1 Renormalization of Yukawa theory

Transcription

1 Quantum eld Theory-II Problem Set n. 5 - Solutons UZH and ETH, S-2016 Prof. G. Isdor Assstants: A. Greljo, D. Marzocca, J. Shapro Due: Renormalzaton of Yukawa theory The lagrangan densty s gven by L 1 2 µφ µ φ 1 2 m2 Sφ 2 + ψ(/ m )ψ g ψγ 5 ψφ. (1) 1.1 eynman rules The frst two terms are just the free lagrangan for a real scalar feld, whch gves the eynman rule for the propagator p 2 m 2. (2) S + ɛ Then we recognze the free lagrangan for a spnor feld, whose propagator s also well-known ( /p + m) p 2 m 2. (3) + ɛ The nteracton term descrbes a vertex between a fermon, an antfermon and a scalar whch ncludes a γ 5 matrx. In order to get the correct factors we note that, once we wrte the path ntegral W [η, η, J] Dψ D ψ Dφ e ( S[ψ, ψ, φ] + Jφ + ψη + ηψ d 4 x ) e S nt [ 1 δ δ η, 1 δ δη, 1 δ δj ] W 0 [η, η] W 0 [J], (4) the connected three-pont functon at tree level s gven by the term ( g) d 4 x 1 δ δη(x) γ 1 δ 1 δ 5 δ η(x) δj(x) W 0[η, η] W 0 [J], (5) where all dervatves ht only the exponentals. Gven that W 0 [J] e 1 2 [d 4 x]j(x 1 )G S (x 1 x 2 )J(x 2 ), W 0 [η, η] e [d 4 x] η(x 1 )G (x 1 x 2 )η(x 2 ), (6) 1

2 we fnd g [d 4 x] η(x 1 )G (x 1 x)γ 5 G (x x 2 )η(x 2 )G S (x x 3 )J(x 3 ), (7) whch once sources are removed gves 0 T{ ψ(x 2 )ψ(x 1 )φ(x 3 )} 0 C g [d 4 x] G (x 1 x)γ 5 G (x x 2 )G S (x x 3 ) + O(g 2 ). We thus read the eynman rule Superfcally dvergent ampltudes (8) gγ 5. (9) We wll tackle ths problem n a generc number of spacetme dmensons d, and later specfy our results to d 4. The superfcal degree of dvergence of a loop dagram s gven by the number of (postve) powers of the loop momenta k that appear n the loop ntegral, ncludng the dfferentals d d k. In order to dscuss the renormalzablty of the theory, we only need to consder 1PI dagrams where all nternal propagators belong to a loop. Snce, for large k, a loop scalar propagator goes lke k 2 and a loop fermon propagator as k 1 (all external momenta can be gnored n the UV regon where k j p ), the superfcal degree of dvergence of a 1PI dagram s gven by Sup dl P 2P S, (10) where L s the number of loops and P S (P ) the number of scalar (fermon) nternal propagators. The number of loops, on the other hand, s the number of unconstraned nternal momenta k whch s gven by L P + P S V + 1, (11) where V s the number of vertces. Indeed P + P S s the number of nternal lnes (momenta to be constraned) and each vertex provdes a momentum conservaton constrant, ncludng the total momentum conservaton one whch cannot be used 1 Techncally speakng, the dagram wthout sources s related to a scatterng ampltude va the LSZ formula, whch removes external propagators and gves rse to eynman rules for external legs. Snce we are nterested n the vertex here, we can smply gnore ths step. 2

3 to fx nternal momentum flow. urthermore, each lne orgnatng from a vertex must ether be one end of a propagator or be external. Snce each vertex must attach to one scalar and two fermon lnes, we have Ths allows us to solve whch gves V E S + 2P S, 2V E + 2P. (12) P S V 2 E S 2, P V E 2, L V 2 E S 2 E 2 + 1; (13) Sup d d 4 2 V d 1 2 E d 2 2 E S. (14) In 4 dmensons we therefore fnd that the superfcal degree of dvergence does not depend on the nternal structure of a dagram but from ts external legs only: We thus end up wth the followng table: Sup E E S. (15) Sup E S E where we have allowed only for an even number of external fermons. In other words ths means we have 7 superfcally dvergent n-pont functons,.e. dagrammatcally Sup 4, Sup 3, (16) Sup 2, Sup 1, Sup 0, (17) Sup 1, Sup 0, (18) where Sup ndcates the superfcal degree of dvergence. 3

4 1.3 Renormalzaton In order to absorb the dvergences of our theory we redefne felds and bare parameters n terms of renormalzed quanttes as follows: φ Z φ φ R, ψ Z ψ ψ R, m S Z S Z φ m S,R, m Z Z ψ m,r, g Z g Z ψ Zφ g R. Note that the precse arrangement of scale factors n these defntons s arbtrary, as long as there s one for every quantty. 2 The one we chose gves L 1 2 Z φ µ φ R µ φ R 1 2 Z Sm 2 S,Rφ 2 R + ψ R (Z ψ / Z m,r )ψ R Z g g R ψr γ 5 ψ R φ R. We now proceed to determne the modfed eynman rules whch contan counterterms expandng the renormalzaton constants n powers of g, (19) (20) Z 1 + δz + O(g 2 ). (21) Counterterm eynman rules Scalar propagator 3 By observng that e.g. the free part of the equaton of moton for φ R reads ( Z φ 2 Z S m 2 S,R)φ R 0, (22) and, settng the RHS to δ, n momentum space the equaton for Green s functon s the propagator for the scalar feld reads (Z φ p 2 Z S m 2 S,R) G S (p). (23) G S (p) Z φ p 2 Z S m 2 S,R p 2 m 2 S,R + δz φp 2 δz S m 2 S,R ( ) p 2 m (δz φ p 2 δz S m 2 S,R) S,R p 2 m 2 S,R. (24) 2 Techncally speakng, what we should really do s also shft the value of φ as n φ Z 1/2 φ (φ R + v) so that the lagrangan acqures a term +Y φ R where Y s some combnaton of v and the other renormalzaton factors Z. Ths s n general needed to cancel tadpole dagrams for φ. In the theory at hand these dagrams happen to vansh all by themselves so that shftng the VEV of the feld s not necessary. 3 Throughout ths subsecton we are gong to drop ɛ s n order to keep our expressons as short as possble. 4

5 We thus read the eynman rule for the counterterm (δz φ p 2 δz S m 2 S,R). (25) ermon propagator Proceedng along the same lnes, t s easy to see that the equaton for the fermon propagator reads ormally we then fnd 4 G (p) ( (/p m,r ) whch gves Z ψ/p Z m,r 1 + δz ψ/p δz m,r /p m,r (Z ψ/p Z m,r ) G (p). (26) /p m,r + δz ψ/p δz m,r ( ) ) 1 + (δz ψ/p δz m,r ), /p m,r /p m,r (δz ψ/p δz m,r ). (28) Interacton vertex The counterterm eynman rule for the nteracton vertex can be drectly read from the lagrangan snce whch mples Z g g R ψr γ 5 ψ R φ R g R ψr γ 5 ψ R φ R δz g g R ψr γ 5 ψ R φ R, (29) δz g g R γ 5. (30) 4 The dervaton presented here s compact but nvolves wrtng gbbersh formulae wth matrces n denomnators. An alternatve s to expand solutons on a lnear bass of our matrx space, e.g. G (p) A/p + Bm,R, and compute the requred scalar coeffcents. The soluton we wll fnd heurstcally can also be explctly verfed usng the substtuton ( /p + m,r) /p m,r p 2 m 2,,R whch removes all ambgutes. 5 (27)

6 Note that, although we computed all counterterms as a functon of the renormalzed parameters, these are equal to the bare ones up to hgher order n the couplngs, e.g. g g R + O(g 2 R ) + O(g Rλ R ). Ths means that whenever gnorng hgher order terms we can be cavaler about whether we wrte g or g R, m or m R and so forth. 1.4 Dscusson of renormalzablty Now that we have the eynman rules for every counterterm, we are ready to dscuss how to tackle and hopefully reabsorb the dvergences of our theory Vacuum bubbles We saw that vacuum bubbles have a superfcal degree of dvergence of 4. However, we also know that for every correlaton functon 0 ψ(x 1 )... ψ(y 1 )... φ(z 1 ) δ δ η(x 1 )... 1 δ δη(y 1 )... 1 δ δj(z 1 )... W [η, η, J] η ηj0 W [0, 0, 0] (31) vacuum bubbles exactly cancel between numerator and denomnator. Therefore, once nfntes have been approprately regulated, vacuum bubbles cancel properly for all fnte values of the regulator - never enterng any physcal ampltude Self-energy correctons We now swtch to loop correctons to the scalar and fermon propagators. Snce every two-pont ampltude relatve to the same felds has to have the same dmensons, the product of the loop structure tmes one propagator should be dmensonless. Dagrammatcally Σ S (p 2 ) p 2 m 2 1, S (32) where the last dagram s not a tadpole but rather the self-energy wth ts rght propagator removed, as ndcated by the vertex marked on the left of the bubble. Ths means that the self-energy Σ tself, wthout any propagator attached, must have mass dmenson 2 for the scalar and 1 for the fermon. If we take nto account that Σ must be a Lorentz scalar, t can only have the structure Σ S (p 2 ) A S p 2 + B S m 2 S, Σ (/p) A /p + B m. (33), 6

7 Once the necessary dependence on the external (non-loop) scales p and m has been extracted, the UV-dvergent part of the coeffcents A and B s due to the k regon and may therefore be computed neglectng p and m completely. Ths means that the coeffcent of the regulated nfnty (1/ɛ n dmensonal regularzaton, log Λ wth a cutoff, etc...) does not further depend on the momentum p and may be cancelled n all calculatons by fxng one sngle specfc value of the feld and mass renormalzaton constants. More explctly, t wll be suffcent to fx { { δzφ dv(as) + fnte terms, δz ψ dv(a ) + fnte terms, (34) δz S dv(b S ) + fnte terms, δz dv(b ) + fnte terms, to cancel dvergences due to self-energy correctons n every dagram, ndependently of the value of p. How fnte terms are chosen wll defne a renormalzaton scheme Tadpole and three-vertex Whle n general we are allowed to have a counterterm cancellng the dvergent part of the tadpole, we dd not nclude one when renormalzng our lagrangan. A nave motvaton for ths choce s that computng the only dagram that contrbutes to the scalar one-pont functon at one loop gves 5 d 4 g k Tr[γ 5 (/k + m )] (2π) 4 k 2 m 2 0. (35) Indeed, even n 4 dmensons where the value of ths dagram s expected to be nfnte, Tr[γ 5 ] Tr[γ µ γ 5 ] 0 and the ntegrand vanshes dentcally. Now one could wonder f ths s just a lucky concdence or there s some deep reason why the tadpole vanshes. Ths ssue should not be gnored for two reasons: 1. we would lke to dscuss renormalzablty at all orders n perturbaton theory, and the result we just found only holds at one loop; 2. t mght turn out that other n-pont functons vansh because of some property of our theory, and f ths s the case we do not want to be computng very complcated zeroes. The fact that a γ 5 s nvolved n the explct example at hand should already pont towards the soluton. A quck look at our lagrangan reveals that the spnor 5 Here we are allowed to gnore dmensonal regularzaton, as wll be dscussed later. 7

8 blnear nvolved n the nteracton term s ψγ 5 ψ, whch we know to transform as a pseudoscalar under party transformatons: 6 P ψ(x)γ 5 ψ(x)p 1 ψ(px)γ 5 ψ(px), (36) where P(t, x) (t, x). Therefore f we assgn negatve party to our scalar feld, Pφ(x)P 1 φ(px), (37) our lagrangan satsfes PL(x)P 1 L(Px) and the acton s nvarant under party transformatons. Ths symmetry has to be respected by correlaton functons, whch means that n our nteractng theory 0 φ(k) 0! 0 Pφ(k)P φ(pk) 0. (38) Snce for a massve partcle we are always allowed to compute n ts rest frame, 0 φ(m S, 0) 0 0 φ(m S, 0) 0 0. (39) Ths mples that the scalar one-pont functon vanshes at all orders n perturbaton theory. Smlarly, 0 φ(m S, 0)φ(E p, p)φ(e p, p) 0! 0 Pφ(m S, 0)φ(E p, p)φ(e p, p)p 1 0 ( ) 3 0 φ(m S, 0)φ(E p, p)φ(e p, p) 0 0. (40) Thus, we do not need to compute the three vertex because we have the exact result 0. (41) ermon-fermon-scalar vertex The fermon-fermon-scalar vertex s expected to be dvergent. However, the counterterm assocated wth δz g n the lagrangan has exactly the same form as the tree-level vertex and may be used to remove ths dvergency by requrng fnte. (42) 6 See the secton about spnor blnears and dscrete transformatons of any QT book for detals, e.g. Peskn sec. 3.6, eq. (3.131). 8

9 Agan we note that, for the computaton of the dvergent part, all external momenta n the ntegral representaton of the loop dagrams can be gnored: t s thus suffcent to mpose the condton wth a specfc momentum assgnment to cancel all nfntes assocated wth 1PI loop correctons to the fermon-fermon-scalar vertex scalar vertex Last, but not least, we come to the most trcky vertex. So far we found no reason to argue that the φ 4 Green functon should vansh. Nether can we fnd an argument for a softenng of ts dvergence, whch s expected to be logarthmc due to power countng (Sup 0). In order to check that ths vertex s ndeed dvergent we can wrte down ts expresson at one loop: + permutatons + O(g 5 ), (43) where the omtted terms nclude all other nequvalent cyclc permutatons of the four external momenta (n total, there are 3! of these confguratons). We thus have d g 4 4 k Tr[γ 5 (/k + m )γ 5 (/k + /p 1 + m )γ 5 (/k + /p 12 + m )γ 5 (/k + /p m (2π) 4 [k 2 m 2 ][(k + p 1) 2 m 2 ][(k + p 12) 2 m 2 ][(k + p 123) 2 m 2 (44) where p 1...n p p n. or k µ we are allowed to neglect all of the components of the external momenta and all masses at the numerator; however we keep fermon masses n the denomnator because there s stll a chance for k 2 to be small even f all components of k are large. 7 Rememberng that the matrx γ 5 n 4 dmensons antcommutes wth all other γ matrces and that /k/k k 2 we fnd g 4 d 4 k k 4 (2π) 4 [k 2 m 2 g4 ]4 d 4 k E (2π) 4 k 4 E [ke 2 + m2 g4 ]4 where we have performed a Wck rotaton k 0 ke 0 to get rd of the dfferent sgn n the metrc (k E s eucldean) and we rescaled to x k E /m n the last step. 7 Ths ndeed generates IR dvergences f m 0. d 4 x (2π) 4 x 4 (x 2 + 1) (45) 9

10 Swtchng to sphercal coordnates n 4 dmensons one fnds g 4 Ω4 r 7 (2π) 4 0 (r 2 + 1) 4dr, (46) whch dverges logarthmcally. Ths s a dsaster. Our theory contans genune nfntes that cannot be canceled through the renormalzaton program, and wll propagate to physcal observables. If we want to keep gong, we need an extra counterterm to fx ths problem. Such a counterterm need have the form δz φ 4 (x), (47) n our lagrangan. We observe that ths arses naturally f the theory contans a four-pont nteracton,.e. f we start from L 1 2 µφ µ φ 1 2 m2 Sφ 2 + ψ(/ m )ψ g ψγ 5 ψφ + λ 4! φ4, (48) namely from the lagrangan densty that contans all possble nteractons wth dmensonless couplngs n d Modfcatons due to the φ 4 term rom now on we shall work wth the lagrangan specfed by eq. (48). Due to the φ 4 nteracton, the theory acqures a new eynman rule λ. (49) as can be readly verfed by takng the functonal dervatves n λ [ ] 1 d 4 δ x W 0 [η, η] W 0 [J]. (50) 4! δj(x) Once one renormalzes defnng the counterterm reads λ Z λ Zφ 2 λ R, (51) λ R δz λ. (52) 10

11 nally, we note that the presence of the new nteracton does not change our rules for power countng. Indeed eq. (10) holds unaltered and although now the graph equatons read L P + P S V g V λ + 1, 4V λ + V g E S + 2P S, 2V g E + 2P, (53) the result for Sup n terms of L, E S and E s the same as before loop renormalzaton We now proceed to determne the dvergent parts of the counterterms at one loop. The two couplng constants are ndependent expanson parameters. Therefore, once the counterterms for both of them are present n the lagrangan, the value of the δz s can be computed at each order λ n g m separately. Throughout the calculatons that follow, we are gong to use dmensonal regularzaton. More precsely, we are gong to promote our momenta k µ to have d components and consder scatterng ampltudes as analytc functons of d. Loop ntegrals wll thus be regulated by carryng out the substtuton d 4 k d d (2π) k 4 (2π). (54) d Settng d 4 2ε, sngulartes wll be manfest as poles n ε that make ntegrals blow up when d 4 (ε 0). As suggested by the exercse and prevously announced, we are not gong to compute them terms that are fnte for ε 0 at all. Settng counterterms to exactly cancel the ε poles (.e. all we compute) and nothng more, s known n the lterature as mnmal subtracton or MS renormalzaton scheme. When usng dmensonal regularzaton n theores that contan fermons one should worry about extendng the γ matrx formalsm to d-dmensons. Whle ths s relatvely straghtforward for γ µ matrces, more than one soluton s avalable. Moreover, t s possble to show that one s forced to gve up some of the fundamental propertes of the γ 5 matrx when workng n d-dmensons. The dfference among all of these possbltes however shows up as terms that are hgher order n ε wth respect to the leadng poles ermon propagator We start wth one-loop correctons to the fermon propagator, whch are gven by +. (55) 11

12 The relevant dagram wth both propagators amputated reads d d k (/k + m ) g2 (2π) dγ 5 k 2 m 2 γ 5 (k p) 2 m 2. (56) S In order to compute the dvergent part we may use {γ 5, γ µ } 0, γ5 2 1 to fnd g2 Introducng eynman parameters 1 d d k (2π) d /k m [k 2 m 2 ][(k p)2 m 2 S ]. (57) /k m [k 2 m 2 ][(k p)2 m 2 S ] /k m dx 0 [k 2 2xkp + xp 2 (1 x)m 2, (58) xm2 S ]2 and wth the change of varables k k + xp we fnd 1 d d k /k + x/p m g2 dx, (59) 0 (2π) d [k 2 ] 2 where xm 2 S + (1 x)m 2 x(1 x)p 2. (60) Now the ntegrand n /k s odd and vanshes, whle the rest can be evaluated accordng to standard procedures descrbed e.g. n appendx (A.4) of Peskn&Schröder: d d k 1 (2π) d [k 2 ] 2 (4π) d/2γ(2 d/2) d/2 2 Expandng at leadng order around ε 0 gves Thus settng (4π) 2 εγ(ε) ε. (61) Γ(ε) 1 ε + O(1), x ε 1 + O(ε), (62) g dx(x/p m (4π) 2 ) + O(1) ε 0 ( ) g2 1 p / (4π) 2 ε 2 m + O(1). (63) δz ψ g2 1 (4π) 2 2ε, δz g2 1 (4π) 2 ε, (64) makes the fermon propagator fnte at one loop. 12

13 1.5.2 Scalar propagator One-loop correctons to the scalar propagator are of order g 2 and λ, namely ++. (65) The fermon loop correcton reads d ( )g 2 d k Tr[γ 5 (/k + m )γ 5 (/k + /p + m )] (2π) d [k 2 m 2 ][(k + p)2 m 2 ]. (66) Once more we antcommute γ 5 and perform 4-dmensonal Drac algebra to fnd Tr[γ 5 (/k + m )γ 5 (/k + /p + m )] Tr[(/k m )(/k + /p + m )] 4(k 2 + kp m 2 ) ; (67) ntroducng eynman parameters one then has 1 4g 2 0 We then shft k k xp to get d d k (2π) d dx k 2 + kp m 2 [k 2 + 2xkp + xp 2 m 2 ]2 d d k (2π) d k 2 + kp m 2 [k 2 + 2xkp + xp 2 m 2. (68) ]2 d d k (2π) d k 2 x(1 x)p 2 m 2 [k 2 ] 2, (69) where we gnored the lnear terms n k n the last step because they ntegrate to zero, and we set m 2 x(1 x)p 2. (70) Thus usng the conventonal results for d-dmensonal ntegrals d d k k 2 d (2π) d [k 2 ] [x(1 2 x)p2 + m 2 d k 1 ] (2π) d [k 2 ] 2 d (4π) d/2 2 Γ(1 d/2) d/2 1 [x(1 x)p 2 + m 2 ] (4π) d/2γ(2 d/2) d/2 2, (71) whch usng the Γ functon property Γ(1 + x) xγ(x) Γ(2 d/2) (1 d/2)γ(1 d/2), (72) 13

14 may be rewrtten as (4π) d/2γ(2 d/2) d/2 2 [ ] d/2 1 d/2 + x(1 x)p2 + m 2 Everywhere except n the argument of Γ we may now set ε 0 to fnd 1. (73) g dx[ 2 + x(1 x)p 2 + m 2 (4π) 2 ] ε 0 g dx[3x(1 x)p 2 m 2 g 2 1 (4π) 2 ] + O(1) ε 0 (4π) 2 ε 2[p2 2m 2 ] + O(1). (74) The snal dagram for the scalar s very smple to compute: 1 λ 2 d d k (2π) d k 2 m 2 S λ Γ(1 d/2) (m 2 (4π) d/2 2 S) d/2 1 λ 1 (4π) 2 2ε m2 S + O(1). (75) The renormalzaton constants can therefore be computed by requrng the counterterm to cancel the loop dvergences:! + O(1), (76) δz φ Yukawa couplng g2 2 (4π) 2 ε, δz S λ 1 (4π) 2 2ε g2 4 (4π) 2 ε. (77) The fermon-fermon-scalar vertex receves only one 1PI correcton of order g 2 (wth respect to the orgnal order g); dagrams contanng λ need to attach through extra g-lke vertces n order to be rreducble. Thus we only need compute one correcton: +. (78) 14

15 Agan we amputate the dagram and make ts ntegral representaton explct d g 3 d k (2π) γ (/p 1 /k + m ) (/p 2 /k + m ) d 5 (p 1 k) 2 m 2 γ 5 (p 2 k) 2 m 2 γ 5 k 2 m 2. (79) S Now we note that, from power countng, expandng the product at the numerator only the term wth two /k has a chance to dverge. Thus, mpunely antcommute the γ 5 s, d g 3 d k k D 2 1 (k p 1 ) 2 m 2, γ 5 (2π) d D 1 D 2 D + O(1) where D 2 (k p 2 ) 2 m 2, D k 2 m 2 S, (80) and addng to the ntegrand a non-dvergent term m 2 S /(D 1D 2 D) we get d g 3 d k 1 γ 5 + O(1). (81) (2π) d D 1 D 2 Introducng eynman parameters and completng the square yelds d d k 1 d d k dx 1 dx 2 δ(1 x 1 x 2 ) (2π) d D 1 D 2 (2π) d [k 2 2k(x 1 p 1 + x 2 p 2 ) + x 1 p x 2p 2 2 m2 ]2 d d k dx 1 dx 2 δ(1 x 1 x 2 ), (82) (2π) d [k 2 ] 2 where m 2 x 1 (1 x 1 )p 2 1 x 2 (1 x 2 )p 2 2 2x 1 x 2 p 1 p 2. (83) Carryng out the ntegral and expandng n ε d d k 1 (2π) d D 1 D 2 (4π) d/2γ(2 d/2) dx 1 dx 2 δ(1 x 1 x 2 ) d/2 2 1 (4π) 2 ε + O(1), (84) whch allows us to conclude g 2 1 gγ 5 + O(1). (85) (4π) 2 ε The counterterm s readly determned by δz g g2 1 (4π) 2 ε. (86) 15

16 1.5.4 φ 4 vertex nally we come to the last pece of our calculaton: + 5 permutatons (87) It s apparent that ths would be also by far the hardest, but for the fact that we know t to be just dvergent enough to generate a sngularty. Ths mples that for the dagram contanng the fermon loop we may mmedately consder only loop momenta n the numerators. We can gnore external momenta n denomnators as well: ultmately they would end up n a that would be expanded to 1. Thus all permutatons of external momenta contrbute wth the same pole g 4 4g 4 d d k Tr[γ 5 /kγ 5 /kγ 5 /kγ 5 /k] (2π) d [k 2 m 2 4g 4 ]4 d d k [k 2 m 2 ]2 (2π) d [k 2 m 2 4g4 ]4 d d k (2π) d k 4 [k 2 m 2 ]4 d d k 1 (2π) d [k 2 m 2 1 ]2 4g4 (4π) 2 ε. (88) The three dagrams wth scalars only also gve the same contrbuton to the pole, namely (λ) 2 Thus the counterterm reads d d [ k (2π) d δz λ 3 1 (4π) 2 ε k 2 m 2 S ) (8 g4 λ λ ] 2 λ λ (4π) 2 1 ε. (89). (90) 16