Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/comphep_html/tutorial/node94.html
Дата изменения: Tue Aug 15 14:38:18 2000
Дата индексирования: Mon Oct 1 22:47:20 2012
Кодировка:

Parameterization via decay scheme

The element of phase space volume for a $n$-particle state is equal to [Particles&Fields]

$$d\Gamma_n(q) = (2\pi)^4 \delta^4(q-p_1-p_2-p_3-...-p_n)\prod_{i=1}^{n}\frac{\delta(p_i^2-m_i^2)} {(2\pi)^3}d^4p_i\;.$$ (1)

The same expression is valid for both the decay of unstable particle with momentum $q$ and the interaction of two particles with momenta $q_1$ and $q_2$ such that $q_1+q_2=q$. For further discussion we need a designation for a phase space volume of some subset $S$ of the full $n$-particle set. According to (1)

$$d\Gamma(q,S)= (2\pi)^4 \delta^4(q- \sum_{i \in S}p_i)\prod_{i \in S}\frac{\delta(p_i^2-m_i^2)} {(2\pi)^3}d^4p_i\;.$$ (2)

Let $S_1$ and $S_2$ be two disjoint particle subsets, then

$${d\Gamma(q,S_1\cup S_2) =}$$

$$\int ds_1ds_2\Bigl((2\pi)^4 \delta^4(q- q_1 -q_2 ) \frac{\delta(q_1^2-s_1)}{(2\pi)^3}d^4q_1 \frac{\delta(q_2^2-s_2)}{(2\pi)^3}d^4q_2\Bigr)$$

$$\times \frac{d\Gamma(q,S_1)}{2\pi}\times\frac{ d\Gamma(q,S_2)}{2\pi}\;.$$ (3)

The above formula expresses a multi-particle volume in terms of two-particle one, the volumes $d\Gamma(q_1,S_1)$ and $d\Gamma(q_2,S_2)$ with a reduced number of particles, and the virtual squared masses $s_1,\;s_2$ of clusters $S_1,\;S_2$.

Recursive application of this formula allows one to express the multi-particle phase space in terms of two-particle phase space. In its turn the two-particle phase space is explicitly described by spherical angle $\Omega$ of motion of the first decaying particle in the rest frame of initial state [Particles&Fields].

$$\frac{d\Gamma(q,[1,2])}{2\pi}=\frac{k d\Omega}{4(2\pi)^3 \sqrt{q^2}}\;,$$ (4)

where $k$ is the absolute value of three-dimensional momentum of outgoing particles in the rest frame. Thus, applying recursively (3) and (4) to (1) we obtain an explicit expression for the phase space volume in terms of the squared masses $s_j$ of virtual clusters and the two-dimensional spherical angles $\Omega_j$, where $j$ is an ordinal number of decay:

$$d\Gamma _n(q) = \frac{k_1 d^2\Omega_1 }{4(2\pi)^2 \sqrt{q^2... ... d^2\Omega_j }{4(2\pi)^3 \sqrt{s_j}}\;. \prod_{j=2}^{n-1} ds_j$$ (5)

Here $k_j$ is a momentum of outgoing clusters produced by decay of the $j^{th}$ cluster in its center-of-mass.

The expression (5) means some sequential 1->2 decay scheme which starts from incoming state and finishes with outgoing particles of the process. For example, the integration domain for $s_j$ parameters depends on this scheme. Below we present two such schemes for a process with four outgoing particles:

        

In the case of CompHEP  project such decay scheme is defined by the user via the `Kinematics' menu (see Section ).