Exclusive double diffractive events: general framework and prospects

Abstract

We consider the general theoretical framework to study exclusive double diffractive events (EDDE). It is a powerful tool to explore the picture of the pp interaction. Basic kinematical and dynamical properties of the process, and also normalization of parameters via standard processes like the exclusive vector meson production (EVMP), are considered in detail. As an example, calculations of the cross-sections in the model with three pomerons for the process p+p→p+M+p are presented for Tevatron and LHC energies.

Appendices

Appendix A

Here we collect basic expressions for gluon–gluon fusion partonic cross-sections of the type g+g→a+b. Some of them can be found, for example, in [90] (gg→gg, $gg\to Q\bar{Q}$) and [75] (gg→γγ).

(A.1)

(A.2)

(A.3)

where η=(η _a −η _b )/2, M is the invariant mass of the system, α _e and α _s are electromagnetic and strong couplings, respectively, m _Q is the quark mass.

We use the following formulas for the widths of resonances:

$$ \everymath{\displaystyle} \begin{array}{@{}l} \varGamma_{H\to gg}= \frac{M_H^3}{4\pi} \frac{G_F}{\sqrt{2}} \biggl( \frac{\alpha_s(M_H/2)}{2\pi} \biggr)^2 \biggl\vert f_H \biggl( \frac{M_H^2}{4m_t^2} \biggr) \biggr\vert^2K_H, \\[4mm] K_H\simeq1+\frac{\alpha_s(M_H/2)}{\pi} \biggl( \pi^2+ \frac{11}{2} \biggr)+0.2, \\[4mm] f_H(x)=\frac{1}{x} \biggl( 1+\frac{1}{2} \biggl( 1-\frac{1}{x} \biggr) [ L_+ +L_- ] \biggr), \\[4mm] L_{\pm}=Li_2 \biggl( \frac{2}{1\pm\sqrt{1-\frac {1}{x}}\pm\imath0} \biggr), \end{array} $$

(A.4)

(A.5)

(A.6)

where G _F is the Fermi constant and m _t is the top quark mass.

Appendix B

Here we present the calculation of the “soft survival probability” using (65) in the simple case, when the amplitude has the form

$$ \mathcal{M}\sim\bar{\mathcal{M}}=\mathrm {e}^{-B(\boldsymbol {\varDelta}_1^2+\boldsymbol {\varDelta}_2^2)}. $$

(B.1)

When y=0 we can take the above form of the amplitude. In this case

$$ \everymath{\displaystyle} \begin{array}{@{}l} \int\vert\bar{\mathcal{M}}\vert \,d^2\boldsymbol { \varDelta}\,d^2\boldsymbol {\delta}=\frac{\pi^2}{16B^2}, \\[4mm] \boldsymbol {\varDelta}=\frac{\boldsymbol {\varDelta}_2+\boldsymbol {\varDelta}_1}{2}, \qquad \boldsymbol {\delta }=\frac{\boldsymbol {\varDelta}_2-\boldsymbol {\varDelta}_1}{2}, \end{array} $$

(B.2)

and

(B.3)

where κ=q+q′+δ, b=|b|, b′=|b′|,

(B.4)

where the function h is presented in (77). Finally we have

$$ \int\bigl\vert\bar{\mathcal{M}}^U\bigr\vert \,d^2\boldsymbol {\varDelta}\,d^2\boldsymbol {\delta}= \frac{\pi^2}{16B^2} \frac{1}{4B}\int _0^{\infty}\bigl\vert h(\delta) \bigr \vert^2 \,d\delta^2, $$

(B.5)

and

$$ \bigl\langle S^2\bigr\rangle \simeq \bigl\langle S^2\bigr\rangle _{y=0}=\frac{\int\vert\bar{\mathcal{M}}^U\vert \,d^2\boldsymbol {\varDelta}\,d^2\boldsymbol {\delta}}{\int\vert\bar{\mathcal {M}}\vert\,d^2\boldsymbol {\varDelta}\,d^2\boldsymbol {\delta}}, $$

(B.6)

which leads to the expression (76). The accuracy of this approximation is about 1 %.