Doubly heavy baryons at LHC

The theoretical analysis of production, lifetime, and decays of doubly heavy baryons is presented. The lifetime of $\Xi_{cc}^{++}$ baryon recently measured by the LHCb Collaboration is used to estimate the lifetimes of other doubly heavy baryons. The production and the possibility of observation of $\Xi_{bc}$ baryon at LHC are discussed.


I. INTRODUCTION
Doubly heavy baryons are extremely interesting objects that allow us to take a fresh look at the problems of the production and hadronization of heavy quarks. These baryons consist of two heavy and one light quarks and therefore, unlike ordinary heavy baryons, are characterized by several scales at once: where m Q 1 , m Q 2 are masses of heavy quarks, and v is there velocity inside the quarkonium.
For clarity, one can go to the coordinate representation and select a specific family of baryons.
Thus, for a baryon Ξ bc containing b -and c -quarks simultaneously, the scales are ordered as follows: λ b : λ c : r bc : r QCD ≈ 1 : 3 : 9 : 27, where λ Q = 1/m Q is a Compton length of quark, r bc ∼ 1/(v · m Q ) is heavy quark size, r QCD = Λ QCD is a scale of nonperturbative confinement [1].
It is worth to mention, that a baryon with one heavy quark is characterized by only two scales, namely, the mass of the heavy quark and Λ QCD . In the limit m Q 1 , m Q 2 → ∞ a heavy diquark interacts with a light quark as heavy anti-quark and, therefore, it is quite natural to subdivide calculating the characteristics of doubly heavy quarkonium in two stages: the calculation of the properties of the heavy diquark and the subsequent calculation of the properties of the system of quark-diquark 1 .
The problems of production and decays of such systems was of interest to researchers for many years. But the last year was special because it was marked by the discovery of the doubly charmed Ξ ++ cc baryon in the decay mode Λ + c K − π + π + [4]. The LHCb Collaboration observes hundreds of such particles. This discovery was confirmed by the observation of decay Ξ ++ cc → Ξ + c π + [5]. This circumstance greatly revived the research activities in this direction. In this article we discuss the perspectives of further research of doubly heavy baryon states: there decays, productions and possibility of observation of excited states.
The rest of the paper is organized as follows. In the next section production of doubly heavy baryons is considered. Section III is devoted to theoretical calculation of the lifetimes of the considered particles. Observation probability of these baryons is discussed in section IV and finally the Conclusion will be given.

II. DOUBLY HEAVY BARYON PRODUCTION
It is natural to use a two-step procedure to produce a doubly heavy baryon. In the first calculation step a doubly heavy diquark is produced perturbatively in the hard interaction.
In the second step a doubly heavy diquark is transformed to the baryon within the soft hadronization process.
Our calculation of doubly heavy diquark production were done within the following approach: 1. the color singlet model for doubly heavy mesons and the color triplet model for doubly heavy baryons; 2. the contribution from scattering of sea heavy quark and gluon (Q 1 g → Q 1 + Q 2 +Q 2 ) does not take into account to avoid double counting; 2 3. the contribution of color sextet state to baryon production is neglected.
Quarks in color antitriplet3 c attract each other and their interaction can be described by the wave function in the framework of potential model, as well as the quark-antiquark interaction in quarkonium. By analogy with quarkonium one can write for the production amplitude of doubly heavy diquark: where T Ssz Q 1Q1 Q 2Q2 is an amplitude of the hard production of two heavy quark pairs; Ψ Llz is the diquark wave function (color antitriplet); J and j z are the total angular momentum and its projection on z-axis in the [Q 1 Q 2 ]3 c rest frame; L and l z are the orbital angular momentum of bc-diquark and its projection on z-axis; S and s z are Q 1 Q 2 -diquark spin and its projection; C Jjz szlz are Clebsh-Gordon coefficients; p i are four momenta of diquark,Q 1 quark andQ 2 quark; q is three momentum of Q 1 -quark in the Q 1 Q 2 -diquark rest frame (in this frame (0, q) = k( q)).
Under assumption of small dependence of T Ssz bbcc on k( q) and, particularly, for the S-wave states where R S (0) is a value of radial wave function at origin.
In our early work [6] we discussed the similarity of the production mechanisms of doubly charmed baryons and the associative J/ψ and the open charm in hadronic interactions.
Indeed, both processes within a single parton scattering approach are described by the similar sets of diagrams, because both ones involve the production of four heavy quarks (see diagram examples in Fig. 1). However, the experimental data indicate the presence of contribution of double parton scattering (DPS), which dominates at LHC energies [7].
Within the DPS mechanism two cc pairs are produced independently in the different parton Figure 1: The example of analogous diagrams for (Q 1Q2 )-quarkonium production and for (Q 1 Q 2 q)baryon production.
interactions. Such mechanism can contribute to the associative J/ψ + c production but one can hardly contribute to the process Ξ cc production, because to produce doubly charmed baryon c charm quark from different pairs are needed. 3 Thus we currently tend to think, that DPS mechanism contributes only to J/ψ + c production. This is why the yield of Ξ cc is essentially smaller, than the yield of the associative production of J/ψ-meson and open charm, whereas the yields of B c mesons and Ξ bc baryons should be comparable. Also it is worth to mention that J/ψ + c cross section and Ξ cc cross section should have different dependence on the pp interaction energy: DPS cross section increases faster than SPS.
It should be noted that the doubly heavy diquark production can not be described within the fragmentation model due to the large contribution of non-fragmentation diagrams, which can not be interpreted as b-quark production followed by the fusion of b-quark into bcdiquark. The same feature is inherent in the process of B c -meson production. This is not surprising because the production processes of bc-diquark production and B c production are described by the same set of the diagrams. The difference comes from different color coefficients and different choice of values for c and b quark masses.
The dominant contribution to the production cross under LHCb kinematics conditions comes from gluonic interaction, as well as for the B c meson: gg → Ξ bc +bc.
Our estimations for that process show that difference of yields of Ξ bc and B c is mostly 3 However these is a research, where it was made an attempt to expand the DPS model to the case of Ξ cc production [8] using quark-hadron duality approach. determined by the difference of wave functions: Indeed, if one choose the same quark mass values for the subprocesses gg → [bc]3 +bc and gg → B c +bc and put R 2 [bc]3 = R 2 Bc one can see that this process have very similar behavior on transverse momenta of doubly heavy system, as it is shown in Fig. 2, where we put |R Bc (0)| 2 and |R [bc]3 (0)| 2 equal for convenience of comparison.
Of course, a color antitriplet of bc system is not a Ξ bc yet. It should be somehow transformed to the bcq baryon. The transverse momentum of light quark q with mass m q is about mq is a transverse momentum of Ξ bc . For LHCb kinematical conditions such quark always exits in the quark sea. This is why we assume, that a doubly heavy is hadronized by joining with a light quarks u, d and s in proportion 1 : 1 : 0.3. We also assume that it is hadronized with probability equal 1. It is worth to note, that the latter assumption is pretty much a guess, because diquark has a color charge and therefore strongly interacts with its environment, that could lead to the diquark dissociation. Thus, (3) can be considered as an upper limit for ratio of yields of Ξ bc and B c .
We estimate the ratio of yields Ξ bc and B c for hadronic interactions at √ s = 13 TeV for several scales (µ R = µ F = 10 GeV, T ) and find, that the dependence of this value on scale choice is unessential. The main uncertainties come from wave functions and from choice of mass values for b and c quarks. There are many estimations for R [bc] 3 (0) value, as well as for R Bc (0) (see, for example [1,[10][11][12][13]). However, to obtain the ratio, it is rational to use values extracted within the similar framework. From [1] and [10], where the non-relativistic model with Buchmüller-Tye wave function was used, we obtain that From [12] and [11], where the relativistic potential model was applied and relativistic correction have been accounted perturbatively, we obtain for the same ratio In [14,15] the corrections to the relativistic potential model predictions had been taken into account non-perturbatively, that leads to the noticeable difference of wave function values for different spin states. However the cross section ratio value remains the same: Therefore, one can conclude that It is worth to note that both the numerator and the denominator in (4) will be modified by the feed-down from excitations. However we believe, that in ratio these contributions will approximately canceled out. The obtained ratio value σ Ξ bc /σ Bc coincides with that used in talk [16].
As it was mentioned before an analogous ratio can not be valid for J/ψ + c and Ξ cc due to the large contribution of DPS to the associative J/ψ and c production.

A. Method description
In accordance with Operator Product Expansion (OPE) and optic theorem the life time of doubly heavy baryon B can be represented as where operator T is with In the above expression Wilson coefficients C ± (µ) equal where α s (µ) is a running strong coupling constant calculated within two-loop approximation and n f is a number of active flavors. The operators O ± in (7) are determined as follows: where α, β, γ, δ are color indices of quarks.
For large energy of heavy quark decay one can represent T (6) a set of local operators ordered by increasing of their dimension. The contribution of high dimension term are suppressed by inverse powers of heavy quark mass m Q , and therefore only several first terms contribute to the decay value. This method was broadly used for the calculation of lifetimes of heavy hadrons [6,[17][18][19][20][21][22][23], as well as doubly heavy hadrons [24,25]. It was shown in the cited papers the operators of dimension 3 and 5 correspond to the spectator decay of heavy quark and give the main contribution to the value (5). The following operator of dimension 6 can also give noticable contribution to the decay process: and Pauly interference (c,d).
The other operators of dimension: tribute insignificantly comparing with (11).
Typical Feynman diagrams for the discussed processes are shown in Fig. 4. In accordance with OPE method the following mechanisms can contribute to the total decay width: • Spectator mechanism ( the operator (10) and the diagram 4(a)), • Weak scattering,WS (the operator (11) and the diagram 4(b)), • Pauli-interference, PI (the operator (11) and the diagrams 4(c), (d)), The decay amplitudes for doubly charmed baryons Ξ ++ cc and Ξ + cc can be performed as follows: In these equations the contribution of operators with dimension 3 and 5 can be determined where and the width of spectator mechanism was estimated in papers [24,[26][27][28][29][30][31].
As it was mention above the contribution values of PI and WS mechanisms depend on the baryon composition. For example, it is clear from diagrams in Fig. 4 that for Ξ ++ cc = (ccu) and Ω + cc = (ccs) the WS is forbidden and PI destructively contributes to the width. Contrary, for the Ξ + cc the PI is forbidden. Taking this in mind one can perform the contributions of operators of 6 dimension as follows: where (see, e.g, [24,[32][33][34]) and (13) : : . : In these relations we also introduce the notations The hadronic matrix elements are determined as follows: where Q = c, b is a heavy quark, q = u, d is light quark, and |Ψ dl (0)| 2 is a wave function at origin. The wave function structure leads to the following relation: where T µ is an arbitrary spinor matrix.

C. Lifetimes of doubly beauty baryons
For the double beauty baryons Ξ 0 bb = (bbu), Ξ − bb = (bbd) and Ω − bb = (bbs) WS mechanism contributes only to the width of neutral states, whereas for charge states the PI mechanism contribution must be accounted for the charged states: The spectator mechanism of b-quark decay is described by the following operators with dimensions 3 and 5: where (17) : : Lifetimes of Ξ + bc , Ξ 0 bc , Ω 0 bc baryons It can be easily seen that in the case of Ξ + bc = (bcu), Ξ 0 bc = (bcd), and Ω 0 bc = (bcs) baryons both PI and WS channels are opened. As a result, the corresponding transition amplitudes are equal to where the contributions of c and b quarks' spectator decays are given in the previous subsections and PI, WS amplitudes are equal to In these expressions [25] T b PI,sc = − : : .
The other functions are defined earlier.

E. Numerical results
From presented above results it is clear that in OPE formalism theoretical predictions of doubly heavy baryons' lifetimes depend on such input parameters as quark masses, wave function at the origin, etc. In paper [35] the following values of these parameters were used:   Figure 5: Lifetimes in ps for Ξ ++ cc (solid black curve), Ξ + cc (blue dashed curve) and Ω + cc (red dotted curve) as a function of the model parameters. The results of [35] are shown by dots. it was proposed that a slightly different masses should be used in the case of doubly heavy baryons. We will discuss the results of these papers in the next subsection, while here we consider quark masses as free and check the dependence of doubly heavy baryons lifetimes on the variation of these parameters.
In Figure 5 we show model parameter dependence of Ξ ++ cc lifetime, while Fig. 6a shows m c dependence of different channels that contribute to this lifetime. It can be seen from these figures that τ (Ξ ++ cc ) is most sensitive to change of c-quark mass. Our analysis shows that experimental value (27) is restored with the following values: With these masses we have τ (Ξ ++ cc ) = 0.26 ± 0.03 ps. In the second column of table I we show calculated with these masses contributions of different decay channels to Ξ ++ cc baryon lifetime in comparison with that presented in [35]. One can see from this  mentioned in the previous sections, the spectator decay channel gives the main contribution and it increases with the increase of charm quark mass. In addition, PI channel gives destructive contribution in this case, which leads to increase of the lifetime. As for weak scattering mechanism, it is forbidden for Ξ ++ cc decay. Using the approach described above, it is easy to calculate also lifetimes of Ξ + cc and Ω + cc baryons: τ (Ξ + cc ) = 0.14 ± 0.01 ps, τ (Ω + cc ) = 0.18 ± 0.02 ps.
Lifetime and decay width dependences on parameters are shown in figures 5, 6. The numerical estimations for parameter values (25) and (28) can be found in the third and fourth columns of table I. In the case of Ξ + cc baryon the PI channel is forbidden, thus only the spectator decay and the weak scattering give contributions. For for Ω + cc baryon the spectator and PI channels are important. The contribution of the last one is positive. As a result theoretical predictions for the lifetimes of Ξ + cc and Ω + cc are smaller than for Ξ ++ cc particle.   One can find in the literature some other theoretical works devoted to analysis of doubly heavy baryons lifetimes. In the current subsection we will discuss these papers and compare presented there results with ours.
As it was mentioned above, in papers [37,38] it was assumed that quark masses used for doubly heavy baryons analysis could be a little bit different from constituent quark masses obtained from analysis of meson spectroscopy. In particular, in paper [37]  One can see that the mass of the baryon is more close to the experimental value (27), while the lifetime is even smaller. We would like, however, make some comments considering the last result. Presented in [37] analytical expression for Ξ ++ cc decay width reads From this expression it is clear that in [37] only spectator decays of the valence c quark contribute. Indeed, the prefactor 10 = 2 × (3 + 1 + 1) in relation (32) shows that only c → sud, c → seν c , and c → µν mu channels were taken into account and the final result is doubled because of two valence quarks in Ξ cc baryon Fock state. It seems to us, that such an approach is not reliable.
First of all, as it can be clearly seen from comparison with neutron's total width, mentioned above factor 2 should be avoided. Indeed, since only one spectator decay d → ueν e is possible in this case and there are two valence d quarks in the neutron, used in [37] approach would give us the lifetime which is almost three times smaller than the experimental result τ exp n = 939 s. Without the factor 2 in relation (33) this disagreement is partially removed. In addition, in paper [37] contributions of any form factors are neglected. It is clear that the energy deposit in Ξ cc baryon decay is much larger than for neutron β-decay. It is well known, however, that even in the latter case n → peν e such form factors are important (actually, the axial form factor helps us to obtain the experimental value of the considered lifetime), so it seems strange to forget about them in the case of Ξ cc lifetime.
The other point is that PI and WS contributions are completely ignored in [37]. As a result, one can expect that lifetimes of all ccq, ccs baryons should be equal to each other.
For some reason, however, the authors of paper [37] use completely different approach to calculate Ξ + cc baryon lifetime and the value τ Ξ + cc ≈ τ (Ξ ++ cc )/2 is given there. No detailed explanation for such difference in calculation methods is presented in [37].
If we use the presented in [KR14] values in described above OPE calculations, the lifetime of Ξ ++ cc baryon is equal to 0.32ps, that is a little bit larger than the experimental result (27). In paper [38] ([KR18]) another set of quark masses was presented, that describe both meson and baryon masses: No predictions for the lifetimes can be found in this paper, but OPE approach gives the value τ (Ξ ++ cc ) ≈ 0.37 ps, which is also larger than the experimental one. In a series of papers [33,[39][40][41] the lifetimes of heavy and doubly heavy baryons are considered in the framework of operator product expansion with PI and WS channels taken into account. The result of these works agrees qualitatively with ours (for example, the hierarchy of cc-baryons lifetimes is the same), but the numerical values of the lifetimes are somewhat larger. The reason for the difference is that used in these papers values of quark masses are smaller (for example, m c = 1.35 GeV in these papers).
It should be noted that the mass of c quark is not really large, so higher order contributions in operator product expansion could also give significant contributions. In the recent article [42] the authors show that the experimental value of Ξ ++ cc baryon lifetime can be explained if contributions of higher dimension operators are taken into account. It is interesting to note, that the lifetimes of other doubly charmed baryons are changed in different way in comparison with our results: τ (Ξ + cc ) decreases only slightly, while the lifetime of Ω + cc baryon increases and is comparable with τ (Ξ ++ cc ). It is clear that a detailed theoretical and experimental investigation of the lifetimes of these particles is highly desirable.

IV. OBSERVATION PERSPECTIVES
Here we briefly discuss the observation possibilities of doubly heavy baryons at LHC.
As it was already mentioned the observation of Ξ ++ cc baryon has been done by the LHCb Collaboration in the decay mode Λ + c K − π + π + [4] and confirmed in the decay mode Ξ + c π + [5]. The next step is the observation of Ξ cb baryon. In spite of large number of theoretical predictions for branching fractions (see, for example, [1,[43][44][45][46][47][48] and Table IV), the "golden mode" is not found yet. Of course, the greater branching fraction value, the more chances for the decay mode to be observed. But the decay branchings of intermediate particles are also very important. In addition, as it is shown in [16], the possibility of the experiment also must be taken into account. For example, each extra track in final state decreases the registration efficiency. That is why understanding the experiment features is very important for searching the most promising decay modes. We share cautious optimism of [16] about the observation of particle in the LHCb data of Run I and Run II, and also think that in any case Ξ cb will be observed in the LHCb data of Run III.
As for the observation of the Ξ bb , we doubt its possibility at the LHC because of the very small production rate. This article is devoted to theoretical study of total widths, production rates, and observation probabilities of the doubly heavy baryons.
We briefly discussed the production and the possibility of observation of Ξ bc baryon at LHC, and showed that the kinematical features of Ξ bc baryon production and B c meson production are very similar.