Heavy Quarks Production in Hadronic Processes: Qualitative Study of Higher-Order Fock States

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1 Commun. Theor. Phys. (Beijing, China) 51 (009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 4, April 15, 009 Heavy uarks Production in Hadronic Processes: ualitative Study of Higher-Order Fock States N. Mebarki, 1 K. Benhizia, 1 Z. Belghobsi, and D. Bouaziz 1 Laboratoire de Physique Mathematique et Subatomique, Mentouri University, Constantine, Algeria Département de Physique Université de Jijel, Jijel, Algeria (Received March, 008; Revised August 8, 008) Abstract The contribution of the two particles Fock states for the production of a heavy quark in proton-pion and photon-pion collisions is studied. It is shown that the effect depends strongly on the produced heavy quark mass, and the choice of the factorization scale. PACS numbers: p, 1.85.Hd, a Key words: jets in large scattering, higher twist effect, heavy quarks 1 Introduction During the last few years, many theoretical as well as experimental interest has been devoted to the study of heavy quarks production in hadronic collisions at the new colliders machines. [1 8] In fact, since hadronic processes involving heavy flavors take place with a large transferred momenta, the CD running coupling constant becomes small, and consequently perturbation theory is applicable. Thus, they can be used to understand the dynamics of the strong interactions and probe of the hadronic matter. Moreover, with the heavy quarks, one can expect new phenomena becoming important and their contribution may be comparable to that of the higher-order (H.O.C) radiative corrections. This is a subject of considerable importance for future p p and pp (LHC) accelerators. [4 6] The goal of this paper is to study a phenomenon called higher-twist effect, in which at the subprocess level the partons of one of the hadrons involved in the collision constitute higher Fock states and interact directly with one parton of the other hadron. [9 1] To get qualitative results, understand this phenomenon, and simplify the formalism, we take one of the hadrons in the initial state a meson (pion), their partons constitute a higher Fock state (more general cases relevant to the LHC physics are under investigations). In Sec., we present the formalism for inelastic hadronic processes p + π 0 + X and γ + π 0 + X ( stands for the heavy quark); and finally in Sec., we give some qualitative results and draw our conclusions. Formalism There have been several studies of higher-twist (HTE) effects in DIS experiments. If power law corrections to a leading twist perturbative CD are needed, they will introduce additional nonperturbative degrees of freedom in the global analysis. This will complicate partons distribution functions (pdfs) analysis considerably, because the extracted pdfs would then depend on the HTE model. Since actually there is no accepted theory of HTE, terms are probably process-dependent, pdfs obtained with the inclusion of HTE terms would no longer be universal. In the absence of a firm theoretical guidance, we will be limited (as it will be clear later) to a factorizable form (convolution product) of the transition amplitude consisting of nonpertubative and perturbative parts ralated to the wave function of the hadron and subprocess matrix elements respectively. In what follows, we will present a qualitative study of the HTE effect and show its strong dependence to the heavy quarks mass m and factorization scale. To keep our results transparents, we will concentrate only on the two inelastic hadronic processes p + π 0 + X and γ + π 0 + X and taking into account just the Fock states of the form q q and/or gg of the incident pion π 0. The contributions come from higher-order Fock states of the light quarks q, q, q and the gluon g such that qq q, q qq q, q qgg, etc., are negligeable. Here p and denote the proton and the produced heavy quark respectively. Moreover, in our work, we have restricted ourselves only to the lowest Fock state for the meson. By contrast, the proton is depicted in terms of the regular parton distribution. [14,15] Of course, this is not always true for all processes. This is justified only for the heavy quark production in proton-pion collisions (case of our study). If we do not consider the pion higher twist effect and take just the (uud) proton leading Fock state, we cannot have at the final state (in the proton-pion collision) only the Heavy quark and antiquark but rather other jets (at least two). Thus, in this case, we are dealing with at least 4 body reaction. This, has to be compared with the body contribution to the heavy quark production in proton-pion transition amplitude we consider just the π 0 leading higher twist effect. Furthermore, from a phase space argument the former is highly suppressed with respect to the latter. Moreover, the hard scattering amplitude in the former is of O(g 4 s), as for the latter is of O(g s) (here g s stands for the strong CD coupling). To be more specific, if we consider for

2 718 N. Mebarki, K. Benhizia, Z. Belghobsi, and D. Bouaziz Vol. 51 example only the proton leading higher twist configuration (uud) in the subprocess (uud) + d + + u + u contributing to the physical process p+π jets, we will notice that this leads to a contribution smaller than the one if we take into account only the π 0 leading higher twist in the subprocess: g + (q q) + (g is the gluon inside the proton) of the physical process p+π 0 +. Of course, if we consider the two leading higher twist effects of both the proton and the π 0 at the same time and because of the proton and pion form factors, the corresponding cross section is highly suppressed by powers of O(1/ 4 ). Moreover, we will have in addition to the heavy quarks pair, multipartons final states the phase space contribution is much smaller. It is worth to mention also that in other processes there are no heavy quarks at the final states, the proton higher twist effect may not be neglected. For example, in the physical process, p + π 0 γ + jets, the contribution from the proton higher twist effect in the subprocess (uud) + ū γ + u + d + g is as important as the one coming from the pion higher twist effect in the subprocess u + (uū) γ + g + g + u. The higher twist gauge invariant amplitude M(ŝ, ˆt, û) of the corresponding dominant subprocesses can be factorized into two parts. [] (i) A non perturbative distribution amplitude Ψ π 0 (x, ) related to the pion wave function, which gives the probability density to find a quark or a gluon carrying a momentum fraction x inside the pion at a factorization scale. It is very important to mention that all contributions of the form [α s ln( /µ )] n, α s is the CD running coupling, have been absorbed into Ψ π 0(x, ). This distribution amplitude have the following product form [] Ψ π 0(x, ) = B( )x(1 x), (1) B( ) is a function of the factorization scale. Moreover, Ψ π 0(x, ) is related to the pion form factor F π 0( ) through the relation [] F π 0( ) = 1 0 dx 1 0 dy Ψ π 0 (y, ) y T(x, y, ) Ψ π 0(x, ), () x the kernel T(x, y, ) is given by [] T(x, y, ) = α s ( 1πC F ) (1 x)(1 y), () C F is a CD color factor (C F = 4/ in SU c ()). At very high energies Λ CD (Λ CD is the CD parameter or cut off), the pion form factor F π 0( ) has the following approximate asymptotic form [] F π 0( ) 4πC FC 0 α s( ), (4) C 0 = f π, (5) and f π is the pion decay constant given by f π 9 MeV. (6) The constant C 0 is determined from the weak decay amplitude of the process: π µ + ν µ. (ii) A perturbative transition amplitude M(ŝ, ˆt, û, x) calculated at the subprocesse level with standard Feynman rules techniques. Here ŝ, ˆt, and û are the Mandelstam variables. Thus, the final higher twist (Fock states) transition amplitude M(ŝ, ˆt, û) takes the form: M(ŝ, ˆt, û) = Ψ π 0(x, ) M(ŝ, ˆt, û, x), (7) stands for the convolution product..1 Inelastic Hadronic Process p + π 0 + X (i) Born Term Kinematics At the lowest order of the perturbation theory, the dominant contributing subprocesses to the physical inelastic process p + π 0 + X are q + q + and g + g +. The square of the Born transition amplitudes M B of these subprocesses are given by [16] with M B (q + q + ) = M 1 = 64π αs 9 ŝ [L 1 + L + m ŝ ], (8) M B (g + g + ) = M = π αs ) (4 ŝ 9 L 1 L [ L 1 + L ŝ + 4m ŝ 4m4 ] (9) L 1 L L 1 = ˆt m, (10) L = û m. (11) Here ŝ, ˆt, and û are the Mandelstam variables of the subprocess a + b +, a and b stands for quarks, antiquarks or gluons. They are related to the physical process Mandelstam variables s, t, and u through the following relations: ŝ = x 1 x s, (1) ˆt = x 1 (t m ) + m, (1) û = x (u m ) + m, (14) x 1 (resp. x ) is the momentum fraction carried by the parton a (resp. b) inside the proton (resp. pion) and m the mass of the heavy quark. Now, with the help of the factorization theorem, the Born differential cross section d σ B E dp (p + π 0 + X) of the hadronic process p + π 0 + X can be written as:

3 No. 4 Heavy uarks Production in Hadronic Processes: ualitative Study of Higher-Order Fock States d σ 1 B E dp (p + π X) = dx 1 dx π ŝ δ(ŝ + ˆt + û m ) ( ) M 1 G q/p (x 1, )G q/π 0(x, ) + M G g/p (x 1, )G g/π 0(x, ), (15) q G a/p (x 1, ) and G b/π 0(x, ) are the partons distribution functions inside the proton and pion respectively. Using the properties of the delta function, it is easy to show that Eq. (15) can be rewritten as: d σ B E dp (p + π 0 + X) = 1 1 ( σ1 + σ ) 16π dx 1 s x 0 x, (16) 1 (m σ 1 = M B (q + q + ) = dx 1 16π ŝ δ(ŝ + ˆt + û m ) q G g/p (x, ) M g+π0 + (ŝ, ˆt, û), (1) G q/p (x 1, )G q/π 0(x, ), (17) σ = M B (g + g + ) G g/p (x 1, )G g/π 0(x, ), (18) x = x 1(m x 1 s + u m, (19) x 0 = 1 s (m u). (0) (ii) Higher Fock States Kinematics The dominant diagrams are those corresponding to the subprocess g + π 0 + (see Fig. 1). The higher Fock states differential cross section d σ HTE E dp (p + π 0 + X) of the hadronic process p + π 0 + X can be written as: E d σ HTE d (p + π0 + X) ŝ, ˆt, and û are given by: ŝ = xs, () ˆt = xt + m (1 x), () û = u, (4) and M g+π 0 + (ŝ, ˆt, û) is the higher twist effect transition amplitude corresponding to the subprocess g + π 0 +. Using the properties of the Dirac delta function, equation (1) can be rewritten as: d σ HTE E dp (p + π 0 + X) = M g+π 0 + (ŝ, ˆt, û) G g/p (x, ) 16π s(m u) x takes the value, (5) x = m u s + t m. (6) Fig. 1 Dominant higher twist Feynman diagrams of the physical process g + π 0 +. Now, using the fact that the two partons inside the pion constitute a pseudo scalar and color singlet state, one can show easily that: M g+π 0 + (ŝ, ˆt, û) = M 1 (ŝ, ˆt, û) + M 1(ŝ, ˆt, û) + M (ŝ, ˆt, û) + M (ŝ, ˆt, û) + M 4 (ŝ, ˆt, û), (7)

4 70 N. Mebarki, K. Benhizia, Z. Belghobsi, and D. Bouaziz Vol. 51 M 1 (ŝ, ˆt, û) + M 1(ŝ, ˆt, û) = 56π αs )( 0 C1 48 M (ŝ, ˆt, û) = 64π αs )( 0 C )( 1 )[ ŝ M (ŝ, ˆt, û) = 64π αs )( 0 C )[ 1 ] 51 ŝ (m û) [ ( 4m 19m 4 19m (û+ˆt)+ 11 M 4 (ŝ, ˆt, û) = 64π α s )( 1 )[ ŝ 5m ŝ + 1 ] (û + ˆt ) m 4, (8) 7 (m ˆt)(m û) ((m ˆt) +(m û) ) ŝ m ŝ ], (9) m ŝ 7(û +ˆt )+ŝ ++5ˆtû+ ) ŝû+4ŝˆt +ŝ(û +ˆt )+ ŝˆtû 8 ], (0) )( 0 C4 )[ m ] [ 08m 51 (m û) (m ˆt) (û + ˆt) + 56ûˆt 59m 4 ]. (1) The color factors C 1, C, C, and C 4 are given by C 1 = 1 (T b ijtjit a ij a T j b i) = 8, C = 1 ijj ab C = f bac f bac Tr(T b T c T c T b ) = 6 9, a,b,c,c C 4 = a,b a,b,c,c,d (f bc d f bac f bcd f bac ) = , Tr (T b T a T b T b T a T b ) = 5 6. () Here Tr, T a, and f bac denote the trace, generators (in the fundamental representation) and the structure constants of the SU() gauge group.. Inelastic Hadronic Process γ + π 0 + X (i) Born term Kinematics At the lowest order of the perturbation theory, the dominant subprocess contributing to the physical process γ + π 0 + X is γ + g +. The square of the transition amplitude M B (γ + g + ) is given by [17,18] M B (γ + g + ) [ = π e 1 ( m αα s ( 1 + m 1 ŝ + 8m 4 ) m + ˆt + m + û)] 1, () e and α are the charge of the produced heavy quark and the ED running coupling respectively. The functions 1 and are such that 1 = ˆt m, (4) = û m. (5) In this case, the subprocess Mandelstam variables ŝ, ˆt, and û are given in terms of those of the physical process by the following relations: ŝ = xs, (6) ˆt = t, (7) û = x(u m ) + m. (8) Here x represents the momentum fraction carried by the gluon inside the pion. With the help of the factorization theorem and Dirac delta function, the Born differential cross section has as a simplified expression: d σ B E dp (γ + π 0 + X) again = M(γ + g + ) G g/π 0(x, ) 16π s(m, (9) x = m t s + u m. (40) (ii) Higher Fock States Kinematics The dominant subprocesses diagrams contributing to the physical cross section are those of γ + (q q) + and γ + (gg) + (see Fig. ). Of course the higher Fock states q q and gg constitute a pseudo scalars and singlets color states. Tedious but straightforward calculations lead to the following form of the transition amplitude M γ+π 0 + : M γ+π 0 + = M 5 (s, t, u) + M 5(s, t, u) + M 6 (s, t, u), (41) M 5 (s, t, u)+m 5 (s, t, u) and M 6 (s, t, u) have as expressions: M 5 (s, t, u) + M 5(s, t, u) = 56π 0 )( 5 )( αα ) s 108 s [ 5m s + 1 ] (u + t ) m 4, (4) and M 6 (s, t, u) = 64π 0 )( 5 )[ ααsm ] 576 (m u) (m [ 08m (u + t) + 56ut 59m 4 ]. (4) Finally, the corresponding higher twist effect differential cross section takes the form: d σ HTE E dp (γ + π 0 + X) = M γ+π π s(s + u m ).(44)

5 No. 4 Heavy uarks Production in Hadronic Processes: ualitative Study of Higher-Order Fock States 71 Ω 1 = 1 s (5m s + 1 u + 1 t m 4 ), (47) Fig. Dominant higher twist Feynman diagrams of the physical process γ + π 0 +. Results and Conclusions In what follows, we consider the ratio R of the higher twist effect over the Born differential cross sections: R = E d σ HTE /dp E d σ B /d. (45) P To keep our results clear, we consider only the inelastic process γ + π 0 + X. After direct simplifications, the ratio R can be written as: ( 5π )( fπ ) R 88 e (Ω 1 / + 4Ω )(m Ω (s + u m )G g/π 0(x, ) ln( /Λ CD ). (46) and Ω = m [ 08m (u + t) + 56ut 59m 4 ] 1, (48) Ω = 1 ( 1 + m 1 ŝ + 8m 4 ) ( m m + ˆt + m + û) 1. (49) We remind the reader that in the lab system the pion π 0 is at rest, the Mandelstam variables s, t, and u are expressed in terms of the photon energy E γ and the produced heavy quark scattering angle θ as: s = m π 0E γ, (50) t = m E γ Φ(m π 0, m, θ, E γ ) + E γ cos θ [Φ(m π 0, m, θ, E γ )] m, (51) u = m m π 0Φ(m π 0, m, θ, E γ ), (5) Φ(m π 0, m, θ, E γ ) = m π 0E γ + (m π 0E γ ) (E γ sin θ + m π 0)(m π + m 0 cos θ)e γ E γ sin θ + m π 0, (5) together with the following conditions: E γ m π 0(m π 0 + m cos θ) (m π 0 m sin θ) cos θ, (54) sin θ < m π 0 m. (55) It is worth to mention that the relations (54) and (55) put some constraints in order that the higher twist effect (HTE) takes place. Notice that from Eq. (55), HTE is possible only for very small angles (almost θ 0) that is the produced heavy quark is almost forward and collinear to the incoming photon and this can be a good signal (test from the experimental point of view). Moreover, for the top quark production (m t = 175 GeV), the higher twist effect is relevant, it takes place with small scattering angle (sinθ < 0.15/175 0) and requires very high energy (E γ 454 TeV); which is not accessible with the actual acceleretors. However, we need just E γ 71 GeV and E γ 4 GeV for the bottom (m b = 5 GeV) and the charm (m c = 1.5 GeV) quarks production respectively with almost θ 0 (forward direction). Figure displays the ratio R as a function of E γ for the bottom (b) (strong line) and charm (c) (dotted line) quark with a factorization scale = 75 (GeV) and using the Duke Owens parametrization of the partons structure functions (pdf) with Λ CD = 0. GeV (SET1). [19] Our analysis and features remain almost unchanged with the MRST or CTE pdfs parametrizations. [0,1] Notice that, R is more important for the (b) quark than for the (c) quark. This is due essentially to the fact that the differential cross section of the higher twist effect (as it is clear from the previous expressions) is an increasing function of the produced heavy quark mass (see e.g. the overall factor 1/(s + u m ), as m increases s+u m decreases). Furthermore, from the phase space we expect that the Born term differential cross section decreases as the mass of the produced heavy quark increases. Notice also that the momentum fraction x of the gluon inside the pion increases (see Eq. (40)) and therefore the gluon distribution function decreases leading to an overall increase of the ratio R. For example, with E γ = 550 GeV, R takes the values 0.48 and 0.17 for the b and c quarks respectively. Figure 4 represents, the factorization scale -dependence of the ratio R for the production of a b quark. Notice that, R is very sensitive to the values of. For example, for E γ = 500 GeV and = 75 (GeV), R = 0.451; however, for = 50 (GeV) and for the same value of E γ, R = This is due essentially, to the fact that R is inversely proportional to the gluon distribution function inside the pion G g/π 0(x, ) and as increases the latter decreases and consequently R increases. It is worth to mention that the quantity G g/π 0(x, ) ln[ /Λ ] is also a decreasing function of.

6 7 N. Mebarki, K. Benhizia, Z. Belghobsi, and D. Bouaziz Vol. 51 Fig. The ratio R for (b) and (c) quarks as a function of E γ for = 75 (GeV). Fig. 4 The ratio R for a (b) quark as a function of E γ for various values of. Regarding the behavior of the ratio R as a function of the energy, we notice that R increases as E γ increases till a certain maximum value E γ max corresponding to R = R max and than it starts decreasing as E γ reachs larger values. This is due essentially to the function Ω in Eq. (48), which has a maximum value at E γ max.94m /m π 0. For E γ E γ max or at small values of the momentum fraction x of the gluon inside the pion, there are two competitive asymptotic behaviors: The increase of the gluon distribution function inside the pion at small values of x (large values of E γ ), it becomes important and also the decrease of the quantity RG g/π 0(x, ) ln( /Λ CD ) at high energies ( 1/E γ ). This means that the higher twist effect is negligeable at very high energies. We conclude from our previous qualitative study that the higher twist effect (HTE) is very important for the heavy quarks production in hadrons or photon-hadrons collisions and has the following features: (i) Its contribution increases with the increase of the produced heavy quark mass. (ii) It is more significant at smaller values of the scattering angle or larger pseudo rapidity (forward direction). (iii) It is very sensitive to the choice of the factorization scale. Acknowledgments We are very grateful to the Algerian Ministry of education and research for the financial support. One of us (N.M.) would like to thank Prof. Goran Sanjanovic and Dr. Lotfi Boubekeur for their kind hospitality during my visit to the ICTP, part of this work has been completed. References [1] R.K. Ellis, Strong Interactions and Gauge Theories, ed. J. Tran Thanh Van, Editions Frontieres, Gif-sur-Yvette (1986) 9. [] J.C. Collins, D.E. Soper, and G. Sterman, Nucl. Phys. B 6 (1986) 7. [] S.J. Brodsky, J.C. Collins, S.D. Ellis, J.F. Gunion, and A.H. Mueller, DOE/ER/ P4, Proc. of 1984 Summer Study on the SSC, Snowmass, CO, Jun. -Jul. 1, [4] S.J. Brodsky, J.F. Gunion, and D.E. Soper, Phys. Rev. D 6 (1987) 710. [5] P. Nason, S. Dawson, and R.K. Ellis, Nucl. Phys. B 0 (1988) 607. [6] W. Beenakker, H. Kuijf, W.L. Van Neerven, and J. Smith, Phys. Rev. D 40 (1989) 54. [7] A.H. Mueller and P. Nason, Phys. Lett. B 157 (1985) 6. [8] G. Altarelli and G. Parisi, Nucl. Phys. B 08 (1988) 74. [9] E. Leader, A.V. Sidorov, and D.B. Stamenov, arxiv:hepph/050918; Proc. of First Workshop on uark-hadron Duality and the Transition to P CD, Frascati, June 6-8, 005. [10] K. Djagouri, J.J. Dugne, C. Carimalo, and P. Kessler, Z. Phys. C 45 (1989) 141. [11] A.I. Ahmadov, I. Boztosun, A. Soylu, and E.A. Dadashov, arxiv:hep-ph/ [1] J.A. Bagger and J.F. Gunion, Phys. Rev. D 9 (1984) 40. [1] J.A. Hassan and J.K. Storrow, Z. Phys. C 14 (198) 65. [14] S.D. P. Vlassopulos, Phys. Lett. B 166 (1986) 449. [15] J.A. Bagger and J.F. Gunion, Phys. Rev. D 5 (198) 87. [16] M.L. Mangano, arxiv:hep-ph/97117v1. [17] J. Smith and W.L. Van Neerven, Nucl. Phys. B 74 (199) 6. [18] R.K. Ellis and P. Nason, Nucl. Phys. B 1 (1989) 551. [19] D.W. Duke and J.F. Owens, Phys. Rev. D 0 (198) 160. [0] A.D. Martin, W.J. Stirling, and R.S. Thorne, Phys. Lett. B 66 (006) 59. [1] H.L. Lai, J. Huston, S. Kuhlmann, F. Olness, J. Owens, D. Soper, W.K. Tung, and H. Weerts, Phys. Rev. D 55 (1997) 180.