
2
. How-
ever,
Σ
2
(m
e
)=
m
e
1−2log
m
e
2
Λ2
(19.32)
whichisnotzero. Sothatmeans s thatG
2
(q
)hasadoublepoleatq
=m
e
. Whatisthephysicalinterpreta-
tionofthat?
Perhaps we e could d remove e it t with the e wave e function renormalization. . Putting g in the Z Z factor, , and
writing
Z
2
=1+δ
2
(19.33)
withδ
2
=O(e
2
),theonlyeffectisfromtheleadingpropagator:
1
Z
2
i
q
−m
e
=
i
q
−m
e
+
i
q
−m
e
2
(q
−m
e
)
i
q
−m
e
(19.34)
Thenweget
Σ(q
)=Σ
2
(q
)−δ
2
q
−m
e
(19.35)
Andwecanseethatthereisnowaytochooseanumberδ
2
sothatΣ(m
e
)=0.
TheresolutionisthattheelectronmassintheLagrangianisnotnecessarilythephysicalelectronmass
(thelocationofthe pole). SoletustreattheLagrangianmass as afreeparameteranduseit’srenormal-
izationtoremovethedoublepole.Wewrite
m
e
=
1
Z
2
(m
R
m
)
(19.36)
wherem
e
istakentobeinfinite,butm
R
isthephysicalfinite,renormalizedelectronmass.
Then
1
Z
2
i
q
−m
e
=
1
1+δ
2
i
q
−m
R
2
m
R
−δ
m
(19.37)
=
i
q
−m
e
+
i
q
−m
e
−δ
2
(q
−m
R
)−m
R
δ
2
m
1
q
−m
e
(19.38)
=
i
q
−m
e
+
i
q
−m
e
−i(−δ
2
q
m
)
i
q
−m
e
(19.39)
Notethatthem
R
δ
2
pieceshavecanceled,whichiswhywedefinedm
e
withafactorof Z
2
.
So,
Σ(q
)=Σ
2
(q
)−δ
2
q
m
(19.40)
192
Systematic Renormalization
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Nowthere is avalueof δ
2
andδ
m
for whichΣ(m
R
)=0. We e canusethis toenforce thatthereis onlya
singlepoleinthepropagator. Letuspressonabitbeforesettingtheotherconditiontodetermineδ
m
and
δ
2
.
19.3.4 one-particleirreduciblegraphs
We only y includedone e particular r self-energy correction n in the e series s we e summed d to correct the electron
propagator. There e are e lots more e graphs, and d the e full l non-perturbative propagator, , should d properly be
definedasthe sum ofallthesegraphs. . To o make sure we are not t countinganythingtwice, we must only
includegraphs whichcannot becut intwoby cuttingasinglepropagator. . Those wouldformadifferent
series,whichwecouldalsosum. Soanythingwhichcannotbecutthatconnectsoneincomingelectronto
oneoutgoingelectronarecalledone-particleirreduciblegraphs.
Thenwedefine
Σ(q
)=
1PI
2
(q
)+
(19.41)
Each1PIinsertionwillgiveacontributiontotheGreen’sfunction
G
2
(q
)=
+
1PI
+
(19.42)
=
i
q
−m
+
i
q
−m
[−iΣ(q
)]
i
q
−m
+
(19.43)
=
i
q
−m
1+
Σ(q
)
q
−m
+
Σ(q
)
q
−m
2
+
=
i
q
−m
1
1−
Σ(q
)
q
−m
(19.44)
=
i
q
−m−Σ(q
)
(19.45)
Forthefullnon-perturbativepropagator.
Wefoundthatifweshiftthebaremassm
e
andaddarenormalizationfactortoψ,then
Σ(q
)=Σ
2
(q
)−δ
2
q
m
+O(e
4
)
(19.46)
19.3.5 pole-massrenormalizationconditions
Atthispoint,thereappearsanaturalconditionwhichwecanimposetofixtheδ
m
andδ
2
:wecandemand
thatthepoleinthepropagatorremainatq
=m
R
. Recallthattouse(orreally,tointerpret)theLSZthe-
orem,weneededtoprojectoutthe1-particlestatesbymultiplyingbytermslikep
−m. Sincethephysical
oneparticle states havepoles attheirphysical masses,thisshouldbe projectingout p
−m
R
. Thisgives
zerounlessthereisapoleinthecorrespondingmatrixelement,soitiscriticalthatthepoleremainthere.
SotherenormalizedGreen’sfunctionhadbetterhaveapoleatp
=m
R
.
Sowecandefinetherenormalizedmassby
Σ(m
R
)=0
(19.47)
Thisimpliestoordere
R
2
that
Σ
2
(m
R
)=m
R
δ
2
−δ
m
(19.48)
Thus,atordere
R
2
(withPVregulator)
m
R
δ
2
−δ
m
=
m
e
1−2log
m
R
2
Λ2
(Pauli−Villars)
(19.49)
19.3 Electronself-energy
193
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Anotherreasonablerequirementwemightimposeisthattheresidueofthepoleofthepropagatorbe1
(ori). Insomesensethis s is just aconventionforwhatwe multiply byinLSZ,butwe havetobeconsis-
tentwiththatconvention! So,
1= lim
q
→m
R
q
−m
R
q
−m
R
−Σ(q
)
= lim
q
→m
R
1
1−
d
dq
Σ(q
)
(19.50)
wherewehaveusedL’Hospital’srule.Thisimplies
d
dq
Σ(q
)
|
q
=m
R
=0
(19.51)
whichmeans,toordere
R
2
δ
2
=
d
dq
Σ
2
(q
)
|
q
=m
R
(19.52)
Recalling
Σ
2
(q
)=
α
0
1
dx(4m
e
−2xq
)log
2
(1−x)m
e
2
+xm
γ
2
−x(1−x)q2
(PV)
(19.53)
Thisleadsto
δ
2
=
d
dq
Σ(q
)
|
q
=m
R
=
α
dx
−xlog
2
(1−x)m
e
2
+xm
γ
2
+2
(2−x)x(1−x)m
R
2
(1−x)2m
R
2
+xm
γ
2
(PV)
(19.54)
=−
α
logΛ
2
+
(19.55)
NotethatthedivergentlogΛpartisthesameas wefoundforamasslesselectron,whichmakes senseas
theUVdivergencesshouldn’tbesensitivetoanyfinitemass.And
δm=
α
m
e
0
1
dx(4−2x)log
2
(1−x)2m
e
2
+xm
γ
2
(PV)
(19.56)
Ifwerepeatthecalculationindimensionalregularization,wewouldfind
Σ
2
(q
)=
e
R
2
(4π)
d/2
µ4−dΓ(2−
d
2
)
0
1
dx
(4−ε)m
R
−(2−ε)xq
((1−x)(m
R
−xq
2)+xm
γ
2)
2−d/2
, (DR)
(19.57)
So,
δ
2
=
d
dq
Σ(q
)
|
q
=m
R
=
e
R
2
16π2
4
ε
−1−log
µ˜2
m
R
2
+4
dx
2−x
1−x
x
(1−x)2m
R
2
(1−x)2m
R
2
+xm
γ
2
, (DR)
(19.58)
δ
m
=m
R
δ
2
2
(m
R
)=m
R
δ
2
+
e
R
2
16π2
m
R
4
ε
+
3
2
+3log
µ˜2
m
R
2
, (DR)
(19.59)
19.3.6 Summaryofelectronself-energyrenormalization
Insummary,thetwoconditions
Σ(m
R
)=0
(19.60)
d
dq
Σ(q
)
|
q
=m
R
=0
(19.61)
allowustouniquelyfixthecountertermsδ
2
andδ
m
. Theseareknownasrenormalizationconditions.The
conditions we havechosen n force e the physicalelectronmass m
R
andthe residue of thepropagator to be
constant toallorders inperturbationtheory. Thevalues ofthe counterterms dependontheUVregular-
izationschemeandcandependonaninfraredregularizationscheme(e.g.photonmass)aswell.Thecoun-
tertermsδareformallyinfinite.
194
Systematic Renormalization
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Bytheway,noticethatinthisschemewecannotaskaboutradiativecorrectionstotheelectronmass.
Thatquestionhas nomeaninginthispole-massscheme,whichdefinesmass as thelocationofthepolein
the propagator. . Thereareother r ways todefinemass,suchas theMS
mass,for whichyoucanaskabout
radiativecorrections,whichwediscussbelow,orthePythiamass,whichis relatedtoaparticular Monte
Carlo event t generator. . Inthose e cases,radiativecorrections to masses dohave meaning. . If f youfindthis
confusing,thinkabouthowyoucouldmeasuretheradiativecorrectiontoamass. Notsoeasy!
19.4 Amputation
Thetwopointfunctionconstructedoutofrenormalizedfieldsis
G
2
R
(p
)=
0|T{ψ
R
ψ
R
|0
=
i
p
−m
R
+regularatp
=m
R
(19.62)
Intermsofthebarefieldsψ0=
Z
2
ψitis
G
2
bare
(p
)=
0|T{ψ
0
ψ
0
|0
=Z
2
i
p
−m
R
=
i
p
−m
R
−Σ(p
)
(19.63)
whereΣincludescountertermcorrectionsfromtheexpansionof Z
2
.
RecallthattheLSZ theoremprojectedout thephysicalone-particleasymptoticstates intheSmatrix
bymultiplyingbyfactorsof p
2
−m
e
2
:
f|S|i=(p
f
−m
e
)
(p
i
−m
e
)
0|T{ψ
0
ψ
0
}|0
(originalLSZ)
(19.64)
Whichmassshouldthisbe? Attreelevelit t doesn’t matter,butincludingtheloopeffects,wehadbetter
makesureweareprojectingout thephysicalelectronmass,byaddingfactors of f p
2
−m
R
2
withm
R
being
thelocationofthepole.So
f|S|i=(p
f
−m
R
)
(p
i
−m
R
)
0|T{ψ
0
ψ
0
}|0
(renormalizedLSZ)
(19.65)
Notethatthisisstillexpressedintermsoftimeorderedproductsofthebarefields.
Now,ifweareto use thisexact t propagator, wemustalsotruncate externallines. . Thatis,we e should
computethe1PI bubbles onexternallines separately,thenaddthem to the diagrams without 1PI bub-
bles. Thisisalsoknownasamputation. Also,notabigdeal.Itactuallymakesthingseasier. Forexample,
consider
(19.66)
Thisgraphjustsets the counterterms δ
2
andδ
m
butdoesn’tchangethephysicalelectronmasse
R
orthe
residueoftheelectronpropagator.Thusaslongasweusethephysicalelectronmassandcanonicallynor-
malizedinteractioninCoulomb’slaw,wecanignorethisgraph. Anotherwaytoseeit,isthattheexternal
electronhereisonshell.Soq
=m
R
andthenΣ(q
)=0bytherenormalizationcondition. Theresultisthat
theexternallinesinthegraphwouldgivefactorsof
0|T{ψ
0
ψ
0
|0
=Z
2
i
p
−m
R
(19.67)
Soitisthesepoleswithfactorsof Z
2
that get canceledbyLSZ. . Thenifwelookatamputateddiagrams,
LSZsimplifies
f|S|i=Z
2
n
0|T{ψ
0
ψ
0
}|0
amputated
=
0|T{ψ
R
ψ
R
}|0
amputated
(19.68)
SotocomputeS-matrixelements,we canjust usethe renormalizedpropagators,anddropallcorrections
toexternallines.
19.4 Amputation
195
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S-matrixelements canbethought ofasGreen’sfunctions inwhichthe externalparticlesareon-shell.
Since they areonshell, thereare poles, andthese poles must be removed d tomake e sense of the answer.
Thepointofamputationis
• Toremovethepolessotheexternallinescanbetakenon-shell
• Sincetheexternallinesareon-shell,theyarealwaysevaluatedattherenormalizationpoint–there
isneveranyadditionalmeasurableeffectoftherenormalizationofexternallines.
TherearemoregeneralGreen’sfunctionswecancalculate,withoff-shellexternalparticles. Usually,these
can be e extracted from on-shell states. . For r example, the e Green’s function for an off-shell photon n with
momentum p
µ
(p
2
0)canbe calculatedfrom theGreen’sfunctionfor anon-shelle
+
e
whosecenter of
massenergyis p
2
.
19.5 Renormalizedperturbationtheory
Theideabehindrenormalizationisthat forevery infinity,thereshouldbeafreeparameter toabsorbit.
We startedby absorbing infinities into a constant vacuum energy density(the Casimir r force),and then
intotheelectriccharge(renormalizedCoulombpotential). Nowwehavebegunasystematicstudyofcor-
relationfunctions andfoundwehadtwonewparameters,theelectronmass andthenormalizationofthe
electronfieldwhichwecouldalsouse. Arewealwaysgoingtogetanewparameterforeverycalculation?
LookingattheQEDpathintegral
Z=
DA
µ
DψDψ
¯
exp
i
d
4
x
1
4
F
µν
2
¯
(i∂
+eA
µ
+m)ψ+ρ

(19.69)
itseemsthatthereare onlythreeparameters m,e,andthevacuumenergydensity ρ. However,thepath
integralwillgivethesameanswerforcorrelationfunctionsifwerescalethefields. Thatis,thenormaliza-
tionofthepathintegraldropsoutofanyphysicalcalculation.
Letusadda0tothefieldstodenotetheirbare(unrenormalized)values:
L=−
1
4
F
µν
2
¯
0
(i∂
+e
0
A
µ
0
+m
0
0
(19.70)
WesawthatGreen’sfunctionsofthesebarefieldsareambiguousuptonormalization
0|T{ψ
0
ψ
0
}|0
=
iZ
2
q
−m
R
(19.71)
Soifwerescalethefieldsby
ψ0=
Z
2
ψR
(19.72)
A
µ
0
=
Z
3
A
µ
R
(19.73)
then
0|T{ψ
R
ψ
R
}|0
=
i
q
−m
R
(19.74)
whichisnice.
TheLagrangianbecomes
L=−
1
4
Z
3
F
µν
2
+iZ
2
ψ
¯
R
ψ
R
+Z
2
m
0
ψ
¯
R
ψ
R
+e
0
Z
2
Z
3
ψ
¯
R
A
R
ψ
R
R
(19.75)
Sowe seethat QED has5terms init’sLagrangianand5parameters:m
0
,e
0
,Z
2
,Z
3
andρ. Allofthese
are formally infinite,and d we e can n absorbadditional infinities into any y of them as required. . The e vacuum
energy ρislargelyexternaltoQED,sowewilljust dropitfromnowon. Theimportantpointisthat all
wehavearethese5parameters,andtheymustbesufficienttoabsorbeveryinfinity. Therearemanymore
than5correlationfunctionswecancompute!
196
Systematic Renormalization
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Also,itisconventionaltodefine
Z
1
e
0
e
R
Z
2
Z
3
(19.76)
andtouseZ
1
insteadofe
0
.Then(droppingthesubscriptRonfields)
L=−
1
4
Z
3
F
µν
2
+iZ
2
ψ
¯
ψ+Z
2
m
0
ψ
¯
ψ+e
R
Z
1
ψ
¯
A
ψ
(19.77)
Atleadingorder,weknowwhatalltheconstantsare:Z
3
=Z
2
=1andthemassandchargearedetermined
bytheirphysical,renormalized,valuesm
0
=m
R
ande
0
=e
R
. The0subscripts s arecalledbare parameters,
andtheR subscript is for renormalized. . At t next-to-leading g order,these constants s areall l infinite. But if
wedoourperturbationtheory expansioninpowersof e
R
,thisisacalculableseries,andphysicalquanti-
tieswillcomeoutfinite.
Sowecanexpandperturbatively
Z
3
=1+δ
3
(19.78)
Z
2
=1+δ
2
(19.79)
Z
1
=1+δ
1
=1+δ
e
2
+
1
2
δ
3
(19.80)
Z
2
m
0
=m
R
m
(19.81)
wherealltheδ
sstartatO(e
2
). ThentheLagrangianis
L=−
1
4
F
µν
2
+iψ
¯
ψ−m
R
ψ
¯
ψ+e
R
ψ
¯
A
ψ
(19.82)
1
4
δ
3
F
µν
2
+iδ
2
ψ
¯
ψ−δ
m
ψ
¯
ψ+e
R
δ
1
ψ
¯
A
ψ
(19.83)
Anicefeatureofthisapproachisthatsincetheδ
sareallinfinite,buthigher orderinpowersof e
R
they
canbe simplyusedasvertices intheLagrangian. . Thesecounterterms s subtract off the infinities directly,
leavingfinitequantities. Moreover,wecandotheperturbationexpansiondirectly y intermsof e
R
instead
ofe
0
,whichisnicebecausee
0
isinfinite.
Notethatthecountertermsδmustbenumbers(orfunctionsof e
R
andm
R
)–theycannot dependon
derivativesorfactorsofmomentum. Weassumedthiswhenwecommutedthefactorsof Z throughevery-
thing. Iftheydiddependonmomentum,thatwouldcorrespondtotherenormalizationofadifferenttree
level operator. . For r example, , if f δ
m
= p
2
thenthat wouldhaverenormalizedψ
¯
ψ,whichis not whatwe
startedwith. Youcanhavecountertermslikethis,butthenyouneedtostartfromadifferenttheorythan
QED(i.e.onewhoseLagrangianhasatermlikeψ
¯
ψ). Wewillcomebacktothispointwhenwediscuss
renormalizablevsnon-renormalizabletheories.
TheoriginalLagrangianEq(19.70)is consideredbareandtheonewiththeZsEq(19.75)considered
renormalized. Thatis,inthebareLagrangianthefieldsareinfinite. Inthe renormalizedLagrangian,the
fieldsarefiniteandtheinfinitiesareallinthecounterterms.SometimeswedropthesubscriptR.
19.5.1 2-pointfunctioninrenormalizedperturbationtheory
Let’s comparethecalculationofthetwopoint functioninbare perturbationtheory andinrenormalized
perturbationtheory.
Inrenormalizedperturbationtheory,theFeynmandiagramsgive
0|T{ψ
R
ψ
R
}|0
=
+
q
q−k
k
q
=
i
q
−m
R
+
i
q
−m
R
[−iΣ
2
(q
)]
i
q
−m
R
(19.84)
andthereisalsoacountertermgraph
=
i
q
−m
R
i(q
δ
2
−δ
m
)
i
q
−m
R
(19.85)
19.5 Renormalizedperturbationtheory
197
So,
0|T{ψ
R
ψ
R
}|0
=
i
q
−m
R
+
i
q
−m
R
Σ
2
(q
)−q
δ
2
m
q
−m
R
(19.86)
Thewaywediditoriginally(bareperturbationtheory),
0|T{ψ
R
ψ
R
}|0
=
1
Z
2
0|T{ψ
0
ψ
0
}|0
(19.87)
and
0|T{ψ
0
ψ
0
}|0
=
+
q
q−k
k
q
=
i
q
−m
0
+
i
q
−m
0
[−iΣ
2
(q
)]
i
q
−m
0
(19.88)
Thenexpandingm
0
=m
R
m
+m
R
δ
2
andZ
2
=1+δ
2
0|T{ψRψR}|0
=
1
1+δ
2
i
q
−m
R
m
+m
R
δ
2
+
i
q
−m
R
[−iΣ
2
(q
)]
i
q
−m
R
(19.89)
=
i
q
−m
R
+
i
q
−m
R
Σ
2
(q
)−q
δ
2
m
q
−m
R
(19.90)
Anoteonsigns. Let’sdoublecheck thesigns s here:inbareperturbationtheorythefactor
1
Z
2
=
1
1+δ
2
leadsto −δ
2
iq
iq
=+δ
2
iq
i
q
. Inrenormalizedperturbationtheory,thekineticterm givesZ
2
i∂
=(1+δ
2
)q
,
whichleads toanδ
2
iq
vertex,andtothesame correction n +δ
2
iq
i
q
. This s worksfor the2-point function,
whichisaspecialcase–itisthethingweamputate
Ingeneral,withanamputatedamplitude,weuse
0|T{ψ
R
ψ
R
}|0
amputated
=Z
2
n
0|T{ψ
0
ψ
0
}|0
amputated
(19.91)
ThentheZ factors s ontherightareallthat’sleft over. . Forexample,foranamputatedthree pointfunc-
tion
0|T{A
µ
R
ψ
¯
RψR
|0
=
Z
3
Z
2
0|T{A
µ
0
ψ
¯
0
ψ0
|0
. Recallingthat t Z
1
=
Z
3
Z
2
we see wewill get the
same counterterms s automatically using the two methods. . Thus s they y are really two ways s of f saying the
samething.
Thedifferenceis
• Renormalizedperturbationtheory:allcountertermscomefromtheLagrangian
• Bareperturbationtheory:fieldZ-factorcountertermscomefromexternalstatesincorrelationfunc-
tions, mass/charge/interactioncounterterms s comes from replacing g bare parameters s by their phys-
icalvalues.
Therenormalizedperturbationtheoryapproachissimplerinmanyways,but wewillgobackandfortha
bittocheckconsistency.
Wehavealreadyfoundreasonableconditionstofixδ
2
andδ
m
,solet’scontinueourstudyofcorrelation
functiontofixδ
1
andδ
3
.
19.6 Photonself-energy2-pointfunction:vacuumpolarization
Theothernon-vanishing2-pointfunctioninQEDisthephotonpropagator
Π
µν
(p)=0|T{A
R
µ
A
R
ν
}|0
(19.92)
Wealreadycalculatedthevacuumpolarizationgraph
p
p
k
p−k
=−i(g
µν
p
µ
p
ν
p2
2
(p
2
)
198
Systematic Renormalization
with
Π
2
(p
2
)=−8
e
2
(4π)
d/2
Γ(2−
d
2
4−d
0
1
dxx(1−x)
1
m
e
2−p
2x(1−x)
2−
d
2
(19.93)
=−
1
ε
π
0
1
dxx(1−x)log
µ˜2
m
e
2
−p2x(1−x)
(19.94)
Actually,weonlycomputedthe p2gµν part of theexpression. PeskinandSchroedercomputetherest. In
anycase,wecouldhaveguessedtheformbytheWardidentity. Notethat
(p
µ
p
ν
−p
2
g
µν
)
2
=p
2
(p
µ
p
ν
−p
2
g
µν
)
(19.95)
Sothatthisisaprojector. Forsimplicity,wewilljustdropthe p
µ
p
ν
terms,sincewecanalwaysputthem
backintheendbyaskingthattheWardidentityberestored.
Before,wehadclaimedweclaimedwecouldabsorbinfinitiesofthevacuumpolarizationgraphintothe
renormalized electric c coupling e
R
2
. That t was s for r Coulombscattering, , where the photon was s an n internal
line.Here,thisisjustthe2pointfunctionofthephoton. Forthe2-pointfunction,theleadingorderpiece
isnotproportionaltoe
R
2
,thus wecannotabsorbtheinfinityintoe
R
. However,sincetheGreen’sfunction
doesinvolvetheexternalstates,wedohavetheZ
3
countertermtoplaywith.Itgives:
=
(−i)g
µν
p2
3
p
2
g
µν
(−i)g
µν
p2
Nowresummingtheseriestoallorders,includingtheother1PIgraphsgives
Π
µν
(p
2
)=
+
p
p
k
p−k
+
+
=−i
gµν
p2(1−Π(p2))
(19.96)
with
Π(p
2
)=Π
2
(p
2
)+δ
3
+
(19.97)
We have dropped gauge dependent t terms. . (You u can actually treat t the gauge as s a parameter r in n the
Lagrangian,introduceacountertermforit,andcalculateradiativecorrections. Thisissillyinpractice,as
they always dropout of physicalcalculations. . But t itis usefulfor proving g the renormalizability ofgauge
theories.)
Notethatthephotonautomaticallyhasapoleatp=0. Thisisbecausethefunctionvacuumpolariza-
tionloopisproportionaltop
2
. Thus,theonlyconditionwerequireisthattheresidueofthepoleatp=0
be −ig
µν
. Whichmeans
Π(0)=0
(19.98)
Atordere
R
2
,thisimplies
δ
3
=−Π
2
(0)=
4
3
e
2
(4π)
d/2
Γ(2−
d
2
)
µ
m
e
4−d
(19.99)
=
1
ε
+
α
log
µ˜2
m
e
2
(19.100)
SothenthecompleteΠ-functionis
Π(p
2
)=−
π
0
1
dxx(1−x)log
m
e
2
m
e
2
−p2x(1−x)
(19.101)
whichistotallyfinite(andalsoµ˜independent!).
Althoughthereisnoinfrareddivergenceshere,therewouldbeifwetookm
e
→0. Sothisis s verysim-
ilartotheinfrareddivergenceintheelectronself-energydiagram.
19.6 Photonself-energy2-point t function: vacuumpolarization
199
19.6.1 gaugeinvarianceandcounterterms
Fortheelectronpropagator,weneededtwoconditions,onetofixthelocationofthepole,andtheotherto
fix its residue. Forthe photon, the locationis fixedbygaugeinvariance,so we only needone condition.
That is especially convenient because there is s only y one free parameter, the normalizationZ
3
for the A
µ
field.
Itwouldhavebeenpossible,apriori,thattheloopmighthavegiven
p
p
k
p−k
=−i(p
µ
p
ν
−p
2
g
µν
)g
µν
Π
2
(p
2
)+M
2
Π
M
(p
2
)
WiththeadditionaltermproportionaltosomedimensionfulquantityM (presumablyproportionaltothe
electronmass).Thiswouldhaveledto
Πµν=−i
gµν
p2(1−Π(p2))+M2Π
M
(p2)
(19.102)
Then we e would d have needed a counterterm m so that t we could renormalize the photon mass s back k to o it’s
physicallocation. However,thereisnosuchcountertermavailable. Togetone,wewouldhaveneeded
L=−
1
4
Z
3
F
µν
2
+Z
3
m
γ
2
A
µ
2
(19.103)
Whichallows for thecountertermtoappearintheredefinitionof m
γ
. Instead,noM
2
term is generated,
andnot coincidentally this additiontotheLagrangianis forbiddenby gaugeinvariance. . We e are seeinga
veryimportantandgeneralphenomenon:theconnectionbetweensymmetriesandFeynmandiagrams. We
willbecomingbacktothisthemeagainandagain.
It is alsoworth h pointingout t that the generationof aphotonmass is somethingthat youhavetobe
verycarefulabout. If f wedidn’t use dimensional regularization,we might have endedupwithaphoton
mass. Dimensional l regularizationis nice because you can define gauge e invariance in any y dimension. . In
contrastwithaPauli-Villarsregulator,orahardcutoff,theexplicitscaledestroysgaugeinvarianceandin
factaspuriousM
2
termdoesappear.
19.7 3-pointfunctions
Somuchfor 2-point functions. . Wehavemadeeverythingfinite. . Nowlet’s s lookat3-pointfunctions. . The
simplestoneissimply
G
3
=
0|T{ψ
¯
(q
1
)A
µ
(p)ψ(q
2
)}|0
(19.104)
Thisisnon-zeroattreelevelduetotheQEDvertex.
G
3
=ieγ
µ
(19.105)
Atnextorder,weneedtocomputealoop
q
1
k
k−q
1
p+k
p
q
2
2
µ
=ie
R
F
1
(p
2
µ
+
µν
2m
e
p
ν
F
2
(p
2
)
(19.106)
Thereisalsothecountertermcontributiontothevertexcorrection,
q
1
q
2
p
=ie
R
δ
1
γ
µ
(19.107)
200
Systematic Renormalization
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