Does QFT make sense in non-integer dimensions?

Technical Summary: "Does QFT make sense in non-integer dimensions?"

Problem Statement
The paper addresses the long-standing problem of analytically continuing Quantum Field Theories (QFTs) from integer spacetime dimensions $d \in \mathbb{N}$ to fractional or complex dimensions $d \in \mathbb{C}$. While dimensional regularization (dim-reg) successfully treats vector-like theories in an infinitesimal neighborhood of an integer $d$, the authors argue that a rigorous, non-perturbative continuation to a finite open set in the complex plane (containing multiple distinct integers) is obstructed. The central issue is that common theories, such as QCD and QED, exhibit "low-rank exceptions" where observables undergo qualitative, discrete changes at specific integer dimensions. These discontinuities prevent the existence of a single, uniform analytic function describing the theory across all $d$.

Methodology
The authors approach the problem by disentangling kinematics (representation theory of the Lorentz group) from dynamics.

1. Kinematic Analysis: They utilize Deligne categories $\overline{\text{Rep}}(O(d))$ to define stable representations at fractional $d$. They demonstrate that while "true tensors" (stable sequences of representations) admit a unique analytic continuation, "pseudo-tensors" and representations involving spinors (pinors) do not. Specifically, the trace of gamma matrices $\text{tr}[\gamma_{\mu_1 \dots \mu_n}]$ vanishes for generic large $d$ but becomes non-zero at specific low-rank exceptions (e.g., $\text{tr}[\gamma_{\mu\nu\rho}] \neq 0$ only at $d=3$).
2. Perturbative Computation: To illustrate these obstructions, the authors perform explicit calculations in the large-$N_f$ limit of vector-like massless QCD (and QED). They compute scaling dimensions ($\Delta$) of operators such as the fermion $\psi$ and the scalar bilinear $\bar{\psi}\psi$ to leading order in $1/N_f$.
3. Branch Structure Analysis: They analyze the correlation functions and scaling dimensions by comparing the "honest" results computed directly at integer $d$ (using rigorous gamma matrix traces) against "analytic" results derived by assuming the stable limit (setting odd traces to zero).

Key Contributions and Results

• Obstruction to Uniform Analyticity: The paper establishes that for generic operators, particularly parity-odd scalars like $\bar{\psi}\psi$, the scaling dimension $\Delta(d)$ is not a single analytic function of $d$. Instead, the theory exhibits "low-rank exceptions" where operators of different spin become isomorphic (e.g., a rank-3 antisymmetric tensor becomes a scalar at $d=3$). This leads to level crossing and operator mixing that is discontinuous as a function of $d$.
• The Branched Structure: The authors propose that the analytic continuation of QFT data does not live on the complex plane $\mathbb{C}$, but rather on an infinite cover of it.
  • For the operator $\bar{\psi}\psi$, there exists an infinite family of analytic branches $\hat{\Delta}(\mathcal{O}_i)$, where $\mathcal{O}_i$ are operators that become isomorphic to $\bar{\psi}\psi$ at specific integer dimensions (e.g., $\bar{\psi}\gamma_{\mu\nu\rho}\psi$ at $d=3$, $\bar{\psi}\gamma_{\mu\nu\rho\sigma\tau}\psi$ at $d=5$).
  • Each branch $\hat{\Delta}$ is analytic in $d$ and matches the correct physical scaling dimension only in the neighborhood of its specific integer $d$. No single branch is correct for all integers.
  • This structure arises because the "stable limit" (ignoring low-rank exceptions) yields an analytic function that fails at small $d$, while the "honest" calculation yields a discontinuous function.
• Explicit Calculations in QCD/QED:
  • Wilson Lines: The expectation value of Wilson loops is shown to admit a well-defined analytic continuation, as they depend on stable tensor structures.
  • Fermion Scaling Dimension: The scaling dimension of $\psi$ is found to be analytic in $d$ (at leading order), as it does not suffer from the same low-rank mixing issues as composite scalars.
  • Scalar Bilinear $\bar{\psi}\psi$: The calculation of $\Delta(\bar{\psi}\psi)$ reveals a term proportional to $\text{tr}[\gamma_{\mu\nu\rho}]\text{tr}[\gamma^{\mu\nu\rho}]$. This term vanishes for all $d \neq 3$ in the stable limit but is non-zero at $d=3$. Consequently, the naive analytic continuation gives the wrong answer at $d=3$. The correct value is recovered only by switching to the branch corresponding to the operator $\bar{\psi}\gamma_{\mu\nu\rho}\psi$, which is the isomorphic partner at $d=3$.
• Gross–Neveu–Yukawa Model: The authors extend these findings to the QED-Gross–Neveu–Yukawa model, showing that the addition of scalar interactions does not remove the branch structure; the low-rank obstructions persist.

Significance and Claims
The paper claims that the intuition of QFTs as a smooth family connected by continuous parameters $d$ and $N$ is fundamentally flawed due to these discrete, non-perturbative effects in $1/d$.

• On Dimensional Regularization: The authors suggest that the standard 't Hooft-Veltman prescription (which effectively sets odd traces to zero) is only valid in an infinitesimal neighborhood of a specific integer $d$ (e.g., $d=4$). It fails to provide a global analytic continuation that correctly reproduces results at other integers (e.g., $d=3$).
• On Uniqueness: They argue that a unique analytic continuation is only possible if one imposes "nice behavior" at $d \to \infty$. However, since the large-$N_f$ expansion breaks down at sufficiently large $d$ for fixed $N_f$, the uniqueness of the continuation cannot be established perturbatively.
• On Chiral Theories: The paper notes that chiral theories (involving $\gamma_*$) face even more severe obstructions, with discontinuities expected at even integers due to traces involving $\gamma_*$.
• Conclusion: The paper concludes that while a meaningful continuation exists, it requires navigating a branched structure where one must choose the correct analytic branch adapted to the specific integer dimension of interest. There is no single formula that is both analytic in $d$ and correct for all $d \in \mathbb{N}$. This challenges the feasibility of using analytic continuation as a tool to interpolate between distinct integer-dimensional dualities (e.g., between $d=3$ and $d=4$ Seiberg-like dualities) without encountering these structural discontinuities.