Three-loop contribution of the Faddeev-Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories and the NSVZ relation

I.E.Shirokov; K.V.Stepanyantz; M.D.Kuzmichev; N.P.Meshcheriakov; S.V.Novgorodtsev

REVIEW 3 major objections 4 minor 70 references

Three-loop contribution of the Faddeev-Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories and the NSVZ relation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Three-loop supergraphs containing Faddeev–Popov ghost loops satisfy the NSVZ beta-function relation term by term, giving the first nontrivial check of the ghost anomalous-dimension contribution in a scheme-dependent approximation.

desk verdict A real three-loop ghost-loop beta-function calculation that checks the gamma_c term of NSVZ; the main caveat is reliance on the unproven Ref. [55] algorithm, but the comparison is not circular. read the letter →

arxiv 1908.10586 v2 pith:FXKKSLLW submitted 2019-08-28 hep-th

M.D.Kuzmichev , N.P.Meshcheriakov , S.V.Novgorodtsev , I.E.Shirokov , K.V.Stepanyantz This is my paper · ORCID

classification hep-th

keywords N=1supersymmetricgaugetheoryNSVZrelationbeta-functionFaddeev-Popovghostshighercovariantderivativeregularizationanomalousdimensionnonlinearrenormalizationdoubletotalderivatives

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 checks a previously untested piece of the NSVZ relation, an exact formula connecting the $\beta$-function of an ${\cal N}=1$ supersymmetric gauge theory to the anomalous dimensions of its quantum superfields. Using the higher covariant derivative regularization and a recently proposed algorithm, the authors compute all three-loop contributions to $\beta/\alpha_0^2$ coming from supergraphs with Faddeev–Popov ghost loops and write them as integrals of double total derivatives. Cutting internal ghost and matter lines in those integrands reproduces, term by term, the two-loop ghost anomalous dimension $\gamma_c$ and the relevant matter-field pieces. The equality holds at the level of loop integrals, so the NSVZ relation in its alternative form is verified for the ghost sector in a scheme-dependent approximation, and the nonlinear renormalization of the quantum gauge superfield is shown to be necessary.

What carries the argument

The object that carries the argument is the representation of every $\beta$-function contribution as an integral of double total momentum derivatives. The algorithm from earlier work constructs this representation by taking a vacuum supergraph, inserting a factor $\theta^4(v_B)^2$ at a full-superspace point, applying the $D$-algebra, and replacing marked propagator delta functions by second derivatives with respect to loop momenta. The identity $\partial^2/\partial Q_\mu^2\,(1/Q^2) = -4\pi^2\delta^4(Q)$ then cuts internal lines, generating the two-point superdiagrams that define $\gamma_c$ and $(\gamma_\varphi)^i_j$. Matching each $\beta$-function graph $B_1,\dots,B_{13}$ to such cuts produces Eqs. (34)–(45), whose sum is exactly Eq. (46). This machinery turns the $\beta$-function calculation into an algebraic check of the NSVZ relation at integrand level, with the remaining integrals left unevaluated.

What would settle it

Recompute one of the thirteen graphs, say $B_{13}$, by direct superspace Feynman rules without the algorithm and keep all longitudinal pieces; if the result differs from the double-derivative expression in Eqs. (62)–(63) by anything beyond terms that vanish after $d/d\ln\Lambda$, then Eq. (46) is not an identity at the loop-integral level and the NSVZ check fails.

Watch

Extended reading notes

Core claim

The central result is Eq. (46): for the sum of all three-loop supergraphs containing ghost loops, $$\$\Delta$\!\left(\frac{\$\beta$}{\$alpha_0^{2}$}\right) = \frac{C_2}{\pi}\,\$\Delta$\gamma_c - \frac{1}{2\pi r}\,C(R)^i_j\,(\$\Delta$\gamma_\varphi)^j_i + \dots,$$ where the dots stand for terms produced by cuts of internal gauge lines. Each equality (34)–(45) that builds this sum holds before the momentum integrals are evaluated, as an identity among loop integrands. The paper establishes that the NSVZ equation written in the form $\beta/\alpha^2 = -(1/2\pi)\bigl(3C_2 - T(R) - 2C_2\gamma_c - 2C_2\gamma_V + C(R)^i_j(\gamma_\varphi)^j_i/r\bigr)$ is satisfied by the considered ghost-loop contributions for renormalization group functions defined in terms of bare couplings. This is the first verification of the $\gamma_c$ term in an order where the scheme dependence is essential, and it confirms that the nonlinear renormalization of the quantum gauge superfield must be included.

Load-bearing premise

The calculation leans on a previously proposed method, cited rather than re-derived, that rewrites every beta-function graph as a derivative of a loop integral; if that rewrite is wrong, the term-by-term match with anomalous dimensions collapses.

Editorial extensions

If this is right

For the ghost-loop sector, the three-loop $\beta$-function is fixed by two-loop ghost and matter anomalous dimensions; no further integration is needed to verify those terms.
The previously unverified $\gamma_c$ term in the NSVZ equation is confirmed in a scheme-dependent approximation, using renormalization group functions defined through bare couplings.
The equality holds at the level of loop integrals, which means regularization dependence cancels inside the integrands rather than only after integration.
The nonlinear renormalization of the quantum gauge superfield is required: the parameter $y_0$ entering $\gamma_c$ is essential for the renormalization group equations to close.
The result supports the program of deriving the NSVZ equation by summing singular contributions from cuts of internal lines in the higher-covariant-derivative regularization.

Reading between the lines

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

The same cut-and-match logic could be applied to the omitted gauge-line terms: including purely gauge supergraphs should make the sum transversal and reproduce the $2C_2\gamma_V$ contribution, completing Eq. (46) for all three-loop graphs without evaluating integrals.
Because Eq. (46) holds for arbitrary regulator functions $R$ and Pauli–Villars masses, the method turns the scheme dependence of NSVZ into an algebraic identity; testing another regulator, for instance $R(x)=1+x^n$ with a different $n$, would only change finite parts and should leave the relation intact.
The appearance of $y_0$ suggests that any all-order formulation of the bare-coupling NSVZ relation should treat the full set of bare parameters, including nonlinear-renormalization parameters, as part of the scheme data.
A natural next step is to compute the purely gauge three-loop supergraphs with the same algorithm; if the omitted dots in Eq. (46) assemble into $2C_2\gamma_V$, the full three-loop NSVZ check would be complete.

Share X Bluesky LinkedIn Reddit HN

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

Nothing here is a breakthrough, but the calculation is real. The authors compute the three-loop ghost-loop contribution to the beta function in N=1 SYM with higher covariant derivative regularization, and match it against the two-loop ghost anomalous dimension from their earlier work. The check is nontrivial: scheme dependence enters at this order, and the nonlinear renormalization parameter y0 appears in the game. They also put explicit integral expressions for all thirteen supergraphs in Appendix A, so the calculation can be followed.

The stress-test note goes too far when it calls this circular. The gamma_c side is not computed with the algorithm of Ref. [55]; it is taken from Ref. [58], a direct calculation of the ghost two-point function. So Eqs. (34)-(45) are a genuine consistency test between the algorithm's beta-side output and an independent ghost-side calculation. The algorithm is cited, not re-derived, and the paper itself says it is testing that algorithm. That is a real dependency, but it is not fitting the result to the target.

Soft spots, in proportion. First, the matter term (50) is ill-defined until all matter graphs are summed; the authors admit it. Second, the gamma_V terms are omitted, so Eq. (46) only checks the gamma_c part of NSVZ. Third, the two-loop gamma_c is only fully known in the gauge y0=0, because the nonlinear renormalization parameters were only included at one loop. These limitations are stated plainly, not hidden. For the claim being made—that the ghost-loop contribution is consistent with the NSVZ relation in this scheme—the evidence is solid conditional on the Ref. [55] algorithm.

The paper is for people working on the NSVZ program and higher covariant derivative regularization. It is a subfield-level step, not an epochal result. But it is a careful, honest calculation that deserves a serious referee rather than a desk rejection.

Referee Report

3 major / 4 minor

Summary. The paper computes the three-loop contribution to the beta-function of N=1 supersymmetric gauge theories, regularized by higher covariant derivatives, from supergraphs containing Faddeev–Popov ghost loops. Using an algorithm proposed in Ref. [55], the authors express each such contribution as an integral of double total derivatives in momentum space. They then cut internal ghost and matter propagators using identity (30) and match the resulting terms to the two-loop ghost anomalous dimension of Ref. [58] and to matter-superfield anomalous-dimension contributions. The main result, Eq. (46), states that the ghost-loop part of beta/alpha_0^2 equals (C2/pi) Delta gamma_c minus (1/(2 pi r)) C(R) Delta gamma_phi, plus omitted gauge-line terms, at the level of loop integrals. The paper claims this verifies the NSVZ relation in the form of Eq. (3) for the considered ghost-loop supergraphs and confirms the necessity of nonlinear renormalization of the quantum gauge superfield.

Significance. If the calculation is correct, it provides a nontrivial, scheme-dependent test of the NSVZ relation in a higher-derivative regularization, going beyond previous checks and specifically addressing the gamma_c term that had not been verified before. The explicit integral expressions for all thirteen supergraphs in Appendix A are a useful resource for independent checks, and the matching of the beta-function cuts to gamma_c and gamma_phi is conceptually interesting. The paper also gives due credit to the machinery of Refs. [2] and [55] and is honest about the omitted gamma_V terms. However, the central result is conditional on the correctness of the algorithm of Ref. [55], which is cited rather than derived or proved here, and the matter contribution in Eq. (50) is admitted to be ill-defined. The significance is therefore real but more limited than the abstract suggests.

major comments (3)

[Section 3 and Eqs. (34)–(45)] The central derivation relies entirely on the algorithm of Ref. [55], described in steps 1–6 of Sect. 3, but that algorithm is neither proved nor independently checked in this manuscript. The beta-function contributions in Appendix A and the matching identities (34)–(45) are both outputs of this algorithm, so the comparison in Eq. (46) does not independently confirm the NSVZ relation unless the algorithm itself is established. Please state the precise theorem from Ref. [55] (including its hypotheses and proof or a detailed derivation) and explain why it applies to the graphs in Fig. 1. Without this, Eq. (46) is a consistency check between two computations that share the same untested input.
[Section 4, Eq. (50)] The matter-field contribution in Eq. (50) is explicitly admitted to be not well-defined, yet it appears in the main result Eq. (46) and is used to claim verification of the NSVZ relation. As written, the paper verifies only the gamma_c part of Eq. (3), with gamma_V omitted and the gamma_phi term uncomputed. The abstract and conclusion state that the NSVZ equation is satisfied in the considered approximation, which overstates what is actually shown. The authors should either compute the well-defined sum of the relevant matter superdiagrams or restrict their claim to the gamma_c contribution and explain why the ill-defined terms in Eq. (50) are expected to cancel against other graphs.
[Section 4, Eqs. (34)–(45)] The matching identities are asserted rather than derived. For example, Eq. (42) combines the sum B9+B10 with contributions from A14, A15, M1, M2, M3, and M4, including the subtleties of double counting and the factor 1/2 described in Appendix A; Eqs. (37), (43), and (44) invoke the logarithm expansion (32) for non-1PI cuts. A reader cannot verify the signs, coefficients, or treatment of non-1PI contributions from Appendix A alone. Please provide at least one representative derivation in detail (e.g., for B1 and for B9+B10) and state explicitly how the double total derivatives act to produce the cut diagrams and the logarithms in Eq. (32).

minor comments (4)

[Appendix A, Eqs. (59)–(60)] There are typographical errors in the integration measures: "d4Q/(2π4)" should read "d4Q/(2π)^4", and similarly for d4K and d4L.
[Section 4, around Eq. (48)] The statement that the two-loop result for gamma_c can be written only in the gauge y0=0, apart from a one-loop y0 term, is presented briefly. Clarifying which terms in Eq. (48) are complete would help the reader assess the role of the nonlinear renormalization parameters.
[Fig. 1 caption] The factor 1/2 for supergraphs containing two ghost loops is mentioned in the text but not in the caption; adding it to the caption would prevent misreading of the figure.
[References] Reference [55] is cited as an arXiv preprint (1908.04108). If a published or updated version exists, the citation should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NSVZ check compares independently computed beta-function contributions with anomalous dimensions, with no fitted parameters or definitions that presuppose the target relation.

full rationale

The paper's central result, Eq. (46), is a comparison between three-loop ghost-loop contributions to beta/alpha_0^2 (computed in Appendix A via the algorithm of Ref. [55]) and the two-loop ghost anomalous dimension Delta gamma_c (Eq. (47), from Ref. [58]) plus matter contributions (Eq. (50)). The algorithm is self-published by the same group, but it is explicitly described in Sect. 3 and is not fitted to the NSVZ relation; an error in it would make the matching identities (34)-(45) fail rather than force them to hold. The anomalous dimension side is taken from a separate calculation (Ref. [58]), and the nontrivial equalities (34)-(45) are stated as loop-integral identities that could in principle be violated. No parameter is adjusted to make Eq. (46) come out; the only input from the target relation (3) is the choice of which line cuts to compare with gamma_c and gamma_phi, which is a legitimate testing strategy. The paper explicitly omits gamma_V terms and notes that Eq. (50) is not well-defined until all supergraphs are summed, so the verification is partial; but incompleteness and reliance on a cited algorithm are correctness risks, not circularity. The derivation is therefore self-contained in the sense required by the circularity test.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new particles or forces are introduced; the paper is a quantum field theory computation within an existing framework.

free parameters (3)

Pauli-Villars mass coefficients a_phi, a
Introduced in Eq. (15) to set the regulator masses proportional to the cutoff Lambda. They are scheme parameters, not fitted to data, and appear in the ghost anomalous dimension as ln a_phi and ln a.
Gauge-fixing parameter xi0
Appears in the gauge-fixing action (12) and all beta-function and gamma_c expressions. It is a free parameter of the gauge-fixing scheme.
Nonlinear renormalization parameter y0
Parameter of the nonlinear renormalization function F(V) in Eq. (5). It appears in Eq. (17) and in the ghost anomalous dimension. Its necessity is one of the results, not a fitted number.

assumptions (4)

ad hoc to paper The algorithm of Ref. [55] correctly computes beta-function contributions from vacuum supergraphs as integrals of double total derivatives.
Invoked in Sect. 3 and used for all supergraphs B1-B13. It is from a companion paper by the same group and is not re-derived here.
domain assumption The higher covariant derivative regularization with Pauli-Villars fields is a valid regularization for N=1 supersymmetric gauge theories.
Used throughout; the Pauli-Villars fields are introduced in Sect. 2 following Ref. [55].
domain assumption The non-renormalization theorem for the triple gauge-ghost vertices (Ref. [2]) is valid, allowing the NSVZ equation to be rewritten as Eq. (3).
Stated in the Introduction; this rewriting is the basis for comparing beta with gamma_c and gamma_V.
standard math Standard superspace Feynman rules and D-algebra manipulations are correct.
Used implicitly in all calculations.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Three-loop contribution of the Faddeev-Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories and the NSVZ relation." pith.science (2026). https://pith.science/paper/FXKKSLLW

@misc{pith2026190810586,
  author       = {Pith},
  title        = {Pith review of: Three-loop contribution of the Faddeev-Popov ghosts to the $\beta$-function of $\cal N=1$ supersymmetric gauge theories and the NSVZ relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXKKSLLW}},
  note         = {Machine review of arXiv:1908.10586}
}

abstract

We find the three-loop contribution to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev--Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $\beta$-function is compared with the two-loop anomalous dimension of the Faddeev--Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $\beta$-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.

Figures

Figures reproduced from arXiv: 1908.10586 by the authors.

**Figure 2.** Superdiagrams contributing to the one-loop polar [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** One- and two-loop superdiagrams contributing to t [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Two-loop superdiagrams with a ghost loop contribu [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 68 canonical work pages

[55]

The $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

K. V. Stepanyantz, “The β -function of N = 1 supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double tot al derivatives,” arXiv:1908.04108 [hep-th]

work page Pith review arXiv 1908
[58]

A. E. Kazantsev, M. D. Kuzmichev, N. P. Meshcheriakov, S . V. Novgorodtsev, I. E. Shi- rokov, M. B. Skoptsov and K. V. Stepanyantz, JHEP 1806 (2018) 020

work page 2018
[2]

K. V. Stepanyantz, Nucl. Phys. B 909 (2016) 316

work page 2016
[1]

M. T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B 159 (1979) 429

work page 1979
[3]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, Nucl. Phys. B 229 (1983) 381

work page 1983
[4]

D. R. T. Jones, Phys. Lett. 123B (1983) 45

work page 1983
[5]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, Phys. Lett. 166B (1986) 329 [Sov. J. Nucl. Phys. 43 (1986) 294] [Yad. Fiz. 43 (1986) 459]

work page 1986
[6]

M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 277 (1986) 456 [Sov. Phys. JETP 64 (1986) 428] [Zh. Eksp. Teor. Fiz. 91 (1986) 723]

work page 1986

Show all 70 references

[7]

M. A. Shifman and A. I. Vainshtein, In *Shifman, M.A.: ITE P lectures on particle physics and ﬁeld theory, vol. 2* 485-647 [hep-th/9902018]

arXiv
[8]

M. A. Shifman, Int. J. Mod. Phys. A 14 (1999) 5017

1999
[9]

Shifman, Int

M. Shifman, Int. J. Mod. Phys. A 33 (2018) no.12, 1830009

2018
[10]

I. L. Buchbinder and K. V. Stepanyantz, Nucl. Phys. B 883 (2014) 20

2014
[11]

Galperin, E

A. Galperin, E. Ivanov, V. Ogievetsky and E. Sokatchev, JETP Lett. 40 (1984) 912 [Pisma Zh. Eksp. Teor. Fiz. 40 (1984) 155]

1984
[12]

Galperin, E

A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E . Sokatchev, Class. Quant. Grav. 1 (1984) 469 Erratum: [Class. Quant. Grav. 2 (1985) 127]

1984
[13]

Harmonic superspace

A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. So katchev, “Harmonic superspace”, Cambridge, UK: Univ. Pr. (2001) 306 p

2001
[14]

E. I. Buchbinder, B. A. Ovrut, I. L. Buchbinder, E. A. Iva nov and S. M. Kuzenko, Phys. Part. Nucl. 32 (2001) 641-674 [Fiz. Elem. Chast. Atom. Yadra 32 (2001) 1222-1264]

2001
[15]

I. L. Buchbinder, N. G. Pletnev and K. V. Stepanyantz, Ph ys. Lett. B 751 (2015) 434

2015
[16]

M. T. Grisaru and W. Siegel, Nucl. Phys. B 201 (1982) 292 Erratum: [Nucl. Phys. B 206 (1982) 496]

1982
[17]

P. S. Howe, K. S. Stelle and P. K. Townsend, Nucl. Phys. B 236 (1984) 125. 18

1984
[18]

I. L. Buchbinder, S. M. Kuzenko and B. A. Ovrut, Phys. Let t. B 433 (1998) 335

1998
[19]

Mandelstam, Nucl

S. Mandelstam, Nucl. Phys. B 213 (1983) 149

1983
[20]

Brink, O

L. Brink, O. Lindgren and B. E. W. Nilsson, Nucl. Phys. B 212 (1983) 401

1983
[21]

L. V. Avdeev, O. V. Tarasov and A. A. Vladimirov, Phys. Le tt. 96B (1980) 94

1980
[22]

I. O. Goriachuk, A. L. Kataev and K. V. Stepanyantz, Phys . Lett. B 785 (2018) 561

2018
[23]

I. Jack, D. R. T. Jones and C. G. North, Phys. Lett. B 386 (1996) 138

1996
[24]

I. Jack, D. R. T. Jones and C. G. North, Nucl. Phys. B 486 (1997) 479

1997
[25]

I. Jack, D. R. T. Jones and A. Pickering, Phys. Lett. B 435 (1998) 61

1998
[26]

R. V. Harlander, D. R. T. Jones, P. Kant, L. Mihaila and M. Steinhauser, JHEP 0612 (2006) 024

2006
[27]

A. A. Slavnov, Nucl. Phys. B 31 (1971) 301

1971
[28]

A. A. Slavnov, Theor.Math.Phys. 13 (1972) 1064 [Teor. Mat. Fiz. 13 (1972) 174]

1972
[29]

A. A. Slavnov, Theor. Math. Phys. 33 (1977) 977 [Teor. Mat. Fiz. 33 (1977) 210]

1977
[30]

V. K. Krivoshchekov, Theor. Math. Phys. 36 (1978) 745 [Teor. Mat. Fiz. 36 (1978) 291]

1978
[31]

P. C. West, Nucl. Phys. B 268 (1986) 113

1986
[32]

A. L. Kataev and K. V. Stepanyantz, Nucl. Phys. B 875 (2013) 459

2013
[33]

A. L. Kataev and K. V. Stepanyantz, Phys. Lett. B 730 (2014) 184

2014
[34]

A. L. Kataev and K. V. Stepanyantz, Theor. Math. Phys. 181 (2014) 1531

2014
[35]

Siegel, Phys

W. Siegel, Phys. Lett. 84B (1979) 193

1979
[36]

A. A. Soloshenko and K. V. Stepanyantz, Theor. Math. Phy s. 140 (2004) 1264 [Teor. Mat. Fiz. 140 (2004) 437]

2004
[37]

A. V. Smilga and A. Vainshtein, Nucl. Phys. B 704 (2005) 445

2005
[38]

K. V. Stepanyantz, Nucl. Phys. B 852 (2011) 71

2011
[39]

K. V. Stepanyantz, JHEP 1408 (2014) 096

2014
[40]

A. L. Kataev, A. E. Kazantsev and K. V. Stepanyantz, Eur. Phys. J. C 79 (2019) no.6, 477

2019
[41]

Hisano and M

J. Hisano and M. A. Shifman, Phys. Rev. D 56 (1997) 5475

1997
[42]

Jack and D

I. Jack and D. R. T. Jones, Phys. Lett. B 415 (1997) 383

1997
[43]

L. V. Avdeev, D. I. Kazakov and I. N. Kondrashuk, Nucl. Ph ys. B 510 (1998) 289

1998
[44]

I. V. Nartsev and K. V. Stepanyantz, JHEP 1704 (2017) 047

2017
[45]

I. V. Nartsev and K. V. Stepanyantz, JETP Lett. 105 (2017) no.2, 69

2017
[46]

Shifman and K

M. Shifman and K. Stepanyantz, Phys. Rev. Lett. 114 (2015) no.5, 051601. 19

2015
[47]

Shifman and K

M. Shifman and K. V. Stepanyantz, Phys. Rev. D 91 (2015) 105008

2015
[48]

A. L. Kataev, A. E. Kazantsev and K. V. Stepanyantz, Nucl . Phys. B 926 (2018) 295

2018
[49]

S. S. Aleshin, A. L. Kataev and K. V. Stepanyantz, JHEP 1903 (2019) 196

2019
[50]

A. B. Pimenov, E. S. Shevtsova and K. V. Stepanyantz, Phy s. Lett. B 686 (2010) 293

2010
[51]

K. V. Stepanyantz, Proc. Steklov Inst. Math. 272 (2011) no.1, 256

2011
[52]

K. V. Stepanyantz, “Factorization of integrals deﬁnin g the two-loop β -function for the general renormalizable N=1 SYM theory, regularized by the h igher covariant derivatives, into integrals of double total derivatives,” arXiv:1108.1 491 [hep-th]
[53]

V. Y. Shakhmanov and K. V. Stepanyantz, Nucl. Phys. B 920 (2017) 345

2017
[54]

A. E. Kazantsev, V. Y. Shakhmanov and K. V. Stepanyantz, JHEP 1804 (2018) 130

2018
[56]

V. Y. Shakhmanov and K. V. Stepanyantz, Phys. Lett. B 776 (2018) 417

2018
[57]

A. A. Slavnov and K. V. Stepanyantz, Theor. Math. Phys. 139 (2004) 599 [Teor. Mat. Fiz. 139 (2004) 179]

2004
[59]

Piguet and K

O. Piguet and K. Sibold, Nucl. Phys. B 197 (1982) 257

1982
[60]

Piguet and K

O. Piguet and K. Sibold, Nucl. Phys. B 197 (1982) 272

1982
[61]

I. V. Tyutin, Yad. Fiz. 37 (1983) 761

1983
[62]

J. W. Juer and D. Storey, Phys. Lett. 119B (1982) 125

1982
[63]

J. W. Juer and D. Storey, Nucl. Phys. B 216 (1983) 185

1983
[64]

Dynamical theory of groups and ﬁelds,

B. S. DeWitt, “Dynamical theory of groups and ﬁelds,” Go rdon and Breach, New York, 1965

1965
[65]

L. F. Abbott, Nucl. Phys. B 185 (1981) 189

1981
[66]

L. F. Abbott, Acta Phys. Polon. B 13 (1982) 33

1982
[67]

S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Front. Phys. 58 (1983) 1 [hep-th/0108200]

1983 arXiv
[68]

S. S. Aleshin, A. E. Kazantsev, M. B. Skoptsov and K. V. St epanyantz, JHEP 1605 (2016) 014

2016
[69]

L. D. Faddeev and A. A. Slavnov, Front. Phys. 50 (1980) 1 [Front. Phys. (1991) 1]

1980
[70]

A. E. Kazantsev, M. B. Skoptsov and K. V. Stepanyantz, Mo d. Phys. Lett. A 32 (2017) no.36, 1750194. 20

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [55]

The $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

K. V. Stepanyantz, “The β -function of N = 1 supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double tot al derivatives,” arXiv:1908.04108 [hep-th]

work page Pith review arXiv 1908

[2] [58]

A. E. Kazantsev, M. D. Kuzmichev, N. P. Meshcheriakov, S . V. Novgorodtsev, I. E. Shi- rokov, M. B. Skoptsov and K. V. Stepanyantz, JHEP 1806 (2018) 020

work page 2018

[3] [2]

K. V. Stepanyantz, Nucl. Phys. B 909 (2016) 316

work page 2016

[4] [1]

M. T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B 159 (1979) 429

work page 1979

[5] [3]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, Nucl. Phys. B 229 (1983) 381

work page 1983

[6] [4]

D. R. T. Jones, Phys. Lett. 123B (1983) 45

work page 1983

[7] [5]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, Phys. Lett. 166B (1986) 329 [Sov. J. Nucl. Phys. 43 (1986) 294] [Yad. Fiz. 43 (1986) 459]

work page 1986

[8] [6]

M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 277 (1986) 456 [Sov. Phys. JETP 64 (1986) 428] [Zh. Eksp. Teor. Fiz. 91 (1986) 723]

work page 1986