Fully-Unintegrated Parton Distribution and Fragmentation Functions at Perturbative k_T

We define and study the properties of generalized beam functions (BFs) and fragmenting jet functions (FJFs), which are fully-unintegrated parton distribution functions (PDFs) and fragmentation functions (FFs) for perturbative k_T. We calculate at one loop the coefficients for matching them onto standard PDFs and FFs, correcting previous results for the BFs in the literature. Technical subtleties when measuring transverse momentum in dimensional regularization are clarified, and this enables us to renormalize in momentum space. Generalized BFs describe the distribution in the full four-momentum k_mu of a colliding parton taken out of an initial-state hadron, and therefore characterize the collinear initial-state radiation. We illustrate their importance through a factorization theorem for pp -> l^+ l^- + 0 jets, where the transverse momentum of the lepton pair is measured. Generalized FJFs are relevant for the analysis of semi-inclusive processes where the full momentum of a hadron, fragmenting from a jet with constrained invariant mass, is measured. Their significance is shown for the example of e^+ e^- -> dijet+h, where the perpendicular momentum of the fragmenting hadron with respect to the thrust axis is measured.Comment: Journal versio

Non-global Structure of the O({\alpha}_s^2) Dijet Soft Function

High energy scattering processes involving jets generically involve matrix elements of light- like Wilson lines, known as soft functions. These describe the structure of soft contributions to observables and encode color and kinematic correlations between jets. We compute the dijet soft function to O({\alpha}_s^2) as a function of the two jet invariant masses, focusing on terms not determined by its renormalization group evolution that have a non-separable dependence on these masses. Our results include non-global single and double logarithms, and analytic results for the full set of non-logarithmic contributions as well. Using a recent result for the thrust constant, we present the complete O({\alpha}_s^2) soft function for dijet production in both position and momentum space.Comment: 55 pages, 8 figures. v2: extended discussion of double logs in the hard regime. v3: minor typos corrected, version published in JHEP. v4: typos in Eq. (3.33), (3.39), (3.43) corrected; this does not affect the main result, numerical results, or conclusion

Parton Fragmentation within an Identified Jet at NNLL

The fragmentation of a light parton i to a jet containing a light energetic hadron h, where the momentum fraction of this hadron as well as the invariant mass of the jet is measured, is described by "fragmenting jet functions". We calculate the one-loop matching coefficients J_{ij} that relate the fragmenting jet functions G_i^h to the standard, unpolarized fragmentation functions D_j^h for quark and gluon jets. We perform this calculation using various IR regulators and show explicitly how the IR divergences cancel in the matching. We derive the relationship between the coefficients J_{ij} and the quark and gluon jet functions. This provides a cross-check of our results. As an application we study the process e+ e- to X pi+ on the Upsilon(4S) resonance where we measure the momentum fraction of the pi+ and restrict to the dijet limit by imposing a cut on thrust T. In our analysis we sum the logarithms of tau=1-T in the cross section to next-to-next-to-leading-logarithmic accuracy (NNLL). We find that including contributions up to NNLL (or NLO) can have a large impact on extracting fragmentation functions from e+ e- to dijet + h.Comment: expanded introduction, typos fixed, journal versio

The Quark Beam Function at NNLL

In hard collisions at a hadron collider the most appropriate description of the initial state depends on what is measured in the final state. Parton distribution functions (PDFs) evolved to the hard collision scale Q are appropriate for inclusive observables, but not for measurements with a specific number of hard jets, leptons, and photons. Here the incoming protons are probed and lose their identity to an incoming jet at a scale \mu_B << Q, and the initial state is described by universal beam functions. We discuss the field-theoretic treatment of beam functions, and show that the beam function has the same RG evolution as the jet function to all orders in perturbation theory. In contrast to PDF evolution, the beam function evolution does not mix quarks and gluons and changes the virtuality of the colliding parton at fixed momentum fraction. At \mu_B, the incoming jet can be described perturbatively, and we give a detailed derivation of the one-loop matching of the quark beam function onto quark and gluon PDFs. We compute the associated NLO Wilson coefficients and explicitly verify the cancellation of IR singularities. As an application, we give an expression for the next-to-next-to-leading logarithmic order (NNLL) resummed Drell-Yan beam thrust cross section.Comment: 54 pages, 9 figures; v2: notation simplified in a few places, typos fixed; v3: journal versio

An effective theory for jet propagation in dense QCD matter: jet broadening and medium-induced bremsstrahlung

Two effects, jet broadening and gluon bremsstrahlung induced by the propagation of a highly energetic quark in dense QCD matter, are reconsidered from effective theory point of view. We modify the standard Soft Collinear Effective Theory (SCET) Lagrangian to include Glauber modes, which are needed to implement the interactions between the medium and the collinear fields. We derive the Feynman rules for this Lagrangian and show that it is invariant under soft and collinear gauge transformations. We find that the newly constructed theory SCET$_{\rm G}$ recovers exactly the general result for the transverse momentum broadening of jets. In the limit where the radiated gluons are significantly less energetic than the parent quark, we obtain a jet energy-loss kernel identical to the one discussed in the reaction operator approach to parton propagation in matter. In the framework of SCET$_{\rm G}$ we present results for the fully-differential bremsstrahlung spectrum for both the incoherent and the Landau-Pomeranchunk-Migdal suppressed regimes beyond the soft-gluon approximation. Gauge invariance of the physics results is demonstrated explicitly by performing the calculations in both the light-cone and covariant $R_{\xi}$ gauges. We also show how the process-dependent medium-induced radiative corrections factorize from the jet production cross section on the example of the quark jets considered here.Comment: 52 pages, 15 pdf figures, as published in JHE

Jet Shapes and Jet Algorithms in SCET

Jet shapes are weighted sums over the four-momenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its partonic origin. In this work we make predictions for quark and gluon jet shape distributions in N-jet final states in e+e- collisions, defined with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of Soft-Collinear Effective Theory, we prove a factorization theorem for jet shape distributions and demonstrate the consistent renormalization-group running of the functions in the factorization theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size R of the jets, as long as R is much smaller than the angular separation between jets. We calculate the jet and soft functions for angularity jet shapes \tau_a to one-loop order (O(alpha_s)) and resum a subset of the large logarithms of \tau_a needed for next-to-leading logarithmic (NLL) accuracy for both cone and kT-type jets. We compare our predictions for the resummed \tau_a distribution of a quark or a gluon jet produced in a 3-jet final state in e+e- annihilation to the output of a Monte Carlo event generator and find that the dependence on a and R is very similar.Comment: 62 pages plus 21 pages of Appendices, 13 figures, uses JHEP3.cls. v2: corrections to finite parts of NLO jet functions, minor changes to plots, clarified discussion of power corrections. v3: Journal version. Introductory sections significantly reorganized for clarity, classification of logarithmic accuracy clarified, results for non-Mercedes-Benz configurations adde

Factorization and NNLL Resummation for Higgs Production with a Jet Veto

Using methods of effective field theory, we derive the first all-order factorization theorem for the Higgs-boson production cross section with a jet veto, imposed by means of a standard sequential recombination jet algorithm. Like in the case of small-q_T resummation in Drell-Yan and Higgs production, the factorization is affected by a collinear anomaly. Our analysis provides the basis for a systematic resummation of large logarithms log(m_H/p_T^veto) beyond leading-logarithmic order. Specifically, we present predictions for the resummed jet-veto cross section and efficiency at next-to-next-to-leading logarithmic order. Our results have important implications for Higgs-boson searches at the LHC, where a jet veto is required to suppress background events.Comment: 28 pages, 5 figures; v2: published version; note added in proo

Factorization and resummation of s-channel single top quark production

In this paper we study the factorization and resummation of s-channel single top quark production in the Standard Model at both the Tevatron and the LHC. We show that the production cross section in the threshold limit can be factorized into a convolution of hard function, soft function and jet function via soft-collinear-effective-theory (SCET), and resummation can be performed using renormalization group equation in the momentum space resummation formalism. We find that in general, the resummation effects enhance the Next-to-Leading-Order (NLO) cross sections by about $3$ at both the Tevatron and the LHC, and significantly reduce the factorization scale dependence of the total cross section at the Tevatron, while at the LHC we find that the factorization scale dependence has not been improved, compared with the NLO results.Comment: 29 pages, 7 figures; version published in JHE

A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory

Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the rele- vant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group". That is, a Collin-Soper like equation is realized as a renormalization group equation, but has a more universal applicability to observables beyond the traditional transverse momentum dependent parton distribution functions (TMDPDFs) and the Sudakov form factor. This formalism has the feature that it allows one to track the (non-standard) scheme dependence which is inherent in any scenario where one performs a resummation of rapidity divergences. We present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form factor. The formalism is then used to study observables of current interest. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. Our formalism allows one to define gauge invariant TMDPDFs which are independent of both the hard scattering amplitude and the soft function, i.e. they are uni- versal. We present details of the factorization and resummation of the jet broadening cross section including a renormalization in pT space. We furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.Comment: Typos in Appendix C corrected, as well as a typo in eq. 5.6

Double Non-Global Logarithms In-N-Out of Jets

We derive the leading non-global logarithms (NGLs) of ratios of jet masses m_{1,2} and a jet energy veto \Lambda due to soft gluons splitting into regions in and out of jets. Such NGLs appear in any exclusive jet cross section with multiple jet measurements or with a veto imposed on additional jets. Here, we consider back-to-back jets of radius R produced in e^+e^- collisions, found with a cone or recombination algorithm. The leading NGLs are of the form \alpha_s^2 \ln^2(\Lambda/m_{1,2}) or \alpha_s^2\ln^2(m_1/m_2). Their coefficients depend both on the algorithm and on R. We consider cone, \kt, anti-\kt, and Cambridge-Aachen algorithms. In addition to determining the full algorithmic and R dependence of the leading NGLs, we derive new relations among their coefficients. We also derive to all orders in \alpha_s a factorized form for the soft function S(k_L,k_R,\Lambda) in the cross section \sigma(m_1,m_2,\Lambda) in which dependence on each of the global logs of \mu/k_L, \mu/k_R and \mu/\Lambda determined by the renormalization group are separated from one another and from the non-global logs. The same kind of soft function, its associated non-global structure, and the algorithmic dependence we derive here will also arise in exclusive jet cross sections at hadron colliders, and must be understood and brought under control to achieve precise theoretical predictions.Comment: 19 pages, 10 figures. v2: minor edits, additional discussion in Introduction. v3: version published in JHE