Doubly-Heavy Baryon Weak Decays: \Xi_{bc}^{0}\to pK^{-} and \Xi_{cc}^{+}\to\Sigma_{c}^{++}(2520)K^{-}

Doubly-Heavy Baryon Weak Decays: and

Run-Hui Li 1, Cai-Dian Lü 2, Wei Wang 3, Fu-Sheng Yu 4, Zhi-Tian Zou 5 School of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, China;
Institute of High Energy Physics, YuQuanLu 19B, Beijing 100049, China;
School of Physics, University of Chinese Academy of Sciences, YuQuanLu 19A, Beijing 100049, China;
INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai 200240, China;
School of Nuclear Science and Technology, Lanzhou University, Lanzhou, 730000, China;
Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou, 730000, China;
Department of Physics, Yantai University, Yantai 264055, China
11Email:lirh@imu.edu.cn
22Email:lucd@ihep.ac.cn
33Email:wei.wang@sjtu.edu.cn
44Email:yufs@lzu.edu.cn
55Email:zouzt@ytu.edu.cn
Abstract

Doubly-heavy baryons, with two heavy and one light quarks, are expected to exist in QCD and their masses have been predicted in the quark model. However their existence is not well established so far in experiment. In this work, we explore the possibility of searching for and in the -exchange processes, and . On the basis of perturbative calculations, we estimate the branching ratio of the first decay as , where () are the ratios of the decay constants (lifetimes) of and . The branching ratio of is related to that of , and thereby a conjectured topology analysis leads to the range for the branching ratio as: . The decay would be reconstructed in the final state which is easy to access even at a hadron collider. Based on the two facts that abundant heavy quarks can be produced at a hadron collider like LHC, and the branching ratios of and are sizable, we urge our experimental colleagues to perform a search at LHCb. This will presumably lead to the discovery of the and , and precision measurements of the branching ratios in the future are helpful to investigate their decay mechanism.
Keywords: Doubly-heavy baryons; Electro-weak decays; Branching ratios; -exchange process.

I Introduction

In the quark model, two heavy quarks (bottom and/or charm) can bound together with a light quark to form the so-called doubly-heavy baryons. The study of doubly-heavy baryons is of great interest for the understanding of the hadron spectroscopy of various systems. The study of their decays can also shed light on the nonperturbative dynamics in the transition with two heavy quarks.

The mass spectra of ground states of doubly-heavy baryons have already been studied in various versions of the quark model Kiselev:2001fw ; Karliner:2014gca ; DeRujula:1975qlm ; Anikeev:2001rk ; Fleck:1989mb ; Korner:1994nh ; Roncaglia:1995az ; Lichtenberg:1995kg ; Ebert:1996ec ; Gerasyuta:1999pc ; Itoh:2000um ; Narodetskii:2002ib ; Ebert:2002ig ; He:2004px ; Richard:2005jz ; Migura:2006ep ; Albertus:2006ya ; Roberts:2007ni ; Weng:2010rb ; Zhang:2008rt ; Lewis:2001iz ; Flynn:2003vz ; Liu:2009jc ; Namekawa:2012mp ; Alexandrou:2012xk ; Briceno:2012wt ; Alexandrou:2014sha ; Gerasyuta:2008zy ; Garcilazo:2016piq ; Valcarce:2008dr ; meanwhile their lifetimes Karliner:2014gca ; Anikeev:2001rk ; Kiselev:2001fw ; Moinester:1995fk ; Chang:2007xa ; Guberina:1999mx and production  Kiselev:2001fw ; Karliner:2014gca ; Chang:2006eu ; Chang:2006xp ; Gunter:2001qy ; Kiselev:1994pu ; Koshkarev:2016rci ; Koshkarev:2016acq ; Ma:2003zk ; Chang:2009va ; Chang:2007pp ; Zhang:2011hi ; Chen:2014hqa are also investigated in phenomenological ways. However the doubly-heavy baryons like and are not well established in experiment. The only evidence for , found by the SELEX collaboration Mattson:2002vu ; Ocherashvili:2004hi , is not confirmed by other experiments Kato:2013ynr ; Aaij:2013voa . Actually, the search for doubly-heavy baryons depends on two factors, the production and decay. At a hadron collider like LHC, abundant heavy quarks can be generated, which means plenty of doubly-heavy hadrons due to quark-hadron duality. For instance the properties of the have recently been studied by the LHCb collaboration in great detail Aaij:2016qlz ; Aaij:2017kea ; Aaij:2013cda ; Aaij:2016xas ; Abazov:2008kv ; Aaij:2014ija . Another factor, the decay of the doubly-heavy baryons, is the main focus of this paper and subsequent ones.

In this work, we will explore the possibility of searching for the and through the and decays. Both channels are dominated by the -exchange contribution in theory, but as we will demonstrate later that their branching ratios are likely sizable. These channels have an advantage because of the all charged final states and thus can be accessed straightforwardly in experiment.

So far, our knowledge of the doubly-heavy baryon decays is limited; for example even the decay constants are still not well known, which prevents us from making reliable predictions of decay branching ratios in a QCD-rooted approach. Fortunately one can make use of the analogue in and decays, which can result in an estimation of decay branching ratios. Since the bottom quark is annihilated, there is a large energy release in the decay . As a result, the proton and kaon move very fast in the rest frame of . The large momentum transfer in this process guarantees the applicability of QCD perturbation theory. It enables us to relate this channel to which has been calculated in perturbative QCD Lu:2009cm . The decay is governed by the transition at the quark level, where the momentum transfer is limited and thus nonperturbative dynamics is dominating. The decay  Olive:2016xmw , which is also a pure -exchange process with exactly the same polarization contributions, provides an opportunity to estimate the branching ratio of .

In the rest of this paper, we will analyze the decay in Sec. II and decay in Sec. III. A brief summary is given in the last section.

Ii The study of decay

Figure 1: The leading -exchange diagram contributions for and decays.

The decay can proceed either through a exchange between and transitions or by double flavor changing neutral current (FCNC) processes of and . The latter one is highly suppressed by loop effects and thus can be neglected. Therefore the -exchange mechanism depicted in Fig. 1(a) is the only tree level contribution to the decay . The amplitude of decay can be decomposed into two different structures with corresponding coefficients and Lu:2009cm :

(1)

where and are the respective spinors of the proton and baryon with and being the momenta. Since is very heavy,  Kiselev:1999zj , the energy release of this decay is expected to be large. Therefore the proton and kaon in the final state can be approximately treated as light-like particles, and in the light cone coordinates the momenta and are defined as

(2)

and in Eq. (1) can be extracted from the calculation of Fig. 1(a). Firstly a very simple picture in the heavy quark limit is adopted, in which -quark is the only heavy one while , , and quarks are massless. At the leading power in , one can find a very similar diagram, depicted by Fig. 1(b), in the decay . Factoring out the decay constants, masses and CKM matrix elements, one can find that the two diagrams in Fig. 1 should contribute equally. Fig. 1(b), called Bow-tie in Ref. Lu:2009cm but named as the W-exchange ( for short) contribution in this paper, has been calculated in the conventional perturbative QCD approach. For the pQCD approach, see Ref. Keum:2000ph ; Lu:2000em . For the decay , the diagrams give Lu:2009cm

(3)

In contrast with the situation in decay, both tree (shown in Fig. 1(b)) and penguin operators contribute to the topological diagrams of decay. One can see from Table I of Ref. Lu:2009cm that the penguin contributions are suppressed by at least one order of magnitude due to the Wilson coefficients. Therefore, one can take the values of and in Eq. (3) approximately as the contribution due to Fig. 1(b). The differences between diagram (a) and (b) of Fig. 1 are listed as follows.

  • The CKM matrix elements, which is in diagram (a) and in diagram (b), are approximately the same in magnitude.

  • The kaon and pion are in the same octet in the symmetry. In this paper the symmetry breaking effect arises from the decay constants which are used as  Olive:2016xmw .

  • The difference between and resides in the decay constants, masses and lifetimes. and will appear as unknown parameters because of the absence of ’s decay constant and the large ambiguity of its predicted lifetime in the literature. According to the structure of pseudoscalar meson wave functions, the terms in the magnitude can be divided into two groups: one with the chiral mass of pseudoscalar meson, the other without. The former group is proportional to with , and the latter one is proportional to . Here is the chiral mass of the pseudoscalar meson, where and are the masses of the valence quarks. Neglecting the small difference between and , one can treat the total magnitude as simply being proportional to .

Combining all the pieces, the for decay are given as

(4)

and the branching ratio is predicted as

(5)

The predicted ranges from to Karliner:2014gca ; Anikeev:2001rk ; Kiselev:2001fw , and Olive:2016xmw . Assuming the decay constants of and are of the same order, it is expected that the branching ratio of is of the order of .

It should be stressed that the perturbative QCD calculation bases on the leading order analysis in the expansion. Above all, the study of decays shows that only considering the perturbative contribution Lu:2009cm will undershoot the data, and it indicates that the nonperturbative mechanism might also contribute sizably. If it were also the situation in , the branching ratio will be greatly enhanced compared to the value given in Eq. (5).

Iii The Study of decay

Figure 2: Leading Feynman diagrams of and decays.

Another decay with only the contribution at tree level is (shown in Fig. 2(a)). The with two identical heavy quarks is expected to be quite different from . Nonperturbative contributions are much more important because of the low energy scale. Fortunately, there exists a twin process, , which is depicted in Fig. 2(b). Both of the processes are decays of a baryon to a baryon and a kaon. The only difference is the spectator quark. Investigating these two decays in the rest frame of the initial baryons, one finds that either or in the final states moves slowly, since the energy release is very small. It means that the spectator -quark almost keeps static in the decay . Meanwhile, the spectator -quark moves with the similar energy of in both initial and final states. Both the spectators exchange little energy with the weak transitions, which indicates that the magnitudes of the diagrams are not sensitive to the spectators. With this viewpoint, one can reach the conclusion that Fig. 2(a) and (b) have nearly equal magnitudes. However, if comparing their branching ratios, one should notice that each charm quark in can decay to a strange one. Combing the differences of phase space and the lifetimes, the branching ratio is given by

(6)

where is from the phase space difference, from the symmetry of exchanging identical particles -quarks in the , and is the ratio of and lifetimes. There are a lot of studies of the mass of with small deviations from each other, and  Karliner:2014gca is adopted in this paper. The masses which are not listed here are all from the Particle Data Group Olive:2016xmw . Using the measured value of branching ratio  Olive:2016xmw , we have

(7)

is predicted in the literature with large uncertainty, ranging from 0.5 to 2.5  Karliner:2014gca ; Anikeev:2001rk ; Kiselev:2001fw ; Moinester:1995fk ; Chang:2007xa ; Guberina:1999mx , and  Olive:2016xmw . With the experimental errors neglected, the branching ratio is given by

(8)

Experimentally, is found to decay into with branching ratio . This would make the branching ratio of the three-body decay the same order as that of the two body decay shown in Eq. (8). Therefore, this decay mode is an ideal discovery channel. Actually, the SELEX collaboration found the evidence for in this process Mattson:2002vu .

Iv Summary

The existence of doubly-heavy baryons which consist of two heavy quarks and a light one is undoubted in QCD but has never been confirmed in experiment. Searching for these baryons is of great interest in hadron physics, and we believe that it is only a problem of time to establish their existence. To improve the efficiency of experimental searching, it becomes urgent to study theoretically the doubly-heavy baryon decays. In this work we present an estimate of the branching ratio of and decays, which may be ideal channels for the and reconstruction.

On basis of perturbative calculation, the first branching ratio is given by , where () are the ratios of the decay constants (lifetimes) of and . Considering that the lifetime of is smaller than that of by one order of magnitude, and assuming their decay constants are at the same order, the branching ratio of is of order of . The analysis of decays indicates that the decay magnitudes are probably underestimated if only considering the perturbative contribution. Including the nonperturbative contribution, one may get a larger result for the branching ratio.

is dominated by the nonperturbative dynamics because of small energy release. By analyzing an analogous decay , and utilizing its experimental result, we estimate that , where is the ratio of and lifetimes. Considering the rough predicted value for the lifetime of , one can specify that .

Acknowledgement

We are grateful for Ji-Bo He for enlightening discussions which initiated this project. We thank Yu-Ming Wang and Dan Zhang for fruitful discussions. This work was supported in part by the National Natural Science Foundation of China under the Grant Nos. 11505098, 1164700375, 11347027, 11505083, 11575110, 11375208, 11521505, 11621131001, 11235005, 11655002, and 11447032, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100 and No. 15ZR1423100, by the Young Thousand Talents Plan, and Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, and Shanghai Key Laboratory for Particle Physics and Cosmology(SKLPPC), and the Natural Science Foundation of Shandong province ZR2014AQ013, and the Doctoral Scientific Research Foundation of Inner Mongolia University.

References

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