Scale uncertainties in ggF->Higgs(+jets) J. Huston, S. Ellis, B. Mellado

Scale uncertainty The Higgs cross section depends on the renormalization scale  R and factorization scale  F Consider default values for these two scales,  o,F and  o,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  R,  F 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 saddle point region typical scale choice (p T jet ) is not at the saddle point but scale uncertainty choices include it

Saddle points For c F >0,c R >|c RF |, the saddle point axes are aligned with the plot axes, as shown at the top right At higher p T values, c RF <0 and c F,|c R |<<|c RF |, the saddle position rotates by about 45 o 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 p T 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 p T inclusive jet production, steep at low values of  R, shallow in  F 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 ~(0.15m H,0.24m H ), i.e. outside of the range of uncertainties typically taken into account when using a scale of either m H or 0.5 m H Saddle point ~23.1pb compared to 20.7pb for m H /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  F

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 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  R and  F 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

 F dependence As we have seen, the  F dependence is much flatter than the  R 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 p T 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 p T requirement increases

Higgs+1 jet: y jet Apply selection cuts on photons Require |y jet |<4.5 p T jet >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 p T jet >20 GeV/c; |y jet |<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 (m H,m H )  ~3400 fb  peak ~4000 fb

gg->Higgs + >= 2 jets red=  y jj >1 green=  y jj >2 blue=  y jj >3 from top to bottom for each  y, lines show m jj >0,100, 200,300,400, 500 GeV This is  y>3,m jj >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 m Higgs lead to peak cross section Cross section uncertainties on the order of 20% p T jet >25 GeV/c

Higgs + 2 jets (after VBF cuts) Cross section again peaks at a scale of m Higgs, 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 Red squares Green triangles Blue triangles Magenta crosses

 R increases with y*/y max Black circles Red squares Green triangles Blue triangles Magenta crosses y*=(y j1 -y j2 )/2

 F increases with y*/y max Black circles Red squares Green triangles Blue triangles Magenta crosses Note: maybe no true saddle points at high y* and high mass, so script has trouble finding them; there are still flat places