Probing proton structure with c ¯ c correlations in ultraperipheral pA collisions

We study the exclusive diffractive c ¯ c photoproduction in ultraperipheral pA collisions. The formalism makes use of off-diagonal generalizations of the unintegrated gluon distribution, the so-called generalized transverse momentum dependent distributions (GTMDs). We present two different formulations. The first one is based directly on gluon GTMD parametrizations in momentum space. Another option is the calculation of the GTMD as a Fourier transform of the dipole-nucleon scattering amplitude N ( Y,⃗r ⊥ ,⃗b ⊥ ). The latter approach requires some extra regularization discussed in the paper. Different dipole amplitudes from the literature are used. Compared to previous calculations in the literature, we integrate over the full phase space and therefore cross sections for realistic conditions are obtained. We present distributions in rapidity of c or ¯ c , transverse momentum of the c ¯ c pair, four-momentum transfer squared as well as the azimuthal correlation between a sum and a difference of the c and ¯ c transverse momenta. The azimuthal correlations are partially due to the so-called elliptic gluon Wigner distribution. Different models lead to different modulations in the azimuthal angle. The modulations are generally smaller than 5%. They depend on the range of transverse momentum selected for the calculation.


I. INTRODUCTION
The diffractive production of high momentum particles, such as quark-antiquark dijets, serves as a probe of various aspects of the hadron structure [1].For example, in the diffractive dissociation of photons or pions into dijets, the forward cross section at large jet transverse momenta maps out the unintegrated gluon distribution of the target [2,3].Recently, there has been much interest in a generalization of unintegrated (or transverse momentum dependent) parton distributions to a five-dimensional quasi-probability phase space distribution, known as the Wigner distribution [4,5], which depends on both the transverse momentum and impact parameter of a parton in the proton or nucleus.When transformed fully to momentum space these are equivalent to the generalized transverse momentum distributions (GTMDs), see e.g.[6][7][8].In particular, these distributions encode the dependence on the transverse momentum transfer ⃗ ∆ ⊥ to the target and, therefore, are of relevance for the description of the forward cone in diffractive processes.
The Wigner distribution depends on the transverse momentum ⃗ k ⊥ of partons as well as the impact parameter ⃗ b ⊥ , including a dependence on the azimuthal angle between the two transverse vectors.In the GTMD approach this translates into a dependence on ⃗ k ⊥ and ⃗ ∆ ⊥ .This dependence gives rise to a so-called elliptic Wigner function or GTMD [9].In the region of small-x which is of interest in this paper, the equivalent color dipole approach for diffractive processes [10,11] can be used.Here, the dipole amplitude depending on dipole size ⃗ r ⊥ and impact parameter ⃗ b ⊥ contains the same information as the Wigner function/GTMD.The azimuthal angle correlation discussed above translates then into a dependence on the dipole orientation with respect to the background color field of the target.In inclusive processes this dependence can give rise to flow-like azimuthal correlations, for example in prompt photons production [12], or in inclusive two-particles' production [13].
Here, we revisit the exclusive production of cc pairs in proton-lead collisions at LHC energies.The dominant reaction mechanism here is the diffractive photoproduction of the cc pair on FIG. 1. Feynman diagrams for the diffractive photoproduction of q q pairs in nucleus-proton collisions, discussed in the present paper.
a proton by a Weizsäcker-Williams photon emitted by the lead nucleus, see the diagrams in Fig. 1.One is then interested in the correlations of and the momentum transfer to the proton, which for the exclusive reaction fulfills In this work we account only for the contribution where the ion serves as a source of the photon.Here we exploit the fact, that the photon flux scales with the charge of the ion as Z 2 .This factor will be missing when the photon is emitted by the proton.On the other hand, the integrated diffractive process on the nucleus will display a dependence σ ∝ A α , with roughly α ∼ 1.4 ÷ 1.5 (see e.g.[1,3]).While the diffractive photoproduction on the nucleus may therefore still be a sizeable effect, the extraction of the process discussed in this paper is possible if proton tagging [21][22][23] is available, due to the very sharp transverse momentum dependence of the electromagnetically scattered proton.
In distinction to dijet production, for the heavy quarks already the quark mass m Q can play the role of a hard scale, and also intermediate and small values of ⃗ P ⊥ are tractable.Here, we will calculate cross sections and distributions for realistic kinematic conditions at the LHC.We also discuss some subtleties regarding the correlations in the azimuthal angle At large jet momenta P ⊥ the correlations in ϕ expected to be dominantly ∝ cos(2ϕ) and are unambiguously expressed in terms of the elliptic Wigner function/GTMD.At lower P ⊥ , however, the azimuthal correlations are better understood as coming from the matrix element (or impact factor) at finite ∆ ⊥ .While formally also these correlations can be absorbed into the Wigner function/GTMD, these would correspond to δ-function terms in the latter.We discuss the relation between the different approaches and the alternative formulation in terms of a off-forward gluon density matrix introduced in Ref. [24].
This paper is organized as follows.In Section II, we explain how to calculate the diffractive cross section of interest and discuss the formalism starting from the dipole representation of the amplitude.Then, in Section III we present our numerical results for differential cross sections as functions of various kinematical variables using a variety of the existing gluon GTMD models.Finally, in Section IV the main conclusions are summarised.

A. Kinematics and cross section
The cross section for the proton-nucleus reaction can be written in the following form Here, x Q , x Q are the fractions of the nucleus' Light-Front plus momentum carried by a heavy quark Q and antiquark Q (with mass m Q ), respectively.The transverse momenta of quark and antiquark are respectively, so that ⃗ ∆ ⊥ is the transverse momentum of the Q Q pair.As the photon is collinear to the incoming nucleus, ⃗ ∆ ⊥ is also equal, up to a sign, to the transverse momentum transfer to the proton target.
The Weizsäcker-Williams photons carry the momentum fraction For the flux of quasireal photons, we use the well-known expression (see for example the review [25]), where Z correspond to the atomic number of the projectile particle, α em is the fine structure constant, ξ jA = x A m p (R j + R A ) involves the target and nucleus radii (R j and R A , respectively) and effectively excludes the overlap of the projectile and the target in impact parameter space, and m p is the proton mass.
Let us also quote a useful expression for the cross section which reads in terms of centerof-mass rapidities y Q , y Q of quarks, where We also need the transverse mass of the Q Q-pair, which square is given by The rapidity of the Q Q-pair in the cm-frame can then be calculated from (2.9) We will be also interested in the laboratory frame rapidities, which are obtained from the shift where for the 208 Pb nucleus, we have A = 208, Z = 82.

B. Color dipole representation of the diffractive amplitude
We start our discussion from a basic description of the color dipole approach to diffractive processes [10,11].Here, the cross section for the γp → Q Q diffractive dissociation process is written as (2.11) Above, z, 1 − z are the Light-Front momentum fractions carried by quark/antiquark in the γ → Q Q transition.The corresponding Light-Front wave function for the depends on the Light-Front helicities of quarks, λ/2, λ/2 and photon, λ γ .Its explicit form in transverse momentum space can be found for example in Ref. [26].
Our main interest is in the dipole scattering amplitude N (Y, ⃗ r ⊥ , ⃗ b ⊥ ).Its energy dependence is encoded through the "rapidity" Y .We define the latter as Y = ln(x 0 /x IP ), with x 0 = 0.01.Here, where M ⊥ is the transverse mass of the Q Q system.The dipole amplitude is related to the familiar color dipole cross section (see e.g. the textbook [27]) via It is related to an off-diagonal generalization of the unintegrated gluon distribution -a gluon density matrix through the relation [24]: (2.15) Quark and antiquark move at impact parameters, which allow us to relate the four phase factors in the curly bracket in Eq. (2.15) to the four diagrams of Fig. 1 in an obvious fashion.We write the unintegrated gluon density matrix in the following form, where N c = 3 for the number of colors in QCD, and we put in evidence the strong coupling constant α s and gluon propagators here.Integrating Eq. (2.15) over ⃗ b ⊥ , we obtain for the dipole cross section the well-known representation [28] so that indeed F(x, ⃗ κ ⊥1 , ⃗ κ ⊥2 ) is the proper off-forward generalization of the standard unintegrated gluon distribution, which relates to its collinear counterpart via Below, we will also use the non-perturbative parameter, which has the interpretation of the dipole cross section for large dipoles.Now, inserting the representation of the dipole amplitude given in Eq. (2.15) into Eq.(2.11), we obtain for the diffractive amplitude the following convolution structure: (2.21) Here, in the terminology of small-x (or BFKL) factorization, one would refer to the structure in brackets as the impact factor for the coupling of two off-shell gluons to the γ → Q Q amplitude.The two t-channel gluons carry the transverse momenta, and the impact factor has the property that it vanishes when either of the gluon transverse momenta goes to zero, i.e. for ⃗ κ ⊥ = ± ⃗ ∆ ⊥ /2.Finally, we obtain for our diffractive photoproduction cross section: Here, M 1 and ⃗ M 0 are amplitudes for the sum of quark helicities equal to one or zero, respectively.Explicitly, we find (see e.g.Ref. [29]): In order to understand the origin of azimuthal correlations, it is useful to decompose our amplitude.To this end, let us introduce Then, our matrix elements take the form: (

2.26)
There emerges a rapidity-dependent form factor, which is a non-perturbative parameter, as is obvious from its form in the forward limit as an integral, that converges at soft, non-perturbative values of k ⊥ .Indeed, as can be seen from Eq. (2.20), it is directly proportional to the dipole cross section for large dipoles, σ 0 (x IP ).

C. GTMD representation
In the literature, often a different momentum-space representation of the diffractive amplitude is used (see, for example, Refs.[9,19,20]).Namely, one introduces the Fourier transform of the dipole amplitude (using the normalization and notation of Ref. [20]), Here, T (Y, ⃗ k ⊥ , ⃗ ∆ ⊥ ) is often referred to as the generalized transverse momentum distribution (GTMD) of gluons in the proton target.Certainly, just like the gluon density matrix it encodes the same information as the dipole amplitude.What is the relation between these two momentum space distributions?
To answer this question, let us perform the Fourier transform by inserting Eq. (2.15) into Eq.(2.29), which yields where C(Y, ⃗ ∆ ⊥ ) of Eq. (2.27) multiplies a combination of δ-functions.Now, evidently T is essentially equal to the f , up to the term containing δ-functions.The latter, however, will not contribute when convoluted with the impact factor in Eq. (2.21).Formally, we might therefore replace In practical numerical applications, however, this equivalence is not that obvious.In diffractive interactions the values of ∆ ⊥ are bounded by the diffractive slope B D , so that at large values of k ⊥ the delta-functions are irrelevant, and the two gluon distributions, f and T are equal to each other, up to a factor of 1/2.An immediate corollary of the representation in Eq. (2.30) is the sum rule which encodes the fact, that N (Y, ⃗ r ⊥ , ⃗ b ⊥ ) → 0 for r ⊥ → 0. Generally speaking, as evidenced by the presence of δ-function terms, the Fourier transform is a non-convergent integral and does not exist as a function.One therefore needs to regularize the Fourier transform which is often done by inserting a Gaussian cutoff function [9,19,20]: Now, inserting the representation in Eq. (2.15), we obtain the cutoff-dependent T as where is a "smeared out" delta-distribution, and is a smeared out version of the gluon density matrix.The regularized T -matrix also fulfills the sum rule of Eq. (2.32).
Let us finally quote the expressions of matrix elements ⃗ M 0 , M 1 in terms of T , which read: In its derivation we made use of the sum rule of Eq. (2.32).Notice, that in effect here we use the impact factor for forward scattering, and all dependence on ⃗ ∆ ⊥ has been absorbed into the GTMD T (Y, ⃗ k ⊥ , ⃗ ∆ ⊥ ).

D. Benchmark gluon GTMDs
Let us briefly discuss the different GTMD models used in this work.We follow both approaches presented above, and consider a total of five different models as benchmarks in our analysis of the differential cross section of charm photoproduction in pA UPCs.
First, we use two different parametrizations of the off-forward gluon density matrix f of Eq. (2.17).Here, we choose to write Such a form has been suggested in Ref. [30] for the case of vector meson production (for a recent use, see Ref. [31]).The same approach has also been taken in Ref. [12].For the diffractive slope, we use B = 4 GeV −2 , and the diagonal unintegrated gluon distribution F is taken from two different models: the Golec-Biernat-Wüsthoff (GBW) model [32], and the Moriggi-Peccini-Machado (MPM) parametrization [33].
We also consider models based on the regularized Fourier transform of dipole amplitudes as in Eq. (2.33).The first model is based on a numerical solution of the Balitsky-Kovchegov equation for the dipole S-matrix with impact parameter dependence [34,35].Such solutions have been obtained, for example in Refs.[36][37][38].In our work we use the results of Ref. [9].This solution is based on exploiting a symmetry first observed in [39].We label it as HHU below.The second model, based on the original effective McLerran-Venugopalan model of Ref. [40], that has been generalised for the proton target and extended to incorporate inhomogeneities in the transverse-plane distribution of gluons in Ref. [13].The latter model has been applied for exclusive diffractive light and heavy quarks' photoproduction in Refs.[19] and [20], respectively.We denote this model as MV-IR in what follows.A third model is the so-called bSat model of Kowalski and Teaney [41].The Kowalski-Teaney (KT) model has been adjusted to the proton structure function data and hence it gives a rather realistic dipole amplitude.We use parameters from Table I fit 3 of [41] for the gluon distribution.In this regard, the HHU and MV-IR models can be considered as rather toy models, they do, however, incorporate non-trivial dipole orientation effects, which is not the case for the KT model.
The leading dependence on dipole orientation is quantified by the elliptic part N ϵ of the dipole amplitude in the Fourier expansion, We translate the isotropic and elliptic parts of the dipole amplitude to GTMDs by the appropriate Fourier-Bessel transforms [14,19,20]: (2.40) The explicit form of the matrix element for the elliptic glue for arbitrary quark mass m Q has been found in Ref. [20].We briefly review its derivation in Appendix A.
It is interesting to note that the gluon density matrices, constructed according to the prescription of Eq. (2.38), do also lead to a dipole amplitude that depends on dipole orientation, as can be seen by plugging the expression of Eq. (2.38) into Eq.(2.15).It gives us the expression where (2.42) Again, we see that the dipole orientation dependence appears together with the nonperturbative parameter σ 0 -the dipole cross section for large dipoles.We want to stress, that the correlation between ⃗ r ⊥ and ⃗ b ⊥ emerges, although Eq. (2.38) does not contain any correlation between ⃗ k ⊥ and ⃗ ∆ ⊥ !The ⃗ r ⊥ • ⃗ b ⊥ correlation is in fact of a simple geometric origin, as it singles out the contributions of diagrams where only the quark or antiquark interact and probe the matter density at their respective impact parameters.There is no obvious way how to construct an off-forward glue that leads to totally isotropic dipole amplitude.For this to happen, the relevant dipole sizes simply have to be small enough for the matter density to be constant over distances ∼ r ⊥ .Then the shifts in the curly brackets do not matter and only the last term in Eq. (2.41) effectively contributes.Of course, in general one would expect Eq. (2.17) to have nontrivial azimuthal correlations between ⃗ k ⊥ and ⃗ ∆ ⊥ , in which case the dipole amplitude will have a genuinely dynamical elliptic piece that can contribute also at hard momenta ⃗ P ⊥ .Finally, to wrap up this section, in Fig. 2 we show the ( ⃗ k ⊥ , ⃗ ∆ ⊥ ) maps for various dipolenucleon amplitudes T (Y, ⃗ k ⊥ , ⃗ ∆ ⊥ ) for comparison.In Fig. 3 we show for x IP = 0.01 and ∆ ⊥ = 0.01 GeV the function of Eq. (2.38) for the GBW and MPM models, and the result of the Fourier transform Eq (2.40).Notice that the latter do change sign, in accordance with the sum rule Eq. (2.32).

III. NUMERICAL RESULTS
We now turn to our numerical results for the production of cc pairs.In this work, we use the charm quark mass value of m c = 1.5 GeV.In this section, we show the crosssection distributions for cc photoproduction in pA UPCs differential in y LAB c , Y LAB pair , x IP , as well as P ⊥ , ∆ ⊥ , t and ϕ, for several benchmark models of the gluon GTMD in the proton discussed above.We will show numerical results for differential distributions integrated over 0.01 < P ⊥ < 10 GeV, as well as over 5 < P ⊥ < 10 GeV domains of the phase space.For all distributions calculated as Fourier transforms of N (Y, ⃗ r ⊥ , ⃗ b ⊥ ) (see Eq. (2.33)) we use the same regularization parameter ε = (0.5 fm) −2 as in Ref. [20].In Table 1, we show our results for the integrated cross section for P ⊥ < 10 GeV as well as for 5 < P ⊥ < 10 GeV.In addition we show results for x IP < 0.05.The results with the extra cut are only slightly smaller, except of the MPM model.In Table 1 we also varied the parameter ε by a factor two in order to give an estimate on the dependence on ε, which turns out to be rather large.We notice considerable differences for the different GTMDs at large transverse momenta.For instance, the GBW distribution drops off with transverse momentum uch faster than other GTMDs.This is due to the fact that the GBW UGD does not possess the perturbative power-law tail at large momenta, but rather features a Gaussian distribution in transverse momenta with an x-dependent width.It should also be noted that the original MV-IR distribution is formulated in Ref. [13] without x IP (or Y ) dependence.To get semi-realistic results for the differential distributions, the original MV-IR has been modified as with λ = 0.277.It should be noted that more realistic extensions of the MV-IR model have been proposed in the literature, see e.g.[42].We start our presentation from the lab-frame rapidity distributions of charm quarks in Fig. 4. Here, the incoming nucleus has a large positive rapidity, and the proton -a large negative rapidity.The range of the rapidity distribution at large y LAB c is essentially controlled by the photon flux.We see that the HHU GTMD leads to a significantly higher peak in the charm rapidity distribution than other models, and extends at the negative side only to y LAB c ∼ −4.This is due to the fact that it has a support only for small x IP < 0.01.The GBW distribution gives results consistent with an earlier calculation of Gonçalves at al [43], where only rapidity distributions were studied.In addition, the distributions in lab-frame rapidity of the cc-pair are shown in Fig. 5.They rather closely resemble the single-quark rapidity distributions.The related x IP distributions are presented in Fig. 6.Values of x IP as small as 10 −5 enter the calculation.Note that in the case of HHU GTMD, we assumed that this distribution only applies for x IP < 0.01.
A comment on the range of the x IP distributions is in order.Notice, that the diffractively scattered proton will carry a fraction ξ ≈ 1 − x IP of the beam momentum.Also, the rapidity gap between the products of photon dissociation rises at small x IP as ∝ log(1/(1 − ξ)) ∼ log(1/x IP ).It is therefore clear that an experiment, which tags protons carrying a sizeable fraction of the beam momentum, would provide an upper limit on x IP .For a discussion of relevant forward detectors at the LHC, see [21][22][23].This goes together with considerations on the applicability of the dipole amplitudes used in this paper.This will in general require a large rapidity gap as mentioned above.There will be no sharp value of x IP where the dipole approach breaks down, but one would expect it to require x IP ≪ 0.1, with a conservative upper limit of x IP < 0.05.In absence of concrete experimental conditions, below we will integrate over the whole phase space, i.e. up to x max IP = 1, expecting that the impact from large x IP to the integrals is in general small.An exception are the distributions in rapidity, where large negative rapidities are close to the proton beam, and therefore associated with small gaps.We return to this issue at the end of this section, where we demonstrate the role of a cutoff x max IP on the various distributions.We observe that the cut due to ε in Eq.(2.40) has a different impact for different models.In particular, it has a strong effect on the KT model in the region of small P T , where the bulk of the cross section resides.This makes the distributions integrated over P T for the KT model very small.For P T > 2 GeV, however the results are comparable to other models.Eventually the value of the ε could be adjusted to experimental data.
Let us now term to transverse momentum distributions of the produced charm quarks.In Fig. 7 we show the differential distributions in P ⊥ .We notice a considerable difference in the tail of the distributions, with the GBW glue giving the softest behaviour as expected.In Fig. 8 we present the charm distribution in ∆ ⊥ .As generically anticipated for diffractive scattering, such a distribution is peaked at low values of ∆ ⊥ ∼ 1/ √ B, with B being the diffractive slope.The HHU model somewhat stands out as it gives rise to a peak in ∆ ⊥ distribution that is shifted to a softer value than that for the other GTMD models considered in this work.These distributions are the result of the dynamics of gluons encoded in the BK equation.Presumably the behaviour of the ∆ ⊥ distribution could be made more realistic by introducing a similar regulator function in the impact parameter variable b ⊥ as in Eq. (2.40) for the dipole size r ⊥ .Here however we restrain from introducing additional parameters.
Next, in Fig. 9, we show the distribution in the Mandelstam variable at the proton side.We see the typical diffractive peak for the P ⊥ -integrated case.Again, the HHU model stands out with a noticeable curvature and very sharp forward peaking.In the large-P ⊥ tail, the t-distribution flattens out considerably for all considered GTMD benchmark models.Notice that this case corresponds to a rather large diffractive mass and, therefore, to a sizeable longitudinal momentum transfer.We now turn to the angular correlations in azimuthal angle ϕ between ⃗ P ⊥ and ⃗ ∆ ⊥ , shown in Fig. 10, which are one of the main results of this work.There are in general clearly visible correlations.Notice that for the GBW and MPM GTMDs these correlations are of the "geometric" origin, such that in momentum space they are fully generated by the matrix element.The angular modulations obtained from the GBW or MPM GTMDs are of order 10 %.We observe that at large P ⊥ they quickly drop and change shape.Notice, that the calculations with these two GTMDs include all possible harmonics, not only cos 2ϕ.In the case when these correlations are computed accounting for the elliptic gluon distribution only (see Eq. (2.33)), i.e. when only cos 2ϕ modulations are included, they appear to be at the level of about 1 -5%.
In order to better visualize the strength of the azimuthal correlations in Fig. 11, we present also the angular distributions divided by the integrated cross section for a given  GTMD model.Such normalised distributions are easier to compare for different GTMDs.Again, we show the results for two ranges of P ⊥ as explained in the figure.The azimuthal modulations are of the order of a few percent.Let us return to the issue on a cutoff in x IP .In Figs. 12, 13, 14 we show the effect of introducing an upper limit x max IP into our integration by example of three distributions.Here we used the MPM model for the GTMD.In Fig. 12 we clearly see the expected effect of the cutoff on the distribution at large negative rapidities, beyond y LAB c ∼ −4 the cross section quickly drops.The effect of the cutoff is more marked in the high P ⊥ domain as seen from the right panel.The ∆ ⊥ distribution shown in Fig. 13 is rather mildly affected, with again a stronger effect at large P ⊥ .Finally, the distribution in the azimuthal angle ϕ shown in Fig. 14 is practically unaffected.
As these are parton-level observables, these angular distributions are not directly measurable.For the case of dijets, one would expect soft-gluon corrections to have an impact,  see e.g.Ref. [44].For the more relevant case of exclusive (or inclusive diffractive) pairs of open heavy flavor mesons (D-mesons), a thorough study of hadronization corrections would be required.Such a study however goes beyond the scope of the present work.Recently, the LHCb Collaboration was able to measure the inclusive cc dijets [45] (see also [46]), whether such a measurement would also be possible in the kinematics discussed here is an open issue.The different models give slightly different azimuthal angle distributions.Specially interesting are distributions for 5.0 < P ⊥ < 10.0 GeV where the localization of maxima for some models are reversed.where Here, we used the identity

FIG. 2 .
FIG.2.Different parametrizations for the gluon distribution in the proton.

FIG. 3 .
FIG.3.Different parametrizations for gluon distribution in the proton for ∆ = 0.01 GeV and x IP = 0.01 (on the left side) as well as x IP = 0.0001 (on the right side).For the HHU, KT and MV-IR models, the regularization parameter ε = (0.5 fm) −2 was chosen.

FIG. 4 .
FIG. 4. Distributions in lab-frame rapidity of the charm quark y LAB c for 0.01 < P ⊥ < 10.0 GeV on the left and for 5.0 < P ⊥ < 10.0 GeV on the right.

FIG. 5 .
FIG.5.Distributions in rapidity of the cc pair Y pair for 0.01 < P ⊥ < 10.0 GeV on the left and for 5.0 < P ⊥ < 10.0 GeV on the right.

FIG. 11 .
FIG. 11.Distributions in the azimuthal angle ϕ between ⃗ P ⊥ and ⃗ ∆ ⊥ normalised to the total cross section for 0.01 < P ⊥ < 10.0 GeV on the left and for 5.0 < P ⊥ < 10.0 GeV on the right.

TABLE 1
. Total cross section for 0.01 < P ⊥ < 10.0 GeV and for 5.0 < P ⊥ < 10.0 GeV and different approaches.In this table, the cross sections are integrated over ∆ ⊥ < 3 GeV and −8 < y c < 8.
Distributions in ∆ ⊥ for 0.01 < P ⊥ < 10.0 GeV on the left and for 5.0 < P ⊥ < 10.0 GeV on the right.
FIG.10.Distributions in the azimuthal angle ϕ between ⃗ P ⊥ and ⃗ ∆ ⊥ for 0.01 < P ⊥ < 10.0 GeV on the left and for 5.0 < P ⊥ < 10.0 GeV on the right.