Strange Axial-vector Mesons in D Meson Decays
aa r X i v : . [ h e p - ph ] J a n Strange Axial-vector Mesons in D Meson Decays
Peng-Fei Guo, Di Wang, Fu-Sheng Yu
School of Nuclear Science and Technology,Lanzhou University, Lanzhou 730000, China
Abstract
The nature of strange axial-vector mesons are not well understood and can be investigated in D mesondecays. In this work, it is found that the experimental data of D → K ± K ∓ (1270)( → ρK or K ∗ π ) inthe D → K + K − π + π − mode, disagree with the equality relation under the narrow width approximationand CP conservation of strong decays. Considering more other results of K (1270) decays, the data of B ( D → K − K +1 (1270)( → K ∗ π + )) is probably overestimated by one order of magnitude. We then calculatethe branching fractions of the corresponding processes with K (1400) in the factorization approach, and find B ( D → K − K +1 (1400)( → K ∗ π + )) is comparable to the predicted B ( D → K − K +1 (1270)( → K ∗ π + )) usingthe equality relation. Besides, we suggest to measure the ratios between K (1270) → ρK and K ∗ π or totest the equality relations in other D meson decay modes. . INTRODUCTION In the quark model, there are two nonets of axial-vector ( J P = 1 + ) mesons, namely, P and P in the spectroscopic notation S +1 L J , which correspond to the charge parity of C = + and C = − ,respectively, for the neutral mesons with isospin I = 0 in each nonet. The strange axial-vectormesons in these two nonets are called as K A and K B , respectively. They can mix with each otherto construct the mass eigenstates, K (1270) and K (1400), by the mixing angle θ K : | K (1270) i| K (1400) i = sin θ K cos θ K cos θ K − sin θ K | K A i| K B i . (1)The experimental measurements on K (1270) and K (1400) have been performed in Kp scattering[1, 2], τ ± decays [3–6], B -meson decays [7–12] and D -meson decays [13–18]. However, the mixingangle θ K has not yet been well determined. Many phenomenological analysis indicate that thevalue of θ K is around either 35 ◦ or 55 ◦ through the strong decays of K (1270) and K (1400)[19, 20], τ → K (1270) , K (1400) ν [19], B → K (1270) , K (1400) γ [21] and the mass relation[22], θ K ∼ ◦ in the relativized quark model [23] and the modified Godfrey-Isgur model [24], or θ K ∼ ◦ based on the P quark-pair-creation model for decays of K (1270) and K (1400) [25].35 ◦ . θ K . ◦ are obtained in some other analysis [26–28].The mixing angle θ K can also be investigated in heavy flavor decays. The difference be-tween the production rates of K (1270) and K (1400) may provide the indication on the valueof θ K . It has been widely studied in B -meson decays, such as hadronic decays of B → K (1270) , K (1400) P ( V ) [29–38], with P = π, K, η ( ′ ) , and V = ρ, ω, K ∗ , φ, J/ Ψ, semi-leptonic de-cays of B → K (1270) , K (1400) ℓ + ℓ − [39–43], and radiative decays of B → K (1270) , K (1400) γ [21, 44–46]. The two-body hadronic D -meson decays with an axial-vector meson in the final stateshave been studied in [47–54]. The large non-perturbative contributions in charm decays alwayspollute the analysis on the K (1270) and K (1400) productions. On the other hand, at the LHCb,more data of D decays are obtained than B decays, due to the larger production cross sections of D mesons and the larger branching fractions of D decays. Besides, the running BESIII and theupcoming Belle II experiments will provide large data of D decays as well. For example, K (1270)and K (1400) have been analyzed in the D → K − π + π + π − mode at the BESIII [17] and LHCb[18] very recently. With the large data and thus high precision of measurements in the near future,the processes of D decaying into K (1270) and K (1400) are worthwhile to be studied with moreefforts.Among the exclusive D → K (1270) , K (1400) decays, the D → K + K − π + π − mode is of par-2 ABLE I: List of the fractions for the K ± (1270)-involved cascade modes in the D → K + K − π + π − decaymeasured by CLEO [15], Γ( D → K ± K ∓ (1270)( → ρK, K ∗ π → K ∓ π ± π ∓ )) / Γ( D → K + K − π + π − ). Thefirst and second uncertainties are statistical and systematic respectively.Modes Fractions (%) K − K (1270) + ( → π + K ∗ ( → K + π − )) 7 . ± . ± . K + K (1270) − ( → K − ρ ( → π + π − )) 6 . ± . ± . K − K +1 (1270)( → K + ρ ( → π + π − )) 4 . ± . ± . K + K (1270) − ( → π − K ∗ ( → K − π + )) 0 . ± . ± . ticular interest since there are more cascade channels involving K − K +1 (1270)( → K + ρ ( → π + π − )), K − K (1270) + ( → π + K ∗ ( → K + π − )), K + K (1270) − ( → K − ρ ( → π + π − )), K + K (1270) − ( → π − K ∗ ( → K π + )), and the corresponding ones with K ± (1400) instead of K ± (1270). Besides,all the particles in the final states are charged and thus easier to be measured in experiments.So far the relevant measurements have been performed by the E791 [13], FOCUS [14] and CLEO[15] collaborations. In [15], only K ± (1270) are involved but with K ± (1400) neglected. The frac-tions of decay widths of D → K ± K ∓ (1270)( → ρK, K ∗ π → K ∓ π ± π ∓ ) compared to that of D → K + K − π + π − are shown in Table 1. We find a puzzle in the fractions given in Table 1 . In the narrow width approximation and the CP conservation of strong decays, the four partial widths satisfy a relation ofΓ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + )Γ( D → K − K +1 (1270) , K +1 (1270) → ρ K + )= Γ( D → K + K − (1270) , K − (1270) → K ∗ π − )Γ( D → K + K − (1270) , K − (1270) → ρ K − ) , (2)in which the weak-decay parts are canceled and it retains only the strong decays of the K (1270).However, from Table 1, the left-hand side of the above relation is 1.55 ± ± σ . The central valuesare even different by a factor of 10.We calculate the branching fractions of D → K ± K ∓ (1400) considering the finite-width effect in the factorization approach. It is found that the branching fraction of D → K − K +1 (1400) , K +1 (1400) → K ∗ π + , K ∗ → K + π − ) is comparable to D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − ). Thus the inclusion of K (1400) in 1 + state Very recently, PDG[55] reversed these decay modes according to the re-analysis on the CLEO data by [16]. Wewill discuss on it in Sec.4. ABLE II: Masses and widths of K (1270) and K (1400). Data are from PDG [55].Mass Width K (1270) 1272 ± ±
20 MeV K (1400) 1403 ± ±
13 MeV may contribute to the overestimation of the latter process. Besides, we propose to test somerelations of D mesons decaying into K (1270) processes in the subsequent measurements.This paper is organized as follows. In Sec. 2, we discuss the puzzle of the experimental dataof D → K + K − π + π − decays with K (1270) resonances. In Sec. 3, the branching fractions of D → K (1400) transitions are estimated. Some relations about D decays into K (1270) are listedin Sec. 4. And Sec. 5 is the conclusion. II. K PUZZLE IN D → K + K − π + π − The puzzle introduced above is based on the narrow width approximation in the chain decaysof heavy mesons. Taking the process of D → f f f with a resonant contribution of R → f f asan example, the branching fraction of D → f R → f f f is the product of branching fractions of D → f R and R → f f : B ( D → f R → f f f ) = B ( D → f R ) B ( R → f f ) . (3)The narrow width approximation is valid in the decay of D → KK (1270) , K (1270) → Kππ where the first decay is kinematically allowed and the width of K (1270) is much smaller than itsmass, Γ K (1270) ≪ m K (1270) , as seen in Table 2.Therefore, the ratios of branching fractions of the processes in Eq. (2) are thus B ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + ) B ( D → K − K +1 (1270) , K +1 (1270) → ρ K + )= B ( D → K − K +1 (1270)) B ( K +1 (1270) → K ∗ π + ) B ( D → K − K +1 (1270)) B ( K +1 (1270) → ρ K + )= B ( K +1 (1270) → K ∗ π + ) B ( K +1 (1270) → ρ K + ) , (4)4nd B ( D → K + K − (1270) , K − (1270) → K ∗ π − ) B ( D → K + K − (1270) , K − (1270) → ρ K − )= B ( D → K + K − (1270)) B ( K − (1270) → K ∗ π − ) B ( D → K + K − (1270)) B ( K − (1270) → ρ K − )= B ( K − (1270) → K ∗ π − ) B ( K − (1270) → ρ K − ) . (5)The equality relation in Eq. (2) can then be obtained from Eqs. (4) and (5), due to the CP conservation of the strong interaction.The branching fractions of the cascade decays involving K (1270) are obtained from the fractionsby CLEO [15] shown in Table 1 and the data of B ( D → K + K − π + π − ) = (2 . ± . × − [55], B = B ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − ) = (1 . ± . × − , (6) B = B ( D → K − K +1 (1270) , K +1 (1270) → ρ K + , ρ → π + π − ) = (1 . ± . × − , (7) B = B ( D → K + K − (1270) , K − (1270) → K ∗ π − , K ∗ → K − π + ) = (2 . ± . × − , (8) B = B ( D → K + K − (1270) , K − (1270) → ρ K − , ρ → π + π − ) = (1 . ± . × − . (9)The narrow width approximation indicates B B = B B , (10)while the data in (6) − (9) give B B = 1 . ± . , and B B = 0 . ± . , (11)which have large discrepancy with more than 2 standard deviations. The central values of B / B and B / B are even different by a factor of 10. This is the K puzzle that the data measured byCLEO are inconsistent with the equality relation of the narrow with approximation.From Eqs. (4) and (5), it can be found that only the strong decays of K (1270) are left.There are some other measurements on the K (1270) decays. It would be useful to compareamong the measurements, to give some implications on the solution of the K puzzle. Beforethe comparison, it is more convenient to define a parameter, η , describing the ratio of branchingfractions of K (1270) → K ∗ π and K (1270) → ρK , η ≡ B ( K (1270) → K ∗ π ) B ( K (1270) → Kρ ) , (12)where the branching fractions are the sums of all the possible charged and neutral final states.For example, B ( K +1 (1270) → K ∗ π ) = B ( K +1 (1270) → K ∗ π + ) due to the isospin relation of5 ( K +1 (1270) → K ∗ π + ) = −√ A ( K +1 (1270) → K ∗ + π ). Similarly, B ( K +1 (1270) → ρK ) =3 B ( K +1 (1270) → ρ K + ), Γ K ∗ = Γ( K ∗ → K + π − ). Therefore, the values of η obtained fromEq. (11) are then η = 34 B B = 1 . ± . , and η = 34 B B = 0 . ± . . (13)The K puzzle can be taken as the discrepancy between η and η .In the following, we discuss on the other measurements which can provide the information onthe value of η . Except for the singly Cabibbo-suppressed mode of D → K + K − π + π − , K (1270) → K ∗ π and ρK are also measured in the Cabibbo-favored D → K − π + π + π − decay by BESIII [17]and LHCb [18]. With 1 . × signal events of D → K − π + π + π − and fixing the mass and widthof K (1270) as the PDG values, BESIII obtains the branching fractions of [17] B = B ( D → π + K − (1270) , K − (1270) → K ∗ π − , K ∗ → K − π + ) = (0 . ± . , (14) B = B ( D → π + K − (1270) , K − (1270) → ρ K − , ρ → π + π − ) = (0 . ± . . (15)Similarly to Eq. (13), we have η = 34 B B = 0 . ± . , (16)which is consistent with η .At the LHCb with even more data of D → K − π + π + π − with 9 × signal events [18],more discoveries and higher precisions are obtained. K (1270) → ρ (1450) K is observed and hasa relatively large branching fraction. They also find the D -wave K ∗ π with a high significance.The interference between amplitudes are considered in [18]. The results of partial fractions are(96 . ± . ± . K − (1270) → ρ K − , (27 . ± . ± . S -wave K ∗ π − and(3 . ± . ± . D -wave K ∗ π − . The phases of the amplitudes of the S-wave and D-waveare ( − . ± . ± . ◦ and ( − . ± . ± . ◦ , respectively. Then, it is obtained that η ′ = 0 . ± . . (17)The decays of K (1270) are also studied in B + → J/ Ψ K + π + π − by Belle [11]. Two amplitudeanalysis have been performed with the mass and width of K (1270) fixed or floated, named as Fit1 and Fit 2, respectively. The analysis are based on the assumption of K (1270) decaying only to K ∗ π , Kρ , Kω and K ∗ (1430) π , and neglect the interference between decay channels. The resultsare thus not reliable. We just list them here: 6 ABLE III: Values of observable η extracted from different experiments. η Processes Experiments η = 1 . ± . D → K + K − π + π − CLEO [15] η = 0 . ± . D → K + K − π + π − CLEO [15] η = 0 . ± . D → K − π + π + π − BESIII [17] η ′ = 0 . ± . D → K − π + π + π − LHCb [18] η = 0 . ± . B + → J/ Ψ K + π + π − Belle [11] (Fit 1) η ′ = 0 . ± . B + → J/ Ψ K + π + π − Belle [11] (Fit 2) η = 0 . ± . K − p → K − π − π + p ACCMOR [1]
The values of branching fractions of K (1270) decays in PDG are obtained from the K − p → K − π − π + p scattering experiment by the ACCMOR collaboration in 1981 [1], with B ( K (1270) → Kρ ) = (42 ± , B ( K (1270) → K ∗ π ) = (16 ± , (18)and thus η = 0 . ± . . (19)All the values of η obtained from different experiments are listed in Table 3 for comparison.We can find that except η , all the other η ’s indicate a smaller value of η ≪
1, especially η , , = O (0 . − .
2) in D decays. Thus it is of a large probability that η = 1 . ± .
43 is overestimated.Due to its large uncertainty, η can be decreased by about 2 standard deviations to be consistentwith other values of η .Using the equality relation of Eq. (10) and the measured values of B , , , in Eqs. (6)-(9), itcan be estimated that B ′ = B ′ ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − )= B B B = (1 . ± . × − , (20)if B = (1 . ± . × − was overestimated, or B ′ = B ′ ( D → K − K +1 (1270) , K +1 (1270) → ρ K + , ρ → π + π − ) = B B / B = (1 . ± . × − , if B = (1 . ± . × − was underestimated.That means, under the equality relation, either B should be reduced to be one-order smaller, or B to be one-order larger. However, with an uncertainty of 20%, the measured value of B deviates toomuch from the central value of B ′ . Considering the large uncertainty of B ′ , the lower bound of B ′ is close to B . Therefore, the true value of B ( D → K − K +1 (1270) , K +1 (1270) → ρ K + , ρ → π + π − )7ould be around B . On the contrary, the value of B ′ deviates from the measured B by about3 σ . It is of large possibility that B is overestimated.Recall that in the CLEO analysis [15], only K ± (1270) are considered as the 1 + states but with K ± (1400) neglected. It deserves to test whether K (1400) contributes to the overestimation of B ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − ).Note in the end of this section that, we have tested the finite width effect of K (1270) in thefactorization approach, and find that this effect shifts the branching fractions from the narrowwidth approximation by less than 10%. From Table 3, any uncertainty of the η ’s is larger than10%. Therefore, the finite width effect can be neglected. The narrow width approximation is validin the discussions. III. D → K (1400) TRANSITIONS
The contributions from K ± (1400) in the D → K + K − π + π − decay are studied in this sec-tion. The branching fractions of D → K ± K ∓ (1400)( → ρK, K ∗ π ) decays are calculated inthe factorization approach. Note that the above processes are kinematically forbidden due to m D < ( m K (1400) + m K ). However, the chain decays of D → K ± K ∓ (1400)( → ρK, K ∗ π ) can stillhappen considering the finite width of K (1400). From Table 2, m K (1400) + m K − m D = 32 ± < Γ K (1400) = 174 ±
13 MeV.The decay constant of axial-vector meson ( A ) and the form factors D → A transition are definedas h A ( p, ε ) | A µ | i = f A m A ǫ ∗ µ , h A ( p, ε ) | A µ | D ( p D ) i = 2 m D − m A ǫ µναβ ǫ ∗ ν p αD p β A D → A ( q ) , h A ( p, ε ) | V µ | D ( p D ) i = − i (cid:26) ( m D − m A ) ǫ ∗ µ V D → A ( q ) − ( ǫ ∗ · p D )( p D + p ) µ V D → A ( q ) m D − m A − m A ǫ ∗ · p D q q µ (cid:2) V D → A ( q ) − V D → A ( q ) (cid:3)(cid:27) , (21)in which q µ = ( p D − p ) µ . The decay constant of pseudoscalar meson ( P ) and the form factors of D → P transition are h P ( p ) | A µ | i = if P p µ , h P ( p ) | V µ | D ( p D ) i = (cid:18) ( p D + p ) µ − m D − m P q ′ µ (cid:19) F D → P ( q ′ ) + m D − m P q ′ q ′ µ F D → P ( q ′ ) , (22)8 ABLE IV: The form factors of D → K, K A , K B transitions under the parametrization of Eq.(25), takenfrom the covariant light-front quark model [30].F F(0) a b V DK A V DK B F DK with q ′ µ = ( p D − p ) µ . In the factorization approach, the amplitudes of D → K − K +1 (1400) and D → K + K − (1400) are expressed as M ( D → K − K +1 (1400)) = − G F √ V ∗ cs V us × (cid:2) a ( µ ) p q f K (1400) F D → K ( q ) (cid:3) ( ǫ ∗ · p D ) , (23) M ( D → K + K − (1400)) = G F √ V ∗ cs V us × (cid:2) a ( µ ) p q f K (cos θ K V D → K A ( m K ) − sin θ K V D → K B ( m K )) (cid:3) ( ǫ ∗ · p D ) , (24)where ǫ ∗ is the polarization vector of K (1400) and the effective Wilson coefficient a ( µ ) = C ( µ ) + C ( µ ) /
3. In this work, we take µ = µ c = m c , so that a ( µ c ) = 1 .
08 [56]. Note that, to consider thefinite-width effect [47, 48], a running mass p q for the unstable particle K (1400) is considered inEqs. (23) and (24). According to [30], the form factors of charm decays are parameterized as F ( q ) = F (0)1 − a ( q /m D ) + b ( q /m D ) . (25)In this work, the values of form factors of D → K A, B and K are taken from [30] in the covariantlight-front quark model, as shown in Table 4. The decay constant of K (1400) is taken as 139 . +41 . − . MeV obtained from the τ → K (1400) ν decay [54]. The decay constant of K meson is from [55].Considering the finite-width effect, the decay widths of the chain decay of D → K ± K ∓ (1400)( → ρ K ∓ or K ∗ π + , K ∗ π − ) can be expressed asΓ( D → K − K +1 (1400)( → K ∗ π + )) = Z ( m D − m K ) ( m K ∗ + m π ) dq π Γ( q )( D → K − K +1 (1400)) × B ( K +1 (1400) → K ∗ π + ) × p q Γ( q )( q − M ) − M Γ ( q ) , (26) Note that from the τ → K (1400) ν decay the decay constant of K (1400) is actually obtained as | f K (1400) | =139 . +41 . − . MeV. Its sign cannot be determined from an individual process. However, in this work our results areindependent on the sign of f K (1400) , since in the factorization approach the decay width of D → K − K +1 (1400)is the squared magnitude of the amplitude in Eq. (23). D → K − K +1 (1400)( → ρ K + )) = Z ( m D − m K ) ( m ρ + m K ) dq π Γ( q )( D → K − K +1 (1400)) × B ( K +1 (1400) → ρ K + ) × p q Γ( q )( q − M ) − M Γ ( q ) , (27)Γ( D → K + K − (1400)( → K ∗ π − )) = Z ( m D − m K ) ( m K ∗ + m π ) dq π Γ( q )( D → K + K − (1400)) × B ( K − (1400) → K ∗ π − ) × p q Γ( q )( q − M ) − M Γ ( q ) , (28)Γ( D → K + K − (1400)( → ρ K − )) = Z ( m D − m K ) ( m ρ + m K ) dq π Γ( q )( D → K + K − (1400)) × B ( K − (1400) → ρ K − ) × p q Γ( q )( q − M ) − M Γ ( q ) , (29)where p q is the invariant masses of the K ∗ π and Kρ final states, and M and Γ are the mass andwidth of K (1400), respectively. The q -dependent width of K (1400) is [57]:Γ( q ) = Γ K (1400) M K (1400) p q (cid:18) p ( q ) p ( M K (1400) ) (cid:19) F R ( q ) , (30)in which F R ( q ) = q R p ( M K (1400) ) p R p ( q ) , (31)and p ( q ) = λ / ( q , m , m ) (cid:14) (2 p q ), λ ( q , m , m ) = ( q − ( m − m ) )( q − ( m + m ) ), m , are the masses of K ∗ and π or ρ and K . The radius of the axial meson is taken as R =1.5GeV − [58]. The branching fractions of K (1400) decays are [55] B ( K (1400) → K ∗ π ) = (94 ± , and B ( K (1400) → Kρ ) = (3 . ± . . (32)To calculate the branching fractions, the mixing angle of θ K has to be fixed. We test thevalues of 35 ◦ , 45 ◦ , 55 ◦ and 60 ◦ which are usually predicted in literatures as shown in the INTRO-DUCTION. The numerical results of D → K ± K ∓ (1400)( → ρ K ± or K ∗ π + , K ∗ π − ) decaysare listed in Table 5. The finite width effect allow the D → K ± K ∓ (1400) processes to hap-pen. In principle, the branching fractions depend on the K mixing angle. The predictions on10 ABLE V: Branching fractions of D → K ± K ∓ (1400)( → ρ K ± or K ∗ π + , K ∗ π − ) decays with mixingangles θ K = 35 ◦ , 45 ◦ , 55 ◦ and 60 ◦ .Modes B ( θ K = 35 ◦ ) B ( θ K = 45 ◦ ) B ( θ K = 55 ◦ ) B ( θ K = 60 ◦ ) K − K +1 (1400)( → π + K ∗ ( → K + π − )) (1 . ± . × − (1 . ± . × − (1 . ± . × − (1 . ± . × − K − K +1 (1400)( → K + ρ ( → π + π − )) (6 . ± . × − (6 . ± . × − (6 . ± . × − (6 . ± . × − K + K − (1400)( → π − K ∗ ( → K − π + )) (1 . ± . × − (3 . ± . × − (2 . ± . × − (5 . ± . × − K + K − (1400)( → K − ρ ( → π + π − )) (6 . ± . × − (1 . ± . × − (1 . ± . × − (2 . ± . × − B ( D → K − K +1 (1400)( → ρ K + and K ∗ π + )) are, nevertheless, invariant for different values of θ K , since the mixing angle is involved in the decay constant of K (1400) which is however takenas a constant from the τ → K (1400) ν decay, seen in Eq. (23). The branching fractions of theprocesses associated with K (1400) → K ∗ π and ρK differ by about two orders of magnitude, dueto the hierarchy of branching fractions of K (1400) decays in Eq. (32), and the difference of inte-gral lower limits in Eqs. (26) − (29). The branching fractions of the K − K +1 (1400) modes are largerthan those of the K + K − (1400) modes by two or three orders of magnitude, since the transitionform factor of D → K (1400) is destructive and suppressed as (cos θ K V D → K A − sin θ K V D → K B )with θ K in the range between 35 ◦ and 60 ◦ , given in Eq. (24). The uncertainties in our calculationinclude errors of the width Γ K (1400) , the decay constant f K (1400) and the branching fractions of K (1400) → K ∗ π and ρK decays.From Table 5, it is found that the branching fraction of D → K − K +1 (1400)( → K ∗ π + ) isof the order of 10 − , same order as our prediction of B ′ ( D → K − K +1 (1270)( → K ∗ π + )) inEq. (20). The branching fraction of D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − isalso estimated in the naive factorization in which the width of K (1270) is considered as m D − m K ± − m K (1270) ∼
100 MeV. Its value is (2 . ± . × − , and again, being as same order asthe branching fraction of B ( K − K +1 (1400)( → π + K ∗ ( → K + π − ))) = (1 . ± . × − . In orderto estimate how large the interference between D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − and D → K − K +1 (1400)( → K ∗ π + ) could be, we assume that the two chain decays havesame phase space, m K (1270) ∼ m K (1400) , for simplification, since the amplitudes of the strongdecays and their relative phase are unknown. Then the total branching fraction of the two chaindecays and the maximal interference between them are expected to be ( p (2 . ± . × − + p (1 . ± . × − ) = (6 . ± . × − and 2 × p (2 . ± . × − × p (1 . ± . × − =(3 . ± . × − , respectively. Therefore, D → K − K +1 (1400)( → K ∗ π + ) might contribute to11he overestimation of B ( D → K − K +1 (1270)( → K ∗ π + )). The contribution of K (1400) cannotbe neglected in the experimental analysis.The estimation of charm decays in the naive factorization approach is not very reliable. Forexample, the non-factorizable W -exchange diagram E is missed in the above calculation, but isusually large and non-negligible as seen in D → P P and
P V modes [56, 59, 60]. If more dataof D → P A decays are obtained by experiments, their branching fractions can be calculatedin the factorization-assisted topological amplitude (FAT) approach [56, 59] in which some globalparameters are extracted from data. More experimental data of D → P A decays are beneficial tounderstand the charmed meson decays into axial-vector mesons.Although K (1400) might contribute to the overestimation of B , we still cannot concludewhether the K puzzle is solved by the consideration of K (1400), due to the rough understandingof D → P A decays. It has to be tested by the experimental measurements with higher precision,and cross checks from other processes.
IV. EXPERIMENTAL POTENTIALS
The K puzzle is found in the D → K + K − π + π − decay measured by the CLEO collaboration[15], based on 3 × signal events. With such limited data set, the amplitude analysis heavilydepends on the model. Recently, the CLEO data is re-analyzed with improved lineshape parameter-izations [16]. With B ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − ) = (1 . ± . × − and B ( D → K − K +1 (1270) , K +1 (1270) → ρ K + , ρ → π + π − ) = (2 . ± . × − in [16], we canobtain η ′ = 0 . ± .
32, which is smaller than η = 1 . ± .
43, but larger than η = 0 . ± .
06. Thecentral value of the branching fraction of D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − is larger by one order of magnitude than our prediction in Eq. (20) based on the equality relationand the previous CLEO result. Besides, it is found a large contribution from K (1400) in [16], with B ( D → K − K +1 (1400) , K +1 (1400) → K ∗ π + , K ∗ → K + π − ) = (3 . ± . × − with its centralvalue larger by one order than our prediction in Table 5 under the naive factorization approach, andalso larger than B ( D → K − K +1 (1270) , K +1 (1270) → K ∗ π + , K ∗ → K + π − ) = (1 . ± . × − .It is a challenge to be understood, since the K (1400)-involved mode should be suppressed byits phase space from the finite-width effect in this kinematically forbidden decay. All the relatedresults are of large uncertainties. The additional four models in [16] also provide different results.A more precise analysis is required to understand the D → K + K − π + π − decay.LHCb is collecting the data of D decays. In [18], LHCb measured the mode of D → − π + π + π − with 9 × signal events. Considering the ratio of branching fractions B ( D → K + K − π + π − ) / B ( D → K − π + π + π − ) = (3 . ± . D → K + K − π + π − could be as large as 3 × at LHCb, since all the final particles of charged kaonsor pions are of similar detecting efficiencies. With the much larger data of the D → K + K − π + π − decay at LHCb compared to 3 × events at CLEO, the equality relation in Eq. (2) and theimportance of K (1400) could be tested.The equality relation in (2) is given by the ratios between the same weak decays, such as D → K − K +1 (1270) , K +1 (1270) → K ∗ π + v.s. D → K − K +1 (1270) , K +1 (1270) → ρ K + . In thisway, the weak decay parts are cancelled in the narrow width approximation. On the other hand,the equality relation can also be expressed asΓ( D → K − K +1 (1270)( → π + K ∗ ( → K + π − )))Γ( D → K − K +1 (1270)( → K + ρ ( → π + π − )))= Γ( D → K + K − (1270)( → π − K ∗ ( → K − π + )))Γ( D → K + K − (1270)( → K − ρ ( → π + π − ))) . (33)Experimental measurements can use the equality relation in the formula as either Eq. (2) or Eq.(33).Except for testing the equality relation in the D → K + K − π + π − decay, it is also helpful tomeasure the ratios or test the relations in other four-body D decays, such as D → K S K S π + π − , D + → K S π + π π , D + s → K S π + π + π − , etc. The K (1270) resonance exists in such processes. Allof the ratios or relations are listed in Tables 6 and 7, for the Cabibbo-favored and singly Cabibbo-suppressed modes, respectively. The ratios are given by the η parameter defined in Eq. (12), withthe factors from the isospin analysis of strong decays of K (1270), ρ and K ∗ . Any ratio in Tables6 and 7 can be measured to be compared with those in Table 3. More measurements on η will helpto solve the K puzzle.Note that all the processes listed in Tables 6 and 7 satisfy that m D ( s ) − ( m K (1270) + m π,K ) & Γ K (1270) , so that the narrow width approximation is still valid in these processes. Besides, in the K S involved modes in Table 6, the doubly Cabibbo-suppressed amplitudes are neglected due totheir smallness.In Tables 6 and 7, we only list the observables associated with K (1270) → K ∗ π and ρK , whichare relevant to the K puzzle. Actually, the ratios could be between any decay modes of K (1270),for example, the fractions between the D -wave and S -wave widths of K (1270) → K ∗ π and ρK .More precise measurements on K (1270) decays are helpful for the determination of the mixingangle θ K [19, 20, 24, 25] 13ome of the processes in Tables 6 and 7 are more preferred in the experimental measurements.Firstly, the branching fractions of the Cabibbo-favored modes are usually large, and hence easier tobe measured. In the decay of D + s → K + K S π + π − with a large branching fraction of (1 . ± . K (1270) related processes. Thus the equality relation can be directly testedwith the ratios in D + s → K S K +1 (1270) and D + s → K + K (1270). The D → K S π + π − π decay,with B = (5 . ± . K (1270) related processes to test the equality relation. Theobservables in Tables 6 and 7 can be measured and tested by BESIII, Belle II and LHCb in thenear future. V. CONCLUSIONS
Charmed meson decays can provide much useful information about strange axial-vector mesons.In this work, it is found that the data of K (1270) related processes in the D → K + K − π + π − modeare inconsistent with the equality relation under the narrow width approximation and CP con-servation of strong decays. The ratio between B ( D → K − K +1 (1270)( → π + K ∗ ( → K + π − ))) and B ( D → K − K +1 (1270)( → K + ρ ( → π + π − ))), with a value of 1 . ± .
57, deviates by about 2 σ fromthe one between B ( D → K + K − (1270)( → π − K ∗ ( → K − π + ))) and B ( D → K + K − (1270)( → K − ρ ( → π + π − ))) with a value of 0 . ± .
09. In the amplitude analysis by CLEO of the above mea-surement, K (1400) was neglected. We calculate the branching fractions of the D → K ± (1400)( → ρ K ± or K ∗ π + , K ∗ π − ) K ∓ modes using the factorization approach considering the finite-widtheffect. It is found that the branching fraction of D → K − K +1 (1400)( → π + K ∗ ( → K + π − )) iscomparable to D → K − K +1 (1270)( → π + K ∗ ( → K + π − )), and hence might contribute to theoverestimation of the latter process. Thus K (1400) could not be neglected in the analysis. Inaddition, some relations in other D decay modes to study K (1270) decays are proposed to betested by BESIII, Belle (II) and LHCb. 14 ABLE VI: The relations of the branching fractions of the Cabbibo-favored cascade decays listed in thetable, in which η is defined by Eq. (12).Four-body decays Resonant processes Relations D → K + K − π + π − B = B ( D → K +1 (1270) K − , K +1 → K ∗ π + , K ∗ → K + π − ) B = B ( D → K +1 (1270) K − , K +1 → ρ K + , ρ → π + π − ) B = B ( D → K − (1270) K + , K − → K ∗ π − , K ∗ → K − π + ) B = B ( D → K − (1270) K + , K − → ρ K − , ρ → π + π − ) B / B = 4 η/ , B / B = 4 η/ D → K S K S π + π − B = B ( D → K (1270) K S , K → K ∗ + π − , K ∗ + → K S π + ) B = B ( D → K (1270) K S , K → ρ K S , ρ → π + π − ) B = B ( D → K (1270) K S , K → K ∗− π + , K ∗− → K S π − ) B = B ( D → K (1270) K S , K → ρ K S , ρ → π + π − ) B / B = 4 η/ , B / B = 4 η/ D → K − K S π + π B = B ( D → K +1 (1270) K − , K +1 → K ∗ π + , K ∗ → K S π ) B = B ( D → K +1 (1270) K − , K +1 → ρ + K S , ρ + → π + π ) B = B ( D → K (1270) K S , K → K ∗− π + , K ∗− → K − π ) B = B ( D → K (1270) K S , K → ρ + K − , ρ + → π + π ) B / B = η/ , B / B = η/ D → K + K S π − π B = B ( D → K − (1270) K + , K − → K ∗ π − , K ∗ → K S π ) B = B ( D → K − (1270) K + , K − → ρ − K S , ρ − → π − π ) B = B ( D → K (1270) K S , K → K ∗ + π − , K ∗ + → K + π ) B = B ( D → K (1270) K S , K → ρ − K + , ρ − → π − π ) B / B = η/ , B / B = η/ D + → K + K S π + π − B = B ( D + → K +1 (1270) K S , K +1 → K ∗ π + , K ∗ → K + π − ) B = B ( D + → K +1 (1270) K S , K +1 → ρ K + , ρ → π + π − ) B = B ( D + → K (1270) K + , K → K ∗− π + , K ∗− → K S π − ) B = B ( D + → K (1270) K + , K → ρ K S , ρ → π + π − ) B / B = 4 η/ , B / B = 4 η/ D + → K S K S π + π B = B ( D + → K +1 (1270) K S , K +1 → K ∗ π + , K ∗ → K S π ) B = B ( D + → K +1 (1270) K S , K +1 → ρ + K S , ρ + → π + π ) B / B = η/ D + → K + K − π + π B = B ( D + → K (1270) K + , K → K ∗− π + , K ∗− → K − π ) B = B ( D + → K (1270) K + , K → ρ + K − , ρ + → π + π ) B / B = η/ D + s → K + π + π − π B = B ( D + s → K (1270) π + , K → K ∗ + π − , K ∗ + → K + π ) B = B ( D + s → K (1270) π + , K → ρ − K + , ρ − → π − π ) B = B ( D + s → K +1 (1270) π , K +1 → K ∗ π + , K ∗ → K + π − ) B = B ( D + s → K +1 (1270) π , K +1 → ρ K + , ρ → π + π − ) B / B = η/ , B / B = 4 η/ D + s → K S π + π + π − B = B ( D + s → K (1270) π + , K → K ∗ + π − , K ∗ + → K S π + ) B = B ( D + s → K (1270) π + , K → ρ K S , ρ → π + π − ) B / B = 4 η/ D + s → K S π + π π B = B ( D + s → K +1 (1270) π , K +1 → K ∗ π + , K ∗ → K S π ) B = B ( D + s → K +1 (1270) π , K +1 → ρ + K S , ρ + → π + π ) B / B = η/ ABLE VII: Same as Table 6 but for singly Cabibbo-suppressed modes.Four-body decays Resonant processes Relations D → K S π + π − π B = B ( D → K − (1270) π + , K − → K ∗ π − , K ∗ → K S π ) B = B ( D → K − (1270) π + , K − → ρ − K S , ρ − → π − π ) B = B ( D → K (1270) π , K → K ∗− π + , K ∗− → K S π − ) B = B ( D → K (1270) π , K → ρ K S , ρ → π + π − ) B / B = η/ , B / B = 4 η/ D → K − π + π + π − B = B ( D → K − (1270) π + , K − → K ∗ π − , K ∗ → K − π + ) B = B ( D → K − (1270) π + , K − → ρ K − , ρ → π + π − ) B / B = 4 η/ D → K − π + π π B = B ( D → K (1270) π , K → K ∗− π + , K ∗− → K − π ) B = B ( D → K (1270) π , K → ρ + K − , ρ + → π + π ) B / B = η/ D + → K S π + π + π − B = B ( D + → K (1270) π + , K → K ∗− π + , K ∗− → K S π − ) B = B ( D + → K (1270) π + , K → ρ K S , ρ → π + π − ) B / B = 4 η/ D + → K − π + π + π B = B ( D + → K (1270) π + , K → K ∗− π + , K ∗− → K − π ) B = B ( D + → K (1270) π + , K → ρ + K − , ρ + → π + π ) B / B = η/ D + s → K + K S π + π − B = B ( D + s → K +1 (1270) K S , K +1 → K ∗ π + , K ∗ → K + π − ) B = B ( D + s → K +1 (1270) K S , K +1 → ρ K + , ρ → π + π − ) B = B ( D + s → K (1270) K + , K → K ∗− π + , K ∗− → K S π − ) B = B ( D + s → K (1270) K + , K → ρ K S , ρ → π + π − ) B / B = 4 η/ , B / B = 4 η/ D + s → K S K S π + π B = B ( D + s → K +1 (1270) K S , K +1 → K ∗ π + , K ∗ → K S π ) B = B ( D + s → K +1 (1270) K S , K +1 → ρ + K S , ρ + → π + π ) B / B = η/ D + s → K + K − π + π B = B ( D + s → K (1270) K + , K → K ∗− π + , K ∗− → K − π ) B = B ( D + s → K (1270) K + , K → ρ + K − , ρ + → π + π ) B / B = η/ et al. 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