A Proof of Factorization Theorem of Drell Yan Process at Operator Level

Transcription

1 Commu. Theor. Phys. 65 ( Vol. 65, No. 2, February, 206 A Proof of Factorzato Theorem of Drell Ya Process at Operator Level Gao-Lag Zhou ( Ô Key Laboratory of Froters Theoretcal Physcs, Isttute of Theoretcal Physcs, Chese Academy of Sceces, Beg 0090, Cha (Receved October 2, 205 Abstract A alteratve proof of factorzato theorem for Drell Ya process that works at operator level s preseted ths paper. Cotrbutos of teractos after the hard collso for such clusve processes are proved to be caceled at operator level accordg to the utarty of tme evoluto operator. After ths cacellato, there are o loger leadg pch sgular surface Glauber rego the tme evoluto of electromagetc currets. Effects of soft gluos are absorbed to Wlso les of scalar-polarzed gluos. Cacelato of soft gluos s attrbute to utarty of tme evoluto operator ad such Wlso les. PACS umbers: 2.39.St, t Key words: QCD factorzato, Drell Ya process Itroducto Factorzato theorem of Drell Ya process used to be a challegg questo for a log tme past several decades. [ 3] Beg dfferet from the cases deep elastc scatterg process [4 5] ad ahlato process of e + e, [6 8] Glauber gluos, that s, space-lke soft gluos wth trasverse mometa much larger tha plus ad mus mometa, ca cause partcular dffculty the proof of factorzato theorem of Drell Ya process. [,9, 3] It was proved that leadg pch sgular surfaces (LPSS caused by Glauber gluos wll cacel out covarat gauge after oe sums over all possble cuts accordg to utarty. [ 3] The proofs appeared Refs. [ 3] work dagram level ad tur out to be complcated. The hadroc tesor of Drell Ya process s: H µν = d 4 xe q x p p 2 J µ (xj ν (0 p p 2, ( where J µ (x s the electromagetc curret, p ad p 2 deote the tal hadros, q represets the mometum of the vrtual photo. It s coected wth the dfferetal cross secto of Drell Ya process by the formula: dσ dq 2 dy = 4πα2 Q 2 q 3s d 2 q (2π 4 Hµν q 2 ( qµ q ν q 2 g µν, (2 where y = (/2lq + /q, Q q represets the electrc charge of quarks. Mometa of tal hadros become lght-lke the hgh eergy lmt Q, where Q = (q 2 /2. They are deoted as p µ /Q = (p+, p, p /Q (, 0, 0, ad p µ 2 /Q (0,, 0 such lmt. The degrees of freedom that cotrbute to LPSS of Drell Ya process are: ( Collear partcles, that s, partcles wth mometa scale as p 2 /Q 2 0 ad l (p 0 + p /(p 0 p at the lmt Q ; ( Hard partcles, that s, partcles wth mometa scale as p 2 /Q 2 0 at the lmt Q ; ( Soft partcles, that s, partcles wth mometa scale as p µ /Q 0 at the lmt Q. Dagrams that gve leadg order cotrbutos to Drell Ya process have the same topology as Fg.. [ 3] There are arbtrary umber of soft gluos exchaged betwee collear partcles, whch s deoted as S. Soft gluos do ot coect to the hard vertex, whch s deoted as H. There are arbtrary umber of scalarpolarzed gluos coected to the hard vertex for each et covarat gauge, whch s proved to be approprate for Drell Ya process. [ 3] There s oly oe physcal collear partcle (that s quark or trasverse polarzed gluo coect to the hard vertex for each collear et. Fg. The topology of dagrams that cota leadg pch sgulartes Drell Ya process. To prove the factorzato theorem oe has to show that: Supported by the Natoal Natural Scece Foudato of Cha uder Grat No c 206 Chese Physcal Socety ad IOP Publshg Ltd

2 94 Commucatos Theoretcal Physcs Vol. 65 ( Soft gluos exchaged betwee collear partcles do ot volate factorzato; ( Scalar polarzed gluos coected to the hard vertex do ot volate factorzato. Scalar polarzed collear gluos are absorbed to parto dstrbuto fucto (PDF accordg to Ward detty. [ 3,9] For the couplg betwee a soft gluo wth mometum q ad a partcle collear-to-plus wth mometum p ad mass m, people take the Grammer Yee: [5,8,20] A µ s A s, (p + q2 m 2 2p + q + p 2 m 2 (3 to factorze the soft gluo from the collear partcle ad fally prove the cacellato of soft gluos, where A s deotes the soft gluo. There are specal kds of soft gluos that volate such approxmato, whch are amed as Glauber gluos lteratures. [] They are space lke soft gluos wth Λ 2 QCD q2 Q 2 ad q 0 E q. For example, oe may cosder the couplg betwee a partcle collear-to-plus ad a soft gluo wth mometum scale as (q +, q, q /Q (λ 2, λ 2, λ where λ Λ QCD /Q. The Grammer Yee approxmato does ot work such couplg as 2p + q + p 2 m 2 q 2 2 p q. For processes wthout tal hadros movg dfferet drecto lke the ahlato of e + e, there are ot leadg pch sgulartes caused by Glauber gluos ad oe ca deform the tegral path to avod such rego. [6 8] For processes wth tal collear hadros movg dfferet drectos lke the Drell Ya process, leadg pch sgulartes caused by Glauber gluos [] do exst ad ths ca cause partcular dffculty the proof of factorzato theorem. It s well kow that QCD factorzato s destroyed by Glauber gluos hadro-hadro collso process wth detected hadros the target mometa rego. [2 25] To over come ths dffculty Drell Ya process, people [ 3] sum over all possble fal cuts ad show that LPSS caused by Glauber gluos cacel out after ths summato. Oe ca the deform the tegral path so that the Grammer Yee approxmato works. Soft gluos factorze from collear partcles ths approxmato ad wll cacel out. Therefore utarty plays a mportat role such proof. I ths paper, we otce that electromagetc currets are color sglets ad the summato over all possble fal hadros equals to summato over all possble fal states ths process. Cosequetly, utarty ca be mplemeted at operator level for Drell Ya process. Wth ths sght, we preset a alteratve proof of factorzato theorem of Drell Ya process at operator level. Ths proof does ot deped o explct processes as t works at operator level ad ca be easly exted to some other processes. It also hghlghts theoretcal aspects ad physcal pctures of factorzato theorem more clearly compared to covetoal dagram level approach. We frst show the cacellato of effects of teractos after the hard collso Eq. (. Ths s the drect result of of tme evoluto operator of QCD. After such cacellato, the δ-fucto of eergy coservato at the vertex x (x 0 x0 becomes x 0 dx0 e p0 x0 e p 0 x0 /(p 0 + ǫ, where p represets the total mometa that flow to the vertex x, x deotes the vertex at whch the hard photo s produced. If a soft gluo wth mometum q s couple to partcles collear-to-plus at the vertex x, the we ca drop the plus mometa of q s + the collear teral les. That s qs 0 q the term p 0 + ǫ. Thus sgular vertexs of qs produced by teral les collear-to-plus all locate the lower half plae f we defe q s as flow to the vertex x. Ths s our ma strategy to prove the cacellato of LPSS caused by Glauber gluos. We wll show ths explctly ths paper. The cocluso s that we ca deform the tegral path of qs to the upper half plae so that the Grammer Yee approxmato works, where q s s defed as flow to the collear partcles. We work the covarat gauge such proof. Couplgs betwee soft gluos ad collear partcles ca the be absorbed to Wlso les of scalarpolarzed gluos. Dagrams wth dscoected hard parts do ot cotrbute to the process at leadg order Λ QCD /Q, [26 27] we brg effectve hard vertex to descrbe the hard sub-processes. Effects of scalar-polarzed collear gluos are the absorbed to Wlso les the effectve hard vertex accordg to gauge varace. These Wlso les ca be foud lteratures, e.g. Refs. [ 3, 8 9, 28 30]. We work wth the boudary codto A µ ( = 0 ths paper, thus the trasverse Wlso les Ref. [3] do ot dsturb us. We the factorze ad cacel effects of soft gluos at operator level the hadroc tesor. Factorzato theorem s the proved based o these results. The paper s orgazed as follows. We prove the cacelato of effects of teractos after the collso the hadroc tesor at operator level Sec. 2. I Sec. 3, we prove that oe ca deform the tegral path so that the Grammer Yee approxmato works the hadroc tesor (. I Sec. 4, we brg the Wlso les of scalar polarzed gluos to absorb the effects of soft gluos. We also costruct effectve acto that descrbe the electromagetc scatterg processes betwee dfferet ets ths secto. I Sec. 5, we fsh the proof of factorzato theorem of Drell Ya process. Some dscussos are preseted Sec Cacellato of Effects of Iteractos After the Collso I ths secto we prove the cacellato of cotrbutos of teractos after the collso the hadroc tesor (. To make the proof clear, we start from the S-matrx of a explct process: [32] S βα = Ψ + β Ψ α = lm Φ β e H freet e H(2T e H freet Φ α, (4

3 No. 2 Commucatos Theoretcal Physcs 95 where Ψ α ad Ψ+ β stad for the ad out states respectvely, α ad β stad for all possble degrees of freedom of tal ad fal states, H deotes the full Hamltoa of the system, H free deotes the free Hamltoa of leptos ad hadros (The weak teracto s ot cocered ths paper, Φ α ad Φ β are egestates of H free. Strctly speakg, both tal ad fal states are wake packets wth arrow wdths. Thus oe should cosder Φ α ad Φ β as the lmt whe the wdths of the wave packets ted to 0. The Hamltoa H free s otrval, however, t does ot affect the perturbatve parts of the cross secto as show by the followg texts. We ow cosder the Drell Ya process, for whch we have: Φ α = p,p 2 Φ(p, p 2 p p 2, (5 Φ β = L,H Φ (L, H LH, (6 H = H 0 + (H t QCD + (H t QED, (7 where p ad p 2 represet the tal hadro states, L represets the fal lepto states, H represets the fal hadro states, Φ ad Φ stad for the wave fuctos of tal ad fal wave packets, H 0 s the free Hamltoa of fermos ad gauge partcles, (H t QCD ad (H t QED are the teracto Hamltoa of QCD ad QED respectvely. We keep the tree level result of QED ad full result of QCD at ths step ad have: S βα = Q q e 2 lm d 4 x p,p 2 Φ(p, p 2 L G µν (q ν (q 0 H e H freet e HQCD(T x0 d 4 q e q x (2π 4 (Φ (L, H L,H J µ ( xe HQCD(x0 +T e H freet p p 2, (8 where Q q represets the electrc charge of the quarks, J µ ad µ represet the quark ad lepto currets respectvely, G µν (q s the propagator of photos wth q the mometa of the vrtual photo. The fal leptos are detected. We take the lmt that fal leptos are separate from other partcles ad have: S βα = Q q e 2 lm d 4 x p,p 2 Φ(p, p 2 L G µν (q ν (q 0 H µν (q = lm X d 4 q e q x (2π 4 (Φ (H H H e H freet e HQCD(T x0 J µ ( x e HQCD(x0 +T e H freet p p 2. (9 We the take the modular square of ths S-matrx ad sum over all possble fal hadro states. After ths, we ca wrte the hadroc tesor ( as: H µν (q = P,p,k Φ (P + k, P kφ(p + p, P p lm d 4 xe q x P + k, P k e H freet e HQCD(x0 +T J µ ( x e HQCDx0 J ν ( 0e HQCDT e H freet P + p, P p, (0 where we have made use of the utarty of the operator: ( lm eh freet e HQCDT. ( To see ths oe oly eeds to otce that: δ H,H = (Ψ + (H, Ψ + (H d 4 xe q x p p 2 e H freet e H0T U QCD (x0, J µ I (x X X J ν I (0U QCD (0, e H0T e H freet p p 2 = lm = (Φ(H( lm eh freet e HQCDT, ( lm eh freet e HQCDT Φ(H = (Φ(H, Φ(H, (2 where Ψ + (H ad Ψ + (H are out states formed by hadros, Φ(H ad Φ(H are egestates of H free formed by hadros. We the take the wdth of tal wave packets to be 0 ad sert the complete base formed by all possble quark ad gluo states to above formula ad have: H µν (q = lm d 4 xe q x p p 2 e H freet X e HQCD(x0 +T J µ ( xe HQCD x 0 X X J ν ( 0 e HQCDT e H freet p p 2, (3 where X deotes arbtrary quark ad gluo states. Oe should cosder X as wave packets wth very arrow wdths. We wrte the hadroc tesor the teracto pcture, whch: J µ I (x eh0x0 J µ ( x e H0x0, (4 U QCD (t, t 2 e H0t e HQCD(t t2 e H0t2 { = T exp t wth H 0 the free Hamltoa. We have:,m p,k,x t 2 } dt(h I QCD (t, (5 d 4 xe q x p p m e H0T e H freet p p 2 p m2 2 e H0T e H freet p 2 p e H freet e H0T k k p 2 e H freet e H0T k 2 k 2 2

4 96 Commucatos Theoretcal Physcs Vol. 65 k k k 2 k2 2 U QCD (x0, J µ I (x X X Jν I (0U QCD(0, p pm p 2 pm2 2, (6 where p ad k deote arbtrary oe-quark state or oe-gluo state. We see that H free does ot affect the matrx-elemets betwee parto states above hadroc tesor. Cotrbutos of teractos after the hard collso cacel out the hadroc tesor (6. 3 Deformato of Itegral Path We ow prove that we ca deform the tegral path so that the Grammer Yee approxmato works Eq. (. Our cocluso s that whle dealg wth the couplg betwee a soft gluo exchage betwee dfferet ets wth mometum q ad partcles collear-to-plus, oe ca deform the tegral path of q to the upper half plae so that the Grammer Yee approxmato works, where q s defed as flow to the partcles collear to the plus drecto. We work covarat gauge ths secto. I the couplg betwee a partcle collear-to-plus wth mometum p ad a soft gluo wth mometum q whch s deoted as A s, there s a specal rego of q: q q + q Λ QCD, (7 whch s called the Glauber rego. [] The Glauber gluos are resposble for elastc scatterg process betwee collear partcles movg dfferet drectos. The Grammer Yee approxmato: A µ s A s, (p + q2 m 2 2p + q + p 2 m 2, (8 whch s of great mportace the proof of QCD factorzato s broke by the Glauber gluos. However, we wll show that LPSS caused by Glauber gluos cacel out Eq. (6 ad oe ca deform the tegra path so that Grammer Yee approxmato does work. We start from the matrx-elemet: X J µ I (xu QCD(x 0, p pm p 2 pm2 2 = X T {J µ I (xu QCD(x 0, } p p m p 2 p m2 2, (9 where T deotes the tme orderg operator, x 0 /Q. We the expad the matrx elemet accordg to the couplg costat g ad cosder the perturbatve seres. Such perturbatve seres ca be calculated accordg to the usual dagram skll. The dfferece s that the tme at whch QCD teractos occur s costraed to be before the tme x 0 these seres. We cosder a cocrete vertex V. Mometa of les that coect to V are deoted as l, whch are defed as flow to the vertex V. Whle calculatg the matrx-elemet (9 accordg to perturbato theory, oe should otce that QCD teractos occur before the hard collso Eq. (9. Thus oe should make the substtuto: d 4 x G (x, x 2 V (x x 0 dx 0 d 3 x G (x, x V (x, (20 at each vertex x the perturbatve seres, where G (x, x 2 deotes possble propagators whch coect x ad other pots, V (x deotes the vertex. Ths s equvalet to substtute the δ-fucto of mometa coservato at each vertex wth the formula: (2π 4 δ (4( l (2π 3 δ (3( e ( l0 +ǫ x 0 l ( ( (2 l0 + ǫ, to calculate the perturbatve seres mometa space, where l are defed as flow to the vertex V. We cosder the case that soft gluos couple to partcles collear-to-plus at a vertex V. Mometa of les that coect to V are deoted as l, whch are defed as flow to the vertex V. We have: δ( ( l3 e l0 +ǫ x 0 ( ( l0 + ǫ = δ ( l3 e ( l0 +ǫ x 0 2 ( ( l + ǫ. (22 We otce that x 0 /Q ad take the approxmato: [ ( exp l 0 + ǫ x 0] [ ( l + exp + ǫ x 0], (23 2 where l µ = (l +, l, l for l s collear to plus ad l = (0, l, l for l s soft. We deote oe of collear teral les that coect to the vertex V as l. We take l as depedet varables. We also take l+ ( as depedet varables. The accordg to the coservato of the thrd compoet of the mometa, we have: l + = l l +, (24 l s assumed to be collear to plus, thus we ca drop the small cotrbutos of above formula to l + plus propagators of partcles collear to plus. Ths s equvalet to make the further approxmato: ( δ l 3 = ( 2δ l + l ( 2δ l+. (25 After the substtuto (22 ad the approxmatos (23 ad (25, we ca wrte the substtuto (2 as: (2π 4 δ (4( l ( (2π 3 δ l+ δ (2( l e ( l+ / 2+ǫ x 0 ( ( l +ǫ.(26

5 No. 2 Commucatos Theoretcal Physcs 97 We should emphasze here that although we start from the hadroc tesor (6 whch correspods to the tmeordered perturbato sklls Ref. [2] as effects of fal state teractos are cacelled Eq. (6, the substtuto (26, however, correspods to the sklls Ref. [3]. It s because that fal poles of ls Eq. (9 cacel out after the substtuto (26 wth l s the mometum of a soft gluo that coect to the vertex V. Ths wll be clear the followg paragraphs. We deote the mometum of a soft gluo that coect to V as l s. There are ot δ-fucto correspodg to coservato of the mus mometa at the vertex V after the substtuto (26. Thus other parts of the et collearto-plus s depedet of ls the matrx-elemet (9. We see that the matrx-elemet (9 ca rely o l s + oly through the propagator of l s ad the other ed of l s. I addto, sgular pots of ls that locate the upper half plae ca be produced oly through the propagator of l s ad the other ed of l s. We the cosder the other ed of l s. If partcles at that ed are all soft (cludg Glauber, the mometa of whch we deote as k, the the part of that ed that deped o l s µ reads: δ 3 ( k l s k0 l0 s + ǫ = δ 3 ( k l s 2 k+ l s + + ǫ. (27 That ed ca rely o ls oly through the fucto: δ 3( k l s. (28 Propagators of k themselves do ot produce sgular pot of ls that locate Glauber rego. Ths s because that all compoets of k are much smaller tha Q ad sgular pots of the type ls l s 2 /Q ca ot be produced by these propagators. Wthout loss of geeralty, we choose k µ ( ad k+ as depedet varables. We have k = l s k, (29 k = k+ (l+ s l s + (k + k. (30 Thus that ed ca rely o ls oly through k. We ext cosder the other ed of k ad repeat the above procedure utl we meet a et collear to a specfc drecto µ = (/ 2(, s(θ, 0, cos(θ (Oe ca always choose the coordate system so that the y-compoet of equal to 0. If θ = π, that s the two ets are back-to-back. The the et collear to the mus drecto s depedet of ls. Thus ls s ot pched the Glauber rego. We ca deform the tegral path of ls to the upper half plae wth radus of order l s to avod the Glauber rego. If θ π ad θ, the we ca boost the referece system to the ceter of mass system of the two ets, whch produce a Loretz factor of order /(s(θ/2. Such operato chages the drectos of the two ets to those: µ = (/ 2(, 0, 0, µ = Λµ ν ν ad µ = (/ 2(, s(θ, 0, cos(θ µ 2 = Λµ ν ν, where Λ µ ν represets the boost operato, = 2. It also chages l s µ to l s µ = Λ µ νls. ν Oe should otce that such operato does ot chage the soft partcles to collear partcles as s(θ/2. I the ew referece system, we ca repeat above procedure. After ths, we boost the referece system to the orgal oe. Such operato chage the quatty l s to ls. Thus ths s equvalet to deform the tegral path of ls to the upper half plae wth radus of order l s to avod the Glauber rego. If θ, the collear partcles at the two eds are both collear to the plus drecto. We oly eed to cosder the case that the soft subgraph that cota l s oly coect to the et collear to plus. Ths s because that f there s at least oe le that coect to other et the subgraph the we ca choose mometa of other exteral les of the subgraph as depedet varables ad repeat the procedure above two cases. I ths case, the soft subgraph does ot break factorzato ad does ot eed to be further cosdered. We the prove that oe ca take the Grammer Yee approxmato couplgs betwee soft gluos ad partcles collear to the plus drecto after the deformato of tegral path. We deote mometa of teral les wth mometa collear to the plus drecto as k ad those of soft gluos as l, where l are defed as flow to the et collear to the plus drecto. Accordg to the substtuto (26, k + ad k are fuctos of the plus ad trasverse compoets of l ad mometa of exteral les, whch are determed by the δ-fuctos (26. Whle k are depedet varables, we ca tegrate out k usg the sklls of x + -order perturbato theory. After ths operato, the perturbato seres deped o l through the propagators: l + [( K + + l 2 + M 2 ]/2K + + ǫ, (3 at leadg order, where K µ represets the total mometa of exteral les that flow to oe tegrated teral le, M represets the mass of the teral le. Oe ca drop l above propagators wth correctos of order l /Q. However, there are propagators depedg o l but depedet of l. x+ of these sates are smaller tha that of the vertex at whch l couple to the et collear to the plus drecto. For these propagators, we ca ot smply drop the l term. Istead, we choose the states x + of whch s the smallest the perturbatve seres. Geerally speakg, such states deped o the x + order the pertubatve seres. However, we ca always choose such state for each fxed x + -order. We shft the tegrato varables, so that the trasverse mometa of such states are depedet varables. Ths ca be doe as we tegral over the trasverse mometa of exteral ad teral les the process cosdered here. After ths operato, propagators of states wth x + smaller tha that of the vertex at whch l couple to the et collear to the plus drecto are

6 98 Commucatos Theoretcal Physcs Vol. 65 depedet of l. There are o loger propagators depedg o l but depedet of l after ths operato. We ca thus drop l the couplg betwee l ad partcles collear to the plus drecto. Ths s equvalet to say that oe ca take the Grammer Yee approxmato couplgs betwee soft gluos ad partcles collear to the plus drecto. For couplgs betwee soft gluos ad partcles collear to other drectos, oe ca take the smlar proofs ad show that the Grammer Yee approxmato works these couplgs. We thus coclude that whle dealg wth the couplg betwee a soft gluo exchage betwee dfferet ets wth mometum l s ad partcles collear a lght lke drecto µ, oe ca deform the tegral path of l s to the upper half plae so that the Grammer Yee approxmato works, where l s s defed as flow to the partcles collear to µ. 4 Wlso Les of Collear ad Soft Gluos We have proved that the Grammer Yee approxmato works the couplg betwee soft gluos ad collear partcles the hadroc tesor (6. We brg Wlso les of scalar-polarzed gluos to absorb the effects of soft gluos ths secto. We also brg effectve acto equpped wth Wlso les of scalar-polarzed collear gluos to descrbe the hard process ths secto. We frst determe the power coutg for dfferet modes. For a massless scalar feld φ(p wth p µ Qλ, where λ, we have: [φ(p, φ(p ] = Power coutg for φ(p s: p 2 + ǫ δ(4 (p p. (32 φ(p (Qλ 3. (33 Power coutg for other modes ca be determed accordg to smlar method. We dsplay the result covarat gauge Table, where µ = (/ 2(, s a lght-lke drecto wth 2 =, p = 0, µ = 2(, 0 µ, λ, λ ad λ s are parameters much smaller tha, λ m = m{λ, λ 2 }, λ max = max{λ, λ 2 }. There are ambgutes power coutg for collear gluos. If the parameter ξ s of order, where ξ s the gauge parameter, the we get the result Table. If ξ, the we have g = g = 0 ad g = g =. Thus, power coutg for A ad A s ot defte ths case. We have determed the power coutg by requrg that power coutg for couplgs betwee collear fermos ad collear gluos gves d 4 x ψa ψ. The summato over µ, s of order θ 2 /θ 2 µ, where θ s the agle resoluto of the drectos µ, θ s the typcal scatterg agle of the process. For Drell Ya process, we have θ. We also otce that for the cotracto of two felds wth mometa the collear rego to be ozero, the mometa of these two felds should be collear to the same drecto. Power coutg for µ the gves θ/θ µ. We do ot absorb ths to power coutg of felds, as there s oly oe such summato for each collear et. Table Power coutg for varous modes covarat gauge. Modes Mometa ( p, p, p Power coutg ψ(p Q(, λ, λ Q 5/2 λ /2 m λ max A µ (p Q(, λ, λ Q 3 λ /2 m λ max(λ /2 max,, λ /2 max ψ(p p µ Qλ s (Qλ s 5/2 A µ (p p µ Qλ s (Qλ s 3 We ow cosder the power coutg for the hard subprocess. For the hard teral les, we have k 2 Q 2. For exteral les collear to µ, we have k 2 Q 2 λ 2 wth λ. For soft exteral les, we have q 2 Q 2 λ 2 s ad λ s. We are cosderg the hard sub-process at ths step, thus we do ot cosder couplgs betwee collear partcles ad soft partcles. The power of λ s s produced by the term d 4 qa s (q ad s λ S s wth S the umber of soft partcles that coect to the hard teral les. The power of λ s produced by the term d 4 pψ(p ad d 4 pa(p ad the summato µ. The result s λ d = λ d, where d = for collear fermos, d = 0,, 2 for collear gluos polarzed the drectos µ, µ ad µ respectvely. We do ot cosder the back-to-back rego [8] ths paper, thus the power coutg for the δ- fucto of mometa coservato of the whole process do ot dsturb us. Power coutg for the hard sub-process µ λd gves λ S s. If d the there ot super leadg powers produced by partcle collear to µ. If all partcles collear to µ are scalar polarzed gluos, the there s a addtoal power of θ 2 λ 2 accordg to Ward detty ad the super-leadg powers cacel out. We the cosder couplgs betwee collear ad soft partcles. Couplgs betwee partcles collear to µ ad soft fermos are of order ( p q µ /2 /( p accordg to the power coutg dsplayed Table, where p ad q deote the mometa of collear partcles ad soft fermos. We eglect such couplg ths paper. Couplgs betwee soft gluos ad partcles collear to µ are of order ad should be summed to all orders the

7 No. 2 Commucatos Theoretcal Physcs 99 couplg costat g. We also see that couplgs betwee partcles collear to the same drectos ad those betwee soft gluos are also order ad should be summed to all orders g. We see that such couplgs do ot produce super-leadg powers. The hard collso s early local coordate space wth ucertaty of order /Q. We deote the space tme rego whch the hard collso occur as H(x, /Q, where x s the pot at whch the hard photo s produced. After or before the hard collso, there are teractos betwee collear ad soft partcles whch are resposble for the formato of hadros or the producto of tal actve partos. The space-tme ucertaty of such teractos are of order /Λ QCD. We deote the spacetme rego of such teracto as S(x, /Λ QCD where H(x, /Q S(x, /Λ QCD. We also deote the other space-tme rego as C(x. We ow cosder the classcal cofgurato of quark ad gluo felds the rego S(x, /Λ QCD H(x, /Q. Dfferet ets separate from each other ths rego although they exchage soft gluos. We deote the rego whch the et collear to µ locate as y. We also deote the rego whch there are ot collear ets as y s. (y S(x, /Λ QCD H(x, /Q, y s S(x, /Λ QCD H(x, /Q. I the rego y s, we ca defe the soft felds: ψ s (y s ψ(y s, A µ s (y s A µ (y s. (34 We exted the defto of soft felds to the rego y accordg to the maer: where (D sµ G µν s a (y = g ψ s γ µ t a ψ s (y, (35 A µ s (y s = A µ (y s, (36 D sψ s (y = 0, ψ s (y s = ψ(y s, (37 D sµ µ ga µ s, G µν s g [Dµ s, D ν s]. (38 The equato should be solved perturbatvely, that s, we defe soft felds accordg to perturbato theory. Mometa of the classcal felds the rego y s are soft. Mometa of the classcal felds ψ s ad A s the rego y are also soft at lowest order the couplg costat g as mometa of propagators do ot chage the prorogato. At hgher order, we assume that dscrepacy betwee rapdty of collear ad soft felds are so large that couplgs betwee soft felds do ot produce collear felds. Thus mometa of the classcal felds ψ s ad A s the rego y are soft hgher order g. We the defe the effectve felds: ψ (y ψ(y ψ s (y, (39 ψ (y m = ψ (y s = 0(m µ µ, (40 A µ (y A µ (y A µ s (y, (4 A µ (y m = A µ (y m = 0(m µ µ. (42 We ca the wrte the classcal Lagraga desty of QCD as: L(y = µ L (y + L s (y = µ ψ ( ga ga sψ + 2g 2 µ tr c {[ µ ga µ gaµ s, ν ga ν gaν s ]2 } + ψ s ( g A s ψ s + 2g 2 tr c{[ µ ga µ s, ν ga ν s] 2 }. (43 Mometa of the effectve felds ψ ad A ca be soft. However, couplgs betwee soft felds costraed the rego y are power suppressed. To see ths, we otce that ( y /Q for the rego y, whle for couplgs betwee soft felds we have y µ s /Λ QCD. Thus couplgs betwee soft felds costraed the rego y are suppressed as power of Λ QCD /Q. Couplgs betwee soft fermos ad collear fermos are also power suppressed ad ca be dropped. Whle dealg wth the couplg betwee A s ad ψ (A, we ca treat the mometa of ψ ad A as collear to µ. We the take the Grammer Yee approxmato such couplg. We have: µ ga µ gaµ s µ g A s µ, (44 where µ ψ = µ µ ψ, A ν = µ A ν. (45 µ A s = µ A s. (46 We the have: µ gaµ g A s µ µ = Y Y gaµ, (47 where ( 0 Y (x P exp g ds A s (x µ + s µ. (48 The Wlso le Y travels from to x, ths s accordace wth our deformatos of tegral paths last secto. We redefe the felds: ψ (0 = Y ψ, A (0µ = Y A µ Y, (49 ad wrte L as: L (0 (0 = ψ ( (0 ga ψ(0 + µ tr {([ 2g2 ga (0µ, ν It s coveet to brg the otato ga (0ν ] 2 }. (50 L Λ µ L (0 + L s, (5 the ψ s ad A s decouple from ψ (0 ad A (0 L Λ. L Λ s varat uder the gauge trasformato U s (y: ( ψ s (y U s ψ s (y, A µ s (y U s A µ s + g µ U s (y, (52 ψ (0 (y ψ (0 (y, A (0µ (y A (0µ (y, (53

8 200 Commucatos Theoretcal Physcs Vol. 65 where we have costraed that U s ( =. Ths s coected wth the usual gauge varace of QCD. To see ths, oe may cosder the trasformato of ψ ad A : ψ U s ψ (0, A µ U s A µ U s. (54 We the cosder the felds ψ = µ ψ + ψ s ad A = µ A + A s ad have: ψ U s ψ, A µ U s (A µ + g µ U s, (55 ths s ust the usual gauge trasformato of QCD. L Λ s also varat uder the trasformato: ψ s ψ s, A µ s Aµ s, (56 ( ψ (0 U cψ (0, A(0µ U c A (0µ + g µ U c. (57 Ths s also coected wth the gauge varace of QCD. To see ths we cosder the specal cofgurato ψ s = A s = 0, whch s varat uder the trasformato U c ad have: ψ U c ψ, A µ U c (A µ + g µ U c. (58 We ow cosder the quatzato of L Λ. We frst spect the boudary codto provded by the rego C(x. I C(x, teractos betwee dfferet ets ad soft hadros ca be eglected as they are color sglets ad separate from each other far eough. We deote the rego whch the et collear to µ ad soft hadros locate as x ad x s. We have ψ s (x = A s (x = 0 ths case. That s to say ψ(x = ψ (0 (x ad A = A (0 (x. Thus boudary codtos provded by the rego C(x are costrats o the cofgurato of the classcal felds ψ (0 A (0,, ψ s ad A s. We ca thus quatze such felds stead of the felds ψ ad A. Whle dealg wth the sub-process betwee soft partcles ad partcles collear to µ Drell Ya process, our quatzato scheme gves the same result as that QCD at leadg order Λ QCD /Q. At tree level, ths s obvous accordg to the costructo of the classcal lagraga desty L Λ. At oe loop level, the loop mometa are hard, collear to µ or soft o LPSS. We cosder these three cases respectvely: ( If the loop mometa are soft, the oe ca smply take the Grammer Yee approxmato the couplg betwee the soft teral les ad the collear les. Thus such part ca be descrbed by L Λ. ( If the loop mometa are collear to µ, the oe ca take the Grammer Yee approxmato the couplg betwee the soft exteral les ad the collear teral les. Such part ca also be descrbed by L Λ. ( If the loop mometa are hard, the the soft partcles ca ot be fermos at leadg order accordg to the power coutg dsplayed Table. There ca be o more tha oe soft gluo ad two physcal collear felds coected to the loop at leadg order. The soft gluo felds A µ s should be cotracted wth other vectors to make the couplg Loretz varat. At leadg order, they should be cotracted wth p µ, γ µ or A µ where p represets the mometum of a collear partcle, A µ represets collear gluos. Whle p µ p µ, A s ψ A s ψ. Thus the soft gluo should be polarzed the drecto µ ths case. We the drop the mometa compoets q ad q hard ad collear les, where q deotes the mometum of the soft gluo. That s, oe ca take the Grammer Yee approxmato such couplg. We see that such part ca also be descrbed by L Λ. Oe ca exted such procedure to hgher order the couplg costat g. There are hard process the space tme rego H(x, /Q ad L Λ s ot eough to descrbe physcs ths rego. LPSS wth dscoected hard vertexes cacel out after the summato over gauge varat set of graphs s performed. [27] Thus hard process betwee dfferet ets ca be descrbed by a effectve hard vertex. We brg the effectve operator Γ µ (xb µ (x to descrbe such effects, where B µ deote the photo feld. Γ µ s determed by requrg that t ca gve the equvalet hadroc tesor as [27] at leadg order Λ QCD /Q. Accordg to the gauge varace uder U s, we have: Γ µ (x = Γ µ (Y ψ (0 (x,..., Y m A (0µ m Y m (x m(x, (59 where x 0 x 0, x 0 m x 0 as effects of teractos after the producto of hard photo cacel out Eq. (6. Couplgs betwee soft partcles ad modes wth k 2 Q 2 are suppressed as power of Λ QCD /Q, thus we ca drop the ψ(x s ad A(x s terms Γ µ. It s coveet to work wth the boudary codto A µ ( = 0. We the expad Γ µ accordg to the small mometa compoets p ad (p, where p represets mometa of partcles collear to µ. Ths gves: Γ µ (x = d( x d(m x m µ,...,m µ J µ (ψ (x + x,..., A µ m(x + m x m m(x + O(Λ QCD /Q. (60 The codto (x + x 0 x 0 s equvalet to x 0 ths case as x = 0. Scalar polarzed collear gluos decouple from physcal processes accordg to gauge varace. We defe the felds: Ã (0µ,x (x A (0µ ( x, x, x, (6 Ã µ,x(x A µ ( x, x, x. (62 We also brg the Wlso les: ( W,x (0 (x P exp g ( W,x (x P exp g 0 0 ds Ã(0,x (x + s, (63 ds Ã,x(x + s. (64 They travel from to x as teractos after the hard collso do ot cotrbute to the hadroc tesor (6.

9 No. 2 Commucatos Theoretcal Physcs 20 Smlar Wlso les ca be foud lteratures, e.g. Refs. [ 3, 8 9, 28 30]. For future coveece, we defe the felds: ˆψ,x(x (0 = W,x (0 (x ψ (0 (x, ( µ gâ(0µ,x = W,x (0 ( µ ga (0µ W,x (0, (65 ˆψ,x (x = W,x (x ψ (x, ( µ gâµ (x = W,x ( µ ga µ,x W,x. (66 We cosder a specal part of Γ µ, whch all felds are physcal felds. We deote such part as Γ µ phy. Accordg to gauge varace, we have: Γ µ phy = Γµ phy (ψ,..., m ga m m. (67 Cotrbutos of the felds A to Γ µ are power suppressed ad ca be dropped accordg to the power coutg dsplayed Table. We ca expad Γ µ accordg to the felds l A l for all drectos l µ. The the lowest order result equal to Γ µ phy. Accordg to such expaso, f we take the lowest order perturbato of A for a specal drecto µ, we have: Γ µ A=0 = Γ µ (ψ, ga,...,a m, (68 where A = 0 should be treated as the lowest order perturbato of A ot the axal gauge. We the choose a trasformato U c so that U c( = ad A(x = 0 at fte pot ths gauge. I ths specal gauge, we have: Γ µ = Γ µ ( ˆψ,x,..., A m, (69 ˆψ,x (x + x ad ( gâ,x (x + x are varat uder (U c. Accordg to the gauge symmetry of Γ µ, we have: Γ µ = Γ µ ( ˆψ,x,...,A m (70 ay gauge wth the boudary codto A( = 0. We repeat the argumets ad have: Γ µ A( =0 = d( x d(m x m µ,...,m µ J µ ( ˆψ m,x (x+ x,..., ( Âm m,x (x+m x m m.(7 We the extract large mometa compoets of collear felds. That s: ψ (0 (y = p ψ, p(ye (0 p y, (72 A (0µ (y = p A (0µ, p(ye p y. (73 The the large mometa compoets become labels o the effectve felds. Ths s smlar wth the defto of collear felds as Ref. [28], but we oly extract large mometa compoets. After ths extracto, we ca take that ψ, p (x + x ψ, p (x ad A µ, p(x + x A, p (x Γ µ as x /Q. Γ µ ca the be wrtte as: Γ µ A( =0 = Γ µ (0 (Y ( ˆψ,x p,...,, p,...,m, m p Y m ( m gâ(0m,x m p Y m (x, (74 where the short dstace coeffcets Γ µ are fuctos of the large mometa compoets p. If we cosder the cofgurato Ã,x = 0 (the lowest order perturbato of Ã,x ot the axal gauge, the Γ µ ca be matched perturbatvely the o-shell scheme. We always make such extracto ext secto, although we do ot dspaly t explctly. Cotrbutos of hard dagrams wth several physcal exteral les early collear to the same drecto µ are suppressed as powers of θ accordg to the power coutg dsplayed Table, where θ deotes the agle betwee the drectos of these collear partcles. If these exteral les coect to the same et the oe ca cotract the teral les wth mometa square k 2 Q 2 θ 2 to hard subdagram. Correctos are suppressed as powers of Λ QCD /(Qθ ad of order θ Λ QCD /(Qθ = Λ QCD /Q. We repeat ths procedure (ths s ust the usual matchg procedure ad get the cocluso that physcal collear felds effectve operators Γ µ should coect to dfferet ets at leadg order wth correctos of order Λ QCD /Q. Ths s accordace wth the result Ref. [6]. 5 Factorzato I ths secto we fsh the proof of QCD factorzato of Drell Ya process. We wrte the hadroc tesor the frame of effectve theory as: H µν (q= d 4 xe q x p p 2 TΓ µ (xtγ ν (0 p p 2, (75 X where we have made the summato over all possble pots at whch the hard photo s produced. We wrte Γ µ as: Γ µ (x = color dces Γ µ c,γ s Γ µ c (0 (0 ( ˆψ,x, ˆψ m,x, l gâ(0l l,x Γ s (Y, Y m, Y l (x, (76 where ( =, 2 deote the drecto of tal collear partcles, m ad l deote the drecto of fal collear partcles. We have omtted possble color dces for smplcty. Γ µ (0 (0 c s mult-lear wth ˆψ,x, ˆψ m,x ad (Âl,x (0l as physcal collear felds Γ µ coect to dfferet ets at leadg order. Wlso les that appear Γ s deped o types of felds Γ c. We otce that (s µ s 2 m µ 2 < 0 for s < 0, s 2 < 0 ad µ m µ. Thus, the order of Wlso les Γ s do ot affect the result. (They are matrx-elemets ot matrxes color space at ths step I addto, we ca choose µ as slghtly space-lke so that the tme orderg ad at-tme orderg operators Wlso les ca be dropped. We ca drop the mometa of soft partcles ad small mometa compoets of collear partcles the δ- fucto of mometa coservato for the whole process. Thus we ca set that x µ (0 = ( x, 0, 0 the felds ˆψ,x ad

10 202 Commucatos Theoretcal Physcs Vol. 65 Â (0,x. We ca also set that x = 0 the Wlso les Y (x. Oe should otce that there are ot detected hadros the process cosdered ths paper ad termedate eergy scale that prevet we make such approxmato do ot exst. The part of H µν that deped o soft felds are: H s (0 0 Γ s(0γ s (0 0. (77 Hadros are all color sglets ad we have: (Y (Y = δ, (Y (Y = δ, (78 (Y t a Y (Y t a Y kl = t a ta kl. (79 a a We the have: H µν (q = d 4 xe q x p p 2 TΓ µ (x X TΓ ν (0 p p 2 ψs=a s=0. (80 We defe the ahlato operators â s 2Ep,x,p d 3 x ū s (p 2m ˆψ,x ( x e p x, (8 â s,x,p 2Ep d 3 x ˆψ,x ( x v s (pe p x, (82 2m for collear fermos ad at-fermos respectvely. Oe ca verfy that â s,p 0 = 0. For collear gluos, we defe that: â,x,p 2Ep d 3 x e p x ǫ (p p Â,x( x = 2Ep d 3 x e p x ǫ µ p (p W,xG µ W,x ( x, (83 wth deotes dfferet polarzatos, where G µ g [ ( ga, µ ga µ ], (84 we have made use of the fact Â,x(x = 0. Oe ca also verfy that â,x,p 0 = 0. Hadro states ca the be expaded accordg to the parto states ˆp x whch are created by the creato operator â,x,p. That s: ˆp x = 2E p a,x,p 0, (85 where â,x,p deote the cougato of the operators (8, (82 or (83. Γ µ s mult-lear wth ˆψ,x ad µ gâµ m,x wth m. There ca be o more tha oe parto states created by the operator â that cotract wth,x,p Γµ for each tal hadro p. We deote these actve partos as ˆp x ad have: H µν (q = lm p,p 2 2E p 2 2E p N 2 c D(G 2 p e HfreeT e H (x0 +T = tr c {â,x,p e H x0 â,0,p } Γ d 4 xe q x e H T e H freet p ψs=a s=0 tr c x ˆp ˆp 2 Γµ ( x e, 2 µ Hx 0 Γ ν ( 0 ˆp ˆp 2 0 ψs=a s=0, (86 where /D(G deotes the color factors produced by felds that do ot collear to tal hadros, H free represets the free Hamltoa of hadros. We otce that the states ˆp x are varat uder the gauge trasformato U (x (U ( = : ( ψ U ψ, A µ U A µ + g µ U, (87 Γ µ s also varat uder such gauge trasformato eve f we choose dfferet U for dfferet drectos µ. Especally, we ca choose that U = for ( =, 2. The matrx-elemet betwee parto states Eq. (86 s varat uder such gauge trasformato. We ca take the lowest order perturbato of the felds Ã,x such matrx-elemet, as these felds are scalar polarzed. We the have: H µν (q = lm p,p 2 Γ 2E p 2 2E p N 2 c 2 d 4 xe q x p = e H freet e H (x0 +T D(G tr c {â,x,p e H x0 â,0,p } e H T e H freet p ψs=a s=0 tr c p p 2 Γµ ( xe, 2 µ H x 0 Γ ν ( 0 p p 2 ψs=a s=, (88 Ã,x= Ã,0=0 where p are the usual parto states produced by the operator a p = â,x,p Ã,x=0. The codto Ã,x = 0 should be treated as the lowest order perturbato of the felds Ã,x ot the axal gauge. The formula (88 s our fal result. We see that soft gluos ad scalar polarzed gluos that collear to tal hadros decouple from the matrx-elemet betwee parto states Eq. (88. Such matrx-elemet ca be calculated accordg to perturbato theory. 6 Cocluso We have fshed the proof of factorzato theorem for Drell Ya process at operator level. Beg dfferet from covetoal dagram-level approach, [ 3] we use the utarty of the tme evoluto operator of QCD drect to cacel teractos after the collso the hadroc tesor. The hadroc tesor of LPSS caused by Glauber gluos after such cacellato. Decouplg of soft gluos from collear ets s realzed by defg collear felds that decouple from soft gluos. We gve some dscusso here: ( We treat partos of hadros dfferet ets as dfferet partcles. Ths s the cosequece of the fact that

11 No. 2 Commucatos Theoretcal Physcs 203 dfferet ets movg dfferet drectos ad separate from each other coordate space. The terferece effects betwee dfferet ets are descrbed by the teractos betwee these ets ad teractos betwee collear ad soft partcles. ( We have dropped the coordates x ad x of felds collear to µ Eq. (88. Thus the parto dstrbuto fuctos of the tal partos that collear to µ ( =, 2 are depedet of x ad x respectvely. There are ot teral les wth mometa collear to µ the matrx-elemet betwee parto states Eq. (88. Such matrx-elemet ca rely o ( p ad p oly through the tal exteral les. We ca drop these terms such matrx-elemet wth correctos of order Λ QCD /Q. ( To get the smlar form as Refs. [ 3], oe should factorze the spor structure ad cosder the reormalzato of the parto dstrbuto fuctos. We do ot show ths here. Ackowledgmets We would lke to ackowledge G.T. Bodw, Y.Q. Che, ad J. Qu for helpful dscussos ad mportat suggestos about the artcle. Refereces [] G.T. Bodw, S.J. Brodsky, ad G.P. Lepage, Phys. Rev. Lett. 47 ( [2] A.H. Mueller, Phys. Lett. B 08 ( [3] J.C. Colls, D.E. Soper, ad G. Sterma, Phys. Lett. B 09 ( [4] P.V. Ladshoff ad W.J. Strlg, Z. Phys. C 4 ( [5] K. Harada, T. Kaeko, ad N. Saka, Nucl. Phys. B 55 (979 69; B. Humpert ad W.L. va Neerve, Phys. Lett. B 89 (979 69; G.T. Bodw, C.Y. Lo, J.D. Stack, ad J.D. Sullva, Phys. Lett. B 92 ( [6] W.W. Ldsay, D.A. Ross, ad C.T. Sachrada, Nucl. Phys. B 24 ( [7] A. Se ad G. Sterma, Nucl. Phys. B 229 ( [8] J.C. Colls, D.E. Soper, ad G. Sterma, Phys. Lett. B 26 ( ; Nucl. Phys. B 223 ( [9] J.C. Colls, D.E. Soper, ad G. Sterma, Phys. Lett. B 34 ( [0] J. Frekel, J.G.M. Gatheral, ad J.C. Taylor, Nucl. Phys. B 233 ( [] G.T. Bodw, Phys. Rev. D 3 ( ; Erratumbd. D 34 ( [2] J.C. Colls, D.E. Soper, ad G. Sterma, Nucl. Phys. B 26 ( [3] J.C. Colls, D.E. Soper, ad G. Sterma, Nucl. Phys. B 308 ( [4] N.H. Chrst, B. Hasslacher, ad A.H. Mueller, Phys. Rev. D 6 ( ; S. Lbby ad G. Sterma, Phys. Rev. D 8 ( ; A.H. Mueller, Phys. Rev. D 8 ( [5] R.K. Ells, H. Georg, M. Machacek, H.D. Poltzer, ad G.G. Ross, Nucl. Phys. B 52 ( ; D. Amat, R. Petrozo, ad G. Vezao, Nucl. Phys. B 40 (978 54; bd. B 46 ( [6] G. Sterma, Phys. Rev. D 7 ( ; [7] J.C. Colls ad G. Sterma, Nucl. Phys. B 85 ( [8] J.C. Colls ad D.E. Soper, Nucl. Phys. B 93 ( [9] G. Curc, W. Furmask, ad R. Petrozo, Nucl. Phys. B 75 (980 27; J.C. Colls ad D.E. Soper, Nucl. Phys. B 94 ( [20] G. Grammer ad D.R. Yee, Phys. Rev. D 8 ( [2] J.C. Colls, L. Frakfurt, ad M. Strkma, Phys. Lett. B 307 ( [22] F. Abe, et al., [CDF Collaborato], Phys. Rev. Lett. 78 ( [23] L. Alvero, J.C. Colls, J. Terro, ad J. Whtmore, hepph/ [24] K. Goulaos, hep-ph/ [25] A. Aktas, et al., [H Collaborato], Eur. Phys. J. C 5 ( [26] H.D. Poltzer, Nucl. Phys. B 72 ( ; N. Paver ad D. Trelea, Nouvo. Cm. 70 ( [27] J.M.F. Labastda ad G. Sterma, Nucl. Phys. B 254 ( [28] C.W. Bauer, S. Flemg, D. Prol, ad I.W. Stewart, Phys. Rev. D 63 ( ; C.W. Bauer, D. Prol, ad I.W. Stewart, Phys. Rev. D 65 ( ; Chrsta W. Bauer, Sea Flemg, Da Prol, Ira Z. Rothste, ad Ia W. Stewart, Phys. Rev. D 66 ( [29] J. Qu ad G. Sterma, Nucl. Phys. B 353 ( [30] C.J. Bomhof, P.J. Mulders, ad F. Plma, Eur. Phys. J. C 47 ( [3] A.V. Beltsky, X. J, ad F. Yua, Nucl. Phys. B 656 ( [32] See, e.g., S. Weberg, The Quatum Theory of Felds, Cambrdge Uversty Press, Cambrdge (995, Volume, Chapter 3.