Dynamical parton distributions and weak-gauge and Higgs boson production at hadron colliders at NNLO of QCD
DDynamical parton distributions and weak-gauge andHiggs boson production at hadron colliders atNNLO of QCD
P. Jimenez-Delgado ∗ † University of ZurichE-mail: [email protected]
E. Reya ‡ TU DortmundE-mail: [email protected]
Utilizing recent DIS measurements and data on hadronic dilepton production we determine atNNLO (3-loop) of QCD the dynamical parton distributions of the nucleon generated radia-tively from valencelike positive input distributions at an optimally chosen low resolution scale( Q < ) by employing the “fixed flavor number factorization scheme” (FFNS). These arecompared with “standard” NNLO distributions, generated at some fixed and higher resolutionscale ( Q > ). The NNLO corrections imply in both approaches an improved value of χ ,typically χ NNLO ∼ . χ NLO . The dynamical NNLO uncertainties are somewhat smaller than theNLO ones and both are, as expected, smaller than those of their “standard” counterparts. The dy-namical predictions for F L ( x , Q ) become perturbatively stable already at Q = , whereprecision measurements could even delineate NNLO effects in the very small-x region. We obtain α s ( M Z ) = ± ± σ ) than the NLO ones, and rates atLHC energies can be predicted with an accuracy of about 5%, whereas at Tevatron they are morethan 2 σ above the NLO ones. The NNLO predictions for SM Higgs boson production via thedominant gluon fusion process have a total (PDFs and scale) uncertainty of about 10% at LHCwhich almost doubles at the lower Tevatron energies; these predictions are typically about 20%larger than the ones at NLO but the total uncertainty bands overlap. 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. ‡ Supported in part by the “Bundesministerium für Bildung und Forschung”, Berlin. 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 ynamical parton distributions and weak-gauge and Higgs boson ... 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 correspond-ing input distributions are less restricted; for example, the mentioned steep small- x behavior has tobe fitted .Following the radiative approach, the well-known LO/NLO GRV98 dynamical parton dis-tribution functions of [2] have been updated in [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. The associateduncertainties encountered in the determination of the parton distributions turn out, as expected, tobe larger in the “standard” case, particularly in the small- x region, than in the more restricted dy-namical radiative approach where, moreover, the “evolution distance” (starting at Q < ) issizably larger [3, 4].The NNLO corrections imply in both approaches an improved value of χ , typically χ NNLO ∼ . χ NLO . The dynamical NNLO uncertainties are somewhat smaller than the NLO ones and bothare smaller than those of their “standard” counterparts. The strong coupling constant α s ( M Z ) isdetermined in our analyses together with the parton distributions, in particular it is closely relatedto the gluon distribution which drives the QCD evolution and consequently its uncertainty is alsosmaller in the dynamical case. We obtain α s ( M Z ) = ± ± α s ( M Z ) = ± ± . The dynamical predictions for F L ( x , Q ) become perturbatively stable already at Q = , where precision measurements couldeven delineate NNLO effects in the very small-x region. This is in contrast to the results in thecommon “standard” approach, but NNLO/NLO differences are there less distinguishable due tothe larger uncertainty bands .With the LHC and Tevatron running and having in mind that parton distributions are one ofthe major sources of uncertainty in the predictions at hadron colliders, we will focus in this talkon the implications of our NNLO distributions, and specially of their uncertainties, for importantprocess like weak gauge boson production and the production of the standard model (SM) Higgsboson itself. These results have been published in [5], where more details and further necessaryreferences have been given.The analyses in [3, 4] were performed within the framework of the so-called “fixed flavornumber scheme” (FFNS) where, besides the gluon, only the light quark flavors q = u , d , s areconsidered as genuine, i.e., massless partons within the nucleon. This factorization scheme isfully predictive in the heavy quark h = c , b , t sector where the heavy quark flavors are producedentirely perturbatively as part of the final state. Here the full heavy quark mass m h dependence is 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 . See [4] for a more detailed discussion. ynamical parton distributions and weak-gauge and Higgs boson ... P. Jimenez-Delgado taken into account in the production cross sections, as required experimentally, in particular, in thethreshold region. Even for very large values of Q (cid:29) m c , b , the FFNS predictions are in remarkableagreement with deep inelastic scattering data and, moreover, are perturbatively stable despite thecommon belief that “non-collinear” logarithms ln Q m h have to be resummed. This agreement withexperiment even at Q (cid:29) m h indicates that there is little need to resum these supposedly “largelogarithms”, which is of course in contrast to the genuine collinear logarithms appearing in light(massless) quark and gluon hard scattering processes.However, in many situations calculations within the FFNS become unduly complicated, thusit is of practical advantage to consider the so-called “variable flavor number scheme” (VFNS) inwhich the heavy quarks are considered to be (massless) partons within the nucleon as well. Thisfactorization scheme is characterized by increasing the number of flavors n f of massless partonsby one unit at Q = m h starting from n f = Q = m c . Hence the n f > n f − Q -evolution; a comparative qualitative and quantitative discussion of this (zero-mass)VFNS and the FFNS has been recently presented in [6]. Eventually one has to assume that thesemassless “heavy” quark distributions are relevant asymptotically, i.e., that they correctly describethe asymptotic behavior of DIS structure functions for scales Q (cid:29) m h . However, for most exper-imentally accessible values of Q , in particular around the threshold region of heavy quark ( h ¯ h ) production, effects due to finite heavy quark masses m h can not be neglected. One therefore needseither to stick to the FFNS or an improvement of this zero-mass VFNS which maintain heavy quarkmass-dependent corrections in the hard cross-sections and interpolate between the zero-mass VFNS(assumed to be correct asymptotically) and the (experimentally required for most data) FFNS. Suchimprovements are often referred to as general-mass VFNS and there exist various different model-dependent ways of implementing the required m h dependence (see, e.g. [6] for references).In order to avoid any such model ambiguities we generate “heavy”-quark zero-mass VFNSdistributions using our unique NNLO dynamical FFNS distributions as input at Q = m c . Thisconsiderably eases the otherwise unduly complicated calculations in the FFNS of weak gauge- andHiggs-boson production at hadron collider energies. It has been shown [6] that for situations wherethe invariant mass of the produced system ( cW , tW , t ¯ b , Higgs-bosons, etc.) exceeds by far the massof the participating heavy flavor, the VFNS predictions deviate rather little from the FFNS ones,typically by about 10%, which is within the margins of renormalization and factorization scaleuncertainties and ambiguities related to presently available parton distributions. Within the presentintrinsic theoretical uncertainties, we can therefore rely on our uniquely generated NNLO VFNSparton distribution functions where, moreover, the required NNLO cross sections for masslessinitial-state partons are, in contrast to the fully massive FFNS, available in the literature for avariety of important production processes.Our NNLO predictions [5] for σ ( p ¯ p → W ± X ) and σ ( p ¯ p → Z X ) are compared with ourNLO ones [6] in Fig. 1, where also the predictions of Alekhin [7, 8] and some data points (see[5] for the appropriate references) are shown for comparison. The vector boson production ratesat NNLO are typically slightly larger (by more than 1 σ ) than at NLO with a K ≡ NNLO/NLOfactor of K W + + W − = .
04 and K Z = .
06 at Tevatron energies. This confirms the fast perturbativeconvergence at NNLO since the NLO/LO K -factor is of about 1.3 [6]. The scale uncertainties of3 ynamical parton distributions and weak-gauge and Higgs boson ... P. Jimenez-Delgado s ( nb ) (cid:214) s (TeV) - W + + W - Z NNLO NLO Alekhin NNLOAlekhin NLOUA1 UA2 D0 CDF 0 5 10 15 20 25 30 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s ( nb ) (cid:214) s (TeV) - W + + W - Z Figure 1:
Predictions for the total W + + W − and Z production rates at p ¯ p colliders. The shaded bandaround our central results are due to the ± σ PDF uncertainties. See [5] for more details and references. our NNLO predictions, due to M V ≤ µ F ≤ M V , amount to less than 0.5% at √ s = .
96 TeV, i.e.,is four times less than at NLO [6]. Our results at √ s = .
96 TeV are similar to the ones of MSTW[9] and about 4% smaller than those of ABKM [10].Our NNLO expectations for W ± and Z production at the LHC at √ s =
14 TeV are: σ ( pp → W + + W − + X ) = . ± . pdf + . − . | scale nb (1) σ ( pp → Z + X ) = . ± . pdf + . − . | scale nb . (2)Here the scale uncertainties amount to less than 1.7%, i.e., are about half as large than the statedPDF uncertainties and than the scale uncertainties at NLO [6]. These results are about 5% smallerthan the ones of MSTW [9] and about 10% smaller than the obtained by ABKM [10]. For compar-ison we note that within the FFNS the W + + W − production rate has been estimated [6] to be about192.7 nb at NLO with a total (PDF as well as scale) uncertainty of about 5%. In general the NLO-VFNS prediction falls somewhat below that estimate but remains well within its total uncertaintyof about 6% [6]. Due to the reduced scale ambiguity at NNLO and due to the slightly differentNNLO estimates obtained by other groups, we conclude that the rates for gauge boson productionat LHC energies can be rather confidently predicted with an accuracy of about 5% irrespective ofthe factorization scheme.We turn now to the hadronic production of the SM Higgs boson, where the dominant pro-duction mechanism proceeds via gluon-gluon fusion. Our NNLO and NLO results are shown inFig. 2, where the shaded regions around the central predictions are due to the ± σ PDF uncertain-ties and the outer lines are obtained by varying the scale by a factor of 2 around its nominal value µ F = M H (see [5] for a more explicit illustration of the scale ambiguities). Despite the fact that theNLO and NNLO total uncertainty bands overlap in Fig. 2, the predicted NNLO production ratesare typically about 20% larger than at NLO. The insensitivity of these predictions with respect to In our calculations we always set µ R = µ F , as dictated by all presently available PDFs. ynamical parton distributions and weak-gauge and Higgs boson ... P. Jimenez-Delgado s pp fi H X ( pb ) M H (GeV) NNLO NNLO NLO 0 10 20 30 40 50 60 70 100 150 200 250 300 s pp fi H X ( pb ) M H (GeV) (NLO pdfs) gluon fusion (cid:214) s = 14 TeV - Figure 2:
Predictions for SM Higgs boson production at LHC via the dominant gluon-gluon fusion process.The shaded bands around the central values are due to the ± σ PDF uncertainties only, while the outercurves include both the PDF uncertainties and scale variations. See [5] for more details. the appropriate choice of the PDFs is illustrated by the dashed curve, which has been obtained byusing NNLO matrix elements and (inconsistently) NLO PDFs. Our central predictions in Fig. 2 areabout 10% smaller than the ones of MSTW [9], and are 5-8% smaller than those of ABKM [10]for M H < ∼
150 GeV, but agree with their predictions for larger Higgs masses [10].Higgs boson production at Tevatron have similar features than those shown in Fig. 2 but withmuch larger uncertainty bands; for instance, the uncertainties of our expectations at √ s = .
96 TeValmost double at NNLO and NLO as compared to the ones at LHC [5]. The rates obtained byABKM [10] are 12-30% smaller than our ones for M H =
100 - 200 GeV in this case. We concludethat SM Higgs boson production at LHC ( √ s =
14 TeV) can be predicted with an accuracy ofabout 10% at NNLO (with the total uncertainty being almost twice as large at NLO), whereas theuncertainty almost doubles at Tevatron ( √ s = .
96 TeV).
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