Parton-parton luminosity functions for the LHC Alexander Belyaev, Joey Huston, Jon Pumplin Michigan State University, East Lansing, Michigan, USA Abstract In this short writeup, we discuss one of the LHC Standard Model benchmarks initiated at the Les Houches 2005 workshop, the parton-parton luminosity distributions and their related uncertainties. 1. Introduction The number of events anticipated at the LHC for a process with a cross section σ can be calculated by multiplying the cross section times the beam-beam luminosity. There are a number of programs available to calculate cross sections for processes of interest at leading-order, next-to-leading order and next-tonext-to-leading order, and in some cases with parton showering and hadronization effects included [?]. But it is sometimes also useful to be able to make quick order-of-magnitude estimates for the sizes of cross sections. For hard interactions, the collision is not between the protons per se but between the partons in the two protons, carrying fractions x 1 and x 2 of their parent proton s momentum. A plot showing the parton kinematics at the LHC is shown in Fig. 1, indicating the relationship between the two parton x values and the mass M = ŝ and rapidity y = 1 2 ln(x 1/x 2 ) of the produced system Thus, for example, a final state with a mass M = 100 GeV and a rapidity y = 4 is produced by two partons with x values of approximately 0.00015 and 0.35. 10 9 LHC parton kinematics 10 8 x 1,2 = (M/14 TeV) exp(±y) Q = M M = 10 TeV 10 7 10 6 M = 1 TeV Q 2 (GeV 2 ) 10 5 10 4 10 3 M = 100 GeV 10 2 y = 6 4 2 0 2 4 M = 10 GeV 6 10 1 10 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10 0 Fig. 1: Parton kinematics for the LHC. x
Because the interacting partons carry only a fraction of the parent proton s momentum, it is useful to define the differential parton-parton luminosity dl ij /dŝ dy and its integral dl ij /dŝ: dl ij dŝ dy = 1 s 1 1 + δ ij [f i (x 1, µ)f j (x 2, µ) + (1 2)]. (1) The prefactor with the Kronecker delta avoids double-counting in case the partons are identical. The generic parton-model formula σ = i,j 1 0 dx 1 dx 2 f i (x 1, µ) f j (x 2, µ) ˆσ ij (2) can then be written as σ = i,j ( ) ( ) dŝ ŝ dy dlij (ŝ ˆσ ij ). (3) dŝ dy (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 3 can be used to estimate the production rate for a hard scattering process at the LHC as follows. Figure 2(left) shows a plot of the luminosity function integrated over rapidity, dl ij /dŝ = (dlij /dŝ dy) dy, at the LHC value s = 14 TeV for various parton flavor 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. (Similar plots made with earlier PDFs are shown in Ellis, Stirling, Webber [?].) On the other hand, Fig. 2(right) presents the second product, [ŝˆσ ij ], for various 2 2 partonic processes. The parton level cross sections are given for a parton p T > 0.1 ŝ cut and for fixed α s = 0.118. We have used the CalcHEP package [?] to estimate these cross sections. As expected, the gg luminosity is large at low ŝ but falls rapidly with respect to the x S^ σ^ 10 gg gg gq gq qq qq qq qq, qq qq qq qq 1 10-1 qq gg gg qq [ Fig. 2: Left: luminosity 1 ŝ 10-2 qq q q 0 2 4 6 8 10 S^ (TeV) ] dl ij in pb integrated over y. Green=gg, Blue=g(d + u + s + c + b) + g( d + ū + s + c + b) + dτ (d + u + s + c + b)g + ( d + ū + s + c + b)g, Red=d d + uū + s s + c c + b b + dd + ūu + ss + cc + bb. Right: parton level cross sections [ŝˆσ ij] for various processes other parton luminosities. The gq luminosity is large over the entire kinematic region plotted.
One can use Equation 3 in the form σ = ŝ ŝ ( dlij dŝ ) (ŝ ˆσ ij ). (4) and Fig. 2 to estimate the production cross sections for QCD jets for a given ŝ interval. For example, for the gluon pair production rate for ŝ=1 TeV and ŝ = 0.01ŝ, we have dlgg dŝ 10 3 pb and ŝ ˆσ gg 20 leading to σ 200 pb (for the p g T > 0.1 ŝ cut we have assumed above). Note that for a given small ŝ/ŝ interval, the corresponding invariant mass ŝ/ ŝ interval, is ŝ/ ŝ 1 2 ŝ/ŝ. One should also mention that all hard cross sections presented in Fig.2 are proportional to α 2 s and have been calculated for α s = 0.118, so production rates can be easily rescaled for a certain α s at a given scale. One can further specify the parton-parton luminosity for a specific rapidity y and ŝ, dl ij /dŝ dy. If one is interested in a specific partonic initial state, then the resulting differential luminosity can be displayed in families of curves as shown in Fig. 3, where the differential parton-parton luminosity at the LHC is shown as a function of the subprocess center-of-mass energy ŝ at various values of rapidity for the produced system for several different combinations of initial state partons. One can read from the curves the parton-parton luminosity for a specific value of mass fraction and rapidity. (It is also easy to use the Durham PDF plotter to generate the pdf curve for any desired flavor and kinematic configuration [?].) It is also of great interest to understand the uncertainty for the parton-parton luminosity for specific kinematic configurations. Some representative parton-parton luminosity uncertainties are shown in Figs. 4-7. The PDF uncertainties were generated from the CTEQ6.1 Hessian error analysis using the standard χ 2 100 criterion. Except for kinematic regions where one or both partons is a gluon at high x, the pdf uncertainties are of the order of 5-10%. Even tighter constraints will be possible once the LHC Standard Model data is included in the global pdf fits. Again, the uncertainties for individual PDF s can also be calculated online using the Durham pdf plotter. CONCLUSIONS Some representative parton-parton luminosity and luminosity uncertainty plots have been presented. A more complete set will be maintained at the Standard Model benchmark website started at Les Houches 2005: www.pa.msu.edu/ huston/les_houches_2005/les_houches_sm.html and will also be included in a review article to be published in the near future. ACKNOWLEDGEMENTS We would like to thank J. Campbell, W.J. Stirling and W.K. Tung for useful discussions and comments.
Fig. 3: dluminosity/dy at y = 0, 2, 4, 6. Green=gg, Blue=g(d + u + s + c + b) + g( d + ū + s + c + b) + (d + u + s + c + b)g + ( d + ū + s + c + b)g, Red=d d + uū + s s + c c + b b + dd + ūu + ss + cc + bb.
Fig. 4: Fractional uncertainty of gg luminosity integrated over y. Fig. 5: Fractional uncertainty of gg luminosity at y = 0.
Fig. 6: Fractional uncertainty for Luminosity at y = 2 for gg. Fig. 7: Fractional uncertainty for Luminosity at y = 4 for gg.
Fig. 8: Fractional uncertainty for Luminosity at y = 6 for gg. Fig. 9: Fractional uncertainty for Luminosity integrated over y for g(d + u + s + c + b) + g( d + ū + s + c + b) + (d + u + s + c + b)g + ( d + ū + s + c + b)g,
Fig. 10: Fractional uncertainty for Luminosity at y = 0 for gq + g q + qg + qg. Fig. 11: Fractional uncertainty for Luminosity at y = 2 for gq + g q + qg + qg.
Fig. 12: Fractional uncertainty for Luminosity at y = 4 for gq + g q + qg + qg. Fig. 13: Fractional uncertainty for Luminosity at y = 6 for gq + g q + qg + qg.
Fig. 14: Fractional uncertainty for Luminosity integrated over y for d d + uū + s s + c c + b b + dd + ūu + ss + cc + bb. Fig. 15: Fractional uncertainty for Luminosity at y = 0 for q q.
Fig. 16: Fractional uncertainty for Luminosity at y = 2 for q q. Fig. 17: Fractional uncertainty for Luminosity at y = 4 for q q.
Fig. 18: Fractional uncertainty for Luminosity at y = 6 for q q.