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virtual.f
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c returns 2 Re(M_B * M_V)/(as/(2pi)),
c where M_B is the Born amplitude and
c M_V is the finite part of the virtual amplitude
c The as/(2pi) factor is attached at a later point
subroutine setvirtual(p,vflav,virtual)
implicit none
include 'nlegborn.h'
include 'pwhg_math.h'
include 'pwhg_st.h'
include 'PhysPars.h'
integer nleg
parameter (nleg=nlegborn)
real * 8 p(0:3,nleg)
integer vflav(nleg)
real * 8 virtual
ccccccccccccccccccccc
integer mu,ileg
real *8 kblab(0:3,nleg)
real *8 ewcoupl,s,t,u
real *8 fvirt_udx,fvirt_dxu
real *8 dotp
external dotp
integer three_ch(-6:6)
data three_ch /-2,1,-2,1,-2,1,0,-1,2,-1,2,-1,2/
real *8 ckm_b_s
ccccccccccccccccccccc
ccccccccccccccc
include 'pwhg_flst.h'
integer i_fb
c To pass the already subtracted virtuals to sigsoftvirt
real *8 fksfinite(1000)
common/cfksfinite/fksfinite
ccccccccccccccc
integer vflav_loc(nleg)
c check
if (abs(vflav(3)).ne.6) then
write(*,*) 'setvirtual: ERROR in flavor assignement'
call exit(1)
endif
ccccccccccccccccccccccccccccccccccccccc
c charge conjugation
c if ttype=-1, then bflav has been filled with tbar-production flavours.
c Subroutines here work for t-production flavour assignment.
c Therefore, invert the sign of local flavours.
do ileg=1,nleg
vflav_loc(ileg)= ttype *vflav(ileg)
enddo
ccccccccccccccccccccccccccccccccccccccc
c local copy of variables
do ileg=1,nleg
do mu=0,3
kblab(mu,ileg)=p(mu,ileg)
enddo
enddo
c ew coupling
ewcoupl=4d0*pi*alphaem_pow/sthw2_pow
c Mandelstam variables
s=2d0*dotp(kblab(0,1),kblab(0,2))
t=dotp(kblab(0,3),kblab(0,3)) - 2d0*dotp(kblab(0,1),kblab(0,3))
u=dotp(kblab(0,3),kblab(0,3))-s-t
ccccccccccccccccccccccccccccccccccccccccccc
c >>> S CHANNEL <<<
ccccccccccccccccccccccccccccccccccccccccccc
c Evaluate finite part of virtual contribution
call virt_finite(s,t,u,dotp(kblab(0,3),kblab(0,3)),
#fvirt_udx,fvirt_dxu,st_muren2)
! write(*,*) 'virtual: ',s,t,fvirt_udx,fvirt_dxu
c udx
if ((three_ch(vflav_loc(1)).eq.2).and.
#(three_ch(vflav_loc(2)).eq.1)) then
virtual=fvirt_udx
ckm_b_s=
#CKM(abs(vflav_loc(1)),abs(vflav_loc(2)))**2
#* CKM(abs(vflav_loc(3)),abs(vflav_loc(4)))**2
c dxu
elseif ((three_ch(vflav_loc(1)).eq.1).and.
#(three_ch(vflav_loc(2)).eq.2)) then
virtual=fvirt_dxu
ckm_b_s=
#CKM(abs(vflav_loc(1)),abs(vflav_loc(2)))**2
#* CKM(abs(vflav_loc(3)),abs(vflav_loc(4)))**2
else
write(*,*) 'Error in setvirtual, (s)'
call exit(1)
endif
ccccccccccccccccccccccc
c assign output
virtual=virtual *ewcoupl**2 *ckm_b_s
cccccccccccccccccccccc
end
c Output the finite part of virtual contribution.
c Finite parts for s-channel single-top.
c Virtual contributions are taken from Laenen-Weinzierl-Sullivan...
c paper, after a manipulation analogous to the
c one described in the t-channel note section.
c An analytic continuation is also needed in this case.
c Prefactor of FKS is different, so an expansion of
c EulerGamma functions was needed.
c Moreover, nontrivial logaritms arise because of
c the presence of (mu^2/s) in front of the result.
subroutine virt_finite(s,t,u,m2,fvirt_udx,fvirt_dxu,mur2)
implicit none
real *8 s,t,u,m2,fvirt_udx,fvirt_dxu,mur2
include 'pwhg_math.h'
include 'PhysPars.h'
real *8 lambda,q2,prop
real *8 amp2_udx,amp2_dxu
real *8 vcf
parameter (vcf=4.d0/3.d0)
real *8 ddilog,A0,A1
real *8 A0h_s,A0l_s,A1h_s,A1l_s,A2h_s,A2l_s
real *8 cv0,cv,prop_mcfm,prop_ztop
double complex c1
logical check_virt
parameter (check_virt=.false.)
if(dabs(m2/topmass_pow**2 -1.).gt.1d-6) then
write(*,*) 'virt_finite: problem with top offshelness'
call exit(1)
endif
amp2_udx=u*(u-topmass_pow**2)/(s-wmass_pow**2)**2/4
amp2_dxu=t*(t-topmass_pow**2)/(s-wmass_pow**2)**2/4
if(s+t+u-m2.gt.1d-7) then
print*, 'virt_finite: problem with invariants'
call exit(1)
endif
cccccccccccccccccccccccc
c s-channel
cccccccccccccccccccccccc
lambda=s/(s-m2)
ccccccccccccccccccccccccccccc
c This result has been obtained by crossing
c the result in the paper by Laenen&al.
c Several checks have been performed.
c A2l=-2.
c A1l=-3.
c A0l=pi**2-8.-pi**2/3.
c After having factorized the same prefactor
c of Altarelli-Ellis-Martinelli, that corresponds
c to 1/Ga[1-ep] up to order O(ep^3), we have
A2l_s=-2.
A1l_s=-3.
A0l_s=(pi**2 -8. -pi**2/3.)
$ + pi**2/3. !from 'EulerGamma prefactor'
c A2h=-1.
c A1h=-5./2. -2.*log(m2/(s-m2)) -log(s/m2)
c A0h=....
c After having factorized the same prefactor
c of Altarelli-Ellis-Martinelli, that corresponds
c to 1/Ga[1-ep] up to order O(ep^3), we have
A2h_s=-1.
A1h_s=-5./2. -2.*log(m2/(s-m2)) -log(s/m2)
A0h_s=(
$ -0.5*(log(s/m2))**2 -5./2.*log(s/m2)
$ - 2*log(m2/(s-m2))*log(s/m2) -6.
$ -(s-m2)/s*log(m2/(s-m2)) -(log(m2/(s-m2)))**2
$ +pi**2 -2*log(m2/(s-m2))
$ +2*ddilog(lambda) -pi**2/3.)
$ +pi**2/6. !from 'EulerGamma prefactor'
prop= (A0l_s+A0h_s)
$ +(A1l_s+A1h_s) *log(mur2/s)
$ +(A2l_s+A2h_s) *0.5*(log(mur2/s))**2
ccccccccccccccccccccccccccccc
c$$$ if(check_virt) then
c$$$c A0 from ZTOP, s-channel
c$$$ A0=
c$$$ $ + (-0.5*(log(s/m2))**2 -5./2.*log(s/m2)
c$$$ $ - 2*log(m2/(s-m2))*log(s/m2) -3.*log(m2/(s-m2))
c$$$ $ + m2/s*log(m2/(s-m2)) -(log(m2/(s-m2)))**2
c$$$ $ +2*ddilog(lambda))
c$$$ $ - 14. +pi**2 +pi**2/3.
c$$$c A1, by crossing the Laenen&al result
c$$$ A1= -11./2. -2.*log(m2/(s-m2)) -log(s/m2)
c$$$c from previous A0 and A1
c$$$ prop_ztop=
c$$$ $ A0
c$$$ $ + pi**2/2. !from different EulerGamma
c$$$ $ + A1*log(mur2/s)
c$$$ $ - 3./2.*(log(mur2/s))**2
c$$$
c$$$c from Campbell-Ellis, MCFM
c$$$c Notice that the use of lnrat is crucial in order
c$$$c to get all pi's from analytical continuation
c$$$ call coefs(s,m2,cv0,cv,c1)
c$$$ prop_mcfm=cv0+cv
c$$$
c$$$ if(dabs(prop/prop_mcfm-1).gt.1d-6) then
c$$$ write(*,*) '>>>>> POW/MCFM, light ',
c$$$ $ (A0l_s+A1l_s*log(mur2/s)+A2l_s*0.5*(log(mur2/s))**2)/cv0
c$$$
c$$$ write(*,*) '>>>>> POW/MCFM, heavy ',
c$$$ $ (A0h_s+A1h_s*log(mur2/s)+A2h_s*0.5*(log(mur2/s))**2)/cv
c$$$
c$$$ write(*,*) 'POW/MCFM ',prop/prop_mcfm
c$$$ call exit(1)
c$$$ endif
c$$$ if(dabs(prop/prop_ztop-1).gt.1d-6) then
c$$$ write(*,*) 'POW/ZTOP ',prop/prop_ztop
c$$$ call exit(1)
c$$$ endif
c$$$ endif
prop=prop*vcf
fvirt_udx=prop*amp2_udx !1
fvirt_dxu=prop*amp2_dxu !3
c add part not proportional to born
fvirt_udx=fvirt_udx +vcf *m2*t*u/s *log((s-m2)/m2) /4.
$ * (s- wmass_pow**2)**(-2)
fvirt_dxu=fvirt_dxu +vcf *m2*u*t/s *log((s-m2)/m2) /4.
$ * (s- wmass_pow**2)**(-2)
end
c$$$ subroutine coefs(s12,mtsq,cv0,cv,c1)
c$$$ implicit none
c$$$!: include 'constants.f'
c$$$ include 'pwhg_math.h'
c$$$!: include 'epinv.f'
c$$$
c$$$!: include 'epinv2.f'
c$$$
c$$$!: include 'scale.f'
c$$$ include 'pwhg_st.h'
c$$$
c$$$!: include 'scheme.f'
c$$$
c$$$
c$$$ double precision cv,cv0,Li2la
c$$$ double precision s12,mtsq,taucs,ddilog,eta,la,oml
c$$$ double complex lnrat,logoml,logla,xl12,logsca,Kfun,c1
c$$$
c$$$cccccccccccccccccccccccccccccccccccccc
c$$$c !:
c$$$ double precision epinv
c$$$ double precision epinv2
c$$$ real *8 musq
c$$$ double complex cone
c$$$ real *8 pisqo6
c$$$ character*4 scheme
c$$$ external lnrat
c$$$
c$$$
c$$$ cone=(1d0,0d0)
c$$$ pisqo6=pi**2/6.
c$$$ scheme='tH-V'
c$$$ epinv=0d0
c$$$ epinv2=0d0
c$$$ musq=st_muren2
c$$$cccccccccccccccccccccccccccccccccccccc
c$$$
c$$$ if (scheme .eq.'dred') then
c$$$C------ eta=0 4d-hel
c$$$ eta=0d0
c$$$ elseif (scheme .eq. 'tH-V') then
c$$$C------ eta=1 t'Hooft Veltman
c$$$ eta=1d0
c$$$ endif
c$$$
c$$$C**********************************************************************
c$$$C Massless case
c$$$C Taken from
c$$$C %\cite{Altarelli:1979ub}
c$$$C \bibitem{Altarelli:1979ub}
c$$$C G.~Altarelli, R.~K.~Ellis and G.~Martinelli,
c$$$C %``Large Perturbative Corrections To The Drell-Yan Process In QCD,''
c$$$C Nucl.\ Phys.\ B {\bf 157}, 461 (1979).
c$$$C %%CITATION = NUPHA,B157,461;%%
c$$$C Using Eqn(58) with normalization changed to
c$$$C as/2/pi*cf*(4*pi)^ep/Gamma(1-ep)
c$$$C Taking account that Gamma(1-ep)^2/Gamma(1-2*ep)=1-ep^2*pi^2/6
c$$$C**********************************************************************
c$$$ xl12=lnrat(-s12,musq)
c$$$ cv0=-2d0*epinv*(epinv2-dble(xl12))-dble(xl12**2)
c$$$ . -3d0*(epinv-dble(xl12))-7d0-eta
c$$$
c$$$
c$$$
c$$$C---- this routine has been constructed following closely
c$$$C---- the notation of
c$$$C---- %\cite{Gottschalk:1980rv}
c$$$C---- \bibitem{Gottschalk:1980rv}
c$$$C---- T.~Gottschalk,
c$$$C---- %``Chromodynamic Corrections To Neutrino Production Of Heavy Quarks,''
c$$$C---- Phys.\ Rev.\ D {\bf 23}, 56 (1981).
c$$$C---- %%CITATION = PHRVA,D23,56;%%
c$$$C----- Adapted from Eqs.(A8,A9)
c$$$
c$$$ taucs=s12-mtsq
c$$$ la=-s12/(mtsq-s12)
c$$$ oml=1d0-la
c$$$C-----oml=mtsq/(mtsq-s12)
c$$$ logoml=-lnrat(-taucs,mtsq)
c$$$ logsca=lnrat(-taucs,musq)
c$$$ Kfun=dcmplx(oml/la)*logoml
c$$$
c$$$c--- Minus sign relative to Gottschalk since incoming b has momentum
c$$$c--- vector reversed for the t-channel process
c$$$c--- s-channel process follows by crossing
c$$$ c1=-dcmplx(2d0)*Kfun
c$$$
c$$$ if (la .lt. 1d0) then
c$$$ Li2la=ddilog(la)
c$$$ else
c$$$ logla=lnrat(-s12,-taucs)
c$$$ Li2la=pisqo6-ddilog(oml)-dble(logla*logoml)
c$$$ endif
c$$$ cv=-epinv*epinv2
c$$$ . -epinv*(2.5d0+dble(logoml-logsca))
c$$$ . -0.5d0*(11d0+eta)-pisqo6+2d0*Li2la-dble(Kfun)
c$$$ . -0.5d0*dble(logoml*(cone-logoml))
c$$$ . +2.5d0*dble(logsca)+dble(logsca*logoml)-0.5d0*dble(logsca**2)
c$$$
c$$$ return
c$$$ end
c$$$
c$$$
c$$$ double complex function Lnrat(x,y)
c$$$************************************************************************
c$$$* Author: R.K. Ellis *
c$$$* August, 1998. *
c$$$c Lnrat(x,y)=log(x-i*ep)-log(y-i*ep) *
c$$$c this function is hard-wired for sign of epsilon we must adjust *
c$$$c sign of x and y to get the right sign for epsilon *
c$$$************************************************************************
c$$$ implicit none
c$$$!: include 'constants.f'
c$$$ include 'pwhg_math.h'
c$$$ double precision x,y,htheta
c$$$C--- define Heaviside theta function (=1 for x>0) and (0 for x < 0)
c$$$ htheta(x)=0.5+0.5*sign(1d0,x)
c$$$
c$$$
c$$$c$$$ real *8 half,one
c$$$c$$$ one=1d0
c$$$c$$$ half=0.5d0
c$$$
c$$$ double complex impi
c$$$ impi=(0d0,1d0)*pi
c$$$
c$$$
c$$$ Lnrat=dcmplx(dlog(abs(x/y)))-impi*dcmplx((htheta(-x)-htheta(-y)))
c$$$ return
c$$$ end