Search

Energy correlators provide a powerful observable to study fragmentation dynamics in QCD. We demonstrate that the leading nonperturbative corrections for projected $N$-point energy correlators are described by the same universal parameter for any $N$, which has already been determined from other event shape fits. Including renormalon-free nonperturbative corrections substantially improves theoretical predictions of energy correlators, notably the transition into the confining region at small angles. Nonperturbative corrections are shown to have a significant impact on $\alpha_s$ extractions., Comment: 5 pages, 4 figures

We update and extend a previous N$^3$LL$^\prime$+${\cal O}(\alpha_s^3)$ strong coupling determination from thrust data. In particular, we carry out a fit with data fully restricted to the dijet region seeking to minimize the potential impact of power corrections that go beyond dijet configurations. In addition, we parametrize deviations from the dijet power correction in order to add an additional source of uncertainty in the result for $\alpha_s(m_Z)$. We also show that the inclusion of resummation is important to achieve stability with respect to varying the fit region., Comment: 4 pages, 3 figures. Contribution to the 2024 QCD session of the 58th Rencontres de Moriond

The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: Bankruptcy Minimization/Maximization, in which the aim is to produce a payment schedule with at most/at least a given number of bankruptcies; Perfect Scheduling, the special case of the minimization variant where the aim is to produce a schedule with no bankruptcies (that is, a perfect schedule); and Bailout Minimization, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. We show that each of these problems is NP-complete, in many cases even on very restricted input instances. On the positive side, we provide for Perfect Scheduling a polynomial-time algorithm on (rooted) out-trees although in contrast we prove NP-completeness on directed acyclic graphs, as well as on instances with a constant number of nodes (and hence also constant treewidth). When we allow non-integer payments, we show by a linear programming argument that the problem Bailout Minimization can be solved in polynomial time., Comment: 30 pages, 17 figures

The Reconfigurable Routing Problem (RRP) in hybrid networks is, in short, the problem of finding settings for optical switches augmenting a static network so as to achieve optimal delivery of some given workload. The problem has previously been studied in various scenarios with both tractable and NP-hardness results obtained. However, the data center and interconnection networks to which the problem is most relevant are almost always such that the static network is highly structured whereas all previous results assume that the static network can be arbitrary (which makes existing computational hardness results less technologically relevant and also easier to obtain). In this paper, and for the first time, we prove various intractability results for RRP where the underlying static network is highly structured, for example consisting of a hypercube, and also extend some existing tractability results., Comment: 30 pages, 6 figures

The high energy (Regge) limit provides a playground for understanding all loop structures of scattering amplitudes, and plays an important role in the description of many phenomenologically relevant cross-sections. While well understood in the planar limit, the structure of non-planar corrections introduces many fascinating complexities, for which a general organizing principle is still lacking. We study the structure of multi-reggeon exchanges in the context of the effective field theory for forward scattering, and derive their factorization into collinear operators (impact factors) and soft operators. We derive the structure of the renormalization group consistency equations in the effective theory, showing how the anomalous dimensions of the soft operators are related to those of the collinear operators, allowing us to derive renormalization group equations in the Regge limit purely from a collinear perspective. The rigidity of the consistency equations provides considerable insight into the all orders organization of Regge amplitudes in the effective theory, as well as its relation to other approaches. Along the way we derive a number of technical results that improve the understanding of the effective theory. We illustrate this collinear perspective by re-deriving all the standard BFKL equations for two-Glauber exchange from purely collinear calculations, and we show that this perspective provides a number of conceptual and computational advantages as compared to the standard view from soft or Glauber physics. We anticipate that this formulation in terms of collinear operators will enable a better understanding of the relation between BFKL and DGLAP in gauge theories, and facilitate the analysis of renormalization group evolution equations describing Reggeization beyond next-to-leading order., Comment: 52 pages, 8 figures

The heavy jet mass event shape has large perturbative logarithms near the leading order kinematic threshold at $\rho = \frac{1}{3}$. Catani and Webber named these logarithms Sudakov shoulders and resummed them at double-logarithmic level. A resummation to next-to-leading logarithmic level was achieved recently. Here, we extend the resummation using an effective field theory framework to next-to-next-to-leading logarithmic order and show how to combine it with the resummation of dijet logarithms. We also solve the open problem of an unphysical singularity in the resummed momentum space distribution, in a way similar to how it is resolved in the Drell-Yan $q_T$ spectrum: through a careful analysis of the kinematics and scale-setting in position space. The heavy jet mass Sudakov shoulder is the first observable that does not involve transverse momentum for which position space resummation is critical. These advances may lead to a more precise extraction of the strong coupling constant from $e^+ e^-$ data., Comment: 33 pages + appendices, 10 figures

The energy-energy correlator (EEC) is an observable of wide interest for collider physics and Standard Model measurements, due to both its simple theoretical description in terms of the energy-momentum tensor and its novel features for experimental studies. Significant progress has been made in both applications and higher-order perturbative predictions for the EEC. Here, we analyze the nature of the asymptotic perturbative series for the EEC by determining its analytic form in Borel space under the bubble-sum approximation. This result provides information on the leading and subleading nonperturbative power corrections through renormalon poles. We improve the perturbative convergence of the $\overline{\mathrm{MS}}$ series for the EEC by removing its leading renormalon using an R scheme, which is independent of the bubble-sum approximation. Using the leading R-scheme power correction determined by fits to thrust, we find good agreement with EEC OPAL data already at ${\mathcal O}(\alpha_s^2)$., Comment: 19 pages + references, 6 figures; v2: update to fix a typo in the plotting routine for Figure 6

We present a description of saturation in small $x$ deep inelastic scattering from power counting in a top-down effective theory derived from QCD. A factorization formula isolates the universal physics of the nucleus at leading power in $x$. The onset of saturation is then understood as a breakdown in the expansion in an emergent power counting parameter, which is defined by the matrix element of a gauge invariant operator. We identify a new radiation mode, which enables us to extend previous literature by distinguishing the appearance of the saturation scale from the transition to non-linear evolution., Comment: 5 pages, 2 figures, Version to appear in PRD letters

We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize this ambitious and impactful physics program, including how to engage a diverse and inclusive workforce. In order to address these many-fold challenges, we propose a coordinated effort involving theory groups with differing expertise is needed. We discuss the scientific goals and scope of such an EIC Theory Alliance., Comment: 44 pages, ReVTeX, White Paper on EIC Theory Alliance

This handbook provides a comprehensive review of transverse-momentum-dependent parton distribution functions and fragmentation functions, commonly referred to as transverse momentum distributions (TMDs). TMDs describe the distribution of partons inside the proton and other hadrons with respect to both their longitudinal and transverse momenta. They provide unique insight into the internal momentum and spin structure of hadrons, and are a key ingredient in the description of many collider physics cross sections. Understanding TMDs requires a combination of theoretical techniques from quantum field theory, nonperturbative calculations using lattice QCD, and phenomenological analysis of experimental data. The handbook covers a wide range of topics, from theoretical foundations to experimental analyses, as well as recent developments and future directions. It is intended to provide an essential reference for researchers and graduate students interested in understanding the structure of hadrons and the dynamics of partons in high energy collisions., Comment: 471 pages, many figures

We derive a factorization theorem that allows for resummation of small-$x$ logarithms by exploiting Glauber operators in the soft collinear effective field theory. Our analysis is carried out for the hadronic tensor $W^{\mu\nu}$ in deep inelastic scattering, and leads to the definition of a new gauge invariant soft function $S^{\mu\nu}$ that describes quark and gluon emission in the central region. This soft function provides a new framework for extending resummed calculations for coefficient functions to higher logarithmic orders. Our factorization also defines impact factors by universal collinear functions that are process independent, for instance being identical in small-$x$ DIS and Drell-Yan., Comment: 43 pgs

This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed., Comment: Invited volume for the EJPC; 567 pages if text/figures and 4783 references occupying about 160 additional pages. arXiv abstract abridged, for the complete abstract please see the full text

We compute the soft-drop jet-mass distribution from $pp$ collisions to NNLL accuracy while including nonperturbative corrections through a field-theory based formalism. Using these calculations, we assess the theoretical uncertainties on an $\alpha_s$ precision measurement due to higher order perturbative effects, nonperturbative corrections, and PDF uncertainty. We identify which soft-drop parameters are well-suited for measuring $\alpha_s$, and find that higher-logarithmic resummation has a qualitatively important effect on the shape of the jet-mass distribution. We find that quark jets and gluon jets have similar sensitivity to $\alpha_s$, and emphasize that experimentally distinguishing quark and gluon jets is not required for an $\alpha_s$ measurement. We conclude that measuring $\alpha_s$ to the 10% level is feasible now, and with improvements in theory a 5% level measurement is possible. Getting down to the 1% level to be competitive with other state-of-the-art measurements will be challenging., Comment: 53 pages + appendices, 15 figures

We propose an observable $q_*$ sensitive to transverse momentum dependence (TMD) in $e N \to e h X$, with $q_*/E_N$ defined purely by lab-frame angles. In 3D measurements of confinement and hadronization this resolves the crippling issue of accurately reconstructing small transverse momentum $P_{hT}$. We prove factorization for $\mathrm{d} \sigma_h / \mathrm{d}q_*$ for $q_*\ll Q$ with standard TMD functions, enabling $q_*$ to substitute for $P_{hT}$. A double-angle reconstruction method is given which is exact to all orders in QCD for $q_*\ll Q$. $q_*$ enables an order-of-magnitude improvement in the expected experimental resolution at the EIC., Comment: 5 pages + 4 figures, 9 pages of supplemental material

Using an effective field theory (EFT) formalism for forward scattering, we reconsider the factorization of $2\to 2$ scattering amplitudes in the Regge limit. Expanding the amplitude in gauge invariant operators labelled by the number of Glauber exchanges, allows us to further factorize the standard impact factors into separate collinear and soft functions. The soft functions are universal, and describe radiative corrections to the Reggeized gluon states exchanged by the collinear projectiles. Remarkably, we find that the one-loop soft function for the single Reggeized gluon state is given to $\mathcal{O}(\epsilon)$ in terms of the two-loop cusp and two-loop rapidity anomalous dimensions. We argue that this iterative structure follows from the simple action of crossing symmetry in the forward scattering limit, which in the EFT allows us to replace the divergent part of a soft loop by a much simpler Glauber loop. We use this correspondence to provide a simple calculation of the two-loop Regge trajectory using the EFT. We then explore its implications at higher perturbative orders, and derive the maximally matter dependent contributions to the Regge trajectory to all loop orders, i.e.~the terms $\sim \alpha_s^{k+1}n_f^k$ for any $k$, where $n_f$ is the number of massless flavors. These simplifications suggests that the EFT approach to the Regge limit will be helpful to explore and further understand the structure of the Regge limit., Comment: 29 pages, 5 figures

Transverse-momentum-dependent parton distributions (TMDs) can be calculated from first principles by computing a related set of Euclidean lattice observables and connecting them via a factorization formula. This work focuses on the leading-power factorization formula connecting the lattice quasi-TMD and continuum Collins TMD for gluons. We calculate the one-loop gluon matching coefficient, which is known to be independent of spin and exhibits no mixing with quarks. We demonstrate that this coefficient satisfies Casimir scaling with respect to the quark matching coefficient at one-loop order. Our result facilitates reliable lattice QCD calculations of gluon TMDs., Comment: 11 pages, 2 figures, v2: updates to match journal

We construct a class of pure quark and gluon observables by using the collinear drop grooming technique. The construction is based on linear combinations of multiple cumulative distributions of the jet mass in collinear drop, whose specific weights are fully predicted perturbatively. This yields observables which obtain their values purely from quarks (or purely from gluons) in a wide region of phase space. We demonstrate this by showing that these observables are effective in two phase space regions, one dominated by perturbative resummation and one dominated by nonperturbative effects. The nonperturbative effects are included using shape functions which only appear as a common factor in the linear combinations constructed. We test this construction using a numerical analysis with next-to-leading logarithmic resummation and various shape function models, as well as analyzing these observables with Pythia and Vincia. Choices for the collinear drop parameters are optimized for experimental use., Comment: 58 pages, 10 figures; v2: published version

Transverse-momentum-dependent parton distribution functions (TMDs) can be studied from first principles by a perturbative matching onto lattice-calculable quantities: so-called lattice TMDs, which are a class of equal-time correlators that includes quasi-TMDs and TMDs in the Lorentz-invariant approach. We introduce a general correlator that includes as special cases these two Lattice TMDs and continuum TMDs, like the Collins scheme. Then, to facilitate the derivation of a factorization relation between lattice and continuum TMDs, we construct a new scheme, the Large Rapidity (LR) scheme, intermediate between the Collins and quasi-TMDs. The LR and Collins schemes differ only by an order of limits, and can be matched onto one another by a multiplicative kernel. We show that this same matching also holds between quasi and Collins TMDs, which enables us to prove a factorization relation between these quantities to all orders in $\alpha_s$. Our results imply that there is no mixing between various quark flavors or gluons when matching Collins and quasi TMDs, making the lattice calculation of individual flavors and gluon TMDs easier than anticipated. We cross-check these results explicitly at one loop and discuss implications for other physical-to-lattice scheme factorizations., Comment: 33 pages + appendices, 6 figures

The extraction of nonperturbative TMD physics is made challenging by prescriptions that shield the Landau pole, which entangle long- and short-distance contributions in momentum space. The use of different prescriptions then makes the comparison of fit results for underlying nonperturbative contributions not meaningful on their own. We propose a model-independent method to restrict momentum-space observables to the perturbative domain. This method is based on a set of integral functionals that act linearly on terms in the conventional position-space operator product expansion (OPE). Artifacts from the truncation of the integral can be systematically pushed to higher powers in $\Lambda_{\rm QCD}/k_T$. We demonstrate that this method can be used to compute the cumulative integral of TMD PDFs over $k_T \le k_T^\mathrm{cut}$ in terms of collinear PDFs, accounting for both radiative corrections and evolution effects. This yields a systematic way of correcting the naive picture where the TMD PDF integrates to a collinear PDF, and for unpolarized quark distributions we find that when renormalization scales are chosen near $k_T^\mathrm{cut}$, such corrections are a percent-level effect. We also show that, when supplemented with experimental data and improved perturbative inputs, our integral functionals will enable model-independent limits to be put on the nonperturbative OPE contributions to the Collins-Soper kernel and intrinsic TMD distributions., Comment: 39 pages + appendices and references, 15 figures; v2: journal version

Differential measurements of the semi-inclusive deep inelastic scattering (SIDIS) process with polarized beams provide important information on the three-dimensional structure of hadrons. Among the various observables are azimuthal asymmetries that start at subleading power, and which give access to novel transverse momentum dependent distributions (TMDs). Theoretical predictions for these distributions are currently based on the parton model rather than a rigorous factorization based analysis. Working under the assumption that leading power Glauber interactions do not spoil factorization at this order, we use the Soft Collinear Effective Theory to derive a complete factorization formula for power suppressed hard scattering effects in SIDIS. This yields generalized definitions of the TMDs that depend on two longitudinal momentum fractions (one of them only relevant beyond tree level), and a complete proof that only the same leading power soft function appears and can be absorbed into the TMD distributions at this order. We also show that perturbative corrections can be accounted for with only one new hard coefficient. Factorization formulae are given for all spin dependent structure functions which start at next-to-leading power. Prospects for improved subleading power predictions that include resummation are discussed., Comment: 118 pages, 1 figure; Update fixing typos and with complex C^(1) [JHEP erratum version]

We develop a factorization framework to compute the double differential cross section in soft drop groomed jet mass and groomed jet radius. We describe the effective theories in the large, intermediate, and small groomed jet radius regions defined by the interplay of the jet mass and the groomed jet radius measurement. As an application we present the NLL$'$ results for the perturbative moments that are related to the coefficients $C_1$ and $C_2$ that specify the leading hadronization corrections up to three universal parameters. We compare our results with Monte Carlo simulations and a calculation using the coherent branching method., Comment: 67 pages + appendices, 12 figures, v2: leading NGLs included and formulae for pp case added

We present results for the 2-jettiness differential distribution for boosted top quark pairs produced in $e^+e^-$ collisions in the peak region accounting for QCD large-logarithm resummation at next-to-next-to-next-to-leading logarithmic (N$^3$LL) order and fixed-order corrections to matrix elements at next-to-next-to-leading order (NNLO) calculated in the framework of soft-collinear effective theory and boosted heavy quark effective theory. Electroweak and finite-width effects are included at leading order. We study the perturbative convergence of the cross section in the pole and MSR mass schemes, with and without soft gap subtractions. We find that there is a partial cancellation between the pole mass and soft function renormalons. When renormalon subtractions concerning the top mass and the soft function are implemented, the perturbative uncertainties are, however, systematically smaller and an improvement in the stability of the peak position is observed. We find that the top MSR mass may be determined with perturbative uncertainties well below $100$\,MeV from the peak position of the 2-jettiness distribution. This result has important applications for Monte Carlo top quark mass calibrations., Comment: Added explanations for expected size of power corrections. Results unchanged

We perform the first global fit to inclusive $B\to X_s\gamma$ measurements using a model-independent treatment of the nonperturbative $b$-quark distribution function, with next-to-next-to-leading logarithmic resummation and $\mathcal{O}(\alpha_s^2)$ fixed-order contributions. The normalization of the $B\to X_s\gamma$ decay rate, given by $\lvert C_7^{\rm incl} V_{tb} V_{ts}^*\rvert^2$, is sensitive to physics beyond the Standard Model (SM). We determine $\lvert C_7^{\rm incl} V_{tb} V_{ts}^* \rvert = (14.77 \pm 0.51_{\rm fit} \pm 0.59_{\rm theory} \pm 0.08_{\rm param})\times 10^{-3}$, in good agreement with the SM prediction, and the $b$-quark mass $m_b^{1S} = (4.750 \pm 0.027_{\rm fit} \pm 0.033_{\rm theory} \pm 0.003_{\rm param})\,\mathrm{GeV}$. Our results suggest that the uncertainties in the extracted $B\to X_s\gamma$ rate have been underestimated by up to a factor of two, leaving more room for beyond-SM contributions., Comment: 21 pages, 8 figures

We consider Drell-Yan production $pp\to V^* X \to L X$ at small $q_T \ll Q$. Experimental measurements require fiducial cuts on the leptonic state $L$, which introduce enhanced, linear power corrections in $q_T/Q$. We show that they can be unambiguously predicted from factorization, and resummed to the same order as the leading-power contribution. We thus obtain predictions for the fiducial $q_T$ spectrum to N3LL and next-to-leading-power in $q_T/Q$. Matching to full NNLO ($\alpha_s^2$), we find that the linear power corrections are indeed the dominant ones, and the remaining fixed-order corrections become almost negligible below $q_T \lesssim 40$ GeV. We also discuss the implications for more complicated observables, and provide predictions for the fiducial $\phi^*$ spectrum at N3LL+NNLO. We find excellent agreement with ATLAS and CMS measurements of $q_T$ and $\phi^*$. We also consider the $p_T^\ell$ spectrum. We show that it develops leptonic power corrections in $q_T/(Q - 2p_T^\ell)$, which diverge near the Jacobian peak $p_T^\ell \sim Q/2$ and must be kept to all powers to obtain a meaningful result there. Doing so, we obtain for the first time an analytically resummed result for the $p_T^\ell$ spectrum around the Jacobian peak at N3LL+NNLO. Our method is based on performing a complete tensor decomposition for hadronic and leptonic tensors. In practice this is equivalent to often-used recoil prescriptions, for which our results now provide rigorous, formal justification. Our tensor decomposition yields nine Lorentz-scalar hadronic structure functions, which directly map onto the commonly used angular coefficients, but also holds for arbitrary leptonic final states. In particular, for suitably defined Born-projected leptons it still yields a LO-like angular decomposition even when including QED final-state radiation. We also discuss the application to $q_T$ subtractions., Comment: 81 pages + appendices, beautiful figures [abstract abridged]; v2: more beautiful figures (illustration of Born lepton projection, comparisons to normalized CMS spectra); v3: journal version (added figure with uncertainty breakdown)

In Proof-of-Work blockchains, difficulty algorithms serve the crucial purpose of maintaining a stable transaction throughput by dynamically adjusting the block difficulty in response to the miners' constantly changing computational power. Blockchains that may experience severe hash rate fluctuations need difficulty algorithms that quickly adapt the mining difficulty. However, without careful design, the system could be gamed by miners using coin-hopping strategies to manipulate the block difficulty for profit. Such miner behavior results in an unreliable system due to the unstable processing of transactions. We provide an empirical analysis of how Bitcoin Cash's difficulty algorithm design leads to cyclicality in block solve times as a consequence of a positive feedback loop. In response, we mathematically derive a difficulty algorithm using a negative exponential filter which prohibits the formation of positive feedback and exhibits additional desirable properties, such as history agnosticism. We compare the described algorithm to that of Bitcoin Cash in a simulated mining environment and verify that the former would eliminate the severe oscillations in transaction throughput.

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 figures

There has been recent interest in understanding the all loop structure of the subleading power soft and collinear limits, with the goal of achieving a systematic resummation of subleading power infrared logarithms. Most of this work has focused on subleading power corrections to soft gluon emission, whose form is strongly constrained by symmetries. In this paper we initiate a study of the all loop structure of soft fermion emission. In $\mathcal{N}=1$ QCD we perform an operator based factorization and resummation of the associated infrared logarithms, and prove that they exponentiate into a Sudakov due to their relation to soft gluon emission. We verify this result through explicit calculation to $\mathcal{O}(\alpha_s^3)$. We show that in QCD, this simple Sudakov exponentiation is violated by endpoint contributions proportional to $(C_A-C_F)^n$ which contribute at leading logarithmic order. Combining our $\mathcal{N}=1$ result and our calculation of the endpoint contributions to $\mathcal{O}(\alpha_s^3)$, we conjecture a result for the soft quark Sudakov in QCD, a new all orders function first appearing at subleading power, and give evidence for its universality. Our result, which is expressed in terms of combinations of cusp anomalous dimensions in different color representations, takes an intriguingly simple form and also exhibits interesting similarities to results for large-x logarithms in the off diagonal splitting functions., Comment: 21 pages, many beautiful figures. v2: journal version, added appendices

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 version

We introduce collinear drop jet substructure observables, which are unaffected by contributions from collinear radiation, and systematically probe soft radiation within jets. These observables can be designed to be either sensitive or insensitive to process-dependent soft radiation originating from outside the jet. Such collinear drop observables can be exploited as variables to distinguish quark, gluon, and color neutral initiated jets, for testing predictions for perturbative soft radiation in Monte Carlo simulations, for assessing models and universality for hadronization corrections, for examining the efficiency of pileup subtraction methods, and for any other application that leaves an imprint on soft radiation. We discuss examples of collinear drop observables that are based both on clustering and on jet shapes. Using the soft-collinear effective theory we derive factorization expressions for collinear drop observables from QCD jets, and carry out a resummation of logarithmically enhanced contributions at next-to-leading-logarithmic order. We also identify an infinite class of collinear drop observables for which the leading double logarithms are absent., Comment: 60 pages, 20 figures

We provide a quantum field theory based description of the nonperturbative effects from hadronization for soft drop groomed jet mass distributions using the soft-collinear effective theory and the coherent branching formalism. There are two distinct regions of jet mass $m_J$ where grooming modifies hadronization effects. In a region with intermediate $m_J$ an operator expansion can be used, and the leading power corrections are given by three universal nonperturbative parameters that are independent of all kinematic variables and grooming parameters, and only depend on whether the parton initiating the jet is a quark or gluon. The leading power corrections in this region cannot be described by a standard normalized shape function. These power corrections depend on the kinematics of the subjet that stops soft drop through short distance coefficients, which encode a perturbatively calculable dependence on the jet transverse momentum, jet rapidity, and on the soft drop grooming parameters $z_{\rm cut}$ and $\beta$. Determining this dependence requires a resummation of large logarithms, which we carry out at LL order. For smaller $m_J$ there is a nonperturbative region described by a one-dimensional shape function that is unusual because it is not normalized to unity, and has a non-trivial dependence on $\beta$., Comment: 61 pages + Appendices, 26 figures

The study of amplitudes and cross sections in the soft and collinear limits allows for an understanding of their all orders behavior, and the identification of universal structures. At leading power soft emissions are eikonal, and described by Wilson lines. Beyond leading power the eikonal approximation breaks down, soft fermions must be added, and soft radiation resolves the nature of the energetic partons from which they were emitted. For both subleading power soft gluon and quark emissions, we use the soft collinear effective theory (SCET) to derive an all orders gauge invariant bare factorization, at both amplitude and cross section level. This yields universal multilocal matrix elements, which we refer to as radiative functions. These appear from subleading power Lagrangians inserted along the lightcone which dress the leading power Wilson lines. The use of SCET enables us to determine the complete set of radiative functions that appear to $\mathcal{O}(\lambda^2)$ in the power expansion, to all orders in $\alpha_s$. For the particular case of event shape observables in $e^+e^-\to$ dijets we derive how the radiative functions contribute to the factorized cross section to $\mathcal{O}(\lambda^2)$., Comment: 62 pages + appendices, many pretty and colorful figures. v2: journal version

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 version