The Quantum EM Felds and the Photon Propagator Quantzng the free electromagnetc tenson felds E and B s farly straghtforward. The tme-ndependent Maxwell equatons ˆB(x) = 0, Ê(x) = Ĵ0 (x) 0 (for the free felds) (1) are mposed as operatoral constrants n the Hlbert space, whle the tme-dependent Maxwell equatons follow from the free Hamltonan t ˆB = Ê, Ĥ = tê = + ˆB (2) d x( 3 12 Ê 2 (x) + 1 ˆB ) 2 2 (x) (3) and the equal-tme commutaton relatons [Ê (x,t),êj (y,t) ] = 0, [ˆB (x,t), ˆB j (y,t) ] = 0, [Ê (x,t), ˆB j (y,t) ] = ǫ jk x kδ(3) (x y). (4) In lght of eqs. (1), we may expand the vector felds nto momentum/polarzaton modes usng only the transverse polarzatons, for example the helcty modes λ = ±1 only, thus k e k,λ = λ k e k,λ, e k,λ k for λ = ±1, d 3 k Ê(x) = (2π) 3 ˆB(x) = d 3 k (2π) 3 λ=±1 only λ=±1 only e xk e k,λ Êk,λ, Ê k,λ = Ê k,λ, e xk e k,λ ˆB k,λ, ˆB k,λ = ˆB k,λ, [Êk,λ,Êk,λ ] = 0, [ˆBk,λ, ˆB k,λ ] = 0, [Êk,λ, ˆB k,λ ] = λ k δλλ (2π) 3 δ (3) (k+k ). (5) Consequently, the photonc creaton and annhlaton operators are defned for the transverse polarzatons only, â k,λ = λˆb k,λ + Êk,λ, â k,λ = λˆb k,λ Ê k,λ, λ = ±1 only, (6) 1
[âk,λ,â k,λ ] = 0, [â k,λ,â k,λ ] = 0, [âk,λ,â k,λ ] = δλλ 2 k (2π) 3 δ (3) (k k ). (7) In terms of these operators, the free EM Hamltonan becomes Ĥ d 3 k (2π) 3 1 2ω k λ=±1 only ω k â k,λâk,λ + const for ω k = + k, (8) whch makes for the usual tme-dependence n the Hesenberg pcture: â k,λ â k,λ e ω kt, â k,λ â k,λ e+ω kt, (9) and hence (after a bt of algebra) d 3 k 1 Ê(x,t) = (2π) 3 2ω d ˆB(x,t) 3 k 1 = (2π) 3 2ω λ=±1 only λ=±1 only ( ) e ωt+kx ( ωe k,λ )â k,λ + e +ωt kx (+ωe k,λ )â k,λ, ( ) e ωt+kx ( k e k,λ )â k,λ + e +ωt kx (+k e k,λ )â k,λ. Up to now I have focused on the EM tenson felds ˆF µν (x), but to couple the EM to come charged felds lke theelectron feld ˆΨ(x) Iwould also need thequantum potental felds µ(x) to spell out the nteractons L J µ A µ. Classcally, the potental felds are subject to gauge symmetres (10) A µ old (x) Aµ new(x) = A µ old (x) µ Λ(x), (11) but mplementng such symmetres n the quantum theory s hghly non-trval snce a transform changng the tme-dependence of quantum felds must change the Hamltonan operator. There are more problems wth quantum gauge theores, and I ll deal wth them next semester. For the moment, let me smply say that the canoncal quantzaton of the potental felds A µ (x) requres fxng a gauge. That s, we should remove the gauge-redundancy of the potental felds by mposng a local lnear constrant at each x, for example A(x) 0 (the Coulomb gauge), or µ A µ (x) 0 (the Landau gauge), or A 3 (x) 0 (the axal gauge). 2
Once we mpose such a constrant at the Lagrangan level, we fnd the canoncal conjugates of the remanng ndependent felds, buld the classcal Hamltonan, and then quantze the theory n the usual way to buld the quantum felds, the Hlbert space where they act, and the Hamltonan operator. In general, dfferent gauge-fxng constrants gve rse to dfferent quantum theores, each havng ts own Hlbert space and the Hamltonan operator. However, all such theores are physcally equvalent to each other! For the free electromagnetc felds, I do not have to re-quantze the theory startng from the Lagrangan and some gauge-fxng constrant for the A µ (x). Snce I already have the quantum Ê(x) and ˆB(x) felds as n eqs. (10), I can can obtan the free potental felds µ (x) by smply solvng the equatons µ  ν (x) ν  µ (x) = ˆF µν (x) combned wth the gauge-fxng constrant. By lnearty and translaton nvarance, the result has form  µ free (x) = d 3 k (2π) 3 1 2ω λ=±1 only ( ) e kx E µ (k,λ) â k,λ + e +kx E µ (k,λ) â k,λ (12) where the plane waves A µ (x) = e kx E µ (k,λ) and A µ (x) = e +kx E µ (k,λ) for k 0 = + k (13) obey the gauge-fxng condtons as well as the Maxwell equatons. For example, n the Coulomb gauge  = 0, the polarzaton vectors Eµ have smple form E µ (k,λ) = ( 0,e(k,λ) ) (14) whle n the axal gauge n A = 0 (where n s a fxed unt 3-vector) we have much messer formulae E(k,λ) = e(k,λ) n e(k,λ) n k k, E 0 (k,λ) = n e(k,λ) n k ω k. (15) 3
Photon Propagator The photon propagator G µν F (x y) = 0 Tµ (x)âν (y) 0 (16) depends on the gauge-fxng condton for the quantum potental felds µ (x). So let me frst calculate t for the Coulomb gauge  0, and then I ll deal wth the other gauges. Instead of calculatng the propagator drectly from eqs. (16), (12), and (14), let me use the fact that G µν (x y) s a Green s functon of the Maxwell equatons for the A µ felds, 2 A ν (x) ν ( µ A µ (x)) = J ν (x) = A µ (x) = d 4 y( )G µν (x y) J ν (y). (17) Let s start wth the scalar potental A 0 (x,t). In the Coulomb gauge we have µ A µ = 0 A 0 + A = 0 A 0 + 0, (18) whch makes the equaton for the A 0 (x) tme-ndependent. Indeed, J 0 = 2 A 0 (x) 0 ( µ A µ (x)) = ( 2 0 2 )A 0 0 ( 0 A 0 ) = 2 A 0 (x) (19) whle the terms contanng the tme dervatves 0 cancel out. Consequently, the A 0 (x,t) s the nstantaneous Coulomb potental for the electrc charge densty J 0 (y,t), A 0 (x,t) = d 3 y J0 (y, samet) 4π x y (20) whch s why ths gauge s called the Coulomb gauge. In terms of the Green s functon, ths means G 0 0, G 00 (x y) = δ(x0 y 0 ) 4π x y of after Fourer transform to the momentum space,, (21) G 00 (k) = k 2 ndependent of k 0. (22) 4
As to the vector potental A, t obeys where n the Coulomb gauge 2 A + ( µ A µ ) = J (23) µ A µ = 0 A 0 = 1 2 0J 0 = + 1 ( J) (24) 2 where the last equalty follows from the electrc current conservaton, 0 J 0 = J. Consequently, or n momentum bass ( 0 2 2 )A = J 1 ( J), (25) 2 (k 2 0 k2 ) A (k) = J k k j In terms of the Green s functon, ths means G 0 = 0, G j (k) = k 2 0 k2 k 2 Jj. (26) (δ j k k j Altogether, all the components of the G µν (k) can be summarzed as k 2 ). (27) G µν (k) = k 2 0 k2 Cµν (k) (28) where C j = δ j k k j k 2, C0 = C 0 = 0, C 00 = k2 0 k2 k 2 = k2 0 1. (29) k2 A convenent way to summarze ths C µν (k) tensor s C µν (k) = g µν + k µ c ν (k) + k ν c µ (k) for c µ (k) = (k0, k) 2k 2. (30) In the coordnate space G µν (x y) = d 4 k (2π) 4 e k(x y) k 2 C µν (k) (31) where the denomnator has poles at k 0 = ± k. As usual, these poles should be regularzed by movng them off the real axs, and dfferent regularzatons lead to dfferent Green s 5
functons. We are nterested n the specfc Green s functon the Feynman propagator so we should move the poles to k 0 = + k 0 and k 0 = k +0, thus G µν d 4 F (x y) = k e k(x y) ( ) (2π) 4 k 2 +0 C µν (k) = g µν + k µ c ν (k) + k ν c µ (k). (32) Now consder photon propagators for dfferent gauge condtons for the EM potental felds µ (x). When we change a gauge condton, the potental felds change as  µ new (x) = µ old (x) µˆλ(x) (33) for some ˆΛ(x) that depends on the old µ (x) and the new gauge condton. For lnear gauge condtons, ˆΛ(x) = d 4 yl µ (x y)âµ old (y) (34) for some kernel L µ (x y), or n momentum space ˆΛ(k) = L µ (k)âµ old (k). (35) Consequently, n the new gauge the Feynman propagator becomes G µν new(x y) = 0 Tµ new(x)âν new(y) 0 = 0 Tµ old (x)âν old (y) 0 x µ 0 TˆΛ(x)Âν old (y) 0 y ν 0 Tµ old (x)ˆλ(y) 0 + 2 x µ y ν 0 TˆΛ(x)ˆΛ(y) 0 = G µν old (x y) + x µ f ν (x y) + y ν f µ (y x) + 2 x µ y ν h(x y) for some functons f µ (x y) and h(x y). Fourer transformng to the momentum space, we obtan G µν new (k) = Gµν old (k) + kµ f ν (k) k ν f µ (k) + k µ k ν h(k) (36) for some functons f ν (k) = and h(k) of the off-shell momenta. (In terms of the L µ (k) from eq. (35), f µ (k) = L λ (k) G λν old (k) whle h(k) = L κ(k)l λ (k)g κλ (k).) Consequently, f n old 6
the old gauge we had G µν old (k) = ( ) k 2 +0 g µν + k µ c ν old (k) + kν c µ old (k) (37) then n the new gauge we also have G µν new (k) = ( ) k 2 +0 g µν + k µ c ν new (k) + kν c µ new (k) (38) for c ν new (k) = cν old (k) + k2 ( f ν (k) + 1 2 h(k)kν). (39) Now, back n eq. (32) we saw that n the Coulomb gauge the photon propagator ndeed has form (37) for some c ν (k). Consequently, n any gauge, the photon s Feynman propagator has form G µν d 4 F (x y) = k e k(x y) ( ) (2π) 4 k 2 +0 g µν + k µ c ν (k) + k ν c µ (k) (40) where the vector-valued functon c ν (k) depends on the partcular gauge condton, but everythng else s the same n all gauges. In the Feynman rules for QED or for any other QFT contanng the EM felds the photon propagators depend on the gauge-fxng condton va the c ν (k) functons. However, all the physcal scatterng ampltudes turn out to be gauge-ndependent. We shall see a few examples of such gauge-nvarant ampltudes ths semester, and I shall prove the general theorem n the Sprng. In practce, one uses whatever gauge condton would smplfy the calculaton at hand. Usually, ths means one of the Lorentz-nvarant gauges where c ν (k) s parallel to the k ν and hence G µν d 4 F (x y) = k e k(x y) ( g (2π) 4 k 2 +0 µν + ξ kµ k ν ) k 2 +0 for some real constant ξ. For ξ = 1 ths photon propagator corresponds to the Landau gauge µ A µ 0. The propagators wth other values of ξ do not correspond to any sngle (41) 7
gauge condton for the A µ (x); nstead, they obtan from a gauge-averagng procedure I shall explan n Aprl; for the moment, let me smply say that these gauges work fne. Of partcular mportance s the Feynman gauge ξ = 0, whch makes for a partcularly smple photon propagator G µν F (x y) = d 4 k (2π) 4 g µν k 2 +0 e k(x y). (42) For the rest of ths semester and also through most of the Sprng semester I shall be usng ths gauge. QED Feynman rules Quantum Electro-Dynamcs or QED s the theory of EM feld A µ (x) coupled to the electron feld Ψ(x) (and optonally other charged fermon felds). The Lagrangan s L = 1 4 F µνf µν + Ψ( D m)ψ = 1 4 F µνf µν + Ψ( m)ψ+ ea µ Ψγ µ Ψ (43) where the frst 2 terms on the last lne descrbe free photons and electrons e ±, and the thrd term s treated as a perturbaton. Thetwo dfferent feldtypeshavedfferent propagatorss F αβ (x y) = 0 TΨ α(x)ψ β (y) 0 and G µν F (x y) = 0 TAµ (x)a ν (y) 0. In QED Feynman rules, these propagators are denoted by two dfferent types of nternal lnes: The electron propagator s drawn as a sold lne wth an arrow ndcatng whch end of the lne belongs to a Ψ feld and whch to a Ψ, Ψ α q Ψ β = [ ] q m+0 αβ. (44) The smaller arrow near q ndcates the drecton of the momentum flow. Both arrows should have the same drecton; otherwse we would have Ψ α q Ψ β = [ ] q m+0 αβ. (45) 8
The photon propagator s drawn as a wavy lne wthout arrow, A µ q A ν = usually = ( ) q 2 +0 g µν +q µ c ν (q) + q ν c µ (q) for some c ν (q), ( q 2 +0 g µν ξ qµ q ν ) q 2 for some constant ξ. +0 (46) In ths class I shall work n the Feynman gauge ξ = 0 where the photon propagator s smply A µ q A ν = gµν q 2 +0. (47) The physcal ampltudes do not depend on the gauge-fxng parameters ξ or c ν (q), but the ntermedate results often do. Make sure to use the same gauge for all the photon propagators n a dagram and also for all the dagrams contrbutng any partcular process. The vertces of Feynman dagrams follow from the nteracton terms n the Lagrangan that nvolve 3 or more felds. The QED Lagrangan has only one nteracton term ea µ Ψγ µ Ψ, so there s only one vertex type, namely β µ = (+eγ µ ) βα. (48) α Ths vertex has valence = 3, and the 3 lnes t connects must be of specfc types: one wavy (photonc) lne, one sold lne wth ncomng arrow, and one sold lne wth outgong arrow. Now consder the external lnes. When the quantum EM feld  µ (x) = d 3 k (2π) 3 1 2ω k λ=±1 ( e kx E µ (k,λ) â k,λ + e +kx E µ (k,λ) â k,λ) k 0 =+ω k (49) s contracted wth an ncomng or an outgong photon, we end up wth an external-lne factor accompanyng the matchng â k,λ or â k,λ operator, namely e kx E µ (k,λ) for an ncomng photon or e +k x E µ (k,λ ) for an outgong photon. The momentum space Feynman rules take care of the e kx factors, whch leaves us wth the polarzaton vectors E µ (k,λ) or E µ (k,λ ) for the external photon lnes: k = E µ (k,λ) for an ncomng photon, (50) 9
k = Eµ (k,λ ) for an outgong photon. (51) In general, the Feynman rules for any theory has smlar external-lne factors for all the ncomng and outgong partcles. For the scalar partcles such factors are trval that s why we have not seen them before but for all other types of partcles there are non-trval external lne factors we should not forget. In partcular, the fermonc external lnes carry plane-wave Drac spnors u(p,s), v(p,s), ū(p,s), or v(p,s), dependng on the charge of the fermon and whether t s ncomng or outgong. Specfcally, an ncomng electron e carres an outgong electron e carres an ncomng postron e + carres an outgong postron e + carres p p p p = u α (p,s), (52) = ū α (p,s), (53) = v α (p,s), (54) = v α (p,s). (55) Note that for the ncomng / outgong electrons, the arrow on the external lne has the same drecton as the partcle ncomng for an ncomng e and outgong for an outgong e, but for the postrons the arrow ponts n the opposte drecton from the partcle and ts momentum: An ncomng e + has an outgong lne (but n-flowng momentum) whle an outgong e + has an ncomng lne (but out-flowng momentum). In general, the arrows n the fermonc lnes follow the flow of the electrc charge (n unts of e), hence opposte drectons for the electrons and the postrons. Each QED vertex (48) has one ncomng fermonc lne and one outgong, and we may thnk of them as beng two segments of sngle contnuous lne gong through the vertex. From ths pont of vew, a fermonc lne enters a dagram as an ncomng e or an outgong e +, goes through a sequence of vertces and propagators, and eventually exts the dagram as an outgong e or an ncomng e +, e e + out } { n e out e + n (56) (The photonc lnes here may be external or nternal; f nternal, they connect to some other fermonc lnes, or maybe even to the same lne at another vertex.) Alternatvely, a fermonc 10
lne may form a closed loop, for example (57) The contnuous fermonc lnes such as (56) or (57) are convenent for handlng the Drac ndces of vertces, fermonc propagators, and external lnes. For an open lne such as (56), the rule s to read the lne n the drecton of the arrows, from the lne s begnnng to ts end, spell all the vertces, the propagators, and the external lne factors n the same order rghtto-left, them multply them together as Drac matrces. For example, consder a dagram where an ncomng electron and ncomng postron annhlate nto 3 photons, real or vrtual. Ths dagram has a fermonc lne whch starts at the ncomng e, goes through 3 vertces and 2 propagators, and exts at the ncomng e + as shown below: ν µ λ (58) n e + (p +,s + ) k 3 k 2 k 1 n e (p,s ) q 2 q 1 The propagators here carry momenta q 1 = p k 1 and q 2 = q 1 k 2 = k 3 p +. The fermonc lne (58) carres the followng factors: u(p,s ) for the ncomng e ; +eγ λ for the frst vertex (from the rght); for the frst propagator; q 1 m+0 +eγ µ for the second vertex; for the second propagator; q 2 m+0 +eγ ν for the thrd vertex; 11
v(p +,s + ) for the ncomng e +. Readng all these factors n the order of the lne (58), tal-to-head, and multplyng them rght-to-left, we get the followng Drac sandwch v(p +,s + ) (+eγ ν ) q 2 m+0 (+eγµ ) q 1 m+0 (+eγλ ) u(p,s ). (59) In ths formula, all the Drac ndces are suppressed; the rule s to multply all factors as Drac matrces (or row / column spnors) n ths order. For a closed fermonc loop such as (57), the rule s to start at an arbtrary vertex or propagator, follow the lne untl one gets back to the startng pont, multply all the vertces and the propagators rght-to-left n the order of the lne, then take the trace of the matrx product. For example, the loop κ λ q 1 q 4 q 2 (60) q 3 ν µ produces the Drac trace [ tr (+eγ κ ) q 1 m+0 (+eγλ ) q 2 m+0 (61) (+eγ µ ] ) q 3 m+0 (+eγν ). q 4 m+0 Note that a trace of a matrx product depends only on the cyclc order of the matrces, tr(abc YZ) = tr(bc YZA) = tr(c YZAB) = = tr(zabc Y). Thus, n eq. (61), we may start the product wth any vertex or propagator as long as we multply them all n the correct cyclc order, the trace wll be the same. 12
As to the Lorentz vector ndces λ,µ,ν,..., the ndex of a vertex should be contracted to the ndex of the photonc lne connected to that vertex. For example, the followng dagram for the e +e e +e scatterng 1 2 (62) evaluates to M = 1 2 ( ) ( ū(p 1,s 1 ) (+eγ µ) u(p 1,s 1 ) ū(p 2,s 2 ) (+eγ ν) u(p 2,s 2 ) ) gµν q 2. (63) Here I have used the Feynman gauge for the photon propagator, but any other gauge would produce exactly the same ampltude M = ū 1(eγ µ )u 1 ū 2(eγ ν )u 2 ( gµν + c µ q ν + q µ c ν ) q 2 = ū 1(eγ µ )u 1 ū 2(eγ ν )u 2 gµν q 2 (64) because of Gordon denttes ū 1 (eγ µ)u 1 q µ = ū 2 (eγ ν)u 2 q ν = 0. (65) To prove these denttes, we note that the spnors u 1 u(p 1,s 1 ) and ū 1 ū(p 1,s 1 ) obey the Drac equatons for the approprate momenta, p 1 u 1 = mu 1, ū 1 p 1 = mū 1. (66) Moreover, q = p 1 p 1 and hence ū 1γ µ u 1 q µ = ū 1 qu 1 = ū 1( p 1 p 1 )u 1 = (mū 1)u 1 ū 1(mu 1 ) = 0. (67) Smlarly, q = p 2 p 2 and hence ū 2 γ µu 2 q µ = ū 2 qu 2 = ū 1 ( p 2 p 2 )u 1 = ū 2 (mu 2)u 2 (mū 2 ) = 0. (68) For other combnatons of ncomng or outgong electrons or postrons connected to the 13
same vertex we have smlar Gordon denttes: v(p) qv(p ) = 0 for q = p p, v(p 2 ) qu(p 1 ) = 0 for q = p 1 +p 2, (69) ū(p 2) qv(p 2) = 0 for q = p 1 +p 2, whch provde for gauge nvarance of the Feynman dagrams lke 1 2 1 2 or (70) 1 2 1 2 In general, an ndvdual Feynman dagram s not always gauge-ndependent. However, when one sums over all dagrams contrbutng to some scatterng process at some order, the sum s always gauge nvarant. We shall return to ths ssue later ths semester. To complete the QED Feynman rules, we need to keep track of the sgns arsng from re-orderng of the fermonc felds and creaton / annhlaton operators. To save tme, I wll not go through the gory detals of the perturbaton theory. Instead, let me smply state the rules for the overall sgn of a Feynman dagram n terms of the contnuous fermonc lnes: There s a sgn for every closed fermonc loop. There s a sgn for every open fermonc lne whch begns at an outgong postron and ends at an ncomng postron. There s a sgn for every crossng of the fermonc lnes. Although the number of such crossng depends on how we draw the dagram on a 2D sheet of paper, for example 1 2 1 2 versus (71) 1 2 1 2 14
ts party #crossngs mod 2 s a topologcal nvarant, and that s all we need to determne the overall sgn of the dagram. If multple Feynman dagrams contrbute to the same process, then the external legs should stck out from the dagram n the same order for all the dagrams. Or at least all the fermonc external legs should stck out n the same order, whch should also agree wth the order of fermons n the bra and ket states of the S matrx element e,...,e,e +,...,e +,γ,...,γ M e,...,e,e +,...,e +,γ,...,γ for the process n queston. Fnally, the QED s usually extended to nclude other charged fermons besdes e. The smplest extenson ncludes the muons µ and the tau leptons τ whch behave exactly lke the electrons, except for larger masses: whle m e = 0.51100 MeV, m µ = 105.66 MeV and m τ = 1777 MeV. In terms of the Feynman rules, the muons and the taus have exactly the same vertces, propagators, or external lne factors as the electrons, except for a dfferent mass m n the propagators. To dstngush between the 3 lepton speces, one should label the sold lnes wth e, µ, or τ. Dfferent speces do not mx, so a label belongs to the whole contnuous fermonc lne; for an open lne, the speces must agree wth the ncomng / outgong partcles at the ends of the lne; for a closed loop, one should sum over the speces l = e,µ,τ. Coulomb Scatterng As an example of QED Feynman rules, consder the elastc scatterng of two electrons, e +e e +e. There are two tree dagrams contrbutng to ths process, 1 2 1 2 q + q (72) 1 2 1 2 whch are related by exchangng the fnal-state electrons, 1 2. The frst dagram (72) 15
was evaluated back n eq. (63) as M 1 = gµν q 2 = t ū 1(eγ µ )u 1 ū 2(eγ ν )u 2. (73) For the second dagram, we exchange u 1 u 2, change the momentum of the vrtual photon from q = p 1 p 1 to q = p 2 p 1 and hence q2 = t n the denomnator to q 2 = u, and there s an overall mnus sgn due to electron lne crossng, thus M 2 = gµν q 2 = u ū 2(eγ µ )u 1 ū 1(eγ ν )u 2. (74) Combnng the two dagrams, we obtan the net tree-level scatterng ampltude as M tree = M 1 + M 2 = e2 t ū 1γ µ u 1 ū 2γ µ u 2 e2 u ū 2γ µ u 1 ū 1γ µ u 2. (75) Now let s take the non-relatvstc lmt of ths scatterng ampltude. A non-relatvstc electron wth 3-momentum p m and energy p 0 m has plane-wave Drac spnor u(p,s) ( ) mξ mξ +O( p / m). (76) Consequently, the Drac sandwches ū γ µ u between non-relatvstc electron spnors are approxmately ū(p,s )γ 0 u(p,s) = u (p,s )u(p,s) 2m ξ s ξ s = 2m δ s,s, ( ) + σ 0 ū(p,s ) γu(p,s) = u (p,s ) u(p,s) = 0 m + O(p,p ) m, 0 σ (77) so the scatterng ampltude (75) s domnated by the µ = 0 terms. Specfcally, M non.rel. tree e2 t 2mδ s 1,s1 2mδ s 2,s2 e2 u 2mδ s 2,s1 2mδ s 1,s2 = 4m2 e 2 q 2 δ s 1,s 1 δ s 2,s 2 + 4m2 e 2 q 2 δ s 2,s 1 δ s 1,s 2, (78) where on the second lne I have used t (q = p 1 p 1 )2 and u ( q = p 2 p 1 )2 for the non-relatvstc electrons (snce q 0 = E 1 E 1 = O(p 2 /m) q and lkewse q 0 q ). 16
Note that despte the non-relatvstc lmt, the ampltude (78) s relatvstcally normalzed. In terms of the non-relatvstcally normalzed scatterng ampltude f = M 8πE cm (79) where E cm 2m, we have f non.rel. tree me2 4πq 2 δ s,s1δ 1 s + me2 2,s2 4π q 2 δ s,s1δ 2 s 1,s2. (80) Now let s compare the QED ampltude (80) to the non-relatvstc ampltude n potental scatterng. In the non-relatvstc lmt, the perturbatve expanson of QED n powers of e 2 roughly corresponds to the Born seres n potental scatterng, so the tree-level ampltude(78) should be compared to the frst Born approxmaton f B (p p ) = M red 2π Ṽ(q = p 1 p 1 ) = M red 2π d 3 x rel V(x rel )e q x rel. (81) Or rather, ths s the Born ampltude for dstnct spnless partcles wth a reduced mass M red. For partcles wth spn but nteractng wth a spn-blnd potental V(x 1 x 2 ), the Born ampltude s f B = M red 2π Ṽ(q) δ s 1,s1δ s 2,s2 (82) f the two partcles are dstnct, and f they are dentcal fermons then there are two such terms related by partcle permutaton, f B = M red = 1 2 m 2π Ṽ(q) δ s 1,s1δ s 2,s2 + M red = 1 2 m 2π Ṽ( q) δ s,s1δ 2 s 1,s2. (83) Comparng ths Born ampltude to the non-relatvstc lmt of the tree-level QED ampltude(80), wemmedatelyseethattheqedampltudesaspecalcaseofthebornampltude for Ṽ(q) = + e2 q 2. (84) Fourer transformng ths formula back to the coordnate space gves us the good old Coulomb 17
potental for the two electrons, V(x 1 x 2 ) = d 3 q (2π) 3 e(x1 x2) q +e2 q 2 = +e 2 4π x 1 x 2. (85) Now consder the electron-postron elastc scatterng e + e + e + e + and ts nonrelatvstc lmt. Agan, there are two tree dagrams contrbutng to ths process 1 2 1 2 + (86) 1 2 1 2 whch evaluate to M tree = ū 1 (eγ µ)u 1 v 2 (eγ ν )v 2 gµν t + v 2 (eγ µ )u 1 ū 1 (eγ ν)v 2 gµν s. (87) The overall mnus sgn of the frst term here s due to the outgong-postron-to-ncomngpostron lne n the frst dagram; n the second dagram, the ncomng electron-to-ncomngpostron or the outgong-postron-to-outgong-electron lnes do not carry mnus sgns. Ths tme, there s no symmetry between the two dagrams (86), and the correspondng ampltudes have rather dfferent non-relatvstc lmts. In partcular, the denomnator of the frst dagram becomes t q 2 m 2 (n absolute value) whle the second dagram s denomnator has a much larger value s (2m) 2. Consequently, the non-relatvstc electron-postron scatterng s domnated by the t-channel dagram, thus M non.rel. tree e2 t ū 1 γ µu 1 v 2 γ µ v 2. (88) Moreover, n the non-relatvstc lmt ū 1 γ0 u 1 +2m δ s 1,s 1, v 2 γ 0 v 2 +2m δ s 2,s2, (89) 18
whle for µ 0 ū 1 γu 1, v 2 γv 2 = O(p) m, (90) hence M non.rel. tree e2 t 2mδ s 1,s1 2mδ s 2,s2 + 4m2 e 2 q 2 δ s 1,s 1 δ s 2,s 2, (91) or n the non-relatvstc normalzaton f non.rel. tree + me2 4πq 2 δ s,s1δ 1 s 2,s2. (92) Comparng ths QED ampltude to the Born ampltude (82) for dstnct fermons (snce an e s dstnct from an e + ), we see that they agree for Ṽ(q) = e2 q 2. (93) In coordnate space terms, ths means the attractve Coulomb potental V(x 1 x 2 ) = e 2 4π x 1 x 2. (94) Thus QED perturbaton theory confrms the oldest law of electrostatcs: the lke-sgn charges repel, whle the unlke-sgn charges attract. However, ths rule works only for the forces arsng from exchanges of vrtual odd spn bosons such as photons. The forces arsng from exchanges of vrtual even spn bosons such as scalar mesons, or gravtons do not change sgn when one of the two partcles s replaced wth ts ant-partcle. Thus, the gravty force s always attractve. Lkewse, the Yukawa force due to an soscalar scalar meson s attractve for all combnatons of nucleons and antnucleons NN, NN, or NN. 19
Yukawa Potental The Yukawa theory and theyukawa potental aredscussed n detal n 4.7 of the Peskn and Schroeder textbook, so n these notes let me smply hghlght the dfferences between the Yukawa theory and the QED. Instead of the EM feld, the Yukawa theory has a scalar feld φ, thus L = 2 1 ( µφ) 2 1 2 m2 φ 2 + Ψ( M)Ψ gφ ΨΨ. (95) (For smplcty, I assume a sngle fermon speces.) The Feynman rules of the Yukawa theory have the same fermonc propagators, external lne factors, and sgn rules as QED, but nstead of photon propagators t has scalar propagators drawn as dotted lnes φ q φ = + q 2 m 2 +0, (96) and nstead of the fermon-antfermon-photon vertces of QED the Yukawa theory has fermon-antfermon-scalar vertces β = gδ βα. (97) α wthout the γ µ matrces. Consequently, evaluatng the scalar analogues of the dagrams (72) and (86) for the ff ff and f f f f scatterng processes, we obtan M(ff ff) = g2 t m 2 ū 1 u 1 ū 2 u 2 + g 2 u m 2 ū 2 u 1 ū 1 u 2. M(f f f f) = + g2 t m 2 ū 1 u 1 v 2 v 2 g 2 s m 2 v 2 u 1 ū 2 v 2. (98) In the non-relatvstc lmt ū(p,s )u(p,s) +2Mδ s,s but v(p,s)v(p,s ) 2Mδ s,s, (99) 20
whle hence v(p 2,s 2 )u(p 1,s 1 ), ū(p 1,s 1)v(p 2,s 2) = O(p) M, (100) M(ff ff) 4M2 g 2 M(f f f f) 4M2 g 2 t m 2 δ s 1,s1δ s 2,s2 + 4M2 g 2 u m 2 δ s 2,s1δ s 1,s2, t m 2 δ s 1,s1δ s 2,s2 + g 2 s m 2 O(p2 ). (101) Assumng the scalar s much lghter than the fermon, m M, and takng the fermons 3-momenta p,p to be O(m) M, we have 1 t m 2 hence 1 q 2 +m 2 1 M 2, 1 u m 2 1 q 2 +m 2 1 M 2, but 1 s m 2 +1 4M 2, M(ff ff) + 4M2 g 2 q 2 +m 2 δ s,s1δ 1 s 4M2 g 2 2,s2 q 2 +m 2 δ s,s1δ 2 s,s2, 1 M(f f f f) + 4M2 g 2 q 2 +m 2 δ s,s1δ 1 s + 0. 2,s2 In the non-relatvstc normalzaton, these ampltudes become (102) (103) Mg 2 f(ff ff) + 4π(q 2 +m 2 ) δ s,s1δ 1 s 2,s2 f(f f f f) Mg 2 + 4π(q 2 +m 2 ) δ s,s1δ 1 s,s2, 2 whch agree wth Born ampltudes for Mg 2 4π( q 2 +m 2 ) δ s 2,s1δ s 1,s2, (104) Ṽ ff (q) = Ṽf f = g2 q 2 +m 2. (105) Fourer transformng ths formula back too coordnate space gves us the Yukawa potental V ff (x 1 x 2 ) = V f f(x 1 x 2 ) = g2 4π e mr r for r = x 1 x 2. (106) Note the sgns of these potentals two fermons attract each other, and a fermon and an antfermon also attract each other! 21