A few remarks on the use of LO or NLO PDFs with LO Mc’s A M Cooper- Sarkar

1. A few remarks on the use of LO or NLO PDFs with LO Mc’s A M Cooper- Sarkar Oxford

2. I have read TJorBjorn’s presentation What I have to say is practical not ideological and applies in a different kinematic region: Concerning di –lepton production from low ~ 50 GeV to high ~ 850 GeV invariant mass I have made LO and NLO QCD calculations of the cross-section for this process using a programme from James Stirling, which includes the contribution of the Z* as well as the γ* (and of course the γ/Z* interference term).

3. The results are given below using the SAME input NLO PDF ( CTEQ6.1M- but very similar for MRST2004 or ZEUS2005). For y=0, the results are: l+ l- mass d2б/dMdy(NLO) d2б/dMdy(LO)k =d2б/dMdy(NLO/LO). 50.0000 0.780191 0.689268 1.13191 150.0000 0.358557E-01 0.315450E-01 1.13665 250.0000 0.382462E-02 0.333211E-02 1.14781 350.0000 0.103442E-02 0.895811E-03 1.15473 450.0000 0.403739E-03 0.348296E-03 1.15918 550.0000 0.193976E-03 0.166892E-03 1.16228 650.0000 0.106464E-03 0.914202E-04 1.16456 750.0000 0.640727E-04 0.549178E-04 1.16670 850.0000 0.411852E-04 0.352451E-04 1.16854 950.0000 0.278270E-04 0.237786E-04 1.17025 If we define the 'truth' as the cross-section using an NLO(calculation) with NLO(PDF) then this is the truth and this LO(calculation) with NLO(PDF) is a half-truth

4. This can also be done for the same input LO PDFs (CTEQ6.1L) (for which αs is calculated at NLO) For y=0, the results are: l+ l- mass d2б/dMdy(NLO) d2б/dMdy(LO) k =d2б/dMdy(NLO/LO). 50.0000 0.665185 0.611352 1.08806 150.0000 0.302169E-01 0.269896E-01 1.11958 250.0000 0.322880E-02 0.283605E-02 1.13849 350.0000 0.875813E-03 0.762209E-03 1.14905 450.0000 0.342926E-03 0.296804E-03 1.15540 550.0000 0.165343E-03 0.142603E-03 1.15947 650.0000 0.911521E-04 0.784304E-04 1.16220 750.0000 0.551402E-04 0.473538E-04 1.16443 850.0000 0.356573E-04 0.305770E-04 1.16615 950.0000 0.242529E-04 0.207711E-04 1.16762 SO here we have NLO(calculation) with LO(PDF) which isn’t very interesting and here we have LO(calculation) with LO(PDF) -which is a different kind of half-truth. The point is we are closer to the truth using LO(calculation) with NLO(PDF) (previous page) than we are using LO(calculation) with LO(PDF) (this page).

5. In case you are worried that this is because I used CTEQ6.1L here are the same calculations using LO PDFs CTEQ6.1LL (for which αs is calculated at LO) For y=0, the results are: l+ l- mass d2б/dMdy(NLO) d2б/dMdy(LO) k =d2б/dMdy(NLO/LO). 50.0000 0.725822 0.655623 1.10707 150.0000 0.324340E-01 0.285309E-01 1.13680 250.0000 0.343042E-02 0.297184E-02 1.15431 350.0000 0.921696E-03 0.791999E-03 1.16376 450.0000 0.357561E-03 0.305763E-03 1.16941 550.0000 0.170868E-03 0.145671E-03 1.17297 650.0000 0.934141E-04 0.794778E-04 1.17535 750.0000 0.560761E-04 0.476296E-04 1.17734 850.0000 0.360077E-04 0.305440E-04 1.17888 950.0000 0.243338E-04 0.206178E-04 1.18023 The LO(calculation) with LO(PDF) is closer to the ‘truth’ for CTEQ6.1LL but it is still not as good as LO(calculation) with NLO(PDF)

6. Further comments: These illustrations were made with y=0, but the same applies right out to y=2.7 Finally James Stirling comments: Note that all "LO" pdfs result from global fits with very poor total χ2 and therefore all resulting LO predictions (in particular cross section normalisations) must be taken with a pinch of salt.

7. Extras on shape of k-factors

8. The k-factors rise with lepton mass and Stirling commented " The interpretation here is that the (negative) quark-gluon order alphas contribution dies away for high masses, leaving the positive (and fairly constant) quark-antiquark contribution. " But Thorne added “I think there is slightly more going on here. At the highest masses the quark-gluon contribution has vanished, but the total is still growing with M and indeed the qq contribution alone is growing, despite the fact that αs is falling. In this region of high masses I think we are starting to see the effect of the ln(1-x)/(1-x) terms in the NLO quark coefficient function, with the x of the incoming quarks becoming rather large, and increasing as M increases. At lower masses we are insensitive to these terms, and each individual contribution falls with M, as expected, but the opposite sign of the two contributions does indeed lead to the increase in the total. At high M everything is determined by the quark-antiquark since the quark-gluon coefficient functions do not have the ln(1-x)/(1-x) type contributions”

9. Thorne’s explanation would make the rise with M for the Tevatron faster than that for the LHC since x values are closer to 1 -bringing in the ln(1-x)/(1-x) terms. I have checked this out and it is true that the LHC k-factors are much flatter than those of the Tevatron as a function of mass. • The Tevatron k-factors are also larger as expected (larger αs) • If we move to higher y then the k-factors are relatively larger for both LHC and Tevatron. • The trend for the k-factor to be larger as mass increases is also enhanced at higher y (presumably due to larger x values for one of the partons The stronger effect for Tevatron than LHC remains. • Illustrative numbers can be supplied on demand!