Lowest-Order e + e l + l Processes in Quantum Electrodynamics. Sanha Cheong

Transcription

1 Lowest-Order e + e + Processes n Quantum Eectrodynamcs Sanha Cheong

2 Introducton In ths short paper, we w demonstrate some o the smpest cacuatons n quantum eectrodynamcs (QED), eadng to the owest-order expermenta predctons or crosssectons. In partcuar, we w rst dene what (derenta) cross-sectons are. Then, we w dscuss spn/poarzaton sums and averages, whch are mportant or expermenta predctons. Ths arses because our expermenta set-ups are agnostc to the spn/poarzaton states o the externa partces. Aterwards, we w cacuate n deta the cross-sectons or one o the smpest and the most common processes n QED: e + e +. We treat the case where e separatey, snce that requres much more cacuatons. 2

3 2 QED Cross-Secton, Spn, and Poarzaton Let us abe the nta and na states o a quantum system as and respectvey. Then, the transton probabty rom to s gven by where the S-matrx eement S s dened as P ( ) S 2, () S S (2) and sgnes the transton amptude rom a state to a state wthn a tme duraton T and a voume V, not expcty ncuded n eq.. More speccay, consder a process where we begn wth an nta state wth two partces whose -momenta are p (E, p ),, 2, ) and end up wth a na state wth N partces whose -momenta are p (E, p,, 2,..., N. In ths paper, we w ony dscuss QED processes, and hence the partces nvoved are eptons and photons ony. Then, we specy the tme T and the spata voume V under consderaton, the S-matrx eement s gven by S,T V δ + δ T V p ) p ( ) ( ) /2 /2 (2m 2V E 2V E ) /2 M, where the ndex runs over a externa eptons n the process and δ T V s dened as δ T V p ) T/2 p dt d 3 x exp x µ p ) p. T/2 V M s caed the nvarant amptude because ths quantty s Lorentz-nvarant (or the Feynman amptude); t s determned by the reevant Feynman dagrams or specc processes under consderaton. Now, suppose T and V are very arge. Then, δ T V p ) p (2π) δ () p ) p µ due to Fourer transorm, and δ T V p p )] 2 T V (2π) δ () p p ). 3

4 Then, the transton probabty per unt tme s w S,T V 2 T V (2π) δ () p ) p ( ) ( ) ( ) 2m 2V E 2V E M 2. (3) Ths resut ony hods or one exact na state. Practcay, we( are nterested ) n the transton rate to a set o na states wthn some momenta range p, p + d3 p. Ths gves eq. 3 an addtona actor o V d 3 p. Furthermore, t s useu to normaze (2π) 3 the transton rate to one scatterng/codng center (reca that we have ony 2 partces ntay) n the voume and unt ncdent ux; ths requres an addtona actor o V/v re, where v re s the reatve veocty o the two nta partces. Then, we obtan a quantty caed the derenta cross-secton whch equas dσ w V v re V d 3 p (2π) 3 (2π) δ () p ) ( ) ( p 2m E E 2 v re d 3 p (2π) 3 2E ) M 2. () Note that the two actors o V rom the nta-state product are canceed wth one actor n w and another n the normazaton actor V/v re. Assumng that the two nta partces are movng co-neary, one partcuary useu rame o reerence s the center-o-momentum rame (CoM) t s aso oten caed the center-o-mass rame, whch s a msnomer. In ths rame, p p 2, so the reatve veocty o the two nta partces s v re p E + p 2 E 2 E + E 2 E E 2 p. (5) Now, we consder the specc case n whch there are ony two partces n the na state as we. Then, eq. becomes dσ (p, p 2) δ () (p + p 2 p p 2 ) d 3 p d3 p 2, (6) where (p, p 2) M 2 (2m 6π 2 v re E E 2 E E 2 ). (7)

5 However, due to energy- and momentum-conservatons, the na state -momenta are not truy ndependent. For nstance, n the CoM rame, p p 2, and p p 2. To wrte the cross-secton n more useu ndependent varabes, we rst ntegrate eq. 6 over p 2 and obtan dσ (p, p 2) δ(e + E 2 E E 2 ) p 2 d p dω, where we now have the momentum-conservaton condton p 2 p + p 2 p. Integratng over p, we obtan dσ (p, p 2) ] p 2 dω (E + E 2 E E 2 ) p, (8) where we have the momentum- and energy-conservaton condton p 2 p + p 2 p. As mentoned, n the CoM rame, the na-state momenta are not ndependent: p p 2. Hence, (E + E 2 E E 2 ) p (E + E 2) p p p + E + E 2 p E E 2, (9) where we have used the CoM condton, p p 2, and the energy-conservaton condton, E + E 2 E + E 2. Substtutng eqs. 5, 7, and 9 nto eq. 8, we obtan the CoM derenta cross-secton ( ) ( ) dσ p 6π 2 (E + E 2 ) 2 2m M 2. (0) p dω CoM However, even ths resut s too specc or emprca purposes. That s, the nta and the na states o the partces are st speced competey. In partcuar, the spn and the poarzaton states o the reevant eptons and photons must be speced; these are ncuded wthn the nvarant amptude M. However, n most ( not a) practca expermenta settngs, the ncomng partces are typcay not poarzed, and the poarzatons o the outgong partces are not detected ether hereon, the term poarzaton w be used oosey to reer the o eptons as we as the photon poarzatons. Hence, n order to obtan practca cross-sectons or expermenta predctons, we need to average over the derent poarzatons o the ncomng partces, to whch we are agnostc. Smary, we need to sum up the derent outgong partces poarzaton states; snce processes wth derent na poarzaton states are dsjont, the probabtes can be smpy added. 5 E E 2

6 For nstance, consder a QED process that has one nta-state epton wth - momentum p & spn r and one na-state epton wth -momentum p & spn s. Then, ts nvarant amptude s o the orm M rs ū s (p )Γu r (p), () where u and ū are the postve-energy Drac spnors and Γ s a -by- matrx made out o γ-matrces. The specc orm o Γ depends on the specc QED process under consderaton. As mentoned, eq. s too specc and mted to xed spn states r, s. I we want to obtan the expermentay useu unpoarzed cross-secton, we need to average over r and sum over s. Hence, we get a cross-secton proportona to the quantty X 2 r s M rs 2. (2) Usng the act that γ 0 s rea & symmetrc and that (γ 0 ) 2 I, we can compute where we have dened (M rs ) ū s (p ) Γ u r (p)] u s(p )γ 0 Γ u r (p) ] u T s (p )γ 0 Γ u r(p) u r(p) Γ γ 0 u s (p ) u r(p) ( γ 0) 2 Γ γ 0 u s (p ) ū r (p) ( γ 0 Γ γ 0) u s (p ) ū r (p) Γ u s (p ), (3) Γ γ 0 Γ γ 0. Usng eqs. and 3 and expcty wrtng out the spnor ndces as α, β, γ, δ, we obtan X ū α s (p )Γ αβ u β s (p)] ū γ r (p) Γ γδ u δ 2 s(p )] r s ( ) ( ) u δ 2 s(p )ū α s (p ) Γ αβ u β s (p)ū γ s(p) Γ γδ s r / 2 Tr p ] + m 2m Γ/ p + m 2m Γ, () where we have used the competeness reaton r u s (p)ū s (p) / p + m 2m. 6

7 Smar cacuatons can be apped to processes nvovng ant-eptons usng the dentty p v s (p) v s (p) / m 2m. r Smary, we shoud consder the poarzatons o externa photons as we as eptons. For nstance, consder a process wth one na externa photon. Then, the unpoarzed cross-secton s proportona to the quantty X r M r 2 r ɛ µ r (k)m µ 2, (5) where ɛ µ r (k) s the poarzaton -vector correspondng to a poarzaton state r and a wave -vector k. Note that k 0 k. Snce the quanttes M and thereore X are gauge-ndependent, t s convenent to work wth the Lorenz gauge. That s, we et the -potenta A µ o the externa photon satsy the condton µ A µ 0, (6) whch descrbes a smpe transverse wave, n agreement wth the common ntuton o ree rea radaton. Because the quantty M r ɛ µ r M µ s gauge-nvarant and we chose a transversewave gauge, we have the property k µ M µ 0, (7) and we can choose a coordnate system such that k µ (κ, 0, 0, κ). Then, rom eq. 5, we can concude that the unpoarzed cross-secton must be proportona to M r 2 M µ M ν r ɛ µ r ɛ ν r r M µ M µ, (8) where we have used the gauge condton eq. 7 and the competeness reaton or rea photons,.e., ɛ µ r ɛ ν r g µν + (k µ kν + 2 k ) µ k ν, r where k µ (κ, 0, 0, κ ). Now, we can use these resuts to cacuate some basc QED cross-sectons. 7

8 3 e + e + Process ( e) Consder the coson process n whch an eectron-postron codes and annhates to produce a epton-ant-epton par. I the poarzatons o the externa eptons are a speced, ths can be wrtten as e + (p, r ) + e (p 2, r 2 ) + (p, s ) + (p 2, s 2 ) (9) where the p s denote 3-momenta, r s and s s denote spn states, and the prmes abe the na state. For smpcty, we w et the na state eptons (a epton and an ant-epton) be muons or tauons, but not eectrons. Then, the rst-order Feynman dagram correspondng to the process 9 s gven by g.. Its nvarant amptude s M(r, r 2, s, s 2 ) e 2 ū (p 2, s 2 )γ µ v +(p, s )] }{{} and the compex conjugate s na epton vertex M (r, r 2, s, s 2 ) e 2 v +(p, s )γ µ u (p 2, s 2 )] (p + p 2 ) 2 }{{} photon propagator v e +(p, r )γ µ u e (p 2, r 2 )], }{{} nta eectron vertex (20) (p + p 2 ) 2 ū e (p 2, r 2 )γ µ v e +(p, r )], where we have used the γ-matrx dentty: γ µ γ 0 γ µ γ 0. Averagng over r s and summng over s s (generazng eq. ), we obtan the quantty proportona to the unpoarzed cross-secton X M(r, r 2, s, s 2 ) 2 s 2 r r 2 s e (p + p 2 ) A µνb e µν e (p + p 2 ) Tr / p 2 + m 2m /p γ m ] p / µ γ ν Tr m e γ µ / ] p 2 + m e γ ν. (2) 2m 2m e 2m e e + r + s p p 2 γ p + p 2 p p 2 e r 2 s 2 Fgure : Lowest-order Feynman dagram or the process e + e + wth compete poarzaton-speccaton. 8

9 Now, we use γ-matrx trace theorems to cacuate the quanttes n eq. 2. partcuar, In Tr(γ µ...γ µn ) 0 n odd, Tr(γ µ γ ν ) g µν, Tr(γ µ γ ν γ ρ γ σ ) (g µν g ρσ g µρ g νσ + g µσ g νρ ). Then, we get p / A 2 µν Tr + m 2m m 2 m 2 m 2 m 2 /p γ m ] µ γ ν 2m ] Tr /p 2 γ µ/p γ ν m 2 γ µ γ ν p ρ 2 p σ Tr (γ ρ γ µ γ σ γ ν ) m 2 Tr (γ µ γ ν ) ] p ρ 2 p σ (g ρµ g σν g ρσ g µν + g ρν g µσ ) m 2 g µν ] p µ p 2ν + p 2µp ν (m 2 + p p 2)g µν ] and smary Substtutng nto eq. 2, we get X B e µν p µ m p ν p µ 2p ν (m 2 e + p p 2 )g µν]. e e ] 2m 2 em 2 (p (p p + p 2 ) )(p 2 p 2) + (p p 2)(p 2 p ) + m 2 ep p 2 + m 2 p p 2 + 2m 2 em 2 Now, to smpy ths expresson urther, we consder a specc rame o reerence: the center-o-momentum rame. In ths rame, p p 2, p p 2, and E E 2 E E 2 E. The knematcs o e + e + process s smpy descrbed by the ange θ between p & p and the magntudes o the 3-momenta p p and p p. In summary, we have the oowng resuts: p p p 2 p 2 E 2 pp cos θ, p p 2 p 2 p E 2 + pp cos θ, p p 2 E 2 + p 2, p p 2 E 2 + p 2, (p + p 2 ) 2 E 2. (22) Thereore, we obtan the na unpoarzed derenta cross-secton or the CoM 9

10 rame: ( ) dσ dω CoM ( ) p 2m 6π 2 (E + E 2 ) 2 X p e p 256π 2 E 2 p 6m2 em 2 32m 2 em 2 E ] 2E + 2p 2 p 2 cos 2 θ + m 2 e(e 2 + p 2 ) + m 2 (E 2 + p 2 ) + 2m 2 em 2 ( ) α2 p E 2 + m 2 6E + p 2 cos 2 θ ], (23) E where α e 2 /(π) and we have made the approxmaton m e m E and thereore p 2 E 2. The tota cross-secton s obtaned by ntegratng over the whoe π steradan sod ange: σ CoM πα2 E ( ) p E 2 + m ] E p 2. (2) In even more hgh-energy cases, we have E m whch aso mpes p E. Thereore, ( ) dσ α2 ( + cos 2 θ ), dω CoM 6E 2 σ CoM πα2 3E 2. Bhabha Scatterng: e + e e + e Process In the prevous secton, we gnored the case where the na state epton par s an eectron-postron par. Ths s because there are two owest-order Feynman dagrams that contrbute to ths process. In addton to the par annhaton-creaton process dscussed prevousy, there s aso eastc scatterng o the eectron and the postron by smpy exchangng a photon. See g. 2. The nvarant amptude or process (a) was aready dscussed n the prevous secton: M a e 2 ū(p 2, s 2 )γ µ v(p, s )] (p + p 2 ) v(p, r 2 )γ µ u(p 2, r 2 )]. (25) The nvarant amptude or process (b) s: M b e 2 ū(p 2, s 2 )γ µ u(p 2, r 2 )] }{{} ower vertex (p p ) 2 }{{} photon propagator 0 v(p, r )γ µ v(p, s )] }{{} upper vertex. (26)

11 e + e p p 2 p p 2 e + e e + e p p γ p + p 2 p 2 p 2 e + e + e + (a) e p p p γ p p 2 p 2 e + e (b) Fgure 2: Two owest-order Feynman dagrams or the e + e e + e process. (a) Annhaton-creaton process, dscussed prevousy. (b) Eastc scatterng by photon exchange. Hence, the unpoarzed derenta cross-secton s now proportona to the quantty X M a + M b 2 ( Ma 2 + M b 2 ) + M a M b + M am b. (27) Agan, we choose to work n the CoM rame so that we have a the denttes n eq. 22 and addtonay p p 2 p p p 2 p, and consder the reatvstc case so that E m e (.e. m e /E 0) and p p E. Then, we get X aa M a 2 e + cos 2 θ ], (28) 6m e whch oows rom the resut that we saw n the prevous secton. Smar process shows that X bb M b 2 e 2m e(p p ) (p p )(p 2 p 2) + (p p 2)(p 2 p )] e 8m e sn + cos θ ]. (29) (θ/2) 2

12 The ast term, whch s the cross-term between processes (a) and (b), s more compcated. Wth hgher-power trace theorems, we obtan X ab M a M b e (p + p 2 ) 2 (p p ) 2 e (p + p 2 ) 2 (p p ) 2 e (p + p 2 ) 2 (p p ) Tr p / + m 2 e 2 2m e ū(p 2, s 2 )γ µ v(p, s )] v(p, r )γ µ u(p 2, r 2 )] v(p, s )γ ν v(p, r )] ū(p 2, r 2 )γ ν u(p 2, s 2 )] ū(p 2, s 2 )γ µ v(p, s )] v(p, s )γ ν v(p, r )] e ] 6m e(p + p 2 ) 2 (p p ) Tr /p γ 2 2 µ/p γ ν/p γ µ /p 2 γ ν v(p, r )γ µ u(p 2, r 2 )] ū(p 2, r 2 )γ ν u(p 2, s 2 )] /p γ m e /p µ γ m e ν γ µ / ] p 2 + m e γ ν 2m e 2m e 2m e e 8m e sn 2 (θ/2) cos θ 2. (30) Thereore, the unpoarzed derenta cross-secton n the hghy-reatvstc CoM rame s ( ) dσ p dω CoM 6π 2 (E + E 2 ) 2 p (2m e) X aa + X bb + X ab + Xab] α2 8E 2 + cos 2 θ cos (θ/2) sn (θ/2) 2 cos (θ/2) sn 2 (θ/2) ]. (3) Note that ths quantty dverges to postve nnty as θ 0 due to the X bb contrbuton. In other words, the derenta-cross secton s nnte n the co-near orward drecton. Ths s because the -momentum o the photon exchanged between the eectron and the postron k µ (p p ) µ goes to zero and the photon propagator n eq. 26 dverges. 2

13 Reerences ] Hazen, F. and Martn, A. D. (98). Quarks & Leptons: An Introductory Course n Modern Partce Physcs. John Wey & Sons, Inc. 2] Mand, F. and Shaw, G. (2009). Quantum Fed Theory. John Wey & Sons, Inc., second edton. 3] Schwartz, M. D. (203). Quantum Fed Theory and the Standard Mode. Cambrdge Unversty Press. 3