Holographic beta functions for the generalized sine-Gordon theory

The sine-Gordon theory is generalized to include many scalar fields and several cosine terms. This is similar to the world sheet description of a string propagating in a tachyon background. This model is studied as a (boundary) 2D Euclidean field theory and also using an ${\mathrm{AdS}}_3$ holographic bulk dual. The beta functions for the cosine vertex of this modified theory are first computed in the boundary using techniques based on the exact RG. The beta functions are also computed holographically using position space and momentum space techniques. The results are in agreement with each other and with earlier computations. The cosine perturbation is of the form $\cos bX$. Due to wave function renormalization the parameter $b$, and thus the dimension of the cosine, get renormalized. The beta function for this parameter is thus directly related to the anomalous dimension of the $X$ field. We compute this beta function in position space. They match with the earlier results in [P. Oak and B. Sathiapalan, Exact renormalization group and sine-Gordon theory, J. High Energy Phys. 07 (2017) 103; P. Oak and B. SathiapalanJ. High Energy Phys. 09 (2017) 77.].

Figure 1

The leading order Witten diagram.

Figure 2

Witten diagram for the subleading contribution.

Figure 3

Witten diagram for the graviton-scalar-scalar vertex for the $\beta$ function of the field strength renormalization.