(( )( )) = = S p S p = S p p m ( )

Transcription

1 36 Chapt 3. Rnoalization Toolit Poof of th oiginal Wad idntity o w nd O p Σ i β = idβ γ is p γ d p p π π π p p S p = id i d = id i S p S p d π β γ γ γ i β i β β γ γ β γ γ γ p = id is p is p d = Λ p, p. β S p = p 3-0 = = S p S p = S p p 0 p p p = δ S p S p = p S p p = S p S p γ. p p p S p S p S p S p S p S p p Taing p p -, 3- bco γ = γ = p S p p γ Thn, with 3-3 ud in th cond lin blow, w hav End of poof 3.. Th Wad Idntiti = S p S p Wad na i aociatd with an additional t of idntiti, which play a y ol in noalization, and alo in catting calculation. Thy a calld Wad idntiti, but to ditinguih th fo 3-9, w calld th ali lation th oiginal Wad idntity. W div th Wad idntiti in thi ction, but bfo that, w nd a bit of bacgound infoation. Gaug Invaianc Man Aplitud Invaianc Local gaug invaianc an ou Lagangian L i ytic in fo und th tanfoation i v ψ ψ = ψ A A = A, 3-5 wh th nuic not opato ld i ou gaug and i not th QED coupling contant. Sinc L = L 0 L I, i unchangd in fo, thn ach of L I and L 0 tain th a functional fo, a wll. S$-36, pg. 9. That i, und a yty tanfoation of th full L, vn though L I alon i not ytic in it own ight, in cobination with L 0, th tanfoation yild two t L I and L 0 that a idntical in fo to th p-tanfoation t L I and L 0. And thu, ou tanition aplitud ut alo b th a in fo, a dpictd ybolically in L y L I unchangd H I unchangd S unchangd S unchangd S unchangd. Effctivly, w can ay that if L i ytic und 3-5, thn o i th aplitud S. o th S opato S ψ,a = S ψ,a, 3-5 i.., it ha th a functional fo in t of unpid o pid tanfod ld. With M, ou ynan aplitud, th tanition aplitud, a w found in Chap.$8, i Poof of oiginal Wad idntity Wad idntiti ditinguihd fo oiginal Wad idntity Tanition aplitud and pobability a gaug invaiant

2 Sction 3. Rlation W ll Nd 37 alltnal all tnal boon fion S = δ π δ Pf P i = V VE n M M M. 3-6 Und th gaug tanfoation, th incoing and outgoing fou-onta P i and P f a unchangd, a a th volu V and th tnal paticl ngi, and E. Thu, M i gaug invaiant if S i. Not that th gaug invaianc appli to th total ynan aplitud fo all diaga fo givn incoing and outgoing tat. o apl, in Bhabha catting th a two way fo it to occu. S Chap. 8, ig.$8-, pg.$. That i, fo a givn od in n n n n 0, M = M M, wh n B M and n B n= B B M ach hav any ub diaga fo n >. Th point i that M n i gaug invaiant, B B n n but th individual M and M nd not b. Rcogniz that if L i gaug invaiant, thn H I ain th a und any uch gaug, and ach t in ou S opato panion ach t contain n facto of H I do alo. Thu, fo ach od of intaction n, S n n i ffctivly gaug invaiant. Hnc, o a S and M n. An Eapl Conid th initial photon of th LHS of ig. 3- to b a al photon ath than vitual, i.., ath than a photon popagato. Th lf ngy ynan aplitud of th al photon i Mγ lf = ε T S p i0γ S p i0γ d p ε = ε ε M γ lf,3-7 π M = i0 Π γ lf wh w pnt th pat of th intaction that do not includ th intaction photon contibution a M. Of cou, w now that = and = but that i not ipotant fo γ lf pnt pupo and w want to gnaliz, o w lav in th pi. Th point i that fo vy intaction having on o o tnal photon, w can pnt th ynan aplitud in two facto, on fo th photon polaization tat and on fo th t, wh th latt ha pacti indic which a ud with tho on th polaization vcto. Gnalization o any intaction having on o o photon a initial o nal paticl, w can pnt th gaug invaiant ynan aplitud fo any od n a n n... M = ε ε ε 3... M,, 3,..., wh w again not that 3-8 i gaug invaiant only whn th aplitud includ th ub aplitud fo vy diaga having th a incoing and outgoing tat. Wad Idntiti A w pov blow, gaug invaianc lad to th Wad idntiti n n n M,,.. = M,,.. = M,,.. =... = Poof of Wad Idntiti Th gaug tanfoation of 3-5 an ut atify Mawll wav quation wh inc A i al, hould b al, inc, if ou Mawll quation ha fo und 3-5, thi bco A 0 A = A = = If w qui which w do, a ou thoy i built upon Mawll quation in thi fo A = 0, 3-93 thn fo 3-9, w ut atify Mawll quation LHS blow. on uch = 0 o = 0 olution olv = 0, 3-9 Thu, ynan aplitud alo gaug invaiant But it ut b th total ynan aplitud fo all diaga fo givn od n An apl howing ybol fo th pat of th aplitud without tnal photon facto Gnalizing th apl to any aplitud Stating th Wad idntiti Poof of Wad idntiti

3 37a 38 Chapt 3. Rnoalization Toolit and can hav ntially th a functional fo a A, th olution to 3-9. o a al photon ld dcibd in th Lontz gaug by th ign tat plan wav ψ, = C ψ, = ψ ψ ψ ψ S A S A d d N A A C und S, C d d N ψ A γ ψ ψ A γ ψ tanf i ψ i i γ γ i ψ, ψ A γ γ ψ ψ, γ γ ψ ψ γ γ ψ ψ γ ψ, ψ γ ψ = fo lcton not poiton, = d d N A = d d N A A = d d N A A S C, ψ γ ψ, ψ γ ψ ψ γ ψ, ψ γ ψ d d N A ψ, γ ψ ψ γ ψ d d N A i i A = ε a ε a 3-30, V a uful fo on paticula gaug fo i i i = ɶ ɶ ɶ, ɶ nub, not opato. 3-3 V, Thu, fo thi ca, th photon gaug tanfoation of 3-5 bco i A A = A = A i ɶ i ɶ V i ɶ i i ɶ = A i i = A ɶ V call thi = =, V, ɶ ε a i ɶ i i ε ɶ d d N a i. 3-3 Conid a typical t of S pd in facto of H I, fo apl, th Copton catting t fo n =, pg.$5, und th yty tanfoation 3-5, with 3-5, Uing ou dnition of ɶ fo 3-3, thi bco ψ ɶ γ ψ, ψ γ ψ ψ ɶ, γ ψ, ψ γ ψ. ɶ,, =,,, C ɶ S ψ A S ψ A d d N ψ A γ ψ ψ γ ψ C,, d d N A d d N W can idiatly, bcau idntical t appa on both id of 3-3, that th lat th t ut u to zo. Mo on that hotly. Altnativly, ou thoy wa dvlopd in th Lontz gaug pg.$, i.., A = 0, o w nd A = 0 alo. Thu, 0 = A = A = A = 0. To p th Lontz gaug und th tanfoation, = 0.

4 o now, uing ou ybol is Sction 3. Rlation W ll Nd 37b 39 fo th lcton popagato w nd ψ γ ɶ γ ψ ɶ S = S d d N A is C, C, ψ γ γ ψ ψ ɶ γ ɶ γ ψ, d d N is A d d N is,, A long a w a ticting oulv to lcton Copton catting, and not poiton Copton catting, w can p th lcton popagato S a th full popagato S wh th lat th t blow u to zo S = S d d ɶ N ψ A γ is γ ψ t way, no initial photon C C S C d d ɶ N ψ γ is A γ ψ, Th two t in th u S nd way, no initial photon C ɶ d d ɶ, N ψ γ is γ ψ. C SC S both way, no initial o nal photon C 3-3 Σ S Σ hav th a tnal paticl, o that u ut qual zo indpndntly of Σ Σ S C, which ha diffnt tnal paticl. Rcall that to nd th aplitud S = S Copton fo Copton catting to nd od on th RHS blow, w cay out tp, a w did in Chap. 8, to valuat Copton n p,, γ, p,, γ, Copton p,, γ, C p,, γ, n S = f S i = S S = S.3-35 Whn w did that, w found π Copton = δ M VE VE V V Copton p p S p p MCopton = MC MC MC = u p ε, γ is q = p ε, γ u p M C = u p ε, γ is q = p ε, γ u p, 3-36 which ult fo th S C t in 3-3. Doing a iila thing with th nd and 3 d t on th RHS of 3-3, wh ou initial tat lac th photon of 3-36, w gt Appndi p,, γ, SC SC p, = 0 0 = π VE V V VE p p ɶ ɶ δ p p i MCopton a a ut = 0 Copton C C C, u M = M M M = ε p γ is p γ u p C, u M = ε p γ is p γ u p Thu wh th RHS of 3-33 follow fo iila analyi of th lat t in 3-3, C C Copton Copton 3-37 M M = M = 0 M =

5 38 Chapt 3. Rnoalization Toolit If w ta and, thn 3-38 and 3-33 abov qual 3-9 fo Copton catting. On hould b abl to viualiz a iila ult fo any aplitud with fion popagato and tnal fion and photon. o th ψ ψ of 3-5, th i and i facto will alway cancl. Th tnal A A, du to th pat, will alway lav a i of t in th S opato panion of fo iila to tho in 3-35 with appopiatly o uch t whn th a o facto of A. And ach of th t ut qual zo bcau a t li S ψ, A C in 3-3 occu on ach id th lationhip ulting fo th tanfoation. id o a photon popagato w hav with iila ult fo id = A,A A, A = A,A 3-3 a nub a nub Thu, any photon popagato in any aplitud p th a fo und th tanfoation, o w gt no ta t fo it, a in 3-3, that ut qual zo. And o, w hav povn th Wad idntiti 3-9 uing local gaug invaianc which aniftd in 3-5, th tating point of ou poof. End of poof Not that 3-9 i a lation fo n = od btwn th photon loop and th vt loop, wha 3-9 i good at any od fo any aplitud involving at lat on tnal photon. Additional Idntiti Th a yt oth idntiti calld Wad-Taahahi idntiti, of which 3-9 i a pcial ca, but w will not tat tho h. In Wad-Taahahi idntiti, th i a not tictd to pnt tnal photon, but can b off hll popagato, and th RHS of 3-9 i, fo intnal photon, not zo. Th Wad idntiti a th Wad-Taahahi idntiti fo al photon. Th Poc o any aplitud lation of fo on th LHS of 3-35 blow, th RHS, pnting th Wad idntiti, i tu. That i, w iply plac th polaization vcto by th aociatd fouontu and th ult qual zo. n n n,..., j,... ε,..., j,... j,..., j,... M M M = = 0 j End of Wad idntiti poof Wad idntiti a pcial ca of Wad-Taahahi idntiti How to apply Wad idntiti Th Mag Local gaug invaianc lad to both chag convation and th Wad idntiti. All th a diffnt way of aying th a thing. Each ipli th oth two. chag convation local gaug invaianc Wad idntiti. 3.3 Wad Idntiti, Rnoalization, and Gaug Invaianc Conid th catting of light by light hown in ig. 3-. Two incoing photon catt via fion vitual paticl to yild two outgoing photon. Thi i calld photon-photon catting, o light-by-light catting, o l coonly, Dlbüc catting. Occaionally, it i fd to a a fou photon vt, but thi i ilading a th a ally fou vtic, not a ingl on with fou photon connctd dictly to it. Light-by-light catting do not occu in claical lctoagnti, but do o in QT du to high od coction. Claical lctoagnti contain only t lina in th photon ld A and copond to ou t lvl diaga. Howv via th Dyon-Wic panion in QT, w hav t contibuting to th catting aplitud byond t lvl, at cond and high od which ffctivly a non-lina contibution. ig. 3- pnt on way fou tnal photon can catt at cond od. Th a oth way th a tat can catt at cond od and Pob. a you to daw th ynan diaga fo at lat th oth poibiliti. Not that in ig. 3- w hav dpictd a ctain ti od fo lft to ight fo th vtic in od to a th intnal lin fou-onta ay to dtin. Dpicting a diffnt vt ti od uch a th upp lft vt bfo, ath than Gaug invaianc & Wad idntiti th a thing Application of Wad idntiti in noalization fo photon-photon catting ca

6 Sction 3.8 Appndi: inding Wad Idntiti fo Copton Scatting 337 Coction to nd od nd 0 0 nd 0 0 nd 0 0 nd 0 0 nd = 0 A Λ id id = id A, Λ Π c is p is p = is p B Λ Σ p 3-7 nd nd c = Λ = Λ u p u p B u p u p u p B u p = Λ = Λ v p v p B v p v p v p B v p ε ε ε 0γ 0γ nd = 0 γ 0 Λ 0 Λc i0γ i i p, p i L p, p 3.8 Appndi: inding Wad Idntiti fo Copton Scatting Sinc w wod though vy tp of nding quit a nub of aplitud th long way in Chap. 8, w will b bif h. If you hav toubl at any point, pla f to th dtaild divation of th Copton catting tanition aplitud on pg.$5-8, which cloly paalll th following. p,, γ, S S = 0 bcau S S = C C p, C C o Copton catting th t way LH of ig. 8-3, pg.$5 without th incoing photon, th pat of 3-77 with S bco p,, γ, C S C p, p,,, ɶ = γ d d N ψ A γ is γ ψ,, ip = p γ, d d c u VE p, p p p, V a i ε, γ i i ɶ ɶ π iq d qis q i a a c u p ip V γ VE p p, p, p p, i ε 3-78,,,,, ip = p γ γ, d d u VE p V, γ,, p p p δpp δ δ δ 3-79 iq i d qis q i aɶ V ɶ i ip a γ u VE p π p = ε, VE V VE u d qis q u p p p p π i i iq ip i iq ip ɶ ɶ i a a d d V = ε, VE V VE u d qis q u p p p p π 3-80 ɶ ɶ iq ip i iq ip i iq ip i d V i a d a d 3-8

7 338 Chapt 3. Rnoalization Toolit = ε, VE V VE u d qis q u p p p p π q p i aɶ q p aɶ q p π δ π δ π δ V = π ε, VE V VE u is p u p p p p = M C ɶ δ ɶ δ i V a p p a p p VEp V V VEp V V = 0 cpt whn = p p, = 0 cpt whn = p p, ngativ th valu in full Copton catting of th valu in full Copton catting = π VEp = π VEp δ p p imc a a ɶ ɶ δ p p imc aɶ aɶ o Copton catting th cond way RH of ig. 8-3, th pat of 3-77 with iila valuation, yild, with th ub aplitud M a hown in 3-37, S p,, γ, C nd p, = π. δ p p i VE V V VE MC a a p p C ɶ ɶ S C, aft and 3-8 ud qual th LHS of 3-77, o thi u qual zo. To do thi, th cofcint of aɶ, which i abitay, in that u ut vanih a ut th cofcint of aɶ. Th only way thi can happn i if C C = Copton = 0 M M M Pobl. Show that = 0 p v p. Hint: ollow tp li w did to gt Daw at lat th way, oth than that hown in ig. 3-, fo which th incoing a two photon tat catt at cond od into th a outgoing two photon tat. 3. R-daw th ynan diaga of ig. 3- with th upp lft vt occuing bfo th low lft vt. Labl th intnal lin fou-onta. Show by witing out th ynan aplitud fo thi diaga uing ynan ul, that th aplitud you gt i th a a w got in 3-36 fo ig. 3-. Hint: -p 3-36 with p, p 3, and p in t of p. Thn p you nw diaga wh all popagato facto a in t of p. Rb that fo anti-paticl intnal lin, th fou-ontu ha oppoit ign fo phyical ality. S Wholn Chat$8-, pg.$3. Raliz that th diaga you dw fo thi pobl i not on of th anw fo Pob... Show that by uing pat b of ig. 3-5 fo ynan diaga to nd od of fion lf i Σ p i Σ p iδ. ngy, you obtain Hint: In 3-5 ta 0 and Show 3-6 uing iila logic to what w ud fo Not that in 3-5, th c 0 Π t i an panion with t in to vaiou pow, but that fo a al photon = 0.