Gauge-invariant Two-point Function Research Articles

Synchronous coordinates and gauge-invariant observables in cosmological spacetimes

We consider the relational approach to construct gauge-invariant observables in cosmological perturbation theory using synchronous coordinates. We construct dynamical synchronous coordinates as non-local scalar functionals of the metric perturbation in the fully non-linear theory in an arbitrary gauge. We show that the observables defined in this dynamical coordinate system are gauge-independent, and that the full perturbed metric has the expected form in these coordinates. Our construction generalises the familiar synchronous gauge in linearised gravity, widely used in cosmological perturbation theory, to the non-linear theory. We also work out the expressions for the gauge-invariant Einstein equation, sourced either by an ideal fluid or a scalar field up to second order in perturbation theory, and give explicit expressions for the Hubble rate—as measured by synchronous observers or by observers co-moving with the matter field—up to that order. Finally, we consider quantised linear perturbations around Minkowski and de Sitter backgrounds, and compute the two-point function of the gauge-invariant metric perturbation in synchronous coordinates, starting with two-point function in a general linear covariant gauge. Although the gauge-fixed two-point function contains gauge modes, we show that the resulting gauge-invariant two-point function only contains the physical tensor modes and it is thus positive, i. e. it has a spectral representation.

Gradient-flowed order parameter for spontaneous gauge symmetry breaking

The gauge-invariant two-point function of the Higgs field at the same spacetime point can make a natural gauge-invariant order parameter for spontaneous gauge symmetry breaking. However, this composite operator is ultraviolet divergent and is not well defined. We propose using a gradient flow to cure the divergence from putting the fields at the same spacetime point. As a first step, we compute it for the Abelian Higgs model with a positive mass squared at the one-loop order in the continuum theory using the saddle-point method to estimate the finite part. The order parameter consistently goes to zero in the infrared limit of the infinite flow time.

Anomalous dimension of Dirac's gauge-invariant nonlocal order parameter in Ginzburg–Landau field theory

In a Ginzburg–Landau theory with n fields, the anomalous dimension of the gauge-invariant nonlocal order parameter defined by the long-distance limit of Dirac's gauge-invariant two-point function is calculated. The result is exact for all n to first order in ε≡4−d, and for all d∈(2,4) to first order in 1/n, and coincides with the previously calculated gauge-dependent exponent in the Landau gauge.

Sphaleron-like processes in a realistic heat bath

We measure the diffusion rate of Chern-Simons number in the (1+1)-dimensional Abelian Higgs model interacting with a realistic heat bath for temperatures between 1/13 and 2/3 times the sphaleron energy. It is found that the measured rate is close to that predicted by the sphaleron approximation at the lower end of the temperature range considered but falls at least an order of magnitude short of the sphaleron estimate at the upper end of that range. We show numerically that the sphaleron approximation breaks down as soon as the gauge-invariant two-point function yields correlation length close to the sphaleron size.

Sphaleron transitions in a realistic heat bath

We measure the diffusion rate of Chern-Simons number in the (1 + 1)-dimensional Abelian Higgs model interacting with a realistic heat bath for temperatures between 1 13 and 1 3 times the sphaleron energy. It is found that the measured rate is close to that predicted by a one-loop calculation at the lower end of the temperature range considered but falls at least an order of magnitude short of the one-loop estimate at the upper end of that range. We show numerically that the sphaleron approximation breaks down as soon as the gauge-invariant two-point function yields a correlation length close to the sphaleron size.

An order parameter distinguishing between different phases of lattice gauge theories with matter fields

We propose an order parameter, a “gauge-invariant two-point function” whose behaviour distinguishes between different phases of gauge theories with matter fields, in particular Higgs theories. We argue that, moreover, the existence or non-existence of “coloured” physical states is reflected in the behaviour of our parameter. We analyze several lattice gauge theories with phase transitions and verify that the behaviour of the “gauge-invariant two-point function” depends indeed on the different phases of the theory.