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