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
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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].
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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
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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
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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)
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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
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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. We do, howe e ,
di e by he nume ically signi ican p e ac o 16
[2(2π)3]2≈6.5×10−5which a ises om he
in a ian phase space o he wo-gluon inal s a e. While he p ecise alue o he p e ac o is
la gely unimpo an o he physical con en o he calcula ion, we no e ha including his
p e ac o is necessa y o ob ain absolu e ag eemen wi h e . [34]. Nume ically, inse ing
his p e ac o supp esses he c oss sec ion by mo e han 4 o de s o magni ude, which will
su ely be impo an o he phenomenology o mul ipa icle p oduc ion.
Open Access. This a icle is dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (CC-BY 4.0), which pe mi s any use, dis ibu ion and ep oduc ion in
any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
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