Report

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 é æm ö æm ö æm ö æm ö æ m ö æ m öù 2 2 R F R F s (mF , m R ) » s (m0,F , m0,R )ê1+ bR ln çç ÷÷ + bF ln çç ÷÷ + cR ln çç ÷÷ + cF ln çç ÷÷ + cRF ln çç R ÷÷ ln çç F ÷÷ú êë è m0,R ø è m0,F ø è m0,R ø è m0,F ø è m0,R ø è m0,F øúû …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 é æm ö æm ö æ m ö æ m öù 2 2 R F s (m F , m R ) » s (mS,F , mS,R )ê1+ cR ln çç ÷÷ + cF ln çç ÷÷ + cRF ln çç R ÷÷ ln çç F ÷÷ú êë è mS,R ø è mS,F ø è mS,R ø è mS,F øúû Consider inclusive jet production NLOJET++ with Applgrid Some 1-D slices 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 é æm ö æm ö æ m ö æ m öù 2 2 R F s (m F , m R ) » s (mS,F , mS,R )ê1+ cR ln çç ÷÷ + cF ln çç ÷÷ + cRF ln çç R ÷÷ ln çç F ÷÷ú êë è mS,R ø è mS,F ø è mS,R ø è mS,F øúû 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 wellbehaved 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 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 illdefined 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 nonmonotonic 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 pTjet>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 speak~4000 fb (mH,mH) s~3400 fb 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) 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 Summary The hope is to incorporate some of this information into Bruce’s note Steve 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 0-0.3 Red squares 0.3-0.8 Green triangles 0.8-1.2 Blue triangles 1.2-.21 Magenta crosses 2.1-2.8 mR increases with y*/ymax y*=(yj1-yj2)/2 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 mF 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 Note: maybe no true saddle points at high y* and high mass, so script has trouble finding them; there are still flat places