Production and sequential decays of charmed hyperons

We investigate production and decay of the $\Lambda_c^+ $ hyperon. The production considered is through the $e^+e^-$ annihilation channel, $e^+e^-\rightarrow\Lambda_c^+ \bar{\Lambda}_c^-$, with summation over the $\bar{\Lambda}_c^- $ anti-hyperon spin directions. It is in this situation that the $\Lambda_c^+ $ decay chain is identified. Two kinds of sequential decays are studied. The first one is the doubly weak decay $B_1\rightarrow B_2 M_2$, followed by $B_2 \rightarrow B_3 M_3$. The other one is the mixed weak-electromagnetic decay $B_1\rightarrow B_2 M_2$, followed by $B_2 \rightarrow B_3 \gamma$. In both schemes $B$ denotes baryons and $M$ mesons. We should also mention that the initial state of the $\Lambda_c^+ $ hyperon is polarized.


I. INTRODUCTION
We shall investigate properties of certain sequential decays of the Λ + c hyperon, but in order to do so we first need to produce them. To this end we consider the reaction e + e − → Λ + cΛ − c , which is analyzed in detail in Refs. [1,2]. In order to describe such an annihilation process two hadronic form factors are needed. They can be parametrized by two parameters, α and ∆Φ, with −1 ≤ α ≤ 1. For their precise definitions we refer to Ref. [2].
The general cross-section distribution of this annihilation reaction depends on six structure functions which themselves are functions of α, ∆Φ, and θ, the scattering angle. In our application, however, we sum over the decay products of the anti-hyperonΛ − c , but identify the decay chain of the hyperon Λ + c , so called single tag events. In this simplified case only two structure functions are relevant, R = 1 + α cos 2 θ, (I.1) The scattering distribution function for the Λ + c hyperon production becomes, according to Refs. [2], proportional to W (n) = R + S N · n, (I. 3) where n is the direction of the hyperon spin vector in the hyperon rest system, N the normal to the scattering plane, and cos θ =p ·k. The momenta k and p are the relative momenta in the initial and final states, in the center of momentum (c.m.) system. The meaning of the spin vector n is explained in Ref. [3] From Eq.(I.3) we deduce for the spin distribution function, S(P) = 1 + P · n (I.5) and P the hyperon polarization, subject to the restriction |P| ≤ 1. For an unpolarized initial state hyperon P = 0.

II. WEAK HYPERON DECAYS
The weak hyperon decay c → dπ, of which Λ → pπ − is an example, is described by two amplitudes, one S-wave and one P-wave amplitude. The decay distribution is commonly described by three parameters, denoted αβγ. They are not independent but fulfill the relation The parametrization of this hyperon decay is discussed in detail in Ref. [4] and also in Ref. [2].
We denote by G c (c, d) the distribution function describing the weak hyperon decay c → dπ, given the spin vectors n c and n d , The vector l d is a unit vector in the direction of motion of the decay baryon d in the rest system of baryon c. The indices on the αβγ parameters remind us they characterize hyperon c.
Since the spin of baryon d is often not measured, the relevant decay density is obtained by averaging over the spin directions n d , Hyperons we study are produced in some reaction, and their states are described by some spin distribution function, Eq.(I.5), The final-state distribution in a production reaction followed by decay is obtained by a folding, pertaining to the intermediate and final hyperon spin directions n c and n d , where V c = α c l d , from Eq.(II.11).
The folding over intermediate spin directions follows the prescription of Ref.
[1], 1 n = 1, n n = 0, n · kn · l n = k · l. (II.14) From Eq.(II.13) it is clear that if the polarization is known the asymmetry parameter α c can be measured, but not the β c or γ c parameters. For that to be possible we must measure the polarization of the decay baryon d. If hyperon c is produced within a cc pair in e + e − annihilation then the polarization can be determined from the cross-section distribution.
An electromagnetic transition c → dγ is described by a transition distribution function similar to that of the weak decay, Eq.(II.8). However, the special feature of the electromagnetic interaction is the photon helicity which takes only two values, λ γ = ±1.
The electromagnetic transition distribution function corresponding to Eq.(II.8) is where l d is a unit vector in the direction of motion of hyperon d in the rest system of hyperon c.
Averaging over photon polarizations the transition distribution takes a very simple form, We notice that when both hadron spins are parallel or anti-parallel to the photon momentum, then the transition probability vanishes, a property of angular-momentum conservation. We also notice that expression (III.16) cannot be written in the αβγ representation of Eq.(II.8).

IV. TWO-STEP WEAK HYPERON DECAY
Now, we apply the above technique to hyperons decaying in two steps, such as b → c → d, accompanied by pions. An example of this decay mode is Λ + c → Λπ + followed by Λ → pπ − .
We denote by G b (b, c) the distribution function describing the hyperon decay b → cπ pertaining to spin vectors n b and n c , The vector l c is a unit vector in the direction of motion of baryon c in the rest system of baryon b.
Folding together the distribution functions G b (b, c) and G c (c, d), averaging over spin vectors n c and n d following the folding prescription (II.14), we get the decay density distribution The result is interesting. In many cases the asymmetry parameter α c for the c hyperon and the polarization P b for the initial-state b hyperon are known. Then, just as in the single-step case of Eq.(II.12), the initial state is described by a spin distribution function For the decay distribution of a polarized hyperon, we obtain This is equivalent to making the replacement n b → P b in Eq.(IV.19).
We conclude that by determining U The cross-section distribution of Eq.(IV.24) applies also to the decay chain, Λ + c → Σ + π 0 and Σ + → pπ 0 , with the corresponding identification of indices b, c, and d.

V. DIFFERENTIAL DISTRIBUTIONS
The cross-section distribution (IV.24) is a function of two unit vectors l 1 = l Λ , the direction of motion of the Lambda hyperon in the rest system of the charmed-Lambda hyperon, and l 2 = l p the direction of motion of the proton in the rest system of the Lambda hyperon. In order to handle these vectors we need a common coordinate system which we define as follows.
The scattering plane of the reaction e + e − → Λ cΛc is spanned by the unit vectorsp = l Λc andk = l e + , as measured in the c.m. system. We assume the scattering to be to the left, with scattering angle θ ≥ 0. If the scattering is to the right we rotate such an event 180 • around the k-axis, so that the scattering appears to be to the left. The scattering plane makes up the xz-plane, with the y-axis along the normal to the scattering plane. We choose a right-handed coordinate system with basis vectors using a set of eight angular functions F k (ξ) defined as: The distributions presented here will hopefully be of value in the analysis of BESIII data.

VI. MIXED WEAK-ELECTROMAGNETIC HYPERON DECAY
Now, we extend the formalism to hyperons decaying in two steps, with one being elec- tromagnetic. An example of such a decay chain is Λ + c → Σ 0 π + followed by Σ 0 → Λγ. As before we employ indices b, c, and d for variables belonging to Λ + c , Σ 0 , and Λ.
The distribution functions for the weak and electromagnetic transitions are given in Eqs.(IV.17) and (III.16), Performing a folding of the product of the distribution functions G b (b, c) and G γ (c, d), i.e. averaging over spin vectors n c and n d following the folding prescription (II.14), we get As noted earlier this is equivalent to making the replacement n b → P b in Eq.(VI.39). We also notice if we manage to determine U b and V b of Eqs.(VI.42), the only parameter that can be fixed is α b , a meager return.
The expression for the cross-section distribution for Λ + c production and subsequent decays Λ + c → Σ 0 π + and Σ 0 → Λγ is The expression to be integrated, Eq.(IV.21), reads U b + P · V b =1 + α b α c l c · l d + α b P · l c + γ b P · l d + (1 − γ b )P · l c l d · l c + β b P · (l d × l c ), (VI.49) with P along the Z-direction.
Now, we note that terms proportional to P · l c or P · l d vanish upon integration over angles α or γ. Therefore,