η_{Q} meson photoproduction in ultrarelativistic heavy ion collisions
Gong-Ming Yu, Gao-Gao Zhao, Zhen Bai, Yan-Bing Cai, Hai-Tao Yang, Jian-Song Wang
aa r X i v : . [ h e p - ph ] J a n η Q meson photoproduction in ultrarelativistic heavyion collisions Gong-Ming Yu ∗ , Gao-Gao Zhao , Zhen Bai , Yan-Bing Cai † ,Hai-Tao Yang ‡ , and Jian-Song Wang § CAS Key Laboratory of High Precision Nuclear Spectroscopy and Center for NuclearMatter Science, Institute of Modern Physics, Chinese Academy of Sciences,Lanzhou 730000, People’s Republic of China Department of physics, Yunnan University, Kunming 650091, People’s Republic of China School of physics and electronic information engineering, Zhaotong University,Zhaotong 657000, People’s Republic of China
Abstract
The transverse momentum distributions for inclusive η c,b meson described by gluon-gluon interactions from photoproduction processes in relativistic heavy ion collisionsare calculated. We considered the color singlet (CS) and color octet (CO) componentswith the framework of non-relativistic Quantum Chromodynamics (NRQCD) intothe production of heavy quarkonium. The phenomenological values of the matrixelements for the color-singlet and color-octet components give the main contributionto the production of heavy quarkonium from the gluon-gluon interaction caused bythe emission of additional gluon in the initial state. The numerical results indicatethat the contribution of photoproduction processes cannot be negligible for mid-rapidity in p-p and Pb-Pb collisions at the Large Hadron Collider (LHC) energies. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected]
Introduction
Heavy quarkonium is a multiscale system which can probe all regimes of Quantum Chro-modynamics (QCD), and present an ideal laboratory for testing the interplay betweenperturbative and nonperturbative QCD within a controlled environment. In the recentyears, many measurement reports have been published by ALICE collaboration[1, 2], CMScollaboration[3, 4], ATLAS collaboration[5, 6], and LHCb collaboration[7, 8] at the LargeHadron Collider (LHC) energies; several theoretical approaches have been proposed suchas the color-singlet (CS) mechanism[9, 10], the color-octet (CO) mechanism[11, 12], thecolor evaporation mechanism[13, 14], the color-dipole mechanism[15, 16, 17, 18], the mixedheavy-quark hybrids mechanism[19], the recombination mechanism[20, 21, 22, 23, 24],the photoproduction mechanism[25, 26, 27, 28, 29, 30, 31], the potential non-relativisticQuantum Chromodynamics (pNRQCD) approach[32, 33, 34], the transverse-momentum-dependent factorization approach[35], the transport approach[36, 37, 38, 39, 40, 41], the k T -factorization approach[42, 43, 44, 45], the fragmentation approach[46, 47, 48, 49, 50, 51, 52],and the non-relativistic Quantum Chromodynamics (NRQCD) approach[53, 54, 55, 56, 57,58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. Among them, the NRQCD approach, which takesinto account contributions of color-singlet component and color-octet components withthe nonperturbative long-distance matrix elements (LDME), is the most successful in phe-nomenological studies. The long-distance matrix elements are process-independent, andcan be classified in terms of the relative velocity for the heavy quarks in the bound state.But, the heavy quarkonium production mechanism is still not fully understood.In this study, we extend the hard photoproduction mechanism[68] to the heavy quarko-nium production and investigate the production of inclusive η c,b meson in p-p and Pb-Pbcollisions at the LHC. According to Ref. [71], the light q ¯ q contributions for heavy quarko-nium production are negligible, therefore in this work we only consider the contributionsof gluon-gluon processes caused by the emission of additional gluons, that is different fromour previous work[26] based on the method developed in Ref. [69, 70]. In high-energycollisions, the partons from the nucleus can emit high-energy photons that can fluctuateinto gluons and then interact with the partons of the other nucleus. Hence we considerthat the hard photoproduction processes of a charged parton of the incident nucleon canemit a high energy photon in high energy nucleus-nucleus collisions.The paper is organized as follows. In section 2 we present the photoproduction of inclusive η c,b from gluon-gluon interactions at LHC. The numerical results for large- p T η c,b mesonproduction in p-p collisions and Pb-Pb collisions at LHC are given in section 3. Finally,the conclusion is given in section 4. In relativistic heavy ion collisions, the production of η Q mesons by the gluon-gluon (g-g)processes from the initial parton interaction can be divided into three processes: direct g-g processes, semi-elastic resolved photoproduction and inelastic resolved photoproduction1rocesses.In direct processes, the parton (gluon) a of the incident nucleus A interacts with the parton(gluon) b of another incident nucleus B by the interaction of gg → η c g . The invariant crosssection of large- p T η Q meson of the process ( A + B → η Q + X ) is described in the pQCDparton model on the basis of the factorization theorem and can be written as dσ LOAB → η Q X dp T dy = Z dx a f g/A ( x a , Q ) f g/B ( x b , Q ) x a x b x a − x d ˆ σd ˆ t ( gg → η Q g ) , (1)where the variables x a and x b = ( x a x − τ ) / ( x a − x ) are the momentum fractions of thepartons, z c is the momentum fraction of the final charmed-meson, x = ( x T +4 τ ) / exp( y ), x = ( x T + 4 τ ) / exp( − y ), x T = 2 p T / √ s , τ = ( M/ √ s ) , and M is the mass of the η Q meson; f g/A ( x a , Q ) and f g/B ( x b , Q ) are the parton distribution functions (PDF) forthe colliding partons a and b carrying fractional momentum x a and x b in the interactingnucleons[72], f g/A ( x, Q ) = R A ( x, Q ) f g ( x, Q ) , (2)where R A ( x, Q ) is the nuclear modification factor[73], and f g ( x, Q ) is the gluon distribu-tion function of nucleon.According to NRQCD scaling rules[74, 75], the color-singlet as well as S -wave and P -wave color-octet components give the main contributions to the production process underconsideration[76] d ˆ σd ˆ t ( gg → η Q g ) = | R (0) | d ˆ σd ˆ t ( gg → Q ¯ Q [ S [1]0 g ])+ h O S i d ˆ σd ˆ t ( gg → Q ¯ Q [ S [8]1 g ])+ h O P i d ˆ σd ˆ t ( gg → Q ¯ Q [ P [8]1 g ]) . (3)The subprocesses cross section of [ S [1]0 ], [ S [8]1 ], and [ P [8]1 ] state are respectively givenby[77, 78] d ˆ σd ˆ t ( gg → Q ¯ Q [ S [1]0 ] g ) = πα s ˆ s M P Q ( Q − M P ) [( P − M ) + 2 M Q ] , (4) d ˆ σd ˆ t ( gg → Q ¯ Q [ S [8]1 ] g ) = πα s M ˆ s ( P − M Q )(19 M − P ) M ( Q − M P ) , (5) d ˆ σd ˆ t ( gg → Q ¯ Q [ P [8]1 ] g ) = 2 πα s M ˆ s Q ( Q − M P ) [179 M Q + 217 M Q − M P + 54 M P − M P +135 P Q +103 M P Q − M P Q − M P Q +43 M P Q +27 P Q ] , (6)2here M = ˆ s + ˆ t + ˆ u , P = ˆ s ˆ t + ˆ t ˆ u + ˆ u ˆ s , and Q = ˆ s ˆ t ˆ u . Here ˆ s , ˆ t , and ˆ u are the Mandelstamvariables. R (0) = [ M H Γ( H → e + e − ) / α e Q ] / is the wave function value of η Q meson forthe color-singlet state at the origin[79, 80, 81, 82, 83, 84], where M H ≈ m Q is the massof the heavy quark pairs. The LDMEs of the color-octet components are used as follows h O S i = h R η Q [ S (8)1 ] i = π h | O η Q [ S ] | i , h O P i = h R η Q [ P (8)1 ] i = π h | O η Q [ P ] | i . (7)For the η c meson they are[56] | R cc (0) | ≈ .
58 GeV , . × − GeV < h O η c S i < . × − GeV , h O η c P i = π × × h O J/ψ ( P [8]0 ) i = π × m c × (1 . ± . × − GeV , (8)and for η b meson they are[85, 86] | R bb (0) | ≈ . , h O η b S i ≈ .
01 GeV , h O η b P i = π × × h O γ (1 s )8 ( P ) i = 5 π × m b × (0 . ± . , (9)where m c ( m b ) is the mass of charm (bottom).In the semi-elastic resolved photoproduction g-g processes, the parton (gluon) a fromresolved photon of the incident nucleus A interacts with the parton (gluon) b of anotherincident nucleus B , and the cross section is given by dσ semi.AB → η Q X dp T dy = Z dx a dx b f γ/N ( x a ) f g/γ ( z a , Q ) f g/B ( x b , Q ) x a x b z a x a x b − x a x d ˆ σd ˆ t ( gg → η Q g ) , (10)where f γ/N ( x a ) is the photon spectrum of the nucleus, and f g/γ ( z a , Q ) is the parton dis-tribution function of the resolved photon[87].For p-p collisions, the photon spectrum function of a proton can be written as[88, 89, 90] f γ/p ( x ) = α πx [1 + (1 − x ) ] (cid:20) ln A p −
116 + 3 A p − A p + 13 A p (cid:21) , (11)where x is the momentum fraction of photon, A p = 1 + 0 .
71 GeV /Q min with Q min = − m p + 12 s (cid:20) ( s + m p )( s − xs + m p ) − ( s − m p ) q ( s − xs − m p ) − m p xs (cid:21) . (12)3
10 15 2010 y=0 c at LHC p-p 7TeV d / dp T d y ( nb / G e V ) p T (GeV/c)(a) (b) c at LHC p-p 14TeV p T (GeV/c)y=0 -2 -1 y=0(c) c at LHC Pb-Pb2.76ATeV p T (GeV/c) -1 y=0(d) c at LHC Pb-Pb 5.5ATeV p T (GeV/c) Figure 1: The invariant cross section of large- p T η c meson production from gluon-gluoninteraction at mid-rapidity in p-p collisions ( √ s = 7 . √ s = 14 . √ s = 2 .
76 TeV and √ s = 5 . m p is the mass of the proton and at high energies Q min is given to a very goodapproximation by m p x / (1 − x ).For Pb-Pb collisions, the photon spectrum obtained from a semiclassical description ofhigh energy electromagnetic collisions for low photon energies is given by[91, 92] f γ/N = 2 Z απω ln (cid:18) γωR (cid:19) , (13)where ω is the photon energy, and R = b min is the nucleus radius.In inelastic resolved photoproduction g-g processes, the parton (gluon) a ′ from resolvedphoton which emitted by the charged parton a of the incident nucleus A interacts with theparton (gluon) b of another incident nucleus B , and the expression of the cross section isgiven by dσ inel.AB → η Q X dp T dy = Z dx a dx b dz a f q/A ( x a , Q ) f γ/q ( z a ) f g/γ ( z ′ a , Q γ ) × f g/B ( x b , Q ) x a x b z a z ′ a x a x b z a − x a z a x d ˆ σd ˆ t ( gg → η Q g ) , (14)where f γ/q ( z ) is the photon spectrum from the charged parton of the incident nucleus. Inrelativistic hadron-hadron and nucleus-nucleus collisions[71] we have, f γ/q ( x ) = απ e Q (cid:26) − x ) x (cid:18) ln Em − (cid:19) + x (cid:20) ln (cid:18) x − (cid:19) +1 (cid:21) + (2 − x ) x ln (cid:18) − x − x (cid:19)(cid:27) , (15)4
10 15 2010 -1 b at LHC p-p 7TeV d / dp T d y ( nb / G e V ) p T (GeV/c)(a) y=0 y=0 b at LHC p-p 14TeV(b) p T (GeV/c) -2 -1 y=0 b at LHC Pb-Pb2.76ATeV(c) p T (GeV/c) -2 -1 y=0 b at LHC Pb-Pb 5.5ATeV(d) p T (GeV/c) Figure 2: The same as Fig. 1 but for large- p T η b meson production from gluon-gluoninteraction at mid-rapidity in p-p and Pb-Pb collisions at the LHC.with x being the photon momentum fraction. In ultrarelativistic high energy nucleus-nucleus collisions, the equivalent photon spec-trum obtained with a semiclassical description of high-energy electromagnetic collisionsfor the nucleus is f γ/N ∝ Z ln γ . At LHC energies, the Lorentz factor γ = E/m N = √ s NN / m N ≫ a cker-Williams approximation for the proton is f γ/p ∝ ln A ∝ ln( s NN /m p ),where m p is the proton mass and √ s NN is the centre-of-mass energy per nucleon pair. Since √ s NN is very high, the photon spectrum function becomes very large. Therefore the contri-bution of η Q meson produced by semielastic hard photoproduction g-g processes cannot benegligible at LHC energies. For the inelastic photoproduction processes, the equivalent pho-ton spectrum function of the charged parton is f γ/q ∝ ln( E/m q ) = ln( √ s NN / m q ) + ln( x ),where m q is the charged parton mass. Hence, the photon spectrum for the charged partonbecomes prominent at LHC energies. The numerical results of our calculations for large- p T η Q mesons produced by the hard photoproduction gluon-gluon processes in relativisticheavy ion collisions are plotted in Fig. 1 and Fig. 2.In Fig. 1 (Fig. 2), we plot the contributions from the hard photoproduction gluon-gluonprocesses to the η c ( η b ) meson at mid-rapidity in p-p and Pb-Pb collisions at LHC energies.Compared with the production of the initial gluon-gluon interaction (the dashed line), thecontribution of η c,b meson produced by semielastic hard photoproduction g-g processes (thedotted line) is not prominent in p-p collisons with √ s NN = 7 . √ s NN = 14 . √ s NN = 2 .
76 TeV and √ s NN = 5 . In summary, we have investigated the production of heavy quarkonium η c,b meson fromthe gluon-gluon interactions in p-p collisions and Pb-Pb collisions at LHC energies. Thecolor singlet and color octet mechanisms have been used for heavy quarkonium productionprocesses. At the early stages of relativistic high energy nucleus-nucleus collisions, theultrarelativistic nucleus (charged parton) can emit hadron-like photons that can fluctuateinto a gluon, then the gluon interacts with a gluon of the other incident nucleus by gluon-gluon interaction. Our results indicate that the contribution of η c,b meson produced by thehard photoproduction processes cannot be negligible in p-p and Pb-Pb collisions at LHCenergies. Acknowledgments
This work has been supported by the National Basic Research Program of China (973Program, 2014CB845405), the China Postdoctoral Science Foundation funded project(2017M610663), and the Applied Basic Research Plan of Yunnan Province (Youth Project,2017FD147).
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