Directly global y3 (kt algorithm, DeltaR, E-scheme)

Definition of the observable

We recall the $k_t$-invariant jet algorithm of ref. [2].

1. One defines, for all $n$ final-state (pseudo)particles still in the event,

   $$d_{kB} = p_{\perp k}^2\,,$$ (29)

   
   
   and for each pair of final state particles

   $$d_{kl} = \min\{{p_{\perp k}^2,p_{\perp l}^2}\} \left((\eta_k-\eta_l)^2+(\phi_k-\phi_l)^2\right)\,.$$ (30)

   
   

2. One determines the minimum over $k$ and $l$ of the $d_{kl}$ and the $d_{kB}$ and calls it $d^{(n)}$. If $d^{(n)}=d_{iB}$ then particle $q_{i}$ is included in the beam and eliminated from the final state particles. If $d^{(n)}=d_{ij}$ then particles $p_{i}$ and $p_{j}$ are recombined into a pseudoparticle (jet) in the E-scheme:

   $$p_{i j} = p_{i}+p_{j}\,.$$ (31)

   
   

3. The procedure is repeated until only 3 pseudoparticles are left in the final state.

We apply the algorithm to all particles in the event and consider

$$y_{23} = \frac{1}{E_{\perp}^2} \max_{n \ge 3}\{d^{(n)}\}\>,$$ (32)

where $E_{\perp}$ is defined by further clustering the event until only two jets remain and taking $E_{\perp}$ as the sum of the two jet transverse energies,

$$E_{\perp} = E_{\perp,1} + E_{\perp,2}\,.$$ (33)

Born event used for the analysis

The hard scale (Q) is taken to be the Born partonic center-of-mass energy.
Rapidity of the center of mass of the outgoing pair: 0.000
Cosine of the angle between outgoing jet and beam (in the partonic CM frame): 0.200

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	$1.042$	0
2	$2.000$	$0.000$	1	$1.042$	0
3	$2.000$	$0.000$	1	$1.042$	0
4	$2.000$	$0.000$	1	$1.042$	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

qq -> qq	${\cal F}_2 =-0.07456\pm 0.01270$
qg -> qg	${\cal F}_2 =-0.15933\pm 0.02429$
qg -> gq	${\cal F}_2 =-0.16162\pm 0.02271$
gq -> gq	${\cal F}_2 =-0.16244\pm 0.02366$
gq -> qg	${\cal F}_2 =-0.14758\pm 0.02179$
qq -> gg	${\cal F}_2 =-0.24593\pm 0.02975$
gg -> qq	${\cal F}_2 =-0.01242\pm 0.00328$
gg -> gg	${\cal F}_2 =-0.09860\pm 0.01456$

For a precise definition of the configurations see [3].

The multiple emission function ${\cal F}$

Number of events used: 256550

Collection of automatically generated results

y3_DeltaRE_HH_dir.tar.gz collects all files produced automatically by Caesar.