Factorised processes

This page describes the different techniques used to simplify the computation of a central exclusive process through the factorisation of its total matrix element into sub-blocks:

• a parton-from-beam particle flux, encompassing all kinematic features of a “physical” emission, possibly leaving the beam particle on-shell (i.e. surviving the emission), or in a diffractive state (possibly dissociating into a hadronic jet) ;

• a parton-parton matrix element, e.g. a \(\gamma\gamma\to X\) process, much easier to compute than the full beam-beam interaction \(S\)-matrix.

CepGen attempts to give the user a relative freedom in its implementation of factorised, two-parton level processes. As of version 1.2.0, two main parton emission types (and central matrix element definitions) are indeed handled:

• a standard, collinear parton emission, with a \(x\) (inelasticity) and \(Q^2\) squared momentum transfer dependence, implying a full momentum transfer along the beam direction (conventionally, the \(z\)-axis) ; this technique is used in many other Monte Carlo generators for the simulation of CEPs ;

• a parton \(\kt\)-dependent emission, also involving a dependence in the transverse parton virtuality, \(\kt\), involving a broader definition of the phase space (i.e. more time-consuming, but more precise).

Collinear parton emission

Warning

Under construction

\(\kt\)-factorised processes

The \(\kt\) factorisation, described in detail in [GdSFP+15] through the \(\ggff\) process listed above, allows to provide a “physical” modelling of this \(\kt\) dependence, commonly introduced as a simple gaussian smearing, e.g. in Pythia 8 (see the BeamRemnants:primordialKT flag).

For instance, a \(pp\to p^{(\ast)}(\ggx)p^{(\ast)}\) matrix element can be factorised through the following formalism:

\[\mathrm d\sigma = \int \frac{\mathrm d^2{\mathbf q_{\mathrm T}^2}_1}{\pi {\mathbf q_{\mathrm T}^2}_1} {\cal F}_{\gamma/p}^{\rm el/inel}(x_1,{\mathbf q_{\mathrm T}^2}_1) \int \frac{\mathrm d^2{\mathbf q_{\mathrm T}^2}_2}{\pi {\mathbf q_{\mathrm T}^2}_2} {\cal F}_{\gamma/p}^{\rm el/inel}(x_2,{\mathbf q_{\mathrm T}^2}_2) ~ \mathrm d\sigma^\ast,\]

where \(\mathcal F_{\gamma/p}^{\rm el/inel}(x_i,\vecqt_i)\) are unintegrated parton densities, and \(\mathrm d\sigma^\ast\) the hard process factorised out of the total matrix element.

Elastic unintegrated photon densities are expressed as functions of the proton electric and magnetic form factors \(G_E\) and \(G_M\):

\[\mathcal F_{\gamma/p}^{\rm el}(\xi,\vecqt^2) = \frac{\alpha}{\pi}\left[(1-\xi)\left(\frac{\vecqt^2}{\vecqt^2+\xi^2 m_p^2}\right)^2 F_E(Q^2)+\frac{\xi^2}{4}\left(\frac{\vecqt^2}{\vecqt^2+\xi^2 m_p^2}\right) F_M(Q^2)\right].\]

The inelastic contribution further requires both the diffractive state four-momentum norm \(M_X\) and a proton structure functions parameterisation as an input:

\[\begin{split}\mathcal F_{\gamma/p}^{\rm inel}(\xi,\vecqt^2) = \frac{\alpha}{\pi}\Bigg[(1-\xi)\left(\frac{\vecqt^2}{\vecqt^2+\xi(M_X^2-m_p^2)+\xi^2 m_p^2}\right)^2\frac{F_2(\xbj,Q^2)}{Q^2+M_X^2-m_p^2}+{}\\ {}+\frac{\xi^2}{4}\frac{1}{\xbj^2} \left(\frac{\vecqt^2}{\vecqt^2+\xi(M_X^2-m_p^2)+\xi^2 m_p^2}\right) \frac{2\xbj F_1(\xbj,Q^2)}{Q^2+M_X^2-m_p^2}\Bigg],\end{split}\]

with \(\xbj = {Q^2}/({Q^2+M_X^2-m_p^2})\) the Bjorken scaling variable.

Implementations

Common options

• mode (integer): kinematic regime to generate and the size of the phase space to perform the integration. It can take the following values:

  • ProcessMode.ElasticElastic := 1: elastic emission of photons from the incoming protons (default value if unspecified),

  • ProcessMode.ElasticInelastic := 2 / ProcessMode.InelasticElastic := 3: elastic scattering of one photon and an inelastic/semi-exclusive emission of the other photon, resulting in the excitation/fragmentation of the outgoing proton state,

  • ProcessMode.InelasticInelastic := 4: both protons fragmented in the final state.

\(\ggff\) process

The photon transverse momentum-dependant description of this process was previously developed in PPtoLL, and described in [GdSFP+15]. Since it now also supports the quark-antiquark production (thus all charged fermions), it is defined as the PPtoFF process in CepGen.

Process-specific options

• method (integer): switch between the two matrix element definitions:

  • 0: on-shell amplitude,

  • 1: off-shell amplitude.

• pair (integer/PDG): PDG identifier of the fermion pair to be produced in the final state. It can take the following values:

  • PDG.electron := 11: \(e^+e^-\) pair production

  • PDG.muon := 13: \(\mu^+\mu^-\) pair production

  • PDG.tau := 15: \(\tau^+\tau^-\) pair production

  • PDG.down, PDG.up, PDG.strange, PDG.charm, PDG.bottom, PDG.top (or equivalently 1-6): quark pair production.

Full object reference

class PPtoFF : public FactorisedProcess

\(\ggww\) process

The two-photon production of gauge boson pairs process, i.e. \(pp \rightarrow p^{(\ast)}(\ggww)p^{(\ast)}\), is implemented through the PPtoWW process object, featuring the on-shell and off-shell matrix elements reviewed in [LuszczakSchaferS18].

Process-specific options

• method (integer): switch between the two matrix element definitions:

  • 0: on-shell amplitude [DDS95],

  • 1: off-shell amplitude [NNPU06a, NNPU06b].

• polarisationStates (int): switch between all combinations of polarisation states to be included in the matrix element. It can take the following values:

  • 0: all contributions,

  • 1: longitudinal-longitudinal,

  • 2: longitudinal-transverse,

  • 3: transverse-longitudinal,

  • 4: transverse-transverse.

Full object reference

class PPtoWW : public FactorisedProcess