Hadronic single inclusive kt distributions inside one jet beyond MLLA

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1 Hadronic single inclusive kt distributions inside one jet beyond MLL François rleo, Redamy Perez Ramos, Bruno Machet To cite this version: François rleo, Redamy Perez Ramos, Bruno Machet. Hadronic single inclusive kt distributions inside one jet beyond MLL. Physical Review Letters, merican Physical Society, 28, 1, pp.522. <1.113/PhysRevLett.1.522>. <hal v2> HL Id: hal Submitted on 7 Jan 28 HL 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 HL, 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.

2 Hadronic single inclusive k distributions inside one jet beyond MLL François rleo, 1 Redamy Pérez-Ramos, 2 and Bruno Machet 3 1 CERN, PH department, TH division, CH-1211 Geneva 23 2 Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, D-885 München 3 Laboratoire de Physique Théorique et Hautes Énergies, BP 126, 4 place Jussieu, F Paris Cedex 5 (Dated: December 3rd 27 The hadronic k -spectrum inside one jet is determined including corrections of relative magnitude O ( α s with respect to the Modified Leading Logarithmic pproximation (MLL, at and beyond the limiting spectrum (assuming an infrared cut-off Q = Λ and Q Λ. The agreement between our results and preliminary measurements by the CDF collaboration is impressive, much better than at MLL, pointing out very small overall non-perturbative contributions. PCS numbers: Cy, a., Fh hal , version 2-7 Jan 28 Jet production a collimated bunch of hadrons in e + e, e p and hadronic collisions is an ideal playground for parton evolution in perturbative (p. One of the major successes of p is the hump-backed shape of inclusive spectra, predicted in [1] within MLL, and later discovered experimentally (see e.g. [2]. Refining the comparison of p calculations with data taken at LEP, Tevatron and LHC will ultimately allow for a crucial test of the Local Parton Hadron Duality (LPHD hypothesis [3] and for a better understanding of color neutralization processes. In this Letter, a class of next-tonext-to-leading logarithmic (NMLL corrections to the single inclusive k -distribution of hadrons inside one jet is determined. Unlike other NMLL corrections, these terms better account for recoil effects and were shown to drastically affect multiplicities and particle correlations in jets [4]. We start by writing the MLL( evolution / equations for the fragmentation function DB h x z, zeθ, Q of a parton B (energy ze and transverse momentum k = zeθ into a gluon (identified as a hadron h with energy xe according to LPHD inside a jet of energy E. s a consequence of angular ordering in parton cascading, partonic distributions inside a quark and gluon jet, Q, G(z = x / ( / z DQ,G h x z, zeθ, Q, obey the system of two coupled equations [5] (the subscript y denotes / y Q y = G y = dz α s dz α s π [ ( π Φg q (z Q(1 z Q ] + G(z, (1 [Φ gg (z(1 z ( G(z + G(1 z G +n f Φ q g (z (2Q(z G ], (2 On leave from Laboratoire d nnecy-le-vieux de Physique Théorique (LPTH, Université de Savoie, CNRS, B.P. 11, F nnecy-le-vieux Cedex, France LPTHE, UMR 7589 du CNRS associée à l Université P. et M. Curie - Paris 6 et à l Université D. Diderot - Paris 7 where Φ B (z denote the DGLP [6] splitting functions, α s = 2π / 4N c β (l + y + λ is the one-loop coupling constant of [13] and l = (1/x, y = ln ( k / Q, λ = ln ( Q /Λ, (Q being the collinear cut-off parameter, and where G G(1 = xd h G(x, EΘ, Q, Q Q(1 = xd h Q (x, EΘ, Q. t small x z, the fragmentation functions behave as ( B(z ρ h B ln z x, ln zeθ = ρ h B (lnz + l, y, Q ρ h B being a slowly varying function of two logarithmic variables ln(z/x and y that describes the humpbacked plateau [1]. In order to better account for recoil effects, the strategy followed in this Letter is to perform Taylor expansions (first advocated for in [7] of the nonsingular parts of the integrands in (1,2 in powers of ln z and ln(1 z, both considered small with respect to l in the hard splitting region z 1 z = O (1 B(z = B(1 + B l (1lnz + O ( ln 2 z ; z 1 z. (3 Each l-derivative giving an extra α s factor (see [5], the terms B l (1lnz and B l (1ln (1 z yield NMLL corrections to the solutions of (2. From (3 and the expressions of the DGLP splitting functions, one gets after some algebra (γ 2 = 2N c α s /π [8] l Q(l, y = δ(l + C y F dl dy γ 2 N (l + y (4 c [ ] 1 ã 1 δ(l l + ã 2 δ(l lψ l (l, y G(l, y, l y G(l, y = δ(l + dl dy γ(l 2 + y (5 [ ] 1 a 1 δ(l l + a 2 δ(l lψ l (l, y G(l, y.

3 2 with ψ l (l, y = G l (l, y/g(l, y. The MLL coefficients ã 1 = 3/4 and a are computed in [5] while at NMLL, we get [14]: ã 2 = C ( F 5 N c 8 π2.42, 6 (6 a 2 = π n f T R C F N c N c (7 Computing the NMLL partonic distributions inside a quark and gluon jet, Q(z and G(z, is the first step to determine the double differential spectrum d 2 N/dxdΘ of a hadron produced with energy xe and at angle Θ with respect to the jet axis identified with the direction of the energy flow (see [8]. s shown in [9], it is given by d 2 N dxdln Θ = d d ln Θ F h (x, Θ, E, Θ, (8 where F h is given by the convolution of two fragmentation functions F h ( x dud (u, EΘ, ueθd h x u, ueθ, Q, (9 u being the energy fraction of the intermediate parton. D describes the probability to emit with energy ue off the parton (which initiates the jet, taking into account the evolution of the jet between Θ and Θ. D h describes the probability to produce the hadron h off with energy fraction x/u and transverse momentum k ueθ Q (see Fig. 1. s it into Eq. (9 leads to xf h du u D (u, EΘ, ueθ D (l, h y ( [ 2 du u lnu D (u, EΘ, ueθ d D h (l, y dl ] d du u ln 2 ud 2 Dh (u, EΘ, ueθ (l, y dl 2. The first two terms in Eq. (11 correspond to the MLL distribution calculated in [9] when D h is evaluated at NLO and its derivative at LO. NMLL corrections arise from their respective calculation at NNLO and NLO, and, mainly in practice, from the third line, which is new. Indeed, since x/u is small, the inclusive spectrum D h (l, y is the solution of the next-to-mll evolution equations (4 and (5. However, because of the smallness of the coefficient a 2 (see (7, G(l, y shows no significant difference from MLL to NMLL. s a consequence, we use the MLL expression for G. It is determined here from a representation in terms of a single Mellin transform of confluent hypergeometric functions (see Eq. (24 of [1], well suited for numerical studies [15]. The NM- LL quark distribution Q(l, y can then be deduced from G(l, y using (4 and (5, which yields Q(l, y = C [ F G(l, y + (a 1 ã 1 G l (l, y (12 N c ] + (a 1 (a 1 ã 1 + ã 2 a 2 G ll (l, y + O(γ 2. The functions F h g and F h q are related to the gluon distribution via the color currents C g,q defined as: Θ Θ D E ue h D xe h (Jet xis xf h g,q = C g,q N c G(l, y. (13 C g,q can be seen as the average color charge carried by the parton due to the DGLP evolution from to. Introducing the first and second logarithmic derivatives of D h, FIG. 1: Inclusive production of hadron h at angle Θ inside a high energy jet of total opening angle Θ and energy E. discussed in [9], the convolution (9 is dominated by u 1 and therefore D (u, EΘ, ueθ is given by DGLP evolution [6]. On the contrary, the distribution D h x u Dh ( x u, ueθ, Q = Dh (l+lnu, y at low x u reduces to the hump-backed plateau, D h x u (l + lnu, y ρ h (l + lnu, Y Θ + lnu, (1 with Y Θ = l + y = lneθ/q. Performing the Taylor expansion of D to the second order in (lnu and plugging ψ,l (l, y = (ψ 2,l + ψ,ll (l, y = 1 d D (l, y D h(l, y = O( α s, dl 1 d 2 D (l, y D h(l, y dl 2 = O(α s, Eq. (11 can now be written as xf h [ u + u lnu ψ,l (l, y with the notation u ln i u u ln2 u (ψ 2,l + ψ,ll(l, y] Dh, (14 du (u ln i u D (u, EΘ, ueθ

4 3 C Q LO MLL NMLL Y = 6.4 l = y δ C Q NMLL-MLL C Q MLL.6 Y = l = 2.2 λ =. -.2 λ =.2 λ = λ = y FIG. 2: The color current of a quark jet with Y Θ = 6.4 as a function of y at fixed l = 2. FIG. 3: NMLL corrections to the color current of a quark jet with Y Θ = 6.4 and l = 2 for various values of λ. du (u ln i u D (u, EΘ, EΘ. (15 The scaling violation of the DGLP fragmentation function neglected in the last approximation is a O(α s correction to u. It however never exceeds 5% [8] of the leading term and is thus neglected in the following. Using (13, the MLL and NMLL contributions to the leading color current of the parton = g, q read δ C MLL LO = N c u lnu g ψ g,l + C F u lnu q ψ q,l, δ C NMLL MLL = N c u ln 2 u g (ψ 2 g,l + ψ g,ll + C F u ln 2 u q (ψ 2 q,l + ψ q,ll. (16 The MLL correction, O ( αs, was determined in [9] and the NMLL contribution, O (α s, to the average color current is new. The latter can be obtained from the Mellin moments of the DGLP fragmentation functions D (j, ξ = du u j 1 D (u, ξ, leading to u ln 2 u = d2 dj 2 D (j, ξ(eθ ξ(eθ. (17 j=2 Plugging (17 into (16, the NMLL color currents for gluon and quark jets are determined analytically [8]. For illustrative purposes, the LO, MLL, and NMLL average color current of a quark jet with Y Θ = 6.4 corresponding roughly to Tevatron energies is plotted in Fig. 2 as a function of y, at fixed l = 2. s discussed in [9], the MLL corrections to the LO color current are found to be large and negative. s expected, the correction O (α s from MLL to NMLL proves much smaller; it is negative (positive at small (large y. This calculation has also been extended beyond the limiting spectrum, λ, to take into account hadronization effects in the production of massive hadrons, m = O (Q [1]. The NMLL (normalized corrections to the MLL result are displayed in Fig. 3 for different values λ =,.5, 1. It clearly indicates that the larger λ, the smaller the NMLL corrections. In particular, they can be as large as 3% at the limiting spectrum (λ = but no more than 1% for λ =.5. This is not surprising since λ (Q Λ reduces the parton emission in the infrared sector and, thus, higher-order corrections. The double differential spectrum d 2 N/dy dl, Eq. (8, can now be determined from the NMLL color currents (16 using the MLL quark and gluon distributions Integrating it over l leads to the single inclusive y-distribution (or k -distribution of hadrons inside a quark or a gluon jet: ( dn dy g,q = ( k dn dk g,q = YΘ y l min dl ( d 2 N. dl dy g,q (18 The MLL framework does not specify down to which values of l (up to which values of x the double differential spectrum d 2 N/dy dl should be integrated over. Since d 2 N/dy dl becomes negative (non-physical at small values of l (see e.g. [9], we chose the lower bound l min so as to guarantee the positiveness of d 2 N/dy dl over the whole l min l Y Θ range (in practice, l g min 1 and l q min 2. Having successfully computed the single k -spectra including NMLL corrections, we now compare the result with existing data. The CDF collaboration at the Tevatron recently reported on preliminary measurements over a wide range of jet hardness, Q = EΘ, in p p collisions at s = 1.96 TeV [12]. CDF data, including systematic errors, are plotted in Fig. 4 together with the MLL predictions of [9] and the present NMLL calculations, both at the limiting spectrum (λ = and taking Λ = 25 MeV; the experimental distributions suffering from large normalization errors, data and theory are normalized to the same bin, ln(k /1 GeV =.1. The agreement between the CDF results and the NMLL dis-

5 4 Q=155 GeV NMLL MLL normalized to bin: ln(k =-.1 CDF preliminary Q=119 GeV 1 normalized to bin: ln(k =-.1 (N CDF preliminary 1/N dn / d ln k Q=9 GeV Q=5 GeV Q=68 GeV Q=37 GeV 1/N dn / d ln k NMLL Q = 119 GeV Λ = 25 MeV λ = λ =.5 λ = 1 Q=27 GeV Q=19 GeV ln (k / 1GeV FIG. 5: CDF preliminary results (Q = 119 GeV for inclusive k distribution compared with NMLL predictions beyond the limiting spectrum. 1 2 ln (k / 1GeV ln (k / 1GeV FIG. 4: CDF preliminary results for the inclusive k distribution at various hardness Q in comparison to MLL and NMLL predictions at the limiting spectrum; the boxes are the systematic errors (their lower limits at large k are cut for the sake of clarity. tributions over the whole k -range is particularly good. In contrast, the MLL predictions prove reliable in a much smaller k interval. t fixed jet hardness (and thus Y Θ, NMLL calculations prove accordingly trustable in a much larger x interval. Despite this encouraging agreement with data, the present calculation still suffers from various theoretical uncertainties, discussed in detail in [8]. mong them, the variation of Λ giving NMLL corrections from the default value Λ = 25 MeV to 15 MeV and 4 MeV affects the normalized k -distributions by roughly 2% in the largest ln(k /1 GeV = 3 GeV-bin at Q = 1 GeV. lso, cutting the integral (18 at small values of l is somewhat arbitrary. However, we checked that changing l g min from 1 to 1.5 modifies the NMLL spectra at large k by 2% only [16]. Finally, the k -distribution is determined with respect to the jet energy flow from 2-particle correlations (which includes a summation over secondary hadrons, while experimentally the jet axis is determined exclusively from all particles inside the jet. The question of the matching of these two definitions at O (α s accuracy goes beyond the scope of this Letter. The NMLL k -spectrum has also been calculated beyond the limiting spectrum, as illustrated in Fig. 5. However, the best description of CDF preliminary data is reached at the limiting spectrum, or at least for small values of λ.5, which is not too surprising since these inclusive measurements mostly involve pions. Identifying produced hadrons would offer the interesting possibility to check a dependence of the shape of k -distributions on the hadron species, such as the one predicted in Fig. 5. To summarize, single inclusive k -spectra inside a jet are determined including higher-order O (α s (i.e. NM- LL corrections from the Taylor expansion of the MLL evolution equations and beyond the limiting spectrum, λ. The agreement between NMLL predictions and CDF preliminary data in p p collisions at the Tevatron is very good, indicating very small overall non-perturbative corrections. The MLL evolution equations for inclusive enough variables prove once more (see e.g. [6] to include reliable information at a higher precision than the one at which they have been deduced. cknowledgments: We gratefully acknowledge enlightening discussions with Yu.L. Dokshitzer, I.M. Dremin, S. Jindariani (CDF, W. Ochs and M. Rubin. [1] Yu.L. Dokshitzer, V.S. Fadin, V.. Khoze, Phys. Lett. B 115 ( ; Ya.I. zimov, Yu.L. Dokshitzer, V.. Khoze, S.I. Troian, Z. Phys. C 31 ( ; C.P. Fong, B.R. Webber, Phys. Lett. B 229 ( [2] V.. Khoze, W. Ochs, Int. J. Mod. Phys. 12 ( , and references therein. [3] Ya.I. zimov, Yu.L. Dokshitzer, V.. Khoze, S.I. Troian, Z. Phys C 27 ( ; Yu.L. Dokshitzer, V.. Khoze, S.I. Troian, J. Phys. G 17 ( [4] F. Cuypers and K. Tesima, Z. Phys. C 54 ( ; Yu.L. Dokshitzer, Phys. Lett. B 35 ( [5] R. Pérez-Ramos, JHEP 6 ( [6] see for example: Yu.L. Dokshitzer, V.. Khoze,.H. Mueller, S.I. Troyan, Basics of Perturbative, Ed.

6 5 Frontières, Gif-sur-Yvette, 1991, and references therein. [7] I.M. Dremin, Phys. Lett. B 313 ( [8] F. rleo, R. Pérez-Ramos, B. Machet, to appear. [9] R. Pérez-Ramos, B. Machet, JHEP 4 ( [1] Yu.L. Dokshitzer, V.. Khoze, S.I. Troian, Int. J. Mod. Phys. 7 ( [11] R. Pérez-Ramos, JHEP 9 ( [12] S. Jindariani,. Korytov,. Pronko, CDF report CDF/NL/JET/PUBLIC/846 (March 27. [13] 2-loop evaluation of the splitting functions and α s would not fit into the present logic of a systematic expansion in powers of α s (see [8]. [14] ssuming Q/G = C F/N c. We checked that O ( α s and O (α s corrections affect marginally these coefficients. [15] It was also given in [5] a compact Mellin representation from which an analytic approximated expression was found using the steepest descent method [11]. [16] The effect of varying l min is more dramatic at MLL.