Bilinear Condensates Research Articles

Kähler-Dirac fermions on Euclidean dynamical triangulations

We study K\"ahler-Dirac fermions on Euclidean dynamical triangulations. This fermion formulation furnishes a natural extension of staggered fermions to random geometries without requring vielbeins and spin connections. We work in the quenched approximation where the geometry is allowed to fluctuate but there is no back-reaction from the matter on the geometry. By examining the eigenvalue spectrum and the masses of scalar mesons we find evidence for a four fold degeneracy in the fermion spectrum in the large volume, continuum limit. It is natural to associate this degeneracy with the well known equivalence in continuum flat space between the K\"ahler-Dirac fermion and four copies of a Dirac fermion. Lattice effects then lift this degeneracy in a manner similar to staggered fermions on regular lattices. The evidence that these discretization effects vanish in the continuum limit suggests both that lattice continuum K\"ahler-Dirac fermions are recovered at that point, and that this limit truly corresponds to smooth continuum geometries. One additional advantage of the K\"ahler-Dirac action is that it respects an exact $U(1)$ symmetry on any random triangulation. This $U(1)$ symmetry is related to continuum chiral symmetry. By examining fermion bilinear condensates we find strong evidence that this $U(1)$ symmetry is not spontaneously broken in the model at order the Planck scale. This is a necessary requirement if models based on dynamical triangulations are to provide a valid ultraviolet complete formulation of quantum gravity.

Living without supersymmetry—the conformal alternative and a dynamical Higgs boson

We show that the key results of supersymmetry can be achieved via conformal symmetry instead. We propose that the Higgs boson be a dynamical fermion–antifermion bound state rather than an elementary scalar field, so that there is then no quadratically divergent self-energy problem for it and thus no need to invoke supersymmetry to resolve the problem. To obtain such a dynamical Higgs boson we study a conformal invariant gauge theory of interacting fermions and gauge bosons. The conformal invariance of the theory is realized via scaling with anomalous dimensions in the ultraviolet, and by a dynamical symmetry breaking via fermion bilinear condensates in the infrared, a breaking in which the dynamical dimension of the composite operator is reduced from three to two. With this reduction in dimension we can augment the gauge theory with a four-fermion interaction made renormalizable by this reduction, and can reinterpret the theory as a renormalizable version of the Nambu–Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being a mean-field theory and the four-fermion interaction being the residual interaction. It is this residual interaction and not the mean field that then generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold, with its width potentially being a diagnostic that could distinguish a dynamical Higgs boson from an elementary one. We couple the theory to a gravity theory, conformal gravity, that is equally conformal invariant, with the interplay between conformal gravity and the four-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a unitary and renormalizable quantum theory of gravity there is no need for string theory with its supersymmetric underpinnings. With the vacuum energy problem being resolved and with conformal gravity fits to phenomena such as galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.

Novel phases in strongly coupled four-fermion theories

We study a lattice model comprising four massless reduced staggered fermions in four dimensions coupled through an $SU(4)$-invariant four-fermion interaction. We present both theoretical arguments and numerical evidence that no bilinear fermion condensates are present for any value of the four-fermi coupling, in contrast to earlier studies of Higgs--Yukawa models with different exact lattice symmetries. At strong coupling we observe the formation of a four-fermion condensate and a mass gap in spite of the absence of bilinear condensates. Unlike those previously studied systems we do not find a ferromagnetic phase separating this strong-coupling phase from the massless weak-coupling phase. Instead we observe long-range correlations in a narrow region of the coupling, still with vanishing bilinear condensates. While our numerical results come from relatively small lattice volumes that call for caution in drawing conclusions, if this novel phase structure is verified by future investigations employing larger volumes it may offer the possibility for new continuum limits for strongly interacting fermions in four dimensions.

Fermion masses through four-fermion condensates

Fermion masses can be generated through four-fermion condensates when symmetries prevent fermion bilinear condensates from forming. This less explored mechanism of fermion mass generation is responsible for making four reduced staggered lattice fermions massive at strong couplings in a lattice model with a local four-fermion coupling. The model has a massless fermion phase at weak couplings and a massive fermion phase at strong couplings. In particular there is no spontaneous symmetry breaking of any lattice symmetries in both these phases. Recently it was discovered that in three space-time dimensions there is a direct second order phase transition between the two phases. Here we study the same model in four space-time dimensions and find results consistent with the existence of a narrow intermediate phase with fermion bilinear condensates, that separates the two asymptotic phases by continuous phase transitions.

From domain wall to overlap in 2 + 1 d

The equivalence of domain wall and overlap fermion formulations is demonstrated for lattice gauge theories in 2+1 spacetime dimensions with parity-invariant mass terms. Even though the domain wall approach distinguishes propagation along a third direction with projectors ${1\over2}(1\pm\gamma_3)$, the truncated overlap operator obtained for finite wall separation $L_s$ is invariant under interchange of $\gamma_3$ and $\gamma_5$. In the limit $L_s\to\infty$ the resulting Ginsparg-Wilson relations recover the expected U($2N_f$) global symmetry up to O($a$) corrections. Finally it is shown that finite-$L_s$ corrections to bilinear condensates associated with dynamical mass generation are characterised by whether even powers of the symmetry-breaking mass are present; such terms are absent for antihermitian bilinears such as $i\bar\psi\gamma_3\psi$, markedly improving the approach to the large-$L_s$ limit.

Massive fermions without fermion bilinear condensates

We study a lattice field theory model containing two flavors of massless staggered fermions with an onsite four-fermion interaction. The model contains a $SU(4)$ symmetry which forbids non-zero fermion bilinear mass terms, due to which there is a massless fermion phase at weak couplings. However, even at strong couplings fermion bilinear condensates do not appear in our model, although fermions do become massive. While the existence of this exotic strongly coupled massive fermion phase was established long ago, the nature of the transition between the massless and the massive phase has remained unclear. Using Monte Carlo calculations in three space-time dimensions, we find evidence for a direct second order transition between the two phases suggesting that the exotic lattice phase may have a continuum limit at least in three dimensions. A similar exotic second order critical point was found recently in a bilayer system on a honeycomb lattice.

O(Nc) and USp(Nc) QCD from String Theory

We propose a holographic dual of large N_c QCD with the gauge group O(N_c) and USp(N_c) and N_f flavors of massless quarks. This is constructed by adding O6-planes to an intersecting D4-D8 system in type IIA superstring theory. The holographic dual description is formulated in the Witten's D4-brane background with D8-branes and O6-planes embedded in it as probes. The D4-brane background gives rise to a smooth interpolation of D8-anti-D8 pairs and an O6-anti-O6 pair. We show that the resultant brane configuration explains geometrically the flavor symmetry breaking patterns in O(N_c) and USp(N_c) QCD that are caused by quark bilinear condensates. We next discuss that baryons can be realized as D4-anti-D4 pairs wrapped on S^4 that intersect with the O6-plane. By analyzing the tachyons on it, we reproduce the stability conditions of the baryons that are expected from the gauge theory viewpoint. The stable baryon configurations are classified systematically by using K-theory. We also give a similar analysis for the flux tubes and again reproduce the results that are consistent with QCD.

A coset space compactification of the field theory limit of a heterotic string

The D=10-E8*E8 field theory limit of the heterotic string is compactified on the non-symmetric coset space Sp(4)/SU(2)*U(1) that is known in the limit of decoupled gravity to give three standard fermion generations, with SU(5)*SU(3)F*U(1)F as a gauge group in D=4. Allowing for non-vanishing fermion bilinear condensates, and assuming the conventional form of the supersymmetry transformations, the authors prove the presence of a family of N=1 supersymmetric background field configurations. This requires the non-compact space to be flat: (Minkowski)4, while the 3-form HMNP is non-vanishing and proportional to the torsion on the internal manifold. All equations of motion, including that of the dilaton, are satisfied.