Hadron-jet correlations in high-energy hadronic collisions at the LHC
Andrèe Dafne Bolognino, Francesco Giovanni Celiberto, Dmitry Yu. Ivanov, Mohammed M.A. Mohammed, Alessandro Papa
HHadron-jet correlationsin high-energy hadronic collisions at the LHC
Andr`ee D. Bolognino,
1, 2
Francesco G. Celiberto,
3, 4
Dmitry Yu.Ivanov,
5, 6
Mohammed M.A. Mohammed, and Alessandro Papa
1, 2 Dipartimento di Fisica dell’Universit`a della CalabriaI-87036 Arcavacata di Rende, Cosenza, Italy INFN - Gruppo collegato di Cosenza,I-87036 Arcavacata di Rende, Cosenza, Italy Instituto de F´ısica Te´orica UAM/CSIC,Nicol´as Cabrera 15, 28049 Madrid, Spain Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia (Dated: October 1, 2018)
Abstract
The inclusive production at the LHC of a charged light hadron and of a jet, featuring a wideseparation in rapidity, is suggested as a new probe process for the investigation of the BFKLmechanism of resummation of energy logarithms in the QCD perturbative series. We present somepredictions, tailored on the CMS and CASTOR acceptances, for the cross section averaged overthe azimuthal angle between the identified jet and hadron and for azimuthal correlations.
PACS numbers: 12.38.Bx, 12.38.-t, 12.38.Cy, 11.10.Gh a r X i v : . [ h e p - ph ] S e p . INTRODUCTION The LHC record energy, as well as the good resolution in azimuthal angles of the particledetectors, offer a unique opportunity to test a wide class of predictions of perturbative QCD.These include the so called Mueller-Navelet jet production [1], i.e. the inclusive production oftwo jets featuring a large rapidity separation between them, for which a wealth of theoreticalanalyses were produced in the last years [2–15], and the somewhat related process where twoidentified, charged light hadrons well separated in rapidity are inclusively produced [16, 17],instead of jets.A common feature to these two processes is that their high-energy behavior is dominatedby those final-state configurations where the produced particles are strongly ordered inrapidity, the tagged objects (jets or identified hadrons) being the two extrema in the rapiditytower, thus yielding a number of energy logarithms growing with the number of producedparticles. Such energy logarithms are so large to compensate the smallness of the coupling α s , so the perturbative series must be properly resummed.The theoretical framework for the resummation of energy logs for these two processes, aswell as for any semi-hard process in perturbative QCD, is provided by the Balitsky–Fadin–Kuraev–Lipatov (BFKL) approach [18], whereby the resummation of all terms proportionalto ( α s ln( s )) n , the so called leading logarithmic approximation or LLA, and that of all termsproportional to α s ( α s ln( s )) n , the next-to-leading approximation or NLA, can be systemat-ically carried out. The bottom line of the BFKL formalism is that azimuthal coefficients ofthe Fourier expansion of the cross section differential in the variables of the tagged objectsover the relative azimuthal angle take the very simple form of a convolution between twoimpact factors, describing the transition from each colliding proton to the respective finalstate tagged object, and a process-independent Green’s function. The BFKL Green’s func-tion obeys an 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) [19, 20] and forany fixed (not growing with energy) momentum transfer t and any possible two-gluon colorstate in the t -channel [21–23].The impact factors for the proton to forward jet transition (the so called “jet vertices”)are known up to the NLO for several jet selection algorithms [24–28]. The jet vertex, in itsturn, can be expressed, within leading-twist collinear factorization, as the convolution of the2arton distribution function (PDF) of the colliding proton, obeying the standard DGLAPevolution [29], with the hard process describing the transition from the parton emitted bythe proton to the forward jet in the final state. Two such jet vertices must be convolutedwith the BFKL Green’s function to theoretically describe the Mueller-Navelet jet production.The main aim is to calculate cross sections and azimuthal angle correlations [30, 31] betweenthe two measured jets, i.e. average values of cos ( nφ ), where n is an integer and φ is theangle in the azimuthal plane between the direction of one jet and the direction opposite tothe other jet, and ratios of two such cosines [32, 33].Also the impact factors for the proton to identified hadron transition are known upto the NLO [34] and can be expressed, within leading-twist collinear factorization, as theconvolution of the parton distribution function (PDF) of the colliding proton with the hardprocess describing the transition from the parton emitted by the proton to a final-stateparton and with the fragmentation function (FF) for that parton to the desired hadron. Twosuch hadron vertices must be convoluted with the BFKL Green’s function to theoreticallydescribe the above-mentioned inclusive hadron-hadron production and finally get predictionsfor cross sections and azimuthal angle correlations, similarly to the case of jets.Within the same formalism, other interesting processes have been proposed as a testfieldfor BFKL dynamics at the LHC, namely the inclusive production of three or four jets, wellseparated in rapidity from each other [35–39], the inclusive detection of two heavy quark-antiquark pairs, separated in rapidity, in the collision of two real (or quasi-real) photons [40],and the inclusive tag of a forward J/ Ψ-meson and a very backward jet at the LHC [41].On the experimental side the situation is as follows: the CMS Collaboration [42] haspresented the first measurements of the azimuthal correlation of the Mueller-Navelet jets at √ s = 7 TeV, but further experimental studies of the Mueller-Navelet jets are expected athigher LHC energies and larger rapidity intervals, including also the effects of using asym-metrical cuts for the jet transverse momenta. No experimental analyses have yet appeared onazimuthal correlation between two rapidity-separated identified light hadrons. The reasonfor that could be that events with identified hadrons in the final state, carrying transversemomenta of the order of, say, 5 GeV or larger, fall into the class of what experimentalistscall “minimum bias events”, which represent the main background in high-luminosity runsat a collider. They would be better studied in low-luminosity, dedicated, runs.In this paper we want to introduce a new process which could serve as a probe of BFKL3 x jet p x ( k J , θ J , y J ) π ± , K ± , p (¯ p )( k H , θ H , y H ) FIG. 1: Inclusive hadroproduction of a charged light hadron and of a jet. dynamics: the inclusive hadron-jet production in proton-proton collisions,proton( p ) + proton( p ) → hadron( k H , y H ) + X + jet( k J , y J ) , (1)when a charged light hadron: π ± , K ± , p (¯ p ) and a jet with high transverse momenta, sep-arated by a large interval of rapidity, are produced together with an undetected hadronicsystem X (see Fig. 1 for a schematic view). The process (1) has many common features withthe inclusive J/ Ψ-meson plus backward jet production, considered recently in Ref. [41].From the experimental side, the detection of the J/ Ψ-meson looks rather appealing. But,from the theory side, there are more uncertainties in this case in comparison to our pro-posal. The J/ Ψ-meson production impact factor was considered in LO; moreover, severalproduction mechanisms in the frame of NRQCD were discussed. Instead, the light hadronimpact factor is well defined in collinear factorization and it is known in NLO. Previousexperience in BFKL calculations for various processes at LHC shows that the account ofNLO corrections to the impact factors leads both to a considerable change of predictionsand to a big reduction of the theoretical uncertainties.The theoretical task to build predictions for cross section and azimuthal correlationsfor our process is embarrassingly simple: one should simply replace one of the two jetimpact factor entering the Mueller-Navelet formulas with the vertex for the proton-to-hadrontransition. From the theoretical point of view, this process is definitely an easy target, sinceall the needed building blocks are available, with NLO accuracy.4et, we believe that there are some good reasons for building numerical predictions forthis process and submitting them to the attention of both experimentalists and theorists: • the BFKL resummation implies certain factorization structure for the predicted ob-servables: the latter are calculated as a convolution of the universal BFKL Green’sfunction with the process dependent impact factors, which resembles the factorizationin Regge theory. It is important to test this picture experimentally, considering allpossible processes for which the full NLO BFKL description is available. • In Refs. [9, 12, 43] it was discussed, in the context of Mueller-Navelet jet production,that using asymmetric cuts for the transverse momenta of the tagged jets suppressesthe Born term, present only for back-to-back jets, thus enhancing the effects of the ad-ditional undetected hard gluon radiation and making therefore more visible the impactof the BFKL resummation, with respect to the fixed-order (DGLAP) contribution. Forthe process we are considering here this kind asymmetry would be naturally imposedby the completely different nature of the two tagged objects: the identified jet shouldhave transverse momentum not smaller than 20 GeV or so, whereas the minimumhadron transverse momentum can be as small as 5 GeV. • For the process under consideration only one hadron in the final state should beidentified, instead of two as in the hadron-hadron inclusive production, the otheridentified object being a jet with a typically much larger transverse momentum. Thisshould facilitate the mining of these events out of the minimum-bias ones. • From the theoretical point of view one can use this process to compare models forFFs or for jet algorithms, handling expressions which are linear in the correspondingfunctions and not quadratic as it would be, respectively, in the hadron-hadron and inthe Mueller-Navelet jet case.The summary of the paper is as follows: in Section 1 we present the theoretical frameworkand sketch the derivation of our predictions; in Section 2 we show and discuss the results ofour numerical analysis; finally, in Section 3, we draw our conclusions and give some outlook.5
I. THEORETICAL FRAMEWORK
The final state configuration of the inclusive process under consideration is schematicallyrepresented in Fig. 1, where a charged light hadron ( k H , y H ) and a jet ( k J , y J ) are detected,featuring a large rapidity separation, together with an undetected system of hadrons. Forthe sake of definiteness, we will consider the case where the hadron rapidity y H is largerthan the jet one y J , so that Y ≡ y H − y J is always positive. This implies that, for most ofthe considered values of Y , the hadron is forward and the jet is backward.The hadron and the jet are also required to possess large transverse momenta, (cid:126)k H ∼ (cid:126)k J (cid:29) Λ . The protons’ momenta p and p are taken as Sudakov vectors satisfying p = p = 0and 2( p p ) = s , so that the momenta of the final-state objects can be decomposed as k H = x H p + (cid:126)k H x H s p + k H ⊥ , k H ⊥ = − (cid:126)k H ,k J = x J p + (cid:126)k J x J s p + k J ⊥ , k J ⊥ = − (cid:126)k J . (2)In the center-of-mass system, the hadron/jet longitudinal momentum fractions x H,J areconnected to the respective rapidities through the relations y H = ln x H s(cid:126)k H , and y J = ln (cid:126)k J x J s ,so that dy H = dx H x H , dy J = − dx J x J , and Y = y H − y J = ln x H x J s | (cid:126)k H || (cid:126)k J | , here the space part of thefour-vector p (cid:107) being taken positive.In QCD collinear factorization the cross section of the process (1) reads dσdx H dx J d k H d k J = (cid:88) r,s = q, ¯ q,g (cid:90) dx (cid:90) dx f r ( x , µ F ) f s ( x , µ F ) d ˆ σ r,s (ˆ s, µ F ) dx H dx J d k H d k J , (3)where the r, s indices specify the parton types (quarks q = u, d, s, c, b ; antiquarks ¯ q =¯ u, ¯ d, ¯ s, ¯ c, ¯ b ; or gluon g ), f r,s ( x, µ F ) denote the initial proton PDFs; x , are the longitudinalfractions of the partons involved in the hard subprocess, while µ F is the factorization scale; d ˆ σ r,s (ˆ s ) is the partonic cross section and ˆ s ≡ x x s is the squared center-of-mass energy ofthe parton-parton collision subprocess.In the BFKL approach the cross section can be presented (see Ref. [4] for the details ofthe derivation) as the Fourier sum of the azimuthal coefficients C n , having so dσdy H dy J d | (cid:126)k H | d | (cid:126)k J | dφ H dφ J = 1(2 π ) (cid:34) C + ∞ (cid:88) n =1 nφ ) C n (cid:35) , (4)6here φ = φ H − φ J − π , with φ H,J the hadron/jet azimuthal angles, while y H,J and (cid:126)k
H,J aretheir rapidities and transverse momenta, respectively. The φ -averaged cross section C andthe other coefficients C n (cid:54) =0 are given by C n ≡ (cid:90) π dφ H (cid:90) π dφ J cos[ n ( φ H − φ J − π )] dσdy H dy J d | (cid:126)k H | d | (cid:126)k J | dφ H dφ J = e Y s (cid:90) + ∞−∞ dν (cid:18) x H x J ss (cid:19) ¯ α s ( µ R ) (cid:40) χ ( n,ν )+¯ α s ( µ R ) (cid:34) ¯ χ ( n,ν )+ β Nc χ ( n,ν ) (cid:34) − χ ( n,ν )+ +2 ln (cid:32) µ R √ (cid:126)k H(cid:126)k J (cid:33)(cid:35)(cid:35)(cid:41) × α s ( µ R ) c H ( n, ν, | (cid:126)k H | , x H )[ c J ( n, ν, | (cid:126)k J | , x J )] ∗ × (cid:40) α s ( µ R ) (cid:34) c (1) H ( n, ν, | (cid:126)k H | , x H ) c H ( n, ν, | (cid:126)k H | , x H ) + (cid:34) c (1) J ( n, ν, | (cid:126)k J | , x J ) c J ( n, ν, | (cid:126)k J | , x J ) (cid:35) ∗ (cid:35) (5)+ ¯ α s ( µ R ) ln (cid:18) x H x J ss (cid:19) β N c χ ( n, ν ) f ( ν ) (cid:27) . Here ¯ α s ( µ R ) ≡ α s ( µ R ) N c /π , with N c the number of colors, β = 113 N c − n f (6)is the first coefficient of the QCD β -function, where n f is the number of active flavors, χ ( n, ν ) = 2 ψ (1) − ψ (cid:18) n iν (cid:19) − ψ (cid:18) n − iν (cid:19) (7)is the leading-order (LO) BFKL characteristic function, c H ( n, ν ) is the LO forward hadronimpact factor in the ν -representation, given as an integral in the parton fraction x , containingthe PDFs of the gluon and of the different quark/antiquark flavors in the proton, and theFFs of the detected hadron, c H ( n, ν, | (cid:126)k H | , x H ) = 2 (cid:114) C F C A ( (cid:126)k H ) iν − / (cid:90) x H dxx (cid:18) xx H (cid:19) iν − × (cid:34) C A C F f g ( x ) D hg (cid:16) x H x (cid:17) + (cid:88) r = q, ¯ q f r ( x ) D hr (cid:16) x H x (cid:17)(cid:35) , (8) In Ref. [17], on the last line of Eq. (5), which is closely related to this formula for C n , it was mistakenlywritten 2 ln (cid:16) (cid:126)k (cid:126)k (cid:17) instead of ln (cid:16) (cid:126)k (cid:126)k (cid:17) , although the numerical results presented there were obtainedusing the correct formula. J ( n, ν ) is the LO forward jet vertex in the ν -representation, c J ( n, ν, | (cid:126)k J | , x J ) = 2 (cid:114) C F C A ( (cid:126)k J ) iν − / (cid:32) C A C F f g ( x J ) + (cid:88) s = q, ¯ q f s ( x J ) (cid:33) (9)and the f ( ν ) function is defined by i ddν ln (cid:18) c H [ c J ] ∗ (cid:19) = 2 (cid:20) f ( ν ) − ln (cid:18)(cid:113) (cid:126)k H (cid:126)k J (cid:19)(cid:21) . (10)The remaining objects are the hadron/jet NLO impact factor corrections in the ν -representation, c (1) H,J ( n, ν, | (cid:126)k H,J | , x H,J ), their expressions being given in Eqs. (4.58)-(4.65) ofRef. [34] and in Eq. (36) of Ref. [4], respectively.
III. RESULTS AND DISCUSSIONA. Integration over the final-state phase space
In order to match the actual LHC kinematic cuts, we integrate the coefficients over thephase space for two final-state objects and keep fixed the rapidity interval, Y , between thehadron and the jet: C n = (cid:90) y max H y min H dy H (cid:90) y max J y min J dy J (cid:90) k max H k min H dk H (cid:90) k max J k min J dk J δ ( y H − y J − Y ) C n ( y H , y J , k H , k J ) . (11)We consider two distinct ranges for the final-state objects: H - y J C [ nb ] NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) ( µ F ) = k H,J µ R = (k H k J ) k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMS scheme H - y J C [ nb ] NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) ( µ F ) = k H,J µ R = (k H k J ) k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMS scheme
FIG. 2: Y -dependence of C for µ R = µ N = (cid:113) | (cid:126)k H || (cid:126)k J | , ( µ F ) , = | (cid:126)k H,J | , for √ s = 7 TeV (left)and √ s = 13 TeV (right), and Y ≤ . CMS-jet configuration). H - y J C [ nb ] NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme
FIG. 3: Y -dependence of C and of several ratios C m /C n for µ F = µ BLM R , √ s = 7 TeV, and Y ≤ . CMS-jet configuration). • both the hadron and the jet tagged by the CMS detector in their typical kinematicconfigurations, i.e. : k min H = 5 GeV, k min J = 35 GeV, y max H = − y min H = 2 . y max J = − y min J = 4 . CMS-jet configuration; • a hadron always detected inside CMS in the range given above, together with a very9 H - y J C [ nb ] NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) ( µ F ) = µ RBLM µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme
FIG. 4: Y -dependence of C and of several ratios C m /C n for µ F = µ BLM R , √ s = 13 TeV, and Y ≤ . CMS-jet configuration). backward jet tagged by CASTOR. In this peculiar,
CASTOR-jet configuration, thejet lies in the typical range of the CASTOR experimental analyses, i.e. k min J = 5 GeV, y max J = − . y min J = − . k max H is constrained by the lower cutoff of the adopted FF parametrizations(see below) and is always fixed at 21.5 GeV. The value of k max J is instead constrained by10 H - y J C [ nb ] NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme H - y J < c o s ϕ > / < c o s ϕ > = C / C NLA AKK08NLA HKNS07LLA AKK08LLA HKNS07proton(p ) + proton(p ) → hadron(k H , y H ) + X + jet(k J , y J )s = (13 TeV) µ F = µ R = µ RBLM k J > 5 GeV; k H > 5 GeV-6.6 < y J < -5.2; |y H | < 2.4 MMHT14 NLO PDF setMOM scheme
FIG. 5: Y -dependence of C and of several ratios C m /C n for µ F = µ BLM R , √ s = 13 TeV, and Y ≤ CASTOR-jet configuration). the requirement that x J ≤ k max J (cid:39)
60 GeV for √ s = 7 TeV and | y J | < . CMS-jet) and k max J (cid:39) .
68 GeV for √ s = 13 TeV ( CASTOR-jet) .The rapidity interval, Y , is taken to be positive: 0 < Y ≤ y max H − y min J . Two center-of-massenergies, √ s = 7 and 13 TeV, are taken into account in the CMS-jet configuration, while wegive predictions for √ s = 13 TeV in the CASTOR-jet case.11 - y -1 C [ nb ] Mueller-Navelethadron-jetdihadronproton(p ) + proton(p ) → O (k , y ) + X + O (k , y )s = (7 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO + AKK08 NLOMOM schemeNLA BFKL - y -1 C [ nb ] Mueller-Navelethadron-jetdihadronproton(p ) + proton(p ) → O (k , y ) + X + O (k , y )s = (13 TeV) µ F = µ R = µ RBLM k J > 35 GeV; k H > 5 GeV|y J | < 4.7; |y H | < 2.4 MMHT14 NLO + AKK08 NLOMOM schemeNLA BFKL
FIG. 6: Comparison of the φ -averaged cross section C in different NLA BFKL processes: Mueller-Navelet jet, hadron-jet and dihadron production, for µ F = µ BLM R , √ s = 7 and 13 TeV, and Y ≤ . CMS-jet configuration).
In our calculations we use the MMHT 2014 NLO PDF set [45] with two different NLOparametrizations for hadron FFs: AKK 2008 [46] and HKNS 2007 [47] (see Section III C fora related discussion). In the results presented below, we sum over the production of forwardcharged light hadrons: π ± , K ± , p, ¯ p . B. Scale optimization
To fix the renormalization scale µ R , which can be arbitrarily chosen within the NLA,we adopt the BLM [48] approach, which has become a quite common choice for semihardprocesses. We first perform a finite renormalization from the MS to the physical MOMscheme, whose definition is related to the 3-gluon vertex being a key ingredient of the BFKLapproach and get α MS s = α MOM s (cid:18) α MOM s π T (cid:19) , (12)with T = T β + T conf , T β = − β (cid:18) I (cid:19) , (13) T conf = 38 (cid:20) I + 32 ( I − ξ + (cid:18) − I (cid:19) ξ − ξ (cid:21) , where I = − (cid:82) dx ln( x ) x − x +1 (cid:39) . ξ is the gauge parameter of the MOM scheme, fixedat zero in the following. Then, the “optimal” BLM scale µ BLM R is the value of µ R that makes12he β -dependent part in the expression for the observable of interest vanish. In Ref. [11]some of us showed that terms proportional to the QCD β -function are present not only inthe NLA BFKL kernel, but also in the expressions for the NLA impact factor. This leadsto a non-universality of the BLM scale and to its dependence on the energy of the process.Finally, the condition for the BLM scale setting was found to be C βn ∝ (cid:90) y max H y min H dy H (cid:90) y max J y min J dy J (cid:90) k max H k min H dk H (cid:90) k max J k min J dk J ∞ (cid:90) −∞ dν e Y ¯ α MOM s ( µ BLM R ) χ ( n,ν ) c H ( n, ν )[ c J ( n, ν )] ∗ (cid:34)
53 + ln ( µ BLM R ) | (cid:126)k H || (cid:126)k J | + f ( ν ) − (cid:18) I (cid:19) + ¯ α MOM s ( µ BLM R ) Y χ ( n, ν )2 (cid:32) − χ ( n, ν )2 + 53 + ln ( µ BLM R ) | (cid:126)k H || (cid:126)k J | + f ( ν ) − (cid:18) I (cid:19)(cid:33)(cid:35) = 0 . (14)The term in the r.h.s. of Eq. (14) proportional to α MOM s originates from the NLA part of thekernel, while the remaining ones come from the NLA corrections to the hadron/jet vertices.In order to find the values of the BLM scales, we introduce the ratios of the BLM tothe “natural” scale suggested by the kinematic of the process, µ N = (cid:113) | (cid:126)k H || (cid:126)k J | , so that m R = µ BLM R /µ N , and look for the values of m R which solve Eq. (14).We finally plug these scales into our expression for the integrated coefficients in the BLMscheme (for the derivation see Ref. [11]): C n = (cid:90) y max H y min H dy H (cid:90) y max J y min J dy J (cid:90) k max H k min H dk H (cid:90) k max J k min J dk J ∞ (cid:90) −∞ dν (15) e Y s e Y ¯ α MOM s ( µ BLM R ) (cid:104) χ ( n,ν )+¯ α MOM s ( µ BLM R ) (cid:16) ¯ χ ( n,ν )+ T conf3 χ ( n,ν ) (cid:17)(cid:105) (cid:0) α MOM s ( µ BLM R ) (cid:1) × c H ( n, ν )[ c J ( n, ν )] ∗ (cid:40) α MOM s ( µ BLM R ) (cid:34) ¯ c (1) H ( n, ν ) c H ( n, ν ) + (cid:34) ¯ c (1) J ( n, ν ) c J ( n, ν ) (cid:35) ∗ + 2 T conf (cid:35)(cid:41) . The coefficient C gives the φ -averaged cross section, while the ratios R n ≡ C n /C = (cid:104) cos( nφ ) (cid:105) determine the values of the mean cosines, or azimuthal correlations, of the pro-duced hadron and jet. In Eq. (15), ¯ χ ( n, ν ) is the eigenvalue of NLA BFKL kernel [49] andits expression is given, e.g. in Eq. (23) of Ref. [4], whereas ¯ c (1) H,J are the NLA parts of thehadron/jet vertices (see Ref. [11]).We set the factorization scale µ F equal to the renormalization scale µ R , as assumed bythe MMHT 2014 PDF. 13ll calculations are done in the MOM scheme. For comparison, we present results forthe φ -averaged cross section C in the MS scheme, as implemented in Eq. (11). In thelatter case, we choose natural values for µ R , i.e. µ R = µ N ≡ (cid:113) | (cid:126)k H || (cid:126)k J | , and two differentvalues of the factorization scale, ( µ F ) , = | (cid:126)k H,J | , depending on which of the two vertices isconsidered. We checked that the effect of using natural values also for µ F , i.e. µ F = µ N , isnegligible with respect to our two-value choice. C. Used tools and uncertainty estimation
All numerical calculations were done using
JETHAD , a
Fortran code we recently de-veloped, suited for the computation of cross sections and related observables for two-bodyfinal-state processes, and offering also support in the study of multi-body final-state reac-tions. In order to perform numerical integrations, we interfaced
JETHAD with specific
CERN program libraries [50] and with
Cuba library integrators [51, 52]. We made extensive use ofthe CERNLIB routines
Dadmul and
WGauss , while the
Cuba ones were mainly used for cross-checks. A two-loop running coupling setup with α s ( M Z ) = 0 . | (cid:126)k H,J | , the hadron rapidity y H , and over ν . Its effect wasdirectly estimated by Dadmul integration routine [50]. The other three sources of uncertainty,which are respectively: the one-dimensional integration over the parton fraction x neededto perform the convolution between PDFs and FFs in the LO/NLO hadron impact factors,14he one-dimensional integration over the longitudinal momentum fraction ζ in the NLOhadron/jet impact factor corrections, and the upper cutoff in the numerical integrationsover | (cid:126)k H,J | and ν , are negligible with respect to the first one. For this reason the error bandsof all predictions presented in this work are just those given by the Dadmul routine.
D. Discussion
In Fig. 2 we present our results at natural scales for the φ -averaged cross section C at √ s = 7 and 13 TeV in the CMS-jet kinematic configuration. We can see that the NLOcorrections become larger and larger at increasing Y , an expected phenomenon in the BFKLapproach.In Figs. 3 and 4, predictions with the BLM scale optimization for C and several R nm ≡ C n /C m ratios with the jet tagged inside the CMS detector are shown for √ s = 7 and 13 TeV,respectively. Here the benefit of the use of BLM optimization appears, since the LLA andNLA predictions for C are now comparable, a sign of stabilization of the perturbative series.The trend of ratios of the form R n is the standard one and indicates increasing azimuthaldecorrelation between the jet and the hadron as Y goes up, with the NLA predictionssystematically above the LLA ones, as it was also observed in Mueller-Navelet jets and inthe hadron-hadron case. The ratios R and R seem to be almost insensitive to the NLOcorrections.Panels in Fig. 5 show results with BLM scale optimization for C and several R nm ratiosin the CASTOR-jet configuration at √ s = 13 TeV. They exhibit some new and, to someextent, unexpected features: (i) the two parametrizations for the FFs lead to clearly distinctpredictions, (ii) (cid:104) cos φ (cid:105) exceeds one at the smaller values for Y , a clearly unphysical effect.The reason for these phenomena could reside in the fact that, the lower values for Y in the CASTOR-jet case are obtained for negative values of the hadron rapidity, i.e. in final-stateconfigurations where both jet and hadron are backward.Finally, in Fig. 6 we compare the φ -averaged cross section C in different NLA BFKLprocesses: Mueller-Navelet jet, hadron-jet and hadron-hadron production, for µ F = µ BLM R ,at √ s = 7 and 13 TeV, and Y ≤ . CMS-jet case. The hadron-hadron cross section,with the kinematical cuts adopted, dominates over the jet-jet one by an order of magnitude,with the hadron-jet cross section lying, not surprisingly, in-between.15
V. SUMMARY
In this paper we have proposed a new candidate probe of BFKL dynamics at the LHCin the process for the inclusive production of an identified charged light hadron and a jet,separated by a large rapidity gap.We have given some arguments that this process, though being a naive hybridization oftwo already well studied ones, presents some own characteristics which can make it worthyof consideration in future analyses at the LHC.In view of that, we have provided some theoretical predictions, with next-to-leadingaccuracy, for the cross section averaged over the azimuthal angle between the identified jetand hadron and for ratios of the azimuthal coefficients.The trends observed in the distributions over the rapidity interval between the jet andthe hadron are not different from the cases of Mueller-Navelet jets and hadron-hadron, whenthe jet is detected by CMS, whereas some new features have appeared when the jet is seenby CASTOR, which deserve further investigation.
V. ACKNOWLEDGMENTS
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