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Validation¶

The phase-space parametrisations in terms of blocks are validated using two differents methods:

Computation of a cross-section: the block is used to compute the cross-section of a given process. The result is then compared to the cross-section computed using MadGraph¹ for the same process.
Computation of a phase-space volume: only the phase-space density is integrated (no PDF or matrix element), and the result is compared with the analytical result for production of $n$ massless particles at a hadronic centre-of-mass energy of $\sqrt{s}$ GeV: $V(\sqrt{s}, n) = \frac{s^{n-3}}{2^{4(n-1)} \pi^{2n-3} (n-1) (n-2)^2 ((n-2)!)^2}$

In both cases, MoMEMta's phase-space parametrisations are not efficient for the problem, but by increasing the number of integrand evaluations to a very large number, sufficient precision can be reached nonetheless.

Block A¶

Cross-section¶

The process used is $pp \rightarrow W^+W^- \rightarrow \mu^+\mu^-\nu\bar{\nu}$

MadGraph cross-section: $0.866(5)$ pb
MoMEMta cross-section:

Phase-space volume¶

Volume computed: production of 3 massless particles at $\sqrt{s} = 1$ TeV

Theoretical phase-space volume: $6.3 \cdot 10^{-15}$
MoMEMta phase-space volume: $6.299(6) \cdot 10^{-15}$

Block B¶

Cross-section¶

The process used is $pp \rightarrow W^+W^- \rightarrow \mu^+\mu^-\nu\bar{\nu}$

MadGraph cross-section: $0.866(5)$ pb
MoMEMta cross-section: $0.868(8)$ pb

Phase-space volume¶

Volume computed: production of 3 massless particles at $\sqrt{s} = 1$ TeV

Theoretical phase-space volume: $6.3 \cdot 10^{-15}$
MoMEMta phase-space volume: $6.297(6) \cdot 10^{-15}$

Block D¶

The process used is $pp \rightarrow t\bar{t}$ fully-leptonic

MadGraph cross-section: $6.73(2)$ pb
MoMEMta cross-section: $6.69(3)$ pb

Phase-space volume¶

Volume computed: production of 6 massless particles at $\sqrt{s} = 1$ TeV

Theoretical phase-space volume : $694 \text{GeV}^6$
MoMEMta phase-space volume: $695(4) \text{GeV}^6$

Block F¶

Cross-section¶

The process used is $pp \rightarrow W^+W^- \rightarrow \mu^+\mu^-\nu\bar{\nu}$

MadGraph cross-section: $0.866(5)$ pb
MoMEMta cross-section: $0.864(4)$ pb

Phase-space volume¶

Volume computed: production of 4 massless particles at $\sqrt{s} = 1$ TeV

Theoretical phase-space volume: $0.0166 \text{GeV}^2$
MoMEMta phase-space volume: $0.0166(3) \text{GeV}^2$

Block G¶

Cross-section¶

The process used is $pp \rightarrow W^+W^- \rightarrow \mu^+\mu^-\nu\bar{\nu}$

MadGraph cross-section: $0.866(5)$ pb
MoMEMta cross-section: $0.864(4)$ pb

Phase-space volume¶

Volume computed: production of 4 massless particles at $\sqrt{s} = 1$ TeV

Theoretical phase-space volume: $0.0166 \text{GeV}^2$
MoMEMta phase-space volume: $0.01659(3) \text{GeV}^2$

Secondary Block C/D¶

Cross-section¶

The process used is $pp \rightarrow W^+W^- \rightarrow \mu^+\mu^-\nu\bar{\nu}$
Main block used for computation: B

MadGraph cross-section: $0.866(5)$ pb
MoMEMta cross-section: $0.868(8)$ pb

Phase-space volume¶

Volume computed: production of 4 massless particles at $\sqrt{s} = 1$ TeV
Main block used for computation: A

Theoretical phase-space volume: $0.0166$ $\text{GeV}^2$
MoMEMta phase-space volume: $0.01663(6) \text{GeV}^2$

J. Alwall et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”, JHEP 07 (2014) 079 ↩