The radiative decay of the axial-vector resonances into a pseudoscalar meson and a photon is studied using the vector meson Lagrangian obtained from the hidden gauge symmetry (HGS) formalism. The formalism is well suited to study this problem since it deals with pseudoscalar and vector mesons in a unified way, respecting chiral invariance. We show explicitly the gauge invariance of the set of diagrams that appear in the approach and evaluate the radiative decay width of the a1(1260) and b1(1235) axial vector meson resonances into pi gamma. We also include the contribution of loops involving anomalous couplings and compare the results to those obtained previously within another formalism.
In this article, we perform a systematic study of the mass spectrum of the axial-vector hidden charmed and hidden bottom tetraquark states using the QCD sum rules, and identify the $Z^+(4430)$ as an axial-vector tetraquark state tentatively.
We discuss a connection between axial anomaly and polarized radiative processes. By comparison with the corresponding unpolarized cases, we consider some physical outputs for the $π^+$ and $Z^0$ polarized radiative decays. We analyse in detail the pattern of mass singularity cancellation.
We evaluate the radiative decay into a pseudoscalar meson and a photon of the whole set of the axial-vector mesons dynamically generated from the vector-pseudoscalar meson ($VP$) interaction. We take into account tree level and loop diagrams coming from the axial-vector decay into a vector and a pseudoscalar meson. We find a large span for the values of the radiative widths of the different axial-vector mesons. In particular, we evaluate the radiative decay into $K γ$ of the two $K_1(1270)$ states, recently claimed theoretically, and discuss the experimental values quoted so far on the assumpt…
We extend the approach of Dolgov and Zakharov to the axial anomaly to study polarized radiative decays. We analyse the pattern of mass singularity cancellation in the corresponding decay rates. We compare polarized and unpolarized cases. The cancellation of the infrared and collinear singularities is verified to all powers of the lepton mass for the $π^+$ and $Z^0$ polarized radiative decays.
The Lagrangian of pseudoscalar, vector, and axial-vector mesons is determined by the explicit global chiral symmetry and hidden local chiral symmetry. There are fourteen interacting terms up to the dimension-four of covariant derivatives rather than the usual eleven interacting terms given in literature from hidden local symmetry approach. There are only two independent parameters. The three new terms are important in phenomenology.
In this work, we use the QCD sum rule method to study the radiative decays of the vector and axial-vector $B_c$ mesons to each of three charmed strange mesons, $D_s$, $D^*_s$ and $D_{s1}$, through their dominant weak annihilation channels. We calculate all relevant transition form factors, which are used to estimate the branching fractions at different channels. The order of branching ratios are obtained to be in the order of $ 10^{-6} - 10^{-5} $, which may be checked via differnet experiments.
Supported by recent lattice results, we consider a scenario in which a U(1)-breaking condensate survives across the chiral transition in QCD. This scenario has important consequences on the pseudoscalar-meson sector, which can be studied using an effective Lagrangian model. In particular, generalizing the results obtained in a previous paper (where the zero-temperature case was considered), we study the effects of this U(1) chiral condensate on the radiative decay eta' --> gamma gamma at finite temperature.
A SU(4) flavor symmetrical Lagrangian is constructed for the interaction of the pseudo-scalar mesons with the vector mesons. SU(4) symmetry is broken to SU(3) by suppression of terms in the Lagrangian where the interaction should be driven by charmed mesons. Chiral symmetry can be restored by setting this new SU(4) symmetry breaking parameters to zero. Unitarization in coupled channels leads to the dynamical generation of resonances. Many known axial resonances can be identified including the new controversial X(3872) and the structure found recently by Belle around 3875 MeV in the hidden char…
In this article, we study the mass spectrum of the scalar hidden charm and hidden bottom tetraquark states which consist of the axial-axial type and the vector-vector type diquark pairs with the QCD sum rules.
A previous model where the low-lying axial-vector mesons are dynamically generated, implementing unitarity in coupled channels in the vector-pseudoscalar ($VP$) meson interaction, is applied to evaluate the decay widths of the a_1(1260)$ and $b_1(1235)$ axial-vector mesons into $πγ$. Unlike the case of the $a_1$, the $b_1$ radiative decay is systematically underestimated at tree level. In this work we evaluate for the first time the loop contribution coming from an initial $VP$ vertex. Despite the large superficial divergence of the loops, the convergence of the relevant loops can be establish…
We study the radiative $ϕ$ decay into $π^0 π^0 γ$ and $π^0 ηγ$ taking into account mechanisms in which there are two sequential vector-vector-pseudoscalar or axial-vector--vector--pseudoscalar steps followed by the coupling of a vector meson to the photon, considering the final state interaction of the two mesons. There are other mechanisms in which two kaons are produced through the same sequential mechanisms or from $ϕ$ decay into two kaons and then undergo final state interaction leading to the final pair of pions or $π^0 η$, this latter mechanism being the leading one. The results of the p…
We study composite systems of light ($ρ$ and $ω$) and heavy ($D^*$) vector mesons by using the interaction within the hidden gauge formalism. We find a strong attraction in the isospin, spin channels $(I,S)=(1/2,0)$; $(1/2,1)$; $(1/2,2)$ with positive parity. The attraction is large enough to strongly bind these mesons in states with these quantum numbers, leading to states which can be identified with $D_2^*(2460)$ and probably with $D^*(2640)$, the last one without experimental spin and parity assignment. In the case of $I=3/2$ one obtains repulsion and thus, no exotic mesons in this sector …
We apply a unitary approach together with a set of hidden-gauge Lagrangians to study the vector-vector interaction. We study the case of the $ρ$($ω$)$D^*$ interaction within the model. In $I=1/2$ we get strong enough attraction to bind the system. Concretely, we get one resonance for each spin $J=0,1,2$. For J=1 and 2 these resonances can be easily identified with the $D^*(2460)$ and $D^*(2640)$. Whereas the state for J=0 with mass $M\sim2600$ MeV is a prediction of the model. In $I=3/2$ we get a repulsive interaction and hence no exotic dynamically generated states.
In this paper, we present the full Lagrangian of mesons (pseudoscalars, vectors and axial-vectors) to $O(p^4)$ by using the explicit global chiral symmetry and hidden local symmetry in the chiral limit. In this approach, we see that there are many other terms besides the usual eleven terms given in the literature from hidden local symmetry approach. Of particular, there are some terms in our full results which are important for understanding the vector meson dominance and $π-π$ scattering and providing consistent predictions on the decay rates of $a_1\toγπ$ and $a_1\toρπ$ as well as for constr…
We present the formalism for the decay of dynamically generated scalar mesons with open- or hidden-charm and give results for the decay of D^*_{s0} (2317) to γD_s^* plus that of a hidden charm scalar meson state predicted by the theory around 3700 MeV decaying into γJ/ψ.
Strange and non-strange axial-vector meson radiative decays contribute to photon production in hadron gas. One- and two-hadron radiative decay modes of $b_{1}(1235)$, $a_{1}(1260)$ and $K_{1}(1270)$ are studied. At 200 MeV temperature and for a narrow range in photon energies they contribute more to the net thermal photon production rate than $πρ\rightarrow πγ$, $ππ\rightarrow ργ$ or $ρ\rightarrowππγ$. They provide significant contribution to the rate for photon energies as high as 1.5--2.0 GeV. For higher energies they are less important.
We propose a radiative seesaw model with a hidden $U(1)$ gauge symmetry. In order to have anomaly cancellations, we need to introduce several new fermions that contribute to muon anomalous magnetic dipole moment $(g-2)_μ$ as well as explain the neutrino oscillation data. We also consider a fermionic dark matter candidate that correlates with $(g-2)_μ$ and neutrino mass matrix at the same time. We show allowed regions in our input parameters satisfying several constraints. Finally, we also briefly discuss the possibility of the resolution of the Hubble tension via neutrino self-interactions med…
We propose two-loop neutrino mass model with gauged hidden $U(1)$ symmetry, and discuss a Majorana type of dark matter candidate that has semi-annihilation processes in the relic density as well as lepton flavor violations and muon anomalous magnetic moment. Also, we demonstrate global analysis to satisfy neutrino oscillation data, lepton flavor violations, and relic density of dark matter candidate and show that semi-annihilation modes play a crucial role in finding observed relic density.
We present a calculation of full one-loop radiative corrections, including the constant term, to the asymmetry parameter of polarised neutron beta decay. This gives the radiative correction to the axial coupling constant $g_A$ extracted from the beta asymmetry so that it ties to $g_A$ that appears in neutron decay lifetime in a consistent renormalisation scheme. We find that the ratio of axial-vector to vector couplings determined from the beta asymmetry, after taking account of the outer radiative correction, is related to the bare value as $G_A/G_V=1.0012 G_A^0/G_V^0$.
We study the properties of axial vector mesons a1 and b1 as relativistic S-wave states which are predicted in the U~(12)-scheme, through the analyses of their radiative and pionic decays. Specifically, partial widths of the strong a1 (b1) -> rho(omega) pi processes, their D/S-wave amplitude ratios, and radiative transition widths of a1(b1) -> pi gamma processes are calculated by using a simple decay interaction model, and made a comparison with the respective experimental values.
We propose a phenomenological Lagrangian for the decay of the SU(3) nonets of the axial-vector mesons of $J^{PC}=1^{+-},1^{++}$ into a vector meson and a pseudoscalar constructed with tensor fields for the vector and axial-vector mesons. The formulation leads to a good reproduction of the different decay branching ratios and assuming vector meson dominance (VMD) it also leads to good results for the radiative decay of the $a_1$ into pion and photon, and in agreement with the structure proposed in the chiral tensor formulation of radiative decay of axial-vector mesons. The two SU(3) parameters …
We present a contribution of higher order to neutrino-electron scattering that is a charged-current counterpart of both the anomalous axial-vector triangle and possible non-standard interaction contributions. It arises in the standard model with massive neutrinos, and renormalizes the nondiagonal axial-vector form-factor at low energies. We show that, due to the small size of radiative corrections, the neutrino-electron scattering still provides a discovery potential for some of the non-standard neutrino interactions proposed in the literature.
This work is an extension of our previous work in \cite{bhatnagar20} to calculate M1 transitions, $0^{-+}\rightarrow 1^{--} γ$, and E1 transitions involving axial vector mesons such as, $1^{+-} \rightarrow 0^{-+}γ$, and $0^{-+}\rightarrow 1^{+-} γ$ for which very little data is available as of now. We make use of the general structure of the transition amplitude, $M_{fi}$ derived in our previous work \cite{bhatnagar20} as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadrons. In the present work,…
With the chiral quark model, we have calculated the axial charges of the pentaquark states with $(I,I_3)=(\frac{1}{2},\frac{1}{2})$ and $J^{P}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}$. The $P_c$ states with the same $J^P$ quantum numbers but different color-spin-flavor configurations have very different axial charges, which encode important information on their underlying structures. For some of the $J^{P}=\frac{3}{2}^{\pm}$ or $\frac{5}{2}^{\pm}$ pentaquark states, their axial charges are much smaller than that of the proton.
We study the vector and axial-vector form factors of radiative kaon decay within the framework of the gauged nonlocal effective chiral action from the instanton vacuum, focusing on the effects of flavor SU(3) symmetry breaking. The general tendency of the results are rather similar to those of radiative pion decays: The nonlocal contributions make the results of the vector form factor increased by about $20\,\%$, whereas they reduce those of the axial-vector form factor by almost $30\,\%$. Suppressing the prefactors consisting of the kaon mass and the pion decay constant, we scrutinize how the…
In this work we present our model that dynamically generates resonances from charmed and light pseudoscalar and vector mesons. The model generates a rich spectrum of charmed and hidden charm scalar and axial resonances, most of them which can be associated with observed states. Other states are predicted, in particular a hidden charm scalar resonance with mass close to the $D\bar D$ threshold. For this resonance we present calculations which show that an observed enhancement in the $D\bar D$ mass spectrum measured by Belle in reactions producing $D\bar D J/ψ$ in the final state could be origin…
We calculate the branching ratios of $B\to K_1γ$ at next-to-leading order (NLO) of $α_s$ where $K_1$ is the orbitally excited axial vector meson. The NLO decay amplitude is divided into the vertex correction and the hard spectator interaction part. The one is proportional to the weak form factor of $B\to K_1$ transition while the other is a convolution between light-cone distribution amplitudes and hard scattering kernel. Using the light-cone sum rule results for the form factor, we have $\calB(B^0\to K_1^0(1270)γ)=(0.828\pm0.335)\times 10^{-5}$ and $\calB(B^0\to K_1^0(1400)γ)=(0.393\pm0.151)\…
We develop a previously proposed gauge-invariant method to integrate out infinite towers of vector and axialvector mesons arising as Kaluza-Klein (KK) modes in a class of holographic models of QCD (HQCD). We demonstrate that HQCD can be reduced to the chiral perturbation theory (ChPT) with the hidden local symmetry (HLS) (so-called HLS-ChPT) having only the lowest KK mode identified as the HLS gauge boson, and the Nambu-Goldstone bosons. The ${\cal O} (p^4)$ terms in the HLS-ChPT are completely determined by integrating out infinite towers of vector/axialvector mesons in HQCD: Effects of highe…
The masses and decays of the scalar D_{s0}^*(2317) and axial-vector D_{s1}^*(2460) charmed strange mesons are calculated consistently in the hadrogenesis conjecture. These mesons decay either strongly into the isospin-violating pi^0 D_s and pi^0 D_s^* channels or electromagnetically. They are generated by coupled-channel dynamics based on the leading order chiral Lagrangian. The effect of chiral corrections to chiral order Q_χ^2 is investigated. We show that taking into account large-N_c relations to determine the strength of these correction terms implies a measurable signal for an exotic axi…
We calculate the leading radiative corrections to the axial current in the chiral separation effect in dense QED in a magnetic field. Contrary to the conventional wisdom suggesting that the axial current should be exactly fixed by the chiral anomaly relation and is described by the topological contribution on the lowest Landau level in the free theory, we find in fact that the axial current receives nontrivial radiative corrections. The direct calculations performed to the linear order in the external magnetic field show that the nontrivial radiative corrections to the axial current are provid…
We propose an inverse seesaw scenario under hidden $U(1)$ gauge symmetry, having rather natural hierarchies among neutral fermion mass scales. The hierarchies are derived from theory and experimental constraints. The theory requires Majorana exotic masses have to be induced at one-loop level. The experimental side requests that the Dirac mass terms has to be highly suppressed to satisfy the constraints from the lepton flavor violations(LFVs) such as $μ\to eγ$. In order to induce such a small Majorana exotic masses we introduce exotic fermions and bosons that also provide us additional intrigui…
We study the light vector and axial-vector mesons. According to the hadrogenesis conjecture the nature of the two types of states is distinct. The axial-vector mesons are generated dynamically by coupled-channel interactions based on the chiral Lagrangian written down in terms of the Goldstone bosons and the light vector mesons. We propose a novel counting scheme that arises if the chiral Lagrangian is supplemented by constraints from large-N_c QCD in the context of the hadrogenesis conjecture. The counting scheme is successfully tested by a systematic study of the properties of vector mesons.…
Decay property of hidden-charm tetra-quark mesons is studied. It is seen that estimated width of iso-triplet odd $\mathcal{C}$ partners of X(3872), although still crude, is compatible with the measured ones of Z_c(3900). It is pointed out that confirmation of an eta-pi^0 peak around 3.2 GeV indicated in gamma-gamma collisions gives a clue to select a realistic model of multi-quark mesons.
The polarization effects in the one-meson radiative decay of the polarized $τ$ lepton, $τ^-\toπ^-γν_τ$, have been investigated. We present the analytical results for the $t-$distribution of the partial differential widths, which responsible for different polarization phenomena, in the case of the photon energy cut: $ω>X$. These quantities depend on the invariant mass squared $t$ of the pseudoscalar meson and photon and the photon energy cut $X.$ Our analytical formulae, in terms of the weak vector and axial-vector form factors, describing the structure-dependent part of the decay amplitude, ar…
Experimentally several charged axial-vector hidden-charm states were reported. Within the framework of the color-magnetic interaction, we have systematically considered the mass spectrum of the hidden-charm and hidden-bottom tetraquark states. It is impossible to accommodate all the three charged states $Z_c(3900)$, $Z_c(4025)$ and $Z_c(4200)$ within the axial vector tetraquark spectrum simultaneously. Not all these three states are tetraquark candidates. Moreover, the eigenvector of the chromomagnetic interaction contains valuable information of the decay pattern of the tetraquark states. The…
We study the $Z^\pm(4430)$ meson within a heavy hadron molecule interpretation where the $Z$ is considered as a bound state of a vector $D^\ast(2010)$ and an axial-vector $D_1(2420)$. We give predictions for the strong hidden-charm $Z^\pm(4430) \to π^\pm ψ$ and $π^\pmψ^\prime$ decay widths and also study the radiative $Z^\pm\;(J^P=1^-) \to π^\pmγ$ decay properties in a phenomenological Lagrangian approach. Our findings are qualitatively in line with the experimental observation that the $π^\pmψ^\prime$ transition dominates over the $π^\pmψ$ decay mode despite a smaller phase space. The width f…
We present two-loop calculations on the axial-vector form factors $F_A$ of semileptonic radiative kaon and pion decays in chiral perturbation theory. The relevant dimension-6 terms of the lagrangian are evaluated from the resonance contribution and the results of the irreducible two-loop graphs of the sunset topology are given in detail. We also explicitly show that the divergent parts in $F_A$ are cancelled exactly as required.
We study the radiative decay properties of the charmonium-like X, Y and Z mesons generated dynamically from vector meson-vector meson interaction in the framework of a unitarized hidden-gauge formalism. In the present work we calculate the one- and two-photon decay widths of the hidden-charm Y(3940), Z(3930) (or X(3915)) and X(4160) mesons in the framework of the vector meson dominance formalism. We obtain good agreement with experiment in case of the two photon width of the X(3915) which we associate with the $2^+$ resonance that we find at 3922 MeV.
Nucleon axial-vector form factor and radius from radiatively-corrected antineutrino scattering data
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）；Nuclear & Chemical Sciences Division, Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室核与化学科学部）；Department of Physics, Imperial College London（伦敦帝国理工学院物理系）；Department of Physics and Astronomy, University of Rochester（罗切斯特大学物理与天文学系）；Department of Physics and Astronomy, York University（约克大学物理与天文学系）；Department of Physics and Astronomy, University of Kentucky（肯塔基大学物理与天文学系）；Fermilab（费米实验室）
We analyze the degree to which parity-violating (PV) electroexcitation of the $Δ(1232)$ resonance may be used to extract the weak neutral axial vector transition form factors. We find that the axial vector electroweak radiative corrections are large and theoretically uncertain, thereby modifying the nominal interpretation of the PV asymmetry in terms of the weak neutral form factors. We also show that, in contrast to the situation for elastic electron scattering, the axial $N\toΔ$ PV asymmetry does not vanish at the photon point as a consequence of a new term entering the radiative corrections…
The first radial excitations of axial-vector mesons are considered in the framework of the extended $U(3)\times U(3)$ Nambu--Jona-Lasinio model. We calculate the mass spectrum of $a_1$, $f_1$, and also strange axial-vector mesons. For description of radially excited states we used the form factors of polynomial type of the second order in transverse quark momentum. For the ground- and the excited-state mesons consisting of light quarks we have calculated the widths of a number of strong and radiative decays. We got satisfactory agreement with experimental data for the ground states. A set of p…
We suggest a rigorous generalization of the pseudoscalar mesons mixing description in SU(3) basis. It is shown that the appearance of extra massive state nicknamed in the paper as glueball is unavoidable in any scheme with more than one angle. In this framework we develop the dispersive approach to Abelian axial anomaly of isoscalar non-singlet current. Combining it with the analysis of experimental data of charmonium radiative decays ratio we get the number of quite strict constraints for mixing parameters. Our analysis favors the equal values of axial currents coupling constants which may be…
We propose a one-loop induced neutrino mass model with hidden $U(1)$ gauge symmetry, in which we successfully involve a bosonic dark matter (DM) candidate propagating inside a loop diagram in neutrino mass generation to explain the $e^+e^-$ excess recently reported by the DArk Matter Particle Explorer (DAMPE) experiment. In our scenario dark matter annihilates into four leptons through $Z'$ boson as DM DM $\to Z' Z' (Z' \to \ell^+ \ell^-)$ and $Z'$ decays into leptons via one-loop effect. We then investigate branching ratios of $Z'$ taking into account lepton flavor violations and neutrino osc…
We investigate the vector and axial-vector form factors for the pion radiative weak decays $π^+\to e^+ν_e γ$ and $π^+\to e^+ν_e e^+ e^-$, based on the gauged effective chiral action from the instanton vacuum in the large $N_c$ limit. The nonlocal contributions, which arise from the gauging of the action, enhance the vector form factor by about $20\,\%$, whereas the axial-vector form factor is reduced by almost $30\,\%$. Both the results for the vector and axial-vector form factors at the zero momentum transfer are in good agreement with the experimental data. The dependence of the form factors…
We show that the radiative corrections containing terms up to order $αq / πM_1$ for unpolarized semileptonic decays of baryons with positron emission can be obtained by simply reversing the sign of the axial-vector form factors in the corresponding final expressions of such decays with electron emission. This rule is valid regardless of the final kinematical variables chosen and of the particular Lorentz frame in which the final results are required.
Whilst the prospect of new $Z'$ gauge bosons with only axial couplings to the Standard Model (SM) fermions is widely discussed, examples of anomaly-free renormalisable models are lacking in the literature. We look to remedy this by constructing several motivated examples. Specifically, we consider axial vectors which couple universally to all SM fermions, as well as those which are generation-specific, leptophilic, and leptophobic. Anomaly cancellation typically requires the presence of new coloured and charged chiral fermions, and we argue that in a large class of models masses of these new s…
Using the standard auxiliary field method, we derive from the extended Nambu-Jona-Lasinio model an effective meson action containing vector and axial-vector mesons in addition to Goldstone bosons. The vector and axial-vector mesons in this effective action transform as gauge fields of hidden local symmetry $G_{local} = {[U(n)_L \times U(n)_R]}_{local}$. Here, the realization of enlarged hidden local symmetry is accomplished via the introduction of two kinds of ``compensating'' fields. For obtaining the intrinsic-parity violating part of the action, we generalize the standard gauged Wess-Zumino…
In this work the analysis of mixing parameters of the system involving eta, eta' mesons and some third massive state G is carried out. We use the generalized mixing scheme with three angles. The framework of the dispersive approach to Abelian axial anomaly of isoscalar non-singlet current and the analysis of experimental data of charmonium radiative decays ratio allow us to get a number of quite strict constraints for the mixing parameters. The analysis shows that the equal values of axial current coupling constants f_8 and f_0 are preferable which may be considered as a manifestation of SU(3)…
The coupling between the spin of a massive Dirac fermion and the angular momentum of the medium, i.e. the gravitomagnetic moment, is shown here to be renormalized by QED interactions at finite temperature. This means that the anomalous gravitomagnetic moment (AGM) does not vanish, and implies that thermal effects can break the Einstein equivalence principle in quantum field theory, as argued previously. We also show that the AGM causes radiative corrections to the axial current of massive fermions induced by vorticity in quantum relativistic fluids, similarly to the previous findings for massl…
The positive parity scalar D$_{s0}^*$(2317) and axial-vector D$_{s1}^*$(2460) charmed strange mesons are generated by coupled-channel dynamics through the s-wave scattering of Goldstone bosons off the pseudoscalar and vector D(D$_s$)-meson ground states. The attraction leading to the specific masses of these states reflects the chiral symmetry breaking scale which characterizes the Weinberg-Tomozawa interaction in the chiral Lagrangian. Chiral corrections to order Q$_χ^2$ are calculated and found to be small. The D$_{s0}^*$(2317) and D$_{s1}^*$(2460) mesons decay either strongly into the isosp…
The large branching ratio for B-> K1 gamma recently measured at Belle implies a large B -> K1 transition form factor and large branching ratios for non leptonic B decays involving an axial-vector meson. In this paper we present an analysis of two-body B decays with an axial-vector meson in the final state using naive factorization and the B -> K1 form factors obtained from the measured radiative decays. We find that the predicted B -> J/psi K1 branching ratio is in agreement with experiment. We also suggest that the decay rates of B -> K1 pi, B -> a1 K and B -> b1 K could be used to test the f…
In this article, we study the axial-vector mesons $Z_b(10610)$ and $Z_b(10650)$ with the $Cγ_μ-Cγ_5$ type and $Cγ_μ-Cγ_ν$ type interpolating currents respectively by carrying out the operator product expansion to the vacuum condensates up to dimension-10. In calculations, we explore the energy scale dependence of the QCD spectral densities of the hidden bottom tetraquark states in details for the first time, and suggest a formula $μ=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_b)^2}$ with the effective mass ${\mathbb{M}}_b=5.13\,\rm{GeV}$ to determine the energy scales. The numerical results favor assigni…
The formulae describing the radiative decays $η'(η)\rta\c\c$ in QCD beyond the chiral limit are derived. The modifications of the conventional PCAC formulae due to the gluonic contribution to the axial anomaly in the flavour singlet channel are precisely described. The decay constants are found to satisfy a modified Dashen formula which generalises the Witten--Veneziano formula for the mass of the $η'$. Combining these results, it is shown how the topological susceptibility in QCD with massive, dynamical quarks may be extracted from measurements of $η'(η)\rta\c\c$.
