#### Resummation improved rapidity spectrum for gluon fusion Higgs production

Received: February
Resummation improved rapidity spectrum for gluon fusion Higgs production
Markus A. Ebert 0
Johannes K.L. Michel 0 1
Frank J. Tackmann 0
Open Access 0
c The Authors. 0
0 Notkestra e 85 , D-22607 Hamburg , Germany
1 Institute for Theoretical Physics, WWU Munster
turbative corrections. These partially arise from large virtual corrections to the gluon form factor, which at timelike momentum transfer contains Sudakov logarithms evaluated at negative arguments ln2( 1) = 2. It has been observed that resumming these terms in the timelike form factor leads to a much improved perturbative convergence for the total cross section. We discuss how to consistently incorporate the resummed form factor into the perturbative predictions for generic cross sections di erential in the Born kinematics, including in particular the Higgs rapidity spectrum. We verify that this indeed improves the perturbative convergence, leading to smaller and more reliable perturbative uncertainties, and that this is not a ected by cancellations between resummed and unresummed contributions. Combining both xed-order and resummation uncertainties, the perturbative uncertainty for the total cross section at N3LO+N3LL0' is about a factor of two smaller than at N3LO. The perturbative uncertainty of the rapidity spectrum at NNLO+NNLL0' is similarly reduced compared to NNLO. We also study the analogous resummation for quark-induced processes, namely Higgs production through bottom quark annihilation and the Drell-Yan rapidity spectrum. For the former the resummation leads to a small improvement, while for the latter it con rms the already small uncertainties of the xed-order predictions.
aTheory Group; Deutsches Elektronen-Synchrotron (DESY)
1 Introduction Calculational setup Gluon fusion 2
Quark annihilation
Resummation framework
Perturbative uncertainties and numerical inputs
Color-singlet production
Inclusive Higgs production in the rEFT scheme
Incorporating quark mass and electroweak e ects beyond rEFT
Higgs rapidity spectrum
Higgs production through bottom-quark annihilation
Drell-Yan rapidity spectrum
A Perturbative ingredients
Master formula for hard Wilson coe cients to three loops
A.2 Anomalous dimensions
A.3 Constant terms to three loops
Gluon matching coe cient
Ct coe cient for Higgs production in the EFT limit
Quark vector-current matching coe cient
Quark scalar-current matching coe cient
A.4 Renormalization group evolution
B Fixed-order estimates from resummed timelike logarithms
Introduction
After the discovery of the Higgs boson [1, 2], the LHC has entered an era of precision Higgs
measurements. One important goal is the precise determination of the Higgs couplings
in order to test the Standard Model and search for evidence of physics beyond it. Other
important color-singlet processes like Drell-Yan production serve as standard candles that
are used, for example, to constrain parton distribution functions (PDFs).
In order to match the ever increasing level of experimental precision, precise theoretical
predictions for the measured cross sections are needed. An important example is the
dominant Higgs production via gluon fusion, which receives large perturbative corrections.
This has led to the calculation of the total production cross section up to N3LO [3{10],
and including the resummation of threshold logarithms up to N3LL0 [11{16]. However, due
to the limited detector acceptance the experimental measurements cannot measure the
cross section fully inclusively but only in a restricted kinematic range, in particular in a
restricted range of Higgs rapidities. The interpretation of the experimental measurements
thus fundamentally requires theoretical predictions di erential in the Higgs kinematics.
The essential nontrivial ingredient is the Higgs rapidity spectrum (or equivalently the cross
section with a rapidity cut), which is so far known to NNLO [17{21].
A speci c class of perturbative corrections to Drell-Yan-like color-singlet production
arises from the associated quark and gluon form factors, which contain Sudakov logarithms
fer, q2 =
Q2 < 0 as in deep-inelastic scattering, these logarithms vanish with the standard
inducing large corrections at each order in the perturbative series. For simplicity, we will
henceforth refer to these as \timelike" logarithms or contributions, as they arise in the
ratio of the timelike and spacelike form factors.1 This e ect was rst observed long ago in
Drell-Yan production in ref. [22], where it was realized that the coe cients of these terms
are directly related to infrared (IR) singularities. Due to the universal structure of IR
singularities, these terms arise to all orders and their resummation is well known [23{26].
As discussed in ref. [27], the timelike logarithms are also present in the soft contributions
to the pion electromagnetic form factor providing an enhancement compared to the
spacelike case in agreement with the measured enhancement. The resummation of the timelike
logarithms for gluon-fusion Higgs production was carried out in refs. [11, 28] in the context
of soft-gluon (threshold) resummation, where it was shown that it substantially reduces
the large perturbative corrections to the total gg ! H cross section.
The resummation of the timelike logarithms originating in the form factors has since
been included in the resummation of various other exclusive color-singlet cross sections
(see e.g. refs. [29{37]), leading to improvements in the perturbative uncertainties. In these
contexts, the use of the resummed form factor is unambiguous, as it explicitly appears as
an ingredient in the corresponding factorized cross section.
In this paper, we study in detail the utility of the resummed timelike form factors
for predictions of inclusive color-singlet production cross sections. In the case of inclusive
cross sections the bene t of the resummation is a priori not obvious, and its applicability
has occasionally been called into question. For this reason, we discuss in some detail the
arguments for it and its consistent application, as well as the potential pitfalls one might
worry about. For our numerical analysis, we consider both gluon-induced and
quark1Since the resummed logarithms ln2n( 1) happen to give factors of (
2)n, their resummation has
been referred to as \ 2-resummation". Since factors of
2 from other (unrelated) sources are typical to
appear in the perturbative coe cients as well, we will always refer to the resummed logarithms as \timelike
logarithms", to avoid any possible confusion as to what is being resummed.
induced processes. The cases we consider include a generic scalar resonance gg ! X as
a function of mX , gg ! H as a function of the Higgs rapidity, bb ! H, and Drell-Yan
qq ! Z as a function of the Z rapidity.
We nd that in all cases the resummation of the timelike logarithms leads to stable
perturbative predictions. For the gluon-induced cases it leads to a signi cantly improved
convergence compared to the xed-order predictions, as rst pointed out in refs. [11, 28]. This
results in perturbative uncertainties that are both smaller and more reliable. In addition to
the total cross section studied previously, we show how the resummation can be easily and
consistently applied to generic inclusive cross sections di erential in the Born kinematics.
This allows us in particular to obtain the currently most precise predictions for the Higgs
rapidity spectrum, or equivalently the inclusive cross section with a rapidity cut, with
perturbative uncertainties that are reduced by almost a factor of two compared to NNLO.
For the quark-induced processes, the improvement is not as dramatic. Here, the resummed
xed-order results have a similar stability. With an optimal choice of F the
resummation still provides some improvement in the perturbative convergence and uncertainties.
This demonstrates that using the resummed form factor is also viable for quark-induced
processes and provides additional con dence in the estimated perturbative uncertainties.
The remainder of the paper is structured as follows: the basic setup how to
consistently incorporate the resummed form factors into the inclusive cross section is discussed in
section 2. The application to gluon-fusion processes is then discussed in section 3, to Higgs
production through bottom quark annihilation in section 4.1, and to Drell-Yan
production in section 4.2. We conclude in section 5. For completeness all required perturbative
ingredients for the resummed form factors are collected in appendix A.
Calculational setup
Resummation framework
We consider the hadronic production gg ! L or qq ! L of a color-singlet nal state L
with total invariant mass Q2 = q
2 > 0. The hard virtual corrections to these processes
are described by the corresponding QCD form factors. The full form factors contain
infrared divergences, which when combined into the full cross section cancel against the
infrared divergences in the real corrections. Hence, what enters in the nal cross section
are the IR- nite parts of the form factor. In the context of soft-collinear e ective theory
(SCET) [38{41], these are equivalent to the Wilson coe cients from matching the QCD
currents de ning the form factors onto the corresponding SCET currents [42{44]. For the
cases we consider, these are the gluon, quark vector, and quark scalar form factors. The
corresponding matching conditions read schematically
where the Bn? and n are collinear gluon and quark elds in SCET. (The exact matching
conditions for the currents can be found e.g. in refs. [30, 45].) The IR divergences in the
full QCD form factors, given by the quark and gluon matrix elements of the left-hand side,
are exactly reproduced by the corresponding matrix elements of the SCET operators on
the right-hand side, such that the hard Wilson coe cients Cij are given in terms of the
IR- nite parts of the form factors.
The relevant object entering the cross section is the hard function given by the square
of the Wilson coe cient, which we write as
where by default we normalize H to unity at leading order, and H(n) denotes the O( sn
term. To all orders in perturbation theory, C and H depend on the hard momentum
transfer q through logarithms L
i0)= 2]. For spacelike processes, q2 =
L = 2 ln( iQ= ).
The Wilson coe cients in SCET obey the renormalization group equation (RGE)
H (q2; ) = cusp[ s( )] ln
where cusp( s) is the cusp anomalous dimension and H ( s) the noncusp term. Integrating
eq. (2.3) yields the solution
H(q2; ) = H(q2; H ) UH ( H ; ) ;
UH ( H ; ) = exp
The explicit result for the evolution kernel UH is given in appendix A.4. By choosing the
imaginary-valued scale
H =
iQ, the hard function H(Q; H ) is free of logarithms and
can be calculated in xed-order perturbation theory, while the evolution kernel UH resums
all logarithms ln( H = ) = ln( iQ= ).
The hard function explicitly appears in calculations of exclusive cross sections as
[1 + O(T =Q)] :
Here T denotes a resolution variable, which resolves additional emissions, such that in the
Q the cross section is restricted to the soft-collinear regime. In this limit it is
dominated by hard virtual corrections contained in H, and soft and collinear contributions
(both real and virtual) at lower scales
T contained in SC, while hard real emissions
are forbidden. At the partonic level, an example for T is the partonic threshold variable (1
z)Q. More physical examples of T are beam thrust or the pT of the leading jet. The precise
form of the soft-collinear contribution SC depends on the de nition of T but is irrelevant for
our discussion. For a given process always the same hard function appears independently
of the precise choice of T . The factorization in eq. (2.6) implies that in the T
Q limit H
appears as a well-de ned perturbative object (namely as a hard matching coe cient), which
is fully factorized from the rest of the cross section. In particular, the only dependence
on the hard timelike momentum transfer Q2 resides in H, while SC only depends on
parametrically smaller soft and collinear scales proportional to T . In practice, eq. (2.6)
eq. (2.4) to evolve H from its natural scale
H =
iQ to the relevant lower scale
We want to apply the resummed form factor to the inclusive cross section for
colorsinglet production. Here, inclusive refers to the fact that the cross section is fully integrated
over any additional QCD emissions, but it can still be di erential in or contain cuts on any
kinematic variables that are present at Born level and describe the produced color-singlet
system, such as its total rapidity Y or total invariant mass Q. To do so, we can factor out
the hard function from the inclusive cross section
(X) = H(Q2; FO)
R(X; FO) ;
which de nes the remainder R(X; FO). Here, X denotes any dependence on Born variables
or cuts. By de nition, H only depends on the Born kinematics via Q, while the remainder
R can depend on X.
We write the perturbative expansion of the remainder as
R(X; FO) =
(0)(X; FO) 1 + R(1)(X; FO) + R(2)(X; FO) +
where for convenience we pulled out the leading-order cross section
dependence on the factorization scale F related to the PDFs entirely cancels within R,
and we will mostly suppress it. The
the renormalization scale
FO scale in eqs. (2.7) and (2.8) is equivalent to
xed-order prediction, and its dependence explicitly
cancels between H and R. The R(n) coe cients depend primarily only on the total
colorsinglet invariant mass and rapidity, while any dependence on additional Born kinematics
or cuts resides primarily in (0). (This becomes exact for a scalar resonance in the
narrowwidth approximation like the Higgs.) In the following we will for simplicity suppress the
dependence on X and Q2.
We also de ne the K factor
K( ) =
= 1 + K(1)( ) + K(2)( ) +
which captures the total perturbative correction relative to the leading-order result.
Expanding eq. (2.7) order by order in
s( ), it is straightforward to obtain the xed-order
coe cients of R from those of K and H. Up to N3LO we have,
R(1)( ) = K(1)( )
R(2)( ) = K(2)( )
R(3)( ) = K(3)( )
To resum the timelike logarithms from the form factor in the cross section we can
simply take the resummed result for the hard function eq. (2.4) and use it in eq. (2.7),
res = H( H ) UH ( H ; FO) R( FO)
+ H(2)( H ) + R(2)( FO) + H(1)( H ) R(1)( FO) + : : : :
As indicated, the xed-order expansions for H( H ) and R( FO) are reexpanded against
each other (but without reexpanding the s( H ) inside the coe cients H(n)( H ) in terms
of s( FO)). This is analogous to the standard treatment in resummed predictions as
FO we exactly
recover the usual xed-order result without inducing any higher-order cross terms between
H and R. Using the de nition of R in eq. (2.7), the resummed cross section in eq. (2.11)
can equivalently be written as
res = UH ( H ; FO)
where the brackets [: : : ]FO indicate the xed-order reexpansion in powers of s( FO) and
s( H ), with
FO the usual xed-order cross section expanded in s( FO). Written in this
way, the ratio of timelike to spacelike form factors is manifest.
Equation (2.11) will be the basis of all our results. For consistency with the
order limit, we always include H( H ) and R( FO) to the same xed order. Furthermore,
we always combine the NnLO
xed-order contributions with the NnLL resummation for
H, which corresponds to the primed resummation counting and ensures consistency with
the exclusive resummations [30, 33] based on eq. (2.6). We will denote the perturbative
accuracy by NnLO+NnLL0', where the subscript indicates that the resummed logarithms
correspond to the complex phase ' of the hard scale in the form factor.
While the remainder R is uniquely de ned by eq. (2.7), one should of course ask
the question to what extent it is justi ed or meaningful to \brute-force" factorize the
perturbative series for the inclusive cross section into those for H and R.
First, one might be worried by the fact that the remaining nonlogarithmic constant
terms in the
xed-order expansion of H( H ) are scheme-dependent, i.e. they depend on
the fact that H is renormalized in the MS scheme and using a di erent scheme would result
in di erent constant terms. However, this xed-order scheme dependence is canceled by
R up to higher orders, and this cancellation is explicitly ensured in our implementation in
eq. (2.11) by the fact that we always reproduce the exact xed-order result, as discussed
above. The cancellation can also be seen explicitly from eq. (2.12). Expanding the ratio
H explicitly drop out. In particular, the nonlogarithmic constant terms at O( sn) cancel
Therefore, the relevant question is whether the series of timelike Sudakov logarithms
present in H can be considered to be independent from the perturbative series in R. This
would not be the case if (and only if) R were to contain contributions at each order
correlated with the timelike Sudakov series in H and of opposite sign, which would then lead
to large cancellations between H and R at each order in perturbation theory. These
cancellations would then be spoiled by resumming the timelike logarithms in H while keeping
the corresponding pieces in R at
xed order. This would imply that the perturbative
corrections for R would be noticeably larger than for the cross section itself, and since
the resummation of H eliminates its large corrections, the larger perturbative corrections
of R would result in the resummed cross section being worse behaved. In other words,
the absence or presence of sizeable cancellations between the resummed terms and the
unresummed xed-order terms, is mathematically equivalent to whether the resummation
improves the perturbative convergence of the cross section or not. This is of course easy
to check up to the available order, and in all our applications we have veri ed that there
are indeed no large cancellations that are being spoiled by the resummation.
The primary reason one could be worried about such cancellations is that this is
actually what happens in the reverse timelike process, namely color-singlet decays such as
! Z ! qq, or hadronic
decays. These processes involve the
same timelike form factor, but their perturbative series is known to not contain timelike
Sudakov logarithms. The relation to these processes was already discussed in some detail
in ref. [11]. In these processes, timelike logarithms only appear as single logarithms (and
thus only at higher orders) through the running of s, for which analogous analytic
continuation methods have been considered, e.g. for e+e
! hadrons in refs. [27, 46{49] and
hadronic -decays e.g. in refs. [50{52] (see also refs. [53, 54] and references therein).
However, the situation is fundamentally di erent when the hard partons appear in
the initial vs. the
nal state. An explicit discussion how the timelike Sudakov logarithms
cancel in the
nal-state case but not in the initial-state case can be found in ref. [27].
nal-state case, the process can be written as the imaginary part of forward
matrix elements summed over all possible cuts, in which case the whole calculation can be
deformed into the Euclidean domain where the timelike logarithms never appear. That
is, the timelike Sudakov logarithms fully cancel between all cuts, or equivalently between
the virtual corrections to the form factor and the real corrections to the corresponding
remainder. The same does not happen if the partons appear in the initial state, which
simply cannot be obtained from cutting a diagram, i.e. the process with incoming partons
is intrinsically more exclusive, which exposes the timelike Sudakov logarithms in the form
factor. Note also that if the same cancellations as in the
nal-state case were present in
the initial-state case, they would have to be present at each order starting at NLO. The
fact that we do not observe this even in the rst several orders of the perturbative series
provides clear evidence that this is indeed not the case.
It is also easy to understand why one nds a substantial numerical improvement for
inclusive Higgs production. Comparing eq. (2.7) with the exclusive cross section in eq. (2.6),
in the soft-collinear limit the remainder R reduces to the soft-collinear contributions times
power corrections,
B SC(T ) [1 + O(T =Q)] :
Therefore, the factorization in eq. (2.7) also becomes formally justi ed when the inclusive
cross section is numerically dominated by soft-collinear contributions. It is well known that
a large portion of the Higgs cross section comes from the partonic threshold limit, in which
the hard function factors out of the cross section as in eq. (2.6). One can also take the
more physical limit and simply veto additional hard radiation (which is also a weaker limit
as it allows both soft and collinear radiation). Going from this exclusive 0-jet region, to
which eq. (2.6) strictly applies, to the inclusive cross section amounts to factoring out the
form factor also from the nonsingular power corrections. As pointed out in refs. [30, 33],
using either beam thrust or the pT of the leading jet to veto hard radiation, one nds that
utilizing the resummed form factor for both singular and nonsingular corrections, and hence
for the full inclusive cross section, is actually important, since not doing so can easily lead to
unphysical results with the inclusive cross section being smaller than the 0-jet cross section.
Finally, we note that it has been argued in ref. [10] on the basis of the coe cient
of the (1
z) term in the partonic cross section that the timelike logarithms are not a
dominant source of higher-order corrections and in particular that their resummation fails
to improve the results beyond NNLO. We need to disagree with this assessment, because
this coe cient is strongly scheme dependent and not a very well-de ned quantity. Rather
the impact or improvement should be judged at the level of the physical cross section. A
more detailed discussion on this is given in appendix B.
Perturbative uncertainties and numerical inputs
We use the
PDF4LHC nnlo 100 [55{60] NNLO PDFs with
s(mZ ) = 0:118. Since we are interested
in the size of the coe cients in the perturbative series we always use this same PDF
independent of the perturbative order in consideration. The numerical value of
obtained with the corresponding three-loop running, except for the total gluon-fusion cross
section known at N3LO, where we use four-loop running (though the numerical di erences
are negligible). For bottom-quark annihilation we use the PDF sets from refs. [61, 62],
which are reevolved from PDF4LHC nnlo mc in order to allow varying the b-quark matching
scale separately from the b-quark mass. The relevant masses entering our predictions are
Since we are primarily interested in investigating the perturbative structure, we do
not consider parametric uncertainties due to PDFs and the value of
s(mZ ), which are
straightforward to evaluate. They are essentially una ected by the resummation of the
form factor, since all PDF dependence, as well as the dominant overall dependence on
s(mZ ) in case of Higgs production, resides in the remainder R.
An important aspect of precision predictions is a reliable assessment of the theory
uncertainties due to missing higher-order corrections. Our predictions in principle involve
three scales that we can vary as a means to estimate the size of higher-order corrections:
the factorization scale F probing collinear logarithms in the PDFs, the renormalization
R probing higher orders in the
xed-order series, and the hard resummation scale
H probing higher orders in the series of timelike Sudakov logarithms. We like to stress
that these scales are unphysical parameters whose variations simply provide a convenient
way to probe the \typical" size of the associated missing higher-order terms. The resulting
variations in the cross section must be interpreted as such. In particular, we do not
assign any meaning to accidentally small one-sided scale variations that yield asymmetric
uncertainties, which are just artifacts of a nonlinear scale dependence, which is frequently
encountered in predictions at higher orders or involving resummation. We therefore always
consider the maximum absolute deviation from the central result at the chosen central
scale as the (symmetric) uncertainty. To be explicit, an observed scale variation of +jxj
jyj in the cross section is interpreted as a perturbative uncertainty of
maxfjxj; jyjg.
We parametrize the three scales as
H =
FO exp( i') ;
R =
F =
The choices for FO and F for the central value depend on the process we consider. For the
=2 , while the xed-order predictions
We explicitly distinguish two di erent sources of perturbative uncertainties, namely
xed-order and resummation uncertainties, that are associated to the two independent
perturbative series involved. The xed-order uncertainty, denoted as
, is obtained via
the conventional variations of
F . This comprises a collective overall variation
of FO by a factor of two around its central value, which is combined with an additional
variation of F by a factor of two around its central value, without considering the extreme
variations where both are varied up or down at the same time. That is, relative to the
central values we consider the set of variations
VFO =
from which the xed-order uncertainty
is obtained as the maximum deviation from the
= max
In the limit where the resummation is turned o , this reproduces the perturbative
uncertainty in the xed-order predictions. For the resummed predictions, the magnitude of the
FO, as illustrated in
on the left, such that the xed-order variations do not change the resummed logarithms
ln( H = FO).
around the central value of ' =
gure 1 on the right. This probes the intrinsic size of the higher-order timelike logarithms.
The phase variation by
' is then obtained as
the maximum observed deviation from the central value (usually happening at one of the
endpoints), such that
' =
'2[ =4; 3 =4]
' =
the overall variations of
FO, which determines
(in conjunction with the variation of F , which
is not shown). Right: the phase variation for H for xed FO, which determines the resummation
This additional resummation uncertainty was not considered in earlier treatments, but has
already been included in the resummed 0/1/2-jet-bin results reported in ref. [63].
The total perturbative uncertainty is obtained by adding the two independent sources,
For bb ! H we follow ref. [62] and consider the low-scale matching at b onto the b-quark
PDFs as a third independent source of uncertainty
b, which is estimated by varying b
by a factor of two.
Gluon fusion
Gluon-fusion processes are well-known to contain large perturbative corrections, which are
partially due to the timelike logarithms in the gluon form factor, as rst demonstrated in
ref. [28]. We rst consider the total production cross section up to N3LO+N3LL0' for a
generic scalar
nal state gg ! X in section 3.1 and for the SM Higgs boson in the rEFT
1 limit in section 3.2. In section 3.3 we discuss how to incorporate quark-mass and
electroweak e ects into the resummed results. In section 3.4 we then present our results for
the Higgs rapidity spectrum and the cross section with a rapidity cut to NNLO+NNLL0'.
Color-singlet production
We rst consider the total production cross section from gluon fusion for a generic
colorsinglet scalar X with mass mX . Its coupling to gluons at the scale
mX can be expressed
in terms of an e ective Lagrangian as
is a suitable high mass scale and CX is the Wilson coe cient from integrating
out heavy particles that mediate the e ective ggX interaction. This e ective operator
arises for SM Higgs production in the mt ! 1 limit, which we discuss in more detail in
section 3.2. Here, we use it as a simple case to study the e ects of the resummation and
its dependence on the mass over a wide range mX 2 [100; 1000] GeV. For this purpose, the
precise values of the e ective coupling CX (
= mX )=
need not be speci ed, as it drops
We obtain the total gg ! X cross section to N3LO from SusHi 1.6.0 [6, 10, 64{69].
Our central scale choices are
FO = mX and
F = 1, such that
R =
F = mX . Away
and N3LO. Their resummation is irrelevant and can be neglected, and they are instead
included in the xed-order cross section [66].
The gluon form factor is known up to three loops [70{75], and the Wilson coe cient
Cgg is explicitly extracted from it in ref. [75] (see also refs. [76, 77]),
Hgg(m2X ; ) = Cgg(m2X ; )
2 = 1 + X
where now Q2 = m2X . The RGE of Cgg reads
Cgg(m2X ; ) = gg(m2X ; ) Cgg(m2X ; ) ;
gg(m2X ; ) =
where cgusp( s) is the gluon cusp anomalous dimension and the last three terms are the
total noncusp contribution. All the relevant ingredients are collected in appendix A.
The separation of the perturbative series for the K factor at xed order into those of
H and R is shown in gure 2 as a function of mX . Half of the large NLO K factor comes
from H and half from R, while beyond NLO the corrections in H are larger than for R.
Hence, the large corrections to the K-factor present at each order are driven to a large
extent (but also not entirely) by the corrections from H. In particular, the remainder R
by itself has a much better behaved perturbative series than K, and there are clearly no
cancellations between H and R. (Otherwise, as already explained in section 2.1, R would
need to have negative corrections that are larger in size than those in K.) This pattern
holds independently of mX . The visible increase in the corrections toward smaller mX is
due to the running of s(mX ).
The large perturbative corrections in Hgg at the real scale
H = mX are absent at
the imaginary scale
H =
imX , as shown by the long-dashed curve in the middle panel
of gure 2. To illustrate this more explicitly, the numerical values for an example mass of
mX = 750 GeV are,2
Hgg(m2X ; H = mX )
Hgg(m2X ; H =
2The value is chosen purely for historical reasons.
= 1 + 0:49279 + 0:13855 + 0:02288 ;
imX ) = 1 + 0:06820
200 400 600 800 1000
200 400 600 800 1000
200 400 600 800 1000
imX (black
long dashed), for which it contains no timelike logarithms.
where each term is the contribution from a subsequent order in
s up to N3LO. Clearly,
the large corrections to the gluon form factor at real scales are almost entirely due to the
timelike Sudakov logarithms that are present for
H = mX and are eliminated by taking
H =
imX . Since the corrections in Hgg at H =
imX are very small, the perturbative
convergence of the resummed cross section will be essentially determined by that of the
remainder R.
In gure 3, we compare the xed-order and resummed cross sections as a function of
mX , with the bands showing the total perturbative uncertainties evaluated as discussed in
section 2.2. (Note that in case of gg ! X and gg ! H, the xed-order uncertainties come
from the variation of R for
F .) All results are normalized to the LO prediction
hard function directly translates into a much faster convergence of the resummed cross
section. Furthermore, the uncertainties in the resummed predictions at lower orders cover
the higher-order bands much better than at xed order, while at the same time being
substantially reduced at higher orders. Hence, even at NNLO and N3LO, where the xed-order
results start to show convergence, the resummation noticeably improves the predictions.
Due to their better convergence, the resummed predictions provide substantially improved
uncertainty estimates both in terms of their reliability and their size. In particular, we can
be reasonably con dent that the result at the next higher order will lie within the small
N3LO+N3LL0' uncertainty band.
resummation of timelike logarithms (right). All results are normalized to the central LO prediction
at FO = mX .
Inclusive Higgs production in the rEFT scheme
We now turn to the case of Higgs production through gluon fusion as an important
application of the singlet production discussed above. For Higgs masses below the top threshold,
mH < 2mt, the gluon-fusion cross section can be well approximated by an e ective
theory where the top quark is integrated out [78{81], giving rise to an e ective Lagrangian
analogous to eq. (3.1),
In this case, the Wilson coe cient Ct itself receives sizable QCD corrections, which have
been calculated to N4LO in refs. [82{84]. The e ective operator in eq. (3.5) is the same as
in eq. (3.1), giving rise to the same gluon form factor and hard function Hgg in eq. (3.2).
Rescaling the cross section
EFT obtained from eq. (3.5) by the LO mt dependence [85]
F0( ) =
one obtains the inclusive cross section in the \rescaled EFT" scheme (rEFT),
rEFT = jF0( )j2 EFT :
This rescaling is known to well reproduce the mt-exact result at NLO, and hence it is
believed to be a useful approximation also at higher orders [5, 86{92]. The inclusion of
further quark mass and electroweak e ects will be discussed in section 3.3.
We use SusHi 1.6.0 [6, 64{67] to compute the total cross section in the rEFT scheme
to NNLO. For the N3LO contribution we use the results of ref. [10] as implemented
in ggHiggs 3.5 [16].3
3In SusHi 1.6.0, the
R dependence at N3LO is threshold expanded consistently with the
independent terms, while it is kept exact in refs. [10, 16]. There is no clear theoretical preference for either
and F = 1 (so
R =
section has the perturbative series
rEFT = (1 + 1:291 + 0:783 + 0:296)
FO
R( FO = mH ) = (1 + 0:672 + 0:148 + 0:012)
where again each term gives the contribution from a subsequent order in s. The remainder
R now includes the corrections to jCtj2. As before, its perturbative series is much better
behaved than that of the cross section, whose large perturbative corrections are thus driven
by the large corrections from timelike logarithms in Hgg.
To illustrate the improved convergence of the resummed form factor, we consider the
hard function Hgg(mH ; H ) at various scales H ,
Hgg(m2H ; H = mH )
Hgg(m2H ; H =
Hgg(m2H ; H = mH =2)
Hgg(m2H ; H =
Hgg(m2H ; H = mH =5)
imH =2) = 1
= 1 + 0:61925 + 0:21878 + 0:04539 ;
= 1 + 0:08408
= 1 + 0:57325
= 1 + 0:08090
For both imaginary-valued scales
H =
imH and
H =
imH =2, the corrections are
drastically reduced compare to the real scale choice. For comparison, choosing a real value
H =
imH still leads to
much larger NNLO corrections.
To examine the dependence on the resummation phase ' of the hard scale,
H =
FO exp( i'), we show in the left panel of gure 4 the resummed cross section as a function
of '. Here, the uncertainty bands only show the xed-order uncertainty
. At ' = 0,
res(') is just the
xed-order cross section. As ' !
=2, the timelike resummation is
turned on, visibly improving the convergence of the cross section and providing better
start to enter again.
In the right panel of gure 4, we compare the xed-order results at the conventional
scales of FO = mH and
as relative corrections to our best prediction at N3LO+N3LL0'. For the resummed results,
the inner uncertainty bars indicate
' alone, while the outer ones show
'. While
' contributes to obtaining a more realistic uncertainty estimate at LO+LL0' (compared
to LO), its impact is strongly reduced at higher orders. The overall picture and conclusions
from the generic color-singlet case are una ected by the presence of the Wilson coe cient
jCtj2 in the cross section. The resummation yields again a clear improvement in convergence
treatment. The resulting numerical di erences away from the canonical values
R =
F = mH are around
0.3%, consistent with the level of systematic uncertainties expected from the threshold expansion [10]. To
ease numerical comparisons we use the numerical values corresponding to the exact running here.
scheme. Left: the cross section as a function of the resummation phase ' of the hard scale
H =
FO exp( i'), with the uncertainty bands corresponding to
only. Right: comparison
uncertainty bars show
xed-order results and
' for the resummed results (with
the inner bars visible at the lower orders showing
' only). The xed LO results are out of range.
and uncertainties, also compared to the xed-order results at
FO = mH =2, which are
already better behaved than those at
already fully covers the highest-order result, which is not the case at xed NLO, and the
precision of the NNLO+NNLL0' result is roughly comparable to the
This gives us good con dence in the small remaining uncertainty at N3LO+N3LL0', which
is reduced by a factor of two compared to N3LO. The explicit numerical results at the
xed N3LO results.
highest order are
2:60 ) pb
1:83 ) pb
(5:59%) ; (N3LO; FO = mH ) ;
(3:82%) ; (N3LO; FO = mH =2) ;
Note that for the N3LO results in ref. [10] the perturbative uncertainties are estimated by
varying FO but keeping
F = 1 xed. Doing so reduces
to 2:21 pb (4:76%) at central
FO = mH and 1:54 pb (3:21%) at central
FO = mH =2.4 Similarly dropping the
variation in the resummed results gives
= 0:67 pb, which combined with
' then yields
a total perturbative uncertainty of 1:44%. Note also that using the threshold-expanded
increases to (48:17 1:99 ) pb (4:14%), with a corresponding increase in
since the result
at FO = mH is una ected.
4Ref. [10] further utilizes the MS top-quark mass mt( FO) in the rescaling factor in eq. (3.7), which
participates in the overall FO scale variation and further reduces its e ect to 2:4%. However, the perturbative
series for the MS top-quark mass entering in the rescaling factor has nothing to do with the perturbative
series of the gg ! H cross section in the mt !
1 limit arising from the e ective Lagrangian eq. (3.5).
Hence, the fact that their FO dependences partially compensate numerically is purely accidental.
While the previous section focused on the QCD corrections to Higgs production in the
mt ! 1 limit, further corrections arise from
nite quark-mass e ects as well as electroweak
contributions. Here we discuss how to consistently combine them with the resummation
of timelike logarithms.
The full dependence of the cross section on the heavy quark masses mt, mb, mc is fully
known at NLO [5, 64, 86, 87, 93{95]. We de ne
result relative to the rEFT result,
(N)LO as the correction of the exact
On top of the exact NLO corrections, top-quark mass e ects are also known in an
asymptotic expansion in 1=mt at NNLO [88{92].
In the following we consider the top-mass e ects in more detail. As discussed in
improve over the mt ! 1 limit. Rather, they can serve to estimate the uncertainty due
to the still unknown full NNLO mt corrections. For this reason we will only take into
account the NLO corrections
NtLO. (The inclusion of the NNLO mt corrections would be
completely analogous.) This is also consistent with our analysis of the rapidity spectrum
in section 3.4, for which the mt-corrections are only known at NLO. For illustration, the
numerical results for
Nt;bL;Oc =
NtLO =
NtLO =
( FO = mH ) ;
( FO = mH =2) :
The nite mt contributions correspond to a correction to the mt ! 1 limit in eq. (3.5),
from which the gluon form factor arises, and so a priori they do not involve the same local
gluon form factor. Therefore, one option to include them in the resummed results is to
simply add them to the rEFT results in eq. (3.10), which yields
NtLO = (46:30
NtLO = (47:74
NtLO = (47:69
2:55 ) pb
1:75 ) pb
(N3LO; FO = mH ) ;
(N3LO; FO = mH =2) ;
The complete results including those at lower orders are collected in table 1.
Alternatively, following ref. [30] we can perform a one-step matching of the full
Standard Model including the top quark onto SCET, simultaneously integrating out both the
sponds to the full SM gg ! H form factor and includes all virtual nite-mt e ects. It takes
the form [30]
Hgtg(mt; m2H ; ) = jF0( )j2j s( )j2
The RGE for Cgtg is given by
where as before
and the contributions from Ct are now moved from the remainder into the hard function.
The F1( ) contains the full virtual mt dependence at NLO and the O(
neglected NNLO virtual mt corrections.5 Although Hgtg is no longer normalized to unity at
leading order, we can continue to use eq. (2.12) to obtain the resummed cross section.
Comratio of hard functions, which replaces the s2( FO) inside the LO cross section by j s( H )j2.
s2) terms denote the
Cgtg(mt; m2H ; ) = gtg(m2H ; ) Cgtg(mt; m2H ; ) ;
gtg(m2H ; ) =
The noncusp terms in
gtg di er from those in gg in eq. (3.3) due to the additional
dependence of
s( )Ct( ), which is now included in the hard Wilson coe cient. The
i FO and then evolved
FO. For the overall s( ) this is largely irrelevant since it is ultimately evolved
starting from
s(mZ ). For Ct( ), which is treated in
xed order, this induces di erent
subleading timelike logarithms starting at NNLO compared to Hgg. This is re ected in the
noncusp terms di ering by t, whose numerical e ect however is not signi cant. Also, the
perturbative convergence of jCt( )j2 at
= mt) is practically the same.
= mH and
imH (and at its natural scale
The perturbative convergence of Hgtg shows the same improvement as seen for Hgg
when evaluated at H =
imH rather than
H = mH ,
Hgtg(mH ; H = mH )
Hgtg(mH ; H =
= j s(mH )j2 jF0j2
imH ) = j s( imH )j2 jF0j2
1 + 0:82152 + 0:36170 + 0:10268 ;
1 + 0:27631 + 0:04244
0:00257 : (3.16)
The main di erence compared to Hgg are the additional constant terms from Ct that
are now included in Hgtg. The
nite-mt corrections have a very small e ect on the NLO
contribution, contributing a +0:005 to the above 0:82152 and 0:27631.
For reference, we rst consider the rEFT limit and drop the nite-mt terms in Hgtg as
NtLO. The rEFT result based on Hgtg at N3LO+N3LL0' then reads
rreEsF;HTt = (47:98
0:24') pb (1:85%) :
This is equivalent to the results at
NLO mt dependence, we obtain
H =
imH reported in ref. [63]. Including the full
NtLO res; Ht = (47:84
0:25') pb (1:77%) :
The full set of results including the lower orders are shown in the last column of table 1.
F1(0) here simply removes the leading mt ! 1 part of F1( ), which is already included via Ct.
We drop all cross terms of F1( )
F1(0) with Cgg, which are of O(
are also not included in the xed-order cross section.
s2) and higher, because these terms
n NnLO; FO = mH
0 13:8 3:2 (23%)
NnLO; FO = m2H
16:0 4:3 (27%)
1 31:4 6:2 (20%)
36:6 8:2 (23%)
2 42:2 4:5 (11%)
46:2 4:6 (10%)
3 46:3 2:5 (5:5%)
47:7 1:7 (3:7%)
NnLO+NnLL0' (Hgg)
NnLO+NnLL0' (Hgtg)
3:5' (27%)
2:7' (15%)
1:0' (6:0%)
3:4' (26%)
2:8' (15%)
1:0' (6:1%)
0:18' (1:7%) 47:8 0:8
0:25' (1:8%)
exact mt dependence
to the total uncertainty
t at NLO. The percent uncertainties for the resummed results correspond
Comparing the last two columns of table 1, the resummed results using the two di
erent ways to include the top-quark contributions are perfectly compatible with each other.
The xed-order uncertainty is essentially una ected, because it is insensitive to the
precise split of the constant terms into H and R due to the reexpansion of their xed-order
contributions [see eq. (2.11)]. The resummation uncertainty
' increases somewhat in the
one-step matching, which re ects the fact that the Ct contributions introduce an additional
dependence and that they are evaluated at
imH rather than their
natuwhich shows that the results are insensitive to the precise treatment of the top
contributions. This also provides nontrivial veri cation that the scheme dependence in how the
nonlogarithmic constant terms are split between H and R at each order is much smaller
than the perturbative uncertainties and hence irrelevant.
A complete numerical inclusion of all known corrections beyond the rEFT limit is
beyond the scope of this paper. The inclusion of b-quark and electroweak e ects can
proceed completely analogously to the treatment of the top contributions. Any multiplicative
contributions can be trivially included, while additive corrections such as the NLO
mbdependent terms can be treated analogously to the nite-mt corrections. For example, the
dominant known electroweak corrections can be included by replacing [96]
Ct ! Ct + EW(1 + C1w s +
where EW is the pure NLO electroweak correction to the LO cross section [97, 98] and C1w
contains the mixed O(
bosons as an estimate of the full O(
s) correction calculated in ref. [96] by integrating out W - and
Z
s) corrections. These additional contributions will not
a ect the bene t of the resummation, in the same way the inclusion of the top corrections
for gg ! H did not a ect the conclusions compared to the generic scalar gg ! X case.
Higgs rapidity spectrum
As discussed in section 2.1, the resummed form factor can be incorporated in the same
way as for the total production cross section into generic cross sections that are di erential
in or contain cuts on the Born kinematics. Here we consider the primary example of the
malized to the LO spectrum
FO = mH =2 (right).
rapidity spectrum as well as the cross section with a rapidity cut. For simplicity we do not
consider additional ducial cuts on the Higgs decay products here, but stress again that
these are straightforward to include.
The rapidity spectrum for gluon-fusion Higgs production is known to NNLO [17{21],
while the N3LO corrections are available in the threshold limit [99, 100]. The resummation
in the small-x limit is also known [101]. We obtain the xed-order bin-integrated rapidity
distribution for gg ! H to NNLO with HNNLO 2.0 [20, 21]. We use a binsize of
Y = 0:25
and for clarity in all plots interpolate the binned results.
We rst consider the rEFT limit and exclude additional quark mass e ects. In gure 5,
we display the perturbative remainder R(Y ) as a function of Y . Although it has some
intrinsic nontrivial rapidity dependence, the overall behavior is as for the total cross section,
namely it exhibits a noticeably better convergence than the full
xed-order spectrum.
Hence, we expect a similar improvement from applying the resummation to the rapidity
spectrum as for the total cross section.
FO =
mH =2, with the bands showing
. The overall K factor at NLO and NNLO is roughly
constant in the central rapidity range and similar to that of the total cross section. This is
consistent with the fact that a large part of the K factor stems from the timelike logarithms
in the gluon form factor, which is independent of the rapidity.
The resummed result including xed-order and resummation uncertainties,
is shown in the bottom panel of gure 6. Clearly, resumming the timelike logarithms
improves the perturbative convergence across the spectrum as it did for the total cross
section. The NNLO+NNLL0' result has perturbative uncertainties that are almost a factor
of two smaller than at NNLO. At the same time, the NNLO+NNLL0' result is well covered
by the lower-order NLO+NLL0' uncertainty band, which is not the case at
Judging from the results for the total cross section, for which the full N3LO is known,
! `+` , normalized to (0)(Y )
the result including only the qq channel.
cancellation in the one-loop matching coe cient
CqVq(Q; ) = 1 +
where the rather large nonlogarithmic constant term of
2=6 partially cancels the
ln2( 1) =
2 when Hqq is evaluated at
H = mZ . As discussed in section 2, the
separation of the nonlogarithmic constant terms between H and R amounts to a scheme
choice and only their sum is ultimately relevant. Hence, this large NLO constant term is a
scheme-dependent artifact and in fact cancels most of the equally large NLO contribution
in Hqq, which gives a contribution of +0:247 to it, even though this is not immediately
obvious from eq. (4.6). This explains the much improved convergence of the resummed
It was already noted in ref. [25] that the constant terms in CqVq are scheme dependent
and hence not physical, unlike the ratio of form factors. Since the constant terms in
H and R are evaluated at di erent scales, there is a residual scheme dependence, which
is analogous to a scale choice in that it a ects the numerical results but is formally of
higher order [see eq. (2.13)]. To check this, we can consider an alternative renormalization
scheme for the Wilson coe cient C~qVq, for which all constant terms exactly vanish. That
imZ ) = 1 + 0 + 0 + 0. Hence,
the constant terms are moved entirely into the remainder. In this scheme, the resummed
0:8 pb is of
the same size as the uncertainties and thus of the typical size we expect for an O( s3) e ect.
We now discuss the e ect of the resummation on the rapidity spectrum. In
we show the remainder R(Y ) normalized to the Born cross section as a function of Y for
! `+` at Q = mZ . The xed-order results are
( xed order) and
' (resummed).
large and of the same size and opposite sign as the NLO contributions, while the NNLO
corrections are almost negligible at
qq and non-qq channels, which individually are very large. In contrast, at
F = mZ =2 also
the individual corrections to the remainder are very small, again supporting this central
choice when including the resummation. (A large part of the rapidity-independent constant
shift at NLO will again be canceled by the constant term in Hqq.) In gure 10, we compare
Overall, we nd that the NNLO and NNLO+NNLL0' predictions provide very similar
results. On the one hand, this is reassuring, as it shows that the good convergence of the
xed-order series is not spoiled by the resummation. On the other hand, given the extreme
reduction of the perturbative uncertainties in the
xed-order results at the conventional
uncertainties are somewhat underestimated, in part due to the accidentally small NNLO
contribution. In this respect, the resummed results provide a useful con rmation and
increased con dence in the very small perturbative uncertainties in the Drell-Yan predictions.
that arise to all orders in perturbation theory and are an important source of
perturbative corrections in s-channel color-singlet production processes, which involve a timelike
hard momentum transfer. These logarithms can be resummed to all orders using the RG
evolution of the corresponding quark or gluon form factors from spacelike to timelike scales.
We have shown how to incorporate the resummed form factor in a completely
straightforward manner into predictions for generic inclusive cross sections with arbitrary
dependence or cuts on the Born kinematics.
We have veri ed that this does not spoil the
perturbative series in all considered cases. We have also discussed the assessment of the
uncertainties intrinsic to the resummation.
We rst revisited the resummation for the total gluon-fusion cross section, for which
it has been discussed before, considering both the production of a generic scalar as well as
the SM Higgs boson in the mt ! 1 limit up to N3LO+N3LL0'. For the latter we have
also shown how to incorporate quark-mass and electroweak e ects. We con rm that the
resummation signi cantly improves the perturbative series, and
nd that it reduces the
perturbative uncertainties at the highest orders by about a factor of two.
For the Higgs rapidity spectrum as well as the cross section with a cut on the Higgs
rapidity we obtain results at NNLO+NNLL0', which provide the currently most precise
predictions with central values close to what might be expected at N3LO, and perturbative
uncertainties of
6%, which are almost a factor of two smaller than at NNLO. Once N3LO
results for the rapidity dependence become available, we project that the corresponding
resummation at N3LO+N3LL0' will provide a similar improvement.
We also studied the resummation of timelike logarithms for quark-induced processes,
namely Higgs production through bottom-quark annihilation and the Drell-Yan rapidity
spectrum. For the former, the resummation provides a small improvement in the
perturbative convergence and resulting uncertainties. For Drell-Yan production, the resummation
provides no clear improvement but also no worsening of the predictions, due to the already
fast convergence of the
xed-order perturbative series. In this case it provides a useful
con rmation of the very small residual perturbative uncertainties.
We conclude that utilizing the resummed timelike quark and gluon form factors is
viable and bene cial for obtaining precise and reliable predictions for s-channel color-singlet
production processes.
Acknowledgments
We like to thank Stefan Liebler for his support with SusHi and for comments on the
manuscript as well as Dirk Rathlev for his expertise on HNNLO. We thank the anonymous
referee for suggesting to also study the individual partonic channels for the Drell-Yan
process. This work was supported by the DFG Emmy-Noether Grant No. TA 867/1-1 and
the PIER Helmholtz Graduate school. J. M. thanks DESY for hospitality and gratefully
acknowledges support by Munster University funds designated for student research.
Perturbative ingredients
Master formula for hard Wilson coe cients to three loops
The hard matching coe cients C satisfy an RGE of the form
C(q2; ) =
which allows us to completely predict the logarithmic structure in terms of the cusp and
noncusp anomalous dimension coe cients. We write the perturbative expansion of the
hard coe cient as
C(q2; ) = X C(n)(L)
L = ln
Cn = C(n)(0) :
eq. (A.1) to N3LO is given by
C(0) = 1 ;
C(1)(L) =
C(2)(L) =
C(3)(L) =
(2C1 0 + C1 0 + 1) + C2 ;
03 + 0(16C1 0 +6C1 0 +6 1 +4 1)+ 1(8 0 +6 0)
2 1) + 2 1 0 + 4 0 1 + 2 0 1
4C2 0 + 2C1 1 + C2 0 + C1 1 + 2 + C3 :
Here, n are the beta-function coe cients, n
in the appropriate quark or gluon cusp
anomalous dimensions coe cients, and n are the coe cients of the total noncusp
anomalous dimension
in eq. (A.1) as appropriate for the hard coe cient of interest. All required
anomalous dimension coe cients are given below in appendix A.2. The results for the
nonlogarithmic constant terms Cn for the di erent Wilson coe cients are given below in
appendix A.3
The full expression for the hard function is obtained by squaring C, accounting for
cross terms. In the case of Hgtg de ned in eq. (3.14) the product of Ct Cgg is reexpanded.
Anomalous dimensions
We expand the function of QCD as
= [ s( )] ;
( s) =
The coe cients up to four loops in the MS scheme are [134{137]
0 =
1 =
2 =
3 =
3 TF nf ;
CA + 4CF TF nf ;
active avors.
The cusp and noncusp anomalous dimensions are expanded as
icusp( s) = X i
( s) = X
The coe cients of the MS cusp anomalous dimension to three loops are [138{140]
gn = CA n ;
(for n = 0; 1; 2) ;
qn = CF n ;
0 = 4 ;
1 = 4 CA
2 = 4 CA2 245
+ CF TF nf
TF nf =
+ CA TF nf
The resummation at N3LL formally also requires the yet unknown four-loop coe cient i3
which we estimate as usual by the Pade approximation
and explicitly verify that a variation
200% only a ects the hard evolution kernel UH (and
thus the resummed cross section) at the sub-permille level. We therefore neglect this source
of theory uncertainty.
The gluon noncusp anomalous dimension Cg enters the RGE for the gluon-to-scalar
matching coe cients Cgg and Cg0g in eqs. (3.3) and (3.15). The coe cients in MS up to
The evolution of Cgtg in the one-step matching also requires the anomalous dimension
t of the Wilson coe cient Ct arising from integrating out the top quark. It is given by
t n =
The quark noncusp anomalous dimension Cq enters the RG eqs. (4.1) and (4.5) for both
quark-induced processes we consider. The coe cients in MS up to three loops are [72, 77,
C 0 =
C 2 =
C 1 = CA CA
C 0 =
C 1 =
C 2 =
The evolution of CqSq also requires the anomalous dimension of the quark Yukawa
coupling, which is equivalent to the quark mass anomalous dimension m,
y( ) = m[ s( )] y( ) :
It is known to ve loops [142{148]. For our main analysis at NNLL we only require the
two-loop result, while the three-loop coe cient m 2 serves to verify our N3LO result for
+ CF TF nf ( 46 + 48 3)
CqSq. The results are
m 0 =
m 1 =
m 2 =
Constant terms to three loops
In the following, we provide the process-speci c nonlogarithmic constant terms Cn for the
various hard matching coe cients. For Cgg, Ct, and CqVq, we can collect the results from
the literature. The result for CqSq we have extracted from the three-loop scalar quark form
factor. By convention, we normalize all coe cients to unity at LO,
Cgg 0 = Ct 0 = CqVq 0 = CgSg 0 = 1 :
Note that for all coe cients quoted here, we closely follow the notation from the original
Gluon matching coe cient
The nite terms of Cgg can be read o from the full result given in ref. [75],
+ CACF nf
The general expression for Ct(mt; ) up to O( s3) is given by
Ct(mt; ) = 1 + Ct 1 4
The constant terms are given by
Ct 1 = 5CA
Ct 2 =
Ct 3 =
The dependence of Hgtg on
= m2H =(4mt2) at NLO is given by [30]
F1( ) = CA 5
659504801 4
where F1(0) = Ct 1. The exact
dependence of F1( ) in terms of harmonic polylogarithms
is known [64, 93, 94]. We use the results expanded in , which are completely su cient
for practical purposes because the corrections are small and the expansion in
converges very quickly.
Quark vector-current matching coe cient
The nite terms of CqVq to three loops can be read o from ref. [75],
CqVq 1 = CF ( 8 + 2) ;
CqVq 2 = CF CA
CqVq 3 = CF CA2
The last term is the three-loop contribution from diagrams where the initial-state quarks
refer to ref. [75] for details of NF;V . Since the full Drell-Yan xed-order cross section is only
available to NNLO, the three-loop coe cient never enters our resummed predictions. For
The explicit three-loop results for CqVq were also extracted in ref. [131] from the
threeloop form factor in ref. [74]. We veri ed that the above results agree with the numerical
Quark scalar-current matching coe cient
As far as we are aware, a result for CqSq has not been given explicitly in the literature so far.
The quark scalar form factor F in QCD has been computed to O( s3) in ref. [103], from
which we can extract CqSq. A slight di culty arises as F is only given at timelike kinematics
from the bare form factor F given in ref. [103] and perform its UV-renormalization at an
arbitrary MS renormalization point . We explicitly checked that the ratio of the timelike
to spacelike form factor is IR- nite as required. We then proceed by subtracting the IR
poles in F in MS by a multiplicative renormalization factor,
CqSq( ) =
of the bare Wilson coe cient, so eq. (A.20) is equivalent to the MS renormalization of CqSq.
Here we have made explicit that the renormalized quark Yukawa coupling y( ) is
excluded from CqSq.
We have also veri ed that the obtained renormalization factor Z
reproduces the correct anomalous dimension for CqSq, i.e. that it satis es
eq. (A.3) (with cusp
= 2 Cq
order-by-order in
s, which provides a strong check on the pole structure of F .
Equivalently, we also checked that the full result for CqSq( ) obtained from eq. (A.20) agrees with
m). For the nonlogarithmic constant terms
of CqSq we obtain
CqSq 1 = CF ( 2 + 2) ;
CqSq 2 = CF CF 6 + 14 2
CqSq 3 = CF CA2
654 3 + 424 5 +
Renormalization group evolution
For reference we collect the explicit expressions needed for the RG evolution of the hard
functions. The evolution factor UH is de ned by eq. (2.5). It is given explicitly by
UH (Q; 0; ) = exp 2 i ( 0; ) ln
K ( 0; ) =
cusp( s) is the relevant quark or gluon cusp anomalous dimension and ( s) the
appropriate noncusp anomalous dimension of the relevant hard matching coe cient.
Their explicit expressions at NNLL are
K ( 0; ) =
( 0; ) =
K ( 0; ) =
r + ln r) +
where r =
; (A.26)
the expressions in eq. (A.25) are truncated accordingly. The relevant expressions at N3LL,
used for the inclusive gg ! H cross sections, can be found in ref. [131].
Fixed-order estimates from resummed timelike logarithms
It is instructive to compare explicitly the xed-order contributions induced purely by the
timelike logarithms in the form factor with the full xed-order result to assess whether
they are indeed a dominant part of the perturbative corrections. However, we also stress
that this is not a good way for judging the usefulness of the resummation as a whole, since
it does not capture the full resummed result and in particular does not take into account
the improvements in perturbative convergence and uncertainties.
In ref. [10], such an analysis was carried out for gg ! H for the coe cient C of the
z) term in the partonic cross section. This coe cient is fully determined by the
and factorizes as in eq. (2.6) into the product of the gluon form factor and purely soft
contributions. Ref. [10] found that C is poorly predicted from the timelike logarithms
alone. However, this can be very misleading since the (1
z) coe cient is strongly scheme
dependent and not a physical quantity. Rather, this type of analysis should be carried out
at the level of the cross section, which is a scheme-independent physical observable. To
illustrate this, we repeat the analysis of ref. [10] and compare it to a di erent convention
for the soft function, as well as considering the hadronic K factor.
against jCtj2.) The relevant hard function is hence Hgg in eq. (3.2). Given the exact hard
function to O( sn), which is fully included in the NnLL0 resummed result, the O( sn+1)
contribution predicted by and included in the resummation is given by
Ha(npp+r1) = 2
RehC(n+1)(L =
= H(n+1)
Cn+1=0
+ cross terms
where all logarithmic terms in the O( sn+1) Wilson coe cient C(n+1) are predicted by the
RGE [see eq. (A.3)], and the only missing ingredient compared to the full result for H(n+1)
is the nonlogarithmic O( sn+1) term Cn+1. Denoting the soft function contribution to the
z) coe cient by S = 1 + S(1) +
, the corresponding approximate result for C at
C(na+pp1r) = Ha(npp+r1) + H(n) S(1) +
The result for S to O( s3) can be obtained from ref. [13], which writes the soft function
in terms of the standard (plus) distributions
Applying eq. (B.2) at each successive order, we nd
LO+LL0':
C = 1 + 14:80
NLO+NLL0':
C = 1 + 9:87
NNLO+NNLL0':
C = 1 + 9:87
C = 1 + 9:87
The last coe cients in the rst three lines are those predicted by the resummation beyond
the included xed-order accuracy. In the last line the N3LO result is given for comparison.
(These numbers agree with those given in ref. [10] except for C(3a)ppr, where they
554:79 rather than our
644:26. We were unable to resolve this numerical di erence, but
it is immaterial for the present discussion.)
From eq. (B.4) it looks like the resummation does a poor job at approximating the
higher xed-order result, which would be in stark contrast to what we have seen in
section 3. The resolution lies in the cross terms with the soft function in eq. (B.2). The
S(n) coe cients depend on the (in principle arbitrary) boundary condition chosen for the
plus distributions. In other words, the distinction between the soft-function cross terms
included in eq. (B.4) and those between H(n) and the remaining soft-function terms is
arbitrary. To illustrate this, we can instead write the soft function in terms of the di erent
set of plus distributions
used e.g. in refs. [11, 14], for which we get a di erent S~ coe cient6 and a corresponding
di erent C~ coe cient of (1
LO+LL0':
NLO+NLL0':
NNLO+NNLL0':
C~ = 1 + 14:80
C~ = 1 + 19:74
C~ = 1 + 19:74
C~ = 1 + 19:74
In this convention, the resummation approximates the higher xed-order terms of C~ very
well. The strong scheme dependence of the (1 z) coe cient is obvious from the completely
di erent coe cients in the exact results for C and C~ in eqs. (B.4) and (B.6).
Instead, it is much more meaningful to consider physical quantities such as the inclusive
hadronic cross section. The approximate result analogous to eq. (B.2) for the total K factor
is given by
Ka(npp+r1) = Ha(npp+r1) + H(n) R(1) +
where the S coe cient is now replaced by the full perturbative remainder R de ned in
eq. (2.7). The scheme dependence in this case is how the nonlogarithmic constant terms are
split between H and R, which as discussed in section 2.1 cancels in their product and by
construction does not enter Ha(npp+r1). The analogous xed-order expansions of the resummed
results for the K factor are given by
LO+LL0':
Kgg!X = 1 + 14:80
NLO+NLL0':
Kgg!X = 1 + 30:52
NNLO+NNLL0':
Kgg!X = 1 + 30:52
Kgg!X = 1 + 30:52
Evidently, the resummed results approximate the higher xed-order terms in the K
factor very well, except at NLO, where H(1) and R(1) each contribute about half of the full
K factor. This is precisely equivalent to our discussion in section 3.1 that the large
corrections to the K factor are primarily driven by the timelike logarithms in H, while the
imX ), are much smaller.
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