Resummed Cross Section for Jet Production at Hadron Colliders
BNL-NT-07/17
hep-ph/yymmnnn
Resummed Cross Section for Jet Production
at Hadron Colliders
Daniel de Florian
Departamento de F́ısica, FCEYN, Universidad de Buenos Aires,
(1428) Pabellón 1 Ciudad Universitaria, Capital Federal, Argentina
Werner Vogelsang
BNL Nuclear Theory, Brookhaven National Laboratory, Upton, NY 11973, USA
Abstract
We study the resummation of large logarithmic perturbative corrections to the single-
inclusive jet cross section at hadron colliders. The corrections we address arise near the
threshold for the partonic reaction, when the incoming partons have just enough energy
to produce the high-transverse-momentum final state. The structure of the resulting
logarithmic corrections is known to depend crucially on the treatment of the invariant
mass of the produced jet at threshold. We allow the jet to have a non-vanishing mass
at threshold, which most closely corresponds to the situation in experiment. Matching
our results to available semi-analytical next-to-leading-order calculations, we derive
resummed results valid to next-to-leading logarithmic accuracy. We present numerical
results for the resummation effects at Tevatron and RHIC energies.
hep-ph/yymmnnn
November 2, 2018
http://arxiv.org/abs/0704.1677v1
1 Introduction
High-transverse-momentum jet production in hadronic collisions, H1H2 → jetX , plays a fun-
damental role in High-Energy Physics. It offers possibilities to explore QCD, for example the
structure of the interacting hadrons or the emergence of hadronic final states, but is also inti-
mately involved in many signals (and their backgrounds) for New Physics. At the heart of all
these applications of jet production is our ability to perform reliable and precise perturbative cal-
culations of the partonic short-distance interactions that generate the high-transverse-momentum
final states. Up to corrections suppressed by inverse powers of the jet’s transverse momentum pT ,
the hadronic jet cross section factorizes into parton distribution functions that contain primar-
ily long-distance information, and these short-distance cross sections. In the present paper, we
address large logarithmic perturbative corrections to the latter.
At partonic threshold, when the initial partons have just enough energy to produce the high-
pT jet and an unobserved recoiling partonic final state, the phase space available for gluon
bremsstrahlung vanishes, so that only soft and collinear emission is allowed, resulting in large
logarithmic corrections to the partonic cross section. To be more specific, if we consider the cross
section as a function of the jet transverse momentum, integrated over all jet rapidities, the par-
tonic threshold is reached when
s = 2pT , where
s is the partonic center-of-mass (c.m.) energy.
Defining x̂T ≡ 2pT/
s, the leading large contributions near threshold arise as αkS(pT ) ln
2m (1− x̂2T )
at the kth order in perturbation theory, where m ≤ k (the logarithms with m = k are leading)
and αS is the strong coupling. Even if pT is large so that αS(pT ) is small, sufficiently close to
threshold the logarithmic terms will spoil the perturbative expansion to any fixed order. Thresh-
old resummation [1, 2, 3, 4, 5, 6, 7], however, allows to reinstate a useful perturbative series by
systematically taking into account the terms αkS ln
2m (1− x̂2T ) to all orders in αS. This is achieved
after taking a Mellin-transform of the hadronic cross section in xT = 2pT/
S, with
S the
hadronic c.m. energy. The threshold logarithms exponentiate in transform space.
Regarding phenomenology, the larger xT , the more dominant will the threshold logarithms be,
and hence the more important will threshold resummation effects be. In addition, because of the
convoluted form of the partonic cross sections and the parton distribution functions (PDFs), the
steep fall-off of the PDFs with momentum fraction x automatically enhances the contributions
from the threshold regime to the cross section, because it makes it relatively unlikely that the
initial partons have very high c.m. energy. This explains why partonic threshold effects often
dominate the hadronic cross section even at not so high xT . Studies of cross sections for [8]
pp → hX (with h a high-pT hadron) and [9, 10, 11, 12] pp → γX in the fixed-target regime,
where typically 0.2 . xT . 0.7, indeed demonstrate that threshold-resummation effects dominate
there and can be very large and important for phenomenology. They enhance the theoretical cross
section with respect to fixed-order calculations.
These observations suggest to study the resummation also for jet production at hadron col-
liders, in particular when xT is rather large. An application of particular interest is the jet cross
section at very high transverse momenta (pT ∼ several hundreds GeV) at the Tevatron [13, 14],
for which initially an excess of the experimental data over next-to-leading order (NLO) theory
was reported, which was later mostly attributed to an insufficient knowledge of the gluon distri-
bution [15]. Similarly large values of xT are now probed in pp collisions at RHIC, where currently√
s = 200 GeV and jet cross section measurements by the STAR collaboration are already ex-
tending to pT & 40 GeV [16]. In both these cases, one does expect threshold resummation effects
to be smaller than in the case of related processes at similar xT in the fixed-target regime, just
because (among other things) the strong coupling constant is smaller at these higher pT . On the
other hand, as we shall see, the effects are still often non-negligible.
Apart from addressing these interesting phenomenological applications, we believe we also
improve in this paper the theoretical framework for threshold resummation in jet production.
There has been earlier work in the literature on this topic [4, 17, 18]. In Ref. [4] the threshold
resummation formalism for the closely related dijet production at large invariant mass of the
jet pair was developed to next-to-leading logarithmic (NLL) order. In [17], these results were
applied to the single-inclusive jet cross section at large transverse momentum, making use of
earlier work [6] on the high-pT prompt-photon cross section, which is kinematically similar. As
was emphasized in [4], there is an important subtlety for the resummed jet cross section related to
the treatment of the invariant mass of the jet. The structure of the large logarithmic corrections
that are addressed by resummation depends on whether or not the jet is assumed to be massless
at partonic threshold, even at the leading-logarithmic (LL) level. This is perhaps surprising at
first sight, because one might expect the jet mass to be generally inessential since it is typically
much smaller than the jet’s transverse momentum pT and in fact vanishes for lowest-order partonic
scattering. However, the situation can be qualitatively understood as follows [4]: let us assume
that we are defining the jet cross section from the total four-momentum deposited in a cone of
aperture R †. Considering for simplicity the next-to-leading order, we can have contributions by
virtual 2 → 2 diagrams, or by 2 → 3 real-emission diagrams. For the former, a single particle
produces the (massless) jet, in case of the latter, there are configurations where two particles in
the final state jointly form the jet.
Then, for a jet forced to be massless at partonic threshold, the contributions with two partons
in the cone must either have one parton arbitrarily soft, or the two partons exactly collinear.
The singularities associated with these configurations cancel against analogous ones in the virtual
diagrams, but because the constraint on the real-emission diagrams is so restrictive, large double-
and single-logarithmic contributions remain after the cancellation. This will happen regardless
of the size R of the cone aperture, implying that the coefficients of the large logarithms will be
independent of R. These final-state threshold logarithms arising from the observed jet suppress
the cross section near threshold. Their structure is identical to that of the threshold logarithms
generated by the recoiling “jet”, because the latter is not observed and is indeed massless at
partonic threshold. The combined final-state logarithms then act against the threshold logarithms
associated with initial-state radiation which are positive and enhance the cross section [1].
If, on the other hand, the jet invariant mass is not constrained to vanish near threshold, far
more final states contribute– in fact, there will be an integration over the jet mass to an upper
limit proportional to the aperture of the jet cone. As the 2 → 3 contributions are therefore much
less restricted, the cancellations of infrared and collinear divergences between real and virtual
diagrams leave behind only single logarithms [4], associated with soft, but not with collinear,
emission. Compared to the previously discussed case, there is therefore no double-logarithmic
suppression of the cross section by the observed jet, and one expects the calculated cross section
to be larger. Also, the single-logarithmic terms will now depend on the jet cone size R.
†Details of the jet definition do not matter for the present argument.
The resummation for both these cases, with the jet massless or massive at threshold, has
been worked out in [4]. The study [17] of the resummed single-inclusive high-pT jet cross section
assumed massless jets at threshold. From a phenomenological point of view, however, we see no
reason for demanding the jet to become massless at the partonic threshold. The experimental jet
cross sections will, at any given pT , contain jet events with a large variety of jet invariant masses.
NLO calculations of single-inclusive jet cross sections indeed reflect this: they have the property
that jets produced at partonic threshold are integrated over a range of jet masses. This becomes
evident in the available semi-analytical NLO calculations [19, 20, 21, 22]. For these, the jet cross
section is obtained by assuming that the jet cone is relatively narrow, in which case it is possible
to treat the jet definition analytically, so that collinear and infrared final-state divergences may be
canceled by hand. This approximation is referred to as the “small-cone approximation (SCA)”.
Section II.E in the recent calculation in [21] explicitly demonstrates for the SCA that the threshold
double-logarithms associated with the observed final-state jet cancel, as described above.
In light of this, we will study in this work the resummation in the more realistic case of jets
that are massive at threshold. We will in fact make use of the NLO calculation in the SCA
approximation of [21] to “match” our resummed cross sections to finite (next-to-leading) order.
Knowledge of analytical NLO expressions allows one to extract certain hard-scattering coefficients
that are finite at threshold and part of the full resummation formula. These coefficients will be
presented and used in our paper for the first time.
We emphasize that the use of the SCA in our work is not to be regarded as a limitation to
the usefulness of our results. First, the SCA is known to be very accurate numerically even at
relatively large jet cone sizes of R ∼ 0.7 [21, 20, 23]. In addition, one may use our results to obtain
ratios of the resummed over the NLO cross sections. Such “K-factors” are then expected to be
extremely good approximations for the effects of higher orders even when one goes away from the
SCA and uses, for example, a full NLO Monte-Carlo integration code that allows to compute the
jet cross section for larger cone aperture and for other jet definitions (see, for example, Ref. [24]).
We will therefore in particular present K-factors for the resummed jet cross section in this paper.
The paper is organized as follows: in Sec. 2 we provide the basic formulas for the single-
inclusive-jet cross section at fixed order in perturbation theory, and discuss the SCA and the role
of the threshold region. Section 3 presents details of the threshold resummation for the inclusive-
jet cross section and describes the matching to the analytical expressions for the NLO cross section
in the SCA. In Sec. 4 we give phenomenological results for the Tevatron and for RHIC. Finally,
we summarize our results in Sec. 5. The Appendix collects the formulas for the hard-scattering
coefficients in the threshold-resummed cross section mentioned above.
2 Next-to-leading order single-inclusive jet cross section
Jets produced in high-energy hadronic scattering, H1(P1)H2(P2) → jet(PJ)X , are typically defined
in terms of a deposition of transverse energy or four-momentum in a cone of aperture R in pseudo-
rapidity and azimuthal-angle space, with detailed algorithms specifying the jet kinematic variables
in terms of those of the observed hadron momenta [13, 25, 26, 27, 28, 29]. QCD factorization
theorems allow to write the cross section for single-inclusive jet production in hadronic collisions in
terms of convolutions of parton distribution functions with partonic hard-scattering functions [30]:
dx1dx2 fa/H1
x1, µ
fb/H2
x2, µ
dσ̂ab(x1P1, x2P2, PJ , µF , µR) , (1)
where the sum runs over all initial partons, quarks, anti-quarks, and gluons, and where µF and µR
denote the factorization and renormalization scales, respectively. It is possible to use perturbation
theory to describe the formation of a high-pT jet, as long as the definition of the jet is infrared-
safe. The jet is then constructed from a subset of the final-state partons in the short-distance
reaction ab → partons, and a “measurement function” in the dσ̂ab specifies the momentum PJ of
the jet in terms of the momenta of the final-state partons, in accordance with the (experimental)
jet definition.
The computation of jet cross sections beyond the lowest order in perturbative QCD is rather
complicated, due to the need for incorporating a jet definition and the ensuing complexity of the
phase space, and due to the large number of infrared singularities of soft and collinear origin at
intermediate stages of the calculation. Different methods have been introduced that allow the
calculation to be performed largely numerically by Monte-Carlo “parton generators”, with only
the divergent terms treated in part analytically (see, for example, Ref. [24]).
A major simplification occurs if one assumes that the jet cone is rather narrow, a limit known as
the “small-cone approximation (SCA)” [19, 20, 21, 22]. In this case, a semi-analytical computation
of the NLO single-inclusive jet cross section can be performed, meaning that fully analytical
expressions for the partonic hard-scattering functions dσ̂ab can be derived which only need to be
integrated numerically against the parton distribution functions as shown in Eq. (1). The SCA
may be viewed as an expansion of the partonic cross section for small δ ≡ R/ cosh η, where η is
the jet’s pseudo-rapidity. Technically, the parameter δ is the half-aperture of a geometrical cone
around the jet axis, when the four-momentum of the jet is defined as simply the sum of the four-
momenta of all the partons inside the cone [19, 21]. At small δ, the behavior of the jet cross section
is of the form A log(δ)+B+O(δ2), with both A and B known from Refs. [19, 21]. Jet codes based
on the SCA have the virtue that they produce numerically stable results on much shorter time
scales than Monte-Carlo codes. Moreover, as we shall see below, the relatively simple and explicit
results for the NLO single-inclusive jet cross section obtained in the SCA are a great convenience
for the implementation of threshold resummation, particularly for the matching needed to achieve
full NLL accuracy.
It turns out that the SCA is a very good approximation even for relatively large cone sizes of up
toR ≃ 0.7 [21, 20, 23], the value used by both Tevatron collaborations. Figure 1 shows comparisons
between the NLO cross sections for single-inclusive jet production obtained using a full Monte-
Carlo code [24] and the SCA code of [21], for pp̄ collisions at c.m. energy
S = 1800 GeV and very
high pT . Throughout this paper we use the CTEQ6M [31] NLO parton distribution functions. We
have chosen two different jet definitions in the Monte-Carlo calculation. One uses a conventional
cone algorithm [25], the other the CDF jet definition [13]. One can see that the differences with
respect to the SCA are of the order of only a few per cent. We note that similar comparisons
in the RHIC kinematic regime have been shown in [21]. In their recent paper [16], the STAR
collaboration used R = 0.4, for which the SCA is even more accurate.
Encouraged by this good agreement, we will directly use the SCA analytical results when
performing the threshold resummation. As stated in the Introduction, this is anyway not a
Figure 1: Ratio between NLO jet cross sections at Tevatron at
S = 1800 GeV, computed with
a full Monte-Carlo code [24] and in the SCA. The solid line corresponds to the jet definition
implemented by CDF [13] (with parameter Rsep = 1.3), and the dashed one to the standard cone
definition [25]. In both cases the size of the jet cone is set to R = 0.7, and the CTEQ6M NLO
[31] parton distributions, evaluated at the factorization scale µF = PT , were used.
limitation, because we will also always provide the ratio of resummed over NLO cross sections
(K-factors), which may then be used along with full NLO Monte-Carlo calculations to obtain
resummed cross sections for any desired cone size or jet algorithm.
A further simplification that we will make is to consider the cross section integrated over all
pseudo-rapidities of the jet. As was discussed in [8], this considerably reduces the complexity
of the resummed expressions. By simply rescaling the resummed prediction by an appropriate
ratio of NLO cross sections one can nonetheless obtain a very good approximation also for the
resummation effects on the non-integrated cross section, at central rapidities [11]. To perform the
NLL threshold resummation for the full rapidity-dependence of the jet cross section remains an
outstanding task for future work.
From Eq. (1), we find for the single-inclusive jet cross section integrated over all jet pseudo-
rapidity η, in the SCA:
p3T dσ
SCA(xT )
dx1 fa/H1
x1, µ
dx2 fb/H2
x2, µ
dx̂T δ
x̂T −
∫ η̂+
x̂4T s
dσ̂ab(x̂
T , η̂, R)
dx̂2Tdη̂
, (2)
where as before xT ≡ 2pT/
S is the customary scaling variable, and x̂T ≡ 2pT/
s with s = x1x2S
is its partonic counterpart. η̂ is the partonic pseudo-rapidity, η̂ = η − 1
ln(x1/x2), which has the
limits η̂+ = −η̂− = ln[(1 +
1− x̂2T )/x̂T ]. The dependence of the partonic cross sections on µF
and µR has been suppressed for simplicity. The perturbative expansion of the dσ̂ab in the coupling
constant αS(µR) reads
dσ̂ab(x̂
T , η̂, R) =α
S(µR)
ab (x̂
T , η̂) + αS(µR) dσ̂
ab (x̂
T , η̂, R) +O(α2S)
. (3)
As indicated, the leading-order (LO) term dσ̂ab has no dependence on the cone size R, because
for this term a single parton produces the jet. The analytical expressions for the NLO terms dσ̂
have been obtained in [19, 21]. It is customary to express them in terms of a different set of
variables, v and w, that are related to x̂T and η̂ by
x̂2T = 4vw(1− v) , e2η̂ =
. (4)
Schematically, the NLO corrections to the partonic cross section for each scattering channel then
take the form
s dσ̂
ab (w, v, R)
dw dv
= Aab(v, R) δ(1− w) +Bab(v, R)
ln(1− w)
+Cab(v, R)
+ Fab(w, v, R) , (5)
where the “plus”-distributions are defined as usual by
dwf(w)[g(w)]+ ≡
dw(f(w)− f(1))g(w) , (6)
and where the Fab(w, v, δ) collect all terms without distributions in w. Partonic threshold corre-
sponds to the limit w → 1. The “plus”-distribution terms in Eq. (5) generate the large logarithmic
corrections that are addressed by threshold resummation. At order k of perturbation theory, the
leading contributions are proportional to αkS
ln(1−w)
)2k−1
. Performing the integration of these
terms over η̂, they turn into contributions ∝ αkS ln
2k (1− x̂2T ), as we anticipated in the Introduc-
tion, and as we shall show below. Subleading terms are down by one or more powers of the
logarithm. We will now turn to the NLL resummation of the threshold logarithms.
3 Resummed cross section
The resummation of the soft-gluon contributions is carried out in Mellin-N moment space, where
they exponentiate [1, 2, 3, 4, 5, 6, 7]. At the same time, in moment space the convolutions between
the parton distributions and the partonic subprocess cross sections turn into ordinary products.
For our present calculation, the appropriate Mellin moments are in the scaling variable xT
σ(N) ≡
)(N−1) p
T dσ(xT )
. (7)
‡We drop the superscript “SCA” from now on.
In Mellin-N space the QCD factorization formula in Eq. (2) becomes
σ(N) =
(µ2F ) f
(µ2F ) σ̂ab(N) , (8)
where the fN
are the moments of the parton distribution functions,
fNa/H(µ
F ) ≡
dxxN−1fa/H(x, µ
F ) , (9)
and where
σ̂ab(N) ≡
dv [4v(1− v)w]N+1 s dσ̂ab(w, v)
dw dv
. (10)
The threshold limit w → 1 corresponds to N → ∞, and the LL soft-gluon corrections contribute
as αmS ln
2mN . The large-N behavior of the NLO partonic cross sections can be easily obtained by
using
[4v(1− v)]N+1 f(v) =
[4v(1− v)]N+1
. (11)
Up to corrections suppressed by 1/N it is therefore possible to perform the v-integration of the
partonic cross sections by simply evaluating them at v = 1/2. According to Eq. (5), when v = 1/2
is combined with the threshold limit w = 1, one has x̂T = 1. It is worth mentioning that in
the same limit one has η̂ = 0, and therefore the coefficients for the soft-gluon resummation for
the rapidity-integrated cross section agree with those for the cross section at vanishing partonic
rapidity. This explains why generally the resummation for the rapidity-integrated hadronic cross
section yields a good approximation to the resummation of the cross section integrated over only
a finite rapidity interval, as long as a region around η = 0 is contained in that interval [11].
The resummation of the large logarithms in the partonic cross sections is achieved by showing
that they exponentiate in Sudakov form factors. The resummed cross section for each partonic
subprocess is given by a formally rather simple expression in Mellin-N space:
(res)
ab (N) =
Cab ∆
GIab→cd∆
(int)ab→cd
(Born)
ab→cd(N) , (12)
where the first sum runs over all possible final state partons c and d, and the second over all possible
color configurations I of the hard scattering. Except for the Born cross sections σ̂
(Born)
ab→cd (which we
have presented in earlier work [8]), each of the N -dependent factors in Eq. (12) is an exponential
containing logarithms in N . The coefficients Cab collect all N -independent contributions, which
partly arise from hard virtual corrections and can be extracted from comparison to the analytical
expressions for the full NLO corrections in the SCA. Finally, the GIab→cd are color weights obeying
ab→cd = 1. The color interferences expressed by the sum over I appear whenever the number
of partons involved in the process at Born level is larger than three, as it is the case here §. Figure 2
gives a simple graphical mnemonic of the structure of the resummation formula, and of the origin
of its various factors, whose expressions we know present.
§In a more general case without rapidity integration, the terms GI
ab→cd × σ̂
(Born)
ab→cd
(N) should be replaced by the
color-correlated Born cross sections σ̂
(Born I)
ab→cd
(N) [4, 7, 17].
(int)
Figure 2: Pictorial representation of the resummation formula in Eq. (12).
Effects of soft-gluon radiation collinear to the initial-state partons are exponentiated in the
functions ∆
N , which read:
ln∆aN =
zN−1 − 1
∫ (1−z)2Q2
Aa(αS(q
2)) , (13)
(and likewise for b) where Q2 = 2p2T . J
N is the exponent associated with collinear, both soft and
hard, radiation in the unobserved recoiling “jet”,
ln JdN =
zN−1 − 1
∫ (1−z)Q2
(1−z)2Q2
Ad(αS(q
2)) +
Bd(αS((1− z)Q2))
. (14)
The function J ′cN describes radiation in the observed jet. As we reviewed in the Introduction,
this function is sensitive to the assumption made about the jet’s invariant mass at threshold [4]
even at LL level. Our choice is to allow the jet to be massive at partonic threshold, which is
consistent with the experimental definitions of jet cross sections and with the available NLO
calculations. In this case, J ′cN is given as
ln J ′cN =
zN−1 − 1
C ′c(αS((1− z)2Q2)) (jet massive at threshold) . (15)
A similar exponent was derived for this case in [4]. The expression given in Eq.(15) agrees with
the one of [4] to the required NLL accuracy. Notice that J ′cN contains only single logarithms, which
arise from soft emission, whereas logarithms of collinear origin are absent. This is explicitly seen
in the NLO calculations in the SCA [20, 21] in which there is an integration over the jet mass up
to a maximum value of O(δ ∼ R), even when the threshold limit is strictly reached. Collinear
contributions that would usually generate large logarithms are actually “regularized” by the cone
size δ and give instead rise to log(δ) terms in the perturbative cross sections.
If, however, the jet is forced to be massless at partonic threshold the jet-function is identical
to the function for an “unobserved” jet given in Eq. (14) [4, 17]:
ln J ′cN = lnJ
N (jet massless at threshold) , (16)
which produces a (negative) double-logarithm per emitted gluon, because it also receives collinear
contributions due to the stronger restriction on the gluon phase space. There is then no dependence
on log(δ) in this case.
Finally, large-angle soft-gluon emission is accounted for by the factor ∆
(int)ab→cd
I N , which reads
(int)ab→cd
I N =
zN−1 − 1
DI ab→cd(αS((1− z)2Q2)) , (17)
and depends on the color configuration I of the participating partons.
The coefficients Aa, Ba, C
a, DI ab→cd in Eqs. (13),(14),(15),(17) are free of large logarithmic
contributions and are given as perturbative series in the coupling constant αS:
F(αS) =
F (1) +
F (2) + . . . (18)
for each of them. For the resummation to NLL accuracy we need the coefficients A
a , A
a , C
a , and D
I ab→cd. The last of these depends on the specifics of the partonic process under
consideration; all the others are universal in the sense that they only distinguish whether the
parton they are associated with is a quark or a gluon. The LL and NLL coefficients A
a , A
a and
a are well known [32]:
A(1) = Ca , A
(2) =
CaK , B
a = γa (19)
K = CA
Nf , (20)
where Cg = CA = Nc = 3, Cq = CF = (N
c − 1)/2Nc = 4/3, γq = −3/2CF = −2, γg = −2π0¯, and
Nf is the number of flavors. The coefficients C
a needed in the case of jets that are massive at
threshold, may be obtained by comparing the first-order expansion of the resummed formula to
the analytic NLO results in the SCA. They are also universal and read
C ′(1)a = −Ca log
. (21)
This coefficient contains the anticipated dependence on log(δ) that regularizes the final-state
collinear configurations. As expected from a term of collinear origin, the exponent (15) hence
provides one power of log(δ) for each perturbative order.
The coefficients D
I ab→cd governing the exponentiation of large-angle soft-gluon emission to NLL
accuracy, and the corresponding “color weights” GI ab→cd, depend both on the partonic process
and on its “color configuration”. They can be obtained from [4, 5, 17, 33, 34] where the soft
anomalous dimension matrices were computed for all partonic processes. The results are given
for the general case of arbitrary partonic rapidity η̂. As discussed above, the coefficients for the
rapidity-integrated cross section may be obtained by setting η̂ = 0. We have presented the full
set of the D
I ab→cd and GI ab→cd in the Appendix of our previous paper [8].
Before we continue, we mention that one expects that a jet cross section defined by a cone
algorithm will also have so-called “non-global” threshold logarithms [35, 36]. These logarithms
arise when the observable is sensitive to radiation in only a limited part of phase space, as is the
case in presence of a jet cone. For instance, a soft gluon radiated at an angle outside the jet cone
may itself emit a secondary gluon at large angle that happens to fall inside the jet cone, thereby
becoming part of the jet [35, 36]. Such configurations appear first at the next-to-next-to-leading
order, but may produce threshold logarithms at the NLL level. One may therefore wonder if our
NLL resummation formulas given above are complete. Fortunately, as an explicit example in [35]
shows, it turns out that these effects are suppressed as R log(R) in the SCA. They may therefore
be neglected at the level of approximation we are making here. Given their only mild suppression
as R → 0, a study of non-global logarithms in hadronic jet cross sections is an interesting topic
for future work.
Returning to our resummed formulas, it is instructive to consider the structure of the leading
logarithms. The LL expressions for the radiative factors are
∆aN = exp
Ca ln
JdN = exp
Cd ln
. (22)
As discussed above, J ′c does not contribute at the double-logarithmic level. Therefore, for a given
partonic channel, the leading logarithms are
(res)
ab→cd(N) ∝ exp
Ca + Cb −
ln2(N)
. (23)
The exponent is positive for each partonic channel, implying that the soft-gluon effects will increase
the cross section. This enhancement arises from the initial-state radiation represented by the ∆a,b
and is related to the fact that finite partonic cross sections are obtained after collinear (mass)
factorization [1, 2]. In the MS scheme such an enhancing contribution is (for a given parton
species) twice as large as the suppressing one associated with final-state radiation in Jd, for which
no mass factorization is needed. For quark or anti-quark initiated processes, the color factor
combination appearing in Eq. (23) ranges from 2CF − CF/2 = 2 for the qq → qq channel to
2CF −CA/2 = 7/6 for qq̄ → gg, while for those involving a quark-gluon initial state one has larger
factors, CF + CA − CF/2 = 11/3 (for qg → qg) or CF + CA − CA/2 = 17/6 (for qg → gq). Yet
larger factors are obtained for gluon-gluon scattering, with 2CA − CA/2 = 9/2 for gg → gg and
2CA−CF/2 = 16/3 for gg → qq̄. Initial states with more gluons therefore are expected to receive
the larger resummation effects. We mention that if the observed jet is assumed strictly massless
at threshold, an extra suppression term proportional to Jc arises (see Eq. (16)).
It is customary to give the NLL expansions of the Sudakov exponents in the following way [2]:
ln∆aN(αS(µ
R), Q
2/µ2R;Q
2/µ2F ) = lnN h
a (λ) + h
a (λ,Q
2/µ2R;Q
2/µ2F ) +O
αS(αS lnN)
, (24)
lnJaN (αS(µ
R), Q
2/µ2R) = lnN f
a (λ) + f
a (λ,Q
2/µ2R) +O
αS(αS lnN)
, (25)
ln J ′aN (αS(µ
R)) =
ln(1− 2λ) +O
αS(αS lnN)
, (26)
(int)ab→cd
I N (αS(µ
R)) =
I ab→cd
ln(1− 2λ) +O
αS(αS lnN)
, (27)
with λ = 0
R) lnN . The LL and NLL auxiliary functions h
(1,2)
a and f
(1,2)
a are
h(1)a (λ) = +
[2λ+ (1− 2λ) ln(1− 2λ)] , (28)
h(2)a (λ,Q
2/µ2R;Q
2/µ2F ) =−
[2λ+ ln(1− 2λ)]−
ln(1− 2λ)
2λ+ ln(1− 2λ) + 1
ln2(1− 2λ)
[2λ+ ln(1− 2λ)] ln Q
, (29)
f (1)a (λ) = −
2πb0λ
(1− 2λ) ln(1− 2λ)− 2(1− λ) ln(1− λ)
, (30)
f (2)a (λ,Q
2/µ2R) = −
2πb30
ln(1− 2λ)− 2 ln(1− λ) + 1
ln2(1− 2λ)− ln2(1− λ)
ln(1− λ)−
ln(1− λ)− ln(1− 2λ)
2π2b20
2 ln(1− λ)− ln(1− 2λ)
2 ln(1− λ)− ln(1− 2λ)
where 0
, b1 are the first two coefficients of the QCD β-function:
(11CA − 2Nf ) , b1 =
17C2A − 5CANf − 3CFNf
. (32)
The N -independent coefficients Cab in Eq. (12), which include the hard virtual corrections,
have the perturbative expansion
Cab = 1 +
ab +O(α
S) . (33)
The C
ab we need to NLL are obtained by comparing the O(αS)-expansion (not counting the
overall factor α2S of the Born cross sections) of the resummed expression with the fixed-order
NLO result for the process, as given in [19, 21]. The full analytic expressions for the C
ab are
rather lengthy and will not be given here. For convenience, we present them in numerical form
in the Appendix. We note that apart from being useful for extracting the coefficients C
a and
ab , the comparison of the O(αS)-expanded resummed result with the full NLO cross section also
provides an excellent check of the resummation formula, since one can verify that all leading and
next-to-leading logarithms are properly accounted for by Eq. (12).
The improved resummed hadronic cross section is finally obtained by performing an inverse
Mellin transformation, and by properly matching to the NLO cross section p3T dσ
(NLO)(xT )/dpT as
follows:
p3T dσ
(match)(xT )
a,b,c
∫ CMP+i∞
CMP−i∞
)−N+1
fNa/H1(µ
F ) f
(µ2F )
(res)
ab→cd(N)−
(res)
ab→cd(N)
(NLO)
p3T dσ
(NLO)(xT )
, (34)
where σ̂
(res)
ab→cd is given in Eq. (12) and (σ̂
(res)
ab→cd)(NLO) represents its perturbative truncation at NLO.
Thus, as a result of this matching procedure, in the final cross section in Eq. (34) the NLO cross
section is exactly taken into account, and NLL soft-gluon effects are resummed beyond those
already contained in the NLO cross section.
The functions h
(1,2)
a (λ) and f
(1,2)
a (λ) in Eqs. (28)-(31) are singular at the points λ = 1/2 and/or
λ = 1. These singularities are related to the divergent behavior of the perturbative running
coupling αS near the Landau pole, and we deal with them by using the Minimal Prescription
introduced in Ref. [2]. In the evaluation of the inverse Mellin transformation in Eq. (34), the
constant CMP is chosen in such a way that all singularities in the integrand are to the left of the
integration contour, except for the Landau singularities, that are taken to lie to its far right. We
note that an alternative to such a definition one could choose to expand the resummed formula
to a finite order, say, next-to-next-to-leading order (NNLO), and neglect all terms of yet higher
order. This approach was adopted in Ref. [17]. We prefer to keep the full resummed formula in
our phenomenological applications since, depending on kinematics, high orders in perturbation
theory may still be very relevant [8]. It was actually shown in [2] that the results obtained within
the Minimal Prescription converge asymptotically to the perturbative series.
This completes the presentation of all ingredients to the NLL threshold resummation of the
hadronic single-inclusive jet cross section. We will now turn to some phenomenological applica-
tions.
4 Phenomenological Results
We will study the effects of threshold resummation on the single-inclusive jet cross section in pp̄
collisions at
S = 1.8 TeV and
S = 630 GeV c.m. energies, and in pp collisions at
200 GeV. These choices are relevant for comparisons to Tevatron and RHIC data, respectively.
Unless otherwise stated, we always set the factorization and renormalization scales to µF = µR =
pT and use the NLO CTEQ6M [31] set of parton distributions, along with the two-loop expression
for the strong coupling constant αS.
We will first analyze the relevance of the different subprocesses contributing to single-jet pro-
duction. The left part of Fig. 3 shows the relative contributions by “qq” (qq, qq′, qq̄ and qq̄′
combined), qg and gg initial states at Born level (dashed lines) and for the NLL resummed case
(without matching, solid lines). Here we have chosen the case of pp̄ collisions at
S = 1.8 TeV.
As can be seen, the overall change in the curves when going from Born level to the resummed case
is moderate. The main noticeable effect is an increase of the relative importance of processes with
gluon initial states toward higher pT , compensated by a similar decrease in that of the qq channels.
In the right part of Fig. 3 we show the enhancements from threshold resummation for each initial
partonic state individually, and also for their sum. At the higher pT , where threshold resummation
is expected to be best applicable, the enhancements are biggest for the gg channel, followed by
the qg one. All patterns observed in Fig. 3 are straightforwardly understood from Eq. (23), which
demonstrates that resummation yields bigger enhancements when the number of gluons in the
initial state is larger. We note that results very similar to those shown in the figure hold also at√
S = 630 GeV, if the same value of xT = 2pT/
S is considered. This remains qualitatively true
even when we go to pp collisions at
S = 200 GeV, except for the larger enhancement in the
Figure 3: Left: relative contributions of the various partonic initial states to the single-inclusive jet
cross section in pp̄ collisions at
S = 1.8 TeV, at Born level (dashed) and for the NLL resummed
case (solid). We have chosen the jet cone size R = 0.7. Right: ratios between resummed and Born
contributions for the various channels, and for the full jet cross section.
Figure 4: Same as Fig. 3, but for pp collisions at
S = 200 GeV and R = 0.4.
quark contribution, due to the dominance of the qq channel (instead of the qq̄ as in pp̄ collisions)
with a larger color factor combination in the Sudakov exponent (see the discussion after Eq. (22)).
Figure 4 repeats the studies made for Fig. 3 for this case. As one can see, if the same xT
as in Fig. 3 is considered, the qq scattering contributions are overall slightly less important. At
the same time, resummation effects are overall somewhat larger because the pT values are now
much smaller than in Fig. 3, so that the strong coupling constant that appears in the resummed
exponents is larger.
Figure 5: Ratio between the expansion to NLO of the (unmatched) resummed cross section and
the full NLO one (in the SCA), for pp̄ collisions at
s = 1.8 TeV (solid) and
s = 630 GeV
(dots), and for pp collisions at
s = 200 GeV (dashed).
Before presenting the results for the matched NLL resummed jet cross section and K-factors,
we would like to identify the kinematical region where the logarithmic terms constitute the bulk
of the perturbative corrections and subleading contributions are unimportant. Only in these is
the resummation expected to provide an accurate picture of the higher-order terms. We can
determine this region by comparing the resummed formula, expanded to NLO, to the full fixed-
order (NLO) perturbative result, that is, by comparing the last two terms in Eq. (34). Figure 5
shows this comparison for both Tevatron energies and for the RHIC case, as function of the
“scaling” variable xT . As can be observed, the expansion correctly reproduces the NLO result
within at most a few per cent over a region corresponding to pT & 200 GeV for the higher Tevatron
energy, and to pT & 30 GeV at RHIC. This demonstrates that, at this order, the perturbative
corrections are strongly dominated by the terms of soft and/or collinear origin that are addressed
by resummation. The accuracy of the expansion improves toward the larger values of the jet
transverse momentum, were one approaches the threshold limit more closely.
Having established the importance of the threshold corrections in a kinematic regime of interest
for phenomenology, we show in Fig. 6 the impact of the resummation on the predicted single-jet
cross section at
S = 1.8 TeV. NLO and NLL resummed results are presented, computed at three
different values of the factorization and renormalization scales, defined by µF = µR = ζpT (with
ζ = 1, 2, 1/2). The most noticeable effect is a remarkable reduction in the scale dependence of
the cross section. This observation was also made in the previous study [17]. If, as customary,
one defines a theoretical scale “uncertainty” by ∆ ≡ (σ(ζ = 0.5) − σ(ζ = 2))/σ(ζ = 1), the
improvement is considerable. While ∆ lies between 20 and 25% at NLO, it never exceeds 8% for
the matched NLL result. The inset plot shows the NLL K-factor, defined as
K(res) =
dσ(match)/dpT
dσ(NLO)/dpT
, (35)
at each of the scales. The corrections from resummation on top of NLO are typically very mod-
erate, at the order of a few per cent, depending on the set of scales chosen. The higher-order
corrections increase for larger values of the jet transverse momentum. These findings are again
consistent with those of [17], even though more detailed comparisons reveal some quantitative
differences that must be related to either the different choice of the resummed final-state jet func-
tion in [17] (see discussion in Sec. 3), or to the fact that [17] uses only a NNLO expansion of the
resummed cross section. The main features of our results remain unchanged when we go to the
Tevatron-run II energy of
S = 1.96 TeV, at which measured jet cross sections are now avail-
able [37]. Quantitatively very similar results are also found for the lower Tevatron center-of-mass
energy, as seen in Fig. 7. In the case of pp collisions at
s = 200 GeV, presented in Fig. 8, a
similar pattern emerges, even though the resummation effects tend to be overall somewhat more
substantial here.
Figure 6: NLO and NLL results for the single-inclusive jet cross section in pp̄ collisions at
S = 1.8
TeV, for different values of the renormalization and factorization scales. We have chosen R = 0.7.
The inset plot shows the corresponding K-factors as defined in Eq. (35).
In Fig. 9 we analyze how the resummation effects build up order by order in perturbation
theory. We expand the matched resummed formula beyond NLO and define the “partial” soft-
gluon K-factors as
dσ(match)/dpT
dσ(NLO)/dpT
, (36)
which for n = 2, 3, . . . give the additional enhancement over full NLO due to the O(α2+nS ) terms
in the resummed formula¶. Formally, K1 = 1 and K∞ = K(res) of Eq. (35). The results for
K2,3,4,∞ are given in the figure, for the case of pp̄ collisions at
S = 1.8 TeV. One can see that
¶We recall that the Born cross sections are of O(α2S), hence the additional power of two in this definition.
Figure 7: Same as Fig. 6, but for
S =630 GeV.
contributions beyond N3LO (n = 3) are very small, and that the O(α6S) result can hardly be
distinguished from the full NLL one.
It is interesting to contrast the rather modest enhancement of the jet cross section by resum-
mation to the dramatic resummation effects that we observed in [8] in the case of single-inclusive
pion production, H1H2 → πX , in fixed-target scattering at typical c.m. energies of
S ∼ 30 GeV.
Even though in both cases the same partonic processes are involved at Born level, there are several
important differences. First of all, the values of pT are much smaller in fixed-target scattering
(even though roughly similar values of xT = 2pT/
S are probed), so that the strong coupling
constant αS(pT ) is larger and resummation effects are bound to be more significant. Furthermore,
for the process H1H2 → πX one needs to introduce fragmentation functions into the theoretical
calculation that describe the formation of the observed hadron from a final-state parton. As the
hadron takes only a certain fraction z & 0.5 of the parent parton’s momentum, the partonic hard-
scattering necessarily has to be at the higher transverse momentum pT/z in order to produce a
hadron with pT . Thus one is closer to partonic threshold than in case of a jet produced with
transverse momentum pT which takes all of a final-state parton’s momentum. In addition, it
turns out [8, 38] that due to the factorization of final-state collinear singularities associated with
the fragmentation functions, the “jet” function J ′cN in the resummation formula Eq. (12) is to be
replaced by a factor ∆cN , which has enhancing double logarithms.
Finally, as one illustrative example, we compare our resummed jet cross section to data from
CDF [13] at
S = 1800 GeV. While so far we have always considered the cross section integrated
over all jet rapidities, we here need to account for the fact that the data cover only a finite
region in rapidity, 0.1 ≤ |η| ≤ 0.7. Also, we would like to properly match the jet algorithm
chosen in experiment, rather than using the SCA. We have mentioned before that both these
issues can be accurately addressed by “rescaling” the resummed cross section by an appropriate
Figure 8: Same as Fig. 6, but for pp collisions at
S =200 GeV and R = 0.4.
ratio of NLO cross sections. We simply multiply our K-factors defined in Eq. (35) and shown
in Fig. 6 by dσ(MC)(0.1 ≤ |η| ≤ 0.7)/dpT , the NLO cross section obtained with a full Monte-
Carlo code [24], in the experimentally accessed rapidity regime. The comparison between the
data and the NLO and NLL-resummed cross sections is shown in Fig. 10 in terms of the ratios
“(data−theory)/theory”. As expected from Fig. 6, the impact of resummation is moderate and
in fact smaller than the current uncertainties of the parton distributions [37]. Nonetheless, it does
lead to a slight improvement of the comparison and, in particular, the plot again demonstrates
the reduction of scale dependence by resummation.
5 Conclusions
We have studied in this paper the resummation of large logarithmic threshold corrections to
the partonic cross sections contributing to single-inclusive jet production at hadron colliders. Our
study differs from previous work [17] mostly in that we allow the jet to have a finite invariant mass
at partonic threshold, which is consistent with the experimental definitions of jet cross sections
and with the available NLO calculations. Moreover, using semi-analytical expressions for the
NLO partonic cross sections derived in the SCA [19, 21], we have extracted the N−independent
coefficients that appear in the resummation formula, and properly matched our resummed cross
section to the NLO one. We hope that with these improvements yet more realistic estimates of
the higher-order corrections to jet cross sections in the threshold regime emerge.
It is well known that the NLO description of jet production at hadron colliders is overall very
successful, within the uncertainties of the theoretical framework and the experimental data. From
that perspective, it is gratifying to see that the effects of NLL resummation are relatively moderate.
Figure 9: Soft-gluon Kn factors as defined in Eq. (36), for pp̄ collisions at
S = 1.8 TeV.
On the other hand, resummation leads to a significant decrease of the scale dependence, and we
expect that knowledge of the resummation effects should be useful in comparisons with future,
yet more precise, data, and for extracting parton distribution functions. Given the general success
of the NLO description, we have mostly focused on K-factors for the resummed cross section
over the NLO one, and only given one example of a more detailed comparison with experimental
data. We believe that these K-factors may be readily used in conjunction with other, more flexible
NLO Monte-Carlo programs for jet production, to estimate threshold-resummation effects on cross
sections for other jet algorithms and possibly for larger cone sizes.
Acknowledgments
We are grateful to Stefano Catani, Barbara Jäger, Nikolaos Kidonakis, Douglas Ross, George
Sterman, and Marco Stratmann for helpful discussions. DdF is supported in part by UBACYT
and CONICET. WV is supported by the U.S. Department of Energy under contract number
DE-AC02-98CH10886.
Appendix: First-order coefficients C
ab in the SCA
In this appendix we collect the process-dependent coefficients C
ab for the various partonic channels
in jet hadroproduction in the SCA. The C
ab are constant, that is, they do not depend on the
Mellin-moment variable N . They may be extracted by expanding the resummed cross section in
Eq. (12) to first order and comparing it to the full NLO cross section in the SCA. For the sake
Figure 10: Ratios “(data−theory)/theory” for data from CDF at 1.8 TeV [13] and for NLO and
NLL resummed theoretical calculations. We have chosen the theory result at scale pT as “default”;
results for other scales are also displayed in terms of their relative shifts with respect to the default
theory.
of simplicity, we provide the C
ab only in numerical form, as the full analytic coefficients have
relatively lengthy expressions. We find:
qq′ = 17.9012 + 1.38763 log
= 19.0395 + 1.38763 log
qq̄ = 13.4171 + 1.6989 log
C(1)qq = 17.1973 + 1.38763 log
C(1)qg = 14.4483 + 2.58824 log
C(1)gg = 14.5629 + 3.67884 log
, (37)
where R is the jet cone size.
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	Introduction
	Next-to-leading order single-inclusive jet cross section 
	Resummed cross section 
	Phenomenological Results 
	Conclusions
