Small Mass Limit Research Articles

Mass and isospin breaking effects in the Skyrme model and in holographic QCD

We discuss how the quark masses and their mass splitting affect the baryons in the Skyrme model as well as the Witten-Sakai-Sugimoto (WSS) model. In both cases, baryons are described by solitonic objects, i.e., Skyrmions and instantons, respectively. After the quantization of their zero modes, the nucleons become quantum states of a rotor. We show how the quark mass affects the moment of inertia and we provide a semianalytic approach valid in the small-mass limit. Additionally, we show how the two lightest quarks’ mass splitting affects the moments of inertia of the Skyrmion and induces an isospin breaking effect. This effect turns out not to be enough to split the degeneracy in the neutron-proton multiplet, but it splits some of the states in the Δ multiplet. Unlike the basic Skyrme model, the WSS model already includes vector mesons and another mechanism to transfer isospin breaking from quark masses to the solitons is known. We compute the splitting of the moment of inertia in the small-mass limit in the WSS model and combine the two effects on the spectrum of baryons, in particular the Δ’s. Published by the American Physical Society 2024

On the Small-Mass Limit for Stationary Solutions of Stochastic Wave Equations with State Dependent Friction

On the Small-Mass Limit for Stationary Solutions of Stochastic Wave Equations with State Dependent Friction

Analytic decay width of the Higgs boson to massive bottom quarks at next-to-next-to-leading order in QCD

The Higgs boson decay to a massive bottom quark pair provides the dominant contribution to the Higgs boson width. We present an exact result for such a decay induced by the bottom quark Yukawa coupling with next-to-next-to-leading order (NNLO) QCD corrections. We have adopted the canonical differential equations in the calculation and obtained the result in terms of multiple polylogarithms. We also compute the contribution from the decay to four bottom quarks which consists of complete elliptic integrals or their one-fold integrals. The result in the small bottom quark mass limit coincides with the previous calculation using the large momentum expansion. The threshold expansion exhibits power divergent terms in the bottom quark velocity, which has a structure different from that in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${e}^{+}{e}^{-}\ o t\\overline{t }$$\\end{document} but can be reproduced by computing the corresponding Coulomb Green function. The NNLO corrections significantly reduce the uncertainties from both the renormalization scale and the renormalization scheme of the bottom quark Yukawa coupling. Our result can be applied to a heavy scalar decay to a top quark pair.

Small mass limit for stochastic interacting particle systems with Lévy noise and linear alignment force.

We study the small mass limit in mean field theory for an interacting particle system with non-Gaussian Lévy noise. When the Lévy noise has a finite second moment, we obtain the limit equation with convergence rate ε+1/εN, by taking first the mean field limit N→∞ and then the small mass limit ε→0. If the order of the two limits is exchanged, the limit equation remains the same but has a different convergence rate ε+1/N. However, when the Lévy noise is α-stable, which has an infinite second moment, we can only obtain the limit equation by taking first the small mass limit and then the mean field limit, with the convergence rate 1/Nα-1+1/Np2+εp/α where p∈(1,α). This provides an effectively limit model for an interacting particle system under a non-Gaussian Lévy fluctuation, with rigorous error estimates.

Stochastic wave equation with Hölder noise coefficient: Well-posedness and small mass limit

We construct unique martingale solutions to the damped stochastic wave equationμ∂2u∂t2(t,x)=Δu(t,x)−∂u∂t(t,x)+b(t,x,u(t,x))+σ(t,x,u(t,x))dWtdt, where Δ is the Laplacian on [0,1] with Dirichlet boundary condition, W is space-time white noise, σ is 34+ϵ -Hölder continuous in u and uniformly non-degenerate, and b has linear growth. The same construction holds for the stochastic wave equation without damping term. More generally, the construction holds for SPDEs defined on separable Hilbert spaces with a densely defined operator A, and the assumed Hölder regularity on the noise coefficient depends on the eigenvalues of A in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume b is Hölder continuous in u, then as μ tends to 0 the solution to the damped stochastic wave equation converges in distribution, on the space of continuous paths, to the solution of the corresponding stochastic heat equation. The latter result is new even in the case of additive noise.

U(1) quasi-hydrodynamics: Schwinger-Keldysh effective field theory and holography

We study the quasi-hydrodynamics of a system with a softly broken U(1) global symmetry using effective field theory (EFT) and holographic methods. In the gravity side, we consider a holographic Proca model in the limit of small bulk mass, which is responsible for a controllable explicit breaking of the U(1) global symmetry in the boundary field theory. We perform a holographic Schwinger-Keldysh analysis, which allows us to derive the form of the boundary effective action in presence of dissipation. We compare our results with the previously proposed EFT and hydrodynamic theories, and we confirm their validity by computing the low-energy quasi-normal modes spectrum analytically and numerically. Additionally, we derive the broken holographic Ward identity for the U(1) current, and discuss the recently proposed novel transport coefficients for systems with explicitly broken symmetries. The setup considered is expected to serve as a toy model for more realistic situations where quasi-hydrodynamics is at work, such as axial charge relaxation in QCD, spin relaxation in relativistic systems, electric field relaxation in magneto-hydrodynamics, or momentum relaxation in condensed matter systems.

On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction

We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent. We also provide the proof of the Smolchowski-Kramers approximation for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion.

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi–linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction–diffusion equation. Together with the uniform geometric ergodicity, we obtain the validity of the small mass limit for the solutions on the infinite time horizon [0,∞). This extends previously known results established for the damped wave equations under Lipschitz nonlinearities.

The Smoluchowski–Kramer approximation of a generalized Langevin equation with state-dependent damping

The small-mass limit is derived for a generalized Langevin equation (GLE) with state-dependent damping and a kernel expressed as an infinite sum of exponentials. The state dependence includes both the current and the past. By some bounded estimates and tightness, as the mass tends to 0, the GLE is shown to converge in distribution to a limit equation with additional drift terms that come from the current and the past state dependence, respectively.

Energizing gamma ray bursts via Z^{prime } mediated neutrino heating

The pair annihilation of neutrinos (nu {overline{nu }}rightarrow e^+e^-) can energize violent stellar explosions such as gamma ray bursts (GRBs). The energy in this neutrino heating mechanism can be further enhanced by modifying the background spacetime over that of Newtonian spacetime. However, one cannot attain the maximum GRB energy (sim 10^{52}~text {erg}) in either the Newtonian background or Schwarzschild and Hartle–Thorne background. On the other hand, using modified gravity theories or the Quintessence field as background geometries, the maximum GRB energy can be reached. In this paper, we consider extending the standard model by an extra U(1)_{{mathrm{B-L}}} gauge group and augmenting the energy deposition by neutrino pair annihilation process including contributions mediated by the Z^{prime } gauge boson belonging to this model. From the observed energy of GRB, we obtain constraints on U(1)_{{mathrm{B-L}}} gauge coupling in different background spacetimes. We find that the bounds on gauge coupling in modified gravity theories and quintessence background are stronger than those coming from the neutrino-electron scattering experiments in the limit of small gauge boson masses. Future GRB observations with better accuracy can further strengthen these bounds.

Mass decomposition of the pion in the ’t Hooft model

We obtain the energy-momentum tensor (EMT) in the 't Hooft model of two-dimensional quantum chromodynamics. The EMT is decomposed into contributions from quark and gluon fields, with all of the (plus component of) the light front momentum being carried by the quark field. The energy is split between quark and gluon fields, with the gluon field carrying the self-energy of the dressed quarks. We consider the pion in the limit of small but non-zero quark masses -- which has previously withstood numerical treatment -- as as a concrete example. We solve for the pion wave function using a variational method and obtain numerical results for its energy breakdown into quark and gluon contributions.

Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation

<p style='text-indent:20px;'>The small mass limit (Smoluchowski–Kramers approximation) of class systems of ordinary differential equations describing motions of small mass particle with state dependent friction and high oscillation is derived by a diffusion approximation approach. In the small mass limit, due to the state dependent damping, one additional term appears in the limit equation, which leads to a stochastic differential equation (SDE) as the highly random oscillation appears as a multiplicative white noise.</p>

Homogenization of Dissipative Hamiltonian Systems Under Lévy Fluctuations

We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian Lévy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a discontinuous noise-induced drift term arises. Firstly, we show that the momentum in the stochastic Hamiltonian system converges to zero when the kinetic energy has polynomial growth. Then, we prove that the stochastic Hamiltonian system with classical kinetic energy converges to the limiting equation in probability, with respect to Skorokhod topology as the mass tends to zero.

Exploring the small mass limit of stationary black holes in theories with Gauss–Bonnet terms

In this work we examine the small mass limit of black holes (BHs), with and without spin, in theories where a scalar field is non-minimally coupled to a Gauss–Bonnet (GB) term. First, we provide an analytical example for a theory where a static closed-form solution with a small mass limit is known, and later use analytical and numerical techniques to explore this limit in standard scalar-GB theories with dilatonic, linear and quadratic-exponential couplings. In most cases studied here, we find an inner singularity that overlaps with the event horizon of the static BH as the small mass limit is reached. Moreover, since solutions in this limit possess a non-vanishing Hawking temperature, a naked singularity is expected to be reached through evaporation, raising questions concerning the consistency of these theories altogether. On the other hand, we provide for the first time in this context an example of a coupling where the small mass limit is never reached, thus preferred from the point of view of cosmic censorship. Finally, we consider BHs with spin and numerically investigate how this changes the picture, using these to place the tightest upper bounds to date on the coupling constant for the dilatonic and linear theories, with km.

Infrared properties of five-point massive amplitudes in mathcal{N} = 4 SYM on the Coulomb branch

We investigate the structure of the five-point W boson scattering amplitude in mathcal{N} = 4 SYM on the Coulomb branch in a small mass limit. We show that up to two loops the IR divergences exponentiate and are controlled by the Γoct anomalous dimension similar to the four-point amplitude case considered recently in the literature. We also make a conjecture regarding the all-loop structure of the five-point amplitude.

Small mass limit in mean field theory for stochastic N particle system

The small mass limit is derived for a stochastic N particle system in mean-field limit theory. In the case of the constant communication weight function and by applying the averaging approach to distribution dependent slow–fast stochastic differential equations (which has an independent interest), the small mass limit model is derived.

The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients

In this paper, we study the small mass limit of the dynamic of stochastic system with state dependent friction and diffusion. We prove the strong convergence of the original system under weak assumptions on the coefficients, where the limiting equation admits an additional noise-induced drift term. Furthermore, the rate of convergence is also obtained. Our results even provide non-trivial improvement of the previous limit theorems for the classical Langevin equation.

Quantum kinetic equation for fluids of spin-1/2 fermions

Fluid of spin-1/2 fermions is represented by a complex scalar field and a four-vector field coupled both to the scalar and the Dirac fields. We present the underlying action and show that the resulting equations of motion are identical to the (hydrodynamic) Euler equations in the presence of Coriolis force. As a consequence of the gauge invariances of this action we established the quantum kinetic equation which takes account of noninertial properties of the fluid in the presence of electromagnetic fields. The equations of the field components of Wigner function in Clifford algebra basis are employed to construct new semiclassical covariant kinetic equations of the vector and axial-vector field components for massless as well as massive fermions. Nonrelativistic limit of the chiral kinetic equation is studied and shown that it generates a novel three-dimensional transport theory which does not depend on spatial variables explicitly and possesses a Coriolis force term. We demonstrated that the three-dimensional chiral transport equations are consistent with the chiral anomaly. For massive fermions the three-dimensional kinetic transport theory generated by the new covariant kinetic equations is established in small mass limit. It possesses the Coriolis force and the massless limit can be obtained directly.

Quasinormal modes and shadow of a Schwarzschild black hole with GUP

We consider quantum corrections for the Schwarzschild black hole metric by using the generalized uncertainty principle (GUP) to investigate quasinormal modes, shadow and their relationship in the eikonal limit. We calculate the quasinormal frequencies of the quantum-corrected Schwarzschild black hole by using the sixth-order Wentzel–Kramers–Brillouin (WKB) approximation, and also perform a numerical analysis that confirms the results obtained from this approach. We also find that the shadow radius is nonzero even at very small mass limit for finite GUP parameter.

Small mass limit and diffusion approximation for a generalized Langevin equation with infinite number degrees of freedom

An approximation of a Generalized Langevin equations with small mass and randomly fast oscillation is derived. By some bounded estimates and a diffusion approximation the limiting equation is shown to be stochastic partial differential equations (SPDEs) driven by white noise.