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Web End = Eur. Phys. J. C (2015) 75:292DOI 10.1140/epjc/s10052-015-3522-6

Regular Article - Theoretical Physics

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Web End = MuellerNavelet jets at LHC: BFKL versus high-energy DGLAP

F. G. Celiberto1,a, D. Yu. Ivanov2,3,b, B. Murdaca1,c, A. Papa1,d

1 Dipartimento di Fisica, Universit della Calabria, and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Cosenza, Arcavacata di Rende, 87036 Cosenza, Italy

2 Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia

3 Novosibirsk State University, 630090 Novosibirsk, Russia

Received: 5 May 2015 / Accepted: 10 June 2015 / Published online: 26 June 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract The production of forward jets separated by a large rapidity gap at LHC, the so-called MuellerNavelet jets, is a fundamental testeld for perturbative QCD in the high-energy limit. Several analyses have already provided us with evidence about the compatibility of theoretical predictions, based on collinear factorization and BFKL resummation of energy logarithms in the next-to-leading approximation, with the CMS experimental data at 7 TeV of center-of-mass energy. However, the question if the same data can be described also by xed-order perturbative approaches has not yet been fully answered. In this paper we provide numerical evidence that the mere use of partially asymmetric cuts in the transverse momenta of the detected jets allows for a clear separation between BFKL-resummed and xed-order predictions in some observables related with the MuellerNavelet jet production process.

1 Introduction

It is widely believed now that the inclusive hadroproduction of two jets featuring transverse momenta of the same order and much larger than the typical hadronic masses and being separated by a large rapidity gap Y , the so-called MuellerNavelet jets [1], is a fundamental testeld for perturbative QCD in the high-energy limit, the jet transverse momenta providing us with the hard scales of the process.

At the LHC energies, the theoretical description of this process lies at the crossing point of two distinct approaches: collinear factorization and BFKL [25] resummation. On one side, at leading twist the process can be seen as the hard scattering of two partons, each emitted by one of the colliding

a e-mail: mailto:[email protected]

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b e-mail: mailto:[email protected]

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c e-mail: mailto:[email protected]

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d e-mail: mailto:[email protected]; [email protected]

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hadrons according to the appropriate parton distribution function (PDF); see Fig. 1. Collinear factorization takes care of systematically resumming the logarithms of the hard scale, through the standard DGLAP evolution [68] of the PDFs and the xed-order radiative corrections to the parton scattering cross section.

The other resummation mechanism at work, justied by the large center-of-mass energy s available at LHC, is the BFKL resummation of energy logarithms, which are so large as to compensate the small QCD coupling and must therefore be accounted for to all orders of perturbation. These energy logarithms are related with the emission of undetected par-tons between the two jets (the larges s, the larger the number of partons), which lead to a reduced azimuthal correlation between the two detected forward jets, in comparison to the xed-order DGLAP calculation, where jets are emitted almost back-to-back.

In the BFKL approach energy logarithms are systematically resummed in the leading logarithmic approximation (LLA), which means all terms (s ln(s))n, and in the next-to-leading logarithmic approximation (NLA), which means resummation of all terms s(s ln(s))n. The process-independent part of such resummation is encoded in the BFKL Greens function, obeying an iterative integral equation, whose kernel is known at the next-to-leading order (NLO) both for forward scattering (i.e. for t = 0 and color

singlet in the t-channel) [9,10] and for any xed (not growing with energy) momentum transfer t and any possible twogluon color state in the t-channel [1117].

To get the cross section for MuellerNavelet jet production and other related observables, the BFKL Greens function must be convoluted with two impact factors for the transition from the colliding parton to the forward jet (the so-called jet vertices). They were rst calculated with NLO accuracy in [18,19] and the result was later conrmed in [20]. A simpler expression, more practical for numerical purposes, was obtained in [21] adopting the so-called small-cone

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methods are those inspired by the principle of minimum sensitivity [41,42], the fast apparent convergence [4345] and the BrodskyLepageMackenzie method (BLM) [46].

This variety of options reects in the large number of numerical studies of the MuellerNavelet jet production process at LHC, both at a center-of-mass energy of14 TeV [4749] and 7 TeV [5053]. All these studies were concerned with the behavior on Y of azimuthal angle correlations between the two measured jets, i.e. average values of cos (n), where n is an integer and is the angle in the azimuthal plane between the direction of one jet and the opposite direction of the other jet, and ratios of two such cosines [54,55]. In particular, one of these analyses [51], based on the use of a collinear improvement and energy scales optimized la BLM, found a nice agreement with CMS data [56].

In a recent paper [53], some of us stressed that another important source of systematics on should be aware of is the representation uncertainty, deriving from the freedom to use different representation of the BFKL cross section, equivalent within the NLO.1 The good agreement between CMS data and BLM-optimized theoretical predictions gives us perhaps a hint toward the right direction. In the same paper, a list of issues was presented which, if considered in the experimental analysis, could help the matching between the way MuellerNavelet are dened in the theory and in the experiment. One of these issue was, for instance, the very measurement of the MuellerNavelet total cross section, C0, which, on the theory side, is strongly sensitive both to the representation of the BFKL amplitude and to the optimization procedure.

In this paper we want to further discuss and expand on another of the issues listed in the Discussion section of [53], related with the choice of the experimental cuts in the values of the forward jet transverse momenta. Since the Born contribution to the cross section C0 is present only for back-to-back jets, its effect is maximized when symmetric cuts are used; on the contrary, in the case of asymmetric cuts, the Born term is suppressed, and the effects of the additional undetected hard gluon radiation is enhanced, thus making more visible the BFKL resummation, in comparison to descriptions based on the xed-order DGLAP approach, in all observables involving C0.

For this purpose, we compare predictions for several azimuthal correlations and their ratios obtained, on one side, by a xed-order DGLAP calculation at the NLO and, on the other side, by BFKL resummation in the NLA.

To avoid misunderstanding we note that in what follows our implementation of the NLO DGLAP calculation will be

1 We remark that the impact of this uncertainty on azimuthal correlations and their ratios is much larger than the one resulting from the adoption of different jet algorithms at the NLO.

x1 k1

Fig. 1 MuellerNavelet jet production process

approximation (SCA) [2224], i.e. for small jet cone aperture in the rapidity-azimuthal angle plane. A critical comparison between the latter result and the exact jet vertex in the cases of kt and cone algorithms and their small-cone versions has been recently carried out in [25]. We stress that, within NLO accuracy, the jet can be formed by either one or two particles and no more, so that the argument given in [25] about the non-infrared-safety of the all-order extension of the jet algorithm used to obtain the SCA jet vertex in [21] does not apply here.

The BFKL approach brings along some extra-sources of systematic uncertainties with respect to the xed-order, DGLAP calculation. First of all, in addition to the renormalization and factorization scales, R and F, which appear also in DGLAP, there is a third, articial normalization scale, usually called s0, which must be suitably xed. Moreover, there is compelling evidence that choosing for these scales the values dictated by the kinematics of the process is not necessarily the best choice when the BFKL resummation is at work. It is well known, indeed, that the NLO BFKL corrections for the n = 0 conformal spin have opposite sign

with respect to the leading order (LO) result and are large in absolute value. This happens both to the NLO BFKL kernel and to the process-dependent NLO impact factors (see, e.g., Refs. [2628], for the case of the vector meson photoproduction). This calls for some optimization procedure, which can consist in (i) including some pieces of the (unknown) next-to-NLO corrections, such as those dictated by renormalization group, as in collinear improvement [2939], or by energy momentum conservation [40], and/or (ii) suitably choosing the values of the energy and renormalization scales, which, though arbitrary within the NLO, can have a sizeable numerical impact through subleading terms. Common optimization

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Eur. Phys. J. C (2015) 75 :292 Page 3 of 9 292

an approximate one. We just use here NLA BFKL expressions for the observables that are truncated to the O 3s

order. In this way we take into account the leading power asymptotic of the exact NLO DGLAP prediction and neglect terms that are suppressed by the inverse powers of the energy of the partonparton collisions. Such approach is legitimate in the region of large Y which we consider here. The exact implementation of NLO DGLAP for MuellerNavelet jets is important, because it allows one to understand better the region of applicability of our approach, but it requires more involved Monte Carlo calculations (some rst results were reported recently in [57]).

To single out the only effect of transverse momentum cuts, we consider just one representation of the MuellerNavelet cross section (the exponentiated one) and just one optimization procedure (the BLM one, in the two variants discussed in [58]).

The paper is organized as follows: in the next section we give the kinematics and the basic formulas for the MuellerNavelet jet process cross section; in Sect. 3 we present our results; nally, in Sect. 4 we draw our conclusions.

2 Theoretical setup

In this section we briey recall the kinematics of the process and the main formulas, referring the reader to previous papers [48,53] for the omitted details.

The process under exam is the production of Mueller

Navelet jets [1] in protonproton collisions

p(p1) + p(p2) jet(kJ1) + jet(kJ2) + X, (1) where the two jets are characterized by high transverse momenta,

k2J1

k2J2 2QCD and large separation in

rapidity; p1 and p2 are taken as Sudakov vectors satisfying p21 = p22 = 0 and 2 (p1 p2) = s.

In QCD collinear factorization the cross section of the process (1) reads

d dxJ1dxJ2d2kJ1d2kJ2 =

i, j=q, q,g

squared center-of-mass energy of the partonparton collision subprocess (see Fig. 1).

The cross section of the process can be represented as

dyJ1dyJ2 d|

kJ1| d|

kJ2|dJ1dJ2

1 (2)2

, (3)

where = J1 J2 , while C0 gives the total cross sec

tion and the other coefcients Cn determine the distribution

of the azimuthal angle of the two jets. In the BFKL approach several NLA-equivalent expressions can be adopted for Cn.

A large list of them and of their features can be found in [53]. For the purposes of the present analysis, we concentrate on one representation, the so-called exponentiated representation, and use it in combination with the BLM optimization procedure. We recall that BLM setting means choosing the scale R such that it makes vanish completely the 0-dependence of a given observable. As discussed in [53], we implement two variants of the BLM method, dubbed (a) and (b), derived in [58]. Moreover, we use a common optimal scale for the renormalization scale R and for the factorization scale F. In [53] it was shown that this setup allows a nice agreement with CMS data for several azimuthal correlations and their ratios in the large Y regime.

Introducing, for the sake of brevity, the denitions

Y = ln

2 cos(n) Cn

xJ1xJ2s

kJ1||

kJ2|

, Y0 = ln

kJ1||

kJ2|

we have then the following expressions for the coefcients

Cn, in the two variants of BLM setting:

CBFKL(a)n =

xJ1 xJ2

kJ1||

kJ2|

d e(YY0)[ s(R)(n,)+

s (R)(

2 (n,) T

CA (n,)

8CA 2(n,))]

2s(R)c1(n, , | kJ1|, xJ1)c2(n, , | kJ2|, xJ2)

2 s (R) T

+s (R) c(1)1(n, , |

kJ1|, xJ1) c1(n, , |

kJ1|, xJ1) +

0 dx1

0 dx2 fi (x1, F)

f j (x2, F)

i,j (x1x2s, F)

dxJ1dxJ2d2kJ1d2kJ2 ,

c(1)2(n, , | kJ2|, xJ2)

c2(n, , |

kJ2|, xJ2)

, (4)

(2)

where the i, j indices specify the parton types (quarks q = u, d, s, c, b; antiquarks q = , d, s, c, b; or gluon g),

fi (x, F) denotes the initial proton PDFs; x1,2 are the longitudinal fractions of the partons involved in the hard subprocess, while xJ1,2 are the jet longitudinal fractions; F is the factorization scale; d

with R xed at the value

(BLMR)2 = kJ1kJ2 exp

2 1 +

3 I

5 3

(5)

i,j (x1x2s, F) is the partonic cross section for the production of jets and x1x2s is the

and

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292 Page 4 of 9 Eur. Phys. J. C (2015) 75 :292

where I = 2

10 dx

ln(x) x2x+1

2.3439 and c(1)1,2(n, , |

kJ2|,

CBFKL(b)n

xJ1 xJ2

= kJ1||

kJ2|

xJ2) are the same as c(1)1,2(n, , |

kJ1,2|, xJ1,2) with the terms

proportional to 0 removed. The scale s0 entering Y0 is articial. It is introduced in the BFKL approach at the time to perform the Mellin transform from the s-space to the complex angular momentum plane and cancels in the full expression, up to terms beyond the NLA. In the following it will always be xed at the natural value Y0 = 0.

In the xed-order DGLAP approach at the NLO, the expressions for the coefcients Cn are nothing but the trunca

tion of the BFKL expressions up to inclusions of NLO terms and read

CDGLAP(a)n =

d e(YY0)[ s(R)(n,)+

s (R)(

2 (n,) T

CA (n,))]

2s (R) c1(n, , | kJ1|, xJ1)c2(n, , | kJ2|, xJ2)

1 + s (R)

4 (n, ) 2

+s (R) c(1)1(n, , |

kJ1|, xJ1) c1(n, , |

kJ1|, xJ1) +

c(1)2(n, , | kJ2|, xJ2)

c2(n, , |

kJ2|, xJ2)

, (6)

with R xed at the value

(BLMR)2 = kJ1kJ2 exp

2 1 +

xJ1 xJ2

kJ1||

kJ2|

3 I

53 +

2 (, n)

(7)

d 2s (R)

c1(n, , | kJ1|, xJ1)c2(n, , | kJ2|, xJ2)

In Eqs. (4) and (6),

s(R) s(R)Nc/, with Nc the

number of colors, and

0 =

3 Nc

2 s (R) T + s (R) (Y Y0) (n, )

+ s (R)

c(1)1(n, , |

kJ1|, xJ1) c1(n, , |

kJ1|, xJ1) +

c(1)2(n, , |

kJ2|, xJ2) c2(n, , |

kJ2|, xJ2)

(12)

3n f (8)

is the rst coefcient of the QCD -function,

(n, ) = 2 (1)

d 2s (R)

c1(n, , | kJ1|, xJ1)c2(n, , | kJ2|, xJ2)

1 + s (R)

CDGLAP(b)n =

xJ1 xJ2

kJ1||

kJ2|

4 (n, ) 2

2 +

2 + i

(9)

n2 +

2 i

+ s (R) (Y Y0) (n, ) + s (R)

c(1)1(n, , |

kJ1|, xJ1) c1(n, , |

kJ1|, xJ1) +

c(1)2(n, , |

kJ2|, xJ2) c2(n, , |

kJ2|, xJ2)

is the LO BFKL characteristic function,

c1(n, , |

k|, x) = 2

(13)

which we will use in the following with the two possible choices (a) and (b) of the optimal scales, given in Eqs. (5) and (7), respectively. It is worth mentioning that our DGLAP expressions, Eqs. (12) and (13), do not actually depend on Y0. The corresponding terms in the r.h.s. of (12) and (13) are canceled by similar terms in c(1)1,2; see [48].

We note that, in our way to implement the BLM procedure (see [58]), the nal expressions are given in terms of s in the MS scheme, although in one intermediate step the MOM scheme was used.

3 Numerical analysis

3.1 Results

In this section we present our results for the dependence on the rapidity separation between the detected jets, Y =

yJ1 yJ2, of ratios Rnm Cn/Cm between the coefcients Cn. Among them, the ratios of the form Rn0 have a simple physical interpretation, being the azimuthal correlations

cos(n) .

CA (

k 2)i1/2

CF fg(x, F) +

a=q, q

fq(x, F)

(10)

and

c2(n, , |

k|, x) =

c1(n, , |

k|, x)

, (11)

are the LO jet vertices in the -representation. The remaining objects are related with the NLO corrections of the BFKL kernel (

(n, )) and of the jet vertices in the SCA (c(1)1,2(n, , |

kJ1,2|, xJ1,2)) in the -representation. Their

expressions are given in Eqs. (23), (36), and (37) of Ref. [48]. Moreover,

T =

0 2

1 +

3 I

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Eur. Phys. J. C (2015) 75 :292 Page 5 of 9 292

Table 1 Ratios Cn/Cm for kJ1,min = 35 GeV and

kJ2,min = 45 GeV

Y BFKL(a) DGLAP(a) BFKL(b) DGLAP(b)

C1/C0 3.0 0.963 (21) 1.003 (44) 0.964 (17) 1.021 (78)6.0 0.7426 (43) 0.884 (61) 0.7433 (30) 0.914 (91)

9.0 0.897 (15) 0.868 (16) 0.714 (10) 0.955 (50)

C2/C0 3.0 0.80 (2) 0.948 (43) 0.812 (15) 0.949 (75)6.0 0.4588 (32) 0.726 (56) 0.4777 (26) 0.702 (81)

9.0 0.4197 (79) 0.710 (15) 0.3627 (50) 0.850 (48)

C3/C0 3.0 0.672 (18) 0.876 (41) 0.684 (13) 0.838 (70)6.0 0.3095 (26) 0.566 (45) 0.3282 (21) 0.435 (68)

9.0 0.2275 (72) 0.558 (13) 0.2057 (29) 0.717 (44)

C2/C1 3.0 0.831 (18) 0.945 (43) 0.842 (16) 0.929 (72)6.0 0.6178 (43) 0.821 (66) 0.6427 (34) 0.768 (91)

9.0 0.4677 (63) 0.817 (18) 0.5079 (56) 0.890 (51)

C3/C2 3.0 0.839 (22) 0.924 (45) 0.843 (17) 0.883 (76)6.0 0.6745 (64) 0.780 (71) 0.6869 (52) 0.62 (11)

9.0 0.542 (15) 0.787 (21) 0.5670 (59) 0.844 (56)

Table 2 Ratios Cn/Cm for kJ1,min = 35 GeV and

kJ2,min = 50 GeV

Y BFKL(a) DGLAP(a) BFKL(b) DGLAP(b)

C1/C0 3.0 0.961 (23) 1.006 (46) 0.964 (15) 1.034 (89)6.0 0.7360 (49) 0.869 (58) 0.7357 (25) 0.89 (12)

9.0 1.0109 (61) 0.857 (16) 0.7406 (46) 0.958 (56)

C2/C0 3.0 0.788 (21) 0.946 (44) 0.801 (14) 0.950 (85)6.0 0.4436 (37) 0.698 (53) 0.4626 (19) 0.611 (98)

9.0 0.4568 (50) 0.695 (15) 0.3629 (23) 0.862 (54)

C3/C0 3.0 0.653 (19) 0.868 (43) 0.669 (12) 0.814 (79)6.0 0.2925 (31) 0.530 (42) 0.3115 (15) 0.320 (57)

9.0 0.2351 (35) 0.551 (17) 0.1969 (17) 0.748 (50)

C2/C1 3.0 0.820 (21) 0.940 (44) 0.832 (15) 0.918 (81)6.0 0.6027 (51) 0.803 (64) 0.6288 (26) 0.69 (12)

9.0 0.4518 (35) 0.811 (18) 0.4900 (24) 0.899 (57)

C3/C2 3.0 0.829 (26) 0.917 (46) 0.835 (17) 0.857 (85)6.0 0.6595 (82) 0.759 (70) 0.6733 (36) 0.52 (11)

9.0 0.5146 (85) 0.793 (23) 0.5426 (38) 0.869 (62)

y2,max = 4.72 and consider Y = 3, 6, and 9. As for the jet

transverse momenta, differently from all previous analyses, we make two asymmetric choices: (1) kJ1,min = 35 GeV,

kJ2,min = 45 GeV and (2) kJ1,min = 35 GeV, kJ2,min = 50

GeV. The jet cone size R entering the NLO-jet vertices is xed at the value R = 0.5, the center-of-mass energy at

s = 7 TeV and, as anticipated, Y0 = 0. We use the PDF

set MSTW2008nlo [59] and the two-loop running coupling with s (MZ) = 0.11707.

2 In [53] it was mistakenly written yi,min = 0, although all numerical

results presented there were obtained using the correct value for yi,min.

In order to match the kinematic cuts used by the CMS collaboration, we will consider the integrated coefcients given by

Cn =

y1,max

y1,min

dy1

y2,max

y2,min

dy2 kJ1,min

dkJ1

dkJ2 (y1 y2 Y ) Cn yJ1, yJ2, kJ1, kJ2

(14)

and their ratios Rnm Cn/Cm. We will take jet rapidities in

the range delimited by y1,min = y2,min = 4.7 and y1,max =

kJ2,min

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C1/C0

C2/C0

1.2

0.8

0.6

0.4

BFKLa BFKLb

DGLAPa DGLAPb

0.4

kJ > 45 GeV

BFKLa BFKLb

DGLAPa DGLAPb

kJ > 45 GeV

0.2

0 2 4 8 10

C3/C0

C2/C1

kJ > 45 GeV

BFKLa BFKLb

DGLAPa DGLAPb

kJ > 45 GeV

BFKLa BFKLb

DGLAPa DGLAPb

0.8

0.6

0.4

0.2

0 2 4 8 10

C3/C2

0.8

0.6

0.4

BFKLa BFKLb

DGLAPa DGLAPb

kJ > 45 GeV

0.2

0 2 4

8 10

Fig. 2 Y -dependence of several ratios Cm/Cn for kJ1,min = 35 GeV and kJ2,min = 45 GeV, for BFKL and DGLAP in the two variants of the BLM

method (data points have been slightly shifted along the horizontal axis for the sake of readability)

We summarize our results in Tables 1, 2 and Figs. 2, 3. We can clearly see that, at Y = 9, BFKL and DGLAP, in both

variants (a) and (b) of the BLM setting, give quite different predictions for all considered ratios except C1/C0; at Y =

6 this happens in fewer cases, while at Y = 3 BFKL and

DGLAP cannot be distinguished with given uncertainties. This scenario is similar in the two choices of the transverse momentum cuts.

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Eur. Phys. J. C (2015) 75 :292 Page 7 of 9 292

C1/C0

C2/C0

1.2

kJ > 50 GeV

0.8

0.6

0.4

BFKLa BFKLb

DGLAPa DGLAPb

BFKLa BFKLb

DGLAPa DGLAPb

kJ > 50 GeV

0.2

0 2 4 8 10

C3/C0

C2/C1

kJ > 50 GeV

0.8

0.6

0.4

BFKLa BFKLb

DGLAPa DGLAPb

BFKLa BFKLb

DGLAPa DGLAPb

0.4

kJ > 50 GeV

0.2

0 2 4 8 10

C3/C2

0.8

0.6

0.4

BFKLa BFKLb

DGLAPa DGLAPb

kJ > 50 GeV

0.2

0 2 4

8 10

Fig. 3 Y -dependence of several ratios Cm/Cn for kJ1,min = 35 GeV and kJ2,min = 50 GeV, for BFKL and DGLAP in the two variants of the BLM

method (data points have been slightly shifted along the horizontal axis for the sake of readability)

3.2 Used tools

All numerical calculations were implemented in Fortran, using the corresponding interfaces for the MSTW 2008

PDFs [59]. Numerical integrations and the computation of the polygamma functions were performed using specic CERN program libraries [60]. Furthermore, we used slightly modied versions of the Chyp [61] and Psi [62] routines

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in order to perform the calculation of the Gauss hypergeo-metric function 2 F1 and of the real part of the function, respectively.

3.3 Uncertainty estimation

The are three main sources of uncertainty in our calculation:

The rst source of uncertainty is the numerical four-

dimensional integration over the variables |

kJ1|, |

kJ2|, yJ1

and and was directly estimated by Dadmul integration routine [60].

The second one is the one-dimensional integration over

the longitudinal momentum fraction entering the expression for the NLO impact factors c(1)1,2(n, , |

kJ1,2|,

xJ1,2) given in Eqs. (36) and (37) of Ref. [48] and used in this work. This integration was performed by using the

WGauss routine [60]. At rst, we xed the best value of the input accuracy parameter EPS by making comparisons between separate Fortran and Mathematica calculations of the impact factor. Then we veried that, under variations by factors of 10 or 1/10 of the EPS parameter, the CBFKLn and CDGLAPn coefcients change by less than 1 permille.

The third one is related with the upper cutoff in the

integrations over |

kJ1|, |

kJ2| and . We xed kJ1max =

kJ2max = 60 GeV as in [50], where it was shown that the

contribution to the integration from the omitted region is negligible. Concerning the -integration, we xed the upper cutoff max = 30 for the calculation of the CBFKLn

coefcients, after verifying that a larger value does not change the result in appreciable way.

The CDGLAPn coefcients show a more pronounced sensitivity to max, due to the fact that the oscillations in the integrand in Eqs. (12) and (13) are not dumped by the exponential factor as in the BFKL expressions (4) and (6). For the same reason, the computational time of CDGLAPn is much larger than for CBFKLn. We found that the best compromise was to set max = 50. We checked in some

sample cases, mostly at Y = 6 and 9, that, putting max

at 60, the ratios Cm/Cn change always less than 1%, in spite of the fact that the single coefcients Cn change in a more pronounced way.

Of the three main sources of uncertainty, the rst one is, by far, the most signicant, therefore the error bars of all data presented in this work are just those given by the Dadmul integration. We checked, however, using some trial functions which mimic the behavior of the true integrands involved in this work, that the error given by the Dadmul integration is a large overestimate of the true one. We are therefore condent that our error estimation is quite conservative.

4 Conclusions

In this paper we have considered the MuellerNavelet jet production process at LHC at the center-of-mass energy of7 TeV and have compared predictions for several azimuthal correlations and ratios between them, both in full NLA BFKL approach and in xed-order NLO DGLAP.

Differently from current experimental analyses of the same process, we have used asymmetric cuts for the trans-verse momenta of the detected jets. In particular, taking one of the cuts at 35 GeV (as done by the CMS collaboration [56]) and the other at 45 or 50 GeV, we can clearly see that predictions from BFKL and DGLAP become separate for most azimuthal correlations and ratios between them, this effect being more and more visible as the rapidity gap between the jets, Y , increases. In other words, in this kinematics the additional undetected parton radiation between the jets which is present in the resummed BFKL series, in comparison to just one undetected parton allowed by the NLO DGLAP approach, makes its difference and leads to more azimuthal angle decorrelation between the jets, in full agreement with the original proposal of Mueller and Navelet.

This result was not unexpected: the use of symmetric cuts for jet transverse momenta maximizes the contribution of the Born term, which is present for back-to-back jets only and is expected to be large, therefore making less visible the effect of the BFKL resummation. This phenomenon could be at the origin of the instabilities observed in the NLO xed-order calculations of [63, 64].

Another important benet from the use of asymmetric cuts, pointed out in [52], is that the effect of violation of the energymomentum conservation in the NLA is strongly suppressed with respect to what happens in the LLA.

In view of all these considerations, we strongly suggest experimental collaborations to consider also asymmetric cuts in jet transverse momenta in all future analyses of Mueller Navelet jet production process.

Acknowledgments We thank G. Safronov for fruitful discussions. D.I. thanks the Dipartimento di Fisica dellUniversit della Calabria and the Istituto Nazionale di Fisica Nucleare (INFN), Gruppo collegato di Cosenza, for warm hospitality and nancial support. The work of D.I. was also supported in part by the Grant RFBR-15-02-05868-a. B.M. thanks the Sobolev Institute of Mathematics of Novosibirsk for warm hospitality during the preparation of this paper. The work of B.M. was supported in part by the Grant RFBR-13-02-90907 and by the European Commission, European Social Fund, and Calabria Region, that disclaim any liability for the use that can be made of the information provided in this paper.

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The production of forward jets separated by a large rapidity gap at LHC, the so-called Mueller-Navelet jets, is a fundamental testfield for perturbative QCD in the high-energy limit. Several analyses have already provided us with evidence about the compatibility of theoretical predictions, based on collinear factorization and BFKL resummation of energy logarithms in the next-to-leading approximation, with the CMS experimental data at 7 TeV of center-of-mass energy. However, the question if the same data can be described also by fixed-order perturbative approaches has not yet been fully answered. In this paper we provide numerical evidence that the mere use of partially asymmetric cuts in the transverse momenta of the detected jets allows for a clear separation between BFKL-resummed and fixed-order predictions in some observables related with the Mueller-Navelet jet production process.

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Title

Mueller-Navelet jets at LHC: BFKL versus high-energy DGLAP

Author

Celiberto, F G; Ivanov, D Yu; Murdaca, B; Papa, A

Pages

1-9

Publication year

2015

Publication date

Jun 2015

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3522-6

ProQuest document ID

1691229312

The Author(s) 2015

Mueller-Navelet jets at LHC: BFKL versus high-energy DGLAP

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