Contact interaction treatment of {\pi} and {\rho} elastic and transition tensor form factors

Chen Chen; Craig D. Roberts; Ke-Xin Zeng; Ke-Yu Guo; Xin-Yu Bai

Contact interaction treatment of {\pi} and {\rho} elastic and transition tensor form factors

T0 review · 0 major / 3 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read A symmetry-preserving contact interaction yields tensor charges of 0.36 for the pion and roughly 80 percent of the proton value for the rho meson.

desk verdict This extends an existing contact-interaction model to tensor form factors for pions and rhos, giving consistent reference numbers but no new method or independent results. read the letter →

arxiv 2606.04255 v1 pith:X3QQJXI7 submitted 2026-06-02 hep-ph hep-exhep-latnucl-exnucl-th

Ke-Yu Guo , Ke-Xin Zeng , Xin-Yu Bai , Chen Chen , Craig D. Roberts This is my paper · ORCID

classification hep-phhep-exhep-latnucl-exnucl-th

keywords tensorformfactorscontactinteractionpionrhomesondiquarkchargeQCD

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 computes elastic and transition tensor form factors for pions and rho mesons together with their scalar and axialvector diquark partners. It employs a vector-vector contact interaction treated in a symmetry-preserving manner and regularized with two different schemes that produce similar outcomes. The resulting form factors are stiff, yet the authors regard their infrared behavior as physically reliable. The work supplies concrete numerical predictions that can act as reference points for later calculations with closer ties to full QCD.

What carries the argument

Symmetry-preserving vector*vector contact interaction (SCI) with two regularization schemes, used to evaluate tensor form factors and charges of mesons and diquarks.

What would settle it

A lattice QCD calculation or experimental extraction that finds the pion tensor charge far from 0.36 while the regularization schemes are held fixed would falsify the central numerical predictions.

Watch

Extended reading notes

Core claim

Predictions for tensor charges and form factors, elastic and transition, involving π- and ρ-mesons and their scalar and axialvector diquark partners, are delivered using a symmetry-preserving treatment of a vector*vector contact interaction. The pion tensor charge is approximately 0.36 and the associated tensor form factor radius is practically the same as the pion charge radius; the ρ-meson tensor charge is roughly 80% of that for the proton; and diquark tensor charges and form factors are semiquantitatively alike with those of their π, ρ partners.

Load-bearing premise

The infrared behaviour of the computed form factors remains physically reliable despite the known stiffness of SCI results.

Editorial extensions

If this is right

The pion tensor form factor radius equals the pion charge radius.
The rho tensor charge reaches about 80 percent of the proton value.
Diquark tensor charges and form factors track those of their meson partners at the semiquantitative level.
The results supply reference values for future calculations that incorporate more of QCD dynamics.

Reading between the lines

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

The approach supplies a computationally light benchmark that can be compared directly against lattice data to gauge how much the contact-interaction truncation affects infrared tensor observables.
Because diquark results track meson results, the same framework may be reused to estimate tensor properties inside baryons built from those diquarks.

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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.

Desk Editor's Note

read the letter

The paper applies the symmetry-preserving vector-vector contact interaction to tensor charges and form factors for the pion, rho, and their scalar and axialvector diquark partners. Two regularization schemes are used and produce similar outputs. The quoted values include a pion tensor charge near 0.36 with a radius close to the charge radius, a rho tensor charge about 80 percent of the proton value, and diquark results that track the meson ones fairly closely. The authors note the usual stiffness of these form factors yet argue the infrared region remains useful, and they present the numbers as baselines rather than exact QCD quantities.

The work maintains the symmetry properties that define this framework and shows little sensitivity to the choice of regularization, which is a clear positive for internal consistency. The direct comparison between mesons and diquarks functions as a useful cross-check within the model.

The limitations are straightforward and acknowledged. Parameters are still fitted to masses and decay constants, so the tensor predictions inherit that dependence rather than standing as independent tests. No new theoretical element or first-principles derivation appears; this is an incremental extension to another observable set. The stiffness remains a built-in constraint, even if the infrared claim holds inside the model. The stress-test note finds no internal contradictions, which matches the abstract.

This is for specialists already working with contact-interaction or similar effective models who need reference values for tensor quantities. It could help when comparing to lattice results but will not change broader understanding of hadron structure.

I would send it for peer review. The calculations are systematic, the limitations are stated plainly, and the results add concrete numbers to the existing series of SCI papers.

Referee Report

0 major / 3 minor

Summary. The manuscript delivers predictions for tensor charges and elastic/transition form factors involving the π and ρ mesons together with their scalar and axialvector diquark partners. These are obtained from a symmetry-preserving treatment of a vector×vector contact interaction (SCI) using two distinct regularization schemes that exhibit little sensitivity to each other. The form factors are characterized as stiff, yet their infrared behaviour is argued to remain physically reliable. Highlighted results include a pion tensor charge ≈0.36 with tensor radius comparable to the charge radius, a ρ-meson tensor charge roughly 80% of the proton value, and semiquantitative similarity between the diquark and meson tensor quantities. The predictions are positioned as baselines for future QCD-connected studies.

Significance. If the central results hold, the work supplies concrete numerical benchmarks from a symmetry-preserving contact-interaction framework for tensor observables that are relevant to hadron structure. Explicit comparison of two regularization schemes and transparent acknowledgment of the model's characteristic stiffness constitute strengths that make the outputs useful reference points rather than claimed QCD-exact quantities. The reported semiquantitative likeness between meson and diquark tensor properties is a concrete observation that can guide more elaborate calculations.

minor comments (3)

Abstract: the statement that 'two distinct SCI regularisation schemes' are employed would benefit from naming the schemes and briefly indicating their principal difference, even if the numerical sensitivity is small.
Abstract and results sections: approximate phrasing ('approximately 0.36', 'roughly 80%') for key charges should be supplemented by the precise central values obtained in each regularization scheme so that readers can judge the quoted similarity directly.
The manuscript notes the stiffness of SCI form factors and asserts that infrared behaviour remains reliable; a short dedicated paragraph comparing the infrared slopes or radii obtained here with those from more elaborate Dyson-Schwinger studies would strengthen the baseline claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its utility as a set of benchmarks, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained within model

full rationale

The manuscript employs a contact-interaction model whose parameters are fixed by standard meson observables (masses, decay constants) and then computes tensor charges and form factors as outputs. No quoted equation or section shows a tensor quantity being fitted and then relabeled as a prediction, nor any self-definitional loop, load-bearing self-citation chain, or ansatz smuggled via prior work that reduces the central results to the inputs by construction. The text explicitly positions the SCI results as baselines rather than QCD-exact quantities and notes their stiffness, satisfying the requirement for independent content against external benchmarks.

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

Abstract alone supplies no explicit list of fitted parameters, background axioms, or invented entities. Typical SCI studies introduce a few scale parameters fitted to meson masses or radii, but none are identified here.

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Cite this review

Pith. "Pith review of Contact interaction treatment of {\pi} and {\rho} elastic and transition tensor form factors." pith.science (2026). https://pith.science/paper/X3QQJXI7

@misc{pith2026260604255,
  author       = {Pith},
  title        = {Pith review of: Contact interaction treatment of \pi and \rho elastic and transition tensor form factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3QQJXI7}},
  note         = {Machine review of arXiv:2606.04255}
}

abstract

Predictions for tensor charges and form factors, elastic and transition, involving $\pi$- and $\rho$-mesons and their scalar and axialvector diquark partners, are delivered using a symmetry-preserving treatment of a vector*vector contact interaction (SCI). Two distinct SCI regularisation schemes are employed, with the results showing little sensitivity. Although, as typical in SCI analyses, the form factors are stiff; their infrared behaviour may be considered physically reliable. Notable amongst related quantities are the following: the pion tensor charge is approximately $0.36$ and the associated tensor form factor radius is practically the same as the pion charge radius; the $\rho$-meson tensor charge is roughly 80% of that for the proton; and diquark tensor charges and form factors are semiquantitatively alike with those of their $\pi$, $\rho$ partners. In addition to being interesting in themselves, the SCI predictions can serve as baselines for future studies with a closer connection to QCD.

Figures

Figures reproduced from arXiv: 2606.04255 by the authors.

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. SCI results for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

109 extracted references · 3 canonical work pages

[1]

Special functions Functions of the following type arise in SCI bound- state equations: n! C iu n (ω) = Γ(n−1, ωτ 2 uv)−Γ(n−1, ωτ 2 ir),(A1) whereτ ir,τ uv are SCI parameters,C iu n (ω) =ω C iu n (ω),n∈ Z≥, with Γ(x, y) being the incomplete gamma function. In connection with these functions, there are useful differentiation and integration rules: Ciu n (ω)...
[2]

The key piece in Eq

SCI background Using RL truncation, the quark + antiquark scattering kernel can be written: K α1α′ 1,α2α′ 2 = ˜G(k 2)P k µν[iγµ]α1α′ 1[iγν]α2α′ 2 ,(A5) wherek=p 1 −p ′ 1 =p ′ 2 −p 2, withp 1,2,p ′ 1,2 being the initial and final momenta, respectively, of the scatterers, andk 2P k µν =k 2δµν −k µkν. The key piece in Eq. (A5) is˜G. Analyses of QCD gauge sec...
[3]

Al- though only mass-degenerate light quarks are considered herein, to provide additional context, Table IV also lists kaon-related results

Gap equation Consider the SCI gap equation for anf-flavour quark: S−1 f (p) =iγ·p+m f + 16π 3 αIR m2 G Z d4q (2π)4 γµSf(q)γµ ,(A8) wherem f is the associated quark current-mass. Al- though only mass-degenerate light quarks are considered herein, to provide additional context, Table IV also lists kaon-related results. Using a Poincar´ e-invariant regular- ...
[4]

Eπ(P) Fπ(P) # = 4αIR 3πm2 G

Meson Bethe-Salpeter amplitudes a.π-meson The pion emerges as a quark +antiquark bound-state, whose structure is described by a Bethe-Salpeter ampli- tude. In the SCI, that amplitude takes the following form (M=M u): Γπ(P) =γ 5 iEπ(P) + γ·P 2M Fπ(P) ,(A12) wherePis the pion total momentum,P 2 =−m 2 π. As stressed elsewhere [39, 89], the axialvector Ward-G...
[5]

Another, less frequently used, ap- proach can be introduced as follows [63]

Regularisation Scheme II The results obtained above were obtained using reg- ularisation Scheme I. Another, less frequently used, ap- proach can be introduced as follows [63]. First consider: C−2α(ω) = Z d4l π2 1 [l2 +ω] 2+α (A25a) = Z d4l π2 Z ∞ 0 dτ τ 1+α Γ(2 +α) e−τ[l 2+ω] (A25b) = Z ∞ 0 dτ τ α−1 Γ(2 +α) e−τ ω .(A25c) Here, each step is well defined fo...
[6]

To that end, consider the following integral Z dt At2 + 4B(u·t) 2 (t2 +ω) 3 ,(A34) which arises in the calculation of tensor form factors and is ln-divergent

Scheme Differences Here, it is worth recording the origin of mismatches between tensor form factors obtained with the two dif- ferent regularisation schemes. To that end, consider the following integral Z dt At2 + 4B(u·t) 2 (t2 +ω) 3 ,(A34) which arises in the calculation of tensor form factors and is ln-divergent. Using Scheme I, Eqs. (A1), (A2a), one fi...
[7]

Scheme I T πXY Ij (Q2) = 3 8π2 Z 1 0 dαN πXY j ¯Ciu 1 (ω) + Z 1 0 dαdβ2α AπXY j ¯Ciu 1 (ωπ) + (BπXY j −ω πAπXY j ) ¯Ciu 2 (ωπ) ωπ ,(B2) withj= 1,2,{X, Y}={E, F}, ω=M 2 +α(1−α)Q 2,(B3a) ωπ =M 2 +Q 2α2β(1−β)−α(1−α)m 2 π ,(B3b) and, forj= 1, AπEE 1 = 0 =N πEE 1 ,(B4a) BπEE 1 = 4M ,(B4b) 12 N πEF 1 = 8/M ,(B4c) AπEF 1 =− 2(3α+ 2) M (B4d) BπEF 1 =− 4 M M 2(α+ ...
[8]

Scheme II T πXY IIj (Q2) = 3 8π2 Z dαdβ2α (AπXY j + 1 4 N πXY j )Ciu 0 (ωπ) + (Bπ,XY j −ω πAπXY j )Ciu −2(ωπ) (B6) withω π given in Eq. (B3b) and, forj= 1, AπEE 1 = 0 =N π,EE 1 ,(B7a) BπEE 1 = 4M ,(B7b) AπEF 1 = 4(2−α) M ,(B7c) N πEF 1 =− 8(α+ 2) M ,(B7d) BπEF 1 =− 4 M M 2(α+ 2) +m 2 πα(1−α 2) +Q 2α3(β−1)β ,(B7e) AπF F 1 =− 12 M ,(B7f) N πF F 1 = 32 M ,(B...

Show all 109 references

[9]

Scheme I T ρ Ijl (Q2) = 3 8π2 Z 1 0 dαN ρ jl ¯Ciu 1 (ω) + Z 1 0 dαdβ2α Aρ jl ¯Ciu 1 (ωρ) + (Bρ jl −ω ρAρ jl) ¯Ciu 2 (ωρ) ωρ .(C3) Coefficient functions. lI = 1. N ρ 11 = 4M mρ ,(C4a) Aρ 11 = 0 =A ρ 21 =B ρ 21 =N ρ 21 ,(C4b) Bρ 11 = 2M(2m 2 ρ +Q 2) mρ .(C4c) lI = 2. N ρ 12 = 16...
[10]

Scheme II T ρ IIjl (Q2) = 3 8π2 Z dαdβ2α h (Aρ jl +N ρ jl/4)Ciu 0 (ωρ) +(Bρ jl −ω ρAρ jl)Ciu −2(ωρ) i .(C9) Coefficient functions. lII = 1. N ρ 11 = 0 =A ρ 21 =B ρ 21 =N ρ 21 ,(C10a) Aρ 11 = 4M mρ ,(C10b) Bρ 11 = 2M mρ 2M 2 −2m 2 ρ α2 −1 +2Q2α2(β−1)β+Q 2 .(C10c) lII = 2. N ρ 1...
[11]

Rho tensor form factor interpolations Withz=Q 2, these interpolations are valid at the resolving scaleζ=ζ 2. F ρI T1 (z) = 0.499 + 0.268z+ 0.020z 2 1 + 0.547z+ 0.032z 2 =F ρII T1 (z) (C15a) F ρI T2 (z) = 1.433 + 0.531z+ 0.024z 2 1 + 0.542z+ 0.028z 2 =F ρII T2 (z),(C15b) F ρI T...
[12]

Scheme I T πρ Ijl (Q2) = 3 8π2 Z 1 0 dαdβ2α Aπρ i,j ¯Ciu 1 (ωπρ) + (Bπρ i,j −ω πρAπρ i,j) ¯Ciu 2 (ωπρ)/ωπρ .(D3) Coefficient functions. lI = 1. Aπρ 11 = 6EπEρ(α−1),(D4a) Bπρ 11 = 4Eρ[EπM 2(α−2) +E πα{−m2 π(α−1) 2β +m 2 ρ(α−1) 2(β−1) +Q 2(α−2)α(β−1)β} +F π{M 2 −m 2 π(α−1)(αβ+ 1...
[13]

Scheme II T πρ IIjl (Q2) = 3 8π2 Z dαdβ2α[(Aπρ jl +N πρ jl /4)Ciu 0 (ωπρ) + (Bπρ jl −ω πρAπρ jl )Ciu −2(ωπρ)].(D7) Coefficient functions. lII = 1. Aπρ 11 = 4EπEρ(α−1),(D8a) N πρ 11 = 2Aπρ 11 ,(D8b) Bπρ 11 = 4Eρ[EπM 2(α−2) +E πα{−m2 π(α−1) 2β+m 2 ρ(α−1) 2(β−1) +Q 2(α−2)α(β−1)β}...
[14]

Pi-rho transition tensor form factor interpolations Withz=Q 2, the following interpolations are valid at the resolving scaleζ=ζ 2. F πρI T1 (z) = −0.859−0.081z−0.001z 2 1 + 1.373z−0.027z 2 ,(D11a) 16 F πρI T2 (z) = −0.155 + 0.156z+ 0.046z 2 1 + 1.937z+ 0.561z 2 ,(D11b) F πρI T...
[15]

(12) and the larger mass of the scalar diquark compared to theπ-meson, these form fac- tors are roughly 5≈m 0+ /mπ-times greater in magnitude thanF π T

Scalar diquark elastic F 0+I T (z) = 1.716 1 + 0.287z+ 0.00117z 2 1 + 1.810z+ 0.278z 2 ,(E1a) F 0+I T (z) = 1.757 1 + 0.111z+ 0.000285z 2 1 + 1.746z+ 0.149z 2 .(E1b) Naturally, owing to Eq. (12) and the larger mass of the scalar diquark compared to theπ-meson, these form fac- ...
[16]

Axialvector diquark elastic F 1+I T1 (z) = 0.435 + 0.169z+ 0.014z 2 1 + 0.520z+ 0.039z 2 =F 1+II T1 (z) (E2a) F 1+I T2 (z) = 1.133 + 1.295z+ 0.244z 2 1 + 1.403z+ 0.451z 2 =F 1+II T2 (z), (E2b) F 1+I T3 (z) = −1.245−0.277z−0.004z 2 1 + 1.680z+ 0.036z 2 ,(E2c) F 1+I T4 (z) = −0....
[17]

Scalar-axialvector diquark transition F 0+1+I T1 (z) = −0.776−0.044z−0.005z 2 1 + 1.370z+ 0.028z 2 ,(E4a) F 0+1+I T2 (z) = 0.016 + 0.161z+ 0.032z 2 1 + 2.132z+ 0.752z 2 ,(E4b) F 0+1+I T3 (z) = 0.662 + 0.019z−0.001z 2 1 + 1.858z+ 0.452z 2 .(E4c) F 0+1+II T1 (z) = −0.766−0.066z−...
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H.-Y. Xing, Z.-N. Xu, Z.-F. Cui, C. D. Roberts, C. Xu, Heavy + heavy and heavy + light pseudoscalar to vector semileptonic transitions, Eur. Phys. J. C 82 (10) (2022) 889

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J. Segovia, C. Chen, C. D. Roberts, S.-L. Wan, Insights into theγ ∗N→∆ transition, Phys. Rev. C 88 (2013) 032201(R)

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Y. Yu, P. Cheng, H.-Y. Xing, D. Binosi, C. D. Roberts, Distribution Functions of Λ and Σ 0 Baryons, Eur. Phys. J. A 61 (9) (2025) 208

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Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

Special functions Functions of the following type arise in SCI bound- state equations: n! C iu n (ω) = Γ(n−1, ωτ 2 uv)−Γ(n−1, ωτ 2 ir),(A1) whereτ ir,τ uv are SCI parameters,C iu n (ω) =ω C iu n (ω),n∈ Z≥, with Γ(x, y) being the incomplete gamma function. In connection with these functions, there are useful differentiation and integration rules: Ciu n (ω)...

[2] [2]

The key piece in Eq

SCI background Using RL truncation, the quark + antiquark scattering kernel can be written: K α1α′ 1,α2α′ 2 = ˜G(k 2)P k µν[iγµ]α1α′ 1[iγν]α2α′ 2 ,(A5) wherek=p 1 −p ′ 1 =p ′ 2 −p 2, withp 1,2,p ′ 1,2 being the initial and final momenta, respectively, of the scatterers, andk 2P k µν =k 2δµν −k µkν. The key piece in Eq. (A5) is˜G. Analyses of QCD gauge sec...

[3] [3]

Al- though only mass-degenerate light quarks are considered herein, to provide additional context, Table IV also lists kaon-related results

Gap equation Consider the SCI gap equation for anf-flavour quark: S−1 f (p) =iγ·p+m f + 16π 3 αIR m2 G Z d4q (2π)4 γµSf(q)γµ ,(A8) wherem f is the associated quark current-mass. Al- though only mass-degenerate light quarks are considered herein, to provide additional context, Table IV also lists kaon-related results. Using a Poincar´ e-invariant regular- ...

[4] [4]

Eπ(P) Fπ(P) # = 4αIR 3πm2 G

Meson Bethe-Salpeter amplitudes a.π-meson The pion emerges as a quark +antiquark bound-state, whose structure is described by a Bethe-Salpeter ampli- tude. In the SCI, that amplitude takes the following form (M=M u): Γπ(P) =γ 5 iEπ(P) + γ·P 2M Fπ(P) ,(A12) wherePis the pion total momentum,P 2 =−m 2 π. As stressed elsewhere [39, 89], the axialvector Ward-G...

[5] [5]

Another, less frequently used, ap- proach can be introduced as follows [63]

Regularisation Scheme II The results obtained above were obtained using reg- ularisation Scheme I. Another, less frequently used, ap- proach can be introduced as follows [63]. First consider: C−2α(ω) = Z d4l π2 1 [l2 +ω] 2+α (A25a) = Z d4l π2 Z ∞ 0 dτ τ 1+α Γ(2 +α) e−τ[l 2+ω] (A25b) = Z ∞ 0 dτ τ α−1 Γ(2 +α) e−τ ω .(A25c) Here, each step is well defined fo...

[6] [6]

To that end, consider the following integral Z dt At2 + 4B(u·t) 2 (t2 +ω) 3 ,(A34) which arises in the calculation of tensor form factors and is ln-divergent

Scheme Differences Here, it is worth recording the origin of mismatches between tensor form factors obtained with the two dif- ferent regularisation schemes. To that end, consider the following integral Z dt At2 + 4B(u·t) 2 (t2 +ω) 3 ,(A34) which arises in the calculation of tensor form factors and is ln-divergent. Using Scheme I, Eqs. (A1), (A2a), one fi...

[7] [7]

Scheme I T πXY Ij (Q2) = 3 8π2 Z 1 0 dαN πXY j ¯Ciu 1 (ω) + Z 1 0 dαdβ2α AπXY j ¯Ciu 1 (ωπ) + (BπXY j −ω πAπXY j ) ¯Ciu 2 (ωπ) ωπ ,(B2) withj= 1,2,{X, Y}={E, F}, ω=M 2 +α(1−α)Q 2,(B3a) ωπ =M 2 +Q 2α2β(1−β)−α(1−α)m 2 π ,(B3b) and, forj= 1, AπEE 1 = 0 =N πEE 1 ,(B4a) BπEE 1 = 4M ,(B4b) 12 N πEF 1 = 8/M ,(B4c) AπEF 1 =− 2(3α+ 2) M (B4d) BπEF 1 =− 4 M M 2(α+ ...

[8] [8]

Scheme II T πXY IIj (Q2) = 3 8π2 Z dαdβ2α (AπXY j + 1 4 N πXY j )Ciu 0 (ωπ) + (Bπ,XY j −ω πAπXY j )Ciu −2(ωπ) (B6) withω π given in Eq. (B3b) and, forj= 1, AπEE 1 = 0 =N π,EE 1 ,(B7a) BπEE 1 = 4M ,(B7b) AπEF 1 = 4(2−α) M ,(B7c) N πEF 1 =− 8(α+ 2) M ,(B7d) BπEF 1 =− 4 M M 2(α+ 2) +m 2 πα(1−α 2) +Q 2α3(β−1)β ,(B7e) AπF F 1 =− 12 M ,(B7f) N πF F 1 = 32 M ,(B...