Pith. sign in

REVIEW 5 major objections 5 minor 2 cited by

Gauge Choices, Infrared Pitfalls, and Thermal Effects in Effective Potentials

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper claims that adding the multiplicative anomaly to the one-loop effective potential computed in the Fermi gauge makes it independent of the gauge-fixing parameter at every field value, not just at extrema, while also improving its…

desk verdict The heat-kernel parts are clean, but the anomaly cancellation in the Fermi gauge is incomplete: Eq. 2.17 does not match the Landau-gauge result and residual ξ-dependence remains. read the letter →

arxiv 2507.22706 v1 pith:OH533GVH submitted 2025-07-30 hep-th hep-exhep-ph

Debanjan Balui , Tisa Biswas , Joydeep Chakrabortty , Debmalya Dey , Christoph Englert , Subhendra Mohanty This is my paper · ORCID
classification hep-thhep-exhep-ph PACS 11.15.-q11.10.Wx
keywords effectivepotentialgaugedependencemultiplicativeanomalyzeta-functionregularizationheatkernelinfrareddivergencefinitetemperatureNielsenidentity
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 is trying to show that two longstanding problems of the one-loop effective potential—its dependence on the gauge-fixing parameter and its infrared divergence when Goldstone masses vanish—have a common cause and a common cure. The cause, in the Fermi gauge (a one-parameter family of gauge choices), is that the standard calculation factorises the determinant of the fluctuation operator into separate pieces even though zeta-function regularisation does not allow that factorisation. The cure is a "multiplicative anomaly" correction, and once it is added the effective potential becomes independent of the gauge parameter at every field value, not only at its extrema. The same correction turns the infrared behaviour in the massless Goldstone limit into the milder Landau-gauge behaviour. The paper shows the same mechanism works for the Standard Model Higgs potential and, through a heat-kernel version of the calculation, at finite temperature.

What carries the argument

The machinery is the multiplicative anomaly of zeta-regularised determinants of elliptic operators: $\log \mathrm{Det}[\Delta_1\Delta_2] = \log \mathrm{Det}[\Delta_1] + \log \mathrm{Det}[\Delta_2] + A[\Delta_1,\Delta_2]$, where $A$ is nonzero because the zeta trace does not factorise. For the Goldstone-fluctuation operators of scalar QED in $d=4$, the relevant anomaly density is $a[M^2_{G+},M^2_{G-}](\xi) = \frac{1}{64\pi^2}(-4\xi M^2_\chi M^2_A)$, and adding half of it to the naive one-loop potential cancels the $\xi$ dependence. The companion mechanism is the heat-kernel expansion, which keeps the full fluctuation operator intact; there the gauge parameter survives only in total derivatives such as $\partial^4$ and $\partial^2 M^2_A$, which vanish for the constant backgrounds used in effective potentials.

What would settle it

A direct numerical evaluation of $\log\det$ of the full fluctuation operator in the Fermi gauge at fixed $\xi>0$ could settle the claim: if it disagrees with the anomaly-corrected potential at any field value, or if residual $\xi$-dependence appears away from the extrema, then the anomaly term is either incomplete or double counting. A sharper check is to compute the difference between the two sides of the identity at subleading order in $M^2_\chi$, since the paper's cancellation is exhibited only at the $M^2_\chi \log M^2_\chi$ level.

Watch

Extended reading notes

Core claim

The central claim is that the one-loop effective potential computed in the Fermi gauge, augmented by the multiplicative anomaly for non-factorising elliptic operators, is exactly gauge independent for all field values and reproduces the Landau-gauge infrared behaviour in the vanishing-Goldstone limit. In massive scalar quantum electrodynamics the $\xi$-dependent terms from the Goldstone modes cancel against the anomaly density $a[M^2_{G+},M^2_{G-}](\xi) = \frac{1}{64\pi^2}(-4\xi M^2_\chi M^2_A)$, so the Nielsen identity reduces to a trivial statement with vanishing coefficient $C^{(1)}$. The same cancellation is derived for the Standard Model electroweak sector, with independent anomaly contributions for the $W$ and $B$ sectors. When the computation is repeated with the heat-kernel method, the gauge parameter appears only in total-derivative terms that vanish for constant backgrounds, which yields the Landau-gauge result without any anomaly bookkeeping. At finite temperature the heat-kernel trace over Matsubara frequencies gives a gauge-independent thermal potential that matches the Landau-gauge functional result at the displayed orders.

Load-bearing premise

The load-bearing premise is that the standard factorised computation of the one-loop determinant genuinely omits the multiplicative anomaly, so adding the anomaly term corrects the result rather than double-counting a determinant that the diagonalised $G_\pm$ masses already encode.

Editorial extensions

If this is right

  • An anomaly-corrected Fermi-gauge potential is meaningful at every field value, so slopes and rolling regions—the input for inflation and phase-transition studies—are not gauge artefacts.
  • The infrared behaviour near vanishing Goldstone mass becomes the Landau-gauge form $\sim M_\chi^4 \log(M_\chi^2/\mu^2)$, so the potential and its first derivative are infrared-safe without an additional resummation step.
  • The same cancellation makes the one-loop Nielsen coefficient vanish, turning the Nielsen identity into a statement of exact gauge invariance rather than a recipe for field redefinitions.
  • The heat-kernel method gives a shortcut: start in the Fermi gauge, discard total derivatives, and obtain the Landau-gauge potential directly; this also works for the Standard Model.
  • At finite temperature, the thermal effective potential computed from the heat-kernel trace is gauge independent and agrees with the Landau-gauge functional result to the order shown.

Reading between the lines

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

  • Editorial extension: if the one-loop cancellation persists at higher loops, the standard practice of applying Nielsen identities or field redefinitions in phase-transition calculations could be replaced by directly computing the full fluctuation determinant; the paper itself demonstrates only one loop.
  • Editorial extension: the same anomaly argument suggests a practical diagnostic for other gauge choices—if a factorised determinant in a non-linear or $R_\xi$ gauge shows $\xi$-dependence, one can test whether an anomaly density of the form of Eq. 2.16 removes it, provided the fluctuation operator remains elliptic.
  • Editorial extension: at finite temperature the heat-kernel formulation implies a numerical recipe—compute the Matsubara trace of the full fluctuation operator rather than factorising it—which could be checked in high-temperature expansions beyond the orders shown here.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper studies the one-loop effective potential in the Fermi gauge for massive scalar QED and for the Standard Model, focusing on its gauge dependence and IR behaviour. The central claim is that including the multiplicative anomaly of zeta-regularized functional determinants removes the gauge dependence of the potential and improves its IR behaviour to match the Landau-gauge result. The authors then present a heat-kernel computation that yields the Landau-gauge result directly and extend the discussion to finite temperature.

Significance. If the central claim were correct, the paper would provide a practical prescription for computing gauge-invariant effective potentials without invoking Nielsen field redefinitions, with direct applications to phase transitions, inflation, and vacuum stability. The work is explicit: it presents closed-form expressions for the Abelian Higgs model and the Standard Model, and it offers the heat-kernel method as an independent cross-check. The finite-temperature extension would also be of interest to the phenomenology community. However, the main anomaly-based argument contains load-bearing technical problems that undermine the central claim; the heat-kernel part, while cleaner, does not rescue the functional-method conclusion.

major comments (5)
  1. [§2.3, Eq. (2.17) vs §2.1, Eq. (2.11)] The anomaly-corrected potential in Eq. (2.17) contains the Goldstone term (M_χ^4/2)(1/2 log(M_χ^2/μ^2) − 3/2), whereas the Landau-gauge result in Eq. (2.11) contains M_χ^4(log(M_χ^2/μ^2) − 3/2). The coefficient of the logarithm differs by a factor of 4 and the constant term by a factor of 2. Thus the claimed reproduction of the Landau-gauge result is not achieved; only the overall power M_χ^4 log M_χ^2 is reproduced, with a different coefficient. The same mismatch appears in the Standard Model extension: Eq. (A.26) gives 3/2 M_χ^4(1/2 log M_χ^2/μ^2 − 3/2), while the Landau-gauge result Eq. (A.19) has 3 M_χ^4(log M_χ^2/μ^2 − 3/2). The heat-kernel result in Eq. (3.16) does match Eq. (2.11) exactly, so the two methods presented as equivalent are mutually inconsistent.
  2. [§2.3, Eqs. (2.12)–(2.17); §2.4, Eqs. (2.20)–(2.22)] The derivation of Eq. (2.17) replaces log(|M_G+|^2/μ^2) by (1/2) log(M_χ^2/μ^2). In the stated regime 4ξM_A^2 ≫ M_χ^2, one has |M_G+|^2 ≃ ξ M_χ^2 M_A^2, so the logarithm contains log ξ and log M_A^2. After the anomaly cancellation, the surviving coefficient M_χ^4/2 multiplies this logarithm, leaving a residual gauge dependence proportional to M_χ^4 log ξ. The Nielsen-identity check in §2.4 only examines terms ∼ M_χ^2 log(M_χ^2/μ^2) and does not control the derivative of log|M_G+|^2. Therefore the conclusion in §2.4 that the total effective potential is gauge invariant at all field values is not established by the calculation shown.
  3. [§2.2–§2.4 and Abstract] The entire calculation is performed under the assumption 4ξM_A^2 ≫ M_χ^2, stated at the start of §2.2. The abstract and §2.4 nevertheless claim gauge independence at all field values. Since M_χ and M_A are field-dependent via Eq. (2.3), the hierarchy 4ξM_A^2 ≫ M_χ^2 may fail for generic field values, and no argument is given that the cancellation persists outside this regime. The claim in the abstract should be restricted to the actual domain of validity or supported by an additional analysis.
  4. [§2.3, Eqs. (2.14)–(2.16)] The multiplicative anomaly term in Eq. (2.16) is constructed from the same masses M_G± that appear in the naive calculation, with a coefficient that precisely cancels the ξ-dependent coefficient of the logarithm. This makes the gauge independence of the coefficient partly an artifact of the input. A more serious concern is that the paper does not justify why the standard determinant of the full fluctuation matrix is missing this anomaly factor; if the diagonalized determinant already accounts for the full operator, the anomaly correction would be a double counting. This is the load-bearing premise of the paper and needs a careful derivation rather than an assertion.
  5. [§4.2, Eqs. (4.11)–(4.17)] The finite-temperature heat-kernel calculation in Eq. (4.17) uses only the mass matrix M^2 in Eq. (4.12) and discards the ξ-dependent U-matrix terms without demonstrating that the total-derivative argument used at zero temperature in Eq. (3.14) applies to the Matsubara-summed trace on R^3 × S^1. The claim that the heat kernel 'extends gauge independence to any value of the expansion in mass over temperature' is therefore not demonstrated. In addition, the functional-method Fermi-gauge result in Eq. (4.9) is explicitly gauge-dependent, and no anomaly correction is applied at finite temperature, so the two methods are not compared on equal footing.
minor comments (5)
  1. [References [36] and [55]] References [36] and [55] are duplicate entries for R. T. Seeley, 'Complex powers of an elliptic operator'; one should be removed or replaced with the appropriate original source.
  2. [Eq. (2.15)] The formula for the anomaly density uses the symbol ⊃ and a sum over n; the notation is unclear because the displayed expression is the d=4 contribution only. Please state explicitly which terms are retained in d=4 and which are dropped.
  3. [Eq. (4.17)] The notation M_i^3 ≡ (M_i^2)^{3/2} is confusing; it should be written consistently as (M_i^2)^{3/2} throughout the equation.
  4. [§4.1] The statement 'we work with the temporal gauge A0 = 0' appears after the Fermi-gauge Lagrangian is defined; the relation between the temporal gauge and the ξ-dependent Fermi gauge should be clarified.
  5. [Abstract and §2.4] The abstract's phrase 'gauge independence ... at all field values' is stronger than what the body actually proves, which is limited to the regime 4ξM_A^2 ≫ M_χ^2 and to the specific terms retained. The abstract and conclusions should be aligned with the actual domain of the calculation.

Circularity Check

2 steps flagged · score 6.0 of 10

The Fermi-gauge 'gauge-invariant' potential is engineered by attaching to the multiplicative anomaly the same logarithm that carries the naive ξ dependence, so the cancellation is built into the input rather than derived; the Nielsen coefficient C=0 is then read off from that already-constructed cancellation.

  1. self definitional [Sec. 2.3, Eq. (2.17)]
    "The one-loop effective potential after incorporating the multiplicative anomaly [27], therefore, becomes V(1)eff|F+MA_MSQED = ˜V(1)eff|F_MSQED − 1/2 a[M^2_G+, M^2_G−](ξ) (log |M^2_G+|/µ^2 − 3/2) = ... + M^4_χ/2 (1/2 log M^2_χ/µ^2 − 3/2), (2.17)"

    The anomaly density a in Eq. (2.16) is the d=4 polynomial from Eq. (2.15), a volume-integrated constant. Eq. (2.14) requires adding it as a constant, not multiplying it by (log |M_G+|^2/µ^2 − 3/2). That logarithm is precisely the factor carrying the ξ-dependent coefficient of the naive Fermi-gauge result Eq. (2.13). Subtracting (1/2)a times that factor removes exactly the ξ-dependent part of the input. The final gauge-independent expression is therefore the input with its ξ dependence cancelled by construction, not a consequence of the multiplicative-anomaly formula.

  2. self definitional [Sec. 2.4, Eqs. (2.22) and (2.23)]
    "ξ ∂V(1)/∂ξ = ξ ∂˜V(1)eff|F_MSQED/∂ξ − ξ d/dξ(1/2 × a[M^2_G+, M^2_G−](ξ) (1/2 log M^2_χ/µ^2)) = 0 (2.22)"

    The equality to zero is the same cancellation that was inserted in Eq. (2.17): the anomaly term was assigned the prefactor (1/2 log M^2_χ/µ^2) so that its ξ derivative is the negative of the retained ξ derivative of the naive potential. Reading off C^(1)∂V^(0)/∂φ = 0 from Eq. (2.22) is therefore not an independent derivation of the Nielsen coefficient; it restates the fact that the correction was engineered to cancel the leading ξ dependence. The subsequent statement that the Nielsen identity reduces to gauge invariance is a consistency check of the construction, not a prediction.

full rationale

The functional-method central claim is partially circular: the anomaly correction is given a logarithmic prefactor that is not present in the zeta-function anomaly formula and that exactly matches the logarithm carrying the naive ξ dependence. The conclusion of gauge invariance at all field values is thereby built into the form of the added term. The Heat Kernel calculation provides a separate, independent route to the Landau-gauge result, and the anomaly formula itself is standard literature, so the paper is not wholly circular. The Nielsen-coefficient step is likewise a restatement of the engineered cancellation. Numerical discrepancies with the Landau result and residual ξ-dependent logarithms are correctness concerns rather than additional circular steps, but they reinforce that the claimed universal gauge independence is not independently established. Score 6 reflects a central 'prediction' that reduces by construction, while acknowledging the partially independent Heat Kernel and standard anomaly input.

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

No fitted parameters and no invented entities. The multiplicative anomaly is a known mathematical object; the free-parameter ledger is empty. The load-bearing inputs are the anomaly prescription and the background-field assumptions listed in axioms.

assumptions (4)
  • ad hoc to paper Including the multiplicative anomaly in the zeta-regularized one-loop determinant is the correct prescription for the effective potential.
    Sec. 2.3: the cancellation of ξ-dependence relies entirely on adding Eq. 2.16 with coefficient 1/2. This is asserted, not derived from BRST or from a first-principles definition of the effective potential.
  • domain assumption The ξ-dependent terms in the Heat Kernel computation are total derivatives and vanish for constant backgrounds, leaving a gauge-independent potential.
    Sec. 3, Eq. 3.14 and App. A.5: the argument that gauge dependence disappears for constant fields.
  • domain assumption The limit 4ξM_A^2 >> M_χ^2 is representative enough to draw conclusions about all field values.
    Sec. 2.2: the explicit cancellation is demonstrated in this limit; extending it to all field values is an unproven leap.
  • domain assumption Temporal gauge A0=0 and the neglect of light-heavy mixing do not affect the finite-temperature conclusion.
    Sec. 4 footnotes: the authors concede temporal gauge is not ideal and heavy mixing is ignored, but assert the conclusion is unchanged.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gauge Choices, Infrared Pitfalls, and Thermal Effects in Effective Potentials." pith.science (2026). https://pith.science/paper/OH533GVH

@misc{pith2026250722706,
  author       = {Pith},
  title        = {Pith review of: Gauge Choices, Infrared Pitfalls, and Thermal Effects in Effective Potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OH533GVH}},
  note         = {Machine review of arXiv:2507.22706}
}
read the original abstract

The evaluation of effective potentials is critical for a range of phenomenological applications, including inflation, vacuum stability, and phase transitions. A drawback arises from the gauge-dependence of the effective potential. Furthermore, in theories with spontaneous symmetry breaking, the effective potential exhibits infrared (IR) divergences in the limit of vanishing Goldstone masses. By considering the multiplicative anomaly that arises due to non-factorisation of elliptic operators in the Fermi gauge when computing the effective potential at one-loop order, we demonstrate that its gauge independence and IR behaviour are improved to the corresponding findings of Landau gauge calculations simultaneously. The latter are straightforwardly and transparently reproduced using an approach that employs the Heat Kernel technique, thereby providing a shortcut to reflect anomaly-related cancellations from the outset. Our findings generalise to the treatment of the effective potential at finite temperature. In particular, the Heat Kernel extends gauge independence to any value of the expansion in mass over temperature.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Polyakov Loops Tame Phase Transitions

    hep-ph 2026-07 conditional novelty 6.0 of 10

    Polyakov loop contributions to the thermal effective potential soften electroweak phase transitions, disfavoring first-order transitions and suppressing gravitational-wave signals.

  2. Beyond the Daisy Chain: Running and the 3D EFT View of Supercooled Phase Transitions

    hep-ph 2025-11 conditional novelty 5.0 of 10

    With renormalisation-group running included, the one-loop high-temperature Daisy-resummed potential at µ=πT reproduces the phase-transition parameters of the two-loop dimensionally reduced EFT, while the no-running on...

Reference graph

Works this paper leans on

59 extracted references · 25 canonical work pages · cited by 2 Pith papers

  1. [1]

    S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7 (1973) 1888–1910

    work page 1973

  2. [2]

    Sher, Electroweak Higgs Potentials and Vacuum Stability , Phys

    M. Sher, Electroweak Higgs Potentials and Vacuum Stability , Phys. Rept. 179 (1989) 273–418

    work page 1989

  3. [3]

    Jackiw, Functional evaluation of the effective potential , Phys

    R. Jackiw, Functional evaluation of the effective potential , Phys. Rev. D 9 (1974) 1686

    work page 1974

  4. [4]

    Dolan and R

    L. Dolan and R. Jackiw, Gauge Invariant Signal for Gauge Symmetry Breaking , Phys. Rev. D 9 (1974) 2904

    work page 1974

  5. [5]

    Kluberg-Stern and J

    H. Kluberg-Stern and J. B. Zuber, Ward Identities and Some Clues to the Renormalization of Gauge Invariant Operators, Phys. Rev. D 12 (1975) 467–481

    work page 1975

  6. [6]

    Piguet and K

    O. Piguet and K. Sibold, Gauge Independence in Ordinary Yang-Mills Theories , Nucl. Phys. B 253 (1985) 517–540

    work page 1985

  7. [7]

    Kluberg-Stern and J

    H. Kluberg-Stern and J. B. Zuber, Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 1. Green Functions , Phys. Rev. D 12 (1975) 482–488

    work page 1975

  8. [8]

    N. K. Nielsen, On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge Theories , Nucl. Phys. B 101 (1975) 173–188

    work page 1975

Show all 59 references
  1. [9]

    O. M. Del Cima, D. H. T. Franco, and O. Piguet, Gauge independence of the effective potential revisited, Nucl. Phys. B 551 (1999) 813–825, [ hep-th/9902084]

    1999 arXiv

  2. [10]

    Gambino and P

    P. Gambino and P. A. Grassi, The Nielsen identities of the SM and the definition of mass , Phys. Rev. D 62 (2000) 076002, [ hep-ph/9907254]

    2000 arXiv

  3. [11]

    P. A. Grassi, B. A. Kniehl, and A. Sirlin, Width and partial widths of unstable particles in the light of the Nielsen identities , Phys. Rev. D 65 (2002) 085001, [ hep-ph/0109228]

    2002 arXiv

  4. [12]

    Goria, G

    S. Goria, G. Passarino, and D. Rosco, The Higgs Boson Lineshape , Nucl. Phys. B 864 (2012) 530–579, [arXiv:1112.5517]

    2012 arXiv

  5. [13]

    N. K. Nielsen, Removing the gauge parameter dependence of the effective potential by a field redefinition , Phys. Rev. D 90 (2014), no. 3 036008, [ arXiv:1406.0788]

    2014 arXiv

  6. [14]

    I. J. R. Aitchison and C. M. Fraser, Gauge Invariance and the Effective Potential , Annals Phys. 156 (1984) 1

    1984

  7. [15]

    Metaxas and E

    D. Metaxas and E. J. Weinberg, Gauge independence of the bubble nucleation rate in theories with radiative symmetry breaking, Phys. Rev. D 53 (1996) 836–843, [ hep-ph/9507381]

    1996 arXiv

  8. [16]

    H. H. Patel and M. J. Ramsey-Musolf, Baryon Washout, Electroweak Phase Transition, and Perturbation Theory, JHEP 07 (2011) 029, [ arXiv:1101.4665]

    2011 arXiv

  9. [17]

    Andreassen, W

    A. Andreassen, W. Frost, and M. D. Schwartz, Consistent Use of Effective Potentials , Phys. Rev. D 91 (2015), no. 1 016009, [ arXiv:1408.0287]

    2015 arXiv

  10. [18]

    L. P. Alexander and A. Pilaftsis, The One-Loop Effective Potential in Non-Linear Gauges , J. Phys. G 36 (2009) 045006, [ arXiv:0809.1580]

    2009 arXiv

  11. [19]

    Garny and T

    M. Garny and T. Konstandin, On the gauge dependence of vacuum transitions at finite temperature , JHEP 07 (2012) 189, [ arXiv:1205.3392]

    2012 arXiv

  12. [20]

    J. R. Espinosa, G. F. Giudice, E. Morgante, A. Riotto, L. Senatore, A. Strumia, and N. Tetradis, The cosmological Higgstory of the vacuum instability , JHEP 09 (2015) 174, [ arXiv:1505.04825]. – 22 –

    2015 arXiv

  13. [21]

    Quiros, Finite temperature field theory and phase transitions , in ICTP Summer School in High-Energy Physics and Cosmology , pp

    M. Quiros, Finite temperature field theory and phase transitions , in ICTP Summer School in High-Energy Physics and Cosmology , pp. 187–259, 1, 1999. hep-ph/9901312

    1999 arXiv

  14. [22]

    Hirvonen, J

    J. Hirvonen, J. L¨ ofgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory , JHEP 07 (2022) 135, [arXiv:2112.08912]

    2022 arXiv

  15. [23]

    L¨ ofgren, M

    J. L¨ ofgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Nucleation at Finite Temperature: A Gauge-Invariant Perturbative Framework , Phys. Rev. Lett. 130 (2023), no. 25 251801, [arXiv:2112.05472]

    2023 arXiv

  16. [24]

    Bernardo, P

    F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen, Higher-dimensional operators at finite-temperature affect gravitational-wave predictions, arXiv:2503.18904

    arXiv

  17. [25]

    Elias-Miro, J

    J. Elias-Miro, J. R. Espinosa, and T. Konstandin, Taming Infrared Divergences in the Effective Potential, JHEP 08 (2014) 034, [ arXiv:1406.2652]

    2014 arXiv

  18. [26]

    J. R. Espinosa, M. Garny, and T. Konstandin, Interplay of Infrared Divergences and Gauge-Dependence of the Effective Potential , Phys. Rev. D 94 (2016), no. 5 055026, [ arXiv:1607.08432]

    2016 arXiv

  19. [27]

    Balui, J

    D. Balui, J. Chakrabortty, D. Dey, and S. Mohanty, Gauge invariant effective potential , Phys. Rev. D 111 (2025), no. 8 085032, [ arXiv:2502.17156]

    2025 arXiv

  20. [28]

    S. W. Hawking, Zeta Function Regularization of Path Integrals in Curved Space-Time , Commun. Math. Phys. 55 (1977) 133

    1977

  21. [29]

    E. Elizalde, Zeta-function regularization techniques for series summations and applications , in Proceedings Leipzig Workshop Quantum field theory under the influence of external conditions , (Leipzig), Universit¨ at Leipzig, Naturwissenschaftlich-Theoretisches Zentrum, 10, 1992

    1992

  22. [30]

    Elizalde, Complete determination of the singularity structure of zeta functions , J

    E. Elizalde, Complete determination of the singularity structure of zeta functions , J. Phys. A 30 (1997) 2735–2744, [hep-th/9608056]

    1997 arXiv

  23. [31]

    Wodzicki, Noncommutative residue Chapter I

    M. Wodzicki, Noncommutative residue Chapter I. Fundamentals , pp. 320–399. Springer Berlin Heidelberg, Berlin, Heidelberg, 1987

    1987

  24. [32]

    Mickelsson, Wodzicki residue and anomalies of current algebras , Lect

    J. Mickelsson, Wodzicki residue and anomalies of current algebras , Lect. Notes Phys. 436 (1994) 123–135, [hep-th/9404093]

    1994 arXiv

  25. [33]

    A. A. Bytsenko, G. Cognola, L. Vanzo, and S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section , Phys. Rept. 266 (1996) 1–126, [ hep-th/9505061]

    1996 arXiv

  26. [34]

    Elizalde, L

    E. Elizalde, L. Vanzo, and S. Zerbini, Zeta function regularization, the multiplicative anomaly and the Wodzicki residue, Commun. Math. Phys. 194 (1998) 613–630, [ hep-th/9701060]

    1998 arXiv

  27. [35]

    Elizalde, A

    E. Elizalde, A. Filippi, L. Vanzo, and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071

    arXiv

  28. [37]

    A. A. Bel’kov, A. V. Lanyov, and A. Schaale, Calculation of heat kernel coefficients and usage of computer algebra, Comput. Phys. Commun. 95 (1996) 123–130, [ hep-ph/9506237]

    1996 arXiv

  29. [38]

    D. V. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. 388 (2003) 279–360, [hep-th/0306138]

    2003 arXiv

  30. [39]

    I. G. Avramidi, Heat kernel approach in quantum field theory , in Nucl. Phys. B Proc. Suppl. , vol. 104, pp. 3–32, 2002. math-ph/0107018

    2002 arXiv

  31. [40]

    Kontsevich and S

    M. Kontsevich and S. Vishik, Geometry of determinants of elliptic operators , hep-th/9406140. – 23 –

    arXiv

  32. [41]

    I. G. Avramidi, The Covariant Technique for Calculation of One Loop Effective Action , Nucl. Phys. B 355 (1991) 712–754. [Erratum: Nucl.Phys.B 509, 557–558 (1998)]

    1991

  33. [42]

    Banerjee, J

    U. Banerjee, J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to dimension eight: integrating out heavy scalar(s) , Eur. Phys. J. Plus 139 (2024), no. 2 159, [arXiv:2306.09103]

    2024 arXiv

  34. [43]

    Chakrabortty, S

    J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to dimension eight: Integrating out heavy fermion(s) , Nucl. Phys. B 1000 (2024) 116488, [ arXiv:2308.03849]

    2024 arXiv

  35. [44]

    Banerjee, J

    U. Banerjee, J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to any mass-dimension for non-degenerate scalars and fermions including light–heavy mixing , Eur. Phys. J. Plus 139 (2024), no. 2 169, [ arXiv:2311.12757]

    2024 arXiv

  36. [45]

    Banerjee, J

    U. Banerjee, J. Chakrabortty, and K. Ramkumar, Renormalization of scalar and fermion interacting field theory for arbitrary loop: Heat–Kernel approach , Eur. Phys. J. Plus 139 (2024), no. 8 714, [arXiv:2404.02734]

    2024 arXiv

  37. [46]

    Adhikary, J

    N. Adhikary, J. Das, and D. Dey, Two-loop dimension Six Effective Action: Integrating Out Heavy Scalar, arXiv:2501.01313

  38. [47]

    Adhikary, T

    N. Adhikary, T. Biswas, J. Chakrabortty, C. Englert, and M. Spannowsky, Electroweak Scalar Effects Beyond Dimension-6 in SMEFT , arXiv:2501.12160

    arXiv

  39. [48]

    Urbano, Inflation without gauge redundancy , JCAP 04 (2020) 040, [ arXiv:2001.05480]

    A. Urbano, Inflation without gauge redundancy , JCAP 04 (2020) 040, [ arXiv:2001.05480]

    2020 arXiv

  40. [49]

    Fukuda and T

    R. Fukuda and T. Kugo, Gauge Invariance in the Effective Action and Potential , Phys. Rev. D 13 (1976) 3469

    1976

  41. [50]

    Andreassen, W

    A. Andreassen, W. Frost, and M. D. Schwartz, Consistent Use of the Standard Model Effective Potential, Phys. Rev. Lett. 113 (2014), no. 24 241801, [ arXiv:1408.0292]

    2014 arXiv

  42. [51]

    J. R. Espinosa, M. Garny, T. Konstandin, and A. Riotto, Gauge-Independent Scales Related to the Standard Model Vacuum Instability , Phys. Rev. D 95 (2017), no. 5 056004, [ arXiv:1608.06765]

    2017 arXiv

  43. [52]

    S. P. Martin, Taming the Goldstone contributions to the effective potential , Phys. Rev. D 90 (2014), no. 1 016013, [ arXiv:1406.2355]

    2014 arXiv

  44. [53]

    A. O. Barvinsky and A. E. Kalugin, Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies , Phys. Rev. D 110 (2024), no. 10 105007, [arXiv:2408.16174]

    2024

  45. [54]

    A. A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, and S. Zerbini, Analytic Aspects of Quantum Fields. World Scientific, Singapore, 2003

    2003

  46. [55]

    R. T. Seeley, Complex powers of an elliptic operator , Proc. Symp. Pure Math. 10 (1967) 288–307

    1967

  47. [56]

    Dolan and R

    L. Dolan and R. Jackiw, Symmetry Behavior at Finite Temperature , Phys. Rev. D 9 (1974) 3320–3341

    1974

  48. [57]

    Megias, E

    E. Megias, E. Ruiz Arriola, and L. L. Salcedo, The Thermal heat kernel expansion and the one loop effective action of QCD at finite temperature , Phys. Rev. D 69 (2004) 116003, [ hep-ph/0312133]

    2004 arXiv

  49. [58]

    Chakrabortty and S

    J. Chakrabortty and S. Mohanty, One Loop Thermal Effective Action , arXiv:2411.14146

    arXiv

  50. [59]

    F. J. Moral-Gamez and L. L. Salcedo, Derivative expansion of the heat kernel at finite temperature , Phys. Rev. D 85 (2012) 045019, [ arXiv:1110.6300]

    2012 arXiv

  51. [60]

    Di Luzio and L

    L. Di Luzio and L. Mihaila, On the gauge dependence of the Standard Model vacuum instability scale , JHEP 06 (2014) 079, [ arXiv:1404.7450]. – 24 –

    2014 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.