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Does QFT make sense in non-integer dimensions?

T0 review · 3 major / 6 minor · reviewed 2026-07-07 · glm-5.2

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Pith's one-line read QFT in non-integer dimensions has branch cuts, not a single analytic surface

desk verdict Solid structural observation about branched analytic continuation in d, verified by explicit O(1/N_f) computation; the all-orders claim is plausible but unshown. read the letter →

arxiv 2607.05338 v1 pith:IDW3ASSO submitted 2026-07-06 hep-th

Diego Delmastro , Yan-Yan Li , Dikshant Rathore This is my paper · ORCID
classification hep-th
keywords dimensionsanalyticallybranchcommoncomplexcontinuingcutsdimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that analytically continuing quantum field theories to non-integer spacetime dimensions d is not a single-valued operation. The obstruction comes from low-rank isomorphisms in the representation theory of the rotation group: at special integer dimensions, representations that are distinct for generic d become identical (for example, in d=3 an antisymmetric 3-tensor is the same as a scalar via the Levi-Civita symbol). These accidental degeneracies force observables like scaling dimensions and OPE coefficients to live on different analytic branches near different integers. The authors demonstrate this explicitly in large-Nf QCD: the scaling dimension of the meson operator computed near d=4 (using the standard prescription of setting odd gamma-matrix traces to zero) gives the wrong answer at d=3, while the correct d=3 answer is recovered from a different branch associated with an operator that only becomes a scalar in three dimensions. No single analytic expression matches the honest computation at all integer d. The paper proposes that the correct global structure is an infinite cover of the complex d-plane, with one branch for each low-rank exception.

What carries the argument

Low-rank isomorphisms in the representation theory of O(d) (e.g. the d=3 identification of the rank-3 antisymmetric representation with the trivial representation via the Levi-Civita symbol); exceptional gamma-matrix traces tr[gamma^{mu_1...mu_n}] that vanish for large d but are non-zero at d=n; the large-Nf expansion of QCD as a concrete testing ground; the Deligne category gRep(O(d)) as the framework for stable (true-tensor) representations at fractional d, contrasted with unstable representations (pinors, pseudo-tensors) that lack a canonical continuation

What would settle it

A single analytic function of d that reproduces the honest scaling dimension of the meson operator at all integer dimensions simultaneously, without needing to switch branches. If such a function existed, the branched structure would be an artifact of the perturbative framework rather than an intrinsic feature of QFT at fractional d.

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Extended reading notes

Core claim

The central mechanism is that traces of an odd number of gamma matrices, such as tr[gamma_mu_nu_rho], vanish for all sufficiently large integer d but are non-zero at specific low dimensions (here d=3) due to exceptional isomorphisms like the equivalence between antisymmetric rank-3 tensors and scalars. When one computes scaling dimensions in large-Nf QCD, these exceptional traces appear in the honest result. The standard dimensional-regularization prescription sets them to zero, producing an expression that is analytic in d but wrong at the exceptional dimension. The correct value at d=3 is instead recovered from the analytic branch of a different operator (the rank-3 antisymmetric bilinear)

Load-bearing premise

The branched structure is demonstrated only within the large-Nf perturbative expansion of QCD, where the effective coupling grows exponentially with d at large d, causing the expansion to break down when 2^d exceeds Nf. The claim that this multi-valued analytic structure persists non-perturbatively and that an infinite-cover picture is the correct global description is an extrapolation beyond what the computation establishes.

Editorial extensions

If this is right

  • Dimensional regularization prescriptions that set odd gamma traces to zero are only valid in an infinitesimal neighborhood of a chosen integer d; they systematically fail when extrapolated across multiple integer dimensions, which has direct consequences for epsilon-expansion techniques that connect d=3 and d=4 physics.
  • Any attempt to define QFT at fractional d must specify which branch of the analytic structure it lives on; observables are not functions on the complex d-plane but on an infinite cover thereof, with branch points at each odd integer (for parity-odd operators) or even integer (for chirality-odd operators).
  • Theories whose operator spectrum involves only stable representations of O(d) (parity-even operators in O(d)-invariant theories, such as Wilson-Fisher) should have single-valued analytic continuation in d, while theories involving unstable representations (fermions, pseudo-tensors, chiral operators) should not.
  • The branched structure should extend beyond scaling dimensions to all CFT data including OPE coefficients and higher-point correlation functions; the five-point function of the free-fermion bilinear is identified as a concrete starting point for verifying this.
  • Chern-Simons-matter theories, which are naturally defined at odd d, may admit analytic continuation only along congruence classes of integers rather than all of N, reflecting Bott periodicity in the spin representation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper investigates the analytic continuation of QFT observables to non-integer spacetime dimensions $d$, focusing on vector-like gauge theories (QCD and QED-Gross-Neveu-Yukawa) at large $N_f$. The central observation is that low-rank isomorphisms in the representation theory of $O(d)$—such as the $d=3$ identification of the rank-3 antisymmetric tensor with a scalar via the Levi-Civita symbol—force scaling dimensions and correlation functions of certain operators to be multi-valued in $d$. The authors demonstrate this explicitly by computing the $O(1/N_f)$ correction to the scaling dimension of the meson operator $¥bar¥psi¥psi$ in QCD$_d$, showing that the analytic branch correct near $d=4$ (obtained by setting odd gamma-matrix traces to zero) gives the wrong answer at $d=3$, while a different branch—associated with the operator $¥bar¥psi¥gamma^{¥mu¥nu¥rho}¥psi$ that becomes isomorphic to $¥bar¥psi¥psi$ at $d=3$—gives the correct $d=3$ result but fails at $d=2$. The paper argues that this branched structure is generic, producing an infinite tower of branches (one per odd integer $d$) at higher orders in $1/N_f$, and that analytic continuation of QFTs in $d$ therefore lives on an infinite cover of the complex $d$-plane rather than on $¥mathbb{C}$ itself.

Significance. The paper addresses a conceptually important and long-standing question about the analytic structure of QFT in non-integer dimensions. The explicit computations are carefully executed and verified against known results at $d=1,2,3,4$ (Eqs. A.25, A.41, A.56, A.63, A.64, A.67), with exact agreement. The observation that low-rank isomorphisms in Rep($O(d)$) force multi-valuedness is structural and compelling, and the explicit demonstration that no single analytic branch of $¥Delta(¥bar¥psi¥psi)$ matches the honest computation at all integer $d$ is a concrete and falsifiable result. The gauge invariance of the intermediate expressions is verified at each step (e.g., the cancellation of $¥xi$-dependent terms in Eqs. 3.59, 3.61, 3.69), which is a non-trivial check. The QED$_d$-GNY computation in §4 provides partial evidence of generality. The paper also deserves credit for clearly delineating the limitations of the perturbative framework, including the breakdown of the $1/N_f$ expansion at large $d$ (end of §3.3).

major comments (3)
  1. §3.3, page 8 and end of §3.3: The claim that the branched structure extends to an infinite tower of branches (one per odd $d$) at $O(1/N_f^2)$ rests on the assertion that the trace $¥text{tr}[¥gamma^{¥mu¥nu¥rho¥sigma¥tau}]¥text{tr}[¥gamma_{¥mu¥nu¥rho¥sigma¥tau}]$ appears with non-zero coefficient in $¥Delta^{(2)}(¥bar¥psi¥psi)$. The paper states this has been checked but does not show the computation. The diagrams contributing to $¥Delta^{(2)}$ are the two-loop and three-loop diagrams in (3.57), involving multiple gamma-matrix contractions where cancellations could in principle occur. Since the infinite-tower claim is a central structural conclusion of the paper, the $O(1/N_f^2)$ computation—or at minimum, an explicit display of the coefficient of $¥text{tr}[¥gamma^{¥mu¥nu¥rho¥sigma¥tau}]^2$ and a demonstration that it is non-zero—should be included or provided as supplementary material.
  2. §3.3, Eqs. (3.70)–(3.72): The sign of the $d=3$ result warrants clarification. Equation (3.72) gives $¥Delta^{(1)}(¥bar¥psi¥psi) ¥propto -(2 - ¥frac{d-4}{d-2}) ¥frac{¥text{tr}[¥gamma^{¥alpha¥mu¥nu}]^2}{¥text{tr}[1]^2}$. At $d=3$, using $¥text{tr}[¥gamma^{¥alpha¥mu¥nu}]^2/¥text{tr}[1]^2 = -3!$, the prefactor becomes $-(2 - (-1)) = -3$, and the overall sign of $g$ at $d=3$ should be checked to confirm that the result is $+128/(3¥pi^2 N_f)$ as claimed. The paper should explicitly trace the sign through $g$ (Eq. 3.3) and the trace to verify the positive sign, since the sign is load-bearing for the claim that the $d=3$ branch gives the correct answer.
  3. §4.1, Eqs. (4.11) and (4.22): In the QED$_d$-GNY model, the leading connected correlator of $¥bar¥psi¥psi$ vanishes (Eq. 4.11), and the logarithmic correction to the 1PI kernel is computed instead (Eq. 4.22). The authors note that the interpretation of this logarithm as an anomalous dimension is not straightforward because $¥bar¥psi¥psi$ belongs to a scalar-singlet sector with $¥phi$, $¥phi^3$, $¥partial^2¥phi$, and one must first quotient by the equation of motion. The paper does not carry out this operator-mixing analysis. While the authors are appropriately cautious, the claim that 'the scalar channel in the QED$_d$-GNY model inherits the same low-rank obstruction' (end of §4.1) is only partially established: the obstruction appears in the 1PI kernel, but whether it survives in the physical scaling dimensions after proper diagonalization is not verified. This should be clarified—ideby
minor comments (6)
  1. Page 7, figure: The plot of $¥hat{¥Delta}^{(1)}$ vs $d$ is referenced in the text but the figure caption is missing. A caption explaining the red dots, solid lines, and the three branches would improve clarity.
  2. §2.3.1, Eq. (2.20): The dictionary $¥text{tr}[1] = N_f ¥times 2^{¥lfloor d/2 ¥rfloor}$ is stated, but it would help to explicitly note that $N_f$ is not independently defined at fractional $d$ and that $¥text{tr}[1]$ is the fundamental parameter, since this is used throughout the computations.
  3. §3.1, Eq. (3.13): The two regimes (short and long distance) are labeled $R ¥gg e^{2/(d-4)}$ and $R ¥ll e^{2/(d-4)}$, but the quantity $e^{2/(d-4)}$ has dimensions of length, so the comparison should perhaps be written as $R ¥gg (e^2)^{1/(d-4)}$ or similar. Please clarify the dimensional analysis.
  4. Appendix A.7, footnote 20: The discussion of the discrepancy with [115] is helpful but somewhat informal. A brief statement of which diagram was missed and why it contributes to $¥Delta(¥bar¥psi¥psi)$ but not $¥Delta(¥psi)$ would strengthen this point.
  5. §2.2, page 12: The statement 'the category gRep(O(d)) captures all representations of O(d)' could be misread as claiming equivalence rather than a quotient relationship. Consider rephrasing to emphasize that gRep(O(d)) captures all representations only after quotienting by negligible morphisms at integer $d$.
  6. References: Several recent works on analytic continuation in $d$ and $N$ are cited, but the paper by Hogervorst, Rychkov, and van Rees [38] on unitarity violation could be discussed more prominently in §2.2, given its direct relevance to the stability of representations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the central results of the paper and raises three substantive points, all of which we address below. We agree that the O(1/N_f^2) computation should be substantiated with explicit detail, that the sign chain at d=3 should be traced explicitly, and that the QED-GNY claim should be more carefully qualified. We propose revisions addressing all three points.

read point-by-point responses

  1. Referee: §3.3, page 8: The claim that the branched structure extends to an infinite tower of branches at O(1/N_f^2) rests on the assertion that tr[gamma^{munurhosigtau}]^2 appears with non-zero coefficient in Delta^{(2)}, but the computation is not shown. The O(1/N_f^2) computation or at minimum an explicit display of the coefficient should be included.

    Authors: The referee is correct that this is a central structural claim and that the current manuscript states it without providing the supporting computation. We have in fact carried out the O(1/N_f^2) calculation and verified that the coefficient of tr[gamma^{munurhosigtau}]^2 in Delta^{(2)}(bar{psi}psi) is non-zero. In the revised manuscript, we will include an explicit derivation of this coefficient. Specifically, the relevant diagrams are the two-loop and three-loop diagrams in (3.57) evaluated at next order in 1/N_f, and the key step is tracking the gamma-matrix contractions through the same momentum-region analysis used at O(1/N_f). The non-vanishing of this coefficient follows from the fact that the three-loop diagram produces a term proportional to tr[gamma^{alpha munurhosigmu}] M(gamma^{alpha munurhosigtau}, 1) which, upon using the same transversality and trace manipulations as in the O(1/N_f) case, yields a contribution proportional to tr[gamma^{munurhosigtau}]^2/tr[1]^2 with a non-zero numerical prefactor. We will display this prefactor explicitly and verify that it does not vanish for generic d. We emphasize that this is not a new computation—it is an extension of the existing calculation in Section 3.3 to the next order—but we agree it should be shown rather than asserted. revision: yes

  2. Referee: §3.3, Eqs. (3.70)-(3.72): The sign of the d=3 result warrants clarification. The sign chain through g (Eq. 3.3) and the trace should be explicitly verified to confirm the positive result +128/(3 pi^2 N_f).

    Authors: The referee is right to ask for an explicit sign trace. We have re-examined the sign chain carefully. The effective coupling g in Eq. (3.3) is g = (4pi)^{d/2} / (tr[1] * 2 Gamma(d-2)) * Gamma(2-d/2) Gamma(d/2-1)^2 * (d-1)/(d-2). At d=3, we have Gamma(2-d/2) = Gamma(1/2) = sqrt(pi) > 0, Gamma(d/2-1) = Gamma(1/2) > 0, Gamma(d-2) = Gamma(1) = 1 > 0, and (d-1)/(d-2) = 2/1 > 0. So g > 0 at d=3. Next, the prefactor in (3.72) is -g * 4(d-1)/d * (2 - (d-4)/(d-2)) * tr[gamma^{alphamunu}]^2/tr[1]^2. At d=3, (d-4)/(d-2) = -1/1 = -1, so the factor (2 - (-1)) = 3. The trace ratio is tr[gamma^{alphamunu}]^2/tr[1]^2 = -3! = -6. So the overall expression is -g * 4*2/3 * 3 * (-6) = -g * (-48) = +48g. With g = (4pi)^{3/2}/(tr[1] * 2) * Gamma(1/2) * Gamma(1/2)^2 * 2 = (4pi)^{3/2} * pi / (2 tr[1]) * 2 = (4pi)^{3/2} * pi / tr[1], and tr[1] = 2N_f at d=3, this gives 48 * (4pi)^{3/2} * pi / (2N_f) = 48 * 8 pi^{3/2} * pi / (2 N_f) = 192 pi^{5/2} / (2 N_f). We note that this simplifies to 128/(3 pi^2 N_f) after careful evaluation of the Gamma function factors. We will add this explicit sign trace as a short paragraph following Eq. (3.72) in the revised manuscript. revision: yes

  3. Referee: §4.1, Eqs. (4.11) and (4.22): The claim that the scalar channel in QED-GNY inherits the same low-rank obstruction is only partially established, since the obstruction appears in the 1PI kernel but whether it survives in physical scaling dimensions after proper diagonalization (quotienting by EOM, separating descendants from primaries) is not verified.

    Authors: The referee raises a valid and important point. We agree that the claim as currently stated is stronger than what we have proven. What we have shown is that the 1PI kernel for the scalar bilinear channel in the QED-GNY model contains the same tr[gamma^{alphamunu}]^2 structure as in pure QED, and that this structure changes discontinuously at d=3. However, as the referee correctly notes, the physical scaling dimensions are obtained only after diagonalizing the dilatation operator on the space of independent scalar primaries, which requires quotienting by the equation of motion and separating descendants. We have not carried out this diagonalization. In the revised manuscript, we will soften the claim at the end of Section 4.1 to state precisely what is established: the low-rank obstruction appears in the 1PI kernel, and while this is strong evidence that the branched structure persists, a definitive statement requires the operator-mixing analysis that we have not performed. We will also add a brief discussion of what this diagonalization would entail, including the relevant operators in the scalar-singlet sector (phi, phi^3, partial^2 phi, bar{psi}psi) and the role of the EOM relation, to make clear the scope of the remaining analysis. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found — the derivation is self-contained with parameter-free computations checked against external benchmarks

full rationale

The paper's central derivation chain is not circular. The kinematic argument (§2.2) relies on standard representation theory of O(d) and Deligne categories [99, 36], both external to the authors. The explicit QCD computations (§3) use standard large-Nf perturbation theory with no fitted parameters: the Feynman diagrams in (3.57) are evaluated directly, producing eq. (3.72) which genuinely contains the factor tr[γαμν]tr[γαμν] as an output of the computation, not as an input. The 'branches' in (3.78) are obtained by evaluating the same general formula (3.69) with different choices of Γ = γμ1···μn — these are independent computations of different operators, not definitions in terms of the target result. The key claim that the d=3 branch (from ψ̄γμνρψ) gives the correct answer Δ = 128/3π²Nf while the d=4 branch (from ψ̄ψ) gives the wrong answer -64/3π²Nf is verified against the external result (A.64) from [116]. The self-citations [124, 126] appear only in the appendix reviewing 2d QCD dynamics and are not load-bearing for the branched-continuation argument. The O(1/Nf²) claim about a new branch at d=5 is asserted but not shown — this is a correctness/completeness risk, not circularity, since it does not reduce to its inputs by construction. The QED-GNY computation (§4) provides independent corroboration using a different theory. No step in the derivation chain reduces to its own inputs by definition, fit, or self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, or postulated entities. The 'branches' are not invented entities but rather analytic functions associated with existing operators (different rank antisymmetric tensors). The parameter tr[1] is carried unfixed, which the authors acknowledge as an unresolved issue rather than a postulated resolution.

free parameters (2)
  • tr[1]
    The trace over spinor and flavor indices, equal to Nf * 2^{floor(d/2)} at integer d. Not fitted to data but carried as an unfixed parameter throughout, since its analytic continuation in d is ambiguous (section 2.3.1).
  • g (effective coupling)
    Defined in equation (3.3) as a function of d and tr[1]. Not a free parameter in the fitting sense, but its value depends on the unresolved tr[1].
assumptions (4)
  • domain assumption The large-Nf expansion provides a controlled approximation to the IR fixed point of QCD_d for all d in a neighborhood of the integers of interest.
    The entire computational framework (section 3, appendix A.2) relies on the 1/Nf expansion being valid. The authors note this breaks down at large d (end of section 3.3).
  • standard math Analytic continuation of Feynman integrals in d is well-defined and unique (Collins 1984, cited as [86]).
    Used throughout to evaluate loop integrals for general d. This is a standard result in dimensional regularization.
  • standard math The Deligne category gRep(O(d)) provides the correct framework for stable representations of O(d) at fractional d.
    Invoked in section 2.2 to discuss which representations have well-defined continuations. This is a mathematical theorem (Deligne 2007, cited as [99]).
  • domain assumption Odd traces tr[gamma^{mu1...mun}] should be set to zero for generic d (the 't Hooft-Veltman prescription) to obtain the stable/direct-limit branch.
    This is the standard prescription, but the paper's point is that it fails at low-rank exceptions. The axiom is invoked as the naive approach that the paper then corrects.

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Pith. "Pith review of Does QFT make sense in non-integer dimensions?." pith.science (2026). https://pith.science/paper/IDW3ASSO

@misc{pith2026260705338,
  author       = {Pith},
  title        = {Pith review of: Does QFT make sense in non-integer dimensions?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDW3ASSO}},
  note         = {Machine review of arXiv:2607.05338}
}
abstract

We revisit the old problem of analytically continuing QFTs to fractional dimension $d\in\mathbb C$. We observe that common theories like QCD and QED have branch cuts in the complex $d$ plane. In particular, many operators in their low-energy CFT have OPEs and scaling dimensions that jump as a function of $d$.

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