Recent Belle results related to $π-K$ interactions
Bilas Pal















 


                             
                             
                             




We report the recent results related to  π-K interactions based on the data collected by the Belle experiment at the KEKB collider. This includes the branching fraction and CP asymmetry measurements of B^+→ K^+K^-π^+ decay, search for the Λ_c^+→ϕ pπ^0, Λ_c^+→ P_s^+π^0 decays, branching fraction measurement of Λ_c^+→ K^-π^+pπ^0, first observation of  doubly Cabibbo-suppressed decay Λ_c^+→ K^+π^-p, and the measurement of CKM angle ϕ_3 (γ) with a model-independent Dalitz plot analysis of B^±→ DK^±,D→ K_S^0π^+π^- decay.



Recent Belle results related to π-K interactions
    Bilas Pal
 (On behalf of the Belle Collaboration)
 Brookhaven National Laboratory, Upton, NY & University of Cincinnati, Cincinnati, OH
    December 30, 2023
===========================================================================================================================================







§ INTRODUCTION

In this report, we present some recent results related to π-K interactions based on the  data, collected by the Belle experiment at the KEKB e^+e^- asymmetric-energy collider <cit.>. 
(Throughout this paper charge-conjugate modes are implied.) 
The experiment
took data at center-of-mass energies corresponding
to several Υ(nS) resonances; the total data sample
recorded exceeds 1  ab^-1.

The Belle detector is a large-solid-angle magnetic
spectrometer that consists of a silicon vertex detector
(SVD), a 50-layer central drift chamber (CDC),
an array of aerogel threshold Cherenkov counters
(ACC), a barrel-like arrangement of time-of-flight
scintillation counters (TOF), and an electromagnetic
calorimeter comprised of CsI(Tl) crystals
(ECL) located inside a super-conducting solenoid
coil that provides a 1.5 T magnetic field. An iron
flux-return located outside of the coil is instrumented
to detect K^0_L
mesons and to identify muons
(KLM). The detector is described in detail elsewhere <cit.>.



§ CP ASYMMETRY IN B^+→ K^+K^-Π^+ DECAYS

In the recent years, an unidentified structure has been observed by BaBar <cit.> and LHCb experiments <cit.> in the low K^+K^-
invariant mass spectrum of the B^+→ K^+K^-π^+ decays. The LHCb reported a nonzero inclusive CP asymmetry of -0.123±0.017±0.012±0.007 and a large unquantified local CP asymmetry in the same mass region. These results suggest that final-state interactions may contribute to CP violation <cit.>. In this analysis, we attempt to quantify the CP asymmetry and branching fraction as a function of the K^+K^- invariant mass, using 711  fb^-1 of data, collected at Υ(4S) resonance <cit.>.

The signal yield is extracted by performing a two-dimensional unbinned maximum likelihood fit to the variables: the beam-energy constrained mass M_ bc  and the energy difference Δ E. The resulting branching fraction and CP asymmetry are

    ℬ(B^+→ K^+K^-π^+)   =   (5.38±0.40±0.35)×10^-6,
    
    A_CP   =   -0.170±0.073±0.017,

where the quoted uncertainties are statistical and systematic, respectively.

To investigate the localized CP asymmetry in the low K^+K^-
invariant mass region, we perform the 2D fit (described above) to extract the signal yield and A_CP in bins of M_K^+K^-. The fitted results are shown in Fig. <ref> and Table <ref>. We confirm the excess and local A_CP in the low  M_K^+K^- 
 region, as reported by the LHCb, and quantify the differential branching fraction in each K^+K^- invariant mass bin. We find a 4.8σ evidence for a negative CP asymmetry in the region M_K^+K^-<1.1 GeV/c^2. To understand the origin of the low-mass dynamics, a full Dalitz analysis from experiments with a sizeable data set, such as LHCb and Belle II, will be needed in the future.





§ SEARCH FOR  AND BRANCHING FRACTION MEASUREMENT OF   


The story of exotic hadron  spectroscopy   begins  with the  discovery  of  the
X(3872) by the Belle collaboration in 2003 <cit.>. Since then, many exotic XYZ states have been reported by Belle and other experiments <cit.>. Recent observations of two hidden-charm pentaquark  states P_c^+(4380) and P_c^+(4450) by the LHCb collaboration in the J/ψ p invariant mass spectrum of the  Λ_b^0→ J/ψ pK^- process <cit.> raises the question of whether a hidden-strangeness pentaquark  P_s^+, where the cc̅ pair  in  P_c^+  is replaced by  an ss̅ pair, exists <cit.>. The strange-flavor analogue of the P_c^+ discovery channel is the decay Λ_c^+→ϕ pπ^0 <cit.>, shown in Fig. <ref> (a). The detection of a hidden-strangeness pentaquark could be possible through  the ϕ p invariant mass spectrum within this channel [see Fig. <ref> (b)]
if the underlying mechanism creating the P_c^+ states also holds for P_s^+, independent of the flavor <cit.>, and only if  the mass of P_s^+ is less than M_Λ_c^+-M_π^0.  


In an analogous ss̅ process of ϕ photoproduction (γ p→ϕ p),  a forward-angle  bump structure at √(s)≈2.0 GeV 
has been observed by  the LEPS <cit.> and CLAS collaborations <cit.>.

However, this structure appears only at the most forward angles, 
which is  not   expected for  the decay of a resonance <cit.>.




Previously, the decay Λ_c^+→ϕ pπ^0 has not been studied by any experiment. Here, we report a search for this decay, using 915 fb^-1 of data <cit.>. In addition, we search for the nonresonant decay Λ_c^+→ K^+K^-pπ^0 and measure the branching fraction of the  Cabibbo-favored decay Λ_c^+→ K^-π^+pπ^0.

In order to extract the signal yield, we perform a two-dimensional (2D) unbinned extended maximum likelihood fit to the variables m (K^+K^-pπ^0) and  m(K^+K^-).  Projections of the fit result are shown in Fig. <ref>. 

From the fit, we extract 148.4±61.8 signal events, 75.9±84.8 nonresonant events, 
and 7158.4±36.4 combinatorial background events. 
The statistical significances are found to be 2.4 and 1.0 standard deviations for Λ_c^+→ϕ p π^0 and nonresonant Λ_c^+→ K^+K^- p π^0 decays, respectively. We use the well-established decay Λ_c^+→ p K^-π^+ <cit.> as the normalization channel for the branching fraction measurements. 






Since the significances are below 3.0 standard deviations both for ϕ pπ^0 signal and K^+K^-pπ^0 nonresonant decays, we set upper limits on their branching fractions  at 90% confidence level (CL) using a Bayesian approach. 





The results are

    ℬ(Λ_c^+→ϕ pπ^0)    <    15.3×10^-5 ,
    ℬ(Λ_c^+→ K^+K^-pπ^0)_ NR   <   6.3×10^-5 ,

which are the first limits on these branching fractions. 

To search for a putative P_s^+→ϕ p decay, we select Λ_c^+→ K^+K^-pπ^0 candidates in which m(K^+K^-) is within 0.020 GeV/c^2 of the   ϕ meson mass <cit.>
and plot the  background-subtracted m(ϕ p) distribution (Fig. <ref>). This distribution is obtained by performing 2D fits as discussed above in bins of m(ϕ p). 
The data shows no clear evidence for a P_s^+ state. 
We set an upper limit on the product branching fraction 
ℬ(Λ_c^+→ P_s^+π^0) ×ℬ(P_s^+→ϕ p) by fitting the  distribution of Fig. <ref>
to the sum of a RBW function and a phase space 
distribution determined from a sample of simulated Λ^+_c→ϕ pπ^0 
decays. We obtain 77.6±28.1 P_s^+ events from the fit, which gives an upper limit of 

    ℬ(Λ_c^+→ P_s^+π^0) ×ℬ(P_s^+→ϕ p)  <  8.3× 10^-5

at 90% CL. 
From the fit, we also obtain, 
M_P_s^+=(2.025± 0.005) GeV/c^2 and 
Γ_P_s^+=(0.022± 0.012) GeV,
where the uncertainties are statistical only.


The high statistics decay  Λ_c^+→ K^-π^+pπ^0 is used to adjust the data-MC differences in the ϕ pπ^0 signal and K^+K^-pπ^0 nonresonant decays. For the Λ_c^+→ K^-π^+pπ^0 sample, 
the mass distribution is plotted in Fig. <ref>. We fit this distribution to obtain the signal yield. We find 242 039± 2342 signal candidates and 472 729± 467 background candidates. We measure the ratio of branching fractions,

    ℬ(Λ_c^+→ K^-π^+pπ^0)/ℬ(Λ_c^+→ K^-π^+p)=(0.685±0.007± 0.018),

where the first uncertainty is statistical and the second is systematic. Multiplying this ratio by the world average value of ℬ(Λ_c^+→ K^-π^+p)=(6.46±0.24)% <cit.>, we obtain

    ℬ(Λ_c^+→ K^-π^+pπ^0)=(4.42±0.05± 0.12±0.16)%,

where the first uncertainty is statistical, the second is systematic, and the third reflects the uncertainty due to the branching fraction of the normalization decay mode. This is the most precise measurement of ℬ(Λ_c^+→ K^-π^+pπ^0) to date and is consistent with the recently measured value ℬ(Λ_c^+→ K^-π^+pπ^0)=(4.53±0.23±0.30)% by the BESIII collaboration <cit.>.





§ OBSERVATION OF THE DOUBLY CABIBBO-SUPPRESSED  DECAY 

Several doubly Cabibbo-suppressed (DCS) decays of charmed mesons have been observed <cit.>. Their measured branching ratios with respect to the corresponding Cabibbo-favored (CF) decays play an important role in
constraining models of the decay of charmed hadrons and in the study of flavor- SU(3) symmetry <cit.>. On the other hand, because of the smaller production cross-sections for charmed baryons, DCS decays of charmed
baryons have not yet been observed, and only an upper limit, ℬ(Λ_c^+→ pK^+π^-)/ℬ(Λ_c^+→ pK^-π^+)<0.46% at 90% CL, has been reported by the FOCUS Collaboration <cit.>. Here we present the first observation of the DCS decay Λ_c^+→ pK^+π^- and the measurement of its branching ratio with respect to the CF decay Λ_c^+→ pK^-π^+, using 980  fb^-1 of data <cit.>.


Figure <ref> shows the invariant mass distributions of (a) pK^-π^+ (CF) and (b) pK^+π^- (DCS) combinations. DCS decay events are clearly observed in M(pK^+π^-). In order to obtain the signal yield, a binned least-χ^2 fit is performed. From the mass fit, we extract (1.452±0.015)×10^6 Λ_c^+→ pK^-π^+ events and 3587±380 Λ_c^+→ pK^+π^- events. The latter has a peaking background from the single Cabibbo-suppressed (SCS) decay Λ_c^+→Λ(→ pπ^-)K^+, which has the same final-state topology. After subtracting the SCS contribution, we have 3379±380±78 DCS events, where the first uncertainty is statistical and the second is the systematic due to SCS subtraction. The corresponding statistical significance is 9.4 standard deviations. We measure the branching ratio,

    ℬ(Λ_c^+→ pK^+π^-)/ℬ(Λ_c^+→ pK^-π^+)=(2.35±0.27±0.21)×10^-3,

where the uncertainties are statistical and systematic, respectively.  This measured branching ratio corresponds to (0.82±0.21)tan^4θ_c, where the uncertainty is the total, which is consistent  within 1.5 standard deviations with the 
naïve expectation (∼tan^4θ_c <cit.>). LHCb's recent measurement of ℬ(Λ_c^+→ pK^+π^-)/ℬ(Λ_c^+→ pK^-π^+)=(1.65±0.15±0.05)×10^-3 <cit.> is lower than our ratio at the 2.0σ level.  Multiplying this ratio with the previously measured ℬ(Λ_c^+→ pK^-π^+)=(6.84±0.24^+0.21_-0.27)% by the Belle Collaboration <cit.>, we obtain the 
 the absolute branching fraction of the DCS decay, 
 
    ℬ(Λ_c^+→ pK^+π^-)=(1.61±0.23^+0.07_-0.08)×10^-4,

where the first uncertainty is due to the total uncertainty of the branching ratio and the second is uncertainty due to the branching fraction of the CF decay. 
After subtracting the contributions of Λ^*(1520) and Δ isobar intermediates, which contribute only to the CF decay, the revised ratio,  ℬ(Λ_c^+→ pK^+π^-)/ℬ(Λ_c^+→ pK^-π^+)=(1.10±0.17)tan^4θ_c is consistent with the naïve expectation within 1.0 standard deviation.




§ Φ_3 MEASUREMENT WITH A MODEL-INDEPENDENT DALITZ PLOT ANALYSIS OF B^±→ DK^±,D→ K_S^0Π^+Π^- DECAY

The CKM angle ϕ_3 (also denoted as γ) is one of the least constrained parameters of the CKM Unitary Triangle. Its determination is however theoretically clean due to absence of loop contributions; ϕ_3 can be determined using tree-level processes only, exploiting the interference between b→ uc̅s and b→ cu̅s transitions that occurs when  a process involves a neutral D meson reconstructed in a final state accessible to both D^0 and D̅^0  decays (see Fig. <ref>). Therefore, the angle ϕ_3 provides a SM benchmark, and its precise measurement is crucial in order to disentangle non-SM contributions to other processes, via global CKM fits. The size of the interference also depends on the ratio (r_B) of the magnitudes of the two tree diagrams involved and δ_B, the strong phase difference between them. Those hadronic parameters will be extracted from data together with the angle ϕ_3.


The measurement are performed in three different ways: (a) by utilizing decays of D mesons to CP eigenstates, such as π^+π^-, K^+K^- (CP even) or K_S^0π^0, ϕ K_S^0 (CP odd), proposed by M. Gronau, D. London, and D. Wyler (and called the GLW method <cit.>)(b) by making use of DCS decays of D mesons, e.g., D^0→ K^+π^-, proposed by D. Atwood, I. Dunietz, and A. Soni (and called the ADS method <cit.>) and (c) by exploiting the interference pattern in the Dalitz plot of the D decays such as D^0→ K_S^0π^+π^-, proposed by A. Giri, Y. Grossman, A. Soffer, and J. Zupanc (and called the GGSZ method <cit.>). 

Using a model-dependent Dalitz plot method, Belle's earlier measurement <cit.>  based on a data sample of 605  fb^-1 integrated luminosity yielded ϕ_3=(78.4^+10.8_-11.6±3.6±8.9)^∘ and r_B=0.160^+0.040_-0.038±0.011^+0.050_-0.010, where the uncertainties are statistical, systematic and  Dalitz model dependence, respectively. Although with more data one can squeeze on the statistical part, the result will still remain limited by the model uncertainty.

In a bid to circumvent this problem, Belle has carried out a model-independent analysis <cit.>, using GGSZ method <cit.>,  that is further extended in a latter work <cit.>. The analysis is based on the 711  fb^-1 of data, collected at the Υ(4S) resonance. In contrast to the conventional Dalitz method, where the D^0→ K_S^0π^+π^- amplitudes are parameterized as a coherent sum of several quasi two-body amplitudes as well as a nonresonant term, the model-independent approach invokes study of a binned Dalitz plot. In this approach, the expected number of events in the i^th bin of the Dalitz plan for the D mesons from B^±→ DK^± is given by

    N^±_i=h_B[K_± i+ r^2_BK_∓ i+2√(K_iK_-i)(x_±c_i± y_±s_i)],

where h_B is the overall normalization and K_i is the number of events in the i^th Dalitz bin of the flavor-tagged (whether D^0 or D̅^0) D^0→ K_S^0π^+π^- decays, accessible via the charge of the slow pion in D^*±→ Dπ^±. The terms c_i  and s_i  contain information about the strong-phase difference between the symmetric Dalitz points [m^2(K_S^0π^+), m^2(K_S^0π^-)] and [m^2(K_S^0π^-), m^2(K_S^0π^+)]; they are the external inputs obtained from quantum correlated D^0D̅^̅0̅ decays at the ψ(3770) resonance in CLEO <cit.>. Finally x_± = r_B cos(δ_B ±ϕ_3) and y_± = r_B sin(δ_B ±ϕ_3), where δ_B is the strong-phase difference between B^±→D̅^0K^± and B^±→ D^0K^±.

We perform a combined likelihood fit  to four signal selection variables in all Dalitz bins (16 bins in our case) for 
the  B^±→ DK^± signal and Cabibbo-favored B^±→ Dπ^± control samples; the free parameters of the fit are x_±, y_±, overall normalization (see Eq. 1) and background fraction. Table <ref> summarizes
the results obtained for B^±→ DK^± decays. From these results, we obtain ϕ_3 = (77.3^+15.1_-14.9± 4.1 ± 4.3)^∘ and r_B = 0.145±0.030±0.010±0.011, where the first error is statistical, the second is systematic, and the last error is due to limited precision on c_i and s_i. Although ϕ_3 has a mirror solution at ϕ_3 + 180^∘, we retain the value consistent with 0^∘ < ϕ_3 <180^∘. 

We report evidence for direct CP violation, the fact that ϕ_3 is nonzero, at the 2.7 standard deviations  level. Compared to results of the model-dependent Dalitz method, this measurement has somewhat poorer statistical precision despite a larger data sample used. There are two factors responsible for lower statistical sensitivity: 1) the statistical error for the same statistics is inversely proportional to the r_B value, and the central value of r_B in this analysis is smaller, and 2) the binned approach is expected to have the statistical precision that is, on average, 10–20% poorer than the unbinned one. On the positive side, however, the large model uncertainty for the model-dependent study (8.9^∘) is now replaced by a purely statistical uncertainty due to limited size of the ψ(3770) data sample available at CLEO (4.3^∘). With the use of BES-III data, this error will decrease to 1^∘ or less. 
    
The model-independent approach therefore offers an ideal avenue for Belle II and LHCb  in their pursuits of ϕ_3. We expect that the statistical error of the ϕ_3 measurement using the statistics of a 50  ab^-1 data sample that will be available at Belle II will reach 1-2^∘. We also expect that the experimental systematic error can be kept at the level below 1^∘, since most of its sources are limited by the statistics of the control channels.




Acknowledgement:
The author thanks the organizers of PKI2018 for excellent hospitality and for assembling a nice scientific program. This work is supported by the U.S. Department of Energy.


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