DGLAP-BFKL duality from QCD to quantum computers

Igor Kondrashuk

REVIEW 3 major objections 5 minor 28 references

DGLAP-BFKL duality from QCD to quantum computers

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper aims to establish that the complex-map method that solves DGLAP also solves BFKL, because BFKL is the Regge limit of the dual DGLAP equation.

desk verdict Correct toy calculation, but the paper's main claim about solving BFKL is asserted on the strength of self-citations rather than demonstrated; more conference abstract than research result. read the letter →

arxiv 2509.04327 v1 pith:PVANXNTH submitted 2025-09-04 quant-ph math-phmath.MP

Igor Kondrashuk This is my paper · ORCID

classification quant-phmath-phmath.MP

keywords DGLAPequationBFKLMellinmomentscomplexmapsSchrödingeropticaltheoremdualquantumcommunication

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

This paper tries to establish a transfer of solution techniques between the two central evolution equations of QCD: DGLAP, which governs how parton distributions change with resolution, and BFKL, which governs their high-energy, small-x limit. Its claim is that because BFKL is the Regge limit of the equation dual to DGLAP under the Mellin-space map gamma(N)=M, the complex-map method that solves DGLAP also solves BFKL and the associated Schrödinger equation. The worked example uses a toy anomalous dimension gamma(N)=1/(N+1), where the method yields the closed form x I0(2 sqrt(ln u ln(1/x))). If the transfer holds, one contour-integral technique connects renormalization-group evolution, unitarity, and quantum-mechanical wave equations, with possible use in quantum communication modelling.

What carries the argument

The central object is the duality condition gamma(N)=M, where gamma(N) is the anomalous dimension of the gluon distribution and M is the Mellin variable of the dual representation; its inverse chi(M)=N satisfies chi(gamma(N))=N. The carrying mechanism is a complex diffeomorphism in the Mellin-moment plane: the integration variable is changed N->M so that a contour integral with a complicated exponent becomes a Laplace-type integral weighted by the Jacobian of the map, which is then evaluated on a rectified contour. The concrete example is the map N(M)=[Mw-1+sqrt((Mw+1)^2-4w^2)]/2 with w=sqrt(ln u/ln(1/x)), producing the Bessel function via contour integration.

What would settle it

Apply the claimed dual DGLAP construction to a next-order anomalous dimension beyond 1/(N+1), take its Regge limit, and compare the resulting integral kernel term by term with the known leading-order BFKL kernel; any mismatch in the kernel would break the transfer.

Watch

Extended reading notes

Core claim

Working in the Mellin-moment representation of parton distributions, the note shows for a toy anomalous dimension gamma(N)=1/(N+1) that the inverse Mellin integral for the DGLAP solution can be evaluated exactly by changing variables in the complex plane of the moment variable. The result is the explicit closed form phi(x,u)=x I0(2 sqrt(ln u ln(1/x))). The same contour integral can be re-expressed in a dual variable M defined by gamma(N)=M, with inverse chi(M)=N; this is the 'dual DGLAP' equation. Its Regge limit, the note claims, coincides with the BFKL equation, which is itself the Regge limit of the optical theorem and can be written as a Schrödinger equation. Hence the complex-map/Jacobi

Load-bearing premise

The whole transfer rests on the premise that the high-energy limit of the equation dual to DGLAP is exactly the BFKL equation, and the worked example only covers a simplified model with a fixed coupling rather than the full QCD case.

Editorial extensions

If this is right

The BFKL equation inherits a concrete solution route: rewrite it as a contour integral and evaluate it with the Jacobians of a suitable complex map in the Mellin-moment plane.
The n-parton Schrödinger equation equivalent to DGLAP becomes accessible to the same technique, so wave functions can be obtained from the solved parton distributions.
Because the dual representation comes from an exact change of variable, any answer found in dual variables automatically obeys the original DGLAP equation under chi(gamma(N))=N; the two pictures carry the same information.
The link between renormalization (DGLAP) and unitarity (optical theorem/BFKL) is reduced to a choice of complex coordinate in Mellin space, rather than a separate dynamical input.
In the toy model the unintegrated gluon distribution is the closed form x I0(2 sqrt(ln u ln(1/x))), giving a concrete target for numerical checks.

Reading between the lines

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

If the map method survives beyond the toy anomalous dimension, it would turn integro-differential evolution equations into contour-integral evaluations, a structural shortcut applicable to any kernel whose Mellin transform is invertible in closed form.
The explicit Bessel solution could serve as a benchmarking case for quantum algorithms that simulate parton or Schrödinger evolution, since exact closed forms with slowly convergent series are good stress tests.
The paper only establishes the duality link in the Regge limit; a fuller test would be to construct the dual DGLAP for a running coupling and ask whether the complex-map contours still close without leaving the Riemann surface of gamma(N)=M.
A practical, testable extension is to feed an exact running-coupling solution, for example from a supersymmetric gauge model, into the dual equation and see whether the contour method reproduces the corresponding BFKL kernel at small x.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The honest summary is: this note has one genuinely correct piece of mathematics, and everything that matters rests on unsupported extrapolation. The contour integral for the toy anomalous dimension gamma(N)=1/(N+1) does give x I0(2 sqrt(ln u ln(1/x))), and the dual-map reproduction of the same Bessel function checks out. That part is solid, and it is also not new: the same solution and the same duality relation already appear in [13,14,18]. What is new here is the suggestion that this method solves BFKL and related Schrödinger equations, and that is asserted, not shown.

The load-bearing step is the claim that the Regge limit of the dual DGLAP equation coincides with the BFKL integro-differential equation. That is imported from [13] without derivation. Even if we grant that coincidence, there is no step in this note that applies the complex-map technique to the actual BFKL kernel, which in QCD is a two-dimensional Mellin-Barnes integral with branch cuts for the eigenvalue function. The toy chi(M)=1/M-1 is a single rational branch; QCD is different. The note itself concedes the Riemann surface is more sophisticated, but that concession is the end of the story, not the beginning of a calculation. The quantum communication remark in Section 6 is likewise a hand-wave. None of this is fatal if the goal is a conference proceedings note about a promising direction, but as a claim of a solution method for BFKL it is a placeholder.

The self-citation pattern deserves a mention: citing your own prior work for the one step that carries the result is a real weakness here, because the cited result is not reproduced and the reader cannot see whether the Regge-limit coincidence actually holds in the needed form. That said, I do not see deliberate obscurity; the toy derivation is transparent and the limits are acknowledged, if only in prose.

Who gets value from this? A reader interested in the toy-model technique might read Sections 3 and 4 as a compact summary. The paper would not be a citation for a substantive claim about BFKL. If the author follows through with a real derivation for the BFKL kernel, that would deserve a referee. As it stands, I would not send this to peer review; it is not enough of a paper and the main conclusion is unsupported. If it lands on my desk, I would desk-reject with an invitation to resubmit once the BFKL step is actually worked out.

Referee Report

3 major / 5 minor

Summary. This short note claims that the DGLAP integro-differential equation can be solved by complex maps in the Mellin-moment plane; that the same DGLAP equation can be rewritten as an n-parton Schrödinger equation; and that a dual DGLAP equation has a Regge limit coinciding with the BFKL equation, so that BFKL and its Schrödinger form are solvable by the same method, with applications to quantum communication. The only calculation shown in detail is the toy-model inverse Mellin transform for γ(N)=1/(N+1) with fixed coupling, which yields φ(x,u)=x I0(2√(ln u ln(1/x))). The extension to BFKL is asserted through references to the author's earlier work and to [11,12] rather than derived.

Significance. If the claimed method really solved BFKL and the corresponding n-body Schrödinger equations, it would be an interesting contribution. The self-contained toy-model calculation in Secs. 3–4 is a genuine check: the contour integral is evaluated explicitly and returns the stated Bessel function. However, the paper contains no derivation of the BFKL/dual-DGLAP relation, no explicit BFKL solution, no error estimates, and no concrete quantum-communication problem. Its significance as submitted is prospective rather than established.

major comments (3)

[§5, §6] The central assertion, stated in the abstract and repeated in Sec. 5, is that 'the BFKL IDE is a Regge limit of the dual DGLAP IDE' and hence is solvable by the same complex-map technique. This is not demonstrated: the dual DGLAP equation is never written as an integral equation, its Regge limit is not defined, and the identification is delegated to [13] and [11,12]. Section 6 explicitly records uncertainty ('It is not clear if the dual DGLAP equation ... coincides with the optic theorem completely'). The only self-contained calculation, in Secs. 3–4, is a one-dimensional inverse Mellin transform for a rational toy anomalous dimension; the BFKL Mellin representation is two-dimensional and involves χ(γ)=2ψ(1)−ψ(γ)−ψ(1−γ), with branch cuts and a running-coupling dressing. The note supplies no derivation or bound showing that the Jacobian method transfers. This is the load-bearing step, not
[§3, §4] The toy model has γ(N)=1/(N+1) and fixed coupling. For this model χ(M)=1/M−1 is a rational single-branch function and χ(γ(N))=N is exact. In QCD the duality is only a leading-log small-x relation, and the BFKL eigenvalue function has branch cuts; Sec. 5 itself concedes that the QCD Riemann surface is more sophisticated. No error estimate or numerical check is given to control the extrapolation from the toy to QCD. Therefore the statement that BFKL, or the corresponding Schrödinger equation, 'may be solved' by the proposed method is an extrapolation rather than a result established in this manuscript.
[§2, §6] The claimed equivalence with quantum mechanics is also cited rather than shown. Lipatov's Schrödinger equation (11) is quoted, but the paper does not spell out how Eq. (11) follows from, or is related to, the scale-evolution equation (8) for the specific toy γ(N), nor how a solution of the Mellin-space equation translates into the n-parton wave function. Section 6's reference to quantum communication is programmatic, with no concrete task. For a manuscript whose title and abstract promise a route from QCD to quantum computers, this missing link is central.

minor comments (5)

[Throughout] Typos and wording: 'QUANTUM MECANICS' (Sec. 2), 'dimesion', 'Bethe-Salpete', 'funactions', 'patrons', and 'optic theorem' for 'optical theorem'.
[§2] The symbol N is used both for the Mellin variable and for the gauge group SU(N); the note acknowledges this, but the collision makes Eqs. (4)–(9) confusing, especially in the sentence introducing the Lambert-function solution.
[§3] The contours C, C', C'' and the parameter δ are not defined; Eq. (13) is introduced without a derivation of the map or a discussion of its branches.
[§4] The text refers to 'complex diffeomorphism (15)', but no equation is numbered (15) in the manuscript. The two differential equations for φ(N,u) and φ(x,M) are also not labeled.
[§6] The connection to quantum computers is only a single vague sentence. A concluding section stating precisely what was established and what remains programmatic would make the scope clearer.

Circularity Check

2 steps flagged · score 7.0 of 10

BFKL solvability is carried by a self-citation: the Regge-limit identity between dual DGLAP and BFKL is asserted, not derived, while the only worked example is a change-of-variable rewrite of the same toy DGLAP integral.

self citation load bearing [Sections 5-6, text following Eq. (15) and opening paragraph of Section 6]
"It is not clear if the dual DGLAP equation, that is the IDE constructed from DGLAP equation via complex maps in the complex plane of the Mellin variable, coincides with the optic theorem completely but definitely the Regge limit of the dual DGLAP equation coincides with the Regge limit of the optic theorem [13]."

The bridge from the solved DGLAP toy model to BFKL is entirely this Regge-limit coincidence. The note never writes the BFKL kernel, the QCD eigenvalue function chi(gamma)=2psi(1)-psi(gamma)-psi(1-gamma), or a matching calculation; it imports the coincidence from [13], the author's own prior paper. The conclusion that the BFKL equation 'also may be found via Jacobians of the complex maps' therefore reduces to a self-citation plus the toy integral's change of variables. The paper's own words 'It is not clear if the dual DGLAP equation coincides with the optic theorem completely' concede that the full equivalence is not established, yet the Regge-limit part is then asserted as definite by citation to [13].
self definitional [Section 4 ('EQUATION DUAL TO DGLAP') and Section 5, first sentence]
"The new variable M is related to the original variable N by the duality relation γ(N) = M. If we define the inverse function χ(M) ≡ N, then we model the duality condition χ(γ(N)) =N and γ(χ(M)) =M. ... We have reproduced the same Bessel function, as it should be because any integral does not depend on any change of the variable. ... The dual DGLAP equation taken in the Regge limit under the duality χ(γ(N)) =N and γ(χ(M)) =M may be treated as BFKL equation."

The 'dual DGLAP equation' is introduced as the same DGLAP contour integral evaluated in a new Mellin variable, with the paper explicitly noting that the change of variables reproduces the same Bessel function. Thus the dual is a rewriting of DGLAP, not an independent dynamical equation. The identification of its Regge limit with BFKL is an extra assumption, not a consequence of the complex map. Consequently, the claim that BFKL is solvable by the same method is circular: it presupposes that the re-written DGLAP object is BFKL in the Regge limit, which is exactly the content that the note claims to transfer from the toy calculation.

full rationale

The only fully self-contained calculation is the toy inverse Mellin transform for gamma(N)=1/(N+1) in Section 3, yielding phi(x,u)=x I0(2 sqrt(ln u ln(1/x))), and its duplicate in Section 4 under the change of variables gamma(N)=M. That part is not circular: it is a legitimate evaluation of a contour integral. Circularity enters at the extrapolation step. The paper's central conclusion about BFKL rests on the assertion that the Regge limit of the 'dual DGLAP equation' coincides with the BFKL equation. This assertion is not derived anywhere in the note; it is cited to the author's own [13], and the note itself flags uncertainty about the complete coincidence. The self-contained toy only demonstrates equivalence between two contour representations of the same DGLAP solution, not any connection to BFKL. One should note that [11,12] are external references for small-x GLAP/BFKL duality, so not every link is self-citation, but the specific complex-map-based dual construction and its Regge-limit identification are attributed to [13]. A score of 7, rather than 8 or 10, reflects that the toy computation has independent mathematical content and the conclusion is hedged ('may be solved'); however, the load-bearing BFKL link is neither derived nor independently reproduced here, so the claimed transfer is effectively carried by the author's prior citation and by a definitional rewriting.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central derivation depends on a hand-picked toy anomalous dimension, a fixed coupling, a gluon-only approximation, and a duality condition imported from prior literature. The link between dual DGLAP and BFKL is assumed via [13], not derived in this note.

free parameters (2)

Toy anomalous dimension gamma(N)=1/(N+1) = 1/(N+1)
Chosen ad hoc in Section 3 to make the inverse Mellin integral tractable; all displayed solutions depend on this choice.
Fixed coupling approximation = alpha(u)=constant
Implicit in Section 3: the solution is power-like only when the coupling does not depend on the scale u. Real QCD has a running coupling.

assumptions (5)

ad hoc to paper Scale-independent coupling for the DGLAP evolution equation.
The toy-model solution requires alpha(u) constant in u; the paper discusses realistic running coupling only qualitatively.
domain assumption Gluon-only approximation: the sum over parton types is omitted, leaving a single gluon distribution.
Used to simplify Eq. (6) and to write Eq. (8); justified in the paper by small-x gluon dominance, but not generally valid.
domain assumption Duality condition chi(gamma(N))=N, with M=gamma(N), from Catani and Hautman.
The paper imports this condition from [26] and [13] and uses it to construct the dual DGLAP equation; its global validity in QCD is asserted rather than proven.
domain assumption The Regge limit of the dual DGLAP equation coincides with the BFKL equation.
Stated in Section 5 with a citation to [13]; this is the load-bearing link that lets the author claim BFKL is solvable by the complex-map method. No derivation appears in this note.
domain assumption The DGLAP equation is equivalent to a Schrödinger equation for n partons, following Lipatov.
Cited to [3] and used to extend the method to quantum mechanics; accepted without proof in this note.

invented entities (1)

Dual DGLAP equation
purpose: A re-written DGLAP equation obtained by the complex map M=gamma(N); used to connect DGLAP to BFKL in the Regge limit.
This is a mathematical construct rather than a physical entity. Its relevance relies on the assumed duality condition and on [13]; no independent evidence is given.

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Cite this review

Pith. "Pith review of DGLAP-BFKL duality from QCD to quantum computers." pith.science (2026). https://pith.science/paper/PVANXNTH

@misc{pith2026250904327,
  author       = {Pith},
  title        = {Pith review of: DGLAP-BFKL duality from QCD to quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVANXNTH}},
  note         = {Machine review of arXiv:2509.04327}
}

abstract

DGLAP integro-differential equation can be solved by applying certain map in the complex plane of Mellin moments. It may be re-written as Schr\"odinger equation for $n$ particles. By applying another map in the complex plane of Mellin moment we may re-write the DGLAP equation as a dual DGLAP equation. Regge limit of the dual DGLAP equation coincides with BFKL integro-differential equation which in turn is the Regge limit of optic theorem in quantum field theory and may be re-written as another Schr\"odinger equation. This means that the BFKL equation and the corresponding Schr\"odinger equation may be solved by the proposed method of complex mapping in the complex plane of Mellin moments. This approach may be useful in solving tasks related to quantum communication processes.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [13]

Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type

Kondrashuk, I. Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type. Phys. Part. Nucl. Lett. 2021, 18, 141–147

work page 2021

[2] [1]

Gribov V N and Lipatov L N 1972 Deep inelastic e p scattering in perturbation theory Sov. J. Nucl. Phys. 15 438 [Yad. Fiz. 15 781]

work page 1972

[3] [2]

Gribov V N and Lipatov L N 1972 e+ e- pair annihilation and deep inelastic e p scattering in perturbation theory Sov. J. Nucl. Phys. 15 675 [Yad. Fiz. 15 1218]

work page 1972

[4] [3]

Lipatov L N 1975 The parton model and perturba- tion theory Sov. J. Nucl. Phys. 20 (1975) 94 [Yad. Fiz. 20 (1974) 181]

work page 1975

[5] [4]

Dokshitzer Y L 1977 Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics Sov. Phys. JETP 46 641 [Zh. Eksp. Teor. Fiz. 73 1216]

work page 1977

[6] [5]

Altarelli G and Parisi G 1977 Asymptotic Free- dom in Parton Language Nucl. Phys. B 126 29

work page 1977

[7] [6]

Lipatov L N 1976 Reggeization of the Vector Me- son and the Vacuum Singularity in Nonabelian Gauge Theories Sov. J. Nucl. Phys. 23 338 [Yad. Fiz. 23 642]

work page 1976

[8] [7]

Fadin V S, Kuraev E A and Lipatov L N 1975 On the Pomeranchuk Singularity in Asymptotically Free Theories Phys. Lett. B 60 50

work page 1975