Gauge Invariance Constraints Research Articles

Gauging the bulk: generalized gauging maps and holographic codes

Gauging is a general procedure for mapping a quantum many-body system with a global symmetry to one with a local gauge symmetry. We consider a generalized gauging map that does not enforce gauge symmetry at all lattice sites, and show that it is an isometry on the full input space including all charged sectors. We apply this generalized gauging map to convert global-symmetric bulk systems of holographic codes to gauge-symmetric bulk systems, and vice versa, while preserving duality with a global-symmetric boundary. We separately construct holographic codes with gauge-symmetric bulk systems by directly imposing gauge-invariance constraints onto existing holographic codes, and show that the resulting bulk gauge symmetries are dual to boundary global symmetries. Combining these ideas produces a toy model that captures several interesting features of holography — it exhibits a rudimentary sort of dynamical duality, can be modified to demonstrate the relationship between metric fluctuations and approximate error-correction, and serves as an illustration for certain no-go theorems concerning symmetries in holography. Finally, we apply the generalized gauging map to construct codes with arbitrary transversal gate sets — for any compact Lie group, we use a symmetry-preserving truncation scheme to construct covariant finite-dimensional approximate holographic codes.

New minimal supersymmetric GUT emergence and sub-Planckian renormalization group flow

Consistency of trans-unification renormalization group (RG) evolution is used to discuss the domain of definition of the New Minimal Supersymmetric SO(10)GUT (NMSGUT). We define the 1-loop RGE $\ensuremath{\beta}$ functions, simplifying generic formulae using constraints of gauge invariance and superpotential structure. We also calculate the 2 loop contributions to the gauge coupling and gaugino mass and indicate how to get full 2 loop results for all couplings. Our method overcomes combinatorial barriers that frustrate computer algebra based attempts to calculate SO(10) $\ensuremath{\beta}$ functions involving large irreps. Use of the RGEs identifies a perturbative domain $Q<{M}_{E}$, where ${M}_{E}<{M}_{\text{Planck}}$ is the scale of emergence where the NMSGUT, with GUT compatible soft supersymmetry breaking terms emerges from the strong UV dynamics associated with the Landau poles in gauge and Yukawa couplings. Due to the strength of the RG flows the Landau poles for gauge and Yukawa couplings lie near a cutoff scale ${\mathrm{\ensuremath{\Lambda}}}_{E}$ for the perturbative dynamics of the NMSGUT which just above ${M}_{E}$. SO(10) RG flows into the IR are shown to facilitate small gaugino masses and generation of negative Non Universal Higgs masses squared needed by realistic NMSGUT fits of low energy data. Running the simple canonical theory emergent at ${M}_{E}$ through ${M}_{X}$ down to the electroweak scale enables tests of candidate scenarios such as supergravity based NMSGUT with canonical kinetic terms and NMSGUT based dynamical Yukawa unification.

Gauge coupling unification without leptoquarks

We propose an interpretation of the gauge coupling unification scale which is not related to any new particle threshold. We revisit Grand Unified Theories and show that it is possible to completely eliminate the scalar as well as vector leptoquarks from the particle physics spectrum. As a consequence, in our approach the gauge hierarchy problem is put on different grounds, and the proton may be absolutely stable. In order to achieve that, we employ a number of nonlinear gauge-invariant constraints which only affect the superheavy degrees of freedom. We illustrate our considerations in a model based on the SU(5) group, with the generalization to other groups being straightforward. We discuss how scale or conformal invariance may be added to our proposal.

Color-factor symmetry and BCJ relations for QCD amplitudes

Tree-level n-point gauge-theory amplitudes with n − 2k gluons and k pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude. These symmetries lead to gauge-invariant constraints on the kinematic numerators. They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.

Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang-Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.

Quartet unconstrained formulation for massless higher spin fields

We construct simple unconstrained Lagrangian formulations for massless higher spin fields in flat space of arbitrary dimension and on anti-de Sitter background. Starting from the triplet equations of Francia and Sagnotti, which describe a chain of spin modes, we introduce an auxiliary field and find appropriate gauge invariant constraints that single out the spin- s mode. The resulting quartet of fields, thus describing an irreducible representation of the Poincaré group, is used to construct simple Lagrangian formulations, which are local, free from higher derivative terms and use equal number of auxiliary fields for an unconstrained description of any value of spin. Our method proves to be most efficient for an unconstrained description of massless higher spin fermions in anti-de Sitter space. A relation of the minimal models with the universal BRST approach is discussed.

How to solve the Schwinger-Dyson equations once and for all gauges

Study of the Schwinger-Dyson equation (SDE) for the fermion propagator to obtain dynamically generated chirally asymmetric solution in any covariant gauge is a complicated numerical exercise specially if one employs a sophisticated form of the fermion-boson interaction complying with the key features of a gauge field theory. However, constraints of gauge invariance can help construct such a solution without having the need to solve the SDE for every value of the gauge parameter. Starting from the Landau gauge where the computational complications are still manageable, we apply the Landau-Khalatnikov-Fradkin tansformation (LKFT) on the dynamically generated solution and find approximate analytical results for arbitrary value of the covariant gauge parameter. We also compare our results with exact numerical solutions.

Dynamical fermion masses and constraints of gauge invariance in quenched QED3

Numerical study of the Schwinger–Dyson equation (SDE) for the fermion propagator (FP) to obtain dynamically generated chirally asymmetric solution in an arbitrary covariant gauge ξ is a complicated exercise specially if one employs a sophisticated form of the fermion–boson interaction complying with the key features of a gauge field theory. However, constraints of gauge invariance can help construct such a solution without having the need to solve the Schwinger–Dyson equation for every value of ξ. In this article, we propose and implement a method to carry out this task in quenched quantum electrodynamics in a plane (QED3). We start from an approximate analytical form of the solution of the SDE for the FP in the Landau gauge. We consider the cases in which the interaction vertex (i) is bare and (ii) is full. We then apply the Landau–Khalatnikov–Fradkin transformations (LKFT) on the dynamically generated solution and find analytical results for arbitrary value of ξ. We also compare our results with exact numerical solutions available for a small number of values of ξ obtained through a direct analysis of the corresponding SDE.

Semiclassical gauge theories

We study the properties of a non-abelian gauge theory subjected to a gauge invariant constraint given by the classical equations of motion. The constraint is not imposed by hand, but appears naturally when we study a particular type of local gauge transformations. In this way, all standard techniques to treat gauge theories are available. We will show that this theory lives at one-loop. Also this model retains some quantum characteristic of the usual non-abelian gauge theories as asymptotic freedom.

Renormalization of Wilson operators in the light-cone gauge

We test the renormalization of Wilson operators and the Mandelstam- Leibbrandt gauge in the case when the sides of the loop are parallel to the n, n* vectors used in the M-L gauge. Graphs which in the Feynman gauge are free of ultra-violet divergences, in the M-L gauge show double divergences and single divergences with non-local Si and Ci functions. These non-local functions cancel out when we add all graphs together and the constraints of gauge invariance are satisfied. In Appendix C we briefly discuss the problems of the M-L gauge for loops containing spacelike lines.

Gauge invariance and confinement in a generalized Nambu-Jona-Lasinio model.

We discuss the quark-antiquark loop diagrams of an extended version of the Nambu{endash}Jona-Lasinio (NJL) model that includes a description of confinement. We show how the constraints of gauge invariance affect the calculations of such diagrams, if the {ital q{bar q}} system has the quantum numbers of the rho or omega mesons. In the NJL model without confinement, gauge invariance may be maintained by using a regularization scheme that respects that symmetry. Alternatively, gauge-invariance constraints may be imposed by introducing a substraction procedure when evaluating the {ital q{bar q}} loop integrals (vacuum polarization diagrams). We discuss the merits of such subtraction procedures in the presence of a confining interaction. We also provide a calculation of the vacuum polarization diagrams of the NJL model that includes the effects of confinement and also satisfies the constraints imposed by gauge invariance. Our analysis provides some justification for the subtraction procedure used in several studies of the NJL model. We also show, in a rather clear fashion, how confinement serves to change the analytic properties of the polarization tensors, so that the tensor for the theory with confinement has only a physical cut structure in the region of interest. {copyright} {ital 1996 The American Physicalmore » Society.}« less

Supersymmetry and gauge invariance constraints in a U(1) x U(1)'-Higgs superconducting cosmic string model.

A supersymmetric extension of the $U(1)\times U(1)^{\prime }$-Higgs bosonic superconducting cosmic string model is considered,and the constraints imposed upon such a model due to renormalizability, supersymmetry, and gauge invariance are examined. For a simple model with a single $U(1)$ chiral superfield and a single $% U(1)^{\prime }$ chiral superfield, the Witten mechanism for bosonic superconductivity (giving rise to long range gauge fields outside of the string) does not exist. The simplest model that can accommodate the requisite interactions requires five chiral supermultiplets. This superconducting cosmic string solution is investigated.

Compact and noncompact gauge theories on a lattice

A noncompact formulation of gauge theories on the lattice recently proposed makes use of auxiliary fields. We show that in such a formulation the representation of the gauge group in the case of SU(2) is completely reducible, one of the irreducible multiplets containing the Yang-Mills fields plus one auxiliary field. Elimination of this field by a gauge-invariant constraint leads to Wilson's formulation.

Effective monopoles in noncompact lattice QED.

In currently used lattice formulations of quantum electrodynamics, even though gauge fields are introduced as noncompact degrees of freedom, the constraints of gauge invariance imply that fermion fields are sensitive to features best described by compact or periodic variables such as magnetic monopoles. Results from a quenched simulation are presented, showing the existence of a percolation threshold for monopole current networks near the chiral-symmetry-breaking transition already known to occur from previous studies. Implications for the mechanism driving the chiral transition are discussed.

Unitary theory of the deuteron photodisintegration.

The deuteron photodisintegration in the \ensuremath{\Delta}-resonance region is calculated within a unitary theory of the NN-N\ensuremath{\Delta}-\ensuremath{\pi}NN system coupled to the photon. The photoexcitation amplitude of the \ensuremath{\Delta} is deduced from the ${M}_{1+}$((3/2) pion photoproduction amplitude consistently with its background term and the \ensuremath{\pi}N interaction. The NN-N\ensuremath{\Delta} transition amplitude is obtained from the coupled equations for the NN-N\ensuremath{\Delta}-\ensuremath{\pi}NN system. The one-pion-exchange currents as well as the normal single-nucleon current are included. The polarization parameters are reproduced quantitatively, but the cross section is slightly underestimated. Comparison of our results with those of the NN-N\ensuremath{\Delta} coupled-channel model is made. Effects of relativistic corrections and gauge-invariance constraints are discussed.

A gauge symmetric approach to Fierz identities

Seventy-five years ago Cartan invented spinors by mapping C2 onto isotropic (null) vectors in C3. In recent work this map was extended and it was shown that bispinors are isomorphic to a class of Yang–Mills vector triplets Fk =Ek +iHk which satisfy the following SU(2)×U(1) gauge invariant constraint: Fj ⋅ Fk =s2 δjk, where s2 = (1)/(3) Fk ⋅ Fk (k summed from 1 to 3). Thus bispinors have inherent SU(2) ×U(1) gauge symmetry. In this paper it is shown, using the extended Cartan map and the gauge symmetry of the constrained Yang–Mills fields, that all the Fierz identities reduce to a single equation. Moreover, this equation includes not only the 75 identities recently derived by Takahashi [Y. Takahashi, J. Math. Phys. 24, 1783 (1983)] but an additional 75 which come from interchanging gauge and vector components. It is further shown that the Fierz identities for bispinors can be generalized to any multiplet, Ψ∈C2n, consisting of 2n−1 spinors (n=1 for spinors, n=2 for bispinors, n=3 for bispinor doublets, etc.). The generalized identities can also be used to show that the 2n−1 spinor multiplets are isomorphic to multiplets of constrained Yang–Mills vector fields.

A reduced hamiltonian model for U(∞) gauge theories

A reduced version of the Kogut-Susskind gauge theory is presented for U(∞) gauge theories. The reduced hamiltonian H= 1 2 g 2ϵ iE 2 2−( 1 2 g 2) ϵ i≠jV ij , where V ij =tr U i D i D j D j ( U i D j ) + ( U i D j ) +, is obtained by representing the group of space translations in the diagonal part of U(∞) ⊗ U(∞). The space of states is invariant to the reduced gauge transformation U i → ωU i D i ω + D i + and the scalar product carries a gauge invariant constraint on U i The hamiltonian is also expr essed in terms of the string variables of Bars. We present the linear potential between static quarks at strong coupling and sketch the weak coupling expansion.

Gauge covariance in lattice field theories

We consider in detail the gauge invariance constraints in Hamiltonian lattice gauge theories, focusing mainly on pureSU(2) Yang-Mills theory in 2+1 dimensions. We present matrix and partial differential representations of the Hamiltonian in which all gauge constraints have been taken fully into account. The applicability of this formulation is demonstrated on small lattices.

Low-energy photo- and electroproduction for physical pions. I. Ward-identity and chiral-symmetry-breaking structure

The Ward identities of current algebra are combined with gauge-invariance constraints, on-shell partial conservation of the axial-vector current, and the Bjorken limit to obtain the low-energy expressions of the pion photo- and electroproduction invariant amplitudes for physical pions.

Amplitude structure and gauge-invariance constraints for dynamical models ofγN→π±Δ

An invariant-amplitude formalism is presented which features manifest gauge invariance and provides a convenient separation of the prominent dynamical components for $\ensuremath{\gamma}N\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\Delta}$. Kinematic constraints and low-$t$ theorems are analyzed in terms of $s$- and $t$-channel helicity amplitudes. Various dynamical models are discussed in terms of the general formalism and constraints.