Derivation Notes on Phase Space Integral for Jet

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Derivation Notes on Phase Space Integral for Jet

For a canonical ensemble with :<math>n</math> particles which conserved energy and momentum, one has

<math>\langle n_k \rangle \equiv \langle \frac{d^3n}{dydp_T}|\Delta V_k\rangle_{n,W,P}=\frac{\sum_{\{n_l\}}P(\{n_l\})n_k}{\sum_{\{n_l\}}P(\{n_l\})}\equiv \frac{A}{B}</math>

with

<math>P(\{n_l\})\equiv \frac{n!}{n_1!n_2!\cdots n_N!}q_1^{n_1}\cdots q_N^{n_N}\delta_{n,\sum n_l}\delta\left[W-\sum_{l=1}^N n_l E_l\right]\delta\left[P_L-\sum_{l=1}^N n_l P_{Ll}\right]\delta\left[P_T-\sum_{l=1}^N n_l P_{Tl}\right]</math>

and one notes

<math>q_k \equiv f(y_k,p_{Tk})\Delta V_k \rightarrow f(y_k,p_{Tk})dydp_T</math>

By using the Fourier-Laplace transformation, one may rewrite

<math>A=\frac{-n}{(2\pi)^4}f(y,p_T)dydp_T\int ds_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}dt\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}\int du_T \left[F(s,t,u_T)\right]^{n-1} exp\left[(W-\sqrt{p_T^2+m^2}ch y)s-(P_L-\sqrt{p_T^2+m^2}sh y)t-i(P_T-p_T)\cdot u_T\right]</math>
<math>B=-\frac{1}{(2\pi)^4}\int ds_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}dt\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}\int du_T \left[F(s,t,u_T)\right]^{n} exp\left[Ws-P_Lt-iP_T\cdot u_T\right]</math>

with

<math>F(s,t,u_T) \equiv dydp_Tf(y,p_T) exp\left[-\sqrt{p_T^2+m^2}(s ch y- t sh y)+ip_T \cdot u_T\right]</math>

By ignoring the <math>p_T</math> conservation, it can be show straightforwardly

<math>\langle n_k \rangle \simeq nf(y,p_T)dydp_T\frac{C}{D}</math>

with

<math>C=\int ds_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}dt\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}\left[F(s,t,u_T)\right]^{n-1} exp\left[(W-\sqrt{p_T^2+m^2}ch y)s-(P_L-\sqrt{p_T^2+m^2}shy)t\right]</math>
<math>D=\int ds_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}dt\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}\left[F(s,t,u_T)\right]^{n} exp\left[Ws-P_Lt\right]</math>

3D Massive Case

1D Massless Case

<math>f(p)=\frac{\beta}{2}e^{-\beta|p|}</math>
<math>C=-4\pi^2\beta^{n-1}e^{-\beta(M-|p|)}\frac{1}{(n-2)!}\sum_{r=0}^{n-2}\frac{(n+r-2)!}{2^{n+r-1}r!(r-1)!(n-r-2)!}\times p^{n-r-2}(M-|p|-p)^{r-1}</math>
<math>D=\frac{-\pi^2(2n-2)!\beta^2(\beta M)^{n-2}}{2^{2n-3}[(n-1)!]^2(n-2)!}e^{-\beta M}</math>