D-parameter

Definition of the observable

The D parameter is defined as [6,8]

$$D \equiv \lambda_1 \lambda_2\lambda_3\>,$$ (8)

where $\lambda_\alpha$ $( 0 \leq \lambda_\alpha \leq 1 , \, \sum_\alpha \lambda_\alpha = 1)$ are the eigenvalues of the linearized momentum tensor defined in eq. (5).

The D parameter has been resummed in the 3-jet limit in [9].

Born event used for the analysis

The hard scale (Q) is taken to be the center-of-mass energy.
Born parton energy fractions (2E/Q) are 0.800, 0.700 and 0.500.

Elementary tests on the observable

Test	result
check number of jets	T
all legs positive	T
globalness	T

Single emission properties

leg $\ell$	$a_{\ell}$	$b_{\ell}$	$g_{\ell}(\phi)$	$d_{\ell}$	$\langle \ln g_{\ell}(\phi) \rangle$
1	$1.000$	$1.000$	$\sin^2\phi$	$11.571$	$-2\ln(2)$
2	$1.000$	$1.000$	$\sin^2\phi$	$11.571$	$-2\ln(2)$
3	$1.000$	$1.000$	$\sin^2\phi$	$11.571$	$-2\ln(2)$

Multiple emission tests

Test	result
continuously global	T
exponentiation (condition 1)	T
exponentiation (condition 2a)	T
exponentiation (condition 2b)	T
exponentiation	T
additivity	T
eliminate subleading effects	T
opt. probe region exists	T

Information regarding the presence of possible zeros

No zeroes or small values found.

Multiple emission effects

Second order coefficient ${\cal F}_2$ of the function ${\cal F}$

The observable is additive, therefore ${\cal F}_2 =-\frac{\pi^2}{12}$.

The multiple emission function ${\cal F}$

The observable is additive, therefore ${\cal F}(R')=e{^{-\gamma_E R'}}/\Gamma(1+R')$.

Collection of automatically generated results

dpar_ee.tar.gz collects all files produced automatically by Caesar.