Computation of the 1-loop diagrams

Propagators

We shall work with 3=1-AmalAus())-↑ (x)4(3)=(3) =

P=>-=(↑(-/i--P==ip(x 3)

P--m

my--e

=-+/Pip =m

vertise Sard-AA)(AMA)=-

TzCA" AS-CAn Au]
,

.

iPA4+

GMT (Ap <].

,TA"GD-d * And+A And2S

M↳-+~zC

3-------&xIt3

·Contributions-=(Amogo)Arrbols)]

toim+2,72↓--A P 17.) : F * 4 (72) Arobols) * 2
Auraok/zidzniT-~

move=(1)/dz-

Apoao(k)Amalzi), d zuiP , (x z 1-(1)Pl,F-#useMaMoGo--(vie 417(7) (41745(7) ( Aug(z) Avbols-toSe
-m

*

VIS

-ik, (7 ,zu)~ikz(z2-z ]

I 14

SSei-

joz,

dizn
=v-

daP2,-(2T-2PaiPaEz-y) -Ubrobo(2TP5(P hi + bu) [5P & Chi-be-Pu),·-(24)85(p -42)d(k-Chi- P ).

:weP = 12k,= kP=k=,↳=>

NObTMa( unPrb-L>

bok P-

ipye

inye-Z , fizz

↓Apr jazC]Arbole--=E CAv.

,M=74/diziditu Apoar() Amalzil---Clz) ad)
JadeTry-Arbols-Sittiit ih=>
ModoMa7) J try Jadea--ide&- ubf

Ma-C1P(S3mitymz 77--=VB(71) Vz(n) Arobal3)
Aprodz dzu uB,v↓,x2

add - Tab = (i) p Apa.

=-=Sdz dizu Apoao(k) Apa(zi,M-Ply(_P()Abit)
Plt)
gi=MoarP-YunobNaStry((-ii(k-pe-eiPyX

UsbrE (ich-ik)e )-C(mr((MaArbolub(iSE-An Ar-↓And AnP(7) Arbo (3) +2Appar()(aiz *=Unive--==-2/0z ApaosApalzi) +rVb(e** P(z)P()) AugAl AnbulY)

SaP

ais[MorMaP2)-etreeub&E

zefeub
Ma

iPY

Urba

umuni3 3 2 terms.

X.

z,=Aurar()va:

3**V3 171) V3 (2) Arobo(3) x2z . dizeY572;

L--A ) (Ap Au]*·.

=A [Ap An].

,---AJC Vg (A A AJ() Arobol)
Jaz d Ama) V (A A-=,,V [ti An As] =:

,

Ifwe,

Acput

*=e,

,

A [Aup Aam],d "eiheAmalke),+ permutations[=1, 2 3.

,;(3 ! termsthenVs (Ai Az A3],,ikeA(hi) (eAmanlk) eAvaalks] eikitkusts)-·,>-(ee]=+Fv

M G 1 M2@uMs As.

41 , Rz , kepermnations

Apalt All elites
permations

Amalkil Apnanskas Apsalks] eikirkthis jpagean, eas] gimn

Mia , Man Made

V mNote·

By

RL ,,

=

The inner product:[X 4),andee"
Then&=

zIt is.

antisymmetric(e9) C is[ea e3].

+,exchange

bec

init

Using

thisis

also.

fa=,

-facb

have

f+

f

*0.

=

fab -gabc

fabe is totaly antisymmetric

=>

in.

Te=we,fbac.

zef=

the structure constante. jee %0- fbcaorthonormal basis,

-=>

=

an,

By the adjoint invariance of(e e] e0=fabe=invariant

adjoint

is,Supposeea(ee'],Y [X z)+assuming

arewe.

ofinpermutation ascyclic

exchanges ofininvariant,a,b,.
cfabefbca-fcabe]
property of falaras-easear,

wefind,V Ma, ManAs=↓er

faiGuan[gRM(R -k ]M3 JMaRh GM Mok-k,=+-·um

Back touni
-==>

--

AJC Vg (A A AJ(u) Arobo (3)Jaz d Ama) V (A A,,agipk,,PeiPy

F*,-Xub

No bokC↓ Raman Mabea prube ub(kman-r3bs

My 93inTa-pPrb7m((SMaImm&-p)&-P

5~--JaPzVp()

Apao()=>(4 3 terms) ;Aroboli).

LV: =-CAP AV] [An An]-,E,

VPCA A A A]
,

=

JaPz

--Apar Vp[A A A A](z) Avobol3),

VLAi An As 14]
,,

,

:=

If we put

At

=

*E ;

-en(t A]-CAsm Aqu]
,+,

(4 !

edi like Amalke)=,X,,,permutations

24 +erms)[ 1, 2=,3, 4,

then

Vq[Ai An As Ap],=-,,

En (e 24] Le ] A Chil AanChes Anas (hs/Araalha).

,,,+=permutations

Telee]. (eas e] JrMa GRIMP,Amai(ho)---Appap(ha) eilhit---+permutations

haselhit-has

(12) =134)4=Apa (k) ---Appap(ha) ei(k , +-+↓↓hp)xx2x2,

5 Ce edn] Jea 994] (gMiRes gMuMP MiMaMMsCees] Jean (94] (gMM2gNsMR-SMiMG jMM·,↑+=M,:·[e e44] /eas 992] (JMiMs JMaM2 gRiMn RIMs) Y·,,Ma , --, MaAmalki)--.

Amae(ha)i (h , +e- ..

+ha)x=a., ..., M49p

- Ibangbagap (gMi MMP MMM-Zybaaz fanap (gRMn jMsMRSMiReM-Ifbaapfbasan (gNiNsgMeMLRM] Y&PP

~Noto~

iPy&FMa-ob

Ma , Man , Ma , erobe

k=

-

M2Gz

5et

Mz93

·Contribution

-=S

M

--

(4()#131) oftoCE-

4()(az d iTA4()i4 * 4(z)4(3),+L

2&z , +22

↳4eb

Ma=

Sirek+ eive&M(C)(4)
P

C·Contribution=(C(([(3) oftom.............
C&---*diz d [An CJC & (An C](z)[(3] +2(k)[Jaz,,,-&z , 772
n p--neebin=> in ipad deh( 2)m---------P

c -f...........

(((X) Anobolw) [1317ContributionstoEven(tree diagram (......

=

((u)

Jatz

O TCAp C](z)*.

-=Saz()

aadeApoarlm) [ (3)--Row↑A)

Sip Fighti
Saz+(2πT)"J(p -9 92)vP2-,

=P,

p=.

P+ 9Martin

3↓ 9Ma=SviSPilpasmade-3Ma~C(([e--a-mamanan,------e↑·en...

=-e)

z

--**

*/Ap (714 &@ (App C] (3) Apar(w) [ (3)

A,,x

3 !
&=>

-iPt

permutationof
1 z2 2)
,

eign

Moder

3MaP

Jailpdikeas------

Pleai>-k P-zGety y.

,

gip2

3...

t.......

= ( dz, d Ze dizz---*(()8 *C. (Am CJ(1) VCA A AJ() O 2 CAM2 <](2) Apoao(ws <(3)
.

,,,,

-

*

207 77,i(( 9)x+=>-MrGo-----E-- 89, b,3mafabVh 9 -9+iiik+Ma

99, b,eigw,-,Gabz

kI

39SpilP+/Meal P(p-h)Madeae ...... e......

an(-yfM-----y----Man-) z-..

)iPYuseful facts

Somesimple Lie algebra yon~Seas ToFor

beforeas(trylee")5 ene"-=-

orthonormal basis

product ee

inner

If V isTr daG(V)·V=irreducible

G (adjoint rep)adjointinv(ee] Zef=a:scalar ) quadratic Casimir(each,,irreducible component

TodimG = G(V) dinTy

=((Y)

=:

forah

called theis

suitable normalization of "o"-Jere = (ee)+umJeeO-G(V)

fbda
-d-

fabc fab↳=(Tv E Rzo

dual Coxeter number of of·anfab < totally antisymmetric>-on·=wrt

V of ofany representation=an=(2 G(V))ea--

Vis [ceadeade=-deTydla-huddaWe needtoevaluate

theintegralsof the formi-

whichoftenareregularization[pr(f)=JDR(9)=d=withinwhich

employ the dimensional

We shalldivergent.
thesearepolynomials of Let's

replaced byMinimMish-p2(eventuallyP-E

wetake - +0)

We use.

+xB(2= SBC)
+where:==<(3 7) /R2/320,<,720,

y+z-13

Computation goesI')

asfollows

Mine Socce=dxf(k)Chemic + ocpu-ps+Ms2

h=-zxpk

xp"++?

(1 + )m2 xm+(k-xp)"x(l-e) pi+ (H) m

R

X++/2

--·We

=jMe

expandf(l up)

replaceandf(·+Use=>=1)even+-&sin+,power terms

F(exp).

(eg.

dropoddbyfunction of 12a11

power terms-↑e)(Fley =sj(45)am4(9/)*+.

*a+F(t)[pr(f)WeInMa=↑(1)) ! do(5)thisway[DRH)=we+↑ (p 9)49

49+

findaX

=

xlrp

+

1

=

=,-Ri

h (k p)r*-="Ch-p( (h-p+ja 22=(1-23m2 + xM2-xprp-x()-x)pmpa

dj(exercise) :/xpi

ja=-Similarly(t x)2

cit.dos(EN12-1 jwherei,B() ↑(p)T(9)usedmay

E "de F(t 2)+(1) prp-andTabat(egT(3-4)i)-

Jar(9)=-wherei=(=y (1 3192-,1"wh=(=*:↑ (3 -y):z)1=-d-z)p2+2yz9P

andy9 zpm++

4Note-+( y9" z4m)( 39 744-+-+(2 e) P(2-2) < (4 -d) P(2 2)-=-*Jan(s)=M2(ingi) abatzI-