Production of the Doubly Charmed Baryons at the SELEX experiment -- The double intrinsic charm approach

The high production rate and $\langle x_{F} \rangle>0.33$ of the doubly charmed baryons measured by the SELEX experiment is not amenable to perturbative QCD analysis. In this paper we calculate the production of the doubly heavy baryons with the double intrinsic charm Fock states whose existence is rigorously predicted by QCD. The production rate and the longitudinal momentum distribution are both reproduced. We also show that the production rates of the doubly charmed baryons and double $J/\psi $ production observed by NA3 collaboration are comparable. Recent experimental results are reviewed. The production cross section of the doubly charmed baryons at a fixed-target experiment at the LHC is presented.


Introduction
The SELEX measurements of the production of the doubly charmed baryons at large x F are among the most intriguing and surprising results in modern baryonic physics [1,2,3]. The SELEX experiment is a fixed-target experiment utilizing the Fermilab negative charged beam at 600 GeV/c and positive beam at 800 GeV/c to produce charm particles in a set of thin foil of Cu or in a diamond and operated in the x F > 0.1 kinematic region. The negative beam composition was about 80% Σ − and 20% π − . The positive beam was 90% protons.
In early 2000s the SELEX published first observation of 15.9 signal over 6.1 ± 0.5 background events, at a mass of 3.52 GeV, of the doubly charmed baryons in the charged decay mode Ξ + cc → Λ + c K − π + from 1630 Λ + c → pK − π + events sample [1] which was previously used for precision measurement of Λ + c lifetime [4,5]. Using same search strategy the SELEX reported 20 signal events, at a mass of 3.76 GeV, of Ξ ++ cc → Λ + c K − π + π + decay mode over 1656 Λ + c → pK − π + events sample [2]. In 2005 the SELEX collaboration published an observation of 5.62 signal over 1.38 ± 0.13 background events, at a mass of 3.52 GeV, of Ξ + cc → pD + K − decay mode from 1450 D + → K − π + π + decays to complement the previous results [3]. The SELEX measurements imply that the lifetime of Ξ + cc is less than 33 fs at 90% confidence level [1].
The production cross section has not been provided by the SELEX collaboration. Still the production properties of Ξ + cc and Ξ ++ cc can be compared to that of Λ + c baryon: where N is number of events in the respective sample and the reconstruction efficiencies of Ξ + cc and Ξ ++ cc are ǫ + ≃ 0.11 [1] and 1/ǫ ++ ≃ 3.7 [2] respectively. Such a high production rate with x F > 0.33 and the relatively small mean transverse momentum ≈1 GeV/c is not amenable to perturbative QCD analysis [1,2].
The production of states with two charm quarks with a high fraction of a light hadron's momentum is unexpected if one adopts the conventional assumption that heavy quarks can only arise from gluon splitting as in DGLAP evolution.
However, QCD predicts another source of heavy quarks in the wavefunction of a light hadron -from diagrams where the heavy quarks are multiply attached by gluons to the valence quarks [6,7]. In this case, the frame-independent light-front wavefunction of the light hadron has maximum probability when the Fock state is minimally off-shell. This occurs when all of the constituents are at rest in the hadron rest frame and thus have the same rapidity when the hadron is boosted. Equal rapidity occurs when the light-front momentum fractions of the Fock state constituents are proportional to their transverse mass; i.e. when the heavy constituents have the largest momentum fractions. This feature underlies the Brodsky, Hoyer, Peterson, and Sakai (BHPS) model for the distribution of intrinsic heavy quarks [8,9].
Thus hadrons containing heavy quarks, such as the Λ c , the J/ψ, and even the doubly charmed baryons such as the ccu or ccd, can be produced in a hadronic collision with a high momentum fraction of the beam momentum from the coalescence of the produced heavy and valence quarks. The SELEX doubly charmed baryon results thus signify a significant probability for the existence of Fock states such as |h l cccc , where h l is light quark content of the initial hadron.

The doubly charmed baryons production cross section
In the BHPS model the wavefunction of a hadron in QCD can be represented as a superposition of Fock state fluctuations, e.g. |h ∼ |h l + |h l g + |h l cc . . . , where h l , as above, is light quark content. When the projectile interacts with the target, the coherence of the Fock components is broken and the fluctuation can hadronize. The intrinsic charm Fock components are generated by virtual interactions such as gg → cc where gluon couple to two or more projectile valence quarks. The probability to produce such cc fluctuations scales as α 2 s (m 2 c )/m 2 c relative to leading-twist production.
Following [8,9,10] a general formula for the probability distribution of an n-particle intrinsic charm Fock state as a function of x i and transverse momentum k T,i can be written as: where m T,i denotes m 2 i + k 2 T,i and m h is mass of the initial hadron. Let us denote the probability of |h l cc and |h l cccc Fock states as P ic and P icc . In this paper we will also simplify the formula with replacement m T,i with the effective massm i = m 2 i + k 2 T,i and neglect the mass of the light quarks. This model assumes that the vertex function in the intrinsic charm wavefunction is relatively slowly varying; the particle distributions are then controlled by the light-cone energy denominator and phase space. The Fock states can be materialized by a soft collision in the target which brings the state on shell. The distribution of produced open and hidden charm states will reflect the underlying shape of the Fock state wavefunction.

The double intrinsic charm approach
We assume that all of the doubly charmed baryons are produced from |h l cccc Fock states. In the quark-hadron duality approximation the probability to produce a Ξ cc is proportional to the fraction of cc production below threshold mass m th = m D + ∆m [11], where m D is Dmeson mass and ∆m ≃ 0.5-1 GeV. The fraction of cc pairs can be written as: To obtain the fraction ratio of cc pairs into Ξ cc baryons we have to isolate color-antitriplet states, the fraction ratio of the doubly charmed baryons is where the s c is the color-antitriplet factor. The cc pair has 3 × 3 = 9 color components, 3 color-antitriplet, and 6 color-sixtet. Assuming that cc are unpolarized in the color space in the double intrinsic charm Fock state, there is 1/3 probability for the color-antitriplet possibility. Finally, we get s c ≃ 2 × 1/3. Let us remind the reader that some of c-quarks could produce open charm states so we need to interpret f Ξ cc /h as the upper limit. If we take m c = 1.5 GeV the value of f Ξ cc /p ≈ 0.6. This model also predicts f Ξ cc /Σ − ≃ f Ξ cc /p that is comparable with the SELEX data [1].
There is a simple connection between the intrinsic charm cross section and the inelastic one [10,12,13] where µ 2 ≈ 0.2 GeV 2 denotes the soft interaction scale parameter; P ic ≃ 0.3-2% (see [14] and references therein). In the Ref. [10] it is found that for proton P icc ≈ 20% · P ic . In our calculation we use the following approximation [12,13]: The normalization is fixed to be the same as Eq. 4 at √ s = 20-40 GeV. Combining Eqs. (3), (4) and (5)3, 4 and 5 we may expect the upper limit of the production cross section of the doubly charmed baryons to be: where σ pQCD (cc) ≈ σ(gg → cc) ≃ 5.8 × 10 6 pb is the charm production cross section at 600 GeV/c beam momentum, where most of statistics was collected, calculated with CalcHEP Monte-Carlo tool [15].

The intrinsic charm approach
The intrinsic charm production cross section of the doubly charmed baryons can be written as follows: where f g,c (x, µ) is the gluon [16] or intrinsic charm [17] distribution functions, x is the ratio of the parton momentum to the momentum of the hadron and µ is the energy scale of the interaction. Explicit form ofσ(gc → Ξ + cc ) can be found in [18]. In the SELEX case these calculations have been done in Ref. [19]: σ ic (Ξ + cc ) ≃ 102 pb. The value is relatively small and can be neglected.

The total production cross section
The charm quark fragmentation into the doubly charm baryon and the perturbative approaches give too small a contribution and can be also neglected so the total production cross section of the doubly charmed baryons at the SELEX experiment will be: It is interesting to estimate of the doubly charmed baryon production at a fixed-target experiment at the LHC [20] with √ s ≃ 115 GeV. Following the method described above the production cross section of the doubly charmed baryons will be: where the value of f Ξ cc /p ≈ 0.56 and σ in ( √ s ≃ 115 GeV) ≈ 28.4 mb [21]. It is two order of magnitude bigger than predicted in ref. [22] with the single intrinsic charm approach.

The shape of P icc (Ξ cc ) as a function of x F
As we already mentioned (see Sec. 1) the large mean x F and small mean transverse momentum is not amenable to perturbative QCD analysis. The x F distribution of Ξ cc baryons can be written as: The mean value of x F is Integrating Eq. 7 over the SELEX kinematic region, x F > 0.1, we find x F ≈ 0.33-0.34 that is in agreement with the SELEX data, x F ∼ 0.33 for Ξ + cc → Λ + c K − π + decay [1]. In his PhD thesis, Mattson provide the x F distribution of the Ξ ++ cc candidates into Λ + c K − π + π + decay mode [2]. Integrating Eq. 7 over the region where data presents, x F > 0.175 (see Fig. 3), we find x F ≈ 0.36 that also agrees with the data. The relatively small transverse momentum also is a sign of the intrinsic charm mechanism.

Solving mystery of the SELEX result
As we noted above, the SELEX collaboration did not provide the doubly charmed baryons production cross section but we are still able to compare it to the production properties of the Λ + c baryons. Let us remind the reader the measured ratios: R exp In 3 the leading order perturbative QCD the production cross section of the Λ + c baryons can be approximated as: The SELEX search strategy of the doubly charmed baryons requires minimum value of x Λ + c > 0.15 [1,2,4,5]. Assuming that x Λ + c ∼ x c and using CalcHEP Monte-Carlo tool find σ(gg → cc)| x c >0.15 ∼ 3 · 10 5 pb, fragmentation ratio f (c → Λ + c ) = 0.071 ± 0.003 (exp.) ± 0.018 (br.) [23] so the production cross section of the Λ + c baryons at the SELEX experiment will be: Using the branching ratios predicted by J. Bjorken Br(Ξ + cc → Λ + c K − π + ) = 0.03 and Br(Ξ ++ cc → Λ + c K − π + π + ) = 0.05 [24], one can obtain the ratio of the production cross sections: However this result is not really accurate. Playing with parameters we can change both doubly charmed baryons and charm production cross sections in wide enough range. The most important thing about the ratio (8) is that it has the same order of magnitude as the measured ones against a few order of magnitude gap another predictions provide [11,18,19]. The relatively high production rate of cccc states to charm is not a unique feature of the SELEX experiment. The double J/ψ production properties measured by the NA3 experiment [25,26] have many similar features: the high σ(ψψ)/σ(ψ) = (3 ± 1) × 10 −4 rate, large x ψψ and small average transverse momentum, p T,ψψ = 0.9 ± 0.1 GeV/c. It is interesting to compare the SE-LEX result with the NA3 data on the double J/ψ production. The NA3 experiment is a beam dump experiment at CERN utilizing antiprotons, protons, pions and kaons at 150, 200 and 280 GeV/c to produce charm particles with incident on hydrogen and platinum targets in the x F > 0 kinematic region. The most informative data the NA3 collaboration present is the double J/ψ production with π − beam at 280 GeV/c. It is not possible to compare the SELEX and the NA3 data directly but we are able to compare the following ratios, where R denotes σ(cccc)/σ(cc): where f ψ/π ≈ 0.03 [10] and f J/ψ ≈ 0.06 [27]. Therefore, as we can see, the NA3 data complements the hight production rate at the SELEX experiment.

Review of Belle and LHCb recent results
The Belle experiment [28] presented the upper limit on the σ(e + e − → Ξ + cc X) is 82-500 fb for the decay mode with the Λ + c at √ s = 10.58 GeV using 980 fb −1 . The most realistic calculations [11,29] of the upper limit cross section predict σ(Ξ + cc ) ≃ 35 ± 10 fb what turns out to be at least twice as less as the given limit.
Another recent result from the LHCb experiment [30] provides the upper limits at 95% C.L. on the ratio σ(Ξ + cc ) · Br(Ξ + cc → Λ + c K − π + )/σ(Λ + c ) to be 1.5 × 10 −2 and 3.9 × 10 −4 for lifetimes 100 fs and 400 fs respectively, for an integrated luminosity of 0.65 fb −1 . It is compared with result from Ref. [11,19,18,31] ∼ 10 −4 -10 −3 . However, the LHCb did not reach the lifetime measured by the SELEX experiment yet. Moreover, the LHCb analysis requires that Λ + c candidates have to be significantly displaced from the primary vertex so this requirement potentially cuts down most of the signal region. The contribution from the double intrinsic charm is suppressed due to LHCb experiment kinematics. Assuming that the hadron identification efficiency for pions and kaons is degraded above 100 GeV/c [32] (such that when raised to the fourth power it is negligible), and making the naive assumption that momentum is split evenly between all final-state tracks, the analysis loses sensitivity around p(Ξ cc ) = 500 GeV/c, i.e. x F = 0.14.

Summary
The experimental results (see Sec. 1) and theoretical predictions (see Sec. 4) on the production properties of the Ξ cc in the SELEX experiment have the same order of magnitude accuracy. The predicted mean Feynmanx values (see Sec. 3) agree with the experimental data. The NA3 collaboration result on the double J/ψ production strongly complements the SELEX data. We would like to specially point out the fact that unexpectedly high production rate of Ξ cc baryons is due to the kinematics features of the SELEX experiment, and could not be described by the production mechanism only. We also find that the doubly intrinsic charm approach will be the leading production mechanism of the doubly charmed baryons at high Feynman-x at a future fixed-target experiment at the LHC.