# One-loop algebras and fixed flow trajectories in adjoint multi-scalar gauge theory

**Authors:** Nadia Flodgren, Bo Sundborg  
**arXiv:** [2303.13884](https://arxiv.org/abs/2303.13884)  
**Date:** 2023-03-24  
**Summary page:** https://www.summarizepaper.com/en/paper/2303.13884v1/

## Key points

- Investigate one-loop renormalization of 4-dimensional $SU(N)$ Yang-Mills theory with $M$ adjoint representation scalar multiplets related by an $O(M)$ symmetry
- One-loop beta function of gauge coupling $g^2$ vanishes when $M=22, related to D3 branes in critical bosonic string theory in a spacetime dimension of $D=26$
- Scalar self-couplings $\lambda$ disrupt one-loop conformal invariance in large $N$ limit
- Real fixed flows (fixed points of $\lambda/g^2$) only for values of $M\ge 406, incompatible with one-loop fixed points of gauge coupling and scalar couplings
- Develop and verify algebraic approach to one-loop renormalization group
- Non-associative algebra of marginal couplings characterizes the approach
- Most resulting RG flows suffer from strong coupling in ultraviolet and infrared regimes, weakly coupled solutions exist only for finely tuned solutions with $M\ge 406$
- Limitations to validity of one-loop RG flow analysis revealed
- Connecting multiscalar adjoint gauge theory to bosonic D3-brane model not straightforward
- Initial connection between critical dimension 26 for bosonic string and vanishing one loop gauge coupling beta function in 4 dimensions for M = 22 does not withstand corrections from interactions.

## Summary

In this study, the authors investigate the one-loop renormalization of 4-dimensional $SU(N)$ Yang-Mills theory with $M$ adjoint representation scalar multiplets that are related by an $O(M)$ symmetry. The case of general $M$ is of interest in field theory and has implications for string theory; specifically, the one-loop beta function of the gauge coupling $g^2$ vanishes when $M=22$, which is related to D3 branes in critical bosonic string theory in a spacetime dimension of $D=22+4=26$. This raises the possibility that an RG fixed point could provide a definition for a purely bosonic AdS/CFT correspondence. However, they demonstrate that scalar self-couplings $\lambda$ disrupt one-loop conformal invariance in the large $N$ limit. They find that there are real fixed flows (fixed points of $\lambda/g^2$) only for values of $M\ge 406$, which renders the one-loop fixed points of the gauge coupling and scalar couplings incompatible. To analyze these findings, the authors develop and verify an algebraic approach to the one-loop renormalization group. They discover that this approach is characterized by a non-associative algebra of marginal couplings. In the large $N$ limit, they observe that most resulting RG flows suffer from strong coupling both in the ultraviolet and infrared regimes. Only for finely tuned solutions with $M\ge 406$, weakly coupled solutions exist in the infrared. In conclusion, this study reveals limitations to the validity of one-loop RG flow analysis. The authors find that only very exceptional trajectories can be accurately described by the one-loop approximation asymptotically in either ultraviolet or infrared regimes. Furthermore, their investigation suggests that connecting multiscalar adjoint gauge theory to a bosonic D3-brane model - which could be useful in a bosonic AdS/CFT correspondence - is not straightforward. The initially promising connection between critical dimension 26 for bosonic string and vanishing one loop gauge coupling beta function in 4 dimensions for M = 22 does not withstand corrections from interactions.