$a_1(1260), a_1(1420)$ and the production in heavy meson decays

The $a_1(1420)$ with $I^G(J^{PC})= 1^-(1^{++})$ observed in the $\pi^+ f_0(980)$ final state in the $\pi^-p\to \pi^+\pi^-\pi^- p$ process by the COMPASS collaboration seems unlikely to be an ordinary $\bar qq$ mesonic state. Available theoretical explanations include tetraquark or rescattering effects due to $a_1(1260)$ decays. If the $a_1(1420)$ were induced by the rescattering, its production rates are completely determined by those of the $a_1(1260)$. In this work, we propose to explore the ratios of branching fractions of heavy meson weak decays into the $a_1(1420)$ and $a_1(1260)$, and testing the universality of these ratios would be a straightforward way to validate/invalidate the rescattering explanation. The decay modes include in the charm sector the $D^0\to a_1^-\ell^+\nu$ and $D^0\to \pi^\pm a_1^\mp$, and in the bottom sector $\overline B^0\to a_1^+ \ell^- \bar\nu$, $B\to D a_1, \pi^\pm a_1^\mp$, $B_c\to J/\psi a_1$ and $\Lambda_b\to \Lambda_c a_1$. We calculate the branching ratios for various decay modes into the $a_1(1260)$. The numerical results indicate that there is a promising prospect to study these decays on experiments including BES-III, LHCb, Babar, Belle and CLEO-c, the forthcoming Super-KEKB factory and the under-design Circular Electron-Positron Collider. Experimental analyses in future will lead to a deeper understanding of the nature of the $a_1(1420)$.


I. INTRODUCTION
Since Gell-Mann proposed the concept of quarks in 1964 [1], quark model has achieved indisputable sucesses: most of the established mesons and baryons on experimental side can be well accommodated in the predicted scheme [2].However, recently there have been experimental observations of resonance-like structures with quantum numbers hardly to be placed in the quark-antiquark or three-quark schemes [3][4][5][6][7][8][9].This leads to the suspect that the hadron spectrum is much richer than the simple quark model [10].
Recently the COMPASS collaboration [11,12] has reported the observation of a light resonance-like state with quantum numbers I G (J P C ) = 1 − (1 ++ ) in the P -wave f 0 (980)π final state with f 0 (980) → π + π − .The signal was also confirmed by the VES experiment [13] in the π − π 0 π 0 final state.The new state was tentatively called a 1 (1420) with the mass m a1 ≈ 1.42 GeV and width Γ a1 ≈ 0.14 GeV.The interpretation of this state as a new qq meson is challenging, since it could hardly be accommodated as the radial excitation of the a 1 (1260) which is expected to have a mass above 1650 MeV.Therefore, this state has been interpreted as a tetra-quark [16] or some dynamical effects arising from final state interactions [14,15].An illustration of the rescattering mechanism is shown in Fig. 1.
The deciphering of the internal structure of the a 1 (1420) can proceed not only through the detailed analysis of the pole position, but also through the decay and production characters.In this work, we propose that semileptonic and nonleptonic heavy meson decays can be used to examine the rescattering interpretation.In particular, an intriguing property in the rescattering picture is that the production rates of a 1 (1420) are completely determined by those of a 1 (1260) the a 1 (1260).In this case, the ratios would be insensitive to the production mechanism, and thus would be a constant.The value for the ratios is estimated to be at percent level in Ref. [14].Testing the universality of these ratios will be a straightforward way to validate/invalidate the rescattering interpretation.
In the above equations, the X, Y correspond to certain leptonic/hadronic final states, and more explicitly we suggest to study in the charm sector the D → a 1 ℓ + ν and D 0 → π ± a ∓ 1 , and in the bottom sector the B → a In the following of this work, we will provide the theoretical calculation of branching ratios for the B or D decays into a 1 (1260).In addition, the B/D decays into the a 1 (1260) are also of great interest since they are helpful to understand the dynamics in the decays into an axial-vector meson.Some theoretical studies can be found in the literature [17][18][19][20][21][22][23][24][25][26][27].
The rest of this paper is organized as follows.In Sec.II, we will concentrate on the B → a 1 (1260) decays, including the transition form factors, semileptonic and non-leptonic decay modes.We will subsequently discuss the production of the a 1 (1260) in semileptonic and nonleptonic D/D s decays in Sec.III.The last section contains our summary.

A. Form factors
Unless specified in the following, we will use the abbreviation a 1 to denote the a 1 (1260) for simplicity.The Feynman diagram for semileptonic B0 → a + 1 ℓ − ν decays is given in Fig. 2.After integrating out the off-shell W boson, one obtains the effective Hamiltonian Here the V ub is the CKM matrix element, and G F is the Fermi constant.[17], light-cone sum rules (LCSR) [18] and perturbative QCD approach (PQCD) [19].Hadronic effects are parametrized in terms of the B → a 1 form factors: with q = p B − p a1 , and ǫ 0123 = +1.The B → a 1 (1260) form factors have been studied in the covariant light-front quark model (LFQM) [17], light-cone sum rules (LCSR) [18] and perturbative QCD approach (PQCD) [19].The corresponding results are collected in Table I.In order to access the form factors in the full kinematics region, one has adopted the dipole parametrization [17][18][19]: ( In the PQCD approach [19], the form factor V 2 is parametrized as with η = 1 − q 2 /m 2 B , and Decay amplitudes for the B 0 → a + 1 (1260)ℓ − νℓ can be divided into hadronic and leptonic sectors.Each of them are expressed in terms of the Lorentz invariant helicity amplitudes.The hadronic amplitude is obtained by evaluating the matrix element: with ml = m l / q 2 , β l = (1 − m 2 l /q 2 ) and In the above, ǫ µ (h) with h = 0, ±, t is an auxiliary polarisation vector for the lepton pair system.The polarised decay amplitudes are evaluated as with For the sake of convenience, we use The differential decay width for with the I i having the form: With the above formulas at hand, we present our results for differential branching ratios dB/dq 2 (in units of 10 −4 /GeV 2 ) in Fig. 3.The left and right panel corresponds to ℓ = (e, µ) and ℓ = τ , respectively.The dotted, dashed and solid curves are obtained using the LFQM [17], LCSR [18] and PQCD [19] form factors.The other input parameters are taken from Particle Data Group (PDG) [2] as follows: As for the |V ub |, we have quoted the value extracted from the exclusive B → πℓν ℓ for self-consistent, see Refs.[28,29] for discussions on the so-called |V ub | puzzle.Integrating over the q 2 , one obtains the longitudinal and transverse contributions to branching fractions of B 0 → a + 1 ℓ − ν decays, and our results are given in Table II.Uncertainties shown in the table arise from the ones in the B → a 1 form factors.We can see from this table that branching fractions for the B 0 → a + 1 ℓ − ν are of the order 10 −4 .These values are comparable to the data by Belle collaboration on branching fractions for the semileptonic B decays into a vector meson [31]: Babar collaboration [32] also gives similar results for the B − → ωℓ − ν: Currently, there is no experimental analysis of the B 0 → a 1 (1260) + ℓ − ν, but the two B factories at KEK and SLAC have accumulated about 10 9 events of B 0 and B ± .The branching fractions O(10 −4 ) correspond to about 10 5 events for the signal.The above estimate may be affected by the detector efficiency, but an experimental search would very presumably lead to the observation of this decay mode.In addition, the sizable branching fractions as shown in Table II also indicate a promising prospect at the ongoing LHC experiment [33], the forthcoming Super-KEKB factory [34] and the under-design Circular Electron-Positron Collider (CEPC) [35].

C. Nonleptonic B decays into a1
Since our main goal in this work is to investigate the internal structure of the a 1 (1420), we will focus on the decay modes which can be handled under the factorization approach.These decay modes are typically dominated by tree operators with effective Hamiltonian TABLE II: Integrated branching ratios for the B 0 → a + 1 (1260)ℓ − ν decays (in units of 10 −4 ) with the form factors from the LFQM [17], LCSR [18] and PQCD [19].where C 1 and C 2 are the Wilson coefficients.The α and β are the color indices.V ub , V ud are the CKM matrix elements.The up type quark can also be replaced by the charm quark.
With the definitions of decay constants, we expect the factorization formula to have the form: with ).In the above, the amplitude for the B → J/ψa 1 has been decomposed according to the Lorentz structures The partial decay width of the B → a 1 P , where P denotes a pseudoscalar meson, is given as with | p| being the three-momentum of the a 1 in the B meson rest frame.For the B → a 1 V , the partial decay width is the summation of three polarizations We use the LFQM results [17] for all transition form factors and other inputs are given as [2] τ B − = (1.638× 10 −12 )s, τ Bs = (1.511× 10 −12 )s ( 29) The f π and f J/ψ can be extracted from the π − → ℓ − ν and J/ψ → ℓ + ℓ − data [2]: We use QCD sum rules results for the f a1 [36] The Wilson coefficient a 1 is used as [37] a 1 = 1.07, (33) while under the same factorization hypothesis the a 2 is extracted from the B → J/ψK * data [2] : Then theoretical results for branching ratios are given as where the errors come from the one in the f a1 .For decay modes induced by the B → a 1 transition, we have where the errors arise from those in form factors. Babar [38] and Belle [39] collaborations have reported the observation of B 0 → a ± 1 π ∓ and their results for branching fractions are given as TABLE III: The D → a1 form factors calculated in the covariant LFQM [17].

F F (0) a b
A D→a 1 0.20 0.98 0.20 V D→a 1 0 0.31 0.85 0.49 1.54 −0.05 0.05 0.06 0.12 0.10 The above results have been averaged by PDG as [2] B(B As we can see, the averaged data is consistent with our theoretical results in Eqs.(38) and (39).We also predict the branching ratios for B B(B The numerical results given in Eqs.(35-43) indicate that there is a promising prospect to study these decays by the LHCb, Babar, and Belle collaborations, and on the forthcoming Super-KEKB factory and the CEPC.

III. D → a1 DECAYS
By replacing the corresponding form factors and CKM matrix elements, the analysis of B decays in the last section can be straightforwardly generalized to the D → a 1 decays.The D → a 1 form factors are only available in LFQM [17] and we summarize these results in Table III.We will use other input parameters as follows [2]: Our results for the differential branching ratios of the semileptonic D 0 → a − 1 ℓ + ν are given in Fig. 4, and the integrated branching fractions are presented in Table IV.In Fig. 4, the dotted and solid curve corresponds to ℓ = e and ℓ = µ, respectively.The differences in the two curves arise from the lepton masses and can reach about 10%.
Recently, based on the 2.9f b −1 data of electron-positron annihilation data collected at a center-of-mass energy of √ s = 3.773 GeV, BES-III collaboration has searched for the D + → ωℓ + ν decay [40] and the branching fraction is measured In this procedure, the ω meson is reconstructed by three pions, and it is interesting to notice that the neutral a 1 (1260) should also be reconstructed by the same final state.Extending the analysis in Ref. [40] to higher mass region at round 1.23 GeV may discover the D + → a 0 1 ℓ + ν transition.Actually, BES-III have collected about 10 7 events of D − D. The 10 −4 branching fractions correspond to about 10 3 events for the D → a 1 ℓ + ν, which might be observed in the future analysis.We can also study the nonleptonic D/D s decays into a 1 (1260) with the factorization amplitudes: Our theoretical results are given as: where the errors arise from those in the decay constant f a1 .
The FOCUS collaboration has measured the branching fraction for the D 0 → a ± 1 π ∓ [41]: which is consistent with our theoretical results in Eqs.(49)(50).The LHCb collaboration makes use of the D 0 → a ± 1 π ∓ to study CP violation [42], and it is also feasible to study this mode using the CLEO-c data [43].The BES-III collaboration has accumulated about 10 7 events of the D 0 and will collect about 3f b −1 data at the center-of-mass √ s = 4.17 GeV to produce the D + s D − s [44,45].All these data can be used to study the charm decays into the a 1 .

IV. CONCLUSIONS
Experimental observations of resonance-like states in recent years have invoked theoretical research interest on exotic hadron spectroscopy.In particular, many of the experimentally established structures defy the naive quark model assignment as a qq or qqq state.At the low-energy, the a 1 (1420) with I G (J P C ) = 1 − (1 ++ ) observed in the π + f 0 (980) final state in the π − p → π + π − π − p process by COMPASS collaboration seems unlikely to be an ordinary qq mesonic state.Available theoretical explanations include tetraquark or rescattering effects due to a 1 (1260) decays.If the a 1 (1420) were induced by rescattering effects, its production rates are completely determined by those of the a 1 (1260).
In this work, we have proposed to explore the ratios of branching fractions of heavy meson weak decays into the a 1 (1420) and a 1 (1260), and testing the universality of these ratios would be a straightforward way to validate/invalidate the rescattering explanation.The decay modes include in the charm sector the D 0 → a − 1 ℓ + ν and D → π ± a ∓ 1 , and in the bottom sector B → a 1 ℓν and B → Da 1 , π ± a ∓ 1 , and the B c → J/ψa 1 and Λ b → Λ c a 1 .We have calculated the branching ratios for various decays into the a 1 (1260).Other decay modes like Λ b → Λ c a 1 , and B − c → J/ψa − 1 which has been measured by the LHCb collaboration [46] and CMS collaboration [47], in agreement with theoretical results based on the form factors [48,49], are also of helpful in this aspect.
Our results have indicated that there is a promising prospect to study these decays on experiments including BES-III, LHCb, Babar, Belle and CLEO-c, the forthcoming Super-KEKB factory and the under-design Circular Electron-Positron Collider.Experimental analyses in future will very probably lead to a deeper understanding of the nature of the a 1 (1420).

2 FIG. 4 :
FIG.4: Differential branching fractions dB/dq 2 (in units of 10 −4 /GeV 2 ) for the decay D 0 → a − 1 ℓ + ν.The dotted and solid curve corresponds to ℓ = e and ℓ = µ, respectively.The differences between the two curves arise from the lepton masses and can reach about 10%.

TABLE I :
Results for the B → a1(1260) form factors calculated in the covariant light-front quark model (LFQM)