How to add momentum dependent form factors? in UFO model

Compatible model.

All UFO model are not ready to use form-factor (all recent one should be). In order to see if your model is compatible, look in the file init.py if you see the following lines:

try:
   import form_factors
except ImportError:
   pass
else:
   all_form_factor = form_factors.all_form_factors

Defining your form factor:

create/edit the file form_factors.py and add the following lines:

from object_library import all_form_factors, FormFactor

from function_library import complexconjugate, re, im, csc, sec, acsc, asec, HeavTheta



AAA = FormFactor(name = 'AAA',
                 type = 'real',
                 value = 'HeavTheta(MH - P(-1,1)*P(-1,3),-2*MH*,MH)')

1. HeavTheta is a special function for a IF statement. If the first argument is positive then the function is equal to the second argument otherwise the function is equal to the third argument.
2. Any parameter of the model can be used inside the 'value' expression
3. You can use any type of object define in the ALOHA syntax (see the [ALOHA paper ​http://arxiv.org/abs/arXiv:1108.2041] for the convention). This includes the momenta P(x,y). This first index is the lorentz index which should be contracted, while the second index refers to the particle ordering of the vertex.
4. If you need to add an additional parameter in your model, you can do it in the parameters.py file.

Add the vertex dependencies in your form factor:

Go in the file vertices.py and identify the interactions that you want to modify: Let assume that you want to modify the following interactions:

V_2 = Vertex(name = 'V_2',
             particles = [ P.W__minus__, P.W__plus__, P.H ],
             color = [ '1' ],
             lorentz = [ L.VVS1, L.VVS2 ],
             couplings = {(0,0):C.GC_1,(0,1):C.GC_3})

Then youhave to modify the lorentz structure associate to it:

V_2 = Vertex(name = 'V_2',
             particles = [ P.W__minus__, P.W__plus__, P.H ],
             color = [ '1' ],
             lorentz = [ L.FF1, L.FF2 ],
             couplings = {(0,0):C.GC_1,(0,1):C.GC_3})

Then you have to define those lorentz structure in the file lorentz.py if they were define as:

VVS1 = Lorentz(name = 'VVS1',
               spins = [ 3, 3, 1 ],
               structure = 'Metric(1,2)')

VVS2 = Lorentz(name = 'VVS2',
               spins = [ 3, 3, 1 ],
               structure = 'P(-1,1)*P(-1,2)*Metric(1,2)')

then you need to add the following lines:

FF1 = Lorentz(name = 'FF1',
               spins = [ 3, 3, 1 ],
               structure = 'AAA*Metric(1,2)',
               formfactors=[ForFac.AAA])

VVS2 = Lorentz(name = 'VVS2',
               spins = [ 3, 3, 1 ],
               structure = 'AAA*P(-1,1)*P(-1,2)*Metric(1,2)',
               formfactors=[ForFac.AAA])	

1. You need to be sure to have the line "import formfactors as ForFac" at the beginning of the lorentz.py file
2. You can include more than one formfactor for a single lorentz structure.
3. You are not force to define a new Lorentz structure if the old one will not be use anymore.

For More complex formfactor (Those who cann't be written in one line).

If the formfactor is a very complex function with a lot of "if" statement (like those obtained if when you add a form-factor to have the vertex associated to a full loop). The above procedure is not the most clear one. They are therefore the possibility to code your form-factor directly in Fortran. By providing a unique file containing the definition of the functions. Even if this method is usually easier to implement, this is not compatible with the C++/Python output, and you can not validate your implementation with the "check" command and/or produce any type of code related to the C++/Python output provided by MG5_aMC@NLO.

In that case you need to create a directory Fortran insice the UFO directory and create a file functions.f You are free to define whatever you want in that file (if this doesn't create conflict with another functions of mg5. I advice to start all your routine/function by "mymdl_" to avoid any problem).

Assuming that this Fortran file define the function AAA. Then you can modify the lorentz.py structure to the following:

VVS2 = Lorentz(name = 'VVS2',
               spins = [ 3, 3, 1 ],
               structure = 'AAA(P(-1,1)*P(-1,3))*P(-1,1)*P(-1,2)*Metric(1,2)')	

1. All functions defined liked that need to return a complex number
2. All argument of such function need to be a complex number, or an expression returning a complex number. It is therefore not possible to provide to the function the set of momenta as argument (since this is a vector).

How to add momentum dependent form factors? in v4 model

The easiest way to implement momentum dependent form factors, without changing all the details in the

matrix.f

files in a process is by adding them in the model files. In particular the

couplings.f

can contain momentum dependent form factors. The following steps explain most of the details.

• The first thing you have to do is download the full MG_ME package and untar it. Then, if your model contains new particles and interactions add them to the SM using the UserModel, giving some generic values to the coupling constants. (If your model does not contain new particles and you only want to modify existing interactions, it is enough to make a copy of the

  sm
  

directory to a new directory. )

• Generate the process you would like to study.
• Go to the =./Source/MODEL/= directory and open the

  couplings.f
  

file.

• Add (in the setpara subroutine) before the line

  c Start calculating the couplings for HELAS
  

the following lines to the file:

c Momenta of particles in event
      include '../genps.inc'
      double precision pp(0:3,max_particles)
      common/momenta_pp/pp 

This makes sure that the information of the momenta is available in the

      common/momenta_pp/pp 

This makes sure that the information of the momenta is available in the

couplings.f

file. These momenta can be used to calculate the anomalous (momentum dependent) couplings. The syntax is as follows. All the momenta are saved in an array called pp. The first element is running from 0 to 3, these are the energy, and the x, y and z components of the momentum of a particle (the beam direction is in the z direction) in the center of momentum frame of the event. The second element of the array labels the particles. Only momenta of external particles are in the array. The labeling is as follows. The first two are the incoming particles, and the others the outgoing numbered according to how the process is written down in the

proc_card.dat

. For example, if you are studying

e+e->Zh>bb~mu+mu-

then the first particle will be the positron, the second the electron. The final state particles: b quark, b anti-quark, muon+ and muon- , are 3, 4, 5 and 6 respectively. Hence momentum pp(2,4) is the y-component of the momentum of the b anti-quark in the center of momentum frame of the event. Using these components most anomalous couplings can be implemented by changing the SM values of the couplings in the

couplings.f

file. These momenta can be used to calculate the anomalous (momentum dependent) couplings. The syntax is as follows. All the momenta are saved in an array called pp. The first element is running from 0 to 3, these are the energy, and the x, y and z components of the momentum of a particle (the beam direction is in the z direction) in the center of momentum frame of the event. The second element of the array labels the particles. Only momenta of external particles are in the array. The labeling is as follows. The first two are the incoming particles, and the others the outgoing numbered according to how the process is written down in the

proc_card.dat

. For example, if you are studying

e+e->Zh>bb~mu+mu-

then the first particle will be the positron, the second the electron. The final state particles: b quark, b anti-quark, muon+ and muon- , are 3, 4, 5 and 6 respectively. Hence momentum pp(2,4) is the y-component of the momentum of the b anti-quark in the center of momentum frame of the event. Using these components most anomalous couplings can be implemented by changing the SM values of the couplings in the

couplings.f

file.

• As a final step you have to make sure that the

  setpara
  

subroutine is called on an event-by-event basis, by adding to the

run_card.dat

the line

T        = fixed_couplings  ! if .false. calc. coupl. for every event

just after the line where you can set the scale factor. For an example

run_card.dat

look here​.

Note that it might be interesting to make a new model for every process you want to study. In that case, instead of modifying the

couplings.f

file in the =./Source/MODEL/= directory, you could change the one in the =./Models/newmodel/= directory. Doing it this way has the clear advantage of being able to regenerate your process, without having to change the

couplings.f

file every time, which makes it less error prone. (Make sure that you also save a copy of the

proc_card.dat

in the model directory so that you won't forget for which process you created this model.)

-- Main.RikkertFrederix - 20 Jul 2008