Partial Resummation Research Articles

Limits on an improved action for contact effective field theory in two-body systems

We consider a possible resummation of subleading effects in two-body systems with a large scattering length as described by a short-range effective field theory (EFT). In particular, we investigate the consequences of a resummation of part of the range corrections. Explicit calculations of the two-body phase shifts and charge form factor indicate that, except for extreme choices, resummations do not alter the convergence of the EFT expansion and are often beneficial at the lowest orders. We have considered the expansion when the regulator cutoff is removed as well as when it is finite, and find that the cutoff is not an important factor for resummations. Our results connect with other works where the partial resummation is induced by potentials with finite cutoffs or interaction ranges.

Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth

We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.

Color-octet contributions for J/ψ inclusive production at B factories in soft gluon factorization

We have studied color-octet contributions for J/ψ inclusive production at B factories, i.e., e+e− → J/ψ(3 {P}_J^{left[8right]} ,1 {S}_0^{left[8right]} ) + {X}_{mathrm{non}-coverline{c}} , using the soft gluon factorization (SGF) approach, in which the J/ψ energy spectrum is expressed in a form of perturbatively calculable short-distance hard parts convoluted with one-dimensional soft gluon distributions (SGDs). The series of velocity corrections originated from kinematic effect can be naturally resummed in this approach. Short-distance hard parts have been calculated analytically to next-to-leading order in αs. Renormalization group equations for SGDs have been derived and solved, which resums Sudakov logarithms originated from soft gluon emissions. Our final result gives a upper bound for color-octet matrix elements consistent with that extracted from hadron colliders. This may relieve the well-known universality problem in the NRQCD factorization.As a comparison, we also analytically calculated short-distance hard parts in the NRQCD factorization, with Sudakov logarithms resummed by using soft collinear effective theory. The comparison shows that velocity corrections from kinematic effect, which have been resummed in SGF, are significant for phenomenological study. Furthermore, it is found that Sudakov logarithms originated from soft gluon emissions are very important, while it is not the case for Sudakov logarithms originated from jet function. Therefore, the partial Sudakov resummation in SGF has already captured the main physics.

Resummation-based quantum Monte Carlo for quantum paramagnetic phases

For spin rotational symmetric models with a positive-definite high-temperature expansion of the partition function, a stochastic sampling of the series expansion upon partial resummation becomes logically equivalent to sampling an uncoloured closely-packed loop-gas model in one higher dimension. Based on this, we devise quantum Monte Carlo updates that importance-sample loop configurations for general $SU(N)$ in fundamental and higher-symmetric representations. The algorithmic performance systematically improves with increase in (continuous) $N$ allowing efficient simulation of quantum paramagnets. The underlying reason for the increased efficacy is the correspondence of quantum paramagnetic phases like valence bond solids to short-loop phases on the loop-gas side rather than the particular value of $N$. This also gives a connection between Sandvik's $JQ$ model class and classical loop-gas models in the deconfined universality class.

(F, G) -summed form of the QED effective action

We conjecture that the proper-time series expansion of the one-loop effective Lagrangian of quantum electrodynamics can be summed in all terms containing the field-strength invariants $\mathcal{F}=\frac{1}{4}{F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$, $\mathcal{G}=\frac{1}{4}{\stackrel{\texttildelow{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$, including those also possessing derivatives of the electromagnetic field strength. This partial resummation is exactly encapsulated in a factor with the same form as the Heisenberg-Euler Lagrangian density, except that now the electric and magnetic fields can depend arbitrarily on spacetime coordinates. We provide strong evidence for this conjecture, which is proved to sixth order in the proper time. Furthermore, and as a byproduct, we generate some solvable electromagnetic backgrounds. We also discuss the implications for a generalization of the Schwinger formula for pair production induced by nonconstant electric fields. Finally, we briefly outline the extension of these results in the presence of gravity.

From boundary data to bound states

We introduce a — somewhat holographic — dictionary between gravitational observables for scattering processes (measured at the boundary) and adiabatic invariants for bound orbits (in the bulk), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the two­body problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic orbits (such as the periastron advance ∆Φ) through analytic continuation, via a radial action depending only on boundary data. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, Ω(E), directly from scattering information. As an example, using the results in Bernet al. [36, 37], we readily derive Ω(E) and ∆Φ(J, E) to two-loop orders. We also provide closed-form expressions for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a no-recoil approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy.

High-energy resummation in heavy-quark pair hadroproduction

The inclusive hadroproduction of two heavy quarks, featuring a large separation in rapidity, is proposed as a novel probe channel of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) approach. In a theoretical setup which includes full resummation of leading logarithms in the center-of-mass energy and partial resummation of the next-to-leading ones, predictions for the cross section and azimuthal coefficients are presented for kinematic configurations typical of current and possible future experimental analyses at the LHC.

Long distance behavior of O(N)-model correlators in de Sitter space and the resummation of secular terms

We analyze the long distance behavior of the two-point functions for an interacting scalar O(N) model in de Sitter spacetime. Following our previous work, this behavior is analyzed by analytic continuation of the Euclidean correlators, which are computed by treating the homogeneous zero mode exactly and using a partial resummation of the interactions between the zero and the non-zero modes. We focus on massless fields and present an alternative derivation of our method, which involves a double expansion in 1/N and the coupling constant of the theory. This derivation is simpler than the previous one and can be directly extended for fields with negative squared-mass. We extend our previous results by computing the long wavelength limit of the two-point functions at next-to-leading order in 1/N and at leading order in the coupling constant, which involves a further resummation of Feynman diagrams (needed when the two-point functions are analitically continued). We prove that, after this extra resummation, the two-point functions vanish in the long distance limit.

Reparametrization invariance and partial re-summations of the heavy quark expansion

We extend existing work on reparametrization invariance (RPI) of the heavy-quark expansion (HQE). RPI implies relations between different orders of the HQE. We discuss the total rates of inclusive processes and obtain results which have a manifest RPI and can be expressed through matrix elements of operators and states defined in full QCD. This approach leads to a partial re-summation of higher-order terms in the heavy-quark expansion. It has the advantage that for reparametrization-invariant observables it shows that the number of independent non-perturbative parameters can be reduced.

Probability distribution for neutron density of two energy group point kinetics system

The probability distribution of the number of neutrons and delayed neutron precursors in a multiplying assembly of various types of reactivities is developed for two energy groups. The space independent point reactor kinetics model for six precursors group of delayed neutrons is considered. The problem is formulated in terms of the probability distribution, generating function which satisfies a partial differential equation, derived in this paper. The probability distribution of two energy group delayed neutrons and the density of the precursors are obtained by solving this system of reactor kinetics model by adopting the mathematical methods. At the point, when the system is formulated in an operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the kinetics system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as time-dependent exponential perturbation theory. Every Magnus approximate corresponds in perturbation theory to a partial re-summation of infinite terms with the important additional property of preserving, in any order, certain symmetries of the exact solution. The first, second and third Magnus expansions are described and used to predict the first moment of fast, thermal and multi-group of delayed neutrons precursor for the two-energy point kinetics reactor system. The validity of the presented method is tested with the aid of the eigenvectors and eigenvalues of the kinetics system in the matrix form by comparing with the conventional methods.

Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory

The Luttinger-Ward functional was proposed more than five decades ago and has been used to formally justify most practically used Green's function methods for quantum many-body systems. Nonetheless, the very existence of the Luttinger-Ward functional has been challenged by recent theoretical and numerical evidence. We provide a rigorously justified Luttinger-Ward formalism, in the context of Euclidean lattice field theory. Using the Luttinger-Ward functional, the free energy can be variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.

High-energy resummation in heavy-quark pair photoproduction

We present our predictions for the inclusive production of two heavy quark–antiquark pairs, separated by a large rapidity interval, in the collision of (quasi-)real photons at the energies of LEP2 and of some future electron–positron colliders. We include in our calculation the full resummation of leading logarithms in the center-of-mass energy and a partial resummation of the next-to-leading logarithms, within the Balitsky–Fadin–Kuraev–Lipatov (BFKL) approach.

Determination of the exchange interaction energy from the polarization expansion of the wave function

The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error falling as $3^{-K}$, where $K$ is the order of the applied polarization expansion. The error of the surface-integral formula decreases exponentially as $a^K/(K+1)!$, where $a=\ln2 -\tfrac{1}{2}$. The variational formula performs best, its error decays as $K^{1/2} [a^{ K}/(K+1)!]^2$. These convergence rates are much faster than those resulting from approximating the wave function through the multipole expansion. This shows the efficiency of the partial resummation of the multipole series effected by the polarization expansion. Our results demonstrate also the benefits of incorporating the variational principle into the perturbation theory of molecular interactions.

O(N) model in Euclidean de Sitter space: beyond the leading infrared approximation

We consider an $O(N)$ scalar field model with quartic interaction in $d$-dimensional Euclidean de Sitter space. In order to avoid the problems of the standard perturbative calculations for light and massless fields, we generalize to the $O(N)$ theory a systematic method introduced previously for a single field, which treats the zero modes exactly and the nonzero modes perturbatively. We compute the two-point functions taking into account not only the leading infrared contribution, coming from the self-interaction of the zero modes, but also corrections due to the interaction of the ultraviolet modes. For the model defined in the corresponding Lorentzian de Sitter spacetime, we obtain the two-point functions by analytical continuation. We point out that a partial resummation of the leading secular terms (which necessarily involves nonzero modes) is required to obtain a decay at large distances for massless fields. We implement this resummation along with a systematic double expansion in an effective coupling constant $\sqrt\lambda$ and in 1/N. We explicitly perform the calculation up to the next-to-next-to-leading order in $\sqrt\lambda$ and up to next-to-leading order in 1/N. The results reduce to those known in the leading infrared approximation. We also show that they coincide with the ones obtained directly in Lorentzian de Sitter spacetime in the large N limit, provided the same renormalization scheme is used.

Massless Interacting Scalar Fields in de Sitter space

We present a method to compute the two-point functions for an $O(N)$ scalar field model in de Sitter spacetime, avoiding the well known infrared problems for massless fields. The method is based on an exact treatment of the Euclidean zero modes and a perturbative one of the nonzero modes, and involves a partial resummation of the leading secular terms. This resummation, crucial to obtain a decay of the correlation functions, is implemented along with a double expansion in an effective coupling constant $\sqrt\lambda$ and in $1/N$. The results reduce to those known in the leading infrared approximation and coincide with the ones obtained directly in Lorentzian de Sitter spacetime in the large $N$ limit. The new method allows for a systematic calculation of higher order corrections both in $\sqrt\lambda$ and in $1/N$.

Finite coupling corrections to holographic predictions for hot QCD

Finite 't Hooft coupling corrections to multiple physical observables in strongly coupled $N=4$ supersymmetric Yang-Mills plasma are examined, in an attempt to assess the stability of the expansion in inverse powers of the 't Hooft coupling $\lambda$. Observables considered include thermodynamic quantities, transport coefficients, and quasinormal mode frequencies. Although large $\lambda$ expansions for quasinormal mode frequencies are notably less well behaved than the expansions of other quantities, we find that a partial resummation of higher order corrections can significantly reduce the sensitivity of the results to the value of $\lambda$.

Symmetry Improved 2PI Effective Action and the Infrared Divergences of the Standard Model

Resummations of infinite sets of higher-order perturbative contributions are often needed both in thermal field theory and at zero temperature. For instance, the behaviour of the Standard Model (SM) effective potential extrapolated to very high energies is known to be extremely sensitive to higher-order effects. The 2PI effective action provides a systematic approach to consistently perform such resummations. However, one of its major limitations was that its loopwise expansion introduces residual violations of possible global symmetries, thus giving rise to massive Goldstone bosons in the spontaneously broken phase of the theory. We review the recently developed symmetry-improved 2PI formalism for consistently encoding global symmetries in the 2PI approach, and discuss its satisfactory field-theoretical properties. We then apply the formalism to study the infrared divergences of the SM effective potential due to Goldstone bosons, which may affect the stability analyses of the SM. We present quantitative comparisons, for the scalar sector of the SM, with the approximate partial resummation procedure recently developed to address this problem, and show the quantitative discrepancy of the latter with the more complete 2PI approach, thus motivating further studies in this direction.

Impact on the power spectrum of screening in modified gravity scenarios

We study the effects of screened modified gravity of the $f(R)$, dilaton and symmetron types on structure formation, from the quasi-linear to the non-linear regime, using semi-analytical methods. For such models, where the range of the new scalar field is typically within the Mpc range and below in the cosmological context, non-linear techniques are required to understand the deviations of the power spectrum of the matter density contrast compared to the $\Lambda$-CDM template. This is nowadays commonly tackled using extensive N-body simulations. Here we present new results combining exact perturbation theory at the one loop level (and a partial resummation of the perturbative series) with a halo model. The former allows one to extend the linear perturbative analysis up to $k\lesssim 0.15{\rm h Mpc}^{-1}$ at the perturbative level while the latter leads to a reasonable, up to a few percent, agreement with numerical simulations for $k\lesssim 3{\rm h Mpc}^{-1}$ for large curvature $f(R)$ models, and $k\lesssim 1{\rm h Mpc}^{-1}$ for dilatons and symmetrons, at $z=0$. We also discuss how the behaviors of the perturbative expansions and of the spherical collapse differ for $f(R)$, dilaton, and symmetron models.

PROTON SCATTERING ON AN ELECTRON GAS

It is shown in the case of proton scattering on an electron gas target that the Closed Time Path formalism can handle final state interactions of the target in equilibrium in a simple and natural manner. The leading-order cross-section is proportional to the photon density of states. The scattering needs a partial resummation of the perturbation series when the electron gas forms long living quasi-particles with high density of state during the collision. A strong cancellation between real and virtual electron-hole pairs is found in this case.

Convolution Lagrangian perturbation theory for biased tracers

We present a new formulation of Lagrangian perturbation theory which allows accurate predictions of the real- and redshift-space correlation functions of the mass field and dark matter halos. Our formulation involves a non-perturbative resummation of Lagrangian perturbation theory and indeed can be viewed as a partial resummation of the formalism of Matsubara (2008a,b) in which we keep exponentiated all of the terms which tend to a constant at large separation. One of the key features of our method is that we naturally recover the Zel'dovich approximation as the lowest order of our expansion for the matter correlation function. We compare our results against a suite of N-body simulations and obtain good agreement for the correlation functions in real-space and for the monopole correlation function in redshift space. The agreement becomes worse for higher multipole moments of the redshift-space, halo correlation function. Our formalism naturally includes non-linear bias and explains the strong bias-dependence of the multipole moments of the redshift-space correlation function seen in N-body simulations.