Bakamjian--Thomas construction and light-front zero modes in pseudoscalar transitions: A comparison with the covariant Bethe--Salpeter model

Ho-Meoyng Choi (Kyungpook National University)

REVIEW

Bakamjian--Thomas construction and light-front zero modes in pseudoscalar transitions: A comparison with the covariant Bethe--Salpeter model

Ho-Meoyng Choi (Kyungpook National University)

High Energy Physics - Phenomenology · Submitted 5 Oct 2026 · 60 references

Pith's verdict: A practical pBT/fBT decomposition and a direct timelike contour are the real value here; the advertised class-level proof of current-component independence rests on a one-sentence factorization at Eq. (D.3) that is verified numerically only for the Gaussian, with the claimed power-law checks unreported.

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CitationHo-Meoyng Choi (Kyungpook National University). "Bakamjian--Thomas construction and light-front zero modes in pseudoscalar transitions: A comparison with the covariant Bethe--Salpeter model." arXiv:2610.05727 (2026). https://pith.science/paper/B3ESGC6S

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Editor's note

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The real value here is the pBT/fBT decomposition and the timelike contour; the analytic claim of Eq. (4.31) is the headline but not, I think, the part that will last. Ref. [46] already had the numerical equality. What this paper adds is an attempt to prove it for a specified radial class, an explicit decomposition of the difference from physical-mass projection, and a mass-weighted complex transverse contour that evaluates D-to-K overlaps directly through the physical timelike interval, including zero recoil. That last piece is genuinely practical.

The BS benchmark is well done. The direct nonvalence zero mode is computed and matched by the effective operator, and the paper is unusually careful about scope: Fock truncation, one-body current, the restricted radial class of Eq. (D.1), integration ordering, and the explicit statement that component independence is not a proof of full Lorentz covariance [b69, b443]. Those are acknowledged limitations, not hidden ones.

The soft spot is load-bearing, and the stress-test note lands on it. At Eq. (D.3), the paper asserts in one sentence that with the transformed measure and the kernels of Eqs. (4.18)-(4.19), the residual reduces to a single orientation integral with an x-independent prefactor. The helicity-flip sector of (4.19) carries (m2-m1)(A2-A1)k_perp^2 and (m1-m2)(A2 k_perp·q_perp - A1 k'_perp·q_perp), and H^-_NF carries the squared cross product. In invariant-mass variables, any leftover piece proportional to k_x(x)/sqrt(p(x)) or 1/sqrt(p(x)) would not vanish under the orientation integral; the even part is an elliptic arc. So the class-level theorem stands or falls on an algebraic reduction that is asserted, not displayed. The only direct numerical check is the Gaussian (Sec. IV G), exactly one member of the class. And Appendix D claims power-law checks for n=2,3 as an independent check but reports no numbers; that is the place a reader most wants them, since those are the only non-Gaussian members tested.

To be fair: this is a verification gap, not a demonstrated flaw. I have no positive reason to think (D.3) is wrong; the structure is plausible, the boundary-cancellation mechanism has the right shape, and the author's prior correspondence work found the equality over a wider numerical range. But as written, “analytic proof for the whole radial class” is stronger than the displayed evidence: an asserted factorization plus one numerical member. I would put confidence in the class-level claim below the reader's moderate score, while agreeing the paper deserves refereeing.

For the LFQM form-factor community this is useful and citable; the contour method and the pBT/fBT separation will outlive the theorem. It deserves a serious referee who can push on the (D.3) algebra. If the reduction checks out and the power-law residuals get reported, the class-level claim would be established rather than asserted. I would accept for review.

Referee report

Significance

If the analytic claim holds, this is a structurally valuable result: current-component independence is not a numerical accident of the Gaussian ansatz but an exact property of a well-specified BT realization for a broad radial class, provided the reduced overlaps converge. The paper is unusually explicit about its scope: it states convergence and regularity conditions (Appendix D, E), acknowledges that current-component independence tests the adopted one-body current rather than proving full Lorentz covariance (Sec. II.C, V), and does not claim the numerical sizes transfer between the BS and BT pictures. Strengths include the independent BS benchmark (direct nonvalence calculation reproduced by the effective operator), the stated integration ordering that makes the spacelike minus projection finite, the mass-weighted contour giving direct timelike evaluation without extrapolation, and the quantified Gaussian residual at 10^{-12} level. The main risk is that the class-level proof rests on a one-sentence factorization assertion in Appendix D, with the only quantified numerical check being a single element of the class.

Major objections (2)

[Appendix D, Eq. (D.3)] The factorization that carries the entire radial-independence claim is asserted rather than shown. The text states in one sentence, "Including the transformed measure and the kernels in Eqs. (4.18) and (4.19), the residual for d != 0 becomes Eq. (D.3)." The helicity kernels in Eqs. (4.18)-(4.19) contain terms proportional to (k_perp x q_perp)^2 = kappa^2 p(x) (in H^-_NF), to k_perp.q_perp = kappa k_x(x) (in H^-_HF and H^perp_HF), and to k_perp^2 with x-dependent coefficients. A priori these generate orientation-integrand contributions proportional to sqrt(p(x)), k_x(x)/sqrt(p(x)), and 1/sqrt(p(x)), none of which vanish under integration over [x_-, x_+]; only the combination (2Lx-B)/sqrt(p(x)) integrates to zero as the total derivative in Eq. (D.4). The reduction to the single total-derivative form therefore requires a nontrivial cancellation among these terms after the radial/Jacobian/measure factors are included, and the reader cannot verify that cancellation from the text. Please supply the algebra (or a supplementary material derivation) showing that the sqrt(p(x)) and k_x(x)/sqrt(p(x)) terms cancel, or explicitly restate the analytic claim at the level actually demonstrated.
[Appendix D, power-law checks] The non-Gaussian evidence for the class-level claim is unquantified. The text states that numerical calculations with power-law radial functions u_i proportional to (1 + rho_i/beta_i^2)^{-n} for n = 2, 3 "provide an independent check of this cancellation," but no residual values are reported, in contrast to the Gaussian run where the 2.5e-12 residual is quoted (Sec. IV.G). Since the Gaussian of Eq. (4.8) is exactly one element of the class (D.1), the power-law runs are the only evidence that the identity holds away from the single checked example. Please report the integrated residuals for these runs and, if they are not at the 10^{-12} level, state that explicitly; the inconsistency raised in Appendix D concerns generic members of the class, so quantified power-law residuals are the minimal evidence that would convert the class-level claim from assertion to verification.

Minor objections (4)

[Sec. II.B, IV.F, Appendix E] The sign conventions oscillate between q^2_perp = Q^2 >= 0 (spacelike), q^2 = -Q^2, and t = q^2 > 0 (timelike) across Secs. II, IV.F, V, and Appendix E. A single glossary stating q^2 = -Q^2 = -t on the spacelike side would reduce the risk of reader confusion with the kinematic constraint f_0(0) = f_+(0).
[Sec. IV.D, Eq. (4.31); Sec. IV.G] The notation f^{fBT}_- is used interchangeably with [f^{(-)}_-]_{fBT} in Eq. (4.31) and Table I; a sentence at first use confirming these are the same quantity would help, as the paper elsewhere distinguishes superscripts (perp) and (-) as extraction labels.
[Sec. IV.E, Eq. (4.35)] The mapping in Eq. (4.35) from the BS vertex product to the Gaussian LFQM product is the load-bearing step of the Jaus comparison, and the paper appropriately labels it a model correspondence rather than an equality. It would be helpful to state explicitly that f^{Jaus}_- in Eq. (4.36) is benchmarked only against the fBT result within this mapping, not against the BS covariant form factor.
[Sec. V] The closing statement that the pBT baseline and the BT completion develop opposite kinematic poles that cancel in their sum (Sec. IV.D) is interesting but the cancellation is not displayed. A sentence pointing to the regular limit of Eq. (4.29) would make the claim checkable.

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The paper

The paper claims

In a Bakamjian–Thomas light-front quark model, the transverse- and minus-current component extractions of the semileptonic form factor $f_-(q^2)$ are proven to give exactly the same value, provided the adopted radial class and…

The reading

This paper argues that the Bakamjian–Thomas (BT) construction, which puts the interaction into the invariant mass operator and builds the current from on-shell constituent spinors, makes the two standard light-front extractions of the semileptonic form factor $f_-(q^2)$ exactly equal. Specifically, when the free constituent invariant masses $M_0$ and $M_0'$ are used pointwise in the internal minus-current projection, the $(+,\perp)$ and $(+,-)$ component extractions give the same $f_-(q^2)$ after integration. The equality is proved analytically for rotationally invariant radial wave functions carrying the light-front Jacobian and the adopted pseudoscalar spin coupling, provided the reduced overlaps converge, so it is a structural property of the realization rather than a numerical coincidence. The paper contrasts this with an exactly solvable covariant Bethe–Salpeter model, in which the $(+,\perp)$ extraction needs a finite zero mode from the shrinking nonvalence region; the difference shows that zero-mode assignments depend on the dynamical realization of the current. A mass-weighted complex transverse contour lets the Gaussian $D\to K$ overlaps be evaluated directly throughout the physical timelike region, including zero recoil, without parametrization or extrapolation.

What carries the argument

The machinery is the fBT projection: the internal mass-dependent Lorentz factors $q^-$ and $P^-$ of the minus-current extraction are evaluated at the free constituent invariant masses, $q^-_{\rm on}$ and $P^-_{\rm on}$, point by point inside the overlap integral, so that every constituent configuration carries its own projection instead of a globally averaged hadron mass. The proof of current-component independence rests on the radial class of Eq. (D.1), $\phi_i = \sqrt{\partial k_z/\partial x}\,u_i(\rho_i)$, with a shared pseudoscalar spin coupling and light-front Jacobian; in invariant-mass variables the radial factor separates from an internal-orientation integral whose boundary term (Eq. D.4) vanishes, giving a residual $R(q^2)=0$ for any radial shape that converges. The analysis uses the $x$-first integration ordering at fixed transverse momentum, which yields a finite spacelike minus-current projection, and the timelike region is reached by a mass-weighted complex transverse contour that keeps the physical interval $0<t<(m_1+m_2)^2$ inside the wave-function analyticity domain.

Extended reading notes

Core claim

The central claim is Eq. (4.31): within the BT-based light-front quark model treated here, the fBT prescription — replacing the physical meson masses $M$, $M'$ by the pointwise constituent invariant masses $M_0(x,k_\perp)$, $M_0'(x,k_\perp')$ in the mass-dependent Lorentz factors of the minus-current projection — makes the $(+,\perp)$ and $(+,-)$ extractions of $f_-(q^2)$ identical after integration. The analytic content is a vanishing internal-orientation boundary term (Eq. D.4) that is independent of the radial shape within the convergent class $\phi_i = \sqrt{\partial k_z/\partial x}\,u_i(\rho_i)$. In this realization the on-shell two-body overlap is already complete for both extractions within the one-body impulse approximation, so current-component independence is an internal consistency property of the BT construction and not a proof of a unique interacting current. By contrast, in the covariant Bethe–Salpeter benchmark the same $(+,\perp)$ extraction requires a finite zero-mode contribution, computed directly from the shrinking nonvalence region and reproduced by an effective valence-space operator; the BT completion — the difference between the fBT and the physical-mass pBT minus projections — is distinct from that endpoint contribution and is already contained in the fBT result.

Load-bearing premise

The equality holds only within the radial class of Eq. (D.1) — $\phi_i = \sqrt{\partial k_z/\partial x}\,u_i(\rho_i)$ with the same pseudoscalar spin coupling and light-front Jacobian — under the specified $x$-first integration ordering, and only provided the reduced overlaps converge; the paper acknowledges this condition explicitly.

Editorial extensions

If this is right

The $(+,\perp)$ and $(+,-)$ extractions of $f_-(q^2)$ agree exactly in the BT realization, so the on-shell two-body overlap is complete for both channels within the adopted one-body impulse approximation; no separate zero-mode operator enters the BT result.
The difference between the fBT minus projection and the physical-mass pBT projection defines an interaction-dependent BT completion that is already contained in the fBT result, is distinct from the Bethe–Salpeter endpoint contribution, and vanishes in the free limit.
The practical standard-vertex implementation of Jaus's covariant prescription differs from fBT by mapped instantaneous and mapped zero-mode contributions even when the radial wave functions are identical, so sharing a wave function does not by itself establish equivalence of prescriptions.
Using the mass-weighted complex transverse contour, the Gaussian $D\to K$ overlaps can be evaluated directly at every physical value of the timelike interval up to zero recoil, without form-factor parametrization or extrapolation, provided $(M-M')^2 < (m_1+m_2)^2$.
The equality of extractions propagates to the scalar form factor through $\Delta f_0 = q^2\Delta f_-/(M^2-M'^2)$ for a common $f_+$, leaving the kinematic constraint $f_0(0)=f_+(0)$ unchanged.

Reading between the lines

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

Because the equality is stated to be a property of the adopted BT realization rather than of light-front dynamics in general, a natural test is to repeat the two-extraction comparison in other channels satisfying $(M-M')^2<(m_1+m_2)^2$; agreement there would suggest the projection prescription transfers, while disagreement would locate the boundary of the claimed consistency.
The opposite-sign cancellation between the pBT baseline and the BT completion near the kinematic pole suggests the pBT/completion split is a diagnostic baseline rather than a separation of physical contributions; a corollary is that similar pole cancellation should appear in any channel where the same projection structure is applied.
Nothing in the analytic argument requires the Gaussian shape, since $R(q^2)=0$ holds for any convergent radial function in the class (D.1); substituting power-law wave functions of the same class would provide an independent numerical check of the claimed radial independence.

Original abstract

We examine the role of the Bakamjian--Thomas (BT) construction in achieving current-component independence for pseudoscalar-to-pseudoscalar semileptonic form factors in light-front (LF) dynamics. An exactly solvable covariant Bethe--Salpeter (BS) model provides an off-shell benchmark. Its transverse-current zero-mode contribution is calculated directly from the shrinking nonvalence region and reproduced by an effective operator in the valence convolution. In the BT-based LF quark model, the one-body current kernel is constructed from on-shell constituent spinors, while interactions are encoded in the mass operator and wave functions. In this realization, the $(+,\perp)$ current-component extraction is zero-mode free and is saturated by the on-shell overlap. Using the free constituent invariant masses consistently in the internal projections yields the same $f_-(q^2)$ from the $(+,\perp)$ and $(+,-)$ current-component extractions after integration. We establish this equality analytically for rotationally invariant radial wave functions with the LF Jacobian and the adopted pseudoscalar spin coupling, provided the reduced overlaps converge. The difference from the minus-current projection using physical meson masses defines an interaction-dependent BT completion, distinct from the BS endpoint contribution. A mass-weighted complex transverse contour permits direct evaluation of the Gaussian $D\to K$ overlaps throughout the physical timelike interval, including zero recoil, without a form-factor parametrization or extrapolation. These results establish current-component independence within the specified BT realization and clarify the dynamical dependence of zero-mode assignments.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 7,681 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.73 · human participants 0.01 · controlled comparison 0.49 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We examine the role of the Bakamjian--Thomas (BT) construction in achieving current-component independence for pseudoscalar-to-pseudoscalar semileptonic form factors in light-front (LF) dynamics. An exactly solvable covariant Bethe--Salpeter (BS) model provides an off-shell benchmark. Its transverse-current zero-mode contribution is calculated directly from the shrinking nonvalence region and reproduced by an effective operator in the valence convolution. In the BT-based LF quark model, the one-body current kernel is constructed from on-shell constituent spinors, while interactions are encoded in the mass operator and wave functions. In this realization, the $(+,\perp)$ current-component extraction is zero-mode free and is saturated by the on-shell overlap. Using the free constituent invariant masses consistently in the internal projections yields the same $f_-(q^2)$ from the $(+,\perp)$ and $(+,-)$ current-component extractions after integration. We establish this equality analytically for rotationally invariant radial wave functions with the LF Jacobian and the adopted pseudoscalar spin coupling, provided the reduced overlaps converge. The difference from the minus-current projection using physical meson masses defines an interaction-dependent BT completion, distinct from the BS endpoint contribution. A mass-weighted complex transverse contour permits direct evaluation of the Gaussian $D\to K$ overlaps throughout the physical timelike interval, including zero recoil, without a form-factor parametrization or extrapolation. These results establish current-component independence within the specified BT realization and clarify the dynamical dependence of zero-mode assignments.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We examine the role of the Bakamjian--Thomas (BT) construction in achieving current-component independence for pseudoscalar-to-pseudoscalar semileptonic form factors in light-front (LF) dynamics.	contribution 0.91	0.85	0.08	unclear 0.60
s1	An exactly solvable covariant Bethe--Salpeter (BS) model provides an off-shell benchmark.	method 0.89	0.29	0.03	asserted 0.98
s2	Its transverse-current zero-mode contribution is calculated directly from the shrinking nonvalence region and reproduced by an effective operator in the valence convolution.	result 0.50	0.74	0.06	asserted 1.00
s3	In the BT-based LF quark model, the one-body current kernel is constructed from on-shell constituent spinors, while interactions are encoded in the mass operator and wave functions.	method 0.94	0.45	0.04	asserted 1.00
s4	In this realization, the $(+,\perp)$ current-component extraction is zero-mode free and is saturated by the on-shell overlap.	result 0.83	0.67	0.17	asserted 1.00
s5	Using the free constituent invariant masses consistently in the internal projections yields the same $f_-(q^2)$ from the $(+,\perp)$ and $(+,-)$ current-component extractions after integration.	result 0.98	0.89	0.24	asserted 1.00
s6	We establish this equality analytically for rotationally invariant radial wave functions with the LF Jacobian and the adopted pseudoscalar spin coupling, provided the reduced overlaps converge.	result 0.63	0.91	0.17	asserted 0.97
s7	The difference from the minus-current projection using physical meson masses defines an interaction-dependent BT completion, distinct from the BS endpoint contribution.	interpretation 0.61	0.80	0.16	asserted 1.00
s8	A mass-weighted complex transverse contour permits direct evaluation of the Gaussian $D\to K$ overlaps throughout the physical timelike interval, including zero recoil, without a form-factor parametrization or extrapolation.	method 0.93	0.88	0.10	asserted 0.99
s9	These results establish current-component independence within the specified BT realization and clarify the dynamical dependence of zero-mode assignments.	interpretation 0.86	0.91	0.75	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** shows the weak form factors for the D → Kℓν decay obtained from the direct covariant and LF calculations. Although the physical decay region is 0 ≤ q 2 ≤ q 2 max = (MD − MK) 2 , we extend the displayed range to −1 GeV2 ≤ q 2 ≤ q 2 max to show the continuation from the spacelike region to the physical timelike region. The left panel shows the full results for f+(q 2 ) and f−(q 2 ) = f (⊥) − (q 2 ). The right panel decomposes f (⊥) − (q 2 ) into its LF valence and zero-mode contributions. For f+(q 2 ), the manifestly covariant result in Eq. (3.6) is reproduced by the L

**Figure 2.** FIG. 2. Illustration of the BT organization for

Figures reproduced from arXiv:2610.05727 by the authors.

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