Solution of the linearized Balitsky-Kovchegov equation

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Apr 25, 2017 - We revisited the solution of a linearized form of the leading-order Balitsky-Kovchegov equation (linear in the S-matrix for dipole-nucleus ...
PHYSICAL REVIEW D 95, 074035 (2017)

Solution of the linearized Balitsky-Kovchegov equation Mariyah Siddiqah* and Raktim Abir† Department of Physics, Aligarh Muslim University, Aligarh-202002, India (Received 16 February 2017; published 25 April 2017) We revisited the solution of a linearized form of the leading-order Balitsky-Kovchegov equation (linear in the S-matrix for dipole-nucleus scattering). Here we adopted a dipole transverse-width-dependent cutoff in order to regulate the dipole integral. We also have taken care of all the higher-order terms (higher order in the cutoff) that have been reasonably neglected before. The solution reproduces both the McLerranVenugopalan-type initial condition (Gaussian in the scaling variable) and the Levin-Tuchin solution (Gaussian in the logarithm of scaling variable) in the appropriate limits. It also connects these two opposite limits smoothly with better accuracy for sets of rescaled rapidity when compared to numerical solutions of the full leading-order Balitsky-Kovchegov equation. DOI: 10.1103/PhysRevD.95.074035

I. INTRODUCTION A typical scattering event in any high-energy collider experiment usually involves a rapidly growing cascade of gluons. This is partly because high-energy and/or highvirtuality emitted gluons themselves emit further gluons. At high-enough energy this cascade of gluons may occupy all the available final-state phase space to such an extent that the fusion of multiple gluons to a single gluon begins. This could eventually develop a thermodynamical detail balance with multiple gluons produced from a single gluon, which leads to the origin of gluon saturation with a characteristic momentum scale Qs [1]. This is a dynamically generated and energy-dependent scale below which stochastic (almost) independent multiple-scattering approximations are no longer valid and highly correlated nonlinear gluon interactions dominate the phase space. This gluon recombination also restores the unitarity of the scattering S-matrix which will otherwise be violated by an exponential growth of gluon multiplicity. Consequently, this saturation of gluons also avoids a possible violation of the Froissart bound for the total scattering cross section through the power-law growth of the Balistky-Fadin-Kuraev-Lipatov (BFKL) [2] solution, which encodes the energy evolution of the cross section away from the nonlinear region. Unitary corrections to the BFKL equation in the Regge kinematics were first studied by Gribov et al. [1]. Further studies were done later by Balitsky [3] within a Wilson line formalism [4] and soon after by Kovchegov [5,6] in Mueller’s color dipole approach [7–9]. The Balistky hierarchic chain formed by the Wilson line operators reduced to the closed-form equation derived by Kovchegov in the large-N c limit. The integral kernels in the Balitsky-Kovchegov (BK) equation for both linear and nonlinear terms are identical, which has a simple interpretation of splitting of one parent color dipole into two daughter * †

[email protected], [email protected] [email protected], [email protected]

2470-0010=2017=95(7)=074035(8)

dipoles. A lot of progress has been made since then, including solving the equation both analytically and numerically and extending the equation beyond its leading-order (LO) accuracy [10]. The next-to-leading-order (NLO) BK equation was derived [11], and the running coupling corrections were included in the BK evolution equations [12–14]. The solution of the NLO BFKL equation has been found analytically [15], and the application of the leading-order equation has been extended to jet quenching studies [16]. Recently, the first numerical study for the solution to the NLO BalitskyKovchegov equation in coordinate space has been performed [17]. Large double logarithms have been resummed in the QCD evolution of color dipoles [18] and in accordance with the HERA data [19]. In addition, an analytic BK solution based on the eigenfunctions of the truncated BFKL equation has been proposed recently that reproduces the initial condition and the high-energy asymptotics of the scattering amplitude [20]. In order to have the evolved solution of the BK equation, one usually starts with the initial condition for the evolution from McLerran-Venugopalan model [21–23] or from the phenomenological Golec-Biernat–Wusthoff model [24,25]. The imaginary part of the dipole-nucleus amplitude for deep inelastic scattering of the dipole with a large nucleus takes the following form: N MV ðx10 ; YÞ ¼ 1 − SMV ðx10 ; YÞ ¼ 1 − exp ð−κx210 Q2s ðYÞÞ; ð1Þ where κ ¼ 1=4 or could be fixed from the definition of the saturation scale Qs and x10 being the transverse width of the parent dipole. Equation (1) is taken as the initial condition for the evolution, and expected to be valid for some initial rapidity both inside and outside the saturation region. The S-matrix in Eq. (1) is Gaussian in the scaling variable pffiffiffiτ [τ ¼ x⊥ Qs ðYÞ] with a (model-dependent) variance 1= κ . However in ultra-high-energy limit where the Levin-Tuchin

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© 2017 American Physical Society

MARIYAH SIDDIQAH and RAKTIM ABIR

solution [26–28] of Balitsky-Kovchegov equation is valid, the S-matrix has the following asymptotic expression: N LT ðx10 ; YÞ ¼ 1 − SLT ðx10 ; YÞ   1 þ 2iν0 2 2 2 ¼ 1 − exp − ln ½x10 Qs ðYÞ : ð2Þ 4χð0; ν0 Þ Unlike Eq. (1), the S-matrix in Eq. (2) is Gaussian in ln τ (not in τ). A solution that spans over the full kinematic range of saturation dynamics is expected to be in accordance with both the McLerran-Venugopalan-type initial condition and the Levin-Tuchin solution in their appropriate limits. In this article we have revisited the solution for the linearized LO BK equation. By linearized, we mean linear in S (unlike BFKL, which is linear in N) where the term quadratic in S has not been taken. With a modified x⊥ dependent form of cutoff in the dipole integral, we obtain the general solution as Nðx10 ; YÞ ¼ 1 − Sðx10 ; yÞ   1 þ 2iν0 2 2 ¼ 1 − exp Li ½−λ x Q ðYÞ ; 2χð0; ν0 Þ 2 1 10 s

where Li2 is a dilogarithm function and λ1 ð≈7.22Þ is a parameter that is fixed by the definition of Qs . Interestingly, Eq. (3) as solution of the linearized BK equation reproduces both Eq. (1) [Gaussian in τ ¼ x10 Qs ðYÞ] and Eq. (2) [Gaussian in a logarithm of τ ¼ x10 Qs ðYÞ, i.e., in ln τ] in the limits τ ¼ x10 Qs ðYÞ ≪ 1 and τ ¼ x10 Qs ðYÞ ≫ 1, respectively. It also connects these two opposite limits smoothly, with a better accuracy than numerical solutions of the full LO BK equation. II. THE DIPOLE INTEGRAL One convenient way to address high-energy scatterings in QCD is to express the problem in hand in terms of colordipole degrees of freedom. This approach, originally proposed by Mueller [7–9], is formulated in the transverse coordinate space. It has the added advantage that transverse coordinates of the dipoles are not changed during rapidity (or energy) evolution. This makes it easier to include the saturation effects in the model. Typically one starts with a quark-antiquark pair in order to calculate the probability of emission of a soft gluon off this pair. Both the quark and antiquark are to follow light-cone trajectories, and emitted gluons are calculated in the eikonal approximations (the projectile does not suffer any recoil). Adding contributions coming from the quark and antiquark together with their interference, one gluon part of the onium wave function is found to be proportional to the following integral kernel convoluted over the onium wave function with no soft gluon [7]:

d2 x2

I dip ≡

x210 ≡ x220 x221

Z

ðx − yÞ2 d2 z: ð4Þ ðx − zÞ2 ðz − yÞ2

The above kernel (together with a Sudakov-type form factor) can be interpreted as the emission probability of a soft gluon from the dipole with two poles located at x and y. In the large-N c limit, the emitted gluon can be seen as a quark-antiquark pair and the above formula can be interpreted as the probability of decay of the original parent dipole at ðx; yÞ of transverse size x10 ≡ jx − yj into two new daughter dipoles at ðx; zÞ and at ðz; yÞ with sizes x20 ≡ jx − zj and x21 ≡ jz − yj. In this section we will revisit the derivation of the above integral, which is central to dipole studies. The integral I dip supplemented with the factor α¯ s =2π could be interpreted as the differential probability of decay of one parent dipole of transverse size x10 ð≡x1 − x0 Þ into two daughter dipoles of arbitrary sizes. Noting that d2 x2 is equal to [7] Z 2πx02 x12

ð3Þ

PHYSICAL REVIEW D 95, 074035 (2017)

Z



0

dkkJ0 ðkx10 ÞJ 0 ðkx20 ÞJ0 ðkx21 Þdx20 dx12 ; ð5Þ

where J 0 ðzÞ is a Bessel function of the first kind, we now write Eq. (4) as [10], 2πx210

I dip ¼

Z

×

0

Z





0

Z dkkJ0 ðkx10 Þ

0



dx20 J ðkx Þ x20 0 20

dx21 J ðkx Þ: x21 0 21

ð6Þ

The integral in Eq. (6) over x20 and x21 is ill defined until one specifies a way to regulate the ultraviolet singularities at x20 , x21 → 0. In the dipole model studies, one usually introduces a lower cutoff ρ into the x20 and x21 integrals. This procedure was first adopted by Mueller in [7] and followed in subsequent studies [6], Z 0

∞ dx

x

Z J 0 ðkxÞ ⇒

ρ

∞ dx

x

J0 ðkxÞ ¼ ln

2 − γ þ OðρÞ: kρ

ð7Þ

Using ρ as a cutoff as usually done in the dipole model studies, is one way to regulate the integral. Alternatively, for example, one could replace 1=x2 → 1=ðx2 þ ρ2 Þ for x220 and x221 in the denominator in Eq. (6) which gives an orderzero modified Bessel function of second kind K 0 ðρkÞ. There are also other ways to regulate the integral. All these regularizations should give the same leading-order result as ρ → 0, but the subleading terms would depend on the regularization procedure that was followed. Inside the saturation region ρ is usually identified with the inverse saturation momentum 1=Qs . In this study we have revisited this issue with the following two points: (1) We have considered OðρÞ and all other higher-order terms in Eq. (7) that have been ignored earlier,

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SOLUTION OF THE LINEARIZED BALITSKY-KOVCHEGOV …

Z

ρ

∞ dx

x

J 0 ðkxÞ ¼ ln

2 −γ kρ

the cutoff tends to zero, ρ → 0. Hence, unlike earlier studies where the result is valid only in the limit Qs → ∞, here the final result is expected to be valid and free from regularization-scheme artifacts in both the limits x10 Qs ≫ 1 and x10 Qs ≪ 1. With abovementioned modifications, I dip is found to be (details are in the Appendix)   1 2 2 I dip ¼ 2π lnðλ1 x10 Qs þ λ2 Þ − 2πLi1 : λ1 x210 Q2s þ λ2

  k2 ρ2 1 2 2 þ F 1;1;2;2;2;− k ρ ; 4 8 2 3 2 ¼ ln − γ kρ ∞ X ð−1Þmþ1 1 2m 2m þ k ρ : ð8Þ 22m ðm!Þ2 2m m¼1

ð14Þ

A simple ratio test confirms that the radius of convergence of this series is infinity. We derived a compact closed-form expression of I dip that contains contributions from OðρÞ and all other higherorder terms. (2) In earlier studies, the cutoff ρ usually identified with inverse saturation momentum 1=Qs as x20 ; x21 ≥ ρ ¼

1 ; Qs

Noting that Li1 ðzÞ ¼ − lnð1 − zÞ, one may further simplify as I dip ¼ 2π ln ðλ1 x210 Q2s þ λ2 − 1Þ;

ð9Þ

  1 − λ2 ; ¼ þ 2π ln 1 − λ1 x210 Q2s   1 − λ2 ¼ 2π ln ðλ1 x210 Q2s Þ − 2πLi1 : ð16Þ λ1 x210 Q2s

ð10Þ

Looking at Eq. (15), one could fix λ2 as λ2 ¼ 2 by taking the limit I dip → 0 when x10 → 0. Therefore, we have

In this study we have adopted a similar regularization procedure as done earlier but assumed a general x10 -dependent form of cutoff as x20 ; x21 ≥ ρ ¼

1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Qs λ1 þ λ2 ð1=x10 Qs Þ2

I dip ¼ 2π ln ðλ1 x210 Q2s þ 1Þ:

ð11Þ

III. S-MATRIX INSIDE THE SATURATION REGION ð12Þ

which can be compared with Eq. (10). Here λ1 and λ2 are two positive real parameters that would be fixed in the following ways: (a) Parameter λ2 would be fixed by requiring the fact that in the limit x10 → 0 the dipole integral in Eq. (6) vanishes, i.e., I dip → 0. (b) Parameter λ1 would be fixed by the definition of saturation momentum: at x10 ¼ 1=Qs , numerical value of the S-matrix would be half, 1 Sðx10 ¼ 1=Qs Þ ¼ : 2

ð17Þ

The parameter λ1 would be fixed from the definition of Qs in the next section, as mentioned earlier.

Equation (11) actually implies 1 1 λ 1 ; ≤ λ1 Q2s þ 22 ¼ 2 ; x220 x221 x10 ρ

ð15Þ

2π ln ðλ1 x210 Q2s Þ

or, equivalently, 1 1 1 ; 2 ≤ Q2s ¼ 2 : 2 x20 x21 ρ

PHYSICAL REVIEW D 95, 074035 (2017)

Scattering the S-matrix for the color dipole interacting with a large nuclear target can be expressed as the expectation value of two lightlike path-ordered Wilson lines transversely separated by x10 ð¼ x⊥ − y ⊥ Þ as Sðx10 ; YÞ ¼

ð18Þ

In the large-N c limit, nonlinear energy (or rapidity) evolution of the S-matrix is governed by the BalitskyKovchegov equation, Z ∂ αN x2 Sðx01 ; YÞ ¼ s 2c d2 x2 2 012 ∂Y 2π x02 x21 × ½Sðx02 ; YÞSðx12 ; YÞ − Sðx01 ; YÞ:

ð13Þ

Equation (12) is just an ad hoc ansatz for the UV cutoff as the generalization of Eq. (10). However, this modified form of the cutoff ensures that in both the limits Qs → ∞ ðx10 fixedÞ and x10 → 0 ðQs fixedÞ,

1 hTr½Wðx⊥ ; YÞW † ðy⊥ ; YÞi: Nc

ð19Þ

Within the kinematic domain where Sðx10 Þ ≫ Sðx02 ; YÞSðx12 ; YÞ, one can neglect the term quadratic in S in Eq. (19) and the BK equation, an integro-differential equation in general, becomes a first-order partial differential equation of Sðx01 ; YÞ,

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MARIYAH SIDDIQAH and RAKTIM ABIR

∂ αN Sðx01 ; YÞ ¼ − s 2c ∂Y 2π

Z

x2 d2 x2 2 012 x02 x21

Sðx01 ; YÞ:

PHYSICAL REVIEW D 95, 074035 (2017)

ð20Þ

In general one expects the validity of this linear equation in the limit when x10 > x20 , x21 > 1=Qs ðYÞ. Here we note that when x20 , x21 > x10 , we could also expect Sðx20 ÞSðx21 Þ < Sðx10 Þ; i.e., the quadratic term is smaller than the linear term in the BK equation. Therefore this linearized form should be expected to be valid (at least approximately) in the limiting domains defined by x10 > x20 ; x21 and x10 < x20 ; x21 . In Eq. (20) the integral over dipole size goes over x02 ; x12 > ρ, as discussed in Sec. II with ρ¼

1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Qs λ1 þ 2=ðQ2s x210 Þ

χð0; ν0 Þ=ð1 þ 2iν0 Þ ≈ 2.44 from Eq. (24), (initial-condition-independent) constant S0 as unity, and defining the saturation momentum Qs [at x10 ¼ 1=Qs , the numerical value of the S-matrix would be half, Sðx10 ¼ 1=Qs Þ ¼ 1=2], one could now estimate λ1 ≈ 7.22. Therefore,   1 þ 2iν0 Sðx⊥ ; YÞ ¼ exp Li2 ½−λ1 x2⊥ Q2s ðYÞ ; ð28Þ 2χð0; ν0 Þ with λ1 ≈ 7.22. Equation (28) is the main result of this article. We will next discuss different limits of Eq. (28). (i) In the limit x10 Qs ≪ 1=λ1 ∼ 0.14 < 1, one may retain only the first term in the dilogarithm series, Sðx⊥ ; YÞ ≈ exp ½−1.48x210 Q2s ðYÞ:

ð21Þ

This is Gaussian in the variable τ ¼ x10 Qs ðYÞ in accordance with the McLerran-Venugopalan model [21–23] or the Golec-Biernat–Wusthoff model [24,25] up to a model dependent variance ∼1=3. (ii) τ ¼ x10 Qs ≈ 1: A numerical fit to power-law function aτ2γ (with a ¼ 1=2 to reproduce N ¼ 1=2 at τ ¼ 1) around τ ≡ x10 Qs ∼ 1 reveals

where λ2 have already been fixed at 2. Using Eq. (16), Eq. (20) can now be written as ∂ ln Sðx01 ; YÞ ¼ −α¯ s ln ½λ1 x210 Q2s ðYÞ ∂Y   1 þ α¯ s Li1 − 2 2 : λ1 x10 Qs ðYÞ

ð22Þ

Solution of Eq. (22) can be written straightforwardly as   1 þ 2iν0 1 2 ln ðλ1 x210 Q2s ðYÞÞ S ¼ S0 exp − 2χð0; ν0 Þ 2   1 þ Li2 − 2 2 : ð23Þ λ1 x10 Qs ðYÞ Here we have used following leading-order expression for saturation momentum [1]:   χð0; ν0 Þ Qs ðYÞ ¼ Qs0 exp α¯ s Y ≈ Qs0 e2.44α¯ s Y ; ð24Þ 1 þ 2iν0 where  χð0; νÞ ¼ 2ψð1Þ − ψ

   1 1 þ iν − ψ − iν ; 2 2

ð25Þ

ψ is the digamma function, and S0 is a constant independent of any initial condition. One can further simplify Eq. (23) to   1 þ 2iν0 2 2 S ¼ S0 exp ½Li ð−λ1 x10 Qs ðYÞÞ ; ð26Þ 2χð0; ν0 Þ 2 where we have used the following identity of a dilogarithm for x > 0:   1 π2 1 Li2 ð−xÞ þ Li2 − ð27Þ ¼ − − ln2 x: x 6 2 A factor exp ðκπ 2 =6Þ with κ ¼ð1þ2iν0 Þ=ð2χð0;ν0 ÞÞ has also been absorbed in the normalization constant S0 . Taking

ð29Þ

1 Nðx⊥ ; YÞ ¼ 1 − Sðx⊥ ; YÞ ¼ ðx210 Q2s Þ0.43 ; 2

ð30Þ

which is quite far away from theoretical predictions, Nðx⊥ ;YÞ ¼ N 0 ðx210 Q2s Þ1−γcr ≃ N 0 ðx210 Q2s Þ0.63 ;

ð31Þ

with γ cr ¼ 0.37 [30]. However, as mentioned before, in this article the final result is expected to be valid and free from regularization-scheme artifacts in both the limits x10 Qs ≫ 1 and x10 Qs ≪ 1. The result around x10 Qs ∼ 1 should be taken with the caution that it may involve regularization-scheme artifacts. (iii) In the black-disc limit, τ ¼ x10 Qs ≫ λ1 ∼ 7.2 > 1, Eq. (26) reproduces the Levin-Tuchin solution as   1 þ 2iν0 2 2 2 Sðx⊥ ; YÞ ¼ exp − ln ½x10 Qs ðYÞ : ð32Þ 4χð0; ν0 Þ Here we have used the asymptotic expansion of polylogarithms, Lis ðzÞ in terms of ln ð−zÞ and Bernoulli numbers B2k , as Lis ðzÞ ¼

∞ X ð−1Þk ð1 − 21−2k Þð2πÞ2k k¼0

×

B2k ½lnð−zÞs−2k : ð2kÞ! Γðs þ 1 − 2kÞ

ð33Þ

In Fig. 1 we have plotted the dipole amplitude Nðx10 ; YÞ as a function of scaling variable τ ¼ x10 Qs ðYÞ; the new solution [Eq. (3)] is compared with numerical solutions of the leading-order Balitsky-Kovchegov equation for two sets

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SOLUTION OF THE LINEARIZED BALITSKY-KOVCHEGOV …

0.8

New Solution BK αsY = 1.2 BK αsY = 2.4 BK αs = 0.2, Y = 40 Levin-Tuchin solution McLerran-Venugopalan IC

0.6

0.4

0.2

0 0.01

0.1

1

10

100

FIG. 1. The dipole amplitude Nðx10 ; YÞ as a function of the scaling variable τ ¼ x10 Qs ðYÞ; the new solution [Eq. (3)] is compared with numerical solutions of the leading-order BalitskyKovchegov equation for two sets of rescaled rapidity, αs Y ¼ 1.2, 2.4, one set of fixed coupling αs ¼ 0.2 for Y ¼ 40 [29], and the McLerran-Venugopalan initial condition Eq. (1) (tweaked to reproduce N ¼ 1=2 at τ ¼ 1), with the Levin-Tuchin solution Eq. (2) also displayed for reference.

of rescaled rapidity, αs Y ¼ 1.2, 2.4, and one set of fixed coupling αs ¼ 0.2. The solution is in better agreement with the numerical solutions of the full LO BK equation in a wide kinematic domain inside the saturation region. IV. CONCLUSION AND OUTLOOK The quest to find a general solution that satisfies both the initial condition and the asymptotic analytical solution of evolution equations and also shows geometrical scaling began more than a decade ago. Recently, Contreras et al. [30,31] proposed a solution for the full BK equation in the entire kinematic region that satisfies the McLerramVenugopalan initial condition. It was found that for 0 < z ≪ 1 the solution of the BK equation takes the form N 0 0 and p is either zero or a positive even integer,

m¼1

Z

ðA3Þ

Using

m¼1

4πx210

1 1 : 2m 22m ðm!Þ2

∂ ϵ→0 ∂ϵ

Z

dkkJ 0 ðx10 kÞ ln k ¼ lim



0

dkk1þϵ J 0 ðx10 kÞ ¼ −

1 x210

ðA5Þ

and Z



0

∂2 dkkJ0 ðx10 kÞln k ¼ lim 2 ϵ→0 ∂ϵ 2

Z 0



1þϵ

dkk

  2 x10 J0 ðx10 kÞ ¼ 2 ln þ γE ; 2 x10

ðA6Þ

all of which follows from Z



0

dkkλ−1 J0 ðkxÞ ¼ 2λ−1 x−λ

Γðλ=2Þ : Γð1 − 2λÞ

ðA7Þ

Using Eq. (A5) and Eq. (A6), the integral I 0 can be written as I0 ¼

2πx210

Z



0

2   2 2 x : dkkJðx10 kÞ ln − γ E ¼ 2π ln 10 kρ ρ2

ðA8Þ

Equation (A4) ensures that I 1 vanishes for x10 > 0, I 1 ¼ 2πx210

Z 0



2 X ∞ dkkJðx10 kÞ ð−1Þmþ1 C2m k2m ρ2m ¼ 0;

ðA9Þ

m¼1

and terms containing γ E and ln 2 in the integral I 2 in Eq. (A2) will vanish as well, X   ∞ 2 mþ1 2m 2m dkkJðx10 kÞ ln − γ E ð−1Þ C2m k ρ I2 ¼ kρ 0 m¼1 Z ∞ ∞ X ð−1Þmþ1 C2m dkkJðx10 kÞ lnðkρÞðkρÞ2m ¼ −4πx210 4πx210

¼

Z

−4πx210



m¼1 ∞ X

0

ð−1Þ

m¼1

mþ1

∂ C2m lim ϵ→0 ∂ϵ

Z



0

Equation (A7) could be used to evaluate the integral

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dkkJðx10 kÞðkρÞ2mþϵ :

ðA10Þ

SOLUTION OF THE LINEARIZED BALITSKY-KOVCHEGOV …

I2 ¼ ¼

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  ∞ x210 X ∂ 2mþ1þϵ x10 −2m−2−ϵ Γðm þ 1 þ ϵ=2Þ mþ1 −4π 2 ð−1Þ C2m lim 2 ϵ→0 ∂ϵ Γð−m − ϵ=2Þ ρ ρ m¼1     ∞ m  X ρ 2m ∂ ϵ x10 −ϵ Γð1 þ ϵ=2Þ Y mþ1 2mþ1 m ð−1Þ C2m 2 ð−1Þ lim 2 i −4π ϵ→0 ∂ϵ x10 Γð−ϵ=2Þ i¼1 ρ m¼1

 2m   ∞ X ρ 1 mþ1 2mþ1 m 2 − ¼ −4π ð−1Þ C2m 2 ð−1Þ ðm!Þ x10 2 m¼1   ∞ X ρ 2m C2m 22m ðm!Þ2 ¼ −4π x10 m¼1   ∞ X 1 ρ2 m ¼ −2π m x210 m¼1  2 ρ ¼ −2πLi1 2 : x10

 ϵ 2 þ 2

ðA11Þ

Finally, the dipole integral is I dip ¼ I 0 þ I 1 þ I 2  2  2 x10 ρ ¼ 2π ln 2 − 2πLi1 2 ρ x10 ¼

2π ln ðλ1 x210 Q2s

þ λ2 Þ − 2πLi1



 1 ; λ1 x210 Q2s þ λ2

ðA12Þ

where we have substituted ρ by Qs and x10 using Eq. (12) in the last line.

[1] L. V. Gribov, E. M. Levin, and M. G. Ryskin, Semihard Processes in QCD, Phys. Rep. 100, 1 (1983). [2] L. N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Yad. Fiz. 23, 642 (1976) [Sov. J. Nucl. Phys. 23, 338 (1976)]; E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, The pomeranchuk singularity in nonabelian gauge theories, Zh. Eksp. Teor. Fiz. 72, 377 (1977) [Sov. Phys. JETP 45, 199 (1977)]; I. I. Balitsky and L. N. Lipatov, The pomeranchuk singularity in quantum chromodynamics, Yad. Fiz. 28, 1597 (1978) [Sov. J. Nucl. Phys. 28, 822 (1978)]. [3] I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996). [4] I. Balitsky, High-energy QCD and Wilson lines, in At the Frontier of Particle Physics, edited by M. Shifman (World Scientific, Singapore, 2001), Vol. 2, pp. 1237–1342. [5] Y. V. Kovchegov, Small-x F(2) structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999). [6] Y. V. Kovchegov, Unitarization of the BFKL pomeron on a nucleus, Phys. Rev. D 61, 074018 (2000). [7] A. H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL pomeron, Nucl. Phys. B415, 373 (1994).

[8] A. H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B425, 471 (1994). [9] Z. Chen and A. H. Mueller, The dipole picture of highenergy scattering, the BFKL equation and many gluon compound states, Nucl. Phys. B451, 579 (1995). [10] Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Cambridge University Press, Cambridge, 2012). [11] I. Balitsky and G. A. Chirilli, Next-to-leading order evolution of color dipoles, Phys. Rev. D 77, 014019 (2008). [12] Y. V. Kovchegov and H. Weigert, Triumvirate of running couplings in small-x evolution, Nucl. Phys. A784, 188 (2007). [13] I. Balitsky, Quark contribution to the small-x evolution of color dipole, Phys. Rev. D 75, 014001 (2007). [14] J. L. Albacete, N. Armesto, J. G. Milhano, C. A. Salgado, and U. A. Wiedemann, Numerical analysis of the BalitskyKovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity, Phys. Rev. D 71, 014003 (2005). [15] G. A. Chirilli and Y. V. Kovchegov, Solution of the NLO BFKL equation and a strategy for solving the all-order BFKL equation, J. High Energy Phys. 06 (2013) 055.

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MARIYAH SIDDIQAH and RAKTIM ABIR

PHYSICAL REVIEW D 95, 074035 (2017)

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