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Subsections

C-parameter


Definition of the observable

The $ C$ parameter was initially defined as [6]

$\displaystyle C \equiv 3(\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1)\>,$ (4)

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

$\displaystyle \Theta^{\alpha \beta} = \frac{\sum_i p^\alpha_i {p}^\beta_i /\vert \vec p_i\vert} {\sum_j \vert\vec p_j\vert}\> ,$ (5)

the sums being over all final-state particles. Neglecting particle masses, one can express $ C$ in terms of invariants as

$\displaystyle C=3-\frac{3}{2}\sum_{i,j}\frac{(p_i\cdot p_j)^2}{(p_i\cdot Q)(p_j\cdot Q)}\>,$ (6)

where $ Q^{\mu}$ is the total four-momentum

$\displaystyle Q^{\mu}=\sum_i p_i^{\mu}\> .$ (7)

The $ C$ parameter has been resummed at NLL accuracy in [7].

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 $ 1.000 $ $ 1.000 $ 1 $ 6.000 $ 0
2 $ 1.000 $ $ 1.000 $ 1 $ 6.000 $ 0

Multiple emission tests

Test result
continuously global T
exponentiation (condition 1) T
exponentiation (condition 2a) T
exponentiation (condition 2b) T
exponentiation T
additivity 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

cpar_ee.tar.gz collects all files produced automatically by Caesar.
next up previous
Next: D-parameter Up: Observables in e+e- Previous: Thrust-minor
Giulia Zanderighi 2004-11-19