LPAIR’s \(\ggll\)

The \(pp \rightarrow p^{(\ast)}(\ggll)p^{(\ast)}\) process as previously implemented in the early 1990s in LPAIR [BDSV91] can be reached through the lpair process in CepGen. More generally, this implementation was designed to compute the cross-section and to generate events for the \(\ggll\) process for ee, ep, and pp collisions.

A phenomenological review of both this process and its first implementation in LPAIR may be found in [Ver83].

Process-specific 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.

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

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

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

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

Full object reference

class LPAIR : public Process

Matrix element for the \(\gamma\gamma\to\ell^{+}\ell^{-}\) process as defined in [28].

Private Functions

bool orient()

Calculate energies and momenta of full event content, in the CM system.

inline double periPP() const

Compute the squared matrix element squared for the \(\gamma\gamma\rightarrow\ell^{+}\ell^{-}\) process.

Note

Its expression is of the form:

\[M = \frac{1}{4bt_1 t_2}\sum_{i=1}^2\sum_{j=1}^2 u_i v_j t_{ij} = \frac{1}{4}\frac{u_1 v_1 t_{11}+u_2 v_1 t_{21}+u_1 v_2 t_{12}+u_2 v_2 t_{22}}{t_1 t_2 b}\]

where \(b\) = bb_ is defined in computeWeight as:

\[b = t_1 t_2+\left(w_{\gamma\gamma}\sin^2{\theta^{\rm CM}_6}+4m_\ell\cos^2{\theta^{\rm CM}_6}\right) p_g^2\]

Returns:

Convolution of the form factor or structure functions with the squared central two-photons matrix element (for a pair of spin \(-\frac{1}{2}-\)point particles)

double pickin()

Describe the kinematics of the process \(p_1+p_2\to p_3+p_4+p_5\) in terms of Lorentz-invariant variables.

Note

These variables (along with others) will then be fed into the periPP method (thus are essential for the evaluation of the full matrix element).

Returns:

Value of the Jacobian after the operation

Private Members

const ParticleProperties pair_

const bool symmetrise_

const bool randomise_charge_

double ml_ = {0.}

mass of the outgoing leptons

double ml2_ = {0.}

squared mass of the outgoing leptons

double charge_factor_ = {0.}

mode::Kinematics beams_mode_

double re_ = {0.}

double ep1_ = {0.}

energy of the first proton-like incoming particle

double ep2_ = {0.}

energy of the second proton-like incoming particle

double w12_ = {0.}

\(\delta_2=m_1^2-m_2^2\) as defined in [28]

double ss_ = {0.}

double p12_ = {0.}

\(p_{12} = \frac{1}{2}\left(s-m_{p_1}^2-m_{p_2}^2\right)\)

double sl1_ = {0.}

double e1mp1_ = {0.}

double p_cm_ = {0.}

double mom_prefactor_ = {0.}

double gamma_cm_ = {0.}

double beta_gamma_cm_ = {0.}

std::unique_ptr<formfac::Parameterisation> formfac1_

std::unique_ptr<formfac::Parameterisation> formfac2_

std::unique_ptr<strfun::Parameterisation> strfun_

bool is_strfun_sy_ = {false}

double m_u_t1_ = {0.}

\(t_1\), first parton normalised virtuality

double m_u_t2_ = {0.}

\(t_2\), second parton normalised virtuality

double m_u_s2_ = {0.}

\(s_2\)

double m_w4_ = {0.}

\(w_4\), squared invariant mass of the two-parton system

double m_theta4_ = {0.}

polar angle of the two-photon system

double m_phi6_cm_ = {0.}

\(\phi_6^{\rm CM}\), azimutal angle of the first outgoing lepton xx6 = \(\frac{1}{2}\left(1-\cos\theta^{\rm CM}_6\right)\) definition (3D rotation of the first outgoing lepton with respect to the two-photon centre-of-mass system).

Note

If the nm_ optimisation flag is set this angle coefficient value becomes

\[\frac{1}{2}\left(\frac{a_{\rm map}}{b_{\rm map}}\frac{\beta-1}{\beta+1}+1\right)\]

with \(a_{\rm map}=\frac{1}{2}\left(w_4-t_1-t_2\right)\), \(b_{\rm map}=\frac{1}{2}\sqrt{\left(\left(w_4-t_1-t_2\right)^2-4t_1t_2\right)\left(1-4\frac{w_6}{w_4}\right)}\), and \(\beta=\left(\frac{a_{\rm map}+b_{\rm map}}{a_{\rm map}-b_{\rm map}}\right)^{2x_5-1}\) and the Jacobian element is scaled by a factor \(\frac{1}{2}\frac{\left(a_{\rm map}^2-b_{\rm map}^2\cos^2\theta^{\rm CM}_6\right)}{a_{\rm map}b_{\rm map}}\log\left(\frac{a_{\rm map}+b_{\rm map}}{a_{\rm map}-b_{\rm map}}\right)\)

double m_x6_ = {0.}

double s1_ = {0.}

double s2_ = {0.}

double sa1_ = {0.}

double sa2_ = {0.}

double p1k2_ = {0.}

double p2k1_ = {0.}

double ec4_ = {0.}

central system energy

double pc4_ = {0.}

central system 3-momentum norm

double pt4_ = {0.}

central system transverse momentum

double mc4_ = {0.}

central system invariant mass

double cos_theta4_ = {0.}

central system polar angle cosine

double sin_theta4_ = {0.}

central system polar angle sine

double q2dq_ = {0.}

double epsilon_ = {0.}

double alpha4_ = {0.}

double beta4_ = {0.}

double gamma4_ = {0.}

double alpha5_ = {0.}

double gamma5_ = {0.}

double alpha6_ = {0.}

double gamma6_ = {0.}

double bb_ = {0.}

double gram_ = {0.}

double dd5_ = {0.}

std::array<double, 2> deltas1_

std::array<double, 2> deltas2_

double delta_ = {0.}

Invariant used to tame divergences in the matrix element computation.

Note

Defined as

\[\Delta = \left(p_1\cdot p_2\right)\left(q_1\cdot q_2\right)-\left(p_1\cdot q_2\right)\left(p_2\cdot q_1\right)\]

with \(p_i, q_i\) the 4-momenta associated to the incoming proton-like particle and to the photon emitted from it.

double eph1_ = {0.}

double eph2_ = {0.}

Private Static Attributes

static constexpr double constb_ = 0.5 * M_1_PI * M_1_PI * M_1_PI