Energy evolution of the soft parton-to-hadron fragmentation functions

Transcription

1 α s from soft parton-to-hadron fragmentation functions David d Enterria and Redamy Pérez-Ramos,, CERN, PH Department, CH- Geneva, Switzerland Sorbonne Universités, UPMC Univ Paris, UMR 789, LPTHE, F-7, Paris, France CNRS, UMR 789, LPTHE, BP, place Jussieu, F-7 Paris Cedex, France Abstract: The QCD coupling α s is extracted at approximate next-to-next-to-leading-order (NNLO ) accuracy from the energy evolution of the first two moments (multiplicity and mean) of the partonto-hadron fragmentation functions at low fractional hadron momentum z. Comparisons of the experimental e + e and DIS e ± p jet data to our NNLO +NNLL predictions, allow us to obtain α s (m ) =.±.+.., in excellent agreement with the current world average. Introduction For massless quarks and fixed number of colours N c, the only fundamental parameter of quantum romodynamics (QCD), the theory of the strong interaction, is its coupling constant α s. Starting from a value of Λ QCD. GeV, where the perturbatively-defined coupling diverges, α s decreases with increasing energy Q following a / ln(q /Λ ) dependence. The current uncertainty on α QCD s evaluated at the mass, α s (m ) =.8±., is ±.% [], making of the strong coupling the least precisely known of all fundamental constants in nature. Improving our knowledge of α s is a prerequisite to reduce the uncertainties in perturbative QCD calculations of all partonic cross sections at hadron colliders, notably in Higgs physics [], and for precision fits of the Standard Model. It has also far-reaing implications including the stability of the electroweak vacuum [] or the scale at whi the interaction couplings unify. Having at hand new independent approaes to determine α s from the data, with experimental and theoretical uncertainties different from those of the other methods currently used, is crucial to reduce the overall uncertainty in the combined α s world-average value []. In Refs. [,] we have presented a novel tenique to extract α s from the energy evolution of the moments of the partonto-hadron fragmentation functions (FF) computed at approximate next-to-leading-order (NLO ) accuracy including next-to-next-to-leading-log (NNLL) resummation corrections. This approa has been extended to include full NLO plus a set of NNLO corrections []. Our new NNLO +NNLL theoretical results for the energy dependence of the hadron multiplicity and mean value of the FF are compared to jet fragmentation measurements in e + e and deep-inelastic e ± p collisions, and a high-precision value of α s is extracted. Energy evolution of the soft parton-to-hadron fragmentation functions The distribution of hadrons in a jet is encoded in a fragmentation function, D i h (z, Q), whi describes the probability that parton i fragments into hadron h carrying a fraction z = p hadron /p parton of the parent parton momentum. Usually one writes the FF as a function of the log of the inverse of z, ξ = ln(/z), emphasizing the region of soft momenta that dominates the jet hadronic fragments. Indeed, due to colour coherence and gluon radiation interference, not the softest partons but those with intermediate energies multiply most ectively in QCD cascades, leading to a final FF with

2 a typical hump-backed plateau (HBP) shape as a function of ξ (Fig. ), whi can be expressed in terms of a distorted Gaussian (DG): D(ξ, Y, λ) = N /(σ π) e [ 8 k sδ (+k)δ + sδ + kδ ], with δ = (ξ ξ)/σ, () where N is the average hadron multiplicity inside a jet, and ξ, σ, s, and k are respectively the mean peak position, dispersion, skewness, and kurtosis of the distribution. /σ dσ/dξ s =. GeV [BES] DG fits to e+e- jet hadron data s=. GeV [BES] s=. GeV [BES] 9 (Limiting spectrum Q =Λ s=. GeV [BES] QCD ; m = MeV) s. GeV [BES] s =.8 GeV [BES] s=. GeV [MARK-II] 8 s=. GeV [MARK-II] s. GeV [BaBar] s = GeV [TASSO] s= GeV [TASSO] 7 s= GeV [TASSO] s= GeV [TASSO] s= 9 GeV [MARK-II] s 9 GeV [TPC] s = 9 GeV [HRS] s= 8 GeV [TOPA] s= 9. GeV [OPAL] s= 9. GeV [L] s= 9. GeV [OPAL] s 9. GeV [ALEPH] s = GeV [DELPHI] s= GeV [OPAL] s= GeV [ALEPH] s= GeV [ALEPH] s= GeV [OPAL] s GeV [ALEPH] s = 7 GeV [OPAL] s= 7 GeV [ALEPH] s 8 GeV [OPAL] s = 8 GeV [ALEPH] s= 89 GeV [OPAL] s= 89 GeV [ALEPH] s= 9 GeV [ALEPH] s= GeV [ALEPH] s= GeV [OPAL] s GeV [ALEPH] ξ = ln(/x) /σ dσ/dξ 7 DG fit to DIS (Breit frame) jet hadron data s=.8 GeV [EUS'9] (Limiting spectrum Q =Λ QCD ; m = MeV) s=. GeV [EUS'9] s= 7. GeV [EUS'9] s=. GeV [EUS'9] s=. GeV [EUS'9] s=. GeV [EUS'9] s= 9. GeV [EUS'9] s=. GeV [EUS'] s=. GeV [EUS'] s=. GeV [EUS'] s=. GeV [EUS'] s=. GeV [EUS'] s= 8. GeV [EUS'] s=. GeV [EUS'] s= 7.7 GeV [EUS'] ξ = ln(/x) Figure : Charged-hadron spectra in jets as a function of ξ = ln(/z) in e + e at s GeV (left), and e ±, ν-p (Breit frame, scaled for the full hemisphere) at s 8 GeV (right), individually fitted to Eq. () with the hadron mass corrections (m =, MeV) quoted. Starting with a parton at a given energy Q, its evolution to another energy scale Q is driven by a braning process of parton radiation and splitting, resulting in a jet shower, whi can be computed perturbatively using the DGLAP [7] equations at large z., and the Modified Leading Logarithmic Approximation (MLLA) [8], resumming soft and collinear singularities at small z. As for the Srödinger equation in quantum meanics, the system of equations for the FFs D i h (z, Q) can be written as an evolution Hamiltonian whi mixes gluon and (anti)quark states expressed in terms of DGLAP splitting functions for the branings g gg, q(q) gq(q) and g qq, where g, q and q label a gluon, a quark and an anti-quark respectively. Analytical solutions can only be obtained from the Mellin transform of the full-resummed regularized NNLL splitting functions [9]. The set of integro-differential equations for the FF evolution combining hard (DGLAP, MLLA, nextto-mlla) and soft (DLA) radiation can be solved by expressing the[ Mellin-transformed hadron ] t distribution in terms of the anomalous dimension γ: D C(α s (t)) exp γ(α s (t ))dt for t = ln Q, leading to a perturbative expansion in half powers of α s : γ O(α / s ) + O(α s ) + O(α / s ) + O(αs) + O(α / s )+. The anomalous dimension γ allows one to calculate the moments of the DG through: N = K, ξ = K, σ = K, s = K σ, k = K Y ( σ ; with K n (Y, λ) = dy ) n γ ω, () ω ω= whi are then inserted into Eq. (). Corrections of γ up to order α / s were computed in Refs. [,], followed by the full set of NLO O(α s ) terms, including the two-loop splitting functions, in Ref. [].

3 At NLO, the diagonalisation of the evolution Hamiltonian results in two eigenvalues γ ±± in the D ± basis, where the relevant one for the calculation of the FF moments γ ++ γω NLO+NNLL, reads: γω NLO+NNLL = [ ω(s ) + γ N c a ( + s ) + β ] ( s ) + γ Nc (ωs) [a ( s ) + 8a β ( s ) + β( s )( + s ) ] β N c ln (Y + λ) β + ] γ ω [a ( + s + s) + a (s ) a ( s ) a ( s ) a, () where γ = Ncαs π = Nc β (Y +λ) is the LL anomalous dimension, s = + γ, β ω i the QCD β-function coicients, a, and hard constants obtained in [], and a,,, are new constants obtained from the full-resummed NNLL splitting functions [9]. In addition, a fraction of the O(α / s ) terms from the NNLO α s running expression, have been now added []. Upon inverse-mellin transformation, one obtains the energy evolution of the FF, and its associated moments, at NNLO +NNLL accuracy as a function of Y = ln(e/λ QCD ), for an initial parton energy E, down to a shower cut-off scale λ = ln(q /Λ QCD ) for N f =,, quark flavors. The resulting formulae for the energy evolution of the moments depend on Λ QCD as single free parameter. Relatively simple expressions are obtained in the limiting-spectrum case (λ =, i.e. evolving the FF down to Q = Λ QCD ) motivated by the local parton hadron duality hypothesis for infrared-safe observables whi states that the HBP distribution of partons in jets is simply renormalized in the hadronization process without anging its shape. Thus, by fitting the experimental hadron jet data at various energies to Eq. (), one can determine α s from the corresponding energy-dependence of its FF moments. multiplicity jet N LO*+LL NLO*+NNLL NLO+NNLL E jet (GeV) jet FF(ξ) peak position..... LO*+LL NLO*+NNLL NLO+NNLL E jet (GeV) Figure : Comparison of theoretical predictions at increasing level of accuracy (LO to NNLO ) for the energy evolution of the jet arged-hadron multiplicity (left) and FF peak position (right). Figure shows the energy evolution of the zeroth (multiplicity) and first (peak position, closely connected to the mean of the distribution) moments of the FF, at four levels of accuracy (LO +LL, NLO +NLL, NLO+NNLL, and NNLO +NNLL). The hadron multiplicity and FF peak increase exponentially and logarithmically with energy, and the theoretical convergence of their evolutions are very robust as proven by the small anges introduced by incorporating higher-order terms. 7

4 Data-theory comparison and α s extraction The first step of our procedure is to fit all existing jet FF data measured in e + e and e ±, ν- p collisions at s GeV (Fig. ) to Eq. (), in order to obtain the corresponding FF moments at ea jet energy. Finite hadron-mass ects in the DG fit are accounted for through a rescaling of the theoretical (massless) parton momenta with an ective mass m as discussed in Refs. [,]. The overall normalization of the HBP spectrum (K ), whi determines the average arged-hadron multiplicity of the jet, is an extra free parameter in the DG fit whi, nonetheless, plays no role in the final Λ QCD value given that its extraction just depends on the evolution of the multiplicity, and not on its absolute value at any given energy. multiplicity jet N Multiplicity DG limiting-spectrum (m = MeV) World e e and DIS jet data α s (m )=.7 ±. K =. ±. χ /ndf = 9./ jet FF(ξ) peak position Max. peak DG limiting-spectrum (m = MeV) World e e and DIS jet data α s (m )=. ±. χ /ndf = 8.7/8.. s e+e-,q (GeV) DIS.8 s e+e-,q (GeV) DIS Figure : Energy evolution of the jet arged-hadron multiplicity (left) and FF peak position (right) in e + e and DIS jet data, fitted to the NNLO +NNLL predictions. The obtained K normalization constant, the individual NNLO α s (m ) values, and χ /ndf of the two fits, are quoted. Once the FF moments have been obtained, we perform a combined fit of them as a function of the original parton energy (i.e. s/ in the case of e + e, and the invariant -momentum transfer Q DIS for DIS). The individual fits for the first two FF moments are shown in Fig.. The NNLO +NNLL limiting-spectrum (λ = ) predictions for N f = active quark flavours, leaving Λ QCD as a free parameter, reproduce very well the data. The most robust FF moment for the determination of Λ QCD is the peak position ξ max whi proves quite insensitive to most of the uncertainties associated with the extraction method [] as well as to higher-order corrections. The hadron multiplicities measured in DIS jets appear somewhat smaller (especially at high energy) than those measured in e + e, due to limitations in the FF measurement only in half (current Breit) e ± p hemisphere and/or in the determination of the relevant Q scale []. The value of α s (m ) obtained from the combined multiplicity+peak fit yields α s (m ) =.±., where the error includes all uncertainties discussed in Ref. []. A conservative theoretical scale uncertainty of +.. (obtained in [] at NLO accuracy only) is added. In Fig. we compare our α s (m ) value to all other NNLO results from the PDG compilation [], plus that obtained from the π-decay factor [], and the top-quark pair cross sections at the LHC []. The precision of our result ( +% % ) is clearly competitive with the other measurements, with a totally different set of experimental and theoretical uncertainties. A simple weighted average of all these NNLO values yields: α s (m ) =.8 ±., in perfect Few-% corrections are applied to deal with slightly different N f =, evolutions below arm,bottom thresholds. 8

5 agreement with the world-average, α s (m ) =.8±., but with a % smaller uncertainty. Upcoming full-nnlo corrections of the energy evolution of the FF moments will allow the inclusion of our α s result into the PDG world-average. World NNLO average: α s =.8 ±. e e,dis jet FFs () Pion decay factor (NNLO) tt cross sections CMS (NNLO) Lattice QCD "data" (NNLO) τ hadronic decays (NNLO) e e : evt shapes/thrust/jets x-sections (NNLO) DIS PDFs (NNLO) α S (m ) Figure : Summary of α s determinations using different methods at NNLO ( ) accuracy. The dashed line and shaded (yellow) band indicate the world-average and uncertainty (listed on top) []. References [] K. A. Olive et al. [Particle Data Group Collab.], Chin. Phys. C 8 () 9. [] L. Mihaila, these proceedings, p.. [] D. Buttazzo et al., JHEP () 89. [] R. Pérez-Ramos, D. d Enterria, JHEP 8 () 8; and Proceeds. Moriond QCD, pp. ; arxiv:8.8 [hep-ph]. [] D. d Enterria, R. Pérez-Ramos, Nucl. Phys. B Proc. Suppl., arxiv:.88 [hep-ph]; and EPJ Web Conf. 9 () ; arxiv:.. [] D. d Enterria, R. Pérez-Ramos, Proceeds. Moriond QCD ; arxiv:. [hep-ph]. [7] V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. (97) 8; G. Altarelli, G. Parisi, Nucl. Phys. B (977) 98; Y.L. Dokshitzer, Sov. Phys. JETP (977). [8] Y.L. Dokshitzer, V.A. Khoze and S. Troian, Int. J. Mod. Phys. A7 (99) 87. [9] C.-H. Kom, A. Vogt and K. Yeats, JHEP (). [] R. Pérez-Ramos and D. d Enterria, in preparation. [] J. L. Kneur, A. Neveu, Phys. Rev. D 88 () 7, 7. [] S. Chatryan et al. [CMS Collab.], Phys. Lett. B 78 () 9. 9