First, let us recall that the plots presented show the comparison,
in a chosen framework (POWHEG, MC@NLO, analytic resummation) of the
shapes obtained in the approximation mt -> oo versus the case with full
mass dependence included.
Then there are some basic features of the various calculations that we
think we all agree upon:
-the numerical predictions in the different approaches are consistent with
their respective underlying formulation; we have no evidence of bugs in
the different implementations;
-all the approaches share NLO-accuracy when predicting the inclusive Higgs
production total cross section;
they differ in the way higher order effects are included, in particular
in the description of multiple gluon emission;
-the formalism of the different approaches is adequate to describe a model
with only the top quark (and no bottom), also with finite mass;
indeed if we look at the POWHEG and MC@NLO plots for the top-only case
they look very similar.
In fact in this case there are two scales: the resummation
scale and the scale at which the finite mass effects become relevant;
for the resummation scale it is customary to choose a value like MH/2,
i.e. half of the invariant mass of the final state boson;
from NLO calculations we know that the top quark mass effects start to be
relevant at pt^Higgs ~ mtop;
the conclusion is that the resummation is used for pt^Higgs \in [0, MH/2]
and that in this interval, with MH/2 << mtop, the dominant part of the
amplitude is given indeed by triangle diagrams and not by the boxes;
in other words we are resumming contributions that are not
only formally, but also numerically leading;
Let us come to case top+bottom. Here, if we look at the POWHEG and MC@NLO
plots it seem there are large differences, however there are also some
common features.
First the point where the curve top+bottom crosses 1 (i.e. mt -> oo) is in
both plots at pt~ 30 GeV while its maximum is, in both plots, at pt ~120
+- 20 GeV.
What seems to happen is that the POWHEG method amplifies the bottom
contribution with respect to MC@NLO. Indeed for pt < 30 GeV in POWHEG
one obtains differences with respect to mt -> oo up to 10% while in MC@NLO
the same effect is 1-2%. Above pt~30 GeV in POWHEG one still gets effects
up to O(10%) while in MC@NLO up to O(6%).
Similar results to MC@NLO are obtained by Mantler and Wiesemann (and, it
seems, also Grazzini).
We know that the POWHEG, MC@NLO and the analytic resummation,
Grazzini-Mantler-Wiesemann (GMW), approaches are different.
Since all the three approaches combine fixed-order and all-orders results,
a prescription must be provided to specify the interval where multiple
gluon radiation is included and the interval where the fixed order results
should be recovered.
To our understanding, in the GMW case, the resummation scale plays this
role and is chosen to be Q=MH/2.
In MC@NLO different possibilities are available,
while in POWHEG, on a event-by-event basis, the virtuality of the first
emission fixes the maximum scale at which the Parton Shower can emit extra
partons.
Thus, the GMW and the MC@NLO approaches share the additive treatment of
the exact matrix element corrections and, in this respect,
the numerical agreement between them is not a surprise,
because of the underlying similarities of the two methods.
In contrast, the POWHEG approach uses a factorized formulation,
where the finite quark mass effects are propagated also to higher orders,
multiplying the exact treatment of the first emission with the weights
associated to all the subsequent Parton Shower emissions.
Said this, and knowing that POWHEG includes higher order effects,
we may suspect that the differences for pt >~ 30 GeV can be ascribed to
higher order effects, while the case pt <~ 30 GeV,
where the difference is more pronounced, seems to be more complicated.
Our feeling about the latter is that it can be viewed as a signal that,
in the case of the bottom, a resummation scale MH/2 is not adequate,
because the bottom at this scale is well resolved.
In particular we suspect that, for pt ~ mb an important role is played by
the bottom box diagrams.
They are formally subleading with respect to the triangle diagrams,
i.e. to the LO contribution, because they are not collinearly divergent;
however at pt~mb they are probably numerically comparable or larger than
the triangles. We are planning to check this statement soon.
The bottom contributions enter via the interference with the top part of
the amplitude, because the modulus squared of the botton part suffers an
extra suppression factor due to the Yukawa coupling of the Higgs to the
bottom. The presence of this interference term represents a novelty with
respect to the canonical approach to the ptH resummation, and should be
studied in detail.
A fair conclusion is that at present we have evidence that bottom mass
effects can be numerically not negligible,
but we do not have an adequate framework to matchNLO fixed order result
with multiple hard scales
with the resummation of effects to all orders.
We can consider the difference of the size of the effects obtained with
POWHEG and with MC@NLO
as a possible estimate of the uncertainty that should be applied to the
predictions, until the issue of the inclusion of finite quark mass effects
will be studied in greater detail.
Giuseppe Degrassi and Alessandro Vicini
