Using Effective Field Theory (EFT) methods we present a Lagrangian formalism which describes the dynamics of non-relativistic extended objects coupled to gravity. The formalism is relevant to understanding the gravitational radiation power spectra emitted by binary star systems, an important class of candidate signals for gravitational wave observatories such as LIGO or VIRGO. The EFT allows for a clean separation of the three relevant scales: rs, the size of the compact objects, r the orbital radius and r/v, the wavelength of the physical radiation (where the velocity v is the expansion parameter). In the EFT radiation is systematically included in the v expansion without need to separate integrals into near zones and radiation zones. Using the EFT, we show that the renormalization of ultraviolet divergences which arise at v 6 in post-Newtonian (PN) calculations requires the presence of two non-minimal worldline gravitational couplings linear in the Ricci curvature. However, these operators can be removed by a redefinition of the metric tensor, so that the divergences at arising at v 6 have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of non-minimal couplings, this implies a simple proof of the decoupling of internal structure for spinless objects to at least order v 6 . Neglecting absorptive effects, we find that the power counting rules of the EFT indicate that the next set of short distance operators, which are quadratic in the curvature and are associated with tidal deformations, do not play a role until order v 10 . These operators, which encapsulate finite size properties of the sources, have coefficients that can be fixed by a matching calculation. By including the most general set of such operators, the EFT allows one to work within a point particle theory to arbitrary orders in v.

In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulas in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the -␥ form factor (␥␥*→ 0 ), light meson form factors (␥*M →M ), as well as deep inelastic scattering (e Ϫ p→e Ϫ X), the Drell-Yan process (pp →Xl ϩ l Ϫ ), and deeply virtual Compton scattering

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 relevant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. Such observables include: transverse momentum distributions at p T much less then the high energy scattering scale, jet broadening, exclusive hadroproduction and decay, as well as the Sudakov form factor. 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 universal. We present details of the factorization and resummation of the jet broadening cross section including a renormalization in p ⊥ 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.

We combine tools from effective field theory and generalized unitarity to construct a map between on-shell scattering amplitudes and the classical potential for interacting spinless particles. For general relativity, we obtain analytic expressions for the classical potential of a binary black hole system at second order in the gravitational constant and all orders in velocity. Our results exactly match all known results up to fourth post-Newtonian order, and offer a simple check of future higher order calculations. By design, these methods should extend to higher orders in perturbation theory.

We discuss the matching conditions and renormalization group evolution of non-relativistic QCD. A variant of the conventional MS-bar scheme is proposed in which a subtraction velocity nu is used rather than a subtraction scale mu. We derive a novel renormalization group equation in velocity space which can be used to sum logarithms of v in the effective theory. We apply our method to several examples. In particular we show that our formulation correctly reproduces the two-loop anomalous dimension of the heavy quark production current near threshold.Comment: (27 pages, revtex

We introduce a systematic approach for the resummation of perturbative series which involves large logarithms not only due to large invariant mass ratios but large rapidities as well. A series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next-to-leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to-leading-log cross section are presented. The result agrees with the data to within errors.

Using the soft-collinear effective theory we derive the factorization theorem for the decays B ! M 1 M 2 with M 1;2 ; K; ; K , at leading order in =E M and =m b . The results derived here apply even if s E M is not perturbative, and we prove that the physics sensitive to the E scale is the same in B ! M 1 M 2 and B ! M form factors. We argue that c c penguins could give long-distance effects at leading order. Decays to two transversely polarized vector mesons are discussed. Analyzing B ! we find predictions for B 0 ! 0 0 and jV ub jf B! 0 as a function of .Decays of B mesons to two light mesons are important for the study of CP violation in the standard model. In [1] it was suggested that since m b ; E M ; m M the amplitudes should factorize into simpler nonperturbative objects, and the proposed factorization theorem was checked at one-loop. This approach is often referred to as ''QCD factorization'' (QCDF). Factorization has also been considered in the ''perturbative QCD'' (pQCD) approach [3]. These approaches rely on a perturbative expansion in s E M . The results obtained from factorization are quite predictive and may allow us to answer fundamental questions about the standard model. At the current time several important issues remain to be answered. These include (i) the extent to which the results are model-independent consequences of QCD (since QCD is a predictive theory any model-independent limit must give the same answer in different approaches). A complete proof of a factorization theorem will answer this question. (ii) Unambiguous definitions of any nonperturbative hadronic parameters which appear are required. This allows the universality of parameters to be understood, as well as making clear the extent to which predictions rely on model dependent assumptions about parameter values. (iii) Does the power expansion converge? If power suppressed contributions really compete with leading order contributions as some studies [4,5] suggest then the expansion cannot be trusted. In this case the only hope is a systematic modification of the power counting to promote these effects to leading order, or an identification of certain observables that are free from this problem.The soft collinear effective theory (SCET) [6,7] provides the necessary tools to address these issues. A first study of SCET factorization for B ! has been made in [8]. In this paper we go beyond Refs. [1,3,8] in several ways. We first reduce the SCET operator basis to its minimal form and extend it to allow for all B ! M 1 M 2 decays (including two vectors). Our results show that all of the so-called ''hard spectator'' contributions are already present in the form factors, just with different hard Wilson coefficients. We also derive a form of the factorization theorem which does not rely on a perturbative expansion in s E M , and show that the nonperturbative parameters are still the same as those in the B ! M form factors. In our analysis long distance c c penguins [9,10] are investigated, but are left unfactorized. For the values of m b ...

Starting with QCD, we derive an effective field theory description for forward scattering and factorization violation as part of the soft-collinear effective field theory (SCET) for high energy scattering. These phenomena are mediated by long distance Glauber gluon exchanges, which are static in time, localized in the longitudinal distance, and act as a kernel for forward scattering where |t| s. In hard scattering, Glauber gluons can induce corrections which invalidate factorization. With SCET, Glauber exchange graphs can be calculated explicitly, and are distinct from graphs involving soft, collinear, or ultrasoft gluons. We derive a complete basis of operators which describe the leading power effects of Glauber exchange. Key ingredients include regulating light-cone rapidity singularities and subtractions which prevent double counting. Our results include a novel all orders gauge invariant pure glue soft operator which appears between two collinear rapidity sectors. The 1-gluon Feynman rule for the soft operator coincides with the Lipatov vertex, but it also contributes to emissions with ≥ 2 soft gluons. Our Glauber operator basis is derived using tree level and one-loop matching calculations from full QCD to both SCET II and SCET I . The one-loop amplitude's rapidity renormalization involves mixing of color octet operators and yields gluon Reggeization at the amplitude level. The rapidity renormalization group equation for the leading soft and collinear functions in the forward scattering cross section are each given by the BFKL equation. Various properties of Glauber gluon exchange in the context of both forward scattering and hard scattering factorization are described. For example, we derive an explicit rule for when eikonalization is valid, and provide a direct connection to the picture of multiple Wilson lines crossing a shockwave. In hard scattering operators Glauber subtractions for soft and collinear loop diagrams ensure that we are not sensitive to the directions for soft and collinear Wilson lines. Conversely, certain Glauber interactions can be absorbed into these soft and collinear Wilson lines by taking them to be in specific directions. We also discuss criteria for factorization violation.