I

Introduction to perturbation theory
Considerwiththeory ofthedomeasureSe(p)single real variable O

aactionand+ap +=4

partition function

zSz(p)(dde==Jap tapt) " pIRItIts

perturbativeZ

expansionis= Jadetappin

pert

IR

It is(not.
convergent

If it hadthe integral

Instead

X-

IKR

for

*,

it(z(x)convergence radius R30

ais-but,

isanZ(x) is

notbadly divergent,

Zpert(x)z(x)

0& truncation

asatz(x)obviously analytic at/for

realXX0order= 0:

negative X.

asymptotic expansion of Z(X)
+=.

&N:

See the additionalnote for a proof

2

Each term ofthefree

Epert

Eperttheory
Zree=

Similarly forwithacorrelation

actionfunction ofSE free(P) tap2=:) < P )free*

correlation functions

Selfe

period=:,(f(a)) =(f(p)

isZfree-me (p)" (f(p)PFree*)free*< P )free

Digression

Free theory&free=

Se free(p) = tap,(dap< >free ==: G

Gaussian

integralWick contraction

S[Q0i-- freee g.

sum.

(pdbb)freeof all possible Wickcontractions==

see the additional notes if you are not familiar with this.

Using thisZa /Zee+findwe,1=+(4))

7- )freefree

7 Pfee-Pox3A(7)

free

=

Eipp 1973+W:

B

00PX()22cLpp)

t+

4!

3D

4

Diagramatic presentation-apropagator

60

interaction

-

A:
C

=:

line

us

Vertexo,

Bitco
Zar /Ze+

=

Ciao
1+of

Dif"

00 +
sum,vacuum

200+

---,+

A

+...

diagrams↑noexternal line#=-

See the additionalnote for a proofexp(0+000+Sumofconnectedvacuumnu< PPXpert=...

(PPfree + <PP"))free
1+1- )free*+--

diagrams+--)=A+-som of+diagrams

2A+withtwo

5-++external lines(Pb)

conSee the additionalnote for a proof< POPP) port=

Sum<PPPPCSee the additionalnote for a proof

diagrams

ofconn+with

(PP)

·conn

=

part

X

X+B

=+...

+(xX X++X++Q+...

disconnected part(( q -) (( q -)++++++((-+2+-+N p ) (x+3B

connected+<PP--&&

4 external lines+...

+++-.

.

- -1)x)6Generalizations
more interactions

Se(p) = ap +()

...

-free

Zee/Zee(Pm=< PP(petexp)Af==

interactions++<PP)

conn

00+0-0+00+

. . .

)

( (b))=. . . .

+<PYone (Pconn

im

6f- -++. . .

1969 Xpert <PPPYonnt <Pconn (PP)

conun

Cx
i++Q+....

*

3+10

7nvariables

Se(p =A j-+-free

Epert/Eree=interaction

exp(08+000++-=i(P)=imj&i-(...

inj+=fj ++(-+,=Elij=& (P P; ).

conn-Putut(a)Pj x4 3.

=in An As,i
D=- >Pete; 24e Aintre) es,

S(PiPinPisPia)pert=(PisPinPisPatconnt (PiPin) conn <Pin Pipcommm↑(iii)( : :4) + (4) Liz is)

&i
/
Vip I
I /
I i
Lis -"isi++12i4=-inis&,D&

54

14i isO+iO-in14"VisOri"Si& =inisD2is+im i4LiO+-,

PisEli x4 !

bil2bine~Pipx[j As j jO =t10.

P= As-CPartit-Pizx

Pip

Aris Aria

2 (4.

.

3)? 2.

9

Scalar field in 4-dimensions

SE(PT=Sai (t(0)2+me=P-+ 4)-freeexp)CO 200
00 = x(Pipas)"

Epert/Zree+=

14

interaction

+

D+...

↑P-theory"(iD&
CaspertiAtt
D2.

-=-(2Plis Plan---&Jax &-cIO-·i.

=(a) blas Plasplas Play(a)

? Peo
pinp
Papie
Jarding18[P( ) Pl / Pla)(44) += (P()P)11) Pla-,arcom

m+ (P(4 )b())

SI ."+ S "+I

I

4D&23&t

4&+&&

X+&on&S

3O&Z

3V[P(43)P(K4)I

324+- 44&2Sx)a xp(x ba)+-=

PQ S=91:)

disdiy Pitti dM

plus

2(pip(s)--iconn((3)(24) + (14/(23)4Do
I/&,plan) &(3) (4)Il

Scalarpropagator

SE free(p]=,1Plp(2)+(dp()(7m==- + my b(x)-iP(x)

PE+-ym2-

reversex4+1P7 P

Wick rotationi),yP --iy ;↳i m2

Pp+-iPo
PoP4C-IPEM2Xu-⑨&

IP= M2

N-Vim-P(x)p(y)-i-ilpo(X -y)-iPo y9edPIP(idPo)(2πT)4Ip2-

Po + m2

3xC

Po

=·-p2

-

iP (x y).

-m2 +-P-1p2

i 0.

M + iO-&#M- iO

12Gauge theory (egof·Diracfermion

0 A

gangefixing byTeYe=Le,

Le free =,-LE int =P,-=.

I-CE free

representation R

aof G+0z(dA)"

+ c.

& D.
.

LE int (rescale Ap-eAr)
,

10 A)
I
(An-GAp .

iF@4 + mπ4+joc(dAn-JuAm) [Am Au]
.

.

ieFA4

perturbative expansion ofdecanin

FE-iFa4 + mπ4&e ==QCD (An gauge potential of garge group G

variables.

-.

computed+ E.

+

3[Ap An]·&C,

OCAnC]

partition/correlation functions
using13

Propagators

S

se's <Ap=

4=CApaC =Ca,,(Orthonormal bases

Sei] <R,

J=e'Pi~=Apalks Arb (3)=Jaber (d 15-1-iPE(x 3)-+M>-=4 : (F(3)-=iPE(k 3)

E(P5)DE PE==-0(.

.

.

..

=Ca(k)[b(3)

+.

.....

(4z m)+ mi-

iPE(s73)me(mic+iPE(x 3)--=Gab)(14Vertices
3~ =4M↓-=-=.

-(Ape/ai==(GAn-JuAn) [Apitu)-Ac] (An Au],-,* 4,e(axe5 [d(tr,,15e.

g.

[Apk)An(3s)-minm-

port+mi+=mu &[-=t+iE

3C

3M 11S>S~↓&-

3[Sz+&+↓S,S--E

3 ME14~Dr↓

Sx+33++=3S+...

&< 4(4)[(3) Ap(z) perttt~((4(5(3))prB&mS

16

Minkowski limit(Note:Z~Apak) Any(3)=M+=

4 (4) (3):-&=------

Calcs[b(3)=dab:

J-~&M↓-iJ)( P -(13)4d) P
4,-=Gab) dip

ip(x 3)-tio

4,3-=-(AA) CAA,=- d (AAZCAR At=ie(di 4=ie/did,*.

,(Ap C.

.