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DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

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A. V. Kotikov, L. N. Lipatov

High Energy Physics - Phenomenology · Submitted 23 Aug 2002 · 57 references

Pith's verdict: A genuinely important N=4 paper whose central new formula is assembled from heuristic constraints rather than derived, and the authors mostly say so.

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arXiv IDhep-ph/0208220 (version 3)
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CitationA. V. Kotikov, L. N. Lipatov. "DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory." arXiv:hep-ph/0208220 (2002). https://pith.science/paper/EWIOJFXF
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The paper

The paper claims

In the maximally supersymmetric gauge theory, the small-$x$ and $Q^2$ evolution equations are shown to encode the same anomalous dimensions, at least at the leading singularities.

The reading

The paper asks whether the two standard descriptions of how parton distributions change with energy — the BFKL equation for small Bjorken $x$ and the DGLAP equation for the $Q^2$ evolution of Mellin moments — are two faces of one structure in the maximally supersymmetric $N=4$ gauge theory. The authors compute the next-to-leading correction to the BFKL kernel in that model and observe that its eigenvalue is an analytic function of the conformal spin $|n|$; continuing it to negative $|n|$ lets them read off the singularities of twist-2 anomalous dimensions at the non-physical points $j=0,-1,-2$ directly from the BFKL side. Those singularities match what the DGLAP equation gives, at least for the leading poles. Along the way they write down a closed expression for the universal next-to-leading anomalous dimension of the model, $16Q(j-2)$ in terms of polylogarithms and associated functions, and show that the next-to-leading Bethe-Salpeter kernel is hermitian-separable even though strict holomorphic separability is lost. If the picture holds, the Regge and Bjorken asymptotics of this theory are governed by one function of the Lorentz spin, which is exactly the structure one would want in a theory that is conjectured to be integrable in the multi-colour limit.

What carries the argument

The load-bearing device is the hermitian separability of the next-to-leading Bethe-Salpeter kernel: the correction $delta(n,gamma)$ splits into a function of $M=gamma+|n|/2$ plus the same function of $1-tilde{M}$, minus a term proportional to the Born eigenvalue times $rho(M)+rho(1-tilde{M})$, with $rho(M)=beta'(M)+zeta(2)/2$ and $phi(M)=3zeta(3)+Ψ''(M)-2Phi_2(M)+2beta'(M)(Ψ(1)-Ψ(M))$. This structure guarantees reality of the spectrum and replaces the holomorphic separability of the leading order, which the authors show is violated by a term proportional to $(1+(-1)^n)$ that vanishes for odd conformal spin. The second device is the construction of the universal next-to-leading anomalous dimension from a list of consistency requirements — transcendentality level three, cancellation of $zeta(2)$ at even $j$ in the unpolarized case, absence of double poles of the form $S_{pm l}(j-2)/(j pm m)^k$, and equality of the highest-transcendentality pieces with the corresponding QCD anomalous dimensions at $C_F=C_A=N_c$ — yielding $gamma_{NLO}(j)=16Q(j-2)$ with $Q(j)$ given in Eq. (68). A footnote records that the published form of the double-logarithmic QED partial wave used as an analogy was incorrect and is replaced by the corrected expression derived in Appendix D.

Extended reading notes

Core claim

The central claim is that in $N=4$ super Yang-Mills the BFKL eigenvalue $w(n,nu)$ is analytic in the conformal spin $|n|$, and that this analyticity is not a technicality but a bridge: continuing $|n|$ to $-r-1$ while taking $w to 0$ reproduces the residues of the twist-2 anomalous dimensions $gamma$ at $j=-r$. In leading logarithmic accuracy this continuation reproduces $gamma(j)=4a(Ψ(1)-Ψ(j-1))$ and its poles at $j=0,-1,dots$ in agreement with the direct DGLAP calculation. At next-to-leading order the authors show that the leading singularities $1/(j+r)$ and $1/(j+r)^3$ obtained from the two equations coincide, while the residues of the subleading poles $1/(j+r)^2$ and $1/(j+r)$ depend on parameters of the kernel that their present accuracy does not fix — a limitation they state explicitly. They also establish that the next-to-leading Pomeron hamiltonian loses holomorphic separability but the corresponding Bethe-Salpeter kernel retains a hermitian separability, meaning the eigenvalue stays real for real $nu$ and symmetric under $nu to -nu$.

Load-bearing premise

The next-to-leading anomalous dimension is fixed by a list of consistency conditions — transcendentality level three, cancellation of $zeta(2)$ terms, absence of double poles, and agreement with the most complicated QCD pieces — and the paper does not prove that these conditions single out one answer rather than merely one convenient answer.

Editorial extensions

If this is right

  • The Regge and Bjorken asymptotics of $N=4$ super Yang-Mills are controlled by a single function of the Lorentz spin, so a calculation on one side constrains the other.
  • The next-to-leading anomalous dimensions of all twist-2 operators in the model follow from $gamma_{uni}(j)$ by integer shifts of the argument, which is what makes the multi-colour evolution equation integrable in the leading logarithmic approximation.
  • The hermitian separability of the Bethe-Salpeter kernel provides a practical route to the spectrum at next-to-leading order, since it reduces the problem to functions of a single variable.
  • The size of the next-to-leading correction to the Pomeron intercept in this model, $c(0,1/2) approx 7.58$ against roughly $3.5$ times that in QCD, supports reading the large QCD correction as an effect of coupling-constant renormalization rather than of the kernel's structure.

Reading between the lines

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

  • If the analytic continuation in $|n|$ is really a symmetry of the model rather than an accident of the leading order, one would expect the exact anomalous dimension to be a single analytic function whose poles at all negative integers are fixed by the same continuation; that is a sharper statement than the paper proves and would be worth testing at three loops.
  • The failure to pin down the subleading pole residues is not merely a calculational gap: the parameters $a(r),b(r),d(r)$ that the authors leave free are exactly the place where a scheme-dependent finite renormalization could hide, so a future determination of them would double as a check that the DRED-scheme coupling used here is the one in which the universal form holds.
  • The connection drawn between the loss of holomorphic separability and the appearance of double-logarithmic terms suggests that integrability in this model survives at next-to-leading order only in the weakened hermitian sense; whether a Baxter-type equation still exists at that order is left open by the paper and is a natural next question.
Original abstract

We derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions \gamma of twist-2 operators in the non-physical points j=0,-1,... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N=4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. The main singularities of anomalous dimensions \gamma at j=-r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,181 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions \gamma of twist-2 operators in the non-physical points j=0,-1,... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N=4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. The main singularities of anomalous dimensions \gamma at j=-r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.

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 derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the next-to-leading approximation. contribution 1.00 0.97 0.24 asserted 1.00
s1 The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. result 1.00 0.90 0.13 asserted 1.00
s2 Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions \gamma of twist-2 operators in the non-physical points j=0,-1,... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. result 0.81 0.93 0.18 asserted 0.98
s3 Moreover, in the multi-color limit of the N=4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. result 1.00 0.83 0.17 asserted 1.00
s4 In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. result 1.00 0.88 0.15 asserted 1.00
s5 The main singularities of anomalous dimensions \gamma at j=-r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles. limitation 0.84 0.93 0.22 asserted 0.95

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
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

References
How later papers used it

154 later papers cite it

Evidence: substantial. 151 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 269 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 2 of 3 papers with this label
Uses its data or values 1 of 6 papers with this label
Cites it as support 1 of 3 papers with this label

  • Said its small fitted value of alpha-prime is compatible with N=4 SYM in accordance with this paper.

    Soft interaction model and the LHC data (2012) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

146 arXiv papers cite this in 246 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 169 sentences without a confident label

Contests it 1 paper
Builds on it 2 papers
  • Lipatov in QCD [ref] , to the maximally extended $=4$ supersymetric Yang-Mills (SYM) theory shows [ref] , that a lot of terms in the QCD result ([ref]) are cancelled and the final result contains the functions, which have the same property called later as a transcedentality[Their large $γ$ limit coincides with the special transcendental numbers such as zeta-numbers $ζ_i$]: [equation] Using the same suggestion the results for the anomalous dimension of the twist-2 operators in $=4$ SYM theory was obtained without any computation [ref] , but argued from the relation between BFKL and Dokshitzer-Gribov-Lipatov-Altarelli-Parizi (DGLAP) [ref] equations.

    BFKL pomeron in the next-to-next-to-leading approximation in the planar N=4 SYM theory · Introduction · 0.92

  • Lipatov in QCD [ref] , to the maximally extended $=4$ supersymetric Yang-Mills (SYM) theory shows [ref] , that a lot of terms in the QCD result ([ref]) are cancelled and the final result contains the functions, which have the same property called later as a transcedentality[Their large $γ$ limit coincides with the special transcendental numbers such as zeta-numbers $ζ_i$]: [equation] Using the same suggestion the results for the anomalous dimension of the twist-2 operators in $=4$ SYM theory was obtained without any computation [ref] , but argued from the relation between BFKL and Dokshitzer-Gribov-Lipatov-Altarelli-Parizi (DGLAP) [ref] equations.

    NNNLLA BFKL pomeron eigenvalue in the planar N=4 SYM theory · Introduction · 0.86

Uses its method or tool 3 papers
Uses its data or values 5 papers
  • By the maximal-transcendentality principle [ref] , the $w=4$ terms in the coefficients of $C_A^4$ and $d_AA^(4)/n_a$ may be gleaned respectively from the planar [ref] and nonplanar [ref] universal anomalous dimensions of the twist-two operators with general Lorentz spin $N$ at N$^3$LO in $𝒩=4$ SYM, which serves as welcome input.

    Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part · Introduction · 0.80

  • Still, one can define $ΔG$ as the first moment of a gauge–invariant non–local operator and consider the evolution ∂/∂lnQ^2[ ΔΣ; ΔG ] = λ/16π^2[ -6 0; 3 0 ] [ ΔΣ; ΔG ] , where, as an illustration, we show the one–loop result from [ref] .

    Polarized DIS in N=4 SYM: Where is spin at strong coupling? · Polarized DIS in $𝒩=4$ SYM · 0.88

  • Let us first transcribe the result presented in [ref] [Our definition of $ν$ differs from that used in [ref] by a factor of 2.] [equation] with [equation] where $Ψ(x)=Γ'(x)/Γ(x)$ is the Euler $Ψ$-function and the function $Φ$ [ In the BFKL literature there are two different functions that are usually denoted by $Φ$.

    Conformal Regge theory · Weak coupling · 0.83

Compares with it 5 papers
  • Furthermore, the highest transcendental-weight terms of the trajectory agree with the $𝒩=4$ SYM result [ref] , as predicted by the maximal transcendentality principle [ref] .

    Three-loop gluon scattering in QCD and the gluon Regge trajectory · High Energy Limit and the Gluon Regge trajectory · 1.00

  • The result agrees at two-loop order with the one obtained from an explicit field-theory calculation [ref] , and at three-loop order with a solid conjecture [ref] extracted by the principle of maximum transcendentality [ref] from a rigorous field theory calculation in QCD [ref] .

    Dressing and Wrapping · Introduction and Verdict · 0.96

  • This is in contrast to more familiar operator mixing such as the well studied BFKL-DGLAP-Shadow mixing [ref] , for which branch points can only be accessed at finite coupling.[In $𝒩=4$ SYM the Riemann surface realizing this mixing has been constructed at finite coupling for the twist $2$ trajectory in [ref] , and recently in [ref] for the case of the leading twist three trajectory.

    Light-Ray Wave Functions and Integrability · Riemann surface, expectation and result · 0.80

Cites it as support 3 papers
Cites it as background 49 papers

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