Large charge sector in the theory of a complex scalar field with quartic self-interaction

          @misc{ pirsa_PIRSA:24090200,
            doi = {10.48660/24090200},
            url = {https://pirsa.org/24090200},
            author = {Sanjurjo, Sergio},
            keywords = {},
            language = {en},
            title = {Large charge sector in the theory of a complex scalar field with quartic self-interaction},
            publisher = {Perimeter Institute},
            year = {2024},
            month = {sep},
            note = {PIRSA:24090200 see, \url{https://pirsa.org}}
          }
          

Abstract

The theory $S = \int\text{d}^{4-\epsilon}x\left(\frac{1}{2}|\partial\phi|^2 - \frac{m^2}{2}|\phi|^2-\frac{g}{16}|\phi|^4\right)$ exhibits a global $U(1)$ symmetry, and the operators $\phi^n$ ($\bar\phi^n$) have charge $n$ ($-n$) with respect to this symmetry. By rescaling the fields and the coupling constant, it is possible to work in a double limit $n\to\infty$, $g\to 0$ with $\lambda = gn$ kept constant. In this way, it is possible to compute 2-point functions of the form $\langle \phi^n(x) \bar\phi^n(0) \rangle$ in the large $n$ limit, either diagrammatically by a resummation of the leading contribution at all orders in $g$, or using semiclassical methods through the saddle point approximation. This second approach is particularly powerful because it can also be applied to the theory on a curved background. This allows obtaining the form of the 2-point function for an arbitrary metric, and by functionally differentiating with respect to it, it is also possible to obtain, in the flat theory, the 3-point function $\langle T^{ij}(z) \phi^n(x) \bar \phi^n(0) \rangle$ in which an energy-momentum tensor has been inserted. This allows for a non-trivial check of the conformal symmetry of this sector of the theory by verifying the Ward identities that this 3-point function should satisfy.