pith. sign in

arxiv: 2509.06799 · v2 · submitted 2025-09-08 · ✦ hep-th · hep-ph

Three-Loop Gauge Beta Functions in Supersymmetric Theories with Exponential Higher Covariant Derivative Regularization

Swapnil kumar Singh This is my paper

Pith reviewed 2026-05-18 18:20 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords supersymmetric gauge theoriesbeta functionshigher covariant derivativesthree-loop calculationsNSVZ relationexponential regulatorsPauli-Villars subtraction
0
0 comments X

The pith

Exponential higher covariant derivative regulators yield explicit three-loop gauge beta functions in N=1 supersymmetric theories.

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

The paper takes the known general three-loop form of the gauge beta function in the higher covariant derivative framework, which depends on regulator parameters, and evaluates those parameters explicitly when the regulators are chosen to be exponential functions R(x) = e^{x^n} and F(x) = e^{x^m}. Closed-form expressions for the constants A(n) and B(m) are derived together with their large-n and large-m expansions. These are substituted back to produce fully explicit, regulator-parameterized beta functions that admit a systematic expansion in powers of 1/n and 1/m. Finite coupling redefinitions are then constructed that map the result to a scheme consistent with the Novikov-Shifman-Vainshtein-Zakharov relation while preserving that relation at the bare level.

Core claim

We obtain the constants A(n) and B(m) in closed form, together with their large-n,m asymptotics, and substitute them into the general three-loop expressions. This yields fully explicit, regulator-parameterized beta-functions and a systematic expansion in 1/n and 1/m that organizes finite, scheme-dependent terms. Finite coupling redefinitions are exhibited that map the renormalized DR-bar result to a scheme compatible with the Novikov-Shifman-Vainshtein-Zakharov relation. The analysis clarifies how exponential higher-derivative regulators preserve this relation at the bare level.

What carries the argument

The regulator-dependent constants A(n) and B(m) obtained from the exponential regulators R(x)=e^{x^n} and F(x)=e^{x^m}, which enter the general three-loop beta-function expressions and control the finite scheme-dependent contributions.

If this is right

  • The beta functions become fully explicit functions of the regulator exponents n and m.
  • A systematic 1/n and 1/m expansion organizes all finite, scheme-dependent contributions.
  • Finite redefinitions of the coupling exist that restore compatibility with the NSVZ relation.
  • The NSVZ relation holds at the bare level for this class of regulators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same substitution procedure could be applied to other regulator shapes once their corresponding constants are known.
  • The explicit 1/n and 1/m expansions may simplify numerical studies of renormalization-group trajectories in specific models.
  • The approach suggests a route to organize scheme dependence across different regularization methods in supersymmetric theories.

Load-bearing premise

The all-structure three-loop form of the beta functions in the higher covariant derivative framework is taken to be known and complete, including every regulator-dependent term.

What would settle it

An independent three-loop calculation of the gauge beta function in a concrete N=1 supersymmetric model, performed directly with the exponential regulators, compared term-by-term against the substituted expressions for A(n) and B(m).

Figures

Figures reproduced from arXiv: 2509.06799 by Swapnil kumar Singh.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustrative RG evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Convergence of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Convergence of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Regulator dependence of the mixed gauge–Yukawa coefficient [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Regulator dependence of individual contributions to [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of the truncated expansion [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

We study the three-loop gauge $\beta$-functions in general $\mathcal{N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives (HCD) supplemented with Pauli--Villars subtraction. The all-structure three-loop form of the $\beta$-functions in the HCD framework is known and involves regulator-dependent parameters. Here we evaluate these parameters explicitly for the exponential regulators $R(x)=e^{x^n}$ and $F(x)=e^{x^m}$. We obtain the constants $A(n)$ and $B(m)$ in closed form, together with their large-$n,m$ asymptotics, and substitute them into the general three-loop expressions. This yields fully explicit, regulator-parameterized $\beta$-functions and a systematic expansion in $1/n$ and $1/m$ that organizes finite, scheme-dependent terms. We then exhibit finite coupling redefinitions that map the renormalized $\overline{\mathrm{DR}}$ result to a scheme compatible with the Novikov--Shifman--Vainshtein--Zakharov relation. Our analysis clarifies how exponential higher-derivative regulators preserve this relation at the bare level and illustrates the regulator-driven structure of supersymmetric renormalization group flows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript computes the three-loop gauge beta functions in general N=1 supersymmetric gauge theories regularized by higher covariant derivatives supplemented with Pauli-Villars subtraction, using exponential regulators R(x)=exp(x^n) and F(x)=exp(x^m). It evaluates the regulator-dependent constants A(n) and B(m) in closed form along with their large-n,m asymptotics, substitutes them into known general three-loop expressions to obtain explicit regulator-parameterized beta functions and 1/n,1/m expansions, and exhibits finite coupling redefinitions that map the renormalized DR-bar result to an NSVZ-compatible scheme.

Significance. If the closed-form results for A(n) and B(m) hold, the work supplies fully explicit three-loop beta functions in a specific higher-derivative scheme and organizes finite scheme-dependent contributions via systematic expansions in 1/n and 1/m. It also clarifies how exponential regulators preserve the NSVZ relation at the bare level. The provision of closed forms and asymptotics for the regulator integrals constitutes a concrete technical contribution to supersymmetric renormalization-group calculations.

major comments (1)
  1. [Abstract (paragraph 2) and the section presenting A(n), B(m)] The central claim rests on the accuracy of the closed-form expressions for A(n) and B(m) extracted from the three-loop diagrams with the exponential regulators. The manuscript states these closed forms and their asymptotics but does not display the intermediate momentum-integral reductions or supply numerical verifications for sample integer values of n and m. Because these constants are directly substituted into the general three-loop beta-function formulas, any algebraic error in their evaluation would propagate to the explicit beta functions, the 1/n,1/m expansions, and the finite redefinitions to the NSVZ scheme.
minor comments (2)
  1. Clarify the precise reference for the 'all-structure three-loop form' of the beta functions that is being used; if it is from prior work, include an explicit citation and a brief statement of the assumptions under which it was derived.
  2. In the large-n,m asymptotic expansions, state the order to which the expansions are carried and indicate whether higher-order terms affect the finite scheme-dependent contributions discussed later.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment on the presentation of the regulator-dependent constants. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract (paragraph 2) and the section presenting A(n), B(m)] The central claim rests on the accuracy of the closed-form expressions for A(n) and B(m) extracted from the three-loop diagrams with the exponential regulators. The manuscript states these closed forms and their asymptotics but does not display the intermediate momentum-integral reductions or supply numerical verifications for sample integer values of n and m. Because these constants are directly substituted into the general three-loop beta-function formulas, any algebraic error in their evaluation would propagate to the explicit beta functions, the 1/n,1/m expansions, and the finite redefinitions to the NSVZ scheme.

    Authors: We agree that the absence of intermediate steps and numerical checks in the current version limits independent verification of the closed forms for A(n) and B(m). In the revised manuscript we will add a dedicated appendix that outlines the principal momentum-integral reductions performed with the exponential regulators R(x)=exp(x^n) and F(x)=exp(x^m), including the key substitutions and symmetry arguments that lead to the closed expressions. We will also include a new subsection with direct numerical evaluations of the relevant integrals for several integer values (n=2,3,4 and m=2,3,4) and compare them to the analytic results, thereby confirming the accuracy of the closed forms and the subsequent 1/n,1/m expansions. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constants derived independently and substituted into externally known general form

full rationale

The paper states that the all-structure three-loop β-function form in the HCD framework is already known and contains regulator-dependent parameters. It then evaluates the specific constants A(n) and B(m) for the exponential regulators R(x)=e^{x^n} and F(x)=e^{x^m} by direct computation of the relevant integrals, obtains closed forms and asymptotics, and substitutes these into the pre-existing general expressions. This produces explicit β-functions without any reduction of the final result to a quantity defined or fitted inside the present work. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the derivation are present in the described chain. The computation is therefore self-contained against external benchmarks for the general form.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central results rest on the validity of the previously published general three-loop beta-function structure in the HCD+Pauli-Villars scheme and on the assumption that the exponential regulators can be inserted without introducing new divergences or breaking the supersymmetry-preserving properties at three loops.

axioms (2)
  • domain assumption The all-structure three-loop form of the gauge beta functions in the higher covariant derivative framework is known and regulator-dependent only through two constants A and B.
    Invoked in abstract paragraph 2 as the starting point for substitution.
  • domain assumption Exponential regulators R(x)=e^{x^n} and F(x)=e^{x^m} are admissible within the HCD+Pauli-Villars subtraction procedure and preserve the necessary supersymmetry properties.
    Used to define the specific regulators whose parameters are evaluated.

pith-pipeline@v0.9.0 · 5739 in / 1578 out tokens · 51124 ms · 2026-05-18T18:20:21.626668+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 13 internal anchors

  1. [1]

    Introduce a complex regulators, Ip(s)≡ ∫ ∞ 0 xs−1e−xp dx= 1 p Γ (s p ) ,Re(s)>0

    Mellin regularization of the master integral A recurring object is the logarithmically divergent in- tegral Ip ≡ ∫ ∞ 0 dx x e−xp ,(A1) which we define by analytic continuation. Introduce a complex regulators, Ip(s)≡ ∫ ∞ 0 xs−1e−xp dx= 1 p Γ (s p ) ,Re(s)>0. (A2) Expanding nears= 0using Γ(ε) = 1 ε−γE + π2 12ε+O(ε2),(A3) we obtain Ip(s) = 1 s−γE p + π2 12p2...

    work page

  2. [2]

    Evaluation ofA(n)forR(x) =e xn The gauge-sector constant is defined by A≡ ∫ ∞ 0 dxlnx d dx ( 1 R(x) ) , R(x) =e xn .(A6) Thus, A(n) = ∫ ∞ 0 lnx d dx ( e−xn ) dx=−n ∫ ∞ 0 xn−1lnxe−xn dx t=xn −−−−→− ∫ ∞ 0 lnte−tdt=γE =⇒A(n) = γE n , (A7) where integrability at the endpoints (n≥2) ensures the boundary term vanishes

    work page

  3. [3]

    BothA,Bagree with the finite constants entering the compact three-loop HCD formulas [1–3]

    Evaluation ofB(m)forF(x) =e xm The matter-sector constant is B≡ ∫ ∞ 0 dxlnx d dx ( 1 F(x) 2 ) , F(x) =e xm ,(A8) 16 so that B(m) = ∫ ∞ 0 lnx d dx ( e−2xm ) dx=−2m ∫ ∞ 0 xm−1lnxe−2xm dx u=2xm −−−−−→− ∫ ∞ 0 e−u u (lnu−ln 2)du=γE + ln 2 =⇒B(m) = γE + ln 2 m .(A9) Each term in the intermediate line is separately diver- gent; their Mellin-regularized combinati...

    work page

  4. [4]

    Scaled exponential profiles For scaled profilesR(x) =e cxp ,F(x) =e cxq withc > 0, ∫ ∞ 0 xs−1e−cxp dx= 1 p Γ ( s p ) c−s/p= 1 s−γE + lnc p +···, (A10) so that A(p;c) = γE + lnc p , B(q;c) = γE + ln(2c) q .(A11) These constants differ only by finite, scheme-tagginglnc shifts, as expected

    work page

  5. [5]

    Auxiliary identity (finite-part form) A useful auxiliary identity is ∫ ∞ 0 dx x1−s ( 1−e−xp ) = ( 1 s−1 s ) + γE p −s 2p2 +O(s 2), (A12) where the1/spoles cancel explicitly; the finite part is γE/p, consistent with Eq. (A5). B. Pauli–Villars Masses In HCD, Pauli–Villars superfields are introduced to eliminate residual one-loop divergences [16, 21]. Their ...

    work page

  6. [6]

    Three-loop anomalous dimensions forN= 1supersymmetric gauge theories regularized by higher derivatives,

    A. Kazantsev and K. Stepanyantz, “Three-loop anomalous dimensions forN= 1supersymmetric gauge theories regularized by higher derivatives,”JHEP06 (2020) 108,arXiv:2004.00330

    work page arXiv 2020

  7. [7]

    Three-loopβ-functions and two-loop anomalous dimensions for mssm regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme,

    O. V. Haneychuk, V. Y. Shirokova, and K. V. Stepanyantz, “Three-loopβ-functions and two-loop anomalous dimensions for mssm regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme,”JHEP09(2022) 189, arXiv:2207.11944

    work page arXiv 2022

  8. [8]

    General expression for three-loop β-functions ofN= 1supersymmetric theories with multiple gauge couplings regularized by higher covariant derivatives,

    O. Haneychuk, “General expression for three-loop β-functions ofN= 1supersymmetric theories with multiple gauge couplings regularized by higher covariant derivatives,”JETP Lett.121no. 5, (2025) 320–323, arXiv:2501.05174

    work page arXiv 2025

  9. [9]

    Probing the desert using gauge coupling unification,

    J. R. Ellis, S. Kelley, and D. V. Nanopoulos, “Probing the desert using gauge coupling unification,”Phys. Lett. B260(1991) 131–137

    work page 1991

  10. [10]

    Comparison of grand unified theories with electroweak and strong coupling constants measured at lep,

    U. Amaldi, W. de Boer, and H. Furstenau, “Comparison of grand unified theories with electroweak and strong coupling constants measured at lep,”Phys. Lett. B260(1991) 447–455

    work page 1991

  11. [11]

    Implications of precision electroweak experiments formt,ρ0,sin 2θw, and grand unification,

    P. Langacker and M.-x. Luo, “Implications of precision electroweak experiments formt,ρ0,sin 2θw, and grand unification,”Phys. Rev. D44(1991) 817–822

    work page 1991

  12. [12]

    More on the axial anomaly in supersymmetric yang–mills theory,

    D. R. T. Jones, “More on the axial anomaly in supersymmetric yang–mills theory,”Phys. Lett. B123 (1983) 45–47

    work page 1983

  13. [13]

    Beta function in supersymmetric gauge theories: instantons versus traditional approach,

    V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Beta function in supersymmetric gauge theories: instantons versus traditional approach,”Phys. Lett. B166(1986) 329–333. 17

    work page 1986

  14. [14]

    Snowmass Benchmark Points and Three-Loop Running

    I. Jack, D. R. T. Jones, and A. F. Kord, “Three loop soft running, scalar anomalous dimensions and β-functions,”Annals Phys.316(2005) 213–233, hep-ph/0408128

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  15. [15]

    On Gauge Invariant Wilsonian Flows

    D. I. Kazakov, “FiniteN= 1supersymmetric gauge theories,”Phys. Rept.320(1999) 187–260, hep-th/9901063

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  16. [16]

    Exact gell-mann–low function of supersymmetric yang–mills theories from instanton calculus,

    V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Exact gell-mann–low function of supersymmetric yang–mills theories from instanton calculus,”Nucl. Phys. B229(1983) 381–393

    work page 1983

  17. [17]

    Solution of the anomaly puzzle in susy gauge theories and the wilson operator expansion,

    M. A. Shifman and A. I. Vainshtein, “Solution of the anomaly puzzle in susy gauge theories and the wilson operator expansion,”Nucl. Phys. B277(1986) 456–486

    work page 1986

  18. [18]

    F-theory and Orientifolds

    M. A. Shifman, “Exact results in gauge theories: putting supersymmetry to work,”Int. J. Mod. Phys. A 11(1996) 5761–5885,hep-th/9605150

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  19. [19]

    Exact Results on the Space of Vacua of Four Dimensional SUSY Gauge Theories

    N. Seiberg, “Exact results on the space of vacua of four-dimensional susy gauge theories,”Phys. Rev. D49 (1994) 6857–6863,hep-th/9402044

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  20. [20]

    Holomorphy, Rescaling Anomalies and Exact beta Functions in Supersymmetric Gauge Theories

    N. Arkani-Hamed and H. Murayama, “Holomorphy, rescaling anomalies and exactβfunctions in supersymmetric gauge theories,”JHEP06(2000) 030, hep-th/9707133

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  21. [21]

    The all-loop nsvz relation inN= 1 sqed withnf flavors regularized by higher derivatives,

    K. V. Stepanyantz, “The all-loop nsvz relation inN= 1 sqed withnf flavors regularized by higher derivatives,” Eur. Phys. J. C80(2020) 911,arXiv:2007.11935

    work page arXiv 2020

  22. [22]

    $N=1$ supersymmetry and the three loop gauge beta function

    I. Jack, D. R. T. Jones, and C. G. North, “Scheme dependence and the nsvzβ-function,”Phys. Lett. B 386(1996) 138–142,hep-ph/9606323

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  23. [23]

    Scheme dependence and the NSVZ $\beta$-function

    I. Jack, D. R. T. Jones, and C. G. North, “Scheme dependence and the nsvzβ-function: A comment,” Nucl. Phys. B486(1997) 479–499,hep-ph/9609325

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  24. [24]

    Invariant regularization of gauge theories,

    A. A. Slavnov, “Invariant regularization of gauge theories,”Nucl. Phys. B31(1971) 301–315

    work page 1971

  25. [25]

    Invariant regularization of nonlinear chiral theories,

    A. A. Slavnov, “Invariant regularization of nonlinear chiral theories,”Theor. Math. Phys.13(1972) 174–177

    work page 1972

  26. [26]

    The pauli–villars regularization for nonabelian gauge theories,

    A. A. Slavnov, “The pauli–villars regularization for nonabelian gauge theories,”Theor. Math. Phys.33 (1977) 977–981

    work page 1977

  27. [27]

    NSVZ scheme with the higher derivative regularization for N=1 SQED

    A. L. Kataev and K. V. Stepanyantz, “Nsvz scheme with the higher derivative regularization forN= 1 sqed,”Nucl. Phys. B875(2013) 459–479, arXiv:1305.7094

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  28. [28]

    E. T. Whittaker and G. N. Watson,A Course of Modern Analysis. Cambridge University Press, 4 ed., 1927

    work page 1927

  29. [29]

    Supersymmetric dimensional regularization via dimensional reduction,

    W. Siegel, “Supersymmetric dimensional regularization via dimensional reduction,”Phys. Lett. B84(1979) 193–196

    work page 1979

  30. [30]

    Resurgence of the cusp anomalous dimension,

    I. Aniceto, M. Honda, and R. Schiappa, “Resurgence of the cusp anomalous dimension,”Phys. Rept.932(2021) 1–81

    work page 2021

  31. [31]

    What is qft? resurgent trans-series, lefschetz thimbles, and new exact saddles,

    G. V. Dunne and M. Ünsal, “What is qft? resurgent trans-series, lefschetz thimbles, and new exact saddles,” PoS LATTICE2015(2016) 010

    work page 2016

  32. [32]

    Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

    M. Mariño, “Nonperturbative effects and nonperturbative definitions in matrix models and topological strings,”JHEP12(2008) 114, arXiv:0805.3033

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  33. [33]

    Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories

    D. Anselmi, D. Z. Freedman, M. T. Grisaru, and A. A. Johansen, “Nonperturbative formulas for central functions of supersymmetric gauge theories,”Nucl. Phys. B526(1998) 543–571,hep-th/9708042

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  34. [34]

    Superspace, or One thousand and one lessons in supersymmetry

    S. J. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, “Superspace, or one thousand and one lessons in supersymmetry,”Front. Phys.58(1983) 1–548, hep-th/0108200

    work page internal anchor Pith review Pith/arXiv arXiv 1983

  35. [35]

    P. C. West,Introduction to Supersymmetry and Supergravity. World Scientific, 1990

    work page 1990

  36. [36]

    I. L. Buchbinder and S. M. Kuzenko,Ideas and Methods of Supersymmetry and Supergravity. IOP Publishing, 1998

    work page 1998

  37. [37]

    V. A. Smirnov,Evaluating Feynman Integrals. Springer, 2004

    work page 2004

  38. [38]

    Two-loop renormalization group equations for soft SUSY breaking scalar interactions: supergraph method

    Y. Yamada, “Two-loop renormalization group equations for soft susy-breaking scalar interactions: Supergraph method,”Phys. Rev. D50(1994) 3537–3545, hep-ph/9401241

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  39. [39]

    Three-loopβ-functions forN= 1 supersymmetric qed regularized by higher derivatives,

    D. Korneev, D. Plotnikov, K. Stepanyantz, and N. Tereshina, “Three-loopβ-functions forN= 1 supersymmetric qed regularized by higher derivatives,” JHEP10(2021) 046,arXiv:2108.10366

    work page arXiv 2021

  40. [40]

    Volkov-Akulov theory and D-branes

    D. I. Kazakov, “Finiten= 1susy gauge field theories,” Surveys in High Energy Physics13(1998) 105–146, arXiv:hep-th/9705118

    work page internal anchor Pith review Pith/arXiv arXiv 1998