#### The effect on PDFs and \(\alpha _S(M_Z^2)\) due to changes in flavour scheme and higher twist contributions

R. S. Thorne
0
0
Department of Physics and Astronomy, University College London
, Gower Place, London WC1E 6BT,
UK
I consider the effect on MSTW partons distribution functions (PDFs) due to changes in the choices of theoretical procedure used in the fit. I first consider using the 3-flavour fixed flavour number scheme instead of the standard general mass variable flavour number scheme used in the MSTW analysis. This results in the light quarks increasing at all relatively small x values, the gluon distribution becoming smaller at high values of x and larger at small x , the preferred value of the coupling constant S(MZ2 ) falling, particularly at NNLO, and the fit quality deteriorates. I also consider lowering the kinematic cut on W 2 for DIS data and simultaneously introducing higher twist terms which are fit to data. This results in much smaller effects on both PDFs and S(MZ2 ) than the scheme change, except for quarks at very high x . I show that the structure function one obtains from a fixed input set of PDFs using the fixed flavour scheme and variable flavour scheme differ significantly for x 0.01 at high Q2, and that this is due to the fact that in the fixed flavour scheme there is a slow convergence of large logarithmic terms of the form (S ln(Q2/mc2))n relevant for this regime. I conclude that some of the most significant differences in PDF sets are largely due to the choice of flavour scheme used.
1 Introduction
There have recently been various improvements in the PDF
determinations by the various groups (see e.g. [16])
generally making the predictions using different PDF sets more
consistent with each other. However, there still remain some
large differences which are occasionally much bigger than
the individual PDF uncertainties [79]. This is particularly
the case for cross sections depending on the high-x gluon
or on higher powers of the strong coupling constant S. In
this article I investigate potential reasons for these
differences, based on alternative theoretical procedures that can
be chosen for a PDF fit. The two main potential sources of
differences which may affect rather generic features such as
the general form of the gluon distribution and the preferred
value of S(MZ2 ), (rather than more detailed features such as
quark flavour decomposition), are the choice of active flavour
number used and whether or not higher twist corrections are
applied to theory calculations, and related to this whether
low Q2 and W 2 data are used in a PDF fit. I discover that the
issue of heavy flavours is by far the more important of these,
and explain the reason why the differences between PDFs
obtained using fixed flavour number scheme (FFNS) and
those using a general mass variable flavour number scheme
(GM-VFNS) is so great at finite order in perturbative QCD.
This study builds on some initial results in [10] and in many
senses is similar to the NNPDF study in [11] and reaches
broadly the same conclusions. However, there are a variety
of differences to the NNPDF study, not least the investigation
of the S dependence, and also a much more detailed
discussion of the theoretical understanding of the conclusions. A
very brief summary of the results here have been presented
in [12].
2 Flavour number
I first examine the number of active quark flavours used in
the calculation of structure functions. There are essentially
two different choices for how one deals with the charm and
bottom quark contributions, the former being of distinct
phenomenological importance as the charm contribution to the
total F2(x , Q2) at HERA can be of order 30 %. Hence, I will
concentrate on the charm contribution to structure functions
F c(x , Q2), but all theoretical considerations are the same for
the bottom quark contribution. In the n f = 3 Fixed Flavour
Number Scheme (FFNS) we always have
MMSSTTWW0088FFNS
0 1
0 1
0 1
0 1
MMSSTTWW0088FFNS
MMSSTTWW0088FFNS
0 1
0 1
0 1
MMSSTTWW0088FFNS
i.e. for Q2 mc2 massive quarks are only created in the
final state. This is exact (up to nonperturbative corrections)
but does not sum Sn lnn Q2/mc2 terms in the perturbative
expansion. The FFNS has long been fully known at NLO
[13], but this is not yet the case at NNLO (O(S3)).
Approximate results can be derived [14], and are sometimes used
in fits, e.g. [15]). However, it turns out that these NNLO
corrections are not actually very large, except near
threshold and at very low x , being generally of order 10 % or less
away from these regimes. (Perhaps surprisingly, the
approximate NNLO corrections also do not reduce the scale
dependence by much compared to NLO, see e.g. Figs. 12 and 13
of [14].) Hence, the use of approximate NNLO corrections
to F c(x , Q2) has not led to significant changes compared to
NNLO PDFs which used the simpler approximation of only
going to NLO in F c(x , Q2), e.g [16].
In a variable flavour scheme one uses the fact that at Q2
mc2 the heavy quarks behave like massless partons and the
ln(Q2/mc2) terms are automatically summed via evolution.
PDFs in different number regions are related perturbatively,
where the perturbative matrix elements A jk (Q2/mc2) are
known exactly to NLO [17,18].1 The original Zero Mass
Variable Flavour Number Scheme (ZM-VFNS) ignores all
O(mc2/Q2) corrections in cross sections, i.e. for structure
functions
1 NNLO contributions are being calculated [19]-[26] and are used in
the approximate NNLO expressions for F2c(x, Q2) in [14].
F (x , Q2) = C Zj M,4 f j4(Q2),
but this is an approximation at low Q2. The majority of
PDF groups use a General-Mass Variable Flavour Number
Scheme (GM-VFNS). This is designed to take one from the
well-defined limits of Q2 mc2 where the FFNS
description applies to Q2 mc2 where the variable flavour number
description is more applicable in a well defined theoretical
manner. Some of the variants are reviewed and compared
in [27], and for specific examples see e.g. [2833] There is
an ambiguity in precisely how one defines a GM-VFNS at
fixed order in perturbation theory (in the same way there
is a renormalisation and factorisation scale uncertainty), but
this is always formally higher order than that at which one
is working. A study of the variation of both F c(x , Q2) and
extracted PDFs was made in [10], and both reduced
significantly at NNLO. PDFs and predictions for LHC cross
sections could vary by amounts of order the experimental PDF
uncertainty at NLO, i.e. 2 % but this reduced to generally
fractions of a percent at NNLO. In both cases there was little
variation in the preferred values of S(MZ2 ). Some results of
variations in GM-VFNS definition can also be found in [34].
The predictions for F2c(x , Q2) using the TR GM-VFNS
[32] and the MSTW2008 PDFs [35] are compared to those
using the FFNS and three-flavour PDFs generated using the
MSTW2008 input distributions [36], and are shown in Fig. 1.
At LO there is a very big difference between the two,
particularly for x 0.05 where the GM-VFNS result is larger
than the FFNS result, but also at very low x where the FFNS
is larger. At NLO F2c(x , Q2) at high Q2 for the FFNS is
NNLO
MSTW08
MSTW08FFNS
Fig. 2 The ratio of F2(x, Q2) using the FFNS to that using the
GMVFNS
nearly always lower than for the GM-VFNS, significantly so
at higher x 0.05. For FFNS at NNLO only NLO
coefficient functions are used, but (various choices of)
approximate O(S3) corrections give only small increases that would
not change the plots in any qualitative manner. There is no
dramatic improvement in the agreement between FFNS and
GM-VFNS at NNLO compared to NLO, contrary to what one
might expect. This suggests that logarithmic terms beyond
O(S3 ln3(Q2/mc2)) are still important.
This 2040 % difference between FFNS and GM-VFNS in
F2c(x , Q2) can lead to over 4 % changes in the total inclusive
structure function F2(x , Q2), see Fig. 2 for an illustration at
NNLO, with the GM-VFNS result usually being above the
FFNS result. At x 0.01 this is mainly due to the difference
in F2c(x , Q2) itself. However, at lower x there is a
contribution to the difference from the light quarks evolving slightly
more slowly in the FFNS, mainly due to the strong coupling in
the FFNS falling below that in the GM-VFNS as Q2 increases
above mc2. For x > 0.1 the FFNS and GM-VFNS are very
similar largely because the charm contribution is becoming
very small, and the valence quark contribution dominates. In
order to test the importance of this difference between FFNS
and GM-VFNS in inclusive F2(x , Q2) I have extended an
investigation begun in [10] and performed fits using the FFNS
scheme in order to compare the fit quality and resulting PDFs
and S(MZ2 ) to those obtained from fits using the GM-VFNS.
At NNLO O(S2) heavy flavour coefficient functions are used
as default (which has been done until quite recently in other
FFNS fits, e.g. [16]). It has been checked, however, that
approximate O(S3) expressions change the results very little.
In order to make comparison to the existing MSTW2008
PDFs, which have been very extensively used in LHC studies,
I perform the fits within the framework of the MSTW2008
PDFs [35], i.e. data sets and treatment are the same, as is the
definition of the GM-VFNS, quark masses, etc.. (The effect
on the MSTW2008 PDFs due to numerous improvements in
both theory and inclusion of new data sets (see [1,37,38]) has
been studied and so far only received corrections of any real
significance in the small-x valence quarks from the improved
parameterisation and deuteron corrections in [1].) For the
fixed target Drell-Yan data the contribution of heavy flavour
is negligible, and has been omitted in the FFNS fits. This
study also maintains continuity with the previous results in
[10]. I first perform fits to only DIS and fixed target
DrellYan data (charged current HERA DIS data is omitted due
to the absence of full O(S2) calculations for these,2 though
these run I data carry very little weight in the fit), but this
is also extended to the additional inclusion of Tevatron jet
and Z boson production data, where the 5-flavour calculation
scheme is used in these cases, with the PDFs being converted
appropriately for combination with these hard cross sections.
At NNLO the fit to Tevatron jet data uses the NNLO threshold
corrections that are available [40] (though more complete
calculations which take into account the dependence on the
jet radius R have just appeared in [41] these are not available
for use yet). As argued in [38] the precise form of these is
not very important to the results.
The results of the fit quality for various different fits are
shown in Table 1 for NLO and Table 2 for NNLO, along with
the value of S(MZ2 ), evaluated for 5 quark flavours. The fit
quality for DIS and Drell-Yan data are at least a few tens of
units higher in 2 in the FFNS fit than in the MSTW2008
fit, with the difference being greater at NNLO than at NLO.
The results appear similar to those in Table 1 of [11], though
there S(MZ2 ) was kept fixed. The FFNS fit is often slightly
better for the F2c(x , Q2) itself, but the total F2(x , Q2) is
flatter in Q2 for x 0.01, and this worsens the fit to HERA
inclusive structure function data. For both GM-VFNS and
FFNS, and at both NLO and NNLO, the fit quality to DIS data
deteriorates by about 30 units when the fixed target Drell Yan
data is added, showing that there is some tension in
quarkantiquark decomposition between DIS and fixed-target Drell
Yan data. Although there is no difficulty in obtaining a good
fit to Tevatron jet data when using the the FFNS for structure
functions the fit quality for DIS and Drell Yan deteriorates by
2 There is a very recent calculation of the O(S2) results for charm
production in the large Q2 limit [39].
MSTW2008 1876
MSTW2008 (DIS only) 1845
MSTW2008 (no jets) 1875
MSTWn f = 3 (DIS only) 1942
2000
MSTWn f = 3
(DIS + ftDY)
MSTWn f = 3 (jets)
MSTWn f = 3 (jets+Z )
MSTW2008 1864
MSTW2008 (DIS only) 1822
MSTW2008 (no jets) 1855
MSTWn f = 3 (DIS only) 2003
2032
MSTWn f = 3
(DIS + ftDY)
MSTWn f = 3 (jets)
MSTWn f = 3 (jets+Z )
Table 2 The 2 values for DIS data, fixed target Drell Yan (ftDY) data
and Tevatron jet data for various NNLO fits performed using the
GMVFNS used in the MSTW 2008 global fit and using the n f = 3 FFNS
for structure functions. The bracketed numbers denote the 2 values for
jet data when not included in the fit
2 DIS 2 ftDY 2 jets Sn f =5(MZ2 )
2073pts 199pts 186pts
Table 1 The 2 values for DIS data, fixed target Drell Yan (ftDY) data
and Tevatron jet data for various NLO fits performed using the
GMVFNS used in the MSTW 2008 global fit and using the n f = 3 FFNS
for structure functions. The bracketed numbers denote the 2 values for
jet data when not included in the fit
2 DIS 2 ftDY 2 jets Sn f =5(MZ2 )
2073pts 199pts 186pts
50 units when both Tevatron jet and Z data are included,
as opposed to 10 units or less when using a GM-VFNS. It is
important to add the Tevatron Z rapidity data as well as the
jet data since the former fixes the luminosity at the Tevatron
quite precisely, and makes the jet data more difficult to fit
than when the luminosity is left free [42] and vector boson
production ignored. The preferred S(MZ2 ) values in each
fit are also shown. These do not vary much for the
GMVFNS fits, though for DIS only fits there is in fact very little
variation in fit quality with a wide range of S(MZ2 ) and it
is quite difficult to obtain a definite best fit. For the FFNS
fits there is a very distinct increase when Tevatron jet data
is added. The values of S(MZ2 ) are lower than for the
GMVFNS fits for the DIS and DIS plus Drell Yan fits, but higher
when the jet data is added, though the NNLO FFNS values
are relatively slightly lower compared to GM-VFNS than the
NLO values.
The PDFs resulting from the fits, evolved up to Q2 =
10,000 GeV2 (using variable flavour evolution for consistent
comparison) are shown in Fig. 3. The PDFs are consistently
different in form to the MSTW2008 PDFs. There are larger
light quarks for all the FFNS fit variants, due to the need to
make up for the smaller values of F2c(x , Q2) at high Q2. The
effect is very slightly reduced at NNLO compared to NLO.
The FFNS fits produce a gluon which is bigger at low x
that when using the GM-VFNS, and much smaller at high x .
The effect is somewhat reduced when the Tevatron jet data is
included in the fit, but not removed. Some similar differences
have been noted in [11], though S(MZ2 ) was not left free,
and also earlier in [43]. Hence it is clear that using FFNS
rather than GM-VFNS leads to significant changes in PDFs,
and much larger changes than any variation in choice of
GMVFNS [10], particularly at NNLO. In Fig. 4 I show the same
type of plot for a different PDF set obtained using FFNS for
the structure function calculations, i.e. the ABKM set from
[16], which was obtained fitting to DIS and fixed target
DrellYan data, and which obtained values of S(MZ2 ) of 0.1179
and 0.1135 at NLO and NNLO respectively. I compare to this
set, despite the fact that there have been more recent updates,
since the data fit and the FFNS definition used at NNLO
are most similar to the data used in the MSTW2008 fit and
to the heavy flavour calculations used in this article. (More
recent updates of the ABM fits have not led to very significant
changes in the most striking features of the comparison of
FFNS to GM-VFNS PDFs, i.e. FFNS has larger light quarks,
a different shape gluon and lower S(MZ2 ).) There are
considerable additional differences between the fits of the two
groups though, for instance the issue of higher twist, which is
a topic to be discussed later. However, first I will explore the
origin of the differences between the FFNS and GM-VFNS
results.
3 Perturbative convergence of heavy flavour evolution
The fact that there is a considerable difference between the
FFNS and GM-VFNS results for F c(x , Q2) for some values
of x , mainly x 0.05 at NLO, with little apparent
improvement at NNLO, might seem surprising. It has generally been
assumed that differences between the two flavour schemes
would diminish quickly at higher orders, and hence thought
unlikely that it could be a major source of difference between
PDF sets. However, the results of the previous section, plus
those in [10,11,43] demonstrate that differences are indeed
significant, and the origin of this needs to be understood.
In order to explain the differences between the results of
FFNS and GM-VFNS evolution it is useful to concentrate
on the relative size of (d F2c(x , Q2)/d ln Q2) rather than on
the absolute value of F2c(x , Q2), though differences in the
former clearly lead to differences in the latter as at very low
MSTW08NLO-0.1202
FFNSDIS-0.1187
FFNS+DY-0.1185
FFNSjet-0.1222
FFNSjetZ-0.1225
MSTW08NLO
FFNSDIS
FFNS+DY
FFNSjet
FFNSjetZ
MSTW08NNLO-0.1171
FFNSDIS-0.1144
FFNS+DY-0.1152
FFNSjet-0.1181
FFNSjetZ-0.1184
Q2 the inputs are the same in the two schemes. I show the
ratio of (d F2c(x , Q2)/d ln Q2) in FFNS to that in GM-VFNS
at LO, NLO and NNLO, using MSTW2008 PDFs, for Q2 =
500 GeV2 in Fig. 5. As one can see the results mirror those for
the values of F2c(x , Q2) in Fig. 1 with all orders lower using
FFNS for x > 0.001, but FFNS and GM-VFNS being similar
at NLO and NNLO for very small x , and the LO FFNS being
greater in this regime.3 These results in the relative speed of
evolution can be understood analytically.
Let us begin at leading order. At LO in the FFNS (setting
all scales to be Q2, which is appropriate at Q2 mc2)
= S ln(Q2/mc2) pq0g g + O(S g)
S A1,1
H g g + O(S g),
3 Note that these results are consistent with those in Fig. 5 of [45], which
shows the difference between the heavy quark evolution calculated at
finite order via the matrix elements and from full evolution. For example,
at x = 0.02 this difference is negative and hardly diminished at all
at approximate NNLO compared to NLO. At lower values of x the
difference changes sign, but may be seen to be a smaller fraction of the
total evolution. Exact details depend on the PDFs and S values used.
where the term not involving the logarithm ln(Q2/mc2) can
easily be seen to be very sub-dominant at high Q2.
Calculating the rate of change of evolution
0
= S pqg g + ln(Q2/mc2) d (dS lpnqQg2 g) +
0
= S pqg g + ln(Q2/mc2)S2 ( pqg pgg g
0 0 0
0 pq0g g) + , (5)
where 0 = 9/(4 ) = 0.716. A quark dependent term of
O(S2 ) (i.e. ln(Q2/mc2)S2 ( pq0g pg0q ) is deemed to be
subleading. At small x this is an excellent approximation
due to the smallness of the quark distribution compared to
the gluon and the fact that in this limit pg0q = 4/9 pg0g . At high
x the quark distributions begin to dominate and the
approximation is not as good. However, even this is not a major issue
until very high x , where valence quarks are completely
dominant, since the effect of pg0q is small compared to that of pg0g ,
e.g. the fifth moment of pg0q is only about 0.03 that of pg0g .
At LO in the GM-VFNS, where F2c,1,V F = (c + c) = c+,
to a very good approximation at high Q2 we have
d ln Q2
d c+ 0 0
= d ln Q2 = S pqg g + S pqq c+,
MSTW08NLO-0.1202
so the second term in (6) is formally O(S2 ln(Q2/mc2)). The
first terms in Eqs. (5) and (6) are of order S and they are
equivalent, as they must be. The difference between the two
LO expressions is O(S2 ln(Q2/mc2)) and is
d(F2c,1,V F F2c,1,F F )
d ln Q2
( pq0q + 0 pg0g) g +
PVLOFF F g + .
The effect of pg0g is positive at high x and negative at small
x . That of pq0q is negative at high x , but smaller than pg0g,
and that of 0 is always positive. Hence, the difference is
large and positive at high x and becomes large and negative
at small x . This explains the features observed in Fig 5, which
plots the ratio of the evolution using the FFNS to that using
the GM-VFNS. Hence, the difference between FFNS and
GM-VFNS evolution is fully explained.
The subleading terms providing the difference between
FFNS and GM-VFNS evolution at LO then provide
important information about the NLO FFNS expressions. This
formally NLO difference between the two forms of evolution
Fig. 5 The ratio of d F2c/d ln Q2 using the FFNS to that using the
GMVFNS at LO, NLO and NNLO
MSTW08NNLO-0.1171
approx NNLO
Q2=500GeV2
must be eliminated in the full NLO expressions by
defining the leading-log term in the FFNS expression to provide
cancellation, i.e. it requires that
up to quark mixing corrections and sub-dominant terms. With
this definition all previous O(S2 ln(Q2/mc2)) terms in the
NLO evolution cancel between the GM-VFNS and FFNS
expressions. However, the derivative of F2c,2,F F contains
which does not cancel with anything in the NLO GM-VFNS
expression. This leads to
where again the pq0q comes form the contribution in Eq. (6)
but using the O(S2 ln2(Q2/mc2)) contribution to c+ in
2 A2,2
S Hg g. The additional factor of ( pq0q + 20 pg0g)
is large and positive at high x and negative at small x , but
not until smaller x than at LO. Therefore, PVNFLOF F is large
and positive at high x , negative for smaller x and positive
for extremely small x . This explains the difference in the
evolution between GM-VFNS and FFNS at NLO correctly.
The pattern is now established. In order to cancel this
difference between the evolutions at NLO then at NNLO
the dominant part of F c,2,F F at leading-log is (up to
quark2
mixing and scheme-dependent terms)
S3 A3H,3g g = 61 S3 ln3
Repeating the previous arguments, at NNLO the dominant
high-Q2 uncancelled term between GM-VFNS and FFNS
evolution is
This remains large and positive at high x , then changes sign
twice but stays small until becoming negative at tiny x . Again
this explains the behaviour at NNLO correctly. The
expression can be straightforwardly generalised to higher orders. It
is similar in some sense to the results for the bottom quark of
Eq. (3.5) in [46], but this neglected the evolution of the gluon
and hence the pg0g terms, which as shown here are actually
the dominant effect at lowish orders.
The extent to which these relatively simple analytic
results, true at leading log and ignoring quark mixing,
NLO approx NNLO
Q2=500GeV2
Fig. 6 The ratio of the analytic leading-log approximation to the
evolution difference between FFNS and the full GM-VFNS evolution at
LO, NLO and NNLO, i.e. PF FV F g at each order
describe the true detailed difference between the GM-VFNS
and FFNS evolution can be tested by calculating the ratio
PFx xFLOV F g
(d F c,x x L O,V F (x , Q2)/d ln Q2)
(d F c,x x L O,F F (x , Q2)/d ln Q2)
at LO, NLO and NNLO. With the addition of unity this should
be the same as the result of FFNS to GM-VFNS evolution
shown in Fig. 5. The ratio is shown in Fig. 6. Indeed the
comparison to Fig. 5, though not exact is generally very good,
with the most important feature of a suppression of FFNS
evolution compared to GM-VFNS of at least 20 % for x
0.01, with slow convergence at higher orders, explained well
by the simple expression.
In order to look at the effect of this dominant high-Q2
difference between GM-VFNS and FFNS evolution, and in
particular to understand the rate of convergence between the
two, it is useful to define the moment space effective
anomalous dimension V FF F obtained from from the effective
splitting function PV FF F by
V FF F (N , Q2) =
x N PV FF F (x , Q2).
This is shown at LO, NLO and NNLO for Q2 = 500 GeV2
in Fig. 7. Since the expression depends only on leading logs
Fig. 7 The effective anomalous dimension V FF F (N ) for Q2 =
500 GeV2 at LO (purple), NLO (brown) and NNLO (green). Also shown
(blue) is the NNNLO expression
it can actually be expressed at any order, so NNNLO is also
shown. At high Q2, values of x 0.05 correspond to N 2,
where V FF F only tends to zero slowly as the
perturbative order increases. This explains why FFNS evolution for
x 0.05 only slowly converges to the GM-VFNS result
with increasing order, very roughly like 1/n where n is the
power of S(Q2) ln(Q2/mc2). For N 0.5 which is
applicable to x 0.0001 there is good convergence, and in fact
very little difference between FFNS and GM-VFNS
evolution. For N 0, there is poor convergence, but this only
affects extremely low values of x indeed. It is the slow
convergence relevant for x 0.05 that is of phenomenological
importance, as there is a great deal of very precise HERA
inclusive structure function data that is sensitive to this.
4 Higher twist
Another difference in theoretical assumptions made when
performing fit to data in order to extract PDFs is how to
deal with the low Q2 and low W 2 DIS data which is
potentially susceptible to higher twist corrections to the
factorisation theorem. The majority of analyses choose a set of cuts
which they deem to be large enough to eliminate the effect of
higher twist effects, and in the case of MSTW this is chosen
to be Q2min = 2 GeV2 and Wm2 in = 15 GeV2 (with the higher
choice Wm2 in = 25 GeV2 for the small amount of F3(x , Q2)
data which is more likely to have large higher twist
corrections) where it has been checked in previous studies, e.g.
[47], that the PDFs and fit quality obtained are insensitive to
smooth increases of the cuts in the upwards direction.
However, some studies, e.g. [16] use lower cuts and parametrise
the higher twist corrections as functions of x and Q2.
In order to check the sensitivity of the PDFs to this
choice I have investigated the effect of lowering the W 2 cut
for F2(x , Q2) and FL (x , Q2) to 5 GeV2 (keeping that for
F3(x , Q2) unchanged) and parameterising higher twist
corrections in the form (Di /Q2)Fi (x , Q2), where
Table 3 The values of the higher-twist coefficients Di of (16), in the
chosen bins of x, extracted from the NLO and NNLO GM-VFNS global
fits and the NLO and NNLO FFNS fits to DIS data
Fi (x , Q2) = FiLT(x , Q2) 1 +
in 13 bins of x , and then fitting the Di and PDFs
simultaneously, as in [47]. This is similar to the procedure in [16] and
more recent PDF fits by the same group. It is less
sophisticated than these fits, but the aim is simply to investigate the
major changes in PDFs from including higher twist
corrections, not to produce an official new set of PDFs. It is checked
that results are insensitive to the treatment of longitudinal
structure functions, which carry extremely little weight in
the fit. The higher twist analysis differs significantly from
that in [11] which took fixed higher twist parameterisations
and kept the cuts of Q2min = 3 GeV2 and Wm2 in = 12.5 GeV2
used as default by the NNPDF group, though variations, e.g.
reversing the sign of the correction or doubling it were
performed and the impact of these large changes investigated.
The Di extracted in this study are shown in Table 3. They are
similar to the older MRST study in [47], though larger at the
smallest x . The effect on the PDFs and S(MZ2 ) compared
to the default MSTW fit using GM-VFNS and all the same
data sets is small, except for very high-x quarks, as shown in
Fig. 8. The value of S(MZ2 ) decreases slightly from 0.1202
to 0.1189 at NLO but actually increases slightly from 0.1171
to 0.1175 at NNLO. The fit quality is shown at NLO in Table 4
and at NNLO in Table 5. The 2 for the nuclear target
structure function data is omitted here, as I will later consider a
variety of fits where these data are left out.
I have also repeated the higher twist study for fits using
the FFNS for heavy flavour production, fitting to DIS data
only. Again the results are shown in Fig. 8. The value of
S(MZ2 ) only changes from from 0.1187 to 0.1188 at NLO
and increases from 0.1144 to 0.1152 at NNLO. The change in
MSTW08NLO-0.1202
MSTW08HT-0.1189
FFNSDIS-0.1187
FFNSHT-0.1188
MSTW08NLO
MSTW08HT
FFNSDIS
FFNSHT
MSTW08NNLO-0.1171
MSTW08HT-0.1175
FFNSDIS-0.1144
FFNSHT-0.1152
MSTW08NNLO
MSTW08HT
FFNSDIS
FFNSHT
Table 4 The 2 values for DIS data, fixed target Drell Yan (ftDY) data
and Tevatron jet data for various NLO fits performed using the
GMVFNS used in the MSTW 2008 global fit and using the n f = 3 FFNS
for structure functions with reduced cuts and higher twist terms added
Table 5 The 2 values for DIS data, fixed target Drell Yan (ftDY) data
and Tevatron jet data for various NNLO fits performed using the
GMVFNS used in the MSTW 2008 global fit and using the n f = 3 FFNS
for structure functions with reduced cuts and higher twist terms added
MSTW2008 HT
MSTW2008 HT*
(DIS+ftDY)
MSTWn f = 3 HT
(DIS only)
MSTWn f = 3 HT*
(DIS only)
MSTWn f = 3 HT*
(DIS + ftDY)
MSTWn f = 3 HT*
(jets)
MSTWn f = 3 HT*
(jets+Z )
MSTWn f = 3 HT*
(DIS+ftDY)
(>300) 0.1188
(>300) 0.1175
(>300) 0.1179
MSTW2008 HT
(>300) 0.1152
(>300) 0.1132
(>300) 0.1136
(>300) 0.1171
PDFs is fairly small and similar to that using the GM-VFNS
and all global fit data. The extracted higher twist terms are
shown in Table 3. These are similar to the GM-VFNS fit, but
a little bigger, particularly NLO at small x . The fit quality
is also shown at NLO in Table 4 and at NNLO in Table 5.
There is less change in going from GM-VFNS to FFNS when
higher twist terms are included. In fact at NLO the FFNS DIS
data only fit gives a slightly better fit to the DIS data than the
MSTW08NLO-0.1202
FFNS+DYHT*-0.1179
Q2=10,000GeV2
MSTW08NNLO-0.1171
FFNS+DYHT*-0.1136
FFNS+DYHT* 0.1136
full higher twist MSTW2008 fit. However, this is no longer
quite true for a DIS only GM-VFNS higher twist fit.
However, the compatibility of the resultant PDFs with Tevatron
jet data is far worse for the FFNS fit that the GM-VFNS fit.
Although the value of S(MZ2 ) obtained from the FFNS
fits with higher twist corrections is generally lower than that
obtained in the GM-VFNS fits, particularly at NNLO, it is not
as low as that obtained by other PDF groups which perform
fits using the FFNS, e.g. [5,48]. In the latter of these there is
sensitivity to the input scale of the PDFs, with values of Q2
0
lower than 1 GeV2 leading to lower values of S(MZ2 ). I do
not investigate this possibility since the MSTW PDF
parameterisation is already such as to make the input gluon
distribution rather different at any low scale. However, another
difference in these fits compared to MSTW2008 is the absence
of nuclear target inclusive structure function data [49,50]
which are dependent on nuclear corrections, but where the
non-singlet F3(x , Q2) data do favour high S(MZ2 ) values,
as shown in [35]. Also in many higher twist studies the higher
twist corrections are only included for x > 0.01 Hence, I
perform FFNS fits which restrict the higher twist from the three
lowest x bins and simultaneously omit the less theoretically
clean nuclear target data (except for dimuon cross sections,
which constrain the strange quark). This results a series of
fits labelled HT*. The fit quality for fits to only DIS data, DIS
plus Drell Yan data and with the addition of Tevatron jet data
and Tevatron Z rapidity data is shown in Tables 4 and 5. As
mentioned earlier, in these tables the 2 for DIS data does
not include that for the nuclear target data, although the data
has been included in the fits except for those labelled HT*.
Removal of these data generally allow a slight improvement
to the rest of the data, but this is compensated for by a
(usually slightly larger) deterioration when the higher twist below
x = 0.01 is removed. As well as the FFNS fits I also show
the fit quality for a GM-VFNS fit with S(MZ2 ) fixed to the
same value as the full MSTW2008 higher twist fit, but the
same data as the FFNS DIS plus Drell Yan fit is used. This
is labelled MSTW2008HT*. For this approach the fit quality
for the DIS plus Drell Yan data is the best exhibited, and the
prediction for the Tevatron jets is quite good. The PDFs for
the fits containing DIS plus fixed target Drell Yan data are
compared to MSTW2008 for two variants of the FFNS fit in
Fig. 9 and the full range of HT* fits are shown in Fig. 10.
The additional changes in the HT* fits do result in slightly
lower values of S(MZ2 ), particularly at NNLO, with values
of S of S(MZ2 ) = 0.1179 at NLO and S(MZ2 ) = 0.1136
at NNLO for the fits without Tevatron data. These are very
close to those in [16], where the FFNS scheme choice, data
types, and form of higher twist (and the resulting PDFs) are
similar. The change in the PDFs in going from the FFNS fits
MSTW08NLO-0.1202
FFNSHT*DIS-0.1175
FFNS+DYHT*-0.1179
FFNSjetHT*-0.1199
FFNSjetZHT*-0.1215
MSTW08NNLO-0.1171
FFNSDISHT*-0.1132
FFNS+DYHT*-0.1136
FFNSjetHT*-0.1155
FFNSjetZHT*-0.1174
MSTW08NLO
FFNSDISYHT*
FFNS+DYHT*
FFNSjetHT*
FFNSjetZHT*
MSTW08NNLO
FFNSDISHT*
FFNS+DYHT*
FFNSjetHT*
FFNSjetZHT*
to FFNSHT* fits is not large at all, as seen in Fig. 9, with
the essential features of the differences between FFNS and
GM-VFNS PDFs being fully maintained.
I have also made some further checks on the general
validity of the results. It was noted in [36] that when using the
default GM-VFNS for the MSTW2008 fit the best fit
quality was obtained for values of the pole mass mc different
to the default mc = 1.4 GeV. At NLO the global 2 could
decrease by just a couple of units with a very slightly larger
value mc = 1.45 GeV, but at NNLO the global 2 could
decrease by 24 units if the lower value of mc = 1.26 GeV is
used. In the FFNS fits a very slight decrease in 2 of a few
units is obtained at NLO if mc lowers by 0.1 GeV or less
and at NNLO an improvement in 2 of up to 30 units can
be achieved for mc = 1.21.25 GeV. Hence, the
improvements in fit quality possible using the GM-VFNS and FFNS
are very similar, perhaps marginally better for FFNS, and
FFNS prefers a slight lower optimum mc value. None of
this has any significant effect on the relative differences in
PDFs or S(MZ2 ). Also, as demonstrated in Sect. 3, the
differences between FFNS and GM-VFNS can be very largely
understood in terms of the leading ln(Q2/mc2) terms in the
perturbative expansions. These are completely unaltered by
a change in quark mass scheme of mc mc(1 + cS + ).
Indeed, there is only a fairly minor change in PDFs from [16]
to [15], and almost no change in S(MZ2 ), despite the change
from the pole mass to M S mass schemes. Perhaps the most
striking change, an increase in sea quarks near x = 0.01 is
due to the inclusion of the combined HERA data [51], an
effect noticed elsewhere, e.g. [37]. As a final check, fits were
performed using approximations to the full NNLO heavy
flavour DIS coefficients. Wider variations in coefficient
functions were allowed than options A and B in [14]. At best the
NNLO FFNS fits improved quality by about 40-50 units -
significant but still leaving them some way from the GM-VFNS
fit quality at NNLO. The change in PDFs and S(MZ2 ) is
never very large, and the very best fits actually preferred a
marginally lower S(MZ2 ) value. Hence, the conclusions on
fit quality, the PDF shape and S (MZ2 ) values are stable under
a variety of variation in the full details of the fit. The general
features of the FFNS fits producing gluon distributions which
are about 10 % lower at x 0.1 at Q2 = 10,000 GeV2 than
when using GM-VFNS, but rising to 5 % (or more) greater
below x = 0.01, along with a light quark distribution which
is a few percent bigger at most x values seems to be largely
insensitive to any other variations in procedure or data fit.
The reduction of S(MZ2 ) also seems to be a stable feature,
but the precise difference is more sensitive to details of the fit.
Q2=25GeV2
Q2=10,000GeV2
Fig. 11 The ratio of FFNS PDFs from NNLO fits with both free (red) and fixed S(MZ2 ) (blue) to the MSTW2008 PDFs at 25 GeV2 (left) and at
10,000 GeV2 (right)
5 Fixed coupling
Finally, in order to investigate why the value of S(MZ2 )
obtained in FFNS fits is lower than in GM-VFNS fits I also
perform a NNLO fit to DIS and low-energy DY data where
S(MZ2 ) is fixed to the higher value obtained in the
GMVFNS. I also perform a fit with S(MZ2 ) = 0.120 at NLO,
though the relative change in the coupling is less significant
at NLO. This fixed coupling results in the FFNS gluon being
a little closer to that using GM-VFNS, as shown at NNLO
in Fig. 11 for Q2 = 25 GeV2 and Q2 = 10,000 GeV2, and
very similar to the gluon in [11], where studies are performed
with fixed S(MZ2 ). There is little change in the light quarks
in the FFNS fit when the coupling is held fixed. The fit
quality is shown in Tables 4 and 5 The FFNS fit is 8 units worse
when S(MZ2 ) = 0.1171 than for 0.1136. (The deterioration
at NLO is very slightly less.) The fit to HERA data is better,
but it is worse for fixed target data.
By examining the change in the gluon in the FFNS fit
when S(MZ2 ) is fixed one can understand the need for S to
be smaller in FFNS. To compensate for smaller F2c(x , Q2) at
x 0.05 the FFNS gluon must be bigger in this region, and
from the momentum sum rule, is therefore smaller at high
x . The correlation between the high-x gluon and S(MZ2 )
when fitting high-x fixed target DIS data drives S down (for
reduced gluon the quarks fall with Q2 more quickly, hence
the need to lower S to slow evolution), requiring the small
x gluon to even bigger. As the fit undergoes iterations this
pattern is repeated until the best fit is reached with a lower
S(MZ2 ) value and significantly modified gluon shape.
6 Conclusions
In this article I have investigated whether the different
theoretical choices in fits to data in order to determine partons
distribution functions (PDFs) can influence the PDFs, the
value of S(MZ2 ) and the fit quality. I come to the strong
conclusion that within the context of the MSTW2008 global fit
the choice of a FFNS for heavy flavour production in deep
inelastic scattering, as opposed to a GM-VFNS, leads to a
lower S(MZ2 ), a gluon distribution which is much lower at
very high-x but smaller at small x , and larger light quarks
over most x values. In contrast, making the Q2 and W 2 cuts
on the data less conservative and introducing higher twist
corrections which are fit to the data makes little difference
to PDFs, except at very high x and also little difference to
S (MZ2 ), particularly at NNLO.
This result concerning the importance of the choice of
heavy flavour scheme used might seem surprising. It is known
that the FFNS and a well-defined GM-VFNS will converge
towards each other as the perturbative order is increased.
At higher orders more and more large logs in Q2/mc2 are
included in the FFNS and the ambiguities in the GM-VFNS
definition near threshold are shifted to higher and higher
order. Indeed, it has often been suggested, e.g. [52], that
the omission of Tevatron jet data is the likely source of the
smallness of the high-x gluon in some PDF sets. This is
undoubtedly partially true. It is seen in Fig. 3 of this article
that when fitting using FFNS the inclusion of jet data raises
the gluon for x > 0.1 and S (MZ2 ) (in [15] top pair
production cross sections are raised when Tevatron jet data is
included). However, GM-VFNS fits without jet data do not
automatically have a lower high-x gluon or s (MZ2 ) value
it is simply that constraints on both are loosened. For
example, it is not really clear why for the HERAPDF1.5 PDFs in
[4], which fit HERA DIS data only, the NNLO high-x gluon
is harder than NLO. Hence, the inclusion of jet data or not
is only part of reason for significant PDF differences. It has
also been argued, e.g. [5], that it is the absence of NNLO
corrections to jet production that leads to differences in the
gluon in different PDF sets at NNLO, i.e. the NNLO high-x
gluon is being overestimated due to missing positive NNLO
corrections. I find this unconvincing. In the MSTW2008 fits
threshold corrections of 20 % from [40] are used in NNLO
fits. It was shown recently [53] that the absence of jet radius
R dependence in these terms leads to an underestimate of the
full NLO result in the threshold approximation of [40].
However, improved threshold calculations in [41] shown little R
dependence at NNLO, and the size of corrections at NNLO
inferred from [41] is quite similar to that used in MSTW fits.
Additionally, in [38] extreme changes in the assumed NNLO
corrections for Tevatron jets are considered and changes in
PDFs and S (MZ2 ) are considerably smaller than those seen
from changing the flavour scheme in this article. Hopefully
a full NNLO calculation of jet cross sections [54, 55] will
settle this dispute soon. Furthermore, the issue of NNLO
jet cross sections only affects NNLO PDFs, and the general
features of the differences between different PDF sets are
all very similar at NLO and at NNLO, so attributing them
to effects unique to NNLO seems rather unlikely to be
correct.
In fact the study in this article began at NLO in [10], where
significant differences between FFNS and GM-VFNS was
seen. As well as building on the phenomenological results
of this initial study by showing a similar effect is indeed
present at NNLO, and is consistent with results comparing
FFNS and GM-VFNS in [43] and [11], this article shows
exactly why this effect exists by studying the form of the
leading logarithmic contribution to (d F2c(x , Q2)/d ln Q2)
in FFNS and GM-VFNS. It is shown in Sect. 3 that one can
understand exactly why evolution at high Q2 is considerably
slower in FFNS than in GM-VFNS for x 0.05, and that
the difference between the two will only converge at very
high perturbative order. This has an important impact on the
fit to inclusive DIS data since there is a very large amount
of F2(x , Q2) HERA data at high Q2 for 0.1 < x < 0.01,
and F2c(x , Q2) is a large contribution to this. Since the charm
contribution in FFNS is lower at high-Q2 it is clear that light
quarks will be higher to compensate. The change in the gluon
and S (MZ2 ) is less obvious, but an argument for their form
is put forward in Sect. 5.
Hence, I conclude that the use of GM-VFNS and FFNS
will result in significantly different PDFs and S (MZ2 ) up to
NNLO, whereas higher twist corrections are not important
so long as their absence is accompanied by sufficiently high
cuts on W 2 and Q2. The difference between FFNS and
GMVFNS PDFs will be moderated as the fit becomes more global
and more data types are added, but the fit quality seems to be
better using a GM-VFNS and less tension between different
data sets is observed. Indeed, PDFs which are obtained using
a GM-VFNS are already seen to match LHC jet data very
well [2, 38]. Additionally, one may feel that if there is slow
convergence of a expansion which contains finite orders of
Sn lnn (Q2/mc2) to the result of a fully resummed series of
these terms then it is theoretically preferable to use the latter.
Therefore, I advocate the use of a GM-VFNS in PDF fits to
data.
Acknowledgments I would like to thank A. D. Martin, W. J.
Stirling and G. Watt for numerous discussions on PDFs, and A.M Cooper
Sarkar, S. Forte, P. Nadolsky and J. Rojo for discussions on heavy quarks
schemes. 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. I would also like
to thank the Science and Technology Facilities Council (STFC) for
support.
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