Abstract
Holographic model of massive scalar field with its self-interaction \(\lambda \phi ^n\) in AdS space is able to give a logarithmic scale dependence to marginal multi-trace deformation couplings on its dual conformal field theory, where \(\lambda \) is the self-interaction coupling of the scalar field, \(\phi \), and n is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model to see scale dependence of marginal deformations on the dual conformal field theory by employing completely different method: holographic Wilsonian renormalization group. We solve Hamilton–Jacobi equation derived from the holographic model of massive scalar with \(\lambda \phi ^n\) interaction and obtain the solution of marginal multi-trace deformations up to the leading order in \(\lambda \). It turns out that the solution of marginal multi-trace deformation also presents logarithmic behavior in energy scale near UV region.
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Notes
We note that in [6], the \(K=-\frac{1}{(-2\nu )^n}\), where \(\nu =\frac{d}{2}-\Delta \). This is due to the different definitions of multi-trace couplings.
We note that in [11], the authors employ holographic Wilsonian renormalization group to study classically marginal deformations. They investigate marginally relevant or irrelevant cases in quantum level. We also note that in [17], the authors discuss holographic Wilsonian renormalization group for massive scalar theory with self-interaction, whose form is quite general, which is given by \(\sum\nolimits _{n=3}^\infty \lambda _n \phi ^n\). They discuss general features of the holographic Wilsonian renormalization group and its similarity with Callan–Symansik equation. However, they do not give detailed discussion on the reproduction of logarithmic behaviors of marginal deformations on the boundary conformal field theory.
For \( m \ge 2 \), the most general form of Hamilton–Jacobi equation is given by
$$\begin{aligned}& \delta ^{(d)}\left( \sum\limits ^{m}_{B=1} \overrightarrow {{p_{B}}}\right) \partial _\epsilon [ \sqrt{\gamma }D^{(m)}(p_1,\ldots ,p_{n'};\epsilon )] \nonumber \\& = -\int d^dp \left. \frac{(m+1)}{\sqrt{g}g^{rr}}J(\epsilon ,-p)\sqrt{\gamma }D^{(m+1)}(p_1,\ldots ,p_{m+1},p)\delta ^{(d)}\left( \sum\limits ^{m}_{B=1} \overrightarrow {{p_{B}}}\right) \right. \nonumber \\& - \frac{\gamma }{2\sqrt{g}g^{rr}}\sum\limits ^{m}_{m'=2}m'(m+2-m') \mathcal Per \left\{ D^{(m')}\left( p_1,\ldots ,p_{m'-1},-\sum\limits ^{m'-1}_{B=1}p_B;\epsilon \right) D^{(m+2-m')}\left( p_{m'},\ldots ,p_{m},\sum\limits ^{m'-1}_{A=1}p_A;\epsilon \right) \right\} \delta ^{(d)}\left( \sum\limits ^{m}_{B=1} \overrightarrow {{p_{B}}}\right) \nonumber \\& - \frac{1}{2}\sqrt{g}(-g^{ij}p_{1i}p_{2j}+m^2)\delta ^{(d)}( {p_1}+ {p_2})\delta _{m,2} + \left. \sqrt{g}\ \frac{\lambda }{n}\delta _{n,m}(2\pi )^{d(1-\frac{n}{2})}\delta ^{(d)}\left( \sum\limits ^m_{B=1} \overrightarrow {{p_{B}}}\right) . \right. \end{aligned}$$(24)
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Acknowledgements
J.H.O thank his W.J. and Y.J. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1C1B1010107).
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Kim, G., Oh, JH. On holographic Wilsonian renormalization group of massive scalar theory with its self-interactions in AdS. J. Korean Phys. Soc. 80, 30–36 (2022). https://doi.org/10.1007/s40042-021-00357-y
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DOI: https://doi.org/10.1007/s40042-021-00357-y