Durham-P 3-jet resolution y3

Definition of the observable

It is useful to start by recalling the definition of the Durham jet finding algorithm [13], given in terms of a resolution parameter $y_\mathrm{cut}$ as follows:

1. For all pairs of (pseudo-)particles $i,j$ calculate

   $$y_{ij} = \frac{2\mathrm{min} (E^2_i, E_j^2) (1 - \cos \theta_{ij})}{E_{vis}^2}\,,$$ (22)

   
   
   with $E_{vis}$ is the visible energy defined as $E_{vis} = \sum_i E_i$, here $E_i$ are the energies before any recombination.
2. Go back to step 1.
3. If all $y_{ij} > y_\mathrm{cut}$ stop. The number of jets is then defined to be equal to the number of (pseudo-)particles left.
4. Otherwise recombine the pair with the smallest $y_{ij}$ into a single pseudoparticle of momentum $p$ according to the P-recombination scheme:

   $$\vec{p} = \vec{p}_i + \vec{p}_j \qquad \qquad E_p = \vert\vec{p}\vert\>;$$ (23)

   
   

5. Go back to step 1.

The three-jet resolution parameter $y_3$ is defined as the maximum value of $y_\mathrm{cut}$ that leads to a $3$-jet event.

This observable was fully resummed at NLL accuracy for the first time in [3].

Born event used for the analysis

The hard scale (Q) is taken to be the center-of-mass energy.

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	2.000	$0.000$	1	1.000	0
2	2.000	$0.000$	1	1.000	0

Multiple emission tests

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

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 ${\cal F}$

Number of events used: 1000000

Result for each colour configuration

${\cal F}_2 =-0.31120\pm 0.00273$

For a precise definition of the configurations see [5].

The multiple emission function ${\cal F}$

Number of events used: 4050

Collection of automatically generated results

y3_DurP_ee.tar.gz collects all files produced automatically by Caesar.