Constraints on the 1PI functions
Here↑

of()

deriveweMr ab(p p),constraints

MpGp 1),.

effective

the 1PIPgh(4),Lorentz(or Euclidean)

EuclideanrotationAn(p)transforms

S(1)Invariance of N↑,.

the fields

asisunderab(1p (p)=-,Tgh) 1p (p)-F(p(p)S(n)-spin representation

the1 r).

=thisrequires

1 14 x ab(p 9)+=,

Tp( 1p 1p),,(SUNS(n)po,,C(1p) <(1p)>-,+TGP-9 9 1)

invariance

S(1)" 4(p)C(p) ((p)
1,Ar(1p)MY-4(p) +whereI

actioncoefficient functions

theon,,S(r)Tp( p p)S(1),-,Tgh( P p)=-,,,

↑ (-1p-19 19 1p) = -9 9 ,,,,

The solution

thistois

pr ab(p p):

29b(42)85,=,[p( P 1)=-,

Tgh(1 1),=Pgu)pY,=-whereAlpY

a("P 3)(2)

Rigid.

BlpYa,

B (4p D+(P g) ((p 9) + 94D9(p 9),+are,,

identities

D"IP 9) depends on 1P 9)

.
p g.

,

P under rigid G-transformations
gAng"

for matrix Lie algebra

An-AdgA,n (=445 - Adg6

94,requires& (pyAcpy↑

P2,92via,,,
Spinor factor

theinG-invariance

Invariance of+"A(pY4 + B(PY ,Tsu( 9 9 1),

*gulpYjax(pY=BlpY)

a=,B (pYcommute=&with*i-JAdgi

B(pY ,G,fgh(pY ida

Ads ((0 9) Adg",*,=(Adg)y * (P 2)

same.

,for D (P 9),.

Herewehave usedthat theadjoint

By Schur'sanyGis

D-linear*,requires(B (PY)&*,Pgh(p),that G<Y &simple

isirreducible

ismap,

which.

f : -Je

proportional to theG-invariance of Nthe( * PY)(3)assumption

representation

lemma,
withcommutes

Thus

the

that

idy,identity

which

implies that

commutewith AdG,

pay &8 Eguspy&

BRST symmetry

RecallIfthetheSlavnov-Taylor identity

system

IPTthehaseffective

P +p+ pLet

usasymmetry

actionp;

PCP]=:$1 P + Op,

isinvariant(807516

apply this to the BRST symmetryMX]=under:0.

(This is nothing but the Zinn-Justin equ at K=.
0)

idimpliesByphost # symmetry
we& Ap(p)
⑬C(1)

JBX

possible forms ofknow* 4(p)

Lorentz invariance and fermion # symmetry,,f(p2)PmC(p)==quadraticor=quadraticor+:quadraticor

higherinfieldshigher in fields
higherfieldsin(p)XiB(y)

B(p)

0

=

(To be precise

these,

arefor bare fields,

renormalization , these hold also for
Thencomeand

EBT whichthe terms of,from G

onlythatbutby the form ofrenormalized fields )the.

linearare

Apatps ↑inAna***( p 1) Any(p)-,isab(f(pY( P(p(a(-p)4m2 (p 1) Aub(p),-,(We usedPro ab5-p 1),,

PUR ba(p - p),=,.

which wemaysea

Vanishing of

this term

For Thv ab) p p),-=,

PrPMY ab (p p),requires&:(4(p2)d*x(p2p + B(pz)p2pie.

x(py.

Thuswe,=,=,A

Tghtp p)Any

1P[B(p2)pMp)=0

this--=

construct,Pp2diagram-isoft xt 1Pz - x -m-..............
Something
Dthe form

proportionalP.

--- 77-1PI ---- ,vanishes:Ngh(pp)fgulpsidy

Eghtp 1).

--==.

Ugh(-P P)

on

↑

:,.

&(p2g pmpv) iT(Pdiagramatic,+0find↑ M ab (p p)

(4)

pB(q)

-=,if

p2Tigulpy idy

pp=.

=

toPM0

fgulpy

=0

atpeo

More
Tah(P-9 4)
1)
on

Recall (1) (i).

↑(p-9(P 9)((P 2)X Cp a) oD (p 9) & Endeobey:a, 9 .

1)

*+=,

*

where=,,AdgoX" Adg"
Recall(Ad)s"isS

withrespect toNote9,X"

thatAdg

ad ed===theaX

Obey (* )Adge"Adg(e9 Adg"X]
=,Adg
=e=

*:o>
-%

Adgly"(Adge" X]

ade X Adglo"

Are there others?

End Yerepresentation of Ga,:,(e Adg)s X]

isofmatrixbasis Se CY ,=,(A)(Ady) barepresenting

adea

Adg"X

*"

9 D (P 9)↑+,.

by>ge 6 : fe EndyYeAdgo fo Adg ,anditbecan

SupposeEnd

decomposedinto

irreducible representations.

includesM-copies of

Ye -otherthisDm

theadjoint representation

representations

Examples·G

EndY·2SU(2)==End BE B
&m= 1

SU(3)

=IDelD

EndJD

Lead.

&
m= 2

X - End Ja be the image ofLet(10, 1=Xo=,...,

m-1)

ade ?

Cyp 9).

=We.

Then ,

eacy

regard adYy
we,the 0-ththe E-th

copy

Copy

so,

expandcan[X & Clp e)

asintoDapis),=EX* Delp 9).
,

Comment on renormalization condition
Renormalization condition

Co(p19)/p= (19s=r
We should

G(1 9).

notimposed just

is=-i.

impose any condition

With 20aon

Delp 9) Fl.
,on

Co(p 9) :,thatThus,

thecondition presented"

C (p 9),atthe class/p == (p 95 m2

was

too

strongThat

iscorrected+in-=

theinone

thenote)ie=general (itin=(wasOK if

M= 1=.
eg. G SUlzs)

the revised version of the note.