Unequal rapidity correlators in the dilute limit of JIMWLK

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY-NC-ND 4.0 h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ Unequal apidi y co ela o s in he dilu e limi o JIMWLK © The Au ho (s) 2019 Published e sion Lappi, Tuomas; Ramna h, And ecia Lappi, T., & Ramna h, A. (2019). Unequal apidi y co ela o s in he dilu e limi o JIMWLK. In DIS 2019 : P oceedings o he XXVII In e na ional Wo kshop on Deep-Inelas ic Sca e ing and Rela ed Subjec s (A icle 068). Sissa. POS P oceedings o Science, 352. h ps://doi.o g/10.22323/1.352.0068 2019 PoS(DIS2019)068 Unequal apidi y co ela o s in he dilu e limi o JIMWLK T. Lappi Depa men o Physics, P.O. Box 35, 40014 Uni e si y o Jy äskylä, Finland Helsinki Ins i u e o Physics, P.O. Box 64, 00014 Uni e si y o Helsinki, Finland E-mail: [email p o ec ed] A. Ramna h∗ Depa men o Physics, P.O. Box 35, 40014 Uni e si y o Jy äskylä, Finland E-mail: [email p o ec ed] We s udy unequal apidi y co ela o s in he s ochas ic Lange in pic u e o Jalilian-Ma ian– Iancu–McLe an–Weige –Leonido –Ko ne (JIMWLK) e olu ion in he Colo Glass Conden- sa e e ec i e ield heo y. By sepa a ely e ol ing he Wilson lines in he di ec and complex conjuga e ampli udes, we use he o malism o s udy wo-pa icle p oduc ion a la ge apidi y sepa a ions. We show ha he e olu ion be ween he apidi ies o he wo p oduced pa icles can be exp essed as a linea equa ion, e en in he ull nonlinea limi . We also show how he Lange in o malism o wo-pa icle co ela ions educes o a BFKL pic u e in he dilu e limi and in mo- men um space, p o iding an in e p e a ion o BFKL e olu ion as a s ochas ic p ocess o colo cha ges. XXVII In e na ional Wo kshop on Deep-Inelas ic Sca e ing and Rela ed Subjec s - DIS2019 8-12 Ap il, 2019 To ino, I aly ∗Speake . c Copy igh owned by he au ho (s) unde he e ms o he C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 In e na ional License (CC BY-NC-ND 4.0). h ps://pos.sissa.i / PoS(DIS2019)068 Unequal apidi y co ela o s A. Ramna h 1. In oduc ion The Colo Glass Condensa e (CGC, see e.g. [1,2]) is an e ec i e heo y o QCD o high ene gy p ocesses. The JIMWLK1e olu ion equa ion [3,4,5,6,7], can be used o esum leading loga i hmic (in ene gy o x) co ec ions o QCD sca e ing c oss sec ions. In addi ion o p o iding a mo e di ec physical pic u e o he e olu ion, he Lange in o mula ion is he basis o nume ical solu ions o he JIMWLK equa ion [8,9]. The mos common phenomenological applica ions o he CGC amewo k in ol e p ocesses in which one needs only he Wilson lines a one apidi y. The si ua ion becomes mo e complica ed i one is in e es ed in he co ela ions be ween pa icles ha a e sepa a ed by a pa ame ically la ge apidi y in e al ∆Y&1/αs. Fo his pu pose, a o malism based on he Lange in desc ip ion o JIMWLK e olu ion was de eloped by Iancu and T ian a yllopoulos (IT) in [12] (see also ea lie , e y simila wo k in [13,14]). Ou in en ion in his pape , ollowing he mo e de ailed discussion in [15], is o analyze his u he . 2. JIMWLK e olu ion and pa icle p oduc ion a equal apidi y We conside a high ene gy in e ac ion o a dilu e colo ed p obe wi h he colo ield o a dense a ge . The expec a ion alue o an obse able ˆ Ois gi en by Dˆ OEY≡R[DU]WY[U]ˆ O, whe e WY[U]is he CGC weigh unc ion desc ibing he densi y dis ibu ion a Yo he Wilson lines U† x≡PexpigRdx+αa x(x+) ain he a ge . The dependence o he a ge colo ield on apidi y is desc ibed by JIMWLK e olu ion. The CGC weigh unc ion e ol es om an ini ial condi ion Yin o a inal Yacco ding o he JIMWLK equa ion ∂ ∂YWY[U] = HWY[U]. The JIMWLK Hamil onian is H≡1 8π3Ru z Ku z(La u−˜ U†ab zRb u)(La −˜ U†ac zRc ), whe e ildes deno e he adjoin ep esen a ion. The JIMWLK ke nel is Ku z ≡Ki uzKi z, whe e Ki uz =(u−z)i (u−z)2is he Weizsäcke -Williams so gluon emission ke nel. The Land Ra e “le ” and “ igh ” Lie de i a i es ha ac o colo - o a e he Wilson lines. They a e de ined as La u≡ −ig(Uu a)αβ δ δUu,αβ and Ra u≡ −ig( aUu)αβ δ δUu,αβ . In he Lange in o mula ion, e olu ion is ea ed as a andom walk in he unc ional space o Wilson lines. Rapidi y is disc e ized as Y−Y0=εNwi h Z3N→∞,ε→0, whe e each e olu ion s ep is labelled by n∈ {0,1,...,N}. The noise is in oduced wi hin e ms we can call, espec i ely, “le ” and “ igh ” ( aceless, He mi ian) colo ields αL x,n≡RzKi xzνi z,n/p4π3and αR x,n≡RzKi xzUz,nνi z,nU† z,n/p4π3, whe e νi z,n≡νi,a z,n a.νi,a z,m∈Rand sa is ies Dνi,a x,mνj,b y,nE= 1 εδi jδabδmnδxy. The Lange in equa ion desc ibing he e olu ion o a Wilson line is U† x,n+1= eiεgαL x,nU† x,ne−iεgαR x,n. Since εis in ini esimal, we may use i as an expansion pa ame e o ob ain U† x,n+1=U† x,n+Zz iεg p4π3 Ki xzνi,a z,n−εg2 4π3Kxxz a!( aU† x,n−U† x,n˜ U†ab z,n b)+O(ε3/2).(2.1) The Bali sky-Ko chego [17,10] (BK) equa ion can be ob ained om his by i s calcula ing he dipole ˆ Sx¯x,n+1, hen using he Fie z iden i y and inally aking he mean ield app oxima ion. 1The ac onym s ands o Jalilian-Ma ian–Iancu–McLe an–Weige –Leonido –Ko ne . 1 PoS(DIS2019)068 Unequal apidi y co ela o s A. Ramna h Nex , conside a single qua k p oduced in a p o on-nucleus collision. I is desc ibed ma h- ema ically by a undamen al ep esen a ion dipole ˆ Sx¯x≡ nU† x¯ U¯xo/Nc. The ba s on bo h he Wilson line and he coo dina e in ¯ U¯xdeno e ha his Wilson line is in he CCA. The c oss sec ion o inclusi e qua k p oduc ion in a p o on-nucleus collision is hen dσq dηpd2p=xq(x)1 (2π)2Zx¯x e−ip·(x−¯x)ˆ Sx¯x¯ U=UY.(2.2) He e, Yis he ela i e apidi y o he p oduced qua k wi h espec o he a ge , xis he longi udinal momen um ac ion o he p ojec ile, xq(x)is he qua k dis ibu ion in he p o on, and pand ηpa e he ans e se momen um and apidi y, espec i ely, o he qua k. Fo inclusi e qua k-gluon p oduc ion, he c oss sec ion can be w i en compac ly in e ms o a “p oduc ion Hamil onian” [13,14,12] ope a ing on he qua k c oss sec ion: dσqg dηpd2pdηkd2k=1 (2π)4Zx¯x e−ip·(x−¯x)DHp od(k)ˆ Sx¯x¯ U=UEY.(2.3) He e, he qua k has ans e se momen um pand pseudo- apidi y ηp, and he gluon has ans e se momen um kand pseudo- apidi y ηk. The p oduc ion Hamil onian is gi en by [12]Hp od(k) = 1 4π3Ry¯ye−ik·(y−¯y)Ru¯uKi yuKi ¯y¯u(La u−˜ U†ab yRb u)(¯ La ¯u−¯ ˜ U†ac ¯y¯ Rc ¯u). 3. Dilu e limi : s ochas ic pic u e o BFKL e olu ion We s a wi h he undamen al ep esen a ion Wilson line U† x,n=eiλx,n=1+iλx,n−1 2λ2 x,n+ O(λ3), whe e each eal ma ix λis an elemen o he algeb a o SU(Nc)and deno es a one-gluon in e ac ion be ween p ojec ile and a ge . The ull Lange in s ep o linea o de is hen λx,n+1=λx,n+Zz iεg p4π3 Ki xzνi,a z,n−εg2 4π3Kxxz a!i abc c(λb x,n−λb z,n)+ O(ε3/2,λ2).(3.1) The BFKL equa ion can be ob ained by i s expanding he Wilson lines in he dilu e limi , and looking a he e olu ion o a quan i y ha is quad a ic in he expansion pa ame e λ. We i s squa e Eq. (3.1) o ob ain an i e a i e equa ion o λa x¯ λa ¯x. F om his basic equa ion, one can de ine wo di e en e sions o he BFKL equa ion. Fo he i s , we de ine he unin eg a ed gluon dis ibu ion φn x¯x≡ hλa x,n¯ λa ¯x,ni. A e Fou ie ans o ming, λa x,n+1¯ λa ¯x,n+1 akes he amilia o m o φn+1(q) = φn(q)+ 4NcεαsZp 1 (q−p)2φn(p)p2 q2−1 2 φn(q)q2 p2+O(ε3/2,φ3/2), he (colo single , ze o momen um ans e ) ex book e sion o he BFKL equa ion [19]. The o he (Muelle ’s) e sion o he BFKL equa ion [20] is ob ained when one looks a he expansion o he dipole ope a o as nU† xUyo/Nc=1−1 4Nc(λa x−λa y)(λa x−λa y) + O(λ3). The na u al de ini ion o he gluon dis ibu ion based on his expansion is hen he so-called “BFKL pome on” [18]ϕxy ≡(λa x−λa y)(λa x−λa y), which we can w i e in e ms o φby se ing ¯ λ= λ:φxx +φyy −2φxy ¯ λ=λ =ϕxy. One hen a i es a he Muelle e sion o he BFKL equa ion: ϕn+1 xy −ϕn xy =−Nc 2 εαs π2Rz˜ Kxyz[ϕn xy −ϕn xz −ϕn zy]. 2 PoS(DIS2019)068 Unequal apidi y co ela o s A. Ramna h 4. Unequal apidi y co ela o s in JIMWLK Nex , we wan o calcula e he double inclusi e c oss sec ion o he simul aneous p oduc- ion wo pa icles, sepa a ed in apidi y such ha αs(Y−YA)1. The expec a ion alue o he c oss sec ion o p oducing a qua k a some apidi y Yis calcula ed as an a e age o e he noise νa he end o he s ochas ic p ocess: ˆ Sx¯xY−YA=ˆ Sx¯x,Nν. Fo he expec a ion alue o an ope a o a he la e apidi y Y, we now ha e Dˆ OEY−YA ≡R[DUD ¯ U]WY−YA[U,¯ U|UA,¯ UA]ˆ O. We need o ha e a new condi ional weigh unc ion WY−YA[U,¯ U|UA,¯ UA][11], which obeys he di e - en ial equa ion ∂ ∂YWY−YA[U,¯ U|UA,¯ UA] = He olWY−YA[U,¯ U|UA,¯ UA]. The ini ial condi ion a YA o he condi ional weigh unc ion se s Wilson lines o bo h he DA and he CCA: WYA[U,¯ U|UA,¯ UA] = δ[U−UA]δ[¯ U−¯ UA]. Fo a gluon emi ed om a qua k p ojec ile a apidi y YA, we mus now ope a e wi h he p oduc ion Hamil onian ac ing on he Wilson lines a YA[12]. The inal esul is dσqg dYd2pdYAd2kA =1 (2π)4 1 4π3 1 NcZx¯xy ¯y e−ip·(x−¯x)e−ikA·(y−¯y)Zu Ki yuKi ¯y hhINiνiYA,(4.1) In:= n¯ La ¯u,0¯ U¯x,nLa u,0U† x,no−¯ ˜ U†ac ¯y,0 n¯ Rc ¯u,0¯ U¯x,nLa u,0U† x,no−˜ U†ab y,0 n¯ La ¯u,0¯ U¯x,nRb u,0U† x,no +˜ U†ab y,0¯ ˜ U†ac ¯y,0 n¯ Rc ¯u,0¯ U¯x,nRb u,0U† x,no.(4.2) To ind he exp essions o RU†,RU,LU†and LU, one ac s wi h he Lie de i a i es on he e olu ion equa ions o he Wilson lines. Howe e , he ou equa ions a e no independen o each o he . Fo example, we may s a by inding he equa ion o RU†. The He mi ian conjuga e will gi e he equa ion o RU, and he ela ion La u,0=˜ U†ab u,0Rb u,0can be used o ge he equa ions o LU†and LU. Ins ead o an equa ion o Ra u,0U† x,n+1, i is mo e na u al o de ine a quan i y Ra ux,n≡Ux,nRa u,0U† x,n, so we can w i e he Lange in s ep compac ly as Ra ux,n+1=eiεgαR x,nRa ux,ne−iεgαR x,n−iεg p4π3eiεgαR x,nZz Ki xz[˜ νi z,n,Ra uz,n],(4.3) whe e ˜ νi z,n=Uz,nνi z,nU† z,n. This is linea and independen o he Wilson lines, and we can he e o e exp ess he e olu ion be ween he wo apidi ies in e ms o linea BFKL-like dynamics. The whole c oss sec ion, howe e , is no gi en by a “kT- ac o ized” exp ession (unlike he dilu e case ha we discuss in he nex sec ion), due o he explici appea ance o he Wilson lines in he c oss sec ion. 5. Two-pa icle co ela o s in he dilu e limi The essen ial pa o he c oss sec ion (4.1) is gi en by Inas de ined in Eq. (4.2). This equi es he ope a ions o he Lie de i a i es in he dilu e limi . Using his, we ob ain an equa ion o Ra u,0λx,n+1. The equa ion o LU†is iden ical, wi h R→L. Combining hese in o he linea ized p oduc ion Hamil onian and he linea ized dipole, we ge In=g2 2Nc abc ade(¯ λe ¯u,0−¯ λe ¯y,0)(λc u,0−λc y,0)δ δ¯ λd ¯u,0 δ δλb u,0 ¯ λ ¯x,nλ x,n+O(λ3).(5.1) 3 PoS(DIS2019)068 Unequal apidi y co ela o s A. Ramna h The ela ion be ween λb x,nand λa u,0is linea . Thus he G een’s unc ion, de ined by Fn x,¯x,u,¯u≡ δ δ¯ λa ¯u,0 δ δλa u,0 ¯ λb ¯x,nλb x,n, does no depend on λ. Using his we ge hINi=g2 2(φ0 ¯uu −φ0 ¯uy −φ0 ¯yu + φ0 ¯yy)FN x,¯x,u,¯u+O(φ3/2)in e ms o he gluon dis ibu ion, which we can pu in o he equa ion o he wo-pa icle c oss sec ion o ob ain a kT- ac o ized exp ession: dσqg dYd2pdYAd2kA =1 (2π)4 1 2Nc αs π2Zx¯xy ¯yu ¯u Ki yuKi ¯y¯ue−ip·(x−¯x)−ikA·(y−¯y) ×(φ0 ¯uu −φ0 ¯uy −φ0 ¯yu +φ0 ¯yy)FN x,¯x,u,¯u+O(φ3/2).(5.2) We can inally ake he colo ield o be equal in he DA and he CCA. The ac ha we a e aking de i a i es wi h espec o ¯ λ¯u,0and λu,0does no in e e e wi h he BFKL e olu ion o he gluon densi y ¯ λb ¯x,nλb x,n. So Fn x,¯x,u,¯usa is ies he same equa ion as ¯ λb ¯x,nλb x,nwi h espec o index n. Fou ie ans o ming e e y hing, we ge dσqg dYd2pdYAd2kA =−αs NcZq q2 (q−kA)2k2 A FN(−p,p,q−kA,−q+kA)ϕ0(−q)+O(ϕ3/2). wi h he (ze o momen um ans e ) BFKL G een’s unc ion Fsa is ying he usual BFKL equa ion. We ha e hus shown ha he IT Lange in equa ion o malism educes, in he dilu e limi , o a con en ional co ela ion be ween wo pa icles p oduced om he same BFKL ladde . The ini ial condi ion o he e olu ion is F0(P,¯ P,m,¯m) = (Nc 2−1)δ(2)(P+m)δ(2)(¯ P+¯m). Using his in he gene al exp ession educes he equal apidi y c oss sec ion o a kT- ac o ized exp ession: dσqg dYd2pdYAd2kAY=YA =−αs (2π)2 (p+kA)2 p2k2 A ϕ0(p+kA).(5.3) 6. Conclusions We ha e a emp ed o cla i y he Lange in o mula ion [12] o wo-pa icle co ela ions in JIMWLK e olu ion, in he case o a dilu e p obe sca e ing o a dense colo ield a ge . Al hough JIMWLK e olu ion o he Wilson lines is nonlinea , he e olu ion o he Lie de i a i es encoding he co ela ion be ween he wo apidi ies, is in ac no . I can be exp essed as a linea equa ion ha is independen o he Wilson lines. This obse a ion seems o con i m he esul ob ained ea lie (in a a he di e en language) in [16]. We ha e also calcula ed explici ly he dilu e limi o he Lange in o mula ion, whe e he deco ela ions in azimu hal angle be ween he wo pa icles a e gi en by a BFKL G een’s unc ion be ween he wo apidi ies. JIMWLK e olu ion as a unc ion o he qua k apidi y Y“commu es” wi h he p oduc ion Hamil onian and only ope a es on he dipole ope a o a Y. The e olu ion o he double inclusi e c oss sec ion wi h Yis he e o e de e mined by he e olu ion o he single inclusi e c oss sec ion, bu wi h a mo e complica ed ini ial condi ion. Acknowledgmen s We a e g a e ul o R. Boussa ie, M. Lublinsky, E. Iancu and D. T ian a yllopoulos o discus- sions. T. L. has been suppo ed by he Academy o Finland, p ojec s No. 267321 and No. 303756. 4 PoS(DIS2019)068 Unequal apidi y co ela o s A. Ramna h A. R. is suppo ed by he Na ional Resea ch Founda ion o Sou h A ica. This wo k has been suppo ed by he Eu opean Resea ch Council, g an ERC-2015-CoG-681707. Re e ences [1] H. 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