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The narrow D s0 * (2317) and broad D 0 * (2300-2400) charmed scalar mesons and their radial excitations are described in a coupled-channel quark model that also reproduces the properties of the light scalar nonet. All two-meson channels... more
The narrow D s0 * (2317) and broad D 0 * (2300-2400) charmed scalar mesons and their radial excitations are described in a coupled-channel quark model that also reproduces the properties of the light scalar nonet. All two-meson channels containing ground-state pseudoscalars and vectors are included. The parameters are chosen fixed at published values, except for the overall coupling constant λ, which is fine-tuned to reproduce the D s0 * (2317) mass, and a damping constant α for subthreshold contributions. Variations of λ and D 0 * (2300-2400) pole postions are studied for different α values. Calculated cross-sections for S-wave DK and Dπ scattering, as well as resonance pole positions, are given for the value of α that fits the light scalars. The thus predicted radially excited state D s0 *‧(2850), with a width of about 50MeV, seems to have been observed already.
We analyse electron-positron data for b-bbar production published by the BABAR Collaboration, in the invariant-mass interval delimited by the B-Bbar and Lambda(b)-Lambda(b) thresholds. In particular, we describe the Upsilon(10580)... more
We analyse electron-positron data for b-bbar production published by the BABAR Collaboration, in the invariant-mass interval delimited by the B-Bbar and Lambda(b)-Lambda(b) thresholds. In particular, we describe the Upsilon(10580) enhancement, not as a b-bbar resonance, but rather as a threshold phenomenon due to the opening of the B-Bbar decay channel and enhanced by the Upsilon(2D) bound-state pole not far below this threshold. The same data provide evidence for the true Upsilon(4S) resonance, which we find at 10.735 GeV with a width of 38 MeV. At higher energies, two more known Upsilon resonances are observed by us in the data and classified. The vital role played in our analysis by the universal confinement frequency omega is again confirmed, via a comparison with the charmonium spectrum. Comment: 5 pages, plain Latex, 1+3 figures
Research Interests:
We reply to a very recent Comment arxiv:1204.2349 by Bernhard, Friedrich, Schlueter, and Schoenning for the COMPASS Collaboration, in which it is stated that Monte-Carlo simulations of COMPASS data do not support any interpretation of a... more
We reply to a very recent Comment arxiv:1204.2349 by Bernhard, Friedrich, Schlueter, and Schoenning for the COMPASS Collaboration, in which it is stated that Monte-Carlo simulations of COMPASS data do not support any interpretation of a peaked structure in diphoton invariant-mass distributions below the neutral pion mass in terms of a new resonance. Here we show, by directly comparing the simulations to the COMPASS data themselves, that the authors' claim is not substantiated by the Monte-Carlo results presented in the Comment.
Research Interests:
Data on B+ --> J/psi phi K+ and the Y(4140) enhancement recently reported by the CDF collaboration [arxiv:0903.2229] are analysed. The threshold behaviour, as well as traces of the X(4260) enhancement, the known c-cbar resonances psi(2D),... more
Data on B+ --> J/psi phi K+ and the Y(4140) enhancement recently reported by the CDF collaboration [arxiv:0903.2229] are analysed. The threshold behaviour, as well as traces of the X(4260) enhancement, the known c-cbar resonances psi(2D), psi(4S), and a tentative psi(3D) state, as observed in the mass distribution, suggest that the J/psi+phi system has quantum numbers JPC=1--. It is then argued that the Y(4140) enhancement does not represent any kind of resonance, but instead is a natural consequence of the opening of the J/psi+phi channel.
), which were recently reported to have been observed in experiment, are non-resonant manifestations of the Regge zeros that appear in the production amplitude of the Resonance-Spectrum Expansion. Charmonium c-cbar states are visible on... more
), which were recently reported to have been observed in experiment, are non-resonant manifestations of the Regge zeros that appear in the production amplitude of the Resonance-Spectrum Expansion. Charmonium c-cbar states are visible on the slopes of these enhancements.
Light constituent quark masses and the corresponding dynamical quark masses are determined by data, the Quark-Level Linear σ Model, and infrared QCD. This allows to define effective nonstrange and strange current quark masses which... more
Light constituent quark masses and the corresponding dynamical quark masses are determined by data, the Quark-Level Linear σ Model, and infrared QCD. This allows to define effective nonstrange and strange current quark masses which reproduce the experimental pion and kaon masses very accurately, by simple additivity. Moreover, the masses of the light scalar mesons σ(600) and κ(800) can be obtained straightforwardly from the constituent quark masses. In contrast, the usual nonstrange and strange current quark masses employed by Chiral Perturbation Theory do not allow a simple quantitative explanation of the pion and kaon masses.
We study the structures in cross sections and R ratios concerning e{sup +}e annihilation reactions, for open-strangeness, open-charm, and open-beauty hadron-pair production, from the perspective of a recently developed production... more
We study the structures in cross sections and R ratios concerning e{sup +}e annihilation reactions, for open-strangeness, open-charm, and open-beauty hadron-pair production, from the perspective of a recently developed production formalism. Special attention is paid to nonresonant contributions. Known, tentative, and yet unknown resonances are spotted, with estimates for their masses and widths.
An outline is given how to formulate a relativistic unitarized constituent quark model of mesons in momentum space, employing harmonic quark confinement. As a first step, the momentum-space harmonic-oscillator potential is solved in a... more
An outline is given how to formulate a relativistic unitarized constituent quark model of mesons in momentum space, employing harmonic quark confinement. As a first step, the momentum-space harmonic-oscillator potential is solved in a relativistically covariant, three-dimensional quasipotential framework for scalar particles, using the spline technique. Then, an illustrative toy model with the same dynamical equations but now one $q\bar{q}$ and one meson-meson channel, coupled to one another through quark exchange describing the $^3P_0$ mechanism, is solved in closed form on a spline basis. Conclusions are presented on how to generalize the latter to a realistic multichannel quark/meson model.
By comparing SU (3)-breaking scales of linear mass formulae, it is shown that the lowest vector, axial-vector, and scalar mesons all have a q¯q configuration, while the ground-state octet and decuplet baryons are qqq. Also, the... more
By comparing SU (3)-breaking scales of linear mass formulae, it is shown that the lowest vector, axial-vector, and scalar mesons all have a q¯q configuration, while the ground-state octet and decuplet baryons are qqq. Also, the quark-level linear σ model is employed to predict similar q¯q and qqq states. Finally, the approximate mass degeneracy of the scalar a0(980) and f0(980) mesons is demonstrated to be accidental.
Masses and widths of the axial-vector charmed mesons D1(2420), D1(2430), Ds1(2536), and Ds1(2460) are calculated nonperturbatively in the resonance-spectrum-expansion model, by coupling various open and closed meson-meson channels to the... more
Masses and widths of the axial-vector charmed mesons D1(2420), D1(2430), Ds1(2536), and Ds1(2460) are calculated nonperturbatively in the resonance-spectrum-expansion model, by coupling various open and closed meson-meson channels to the bare JP=1+ cq¯ (q=u, d) and cs¯ states. The coupling to two-meson channels dynamically mixes and lifts the mass degeneracy of the spectroscopic P13 and P11 states, as an alternative to the usual spin-orbit splitting. Of the two resulting S-matrix poles in either case, one stays very close to the energy of the bare state, as a quasibound state in the continuum, whereas the other shifts considerably. This is in agreement with the experimental observation that the D1(2420) and Ds1(2536) have much smaller widths than one would naively expect. The whole pattern of masses and widths of the axial-vector charmed mesons can thus be quite well reproduced with only two free parameters, one of which being already strongly constrained by previous model calculations. Finally, predictions for pole positions of radially excited axial-vector charmed mesons are presented.
We study meson (pi,Kappa) form factors in general, and specialize at a later stage to a specific scheme, namely the quark-level linear sigma model (LsigmaM). We compute a variety of electromagnetic and weak observables of light mesons,... more
We study meson (pi,Kappa) form factors in general, and specialize at a later stage to a specific scheme, namely the quark-level linear sigma model (LsigmaM). We compute a variety of electromagnetic and weak observables of light mesons, including pion and kaon form factors and charge radii, charged-pion polarizabilities, semileptonic weak Kappal3 decay, semileptonic weak radiative pion and kaon form factors, radiative decays of vector mesons, and nonleptonic weak Kappa2pi decay. We find good to very good agreement with experiment of all these predicted observables.
The constraint implied by the unitarity relation \Im m(A)=T^{\ast}A is derived for a three-body production amplitude A that consists of a complex linear combination of elements of the two-body scattering amplitude T. We conclude that the... more
The constraint implied by the unitarity relation \Im m(A)=T^{\ast}A is derived for a three-body production amplitude A that consists of a complex linear combination of elements of the two-body scattering amplitude T. We conclude that the unitarity relation does not impose a realness condition on the coefficients in the expansion of A in terms of T.
We present evidence for small oscillations we observe in electron-positron and proton-antiproton annihilation data, with a periodicity of 76+-2 MeV, independent of the beam energy. We discuss some possible scenarios to explain the... more
We present evidence for small oscillations we observe in electron-positron and proton-antiproton annihilation data, with a periodicity of 76+-2 MeV, independent of the beam energy. We discuss some possible scenarios to explain the phenomenon.
On assuming that Weak substructure has a dynamics which is similar to quantum chromodynamics but much stronger, we conclude that unquenching is indispensable for predictions on the spectrum of Weak-substructure resonances.
We present further and more compelling evidence of the existence of E(38), a light boson that most probably couples exclusively to quarks and gluons. Observations presented in a prior paper will be rediscussed for completeness.
We study S- and P-wave pion-pion and kaon-pion distributions for the three-body production processes $J/\psi\to\omega\pi^{+}\pi^{-}$, $D^{+}\to K^{-}\pi^{+}\pi^{+}$, $D^{+}\to K^{+}\pi^{+}\pi^{-}$, and $D^{+}_{s}\to K^{+}\pi^{+}\pi^{-}$,... more
We study S- and P-wave pion-pion and kaon-pion distributions for the three-body production processes $J/\psi\to\omega\pi^{+}\pi^{-}$, $D^{+}\to K^{-}\pi^{+}\pi^{+}$, $D^{+}\to K^{+}\pi^{+}\pi^{-}$, and $D^{+}_{s}\to K^{+}\pi^{+}\pi^{-}$, by applying the Resonance-Spectrum Expansion to the two-body subamplitudes. The results are compared to the corresponding signals reported by the BES, E791 and FOCUS collaborations. We find that the studied S- and P-wave channels can largely account for the observed structures in the data up to 1 GeV.
We present evidence for the existence of a light scalar particle that most probably couples exclusively to gluons and quarks. Theoretical and phenomenological arguments are presented to support the existence of a light scalar boson for... more
We present evidence for the existence of a light scalar particle that most probably couples exclusively to gluons and quarks. Theoretical and phenomenological arguments are presented to support the existence of a light scalar boson for confinement and quark-pair creation. Previously observed interference effects allow to set a narrow window for the scalar's mass and also for its flavor-mass-dependent coupling to quarks. Here, in order to find a direct signal indicating its production, we study published BABAR data on leptonic bottomonium decays. We observe a clear excess signal in the invariant-mass projections of electron-positron and muon-antimuon, which may be due to the emission of a so far unobserved scalar particle with a mass of about 38 MeV. In the process of our analysis, we also find an indication of the existence of a bottomonium-like hybrid state at about 10.061 GeV. Further signals might be interpreted as replicas of the light scalar, with masses roughly two and three times as large.
We study the structures in cross sections and R ratios concerning electron-positron annihilation reactions, for open-strangeness, open-charm and open-beauty hadron-pair production, from the perspective of a recently developed production... more
We study the structures in cross sections and R ratios concerning electron-positron annihilation reactions, for open-strangeness, open-charm and open-beauty hadron-pair production, from the perspective of a recently developed production formalism. Special attention is paid to non-resonant contributions. Known, tentative, and yet unknown resonances are spotted, with estimates for their masses and widths.
A multichannel calculation of excited JPC=1--ϕ states is carried out within a generalization of the resonance-spectrum expansion, which may shed light on the classification of the ϕ(2170) resonance, discovered by BABAR and originally... more
A multichannel calculation of excited JPC=1--ϕ states is carried out within a generalization of the resonance-spectrum expansion, which may shed light on the classification of the ϕ(2170) resonance, discovered by BABAR and originally denoted X(2175). In this framework, a complete spectrum of bare ss¯ states is coupled to those Okubo-Zweig-Iizuka-allowed decay channels that should be most relevant for the considered energy range. The included S- and P-wave two-meson channels comprise the lowest pseudoscalar, vector, scalar, and axial-vector mesons, while in the qq¯ sector both the S13 and D13 states are coupled. The only two free parameters are tuned so as to reproduce mass and width of the ϕ(1020), but come out reasonably close to previously used values. Among the model’s T-matrix poles, there are good candidates for observed resonances, as well as other ones that should exist according to the quark model. Besides the expected resonances as unitarized confinement states, a dynamical resonance pole is found at (2186-i246)MeV. The huge width makes its interpretation as the ϕ(2170) somewhat dubious, but further improvements of the model may change this conclusion.
We argue that the X(4260) enhancement contains a wealth of information on 1 cc spectroscopy. We discuss the shape of the X(4260) observed in the Okubo-Zweig-Iizuka-forbidden process e{sup +}e{sup +}J/, in particular, at and near vector... more
We argue that the X(4260) enhancement contains a wealth of information on 1 cc spectroscopy. We discuss the shape of the X(4260) observed in the Okubo-Zweig-Iizuka-forbidden process e{sup +}e{sup +}J/, in particular, at and near vector charmonium resonances as well as open-charm threshold enhancements. The resulting very broad X(4260) structure does not seem to classify itself as a 1 cc resonance, but its detailed shape allows us to identify new vector charmonium states with higher statistics than in open-charm decay. Here, we estimate the resonance parameters of the (3D). Our approach also provides an explanation for the odd dip in the {sup +}J/ data precisely at the (4415) resonance.
We comment on the recent observation of the decay mode DsJ*(2860)+-->D*K by the BABAR Collaboration [B. Aubert et al. (BABAR Collaboration), arXiv:0908.0806], and contest their conclusion that the data exclude a 0+ assignment for the... more
We comment on the recent observation of the decay mode DsJ*(2860)+-->D*K by the BABAR Collaboration [B. Aubert et al. (BABAR Collaboration), arXiv:0908.0806], and contest their conclusion that the data exclude a 0+ assignment for the DsJ*(2860)+. In particular, we argue that the observed branching fraction B(DsJ*(2860)+-->D*K)/B(DsJ*(2860)+-->DK)=1.1±0.15±0.19 supports the existence of two largely overlapping resonances at about 2.86 GeV, namely, a pair of radially excited tensor (2+) and scalar (0+) cs¯ states. This scenario is further justified by comparing with the corresponding excited charmonium states. Also other aspects of the charm-strange spectrum are discussed.
We briefly review the general status of meson spectroscopy, especially in light of the often made claim that there are too many observed resonances to be accounted for as $q\bar{q}$ states. Also, the adequacy of the usual... more
We briefly review the general status of meson spectroscopy, especially in light of the often made claim that there are too many observed resonances to be accounted for as $q\bar{q}$ states. Also, the adequacy of the usual Coulomb-plus-linear, alias "funnel", confining potential for reproducing the experimental spectra of light, heavy-light, and heavy mesons is critically analysed. Thus, many serious discrepancies are observed and discussed. As possible causes, we suggest the neglect of unitarisation and other coupled-channel effects, as well as the deficiency of the funnel potential itself. In order to illustrate our alternative, "unquenched" approach, we present some recent examples of successfully described puzzling mesonic enhancements and resonances, such as the charmonium states X(4260) and X(3872), as well as the axial-vector charmed mesons $D_1(2420)$, $D_1(2430)$, $D_{s1}(2536)$, and $D_{s1}(2460)$.
... Authors: Buccella, F.; Bugg, DV; Kalashnikova, Yu. S.; Liu, K.-F.; Llanes-Estrada, F.; Matsuki, T.; Oller, JA; Rosner, JL; Rupp, G.; Scadron, MD. ... Publication: SCADRON70: Workshop on Scalar Mesons and Related Topics Honoring... more
... Authors: Buccella, F.; Bugg, DV; Kalashnikova, Yu. S.; Liu, K.-F.; Llanes-Estrada, F.; Matsuki, T.; Oller, JA; Rosner, JL; Rupp, G.; Scadron, MD. ... Publication: SCADRON70: Workshop on Scalar Mesons and Related Topics Honoring Michael Scadron's 70th Birthday. ...
ABSTRACT
Recent data on e(-)+e(+)-->b+anti-b by the BABAR Collaboration [B. Aubert et al, Phys. Rev. Lett. 102, 012001 (2009), arXiv:0809.4120] in the energy range delimited by the B+anti-B and Lambda(b)(+)+Lambda(b)(-) thresholds are analyzed in... more
Recent data on e(-)+e(+)-->b+anti-b by the BABAR Collaboration [B. Aubert et al, Phys. Rev. Lett. 102, 012001 (2009), arXiv:0809.4120] in the energy range delimited by the B+anti-B and Lambda(b)(+)+Lambda(b)(-) thresholds are analyzed in a multichannel formalism that incorporates the usual Breit-Wigner resonances, but interfering with a background signal due to the opening of open-bottom thresholds. In particular, the Upsilon(4S) resonance is determined to have a mass of 10.735 GeV and a width of 38 MeV. Also two higher Upsilon resonances are identified, parametrized, and classified. Moreover, it is found that near the B+anti-B threshold open-bottom production in electron-positron annihilation is dominated by the reaction chain e(-)+e(+)-->n+anti-n-->B+anti-B (n=u/d) rather than e(-)+e(+)-->b+anti-b-->B+anti-B whereas near the B(s)+anti-B(s) threshold the reaction chain e(-)+e(+)-->s+anti-s-->B+anti-B dominates the production amplitude. The vital role played in this analysis by the universal confinement frequency, defined in 1980 [E. van Beveren, C. Dullemond, and G. Rupp, Phys. Rev. D21, 772 (1980)] and accurately determined in 1983 [E. van Beveren, G. Rupp, T.A. Rijken, and C. Dullemond, Phys. Rev. D27, 1527 (1983)], is further confirmed.
] are analysed by modelling missing signals that should be present in the complete production amplitude with vector quantum numbers in the charmonium region. Thus, it is shown that the data contain evidence for the existence of the... more
] are analysed by modelling missing signals that should be present in the complete production amplitude with vector quantum numbers in the charmonium region. Thus, it is shown that the data contain evidence for the existence of the psi(5S), the psi(4D), the psi(6S) and the psi(5D) c-cbar vector states, and furthermore a clear indication for the mass and width of the psi(3D). Moreover, it is shown that signs of the psi(3S), psi(2D) and psi(4S) can be observed in the same data. Finally, it is argued that the X(4260) enhancement is not a resonance, but rather a phenomenon connected with the opening of the D(s)(*)-Dbar(s)(*) threshold and the coupling to the J/psi f(0)(980) channel.
The nature of the X(3872) enhancement is analyzed in the framework of the Resonance-Spectrum Expansion, by studying it as a regular J PC =1++ charmonium state, though strongly influenced and shifted by open-charm decay channels. The... more
The nature of the X(3872) enhancement is analyzed in the framework of the Resonance-Spectrum Expansion, by studying it as a regular J PC =1++ charmonium state, though strongly influenced and shifted by open-charm decay channels. The observed but Okubo-Zweig-Iizuka-forbidden ρ 0 J/ ψ and ωJ/ ψ channels are coupled as well, but effectively smeared out by using complex ρ 0 and ω masses, in order to account for their physical widths, followed by a rigorous algebraic procedure to restore unitarity. A very delicate interplay between the D 0 D ∗0, ρ 0 J/ ψ, and ωJ/ ψ channels is observed. The data clearly suggest that the X(3872) is a very narrow axial-vector cbar{c} resonance, with a pole at or slightly below the D 0 D ∗0 threshold.
The contribution of strange sea quarks to the proton mass and spin, as well as the related pion-nucleon sigma term, are briefly revisited, in the light of new experimental and lattice results. Also the predictions of chiral perturbation... more
The contribution of strange sea quarks to the proton mass and spin, as well as the related pion-nucleon sigma term, are briefly revisited, in the light of new experimental and lattice results. Also the predictions of chiral perturbation theory for these quantities are discussed.
Masses and widths of the four light scalar mesons σ, κ  , a0(980)a0(980) and f0(980)f0(980) may be reproduced in a model where mesons scatter via a qq¯ loop. A transition potential is used to couple mesons to qq¯ at a radius of... more
Masses and widths of the four light scalar mesons σ, κ  , a0(980)a0(980) and f0(980)f0(980) may be reproduced in a model where mesons scatter via a qq¯ loop. A transition potential is used to couple mesons to qq¯ at a radius of ∼0.57 fm∼0.57 fm. Inside this radius, there is an infinite bare spectrum of confined qq¯ states, for which a harmonic oscillator is chosen here. The coupled-channel system approximately reproduces the features of both light and heavy meson spectroscopy. The generation of σ, κ  , a0(980)a0(980) and f0(980)f0(980) is a balance between attraction due to the qq¯ loop and suppression of the amplitudes at the Adler zeros. Phase shifts increase more rapidly as the coupling constant to the mesons increases. This leads to resonant widths which decrease with increasing coupling constant—a characteristically non-perturbative effect.
A multichannel Schr\"odinger equation with both quark-antiquark and meson-meson components, using a harmonic-oscillator potential for $q\bar{q}$ confinement and a delta-shell string-breaking potential for decay, is applied to the... more
A multichannel Schr\"odinger equation with both quark-antiquark and meson-meson components, using a harmonic-oscillator potential for $q\bar{q}$ confinement and a delta-shell string-breaking potential for decay, is applied to the axial-vecor $X(3872)$ and lowest vector charmonia. The model parameters are fitted to the experimental values of the masses of the $X(3872)$, $J/\psi$ and $\psi(2S)$. The wave functions of these states are computed and then used to calculate the electromagnetic decay widths of the $X(3872)$ into $J/\psi\gamma$ and $\psi(2S)\gamma$.
We analyse electron-positron data for b-bbar production published by the BABAR Collaboration, in the invariant-mass interval delimited by the B-Bbar and Lambda(b)-Lambda(b) thresholds. In particular, we describe the Upsilon(10580)... more
We analyse electron-positron data for b-bbar production published by the BABAR Collaboration, in the invariant-mass interval delimited by the B-Bbar and Lambda(b)-Lambda(b) thresholds. In particular, we describe the Upsilon(10580) enhancement, not as a b-bbar resonance, but rather as a threshold phenomenon due to the opening of the B-Bbar decay channel and enhanced by the Upsilon(2D) bound-state pole not far below this threshold. The same data provide evidence for the true Upsilon(4S) resonance, which we find at 10.735 GeV with a width of 38 MeV. At higher energies, two more known Upsilon resonances are observed by us in the data and classified. The vital role played in our analysis by the universal confinement frequency omega is again confirmed, via a comparison with the charmonium spectrum.
We present flavour-symmetric couplings for the OZI-allowed three-meson vertices of effective meson theories, which for the case of the two-meson channels to which the flavour-singlet scalar meson couples, are endowed with a correction... more
We present flavour-symmetric couplings for the OZI-allowed three-meson vertices of effective meson theories, which for the case of the two-meson channels to which the flavour-singlet scalar meson couples, are endowed with a correction factor with respect to the standard formula.
We argue that the spectra of quark-antiquark systems should better be studied from higher radial excitations and, in particular, from configurations with well-defined quantum numbers, rather than from ground states and lower radial... more
We argue that the spectra of quark-antiquark systems should better be studied from higher radial excitations and, in particular, from configurations with well-defined quantum numbers, rather than from ground states and lower radial excitations, the most suitable system being charmonium.
Data indicate that $\Delta I=1/2$ transitions account for 4.5-4.7% of both CP conserving and CP violating $K \to 2\pi$ decays, as well as CP conserving radiative $K \to \pi\pi\gamma$ processes. Observed $K \to \pi\pi\gamma/\pi\pi$... more
Data indicate that $\Delta I=1/2$ transitions account for 4.5-4.7% of both CP conserving and CP violating $K \to 2\pi$ decays, as well as CP conserving radiative $K \to \pi\pi\gamma$ processes. Observed $K \to \pi\pi\gamma/\pi\pi$ branching ratios are shown to scale near $\alpha/\pi$ or $\alpha/2\pi$. The $K_L$-$K_S$ mixing angle $\phi$ and the semileptonic weak-rate asymmetry $\delta$ are reviewed, and theory is shown to be consistent with data. Also, $K \to 2\pi$ $\Delta I=1/2$ dominance is studied in the context of the chiral constituent quark model, displaying again excellent agreement with data. Finally, indirect and direct kaon CP violation are successfully described in the framework of photon-mediated loop graphs. This suggests that kaon CPV can be understood via second-order weak transitions, radiatively corrected.

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