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Revisiting $\bar B^0 \rightarrow \Lambda_c^+ \bar p$ decay with higher twist corrections

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Higher-twist corrections and W-exchange interference bring the predicted branching fraction of B-bar0 to Lambda_c+ p-bar into agreement with experiment.

desk verdict Solid PQCD update that restores agreement for the best-measured two-body baryonic B decay by adding W-exchange and higher-twist LCDAs, plus first numbers for the suppressed mode and asymmetries. read the letter →

arxiv 2607.02876 v1 pith:IND7QMRW submitted 2026-07-03 hep-ph hep-ex

Zhou Rui , Zhi-Tian Zou , Ying Li This is my paper · ORCID
classification hep-phhep-ex PACS 13.25.Hw12.38.Bx14.20.Lq
keywords baryonicBdecaysperturbativeQCDlight-conedistributionamplitudeshigher-twistcorrectionsW-exchangedecayasymmetryparametersLambda_c
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

This paper recalculates the single-charmed baryonic decay B-bar0 to Lambda_c+ p-bar in the perturbative QCD framework, now including both W-emission and W-exchange topologies and higher-twist light-cone distribution amplitudes of the B meson, the Lambda_c, and the proton. Earlier calculations that kept only W-emission over-predicted the rate; the authors find a sizable destructive interference between the two topologies that lowers the branching fraction into the measured 10^{-5} range. The same machinery is applied for the first time to the doubly Cabibbo-suppressed channel B-bar0 to Lambda_c-bar p, whose rate is predicted at the 10^{-8} level, and first predictions are given for the decay asymmetry parameters of both modes. The work argues that higher-twist baryon structure cannot be neglected and that helicity suppression is milder once a heavy charm quark is present, giving a more coherent QCD picture of two-body baryonic B decays.

What carries the argument

The PQCD factorization formula that convolves hard kernels for all emission and exchange diagrams with B-meson, Lambda_c (twists 2-4) and proton (twists 3-6) light-cone distribution amplitudes, including the subleading B-meson LCDA and three heavy-quark-symmetry models for the Lambda_c.

What would settle it

A high-statistics measurement of the branching fraction of B-bar0 to Lambda_c-bar p near 10^{-8}, or a measurement of the up-down asymmetry alpha for B-bar0 to Lambda_c+ p-bar that is not close to +1, would directly test the predicted rates and interference pattern.

Watch

Extended reading notes

Core claim

When both W-emission and W-exchange topologies are retained and higher-twist LCDAs of the initial and final hadrons are included, the net interference is destructive. The resulting branching fraction of B-bar0 to Lambda_c+ p-bar falls to (1.64^{+0.58+0.06+0.59+0.46}_{-0.42-0.10-0.43-0.24}) times 10^{-5} (Exponential Lambda_c model), matching the experimental average (1.52 plus or minus 0.17) times 10^{-5}. The same framework yields a first prediction of order 10^{-8} for the suppressed mode B-bar0 to Lambda_c-bar p and first predictions for the asymmetry parameters of both channels.

Load-bearing premise

The three models used for the Lambda_c light-cone amplitudes are taken over from bottom-baryon forms under the assumption that heavy-quark symmetry still holds for charm, with free shape parameters that are not fixed by charm data.

Editorial extensions

If this is right

  • The previously omitted W-exchange amplitude is roughly one-quarter the size of the emission amplitude and must be kept in future charmed baryonic B-decay calculations.
  • Higher-twist LCDAs of both heavy and light baryons contribute at the same order as typical next-to-leading corrections and cannot be dropped.
  • The suppressed channel B-bar0 to Lambda_c-bar p should become accessible at future high-luminosity B factories or LHCb upgrades.
  • The large positive asymmetry parameter alpha near unity for the favored mode, together with small beta and gamma, supplies a clean experimental handle on the relative S- and P-wave phases.

Reading between the lines

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

  • If the same destructive emission-exchange pattern persists in other single-charmed modes, many existing upper limits and pole-model estimates will need downward revision.
  • The sensitivity of the asymmetry parameters to the poorly known proton higher-twist amplitudes makes those observables a practical target for lattice or sum-rule determinations of baryon LCDAs.
  • Because penguins are absent, any future observation of direct CP violation in either channel would be a clean beyond-Standard-Model signal.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript presents a leading-order PQCD analysis of the two-body baryonic decays \(\bar B^0\to\Lambda_c^+\bar p\) and the doubly Cabibbo-suppressed mode \(\bar B^0\to\bar\Lambda_c^-p\). Both W-emission and W-exchange topologies are retained, and higher-twist light-cone distribution amplitudes (LCDAs) of the B meson (including the subleading \(\bar\phi_B\)), of \(\Lambda_c\) (up to twist 4, with three phenomenological models constructed via heavy-quark symmetry), and of the proton (up to twist 6) are systematically included. The central numerical result is that a sizable destructive interference (|E/C|\approx0.25) between the two topologies, together with the higher-twist contributions, reduces the predicted branching fraction of the favored mode to (1.64–2.08)\times10^{-5} (depending on the \(\Lambda_c\) LCDA model), in agreement with the PDG average (1.52\pm0.17)\times10^{-5}. The suppressed mode is predicted at O(10^{-8}), and the first theoretical values for the decay asymmetry parameters \(\alpha,\beta,\gamma\) of both channels are given. Explicit factorization formulas, hard-kernel virtualities, and Wilson-coefficient combinations appear in Appendix A and Tables VII–IX; amplitude decompositions by topology and by twist are provided in Tables II–IV.

Significance. If the results hold, the work supplies a coherent QCD-based resolution of a long-standing discrepancy for the most precisely measured two-body baryonic B decay, demonstrates that helicity suppression of W-exchange is alleviated by the heavy charm quark, and furnishes the first predictions for a Cabibbo-suppressed channel and for the full set of angular observables. These predictions are falsifiable at current and future high-luminosity experiments (LHCb, Belle II). Strengths that strengthen the claim include the explicit, reproducible factorization formulas, the multi-model assessment of \(\Lambda_c\) LCDA uncertainty, the hierarchical twist decomposition that supports convergence of the baryonic expansion, and the transparent uncertainty budget (\(\omega_b\), proton \(\lambda_1\), hard-scale variation, LCDA shape parameters).

major comments (2)
  1. [Sec. II, Eqs. (12)–(14)] Sec. II, Eqs. (12)–(14): the three \(\Lambda_c\) LCDA models are obtained by transplanting bottom-baryon forms under the heavy-quark limit and introducing free shape parameters (\(\omega_0=0.4\pm0.1\) GeV, A=0.5\pm0.2, Borel window, etc.) that are not constrained by charm-sector data. While the paper shows that the Exponential and QCDSR models both reproduce the experimental branching fraction and that the |E/C| interference pattern is stable, a quantitative estimate of the residual O(\(\Lambda_{\rm QCD}/m_c\)) corrections (or a comparison with any available lattice/QCDSR moments for the charm system) would strengthen the claim that the model dependence is fully under control.
  2. [Appendix A] Appendix A and the paragraph preceding Eq. (A1): only the hard kernel for the single dominant diagram Fig. 1(d5) is written out; the remaining lengthy expressions are said to be “obtained analogously.” Given that the numerical results rest on the coherent sum of dozens of diagrams (Tables VII–IX), the absence of the full set of H_{Rij}^{A/B} (or a public repository of the integrands) limits independent verification of the interference pattern that drives the central claim.
minor comments (5)
  1. [Table I] Table I: the comparison column for Refs. [13,16,17] is listed as “\sim100”; a more precise range or a footnote clarifying that these are early pole/diquark/sum-rule estimates would improve readability.
  2. [Eq. (32), Tables V–VI] Eq. (32) and Tables V–VI: the four-source uncertainty breakdown is useful, but the ordering of the error bars is not stated in the caption; a short sentence defining the sequence (\(\omega_b\), \(\Lambda_c\) shape, proton moments, hard scale) would eliminate ambiguity.
  3. [Figs. 1–2] Figs. 1 and 2: the diagrams are densely labeled (a1–g4, etc.). Adding a short legend or color-coding the gluon attachments that distinguish emission from exchange would help the reader navigate the topology classification used in Tables II and VII–IX.
  4. [Sec. III] Sec. III, paragraph after Table II: the statement that switching off W-exchange yields 2.2\times10^{-5} is slightly lower than the earlier PQCD range (2.3–5.1)\times10^{-5} of Ref. [24]; a one-sentence attribution to the newly included higher-twist pieces would make the comparison self-contained.
  5. [Appendix A] Throughout: several long multi-line expressions (e.g., DS1, DT1) contain nested parentheses that are hard to parse; breaking them into intermediate definitions or supplying a Mathematica notebook would aid readability without changing the physics.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: standard PQCD evaluation of diagrams with phenomenological LCDAs; BF agreement is a computed outcome, not forced by construction or self-fit.

full rationale

The derivation is a conventional leading-order PQCD factorization (Eq. 21) of the effective Hamiltonian (Eq. 18) into hard kernels H_Rij, Sudakov factors, and hadronic LCDAs. Branching fractions and asymmetries are obtained by coherent summation of all W-emission and W-exchange diagrams (Figs. 1–2, Tables II–IV) after numerical integration over the LCDAs; the destructive interference |E/C|~0.25 and the reduction of B from ~2.2e-5 (C only) to ~1.64e-5 emerge from the explicit amplitudes, not from any definitional identity or fit to the PDG value. The three Lambda_c LCDA models (Eqs. 12–14) are phenomenological extrapolations from bottom-baryon forms under heavy-quark symmetry, with free shape parameters varied for uncertainty; they are not tuned to the present BF. Self-citations (e.g. [7] for W-exchange formulas and Sudakov, [24] for earlier emission-only result) supply reusable technical ingredients that are independently re-derived or tabulated in the Appendix and are not uniqueness theorems that force the central claim. Residual model dependence is quantified and does not close a logical loop. Predictions for the suppressed mode and all asymmetry parameters are genuine outputs. Score 1 only for the minor, non-load-bearing self-citations of prior technical machinery.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central numerical claims rest on a standard but non-trivial set of non-perturbative inputs and modeling choices that are not derived inside the paper. Free parameters control the shapes of all three hadron LCDAs and the hard-scale variation; domain assumptions include the validity of kT factorization for baryonic decays and the heavy-quark-symmetry mapping of bottom-baryon LCDAs onto charm. No new dynamical entities are postulated.

free parameters (5)
  • ω_b (B-meson shape parameter) = 0.40 ± 0.05 GeV
    Fixed at 0.40 GeV and varied by ±0.05 GeV; produces the largest single uncertainty (~30 %) on the branching fraction.
  • ω0 / A / τ (Λc LCDA shape parameters) = model-dependent
    Control the three alternative models (Exponential ω0=0.4±0.1 GeV, Gegenbauer A=0.5±0.2, QCDSR Borel τ∈[0.4,0.8] GeV); varied to estimate model dependence.
  • f_Λc, fN, λ1, λ2 (baryon decay constants and higher-twist moments) = f_Λc=0.022±0.008 GeV³; fN=(5.3±0.5)×10^{-3} GeV²; λ1=−(2.7±0.9)×10^{-2} GeV²; λ2=(5.1±1.9)×10^{-2} GeV²
    Taken from QCD sum rules; λ1 dominates the proton-LCDA uncertainty and therefore the asymmetry-parameter errors.
  • hard-scale variation factor = 0.8–1.2
    t varied in [0.8t,1.2t] to estimate missing higher-order αs corrections; contributes the last quoted error bar.
  • c (Sudakov infrared cut-off parameters) = 1.05 / 1.14
    c=1.05 for Λc, c=1.14 for proton; residual resummation uncertainty.
assumptions (4)
  • domain assumption kT factorization and PQCD hard-scattering kernels remain valid for two-body baryonic B decays at the physical b-quark mass.
    Invoked throughout Sec. II and Appendix A; not proved from first principles for baryons.
  • domain assumption In the heavy-quark limit the light-cone distribution amplitudes of Λc can be approximated by those of bottom baryons.
    Explicitly stated in Sec. II before Eqs. (12)–(14); used to construct all three Λc models.
  • domain assumption Penguin operators do not contribute because of flavor structure; only tree operators O1,O2 enter.
    Stated after Eq. (18); standard for b→cūd and b→u c̄d.
  • domain assumption Three-parton contributions to the B-meson equation of motion can be neglected when relating φB+ and φB−.
    Used to obtain φB+ from Eq. (9).

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Cite this review

Pith. "Pith review of Revisiting $\bar B^0 \rightarrow \Lambda_c^+ \bar p$ decay with higher twist corrections." pith.science (2026). https://pith.science/paper/IND7QMRW

@misc{pith2026260702876,
  author       = {Pith},
  title        = {Pith review of: Revisiting $\bar B^0 \rightarrow \Lambda_c^+ \bar p$ decay with higher twist corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IND7QMRW}},
  note         = {Machine review of arXiv:2607.02876}
}
abstract

We investigate the single-charmed baryonic decays $\bar B^0 \to \Lambda_c^+ \bar p$ and $\bar B^0 \to \bar\Lambda_c^- p$, which receive contributions from both $W$-emission and $W$-exchange topologies, within the framework of perturbative QCD (PQCD). Higher-power corrections associated with the hadronic light-cone distribution amplitudes (LCDAs) of both the initial- and final-state hadrons are systematically taken into account. We find that these higher-twist contributions play an important role in baryonic $B$ decays and cannot be neglected. A sizable destructive interference between the $W$-emission and $W$-exchange amplitudes is observed, which significantly reduces the predicted branching fraction of $\bar B^0 \to \Lambda_c^+ \bar p$ and leads to improved agreement with experimental measurements. The doubly Cabibbo-suppressed decay $\bar B^0 \to \bar\Lambda_c^- p$ is studied for the first time. Its branching fraction is predicted to be of order $10^{-8}$, placing it within the reach of future high-luminosity experiments. We further present the first theoretical predictions for the decay asymmetry parameters of both channels, which provide additional observables for testing the underlying decay dynamics and can be confronted with future experimental data.

Figures

Figures reproduced from arXiv: 2607.02876 by the authors.

Figure 1
Figure 1. W-emission diagrams for the B¯0 → Λ + c p¯ decay at leading order. The solid black blobs represent the vertex of the effective weak interaction, while the red ones indicate the possible connections of the gluon (red) attached to the spectator ¯d quark. The heavy quarks b and c are shown in bold lines [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. W-exchange diagrams for the B¯0 → Λ + c p¯ decay at leading order. To evaluate the hard kernels for B¯0 → Λ + c p¯, we parametrize the valence-quark momenta in the first diagram of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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Reference graph

Works this paper leans on

78 extracted references · 45 linked inside Pith

  1. [1]

    The coefficient aRij denotes the product of CKM matrix elements and Wilson coefficients associated with a gi ven diagram Rij

    (22) The δ functions enforce momentum conservation among the valence quarks. The coefficient aRij denotes the product of CKM matrix elements and Wilson coefficients associated with a gi ven diagram Rij. The function HRij represents the hard kernel, encoding the spinor and Dirac structure of the short-distan ce interaction. The factor Ω Rij (bl, b ′ l, b q) ...

  2. [2]

    Table VII lists the combinations of Wilson coefficients aRij

    for the process ¯B0 → Λ + c ¯p. Table VII lists the combinations of Wilson coefficients aRij . The virtualities of the internal propagators for the W -emission and W -exchange topologies are given in Tables VIII and IX, respectively. The expressions for HRij (xl, x ′ l, y ) in this channel are rather lengthy due to the inclusion of num erous higher-twist c...

  3. [3]

    + (( ¯φ B(− x′ 2r − r + ¯r(rx2 + x′ 2 + x1(− x′ 2r + 2r + 2) + 1) + 1) +φ B(x′ 2r + r + ¯r(− rx2 + x2 − x′ 2 + x1(− 5r + (2r + 1)x′ 2 − 4) − 1) − 1))φ a 3 +(φ B(x′ 2r + r + ¯r((r − 1)x2 − x′ 2 + x1(r + (2r + 1)x′ 2 + 2) − 1) + 1) − ¯φ B(x′ 2r + r + ¯r(rx2 − x′ 2 + x1(rx′ 2 + 2) − 1) + 1))φ s 3)f +), (A2) DS2(xl, x ′ l, y ) = 8M 4 (f − − f +)f + ¯r2(− ¯rφ ...

  4. [4]

    − 1) + (f − − 1)rx′ 2 − 2f − + r + 2) + φ a 3 φ B(2¯r((f − − 1)x′ 2 + 1) − 2(f − − 1)rx′ 2 + 3f − − 2r − 3) + φ Bφ s 3(2¯r((f − − 1)x′ 2 + 1) − 2(f − − 1)rx′ 2 − 3f − − 2r + 3) + φ s 3 ¯φ B(¯r((f − − 1)(− x′

  5. [5]

    − 1) + (f − − 1)rx′ 2 + 2f − + r − 2)) 13 Table VIII: The virtualities of the internal propagators tA,B,C,D for theW -emission diagrams with ¯x(′) l = 1 − x(′) l . Rij tA M 2 tB M 2 tC M 2 tD M 2 Ca1 f +x1y ( f − − 1 ) f +x2x′ 2 f + (x1 + x3) y f + (( f − − 1 ) x′ 2 + y ) Ca2 f +x1y (f − − 1) f +x2x′ 2 (f − − 1) f + ¯x1x′ 2 f + ((f − − 1) x′ 2 + y) Ca3 f ...

  6. [6]

    + ((φ B(x′ 2r + r + ¯r((r + 1)x2 − x′ 2 +x1(5r + (1 − 2r)x′ 2 − 4) − 1) + 1) + ¯φ B(− x′ 2r − r + ¯r(− rx2 + x′ 2 + x1(x′ 2r − 2r + 2) + 1) − 1))φ a 3 +( ¯φ B(− x′ 2r − r + ¯r(rx2 + x′ 2 + x1(rx′ 2 − 2) + 1) + 1) − φ B(− x′ 2r − r + ¯r((r + 1)x2 + x′ 2 + x1(r + (2r − 1)x′ 2 − 2) + 1) + 1))φ s 3)f +), (A4) DP2(xl, x ′ l, y ) = 8M 4 (f − − f +)f + ¯r2(¯rφ B...

  7. [7]

    − 1) + (f − − 1)rx′ 2 + 2f − + r − 2) + φ s 3 ¯φ B(¯r((f − − 1)(− x′

  8. [8]

    − 1) +(f − − 1)rx′ 2 − 2f − + r + 2) + φ Bφ s 3(2¯r((f − − 1)x′ 2 + 1) − 2(f − − 1)rx′ 2 + 3f − − 2r − 3)) +(f − − 1)x2φ B(φ a 3 − φ s 3)) + f +(x1 ¯rφ a 3 ¯φ B((f − − 1)x′ 2 − 2r + 1) + φ a 3φ B(¯r(x1(− 2(f − − 1)x′ 2 + 3r − 2) − f − x′ 2 − rx2 + x′ 2 + 1) + (f − − 1)rx′ 2 + f − − r − 1) + x1¯rφ s 3 ¯φ B((f − − 1)x′ 2 + 2r + 1) +φ Bφ s 3(¯r(− x1(2(f − − ...

Show all 78 references
  1. [9]

    − 2x′ 2 + 2) + ¯φ B(r + x′ 2 + ¯r(rx2 − rx′ 2 − 1) − 1)))x1 − (x2 + 1)φ 4(φ B − ¯φ B) − rφ 2(φ B − ¯φ B)(x2 − x′

  2. [10]

    + ¯rφ 2(φ B − ¯φ B)(x2 − x′ 2) − (φ 2φ B + 2φ 4( ¯φ B − φ B))x′ 2)(f +)2 + (φ 2( ¯φ B(¯r(− (r − 1)x1 + x2 − x′ 2 + 1) − r) +φ B(r + x′ 2 + ¯r(3rx1 + (r − 1)x2 − rx′ 2 + x′ 2 − 1))) + φ 4(¯r ¯φ B((r + 1)x1 + r(x2 − x′ 2)) +φ B(r(x′ 2 − 1) + ¯r(− (2r + 1)x1 − rx2 + x2 + rx′ 2 − ...

  3. [11]

    + φ B(− rφ 2x′ 2 − φ 4(x′ 2 − 1))) + ¯rφ B(rφ 4(x′ 2 − 1) + φ 2x′ 2)) + f − ((f +)3x1φ 2(¯r − r) (− x′ 2 + x1 + x2)(2φ B − ¯φ B) + (f +)2(x1(φ 2( ¯φ B(¯r(− x′ 2 + x2 + 1) − r) +φ B(¯r(2x′ 2 − 2x2 − 3) + 3r)) + φ 4( ¯φ B − φ B)) + φ B(φ 2(¯r − r)x′ 2 + x2(φ 2(r − ¯r) + φ 4) + φ...

  4. [12]

    − 2x′ 2 + 2) − rφ B (x1 + x2 − x′

  5. [13]

    + ¯φ B(x′ 2 − 1))(f +)3 − (¯rφ 2((1 − 2r)φ B + r ¯φ B)x2 1 + (φ 4( ¯φ B − 2φ B) + φ 2(φ B(− (2r − 1)¯r(x2 − x′

  6. [14]

    − 2x′ 2 + 2) + ¯φ B(− r + x′ 2 + ¯r(rx2 − rx′ 2 + 1) − 1)))x1 − (x2 + 1)φ 4(φ B − ¯φ B) + rφ 2(φ B − ¯φ B)(x2 − x′ 2) − ¯rφ 2(φ B − ¯φ B)(x2 − x′

  7. [15]

    − (φ 2φ B + 2φ 4( ¯φ B − φ B))x′ 2)(f +)2 + (φ 4(¯r ¯φ B(r(x′ 2 − x2) − (r − 1)x1) +φ B(r(x′ 2 − 1) + ¯r((2r − 1)x1 + (r + 1)x2 − rx′ 2 − 2x′ 2 + 1))) + φ 2( ¯φ B(¯r((r + 1)x1 + x2 − x′ 2 + 1) − r) − φ B(− r + x′ 2 + ¯r(3rx1 + (r + 1)x2 − rx′ 2 − x′ 2 + 1))))f + + ¯rφ 2((r + 1...

  8. [16]

    − (φ B − ¯φ B)(φ 4(x′ 2 − 1) − rφ 2x′ 2))f + + ¯r(φ 2((r + 1)φ B − ¯φ B)x′ 2 − φ 4((r + 1)φ B − r ¯φ B )(x′ 2 − 1)))), (A12) DA2 (xl, x ′ l, y ) = − 8M 4 (f − − f +)f + ¯r(f +((f +)2x1φ 2(r − ¯r)(x′ 2 − 1)(2φ B − ¯φ B) + f +(φ 2 ¯r(x1(x′ 2 − 1)(2φ B − ¯φ B) +φ Bx′

  9. [17]

    + φ B(φ 4(x′ 2 − 1) − rφ 2x′ 2)) + ¯rφ B(rφ 4(x′ 2 − 1) − φ 2x′ 2)) + f − ((f +)3x1φ 2(r − ¯r) (− x′ 2 + x1 + x2)(2φ B − ¯φ B) + (f +)2(x1(φ 2(φ B(¯r(− 2x′ 2 + 2x2 + 3) − 3r) + ¯φ B(¯r(x′ 2 − x2 − 1) + r)) +φ 4( ¯φ B − φ B)) + φ B(φ 2(r − ¯r)x′ 2 + x2(φ 2(¯r − r) + φ 4) + φ 4(...

  10. [18]

    − (φ a 3 + φ s 3)x′ 2)(f +)3 + (2(r − ¯r)(φ a 3 − φ s 3) (2φ B − ¯φ B)x2 1 + (( ¯φ B(− 2(r − ¯r)x2 + (r − ¯r)x′ 2 − 2) + φ B(4(r − ¯r)x2 + (− 2r + 2¯r + 1)x′ 2 + 1))φ a 3 +(φ B(− 4(r − ¯r)x2 + (− 2r + 2¯r + 1)x′ 2 − 1) − ¯φ B(− 2(r − ¯r)x2 + (¯r − r)x′ 2 − 2))φ s 3)x1 +x2(φ a ...

  11. [19]

    − φ B(¯r(2x2 + x′ 2 + 2x1(x′ 2 + 3)) − 3(x′ 2 + 1)))φ s 3)f + +φ B((2r + ¯r(x′ 2 − 2))φ a 3 + (¯r(x′ 2 + 2) − 2r)φ s 3))(f − )2 − (x1φ B(φ a 3(2x1 + 2x2 − x′ 2 + 1) − φ s 3(2x1 + 2x2 + x′ 2 − 1))(f +)3 − ( ¯φ B(− 2¯rx2 1 + (− x′ 2r + 2r + ¯r(− 2x2 + x′ 2 − 2) + 2)x1 +(r − ¯r +...

  12. [20]

    − 2)x1 + (− r + ¯r − 2)x2 − 2rx′ 2 + 2¯rx′ 2 − x′ 2 + 1)φ s 3 − φ B(− 2(r − 2)¯rx2 1 + (r + (1 − 2r)x′ 2 − ¯r(2(r − 2)x2 + (r − 2)x′ 2 + 1) − 2)x1 − rx2 +(¯r − 3)x2 − 2rx′ 2 + 2¯rx′ 2 − 3x′ 2 + 3)φ s 3)(f +)2 + ((φ B((r + 3)(x′ 2 − 2) + ¯r((3r + 2)x2 +x1(r − 2x′ 2 + 8) − 3rx′ ...

  13. [21]

    − 1) + 1)φ a 3 + ( ¯φ B(− x′ 2r + r + ¯r(rx2 − rx1(x′ 2 − 2) +x′ 2 − 1) − 1) + φ B(x′ 2r − r + ¯r(− rx2 + x2 − x′ 2 + x1((2r + 1)x′ 2 − 3r) + 1) + 1))φ s 3)f +), (A19) DT3(xl, x ′ l, y ) = 16M 4 f − − f + ¯r((f +)2x1φ B(− ¯rφ a 3(− x′ 2 + 2x1 + 2x2 + 1) + ¯rφ s 3(x′ 2 + 2x1 + ...

  14. [22]

    − ¯r(φ a 3((x′ 2 − 2x1 − 2x2 − 1) ¯φ B + φ B((x1 − 3)x′ 2 + 3x2 + 3)) +φ s 3((x′ 2 + 2x1 + 2x2 − 1) ¯φ B + φ B(x1(x′ 2 − 2) − 3(x′ 2 + x2 − 1))))) + f − (¯r(φ a 3( ¯φ B((f + − 1)x′ 2 − 2(f +x1 + f +x2 − 1)) + φ B((2f +x1 + 3)(f +x1 + f +x2 − 1) − (f + − 1)(f +x1 + 3)x′ 2)) − φ...

  15. [23]

    + 3) +φ B(− r − 2x′ 2 + ¯r(8rx2 + (2r − 1)x′ 2 + 1) − 10))x1 + (− r + ¯r − 4)x2φ B + ¯φ B − 5rφ Bx′ 2 +5¯rφ Bx′ 2 − φ Bx′ 2 + r ¯φ Bx′ 2 − ¯r ¯φ Bx′ 2)φ s 3)(f +)2 + ((¯r ¯φ B(x′ 2 − 4rx1) +φ B(− r + x′ 2 + ¯r(12rx1 + 4rx2 − rx′ 2 − 5x′ 2 + 1) − 5))φ a 3 + (¯r ¯φ B(4rx1 + x′ 2...

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    (A25) The corresponding formulas for the B one can be obtained by the replacement B = A|r→− r

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