The ABM parton distributions tuned to LHC data
aa r X i v : . [ h e p - ph ] O c t DESY 13-183DO-TH 13 / / CPP-13-71October 2013
The ABM parton distributions tuned to LHC data
S. Alekhin a , b , , J. Blümlein a , , and S. Moch a , c , a Deutsches Elektronensynchrotron DESYPlatanenallee 6, D–15738 Zeuthen, Germany b Institute for High Energy Physics142281 Protvino, Moscow region, Russia c II. Institut für Theoretische Physik, Universität HamburgLuruper Chaussee 149, D–22761 Hamburg, Germany
Abstract
We present a global fit of parton distributions at next-to-next-to-leading order (NNLO) in QCD.The fit is based on the world data for deep-inelastic scattering, fixed-target data for the Drell-Yanprocess and includes, for the first time, data from the Large Hadron Collider (LHC) for the Drell-Yan process and the hadro-production of top-quark pairs. The analysis applies the fixed-flavornumber scheme for n f = , ,
5, uses the MS scheme for the strong coupling α s and the heavy-quark masses and keeps full account of the correlations among all non-perturbative parameters.At NNLO this returns the values of α s ( M Z ) = . ± . m t (pole) = . ± . ff erences can be linked to di ff erent theoretical descriptions of theunderlying physical processes. e-mail : [email protected] e-mail : [email protected] e-mail : [email protected] Introduction
Our knowledge of the proton structure builds on the accumulated world data from the deep-inelastic scattering (DIS) experiments, which cover a broad kinematic range in terms of the scalingvariable x and the momentum Q transferred to the proton [1]. These data have been gathered ina variety of di ff erent scattering experiments, either on fixed targets or through colliding beams,and in the past two decades, especially the HERA electron-proton collider has contributed signifi-cantly with very accurate measurements spanning a wide range in x and Q . Thus, DIS world dataform the backbone for the determination of the parton distribution functions (PDFs) in the QCDimproved parton model.Modern PDFs, however, are expected to provide an accurate description of the parton contentof the proton not only in a kinematic region for x and Q as wide as possible, but to deliveralso information on the flavor composition of the proton as well as on other non-perturbativeparameters associated to the observables under consideration, such as the strong coupling constant α s or the masses of the heavy quarks charm, bottom and top. In the theoretical predictions thevalues for these quantities are often correlated with the PDFs and, therefore, have to be determinedsimultaneously in a fit.A comprehensive picture of a composite object such as the proton does not emerge withoutthe need for additional assumptions by relying, e.g., on DIS data from the HERA collider alone.Therefore, global PDF fits have to include larger sets of precision data for di ff erent processes,which have to be compatible, though. The release of the new data for so-called standard candleprocesses, i.e., precisely measured and theoretically well-understood Standard Model (SM) scat-tering reactions, initiates three steps in the analysis:i) check of compatibility of the new data set with the available world dataii) study of potential constraints due to the addition of the new data set to the fitiii) perform a high precision determination of the non-perturbative parameters: PDFs, α s ( M Z )and heavy-quark masses.Of course, at every step QCD precision analyses have to provide a detailed account of the system-atic errors and have to incorporate all known theoretical corrections. At the Large Hadron Collider(LHC) PDFs are an indispensable ingredient in almost every experimental analysis and the publi-cation of data for W ± - or Z -boson, top-quark pair or jet-production from the runs at √ s = O (10%) cannot beachieved without recourse to predictions at next-to-next-to-leading order (NNLO) in QCD [2, 3]which has thus become the standard paradigm of QCD precision analyses of the proton’s partoncontent [4]. The PDF fits ABKM09 [5] and, subsequently, ABM11 [6] on which the currentanalysis is building, have been performed precisely in this spirit. At the same time, the NNLOparadigm has motivated continuous improvements in the theory description of processes whereonly next-to-leading order (NLO) corrections are available, such as the hadro-production of jets.In the current article, we are, for the first time, tuning the ABM PDFs to the available LHCdata for a number of standard candle processes including W ± - and Z -boson production as wellas t ¯ t -production. We are demonstrating overall very good consistency of the ABM11 PDFs withthe available LHC data. Particular aspects of these findings have been reported previously [7–11].1ubsequently, we perform a global fit to obtain a new ABM12 PDF set and we discuss in detail theobtained results for the PDFs, α s ( M Z ) and the quark masses along with their correlations and thegoodness of fit.The outline of the article is as follows. We recall in Sec. 2 the footing of our fit and present thebasic improvements in the theory description and the new data sets included. These encompassthe charm-production and high- Q neutral-current HERA data discussed in Sections. 2.1 and 2.2,the W ± - and Z -boson production data from the LHC investigated in Section 2.3 and, likewise, inSec. 2.4 data for the total cross section of t ¯ t -production. The results for ABM12 PDFs are discussedin Section 3 in a detailed comparison with the ABM11 fit in Sec. 3.1 and with emphasis on thestrong coupling constant and the charm quark mass, cf. Section 3.2. Finally, in Section 3.3 weprovide cross section predictions of the ABM12 PDFs for a number of standard candle processesand the dominant SM Higgs production channel. Appendix A describes a fast algorithm for dealingwith those iterated theoretical computations in the PDF fit, which are very time-consuming. The present analysis is an extension of the earlier ABM11 fit [6] based on the DIS and DY dataand performed in the NNLO accuracy. The improvements are related to adding recently publisheddata relevant for the PDF determination: • semi-inclusive charm DIS production data obtained by combination of the H1 and ZEUSresults [12]. This data set provides an improved constraint on the low- x gluon and sea-quarkdistribution and allows amended validation of the c -quark production mechanism in the DIS. • the neutral-current DIS inclusive data with the momentum transfer Q > obtainedby the HERA experiments [13]. These data allow to check the 3-flavor scheme used inour analysis up to very high momentum transfers and, besides, to improve somewhat thedetermination of the quark distributions at x ∼ . • the DY data obtained by the LHC experiments [14–17] improve the determination of thequark distribution at x ∼ .
1, and in particular provide a constraint on the d -quark distribu-tion, which is not sensitive to the correction on the nuclear e ff ects in deuteron. • the total top-quark pair-production cross section data from LHC [18–22] and the Tevatroncombination [23] provide the possibility for a consistent determination the top-quark masswith full account of the correlations with the gluon PDF and the strong coupling α s .The theoretical framework of the analysis is properly improved as compared to the ABM11 fitin accordance with the new data included. In this Section we describe details of these improve-ments related to each of the processes and the data sets involved, check agreement of the new datawith the ABM11 fit, and discuss their impact and the goodness of fit. The HERA data on the c -quark DIS production [12] are obtained by combination of the ear-lier H1 and ZEUS results. The combined data span the region of Q = . ÷ and x = · − ÷ .
05. The dominating channel of the c -quark production at this kinematics is the2hoton-gluon fusion. Therefore it provides an additional constraint on the small- x gluon dis-tribution. Our theoretical description of the HERA data on charm-production is based on thefixed-flavor-number (FFN) factorization scheme with 3 light quarks in the initial state and theheavy-quarks appearing in the final state. The 3–flavor Wilson coe ffi cients for the heavy-quarkelectro-production are calculated in NLO [24, 25] and approximate NNLO corrections have beenalso derived recently [26]. The latter are obtained as a combination of the threshold resummationcalculation [27] and the high-energy asymptotics [28] with the available Mellin moments of themassive operator matrix elements (OMEs) [29–32], which provide matching of these two. Twooptions of the NNLO Wilson coe ffi cient’s shape, A and B, given in Ref. [26] encode the remaininguncertainty due to higher Mellin moments than given in [31]. In the present analysis, the NNLOcorrections are modeled as a linear combination of the option A and B of Ref. [26] with the in-terpolation parameter d N with the values of d N = , d N = − . ± .
15. The same approach was also used in our earlier determination of the c -quark mass from the DIS data including the HERA charm-production ones [33] with a similarvalue of d N obtained. In our analysis we also employ the running-mass definition for the DIS struc-ture functions [34]. For comparison, the ABM11 fit is based on the massive NNLO correctionsstemming from the threshold resummation only [27] and their uncertainty is not considered.The description of the HERA charm data within the ABM12 framework is quite good with thevalue of χ / NDP = /
52, where
NDP stands for the number of data points. The pulls for thisdata set also do not demonstrate any statistically significant trend with respect to either x or Q ,cf. Fig. 2.1. In particular, this gives an argument in favor of using the 3-flavor scheme over the fullrange of existing DIS data kinematics. Q neutral-current HERA data The HERA data for Q > newly added to our analysis are part of the combined in-clusive sample produced using the H1 and ZEUS statistics collected during Run-I of the HERAoperation [13]. Due to kinematic constraints of DIS these data are localized at relatively largevalues of x , where they have limited statistical potential for the PDF constraint as compared to thefixed-target DIS data used in our analysis. For this reason this piece was not used in the ABM11fit. In the present analysis we fill this gap for the purpose of completeness. At large Q the DIScross section gets non-negligible contributions due to the Z -exchange, in addition to the photon-exchange term su ffi cient for the accurate description of the data at Q ≪ M Z , where M Z is the Z -boson mass. The Z -boson contribution is taken into account using the formalism [35, 36] withaccount of the correction to the massless Wilson coe ffi cients up to NNLO [37]. In accordancewith [35] the contribution due to the photon- Z interference term dominates over the one for thepure Z -exchange at HERA kinematics . The values of χ / NDP obtained in our analysis for thewhole inclusive HERA data set and for its neutral-current subset are 694 /
608 and 629 / Q covered by the data. This is illustrated in Fig. 2.2 with the example of theneutral-current e + p HERA data sample, which contains the most accurate HERA measurements atlarge Q . For the e − p sample the picture is similar and the total value of χ / NDP obtained for thenewly added neutral-current data with Q > is 147 / The version 1.6 of the
OPENQCDRAD code used in our analysis to compute the DIS structure functions includingthe contribution due to the Z -exchange is publicly available online [38]. cc (HERA RunI+II combined) d a t a / t h e o r y - Q =2.5 GeV
12 GeV
18 GeV
32 GeV
60 GeV
120 GeV
200 GeV
350 GeV x
650 GeV x x Figure 2.1:
The pulls versus Bjorken x for the HERA combined data on the charm production [12] binnedin momentum transfer Q with respect to our NNLO fit. of Q >
100 GeV and Q >
10 GeV we get for the same sample the values of χ / NDP = / / ffi cient for the description of the existing HERA data in the whole kinematical range (cf. [39,40]for more details). Data on the Drell-Yan (DY) process provide a valuable constraint on the PDFs extracted from aglobal PDF fit allowing to disentangle the sea and valence quark distributions. At the LHC thesedata are now available in the form of the rapidity distributions of charged leptons produced inthe decays of the W -bosons and / or charged-lepton pairs from the Z -boson decays [14–17]. Due4 a t a / f i t- x=2.5E-05 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ Q (GeV ) x=0.32 ÷ Q (GeV ) x=0.51 ÷ Figure 2.2:
The same as in Fig. 2.1 for the pulls of the HERA inclusive combined data [13] binned inBjorken x versus momentum transfer Q . to limited detector acceptance and the W / Z event selection criteria the LHC data are commonlyobtained in a restricted phase space with a cut on the lepton transverse momentum P lT imposed.Therefore, taking advantage of these data to constrain the PDFs requires fully exclusive calcu-lations of the Drell-Yan process. These are implemented up to NNLO in two publicly availablecodes, DYNNLO [41] and
FEWZ [42]. Benchmarking these codes we found good mutual agreementfor the LHC kinematics. We note that with the version 1.3 of
DYNNLO the numerical convergenceis achieved faster than for version 3.1 of
FEWZ , although even in the former case a typical CPUtime required for computing rapidity distribution with the accuracy better than 1% is 200 hours forthe Intel model P9700 / FEWZ (version 3.1) provides a convenient capabilityto estimate uncertainties in the cross sections due to the PDFs. Therefore we use in our analy-sis the benefits of both codes combining the central values of
DYNNLO (version 1.3) and the PDF5
TLAS (7 TeV, 35 1/pb) η l d σ / d η l ( pb ) W - -- > l - ν P Tl >
20 GeVP T ν >
25 GeVM T >
40 GeVNNLO ABM11NNLO ABM12 500525550575600625650675700725 0 1 2 η l d σ / d η l ( pb ) W + -- > l + ν P Tl >
20 GeVP T ν >
25 GeVM T >
40 GeV 405060708090100110120130140 0 2 η ll d σ / d η ll ( pb ) Z -- > l + l - P Tl >
20 GeV
Figure 2.3:
The ATLAS data [14] on the rapidity distribution of charged leptons produced in the decays of W − - and W + -boson (left and central panel, respectively) and charged lepton pairs from the decays of Z -boson(right panel) in comparison with the NNLO calculations based on the ABM11 PDFs (solid curves) takinginto account the uncertainties due to PDFs (grey area). The dashed curves display the ABM12 predictions.The cuts on the lepton transverse momentum P lT and the transverse mass M T imposed to select a particularprocess signal are given in the corresponding panels. LHCb (7 TeV, 37 1/pb) η µ d σ / d η µ ( pb ) W + -- > µ + ν P T µ >
20 GeVNNLO ABM11NNLO ABM12 050100150200250300350400450500 2 3 4 η µ d σ / d η µ ( pb ) W - -- > µ - ν P T µ >
20 GeV 010203040506070 2 3 4 η µµ d σ / d η µµ ( pb ) Z -- > µ + µ - P T µ >
20 GeV2 <η µ < Figure 2.4:
The same as in Fig. 2.3 for the charged muons rapidity distributions obtained by LHCb [15]. uncertainties of
FEWZ (version 3.1).The predictions obtained in such a way with the ABM11 PDFs [6] are compared to the LHCDY data [14–17] in Figs. 2.3, 2.4 and 2.5. The predictions systematically overshoot the ATLASdata [14]. However the o ff set is within the experimental uncertainty, which is dominated by theone of 3.5% due to the luminosity, cf. Fig 2.3. On the other hand, a good agreement is observed6 HCb (7 TeV, 940 1/pb) η ee d σ / d η ee ( pb ) Z -- > e + e - P Te >
20 GeV2 <η e < CMS (7 TeV, 840 1/pb) η e e ± a s y mm e t r y W ± -- > e ± ν P Te >
35 GeVNNLO ABM11NNLO ABM12
Figure 2.5:
The same as in Fig. 2.3 for the LHCb data [17] on the rapidity distribution of the e + e − pairsproduced in the Z -boson decays (left panel) and the CMS data [16] on the charge asymmetry of electronsproduced in the W ± -boson decays (right panel). for the Z -boson data by LHCb [17] in the region overlapping with the ATLAS kinematics, cf.Fig 2.5. This signals some discrepancy between these two sets of data, which is most likely relatedto the general experimental normalization. In any case the normalization o ff -set cancels in theratio and the ATLAS data on the charged-lepton asymmetry are in a good agreement with ourpredictions [14]. This is in some contrast to the CMS results where a few data points go lower thanthe ABM11 predictions, cf. Fig 2.5.Agreement between the LHC data and the ABM11 predictions is quantified by the following χ functional χ = X i , j ( y i − t (0) i )[ C − ] i j ( y j − t (0) j ) , (2.1)where y i and t (0) i stand for the measurements and predictions, respectively, and C i j is the covari-ance matrix with the indices i , j running over the points in the data set. The covariance matrix isconstructed as follows C i j = C expi j + N unc X k = ∆ t ( k ) i ∆ t ( k ) j , (2.2)where the first term contains information about the experimental errors and their correlations andthe second term comprises the PDF uncertainties in predictions. The later are quantified as shiftsin the predictions due to the variation between the central PDF value and the ones encoding thePDF uncertainties. For ABM11 the latter appear primarily due to the variation of the fitted PDFparameters and, besides, due to the uncertainty in the nuclear correction applied to the deuteronDIS data. Therefore, the total number of PDF uncertainty members is N unc = N p +
1, where N p = C expi j = δ i j σ i + f (0) i f (0) j X l = s ki s kj , (2.3)where σ i are the statistical errors in the data combined in quadrature with the uncorrelated errors.Here s li are the correlated systematic uncertainties representing 31 independent sources includingthe normalization, and δ i j stands for the Kronecker symbol. In view of the small background forthe W - and Z -production signal all systematic errors are considered as multiplicative. Therefore,they are weighted with the theoretical predictions f (0) i . The experimental covariance matricesfor the CMS and LHCb data of Refs. [15–17] are employed directly as published in Eq. (2.2)after re-weighting them by the theoretical predictions similarly to Eq. (2.3) with the normalizationuncertainty taken into account in the same way as for the ATLAS data. Experiment ATLAS [14] CMS [16] LHCb [15] LHCb [17]Final states W + → l + ν W + → e + ν W + → µ + ν Z → e + e − W − → l − ν W − → e − ν W − → µ − ν Z → l + l − Luminosity (1 / pb) 35 840 37 940 NDP
30 11 10 9 χ (ABM11) 35 . .
7) 10 . .
7) 13.1(4.5) 11.3(4.2) χ (ABM12) 35.6 9.3 14.4 13.4 Table 2.1:
The value of χ obtained for di ff erent samples of the Drell-Yan LHC data with the NNLOABM11 PDFs in comparison with the one obtained in the ABM12 fit. The figures in parenthesis give onestandard deviation of χ equal to √ NDP . The values of χ computed according to Eq. (2.1) for each of the LHC DY data sets obtainedwith the ABM11 PDFs are given in Tab. 2.1. The description quality is somewhat worse for theATLAS and LHCb muon data, however, in general the agreement between the data and predictionsis still good. The values of χ / NDP are comparable to 1 within the statistical fluctuations in χ .Therefore, the data can be easily accommodated in the ABM fit. Furthermore, in this case thePDF variation is expected to be within the ABM11 PDF uncertainties. This allows to optimize thecomputation of the involved NNLO Drell-Yan corrections in the fit by extrapolation of the gridwith the pre-calculated predictions for the ABM11 eigenvector basis (cf. App. A for the detailson the implementation of this approach). The values of χ obtained for the LHC DY data sets inthe ABM12 fit are quoted in Tab. 2.1. In this case the PDF uncertainties are irrelevant since thePDFs have been tuned to the data. Therefore, they are not included into the second term in thecovariance matrix Eq. (2.2). Despite the di ff erence in the definition, the ABM12 values of χ forthe LHC DY data are in a good agreement with the ABM11 ones giving additional evidence forthe compatibility of these data with the ABM11 PDFs. t ¯ t production in the ABM12 fit At the LHC t ¯ t -pair production proceeds predominantly through initial gluon-gluon scattering.Thus, the total t ¯ t cross section is sensitive to the gluon distribution at e ff ective x values of h x i ≃ pp → tt [pb] at LHC8 - NNLONLOLOm t (m t )/GeV50100150200250300350400450 140 150 160 170 180 σ pp → tt [pb] at LHC8 - NNLONLOLOm t (pole)/GeV50100150200250300350400450 150 160 170 180 190 Figure 2.6:
The LO, NLO and NNLO QCD predictions for the t ¯ t total cross section at the LHC ( √ s = m t ( m t ) at the scale µ = m t ( m t ) (left) and in theon-shell scheme m t (pole) at the scale µ = m t (pole) (right) with the ABM12 PDFs. σ pp → tt [pb] at LHC8 - m t (m t ) = 162 GeV µ /m t (m t )100120140160180200220240260280 1 σ pp → tt [pb] at LHC8 - m t (pole) = 171 GeV µ /m t (pole)100120140160180200220240260280 1 Figure 2.7:
The scale dependence of the LO, NLO and NNLO QCD predictions for the t ¯ t total crosssection at the LHC ( √ s = m t ( m t ) =
162 GeV in the MS scheme (left) and m t (pole) =
171 GeV in the on-shell scheme (right) with the ABM12 PDFs and the choice µ = µ r = µ f .The vertical bars indicate the size of the scale variation in the standard range µ/ m t (pole) ∈ [1 / ,
2] and µ/ m t ( m t ) ∈ [1 / , m t / √ s ≃ . . . . .
05 for the runs at √ s = x which is wellconstrained by data from the HERA collider, though.The available data for the total t ¯ t cross section from ATLAS and CMS at √ s = √ s = √ s = O (5%) while the ATLASmeasurements [18, 20] have an error slightly larger than O (10%).The QCD corrections for inclusive t ¯ t -pair production are complete to NNLO [43–46], so thatthese data can be consistently added to the ABM11 PDF fit at NNLO. The theory predictions areavailable for the top-quark mass in the MS scheme with m t ( µ r ) being the running mass [47] aswell as for the pole mass m t (pole) in the on-shell renormalization scheme [43–46]. The distinc-9ion is important, because the theory predictions as a function of the running mass m t ( µ r ) displaymuch improved convergence and better scale stability of the perturbative expansion [47]. This isillustrated in Figs. 2.6 and 2.7 for the total t ¯ t -cross section computed with the program Hathor (version 1.5) [48]. In Fig. 2.6 we show the size of the higher order perturbative corrections fromLO to NNLO taking the PDFs order independent, i.e., the ABM11 set at NNLO, as a function ofthe top-quark mass for the LHC at √ s = m t ( m t ) =
162 GeV and m t (pole) =
171 GeV.Figs. 2.6 and 2.7 imply a small residual theoretical uncertainty for the t ¯ t -cross section predictionsif expressed in terms of the running mass.We have performed a variant of the ABM12 fit, adding the combined t ¯ t cross section data fromLHC and Tevatron [18–23] to test the impact of these data on the gluon PDF, on the strong coupling α s and on the value and scheme choice for the top-quark mass. It is strictly necessary to considerthese three parameters together, since they are strongly correlated in theory predictions for the t ¯ t -cross section at the LHC. In Figs. 2.8 and 2.9 we present the χ profile versus the top-quark massfor the variants of the ABM12 fit with the t ¯ t cross section data included and for the two di ff erenttop-quark mass definitions, i.e., the MS mass m t ( m t ) and the pole mass m t (pole). Fig. 2.8 displays asteeper χ profile for the pole-mass definition. This implies a bigger impact of the t ¯ t -cross sectiondata in the fit and, as a consequence, greater sensitivity to the theoretical uncertainty at NNLOand uncalculated higher order corrections to the cross section beyond NNLO. In contrast, the χ profile for the MS mass is markedly flatter. Fig. 2.9 shows the χ profile for the subset of the t ¯ t -cross section data with NDP = ∆ χ t =
1, the value for the MS mass is obtained at NNLO m t ( m t ) = . ± . , (2.4)where we define the error in m t ( m t ) due the experimental data, the PDFs and the value of α s ( M Z )as the di ff erence between the value for m t ( m t ) at ∆ χ t = χ -profile inFig. 2.9. The additional theoretical uncertainty from the variation of the factorization and renor-malization scales in the usual range is small, ∆ m t ( m t ) = ± . m t (pole) = . ± . , (2.5)using the known perturbative conversion relations [50–52]. Eq. (2.5) can be compared to the valueof m t (pole) = . ± . ff from Fig. 2.9. This indicates good consistency of theprocedure and also with the top-quark mass values obtained from other determinations .Having established the sensitivity to the value of the top-quark mass, we have performed furthervariants of the ABM12 fit by fixing m t ( m t ) and m t (pole) in order to quantify the impact on thegluon PDF and on α s . The values for α s ( M Z ) which are obtained in these variants span the range α s ( M Z ) = . . . . . m t ( m t ) = . . .
163 GeV and α s ( M Z ) = . . . . . m t (pole) = . . . . µ = t ¯ t -cross section data in the fit andfixing m t ( m t ) and m t (pole) to the values indicated. For the running-mass definition the changesin the gluon PDF are within the uncertainties of the nominal ABM12 fit. In particular, we find The values in Eqs. (2.4) and (2.5) supersede the top-quark mass determination in [49], because full account ofthe correlations among all non-perturbative parameters is kept. t (m t )/GeV χ running mass m t (pole)/GeVpole mass Figure 2.8:
The χ profile versus the t -quark mass for the variants of ABM12 fit with the t ¯ t cross sectiondata included and di ff erent t -quark mass definitions: running mass (left) and pole mass (right). m t (m t )/GeV χ t NDP=5running mass m t (pole)/GeVpole mass Figure 2.9:
The same as in Fig. 2.8 for the t ¯ t cross section data subset. The NDP = α s ( M Z ) = . m t ( m t ) =
162 GeV fixed and with the CMS [19, 21, 22] and the Tevatron [23] data included, i.e.,leaving out the ATLAS data due to the larger experimental uncertainties. This is to be compared11 =3 GeV, n f =3x ∆ G ( % ) running massABM11ABM12 + tt - datam t =161 GeVm t =162 GeVm t =163 GeV x pole massm t =171 GeVm t =172 GeVm t =173.3 GeV Figure 2.10:
The relative uncertainty in the ABM12 gluon distribution in the 3-flavor scheme at the factor-ization scale of µ = t ¯ t crosssection data with the di ff erent mass definitions, running mass (left), pole mass (right), and the t -quark masssettings as indicated in the plot. with α s ( M Z ) = . t ¯ t cross section data and obtains the valueof α s ( M Z ) = . m t (pole) = . Hathor [48], though, which returns a slightly di ff erent the central value( O (1%) change) for the cross section compared to version 1.5. The sensitivity to α s is deter-mined from fits to sets of PDFs for varying values of α s ( M Z ), i.e., using the ABM11 set atNNLO ( abm11_5n_as_nnlo.LHgrid in the LHAPDF library [54, 55]) which covers the range α s = . . . . .
12. As a main caveat, the analysis of Ref. [53] misses the PDFs uncertaintiesfor the PDF sets with varying values of α s ( M Z ) and the correlations of the parameters, i.e., thegluon PDF, α s and m t (pole) discussed above.Ref. [56] explores the constraints on the gluon PDF from the same set of LHC and Tevatron t ¯ t cross section data [18–23] considered here. The analysis of Ref. [56] uses fixed values for α s and the pole mass m t (pole) and, thereby, disregards the correlation of these parameters with thegluon PDF. As illustrated in Fig. 2.10 this introduces a significant bias so that the fit results ofRef. [56] are a direct consequence of those assumptions. Ref. [56] also compares the ABM11PDFs [6] to those data [18–23] and quotes a value of χ = . NDP = χ -value is incomplete, since it is obtained by neglectingthe PDF uncertainties, the uncertainty in the value of m t (pole) as well as other uncertainties, whichmay have an impact on the χ -value such as the uncertainty in the beam energy, currently estimatedto be 1%. The χ profile in Fig. 2.9 shows that a faithful account of all sources of uncertainties andtheir correlation leads to a very good description of the t ¯ t cross section data.12 =3 GeV, n f =4 -20-100102010 -4 -3 -2 -1 x ∆ G ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 ABM11ABM12-ABM11 -4-202410 -4 -3 -2 -1 x ∆ u ( x , µ ) ( % ) -4-2024-5-2.502.5510 -4 -3 -2 -1 x ∆ d ( x , µ ) ( % ) -5-2.502.5510 -4 -3 -2 -1 -2002010 -4 -3 -2 -1 x ∆ s ( x , µ ) ( % ) -2002010 -4 -3 -2 -1 -20-100102010 -4 -3 -2 -1 x ∆ [ u - + d - ] ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 -5005010 -4 -3 -2 -1 x ∆ [ d - - u - ] ( x , µ ) ( % ) -5005010 -4 -3 -2 -1 Figure 3.1:
The 1 σ band for the 4-flavor NNLO ABM11 PDFs [6] at the scale of µ = x (shaded area) compared with the relative di ff erence between ABM11 PDFs and the ABM12 ones obtainedin this analysis (solid lines). The dotted lines display 1 σ band for the ABM12 PDFs. In this Section the results of the ABM12 fit are discussed in detail and compared specifically withthe previous ABM11 PDFs. Regarding the strong coupling constant α s ( M Z ) we also review thecurrent situation for α s -determinations from other processes, where the NNLO accuracy in QCDhas been achieved. Finally, we apply the new ABM12 PDF grids in the format for the LHAPDF library [54, 55] to compute a number of benchmark cross sections at the LHC.13 =3 GeV, n f =4 -20-100102010 -4 -3 -2 -1 x ∆ G ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 ABM12 (no LHC)+ LHC (1 iter.)+ LHC (2 iter.) -4-202410 -4 -3 -2 -1 x ∆ u ( x , µ ) ( % ) -4-2024-5-2.502.5510 -4 -3 -2 -1 x ∆ d ( x , µ ) ( % ) -5-2.502.5510 -4 -3 -2 -1 -2002010 -4 -3 -2 -1 x ∆ s ( x , µ ) ( % ) -2002010 -4 -3 -2 -1 -20-100102010 -4 -3 -2 -1 x ∆ [ u - + d - ] ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 -5005010 -4 -3 -2 -1 x ∆ [ d - - u - ] ( x , µ ) ( % ) -5005010 -4 -3 -2 -1 Figure 3.2:
The same as in Fig. 3.1 for the 1 σ band obtained in the variant of the ABM12 fit without theLHC DY data included (shaded area) and the relative change in the ABM12 PDFs due to the LHC DY dataobtained with one (solid line) and two (dashes) iterations of the fast algorithm used to take into account theDY NNLO corrections. The dotted lines display 1 σ band for the ABM12 PDFs obtained with one iterationof the algorithm. The PDFs obtained in the present analysis are basically in agreement with the ABM11 ones ob-tained in the earlier version of our fit [6] within the uncertainties, cf. Fig. 3.1. The strange quarkdistribution is particularly stable since in our analysis it is constrained by the neutrino-induceddimuon production that was not updated neither from the experimental nor from the theoreticalside. It is still significantly suppressed as compared to the non-strange sea and this contrastswith the strangeness enhancement found in the ATLAS PDF analysis based on the collider dataonly [14]. The change in the gluon distribution happens in particular due to impact of the HERA14 =3 GeV, n f =4 -20-100102010 -4 -3 -2 -1 x ∆ G ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 ABM12 (no LHC)+ LHCb (W + , W - )+ LHCb (Z)+ CMS (W + / W - )+ ATLAS (W + , W - , Z) -4-202410 -4 -3 -2 -1 x ∆ u ( x , µ ) ( % ) -4-2024-5-2.502.5510 -4 -3 -2 -1 x ∆ d ( x , µ ) ( % ) -5-2.502.5510 -4 -3 -2 -1 -2002010 -4 -3 -2 -1 x ∆ s ( x , µ ) ( % ) -2002010 -4 -3 -2 -1 -20-100102010 -4 -3 -2 -1 x ∆ [ u - + d - ] ( x , µ ) ( % ) -20-100102010 -4 -3 -2 -1 -5005010 -4 -3 -2 -1 x ∆ [ d - - u - ] ( x , µ ) ( % ) -5005010 -4 -3 -2 -1 Figure 3.3:
The same as in Fig. 3.2 for the variants of ABM12 fit including separate LHC DY data sets(sold line: LHCb [15], dots: LHCb [17], dashes: CMS [16], dashed dots: ATLAS [14]). charm data and improvements in the heavy-quark electro-production description, cf. Ref. [33] fordetails. At the same time the ABM12 quark distributions di ff er from the ABM11 ones at mostdue to the LHC DY data. This input contributes to a better separation of the non-strange sea andthe valence quark distributions. As a result, at the factorization scale µ = x ∼ . d -quark distribution goes up bysome 2%, cf. Fig. 3.2. In turn, this improvement allows for a better accuracy of both, the seaand the valence distributions, in particular, of the d -quark one. This improvement is particularlyvaluable since the accuracy of the latter is limited in the case of DIS data due to the uncertaintyin the nuclear correction employed to describe the deuterium-target data. The LHCb data on W + and W − production [15] provide the biggest impact on the PDFs as compared to other LHC data,cf. Fig. 3.3, due to the forward kinematics probed in this experiment. It is also worth noting that15 =2 GeV, n f =4 -5 -4 -3 -2 xG x xG x -5 -4 -3 -2 x(u - +d - )/2 x x(u - +d - )/2 x xxuxd -4 -3 -2 -1 x(d - -u - ) x -0.0200.020.040.0600.511.5210 -5 -4 -3 -2 xx(s+s - )/2 xx(s+s - )/2 Figure 3.4:
The 1 σ band for the 4-flavor NNLO ABM12 PDFs at the scale of µ = x obtainedin this analysis (shaded area) compared with the ones obtained by other groups (solid lines: JR09 [57],dashed dots: MSTW [58], dashes: NN23 [59], dots: CT10 [60]). the gluon distribution is also sensitive to the existing LHC DY data and in the ABM fit they pullit somewhat up (down) at small (large) x . However, in general, the changes are within the PDFuncertainties. This justifies our approach of using the set of PDF uncertainties to pre-calculatethe NNLO DY cross-section grid and then to compute those cross sections by grid interpolationin minimal time. To provide the best accuracy of this algorithm the ABM12 PDFs are producedtaking the DY cross-section grid calculated for the PDFs obtained in the variant of ABM12, whichdi ff ers from the nominal ABM12 one by inclusion of the LHC data only. Furthermore, to checkexplicitly the stability of the algorithm we perform a second iteration of the fit based on the DYcross-section grids prepared with the PDFs obtained in the first iteration. The iterations demon-strate nice convergence and the first iteration su ffi ces to obtain an accurate result, cf. Fig. 3.2.16he NNLO PDFs obtained in this analysis are compared to the results of other groups inFig. 3.4. Our PDFs are in reasonable agreement with the newly released CT10 PDFs [60]. Themost striking di ff erence is observed for the large- x gluon distribution, which is constrained bythe Tevatron jet data in the CT10 analysis. It is worth noting that this constraint is obtained forCT10 using the NLO corrections only, while the NNLO corrections may be as big as 15-25% [61].Therefore the discrepancy between CT10 and our result should decrease once the NNLO correc-tions to the jet production are taken into account. Comparison of the ABM12 PDFs with the onesobtained by other groups demonstrate the trend similar to the ABM11 case [6]. The most essentialdi ff erence appears in the large- x gluon distribution. It is also constrained by the Tevatron jet datafor MSTW08 [58] and NN23 [59], with the NNLO corrections due to the threshold resummationtaken into account in this case. However, the threshold resummation terms used in Refs. [58, 59]introduce additional theoretical uncertainties [62]. Therefore, a conclusive comparison with ourresults is still impractical. The spread in the small- x gluon distribution obtained by di ff erent groupscan be consolidated with the help of the H1 data on the structure function F L [63] being sensitive inthis region. Similarly, di ff erences in the estimates of the non-strange sea distribution at x ∼ . ff erence in the theoretical accuracy of the analyses since all the groups use the CCFR andNuTeV data on the neutrino-induced dimuon production [64] as a strange sea constraint and takeinto account the NLO corrections to this process [65, 66]. The very recent precise data on theneutrino-induced dimuon production by NOMAD [67] are still not included in the present anal-ysis. However, they demonstrate good agreement with the ABM11 prediction and may help toconsolidate di ff erent estimates of the shape of the strange sea. The strong coupling constant α s ( M Z ) is measured together with the parameters of the PDFs, theheavy-quark mass m c and the higher twist parameters within the analysis. The present accuraciesof the scaling violations of the deep-inelastic world data make the use of NNLO QCD correctionsmandatory. At NLO the scale uncertainties typically amount to O (5%), cf. [68], and, therefore, aresimply too large.The value of α s ( M Z ) obtained in the present analysis is α NNLO s ( M Z ) = . ± . . (3.1)This result is in excellent agreement with those given by other groups and by us in Refs. [5, 6, 57,69–71], see Tab. 3.1. As has been shown in [6] in detail the α s -values obtained upon analyzingthe partial data sets from BCDMS [72, 73], NMC [74, 75], SLAC [76–81], HERA [13], and theDrell-Yan data [82, 83] both at NLO and NNLO do very well compare to each other and to thecentral value within the experimental errors.Fits including jet data have been carried out before both by JR [84] and ABM [6, 85], alongwith other groups, performing systematic studies including both jet data from the Tevatron andin [6] also from LHC . We would like to note that it is very problematic to call present NNLOfits of the world DIS data including jet data NNLO analyses, since the corresponding jet scatteringcross sections are available at NLO only . The complete NNLO results for the corresponding jet Contrary statements given in Refs. [86, 87] are incorrect; see Ref. [6] for further details. s ( M Z ) ABM JR CT MSTW NNPDFDISDIS + Tevatron jets
Figure 3.5:
The values of α s ( M Z ) at NNLO obtained in the PDF fits of ABM (solid bars: this analysis,dashed bars: ABM11 [6]) in comparison with the CT [60], JR [57], MSTW [58] and NNPDF [88] results. cross sections have to be used in later analyses, since threshold resummations are not expected todeliver a su ffi cient description [62] .The α s ( M Z ) values for some PDF groups are illustrated in Fig. 3.5. In Tab. 3.1 a generaloverview on the values of α s ( M Z ) at NNLO is given, with a few determinations e ff ectively atN LO in the valence analyses [69, 70], and the hadronic Z -decay [89]. The BBG, BB, GRS,ABKM, JR, ABM11, CTEQ analyses and the present analysis find lower values of α s ( M Z ) witherrors at the 1–2% level, while NN21 and MSTW08 find larger values analyzing the deep-inelasticworld data, Drell-Yan data, and partly also jet data with comparable accuracy to the former ones. Reasons for the higher values being obtained by NN21 and MSTW08 were given in [6] before.As has been shown in [91] a consistent F L -treatment for the NMC data and the BCDMS-data,cf. [69], is necessary and leads to a change of the value of α s ( M Z ) to lower values. Furthermore,the sensitivity on kinematic cuts applied to remove higher twist e ff ects has been studied. In theflavor non-singlet case this can be achieved by cutting for W > . , cf. [69]. In the singletanalysis there are also higher twist contributions in the lower x -region to be removed by applyingthe additional cut of Q >
10 GeV , which, however, is not used by NN21 and MSTW08. Weperformed a fit without accounting for the higher twist terms and allowed for the range of datadown to values of Q > . at W > . , [6]. One obtains α s ( M Z ) = . ± . α s values in the fitsby NN21 and MSTW08, furthermore, show strong variations with respect to di ff erent DIS datasets [6], despite of the similar final value.The analyses of thrust in e + e − data by two groups also find low values, also with errors at the1% level. Higher values of α s ( M Z ) are obtained for the e + e − Z -decay, and τ -decay within various analyses. The value of α s ( M Z ) has also been determined in di ff erent latticesimulations to high accuracy. The N LO values for α s ( M Z ) in the valence analyses [69, 70] yieldslightly larger values than at NNLO. They are fully consistent with the NNLO values within errors. Partial NNLO results on the hadronic di-jet cross section are available [61]. Very recently MSTW [90] reported lower values for α s ( M z ) also related to the LHC data. α s ( M Z ) at NLO using the jetdata [92, 93]. The ATLAS and CMS jet data span a wider kinematic range than those of Teva-tron and will allow very soon even more accurate measurements. In the analysis [93] α s ( M Z ) isdetermined scanning grids generated at di ff erent values of the strong coupling constants by thedi ff erent PDF-fitting groups. These are used to find a minimum for the jet data. Including the scaleuncertainties the following NLO values are obtained for the 3 / α s ( M Z ) = . ± . . ) ± . + . − . (scale) NNPDF21 [94] , (3.2) α s ( M Z ) = . ± . . ) [0 .
180 (favored value)] CT10 [60] , (3.3) α s ( M Z ) = . ± . . ) [0 . + . − . ] MSTW08 [95] . (3.4)A comparable NLO value has been reported using ATLAS jet data [92] α s ( M Z ) = . ± . . ) + . − . (th . ) . (3.5)Interestingly, rather low values are obtained already at NLO. In parenthesis we quote the NLOvalues for α s ( M Z ) in Eqs. (3.2)-(3.4) which are obtained by the fitting groups at minimal χ .Obviously the values found in the jet-data analysis do not correspond to these values. Yet, the NLOscale uncertainty in this analysis is large. Recently the jet energy-scale error has been improvedby CMS [93], leading to a significant reduction of the experimental error. The gluonic NNLO K -factor is positive; as shown in Fig. 2 of Ref. [61] the scale dependence for µ = µ F = µ R behavesflat over a wide range of scales. It is therefore expected that also the error due to scale variationwill turn out to be very small in the NNLO analysis. It will be important to repeat this analysis andto fit the LHC jet data together with the world deep-inelastic data, which will be also instrumentalfor the determination of the gluon distribution at large scales.The present DIS world data together with the F c ¯ c ( x , Q ) data, are competitive in the determi-nation of the charm quark mass in a correlated fit with the PDF-parameters and α s ( M Z ). For theMS mass the value of m c ( m c ) = . ± .
03 (exp . ) + . − . (th . ) GeV (3.6)is obtained at NNLO, see also [33]. At present this analysis is the only one, in which all knownhigher order heavy-flavor corrections to deep-inelastic scattering have been considered. This valuestill should be quoted as of approximate NNLO, since the NNLO-corrections are only mod-eled [26] combining small- x and threshold resummation e ff ects with information of the 3-loopmoments of the heavy-flavor Wilson coe ffi cients [31] at high values of Q . Two scenarios havebeen considered in [26] to parameterize the Wilson coe ffi cients accounting for an estimated er-ror. Here the fit favors a region of the parameter d N ∈ [ − . , . . The value in Eq. (3.6) compares well to the present world average of m c ( m c ) = . ± .
025 GeV [1].It is needless to say that the determination of a fundamental parameter of the SM, such as m c , has to follow a thorough quantum field-theoretic prescription, rather than specific models also The calculation of the exact NNLO heavy-flavor Wilson coe ffi cients is underway [32, 109–111]. s ( M Z )BBG 0 . + . − . valence analysis, NNLO [69]BB 0 . ± . .
112 valence analysis, NNLO [71]ABKM 0 . ± . n f = . ± . . ± . . ± . . ± . . ± . . ± . . − . . ± . . ... . . . + . − . valence analysis, N LO( ∗ ) [69]BB 0 . ± . LO( ∗ ) [70] e + e − thrust 0 . ± . e + e − thrust 0 . + . − . Gehrmann et al. [98]3 jet rate 0 . ± . . ± . /
12, N LO [89, 100] τ decay 0 . ± . τ decay 0 . ± . τ decay 0 . ± . . ± . + . ± . . ± . + + . ± . + . ± . . ± . Summary of recent NNLO and N LO QCD analyses of the DIS world data, supplemented byrelated measurements using a series of other processes and lattice determinations. In case that jet data fromhadron colliders are used in the analysis the values of α s ( M Z ) cannot be considered NNLO values. being found in the literature, cf., e.g. [112]. Despite the correct renormalization procedure of the20eavy-flavor Wilson coe ffi cients to 3-loop order were known [31] to 3-loop order, there are stilleven massless scenarios to this level, cf. e.g. [113], which ignore the exact theoretical description. In this Section we quantify the impact of the new PDF set on the predictions for benchmark crosssections at the LHC for various c.m.s. energies. To that end, we confine ourselves to (mostly)inclusive cross sections which are known to NNLO in QCD, see [6, 7] for previous benchmarknumbers, since the NNLO accuracy is actually the first instance, where meaningful statementsabout the residual theoretical uncertainty are possible given the precision of present collider dataand the generally large residual variation of the renormalization and factorization scale at NLO.In detail, we consider the following set of inclusive observables at NNLO in QCD: hadronic W -and Z -boson production [114, 115], the cross section for Higgs boson production in gluon-gluonfusion [115–118], and the cross section for top-quark pair production [43–47]. We have used the LHAPDF library [54, 55] for the cross section computations to interface to our PDFs provided in theform of data grids for n f = , LHAPDF library , abm12lhc_3_nnlo.LHgrid (0+28),abm12lhc_4_nnlo.LHgrid (0+28),abm12lhc_5_nnlo.LHgrid (0+28), which contains the central fit and 28 additional sets for the combined symmetric uncertainty on thePDFs, on α s and on the heavy-quark masses. All PDF uncertainties quoted here are calculated inthe standard manners, i.e., as the ± σ -variation. W - and Z -boson production We start by presenting results for W - and Z -boson production at the LHC. For the electroweak pa-rameters, we follow [6,7] and choose the scheme based on the set ( G F , M W , M Z ). According to [1],we have G F = . × − GeV − , M W = . ± .
015 GeV, M Z = . ± . Γ ( W ± ) = . ± .
042 GeV and Γ ( Z ) = . ± . s Z = − M W ˆ ρ M Z = . ± . , (3.7)and ˆ ρ = . ± . θ c yields the value of sin θ c = . W - and Z -boson production, namely the choice of PDF setswith n f = n f = ff erences are less than 1 σ in the PDF uncertainty and become succes-sively smaller as perturbative corrections of higher order are included. The
LHAPDF library can be obtained from http://projects.hepforge.org/lhapdf together with installationinstructions. HC7 W + W − W ± Z ABM11 59 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 58 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.2:
The total cross sections [pb] for gauge boson production at the LHC with √ s = n f = µ = M W / Z / µ = M W / Z and, respectively, the 1 σ PDF uncertainty.LHC8 W + W − W ± Z ABM11 68 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 67 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.3:
The same as Tab. 3.2 for the LHC with √ s = W + W − W ± Z ABM11 110 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 108 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.4:
The same as Tab. 3.2 for the LHC with √ s =
13 TeV.LHC14 W + W − W ± Z ABM11 119 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 116 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.5:
The same as Tab. 3.2 for the LHC with √ s =
14 TeV.
Let us now discuss the cross section for the SM Higgs boson production in the gluon-gluon fu-sion channel, which is predominantly driven by the gluon PDF and the value of α s ( M Z ) from thee ff ective vertex. The known NNLO QCD corrections [115–118] still lead to a sizable increase in LHC7 LHC8 LHC13 LHC14ABM11 13 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 13 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.6:
The total Higgs production cross sections [pb] in gluon-gluon fusion for the PDF sets ABM11and ABM12 at NNLO accuracy using a Higgs boson mass m H =
125 GeV. The errors shown are the scaleuncertainty based on the shifts µ = m H / µ = m H and, respectively, the 1 σ PDF uncertainty. HC7 LHC8 LHC13 LHC14HiggsXSWG [119] 15 . + . − . + . − . . + . − . + . − . − . + . − . + . − . Table 3.7:
The total Higgs production cross sections [pb] in gluon-gluon fusion of [119] used by ATLASand CMS for a Higgs boson mass m H =
125 GeV. The errors shown are the scale uncertainty and, respec-tively, the PDF + α s uncertainty. the cross section at nominal values of the scale, i.e. µ = m H , and it is well established that afurther stabilization beyond NNLO may be achieved on the basis of soft gluon resummation, seee.g., [120]. At NNLO accuracy in QCD the theoretical uncertainty from the scale variation isdominating by far over the PDF uncertainty. Using a Higgs boson mass m H =
125 GeV in Tab. 3.6we observe again only rather small changes between the ABM11 and the ABM12 predictions.This demonstrates that the gluon PDF is well constrained from existing data and that the ABM11results are consistent with the new fit based on including selected LHC ones.It is therefore interesting to compare the ABM predictions in Tab. 3.6 to the cross section valuesrecommended for use in the ongoing ATLAS and CMS Higgs analyses [119], cf., Tab. 3.7 . Thecentral values of the ABM predictions are significantly lower by some 11-14 %. Only a smallfraction of this di ff erence can be attributed to the inclusion of soft gluon resummation beyondNNLO, which typically does reduce the scale dependence, though, as is obvious from Tab. 3.7,and to the inclusion of other quantum corrections in [119], e.g., the electro-weak ones. Muchlarger sensitivity of the Higgs cross section predictions arise from theory assumptions made in theanalyses, e.g., for constraints from higher orders in QCD due to the treatment of fixed-target DISdata, see [91]. The most interesting aspect is the fact, that the PDFs + α s error in [119] is inflatedroughly by a factor of 4 in comparison to our predictions in Tab. 3.6, where we quote the 1 σ PDF(and α s of course) error entirely determined from the correlated experimental uncertainties in thefitted data. In summary, the cross section predictions [119] used in the current Higgs analyses atthe LHC are subject to both, a bias due to specific theory assumptions made in PDF and α s fitsas well as largely overestimated uncertainties of the relevant non-perturbative input. Thus, checksof correlations between experimental data for di ff erent scattering processes at the LHC and theirsensitivity to PDFs along the lines of Sec. 2 are urgently needed to consolidate this issue, cf. [121]. Finally, we present predictions for the total cross section for t ¯ t -pair hadro-production in Tabs. 3.8and 3.9. Using the program Hathor (version 1.5) [48] which incorporates the recently completedQCD corrections at NNLO [43–46], we give numbers for two representative top-quark masses,that is the running mass m t ( m t ) =
162 GeV in the MS mass scheme and the pole mass m t (pole) =
171 GeV in the on-shell scheme.At NNLO accuracy in QCD, the PDF uncertainties given in Tabs. 3.8 and 3.9 are dominating incomparison to the theory uncertainties based on scale variation. As discussed at length in Sec. 2.4the LHC data for t ¯ t -pair production included in the ABM12 fit predominantly constrains the top-quark mass and has little impact on the gluon PDF and on the value of the strong coupling constant α s ( M Z ). Therefore the cross section predictions of the ABM11 and ABM12 PDFs largely coincide. See also https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CERNYellowReportPageAt7TeV for de-tails. HC7 LHC8 LHC13 LHC14ABM11 141 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 143 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.8:
The total cross section for top-quark pair-production at NNLO [pb] using a pole mass m t (pole) =
171 GeV and the PDF sets ABM11 and ABM12 and with the errors shown as σ + ∆ σ scale + ∆ σ PDF . The scaleuncertainty ∆ σ scale is based on maximal and minimal shifts for the choices µ = m t (pole) / µ = m t (pole)and ∆ σ PDF is the 1 σ combined PDF + α s error.LHC7 LHC8 LHC13 LHC14ABM11 148 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . ABM12 150 . + . − . + . − . . + . − . + . − . . + . − . + . − . . + . − . + . − . Table 3.9:
The same as Tab. 3.8 for a running mass m t ( m t ) =
162 GeV in the MS scheme.
We have presented the PDF set ABM12, which results from a global analysis of DIS and hadroncollider data including, for the first time, the available LHC data for the standard candle processessuch as W ± - and Z -boson and t ¯ t -production. The analysis has been performed at NNLO in QCDand along with the new data included also progress in theoretical predictions has been reflectedaccordingly. The new ABM12 analysis demonstrates very good consistency with the previousPDF sets (ABM11, ABKM09) regarding the parameter values for PDFs as well as the strongcoupling constant α s ( M Z ) and the quark masses. Continuous checks for the compatibility of thedata sets along with a detailed account of the systematic errors and of the correlations among thefit parameters have been of paramount importance in this respect.In detail, we have considered new HERA data sets on semi-inclusive charm production in DISin Sec. 2.1 which have allowed to validate the c -quark production mechanism in the FFN schemerelying on 3 light flavors in the initial state and leading to a precise determination of the running c -quark mass. As another new DIS data set, the neutral-current inclusive data at high Q fromHERA has been included, which exhibits sensitivity to the exchange of photons, Z -bosons as wellas to γ - Z -interference. Our analysis in Sec. 2.2 corroborates again the fact, that even at high scalesthe FFN scheme is su ffi cient for description of the DIS data.The fit of LHC precision data on W ± - and Z -boson production improves the determinationon the quark distributions at x ∼ . d -quark distribution. The fitshows good consistency and a further reduction of the experimental systematic uncertainties wouldcertainly strengthen the impact of the LHC DY data in global fits. On the technical side, we remarkthat the fit of DY data has been based on the exact NNLO di ff erential cross section predictions,expanded over the set of eigenfunctions spanning the basis for the ABM PDF uncertainties. Thishas served as a starting point for a rapidly converging fit including the LHC DY data with accountof all correlations.Also data for the total t ¯ t -cross section has been smoothly accommodated into the fit. A propertreatment of the correlation between the gluon PDF, the strong coupling constant α s ( M Z ) andthe top-quark mass has been crucial here. Moreover, the running-mass definition for the top-24uark provides a better description of data as compared to the pole mass case, the latter showingstill sizable sensitivity to perturbative QCD corrections beyond NNLO accuracy. Our analysis inSec. 2.4 yields a precise value with an uncertainty of roughly 1.5 % for the MS mass m t ( m t ) whichhas been used to extract m t (pole) at NNLO.In summary, the new ABM12 fit demonstrates, that a smooth extension of the ABM global PDFanalysis to incorporate LHC data is feasible and does not lead to large changes in the fit results.As we have shown in Sec. 3.1 di ff erences with respect to other PDFs sets remain. However, thesedi ff erences are based either on a di ff erent treatment of the data sets or on di ff erent theoretical de-scriptions of the underlying physical processes and we have commented on the correctness of someof those procedures. In particular, the value of strong coupling constant α s ( M Z ) in our analysis re-mains largely unchanged as documented in Sec. 3.2 and the theoretical predictions for benchmarkcross section at the LHC are very stable. This particularly applies to the cross section for Higgsproduction in the gluon-gluon fusion shown in Sec. 3.3. We commented on the implications forthe ongoing Higgs analyses at the LHC.The precision of the currently available experimental data make global analyses at NNLOaccuracy in QCD mandatory. This o ff ers the great opportunity for high precision determinationsof the non-perturbative parameters relevant in theory predictions of hadron collider cross section.At the same time, the great sensitivity to the underlying theory allows to test and to scrutinizeremaining model prescriptions and, eventually, to reject wrong assumptions. Acknowledgments
We would like to thank H. Böttcher for discussions, P. Jimenez-Delgado and E. Reya for a pri-vate communication prior to publication. We gratefully acknowledge the continuous support ofM. Whalley to integrate the results of the new ABM fit into the
LHAPDF library [54, 55].J.B. acknowledges support from Technische Universität Dortmund. This work has been sup-ported by Helmholtz Gemeinschaft under contract VH-HA-101 (
Alliance Physics at the Teras-cale ), by Deutsche Forschungsgemeinschaft in Sonderforschungsbereich / Transregio 9, by Bun-desministerium für Bildung und Forschung through contract (05H12GU8), and by the EuropeanCommission through contract PITN-GA-2010-264564 (
LHCPhenoNet ). Note added:
While this work was being finalized, a new combination of measurements of the top-quark pair production cross section from the Tevatron appeared [122], which carries a combinedexperimental uncertainty of 5.4%. This measurements yields σ pp → t ¯ t = . ± .
42 pb for a valueof m t (pole) =
171 GeV for the top-quark pole mass, which is consistent with the NNLO crosssection prediction σ pp → t ¯ t = . + . − . + . − . pb based on the ABM12 PDFs at NNLO within theuncertainties. A A fast algorithm for involved computations in PDF fits
The accommodation of the di ff erent data sets for the PDF fit demands very involved computationsof the QCD corrections to the Wilson coe ffi cients. In particular this applies to the calculation of therapidity distribution of the W - and Z -boson decay products produced in hadronic collisions, whichare based on the fully exclusive NNLO codes DYNNLO [41] and
FEWZ [42]. The typical CPU run-time needed to achieve a calculation accuracy of much better than the uncertainty of the presentdata using the codes [41, 42] amounts to O (100) hours. Therefore an iterative use of the availablefully exclusive DY codes in the QCD fit is widely impossible. Instead, these codes are commonly25un in advance for the variety of PDF sets, covering the foreseeable spread in the PDF variation,the results of which are stored grids. Afterwards the cross section values for a given PDF set canbe computed in a fast manner using linear grid interpolations. For the first time this approachwas implemented in the code fastNLO [123] for the NLO corrections to the jet productions crosssections. A similar approach is also used in the code AppleGrid [124] which provides a tool forgenerating the cross section grids of di ff erent processes, including the DY process. Since fastNLO and AppleGrid are tools of general purpose, the PDF basis used to generate those grids need to besu ffi ciently wide to cover the di ff erences between the existing PDF sets. Meanwhile the possiblevariations of the PDFs in a particular fit are not very large, i.e. if a new fit is aimed to accommodatea new data set being in su ffi cient agreement with those used in earlier versions of the fit, one mayexpect variations of the PDFs being comparable to their uncertainties. In this case the PDF basisused to generate the grids for the cross section can be reliably selected as a PDF-bunch, whichencodes the uncertainties in a given PDF set. For the PDF uncertainties estimated with the Hessianmethod this bunch is provided by the PDF set members corresponding to the 1 σ variation in thefitted parameters. This allows to minimize the size of the pre-calculated cross section grids andreduces the CPU time necessary to generate these grids correspondingly. Moreover, the structureof the calculation algorithm in using these grids for the PDF fit turns out to be simple. In thisappendix we describe, how this approach is implemented in the present analysis.Firstly, we remind the basics of the PDF uncertainty handling, see Ref. [6] for details. Let ~ q ( P i )be the vector of parton distributions encoding the gluon and quark species. It depends on the PDFparameters P i with the index i ∈ [1 , N p ] and N p the number of parameters. P i denote the parametervalues obtained in the PDF fit and ∆ P i are their standard deviations. In general the errors in theparameters are correlated, which is expressed by a non-diagonal covariance matrix C i j . However,it is diagonal in the basis of the covariance matrix’ eigenvectors which makes this basis particularconvenient for the computation of the PDF error. The vector of the parameters P i transformed intothe eigenvector basis reads ˜ P i = N p X k = (cid:16) √ C (cid:17) − ik P k , (A.1)where √ C i j = N p X k = A ik √ D k j . (A.2)Here A ik denotes the matrix with the columns given by the orthonormal eigenvectors of C i j , √ D k j = δ jk √ e k , e k are the eigenvalues of C i j , and δ jk is the Kronecker symbol. The PDF uncertainties arecommonly presented as the shifts in ~ q due to variation of the parameters ˜ P i by their standarddeviation. Since the latter are equal to one the shifts are given by d ~ qd ˜ P i = N p X k = d ~ qdP k (cid:16) √ C (cid:17) ik . (A.3)Moreover, the parameters ˜ P i are uncorrelated. Therefore the shifts in Eq. (A.3) can be combinedin quadrature to obtain the total PDF uncertainty. In a similar way the uncertainty in a theoreticalprediction t ( ~ q ) due to the PDFs can be obtained assuming its linear dependence on the PDFs as acombination of the variations ∆ t ( k ) = t " ~ q ( P k ) + d ~ qd ˜ P k − t h ~ q ( P k ) i (A.4)26n quadrature.Now we show how new data on the hadronic hard-scattering process can be consistently ac-commodated into the PDF fit avoiding involved cross section computations. Let P fit i the currentvalues of the PDF parameters in the fit with the new data set included and δ P i = P fit i − P i , where P i stands for the PDF parameter values obtained in the earlier version of the fit performed withoutthe new data-set. The current PDF value can be expressed in terms of δ P i and the PDF variation inthe eigenvector basis as follows ~ q fit = ~ q ( P i ) + d ~ qd ˜ P i δ ˜ P i , (A.5)where δ ˜ P i = N p X k = (cid:16) √ C (cid:17) − ik δ P i . (A.6)A shift in the hard-scattering cross section corresponding to the variation of the i -th PDF parameterin the fit reads δ t ( k ) = t " ~ q ( P k ) + d ~ qd ˜ P k δ ˜ P k − t h ~ q ( P k ) i ≈ ∆ t ( k ) N p X l = (cid:16) √ C (cid:17) − il δ P i (A.7)and the total change in t is the sum of terms in Eq. (A.7) over all parameters being fitted. Theapproximation Eq. (A.7) allows fast calculations of the cross section for the new data added to thePDF fit since the values of σ h ~ q ( P i ) i and ∆ σ i can be prepared in advance. This approach is justifiedif the variation of the parameters in the new fit is localized within their uncertainties obtained inthe previous fit or in case of su ffi cient linearity of the PDFs with respect to the fitted parametersand the cross sections depending on the PDFs. Furthermore, if the algorithm does not seem toguarantee su ffi cient accuracy, it can be applied iteratively, with the update of the σ h ~ q ( P i ) i and ∆ t i values at each iteration. 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