Multiparticle production at mid-rapidity in the color-glass condensate

JHEP02(2019)024 Published o SISSA by Sp inge Recei ed:Sep embe 26, 2018 Re ised:Decembe 6, 2018 Accep ed:Janua y 4, 2019 Published:Feb ua y 5, 2019 Mul ipa icle p oduc ion a mid- apidi y in he colo -glass condensa e Mau icio Ma inez,aMa hew D. Sie e band Douglas E. We epnyc aDepa men o Physics, No h Ca olina S a e Uni e si y, Raleigh, NC 27695, U.S.A. bTheo e ical Di ision, Los Alamos Na ional Labo a o y, Los Alamos, NM 87545, U.S.A. cDepa amen o de F´ısica de Pa ´ıculas and IGFAE, Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Galicia-Spain E-mail: [email p o ec ed],[email p o ec ed],[email p o ec ed] Abs ac : In his pape , we compu e a numbe o c oss sec ions o he p oduc ion o mul iple pa icles a mid- apidi y in he semi-dilu e / dense egime o he colo -glass condensa e (CGC) e ec i e ield heo y. In pa icula , we p esen new esul s o he p oduc ion o wo qua k-an iqua k pai s (whe he he same o di e en la o s) and o he p oduc ion o one qua k-an iqua k pai and a gluon. We also demons a e he exis ence o a simple mapping which ans o ms he c oss sec ion o p oduce a qua k-an iqua k pai in o he co esponding c oss sec ion o p oduce a gluon, which we use o ob ain a ious esul s and o c oss-check hem agains he li e a u e. We also discuss had oniza ion e ec s in he hea y la o sec o , w i ing explici exp essions o he p oduc ion o a ious combina ions o Dand ¯ Dmesons, J/ψ mesons, and ligh had ons. The a ious mul ipa icle c oss sec ions p esen ed he e con ain a weal h o in o ma ion and can be used o s udy hea y la o p oduc ion, cha ge-dependen co ela ions, and “collec i e” low phenomena a ising om ini ial-s a e dynamics. Keywo ds: Pe u ba i e QCD, Qua k-Gluon Plasma, Resumma ion A Xi eP in : 1808.04896 Open Access,c The Au ho s. A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP02(2019)024 JHEP02(2019)024 Con en s 1 In oduc ion 1 2 P oduc ion ampli udes o (an i)qua k pai s and gluons 5 2.1 Qua k / an iqua k pai p oduc ion ampli ude 5 2.2 Gluon p oduc ion ampli ude 8 3 C oss sec ions o single pai s and gluons 10 3.1 C oss sec ion o single-pai p oduc ion 10 3.2 C oss sec ion o single-gluon p oduc ion 12 4 Double-inclusi e c oss sec ions o pai s and gluons 14 4.1 C oss sec ion o double-pai p oduc ion 14 4.1.1 Case 1: no e mion en anglemen 15 4.1.2 Case 2: e mion en anglemen 18 4.2 C oss sec ion o qua k + an iqua k + gluon p oduc ion 20 4.3 Double-gluon p oduc ion 23 5 Had oniza ion in he hea y la o sec o 27 6 Conclusions 32 A Colo a e aging in he p ojec ile and a ge 34 B Wilson line colo algeb a in momen um space 36 C Compa ison wi h e . [38]38 1 In oduc ion Co ela ions in he p oduc ion o mul iple so o semi-ha d pa icles in he mid- apidi y egion o had onic collisions a e impo an p obes o no el phenomena in quan um ch o- modynamics (QCD). Whe he in p o on-p o on (pp), p o on-nucleus (pA), o hea y- ion (AA) collisions, mul ipa icle p oduc ion e lec s he many-body co ela ions gene - a ed by QCD. In pp collisions, such co ela ions may be p oduced by quan um e olu- ion h ough Dokshi ze -G ibo -Lipa o -Al a elli-Pa isi (DGLAP) e olu ion [1–3] o by a ange o highe -o de co ec ions o he ha d pa (see, e.g. [4,5]) o small-xe olu- ion, including linea Bali sky-Fadin-Ku ae -Lipa o (BFKL) e olu ion [6,7], nonlinea Bali sky-Ko chego (BK) e olu ion [8,9] and Jalilian-Ma ian-Iancu-McLe an-Weige - Leonido -Ko ne (JIMWLK) e olu ion [10–12]. The esul ing co ela ions a e sensi i e p obes o he pe u ba i e ha d e ex and o he s ongly-o de ed emission s uc u e o he e olu ion equa ions. In pA collisions, hese highe -o de and e olu ion co ec ions a e augmen ed by a new se o dynamical co ela ions a ising om he enhancemen o mul- iple sca e ing in he high cha ge densi ies o he hea y nucleus, cha ac e ized by he – 1 – JHEP02(2019)024 colo -glass condensa e (CGC) e ec i e ield heo y (see e.g. [13] and e e ences he ein). The esul ing co ela ions a e sensi i e p obes o he mul iple sca e ing dynamics, includ- ing he signi ican e ec s o Bose enhancemen in he s ong gluon ields [14,15]. Finally, in AA collisions (as well as po en ially in high-mul iplici y pp and pA collisions), all hese ini ial s a e co ela ions a e modi ied and complemen ed by he inal-s a e dynamics o a s ongly-coupled qua k-gluon plasma (QGP) phase. A de ailed cha ac e iza ion o mul ipa icle p oduc ion in he s ong colo ields o he CGC is especially impo an in ying o di e en ia e in ial-s a e e ec s om he inal- s a e dynamics o he QGP, whe e s ongly-coupled in e ac ions lead o subs an ial many- body co ela ions among so and semi-ha d pa icles. The canonical measu es o his collec i e low a e he cumulan s o azimu hal aniso opies n{m}[16], wi h co ela ions among inc easing numbe s o pa icles e lec ed in highe alues mo he cumulan s. O he landma k p ope ies belie ed o be possible in he QGP phase include he onse o no el anspo mechanisms associa ed wi h he axial anomaly: he chi al magne ic e ec , he chi al sepa a ion e ec , he chi al o ical e ec , and he chi al magne ic wa e.1Signa u es o all o hese no el chi al dynamics a e encoded in mul ipa icle co ela ions, o en cha ge dependen , such as he same-sign and opposi e-sign co ela o s γ112 and γ123 [18]. Fo all o hese c i ical signa u es o he qua k-gluon plasma, i is essen ial o disen angle he “backg ound” con ibu ions coming om ini ial-s a e mechanisms o be e quan i y he p ope ies o he QGP and imp o e he chances o disco e ing such no el anomalous dynamics. Acco dingly, a subs an ial e o has been made in ecen yea s o compu e mul ipa icle p oduc ion in he CGC amewo k. The pu es ealiza ion o he CGC o malism is in he “dilu e / dense” amewo k, in which densi y-enhanced e ec s o he “dilu e p ojec ile” a e kep only o lowes o de , while densi y-enhanced co ec ions in he “dense a ge ” a e esummed o all o de s. Few- pa icle p oduc ion has been s udied in he dilu e / dense amewo k om he ea lies days o he CGC o malism, s a ing wi h he inclusi e single-gluon p oduc ion c oss sec ion dσG[19–24] a mid- apidi y and ollowed sho ly he ea e by he inclusi e c oss sec ion dσq¯qo a single q¯qpai ia gluon pai p oduc ion [25–33]. Co ec ions o hese p oduc ion channels we e also conside ed in he o m o small-xe olu ion co ec ions [27,29]. Howe e , a de ailed compu a ion o highe mul ipa icle p oduc ion c oss sec ions in he dilu e / dense amewo k becomes inc easingly di icul due o he p oli e a ion o ways ano he so pa icle could be adia ed om a p e-exis ing one. A signi ican s ep owa d o e coming his ba ie was made h ough he de elopmen o he “semi-dilu e / dense” amewo k [34]. This egime is designed o ill he gap be ween he dilu e / dense egime, in which he p ojec ile cha ge densi y is kep only o lowes o de , and he dense / dense egime, whe e bo h p ojec ile and a ge densi ies mus be simul aneously esummed o all o de s. The semi-dilu e / dense amewo k is app op ia e o “hea y-ligh ion collisions” in e media e o, say, pPb and PbPb collisions. Fo a collision be ween one ligh ion and one hea y ion, such as CuAu collisions, i is possible o cons uc a egime in which he la ge a ge densi y is esummed o all o de s while co ec ions om 1Fo a e iew on chi al magne ic and o ical e ec s in high-ene gy nuclea collisions we e e o he eade o e . [17]. – 2 – JHEP02(2019)024 he p ojec ile densi y a e calcula ed o de by o de in pe u ba ion heo y. In he semi- dilu e / dense amewo k, highe -o de co ec ions which a e enhanced by he p ojec ile densi y a e mo e impo an han genuine quan um co ec ions. Fo mally, o a dense a ge nucleus wi h Anucleons and a semi-dilu e p ojec ile nucleus wi h anucleons, he semi- dilu e / dense egime can be quan i ied by he hie a chy o scales α2 sA1/3∼ O(1) (1.1a) αsα2 sa1/31,(1.1b) o equi alen ly, in e m o he sa u a ion momen a Qs,p and Qs, o he p ojec ile and a ge espec i ely, Λ2 QCD Q2 s,p Q2 s, .(1.2) Wi h he help o he semi-dilu e / dense amewo k, a numbe o signi ican s eps ha e been aken in ecen yea s owa d he calcula ion o genuine mul ipa icle p oduc ion in he CGC amewo k. The key simpli ica ion ha makes his possible in he semi-dilu e / dense amewo k is ha he independen emission o new so pa icles om he high- densi y p ojec ile becomes dominan o e emission om he p e-exis ing sys em o so pa icles. As such, he i s obse able compu ed in he semi-dilu e / dense amewo k was he p oduc ion c oss sec ion dσGG o wo so gluons [34,35]. A simila e o was made owa d de e mining he p oduc ion c oss sec ion dσqq o wo qua ks coming om sepa a e q¯qpai s, wi h a pa ial calcula ion ha ing been pe o med in e . [36] emphasizing he new ole played by Fe mi-Di ac quan um s a is ics among he wo pai s. This calcula ion la e o med he basis o he pa ial calcula ion o he c oss sec ion dσqqG o wo qua k / an i- qua k pai s plus a gluon, wi h he in en o s udying he CGC con ibu ion o he same-sign co ela o s γ112 , γ123 [37]. I should be emphasized ha in his impo an calcula ion [37], only co ela ions gene a ed a he le el o he wa e unc ions we e aken in o accoun , wi hou including he e ec s o mul iple sca e ing ha ansla e hese wa e unc ions in o ac ual p oduc ion c oss sec ions. And e y ecen ly, a new a emp has been made o ex end hese calcula ions o he hi d o de in he p ojec ile cha ge densi y h ough he compu a ion o he iple-gluon p oduc ion c oss sec ion dσGGG [38]. O he no able de el- opmen s in so mul ipa icle p oduc ion include he iden i ica ion o Bose enhancemen as a d i ing mechanism o he Ridge [14] in double-gluon p oduc ion [14,15], he calcula ion o he so double-pho on c oss sec ion dσγγ [39], and he ealiza ion ha so double-pai p oduc ion can be used o p obe he gluon Wigne dis ibu ion wi h Weizs¨acke -Williams gauge s uc u e [40]. A a ian o he semi-dilu e / dense powe coun ing can be ound in he o m o he lowes -o de “glasma g aph” calcula ions, which ha e been used o calcula e mul ipa icle co ela ions such as he iple-gluon c oss sec ion [41]. O he impo an de elopmen s in he calcula ion o mul ipa icle p oduc ion in he CGC o malism ha e emphasized p oduc ion in he o wa d egime, whe e he “hyb id ac- o iza ion” amewo k makes i possible o igo ously ela e he pa icle p oduc ion c oss sec ions o collinea pa on dis ibu ion unc ions in he (semi-)dilu e p ojec ile, d essed wi h he e ec s o mul iple sca e ing in he dense a ge [42–45]. In his app oach, obse - ables such as o wa d double alence-qua k p oduc ion c oss sec ions dσq q [46], o wa d – 3 – JHEP02(2019)024 iple alence-qua k p oduc ion c oss sec ions dσq q q [47], and o wa d alence-qua k + pho on + gluon p oduc ion [48] ha e been calcula ed. Simila s udies o quad uple alence- qua k p oduc ion c oss sec ions dσq q q q ha e also been conside ed in a “pa on model” desc ip ion [49,50] wi hou he bene i o an unde lying hyb id ac o iza ion. O he ecen wo k on he subjec also includes he demons a ion [51] ha wo-gluon co ela ions can b eak he “acciden al” back- o-back symme y which occu s a lowes o de and ela ed phenomenology [52,53]. And inally, in a ecen wo k [54], we ha e conside ed single- and double-pai p oduc ion dσq¯qand dσ(q¯q)(q¯q)in coo dina e space as a means o ini ializing spa ial co ec ions o conse ed cha ges in he qua k-gluon plasma. In his pape , ou p ima y goal is o sys ema ically ex end he calcula ion o mul ipa - icle p oduc ion a mid- apidi y in he semi-dilu e / dense amewo k o highe o de s. One o he key esul s we will de i e he e o he i s ime is he comple e exp ession a exac Nc o he double-pai p oduc ion c oss sec ion dσ(q¯q)(q¯q)in momen um space, as w i en in eqs. (4.5), (4.11), and (4.16). This exp ession signi ican ly gene alizes he esul ob ained in e . [36] by including con ibu ions ha we e in en ionally omi ed he e, by wo king wi h exac Nc, and by keeping he mul iple sca e ing co ec ions o all o de s. In a key concep ual de elopmen , we show in eq. (2.19) ha i is possible o map he ampli ude (and he e o e, he c oss sec ion) o p oducing q¯qpai s in o he co esponding quan i ies o p oducing gluons. Thus, we a e able o di ec ly map he double-pai c oss sec ion in o he co esponding c oss sec ion dσ(q¯q)G o p oduce a qua k / an iqua k pai and a gluon. This exp ession, as w i en in eq. (4.19), is also a new esul . We also pe o m a numbe o alida ions o his gluonic mapping, e i ying explici ly ha i co ec ly ep oduces he known esul s o single- and double-gluon p oduc ion om he li e a u e. While he p eceding esul s all e lec he inal-s a e p oduc ion o mul iple pa ons, hey also open he doo o a subs an ial p og am o compu ing had onic-le el obse ables de i ed om hem. By con olu ing he pa onic-le el esul s wi h he app op ia e ag- men a ion unc ions o p ojec ion ope a o s and long-dis ance ma ix elemen s, we can ansla e hese pa onic-le el c oss sec ions o ull had onic c oss sec ions. The esul ing had onic obse ables can be used o igo ously s udy he co ela ions among same-sign and opposi e-sign cha ged had ons, open and hidden hea y- la o had ons, hea y- la o s ligh had ons, and mo e. The phenomenology based on hese had onic obse ables will p o ide c i ical new insigh in o ini ial-s a e mechanisms o collec i e low, qua konium co ela ions, and cha ge-dependen co ela ions which o m he backg ound o anomalous chi al dynamics in he QGP. This pape is o ganized as ollows. In sec ion 2we cons uc he sca e ing ampli udes o he p oduc ion o so pa icles in momen um space, s a ing wi h he qua k/an iqua k p oduc ion ampli ude in sec ion 2.1 and de i ing he mapping o he gluon p oduc ion ampli ude in sec ion 2.2. Then in sec ion 3we compu e he p oduc ion c oss sec ion o a single q¯qpai in sec ion 3.1 and map i in sec ion 3.2 o he well-known gluon p oduc ion c oss sec ion o alida e he gluonic mapping. Then in sec ion 4we p oceed o calcula e he new c oss sec ions o he p oduc ion o wo se s o so pa icles: double q¯qpai p oduc ion in sec ion 4.1, mixed q¯qG p oduc ion in sec ion 4.2, and double gluon p oduc ion in sec ion 4.3. The success ul c oss-check agains he double-gluon p oduc ion c oss sec ion – 4 – JHEP02(2019)024 k ¯ k=q−k q b u y x b σ σ′ Figu e 1. The ligh - on wa e unc ions o adia e a so q¯qpai a mid- apidi y om a alence sou ce, shown he e as a qua k. in sec ion 4.3 ep esen s ano he alida ion o he gluonic mapping de i ed in sec ion 3.2. In sec ion 5we u ilize he echniques enume a ed in e . [55] o ansla e ou pa onic- le el c oss sec ions in o had onic c oss sec ions o he p oduc ion o open and hidden hea y la o as an illus a ion o how o s aigh o wa dly apply he esul s de i ed he e o had onic obse ables. Finally, we conclude in sec ion 6by ei e a ing he p ima y new esul s and explo ing he many oppo uni ies o phenomenological applica ions and u he heo e ical de elopmen which his wo k p o ides. In appendix Awe p o ide de ails o he Gaussian colo a e aging used o he (semi-)dilu e p ojec ile, in appendix Bwe o mula e some use ul algeb aic p ope ies o Wilson lines in momen um space, and in appendix Cwe poin ou he di e ences in he no maliza ion o he double-gluon c oss sec ion agains e . [38]. Th oughou his pape , we deno e longi udinal momen a in ligh - on coo dina es ±≡qg+− 2( 0± 3) and ans e se ec o s by ≡( 1 ⊥, 2 ⊥) wi h magni udes T≡ | |. Di e en au ho s use di e en con en ions o he ligh - on me ic g+−; we will use g+−= 1, bu i is also common o encoun e g+−= 2. 2 P oduc ion ampli udes o (an i)qua k pai s and gluons 2.1 Qua k / an iqua k pai p oduc ion ampli ude The ampli ude o adia e a so qua k/an iqua k pai a mid- apidi y has been de i ed many imes in he li e a u e [26,27]. In he no a ion o ou p e ious wo k [54] as illus a ed in igu e 1, we deno e he ligh - on wa e unc ions [56,57] o adia e a so q¯qpai as ψ1, ψ2, ψ3co esponding o he a ious ime o de ings o he sca e ing in he a ge ields. The e m ψ1co esponds o sca e ing a e he pai is c ea ed, ψ2 o sca e ing a e he gluon is emi ed bu be o e he pai is c ea ed, and ψ3 o sca e ing be o e he pai is c ea ed. The h ee wa e unc ions a e no all independen , bu sa is y ψ1+ψ2+ψ3= 0, and he explici exp essions a e gi en by ψ1(q,κ) = −2gpα(1−α) κ2 T+m2+α(1−α)q2 T ×(δσ ,−σ02(1−α)+(1−2α)q·κ q2 T−iσ0q×κ q2 T−mσ0δσσ0"q1 ⊥ q2 T−iσ0q2 ⊥ q2 T#) (2.1a) – 5 – JHEP02(2019)024 ψ2(q,κ) = −2gpα(1−α) κ2 T+m2(δσ ,−σ0−(1−2α)q·κ q2 T +iσ0q×κ q2 T+mσ0δσσ0"q1 ⊥ q2 T−iσ0q2 ⊥ q2 T#) (2.1b) ψ3(q,κ) = −ψ1(q,κ)−ψ2(q,κ),(2.1c) whe e α≡k+ k++¯ k+is he ac ion o he pai longi udinal momen um ca ied by he qua k, qis he cen e -o -mass ans e se momen um o he q¯qpai (i.e., he gluon), and κis he in insic ans e se momen um o he qua k spli ing. In (2.1), we ha e omi ed he explici dependence o he wa e unc ions on he momen um ac ion α o b e i y. No e also ha , in compa ison o eqs. (21) o [54], we ha e emo ed a ac o o he coupling g om he de ini ion o he wa e unc ions. This co esponds o abso bing his coupling cons an in o he scale µ2de ined in (A.3) cha ac e izing he sou ces o so gluons. In e ms o hese wa e unc ions, he single-pai ampli ude summed o e all ime o de ings is gi en in coo dina e space by (see eqs. (30 - 31) o [54]) A(x, y, b)=(Vb a)hVx aV† y−Vb aV† bψ1(u−b, x −y) +Vu aV† u−Vb aV† bψ2(u−b, x −y)i,(2.2) whe e, as labeled in igu e 1,x,y, and ba e he inal-s a e posi ions o he qua k, an iqua k, and alence qua k, espec i ely, and u≡αx + (1 −α)yis he cen e -o -mass posi ion o he q¯qpai (equal o he gluon posi ion). The sca e ing o pa ons in he colo ields o he a ge a e desc ibed by Wilson lines in he undamen al o adjoin ep esen a ions, Vx≡ Pexp ig Zdx+A−(x+,0−, x)(2.3a) Uab x≡Pexp ig Zdx+A− adj(x+,0−, x)ab ,(2.3b) whe e we wo k in he A+= 0 ligh cone gauge. Wi h (2.2) w i en his way, he Wilson line Vbassocia ed wi h he alence qua k will always cancel agains a co esponding one in he complex-conjuga e ampli ude. I is con enien o ansla e he speci ic model o he p ojec ile as a dis ibu ion o alence qua ks in o a gene ic con inuous cha ge densi y. This can be accomplished by in oducing he quan i y ρa(b), which loosely co esponds o he wa e unc ion o a colo sou ce in he p ojec ile a posi ion bwhich adia es a so gluon wi h colo a. We can ansla e om he alence qua k model o he p ojec ile o he con inuous colo cha ge densi y by e ec i ely eplacing (Vb a)→ρa(b). (Fo ano he discussion o he ansla ion be ween disc e e and con inuous cha ge dis ibu ions, see e.g. [51].) Wi h his change o – 6 – JHEP02(2019)024 no a ion, we can Fou ie ans o m he buildling block (2.2) in o momen um space o ob ain A(k, ¯ k) = Zd2x d2y d2b e−ik·xe−i¯ k·yρa(b) ×hVx aV† y−Vb aV† bψ1(u−b, x −y) + Vu aV† u−Vb aV† bψ2(u−b, x −y)i,(2.4) whe e he momen a o he inal-s a e qua k and an iqua k a e kand ¯ k, espec i ely. No e ha he Fou ie ac o o he alence qua k cancels because i s posi ion bis he same in he ini ial and inal s a es unde he eikonal app oxima ion.2 Inse ing he in e se ans o ma ion o he wa e unc ions (2.1), Wilson lines, and sou ce densi y ψi(u−b, x −y) = Zd2κ (2π)2 d2q (2π)2eiκ·(x−y)eiq·(u−b)ψi(q, κ) (2.5a) Vx=Zd2κ (2π)2eiκ·xV(κ) (2.5b) V† y=Zd2κ0 (2π)2e−iκ0·yV†(κ0) (2.5c) ρ(b) = Zd2` (2π)2ei`·bρ(`) (2.5d) gi es A(k, ¯ k) = Zd2k0 (2π)2 d2q0 (2π)2ρa(q0)hV(k−k0) aV†(k−k0−q+q0)iψ1(q0,k0−αq0) +Zd2κ (2π)2 d2q0 (2π)2ρa(q0)hV(κ) aV†(κ−q+q0)iψ2(q0,k−αq) (2.6) −Zd2κ (2π)2 d2κ0 (2π)2ρa(q−κ+κ0)hV(κ) aV†(κ0)iψ1(q,k−αq)+ψ2(q,k−αq), wi h q=k+¯ k o b e i y. We can combine all h ee diag ams by ede ining he dummy in eg a ion a iables, ob aining he compac o m A(k, ¯ k) = Zd2k0 (2π)2 d2¯ k0 (2π)2ρa(k0+¯ k0)hV(k−k0) aV†(¯ k0−¯ k)iΨ(k, ¯ k;k0,¯ k0),(2.7) whe e he di e ences among he h ee diag ams a e all encoded in he combined wa e unc ion Ψ(k, ¯ k;k0,¯ k0)≡ψ1k0+¯ k0,(1 −α)k0−α¯ k0+ψ2k0+¯ k0,(1 −α)k−α¯ k −ψ1k+¯ k , (1 −α)k−α¯ k−ψ2k+¯ k , (1 −α)k−α¯ k.(2.8) 2A i s glance, he ampli ude (2.4) may appea o be p oblema ic, because i con ains an impac o e impac pa ame e s bo he sou ce a he ampli ude le el, leading o wo such impac pa ame e in eg als in he c oss sec ion. This is ue; howe e , when a e aged o e colo s a es o he p ojec ile as in (A.3), he co ela o o wo ρ’s possesses a del a unc ion which se s hese wo posi ions equal. Thus he con inuous cha ge dis ibu ion leads o one in eg al o e d2bpe sou ce a he c oss sec ion le el, as wi h he model o disc e e alence qua ks. – 7 – JHEP02(2019)024 q b x b Figu e 2. The ligh - on wa e unc ion o adia e a so gluon a mid- apidi y om a alence sou ce, shown he e as a qua k. Wi h his exp ession, i is easy o do he manipula ions o e all diag ams a once and pa icula ly o s udy hei colo s uc u e, since he Wilson lines en e in exac ly he same o m o all diag ams. As such, when we cons uc c oss sec ions o he p oduc ion o mul iple pai s, we will only ha e o pe o m one calcula ion pe diag amma ic opology, a he han ha ing o epea he calcula ion o many possible ime o de ings. These a ious opologies will co espond o di e en ways o con ac he diag ams, including bo h he colo ma ix V aV†and he wa e unc ion Ψ, which is a ma ix in he 2 ×2 spin space o he pai . 2.2 Gluon p oduc ion ampli ude In compa ison wi h (2.2) o he p oduc ion ampli ude o a so q¯qpai in coo dina e space, he co esponding ampli ude o emi a so gluon is illus a ed in igu e 2and is gi en by Aa glue(x, b) = (Vb b) (Ux)ab φ(x−b)−( aVb)φ(x−b),(2.9) whe e φis he ligh - on wa e unc ion φ(q)=2∗ λ·q q2 T δσ σ0 (2.10a) φ(x−b) = i π ∗ λ·(x−b) (x−b)2 T δσ σ0 (2.10b) o adia e a so gluon om a alence qua k p ojec ile. He e σ and σ0 a e he spin s a es o he alence qua k be o e and a e gluon emission, and λis he spin o he emi ed gluon. When we w i e he ace o e he squa e o hese wa e unc ions, we mean he a e aging o e he quan um numbe s o he ini ial s a e, oge he wi h a sum o e he quan um numbe s o he inal s a e: D[φ(q1)φ†(q2)] ≡1 2X λσ σ0 φ(q1)φ∗(q2)=4q1·q2 q2 1Tq2 2T (2.11a) D[φ(x)φ†(y)] ≡1 2X λσ σ0 φ(x)φ∗(y) = 1 π2 x·y x2 Ty2 T .(2.11b) The i s e m o (2.9) co esponds o he shockwa e passing h ough he gluon, he second e m co esponds o he shockwa e passing h ough he alence qua k be o e he gluon is – 8 – JHEP02(2019)024 k2 ¯ k2 k′ 2 ¯ k′ 2 k′′ 2 ¯ k′′ 2 ρbρd k′ 2+¯ k′ 2k′′ 2+¯ k′′ 2 k1 ¯ k1 k′ 1 ¯ k′ 1 k′′ 1 ¯ k′′ 1 ρaρc k′ 1+¯ k′ 1k′′ 1+¯ k′′ 1 Figu e 4. Double-pai p oduc ion opologies wi hou e mion en anglemen , as calcula ed in eq. (4.6). dummy colo indices a↔b. Squa ing he ull symme ized ampli ude (4.3) and con e ing o he c oss sec ion yields dσ(q¯q) (q¯q) d2k1dy1d2¯ k1d¯y1d2k2dy2d2¯ k2d¯y2 =1 2(2π)34 (4.5) ×DA(k1,¯ k1, k2,¯ k2) 2E−DA(k1,¯ k1, k2,¯ k2)A†(k1,¯ k2, k2,¯ k1)E+ (¯ k1↔¯ k2), whe e he no a ion +(¯ k1↔¯ k2) applies o bo h p eceding e ms. The ask has now been educed o calcula ing he wo con ibu ions in b acke s: he case wi hou e mion en an- glemen in he i s e m and he case wi h e mion en anglemen in he second e m. Bo h exe cises a e s aigh o wa d, and we calcula e hem in he ollowing subsec ions. Fo max- imum gene ali y, we ha e conside ed he e he case in which bo h p oduced pai s ha e he same la o ; i he la o o he qua k pai s is di e en , hen all pa icles a e dis inguishable and only he i s e m o eq. (4.5) con ibu es. 4.1.1 Case 1: no e mion en anglemen Squa ing (4.4) o opologies wi h no e mion en anglemen , as in igu e 4, leads di ec ly o DA(k1,¯ k1,k2,¯ k2) 2E=Z d 2{k0 1¯ k0 1k0 2¯ k0 2k00 1¯ k00 1k00 2¯ k00 2} ×Dρa(k0 1+¯ k0 1)ρb(k0 2+¯ k0 2)ρc∗(k00 1+¯ k00 1)ρd∗(k00 2+¯ k00 2)Ep oj × DhΨ(k1,¯ k1;k0 1,¯ k0 1)Ψ†(k1,¯ k1;k00 1,¯ k00 1)i DhΨ(k2,¯ k2;k0 2,¯ k0 2)Ψ†(k2,¯ k2;k00 2,¯ k00 2)i ×D chV(k1−k0 1) aV†(¯ k0 1−¯ k1)V(¯ k00 1−¯ k1) cV†(k1−k00 1)i × chV(k2−k0 2) bV†(¯ k0 2−¯ k2)V(¯ k00 2−¯ k2) bV†(k2−k00 2)iE g .(4.6) – 15 – JHEP02(2019)024 The e a e now 3 possible “con ac ions” o he sou ce colo s, ob ained in e ms o (A.3): Dρa(k0 1+¯ k0 1)ρb(k0 2+¯ k0 2)ρc∗(k00 1+¯ k00 1)ρd∗(k00 2+¯ k00 2)Ep oj = =δabδcd µ2k0 1+¯ k0 1+k0 2+¯ k0 2, k+ 1+¯ k+ 1+k+ 2+¯ k+ 2 ×µ2−k00 1−¯ k00 1−k00 2−¯ k00 2,−k+ 1−¯ k+ 1−k+ 2−¯ k+ 2 +δacδbd µ2k0 1+¯ k0 1−k00 1−¯ k00 1,0+µ2k0 2+¯ k0 2−k00 2−¯ k00 2,0+ +δadδbc µ2k0 1+¯ k0 1−k00 2−¯ k00 2, k+ 1+¯ k+ 1−k+ 2−¯ k+ 2 ×µ2k0 2+¯ k0 2−k00 1−¯ k00 1, k+ 2+¯ k+ 2−k+ 1−¯ k+ 1.(4.7) No e ha only in he second e m do he plus momen a combine o gi e 0+, e en o his opology wi h no e mion en anglemen . Fo a gi en se o colo con ac ions, we will need o Fie z educe he Wilson line aces o (4.6) wice. The algeb a is s aigh o wa d, bu i is con enien o de ine he ollowing abb e ia ed no a ion: V1≡V(k1−k0 1)V5≡V(k2−k0 2) V† 2≡V†(¯ k0 1−¯ k1)V† 6≡V†(¯ k0 2−¯ k2) V3≡V(¯ k00 1−¯ k1)V7≡V(¯ k00 2−¯ k2) V† 4≡V†(k1−k00 1)V† 8≡V†(k2−k00 2).(4.8) Wi h his sho hand, he Wilson line enso en e ing (4.6) is Ωabcd 1= chV1 aV† 2V3 cV† 4i hV5 bV† 6V7 dV† 8i,(4.9) and we can s aigh o wa dly compu e he a ious con ac ion o (4.7): δabδcd Ωabcd 1=N2 c 4Dˆ D4(1674) ˆ D4(5238)E−1 4D8(16785234) −1 4D8(12385674) + 1 4Dˆ D4(1234) ˆ D4(5678)E,(4.10a) δacδbd Ωabcd 1=N4 c 4Dˆ D2(32) ˆ D2(14) ˆ D2(76) ˆ D2(58)E−N2 c 4Dˆ D2(32) ˆ D2(14) ˆ D4(5678)E −N2 c 4Dˆ D4(1234) ˆ D2(76) ˆ D2(58)E+1 4Dˆ D4(1234) ˆ D4(5678)E,(4.10b) δadδbc Ωabcd 1=N2 c 4Dˆ D4(1854) ˆ D4(7236)E−1 4D8(34185672) −1 4D8(12367854) + 1 4Dˆ D4(1234) ˆ D4(5678)E,(4.10c) wi h he numbe s in pa en heses deno ing he a gumen s o he co esponding Wilson lines in (4.8). These a ious aces a e ilus a ed in igu e 5. Fan as ically, we only ha e o do one calcula ion pe opology (i.e., sou ce con ac ion) because all o he di e en ime o de ings en e on he same oo ing. Combining all hese e ms back in o (4.6) yields he – 16 – JHEP02(2019)024 h ˆ D2 ˆ D2 ˆ D2 ˆ D2i h ˆ D2 ˆ D2 ˆ D4i h ˆ D2 ˆ D6i h ˆ D4 ˆ D4iD8 Figu e 5. Illus a ion o he Wilson line aces hˆ D2ˆ D2ˆ D2ˆ D2i,hˆ D2ˆ D2ˆ D4i,hˆ D2ˆ D6i,hˆ D4ˆ D4i, and D8con ibu ing o double-pai p oduc ion in eqs. (4.11) and (4.16). comple e esul o opologies wi h no e mion en anglemen : DA(k1,¯ k1,k2,¯ k2) 2E=N4 c 4Z d 2{k0 1¯ k0 1k0 2¯ k0 2k00 1¯ k00 1k00 2¯ k00 2} × DhΨ(k1,¯ k1;k0 1,¯ k0 1)Ψ†(k1,¯ k1;k00 1,¯ k00 1)i DhΨ(k2,¯ k2;k0 2,¯ k0 2)Ψ†(k2,¯ k2;k00 2,¯ k00 2)i ×µ2(k0 1+¯ k0 1+k0 2+¯ k0 2, k+ 1+¯ k+ 1+k+ 2+¯ k+ 2)µ2(−k00 1−¯ k00 1−k00 2−¯ k00 2,−k+ 1−¯ k+ 1−k+ 2−¯ k+ 2) ×1 N2 cDˆ D4(k1−k0 1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)ˆ D4(k2−k0 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2)E −1 N4 cD8(k1−k0 1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2,k2−k0 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) −1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1) +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)ˆ D4(k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2)E +µ2(k0 1+¯ k0 1−k00 1−¯ k00 1,0+)µ2(k0 2+¯ k0 2−k00 2−¯ k00 2,0+) ×Dˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(k1−k0 1,k1−k00 1)ˆ D2(¯ k00 2−¯ k2,¯ k0 2−¯ k2)ˆ D2(k2−k0 2,k2−k00 2)E −1 N2 cDˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(k1−k0 1,k1−k00 1)ˆ D4(k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2)E −1 N2 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)ˆ D2(¯ k00 2−¯ k2,¯ k0 2−¯ k2)ˆ D2(k2−k0 2,k2−k00 2)E +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)ˆ D4(k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2)E +µ2(k0 1+¯ k0 1−k00 2−¯ k00 2, k+ 1+¯ k+ 1−k+ 2−¯ k+ 2)µ2(k0 2+¯ k0 2−k00 1−¯ k00 1, k+ 2+¯ k+ 2−k+ 1−¯ k+ 1) ×1 N2 cDˆ D4(k1−k0 1,k2−k00 2,k2−k0 2,k1−k00 1)ˆ D4(¯ k00 2−¯ k2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,¯ k0 2−¯ k2)E −1 N4 cD8(¯ k00 1−¯ k1,k1−k00 1,k1−k0 1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,¯ k0 1−¯ k1) −1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2,k2−k0 2,k1−k00 1) +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)ˆ D4(k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2)E (4.11) – 17 – JHEP02(2019)024 k1 k2 ¯ k2 ¯ k′′ 1 k′′ 2 ρc k′′ 2+¯ k′′ 1 ¯ k1 k′ 2 ¯ k′ 2 k′ 2+¯ k′ 2 ρb k′ 1 ¯ k′ 1 k′ 1+¯ k′ 1 ρa k′′ 1 ¯ k′′ 2 k′′ 1+¯ k′′ 2 ρd Figu e 6. Double-pai p oduc ion “Pac Man” opologies wi h e mion en anglemen , as calcula ed in eq. (4.12). 4.1.2 Case 2: e mion en anglemen The in e e ence e m in (4.5) con ains he opologies wi h he e mions being en angled, such ha he pai s in he ampli ude “swap owne ship” o one o he e mions in going o he complex-conjuga e ampli ude. This con ibu ion a ises om he Fe mi-Di ac s a is ics o iden ical pa icles and he e o e does no con ibu e i he pai s ha e di e en la o s. Taking he in e e ence o (4.4) as shown in he “Pac Man” ype diag am o igu e 6we ha e di ec ly DA(k1,¯ k1, k2,¯ k2)A†(k1,¯ k2, k2,¯ k1)E=Z d 2{k0 1¯ k0 1k0 2¯ k0 2k00 1¯ k00 1k00 2¯ k00 2} ×Dρa(k0 1+¯ k0 1)ρb(k0 2+¯ k0 2)ρc∗(k00 2+¯ k00 1)ρd∗(k00 1+¯ k00 2)Ep oj × DhΨ(k1,¯ k1;k0 1,¯ k0 1) Ψ†(k2,¯ k1;k00 2,¯ k00 1) Ψ(k2,¯ k2;k0 2,¯ k0 2)Ψ†(k1,¯ k2;k00 1,¯ k00 2)i ×D chV(k1−k0 1) aV†(¯ k0 1−¯ k1)V(¯ k00 1−¯ k1) cV†(k2−k00 2) ×V(k2−k0 2) bV†(¯ k0 2−¯ k2)V(¯ k00 2−¯ k2) dV†(k1−k00 1)iE g .(4.12) In he same way as (4.7), we o m he 3 con ac ions o he p ojec ile sou ces: Dρa(k0 1+¯ k0 1)ρb(k0 2+¯ k0 2)ρc∗(k00 2+¯ k00 1)ρd∗(k00 1+¯ k00 2)Ep oj = (4.13) =δabδcd µ2(k0 1+¯ k0 1+k0 2+¯ k0 2, k+ 1+¯ k+ 1+k+ 2+¯ k+ 2) ×µ2(−k00 2−¯ k00 1−k00 1−¯ k00 2,−k+ 2−¯ k+ 1−k+ 1−¯ k+ 2) +δacδbdµ2(k0 1+¯ k0 1−k00 2−¯ k00 1, k+ 1−k+ 2)µ2(k0 2+¯ k0 2−k00 1−¯ k00 2, k+ 2−k+ 1) +δadδbcµ2(k0 1+¯ k0 1−k00 1−¯ k00 2,¯ k+ 1−¯ k+ 2)µ2(k0 2+¯ k0 2−k00 2−¯ k00 1,¯ k+ 2−¯ k+ 1). – 18 – JHEP02(2019)024 Using he same sho hand no a ion as (4.8), he Wilson line enso en e ing (4.12) is Ωabcd 2= c[V1 aV† 2V3 cV† 8V5 bV† 6V7 dV† 4],(4.14) and we can compu e he a ious colo con ac ions in he same way: δabδcd Ωabcd 2=Nc 4D8(16785234) −Nc 4Dˆ D4(3852) ˆ D4(1674)E −Nc 4Dˆ D4(5678) ˆ D4(1234)E+1 4NcD8(12385674).(4.15a) δacδbd Ωabcd 2=N3 c 4Dˆ D2(32) ˆ D2(76) ˆ D4(1854)E−Nc 4Dˆ D2(32) ˆ D6(185674)E −Nc 4Dˆ D2(76) ˆ D6(123854)E+1 4NcD8(12385674).(4.15b) δadδbc Ωabcd 2=N3 c 4Dˆ D2(58) ˆ D2(14) ˆ D4(7236)E−Nc 4Dˆ D2(14) ˆ D6(385672)E −Nc 4Dˆ D2(58) ˆ D6(123674)E+1 4NcD8(12385674).(4.15c) Combining hese back in o (4.12) yields he comple e esul o opologies wi h e mion en anglemen : DA(k1,¯ k1,k2,¯ k2)A†(k1,¯ k2,k2,¯ k1)E=N3 c 4Z d 2{k0 1¯ k0 1k0 2¯ k0 2k00 1¯ k00 1k00 2¯ k00 2} × DhΨ(k1,¯ k1;k0 1,¯ k0 1)Ψ†(k2,¯ k1;k00 2,¯ k00 1)Ψ(k2,¯ k2;k0 2,¯ k0 2)Ψ†(k1,¯ k2;k00 1,¯ k00 2)i ×µ2(k0 1+¯ k0 1+k0 2+¯ k0 2, k+ 1+¯ k+ 1+k+ 2+¯ k+ 2)µ2(−k00 2−¯ k00 1−k00 1−¯ k00 2,−k+ 2−¯ k+ 1−k+ 1−¯ k+ 2) ×h1 N2 cD8(k1−k0 1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2,k2−k0 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) −1 N2 cDˆ D4(¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 1−¯ k1)ˆ D4(k1−k0 1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)E −1 N2 cDˆ D4(k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k2−k00 2)ˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)E +1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)i +µ2(k0 1+¯ k0 1−k00 2−¯ k00 1, k+ 1−k+ 2)µ2(k0 2+¯ k0 2−k00 1−¯ k00 2, k+ 2−k+ 1) ×hDˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(¯ k00 2−¯ k2,¯ k0 2−¯ k2)ˆ D4(k1−k0 1,k2−k00 2,k2−k0 2,k1−k00 1)E −1 N2 cDˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D6(k1−k0 1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)E −1 N2 cDˆ D2(¯ k00 2−¯ k2,¯ k0 2−¯ k2)ˆ D6(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,k1−k00 1)E +1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)i +µ2(k0 1+¯ k0 1−k00 1−¯ k00 2,¯ k+ 1−¯ k+ 2)µ2(k0 2+¯ k0 2−k00 2−¯ k00 1,¯ k+ 2−¯ k+ 1) ×hDˆ D2(k2−k0 2,k2−k00 2)ˆ D2(k1−k0 1,k1−k00 1)ˆ D4(¯ k00 2−¯ k2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,¯ k0 2−¯ k2)E −1 N2 cDˆ D2(k1−k0 1,k1−k00 1)ˆ D6(¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,¯ k0 1−¯ k1)E −1 N2 cDˆ D2(k2−k0 2,k2−k00 2)ˆ D6(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)E +1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k2−k00 2,k2−k0 2,¯ k0 2−¯ k2,¯ k00 2−¯ k2,k1−k00 1)i(4.16) – 19 – JHEP02(2019)024 The exp ession o he c oss sec ion (4.5), oge he wi h he wo classes o opologies (4.11) and (4.16) cons i u e he i s comple e and exac solu ion o he 4-pa icle inclusi e (q¯q) (q¯q) c oss sec ion a his o de . These exp essions a e one o he p ima y esul s o his pape . 4.2 C oss sec ion o qua k + an iqua k + gluon p oduc ion In he same way as in sec ion 3.2, we can now ake he “gluonic limi ” o he q¯qpai k2,¯ k2→1 2q2in he double-pai exp ession (4.11). No e ha , once we eplace one o he pai s wi h a gluon, he e a e no longe any possible e mion en anglemen opologies, since all h ee inal s a e pa icles q¯qG a e now dis inguishable. Thus we need only ake he limi o he opologies in (4.11), immedia ely ob aining: dσ(q¯q)G d2k1dy1d2¯ k1d¯y1d2q2dy2=1 2(2π)33N4 c 4Z d 2{k0 1¯ k0 1k00 1¯ k00 1q0 2q00 2δk0 2δk00 2} × DhΨ(k1,¯ k1;k0 1,¯ k0 1)Ψ†(k1,¯ k1;k00 1,¯ k00 1)i DhΦ(q2;q0 2)Φ†(q2;q00 2)i ×µ2(k0 1+¯ k0 1+q0 2, k+ 1+¯ k+ 1+q+ 2)µ2(−k00 1−¯ k00 1−q00 2,−k+ 1−¯ k+ 1−q+ 2) ×1 N2 cDˆ D4(k1−k0 1,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,k1−k00 1) ׈ D4(1 2q2−1 2q0 2−δk0 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,1 2q2−1 2q00 2−δk00 2)E −1 N4 cD8(k1−k0 1,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2, 1 2q2−1 2q0 2−δk0 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) −1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,1 2q2−1 2q00 2−δk00 2, 1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,k1−k00 1) +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) ׈ D4(1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2)E +µ2(k0 1+¯ k0 1−k00 1−¯ k00 1,0+)µ2(q0 2−q00 2,0+) ×Dˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(k1−k0 1,k1−k00 1)ˆ D2(1 2q00 2−1 2q2−δk00 2,1 2q0 2−1 2q−δk0 2) ׈ D2(1 2q2−1 2q0 2−δk0 2,1 2q2−1 2q00 2−δk00 2)E −1 N2 cDˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(k1−k0 1,k1−k00 1) ׈ D4(1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2)E −1 N2 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1)ˆ D2(1 2q00 2−1 2q2−δk00 2,1 2q0 2−1 2q2−δk0 2) ׈ D2(1 2q2−1 2q0 2−δk0 2,1 2q2−1 2q00 2−δk00 2)E +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) ׈ D4(1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2)E – 20 – JHEP02(2019)024 +µ2(k0 1+¯ k0 1−q00 2, k+ 1+¯ k+ 1−q+ 2)µ2(q0 2−k00 1−¯ k00 1, q+ 2−k+ 1−¯ k+ 1) ×1 N2 cDˆ D4(k1−k0 1,1 2q2−1 2q00 2−δk00 2,1 2q2−1 2q0 2−δk0 2,k1−k00 1) ׈ D4(1 2q00 2−1 2q2−δk00 2,¯ k0 1−¯ k1,¯ k00 1−¯ k1,1 2q0 2−1 2q2−δk0 2)E −1 N4 cD8(¯ k00 1−¯ k1,k1−k00 1,k1−k0 1,1 2q2−1 2q00 2−δk00 2, 1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,¯ k0 1−¯ k1) −1 N4 cD8(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,1 2q0 2−1 2q2−δk0 2, 1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2,1 2q2−1 2q0 2−δk0 2,k1−k00 1) +1 N4 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1,k1−k00 1) ׈ D4(1 2q2−1 2q0 2−δk0 2,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2,1 2q2−1 2q00 2−δk00 2)E, (4.17) whe e as be o e we ha e changed a iables in o q0 2≡k0 2+¯ k0 2and δk0 2≡1 2(k0 2−¯ k0 2), and simila ly o q00 2and δk00 2. A e in eg a ion o e δk0 2, δk00 2, mos o hese e ms anish. These in eg als ac only on he in e ac ion e ms, and as we saw in eqs. (3.8) and (3.9), any ime he same momen um δk(0,00) 2appea s in adjacen a gumen s o he same ace, hose Wilson lines will cancel. In canceling, hey lead o del a unc ions o he momen um di e ence ollowing (B.5), which is always ei he δ2(q0 2−q2) o δ2(q00 2−q2), and hese e ms d op ou due o he anishing o he wa e unc ion as in (3.9). The only in e ac ion e ms which do no anish co espond o lines 1, 5, 7, and 9 ou o he 12 e ms in b aces. As in (3.10), hese e ms ins ead simpli y he Wilson line aces by causing some o he coo dina e-space a gumen s o be epea ed, wi h he composi e objec being Fou ie ans o med o momen um space. One o he wo su i ing ope a o s was al eady calcula ed in (3.10): he squa e o he dipole ampli ude. The o he non anishing ope a o is a pa ial educ ion o he double quad upole (see igu e 7): Z d 2{δk0 2δk00 2}Dˆ D4(p1,1 2q0 2−1 2q2−δk0 2,1 2q00 2−1 2q2−δk00 2, p2) ׈ D4(1 2q2−1 2q0 2−δk0 2, p3, p4,1 2q2−1 2q00 2−δk00 2)E =Zd2{x1y1x2y2z1w1z2w2}Z d 2{δk0 2δk00 2} ×e−ip1·x1ei1 2q0 2−1 2q2−δk0 2·y1e−i1 2q00 2−1 2q2−δk00 2·x2eip2·y2 ×e−i1 2q2−1 2q0 2−δk0 2·z1eip3·w1e−ip4·z2ei1 2q2−1 2q00 2−δk00 2·w2 ×Dˆ D4(x1, y1, x2, y2)ˆ D4(z1, w1, z2, w2)E =Zd2{x1y1x2y2w1z2}e−ip1·x1ei(q0 2−q2)·y1e−i(q00 2−q2)·x2eip2·y2eip3·w1e−ip4·z2 ×Dˆ D4(x1, y1, x2, y2)ˆ D4(y1, w1, z2, x2)E ≡D4,4(p1, q0 2−q2, q00 2−q2, p2;p3p4).(4.18) – 21 – JHEP02(2019)024 h ˆ D2 ˆ D2| ˆ D2|2iD4,4 h ˆ D4| ˆ D2|2i Figu e 7. Illus a ion o he Wilson line aces hˆ D2ˆ D2|ˆ D2|2i,hˆ D4|ˆ D2|2iand D4,4con ibu ing o q¯qG p oduc ion in eq. (4.19). The esul is he comple e exp ession o he q¯qG c oss sec ion in momen um space, wi h he co esponding ope a o s illus a ed in igu e 7: dσ(q¯q)G d2k1dy1d2¯ k1d¯y1d2q2dy2=1 2(2π)33N4 c 4Z d 2{k0 1¯ k0 1k00 1¯ k00 1q0 2q00 2} × DhΨ(k1,¯ k1;k0 1,¯ k0 1) Ψ†(k1,¯ k1;k00 1,¯ k00 1)i DhΦ(q2;q0 2) Φ†(q2;q00 2)i ×µ2(k0 1+¯ k0 1+q0 2, k+ 1+¯ k+ 1+q+ 2)µ2(−k00 1−¯ k00 1−q00 2,−k+ 1−¯ k+ 1−q+ 2) ×h1 N2 cD4,4(k1−k0 1, q0 2−q2, q00 2−q2, k1−k00 1;¯ k0 1−¯ k1¯ k00 1−¯ k1)i +µ2(k0 1+¯ k0 1−k00 1−¯ k00 1,0+)µ2(q0 2−q00 2,0+) ×Dˆ D2(¯ k00 1−¯ k1,¯ k0 1−¯ k1)ˆ D2(k1−k0 1, k1−k00 1) ˆ D2 2(q00 2−q2, q0 2−q2)E −1 N2 cDˆ D4(k1−k0 1,¯ k0 1−¯ k1,¯ k00 1−¯ k1, k1−k00 1) ˆ D2 2(q00 2−q2, q0 2−q2)E +µ2(k0 1+¯ k0 1−q00 2, k+ 1+¯ k+ 1−q+ 2)µ2(q0 2−k00 1−¯ k00 1, q+ 2−k+ 1−¯ k+ 1) ×1 N2 cD4,4(k1−k0 1, q2−q00 2, q2−q0 2, k1−k00 1;¯ k0 1−¯ k1¯ k00 1−¯ k1).(4.19) To ou knowledge, eq. (4.19) ep esen s he i s comple e calcula ion o q¯qG p oduc ion in he CGC amewo k, and his compac o m in momen um space comp ises an exac solu ion a his o de , a ini e Nc. We emphasize, howe e , ha his c oss sec ion applies only o he p oduc ion o a qua k and an iqua k o he same la o . This new exp ession is he second p ima y esul o his pape . – 22 – JHEP02(2019)024 4.3 Double-gluon p oduc ion As a inal c oss-check o he p eceding calcula ions, le us use he mapping (2.19) on he emaining q¯qpai in (4.19) o ob ain he double-gluon p oduc ion c oss sec ion, which we can compa e wi h explici esul s in he li e a u e. As be o e, we ake he limi k1,¯ k1→1 2q1 in (4.19), adjus he coun o 1 2(2π)3in he p e ac o , and change in eg a ion a iables o q0 1≡k0 1+¯ k0 1and δk0 1≡1 2(k0 1−¯ k0 1), and simila ly o q00 1and δk00 1. This gi es dσGG d2q1dy1d2q2dy2=1 2(2π)32N4 c 4Z d 2{q0 1q00 1q0 2q00 2δk0 1δk00 1}(4.20) × DhΦ(q1;q0 1)Φ†(q1;q00 1)i DhΦ(q2;q0 2)Φ†(q2;q00 2)i ×µ2(q0 1+q0 2, q+ 1+q+ 2)µ2(−q00 1−q00 2,−q+ 1−q+ 2) ×h1 N2 cD4,4(1 2q1−1 2q0 1−δk0 1,q0 2−q2,q00 2−q2,1 2q1−1 2q00 1−δk00 1;1 2q0 1−1 2q1−δk0 11 2q00 1−1 2q1−δk00 1)i +µ2(q0 1−q00 1,0+)µ2(q0 2−q00 2,0+) ×Dˆ D2(1 2q00 1−1 2q1−δk00 1,1 2q0 1−1 2q1−δk0 1)ˆ D2(1 2q1−1 2q0 1−δk0 1,1 2q1−1 2q00 1−δk00 1) × ˆ D2 2(q00 2−q2,q0 2−q2)E −1 N2 cDˆ D4(1 2q1−1 2q0 1−δk0 1,1 2q0 1−1 2q1−δk0 1,1 2q00 1−1 2q1−δk00 1,1 2q1−1 2q00 1−δk00 1) × ˆ D2 2(q00 2−q2,q0 2−q2)E +µ2(q0 1−q00 2, q+ 1−q+ 2)µ2(q0 2−q00 1, q+ 2−q+ 1) ×h1 N2 cD4,4(1 2q1−1 2q0 1−δk0 1,q2−q00 2,q2−q0 2,1 2q1−1 2q00 1−δk00 1;1 2q0 1−1 2q1−δk0 11 2q00 1−1 2q−δk00 1)i. O he ou in e ac ion e ms emaining in he b aces, he hi d one (quad upole ace) anishes a e in eg a ion o e δk0 1δk00 1due o epea ed adjacen a gumen s; he second one (double dipole) is he same as (3.10); and he i s and las ones a e u he educ ions o he double-quad upole (4.18): Z d 2{δk0 1δk00 1}D4,4(1 2q1−1 2q0 1−δk0 1,p1,p2,1 2q1−1 2q00 1−δk00 1;1 2q0 1−1 2q1−δk0 11 2q00 1−1 2q1−δk00 1) =Zd2{x1y1x2y2w1z2}Z d 2{δk0 1δk00 1}e−i1 2q1−1 2q0 1−δk0 1·x1eip1·y1e−ip2·x2ei1 2q1−1 2q00 1−δk00 1·y2 ×ei1 2q0 1−1 2q1−δk0 1·w1e−i1 2q00 1−1 2q1−δk00 1·z2Dˆ D4(x1,y1,x2,y2)ˆ D4(y1,w1,z2,x2)E =Zd2{x1y1x2y2w1z2}e−i(q1−q0 1)·x1eip1·y1e−ip2·x2ei(q1−q00 1)·y2 ˆ D4(x1,y1,x2,y2) 2 ≡|D4|2(q1−q0 1,p1,p2,q1−q00 1),(4.21) – 23 – JHEP02(2019)024 which is jus he squa e o he quad upole. Thus he double-gluon c oss sec ion akes he especially compac o m in momen um space dσGG d2q1dy1d2q2dy2=1 2(2π)32N4 c 4Z d 2{q0 1q00 1q0 2q00 2}(4.22) × DhΦ(q1;q0 1)Φ†(q1;q00 1)i DhΦ(q2;q0 2)Φ†(q2;q00 2)i ×µ2(q0 1+q0 2, q+ 1+q+ 2)µ2(−q00 1−q00 2,−q+ 1−q+ 2)h1 N2 c|D4|2(q1−q0 1,q0 2−q2,q00 2−q2,q1−q00 1)i +µ2(q0 1−q00 1,0+)µ2(q0 2−q00 2,0+)D ˆ D2 2(q00 1−q1,q0 1−q1) ˆ D2 2(q00 2−q2,q0 2−q2)E +µ2(q0 1−q00 2, q+ 1−q+ 2)µ2(q0 2−q00 1, q+ 2−q+ 1)h1 N2 c|D4|2(q1−q0 1,q2−q00 2,q2−q0 2,q1−q00 1)i). A his poin , we can di ec ly compa e he c oss sec ion (4.22) agains he known exp essions in he li e a u e; e . [38] gi es he GG c oss sec ion in momen um space, and e . [34] gi es he c oss sec ion in coo dina e space. He e we will pe o m he ans o ma ion o coo dina e space o demons a e exac ag eemen wi h e . [34], and in appendix Cwe p esen he addi ional compa ison wi h e . [38] in momen um space. Mos o he wo k in pe o ming he c oss-check agains e . [34] comes om un olding he compac exp ession (4.22) back in o coo dina e space. Inse ing he Fou ie ans o ms o he a ious quan i ies gi es dσGG d2q1dy1d2q2dy2 =1 2(2π)32N4 c 4Zd2{x1y1x2y2x0 1y0 1x0 2y0 2b1b2}(4.23) × DhΦ(x1−b1;x0 1−b1)Φ†(y1−b1;y0 1−b1)i DhΦ(x2−b2;x0 2−b2)Φ†(y2−b2;y0 2−b2)i ×D ˆ D2 2(x0 1,y0 1) ˆ D2 2(x0 2,y0 2)E ×e−iq1·(x0 1−y0 1+x1−y1)e−iq2·(x0 2−y0 2+x2−y2)µ2(b1,0+)µ2(b2,0+) + DhΦ(x1−b1;x0 1−b1)Φ†(y2−b2;y0 2−b2)i DhΦ(x2−b2;x0 2−b2)Φ†(y1−b1;y0 1−b1)i ×1 N2 c|D4|2(x0 1,y0 1,x0 2,y0 2) ×e−iq1·(x0 1−y0 2+x1−y2−b1+b2)e−iq2·(x0 2−y0 1+x2−y1−b2+b1)µ2(b1, q+ 1−q+ 2)µ2(b2, q+ 2−q+ 1) +e−iq1·(x0 1−y0 2+x1−y2−b1+b2)eiq2·(x0 2−y0 1+x2−y1+b1−b2)µ2(b1, q+ 1+q+ 2)µ2(b2,−q+ 1−q+ 2). In a i ing a (4.23), we ha e used symme y p ope ies o he squa ed dipole and he gluon wa e unc ions (2.11). Inse ing (3.13) in he ou wa e unc ions gene a es 16 e ms, each con aining 4 del a unc ions. As be o e, a e in eg a ing o e hose del a unc ions, we d op any emaining p imes on he in eg a ion a iables; his leads o all o he a ious wa e unc ions in a gi en se o e ms ha ing he same a gumen s, wi h he di e ences esiding in he Wilson line in e ac ions. – 24 – JHEP02(2019)024 he in insic ans e se momen um o he pojec ile is cha ac e ized by he p ojec ile sa u- a ion scale Qs,p and is compu ed o de by o de in pe u ba ion heo y; and he in insic ans e se momen um o had oniza ion is simply o o de ΛQCD and is no enhanced by any high densi y scales. Thus o he semi-dilu e / dense egime, we ha e he hie a chy o scales (1.2), o which we can neglec he in insic ans e se momen um cha ac e ized by he agmen a ion unc ions. This hie a chy o scales is in he same spi i as he amewo k o hyb id ac o iza ion, in which he p oduc ion o wo dis inguishable hea y qua konia has ecen ly been calcula ed [40]. One po en ial d awback o he asymme ic ea men o he p ojec ile, a ge , and agmen a ion sec o s is ha he agmen a ion unc ions employed abo e do no possess an unambiguous ac o iza ion scale µF. This ea u e also applies o he desc ip ion o agmen a ion employed in e . [55]. While he pa icula agmen a ion unc ions gi en in e . [61] do no explici ly e e o a ac o iza ion scale µF, agmen a ion unc ions in gene al — and he a ailable agmen a ion unc ions o ligh had ons in pa icula — ca y such a dependence on an a bi a y scale. This a bi a y scale dependence in he agmen a ion sec o is compensa ed by he scale dependence o he pa on dis ibu ion unc ions o he p ojec ile and a ge , such ha he obse able c oss sec ion is o e all in- a ian unde he eno maliza ion-g oup e olu ion o i s a ious nonpe u ba i e pieces. In ou case, he p ojec ile, a ge , and agmen a ion sec o s ha e all been ea ed e y di e - en ly, such ha he scale dependence coming om he p ojec ile and a ge dis ibu ions is ambiguous.4In a mo e comple e ea men which pu s he nonpe u ba i e p ojec ile, a - ge , and agmen a ion sec o s on compa able oo ing, he cancella ion o his ac o iza ion scale dependence would become explici . This is wha is seen explici ly in he s anda d hyb id ac o iza ion amewo k o he dilu e / dense egime [64,65], and we expec ha he same would be ue o he semi-dilu e / dense egime in a ea men such as ha o e . [40], which o mula es hyb id ac o iza ion in e ms o double pa on dis ibu ions o he p ojec ile. Finally, le us no e ha while he ICEM is a pa icula ly simple and con enien model o desc ibing he had oniza ion o a qua konium s a e om a q¯qpai , i is by no means unique. A a ie y o o he desc ip ions o his had oniza ion p ocess exis , in pa icula he e ec i e ield heo y o Non-Rela i is ic Quan um Ch omodynamics (NRQCD) [66]. While di e en had oniza ion o malisms ha e hei own ad an ages and disad an ages, NRQCD has he pa icula ad an age o being a sel -consis en e ec i e ield heo y o QCD. Employing i equi es a mo e de ailed ea men han jus a simple con olu ion o he pa onic c¯cc oss sec ion as done abo e, using a se ies o p ojec ion ope a o s o selec ou he quan um numbe s o he c¯cs a e app op ia e o a gi en had oniza ion channel. O hese p ojec ion ope a o s, he colo p ojec ions on o single and oc e c¯cs a es will equi e a mo e de ailed implemen a ion because hey will modi y he Wilson lines which en e he mul ipole aces. These a ious p ojec ions a e in p inciple s aigh o wa d, bu hey a e beyond he scope o his pape ; we lea e he inco po a ion o an NRQCD-based app oach 4This is no o be con used wi h he scale dependence on a apidi y egula o ; he RG e olu ion in apidi y is con ained wi hin he apidi y dependence o he Wilson line aces. This dependence is gene ally cha ac e ized by he JIMWLK unc ional e olu ion equa ion. – 31 – JHEP02(2019)024 o qua konium p oduc ion o u u e wo k. I is in e es ing o no e, howe e , ha inclu- si e J/ψ p oduc ion a small xin he CGC amewo k was s udied in e . [55] using bo h NRQCD and he ICEM o had oniza ion, concluding ha he ICEM is a good app oxi- ma ion o he NRQCD app oach due o he dominance o he 3S[8] 1p oduc ion channel. 6 Conclusions In his pape , we ha e compu ed a numbe o c oss sec ions o he p oduc ion o mul iple pa icles a mid- apidi y in he semi-dilu e / dense egime o he colo -glass condensa e e ec i e ield heo y. A he pa onic le el, we ha e compu ed o he i s ime he p oduc- ion c oss sec ions o wo qua k/an iqua k pai s (q¯q) (q¯q) (eqs. (4.5), (4.11), and (4.16)) and o one qua k/an iqua k pai plus a gluon (q¯q)G(eq. (4.19)). The double-pai ex- p ession signi ican ly ex ends p e ious wo k [36] in ha i is ully di e en ial in all ou pa icles, includes all ime o de ings and all o de s o mul iple esca e ing in he a ge ields, and is e alua ed wi h exac Nc. These new pa onic c oss sec ions a e one o he p ima y new esul s o his pape . Addi ionally, we p o ed a simple mapping (2.19) be ween he p oduc ion ampli ude o a q¯qpai and he p oduc ion ampli ude o a gluon, which we used o ob ain he (q¯q)Gc oss sec ion om he (q¯q) (q¯q) c oss sec ion, and which we alida ed by c oss- checking he single-gluon (3.11) and double-gluon (4.22) p oduc ion c oss sec ions agains he li e a u e [13,34,38]. The mapping (2.19) and i s applica ion o de i e whole classes o c oss sec ions om he mul i-pai c oss sec ion is he second p ima y esul o his pape . Finally, in sec ion 5we discussed how o ansla e he pa onic c oss sec ions compu ed he e in o had onic ones in he hea y la o sec o h ough he use o collinea agmen- a ion unc ions o open hea y la o and he Imp o ed Colo E apo a ion Model [63] o hea y qua konia. This p ocedu e ansla es each o he pa onic c oss sec ions in o a ange o had onic obse ables, allowing us o w i e down exp essions o he p oduc ion o : (D¯ D) (D¯ D) — eq. (5.2); (D¯ D) (J/ψ) — eq. (5.6); (J/ψ) (J/ψ) — eq. (5.5); (D¯ D)h— eq. (5.3); and (J/ψ)h— eq. (5.7), whe e his any ligh had on. These exp essions open he doo o a wide ange o phenomenology o s udy he p oduc ion and co ela ions o many pa icles in he hea y la o sec o , and hey a e he hi d p ima y esul o his pape . The abili y o pe o m a small numbe o ab ini io calcula ions in he CGC o malism a he pa onic le el o simul aneously p edic he p oduc ion c oss sec ions and co ela ions o a wide a ie y o had onic obse ables has he po en ial o b oadly es he ini ial-s a e mechanisms as an explana ion o he obse ed co ela ions in hea y- and hea y-ligh ion collisions. Genuine mul ipa icle p oduc ion compu ed wi hin he CGC amewo k makes i possible o sel -consis en ly compu e highe cumulan s such as 2{4}and cha ge-dependen co ela ions like γ112 om pu ely ini ial-s a e mechanisms. Simila ly, he coo dina e-space p og am begun in e . [54] aspi es o ake pa onic co ela ions such as hese as inpu s o he ini ial condi ions o subsequen hyd odynamic e olu ion, including con ibu ions om conse ed cha ges due o qua k p oduc ion. As we con inue o ex end his p og am o genuine mul ipa icle p oduc ion, beyond simple app oxima ions like he so-called “dilu e / dilu e glasma g aphs” o he la ge-Ncapp oxima ion, we an icipa e ha i will open up – 32 – JHEP02(2019)024 b oad oppo uni ies o es he e ec s o ini ial-s a e co ela ions, wi h and wi hou he impac o s ongly-coupled inal-s a e in e ac ions. One po en ial ba ie o such a comp ehensi e p og am o mul i-had on phenomenology in he CGC app oach is ha mul ipa icle c oss sec ions, like he ones calcula ed he e, in oke highe and highe n-poin co ela o s o Wilson lines, up o he oc upole D8 o double-pai p oduc ion. These ope a o s become inc easingly di icul o e alua e, e en in simple models like he MV model, o which analy ic exp essions a e only a ailable o he 4-poin unc ions [67]. Howe e , a compelling a gumen summa ized in e . [38] and a ibu ed o e . [47] sugges s ha , up o co ec ions supp essed by he la ge a ea o he a ge , n-poin co ela o s can in gene al be ac o ized in o p oduc s o 2-poin co ela o s (dipoles), which a e well-cons ained in heo y and phenomenology. I his a gumen holds, hen he inc easingly complex Wilson line s uc u e is no obs acle o he pu sui o mul i- had on phenomenology. Aside om he applica ions al eady discussed abo e, he e a e a numbe o o he di ec ex ensions o his me hod we can pu sue in u u e wo k. One is o epea he double-pai calcula ion o sec ion 4.1 in coo dina e space o s udy he spa ial co ela ions among qua ks and an iqua ks; as discussed in e . [54], hese double-pai co ela ions a e he dominan e ec o same-sign cha ged pa icles and o opposi e-sign cha ged pa icles a dis ances la ge han he in e se qua k mass 1/m. Ano he is o ex end he calcula ions pe o med he e o double-pai p oduc ion o iple-pai p oduc ion: (q¯q) (q¯q) (q¯q). The numbe o pe mu a ions will inc ease subs an ially in going o iple-pai p oduc ion, bu he unda- men al mechanics o he calcula ion will no change, and he compac exp ession (2.7) in momen um space makes such an ex ension ac able. Mo eo e , he gluonic mapping (2.19) will make i possible o immedia ely ansla e he iple-pai c oss sec ion in o a whole am- ily o ela ed c oss sec ions:(q¯q) (q¯q) (q¯q); (q¯q) (q¯q)G; (q¯q)G G; and G G G. Las , we no e ha he exp essions o had oniza ion in he hea y la o sec o we explo e in sec ion 5can be signi ican ly imp o ed and ex ended. The hea y la o sec o is con enien as a jus i ica ion o he assump ion ha a gi en had on (like a Dmeson) in he inal s a e is domina ed by agmen a ion om a gi en pa on (like a cqua k). In p inciple, a sum o e all pa onic channels wi h app op ia e agmen a ion unc ions will elax his assump ion and make i possible o s udy agmen a ion in o se e al iden i ied had ons. As we ex end he p og am o compu e mul ipa icle p oduc ion in he CGC amewo k, we will include mo e and mo e o hese pa onic channels, allowing a comple e calcula ion o co ela ions in inclusi e had on p oduc ion. In pa icula , he CGC con ibu ion o he cha ge-dependen co ela ions γ112 and γ123 which o m he backg ound o he signal o he chi al magne ic e ec is o special impo ance. While some explo a o y wo k on his subjec was done in e . [37], i includes nei he sca e ing in he a ge ields no had oniza ion, bo h o which a e likely o s ongly modi y he cha ge-dependen co ela ions. Howe e , wi h he calcula ion o a ange o pa onic channels and app op ia e cha ge-dependen agmen a ion unc ions, a obus compu a ion o he had onic cha ge co ela ions becomes possible. Fo all o hese easons, we belie e ha he heo e ical ad ancemen s p esen ed in his pape ep esen a signi ican s ep owa d implemen ing a comp ehensi e p og am o phenomenology o s udy mul i-had on co ela ions om he ini ial s a e. – 33 – JHEP02(2019)024 Acknowledgmen s The au ho s wish o hank N. A mes o, D. Pi onyak, J. No onha-Hos le , V. Skoko , P. T ibedy, and R. Venugopalan o use ul discussions. This wo k is suppo ed in pa by he U.S. Depa men o Ene gy g an DE-FG02-03ER41260 and he BEST (Beam En- e gy Scan Theo y) DOE Topical Collabo a ion (MM), DOE Con ac No. DE-AC52- 06NA25396 and he DOE Ea ly Ca ee P og am (MS), he Eu opean Resea ch Council 39 g an Ho LHC ERC-2011-S G-279579, Minis e io de Ciencia e Inno aci´on o Spain unde p ojec FPA2014-58293-C2-1-P and Unidad de Excelencia Ma ´ıa de Maez u unde p ojec MDM-2016-0692, Xun a de Galicia (Conselle ´ıa de Educaci´on) and FEDER (DW). A Colo a e aging in he p ojec ile and a ge In calcula ing c oss sec ions, we will need o compu e squa es and in e e ences o he elemen a y building block (2.7) and a e age hem o e a ious luc ua ing quan i ies. Aside om he a e aging o e he quan um numbe s o he p oduced pa icles, which is pe o med in he usual way, he e en a e aging co e s h ee dis inc ypes o luc ua ions. These a e luc ua ions in he colo ields o he p ojec ile, luc ua ions in he colo ields o he a ge , and luc ua ions o e he global collision geome y. These h ee ypes o a e aging ac o ize om one ano he , such ha we can a e age he sou ces ρin he colo ields o he p ojec ile, which we deno e h···ip oj, sepa a ely om he a e aging o he Wilson lines o e he colo ields in he a ge , which we deno e h···i g o simply h···iwhen he e is no ambigui y. The collision geome y is cha ac e ized by an impac pa ame e Bbe ween he cen e s o he p ojec ile and a ge , which can be ei he held ixed a he c oss sec ion le el o in eg a ed ou a he end o he calcula ion. Simila ly, he e may be o he pa ame e s desc ibing he o e all collision geome y, such as he angula o ien a ion o a non-sphe ical nucleus like u anium; hese global pa ame e s will also be in eg a ed ou a he c oss sec ion le el. In his pape , we make no pa icula assump ion abou he na u e o he a e aging in he colo ields o he a ge ; ou inal exp essions o he c oss sec ions will in ol e a ious aces o Wilson lines (2.3) co esponding o colo dipole, quad upole, sex upole, and oc upole ope a o s. We deno e hose co esponding ope a o s by ˆ D2(x, y)≡1 Nc hVxV† yi(A.1a) ˆ D4(x, y, z, w)≡1 Nc hVxV† yVzV† wi(A.1b) ˆ D6(x, y, z, w, u, )≡1 Nc hVxV† yVzV† wVuV† i(A.1c) ˆ D8(x, y, z, w, u, , , s)≡1 Nc hVxV† yVzV† wVuV† V V† si.(A.1d) Fo quan i ies ha ha e al eady been a e aged we d op he ha o e he ope a o , w i ing e.g. D2(x, y) = hˆ D2(x, y)i. When compu ing simila aces wi h adjoin Wilson lines, we will deno e he ope a o wi h he supe sc ip “adj”, w i ing e.g. ˆ Dadj 2(x, y)≡1 N2 c−1Uab xU† yba . – 34 – JHEP02(2019)024 And we will main ain he same no a ion whe he in oking Wilson lines in coo dina e space o momen um space, w i ing e.g. ˆ D2(p, q) = 1 Nc [V(p)V†(q)] in momen um space. I is also impo an o no e ha he (a e aged) Wilson line aces om eqs. (A.1) implici ly depend on a apidi y scale Y. This apidi y scale egula es he ligh -cone di e - gences associa ed wi h highe -o de co ec ions o hese ope a o s, and he e a e a ange o di e en schemes a ailable o egula e hese di e gences. Physically, we can hink o his scale as being se by he o al apidi y in e al Y∝ln s ⊥2o he collision, and he quan um e olu ion wi h he unning o his scale is gi en by he JIMWLK e olu ion equa- ions [10–12] (o he la ge-Ncanalogue, he BK equa ion [8,9]). This e olu ion is igge ed when he apidi y in e al is pa ame ically la ge, Y∼1 αs. On he o he hand, we es ic ou sel es he e o he case when he p oduced pa icles a e close enough in apidi y ha we do no need o conside quan um e olu ion in he apidi y in e al be ween he pa icles. Fo mally, his means ∆yij <1 αs o he apidi y in e al ∆yij be ween any wo pa icles i, j agged a mid- apidi y. Fo he a e age h···ip oj o e p ojec ile colo ields, we’ll use a Gaussian a e aging p ocedu e inspi ed by he McLe an-Venugopalan (MV) model [68]. Gaussian a e aging co esponds o limi ing he in e ac ion o a sou ce pa icle o wo gluons, which is he lowes o de in pe u ba ion heo y ha p ese es colo neu ali y. Fo Gaussian colo a e aging, he expec a ion alue o p oduc s o se e al ρ’s ac o izes in o a sum o e all possible pai wise “con ac ions,” such ha i is only necessa y o speci y he wo-poin unc ion hρ ρi o ully speci y he esul o he a e aging p ocedu e: hρa(x)ρb(y)ρc∗(z)ρd∗(w)ip oj =Dρa(x)ρb(y)Ep oj Dρc∗(z)ρd∗(w)Ep oj (A.2) +hρa(x)ρc∗(z)ip oj Dρb(y)ρd∗(w)Ep oj +Dρa(x)ρd∗(w)Ep oj Dρb(y)ρc∗(z)Ep oj . The o iginal MV model [68] has been gene alized in a numbe o di e en ways in he li e a u e; o ou pu poses, i is use ul o enume a e h ee dis inc physical assump ions abou he wo-poin unc ion: Dρa(x)ρb∗(y)Ep oj =       δab δ2(x−y)δ(x−−y−)µ2(x, x−) Locali y δab δ(x−−y−)µ2(|x−y|2 T, x−) 2D T ansl.In . δab δ2(x−y)δ(x−−y−)µ2(x−) Bo h (A.3a) Dρa(q1)ρb∗(q2)Ep oj =       δab µ2(q1−q2, q+ 1−q+ 2) Locali y δab (2π)2δ2(q1−q2)µ2(q2 1T, q+ 1−q+ 2) 2D T ansl.In . δab (2π)2δ2(q1−q2)µ2(q+ 1−q+ 2) Bo h. (A.3b) Fo a e ages wi h bo h o he colo ields in he ampli ude hρ ρi, he momen um o he second sou ce is e e sed: q2→ −q2, and o a e ages wi h bo h o he colo ields in he complex conjuga e ampli ude hρ∗ρ∗i, he momen um o he i s sou ce is e e sed: q1→ −q1. The i s case in eqs. (A.3), “Locali y,” is he one we will employ h oughou his pape . This assump ion desc ibes colo luc ua ions cha ac e ized by Gaussian andom noise: hey – 35 – JHEP02(2019)024 a e only co ela ed locally a he same poin , wi h di e en poin s in space ha ing o ally unco ela ed andom luc ua ions. The second case, “2D T ansla ional In a iance,” does no equi e locali y, bu does assume ha he a e age dis ibu ion o sou ce cha ges is uni o m in he ans e se plane; his assump ion is employed in e e ences such as [14] and [36]. The simples case uses “Bo h” locali y and 2D ansla ional in a iance; his assump ion is he one employed by he o iginal MV model [68] and gene alized somewha in e . [69]. An in e es ing a gumen abou he gene al s uc u e o he wo-poin unc ion equi ing only e y weak assump ions abou colo neu ali y was ecen ly gi en in e . [46]; o addi ional discussion abou he p ope ies o he wo-poin unc ion see e.g. e s. [69] and [38]. Wha e e physical assump ions a e made abou he wo-poin unc ion in he Gaussian a e aging, he colo cha ge densi y luc ua ions a e cha ac e ized by he scale µ2, which is ela ed o he sa u a ion scale o he p ojec ile Q2 s,p oj ∝µ2[13,70]. As de ined in (A.3), he scale µ2e ec i ely con ains a ac o o he coupling g2associa ed wi h adia ing a so gluon om he colo sou ces; some e e ences p e e o w i e his ac o explici ly, bu we will use he con en ions o (A.3) in which ha coupling is con ained wi hin µ2. No e also ha he dimensions o µ2as w i en in (A.3) a e di e en among he di e en physical assump ions. I should also be emphasized ha he a e aging pe o med he e o he p ojec ile, deno ed h···ip oj, e e s only o a e aging o e colo con igu a ions a a ixed collision geome y. The scale µ2in (A.3) is w i en wi h a dependence on he ans e se posi ion x, which implici ly keeps ixed he global impac pa ame e Bbe ween he p ojec ile and a ge . In ansla ing back om he con inuous colo cha ge densi ies used in (A.3) o he disc e e alence qua k dis ibu ions used o example in e . [54], he co esponding dic iona y is µ2(b1)···µ2(bn)→Zd2Bg2 2Nc Tp oj(b1−B)···g2 2Nc Tp oj(bn−B),(A.4) whe e he ac o o g2accoun s o he coupling cons an in he emission o he so gluon om he alence qua k sou ce and he 1 2Nca ises om a e aging o e he colo s a es o he alence qua k: 1 Nc c[ a b] = 1 2Ncδab. B Wilson line colo algeb a in momen um space The cancella ion o Wilson lines which is i ial in coo dina e space akes on a mo e sub le o m in momen um space: 1 = VxV† x=Zd2p (2π)2 d2q (2π)2ei(p−q)·xV(p)V†(q) (B.1) Clea ly we can’ jus cancel momen um-space Wilson lines o equal a gumen on he igh - hand side: V(p)V†(p)6= 1. In ac , he e a e wo ac ions aking place on he momen um- space Wilson lines esul ing in he cancella ion: a Fou ie ans o m o e he ela i e momen um (p−q) and in eg al o e he cen e -o -mass momen um p+q 2. The only way – 36 – JHEP02(2019)024 o hese wo in eg als o lead o uni y on he le -hand is i one o hese ac ions esul s in a del a unc ion, which he o he one picks up. Bu a his le el i is no clea which ope a ion should gene a e he del a unc ion. On he o he hand, we can enginee a cancella ion o Wilson lines in momen um space by explici ly gene a ing Wilson lines in coo dina e space wi h he same a gumen : V(p)V†(p+q) = Zd2x d2y e−ip·xei(p+q)·yVxV† y.(B.2) Clea ly i we in eg a e o e he sha ed momen um p, we gene a e a del a unc ion ha se s x=yand cancels he Wilson lines. The emaining in eg al o e d2y hen gene a es a new del a unc ion: Zd2p (2π)2V(p)V†(p+q) = Zd2y eiq·yVyV† y= (2π)2δ2(q).(B.3) This condi ion is necessa y, and as i u ns ou , i is also su icien o gua an ee (B.1). Using (B.3) in (B.1) we ob ain VxV† x=Zd2p (2π)2 d2q (2π)2e−iq·xV(p)V†(p+q) =Zd2q (2π)2e−iq·x(2π)2δ2(q) = 1,(B.4) so we can conclude ha he wo condi ions a e in ac equi alen : Zd2p (2π)2V(p)V†(p+q) = (2π)2δ2(q)↔hVxV† x= 1i.(B.5) Ano he impo an ea u e is he con e sion be ween Wilson line aces in he un- damen al and adjoin ep esen a ions. In coo dina e space, he squa es o undamen al dipoles and quad upoles can be di ec ly con e ed in o adjoin dipoles and quad upoles, plus a cons an e m which is Ncsupp essed: ˆ Dadj 2(x,y)≡1 N2 c−1Uab xU† yba =Nc 2CF ˆ D2(x,y) 2−1 N2 c−1(B.6a) ˆ Dadj 4(x,y,z,w)≡1 N2 c−1Uab xU† ybc Ucd zU† wda =Nc 2CF ˆ D4(x,y,z,w) 2−1 N2 c−1.(B.6b) In going o momen um space, ha addi i e cons an becomes ins ead p opo ional o del a unc ions: ˆ Dadj 2(p, q) = Nc 2CF ˆ D2 2(p, q)−1 N2 c−1(2π)4δ2(p)δ2(q) (B.7a) ˆ Dadj 4(p, q, p0, q0) = Nc 2CF ˆ D4 2(p, q, p0, q0)−1 N2 c−1(2π)8δ2(p)δ2(q)δ2(p0)δ2(q0).(B.7b) – 37 – JHEP02(2019)024 C Compa ison wi h e . [38] Gi en he success ul c oss-check in sec ion 4.3 o ou double-gluon c oss sec ion (4.22) agains e . [34] in coo dina e space, i is also desi able o compa e agains he exp essions gi en in eqs. (11 - 13) o [38] in momen um space. The e, hey keep maximally gene al µ2=µ2(q1, q2) o he p ojec ile, he Lipa o e ex in hei eq. (14) co esponds o ou wa e unc ions (2.18) and (2.11) [Φ(q;q0) Φ†(q;q00)] = 8Li ⊥(q, q −q0)Li ⊥(q, q −q00),(C.1) and hey use he exp essions o he adjoin dipole and quad upole aces om (B.7). When used in (4.22), he del a unc ion e ms o (B.7) always anish, because hey lead o a anishing wa e unc ion Φ(q;q) = 0. Using his in (4.22) gi es dσ d2q1dy1d2q2dy2 =16 [2(2π)3]2Z d 2{q0 1q00 1q0 2q00 2} ×Li ⊥(q1, q1−q0 1)Li ⊥(q1, q1−q00 1)Lj ⊥(q2, q2−q0 2)Lj ⊥(q2, q2−q00 2) ×((N2 c−1) µ2(q0 1+q0 2)µ2(−q00 1−q00 2)Dadj 4(q1−q0 1, q0 2−q2, q00 2−q2, q1−q00 1) + (N2 c−1)2µ2(q0 1−q00 1)µ2(q0 2−q00 2)Dˆ Dadj 2(q00 1−q1, q0 1−q1)ˆ Dadj 2(q00 2−q2, q0 2−q2)E + (N2 c−1) µ2(q0 1−q00 2)µ2(q0 2−q00 1)Dadj 4(q1−q0 1, q2−q00 2, q2−q0 2, q1−q00 1)),(C.2) which can be compa ed wi h hei eqs. (11-13). The h ee lines o (C.2) a e e e ed o as A, B, and C, and a e gi en in eqs. (11), (12), and (13) o [38], espec i ely. A di ec compa ison is ob ained by changing a iables o co espond o he a gumen s o he Wilson line aces, and he inal esul is dσ d2q1dy1d2q2dy2 = (4παs)216 [2(2π)3]2Z d 2{`1`2`3`4} ×((N2 c−1)Dadj 4(`1,`2,`3,`4)µ2(q1+q2−`1+`4) g2 µ2(−q1−q2−`3+`4) g2 ×Li ⊥(q1,`1)Li ⊥(q1,`2)Lj ⊥(q2,−`3)Lj ⊥(q2,−`4) +(N2 c−1)2Dˆ Dadj 2(`1,`2)ˆ Dadj 2(`3,`4)Eµ2(`2−`1) g2 µ2(`4−`3) g2 ×Li ⊥(q1,`1)Li ⊥(q1,`2)Lj ⊥(q2,`3)Lj ⊥(q2,`4) +(N2 c−1)Dadj 4(`1,`2,`3,`4)µ2(q1−q2−`1+`4) g2 µ2(q2−q1+`2−`3) g2 ×Li ⊥(q1,`1)Li ⊥(q1,`2)Lj ⊥(q2,`3)Lj ⊥(q2,`4)),(C.3) whe e we ha e used a ious symme y p ope ies o he momen um-space adjoin aces. Eq. (C.3) ag ees pe ec ly wi h eqs. (11-13) o e . [38], excep o he p e ac o . The – 38 – JHEP02(2019)024 con e sion ac o (4παs g2)2is jus a i ial di e ence o con en ion: we ake ou µ2 o include he gluon-emission coupling g2, while hey inse he coupling explici ly. 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