CALCULATIONS IN QFT : TOOLSFROM £ TO OBSERVABLES
④ sisters {¥¥wsif¥④EftFeymarIrules
Mfiice -f) L !Mfil
"
⑤ ↳ {octet )µ EXAMPLES
⑤ QED : E'e-→ pipe- ( em) - DinohgyE'e- Tete
-
- Polarizations
eraser
⑦④ : eteyutpr cetectroweak)
µ → eieyuIt → FI , VV, . . .
FEYNMAN RULES ( to compute amplitudes Mfi)
① Draw all diagramsconnected topologies , amputated)
at the given order in Perturbation Theory
e.g .QED : µ#e-→ E'e- )
From ?.
c propagators) ¥m (vertices)Mr
- at leading order ( tree level) , Old) (x -- Eff)e- e- - e
-
e-
** ÷:*t t•
te et e e
- next order ( one hoop) , ①(L2 )x
¥←E HEE YEE
x.W.IT#7Itonne • •
A•s a
s i
r,
- etc.
FEYNMAN RULES ( to compute amplitudes Mfi)
② Assign momenta to external linesand L internal moments to the L 60ps( preserving momentum conservation in every vertex )ers .
*g±y¥B .EE"p 7
b' Pg w Pu2 9-1
9-f- PitPEBt Py 9-2=9-Pf Ph- PzK PHK
÷o÷÷:÷÷:±±KK - Pztk
③ Assign propagators to internal lines :
scalar :-
Ee- - =
(Spino) pZMIP
fermion →
( spiny ):X = pain = ICHI
.
PZm2p
vector boson → PµPu( spine )
:
Hmu ; pizml-
Spurt H-E)Enz)
gage boson ( Rg gorge)
FEYNMAN RULES ( to compute amplitudes Mfi)
④ Assign the following factors to every vertexaccording to :
(a) The coupling constant inIL
f!! Gu→- ippe ( incoming momentum ) } Fanpies innext pages
A symmetry factor if there areidentical particles
e.g . ZZH → x2
ZZHH → 82×2--84+13 → 83 ! = X6
+14 → x 4 ! = x 24
check for instance#,2 h
is ⇒ - i tffhex 24 = -si MH - ie# V= T2 =p 4MWSF-
t
-
Mtf 4 vZ±g§=4MTsI£7 foil p ez
e -- gsw
⑤ Forevery internal momentum K not fixed by
momentum conservation ( hoops) introduce & factor :
J fI¥, ( integrate , and regularize it needed)
FEYNMAN RULES ( to compute amplitudes Mfi)
⑥ The amplitude ill of each diagram is obtained bytaking external lines on-shell ( p? =Mf ) and
adding the corresponding spinors ( fermions) and
polarization vectors ( vector bosons) as follows:
$ momentum , spin=. = up
,
s ) momentum pdoittimp
P y f
•-5 = Icp,s)n• Epdp, 'd)
P p→→ *
← = vcp, s) •run Epdp, 2)II. = Icp, s)
Creed diagrams starting by outgoing fermion lines)p2.
;m = Topp Tucpz) . . .Ipy
FEYNMAN RULES ( to compute amplitudes Mfi)
⑦ Multiply the contribution of every diagram by :
(a) A factor C- t) ifor diagrams differing by anodd permutation of identical fermions
e.g . etee :
② ③ e e ② ③ e
eye - Yi✓ r
<⑥s
e '
① ④ e e ① ④ e
①②③④ → ①④③②i
(b) A factor C-r) for every fermion loop
e.g .
mm x C- t)
(c) A symmetry factor Yg where S is the number
of permutations of internal lines and vertices
that leave the diagram invariant when external lines
are fixed
7.oook.FII
e.g .i t x I
÷!y 2
SM FEYNMAN RUES (id , Gut - ipu incoming)
VERTICES LFFV#Lo - eoyfij4.IM/fApe ( QED )
£j¥nEµ= - ieQ8k8ijfi f
Lo e Mgr- gars)4Vµ ( more general)i j
8,7WI!¥.mµ=ie8Mgr- gars)
(e --gsw )( Vµ=Aµ⇒gv= - QfsijiSIE 8am(Vµ=Wµ⇒q=g*= Vij ,ti=uiifj=dj )
W
( Yu -Zr ⇒gr-TEZEEfdij.SE#Iwansij)(8r-gAr5)=8cR-Spen
⇐ s:¥¥E¥: ⇒ FEET's.[12=24-857 , REICHE)
SM FEYNMAN RULES (id , Gut - ipu incoming)
VERTICES (FFS)Is eticgsgptstuljo
44ft§ ! ! - - - - = iecgs-gptd-ielqptcp.tk)T
q=gst8p
VERTICES [511µg92=85812
La EKG II.Iii.
→ giauque; r
E.Type = iekgiuuVI
detail,lot } , V. V'Edt,Z ,
W't}
VERTICES [ 549154%4 Rip incoming ! !
£2 - ie Glo Eino vie1 2-
lo k%9u§) - (9uf.kz
"
inner = ieGcp - PyuT T 7 2
19 R is⇒ -ie Gfipztipglpn
SM FEYNMAN RULES (id , Gut - ipu incoming)
VERTICES flfuckillflkrdvplks)]
so - ie-cwmwuvu-wtuuwrw-wfuw.vn,+
wiuu-oiiwu-owtgvtkomvo-ovM.VE#Z3W Enkete v V
thump =ieJLgpwlkikhptguplkz-kzyutgpeplkikzDEz.KZ/EzK3w- u
←
is e-stick'¥h-Keitaroticket - k¥HEzpE3otickets -k¥5)EyE2u ]
=
ieJL-lkz-CXHEI-lkzEDHED-lkfz.VE#3)-CkyEz)CEf2)+ (Kofi) Ez)- CkzEz)fE3G))
= Elite ieJ L- k2µ8vptk2p9µo+ Kq8µf - k1p8µ0+ k3µ8vp - k3u8µp ]
= Effigies [quulkz-kdptguplkz-kzpiquplkjkgH.TV
SM FEYNMAN RUES (id , Gut - ipu incoming)
VERTICES [VµCkiNulkDYlk3)Vlk4))s- e'cfzwiiwivuv"-www.vkwiivirw.ru)+
W OV'!Yi÷%÷;4 qiea.kqaise-sresuo-gu.ae)
W V
is ⇒ iedgf2CGEzEzEy) -HEDGED- Cefn)HED)=EiEzEzeEieE[2guu9pj8µegy-Quand
9
VERTICES LSSYUV,]
iii.of"- ieczsiuu
i V
VERTICES (SSS]'
'a =
i.
'• - -- . iecz
VERTICES [SSSS]'
n
.
• - ie's,
pages
• CROSS- SECTION & DECAY WIDTH 135 - 138
• EXAMPLE : E'e-→ tipi (QED at tree level) 139-152
• COMMENTS ( signs, identical particles , polarizations) 153 - 161
• PHENOMENOLOGY- observables & experiments 74-85
- precision 86 - go- fits 91 - 95
- NEUTRINOS 96 - 128
• EXERCISES :
[3) T ( µ→eJe4u) muon lifetime
(4) 6 Ceteesffl Cte) EWSM I ¥=all -
IE#arial ocean)an
(5) Z Pole observables- T (Z→FI, = THI ) Tz ⇒ Z lifetime- TCZ→all) = Tzu Nu ( number of light active neutrinos)- Forward-backward asymmetry AFB
Tete)Tchad)- Z lineshape : shade 121T-
G) Higgs partial widths←check ! METE
I T CHOtf ) ← fI=b5 dominant → H' lifetime
- FYI,÷→w¥¥, } assuming Mito > 2Mz
