Momentum Cutoff Research Articles

Generalized virial theorem for massless electrons in graphene and other Dirac materials

The virial theorem for a system of interacting electrons in a crystal, which is described within the framework of the tight-binding model, is derived. We show that, in particular case of interacting massless electrons in graphene and other Dirac materials, the conventional virial theorem is violated. Starting from the tight-binding model, we derive the generalized virial theorem for Dirac electron systems, which contains an additional term associated with a momentum cutoff at the bottom of the energy band. Additionally, we derive the generalized virial theorem within the Dirac model using the minimization of the variational energy. The obtained theorem is illustrated by many-body calculations of the ground state energy of an electron gas in graphene carried out in Hartree-Fock and self-consistent random-phase approximations. Experimental verification of the theorem in the case of graphene is discussed.

Cutoff regulators in chiral nuclear effective field theory

Three-dimensional cutoff regulators are frequently employed in multi-nucleon calculations, but they violate chiral symmetry and Lorentz invariance. A cutoff regularization scheme is proposed to compensate systematically at subleading orders for these symmetry violations caused by regulator artifacts. This is especially helpful when a soft momentum cutoff has to be used for technical reasons. It is also shown that dimensional regularization can still be used for some Feynman (sub)diagrams while cutoff regulators are used for the rest.

Can X (5568) be described as a B s π ,BK¯resonant state?

The D0 Collaboration has recently seen a resonant-like peak in the $B_s\pi$ invariant mass spectrum, claimed to be a new state called $X(5568)$. Using a $B_s\pi$--$B\bar{K}$ coupled channel analysis, implementing unitarity, and with the interaction derived from Heavy Meson Chiral Perturbation Theory, we are able to reproduce the reported spectrum, with a pole that can be associated to the claimed $X(5568)$ state, and with mass and width in agreement with the ones reported in the experimental analysis. However, if the $T$-matrix regularization is performed by means of a momentum cutoff, the value for the latter needed to reproduce the spectrum is $\Lambda = 2.80 \pm 0.04\ \text{GeV}$, much larger than a "natural" value $\Lambda \simeq 1\ \text{GeV}$. In view of this, it is difficult to interpret the nature of this new state. This state would not qualify as a resonance dynamically generated by the unitarity loops. Assuming the observed peak to correspond to a physical state, we make predictions for partners in the $D$, $D^\ast$, and $B^\ast$ sectors. Their observation (or lack thereof) would shed light into this issue.

Subtractive Renormalization of the NJL model *

In this work, we apply a subtractive renormalization method to the SU(2) Nambu-Jona-Lasinio (NJL) model in Born approximation and compare the results obtained here with those achieved by the standard momentum cutoff regularization. We have computed the dynamical quark mass, the chiral condensate, the pion mass and the quark-antiquark T-matrix as a function of the cutoff Λ and the subtraction scale ξ. We have shown that both approaches give similar results and the dependence of the physical quantities on the subtraction point ξ is much weaker than the dependence on the cutoff Λ.

Abinitio Calculations of Charge Symmetry Breaking in the A=4 Hypernuclei.

We report on abinitio no-core shell model calculations of the mirror Λ hypernuclei _{Λ}^{4}H and _{Λ}^{4}He, using the Bonn-Jülich leading-order chiral effective field theory hyperon-nucleon potentials plus a charge symmetry breaking Λ-Σ^{0} mixing vertex. In addition to reproducing rather well the 0_{g.s.}^{+} and 1_{exc}^{+} binding energies, these four-body calculations demonstrate for the first time that the observed charge symmetry breaking splitting of mirror levels, reaching hundreds of keV for 0_{g.s.}^{+}, can be reproduced using realistic theoretical interaction models, although with a non-negligible momentum cutoff dependence. Our results are discussed in relation to recent measurements of the _{Λ}^{4}H(0_{g.s.}^{+}) binding energy at the Mainz Microtron [A. Esser etal. (A1 Collaboration), Phys. Rev. Lett. 114, 232501 (2015)] and the _{Λ}^{4}He(1_{exc}^{+}) excitation energy [T.O. Yamamoto etal. (J-PARC E13 Collaboration), Phys. Rev. Lett. 115, 222501 (2015)].

Regularization of ultraviolet divergence for a particle interacting with a scalar quantum field

When a nonrelativistic particle interacts with a scalar quantum field, the standard perturbation theory leads to a dependence of the energy of its ground state on an undefined parameter---"momentum cutoff"---due to the ultraviolet divergence. We show that the use of nonasymptotic states of the system results in a calculation scheme in which all observable quantities remain finite and continuously depend on the coupling constant without any additional parameters. It is furthermore demonstrated that the divergence of traditional perturbation series is caused by the energy being a function with a logarithmic singularity for small values of the coupling constant.

Investigation of thennΛbound state in pionless effective theory

The possibility of an $nn\Lambda$ bound state is investigated in the framework of pionless effective field theory at leading order. A system of coupled integral equations are constructed in the spin-isospin basis, of which numerical solutions are investigated. In particular, we make use of the limit cycle behavior, i.e., cyclic singularities of coupled integral equations of the system, which would be associated with the formation of a three-body bound state, so-called the Efimov state, in the unitary limit. Furthermore, we find that, when the sharp momentum cutoff introduced in the integral equations is taken significantly larger than the hard scale of the effective theory, the coupling of a three-body contact interaction becomes cyclically singular indicating the onset of Efimov-like bound state formation. However, the paucity of empirical information to determine the parameters of the theory precludes a definitive conclusion on the existence of such a bound state. As a simple test of the feasibility of the $nn\Lambda$ bound system in nature, we explore the cutoff dependence of the theory, and uncertainties of the present study are discussed as well.

Quantum phase transitions of the Dirac oscillator in the anti-Snyder model

We obtain exact solutions of the ($2+1$) dimensional Dirac oscillator in a homogeneous magnetic field within the anti-Snyder modified uncertainty relation characterized by a momentum cutoff ($p\ensuremath{\le}{p}_{\mathrm{max}}=1/\sqrt{\ensuremath{\beta}}$). In ordinary quantum mechanics ($\ensuremath{\beta}\ensuremath{\rightarrow}0$) this system is known to have a single left-right chiral quantum phase transition (QPT). We show that a finite momentum cutoff modifies the spectrum introducing additional quantum phase transitions. It is also shown that the presence of momentum cutoff modifies the degeneracy of the states.

Renormalization group analysis of a neck-narrowing Lifshitz transition in the presence of weak short-range interactions in two dimensions

We study a system of weakly interacting electrons described by the energy dispersion $\xi(\mathbf{k}) = k_x^2 - k_y^2 - \mu$ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential $\mu_c=0$ where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of nonlocal terms in the Wilsonian effective action. We demonstrate that non-locality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale $E$ is lowered, an attractive four-fermion interaction grows as $\log^2 E$ for $E > \mu$, whereas it retains the usual BCS growth, $-\log E$, for $E < \mu$. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.

Solution of the relativistic Schrödinger equation for theδ′-Function potential in one dimension using cutoff regularization

We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.

Coordinate Space Solution of the Generalized Canonical Quantization Relation and the Momentum Cutoff

Coordinate Space Solution of the Generalized Canonical Quantization Relation and the Momentum Cutoff

Quantum design using a multiple internal reflections method in a study of fusion processes in the capture of alpha-particles by nuclei

A high precision method to determine fusion in the capture of α-particles by nuclei is presented. For α-capture by 40Ca and 44Ca, such an approach gives (1) the parameters of the α–nucleus potential and (2) fusion probabilities. This method found new parametrization and fusion probabilities and decreased the error by 41.72 times for α+Ca40 and 34.06 times for α+Ca44 in a description of experimental data in comparison with existing results. We show that the sharp angular momentum cutoff proposed by Glas and Mosel is a rough approximation, Wong's formula and the Hill–Wheeler approach determine the penetrability of the barrier without a correct consideration of the barrier shape, and the WKB approach gives reduced fusion probabilities. Based on our fusion probability formula, we explain the difference between experimental cross-sections for α+Ca40 and α+Ca44, which is connected with the theory of coexistence of the spherical and deformed shapes in the ground state for nuclei near the neutron magic shell N=20. To provide deeper insight into the physics of nuclei with the new magic number N=26, the cross-section for α+Ca46 is predicted for future experimental tests. The role of nuclear deformations in calculations of the fusion probabilities is analyzed.

(d¯)-(u¯) asymmetry in the proton in chiral effective theory.

We compute the d[over ¯]-u[over ¯] asymmetry in the proton in chiral effective theory, including both nucleon and Δ degrees of freedom, within the relativistic and heavy baryon frameworks. In addition to the distribution at x>0, we compute the corrections to the asymmetry from zero momentum contributions from pion rainbow and bubble diagrams at x=0, which have not been accounted for in previous analyses. We find that the empirical x dependence of d[over ¯]-u[over ¯] as well as the integrated asymmetry can be well reproduced in terms of a transverse momentum cutoff parameter.

Diagrammatic Monte Carlo study of the acoustic and the Bose–Einstein condensate polaron

We consider two large polaron systems that are described by a Fröhlich type of Hamiltonian, namely the Bose–Einstein condensate (BEC) polaron in the continuum and the acoustic polaron in a solid. We present ground-state energies of these two systems calculated with the Diagrammatic Monte Carlo (DiagMC) method and with a Feynman all-coupling approach. The DiagMC method evaluates up to very high order a diagrammatic series for the polaron’s self-energy. The Feynman all-coupling approach is a variational method that has been used for a wide range of polaronic problems. For the acoustic and BEC polaron both methods provide remarkably similar non-renormalized ground-state energies that are obtained after introducing a finite momentum cutoff. For the renormalized ground-state energies of the BEC polaron, there are relatively large discrepancies between the DiagMC and the Feynman predictions. These differences can be attributed to the renormalization procedure for the contact interaction.

Critical behaviors of a black hole in an asymptotically safe gravity with cosmological constant

We study the P−V/r+ criticality and phase transition of a quantum-corrected black hole in asymptotic safety (AS) gravity in the extended phase space. For a black hole, the cosmological constant is dependent on the momentum cutoff or energy scale; therefore, one can naturally treat it as a variable and connect it with thermodynamic pressure. We find that for a quantum-corrected black hole there is a first-order phase transition similar to that of the van der Waals liquid–gas system. We also analyze the types of phase transition between the smaller and larger black hole phases according to Ehrenfestʼs classification. It is shown that they are second-order phase transitions.

Quark ensembles with the infinite correlation length

A number of exactly integrable (quark) models of quantum field theory with the infinite correlation length have been considered. It has been shown that the standard vacuum quark ensemble—Dirac sea (in the case of the space-time dimension higher than three)—is unstable because of the strong degeneracy of a state, which is due to the character of the energy distribution. When the momentum cutoff parameter tends to infinity, the distribution becomes infinitely narrow, leading to large (unlimited) fluctuations. Various vacuum ensembles—Dirac sea, neutral ensemble, color superconductor, and BCS state—have been compared. In the case of the color interaction between quarks, the BCS state has been certainly chosen as the ground state of the quark ensemble.

Ultraviolet extrapolations in finite oscillator bases

The use of finite harmonic oscillator spaces in many-body calculations introduces both infrared (IR) and ultraviolet (UV) errors. The IR effects are well approximated by imposing a hard-wall boundary condition at a properly identified radius L_eff. We show that duality of the oscillator implies that the UV effects are equally well described by imposing a sharp momentum cutoff at a momentum Lambda_eff complementary to L_eff. By considering two-body systems with separable potentials, we show that the UV energy corrections depend on details of the potential, in contrast to the IR energy corrections, which depend only on the S-matrix. An adaptation of the separable treatment to more general interactions is developed and applied to model potentials as well as to the deuteron with realistic potentials. The previous success with a simple phenomenological form for the UV error is also explained. Possibilities for controlled extrapolations for A > 2 based on scaling arguments are discussed.

The parton model and its applications

This is a review of the program we started in 1968 to understand and generalize Bjorken scaling and Feynman's parton model in a canonical quantum field theory. It is shown that the parton model proposed for deep inelastic electron scatterings can be derived if a transverse momentum cutoff is imposed on all particles in the theory so that the impulse approximation holds. The deep inelastic electron–positron annihilation into a nucleon plus anything else is related by the crossing symmetry of quantum field theory to the deep inelastic electron–nucleon scattering. We have investigated the implication of crossing symmetry and found that the structure functions satisfy a scaling behavior analogous to the Bjorken limit for deep inelastic electron scattering. We then find that massive lepton pair production in collisions of two high energy hadrons can be treated by the parton model with an interesting scaling behavior for the differential cross-sections. This turns out to be the first example of a class of hard processes involving two initial hadrons.

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order $(\lambda^3 \rho_0)^{-1/2}$ (here $\lambda$ is the interaction strength and $\rho_0$ the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.

Spurious divergences in Dyson-Schwinger equations

We revisit the treatment of spurious ultraviolet divergences in the equation of motion of the gluon propagator caused by a momentum cutoff and the resulting violation of gauge invariance. With present continuum studies of the gluon propagator from its Dyson-Schwinger equation reaching the level of quantitatively accurate descriptions, it becomes increasingly important to understand how to subtract these spurious divergences in an unambiguous way. Here we propose such a method. It is based entirely on the asymptotic perturbative behavior of the QCD Green's functions without affecting non-perturbative aspects such as mass terms or the asymptotic infrared behavior. As a particular example, this allows us to assess the possible influence of the tadpole diagram beyond perturbation theory. Finally, we test this method numerically by solving the system of Dyson-Schwinger equations of the gluon and ghost propagators.