Positivity in the Renormalization of Effective Field Theory

Technical Summary: Positivity in the Renormalization of Effective Field Theory

Problem Statement
Renormalization group (RG) flow governs the evolution of physical parameters with energy scale, yet determining the sign of this flow for general parameters in effective field theories (EFTs) typically requires explicit, complex calculations. While specific results exist in conformal field theory (e.g., the $a$-theorem), a general principle constraining the direction of RG flow in broader EFTs has been elusive. This is particularly relevant for the Standard Model Effective Field Theory (SMEFT) and chiral perturbation theory ($\chi$PT), where tree-level positivity bounds on couplings (derived from UV unitarity, analyticity, and Lorentz invariance) are susceptible to loop corrections. A critical open question is whether these loop effects preserve or violate the tree-level positivity bounds, and if a general theorem can predict the sign of the one-loop running for specific operator insertions without assuming a specific UV completion.

Methodology
The authors employ an on-shell formalism to derive the one-loop RG equations, leveraging the connection between scattering amplitudes and fundamental infrared (IR) principles.

1. On-Shell RG Extraction: The RG flow is extracted from the scale independence of the full amplitude, $d A_{\text{full}}/d \ln \mu = 0$. At one-loop order, the running of tree-level couplings is determined by the discontinuities (branch cuts) of the one-loop amplitude across the $s$, $t$, and $u$ channels.
2. Forward Limit (FL) Analysis: The analysis focuses on the forward limit where incoming and outgoing momenta are identical ($t \to 0$). In this limit, the tree-level amplitude for a dimension-$4n$ operator behaves as $A^{(0)} = \sum c_{2m+4} s^m$.
3. Unitarity and Analyticity: The discontinuity of the amplitude is recast using the optical theorem into an integral over the phase space of intermediate states, involving products of tree-level amplitudes.
4. Channel Separation: The authors analyze the contributions from the $s$, $u$, and $t$ channels separately:
   • $s$ and $u$ channels: In the forward limit, these channels involve identical initial and final states. The integrand is positive definite due to unitarity. However, the sign of the RG flow for a specific coupling depends on the interference between these channels and the power of $s$ in the expansion.
   • $t$ channel: The authors prove that for higher-dimensional operators (dimension $>4$), the $t$-channel discontinuity vanishes in the forward limit due to Lorentz invariance and the kinematic constraints of the phase space integration, provided the operators are local and the theory lacks cubic interactions (or specific cancellations occur).

Key Contributions and Results
The paper establishes a theorem dubbed "RG Positivity," which constrains the sign of the one-loop running of couplings in the forward limit.

1. The RG Positivity Theorem: For an EFT coupling $c_{4n}$ associated with a dimension-$4n$ operator, the one-loop running induced by the insertion of two operators of identical, even mass dimension ($2n+2$) satisfies:
   $$\left. \frac{d c_{4n}}{d \ln \mu} \right|_{(\text{dim-}(2n+2))^2} \leq 0$$
   This implies that the difference between the IR and UV values of the coupling is non-positive: $(c_{4n, \text{IR}} - c_{4n, \text{UV}}) \leq 0$.

   • Conditions: The theorem holds for arbitrary spins and mass dimensions, provided the theory contains massless particles without cubic interactions (or where such interactions do not spoil the forward limit analysis). It relies solely on IR principles (unitarity, analyticity, Lorentz invariance) and is agnostic to the UV completion.
   • Mechanism: The sign is fixed because the $s$ and $u$ channel contributions constructively interfere with a definite sign when the inserted operators have the same even dimension, while the $t$-channel contribution vanishes for $n \geq 2$.

2. Application to SMEFT and $\chi$PT:

   • SMEFT: The theorem applies to the running of dimension-8 operators ($c_8$) induced by the double insertion of dimension-6 operators ($c_6$). Specifically, for the combination $c^{(1)}_{H^4D^4} + c^{(2)}_{H^4D^4}$, the RG flow is negative definite. This explains and generalizes previous observations in the literature regarding the sign of these specific anomalous dimensions.
   • $\chi$PT: The theorem correctly predicts the negative sign of the RG flow for the $O(p^4)$ couplings ($c_1, c_2$) induced by the $O(p^2)$ Lagrangian, consistent with explicit calculations.
   • Dimension-6 Operators: The theorem generally does not apply to the running of dimension-6 operators from dimension-4 insertions due to potential cancellations between $s$ and $u$ channels. However, it does yield definite signs for specific mixings where one channel is absent (e.g., Weinberg operator mixing into $H^4D^2$ or lepton operators).

3. Non-Renormalization Theorems: A corollary of the vanishing $t$-channel discontinuity is a new class of non-renormalization theorems. Higher-dimensional operators that survive the forward limit do not receive RG contributions from operators that vanish in the forward limit (if only the $t$-channel connects them).

4. Violation of Tree-Level Positivity: The paper clarifies that while RG positivity constrains specific sectors (strongly coupled EFTs where dimension-6 insertions dominate), it does not guarantee the preservation of tree-level positivity bounds in all scenarios. In weakly coupled UV completions, the running of dimension-8 couplings can be dominated by interference with Standard Model interactions or dimension-4 terms, potentially driving the coupling to negative values in the IR, thus violating tree-level positivity bounds. This is consistent with dispersion relations when higher-order terms and mass gaps are considered.

Significance
The paper provides a fundamental, model-independent constraint on the direction of RG flow in EFTs, analogous to the $a$-theorem in CFT. By deriving these constraints solely from IR principles, the authors offer a robust tool for:

• Phenomenology: Providing theoretical priors for experimental analyses in SMEFT and $\chi$PT, specifically identifying which RG flows preserve positivity bounds and which do not.
• Theoretical Understanding: Unifying the understanding of non-renormalization theorems and positivity bounds through the lens of on-shell amplitudes and unitarity cuts.
• UV Independence: Demonstrating that the sign of specific RG contributions is fixed regardless of the UV completion, distinguishing between "strongly coupled" EFT regimes (where positivity is preserved) and "weakly coupled" regimes (where violations may occur).

The authors emphasize that their results do not exclude the possibility of positivity violations in the IR but rather delineate the conditions under which such violations are expected, offering a refined framework for interpreting experimental constraints on EFT couplings.