Anomalous Higgs Vector Boson Couplings

In order to study the CP-structure of the Higgs boson it is interesting to look at effective interactions of the Higgs to the \(W^+W^-\) and \(Z^0Z^0\) bosons which have a definite CP-structure, see for example hep-ph/0105325.

This can easily be implemented in Herwig++ in the following way.

1. There is already a class GeneralVVSVertex which implements the interactions of two vector particles with a scalar using a general decomposition in terms of form factors. In order to implement the CP-even and CP-odd Higgs interactions we need to implement a Vertex class which inherits from GeneralVVSVertex and implements the setCoupling() member to calculate the relevant form-factors. The template of the class can easily be create using the ThePEG-class-files emacs macro to create the class, this class will need to be persistent and concrete. An example created using ThePEG-class-files

   • Class  Name: Herwig::AnomalousWWHVertex
   • Base Class Name: Helicity::GeneralVVSVertex
   • #include file for the base class: ThePEG/Helicity/Vertex/Scalar/GeneralVVSVertex.h
   • Will this class be persistent (y or n) y
   • Will this class be concrete (y or n) y

   can be found in the directory contrib/AnomalousHVV in the Herwig++ release.

2. The virtual member which calculates the couplings for a specific combination of particles,

   virtual void setCoupling(Energy2 q2,tcPDPtr part1, tcPDPtr part2, tcPDPtr part3);
   

   must be implemented. In order to calculate the couplings in is also useful to implement the doinit member to perform some calculations during the initialisation of the event generator.]

3. The example provided in contrib/AnomalousHVV includes the option of either the Standard Model form of the coupling, a CP-even or CP-odd form, controlled using a Switch created using the ThePEG-switch emacs macro. A scale Lambda is used for the effective CP-even and CP-odd interactions created using the ThePEG-parameter macro.

4. The form of the coupling is taken to be

   \(c(a_{00}p_{1}.p_{2}g^{ab}+a_{11}p_{1}^ap_{1}^b+a_{12}p_{1}^ap_{2}^b+a_{21}p_{2}^ap_{1}^b+a_{22}p_{2}^ap_{2}^b+a_{Ep}e^{cadb}p_{1c}p_{2d})V_{a}V_{b}\)

   The CP-even coupling can therefore be obtained using a overall normalization of UnitRemoval::E/Lambda, \(a_{00}=1, a_{21}=-1\) and all other couplings zero. The CP-odd coupling is obtained using the same normalization and \(a_{Ep}=1\).

5. It is also important that the kinematic invariants for the vertex are calculated using kinematics(true) in the constructor.

6. The vertex can then be built using the normal Makefile for user code and used instead of the Standard Model version with the Herwig++ StandardModel object using

   create Herwig::AnomalousWWHVertex /Herwig/Vertices/AnonWWHVertex AnomalousWWHVertex.so
   set /Herwig/Vertices/AnonWWHVertex:InteractionType SM
   set /Herwig/Model:Vertex/WWH /Herwig/Vertices/AnonWWHVertex
   

   This means it is used instead of the Standard Model version in all the Standard Model production matrix elements.

The example in contrib/AnomalousHVV contains the full implementation together with an AnalysisHandler SimpleVBFAnalysis and input file designed to reproduce figure 2, for VBF Higgs production at the LHC, in hep-ph/0105325. An example of the output distributions is here.