Massive Majorana Fermions Research Articles

Gravitational action for a massive Majorana fermion in 2d quantum gravity

We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.

Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m2. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.Besides the Liouville action, at order m0, which only involves the background metric and the conformal factor σ, the various contributions to the effective gravitational action at higher orders in m2 can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m2, these contributions can be re-written as a term ∫ e2σσ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.

Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.

Supersymmetry in QCD2 coupled to fermions

We consider $1+1$ dimensional Yang-Mills theory with gauge group $G$ coupled to a massive Majorana fermion field in an adjoint representation and a number of massless Dirac or Majorana fermions transforming in arbitrary representations of the gauge group $G$. It is shown that the spectrum of the massive sector of this theory becomes supersymmetric at particular mass of adjoint fermion. This mass is independent of the detailed structure of the massless sector of the model and depends only on the gauge group $G$ and integer $k$ measuring the total anomaly. The massless sector of the model is shown to be not necessarily supersymmetric.

Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: A Majorana fermion approach

The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank N of the SU(N) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [A. M. Tsvelik, Phys. Rev. B 42, 10 499 (1990)] to describe the low-energy properties of the Haldane phase of the spin-1 Heisenberg chain from three massive Majorana fermions. We find for all N the emergence of a non-degenerate gapped phase with edge states whose topological protection depends on the parity of N. While for N odd there is no such protection, the phase with even N is shown to be topologically protected. We find that the phase belongs to the same topological class as the phase with edge states living in self-conjugate fully antisymmetric representation of the SU(N) group.

Finite-size scaling at a topological transition: Bilinear-biquadratic spin-1 chain

We consider a finite size scaling function across a topological phase transition in 1D models. For models of non-interacting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition (Gulden et al 2016). Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/\xi$, where $L$ is the chain length and $\xi$ is the correlation length, coincides with that of three species of non-interacting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge $c=3/2$. We have shown that it still holds away from the conformal point, including the finite size corrections. We have also observed peculiar differences between even and odd size chains, which may be fully accounted for by residual interactions of the edge states.

SmeftFR – Feynman rules generator for the Standard Model Effective Field Theory

We present SmeftFR, a Mathematica package designed to generate the Feynman rules for the Standard Model Effective Field Theory (SMEFT) including the complete set of gauge invariant operators up to dimension 6. Feynman rules are generated with the use of FeynRules package, directly in the physical (mass eigenstates) basis for all fields. The complete set of interaction vertices can be derived including all or any chosen subset of SMEFT operators. As an option, the user can also choose preferred gauge fixing, generating Feynman rules in unitary or Rξ-gauges (the latter include generation of ghost vertices). Further options allow to treat neutrino fields as massless Weyl or massive Majorana fermions. The derived Lagrangian in the mass basis can be exported in various formats supported by FeynRules, such as UFO and FeynArts. Initialisation of numerical values of d=6 Wilson coefficients used by SmeftFR is interfaced to WCxf format. The package also includes dedicated Latex generator allowing to print the result in clear human-readable form. SmeftFR can be downloaded from the address www.fuw.edu.pl/smeft. Program summaryProgram Title:SmeftFR v2.0Program Files doi:http://dx.doi.org/10.17632/yyv9fpmwzv.1Licensing provisions: GPLv3Programming language: Mathematica 11.3 or later (earlier versions were reported to have problems running this code).Nature of problem: Standard Model Effective Field Theory (SMEFT) has become in recent years a commonly used model to parameterise the potential effects of New Physics. Calculation of physical observables within such a complicated theory requires large degree of automatisation, starting from the generation of Feynman rules in various formats, which could be further imported to Monte Carlo generators or symbolic packages calculating transition amplitudes. SmeftFR package generates such Feynman rules for SMEFT in physical (mass) field basis for the user defined subset of dimension 5 and 6 operators, in unitary or linear Rξ-gauges.Solution method: Implementation (as the Mathematica package) of the results of Ref. [1]. SmeftFR generates dynamically “model files” for the FeynRules package [2], based on user-defined set of options — most important are SMEFT operators subset, gauge fixing choice and numerical input for Wilson coefficients in the WCxf format [3]. Subsequently FeynRules engine is used to calculate SMEFT Feynman rules, which can be later manipulated or exported in several formats, such as UFO, FeynArts, Latex, WCxf and others, by SmeftFR auxiliary routines.Additional comments: Running the code requires prior installation of the FeynRules package [2], https://feynrules.irmp.ucl.ac.be.[1] A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek and K. Suxho, “Feynman rules for the Standard Model Effective Field Theory in Rξ-gauges,” JHEP 1706, 143 (2017).[2] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, “FeynRules 2.0 - A complete toolbox for tree-level phenomenology,” Comput. Phys. Commun. 185, 2250 (2014).[3] J. Aebischer, et al., WCxf: an exchange format for Wilson coefficients beyond the Standard Model, Comput. Phys. Commun. 232 (2018) 71–83.

Topological Quantum Phase Transition and Local Topological Order in a Strongly Interacting Light-Matter System

An attempt is made to understand the topological quantum phase transition, emergence of relativistic modes and local topological order of light in a strongly interacting light-matter system. We study this system, in a one dimensional array of nonlinear cavities. Topological quantum phase transition occurs with massless excitation only for the finite detuning process. We present a few results based on the exact analytical calculations along with the physical explanations. We observe the emergence of massive Majorana fermion mode at the topological state, massless Majorana-Weyl fermion mode during the topological quantum phase transition and Dirac fermion mode for the non-topological state. Finally, we study the quantized Berry phase (topological order) and its connection to the topological number (winding number).

(2+1)−d Georgi-Glashow model near critical temperature

We study correlations functions of magnetic vortices $V$ and Polyakov loop $P$ operators in the 2+1 dimensional Georgi-Glashow model in the vicinity of the deconfining phase transition. In this regime the (dimensionally reduced) model is mapped onto a free theory of two massive Majorana fermions. We utilize this fermionic representation to explicitly calculate the expectation values of $V$ and $P$ as well as their correlators. In particular we show that the $VV$ correlator is large, and thus the anomalous breaking of the magnetic $U(1)$ symmetry is order one effect in the near critical region. We also calculate the contribution of magnetic vortices to the entropy and the free energy of the system.

Minimal scalar-less matter-coupled supergravity

We build the minimal supergravity model where the nilpotent chiral goldstino superfield is coupled to a chiral matter superfield, realising a different non-linear representation through a mixed nilpotency constraint. The model describes the spontaneous breaking of local supersymmetry in the presence of a generically massive Majorana fermion, but in the absence of elementary scalars. The sign and the size of the cosmological constant, the spectrum and the four-fermion interactions are controlled by suitable parameters.

Truncated conformal space approach for perturbed Wess–Zumino–Witten [formula omitted] models

We outline the application of the truncated conformal space approach (TCSA) to perturbations of SU(2)k Wess–Zumino–Witten theories. As examples of this methodology, we consider two distinct perturbations of SU(2)1 and one of SU(2)2. SU(2)1 is first perturbed by its spin-1/2 field, a model which is equivalent to the sine-Gordon model at a particular value of its coupling β. The sine-Gordon spectrum is correctly reproduced as well as the corresponding finite-size corrections. We next study SU(2)1 with a marginal current–current perturbation. The TCSA results can be matched to perturbation theory within an appropriate treatment of the UV divergences. We find however that these results do not match field theoretic computations on the same model performed with a Lorentz invariant regulator. Finally, we consider SU(2)2 perturbed by its spin-1 field, which is equivalent to three decoupled massive Majorana fermions. In this case as well the TCSA reproduces accurately the known spectrum.

Neutrinoless double-beta decay and related topics

The fundamental importance of searching for neutrinoless double-beta decay (0νββ-decay) is widely recognized. Observation of the decay would tell us that the total lepton number is not conserved and that, consequently, neutrinos are massive Majorana fermions. A brief history of the double-beta decay is presented. The 0νββ-decay is discussed in context of neutrino oscillation data. The perspectives of the experimental 0νββ-decay searches are analyzed. The importance of reliable determination of the 0νββ-decay nuclear matrix elements is pointed out. The problem of distinguishing of the light-neutrino exchange, heavy-neutrino exchange and the trilinear R-parity breaking supersymmetric (\(\not R_p \) SUSY) mechanisms of the 0νββ-decay is addressed. Further, the process of resonant neutrinoless double-electron capture (0νɛɛ) is revisited. Arguments are presented that an experimental search for the 0νɛɛ might be feasible.

Neutrinoless double-beta decay and double-electron capture

The fundamental importance of searching for neutrinoless double-beta decay is widely recognized. Observation of the decay would tell us that the total lepton number is not conserved and that, consequently, neutrinos are massive Majorana fermions. The same statement could be made in the case of observing neutrinoless double-electron capture. We address the question of the sensitivity of the 0 ν ε ε decay to the effective mass of the Majorana neutrino. According to our estimates, in the case of 152Gd and 164Er the sensitivity can be comparable to the favored 0 ν β β decays of nuclei. The main uncertainty in the prediction of half-lives of the 0 ν ε ε decay stems from the lack of sufficient precision in measuring the mass difference between the parent and daughter atoms. More accurate measurements can be accomplished using the modern high-precision ion traps.

Finite volume phases of large-Ngauge theories with massive adjoint fermions

The phase structure of QCD-like gauge theories with fermions in various representations is an interesting but generally analytically intractable problem. One way to ensure weak coupling is to define the theory in a small finite volume, in this case S3 × S1. Genuine phase transitions can then occur in the large N theory. Here, we use this technique to investigate SU(N) gauge theory with a number Nf of massive adjoint-valued Majorana fermions having non-thermal boundary conditions around S1. For Nf = 1 we find a line of transitions that separate the weak-coupling analogues of the confined and de-confined phases for which the density of eigenvalues of the Wilson line transform from the uniform distribution to a gapped distribution. However, the situation for Nf > 1 is much richer and a series of weak-coupling analogues of partially-confined phases appear which leave unbroken a psubgroup of the centre symmetry. In these p phases the eigenvalue density has p gaps and they are separated from the confining phase and from one-another by first order phase transitions. We show that for small enough mR (the mass of the fermions times the radius of the S3) only the confined phase exists. The large N phase diagram is consistent with the finite N result and with other approaches based on 3 × S1 calculations and lattice simulations.

Holomorphic parafermions in the Potts model and stochastic Loewner evolution

We analyse parafermionic operators in theQ-state Potts model from three different perspectives. First, we explicitly constructlattice holomorphic observables in the Fortuin–Kasteleyn representation,and point out some special simplifying features of the particular caseQ = 2 (Ising model). In particular, away from criticality, we find a lattice generalizationof the massive Majorana fermion equation. We also compare the parafermionicscaling dimensions with known results from conformal field theory and Coulombgas methods in the continuum. Finally, we show that expectation values ofthese parafermions correspond to local observables of the stochastic Loewnerevolution process which is conjectured to describe the scaling limit of theQ-state Potts model.

Massive Majorana fermion coupled to two-dimensional gravity and the random-lattice Ising model

We consider the partition function of the two-dimensional free massive Majorana fermion coupled to the quantized metric of the spherical topology. By adding an arbitrary conformal “spectator” matter, we gain control over the total matter central charge. This provides an interesting continuously parameterized family of critical points and also allows making a connection with the semiclassical limit. We use the Liouville field theory as the effective description of the quantized gravity. The spherical scaling function is calculated approximately, but (we believe) to a good numerical precision, in almost the whole domain of the spectator parameter. An impressive comparison with the predictions of the exactly solvable matrix model yields a more general model of random-lattice statistics, which is most probably not solvable by the matrix-model technique but reveals a more general pattern of critical behavior. We hope that numerical simulations or series extrapolation will be able to reveal our family of scaling functions.

On exact supersymmetry in DLCQ

In recent years a supersymmetric form of discrete light-cone quantization (hereafter `SDLCQ') has emerged as a very powerful tool for solving supersymmetric field theories. In this scheme, one calculates the light-cone supercharge with respect to a discretized light-cone Fock basis, instead of working with the light-cone Hamiltonian. This procedure has the advantage of preserving supersymmetry even in the discretized theory, and eliminates the need for explicit renormalizations in 1+1 dimensions. In order to compare the usual DLCQ prescription with the supersymmetric prescription, we consider two dimensional SU( N) Yang-Mills theory coupled to a massive adjoint Majorana fermion, which is known to be supersymmetric at a particular value of the fermion mass. After studying how singular-valued amplitudes and intermediate zero momentum modes are regularized in both schemes, we are able to establish a precise connection between conventional DLCQ and its supersymmetric extension, SDLCQ. In particular, we derive the explicit form of the (irrelevant) interaction that renders the DLCQ formulation of the theory exactly supersymmetric for any light-cone compactification. We check our analytical results via a numerical procedure, and discuss the relevance of this interaction when supersymmetry is explicitly broken.

Low-energy excitations in two-leg and three-leg quantum spin ladders

Low-energy excitations in spin- 1 2 antiferromagnetic (AF) Heisenberg spin ladders are studied by bosonization and in a gauge-theoretical description. The AF Heisenberg models on ladders are described by spin- 1 2 fermions and the systems are reduced to relativistic gauge theories of fermions by a mean-field type decoupling and linearization of the fermion's dispersion relation near Fermi points. There the gauge field is nothing other that phase degrees of freedom of the “mean field” on links. It is explicitly shown that zero-modes of boson fields for the bosonization of fermions play an essential role in obtaining correct results. In the two-leg case, the lowest-energy excitations are a spin-triplet magnon and they are described by three massive Majorana fermions. On the other hand, in the three-leg system, spin excitations on the top and bottom chains are described by two massless boson fields. It is predicted that if interaction between spins on the top and bottom chains is introduced, a phase transition occurs at some critical coupling. Above the critical coupling, the system acquires an energy gap and low-energy excitations are a spin-triplet magnon and a spin-singlet excitation.

Spin Gap ofS=1/2Heisenberg Ladder: Field Theoretical Study of Magnetic Susceptibility and NMR Relaxation Rate

The temperature dependence of the static spin susceptibility, χ, and NMR relaxation rate, T 1 -1 , of S =1/2 Heisenberg two-leg ladder systems have been studied based on the Majorana fermion representation of the S =1/2 Heisenberg spin Hamiltonian introduced by Shelton, Nersesyan and Tsvelik (Phys. Rev. B 53 (1996) 8521). It is shown that in the case of SrCu 2 O 3 , we can fit the experimental data of χ and T 1 -1 in terms of the massive Majorana fermion excitation with mass m =440 K to resolve the apparent discrepancy of the spin gap activation energies between χ and T 1 -1 .

The renormalization group analysis of the spin-spin correlation function in the two-dimensional Ising model

We consider the scaling limit of the two-dimensional Ising model, which gives a theory of a massive Majorana fermion. By solving the renormalization group equations we determine the possible logarithmic dependence on the mass in the short-distance expansion of the spin-spin correlation function. In this paper we do not attempt to calculate the coefficients of the short-distance expansion explicitly.