#### Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO

Received: July
Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO
Massimiliano Grazzini 0 1 3 4 5
Stefan Kallweit 0 1 2 4 5
Dirk Rathlev 0 1 3 4 5
Marius Wiesemann 0 1 3 4 5
Open Access, c The Authors.
0 Johannes Gutenberg University , D-55099 Mainz , Germany
1 CH-8057 Zu ̈rich , Switzerland
2 PRISMA Cluster of Excellence, Institute of Physics
3 Physik-Institut, Universita ̈t Zu ̈rich
4 [21] T. Binoth , T. Gleisberg, S. Karg, N. Kauer and G. Sanguinetti, NLO QCD corrections to
5 [56] R. Frederix and M. Grazzini , Higher-order QCD effects in the h
We consider the transverse-momentum (pT ) distribution of ZZ and W +W − boson pairs produced in hadron collisions. At small pT , the logarithmically enhanced contributions due to multiple soft-gluon emission are resummed to all orders in QCD perturbation theory. At intermediate and large values of pT , we consistently combine resummation with the known fixed-order results. We exploit the most advanced perturbative information that is available at present: next-to-next-to-leading logarithmic resummation combined with the next-to-next-to-leading fixed-order calculation. After integration over pT , we recover the known next-to-next-to-leading order result for the inclusive cross section. We present numerical results at the LHC, together with an estimate of the corresponding uncertainties. We also study the rapidity dependence of the pT spectrum and we consider pT efficiencies at different orders of resummed and fixed-order perturbation theory.
QCD Phenomenology; Hadronic Colliders
1 Introduction 2 3 Results
Transverse-momentum resummation for vector-boson pair production
Choice of the central resummation scale
Inclusive transverse-momentum distribution
Rapidity dependence of the transverse-momentum distribution
The W +W − cross section and pT -veto efficiencies
Introduction
Run 1 of the Large Hadron Collider (LHC) has been a great success for the Standard
Model (SM). The collected data are in good agreement with the theoretical predictions so
far and led to the discovery [1, 2] of a resonance at a mass of 125 GeV, which appears to be
fully consistent with the SM Higgs boson. Among the most important reactions at hadron
colliders is the production of vector-boson pairs. This class of processes gives access to
the vector-boson trilinear couplings which may be modified in a large set of Beyond the
Standard Model (BSM) theories. Even small deviations in both the production rate and
the shape of distributions could be a signal of new physics. Anomalous couplings related
to vector-boson pair production have been constrained first by LEP2, and later by the
Tevatron for larger invariant masses. ATLAS and CMS will continue to tighten the bounds
on anomalous couplings, especially with increasing sensitivity during Run 2 of the LHC.1
On the other hand, vector-boson pair production constitutes an irreducible background
to new-physics searches as well as Higgs studies. Particularly important are the off-shell
the high-mass tail used to extract the width of the Higgs boson [4–6]. Furthermore, Higgs
rejection through specific categories based on the transverse momenta of final-state
particles, such as the classification into jet bins or in the Higgs transverse momentum and
related variables. An accurate modelling of the respective observables for both signal and
backgrounds is crucial for such analyses.
The first precise predictions for ZZ production at hadron colliders in the SM were
obtained at the next-to-leading order (NLO) already more than 20 years ago for stable Z
bosons [7, 8], and the leptonic decays were added in ref. [9]. The full spin correlations
1See ref. [3] and references therein.
and off-shell effects at the NLO were first included in refs. [10, 11] using the
corresponding one-loop helicity amplitudes [12]. An important loop-induced contribution proceeds
through gluon fusion; it is enhanced by the gluon densities and was first computed for
on-shell Z bosons in refs. [13, 14], while leptonic decays were included later [15–17]. All
the contributions to ZZ production discussed so far are implemented in the numerical
program MCFM [18]. Electroweak (EW) corrections were evaluated in ref. [19, 20], while
on-shell ZZ+jet production is known through NLO QCD [21, 22]. Recently, the inclusive
cross section for the production of ZZ at the next-to-next-to-leading order (NNLO) has
been presented [23].
neutrinos prohibits a reconstruction of mass peaks. Therefore, a precise understanding of
both the signal and the background is required.
gluon-fusion component [13, 26] have been known for decades. Also in this case, spin and
off-shell effects were included in the NLO prediction [10, 11] after the relevant one-loop
helicity amplitudes had been computed [12]. Leptonic decays for the loop-induced
gluonfusion contribution were considered in refs. [27, 28]. More recently, also interference effects
with the Higgs production mode through gluon fusion were determined [29]. Analogously
plemented in MCFM [18]. Furthermore, EW corrections have been evaluated [20, 30, 31].
ation with up to one jet at NLO have been presented in ref. [35]. Recently, the first NNLO
Due to the very recent computation of the qq¯ → V V 0 helicity amplitudes at two-loop
order [37, 38], the inclusion of the off-shell effects and the leptonic decays in the NNLO
cross section is expected in the near future. In the meanwhile, also the calculation of
gg → V V 0 helicity amplitudes has been performed at two-loop order [39, 40]. This renders
the evaluation of NLO QCD corrections to the gluon-fusion channel feasible.
The production of ZZ pairs in hadron collisions has been measured extensively at
section has also been measured already at the Tevatron, see e.g. ref. [48], and at the LHC
both at 7 TeV [49, 50] and 8 TeV [46, 51, 52]. The ATLAS collaboration recently reported
an excess [51] with respect to the SM prediction, which has drawn a lot of attention on
narios [53]. In the meanwhile, the excess has been alleviated to a significant degree by
the recent NNLO computation [36]. The more recent measurement by the CMS
collaboration [52] is in good agreement with the NNLO prediction.
important differential observables for these processes. The pT spectrum has already been
measured in the case of ZZ production [47] at the LHC. Transverse-momentum
resummations the resummed computation is essentially performed up to next-to-leading logarithmic
(NLL) accuracy (next-to-next-to-leading logarithmic (NNLL) effects in the Sudakov
exponent are considered in refs. [55, 57, 58]) and matched to the fixed-order result up to the
In this paper we consider transverse-momentum resummation for the production of
result valid at large pT . Although our focus is on the inclusive pT spectrum of the ZZ
the vector bosons and allows us to include their EW decays, once the helicity amplitudes
are implemented.2
The pT -resummation formalism of ref. [59] is closely related to the
subtraction method of ref. [63], which was used to compute the NNLO cross section for
and remove all contributions with final-state bottom quarks from our computation of the
tt¯ and W t production. The difference to the prediction in the five-flavour scheme (5FS),
where such terms have to be consistently subtracted, has been shown to be small for the
NNLO inclusive cross section [36]. Furthermore, we neglect the loop-induced gluon-fusion
We note that in the CMS measurement reported in ref. [52] an approximate NNLL
prediction [58] of the pT spectrum has been used to correct the spectrum from the Monte
Carlo simulation. Along these lines, the computation reported in this paper will be useful
in order to validate the predictions obtained from Monte Carlo simulations for both ZZ
of ref. [59].
The manuscript is organized as follows. In section 2 we review the
transversemomentum resummation formalism applied to vector-boson pair production. In section 3
we report our numerical results, starting with remarks on the choice of the resummation
scale in section 3.1. In section 3.2 we present our numerical predictions for the inclusive
pT spectrum and study the ensuing uncertainties. In section 3.3 we analyse the behaviour
of the spectrum at different rapidities of the vector-boson pair. In section 3.4 we
investigate pT efficiencies at different orders in resummed and fixed-order perturbation theory.
In section 4 we summarize our results.
Transverse-momentum resummation for vector-boson pair production
In this section we recall the main points of the transverse-momentum resummation
formalism we use in this paper. For a more detailed discussion the reader is referred to
refs. [59, 64, 65].
2The analogous computations for Higgs, single vector-boson production, and diphoton production are
presented in refs. [60], [61] and [62], respectively.
We consider the inclusive hard-scattering process
h1(P1) + h2(P2) → V (p3) + V 0(p4) + X ,
where the collision of the two hadrons h1 and h2 with momenta P1 and P2 produces the
two vector bosons of momenta p3 and p4. In the center-of-mass frame the momentum of
The kinematics of the vector bosons is fully determined by the vector-boson pair
mopendent variables that specify the angular distribution of the vector bosons with respect
According to the QCD factorization theorem, the differential cross section can be
writ(y, pT , M, s) =
The rapidity yˆ and the center-of-mass energy sˆ of the partonic scattering process are related
to the corresponding hadronic variables y and s by
yˆ = y − 2
sˆ = x1x2s .
When the transverse momentum pT of the vector-boson pair is of the same order as the
invariant mass M , the QCD perturbative expansion is controlled by a single expansion
M the convergence
of the perturbative expansion is spoiled by the presence of large logarithmic terms of the
The resummation is performed at the level of the partonic cross section, which is
dM 2dp2T dyˆ
dM 2dp2T dyˆ
dM 2dp2T dyˆ
The first term on the right-hand side of eq. (2.4) contains all the logarithmically-enhanced
contributions at small pT and has to be evaluated by resumming them to all orders. The
second term is instead free of such contributions and can thus be evaluated at fixed order
in perturbation theory.
running coupling in the MS renormalization scheme.
Resummation is based on the factorization of soft and collinear radiation and is
viable in the impact-parameter (b) space, where the kinematical constraint of momentum
conservation and the factorization of the phase space can be consistently taken into
account [66–68]. Using the Bessel transformation between the conjugate variables pT and b,
the resummed component is expressed as [59, 68]
dM 2dp2T dyˆ
M 2 Z ∞
where J0(x) is the 0-order Bessel function. In the case of fully inclusive pT resummation, the
rapidity dependence is integrated out in eq. (2.5). In that case it is convenient to consider
rapidity dependence in the resummed cross section we take the ‘double’ (N1, N2) Mellin
V V 0
dz1 z1N1−1
dz2 z2N2−1W
V V 0 (b, yˆ, M, sˆ; αS, μ2R, μ2F ) ,
and organize the structure of WV V 0 in the following exponential form [64],
W(N1,N2)(b, M ; αS, μ2R, μ2F ) = H(N1,N2) M ; αS, M 2/μ2R, M 2/μ2F , M 2/Q2
V V 0 V V 0
where we have defined the logarithmic expansion parameter L as
L = ln
and b0 = 2e−γE (γE = 0.5772 . . . is the Euler number).
The scale Q appearing in
eqs. (2.7), (2.8), named resummation scale in ref. [59], parametrizes the arbitrariness in the
resummation procedure, and has to be chosen of the order of the hard scale M . Variations
of Q around its reference value can be exploited to estimate the size of yet uncalculated
V V 0
parameter b, and therefore includes all the perturbative terms that behave as constants as
b → ∞. It can thus be expanded in powers of αS = αS(μ2R):
H(N1,N2)(M, αS; M 2/μ2R, M 2/μ2F , M 2/Q2)
V V 0
includes the complete dependence on b and, in particular, it contains all the terms that
(2)
where the term L g(1) collects the LL contributions, the function g(N1,N2) includes the NLL
contributions, g((N3)1,N2) controls the NNLL terms and so forth.
The resummation of the large logarithmic terms carried out in eq. (2.7), after
transforming back to pT space, allows us to obtain a well behaved transverse-momentum
spectrum as pT → 0. However, the logarithmic expansion parameter L in eq. (2.8) is divergent
as b → 0. This implies that the resummation produces higher-order contributions also in
the high-pT region, which is conjugated to b → 0 after Fourier transformation. In this
region the fixed-order cross section is perfectly viable and any resummation effect is
necessarily artificial. To reduce the impact of such contributions, the logarithmic variable L
is replaced by [59]
L˜ ≡ ln
The variables L and L˜ are equivalent when Qb
1 but they have a very different behaviour
as b → 0. When Qb
1, L˜ → 0 and G(N1,N2) → 1. Moreover, since the behaviour of the
allows us to enforce a unitarity constraint such that the fixed-order prediction is recovered
upon integration over pT .
A well known property of the formalism of ref. [59] is that the process dependence
(as well as the factorization scale and scheme dependence) is fully encoded in the hard
function HV V 0 . In other words, the functions g(i) are universal: they depend only on the
channel in which the process occurs at Born level (qq¯ annihilation in the case of
vectorof the universal perturbative coefficients A(q1), A(q2), A(q3), B˜q(1,N) , B˜q(2,N) . In particular, the LL
function g(1) depends on the coefficient A(q1), the NLL function g(N1,N2) also depends on A(q2)
(2)
and B˜q(1) [69] and the NNLL function g(N1,N2) also depends on A(q3) [70] and B˜q(2,N) [71–73].
(3)
The hard coefficients H
coefficients H
V V 0 depend on the process we want to consider. The first order
V V 0,(1) are known since long time [72, 73]: they can be obtained from the
coefficient H
V V 0,(2).
one-loop scattering amplitudes qq¯ → V V 0 by using a process independent relation. By
(2) for single Higgs [74] and vector-boson [75] production,
exploiting the expressions of H
amplitude for qq¯ → V V 0 is the only process-dependent information needed to obtain the
We now turn to the finite component of the transverse-momentum spectrum, i.e. the
logarithmic terms in the small-pT region, it can be evaluated by truncating the
perturbative series at a given fixed order. In practice, the finite component is computed starting
from the customary fixed-order (f.o.) perturbative truncation of the partonic cross
section and subtracting the expansion of the resummed cross section in eq. (2.5) at the same
perturbative order:
dM 2dp2T dyˆ
dM 2dp2T dyˆ
dM 2dp2T dyˆ
At least formally, this matching procedure between resummed and finite contributions
guarantees to achieve a uniform theoretical accuracy over the entire range of transverse
momenta. At large values of pT , the resummation procedure cannot improve the
fixedorder result, and the resummation (and matching) procedure is eventually superseded by
the customary fixed-order calculations.
(2)
In summary, the inclusion of the functions g(1), g(N1,N2), H
V V 0,(1) in the resummed
us to perform the resummation at NLL+NLO accuracy. This is the theoretical accuracy of
the calculations of refs. [55, 56]. Including also the functions g((N3)1,N2) and H
the recently computed two-loop amplitudes for qq¯ → V V 0, and the process independent
relation of ref. [65], we are now able to present the complete result for the
transverse
V V 0,(2), together
momentum distribution of the vector-boson pair up to NNLL+NNLO accuracy. We point
out that the NNLL+NNLO (NLL+NLO) result includes the full NNLO (NLO) perturbative
contribution in the entire pT range. In particular, the NNLO (NLO) result for the double
differential cross section at NNLL+NNLO (NLL+NLO) accuracy.
We conclude this section by adding a few comments on the way in which our calculation
is actually performed. The practical implementation of eq. (2.4) is done in the numerical
program Matrix,4 which is an extension of the numerical program applied in the NNLO
calculations of refs. [23, 36, 76, 77] and based on a combination of the qT -subtraction
formalism [63] with the Munich5 code. Since already performed within the qT -subtraction
formalism, the extension of these calculations to compute the resummed cross section is
conceptually quite straightforward, and is obtained by replacing the hard-collinear terms
in the fixed-order computation by the proper all-order resummation formula of eq. (2.5).
This procedure is the same that was applied to perform the NLL+NLO calculations for
production of ref. [60].
To obtain the numerical results presented here, the resummed component of eq. (2.5)
is evaluated with an extension of the numerical program used for the calculation of Higgs
production [60], based on the earlier computations of refs. [59, 64]. The hard-collinear
4Matrix is the abbreviation of “Munich Automates qT subtraction and Resummation to Integrate
X-sections”, by M. Grazzini, S. Kallweit, D. Rathlev, M. Wiesemann. In preparation.
5Munich is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision”—an automated
parton level NLO generator by S. Kallweit. In preparation.
coefficients are obtained by exploiting the implementation of the corresponding virtual
respectively, and the knowledge of the collinear coefficients relevant to quark-initiated
processes [75].6
The finite component of eq. (2.12) is obtained from an NLO calculation of V V 0+jet,
computed with the Munich code, which provides a fully automated implementation of the
Catani-Seymour dipole formalism [80, 81] as well as an interface to the one-loop generator
OpenLoops [82] to obtain all required (spin and color-correlated) tree-level and one-loop
amplitudes. For the numerically stable evaluation of tensor integrals we rely on the
Collier library [83], which is based on the Denner-Dittmaier reduction techniques [84, 85]
and the scalar integrals of [86]. To deal with problematic phase-space points, OpenLoops
provides a rescue system using the quadruple-precision implementation of the OPP method
in CutTools [87], involving scalar integrals from OneLOop [88].
In this section we present our results for the resummed transverse-momentum distributions
NLL+NLO, and discuss the corresponding theoretical uncertainties. Additionally, we also
study the rapidity dependence of the pT cross section as well as the pT -veto efficiency.
split off all contributions related to bottom-quark final states in order to remove the tt¯ and
W t contamination from our computation. This is straightforward in the 4FS, because the
W +W −b¯b process is separately finite.
We consider proton-proton collisions at √
central resummation scale Q0 is discussed in the next subsection.
Choice of the central resummation scale
As discussed in section 2, the resummation scale Q is the scale entering the large logarithmic
terms we are resumming (see eq. (2.8)), and it plays the role of the scale up to which
resummation is effective. In on-shell Higgs [59] and vector-boson [90] production, the scale
of the scale lead to a worse matching at high pT . The natural extension of this choice for
6We note that an independent computation of these coefficients in the framework of Soft Collinear
Effective Theory has been presented in refs. [78, 79].
s = 8 TeV. The central values of the
factorM(W+W-) [GeV]
and NNLO (blue, solid).
the hardness of the process. This is indeed the choice that was adopted in the calculations
of refs. [55, 56].
peaked in the threshold region, and that it quickly decreases as MW W increases. As a
We can compare the transverse-momentum distributions obtained with a dynamical
bands are obtained by varying the resummation scale around the central value by a factor of
two. Considering the ratio of the central curves for pT . 250 GeV, the differences between
a fixed and a dynamical scale are extremely small and remain at the 1-2% level over the
whole range. In this region of transverse momenta the uncertainty bands obtained with the
uncertainties, it appears to be the more conservative choice. Therefore, we can conclude
that either choice of the resummation scale is perfectly valid and indeed consistent with
each other as expected from the discussion of the invariant mass distribution.
Looking further at the comparison of the high-pT tails in figure 2 (pT & 250 GeV),
we observe a very well known feature [59, 60, 90–92] of the applied matching procedure,
namely the fact that for large values of the resummation scale the fixed-order cross section
(black dotted curve) is not recovered in the tail of the distribution. It is important to
recall that transverse-momentum resummation is supposed to improve the perturbative
expansion in the low-pT region. At large pT , any large dependence on the resummation
scale is necessarily artificial and an unwanted remnant of the matching procedure. This
NNLL+NNLO dyn. Qres
NNLL+NNLO fixed Qres
NNLO
are obtained by variation of the resummation scales in the numerator by a factor of two around the
central scale. For reference, we show the fixed-order NNLO curve with the same normalization.
figure 2. With this choice, in fact, the resummed result loses predictivity, as its uncertainty
of the resummed prediction.
scale in what follows.
Inclusive transverse-momentum distribution
We now present our resummed predictions for the inclusive transverse-momentum spectrum
of the vector-boson pair and compare them with the corresponding fixed-order results.
the ZZ case. For completeness, we provide the corresponding reference prediction with
uncertainties for ZZ below.
Before presenting our resummed predictions, we recall the well known fixed-order
retogether with their perturbative uncertainties. The uncertainty bands are obtained by
varyThe lower inset shows the same results normalized to the central NLO curve. The NNLO
region where pT . 300 GeV. This implies that, in this region of transverse momenta, the
ito 1
a
as described in the text. Lower inset: results normalized to the NLO prediction at central values of
size of the band obtained through scale variations at NLO definitely underestimates the
theoretical uncertainty.
We now move on to the resummed results. In figure 4 (a) the NLL+NLO spectrum is
compared to the fixed-order NLO result and to the finite component of the resummed cross
section (see eq. (2.4)) in the region between 0 and 80 GeV. As expected, the NLO diverges
component contributes less than 1% in the peak region, where the result is dominated
NLL+NLO result normalized to NLO. In figure 4 (b) the region between 80 and 400 GeV is
displayed. We see that even at large values of pT the NLL+NLO resummed result does not
match very well the fixed-order NLO result, with a difference of about 5%.
The analogous results at NNLL+NNLO are shown in figure 5. The NNLO has an
unphysical (divergent) behaviour as pT → 0, whereas the resummed spectrum is well
behaved, with a slightly harder peak with respect to the NLL+NLO. The finite component
paring the right panels of figure 4 and figure 5, we see that the quality of the matching
at high pT is significantly improved when going from NLL+NLO to NNLL+NNLO, and we
find that this behaviour is indeed preserved up to very high transverse momenta. The
NNLL+NNLO result thus gives a prediction with uniform accuracy from small to very
large transverse momenta and, in fact, provides a sufficiently large region where a hard
switching to the fixed-order result is feasible. We point out that, thanks to our unitarity
constraint, both at NLL+NLO and at NNLL+NNLO the integral of the resummed spectrum
region and (b) at high transverse momenta. The NLL+NLO result (red, dashed) is compared to the
fixed-order NLO prediction (grey, dash-dotted) and to the finite component of eq. (2.4) (magenta,
dash-double dotted). The lower insets show the NLL+NLO result normalized to NLO.
ito 1
a
t
io 1
t
low-pT region and (b) at high transverse momenta.
The NNLL+NNLO result (red, dashed) is
compared to the fixed-order NNLO prediction (grey, dash-dotted) and to the finite component of
eq. (2.4) (magenta, dash-double dotted). The lower insets show the NNLL+NNLO result normalized
is in excellent agreement with the respective total cross sections; the differences are at the
few-permille level.
We now turn to the scale uncertainties of our resummed results. We start our discussion
by separately considering factorization and renormalization scale variations. In figure 6 we
compare the NLL+NLO (red, dashed) and NNLL+NNLO (blue, solid) predictions with their
cases, the bands are obtained by varying the factorization (renormalization) scale by a
N
to 1
N
to 1
toN 1
N
to 1
NNLL+NNLO (blue, solid); thick lines: central scale choices; bands: uncertainty due to (left)
factor of two around its central value, while keeping the other scales at their default values.
First of all, we notice that when going from NLL+NLO to NNLL+NNLO the pT spectrum
becomes harder. Comparing with the results of ref. [55], where the NNLL resummation
spectrum is a combined effect of both features, i.e. NNLL resummation and NNLO matching
at high pT .
We note that neither in the case of the factorization scale, nor in the case of the
renormalization scale, the NLL+NLO and NNLL+NNLO bands overlap. Actually, in the
case of the factorization scale, there is no reduction in scale dependence when going from
NLL+NLO to NNLL+NNLO, and the uncertainty slightly increases with the perturbative
order, even if it is always well below 10%, except at very low pT . The renormalization
scale dependence instead exhibits the expected reduction when going from NLL+NLO to
NNLL+NNLO.
In figure 7 we present our resummed predictions with uncertainty bands obtained from
find that at NNLL+NNLO the resummation scale uncertainty is reduced roughly by a factor
of two in the region of transverse momenta considered in the figure.
respectively, with an estimate of their full perturbative uncertainty. In order to obtain a
which is the same as applied in figure 3 and figure 7 (left), has the purpose of avoiding
large logarithmic contributions from the evolution of parton densities. Analogously, the
between the NNLL+NNLO and NLL+NLO predictions is driven by the NNLO effects, which
increase the NLO result by about 30%.
For ZZ production the uncertainties have essentially the same pattern in the
smallis entirely driven by the resummation-scale dependence. As previously pointed out, this
behaviour is not particularly worrying since, in the large-pT region, the resummed results
should be replaced by the corresponding fixed-order prediction. Also in the ZZ case the
large enhancement of the NNLL+NNLO distribution in the high-pT tail stems from the
fixed-order cross section.
Rapidity dependence of the transverse-momentum distribution
inclusive in the kinematics of the vector-boson pair. Our numerical program, however,
allows us to compute arbitrary observables that are differential with respect to the V V 0
N
to 1
N
to 1
N
to 1
N
to 1
and Q variations obtained as described in the text; thin lines: borders of bands. (b) detail of the
low-pT region.
In the following we study the behaviour of the transverse-momentum spectrum in
different rapidity regions of the vector-boson pair. In figure 10 we study the shape of
the NNLL+NNLO transverse-momentum distribution, i.e. normalized such that its integral
yields one, for |y| < 0.5 (red, solid), 0.5 < |y| < 1 (blue, dashed), 1 < |y| < 2 (black,
dotted), 2 < |y| < 3 (magenta, dash-dotted) and 3 < |y| (orange, dash-double dotted).
The right panel shows the same results normalized to the fully inclusive distribution. We
clearly see that the pT shapes become softer as the rapidity increases. In the central region
(|y| < 2) the distributions are still quite insensitive to the specific value of the rapidity and
only slightly harder than the inclusive spectrum. In the forward rapidity region, on the
other hand, the shapes become increasingly softer.
T
/dp1.5
t)
T
/dp0.5
t)
1 < |y| < 2 (black, dotted), 2 < |y| < 3 (magenta, dash-dotted), 3 < |y| (orange, double-dash
dotted); and (b) the shape-ratio with respect to the inclusive result.
The observed pattern can be understood in the following way: rapidity and transverse
momentum are two not completely independent phase-space variables. Indeed, they affect
their mutual upper integration bounds. At higher rapidities the kinematically allowed
range of transverse momenta is reduced: this squeezes the pT spectrum which consequently
becomes softer. This effect has been observed also in previous studies in the case of Higgs
boson production [64].
The W +W − cross section and pT -veto efficiencies
signature appears in many new physics scenarios [53]. The inclusion of the recently
computed NNLO corrections [36] considerably reduces the significance of the excess. However,
particular attention must be payed to the modelling of the jet veto [52, 58, 93] when
extrapolating from the fiducial region to obtain the inclusive cross section. Effects of jet veto
resummation have been considered in refs. [94, 95], though still matching to the fixed-order
In this paper we are dealing with transverse-momentum spectra, and we perform a
resummation on a different variable with respect to the jet pT . However, the vector-boson
resummed and fixed-order perturbation theory. We define the pT -veto efficiency as
In figure 11 we show (pvTeto) at the NNLL+NNLO (blue, solid), approximate NNLL+NLO
(magenta, dash-double dotted), NLL+NLO (red, dashed), NNLO (black, dotted) and NLO
(grey, dash-dotted). The lower inset shows the same curves normalized to our reference
o
it 1
a
pT veto [GeV]
NLL+NLO (red, dashed), NNLL+NNLO (blue, solid), NLO (grey, dash-dotted), NNLO (black,
dotted), approximate NNLL+NLO (magenta, dash-double dotted); thick lines: central scale choices;
bands: uncertainty due to combined scale variations; thin lines: borders of bands.
prediction at NNLL+NNLO. Our approximate NNLL+NLO is obtained by simply adding
the g(3) function in the Sudakov exponent in eq. (2.10) at NLL+NLO, and corresponds to
the approximation considered in refs. [55, 58].
For reference, the corresponding numerical values of the efficiencies are given in table 1
mation, factorization and renormalization scales as in figure 8. The first thing we observe
is that the NLO result appears to be well above the others and cannot be really considered
a reliable prediction for the efficiency. This is because it is essentially a LO prediction at
the fixed-order NLO and NNLO predictions diverge and cannot be trusted. Comparing
further the fixed-order results among each other and the resummed results among each
other, we observe that higher-order corrections in fixed-order and resummed perturbation
theory reduce the pT -veto efficiency.
Both effects can be easily understood in the light of the results presented up to now.
As seen in figure 3, the inclusion of the NNLO corrections make the pT distribution harder.
Furthermore, resummation effects generally harden the spectrum. A qualitatively similar
result is obtained when going from NLL+NLO to NNLL+NNLO (see figure 8).
It is interesting to compare the approximated NNLL+NLO result with the NNLO and
NNLL+NNLO predictions. For values of pvTeto
∼ 25 − 30 GeV we see that the approximated
result is in between the NNLO one and our best NNLL+NNLO prediction. This means
−19%
−12%
−9.2%
−7.3%
−5.9%
−4.9%
−4.2%
−3.6%
pveto [GeV]
approx. NNLL+NLO
−9.6%
−6.2%
−4.8%
−3.9%
−3.4%
−3.0%
−2.7%
−2.5%
that the effect of NNLL resummation obtained by the inclusion of the g(3) function in the
Sudakov exponent in eq. (2.10) is quantitatively important. Nonetheless, the efficiency
obtained within this approximation is still about 5% higher than the NNLL+NNLO
prediction. We also notice that in this region of pvTeto, the NNLO and NLL+NLO results differ by
less than 1%.
Comparing the NNLL+NNLO and NLL+NLO results, we find that they are compatible
within the corresponding uncertainties.
out by the CMS collaboration [52]. The result shows good agreement with the NNLO
prediction of ref. [36]. The corresponding analysis, however, is based on a reweighting procedure
Pythia6 [97] were reweighted by using the calculation of ref. [58], which corresponds to
our NNLL+NLO approximation, and includes neither the second-order hard-collinear
coef
W W,(2) in eq. (2.9), nor the NNLO matching. The results in figure 11 show that the
NNLL+NNLO pT -veto efficiency is lower than the efficiency obtained with the approximated
NNLL+NLO calculation. As a consequence, a reweighting to the full NNLL+NNLO
predicIn this paper we have studied the transverse-momentum distribution of vector-boson pairs
in hadronic collisions.
We presented a computation of the pT spectrum in which the
logarithmically enhanced contributions at small pT are resummed up to NNLL accuracy
together with a study of their perturbative uncertainties.
We found that, up to relatively large transverse momenta, when scale variations are
momenta the fixed-order result in the tail of the distribution is nicely recovered with
the fixed-scale choice. Our new NNLL+NNLO results significantly reduce the theoretical
uncertainties obtained through scale variations compared to lower orders in both the peak
region and the tail of the distribution.
We have also studied the rapidity dependence of the resummed transverse-momentum
distribution. The rapidity dependence at NNLL+NNLO is quite flat in the central region
(|y| . 2), but signals a substantially softer spectrum in the forward region. Due to
phasespace suppression, the effect on the inclusive transverse-momentum distribution is very
moderate though.
Finally, we have studied the pT -veto efficiency at different orders in resummed and
fixed-order perturbation theory. Both NNLL resummation and the NNLO effects turned
out to be important to obtain an accurate prediction for this quantity. We observed that
to the approximate NNLL+NLO calculation used in the CMS analysis of ref. [52]. This
result suggests that our NNLL+NNLO predictions will be useful to validate the
transversemomentum spectra obtained from Monte Carlo event generators, similarly to what was
done for the NNLL+NNLO calculation of ref. [59] in the case of Higgs boson production.
In this paper we considered the pT spectrum of stable vector-boson pairs. Exploiting
the two-loop helicity amplitudes for qq¯ → V V 0 → 4 leptons [37, 38] will allow us to extend
the calculation to include the leptonic decay of the vector bosons and off-shell effects. The
computation of the transverse-momentum spectrum with realistic experimental cuts will
then become possible.
We would like to thank Andreas von Manteuffel and Lorenzo Tancredi for providing us with
their private code to evaluate the helicity-averaged on-shell V V 0 amplitudes in the
equalmass case. We would like to thank Giancarlo Ferrera for comments on the manuscript.
This research was supported in part by the Swiss National Science Foundation (SNF)
under contracts CRSII2-141847, 200021-156585 and by the Research Executive Agency
(REA) of the European Union under the Grant Agreement number PITN-GA-2012-316704
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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