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Eur. Phys. J. C (2021) 81:846https://doi.org/10.1140/epjc/s10052-021-09593-9
Regular Article- Theoretical Physics
Exclusive production of ρ-mesons in high-energy factorization atHERA and EIC
A. D. Bolognino1,2,a , F. G. Celiberto3,4,5,b , D. Yu. Ivanov6,c , A. Papa1,2,d , W. Schäfer7,e , A. Szczurek7,8,f
1 Dipartimento di Fisica, Università della Calabria, Arcavacata di Rende, 87036 Cosenza, Italy2 Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza, Arcavacata di Rende, 87036 Cosenza, Italy3 European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villazzano, 38123 Trento, Italy4 Fondazione Bruno Kessler (FBK), Povo, 38123 Trento, Italy5 INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Povo, 38123 Trento, Italy6 Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia7 Institute of Nuclear Physics Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, Poland8 College of Natural Sciences, Institute of Physics, University of Rzeszów, ul. Pigonia 1, 35-310 Rzeszów, Poland
Received: 30 July 2021 / Accepted: 30 August 2021 / Published online: 23 September 2021© The Author(s) 2021
Abstract We study cross sections for the exclusive diffrac-tive leptoproduction of ρ-mesons, γ ∗ p → ρ p, within theframework of high-energy factorization. Cross sections forlongitudinally and transversally polarized mesons are shown.We employ a wide variety of unintegrated gluon distribu-tions available in the literature and compare to HERA data.The resulting cross sections strongly depend on the choice ofunintegrated gluon distribution. We also present predictionsfor the proton target in the kinematics of the BrookhavenEIC.
1 Introduction
The diffractive exclusive electroproduction of vector mesonshas been a very active field of study at the HERA collider atDESY. Especially it has served as a testbed for the perturba-tive QCD description of the Pomeron exchange mechanismwhich drives diffractive and elastic processes [1,2]. New pre-cise data, albeit in a different kinematic regime, are expectedto be taken at the Brookhaven electron-ion collider (EIC) in anot too distant future [3,4]. In this work we focus on the elec-troproduction of ρ-mesons, i.e. on the process γ ∗ p → ρpat large virtuality Q2 of the photon and large γ ∗ p-systemcenter-of-mass energy W , with x � Q2/W 2 � 1.
a e-mail: [email protected] e-mail: [email protected] (corresponding author)c e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
The large photon virtuality Q2 justifies the use of pertur-bation theory. A diagrammatic representation of the pertur-bative QCD factorization structure of the forward amplitudeis shown in Fig. 1. At large Q2 the transverse internal motionof quark and antiquark in the ρ-meson can be integrated out,and all information is contained in the distribution ampli-tude (DA). The transverse momenta of gluons in the protonmust be fully taken into account by utilizing the unintegratedgluon distribution (UGD). In the BFKL-approach the UGD isobtained by convoluting the BFKL two-gluon Green’s func-tion with the proton impact factor (IF).
The IF for the transition γ ∗(λγ ) → ρ(λρ) depends onpolarizations of photon and vector mesons. For the longi-tudinal polarizations λγ = λρ = 0 the dominance of smalldipoles is evident, and the standard leading-twist distributionamplitude appears. In the case of the transverse polariza-tions the perturbative QCD factorization remains valid, buthigher-twist DAs are needed [5,6]. The difference betweenthe impact factors makes the polarization dependence ofdiffractive production a sensitive probe of the UGD [7,8].Previously, for the case of ρ-meson production ratios of theforward amplitudes have been calculated in Ref. [8] and com-pared to data from HERA. For the case of φ mesons the effectof a finite strange quark mass on cross sections has been dis-cussed in Ref. [9], where also relations of the so-called gen-uine higher-twist DAs to weighted integrals over light-frontwave functions have been given.
In this work, we wish to extend the efforts of [7–9] to thecalculation of polarized ρ-meson production cross sections(as opposed to ratios of amplitudes) for a variety of uninte-grated gluon distributions, and compare the results to HERA
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846 Page 2 of 12 Eur. Phys. J. C (2021) 81 :846
Fig. 1 Diagrammatic representation of the amplitude for the exclusiveemission of a ρ meson in high-energy factorization. The off-shell impactfactor (upper part) is built up as a collinear convolution between the hardfactor for the photon splitting to a dipole and the non-perturbative ρ-meson DA (sea-green blob). The UGD is given by the convolution of theBFKL gluon Green’s function (yellow blob) and the non-perturbativeproton impact factor (red blob)
data. We will also give predictions for the kinematics relevantfor the Electron-Ion Collider EIC.
2 Polarized ρ-meson leptoproduction
The H1 and ZEUS collaborations have provided extendedanalyses of the helicity structure in the hard exclusive pro-duction of the ρ meson in ep collisions through the subpro-cess
γ ∗(λγ )p → ρ(λρ)p. (1)
Here λρ and λγ represent the meson and photon helicities,respectively, and can take the values 0 (longitudinal polariza-tion) and ±1 (transverse polarizations). The helicity ampli-tudes Tλρλγ extracted at HERA [10] exhibit the hierarchy
T00 � T11 � T10 � T01 � T−11, (2)
that follows from the dominance of a small-size dipole scat-tering mechanism, as discussed first in Ref. [11], see also[12]. As we previously mentioned, the H1 and ZEUS collab-orations have analyzed data in different ranges of Q2 and W .In what follows we will refer only to the H1 ranges (3),
2.5 GeV2 < Q2 < 60 GeV2,
35 GeV < W < 180 GeV, (3)
illustrating the framework to probe the ρ-meson leptopro-duction as a way to test the UGDs [7,13,14] where polarizedcross sections will be the key observables.
2.1 Theoretical setup
In the high-energy regime, s ≡ W 2 � Q2 � �2QCD, which
implies small x = Q2/W 2, the forward helicity amplitudefor the ρ-meson electroproduction can be written, in high-energy factorization (also known as kT -factorization), as theconvolution of the γ ∗ → ρ IF, �γ ∗(λγ )→ρ(λρ)(κ2, Q2), withthe UGD, F(x, κ2). Its expression reads
Tλρλγ (s, Q2) = is
(2π)2
∫d2κ
(κ2)2 �γ ∗(λγ )→ρ(λρ)(κ2, Q2)F(x, κ2),
x = Q2
s. (4)
Defining α = κ2
Q2 and B = 2παse√2fρ , the expression
for the IFs takes the following forms (see Ref. [5] for thederivation):
• longitudinal case
�γL→ρL (κ, Q; μ2) = 2B
√N2c − 1
Q Nc
∫ 1
0dy ϕ1(y; μ2)
(α
α + y y
),
(5)
where Nc denotes the number of colors and ϕ1(y;μ2)
is the twist-2 DA which, up to the second order in theexpansion in Gegenbauer polynomials, reads [15]
ϕ1(y;μ2) = 6y y
(1 + a2(μ
2)3
2
(5(y − y)2 − 1
));(6)
• transverse case
�γT →ρT (α, Q;μ2) = (εγ · ε∗ρ) 2Bmρ
√N 2c − 1
2NcQ2
×{−
∫ 1
0dy
α(α + 2y y)
y y(α + y y)2
×[(y − y)ϕT
1 (y;μ2) + ϕTA (y;μ2)
]
+∫ 1
0dy2
∫ y2
0dy1
y1 y1α
α + y1 y1
×[
2 − Nc/CF
α(y1 + y2) + y1 y2− Nc
CF
1
y2α + y1(y2 − y1)
]
×M(y1, y2;μ2) −∫ 1
0dy2
∫ y2
0dy1
[2 + Nc/CF
y1+ y1
α + y1 y1
×(
(2 − Nc/CF )y1α
α(y1 + y2) + y1 y2− 2
)− Nc
CF
(y2 − y1)y2
y1
× 1
α y1 + (y2 − y1)y2
]S(y1, y2;μ2)
}, (7)
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Eur. Phys. J. C (2021) 81 :846 Page 3 of 12 846
where CF = N2c −1
2Nc, while the functions M(y1, y2;μ2)
and S(y1, y2;μ2) are defined in Eqs. (12)–(13) of Ref. [6]and are combinations of the twist-3 DAs B(y1, y2;μ2)
and D(y1, y2;μ2) (see Ref. [15]), given by
B(y1, y2;μ2) = −5040y1 y2(y1 − y2)(y2 − y1),
D(y1, y2;μ2) = −360y1 y2(y2 − y1)
×(
1 + ωA{1,0}(μ2)
2(7 (y2 − y1) − 3)
).
(8)
In Eqs. (6) and (8) the functional dependence of a2, ωA{1,0},ζ A
3ρ , and ζ V3ρ on the factorization scale μ2 can be deter-
mined from the corresponding known evolution equations[15], using some suitable initial condition at a scale μ0.
Note that the κ-dependence of the IFs is different in thecases of longitudinal and transverse polarizations and thisposes a strong constraint on the κ-dependence of the UGDin the HERA energy range. The main point will be to demon-strate, considering different models of UGD, that the uncer-tainties of the theoretical description do not prevent us fromsome, at least qualitative, conclusions about the shape of theUGD in κ .
The DAs ϕT1 (y;μ2) and ϕT
A (y;μ2) in Eq. (7) encompassboth genuine twist-3 and Wandzura–Wilczek (WW) contri-butions [6,15]. The former are related to B(y1, y2;μ2) andD(y1, y2;μ2); the latter are those obtained in the approxi-mation in which B(y1, y2;μ2) = D(y1, y2;μ2) = 0, and inthis case read1
ϕT WWA (y; μ2) = 1
2
[−y
∫ y
0dv
ϕ1(v; μ2)
v− y
∫ 1
ydv
ϕ1(v; μ2)
v
],
ϕT WW1 (y; μ2) = 1
2
[−y
∫ y
0dv
ϕ1(v; μ2)
v+ y
∫ 1
ydv
ϕ1(v; μ2)
v
].
(9)
The other interesting point will be the extension of infor-mation, collected by the helicity-amplitude analysis, reach-able from the calculation of the cross section. As a matter offact, the expressions for the polarized cross sections σL andσT , calculated using Eqs. (5), (6) in Eqs. (4) and (7), (8) inEq. (4), respectively, are
σL (γ ∗ p → ρ p) = 1
16πb(Q2)
∣∣∣∣T00(s, t = 0)
W 2
∣∣∣∣2
, (10)
σT (γ ∗ p → ρ p) = 1
16πb(Q2)
∣∣∣∣T11(s, t = 0)
W 2
∣∣∣∣2
, (11)
1 For asymptotic form of the twist-2 DA, ϕ1(y) = ϕas1 (y) = 6y y,
these equations give ϕT WW, asA (y) = −3/2y y and ϕ
T WW, as1 (y) =
−3/2y y(2y − 1).
where b(Q2) in Eqs. (10) and (11) is the diffraction slope,for which we adopt the parametrization [16]:
b(Q2) = β0 − β1 log
[Q2 + m2
ρ
m2J/ψ
]+ β2
Q2 + m2ρ
, (12)
fixing the values of the constants as follows: β0 = 6.5GeV−2, β1 = 1.2 GeV−2 and β2 = 1.6. In Fig. 2, wecompare the parametrization of Eq. (12) with the data ofthe H1-collaboration from Ref. [10]. Above, we took intoaccount only the helicity conserving amplitudes. These dom-inate the diffractive peak at small t , and hence also the inte-grated cross section of interest. While the impact factors forthe helicity flip transitions γ ∗(T ) → ρ(L) with �λ = ±1and γ ∗(T ) → ρ(T ′) with �λ = ±2, are available withinthe higher-twist factorization approach [5,6], a discussionof the t−dependent cross section or the polarization densitymatrix of ρ-mesons would require also a generalization tooff-forward UGDs and goes beyond the scope of this work.
The advantage of considering polarized cross sectionsrelies on the possibility to constrain not only the shapeand the behavior of results, but also the normalization, nowessential and which is clearly irrelevant for the evaluationof the helicity-amplitude ratio T11/T00. Although all UGDsdescribed in Sect. 3 present fixed values for their parameters,the ABIPSW model, one of the first UGD parametrizationadopted in phenomenological analysis [17], was defined up tothe overall normalization. As regards the helicity-amplituderatio T11/T00, results for the ABIPSW UGD model show afair agreement with data [6], this suggesting that the shape,controlled by the M parameter (see Eq. (13)), has beenalready guessed. The best value of the normalization param-eter can be obtained via a simple global fit to experimentaldata of both polarized cross sections, σL and σT .
3 UGD models used in the present study
We have considered a selection of several models of UGD,without pretension to exhaustive coverage, but with the aimof comparing (sometimes radically) different approaches. Werefer the reader to the original papers for details on the deriva-tion of each model and limit ourselves to presenting here justthe functional formF(x, κ2) of the UGD as we implementedit in the numerical analysis.
3.1 An x-independent model (ABIPSW)
The simplest UGD model is x-independent and merely coin-cides with the proton impact factor [6]:
F(x, κ2) = A
(2π)2 M2
[κ2
M2 + κ2
], (13)
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846 Page 4 of 12 Eur. Phys. J. C (2021) 81 :846
Fig. 2 Q2-dependence of the diffractive slope for the exclusive ρ-meson leptoproduction (Eq. (12)) compared to the H1 data from Ref.[10]
where M corresponds to the non-perturbative hadronic scale,fixed as M = 1 GeV. The constant A is irrelevant when weconsider the ratio T11/T00 for the ρ-meson leptoproduction,but it becomes essential to calculate cross sections. There-fore, we determined it by a global fit to all available data forpolarized cross sections, getting A = 148.14 GeV2, with arelative uncertainty below 0.05%.
3.2 Gluon momentum derivative
This UGD is given by
F(x, κ2) = dxg(x, κ2)
d ln κ2 (14)
and encompasses the collinear gluon density g(x, μ2F ), taken
at μ2F = κ2. It is based on the obvious requirement that, when
integrated over κ2 up to some factorization scale, the UGDmust give the collinear gluon density. We have employedthe CT14 parametrization [18], using the appropriate cutoffκmin = 0.3 GeV (see Section III A of Ref. [8] for furtherdetails).
3.3 Ivanov–Nikolaev (IN) UGD: a soft-hard model
The UGD proposed in Ref. [19] is developed with the purposeof probing different regions of the transverse momentum. Inthe large-κ region, DGLAP parametrizations for g(x, κ2) areemployed. Moreover, for the extrapolation of the hard gluondensities to small κ2, an Ansatz is made [20], which describesthe color gauge invariance constraints on the radiation of softgluons by color singlet targets. The gluon density at smallκ2 is supplemented by a non-perturbative soft component,according to the color-dipole phenomenology.
This model of UGD has the following two-componentform:
F(x, κ2) = F (B)soft (x, κ
2)κ2s
κ2 + κ2s
+ Fhard(x, κ2)
κ2
κ2 + κ2h
,
(15)
where κ2s = 3 GeV2 and κ2
h = [1+0.047 log2(1/x)] 12 GeV2.
The soft term reads
F (B)soft (x, κ
2) = asoftCF Ncαs(κ
2)
π
(κ2
κ2 + μ2soft
)2
VN(κ),
(16)
with μsoft = 0.1 GeV. The parameter asoft = 2 gives a mea-sure of how important is the soft part compared to the hardone. On the other hand, the hard component reads
Fhard(x, κ2) = F (B)
pt (κ2)Fpt(x, Q2
c)
F (B)pt (Q2
c)θ(Q2
c − κ2)
+Fpt(x, κ2)θ(κ2 − Q2
c), (17)
where Fpt(x, κ2) is related to the collinear gluon parton dis-tribution function (PDF) as in Eq. (14) and Q2
c = 3.26 GeV2
(see Section III A of Ref. [8] for further details). We referto Ref. [19] for the expressions of the vertex function VN(κ)
and of F (B)pt (κ2). Another relevant feature of this model is
given by the choice of the coupling constant. In this regard,the infrared freezing of strong coupling at leading order (LO)is imposed by fixing �QCD = 200 MeV:
αs(μ2) = min
⎧⎪⎪⎨⎪⎪⎩
0.82,4π
β0 log
(μ2
�2QCD
)⎫⎪⎪⎬⎪⎪⎭
. (18)
We wish to stress that this model was successfully testedin the unpolarized electroproduction of vector mesons atHERA.
3.4 Hentschinski–Sabio Vera–Salas (HSS) model
This model, originally used in the study of DIS structure func-tions [21], takes the form of a convolution between the BFKLgluon Green’s function and a LO proton impact factor. It hasbeen employed in the description of single-bottom quark pro-duction at the LHC [22], to investigate the photoproductionof J/� and ϒ in [23–25] and to study the forward Drell–Yan invariant-mass distribution [26]. We implemented theformula given in Ref. [22] (up to a κ2 overall factor neededto match our definition), which reads
F(x, κ2, Mh) =∫ ∞
−∞dν
2π2 C �(δ − iν − 12 )
�(δ)
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Eur. Phys. J. C (2021) 81 :846 Page 5 of 12 846
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.HERA data
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
Fig. 3 Q2-dependence of the longitudinally polarized cross section,σL , for all the considered UGD models, at W = 75 GeV together withthe HERA data (left upper panel) and at W = 20, 30, 50 GeV for EIC
(the remaining panels). Uncertainty bands represent the effect of vary-ing a2(μ0 = 1 GeV) between 0.0 and 0.6
×(
1
x
)χ(
12 +iν
) (κ2
Q20
) 12 +iν
×{
1 + α2s β0χ0
( 12 + iν
)8Nc
log
(1
x
)
×[−ψ
(δ − 1
2− iν
)− log
κ2
M2h
]},
(19)
where β0 = 11Nc−2N f3 , with N f the number of active quarks
(put equal to four in the following), αs = αs(μ2
)Nc
π, with
μ2 = Q0Mh , and χ0(12 + iν) ≡ χ0(γ ) = 2ψ(1) − ψ(γ ) −
ψ(1 − γ ) is (up to the factor αs) the LO eigenvalue of theBFKL kernel, with ψ(γ ) the logarithmic derivative of theEuler Gamma function. Here, Mh plays the role of the hardscale which in our case can be identified with the photonvirtuality,
√Q2. In Eq. (19), χ(γ ) (with γ = 1
2 + iν) is theNLO eigenvalue of the BFKL kernel, collinearly improvedand BLM optimized. It reads
χ(γ ) = αsχ0(γ ) + α2s χ1(γ )
−1
2α2s χ
′0(γ ) χ0(γ ) + χRG(αs, γ ), (20)
with χ1(γ ) and χRG(αs, γ ) given in Section 2 of Ref. [22].This UGD model is characterized by a peculiar parametriza-
tion for the proton impact factor, whose expression is
�p(q, Q20) = C
2π�(δ)
(q2
Q20
)δ
e− q2
Q20 , (21)
which depends on three parameters Q0, δ and C which werefitted to the combined HERA data for the F2(x) proton struc-ture function. We adopted here the so-called kinematicallyimproved values (see Section III A of Ref. [8] for furtherdetails) and given by
Q0 = 0.28 GeV, δ = 6.5, C = 2.35. (22)
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846 Page 6 of 12 Eur. Phys. J. C (2021) 81 :846
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.HERA data
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSSINWMRGluon mom.
Fig. 4 Q2-dependence of the transversely polarized cross section, σT ,for all the considered UGD models, at W = 75 GeV together with theHERA data (left upper panel) and at W = 20, 30, 50 GeV for EIC (the
remaining panels). Uncertainty bands represent the effect of varyinga2(μ0 = 1 GeV) between 0.0 and 0.6
3.5 Golec–Biernat–Wüsthoff (GBW) UGD
This UGD parametrization derives from the effective dipolecross section σ (x, r) for the scattering of a qq pair off anucleon [27],
σ (x, r2) = σ0
{1 − exp
(− r2
4R20(x)
)}, (23)
through a reverse Fourier transform of the expression
σ0
{1 − exp
(− r2
4R20(x)
)}= 2π
Nc
∫d2κ
κ4 αsF(x, κ2)
× (1 − exp(iκ · r)) (1 − exp(−iκ · r)) , (24)
αsF(x, κ2) = Ncκ4σ0
R20(x)
4π2 e−κ2R20(x), (25)
with
R20(x) = 1
GeV2
(x
x0
)λp
(26)
and the following values
σ0 = 23.03 mb, λp = 0.288, x0 = 3.04 · 10−4. (27)
The normalization σ0 and the parameters x0 and λp > 0 ofR2
0(x) have been determined by a global fit to F2(x) in theregion x < 0.01.
3.6 Watt–Martin–Ryskin (WMR) model
This model is based on the idea that the κ dependence of theUGD comes from the last step of the evolution ladder.
The UGD introduced in Ref. [28] reads
F(x, κ2, μ2) = Tg(κ2, μ2)
αs (κ2)
2π
∫ 1
xdz
×[∑
q
Pgq (z)x
zq
(x
z, κ2
)+ Pgg(z)
x
zg
(x
z, κ2
)�
(μ
μ + κ− z
)],
(28)
123
Eur. Phys. J. C (2021) 81 :846 Page 7 of 12 846
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSS
INWMRGluon mom.HERA data
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSS
INWMRGluon mom.
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSS
INWMRGluon mom.
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
ABIPSWBCRTGBWHSS
INWMRGluon mom.
Fig. 5 Q2-dependence of the polarized cross-section ratio, σL/σT , for all the considered UGD models, at W = 75 GeV together with the HERAdata (left upper panel) and at W = 20, 30, 50 GeV for EIC (the remaining panels). Uncertainty bands represent the effect of varying a2(μ0 = 1 GeV)
between 0.0 and 0.6
where the term
Tg(κ2, μ2) = exp
(−
∫ μ2
κ2dκ2
tαs(κ
2t )
2π
×(∫ z′max
z′min
dz′ z′ Pgg(z′) + N f
∫ 1
0dz′ Pqg(z′)
)),
(29)
gives the probability of evolving from the scale κ to thescale μ without parton emission. Here z′max ≡ 1 − z′min =μ/(μ + κt ); N f is the number of active quarks. This UGDmodel depends on an extra-scale μ, which we fixed at Q.The splitting functions Pqg(z) and Pgg(z) are given by
Pqg(z) = TR [z2 + (1 − z)2],Pgg(z) = 2CA
[1
(1 − z)++ 1
z− 2 + z(1 − z)
]
+(
11
6CA − N f
3
)δ(1 − z),
with the plus-prescription defined as∫ 1
adz
F(z)
(1 − z)+=
∫ 1
adz
F(z) − F(1)
(1 − z)−
∫ a
0dz
F(1)
(1 − z).
(30)
3.7 Bacchetta–Celiberto–Radici–Taels (BCRT) distribution
We include in our analysis the unpolarized distribution partof the set of leading-twist T -even transverse-momentum-dependent (TMD) gluon PDFs calculated in Ref. [29] (seealso Ref. [30]), suited for studies in wider kinematic rangesat new-generation collider machines, such as the EIC [4],HL-LHC [31], and NICA-SPD [32]. One has
F(x, κ2) = κ2∫ ∞
MdMX ρX (MX ) f g1 (x, κ2; MX ), (31)
Here, f g1 (x, κ2; MX ) stands for the distribution of unpo-larized gluons in an unpolarized proton, calculated in theso-called spectator-model approximation, namely where theremainders of the proton after gluon emission are treated asa single spin-1/2 spectator particle of mass MX
123
846 Page 8 of 12 Eur. Phys. J. C (2021) 81 :846
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + protonGBW - fullIN - fullHERA data
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + protonGBW - fullIN - full
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + protonGBW - fullIN - full
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σL(Q
2)[nb]
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + protonGBW - fullIN - full
Fig. 6 Q2-dependence of the longitudinally polarized cross section, σL , for GBW and IN UGD models, at W = 75 GeV together with the HERAdata (left upper panel) and at W = 20, 30, 50 GeV EIC (the remaining panels). Uncertainty bands represent the effect of varying a2(μ0 = 1 GeV)
between 0.0 and 0.6
f g1 (x, κ2; MX ) =[(
2Mxg1(κ2) − x(M + MX )g2(κ
2))2
×[(MX − M(1 − x))2 + κ2]
+ 2κ2 (κ2 + xM2X ) g2(κ
2)2
+2κ2M2 (1 − x) (4g1(κ2)2 − xg2(κ
2)2)]
×[(2π)3 4xM2 (xM2
X − x(1 − x)M2+κ2)2]−1
.
(32)
In Eq. (32) M is the proton mass, whereas g1,2 couplingsdepict the effective proton-gluon-spectator vertex interaction
g1,2(κ2) = G1,2
(1 − x)[κ2 + xM2
X − x(1 − x)M2]
[κ2 + xM2
X − x(1 − x)M2 + (1 − x)�2X
]2 .
(33)
The spectral function ρX in Eq. (31) weighs f1 over MX ina continuous range and its expression reads
ρX (MX ) = μ2a
(A
B + μ2b + C
πσe− (MX−D)2
σ2
), (34)
where μ2 = M2X − M2, and the model parameters in
Eqs. (33) and (34) were obtained by a simultaneous fit ofthe κ-integral of spectator-model unpolarized and helicityfunctions on NNPDF3.1sx [33] and NNPDFpol1.1 [34]collinear PDFs, respectively, and they encode effective small-x effects coming from the BFKL resummation. In our studywe employ values for these parameters obtained by averag-ing over the whole set of replicas provided (see Section 3of Ref. [29] for details). They read: G1 = 1.51 GeV2,G2 = 0.414 GeV2, �X = 0.472 GeV, A = 6.1, a = 0.82,b = 1.43, C = 371, D = 0.548 GeV, and σ = 0.52GeV.
We stress that exploratory studies by using this TMDmodel as a UGD are justified by the fact that we areconsidering observables for the exclusive ρ-meson lepto-production in the forward limit. In a more general, off-forward case one should consider rather models for thegluon generalized parton distribution (GPD), instead ofTMDs.
123
Eur. Phys. J. C (2021) 81 :846 Page 9 of 12 846
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT( Q
2)[nb]
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWGBW - genuineIN - fullIN - WWIN - genuineHERA data
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWGBW - genuineIN - fullIN - WWIN - genuine
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT( Q
2)[nb]
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWGBW - genuineIN - fullIN - WWIN - genuine
100 101 102
Q2 [GeV2]
10−5
10−3
10−1
101
103
105
σT(Q
2)[nb]
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWGBW - genuineIN - fullIN - WWIN - genuine
Fig. 7 Q2-dependence of the transversely polarized cross section, σT ,for GBW and IN UGD models, at W = 75 GeV together with theHERA data (left upper panel) and at W = 20, 30, 50 GeV relevant
for EIC (the remaining panels). Uncertainty bands give the effect ofvarying a2(μ0 = 1 GeV) between 0.0 and 0.6. Full, WW and genuinecontributions are shown separately
4 Results
All the results shown in this section were obtained by mak-ing use of the leptonic-exclusive-amplitudes (LExA) modularinterface as implemented in the JETHAD code [35].
We present predictions for the polarized cross sections σL
and σT and their ratio σL/σT , as obtained with all the UGDmodels presented above, and compare them with HERA data.The behavior of the polarized cross sections σL and σT andthe ratio σL/σT [36] in terms of Q2 for all UGDs, at W = 75GeV (comparison with HERA data) and at W = 20, 30, 50GeV (predictions for the EIC), is shown in Figs. 3, 4, and 5,taking into account the variation of the Gegenbauer coeffi-cienta2(μ0) in the same range used for the helicity-amplituderatio analysis of Ref. [8]. As can be seen from the figures,the uncertainties due to the choice of UGDs are much biggerthan those due to a2 uncertainty which opens a possibility totest UGD models for the reaction under consideration. Thecomparison exhibits a partial agreement with the experimen-tal data, where once again none of the proposed models is
able to describe the whole Q2 region. However, we can spec-ify which UGD model is more suitable for the description ofthe polarized cross sections and which, indeed, for their ratioin the Q2 intermediate range. On one side, observing Figs. 6and 7 , the GBW model, considered in its standard definition,i.e. without the evolution of saturation scale (for a detaileddiscussion about saturation effects see Ref. [37]), and the INone appear to be the UGD models that allow us to match datafor the single polarized cross sections, σL and σT , in the mostaccurate way. Here the genuine twist-3 contribution is con-sidered. On the other side, although the predictions for thecross section ratio σL/σT in Fig. 8 with the GBW model isquite resonable, if we regard increasing values of Q2 ∼ 10GeV2, the IN UGD model2 is able to slightly better catchalso the low-Q2 region of data. We stress, however, that in
2 Regarding the helicity amplitude ratio T11/T00, as a result of a correc-tion in the numerical implementation, we note a significantly improvedagreement between the IN model and the experimental data with respectto the Fig. 2 in Ref. [8]. For the sake of simplicity and recognizing inthe GBW model intriguing features and parameters worthy to be probed
123
846 Page 10 of 12 Eur. Phys. J. C (2021) 81 :846
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 75 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWIN - fullIN - WWHERA data
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 50 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWIN - fullIN - WW
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 30 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWIN - fullIN - WW
100 101 102
Q2 [GeV2]
10−2
10−1
100
101
102
σL/σ
T(Q
2)
W = 20 GeV
0.0 < a2(μ0) < 0.6
μ0 = 1 GeV
LExA v0.4.6
γ∗ + proton → ρ0 + proton
GBW - fullGBW - WWIN - fullIN - WW
Fig. 8 Q2-dependence of the polarized cross-section ratio, σL/σT , forthe GBW and IN UGD models, at W = 75 GeV HERA (left upperpanel) and at W = 20, 30, 50 GeV relevant for EIC (the remaining pan-
els). Uncertainty bands represent the effect of varying a2(μ0 = 1 GeV)
between 0.0 and 0.6. Full and WW contributions are shown separately
this low-Q2 range the validity of our approach to IFs, basedon the use of collinear DAs, could be questionable. There-fore our ability to discriminate among UGD models at smallvirtualities could be limited.
5 Conclusions
We have calculated cross sections for the diffractive elec-troproduction of ρ mesons in the energy range of HERAand EIC. We have used the high-energy factorization formal-ism for the forward amplitude [5,6], utilizing an empiricalparametrization of the diffractive slope to obtain the relevantintegrated cross sections.
The impact factors for longitudinal and transverse mesonsprobe the transverse momentum dependence of the UGDin different ways, so that the polarization dependence of ρ-
Footnote 2 continuedfurther, in that paper an exhaustive phenomenological analysis of thisUGD parametrization was proposed.
production has been proposed as a sensitive probe of theshape of the UGD [7,8]. The impact factor for longitudinallypolarized meson is obtained in terms of the leading twist DA.We find that, for the higher-twist DAs relevant for transversemesons, the WW contributions dominate. We have investi-gated the effect of uncertainty related with the form of leadingtwist DA by varying the Gegenbauer coefficient a2.
We have performed calculations for a representativechoices of UGDs available in the literature. These UGDs fol-low different strategies in their construction. Some explicitlyconnect to solutions of a specific evolution equation [21,28],while others stress the presence of a substantial nonpertur-bative component [19,27], while being adjusted to giving agood description of proton deep inelastic structure functions.
The latter two UGDs in fact do give the best descriptionof the HERA cross sections. While our use of an empiricalparametrization of the diffractive slope introduces an addi-tional model element/uncertainty, the spread of predictionsfrom various UGDs appears to be larger than the uncertaintyin the slope. Some UGDs substantially underestimate the
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total cross section, which may mean that, in the relevant κ-range, their κ-shape and normalization miss some importantinsights on the nonperturbative proton dynamics.
The cross section ratio σL/σT indeed appears to havepotential to discriminate further between UGDs, for the caseat hand the IN UGD gives the best description of HERA data.
New data taken at the Electron-Ion Collider EIC alsohave a potential to further discriminate between the differ-ent UGDs. At the lower range of W an investigation of thematching/correspondence to TMD factorization approacheswith on-shell partons would be interesting.
It has been recently pointed out how the inclusive diffrac-tive tagging of particles with a heavy transverse mass, such asHiggs bosons [38], heavy-flavored jets [39–41] and charmedbaryons [42], leads to a fair stabilization of the high-energyseries under higher-order corrections. Future studies of reac-tions featuring the forward emission of those objects will pro-vide us with additional and possibly clearer probe channelsfor the UGD. Furthermore, the detection of forward quarko-nia (recently studied in the context of high-energy factor-ization in more central directions of rapidity [43–48]) in thelow−pT region will help us (i) to shed light on the quarko-nium production mechanisms and (ii) to investigate kine-matic regions at the frontier between the high-energy and theTMD dynamics.
The analysis presented in this work is at leading orderand an obvious improvement would be its extension to thenext-to-leading order. This calls for the setup of a theoreticalscheme for the convolution of NLO IFs and UGD, which isnot trivial, except for UGD models based on BFKL.
Acknowledgements A.D.B. and A.P. acknowledge support fromthe INFN/QFT@COLLIDERS project. F.G.C. acknowledges supportfrom the Italian Ministry of Education, Universities and Researchunder the FARE grant “3DGLUE” (n. R16XKPHL3N), and from theINFN/NINPHA project. F.G.C. thanks the Università degli Studi diPavia for the warm hospitality. The work of D.I. was carried outwithin the framework of the state contract of the Sobolev Institute ofMathematics (Project No. 0314-2019-0021). This study was partiallysupported by the Polish National Science Center Grant No. UMO-2018/31/B/ST2/03537 and by the Center for Innovation and Transfer ofNatural Sciences and Engineering Knowledge in Rzeszów. The diagramin Fig. 1 was realized via the JaxoDraw 2.0 interface [49].
Data Availability Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: Data produced inthis work are summarized in figures; they can be made available uponrequest].
Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.
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