## VBFNLO utility to calculate form factors, version 1.4.0:

This program belongs to the program package VBFNLO and can calculate input parameters needed for anomalous gauge boson coupling studies with VBFNLO.
As especially the pure operators for anomalous quartic gauge boson couplings lead to a violation of tree-level unitarity within the energy range of the LHC, special care has to be taken to avoid this unphysical behaviour. Within VBFNLO we have opted for the use of a dipole form factor and this tool can calculate the maximal form factor scale Lambda_FF which is allowed for a given input of coupling parameters, assuming the form factor shape

$$FF = \frac{1}{( 1 + \frac{s}{\Lambda_{FF}^2} )^{FFexp}}$$

The form factor is determined by calculating on-shell VV scattering and computing the zeroth partial wave of the amplitude. As unitarity criterion the absolute value of the real part of the zeroth partial wave has to be below 0.5 [1].

Each channel in VV → VV scattering (with V = W / Z / gamma) is checked individually, additionally channels with the same electrical charge Q of the VV system are combined [4] (their definition of the partial wave expansion differs from ours by a factor of 2).

For more details please refer to

• diploma thesis of B. Feigl (in German only)
• diploma thesis of O. Schlimpert (in German only)

The full list of operators with anomalous gauge boson couplings which is implemented in VBFNLO and probed here can be found in the manual of VBFNLO version 2.7 located here .

Further reading on tree-level unitarity with anomalous couplings:

##### [5]

The current release v1.4.0:

Older versions:

### Usage:

• run make in the source directory
• the binary and input file are located in subfolder “run”
• modify run/input.dat
• run ./get-formfactor from the folder “run”

The following compilers have been tested and compile this tool properly: gfortran (default), g77, ifort. If you want to switch the compiler, uncomment the appropriate lines in the makefile.

By running “./batchrun” all operators can be checked sequentially with a given coupling strength. Issuing the command without arguments displays a short manual.

### Input:

In the file “run/input.dat” the following parameters can be set:

• Maximum center-of-mass energy that should be considered
• Desired exponent of the form factor.
You need at least
• FFexp = 1 for the dimension-6 operators and
• FFexp = 2 for the dimension-8 operators.
• Coupling constants of the anomalous couplings

### Output:

The program calculates the energy at which tree-level unitarity would be violated without a form factor and the form factor scale that ensures tree-level unitarity up to the given energy. The form factor scales are also written out to the file “run/ff_output.dat”.

The form factor scales are determined in three ways:

• Considering only the helicity combination with the largest contribution to the zeroth partial wave, separately for each VV → VV channel, with V including W-bosons, Z-bosons and photons.
This is the simplest approach. Only taking the largest helicity combination may be slightly too optimistic. This option is useful for checking the relative importance of different channels.
• Considering all helicity combinations with contributions to the zeroth partial wave by diagonalizing the T-matrix, separately for each VV → VV channel, where V includes W-bosons, Z-bosons and photons.
This option gives form factor scales that are more restrictive than option 1. The values are valid for processes which involve only one channel.
• Considering all helicity combinations with contributions to the zeroth partial wave by diagonalizing the T-matrix for all VV → VV channels with V being a W-boson, Z-boson or photon.
Using this approach gives the most stringent values for form factor and unitarity violation scale. They also hold for processes with more than one quartic vertex contributing.

### Changelog:

Version 1.4.0:

• Add dimension 8 operator O_S,2

Version 1.3.0:

• Bugfix in printout of coupling constants: negative values are now printed as well.
• Include operator O_phiD .

Version 1.2.1:

• Bugfix for form factor output option 1:
When using FFexp larger than minimal value, the form factor scale for some operators was too high. Output of option 2 and 3 was not affected.
• README: Added some guidelines how to use the results from the different calculation options.

Version 1.2.0:

• Include all photon channels for all three calculation options.
• Include W+W+ → W+W+ channel.

Version 1.1.0 beta:

• Some channels with external photons have been included for the first form factor calculation option.

Version 1.0.0:

• First public release.