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# Higgs Cross Section Calculation at 7 and 8 TeV for RUN-1 This is the instruction for Higgs cross section calculations at centre-of-mass energy of 7 and 8 TeV.

Complete inclusive cross sections at LO, NLO and NNLO(+NNLL) should be calculated.

## Higgs mass range and step:

Higgs Mass range step size # of points
[ 80,110] GeV 1 GeV 31 points
[110,140] GeV 0.5 GeV 60 points
[140,160] GeV 1 GeV 20 points
[160,290] GeV 2 GeV 65 points
[290,350] GeV 5 GeV 12 points
[350,400] GeV 10 GeV 5 points
[400,1000] GeV 20 GeV 30 points
+ 450, 550, 650, 750, 850, 950 GeV (6 points)
• 229 points in total

## QCD scale uncertainty

• It should be discussed in each subgroup how to estimate QCD scale uncertainty.
• As these uncertainties change smoothly, it would be enough to do the coarse scan with following points.
• Please carry out the interpolation with spline fit (or 3rd order polynomial fit) for other Higgs mass points.

* Scan example:

Higgs Mass range step size # of points [200,300] GeV 10 GeV 10 points
[80,200] GeV 5 GeV 25 points
[300,1000] GeV 20 GeV 35 points
• 70 points in total

• The factorization and the renormalization scales are process dependent and should be defined by each subgroup.
• The scale uncertainty should also be defined by each group, (e.g.. 1/2MH < μR, μF < 2MH, do 2D scan with 1/2 < μRF < 2)

## PDF+αs uncertainty calculations (a la PDF4LHC prescription)

• How to estimate PDF+αs uncertainty for 8TeV is under discussion by PDF experts.

### PDF sets

1. For NLO calculations, CTEQ6.6, MSTW2008 and NNPDF2.0 are the default sets. Others, e.g. HERAPDF, ABKM and (G)JR are optional, but we recommend to calculate with these PDF sets as well.
2. For NNLO calculations, MSTW2008 is the defalut set. We also recommend to compare MSTW with ABKM and (G)JR at NNLO.

### Cross section central values

1. For NLO, take the midpoint of CTEQ6.6, MSTW2008 and NNPDF2.0 envelope following the recommendation from PDF4LHC group.
2. For NNLO, take MSTW2008 result.

### PDF+αs uncertainties (PDF4LHC group recommendation )

• Δαs=+-0.0012 for 68%C.L. and Δαs=+-0.0020 for 90%C.L.
• If calculations only at 90% C.L. calculation exist (CTEQ), we shall translate (i.e., by dividing 90%C.L. error by 1.645).
• PDF+αs uncertainties should be calculated both at 68% C.L. (priority) and at 90% C.L..

1. The total PDF+αs uncertainty can be evaluated by adding the variations in PDFs due to αs uncertainty in quadrature with the fixed αs PDF uncertainty,
2. For NNPDF, more efficiently taking a Gaussian distribution of PDF replicas corresponding to different values of αs.
3. For MSTW, the PDF+ αs uncertainties should be evaluated using their prescription which better accounts for correlations between the PDF and αs uncertainties when using the MSTW dynamical tolerance procedure for uncertainties. Adding the αs uncertainty in quadrature for MSTW can be used as a simpli cation but generally gives slightly smaller uncertainties.

1. For NLO, use the envelope provided by the central values and PDF+ αs errors from the MSTW08, CTEQ6.6 and NNPDF2.0 PDFs, using each group's prescriptions for combining the two types of errors. Take the symmetric error following the recommendation from PDF4LHC group.
2. For NNLO, multiply the MSTW uncertainty at NNLO by the factor obtained by dividing the full PDF+αs uncertainty obtained from the envelope of MSTW, CTEQ and NNPDF results at NLO by the MSTW PDF+αs uncertainty at NLO (~ factor 2 at 7 TeV).

## Recipe from PDF4LHC The official PDF4LHC recommendations on NLO/NNLO cross sections and PDF uncertainty estimations:

1. Recommendation for LHC cross section calculations (June 22, 2010)

### Practical guide on PDF+ αs error calculation

1. Suggestions on calculating the PDF4LHC prescription (September 21, 2010)

1. MSTW: Check the first two paragraphs of section 6 in arXiv:0905.3531 .
• However adding in quadrature is a pretty good approximation if one is attempting to be quicker.
2. CTEQ: Check section III-B in arXiv:1004.4624 .
3. NNPDF: Check arXiv:1004.0962 .

-- ReiTanaka - 21-May-2010

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