Fermionic Currents Research Articles

Scalar and fermionic vacuum currents in de Sitter spacetime with compact dimensions

Vacuum expectation values (VEVs) of the current densities for charged scalar and Dirac spinor fields are investigated in $(D+1)$-dimensional de Sitter (dS) spacetime with toroidally compactified spatial dimensions. Along compact dimensions we impose quasiperiodicity conditions with arbitrary phases. In addition, the presence of a classical constant gauge field is assumed. The VEVs of the charge density and of the components for the current density along noncompact dimensions vanish. The gauge field leads to Aharonov-Bohm-like oscillations of the components along compact dimensions as functions of the magnetic flux. For small values of the comoving length of a compact dimension, compared with the dS curvature scale, the current density is related to the corresponding current in the Minkowski spacetime by a conformal relation. For large values of the comoving length and for a scalar field, depending on the mass of the field, two different regimes are realized with the monotonic and oscillatory damping of the current density. For a massive spinor field, the decay of the current density is always oscillatory. In supersymmetric models on the background of Minkowski spacetime with equal number of scalar and fermionic degrees of freedom and with the same phases in the periodicity conditions, the total current density vanishes due to the cancellation between the scalar and fermionic parts. The background gravitational field modifies the current densities for scalar and fermionic fields in different ways, and for massive fields there is no cancellation in the dS spacetime.

Exact solution of the Dirac equation withCPviolation

We consider Yukawa theory in which the fermion mass is induced by a Higgs-like scalar. In our model the fermion mass exhibits a temporal dependence, which naturally occurs in the early Universe setting. Assuming that the complex fermion mass changes as a tanh-kink, we construct an exact, helicity-conserving, $CP$-violating solution for the positive- and negative-frequency fermionic mode functions, which is valid in both the weak and strong $CP$ violation cases. Using this solution we then study the fermionic currents in both the initial vacuum and finite-density/-temperature setting. Our result shows that---due to a potentially large state squeezing---fermionic currents can exhibit a large oscillatory magnification. Having in mind applications to electroweak baryogenesis, we then compare our exact results with those obtained in a gradient approximation. Even though the gradient approximation does not capture the oscillatory effects of squeezing, it describes quite well the averaged current, obtained by performing a mode sum. Our main conclusion is that while the agreement with the semiclassical force is quite good in the thick-wall regime, the difference is sufficiently significant to motivate a more detailed quantitative study of baryogenesis sources in the thin-wall regime in more realistic settings.

FIERZ TRANSFORMATION FOR SU(2) AND SU(3) EIGHTH-ORDER FERMION INTERACTIONS

The Fierz transformations (Ft) for SU(2) and SU(3) operators in eighth-order fermion interactions are derived. These terms appear in expansions in terms of fermion currents of the following form: I1 I2 I3 I4 and also the composite [Formula: see text], where Γ(i) are either the SU(2) Pauli matrices or SU(3) Gell-Mann matrices. The calculation is carried out using the exchange projector operators.

Fermionic current from topology and boundaries with applications to higher-dimensional models and nanophysics

We investigate combined effects of topology and boundaries on the vacuum expectation value (VEV) of the fermionic current in the space with an arbitrary number of toroidally compactified dimensions. As a geometry of boundaries we consider two parallel plates on which the fermion field obeys bag boundary conditions. Along the compact dimensions, periodicity conditions are imposed with arbitrary phases. In addition, the presence of a constant gauge field is assumed. The nontrivial topology gives rise to an Aharonov-Bohm effect for the fermionic current induced by the gauge field. It is shown that the VEV of the charge density vanishes and the current density has nonzero expectation values for the components along compact dimensions only. The latter are periodic odd functions of the magnetic flux with the period equal to the flux quantum. In the region between the plates, the VEV of the fermionic current is decomposed into pure topological, single plate, and interference parts. For a massless field the single plate part vanishes and the interference part is distributed uniformly. The corresponding results are generalized for conformally flat spacetimes. Applications of the general formulas to finite-length carbon nanotubes are given within the framework of the Dirac model for quasiparticles in graphene. In the absence of the magnetic flux, two sublattices of the honeycomb graphene lattice yield opposite contributions and the fermionic current vanishes. A magnetic flux through the cross section of the nanotube breaks the symmetry, allowing the current to flow along the compact dimension.

Three point functions in higher spin AdS3 supergravity

In a previous work we have proposed that the Prokushkin-Vasiliev higher spin N=2 supergravity on AdS_3 is dual to a large N limit of the N=(2,2) CP^N Kazama-Suzuki model. There is now strong evidence supporting this proposal based on symmetry and spectrum comparison. In this paper we will give further evidence for the duality by studying correlation functions. We compute boundary three point functions with two fermionic operators and one higher spin bosonic current in terms of the bulk supergravity theory. Then we compare with the results in the dual CFT, where the supersymmetry of the theory turns out to be very helpful. In particular we use it to confirm results conjectured in the bosonic case. Moreover, correlators with a fermionic current can be obtained via supersymmetry.

On-shell recursion for massive fermion currents

Abstract We analyze the validity of BCFW recursion relations for currents of n − 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell. These currents are gauge-dependent and can be used as ingredients in the unitarity-based approach to computing one-loop amplitudes. The validity of BCFW recursion relations is well known to depend on the so-called boundary behavior of the currents as the momentum shift parameter goes to infinity. With off-shell currents, a new potential problem arises, namely unphysical poles that depend on the choice of gauge. We identify conditions under which boundary terms are absent and unphysical poles are avoided, so that there is a natural recursion relation. In particular, we are able to choose a gauge in which we construct a valid shift for currents with at least n − 3 gluons of the same helicity. We derive an analytic formula in the case where all gluons have the same helicity. As by-products, we prove the vanishing boundary behavior of general off-shell objects in Feynman gauge, and we find a compact generalization of Berends-Giele gluon currents with a generic reference spinor.

Non-Perturbative Self-Consistent Model in SU(N) Gauge Field Theory

Non-perturbative quasi-classical model in a gauge theory with the Yang-Mills (YM) field is developed. The self-consistent solutions of the Dirac equation in the SU (N ) gauge field, which is in the eikonal approximation, and the Yang-Mills (YM) equations containing the external fermion current are solved. It shown that the developed model has the self-consistent solutions of the Dirac and Yang-Mills equations at N ≥ 3. In this way, the solutions take place provided that the fermion and gauge fields exist simultaneously, so that the fermion current completely compensates the current generated by the gauge field due to self-interaction of it.

Higgs Mechanism and New Propagator of Massive Vector Bosons

discuss a serious problem related to the Higgs mechanism and show that the unitary gauge which imposes the condition of ф=ф÷ on the Higgs fields does not correspond to a proper gauge fixing. Instead, this is simply a procedure for producing the massive vector boson fields by hand. This suggests that the Lagrangian density of the weak interactions should be reconsidered by starting from the massive vector boson fields which couple to the fermion currents as the initial ingredients. Here, it is shown that the new renormalization scheme with massive vector bosons has no intrinsic problem and the massive vector boson fields do not give rise to any divergences for the physical observables in the renormalization scheme.

Surface states, edge currents, and the angular momentum of chiralp-wave superfluids

The spectra of fermionic excitations, pairing correlations and edge currents confined near the boundary of a chiral p-wave superfluid are calculated to leading order in $\hbar/p_f\xi$. Results for the energy- and momentum-resolved spectral functions, including the spectral current density, of a chiral p-wave superfluid near a confining boundary are reported. The spectral functions reveal the subtle role of the chiral edge states in relation to the edge current and the angular momentum of a chiral p-wave superfluid, including the rapid suppression of $L_z(T)$ for $0 \lesssim T\ll T_c$ in the fully gapped 2D chiral superfluid. The edge current and ground-state angular momentum are shown to be sensitive to boundary conditions, and as a consequence the topology and geometry of the confining boundaries. For perfect specular boundaries the edge current accounts for the ground-state angular momentum, $L_z=(N/2)\hbar$, of a cylindrical disk of chiral superfluid with $N/2$ fermion pairs. Non-specular scattering can dramatically suppress the edge current. In the limit of perfect retro-reflection the edge states form a flat band of zero modes that are non-chiral and generate no edge current. For a chiral superfluid film confined in a cylindrical toroidal geometry the ground-state angular momentum is, in general, non-extensive, and can have a value ranging from $L_z > (N/2)\hbar$ to $L_z < -(N/2)\hbar$ depending on the ratio of the inner and outer radii and the degree of back scattering on the inner and outer surfaces. Non-extensive scaling of $L_z$, and the reversal of the ground-state angular momentum for a toroidal geometry, would provide a signature of broken time-reversal symmetry of the ground state of superfluid \Hea, as well as direct observation of chiral edge currents.

Spectral function of the supersymmetry current

We continue our study of the retarded Green's function of the universal fermionic supersymmetry current ("supercurrent") for the most general class of d=3 N=2 SCFTs with D=10 or D=11 supergravity duals by studying the propagation of the Dirac gravitino in the electrically charged AdS-Reissner-Nordstr\"om black-brane background of N=2 minimal gauged supergravity in D=4. We expand upon results presented in a companion paper, including the absence of a Fermi surface and the appearance of a soft power-law gap at zero temperature. We also present the analytic solution of the gravitino equation in the AdS_2 X R^2 background which arises as the near-horizon limit at zero temperature. In addition we determine the quasinormal mode spectrum.

Relativistic two-body Coulomb–Breit Hamiltonian in an external weak gravitational field

A construction of the Coulomb–Breit Hamiltonian for a pair of fermions, considered as a quantum two-body system, immersed in an arbitrary background gravitational field described by Einsteinʼs General Relativity is presented. Working with Fermi normal coordinates for a freely falling observer in a spacetime region where there are no background sources and ignoring the gravitational back-reaction of the system, the effective Coulomb–Breit Hamiltonian is obtained starting from the S-matrix element corresponding to the one-photon exchange between the charged fermionic currents. The contributions due to retardation are considered up to order (v/c)2 and they are subsequently written as effective operators in the relativistic quantum mechanical Hilbert space of the system. The final Hamiltonian includes effects linear in the curvature and up to order (v/c)2.

The self-consistent quasi-classical model in [formula omitted] gauge field theory

The quasi-classical model in a gauge theory with the Yang–Mills (YM) field is developed. The self-consistent solutions of the Dirac equation in the external SU(N) gauge field and the Yang–Mills equations containing the external fermion current are obtained when the the Yang–Mills field is in the eikonal approximation. The derived solutions are quantized in the quasi-classical approach. It is shown that the solutions take place provided that the dimension of the gauge group is N⩾3. In this way, the fermion and gauge fields exist simultaneously, so that the fermion current completely compensates the current generated by the gauge field due to its self-interaction. The obtained solution are considered in the context of QCD.

Universal quantum computation in a semiconductor quantum wire network

Universal quantum computation (UQC) using Majorana fermions on a 2D topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include \emph{any} two-qubit gate and also the single-qubit $\pi/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, \emph{etc.}), to which the Majorana fermions are bound in a TS medium, can be quantum coherent. We show that these obstacles can be overcome using a semiconductor quantum wire network in the vicinity of an $s$-wave superconductor, by constructing topologically protected two-qubit gates and any arbitrary single-qubit phase gate in a topologically unprotected manner, which can be error corrected using magic state distillation. Thus our strategy, using a judicious combination of topologically protected and unprotected gate operations, realizes UQC on a quantum wire network with a remarkably high error threshold of $0.14$ as compared to $10^{-3}$ to $10^{-4}$ in ordinary unprotected quantum computation.

Fermionic current densities induced by magnetic flux in a conical space with a circular boundary

We investigate the vacuum expectation value of the fermionic current induced by a magnetic flux in a ($2+1$)-dimensional conical spacetime in the presence of a circular boundary. On the boundary the fermionic field obeys the MIT bag boundary condition. For irregular modes, a special case of boundary conditions at the cone apex is considered, when the MIT bag boundary condition is imposed at a finite radius, which is then taken to zero. We observe that the vacuum expectation values for both the charge density and azimuthal current are periodic functions of the magnetic flux with the period equal to the flux quantum whereas the expectation value of the radial component vanishes. For both exterior and interior regions, the expectation values of the current are decomposed into boundary-free and boundary-induced parts. For a massless field the boundary-free part in the vacuum expectation value of the charge density vanishes, whereas the presence of the boundary induces nonzero charge density. Two integral representations are given for the boundary-free part in the case of a massive fermionic field for arbitrary values of the opening angle of the cone and magnetic flux. The behavior of the induced fermionic current is investigated in various asymptotic regions of the parameters. At distances from the boundary larger than the Compton wavelength of the fermion particle, the vacuum expectation values decay exponentially with the decay rate depending on the opening angle of the cone. We make a comparison with the results already known from the literature for some particular cases.

Anomalies, gauge field topology, and the lattice

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I suggest that the fourth power of the naive Dirac operator can provide a natural method to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally “fall through the lattice.” Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.

Induced fermionic current in toroidally compactified spacetimes with applications to cylindrical and toroidal nanotubes

The vacuum expectation value of the fermionic current is evaluated for a massive spinor field in spacetimes with an arbitrary number of toroidally compactified spatial dimensions in presence of a constant gauge field. By using the Abel-Plana type summation formula and the zeta function technique we present the fermionic current in two different forms. Non-trivial topology of the background spacetime leads to the Aharonov-Bohm effect on the fermionic current induced by the gauge field. The current is a periodic function of the magnetic flux with the period equal to the flux quantum. In the absence of the gauge field it vanishes for special cases of untwisted and twisted fields. Applications of the general formulae to Kaluz-Klein type models and to cylindrical and toroidal carbon nanotubes are given. In the absence of magnetic flux the total fermionic current in carbon nanotubes vanishes, due to the cancellation of contributions from two different sublattices of the graphene hexagonal lattice.

Free-field representations and geometry of some Gepner models

The geometry of the k K Gepner model, where k + 2 = 2K, is investigated by a free-field representation known as the “bcβγ” system. Using this representation, we directly show that the internal sector of the model is given by Landau-Ginzburg ℂK/ℤ2K orbifold. Then we consider the deformation of the orbifold by a marginal antichiral-chiral operator. Analyzing the chiral de Rham complex structure in the holomorphic sector, we show that it coincides with chiral de Rham complex of some toric manifold, where toric data are given by certain fermionic screening currents. This allows relating the Gepner model deformed by the marginal operator to a σ-model on the CY manifold realized as a double cover of ℙK − 1 with ramification along a certain submanifold.

Scattering of charge carriers in graphene induced by topological defects

We study the scattering of graphene quasiparticles by topological defects, represented by holes, pentagons and heptagons. For holes, we found that at low concentration they give a negligible contribution to the resistivity. Whenever pentagons or heptagons are introduced we realize that a fermionic current is scattered by defects.

B-L cosmic strings in heterotic standard models

E_{8} X E_{8} heterotic string and M-theory, when compactified on smooth Calabi-Yau manifolds with SU(4) vector bundles, can give rise to softly broken N=1 supersymmetric theories with the exact matter spectrum of the MSSM, including three right-handed neutrinos and one Higgs-Higgs conjugate pair of supermultiplets. These vacua have the SU(3)_{C} X SU(2)_{L} X U(1)_{Y} gauge group of the standard model augmented by an additional gauged U(1)_{B-L}. Their minimal content requires that the B-L symmetry be spontaneously broken by a vacuum expectation value of at least one right-handed sneutrino. The soft supersymmetry breaking operators can induce radiative breaking of the B-L gauge symmetry with an acceptable B-L/electroweak hierarchy. In this paper, it is shown that U(1)_{B-L} cosmic strings occur in this context, potentially with both bosonic and fermionic superconductivity. We present a numerical analysis that demonstrates that boson condensates can, in principle, form for theories of this type. However, the weak Yukawa and gauge couplings of the right-handed sneutrino suggests that bosonic superconductivity will not occur in the simplest vacua in this context. The electroweak phase transition also disallows fermion superconductivity, although substantial bound state fermion currents can exist.

Amplitudes in theN=4supersymmetric Yang-Mills theory from quantum geometry of momentum space

We discuss multiloop MHV amplitudes in the N=4 SYM theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of freedom. Kinematical invariants of external particles yield the moduli spaces of complex or Kahler structures which are the playgrounds for the Kodaira-Spencer(KS) or Kahler type gravity. We suggest fermionic representation of the loop MHV amplitudes in the N=4 SYM theory assuming the identification of the IR regulator branes with KS fermions in the B model and Lagrangian branes in A model. The two-easy mass box diagram is related to the correlator of fermionic currents on the spectral curve in B model or hyperbolic volume in the A model and it plays the role of a building block in the whole picture. The BDS-like anzatz has the interpretation as the semiclassical limit of a fermionic correlator. It is argued that fermionic representation implies a kind of integrability on the moduli spaces. We conjecture the interpretation of the reggeon degrees of freedom in terms of the open strings stretched between the IR regulator branes.