PAYS 673 Problem set 3 Benoit Laurent

# The substruction of the 2 diagramr given taken we don't include the polarisationµ' of the externat photon)

Ë÷é -§-ienTp- k

p

-

- t" HENRYniftp.ff.mn#.kifYpifEnn!-?ulpYpiiz - p˥ .where trumeaux .

We see that at small momentum lkànt,the second term in the barbets gees like

ftp..ie→ prie .

So the second term is negligible and nearly geta

contribution from the first diagram .

At large momentum Hi"Mike →Ë ⇒ PÉTÉ PËÆ.io .

So the second diagram cut off the high momentum from the first one , which iswhat corned the UV divergence . This is hour the Pauli - Villars volumeletuswgnlarizethisdiagrami.ithas no effect on small moment and cut off the large moment which causalthe UV divergence .

If we had instant taken a second set of fermion of mass M»m ,we would have

Àktq fr+8 ¥ËË -⇒mnk pr

7ps-

- toi ¥1# titillait;¥à÷Hnm÷?umm*ai"- aimait!ËËîKiki?"M

I

We can see from dimension analysais that this has not exaetlp the same effectthan the Pauli Villars régularisation seheme . We can see that undptn ptn sincefin ulptùlp) =Ptn . Soif we assume a collision at low energy (pin :p ' '

"M' and q%M'),

the quantités between brochets behaves like

Æi¥¥¥¥¥û - ÆHÆ-ÆÆn4kffNKM )2 z

EM Fptmpînîkp - Mt which does not cancel the UV divergence sinceM»ÆÆmT

Brute ait- off :

The amplitude of the diagram ËÜÏ is

in -_ tietffffqulpttuif.fm#.kilkImtMulpl(÷! Étaiq , pinkie lp-ki

Using Feynman paramétrisation ,we can write

{ UHkkmtietlk-mt-iic-IUP-kttiettt-dj.dxdydzdxtg.gs ,where

D=xkgtkt-mt-iitt-yl.fr?m4ieItzKp-NtieI=kYx+ytzI-2q.kxi-ofx-mkx-yt-2p.kz +paz-iielxty-izl-fi-2k.lgx-pzt-ajx-paz-m.lt- g) + ie,since a-ytz -- t

tktqx-pzt-qw-pajdg.pxztajx-paz-m.lt - z) tie

=Lftp.pzf-lgx-pzttqax-ipaz-maltzltie-l?-Atie ,

where l'= kntxqn -zpnandt-lqx-pzf-qix-paz-maltzt-qiri-txzg.ptmaya _qix-mz.im?ltzt=g'xlx-h-2xzp.g+mYt-zt=-ojxly+z)-2xzp.gt-MYI-zf-xyqi-mktzt-xzg.HN- g)

But m'=p":(p-tqk-pt-tzp.gg?m4g.l2ptgKsq.lap-iq)=o--ss---xyga-malt -zi

⇒ intiti xdgdzdx-ytz-ttfdkulpf.NU#Elp ) , .

HII

where A- tu mlklktmltH-tnktrktu-mtnlkk-KHF-m.tkM

Using 8µVN' = -2K,Hatteras and tutti t'= - ttpt.tv

,we get

A- HAHAHmlkutkil -2mW ]

= -24f -xgtzpdtrl-h-xtqftzpd-tmlkl-xgtzpt-qih.tnNv }We can now change the intégral ouerktoanintgralouerl :

in -_ ditietjqxdgdzdx-ytz-ttfdlulpf.li#Elp )Http

Sinn Dis eveninl, fd "l ¥ = 0

.for the same room

,

the intégral fdklz.lvtaniser when mer , so by Lorentz invariance it must he quattro tgfdlf⇒ gmfd"llµ¥ = Agurgmfd "llÊ

⇒ file --4A fait! ⇒A-4

This impies that we can drop all the toms linear in l in Nr

⇒ Nu = -22kfr#txq-zpdtrkt-xtq-zpd-2mdtaxlgr-kzp.tn%}

Butt = Host-K¥2lettre-- 24bar-hl : taguer-HE -taré

⇒NE-2f#l'A-xg-tzpdtrktxtqazpd-2mdtaxlgr-kzp.tn%}From the Dinar équation , pulpt-mulpl.ulptpt-ulp.IM ⇒ ùlp .IQ/ulpt-ulpYlp-pAulpt-O .⇒NE-2L-Étre :L-xp -ptzptrfltxllptptzpt -2mW -2xkpi-ph-azp.tn% }= -4-ttrltttxmtlztxtptrlltxptlxtz - Dm] -2mW -axlph-kx-itz-hp.ttM'A }= -2 ttatrlitxymatr -xmltxlkpi.pk/-yltylmKprtrpHtyH-xIplapi-pHr )- dmktaxlph-H-aylp.ttM'A }= -statut txymrtr - xmh-xkapi-mtrt-yh-ytmldprkmtkph-ytttxlm-H-ytlt.dk pop'-D'pdt

- dmktaxlph-H-aylp.ttM'A }= -2ff81 ?tmttrlxy + xh-xh-ylty-D-2mpilxltxt-tax-H-ylltxD-tmpdyltytt.dz)-dll -yktxkp.pl/OrtmltytltxK2pr-trpB---2ttzKl?tmNrlxy-xh-xh-ylty-D-2mp.lt/r-xHtyH-xD-2mprlty-yt-ltyH-xD-2lt-ylltHlp.pk -m'4-g)ltxttrb

,fut ftp.pt-pt-zp.p-pt-amt-2p.p#2p:p--qt-ama

= - dttakltmaktxy-xlt-xkyh-yH-H-yxt-xD-dmpil-x-xt-tty.ir -xytdmplty-p-lty-x-xgl-gttyxt.dk}

= -2¥ trltmtrfxytx - xtiyyattztzytzx -zxgtdmpitxt-y-xyI-dmprty-x-xyJ-qalt-ykt.HN}

= -2ff trlttmatrtdxyxttx - xa-Hy-uj-2J-2mpitxlx-iyttytdmptyly-xkxt-gytyH.rs}= -2¥41math- tug)H-yz-H-2mpifxltztytdmpt-yltzttxtqih.HU -xttr}=-21Kff l'tqyt-yktxtm42-4z-taz-zYI-2mltzkpix-p.gl -2mL piytpx)}=-21Kff14 qyt-yktxtmalt-tz-zaD-2mtxlpi-ph-ylpi-pd-pilzx.ly ) - pdzytxD}But pilzx-yttpdzytxt-pifazhat-y-zt.ly#prt-azlyttx-zt-x )

⇒pi-prlzltzD-ttalpi-prllzlx.gl)⇒NE -21Kff l'+ qyt-yktxtmalt-tz-zaD-tmqlxzktzttmlpi-ipdzlz.tt }

The dénomination of the intégral iragmmetweunderxxg.ro the intégral overx andy ofthe term proportionnel togrvanishnsinceitisantisgmmetrirunderxxy.tk

can now use the Gordon ident-tyulpytuulpt-ulp.NET tintât)ulptho moitethe best term :

NE-21Kff l'+ qyt-yxtxtmalt-tz-zaD-mldmk-iomqlzlz.bz= -21kffétait - g) Il -xtmalt-4z-zD-tiomqumzltzB-sirt-ilitietj.dxdgdzdx-y.ir- 1) ftp.s.w.pulplltrttttgk-zk-xkmalt-4ztztttiomgumzt-4¥!,HAI

There are think of intégral over l to compute :

H¥i¥Ë Es .

The second intégral is divergent : I-ffffqyff.jpHowever wecan régularise it by including a cut -off at large l . First , we can minplfyit by goingin Euclidien coordinateur :

l'site and Is È2

⇒ I-ifqfqpq.jp ,we can go in sphoiual cardinales :

±- i saisie:*:÷à¥!h÷Let's change variables for E- lits ⇒ a- = Hello

⇒E-÷Ëæa¥→÷Ë¥¥¥ lent -¥ -¥IË=

, tennistennis - a-¥⇒¥fËHÆH¥¥Il=

:# lent#t' s¥¥÷ -IfIf his my large ,

it gees like

I→ Égalent ,which is exultez the result obtaindrvithlanti-V.ms régularisation if| Ê)we do the transformation l'→ ztt .

We can also régularise I by doing it in 4- c- dimensions,with eso

.

fËËüÈËË¥dÊÇËËq , inaugurait¥soi the angdoimtapd willwe can change the variable for x-1K¥

'⇒ dx = -21¥14

⇒ F- ËËÆ "

x µ s'"( txt" x""This intégral is the definition of the beta function :

± avais

But UE) ¥-8,b-"=Ë? t - Êfnb and Its - E)→atriale

⇒ Iç÷¥ -4f" -"Il ens) ¥as -uns-4 ¥- end -z)If we set c-=L

,this bromes identieal to the result obtained with Pauli - Villars

mn andbrute eut -off régularisation .

The function f. IgM and Fdg') are defined asin-_ - ieùlp' ) (t' ftp.t igmmqfalg ')) ulp) Edy )We prevîously found

in= -4itietjqxdgdzdx-ytz-ttfdlp.s.w.jp/ltrttttgH-yH-xtmall.4z-iztttiomqnmzt-zB.ggHttp

= hé§dydzdxtnz-D-ulptpfjai.hn#A-zi*-ftltHltxttmKHHittorff?ftp.?yg,

This is only the first correction to the tree - level diagram .The correction to the

fi and fa function is then

oflait = Hé§dxdydzdx-iy-iz-hfzizjfhntrttlntxgai.int -sikM¥¥7ËBWe can get rid off the divergent logarithme by snhtraeting the value of this correctionat geo :

Fait -Flot-g¥ !xdgdzdans - Ilftp.gyq.etfailtm?qgf:mjftgKH-- High}f-doit !xdydzdx-yiz-ttqftzt-mfef.dxdydzdxtytz.tth¥j¥→g.We can emily compute this intégral when ga :O :

et falot-¥§xdydzdxty-iz.tt?z=f-fjdzf?dqzg--f-fjd8Efz↳ because x can be

equaltoty-zonlyifyatz.g-2-28f.IQ⇒ g- 2 = ¥ x 0,00232

We see that the choice of the régularisation schemedoesnt change the final resultsince these are only different waysofisolatmgtheinfinituimintegrols.TK finit part ,which has a physico meaning , stars unchange by this choice .

¥2 The diagramme contribution at that outer are

① ② ③ ④

¥. Ii # II.⑧

¥. ¥. ËIf we assîgn to the www.umgandoutgoingeleetrom the moment panda ,

the

incomingandoutgo.mg photons the maranta k and k ' and one of the fermin propagation in eachloop the momentum g ,

the amplitudes are

in- t- ie¥:* ami i§¥:?÷ü:¥iïà:÷:un↳÷: **"

initie"

ff1* ami ifeng.ie?ilqi:tiaili.ni?.i!attire)"#MAG- g-k lp-k.ptit

initial:#iitmrnilgii.in?iiiitgEmnt:aih&:¥:÷?"MË?÷ "Ë"initie" ¥¥¥mii¥::÷¥:÷?Haitink"MË÷. le"""The 4 other amplitudes can be oftainedlyinteuhangingthetmdu.es on the polarisationvector and changing km - toi

.

We can write the first amplitude at

i M = - tiel" àlpttmulpt ETRE*HA

,where

Am ¥H§¥m¥→!¥î¥ii¥Hh÷ iraitHH"

ü¥¥÷H¥ü¥n¥¥H→.t.int#?---*tHnHI*tHtHaa"¥4t.im#.4whereD--xlaf-matie)-lt-xtUp-ik-qfiie)

= q'- m'y -24 - x) g.lp-ikt-U-xkptkf-ic-lg-U-xkp-kba-H-xtlp-kkltxllp.lkt -Maxxie

= la - t -xkpftp.ktpi/ltx-D-mtxeie=lIx4-xXm42p.ktmix-iie--lt-D- ie

with f- g- ltxllpxkl , t.xltxllm42p.ttm'x

⇒tri -*ftp.t#mnf*U!xft*Hi:xsq-1q:amI-ymktmtklp-tht-mt.itSince Diseveninl

,the Integral fd

"

¥ vanishes.

We onlyneedto amputé the integral

¥¥µ¥µ¥ .

We can emily see that it diverges .

We with régularise itusing dimensionnent régularisation :

HÉ "HÉHÉ Mff1,1 ¥. . If we chose d- 4- c- this boom

"÷!¥÷ Kitten tÊH¥Ë*ËDËH¥D

⇒ Était¥1

⇒ tri -¥+1!xËH¥DH¥¥h.HN#kHmItqF?trWe then have

itt-fiej.dxf-t-lnppalpttdqf.pk#.mf-.H-xK#Htmlftp.tulptetkk#TkYiMs---tieYulptAmulpK-YkK-*lk '),where

t.im?ii#mltEm*ofmrttYg-iHIm'EpatantBut 88F89. = -2098'

,OJJEH ' and hit. = -28"

⇒Ami 4h ftp.f#mImafjdxdydzdx-y-iz-Dffq*fsHEHtr#2mldgrkr ) -m'8D ,

where D= xlp-mt-yff-kt-mp-izlp.tk -g)'

=p -m'lxty )-2g .Llytzkxzptyktzlptkl'

=L g- ltxlk -zpf-H-xtktzpfzlp42pk.tk ')-m'ltz)= l' -24-xlzlpktzmt-zmhdzlp.tl -m'4-g)=p - 2zlp.kkt-x-h-myz-ztt-zt-l-2xzlp.fr/-mYtzK-l2-swhere t-q-t-xk-zpandt-mkt-zf-2xzlp.fr )Abu

, Iqtkltvqflg-KIRA-8kt- 29dg -Al - qatrtkgtv=L Llrtltxlkrtzprllltltxlktzp -AI - Lltltxlkxzpjk-klkth-xlktzpt.tv

= 24484ff -xlkrtzp.kz#xkD-lYr-mziK-2zltxlkopK-IzkpK--a-l' 8. tzpdzm-xkt-tk-mmjk-dzh-xkpttvtzkkpr.tn),

where we used pulpt-mulphkrttkt-O.fm#--OandfdYlgf--tdfdlltfg .

⇒ - Lg-HIGHmkg - k . ) -match -4dm - zpdzm-xkt-mazaktdzh-xHR.pk-

zkkpr-trml-dmftlrth-xtkrtzpr-KI-MYr-t-4dlltt.se- zprlzm -xkltdk-dmttlmj-mzktdzh-xllk.pl -mm

⇒AmHtmlqq.fi?mm+!dxdydzdx-y-iz-Dfd!*efspKt4dllVr- zprlzm-xk-idk.am )There are d intégrales to do :

+ lmiitmzktdzltxllkp) -mm

H¥m¥IËÉ and

¥.ph - Hd ) . Aqimwewilluse dimensionnel rogutarizatùntoamputetfisgae :""¥÷ËYË¥f¥¥→=ihËÊII "M¥1"

LHAIHEHD kf1 E-DU - EH#D-yiq.lt#H-4sllE-Ntt-EahH-n-Dte - t - t - ent#D

Ame ftp.fhf?mI.f!xdydzkxtztz-Dffet-t-lnp*Dt # fort zm-xk-i.am )- (miam zktdzh-xllk.pl -mm}

⇒ iMèËÏËËzàMk¥¥! #t.tk#*DtrttIzprlzm-xk-iak-aml-lniz4mzkt2zh-xllkp) - m'④ULPIÉLKIËLK ')

The amplitude ils isexartlythesameasitk.but.in Auvwechangepespandkxklandweinverttheorderoftheoperators.eu

iMEËËdxjjdzulptfff-H.tn#dJkttfzp:lzm-xk-i2kl-2m )-8dm mzk-azltxxpi.pt -m'Btptxkltmltrulpte

À4k¥41 ')

Finaly , thefourth amplitude à

iMy= -èàlpttmulpklkklk,where

J%¥Ît¥÷÷?gE¥¥¥¥¥.it?p..!p:ic.--bs!xdsdadwdx-r-s-tygq.N#.uheref-xllq-pit-ma-iietylqi-ma-iietzkg-kt-mh-iei-wllp.tk -glatiese- gaxpaw -2g.lk#klz+w)tpwIt2p.kw-mYx-iy-tzItie=Lg-pix-klz-iwt-pwj-lpix-iklz-iwkpwf-2p.tw -1mW -m'ltwttie

=L '- ftp.pixlz-iw-2pi.pwx-2koplztw/w-p'wt-i2p.kw-m'kw-D-tic-f-2k.pixlztwt-2kipwx-2k.ph-z-wlw-malpa-2wxwttie-lh-A-tict-g-kx-klz.tw)- pw andsédkpixlztwtpiopwx - dk.pltz-wlw-mhlp.at '

And Nmitdqfkitmltnlgtmttrlgtktmht.#Http ftp.kkm/tngHrH-KHg-AfutrlgtkHgtkHnqN+m'Ltutrlgtktttugtrlg -AHAHAHAH

= ta#d)tnllthltrlltctmltnlltbltr#HAahhh HAHAHAHA)

+m'Ltutdltctttnlltbhtlltatttrtmsttr } T,where

a-- pilxtttklztwtpw , bt-kxtklz.tw/tpwandc=fix+klztw-HtpwWe can use the identités t.HN?4gm,txfutNoT-- -2888'

,

HWAN -

_ Htptntvtottotrtntp) and kfr48.HN?-2K.tptokk:Nm--2ttlHdtdlttbAfullltaDtMLllHH8nlltHAHAHA ttrtrlltattulltbttdl#fille

+ Ntdll HAHAHA) tu#NI-m'#Athlète Mtn tkt#Dttmgµ}

We can now negbeteveryoddtouninl :

Nm-dtckltnl-lkbtd-ltvltna-ekbtnat-mf.ltHANNAH + l'tintant+ Kf8ne + tratnbttvltnlttrbtncxtvtpéttrtnactltnltrbquatr]-m'4kfr trbtnxtrtnatitmignr}

Also,ktnlt-2lnl-tnl-dlulxktnl-ttg.at-que -tatoué

⇒Nµ=2¥NHHHv - tvlbldlrltnttatrtnaltctrbtnatmttatnrlttetnbtv

+ étourneaux - ttrtnéttratnbtttrtnlxtrbtnlthtntttrtnac -tantrlitbfnatsf-m' llctrtnttrbtnttrtnalttmgµ}

⇒ tachette gmbllt-taktnbl-tbtrtnlt-tkbtnlt-ktd.at- Niska+mktnbtncatnktkatnbl-trbtnfttrtndc.to/8natrI-m' llctrtnttrbtnttrtnalttmgµ}

mdrrr= l' llctrtnrldgnr - trtnb - HAK -trbtn-trfuab-2etrbtua-2mtctrtmtrbtm-ktndl.hnmhctnbtrxcatttttvatnbttrbtnlttrtnd ¢ #trahi 4mgµThere are this d intégrale to compute :

ËüË⇒iËË and H¥i¥*iË⇒ Am = bgjdxdydzdwdx-ytzttfffq.it#

˧ËqÜËw KHANNA - blah -trbtmktnatmurrtmtrbrmkrnah-dckbtna.imlctnbtrtcatnttttvatrbttrbtncttrtud ¢ +ÈÙaK) -t 2mg,§ ,

where a-- k' lxtttklztwl,bt-kxtklz.tw/andc=k'x-tklz-w-HandA=2k.pixlztwftp.kw-makw-I)

.

-

There are stitl 3 diagramme that we could considére : g- k'

Iv ¥ and° À

P P'

We can easily see that the first 2 diagramsvanish because they involve theIntegral over the loop

¥¥î¥m¥#¥¥¥÷o

Thethirddiagramhasan amplitudeini - tietàtptuulpttiecilkktlfdtr-EHmk-k-mblitlhttlqf-maxitklq.AT-mtticillgtk - kHmt-Â

We can compute the trace on Mathematica :

trklg.im/8slg-A-imHdqfk-kHmD--4maLkxgm-kpg.rtkrg.s-2kdgmb-2qiltkpg.idk.gr- kkrg.pt#pi.gmtkrg.ptkygpr-kYg.p-kYg.r-2kIa. -kfgxrtkpd.nl/g.p32=4maLkxgm-kpg.rtkrg.p-2kdgmbtdqaskpg.r-kagpr-krgxptk.gs , }= 2lqa-2makkpgar-kagsr-krgxptk.gov }

,

whereweincludedonlytheeventerms.mg.

⇒ IMEC"[dxdydzdlxxytz - t)¥¥,.IQ?2maKkp8ar-kagsr-krgxrtkagsu3 ,

where Ü=iêûÆuHËÎlmÉMk ') andA-

xlqi-ma-itt-yllq-KT-ma.ie/-zLlgtk-kYt-mHitJ--qt-m' -2g.LK#z)-kzItzlpi-2k.pitk'Yxyk'xie=lg-k4-xt-kz)'

- tklt-xt-kzf-2zk.fi -m' + ie

= l'+2ltxtzkk-2zk.pi-maxie-lt-2zxlk.fi/-ma-iie=l2-DtiE,with

f- q-pill-xt-kzandt-2xzk.kt-mtgbaausepe.pe andwecan negbtt.hetY.IE⇒ IMEC"[dxdydzdlxxytz - 1)ftp.y.2lt-2-ma/Lkp8ar-kagsr-krgxrtk'agsu3l'-Stitt

A-gain ,

we have d intégrale to do :

tt:#¥⇒ü÷à and

¥üË⇒÷Æ÷¥÷ËïEH¥¥If we set D= 4-E , we get

ËËÈËD¥ÏË¥. ¥44 - Einstein#

GÈLE - t - abstinent - atf-Ë¥ -at- t -hpËB⇒ iM5-iqpltf.dxdydzdlxxytz-DLkpg.r-k.gr - krgxp-tk.gov }

ME -A- tentant ff7t -x

= i ["Lkpg. . - kagm-krgxmk.gs . } §xfd.zfte-H-tlnlfa.tt ff1]Frit

In this analyses , we didn't include correction to externat legs since theyare not relevant in a weather.my expriment . These corrections have an effect on aspecific partite state . for exemple, they can be used to go from the photon +electronstate in a free theory Ip ,

b. to the équivalent state in the interactive theory lp.kz .However

,they don't have any effect on the seattoimg amplitude in .

Also,we can see that including these corrections in in would result in divergences .

Au easy way to see that is with the following diaporama :

yk

7û_The amplitude would contraire the propagation for the Internal photon of momentumk.But the externat photon is on shell so k:O .

The propagation intrus

àf÷ = - i.qc.ca?Therefoa , the seattuing amplitude would diverge .

The relevant diagramme for Compton seottering with soft photon emission are

ËËï ②Ë: %Fa.ir⑥¥Ëp .

⑤

¥1; o#n

q k'

The amplitude for these diagramme are

intitulait;¥:!÷hifeng.mn?!aulptEMNe*Htiklithi-tietxp¥g¥f¥n:aiftp.ifeng#.aulptEtgK-*tkk-Tklini-tietulNkiffe;¥i?÷ip¥¥î÷.int#tiKtkk*Td

1-gain , the lost amplitudes can be obtained from the first 3 byintoahangimgkx.fi .Wewill also have to tohu the limit q- O at the end of the calculateur aimait représentera soft photon .

We can first see that

lk-tpt-m-k-2p.ktpa-mt-dp.hr , and similarly Hitptn ! tkplqxplm? 2p.gandlq.pt?m:dgp!Also

,lkttptmltxulpt ETkt-ktk-p.tk#tAHmtxIulp-KTkHapx-tdKtmHmHulptÉtkl

=ppa - t.kdulpk.TN

⇒ im-ieulptl-trkp.plNEHA ËlgklkfiéàlpHinault EMMÈNERAi Ma and i Mrs can be obtained from i M

, by intouhanging g- k' and geek respectives :

⇒ in = - iéàlp ) ftp.n-gtnltrlapa-txk/lulptE*YgtE*TpitElkf-ieiilp#i'de# *

Hop ËYKIETKI'

IT

in - ieulptl.kkpg.pk#ptE*MpiteTHETot--ieiilp'Hinault ÉTAIENTËlglin éàlptapageNETAIËLIË = - ièùlptttmaulptîth lot ki

in: ieiilpttdzg.nu#.pxkIulptE*MgteHt-Elkf- ièùlpttiiiiulpt Étroite Ykt ki

in: ieulptlprkpzg.pulpt.MN Était Ëttrièùlp#"maulp tkt Ëtkt Ëlgl

The whale amplitude is nowin = - ie bin ÈHKFCYKIÉTg) àlp#

"

mmAIm-AIK-AY.n-AYm-tih.HNquo

So the équation 5.81 now bécanes

4!Mt- éasmbjçr. ËLKYEUYÉUËtktËlgtÉtaitx-ulpth-I.nt-Im-AI.tt?m-AYm-tHmlulplulpXtiIm- t'II. AÏ TÆ! -Ass! -ÆIt%µ= -¥bjçç grog"j Tr ftp.imtttfxr-AIm-AI?xtAItn-AHm-tIhxKpHm)× ftp.t-t?os-A?otAI'm-AIhs-tYoxD ,

because § ÊTRElp) → - quand t'¥8 - TAILThis gennady diverges for q→ 0 .

But,we can expert that divergence

ts exaetly cancel the infrared divergence from the Compton seatteringwithoutthe soft photon . So , the sum of the 2 squared amplitude rtayr finit .

where D= xtp-kt-mii-I-ylqi-mhiet-zkp-kt-matic-T.tw/lptk-g)Iie ) ¢= tp-ktlx-iz-iwi-galy-wt-dwg.lp.tk) -m'txxyiztietxtytzxw)= tttwlfg

'

-2¥qq.lp-ikhyfwjlp-iktftlptktfty-yy.IM - malt -wttit

= lytwlfq - yfzwylptktjilpttapkttilftp.ffqt-f-mttwtic-ly-iwllt-bxie,where l = g- ¥w lptk ) and

A- m'4- wt-lmttp.tl µ42yqj¥wfqw-yw#= malt -wtlmiédpoklyfgtdw - toutt=mah -

wtlmhthtkttffw.pt#w-Dmpwauseptk--pt-ik'Nm--ftp.ktmlkl#mlttptkHmlK=I2Nntkil-lpHkAtn-mflklqltmltlxp.tk/-HpHkA-imHButk'nE*lk't-k. Elkto so we can ngbetknandkr :

placeuse ta tut? -2fr and fr814

Nm = ftp.n-lm-ik/tntmtnIt2gt4mlfdpr-Klm-AHmtr]= -tapir-KOH¥.gl#Ktamyapr-trklSineeDiseveninl

,the toms in Nur linear inlvanish :

Nur = -2wfpi-kltkptkttdprkktitmkp.nl#ktmltnKapr -km)Wtf

= -2¥,

12pm'#8Mfr lm# - 21pmkdkh-trlp.tk#IHmldpiimtn-dlpmknttfulpHADHAHA)

=-2¥,

lapin-n' tnkdprmttddp.tk/pDt8mprl.Hputkn-kit-t2mtn-HpmkdNnkd y- 4mlklpmku-kittmtn-dpmkrDK-ttxpr.hr)-KARABA

=¥gKpà -HAHAHAmttrldp.tk/mDt8mpukmtnttnmkt4mYnkkl-4mtnkprkttrkp.tk/pD

vu

= -2µg ftp.I-mtn-2lpmknttnlptk/Dldprmttrlap.k-KmDt- lbmaprtn-

4mattrkl-4mtukkp.fr-Km )

= -¥wy # kiimtullaprmttdapuk- kmD-tdp.mfulm-ikt-tnkpnkrttrlp.tk/Ildp.k-km)}tlbmptn -8matnkk-8mlpkltnk-fqzlapmtmzmk.EE-m'ftp.H-tdpmte-tfrmI-ttprlpkltn-fmpnf-xpkltntrlm+montre - ftp.AI-tlbmaptn-8matnkkttmlpkltntr

= -w¥y{4prmtntdmlp.kltuk-2makkklt-4pdp.br/tu-2lpkltntrklftlbnipttm .tk#8mlpkltnk⇒¥¥ÎË=Nm¥**àsÏËËËme"ÜFia¥⇒Ami %dadydzdwdx-y-z-iw-ttfqgq.tt#--yiqf.dxdgdzdw#gxztw-Daffy.¥*¥Ëüü¥¥ .

Since the intégrant enlydepadonyandw, thexintegralcanbedonedûutty :

tu !Ë4z7û¥ X⇒ in-tietulpttmulpk-tkk-HH.ci#.f.dyf!:4jfz! àtrwmutpklkt tilwith Nm

= -w¥y {4pmVu -12mLp.kltuk-2makkklt-4pdp.kltu-2lpklkhkgtlbmtp.tn - 8makkk-smlp.dkk

We can compute the second amplitude with a similor method :

IME -éùlpttnrulp) EURÊKA) , where

ÆÆ¥ïHfH?!?¥mm÷.MIL?mn7i?ip**i-bs!.dsdzdwdxx-z-w-¥¥*f÷D= xllp-kt-mfiettylaf-mi-iettzflg-kt-maxiet.tw/lptk-qttiet-pah-iwttfitx-z-twt-ajly-z-wh-2p.ktx.tw)-2g . klztwftp.gw-m#y-iwt-ic=ttHfgt-zq;lklz-iwttpwI+fklzq)) -tktq.ro#wI-lmt-ApokKxtwt-mH-ztiie=ttHfq-kfzty))

'-

ftp.kwlzjyftma#-mawx2p.khHwttit=ltxK-stit,with t-g-pklzf.IM#andA-2pkwlyq-mwI-niw-apkhaut

Nm =ftp.tkkm/8xlqftmltrlg-kHmlF=8nlp+kHml8xLgtrlq-A) + mtdef-kkmgk.intKIT

=ftp.IH-mttdlf-HKA-4mkg-krttmND ×We now set fr-0 and fin -- O since kEUR1 -_ KIÊHK :O .term linear in q :

Nmitnlptkltmtt HA -AHµ 8mg -2mW

= ftp.mkitlpukltmmtht#-tHl2gr-qktt8mgr-dmNr ]= tapé -1mAHmmHttp#-HAK Kappa 8mg-2mW

tapir -Hinton f4µkb¥¥rwÜHq¥w-kf44kb¥¥Ü*ftp.w#wDkt8mflrtku-zwwf2mNJt - x

As befoa , we can nqgbeteroyh-nearterm.nl and replace lulu by tgml ?

Nrrtapi Huntf-44kt-4frdz¥¥Üklz¥w)+Yithzak -

kfbY.tn#wDkt8mfRdHEwf2mM--kpnHnkHf-eh -4µdq¥¥rxµ¥¥-2Mf ÀHfkw¥¥¥¥Üm§=ppierre petit Yap.k¥ç¥w -map - a¥kkrtmtYk¥¥w)µq¥ - am)}There are now 2 intégrale to do (which are both convergent) :

HËn¥Ëa¥ and

tËuxÏ¥xH¥mÊÏËÏà ×⇒Aµ =6¥dydzdwdx-y-ztw-ttfffy.fikpi-NAY.xdydzdwdx-wz-iw.tt#.sffqfmt-ftp.kwty.YY.int#-mYk-aq*ar-km-yk¥¥¥xµq¥x - am)}ritals!xËËü¥É¥ ftp.hw#::mw--mYrr-aq*ar-nm

-yk¥¥¥Yµw¥¥rI - am)}And again , iME-eiulph-mulptelkk.HNWe can see that we can oftain the third amplitude fromthe second one orlyby Interchange.mg képi and pxp ' in Am and by invertines the ondes of the operationin the numéroter

.

⇒ ils = -êùlptturulptÉTIËUII , with

ÆËËË˥ɥ¥ftp.tianya.myrr-zyiapi-r.mx4k¥,¥iYµ¥ - am Hum

and E- tnpitiwfqtywmwtniw -2pAhaut

Finaly , the fourth amplitude à

i My = -èàlpttmulpklkt ÊTKY,where