Changes between Version 18 and Version 19 of ComplexMassScheme

Timestamp:

Aug 13, 2015, 3:19:42 AM (9 years ago)

Author:

Valentin Hirschi

Comment:

--

ComplexMassScheme

@@ -14 +14 @@
 The difference between these two amplitudes must be higher order. More formally, this means that if we have $\mathcal{A}^{\text{Born}}_{\text{CMS}}\sim \mathcal{A}^{\text{Born}}_{\Gamma=0} \sim \mathcal{O}(\alpha^a)$, then we can write the following:
 
-At LO, we can write $(\mathcal{A}^{\text{Born}}_{\text{CMS}}-\mathcal{A}^{\text{Born}}_{\Gamma=0})/\alpha^a \equiv \Delta^{\text{LO}} = \kappa^{\text{LO}}_0 + \kappa^{\text{LO}}_1\alpha + \mathcal{O}(\alpha^2) $. The statement that the difference is higher order is then equivalent to state that $\kappa^{\text{LO}}_0=0$. 
+At LO, $(\mathcal{A}^{\text{Born}}_{\text{CMS}}-\mathcal{A}^{\text{Born}}_{\Gamma=0})/\alpha^a \equiv \Delta^{\text{LO}} = \kappa^{\text{LO}}_0 + \kappa^{\text{LO}}_1\alpha + \mathcal{O}(\alpha^2) $. The statement that the difference is higher order is then equivalent to state that $\kappa^{\text{LO}}_0=0$. 
 
 At NLO, this relation becomes
 $((\mathcal{A}^{\text{Virtual}}_{\text{CMS}}+\mathcal{A}^{\text{Born}}_{\text{CMS}})-(\mathcal{A}^{\text{Virtual}}_{\Gamma=0}+\mathcal{A}^{\text{Born}}_{\Gamma=0}))/\alpha^{a+1} \equiv \Delta^{\text{NLO}} = \kappa^{\text{NLO}}_0 + \kappa^{\text{NLO}}_1\alpha + \mathcal{O}(\alpha^2) $
 
-In order to check that $\kappa^{\text{LO}}_0$ and $\kappa^{\text{NLO}}_0$ are indeed zero, the test proceeds by scaling down all relevant couplings and widths by the parameter $\lambda$ and to evaluate the expressions of $\Delta$ for many progressively smaller values of \lambda but always on the same offshell kinematic configuration. One can then plot the quantities $\Delta^{\text{NLO|LO}}/\lambda$ and make sure that the asymptot for small values of lambda is the constant $\kappa^{\text{NLO|LO}}_1$. Any divergent behavior would be a manifestation of the presence of the term $\kappa^{\text{NLO|LO}}_0/\lambda$ which reveals an issue with the CMS implementation (most likely one of the two points mentioned above) which spoils the expected cancellation.
+In order to check that $\kappa^{\text{LO}}_0$ and $\kappa^{\text{NLO}}_0$ are indeed zero, the test proceeds by scaling down all relevant couplings and widths by the parameter $\lambda$ and evaluate the expressions of $\Delta$ for many progressively smaller values of $\lambda$, but always on the same offshell kinematic configuration. One can then plot the quantities $\Delta^{\text{NLO|LO}}/\lambda$ and make sure that the asymptot for small values of lambda is the constant $\kappa^{\text{NLO|LO}}_1$. Any divergent behavior would be a manifestation of the presence of the term $\kappa^{\text{NLO|LO}}_0/\lambda$ which reveals an issue with the CMS implementation (most likely one of the two points mentioned above) which spoils the expected cancellation.
 Before we detail the option of this test, here is the expected output (generated automatically, incl. this plot) for the case of QCD and QED corrections to fully decayed top quark pair production: