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Scale uncertainties in ggF ->Higgs(+jets). J. Huston, S. Ellis, B. Mellado. Scale uncertainty. The Higgs cross section depends on the renormalization scale m R and factorization scale m F Consider default values for these two scales, m o,F and m o,R and expand around these values

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scale uncertainties in ggf higgs jets

Scale uncertainties in ggF->Higgs(+jets)

J. Huston, S. Ellis, B. Mellado

scale uncertainty
Scale uncertainty
  • The Higgs cross section depends on the renormalization scale mR and factorization scale mF
  • Consider default values for these two scales, mo,Fand mo,Rand expand around these values
  • Can write the NLO Higgs cross section (actually any NLO cross section) near the reference scales as
  • …where the explicit logarithmic dependences have been factorized out; the b and c variables will depend on the kinematics
  • In general, there will be a saddle point, where the local slope as a function of mR,mF is zero
  • Around the saddle point, can write the scale dependence as
consider inclusive jet production
Consider inclusive jet production

NLOJET++ with Applgrid

Some 1-D slices

use logarithmic scales
Use logarithmic scales

broad saddle

point region

typical scale

choice (pTjet) is not at

the saddle point

but scale

uncertainty choices

include it

saddle points
Saddle points
  • For cF>0,cR<0 and cF,|cR|>>|cRF|, the saddle point axes are aligned with the plot axes, as shown at the top right
  • At higher pT values, cRF<0 and cF,|cR|<<|cRF|, the saddle position rotates by about 45o
  • The saddle position also depends on jet size and on rapidity (somewhat)
  • In any case, the perturbative series is well-behaved for inclusive jet production, leading to stable predictions at NLO, using a scale related to the pT of the jet
  • …except perhaps when you go very far forward
2 d plots for ggf for higgs
2-D plots for ggF for Higgs
  • The NNLO scale dependence looks similar to that for low pT inclusive jet production, steep at low values of mR, shallow in mF
  • Note that there is no saddle point at NLO; it looks similar to LO for inclusive jet production


ggf at nnlo
ggF at NNLO
  • Note that the location of the saddle point is at ~(0.15mH,0.24mH), i.e. outside of the range of uncertainties typically taken into account when using a scale of either mH or 0.5 mH
  • Saddle point ~23.1pb compared to 20.7pb for mH/2
ggf at nnlo1
ggF at NNLO
  • Now consider a 450 GeV Higgs produced by ggF
  • There’s some rotation of the saddle region as you would expect from the jet analysis
  • Saddle point also moves to smaller mF
babis at ggi
Babis at GGI
  • Points out that series is not well-behaved and that even NNLO might not be enough for precision predictions
  • ~N3LO prediction peaks near a scale of mHiggs
  • But normalization has not been determined; likely to have some additional positive corrections
  • I don’t really understand the ~NNNLO
  • curve. Very large change in
  • predicted cross section at low scales.
  • claims that 5% precision might be
  • achievable at NNNLO.
  • good progress in the
  • calculation, so maybe we don’t have too
  • long to wait
now look at higgs 1 jet at nlo
Now look at Higgs+1 jet at NLO
  • This is for inclusive requiring only a 20 GeV/c cut on the jet; behavior is monotonic and no saddle point is present; scale uncertainties are large and ill-defined
higgs 1 jet at nlo
Higgs+1 jet at NLO
  • This plot was generated using MCFM running on a 5X5 grid of scale choices for mR and mF
  • What we’re trying to understand is how well we can define the scale uncertainties for Higgs+jets in a region where ggF dominates, use the measured cross section to pin down that cross section, and then translate that to the region where we are trying to measure the contribution of VBF
  • Can we define a region where ggF dominates and where the scale dependence is better-behaved
m f dependence
mF dependence
  • As we have seen, the mF dependence is much flatter than the mR dependence
  • Mostly because ggF probes the gluon distribution in the region around the inflection point
  • For the higher x values probed in the VBF region, this will change somewhat
higgs 1 jet
Higgs + 1 jet
  • No cuts on photons or jets (other than jet pTcuts shown)
  • I said the scale behavior of the Higgs+1 jet cross section was worrisome
  • The behavior of the NLO cross section becomes non-monotonic as the jet pT requirement increases
higgs 1 jet y jet
Higgs+1 jet: yjet
  • Apply selection cuts on photons
  • Require |yjet|<4.5
  • pTjet>25 GeV/c
  • Non-monotonic behavior only when jet rapidity is large
  • We need Higgs+1 jet at NNLO
  • Luckily that will happen in 2013
what about higgs 2 jets
What about Higgs+2 jets?
  • The 1-D plot is shown here
  • Much better behavior than either inclusive Higgs (at NNLO) or Higgs+1 jet (at NLO)
higgs 2 jets 2d
Higgs + 2 jets-2D
  • pTjet>20 GeV/c; |yjet|<5
higgs 2 jets 2d1
Higgs + 2 jets-2D
  • Cutoff at 2000 fb to look at peak in more detail
higgs 2 jets 2d2
Higgs + 2 jets 2D
  • Add a few cross section points at lower mR scale
higgs 2 jets 2d3
Higgs + 2 jets-2D
  • Cutoff at 2000 fb to look at peak in more detail

speak~4000 fb


s~3400 fb

g g higgs 2 jets
gg->Higgs + >= 2 jets

pTjet>25 GeV/c

  • red=Dyjj>1
  • green=Dyjj>2
  • blue=Dyjj>3
  • from top to bottom for each Dy, lines show mjj>0,100, 200,300,400, 500 GeV
  • This is Dy>3,mjj>400 GeV, closest to VBF cuts
  • Cross sections for scales of 12.6 GeV (and sometimes for 25.2 GeV) are negative
  • For VBF-like cuts, scales of mHiggs lead to peak cross section
  • Cross section uncertainties on the order of 20%
higgs 2 jets after vbf cuts
Higgs + 2 jets (after VBF cuts)
  • Cross section again peaks at a scale of mHiggs, so taking a factor of 2 up or down results in <20% scale uncertainty
  • Still need to look at 2D scale plots
  • The hope is to incorporate some of this information into Bruce’s note
  • Steve Ellis, myself, and PavelStarovoitov are writing a note/paper on scale dependence for inclusive jet production incorporating the detailed information we have for that process
  • Would be nice to try for an analytic understanding of the b and c parameters for both jet production and Higgs(+jets) production
look for saddle point position dijets
Look for saddle point position (dijets)

Position of saddle point

Black circles 0-0.3

Red squares 0.3-0.8

Green triangles 0.8-1.2

Blue triangles 1.2-.21

Magenta crosses 2.1-2.8

m r increases with y y max
mR increases with y*/ymax


Black circles 0-0.3

Red squares 0.3-0.8

Green triangles 0.8-1.2

Blue triangles 1.2-.21

Magenta crosses 2.1-2.8

m f increases with y y max
mFincreases with y*/ymax

Black circles 0-0.3

Red squares 0.3-0.8

Green triangles 0.8-1.2

Blue triangles 1.2-.21

Magenta crosses 2.1-2.8

Note: maybe no true saddle points at high y* and high

mass, so script has trouble finding them; there are still flat places