aa r X i v : . [ h e p - ph ] J u l DESY 11-113
Running heavy-quark masses in DIS
S. Alekhin ∗ ,† and S. Moch ∗ ∗ Deutsches Elektronensynchrotron DESY, Platanenallee 6, D–15738 Zeuthen, Germany † Institute for High Energy Physics, 142281 Protvino, Moscow region, Russia
Abstract.
We report on determinations of the running mass for charm quarks from deep-inelasticscattering reactions. The method provides complementary information on this fundamental param-eter from hadronic processes with space-like kinematics. The obtained values are consistent withbut systematically lower than the world average as published by the PDG. We also address theconsequences of the running mass scheme for heavy-quark parton distributions in global fits todeep-inelastic scattering data.
Keywords:
Deep-inelastic scattering, heavy quark structure function, charm quark mass
PACS:
Quark masses are fundamental parameters of the gauge theory of the strong interac-tions, Quantum Chromodynamics (QCD). They are, however, not directly observabledue to confinement. Rather one has to compute the dependence of cross sections orother measurable quantities on the heavy quark mass including higher order radiativecorrections and renormalization – a procedure which requires the choice of a schemefor the definition the mass parameter. There exists a variety of schemes for heavy-quarkmasses. The most popular ones are the on-shell and the MS scheme. In the former theso-called pole-mass m coincides with the pole of the heavy-quark propagator at eachorder in perturbative QCD. This definition, however, has intrinsic theoretical limitationswith ambiguities of order O ( L QCD ) and a strong dependence of the value of the massparameter on the order of perturbation theory. The MS scheme is one of the so-calledshort-distance masses for heavy quarks. It realizes the concept of a running mass m ( m ) depending on the scale m of the hard scattering in complete analogy to the treatment ofthe running coupling a s ( m ) . As a benefit, predictions for hard scattering cross sectionsin terms of the MS mass display better convergence properties and greater perturbativestability at higher orders.Cross sections for the production of heavy-quarks in deep-inelastic scattering (DIS)are particularly well suited to confront the quark mass dependence of theoretical pre-dictions in perturbative QCD with experimental measurements in space-like kinematics.For the production of charm quarks in neutral (NC) or charged current (CC) DIS thereexists precise data from the HERA collider and fixed-target experiments. In QCD theDIS heavy-quark structure functions F k which parametrize the hadronic cross sectionare subject to the standard factorization F k ( x , Q , m ) = (cid:229) i = q , ¯ q , g (cid:20) f i ( m ) ⊗ C k , i (cid:0) Q , m , a s ( m ) (cid:1)(cid:21) ( x ) , k = , , , (1)where the perturbative coefficient functions C k , i are known to next-to-leading orderNLO) for CC [1, 2] and approximately to next-to-next-to-leading order (NNLO) forNC [3, 4]. Q and x are the usual DIS kinematical variables and m is the heavy-quark(pole) mass. In eq. (1) we also display all dependence on the other non-perturbativeparameters, i.e. the parton distribution functions f i (PDFs) for light quarks and gluons aswell as the strong coupling constant a s . The running mass definition can be implementedin eq. (1) to the respective order in perturbation theory by simply following the standardprocedure for changing the renormalization condition, i.e. m → m ( m ) see ref. [5].The present study has two aspects: First of all we determine the MS charm mass m c ( m c ) from DIS in a variant of the global analysis [6] and compare to the world averageas published by the PDG. Secondly, we investigate the improvements in the uncertaintyof heavy-quark PDFs in a global fit within the fixed-flavor number scheme (FFNS) whenthe running mass scheme is applied.
1. Running charm-quark mass
The parametric dependence of the DIS structure functions F k in eq. (1) on m canbe used for a determination of the heavy-quark mass. The sensitivity of this proce-dure relates directly to the corresponding uncertainty on the measurements of F k . E.g.,for charm production in NC DIS the nucleon structure function F yields D m c / m c ≃ . D F / F , which implies that an experimental accuracy of 8% for F translates intoan uncertainty of 6% for the charm-quark mass [5]. With the precision of current DISdata for charm production this suggests an error on m c ( m c ) of O ( few ) % as the ultimateprecision in the approach based on inclusive structure functions.Starting from eq. (1) we have extracted the MS charm mass m c ( m c ) in a phenomeno-logical study similar to [7], i.e. a global fit including fixed-target (CCFR [8], NuTeV [9])and collider data [10, 11] in the FFNS (with n f =
3) as a variant of the ABKM one [6].We have obtained [5], m c ( m c ) = . ± . ( exp ) ± . ( th ) GeV at NLO , (2) m c ( m c ) = . ± . ( exp ) ± . ( th ) GeV at NNLO approx , (3)to NLO and approximate NNLO in perturbation theory. The quoted experimental un-certainty results from the propagation of the statistical and systematic errors in the datawith account of error correlations whenever available. The theoretical uncertainty is es-timated from the variation of the renormalization and factorization scales for the choice m r = m f = Q + k m c for F with k in the range k ∈ [ , ] . For consistency it has beenchecked that different scale choices do not deteriorate the statistical quality of the fit. Be-sides the charm mass m c quoted in eqs. (2) and (3) and the value of the strong couplingconstant determined to a s ( M Z ) = . ± . m c due to the Born process W ± s → c .At this conference, NOMAD has reported a new analysis of CC fixed-target data. Theincreased precision due to the very high statistics has been used to determine the runningmass m c ( m c ) to NLO exclusively from CC DIS leading to the (preliminary) result [12] m c ( m c ) = . ± . ( exp ) ± . ( th ) GeV at NLO . (4)ne has to compare the numbers in eqs. (2), (3) and (4) with the world average ofthe PDG [13] quoted in the MS scheme as m c ( m c ) = . + . − . GeV , which is entirelybased on lattice computations or analyses of experimental data with time-like kinematicsfrom e + e − -collisions, e.g. with the help of QCD sum rules. It is therefore interesting tonote that the DIS results in eqs. (2), (3) and (4) for hadronic processes with space-likekinematics are consistent with but systematically lower than the PDG value. In orderto understand at least one source of this deviation, one should note the QCD sum rulesanalyses typically assume the Bethke world average for the value of the strong couplingconstant [14], which is a s ( M Z ) = . ± . m c ( m c ) = . ± .
026 GeV using the world average and parametrizes separately the dependenceof m c ( m c ) on value of a s ( M Z ) . Using the ABKM value a s ( M Z ) = . m c ( m c ) = . ± .
026 GeV, i.e. a systematicshift downwards at the level on 1 s statistical uncertainty bringing QCD sum ruleanalysis from e + e − -collisions in better agreement with the DIS results in eqs. (2), (3) and(4). Note that the latter determinations account for the full correlation of the dependenceon m c and a s through a simultaneous fit of the parameters.Including the PDG constraint on m c ( m c ) into the fit of ref. [5] we get the value of m c ( m c ) = . ± .
06 GeV at NNLO. The recent data on the charm structure function F c ¯ c at low Q published by the H1 collaboration [17], which were not included intothe fit of ref. [5], are compared with the running-mass scheme predictions of ref. [5]obtained with this value of m c ( m c ) in fig. 1 (left). The agreement is quite good thereforethese data also prefer somewhat smaller value of m c ( m c ) than the PDG average.
2. Heavy quark PDFs
At asymptotically large scales Q ≫ m c , m b the genuine heavy quark contributionsin a FFNS with n f = a s ( Q ) ln ( Q / m ) and can be resummed by meansof standard renormalization group methods. This procedure leads to so-called heavyquark PDFs in theories with effectively n f = n f = n f =
4- and n f = n f = m c or m b , which appear parametrically in the OMEs.This uncertainty can be significantly reduced through the use of the MS scheme, which,of course, has to be applied also to the massive OMEs.In our global fit [5] m c ( m c ) has been left a free parameter supplemented by the PDGconstraint. In this way we have generated a charm-PDF with comparable uncertaintiesto the one of [6] (which has used the pole mass definition for m c ). For m b the currentlyavailable DIS data displays no sensitivity at all and we have constrained m b ( m b ) directlyto its PDG value [13], i.e. m b ( m b ) = . + . − . GeV. As shown in fig. 1 (right) theuncertainty of the resulting bottom-PDF is greatly reduced. This improvement willcertainly have an impact on LHC phenomenology, e.g. allowing for precise predictionsfor the production of single-top-quarks. cc2 -3 -2 Q = 6.5 GeV -3 -2 Q = 12.0 GeV -3 -2 Q = 18.0 GeV -3 -2 Q = 35.0 GeV x -3 -2 Q = 60.0 GeV x H1 348 1/pbABKM09 (running mass) m =m t -20-15-10-50510152010 -3 -2 x D b(N f =5,x) (%)ABKM09ABKM09 (running mass) FIGURE 1.
Left: The data on the charm structure function F c ¯ c by the H1 collaboration [17] in compar-ison with the running-mass scheme predictions of ref. [5]. Right: The b -quark PDF uncertainties obtainedin the global fits: The dotted (red) lines denote the ± s band of relative uncertainties (in percent) andthe solid (red) line indicates the difference in the central prediction resulting from the change of the massscheme and using m b ( m b ) = .
19 GeV (bottom). For comparison the shaded (grey) area represents theuncertainties in the ABKM fit [6].
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