Normalization and dependence
for the   and processes

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FeynArts

FeynCalc

FeynCalc 3.0.1.3 For help do: ?FeynCalc
Copyright 1997 Mertig Research & Consulting (www.mertig.com)

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-> ->

The three amplitude of -> -> is:

The momentum assignment is (as usual with FA): -> -> .
We use 't Hooft-Feynman gauge.

The Mandelstam variables are defined such that the fermion comes from the proton:

The amplitude squared is:

In the W CM frame:

This agrees with my hand calculations (notes: Total Cross Section) and C.-P.'s notes (Zeroth Order) and with CompHEP.

The averaging factors: initial state color and spin .

CompHEP result

Taking away the initial state color and spin averaging factors and changing notation:

-> ->

The three amplitude of -> -> is:

The momentum assignment is (as usual with FA): -> -> .
We use 't Hooft-Feynman gauge.

The Mandelstam variables are defined such that the fermion comes from the proton:

The amplitude squared is:

In the W CM frame:

The averaging factors: initial state color and spin .

CompHEP result

Taking away the initial state color and spin averaging factors and changing notation:

-> ->

The three amplitude of   -> -> is (the    -> -> process was deleted by hand):

The momentum assignment is (as usual with FA): -> -> .
We use 't Hooft-Feynman gauge. We made tha quark charge explicit.

The Mandelstam variables are defined in the standard way:

The amplitude squared is:

In the W CM frame:

This agrees with CompHEP.

The averaging factors: initial state color and spin .
Summation over colors has not been done yet.

CompHEP result

Taking away the initial state color and spin averaging factors, making the quark charge and the color sum explicit  and changing notation:

-> γ γ

FeynCalc

The three amplitude of -> γ γ is:

The momentum assignment is (as usual with FA): -> .
We use 't Hooft-Feynman gauge. The u quark charge is included by FA and taken made explicit.

This must still be summed over .

In the γγ CM frame:

This agrees with both Ohnemus and Owens and CompHEP.

The averaging factors: initial state color and spin and the identical final state particles factor .

CompHEP

To obtain the uanveraged amplitude square we take away the initial state color and spin averaging factors and the identical final state particles factor , make explicit the quark charges and the sum over colors ), and change notation:

This agrees with both Ohnemus and Owens and FC.

Papageno

In Papageno the following is coded:

      CONST=AEM**2*16.*PISQ
      CONST1=QU4
      CONST2=QD4
      ....
      A4=CONST*2.*(U2+T2)/U/T/3.
      A5=CONST*AL**2*11.**2/9**2.*2./4./8./8.*XMAT_PHOT(S,T,U)
      WT=((SF(23)+SF(24))*CONST1+(SF(27)+SF(28))*CONST2)*A4/2.
     1 +SF(9)*A5/2./16./PISQ

That is

After taking out the initial state color and spin averaging factors and the identical final state particles factor , and making the color sum explicit, for the amplitude square (before averaging) we obtain:

which agrees with both FC and CompHEP.

Berger et.al. NPB239 (1984) 52

On page 56 they have

After taking out the initial state color and spin averaging factors and the identical final state particles factor , and making the color sum explicit, for the amplitude square (before averaging) we obtain:

There's a factor which I don't understand.

g g -> γ γ

FeynCalc

The box contribution to the amplitude of -> γ γ is:

The triangular contribution to the amplitude of -> γ γ is:

For all of the above diagrams:
- each diagram comes with a factor of 2, since the quark can run either in one or in the other direction in the loop,
- all six flavors can run around the loop.

Papageno

We want to find out the normalization of the g g -> γ γ process with respect to the q -> γ γ process from Papageno.
In Papageno the following is coded:

      CONST=AEM**2*16.*PISQ
      CONST1=QU4
      CONST2=QD4
      ....
      A4=CONST*2.*(U2+T2)/U/T/3.
      A5=CONST*AL**2*11.**2/9**2.*2./4./8./8.*XMAT_PHOT(S,T,U)
      WT=((SF(23)+SF(24))*CONST1+(SF(27)+SF(28))*CONST2)*A4/2.
     1 +SF(9)*A5/2./16./PISQ
    
That is

After taking out the initial state color and spin averaging factors and the identical final state particles factor , the factor (without the top quark)  and making the factor from the color sum/traces explicit we obtain:

The factor (without and with the top quark) is

ResBos

In RES.FOR:
- For H production we have:     

- For γγ production we have:    

In ResBos we have:
C Identical particle final state and spin average factors
     &      1.d0/2.d0/2.d0/2.d0*
C Effective matrix element
     &      2.d0*(gWeak*sWeak)**4/(16.0*Pi**2)*
CsB t and b quark masses are non-zero in loop:
     &      GGAA(sH,tH,uH)*(SigS+SigY(0))

-> Z Z

FeynCalc

The three amplitude of -> Z Z is:

The momentum assignment is (as usual with FA): -> .
We use 't Hooft-Feynman gauge. The u quark charge is included by FA and taken made explicit.

This must still be summed over .

In the γγ CM frame:

This agrees with both Ohnemus and Owens and CompHEP.

The averaging factors: initial state color and spin and the identical final state particles factor .

CompHEP

We make the initial state color and spin averaging factors and the identical final state particles factor , and the sum over colors ) explicit ((the quark charges )), and change notation:

This agrees with Ohnemus and Owens.

Notes:

Papageno

Berger et.al. NPB239 (1984) 52

Ratios for initial state

Compared to the process, in the amplitude square of there's an extra factor:

This means: to obtain the resummation formula for from the resummation formula for we have to inclue an extra numerical factor, include extra quark charges , and include an extra factor of .
An extra numerical factor comes from the identical particle final state, which means that the overall extra factor (for compared to ) is 2.
(Note that the initial state is the same, which means that the color and spin averaging factors are the same for both process.)

In RES.FOR:
- the quark charges are adjusted.
In ResBos:

- for γγ:	*2=8is programmed,
- for -> ö½ö½':		is programmed.

This means the constant factor of 2 and the extra is taken into account.

Notes

Compared to the symmetric part of the process in the amplitude square of there's an extra factor:

This means: to obtain the resummation formula for from the resummation formula for we have to set -> , inclue an extra numerical factor, get rid of the W propagator, include the quark charges, replace   with , and include an extra factor of .

Extra numerical factor coming in from the different averaging:

The numerical factors combined give an overall extra factor of:

Ratios for gg initial state

Based on Papageno the ratio of the weights of the g g -> γ γ and q -> γ γ processes is:

From here we see the differences:
- the (unaveraged, un-color-summed) amplitude square (not including the couplings or     quark charges),
- the initial color averaging factors,
- the color factors,
- the couplings.

The factor is:

The factor then should be:

Appendix

-> Z Z

The three amplitude of -> Z Z is:

The momentum assignment is (as usual with FA): -> .
We use 't Hooft-Feynman gauge.

For simplicity we neglect the Z mass

This must be summed over .

Ohnemus and Owens (PRD43,3626(1991)) has in Eq.(4) (for f = u and after taking )