Published for SISSA by Springer
Received: March 23, 2015
Revised: July 20, 2015 Accepted: August 24, 2015 Published: September 16, 2015
Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions
JHEP09(2015)106
P. Kotko,a K. Kutak,b C. Marquet,c E. Petreska,c,d S. Sapetae and A. van Hamerenb
aDepartment of Physics, Penn State University,
University Park, 16803 PA, U.S.A.
bThe H. Niewodniczaski Institute of Nuclear Physics PAN, Radzikowskiego 152, 31-342 Krakw, Poland
cCentre de Physique Thorique,cole Polytechnique,
CNRS, 91128 Palaiseau, France
dDepartamento de Fsica de Partculas and IGFAE,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
eCERN PH-TH,
CH-1211, Geneva 23, Switzerland
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We study forward dijet production in dilute-dense hadronic collisions. By considering the appropriate limits, we show that both the transverse-momentum-dependent (TMD) and the high-energy factorization formulas can be derived from the Color Glass Condensate framework. Respectively, this happens when the transverse momentum imbalance of the dijet system, kt, is of the order of either the saturation scale, or the hard jet momenta, the former being always much smaller than the latter. We propose a new formula for forward dijets that encompasses both situations and is therefore applicable regardless of the magnitude of kt. That involves generalizing the TMD factorization formula for dijet production to the case where the incoming small-x gluon is o -shell. The derivation is performed in two independent ways, using either Feynman diagram techniques, or color-ordered amplitudes.
Keywords: QCD Phenomenology, Heavy Ion Phenomenology
ArXiv ePrint: 1503.03421
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP09(2015)106
Web End =10.1007/JHEP09(2015)106
Contents
1 Introduction 1
2 Forward dijets in p+A collisions 4
3 High energy factorization derived from CGC: the [notdef]p1t[notdef], [notdef]p2t[notdef], [notdef]kt[notdef] Qs limit 7
4 TMD factorization for nearly back-to-back jets: the [notdef]p1t[notdef], [notdef]p2t[notdef] [notdef]kt[notdef], Qs limit 124.1 The qg ! qg channel 13
4.2 The gg ! qq channel 15
4.3 The gg ! gg channel 17
4.4 The [notdef]kt[notdef] Qs limit 225 Unied description of forward dijets in p+A collisions: TMD factorization with o -shell hard factors 235.1 The qg ! qg channel 25
5.2 The gg ! qq channel 25
5.3 The gg ! gg channel 26
6 Helicity method for TMD amplitudes 266.1 Color decompositions 266.2 Gluon TMDs for color ordered amplitudes 276.3 O -shell color-ordered helicity amplitudes 296.4 Hard factors from color-ordered amplitudes 31
7 Conclusions and outlook 33
A O -shell expressions 35
1 Introduction
Forward particle production observables in proton-proton (p+p) and proton-nucleus (p+A) collisions at the Large Hadron Collider (LHC) o er unique opportunities to study the dynamics of QCD at small x, and in particular the non-linear regime of parton saturation [1]. Indeed, in high-energy hadronic collisions, forward particle production is sensitive only to high-momentum partons inside one of the colliding hadrons, which therefore appears dilute. By contrast, for the other hadron or nucleus, it is mainly small-momentum partons, whose density is large, that contribute to the scattering. Such processes, in which a large-x
1
JHEP09(2015)106
projectile is used as a probe to investigate a small-x target, are sometimes called dilute-dense collisions. Since the high-x part of the projectile wave function is well understood in perturbative QCD, forward particle production is indeed ideal to investigate the small-x part of target wave function. This is true both in p+p and p+A collisions, although using a target nucleus does enhance the dilute-dense asymmetry of such collisions.
The separation between the linear and non-linear regimes of the target wave function is characterized by a momentum scale Qs(x), called the saturation scale, which increases as x decreases. Dilute-dense collisions can be described from rst principles, provided Qs QCD. This condition is better realized with higher energies (as they open up the
phase space towards lower values of x), and with nuclear targets (since, roughly, Qs A1/3). Over the years, the Color Glass Condensate (CGC) e ective theory [2] has emerged as the best candidate to approximate QCD in the saturation regime, both in terms of practical applicability and of phenomenological success [3]. In this paper, we focus on forward dijet production in p+A and p+p collisions. We note that the CGC approach has been very successful in describing forward di-hadron production at RHIC [46], in particular it predicted the suppression of azimuthal correlations in d+Au collisions compared to p+p collisions [7], which was observed later experimentally [8, 9].
With forward dijets at the LHC however, the full complexity of the CGC machinery is not needed. Indeed, for the di-hadron process at RHIC energies, no particular ordering of the momentum scales involved is assumed in CGC calculations, while, at the LHC, the presence of particles with transverse momenta much larger than the saturation scale clearly must imply some simplications. On the ip side, there will be other complications since further QCD dynamics, which is not part of the CGC framework but which is relevant at large transverse momenta, must also be considered. There are three important momentum scales in the forward dijet process: a typical transverse momentum of a hard jet, Pt, whose precise denition will be stated in the next section; the transverse momentum of the small-x gluons involved in the hard scattering, kt; and the saturation scale of the small-x target,
Qs. Clearly, Pt is always one of the hardest scales, and it is much bigger than Qs, which is always one of the softest scales. Then, depending on where kt sits with respect to these two, three di erent regimes can be dened.
A rst regime, with Qs kt Pt, corresponds to the domain of applicability of the
so-called high energy factorization (HEF) framework [10, 11], in which the description of forward dijets involves an unintegrated gluon distribution for the small-x target, along with o -shell hard matrix elements. That is explicitly shown in this work, starting form CGC calculations. While such a factorization does not occur when non-linear saturation e ects are accounted for, we shall see that taking the Qs kt Pt limit is tantamount to
restricting the interaction with the small-x target to a two-gluon exchange, therefore allows to indeed write all the CGC correlators in terms of a single gluon distribution. Doing so, the matrix elements of the HEF framework are exactly recovered.
A second regime, with kt Qs Pt, is where the so-called transverse momentum
dependent (TMD) factorization [12] is valid. It involves on-shell matrix elements but several unintegrated gluons distributions. In this regime, non-linear e ects are present, and in the large-Nc limit, equivalence with CGC expressions was shown in [13]. In particular,
2
JHEP09(2015)106
in that case the description of forward dijets involves only two independent unintegrated gluons distributions, each of which can be determined in various other processes [14]. In the present work we shall keep Nc nite, implying, as we show below, that a total of six independent unintegrated gluons distributions are needed.
Finally, the intermediate regime Qs kt Pt, which is naturally obtained from
the two others by taking the appropriate limits, corresponds to the collinear regime, with on-shell matrix elements and the standard integrated gluon distribution.
Separately, the HEF and TMD approaches to dijet production have been extensively studied in the literature [11, 1518] and [12, 1925], but little connection has been made between them so far. The rst result of this paper is to reveal that connection, in the context of dilute-dense collisions, and to show that, in fact, they are both contained in the CGC description. However, as already mentioned, using the CGC approach is unnecessarily complicated and one should take advantage of the fact that Pt Qs to simplify the
theoretical formulation. The second result of the paper is precisely to develop a new formula for forward dijets in dilute-dense collisions that encompasses all three situations described above, meaning that it is applicable regardless of the magnitude of kt. As explained below, this is obtained by extending the TMD factorization framework, more precisely by supplementing it with o -shell matrix elements.
Note that the derivation of our new unied formula is performed in two independent ways: rst using the standard Feynman diagram technique, and second by exploiting the so-called helicity method that employs color-ordered amplitudes [26]. With this second method, the gauge invariance of the results is explicit, and the method will also prove very useful in the future, when processes with more particles in the nal state are considered. As is the case in the CGC framework, our new formulation contains all the relevant limits, but it has the advantage that it is more amenable to phenomenological implementations than CGC calculations. In addition, it is also better suited to be supplemented with further QCD dynamics relevant at high Pt, such as Sudakov logarithms [27, 28] or coherence in the QCD evolution of the gluon density [2931]. These tasks are left for future work.
The plan of the paper is as follows. In section 2, we introduce kinematics and notations, and briey present the HEF and TMD frameworks. In section 3, we show that the HEF framework can be derived from CGC calculations, when the Qs kt Pt limit is
considered; namely we explain how the various CGC correlators reduce to a single gluon distribution in that limit, and show that the o -shell matrix elements of the HEF framework are indeed emerging. Section 4 is devoted to the kt Qs Pt limit, the derivation of
the TMD factorization formula for forward dijets given in [14] is recalled, and extended to the case of nite Nc, implying six independent unintegrated gluons distributions instead of two. The hard factors of the TMD framework are computed again in section 5, but keeping the small-x gluon o -shell, which leads us to our new unied formula for forward dijets in p+A collisions. In section 6, both the TMD factorization formula and the o -shell hard factors are derived again, but using color-ordered amplitudes, instead of Feynman diagram techniques. Finally, section 7 is devoted to conclusions and outlook.
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JHEP09(2015)106
Figure 1. Inclusive dijet production in p+A collision. The blob H represents hard scattering. The solid lines coming out of H represent partons, which can be either quarks or gluons.
2 Forward dijets in p+A collisions
We shall discuss the process of inclusive dijet production in the forward region, in collisions of dilute and dense systems
p(pp) + A(pA) ! j1(p1) + j2(p2) + X . (2.1)
The process is shown schematically in gure 1. The four-momenta of the projectile and the target are massless and purely longitudinal. In terms of the light cone variables, v[notdef] = (v0 [notdef] v3)/p2, they take the simple form
pp =
rs2(1, 0t, 0) , pA =
JHEP09(2015)106
rs2(0, 0t, 1) , (2.2)
where s is the squared center of mass energy of the p+A system.
The energy (or longitudinal momenta) fractions of the incoming parton (either a quark or gluon) from the projectile, x1, and the gluon from the target, x2, can be expressed in terms of the rapidities and transverse momenta of the produced jets as
x1 = p+1 + p+2p+p =
1ps ([notdef]p1t[notdef]ey1 + [notdef]p2t[notdef]ey2) , (2.3a)
x2 = p1 + p2 pA
= 1
ps [notdef]p1t[notdef]ey1 + [notdef]p2t[notdef]ey2
[parenrightbig]
, (2.3b)
where p1t, p2t are transverse Euclidean two-vectors. By looking at jets produced in the forward direction, we e ectively select those fractions to be x1 1 and x2 1. Since the
target A is probed at low x2, the dominant contributions come from the subprocesses in which the incoming parton on the target side is a gluon
qg ! qg , gg ! qq, gg ! gg . (2.4) In dilute-dense collisions, the large-x partons of the dilute projectile are described in terms of the usual parton distribution functions of collinear factorization fa/p, with a scale dependence given by DGLAP evolution equations. By contrast, the small-x gluons of the dense
4
target nucleus are described by a transverse-momentum-dependent distribution, which evolve towards small x according to non-linear equations. Moreover, the momentum k of the incoming gluon from the target, besides the longitudinal component k = x2
ps/2, has in general a non-zero transverse component, kT , which leads to imbalance of transverse momentum of the produced jets
|kt[notdef]2 = [notdef]p1t + p2t[notdef]2 = [notdef]p1t[notdef]2 + [notdef]p2t[notdef]2 + 2[notdef]p1t[notdef][notdef]p2t[notdef] cos , (2.5)
with k2T = [notdef]kt[notdef]2. Here, by kT we mean a four-vector, as opposed to kt = p1t + p2t,
which is a two-dimensional vector in the transverse plane. They are simply related by: kT = (0, kt, 0). Using the notation dened above, the gluons four-momentum can be also parametrized as
k = x2pA + kT . (2.6)
The Mandelstam variables at the partonic level are dened as
= (p + k)2 = (p1 + p2)2 = [notdef]Pt[notdef]2 z(1 z)
JHEP09(2015)106
, (2.7a)
t= (p2 p)2 = (p1 k)2 = [notdef]
p2t[notdef]2 1 z
, (2.7b)
[notdef] = (p1 p)2 = (p2 k)2 = [notdef]
p1t[notdef]2z , (2.7c)
with
z = p+1p+1 + p+2
and Pt = (1 z)p1t zp2t . (2.8)
They sum up to + t+[notdef] = k2T .
Note that we always neglect the transverse momentum of the high-x partons compared with that of the low-x parton [notdef]kt[notdef]. This is justied in view of the asymmetry of the problem,
x1 1 and x2 1, which implies that gluons form the target have a much bigger average
transverse momentum (of the order of Qs) compared to that of the large x partons from the projectile (which of the order of QCD). And even when the transverse momentum imbalance of the dijet system is of the same order as the jet transverse momenta themselves, implying that both parton distributions are probed in their radiative tail, the small x2 (BFKL) evolution implies a 1/kt behavior on the target side, while DGLAP evolution implies a 1/k2t behavior on the projectile side.
To take into account small-x e ects in dijet production, an approach that has been broadly used in phenomenological studies involves the so-called high energy factorization (HEF) formula [15]
dpA!dijets+X
dy1dy2d2p1td2p2t =
1 163(x1x2s)2
Xa,c,dx1fa/p(x1, [notdef]2) [notdef]Mag !cd[notdef]2Fg/A(x2, kt)11 + cd .
(2.9)
This formula makes use of the unintegrated gluon distribution Fg/A that is involved in
the calculation of the deep inelastic structure functions. It is determined from ts to DIS
5
data, and then used in eq. (2.9), along with matrix elements that depend on the transverse momentum imbalance (2.5). Even though the high energy factorization is not strictly valid for dijet production, there exists a kinematic window, the dilute limit Qs [notdef]p1t[notdef], [notdef]p2t[notdef], [notdef]kt[notdef], in which it can be motivated from the CGC approach. We shall demonstrate this explicitly for all channels in the next section.
A second approach, valid in the regime where the transverse momentum imbalance between the outgoing particles, eq. (2.5), is much smaller than their individual transverse momenta, is the so-called transverse momentum dependent (TMD) factorization. This limit, [notdef]p1t + p2t[notdef] [notdef]p1t[notdef], [notdef]p2t[notdef], or [notdef]kt[notdef] [notdef]Pt[notdef], corresponds to the situation of nearly back-
to-back dijets. Even though, in general, there exists no TMD factorization theorem for jet production in hadron-hadron collisions, such a factorization can be established in the asymmetric dilute-dense situation considered here, where only one of the colliding hadrons is described by a transverse momentum dependent gluon distribution. Again, selecting dijet systems produced in the forward direction implies x1 1 and x2 1, which in turn allows
us to make that assumption. The TMD factorization formula reads (so far, this has been obtained in the large-Nc approximation, but this restriction will be lifted in the present work) [13]
dpA!dijets+X
dy1dy2d2p1td2p2t =
JHEP09(2015)106
2s (x1x2s)2
Xa,c,dx1fa/p(x1, [notdef]2)
XiH(i)ag!cdF(i)ag(x2, kt)11 + cd , (2.10)
where several unintegrated gluon distributions F(i)ag with di erent operator denition are
involved and accompanied by di erent hard factors H(i)ag!cd. Those hard factors were
calculated in [13] as if the small-x2 gluon was on-shell (i.e. [notdef]kt[notdef] = 0). The kt dependence
survived only in the gluon distributions.
By restoring the kt dependence of the hard factors inside formula (2.10), we can make the bridge between the HEF and TMD frameworks and obtain a unied formulation which encompasses both the dilute and the nearly back-to-back limit. Note that we follow the conventions used in earlier papers that dealt with these formalisms, such as ref. [15] and [13] respectively. Therefore, contrary to the HEF matrix elements [notdef]Mag !cd[notdef]2, the hard factors
H(i)ag!cd of the TMD factorization are dened without the g4 factor. In addition, the de
nition of the gluon distribution also di er by a factor . The integrated gluon distribution x2fg/A is obtained from
[integraltext]
dk2t Fg/A in the HEF formalism, and from
[integraltext]
d2kt F(i)ag in the
TMD formalism.
Finally, let us point out that, in the frameworks described above, one emits radiation in the transverse direction that one has no control over, as it is part of the small-x gluon distributions and therefore is treated fully inclusively. To be more specic, at this level, transverse momentum conservation is obtained either by several particles of average trans-verse momentum Qs, or by a third hard jet, depending on the magnitude of [notdef]kt[notdef]. Due to
the small-x evolution, that radiation is ordered in rapidity, therefore it does not contribute to the measured forward dijets systems.
6
3 High energy factorization derived from CGC: the [notdef]p1t[notdef], [notdef]p2t[notdef], [notdef]kt[notdef] Qs limit
We shall demonstrate that the high-energy factorization formula for double-inclusive particle production, eq. (2.9), is identical to a result obtained from the CGC formalism in the dilute target approximation. This is a limit where all the momenta involved in the process are much larger than the saturation scale: [notdef]p1t[notdef], [notdef]p2t[notdef], [notdef]kt[notdef] Qs. Here, we show explicitly the equivalence of the HEF and CGC formulas for the qg ! qg channel and only
provide the nal results for the two other channels, as the derivations proceed identically for all of them. We derive the CGC cross sections for the qg ! qg and gg ! qq channels
in the dilute limit following a procedure developed in ref. [32] where only the gg ! gg sub-process was considered.
The amplitude for quark-gluon production is schematically presented in gure 2 as in ref. [7]. In the left diagram, the emission of the gluon from the quark happens before the interaction with the target, and in the right diagram the emission occurs after the quark has interacted with the target. There is a relative minus sign between the two cases as explained in details in ref. [7]. Multigluon interactions of quarks and gluons with a target, in the CGC formalism, enter as Wilson lines in the expression for the amplitude. A quark propagator is represented as a fundamental Wilson line, while a gluon propagator as an adjoint Wilson line. As a result, the cross section involves multipoint correlators of Wilson lines. In particular, the amplitude from gure 2, after squaring, has four terms: a correlator of four Wilson lines, S(4), corresponding to interactions happening after the emission of the gluon, both in the amplitude and the complex conjugate, then a correlator of two Wilson lines, S(2), representing the case when interactions with the target take place before the radiation of the gluon in both amplitude and complex conjugate, and two correlators of three Wilson lines, S(3), for the interference terms. In all the cases the splitting function is the same, and is given by the product of the quark wave functions: (p, p+1, x[prime] b[prime]) (p, p+1, x b). The total expression for the inclusive cross section in
CGC is then given by the following formula [7]:
d(pA ! qgX)
dy1dy2d2p1td2p2t = sCF (1 z)p+1x1fq/p(x1, [notdef]2) [notdef]M(p, p1, p2)[notdef]2 , (3.1)
where the amplitude squared, [notdef]M(p, p1, p2)[notdef]2, has the form:
[notdef]M(p, p1, p2)[notdef]2 = [integraldisplay]
d2x
(2)2
d2x[prime]
(2)2
d2b
(2)2
JHEP09(2015)106
(2)2 eip1t[notdef](xx[prime])eip2t[notdef](bb[prime])
d2b[prime]
[notdef]
X
(p, p+1, x[prime] b[prime]) (p, p+1, x b)
nS(4)qgqg[b, x, b[prime], x[prime]; x2] S(3)qgq[b, x, b[prime] + z(x[prime] b[prime]); x2]
S(3)qgq[b + z(x b), x[prime], b[prime]; x2]+S(2)qq[b + z(x b), b[prime] + z(x[prime] b[prime]); x2][bracerightBig]
, (3.2)
[notdef]
7
p p2
p1
p p2 p1
Figure 2. Amplitude for quark-gluon production in the CGC formalism. Left: the gluon is radiated before the interaction with the target. Right: the gluon is radiated after the interaction with the target. The two terms have a relative minus sign.
where are mixed-space quark wave functions and S(i) are correlators of Wilson lines explained in details below. Following the notation from gure 1 and eq. (2.8), we use the fraction of the plus components of four-momenta, z, with p1 being the four-momentum of the outgoing gluon and p2, the four-momentum of the outgoing quark.
The fundamental, U(x), and adjoint, V (x), Wilson lines are dened as path-ordered exponentials of the gauge eld (written here in the A+ = 0 gauge):
U(x) = P exp
ig
[integraldisplay]
and V (x) = P exp
ig
[integraldisplay]
dx+Aa(x+, x)T a
[bracketrightbigg]
,
(3.3)
where ta and T a are the generators of the fundamental and adjoint representations of SU(N) respectively. The traces of products of Wilson lines appearing in the cross section are dened in the following way:
S(4)qgqg(b, x, b[prime], x) = 1
CF Nc
Tr
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dx+Aa(x+, x)ta
[bracketrightbigg]
U(b)U(b[prime])tdtc[parenrightBig] [bracketleftBig]
V (x)V (x[prime])
icd
x2; (3.4)
S(3)qgq(b, x, z[prime]) = 1
CF Nc
DTr
U(z[prime])tcU(b)td[parenrightBig]
V cd(x)
Ex2 ; (3.5)
S(2)qq(z, z[prime]) = 1
Nc
U(z)U(z[prime])
[parenrightBig][angbracketrightBig]x2 . (3.6)
The CGC average is taken over the background led evaluated at Y = ln(1/x2). The product of wave functions in the massless limit is:
X
(p, p+1, u[prime]) (p, p+1, u) = 82p+1u [notdef] u[prime] |u[notdef]2[notdef]u[prime][notdef]2
DTr
(1 + (1 z)2) . (3.7)
Introducing a change of variables, u = x b and v = zx + (1 z)b (and similar for the
primed coordinates), we get [7]:
[notdef]M(p, p1, p2)[notdef]2 = [integraldisplay]
d2u
(2)2
(2)2 eiPt[notdef](u[prime]u)
X
(p, p+1, u[prime]) (p, p+1, u)
d2u[prime]
[notdef]
[integraldisplay]
d2v
(2)2
(2)2 eikt[notdef](v[prime]v)
hS(4)qgqg(b, x, b[prime], x[prime]) S(3)qgq(b, x, v[prime])
S(3)qgq(v, x[prime], b[prime]) + S(2)qq(v, v[prime])
i. (3.8)
d2v[prime]
8
The conjugate momentum to u[prime] u is Pt = (1 z)p1t zp2t, and the one corresponding to
v[prime] v is the total transverse momentum of the produced particles kt = p1t + p2t. In terms
of fundamental Wilson lines only:
S(4)qgqg(b, x, b[prime], x[prime]) = 1
2CF Nc
Tr
U(b)U(b[prime])U(x[prime])U(x)
[parenrightBig] Tr
U(x)U(x[prime])
[parenrightBig](3.9)
1Nc Tr [parenleftBig]
U(b)U(b[prime])
[parenrightBig]
x2
,
and
S(3)qgq(b, x, v[prime]) = 1
2CF Nc
Tr
U(x)U(v[prime])
[parenrightBig] 1Nc Tr [parenleftBig]
U(b)U(v[prime])
[parenrightBig]x2.
(3.10)
In the dilute target limit we allow for only up to two gluon exchanges between the Wilson line propagators and the nucleus. Accordingly, we expand the Wilson lines to second order in the background eld:
U(x) 1+ig
[integraldisplay]
g2 2
[integraldisplay]
+O(A3) . (3.11)
To this order, the expectation values of the four- and three-point correlators are simply expressed in terms of the dipole operator S(2)qq(v, v[prime]). The dilute target approximation
gives only a leading result in [notdef]v v[prime][notdef]2Q2s for the expectation value of S(2)qq(v, v[prime]), which is
equivalent to taking the limit [notdef]kt[notdef] Qs. Similarly, when all the momenta involved in the
process are much larger than the saturation scale, the correlators entering the cross section get the following expressions:
S(4)qgqg(b, x, b[prime], x[prime]) = 1 g2Nc x2(x x[prime]) g2
N2c 1
2Nc x2(b b[prime])
dx+dy+P
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U(b)U(x)
[parenrightBig] Tr
dx+A(x+, x)
A(x+, x)A(y+, x)
g2Nc
x2(xb) + x2(x[prime]b[prime]) x2(x[prime]b) x2(xb[prime])
[bracketrightbig]
; (3.12)
2 g2Nc
g2Nc
2 x2(x v[prime]) +
S(3)qgq(b, x, v[prime]) = 1
2 x2(b x)
g22Nc x2(b v[prime]) ; (3.13)
S(2)qq(v, v[prime]) = 1 g2
N2c 1
2Nc x2(v v[prime]) . (3.14)
In the above equations:
x2(x y) =
x2(x+, 0) x2(x+, r)
[bracketrightbig]
, (3.15)
where r = xy and x2(x+, r) is related to the expectation value of the two-eld correlator:
Aa(x+, x)Ab(y+, y) x2 = ab (x+ y+) x2(x+, x y) . (3.16)
9
[integraldisplay]
dx+
Using the expressions for the multi-point functions S(i), we get the following result for the amplitude squared:
[notdef]M(p, p1, p2)[notdef]2 = 42g2Nc(1 + (1 z)2)
1 p+1
[integraldisplay]
d2u
(2)2
(2)2 eiPt[notdef](u[prime]u)
u [notdef] u[prime] |u[notdef]2[notdef]u[prime][notdef]2
d2u[prime]
[notdef]
[integraldisplay]
d2v
(2)2
d2v[prime]
x2(x b[prime]) + x2(x[prime] b) + x2(x v[prime])
+ x2(v x[prime]) 2 x2(x x[prime])
N2c 1
N2c x2(b b[prime])
(2)2 eikt[notdef](v[prime]v)[bracketleftbigg]
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N2c 1
N2c x2(v v[prime])
1N2c x2(b v[prime])
1N2c x2(v b[prime])[bracketrightbigg]
.(3.17)
We perform the integrals in the above expression by changing the variables from v and v[prime] to r and B. The integrals over the transverse distances of the type r = vv[prime] are equivalent
to the Fourier transform of eq. (3.15) and give the unintegrated gluon distribution:
fx2(kt) k2t [integraldisplay]
d2r x2(r)eikt[notdef]r = k2t
[integraldisplay]
dx+ x2(x+, kt) . (3.18)
In our approximation, the correlators do not depend on the impact parameter B = (v + v[prime])/2. The integrals over B factorize and give the transverse area of the target:
[integraltext]
d2B = S?.
Finally, the rest two integrations reduce to:
[integraldisplay]
d2u eiPt[notdef]u u
. (3.19)
In terms of the unintegrated gluon distribution, the amplitude squared then gets the form:
[notdef]M(p, p1, p2)[notdef]2 =
2(2)4 g2S?Nc
|u[notdef]2
= 2i
Pt
|Pt[notdef]2
fx2(kt)
k2t
(1 + (1 z)2)
1 p+1
(N2c 1)2N2c1P 2t+ (N2c 1)2N2c1p21t+ 1 p22t
+ 1
N2c
[notdef]
Pt [notdef] p1t P 2tp21t
+ Pt [notdef] p2t
P 2tp22t
+ p1t [notdef] p2t p21tp22t
[bracketrightbigg]
, (3.20)
We want to show that eq. (3.20) reproduces the HEF formula (2.9) with the appropriate unintegrated parton distribution function and o -shell matrix elements. For this purpose, we need to nd a relation between the unintegrated gluon distribution used in the above equation, fx2(kt), and Fg/A(x2, kt), which appears in the HEF formula (2.9). This is easily
done by considering the deep inelastic scattering process, since Fg/A(x2, kt) is precisely the unintegrated gluon distribution involved in the formulation of the + A ! X total cross
section, and is therefore related to the qq dipole scattering amplitude in a straightforward manner (see for instance [16, 33]):
Fg/A(x2, kt) = Nc s(2)3
[integraldisplay]
d2vd2v[prime] eikt[notdef](vv[prime])r2vv[prime] [bracketleftBig]
1 S(2)qq(v, v[prime])[bracketrightBig]
. (3.21)
10
In the weak-eld limit, using formula (3.14), this gives the relation
fx2(kt) = 42
S?(N2c 1)Fg/A
(x2, kt) . (3.22)
Then, the cross section for the qg production channel from eq. (3.1) can be written in a more compact form
d(pA ! qgX)
dy1dy2d2p1td2p2t =
2s2 x1fq/p(x1, [notdef]2)z(1z)
Pgq(z)
[notdef]
1 + (1z)2p 21tP 2t 1N2cz2p 22t P 2t[bracketrightbigg]
Fg/A(x2, kt) p 21t p 22t
, (3.23)
where Pgq(z) is related to the quark-to-gluon splitting function and is given by:
Pgq(z) = 1 + (1z)2z . (3.24)
It turns out that the above expression for the quark-gluon production cross section is identical to the result in the HEF formalism, eq. (2.9), containing the o -shell amplitudes
[notdef]Mag !cd[notdef]2. The latter have been calculated in refs. [11, 34] and [35].
The equivalence of the CGC and HEF formulas in the dilute limit can be shown in a similar way for the cross sections of the other two subprocesses, gg ! qq and gg ! gg.
The CGC results for the cross sections in this limit are:
d(pA ! qqX)
dy1dy2d2p1td2p2t =
2s4CF x1fg/p(x1, [notdef]2)z(1z)
JHEP09(2015)106
Pqg(z)
[notdef]
1N2c +(1z)2p 21t + z2p 22t P 2t
[bracketrightbigg]
Fg/A(x2, kt) p 21t p 22t
(3.25)
and [32]
d(pA ! ggX)
dy1dy2d2p1td2p2t =
2sNc
CF x1fg/p(x1, [notdef]2)z(1z)
Pgg(z)
[notdef]
1 + (1z)2p 21t + z2p 22t P 2t[bracketrightbigg]
Fg/A(x2, kt) p 21t p 22t
. (3.26)
The expressions for Pqg(z) and Pgg(z) have the form:
Pqg(z) = z2 + (1z)2 ,
Pgg(z) = z
1 z
+ 1 z
z + z(1 z) . (3.27)
Again, eqs. (3.25) and (3.26) are equivalent to the HEF formulas for the corresponding cross sections [16].
Therefore, in principle, the HEF formalism should not be employed to include nonlinear e ects, and one should stick to Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution [3638], or Ciafaloni-Catani-Fiorani-Marchesini evolution [2931], when evaluating the gluon distribution. In this spirit, most studies are performed using a gluon density evolved
11
with an improved BFKL equation that includes some higher-order corrections [39], but no non-linear e ects. However, we note that the HEF framework could be used with the Balitsky-Kovchegov (BK) equation [40, 41] in order to investigate the so-called geometric scaling regime, where saturation e ects are felt, even though Qs kt. The full saturation
region, Qs kt, is however, in principle, out of reach of formula (2.9). Along these lines,
an estimate of saturation e ects was obtained in [42, 43], using the BK equation extended to include the same higher-order corrections as included in the linear case [39].
4 TMD factorization for nearly back-to-back jets: the [notdef]p1t[notdef], [notdef]p2t[notdef] [notdef]kt[notdef], Qs limit
In this section we discuss the special case of nearly back-to-back jets, [notdef]p1t+p2t[notdef] [notdef]p1t[notdef], [notdef]p2t[notdef], where the di erential cross section is given by formula (2.10). Several gluon distributions
F(i)ag, with di erent operator denition, are involved here. Indeed, as explained in [12], a
generic unintegrated gluon distribution of the form
F(x2, kt) naive= 2
[integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
+,
[parenrightbig]
F i (0) [bracketrightbig]
, (4.1)
where F i are components of the gluon eld strength tensor, must be also supplemented with gauge links, in order to render such a bi-local product of eld operators gauge invariant.
The gauge links are path-ordered exponentials, with the integration path being xed by the hard part of the process under consideration. Therefore, unintegrated gluon distributions are process-dependent.
In the following, we shall encounter two gauge links U[+] and U[], as well as the loop
U[ ] = U[+]U[] = U[]U[+]. These links are composed of Wilson lines, their simplest
expression is obtained in the A+ = 0 gauge:
U[[notdef]] = U(0, [notdef]1; 0)U([notdef]1, +; ) with U(a, b; x) = P exp [bracketleftbigg]
ig
JHEP09(2015)106
A[notdef]Tr
F i
|A
[angbracketrightbig]
[integraldisplay]
b
a dx+Aa(x+, x)ta[bracketrightbigg]
,
(4.2)
but the expressions of the various gluon distributions given below are gauge-invariant. From now on, F i (+, ) is simply denoted as F (), and the hadronic matrix elements
hA[notdef] . . . [notdef]A[angbracketright] ! [angbracketleft]. . .[angbracketright]. Note however that they are di erent from the CGC averages [angbracketleft][notdef] [notdef] [notdef] [angbracketright]x2 of the previous section. Indeed, the normalization of the hadronic state [notdef]A[angbracketright] is dened as hA[prime][notdef]A[angbracketright] = (2)3 2p+A (p+A p[prime]+A) (2) (pAt p[prime]At), while the CGC averages are normalized
as [angbracketleft]1[angbracketright]x2 = 1. As explained in [13], the two can be related by making the replacement [angbracketleft][notdef] [notdef] [notdef] [angbracketright]x2 ! [angbracketleft]A[notdef]...[notdef]A[angbracketright][angbracketleft]A[notdef]A[angbracketright].
This approach to dijet production in proton-nucleus collisions was analyzed in ref. [13]. The TMD factorization formula (2.10) was derived there in the large-Nc limit, and shown to be equivalent to CGC calculations (e.g. formulas (3.1) and (3.2) in the case of the qA ! qg
channel), after taking the limit [notdef]p1t[notdef], [notdef]p2t[notdef] [notdef]kt[notdef], Qs. In this section, we derive the TMD
factorization formula keeping Nc nite. We obtain corrections to the hard factors H(i)ag!cd
12
Figure 3. Diagrams for qg ! qg subprocess. The mirror diagrams of (3), (5) and (6) give identical
contributions.
previously derived, and we calculate new hard factors corresponding to gluon distributions that were omitted before (as they were vanishing in the large-Nc limit). The nite Nc extension prevents one to make a further simplication, called correlator factorization, essential to relate the TMD factorization and the CGC formalism, but gives completeness to the main result of this paper, i.e. the new factorization formula we propose below is valid for nite Nc. We also check explicitly the gauge invariance of these hard factors by computing them in a gauge di erent from the one used in [13].
An important fact to note is that, as a consequence of the [notdef]kt[notdef] [notdef]p1t[notdef], [notdef]p2t[notdef] limit, the kt dependence in (2.10) survives only in the gluon distributions, and the hard factors are calculated as if the small-x2 gluon was on-shell. That is, looking at the hard partonic interaction represented by the blob H in gure 1, k2 = [notdef]kt[notdef]2 is set to zero, and+t+[notdef] = 0.
4.1 The qg ! qg channelThe complete set of independent cut diagrams contributing to this channel is shown in gure 3 (mirror images of diagrams (3), (5) and (6) give identical expressions).
The cross section for a quark-gluon scattering involves only two di erent TMD gluon distributions as given in ref. [12]:
dpA!qgX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fq/p(x1, [notdef]2)
JHEP09(2015)106
2
Xi=1F(i)qgH(i)qg!qg , (4.3)
with:
F(1)qg = 2 [integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
DTr
hF () U[]F (0) U[+][bracketrightBig][angbracketrightBig]= x2G(2)(x2, kt) , (4.4)
F(2)qg = 2 [integraldisplay]
Nc U[+]F (0) U[+][bracketrightBigg][angbracketrightBigg]
. (4.5)
These are the same gluon distributions as in the large-Nc limit [13], no additional ones are present in this channel. The only di erence in the expression (4.3) when we go to nite
13
d+d2
(2)3pA
eix2p
A+ikt[notdef]
*Tr
"F () Tr
U[ ]
Nc will appear in the hard factor H(1)qg!qg associated with F(1)qg. That gluon distribution is
sometimes also denoted x2G(2), and is called the dipole distribution, since it is the one that enters the formulation of the inclusive and semi-inclusive DIS.
In the CGC approach, x2G(2) can be related to the qq dipole scattering amplitude, and therefore linked to the gluon distribution used in the HEF formalism: Fg/A(x2, kt) =
x2G(2)(x2, kt). That distribution is not su cient however to compute the forward dijet cross section when [notdef]kt[notdef] Qs (i.e. the case considered in this section). For completeness,
we note that a detailed derivation of this relation between formula (3.21), involving a CGC correlation function, and formula (4.4), involving matrix elements dening TMDs, can be found in appendix A of [13].
The exact results for the two hard factors read
H(1)qg!qg =
1
2D1 +
(2 +[notdef]2)
2[notdef]t2 . (4.9)
The hard factors and the TMDs entering the factorization formula (4.3) are all gauge invariant. In principle, that leaves us some freedom and the factorization formula can be rewritten with new hard factors and the corresponding new gluon distributions formed as linear combinations of the the old ones.
1The choice of axial gauge vectors for external gluons corresponds to the choice of the reference momentum for their polarization vectors, see for example [26], and is arbitrary for gauge invariant quantities. Thus, the independence on those gauge vectors can be used to conrm that the result is gauge invariant.
14
JHEP09(2015)106
1
2D1
1
N2c 1
D2 + D4 + 2D5 + 2D6 , (4.6)
H(2)qg!qg =
Nc2CF D2 + 2D3 , (4.7)
where Dis are the squared and interference diagrams corresponding to the qg ! qg channel,
following the numbering of gure 3. Each term Di = Cuihi represents the product of the color factor, Cui, and the hard coe cient, hi. What kind of diagrams enter the hard factors
H(i)qg!qg depends on the type of the gauge links appearing in each of them. As summarized
in table IV of ref. [12], the distribution F(1)qg is present in diagrams (1), (2), (4), (5) and
(6), while the distribution F(2)qg appears in diagrams (1), (2) and (3). The Di components
were computed in ref. [13] (table II) in an axial gauge with the axial vector, n, set to n = p, for both the incoming and the outgoing gluon, where p is the four-momentum of the incoming quark, as dened in gure 1. Formulated di erently, the polarization vector of each external gluon was chosen such that, besides with the momentum of the gluon, their inner product with p vanishes. We recovered the same results for Dis in that gauge and performed the same calculation in a di erent gauge with the axial vector set to n = p for the incoming gluon and n = p2 for the outgoing gluon.1 The results for the hard factors
H(1)qg!qg and H(2)qg!qg at nite Nc are identical in both gauges and they read
H(1)qg!qg =
[notdef](2 +[notdef]2)
2t2 +
1
2N2c
(2 +[notdef]2)
[notdef] , (4.8)
H(2)qg!qg =
K(1)ag!cd K(2)ag!cd
qg ! qg
2 +[notdef]2
2t2[notdef]
[notdef]2 +2 t2 N2c
[bracketrightbigg]
CF
Nc
(2 +[notdef]2)
t2[notdef]
gg ! qq
1
2Nc
(t2 +[notdef]2)2
2t[notdef]
1
2CF N2c
t2 +[notdef]2
2
(2 t[notdef])2
t[notdef]2
Table 1. The new hard factors following from simplied e ective TMD factorization of eqs. (4.13), (4.25) and (4.52) in the case with all partons being on shell.
For reasons that shall be discussed in detail in section 6, let us dene the new hard factors for the qg ! qg subprocess
K(1)qg!qg = H(1)qg!qg +
gg ! gg
2Nc CF
(2 t[notdef])2(t2 +[notdef]2)
t2[notdef]22
2Nc CF
1N2c H(2)qg!qg and K(2)qg!qg =
N2c 1
N2c H(2)qg!qg , (4.10)
and the corresponding new gluon TMDs
(1)qg!qg = F(1)qg , (4.11)
(2)qg!qg =
1 N2c 1
JHEP09(2015)106
F(1)qg + N2cF(2)qg[parenrightBig]
, (4.12)
such that the factorization formula (4.3) now takes the form
dpA!qgX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fq/p(x1, [notdef]2) [bracketleftBig]
(1)qg!qgK(1)qg!qg + (2)qg!qgK(2)qg!qg
[bracketrightBig]
. (4.13)
The explicit expressions for K(1)qg!qg and K(2)qg!qg are given in table 1.
4.2 The gg ! q q channelThe independent cut diagrams contributing to this channel are shown in gure 4.
In addition to the two gluon distributions, F(1)gg and F(2)gg, used in ref. [13], the result to all orders in Nc involves a third distribution [12, 44], F(3)gg (also sometimes denoted
x2G(1) and called the Weizsacker-Williams gluon distribution), and the di erential cross section reads
dpA!qqX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fg/p(x1, [notdef]2)
3
Xi=1F(i)ggH(i)gg!qq , (4.14)
with the three gluon TMDs dened as
F(1)gg = 2 [integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
*Tr
"F () Tr
U[ ]
Nc U[]F (0) U[+][bracketrightBigg][angbracketrightBigg]
, (4.15)
F(2)gg = 2 [integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef] 1
Nc
DTr
hF () U[ ][bracketrightBig] Tr
hF (0) U[ ][bracketrightBig][angbracketrightBig]
, (4.16)
F(3)gg = 2 [integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
DTr
hF () U[+]F (0) U[+][bracketrightBig][angbracketrightBig]= x2G(1)(x2, kt) . (4.17)
15
JHEP09(2015)106
Figure 4. Diagrams for gg ! qq subprocess. The mirror diagrams of (3), (5) and (6) give identical
contributions.
The appropriate hard factors are constructed from the expressions corresponding to the diagrams (1)-(6) depicted in gure 4, using the following formulas
H(1)gg!qq =
Nc2CF D2 + D4 + 2D5 + 2D6 , (4.18)
H(2)gg!qq = 2N2cD3 D4 2D5 2D6 , (4.19)
H(3)gg!qq =
1 N2c 1
Nc2CF D1 +
D1
1 N2c 1
D2 + 2D3 . (4.20)
Again, the components Di = Cuihi were computed in [13] (table III) and they were used there to determine the hard factors H(1,2)gg!qq in the large Nc limit. Here, we generalize
the results of [13] to the full, nite-Nc case. The calculation can be most readily done by exploiting crossing symmetry that relates the qg ! qg and gg ! qq channels. This
allows for identication of the diagrams between gures 3 and 4 and enables one to recycle the Di expressions calculated in the previous subsection. For example, the expression corresponding to the diagram (1) from gure 4, with the incoming and the outgoing legs connected, is identical to the already computed expression for the diagram (4) from gure 3 (modulo a color averaging factor and swapping of the momenta p1 $ p). Similarly for all
the other diagrams. That gives the following set of hard factors for the gg ! qq subprocess:
H(1)gg!qq =
1 4CF
(t2 +[notdef]2)2
2[notdef]t , (4.21)
H(2)gg!qq =
1
2CF
t2 +[notdef]2
2 , (4.22)
H(3)gg!qq =
1 4N2cCF
t2 +[notdef]2
t[notdef] . (4.23)
Of the three hard factors, H(i)gg!qq, only two are independent. The third hard factor,
16
H(1)gg!qq + H(2)gg!qq[parenrightBig]
. (4.24)
Therefore, the cross section for quark-antiquark production can be rewritten with only two hard factors and two gluon distributions that are linear combinations of F(1)gg, F(2)gg and F(3)gg:
dpA!qqX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fg/p(x1, [notdef]2) [bracketleftBig]
JHEP09(2015)106
Figure 5. Set of diagrams for the gg ! gg subprocess involving only 3-gluon vertices. The mirror
diagrams of (3), (5) and (6) give identical contributions.
H(3)gg!qq, can be expressed as2
H(3)gg!qq =
1 N2c
(1)gg!qqK(1)gg!qq + (2)gg!qqK(2)gg!qq [bracketrightBig]
. (4.25)
In the above, we dened the new gluon TMDs as
(1)gg!qq =
1 N2c 1
N2cF(1)gg F(3)gg[parenrightBig]
, (4.26)
(2)gg!qq = N2cF(2)gg + F(3)gg , (4.27)
and the hard factors K(i)gg!qq as:
K(1)gg!qq =
N2c 1
N2c H(1)gg!qq and K(2)gg!qq =
1N2c H(2)gg!qq . (4.28)
The explicit expressions for the latter are given in table 1.
4.3 The gg ! gg channelFinally, the independent cut diagrams for the gg ! gg channel are given in gures 5 and 6,
and the corresponding di erential cross section for two-gluon production reads:
dpA!ggX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fg/p(x1, [notdef]2)
6
Xi=1F(i)ggH(i)gg!gg . (4.29)
2The same relation holds of course already at the level of eqs. (4.18)(4.20).
17
Figure 6. Set of diagrams for the gg ! gg subprocess involving 4-gluon vertex contributions. The
mirror diagrams of (8), (9) and (10) give identical contributions.
The F(1,2,3)gg distributions are the same as the ones introduced in the previous section in
eqs. (4.15)(4.17). The remaining three are [12]:
F(4)gg = 2[integraldisplay]
d+d2
(2)3pA
JHEP09(2015)106
eix2p
A+ikt[notdef]
DTr
hF () U[]F (0) U[][bracketrightBig][angbracketrightBig]
, (4.30)
F(5)gg = 2[integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
DTr
hF () U[ ]U[+]F (0) U[ ]U[+][bracketrightBig][angbracketrightBig]
, (4.31)
F(6)gg = 2[integraldisplay]
d+d2
(2)3pA
eix2p
A+ikt[notdef]
*Tr
hF () U[+]F (0) U[+][bracketrightBig]
Tr
U[ ]
NcTr
U[ ] Nc
+. (4.32)
The associated hard factors are constructed as:3
H(1)gg!gg =
1
2D1 +
1
2D2 + D4 + 2D5 + 2D6 , (4.33)
H(2)gg!gg = 2D3 D4 2D5 2D6 , (4.34)
H(6)gg!gg =
N2c
2 H(3)gg!gg = N2cH(4)gg!gg = N2cH(5)gg!gg =
1
2D2 + 2D3 . (4.35)
The calculation of the gg ! gg subprocess requires inclusion of diagrams with four-
gluon vertex. Therefore, in general, the expressions Di in the above equations contain contributions from both, the 3-gluon and 4-gluon vertex diagrams, the latter shown in gure 6. The corresponding expressions were computed in [13], where they were used to determine the hard factors in the large-Nc limit. Below, we generalize the result of ref. [13]
to the case of nite-Nc , with the help of the exact denitions given in eqs. (4.33)(4.35).
1
2D1 +
3Note that what is called H(3)gg!gg in ref. [13] is now H(6)gg!gg. Out of six hard factors, only H(1)gg!gg, H(2)gg!gg and H(6)gg!gg survive in the large-Nc limit.
18
h(3)i Ci
(1) 46 + 4t5 + 17t24 + 36t33 + 24t42 + 8t5 + 4t6
4t2
Nc 2CF
(2)6 + 2t5 + 33t24 + 60t33 + 44t42 + 16t5 + 4t6
4( + t)2
Nc 2CF
(3)
26 9t5 + 19t24 + 48t33 + 4t42 24t5 8t6
24t( + t)
Nc 4CF
(4) ( + 2t)2
2
Nc 2CF
JHEP09(2015)106
(5) ( + 2
t) 23 3t2 2t2 + 2t3 2 s3t
Nc 4CF
( + 2t)3 7t2 8t2 2t3 2
s3( + t)
Nc 4CF
Table 2. Expressions for the gg ! gg subprocess corresponding to diagrams (1)-(6) of g
ure 5, hence containing only 3-gluon vertices, in gauge (4.39) with non-vanishing 4-gluon vertex contributions.
The six hard factors read
H(1)gg!gg =
(6)
Nc CF
(t2 +[notdef]2)(2 t[notdef])2
[notdef]2t22 , (4.36)
H(2)gg!gg = =
2Nc CF
(2 t[notdef])2
[notdef]t2 , (4.37)
N2c
2 H(3)gg!gg = N2cH(4)gg!gg = N2cH(5)gg!gg =
H(6)gg!gg =
Nc CF
(2 t[notdef])2
[notdef]2t2 . (4.38)
To get further insight into the above results, we have performed an independent calculation in a gauge with non-vanishing 4-gluon vertex contribution, with the axial vectors dened as:n = p for the gluon k , n = k for the gluon p ,
n = p2 for the gluon p1 , n = p1 for the gluon p2 . (4.39)
The contributions to Dis in this gauge, coming from diagrams with 3-gluon vertices only and depicted in gure 5, are given in table 2.
In order to add the 4-gluon vertex contribution and obtain a full result for the Di coe cients, let us consider a general 4-gluon amplitude, shown on the left hand side of gure 7. A 3-gluon vertex brings a single SU(N) structure constant factor. Each amplitude in gure 5 consists of two 3-gluon vertices and that results in three possible color factor products
cs fa1ca4fca2a3 , ct fa1a2cfca3a4 , cu fa1a3cfca4a2 , (4.40)
19
a1 a2
a3
contributions from diagrams with 4-gluon vertex are depicted in gure 6, where the rst row shows the 4-gluon vertex amplitude squared, and the second row gives the interference terms with the three types of M3g amplitudes from eq. (4.41). A 4-gluon
vertex amplitude contains all three color factor products of eq. (4.40) at once
M4g = ctA4gt + csA4gs + cuA4gu . (4.44)
Therefore, all the contributions from gure 6 can be represented in the basis of the color factors dened in eq. (4.43). This allows us to distribute all the pieces of diagrams from gure 6 over the six Di expressions, needed to calculate the hard factors (4.33)(4.35), according to their color factors. Hence, the full expressions are
D1 = C1
h(3)1 + 2A4gtA3gt + A4gtA4gt[parenrightBig]
, (4.45)
D2 = C2
h(3)2 + 2A4guA3gu + A4guA4gu[parenrightBig]
, (4.46)
D3 = C3
h(3)3 + A4gtA4gu + A4gtA3gu + A4guA3gt[parenrightBig]
, (4.47)
D4 = C4
h(3)4 + 2A4gsA3gs + A4gsA4gs[parenrightBig]
, (4.48)
D5 = C5
h(3)5 + A4gtA4gs + A4gtA3gs + A4gsA3gt[parenrightBig]
, (4.49)
D6 = C6
h(3)6 + A4gsA4gu + A4guA3gs + A4gsA3gu[parenrightBig]
. (4.50)
20
b2 b1
b3 b4
Figure 7. Color indices for the cut four-gluon squared matrix element.
for the amplitudes with a gluon exchange in the t-, s- and u-channels, respectively. Each of the above amplitudes can now be written as
M3gi = ci A3gi , (4.41)
where i is either t, s or u, ci is a color factor from eq. (4.40), and A3gi is a corresponding
kinematic expression. The 3g superscript means that only 3-gluon vertices are involved in the given amplitude. Similarly, for the conjugate amplitudes, following the notation of gure 7, we have
cs fb1cb4fcb2b3 , ct fb1b2cfcb3b4 , cu fb1b3cfcb4b2 . (4.42)
That allows us to identify the color coe cients of the 3-gluon diagrams of gure 5 and write them in a compact form
(1) $ ctct , (2) $ cucu , (3) $ ctcu ,
(4) $ cscs , (5) $ csct , (6) $ cscu .
(4.43)
a4
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The O 2s
[parenrightbig]
Di
(1) Nc 24 + 23
t+ 32t2 + 8t3 + 6t4
CF2t2
(2) Nc4 + 43
t+ 152t2 + 16t3 + 6t4
CF2( + t)2
Nc4 +3t+ 72t2 + 12t3 + 6t4
2 CF2t( + t)
(4) Nc( + 2t)2 CF2
(5) Nc( 2t)( + t)( + 2t) 2CF2t
(6)
(3)
Nc t( + 2t)(3 + 2t)
2CF2( + t)
Table 3. Full expressions for the diagrams including three-gluon and four-gluon vertex contributions in the gauge (4.39).
The results for Dis in the gauge (4.39) are summarized in in table 3. Plugging those expressions into the hard factor denitions (4.33)(4.35) leads to the results identical to eqs. (4.36)(4.38).
We have already seen that not all of the six hard factors that arise in the gg ! gg
subprocess are independent. As shown in eq. (4.35), the expressions for H(3)gg!gg, H(4)gg!gg, H(5)gg!gg and H(6)gg!gg di er only by numerical factors. On top of that, when examining
further eqs. (4.33), (4.34) and (4.35), we see that the hard factors H(1)gg!gg, H(2)gg!gg and H(6)gg!gg are linearly dependent, that is
H(6)gg!gg = H(1)gg!gg + H(2)gg!gg . (4.51)
Hence, the cross section for two-gluon production from eq. (4.29) can be written in a much simpler, factorized form, with only two hard factors and two gluon distributions
dpA!ggX
d2Ptd2ktdy1dy2 =
2s(x1x2s)2 x1fg/p(x1, [notdef]2) [bracketleftBig]
JHEP09(2015)106
. (4.52)
In this channel, the new gluon TMDs, gg!gg, are dened as the following linear combi
nations of F(1)gg, F(2)gg, . . . , F(6)gg:
(1)gg!gg =
1
2
(1)gg!ggK(1)gg!gg + (2)gg!ggK(2)gg!gg
[bracketrightBig]
F(1)gg 2N2c F(3)gg +1N2c F(4)gg +1N2c F(5)gg + F(6)gg[parenrightbigg]
, (4.53)
(2)gg!gg = F(2)gg
2N2c F(3)gg +
1N2c F(4)gg +
1N2c F(5)gg + F(6)gg , (4.54)
and the new hard factors are:
K(1)gg!gg = 2H(1)gg!gg , and K(2)gg!gg = H(2)gg!gg . (4.55)
21
The explicit expressions are given in table 1. We note, that the above simplication occurs naturally when utilizing gauge invariance from the start, as we will show in section 6.
Finally, we point out that, in the large-Nc limit, all the distributions that were introduced in this section, F(1)qg F(2)qg, F(1)gg, F(2)gg, and F(6)gg, can be written in terms of xG(1)
and xG(2), and equivalence of formulas (4.13), (4.25) and (4.52) with CGC results is obtained [13].
Let use conclude that this part of our work brings two improvements to the current state of the art for the TMD factorization in forward dijet production. First of all, we have obtained nite-Nc corrections to the hard factors of ref. [13]. More importantly, however, we have eliminated the redundancy in the number of gluon distributions needed to write a factorization formula for this process, which now takes the compact form
dpA!dijets+X
d2Ptd2ktdy1dy2 =
2s (x1x2s)2
Xi=1K(i)ag!cd (i)ag!cd11 + cd , (4.56)
with only two gluon distributions and two hard factors required in each channel. Note that, as we shall discuss now, the incoming, small-x gluon is kept on-shell. Eqs. (4.56) will be further generalized to the case of the o -shell gluon in section 5.
4.4 The [notdef]kt[notdef] Qs limitFinally, let us consider the limit [notdef]kt[notdef] Qs. This is the dilute limit considered in section 3,
with the extra requirement that [notdef]kt[notdef] [notdef]Pt[notdef], needed for the validity of those formula. In that
limit, the transverse separation between the eld operators in the denition of the gluon distribution is restricted to values much smaller than the distance over which the Fourier integrand varies, and the dependence of the gauge links can be neglected. As a result, they simplify, and all the F(i)ag distributions coincide, except F(2)gg which vanishes. In terms
of the (1,2)ag!cd functions, all six distributions also reduce to that one gluon distribution,
which can therefore be identied with Fg/A/.
Then, for all channels, one can easily sum the surviving hard factors. In terms of diagrams, we always obtain D1 +D2 +2D3 +D4 +2D5 +2D6, meaning that we recover the collinear matrix elements. Indeed we have (noting that H(3)gg!gg + H(4)gg!gg + H(5)gg!gg = 0):
H(1)qg!qg + H(2)qg!qg = K(1)qg!qg + K(2)qg!qg =
2 +[notdef]2
JHEP09(2015)106
Xa,c,dx1fa/p(x1, [notdef]2)
2
t2
CF
Nc
2 +[notdef]2
[notdef] =
1g4 [notdef]Mqg!qg[notdef]2 , (4.57)
t2+[notdef]2
t2 +[notdef]2
H(1)gg!qq + H(3)gg!qq = K(1)gg!qq + K(2)gg!qq =
1
2Nc
t[notdef]
1
2CF
2 =
1g4 [notdef]Mgg!qq[notdef]2, (4.58)
1g4 [notdef]Mgg!gg[notdef]2 . (4.59)
Therefore, we recover the HEF formula (2.9), except that, due to the [notdef]kt[notdef] [notdef]Pt[notdef] limit, the
matrix elements are on-shell: the transverse momentum of the incoming gluon, kt, survives only in Fg/A. In other words, we recover the standard high-[notdef]Pt[notdef] limit:dpA!dijets+X
dy1dy2dP 2tdk2t
=
H(1)gg!gg + H(6)gg!gg = K(1)gg!gg + K(2)gg!gg =
2Nc CF
(2 t[notdef])3
2t2[notdef]2 =
Xa,c,d11 + cd x1fa/p(x1, [notdef]2)
d
ag!cddt Fg/A(x2, kt) , (4.60)
22
Figure 8. Four-parton amplitude with the incoming, small-x, o -shell gluon.
with d
ag!cd/dt= [notdef]Mag!cd[notdef]2/[16(x1x2s)2], and where Fg/A(x2, kt) can be identied with @/@k2t x2fg/A(x2, k2t), the derivative of the integrated gluon distribution.
In the following section, we shall restore the kt dependence of the hard factors. This will extend our formulas such that they recover the full HEF formula when the dilute limit is considered. As a result, we will obtain a unied description, valid for generic forward dijet system with [notdef]p1t[notdef], [notdef]p2t[notdef] Qs, without any additional requirement on the magnitude
of the transverse momentum imbalance kt.
5 Unied description of forward dijets in p+A collisions: TMD factorization with o -shell hard factors
We shall now generalize the hard factors that enter the TMD factorization formula (2.10) to the case with one of the incoming gluons being o the mass shell, as illustrated in gure 8. As it has been already stated, the motivation to include the o shellness is to be able to allow for congurations where the dijets are produced at any azimuthal angle (of course before application of a jet algorithm that will suppress very small angles and hence render the results nite).
As can be seen in gure 9 (as an example we chose only purely gluonic matrix element but the same structure occurs for the other channels), the on-shell matrix element misses substantial contributions when the jets are produced at small angles near = 0 and at small rapidity di erences Y = [notdef]y1 y2[notdef] [similarequal] 0. In such congurations, the matrix element
develops a structure that is divergent and it is suppressed only by a jet algorithm, which has to be applied in order to ensure two-jet congurations [16]. The matrix elements squared we are after, i.e. gg ! gg, gg ! qq and qg ! qg, can be extracted from the high energy
limit (or eikonal limit) of q g ! q g g and q g ! q qq and q q[prime] ! q q[prime] g [35]. In this approach the quark q is an auxiliary line to which the initial state o -shell gluon g couples eikonally.
The high energy factorization is a direct procedure where one uses the standard Feynman rules for all vertices and color factors, and xes the light-cone gauge for the on-shell gluons, using a gauge vector given by the longitudinal component of the o -shell, initial-state gluons momentum. In particular, if we apply the high energy factorization to the process we are after, we set the gauge vector to n = pA, where pA is the target four-momentum, as dened in gure 1 and eq. (2.2). Furthermore, the prescription is to associate with the o -shell gluon a longitudinal polarization vector, called nonsense polarization [1], of
23
JHEP09(2015)106
JHEP09(2015)106
Figure 9. Matrix elements squared for gg ! gg scattering with pt1 = pt2 = 4 GeV and s = 0.2.
Left: the on-shell case. Right: the o -shell case. Y and are, respectively, the di erences in rapidity and azimuthal angle of the two outgoing gluons.
the form4
[epsilon1]0 = i
p2 x2
|kt[notdef]
pA . (5.1)
As elaborated in ref. [10], longitudinally polarized gluons provide the dominant contribution to the cross section in the high energy limit. In the square amplitude, this leads to the polarization tensor of the form [10]
[epsilon1]0[epsilon1]0 = 2 x22
k2 pA pA , (5.2)
In the above, x2 = kp/pA p , which follows directly from the denition in eq. (2.6). The sum over polarizations of the on-shell gluons takes the standard form, with the gauge vector given by pA
X =[notdef]
[epsilon1] [epsilon1] = g
pAq + qpA
qpA , (5.3)
where, depending on the channel, q = p, p1 or p2, cf. eq. (4.39).
Let us note that the procedure outlined above denes the hard process in a gauge invariant manner only when a special choice for polarization vectors of the on-shell gluons is taken. In an arbitrary gauge, for internal and external gluon lines, more sophisticated methods have to be used, see e.g. [35, 4548].
To present our results in a compact form, with direct relation to the on-shell formulas from section 4, in addition to the standard Mandelstam variables given by eqs. (2.7), which now, however, sum up to + t+[notdef] = k2T , we introduce their barred versions, dened only
4The p2 factor in eq. (5.1) follows from a convention. It allows for use of the on-shell-like factor 12
in averaging over polarization, while calculating matrix elements squared, even in the case of the o -shell gluon, where the actual number of polarizations in the high energy limit is 1.
24
with the longitudinal component of the o -shell gluon
s = (x2pA + p)2 = [notdef]Pt[notdef]2 z(1 z)
+ [notdef]kt[notdef]2 = x1x2s , (5.4a)
t= (x2pA p1)2 = zs, (5.4b)
= (x2pA p2)2 = (1 z)s, (5.4c)
which are related via the equation
s + t+ = 0 . (5.5)
In the on-shell limit, k2T ! 0, the variables dened above recover the standard Mandelstam
variables from eq. (2.7)
lim
|kt[notdef]!0
(s ) = 0 , lim
|kt[notdef]!0
(t t) = 0 , lim
|kt[notdef]!0
( [notdef]) = 0 . (5.6)
As a consistency check, we have veried that, for all three subprocesses, the o -shell amplitudes that shall be used to build the hard factors in the remaining part of this section are identical to those rst calculated in ref. [11].
From this point onwards, we shall discuss our results only in terms of the new K(i) hard
factors and the new factorization formulas from eqs. (4.13), (4.25) and (4.52). The results for the old hard factors, H(i), in the o -shell case are given in appendix A for completeness.
5.1 The qg ! qg channelThe o -shell hard factors for this channel are obtained using denitions given in eq. (4.10)
and then eqs. (4.6) and (4.7). The corresponding Di expressions are collected in appendix A in table 8. The two hard factors read
K(1)qg !qg =
s2 +2
2tt[notdef]
JHEP09(2015)106
"[notdef] + s tt
N2c[bracketrightBigg]
, (5.7)
K(2)qg !qg =
CF
Nc
s s2 +2
tt[notdef] . (5.8)
In the limit [notdef]kt[notdef] ! 0, simplication given by eq. (5.6) occurs and the above formulas
manifestly recover the on-shell results from table 1.
5.2 The gg ! q q channelThe o -shell hard factors are obtained using denitions given in eq. (4.28) and then eqs. (4.18), (4.19) and (4.20). The corresponding Di expressions are collected in appendix A in table 9. The two hard factors take the following compact form
K(1)gg !qq =
1
2Nc
t2 +2
st[notdef]
[notdef] + tt
[bracketrightbig]
, (5.9)
K(2)gg !qq =
[notdef] + tt s
[bracketrightbig]
. (5.10)
Again, following eq. (5.6), it is manifest that the above hard factors reduce to those given in table 1, in the limit [notdef]kt[notdef] ! 0.
25
1 4N2cCF
t2 +2
st[notdef]
5.3 The gg ! gg channelIn the gauge chosen for our calculation, all the squared diagrams and interference terms that involve a 4-gluon vertex are identically zero. The corresponding Dis are given in table 10 of appendix A. Using the combinations from eqs. (4.33)(4.35) and then the denition from eq. (4.55) leads to the following set of the o -shell hard factors
2Nc
CF
Nc
CF
K(1)gg !gg =
(s2 t)2
tt[notdef]s
[notdef] + tt
[bracketrightbig]
, (5.11)
[notdef] + tt s
[bracketrightbig]
. (5.12)
The on-shell limit is again manifest, with the above equations reducing to those from table 1 as [notdef]kt[notdef] ! 0.
6 Helicity method for TMD amplitudes
In the preceding sections, the hard factors accompanying the gluon densities F(i)ag were calculated from the squared diagrams presented in gures 36. This procedure has certain drawbacks, especially when one would like to consider more complicated processes. For multiparticle processes, the color decompositions and helicity method [26, 49] are now considered as the most e ective ways to deal with them. Moreover, it is not obvious how the gauge invariance comes into play for the separate diagrams from gures 36 contributing to the hard factors. In the color decomposition method, the so-called color ordered amplitudes are gauge invariant from the start and one can use them directly to construct hard factors.
In view of the above, and to cross-check the results from section 5, we will give an alternative procedure to obtain the factorization formulas with o -shell gluon. To this end, we shall need TMD gluon densities corresponding to color decomposition of amplitudes and the color-ordered amplitudes themselves.
6.1 Color decompositions
Let us recall some basic facts about the color decompositions. We refer to [26, 49] for more details.
We rst consider a gluon amplitude Ma1...aN " 11, . . . , " NN
, where a1, . . . , aN are the external, adjoint color quantum numbers, the " ii is a polarization vector for a gluon i having momentum ki and helicity i = [notdef]. The fundamental color decomposition reads
Ma1...aN [parenleftBig]
[parenrightBig]
K(2)gg !gg =
(s2 t)2
tt[notdef]s
JHEP09(2015)106
" 11, . . . , " NN
=
X2SN1
Tr (ta1ta2 . . . taN ) M [parenleftBig]1 1, 22 . . . , NN
[parenrightBig]
, (6.1)
where the sum is over a set SN1 of all non-cyclic permutations of [notdef]1, . . . , N[notdef]. The coe -
cients of the expansion dene color ordered or dual amplitudes. They possess several useful properties. First of all, they are gauge invariant. Second, there are certain relations between dual amplitudes. Indeed, the following adjoint color decomposition involves only
26
(N 2)! di erent amplitudes [50]
Ma1...aN [parenleftBig]
" 11, . . . , " NN
[parenrightBig]
=
X2SN2(F a2 . . . F aN1 )a1aN M [parenleftBig]1 1, 22 , . . . , N1N1 , N N [parenrightBig]
,
(6.2)
where (F a)bc = fabc.
Consider now an amplitude involving a quark anti-quark pair MD1a2...aN1DN where Di, Dj are the color and the anti-color of the quark and the anti-quark, respectively. The color decomposition reads
MD1a2...aN1DN [parenleftBig]
1, " 22, . . . , " N1N1, N[parenrightBig]
=
JHEP09(2015)106
X2SN2(ta2 . . . taN1 )D1DN M [parenleftBig]1 1, 22 . . . , N1N1 , N N [parenrightBig]
. (6.3)
Now 1 and N are helicities of the quark and the anti-quark. For amplitudes involving more quark anti-quark pairs the decomposition is more complicated and we refer to [26] for details.
It is important to note that the above color decompositions work also for the case when one of the gluons is o -shell.
6.2 Gluon TMDs for color ordered amplitudes
Let us now nd the gluon TMDs corresponding to the color ordered amplitudes squared, as dened in the previous subsection. We constraint ourselves to the 2 ! 2 processes case
considered in this paper.
Let us rst consider the g (k4) g (k1) ! g (k3) g (k2) process. For the purpose of
this and next subsections we have assigned a new set of momenta to the partons. This assignment di ers from the one used before but it is more convenient when dealing with color ordered amplitudes. The correspondence is achieved by the following relations: k1 $ k, k2 $ p1, k3 $ p2, k4 $ p. Moreover, for the o -shell momentum we adopt a notation k1 = n1 + kT . (6.4)
The color decomposition of the four gluon amplitude reads
Ma1a2a3a4gg
!gg
n1, " 22, " 33, " 44[parenrightBig]= fa1a2cfca3a4 Mgg !gg
1 , 2 2, 3 3, 4 4
[parenrightBig]
+ fa1a3cfca2a4 Mgg
!gg
1 , 3 3, 2 2, 4 4[parenrightBig]
, (6.5)
where n1 is placed for the o -shell gluon instead of a polarization vector (in fact it plays a similar role). As far as dual amplitudes are concerned, we indicate the o -shell gluon by a star. In table 4. we calculate the gluon TMDs that correspond to the color structures exposed in (6.5) (after squaring). They agree with the gluon TMDs calculated in [12] and listed in rows 1 and 3 of table 8 of [12]. That table denes one more gluon TMD (the row 2) which however is redundant. Clearly, the color decomposition (6.5) gives all the necessary color structures and already incorporates the gauge invariance. In summary, the
27
color-ordered amplitude squared
gluon TMD
[vextendsingle][vextendsingle]M
gg !gg 1 , 2 2, 3 3, 4 4
[parenrightbig][vextendsingle][vextendsingle]2
(1)gg!gg =
1
2N2c N2cF(1)gg 2F(3)gg+F(4)gg + F(5)gg + N2cF(6)gg
[vextendsingle][vextendsingle]M
gg !gg 1 , 3 3, 2 2, 4 4
[parenrightbig][vextendsingle][vextendsingle]2
[parenrightbig]
Mgg !gg 1 , 2 2, 3 3, 4 4
[parenrightbig]
M gg !gg 1 , 3 3, 2 2, 4 4
[parenrightbig]
(2)gg!gg =
1N2c N2cF(2)gg 2F(3)gg
+F(4)gg + F(5)gg + N2cF(6)gg
M gg !gg 1 , 2 2, 3 3, 4 4
Mgg !gg 1 , 3 3, 2 2, 4 4
Table 4. Gluon TMDs accompanying the color-ordered amplitudes for gg ! gg process. It has
been assumed that TMDs are real. The F(i)gg distributions are dened in eqs. (4.15), (4.16), (4.17)
and in eqs. (4.30), (4.31), (4.32).
color-ordered amplitude squared
gluon TMD
[vextendsingle][vextendsingle]M
gg !qq 2 2, 1 , 4 4, 3 3
[parenrightbig][vextendsingle][vextendsingle]2
(1)gg!qq =
1 N2
c
1 N2cF(1)gg F(3)gg[parenrightBig]
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[vextendsingle][vextendsingle]M
gg !qq 2 2, 4 4, 1 , 3 3
[parenrightbig][vextendsingle][vextendsingle]2
Mgg !qq 2 2, 1 , 4 4, 3 3
[parenrightbig]
M gg !qq 2 2, 4 4, 1 , 3 3
[parenrightbig] (2)gg!qq = N2cF(2)gg + F(3)gg
M gg !qq 2 2, 1 , 4 4, 3 3
[parenrightbig]
Mgg !qq 2 2, 4 4, 1 , 3 3
[parenrightbig]
Table 5. Gluon TMDs accompanying the color-ordered amplitudes for gg ! qq process. It has
been assumed that correlators are real. The F(i)gg distributions are dened in eqs. (4.15), (4.16)
and (4.17).
two gluon TMD listed in table 4 are the only relevant TMDs and correspond to the two independent gauge invariant amplitudes squared and their interference.
Now, let us turn to the g (k4) g (k1) ! q (k3) q (k2) process. The color decomposi
tion reads
MD2a1a4D3gg !qq [parenleftBig]
2, n1, " 44, 3
[parenrightBig]
= (ta1ta4)D2D3 Mgg
2 2, 1 , 4 4, 3 3
[parenrightBig]
2 2, 4 4, 1 , 3 3[parenrightBig]
. (6.6)
The gluon TMDs corresponding to the color structures appearing after squaring this equation are gathered in table 5. They correspond to rows 1 and 5 of table 7 in [12]. Again, we have only two independent TMDs that are needed.
28
+ (ta4ta1)D2D3 Mgg
color-ordered amplitude squared
gluon TMD
Mqg !qg 3 3, 1 , 2 2, 4 4
[parenrightbig]
M qg !qg 3 3, 2 2, 1 , 4 4
[parenrightbig] (1)qg!qg = F(1)qg
M qg !qg 3 3, 1 , 2 2, 4
[parenrightbig]
Mqg !qg 3 3, 2 2, 1 , 4 4
[parenrightbig]
[vextendsingle][vextendsingle]M
qg !qg 3 3, 2 2, 1 , 4 4
[parenrightbig][vextendsingle][vextendsingle]2
[vextendsingle][vextendsingle]M
qg !qg 3 3, 1 , 2 2, 4 4
[parenrightbig][vextendsingle][vextendsingle]2
1 F(1)qg + N2cF(2)qg[parenrightBig]
Table 6. Gluon TMDs accompanying the color-ordered amplitudes for qg ! qg process. It has
been assumed that correlators are real. The F(i)qg distributions are dened in eqs. (4.4) and eqs. (4.5).
For the process q (k4) g (k1) ! q (k3) g (k2), the color decomposition reads
MD3a1a2D4qg
!qg
(2)qg!qg =
1 N2
c
3, n1, " 22, 4[parenrightBig]= (ta1ta2)D3D4 Mqg !qg
3 3, 1 , 2 2, 4 4
[parenrightBig]
+ (ta2ta1)D3D4 Mqg
!qg
JHEP09(2015)106
3 3, 2 2, 1 , 4 4[parenrightBig]
. (6.7)
For anti-quarks we need to exchange the indices 3 $ 4. The TMDs corresponding to those
processes are given in table 6. In general, the TMDs for a sub-process with anti-quarks are di erent than for quarks, but they turn out to be the same assuming that the correlators are real. Again, we end up with only two independent TMDs.
6.3 O -shell color-ordered helicity amplitudes
In section 5, we have calculated the o -shell hard factors in a specic axial gauge, with pA chosen as the gauge vector, and using the high energy projector (5.1). As shown in ref. [10], such a procedure yields results which are gauge invariant within a subclass of axial gauges with the gauge vector n = app + bpA, where a and b are arbitrary complex numbers. There are also methods to calculate gauge invariant o -shell amplitudes in any gauge and choice of polarization vectors [34, 35, 47, 48]. In what follows, we shall use those methods and specically the results of [35, 48].
Consider rst the gluon amplitudes. For the purpose of this section only we assume all momenta to be outgoing. For the non-vanishing helicity congurations, in the helicity basis, we have
Mg g!gg 1 , 2, 3+, 4+
[parenrightbig]
= 2g2 1 [angbracketleft]1 2[angbracketright]4
h1 2[angbracketright][angbracketleft]23[angbracketright][angbracketleft]34[angbracketright][angbracketleft]41 [angbracketright]
, (6.8)
Mg g!gg 1 , 2+, 3, 4+
[parenrightbig]
= 2g2 1 [angbracketleft]1 3[angbracketright]4
h1 2[angbracketright][angbracketleft]23[angbracketright][angbracketleft]34[angbracketright][angbracketleft]41 [angbracketright]
, (6.9)
Mg g!gg 1 , 2+, 3+, 4
[parenrightbig]
= 2g2 1 [angbracketleft]1 4[angbracketright]4
h1 2[angbracketright][angbracketleft]23[angbracketright][angbracketleft]34[angbracketright][angbracketleft]41 [angbracketright]
, (6.10)
29
where we adopted a shorthand notation for the spinor products [angbracketleft]ij[angbracketright] = [angbracketleft]ki [notdef]kj+[angbracketright] with
|ki[notdef][angbracketright] = 12 (1 [notdef] 5) u (ki), and where 1 is a, for our purposes irrelevant, phase factor (see
details e.g. in [48]). We also dened [angbracketleft]1 i[angbracketright] = [angbracketleft]n1i[angbracketright] with n1 being the longitudinal component
of k1, cf. eq. (6.4). The other remaining helicity congurations can be obtained from eqs. (6.8)(6.10) using CP invariance
Mgg !gg 1 , 2+, 3, 4
[parenrightbig]
= M gg !gg 1 , 2, 3+, 4+
[parenrightbig]
, (6.11)
and so on. For the other color ordered amplitude, Mgg
!gg (1 , 3, 2, 4), we need to exchange
2 $ 3 in the denominators.
The above helicity amplitudes can be e ciently evaluated and squared numerically, however for the purpose of this paper we shall need analytic expressions. To this end let us introduce [ij] = [angbracketleft]ki + [notdef]kj[angbracketright], which, up to an unimportant phase, is a complex conjugate
of [angbracketleft]ij[angbracketright]. Moreover, we have the following relation
hij[angbracketright][ji] = (ki + kj)2 ij. (6.12)
For the products involving n1 we use the notation
h1 i[angbracketright][i1 ] = (n1 + ki)2 1 i. (6.13)
With this, we get for the required amplitudes squared summed and averaged over helicities
[vextendsingle][vextendsingle]M
gg !gg (1 , 2, 3, 4)
JHEP09(2015)106
[vextendsingle][vextendsingle]2
= 8g441 2 +41 3 +41 41 2233441 , (6.14)
[vextendsingle][vextendsingle]M
= 8g441 2 +41 3 +41 41 3322441 , (6.15)
Mgg !gg (1 , 2, 3, 4) M gg !gg (1 , 3, 2, 4) = 8g4
gg !gg (1 , 3, 2, 4)
[vextendsingle][vextendsingle]2
41 2 +41 3 +41 4
h1 2[angbracketright][angbracketleft]34[angbracketright][1 3][24]2341 , (6.16)
M gg !gg (1 , 2, 3, 4) Mgg
!gg (1 , 3, 2, 4) = 8g4
41 2 +41 3 +41 4
[1 2][34][angbracketleft]1 3[angbracketright][angbracketleft]24[angbracketright]2341
, (6.17)
where we have used overlines to indicate helicity summations. The last two interference terms enter the cross section as a sum. Therefore, we may simplify it as
Mgg !gg (1 , 2, 3, 4) M gg !gg (1 , 3, 2, 4) + M gg !gg (1 , 2, 3, 4) Mgg !gg (1 , 3, 2, 4)
= 8g4
(41 2 +41 3 +41 4)(241 3 231
4 +341 2)
1 2341 3242341 , (6.18)
where we have used
[1 2][34][angbracketleft]1 3[angbracketright][angbracketleft]24[angbracketright] + [angbracketleft]1 2[angbracketright][angbracketleft]34[angbracketright][1 3][24] = [angbracketleft]n1[notdef]p/3p/4p/2[notdef]n1[angbracketright] + [angbracketleft]n1[notdef]p/2p/4p/3[notdef]n1[angbracketright] , (6.19) and applied p/ip/j =ij p/jp/i a few times. The amplitudes for the on-shell limit are simply
obtained by dropping the star in 1 so that the spinor and the scalar products will be with k1 instead of n1.
30
Now let us turn to processes with quarks. We will give only amplitudes for g (k4) g (k1) ! q (k3) q (k2) process, as all the other can be obtained by the crossing sym
metry (taking care of the proper color ow when crossing). We have
Mgg !qq 3, 1 , 4+, 2+
[parenrightbig]
= 2g2 1 [angbracketleft]21 [angbracketright]3[angbracketleft]31 [angbracketright] h21 [angbracketright][angbracketleft]1 4[angbracketright][angbracketleft]43[angbracketright][angbracketleft]32[angbracketright]
, (6.20)
Mgg !qq 3+, 1 , 4+, 2
[parenrightbig]
= 2g2 1 [angbracketleft]31 [angbracketright]3[angbracketleft]21 [angbracketright] h21 [angbracketright][angbracketleft]1 4[angbracketright][angbracketleft]43[angbracketright][angbracketleft]32[angbracketright]
. (6.21)
We note that the above formulas have never been published in the literature and are given here for the rst time.
Similar as before, the two remaining helicity congurations can be obtained thanks to CP symmetry. For the color ordered amplitudes with 1 and 4 interchanged, we need to make a replacement 1 $ 4 in the denominators. The amplitudes squared and summed
over helicities read (the helicity averaging factor is included)
[vextendsingle][vextendsingle]M
gg !qq (3, 1 , 4, 2)
[vextendsingle][vextendsingle]2
= 2g41
321 2 +21 3
~s1 43423 , (6.22)
[vextendsingle][vextendsingle]M
gg !qq (3, 4, 1 , 2)
[vextendsingle][vextendsingle]2
= 2g41
221 2 +21 3
~s1 42423 , (6.23)
Mgg !qq (3, 1 , 4, 2) M gg !qq (3, 4, 1 , 2) = 2g4
1 21 321 2 +21 3
[parenrightbig]
JHEP09(2015)106
h21 [angbracketright][angbracketleft]43[angbracketright][31 ][42]2341 , (6.24)
M gg !qq (3, 1 , 4, 2) Mgg
!qq (3, 4, 1 , 2) = 2g4
1 21 321 2 +21 3
[21 ][43][angbracketleft]31 [angbracketright][angbracketleft]42[angbracketright]2341 . (6.25)
The sum of the last two interference terms simplies to
Mgg !qq (3, 1 , 4, 2) M gg !qq (3, 4, 1 , 2) + M gg !qq (3, 1 , 4, 2) Mgg !qq (3, 4, 1 , 2)
= 2g4
4 +341 2)
1 2341 3242341 . (6.26)
In order to obtain amplitudes for q (k4) g (k1) ! q (k3) g (k2) we can use the crossing symmetry. Specically, we can obtain
[vextendsingle][vextendsingle]M
qg !qg (3, 1 , 2, 4)
1 21 3(21 2 +21 3)(241 3 231
[vextendsingle][vextendsingle]2, [vextendsingle][vextendsingle]M
qg !qg (3, 2, 1 , 4)
[vextendsingle][vextendsingle]2
and
interference terms by making replacement 2 $ 4 in eqs. (6.23), (6.22), (6.26) respectively.6.4 Hard factors from color-ordered amplitudes
Having computed the color ordered amplitudes it is now straightforward to calculate the hard factors K(i). Let us note, that it is the K(i) hard factors that appear naturally within the color-ordered formalism, not the H(i) factors. It also comes naturally that there are two hard factors and two TMDs per each channel, so the the factorization formulas can be written in a unied form:
dpA!dijets+X
d2Ptd2ktdy1dy2 =
2s (x1x2s)2
Xi=1K(i)ag !cd (i)ag!cd11 + cd , (6.27)
where a, c, d are the contributing partons. The explicit expressions for the generalized gluon TMDs (i)ag!cd are listed in tables 46. The hard factors Ki were already given in
31
Xa,c,dx1fa/p(x1, [notdef]2)
2
i
1
2
K(i)gg !gg
Nc CF
(s4 + t4 + u4) u[notdef] + tt
tt[notdef]s
Nc2CF(s4 + t4 + u4)
u[notdef] + tt s tt[notdef]s
K(i)gg !qq
1
2Nc
(t2 + u2) u[notdef] + tt
s st[notdef]
4N2cCF(t2 + u2)
u[notdef] + tt s s st[notdef]
1
CFNcs s2 + u2
t t[notdef]
Table 7. The hard factors accompanying the gluon TMDs (i)ag!cd.
section 5 (we collect them in table 7 for convenience). In the context of this section, they are obtained by multiplying the left column of tables 46 by the corresponding color factors and combining the cells that belong to the same generalized TMD. More precisely, we have
g4 K(1)gg !gg =
1 (2NcCF )2
K(i)qg !qg
u s2 + u2
2 tt
1 + s tt N2c u[notdef] [parenrightbigg]
N3cCF 2
[parenleftBig][vextendsingle][vextendsingle]
Mgg !gg (1 , 2, 3, 4)
[vextendsingle][vextendsingle]2
+
[vextendsingle][vextendsingle]M
gg !gg (1 , 3, 2, 4)
[vextendsingle][vextendsingle]2[parenrightBig]
,
(6.28)
JHEP09(2015)106
g4 K(2)gg !gg =
1 (2NcCF )2
N3cCF 4
Mgg !gg (1 , 2, 3, 4) M gg !gg (1 , 3, 2, 4) + c.c.[parenrightBig]
,
(6.29)
for pure gluon channel, and
g4 K(1)gg !qq =
1(2NcCF )2 NcC2F [parenleftBig][vextendsingle][vextendsingle]
Mgg !qq (3, 1 , 4, 2)
[vextendsingle][vextendsingle]2
+
[vextendsingle][vextendsingle]M
gg !qq (3, 4, 1 , 2)
[vextendsingle][vextendsingle]2[parenrightBig]
,
(6.30)
Mgg !qq (3, 1 , 4, 2) M gg !qq (3, 4, 1 , 2) + c.c.[parenrightBig]
, (6.31)
for gg ! qq channel. For the qg ! qg sub-process we need to use the crossing symmetry
as described in the preceding section. We have
g4 K(1)qg !qg =
1
2CF N2c
g4 K(2)gg !qq =
1 (2NcCF )2
CF
2
NcC2F
[vextendsingle]
Mgg !qq (3, 1 , 4, 2)
[vextendsingle]2[parenrightBig]
2$4
CF 2
Mgg !qq (3, 1 , 4, 2) M gg !qq (3, 4, 1 , 2) c.c.[parenrightBig] 2$4
,
(6.32)
g4 K(2)qg !qg =
1
2CF N2c NcC2F [parenleftBig] [vextendsingle][vextendsingle]M
gg !qq (3, 4, 1 , 2)
[vextendsingle][vextendsingle]2[parenrightBig]
. (6.33)
In all the formulas above, the rst color factor comes from color averaging. The minus signs in front of the amplitudes in (6.32), (6.33) come from the crossing of a fermion line.
32
2$4
Table 7 is easily recovered using the following relations ofij to the kinematic variables from section 5
23 =14 =,34 =12 = t,24 =13 =[notdef] , (6.34)
1 4 = s,1 2 = t,1 3 = . (6.35)
7 Conclusions and outlook
Dijet production is one of the key processes studied at the LHC. Requiring the two jets to be produced in the forward direction creates an asymmetric situation, in which one of the incoming hadrons is probed at large x, while the other is probed at a very small momentum fraction. This kinematic regime poses various challenges, one of the biggest questions being the existence of a theoretically-consistent and, at the same time, practically-manageable factorization formula. The standard collinear factorization is not applicable in this case as the dependence on the transverse momentum of the low-x gluon in the target, kt, cannot be neglected.
In the limit where the jets transverse momenta [notdef]p1t[notdef], [notdef]p2t[notdef] [notdef]kt[notdef] Qs, with the latter being the saturation scale of the target, an e ective transverse-momentum-dependent factorization formula for forward dijet production has been derived in refs. [13, 14] and it has been shown to be consistent with the CGC framework. On the other side, the high energy factorization approach [10, 11] has been also successfully applied for studying forward dijet production at the LHC. In this paper, we have examined the theoretical status of the HEF approach in the context of forward dijet production at hadron colliders and reconciled it with the TMD factorization by creating a unied framework valid in the limit [notdef]p1t[notdef], [notdef]p2t[notdef] Qs with an arbitrary value of [notdef]kt[notdef], as long as it is allowed by phase space
constraints. In particular, we have shown in section 3 that the HEF formula is indeed justied in the kinematic window of [notdef]p1t[notdef], [notdef]p2t[notdef] [notdef]kt[notdef] Qs, where it was explicitly derived
from CGC for all 2 ! 2 channels. This limit corresponds to the dilute target approximation
hence no non-linear e ects are expected.
The second major result of our work is an improvement of the e ective TMD factorization for forward dijet production, rst derived in ref. [13], by taking into account in section 4 all nite-Nc corrections, as well as generalizing the factorization formula to the case with an o -shell incoming gluon in sections 5 and 6. In addition, we were able to simplify the TMD factorization formula by reducing the number of gluon distributions to two independent TMDs for each channel. The main results of this part of our study are summarized in eq. (6.27), which gives the new TMD factorization formula, as well as in table 7, where we collect all the o -shell hard factors. The corresponding gluon distributions are given in tables 4, 5 and 6. The above results were obtained with two independent techniques: a traditional Feynman diagram approach and helicity methods with color ordered amplitudes. The improved TMD factorization formula (6.27) encapsulates both the result of ref. [13] and the HEF framework as its limiting cases.
The results obtained in this paper open several avenues for future research that we plan to follow. First, a natural next steps will be to use eq. (6.27) for phenomenological
33
JHEP09(2015)106
studies. That shall require some input for the six gluon TMDs (1,2)ag!cd(x, kt), which may be di cult in a general case. But in the large-Nc limit, they can all be written in terms of just two functions: xG(1)(x, kt) and xG(2)(x, kt), which in turn can be evaluated within
certain models, as in [5].Another line of possible extension of our framework is to supplement it with high-
|Pt[notdef] e ects such as Sudakov logarithms or coherence in the evolution of the gluon density.
Essentially, this can be done by adding a [notdef]2 dependence to the unintegrated gluon distributions [2931, 5154]. The equations that combine such e ects with the small-x evolution [55, 56] show a nontrivial interplay between the non-linearities and the [notdef]2 dependence and this may, in particular, weaken the saturation e ects. At the linear level, the so-called single step inclusion of the hard-scale e ects (as demonstrated in [17]) helps in the description of forward-central dijet data, therefore this direction seems to be relevant in order to provide complete predictions. Furthermore, rst estimates of azimuthal decorrelations of the forward-forward dijets in the HEF framework, with inclusion of hard scale e ects and non-linearities, show that they are of similar relevance for this process [33].
Last but not least, it remains to be proved that the large logarithms generated by higher-order corrections can indeed be absorbed into evolution equations for the various parton distributions (and jet fragmentation functions) involved, and potentially for additional soft factors [57]. This limitation however is not specic to our work, the same is true at the level of the TMD and HEF regimes independently. In the former case, it is known that TMD factorization generically does not apply for dijet production in hadron-hadron collisions [22, 24]. It is nevertheless expected that, in dilute-dense collisions, initial state interactions originating from a dilute hadron do not interfere with the intrinsic transverse momentum and thus factorization may hold, although there is no formal proof of this statement yet.
In addition, even though it was possible to write formula (4.56) in terms of just two TMDs per channel, this simplication may not survive after small-x evolution is included, as, in general, the non-linear equations mix the original F(i)ag functions. For instance, xG(1)
does not obey a closed equation and, contrary to what happens with xG(2), the large-Nc
limit does not help [58]. We note that any equivalent linear combination of the gluon distributions, such as (2.10) and (4.56), is equally valid, and it may turn out that some alternative choice allows one to write the evolution equations directly in terms of TMDs. By contrast, it is also possible that the inclusion of small-x evolution can only be achieved within the full complexity of the CGC, meaning that the Qs [notdef]kt[notdef] [notdef]Pt[notdef] limit, which allows one to avoid the quadrupole operator in (3.10) and express the cross section in terms of gluon distributions, may not help when small-x evolution is considered.
In the HEF regime, the issues are di erent. The Qs [notdef]kt[notdef] [notdef]Pt[notdef] limit makes things
simpler from the point of view of small-x evolution, since non-linear e ects can be neglected. However, the o -shellness of the hard process is not neglected and thus the standard power counting of the twist expansion becomes useless. One must then resort to di erent methods, such as those of ref. [59]. Any progress towards an all-order proof of either HEF or TMD factorization for forward dijet production in dilute-dense collisions will naturally carry
34
JHEP09(2015)106
over to our improved TMD factorization formula (6.27) that combines both regimes. In the meantime, our results represent a viable alternative to CGC calculations, equivalent to them in the kinematic regime appropriate for dijets Qs [notdef]Pt[notdef] but more practical.
Acknowledgments
The work of K.K. has been supported by Narodowe Centrum Nauki with Sonata Bis grant DEC-2013/10/E/ST2/00656. P.K. acknowledges the support of the grants DE-SC-0002145 and DE-FG02-93ER40771. S.S. acknowledges useful discussions with Gavin Salam and Fabrizio Caola. P.K., K.K., S.S. and A.vH. are grateful for hospitality tocole Polytechnique, where part of this work has been carried out. K.K. thanks for the hospitality of Penn State University, where part of this research was done.
A O -shell expressions
In this appendix, we gather all expressions corresponding to the Di diagrams from gures 3 6 in the case where one of the incoming gluons is o -shell. All calculations were preformed in the axial gauge discussed at the beginning of section 5, with the axial vectors for the on-shell gluons set according to eq. (4.39).
For completeness, we also give here the results for the old hard factors dened in eqs. (4.6), (4.7) (4.18), (4.19), (4.20), (4.33), (4.34) and (4.35), in the case with o -shell incoming gluon.
Table 8 gives the Di expressions for the subprocesses qg ! qg. The two hard factors
in this channel read
JHEP09(2015)106
H(1)qg !qg =
s2 +2
2t
" tt
N2c[notdef]
[bracketrightBigg]
, (A.1)
s s2 +2
2
utt . (A.2)
In the limit, [notdef]kt[notdef] ! 0, simplication given by eq. (5.6) occurs and the above formulas
manifestly recover the on-shell results from eqs. (4.8) and (4.9).The corresponding Di results for the gg ! qq subprocess are given in table 9. The
three old, o -shell hard factors for this channel take the form
H(1)gg !qq =
H(2)qg !qg =
1 4CF
t2 +2
[notdef]ts
[notdef] + tt
[bracketrightbig]
, (A.3)
H(2)gg !qq =
1 4CF
t2 +2
[notdef]ts
s tt[notdef]
[bracketrightbig]
, (A.4)
H(3)gg !qq =
1 4N2cCF
t2 +2
[notdef]t . (A.5)
Again, following eq. (5.6), it is manifest that the above hard factors reduce to eqs. (4.21) (4.23) in the limit [notdef]kt[notdef] ! 0.
35
qg ! qg Di
(1) 2t + t2 + 22
t + t+[notdef]
[parenrightbig]
(2) CF
Nc
2t + t2 + 22
ut + t+[notdef]
(3) 2
t + t2 + 22
[parenrightbig] t s
+ t
+ ( +[notdef])
4 t[notdef]t + t+[notdef]
JHEP09(2015)106
(4)
CF
Nc
(t+) 2t + t2 + 22
st + t+[notdef]
2t + t2 + 22
[parenrightbig] t s
t
+ ( +[notdef])
4 stt + t+[notdef]
(5)
(6) 1
N2c
2t + t2 + 22
[parenrightbig] t s
+ t
+ ( [notdef])
4 s[notdef]t + t+[notdef]
Table 8. Expressions for the qg ! qg subprocess with o -shell incoming gluon corresponding to
diagrams (1)-(6) of gure 3 in gauge described in section 5.
gg ! qq Di
(1) 1
Nc
(s +) 2s + s2 + 22
2 ts + t+[notdef]
(2)
1 Nc
2s + s2 + 22
2
us + t+[notdef]
(3)
1 N2cCF
2s + s2 + 22
[parenrightbig] s s
+ t
[parenrightbig]
+ t[notdef]
[parenrightbig][parenrightbig]
8t[notdef]s + t+[notdef]
[parenrightbig]
(4)
1 CF
2s + s2 + 22
2 + t+[notdef]
[parenrightbig]
(5)
1 CF
2s + s2 + 22
[parenrightbig] s s
t
[parenrightbig]
t+[notdef]
[parenrightbig][parenrightbig]
8ts + t+[notdef]
[parenrightbig]
(6)
1 CF
2s + s2 + 22
[parenrightbig] s s
+ t
[parenrightbig]
+ t+[notdef]
[parenrightbig][parenrightbig]
8[notdef]s + t+[notdef]
[parenrightbig]
Table 9. Expressions for the gg ! qq subprocess with o -shell incoming gluon corresponding to
diagrams (1)(6) of gure 4 in gauge described in section 5.
36
gg ! gg Di
(1) 2Nc
CF
t + t2 +2
2 t (t+) + t+[notdef]
(2) 2Nc
CF
t + t2 +2
2 [notdef]t(t+) + t+[notdef]
[parenrightbig]
(3) Nc
2CF
t + t2 +2
t
2 + t
[parenrightbig]+ ( +[notdef])
t[notdef]t (t+) + t+[notdef]
(4)
JHEP09(2015)106
2Nc CF
t + t2 +2
2 t + t+[notdef]
[parenrightbig]
Nc 2CF
t + t2 +2
t
2 t
[parenrightbig]+ ( +[notdef])
(5)
stt (t+) + t+[notdef]
Nc 2CF
t + t2 +2
t
2 + t
[parenrightbig]+ ( [notdef])
(6)
s[notdef]t (t+) + t+[notdef]
Table 10. Expressions for the gg ! gg subprocess with o -shell incoming gluon in gauge described
in section 5. The numbering (1)(6) corresponds to the color structures as dened in eq. (4.43) and each expression contains contributions from diagrams with both 3- and 4-gluon vertices.
Finally, the Di expressions for the subprocess gg ! gg are given in table 10 and the
six hard factors read
H(1)gg !gg =
Nc CF
(s2 t)2
tt[notdef]s
tt+[notdef]
[bracketrightbig]
, (A.6)
H(2)gg !gg =
Nc CF
(s2 t)2
tt[notdef]s
s tt[notdef]
[bracketrightbig]
, (A.7)
N2c
2 H(3)gg !gg = N2cH(4)gg !gg = N2cH(5)gg !gg =
(s2 t)2
tt[notdef] . (A.8)
The on-shell limit is again manifest, with the above equations reducing to eqs. (4.36), (4.37) and (4.38) as [notdef]kt[notdef] ! 0.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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SISSA, Trieste, Italy 2015
Abstract
Abstract
We study forward dijet production in dilute-dense hadronic collisions. By considering the appropriate limits, we show that both the transverse-momentum-dependent (TMD) and the high-energy factorization formulas can be derived from the Color Glass Condensate framework. Respectively, this happens when the transverse momentum imbalance of the dijet system, k ^sub t^, is of the order of either the saturation scale, or the hard jet momenta, the former being always much smaller than the latter. We propose a new formula for forward dijets that encompasses both situations and is therefore applicable regardless of the magnitude of k ^sub t^. That involves generalizing the TMD factorization formula for dijet production to the case where the incoming small-x gluon is off-shell. The derivation is performed in two independent ways, using either Feynman diagram techniques, or color-ordered amplitudes.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer