F cc _ Q =GeV Q =4GeV Q =6.5G.5-4 -3-4 -3-4.5 Q = GeV Q = GeV Q =35 G -3-3 - -3.5 Q =6 GeV Q =GeV Q =G. -3 - Q =4GeV - - - - - Q = GeV - HERA (pre H D* HERA H D* HERA H LTT HER H LTT HER ZEUS µ 5 ZEUS D + 5 ZEUS D 5 ZEUS D* 99 ZEUS D* 96 x
HERAPDF F cc _. Q = GeV Q = 4 GeV Q = 6..5 Q = GeV Q = GeV Q = 35.5 Q = 6GeV Q =GeV Q =.5 Q =4GeV Q =GeV -4-3 HERA (prel -4-3 - -4-3 - x
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs Modern Physics Letters A c World Scientific Publishing Company arxiv:37.7875v [hep-ph] 3 Jul 3 DETERMINATION OF CHARM QUARK MASS AND α s (M Z ) FROM HERA DATA AMANDA COOPER-SARKAR Dept. of Physics, University of Oxford,Keble Rd, OXFORD, OX 3RH, UK email:a.cooper-sarkar@physics.ox.ac.uk Received (4 May 3) Revised () Charm production data from HERA may be used to determine the charm quark mass and jet production data from HERA may be used to detrmine α s(m Z ). Recent results are summarised.. Introduction HERA was an electron(positron)-proton collider located at DESY, Hamburg. It ran in two phases HERA-I from 99- and HERA-II 3-7. Two similar experiments,, took data. In HERA- running each experiment collected pb of e + p data and 5pb of e p data with electron beam energy 7.5GeV and proton beam energies 8, 9 GeV. In HERA-II running each experiment took 4pb of e + p data and 8pb of e p data with the same electron beam energy and proton beam energies 9 GeV. Deep inelastic lepton-hadron scattering data has been used both to investigate the theory of the strong interaction and to determine the momentum distributions of the partons within the nucleon. The data from the HERA collider now dominate the world data on deep inelastic scattering since they cover an unprecedented kinematic range: in Q, the (negative of the) invariant mass squared of the virtual exchanged boson,.45 < Q < 3 5 ; in Bjorken x, 6 7 < x <.65. Futhermore, because the HERA experiments investigated e + p and e p, charge current(cc) and neutral current (NC) scattering, information can be gained on flavour separated up- and down-type quarks and antiquarks and on the gluon- from its role in the scaling violations of perturbative quantum-chromo-dynamics. The formalism of how the deep inelastic cross sections relate to the parton distributions though the QCD-improved parton model, and how parton didtribution functions are then extracted from the data, is well documented (see for example ref. ) and only a brief description of the HERAPDF formalism will be given here. This contribution concentrates on results of relevance for the fundamental parameters of QCD: α s (M Z ) and the charm qurak mass m c.
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs A M Cooper-Sarkar. HERAPDF Formalism PerturbativeQCD predicts the Q evolutionofthe partondistributions, but not the x dependence. Parton distributions are extracted by performing a direct numerical integration of the DGLAP evolution equations at NLO. A parametrised analytic shape for the parton distributions is assumed to be valid at some starting value of Q = Q. For the HERAPDF the value Q =.9 GeV is chosen such that the starting scale is below the charm mass threshold, Q < m c. Then the DGLAP equations are used to evolve the parton distributions up to a different Q value, where they are convoluted with NLO coefficient functions to make predictions for the structure functions. The heavy quark coefficient functions are usually calculated in the general-mass variable-flavour-number scheme of Thorne. However, for the study of the charm mass various other schemes have been considered, see Section 3. The heavy quark masses for the central HERAPDF fits were chosen to be m c =.4 GeV and m b = 4.75 GeV and the strong coupling constant was fixed to α s (M Z ) =.76. (The values of m c and α s (M Z ) are varied when studies of these parameters are performed.) The predictions are then fitted to the combined HERA data sets for NC and CC e + p and e p scattering. A minimum Q cut of Q min = 3.5GeV wasimposedtoremaininthekinematicregionwhereperturbative QCD should be applicable. The fit parameters are those necessary to specify the input analytic shape. The valence quark distributions xu v, xd v, and the u-type and d-type anti-quark distributions xū, x D (xū = xū, x D = x d + x s) are parametrised at the input scale Q =.9GeV by the generic form xf(x) = Ax B ( x) C (+Dx+Ex ). () The parametrisation for the gluon distribution xg can be extended, to include a term A g x B g ( x) C g, such that the NLO gluon may become negative at low x and low Q (however it does not do so in the kinematic region of the HERA data). The normalisation parameters, A g,a uv,a dv, are constrained by the quark number sum-rules and momentum sum-rule. Constraints are imposed to ensure that ū = d at low-x, and to specify the strangeness fraction in the sea. The full details for all HERAPDFs are given in ref. The experimental uncertainties on the HERAPDF are determined using the conventional χ tolerance, χ =, for 68%C.L. However model uncertainties and parametrisation uncertainties are also considered. The choice of the heavy quark masses is varied, the choice of Q min is varied and the strangeness fraction is varied. Parametrisation variations for which the E and the D parameters for all the PDFs are freed are considered and variation of the starting scale Q in the range,.5 < Q <.5GeV, is also considered. These model and parametrisation uncertainties are an integral part of the HERAPDF uncertainties.
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs Determination of charm quark mass and α s(m Z ) from HERA data 3 σ r,nc (x,q ) +.6.4. x=. x=. HERA I NC e + p ZEUS H x=.8.8.6 x=.3.4. x=.8 x=.5 3 4 Q / GeV Fig.. HERA combined NC e + p reduced cross section as a function of Q forsix x-binscompared to the separate data input to the averaging procedure. The individual measurements are displaced horizontally for a better visibility. 3. Results From 8, the experiments began to combine their data in order to provide the most complete and accurate set of deep-inelastic data as the legacy of HERA. Data on inclusive cross-sections have been combined for the HERA-I phase of running 3 and a preliminary combination has been made also using the HERA-II data 5. HERA-jet data have been added to this combination in order to give more information on α s (M Z ) and the gluon distribution 8. Very recently a combination of data on F c c has been used to give information on the charm quark mass 4. The combination of the data sets takes into account the full correlated systematic uncertainties of the individual experiments such that the total uncertainty of the combined measurement is typically smaller than %, for 3 < Q < 5 GeV, and reaches %, for < Q < GeV. In Fig averaged data are compared to the input data, illustrating the improvement in precision. Because of the reduction in size of the systematic error this improvement is far better than would be expected simply from the rough doubling of statistics which combining the two experiments represents. A combination has also been made of data on F c c 4 from various different methods of tagging charm: using the D, using the vertex detectors to see the displaced decay vertex, using direct D,D + production identified using the vertex detectors, and indentifying semi-leptonic charm decays via muons, also using the vertex detectors. The results of the F c c combination compared to the separate measurements
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs 4 A M Cooper-Sarkar cc _ σ red. H VTX H D* HERA-I ZEUS D* 98- ZEUS D H D* HERA-II ZEUS c µ X ZEUS D* 96-97 ZEUS D + Q =.5 GeV Q =5 GeV Q =7 GeV.5 Q = GeV Q =8 GeV Q =3 GeV.5 Q =6 GeV Q = GeV Q = GeV.5 Q =35 GeV Q =65 GeV Q = GeV HERA -4-3 - -4-3 - -4-3 - x Fig.. The HERA combined measurement of F c c compared to the data sets of used for the combination. these data sets are slightly displaced in x for visibility. red σ cc Q =.5 GeV Q = 5 GeV Q = 7 GeV....5 Q = GeV.5 Q =8 GeV.5 Q =3 GeV.5 Q =6 GeV.5 Q = GeV.5 Q = GeV.5 Q =35 GeV.5 Q =65 GeV.5 Q = GeV HERA HERAPDF.5-4 -3 - -4-3 - -4-3 - x Fig. 3. The HERA combined measurement of F c c compared to the predictions of HERAPDF. which went into it are shown in Fig. The F c c combination is shown compared to the predictions of HERAPDF. in Fig. 3. 3.. Charm quark mass The uncertainty bands on these predictions are dominated by the variation of the charm quark mass,.35 < m c <.65GeV, used for the model uncertainty. The combined F c c data can help to reduce this uncertainty. Fig. 4 (top left) compares the χ, as a function of the charm mass, for a fit which includes charm data to that
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs Determination of charm quark mass and α s(m Z ) from HERA data 5 for the HERAPDF. fit which does not use these data, when using the Thorne- Roberts (RT) variable-flavour-number(vfn) scheme. The charm data have a clear preference for a particular charm quark mass. However, the RT heavy flavour VFN scheme is not unique, specific choices are made for threshold behaviour. In Fig. 4 (top right) the χ profiles for the standard and the optimized versions (optimized for smooth threshold behaviour) of this scheme are compared. The same figure also compares the alternative ACOT VFN schemes and the Zero-Mass VFN scheme. Each of these schemes favours a different value for the charm quark mass, and the fit to the data is good for all the heavy-quark-mass schemes considered. Each of the VFN schemes has been used to predict W and Z production for the LHC and their predictions for W + are shown in Fig. 4 (bottom left) as a function of the charm quark mass. If a particular value of the charm mass is chosen then the spread of predictions is as large as 7%. However this spread is considerably reduced % if each heavy quark scheme is used at its own favoured value of the charm quark-mass. Furher details of this study are given in ref. 4. It is important to note that the schemes sofar consideredaregeneral MassVFN Schemes and that the charm quark mass used in such schemes is the pole-mass. However since the since the relationship between the pole-mass and the runningmass (or MSbar mass) does not converge it is better to use the running-mass. This has been done within the Fixed Flavour Number scheme 6 and the result is m c (m c ) =.6±.5 as shown in Fig. 4 (bottom right). 3.. α s (M Z ) The value of α s (M Z ) is strongly correlated to the gluon PDF shape in fits to inclusive DIS data, because the gluon PDF is determined indirectly from the scaling violations. Jet production cross-sections depend directly on the gluon PDF. This extrainformationreduces the strongcorrelationbetween the gluonand α s (M Z ) and allows competitive measurements of α s (M Z ). Such studies have been made using jet data alone and by making a simultaneous fit of PDFs and α s (M Z ). Two recent studies using jet data alone are reported here. The first uses H normalised inclusive jet, di-jet and tri-jet cross-sections 9 which are well described by NLO QCD predictions from NLOJet++ using CT PDFs with α s (M Z ) =.8, see Fig. 5. A regularized unfolding is performed for all bins and multijet categories simultaneously to correct for detector effects and this yields a complete correlation matrix, which is used when these data are included into QCD fits which fix the PDF used (CT PDF) while varying α s (M Z ). The covariance matrix was used to normalize the different jet rates to the inclusive neutral-current DIS cross section. This normalization reduces both the experimental and the theoretical uncertainties. Values of α s (M Z ) are extracted separately from inclusive jets, di jets and tri-jets samples yielding: α S (M Z ) =.97 ±.8 (exp.) ±.4 (PDF) ±. (hadr.) ±.53 (th.) for inclusive jets
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs 6 A M Cooper-Sarkar ) c (M χ 8 75 7 HERA-I inclusive Charm + HERA-I inclusive opt M c =.5 ±.6 GeV RT standard ) c (M χ 75 7 Charm + HERA-I inclusive RT standard RT optimised ACOT-full S-ACOT-χ ZM-VFNS opt M C 65 65 6 55..4.6.8 M c [GeV] 6..4.6.8 [GeV] M c [nb] + W σ 64 6 6 58 Charm + HERA-I inclusive RT standard RT optimised ACOT-full S-ACOT-χ ZM-VFNS opt M C χ 7 68 66 Charm + HERA-I inclusive FF (ABM) m c (m )=.6 ± c.5 GeV 56 64 54 s = 7 TeV..4.6.8 M c [GeV] 6..4.6 m c (m ) [GeV] c Fig. 4. The χ of the HERAPDF fit as a function of the charm mass parameter m model c. Top left; using the RT-standard heavy-quark-mass scheme, when only inclusive DIS data are included in the fit and when the data for F c c are also included in the fit. Top right; using various heavyquark-mass schemes, when the data for F c c are also included in the fit. Bottom left; predictions for the W + cross-sections at the LHC, as a function of the charm mass parameter m model c, for various heavy-quark-mass schemes. Bottom right: Using the FFN scheme in terms of the running mass m c(m c). α S (M Z ) =.4 ±. (exp.) ±.6 (PDF) ±.9 (hadr.) ±.48 (th.) for dijets and α S (M Z ) =.85 ±.8 (exp.) ±.3 (PDF) ±.6 (hadr.) ±.4 (th.) for tri-jets. There is some tension between the di-jet and inclusive jet measurements, possibly due to missing higher order calculations, so that to make a combined α s (M Z ) extraction the cross-section predictions at NLO and LO were required to agree within 3%. This was fulfilled for 4 out of 65 bins. The combined fit then gives α S (M Z ) =.63 ±. (exp.) ±.4 (PDF) ±.8 (hadr.) ±.39 (th.), where the largest uncertainty is the theoretical uncertainty from scale variation. The second study uses ZEUS photoproduction data, where a quasi-real photon is emitted from the incoming lepton. In direct photoproduction, the photon
i July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs Determination of charm quark mass and α s(m Z ) from HERA data 7 Normalised Inclusive Jet Normalised Dijet Normalised Trijet H Preliminary 8 NLO c had NLOJet++ and fastnlo QCDNUM CT, α s =.8 σ Jet /σ NC 6 4 5 < Q < GeV (i = ) < Q < 7 GeV (i = 8) 7 < Q < 4 GeV (i = 6) 4 < Q < 7 GeV (i = 4) - 7 < Q < 5 GeV (i = ) 5 < Q < 5 GeV (i = ) 7 5 7 5 7 5 P T,Jet P T Dijet P T Trijet [GeV] Fig. 5. Normalised inclusive jet,dijet and tri-jet cross sections measured by the H Collaboration as a function of the transverse momentum in different Q bins compared to the NLO QCD predictions. directly takes part in the interaction, whereas the photon acts as a source of partons in events with resolved photoproduction. The ZEUS Collaboration published new double-differential jet cross sections in photoproduction events with a center-ofmass energy of the photon-proton system between 4 and 93 GeV. Compared to DIS measurements, this analysis has a relatively high reach of transverse jet energies, E T, up to 8 GeV. Overall, a reasonable agreement between data and the NLO predictions (using ZEUS-S PDFs 4 for the proton PDF, GRV-HO for the photon PDF and the programme of Klasen, Kleinwort and Kramer 3 ) is observed as shown in Fig. 6. The ZEUS Collaboration has used these jet cross sections to measure α s (M Z ). The range in dσ/de T was restricted to < E T < 7 GeV in order to reduce potential non-perturbative effects and the impact of the proton PDF uncertainty. An α S (M Z ) value of α S (M Z ) =.6 +.3 +.4. (exp.).35 (th.) was obtained, where scale uncertainties again represent the dominant uncertainty. The α s (M Z ) extraction is also performed for various different ranges of E jet T and this demonstrates the running of α s within a single experiment, as seen in Fig. 6. These analyses still have some dependence on the PDF used to predict the jet cross sections. This can be better accounted for by making a simultaneous fit of PDFs and α s (M Z ). This was done by the HERA experiments by using H and
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs 8 A M Cooper-SarkarZEUS T (pb/gev) dσ/de jet 6 5 4 3 - - -3 ZEUS 3 pb - NLO (ZEUS-S/GRV-HO) Q < GeV < η jet <.5 (x).5 < η jet < (x) < η jet <.5 (x) - < η jet < (x) < η jet < (x) jet energy-scale uncertainty 4 < W γp < 93 GeV k T algorithm 3 4 5 6 7 8 9 E jet T (GeV) α s.7.6.5.4.3. ZEUS ZEUS 3 pb - corr. exp. corr. exp. th. QCD 3 4 5 6 7 E jet T (GeV) Fig. 6. Left: Inclusive jet cross section in photoproduction measured by the ZEUS Collaboration as a function of the transverse jet energy in different rapidity regions and compared to the NLO prediction. Right: the extracted values of α s measured for different ranges of E jet T. ± σ r,nc (x,q ) 3 - - HERA I+II NC e + p (prel.) HERA I+II NC e - p (prel.) x =. (x3.) x =.3 (x7.) HERAPDF.5 e + p HERAPDF.5 e - p x =.5 (x9.) x =.8 (x5.) x =.3 (x.) x =.8 (x8.) x =.5 (x.4) x =.4 (x.7) x =.65 3 4 5 Q / GeV HERA Inclusive Working Group August xf.8.6.4. xs xd v Combined PDF Fit xg HERAPDF. (HERA I) HERAPDF.5 (prel.) (HERA I+II) xu v Q = GeV..4.6.8 x HERA Structure Functions Working Group July Fig. 7. Left: HERA combined data points for the NC e ± p cross-sections as a function of Q in bins of x, for data from the HERA-I and II run periods. The HERAPDF.5 fit to these data is also shown on the plot. Right: Parton distribution functions from HERAPDF. and HERAPDF.5; xu v, xd v,xs = x(ū + D) and xg at Q = GeV. ZEUS jet data together with the combined HERA inclusive data. The preliminary HERA-II data were first combined with the HERA-I data to yield an inclusive data set wih improved accuracyat high Q and high x 5. This new data set is used as the sole input to a PDF fit called HERAPDF.5 7 which uses the same formalism and assumptions as the HERAPDF. fit. Fig. 7 (left) shows the combined data for NC e ± pcross-sectionswiththeherapdf.5fitsuperimposed.thepartondistribution functions from HERAPDF. and HERAPDF.5 are compared in Fig. 7 (right). The improvement in precision at high x is clearly visible. The HERAPDF.5 analysis has been extended to include inclu-
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs Determination of charm quark mass and α s(m Z ) from HERA data 9 sive jet data 5,6,7,8. The new PDF set which results is called HERAPDF.6 8. For these fits the HERAPDF.5 parametrisation of the gluon distribution and the valence distribution is extended. This more flexible parametrisation is called HER- APDF.5f. The extra flexibility does not change the NLO PDFs central values significantly, and the PDF uncertainties are also not much increased. When the jet data areadded the HERAPDF.6 PDFs are alsosimilar, and there is no tension between the jet data and the inclusive data. For the jet data the NLO cross-sectionsare caluclatedusingnlojet+. The impact of the jet data is clearly seen when α S (M Z ) is allowed to be a free parameter of the fit. Fig. 8 shows the PDFs for HERAPDF.5f and HERAPDF.6, each with α S (M Z ) left free in the fit. It can be seen that without jet data the uncertainty on the gluon PDF at low x is large. This is because there is a strong correlation between the low-x shape of the gluon PDF and α S (M Z ). However once jet data are included the extra information on gluon induced processes reduces this correlation and the resulting uncertainty on the gluon PDF is not much larger than it is for fits with α S (M Z ) fixed. xf.8.6.4. -4 HERA I+II PDF Fit xg (.5) xs (.5) -3 HERAPDF.5f (prel.) free α s (M ) Z exp. uncert. model uncert. parametrization uncert. - Q = GeV xu v xd v - x HERAPDF Structure Function Working Group March xf.8.6.4. -4 HERA I+II PDF Fit with Jets xg (.5) xs (.5) -3 HERAPDF.6 (prel.) free α s (M ) Z exp. uncert. model uncert. parametrization uncert. - Q = GeV xu v xd v - x HERAPDF Structure Function Working Group March Fig. 8. The parton distribution functions xu v,xd v,xs = x(ū + D),xg, at Q = GeV, from HERAPDF.5f and HERAPDf.6, both with α S (M Z ) treated as a free parameter of the fit. The experimental, model and parametrisation uncertainties are shown separately. The gluon and sea distributions are scaled down by a factor. Fig.3.showsaχ scanvsα S (M Z ) forthe fitswith andwithoutjets, illustrating how much better α S (M Z ) is determined when jet data are included. The model and parametrisation errors are also much better controlled. The value of α s (M Z ) extracted from the HERAPDF.6 fit is: α S (M Z ) =. ±.3(exp) ±.7(model/param) ±.(had) +.45/.36(scale) Model and parametrisation uncertainties on α S (M Z ) are estimated in the same wayasforthepdfs andtheuncertaintiesonthehadronisationcorrectionsapplied to the jets are also evaluated. The scale uncertainties are estimated by varying the renormalisation and factorisation scales chosen in the jet publications by a factor
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla-amcs A M Cooper-Sarkar min - χ χ 5 5 (prel.) HERAPDF.5f HERAPDF.6.4.6.8...4.6 α S (M ) Z Fig. 9. The difference between χ and its minimum value for the HERAPDF.5f and HERA- PDf.6 fits as a function of α s(m Z ) HERAPDF Structure Function Working Group March of two up and down. The dominant contribution to the uncertainty comes from the jet renormalisation scale variation. 4. Summary Charm production data from the HERA experiments ZEUS and H have been combined and used to determine the charm quark mass within various heavy quark schemes. Jet cross sections at HERA have been used to determine α s (M Z ). References. A.M. Cooper-Sarkar, J.Phys.G 39, 93. R.S. Thorne and R.G. Roberts, Phys. Lett. B 4,33 (998) and hep-ph/6644 3. F. D. Aaron et al. [ Collaboration], JHEP () 9 [arxiv:9.884 [hep-ex]]. 4. F. D. Aaron et al. [ Collaboration], 3 Eur.Phys.J.C733 [arxiv:.8[hep-ex]]. 5. Collaborations, Hprelim--4, ZEUS-prel--7. 6. S. Alekhin an S. Moch, Phys.Lett.B699345 7. Collaborations, Hprelim--4, ZEUS-prel--8. 8. Collaborations, Hprelim--34, ZEUS-prel--. 9. H Collaboration, Hprelim--3. NLOJEt++, Z Nagy, hep-ph/96533, hep-ph/3768. H-L. Lai et al CT, arxiv:7.4, hep-ph. H. Abramowicz et al(zeus Collaboration) Nucl.Phys.B864 3. M. Klasen, T. Kleinwort and G. Kramer, 998 Eur. Phys. J.C 4. S. Chekanov et al (ZEUS Collaboration) 3 Phys.Rev.D67 7 5. F.D. Aaron et al (H Collaboration) Eur.Phys.J.C65 363 6. F.D. Aaron et al (H Collaboration) Eur.Phys.J.C67 7. S. Chekanov et al (ZEUS Collaboration) Phys.Lett.B547 64 8. S. Chekanov et al (ZEUS Collaboration) 7 Nucl.Phys.B765
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla Modern Physics Letters A c World Scientific Publishing Company arxiv:37.7875v [hep-ph] 3 Jul 3 DETERMINATION OF CHARM QUARK MASS AND α s (M Z ) FROM HERA DATA AMANDA COOPER-SARKAR Dept. of Physics, University of Oxford,Keble Rd, OXFORD, OX 3RH, UK email:a.cooper-sarkar@physics.ox.ac.uk Received (4 May 3) Revised () Charm production data from HERA may be used to determine the charm quark mass and jet production data from HERA may be used to detrmine α s(m Z ). Recent results are summarised.. Introduction HERA was an electron(positron)-proton collider located at DESY, Hamburg. It ran in two phases HERA-I from 99- and HERA-II 3-7. Two similar experiments,, took data. In HERA- running each experiment collected pb of e + p data and 5pb of e p data with electron beam energy 7.5GeV and proton beam energies 8, 9 GeV. In HERA-II running each experiment took 4pb of e + p data and 8pb of e p data with the same electron beam energy and proton beam energies 9 GeV. Deep inelastic lepton-hadron scattering data has been used both to investigate the theory of the strong interaction and to determine the momentum distributions of the partons within the nucleon. The data from the HERA collider now dominate the world data on deep inelastic scattering since they cover an unprecedented kinematic range: in Q, the (negative of the) invariant mass squared of the virtual exchanged boson,.45 < Q < 3 5 ; in Bjorken x, 6 7 < x <.65. Futhermore, because the HERA experiments investigated e + p and e p, charge current(cc) and neutral current (NC) scattering, information can be gained on flavour separated up- and down-type quarks and antiquarks and on the gluon- from its role in the scaling violations of perturbative quantum-chromo-dynamics. The formalism of how the deep inelastic cross sections relate to the parton distributions though the QCD-improved parton model, and how parton didtribution functions are then extracted from the data, is well documented (see for example ref. ) and only a brief description of the HERAPDF formalism will be given here. This contribution concentrates on results of relevance for the fundamental parameters of QCD: α s (M Z ) and the charm qurak mass m c.
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla A M Cooper-sarkar. HERAPDF Formalism PerturbativeQCD predicts the Q evolutionofthe partondistributions, but not the x dependence. Parton distributions are extracted by performing a direct numerical integration of the DGLAP evolution equations at NLO. A parametrised analytic shape for the parton distributions is assumed to be valid at some starting value of Q = Q. For the HERAPDF the value Q =.9 GeV is chosen such that the starting scale is below the charm mass threshold, Q < m c. Then the DGLAP equations are used to evolve the parton distributions up to a different Q value, where they are convoluted with NLO coefficient functions to make predictions for the structure functions. The heavy quark coefficient functions are usually calculated in the general-mass variable-flavour-number scheme of Thorne. However, for the study of the charm mass various other schemes have been considered, see Section 3. The heavy quark masses for the central HERAPDF fits were chosen to be m c =.4 GeV and m b = 4.75 GeV and the strong coupling constant was fixed to α s (M Z ) =.76. (The values of m c and α s (M Z ) are varied when studies of these parameters are performed.) The predictions are then fitted to the combined HERA data sets for NC and CC e + p and e p scattering. A minimum Q cut of Q min = 3.5GeV wasimposedtoremaininthekinematicregionwhereperturbative QCD should be applicable. The fit parameters are those necessary to specify the input analytic shape. The valence quark distributions xu v, xd v, and the u-type and d-type anti-quark distributions xū, x D (xū = xū, x D = x d + x s) are parametrised at the input scale Q =.9GeV by the generic form xf(x) = Ax B ( x) C (+Dx+Ex ). () The parametrisation for the gluon distribution xg can be extended, to include a term A g x B g ( x) C g, such that the NLO gluon may become negative at low x and low Q (however it does not do so in the kinematic region of the HERA data). The normalisation parameters, A g,a uv,a dv, are constrained by the quark number sum-rules and momentum sum-rule. Constraints are imposed to ensure that ū = d at low-x, and to specify the strangeness fraction in the sea. The full details for all HERAPDFs are given in ref. The experimental uncertainties on the HERAPDF are determined using the conventional χ tolerance, χ =, for 68%C.L. However model uncertainties and parametrisation uncertainties are also considered. The choice of the heavy quark masses is varied, the choice of Q min is varied and the strangeness fraction is varied. Parametrisation variations for which the E and the D parameters for all the PDFs are freed are considered and variation of the starting scale Q in the range,.5 < Q <.5GeV, is also considered. These model and parametrisation uncertainties are an integral part of the HERAPDF uncertainties.
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla Determination of charm quark mass and α s(m Z ) from HERA data 3 σ r,nc (x,q ) +.6.4. x=. x=. HERA I NC e + p ZEUS H x=.8.8.6 x=.3.4. x=.8 x=.5 3 4 Q / GeV Fig.. HERA combined NC e + p reduced cross section as a function of Q forsix x-binscompared to the separate data input to the averaging procedure. The individual measurements are displaced horizontally for a better visibility. 3. Results From 8, the experiments began to combine their data in order to provide the most complete and accurate set of deep-inelastic data as the legacy of HERA. Data on inclusive cross-sections have been combined for the HERA-I phase of running 3 and a preliminary combination has been made also using the HERA-II data 5. HERA-jet data have been added to this combination in order to give more information on α s (M Z ) and the gluon distribution 8. Very recently a combination of data on F c c has been used to give information on the charm quark mass 4. The combination of the data sets takes into account the full correlated systematic uncertainties of the individual experiments such that the total uncertainty of the combined measurement is typically smaller than %, for 3 < Q < 5 GeV, and reaches %, for < Q < GeV. In Fig averaged data are compared to the input data, illustrating the improvement in precision. Because of the reduction in size of the systematic error this improvement is far better than would be expected simply from the rough doubling of statistics which combining the two experiments represents. A combination has also been made of data on F c c 4 from various different methods of tagging charm: using the D, using the vertex detectors to see the displaced decay vertex, using direct D,D + production identified using the vertex detectors, and indentifying semi-leptonic charm decays via muons, also using the vertex detectors. The results of the F c c combination compared to the separate measurements
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla 4 A M Cooper-sarkar cc _ σ red. H VTX H D* HERA-I ZEUS D* 98- ZEUS D H D* HERA-II ZEUS c µ X ZEUS D* 96-97 ZEUS D + Q =.5 GeV Q =5 GeV Q =7 GeV.5 Q = GeV Q =8 GeV Q =3 GeV.5 Q =6 GeV Q = GeV Q = GeV.5 Q =35 GeV Q =65 GeV Q = GeV HERA -4-3 - -4-3 - -4-3 - x Fig.. The HERA combined measurement of F c c compared to the data sets of used for the combination. these data sets are slightly displaced in x for visibility. red σ cc Q =.5 GeV Q = 5 GeV Q = 7 GeV....5 Q = GeV.5 Q =8 GeV.5 Q =3 GeV.5 Q =6 GeV.5 Q = GeV.5 Q = GeV.5 Q =35 GeV.5 Q =65 GeV.5 Q = GeV HERA HERAPDF.5-4 -3 - -4-3 - -4-3 - x Fig. 3. The HERA combined measurement of F c c compared to the predictions of HERAPDF. which went into it are shown in Fig. The F c c combination is shown compared to the predictions of HERAPDF. in Fig. 3. 3.. Charm quark mass The uncertainty bands on these predictions are dominated by the variation of the charm quark mass,.35 < m c <.65GeV, used for the model uncertainty. The combined F c c data can help to reduce this uncertainty. Fig. 4 (top left) compares the χ, as a function of the charm mass, for a fit which includes charm data to that
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla Determination of charm quark mass and α s(m Z ) from HERA data 5 for the HERAPDF. fit which does not use these data, when using the Thorne- Roberts (RT) variable-flavour-number(vfn) scheme. The charm data have a clear preference for a particular charm quark mass. However, the RT heavy flavour VFN scheme is not unique, specific choices are made for threshold behaviour. In Fig. 4 (top right) the χ profiles for the standard and the optimized versions (optimized for smooth threshold behaviour) of this scheme are compared. The same figure also compares the alternative ACOT VFN schemes and the Zero-Mass VFN scheme. Each of these schemes favours a different value for the charm quark mass, and the fit to the data is good for all the heavy-quark-mass schemes considered. Each of the VFN schemes has been used to predict W and Z production for the LHC and their predictions for W + are shown in Fig. 4 (bottom left) as a function of the charm quark mass. If a particular value of the charm mass is chosen then the spread of predictions is as large as 7%. However this spread is considerably reduced % if each heavy quark scheme is used at its own favoured value of the charm quark-mass. Furher details of this study are given in ref. 4. It is important to note that the schemes sofar consideredaregeneral MassVFN Schemes and that the charm quark mass used in such schemes is the pole-mass. However since the since the relationship between the pole-mass and the runningmass (or MSbar mass) does not converge it is better to use the running-mass. This has been done within the Fixed Flavour Number scheme 6 and the result is m c (m c ) =.6±.5 as shown in Fig. 4 (bottom right). 3.. α s (M Z ) The value of α s (M Z ) is strongly correlated to the gluon PDF shape in fits to inclusive DIS data, because the gluon PDF is determined indirectly from the scaling violations. Jet production cross-sections depend directly on the gluon PDF. This extrainformationreduces the strongcorrelationbetween the gluonand α s (M Z ) and allows competitive measurements of α s (M Z ). Such studies have been made using jet data alone and by making a simultaneous fit of PDFs and α s (M Z ). Two recent studies using jet data alone are reported here. The first uses H normalised inclusive jet, di-jet and tri-jet cross-sections 9 which are well described by NLO QCD predictions from NLOJet++ using CT PDFs with α s (M Z ) =.8, see Fig. 5. A regularized unfolding is performed for all bins and multijet categories simultaneously to correct for detector effects and this yields a complete correlation matrix, which is used when these data are included into QCD fits which fix the PDF used (CT PDF) while varying α s (M Z ). The covariance matrix was used to normalize the different jet rates to the inclusive neutral-current DIS cross section. This normalization reduces both the experimental and the theoretical uncertainties. Values of α s (M Z ) are extracted separately from inclusive jets, di jets and tri-jets samples yielding: α S (M Z ) =.97 ±.8 (exp.) ±.4 (PDF) ±. (hadr.) ±.53 (th.) for inclusive jets
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla 6 A M Cooper-sarkar ) c (M χ 8 75 7 HERA-I inclusive Charm + HERA-I inclusive opt M c =.5 ±.6 GeV RT standard ) c (M χ 75 7 Charm + HERA-I inclusive RT standard RT optimised ACOT-full S-ACOT-χ ZM-VFNS opt M C 65 65 6 55..4.6.8 M c [GeV] 6..4.6.8 [GeV] M c [nb] + W σ 64 6 6 58 Charm + HERA-I inclusive RT standard RT optimised ACOT-full S-ACOT-χ ZM-VFNS opt M C χ 7 68 66 Charm + HERA-I inclusive FF (ABM) m c (m )=.6 ± c.5 GeV 56 64 54 s = 7 TeV..4.6.8 M c [GeV] 6..4.6 m c (m ) [GeV] c Fig. 4. The χ of the HERAPDF fit as a function of the charm mass parameter m model c. Top left; using the RT-standard heavy-quark-mass scheme, when only inclusive DIS data are included in the fit and when the data for F c c are also included in the fit. Top right; using various heavyquark-mass schemes, when the data for F c c are also included in the fit. Bottom left; predictions for the W + cross-sections at the LHC, as a function of the charm mass parameter m model c, for various heavy-quark-mass schemes. Bottom right: Using the FFN scheme in terms of the running mass m c(m c). α S (M Z ) =.4 ±. (exp.) ±.6 (PDF) ±.9 (hadr.) ±.48 (th.) for dijets and α S (M Z ) =.85 ±.8 (exp.) ±.3 (PDF) ±.6 (hadr.) ±.4 (th.) for tri-jets. There is some tension between the di-jet and inclusive jet measurements, possibly due to missing higher order calculations, so that to make a combined α s (M Z ) extraction the cross-section predictions at NLO and LO were required to agree within 3%. This was fulfilled for 4 out of 65 bins. The combined fit then gives α S (M Z ) =.63 ±. (exp.) ±.4 (PDF) ±.8 (hadr.) ±.39 (th.), where the largest uncertainty is the theoretical uncertainty from scale variation. The second study uses ZEUS photoproduction data, where a quasi-real photon is emitted from the incoming lepton. In direct photoproduction, the photon
i July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla Determination of charm quark mass and α s(m Z ) from HERA data 7 Normalised Inclusive Jet Normalised Dijet Normalised Trijet H Preliminary 8 NLO c had NLOJet++ and fastnlo QCDNUM CT, α s =.8 σ Jet /σ NC 6 4 5 < Q < GeV (i = ) < Q < 7 GeV (i = 8) 7 < Q < 4 GeV (i = 6) 4 < Q < 7 GeV (i = 4) - 7 < Q < 5 GeV (i = ) 5 < Q < 5 GeV (i = ) 7 5 7 5 7 5 P T,Jet P T Dijet P T Trijet [GeV] Fig. 5. Normalised inclusive jet,dijet and tri-jet cross sections measured by the H Collaboration as a function of the transverse momentum in different Q bins compared to the NLO QCD predictions. directly takes part in the interaction, whereas the photon acts as a source of partons in events with resolved photoproduction. The ZEUS Collaboration published new double-differential jet cross sections in photoproduction events with a center-ofmass energy of the photon-proton system between 4 and 93 GeV. Compared to DIS measurements, this analysis has a relatively high reach of transverse jet energies, E T, up to 8 GeV. Overall, a reasonable agreement between data and the NLO predictions (using ZEUS-S PDFs 4 for the proton PDF, GRV-HO for the photon PDF and the programme of Klasen, Kleinwort and Kramer 3 ) is observed as shown in Fig. 6. The ZEUS Collaboration has used these jet cross sections to measure α s (M Z ). The range in dσ/de T was restricted to < E T < 7 GeV in order to reduce potential non-perturbative effects and the impact of the proton PDF uncertainty. An α S (M Z ) value of α S (M Z ) =.6 +.3 +.4. (exp.).35 (th.) was obtained, where scale uncertainties again represent the dominant uncertainty. The α s (M Z ) extraction is also performed for various different ranges of E jet T and this demonstrates the running of α s within a single experiment, as seen in Fig. 6. These analyses still have some dependence on the PDF used to predict the jet cross sections. This can be better accounted for by making a simultaneous fit of PDFs and α s (M Z ). This was done by the HERA experiments by using H and
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla 8 A M Cooper-sarkarZEUS T (pb/gev) dσ/de jet 6 5 4 3 - - -3 ZEUS 3 pb - NLO (ZEUS-S/GRV-HO) Q < GeV < η jet <.5 (x).5 < η jet < (x) < η jet <.5 (x) - < η jet < (x) < η jet < (x) jet energy-scale uncertainty 4 < W γp < 93 GeV k T algorithm 3 4 5 6 7 8 9 E jet T (GeV) α s.7.6.5.4.3. ZEUS ZEUS 3 pb - corr. exp. corr. exp. th. QCD 3 4 5 6 7 E jet T (GeV) Fig. 6. Left: Inclusive jet cross section in photoproduction measured by the ZEUS Collaboration as a function of the transverse jet energy in different rapidity regions and compared to the NLO prediction. Right: the extracted values of α s measured for different ranges of E jet T. ± σ r,nc (x,q ) 3 - - HERA I+II NC e + p (prel.) HERA I+II NC e - p (prel.) x =. (x3.) x =.3 (x7.) HERAPDF.5 e + p HERAPDF.5 e - p x =.5 (x9.) x =.8 (x5.) x =.3 (x.) x =.8 (x8.) x =.5 (x.4) x =.4 (x.7) x =.65 3 4 5 Q / GeV HERA Inclusive Working Group August xf.8.6.4. xs xd v Combined PDF Fit xg HERAPDF. (HERA I) HERAPDF.5 (prel.) (HERA I+II) xu v Q = GeV..4.6.8 x HERA Structure Functions Working Group July Fig. 7. Left: HERA combined data points for the NC e ± p cross-sections as a function of Q in bins of x, for data from the HERA-I and II run periods. The HERAPDF.5 fit to these data is also shown on the plot. Right: Parton distribution functions from HERAPDF. and HERAPDF.5; xu v, xd v,xs = x(ū + D) and xg at Q = GeV. ZEUS jet data together with the combined HERA inclusive data. The preliminary HERA-II data were first combined with the HERA-I data to yield an inclusive data set wih improved accuracyat high Q and high x 5. This new data set is used as the sole input to a PDF fit called HERAPDF.5 7 which uses the same formalism and assumptions as the HERAPDF. fit. Fig. 7 (left) shows the combined data for NC e ± pcross-sectionswiththeherapdf.5fitsuperimposed.thepartondistribution functions from HERAPDF. and HERAPDF.5 are compared in Fig. 7 (right). The improvement in precision at high x is clearly visible. The HERAPDF.5 analysis has been extended to include inclu-
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla Determination of charm quark mass and α s(m Z ) from HERA data 9 sive jet data 5,6,7,8. The new PDF set which results is called HERAPDF.6 8. For these fits the HERAPDF.5 parametrisation of the gluon distribution and the valence distribution is extended. This more flexible parametrisation is called HER- APDF.5f. The extra flexibility does not change the NLO PDFs central values significantly, and the PDF uncertainties are also not much increased. When the jet data areadded the HERAPDF.6 PDFs are alsosimilar, and there is no tension between the jet data and the inclusive data. For the jet data the NLO cross-sectionsare caluclatedusingnlojet+. The impact of the jet data is clearly seen when α S (M Z ) is allowed to be a free parameter of the fit. Fig. 8 shows the PDFs for HERAPDF.5f and HERAPDF.6, each with α S (M Z ) left free in the fit. It can be seen that without jet data the uncertainty on the gluon PDF at low x is large. This is because there is a strong correlation between the low-x shape of the gluon PDF and α S (M Z ). However once jet data are included the extra information on gluon induced processes reduces this correlation and the resulting uncertainty on the gluon PDF is not much larger than it is for fits with α S (M Z ) fixed. xf.8.6.4. -4 HERA I+II PDF Fit xg (.5) xs (.5) -3 HERAPDF.5f (prel.) free α s (M ) Z exp. uncert. model uncert. parametrization uncert. - Q = GeV xu v xd v - x HERAPDF Structure Function Working Group March xf.8.6.4. -4 HERA I+II PDF Fit with Jets xg (.5) xs (.5) -3 HERAPDF.6 (prel.) free α s (M ) Z exp. uncert. model uncert. parametrization uncert. - Q = GeV xu v xd v - x HERAPDF Structure Function Working Group March Fig. 8. The parton distribution functions xu v,xd v,xs = x(ū + D),xg, at Q = GeV, from HERAPDF.5f and HERAPDf.6, both with α S (M Z ) treated as a free parameter of the fit. The experimental, model and parametrisation uncertainties are shown separately. The gluon and sea distributions are scaled down by a factor. Fig.3.showsaχ scanvsα S (M Z ) forthe fitswith andwithoutjets, illustrating how much better α S (M Z ) is determined when jet data are included. The model and parametrisation errors are also much better controlled. The value of α s (M Z ) extracted from the HERAPDF.6 fit is: α S (M Z ) =. ±.3(exp) ±.7(model/param) ±.(had) +.45/.36(scale) Model and parametrisation uncertainties on α S (M Z ) are estimated in the same wayasforthepdfs andtheuncertaintiesonthehadronisationcorrectionsapplied to the jets are also evaluated. The scale uncertainties are estimated by varying the renormalisation and factorisation scales chosen in the jet publications by a factor
July 3, 3 :6 WSPC/INSTRUCTION FILE ws-mpla A M Cooper-sarkar min - χ χ 5 5 (prel.) HERAPDF.5f HERAPDF.6.4.6.8...4.6 α S (M ) Z Fig. 9. The difference between χ and its minimum value for the HERAPDF.5f and HERA- PDf.6 fits as a function of α s(m Z ) HERAPDF Structure Function Working Group March of two up and down. The dominant contribution to the uncertainty comes from the jet renormalisation scale variation. 4. Summary Charm production data from the HERA experiments ZEUS and H have been combined and used to determine the charm quark mass within various heavy quark schemes. Jet cross sections at HERA have been used to determine α s (M Z ). References. A.M. Cooper-Sarkar, J.Phys.G 39, 93. R.S. Thorne and R.G. Roberts, Phys. Lett. B 4,33 (998) and hep-ph/6644 3. F. D. Aaron et al. [ Collaboration], JHEP () 9 [arxiv:9.884 [hep-ex]]. 4. F. D. Aaron et al. [ Collaboration], 3 Eur.Phys.J.C733 [arxiv:.8[hep-ex]]. 5. Collaborations, Hprelim--4, ZEUS-prel--7. 6. S. Alekhin an S. Moch, Phys.Lett.B699345 7. Collaborations, Hprelim--4, ZEUS-prel--8. 8. Collaborations, Hprelim--34, ZEUS-prel--. 9. H Collaboration, Hprelim--3. NLOJEt++, Z Nagy, hep-ph/96533, hep-ph/3768. H-L. Lai et al CT, arxiv:7.4, hep-ph. H. Abramowicz et al(zeus Collaboration) Nucl.Phys.B864 3. M. Klasen, T. Kleinwort and G. Kramer, 998 Eur. Phys. J.C 4. S. Chekanov et al (ZEUS Collaboration) 3 Phys.Rev.D67 7 5. F.D. Aaron et al (H Collaboration) Eur.Phys.J.C65 363 6. F.D. Aaron et al (H Collaboration) Eur.Phys.J.C67 7. S. Chekanov et al (ZEUS Collaboration) Phys.Lett.B547 64 8. S. Chekanov et al (ZEUS Collaboration) 7 Nucl.Phys.B765