On the dynamical determination of strange parton distributions
OOn the dynamical determination of strange partondistributions
P. Jimenez-Delgado ∗ † University of ZurichE-mail: [email protected]
The dynamical parton distributions of the nucleon are generated radiatively from positive definite(valencelike) input distributions at an optimally chosen low resolution scale ( Q < ). Forthe strange distribution in particular, it has been assumed that vanishing strange input distributionsat this low scale is an appropriate choice. By confronting predictions derived from our (GJR08)NLO dynamical parton distributions with recent neutrino dimuon production measurement fromNuTeV we show that this is indeed the case, and that little improvement is achieved by usinga more general ansatz. Nevertheless, the data induce an asymmetry in the strange sea which isfound to be small and positive in agreement with previous results. XVIII International Workshop on Deep-Inelastic Scattering and Related SubjectsApril 19 -23, 2010Convitto della Calza, Firenze, Italy ∗ Speaker. † Supported by the Swiss National Science Foundation (SNF) under contract 200020-126691. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] J un n the dynamical determination of strange parton distributions P. Jimenez-Delgado
The dynamical parton distributions of the nucleon at Q (cid:38) are QCD radiatively gen-erated from valencelike positive definite input distributions at an optimally determined low inputscale Q < . Therefore the steep small-Bjorken- x behavior of structure functions, and con-sequently of the gluon and sea distributions, appears within the dynamical (radiative) approachmainly as a consequence of QCD-dynamics at x (cid:46) − [1]. Alternatively, in the common “stan-dard” approach the input scale is fixed at some arbitrarily chosen Q > , and the corre-sponding input distributions are less restricted; for example, the mentioned steep small- x behaviorhas to be fitted here.Following the radiative approach, the well-known LO/NLO GRV98 dynamical parton distri-bution functions of [2] have been recently updated [3], and the analysis extended to the NNLO ofperturbative QCD in [4]. In addition, in [3, 4] a series of “standard” fits were produced in (for therest) exactly the same conditions as their dynamical counterparts. This allows us to compare thefeatures of both approaches and to test the the dependence in model assumptions. These analyseshave been further augmented with appropriate uncertainty estimations and have shown [3, 4] that,as expected, the associated uncertainties encountered in the determination of the parton distribu-tions turn out to be larger in the “standard” case, particularly in the small- x region, than in themore restricted dynamical radiative approach where, moreover, the “evolution distance” (startingat Q < ) is sizably larger [3, 4].Since the data sets used in all these analyses are insensitive to the specific choice of the strangequark distributions, the strange densities of the dynamical distributions in [1, 2, 3, 4] have beengenerated entirely radiatively starting from vanishing strange input distributions: s ( x , Q ) = ¯ s ( x , Q ) = low input scale. In the the “standard” case, where Q > , the strange input distributionswere chosen s ( x , Q ) = ¯ s ( x , Q ) = ( ¯ u ( x , Q ) + ¯ d ( x , Q )) , as is conventional [3, 4]. In order toinvestigate the plausibility of the assumptions in Eq.(1), we confront here predictions derived fromdynamical distributions determined in this way with data which are particularly sensitive to thestrangeness content of the nucleon. For this purpose we have chosen the latest and most precisemeasurements of neutrino dimuon production from ν µ - and ¯ ν µ -iron deep inelastic scattering (DIS)interactions of NuTeV [5].For details on the calculation of the dimuon cross-section, including a direct comparison ofour predictions with the data, and further neccesary references, we refer to the original paper [6].Here it suffices for our purposes to note that the results obtained with the GJR08 distributions arein good agreement with the data, e.g. we get χ =
65 for 90 data points (see [6] for more detailson this). This agreement demonstrates the compatibility of the data with the conditions of Eq. (1)and shows that in the dynamical case, where the NLO input distributions are parametrized at anoptimally chosen low input scale Q = . [3], the strange sea can be generated entirelyradiatively starting from: s + ( x , Q ) ≡ s ( x , Q ) + ¯ s ( x , Q ) = Valencelike refers to a f > all input distributions x f ( x , Q ) ∝ x a f ( − x ) b f , i.e., not only the valence but alsothe sea and gluon input densities vanish at small x . n the dynamical determination of strange parton distributions P. Jimenez-Delgado
Turning now to the asymmetry in the strange nucleon sea, it is well known (see [6] for refer-ences) that the small differences between neutrino and antineutrino data induce a small differencebetween the strange and antistrange parton distributions. In order to evaluate this asymmetry withinour framework, we parametrize a new input distribution: s − ( x , Q ) ≡ s ( x , Q ) − ¯ s ( x , Q ) = Nx a ( − x ) b ( − xx ) (3)where Q = . is (fixed to) the input scale of our GJR08 NLO fit [3] and the function isconstrained by the quark-number sum rule (cid:82) dx s − ( x , Q ) =
0. Eqs. (2) - (3) imply that the strangeinput distributions, s ( x , Q ) and ¯ s ( x , Q ) , will in turn be negative (positive) at the input scale, byconstruction. This is not a problem as long as at perturbative scales, say for Q > , bothstrange distributions s ( x , Q ) and ¯ s ( x , Q ) become manifestly positive due to the QCD evolution,as is the case [6].After introducing the asymmetry, the χ value improves to 60 for 90 data points, although thepredictions from the strange-symmetric GJR08 distributions are rather similar and the differenceslie within the 1 σ bands. Further, the (anti)neutrino data prefer (smaller)larger values, i.e. the datafavor an increase of the s distribution and a decrease of the ¯ s , in other words, a positive asymmetryin the relevant 0 . (cid:46) x (cid:46) . x values larger than about 0 . Q =
16 GeV appropriate for the NuTeV experiment, and can be directly compared with Fig. 3 of [5]. Althoughdue to the large errors both results are in general agreement, the peak of our asymmetry is lowerand placed at a slightly smaller value of x . The results of MSTW2008 [7] are also shown in Fig. 1for comparison. They are rather similar in size to ours, despite the fact that in [7] older (and lessprecise) data have also been included and this tends to reduce the asymmetry [8]. Note, however,that our asymmetry is much more suppressed for large x (cid:38) .
2, where the data are in excellentagreement with our (strange-symmetric) GJR08 distributions.The changes in the strange-asymmetric distributions as compared with the original GJR08 arerather small, e.g. at Q =
100 GeV they reach at most 5% in the relevant 10 − < x < . s + ( x , Q ) . Furthermore,since we continue to generate the strange distributions radiatively starting from Eq. (2), the increasein the uncertainties encountered in common “standard” fits, where s + ( x , Q ) has to be fitted, isavoided in the more constrained dynamical case, which uncertainties should be very similar to theones of GJR08 in most cases.The strangeness asymmetry is however relevant for applications especially sensitive to thestrange content of the nucleon, as has been shown, for instance, in relation with the so-called the“NuTeV anomaly” (see, e.g. [9] and references therein). As indication of the size and sign of theasymmetry it has become conventional to use the value of its second moment at the reference scale Q =
20 GeV , we obtain: S − ≡ (cid:90) dx x ( s − ¯ s ) = . ± . n the dynamical determination of strange parton distributions P. Jimenez-Delgado -0.01-0.005 0 0.005 0.01 0.015 0.02 10 -6 -5 -4 -3 -2 x(s - s)- Q = 16 GeV x 0.1 0.2 0.3 0.4 0.5 0.6 asymmetricMSTW2008 Figure 1:
Out result for the strangeness asymmetry in the nucleon at Q =
16 GeV appropriate for theNuTeV experiment (cf. Fig. 3 of [5]). The results of MSTW2008 [7] are also shown for comparison. which is of the right sign and size as to explain the “anomaly” and furthermore has, as expected, arelatively small error due to the dynamical assumptions. Previous determinations [5, 7, 8] generallyyield a larger value of about 0.0010 to 0.0020 and a typical uncertainty of about 100% or evenlarger.In conclusion, although in our global QCD fits [3, 4] no data with especial sensitivity to thestrange content of the nucleon have been included, our determination of strange parton distribu-tions, in particular Eq. (2), is compatible with particularly sensitive data, e.g. those in [5]. Fur-thermore, these data induce an asymmetry in the strange sea which has been evaluated within ourdynamical framework and found, in agreement with previous results, to be rather small and pos-itive. This being the case, our original strange-symmetric distributions should suffice for mostapplications. The strangeness asymmetry may, however, be relevant for some especially sensitiveapplications; for these cases our results are available on request. References [1] M. Glück, E. Reya and A. Vogt,
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