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A toy model for the quark pole mass

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Marcos Marino, Maximilian Schwick

High Energy Physics - Theory · Submitted 7 Oct 2026 · 30 references

Pith's verdict: Solid toy-model demonstration of renormalon cancellation; the trans-series decoding rests on an analytic continuation the authors flag but never check numerically.

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arXiv ID2610.10059 (version 1)
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CitationMarcos Marino, Maximilian Schwick. "A toy model for the quark pole mass." arXiv:2610.10059 (2026). https://pith.science/paper/EI7P7SMU
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The paper

The paper claims

In the Gross-Neveu model the heavy-quark pole mass has QCD-like renormalons; the exact large-$N$ result is a trans-series whose exponentially small terms cancel the ambiguity, and one class of those terms cannot be reached by any finite…

The reading

This paper studies the pole mass of a heavy quark probe in the Gross-Neveu model, a two-dimensional field theory with $N$ light fermions that is the standard toy model for the quark sector of QCD. The perturbative series for the pole mass, at first non-trivial order in $1/N$, is factorially divergent: its Borel transform has poles on the positive real axis, the infrared renormalons, structurally the same disease QCD has. The exact large-$N$ calculation, however, gives the pole mass as a well-defined function of the renormalized mass $m$ and the dynamical scale $\Lambda$, and the paper decodes its weak-coupling expansion as a resummed trans-series with infinitely many exponentially small corrections $e^{-k/\lambda_m}$. These corrections cancel the renormalon ambiguity term by term, and the first two match the operator-product expansion with two-quark and four-quark condensates. A third class of corrections, coming from fixed residues of the Mellin-Barnes integrand, only emerges when the on-shell condition is imposed before the residue expansion, and cannot be obtained from any finite truncation of the conventional local-condensate expansion; the authors argue this calls for a reorganization of the QCD program that derives the non-perturbative pole mass from condensates.

What carries the argument

The carrying object is the Mellin-Barnes representation of the exact large-$N$ self-energy, a double integral over $s\in\sigma+i\mathbb{R}$ and $t\in(0,\infty)$ whose building block is the elementary Mellin transform $K_{-1}(-s,t)$ of equation (3.22). Closing the $s$-contour to the left yields two families of residues: 'moving' poles at $s=-t-k$, which after expansion in powers of $1/c$ give the condensate-type contributions that the OPE reproduces, and 'fixed' poles at $s=-1/2-k$ plus an isolated pole at $s=-1$, which assemble correctly only when the on-shell condition $x=-c$ is imposed inside the integrand, turning the residue sum into the functions $G(t,k)$ and $Q(s;c)$ that cancel the renormalon. Supporting machinery includes the bubble-summed structure function $F(y)$ of equation (2.25), whose Gamma-function poles at positive odd integers and at $y=2$ generate the renormalons, and Gauss' hypergeometric summation formula (2.32), used as a meromorphic continuation to evaluate the on-shell hypergeometric functions; the residue-cancellation identities (3.38)–(3.39) are what make the trans-series contour-independent.

Extended reading notes

Core claim

At next-to-leading order in $1/N$, the exact large-$N$ solution gives the pole mass as $m_p = m + M + m_p^{(1)}/N + \cdots$, where $M\sim m e^{-1/\lambda_m}$ is the dynamically generated mass. The paper shows that the order-$1/N$ correction is a laterally Borel-resummed trans-series $\Phi_m(\lambda_m)=\sum_{k\ge 0}e^{-k/\lambda_m}\varphi_{m,k}(\lambda_m)$, with the perturbative sector carrying renormalon singularities at $t=1/2,1,3/2,\dots$. The leading singularity produces an ambiguity $\mp i\pi e^{-1/\lambda_m}/2$ that is cancelled by the opposite imaginary part of the first non-perturbative term $\varphi_{m,1}$ (equations 3.61–3.63), and the residue identities (3.38)–(3.39) make the full sum independent of which lateral contour is chosen. The same machinery produces a trans-series for the complete self-energy, and the first two exponentially small corrections to it are reproduced by an operator-product-expansion calculation with the two-quark condensate $\langle\psi\psi\rangle_c$ and the four-quark condensate, including all renormalization-group functions and the scheme matching. Crucially, the contributions of the fixed residues of the Mellin-Barnes representation reorganize completely when the self-energy is put on shell: going on-shell and expanding in residues do not commute, and the resulting terms — including the one that cancels the leading renormalon — cannot be produced by any finite truncation of the conventional condensate expansion.

Load-bearing premise

The load-bearing step is the limitation the authors themselves flag in section 2: Gauss' hypergeometric summation formula (2.32), which converges only for $y<1$, is used as the meromorphic continuation valid for all $y>0$ to evaluate the on-shell self-energy, and the $t=1/2$ renormalon cancellation lives inside that continuation — if the exact function follows a different branch for $y>1$, the cancellation in equations (3.61)–(3.63) could be an artifact of the continuation rather than a property of the exact result.

Editorial extensions

If this is right

  • The pole-mass renormalon is not an obstruction to a non-perturbative definition: the exact function exists, its lateral Borel resummation is unambiguous once the exponential corrections are included, and the same curing mechanism could in principle operate in QCD.
  • The leading non-perturbative shift of the pole mass, $m\to m+M$, is a genuine condensate effect carried by the two-quark condensate $\langle\psi\psi\rangle_c$, with the normalization fixed by the exact mass gap, so the SVZ picture is correct at its first orders.
  • Any attempt to extract the non-perturbative pole mass from a finite truncation of the self-energy OPE will miss the on-shell corrections: the term that cancels the first renormalon only appears after the Mellin-Barnes residues are reorganized with the on-shell condition imposed first.
  • In QCD, gauge invariance of a non-perturbative pole mass obtained from a finite condensate truncation is not guaranteed, and a reorganization of the expansion, possibly including non-gauge-invariant condensates, may be required.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the bubble-chain resummation used here also exists in the large-$N_f$ limit of QCD, the moving-versus-fixed residue dichotomy could be tested directly there: one would predict that the on-shell pole mass contains a piece with no local-condensate expansion, rather than merely receiving a correction of order $\Lambda$.
  • A portable diagnostic emerges: non-perturbative corrections attached to 'moving' Mellin-Barnes poles are condensate-representable, while those attached to 'fixed' poles signal on-shell, non-local effects; checking whether the same split occurs in other exactly solvable large-$N$ models would show whether the pattern is universal among renormalon-bearing observables.
  • The cancellation claim is numerically checkable now: evaluating the convergent integral representation of $A(-c;c)+B(-c;c)$ at moderate $\lambda_m$ and comparing with the two lateral Borel sums of the trans-series truncated at $k=0,1,2$ would expose whether the Gauss-continuation branch is the correct one.
Original abstract

We study the pole mass for a heavy quark probe in the Gross--Neveu model, at the first non-trivial order in the $1/N$ expansion. We show that this mass suffers from a perturbative renormalon ambiguity structurally similar to the one in QCD. The exact large $N$ solution leads however to an explicit expression for the pole mass which can be decoded as a trans-series, and we find that non-perturbative corrections cure the renormalon ambiguity. We also calculate the self-energy of the heavy quark beyond perturbation theory by using the OPE method with condensates, and show that the results are in agreement with the large $N$ solution. The non-perturbative corrections to the pole mass turn out to be determined to a large extent by the condensate corrections to the self-energy. However, there is a class of corrections that has to be reorganized due to on-shell effects and cannot be obtained from the conventional condensate expansion. We discuss possible implications of this toy model for QCD.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-08: 1 request, 5,613 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.88 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.81 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the pole mass for a heavy quark probe in the Gross--Neveu model, at the first non-trivial order in the $1/N$ expansion. We show that this mass suffers from a perturbative renormalon ambiguity structurally similar to the one in QCD. The exact large $N$ solution leads however to an explicit expression for the pole mass which can be decoded as a trans-series, and we find that non-perturbative corrections cure the renormalon ambiguity. We also calculate the self-energy of the heavy quark beyond perturbation theory by using the OPE method with condensates, and show that the results are in agreement with the large $N$ solution. The non-perturbative corrections to the pole mass turn out to be determined to a large extent by the condensate corrections to the self-energy. However, there is a class of corrections that has to be reorganized due to on-shell effects and cannot be obtained from the conventional condensate expansion. We discuss possible implications of this toy model for QCD.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the pole mass for a heavy quark probe in the Gross--Neveu model, at the first non-trivial order in the $1/N$ expansion. contribution 0.96 0.94 0.04 asserted 0.76
s1 We show that this mass suffers from a perturbative renormalon ambiguity structurally similar to the one in QCD. result 0.99 0.92 0.14 asserted 1.00
s2 The exact large $N$ solution leads however to an explicit expression for the pole mass which can be decoded as a trans-series, and we find that non-perturbative corrections cure the renormalon ambiguity. result 0.98 0.96 0.62 asserted 1.00
s3 We also calculate the self-energy of the heavy quark beyond perturbation theory by using the OPE method with condensates, and show that the results are in agreement with the large $N$ solution. contribution 0.46 0.96 0.08 asserted 1.00
s4 The non-perturbative corrections to the pole mass turn out to be determined to a large extent by the condensate corrections to the self-energy. result 0.92 0.94 0.18 asserted 0.96
s5 However, there is a class of corrections that has to be reorganized due to on-shell effects and cannot be obtained from the conventional condensate expansion. limitation 0.72 0.84 0.07 asserted 1.00
s6 We discuss possible implications of this toy model for QCD. interpretation 0.96 0.63 0.04 hedged possibility 0.61

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
  • full text
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  • references and citing sentences
  • relations to other papers
levels run
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Diagrams contributing to the 1/N correction to the heavy quark propagator. The thick lines are massive quark propagators, the thin lines are massless quark propagators, and the dashed lines are σ propagators. The calculation of the self-energy of the heavy quark at leading order in 1/N is very similar to what was done in [17]. The relevant diagrams are of the form shown in
Figure 2. At order 1/N, this diagram in which a heavy quark runs around the loop gives the only correction to the quenched approximation for the self-energy. 3 The non-perturbative quark pole mass 3.1 The non-perturbative propagator for the heavy quark We will now discuss what happens in the exact solution of the theory within the 1/N expansion. At large N we can integrate out the N massless fermions to obtain Leff = − σ 2 2g0 − iN log det −i∂/ − σ  + χ(i∂/ − m0 + σ)χ, (3.1) after rescaling σ → σ/√g0. The VEV of σ is determined by the usual gap equation, so that at large N we have two
Figure 3. The exact function (m (1) p −M1)/m, as a function of λm (red line), and its asymptotic behavior (3.61) as λm → 0, where we kept the first three terms (dashed black line). 3.3 Trans-series for the self-energy It turns out that, with a little bit more effort, we can calculate the trans-series for the full self￾energy, generalizing in this way the calculation of [17], which was made for a massless quark. We – 14 –
Figure 4. The heavy quark propagator with an insertion of a light two-quark condensate. The first condensate correction, as in [17] is due to the quark bilinear operator. The cor￾responding condensate will be denoted by ⟨ψψ⟩c. Composite operators like this have to be renormalized, and we will denote the renormalized composite operator O by [O]. In the case of the quark bilinear, we have ψψ = Zψψ[ψψ]. (4.1) In addition, renormalization group arguments lead to the equation [17] − πλ N
Figure 5. A double insertion of a two-quark condensate on the massive fermion line gives a four-quark condensate correction. We now consider the corrections due to condensates of higher dimension operators involving the N massless fermions. There are two operators of dimension 2 that one can consider, K = iψ(x) · ∂/ψ(x), V = g0
Figure 6. A double insertion of a two-quark condensate on the self-energy diagram of
Figure 7. The condensate corrections to the chain of bubbles. The final contribution to take into account is due to the condensate corrections to the chain of bubbles, or equivalently to the propagator of the σ field at order 1/N. These corrections are due to four types of diagrams. The first two are due to insertions of the condensates in the polarization loops, as shown in the first two diagrams of

Figures reproduced from arXiv:2610.10059 by the authors.

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