arXiv subjects

Sarben Sarkar

Publications and source records attributed to Sarben Sarkar.

At least 37 records · Page 2Linked to original sources

arXiv2023

Wilsonian approach to the interaction $ϕ^2(iϕ)^\varepsilon$

We study the renormalisation of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric scalar field theory with the interaction $ϕ^2(iϕ)^\varepsilon$ using the Wilsonian approach and without any expansion in $\varepsilon$. Specifically, we solve the Wetterich equation in the local potential approximation, both in the ultraviolet regime and with the loop expansion. We calculate the scale-dependent effective potential and its infrared limit. The theory is found to be renormalisable at the one-loop level only for integer values of $\varepsilon$, a result which is not yet established within the $\varepsilon$-expansion. Particular attention is therefore paid to the two interesting cases $\varepsilon=1,2$, and the one-loop beta functions for the coupling associated with the interaction $iϕ^3$ and $-ϕ^4$ are computed. It is found that the $-ϕ^4$ theory has asymptotic freedom in four-dimensional spacetime. Some general properties for the Euclidean partition function and $n$-point functions are also derived.

hep-th↗

arXiv2023

$\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ theory

The scalar field theory with potential $V(φ)=\textstyle{\frac{1}{2}} m^2φ^2-\textstyle{\frac{1}{4}} gφ^4$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it is well defined, and it has a positive real energy spectrum for the case of spacetime dimension $D=1$. While the methods used in the literature do not easily generalize to quantum field theory, in this paper the path-integral representation of a $\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ theory is shown to provide a unified formulation for general $D$. A new conjectural relation between the Euclidean partition functions $Z^{\mathcal{P}\mathcal{T}}(g)$ of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric theory and $Z_{\rm Herm}(λ)$ of the $λφ^4$ ($λ>0$) Hermitian theory is proposed: $\log Z^{\mathcal{P}\mathcal{T}}(g)=\textstyle{\frac{1}{2}} \log Z_{\rm Herm}(-g+{\rm i} 0^+)+\textstyle{\frac{1}{2}}\log Z_{\rm Herm}(-g-{\rm i} 0^+)$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ field theory. A closely related relation is rigorously valid in $D=0$. For $D=1$, using a semiclassical evaluation of $Z^{\mathcal{P}\mathcal{T}}(g)$, this relation is verified by comparing the imaginary parts of the ground-state energy $E_0^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) and $E_{0,\rm Herm}(-g\pm {\rm i} 0^+)$.

hep-th↗

arXiv2022

Schwinger-Dyson equations and mass generation for an axion theory with a PT symmetric Yukawa fermion interaction

A nonperturbative Schwinger-Dyson analysis of mass generation is presented for a non-Hermitian PT-symmetric field theory in four dimensions of an axion coupled to a Dirac fermion.The model is motivated by phenomenological considerations.The axion has a quartic self-coupling $λ$ and a Yukawa coupling $g$ to the fermion. The Schwinger-Dyson equations are derived for the model with generic couplings. In the non-Hermitian case there is an additional nonperturbative contribution to the scalar mass. In a simplified rainbow analysis the solutions for the SD equations, are given for different regimes of the couplings $g$ and $λ$.

hep-ph↗

arXiv2022

PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation

We consider the renormalisation properties of non-Hermitian Yukawa theories involving a pseudoscalar (axion) field at or near $4$ dimensions. The non-Hermiticity is \cPT-symmetric where $\mathcal P$ is a linear idempotent operator (such as parity) and $\mathcal T$ is an anti-linear idempotent operator (such as time-reversal). The coupling constants of the Yukawa and quartic scalar coupling terms reflect this non-Hermiticity. The path integral representing the field theory is used to discuss the Feynman rules associated with the field theory. The fixed point structure associated with the renormalisation group has \cPT- symmetric and Hermitian fixed points. At two loops in the massless theory, we demonstrate the flow from Hermitian to non-Hermitian fixed points. From the one-loop renormalisation of a massive Yukawa theory, a self-consistent Nambu-Jona Lasinio gap equation is established and its real solutions are discussed.

hep-th↗

arXiv2022

Anomalies, CPT and Leptogenesis

We bring together different puzzles in physics beyond the Standard Model of particle physics. Within our model baryogenesis, neutrino mass, strong CP and dark matter puzzles are related. The common ingredient in connecting these puzzles is the Kalb-Ramond field, a two form present in the gravitational multiplet in the theory of closed strings. Leptons are fermions which we need to couple to gravitational degrees of freedom using a vierbein formalism. The presence of torsion provided by a Kalb-Ramond background leads us to firstly an effective model involving the Einstein-Cartan formulation of gravity, a gauge theory, in which coupling to fermions is via a covariant derivative and secondly to a mechanism for CPT violation. This picture emerges from a low energy string effective action obtained from a closed bosonic string theory after compactification to four dimensions. The Kalb-Ramond field in four dimensions can be identified with a pseudoscalar gravitational axion. Because of the presence of an axial anomaly this axion can couple with the gluon field, and in this way allows a connection with the strong CP problem and axionic dark matter.

hep-ph↗

arXiv2021

$\PT$ Symmetry and Renormalisation in Quantum Field Theory

Quantum systems governed by non-Hermitian Hamiltonians with $\PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that $\PT$ symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to $\PT$-symmetric effective Lagrangians. We show how $\PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples $\PT$-symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the $\PT$ norm that occurs in Hamiltonian quantum theory. We discuss examples where $\PT$-symmetric field theories emerge from Hermitian field theories due to effects of renormalization. We also consider the effects of renormalization on field theories that are non-Hermitian but $\PT$-symmetric from the start.

hep-th↗

arXiv2021

Towards perturbative renormalization of $ϕ^2(iϕ)^\varepsilon$ quantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $ϕ$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nablaϕ)^2+\frac{1}{2}μ^2ϕ^2+\frac{1}{2} gμ_0^2ϕ^2\big(iμ_0^{1-D/2}ϕ\big)^\varepsilon-ivϕ$, which now includes renormalization counterterms that are linear and quadratic in the field $ϕ$. The parameter $g$ is a dimensionless coupling strength and $μ_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

hep-th↗

arXiv2020

Curvature and thermal corrections in tree-level CPT-Violating Leptogenesis

In a model for leptogenesis based on spontaneous breaking of Lorentz and CPT symmetry [1-3], we examine the consistency of using the approximation of plane-wave solutions for a free spin-(1/2) Dirac (or Majorana) fermion field propagating in a Friedmann-Lemaitre-Robertson-Walker space time augmented with a cosmic time-dependent (or, equivalently, a temperature-dependent) Kalb-Ramond (KR) background. For the range of parameters relevant for leptogenesis, our analysis fully justifies the use of plane-wave solutions in our study of leptogenesis with Boltzmann equations; any corrections induced by space-time-curvature are negligible. We also elaborate further on how the lepton asymmetry is communicated to the Baryon sector. We demonstrate that the KR background (KRB) does not contribute to the anomaly equations that determine the baryon asymmetry a) through an explicit evaluation of a triangle Feynman graph and b) indirectly, on topological grounds, by identifying the KRB as torsion (in the effective string-inspired low energy gravitational field theory).

hep-ph↗

arXiv2018

Finite-energy dressed string-inspired Dirac-like monopoles

On extending the Standard Model (SM) Lagrangian, through a non-linear Born-Infeld (BI) hypercharge term with a parameter $β$ (of dimensions of [mass]$^2$), a finite energy monopole solution was claimed by Arunasalam and Kobakhidze [1]. We report on a new class of solutions within this framework which was missed in the earlier analysis. This new class was discovered on performing consistent analytic asymptotic analyses of the nonlinear differential equations describing the model; the shooting method used in numerical solutions to boundary value problems for ordinary differential equations is replaced in our approach by a method which uses diagonal Padé approximants. Our work uses the ansatz proposed by Cho and Maison to generate a static and spherically symmetric monopole with finite energy and differs from that used in the solution of [1]. Estimates of the total energy of the monopole are given, and detection prospects at colliders are briefly discussed.

hep-ph↗

arXiv2018

Spontaneous CPT Violation and Quantum Anomalies in a Model for Matter-Antimatter Asymmetry in the Cosmos

We review scenarios of baryogenesis through leptogenesis at early epochs of the universe, in string-inspired minimal extensions of the Standard Model (SM), involving heavy right-handed Majorana neutrinos. Spontaneous violation of CPT symmetry is induced by appropriate (in general, temperature-dependent) backgrounds of the Kalb-Ramond (KR) axion field, which has its origins in the (bosonic) massless string multiplet. As interesting features of the model, we also discuss two issues associated with quantum (chiral) anomalies: (i) the non-contribution of the KR axion background to the (anomalous) chiral magnetic effect, which arises in the presence of external electromagnetic fields and non-zero chiral chemical potentials of charged fermions and (ii) the potential rôle of quantum fluctuations of the KR axion on the (anomalous) radiative generation of a Majorana mass for the right-handed neutrinos themselves.

hep-ph↗

arXiv2018

The role of temperature dependent string-inspired CPT violating backgrounds in leptogenesis and the chiral magnetic effect

In a temperature dependent CPT-Violating (CPTV) axial time-like background (induced by the Kalb-Ramond tensor field of string theory) we discuss leptogenesis by solving the Boltzmann equation.The current work non-trivially modifies the framework of a previous phenomenological approach by the authors where the CPTV axial background was considered to be a constant (with no microscopic justification). The constant background approximation though is shown to capture the main phenomenological features of leptogenesis. On comparing our analysis to the related chiral magnetic effect for axial current condensates, we conclude that the Kalb-Ramond field does not play the role of the chiral chemical potential needed for that effect.

hep-ph↗

arXiv2018

PT-symmetric quantum field theory in D dimensions

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional spacetime, where $ϕ$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.

hep-th↗

arXiv2018

Regularised Kalb-Ramond Magnetic Monopole with Finite Energy

In a previous work we suggested a self-gravitating electromagnetic monopole solution in a string-inspired model involving global spontaneous breaking of a $SO(3)$ internal symmetry and Kalb-Ramond (KR) axions, stemming from an antisymmetric tensor field in the massless string multiplet. These axions carry a charge, which, in our model, also plays the rôle of the magnetic charge. The resulting geometry is close to that of a Reissner-Nordström (RN) black hole with charge proportional to the KR-axion charge. We proposed the existence of a thin shell structure inside a (large) core radius as the dominant mass contribution to the energy functional. The resulting energy was finite, and proportional to the KR-axion charge; however, the size of the shell was not determined and left as a phenomenological parameter. In the current article, we can calculate the mass-shell size, on proposing a regularisation of the black hole singularity via a matching procedure between the RN metric in the outer region and, in the inner region, a de Sitter space with a (positive) cosmological constant proportional to the scale of the spontaneous symmetry breaking of $SO(3)$ . The matching, which involves the Israel junction conditions for the metric and its first derivatives at the inner surface of the shell, determines the inner mass-shell radius. The axion charge plays an important rôle in guaranteeing positivity of the "mass coefficient" of the gravitational potential term appearing in the metric component; so the KR electromagnetic monopole shows normal attractive gravitational effects. This is to be contrasted with the global monopole case (in the absence of KR axions) where such a matching is known to yield a negative "mass coefficient" (and, hence, repulsive gravitational effects). The total energy of the monopole within the shell is calculated.

hep-th↗

arXiv2018

Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation

This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the potential in this partial differential equation is performed. This shift allows a perturbative asymptotic treatment of the differential equation for large values of the infrared cut-off. To leading order in perturbation theory the differential equation becomes a heat equation, where the sign of the diffusion constant changes as the space-time dimension $D$ passes through $2$. When $D<2$, one obtains a forward heat equation whose initial-value problem is well-posed. However, for $D>2$ one obtains a backward heat equation whose initial-value problem is ill-posed. For the special case $D=1$ the asymptotic series for cubic and quartic models is extrapolated to the small infrared-cut-off limit by using Padé techniques. The effective potential thus obtained from the partial differential equation is then used in a Schrödinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Padé procedure distinguishes between a $PT$-symmetric $igϕ^3$ theory and a conventional Hermitian $gϕ^3$ theory ($g$ real). For an $igϕ^3$ theory the effective potential is nonsingular and has a stable ground state but for a conventional $gϕ^3$ theory the effective potential is singular. For a conventional Hermitian $gϕ^4$ theory and a $PT$-symmetric $-gϕ^4$ theory ($g>0$) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.

hep-th↗

arXiv2017

Leptogenesis from Heavy Right-Handed Neutrinos in CPT Violating Backgrounds

We discuss leptogenesis in a model with heavy right-handed Majorana neutrinos propagating in a constant but otherwise generic CPT-violating axial time-like background (which could be motivated by string theory considerations). At temperatures much higher than the temperature of the electroweak phase transition we solve analytically but approximately (using Pade approximants) the corresponding Boltzmann equations, which describe lepton asymmetry generation due to the tree-level decays of the heavy neutrinos into standard model leptons. These leptons are effectively massless at such temperatures. The current work completes in a rigorous way a preliminary treatment of the same system, by some of the present authors. In this earlier work, lepton asymmetry was crudely estimated considering the decay of a right-handed neutrino at rest. Our present analysis includes thermal momentum modes for the heavy neutrino and this leads to a total lepton asymmetry which is bigger by a factor of two as compared to the previous estimate. Nevertheless, our current and preliminary results for the freezeout are found to be in agreement (within a 12.5% uncertainty). Our analysis depends on a novel use of Pade approximants to solve the Boltzmann equations and may be more widely useful in cosmology.

hep-ph↗

arXiv2017

Magnetic Monopoles from Global Monopoles in the presence of Kalb-Ramond Torsion

Possible classical solutions for electromagnetic monopoles induced by gravitational (global) monopoles in the presence of torsion are found. Such solutions rely on a non-zero (Kalb-Ramond) torsion strength, which may characterise low energy theories inspired from string models. Bounds from the current run of the LHC experiments are used to constrain the parameters of the model, but because the production mechanism depends on the details of the model and its ultraviolet completion, such bounds are presently only indicative.

hep-th↗

arXiv2016

Two-dimensional wave propagation without anomalous dispersion

In two space dimensions and one time dimension a wave changes its shape even in the absence of a dispersive medium. However, this anomalous dispersive behavior in empty two-dimensional space does not occur if the wave dynamics is described by a linear homogeneous wave equation in two space dimensions and {\it two} time dimensions. Wave propagation in such a space can be realized in a three-dimensional anisotropic metamaterial in which one of the space dimensions has a negative permittivity and thus serves as an effective second time dimension. These results lead to a fundamental understanding and new approaches to ultrashort pulse shaping in nanostructures and metamaterials.

cond-mat.mes-hall↗

arXiv2015

PT-symmetric interpretation of unstable effective potentials

The conventional interpretation of the one-loop effective potentials of the Higgs field in the Standard Model and the gravitino condensate in dynamically broken supergravity is that these theories are unstable at large field values. A PT-symmetric reinterpretation of these models at a quantum-mechanical level eliminates these instabilities and suggests that these instabilities may also be tamed at the quantum-field-theory level.

hep-th↗