Abstract
We develop a new approach to RG flows and show that one-loop flows in multiscalar theories can be described by commutative but non-associative algebras. As an example related to D-brane field theories and tensor models, we study the algebra of a theory with M SU(N) adjoint scalars and its large N limits. The algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are used to characterise the RG flows. We classify and describe all large N limits of algebras of multiadjoint scalar models: the standard ‘t Hooft matrix theory limit, a ‘multi-matrix’ limit, each with one free parameter, and an intermediate case with extra symmetry and no free parameter of the algebra, but an emergent free parameter from a line of one-loop fixed points. The algebra identifies these limits without diagrammatic or combinatorial analysis.
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References
P. Romatschke, A solvable quantum field theory with asymptotic freedom in (3+1) dimensions, Int. J. Mod. Phys. A 38 (2023) 2350157 [arXiv:2211.15683] [INSPIRE].
P. Romatschke, What if ϕ4 theory in 4 dimensions is non-trivial in the continuum?, Phys. Lett. B 847 (2023) 138270 [arXiv:2305.05678] [INSPIRE].
D. Benedetti, R. Gurau and S. Harribey, Line of fixed points in a bosonic tensor model, JHEP 06 (2019) 053 [arXiv:1903.03578] [INSPIRE].
D. Benedetti, R. Gurau, S. Harribey and K. Suzuki, Hints of unitarity at large N in the O(N)3 tensor field theory, JHEP 02 (2020) 072 [Erratum ibid. 08 (2020) 167] [arXiv:1909.07767] [INSPIRE].
D. Benedetti, R. Gurau and S. Harribey, Trifundamental quartic model, Phys. Rev. D 103 (2021) 046018 [arXiv:2011.11276] [INSPIRE].
J. Berges, R. Gurau and T. Preis, Asymptotic freedom in a strongly interacting scalar quantum field theory in four Euclidean dimensions, Phys. Rev. D 108 (2023) 016019 [arXiv:2301.09514] [INSPIRE].
F. Ferrari, The large D limit of planar diagrams, Ann. Inst. H. Poincare D Comb. Phys. Interact. 6 (2019) 427 [arXiv:1701.01171] [INSPIRE].
L. Markus, Quadratic differential equations and non-associative algebras, in Contributions to the theory of nonlinear oscillations, Vol. V, Princeton University Press, Princeton, N.J. (1960) pp. 185–213.
Y. Krasnov and V.G. Tkachev, Variety of idempotents in nonassociative algebras, in Topics in Clifford analysis — special volume in honor of Wolfgang Sprößig, Trends Math. (2019) 405, [arXiv:1801.00617].
Y. Krasnov, Non-Associative Structures and Their Applications in Differential Equations, Mathematics 11 (2023) 1790.
M.M. Bosschaert, C.B. Jepsen and F.K. Popov, Chaotic RG flow in tensor models, Phys. Rev. D 105 (2022) 065021 [arXiv:2112.09088] [INSPIRE].
W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].
T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
D.F. Litim and F. Sannino, Asymptotic safety guaranteed, JHEP 12 (2014) 178 [arXiv:1406.2337] [INSPIRE].
N. Flodgren and B. Sundborg, One-loop algebras and fixed flow trajectories in adjoint multi-scalar gauge theory, JHEP 04 (2023) 129 [arXiv:2303.13884] [INSPIRE].
L. Michel, Renormalization-group fixed points of general n-vector models, Phys. Rev. B 29 (1984) 2777 [INSPIRE].
C. Jepsen and Y. Oz, RG flows and fixed points of O(N)r models, JHEP 02 (2024) 035 [arXiv:2311.09039] [INSPIRE].
T. Azeyanagi et al., More on the New Large D Limit of Matrix Models, Annals Phys. 393 (2018) 308 [arXiv:1710.07263] [INSPIRE].
S. Carrozza, F. Ferrari, A. Tanasa and G. Valette, On the large D expansion of Hermitian multi-matrix models, J. Math. Phys. 61 (2020) 073501 [arXiv:2003.04152] [INSPIRE].
I.R. Klebanov and G. Tarnopolsky, On Large N Limit of Symmetric Traceless Tensor Models, JHEP 10 (2017) 037 [arXiv:1706.00839] [INSPIRE].
M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings, Nucl. Phys. B 249 (1985) 70 [INSPIRE].
M.-X. Luo, H.-W. Wang and Y. Xiao, Two loop renormalization group equations in general gauge field theories, Phys. Rev. D 67 (2003) 065019 [hep-ph/0211440] [INSPIRE].
Y. Krasnov and V.G. Tkachev, Idempotent Geometry in Generic Algebras, Advances in Applied Clifford Algebras 28 (2018).
S. Kowalevski, Sur une propriété du système d’équations différentielles qui définit la rotation d’un corps solide autour d’un point fixe, Acta Math. 14 (1890) 81.
F. Ferrari, V. Rivasseau and G. Valette, A New Large N Expansion for General Matrix–Tensor Models, Commun. Math. Phys. 370 (2019) 403 [arXiv:1709.07366] [INSPIRE].
Acknowledgments
We thank Vladimir Tkachev for alerting us about his suspicions regarding an earlier version of equations (B.1)–(B.7) for the finite N algebra with consequences for the a = 2 algebra in table 1, and for sharing his knowledge about non-associative algebras. The work of B.S. is supported by the Swedish research council VR, contract DNR-2018-03803.
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Flodgren, N., Sundborg, B. Classifying large N limits of multiscalar theories by algebra. J. High Energ. Phys. 2024, 108 (2024). https://doi.org/10.1007/JHEP06(2024)108
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DOI: https://doi.org/10.1007/JHEP06(2024)108