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

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A few remarks on the use of LO or NLO PDFs with LO Mc’s

A M Cooper- Sarkar

Oxford

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).

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

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).

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)

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.

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”

- 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!