NNLO solution of nonlinear GLR–MQ evolution equation to determine gluon distribution function using Regge like ansatz

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NNLO solution of nonlinear GLR–MQ evolution equation to determine gluon distribution function using Regge like ansatz

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P. Phukan ∗ , M. Lalung, J.K. Sarma

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Department of Physics, Tezpur University, Tezpur, Assam, 784028, India

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Received 17 July 2017; received in revised form 1 September 2017; accepted 5 September 2017

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Abstract

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In this work we have suggested a solution of the Gribov–Levin–Ryskin, Mueller–Qiu (GLR–MQ) nonlinear evolution equation at next-to-next-to-leading order (NNLO). The range of Q2 in which we have solved the GLR–MQ equation is Regge region of the range 6.5 GeV2 ≤ Q2 ≤ 25 GeV2 and so we have incorporated the Regge like behavior to obtain Q2 evolution of gluon distribution function G(x, Q2 ). We have also checked the sensitivity of our results for different values of correlation radius (R) between two interacting gluons, viz. R = 2 GeV−1 and R = 5 GeV−1 as well as for different values of Regge intercept λG . Our results are compared with those of most recent global DGLAP fits obtained by various parametrization groups viz. PDF4LHC15, NNPDF3.0, HERAPDF15, CT14 and ABM12. © 2017 Elsevier B.V. All rights reserved.

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Keywords: Quantum chromodynamics (QCD); Gluon recombination phenomena at small x; GLR–MQ evolution equation

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1. Introduction

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Parton distribution functions (PDFs) are considered as the most significant tool in hadronic collision processes for the calculation of inclusive cross sections. In perturbative QCD, the

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* Corresponding author.

E-mail addresses: [email protected] (P. Phukan), [email protected] (M. Lalung), [email protected] (J.K. Sarma). http://dx.doi.org/10.1016/j.nuclphysa.2017.09.003 0375-9474/© 2017 Elsevier B.V. All rights reserved.

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scale evolution of the PDFs is well predicted by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equation [1–3] at large interaction scale Q2 and the fractional momentum x. At sufficiently large Q2 , the number densities of the partons can be evaluated by solving the DGLAP equations from which the emission of partons during the process can be spotted and then over a broad range of x and Q2 , a comparison with the data is performed. Further for obtaining a good global fit to the data, the initial distributions are iterated. The initial distributions are the non-perturbative inputs, that perturbative QCD cannot predict. The data from lepton–proton deep inelastic scattering (DIS), particularly the DIS data from ep collisions at DESY-HERA play a key role in these analysis especially in the region of small-x. It is evident from the data that there is a sharp growth of the gluon density towards small-x [1,2]. This is also well predicted by the solutions of linear DGLAP equation. However, the gluon density cannot grow forever because hadronic cross-sections comply with the unitary bound known as Froissart Bound [4]. For this purpose a distinguishable effect known as gluon recombination which is supposed to be responsible for the mechanism that unitarize the cross section at high energies or at small-x. In other words, at small-x the number of gluon will be so large that they will spatially overlap, resulting recombination of gluons. But in the derivation of the DGLAP equation, the gluon–gluon interaction terms are overlooked. Thus a modification in the linear DGLAP equation is required in order to take care of the nonlinear corrections due to gluon recombination. The H1   Collaboration [5] at HERA has able to calculate the proton structure function F2 x, Q2 down to x ∼ 10−5 though it is in the perturbative region. These data have been included in the recent global analyses by the CTEQ [6] and MRST [7] collaborations. Though DGLAP equation comply with the experimental data quite accurately in a wide range of x a favorable explanation in fitting the H1 collaboration data towards the and Q2 , it fails to provide   region of large Q2 > 4 GeV−2 and in the region of small Q2 (1.5 GeV−2 < Q2 < 4 GeV−2 ) [8,9] simultaneously. Furthermore, in the NLO treatment of MRST2001 [10] when both these regions were taken into consider, a good fit was obtained but a negative gluon distribution was encountered. Likewise in the NLO set CTEQ6M [11], the problem of negative gluon distribution also appears. This implies that towards smaller values of x and Q2 , constraining that Q2 ≥ 2 ,  being the QCD cutoff parameter, it is possible to observe gluon recombination effects which lead to nonlinear power corrections to the linear DGLAP equation. Gribov, Levin and Ryskin in the ref. [12] and Mueller and Qiu in the ref. [13] have calculated the nonlinear terms and they formulated these shadowing corrections to obtain a new evolution equation  commonly known as GLR–MQ equation. This equation deals with a new quantity G2 x, Q2 interpreted as the two gluon distribution function per unit area of the hadron. In addition to the explanation of gluon saturation phenomena, GLR–MQ equation predicts a critical line which supposed to separate the gluon saturation regime and the perturbative regime valid in this critical line border [14]. Most significantly GLR–MQ equation introduces a characteristic momentum scale Q2s which provides the measure of the density of the saturated gluons. The GLR–MQ equation is regarded as a hypothetical link between perturbative and non perturbative region. There has been some work in recent years inspired by GLR–MQ approach [14,15]. The solution of GLR–MQ equation provides the determination of the saturation momentum that incorporates physics in addition to that of the linear evolution equations commonly used to fit DIS data. In our previous works we obtained a solution of the Q2 dependence of gluon distribution from GLR–MQ in leading order (LO) [16] as well as next-to-next-leading order (NLO) [17]. In the present work, we adopt the Regge-like parametrizations to obtain a solution of the nonlinear GLR–MQ equation up to next-to-next-leading order (NNLO) and a direct

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comparison of our results with those of the global DGLAP fits obtained by various collaborations viz. NNPDF3.0 [18], HERAPDF1.5 [19], CT14 [20], ABM12 [21] and PDF4LHC [22].

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GLR–MQ equation deals with the number of partons increased through gluon splitting as well as the number of partons decreased through gluon recombination in a phase space cell (ln(1/x)lnQ2 ). Therefore the balance equation for emission and recombination of partons can be formulated as [3,7]   ∂ρ x, Q2 αs (Q2 )Nc αs 2 (Q2 )γ = [ρ(x, Q2 )]2 . ρ(x, Q2 ) − 2 π Q2 ∂ln(1/x)lnQ xg(x,Q2 ) , πR 2

Here ρ = where R is the correlation radius between two interacting gluons, is the 81 target area. The factor γ is evaluated by Muller and Quie that is found to be γ = 16 for Nc = 3     [7]. Now in terms of gluon distribution function G x, Q2 = xg x, Q2 the GLR–MQ equation can be written in standard form [23]      1   ∂G x, Q2 ∂G x, Q2 dω 2  x 81 αs 2 Q2 2 = | − G , Q . (1) DGLAP 16 R 2 Q2 ω ω ∂lnQ2 ∂lnQ2 πR 2

x

The first term of the RHS in eq. (1) represents the double-leading logarithmic approximation (DLLA) linear DGLAP term while the second term is the shadowing correction due to the nonlinearity in gluon density. At small-x region, the contribution of quark–gluon diagrams are very negligible. For the correlation radius, R = RH , shadowing correction is negligibly small whereas for R << RH , shadowing correction is expected to be large, where RH is the radius of the hadron [24,25].  2 We introduce a variable t = ln Q , where  is the QCD cutoff parameter. Now Considering 2 the terms up to NNLO, αs (t) takes the following form    4π  2 2 αs (t) = t − lnt − 1 + c , (2) t − blnt + b ln β0 t 2 where b = ββ12 , c = ββ23 , β0 = 11 − 23 Nf , β1 = 102 − 38 3 Nf . Here we consider the number of 2 0 color charges, Nc and the number of quark flavors, Nf as 3 and 4 respectively. Now in terms of the variable t, eq. (1) can be expressed as

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∂G(x, t) ∂G(x, t) 81 αs 2 (t) = |DGLAP − ∂t ∂t 16 R 2 2 et

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dω 2  x  G ,t . ω ω

∂G(x, t) |DGLAP = ∂t

1 Pgg (ω)G

x ω

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(3)

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 , t dω.

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Ignoring the quark contribution to the gluon rich distribution function, we can write the first term of the eq. (3) of the form

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1 x

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2. Theory

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(4)

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Considering up to NNLO terms, the splitting function Pgg (ω) can be expanded as powers of αs (t),

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αs (t) 0 αs (t) 2 2 αs (t) 3 3 Pgg (ω) = Pgg (ω) + Pgg (ω). P (ω) + 2π gg 2π 2π

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NLO splitting function [27]  4 20ω2 1 Pgg =CF Tf −16 + 8ω + + − (6 + 10ω)lnω − 2(1 + ω) ln2 ω 3 3ω

 1 4 20 26 + Nc Tf 2 − 2ω + ω2 − − (1 + ω)lnω − p(ω) 9 ω 3 9



 1 67 25 11ω 44ω2 2 27(1 − ω) 2 + ω − − − + lnω + Nc 2 9 ω 3 3 3

67 π2 + ln2 ω − p(ω) + 4(1 + ω) ln2 ω + 9 3 − 4lnω ln(1 − ω)p(ω) + 2p(−ω)S2 (ω) ,

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1 1+ω

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dz π2 1 − z small 1 2 −→ ln ω − ln + O(ω), ω 2 z z 6

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where D0 =

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+ L0 L1 (7305 + 8757L0 ) + 274.4L0 − 7471L20

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+ 52.75ω3 − L0 L1 (115.6 − 85.25ω + 63.23L0 ) − 3.422L0 32L30 680 2 + 9.68L0 − − , 27 243ω

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− 144L40

− 350.2 + 755.7ω − 713.8ω2 + 559.3ω3 + L0 L1 (26.15 − 808.7L0 ) + 1541L0 832L30 512L40 182.96 157.27L0 2 + 491.3L0 + + + + 9 27 ω ω  16D0 + Nf2 − + 6.463δ(1 − ω) − 13.878 + 153.4ω − 187.7ω2 9

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+ 4848ω  14214 2675.8L0 + + + Nf − 412.172D0 − 528.723δ(1 − ω) − 320L1 ω ω

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2 Pgg =2643.52D0 + 4425.89δ(1 − ω) + 3589L1 − 20852 + 3968ω − 3363ω2

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N2 − 1 1 , Tf = N f , CF = c 2Nc 2 and NNLO splitting function [28]

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where

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(5)

The corresponding splitting functions involved in eq. (5) are LO splitting function [26]



1−ω 11 2Nf ω Pgg 0 (ω) = 6 + ω(1 − ω) + + − δ(1 − ω), ω (1 − ω)+ 2 3

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1 (1−ω)+ , L0

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Now considering all these terms, the DGLAP equation takes up the following form in NNLO

 ∂G(x, t) 3αs (t) 11 Nf |DGLAP = − + ln(1 − x) G(x, t) ∂t π 12 18

1  ωG ωx , t − G(x, t) + dω (1 − ω) (6) x



 x 1−ω G ,t + ω(1 − ω) + ω ω



αs (t) 2 g αs (t) 3 g + I1 (x, t) + I2 (x, t), 2π 2π   1 1     g g 1 (ω)G x , t 2 (ω)G x , t . where I1 (x, t) = x dω Pgg and I2 (x, t) = x dω Pgg ω ω For simplicity in our calculations, we consider two numerical parameters T0 and T1 such that T 2 (t) = T0 T (t) and T 3 (t) = T1 T (t), where T (t) = αs /2π . T0 and T1 are not arbitrary parameters. These numerical parameters are determined by phenomenological analysis. These are obtained from the particular range of Q2 under our study and by a suitable choice of T0 and T1 we can reduce the difference between T 2 (t) and T0 T (t) as well as T 3 (t) and T1 T (t) to minimum such that the consideration of the parameters T0 and T1 doesn’t give any abrupt change in our work. To get an analytical solution of the GLR–MQ equation we incorporate a Regge-like behavior of gluon distribution function. The behavior of structure functions at small-x is well explained in terms of Regge-like ansatz [29]. For small-x, the Regge behavior of the sea quark and antiquarks distribution is given by qsea ∼ x −αP corresponding to a pomeron exchange with an intercept of αP = 1. But the valence-quark distribution for small-x given by qval (x) ∼ x −αR corresponding to a reggeon exchange with an intercept of αR = 0.5. At moderate Q2 , the leading order calculations   in ln(1/x) with fixed value of αs predicts a steep power law behavior of xg x, Q2 ∼ x −λG ,    where λG = 4αs Nc π ln2 ≈0.5 for αs = 0.2, as appropriate for Q2 > 4 GeV2 [30–32]. In DIS processes, particularly the small-x region is considered to have greater possibility towards exploring the Regge limit of perturbative quantum chromodynamics (pQCD). This Regge model provides parametrizations of the DIS distribution functions, fi (x, Q2 ) = Ai (Q2 )x −λi (i = (singlet structure function) and g (gluon distribution)), where λi is Pomeron intercept minus one. This kind of Regge like form for the parton distribution functions are well supported by the description of experimental data. For instance, in the work by authors of Ref. [33] the Zeus DIS data have been successfully described in the range of x < 0.07 and Q2 < 10 by the Regge factorization of the structure function F2 cc¯ . In another work given in Ref. [34] the author has beautifully shown phenomenological analysis for the spin dependent structure function g1 (x, Q2 ) at low x and low Q2 . Using the Regge pole exchange model the author in this work parametrized the spin dependent structure function at (x → 0) as g1i (x, Q2 ) = β(Q2 )x −αi (0) , α(0) being the intercept of Regge trajectory. To determine the gluon distribution function we try to solve GLR–MQ equation by considering a simple form of Regge like behavior given as G(x, t) = H (t)x which implies

−λG

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(7)

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G

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and G

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(11)

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a b2 c alnz 2 b2 lnz b2 ln2 z (x, z) = − − + lnz + − − ρ(x) − z , z z z z z z ζ (x, z) = − b2 + c + z − alnz + b2 ln2 z

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   2 ρ(x) 1+ at − 2bt ρ(x)

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Eq. (10) is a partial differential equation, the solution of which is of the form 

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a = b + b , b = β1 /β0 and c = β2 /β0 . 2

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−λG 2 2λG



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λG +2 1 x λG 2x − 3Af − −x+1 + λG + 2 λG λG 1  1 T0 T1 1 λG + Af dωPgg (ω)ω + Af dωP2gg (ω)ωλG , 2 2 x x

81 2π 2 1 − x λG 4 φ(x) = T0 Af , Af = , 2 2 16 R  2λG β0

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where

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(8)

t2

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ω

 , t = H (t)x −λG ωλG = G(x, t)ωλ

ω = G (x, t)ω , (9) , t = H (t)x ω where λG is the Regge intercept for gluon distribution function while H(t) is a function of t. This type of Regge like form is well supported by various authors of Refs. [31,33,35]. But it is important to note that for the entire kinematic region the Regge factorization cannot be considered as a good ansatz which is clear from the pioneer literature [36]. However, this type of Regge like behavior of gluon density are only valid in the vicinity of the saturation scale Qs , where the scattering amplitude has the functional form dependence on (Q2s (x)/Q2 )γ [37]. The Regge theory is apparent to be applicable if the quantity W 2(= 1/Q2 (1/x − 1)) is much greater than all other variables. Thus, we expect it to be valid when x is small enough, whatever be the value of Q2 . The range in fact which we are considering viz. 10−5 ≤ x ≤ 10−2 and 5 GeV2 ≤ Q2 ≤ 25 GeV2 is actually the Regge regime. According to Regge theory, at small-x, both gluons and sea quarks behaviors are controlled by the same singularity factor in the complex angular momentum plane [29]. At small x, since the Regge intercepts, λG of all spin-independent singlet, non-singlet and gluon structure functions should tend to 0.5 [35], it is also expected that at λG ≈ 0.5, our theoretical results comply with the experimental data and parametrization. Substituting eqs. (7), (8) and (9) in eq. (3) the GLR–MQ equation becomes 2

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Fig. 1. Comparison of T 2 (t) and T0 .T (t) as well as T 3 (t) and T1 .T (t) vs Q2 .

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and ρ(x), φ(x) are defined earlier in eq. (10) and C is a constant to be

determined using initial conditions of the gluon distributions for a given t0 , where t0 = ln

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G (x, t0 ) =

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C=

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dz z12 φ(x)e(x,z) ζ (x, z)G (x, t0 )

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   2 ρ(x) 1+ at − 2bt ρ(x)

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  2 ρ(x) 1+ ta − 2bt ρ(x) 0 0 t0

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(14) Thus by solving GLR–MQ equation semi numerically, we have obtained an expression for the Q2 or t evolution of gluon distribution function  G(x,t) up to NNLO. From this final expression we can easily anticipate the t-evolution of G x, Q2 for a particular value of x by choosing a suitable input.

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3. Results

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G (x, t0 )

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Now substituting C from eq. (13) we obtain the t (or Q2 ) evolution of gluon distribution function G(x,t) for fixed x in NNLO as

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dz z12 φ(x)e(x,z) ζ (x, z) + C

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1+ ta − 2bt

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which implies

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b2 ln2 t0 a b2 c t0 − t0 − t0 − t0



Q20 2

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In this work we have obtained the solution of nonlinear GLR–MQ  evolution equation to deter mine the t (or Q2 ) evolution of gluon distribution function G x, Q2 up to NNLO. Also we have

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Fig. 2. Q2 evolution of G(x, Q2 ) for R = 2 GeV−1 . Solid red lines are our NNLO result while solid blue lines and solid brown lines are our NLO and LO results respectively. Dotted lines are from the PDF4LHC15 set, dot dashed lines are from HERAPDF15 set, dashed lines are from NNPDF30 and absolute dashed lines are from CT14. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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compared our results with those recent global DGLAP fits obtained by various collaborations viz. NNPDF3.0 [18], HERAPDF1.5 [19], CT14 [20], ABM12 [21] and PDF4LHC [22]. The NNPDF3.0 set uses a global dataset which includes various HERA data as well as relevant LHC data. The QCD fit analysis of the combined HERA-I inclusive deep inelastic cross-sections have been extended to include combined HERA-II measurement at high Q2 resulting into HERAPDF1.5 sets. The CT14 includes the data from LHC experiments as well as the new D∅ charged lepton asymmetry data. The ABM12 set results from the global analysis of DIS and hadron collider data including the available LHC data for standard candle processes such as W ± and Z-boson and t¯t production. The PDF4LHC15 set is the updated recommendation of PDF4LHC group for the usage of sets of PDFs and the assessment of PDF and PDF+ αs uncertainties which is suitable for applications at the LHC Run II.

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Fig. 3. Q2 evolution of G(x, Q2 ) for R = 5 GeV−1 . Solid red lines are our NNLO result while solid blue lines and solid brown lines are our NLO and LO results respectively. Dotted lines are from the PDF4LHC15 set, dot dashed lines are from HERAPDF15 set, dashed lines are from NNPDF30 and absolute dashed lines are from CT14. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In this work we have considered the kinematic region to be 6.5 GeV2 ≤ Q2 ≤ 25 GeV2 where all our assumptions look natural and our solution seems to be valid. Fig. 1(a–b) shows the plot of T 2 (t) and T0 T (t) as well as T 3 (t) and T1 T (t) with respect to Q2 . In the range 6.5 GeV2 ≤ Q2 ≤ 25 GeV2 , it is observed that for T0 = 0.0338 and T1 = 0.00115 the difference between T 2 (t) and T0 T as well as T 3 (t) and T1 T becomes negligible. Fig. 2(a–d) and Fig. 3(a–d) represent our best fit results of t evolution of gluon distribution function G(x, Q2 ) for −2 −3 −4 R = 2 GeV−1 and for R = 5 GeV−1 respectively for different  values of x, viz. 10 , 10 2, 10 2 −5 and 10 . We have taken the input distribution G x, Q0 of a given value of initial Q from PDF4LHC dataset and evolve GLR–MQ equation. The input G(x, Q20 ) is taken at an input value of Q20 ≈ 6.76 GeV2 . We have chosen the input from PDF4LHC15 set since this set is based on the LHC experimental simulations, the 2015 recommendations [38] of the PDF4LHC working group

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Fig. 5. Sensitivity of λG in our results of Q2 evolution of G(x, Q2 ).

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and contain combinations of more recent CT14, MMHT2014, and NNPDF3.0 PDF ensembles. By the phenomenological analysis we have taken the average value of  to be 0.2 GeV [30–32]. In Fig. 4(a–b), we have investigated the effect of nonlinearity in our results for two different values of R viz. R = 2 GeV−1 and R = 5 GeV−1 at different values of x viz. 10−2 , 10−3 , 10−4 and 10−5 respectively. The value of R depends on how the gluons are distributed within the proton. If the gluons are distributed over the entire nucleon then R will be of the order of the proton radius (R 5 GeV−1 ). On the other hand, if gluons are concentrated in hot-spots then R will be very small (R 2 GeV−1 ) [39]. We have also performed an analysis (Fig. 5(a–b)) to check sensitivity   of the Regge intercept λG in our result by comparing our result of gluon distribution G x, Q2 for three different values of λG viz. 0.4, 0.5 and 0.6 for x = 10−2 , 10−3 , 10−4 and 10−5 for both the cases R 2 GeV−1 and R 5 GeV−1 .

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In this work, we have solved the GLR–MQ evolution equation up to next-to-next-leading order (NNLO) by considering Regge like behavior of the gluon distribution function. Here we have incorporated the NNLO  terms  into the gluon–gluon splitting function Pgg (ω) and the running coupling constant αs Q2 . We have examined the validness of Regge behavior of the gluon distribution function in our phenomenologically determined moderate Q2 kinematic region 6.5 GeV2 ≤ Q2 ≤ 25 GeV2 and 10−5 < x < 10−2 , where nonlinear effects cannot be neglected and it is found that our results show almost similar behavior to those obtained from various global parameterization groups and global fits. We can conclude that our solution is valid only in the vicinity of the saturation border. Our solution of gluon distribution function increases with inwith crease in Q2 which agree  the perturbative QCD fits at small-x. It is also observed that the  gluon distribution G x, Q2 of our NNLO solution lies slightly above the NLO and LO results as Q2 increases and x decreases. This is because of the inclusion of the NNLO terms in the splitting function Pgg (ω) which in fact gives a better compatibility with different global fits as compared to NLO and LO results. Again when we go on decreasing x, it is observed that G(x, Q2 ) gets more tamed suggesting gluon recombination which appear to be more apparent towards small x. Thus the nonlinear effects are found to play a significant role towards very small x (≤ 10−3 ) for both cases of R = 2 GeV−1 and at R = 5 GeV−1 as seen in Fig. 2(c–d) and Fig. 3(c–d). On the other hand nonlinearities vanish rapidly at larger values of x. However it is found that R = 5 GeV−1 gives better solution towards experimental data than that of R = 2 GeV−1 at very small x (≤ 10−3 ). Again as seen in Fig. 4(a–b) the nonlinearity increases with decrease in the values of R however the differences in the results at R = 2 GeV−1 and at R = 5 GeV−1 increases with decrease in x. It is found that the gluon distribution function G(x, Q2 ) shows steep behavior at R = 5 GeV−1 on the other hand taming of G(x, Q2 ) is more significant at R = 2 GeV−1 . We have also investigated the effect of nonlinearities in our results for different values of λG in Fig. 5(a–b) and found that our solutions are highly sensitive to λG towards decrease in x. Finally from this work, we can conclude that for very small-x (≤ 10−3 ), our solution of NNLO plays more significant role than that of NLO and LO.

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One of the authors (M. Lalung) is grateful to CSIR, New Delhi for financial assistantship in the form of Junior Research Fellowship.

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