On the linearly polarized gluon distributions in the color dipole model

0.22 MB·English

by  Fabio Dominguez

On the linearly polarized gluon distributions in the color dipole model
Fabio Dominguez,1 Jian-Wei Qiu,2,3 Bo-Wen Xiao,4 and Feng Yuan5
1Department of Physics, Columbia University, New York, NY, 10027, USA
2Physics Department, Brookhaven National Laboratory, Upton, NY 11973
3C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA
4Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
We show that the linearly polarized gluon distributions appear in the color dipole model as we
derivethefullcrosssectionsoftheDISdijetproductionandtheDrell-Yandijet(γ∗ jetcorrelation)
process. Together with the normal Weizs¨acker-Williams gluon distribution, the linearly polarized
onewillcontributetotheDISdijetproductioncrosssectionasthecoefficientofthecos(2∆φ)term
in the correlation limit. We also derive the exact results for the cross section of the Drell-Yan
2
dijet process, andfindthatthelinearly polarized dipolegluon distribution which isidenticaltothe
1
normal dipole gluon distribution involves in the cross section. The results obtained in this paper
0
agreewiththeprevioustransversemomentumdependentfactorization study. Wefurtherderivethe
2
small-x evolution of these linearly polarized gluon distributions and find that they rise as x gets
n small at high energy.
a
J
PACSnumbers: 24.85.+p,12.38.Bx,12.39.St,13.88.+e
1
3

I. INTRODUCTION
]
h
p In small-x physics, two different unintegrated gluon distributions [1, 2] (also known as transverse momentum
p- dependent gluon distributions), namely the Weizs¨acker-Williams gluon distribution xG(1) [3, 4] and the dipole gluon
e distribution xG(2), have been widely used in the literature. The Weizs¨acker-Williams gluon distribution can be
h interpreted as the number density of gluons inside dense hadrons in light-cone gauge. The dipole gluon distribution,
[ despite of lacking the probabilistic interpretation, has been thoroughly studied since it appears in many physical

2 processes [5, 6] and it is defined via the Fourier transform of the simple color dipole amplitude. This dipole gluon
v distribution can be probed directly in photon-jet correlations and Drell-Yan dijet measurement in pA collisions.
3 Recent studies [7, 8] on the Weizs¨acker-Williams gluon distribution indicate that it can be directly measured in DIS
9 dijet production and its operator definition is related to color quadrupoles instead of normal color dipoles. Other
2
more complicated dijet processes in pA collisions (e.g., qg or gg dijets) involve both of these gluon distributions
6
through convolution in transverse momentum space. The complete calculations were performed in Ref. [7, 8] in
.
9 both the Transverse Momentum Dependent (TMD) factorization formalism and the color dipole model. The results
0 demonstrate the complete agreement between these two formalisms in the kinematical region where they are both
1
valid.
1 Linearly polarized gluon distributions, denoted as h(i)(x,q )1, where x and q are the active gluon’s longitudinal
: ⊥ ⊥ ⊥
v momentum fraction and its transverse momentum, respectively, were first introduced in Ref. [9]. This new gluon
i
X distribution effectively measures an averaged quantum interference between a scattering amplitude with an active
gluon polarized along the x(or y)-axis and a complex conjugate amplitude with an active gluon polarized along
r
a the y(or x)-axis inside an unpolarized hadron. Because of the unique transverse spin correlation between the two
gluon fields of the distribution, the linearly polarized gluon distribution can contribute to a physical observable with
cos(2∆φ)-type azimuthal angular dependence, or the azimuthally symmetric observables if they come in pairs. As
proposed in Ref. [10], the linearly polarized gluon distributions can be directly probed in dijet and heavy quark
pair production processes in electron-hadron collisions. As expected, this distribution also contributes to the cross
sectioninphotonpairproductions[11,12]andtheStandardModelHiggsbosonproduction[13–15]inhadron-hadron
collisions. SincetheintegratedpartondistributionsforincomingprotonswereusedinthecalculationsofpAcollisions
in Ref. [7, 8], the linearly polarized gluon distribution does not enter the cross section except for the Drell-Yan dijet
processes as we show in the later discussion.
In Ref. [16], the linearly polarized partner of both Weizs¨acker-Williams and dipole gluon distributions inside an
unpolarized nucleus target is studied in Color Glass Condensate (CGC) formalism. The corresponding cross sections
ofdeepinelasticscattering(DIS)dijetproductionandthe Drell-YanprocessesinpAcollisionsarecomputedinterms
1 Normallyitisdenoted ash⊥1g(x,q ). Herethroughout thepaper, inordertoavoidconfusiononthenotation, weuseh(i)(x,q )with
i=1,2torepresentthelinearlypo⊥larizedgluondistributions. ⊥ ⊥
2
of the TMD formalism. In both processes, the linearly polarized gluon distributions appear as the coefficients of the
cos(2∆φ) term in the cross section, where they were found to be consistent with the small-x formalism as well [16].
Inspired by Ref. [16], we perform the detailed calculation in the color dipole model for the DIS dijet production
and the Drell-Yan dijet processes in pA collisions and we find identical results as those with the TMD formalism
for the cross sections in the correlation limit, which is defined as a limit when the final state dijets are almost back-
to-back. For the DIS dijet production, the complete analysis of the quadrupole amplitude shows that the linearly
polarizedgluondistributionoftheWeizs¨acker-Williamstypecomesfromtheoff-diagonalexpansionofthequadrupole
amplitude. Using a hybrid factorization, we obtain the exact results for the cross section of the Drell-Yan processes
in pA collisions. In the correlation limit, this exact result reduces to the TMD cross section obtained in Ref. [16].
Another objective of this paper is to study the small-x evolution of the linearly polarized gluon distributions.
The small-xevolutionofthe dipole type linearlypolarizedgluondistribution is essentially the evolutionofthe dipole
amplitude,whichisgovernedbytheBalitsky-Kovchegovequation[17,18]. Derivedfromtheevolutionofquadrupoles,
the evolution of the linearly polarized Weizs¨acker-Williams gluon distribution is quite complicated. Nevertheless, in
the dilute regime, we find that both linearly polarized gluon distributions receive the exponential enhancement in
terms of rapidity at high energy as the normal unpolarized gluon distributions do due to the small-x evolution.
The rest of the paper is organizedas follows. In Sec. II, we calculate the cross sections of the DIS dijet production
andthe Drell-YanprocessesinpAcollisionsanddemonstratethatthe linearlypolarizedgluondistributionsnaturally
arise in the dipole model. We discuss the small-x evolution equations of the linearly polarized gluon distributions in
Sec. III. The summary and further discussions are given in Sec. IV.
II. THE LINEARLY POLARIZED GLUON DISTRIBUTION IN DIPOLE MODEL
In this section, following Ref. [8], we show that the cross section of the DIS dijet production and the Drell-Yan
dijet process in the color dipole model, namely the CGC approach, involves the linearly polarized gluon distribution
as well. The reasonwhy this does not appear in the originalworkin [8] is that there the azimuthalorientationof the
outgoing partons was averagedover.
A. DIS dijet production
After averagingoverthe photon’s polarizationandsumming overthe quarkandantiquarkhelicities andcolors,the
cross section of the DIS dijet production in the color dipole model can be cast into
dσγT∗,LA→qq¯X =N α e2δ(p+ k+ k+) d2x1 d2x′1 d2x2 d2x′2
d3k d3k c em q − 1 − 2 (2π)2(2π)2(2π)2(2π)2
1 2 Z
×e−ik1⊥·(x1−x′1)e−ik2⊥·(x2−x′2) ψαTβ,Lλ(x1−x2)ψαTβ,Lλ∗(x′1−x′2)
λαβ
X
1+S(4)(x ,x ;x′,x′) S(2)(x ,x ) S(2)(x′,x′) , (1)
× xg 1 2 2 1 − xg 1 2 − xg 2 1
h i
where the two- and four-point functions, which are characterized by the Wilson lines, take care of the multiple
scatterings between the qq¯-pair and the target. They are defined as
1
S(2)(x ,x )= TrU(x )U†(x ) , (2)
xg 1 2 Nc 1 2 xg
(cid:10) 1 (cid:11)
S(4)(x ,x ;x′,x′)= TrU(x )U†(x′)U(x′)U†(x ) , (3)
xg 1 2 2 1 Nc 1 1 2 2 xg
+∞(cid:10) (cid:11)
with U(x)= exp ig dx+TcA−(x+,x) . (4)
P S c
(cid:26) Z−∞ (cid:27)
The notation ... is used for the CGC average of the color charges over the nuclear wave function where x is
h ixg g
the smallest fraction of longitudinal momentum probed, and is determined by the kinematics. The splitting wave
functionofthevirtualphotonwithlongitudinalmomentump+ andvirtualityQ2 intransversecoordinatespacetakes
3
the form,
ψTλ(p+,z,r)=2π 2 iǫfK1(ǫf|r|)r·|ǫr(⊥|1)[δα+δβ+(1−z)+δα−δβ−z], λ=1, (5)
αβ rp+ iǫfK1(ǫf|r|)r·|ǫr(⊥|2)[δα−δβ−(1−z)+δα+δβ+z], λ=2,
4
ψL (p+,z,r)=2π z(1 z)QK (ǫ r)δ . (6)
αβ p+ − 0 f| | αβ
r
where z is the momentum fraction of the photon carried by the quark, λ is the photon polarization, α and β are the
quark and antiquark helicities, r the transverse separation of the pair, ǫ2 =z(1 z)Q2, and the quarks are assumed
f −
to be massless.
In order to take the correlation limit, we introduce the transverse coordinate variables: u = x x and v =
1 2
zx +(1 z)x ,andsimilarlyfortheprimedcoordinates,withrespectiveconjugatemomentaP˜ =(1 −z)k zk 2
1 2 ⊥ 1⊥ 2⊥
− − −
and q . The correlation limit (P˜ k k q ) is therefore enforced by assuming u and u′ are small as
⊥ ⊥ 1⊥ 2⊥ ⊥
compared to v and v′ and then expa≃nding t≃he inte≫grand with respect to these two variables before performing the
Fourier transform. Following the derivation in Ref. [8], one can find that the lowest order expansion in u and u′ of
the last line of Eq. (1) gives
1
u u′ Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v) . (7)
− i jN h ixg
c
(cid:2) (cid:3) (cid:2) (cid:3)
With the help of the identities
(2dπ2u)2(d22πu)′2e−iP˜⊥·(u−u′)uiu′j∇uK0(ǫfu)·∇u′K0(ǫfu′)= (2π1)2 "(P˜2δ+ijǫ2)2 − 4(Pǫ˜2f2P˜+⊥iǫP˜2⊥)4j# , (8)
Z ⊥ f ⊥ f
d2u d2u′ e−iP˜⊥·(u−u′)u u′K (ǫ u)K (ǫ u′)= 1 4P˜⊥iP˜⊥j , (9)
(2π)2(2π)2 i j 0 f 0 f (2π)2(P˜2 +ǫ2)4
Z ⊥ f
one can integrate over u and u′ and obtain the complete differential cross section in the correlationlimit,
dσγdT∗A.→q.q¯X =αeme2qαsδ(xγ∗ −1)z(1−z) z2+(1−z)2 "(P˜2δ+ijǫ2)2 − 4(Pǫ˜2f2P˜+⊥iǫP˜2⊥)4j#
P S ⊥ f ⊥ f
(cid:0) (cid:1)
(16π3) d3vd3v′e−iq⊥·(v−v′)2 Tr Fi−(v) [+]†Fj−(v′) [+] , (10)
× Z (2π)6 D h U U iExg
dσγdL∗A.→q.q¯X =αeme2qαsδ(xγ∗ −1)4z2(1−z)24(Pǫ˜2f2P˜+⊥iǫP˜2⊥)4j
P S ⊥ f
(16π3) d3vd3v′e−iq⊥·(v−v′)2 Tr Fi−(v) [+]†Fj−(v′) [+] . (11)
× Z (2π)6 D h U U iExg
Here we have used the identity
∞
Tr[∂ U(v)]U†(v′)[∂ U(v′)]U†(v) =g2 dv+dv′+ Tr Fi−(v) [+]†Fj−(v′) [+] , (12)
−h i j ixg SZ−∞ D h U U iExg
where the gauge link [+] connects the two coordinate points by means of longitudinal gauge links going to + and
U ∞
a transverse link at infinity which does not contribute when the appropriate boundary conditions are taken.
If one integrates over the orientation of P˜ , one can replace P˜ P˜ by 1δ P˜2.3 This replacement allows us to
⊥ ⊥i ⊥j 2 ij ⊥
reduce the above expressions into Eqs. (30) and (31) in Ref. [8] which only involve the conventional Weizs¨acker-
Williams gluon distribution.
2 Onecouldalsodefinev= 12(x1+x2)inthisprocesssincethevirtualphotondoesnothaveinitialinteractionswiththenucleustarget,
tthheencetnhteerreosfpmecatisvsefrcaomnjeu.gNateevemrtohmeleenstsu,mtheisfoPl⊥low=in21g(dke1r⊥iv−atiko2n⊥r)e≃maP˜in⊥s.tPhe⊥siasmteheinretlhaitsivceasme.omentum of outgoing partons respect to
3 InthederivationofRef.[8],wehaveemployedthisasanunderlyingassumption.
4
NowwearereadytoshowthatthelinearlypolarizedWeizs¨acker-Williamsgluondistributioncanalsoarisenaturally
in the color dipole model. Since the indices i, j are symmetric, we can decompose the operator expressionappearing
in Eqs (10) and (11) into two parts with one part involving only δ and the other part being traceless,
ij
4 d3vd3v′e−iq⊥·(v−v′) Tr Fi−(v) [+]†Fj−(v′) [+]
Z (2π)3 D h U U iExg
1 1 2qi qj
= δijxG(1)(x,q )+ ⊥ ⊥ δij xh(1)(x,q ). (13)
2 ⊥ 2 q⊥2 − ! ⊥ ⊥
HerexG(1)(x,q )isthe conventionalWeizs¨acker-Williamsgluondistributionwhilethecoefficientofthe tracelessten-
⊥
sorxh(1)(x,q ) is the so-calledlinearlypolarizedpartner ofthe conventionalWeizs¨acker-Williamsgluondistribution.
⊥ ⊥
The physical meaning or interpretation of these two gluon distributions can be better represented in a frame in
which the two components of the transverse momentum qj with j =1,2 or j =x,y are the same. With qx =qy in
⊥ ⊥ ⊥
this frame, the two symmetric projection operators in Eq. (13) can be written as,
1 1 1 0 1 1
δij = = eiej +eiej = ε∗iεj +ε∗iεj , (14)
2 2 0 1 2 x x y y 2 + + − −
1 2qi qj 1(cid:18)0 1(cid:19) 1(cid:0) (cid:1) 1h i
⊥ ⊥ δij = = eiej +eiej = ε∗iεj ε∗iεj , (15)
2 q⊥2 − ! 2(cid:18)1 0(cid:19) 2(cid:0) x y y x(cid:1) 2ih + −− − +i
where ei = (1,0) and ei = (0,1) are 2-dimensional unit vectors along x-axis and y-axis, respectively, which could
x y
be interpreted as two orthogonal linear polarization vectors for transversely polarized gluons. As shown in Eqs. (14)
and (15), these two symmetric projection operators can also be expressed in terms of the two orthogonal circular
polarizationvectorsfor transverselypolarizedgluons, εj [ ej iej]/√2. For the comparison,we alsolisthere the
± ≡ ∓ x− y
antisymmetric projection operator for the polarized gluon helicity distribution,
1 1 0 i 1 1
iǫij = = i eiej eiej = ε∗iεj ε∗iεj . (16)
2 ⊥ 2 i 0 2 x y− y x 2 + +− − −
(cid:16) (cid:17) (cid:18)− (cid:19) h i
(cid:0) (cid:1)
From Eqs. (14) and (16), it is natural to interpret G(1) as a probability distribution to find unpolarized gluons,
while the polarized gluon helicity distribution could be interpreted as a difference of two probability distributions to
find positive helicity gluons and negative helicity gluons, respectively. From Eq. (15), it appears that h(1) does not
⊥
have a probability interpretation in terms of the base polarization vectors εj , which are the eigenstates of angular
±
momentum operators. 4 Instead, it could be interpreted as a transverse spin correlation function to find the gluons
in the amplitude and complex conjugate amplitude to be in two orthogonal polarization states. In a general frame,
qj =(qx,qy)=q (cosφ,sinφ), the projection operator for h(1) can be written as,
⊥ ⊥ ⊥ ⊥ ⊥
1 2qi qj 1 cos(2φ) sin(2φ)
⊥ ⊥ δij = , (17)
2 q⊥2 − ! 2(cid:18)sin(2φ) −cos(2φ)(cid:19)
whichincludesthespecialcaseinEq.(15)whenφ=π/4. SincetheprojectionoperatorinEq.(17)isproportionaltoa
rotationmatrixoftheazimuthalangle,theh(1) couldalsobeinterpretedas“azimuthalcorrelated”gluondistributions
⊥
[12,13]. Becausethegluonsintheamplitude andcomplexconjugateamplitudeareindifferenttransversespinstates,
this kind of gluon distributions could contribute to the observables with cos(2∆φ)-type azimuthal dependence, or
azimuthal symmetric observables if they come in pairs.
4 However, if one chooses different base polarization vectors as ei = 1 (1,1) and ei = 1 (1,−1), which are not the eigenstates of
1 √2 2 √2
angularmomentumoperators,onecanfindthatEq.(15)becomes 1 eiej−eiej whichwouldallowustointerpreth(1)asthelinearly
2(cid:16) 1 1 2 2(cid:17)
⊥
polarized gluon density along the direction of the linear polarization. In a general frame, the polarization vectors are found to be
ei =(cosφ,sinφ)andei =(sinφ,−cosφ)whichconvertEq.(17)into 1 eiej−eiej aswell. Thisindicatesthattheinterpretationof
1 2 2(cid:16) 1 1 2 2(cid:17)
thelinearlypolarizedgluondistributionsdependsonthechoiceofthepolarizationvectors.
5
Substitute Eq. (13) into Eqs (10) and (11), we obtain
dσγdT∗A.→q.q¯X =αeme2qαsδ(xγ∗ −1)z(1−z) z2+(1−z)2 (P˜ǫ4f2++Pǫ˜2⊥4)4
P S ⊥ f
(cid:0) (cid:1)
2ǫ2P˜2
xG(1)(x,q ) f ⊥ cos(2∆φ)xh(1)(x,q ) , (18)
×" ⊥ − ǫ4 +P˜4 ⊥ ⊥ #
f ⊥
dσγdL∗A.→q.q¯X =αeme2qαsδ(xγ∗ −1)z2(1−z)2(P˜82ǫ2f+P˜ǫ⊥22)4
P S ⊥ f
xG(1)(x,q )+cos(2∆φ)xh(1)(x,q ) . (19)
× ⊥ ⊥ ⊥
h i
where ∆φ = φ φ with φ and φ being the azimuthal angle of P˜ and q , respectively. This result is in
P˜⊥ − q⊥ P˜⊥ q⊥ ⊥ ⊥
complete agreement with the one obtained in Ref. [16] by using the TMD approach. The coefficient of the cos(2∆φ)
term in the above cross section can provide us the direct information of the linearly polarized Weizs¨acker-Williams
gluon distribution xh(1)(x,q ). It is also easy to see that the xh(1)(x,q ) term vanishes if one averages the cross
⊥ ⊥ ⊥ ⊥
section over the orientation of either P˜ or q due to the factor cos(2∆φ). This is transparent when one uses the
⊥ ⊥
variables P and q since they can be interpreted as the relative transverse momentum with respect to the center of
⊥ ⊥
mass frame of these two outgoing partons and the total transverse momentum of the CM frame, respectively.
Lastbutnotleast,onecanseethatthecontributionfromthelinearlypolarizedgluondistributionvanishesifQ=0,
i.e., the real photon nucleus scattering only involves the conventional Weizs¨acker-Williams gluon distribution. This
is because the real photon cannot generate a cos(2∆φ)-type transverse spin correlation that matches the transverse
spin correlationgenerated by h(1).
⊥
Letusnowstudythebehaviorofxh(1)(x,q )intheMcLerran-Venugopalan(MV)model[19]foralargenucleuswith
⊥ ⊥
Anucleonsinside. Usingthe quadrupoleresultscalculatedinRef.[8],one cancastthe analyticalformofxh(1)(x,q )
⊥ ⊥
into[16]
xh(1)(x,q )= 2 δij 2q⊥i q⊥j d2vd2v′ e−iq⊥·(v−v′) Tr[∂ U(v)]U†(v′)[∂ U(v′)]U†(v)
⊥ ⊥ αs − q⊥2 !Z (2π)2(2π)2 h i j ixg
S N2 1 ∞ J (q r ) 1
= ⊥ c − dr r 2 ⊥ ⊥ 1 exp r2Q2 , (20)
2π3α N ⊥ ⊥r2 ln 1 − −4 ⊥ sg
s c Z0 ⊥ r⊥2Λ2 (cid:20) (cid:18) (cid:19)(cid:21)
where J (q r ) is the Bessel function of the first kind and Q2 = α g2N µ2ln 1 with µ2 = A . For q2 Q2 ,
2 ⊥ ⊥ sg s c r⊥2Λ2 2S⊥ ⊥ ≫ sg
wefindthatxh(1)(x,q ) αsACFNc whichis identicaltoxG(1)(x,q )andagreeswiththeperturbativeQCDresults.
⊥ ⊥ ≃ π2q⊥2 ⊥
It is important to notice that it scales like A since each nucleon contributes additively in the dilute regime. In this
regime,thedominantcontributiontothegluondistributioncomesfromasingletwo-gluonexchangewithatransverse
momentum transfer q in the color dipole picture. For the case Λ2 q2 Q2 one absorbs the ln 1 factor into
⊥ ≪ ⊥ ≪ sg r⊥2Λ2
the definition of the saturation momentum and finds xh(1)(x,q ) αsACFNc which is an approximate constant. It
⊥ ⊥ ≃ π2Q2sg
scales like A2/3 since Q2 A1/3 as a result of strong nuclear shadowing. It is interesting to note that, in the low
sg ∼
q2 region, the effect of multiple scatterings between probes and target nuclei can be viewed as or attributed to a
⊥
singlescatteringwiththemomentumtransferoforderQ2 . Ascomparedtothesmallq2 behavioroftheconventional
sg ⊥
Weizs¨acker-Williams gluon distribution xG(1)(x,q ) S⊥ Nc2−1lnQ2sg, we find that xG(1)(x,q⊥) lnq⊥2 lnQ2sg 1
⊥ ≃ 4π3αs Nc q⊥2 xh(⊥1)(x,q⊥) ≃ Λ2 q⊥2 ≫
where we have replaced r by 1 . These gluon distributions obtained in the MV model can be viewed as an initial
⊥ q⊥
condition for the small-x evolution. In addition, we can also find that xG(1)(x,q ) xh(1)(x,q ) for any value of q
⊥ ≥ ⊥ ⊥ ⊥
which ensures the positivity of the total cross section.
B. Drell-Yan dijet process
Followingthepromptphoton-jetcorrelationcalculationinRef.[8],itisstraightforwardtocalculatethecrosssection
of dijet (qγ∗) production in Drell-Yan processes in pA collisions. The calculation is essentially the same, except for
the slightly different splitting function since the final state virtual photon, which eventually decays into a di-lepton
6
pair, has a finite invariant mass M. By taking into account the photon invariant mass, the splitting wave functions
of a quark with longitudinal momentum p+ splitting into a quark and virtual photon pair in transverse coordinate
space become
ψTλ(p+,k+,r)=2π 2 iǫMK1(ǫM|r|)r·|ǫr(⊥|1)(δα−δβ−+(1−z)δα+δβ+), λ=1, , (21)
αβ 1 sk1+ iǫMK1(ǫM|r|)r·|ǫr(⊥|2)(δα+δβ++(1−z)δα−δβ−), λ=2.
ψL (p+,k+,r)=2π 2 (1 z)MK (ǫ r)δ , (22)
αβ 1 sk1+ − 0 M| | αβ
where ǫ2 = (1 z)M2, λ is the photon polarization, α,β are helicities for the incoming and outgoing quarks, and
M −
k+
z = 1 is the momentum fraction of the incoming quark carried by the photon.
p+
Atthe endofthe day,for the correlationbetweenthe finalstate virtualphotonandquarkinpAcollisions,wehave
dσDpAP→γ∗q+X = x q (x ,µ)αe.m.e2f (1 z)z2S F (q )
dy dy d2k d2k p f p 2π2 − ⊥ xg ⊥
1 2 1⊥ 2⊥
f
X
q2
1+(1 z)2 ⊥
× − P˜2 +ǫ2 (P˜ +zq )2+ǫ2
h i ⊥ M ⊥ ⊥ M
h ih 2 i
 1 1
ǫ2 , (23)
− M"P˜⊥2 +ǫ2M − (P˜⊥+zq⊥)2+ǫ2M# 
withF (q )= d2r⊥e−iq⊥·r⊥ 1 TrU(0)U†(r ) ,q =k +k andP˜ =(1 z)k zk . IntheMVmodel,
xg ⊥ (2π)2 Nc ⊥ xg ⊥ 1⊥ 2⊥ ⊥ − 1⊥− 2⊥
F (q ) 1 Rexp q⊥2 with(cid:10)Q2 = CFQ2 b(cid:11)eing the quark saturation momentum. q (x ,µ) is the integrated
xg ⊥ ≃ πQ2sq −Q2sq sq Nc sg f p
quark distribution w(cid:16)ith flav(cid:17)or f in the proton projectile. Here we used the hybrid factorization which allows us to
use integrated parton distributions since the proton projectile is considered to be dilute as compared to the nucleus
target. The first term in the curly brackets arises solely from the transverse splitting function in Eq. (21) while the
second term is the sum of contributions from both the transverseand longitudinal splitting functions. We would like
to emphasize that the above crosssection in Eq. (23) is an exact result regardlessof the relative size between q and
⊥
P˜ . By taking the correlation limit, namely q P˜ , we arrive at the result which is identical to the one obtained
⊥ ⊥ ⊥
from TMD factorization [16] 5 ≪
dσpA→γ∗q+X
dy1dyD2Pd2k1⊥d2k2⊥(cid:12)(cid:12)(cid:12)q⊥≪P˜⊥ =Xf xpqf(xp,µ)xG(2)(xg,q⊥)(cid:2)Hqg→qγ∗ −cos(2∆φ)Hq⊥g→qγ∗(cid:3), (24)
(cid:12)
with ∆φ=φ φ , xG(2)(x,q(cid:12) )= q⊥2NcS F (q ) and
P˜⊥ − q⊥ ⊥ 2π2αs ⊥ xg ⊥
Hqg→qγ∗ = αsαe.m.eN2f(c1−z)z2 1P˜+⊥2(+1−ǫ2Mz)22 − P2˜z⊥22+ǫ2MǫP2M˜⊥24, (25)
Hq⊥g→qγ∗ = αsαe.m.eN2f(1−z)z2 2hz2ǫ2MP˜⊥24.i h i  (26)
c P˜2 +ǫ2
⊥ M
h i
Inthiscase,therelevantgluondistributionistheso-calleddipolegluondistributionasdemonstratedinRef.[7,8,20].
5 TocomparewithRef.[16],onecancomputetheMandelstamvariablesandfindthatsˆ=(k1+k2)2=M2+(1−z)(Mz2+k12⊥)+(z1k22⊥z)−
2k1⊥·k2⊥= P˜z⊥2(1+−ǫz2M) ,uˆ= P˜⊥2+zǫ2M andtˆ= 1P˜−⊥2z. −
7
As discussed in Ref. [16], according to the operator definition of dipole type gluon distributions [7, 8, 20],
xGij (x,q )=2 dξ−dξ⊥ eixP+ξ−−iq⊥·ξ⊥ P Tr F+i(ξ−,ξ ) [−]†F+j(0) [+] P , (27)
DP ⊥ (2π)3P+ h | ⊥ U U | i
Z h i
qi qj N
= ⊥ ⊥ cS F (q ), (28)
2π2α ⊥ xg ⊥
s
1 1 2qi qj
= δijxG(2)(x,q )+ ⊥ ⊥ δij xh(2)(x,q ), (29)
2 ⊥ 2 q⊥2 − ! ⊥ ⊥
wherethegaugelink [−] iscomposedbylongitudinalgaugelinksgoingto . Thisshowsthatthelinearlypolarized
U −∞
partner of the dipole gluon distribution is exactly the same as the dipole gluon distribution6. From Eq. (29), with
the proper normalization, we can also find that the linearly polarized gluon distribution xh(2)(x,q )=xG(2)(x,q ).
⊥ ⊥ ⊥
Furthermore, one can see that for the prompt photon-jet correlation,the linearly polarizedgluon distribution does
notcontributesinceHq⊥g→qγ∗ vanisheswhenM =0. This isalsoduetothefactthattherealphotoninthefinalstate
cannot generate the transversespin correlationthat matches the transversespin correlationof the incoming gluon in
the qg qγ subprocess. It takes two matched transverse spin correlations to get a nonvanish observable effect.
→
C. Resummation
For the purpose of the Collins-Soper-Sterman resummation[21] discussed in Ref. [14], it is also useful to define the
coordinate expression of the linearly polarized Weizs¨acker-Williams gluon distribution as follows
1 2qi qj
xh˜(1)ij(x,b )= d2q e−iq⊥·b⊥ ⊥ ⊥ δij xh(1)(x,q ), (30)
⊥ ⊥ 2Z ⊥ q⊥2 − ! ⊥ ⊥
and it is straightforwardto find that in the MV model
1 2bi bj S N2 1 1 1
xh˜(1)ij(x,b )= δij ⊥ ⊥ ⊥ c − 1 exp b2Q2 . (31)
⊥ ⊥ 2 − b2⊥ !π2αs Nc b2⊥lnb2⊥1Λ2 (cid:20) − (cid:18)−4 ⊥ sg(cid:19)(cid:21)
This can be compared to the normal Weizs¨acker-Williams gluon distribution in b space defined as xG˜(1)(x,b ) =
⊥ ⊥
d2q e−iq⊥·b⊥xG(1)(x,q ),
⊥ ⊥
R xG˜(1)(x,b )= S⊥ Nc2−1lnb2⊥1Λ2 −2 1 exp 1b2Q2 . (32)
⊥ π2α N b2 ln 1 − −4 ⊥ sg
s c ⊥ b2⊥Λ2 (cid:20) (cid:18) (cid:19)(cid:21)
At small b , xh˜(1)ij(x,b ) is proportional to δij 2bi bj /b2 times a constant, whereas xG(1)(x,b ) behaves as
⊥ ⊥ ⊥ − ⊥ ⊥ ⊥ ⊥
ln 1 duetothe logarithmicterminQ2 . Th(cid:16)esepropertiesare(cid:17)consistentwiththeirperturbativebehaviorsatlarge
Λ2b2⊥ sg
transverse momentum [14].
Similarly for the dipole gluon counterparts, one gets
2
1 2bi bj N S 1 1 b2Q2 1
xh˜(2)ij(x,b )= δij ⊥ ⊥ c ⊥ exp[ Q2 b2]Q2 + ⊥ sq 1 , (33)
⊥ ⊥ 2 ⊥ − b2⊥ !2π2αs −4 sq ⊥ sqlnΛ21b2⊥ 4 − lnΛ21b2⊥! 
 
and
2
N S 1 2 b2Q2 1
xG˜(2)(x,b )= c ⊥ exp[ Q2 b2]Q2 1 ⊥ sq 1 . (34)
⊥ 2π2αs −4 sq ⊥ sq − lnΛ21b2⊥ − 4 − lnΛ21b2⊥ ! 
 
Again, in the small b limit, they behave the same as those Weizs¨acker-Williams gluon distributions, respectively. It
⊥
is interesting to notice that their large b behaviors are different. For the dipole gluon distributions, they decrease
⊥
exponentially whereas the Weizs¨acker-Williams ones have power behaviors. These expressions can be viewed as the
initial conditions of the resummation discussed in Ref. [14].
6 Thereisafactorof2between thesetwodistributionsinRef.[16]duetodifferentnormalization.
8
III. SMALL-x EVOLUTION OF THE LINEARLY POLARIZED GLUON DISTRIBUTIONS
In this section, we discuss the small-x evolution of the linearly polarized gluon distributions. We separate the
discussions into two parts: the first part is on the evolution of the linearly polarized dipole gluon distribution since
it is trivial and it only involves the dipole amplitude; then we derive the evolution equation for the linearly polarized
Weizs¨acker-Williams gluon distribution from the small-x evolution equation of quadrupoles.
A. The evolution of the linearly polarized dipole gluon distribution
Accordingtothedefinitionofthelinearlypolarizeddipolegluondistribution,andtheabovecalculationofthecross
sectionofdijet(qγ∗)productioninDrell-YanprocessesinpAcollisions,weknowthatthelinearlypolarizedpartnerof
the dipole gluon distribution is identical to the normal dipole gluon distribution, i.e., xh(2)(x,q )=xG(2)(x,q ). In
⊥ ⊥ ⊥
general, one can write these distributions in terms of the dipole amplitude, namely, the two point function of Wilson
lines 1 Tr U(x )U†(y ) as follows
Nc ⊥ ⊥
(cid:10) (cid:0) (cid:1)(cid:11)
q2N d2y 1
xh(2)(x,q )=xG(2)(x,q )= ⊥ c d2x ⊥e−iq⊥·(x⊥−y⊥) TrU(x )U†(y ) . (35)
⊥ ⊥ ⊥ 2π2α ⊥ (2π)2 N ⊥ ⊥ Y
s Z Z c
(cid:10) (cid:11)
The small-x evolution of the dipole amplitude follows the well-known Balitsky-Kovchegov equation [17, 18] which
reads
∂ α N (x y )2
Tr U(x)U†(y) = s c d2z ⊥− ⊥
∂Y Y − 2π2 ⊥(x z )2(z y )2
Z ⊥− ⊥ ⊥− ⊥
(cid:10) (cid:2) (cid:3)(cid:11) 1
Tr U(x)U†(y) Tr U(x)U†(z) Tr U(z)U†(y) . (36)
× Y − N Y
(cid:26) c (cid:27)
(cid:10) (cid:2) (cid:3)(cid:11) (cid:10) (cid:2) (cid:3) (cid:2) (cid:3)(cid:11)
In the dilute regime, the Balitsky-Kovchegov equation reduces to the famous BFKL equation which leads to the
exponential growth in terms of the rapidity Y ln 1.
≃ x
B. The evolution of the linearly polarized Weizs¨acker-Williams gluon distribution
The operator definition of the Weizs¨acker-Williams gluon distribution can be obtained from the quadrupole corre-
lator whose initial condition can be throughly calculated in the MV model. In Refs. [22–25], the small-x evolution
equation of the quadrupole has been derived and studied analytically. Similarly to the Balitsky-Kochegov equation
for dipoles, quadrupoles follow BFKL evolution in the dilute regime and reach the saturation regime as a stable
fixed point. In addition, one expects that quadrupoles should also exhibit the same geometrical scaling behavior as
dipoles. Recently,usingtheJIMWLKrenormalizationequation[26,27],thefirstnumericalstudies[28]ofthesmall-x
evolutionofquadrupolesindeedobserveevidenceoftravelingwavesolutionsandgeometricscalingforthequadrupole.
According to [16] and Ref. [7, 8, 20], the Weizs¨acker-Williams gluon distribution can be written as
xGij (x,k )= 2 d2v d2v′ e−ik⊥·(v−v′) Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v) . (37)
WW ⊥ −α (2π)2(2π)2 Y
S Z
δij 1 qi qj (cid:10) (cid:2) (cid:3) (cid:2) (cid:3) (cid:11)
= xG(1)(x,q )+ 2 ⊥ ⊥ δij xh(1)(x,q ). (38)
2 ⊥ 2 q⊥2 − ! ⊥ ⊥
The evolution equation for the correlator Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v) can be obtained from the evolution
Y
equation of the quadrupole 1 Tr U(x )U†(x′)U(x )U†(x′) by differentiating with respect to xi and xj, and
Nc 1 (cid:10) (cid:2)1 2(cid:3) 2 (cid:2) Y (cid:3) (cid:11) 1 2
(cid:10) (cid:0) (cid:1)(cid:11)
9
then setting xi =x′i =vi and xj =x′j =v′j. Then the resulting evolution equation becomes7
1 2 2 1
∂
Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v)
∂Y Y
α(cid:10)N (cid:2) (cid:3) (v (cid:2)v′)2 (cid:3) (cid:11)
= s c d2z − Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v)
− 2π2 ⊥(v z)2(z v′)2 Y
Z − −
α N 1 (v v′)2 (cid:10) (cid:2)(v v′)(cid:3)i (v (cid:2) z)i (cid:3) (cid:11)
s c d2z − − −
− 2π2 ⊥N (v z)2(z v′)2 (v v′)2 − (v z)2
Z c − − (cid:20) − − (cid:21)
Tr U(v)U†(v′) ∂jU(v′) U†(z) Tr U(z)U†(v) Tr U(z)U†(v′) ∂jU(v′) U†(v) Tr U(v)U†(z)
× Y − Y
αs(cid:8)N(cid:10)c (cid:2)d2z 1 (cid:2)(v−v′)2(cid:3) ((cid:3)v′−(cid:2)v)j (v′−(cid:3)(cid:11)z)j (cid:10) (cid:2) (cid:2) (cid:3) (cid:3) (cid:2) (cid:3)(cid:11) (cid:9)
− 2π2 ⊥N (v z)2(z v′)2 (v′ v)2 − (v′ z)2
Z c − − (cid:20) − − (cid:21)
Tr ∂iU(v) U†(z)U(v′)U†(v) Tr U(z)U†(v′) Tr ∂iU(v) U†(v′)U(z)U†(v) Tr U(v′)U†(z)
× Y − Y
αs(cid:8)N(cid:10)c (cid:2)(cid:2)d2z 1 (cid:3) ∂i∂j (v−v′)(cid:3)2 (cid:2) (cid:3)(cid:11) (cid:10) (cid:2)(cid:2) (cid:3) (cid:3) (cid:2) (cid:3)(cid:11) (cid:9)
− 4π2 ⊥N v v′(v z)2(z v′)2
Z c (cid:20) − − (cid:21)
Tr U(v′)U†(z) Tr U(z)U†(v′) + Tr U(v)U†(z) Tr U(z)U†(v)
× Y Y
(cid:8)−(cid:10) Tr(cid:2) U(v′)U†(v(cid:3)) Tr(cid:2) U(v)U†(v′(cid:3))(cid:11) Y −(cid:10)Nc2(cid:2) . (cid:3) (cid:2) (cid:3)(cid:11) (39)
The evol(cid:10)utio(cid:2)n equation (cid:3)of th(cid:2)e Weizs¨acke(cid:3)r(cid:11)-William(cid:9)s gluon distributions can be obtained by contracting the above
correlator with δ and the one for the linearly polarized partner by contracting with 2q⊥iq⊥j δij . Although the
ij q⊥2 −
expression is quite complicated in general, the result gets simplified in the dilute regime(cid:16)as in Ref. [2(cid:17)3]. In the dilute
regime, the correlator which yields the Weizs¨acker-Williams gluon distribution can be reduced to a simple form in
terms of Γ(v,v′)
C
Tr ∂iU(v) U†(v′) ∂jU(v′) U†(v) = F∂i∂j Γ(v,v′) , (40)
Y 2 v v′ Y
where CFΓ(v,v′) is the leadi(cid:10)ng o(cid:2)rder dip(cid:3)ole amp(cid:2)litude wh(cid:3)ich sat(cid:11)isfies the BFKL equation
2
∂ N α (x x )2
Γ(x ,x ) = c s d2z 1− 2 [Γ(x ,z) +Γ(z,x ) Γ(x ,x ) ]. (41)
∂Y 1 2 Y 2π2 (x z)2(x z)2 1 Y 2 Y − 1 2 Y
Z 1− 2−
In the dilute regime where the gluon density is low, we know that the Weizs¨acker-Williams gluon distributions,
xG(1)(x,q ) and xh(1)(x,q ), as well as the dipole gluon distributions all reduce to the same leading twist result.
⊥ ⊥ ⊥
Therefore,despiteofthedistinctbehaviorinthesaturationregime,wefindthatallthesefourtypesofgluondistribu-
tions follow the BFKL equationinthe dilute regime where the gluondensity is low. The physicalconsequenceof this
results is that the linearly polarized gluon distributions should be as important as the normal gluon distributions in
the low-x region since they also receive the exponential rise in rapidity Y due to the BFKL evolution. Furthermore,
according to the discussion in Ref. [29, 30], the BFKL evolution together with a saturation boundary can give rise
to the geometricalscaling behavior [31–33] of the dipole gluon distribution. Since the quadrupole evolution equation
also contains the same property as discussed in Ref. [23, 28], the Weizs¨acker-Williams gluon distribution and its
linearly polarized partner should exhibit geometrical scaling behavior as well, although their evolution equations are
much more complicated in the saturation regime. In terms of the traveling wave picture [29, 34] for the evolution
of dipoles and quadrupoles, the velocities of the traveling waves for dipoles and quadruples are identical, since the
velocity is determined by BFKL evolution. This implies that the energy dependence of the saturation momentum
Q2 Q2(x /x)λ with Q =1GeV, x =3 10−3 and λ=0.29, should be universal for all these four different gluon
s ≃ 0 0 0 0 ×
distributions.
IV. CONCLUSION
We perform the color dipole model calculation of the cross section of DIS dijet and Drell-Yan dijet processes, and
demonstrate that the linearly polarized partners of the Weizs¨acker-Williams and dipole gluon distributions naturally
7 This evolution equation involves derivatives of the Wilson lines and complicated kernels which make it very hard to solve directly.
However,onecanextracttheevolutioninformationbynumericallysolvingtheevolutionequationforquadrupolesfirstandthenmaking
numericaldifferentiationandidentification ofcoordinates.
10
arise in these processes. This result is in complete agreementwith Ref. [16] and implies that the measurement of the
cos(2∆φ) asymmetries in these dijet processes can be a direct probe of these two different linearly polarized gluon
distributions. In addition, the small-x evolutionstudies of the linearly polarizedgluon distributions revealsthat they
also rise exponentially as function of the rapidity at high energy and they should also exhibit the geometricalscaling
behavior as the normal unpolarized gluon distributions do.
ACKNOWLEDGMENTS
We thank Prof. A. H. Mueller for helpful discussions. This work was supported in part by the U.S. Department of
Energy under the DOE OJI grant No. DE - SC0002145and contract number DE-AC02-98CH10886.
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