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Analytic Derivation of Vertical Chromaticity in the Fermilab Muon $g{-}2$ Storage Ring

T0 review · 0 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The vertical chromaticity of the Fermilab muon g-2 storage ring is obtained in closed analytic form by composing per-element maps.

desk verdict This paper supplies a closed analytic formula for vertical chromaticity in the Fermilab muon g-2 ring that matches independent numerics at 10^{-11}. read the letter →

arxiv 2606.09903 v1 pith:CFDWIGXM submitted 2026-06-05 physics.acc-ph math-phmath.MP

Eremey Valetov , Kyoko Makino , Martin Berz This is my paper · ORCID
classification physics.acc-phmath-phmath.MP
keywords verticalchromaticitymuong-2storageringanalyticderivationHamiltonianexpansiontransfermapsFermilabelectrostaticquadrupole
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

The paper expands the Hamiltonian for the ring elements as a Taylor series in the dynamical variables and integrates the equations of motion order by order. This produces the second-order vertical aberrations for the homogeneous dipole and the combined dipole-quadrupole elements. Those per-element maps are then composed along the periodic dispersion orbit to yield an explicit closed-form expression for the vertical chromaticity of the continuous DIQ360 ring model. The same composition reproduces the horizontal chromaticity result from earlier work on the same ring. Direct numerical checks against differential-algebra tracking confirm the analytic result to 10^{-11} relative accuracy for electrostatic-quadrupole voltages between 10 and 26 kV.

What carries the argument

Composition of per-element transfer maps obtained from order-by-order integration of the expanded Hamiltonian, evaluated on the periodic dispersion orbit.

What would settle it

A discrepancy larger than 10^{-10} between the closed-form expression and independent high-order numerical integration of particle trajectories through the full ring lattice at any voltage in the 10-26 kV range would falsify the derivation.

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Extended reading notes

Core claim

Expanding the Hamiltonian as a Taylor polynomial in the dynamical variables and integrating the equations of motion order by order yields the vertical second-order aberrations of the DI and DIQ elements. Composing the resulting per-element maps over the periodic dispersion orbit produces a closed-form expression for the vertical chromaticity ξ_y of the continuous-ring DIQ360 model, in direct functional analogy with the horizontal result obtained previously for the same ring.

Load-bearing premise

Second-order aberrations obtained from the Hamiltonian expansion are sufficient to fix the ring chromaticity, with all higher-order contributions negligible at the voltages considered.

Editorial extensions

If this is right

  • The analytic expression replaces numerical tracking for chromaticity evaluation in the DIQ360 model.
  • The same map-composition procedure applies without change to the modular DIEQ_ON and DIEQ ring descriptions.
  • Chromaticity can now be written as an explicit algebraic function of the electrostatic-quadrupole voltage.
  • The derivation supplies a direct check on any future higher-order extension of the same Hamiltonian.

Reading between the lines

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

  • The closed form may allow direct differentiation with respect to voltage to obtain sensitivity coefficients for ring tuning.
  • Because the horizontal and vertical expressions are obtained by identical steps, their ratio or difference could be examined analytically for possible symmetry relations.
  • The method is in principle portable to any storage ring whose elements admit a similar Hamiltonian expansion and periodic dispersion orbit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The manuscript derives a closed-form analytic expression for the vertical chromaticity ξ_y of the Fermilab Muon g-2 storage ring. Starting from the Hamiltonian, the authors perform a Taylor expansion and integrate the equations of motion order-by-order to obtain the second-order aberrations of the homogeneous dipole (DI) and combined-function dipole-quadrupole (DIQ) elements. These per-element maps are then composed over the periodic dispersion orbit to yield ξ_y for the continuous-ring DIQ360 model, in direct analogy to the authors' prior horizontal result. The expression is validated by direct comparison to independent COSY INFINITY differential-algebra computations, showing agreement at the 10^{-11} level for ESQ voltages in [10, 26] kV across the DIQ360 closed form and the modular DIEQ_ON and DIEQ models.

Significance. If the result holds, the work supplies a parameter-free analytic formula for vertical chromaticity that can be evaluated directly for the g-2 ring geometry and voltage range. The 10^{-11} agreement with an independent differential-algebra code across three models and the full voltage range constitutes strong machine-checked confirmation that the second-order map composition captures the chromaticity to the reported precision. This extends the authors' earlier horizontal-chromaticity derivation and provides a concrete, falsifiable prediction that can be checked against tracking codes without parameter fitting.

minor comments (1)
  1. [abstract, §1] The abstract and §1 refer to the prior horizontal result as Ref. [ChromCPO11]; the reference list entry should be checked for consistency with the journal's citation style.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring response.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation for analogy only; derivation self-contained

full rationale

The paper derives vertical chromaticity analytically from the Hamiltonian via Taylor expansion and order-by-order integration to obtain second-order aberrations of DI and DIQ elements, followed by map composition over the periodic dispersion orbit. This is directly validated against independent COSY INFINITY differential-algebra results at the 10^{-11} level. The sole self-citation (to prior horizontal work) is invoked only for functional analogy and is not load-bearing for the vertical result or any uniqueness claim. No fitted parameters, self-definitional reductions, or ansatzes are present. The central claim remains independent of the citation.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The derivation rests on standard mathematical techniques of Hamiltonian mechanics and differential algebra with no new free parameters, ad-hoc axioms, or postulated entities introduced beyond the ring element models already used in the field.

assumptions (1)
  • standard math The Hamiltonian can be expanded as a Taylor polynomial in the dynamical variables and the equations of motion integrated order by order
    This is the explicit procedure used to obtain the second-order aberrations of the DI and DIQ elements.

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Cite this review

Pith. "Pith review of Analytic Derivation of Vertical Chromaticity in the Fermilab Muon $g{-}2$ Storage Ring." pith.science (2026). https://pith.science/paper/CFDWIGXM

@misc{pith2026260609903,
  author       = {Pith},
  title        = {Pith review of: Analytic Derivation of Vertical Chromaticity in the Fermilab Muon $g-2$ Storage Ring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFDWIGXM}},
  note         = {Machine review of arXiv:2606.09903}
}
abstract

We derive the vertical chromaticity $\xi_y$ of the Fermilab Muon g-2 storage ring in closed analytic form. Expanding the Hamiltonian as a Taylor polynomial in the dynamical variables and integrating the equations of motion order by order, we obtain the vertical second-order aberrations of the homogeneous magnetic dipole ($\mathtt{DI}$) and the combined-function dipole-and-electrostatic-quadrupole element ($\mathtt{DIQ}$) used in the muon $g{-}2$ ring. Composing the per-element maps over the periodic dispersion orbit yields a closed-form expression for the vertical chromaticity $\xichromy$ of the continuous-ring $\mathtt{DIQ360}$ model, in direct functional analogy with the horizontal result of our earlier work on the same ring (Ref.~\refcite{ChromCPO11}). Comparison against COSY INFINITY differential-algebra computation shows agreement at the $10^{-11}$ level across all three ring models ($\mathtt{DIQ360}$ closed form and the modular $\mathtt{DIEQ\_ON}$, $\mathtt{DIEQ}$ via per-element composition) for muon $g{-}2$ electrostatic-quadrupole (ESQ) voltages $\Vesq \in [10, 26]\,\mathrm{kV}$.

Figures

Figures reproduced from arXiv: 2606.09903 by the authors.

Figure 1
Figure 1. Element layout of the three ring models over one 90 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. System diagram of the muon g−2 storage ring. The four electrostatic quadrupole stations Q1–Q4 provide vertical focusing; the fast and slow muon kickers (K1–K3) inject the beam onto the closed orbit. (Adapted from Ref. 5, CC BY 4.0.) where the constant term H0 is omitted because it does not enter the equations of motion, and the first-order term H1 vanishes for an expansion about the refer￾ence orbit. The quadratic p… view at source ↗
Figure 3
Figure 3. Linear vertical chromaticity ξ (p) y of the full modular DIEQ muon g−2 ring vs ESQ voltage, COSY INFINITY DA: hard-edge model (FR 0, circles, solid) and the realistic Enge-function fringe field with EFB extension (FR 3 + EFB, squares, dashed; same Enge coefficients and zEFB = 1.22 cm calibration as the horizontal-counterpart paper1 ). Vertical dashed gridlines mark the storage oper￾ating voltages VESQ = 18.2 kV (Run… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Vertical chromaticity ξ (p) y vs ESQ voltage. Top: the closed-form DIQ360 result (Eq. 18, solid line) and COSY INFINITY DA values for the simplified modular DIEQ ON ring (squares, hard-edge model FR 0), the full modular DIEQ ring (circles, FR 0), and the full modular D…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 8 canonical work pages

  1. [1]

    Valetov, K

    E. Valetov, K. Makino and M. Berz,Microscopy(2026), doi:10.1093/jmicro/dfag010

    work page doi:10.1093/jmicro/dfag010 2026

  2. [2]

    Grangeet al.,Muong–2 Technical Design Report, Fermilab Rep

    J. Grangeet al.,Muong–2 Technical Design Report, Fermilab Rep. FERMILAB-FN- 0992-E, Fermi National Accelerator Laboratory (2015). Fermilab Muon g-2, GM2-doc- 2055

    2015

  3. [3]

    Muon g-2 Collaboration,Phys. Rev. Lett.135, 101802 (2025), doi:10.1103/7clf-sm2v

    work page doi:10.1103/7clf-sm2v 2025

  4. [4]

    Muon g-2 Collab- oration,Phys. Rev. D110, 032009 (2024), doi:10.1103/PhysRevD.110.032009, Also arXiv:2402.15410, FERMILAB-PUB-24-0084-AD-CSAID-PPD

    work page doi:10.1103/physrevd.110.032009 2024

  5. [5]

    Muon g-2 Collaboration,Phys. Rev. Accel. Beams24, 044002 (2021), doi: 10.1103/PhysRevAccelBeams.24.044002

    work page doi:10.1103/physrevaccelbeams.24.044002 2021

  6. [6]

    Weisskopf, Application of rigorous high-order methods and normal forms to non- linear systems, PhD thesis, Michigan State University (2021)

    A. Weisskopf, Application of rigorous high-order methods and normal forms to non- linear systems, PhD thesis, Michigan State University (2021)

    2021

  7. [7]

    K. L. Brown,Adv. Part. Phys.1, 71 (1968), Also SLAC-R-075, SLAC-75

    1968

  8. [8]

    K. L. Brown, R. Belbeoch and P. Bounin,Rev. Sci. Instrum.35, 481 (1964)

    1964

Show all 25 references
  1. [9]

    K. L. Brown,Nucl. Instrum. Meth.187, 51 (1981)

    1981

  2. [10]

    K. L. Brown and R. Servranckx,AIP Conf. Proc.127, 62 (1985), Also SLAC-PUB- 3381

    1985

  3. [11]

    Berz,Modern Map Methods in Particle Beam Physics(Academic Press, San Diego, 1999)

    M. Berz,Modern Map Methods in Particle Beam Physics(Academic Press, San Diego, 1999). Also available at https://www.bmtdynamics.org/pub

    1999

  4. [12]

    M. Berz, K. Makino and W. Wan,An Introduction to Beam Physics(CRC Press, Taylor & Francis Group, London, Boca Raton, 2014)

    2014

  5. [13]

    Valetov, M

    E. Valetov, M. Berz and K. Makino,Direct Calculation of the Transfer Map of a Spherical Deflector Using Simple ODEs and Kepler Theory and Comparison with the Codes COSY INFINITY and GIOS, Tech. Rep. MSUHEP-171023-rev, Michigan State University (2017). Also arxiv.org/abs/1802.06074

    2017 arXiv

  6. [14]

    Valetov and M

    E. Valetov and M. Berz, Derivation, cross-validation, and comparison of analytic formulas for electrostatic deflector aberrations, inThe 10th International Conference June 10, 2026 0:2 WSPC/INSTRUCTION FILE paper Vertical chromaticity of the Fermilab Muon g-2 storage ring21 on...

    2026

  7. [15]

    Makino and M

    K. Makino and M. Berz,Nucl. Instrum. Meth.558, 346 (2006)

    2006

  8. [16]

    Berz and K

    M. Berz and K. Makino,Microscopy(2026), doi:10.1093/jmicro/dfag016

    2026 doi

  9. [17]

    Makino, E

    K. Makino, E. Valetov and M. Berz,Detailed Linear Optics and Tunes of theg−2Ring, Fermilab Rep. FERMILAB-FN-1289-PPD, Fermi National Accelerator Laboratory (2017). Fermilab Muon g-2, E989 Note 104, GM2-doc-5715

    2017

  10. [18]

    Kim and Y

    O. Kim and Y. K. Semertzidis,Phys. Rev. Accel. Beams25, 024001 (2022), doi: 10.1103/PhysRevAccelBeams.25.024001

    2022 doi

  11. [19]

    Berz and K

    M. Berz and K. Makino,COSY INFINITY Version 10.2 Beam Physics Man- ual, Tech. Rep. MSUHEP20221202, Michigan State University (2023). See also https://cosyinfinity.org

    2023

  12. [20]

    Y. K. Semertzidis, G. Bennett, E. Efstathiadis, F. Krienen, R. Larsen, Y. Y. Lee, W. M. Morse, Y. Orlov, C. S. Ozben, B. L. Roberts, L. P. Snydstrup and D. S. Warburton,Nucl. Instrum. Meth. A503, 458 (2003). 21.Opera Simulation Software(Dassault Syst` emes SIMULIA, V´ elizy-Vi...

    2003

  13. [21]

    Wu,The Beam Dynamics and Beam Related Uncertainties in the Fermilab Muong-2Experiment, Ph.D

    W. Wu,The Beam Dynamics and Beam Related Uncertainties in the Fermilab Muong-2Experiment, Ph.D. thesis, University of Mississippi (2018). Fermilab report FERMILAB-THESIS-2018-08

    2018

  14. [22]

    Valetov, M

    E. Valetov, M. Berz and K. Makino,Int. J. Mod. Phys. A34, 1942041 (2019), doi: 10.1142/S0217751X19420417, Also FERMILAB-PUB-19-092-PPD

    2019 doi

  15. [23]

    Valetov, Field modeling, symplectic tracking, and spin decoherence for EDM and Muon g-2 lattices, PhD thesis, Michigan State University (2017)

    E. Valetov, Field modeling, symplectic tracking, and spin decoherence for EDM and Muon g-2 lattices, PhD thesis, Michigan State University (2017). Also FERMILAB- THESIS-2017-21

    2017

  16. [24]

    INTEGRATED Engineering Software,COULOMBIntegrated Engineering Software, Winnipeg, Canada, (2022)

    2022

  17. [25]

    Muon g-2 Collaboration, Injection, storage, and beam dynamics in the Muong−2 storage ring (2026), in preparation

    2026

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