y2 resolution (Durham algorithm, P-scheme)

Definition of the observable

In analogy with [7], one introduces a resolution parameter $y_{cut}$ and uses the following algorithm

1. For each pair of final-state hadrons $i$ and $j$, one defines

   $$y_{ij} \equiv \frac{2\min\{E_i^2,E_j^2\}}{Q^2} (1 - \cos \theta_{ij})\,,$$ (13)

   
   
   and for each combination of a particle $i$ and incoming parton one defines

   $$y_{ib} \equiv \frac{2{E_i^2}}{Q^2} (1 - \cos \theta_{ib})\,,$$ (14)

   
   
   where $Q$ is the photon virtuality.

2. One finds the smallest value among the $\{y_{ij},y_{ib}\}$. If the smallest $y$ corresponds to two particles $p_k$ and $p_l$, and $y_{kl} < y_{cut}$ then these are to be recombined into a single pseudoparticle $p_{kl}$ in the $P$ recombination scheme

   $$\vec{p} = \vec{p}_i + \vec{p}_j \qquad E_p = \vert\vec{p}\vert\>;$$ (15)

   
   
   If the smallest $y$ corresponds to one of the particles $p_k$ with the beams $b$, and $y_{kb} < y_{cut}$, then particle $p_k$ is included in the `beam-jet' $b$.

3. This procedure is repeated until all remaining objects have $y_{ij}, y_{ib} > y_{cut}$. These objects are called jets. To be consistent with the original DIS definition, note that the beam jets are not counted as jets.

The observable that is to be resummed is then the distribution of $y_{2}$, the largest value of $y_{cut}$ such that the event is clustered to two jets.

Born event used for the analysis

The hard scale (Q) is taken to be photon virtuality.

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: 10000

Result for each colour configuration

${\cal F}_2 =-0.32322\pm 0.02766$

For a precise definition of the configurations see [6].

The multiple emission function ${\cal F}$

Number of events used: 3910

Collection of automatically generated results

y2_DurP_dis.tar.gz collects all files produced automatically by Caesar.