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

PoS(DIS2019)068
Unequal apidi y co ela o s A. Ramna h
The ela ion be ween λb
x,nand λa
u,0is linea . Thus he G een’s unc ion, de ined by Fn
x,¯x,u,¯u≡
δ
δ¯
λa
¯u,0
δ
δλa
u,0
¯
λb
¯x,nλb
x,n, does no depend on λ. Using his we ge hINi=g2
2(φ0
¯uu −φ0
¯uy −φ0
¯yu +
φ0
¯yy)FN
x,¯x,u,¯u+O(φ3/2)in e ms o he gluon dis ibu ion, which we can pu in o he equa ion
o he wo-pa icle c oss sec ion o ob ain a kT- ac o ized exp ession:
dσqg
dYd2pdYAd2kA
=1
(2π)4
1
2Nc
αs
π2Zx¯xy ¯yu ¯u
Ki
yuKi
¯y¯ue−ip·(x−¯x)−ikA·(y−¯y)
×(φ0
¯uu −φ0
¯uy −φ0
¯yu +φ0
¯yy)FN
x,¯x,u,¯u+O(φ3/2).(5.2)
We can inally ake he colo ield o be equal in he DA and he CCA. The ac ha we a e aking
de i a i es wi h espec o ¯
λ¯u,0and λu,0does no in e e e wi h he BFKL e olu ion o he gluon
densi y ¯
λb
¯x,nλb
x,n. So Fn
x,¯x,u,¯usa is ies he same equa ion as ¯
λb
¯x,nλb
x,nwi h espec o index n.
Fou ie ans o ming e e y hing, we ge
dσqg
dYd2pdYAd2kA
=−αs
NcZq
q2
(q−kA)2k2
A
FN(−p,p,q−kA,−q+kA)ϕ0(−q)+O(ϕ3/2).
wi h he (ze o momen um ans e ) BFKL G een’s unc ion Fsa is ying he usual BFKL equa ion.
We ha e hus shown ha he IT Lange in equa ion o malism educes, in he dilu e limi , o a
con en ional co ela ion be ween wo pa icles p oduced om he same BFKL ladde . The ini ial
condi ion o he e olu ion is F0(P,¯
P,m,¯m) = (Nc
2−1)δ(2)(P+m)δ(2)(¯
P+¯m). Using his
in he gene al exp ession educes he equal apidi y c oss sec ion o a kT- ac o ized exp ession:
dσqg
dYd2pdYAd2kAY=YA
=−αs
(2π)2
(p+kA)2
p2k2
A
ϕ0(p+kA).(5.3)
6. Conclusions
We ha e a emp ed o cla i y he Lange in o mula ion [12] o wo-pa icle co ela ions in
JIMWLK e olu ion, in he case o a dilu e p obe sca e ing o a dense colo ield a ge . Al hough
JIMWLK e olu ion o he Wilson lines is nonlinea , he e olu ion o he Lie de i a i es encoding
he co ela ion be ween he wo apidi ies, is in ac no . I can be exp essed as a linea equa ion ha
is independen o he Wilson lines. This obse a ion seems o con i m he esul ob ained ea lie
(in a a he di e en language) in [16].
We ha e also calcula ed explici ly he dilu e limi o he Lange in o mula ion, whe e he
deco ela ions in azimu hal angle be ween he wo pa icles a e gi en by a BFKL G een’s unc ion
be ween he wo apidi ies. JIMWLK e olu ion as a unc ion o he qua k apidi y Y“commu es”
wi h he p oduc ion Hamil onian and only ope a es on he dipole ope a o a Y. The e olu ion o
he double inclusi e c oss sec ion wi h Yis he e o e de e mined by he e olu ion o he single
inclusi e c oss sec ion, bu wi h a mo e complica ed ini ial condi ion.
Acknowledgmen s
We a e g a e ul o R. Boussa ie, M. Lublinsky, E. Iancu and D. T ian a yllopoulos o discus-
sions. T. L. has been suppo ed by he Academy o Finland, p ojec s No. 267321 and No. 303756.
4
PoS(DIS2019)068
Unequal apidi y co ela o s A. Ramna h
A. R. is suppo ed by he Na ional Resea ch Founda ion o Sou h A ica. This wo k has been
suppo ed by he Eu opean Resea ch Council, g an ERC-2015-CoG-681707.
Re e ences
[1] H. Weige , E olu ion a small xbj: The colo glass condensa e,P og. Pa . Nucl. Phys. 55 (2005) 461
[2] F. Gelis, E. Iancu, J. Jalilian-Ma ian and R. Venugopalan, The colo glass condensa e,Ann. Re . Nucl.
Pa . Sci. 60 (2010) 463 [a Xi :1002.0333 [hep-ph]].
[3] J. Jalilian-Ma ian, A. Ko ne , L. D. McLe an and H. Weige , The in insic glue dis ibu ion a e y
small x,Phys. Re . D55 (1997) 5414 [a Xi :hep-ph/9606337 [hep-ph]].
[4] J. Jalilian-Ma ian, A. Ko ne , A. Leonido and H. Weige , The Wilson eno maliza ion g oup o low
x physics: Towa ds he high densi y egime,Phys. Re . D59 (1998) 014014
[5] E. Iancu and L. D. McLe an, Sa u a ion and uni e sali y in QCD a small x,Phys. Le . B510 (2001)
145 [a Xi :hep-ph/0103032].
[6] E. Fe ei o, E. Iancu, A. Leonido and L. McLe an, Nonlinea gluon e olu ion in he colo glass
condensa e. II,Nucl. Phys. A703 (2002) 489 [a Xi :hep-ph/0109115].
[7] A. H. Muelle , A simple de i a ion o he JIMWLK equa ion,Phys. Le . B523 (2001) 243
[8] Y. V. Ko chego , J. Kuokkanen, K. Rummukainen and H. Weige , Subleading-Ncco ec ions in
non-linea small-x e olu ion,Nucl. Phys. A823 (2009) 47 [a Xi :0812.3238 [hep-ph]].
[9] T. Lappi and H. Män ysaa i, On he unning coupling in he JIMWLK equa ion,Eu . Phys. J. C73
(2013) 2307 [a Xi :1212.4825 [hep-ph]].
[10] Y. V. Ko chego , Small-x F2 s uc u e unc ion o a nucleus including mul iple pome on exchanges,
Phys. Re . D60 (1999) 034008 [a Xi :hep-ph/9901281].
[11] F. Gelis, T. Lappi and R. Venugopalan, High ene gy ac o iza ion and long ange apidi y co ela ions
in he glasma,Phys. Re . D79 (2008) 094017 [a Xi :0810.4829 [hep-ph]].
[12] E. Iancu and D. T ian a yllopoulos, JIMWLK e olu ion o mul i-pa icle p oduc ion in Lange in
o m,JHEP 1311 (2013) 067 [a Xi :1307.1559 [hep-ph]].
[13] A. Ko ne , M. Lublinsky and H. Weige , T eading on he cu : Semi inclusi e obse ables a high
ene gy,Phys. Re . D74 (2006) 114023 [a Xi :hep-ph/0608258 [hep-ph]].
[14] A. Ko ne and M. Lublinsky, One gluon, wo gluon: Mul igluon p oduc ion ia high ene gy
e olu ion,JHEP 11 (2006) 083 [a Xi :hep-ph/0609227 [hep-ph]].
[15] T. Lappi and A. Ramna h, Unequal apidi y co ela o s in he dilu e limi o JIMWLK,
[a Xi :1904.00782 [hep-ph]].
[16] J. Jalilian-Ma ian and Y. V. Ko chego , Inclusi e wo-gluon and alence qua k-gluon p oduc ion in
dis and p a,Phys. Re . D70 (2004) 114017 [a Xi :hep-ph/0405266].
[17] I. Bali sky, Ope a o expansion o high-ene gy sca e ing,Nucl. Phys. B463 (1996) 99
[18] S. Ca on-Huo , When does he gluon eggeize?,JHEP 05 (2015) 093
[19] J. R. Fo shaw and D. A. Ross, Quan um ch omodynamics and he pome on,Camb idge Lec . No es
Phys. 9(1997) 1.
[20] A. H. Muelle and B. Pa el, Single and double BFKL pome on exchange and a dipole pic u e o
high-ene gy ha d p ocesses,Nucl. Phys. B425 (1994) 471
5