Probing the mass degeneracy of particles with different spins

The spin is an important property of a particle. Although it is unlikely, there is still a possibility that two particles with different spins share similar masses. In this paper, we propose a method to probe this kind of mass degeneracy of particles with different spins. We use the cascade decay B+→X(3872)K+, X(3872)→D+D− to illustrate our method. It can be seen that the possible mass degeneracy of X(3872) can lead to interesting behavior in the corresponding cascade decay.


Introduction
It is always important to determine the spin of a new particle once it is discovered. However, the statistics of events corresponding to the new particle are usually low. As a result, sometimes we find that there are several possibilities for the spin of the new particle. Consequently, it is possible that several particles with different spins share similar masses. In this paper, we propose a method to confirm or exclude this possibility.
We will use the cascade decay B ± → X(3872)K ± , X(3872)→D + D − as an example to explain our method. X(3872) was first discovered by the Belle Collaboration in 2003 and was the first exotic hadron to be discovered [1]. At first, the analysis of the X(3872) angular distributions in the decay to J/ψπ + π − favored J P C = 1 ++ or 2 −+ [2,3], and only in the last year did the LHCb Collaboration find that the spin of X(3872) is 1 [4]. Before this discovery by the LHCb Collaboration, there was a possibility that there were two particles with spin 1 and 2 degeneracy around 3872 MeV. We will show that this kind of mass degeneracy leads to some interesting behavior.

Formalism
To probe the degeneracy of X(3872), we assume that there are two particles with spin 1 and 2, respectively, and with masses about 3872 MeV. We will denote these two particles as X 1 and X 2 respectively.
When the invariant mass of the D + D − pair lies around 3872 MeV, the decay amplitude M for the cascade decay can be expressed as [5] M(s 12 ,s 13 )=a 1 P 1 (g s DD (s DK ))+a 2 P 2 (g s DD (s DK )), (1) where P l (l = 1,2) is the (l+1) th Legendre polynomial, a l P l represents the decay amplitude with X l being the intermediate resonance, and s DD and s DK are the invariant mass squared of the D + D − pair and the D + and K + respectively (D(D) in the subscript represents D + (D − )). The function g s DD (s DK ) is defined as being the kinematically allowed maximum (minimum) value of s DK . The subscript s DD of the function g s DD (s DK ) indicates its dependence on s DD (through . The definition of s max DK and s min DK can be found in Ref. [5]. The Legendre polynomials P 1 (x) and P 2 (x) have one and two zero points respectively, which for P 1 (x) is x 0 =0 and for P 2 (x) are x ± =±1/ √ 3. Correspondingly, the zero point(s) for P 1 g s DD (s DK ) and DK ,s max DK ), and will be denoted as Ω a , Ω b ,Ω b , andΩ a , respectively. As was shown in Refs. [5] and [6], localized CP asymmetry can be used to probe the interference of two resonances with different spins. However, for the cascade decay that we are considering, the weak phase is arg(V tb V * ts /V cb V * cs ), which is too small to have any detectable effects. Instead, we propose some other quantities to probe the degeneracy of X(3872).
Since the weak phase is small, we can safely neglect it. Then, we can redefine a l (l=1, 2) according to so that the new defined a l s are real, and the δ l s are strong phases. The absolute value squared of M can be expressed as where with δ=δ 2 −δ 1 . S and A have a very interesting property. One can see from the above two equations that with the substitution A changes its sign, while S does not: In other words, as a function of x=(s DK −ŝ DK )/Δ DK , S is even, while A is odd. In Fig. 1, we depict S, A and their sum as a function of x for a 1 = a 2 = cosδ = 1. The four regions in phase space for fixed s DD that were mentioned above now correspond to x ∈ (−1, 1/  Inspired by the above property, we define two quantities R a and R b as where r=a or b, and N ω (ω =Ω a , Ω b ,Ω b ,Ω a ) represents the event number of the cascade decay in the region ω. It can be seen for the situation we are considering that where S Ωr = Ωr ds DK S, A Ωr = Ωr ds DK A.
After doing the above integral, one has With the above equations, one can further derive the relations between R a and R b : and

053101-2
It can be seen that if there are two resonances lying around 3872 MeV, we should observe nonzero R a and R b with opposite signs. If there is only one resonance lying around 3872 MeV, this means a 1 or a 2 equals zero. By setting a 1 or a 2 equal to zero, one can immediately see that A=0. As a result, both R a and R b are equal to zero. Thus, by measuring R a and R b , one can confirm or exclude the degeneracy of X(3872). Note that all the above discussions are for fixed √ s DD , which in this case is around 3872 MeV. Practically, √ s DD takes values which are in a small interval around 3872 MeV. The above discussion then should be modified by taking into account a proper integral over s DD .

Conclusion
The upper limit for the branching ratio of the cascade decay is (at 90% confidence level) [7,8] B(B + →X(3872)K + )×B(X→D + D − )<4.0×10 −5 . (19) Thus the upper limit for the cascade decay event number N cas is N B + ×4.0×10 −5 , where N B + is the total number of events for the B + signal. To double the statistics, one can also include the CP conjugate channel. The method that we proposed above fails when the relative strong phase δ is close to π/2.