Hadron-hadron interactions

In hadron-hadron interactions, the total hadronic cross section for $AB \to$ anything, $\sigma^{AB}_{\mathrm{tot}}$, is calculated using the parameterization of Donnachie and Landshoff [Don92]. In this approach, each cross section appears as the sum of one pomeron term and one reggeon one

$$\sigma^{AB}_{\mathrm{tot}}(s) = X^{AB} \, s^{\epsilon} + Y^{AB} \, s^{-\eta} ~,$$ (110)

where $s = E_{\mathrm{cm}}^2$. The powers $\epsilon = 0.0808$ and $\eta = 0.4525$ are expected to be universal, whereas the coefficients $X^{AB}$ and $Y^{AB}$ are specific to each initial state. (In fact, the high-energy behaviour given by the pomeron term is expected to be the same for particle and antiparticle interactions, i.e. $X^{\overline{A}B} = X^{AB}$.) Parameterizations not provided in [Don92] have been calculated in the same spirit, making use of quark counting rules [Sch93a].

The total cross section is subdivided according to

$$\sigma^{AB}_{\mathrm{tot}}(s) = \sigma^{AB}_{\mathrm{el}}(s)... ...\sigma^{AB}_{\mathrm{dd}}(s) + \sigma^{AB}_{\mathrm{nd}}(s) ~.$$ (111)

Here `el' is the elastic process $AB \to AB$, `sd$(XB)$' the single diffractive $AB \to XB$, `sd$(AX)$' the single diffractive $AB \to AX$, `dd' the double diffractive $AB \to X_1 X_2$, and `nd' the non-diffractive ones. Higher diffractive topologies, such as central diffraction, are currently neglected. In the following, the elastic and diffractive cross sections and event characteristics are described, as given in the model by Schuler and Sjöstrand [Sch94,Sch93a]. The non-diffractive component is identified with the `minimum bias' physics already mentioned, a practical but not unambiguous choice. Its cross section is given by `whatever is left' according to eq. (), and its properties are discussed in section .

At not too large squared momentum transfers $t$, the elastic cross section can be approximated by a simple exponential fall-off. If one neglects the small real part of the cross section, the optical theorem then gives

$$\frac{\d\sigma_{\mathrm{el}}}{\d t} = \frac{\sigma_{\mathrm{tot}}^2}{16 \pi} \, \exp(B_{\mathrm{el}} t) ~,$$ (112)

and $\sigma_{\mathrm{el}} = \sigma_{\mathrm{tot}}^2 / 16 \pi B_{\mathrm{el}}$. The elastic slope parameter is parameterized by

$$B_{\mathrm{el}} = B^{AB}_{\mathrm{el}}(s) = 2 b_A + 2 b_B + 4 s^{\epsilon} -4.2 ~,$$ (113)

with $s$ given in units of GeV and $B_{\mathrm{el}}$ in GeV$^{-2}$. The constants $b_{A,B}$ are $b_{\mathrm{p}} = 2.3$, $b_{\pi,\rho,\omega,\phi} = 1.4$, $b_{\mathrm{J}/\psi} = 0.23$. The increase of the slope parameter with c.m. energy is faster than the logarithmically one conventionally assumed; that way the ratio $\sigma_{\mathrm{el}} / \sigma_{\mathrm{tot}}$ remains well-behaved at large energies.

The diffractive cross sections are given by

$$\frac{\d\sigma_{\mathrm{sd}(XB)}(s)}{\d t \, \d M^2} = \frac{g_{3\mathrm{I}\!\mathrm{P}}}{16\pi} \, \beta_{A\mathrm{I}\!... ...thrm{P}}^2 \, \frac{1}{M^2} \, \exp(B_{\mathrm{sd}(XB)}t) \, F_{\mathrm{sd}} ~,$$

$$\frac{\d\sigma_{\mathrm{sd}(AX)}(s)}{\d t \, \d M^2} = \frac{g_{3\mathrm{I}\!\mathrm{P}}}{16\pi} \, \beta_{A\mathrm{I}\!... ...mathrm{P}} \, \frac{1}{M^2} \, \exp(B_{\mathrm{sd}(AX)}t) \, F_{\mathrm{sd}} ~,$$

$$\frac{\d\sigma_{\mathrm{dd}}(s)}{\d t \, \d M_1^2 \, \d M_2^2} = \frac{g_{3\mathrm{I}\!\mathrm{P}}^2}{16\pi} \, \beta_{A\mathrm{I}... ...ac{1}{M_1^2} \, \frac{1}{M_2^2} \, \exp(B_{\mathrm{dd}}t) \, F_{\mathrm{dd}} ~.$$ (114)

The couplings $\beta_{A\mathrm{I}\!\mathrm{P}}$ are related to the pomeron term $X^{AB} s^{\epsilon}$ of the total cross section parameterization, eq. (). Picking a reference scale $\sqrt{s_{\mathrm{ref}}} = 20$ GeV, the couplings are given by $\beta_{A\mathrm{I}\!\mathrm{P}}\beta_{B\mathrm{I}\!\mathrm{P}} = X^{AB} \, s_{\mathrm{ref}}^{\epsilon}$. The triple-pomeron coupling is determined from single-diffractive data to be $g_{3\mathrm{I}\!\mathrm{P}} \approx 0.318$ mb$^{1/2}$; within the context of the formulae in this section.

The spectrum of diffractive masses $M$ is taken to begin 0.28 GeV $\approx 2 m_{\pi}$ above the mass of the respective incoming particle and extend to the kinematical limit. The simple $\d M^2 / M^2$ form is modified by the mass-dependence in the diffractive slopes and in the $F_{\mathrm{sd}}$ and $F_{\mathrm{dd}}$ factors (see below).

The slope parameters are assumed to be

$$B_{\mathrm{sd}(XB)}(s) = 2b_B + 2\alpha' \ln\left(\frac{s}{M^2}\right) ~,$$

$$B_{\mathrm{sd}(AX)}(s) = 2b_A + 2\alpha' \ln\left(\frac{s}{M^2}\right) ~,$$

$$B_{\mathrm{dd}}(s) = 2\alpha' \ln\left(e^4 + \frac{s s_0}{M_1^2 M_2^2} \right) ~.$$ (115)

Here $\alpha' = 0.25$ GeV$^{-2}$ and conventionally $s_0$ is picked as $s_0 = 1 / \alpha'$. The term $e^4$ in $B_{\mathrm{dd}}$ is added by hand to avoid a breakdown of the standard expression for large values of $M_1^2 M_2^2$. The $b_{A,B}$ terms protect $B_{\mathrm{sd}}$ from breaking down; however a minimum value of 2 GeV$^{-2}$ is still explicitly required for $B_{\mathrm{sd}}$, which comes into play e.g. for a $\mathrm{J}/\psi$ state (as part of a VMD photon beam).

The kinematical range in $t$ depends on all the masses of the problem. In terms of the scaled variables $\mu_1 = m_A^2/s$, $\mu_2 = m_B^2/s$, $\mu_3 = M_{(1)}^2/s$ ($=m_A^2/s$ when $A$ scatters elastically), $\mu_4 = M_{(2)}^2/s$ ($=m_B^2/s$ when $B$ scatters elastically), and the combinations

$$C_1 = 1 - (\mu_1 + \mu_2 + \mu_3 + \mu_4) + (\mu_1 - \mu_2) (\mu_3 - \mu_4) ~,$$

$$C_2 = \sqrt{(1 - \mu_1 -\mu_2)^2 - 4 \mu_1 \mu_2} \, \sqrt{(1 - \mu_3 - \mu_4)^2 - 4 \mu_3 \mu_4} ~,$$

$$C_3 = (\mu_3 - \mu_1) (\mu_4 - \mu_2) + (\mu_1 + \mu_4 - \mu_2 - \mu_3) (\mu_1 \mu_4 - \mu_2 \mu_3) ~,$$ (116)

one has $t_{\mathrm{min}} < t < t_{\mathrm{max}}$ with

$$t_{\mathrm{min}} = - \frac{s}{2} (C_1 + C_2) ~,$$

$$t_{\mathrm{max}} = - \frac{s}{2} (C_1 - C_2) = - \frac{s}{2} \, \frac{4C_3}{C_1 + C_2} = \frac{s^2 C_3}{t_{\mathrm{min}}} ~.$$ (117)

The Regge formulae above for single- and double-diffractive events are supposed to hold in certain asymptotic regions of the total phase space. Of course, there will be diffraction also outside these restrictive regions. Lacking a theory which predicts differential cross sections at arbitrary $t$ and $M^2$ values, the Regge formulae are used everywhere, but fudge factors are introduced in order to obtain `sensible' behaviour in the full phase space. These factors are:

$$F_{\mathrm{sd}} = \left( 1 - \frac{M^2}{s} \right) \left( 1 + \frac{c_{\mathrm{res}} \, M_{\mathrm{res}}^2} {M_{\mathrm{res}}^2 + M^2} \right) ~,$$

$$F_{\mathrm{dd}} = \left( 1 - \frac{\left( M_1 + M_2 \right)^2}{s} \right) \left( \frac{s\, m_{\mathrm{p}}^2}{ s\, m_{\mathrm{p}}^2 + M_1^2\, M_2^2} \right)$$

$$\times \left( 1 + \frac{c_{\mathrm{res}} \, M_{\mathrm{res}}^2} {M_{\mat... ...c_{\mathrm{res}} \, M_{\mathrm{res}}^2} {M_{\mathrm{res}}^2 + M_2^2} \right) ~.$$ (118)

The first factor in either expression suppresses production close to the kinematical limit. The second factor in $F_{dd}$ suppresses configurations where the two diffractive systems overlap in rapidity space. The final factors give an enhancement of the low-mass region, where a resonance structure is observed in the data. Clearly a more detailed modelling would have to be based on a set of exclusive states rather than on this smeared-out averaging procedure. A reasonable fit to $\mathrm{p}\mathrm{p}/ \overline{\mathrm{p}}\mathrm{p}$ data is obtained for $c_{\mathrm{res}} = 2$ and $M_{\mathrm{res}} = 2$ GeV, for an arbitrary particle $A$ which is diffractively excited we use $M_{\mathrm{res}}^A = m_A - m_{\mathrm{p}} + 2$ GeV.

The diffractive cross-section formulae above have been integrated for a set of c.m. energies, starting at 10 GeV, and the results have been parameterized. The form of these parameterizations is given in ref. [Sch94], with explicit numbers for the $\mathrm{p}\mathrm{p}/ \overline{\mathrm{p}}\mathrm{p}$ case. PYTHIA also contains similar parameterizations for $\pi\mathrm{p}$ (assumed to be same as $\rho\mathrm{p}$ and $\omega\mathrm{p}$), $\phi\mathrm{p}$, $\mathrm{J}/\psi \mathrm{p}$, $\rho\rho$ ($\pi\pi$ etc.), $\rho\phi$, $\rho\mathrm{J}/\psi$, $\phi\phi$, $\phi\mathrm{J}/\psi$ and $\mathrm{J}/\psi \mathrm{J}/\psi$.

The processes above do not obey the ordinary event mixing strategy. First of all, since their total cross sections are known, it is possible to pick the appropriate process from the start, and then remain with that choice. In other words, if the selection of kinematical variables fails, one would not go back and pick a new process, the way it was done in section . Second, it is not possible to impose any cuts or restrain allowed incoming or outgoing flavours: if not additional information were to be provided, it would make the whole scenario ill-defined. Third, it is not recommended to mix generation of these processes with that of any of the other ones: normally the other processes have so small cross sections that they would almost never be generated anyway. (We here exclude the cases of `underlying events' and `pile-up events', where mixing is provided for, and even is a central part of the formalism, see sections and .)

Once the cross-section parameterizations has been used to pick one of the processes, the variables $t$ and $M$ are selected according to the formulae given above.

A $\rho^0$ formed by $\gamma \to \rho^0$ in elastic or diffractive scattering is polarized, and therefore its decay angular distribution in $\rho^0 \to \pi^+ \pi^-$ is taken to be proportional to $\sin^2 \theta$, where the reference axis is given by the $\rho^0$ direction of motion.

A light diffractive system, with a mass less than 1 GeV above the mass of the incoming particle, is allowed to decay isotropically into a two-body state. Single-resonance diffractive states, such as a $\Delta^+$, are therefore not explicitly generated, but are assumed described in an average, smeared-out sense.

A more massive diffractive system is subsequently treated as a string with the quantum numbers of the original hadron. Since the exact nature of the pomeron exchanged between the hadrons is unknown, two alternatives are included. In the first, the pomeron is assumed to couple to (valence) quarks, so that the string is stretched directly between the struck quark and the remnant diquark (antiquark) of the diffractive state. In the second, the interaction is rather with a gluon, giving rise to a `hairpin' configuration in which the string is stretched from a quark to a gluon and then back to a diquark (antiquark). Both of these scenarios could be present in the data; the default choice is to mix them in equal proportions.

There is experimental support for more complicated scenarios [Ing85], wherein the pomeron has a partonic substructure, which e.g. can lead to high-$p_{\perp}$ jet production in the diffractive system. The full machinery, wherein a pomeron spectrum is convoluted with a pomeron-proton hard interaction, is not available in PYTHIA. (But is found in the POMPYT program [Bru96].)