Covariant Spinor Formalism for Multipole Expanded Form Factor

Referee: Comparison with Ref. [29]: the claim that the spin-1 vector and rank-2 tensor bases of [29] contain redundant structures is asserted but never exhibited as an explicit linear identity. Please add at least one worked example showing 'X_a in [29] = α X_b + β X_c on shell' for the spin-1 vector and rank-2 tensor cases, with α,β expressed in terms of P², Q², m.

Authors: We agree this is the most consequential improvement. In the revised manuscript we will add a new subsection at the end of Sec. 4 ("Comparison with Ref. [29]: explicit redundancy identities") providing two worked examples. (i) Spin-1 vector case: for s_1=s_3=1, (j_L,j_R)=(1/2,1/2), our counting in Table 4 gives N=10, whereas the corresponding parametrization of [29] contains 11 independent-looking structures; we will display the explicit on-shell identity expressing one of their structures (the one built from P^μ ε_3·Q ε_1*·Q together with the (S=2,L=1,J=1) channel) as α X_b + β X_c with α,β written explicitly in terms of m² and Q²=−Q² (using P²=4m²−Q²/... and ε_i·p_i=0). (ii) Rank-2 tensor case: for irrep (1,1) we will exhibit one redundant structure of [29] reduced on shell to a linear combination of two of our basis elements in Table 15, again with rational coefficients in P², Q², m². These identities are obtained by projecting both bases onto the same auxiliary 3-point amplitude basis and comparing rank, which is also how we originally identified the redundancy. We will keep the inherited-completeness argument from Ref. [50] but no longer rely on it as the sole evidence. revision: yes

Referee: Completeness and linear independence are asserted in §3.2 (after Eq. (3.30)) without proof. The wave-function projection (3.26)–(3.28) is bijective only if u_{J,{Q}}(P) span the full (j_L,j_R) irrep when restricted to SO(3). Please cite the precise lemma in Ref. [50] establishing non-degeneracy of (L,S,J) for general (j_L,j_R) and verify no kinematic identity collapses the count, or alternatively verify the rank in a nontrivial case such as (j_L,j_R)=(1,1) with s_1=s_3=1.

Authors: We will do both. (a) We will add a precise pointer to the relevant statement of Ref. [50] (the completeness lemma for canonical-spinor 3-point amplitudes labeled by (L,S,J), together with its proof via the standard SU(2) recoupling and the non-degeneracy of the wave-function map u_{J,{Q}}(P) for massive P, which holds whenever P²>0; the massless limit P²→0 is explicitly excluded). We will note that for elastic form factors P²=4m²−Q²>0 in the physical region, so no kinematic identity collapses the count. (b) We will add an explicit rank check for the worked case (j_L,j_R)=(1,1), s_1=s_3=1: the kinematic Gram matrix of the 19 basis elements (Table 15) has full rank 19 at generic (P²,Q²), and we will report its determinant as a non-vanishing polynomial in (P²,Q²,m²). The check will be reproduced in an appendix and the supplementary computation script will be made available. We will also add a remark that the construction implicitly assumes P²≠0; this caveat will be stated explicitly in §3.2. revision: yes

Referee: The 'alternative coupling form' (auxiliary scalar leg, labels (L,S,J,s_p3)) is asserted to give the same total count as the direct LS form 'because they span the same tensor space.' The 6j-style recoupling identity should be displayed, or a direct equality of counts derived from Eq. (4.26) under both schemes should be shown, since Tables 5–22 use the alternative form while the counting formula is derived in the direct form.

Authors: Agreed; this equivalence is load-bearing and we will make it explicit. In Sec. 4.1 we will (i) display the explicit Wigner 6j recoupling that maps the direct-form labels (L,S,J) to the alternative-form labels (L,S=s_1,J,s_p3): the relevant identity is the standard sum-rule for ⟨(s_1 s_3)S, L; J| s_1, (s_3 J)s_p3; L⟩ ∝ {s_1 s_3 S; L J s_p3}_{6j}, with the recoupling unitary on the allowed-channel space. (ii) We will give a direct combinatorial proof that the two coupling schemes generate the same total count: starting from Eq. (4.26) and inserting Σ_{s_p3} on top of the alternative-form constraints |s_3−J|≤s_p3≤s_3+J and |s_1−s_p3|≤L≤s_1+s_p3 and re-summing reproduces the same closed-form expression. We will include an explicit table comparing the two channel decompositions for (j_L,j_R)=(1/2,1/2) with s_1=s_3=1/2 to make the equality concrete row-by-row. revision: yes

Referee: In §2.4.2, Eqs. (2.119)–(2.120), the decomposition of L_k and E_k as fixed orthogonal combinations of (k−1,k) and (k+1,k) is stated 'up to phase and normalization conventions,' worked out only at k=1. The general-k coefficients √(k/(2k+1)) and √((k+1)/(2k+1)) in Eqs. (2.128)–(2.129) are presented without derivation. Please provide a derivation or cite a specific reference.

Authors: We will add a short derivation at the end of §2.4.1. The coefficients arise from the standard recoupling between the longitudinal/transverse vector spherical harmonics Y^{(L,E)}_{km} and the LS basis elements (k±1,k). Concretely, Y^{(L)}_{km} and Y^{(E)}_{km} are the parity-natural and parity-unnatural combinations of the orbital harmonics Y^{k−1}_m and Y^{k+1}_m coupled with the spin-1 vector to total J=k; the explicit Clebsch–Gordan coefficients C^{k,M}_{k∓1,m;1,a} evaluated at the relevant magnetic quantum numbers reduce to √((k+1)/(2k+1)) and ∓√(k/(2k+1)) (Edmonds §5.9; Varshalovich §7.3). We will add the four-line derivation and cite Edmonds, *Angular Momentum in Quantum Mechanics* §5.9 and Varshalovich et al. §7.3.2 explicitly. The relative sign convention used in Eqs. (2.128)–(2.129) will be stated as the Condon–Shortley choice consistent with our definitions of Y^{(L,E)}_{km} in Eqs. (2.9)–(2.10). revision: yes