# Next-to-leading order QCD predictions for dijet

## Transcript Of Next-to-leading order QCD predictions for dijet

PHYSICAL REVIEW C 102, 065201 (2020)

Next-to-leading order QCD predictions for dijet photoproduction in lepton-nucleus scattering at the future EIC and at possible LHeC, HE-LHeC, and FCC facilities

V. Guzey 1 and M. Klasen 2 1National Research Center “Kurchatov Institute,” Petersburg Nuclear Physics Institute (PNPI), Gatchina, 188300, Russia 2Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany

(Received 24 March 2020; revised 1 October 2020; accepted 18 November 2020; published 4 December 2020)

We calculate cross sections for inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of the future EIC and the possible LHeC, HE-LHeC, and FCC using next-to-leading order (NLO) perturbative QCD and nCTEQ15 and EPPS16 nuclear parton density functions (nPDFs). We make predictions for distributions in the dijet average transverse momentum p¯T , the average rapidity η¯ , the observed nuclear momentum fraction xAobs, and the observed photon momentum fraction xγobs. Comparing the kinematic reaches of the four colliders, we ﬁnd that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in all four considered variables. Notably, the LHeC and HE-LHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC). The ratio of the dijet cross sections on a nucleus and the proton, σA/(Aσp), depends on xAobs in a similar way as the ratio of gluon densities, gA(xA, μ2 )/[Agp(xA, μ2 )], for which current nPDFs predict a strong suppression due to nuclear shadowing in the region xAobs < 0.01. Dijet photoproduction at future lepton-nucleus colliders can therefore be used to test this prediction and considerably reduce the current uncertainties of nPDFs.

DOI: 10.1103/PhysRevC.102.065201

I. INTRODUCTION

Lepton-nucleus scattering at high energies has traditionally been a fruitful way to access and study the structure of nuclei in quantum chromodynamics (QCD). Despite numerous successes and insights, there is an overarching need to continue these studies at progressively higher energies using colliders. While the plans to use nuclear beams in the HERA collider at DESY [1] have not materialized, a high-energy polarized lepton-proton and lepton-nucleus collider at Brookhaven National Laboratory (BNL) [2,3]—an Electron-Ion Collider (EIC)—has recently been approved. Further down the road, one envisions that the Large Hadron Collider (LHC) at CERN will be complemented by a Large Hadron-Electron Collider (LHeC) and its higher energy upgrade (HE-LHeC) [4,5] as well as a future circular collider (FCC) [6].

The core of the physics program at the future leptonnucleus colliders is comprised of deep inelastic scattering (DIS), allowing one to map out various parton distributions in nuclei with high precision; see, e.g., Refs. [7–10]. In addition, as one learned from HERA, photoproduction of jets [11,12] and dijets [13,14] provides useful complementary information on the QCD (and in particular gluon) structure of hadrons. This has recently been exploited at the LHC, where ultrape-

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

ripheral collisions (UPCs) of heavy ions give an opportunity to study photon-nucleus scattering at unprecedentedly high energies [15]. In particular, it was shown that inclusive dijet photoproduction in Pb-Pb UPCs at the LHC can help to reduce the existing uncertainty in nuclear parton distribution functions (nPDFs) at small x by approximately a factor of 2 [16,17].

In this work, we calculate the cross section of inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of the future EIC, LHeC, HE-LHeC, and FCC using the formalism of collinear factorization, nextto-leading order (NLO) perturbative QCD, and nCTEQ15 [18] and EPPS16 [19] nPDFs. We make predictions for the cross-section distributions as functions of the dijet average transverse momentum p¯T , the average rapidity η¯ , the observed nuclear momentum fraction xAobs, and the observed photon momentum fraction xγobs. We compare the kinematic reaches of the four colliders and ﬁnd that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in all four considered variables. Notably, the LHeC and HE-LHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC), which is two (three) orders of magnitude smaller than that at the EIC. We then discuss in detail the implications of future measurements of dijet photoproduction in lepton-nucleus scattering on the determination of nPDFs.

This work continues and extends the analysis of Ref. [20] by making predictions for high-energy lepton-nucleus colliders including LHeC, HE-LHeC, and FCC, comparing them to the case of the EIC, and analyzing relative merits of the four considered colliders.

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V. GUZEY AND M. KLASEN

TABLE I. Energy conﬁgurations of electron-ion colliders considered in this work.

Ee (GeV)

EA (TeV)

√ s (GeV)

EIC

21

LHeC

60

HE-LHeC

60

FCC

60

0.1

92

2.76

812

4.93

1,088

19.7

2,174

The remainder of the paper is structured as follows. In Sec. II, we recap the formalism and the input for the calculation of inclusive dijet photoproduction in NLO perturbative QCD. Our results and their discussion are presented in Sec. III. A summary of our results is given in Sec. IV.

II. DIJET PHOTOPRODUCTION IN NEXT-TO-LEADING ORDER QCD

In the framework of collinear factorization and next-toleading order (NLO) perturbative QCD [21–25], the cross section of dijet photoproduction in eA → e + 2 jets + X

PHYSICAL REVIEW C 102, 065201 (2020)

electron-nucleus scattering reads dσ (eA → e + 2 jets + X )

=

a,b

dy dxγ

dxA fγ /e(y) fa/γ (xγ , μ2 )

× fb/B(xA, μ2 )dσˆ (ab → jets) ,

(1)

where a, b are parton ﬂavors; fγ /e(y) is the ﬂux of equivalent photons of the electron, which depends on the photon light-cone momentum fraction y; fa/γ (xγ , μ2) is the PDF of the photon for the resolved photon case (see below), which

depends on the momentum fraction xγ and the factorization scale μ; fb/B(xA, μ2) is the nuclear PDF with xA being the corresponding parton momentum fraction; and dσˆ (ab → jets) is the elementary cross section for the production of two-parton

and three-parton ﬁnal states emerging as jets in hard scattering

of partons a and b.

The dijet cross section in Eq. (1) receives two types of con-

tributions: the resolved photon contribution, when the photon

interacts with target partons through its quark-gluon structure expressed by fa/γ (xγ , μ2), and the direct photon contribution, when the photon enters directly the hard scattering cross section dσˆ (ab → jets). At leading order (LO), the direct

FIG. 1. NLO QCD predictions for the dijet photoproduction cross section in eA → e + 2 jets + X electron-nucleus scattering at the EIC, LHeC, HE-LHeC, and FCC as a function of the average dijet transverse momentum p¯T , the average rapidity η¯ , and the momentum fractions xAobs and xγobs. The calculation uses nCTEQ15 nPDFs.

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FIG. 2. NLO QCD predictions for the ratio of the cross sections of dijet photoproduction on nuclei and the proton as a function of p¯T , η¯ , xAobs, and xγobs in the EIC kinematics. The calculation uses central values of nCTEQ15 nPDFs (solid lines) and 32 sets of error PDFs (shaded band).

photon contribution has the support exactly at xγ = 1 and fγ /γ (xγ , μ2) = δ(1 − xγ ). At NLO, the separation between the resolved and direct photon contributions depends on the factorization scheme and scale μ. Indeed, by calculating the virtual and real corrections to the matrix elements of interest using massless quarks in dimensional regularization, one can explicitly show that ultraviolet (UV) divergences are renormalized in the MS scheme and infrared (IR) divergences are canceled and factorized into the nucleus (proton) and photon PDFs, respectively; see Ref. [25]. For the latter, this can imply a transformation from the DISγ to the MS scheme. As a result, the direct photon contribution becomes sizable and in practice dominates the cross section at xγ ≈ 1 even at NLO.

In our analysis, we used for the photon ﬂux of the electron

the improved expression derived in the Weizsäcker-Williams

approximation [26]

fγ /e(y) = α 1 + (1 − y)2 ln Qm2 ax(1 − y)

2π

y

me2y2

+ 2m2y 1 − 1 − y ,

(2)

e Qm2 ax me2y2

where α is the ﬁne-structure constant; me is the electron

mass; and Qm2 ax is the maximal photon virtuality. Motivated

by studies of jet photoproduction at HERA, we take Qm2 ax = 0.1 GeV2 and assume that the inelasticity spans the range of 0 < y < 1.

For the photon PDFs, we used the GRV HO parametrization [27], which we transformed as explained above. These photon PDFs have been tested thoroughly at HERA and the Large Electron Positron (LEP) collider at CERN and are very robust, especially at high xγ (dominated by the pQCD photonquark splitting), which is correlated with the low-xA region that is of particular interest for this work. For the nuclear PDFs fb/B(xA, μ2), we employed the nCTEQ15 [18] and EPPS16 [19] parametrizations including both central and error PDFs. The latter are used to evaluate the theoretical uncertainty bands of our predictions.

III. PREDICTIONS FOR DIJET PHOTOPRODUCTION CROSS SECTIONS AT FUTURE ELECTRON-ION COLLIDERS

We performed perturbative NLO QCD calculations of the dijet photoproduction cross section using Eq. (1), which was numerically implemented in an NLO parton-level Monte Carlo [21–25]. This framework has been successfully tested

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to describe the HERA and LEP data on dijet photoproduction on the proton. It implements the anti-kT algorithm [28] with a jet radius of R = 0.4 (we have at most two partons in the jet) and the following generic conditions on ﬁnal-state jets: The leading jet has pT,1 > 5 GeV, while the other jets have a lower cut on pT,i=1 > 4.5 GeV to avoid an enhanced sensitivity to soft radiation in the calculated cross section [29]; all jets have rapidities |η1,2| < 4. The studied energy conﬁgurations of future electron-ion colliders are summarized in Table I, where Ee and EA refer √to the electron and nucleus beam energies, respectively, and s is the center-of-mass collision energy per nucleon.

In general, i.e., beyond leading order (LO) perturbative QCD, the light-cone momentum fractions xγ and xA in Eq. (1) are not directly measurable. Instead, one usually introduces their estimates, which can be deﬁned using the two highest transverse-energy jets,

xobs = pT,1e−η1 + pT,2e−η2 ,

(3)

γ

2yEe

xobs = pT,1eη1 + pT,2eη2 ,

(4)

A

2EA

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where pT,1,2 and η1,2 are the transverse energies and rapidities

of the two jets (pT,1 > pT,2).

Figure 1 summarizes our predictions for the dijet cross

section, Eq. (1), as a function of the dijet average trans-

verse momentum p¯T = (pT,1 + pT,2)/2, the average rapidity η¯ = (η1 + η2)/2, and the momentum fractions xAobs and xγobs. The calculations are performed using the central value of

the nCTEQ15 nPDFs. On a logarithmic y scale, EPPS16

nPDFs give indistinguishable results. We ﬁnd sizable yields

in all four considered variables. In particular, at the EIC the

kinematic coverage spans 5 p¯T 20 GeV, −2 < η¯ 3, 0.03 xγobs 1, and 0.01 xAobs 1; see also Ref. [20]. Comparing the kinematic reaches of the four colliders, one

can see from the ﬁgure that an increase of the collision energy

dramatically expands the kinematic coverage. At the LHeC,

HE-LHeC, and FCC, one probes the dijet cross section in

the wider ranges of 5 p¯T xγobs 1, and 10−4 xAobs

60 GeV, −2 η¯ 4, 10−3 1 (LHeC and HE-LHeC), and

even 10−5 xAobs 1 (FCC).

To quantify the magnitude of nuclear modiﬁcations of the

calculated cross section, we show the ratios of the nuclear

cross section, Eq. (1), to the cross section of dijet photopro-

duction on the proton, dσA/(Adσp), in Figs. 2 and 3 in the EIC

FIG. 3. NLO QCD predictions for the ratio of the cross sections of dijet photoproduction on nuclei and the proton as a function of p¯T , η¯ , xAobs, and xγobs in the EIC kinematics. The calculation uses central values of EPPS16 nPDFs (solid lines) and 40 sets of error PDFs (shaded band).

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FIG. 4. Same as in Fig. 2 in the LHeC kinematics.

kinematics and in Figs. 4 and 5 in the LHeC kinematics. The

results for the HE-LHeC and FCC closely resemble those for

the LHeC. The cross section ratios are shown as functions of

p¯T , η¯ , xAobs, and xγobs. In each bin, the solid lines correspond to the corresponding central value of nPDFs in the calculation

of dσA and dσp; the shaded band shows the theoretical un-

certainty, which has been calculated using 32 nCTEQ15 error

PDFs [18] and 40 EPPS16 error PDF sets [19].

In these ﬁgures, the results of the calculation using the

central value of nPDFs exhibit a clear nuclear dependence

of the presented distributions. At the EIC, the magnitude of

nuclear modiﬁcations of the dijet cross section is of the order

of 10–20% and is compatible to the theoretical uncertainty

due to current uncertainties of nCTEQ15 and EPPS16 nPDFs.

At the same time, nuclear modiﬁcations of dσA/(Adσp) are

more pronounced in the kinematics of LHeC (HE-LHeC,

FCC) so that the predicted nuclear suppression of the η¯ and xAobs distributions is somewhat larger (in the nCTEQ15 case) than the uncertainty band due to nPDFs.

From the point of view of constraining nPDFs at small

x, the distribution in xAobs is the most important one. The shape of dσA/(Adσp) repeats that of the ratio of the nucleus and proton structure functions F2A(x, μ2)/[AF2p(x, μ2)]

and

parton

distributions

f

j A

(

x

,

μ2

)

/[

A

f

j p

(

x

,

μ2

)]

(in

particular,

the ratio of the nucleus and proton gluon distributions): the

nuclear suppression (shadowing) for xAobs < 0.05 is followed by some enhancement (antishadowing) around xAobs ≈ 0.1, which is then followed by the EMC-effect-like suppression for xAobs > 0.2. While the EIC allows one to probe the dijet cross section down to xAobs ≈ 0.01, the LHeC extends the small-x range down to xAobs ≈ 10−4 (down to xAobs ≈ 10−5 at FCC). It signiﬁcantly enhances the sensitivity to nuclear modiﬁcations

of nPDFs at small x.

An inspection of Figs. 1–5 allows one to qualitatively

explain the obtained results. At the EIC, the dijet cross section is peaked around xAobs ≈ 0.1, where nPDFs are somewhat enhanced compared to the free proton case, and hence one

expects that dσA/(Adσp) 1 in the dominant part of the

phase space, and in particular at small p¯T and large xγ . It also reveals the anticorrelation of xγ with xA: dσA/(Adσp) is simultaneously enhanced around xAobs ≈ 0.1 (which corresponds to small xA in the EIC kinematics) and for large values of xγobs.

At the LHeC, the dijet cross section is dominated by small xA, xAobs < 0.01. Hence, one expects that the dσA/(Adσp) cross-section ratio is suppressed in most of the phase space,

which is indeed observed in Figs. 4 and 5. The anticorrelation

of xγ with xA is also clearly seen: dσA/(Adσp) < 1 for small xAobs and large xγobs.

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PHYSICAL REVIEW C 102, 065201 (2020)

FIG. 5. Same as in Fig. 3 in the LHeC kinematics.

Note that the expected statistical uncertainty of measurements of the cross section of dijet photoproduction will be much smaller than the theoretical error bands due to nPDFs shown in Figs. 2–5. Indeed, using the projected integrated luminosity of dt L = 10 fb−1/A for all four considered colliders [5,8], one can readily estimate that the expected statistic uncertainty in each bin in Figs. 1–5 should be better than 1% . The expected systematic uncertainty is expected to be at the level of 2%; see Refs. [8,20]. Hence, dijet photoproduction at future lepton-nucleus colliders can be used to considerably reduce the current uncertainties of nPDFs.

IV. CONCLUSIONS

We calculated the cross section of inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of such future lepton-nucleus colliders as EIC, LHeC, HE-LHeC, and FCC using NLO perturbative QCD and nCTEQ15 and EPPS16 nPDFs. We made predictions for the cross-section distributions as functions of the dijet average transverse momentum p¯T , the average rapidity η¯ , the nuclear momentum fraction xAobs, and the photon momentum fraction xγobs and compared the kinematic reaches of the four colliders. We found that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in

all four considered variables. Notably, the LHeC and HELHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC). We also calculated the ratio of the dijet cross sections on a nucleus and the proton, σA/(Aσp), and showed that it exhibits clear nuclear modiﬁcations. We found that in the important case of the xAobs dependence, the shape of σA/(Aσp) repeats that of the ratio of the nucleus and proton parton distributions and, in particular, the gA(x, μ2)/[Agp(x, μ2)] ratio, and reveals a strong suppression due to nuclear shadowing for xAobs < 0.01. This indicates that dijet photoproduction in lepton-nucleus scattering in the kinematics of the future lepton-nucleus colliders will be very beneﬁcial to reduce current uncertainties of nPDFs.

ACKNOWLEDGMENTS

M.K. would like to thank the Petersburg Nuclear Physics Institute (PNPI), Gatchina, for the kind hospitality extended to him during his research visit. V.G. would like to thank the Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster for hospitality. V.G.’s research is supported in part by RFBR, Research Project No. 17-52-12070. The authors gratefully acknowledge ﬁnancial support of DFG through Grant No. KL 1266/9-1

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within the framework of the joint German-Russian project “New constraints on nuclear parton distribution functions

PHYSICAL REVIEW C 102, 065201 (2020)

at small x from dijet production in γ A collisions at the LHC.”

[1] M. Arneodo, A. Bialas, M. W. Krasny, T. Sloan, and M. Strikman, arXiv:hep-ph/9610423 [hep-ph].

[2] D. Boer, M. Diehl, R. Milner, R. Venugopalan, W. Vogelsang, D. Kaplan, H. Montgomery, S. Vigdor, A. Accardi, E. C. Aschenauer et al., arXiv:1108.1713 [nucl-th].

[3] A. Accardi et al., Eur. Phys. J. A 52, 268 (2016). [4] J. L. Abelleira Fernandez et al. (LHeC Study Group), J. Phys.

G 39, 075001 (2012). [5] O. Brüning et al. (LHeC and PERLE Collaborations), J. Phys.

G 46, 123001 (2019). [6] A. Abada et al. (FCC Collaboration), Eur. Phys. J. C 79, 474

(2019). [7] H. Paukkunen (LHeC study Group), PoS DIS2017, 109 (2018),

arXiv:1709.08342 [hep-ph]. [8] E. C. Aschenauer, S. Fazio, M. A. C. Lamont, H. Paukkunen,

and P. Zurita, Phys. Rev. D 96, 114005 (2017). [9] R. Abdul Khalek et al. (NNPDF), Eur. Phys. J. C 79, 471

(2019). [10] J. J. Ethier and E. R. Nocera, Ann. Rev. Nucl. Part. Sci. 70, 43

(2020). [11] C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 29, 497

(2003). [12] H. Abramowicz et al. (ZEUS Collaboration), Nucl. Phys. B 864,

1 (2012).

[13] C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 25, 13 (2002).

[14] H. Abramowicz et al. (ZEUS Collaboration), Eur. Phys. J. C 70, 965 (2010).

[15] A. J. Baltz et al., Phys. Rept. 458, 1 (2008). [16] V. Guzey and M. Klasen, Phys. Rev. C 99, 065202 (2019). [17] V. Guzey and M. Klasen, Eur. Phys. J. C 79, 396 (2019). [18] K. Kovarik et al., Phys. Rev. D 93, 085037 (2016). [19] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado,

Eur. Phys. J. C 77, 163 (2017). [20] M. Klasen and K. Kovarik, Phys. Rev. D 97, 114013 (2018). [21] M. Klasen and G. Kramer, Z. Phys. C 72, 107 (1996). [22] M. Klasen and G. Kramer, Z. Phys. C 76, 67 (1997). [23] M. Klasen, T. Kleinwort, and G. Kramer, Eur. Phys. J. D 1, 1

(1998). [24] M. Klasen and G. Kramer, Eur. Phys. J. C 71, 1774 (2011). [25] M. Klasen, Rev. Mod. Phys. 74, 1221 (2002). [26] S. Frixione, M. L. Mangano, P. Nason, and G. Ridolﬁ, Phys.

Lett. B 319, 339 (1993). [27] M. Glück, E. Reya, and A. Vogt, Phys. Rev. D 46, 1973

(1992). [28] M. Cacciari, G. P. Salam, and G. Soyez, J. High Energy Phys.

04 (2008) 063. [29] M. Klasen and G. Kramer, Phys. Lett. B 366, 385 (1996).

065201-7

Next-to-leading order QCD predictions for dijet photoproduction in lepton-nucleus scattering at the future EIC and at possible LHeC, HE-LHeC, and FCC facilities

V. Guzey 1 and M. Klasen 2 1National Research Center “Kurchatov Institute,” Petersburg Nuclear Physics Institute (PNPI), Gatchina, 188300, Russia 2Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany

(Received 24 March 2020; revised 1 October 2020; accepted 18 November 2020; published 4 December 2020)

We calculate cross sections for inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of the future EIC and the possible LHeC, HE-LHeC, and FCC using next-to-leading order (NLO) perturbative QCD and nCTEQ15 and EPPS16 nuclear parton density functions (nPDFs). We make predictions for distributions in the dijet average transverse momentum p¯T , the average rapidity η¯ , the observed nuclear momentum fraction xAobs, and the observed photon momentum fraction xγobs. Comparing the kinematic reaches of the four colliders, we ﬁnd that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in all four considered variables. Notably, the LHeC and HE-LHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC). The ratio of the dijet cross sections on a nucleus and the proton, σA/(Aσp), depends on xAobs in a similar way as the ratio of gluon densities, gA(xA, μ2 )/[Agp(xA, μ2 )], for which current nPDFs predict a strong suppression due to nuclear shadowing in the region xAobs < 0.01. Dijet photoproduction at future lepton-nucleus colliders can therefore be used to test this prediction and considerably reduce the current uncertainties of nPDFs.

DOI: 10.1103/PhysRevC.102.065201

I. INTRODUCTION

Lepton-nucleus scattering at high energies has traditionally been a fruitful way to access and study the structure of nuclei in quantum chromodynamics (QCD). Despite numerous successes and insights, there is an overarching need to continue these studies at progressively higher energies using colliders. While the plans to use nuclear beams in the HERA collider at DESY [1] have not materialized, a high-energy polarized lepton-proton and lepton-nucleus collider at Brookhaven National Laboratory (BNL) [2,3]—an Electron-Ion Collider (EIC)—has recently been approved. Further down the road, one envisions that the Large Hadron Collider (LHC) at CERN will be complemented by a Large Hadron-Electron Collider (LHeC) and its higher energy upgrade (HE-LHeC) [4,5] as well as a future circular collider (FCC) [6].

The core of the physics program at the future leptonnucleus colliders is comprised of deep inelastic scattering (DIS), allowing one to map out various parton distributions in nuclei with high precision; see, e.g., Refs. [7–10]. In addition, as one learned from HERA, photoproduction of jets [11,12] and dijets [13,14] provides useful complementary information on the QCD (and in particular gluon) structure of hadrons. This has recently been exploited at the LHC, where ultrape-

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

ripheral collisions (UPCs) of heavy ions give an opportunity to study photon-nucleus scattering at unprecedentedly high energies [15]. In particular, it was shown that inclusive dijet photoproduction in Pb-Pb UPCs at the LHC can help to reduce the existing uncertainty in nuclear parton distribution functions (nPDFs) at small x by approximately a factor of 2 [16,17].

In this work, we calculate the cross section of inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of the future EIC, LHeC, HE-LHeC, and FCC using the formalism of collinear factorization, nextto-leading order (NLO) perturbative QCD, and nCTEQ15 [18] and EPPS16 [19] nPDFs. We make predictions for the cross-section distributions as functions of the dijet average transverse momentum p¯T , the average rapidity η¯ , the observed nuclear momentum fraction xAobs, and the observed photon momentum fraction xγobs. We compare the kinematic reaches of the four colliders and ﬁnd that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in all four considered variables. Notably, the LHeC and HE-LHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC), which is two (three) orders of magnitude smaller than that at the EIC. We then discuss in detail the implications of future measurements of dijet photoproduction in lepton-nucleus scattering on the determination of nPDFs.

This work continues and extends the analysis of Ref. [20] by making predictions for high-energy lepton-nucleus colliders including LHeC, HE-LHeC, and FCC, comparing them to the case of the EIC, and analyzing relative merits of the four considered colliders.

2469-9985/2020/102(6)/065201(7)

065201-1

Published by the American Physical Society

V. GUZEY AND M. KLASEN

TABLE I. Energy conﬁgurations of electron-ion colliders considered in this work.

Ee (GeV)

EA (TeV)

√ s (GeV)

EIC

21

LHeC

60

HE-LHeC

60

FCC

60

0.1

92

2.76

812

4.93

1,088

19.7

2,174

The remainder of the paper is structured as follows. In Sec. II, we recap the formalism and the input for the calculation of inclusive dijet photoproduction in NLO perturbative QCD. Our results and their discussion are presented in Sec. III. A summary of our results is given in Sec. IV.

II. DIJET PHOTOPRODUCTION IN NEXT-TO-LEADING ORDER QCD

In the framework of collinear factorization and next-toleading order (NLO) perturbative QCD [21–25], the cross section of dijet photoproduction in eA → e + 2 jets + X

PHYSICAL REVIEW C 102, 065201 (2020)

electron-nucleus scattering reads dσ (eA → e + 2 jets + X )

=

a,b

dy dxγ

dxA fγ /e(y) fa/γ (xγ , μ2 )

× fb/B(xA, μ2 )dσˆ (ab → jets) ,

(1)

where a, b are parton ﬂavors; fγ /e(y) is the ﬂux of equivalent photons of the electron, which depends on the photon light-cone momentum fraction y; fa/γ (xγ , μ2) is the PDF of the photon for the resolved photon case (see below), which

depends on the momentum fraction xγ and the factorization scale μ; fb/B(xA, μ2) is the nuclear PDF with xA being the corresponding parton momentum fraction; and dσˆ (ab → jets) is the elementary cross section for the production of two-parton

and three-parton ﬁnal states emerging as jets in hard scattering

of partons a and b.

The dijet cross section in Eq. (1) receives two types of con-

tributions: the resolved photon contribution, when the photon

interacts with target partons through its quark-gluon structure expressed by fa/γ (xγ , μ2), and the direct photon contribution, when the photon enters directly the hard scattering cross section dσˆ (ab → jets). At leading order (LO), the direct

FIG. 1. NLO QCD predictions for the dijet photoproduction cross section in eA → e + 2 jets + X electron-nucleus scattering at the EIC, LHeC, HE-LHeC, and FCC as a function of the average dijet transverse momentum p¯T , the average rapidity η¯ , and the momentum fractions xAobs and xγobs. The calculation uses nCTEQ15 nPDFs.

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FIG. 2. NLO QCD predictions for the ratio of the cross sections of dijet photoproduction on nuclei and the proton as a function of p¯T , η¯ , xAobs, and xγobs in the EIC kinematics. The calculation uses central values of nCTEQ15 nPDFs (solid lines) and 32 sets of error PDFs (shaded band).

photon contribution has the support exactly at xγ = 1 and fγ /γ (xγ , μ2) = δ(1 − xγ ). At NLO, the separation between the resolved and direct photon contributions depends on the factorization scheme and scale μ. Indeed, by calculating the virtual and real corrections to the matrix elements of interest using massless quarks in dimensional regularization, one can explicitly show that ultraviolet (UV) divergences are renormalized in the MS scheme and infrared (IR) divergences are canceled and factorized into the nucleus (proton) and photon PDFs, respectively; see Ref. [25]. For the latter, this can imply a transformation from the DISγ to the MS scheme. As a result, the direct photon contribution becomes sizable and in practice dominates the cross section at xγ ≈ 1 even at NLO.

In our analysis, we used for the photon ﬂux of the electron

the improved expression derived in the Weizsäcker-Williams

approximation [26]

fγ /e(y) = α 1 + (1 − y)2 ln Qm2 ax(1 − y)

2π

y

me2y2

+ 2m2y 1 − 1 − y ,

(2)

e Qm2 ax me2y2

where α is the ﬁne-structure constant; me is the electron

mass; and Qm2 ax is the maximal photon virtuality. Motivated

by studies of jet photoproduction at HERA, we take Qm2 ax = 0.1 GeV2 and assume that the inelasticity spans the range of 0 < y < 1.

For the photon PDFs, we used the GRV HO parametrization [27], which we transformed as explained above. These photon PDFs have been tested thoroughly at HERA and the Large Electron Positron (LEP) collider at CERN and are very robust, especially at high xγ (dominated by the pQCD photonquark splitting), which is correlated with the low-xA region that is of particular interest for this work. For the nuclear PDFs fb/B(xA, μ2), we employed the nCTEQ15 [18] and EPPS16 [19] parametrizations including both central and error PDFs. The latter are used to evaluate the theoretical uncertainty bands of our predictions.

III. PREDICTIONS FOR DIJET PHOTOPRODUCTION CROSS SECTIONS AT FUTURE ELECTRON-ION COLLIDERS

We performed perturbative NLO QCD calculations of the dijet photoproduction cross section using Eq. (1), which was numerically implemented in an NLO parton-level Monte Carlo [21–25]. This framework has been successfully tested

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to describe the HERA and LEP data on dijet photoproduction on the proton. It implements the anti-kT algorithm [28] with a jet radius of R = 0.4 (we have at most two partons in the jet) and the following generic conditions on ﬁnal-state jets: The leading jet has pT,1 > 5 GeV, while the other jets have a lower cut on pT,i=1 > 4.5 GeV to avoid an enhanced sensitivity to soft radiation in the calculated cross section [29]; all jets have rapidities |η1,2| < 4. The studied energy conﬁgurations of future electron-ion colliders are summarized in Table I, where Ee and EA refer √to the electron and nucleus beam energies, respectively, and s is the center-of-mass collision energy per nucleon.

In general, i.e., beyond leading order (LO) perturbative QCD, the light-cone momentum fractions xγ and xA in Eq. (1) are not directly measurable. Instead, one usually introduces their estimates, which can be deﬁned using the two highest transverse-energy jets,

xobs = pT,1e−η1 + pT,2e−η2 ,

(3)

γ

2yEe

xobs = pT,1eη1 + pT,2eη2 ,

(4)

A

2EA

PHYSICAL REVIEW C 102, 065201 (2020)

where pT,1,2 and η1,2 are the transverse energies and rapidities

of the two jets (pT,1 > pT,2).

Figure 1 summarizes our predictions for the dijet cross

section, Eq. (1), as a function of the dijet average trans-

verse momentum p¯T = (pT,1 + pT,2)/2, the average rapidity η¯ = (η1 + η2)/2, and the momentum fractions xAobs and xγobs. The calculations are performed using the central value of

the nCTEQ15 nPDFs. On a logarithmic y scale, EPPS16

nPDFs give indistinguishable results. We ﬁnd sizable yields

in all four considered variables. In particular, at the EIC the

kinematic coverage spans 5 p¯T 20 GeV, −2 < η¯ 3, 0.03 xγobs 1, and 0.01 xAobs 1; see also Ref. [20]. Comparing the kinematic reaches of the four colliders, one

can see from the ﬁgure that an increase of the collision energy

dramatically expands the kinematic coverage. At the LHeC,

HE-LHeC, and FCC, one probes the dijet cross section in

the wider ranges of 5 p¯T xγobs 1, and 10−4 xAobs

60 GeV, −2 η¯ 4, 10−3 1 (LHeC and HE-LHeC), and

even 10−5 xAobs 1 (FCC).

To quantify the magnitude of nuclear modiﬁcations of the

calculated cross section, we show the ratios of the nuclear

cross section, Eq. (1), to the cross section of dijet photopro-

duction on the proton, dσA/(Adσp), in Figs. 2 and 3 in the EIC

FIG. 3. NLO QCD predictions for the ratio of the cross sections of dijet photoproduction on nuclei and the proton as a function of p¯T , η¯ , xAobs, and xγobs in the EIC kinematics. The calculation uses central values of EPPS16 nPDFs (solid lines) and 40 sets of error PDFs (shaded band).

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FIG. 4. Same as in Fig. 2 in the LHeC kinematics.

kinematics and in Figs. 4 and 5 in the LHeC kinematics. The

results for the HE-LHeC and FCC closely resemble those for

the LHeC. The cross section ratios are shown as functions of

p¯T , η¯ , xAobs, and xγobs. In each bin, the solid lines correspond to the corresponding central value of nPDFs in the calculation

of dσA and dσp; the shaded band shows the theoretical un-

certainty, which has been calculated using 32 nCTEQ15 error

PDFs [18] and 40 EPPS16 error PDF sets [19].

In these ﬁgures, the results of the calculation using the

central value of nPDFs exhibit a clear nuclear dependence

of the presented distributions. At the EIC, the magnitude of

nuclear modiﬁcations of the dijet cross section is of the order

of 10–20% and is compatible to the theoretical uncertainty

due to current uncertainties of nCTEQ15 and EPPS16 nPDFs.

At the same time, nuclear modiﬁcations of dσA/(Adσp) are

more pronounced in the kinematics of LHeC (HE-LHeC,

FCC) so that the predicted nuclear suppression of the η¯ and xAobs distributions is somewhat larger (in the nCTEQ15 case) than the uncertainty band due to nPDFs.

From the point of view of constraining nPDFs at small

x, the distribution in xAobs is the most important one. The shape of dσA/(Adσp) repeats that of the ratio of the nucleus and proton structure functions F2A(x, μ2)/[AF2p(x, μ2)]

and

parton

distributions

f

j A

(

x

,

μ2

)

/[

A

f

j p

(

x

,

μ2

)]

(in

particular,

the ratio of the nucleus and proton gluon distributions): the

nuclear suppression (shadowing) for xAobs < 0.05 is followed by some enhancement (antishadowing) around xAobs ≈ 0.1, which is then followed by the EMC-effect-like suppression for xAobs > 0.2. While the EIC allows one to probe the dijet cross section down to xAobs ≈ 0.01, the LHeC extends the small-x range down to xAobs ≈ 10−4 (down to xAobs ≈ 10−5 at FCC). It signiﬁcantly enhances the sensitivity to nuclear modiﬁcations

of nPDFs at small x.

An inspection of Figs. 1–5 allows one to qualitatively

explain the obtained results. At the EIC, the dijet cross section is peaked around xAobs ≈ 0.1, where nPDFs are somewhat enhanced compared to the free proton case, and hence one

expects that dσA/(Adσp) 1 in the dominant part of the

phase space, and in particular at small p¯T and large xγ . It also reveals the anticorrelation of xγ with xA: dσA/(Adσp) is simultaneously enhanced around xAobs ≈ 0.1 (which corresponds to small xA in the EIC kinematics) and for large values of xγobs.

At the LHeC, the dijet cross section is dominated by small xA, xAobs < 0.01. Hence, one expects that the dσA/(Adσp) cross-section ratio is suppressed in most of the phase space,

which is indeed observed in Figs. 4 and 5. The anticorrelation

of xγ with xA is also clearly seen: dσA/(Adσp) < 1 for small xAobs and large xγobs.

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FIG. 5. Same as in Fig. 3 in the LHeC kinematics.

Note that the expected statistical uncertainty of measurements of the cross section of dijet photoproduction will be much smaller than the theoretical error bands due to nPDFs shown in Figs. 2–5. Indeed, using the projected integrated luminosity of dt L = 10 fb−1/A for all four considered colliders [5,8], one can readily estimate that the expected statistic uncertainty in each bin in Figs. 1–5 should be better than 1% . The expected systematic uncertainty is expected to be at the level of 2%; see Refs. [8,20]. Hence, dijet photoproduction at future lepton-nucleus colliders can be used to considerably reduce the current uncertainties of nPDFs.

IV. CONCLUSIONS

We calculated the cross section of inclusive dijet photoproduction in electron-nucleus scattering in the kinematics of such future lepton-nucleus colliders as EIC, LHeC, HE-LHeC, and FCC using NLO perturbative QCD and nCTEQ15 and EPPS16 nPDFs. We made predictions for the cross-section distributions as functions of the dijet average transverse momentum p¯T , the average rapidity η¯ , the nuclear momentum fraction xAobs, and the photon momentum fraction xγobs and compared the kinematic reaches of the four colliders. We found that an increase of the collision energy from the EIC to the LHeC and beyond extends the coverage in

all four considered variables. Notably, the LHeC and HELHeC will allow one to probe the dijet cross section down to xAobs ∼ 10−4 (down to xAobs ∼ 10−5 at the FCC). We also calculated the ratio of the dijet cross sections on a nucleus and the proton, σA/(Aσp), and showed that it exhibits clear nuclear modiﬁcations. We found that in the important case of the xAobs dependence, the shape of σA/(Aσp) repeats that of the ratio of the nucleus and proton parton distributions and, in particular, the gA(x, μ2)/[Agp(x, μ2)] ratio, and reveals a strong suppression due to nuclear shadowing for xAobs < 0.01. This indicates that dijet photoproduction in lepton-nucleus scattering in the kinematics of the future lepton-nucleus colliders will be very beneﬁcial to reduce current uncertainties of nPDFs.

ACKNOWLEDGMENTS

M.K. would like to thank the Petersburg Nuclear Physics Institute (PNPI), Gatchina, for the kind hospitality extended to him during his research visit. V.G. would like to thank the Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster for hospitality. V.G.’s research is supported in part by RFBR, Research Project No. 17-52-12070. The authors gratefully acknowledge ﬁnancial support of DFG through Grant No. KL 1266/9-1

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within the framework of the joint German-Russian project “New constraints on nuclear parton distribution functions

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at small x from dijet production in γ A collisions at the LHC.”

[1] M. Arneodo, A. Bialas, M. W. Krasny, T. Sloan, and M. Strikman, arXiv:hep-ph/9610423 [hep-ph].

[2] D. Boer, M. Diehl, R. Milner, R. Venugopalan, W. Vogelsang, D. Kaplan, H. Montgomery, S. Vigdor, A. Accardi, E. C. Aschenauer et al., arXiv:1108.1713 [nucl-th].

[3] A. Accardi et al., Eur. Phys. J. A 52, 268 (2016). [4] J. L. Abelleira Fernandez et al. (LHeC Study Group), J. Phys.

G 39, 075001 (2012). [5] O. Brüning et al. (LHeC and PERLE Collaborations), J. Phys.

G 46, 123001 (2019). [6] A. Abada et al. (FCC Collaboration), Eur. Phys. J. C 79, 474

(2019). [7] H. Paukkunen (LHeC study Group), PoS DIS2017, 109 (2018),

arXiv:1709.08342 [hep-ph]. [8] E. C. Aschenauer, S. Fazio, M. A. C. Lamont, H. Paukkunen,

and P. Zurita, Phys. Rev. D 96, 114005 (2017). [9] R. Abdul Khalek et al. (NNPDF), Eur. Phys. J. C 79, 471

(2019). [10] J. J. Ethier and E. R. Nocera, Ann. Rev. Nucl. Part. Sci. 70, 43

(2020). [11] C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 29, 497

(2003). [12] H. Abramowicz et al. (ZEUS Collaboration), Nucl. Phys. B 864,

1 (2012).

[13] C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 25, 13 (2002).

[14] H. Abramowicz et al. (ZEUS Collaboration), Eur. Phys. J. C 70, 965 (2010).

[15] A. J. Baltz et al., Phys. Rept. 458, 1 (2008). [16] V. Guzey and M. Klasen, Phys. Rev. C 99, 065202 (2019). [17] V. Guzey and M. Klasen, Eur. Phys. J. C 79, 396 (2019). [18] K. Kovarik et al., Phys. Rev. D 93, 085037 (2016). [19] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado,

Eur. Phys. J. C 77, 163 (2017). [20] M. Klasen and K. Kovarik, Phys. Rev. D 97, 114013 (2018). [21] M. Klasen and G. Kramer, Z. Phys. C 72, 107 (1996). [22] M. Klasen and G. Kramer, Z. Phys. C 76, 67 (1997). [23] M. Klasen, T. Kleinwort, and G. Kramer, Eur. Phys. J. D 1, 1

(1998). [24] M. Klasen and G. Kramer, Eur. Phys. J. C 71, 1774 (2011). [25] M. Klasen, Rev. Mod. Phys. 74, 1221 (2002). [26] S. Frixione, M. L. Mangano, P. Nason, and G. Ridolﬁ, Phys.

Lett. B 319, 339 (1993). [27] M. Glück, E. Reya, and A. Vogt, Phys. Rev. D 46, 1973

(1992). [28] M. Cacciari, G. P. Salam, and G. Soyez, J. High Energy Phys.

04 (2008) 063. [29] M. Klasen and G. Kramer, Phys. Lett. B 366, 385 (1996).

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