# Fragmentation-fraction ratio in - and -baryon decays

###### Abstract

We study the ratio of fragmentation fractions, , from the measurement of and with . With the branching fraction obtained under the U-spin symmetry, the fragmentation ratio is determined as =. To reduce the above uncertainties, we suggest to measure the branching fractions of and at BESIII, Belle(II) and LHCb.

## I Introduction

Bottom quarks can be produced at the high energy colliders, such as LHC and Tevatron, and then hadronized into mesons and -baryons. The probability of a bottom quark fragments into a certain weakly decaying -hadron is called the fragmentation fractions, i.e. , , and . As non-perturbative effects, the fragmentation fractions can only be determined by experiments in some phenomenological approaches.

The -meson fragmentation fractions have been measured by LEP, Tevatron and LHC with a relatively high precision Patrignani:2016xqp (); Amhis:2016xyh (). However, the current understanding of -baryon productions is still a challenge. The total fragmentation fraction of -baryon is the sum of all the weakly-decaying -baryons,

(1) |

where the isospin symmetry is assumed as , and is the correction from to . The averages of the total baryon production fractions are Amhis:2016xyh ()

(2) |

which are inconsistent with each other, and of large uncertainties.

The total fraction of -baryons has not been determined by LHCb because of its lack of measurements on and . It has been found that the ratio depends on the of the final states Aaltonen:2008zd (); Aaltonen:2008eu (); Aaij:2011jp (); Aaij:2014jyk (). At LHCb, the kinematic averaging ratio is Aaij:2011jp ()

(3) |

It is required for the information of and to determine the other fragmentation fractions at LHCb due to the constraint of

(4) |

Since the production of is suppressed compared to those of by the production of an additional strange quark, the determination of is essential to understand the productions of -baryons and mesons.

So far, only Refs. Voloshin:2015xxa (); Hsiao:2015txa () have predicted the ratio , both based on the processes of and with the data given by CDF and D0. At LHCb, the productions of and have been measured by the heavy-flavor-conserving process of Aaij:2015yoy (), and the charm-baryon involving decays of and via Aaij:2014esa (). All the results are listed in Table. 1. The production with the charm-baryon involving method is of the most high precision. The ratio can be obtained as long as the related branching fractions are determined. Among them, the absolute branching fraction of has never been measured Patrignani:2016xqp (), thus is of the largest ambiguity. In this work, we determine with obtained under the -spin symmetry.

This article is organized as follows. In Sec. II, we introduce the status of . The branching fraction of and are obtained in Sec. III and IV, respectively. Sec. V is the summary.

Measurements | |
---|---|

Patrignani:2016xqp () (CDF,D0) | Voloshin:2015xxa () |

Patrignani:2016xqp () (CDF,D0) | Hsiao:2015txa () |

Aaij:2015yoy () (LHCb) | (MIT bag model) Cheng:2015ckx () |

(diquark model) Cheng:2015ckx () | |

Aaij:2014esa () (LHCb) | (this work) |

## Ii Status of

In some literatures, it is usually assumed that the difference between the productions of and is from the strange quark and up or down quarks HerediaDeLaCruz:2011yi (); Aaij:2014esa (),

(5) |

However, since the fragmentation fractions are non-perturbative effects, they can only be extracted from experimental data. In this section, we introduce the status of by means of the relevant measurements.

### ii.1 v.s.

So far, the only theoretical analysis on are performed in Refs. Voloshin:2015xxa (); Hsiao:2015txa () based on the experimental data of and . In Ref. Patrignani:2016xqp (), the relevant results averaging the measurements by CDF and D0 Aaltonen:2009ny (); Abazov:2007am (); Abazov:2011wt (); Abe:1996tr (), are given as

(6) |

The fragmentation fraction ratio of can be obtained unless the ratio of branching fractions of and is known.

Both and are the transitions with the spectators of and , respectively. Therefore, the two processes are related to each other under the flavor symmetry. The width relation of

(7) |

is given by Voloshin Voloshin:2015xxa (). Using the experimental data in (6), the ratio of the fragmentation fractions can then be obtained as Voloshin:2015xxa ()

(8) |

Hsiao et ac express the decay amplitudes of and in the factorization approach Hsiao:2015txa (). They relate the form factors of -baryon to light baryon octet transitions based on the symmetry, and obtain the ratio of branching fractions . Utilizing the data in Eq. (6), the authors give a result similar to Eq. (8),

(9) |

### ii.2 Heavy-flavor-conserving decay

The LHCb collaboration has observed the first heavy-flavor-conserving hadronic weak decay Aaij:2015yoy (), with

(10) |

In Ref. Aaij:2015yoy (), is assumed to be bounded between 0.1 and 0.3, and then obtain the branching fraction of lie in the range from to . On the contrary, the fragmentation ratio can be obtained if is determined.

In Ref. Cheng:2015ckx (), the branching fraction of is calculated in the MIT bag model and the diquark model,

(11) |

Subsequently, we can obtain the ratio of fragmentation fractions according to Eq.(10) as,

(12) | ||||

(13) |

### ii.3 v.s. via

In the above two methods, the experimental measurements are of large uncertainties, as seen in Eqs. (6) and (10). In the decay of , the efficiency of reconstruction of with and , is very small in the hadron colliders Abazov:2007am (); Aaltonen:2009ny (). On the other hand, the branching fraction of is expected to be very small.

Compared to the above processes, the relative production ratio between and has been measured by LHCb with much higher precision Aaij:2014esa (),

(14) |

As long as the branching fractions of the relevant - and -baryon decays are known, Eq. (14) could provide a good determination of . In Ref. Aaij:2014esa (), with naively expected values of and , it can be obtained that .

The branching fraction of has not been directly measured in experiment. and are equal to each other in the heavy quark limit and the flavor symmetry. In literatures, only Refs. Cheng:1996cs () and Ivanov:1997hi () calculate both the branching fractions of and . With the transition form factors in the non-relativistic quark model, the ratio of branching fractions involving the factorizable contribution can be obtained in Cheng:1996cs ():

(15) |

where the difference in the lifetimes is neglected since , and is the effective Wilson coefficient. The deviation from unity results from the mass difference between and , i.e. the breaking effect. In the soft-collinear effective theory, the non-factorizable contributions from the color-commensurate and the -exchange diagrams are suppressed by Leibovich:2003tw (). In Ref. Ivanov:1997hi () in a relativistic three-quark model, it is found that the non-factorizable contributions amount up to 30% of the factorizable ones, with the ratio of Therefore, even without a reliable study in a QCD-based approach, it can still be expected that the deviation of the ratio from unity is under control.

The absolute branching fraction of has been well measured by Belle and BESIII Zupanc:2013iki (); Ablikim:2015flg (), with Patrignani:2016xqp (). However, branching fraction of is of large ambiguity. The ratio of used in Aaij:2014esa (), is only naively assumed by the Cabibbo factor. In the next section, we will obtain the branching fraction of via -spin analysis, and then determine .

## Iii Branching fraction of

The understanding of charmed baryon decays are still of high deficiency both in theory and in experiment. So far, there has not been any measurement on the absolute branching fraction of decays Patrignani:2016xqp (). The ratio of has been measured to be Link:2001rn (); VazquezJauregui:2008eg (); Patrignani:2016xqp (). But it is still unknown for the absolute branching fraction of .

It is first found in Ref. Yu:2017zst () that can be obtained from the measured ratio of Link:2001rn (), and the -spin relation between and . We show the -spin analysis in detail in the present work.

The decays of and are both singly Cabibbo-suppressed modes, with the transition of where the minus sign between and comes from the Cabibbo-Kobayashi-Maskawa matrix elements, . Note that the -spin doublets are and . The effective Hamiltonian of changes the -spin by , , i.e. . and form a -spin doublet of . We have

(16) | ||||

(17) |

The -spin representations of the and states are

(18) | |||

(19) |

The decay amplitudes are then

(20) | ||||

(21) |

where and are the amplitudes of -spin of and , respectively. It is clear that the amplitudes satisfy

(22) |

This relation can also be seen from the topological diagrams in FIG.1.

According to the relation in Eq.(22), the branching ratio of can be obtained by

(23) |

where . The data of masses and lifetimes are taken from PDG Patrignani:2016xqp (): MeV, MeV, MeV, MeV, MeV, s, s. Besides, Patrignani:2016xqp (); Ablikim:2015flg (), and the branching ratios are Link:2002zx (); Link:2001rn ()

(24) | ||||

(25) |

Then we can obtain

(26) |

The uncertainty is dominated by the ratios of branching fractions of and decays in Eqs. (24) and (25).

The central value of at the order of percent, is larger than the typical order of of the ordinary singly Cabibbo-suppressed processes, such as . This can be clarified by Eq. (23). Firstly, the lifetime of is two times larger than that of . Secondly, the phase space of is larger than that of by another factor of two, i.e. MeV and MeV. Due to the larger lifetime and phase space, the branching fraction of is then at the order of percent, .

The understanding of the dynamics of charmed baryon decays is still a challenge at the current stage. Recent theoretical studies are mostly based on the flavor analysis Lu:2016ogy (); Wang:2017gxe (); Geng:2018plk (); Geng:2017mxn (); Geng:2017esc () and the current algebra Cheng:2018hwl (). They have not yet been applied to the singly Cabibbo-suppressed charmed baryon decays into a light baryon and a vector meson. Therefore, it is not available to estimate the -spin breaking effects in the above analysis of Eq. (26). In the -meson decays, the -spin breaking effects, or say the breaking effects, are mainly from the transition form factors and decay constants in the factorizable amplitudes, the difference between , and produced from vacuum in the -exchange and -annihilation amplitudes, and the Glauber strong phase with pion in the non-factorizable contributions Li:2012cfa (); Li:2013xsa (). In Fig. 1, both amplitudes in the and decay are non-factorizable. The vacuum production of and in the -exchange diagrams would be a main source of -spin breaking. In the modes involving a vector meson and a pseudoscalar meson in the final states of -meson decays, the difference between and production in the -exchange diagrams can be seen from and Jiang:2017zwr () where and are the magnitude and strong phase of the non-perturbative parameters in the -exchange diagrams, and the subscripts and denotes the quark flavor of produced from the vacuum. It can be found that the -spin breaking effects are not large in modes. The -exchange diagrams in charmed baryon decays are similar to the ones in charmed meson decays, with an additional spectator quark. It can be expected that the -spin breaking effects between and would not be large, and thus the prediction of would be under control.

The process of with all the charged final particles is widely used to study the properties of, or to search for some heavier baryons. The mass and lifetime of are measured with the most high precision via , Aaij:2014esa (). New states of and are observed in the spectrum with , Aaij:2014yka (). Five new resonances are observed in the final states of with Aaij:2017nav (). It is suggested to search for the doubly charmed baryons in the decay of with Yu:2017zst (); Wang:2017mqp (). In this work, the ratio of fragmentation fractions can be obtained as long as the branching fraction of is determined by Eq. (14).

## Iv and its implications

Utilizing the prediction of in Eq. (26), the measured value of Patrignani:2016xqp () and the reasonable theoretical ratio , the ratio of the fragmentation fraction for quark into and can be obtained from Eq.(14) as

(27) |

The uncertainty is mainly from the branching fraction of in Eq. (26). This result of is much smaller than the naive estimation of or 0.2 in Eq. (5), and the MIT bag model for the branching fraction of in Eq. (12). The central value of our result in Eq. (27) is one half of those obtained via in Eqs. (8) and (9). Only the prediction in the diquark model for in Eq. (13) is consistent with our result within the uncertainties, while the central value is larger as well.

The total -baryon fraction can be obtained by the ratio in Eq. (27). The production of is doubly suppressed by two strange quarks, estimated as 15% of the Aaij:2016avz (). It is smaller than the error of in Eq. (27), and thus can be neglected in the total fraction of -baryons. We then have

(28) |

It is equivalent that the correction in Eq. (1) is , which is smaller than the estimation of in Ref. Aaij:2016avz ().

With the result of in Eq. (27), the branching fraction of can be determined from Eq. (10),

(29) |

This is consistent with the diquark model, but larger than the MIT bag model, seen in Eq. (11).

The precision of our result of in Eq. (27) can be significantly improved by the measurements of and at LHCb, BESIII and Belle II. The large uncertainty of in Eq. (26), inducing the major uncertainty of , is dominated by two ratios of branching fractions: by FOCUS in 2001 Link:2001rn () and measured by FOCUS in 2002 Link:2002zx (). For the former, a more precise measurement can be performed by LHCb with partial wave analysis. At LHCb with the data of 3.3 fb, there are already events of Aaij:2017nav (), which is four orders of magnitude larger than 200 events in Ref. Link:2001rn (). The latter can be improved by the BESIII or Belle(II) experiments, which have recently performed a dozen measurements of decays Pal:2017ypp (); Ablikim:2016vqd (); Ablikim:2017iqd (); Zupanc:2013iki (); Ablikim:2016mcr (); Ablikim:2015prg (); Ablikim:2015flg (); Yang:2015ytm (); Ablikim:2016tze (); Ablikim:2017ors (), especially the observation of some singly or doubly Cabibbo-suppressed processes Yang:2015ytm (); Ablikim:2016tze (); Ablikim:2017ors (). With the updated measurements of and in the near future, could be of higher precision.

## V summary

In this work, we study the ratio of fragmentation fractions with the data of and , at LHCb, which is the most precise measurement related to the fragmentations of and , seen in Table. 1. The least known branching fraction of is obtained under the -spin symmetry, . The ratio is then determined to be =. This is the first analysis of using the LHCb data. It helps to understand the production of -baryons. To improve the precision, we suggest to measure the ratios of branching fractions and at BESIII, Belle(II) and LHCb using the current data set or in the near future.

###### Acknowledgements.

This research was supported by the National Natural Science Foundation of China under the Grant No. 11505083 and U1732101.## References

- (1) C. Patrignani et al. [Particle Data Group], Chin. Phys. C 40, no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001
- (2) Y. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C 77, no. 12, 895 (2017) doi:10.1140/epjc/s10052-017-5058-4 [arXiv:1612.07233 [hep-ex]].
- (3) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 77, 072003 (2008) doi:10.1103/PhysRevD.77.072003 [arXiv:0801.4375 [hep-ex]].
- (4) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 79, 032001 (2009) doi:10.1103/PhysRevD.79.032001 [arXiv:0810.3213 [hep-ex]].
- (5) R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 85, 032008 (2012) doi:10.1103/PhysRevD.85.032008 [arXiv:1111.2357 [hep-ex]].
- (6) R. Aaij et al. [LHCb Collaboration], JHEP 1408, 143 (2014) doi:10.1007/JHEP08(2014)143 [arXiv:1405.6842 [hep-ex]].
- (7) M. B. Voloshin, arXiv:1510.05568 [hep-ph].
- (8) Y. K. Hsiao, P. Y. Lin, L. W. Luo and C. Q. Geng, Phys. Lett. B 751, 127 (2015) doi:10.1016/j.physletb.2015.10.013 [arXiv:1510.01808 [hep-ph]].
- (9) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 115, no. 24, 241801 (2015) doi:10.1103/PhysRevLett.115.241801 [arXiv:1510.03829 [hep-ex]].
- (10) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 113, 032001 (2014) doi:10.1103/PhysRevLett.113.032001 [arXiv:1405.7223 [hep-ex]].
- (11) H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin, T. M. Yan and H. L. Yu, JHEP 1603, 028 (2016) doi:10.1007/JHEP03(2016)028 [arXiv:1512.01276 [hep-ph]].
- (12) I. Heredia-De La Cruz [D0 Collaboration], arXiv:1109.6083 [hep-ex].
- (13) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 80, 072003 (2009) doi:10.1103/PhysRevD.80.072003 [arXiv:0905.3123 [hep-ex]].
- (14) V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 99, 052001 (2007) doi:10.1103/PhysRevLett.99.052001 [arXiv:0706.1690 [hep-ex]].
- (15) V. M. Abazov et al. [D0 Collaboration], Phys. Rev. D 84, 031102 (2011) doi:10.1103/PhysRevD.84.031102 [arXiv:1105.0690 [hep-ex]].
- (16) F. Abe et al. [CDF Collaboration], Phys. Rev. D 55, 1142 (1997). doi:10.1103/PhysRevD.55.1142
- (17) H. Y. Cheng, Phys. Rev. D 56, 2799 (1997) doi:10.1103/PhysRevD.56.2799 [hep-ph/9612223].
- (18) M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and A. G. Rusetsky, Mod. Phys. Lett. A 13, 181 (1998) doi:10.1142/S0217732398000231 [hep-ph/9709325].
- (19) A. K. Leibovich, Z. Ligeti, I. W. Stewart and M. B. Wise, Phys. Lett. B 586, 337 (2004) doi:10.1016/j.physletb.2004.02.033 [hep-ph/0312319].
- (20) A. Zupanc et al. [Belle Collaboration], Phys. Rev. Lett. 113, no. 4, 042002 (2014) doi:10.1103/PhysRevLett.113.042002 [arXiv:1312.7826 [hep-ex]].
- (21) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 116, no. 5, 052001 (2016) doi:10.1103/PhysRevLett.116.052001 [arXiv:1511.08380 [hep-ex]].
- (22) J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B 512, 277 (2001) doi:10.1016/S0370-2693(01)00590-1 [hep-ex/0102040].
- (23) E. Vazquez-Jauregui et al. [SELEX Collaboration], Phys. Lett. B 666, 299 (2008) doi:10.1016/j.physletb.2008.07.072 [arXiv:0804.2298 [hep-ex]].
- (24) F. S. Yu, H. Y. Jiang, R. H. Li, C. D. Lü, W. Wang and Z. X. Zhao, arXiv:1703.09086 [hep-ph].
- (25) J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B 540, 25 (2002) doi:10.1016/S0370-2693(02)02103-2 [hep-ex/0206013].
- (26) C. D. Lü, W. Wang and F. S. Yu, Phys. Rev. D 93, no. 5, 056008 (2016) doi:10.1103/PhysRevD.93.056008 [arXiv:1601.04241 [hep-ph]].
- (27) D. Wang, P. F. Guo, W. H. Long and F. S. Yu, arXiv:1709.09873 [hep-ph].
- (28) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, arXiv:1801.03276 [hep-ph].
- (29) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, JHEP 1711, 147 (2017) doi:10.1007/JHEP11(2017)147 [arXiv:1709.00808 [hep-ph]].
- (30) C. Q. Geng, Y. K. Hsiao, Y. H. Lin and L. L. Liu, Phys. Lett. B 776, 265 (2018) doi:10.1016/j.physletb.2017.11.062 [arXiv:1708.02460 [hep-ph]].
- (31) H. Y. Cheng, X. W. Kang and F. Xu, arXiv:1801.08625 [hep-ph].
- (32) H. n. Li, C. D. Lu and F. S. Yu, Phys. Rev. D 86, 036012 (2012) doi:10.1103/PhysRevD.86.036012 [arXiv:1203.3120 [hep-ph]].
- (33) H. n. Li, C. D. Lü, Q. Qin and F. S. Yu, Phys. Rev. D 89, no. 5, 054006 (2014) doi:10.1103/PhysRevD.89.054006 [arXiv:1305.7021 [hep-ph]].
- (34) H. Y. Jiang, F. S. Yu, Q. Qin, H. n. Li and C. D. Lü, arXiv:1705.07335 [hep-ph].
- (35) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 114, 062004 (2015) doi:10.1103/PhysRevLett.114.062004 [arXiv:1411.4849 [hep-ex]].
- (36) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 118, no. 18, 182001 (2017) doi:10.1103/PhysRevLett.118.182001 [arXiv:1703.04639 [hep-ex]].
- (37) W. Wang, F. S. Yu and Z. X. Zhao, Eur. Phys. J. C 77, no. 11, 781 (2017) doi:10.1140/epjc/s10052-017-5360-1 [arXiv:1707.02834 [hep-ph]].
- (38) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 118, no. 5, 052002 (2017) Erratum: [Phys. Rev. Lett. 119, no. 16, 169901 (2017)] doi:10.1103/PhysRevLett.119.169901, 10.1103/PhysRevLett.118.052002 [arXiv:1612.05140 [hep-ex]].
- (39) B. Pal et al. [Belle Collaboration], Phys. Rev. D 96, no. 5, 051102 (2017) doi:10.1103/PhysRevD.96.051102 [arXiv:1707.00089 [hep-ex]].
- (40) M. Ablikim et al. [BESIII Collaboration], Phys. Lett. B 767, 42 (2017) doi:10.1016/j.physletb.2017.01.047 [arXiv:1611.04382 [hep-ex]].
- (41) M. Ablikim et al. [BESIII Collaboration], Phys. Lett. B 772, 388 (2017) doi:10.1016/j.physletb.2017.06.065 [arXiv:1705.11109 [hep-ex]].
- (42) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 118, no. 11, 112001 (2017) doi:10.1103/PhysRevLett.118.112001 [arXiv:1611.02797 [hep-ex]].
- (43) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 115, no. 22, 221805 (2015) doi:10.1103/PhysRevLett.115.221805 [arXiv:1510.02610 [hep-ex]].
- (44) S. B. Yang et al. [Belle Collaboration], Phys. Rev. Lett. 117, no. 1, 011801 (2016) doi:10.1103/PhysRevLett.117.011801 [arXiv:1512.07366 [hep-ex]].
- (45) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 117, no. 23, 232002 (2016) Addendum: [Phys. Rev. Lett. 120, no. 2, 029903 (2018)] doi:10.1103/PhysRevLett.117.232002, 10.1103/PhysRevLett.120.029903 [arXiv:1608.00407 [hep-ex]].
- (46) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 95, no. 11, 111102 (2017) doi:10.1103/PhysRevD.95.111102 [arXiv:1702.05279 [hep-ex]].
- (47) R. Aaij et al. [LHCb Collaboration], JHEP 1605, 081 (2016) doi:10.1007/JHEP05(2016)081 [arXiv:1603.00413 [hep-ex]].