#### Charm and beauty quark masses in the MMHT2014 global PDF analysis

Eur. Phys. J. C
Charm and beauty quark masses in the MMHT2014 global PDF analysis
L. A. Harland-Lang 1
A. D. Martin 0
P. Motylinski 1
R. S. Thorne 1
0 Institute for Particle Physics Phenomenology, Durham University , Durham DH1 3LE , UK
1 Department of Physics and Astronomy, University College London , London WC1E 6BT , UK
We investigate the variation in the MMHT2014 PDFs when we allow the heavy-quark masses mc and mb to vary away from their default values. We make PDF sets available in steps of mc = 0.05 GeV and mb = 0.25 GeV, and present the variation in the PDFs and in the predictions. We examine the comparison to the HERA data on charm and beauty structure functions and note that in each case the heavy-quark data, and the inclusive data, have a slight preference for lower masses than our default values. We provide PDF sets with three and four active quark flavours, as well as the standard value of five flavours. We use the pole mass definition of the quark masses, as in the default MMHT2014 analysis, but briefly comment on the MS definition.
1 Introduction
Over the past few years there has been a significant
improvement both in the precision and in the variety of the data for
deep-inelastic and related hard-scattering processes. Since
the MSTW2008 analysis [
1
] we have seen the appearance
of the HERA combined H1 and ZEUS data on the total [
2
]
and also on the charm structure functions [
3
], together with
a variety of new hadron-collider data sets from the LHC,
and in the form of updated Tevatron data (for full references
see [
4
]). Additionally, the procedures used in the global PDF
analyses of data have been improved, allowing the parton
distributions of the proton to be determined with more
precision and with more confidence. This allows us to improve
predictions for Standard Model signals and to model
Standard Model backgrounds to possible experimental signals of
New Physics more accurately. One area that now needs
careful attention, at the present level of accuracy, is the treatment
of the masses of the charm and beauty quarks, mc and mb, in
the global analyses. Here we extend the recent MMHT2014
global PDF analysis [
4
] to study the dependence of the PDFs,
and the quality of the comparison to data, under variations of
these masses away from their default values of mc = 1.4 GeV
and mb = 4.75 GeV, as well as the resulting predictions for
processes at the LHC. We make available central PDF sets for
a variety of masses, namely mc = 1.15–1.55 GeV in steps of
0.05 GeV and mb = 4.25–5.25 GeV in steps of 0.25 GeV. We
also make available the standard MMHT2014 PDFs, and the
sets with varied masses in the three and four flavour number
schemes.
2 Dependence on the heavy-quark masses
2.1 Choice of range of heavy-quark masses
In the study of heavy-quark masses that accompanied the
MSTW2008 PDFs [
5
] we varied the charm and beauty quark
masses, defined in the pole mass scheme, from mc = 1.05 to
1.75 GeV and mb = 4 to 5.5 GeV. This was a very generous
range of masses, and it was not clear that there was a demand
for PDFs at the extreme limits. Hence, this time we are a little
more restrictive, and study the effects of varying mc from
1.15 to 1.55 GeV, in steps of 0.05 GeV, and of varying mb
from 4.25 to 5.25 GeV in steps of 0.25 GeV. Part of the reason
for this is that the values are constrained by the comparison to
data, though for both charm and beauty the preferred values
are at the lower end of the range, as we will show. However,
there is also the constraint from other determinations of the
quark masses. These are generally quoted in the MS scheme,
and in [
6
] are given as mc(mc) = (1.275 ± 0.025) GeV and
mb(mb) = (4.18 ± 0.03) GeV. The transformation to the
pole mass definition is not well defined due to the diverging
series, i.e. there is a renormalon ambiguity of ∼0.1–0.2 GeV.
The series is less convergent for the charm quark, due to the
lower scale in the coupling, but the renormalon ambiguity
cancels in difference between the charm and beauty masses.
Indeed, we obtain mpbole −mcpole = 3.4 GeV with a very small
uncertainty [
7,8
]. Using the perturbative expression for the
conversion of the beauty mass, and the relationship between
the beauty and charm mass, as shown in [5], we obtain
mcpole = 1.5 ± 0.2 GeV and mpbole = 4.9 ± 0.2 GeV. (1)
This disfavours mc ≤ 1.2–1.3 GeV and mb ≤ 4.6–4.7 GeV.
As the fit quality prefers values in this region, or lower, we
allow some values a little lower than this. In the upper
direction the fit quality clearly deteriorates, so our upper values
are not far beyond the central values quoted above. There is
some indication from PDF fits for a slightly lower mpole than
that suggested by the use of the perturbative series out to the
order at which it starts to show lack of convergence. We now
consider the variation with mc and mb in more detail.
2.2 Dependence on mc
We repeat the global analysis in [
4
] for values of mc = 1.15–
1.55 GeV in steps of 0.05 GeV. As in [
4
] we use the “optimal”
version [
9
] of the TR’ general-mass variable-flavour-number
scheme GM-VFNS [
10
]. This is smoother near the
transition point, which we define to be at Q2 = μ2 = mc2, than
the original version, so has a slight tendency to prefer lower
masses—the older version growing a little more quickly at
low scales, which could be countered by increasing the mass.
We also assume all heavy flavour is generated by evolution
from the gluon and light quarks, i.e. there is no intrinsic heavy
flavour. We perform the analysis with αS(MZ2 ) left as a free
parameter in the fit at both NLO and NNLO, but also use
our fixed default values of the coupling of αS(MZ2 ) = 0.118
and 0.120 at NLO and αS(MZ2 ) = 0.118 at NNLO. Unlike
the MSTW2008 study [
5
] we will concentrate on the results
and PDFs with fixed coupling, as the standard MMHT PDFs
were made available at these values.
We present results in terms of the χ 2 for the total set of
data in the global fit and for just the data on the reduced cross
section, σ˜ cc¯, for open charm production at HERA [
3
]. This is
shown at NLO with αS(MZ2 ) = 0.120 in Fig. 1. The variation
in the quality of the fit to the HERA combined charm cross
section data is relatively slight, less than the variation in the fit
to the separate H1 and ZEUS data used in [
5
]. This is
presumably due to the use of the full information now available on
correlated systematics, which allows movement of the data
relative to the theory with only a moderate penalty in χ 2.
The lower variation is also likely due in part to the improved
flavour scheme. Despite the fairly small variation in χ 2 the
charm data clearly prefer a value close to mc = 1.35 GeV,
near our default value of mc = 1.4 GeV. However, there is
more variation in the fit quality to the global data set, with a
clear preference for values close to mc = 1.2 GeV. The
deterioration is clearly such as to make values of mc > 1.5 GeV
strongly disfavoured. The main constraint comes from the
inclusive HERA cross section data, but there is also some
preference for a low value of the mass from NMC structure
function data, where the data for x ∼ 0.01 and Q2 ∼ 4 GeV2
is sensitive to the turn-on of the charm contribution to the
structure function. Overall, there is some element of tension
between the preferred value for the global fit and the fit to
charm data. We do not attempt to make a rigorous
determination of the best value of the mass or its uncertainty, as
provided in [
11
] for example, as we believe there are more
precise and better controlled methods for this. However, a
rough indication of the uncertainty could be obtained from
the χ 2 profiles by treating mc in the same manner as the
standard PDF eigenvectors and applying the dynamic tolerance
procedure. In this case the appropriate tolerance, obtained
110 χ2σ˜cc (52 pts), NLO, αS(MZ2) = 0.120
3360 χ2global (2996 pts), NLO, αS(MZ2 ) = 0.120
100
90
80
70
χ2σ˜cc (52 pts), NLO, αS(MZ2 ) = 0.118
by assuming the charm cross section data is the dominant
constraint, would be of the order T = χ 2 ≈ 2.5.1
The analogous results for αS (MZ2 ) = 0.118 and αS (MZ2 )
left free are shown in Fig. 2 and Table 1, respectively, where
in the latter case the corresponding αs (MZ2 ) values are shown
as well. For αS (MZ2 ) = 0.118 the picture is much the same
as for αS (MZ2 ) = 0.120 except that the fit to charm data is
marginally better, while the global fit is a little worse, though
more-so for higher masses. The results with free αS (MZ2 )
are consistent with this, with the preferred value of αS (MZ2 )
falling slightly with lower values of mc. However, the values
of mc preferred by charm data and the full data sets are much
1 As discussed in [
1
], for a 68 % confidence level uncertainty we insist
the fit quality to a given data set deteriorates by no more than the width
of the χ 2 distribution for N points, roughly √N /2 multiplied by the
χ 2 per point for the best fit. For the charm cross section data this is
≈√52/2 × 1.3.
110
100
90
80
70
60
the same as for fixed coupling—the values of the χ 2 just
being a little lower in general.
The results of the same analysis at NNLO are shown for
αS (MZ2 ) = 0.118 and αS (MZ2 ) left free in Fig. 3 and Table 2,
respectively, where again in the latter case the corresponding
αs (MZ2 ) values are shown. Broadly speaking, the results are
similar to those at NLO, but with lower values of mc preferred
and where the χ 2 variation is greater for the inclusive data
than for the charm cross section data. However, in this case
there is essentially no tension at all between the inclusive
and charm data, with both χ 2 values minimising very near to
mc = 1.25 GeV—this lower preferred value for the charm
data meaning that the fit quality at mc = 1.55 GeV has
deteriorated more than at NLO. The picture is exactly the
same for fixed and free strong coupling, with the values of
χ 2 simply being a little lower when αS (MZ2 ) is left free,
since the best fit value of the coupling is a little below 0.118,
particularly for low mc.
2.3 Dependence on mb
We repeat essentially the same procedure for varying
values of mb in the range 4.25–5.25 GeV in steps of 0.25 GeV.
However, this time there were no data on the beauty
contribution to the cross section included in the standard global fit
[
4
]. In the previous heavy-quark analysis [
5
] we compared to
beauty cross section data from H1 [
12
]. This placed a weak
constraint on the value of mb but had negligible constraint
on the PDFs for fixed mb. Hence, we did not include these
data in the updated global fit [
4
]. There are now also data of
comparable precision from ZEUS [
13
], and we will include
both these data sets in future global fits. In this article we
study the quality of the comparison to these data to
predicχ2σ˜cc (52 pts), NNLO, αS(MZ2 ) = 0.118
tions obtained using the MMHT PDFs with different values
of mb. The data themselves are not included in the fit, i.e. we
use predictions from the PDFs, as they still provide negligible
direct constraint.
The results for the NLO PDFs with αS (MZ2 ) = 0.120 and
0.118 are shown in Figs. 4 and 5, respectively. The picture
for the data in the global fit (not including the σ˜ bb¯ data)
is slightly different in the two cases: for αS (MZ2 ) = 0.120
there is a fairly weak tendency to prefer lower values of mb,
similar to the results in [
5
], but for αS (MZ2 ) = 0.118 the
global fit prefers a value of between 4.5 and 5.0 GeV. For
the predictions for the beauty cross section data, however,
the picture is similar in the two cases, and low values of
mb ∼ 4.4–4.5 GeV are preferred.
The results for the NNLO fit with αS (MZ2 ) = 0.118 are
shown in Fig. 6. As can be seen the global fit is fairly weakly
dependent on mb, though more than for αS (MZ2 ) = 0.120 at
60
1.15
NLO, and prefers a value lower than mb = 4.25 GeV. As
in the NLO case the χ 2 for the prediction for σ˜ bb¯ is better
for lower values of mb. The slightly larger variations in the
quality of the global fit with varying mb compared to [
5
] is
perhaps due to the greater precision of the inclusive HERA
cross section data used in this analysis, and to the fact that
the CMS double-differential Drell–Yan data [
14
] has some
sensitivity to the value of mb due to the induced variation in
sea quark flavour composition for low scales. The previous
analysis preferred a value of mb ∼ 4.75 GeV for the
comparison to the H1 beauty data. However, the definition of the
general-mass variable-number scheme has improved since
this previous analysis, being smoother near to the transition
point Q2 = m2b, and including an improvement to the
approximation for the O(αS3 ) contribution at low Q2 at NNLO, so
some changes are not surprising. Another important
difference is in the treatment of the correlated experimental errors,
which we now take as being multiplicative. The result within
exactly the same framework, but with the experimental errors
on the HERA beauty data instead treated as additive is also
shown in Fig. 6 and a higher value of mb ∼ 4.75 GeV is
clearly preferred. Similar results are seen in the NLO fits.
In Fig. 7 the comparison to the (unshifted) HERA beauty
data for different values of mb at NNLO is shown. At
low Q2 and for ZEUS data in particular, the curves for
lower mb are clearly a better fit to unshifted data.
However, the low-m2b predictions do significantly overshoot some
of the unshifted data points. These predictions will work
better with the multiplicative definition of uncertainties as
the size of the correlated uncertainties then scales with
the prediction, not the data point (as would be the case
in the additive definition), or equivalently, if data are
normalised up to match theory, then so is the uncorrelated
uncertainty.
χ2σ˜bb (29 pts), NLO, αS(MZ2 ) = 0.120
χ2global (2996 pts), NLO, αS(MZ2 ) = 0.120
3310
3300
3290
3280
3270
3260
3310
3300
3290
3280
3270
3260
We show how the NLO PDFs for mc = 1.25 and 1.55 GeV
compare to the central PDFs in Figs. 8 and 9. Results are
very similar at NNLO, though more complicated to
interpret for the charm distribution at low Q2 due to the non-zero
transition matrix element at Q2 = mc2 in this case. We see at
Q2 = 4 GeV2 (that is, close to the transition point Q2 = mc2)
that the change in the gluon is well within its uncertainty
band, though there is a slight increase at smaller x with higher
mc (and vice versa) such that extra gluon quickens the
evolution of the structure function which is suppressed by larger
mass. Similarly the light quark singlet distribution increases
slightly near the transition point for larger mc to make up for
the smaller charm contribution to structure functions, and this
is maintained, helped by the increased gluon, at larger scales.
In both cases, however, the changes are within uncertainties
for these moderate variations in mc. The charm distribution
increases at low Q2 for decreasing mc, and vice versa,
simply due to increased evolution length ln(Q2/mc2). As
mentioned before we have identified the transition point at which
heavy flavour evolution begins with the quark mass. This has
the advantage that the boundary condition for evolution is
zero up to NLO (with our further assumption that there is
no intrinsic charm), though there is a finite O(αS2 ) boundary
condition at NNLO in the GM-VFNS, available in [
15
]. In
principle the results on the charm distribution at relatively
low scales, such as that in Fig. 8 are sensitive to these
definiχ2σ˜bb (29 pts), NNLO, αS(MZ2 ) = 0.118
Fig. 6 The quality of the fit versus the quark mass mb at NNLO with
αS (MZ2 ) = 0.118 for (left) the reduced cross section for beauty
production σ˜ bb¯ for the H1 and ZEUS data and (right) the global fit, not
including the beauty data. Recall that in the MMHT analysis the
experimental errors are treated multiplicatively. The lower plot shows the χ 2
profile if the errors in the HERA beauty data were to be treated additively
tions at finite order, though as the order in QCD increases the
correction for changes due to different choices of the
transition point arising from the corresponding changes in the
boundary conditions become smaller and smaller,
ambiguities always being of higher order than the calculation. At
scales typical of most of LHC physics, however, the
relative change in evolution length for the charm distribution is
much reduced, as are the residual effects of choices relating
to the choice of the transition point and intrinsic charm. At
these scales the change in the charm distribution is of the
same general size as the PDF uncertainty for fixed mc, as
seen in Fig. 9. We also note that the charm structure
function at these high scales is reasonably well represented by the
charm distribution, while at low scales, certainty including
Q2 = 4 GeV2, this is not true. Indeed at NNLO the boundary
condition for the charm distribution is negative at very low x
if the transition point is mc2, but this is more than compensated
for by the gluon and light quark initiated cross section. As
noted in [
9
], use of a zero mass scheme becomes unfeasible
at NNLO. The dependence on the heavy-quark cross section
at low scales relative to the mass is much better gauged from
Fig. 7.
The relative changes in the gluon and light quarks for
variations in mb are significantly reduced due to the much
smaller impact of the beauty contribution to the structure
functions from the charge-squared weighting, as can be seen
in Figs. 10 and 11, where we show NLO PDFs for mb = 4.25
and 5.25 GeV. At Q2 = 40 GeV2 ∼ 2m2b the relative change
0.004 σ˜bb, Q2 = 5 GeV2
0.006 σ˜bb, Q2 = 6.5 GeV2
0.016 σ˜bb, Q2 = 30 GeV2
in the beauty distribution for a ∼10 % change in the mass is
similar to that for the same type of variation for mc.
However, the extent to which this remains at Q2 = 104 GeV2 is
much greater than the charm case due to the smaller evolution
length.
3 Effect on benchmark cross sections
In this section we show the variation with mc and mb for
cross sections at the Tevatron, and for 7 and 14 TeV at the
LHC. Variations for 8 and 13 TeV will be very similar to
20
10
0
Gluon (NLO), percentage difference at Q2 = 4 GeV2
Light quarks (NLO), percentage difference at Q2 = 4 GeV2
20
10
0
−10
those at 7 and 14 TeV, respectively. We calculate the cross
sections for W and Z boson, Higgs boson via gluon–gluon
fusion and top-quark pair production. To calculate the cross
section we use the same procedure as was used in [
4,16
].
That is, for W, Z and Higgs production we use the code
provided by W.J. Stirling, based on the calculation in [
17–
19
], and for top pair production we use the procedure and
code of [20]. Here our primary aim is not to present definitive
predictions or to compare in detail to other PDF sets, as both
these results are frequently provided in the literature with
very specific choices of codes, scales and parameters which
may differ from those used here. Rather, our main objective
is to illustrate the relative influence of varying mc and mb for
these benchmark processes.
We show the predictions for the default MMHT2014
PDFs, with PDF uncertainties, and the relative changes due
to changing mc from 1.25 to 1.55 GeV, and mb from 4.25 to
5.25 GeV, i.e. changing the default values by approximately
10 % in each case. The dependence of the benchmark
predictions on the value of mc in Tables 3, 4 and 5 reflects the
behaviour of the gluon with √s. The changes in cross
section generally scale linearly in variation of masses away from
the default values to a good approximation, although for mb,
where the cross section sensitivity to the mass choice is often
small, this is less true, and in some cases the cross section is
even found to decrease or increase in both directions away
from the best fit mass.
We begin with the predictions for the W and Z production
cross sections. The results at NNLO are shown in Table 3. The
PDF uncertainties on the cross sections are 2 % at the
Tevatron and slightly smaller at the LHC—the lower beam energy
at the Tevatron meaning the cross sections have more
contribution from higher x where the PDF uncertainties increase.
The mc variation is at most about 0.4 % at the Tevatron and
is 0.5–1 % at the LHC, being larger at 14 TeV. The results at
NLO are very similar.
In Table 4 we show the analogous results for the top-quark
pair production cross section. At the Tevatron the PDFs are
probed in the region x ≈ 0.4/1.96 ≈ 0.2, and the main
production is from the qq¯ channel. At the LHC the dominant
production at higher energies (and with a proton–proton rather
than proton–antiproton collider) is gluon–gluon fusion, with
the central x value probed being x ≈ 0.4/7 ≈ 0.06 at 7 TeV,
and x ≈ 0.4/14 ≈ 0.03 at 14 TeV. The PDF uncertainties
on the cross sections are nearly 3 % at the Tevatron, similar
for 7 TeV at the LHC, but a little smaller at 14 TeV as there
is less sensitivity to the high-x gluon. The mc variation are
slightly less than 1 % at the Tevatron and for 7 TeV at the
10
0
LHC, but rather lower at 14 TeV since the x probed is near
the fixed point for the gluon (see Fig. 9).
In Table 5 we show the uncertainties in the rate of Higgs
boson production from gluon–gluon fusion. At the Tevatron
the dominant x range probed, i.e. x ≈ 0.125/1.96 ≈ 0.06,
corresponds to a region where the gluon distribution falls as
mc increases and at the LHC where x ≈ 0.01–0.02 at central
rapidity the gluon increases as mc increases, though at 7 TeV
we are only just below the fixed point. At the Tevatron the
resultant uncertainty is ∼0.7 %. At the LHC at 7 TeV it is in
the opposite direction but only ∼0.1 %, whereas at 14 TeV
it has increased to near 0.5 %.
As in [
5
] we recommend that in order to estimate the total
uncertainty due to PDFs and the quark masses it is best to add
the variation due to the variation in quark mass in quadrature
with the PDF uncertainty, or the PDF+αS uncertainty, if the
αS uncertainty is also used.
4 PDFs in three- and four-flavour-number schemes
In our default studies we work in a general-mass
variableflavour-number scheme (GM-VFNS) with a maximum of
five active flavours. This means that we start at our input
scale of Q02 = 1 GeV2 with three active light flavours. At
the transition point mc2 the charm quark starts evolution and
then at m2b the beauty quark also starts evolution. The
evolution is in terms of massless splitting functions, and at high
Q2 the contribution from charm and bottom quarks lose all
mass dependence other than that in the boundary conditions
at the chosen transition point. The explicit mass dependence
is included at lower scales, but falls away like inverse powers
as Q2/mc2,b → ∞. We do not currently ever consider the top
quark as a parton.
We could alternatively keep the information about the
heavy quarks only in the coefficient functions, i.e. the heavy
quarks would only be generated in the final state. This is
called a fixed-flavour-number scheme (FFNS). One example
would be where neither charm and beauty exist as partons.
This would be a three-flavour FFNS. An alternative would
be to turn on charm evolution but never allow beauty to be
treated as a parton. This is often called a four-flavour FFNS.
We will use this notation, but strictly speaking it is a
GMVFNS with a maximum of four active flavours.
One might produce the partons for the three- and
fourflavour FFNS by performing global fits in these schemes.
−100.0001
0.001
Gluon (NLO), percentage difference at Q2 = 40 GeV2
Light quarks (NLO), percentage difference at Q2 = 40 GeV2
However, it was argued in [
21
] that the fit to structure
function data is not optimal in these schemes. Indeed, evidence
for this has been provided in [
9,22,23
]. Moreover, much of
the data (for example, on inclusive jets and W, Z production
at hadron colliders) is not known to NNLO in these schemes,
and is very largely at scales where mc,b are relatively very
small. So it is clear that the GM-VFNS are more
appropriate. Hence, in [
24
] it was decided to make available PDFs
in the three- and four-flavour schemes simply by using the
input PDFs obtained in the GM-VFNS, but with evolution
of the beauty quark, or both the beauty and the charm quark,
turned off. This procedure was continued in [
5
] and is the
common choice for PDF groups who fit using a GM-VFNS
but make PDFs available with a maximum of three or four
active flavours. Hence, here, we continue to make this choice
for the MMHT2014 PDFs.
We make PDFs available with a maximum of three or
four active flavours for the NLO central PDFs and their
uncertainty eigenvectors for both the standard choices of
αn f,max=5(MZ2 ) of 0.118 and 0.120, and for the NNLO central
S
PDF and the uncertainty eigenvectors for the standard choice
of αSn f,max=5(MZ2 ) of 0.118. We also provide PDF sets with
αS(MZ2 ) displaced by 0.001 from these default values, so as to
assist with the calculation of αs uncertainties in the different
flavour schemes. Finally, we make available PDF sets with
different values of mc and mb in the different fixed-flavour
schemes.
By default, when the charm or beauty quark evolution
is turned off, we also turn off the contribution of the same
quark to the running coupling. This is because most
calculations use this convention when these quarks are entirely
final state particles. This results in the coupling running
more quickly. So if the coupling at Q20 is chosen so that
αn f,max=5(MZ2 ) ≈ 0.118, then we find that αn f,max=3(MZ2 ) ≈
S S
0.105 and αn f,max=4(MZ2 ) ≈ 0.113. There are sometimes
S
cases where a set of PDFs with no beauty quark but with
five-flavour running coupling is desired, e.g. [
25
]. After the
publication of [
5
], PDF sets with this definition were made
available. Here we make available PDFs for the central sets
together with their eigenvectors with a maximum of four
active flavours, but the beauty quark included in the running
of the coupling. This type of PDF has also been considered
very recently in [
26
].
The variation of the PDFs defined with a maximum
number of three and four flavours, compared to our default of five
flavours, is shown at Q2 = 104 GeV2 in Fig. 12 for NNLO
10
The PDF uncertainties and mc and mb variations are also shown, where
responds to ±0.5 GeV, i.e. about 10 % in each case
the mc variation corresponds to ±0.15 GeV and the mb variation
corPDFs. The general form of the differences are discussed in
quarks, so evolution is generally slower, which means
pardetail in section 4 of [
5
] and are primarily due to two effects.
tons decrease less quickly for large x and grow less quickly
For fewer active quarks there is less gluon branching, so
at small x . The latter effect dominates for quark evolution,
the gluon is larger if the flavour number is smaller. Also,
as Q2 increases the coupling gets smaller for fewer active
while for the gluon the two effects compete at small x . For
the case where the maximum number of flavours is 4, but
t t Tevatron (1.96 TeV)
t t LHC (7 TeV)
t t LHC (14 TeV)
Higgs Tevatron (1.96 TeV)
Higgs LHC (7 TeV)
Higgs LHC (14 TeV)
Gluon
Up
Down
Strange
Charm
Anti–up
Anti–down
= 104 GeV2. The three and four flavour
Fig. 12 The ratio of the different fixed-flavour PDFs to the standard five flavour PDFs at NNLO and at Q2
schemes are show in the top left and right plots, while the four flavour scheme with five flavours in the running of αS is shown in the bottom plot
the coupling has five-flavour evolution, the overwhelming
effect is that the gluon is larger—effectively replacing the
missing beauty quarks in the momentum sum rule. However,
the increase in the gluon is maximal at small x , where the
increased coupling compared to the case where we use the
four-flavour coupling leading to increased loss of gluons at
high x from evolution.
5 Renormalisation schemes
At present most PDF fitting groups, including the most recent
updates [
4,27–29
], use the pole mass definition for the heavy
quarks. Hence, we have remained with this definition in
our investigation of quark mass dependence in this article.
The analyses in [
30,31
] use the MS definition, following the
developments in [
32
]. The latter analyses perform their fits in
the fixed-flavour-number scheme (FFNS), while all the others
groups use a general-mass variable-flavour-number scheme.
There is no fundamental obstacle to switching between the
two renormalisation schemes using either approach. The
mass dependence in a GM-VFNS appears in entirety from
the FFNS coefficient functions and in the transition matrix
elements which set the boundary conditions for the
(massless) evolution of the charm and beauty quarks. These are
used along with the FFNS coefficient functions to define the
GM-VFNS coefficient functions which tend to the massless
versions as mc2,b/Q2 → 0. Under a change in the definition
of the quark mass2
mpole = m(μR )(1 + αS(μ2R )d1(μ2R ) + · · · )
(2)
the coefficient functions and transition matrix elements can
be transformed from one mass scheme to the other
straightforwardly, as illustrated in Eq. (8) of [
32
], and the mass in
GM-VFNS defined in the MS renormalisation scheme.
However, there is more sensitivity to the definition of the
mass in a FFNS at given order than in a GM-VFNS. At LO
there is no mass scheme dependence in the same way that
there is no renormalisation scheme dependence of any sort.
At NLO in the FFNS the variation of the LO O(αS)
coefficient function under the change in Eq. (2) leads to a change in
the NLO O(αS) coefficient function. Some NLO GM-VFNS
definitions (e.g. the SACOT(χ ) [
33
] and the FONLL-A [
34
])
only use the FFNS coefficient functions at O(αS). The
transition matrix element for heavy-quark evolution in an NLO
GM-VFNS is also defined at O(αS) (and indeed is zero with
the standard choice μF = mc,b), so neither depend on the
mass definition, and the NLO GM-VFNS is independent of
the mass scheme [
35
].
Some NLO GM-VFNS definitions do use the O(αS2)
FFNS coefficient functions. Hence, these will contain some
2 Note that d1(μ2R) = 4/3π if μR = m.
dependence on the mass scheme. However, in the original TR
[
36
] and then the TR’ [
10
] schemes this contribution is frozen
at Q2 = mc2,b, so becomes relatively very small at high Q2.
In the “optimal” TR’ scheme [
9
], and in the FONLL-B, the
dependence falls away like mc2,b/Q2 (in the former case the
whole O(αS2) coefficient function is weighted by mc2,b/Q2,
while in the FONLL-B scheme the subtraction means that
only the massless limit of the O(αS2) coefficient function
remains as mc2,b/Q2 → 0). Hence, the dependence on the
mass scheme is more limited than in the FFNS at NLO, and
is particularly small. Indeed, in all but the original TR and
TR’ schemes, there is no dependence at high Q2.
At NNLO the mass scheme dependence in the FFNS enters
in the O(αS2) and O(αS3) coefficient functions. In a
GMVFNS it now enters in the O(αS2) coefficient functions at low
scales, and in boundary conditions for evolution, which gives
effects which persist to all scales. If the GM-VFNS uses the
O(αS3) coefficient functions these will also give mass scheme
dependent effects at low Q2. However, the expressions for
the O(αS3) coefficient functions are themselves still
approximations [
37
].
Hence, at present it does not seem too important whether
the pole mass or MS renormalisation scheme is used in a
GM-VFNS (indeed in [
27
] the pole mass scheme is used,
but the MS values for the masses are taken). Nevertheless,
in the future it is probably ideal to settle on the MS mass,
since the value of this is quite precisely determined in many
experiments, which is not true of the pole mass. At the same
time it will also be desirable for different PDF groups to agree
on a common value of mc and mb (there is no agreement at
present).
6 Conclusions
The main purpose of this article is to present and make
available PDF sets in the framework used to produce the
MMHT2014 PDFs, but with differing values of the charm
and beauty quark mass. We do not make a determination of
the optimum values of these masses, but we do investigate
and note the effect the mass variation has on the quality of
the fits to the data, concentrating on the HERA cross section
data with charm or beauty in the final state. We note that
for both the charm and beauty quarks a lower mass than our
default values of mc = 1.4 GeV and mb = 4.75 GeV is
preferred, although these are roughly the values of pole masses
one would expect from conversion from the values measured
in the MS scheme. This suggests that in the future it may be
better to use the MS definition, though this is currently not the
practice in global fits using a GM-VFNS—perhaps because,
as we discuss, the mass scheme dependence has less effect in
these schemes than for the FFNS. We also make PDFs
available with a maximum of three or four active quark flavours.
The PDF sets obtained for different quark masses and for
different active quark flavours can be found at [
38
] and will
be available from the LHAPDF library [
39
].
We investigate the variation of the PDFs and the predicted
cross sections for standard processes at the LHC (and
Tevatron) corresponding to these variations in heavy-quark mass.
For reasonable variations of mc the effects are small, but
not insignificant, compared to PDF uncertainties. For
variations in mb the effect is smaller, and largely insignificant,
except for the beauty distribution itself, which can vary more
than its uncertainty at a fixed value of mb; see, in
particular Fig. 10. Hence, currently the uncertainties on PDFs due
to quark masses are not hugely important, but need to be
improved in the future for very high precision predictions at
hadron colliders.
Acknowledgments We particularly thank W. J. Stirling and G. Watt
for numerous discussions on PDFs and for previous work without which
this study would not be possible. We would like to thank Mandy
CooperSarkar, Albert de Roeck, Stefano Forte, Joey Huston, Pavel Nadolsky
and Juan Rojo for various discussions on the relation between PDFs
and quark masses. We would like to thank A. Geiser, A. Gizhko and K.
Wichmann for help regarding the treatment of uncertainties for ZEUS
beauty cross sections. This work is supported partly by the London
Centre for Terauniverse Studies (LCTS), using funding from the European
Research Council via the Advanced Investigator Grant 267352. RST
would also like to thank the IPPP, Durham, for the award of a Research
Associateship held while most of this work was performed. We thank
the Science and Technology Facilities Council (STFC) for support via
Grant Awards ST/J000515/1 and ST/L000377/1.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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