BRST in the exact renormalization group

Abstract

We show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.

The continuum limit of quantum gravity at first order in perturbation theory

“…For example for (non-Abelian) gauge theories the continuum solution can be chosen to respect the corresponding Σ = 0 identities at all scales, e.g. [7], in fact the gauge invariance can even be manifestly respected through e.g. lattice regularisation [8] or directly in the continuum (e.g.…”

“…[4] in terms of the Wilsonian effective action and a regularised Quantum Master Equation. Although this allows for an elegant analysis since the latter effectively leaves BRST invariance unmodified, in section 2 we switch to an equivalent [15,16] description in terms of the infrared regulated Legendre effective action [15,17,18] and the mST [6,7]. Although more cumbersome, this then gives us direct access to the one-particle irreducible amplitudes in the physical limit, and leads to useful simplifications at higher orders [1,19,20].…”

“…Although more cumbersome, this then gives us direct access to the one-particle irreducible amplitudes in the physical limit, and leads to useful simplifications at higher orders [1,19,20]. Furthermore it can still be solved in terms of the total free quantum BRST chargeŝ 0 [7] that naturally incorporates a regularised Batalin-Vilkovisky measure operator ∆ [21,22].…”

“…In sections 2 and 3 we review the development of this free BRST algebra and how computations can be couched in minimal gauge invariant basis [7]. Then in section 3.1, we choose the first order non-trivial quantum BRST cohomology representative on which to build the continuum limit to first order (in the new quantisation these two are not the same).…”

“…Naïvely one thinks of this cutoff as breaking the diffeomorphism invariance. However the Slavnov-Taylor identities get replaced by modified Slavnov-Taylor identities (mST), "Σ = 0", that reduce to the usual ones in the limit that we integrate out all the modes [6,7]. This limit is Λ → 0, which is the limit we need, in order to compute the desired physical observables.…”