Changes between Version 3 and Version 4 of ComplexMassScheme

Timestamp:

Aug 12, 2015, 11:28:12 PM (9 years ago)

Author:

Valentin Hirschi

Comment:

--

ComplexMassScheme

@@ -1 @@
-Testing the complex mass scheme implementation
+=== Testing the complex mass scheme implementation ===
+
+The complex mass scheme is the modern way of handling finite width effects of unstable particles in perturbation theory. In a nutshell, it consists in redefining the mass of unstable particles as complex with an imaginary part proportional to its width.
+The denominator of unstable propagators remains identical, but contrary to the naive implementation of the width, the complex mass will also appear in the numerator of fermion propagators and in all couplings proportional to the unstable particle mass. In doing so, the Lagrangian is simply continued to complex masses and the final result remains gauge invariant.
+
+The Complex Mass Scheme (CMS henceforth) can be activated by simply typing
+
+
+in the interactive shell.
+The CMS implementation is rather straightforward at leading-order (LO) but it becomes more involved at next-to-leading-order (NLO) because of mainly two points
+ * The widths must be LO accurate at least in the offshell region and NLO accurate in the onshell region.
+ * The logarithms appearing in the UV wavefunction renormalization must be evaluated in the correct Riemann sheet.
+
+The details of these issues will be discussed in a forthcoming publication;  This wiki page is mainly to describe the various options to the command '{{{check cms}}}' which automatically tests the consistency of the CMS implementation. The core idea of the test is to compare amplitudes in the CMS scheme ($\mathcal{A}_{\text{CMS}}$) and in the case of widths set to zero ($\mathcal{A}_{\Gamma=0}$) for a given kinematic configuration where all resonances are far off-shell.
+The difference between these two amplitudes must be higher order. More formally, this means $\mathcal{A}^{\text{Born}}_{\text{CMS}}\sim \mathcal{A}^{\text{Born}}_{\Gamma=0} \sim \mathcal{O}(\alpha^a)$. 
+
+Then, if we 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 \cdot \alpha + \mathcal{O}(\alpha^2) $, the statement that difference is higher order is equivalent to state that $\kappa^{\text{LO}}_0=0$. At NLO, this relation translates to :
+
+$((\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 \cdot \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.
+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:
+
+[[Image(CMS_result.jpg,629)]]
+
+We now focus on the description of the command, whose main syntax is
+
+{{{ MG5_aMC> check cms [-reuse] <process_definition> <options> }}}