BNL-NT-07/15
RBRC-665
Asymmetric Di-jet Production in Polarized Hadronic Collisions
Jian-Wei Qiu,1, 2 Werner Vogelsang,2 and Feng Yuan3
1Department of Physics and Astronomy,
Iowa State University, Ames, IA 50011
2Physics Department, Brookhaven National Laboratory, Upton, NY 11973
3RIKEN BNL Research Center, Building 510A,
Brookhaven National Laboratory, Upton, NY 11973
(Dated: October 25, 2018)
Abstract
Using the collinear QCD factorization approach, we study the single-transverse-spin dependent
cross section ∆σ(S⊥) for the hadronic production of two jets of momenta P1 = P + q/2 and P2 =
−P + q/2. We consider the kinematic region where the transverse components of the momentum
vectors satisfy P⊥ ≫ q⊥ ≫ ΛQCD. For the case of initial-state gluon radiation, we show that at the
leading power in q⊥/P⊥ and at the lowest non-trivial perturbative order, the dependence of ∆σ(S⊥)
on q⊥ decouples from that on P⊥, so that the cross section can be factorized into a hard part that
is a function only of the single scale P⊥, and into perturbatively generated transverse-momentum
dependent (TMD) parton distributions with transverse momenta k⊥ = O(q⊥).
PACS numbers: 12.38.Bx, 13.88.+e, 12.39.St
Keywords: QCD factorization, single transverse-spin asymmetry, di-jet correlation
http://arxiv.org/abs/0704.1153v1
1. Introduction. Single-transverse-spin asymmetries (SSAs) in high-energy hadronic
reactions with one transversely polarized hadron were first observed more than three decades
ago [1]. The SSA is defined as AN ≡ (σ(S⊥)− σ(−S⊥))/(σ(S⊥) + σ(−S⊥)), the ratio of the
difference and the sum of (differential) cross sections when the hadron’s spin vector, S⊥, is
flipped. Recent experimental measurements of SSAs both in polarized hadronic collisions
[2, 3] and in semi-inclusive lepton-nucleon deep inelastic scattering (SIDIS) [4] have renewed
the interest in investigating the origin of SSAs in Quantum Chromodynamics (QCD) [5].
It is believed that some SSAs are a consequence of the partons’ transverse motion inside
the polarized hadron. The momentum scale of this transverse motion is a typical hadronic
scale, 〈k⊥〉 ∼ 1/fm ∼ ΛQCD. For observables with only one hard scale Q≫ ΛQCD, the SSA
should be proportional to 〈k⊥〉/Q [6, 7]. Such observables only probe an averaged effect of
the partons’ transverse motion. However, for observables characterized by more than one
physical scale, SSAs may directly probe the partons’ transverse motion. For example, in
the case of Drell-Yan hadronic production of a lepton pair of large invariant mass Q and
transverse momentum q⊥ ≪ Q, the pair probes the (anti-) quark’s transverse motion at the
scale q⊥, while the invariant mass Q of the pair sets the hard scale of the collision [8]. In
this letter, we study the SSA in hadronic production of two jets: A(PA, S⊥) + B(PB) →
J1(P1) + J2(P2) +X , with the jet momenta P1 ≡ P + q/2 and P2 ≡ −P + q/2 [9, 10, 11].
Unlike in the Drell-Yan process or in SIDIS, the SSA in di-jet production can be generated
by both initial- and final-state interactions. We emphasize that measurements of the SSA
for di-jet production have begun at RHIC [12], complementing the measurements in SIDIS.
We are interested here in deriving a QCD formalism for the single transverse-spin de-
pendent cross section, ∆σ(S⊥) = (σ(S⊥)− σ(−S⊥))/2, that is valid in the kinematic region
P⊥ ≫ q⊥ & ΛQCD, where P⊥ and q⊥ are the transverse components of the momenta P and q,
respectively, so that the SSA provides direct information on the partons’ transverse motion.
We first consider the region P⊥ ≫ q⊥ ≫ ΛQCD, where both observed momentum scales are
much larger than the typical hadronic scale ΛQCD. We calculate ∆σ(S⊥) in terms of the
collinear QCD factorization approach, which is expected to be valid in this region [13]. In
this approach, the incoming partons are approximated to be collinear to the corresponding
initial hadrons, and the leading-order partonic processes produce two back-to-back jets with
zero momentum imbalance. The di-jet momentum imbalance, ~q⊥ = ~P1⊥ + ~P2⊥, has to be
perturbatively generated by radiating an additional hard parton.
PA, S⊥
PA, S⊥
PA, S⊥
kg kg kg
(a) (b) (c)
FIG. 1: Sample diagrams for quark-quark scattering contributing to ∆σ(S⊥) through an initial-state
interaction (a), and through final-state interactions with jet P1 (b) and jet P2 (c).
In this letter, we concentrate on the physics issues related to di-jet production, and re-
strict ourselves to the case that the leading contribution in the expansion of the partonic
scattering in q⊥/P⊥ involves a hard qq
′ → qq′ subprocess. We consider the jet imbalance
q⊥ to be generated by gluon radiation off the initial quarks. We derive the corresponding
leading-order contribution to ∆σ(S⊥), and demonstrate that the perturbatively calculated
partonic parts can be further factorized into a single-scale (P⊥) hard part and perturbatively
generated transverse-momentum dependent (TMD) parton distributions with transverse mo-
menta k⊥ = O(q⊥). We find that the complete contributions from all other partonic sub-
processes at the leading order have the same factorization property. These will be discussed
in a forthcoming publication [14]. The factorization of the physics at scale P⊥ from that at
scale q⊥ that we find when q⊥ ≪ P⊥, is consistent with a more general TMD factorization
formula for the SSA in the di-jet momentum imbalance.
2. Single transverse-spin dependent cross section. When both P⊥ and q⊥ are much
larger than ΛQCD, a nonvanishing single transverse-spin dependent cross section ∆σ(S⊥) is
generated by the Efremov-Teryaev-Qiu-Sterman (ETQS) mechanism [6, 7] in the collinear
factorization approach. The calculation of ∆σ(S⊥) then requires to evaluate partonic pro-
cesses with 3-parton initial- and final-states [13]. In Fig. 1 we show generic diagrams for
the quark-quark scattering channel that contribute to ∆σ(S⊥) through initial- and final-
state interactions with the gluon of momentum kg, which is needed for generating the phase
required for a nonvanishing SSA [6, 7]. Radiation of a hard gluon of momentum k′ into
the final state generates the jet imbalance q⊥. The blob in the center represents tree-level
Feynman diagrams with the given initial- and final-state partons. In the ETQS formalism,
the contribution of the subprocess (g)qq′ → qq′g to ∆σ(S⊥), shown in Fig. 1, is generically
given by
d∆σ(S⊥)(qq′)
dy1dy2dP
dx1dx2 TF (x1, x2) q
′(x′)
16s(2π)4
δ((k′)2)H(g)qq′→qq′g , (1)
where y1 and y2 are the rapidities of the two jets, s = (PA + PB)
2, and H represents a
partonic hard part. q′(x′) is the usual quark distribution at momentum fraction x′ in the
incoming hadron B. x1 and x2 are the momentum fractions of the quarks from the polarized
hadron A on the two sides of the cut shown in Fig. 1, and TF (x1, x2) is the corresponding
twist-three quark-gluon correlation function, extracted from the lower blob in the figure
[7, 15]:
TF (x1, x2) ≡
dζ−dη−
ei(x1P
η−+(x2−x1)P+A ζ
× ǫβα⊥ S⊥β
PA, S|ψ(0)L(0, ζ−)γ+ (2)
× gFα+(ζ−)L(ζ−, η−)ψ(η−)|PA, S
where L is the proper gauge link to make the matrix element gauge invariant, and where
the sums over color and spin indices are implicit. In Eq. (1) and the rest of this paper, the
dependence on factorization and renormalization scales is suppressed.
Equation (1) applies when q⊥ ∼ P⊥. Our goal is now to investigate the leading structure
that emerges from Eq. (1) when q⊥ ≪ P⊥. In this limit the gluon of momentum k′ is
radiated either nearly collinearly from one of the external quark legs and/or is soft. In the
present work, we only discuss collinear emission by one of the initial quarks. This radiation
is the most interesting from the point of view of studying the factorization properties of
the cross section at small q⊥, because TMD factorization can only hold if the initial-state
collinear radiation leads to a certain specific structure, as we shall discuss below. Since
we are considering the production of jets (as opposed to that of two specific hadrons),
collinear radiation from final-state quarks becomes part of the jet and will not produce
leading behavior in q⊥/P⊥. On the other hand, large-angle soft gluons produced by the
interference of initial- and final state radiation may give leading contributions, through a
so-called soft factor. We leave the detailed study of the soft factor to future work, but will
briefly return to it later.
As we mentioned above, the strong interaction phase necessary for a nonvanishing ∆σ(S⊥)
arises from the interference between the imaginary part of the partonic scattering amplitude
(4) (5) (6)
(1) (2) (3)
FIG. 2: Sample diagrams for the soft-pole (1-3) and hard-pole (4-6) contributions. We have indi-
cated the momentum k′ of the radiated gluon that produces the jet imbalance q⊥, and the momentum
kg of the additional gluon from the polarized proton. The pole of each diagram is taken from the
propagator with a short bar. Diagram (1) is an example of an initial-state interaction, and (2) and
(3) show final-state interactions. Diagrams (4-6) show the complete set of diagrams for one hard
pole, indicated again by the short bar.
with the extra polarized gluon of momentum kg = xgPA and the real scattering amplitude
without the gluon in Fig. 1. The imaginary part comes from taking the pole of the parton
propagator associated with the integration over the gluon momentum fraction xg = x2−x1.
For a process with two physical scales, P⊥ and q⊥, tree scattering diagrams in Fig. 1 have two
types of poles, corresponding to xg = 0 (“soft-pole”) [7] and xg 6= 0 (“hard-pole”) [8]. With
the extra initial-state gluon attachment of momentum k′, there are many more diagrams
that contribute to ∆σ(S⊥) in comparison to the spin-averaged cross section [14]. In Fig. 2,
we show some sample diagrams that give the leading soft-pole (1-3) and hard pole (4-6)
contributions to ∆σ(S⊥) at q⊥ ≪ P⊥. By using the power counting technique [16], we are
able to classify all Feynman diagrams into different groups [14]. For example, diagrams (1-3)
only make leading contributions when the momentum k′ of the radiated gluon is parallel to
PA, whereas diagrams (4-6) can give a leading contribution when k
′ is either parallel to PA or
to PB. Beyond that, the calculation of the soft-pole and hard-pole contributions to ∆σ(S⊥)
follows the same procedure as introduced in Ref. [8] for the SSA in Drell-Yan and SIDIS,
because the kinematic limit considered is similar. In order to extract the contributions to
∆σ(S⊥) in Eq. (1), we need to convert the extra gluon field operator in the hadronic matrix
element of the polarized hadron in Fig. 1 to a field strength operator in the definition of
TF (x1, x2) in Eq. (2). Working in Feynman gauge, we first give the initial-state collinear
partons from the polarized hadron a small transverse momentum, ki = xiPA + ki⊥ with
i = 1, 2, and then expand the calculated partonic scattering amplitudes around ki⊥ = 0, or
equivalently, kg⊥ = k2⊥ − k1⊥ = 0. The contribution to ∆σ(S⊥) arises from terms linear in
kg⊥. After summing up all contributions [14], we obtain the total leading-power contribution
to ∆σ(S⊥) from the (g)qq
′ → qq′g partonic subprocess in the q⊥/P⊥ expansion:
d∆σ(S⊥)(qq′)
dy1dy2dP
P⊥≫q⊥≫ΛQCD
ǫαβSα⊥q
(~q2⊥)
HSiversqq′→qq′
q′(xb)
A+ TF (xa, xa)
, (3)
where
TF (x, x)
1 + ξ2
+TF (x, x)
2ξ3 − 3ξ2 − 1
+TF (x, ξx)
1 + ξ
B = q′(x′)CF
1 + ξ′2
1− ξ′
, (5)
where ξ = xa/x and ξ
′ = xb/x
′ with xa =
(ey1 + ey2) and xb =
(e−y1 + e−y2). The
above relations are regularized at the integration limits by “plus”-distributions [8]. The
single-scale partonic hard part is given by
HSiversqq′→qq′(ŝ, t̂, û) =
N2c − 5
2(ŝ2 + û2)
, (6)
where the use of the superscript “Sivers” will become clear in the next section, and
where the partonic Mandelstam variables are given as ŝ = xaxbs, t̂ = −P 2⊥(ey2−y1 + 1),
and û = −P 2⊥(ey1−y2 + 1). We note that to the leading power in q⊥/P⊥, we have
P1 = P⊥ (e
2, e−y1/
2, 1) and P2 = P⊥ (e
2, e−y2/
2,−1) for the jet momenta in
light-cone coordinates.
The hard part for the partonic channel we have considered, HSiversqq′→qq′, is very similar to the
spin-averaged partonic differential cross section dσ̂/dt̂ [14]. The only difference is the color
factor in square brackets. In fact, the color factor for HSiversqq′→qq′ is equal to a sum of three
color factors: CI + CF1 + CF2, corresponding to color factors of scattering amplitudes when
the initial-state gluon of momentum kg is attached to the initial-state incoming quark of
momentum kb, the final-state quark with P1, or the final-state quark with P2, respectively.
For the (g)qq′ → qq′g subprocess, we have [14] CI = − 12N2
, CF1 = − 14N2
, and CF2 =
N2c−2
As a result, the final-state interaction with the jet of momentum P2 dominates, and the
overall color factor CI + CF1 + CF2 has a sign opposite to that of CI alone.
3. Factorization in terms of TMD distributions. When q⊥ ≪ P⊥, the di-jet
production and Drell-Yan process at low transverse momentum (qT ≪ Q) considered in
Ref. [8] share very similar kinematics. The difference is that the Drell-Yan process has only
initial-state interactions while for di-jet production both initial- and final-state interactions
are present. It is known [17, 18] that when q⊥ ≪ Q, the Drell-Yan cross section in leading
order in q⊥/Q can be calculated from a generalized QCD factorization formula involving the
TMD parton distributions. It is natural to ask if the generalized QCD factorization formula
can be extended to the di-jet cross section for q⊥ ≪ P⊥.
When q⊥ ≫ ΛQCD, TMD parton distributions can be calculated in perturbative QCD
(pQCD) from hard radiation and parton splitting [8]. The unpolarized TMD quark distri-
bution is well known and given by [8]
q(xb, q⊥) =
[B + · · · ] , (7)
where B is given in Eq. (5), and where the ellipses denote terms proportional to δ(1−ξ) and
terms generated by gluon splitting. The transverse-spin dependent TMD quark distribution,
qT (xa, q⊥), known as the Sivers function [19], can also be calculated in pQCD and is given
in terms of the twist-3 quark-gluon correlation function TF as [8]:
qSIDIST (xa, q⊥) = −
(~q2⊥)
[A+ . . . ] , (8)
where A is given in Eq. (4), MP is a hadron mass scale introduced to keep q(xb, q⊥) and
qT (xa, q⊥) at the same dimension, and where the ellipses as above denote terms proportional
to δ(1 − ξ) and terms not relevant to the following discussion. In Eq. (8), the superscript
“SIDIS” indicates the Sivers function for the SIDIS process, which has an opposite sign from
that for the Drell-Yan process given in Ref. [8], due to the difference in the directions of the
gauge link that defines the TMD quark distributions [20, 21, 22, 23].
One of the important features of the di-jet cross section calculated above to leading order
in q⊥/P⊥ is the separation of two observed physical scales, P⊥ and q⊥. We can use Eqs. (7)
and (8) to rewrite the single transverse-spin dependent di-jet cross section in Eq. (3) in terms
of a single-scale hard factor, HSivers, which is a function of P⊥, and of the q⊥-dependent
perturbatively generated TMD parton distributions:
d∆σ(S⊥)(qq′)
dy1dy2dP
P⊥≫q⊥≫ΛQCD
= ǫαβSα⊥q
Sivers
qq′→qq′
′(xb))
SIDIS
T (xa, q⊥)
(xb q
′(xb, q⊥)) (xa TF (xa, xa))
Using the leading order relation [23]
d2k⊥ ~k
SIDIS
T (x, k⊥) = −TF (x, x) , (10)
we find that our result in Eq. (9), when the contributions from all other partonic subprocesses
at the same order [14] are added, is consistent with the leading-order term of a more general
factorization formula in terms of TMD parton distributions:
d∆σ(S⊥)
dy1dy2dP
ǫαβSα⊥q
d2k1⊥d
2k2⊥d
~k1⊥ · ~q⊥
SIDIS
Ta (xa, k1⊥) xb f
SIDIS
b (xb, k2⊥) (11)
Sab→cd(λ⊥)H
Sivers
ab→cd(P
δ(2)(~k1⊥ + ~k2⊥ + ~λ⊥ − ~q⊥) ,
where apart from the functions already given, fSIDISb denotes the unpolarized TMD quark
distribution and Sab→cd is the soft factor mentioned above [8, 14]. Because of the color
flow into the jets, the product of the soft and hard factors will involve a sum over separate
color amplitudes in the full factorization formalism [24, 25], which has been represented by
a trace [ ]c in color space in the above equation. Our calculation of initial-state collinear
gluon radiation described above would not be sensitive to this complexity of the color flow,
but we emphasize that the definition of the parton distributions cannot be affected by it
[14, 25].
In Eq. (11), we have chosen TMD parton distributions defined in SIDIS because of the
dominance of final-state interactions. Choosing TMD parton distributions defined according
to the Drell-Yan process would change the sign of the partonic hard factors, but not affect
the overall sign of the physical cross section. Based on our explicit calculation here and the
generalized factorization property of the Drell-Yan process at low q⊥ [17, 18], and because of
the similarity in kinematics between two processes, we expect the generalized factorization
formula in Eq. (11) to be valid for describing the single-transverse-spin dependent cross
section for the di-jet momentum imbalance in hadronic collisions in the kinematic region
where P⊥ ≫ q⊥ & ΛQCD.
A key feature of the factorization is that the perturbatively calculated short-distance hard
factors should not be sensitive to details of the factorized long distance physics. We tried, as
a test, to derive all short-distance hard factors by using this factorization formula and the
Brodsky-Hwang-Schmidt model for SSAs [20]. We were able to recover all hard-scattering
factors in this way. We note that the same hard factors HSiversab→cd as above were also found for
the weighted (integrated) SSA for di-jet production in [10]. A detailed comparison between
the approach of [10] and ours will be presented in Ref. [14]. An all-order proof of the above
factorization formula, if correct, is still needed and is beyond the scope of this paper.
4. Summary. We have studied the single-transverse-spin dependent cross section
∆σ(S⊥) for di-jet production momentum imbalance in high-energy hadronic collisions in
a kinematic region where P⊥ ≫ q⊥ ≫ ΛQCD, and calculated contributions from both initial-
and final-state interactions. At the leading order in q⊥/P⊥, the q⊥ and P⊥ dependences in
our calculated results are decoupled and can be factorized into a single-scale hard factor that
depends on P⊥, and into perturbatively generated TMD parton distributions. This factor-
ization occurs for each partonic channel. Overall, final-state interactions turn out to give the
dominant contribution to ∆σ(S⊥) [14]. We therefore expect the SSA in di-jet production to
have the same sign as the Sivers asymmetry in SIDIS.
We have found that our results are consistent with a more general TMD factorization
formula, given in Eq. (11), which we propose to be the correct approach to describing the
single-transverse spin asymmetry in di-jet production at hadron colliders when P⊥ ≫ q⊥.
We emphasize that obviously the result of our first-order calculation is not able to actually
prove this factorization, but should rather be regarded as a “necessary condition” for such
a factorization to hold. A full proof remains an important challenge for future work. Also,
we recall that we have limited ourselves to the case of collinear initial-state gluon radiation.
The effects of large-angle soft-gluon emission have been neglected, even though we have
indicated their likely role in Eq. (11). A proof of TMD factorization in this process would
naturally incorporate a study of this soft factor.
If the proposed factorization formula is valid, the di-jet momentum imbalance at RHIC
could be described by the same TMD parton distributions as those used to describe the SIDIS
and Drell-Yan processes. Because both initial- and final-state interactions are present, the
di-jet momentum imbalance is sensitive to different short distance dynamics, and it will
be an excellent process to test QCD factorization and the universality of the TMD parton
distributions. In addition it should give valuable information on the partons’ transverse
motion in the nucleon.
Acknowledgments
We thank C.J. Bomhof, S.J. Brodsky, J.C. Collins, X. Ji, A. Metz, P.J. Mulders, and G.
Sterman for useful discussions. We are grateful to RIKEN, Brookhaven National Labora-
tory and the U.S. Department of Energy (grant number DE-FG02-87ER40371 and contract
number DE-AC02-98CH10886) for providing the facilities essential for the completion of
this work. J.Q. thanks high energy theory group at Argonne National Laboratory for its
hospitality during the writing of this work.
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http://arxiv.org/abs/hep-ph/0701277
http://arxiv.org/abs/hep-ex/0612036
	Acknowledgments
	References

  Using the collinear QCD factorization approach, we study the
single-transverse-spin dependent cross section Delta sigma(S_perp) for the
hadronic production of two jets of momenta P_1=P+q/2 and P_2=-P+q/2. We
consider the kinematic region where the transverse components of the momentum
vectors satisfy P_perp >> q_perp >> Lambda_QCD. For the case of initial-state
gluon radiation, we show that at the leading power in q_perp/P_perp and at the
lowest non-trivial perturbative order, the dependence of Delta sigma(S_perp) on
q_perp decouples from that on P_perp, so that the cross section can be
factorized into a hard part that is a function only of the single scale P_perp,
and into perturbatively generated transverse-momentum dependent (TMD) parton
distributions with transverse momenta k_perp = O(q_perp).

Introduction. Single-transverse-spin asymmetries (SSAs) in high-energy hadronic
reactions with one transversely polarized hadron were first observed more than three decades
ago [1]. The SSA is defined as AN ≡ (σ(S⊥)− σ(−S⊥))/(σ(S⊥) + σ(−S⊥)), the ratio of the
difference and the sum of (differential) cross sections when the hadron’s spin vector, S⊥, is
flipped. Recent experimental measurements of SSAs both in polarized hadronic collisions
[2, 3] and in semi-inclusive lepton-nucleon deep inelastic scattering (SIDIS) [4] have renewed
the interest in investigating the origin of SSAs in Quantum Chromodynamics (QCD) [5].
It is believed that some SSAs are a consequence of the partons’ transverse motion inside
the polarized hadron. The momentum scale of this transverse motion is a typical hadronic
scale, 〈k⊥〉 ∼ 1/fm ∼ ΛQCD. For observables with only one hard scale Q≫ ΛQCD, the SSA
should be proportional to 〈k⊥〉/Q [6, 7]. Such observables only probe an averaged effect of
the partons’ transverse motion. However, for observables characterized by more than one
physical scale, SSAs may directly probe the partons’ transverse motion. For example, in
the case of Drell-Yan hadronic production of a lepton pair of large invariant mass Q and
transverse momentum q⊥ ≪ Q, the pair probes the (anti-) quark’s transverse motion at the
scale q⊥, while the invariant mass Q of the pair sets the hard scale of the collision [8]. In
this letter, we study the SSA in hadronic production of two jets: A(PA, S⊥) + B(PB) →
J1(P1) + J2(P2) +X , with the jet momenta P1 ≡ P + q/2 and P2 ≡ −P + q/2 [9, 10, 11].
Unlike in the Drell-Yan process or in SIDIS, the SSA in di-jet production can be generated
by both initial- and final-state interactions. We emphasize that measurements of the SSA
for di-jet production have begun at RHIC [12], complementing the measurements in SIDIS.
We are interested here in deriving a QCD formalism for the single transverse-spin de-
pendent cross section, ∆σ(S⊥) = (σ(S⊥)− σ(−S⊥))/2, that is valid in the kinematic region
P⊥ ≫ q⊥ & ΛQCD, where P⊥ and q⊥ are the transverse components of the momenta P and q,
respectively, so that the SSA provides direct information on the partons’ transverse motion.
We first consider the region P⊥ ≫ q⊥ ≫ ΛQCD, where both observed momentum scales are
much larger than the typical hadronic scale ΛQCD. We calculate ∆σ(S⊥) in terms of the
collinear QCD factorization approach, which is expected to be valid in this region [13]. In
this approach, the incoming partons are approximated to be collinear to the corresponding
initial hadrons, and the leading-order partonic processes produce two back-to-back jets with
zero momentum imbalance. The di-jet momentum imbalance, ~q⊥ = ~P1⊥ + ~P2⊥, has to be
perturbatively generated by radiating an additional hard parton.
PA, S⊥
PA, S⊥
PA, S⊥
kg kg kg
(a) (b) (c)
FIG. 1: Sample diagrams for quark-quark scattering contributing to ∆σ(S⊥) through an initial-state
interaction (a), and through final-state interactions with jet P1 (b) and jet P2 (c).
In this letter, we concentrate on the physics issues related to di-jet production, and re-
strict ourselves to the case that the leading contribution in the expansion of the partonic
scattering in q⊥/P⊥ involves a hard qq
′ → qq′ subprocess. We consider the jet imbalance
q⊥ to be generated by gluon radiation off the initial quarks. We derive the corresponding
leading-order contribution to ∆σ(S⊥), and demonstrate that the perturbatively calculated
partonic parts can be further factorized into a single-scale (P⊥) hard part and perturbatively
generated transverse-momentum dependent (TMD) parton distributions with transverse mo-
menta k⊥ = O(q⊥). We find that the complete contributions from all other partonic sub-
processes at the leading order have the same factorization property. These will be discussed
in a forthcoming publication [14]. The factorization of the physics at scale P⊥ from that at
scale q⊥ that we find when q⊥ ≪ P⊥, is consistent with a more general TMD factorization
formula for the SSA in the di-jet momentum imbalance.
2. Single transverse-spin dependent cross section. When both P⊥ and q⊥ are much
larger than ΛQCD, a nonvanishing single transverse-spin dependent cross section ∆σ(S⊥) is
generated by the Efremov-Teryaev-Qiu-Sterman (ETQS) mechanism [6, 7] in the collinear
factorization approach. The calculation of ∆σ(S⊥) then requires to evaluate partonic pro-
cesses with 3-parton initial- and final-states [13]. In Fig. 1 we show generic diagrams for
the quark-quark scattering channel that contribute to ∆σ(S⊥) through initial- and final-
state interactions with the gluon of momentum kg, which is needed for generating the phase
required for a nonvanishing SSA [6, 7]. Radiation of a hard gluon of momentum k′ into
the final state generates the jet imbalance q⊥. The blob in the center represents tree-level
Feynman diagrams with the given initial- and final-state partons. In the ETQS formalism,
the contribution of the subprocess (g)qq′ → qq′g to ∆σ(S⊥), shown in Fig. 1, is generically
given by
d∆σ(S⊥)(qq′)
dy1dy2dP
dx1dx2 TF (x1, x2) q
′(x′)
16s(2π)4
δ((k′)2)H(g)qq′→qq′g , (1)
where y1 and y2 are the rapidities of the two jets, s = (PA + PB)
2, and H represents a
partonic hard part. q′(x′) is the usual quark distribution at momentum fraction x′ in the
incoming hadron B. x1 and x2 are the momentum fractions of the quarks from the polarized
hadron A on the two sides of the cut shown in Fig. 1, and TF (x1, x2) is the corresponding
twist-three quark-gluon correlation function, extracted from the lower blob in the figure
[7, 15]:
TF (x1, x2) ≡
dζ−dη−
ei(x1P
η−+(x2−x1)P+A ζ
× ǫβα⊥ S⊥β
PA, S|ψ(0)L(0, ζ−)γ+ (2)
× gFα+(ζ−)L(ζ−, η−)ψ(η−)|PA, S
where L is the proper gauge link to make the matrix element gauge invariant, and where
the sums over color and spin indices are implicit. In Eq. (1) and the rest of this paper, the
dependence on factorization and renormalization scales is suppressed.
Equation (1) applies when q⊥ ∼ P⊥. Our goal is now to investigate the leading structure
that emerges from Eq. (1) when q⊥ ≪ P⊥. In this limit the gluon of momentum k′ is
radiated either nearly collinearly from one of the external quark legs and/or is soft. In the
present work, we only discuss collinear emission by one of the initial quarks. This radiation
is the most interesting from the point of view of studying the factorization properties of
the cross section at small q⊥, because TMD factorization can only hold if the initial-state
collinear radiation leads to a certain specific structure, as we shall discuss below. Since
we are considering the production of jets (as opposed to that of two specific hadrons),
collinear radiation from final-state quarks becomes part of the jet and will not produce
leading behavior in q⊥/P⊥. On the other hand, large-angle soft gluons produced by the
interference of initial- and final state radiation may give leading contributions, through a
so-called soft factor. We leave the detailed study of the soft factor to future work, but will
briefly return to it later.
As we mentioned above, the strong interaction phase necessary for a nonvanishing ∆σ(S⊥)
arises from the interference between the imaginary part of the partonic scattering amplitude
(4) (5) (6)
(1) (2) (3)
FIG. 2: Sample diagrams for the soft-pole (1-3) and hard-pole (4-6) contributions. We have indi-
cated the momentum k′ of the radiated gluon that produces the jet imbalance q⊥, and the momentum
kg of the additional gluon from the polarized proton. The pole of each diagram is taken from the
propagator with a short bar. Diagram (1) is an example of an initial-state interaction, and (2) and
(3) show final-state interactions. Diagrams (4-6) show the complete set of diagrams for one hard
pole, indicated again by the short bar.
with the extra polarized gluon of momentum kg = xgPA and the real scattering amplitude
without the gluon in Fig. 1. The imaginary part comes from taking the pole of the parton
propagator associated with the integration over the gluon momentum fraction xg = x2−x1.
For a process with two physical scales, P⊥ and q⊥, tree scattering diagrams in Fig. 1 have two
types of poles, corresponding to xg = 0 (“soft-pole”) [7] and xg 6= 0 (“hard-pole”) [8]. With
the extra initial-state gluon attachment of momentum k′, there are many more diagrams
that contribute to ∆σ(S⊥) in comparison to the spin-averaged cross section [14]. In Fig. 2,
we show some sample diagrams that give the leading soft-pole (1-3) and hard pole (4-6)
contributions to ∆σ(S⊥) at q⊥ ≪ P⊥. By using the power counting technique [16], we are
able to classify all Feynman diagrams into different groups [14]. For example, diagrams (1-3)
only make leading contributions when the momentum k′ of the radiated gluon is parallel to
PA, whereas diagrams (4-6) can give a leading contribution when k
′ is either parallel to PA or
to PB. Beyond that, the calculation of the soft-pole and hard-pole contributions to ∆σ(S⊥)
follows the same procedure as introduced in Ref. [8] for the SSA in Drell-Yan and SIDIS,
because the kinematic limit considered is similar. In order to extract the contributions to
∆σ(S⊥) in Eq. (1), we need to convert the extra gluon field operator in the hadronic matrix
element of the polarized hadron in Fig. 1 to a field strength operator in the definition of
TF (x1, x2) in Eq. (2). Working in Feynman gauge, we first give the initial-state collinear
partons from the polarized hadron a small transverse momentum, ki = xiPA + ki⊥ with
i = 1, 2, and then expand the calculated partonic scattering amplitudes around ki⊥ = 0, or
equivalently, kg⊥ = k2⊥ − k1⊥ = 0. The contribution to ∆σ(S⊥) arises from terms linear in
kg⊥. After summing up all contributions [14], we obtain the total leading-power contribution
to ∆σ(S⊥) from the (g)qq
′ → qq′g partonic subprocess in the q⊥/P⊥ expansion:
d∆σ(S⊥)(qq′)
dy1dy2dP
P⊥≫q⊥≫ΛQCD
ǫαβSα⊥q
(~q2⊥)
HSiversqq′→qq′
q′(xb)
A+ TF (xa, xa)
, (3)
where
TF (x, x)
1 + ξ2
+TF (x, x)
2ξ3 − 3ξ2 − 1
+TF (x, ξx)
1 + ξ
B = q′(x′)CF
1 + ξ′2
1− ξ′
, (5)
where ξ = xa/x and ξ
′ = xb/x
′ with xa =
(ey1 + ey2) and xb =
(e−y1 + e−y2). The
above relations are regularized at the integration limits by “plus”-distributions [8]. The
single-scale partonic hard part is given by
HSiversqq′→qq′(ŝ, t̂, û) =
N2c − 5
2(ŝ2 + û2)
, (6)
where the use of the superscript “Sivers” will become clear in the next section, and
where the partonic Mandelstam variables are given as ŝ = xaxbs, t̂ = −P 2⊥(ey2−y1 + 1),
and û = −P 2⊥(ey1−y2 + 1). We note that to the leading power in q⊥/P⊥, we have
P1 = P⊥ (e
2, e−y1/
2, 1) and P2 = P⊥ (e
2, e−y2/
2,−1) for the jet momenta in
light-cone coordinates.
The hard part for the partonic channel we have considered, HSiversqq′→qq′, is very similar to the
spin-averaged partonic differential cross section dσ̂/dt̂ [14]. The only difference is the color
factor in square brackets. In fact, the color factor for HSiversqq′→qq′ is equal to a sum of three
color factors: CI + CF1 + CF2, corresponding to color factors of scattering amplitudes when
the initial-state gluon of momentum kg is attached to the initial-state incoming quark of
momentum kb, the final-state quark with P1, or the final-state quark with P2, respectively.
For the (g)qq′ → qq′g subprocess, we have [14] CI = − 12N2
, CF1 = − 14N2
, and CF2 =
N2c−2
As a result, the final-state interaction with the jet of momentum P2 dominates, and the
overall color factor CI + CF1 + CF2 has a sign opposite to that of CI alone.
3. Factorization in terms of TMD distributions. When q⊥ ≪ P⊥, the di-jet
production and Drell-Yan process at low transverse momentum (qT ≪ Q) considered in
Ref. [8] share very similar kinematics. The difference is that the Drell-Yan process has only
initial-state interactions while for di-jet production both initial- and final-state interactions
are present. It is known [17, 18] that when q⊥ ≪ Q, the Drell-Yan cross section in leading
order in q⊥/Q can be calculated from a generalized QCD factorization formula involving the
TMD parton distributions. It is natural to ask if the generalized QCD factorization formula
can be extended to the di-jet cross section for q⊥ ≪ P⊥.
When q⊥ ≫ ΛQCD, TMD parton distributions can be calculated in perturbative QCD
(pQCD) from hard radiation and parton splitting [8]. The unpolarized TMD quark distri-
bution is well known and given by [8]
q(xb, q⊥) =
[B + · · · ] , (7)
where B is given in Eq. (5), and where the ellipses denote terms proportional to δ(1−ξ) and
terms generated by gluon splitting. The transverse-spin dependent TMD quark distribution,
qT (xa, q⊥), known as the Sivers function [19], can also be calculated in pQCD and is given
in terms of the twist-3 quark-gluon correlation function TF as [8]:
qSIDIST (xa, q⊥) = −
(~q2⊥)
[A+ . . . ] , (8)
where A is given in Eq. (4), MP is a hadron mass scale introduced to keep q(xb, q⊥) and
qT (xa, q⊥) at the same dimension, and where the ellipses as above denote terms proportional
to δ(1 − ξ) and terms not relevant to the following discussion. In Eq. (8), the superscript
“SIDIS” indicates the Sivers function for the SIDIS process, which has an opposite sign from
that for the Drell-Yan process given in Ref. [8], due to the difference in the directions of the
gauge link that defines the TMD quark distributions [20, 21, 22, 23].
One of the important features of the di-jet cross section calculated above to leading order
in q⊥/P⊥ is the separation of two observed physical scales, P⊥ and q⊥. We can use Eqs. (7)
and (8) to rewrite the single transverse-spin dependent di-jet cross section in Eq. (3) in terms
of a single-scale hard factor, HSivers, which is a function of P⊥, and of the q⊥-dependent
perturbatively generated TMD parton distributions:
d∆σ(S⊥)(qq′)
dy1dy2dP
P⊥≫q⊥≫ΛQCD
= ǫαβSα⊥q
Sivers
qq′→qq′
′(xb))
SIDIS
T (xa, q⊥)
(xb q
′(xb, q⊥)) (xa TF (xa, xa))
Using the leading order relation [23]
d2k⊥ ~k
SIDIS
T (x, k⊥) = −TF (x, x) , (10)
we find that our result in Eq. (9), when the contributions from all other partonic subprocesses
at the same order [14] are added, is consistent with the leading-order term of a more general
factorization formula in terms of TMD parton distributions:
d∆σ(S⊥)
dy1dy2dP
ǫαβSα⊥q
d2k1⊥d
2k2⊥d
~k1⊥ · ~q⊥
SIDIS
Ta (xa, k1⊥) xb f
SIDIS
b (xb, k2⊥) (11)
Sab→cd(λ⊥)H
Sivers
ab→cd(P
δ(2)(~k1⊥ + ~k2⊥ + ~λ⊥ − ~q⊥) ,
where apart from the functions already given, fSIDISb denotes the unpolarized TMD quark
distribution and Sab→cd is the soft factor mentioned above [8, 14]. Because of the color
flow into the jets, the product of the soft and hard factors will involve a sum over separate
color amplitudes in the full factorization formalism [24, 25], which has been represented by
a trace [ ]c in color space in the above equation. Our calculation of initial-state collinear
gluon radiation described above would not be sensitive to this complexity of the color flow,
but we emphasize that the definition of the parton distributions cannot be affected by it
[14, 25].
In Eq. (11), we have chosen TMD parton distributions defined in SIDIS because of the
dominance of final-state interactions. Choosing TMD parton distributions defined according
to the Drell-Yan process would change the sign of the partonic hard factors, but not affect
the overall sign of the physical cross section. Based on our explicit calculation here and the
generalized factorization property of the Drell-Yan process at low q⊥ [17, 18], and because of
the similarity in kinematics between two processes, we expect the generalized factorization
formula in Eq. (11) to be valid for describing the single-transverse-spin dependent cross
section for the di-jet momentum imbalance in hadronic collisions in the kinematic region
where P⊥ ≫ q⊥ & ΛQCD.
A key feature of the factorization is that the perturbatively calculated short-distance hard
factors should not be sensitive to details of the factorized long distance physics. We tried, as
a test, to derive all short-distance hard factors by using this factorization formula and the
Brodsky-Hwang-Schmidt model for SSAs [20]. We were able to recover all hard-scattering
factors in this way. We note that the same hard factors HSiversab→cd as above were also found for
the weighted (integrated) SSA for di-jet production in [10]. A detailed comparison between
the approach of [10] and ours will be presented in Ref. [14]. An all-order proof of the above
factorization formula, if correct, is still needed and is beyond the scope of this paper.
4. Summary. We have studied the single-transverse-spin dependent cross section
∆σ(S⊥) for di-jet production momentum imbalance in high-energy hadronic collisions in
a kinematic region where P⊥ ≫ q⊥ ≫ ΛQCD, and calculated contributions from both initial-
and final-state interactions. At the leading order in q⊥/P⊥, the q⊥ and P⊥ dependences in
our calculated results are decoupled and can be factorized into a single-scale hard factor that
depends on P⊥, and into perturbatively generated TMD parton distributions. This factor-
ization occurs for each partonic channel. Overall, final-state interactions turn out to give the
dominant contribution to ∆σ(S⊥) [14]. We therefore expect the SSA in di-jet production to
have the same sign as the Sivers asymmetry in SIDIS.
We have found that our results are consistent with a more general TMD factorization
formula, given in Eq. (11), which we propose to be the correct approach to describing the
single-transverse spin asymmetry in di-jet production at hadron colliders when P⊥ ≫ q⊥.
We emphasize that obviously the result of our first-order calculation is not able to actually
prove this factorization, but should rather be regarded as a “necessary condition” for such
a factorization to hold. A full proof remains an important challenge for future work. Also,
we recall that we have limited ourselves to the case of collinear initial-state gluon radiation.
The effects of large-angle soft-gluon emission have been neglected, even though we have
indicated their likely role in Eq. (11). A proof of TMD factorization in this process would
naturally incorporate a study of this soft factor.
If the proposed factorization formula is valid, the di-jet momentum imbalance at RHIC
could be described by the same TMD parton distributions as those used to describe the SIDIS
and Drell-Yan processes. Because both initial- and final-state interactions are present, the
di-jet momentum imbalance is sensitive to different short distance dynamics, and it will
be an excellent process to test QCD factorization and the universality of the TMD parton
distributions. In addition it should give valuable information on the partons’ transverse
motion in the nucleon.
Acknowledgments
We thank C.J. Bomhof, S.J. Brodsky, J.C. Collins, X. Ji, A. Metz, P.J. Mulders, and G.
Sterman for useful discussions. We are grateful to RIKEN, Brookhaven National Labora-
tory and the U.S. Department of Energy (grant number DE-FG02-87ER40371 and contract
number DE-AC02-98CH10886) for providing the facilities essential for the completion of
this work. J.Q. thanks high energy theory group at Argonne National Laboratory for its
hospitality during the writing of this work.
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http://arxiv.org/abs/hep-ph/0701277
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