Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

Abstract

In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom

“…However, similar to local theories, a phase of broken global symmetry, i.e. a condensate phase, can only be generated when the domain of the group field is non-compact (or its volume is sent to infinity in a thermodynamic limit), since otherwise the symmetry will be restored upon flow towards the IR [34,35]. This had already been anticipated from corresponding mean-field analyses [32] and happens due to large fluctuations essentially generated by r-fold zero-modes in the spectrum on compact domain, see also [42].…”

“…where ζ i is the i'th root of unity. Note that the presence of the volume factor in front of Φ 0 is completely natural since the relevant argument of the interaction potential in the full renormalization group flow is [34,35]…”

“…Still, both are of the same form and differ only in that here the couplings λ γ are fixed, while there they are dynamically depending on the renormalization group scale k. In particular, the example eq. ( 2.20) has already been calculated in [34,35] as it is the particular quartic case of a so-called cyclic-melonic interaction (a closed chain of open melons). The structure is much more general though.…”

“…To have a diagonal form, F [Φ 0 ] would need to be proportional to c δ(g c /h c ). This is already known from the FRG analysis [34,35]. The standard way to diagonalize K is to transform to momentum (representation) space.…”

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

“…On the other hand, one main application of theories with tensor fields of higher rank r > 2 are models of quantum gravity as their perturbative series is a sum over r-dimensional combinatorial (pseudo) manifolds [24] and one can build models which have gravitational amplitudes on such discrete manifolds as lattice gauge theories [13,20,37], known as spin foam models [43]. Then, the main challenge is to recover continuous D = 4 dimensional spacetime in some critical regime of the theory [38,46] in the sense of continuous random geometries [1,33,36]; to this end, nonperturbative results like [44] are necessary. However, also perturbation theory might be the right starting point towards this goal, either by identification of appropriate linear combinations of amplitudes in the perturbative series related to non-perturbative structures as found for example via topological recursion [10,11] or exploiting the Hopf-algebraic structure of perturbative renormalization leading to Kreimer's combinatorial Dyson-Schwinger equations [5,12,31,32].…”