Derivation Notes on Phase Space Integral for Jet

Derivation Notes on Phase Space Integral for Jet

For a canonical ensemble with :$$n$$ particles which conserved energy and momentum, one has


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

with


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

and one notes


 * $$q_k \equiv f(y_k,p_{Tk})\Delta V_k \rightarrow f(y_k,p_{Tk})dydp_T$$

By using the Fourier-Laplace transformation, one may rewrite


 * $$A=\frac{-n}{(2\pi)^4}f(y,p_T)dydp_T\int_{\epsilon_0-i\infty}^{\epsilon_0+i\infty} ds\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}dt\int du_T \left[F(s,t,u_T)\right]^{n-1} exp\left[(W-\sqrt{p_T^2+m^2}ch y)s-(P_L-\sqrt{p_T^2+m^2}sh y)t-i(P_T-p_T)\cdot u_T\right]$$


 * $$B=-\frac{1}{(2\pi)^4}\int_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}ds\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}dt\int du_T \left[F(s,t,u_T)\right]^{n} exp\left[Ws-P_Lt-iP_T\cdot u_T\right]$$

with


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

By ignoring the $$p_T$$ conservation, it can be show straightforwardly


 * $$\langle n_k \rangle \simeq nf(y,p_T)dydp_T\frac{C}{D}$$

with


 * $$C=\int_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}ds\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}dt\left[F(s,t,u_T)\right]^{n-1} exp\left[(W-\sqrt{p_T^2+m^2}ch y)s-(P_L-\sqrt{p_T^2+m^2}shy)t\right]$$


 * $$D=\int_{\epsilon_0-i\infty}^{\epsilon_0+i\infty}ds\int_{\epsilon_1-i\infty}^{\epsilon_1+i\infty}dt\left[F(s,t,u_T)\right]^{n} exp\left[Ws-P_Lt\right]$$

1D Massless Case

 * $$\langle n \rangle = nf(p)dp\frac{C}{D}$$

with


 * $$f(p)=\frac{\beta}{2}e^{-\beta|p|}$$


 * $$C=-4\pi^2\beta^{n-1}e^{-\beta(M-|p|)}\frac{1}{(n-2)!}\sum_{r=0}^{n-2}\frac{(n+r-2)!}{2^{n+r-1}r!(r-1)!(n-r-2)!}\times p^{n-r-2}(M-|p|-p)^{r-1}$$


 * $$D=\frac{-\pi^2(2n-2)!\beta^2(\beta M)^{n-2}}{2^{2n-3}[(n-1)!]^2(n-2)!}e^{-\beta M}$$

By considering jet distribution function


 * $$xxx$$

and the experimental fit on the number of hadrons emitted from the jet


 * $$xxx$$

one obtains


 * $$\frac{dn}{dp}=C\int_0^\infty dp_T \frac{2^n(n-1)n p^{n-3}p_T^{3-n-npower}(p_T-0.5\sqrt{s})^{2g}(n-2)!((n-1)!)^2 _2F_1(n-3,n;2,1-\frac{p_T}{2p})}{(\sqrt{s})^{2g}(2n-2)!\Gamma(n-2)(1.38629+ln(p_T))^2}$$