Inclusive hadronic distributions at small x

Inclusive hadronic distributions at small x Redamy Perez Ramos To cite this version: Redamy Perez Ramos. Inclusive hadronic distributions at small x. QCD and Hadronic Interactions, La Thuile, Italy, 18-25 Mar 2006., Mar 2006, La Thuile, Val d’Aosta, Italy. ฀hal-00068938฀ HAL Id: hal-00068938 https://hal.archives-ouvertes.fr/hal-00068938 Submitted on 15 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. May 2006 Inclusive hadronic distributions at small x inside jets Redamy Perez-Ramos 1 Laboratoire de Physique Théorique et Hautes Energies 2 Unité Mixte de Recherche UMR 7589 Université Pierre et Marie Curie-Paris6; CNRS; Université Denis Diderot-Paris7 Abstract: After giving their general expressions valid at all x, double differential 1-particle inclusive distribution inside a quark and a gluon jet produced in a hard process, together with the inclusive k⊥ distribution, are given at small x in the Modified Leading Logarithmic Approximation (MLLA), as functions of the transverse momentum k⊥ of the outgoing hadron. ccsd-00068938, version 1 - 15 May 2006 Introduction This work concerns the production of two hadrons inside a high energy jet (quark or gluon); they hadronize out of two partons at the end of a cascading process that we calculate in pQCD[1]. Considering this transition as a “soft” process is the essence of the “Local Parton Hadron Duality” (LPHD) hypothesis[2]. We give indeed, in the MLLA scheme of resummation, the double differential inclusive 1-particle distribution and the inclusive k⊥ distribution as functions of the transverse momentum of the emitted hadrons in the limit that proved successful when describing energy spectra of particles in jets, that is Q0 ≈ ΛQCD , the so called “limiting spectrum”[2]. The process under consideration In a hard collision (pp, pp̄ collisions, e+ e− annihilation) we consider a jet of half opening angle Θ0 initiated by a parton A0 , which can be a quark or a gluon, see Fig.1. A0 , by a succession of partonic emissions (quarks, gluons), produces a jet of half opening angle Θ0 , which, in particular, contains the parton A; A splits into B and C, which hadronize respectively into the two hadrons h1 and h2 . Θ is the angle between B and C. Because the virtualities of B and C are much smaller than that of A, Θ can be considered to be close to the angle between h1 and h2 ; angular ordering (AO) is also a necessary condition for this property to hold. A0 carries the energy E and gives rise to the (virtual) parton A, which carries the fraction u of the energy E. A → B + C occurs with probability ∝ Φ, Φ is the corresponding DGLAP splitting function, B and C carry respectively the fractions uz and u(1 − z) of E; h1 carries the fraction x1 of E; h2 carries the fraction x2 of E. One sets Θ ≤ Θ0 . h1 A B Θ0 A 0 D uE E Φ xE 1 D uzE Θ C h D u(1−z)E 2 x2E Fig. 1: process under consideration: two hadrons h1 and h2 inside one jet. Since we are concerned with observables depending on the transverse momentum of outgoing hadrons, 1 2 E-mail: perez@lpthe.jussieu.fr LPTHE, tour 24-25, 5ème étage, Université P. et M. Curie, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France) 1 k⊥ should be defined with respect to the jet axis which is identified with the direction of the energy flow[1]. Double differential 1-particle inclusive distribution at all values of x1 The energy conservation sum rule, considered together with DGLAP evolution equations and AO lead the general expression for the double differential 1-particle inclusive distribution d X d2 N = dx1 d ln Θ d ln Θ A Z 1 x1 h1 A duDA (u, EΘ0 , uEΘ) DA 0 x 1 u , uEΘ, Q0  (1) 2 which is valid at all x1 . dx1d dNln Θ in (1) is described indeed by the convolution of two fragmentation A and D h1 , over the energy fraction u of A. Namely, D A is purely partonic and describes functions, DA A0 A 0 the probability to emit parton A with energy uE off parton A0 , moreover, it takes into account the h1 evolution of the jet between Θ0 and Θ; DA describes the probability to produce hadron h1 of energy fraction x1 /u at angle Θ with respect to the direction of the energy flow inside the subjet A. On the other hand, the transverse momentum of h1 should be bigger than the collinear cut-off parameter Q0 (uEΘ ≥ Q0 ). Consequently, a lower bound for Θ is obtained: Θ ≥ Θmin ≈ Q0 /x1 E. Perturbative expansion parameter and kinematics We conveniently define the variables ℓ1 = ln x11 , y1 = ln x1QEΘ . In what follows, we use the anomalous 0 dimension γ0 defined through the running coupling constant αs as 2 γ02 (k⊥ )= 2) 2Nc αs (k⊥ = π 2 β ln 2 k⊥ Λ2QCD ≡ γ02 (ℓ1 +y1 ) = 1 1 = , β(ℓ1 +y1 +λ) β(YΘ +λ) λ = ln Q0 ΛQCD . (2) nf =3 1 11 It determines the rate of multiplicity growth with energy. Nc = 3 for SU (3), β = 4N ( 3 Nc − 43 TR ) = c 1 0.75 with TR = 2 nf where nf = 3 is the number of light quarks, ΛQCD is the QCD scale, YΘ = ℓ1 + y1 = EΘ0 ln EΘ Q0 and YΘ0 = ln Q0 . For instance, in the LHC environment we can take the typical value YΘ0 = 7.5 ⇒ γ0 ≃ 0.4 by setting λ = 0 (Q0 = ΛQCD ), Θ = Θ0 and Q0 = 250 MeV in (2). γ0 can be therefore treated as the small parameter of the perturbative expansion at MLLA. Double differential 1-particle inclusive distribution at small x1 , x1 ≪ 1 The convolution integral (1) is dominated by u ≈ 1. In order to obtain an analytical expression of (1), since x1 /u ≪ 1, we perform a perturbative expansion in γ0 such that (1) gets factorized and derived in the region of soft multi-particle production[1] < C >q,g d 1 d d2 N = D̃g (ℓ1 , y1 ) + D̃g (ℓ1 , y1 ) < C >q,g , dℓ1 d ln k⊥ Nc dy1 Nc dy1 √ d d D̃g (ℓ1 , y1 ) = O(γ0 ) = O( αs ), < C >q,g = O(γ02 ) = O(αs ). dy1 dy1 (3) (4) The first term in (3) is the main contribution to the double differential 1-particle inclusive distribution h1 while the second one constitutes its MLLA correction of relative order O(γ0 ) (4). DA in (1) has been replaced by the distribution Dg (ℓ1 , y1 ) in (3) that describes the MLLA “hump-backed plateau” in the limit where Q0 can be taken down to ΛQCD (“limiting spectrum”)[2]. < C >q,g is the total colour current that describes the evolution of the jet between Θ0 and Θ. It decomposes into its leading term < C >0q,g and the MLLA correction δ < C >q,g = O(γ0 )[1]. In Fig.2 we represent < C >0q,g (straight line) and the full expression < C >q,g =< C >0q,g +δ < C >q,g (curve) for two values of ℓ1 ; ℓ1 = 2 2.5(left), 3.5(right) in function of y1 . Two types of MLLA corrections are displayed in this figure. Namely, δ < C >q,g < 0 is given by the vertical difference between the straight and curved lines. The other correction is given by the slope of the curve, that is dyd1 < C >q,g in (4), this one is large and positive for y1 ≥ 1.5. For ℓ1 = 2.5 and y1 ≈ 1.5, δ < C >q,g represents 50% of < C >q,g while for ℓ1 = 3.5 > 2.5 it gets under control. We are thus allowed to set the range of applicability of our soft approximation to ℓ1 ≥ ℓ1,min ≈ 2.5 (x1 . 0.08) ⇒ y1 ≤ y1,max = YΘ0 − ℓ1,min = 5.0. YΘ = 7.5 YΘ = 7.5 0 4 0 4 0 <C> G, l = 2.5 <C>0G+δ<C>G, l = 2.5 <C>0Q, l = 2.5 <C>0Q+δ<C>Q, l = 2.5 3.5 3 0 <C> G, l = 3.5 <C>0G+δ<C>G, l = 3.5 <C>0Q, l = 3.5 <C>0Q+δ<C>Q, l = 3.5 3.5 3 2.5 2.5 2 2 1.5 1.5 1 1 0 1 2 3 4 5 0 0.5 1 1.5 2 y y 2.5 3 3.5 4 Fig. 2: < C >0A0 and < C >0A0 +δ < C >A0 for quark and gluon jets, as functions of y, for YΘ0 = 7.5, ℓ = 2.5 on the left and ℓ = 3.5 on the right. Furthermore, one should stay in the perturbative regime, which needs y1 ≥ 1 (k⊥ > 2.72ΛQCD ≈ 0.7 GeV). We finally get an estimate of the range of applicability of the MLLA scheme of resummation to be 1.0≤ y1 ≤ 5.0 in the LHC environment. In Fig.3 below, we represent the double differential 1particle inclusive distribution (3) for ℓ1 = 3.5 in function of y1 . We compare our results with a naive DLA-inspired case where one does not take into account the evolution of the jet between Θ0 and Θ but sets < C >q = CF (quark jet) and < C >g = Nc (gluon jet). YΘ = 7.5 YΘ = 7.5 0 0 4 3 naive case, l = 3.5 cone Θ, l = 3.5 3.5 naive case, l = 3.5 cone Θ, l = 3.5 2.5 3 2 2.5 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 Fig. 3: 1 1.5 d2 N dℓ1 d ln k⊥ 2 y 2.5 3 3.5 4 0 0.5 1 1.5 2 y 2.5 3 3.5 4 for a gluon jet (left) and for a quark jet (right) at fixed ℓ1 = 3.5, MLLA and naive approach. For both, the quark and gluon jets, new MLLA corrections arising from (4) push up the distribution when y1 increases as compared with the naive case. On the other hand, it is enhanced when y1 → 0 by the running of αs (k⊥ ) at k⊥ → ΛQCD . Transverse momentum inclusive k⊥ distribution Integrating (3) over the whole phase space in the logarithmic sense (ℓ1 = ln(1/x1 )) we obtain the inclusive k⊥ distribution inside a quark and a gluon jet[1]    Z YΘ −y1  0 dN d2 N dℓ1 = . (5) d ln k⊥ q,g dℓ1 d ln k⊥ q,g 0 We give results for YΘ0 = 7.5 in Fig.4 below and compare with the naive case. 3 YΘ = 7.5 YΘ = 7.5 0 0 (dN/dy)G 25 (dN/dy) 14 (dN/dy)Q 12 (dN/dy)naiveQ naive G 20 10 15 8 6 10 4 5 2 0 0 1 2 3 4 5 6 1 2 3 y Fig. 4: dN d ln k⊥ 4 5 6 y for a gluon jet, MLLA and naive approach with enlargement We observe in particular that the positivity of the distribution when y1 increases is restored as compared with the naive case. This stems from the global role of MLLA corrections in the range. YΘ = 15 0 100 MLLA (dN/dy) G (dN/dy)MLLAQ 80 60 40 20 0 0 Fig. 5: dN d ln k⊥ 2 4 6 y 8 10 12 for a gluon jet (blue) and for a quark jet (green) at YΘ0 = 15.0 In Fig.5 above, we represent the evidence of two competing effects at the unrealistic value YΘ0 = 15.0. One observes that as y1 → 0 the distribution is depleted by QCD coherence effects. Indeed, in this region of the phase space (k⊥ → Q0 ≈ ΛQCD ) gluons are pushed at larger angles, they are thus emitted independently from the rest of the partonic ensemble. At the opposite, when the value k⊥ ≈ Q0 ≈ ΛQCD is reached, the distribution is enhanced by the running of αs . Conclusion Results for the double 1-particle inclusive distribution and the inclusive k⊥ distribution at small x have been discussed and displayed. Sizable differences with the naive approach in which one forgets the evolution of the jet between its half opening angle Θ0 and the emission angle Θ have been found. The global role of new MLLA corrections is emphasized to recover the positivity of the inclusive k⊥ distribution. On the other hand, at realistic energy scales (LHC, Tevatron, LEP), QCD coherence effects are screened by the running of αs (see Fig.4) which, furthermore, forbids extending the confidence domain for y1 ≤ 1. The range of applicability of the soft approximation has been discussed from the analysis of corrections not to exceed ymax = YΘ0 − ℓmin = 5.0; it is indeed smaller at LEP and Tevatron energies (smaller YΘ0 ). Our results will be compared with forthcoming data from CDF. References [1] R. Perez-Ramos & B. Machet: “MLLA inclusive hadronic distributions inside one jet at high energy colliders”, hep-ph/0512236, JHEP 04 (2006) 043. [2] Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan and A.H. Mueller: Rev. Mod. Phys. 60 (1988) 373-388. 4