Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD

At small transverse momentum $q_T$, transverse-momentum dependent parton distribution functions (TMDPDFs) arise as genuinely nonperturbative objects that describe Drell-Yan like processes in hadron collisions as well as semi-inclusive deep-inelastic scattering. TMDPDFs naturally depend on the hadron momentum, and the associated evolution is determined by the Collins-Soper equation. For $q_T \sim \Lambda_\mathrm{QCD}$ the corresponding evolution kernel (or anomalous dimension) is nonperturbative and must be determined as an independent ingredient in order to relate TMDPDFs at different scales. We propose a method to extract this kernel using lattice QCD and the Large-Momentum Effective Theory, where the physical TMD correlation involving light-like paths is approximated by a quasi TMDPDF, defined using equal-time correlation functions with a large-momentum hadron state. The kernel is determined from a ratio of quasi TMDPDFs extracted at different hadron momenta.Comment: 9 pages, 2 figures; v2: extended the review of TMDPDF commonalities, version submitted to PRD; v3: minor changes, journal versio

Resummation Improved Rapidity Spectrum for Gluon Fusion Higgs Production

Gluon-induced processes such as Higgs production typically exhibit large perturbative corrections. These partially arise from large virtual corrections to the gluon form factor, which at timelike momentum transfer contains Sudakov logarithms evaluated at negative arguments $\ln^2(-1) = -\pi^2$. It has been observed that resumming these terms in the timelike form factor leads to a much improved perturbative convergence for the total cross section. We discuss how to consistently incorporate the resummed form factor into the perturbative predictions for generic cross sections differential in the Born kinematics, including in particular the Higgs rapidity spectrum. We verify that this indeed improves the perturbative convergence, leading to smaller and more reliable perturbative uncertainties, and that this is not affected by cancellations between resummed and unresummed contributions. Combining both fixed-order and resummation uncertainties, the perturbative uncertainty for the total cross section at N$^3$LO$+$N$^3$LL$^\prime_\varphi$ is about a factor of two smaller than at N$^3$LO. The perturbative uncertainty of the rapidity spectrum at NNLO$+$NNLL$^\prime_\varphi$ is similarly reduced compared to NNLO. We also study the analogous resummation for quark-induced processes, namely Higgs production through bottom quark annihilation and the Drell-Yan rapidity spectrum. For the former the resummation leads to a small improvement, while for the latter it confirms the already small uncertainties of the fixed-order predictions.Comment: 30 pages + 17 pages in Appendices, 10 figures; v2: journal version; references added, discussed individual partonic channels for Drell-Ya

Transverse momentum dependent PDFs at N3^3LO

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. Our calculation is based on an expansion of the differential Higgs boson and Drell-Yan production cross sections about their collinear limit. This method allows us to employ cutting edge techniques for the computation of cross sections to extract the universal building blocks in question. The corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N$^3$LO. Our results are the last missing ingredient to extend the $q_T$ subtraction methods to N$^3$LO and to obtain resummed $q_T$ spectra at N$^3$LL$^\prime$ accuracy both for gluon as well as for quark initiated processes.Comment: 12 pages + appendices, 4 awesome figures, important ancillary files. v2: journal versio

TMD Fragmentation Functions at N3^3LO

We compute the unpolarized quark and gluon transverse-momentum dependent fragmentation functions (TMDFFs) at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. The calculation is based on a relation between the TMDFF and the limit of the semi-inclusive deep inelastic scattering cross section where all final-state radiation becomes collinear to the detected hadron. The required cross section is obtained by analytically continuing our recent computation of the Drell-Yan and Higgs boson production cross section at N$^3$LO expanded around the limit of all final-state radiation becoming collinear to one of the initial states. Our results agree with a recent independent calculation by Luo et al.Comment: 20 pages + appendices, 3 figures, 6 ancillary files; v2: journal versio

Collinear expansion for color singlet cross sections

We demonstrate how to efficiently expand cross sections for color-singlet production at hadron colliders around the kinematic limit of all final state radiation being collinear to one of the incoming hadrons. This expansion is systematically improvable and applicable to a large class of physical observables. We demonstrate the viability of this technique by obtaining the first two terms in the collinear expansion of the rapidity distribution of the gluon fusion Higgs boson production cross section at next-to-next-to leading order (NNLO) in QCD perturbation theory. Furthermore, we illustrate how this technique is used to extract universal building blocks of scattering cross section like the N-jettiness and transverse momentum beam function at NNLO.Comment: 39 pages, 1 awesome figure; v2: journal versio

Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice

Transverse momentum dependent parton distributions (TMDPDFs) which appear in factorized cross sections involve infinite Wilson lines with edges on or close to the light-cone. Since these TMDPDFs are not directly calculable with a Euclidean path integral in lattice QCD, we study the construction of quasi-TMDPDFs with finite-length spacelike Wilson lines that are amenable to such calculations. We define an infrared consistency test to determine which quasi-TMDPDF definitions are related to the TMDPDF, by carrying out a one-loop study of infrared logarithms of transverse position $b_T\sim \Lambda_{\rm QCD}^{-1}$, which must agree between them. This agreement is a necessary condition for the two quantities to be related by perturbative matching. TMDPDFs necessarily involve combining a hadron matrix element, which nominally depends on a single light-cone direction, with soft matrix elements that necessarily depend on two light-cone directions. We show at one loop that the simplest definitions of the quasi hadron matrix element, the quasi soft matrix element, and the resulting quasi-TMDPDF all fail the infrared consistency test. Ratios of impact parameter quasi-TMDPDFs still provide nontrivial information about the TMDPDFs, and are more robust since the soft matrix elements cancel. We show at one loop that such quasi ratios can be matched to ratios of the corresponding TMDPDFs. We also introduce a modified "bent" quasi soft matrix element which yields a quasi-TMDPDF that passes the consistency test with the TMDPDF at one loop, and discuss potential issues at higher orders.Comment: 39 pages + appendices, 13 figures; v2: journal versio

Exploiting jet binning to identify the initial state of high-mass resonances

If a new high-mass resonance is discovered at the Large Hadron Collider, model-independent techniques to identify the production mechanism will be crucial to understand its nature and effective couplings to Standard Model particles. We present a powerful and model-independent method to infer the initial state in the production of any high-mass color-singlet system by using a tight veto on accompanying hadronic jets to divide the data into two mutually exclusive event samples (jet bins). For a resonance of several hundred GeV, the jet binning cut needed to discriminate quark and gluon initial states is in the experimentally accessible range of several tens of GeV. It also yields comparable cross sections for both bins, making this method viable already with the small event samples available shortly after a discovery. Theoretically, the method is made feasible by utilizing an effective field theory setup to compute the jet cut dependence precisely and model independently and to systematically control all sources of theoretical uncertainties in the jet binning, as well as their correlations. We use a 750 GeV scalar resonance as an example to demonstrate the viability of our method.Comment: 6 pages, 2 figures, v2: journal versio

Electronic structure of fully epitaxial Co2TiSn thin films

In this article we report on the properties of thin films of the full Heusler compound Co2TiSn prepared by DC magnetron co-sputtering. Fully epitaxial, stoichiometric films were obtained by deposition on MgO (001) substrates at substrate temperatures above 600{\deg}C. The films are well ordered in the L21 structure, and the Curie temperature exceeds slightly the bulk value. They show a significant, isotropic magnetoresistance and the resistivity becomes strongly anomalous in the paramagnetic state. The films are weakly ferrimagnetic, with nearly 1 \mu_B on the Co atoms, and a small antiparallel Ti moment, in agreement with theoretical expectations. From comparison of x-ray absorption spectra on the Co L3/L2 edges, including circular and linear magnetic dichroism, with ab initio calculations of the x-ray absorption and circular dichroism spectra we infer that the electronic structure of Co2TiSn has essentially non-localized character. Spectral features that have not been explained in detail before, are explained here in terms of the final state band structure.Comment: 11 pages, 8 figure

One-loop Matching for Spin-Dependent Quasi-TMDs

Transverse momentum dependent parton distribution functions (TMDPDFs) provide a unique probe of the three-dimensional spin structure of hadrons. We construct spin-dependent quasi-TMDPDFs that are amenable to lattice QCD calculations and that can be used to determine spin-dependent TMDPDFs. We calculate the short-distance coefficients connecting spin-dependent TMDPDFs and quasi-TMDPDFs at one-loop order. We find that the helicity and transversity distributions have the same coefficient as the unpolarized TMDPDF. We also argue that the same is true for pretzelosity and that this spin universality of the matching will hold to all orders in $\alpha_s$. Thus, it is possible to calculate ratios of these distributions as a function of longitudinal momentum and transverse position utilizing simpler Wilson line paths than have previously been considered.Comment: 24 pages, 4 figure

Renormalization and Matching for the Collins-Soper Kernel from Lattice QCD

The Collins-Soper kernel, which governs the energy evolution of transverse-momentum dependent parton distribution functions (TMDPDFs), is required to accurately predict Drell-Yan like processes at small transverse momentum, and is a key ingredient for extracting TMDPDFs from experiment. Earlier we proposed a method to calculate this kernel from ratios of the so-called quasi-TMDPDFs determined with lattice QCD, which are defined as hadronic matrix elements of staple-shaped Euclidean Wilson line operators. Here we provide the one-loop renormalization of these operators in a regularization-independent momentum subtraction (RI$^\prime$/MOM) scheme, as well as the conversion factor from the RI$^\prime$/MOM-renormalized quasi-TMDPDF to the $\overline{\rm MS}$ scheme. We also propose a procedure for calculating the Collins-Soper kernel directly from position space correlators, which simplifies the lattice determination.Comment: 27 pages + appendices, 5 figures; v2: journal versio