pith. sign in

arxiv: 2605.25827 · v1 · pith:GGYXB5V6new · submitted 2026-05-25 · ✦ hep-th · hep-ph

Hard cutoff and gauge theories

V. Branchina , F. Contino , R. Gandolfo , A. Pernace This is my paper

Pith reviewed 2026-06-29 20:37 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords hard UV cutoffgauge invarianceWilsonian EFTEuler-Heisenberg Lagrangianquantum electrodynamicsrenormalization group
0
0 comments X

The pith

A hard UV cutoff can be introduced in gauge theories while preserving quantum gauge invariance.

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

The paper develops a prescription for imposing a hard Wilsonian cutoff at scale Λ in gauge theories that maintains gauge invariance even after quantization. Using the Euler-Heisenberg effective action correction in QED as a test, both scalar and fermionic versions recover the standard result from proper-time regularization, except for additional terms suppressed by powers of the cutoff. These periodic terms in the inverse background field could matter when fields probe near the cutoff scale. The approach aims at a Wilsonian renormalization group closer to the original Wegner-Houghton idea for gauge theories.

Core claim

We present a way to introduce the Wilsonian hard UV cutoff Λ that preserves gauge invariance at the quantum level. For both scalar and fermionic QED, we recover the well-known Euler-Heisenberg result obtained within proper-time regularization, apart from terms that are generically cutoff-suppressed. These terms, periodic in the inverse background field, might become relevant in regimes where the latter probes scales not much smaller than Λ.

What carries the argument

The gauge-invariant hard cutoff prescription applied to the Euler-Heisenberg correction to the Maxwell action.

If this is right

  • Gauge theories become compatible with a physical hard scale Λ in the Wilsonian sense.
  • The Euler-Heisenberg result is reproduced up to cutoff-suppressed corrections.
  • A step is taken toward a Wegner-Houghton-style renormalization group for gauge theories.
  • Periodic cutoff-dependent terms can appear when background fields approach the cutoff scale.

Where Pith is reading between the lines

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

  • The same cutoff prescription could be tested in non-Abelian gauge theories to check invariance preservation.
  • Numerical simulations of gauge theories might adopt this hard cutoff directly instead of lattice discretizations.
  • The periodic terms suggest possible oscillatory behavior in observables when strong fields probe near Λ.

Load-bearing premise

Demonstrating preservation of gauge invariance for the Euler-Heisenberg correction alone is sufficient to establish the result for the full theory.

What would settle it

An explicit check of a gauge-variant observable such as a nonzero photon mass or broken Ward identity in a higher-order calculation with this cutoff.

read the original abstract

According to usual calculations, the use of a hard cutoff $\Lambda$ in gauge theories leads to a violation of gauge invariance. This seems to generate a tension between gauge theories and the Wilsonian effective field theory (EFT) paradigm, where $\Lambda$ has the physical meaning of ultimate scale of the theory, the scale above which the latter has to be replaced by its UV completion. In the present work, considering the Euler-Heisenberg correction to the free Maxwell action, we present a way to introduce the Wilsonian hard UV cutoff $\Lambda$ that preserves gauge invariance at the quantum level. For both scalar and fermionic QED, we recover the well-known Euler-Heisenberg result obtained within proper-time regularization, apart from terms that are generically cutoff-suppressed. These terms, periodic in the inverse background field, might become relevant in regimes where the latter probes scales not much smaller than $\Lambda$. On the theoretical side, the methods developed in the present work represent a first step towards a new (closer in spirit to the Wegner-Houghton construction) realization of the Wilsonian renormalization group program in gauge theories.

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

2 major / 1 minor

Summary. The paper proposes a method to introduce a hard Wilsonian UV cutoff Λ in scalar and fermionic QED that preserves gauge invariance at the quantum level. Considering the Euler-Heisenberg correction to the free Maxwell action, the authors recover the standard result from proper-time regularization, up to generically cutoff-suppressed periodic terms in the inverse background field. The construction is presented as a first step toward a Wegner-Houghton-style RG in gauge theories.

Significance. If the cutoff construction preserves gauge invariance beyond the tested case, it would address the apparent tension between hard cutoffs and gauge invariance in Wilsonian EFTs and enable a more direct RG implementation in gauge theories. The explicit recovery of a known one-loop result provides a concrete positive check, though the restriction to the free Maxwell Euler-Heisenberg term limits the immediate scope.

major comments (2)
  1. [Abstract and the Euler-Heisenberg calculation section] The central claim that the hard-cutoff procedure preserves gauge invariance for gauge theories rests on recovery of the Euler-Heisenberg effective action for the free Maxwell field. No explicit verification is given for preservation of Ward identities when dynamical matter fields are retained, for the photon propagator, vertex functions, or multi-loop diagrams (as required for the full theory).
  2. [Abstract] The abstract states that the known Euler-Heisenberg result is recovered apart from cutoff-suppressed terms, but the provided information contains no derivation steps, explicit gauge-invariance checks, or error estimates; this makes it impossible to assess whether the construction is free of hidden violations at the quantum level.
minor comments (1)
  1. The periodic cutoff-suppressed terms are mentioned but not quantified or plotted; a figure or estimate of their magnitude relative to the leading term would clarify the regime of validity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the scope of our gauge-invariance verification. We address the major comments point by point below. Our work is explicitly framed as a first step toward a Wilsonian hard-cutoff construction in gauge theories, with the Euler-Heisenberg computation serving as a concrete consistency check rather than a complete proof for the full theory.

read point-by-point responses

  1. Referee: [Abstract and the Euler-Heisenberg calculation section] The central claim that the hard-cutoff procedure preserves gauge invariance for gauge theories rests on recovery of the Euler-Heisenberg effective action for the free Maxwell field. No explicit verification is given for preservation of Ward identities when dynamical matter fields are retained, for the photon propagator, vertex functions, or multi-loop diagrams (as required for the full theory).

    Authors: We agree that the explicit check is restricted to the one-loop Euler-Heisenberg effective action obtained by integrating out scalar or fermionic matter in a constant background electromagnetic field. This yields the standard gauge-invariant result (up to cutoff-suppressed periodic corrections), which constitutes a non-trivial test because any violation of gauge invariance would generically produce non-invariant structures in the effective action. However, we do not perform explicit Ward-identity checks for the photon propagator, three- and four-point vertices, or multi-loop diagrams with fully dynamical matter and gauge fields. The manuscript presents the construction as an initial step toward a Wegner-Houghton-style RG flow; extending the verification to those quantities lies beyond the present scope and would require additional technical development. We will revise the abstract and introduction to state this limitation more explicitly. revision: partial

  2. Referee: [Abstract] The abstract states that the known Euler-Heisenberg result is recovered apart from cutoff-suppressed terms, but the provided information contains no derivation steps, explicit gauge-invariance checks, or error estimates; this makes it impossible to assess whether the construction is free of hidden violations at the quantum level.

    Authors: Abstracts are by design concise and cannot contain full derivations. The explicit construction of the cutoff, the mode integration, the resulting effective action, and the comparison to the proper-time result (including the origin of the periodic corrections) are given in the Euler-Heisenberg calculation section of the manuscript. The gauge-invariance check is implicit in the matching to the known invariant Euler-Heisenberg Lagrangian; any hidden violation would have produced additional structures not present in the standard result. We will add a short clarifying sentence to the abstract that points to this section and mentions the one-loop nature of the test. revision: yes

Circularity Check

0 steps flagged

Explicit construction for cutoff; recovers known Euler-Heisenberg result without definitional reduction

full rationale

The paper's central claim is an explicit procedure for imposing a hard UV cutoff while preserving gauge invariance, verified by matching the standard one-loop Euler-Heisenberg effective action (apart from cutoff-suppressed periodic terms). No equations define a parameter or ansatz in terms of the target result, no fitted inputs are relabeled as predictions, and no load-bearing uniqueness theorems or ansatze are imported via self-citation. The derivation chain is self-contained against the external benchmark of proper-time regularization; the single test case is a limitation on scope but does not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; full text required for ledger.

pith-pipeline@v0.9.1-grok · 5726 in / 1077 out tokens · 30496 ms · 2026-06-29T20:37:23.153952+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

66 extracted references · 41 canonical work pages · 27 internal anchors

  1. [1]

    K. G. Wilson,Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B4, 3174 (1971)

    1971

  2. [2]

    K. G. Wilson,Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B4, 3184 (1971)

    1971

  3. [3]

    K. G. Wilson and M. E. Fisher,Critical exponents in 3.99 dimensions, Phys. Rev. Lett.28, 240 (1972)

    1972

  4. [4]

    F. J. Wegner and A. Houghton,Renormalization group equation for critical phenomena, Phys. Rev. A8, 401 (1973)

    1973

  5. [5]

    K. G. Wilson and J. B. Kogut,The Renormalization group and the epsilon expansion, Phys. Rept.12, 75 (1974)

    1974

  6. [6]

    K. G. Wilson,The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys.47, 773 (1975)

    1975

  7. [7]

    J. C. Ward,An Identity in Quantum Electrodynamics, Phys. Rev.78, 182 (1950)

    1950

  8. [8]

    Takahashi,On the generalized Ward identity, Nuovo Cim.6, 371 (1957)

    Y. Takahashi,On the generalized Ward identity, Nuovo Cim.6, 371 (1957). 18

    1957

  9. [9]

    J. C. Taylor,Ward Identities and Charge Renormalization of the Yang-Mills Field, Nucl. Phys. B33, 436 (1971)

    1971

  10. [10]

    A. A. Slavnov,Ward Identities in Gauge Theories, Theor. Math. Phys.10, 99 (1972)

    1972

  11. [11]

    Pauli and F

    W. Pauli and F. Villars,On the Invariant regularization in relativistic quantum theory, Rev. Mod. Phys.21, 434 (1949)

    1949

  12. [12]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino and N. Darvishi,Dimensional regularization, Wilso- nian RG, and the naturalness and hierarchy problem, Phys. Rev. D106, no.6, 065007 (2022), arXiv:2204.10582

    work page arXiv 2022

  13. [13]

    Branchina, V

    C. Branchina, V. Branchina and F. Contino,Physical tuning and naturalness, Phys. Rev. D 107(2023) no.9, 096012, arXiv:2208.05431

    work page arXiv 2023

  14. [14]

    Ward identities and Wilson renormalization group for QED

    M. Bonini, M. D’Attanasio and G. Marchesini,Ward identities and Wilson renormalization group for QED, Nucl. Phys. B418, 81 (1994), arXiv:hep-th/9307174

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  15. [15]

    Renormalization group flow for SU(2) Yang-Mills theory and gauge invariance

    M. Bonini, M. D’Attanasio and G. Marchesini,Renormalization group flow for SU(2) Yang- Mills theory and gauge invariance, Nucl. Phys. B421, 429 (1994), arXiv:hep-th/9312114

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  16. [16]

    BRS symmetry for Yang-Mills theory with exact renormalization group

    M. Bonini, M. D’Attanasio and G. Marchesini,BRS symmetry for Yang-Mills theory with exact renormalization group, Nucl. Phys. B437, 163 (1995), arXiv:hep-th/9410138

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  17. [17]

    BRS symmetry from renormalization group flow

    M. Bonini, M. D’Attanasio and G. Marchesini,BRS symmetry from renormalization group flow, Phys. Lett. B346, 87 (1995), arXiv:hep-th/9412195

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  18. [18]

    Gauge Invariance, the Quantum Action Principle, and the Renormalization Group

    M. D’Attanasio and T. R. Morris,Gauge invariance, the quantum action principle, and the renormalization group, Phys. Lett. B378, 213 (1996), arXiv:hep-th/9602156

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  19. [19]

    Beta function and infrared renormalons in the exact Wilson renormalization group in Yang-Mills theory

    M. Bonini, G. Marchesini and M. Simionato,Beta function and infrared renormalons in the exact Wilson renormalization group in Yang-Mills theory, Nucl. Phys. B483, 475-494 (1997), arXiv:hep-th/9604114

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  20. [20]

    Gauge Consistent Wilson Renormalization Group I: Abelian Case

    M. Simionato,Gauge consistent Wilson renormalization group: Abelian case, Int. J. Mod. Phys. A15, 2121 (2000), arXiv:hep-th/9809004

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  21. [21]

    Gauge Consistent Wilson Renormalization Group II: Non-Abelian Case

    M. Simionato,Gauge consistent Wilson renormalization group. 2: NonAbelian case, Int. J. Mod. Phys. A15, 2153 (2000), arXiv:hep-th/9810117

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  22. [22]

    T. R. Morris,A Gauge invariant exact renormalization group. 1, Nucl. Phys. B573, 97 (2000), arXiv:hep-th/9910058

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  23. [23]

    Simionato,On the consistency of the exact renormalization group approach applied to gauge theories in algebraic noncovariant gauges, Int

    M. Simionato,On the consistency of the exact renormalization group approach applied to gauge theories in algebraic noncovariant gauges, Int. J. Mod. Phys. A15, 4811 (2000), arXiv:hep- th/0005083

    work page arXiv 2000

  24. [24]

    T. R. Morris,A Gauge invariant exact renormalization group. 2, JHEP12, 012 (2000), arXiv:hep-th/0006064

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  25. [25]

    Renormalization Flow of Bound States

    H. Gies and C. Wetterich,Renormalization flow of bound states, Phys. Rev. D65(2002), 065001, arXiv:hep-th/0107221. 19

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  26. [26]

    Running coupling in Yang-Mills theory - a flow equation study -

    H. Gies,Running coupling in Yang-Mills theory: A flow equation study, Phys. Rev. D66(2002), 025006, arXiv:hep-th/0202207

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  27. [27]

    A proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory

    S. Arnone, A. Gatti and T. R. Morris,A Proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory, Phys. Rev. D67, 085003 (2003), arXiv:hep-th/0209162

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  28. [28]

    Renormalization flow of QED

    H. Gies and J. Jaeckel,Renormalization flow of QED, Phys. Rev. Lett.93, 110405 (2004), arXiv:hep-ph/0405183

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  29. [29]

    Chiral phase structure of QCD with many flavors

    H. Gies and J. Jaeckel,Chiral phase structure of QCD with many flavors, Eur. Phys. J. C46 (2006), 433, arXiv:hep-ph/0507171

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  30. [30]

    Manifestly gauge invariant QED

    S. Arnone, T. R. Morris and O. J. Rosten,Manifestly gauge invariant QED, JHEP10, 115 (2005), arXiv:hep-th/0505169

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  31. [31]

    J. M. Pawlowski,Aspects of the functional renormalisation group, Annals Phys.322(2007), 2831, arXiv:hep-th/0512261

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  32. [32]

    T. R. Morris and O. J. Rosten,Manifestly gauge invariant QCD, J. Phys. A39, 11657 (2006), arXiv:hep-th/0606189

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  33. [33]

    A Generalised Manifestly Gauge Invariant Exact Renormalisation Group for SU(N) Yang-Mills

    S. Arnone, T. R. Morris and O. J. Rosten,A Generalised manifestly gauge invariant exact renor- malisation group for SU(N) Yang-Mills, Eur. Phys. J. C50, 467 (2007), arXiv:hep-th/0507154

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  34. [34]

    B. J. Schaefer, J. M. Pawlowski and J. Wambach,The Phase Structure of the Polyakov–Quark- Meson Model, Phys. Rev. D76(2007), 074023, arXiv:0704.3234

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  35. [35]

    C. S. Fischer, A. Maas and J. M. Pawlowski,On the infrared behavior of Landau gauge Yang- Mills theory, Annals Phys.324(2009), 2408, arXiv:0810.1987

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  36. [36]

    Quark Confinement from Color Confinement

    J. Braun, H. Gies and J. M. Pawlowski,Quark Confinement from Color Confinement, Phys. Lett. B684(2010), 262, arXiv:0708.2413

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  37. [37]

    Introduction to the functional RG and applications to gauge theories

    H. Gies,Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852(2012), 287, arXiv:hep-ph/0611146

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  38. [38]

    W. j. Fu, J. M. Pawlowski and F. Rennecke,QCD phase structure at finite temperature and density, Phys. Rev. D101(2020) no.5, 054032, arXiv:1909.02991

    work page arXiv 2020

  39. [39]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rept.910 (2021), 1, arXiv:2006.04853

    work page arXiv 2021

  40. [40]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino and A. Pernace,Does the cosmological constant really indicate the existence of a dark dimension?, Int. J. Geom. Meth. Mod. Phys.22, no.04, 2450305 (2025), arXiv:2308.16548

    work page arXiv 2025

  41. [41]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino and A. Pernace,Dark dimension and the effective field theory limit, Int. J. Geom. Meth. Mod. Phys.22, no.04, 2450303 (2025), arXiv:2404.10068

    work page arXiv 2025

  42. [42]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino and A. Pernace,Path integral measure and the cos- mological constant, Phys. Rev. D111, no.10, 105018 (2025), arXiv:2412.10194. 20

    work page arXiv 2025

  43. [43]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino and A. Pernace,Path integral measure and RG equa- tions for gravity, Phys. Rev. D111, no.12, 125021 (2025), arXiv:2412.14108

    work page arXiv 2025

  44. [44]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino, R. Gandolfo and A. Pernace,Diffeomorphism invariance of the effective gravitational action, Phys. Rev. D112, no.4, 045002 (2025), arXiv:2506.05100

    work page arXiv 2025

  45. [45]

    Branchina, V

    C. Branchina, V. Branchina, F. Contino, R. Gandolfo and A. Pernace,Gravity and the Higgs boson mass, arXiv:2507.13832

    work page arXiv

  46. [46]

    Giacometti, D

    G. Giacometti, D. Rizzo and D. Zappala,Universal content of the proper time flow in scalar and Yang-Mills theories, Phys. Rev. D113(2026) no.4, 045020, arXiv:2510.04896

    work page arXiv 2026

  47. [47]

    Quantum gravity contributions to the gauge and Yukawa couplings in proper time flow

    G. Giacometti, K. Kowalska, D. Rizzo, E. M. Sessolo and D. Zappala,Quantum gravity con- tributions to the gauge and Yukawa couplings in proper time flow, arXiv:2604.03033

    work page internal anchor Pith review Pith/arXiv arXiv

  48. [48]

    Branchina, F

    V. Branchina, F. Contino, R. Gandolfo and A. Pernace, work in progress

  49. [49]

    Consequences of Dirac Theory of the Positron

    W. Heisenberg and H. Euler,Consequences of Dirac’s theory of positrons, Z. Phys.98, no.11-12, 714 (1936), arXiv:physics/0605038

    work page internal anchor Pith review Pith/arXiv arXiv 1936

  50. [50]

    Weisskopf,The electrodynamics of the vacuum based on the quantum theory of the electron, Kong

    V. Weisskopf,The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dan. Vid. Sel. Mat. Fys. Med.14N6(1936) no.6, 1

    1936

  51. [51]

    J. S. Schwinger,On gauge invariance and vacuum polarization, Phys. Rev.82, 664 (1951)

    1951

  52. [52]

    Dittrich,One Loop Effective Potentials in QED, J

    W. Dittrich,One Loop Effective Potentials in QED, J. Phys. A9(1976), 1171

    1976

  53. [53]

    G. V. Dunne,Heisenberg-Euler effective Lagrangians: Basics and extensions, arXiv:hep- th/0406216

    work page arXiv

  54. [54]

    G. V. Dunne,The Heisenberg-Euler Effective Action: 75 years on, Int. J. Mod. Phys. A27, 1260004 (2012), arXiv:1202.1557

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  55. [55]

    Gies and F

    H. Gies and F. Karbstein,An Addendum to the Heisenberg-Euler effective action beyond one loop, JHEP 03, 108 (2017), arXiv:1612.07251

    work page arXiv 2017

  56. [56]

    W. J. de Haas and P. M. van Alphen,The dependence of the susceptibility of diamagnetic metals upon the field, Proc. Acad. Sci. Amsterdam33, 1106 (1930)

    1930

  57. [57]

    Shoenberg,Magnetic Oscillations in Metals, Cambridge University Press (1984)

    D. Shoenberg,Magnetic Oscillations in Metals, Cambridge University Press (1984)

    1984

  58. [58]

    D. R. Hofstadter,Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14, 2239 (1976)

    1976

  59. [59]

    Janecek, E

    S. Janecek, E. Krotscheck and Y. Zhang,Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum, Phys. Rev. B87, 235429 (2013)

    2013

  60. [60]

    C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, P. Kim,Hofstadter’s Butterfly and the Fractal Quantum Hall Effect in Moir´ e Superlattices, Nature 497, 598 (2013). 21

    2013

  61. [61]

    B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, R. C. Ashoori,Massive Dirac Fermions and Hofstadter But- terfly in a van der Waals Heterostructure, Science 340, 1427 (2013)

    2013

  62. [62]

    Aidelsburger, M

    M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, I. Bloch,Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices, Phys. Rev. Lett. 111, 185301 (2013)

    2013

  63. [63]

    Bocarsly, M

    M. Bocarsly, M. Uzan, I. Roy, S. Grover, J. Xiao, Z. Dong, M. Labendik, A. Uri, M. E. Huber, Y. Myasoedov, K. Watanabe, T. Taniguchi, B. Yan, L. S. Levitov, E. Zeldov,De Haas–van Alphen Spectroscopy and Magnetic Breakdown in Moir´ e Graphene, Science 383, 42 (2024)

    2024

  64. [64]

    V. S. Adamchik,Polygamma functions of negative order, Journal of Computational and Applied Mathematics,100(1998) no.2, 191

    1998

  65. [65]

    Oleszczuk,A Symmetry preserving cutoff regularization, Z

    M. Oleszczuk,A Symmetry preserving cutoff regularization, Z. Phys. C64(1994), 533

    1994

  66. [66]

    S. B. Liao,On connection between momentum cutoff and the proper time regularizations, Phys. Rev. D53(1996), 2020, arXiv:hep-th/9501124. 22

    work page internal anchor Pith review Pith/arXiv arXiv 1996