Advanced Quantum Field Theory (Version of November 2015) Jorge Crispim Romão

Transkript

1 Advanced Quantum Feld Teory (Verson of November 015) Jorge Crsm Romão Pyscs Deartment 015

2 Aendx D Feynman Rules for te Standard Model D.1 Introducton To do actual calculatons t s very mortant to ave all te Feynman rules wt consstent conventons. In ts Aendx we wll gve te comlete Feynman rules for te Standard Model n te general R ξ gauge. D. Te Standard Model One of te most dffcult roblems n avng a consstent set of of Feynman rules are te conventons. We gve ere tose tat are mortant for buldng te SM. We wll searate tem by gauge grou. D..1 Gauge Grou SU(3) c Here te mortant conventons are for te feld strengts and te covarant dervatves. We ave G a µν = µ G a ν ν G a µ + gf abc G b µg c ν, a = 1,..., 8 (D.1) were f abc are te grou structure constants, satsfyng [ T a, T b] = f abc T c (D.) and T a are te generators of te grou. Te covarant dervatve of a (quark) feld q n some reresentaton T a of te gauge grou s gven by D µ q = ( µ g G a µt a) q (D.3) In QCD te quarks are n te fundamental reresentaton and T a = λ a / were λ a are te Gell-Mann matrces. A gauge transformaton s gven by a matrx U = e T a α a (D.4) 375

3 376 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL and te felds transform as q e T a α a q δq = T a α a q G a µt a UG a µt a U 1 g µuu 1 δg a µ = 1 g µα a + f abc α b G c µ (D.5) were te second column s for nfntesmal transformatons. Wt tese defntons one can verfy tat te covarant dervatve transforms lke te feld tself, δ(d µ q) = T a α a (D µ q) (D.6) ensurng te gauge nvarance of te Lagrangan. D.. Gauge Grou SU() L Ts s smlar to te revous case. We ave W a µν = µ W a ν ν W a µ + gǫ abc W b µw c ν, a = 1,..., 3 (D.7) were, for te fundamental reresentaton of SU() L we ave T a = σ a / and ǫ abc s te comletely ant-symmetrc tensor n 3 dmensons. Te covarant dervatve for any feld ψ L transformng non-trvally under ts grou s, D µ ψ L = ( µ g W a µ T a) ψ L (D.8) D..3 Gauge Grou U(1) Y In ts case te grou s abelan and we ave B µν = µ B ν ν B µ (D.9) wt te covarant dervatve gven by D µ ψ R = ( µ + g Y B µ ) ψr (D.10) were Y s te yercarge of te feld. Notce te dfferent sgn conventon between Eq. (D.8) and Eq. (D.9). Ts s to ave te usual defnton 1 Q = T 3 + Y. (D.1) It s useful to wrte te covarant dervatve n terms of te mass egenstates A µ and. Tese are defned by te relatons, { { W 3 µ = cos θ W A µ sn θ W Zµ = Wµ 3, cos θ W + B µ sn θ W B µ = sn θ W + A µ cos θ W A µ = Wµ 3 sn θ. (D.13) W + B µ cos θ W 1 For ts to be consstent one must also ave, under yercarge transformatons, for a feld of yercarge Y, ψ = e +Y α Y ψ, B µ = B µ 1 g µαy. (D.11) Ts s mortant wen fndng te gost nteractons. It would ave been ossble to ave a mnus sgn n Eq. (D.10), wt a defnton θ W θ W + π. Ts would also mean reversng te sgn n te exonent of te yercarge transformaton n Eq. (D.11) mantanng te smlarty wt Eq. (D.5).

4 D.. THE STANDARD MODEL 377 Feld l L l R ν L u L d L u R d R φ + φ 0 T Y Q Table D.1: Values of T f 3, Q and Y for te SM artcles. For a feld ψ L, wt yercarge Y, we get, [ D µ ψ L = µ g ( τ + W µ + + τ W ) g µ τ 3W 3 µ + g Y B µ ] ψ L (D.14) [ = µ g ( τ + W µ + + τ W ) g ( τ3 ) ] µ + e Q Aµ cos θ W Q sn θ W ψ L were, as usual, τ ± = (τ 1 ± τ )/ and te carge oerator s defned by 1 Q = + Y Y, (D.15) and we ave used te relatons, e = g sn θ W = g cos θ W, (D.16) and te usual defnton, For a snglet of SU() L, ψ R we ave, = W 1 µ W µ. (D.17) D µ ψ R = [ µ + g ] Y B µ ψr [ ] g = µ + e Q A µ + Q sn θ W ψ R. cos θ W (D.18) We collect n Table D.1 te quantum number of te SM artcles. D..4 Te Gauge Feld Lagrangan For comleteness we wrte te gauge feld Lagrangan. We ave L gauge = 1 4 Ga µν Gaµν 1 4 W a µν W aµν 1 4 B µνb µν (D.19) were te feld strengts are gven n Eqs. (D.1), and (D.9).

5 378 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL D..5 Te Fermon Felds Lagrangan Here we gve te knetc art and gauge nteracton, leavng te Yukawa nteracton for a next secton. We ave L Fermon = qγ µ D µ q + ψ L γ µ D µ ψ L + ψ R γ µ D µ ψ R (D.0) ψ L ψ R quarks were te covarant dervatves are obtaned wt te rules n Eqs. (D.3), (D.14) and (D.18). D..6 Te Hggs Lagrangan In te SM we use an Hggs doublet wt te followng assgnments, φ + Φ = v + H + ϕ Z (D.1) Te yercarge of ts doublet s 1/ and terefore te covarant dervatve reads [ D µ Φ = µ g ] ( τ + W µ + + τ Wµ ) g τ 3Wµ 3 + g B µ Φ (D.) [ = µ g ( τ + W µ + τ W ) µ + e Q Aµ Te Hggs Lagrangan s ten L Hggs = (D µ Φ) D µ Φ + µ Φ Φ λ If we exand ts Lagrangan we fnd te followng terms g cos θ W ( τ3 Q sn θ W ) ] Φ ( Φ Φ) (D.3) L Hggs = g v W 3 µw µ g v B µ B µ gg v W 3 µb µ g v W + µ W µ + 1 v ( µ g B µ + gwµ 3 ) + gvw µ µ gvw µ + µ (D.4) Te frst tree terms gve, after dagonalzaton, a massless feld, te oton, and a massve one, te Z, wt te relatons gven n Eq. (D.13), wle te fourt gves te mass to te carged W µ ± boson. Usng Eq. (D.13) we get, L Hggs = + 1 M Z + M W W + µ W µ + M Z µ + M W ( W µ µ W + µ µ ) (D.5) were M W = 1 gv, M 1 1 Z = cos θ W gv = 1 M W (D.6) cos θ W By lookng at Eq. (D.5) we realze tat besdes fndng a realstc sectra for te gauge bosons, we also got a roblem. In fact te terms n te last lne are quadratc n te felds and comlcate te defnton of te roagators. We now see ow one can use te needed gauge fxng to solve also ts roblem.

6 D.. THE STANDARD MODEL 379 D..7 Te Yukawa Lagrangan Now we ave to sell out te nteracton between te fermons and te Hggs doublet tat after sontaneous symmetry breakng gves masses to te elementary fermons. We ave, L Yukawa = Y l L Φ l R Y d Q Φ d R Y u Q Φ u R +.c. (D.7) were sum s mled over generatons, L (Q) are te leton (quark) doublets and, v + H Φ = σ Φ = (D.8) D..8 Te Gauge Fxng As t s well known, we ave to gauge fx te gauge art of te Lagrangan to be able to defne te roagators. We wll use a generalzaton of te class of Lorenz gauges, te so-called R ξ gauges. Wt ts coce te gauge fxng Lagrangan reads L GF = 1 ξ F G 1 ξ F A 1 ξ F Z 1 ξ F F + (D.9) were F a G = µ G a µ, F A = µ A µ, F Z = µ ξm Z F + = µ W + µ ξm W, F = µ W µ + ξm W (D.30) One can easly verfy tat wt tese defntons we cancel te quadratc terms n Eq. (D.5). D..9 Te Gost Lagrangan Te last ece n wrtng te SM Lagrangan s te gost Lagrangan. As t s well known, ts s gven by te Fadeev-Poov rescrton, L Gost = 4 =1 + [ ] (δf + ) (δf + ) (δf Z ) (δf A ) c + α + c α + c Z α + c A α c 8 a,b=1 ω a (δf a G ) β b ω b (D.31) were we ave denoted by ω a te gosts assocated wt te SU(3) c transformatons defned by, U = e T a β a, a = 1,..., 8 (D.3) and by, c A, c Z te electroweak gosts assocated wt te gauge transformatons, U = e T a α a, a = 1,..., 3, U = e Y α4 (D.33)

7 380 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL For comleteness we wrte ere te gauge transformatons of te gauge fxng terms needed to fnd te Lagrangan n Eq. (D.31). It s convenent to redefne te arameters as α ± = α1 α α Z =α 3 cos θ W + α 4 sn θ W α A = α 3 sn θ W + α 4 cos θ W (D.34) We ten get δfg a = µβ a + g s f abc β b G c µ δf A = µ α A δf Z = µ (δ ) M Z δ δf + = µ (δw µ + ) M W δ δf = µ (δw µ ) + M W δ (D.35) Usng te exlct form of te gauge transformatons we can fnally fnd te mssng eces, ( δ = µ α Z + g cos θ W W + µ α Wµ α +) (D.36) δw µ + = µ α + + g [ α + ( cos θ W A µ sn θ w ) (α Z cos θ w α A sn θ W ) W µ + ] δwµ = µ α g [ α ( cos θ W A µ sn θ w ) (α Z cos θ w α A sn θ W ) Wµ ] and δ = 1 g ( α + α + ) + g cos θ W α Z (v + H) δ = g (v + H + )α + g cos θ W α Z + e α A cos θ W δ = g (v + H )α + g cos θ W α Z e α A cos θ W (D.37) D..10 Te Comlete SM Lagrangan Fnally te comlete Lagrangan for te Standard Model s obtaned uttng togeter all te eces. We ave, L SM = L gauge + L Fermon + L Hggs + L Yukawa + L GF + L Gost (D.38) were te dfferent terms were gven n Eqs. (D.19), (D.0), (D.3), (D.7), (D.9), (D.31).

8 D.3. THE FEYNMAN RULES FOR QCD 381 D.3 Te Feynman Rules for QCD We gve searately te Feynman Rules for QCD and te electroweak art of te Standard Model. D.3.1 Proagators g µ, a ν, b δ ab [ gµν k + ǫ (1 ξ) k ] µk ν (k ) (D.39) a ω b δ ab k + ǫ (D.40) D.3. Trle Gauge Interactons ρ, c 3 1 µ, a ν, b gf abc [ g µν ( 1 ) ρ + g νρ ( 3 ) µ +g ρµ ( 3 1 ) ν ] = 0 (D.41) D.3.3 Quartc Gauge Interactons ) Vértce quártco dos bosões de gauge σ, d 4 ρ, c 3 g [ f eab f ecd (g µρ g νσ g µσ g νρ ) 1 µ, a ν, b +f eac f edb (g µσ g ρν g µν g ρσ ) ] +f ead f ebc (g µν g ρσ g µρ g νσ ) (D.4) = 0 D.3.4 Fermon Gauge Interactons µ, a β, 3 1 α, j g(γ µ ) βα T a j (D.43)

9 38 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL D.3.5 Gost Interactons µ, c 3 a 1 b g C abc µ = 0 (D.44) D.4 Te Feynman Rules for te Electroweak Teory D.4.1 Proagators γ µ ν [ gµν k + ǫ (1 ξ) k ] µk ν (k ) (D.45) W µ ν g µν k M W + ǫ (D.46) Z µ ν g µν k M Z + ǫ (D.47) (/ + m f ) m f + ǫ (D.48) M + ǫ (D.49) ξm Z + ǫ (D.50) ϕ ± ξm W + ǫ (D.51)

10 D.4. THE FEYNMAN RULES FOR THE ELECTROWEAK THEORY 383 D.4. Trle Gauge Interactons W α k q A µ e [g αβ ( k) µ + g βµ (k q) α + g µα (q ) β ] (D.5) W + β W α k q g cos θ W [g αβ ( k) µ + g βµ (k q) α + g µα (q ) β ] (D.53) W + β D.4.3 Quartc Gauge Interactons W + α W β e [g αβ g µµ g αµ g βν g αν g βµ ] (D.54) A µ A ν W + α W β g cos θ W [g αβ g µν g αµ g βν g αν g βµ ] (D.55) Z ν W + α W β eg cos θ W [g αβ g µν g αµ g βν g αν g βµ ] (D.56) A µ Z ν W + α W β g [g αµ g βν g αβ g µν g αν g βµ ] (D.57) W + µ W ν

11 384 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL D.4.4 Carged Current Interacton ψ u,d ψ d,u g γ µ 1 γ 5 (D.58) D.4.5 Neutral Current Interacton were ψ f ψ f g ( ) A µ γ µ g f V cos θ gf A γ 5 W ψ f ψ f eq f γ µ (D.59) g f V = 1 T 3 f Q f sn θ W, g f A = 1 T 3 f. (D.60) D.4.6 Fermon-Hggs and Fermon-Goldstone Interactons f f g m f m W (D.61) f f g T 3 f m f m W γ 5 (D.6) ψ d,u ϕ g ( mu P R,L m ) d P L,R m W m W (D.63) ψ u,d D.4.7 Trle Hggs-Gauge and Goldstone-Gauge Interactons + A µ e ( + ) µ (D.64)

12 D.4. THE FEYNMAN RULES FOR THE ELECTROWEAK THEORY g cos θ W cos θ W ( + ) µ (D.65) g (k ) µ (D.66) ϕ k g (k ) µ (D.67) ϕ k g cos θ (k ) µ (D.68) k ϕ A µ e m W g µν (D.69) W ν ± ϕ g m Z sn θ W g µν (D.70) W ± ν g m W g µν (D.71) W ν

13 386 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL g cos θ W m Z g µν (D.7) Z ν D.4.8 Quartc Hggs-Gauge and Goldstone-Gauge Interactons g g µν (D.73) W ν g g µν (D.74) W ν cos g µν θ W g (D.75) Z ν cos g µν θ W g (D.76) Z ν

14 D.4. THE FEYNMAN RULES FOR THE ELECTROWEAK THEORY 387 A µ e g µν (D.77) A ν ( ) g cos θw g µν (D.78) cos θ W Z ν W + µ g g µν (D.79) W ν ϕ g sn θ W cos θ W g µν (D.80) Z ν ϕ ± W µ g sn θ W cos θ W g µν (D.81) Z ν ϕ ± W µ eg g µν (D.8) A ν

15 388 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL ϕ ± 1 eg g µν (D.83) A ν eg cos θ W cos θ W g µν (D.84) A ν D.4.9 Trle Hggs and Goldstone Interactons g m m W (D.85) 3 g m m W (D.86) g m m W (D.87)

16 D.4. THE FEYNMAN RULES FOR THE ELECTROWEAK THEORY 389 D.4.10 Quartc Hggs and Goldstone Interactons g m m W (D.88) 4 g m m W (D.89) 4 g m m W (D.90) 3 4 g m m W (D.91) 4 g m m W (D.9)

17 390 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL 3 4 g m m W (D.93) D.4.11 Gost Proagators c A k + ǫ (D.94) k ξm W + ǫ (D.95) c Z k ξm Z + ǫ (D.96) D.4.1 Gost Gauge Interactons A µ e µ (D.97) ±g cos θ W µ (D.98) g cos θ W µ (D.99) c Z

18 D.4. THE FEYNMAN RULES FOR THE ELECTROWEAK THEORY 391 ± e µ (D.100) c A c Z W µ g cos θ W µ (D.101) c A W µ ± e µ (D.10) D.4.13 Gost Hggs and Gost Goldstone Interactons ± g ξ m W (D.103) g ξ m W (D.104) c Z g ξ m Z (D.105) cos θ W c Z

19 39 APPENDIX D. FEYNMAN RULES FOR THE STANDARD MODEL c Z ϕ g ξ m Z (D.106) ϕ ± g cos θ W cos θ W ξ m W (D.107) c Z ϕ ± e ξ m W (D.108) c A