W-exchange contribution to the decays $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ using light-cone sum rules

We calculate the W-exchange contribution to the $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay using light-cone sum rules. The two-particle light-cone distribution amplitudes of the pion are used as non-perturbative input for the sum rules calculation, and the perturbative kernel is calculated at the leading order. We obtain the corresponding decay branching fractions by combining our W-exchange amplitudes with the factorizable amplitudes given by various theoretical methods from the literature. It is shown that with the factorizable amplitudes from heavy quark effective theory, we obtain the branching fraction ratio ${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+})/{\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})= 1.42\pm 0.78$, which is consistent with the experimental value of $1.41\pm 0.17\pm 0.1$.

at the leading order. We obtain the corresponding decay branching fractions by combining our W-exchange amplitudes with the factorizable amplitudes given by various theoretical methods from the literature. It is shown that with the factorizable amplitudes from heavy quark effective theory, we obtain the branching fraction ratio B(Ξ ++ cc → Ξ + c π + )/B(Ξ ++ cc → Ξ + c π + ) = 1.42 ± 0.78, which is consistent with the experimental value of 1.41 ± 0.17 ± 0.1.

I. INTRODUCTION
The conventional quark model predicted the existence of doubly-heavy baryons consisting of two heavy quarks (bottom or charm quarks) [1][2][3][4][5][6]. After pursuing the doubly charmed baryons for decades, in 2017 the LHCb collaboration announced the observation of the lowest-lying state Ξ ++ cc with mass 3620.6 ± 1.5(stat) ± 0.4(syst) ± 0.3(Ξ + c )MeV/c 2 [7]. This new baryon was observed via the decay channel Ξ ++ cc → Λ + c K − π + π + which is consistent with the prediction given by Ref. [8]. One year later, in 2018, a two-body decay channel Ξ ++ cc → Ξ + c π + was observed [9], which further confirms the existence of this doubly charmed baryon. Recently, the LHCb collaboration has observed a similar decay channel, Ξ ++ cc → Ξ + c π + , and measured the branching fraction ratio [10]: which means that the branching fraction of the decay into Ξ + c is larger than that into Ξ + c . Theoretically, the weak decay Ξ ++ cc → Ξ +( ) c π + receives contributions from the two topological diagrams as shown in Fig. 1, the W-emission diagram (left) and the W-exchange diagram (right).
Unlike the weak decays of mesons, the W-exchange diagrams in the baryon decays are generally non-factorizable, and this difficulty increases when we are facing the decays of doubly heavy baryons. Nowadays, except an SU(3) symmetry analysis [32,33] and a phenomenological study [34], there is no QCD based or model-independent study on the W-exchange contribution in doubly charmed baryon decays. The mostly used approach to evaluate such a contribution is the pole-model [20,21,25,29,31]. Recently, the combination of the factorizable contribution and the non-factorizable contribution from the pole-model leads to the ratio B /B between 0.81 and 0.83 [20,21], which means that the branching fraction of the decay into Ξ + c is smaller, which is obviously contrary to the experimental result. Furthermore, if one includes the interference between the W-emission and the W-exchange contributions, the B /B will become much larger, namely 6.74 [25].
This deviation of the theoretical prediction from the experimental measurement implies that a more precise theoretical calculation for the W-exchange contribution in the Ξ ++ cc → Ξ +( ) c π + decay is necessary. In this work, we will use the method of LCSR to solve this problem. LCSR were firstly proposed to study the transition form factors of the radiative or semi-leptonic hadron decays [35][36][37]. In the framework of LCSR, the required transition matrix element can be extracted from a suitable correlation function at the hadron level. The quark-hadron duality enables us to relate Following Ref. [38], we have temporarily included the pion in the initial state instead of the final state as in the real decay. The advantage of this is to enable us to factorize out the matrix element 0|J Ξ c |Ξ +( ) c without any ambiguity after inserting the Ξ +( ) c state into the right-hand side of J Ξ c . Otherwise, the final pion state has to be moved to the right over J Ξc firstly before this factorization can be performed. Actually, during the calculation at the hadron level, by suitable analytic continuation this initial pion state will be moved to the final state. We have also introduced an auxiliary momentum k in the correlation function. This momentum is unphysical and should be set to zero at the end of the calculation. The reason why it must be introduced is closely related to the analytic continuation mentioned above, which will be explained in the next section.

III. HADRON LEVEL CALCULATION IN THE LCSR
In this section, we derive the calculation of the correlation function in Eq. (4) at the hadron level. For simplicity, we take the case of the Ξ + c as example, the derivation for the decays into Ξ + c is similar. We insert a complete set of states with the same quantum numbers as J Ξc (y) into the correlation function, where the integration over ρ Ξc represents the contribution from the continuous spectrum, ± corresponds to the positive or negative parity of the Ξ c states, namely Ξ c ( 1 2 ± ), and σ denotes the spin of the Ξ c . The momentum of the one-particle state in the complete set l should be on-shell, Ξc . This means that the first matrix element in Eq. (6) can be simply parameterized by the Ξ c decay constants λ ± Ξc : 0|J Ξc (y)|l, σ , while the second matrix element in Eq. (6) is a function of p 2 , q 2 , k 2 , (p−q) 2 and P 2 = (p−k −q) 2 .
Generally speaking, a matrix element like this should also depend on l 2 , however, the on-shell condition has reduced such a dependence. Further, since one can replace the d 3 l integration by a four-momentum integration with the use of residue theorem, after integrating over d 4 y, one arrives at where we have defined u + = u, u − = iγ 5 u. To simplify the calculation we have chosen p 2 = k 2 = 0 and q 2 = m 2 π ≈ 0. Now the correlation function depends on three Lorentz invariants (p−k) 2 , (p−q) 2 and P 2 , while the matrix element p − k, σ , ± |O i (0)J Ξcc (x)|π − (q) only depends on (p − q) 2 and Further, the same correlation function can be calculated at the quark-hadron level by the OPE in the deep Euclidean region, (p − k) 2 ∼ (p − q) 2 ∼ P 2 0, which will be explicitly done in the next section. In principle, the expression of the same correlation function at these two levels should be equivalent: where the (p − k) 2 dependence of Π O i QCD has been described as in the form of a dispersion integral. We have omitted the u quark mass and (m c + m s ) 2 is the quark level threshold to produce a Ξ c baryon. According to the quark-hadron duality, the integration over the continuous spectrum at the hadron level is canceled by the corresponding integration above a certain threshold s Ξc at the quark-gluon level, which leads to the equation: Generally, this threshold parameter should be slightly larger than the mass squared of the corresponding hadron state. Here, it is chosen to be the same as in Ref. [14], where a QCDSR was used to study the semi-leptonic decay of the Ξ cc → Ξ c . It should be mentioned that in principle s Ξc is a universal parameter which is process-independent so that this procedure should be reasonable.
Note that since ImΠ O i QCD (s , (p − q) 2 , P 2 ) WE is an analytic function of P 2 , the left-hand side of Eq. (11) shares this property. Therefore one can extend P 2 to the physical region, namely P 2 > 0, which allows us to replace the initial state |π − (q) of the matrix element on the left-hand side by a final state π + (−q)|. After that, a Borel transformation for (p − k) 2 on both sides of Eq. (11) is performed, and we obtain Next, we insert another complete set of states with the same quantum numbers asJ Ξcc into the left-hand side of Eq. (12), which becomes For the same reason as discussed before, the matrix element p − k, s , ± ; π + (−q)|O i (0)|p − q, s, ± WE only depends on P 2 . The (p − q) 2 dependence on the right-hand side of Eq. (12) can be further expressed as a dispersion integral, Now using the quark-hadron duality again to cancel out the ds integration above a certain threshold s Ξcc corresponding to the lowest Ξ cc state, and performing the Borel transformation for (p − q) 2 , we arrive at Due to the existence of the auxiliary momentum k, unlike Eq. (3), the matrix element appearing in Eq. (15) must be parameterized by four terms: The form factors A ± ± 1,2,i and B ± ± 1,2,i are functions of P 2 . Using the sum rules equation given in Eq. (15), we can extract these four functions. Summing up the spin indices, we have with p 1 = p − q and p 2 = p − k. Note that there are 16 independent spinor structures, and this number is equal to that of the form factors A ± ± 1,2 and B ± ± 1,2 . This matching enables us to solve all of these form factors from the equation above. Finally, what we really care about is the transition matrix element of positive parity baryons with k = 0 and P 2 = m +2 Ξcc . Therefore, considering the k → 0 limit, and using the equation of motion, the matrix element in Eq. (16) can be simplified as Now the four unknown form factors are reduced to two constants A i WE and B i WE (A i WE and B i WE for the case of Ξ c ), which has the same form as Eq. (27).

IV. QUARK-GLUON LEVEL
In this section, we use the OPE to calculate the correlation function in Eq. (4) at the quark-gluon level. In the deep Euclidean region, to the correlation function can be expressed as a convolution of the perturbative kernel and a nonperturbative matrix element of the pion: where α, β, ρ, σ are spinor indices and S Q,u,s are the free propagators of the c, u, s quarks. The superscript "T " denotes the transposed in spinor space. The last matrix element can be expressed du e −iūq·x ( / pγ 5 ) βρ ϕ π (u) + (γ 5 ) βρ µ π φ p 3π (u) where ϕ π (u) = 6uū 1 + a 2 C are the twist-2, twist-3p and twist-3σ LCDAs respectively. a 2 = 0.27, a 4 = 0.179, µ π = 2.87 GeV, f 3π = 0.0045 GeV 2 , ω π = −1.5 and f π = 0.13 GeV [43]. C α n are the Gegenbauer polynomials. Now the derivation becomes straightforward, and we take the twist-2 LCDA as an example. Its contribution to the correlation function is with N c the number of colors.
The double imaginary part of the correlation function is related to its double discontinuity, which can be extracted by the cutting rules. Setting the momentum of each propagator on-shell, we have Here we have introduced an extra integration on m 2 34 = (k 3 +k 4 ) 2 to express the double discontinuity of the correlation function as a convolution of a two-body phase space integration dΦ 2 (m 2 34 ) and a triangle diagram integration dΦ ∆ (P 2 1 , p 2 2 ), where P 1 = p − uq = p 1 +ūq, and p 2 1 = s, p 2 2 = s 2 . This factorization for the momentum flows can also be understood intuitively by Fig. 2. It is seen that the momentum flowing into the left-lower corner of the triangle diagram is not p 1 itself but P 1 = p−uq = p 1 +ūq instead, and these momenta are related as P 2 1 = up 2 1 = us. For the contribution from the twist-3p and twist-3σ LCDAs the calculation is similar. The only difference is that the twist-3σ LCDA contains a term proportional to the coordinate x ν . Note that from Eq. (22) there is an exponential term exp(iuq · x) in the correlation function. One can use it to express x ν as (−i/u)(∂/∂q ν )exp(iuq · x) so that the calculation can still be done in momentum space.

V. NUMERICAL RESULTS
In this section, we first give the numerical results for the amplitudes of the Ξ ++ cc → Ξ +( ) c π + decays, namely the two constants A   [47] of the u quark and the pion are neglected. The masses and decay constants of the charm baryons with positive or negative parity are listed in Table I, where the decay constants are defined as in Eq. (7).
On the other hand, the LCSR contains two kinds of extra parameters, the thresholds s Ξcc , s Ξc , s Ξ c and the Borel parameters T 2 , T 2 . For the threshold parameters, we have argued that they are process-independent and will be taken from Ref. [38], s Ξcc = (4.1 ± 0.1) 2 GeV 2 , s Ξc = (3.2 ± 0.1) 2 GeV 2 and s Ξ c = (3.3 ± 0.1) 2 GeV 2 . These are about 0.5 2 GeV 2 larger than the squared masses of the corresponding baryons. Since this difference to the squared mass is only an empirical value proposed in Ref. [42], we will consider the uncertainty of the thresholds when evaluating the error of numerical results.
What concerns the Borel parameters, to simplify the problem we apply the following equation to relate the two Borel parameters corresponding to the s and s channels [51]: where M 1(2) is the mass of the initial (final) baryon and m 1(2) is the mass of the quark before (after) the weak decay. In Fig. 3 and Fig. 4 we present the Borel parameter dependence of the Ξ ++ cc → Ξ + c π + decay amplitudes A ( )(2) (B ( )(2) ), A ( )(3p) (B ( )(3p) ) and A ( )(3σ) (B ( )(3σ) ) contributed from the twist-2, twist-3p and twist-3σ LCDAs, respectively. In each diagram, the blue band denotes the uncertainty from the error of the threshold. The upper and lower red bands denote the uncertainty from the error of the Monte Carlo integrations. Generally, the window of Borel parameters is chosen to satisfy three requirements. The first one is that they must be small enough so that the contribution from the continuous spectrum can be suppressed, which determines their upper bound. The second one is that they must be large enough to ensure the OPE convergence, which determines their lower bound. The last one is that the result must be stable in this window.
To determine the upper bound of the Borel parameter, we require that the pole contribution must be larger than the continuous spectrum contribution, namely: The numerator represents the pole contribution, which is just the integral on the right-hand side of Eq. (15). The denominator is the same integral but the upper limits of s and s are extended to infinity, so that it contains both pole and continuous spectrum contributions. This requirement
shows that the upper bound of the Borel parameter is around T 2 = 7 GeV 2 for Ξ ++ cc → Ξ + c π + and T 2 = 6 GeV 2 for Ξ ++ cc → Ξ + c π + . The lower bounds of the Borel parameters are determined in principle by the ratio between the contribution from the leading order and next-to-leading order QCD corrections to the perturbative kernel of OPE. However, in this work we have only considered the leading order contribution so that this method cannot be used. From Fig. 3 and Fig. 4, it can be seen that the upper bounds given above are in a relatively stable region. Therefore, although we cannot determine the lower bound quantitatively, we can take a range below the upper bound of the Borel parameter. Here, FIG. 4: The Borel parameter dependence of the Ξ ++ cc → Ξ + c π + decay amplitudes A (2) (B (2) ), A (3p) (B (3p) ) and A (3σ) (B (3σ) ) from the twist-2, twist-3p and twist-3σ LCDAs respectively. In each diagram, the blue band denotes the uncertainty from the error of the threshold: s Ξcc = (4.1±0.1) 2 GeV 2 and s Ξ c = (3.3±0.1) 2 GeV 2 . The upper and lower red bands denote the uncertainty from the error of Monte-Carlo integrations.
we set the window as 6 < T 2 < 8 GeV 2 for Ξ ++ cc → Ξ + c π + and 5 < T 2 < 7 GeV 2 for Ξ ++ cc → Ξ + c π + . The amplitudes and the corresponding errors from the uncertainties of s Ξcc , s Ξ c and T 2 are listed in Table II. Note that most of the contributions from the twist-3 LCDAs are larger than that from twist-2. This feature is common in the LCSR studies on the heavy-to-light decays. For example, in the LCSR calculation of the B → π form factors [41,43], the contribution of twist-3 is generally of the same order or even larger than that of twist-2. The highly suppressed contributions should come from the LCDAs of twist-4 or higher. Table III shows the comparison of our results with those from the literature. Here, we have unified the definition of the amplitudes from all of these works. The amplitudes presented in the π + from the W-exchange contribution. The Borel parameters are set in the region 6 < T 2 < 8 GeV 2 for Ξ ++ cc → Ξ + c π + and 5 < T 2 < 7 GeV 2 for Ξ ++ cc → Ξ + c π + . Ξ ++ cc → Ξ + c π + Twist-2   [20,25,31] and three-loop quark model (3LQM) [29]. We also list the factorizable amplitudes from QCD sum rules (QCDSR) [14], light-front quark model (LFQM) [25], nonrelativistic quark model (NRQM) [20,31] and heavy quark effective theory (HQET) [20,31]. The notation LFQM + PM for example in the first column means the factorizable amplitude is from LFQM while the W-exchange amplitude is from PM, otherwise both of them are from the same theoretical approach.
The notation LFQM + PM for example in the first column means that the factorizable amplitude is from LFQM while the W-exchange amplitude is from PM. On the other hand, 3LCQM alone means that the factorizable and non-factorizable amplitudes are both from 3LCQM. From the comparison in Table III, one can find that our results for the W-exchange amplitudes of Ξ ++ cc → Ξ + c π + are consistent with most of the results from literatures. However, in terms of the W-exchange amplitudes of Ξ ++ cc → Ξ + c π + , our results are much larger than those from the literature, which provides a possibility to explain the anomaly of B /B. Two issues deserve further discussion:: • We have only used the two-particle LCDAs of the pion for the calculation up to the leading order. Although the contribution from the higher-twist LCDAs or QCD loop corrections is expected to be suppressed, in principle they are still necessary for improving the accuracy of the W-exchange contribution, which will be included in a future study.
• From Table III, we see that the various predictions on the factorizable contribution from the literature are not totally consistent with each other. These works assume the naive factorization which is based on color transparency [11]. It states that in the bottom hadron decays, the b quark is heavy enough so that the emitted light meson flies quickly and decouples from other hadrons before it is caught up by the soft gluons. However, since the charm quark is lighter, in the charmed hadron decays the effect of soft gluon exchange may be important.
Therefore, such effects need to be worked out in the future. Now we calculate the decay branching fractions by combining each factorizable amplitude from the literature and the non-factorizable amplitudes from this work, and make a comparison among them. The decay width is expressed as where p c is the magnitude of the pion three-momentum in the rest-frame of the Ξ cc . The lifetime of the Ξ ++ cc is chosen as τ (Ξ ++ cc ) = 2.56 × 10 −13 s, the Wilson coefficients are chosen as C 1 = 1.35 and C 2 = −0.64 [25], G F = 1.166 × 10 −5 GeV −2 and V cs = 0.975, V ud = 0.973 [45]. The branching fractions are listed in Table IV which agrees amazingly well with the experimental value. Although there exists non-negligible uncertainty for this theoretical result, it still implies that (B /B) exp = 1.41 > 1 can be understood and realized theoretically without introducing any physics beyond the Standard Model. The absolute branching fraction of Ξ ++ cc → Ξ + c π + from HQET+LCSR is B(Ξ ++ cc → Ξ + c π + ) HQET+LCSR = 6.22 ± 1.94 %.
Up to now, there is no experimental announcement on this absolute branching fraction. Instead, in Ref. [25] the authors have given an evaluation of it by phenomenological methods: B(Ξ ++ cc → Ξ + c π + ) = 1.83 ± 1.01 %, which is consistent with our result within the uncertainties. For the details on this phenomenological evaluation we refer to the discussion around the Eq. (38) in Ref. [25]. However, due to the lack of experimental measurements all the theoretical predictions given above are still waiting to be tested by future experiments.

VI. CONCLUSION
We have calculated the W-exchange contribution in the Ξ ++ cc → Ξ +( ) c π + decay with the use of the LCSR. The two-particle LCDAs of pion are used as the non-perturbative inputs for the sum rules calculation, and the perturbative kernel is calculated at the leading order. We obtain the relative decay branching fraction B /B of Ξ ++ cc → Ξ +( ) c π + by combining our W-exchange amplitudes with the factorizable amplitudes from various theoretical methods in the literature.
We find that using the factorizable amplitudes from heavy quark effective theory, we obtain the ratio B /B = 1.42 ± 0.78, which is consistent with the experimental value, 1.41 ± 0.17 ± 0.1. The corresponding absolute decay branching fraction is consistent with that evaluated in the literature, which should be tested by future experiments.