Probing the Small-$x$ Gluon Tomography in Correlated Hard Diffractive Dijet Production in DIS

0.15 MB·

by  Yoshitaka Hatta

YITP-16-1
Probing the Small-x Gluon Tomography in Correlated Hard Diffractive Dijet
Production in DIS
Yoshitaka Hatta,1 Bo-Wen Xiao,2 and Feng Yuan3
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
2Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China
3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
6 We investigate the close connection between the quantum phase space Wigner distribution of
1 small-xgluonsandthecolordipolescatteringamplitude,andproposetostudyitexperimentallyin
0 the hard diffractive dijet production at the planned electron-ion collider. The angular correlation
2 between the nucleon recoiled momentum and the dijet transverse momentum will probe the non-
trivial correlation in the phase space Wigner distribution. This experimental study will not only
n
provide us with three-dimensional tomographic pictures of gluons inside high energy proton, but
a
J also give a uniqueand interesting signal for thesmall-x dynamicswith QCD evolution effects.

7

Introduction. Therehavebeenstronginterestsinhadronphysicscommunity[1–3]toexplorethe partonicstructure
]
h ofthenucleon,inparticular,aimingatatomographypicturefromwhichwecanimagethepartonsinthree-dimensional
p fashion. Thiscanprovidefruitfulanddetailedinformationonthesub-atomicstructureofthebaryonicbuildingblocks
- oftheuniverse,anddeepenourunderstandingofthestronginteractionfactsinconstructingthefundamentalparticles.
p
e Amongthesetomographydistributions,theso-calledquantumphasespaceWignerdistributions[4,5]ofpartonshave
h beenreckonedasthemotherdistributionsofall,sincetheyingeniouslyencodeallquantuminformationofhowpartons
[ are distributed inside hadrons.

1 The key question now is to find experimental probes to measure these distributions. The goal of this paper is
v to pioneer this direction, by pointing out that we can have access to the gluon Wigner distributions at small-x.
5 The proposed new observables will stimulate further developments from both experiment and theory sides for the
8 planned electron-ion colliders (EIC). In general, it is believed that the parton Wigner distributions normally are
5
not directly measurable in high energy scatterings. Due to the uncertainty principle, Wigner distributions, which
1
are not positive definite, are only quasi-probabilistic. As we will demonstrate later in this Letter, one can use the
0
. diffractive dijetproduction(or morecomplicatedprocesses),whichhas been asubject ofstudy inthe small-xphysics
1
and the generalizedparton distribution approach[6–11], to directly probe the Fourier transform of the gluon Wigner
0
distribution at the EIC.
6
1 The phase space distributions [12] of quarks and gluons are often used in small-x literatures, and they are believed
: to be possibly related to Wigner distributions [4], although the exact connection was not known. We will show
v
that the gluon Wigner distributions at small-x can be simplified and written as the Fourier transform of well-known
i
X impactparameterdependent dipole amplitudes, whichhelps us to buildintimate connectionsto small-xfactorization
r framework developed in the last few decades. This will not only provide the motivation to pursue the gluon Wigner
a
distributions in the future EIC, but also prompt further studies to investigate non-trivial correlations in the small-x
dipole scattering amplitude. The latter has become one of the most important elements of the phenomenological
studies in heavy ion collisions and deep inelastic scatterings [13, 14].
One of the nontrivial phenomena is the angular correlation between the traverse momentum of the produced dijet
and the recoiledmomentum of the nucleon, which provides vital informationon the gluon Wigner distributions. It is
important to emphasize that this correlationcan help us test and measure the unique feature of angular correlations
between impact parameter and dipole size predicted by small-x evolutions.
Therestofthepaperisorganizedasfollows. WefirstintroducethegluonWignerdistributionsandtakethesmall-x
limit, whichcanbe connectedtothe dipole scatteringamplitudes. We thenexplorethe small-xdynamicsbyinvoking
the analytical solution to the BFKL equation [15] to show there exist nontrivial correlation in these gluon Wigner
distributions. Last, we apply these results to demonstrate that we will be able to observe these novel correlations in
the future EIC. Finally, we summarize our paper in the end.
Gluon Wigner Distributions at Small-x. The parton Wigner distributions are introduced to describe the quan-
tum phase space distributions of partons inside the nucleon. They unify the two common languages of transverse
momentum dependent and the generalized parton distributions in parton distributions framework.
We focus on the gluon Wigner distributions. Similar study can be done for the quark part. The gluon Wigner
2
distributions are defined through the following matrix elements,
xWT(x,~q ;~b )= dξ−d2ξ⊥ d2∆⊥e−ixP+ξ−−iq⊥·ξ⊥ P + ∆⊥ F+i ~b + ξ F+i ~b ξ P ∆⊥ ,(1)
g ⊥ ⊥ (2π)3P+ (2π)2 2 ⊥ 2 ⊥− 2 − 2
Z Z (cid:28) (cid:12) (cid:18) (cid:19) (cid:18) (cid:19)(cid:12) (cid:29)
(cid:12) (cid:12)
where Fµν represents the field strength tensor, x and q for the longit(cid:12)udinal momentum fraction and(cid:12) the transverse
⊥ (cid:12) (cid:12)
momentum for the gluon, ~b for the coordinate space variable. The above Wigner distributions not only provide
⊥
us with the momentum distribution of the corresponding parton, but also give the average location of that parton
w.r.t. the center oftargethadron. In the following,we willworkinthe small-xlimitwhere wecansimplify the above
expression and relate it to the dipole scattering amplitudes commonly used in the small-x factorization approach.
The Fourier transform of the Wigner distribution w.r.t. the impact parameter b is also referred as the generalized
⊥
transverse momentum dependent (GTMD) gluon distribution [16, 17].
In Ref. [18, 19], it has been demonstrated that TMD gluon distributions are related to small-x unintegrated gluon
distributions. TheWeizs¨acker-Williams(WW)andthedipolegluondistributionusedinsmall-xformalismcorrespond
to twogaugeinvariantbuttopologicallydifferentoperatordefinitions. Inorderto pursuedeeperconnectionsbetween
Wigner distributions and small-x impact parameter dependent gluon distributions, we first use the dipole gluon
distributionas anexample,andwe willcommentonthe caseof the WW gluondistribution inthe end. Followingthe
convention in Ref. [20], we write down the GTMD dipole gluon distribution as
dξ−d2ξ e−iq⊥·ξ⊥−ixP+ξ− ∆ ∆
xG (x,q ,∆ )=2 ⊥ P + ⊥ Tr F+i(ξ/2) [−]†F+i( ξ/2) [+] P ⊥ , (2)
DP ⊥ ⊥ (2π)3P+ 2 U − U − 2
Z (cid:28) (cid:12) h i(cid:12) (cid:29)
(cid:12) (cid:12)
where [±] are the future/past-pointing U-shaped Wilson lines(cid:12)which make the operator gauge inva(cid:12)riant. Its Fourier
U
transform d2∆⊥ei∆⊥·b⊥xG (x,q ,∆ ) can be identified as the Wigner distribution xWT(x,q ,b ). Following
(2π)2 DP ⊥ ⊥ g ⊥ ⊥
similar derivation used in Ref [19–21] in the small-x limit, one can show that Eq. (2) reduces to
R
xGDP(x,q⊥,∆⊥)= 2αNc d2R(2⊥πd)24R⊥′ eiq⊥·(R⊥−R′⊥)+i∆2⊥·(R⊥+R′⊥) ∇R⊥ ·∇R′⊥ N1 Tr U(R⊥)U†(R⊥′ ) x , (3)
s Z (cid:16) (cid:17) c (cid:10) (cid:2) (cid:3)(cid:11)
wherewecanrecognizetheimpactparameterdependentdipoleamplitude. Letus defineits doubleFouriertransform
1 r r
Tr U b + ⊥ U† b ⊥ d2q d2∆ e−iq⊥·r⊥−i∆⊥·b⊥ (q ,∆ ). (4)
⊥ ⊥ ⊥ ⊥ x ⊥ ⊥
N 2 − 2 ≡ F
c h (cid:16) (cid:17) (cid:16) (cid:17)i Z
ThenwecansuccinctlywritexG (x,q ,∆ )=(q2 ∆2/4)2Nc (q ,∆ ). Settingr =0intheaboveexpression,
DP ⊥ ⊥ ⊥− ⊥ αs Fx ⊥ ⊥ ⊥
we obtain the normalization condition for (q ,∆ ) as d2q d2∆ e−i∆⊥·b⊥ (q ,∆ )=1.
x ⊥ ⊥ ⊥ ⊥ x ⊥ ⊥
F F
Gluon Tomography Induced by Small-x Dynamics. In the small-x literature, the dipole scattering amplitudes have
R
been widely used to describe the relevant processes such as the inclusive DIS, inclusive hadron production in pA
collisions. In these calculations, the dipole amplitude is averaged over the azimuthal angle of impact parameter~b .
⊥
This is because most of the observables discussed in phenomenology so far are not sensitive to the~b -dependence.
⊥
On the other hand, there have been theoretical investigations [22–24] on the correlation between~b and the dipole
⊥
size~r , which eventually leads to nontrivialcorrelationsbetween~b and the transversemomentum~q in the gluonic
⊥ ⊥ ⊥
Wigner distributions. In the following, as a simple example, we illustrate how these correlations are generated at
small-x in the BFKL approximation. This provides an intuitive picture of small-x gluon distributions in the nucleon.
Let us considerthe dipole scattering amplitude offa dipole x (quark at~x /2,antiquarkat ~x /2)evolvedupto
⊥ ⊥ ⊥
rapidity Y =ln1/x. Define the dipole T-matrix in impact parameter space as 1 trU b + r−⊥ U† b r⊥ =
Nc ⊥ 2 ⊥− 2 x
1 T(r ,b ,Y). In the BFKL approximation, T is given by [22–24]
− ⊥ ⊥ (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11)
(1+( 1)n) ν2+ n2
T(r ,b ,Y)=2πα2 dν − 4 eχ(n,ν)Y
⊥ ⊥ s (2π)3 ν2+ n−1 2 (cid:16)ν2+ n+(cid:17)1 2
Xn Z 2 2
d2ω E1−(cid:16)h,1−h¯((cid:0)b +(cid:1)r⊥(cid:17)(cid:16) ω ,(cid:0)b (cid:1)r⊥(cid:17) ω )Eh,h¯(x⊥ ω , x⊥ ω ), (5)
⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
× 2 − − 2 − 2 − − 2 −
Z
h h¯
where Eh,h¯(a ω,b ω) = ( 1)n zab z¯ab is the BFKL holomorphic eigenfunction with h = 1−n +iν
− − − zaωzbω z¯aωz¯bω 2
and ¯h= 1+n+iν. χ(n,ν) 2αsNc(cid:16)ψ(1) (cid:17)Re(cid:16)ψ(|n|+1(cid:17)+iν) is the BFKL characteristicfunction. In the highenergy
2 ≡ π − 2
h i
3
limit, the n = 0 part of the solution gives the leading contribution. Using the saddle point approximation around
ν =0 and taking the limit x b ,r , we can cast Eq. (5) into
⊥ ⊥ ⊥
≪
α2 ρ ln 16 ln2 16
T(r ,b ,Y) s| | |ρ| exp 4α¯ Y ln2 |ρ| , (6)
⊥ ⊥ ≈ √π 7α¯ ζ(3)Y 3/2 s − 14α¯sζ(3)Y !
2 s
where (cid:0) (cid:1)
x2r2 x2r2
ρ2 ⊥ ⊥ ⊥ ⊥ . (7)
| | ≡ b + r⊥ x⊥ 2 b r⊥ + x⊥ 2 ≈ b4 + r⊥4 b2⊥r⊥2 cos2φ
⊥ 2 − 2 ⊥− 2 2 ⊥ 16 − 2 br
Clearly,oneseesthatthereisn(cid:0)ontrivialangula(cid:1)rc(cid:0)orrelationbetw(cid:1)eenb andr containedinEqs.(5)and(6). Whenb
⊥ ⊥ ⊥
isparalleltor ,thescatteringisstrongerthanthecasewhenb isperpendiculartor . Thisisaknownphenomenon,
⊥ ⊥ ⊥
seeforexampleRef.[29]. Suchacorrelationis expectedtosurvivenearthe nonlinearsaturatedregime. Indeed,away
from the BFKL saddle point, the saturation momentum Q is defined by the condition T(r = 1/Q ,b ) = const.
s ⊥ s ⊥
which leads to
1 b4⊥+ r1⊥46 − b2⊥2r⊥2 cos2φbr eχ(γγss)Y , (8)
|ρ|2 ≈ x2⊥r⊥2 (cid:12)(cid:12)r⊥=1/Qs ∼
(cid:12)
where γs = 12 +iνs =0.628. If we look for a solution in the regim(cid:12)(cid:12) e b⊥ ≫r⊥ ≃1/Qs, we find
Q2s ∼ xb42⊥eχ(γγss)Y + co2sb22φbr. (9)
⊥ ⊥
This isconsistentwiththe numericalstudyofthe nonlinearsmall-xevolution(e.g.,the Balitsky-Kovchegovevolution
[25, 26]) in Ref. [27, 28] where it was observed that the angular correlation exists even when b and r are of the
⊥ ⊥
same order. These features should be a guiding principle when building saturation models with angular correlations.
The elliptic ( cos2φ) angular correlation can be seen also in the momentum space. After averaging over the
∼
angular orientation of the target dipole x , we find that the Fourier transform of T(r ,b ,Y) w.r.t. b and r is
⊥ ⊥ ⊥ ⊥ ⊥
d2r d2b
(q ,∆ ,Y) ⊥ ⊥eiq⊥·r⊥+i∆⊥·b⊥T(x ,r ,b ,Y)
T ⊥ ⊥ ≡ (2π)4 ⊥ ⊥ ⊥
Z
α2sx⊥ e4α¯sY ln2 π/2dθJ sinθ∆⊥x⊥ K cosθ∆⊥x⊥
≃ (2π)2∆3⊥ 72α¯sζ(3)Yπ 3/2 Z0 0(cid:18) 2 (cid:19) 0(cid:18) 2 (cid:19)
1 (cid:0)dα (cid:1)3 3 ~q⊥+(1/2 α)∆~⊥ 2
F , ,1, | − | , (10)
×Z0 α2(1−α)2 2 1 2 2 − ∆2⊥α(1−α) !
inthehighenergylimit. Dependingonrelativesizeofq and∆ , (q ,∆ ,Y)couldhavesizableangularcorrelations
⊥ ⊥ ⊥ ⊥
T
with only even harmonics (it is not hard to show that all odd harmonics vanish). In the case of BFKL linear
approximation, we have xG (x,q ,∆ ) = (q2 ∆2/4)2Nc (q ,∆ ,Y) for the case with finite momentum
DP ⊥ ⊥ − ⊥ − ⊥ αs T ⊥ ⊥
transfer.
Correlated Hard Diffractive Dijet Production in DIS. Now let us discuss diffractive dijet production in electron-ion
collisions,whichhasbeen studiedquite recentlyinRef. [30], anddemonstratethatit directly probesthe dipole gluon
GTMD.Diffractiveeventsimplythatacolorneutralexchangemustoccurinthet-channelbetweenthevirtualphoton
andthetargethadronoverseveralunitsinrapidity. FollowingthesameframeworkdevelopedinRef[19],byrequiring
that the final state quark-antiquarkpair forms acolorsinglet state,we canwrite the crosssectionfor diffractivedijet
production as illustrated in Fig. 1 as follows
dy dd2σkγT∗Ad→yqq¯dX2k =2Ncαeme2qδ(xγ∗ −1)z(1−z)[z2+(1−z)2] d2q⊥d2q⊥′ Fx(q⊥,∆⊥)Fx(q⊥′ ,∆⊥)
1 1⊥ 2 2⊥ Z
P P q P P q′
⊥ ⊥− ⊥ ⊥ ⊥− ⊥ , (11)
×"P⊥2 +ǫ2f − (P⊥−q⊥)2+ǫ2f#·"P⊥2 +ǫ2f − (P⊥−q⊥′ )2+ǫ2f#
for the transversely polarized photon. A similar cross section formula can be written for the longitudinally polarized
photon. In Eq.(11), y and k arerapidities andtransversemomenta ofthe final state quark and antiquarkjets,
1,2 1,2⊥
respectively, defined in the center of mass frame of the incoming photon and nucleon. P~ 1(~k ~k ) represents
⊥ ≡ 2 2⊥− 1⊥
4
k1
k2
q ∆ q ∆
⊥− 2⊥ − ⊥− 2⊥
p p
′
FIG. 1. Diffractive dijet production in electron-ion collisions. Here we assume that the incoming virtual photon has only the
longitudinalmomentum. Thesignatureofthediffractiveprocessistherapiditygapbetweentheproduceddijet andthetarget
hadron which remains intact.
the typical dijet transverse momentum and ∆ is the nucleon recoiled momentum which equals to (k +k ) at
⊥ 1⊥ 2⊥
−
leading order. We are interested in the back-to-backkinematic regionfor the two final state jets where P ∆ .
⊥ ⊥
| |≫| |
Suppose ǫ2 z(1 z)Q2 is not too large. Then we expect that the above q integrals are dominated by the region
f ≡ − ⊥
q P and the cross sections are roughly proportional to 2(P ,∆ ) for back-to-back dijet configurations. Thus,
⊥ ∼ ⊥ Fx ⊥ ⊥
the diffractive dijet production will be sensitive to the correlationbetween P and ∆ as mentioned in Ref. [30].
⊥ ⊥
With the detector capability at the future EIC [3], we will be able to identify both P~ and ∆~ and measure the
⊥ ⊥
angular correlationbetween them. In particular,the elliptic angular correlation cos2(φ φ ) can be observed
h P⊥ − ∆⊥ i
in this process. By plugging inthe cos2φ asymmetry inthe gluonWigner distributions fromthe numericalstudies of
Balitsky-Kovchegov evolution with impact parameter dependence [27, 28], we find that it will lead to a few percent
cos2(φ φ ) asymmetries in the typicalEIC kinematics. More sophistic calculations shall follow to generalize
h P⊥ − ∆⊥ i
thesaturationmodels[11,31–33]toincorporatethisparticularangularcorrelationfeature. Weleavethatforafuture
study. Comparingthe theoreticalcomputationswiththefuture experimentaldatawillprovideusmuchmoreinsights
on the experimental signature of small-x dynamics.
Summary and Discussions. To conclude, let us make some further but brief comments on the consequence of this
work, while we will leave the detailed discussion for a future publication.
In principle, as far as the gluon Wigner distribution is concerned, there should be correlation between the two
•
vectors~q and~b , which can be shownin theoretical studies for the dipole scattering amplitude in the small-x
⊥ ⊥
region. In order to demonstrate this non-trivial correlation,we parametrize the above Wigner distribution as
xWT(x,~q ;~b )=x T(x, ~q , ~b )+2cos(2φ)x ǫ(x, ~q , ~b )+ , (12)
g ⊥ ⊥ Wg | ⊥| | ⊥| Wg | ⊥| | ⊥| ···
where φ is the azimuthal angle between ~q and~b . The first term above represents the azimuthal symmetric
⊥ ⊥
distribution, whereas the rest of other terms stand for the azimuthal asymmetric distribution. For example,
due to the cos(2φ) nature of the second term, we call it the Elliptic Gluon Wigner Distribution, and in short,
elliptic gluon distribution. This is quite similar to the elliptic flow phenomena observed in heavy ion collisions.
Let us further comment on the WW gluon distribution case. Following the same technique used above for the
•
dipole gluon Wigner distribution, we generalize the WW gluon distribution at small-x as follows
dξ−d2ξ e−iq⊥·ξ⊥−ixP+ξ− ∆ ξ ξ ∆
xG (x,q ,∆ )=2 ⊥ P + ⊥ Tr F [+]†F [+] P ⊥ ,
WW ⊥ ⊥ (2π)3P+ 2 2 U −2 U − 2
Z (cid:28) (cid:12) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)(cid:12) (cid:29)
(cid:12) (cid:12) (13)
(cid:12) (cid:12)
which allows us to find (cid:12) (cid:12)
xGWW(x,q⊥,∆⊥)= 2αNc d(22πR)⊥2 d(22πR)⊥′2 eiq⊥·(R⊥−R′⊥)+i∆2⊥·(R⊥+R′⊥)
S Z
1
Tr[i∂ U(R )]U†(R′ )[i∂ U(R′ )]U†(R ) . (14)
×N i ⊥ ⊥ i ⊥ ⊥ x
c
(cid:10) (cid:11)
Due to the known connection between the WW gluon distribution and color quadrupoles at small-x [19], it is
expectedthatoneneedstogenerateacolorquadrupoleattheamplitudelevelinordertoprobetheWWWigner
5
distribution. This requires two incoming photons at once which produce four-jet diffractive events in the final
states. It seems to be very challenging to measure this type of events at EIC. Nevertheless, it is more probable
to perform such measurement in ultra-peripheral diffractive AA collisions at the LHC where photons are much
more abundant in the wavefunction of colliding nuclei.
Itisalsointerestingtonotethatonecangeneralizetheabovederivationtoobtainthelinearlypolarizedpart[34–
•
41] of the WW and dipole gluon Wigner distribution when the indices of derivatives are off-diagonal,instead of
diagonal as in Eqs. (1,2). The cross sections for dijet and four-jet productions depend on both the unpolarized
and linearly polarized gluon distributions, which are related in the small-x formalism [37, 38].
In addition, when integrating over q in Eqs. (1,2) with off-diagonal indices, the gluon Wigner distributions
⊥
will reduce to the so-called helicity flip gluon GPDs (also called gluon transveristy), which have been exten-
sively discussed in the collinear GPD framework [42–45]. The nontrivial correlations between q and ∆ play
⊥ ⊥
important roles in the integral to obtain the helicity flip gluon GPDs.
The parton Wigner distributions, which contain the most complete information, are the cornerstones of all parton
distributions. WedemonstratethatgluonWignerdistributionsarecloselyrelatedtotheimpactparameterdependent
dipole and quadrupole scattering amplitudes, and point out that they can be measured in diffractive type events at
EIC and the LHC.
Acknowledgments: ThismaterialisbaseduponworksupportedbytheU.S.DepartmentofEnergy,OfficeofScience,
OfficeofNuclearPhysics,undercontractnumberDE-AC02-05CH11231,andbytheU.S.NationalScienceFoundation
under Grant No. PHY-0855561 and PHY-1417326. B.X. acknowledges interesting discussion with A. Mueller, and
wishes to thank Dr. X.N. Wang and the nuclear theory group at LBNL for hospitality and support during his visit
when this work is finalized.
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