Renormalisation Group Flow of Non-Local Effective Average Actions
I wrote the thesis entitled "Renormalization Group Flow of Non-Local Effective Average Actions" for my Master degree in Theoretical Physics.
In a Quantum Field Theory, all the physically relevant information is contained in the correlation functions, which can be conveniently obtained from the generating functional using the functional integration.
The functional integration can be interpreted as a d-dimensional generalization of the path-integral, and from the beginning scientist approached to this task with a perturbative expansion of the field fluctuations around the fundamental configuration. In such a way, we became able to deal with many little-interacting models, and we developed some advanced techniques in order to solve the technical and conceptual issue of divergences arising in the calculations, through an intuitive redefinition of the physical quantities. But as many theories revealed to be easily manageable using renormalization techniques, others manifested pathological behaviors making them impracticable, such as it is the case of a QFT concerning General Relativity.
Luckily, the perturbative expansion is not the only way of integrating out the field fluctuations, and an alternative approach to the functional integration, first developed by K.G. Wilson and reformulated by J. Polchinsky and by C. Wetterich in a different form, allows to achieve non-perturbative results and is particularly useful to introduce a new definition of renormalizability called Asymptotic Safety, that includes a wider range of theories. Such a new approach is based on the concept of the Effective Average Action, that can be interpreted as the effective action we would achieve integrating out all the field fluctuations averaged over a finite volume, and allows to perform the functional integration momentum shell by momentum shell through the Exact Renormalization Group Equation, a non-perturbative integro-differential equation which describe the RG flow of the EAA.
We can interpret the EAA also as the effective action describing our system at a particular energy scale k, providing us an alternative tool to compute the quantum effective action as the result of a flow in an infinite-dimensional theory space.
But we are evidently unable to work with infinite many base elements, and if we are interested in non-numerical results we are forced to project this infinite dimensional space on an arbitrary finite dimensional subspace. Such a process is called truncation, and despite it is a non-perturbative approximation and it allows to work with a wider range of theories than the perturbative expansion do, it has the disadvantage of being arbitrary, and there is in principle no way to know a priori if a truncation is accurate. When we set a truncation, we must try to keep all the physically relevant degrees of freedom neglecting only the basis elements which do not considerably modify the flow of the investigated ones, and the only way we can check the accuracy of a truncation is to compare its outcomes with the ones achieved using a wider truncation or other asserted techniques.
Since now, people mainly investigated local truncations, by expanding the EAA in powers of the fields and focusing on the small momenta behavior of the result.
This is the case of the Local Potential Approximation and of the Derivative Expansion. Such a class of truncations reveals to be accurate because, as a consequence of the particular structure of the ERGE, only the field fluctuations with momentum smaller than the reference energy scale contributes to the RG flow of the EAA.
But we could also be interested in the non-local structure of the effective average n-point vertices, as they are necessary for the calculation of the cross sections and of the scattering amplitudes, and the local truncations reveal to be unsuitable to achieve such a results for arbitrary momenta. The most useful approach for a non-perturbative evaluation of the momentum dependent n-point vertices has been developed by J.-P. Blaizot, R. Mendez-Galain and N. Wschebor, but it requests the numerical evaluation of an integro-differential equation and exploit many approximations.
Therefore, it would be interesting to find an appropriate non-local truncation able to describe the momentum-dependent structure of the correlation functions in the theory under investigation. Moreover, there are many situations in which the non-local structure of the EAA is fundamental in order to achieve meaningful results, such as it is the case of low energy QCD correlators close to the confinement phase or the computation of the contribution to the vacuum energy flow for interacting massless fields. In fact, in both cases the non-local structure of the 2-point function plays a fundamental role.
Finally, an adequate non-local truncation would allow to obtain more accurate results than the LPA without introducing an high order derivative expansion, which reveals to be quite complicate.
In order to achieve all these results, we need to find a non-polynomial analytical function dependent from few k-dependent parameters able to accurately fit the non-local structure of the n-point vertices for every energy scale k, in order to project the exact EAA flow on a subspace as close as possible to the real trajectory. We tried to find such a function and we developed two non-local truncations for a d-dimensional real scalar field theory, comparing the outcomes with the results achieved using other asserted techniques. Finally, we numerically implemented the BMW technique in order to test the 2 and 4 point vertices structure obtained using the non-local truncation we developed.
In Chapter 1, the main concepts about the non-perturbative approach to the RG are introduced. The Average Effective Action is defined and it's flow equation (ERGE) is derived. Finally, the concept of Asymptotic Safety is introduced. All the results are compared with their analogues in perturbation theory, in order to check their validity and to better understand the physical meaning of this approach.
In Chapter 2 the LPA truncation is introduced in a d-dimensional interacting real scalar field framework and the RG flow equations for the couplings are derived. The anomalous dimension of the scalar field is investigated together with the flow of the VEV and, finally, for the 1-dimensional case corresponding to the Quantum Mechanical anharmonic oscillator, the vacuum energy outcomes are compared with the corresponding results in perturbation theory.
In Chapter 3 most of our original work is collected. A non-local ansatz for the 2-point vertex is introduced in a real scalar field framework, supported by the 1-loop results given in appendix I, and the RG flow equations for the couplings are derived. An approximation scheme is developed in order to define a consistent non-local truncation also for a Z-2 invariant real scalar field theory and, under these approximations, the differential equations for the couplings are derived under the new truncation. Finally, the vacuum energy results for the QM anharmonic oscillator are compared with the LPA results and with the most accurate numerical ones.
Finally, in Chapter 4, the BMW integro-differential equation describing the RG flow of the momentum dependent 2-point vertex is derived through some approximations. The case of a real scalar field theory is numerically solved by us for different bare actions both in 1 and 4 dimensions and the results are compared to the ones achieved using the non-local truncations, in order to test the validity of the ansatz we introduced.
Some analytical results concerning the BMW technique are given in appendix II.
In the Conclusions the main achieved results are summarized, some improvements and further tests are suggested and some suggestive future applications for this technique are pointed.