Uncertainties on the determination of the strong coupling alpha_s from the energy evolution of jet fragmentation functions at low z

The QCD coupling alpha_s is determined at NLO*+NMLLA accuracy from the comparison of experimental jet data to theoretical predictions of the energy-evolution of the parton-to-hadron fragmentation function moments (multiplicity, peak, width, skewness) at low fractional hadron momentum z. From the existing e+e- and e-p jet data, we obtain alpha_s(m_Z^2) = 0.1195 +/- 0.0021 (exp.) {+0.0015}_{-0.0} (scale) at the Z mass. The uncertainties of the extracted alpha_s value are discussed.


Introduction
In the chiral (massless quark) limit, the theory of the strong interaction -quantum chromodynamics (QCD)has a single fundamental parameter: its coupling α s which decreases logarithmically with increasing energy scale Q, i.e. α s ∝ ln(Q 2 /Λ 2 QCD ), starting from a value Λ QCD ≈ 0.2 GeV where the perturbatively-defined coupling diverges. The running coupling α s enters in the perturbative expansion of theoretical cross section for any hard process involving quarks and gluons, and the uncertainty in its value is a key (sometimes dominant) component of the theoretical error in all perturbative QCD (pQCD) predictions for any collision process involving hadrons. Many precision fits of the Standard Model as well as searches for new physics depend on α s , whose value needs, thus, to be determined with good accuracy and precision.
The determination of α s relies on the comparison of theoretical predictions for various observables, obtained perturbatively at a given level of accuracy (next-to-nextto-leading order, NNLO, in most cases), with the corresponding experimental measurements. The extracted α s values at different energy scales are then compared to each other by translating the result in terms of the value at the Z mass pole: α s (m 2 Z ). The current value of α s (m 2 Z ) = 0.1185±0.0006, has been obtained from a combination of measurements at e + e − , deep-inelastic scattering (DIS) e,ν-p, and hadron-hadron colliders [1]: (i) hadronic τ decays (N 3 LO), (ii) hadronic W,Z decays (N 3 LO), (iii) radiative heavy-quarkonia decays (NLO), and (iv) event shapes and jet rates (NNLO), in e + e − collisions; (v) scaling violations in parton distribution functions (NNLO), and jet cross sections (NLO), in e,ν-p DIS; (vi) jet cross sections and angular correlations (NLO), and top-quark cross sections (NNLO), in p-p,p-p collisions.
In addition, comparisons of the predictions for different short-distance observables computed through lattice and pQCD methods provide extra "data points" for the determination of the α s (m 2 Z ) world-average, although the low systematic uncertainty assigned to such an approach has been questioned [2]. The current α s uncertainty is of order ±0.5%, although a more conservative estimate sets it at the ±1% level [2], making arXiv:1410.4818v1 [hep-ph] 17 Oct 2014 of α s the least precisely known of all fundamental couplings in nature. In this context, having at hand extra independent approaches to determine α s -with experimental and theoretical uncertainties different than those of the methods currently used-would be an obvious asset.
Attempts were carried out in the past (see e.g. [3]) to try to measure Λ QCD (or, equivalently, α s ) using partonto-hadron fragmentation functions (FFs) in the region of low fractional momenta z = p h /p parton via the Modified Leading Logarithmic Approximation (MLLA) [4] which resums the soft and collinear singularities present in this region of phase-space. However, these older calculations were limited to LO approximations with a number of simplifying assumptions: (i) ad hoc cuts in the experimental distributions, (ii) simple fits in a restricted FF range, (iii) number of quark-flavours fixed to N f = 3, and (iv) use of only one or two FF moments (Gaussian approximation). As a result, very imprecise (or even inconsistent) values of Λ QCD ≈ 80-600 MeV were obtained. In Ref. [5] we have presented a novel extraction of α s from the energy evolution of the first four moments of the parton-to-hadron FFs including, for the first time, resummations of higher-order (NNLL or NMLLA) logarithms and NLO running-coupling corrections. The approach has been successfully tested with the world experimental jet data from e + e − annihilation [5] and deep-inelastic e ± p collisions [6]. We provide here a more detailed assessment of the systematic uncertainties involved in our α s extraction.

NLO*+NMLLA parton-to-hadron fragmentation functions
The parton-to-hadron FF, D i→h (z, Q), encodes the probability that parton i fragments into a hadron h carrying a fraction z of the parent parton's momentum. Due to colour coherence and gluon-radiation interference inside a parton shower, not the softest partons but those with intermediate energies (E h ∝ E 0.3 jet ) multiply most effectively in QCD cascades, leading to a single-particle spectrum with a typical "hump-backed plateau" (HBP) shape as a function of ξ = ln(1/z) (Fig. 1). Such an HBP shape can be parametrized, without any loss of generality, as a distorted Gaussian (DG) which depends on the original energy of the parton, Y = ln Eθ/Q 0 , evolved down to a shower cut-off scale λ = ln(Q 0 /Λ QCD ): where δ = (ξ −ξ)/σ, with moments: N (hadron multiplicity inside the jet),ξ (DG peak position), σ (DG width), s (DG skewness), and k (DG kurtosis). Fig. 1 shows a typical example of FFs fitted to Eq. (1) with the corresponding extracted moments. Since the measured final-state hadrons are massive and the theoretical calculations assume massless partons/hadrons (i.e. ξ p = ξ E ), the expression (1) for ξ needs to be modified by introducing an effective mass, via E = p 2 + m eff 2 , with the corresponding Jacobian determinant correction. Such a change of variable only affects the DG fit for the softest hadrons (i.e. for the highest ξ values) [5].  The process of parton radiation and splitting occurring in a jet shower, i.e. the evolution of the FF from a scale Q to Q' before the final parton-to-hadron transition, is governed by the DGLAP [8] equations (at large z 0.1, i.e. low ξ 2) and by MLLA approaches (at small z, i.e. large ξ 2). In Ref. [5] we have solved the set of integro-differential equations for the FF evolution including 1 NLO* α s and next-to-MLLA corrections, by expressing the Mellin-transformed hadron distribution in terms of the γ anomalous dimension:   HBP moments which depend only on Λ QCD . Such an approach is justified for infrared-safe observables by the "local parton-hadron duality" hypothesis which states that the distribution of partons in jets is identical to that of the final hadrons (up to a constant K ch which only affects the absolute normalization of the HBP distribution). Thus, by fitting the experimental hadron jet data at various energies to the DG parametrization (1), one can determine α s from the corresponding energydependence of its fitted moments. Figure 2 shows the DG fits of the FFs measured in e + e − (32 datasets, left plot) [5] and DIS (15 datasets, right plot) [6]. We note that the DIS FFs need to be scaled up by a factor of 2 since only half of the total hadron production (in the current hemisphere of the Breit frame) is measured. For increasing jet energies the FF peak shifts to higher ξ, and its width broadens.

Experimental uncertainties on α s
The extraction of the FF moments in the DG fit has two possible sources of uncertainty: (i) the choice of the m eff factor to account for finite hadron-mass corrections, and (ii) the use of experimental FFs with slightly different final charged-hadron definitions (the different experiments require tracks coming from within 1-10 cm from the primary interaction vertex, including or not a fraction of secondary hadrons from weak K 0 s and Λ decays). To estimate both effects we have taken the recent high-quality prompt and inclusive pion/kaon/proton FFs measured by the BaBar experiment [7], and fitted their total (summed) distribution to DGs with varying values of m eff . The resulting fitted moments for m eff = 0-0.2 GeV are listed in Fig. 1. In addition, since some older FF studies extracted the peak position and width of the single-hadron distributions assuming a pure Gaussian shape, we list also the extracted values of ξ max and σ in the Gaussian approximation in the top-right legend of Fig. 1. The main findings are as follows: (ii) The DG moments obtained for the prompt and inclusive charged-hadrons FFs are all consistent within their associated uncertainties except, as expected, for the total multiplicity N ch which is a factor of 8-10% smaller for the primary-hadrons FFs.
(iii) The obtained FF maximum ξ max and width σ assuming a Gaussian distribution around the peak, overestimate their values by 6% and 20% respectively compared to the more realistic DG fits.
The conclusion is that our FF analysis is robust with respect to hadronization corrections, and that only the extracted value of N ch needs to be reduced by a factor of (9±1)% if the experimental distributions have not been corrected for secondary weak decays. In addition, the FF peaks and widths extracted in Gaussian fits in the oldest data analyses overestimate these parameters by 6% and 20% respectively compared to the DG fits.
From the parton-to-hadron FF moments obtained for jets of various energies, one can extract α s by fitting  their combined evolutions to our NLO*+NMLLA predictions, which include also corrections to account for the increasing number of flavours, N f = 3,4,5, at the corresponding heavy-quark production thresholds: E jet > m charm,bottom ≈ 1.3,4.2 GeV. The energy dependence of the multiplicity, peak, width and skewness of the jet FFs are shown in Fig. 3 for e + e − annihilation at √ s ≈ 2-200 GeV [5] (top panels), and for DIS e ± p and ν p collisions in the range of four-momentum transfers Q ≈ 4-180 GeV [6] (lower panels). We note that the DIS plot includes more moments (published in previous studies) than those obtained directly fitting the data of Fig. 2, right. The hadron multiplicities measured in DIS jets (scaled by a factor of two to cover both hemispheres) are smaller (especially at high energy) than those measured in e + e − collisions [5], a fact pointing likely to limitations in the FF measurement only in half (current Breit) e ± p hemisphere and/or in the determination of the relevant Q kinematic variable. The DIS HBP widths show also larger scatter than observed in e + e − , which is not unexpected as we are including moments fitted in the Gaussian approximation which overestimates their value by about 20% as discussed in the previous section. The skewness has the largest uncertainties of all moments, and the kurtosis (not shown) is very close to zero and not properly reproduced by the calculations [5]. The most robust FF moment for the determination of Λ QCD is the peak position ξ max which has the simplest theoretical expression for its NLO*+NMLLA energy evolution [5], and it is largely insensitive to the uncertainties (finite-mass corrections, secondary decays, DG fit) associated with our method.
The fits of the energy-dependencies of the HBP moments allow one to extract QCD coupling values: α s (m 2 Z ) = 0.1195 ± 0.0022 (from the e + e − data) [5], and α s (m 2 Z ) = 0.119 ± 0.010 (from DIS jets alone) [6], in perfect agreement with the current world average. The quoted propagated α s (m 2 Z ) uncertainties have been obtained following the "χ 2 averaging" method [1]. Namely, we fit first the energy-dependence for each individual moment to its corresponding theoretical prediction and if the goodness-of-fit χ 2 is larger than the number of degrees of freedom (ndof), all the data-points of the moment are enlarged by a common factor such that χ 2 /ndof equals unity. We then perform the combined fit of all the energy-evolutions of the four moments letting Λ QCD as the single free parameter. Such an approach takes into account in a well defined manner any correlations among the four extractions of α s , as well as any missing extra systematic uncertainties. Our discussion of the experimental uncertainties indicates that such an error assignment covers perfectly well the range of α s variations induced by finite hadron-mass corrections, experimental final-hadron definitions, and DG fits.

Theoretical uncertainties on α s
The advantages of our α s determination compared to all other methods used so far are two-fold: (i) Our approach relies on a theoretically convergent resummation of soft and collinear logs down to Q 0 → Λ QCD and, thus, it is much more robust with respect to corrections due to the transition from partons to hadrons. Hadronization uncertainties are (much) smaller and limited to the choice of the m eff factor accounting for finite-hadron mass effects as discussed in the previous Section.
(ii) Our expansion in half-powers of α s includes almost the full set of α s NLO corrections plus NNLL terms. The latter log corrections partially account for contributions from missing (NNLO) terms and, in this sense, our approach is more complete than existing α s extractions at NLO accuracy. Notwithstanding this last item, the state-of-the-art α s determinations are at NNLO accuracy [1] whereas our method is NLO*+NNLL. In order to estimate the size of theoretical uncertainties associated with missing fixed-order terms in our truncation of the α s expansion, we have redone the fit of the moments versus jet energy for a different value of the scale λ = ln(Q 0 /Λ QCD ) at which the parton evolution is stopped. We vary λ from the default limiting-spectrum value (λ = 0, i.e. Q 0 = Λ QCD ) to λ = 1.4 (i.e. Q 0 = 4 Λ QCD ≈ 1 GeV) and fit the experimental data for jets energies in the range Q = 10-200 GeV, so as to leave some room for parton QCD evolution, i.e. avoiding jet data with energies too close to the shower cutoff at 1 GeV. In the relevant range of experimental jet energies E jet max ≈ 2,10-200 GeV, the average scale at which α s is effectively evaluated in our approach is given by the geometric mean: Q = E jet max × Q 0 ≈ 0.6-2 GeV (for the limiting spectrum) and Q = 3-14 GeV (for the λ = 1.4 case). The results of the fits for the e + e − data are shown in Fig. 4. We obtain α s (m 2 Z ) = 0.1210 ± 0.0025, which is consistent with the value determined in the limiting spectrum case, α s (m 2 Z ) = 0.1195 ± 0.0022, although larger by +0.0015. We assign this difference as a (positive) source of systematic error associated with the scale uncertainty of our calculations.

Conclusions
The QCD coupling has been determined at NLO*+NMLLA accuracy from the energy evolution of the moments of the parton-to-hadron fragmentation functions measured in the E jet = 2-200 GeV range in e + e − and DIS collisions: α s (m 2 Z ) = 0.1195 ± 0.0022(fit), and α s (m 2 Z ) = 0.119 ± 0.010(fit), respectively. We have studied in detail the experimental and systematic uncertainties associated with our α s extraction, finding that the fit uncertainties obtained through the "χ 2 averaging" method fully cover the range of α s (m 2 Z ) variations driven by hadronization (finite hadron-mass) corrections, different experimental final-hadron definitions, as well as overall fit procedure. An additional theoretical-scale uncertainty of +0.0015 has been calculated by redoing the fit of the moments using a different shower cut-off value (i.e. relaxing the limitingspectrum criterion). The final value of α s (m 2 Z ) obtained from a simple weighted-average of the couplings extracted independently from the e + e − and DIS jet data, yields: α s (m 2 Z ) = 0.1195 ± 0.0021 (exp) +0.0015 −0.0 (scale), which is in perfect accord with the current worldaverage of α s (m 2 Z ) = 0.1185 ± 0.0006 (obtained at NNLO accuracy). Work is in progress to combine the e + e − and DIS fragmentation-function moments into a single global fit [9], which will better account for any residual correlated uncertainty. In Fig. 5 we compare our α s (m 2 Z ) value to all other existing results at NLO accuracy. All the strong coupling values plotted are from the latest PDG compilation [1] except for the most recent jet results obtained from the CMS [10] and ATLAS [11] data. Our result is the most precise of all approaches with a totally different set of experimental and theoretical uncertainties. The weighted average of all the NLO values yields: α s (m 2 Z ) = 0.1188 ± 0.0012. The methodology presented here provides a new promising approach for the determination of the QCD coupling constant complementary to other existing jet-based methods -such as jet shapes, and inclusive jet production cross sections in e + e − , DIS and p-p collisions-and can be used to reduce the overall final uncertainty of the least well-known interaction coupling in nature.