Higgs Cross Sections for Early Data Taking Abstract

Transcription

1 Draft version x.y ATLAS NOTE January 24, 20 1 Higgs Cross Sections for Early Data Taking N. Andari a, K. Assamagan b, A.-C. Bourgaux a, M. Campanelli c, G. Carrillo d, M. Escalier a, M. Flechl e, J. Huston f, S. Muanza g, B. Murray h, B. Mellado d, A. Nisati i, J. Qian j, D. Rebuzzi k, M. Schram l, R. Tanaka a, T. Vickey d, M. Warsinsky m, H. Zhang g a LAL-Orsay b Brookhaven National Laboratory c University College London d University of Wisconsin, Madison e University of Uppsala f Michigan State University g CPPM-Marseille h Rutherford Appleton Laboratory i Sezione di Roma I and INFN j University of Michigan k Università di Pavia and INFN, Sezione di Pavia l McGill University m Freiburg University This is the abstract Abstract

2 Introduction Search for and potential discovery of Higgs bosons, both in standard model (SM) or minimal supersymmetric standard model (MSSM), is perhaps one of the most important goals of the ATLAS physics program. The discovery potentials have been the subject of extensive investigations over the last decade. With an integrated luminosity of O(1) fb 1, a standard model Higgs boson can be observed over a wide range of Higgs boson mass. Similarly, a large region of the MSSM parameter space can be explored. All these investigations depend on our precise knowledge of Higgs production cross sections at the LHC. As such there are concerted effort in the theoretical community to calculate both higher order QCD and electroweak corrections. In this note, we survey the most recent theoretical tools available for the calcuations of Higgs production cross sections and also present our current best estimates of cross sections at several center-of-mass energies. The standard model Higgs boson has so far evaded observation despite of extensive pursuit. Experimental results and theoretical arguments favor a relatively light Higgs boson. At the LHC, the Higgs boson can be produced through the following four main mechanisms: gluon-gluon fusion through a heavy quark triangular loop (gg H), vector boson fusion (VBF), associated production with vector bosons W or Z (VH), and production in association with a top quark pair (tth). Significant progresses have been made in recent years in the calculation of higher order corrections to the production cross sections of these processes. Next-Leading-Order calculations are now available for all of them. The dominant gg H process has been calculated up to next-next-leading-order with soft gluon resummation up to next-next-leading-log. Moreover, next-leading-order electroweak corrections are also available for gg H, V BF and V H. These improvements significantly reduce renormalization and factorization scale dependences of the cross sections. Five Higgs bosons are predicted in MSSM: two neutral CP-even h and H, one CP-odd A and two charged H ±. At tree level, the Higgs sector is described by two parameters, usually chosen to be the mass of CP-odd Higgs boson mass M A and the ratio of vacuum expectation values tanβ. The neutral Higgs bosons, two of them being generally degenerate in mass, are produced mainly through two processes at the LHC: in association with a b-quark and through gluon-gluon fusion. The cross section of b-associated process can be calculated either in 4-flavor scheme as gg b bh or in 5-flavor scheme as b b H. Higher order QCD corrections have been calculated up to NLO for the 4-flavor scheme and NNLO for the 5- flavor scheme. The cross section for the gluon-gluon fusion process is calculated in the same way as the standard model gg H process. The charged Higgs bosons can be either directly produced through gb th ± process or indirectly from the top quark decays (t H ± b) if m H ± < m t. Higher order QCD calculations are available up to NLO for gb th ± and NNLO+NNLL for the t t process, the dominant top quark source for t H ± b decays. Along with the improvement of theoretical calculations, recent advancements in constraining parton distribution functions (PDF) have also contributed significantly to our improved knowledge of the Higgs production cross sections. The release of MSTW2008 PDF at NNLO and refined CT09 PDF are two primary examples. The MSTW2008 PDF set incorporates the latest calculation of the NNLO splitting functions and a more sophisticated treatment of heavy-quark threshold effect. CT09... The production cross sections and decay branching ratios of the Higgs bosons depend on a large number of standard model parameters. Unless otherwise specified, the following default parameter sets are used: m uds = 190 MeV, m c = 1.40 GeV, m b = 4.75 GeV, m t = GeV, M W = GeV and M Z = GeV, G F = GeV 2. Pole quark mass values are quoted here. The strong coupling constant α s is in general set to its value of the corresponding PDF fit. For MSTW2008 PDF set, α s (M Z ) = at LO, at NLO and at NNLO. Previous compilations of Higgs cross sections and decay branching ratios were done for CSC and Charmonix studies. The cross section group was setup by Higgs physics group conveners in May 2009

3 charged with the task of updating the Higgs cross sections for first data analysis assuming s = TeV, and for preparing all the tools we need to access the estimate of the cross sections as a function of the center-of-mass energy in a reasonable amount time.. Since then, we have learnt that the LHC will likely operate with s = 7 TeV at least initially. Therefore cross sections are tabulated for s = 7,, 14 TeV in this note. Compared with previous studies, this latest study benefits significantly from the latest theoretical progresses as well as the new state-of-art parton distribution functions. Equal importantly, it compares the results from various programs to arrive at realistic estimations of theoretical uncertainties..mine 2 Parton Distribution Functions Authors: Bruce, Joey 2.1 Introduction Cross sections for Higgs production at the LHC are sensitive to quark, anti-quark and gluon distributions in the x range from approximately to 0.1, at virtualities of the order of 4 GeV 2 or greater. The parton distributions (PDFs) have been determined through global fits to deep-inelastic scattering, Drell- Yan and Tevatron collider cross sections over a wide range of parton x and Q 2. Until recently, there have been two main groups that have provided regular PDF updates, CTEQ (with the latest PDFs being CTEQ6.6 and CT09 1 and MSTW (with the latest PDFs being MSTW2008). More recently, the Neural Net PDF collaboration has been publishing regular updates to their global PDF analysis, with the latest being the NNPDF1.2 set. Only the most modern PDFs should be used for any predictions at the LHC. In particular, the use of a general mass-variable flavor number (GM-VFNS) scheme is essential for accurate predictions at the LHC; such a scheme is present in CTEQ PDFs from CTEQ6.5 and on and for MSTW2008. The schemes used by these two PDF groups are now very similar, leading for example to equivalent predictions for W and Z cross sections at the LHC. The NNPDF group is in the process of developing a GM-VFNS for their global fits (using the FONLL framework). As a result, their current predictions for cross sections such as W and Z at the LHC may be slightly off, but their PDFs will be useful for checks on uncertainties. PDFs are available at leading order (LO), modified leading order (LO*), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in order to match the level of the matrix element calculations. NLO is the first order at which both the matrix elements and the PDFs can be stated to have any level of precision. Global fits at NLO provide a good description to the approximately 3000 data points typically included in modern global PDF analyses, with χ 2 /DOF on the order of unity. Modest improvement in the global fits are obtained at NNLO. A comparison of gluon distributions at LO, NLO and NNLO at a Q 2 of 4 GeV 2 is shown in in Figure 13(left) There is a noticeable change in shape in going from LO to NLO, and little change in going from NLO to NNLO 2. In particular, the LO gluon distribution is larger than the NLO one at small x, and considerably smaller than the NLO gluon at high x. A similar comparison is shown in in Figure 13(right) for the up quark distribution, with similar conclusions. In this case, the LO up quark distribution is higher than the NLO up quark distribution for both low x and high x. The differences observed for LO PDFs result from deficiencies in the LO matrix elements for DIS used in the LO global fits, and can result in 1 An update is currently in the works for CT09( CT09.1) incorporating some new tools and analysis techniques that have been recently developed. These PDFs will be distributed to LHAPDF once complete. 2 Thus, a NLO prediction for the Higgs cross section through gg fusion would be very similar if either a NLO gluon or NNLO gluon were used, and vice versa for a NNLO prediction.

4 significant shapes changes for LO predictions, compared to NLO or NNLO predictions. Given that LO predictions, in the context of parton shower Monte Carlo programs, are widely used by experimentalists, LO* PDFs have been developed by both CTEQ [?] and MSTW [?] that seek to lessen the differences between LO predictions using the LO* PDFs and full NLO predictions. The modified LO PDFs have a similar behavior as the normal LO PDFs at low x, but, in the case of the CT09MC2 PDFs, have closer to a NLO behavior at high x. Figure 1: (left)a comparison of the gluon distributions from the MSTW2008 PDF sets, at LO, NLO and NNLO, at a at a Q 2 value of 4 GeV 2. (right) A comparison of the up quarks distributions from the MSTW2008 PDF sets, at LO, NLO and NNLO, at a at a Q 2 value of 4 GeV 2. The NLO distributions for CTEQ6.6 are also shown for comparison. This figure comes out faint. Will try to improve PDF Uncertainties PDF uncertainties are determined by allowing excursions from the best fit parameters that result in increases in the total χ 2 values within a pre-determined range. This range is difficult to determine from a rigorous statistical basis, given the wide range of processes and data included in the global fits, and tends to be a matter of choice on the part of the global fitting groups: χ 2 has been on the order of 50 (MSTW) or 0 (CTEQ), rather than the canonical value of χ 2 of 1 for a 1-sigma uncertainty and 2.3 for a 90%confidence level 3. Thus, PDF uncertainties for CTEQ have tended to be a factor of 2 larger than those determined from MSTW 4. Such increases in χ 2 correspond to a 90% confidence level (CL) limit. The fact that the tolerance criteria for CTEQ and MSTW differ by a factor of two emphasize that it is difficult to assign a precise statistical meaning to the PDF uncertainties 5. PDF uncertainties corresponding to a 68% CL limit (1-sigma) can be approximately calculated by the appropriate scaling of the 90% CL errors. The MSTW2008 PDFs have specific error PDFs corresponding to a 68% CL in addition 3 Some recent studies have indicated that the need for a larger χ 2 can be tied to parametrization biases inherent in the global fits. 4 CTEQ (and the NNPDF collaboration) uses the correlated systematic error information for all experiments for which this information is available; MSTW uses the correlated information only for the Tevatron jet experiments (and the Z rapidity distribution) and adds all other experimental systematic errors in quadrature with the statistical error. This may be partially responsible for the difference in the assumed χ 2 values. 5 Both groups have recently been developing more sophisticated treatments for the determination of the error PDFs, but the fundamental difference in the relative size of the PDF errors remains.

5 to those corresponding to 90% CL. The approximations involved can be observed in comparing cross section uncertainties using both sets of error PDFs; in many cases, the scaling between them is larger than the factor of 1.6 expected from statistical considerations. A large number of parameters are needed in order to fully specify the behavior of each flavor of PDF over the full x-range. Some parameters can be fixed at reasonable values while others can be determined from sum rules: the net result is that the CTEQ6.6 (CT09) PDFs have 22 (24) free parameters, while the MSTW2008 PDFs have 20 free parameters. Typically, error PDFs are determined by diagonalizing the Hessian matrix corresponding to the free fit parameters (22X22 for CTEQ6.6). The resulting eigenvectors form an orthonormal basis set from which the uncertainty for any cross section can be determined using the Master Equation: X + max = X max = N i=1 N i=1 [max(x + i X 0,X i X 0,0)] 2 [max(x 0 X i +,X 0 Xi,0)] 2 (1) X + adds in quadrature the pdf error contributions that lead to an increase in the observable X and X the pdf error contributions that lead to a decrease. The addition in quadrature is justified by the eigenvectors forming an orthonormal basis. The sum is over all N eigenvector directions, or 22(24) in the case of CTEQ6.6(CT09). Ordinarily, X i + X 0 will be positive and Xi X 0 will be negative, and thus it is trivial as to which term is to be included in each quadratic sum. For the higher number eigenvectors, however, we have seen that the + and contributions may be in the same direction. In this case, only the most positive term will be included in the calculation of X + and the most negative in the calculation of X. Thus, there may be less than N terms for either the + or directions 6. Some programs, such as MCFM, will automatically calculate the PDF uncertainty of a given cross section using the specified error PDF set and the Master Formula given above. The results of the NLO calculation (including the error PDF weights) can also be stored in ROOT ntuples, allowing for the PDF uncertainty for any cross section with cuts to be calculated at a later time. The smaller number eigenvectors correspond to the PDF directions that are better determined; the larger number eigenvector correspond to the poorly determined directions. The eigenvectors have contributions from each of the free PDF parameters. In some cases, an eigenvector may have a large component from a particular PDF parameter, but in most cases, the contributions are widely distributed among different PDF parameters and no specific meaning can be ascribed to each eigenvector. Consider the example of Higgs production through gg fusion at a center-of-mass energy of 14 TeV. The most important contributions to the uncertainty in the cross section for a mass of 120 GeV come from (within the CTEQ6.6 PDF set) eigenvectors 4, 6, 11 and 16. These eigenvectors continue to be important for Higgs masses of 200 and 300, but other eigenvectors become important as well, especially at 300 GeV (5 and 7). A new technique has been developed by CTEQ, termed Data Set Diagonalization. The Hessian matrix is originally diagonalized in order to determine the error PDFs. An additional diagonalization is possible is then possible, taking into account a new quantity, such as the Higgs cross section at a particular mass. The result will be a particular eigenvector(s) which directly probes the uncertainty along this direction (the uncertainty of the Higgs cross section at that mass). Other variables, such as the 2nd moment of the Higgs rapidity distribution, are also possible. In Figure 14(left), the up quark distribution for MSTW2008 (at a Q 2 value of 4 GeV 2 ) is compared to the uncertainty on the up quark distribution for CTEQ6.6; on the right, the CTEQ6.6 up quark distri- 6 Note that MSTW uses the symmetric form of the Master Equation, i.e. there will always be exactly N terms for both the +?? and -?? directions. The net result should be very similar to the use of the asymmetric form described above for most calculations.

6 bution is compared to the up quark uncertainty for MSTW2008. In Figure 15(left), the gluon distribution for MSTW2008 (at a Q 2 value of 4 GeV 2 ) is compared to the uncertainty on the gluon distribution for CTEQ6.6; on the right, the CTEQ6.6 gluon distribution is compared to the up quark uncertainty for MSTW2008. It can be seen that the PDF uncertainties are less than 5% for the relevant PDFs over the range of interest for Higgs production at the LHC. It is also evident from the plots that the MSTW2008 PDFs lie within the CTEQ6.6 PDF error band; the inverse is not always true as the MSTW2008 error bands are about half of the size of the CTEQ6.6 error bands. This may be considered as a minimal requirement for the estimation of a PDF uncertainty: that the central value of the competing PDF lay within the error band of the original PDF. Figure 2: A comparison of the u quark distributions from the CTEQ6.6 and the MSTW2008 PDF sets, at a at a Q 2 value of 4 GeV 2. On the left the u quark distribution from MSTW2008 is compared to the PDF error band from CTEQ6.6; the reverse is plotted on the right. Comes out a bit faint. Will try to improve Parton-parton Luminosities It is useful to define differential parton-parton luminosities. Such luminosities, when multiplied by the dimensionless cross section ŝ ˆσ for a given process, provide a useful estimate of the size of an event cross section at the LHC. Below we define the differential parton-parton luminosity dl i j /dŝdy and its integral dl i j /dŝ: dl i j dŝdy = 1 1 [ f i (x 1, µ) f j (x 2, µ) + (1 2)]. (2) s 1 + δ i j The prefactor with the Kronecker delta avoids double-counting in case the partons are identical. The generic parton-model formula 180 can then be written as 1 σ = dx 1 dx 2 f i (x 1, µ) f j (x 2, µ) ˆσ i j (3) i, j 0 ( ) ( ) dŝ σ = i, j ŝ dy dli j (ŝ ˆσ i j ). (4) dŝdy

7 Figure 3: A comparison of the gluon distributions from the CTEQ6.6 and the MSTW2008 PDF sets, at a at a Q 2 value of 4 GeV 2. On the left the gluon distribution from MSTW2008 is compared to the PDF error band from CTEQ6.6; the reverse is plotted on the right. Comes out a bit faint. Will try to improve (Note that this result is easily derived by defining τ = x 1 x 2 = ŝ/s and observing that the Jacobian (τ,y)/ (x 1,x 2 ) = 1.) Equation eq:xseclum can be used to estimate the production rate for a hard scattering process at the LHC as follows. Figure 16 shows a plot of the luminosity function integrated over rapidity, dl i j /dŝ = (dli j /dŝdy)dy, at the LHC s = 14TeV for various parton flavour combinations, calculated using the CTEQ6.1 parton distribution functions [?]. The widths of the curves indicate an estimate for the pdf uncertainties. We assume µ = ŝ for the scale. As expected, the gg luminosity is large at low ŝ but falls rapidly with respect to the other parton luminosities. The gq luminosity is large over the entire kinematic region plotted. It is also of great interest to understand the uncertainty in the parton-parton luminosity for specific kinematic configurations. Some representative parton-parton luminosity uncertainties, integrated over rapidity, are shown in Figures 17, 18 and 19. The pdf uncertainties were generated from the CTEQ6.1 Hessian error analysis using the standard χ 2 = 0 criterion. Except for kinematic regions where one or both partons is a gluon at high x, the pdf uncertainties are of the order of 5 %. Luminosity uncertainties for specific rapidity values are available at the SM benchmark website. Even tighter constraints will be possible once the LHC Standard Model data is included in the global pdf fits. Again, the uncertainties for individual pdfs can also be calculated online using the Durham pdf plotter. It is useful to compare the gg PDF luminosity uncertainty obtained from CTEQ with those from the NNPDF group. Such a comparison is shown in Figure 20. Over the range of Higgs masses accessible through the gg channel, roughly shown by the red lines, the two uncertainties are basically equivalent. The NNPDF uncertainty is larger at low x, where there is little to no data, as expected. Often it is not the PDF uncertainty for a cross section that is required, but rather the PDF uncertainty for an acceptance for a given final state. The acceptance for a particular process may depend on the input pdfs due to the rapidity cuts placed on the jets, leptons, photons, etc. and the impacts of the varying longitudinal boosts of the final state caused by the different pdf pairs. An approximate rule-of-thumb is that the pdf uncertainty for the acceptance is a factor of 5 times smaller than the uncertainty for the cross section itself. For the first year or so of LHC operation, the total inelastic cross section, and thus the total delivered

8 Figure 4: The parton-parton luminosity [ dli j dτ ] in picobarns, integrated over y. Green=gg, Blue= i (gq i + g q i + q i g + q i g), Red= i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b integrated luminosity, will not be well-known. Thus, it has been suggested that all physics cross sections at the LHC be normalized to standard model benchmark cross sections, such as W and Z production. From the pdf luminosity figures, it can be seen that the uncertainty for W and Z production is on the order of 5-6%. It is interesting to note that the smallest parton-parton PDF uncertainties at the LHC are obtained for gq scattering at ŝ=0.50 TeV. gq processes in this kinematic region may also prove to be useful as a normalization tool at the LHC. Z production at high p T is one example of such a process Correlations It is often instructive to determine whether a particular collider observable shares common degrees of freedom with another collider cross section through the non-perturbative PDF parametrizations. Such correlations can be useful in providing ratios of cross sections that have smaller theoretical uncertainties than either cross section by itself. A quantity, cosϕ (the correlation cosine), can be defined that represents the degree to which two cross sections are correlated; a value of this quantity near 1 implies a very strong correlation, near zero, no correlation, and near -1, a very strong anti-correlation. Here ϕ represents the angle between the gradient vectors of the two cross sections in the Hessian eigenvector space, and can be defined as shown in the equation below (for two cross sections X and Y: cos ϕ = X Y X Y = 1 4 X Y N i=1 ( X (+) i )( X ( ) i Y (+) i ) Y ( ) i. (5) Thus, cosϕ can easily be defined with knowledge of the cross sections for X and Y for each of the Hessian eigenvectors. 7 To appear in the Les Houches 2009 proceedings.

9 Figure 5: Fractional uncertainty of the gg luminosity integrated over y For example, one would expect (and find) the W and Z cross sections to be very strongly correlated, as both are produced by similar initial states in the same x range. A plot of the correlation cosines for various processes is shown in the figure below. It can be observed, for example, for Higgs production through gg fusion, the process goes from a mild correlation at low Higgs mass (120 GeV) with the W cross section at the LHC, to a very strong anti-correlation for high Higgs masses (500 GeV). Conversely, the correlation of the Higgs boson cross section with respect to the t t cross section is the mirror image of that to the W cross section α s An additional uncertainty regarding Higgs cross section predictions at the LHC arises from the uncertainty in the strong coupling constant α s. It is customary now to present α s at its value at the Z mass, with the NLO world average currently being ± (LO average = and NNLO average = 0.117). This is the value currently used by both CTEQ and the Neural Net Collaboration. The philosophy of the CTEQ and NNPDF groups is that α s (m Z ) is a global parameter and that all inputs (i.e. the world average) should be taken into account in the determination of the strong coupling constant. Other PDF fitting groups, such as MSTW, determine the value of α s from their own global PDF fits. This can introduce an additional variability, as (1) the value obtained can vary with PDF version and (2), although the values obtained from the global fits are compatible with the world average, they can be noticeably different. For MSTW2008, the value of α s (m Z ) is (0.117) at NLO (NNLO). This difference in α s (m Z ) at NLO between CTEQ and MSTW can lead to disagreements in predictions for cross sections (such as Higgs production) that have a strong dependence on α s, even if the partonic 8 The 2007 PDG average for α s (m Z ) is ± The world average for 2009, recently compiled by Siggi Bethke is ±

10 Figure 6: Fractional uncertainty for the parton-parton luminosity integrated over y for i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b luminosities are in good agreement 1. If α s is set as a free parameter in the CTEQ global fit, the value of al pha s (m Z ) obtained is consistent with the world average. Setting it equal to the value obtained in the MSTW2008 PDFs results in a modest increase in χ 2. (need to check. This is true for previous generations of CTEQ PDFs.) For example, the cross section for the production of a Higgs boson (though gg fusion) of mass 120 GeV at the LHC (14 TeV) is 36.3 pb using CTEQ6.6 and 38.4 pb using MSTW2008, a difference larger than might be expected given that the NLO gluon distributions are basically the same in the x range relevant for Higgs production (See Figure??.) The higher cross section obtained with MSTW PDFs is largely through the larger value of α s that comes along with those PDFs, and the substantial NLO (αs 3 ) and NNLO (αs 4 ) contributions to the Higgs cross section 2. In the past, variations in α s have been taken into account in global PDF fitting by providing alternate PDFs with different (fixed) values of α s. For example, the CTEQ6.6 PDFs come with such a set, for α s values of 0.112,0.114,0.122, and Depending on the gluon x range, there is either a correlation or anti-correlation between the value of α s (m Z ) and the value of the gluon distribution. For the range relevant for Higgs production at the LHC, there is an anti-correlation, i.e. a larger value of α s (than the world average) leads to a decrease in the gluon distribution (by a few percent). MSTW provides eigenvector sets with different values of α s (m Z ) about their central fit value. They point out that away from their best value of α s (m Z ), the fit quality deteriorates (as expected), so that the allowed uncertainty of the PDF variation is smaller than with the central value. Within a variation in α s of around the central MSTW value, there is little change in χ 2 ; this corresponds to an approximately 1-σ uncertainty for them. For a 90% CL variation of α s 1 There is also an ambiguity as to how α s is defined at NLO and NNLO, which can lead to noticeable differences in the resultant PDFs; luckily, CTEQ, MSTW and NNPDF now use the same convention. 2 As another example, the cross section for t t production at the LHC (14 TeV) is 829 pb for CTEQ6.6 and 902 pb for MSTW. Production is primarily through the gg subprocess, and while the gluon distributions are similar to within a few percent in the relevant x range, the difference in α s is emphasized by the large order α 3 s contributions.

11 Figure 7: Fractional uncertainty for the luminosity integrated over y for i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b (approximately ), the change in chi 2 becomes significant, on the order of the MSTW tolerance. Here is where it becomes tricky; a large variation of α s will cause significant changes in the PDFs, and thus significant changes in the predicted Higgs cross sections. The uncertainty given to α s (m Z ) in the latest global α s fit is less than than that of the current PDG world average (and less than obtained from the MSTW internal variation). This is the difference between α s variation being relatively unimportant and being the dominant uncertainty factor. The error for the new world average (0.0007) should be taken as more than a 1-sigma error 3 ; if we allow for a variation of as a 90%CL, a generous overestimate, then we can quote an error range of ±0.002 around the central value of As part of this exercise, a new set of CTEQ6.6 PDFs with a smaller step size variation of α s (m Z ) were generated. In these fits, α s is varied in steps from to 0.120, comparable to the world average uncertainty. A comparison of the gluon PDF uncertainty due to the variation in α s (in green) is overlayed on the PDF uncertainty due to the Hessian matrix (blue) in Figure 24. Given this range of variation of α s, and the larger intrinisic level of uncertainty in the CTEQ6.6 PDFs, the additional impact of varying α s is reasonably small. ======= 4 Parton Distribution Functions 4.1 Introduction Cross sections for Higgs production at the LHC are sensitive to quark, anti-quark and gluon distributions in the x range from approximately to 0.1, at virtualities of the order of 4 GeV 2 or greater. The 3 S. Bethke, private communication

12 Figure 8: A comparison of the gluon-gluon luminosity uncertainties between CTEQ66 and the NNPDF group. Just copied off one of my talks. Will have to remake.

13 Figure 9: Dependence on the correlation ellipse formed in the δx δy plane on the value of cosϕ.

14 Figure : The correlation cosine cos ϕ for Higgs boson searches at the LHC with respect to Z boson production at the LHC (solid) and Tevatron (dots), and t t production at the LHC (dashes), plotted as a function of Higgs mass. Separate markers denote correlations of W, t-channel single-top cross sections at the LHC and Z cross section at the Tevatron with respect to Z and t t cross sections at the LHC.

15 Figure 11: A comparison of the CTEQ66 and MSTW2008 gluon distributions in the x and Q 2 range relevant for Higgs production through gg fusion at the LHC.

16 Figure 12: A comparison of the PDF uncertainty for the gluon distribution at a Q value of 85 GeV from the variation in α s (green) compared to the error determined from the Hessian method (blue).

17 parton distributions (PDFs) have been determined through global fits to deep-inelastic scattering, Drell- Yan and Tevatron collider cross sections over a wide range of parton x and Q 2. Until recently, there have been two main groups that have provided regular PDF updates, CTEQ (with the latest PDFs being CTEQ6.6 [?] and CT09 [?] 4 and MSTW (with the latest PDFs being MSTW2008 [?]). More recently, the Neural Net PDF collaboration has been publishing regular updates to their global PDF analysis, with the latest being the NNPDF1.2 (and NNPDF2.0) sets [?]. The Neural Net PDF collaboration uses a Monte Carlo method for uncertainty estimation, rather than the Hessian technique primarily used by CTEQ and MSTW. Only the most modern PDFs should be used for any predictions at the LHC. In particular, the use of a general mass-variable flavor number (GM-VFNS) scheme is essential for accurate predictions at the LHC [?]; such a scheme is present in CTEQ PDFs from CTEQ6.5 and on and for MSTW2008. The schemes used by these two PDF groups are now very similar, leading for example to equivalent predictions for W and Z cross sections at the LHC. The NNPDF group has recently developed a GM- VFNS for their global fits (using the FONLL framework) 5,but current sets still use a zero-mass variable flavor number scheme (ZM-VFNS). As a result, their current predictions for cross sections such as W and Z at the LHC may be slightly off, but their PDFs will be useful for checks on uncertainties. PDFs are available at leading order (LO), modified leading order (LO*), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in order to match the level of the matrix element calculations. NLO is the first order at which both the matrix elements and the PDFs can be stated to have any level of precision. Global fits at NLO provide a good description to the approximately 3000 data points typically included in modern global PDF analyses, with χ 2 /DOF on the order of unity. Modest improvement in the global fits are obtained at NNLO. A comparison of gluon distributions at LO, NLO and NNLO at a Q 2 of 4 GeV 2 is shown in Figure 13(left) There is a noticeable change in shape in going from LO to NLO, and little change in going from NLO to NNLO 6 (except at very low x and very high x where the pertubative expansion is unstable and resummation may be necessary). In particular, the LO gluon distribution is larger than the NLO one at small x, and considerably smaller than the NLO gluon at high x. A similar comparison is shown in Figure 13(right) for the up quark distribution, with similar conclusions. In this case, the LO up quark distribution is higher than the NLO up quark distribution for both low x and high x. The differences observed for LO PDFs result from deficiencies in the LO matrix elements for DIS used in the LO global fits, and can result in significant shapes changes for LO predictions, compared to NLO or NNLO predictions. Given that LO predictions, in the context of parton shower Monte Carlo programs, are widely used by experimentalists, LO* PDFs have been developed by both CTEQ [?] and MSTW [?] that seek to lessen the differences between LO predictions using the LO* PDFs and full NLO predictions. The modified LO PDFs have a similar behavior as the normal LO PDFs at low x, but, in the case of the CT09MC2 PDFs, have closer to a NLO behavior at high x. 4.2 PDF Uncertainties PDF uncertainties are determined by allowing excursions from the best fit parameters that result in increases in the total χ 2 values within a pre-determined range. This range is difficult to determine from a rigorous statistical basis, given the wide range of processes and data included in the global fits, and tends to be a matter of choice on the part of the global fitting groups: χ 2 has been on the order of 50 4 An update is currently in the works for CT09( CT09.1) incorporating some new tools and analysis techniques that have been recently developed. These PDFs will be distributed to LHAPDF once complete. 5 S. Forte, private communication. Note that the differences between the heavy quark schemes are due to sub-leading terms, and thus should be considered part of the theoretical uncertainties. 6 Thus, a NLO prediction for the Higgs cross section through gg fusion would be very similar if either a NLO gluon or NNLO gluon were used, and vice versa for a NNLO prediction.

18 Figure 13: (left)a comparison of the gluon distributions from the MSTW2008 PDF sets, at LO, NLO and NNLO, at a Q 2 value of 4 GeV 2. (right) A comparison of the up quarks distributions from the MSTW2008 PDF sets, at LO, NLO and NNLO, at a Q 2 value of 4 GeV 2. The NLO distributions for CTEQ6.6 are also shown for comparison. This figure comes out faint. Will try to improve (MSTW) or 0 (CTEQ), rather than the canonical value of χ 2 of 1 for a 1-sigma uncertainty and 2.3 for a 90%confidence level 7. Thus, PDF uncertainties for CTEQ have tended to be larger than those determined from MSTW 8. Such increases in χ 2 correspond to a 90% confidence level (CL) limit. The fact that the tolerance criteria for CTEQ and MSTW differ by a factor of two emphasize that it is difficult to determine PDF uncertainties in a statistically rigorous way 9. PDF uncertainties corresponding to a 68% CL limit (1-sigma) can be approximately calculated by the appropriate scaling of the 90% CL errors. The MSTW2008 PDFs have specific error PDFs corresponding to a 68% CL in addition to those corresponding to 90% CL. The approximations involved can be observed in comparing cross section uncertainties using both sets of error PDFs; in many cases, the scaling between them is larger than the factor of 1.6 expected from statistical considerations. A large number of parameters are needed in order to fully specify the behavior of each flavor of PDF over the full x-range. Some parameters can be fixed at reasonable values while others can be determined from sum rules: the net result is that the CTEQ6.6 (CT09) PDFs have 22 (24) free parameters, while the MSTW2008 PDFs have 20 free parameters. Typically, error PDFs are determined by diagonalizing the Hessian matrix corresponding to the free fit parameters (22X22 for CTEQ6.6). The resulting eigenvectors form an orthonormal basis set from which the uncertainty for any cross section can be determined using the Master Equation: X + max = N i=1 [max(x + i X 0,X i X 0,0)] 2 7 Some recent studies have indicated that the need for a larger χ 2 can be tied to parametrization biases inherent in the global fits. 8 CTEQ (and the NNPDF collaboration) uses the correlated systematic error information for all experiments for which this information is available; MSTW uses the correlated information only for the Tevatron jet experiments (and the Z rapidity distribution) and adds all other experimental systematic errors in quadrature with the statistical error. This may be partially responsible for the difference in the assumed χ 2 values. 9 Both groups have recently been developing more sophisticated treatments for the determination of the error PDFs, but the fundamental difference in the relative size of the PDF errors remains.

19 X max = N i=1 [max(x 0 X i +,X 0 Xi,0)] 2 (6) X + adds in quadrature the PDF error contributions that lead to an increase in the observable X and X the PDF error contributions that lead to a decrease. The addition in quadrature is justified by the eigenvectors forming an orthonormal basis. The sum is over all N eigenvector directions, or 22(24) in the case of CTEQ6.6(CT09). Ordinarily, X i + X 0 will be positive and Xi X 0 will be negative, and thus it is trivial as to which term is to be included in each quadratic sum. For the higher number eigenvectors, however, we have seen that the + and contributions may be in the same direction. In this case, only the most positive term will be included in the calculation of X + and the most negative in the calculation of X. Thus, there may be less than N terms for either the + or directions. Note that MSTW uses the symmetric form of the Master Equation, i.e. there will always be exactly N terms for both the positive and negative directions. The net result should be very similar to the use of the asymmetric form described above for most calculations. X = 1 2 N i=1 [X + i X i ] 2 (7) Some programs, such as MCFM, will automatically calculate the PDF uncertainty of a given cross section using the specified error PDF set and the Master Formula given above. The results of the NLO calculation (including the error PDF weights) can also be stored in ROOT ntuples, allowing for the PDF uncertainty for any cross section with cuts to be calculated at a later time. The smaller number eigenvectors correspond to the PDF directions that are better determined; the larger number eigenvector correspond to the poorly determined directions. The eigenvectors have contributions from each of the free PDF parameters. In some cases, an eigenvector may have a large component from a particular PDF parameter, but in most cases, the contributions are widely distributed among different PDF parameters and no specific meaning can be ascribed to each eigenvector. Consider the example of Higgs production through gg fusion at a center-of-mass energy of 14 TeV. The most important contributions to the uncertainty in the cross section for a mass of 120 GeV come from (within the CTEQ6.6 PDF set) eigenvectors 4, 6, 11 and 16. These eigenvectors continue to be important for Higgs masses of 200 and 300, but other eigenvectors become important as well, especially at 300 GeV (5 and 7). A new technique has been developed by CTEQ, termed Data Set Diagonalization [?]. The Hessian matrix is originally diagonalized in order to determine the error PDFs. An additional diagonalization is then possible, taking into account a new quantity, such as the Higgs cross section at a particular mass. The result will be a particular eigenvector(s) which directly probes the uncertainty along this direction (the uncertainty of the Higgs cross section at that mass). Other variables, such as the 2nd moment of the Higgs rapidity distribution, are also possible. In Figure 14(left), the up quark distribution for MSTW2008 (at a Q 2 value of 4 GeV 2 ) is compared to the uncertainty on the up quark distribution for CTEQ6.6; on the right, the CTEQ6.6 up quark distribution is compared to the up quark uncertainty for MSTW2008. In Figure 15(left), the gluon distribution for MSTW2008 (at a Q 2 value of 4 GeV 2 ) is compared to the uncertainty on the gluon distribution for CTEQ6.6; on the right, the CTEQ6.6 gluon distribution is compared to the up quark uncertainty for MSTW2008. It can be seen that the PDF uncertainties are less than 5% for the relevant PDFs over the range of interest for Higgs production at the LHC. It is also evident from the plots that the MSTW2008 PDFs lie within the CTEQ6.6 PDF error band; the inverse is not always true as the MSTW2008 error bands are about half of the size of the CTEQ6.6 error bands. If the central values of competing PDFs

20 do not lay within the error bands of each other, this may be an indication of of a basic disagreement or difficulty which should be understood. Figure 14: A comparison of the u quark distributions from the CTEQ6.6 and the MSTW2008 PDF sets, at a Q 2 value of 4 GeV 2. On the left the u quark distribution from MSTW2008 is compared to the PDF error band from CTEQ6.6; the reverse is plotted on the right. Comes out a bit faint. Will try to improve. Figure 15: A comparison of the gluon distributions from the CTEQ6.6 and the MSTW2008 PDF sets, at a Q 2 value of 4 GeV 2. On the left the gluon distribution from MSTW2008 is compared to the PDF error band from CTEQ6.6; the reverse is plotted on the right. Comes out a bit faint. Will try to improve Parton-parton Luminosities It is useful to define differential parton-parton luminosities. Such luminosities, when multiplied by the dimensionless cross section ŝ ˆσ for a given process, provide a useful estimate of the size of an event cross section at the LHC [?]. Below we define the differential parton-parton luminosity dl i j /dŝdy and its

21 Figure 16: The parton-parton luminosity [ dli j dτ ] in picobarns, integrated over y. Green=gg, Blue= i (gq i + g q i + q i g + q i g), Red= i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b integral dl i j /dŝ: dl i j dŝdy = 1 s δ i j [ f i (x 1, µ) f j (x 2, µ) + (1 2)]. (8) The prefactor with the Kronecker delta avoids double-counting in case the partons are identical. The generic parton-model formula 1 σ = dx 1 dx 2 f i (x 1, µ) f j (x 2, µ) ˆσ i j (9) i, j can then be written as ( ) ( ) dŝ σ = i, j ŝ dy dli j (ŝ ˆσ i j ). () dŝdy (Note that this result is easily derived by defining τ = x 1 x 2 = ŝ/s and observing that the Jacobian (τ,y)/ (x 1,x 2 ) = 1.) Equation can be used to estimate the production rate for a hard scattering process at the LHC as follows. Figure 16 shows a plot of the luminosity function integrated over rapidity, dl i j /dŝ = (dli j /dŝdy)dy, at the LHC s = 14TeV for various parton flavour combinations, calculated using the CTEQ6.1 parton distribution functions [?]. The widths of the curves indicate an estimate for the PDF uncertainties. We assume µ = ŝ for the scale. As expected, the gg luminosity is large at low ŝ but falls rapidly with respect to the other parton luminosities. The gq luminosity is large over the entire kinematic region plotted. These should provide similar parton luminosities as with a more modern PDF such as CTEQ6.6 to within 5-%

22 Figure 17: Fractional uncertainty of the gg luminosity integrated over y It is also of great interest to understand the uncertainty in the parton-parton luminosity for specific kinematic configurations. Some representative parton-parton luminosity uncertainties, integrated over rapidity, are shown in Figures 17, 18 and 19. The PDF uncertainties were generated from the CTEQ6.1 Hessian error analysis using the standard χ 2 = 0 criterion. Except for kinematic regions where one or both partons is a gluon at high x, the PDF uncertainties are of the order of 5 %. Luminosity uncertainties for specific rapidity values are available at the SM benchmark website. Even tighter constraints will be possible once the LHC Standard Model data is included in the global PDF fits. Again, the uncertainties for individual PDFs can also be calculated online using the Durham PDF plotter. It is useful to compare the gg PDF luminosity uncertainty obtained from CTEQ (and MSTW) with those from the NNPDF group. A comparison of the CTEQ6.6 and NNPDF (versions 1.2 and 2.0) uncertainties is shown in Figure 20(right). Over the range of Higgs masses accessible through the gg channel, the uncertainties are basically equivalent, especially for the NNPDF2.0 set, which includes Drell-Yan and Tevatron inclusive jet data. (The PDF luminosity uncertainties from MSTW2008 are smaller.) The left-hand side of Figure 20 shows the uncertainty for the gq PDF luminosity. It is interesting that the pinch observed in CTEQ66 (and MSTW2008) for ŝ between 300 GeV and 1 TeV is also present in the NNPDF sets, although not as pronounced. Processes in this kinematic region, such as high p T Z production may serve as a useful benchmark with very little PDF uncertainty. Often it is not the PDF uncertainty for a cross section that is required, but rather the PDF uncertainty for an acceptance for a given final state. The acceptance for a particular process may depend on the input pdfs due to the rapidity cuts placed on the jets, leptons, photons, etc. and the impacts of the varying longitudinal boosts of the final state caused by the different PDF pairs. An approximate rule-of-thumb is that the PDF uncertainty for the acceptance is a factor of 5 times smaller than the uncertainty for the cross section itself [?].

23 Figure 18: Fractional uncertainty for the parton-parton luminosity integrated over y for i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b For the first year or so of LHC operation, the total inelastic cross section, and thus the total delivered integrated luminosity, will not be well-known. Thus, it has been suggested that it would be useful for all physics cross sections at the LHC be normalized to standard model benchmark cross sections, such as W and Z production. From the PDF luminosity figures, it can be seen that the uncertainty for W and Z production is on the order of 5-6%. As noted previously, the smallest parton-parton PDF uncertainties at the LHC are obtained for gq scattering at ŝ=0.50 TeV. gq processes in this kinematic region may also prove to be useful as a normalization tool at the LHC. 4.4 Correlations It is often instructive to determine whether a particular collider observable shares common degrees of freedom with another collider cross section through the non-perturbative PDF parametrizations. Such correlations can be useful in providing ratios of cross sections that have smaller theoretical uncertainties than either cross section by itself. A quantity, cosϕ (the correlation cosine), can be calculated that represents the degree to which two cross sections are correlated; a value of this quantity near 1 implies a very strong correlation, near zero, no correlation, and near -1, a very strong anti-correlation [?]. Here ϕ represents the angle between the gradient vectors of the two cross sections in the Hessian eigenvector space, and can be calculated as shown in the equation below (for two cross sections X and Y 11 : 11 More generally, it can be defined in terms of the correlation between two quantities; it does not require the use of the Hessian technique.

24 Figure 19: Fractional uncertainty for the luminosity integrated over y for i (q i q i + q i q i ), where the sum runs over the five quark flavours d, u, s, c, b cos ϕ = X Y X Y = 1 4 X Y N i=1 ( X (+) i )( X ( ) i Y (+) i ) Y ( ) i. (11) Thus, cosϕ can easily be defined with knowledge of the cross sections for X and Y for each of the Hessian eigenvectors. For example, one would expect (and find) the W and Z cross sections to be very strongly correlated, as both are produced by similar initial states in the same x range. A plot of the correlation cosines for various processes is shown in Figure??. It can be observed, for example, for Higgs production through gg fusion, the process goes from a mild correlation at low Higgs mass (120 GeV) with the W cross section at the LHC, to a very strong anti-correlation for high Higgs masses (500 GeV). Conversely, the correlation of the Higgs boson cross section with respect to the t t cross section is the mirror image of that to the W cross section α s An additional uncertainty regarding Higgs cross section predictions at the LHC arises from the uncertainty in the strong coupling constant α s. It is customary now to present α s at its value at the Z mass, with the NLO world average currently being ± (LO average = and NNLO average = 0.117). This is the value currently used by CTEQ; the Neural Net Collaboration uses a slightly higher 12 The 2007 PDG average for α s (m Z ) is ± The world average for 2009, recently compiled by Siggi Bethke is ± [?].