Entropies, volumes, and Einstein metrics

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstruction to the existence of Einstein metrics on four-manifolds involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe inequality and Sambusetti's obstruction.Comment: This is a substantial revision and expansion of the 2004 preprint, which I prepared in spring of 2010 and which has since been published. The version here is essentially the published one, minus the problems introduced by Springer productio

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

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

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

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

Resummation of heavy jet mass and comparison to LEP data

The heavy jet mass distribution in e+e- collisions is computed to next-to-next-to-next-to leading logarithmic (NNNLL) and next-to-next-to leading fixed order accuracy (NNLO). The singular terms predicted from the resummed distribution are confirmed by the fixed order distributions allowing a precise extraction of the unknown soft function coefficients. A number of quantitative and qualitative comparisons of heavy jet mass and the related thrust distribution are made. From fitting to ALEPH data, a value of alpha_s is extracted, alpha_s(m_Z)=0.1220 +/- 0.0031, which is larger than, but not in conflict with, the corresponding value for thrust. A weighted average of the two produces alpha_s(m_Z) = 0.1193 +/- 0.0027, consistent with the world average. A study of the non-perturbative corrections shows that the flat direction observed for thrust between alpha_s and a simple non-perturbative shape parameter is not lifted in combining with heavy jet mass. The Monte Carlo treatment of hadronization gives qualitatively different results for thrust and heavy jet mass, and we conclude that it cannot be trusted to add power corrections to the event shape distributions at this accuracy. Whether a more sophisticated effective field theory approach to power corrections can reconcile the thrust and heavy jet mass distributions remains an open question.Comment: 33 pages, 14 figures. v2 added effect of lower numerical cutoff with improved extraction of the soft function constants; power correction discussion clarified. v3 small typos correcte

Two-Loop Soft Corrections and Resummation of the Thrust Distribution in the Dijet Region

The thrust distribution in electron-positron annihilation is a classical precision QCD observable. Using renormalization group (RG) evolution in Laplace space, we perform the resummation of logarithmically enhanced corrections in the dijet limit, $T\to 1$ to next-to-next-to-leading logarithmic (NNLL) accuracy. We independently derive the two-loop soft function for the thrust distribution and extract an analytical expression for the NNLL resummation coefficient $g_3$. To combine the resummed expressions with the fixed-order results, we derive the $\log(R)$-matching and $R$-matching of the NNLL approximation to the fixed-order NNLO distribution.Comment: 50 pages, 12 figures, 1 table. Few minor changes. Version accepted for publication 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

Factorization Properties of Soft Graviton Amplitudes

We apply recently developed path integral resummation methods to perturbative quantum gravity. In particular, we provide supporting evidence that eikonal graviton amplitudes factorize into hard and soft parts, and confirm a recent hypothesis that soft gravitons are modelled by vacuum expectation values of products of certain Wilson line operators, which differ for massless and massive particles. We also investigate terms which break this factorization, and find that they are subleading with respect to the eikonal amplitude. The results may help in understanding the connections between gravity and gauge theories in more detail, as well as in studying gravitational radiation beyond the eikonal approximation.Comment: 35 pages, 5 figure