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Gradient Flows and the Curvature of Theory Space
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Gradient Flows and the Curvature of Theory Space
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The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in $d=4-\varepsilon$ dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal operators on the sphere. An explicit form for the associated Ricci scalar in $d=4-\varepsilon$ is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity $\widetilde{F}$ that was previously proposed as a weakly monotonic function interpolating between the $a$-theorem in four dimensions and the $F$-theorem in three dimensions. This implies that the $\widetilde{F}$-theorem can be extended perturbatively to a theorem about gradient flow in $d=4-\varepsilon$.
Forward citations
Cited by 8 Pith papers
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Local CFTs extremise $F$
Local CFTs lie at the extrema of the sphere free energy tilde F for nonlocal CFT lines, and maximize it when unitary.
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Gradient RG Flow in Scalar-Fermion QFTs
RG flow in scalar-fermion theories is gradient through four loops only when the 'beta shift' is included, via over a thousand scheme-independent constraints that hold wherever current data allow a check.
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Matching $A$ with $F$ in long-range QFTs
RG flow in long-range φ⁴ theories obeys gradient structure ∂_I A = G_IJ β^J up to three loops, with A matching F-tilde and G matching C_IJ at leading nontrivial order.
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Matching $A$ with $F$ in long-range QFTs
In long-range non-unitary φ^4 models the RG flow obeys a gradient structure up to three loops, with A matching the sphere free energy F̃ at leading order.
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$\phi^6$ at $6$ (and some $8$) loops in $3d$
Six-loop beta-function graphs for general ϕ⁶ theory in 3d are recalculated (differing from Hager, agreeing with recent work), with large-N eight-loop results, O(ε³) exponents, and gradient-flow linear relations.
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Matching $A$ with $F$ in long-range QFTs
Long-range φ⁴ theories have RG beta functions that satisfy a gradient flow with A matching the sphere free energy F̃ at leading nontrivial order.
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$\phi^6$ at $6$ (and some $8$) loops in $3d$
Recalculation of individual six-loop graph contributions to the β-function in 3d φ⁶ theory with arbitrary potential, plus large-N eight-loop diagrams and O(ε³) critical exponents at the O(N) fixed point.
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$\phi^6$ at $6$ (and some $8$) loops in $3d$
Recalculation of individual six-loop graph contributions to the beta function in 3d phi^6 theory with arbitrary potential, plus large-N eight-loop terms and O(epsilon^3) critical exponents at the O(N) fixed point.
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