Colloquia: LaThuile09
Production of jets at forward rapidities in hadronic collisions
F. HautmannDepartment of Theoretical Physics, University of Oxford - Oxford OX1 3NP, UK
(ricevuto il 10 Novembre 2009; pubblicato online il 7 Gennaio 2010)
Summary. — We discuss high-pT production processes at forward rapidities in
hadron-hadron collisions, and describe recent results on using QCD high-energy factorization in forward jet production at the LHC.
PACS 12.38.-t – Quantum chromodynamics.
PACS 13.85.-t – Hadron-induced high- and super-high-energy interactions (energy
> 10 GeV).
PACS 13.87.-a – Jets in large-Q2 scattering.
1. – Introduction
Experiments at the Large Hadron Collider (LHC) will explore the region of large rapidities both with general-purpose detectors and with dedicated instrumentation, in-cluding forward calorimeters and proton taggers [1-7]. The LHC forward-physics pro-gram involves a wide range of topics, from new particle discovery processes [3, 8, 9] to new aspects of strong-interaction physics [7, 10] to heavy-ion collisions [11, 12]. Owing to the large center-of-mass energy and the unprecedented experimental coverage at large rapidities, it becomes possible for the first time to investigate the forward region with high-p⊥ probes.
In this article we report on studies of forward production of jets [13] based on QCD high-energy factorization at fixed transverse momentum [14]. This theoretical framework serves to take into account consistently both the higher-order logarithmic corrections in the large rapidity interval and those in the hard jet transverse energy. In sect. 2 we introduce the basic structure of jet production in the LHC forward region. In sect. 3 we consider associated parton showering effects. In sect. 4 we consider effects from the short-distance matrix elements that control the resummation of logarithmically enhanced corrections in√s/ET, where ET is the hard jet transverse energy. We give concluding
remarks in sect. 5.
c
B p X s 1 s 2 >> s1 q A B q
Fig. 1. – Jet production in the forward rapidity region in hadron-hadron collisions.
2. – Forward jets at the LHC
The hadroproduction of a forward jet associated with hard final state X is pictured in fig. 1. The kinematics of the process is characterized by the large ratio of sub-energies
s2/s1 1 and highly asymmetric longitudinal momenta in the partonic initial state,
qA· pB qB· pA. At the LHC the use of forward calorimeters allows one to measure
events where jet transverse momenta p⊥ > 20 GeV are produced several units of rapidity
apart, Δy 4–6 [1, 5, 7]. Working at polar angles that are small but sufficiently far from the beam axis not to be affected by beam remnants, one measures azimuthal-plane correlations between high-p⊥ events (fig. 2) widely separated in rapidity [7, 13].
The presence of multiple large-momentum scales implies that, as recognized in [15-17], reliable theoretical predictions for forward jets can only be obtained after summing log-arithmic QCD corrections at high energy to all orders in αs(1). This motivates
ef-forts [22-25] to construct new, improved algorithms for Monte Carlo event generators capable of describing jet production beyond the central rapidity region.
In the LHC forward kinematics, realistic phenomenology of hadronic jet final states requires taking account of both logarithms of the large rapidity interval (of high-energy type) and logarithms of the hard transverse momentum (of collinear type). The theo-retical framework to resum consistently both kinds of logarithmic corrections in QCD calculations is based on high-energy factorization at fixed transverse momentum [14].
Reference [13] investigates forward jets in this framework. It presents the short-distance matrix elements needed to evaluate the factorization formula, including all par-tonic channels, in a fully exclusive form. On the one hand, once convoluted with the BFKL off-shell gluon Green’s function according to the method of [14], these matrix elements control the summation of high-energy logarithmic corrections to the jet cross-sections. They contain contributions both to the next-to-leading-order BFKL kernel [26] and to the jet impact factors [27, 28]. On the other hand, they can be used in a shower Monte Carlo generator implementing parton-branching kernels at unintegrated level (see,
e.g., [29, 30] for recent works) to generate fully exclusive events.
(1) Analogous observation applies to forward jets associated to deeply inelastic scattering [18, 19]. Indeed, measurements of forward jet cross-sections at Hera [20] have illustrated that either fixed-order next-to-leading calculations or standard shower Monte Carlos [20-22], e.g. Pythia or Herwig, are not able to describe forward jet ep data.
Fig. 2. – Left: High-p⊥ events in the forward and central detectors; right: azimuthal plane segmentation.
The high-energy factorized form [13, 14, 27] of the forward-jet cross-section is repre-sented in fig. 3a. Initial-state parton configurations contributing to forward production are asymmetric, with the parton in the top subgraph being probed near the mass shell and large x, while the parton in the bottom subgraph is off-shell and small-x. The jet cross-section differential in the final-state transverse momentum Qtand azimuthal angle ϕ is given schematically by [13, 14, 27] (1) dσ dQ2 tdϕ = a φa/A ⊗ dσ dQ2 tdϕ ⊗ φg∗/B,
where ⊗ specifies a convolution in both longitudinal and transverse momenta, σ is the hard-scattering cross-section, calculable from a suitable off-shell continuation of per-turbative matrix elements, φa/A is the distribution of parton a in hadron A obtained
from near-collinear shower evolution, and φg∗/B is the gluon unintegrated distribution
in hadron B obtained from non-collinear, transverse-momentum–dependent shower evo-lution.
In the next section we comment on the initial-state shower evolution. In sect. 4 we turn to hard-scattering contributions.
3. – Parton shower evolution
Parton distributions can be obtained by parton shower Monte Carlo methods via branching algorithms based on collinear evolution of the jets developing from the hard
(b) a/A Φg*/B 00000000000000000000000 00000000000000000000000 00000000000000000000000 00000000000000000000000 11111111111111111111111 11111111111111111111111 11111111111111111111111 11111111111111111111111 00000000000000000000000 00000000000000000000000 00000000000000000000000 00000000000000000000000 11111111111111111111111 11111111111111111111111 11111111111111111111111 11111111111111111111111 0000 0000 0000 0000 1111 1111 1111 1111 0000 0000 0000 0000 1111 1111 1111 1111 0000000 0000000 1111111 1111111 0000000 0000000 1111111 1111111 00 1101 000 111 00001111 000 111 00001111 (a) σ a g p 6 p p p p3 4 p 2 1 5 Φ
Fig. 3. – (a) Factorized structure of the cross-section; (b) a typical contribution to the qg channel matrix element.
Sudakov
form factors Δ
Fig. 4. – Parton branching in terms of splitting probabilities and form factors.
event [31]. The branching probability can be given in terms of two basic quantities (fig. 4), the splitting functions at the vertices of the parton cascade and the form factors to go from one vertex to the other. An important ingredient of this approach is the inclusion of soft-gluon coherence effects [31-33] through angular ordering of the emissions in the shower.
Corrections to collinear-ordered showers, however, arise in high-energy processes with multiple hard scales [7, 34, 35], as is the case with the production of jets at forward rapidities in fig. 1. In particular, new color-coherence effects set in this regime due to emissions from internal lines in the branching decay chain [7,27,36] that involve space-like partons carrying small longitudinal momentum fractions. The picture of the coherent branching is modified in this case because the emission currents become dependent on the total transverse momentum transmitted down the initial-state parton decay chain [14, 27, 34, 35, 37]. Correspondingly, one needs to work at the level of unintegrated splitting functions and partonic distributions [38, 39] in order to take into account color coherence not only for large x but also for small x in the angular region (fig. 5)
(2) α/x > α1> α,
where the angles α for the partons radiated from the initial-state shower are taken with respect to the initial beam jet direction, and increase with increasing off-shellness.
The case of LHC forward jet production is a multiple-scale problem where coherence effects of the kind above enter, in the factorization formula (1), both the short-distance factorσ and the long-distance factor φ. Contributions from the coherence region (2) are potentially enhanced by terms αn
slnm √
s/p⊥, where√s is the total center-of-mass energy
α
1
α
p
xp
P
P
P
+
+ ..
,
Fig. 5. – Left: coherent radiation in the space-like parton shower for x 1; right: the uninte-grated splitting functionP, including small-x virtual corrections.
and p⊥ is the jet transverse momentum(2). These contributions represent corrections to the angular ordering implemented in collinear showers and are not included at present in standard Monte Carlo generators [31]. Work to develop methods for unintegrated shower evolution, capable of including such corrections, is underway by several authors.
The proposal [29] incorporates NLO corrections to flavor non-singlet QCD evolution in an unintegrated-level Monte Carlo. The approach is based on the generalized ladder expansion of [41], which is extended to the high-energy region in [40]. This approach could in principle be applied generally, including flavor singlet evolution, and used to treat also forward hard processes.
Shower Monte Carlo generators based on small-x evolution equations, on the other hand, have typically included the unintegrated gluon distribution only [35, 37]. We ob-serve that unintegrated quark contributions can be incorporated for sea quarks via the transverse-momentum–dependent but universal splitting kernel given in [40], which has the structure (3) Pg→q(z; q⊥, k⊥) = Pqg(0)(z) 1 + ∞ n=1 bn(z)(k⊥2/q⊥2)n ,
where P(0) is the DGLAP splitting function, and all coefficients b
n are known. The
kernel (3) has been used for inclusive small-x calculations [42]. Its Monte Carlo imple-mentation is relevant to take into account effects from the unintegrated quark distribu-tion in the simuladistribu-tion of exclusive final states [43]. We note that quark contribudistribu-tions to evolution at the fully unintegrated level will also enter the treatment of the subleading high-energy corrections that are being discussed for jet production [28, 44].
Analyses of forward jet hadroproduction including parton showering effects are in progress [45].
4. – The factorizing hard cross-sections
Logarithmic corrections for large rapidity y ∼ ln s/p2
⊥ are resummed to all orders
in αs via eq. (1), by convoluting (fig. 3) unintegrated distribution functions with
well-prescribed short-distance matrix elements, obtained from the high-energy limit of higher-order scattering amplitudes [13, 27]. With reference to fig. 3b, in the forward production region we have (p4+ p6)2 (p3+ p4)2and longitudinal-momentum ordering, so that
(4) p5 (1 − ξ1)p1, p6 (1 − ξ2)p2− k⊥, ξ1 ξ2.
Here ξ1 and ξ2 are longitudinal-momentum fractions, and k⊥ is the di-jet transverse
momentum in the laboratory frame. It is convenient to define the rapidity-weighted average Q⊥ = (1− ν)p⊥4− νp⊥3, with ν = (p2· p4)/p2· (p1− p5). In fig. 3b eq. (1)
factorizes the high-energy qg amplitude in front of the (unintegrated) distribution from the splitting in the bottom subgraph. The factorization in terms of this parton splitting distribution is valid at large y not only in the collinear region but also in the large-angle
(2) Terms with m > n are known to drop out from inclusive processes due to strong cancellations associated with coherence, so that, for instance, the anomalous dimensions γij for space-like evolution receive at most single-logarithmic corrections at high energy [26, 40]. This need not be the case for exclusive jet distributions, where such cancellations are not present and one may expect larger enhancements.
10 10 10 10 kt 2Q t 2 1 10 2 10 3 10 4 10 5 10 Qt 4d 2 dφ dQ t 2 −5 −5 −5 −5 . . . . . −5 −1 0 −2 1 σ 10 10 10 10 kt2Qt2 Qt 4d 2σ dφ dQ t 2 −2 −1 0 −2 −2 −2 1 10 2 10 3 10 . . . 1
Fig. 6. – The (kt/Qt)-dependence of the factorizing qg hard cross-section at high energy [13]:
C2
F term (left); CFCA term (right).
emission region [14]. As a result the rapidity resummation is carried out consistently with perturbative high-Q⊥ corrections [14, 27] at any fixed order in αs.
The explicit expressions for the relevant high-energy amplitudes are given in [13]. Figures 6 and 7 illustrate features of the factorizing matrix elements, partially integrated over final states. We plot distributions differential in Q⊥and azimuthal angle ϕ (cos ϕ =
Q⊥· k⊥/|Q⊥||k⊥|) for the case of the qg channel. Figure 6 shows the dependence on
k⊥, which measures the distribution of the third jet recoiling against the leading di-jet system. Figure 7 shows the energy dependence.
The region k⊥/Q⊥ → 0 in fig. 6 corresponds to the leading-order process with
two back-to-back jets. The resummation of the higher-order logarithmic corrections for large y ∼ ln s/p2
⊥ is precisely determined [14, 27] by integrating the u-pdfs over the k⊥-distribution in fig. 6. So the results in fig. 6 illustrate quantitatively the significance of contributions with k⊥ Q⊥in the large-y region. The role of coherence from multi-gluon emission is to set the dynamical cut-off at values of k⊥of order Q⊥. Non-negligible effects arise at high energy from the finite-k⊥ tail. These effects are not included in collinear-branching generators (and only partially in fixed-order perturbative calculations), and become more and more important as the jets are observed at large rapidity separations. The dependence on the azimuthal angle in figs. 6 and 7 is also relevant, as forward-jet measurements will rely on azimuthal plane correlations between jets far apart in rapidity (fig. 2). 5. 6. 7. 8. 9. 10.sQt2 0.001 0.002 0.003 0.004 0.005 0.006 Qt4d2Σ d dQt 2 CF2term qg channelΑ4 2 1.5 1 4. 5. 6. 7. 8. 9. 10.sQt 2 0.005 0.010 0.015 0.020 Qt 2d2Σ d dQt 2 CACFterm qg channelΑ4 2 1.5 1
Results for all other partonic channels are given in [13]. It is worth stressing that quark and gluon contributions are of comparable size in the LHC forward kinematics [45]: realistic phenomenology requires including all channels. Note also that since the forward kinematics selects asymmetric parton momentum fractions, effects due to the x → 1 endpoint behavior [46] at the fully unintegrated level may become relevant as well.
Let us finally recall that if effects of high-density parton dynamics [10, 47] show up at the LHC, they will influence forward-jet event distributions. In such a case, the uninte-grated formalism discussed above would likely be the natural framework to implement this dynamics at parton shower level.
5. – Conclusion
Forward + central detectors at the LHC allow jet correlations to be measured across rapidity intervals of several units, Δy 4–6. Such multi-jet states can be relevant to new-particle discovery processes as well as new aspects of standard model physics.
Existing sets of forward-jet data in ep collisions, much more limited than the potential LHC yield, indicate that neither conventional parton-showering Monte Carlos nor next-to-leading-order QCD calculations are capable of describing forward-jet phenomenology. Improved methods to evaluate QCD are needed to treat the multi-scale region implied by the forward kinematics.
In this article we have discussed ongoing progress, examining in particular factoriza-tion properties of multi-parton matrix elements in the forward region, and prospects to include parton-showering effects with gluon coherence not only in the collinear region but also in the large-angle emission region.
∗ ∗ ∗
I thank M. Greco, the conference organizers and the conference staff for the kind invitation and for the nice atmosphere at the meeting. The results presented in this article have been obtained in collaboration with M. Deak, H. Jung and K. Kutak.
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