Transverse momentum dependent distributions in dijet and heavy hadron pair production at EIC

JHEP03(2022)047
Published o SISSA by Sp inge
Recei ed:No embe 18, 2021
Accep ed:Feb ua y 21, 2022
Published:Ma ch 8, 2022
T ans e se momen um dependen dis ibu ions in
dije and hea y had on pai p oduc ion a EIC
Ra ael F. del Cas illo,aMiguel G. Eche a ia,b,c Yiannis Mak isdand Ignazio Scimemia
aDp o. de Física Teó ica & IPARCOS, Uni e sidad Complu ense de Mad id,
E-28040 Mad id, Spain
bDepa men o Physics, Uni e si y o he Basque Coun y UPV/EHU,
Apa ado 644, 48080 Bilbao, Spain
cUni e si y o Alcalá, Dep. o Physics and Ma hema ics,
28805 Alcalá de Hena es (Mad id), Spain
dINFN — Sezione di Pa ia,
ia Bassi 6, I-27100 Pa ia, I aly
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed],[email p o ec ed]
Abs ac : We discuss he measu emen o gluon ans e se momen um dis ibu ion
(TMD) in dije and hea y had on pai (HHP) p oduc ion in semi-inclusi e deep inelas-
ic sca e ing. The ac o iza ion o hese p ocesses in posi ion space shows he appea ance
o a speci ic new so ac o ma ix elemen on op o angula and complex alued anoma-
lous dimensions. We show in de ail how hese ea u es can be ea ed consis en ly and we
discuss a scale p esc ip ion o he e olu ion ke nel o he dije so unc ion. As a esul we
ob ain phenomenological p edic ions o unpola ized and angula modula ed c oss-sec ions
o he elec on-ion collide (EIC) using cu en a ailable in o ma ion on unpola ized TMD.
Keywo ds: Je s, QCD Phenomenology
A Xi eP in : 2111.03703
Open Access,c
The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP03(2022)047
JHEP03(2022)047
Con en s
1 In oduc ion 1
2 Fac o iza ion heo em, ame choice and modula ions 3
2.1 No a ion and kinema ics 3
2.2 Fac o iza ion heo em o dije and hea y had on pai p oduc ion 4
3 C oss-sec ions used in phenomenology 7
3.1 Ex ac ing he Bo n-le el c oss-sec ions 7
3.2 Angle in eg a ed and azimu hally modula ed c oss-sec ion 8
4 E olu ion ke nels wi h angula dependen anomalous dimensions 9
4.1 Dije so unc ion and angle dependen anomalous dimensions 9
4.2 T ea men o angula dependen anomalous dimensions and esumma ion 10
5 E olu ion ke nels and scale choices 16
5.1 ζ-p esc ip ion o dije e olu ion ke nel 17
6 Dije and hea y had on pai (HHP) p oduc ion a EIC 20
6.1 Resul s 22
6.1.1 Resul s o dije p oduc ion 22
6.1.2 Resul s o hea y had on p oduc ion 23
7 Conclusions 25
A Ha d p e ac o s 26
B Anomalous dimensions 27
1 In oduc ion
The access o non-pe u ba i e gluon dis ibu ions om expe imen s is no o iously chal-
lenging. This is also he case o gluon ans e se momen um dis ibu ions (TMDs). Gluons
en e di ec ly in Higgs p oduc ion in had onic collide s [1–5] ha has a ela i ely high mass
and low p oduc ion a es, and qua konium p oduc ion bo h a EIC and LHC [3,6–25] ha
is sensi i e also o he hea y qua k had oniza ion e ec s [20,21]. Recen s udies (see o
example [26,27]) sugges ha he expe imen al obse a ion o he dije imbalance is pos-
sible a he u u e EIC. In a ecen wo k [28] we ha e p oposed he dije and had on pai
p oduc ion a elec on-ion collide s (EIC) o p obe gluon TMD. P e ious s udies on hese
– 1 –
JHEP03(2022)047
dije LO p ocess:
hea y meson pai a LO:
(⇤g)
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(⇤ )
<la exi sha1_base64="hxhjwwKSLw AqWdYud89cC8OwLk=">AAAB83icbVBNSwMxEM3W 1q/qh69BI SPZTdKuix4MVjB sB3bXMp k2NMkuSVYoS/+GFw+KePXPePP mLZ70OqDgcd7M8zMCxPO HHdL6ews q2 lHcLG1 7+zul cP2jpOFaE E NYdUPQlDNJW4YZT uJoiBCTj h+Gbmdx6p0iyW92aS0EDAULKIETBW8q +EISAh3Mcn XLFb mzoH/Ei8nFZSj2S9/+oOYpIJKQzho3 PcxAQZKMMIp9OSn2qaABnDkPYslSCoD L5zVN8YpUBjmJlSxo8V39OZCC0nojQdgowI73szcT/ F5qousgYzJJDZVksShKOTYxngWAB0xRY jEEiCK2VsxGYECYmxMJRuC /zyX9Ku17yLW 3us I4zeMooiN0jK IQ1eogW5RE7UQQQl6Qi/o1UmdZ+ NeV+0Fpx85hD9g PxDYHXkJg=</la exi >
+ + . . .
+ . . .
pµ
1
<la exi sha1_base64=" Gojq U+NYcCLWa xz1q7RgcSio=">AAAB8HicdVBNSwMxEJ2 X7V+VT16CRbF07JbRT0W HisYD+kXUs2zbahSXZJskJZ+iu8eFDEqz/Hm//G F1Biz4YeLw3w8y8MOFMG8/7dApLyyu a8X10sbm1 ZOeXe qeNUEdogMY9VO8SaciZpwzDDaT RFIuQ01Y4up 6 QeqNI l RknNBB4IFnECDZWuk 6/n3WFemkV654 jcD8 zzBeLnVgVy1H lj24/Jqmg0hCO e74XmKCDC DCKeTUj VNMFkhAe0Y6nEguogmx08QUdW6aMoV akQTP150SGhdZjEdpOgc1QL3pT8S+ k5 oMsiYTFJDJZk ilKOTIym36M+U5QYP YEE8Xs YgMscLE2IxKNoT T9H/pFl1/VO3enNWqR3ncRThAA7hBHy4gBpcQx0aQEDAIzzDi6OcJ+ VeZu3Fpx8Zh9+wXn/A nkFw=</la exi >
pµ
2
<la exi sha1_base64="ZTWNniBWan 5aeqn0 GMTiqUh0U=">AAAB8HicdVBNSwMxEJ2 X7V+VT16CRbF07JbRT0W HisYD+kXUs2zbahSXZJskJZ+iu8eFDEqz/Hm//G F1Biz4YeLw3w8y8MOFMG8/7dApLyyu a8X10sbm1 ZOeXe qeNUEdogMY9VO8SaciZpwzDDaT RFIuQ01Y4up 6 QeqNI l RknNBB4IFnECDZWuk 61 usK9JJ 1zxXG8G5LnnC8TP Q kqP KH91+TFJBpSEca93x cQEGVaGEU4npW6qaYLJCA9ox1KJBdVBNj 4go6s0kdR GxJg2bqz4kMC63HI SdApuhX Sm4l9eJzXRZZAxmaSGSjJ FKUcmRhN 0d9pigx GwJJo ZWxEZYoWJsRmVbAj n6L/SbPq+qdu9easUj O4yjCARzCC hwATW4hjo0gICAR3iGF0c5T86 8zZ LTj5zD78g P+Bd1xkF0=</la exi >
qµ=Q
p2(nµ¯nµ)=(0,0,0,Q)
<la exi sha1_base64="7+ aBURMonmc9B3AUsQj CH/RUI=">AAACJHicdVDLSgMxFM34 PVVdekmWJQKWmZq YIUBDcuLVhb6NSSSTN aCYzTTJCGeZj3Pg blz4wIUb 8XMOIKKnnDh5Jx7Se5xAkalMs03Y2p6ZnZuP eQX1xaXlk K1 ST8UmDSxz3zRdpAkjHLSVFQx0g4EQZ7DSMsZnSV+64YISX1+qSYB6XpowKlLMVJa6hVOx e2F8K67QqEo0Yc2XIsVFSJY1jiqbUPbQeJiM JbR WYcncS05j 1coWmUzBTTLh7Wjas3SJFO+ CLIcNE PN 9H4ce4QozJGXHMgPVjZBQFDMS5+1QkgDhERqQjqYceUR2o3TJGG5 pQ9dX+jiCqbq94kIeVJOPEd3ekgN5W8 E /yOqFyj7sR5UGoCMe D7khg8qHSWKwTwXBik00QVhQ/VeIh0jHpXSu+e8h/E+uKmX oFxpVIunO1kcObAJ kAJWKAGTsE5uABNgME uAeP4Mm4Mx6MF+P1s3XKyGY2wA8Y7x9xE6Kh</la exi >
kµ=⇠
p2xQ¯nµ=⇠
2x(Q, 0,0,Q)
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Figu e 1. Example LO diag ams o he wo p ocesses. The momen a qµand kµ(co esponding
o he pho on and incoming pa on momen a espec i ely) a e exp essed in he B ei ame.
p ocesses we e pe o med in e s. [29,30] o dije and in e s. [31–33] o hea y-meson pai
p oduc ion in an elec on-had on collide . The ele an p ocesses o ou case a e
`+h→`0+J1+J2+X, and `+h→`0+H+¯
H+X , (1.1)
whe e `and `0a e he ini ial and inal s a e lep ons, his he colliding had on, Jiand
H/ ¯
Ha e he je s and hea y mesons espec i ely and X ep esen unde ec ed pa icles.
A leading o de (LO), and igno ing he in insic momen um o pa ons inside he a ge
had on, he wo ha d-sca e ing p ocesses a e schema ically shown in igu e 1. We ha e
shown ha hese p ocesses a e ac o izable when conside ing he c oss-sec ion
dσ
dxdη1dη2dpTd T
,(1.2)
whe e xis he Bjo ken a iable, and ηi, Tand pTa e espec i ely he apidi y, he
sum o he ans e se momen a (wi h espec o he beam axis) and he a e age scala
ans e se momen a o he wo inal je s. The ac o iza ion condi ion is | T|  pTin he
B ei ame, as he i ual pho on and a ge -had on di ec ions a e back- o-back. These
condi ions a e ce ainly ul illed in he B ei ame o pT∈[5,40] GeV and in he cen al
apidi y egion. We also demand ha he e a e no hie a chies among pa onic Mandels am
a iables, ˆs∼ |ˆ
| ∼ |ˆu|. On he expe imen al side ecen s udies [26] sugges ha he
measu emen o dije imbalance is possible a EIC. Fo he hea y-meson case mon e-ca lo
gene a o s udies sugges ha cha med mesons can be econs uc ed [34,35]. The cha m
p oduc ion a es ha e been conside ed a he LO and NLO QCD o ep →c/¯c+Xin
e . [36]. Fo ou case in p inciple we equi e he ans e se momen a o he hea y mesons,
pH/ ¯
H
T, be pa ame ically la ge han hei mass, mH, i.e. pH/ ¯
H
TmH.
A leading o de (LO) he je p ocesses can be ini ia ed by ei he a gluon o a qua k,
while in he hea y-meson case only he gluon ini ial s a e is ele an . The ac o ized
c oss-sec ion esul s in p oduc s/con olu ions o se e al undamen al unc ions as TMDs,
je /hea y qua k dis ibu ions, so unc ion and a new e olu ion ke nel, de ailed in [28].
The pu pose o he p esen pape is o p o ide a phenomenological s udy o hese p ocesses,
– 2 –
JHEP03(2022)047
including heo e ical e o s. In o de o achie e his, we ha e used he code A emide [37,
38], in oducing new moduli necessa y o desc ibe he p esen cases. The code al eady
includes qua k TMDPDF and he TMD e olu ion ke nel ex ac ed om D ell-Yan and
semi-inclusi e DIS expe imen s [39].
The ac o iza ion o he c oss-sec ion is p oduced in posi ion space, desc ibed by he
a iable b, conjuga e o he momen um T. In o de o Fou ie ans o m he ac o ized
c oss-sec ion om bspace o momen um space one has o pe o m an angula in eg a ion
on φb(i.e. d2b=b db dφband J·b= Jbcos φb) ha esul s non i ial because he
anomalous dimensions o se e al unc ions depend on his angle and a e complex alued.
We show ha his in eg a ion can be pe o med in esummed pe u ba ion heo y and ha
his add esses he p oblem o complex alues in all anomalous dimensions. As a esul he
φb-angle in eg a ed c oss-sec ion is hen ac o ized in o dis ibu ions ha a e b- o a ion
in a ian . The φb-angle in eg a ed dije e olu ion ke nel is de i ed in eg a ing a sys em
o coupled di e en ial equa ions simila o he TMD case. We show and discuss he e a
speci ic scale choice p esc ip ion ha is analogue o he ζ-p esc ip ion al eady discussed
in [40], which was al eady implemen ed in A emide.
The ac o iza ion heo em and he de ailed de ini ion o he obse ables is p o ided
in sec ion 2. The c oss-sec ions sub le ies disco e ed when compa ing o o he g oups a e
discussed in sec ion 3. The esumma ion o logs in ol es angula in eg a ions as explained
in sec ion 4, which leads o he e olu ion ke nels de ailed in sec ion 5. The phenomenolog-
ical esul s ob ained wi h he codes ha we ha e de eloped a e summa ized in sec ion 6,
a e which he conclusions a e d awn.
2 Fac o iza ion heo em, ame choice and modula ions
2.1 No a ion and kinema ics
In o de o de ine angles and o deduce a ac o ized c oss-sec ion we need o es ablish some
kinema ics. The di ec ion o he beam is ixed along he ˆzaxis. I is use ul o de ine he
ou - ec o s nµ=1
√2(1,0,0,1),¯nµ=1
√2(1,0,0,−1), so ha n2= ¯n2= 0,¯n·n= 1. Then
any o he ou ec o can be decomposed in o i s ligh -cone componen s,
pµ=p+¯nµ+p−nµ+pµ
⊥= (p+, p−, p⊥)n,(2.1)
wi h
p+=n·p, p−= ¯n·p, p2= 2p+p−+p2
⊥= 2p+p−−p2.(2.2)
The di ec ion o he wo je s a e 1and 2, no malized as
2
J= ¯ 2
J= 0, J·¯ J= 1,wi h J= 1,2,(2.3)
and ¯ Ja e de ined by e e sing he sign o he spacial componen s. We ha e hen he
s anda d Lo en z-in a ian s,
Q2=−q2, x =Q2
2P·q,(2.4)
– 3 –
JHEP03(2022)047
whe e qµis he momen um o he i ual pho on, Pµis he momen um o he a ge
had on. In he B ei ame we ha e qµ= (0,0,0, Q)and neglec ing mass co ec ions
Pµ=1
2x(Q, 0,0,−Q).The a io o he longi udinal momen a o he incoming pa on
and he a ge had on is ξ=k+
P+.wi h kµ he momen um o he pa on en e ing ha d
p ocess. We can hen exp ess he a iables Qand ξin e ms o he Bo n le el kinema ics
using he pseudo- apidi ies, η1and η2, and he ans e se momen um, pT, o he wo
ou going pa ons,
Q= 2pTcosh(η−) exp(η+), ξ = 2xcosh(η+) exp(−η+),(2.5)
whe e, neglec ing co ec ions om he a ge had on mass, η±=η1±η2
2.The pa onic
Mandels am a iables can be w i en using he same a iables,
ˆs= (q+k)2= 4p2
Tcosh2(η−),
ˆ
= (q−p2)2=−4p2
Tcosh(η−) cosh(η+) exp(η1),
ˆu= (q−p1)2=−4p2
Tcosh(η−) cosh(η+) exp(η2),(2.6)
wi h pµ
1and pµ
2 he momen a o he ou going pa ons. A pa onic le el hey sa is y
ˆs+ˆ
+ ˆu=−Q2.(2.7)
Finally, he ans e se momen um imbalance o he wo je s, T, and he ha d ans-
e se momen um, pT, a e de ined h ough
T=p1T+p2T,pT=p1T−p2T
2,(2.8)
whe e he sub-index 1, 2 e e s o he inal je s. A Bo n le el p1T=−p2Tand hus
T= 0. I mus be aken in o accoun ha he had oniza ion o he ou going pa ons
will o m je -like con igu a ions along simila di ec ions and wide angle adia ion ha can
escape he je clus e ing algo i hm, a ec ing he imbalance.
2.2 Fac o iza ion heo em o dije and hea y had on pai p oduc ion
The ac o iza ion o dije and hea y had on pai p oduc ion a leading powe (LP) o semi-
inclusi e deep inelas ic expe imen s has al eady been p o ided in [28]. The c oss-sec ions
epo ed he e do no ake in o accoun any lep onic iducial cu s, which howe e could be
implemen ed once he expe imen al condi ions a e es ablished (especially a EIC). In his
sec ion we ecall he main o mulas ha a e used in ou phenomenological desc ip ion. We
s a wi h he dije c oss-sec ion which can be w i en as a sum o e ms depending on he
pa on ha ini ia es he ha d p ocess (qua k o gluon)
dσ2J=dσ(γ∗g) + dσU(γ∗ ),(2.9)
dσ(γ∗g) = dσU(γ∗g) + dσL(γ∗g).(2.10)
– 4 –

JHEP03(2022)047
Fo an unpola ized had onic p ocess, he c oss-sec ion is made ou o ha d con ibu ions
om unpola ized ini ial qua ks dσU(γ∗ ), unpola ized ini ial gluons dσU(γ∗g), and linea ly
pola ized gluons dσL(γ∗ ). The qua k con ibu ion o he c oss-sec ion is
dσU(γ∗ )
dxdη1dη2dpTd T
=X
σ U
0HU
γ∗ →g (ˆs, ˆ
, ˆu, µ)Zd2b
(2π)2exp(ib· T)
1(ξ, b, µ, ζ1)(2.11)
×Sγ (b, ζ2, µ)Cg(b, R, µ)Jg(pT, R, µ)C (b, R, µ)J (pT, R, µ).
In his o mula
1is he unpola ized qua k TMDPDF o la o ,HU he ha d ac o o
he unpola ized qua k case. The pe u ba i e calcula ions o TMDPDF has been pe o med
ecen ly a NNLO [5,41–44] and N3LO [45,46]. The je s a e desc ibed by he p oduc o a
collinea -so unc ion C( ,g)and a je shape unc ion J( ,g)speci ic o each pa onic la o .
The calcula ion a NLO o hese unc ions can be ound in [47,48] o gene ic kT- ype and
cone je algo i hms.
The ac o Sγ is he dije so unc ion o he undamen al ep esen a ion o SU(3)C
and calcula ed in [28] up o NLO. A co esponding so ac o , Sγg, o he adjoin ep e-
sen a ion o SU(3)Cis also necessa y o he incoming gluon con ibu ion.
dσ(γ∗g)
dxdη1dη2dpTd T
=X
Hµν
γ∗g→ ¯
(ˆs, ˆ
, ˆu, µ)Zd2b
(2π)2exp(ib· T)Fg,µν(ξ, b, µ, ζ1)(2.12)
×Sγg(b, η1, η2, µ, ζ2)C (b, R, µ)J (pT, R, µ)C¯
(b, R, µ)J¯
(pT, R, µ),
The ha d ac o Hµν(µ)accoun s o con ibu ions o unpola ized and linea ly pola ized
gluons,
Hµν
γ∗g→ ¯
=σgU
0HU
γ∗g→ ¯
gµν
T
d−2+σgL
0HL
γ∗g→ ¯
−gµν
T
d−2+ µ
1T ν
2T+ µ
2T ν
1T
2 1T· 2T.(2.13)
The TMD enso Fg,µν can be also decomposed in e ms o unpola ized and linea ly pola -
ized pa s,
Fµν
g(ξ, b) = g
1(ξ, b)gµν
T
d−2+h⊥
1(ξ, b)gµν
T
d−2+bµbν
b2,(2.14)
wi h gµν
T=gµν −nµ¯nν−¯nµnν. The ha d ac o s a e e alua ed up o NNLO in he
unpola ized case in [49,50] and a LO o he linea ly pola ized case [51]. In hese equa ions
g
1and h⊥
1 ep esen he unpola ized and linea ly pola ized gluon TMD. Bo h o hem a e
known pe u ba i ely up o NNLO [5,42,44]. Combining eq. (2.10), (2.12), (2.13), (2.14)
– 5 –
JHEP03(2022)047
one ob ains
dσU(γ∗g)
dxdη1dη2dpTd T
=σgU
0X
HU
γ∗g→ ¯
(ˆs, ˆ
, ˆu, µ)Zd2b
(2π)2exp(ib· T) g
1(ξ, b, µ, ζ1)(2.15)
×Sγg(b, ζ2, µ)C (b, R, µ)J (pT, R, µ)C¯
(b, R, µ)J¯
(pT, R, µ),
dσL(γ∗g)
dxdη1dη2dpTd T
=σgL
0X
HL
γ∗g→ ¯
(ˆs, ˆ
, ˆu, µ)Zd2b
(2π)2exp(ib· T)h⊥
1(ξ, b, µ, ζ1)(2.16)
×s2
b−c2
b
2Sγg(b, ζ2, µ)C (b, R, µ)J (pT, R, µ)C¯
(b, R, µ)J¯
(pT, R, µ).
We use sb= sin φband cb= cos φb o he sine and cosine o he angle φbbe ween he
ec o s band 1T, espec i ely. Each o dσ has a ha d ac o ha desc ibes he ini ia ing
in e ac ion. The coe icien s σ( ,g),(U,L)
0a e in oduced such ha he leading o de ha d
unc ions a e no malized o he uni y, i.e. HU(L)
LO = 1 + O(αs).
The case o hea y had on pai is e y simila . The measu ed imbalance Tis
T=pH
T+p¯
H
T,(2.17)
whe e he supe sc ip Hindica es a gene ic hea y meson and ¯
H he co esponding an i-
pa icle. The imbalance is measu ed in he B ei ame and assuming he TMD ac o iza-
ion scaling, i.e., | T|  pH, ¯
H
T. We also assume ha he wo hea y mesons a e agmen ed
nea he kinema ic end-poin and ca y mos o he ene gy o he hea y qua k coming om
he ha d p ocess. The c oss-sec ion eads
dσ(γ∗g)
dxdηHdη ¯
HdpTd T
=Hµν
γ∗g→Q¯
Q(ˆs, ˆ
, ˆu, µ)Zdb
(2π)2exp(ib· T)Fg,µν(ξ, b, µ, ζ1)
×Sγg(b, µ, ζ2)JQ→H(b, pT, mQ, µ)J¯
Q→¯
H(b, pT, mQ, µ).(2.18)
wi h ηHand η¯
H he pseudo- apidi ies o he hea y mesons, JQ→H he hea y qua k je -
unc ions [52,53]. The ha d, so , and beam unc ions a e he same as in he dije case. In
he ha d unc ion we do no conside co ec ions due o he qua k mass and we de ine
pT=|pH
T|+|p¯
H
T|
2,(2.19)
The hea y qua k je unc ions, JQ→H, can be pa ially e alua ed in pe u ba ion heo y
as shown in [28]. We wo k in he limi pTmHΛQCD and he hea y qua k je unc ion
can be e- ac o ized using bHQET. We also ha e TpTso ha i is possible o ind
la ge logs o wo pa ame ically di e en scales in he agmen a ion p ocess,
µ+=mQ,and µJ=mQ
T
pT
,(2.20)
– 6 –
JHEP03(2022)047
ha need o be esummed o ensu e he con e gence o he expansion. Following [28]
he je unc ion can be i s ly ac o ized in o a sho dis ance ma ching coe icien and a
bHQET ma ix elemen ,
JQ→H(b, pT, mQ, µ) = H+(mQ, µ)JQ→Hb,mQ
pT
, µ,(2.21)
whe e he coe icien H+is
H+(mQ, µ) = |C+(mQ, µ)|2.(2.22)
and he wo-dimensional shape unc ion is de ined in momen um space as
JQ→H( ) = 1
2p−
HNCX
Xh0|δ(2) −i (¯ ·∂)W†
h β+|XHihXH|¯
h ,β+W /
¯ |0i.(2.23)
No ice ha is a Euclidean, wo dimensional, ans e se componen o he ligh -like ou -
ec o µpoin ing along he di ec ion o he boos ed hea y meson. In posi ion space JQ→H
is ob ained by Fou ie ans o ma ion
JQ→Hb,mQ
pT
, µ=Zd exp(ib· )JQ→H( ).(2.24)
The one-loop exp ession o hese quan i ies a e calcula ed in [28].
3 C oss-sec ions used in phenomenology
The c oss-sec ions p esen ed in p e ious sec ion a e usually pa ially in eg a ed in phe-
nomenological obse ables. We discuss he e hese in eg a ions, which also allow us o
ela e he no maliza ion o ou c oss-sec ion wi h he ones ob ained in he li e a u e.
3.1 Ex ac ing he Bo n-le el c oss-sec ions
The ee le el c oss-sec ions o he dije and had on pai p oduc ion we e conside ed a
ee le el in e . [54]. We s a conside ing he gluon case, om which one can easily deduce
also he qua k case. The gluon ha d con ibu ion o he c oss-sec ion is desc ibed by
dσ(γ∗g)
dxdη1dη2dpTd T
=N
xshA0+A1cos 2φ0
p+···+B0cos 2φ0
+···i,(3.1)
and he azimu hal angles (φ0
,φ0
p) o ec o s pT, Ta e measu ed wi h espec o he lep on
plane. Howe e , ou p e e ed ame is he one whe e he φ`angle is measu ed in he plane
de ined by pTand qT, he sum o he lep on momen a, and φ is he azimu hal angle
be ween Tand pT. In his ame and in eg a ing o e he angle φ`we a e le wi h:
dσ(γ∗g)
dxdη1dη2dpTd T
= 2πpTN
xshA0+B2cos(2φ )i,(3.2)
– 7 –
JHEP03(2022)047
wi h he ac o 2πcoming om φ`in eg a ion. The LO exp essions a e ob ained by sepa-
a ing he unpola ized and linea ly pola ized gluon con ibu ions and Fou ie ans o ming.
The unpola ized pa onic pa has a simila o m also o qua ks, so ha we ind
dσU(γ∗g)
dxdη1dη2dpTd TLO =σgU
0Zd2b
(2π)2exp(ib· T) g
1(ξ, b) = σgU
0 g
1(ξ, T),(3.3)
dσU(γ∗ )
dxdη1dη2dpTd TLO =σ U
0Zd2b
(2π)2exp(ib· T)
1(ξ, b) = σ U
0
1(ξ, T),(3.4)
The same o he linea ly pola ized gluons gi es
dσL(γ∗g)
dxdη1dη2dpTd TLO =σgL
0Zd2b
(2π)2exp(i T·b)sin2φb−cos2φb
2h⊥
1(ξ, b).
=−σgL
0Zb db dφb
8π2exp i Tbcos(φb−φ )cos(2φb)h⊥
1(ξ, b)
= cos(2φ )σgL
0Zb db
4πJ2( Tb)h⊥
1(ξ, b)
=−cos(2φ )
2σgL
0h⊥
1(ξ, T),(3.5)
whe e no ice ha h⊥
1(ξ, T)is no he di ec Fou ie ans o m o h⊥
1(ξ, b)and bo h unc ions
can be ela ed h ough eq. (2.20) in [3]. We ob ain he σ(g, )(U,L)
0p e ac o s om he
s uc u e unc ions gi en in eqs. (3.3, 3.5) in [54] and we lis hem in appendix A.
3.2 Angle in eg a ed and azimu hally modula ed c oss-sec ion
The scala c oss-sec ion ha we inally conside in he phenomenological s udies is ob ained
by in eg a ing o e he φ angle
dσ
dΠd T
= TZ+π
−π
dφ
dσ
dΠd T
,(3.6)
whe e dΠ = dxdη1dη2dpT. Because he ac o ized c oss-sec ion is always exp essed in
posi ion space one can w i e (he e J0,2a e Bessel unc ions)
dσ
dΠd T
= TZ+π
−π
dφ Zdb
(2π)2exp hi Tbcos(φb−φ )id˜σ(b)
dΠdb
= TZ∞
0
b db
2πJ0( Tb)Z+π
−π
dφb
d˜σ(b)
dΠdb
= TZ∞
0
b db
2πJ0( Tb)Z+π
−π
dφbd˜σU(b)
dΠdb−cos 2φb
2
d˜σL(b)
dΠdb,(3.7)
whe e dσU=dσU(γ∗ ) + dσU(γ∗g),dσL=dσL(γ∗g) o he dije case and dσU,L =
dσU,L(γ∗g) o he hea y had on pai case.
In ou phenomenological analysis we conside also he azimu hal angle a e age
hcos 2φ i ≡ "Z+π
−π
dφ cos 2φ
dσ
dΠd T#, dσ
dΠd T
.(3.8)
– 8 –
JHEP03(2022)047
The pa o he c oss-sec ion ela i e o linea ly pola ized gluons can be ea ed sim-
ila ly. In his case we need o inco po a e an addi ional cos 2φb e m in he in eg als
bu his is he only change since he so and collinea -so unc ions ha appea in he
wo con ibu ions a e he same as o he dije case. Using he igonome ic iden i y
cos 2φb= 2 cos2φb−1 he in eg als o his case can be deduced om he discussion o he
unpola ized c oss-sec ion by he eplacemen
In(A)−→ −In(A+ 1) + 1
2In(A).(4.38)
Equi alen ly o he case o angula modula ion in eq. (3.8) he ollowing ans o ma ions
ha e o be pe o med,
d˜σU(b) : In(A)−→ −In(A)+2In(A+ 1)
d˜σL(b) : In(A)−→ −In(A)+2In(A+ 1) −2In(A+ 2) .(4.39)
In all hese cases one ob ains a cancella ion o he imagina y pa o he c oss-sec ion.
T ea ing pe u ba i ely he angula in eg a ion as discussed in his sec ion leads o
w i e eq. (3.7) o he dije case as
dσ
dΠd T
=dσU(γ∗g)
dΠd T
+dσU(γ∗ )
dΠd T
+dσL(γ∗g)
dΠd T
,(4.40)
whe e
dσU(γ∗g)
dΠd T
=X
σgU
0HU
γ∗g→ ¯
(ˆs, ˆ
, ˆu, µ =pT)J (pT, R, µJ)J¯
(pT, R, µJ)
×Z+∞
0
bdb J0(b T) g
1(ξ, b)Rg({µk}, ζ1,0, ζ2,0)→(pT, p2
T,1)ˆσU
g(b, R, {µi}),
(4.41)
dσU(γ∗ )
dΠd T
=X
, ¯
σ U
0HU
γ∗ →g (ˆs, ˆ
, ˆu, µ =pT)J (pT, R, µJ)Jg(pT, R, µJ)
×Z+∞
0
bdb J0(b T)
1(ξ, b)Rq({µk}, ζ1,0, ζ2,0)→(pT, p2
T,1)ˆσU
(b, R, {µi}),
(4.42)
dσL(γ∗g)
dΠd T
=X
σgL
0HL
γ∗g→ ¯
(ˆs, ˆ
, ˆu, µ =pT)J (pT, R, µJ)J¯
(pT, R, µJ)
×Z+∞
0
bdb J0(b T)h⊥
1(ξ, b)Rg({µk}, ζ1,0, ζ2,0)→(pT, p2
T,1)ˆσL
g(b, R, {µi}),
(4.43)
whe e R ,g a e p oduc s o e olu ion ke nels o be desc ibed in he nex sec ion, and ˆσU,L
,g
a e he esul o φbangula in eg a ion and can be w i en as
ˆσU
g=IgU
cons .+as(µC)CU
(b, R, µC) + as(µC)CU
¯
(b, R, µC) + as(µ0)SU
γg(b, ζ2, µ0),(4.44)
ˆσU
=I U
cons .+as(µC)CU
(b, R, µC) + as(µC)CU
g(b, R, µC) + as(µ0)SU
γ (b, ζ2, µ0),(4.45)
ˆσL
g=IgL
cons .+as(µC)CL
(b, R, µC) + as(µC)CL
¯
(b, R, µC) + as(µ0)SL
γg(b, ζ2, µ0).(4.46)
– 15 –

JHEP03(2022)047
The unc ions Cand Sin eq. (4.44)–(4.46) a e he esul o he φbin eg a ion in collinea -
so and dije so unc ions. Fo he hea y meson case we ha e jus con ibu ions om
gluon sca e ing,
dσ
dΠd T
=dσU(γ∗g)
dΠd T
+dσL(γ∗g)
dΠd T
,(4.47)
and we ha e o change J , ¯
→H+and C → JQ→Hin eq. (4.41)–(4.43). In he case o
angula modula ion he c oss-sec ions can also be w i en as in eq. (4.40)–(4.47), wi h he
co ec alues o he unc ions Icons .,Cand S. The non-pe u ba i e e ec s a e in all cases
encoded in he e olu ion ke nels, TMD and je unc ions. In he nex sec ion we desc ibe
how he e olu ion ke nels a e de ined.
5 E olu ion ke nels and scale choices
The e olu ion ke nels appea ing in eq. (4.40)–(4.47) a e
Rg({µk}, ζ1,0, ζ2,0)→(pT, p2
T,1)
=RJ (µJ→pT)2RC (µC→pT)2
×Rg
F(µ0, ζ1,0)→(pT, p2
T)Rq
S(µ0, ζ2,0)→(pT,1),(5.1)
Rq({µk}, ζ1,0, ζ2,0)→(pT, p2
T,1)
=RJ (µJ→pT)RJg(µJ→pT)RC (µC→pT)RCg(µC→pT)
×Rq
F(µ0, ζ1,0)→(pT, p2
T)Rg
S(µ0, ζ2,0)→(pT,1),(5.2)
whe e RJ ,g is a je unc ion ke nel, RC ,g is he one o collinea -so unc ions, Rq,g
F he
one o TMD and inally Rqg
Sis he one o he dije so unc ion. In he hea y qua k case
he e olu ion ke nels a e pa ame e ized like in eq. (5.1) wi h he usual changes J →H+
and C → JQ→H. The ke nels o single-scale e olu ion ha e a s anda d o m and a e iew
up o NLL is gi en in [58],
Ri(µi→pT) = eKi(µi→pT)µi
miωi(µi→pT)
, i ={C ,Cg, J , Jg,JQ→H, H+}(5.3)
whe e
ωi(µi→pT)NLL =−Γ0
i
β0ln +Γ1
Γ0−β1
β0αs(µi)
4π( −1),(5.4)
Ki(µi→pT)NLL =−γ0
i
2β0
ln −2πΓ0
i
(β0)2 −1− ln
αs(pT)
+Γ1
Γ0−β1
β01− + ln
4π+β1
8πβ0
ln2 ,(5.5)
– 16 –
JHEP03(2022)047
wi h =αs(pT)/αs(µi)and
Γ0
C =−4CF,Γ0
Cg=−4CA, γ0
C /g = 0, mC /g =Re−γE
b,
Γ0
J = 4CF,Γ0
Jg= 4CA, γ0
J = 6CF, γ0
Jg= 2β0, mJ /g =pTR,
Γ0
J=−4CF, γ0
J= 4CF, mJ=mQ/pTe−γE
b,
Γ0
+= 4CF, γ0
+= 2CF, m+=mQ,(5.6)
Ini ial scales µichoice is gi en in sec ion 6. The TMD ke nel is conside ed he e in he
ζ-p esc ip ion desc ibed in [40] and implemen ed in he code A emide [37,38] ha we
use,
Rq,g
F({µ0, ζ0}→{µ , ζ }) = ζ
ζµ(b, µ )−Dq,g(b,µ )
.(5.7)
In he nex pa ag aph we de ine a ζ-p esc ip ion also o he dije e olu ion ke nel
Rg
S(µ0, ζ2,0)→(pT,1), which is he only missing pa .
5.1 ζ-p esc ip ion o dije e olu ion ke nel
The angula independen ke nel o he dije so unc ion is ob ained as a solu ion o a
coupled sys em o di e en ial equa ions, epo ed in eq. (4.13), ha a e o mally e y
simila o he TMD ones [59,60]. The anomalous dimensions a e gi en by
¯γSγg (µ, ζ) = γcusp2CFln µ2
µ2
0−CAln ζ
ζγg
2,0+δγγg
S,(5.8)
¯γSγ (µ, ζ) = γcusp(CF+CA) ln µ2
µ2
0−CFln ζ
ζγ
2,0+δγγ
S,(5.9)
whe e
µ0=2
beγE, ζγg
2,0=4p2
T
ˆs
2CF
CA, ζγ
2,0=4p2
T
ˆs
CF+CA
CFˆ
ˆu
CF−CA
CF,(5.10)
and δγSa e he non-cusp SF anomalous dimension, which is known up o h ee-loops o
he gluon-channel and up o one-loop o he qua k-channel and a e epo ed in appendix.
The anomalous dimension and he apidi y anomalous dimension (RAD) in eq. (4.6), (4.7)
sa is y also
−d
dln ζ¯γSγi (µ, ζ) = d
dln µDi(µ, b)=Γcusp(µ)(5.11)
The e olu ion o he SF akes he gene al o m
Ri
S({µi, ζi}→{µ , ζ }) = exp ZP¯γSγi(µ, ζ)dln µ−Di(µ, b)dln ζ(5.12)
wi h i=q, g and {µi, ζi}and {µ , ζ }being he ini ial and inal poin s o ac o iza ion and
apidi y scales. The in eg a ion pa h Pis an a bi a y pa h in he {µ, ζ}-plane. Eq. (5.11)
– 17 –
JHEP03(2022)047
ensu es ha he e olu ion ke nel is pa h only independen when one knows he comple e
pe u ba i e expansion o he anomalous dimensions. Since his is no he case he pa h
independence is b oken. In o de o pa ially es o e he pa h independence we p oceed
as in [40] de ining a ζ-p esc ip ion also o he dije so unc ion e olu ion ke nel. The
ζ-p esc ip ion p o ides a way o choose he ini ial scale ζio he e olu ion ke nel as a
unc ion o µand bso ha he SF does no depend on he ini ial scale µi. This is done
by aking he in eg a ion pa h h ough a null-e olu ion line in he {µ, ζ}-plane and hen
aking a ixed-µe olu ion.
To ind he null-e olu ion line we in e p e he pai o di e en ial equa ions (5.11) as
a wo-dimensional g adien equa ion ∇F=EF, whe e E= (γS(µ, ζ),−DS(µ, b)). The
null-e olu ion line is hen an equipo en ial line o he ield E. In pa icula , he e is
a special null-e olu ion line ha passes h ough he saddle-poin {µsaddle, ζsaddle}o he
e olu ion ield. We ind ha he saddle poin is exac ly µsaddle =µ0and ζγi
saddle =ζγi
0. I
we pa ame e ize he null-e olu ion line as {µ, ζµ(b)}, he alue o ζµis gi en by
γSγi (µ, ζµ(b)) = 2DSγi (µ, b)dln ζµ(b)
dln µ2,(5.13)
which is sol ed pe u ba i ely o de by o de in αs. The pe u ba i e solu ion akes he
o m
ζγg
µ, pe (b) = µ
µ0
2CF
CAζγg
0e γg(µ,b),(5.14)
ζγ
µ, pe (b) = µ
µ0
CF+CA
CFζγ
0e γ (µ,b),(5.15)
whe e
γi(µ, b) = ∞
X
n=0
an
s(µ) γi
n(Lµ),Lµ= ln(Bµ2e2γE),
γg
0(Lµ)=0,(5.16)
γg
1(Lµ) = 2CF
CA"−β0
12L2
µ+
γ2
2CF−d(2,0)
Γ0#,(5.17)
γg
2(Lµ) = 2CF
CA"−β2
0
24L3
µ−β1
12 +β0Γ1
12Γ0L2
µ+ β0γ2
2CF
2Γ0−4β0d(2,0)
3Γ0!Lµ
−
γ2
2CFΓ1−d(2,0)Γ1
Γ2
0
+
γ3
2CF−d(3,0)
Γ0#,(5.18)
γ
0(Lµ)=0,(5.19)
– 18 –
JHEP03(2022)047
and we a e using he ollowing no a ion
Di(µ, b) = Ci
∞
X
n=1
an
s(µ)
n
X
k=0
Lk
µd(n,k), δγS(µ) = ∞
X
n=1
an
s(µ)γn,(5.20)
β(as) = −∞
X
n=0
an+2
sβn,Γcusp(µ) = Ciγcusp(µ) = Ci
∞
X
n=0
an+1
s(µ)Γn,(5.21)
wi h Ci=CF, CA o qua k and gluon channel espec i ely. No ice ha 0 anishes as i
is p opo ional o he LO non-cusp AD, which is ze o o he SF. The non-cusp AD is no
known beyond LO o he qua k-channel.
The RAD is a unc ion o band he e o e has impo an non-pe u ba i e co ec ions
in he la ge-b egion. These co ec ions can be implemen ed as a model. The way o
p oceed is o sol e (5.13) o a gene ic non-pe u ba i e RAD. The equa ion is sol able bu
i is di icul o ob ain he cancella ion o pe u ba i e loga i hms in he small-b egion.
Following [40] we use he pe u ba i e solu ion o he small-b egion and mo e o he exac
(gene ic RAD) solu ion o la ge-b:
ζµ(b) = ζpe
µ(b)e−b2/B2
NP +ζexac
µ(b)1−e−b2/B2
NP ,(5.22)
wi h BNP being he b alue whe e non-pe u ba i e (NP) e ec s become impo an (∼2.5
GeV−1). We ha e al eady discussed he pe u ba i e solu ion o eq. (5.13). Fo he exac
solu ion we ind
ζγg
µ, exac (b) = µ2
µ2
0
2CF
CAζγg
0e−gγg(as,DS)/DS,(5.23)
ζγ
µ, exac (b) = µ2
µ2
0
CF+CA
CFζγ
0e−gγ (as,DS)/DS,(5.24)
whe e
gγi(as,DS) = 1
as
Γ0
2β2
0
∞
X
n=0
an
sgγi
n(DS),
gγg
0=2CF
CAhe−p−1 + pi,(5.25)
gγg
1=2CF
CAβ1
β0e−p−1 + p−p2
2−Γ1
Γ0e−p−1 + p,(5.26)
gγ
0=CF+CA
CFhe−p−1 + pi,(5.27)
gγ
1=CF+CA
CFβ1
β0e−p−1 + p−p2
2−Γ1
Γ0e−p−1 + p,(5.28)
and p= 2β0DS/Γ0.
Finally, he e olu ion ke nel ha p o ides he e olu ion om he null-e olu ion line
and ha passes h ough he saddle-poin o he inal ζpoin is gi en by
Rq,g
S({µ0, ζ0}→{µ , ζ }) = ζ
ζµ(b, µ )−Dq,g(b,µ )
,(5.29)
– 19 –
JHEP03(2022)047
and i we conside he e olu ion om an a bi a y ini ial scale we ake
Ri
S({µi, ζi}→{µ , ζ }) = Ri
S({µ0, ζ0}→{µ , ζ })
Ri
S({µ0, ζ0}→{µi, ζi}).(5.30)
wi h i=q, g. This discussion concludes he analysis o all e ms ha appea in he
ac o iza ion heo em and he scale p esc ip ion. We a e now eady o he implemen a ion
in he code A emide [37,38].
6 Dije and hea y had on pai (HHP) p oduc ion a EIC
In o de o es he phenomenology de eloped in he p e ious sec ions we conside he
case o he EIC. In [28] we al eady s udied he co e age o he EIC and we concluded ha
he mos a ou able case is gi en o a alue o mass ene gy o dije p oduc ion a ound
√s= 140 GeV and cen al apidi y, η1=η2= 0. Typical alues o je adii and momen a
a EIC a e espec i ely R∼0.7and pT∼Q/2∼20 GeV. In o de o simpli y he discussion
we show plo s in eg a ed o e Bjo ken a iable x( he longi udinal ac ion o momen um
ξ ha en e s in he TMDPDFs is ξ∼2x) in he allowed kinema ic in e als. Fo he case
o cen al apidi y we ha e x∈(0.0859,0.5). The c oss-sec ions ha we plo a e
Zxmax
xmin
dx dσ
dxdη1dη2dpTd Tη1, η2, pT
(6.1)
and i s alue is p esen ed as a unc ion o he small ans e se momen um T. The c oss-
sec ions and he e o bands a e ob ained by using and p epa ing speci ic moduli o he
code A emide [37,38]. In pa icula we use he TMD and he TMD e olu ion ke nels
al eady coded in A emide, ha come om he i [39], while he new unc ions s udied
in his wo k a e included in his code o he i s ime. The gluon TMD is no i ed ye ,
howe e in A emide he e is a pa ame e iza ion o i . The code akes in o accoun ha
he con ibu ion o linea ly pola ized gluons is highly supp essed because in he small-b
egime he ma ching o he linea ly pola ized gluon TMD on o he gluon PDF s a s a
o de α1
sand no a o de α0
slike o he dis ibu ions. In e . [5] he c oss sec ion ob ained
in his way ag ees wi h Py hya 8 and cu en expe imen al esul s o he Higgs ans e se
momen um spec um, which a e howe e no e y p ecise. The non-pe u ba i e e ec s a e
expec ed o be impo an in he high-b egion and hey should no al e he small-bbeha io
o his dis ibu ion. No ice also ha he non-pe u ba i e e ec s play a ole o con ol he
beha io o he dis ibu ion a ound he Landau pole a la ge-b, which means a u he
supp ession e ec a la ge-b(as we also obse e in he case o unpola ized dis ibu ions).
Summing up, gi en he cu en pe u ba i e and non-pe u ba i e knowledge o TMDs, a
his s age we p e e no o push o a hypo he ical non-pe u ba i e enhancemen o he
con ibu ion o linea ly pola ized gluon TMD.
The ac o iza ion ha we p opose in gene al needs in o ma ion o he non-pe u ba i e
e ec s in se e al unc ions. Fo he dije case we ha e
C(b, R;pT) = RC(b, R;pT, µC)Cpe (b, R;µC) NP
C(b, R),(6.2)
Sγi(b;pT,1) = RS({µ0, ζ0}→{pT,1})Spe
γi (b;µ0, ζ0) NP
S(b),(6.3)
– 20 –

JHEP03(2022)047
C J S
Bi
NP (GeV−1) 2.5 2.5 2.5 C J
bmax (GeV−1) 0.5 0.3
Table 1. Values o non-pe u ba i e pa ame e BNP and bmax p esc ip ion chosen o collinea -so
unc ion, hea y meson je unc ion and dije so unc ion. Impac o he a i ion o BNP is shown
in igu e 2.
whe e he unc ions wi h su ix pe e e o hei pe u ba i e pa in he MS scheme which
is cu en ly known a one loop. Simila ly o he HHP case we need
J(b, mQ/pT;pT) = RJ(b, mQ/pT;pT, µJ)Jpe (b, mQ/pT;µJ) NP
J(b;mQ).(6.4)
The non-pe u ba i e e ec s a e pa ame e ized as
NP
i(b) = exp −b2
(Bi
NP)2, i =C,J, S. (6.5)
The alues o Bi
NP de ine hee non-pe u ba i e model and we ha e es ed se e al combi-
na ions as shown in igu e 2. Highe alues o Bi
NP a e mo e sensi i e o he pe u ba i e
se ies in he low ans e se momen um spec um, and in gene al p o ide highe alues o
he obse ables. In unpola ized TMD cases we ha e usually ha ypical alues o Bi
NP a e
a ound 1-3 GeV−1so we ha e ound easonable o ix hei alues as in able 1.
The ac o iza ion scales µC o dije and µJ o HHP a e chosen o minimize pe u -
ba i e loga i hms and o no hi he Landau pole o he s ong coupling cons an ,
µC= 2e−γE1
b+1
bmax ,(6.6)
µJ=1
2e−γE1
b+1
bmax .(6.7)
This scale choice dese es some commen s. In he dije case he scale choice does no
include he dependence on he je adius R. Simila ly, he mass o he a io mQ/pTdoes
no en e he collinea -so unc ion and hea y meson je . In all cases, his means ha
he e is no a comple e cancella ion o he loga i hms o hese unc ions. The eason is
ha he φbin eg a ion imposes some cons ain s on he choice o scales. In ac , he
unc ion A({µi})de ined in eq. (4.21), which depends on he ini ial scale choice o he
so unc ion, collinea -so unc ion and hea y meson je , needs o be A>−1/2in o de
o ha e a well de ined angula in eg a ion. Because o his cons ain some scale choices
which could be conside ed like o ins ance
µC=Re−γE
b, µJ=mQ/pTe−γE
b,(6.8)
can no be used. As a esul in ou app oach we only pa ially esum he logs in he
collinea -so unc ion and he hea y meson je in o de o main ain he s uc u e o ζ-
p esc ip ion and double scale e olu ion in he so unc ion ha is desc ibed in sec ion 5.
This leads o he ini ial scales in eq. (6.6), (6.7).
– 21 –
JHEP03(2022)047
Finally o he dije so unc ion we use he b∗-p esc ip ion in he same way as o he
TMDPDF:
µS=2e−γE
b∗, b∗=b
p1 + b2/b2
max
.(6.9)
Conce ning he heo e ical e o s, he scale a ia ions in collinea -so and hea y meson
je unc ion a e he main sou ce. This is due o he non-cancella ion o logs in he unc ions
by he choice o he ini ial scales. The choice o he alues bmax o collinea -so unc ion
and hea y meson je accoun o he con e gence o ou pe u ba i e esul . A mo e
consis en way o ea he esumma ion o hese scales is le o a u u e wo k, in ol ing
he e ac o iza ion o hese unc ions.
Fo unc ions ha do no depend on b he ini ial scale choice does no equi e a
p esc ip ion o NP-model and i is dic a ed by he cancella ion o he loga i hms. Fo he
je unc ion and he H+ma ching coe icien we ha e
µJ=pTR, µ+=mQ.(6.10)
We use a NP-model o he apidi y anomalous dimension ha en e s he exac solu ion
o he null-e olu ion ζµline as i is explained in sec ion 5. In pa icula , we use he same
model ha has been used o TMDPDF in [39]
DNP
F,S =c0bb∗, c0= 2.5·10−2.(6.11)
This model dic a es how he apidi y anomalous dimension beha es in he la ge-b egion and
is used o bo h dije so unc ion and TMDPDF when pe o ming double scale e olu ion.
While a colo e-scaling o he non-pe u ba i e models o gluon TMDPDF and gluon
channel so unc ion wi h espec o hei qua k analogues is possible, we obse e ha his
change does no ha e a signi ican impac on he c oss-sec ion and, he e o e, we choose o
keep he same model o bo h qua ks and gluons.
6.1 Resul s
In his sec ion we show ou esul s o he di e en ial c oss-sec ion o bo h dije and
hea y had on pai p oduc ion p ocesses. Di e en ial c oss-sec ions a e shown wi h e o
bands coming om scale a ia ion o he di e en inal and ini ial scales o he unc ions
appea ing in ou ac o iza ion o mulas. Scale a ia ion bands a e ob ained by changing
he conside ed scale by a ac o o 2 up and down ela i e o i s cen al alue.
6.1.1 Resul s o dije p oduc ion
In igu e 3we show he impac o he change o je adius, je ans e se momen um (ha d
scale) and je pseudo apidi y o e o al dije c oss-sec ion. We show ha o he a ia ion
o he je adius we see a change o a ound 20% on he c oss-sec ion om he cen al alue
when aking he je adius o be ±0.2 om R= 0.7. Fo pT he e is a a ia ion o an
o de o magni ude in he o al c oss-sec ion when aking ±5GeV om 20 GeV. This
co esponds o Q= 30,40,50 GeV espec i ely. Finally, o pseudo apidi y a ia ion we
ob ain an o de o magni ude di e ence abo e and below when compa ed o he cen al
– 22 –
JHEP03(2022)047
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Figu e 2. Impac o BNP a ia ion o e dije s and hea y meson o al c oss-sec ion. Legend
co espond o (BS
NP, BC
NP)and (BS
NP, BJ
NP) o dije and HHP p oduc ion espec i ely
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���
���
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(a) R a ia ion
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(b) pT a ia ion
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(c) η a ia ion (η1, η2)
Figu e 3. Impac o he a ia ion o he je adius (R), ha d scale je ans e se momen um
(pT) and je pseudo apidi y (ηi) o dije p oduc ion. Fo pseudo apidi y a ia ion legend is shown
e e ing o (η1, η2)pai , dashed and do ed lines co espond o nega i e and posi i e apidi y
espec i ely.
apidi y case. Posi i e apidi ies (ηi= 0.5) co espond o Q≃53 GeV while nega i e
apidi ies (ηi=−0.5) co espond o Q≃32 GeV, so bo h pTand ηplo s a e consis en .
No ice ha o al dije c oss-sec ion is no symme ical o bo h je apidi ies as o qua k
channel we ha e bo h a qua k and gluon je in he inal s a e. E e y o he plo is ob ained
aking R= 0.7,pT= 20 GeV and ηi= 0.
In igu e 4 he esul o he c oss-sec ion including qua k and gluon channels is shown.
We conside he con ibu ion o linea ly pola ized gluons in a sepa a e panel o show ha
hei con ibu ion is comple ely negligible, being a ac o 103-104smalle . This leads o
he conclusion ha he con ibu ion om he linea ly pola ized gluons can be neglec ed
when conside ing he unpola ized c oss-sec ion.
The angula modula ion asymme y is shown in igu e 6, being a ound 5%.
6.1.2 Resul s o hea y had on p oduc ion
The analysis o HHP has ollowed simila s eps o he dije case when possible. The
di e en ial c oss-sec ion including all channels is plo ed in igu e 5. A sepa a e analysis o
– 23 –
JHEP03(2022)047
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Figu e 4. C oss-sec ions o dije p oduc ion a EIC wi h e o -bands coming om scale de-
pendence in collinea -so ac o (CSF), ha d ac o (Ha d), je dis ibu ions (Je ) and Wilson
coe icien s (OPE). Rows co espond o con ibu ions om linea ly pola ized gluons ( op) and o al
c oss-sec ion (bo om). √s= 140 GeV, R= 0.7,pT= 20 GeV, η1=η2= 0.
�
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Figu e 5. C oss-sec ions o HHP p oduc ion a EIC wi h e o -bands coming om scale de-
pendence in ha d ac o (Ha d), hea y meson je unc ion (bHQET), hea y meson je unc ion
ma ching coe icien (Ha d+) and Wilson coe icien s (OPE). The ows co espond o con ibu ions
om o al c oss-sec ion ( op) and linea ly pola ized gluons (bo om). √s= 140 GeV, pT= 20 GeV,
η1=η2= 0.
he con ibu ion o linea ly pola ized gluons show also in his case ha hey a e comple ely
negligible being supp essed by a ac o 102-103. The angula modula ion asymme y is
shown in igu e 7, being a ound 5%.
– 24 –
JHEP03(2022)047
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