SM(+hgg) in Calchep

In any model you can use for hgg coupling the same functions as in SM.

In CalcHEP hgg interaction is tuned for h decay. So, we take into
account that h-gg channel indeed includes process with virtual glu which
decays on  quarks.  In CalcHEP model file it is presented by factor
RQCDh=1.24. Squaring this factor you'll get 1.53. So, 35fp cross section
is transformed to 53fb, which is close to CalcHEP result.  I guess it
explains main part of difference. I do not know details of hgg
implementation  in FeynRules. It should be nice to look. Some difference
may be caused by loop QCD correction. CalcHEP does it in the same manner
as Hdecay.

Surely if we consider g,g->h, we have to remove   RQCDh factor. But here
there are another QCD corrections. As I remember loop calculation in QCD
gives cross section larger than CalCHEP. It should be confirmed by
Alexander Belyaev.

Best

    Alexander Pukhov

On 10.09.2019 21:37, Nurhan Karahan wrote:
> New question #683828 on CalcHEP:
> https://answers.launchpad.net/calchep/+question/683828
>
> Dear Calchep team,
>
> We are studying on an extended SM with a real singlet scalar field. We modified the SM.fr model file in accordance with our model and get Calchep outputs via Mathematica. However, this SM.fr file does not include hgg and hhgg effective vertex and interaction terms. We added these vertex and interaction terms by using the definitions on
>
> http://feynrules.irmp.ucl.ac.be/attachment/wiki/HiggsCharacterisation/HC.fr
>
> (loop coefficients lines 108-114, gHgg, gAgg, gHHgg, gAAgg couplings lines 474-492 and some interested parts of the Lagrangians L0v line 548 and L0v6 line 563 by taking sa=0, ca=1). Because we are interested in the pp-> hh process, we added only following parts
>
> -1/4 ( gHgg FS[G,mu,nu,a] FS[G,mu,nu,a] ) H. of L0v
> -1/8 ( gHHgg FS[G,mu,nu,a] FS[G,mu,nu,a]) H H of L0v6
>
> Our extended model works. But the results coming from pp-> h_1h_1 (h_1 is physical Higgs field after mixing) process (by removing the other new physical scalar field h_2) is not compatible with the SM(+hgg) model in Calchep. The dominant process is gg->hh. For this process, at 13 TeV our model obtained via modification of SM.fr file and addition of hgg and hhgg vertex, gives ~35 fb, on the other hand the model SM(+hgg) in Calchep gives ~60 fb. There is a problem. But we could not understand where we make mistake. How do we solve this inconsistency?
>

Dear Prof. Pukhov,

Thank you for your response. We have one more problem. In our FR model file, we define the hgg vertex  GH= aS/(3*Pi*vev) and the effective Lagrangian LCPEven= (-1/4) GH FS[G, mu, nu, aa] FS[G, mu, nu, aa] H

Here; H=cos(th)h1+sin(th)h2  where h1 is physical Higgs with mass 125 GeV and h2 is physical heavy singlet scalar. Since h2 has the mass above t-tbar threshold we should take care of the different limiting cases for hgg vertex (as in page 5 Eq. (4.2 and 4.3) of https://link.springer.com/content/pdf/10.1007%2FJHEP06%282015%29004.pdf). As we see, this is defined in Calchep hgg.c file. Our question is that when we  run gg-> h1 or gg->h2 process by using Calchep outputs obtained from our .fr model file, how Calchep consider this limiting cases. Should we define this effective vertex in .fr model file different from the one as mentioned above or Calchep can see it directly ?

If you  use CalcHEP functions  lAAhiggs, lAA5higgs, lGGhiggs,
lGG5higgs,  then  all thresholds and loops are taken into account
automatically.

In general they  are complex functions (above thershold). In CalcHEP we
use their absolute values assuming that they  should be  squared.

Best

    Alexander Pukhov

On 15.09.2019 12:43, Nurhan Karahan wrote:
> Question #683828 on CalcHEP changed:
> https://answers.launchpad.net/calchep/+question/683828
>
> Status: Answered => Open
>
> Nurhan Karahan is still having a problem:
> Dear Prof. Pukhov,
>
> Thank you for your response. We have one more problem. In our FR model
> file, we define the hgg vertex  GH= aS/(3*Pi*vev) and the effective
> Lagrangian LCPEven= (-1/4) GH FS[G, mu, nu, aa] FS[G, mu, nu, aa] H
>
> Here; H=cos(th)h1+sin(th)h2  where h1 is physical Higgs with mass 125
> GeV and h2 is physical heavy singlet scalar. Since h2 has the mass above
> t-tbar threshold we should take care of the different limiting cases for
> hgg vertex (as in page 5 Eq. (4.2 and 4.3) of
> https://link.springer.com/content/pdf/10.1007%2FJHEP06%282015%29004.pdf).
> As we see, this is defined in Calchep hgg.c file. Our question is that
> when we  run gg-> h1 or gg->h2 process by using Calchep outputs obtained
> from our .fr model file, how Calchep consider this limiting cases.
> Should we define this effective vertex in .fr model file different from
> the one as mentioned above or Calchep can see it directly ?
>

We use the Calchep outputs (.mdl) obtained from modified SM.fr file. Therefore, we could not use CalcHEP functions. According to your response, In this case, we can not use Calchep for our extended SM model. Isn't it? Is there a way to make ggh vertexes compatible with calchep in feynrules model file? Could you suggest a way to overcome this problem?

1) You can use CalcHEP function  -cabs(lAAhiggs(Mh, "h")) in any model
and for any scalar.

2)  if you remove RQCDh  from vertex, result should be close to function
proposed by FeynRules. If not, let me know and I'll check what is a problem.

3)  I guess, cross section of Higgs production will be wrong in both cases.

Best

    Alexander Pukhov

On 15.09.2019 20:08, Nurhan Karahan wrote:
> Question #683828 on CalcHEP changed:
> https://answers.launchpad.net/calchep/+question/683828
>
> Status: Answered => Open
>
> Nurhan Karahan is still having a problem:
> We use the Calchep outputs (.mdl) obtained from modified SM.fr file.
> Therefore, we could not use CalcHEP functions. According to your
> response, In this case, we can not use Calchep for our extended SM
> model. Isn't it? Is there a way to make ggh vertexes compatible with
> calchep in feynrules model file? Could you suggest a way to overcome
> this problem?
>

Dear Prof. Pukhov,

Thanks for your help. Unfortunately, we could not achieve to obtain compatible FR model file with Calchep (SM+hgg).

We understood that lanhep is more compatible with Calchep. We obtained our extended model by using lanhep SM model file. Now our model gives same results with Calchep (SM+hgg) for higgs interactions.

However, In our model, there is an extra real singlet scalar s with mass > 500 GeV. This scalar is mixed with higgs field. As you know, in ggh and gghh vertex Wilson coefficient;

C(\tau) -> - (2/mh^2)arcsin^2(1/sqrt(1/tau1)) when tau>1

and

C(\tau)-> (1/ms^2)(log((1+sqrt(1-\tau))/(1-sqrt(1-\tau)))-i\pi)^2 when tau<1

where tau=4*m_{t}^2/m_{x}^2 (x=h or s ).

In Calchep ggh and gghh vertex depend on external functions which we can not find explicit forms;

         LAAh=-cabs(lAAhiggs(Mh,str(h1))),
          LGGh=-cabs(lGGhiggs(Mh,str(h1))),
          aQCDh =alphaQCD(Mh)/acos(-1),
          RQCDh=sqrt(1+149/12*aQCDh+68.6482*aQCDh**2-212.447*aQCDh**3).

When we consider m_{s} and write these functions such as

          LAAs=-cabs(lAAhiggs(Ms,str(s))),
          LGGs=-cabs(lGGhiggs(Ms,str(s))),
          aQCDs =alphaQCD(Ms)/acos(-1),
          RQCDs=sqrt(1+149/12*aQCDs+68.6482*aQCDs**2-212.447*aQCDs**3

and define effective lagrangian

lterm LAAs*(smix*vevs*s+ smix**2*s*s/2)/vevs*F**2 where
            F=deriv^mu*A^nu-deriv^nu*A^mu.

lterm LGGs*RQCDs*(smix*vevs*s+smix**2*s*s/2) /vevs*F**2 where
           F=deriv^mu*G^nu^a-deriv^nu*G^mu^a.

do these external functions take true values considering Wilson coefficients defined above?

Dear Nurhan,

answers, questions and comments to you are below:

1. what is "smix" it is not defined in your e-mail,

what it is?

2.  lAAhiggs and lGGhiggs are standard triangle loop functions
normalised such a way

that for  LAAh and LGGh defined as

           LAAh=-cabs(lAAhiggs(Mh,str(h))),
           LGGh=-cabs(lGGhiggs(Mh,str(h))),

lterm  LAAh/vevh*F**2

or

lterm  LGGh/vevh*shd*shD*F**2

give you the correct interactions for AAH or GGH effective vertex

Please note that

lAAhiggs(Mh,str(h)) and lGGhiggs(Mh,str(h))

are universal function and they will work for any model:

these functions will check all possible interactions of the "h" filed
and will take this into account for the

HAA or HGG effective vertex

3.

You have the singlet which mixes with the Higgs field,

you need consistently take this into account, which I do not see since
you did not take into account the modification of the HGG etc vertices
according to this mixing.

Regards,

Alexander

On 30/09/2019 08:47, Nurhan Karahan wrote:
> Question #683828 on CalcHEP changed:
> https://answers.launchpad.net/calchep/+question/683828
>
> Status: Answered => Open
>
> Nurhan Karahan is still having a problem:
> Dear Prof. Pukhov,
>
> Thanks for your help. Unfortunately, we could not achieve to obtain
> compatible FR model file with Calchep (SM+hgg).
>
> We understood that lanhep is more compatible with Calchep. We obtained
> our extended model by using lanhep SM model file. Now our model gives
> same results with Calchep (SM+hgg) for higgs interactions.
>
> However, In our model, there is an extra real singlet scalar s with mass
>> 500 GeV. This scalar is mixed with higgs field. As you know, in ggh
> and gghh vertex Wilson coefficient;
>
> C(\tau) -> - (2/mh^2)arcsin^2(1/sqrt(1/tau1)) when tau>1
>
> and
>
> C(\tau)-> (1/ms^2)(log((1+sqrt(1-\tau))/(1-sqrt(1-\tau)))-i\pi)^2 when
> tau<1
>
> where tau=4*m_{t}^2/m_{x}^2 (x=h or s ).
>
> In Calchep ggh and gghh vertex depend on external functions which we
> can not find explicit forms;
>
> LAAh=-cabs(lAAhiggs(Mh,str(h1))),
> LGGh=-cabs(lGGhiggs(Mh,str(h1))),
> aQCDh =alphaQCD(Mh)/acos(-1),
> RQCDh=sqrt(1+149/12*aQCDh+68.6482*aQCDh**2-212.447*aQCDh**3).
>
> When we consider m_{s} and write these functions such as
>
> LAAs=-cabs(lAAhiggs(Ms,str(s))),
> LGGs=-cabs(lGGhiggs(Ms,str(s))),
> aQCDs =alphaQCD(Ms)/acos(-1),
> RQCDs=sqrt(1+149/12*aQCDs+68.6482*aQCDs**2-212.447*aQCDs**3
>
> and define effective lagrangian
>
> lterm LAAs*(smix*vevs*s+ smix**2*s*s/2)/vevs*F**2 where
> F=deriv^mu*A^nu-deriv^nu*A^mu.
>
> lterm LGGs*RQCDs*(smix*vevs*s+smix**2*s*s/2) /vevs*F**2 where
> F=deriv^mu*G^nu^a-deriv^nu*G^mu^a.
>
> do these external functions take true values considering Wilson
> coefficients defined above?
>
--
______________________________________________________________________
Prof. Alexander S Belyaev (<email address hidden>)
https://www.hep.phys.soton.ac.uk/content/alexander-belyaev

School of Physics & Astronomy, University of Southampton
Office 5047, SO17 1BJ, TEL: +44 23805 98509, FAX: +44 23805 93910
.....................................................................
Particle Physics Department, Rutherford Appleton Laboratory
Didcot, OX11 0QX, TEL: +44 12354 45562, FAX: +44 12354 46733
.....................................................................
CERN, CH-1211 Geneva 23, Switzerland
Office 40/1-B20, Mailbox: E27910, TEL: +41 2276 71642
______________________________________________________________________

So Sorry,

Actually, smix is the sinus of the mixing angle theta between higgs and real singlet scalar.
H= cmix h + smix s
S= -smix h + cmix s

In effective vertex, when we write this rotated physical fields h and s, we obtained the same vertex factor for the scalar s (only difference is the smix terms which appears in front of the vertex factors). But, as I mentioned previously, s has mass above the ttbar threshold, so the Wilson coefficient in its vertex factor should be C(\tau)-> (1/ms^2)(log((1+sqrt(1-\tau))/(1-sqrt(1-\tau)))-i\pi)^2 rather than - (2/mh^2)arcsin^2(1/sqrt(1/tau1)). Because of this reason, we think that we should seperate the effective lagrangians such that

do_if hgg==On.

external_func(alphaQCD,1).
external_func(lGGhiggs,2).
external_func(lAAhiggs,2).

parameter LAAh=-cabs(lAAhiggs(Mh,str(h1))),
          LGGh=-cabs(lGGhiggs(Mh,str(h1))),
          aQCDh =alphaQCD(Mh)/acos(-1),
          RQCDh=sqrt(1+149/12*aQCDh+68.6482*aQCDh**2-212.447*aQCDh**3).

lterm LAAh*('W+.f'*'W-.f'+'Z.f'*'Z.f'/2+cmix*vevh*h+ cmix**2*h*h/2)/vevh*F**2 where
             F=deriv^mu*A^nu-deriv^nu*A^mu.
lterm LGGh*RQCDh*('W+.f'*'W-.f'+'Z.f'*'Z.f'/2+cmix*vevh*h+cmix**2*h*h/2) /vevh*F**2 where
             F=deriv^mu*G^nu^a-deriv^nu*G^mu^a.
**(Here, we use h instead of shd because we want to seperate the h and s fields in effective lagrangians. (shd includes H=cmix*h+smix*s). At this stage maybe we should not use the mixing angles ? Because we add this effective lagrangian manually, and we use the physical fields directly. What is your opinion?)

Then we define these external functions depending on the mass of s physical field.
parameter LAAh2=-cabs(lAAhiggs(Ms,str(h2))),
          LGGs=-cabs(lGGhiggs(Ms,str(h2))),
          aQCDs =alphaQCD(Ms)/acos(-1),
          RQCDs=sqrt(1+149/12*aQCDs+68.6482*aQCDs**2-212.447*aQCDs**3).

lterm LAAs*(smix*vevs*s+ smix**2*s*s/2)/vevs*F**2 where
             F=deriv^mu*A^nu-deriv^nu*A^mu.

lterm LGGs*RQCDs*(smix*vevs*s+smix**2*s*s/2) /vevs*F**2 where
             F=deriv^mu*G^nu^a-deriv^nu*G^mu^a.

**Also at this stage we should not use the mixing angles ?

for sgg and ssgg vertex Could these external function definitions
         LAAh2=-cabs(lAAhiggs(Ms,str(h2))),
          LGGs=-cabs(lGGhiggs(Ms,str(h2))),
          aQCDs =alphaQCD(Ms)/acos(-1),
          RQCDs=sqrt(1+149/12*aQCDs+68.6482*aQCDs**2-212.447*aQCDs**3).
give us the effect originating from the Wilson coefficient? Or Are these completely wrong?

If they are wrong, how we can define the effective vertex for s scalar field (which has the mass above ttbar threshold)?

Thanks Alexander Pukhov, that solved my question.