Scale uncertainties in ggF->Higgs(+jets)

J. Huston, S. Ellis, B. Mellado

Scale uncertainty The Higgs cross section depends on the renormalization scale mR

and factorization scale mF

Consider default values for these two scales, mo,F and mo,R and 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

Some 1-D slices

NLOJET++ with Applgrid

Use logarithmic scales

broad saddlepoint region

typical scale choice (pT

jet) is not atthe saddle pointbut scale uncertainty choicesinclude it

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 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 ihixs

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 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 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 ~NNNLOcurve. 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 toolong to wait

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 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

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 No cuts on photons or

jets (other than jet pT

cuts 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: 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? 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 pT

jet>20 GeV/c; |yjet|<5

Higgs + 2 jets-2D Cutoff at 2000 fb to look at peak in more detail

Higgs + 2 jets 2D Add a few cross section points at lower mR scale

Higgs + 2 jets-2D Cutoff at 2000 fb to look at peak in more detail

(mH,mH)s~3400 fb

speak~4000 fb

gg->Higgs + >= 2 jets 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%

pTjet>25 GeV/c

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

SummaryThe hope is to incorporate some of this information into

Bruce’s noteSteve Ellis, myself, and Pavel Starovoitov 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)Position of saddle point

Black circles 0-0.3Red squares 0.3-0.8Green triangles 0.8-1.2Blue triangles 1.2-.21Magenta crosses 2.1-2.8

mR increases with y*/ymax

Black circles 0-0.3Red squares 0.3-0.8Green triangles 0.8-1.2Blue triangles 1.2-.21Magenta crosses 2.1-2.8

y*=(yj1-yj2)/2

mF increases with y*/ymax

Black circles 0-0.3Red squares 0.3-0.8Green triangles 0.8-1.2Blue triangles 1.2-.21Magenta 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