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

Gauge links for transverse momentum dependent correlators at tree-level

In this paper we discuss the incorporation of gauge links in hadronic matrix elements that describe the soft hadronic physics in high energy scattering processes. In this description the matrix elements appear in soft correlators and they contain non-local combinations of quark and gluon fields. In our description we go beyond the collinear approach in which case also the dependence on transverse momenta of partons is taken into consideration. The non-locality in the transverse direction leads to a complex gauge link structure for the full process, in which color is entangled, even at tree-level. We show that at tree-level in a 1-parton unintegrated (1PU) situation, in which only the transverse momentum of one of the initial state hadrons is relevant, one can get a factorized expression involving transverse momentum dependent (TMD) distribution functions. We point out problems at the level of two initial state hadrons, even for relatively simple processes such as Drell-Yan scattering.Comment: 25 pages, corrected typos and updated reference

On Glauber modes in Soft-Collinear Effective Theory

Gluon interactions involving spectator partons in collisions at hadronic machines are investigated. We find a class of examples in which a mode, called Glauber gluons, must be introduced to the effective theory for consistency.Comment: 19 pages, three figures. Uses JHEP3.cl

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

From weak coupling to spinning strings

We identify the gauge theory dual of a spinning string of minimal energy with spins S_1, S_2 on AdS_5 and charge J on S^5. For this purpose we focus on a certain set of local operators with two different types of covariant derivatives acting on complex scalar fields. We analyse the corresponding nested Bethe equations for the ground states in the limit of large spins. The auxiliary Bethe roots form certain string configurations in the complex plane, which enable us to derive integral equations for the leading and sub-leading contribution to the anomalous dimension. The results can be expressed through the observables of the sl(2) sub-sector, i.e. the cusp anomaly f(g) and the virtual scaling function B_L(g), rendering the strong-coupling analysis straightforward. Furthermore, we also study a particular sub-class of these operators specialising to a scaling limit with finite values of the second spin at weak and strong coupling.Comment: 23 pages, 3 figures, minor changes, references adde

Soft-Gluon-Pole Contribution in Single Transverse-Spin Asymmetries of Drell-Yan Processes

We use multi-parton states to examine the leading order collinear factorization of single transverse-spin asymmetries in Drell-Yan processes. Twist-3 operators are involved in the factorization. We find that the so-called soft-gluon-pole contribution in the factorization must exist in order to make the factorization correct. This contribution comes from the corresponding cross-section at one-loop, while the hard-pole contribution in the factorization comes from the cross-section at tree-level. Although the two contributions come from results at different orders, their perturbative coefficient functions in the factorization are at the same order. This is in contrast to factorizations only involving twist-2 operators. The soft-gluon-pole contribution found in this work is in agreement with that derived in a different way. For the hard-pole contributions we find an extra contribution from an extra parton process contributing to the asymmetries. We also solve a part of discrepancy in evolutions of the twist-3 operator. The method presented here for analyzing the factorization can be generalized to other processes and can be easily used for studying factorizations at higher orders, because the involved calculations are of standard scattering amplitudes.Comment: typos eliminated. Published in JHEP 1104:062,201

Renormalisation of heavy-light light ray operators

We calculate the renormalisation of different light ray operators with one light degree of freedom and a static heavy quark. Both $2\to2$- and $2\to3$-kernels are considered. A comparison with the light-light case suggests that the mixing with three-particle operators is solely governed by the light degrees of freedom. Additionally we show that conformal symmetry is already broken at the level of the one loop counterterms due to the additional UV-renormalisation of a cusp in the two contributing Wilson-lines. This general feature can be used to fix the $2\to2$-renormalisation kernels up to a constant. Some examples for applications of our results are given.Comment: 23 pages, 5 figures; v2: changed some wording, added a few references and one appendix concerning some subtleties related to gauge fixing and ghost terms; v3: clarified calculation in section 3.2., added an explicit calculation in section 5.2, corrected a few typos and one figure, added a few comments, results unchanged, except for typesetting matches version to appear 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

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

The COL5A1 gene and musculoskeletal soft-tissue injuries

Background. It has been shown that there is an association between various genetic variants and Achilles tendon injuries as well as anterior cruciate ligament (ACL) ruptures. Among other variants the BstUI restriction fragment length polymorphism (RFLP) within the COL5A1 gene has been shown to be over-represented in asymptomatic participants when compared with those with chronic Achilles tendinopathy, and in asymptomatic female participants when compared with those with ACL ruptures. The male asymptomatic control participants in the ACL study, which were 10 years younger than previously investigated cohorts, had a distinctly different genotype frequency. Aim. The aim of this study was therefore to determine whether the distribution of the COL5A1 BstUI RFLP in the combined asymptomatic participants without any known history of tendon injuries is age dependent, particularly among males. Results. When the 265 male asymptomatic participants from all studies were pooled and divided into age-group tertiles, there was a significant linear increase in the CC genotype frequency (p=0.032) among the male age groups, with the youngest group having the lowest frequency (CC genotype frequency, 13%) and the oldest group having the highest (CC genotype frequency, 27%) frequency. There was however a similar CC genotype content in all three female (N=231) age groups (CC genotype frequency, 24 - 27%; p=0.795). Conclusion. The practical implication is that the selection of asymptomatic groups is of critical importance when future studies of this nature are designed. Future research investigating this genetic variant as a risk factor for soft-tissue injuries should consider these findings when selecting asymptomatic participants