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setlocalscales.f
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setlocalscales.f
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subroutine setlocalscales(iuborn,imode,rescfac)
c returns the rescaling factor including sudakov form factors and
c coupling rescaling, for born (imode=1) and NLO corrections (imode=2)
implicit none
integer iuborn,imode
real * 8 rescfac
include 'nlegborn.h'
include 'pwhg_flst.h'
include 'pwhg_kn.h'
include 'pwhg_flg.h'
include 'pwhg_par.h'
include 'pwhg_st.h'
integer j,k,mu
logical samekin,sameflv
integer flav(nlegborn)
real * 8 op(0:3,nlegborn),basicfac,bornfac,nlofac
real * 8 savbasicfac(maxprocborn),savbornfac(maxprocborn),
1 savnlofac(maxprocborn),savmuf2(maxprocborn)
logical valid(maxprocborn)
save savbasicfac,savbornfac,savnlofac,savmuf2,valid
save op
data valid/maxprocborn*.false./
do j=1,nlegborn
do mu=0,3
if(op(mu,j).ne.kn_cmpborn(mu,j)) then
do k=1,flst_nborn
valid(k)=.false.
enddo
op=kn_cmpborn
endif
enddo
enddo
33 continue
if(valid(iuborn)) then
basicfac=savbasicfac(iuborn)
nlofac=savnlofac(iuborn)
bornfac=savbornfac(iuborn)
st_mufact2=savmuf2(iuborn)
else
flav=flst_born(:,iuborn)
call setlocalscales0(flav,kn_cmpborn,
1 basicfac,bornfac,nlofac)
savbasicfac(iuborn)=basicfac
savnlofac(iuborn)=nlofac
savbornfac(iuborn)=bornfac
savmuf2(iuborn)=st_mufact2
valid(iuborn)=.true.
endif
if(imode.eq.1) then
if(flg_novirtual) then
c basicfac includes the Sudakov subtraction and the scale
c compensation term for setting the scale in the virtual term.
c If the virtual is not included, it must be omitted.
rescfac=basicfac
else
rescfac=basicfac*bornfac
endif
elseif(imode.eq.2) then
rescfac=basicfac*nlofac
endif
end
subroutine setlocalscales0
1 (flav,pin,basicfac,bornfac,nlofac)
implicit none
include 'nlegborn.h'
include 'pwhg_st.h'
include 'pwhg_math.h'
integer flav(nlegborn)
real * 8 pin(0:3,nlegborn),basicfac,bornfac,nlofac
integer onem
parameter (onem=1000000)
real * 8 scales(nlegborn),p(0:3,nlegborn),ptot(0:3),
1 lscalej,lscalek
integer j,k,l,lflav(nlegborn),jmerge,kmerge,inlofac
integer mergedfl
real * 8 q2merge,q2merge0,renfac2,facfact2,alphas,mu2,muf2
real * 8 sudakov,expsudakov,pwhg_alphas,b0,powheginput
external sudakov,expsudakov,pwhg_alphas,powheginput
real * 8 q2mergeMAX
logical dijetflag
common/cdijetflag/dijetflag
logical raisingscales,ini
save raisingscales,ini
data ini/.true./
if(ini) then
if(powheginput("#raisingscales").eq.0) then
raisingscales = .false.
else
raisingscales = .true.
endif
ini = .false.
endif
renfac2=st_renfact**2
facfact2=st_facfact**2
lflav=flav
p=pin
scales=0
q2mergeMAX=-1d10
do l=1,nlegborn
call findNearestNeighbours(p,lflav,jmerge,kmerge,mergedfl,
$ q2merge)
if(q2merge.lt.1d10) then
c perform the merging
if(q2merge.gt.q2mergeMAX) q2mergeMAX=q2merge
lscalej=scales(jmerge)
lscalek=scales(kmerge)
scales(jmerge)=q2merge
if(lscalej.eq.0) then
c This is the first merge; it sets the low scale for
c all partons; no Sudakov factor or reweighting is introduced
do j=1,nlegborn
scales(j)=q2merge
enddo
c save this scale; it is the Q_0 scale that appears in all Sudakovs
q2merge0=q2merge
bornfac=0
c Provide alpha_S reweighting for the first merge
alphas=pwhg_alphas(max(q2merge*renfac2,1d0),
1 st_lambda5MSB,st_nlight)
basicfac=alphas/st_alpha
nlofac=basicfac
mu2=max(q2merge*renfac2,1d0)
muf2=max(q2merge*facfact2,1d0)
inlofac=1
else
c provide Sudakov
basicfac=basicfac*
1 sudakov(q2merge0,q2merge,lscalej,lflav(jmerge))
basicfac=basicfac*
1 sudakov(q2merge0,q2merge,lscalek,lflav(kmerge))
bornfac=bornfac+
1 expsudakov(q2merge0,q2merge,lscalej,lflav(jmerge))
bornfac=bornfac+
1 expsudakov(q2merge0,q2merge,lscalek,lflav(kmerge))
c provide alpha_S reweighting
alphas=pwhg_alphas(max(q2merge*renfac2,1d0),
1 st_lambda5MSB,st_nlight)
basicfac=basicfac*alphas/st_alpha
mu2=mu2*max(q2merge*renfac2,1d0)
nlofac=nlofac+alphas/st_alpha
inlofac=inlofac+1
endif
if(jmerge.gt.2) then
p(:,jmerge)=p(:,jmerge)+p(:,kmerge)
else
p(3,jmerge)=p(3,jmerge)-p(3,kmerge)
p(0,jmerge)=abs(p(3,jmerge))
endif
lflav(kmerge)=onem
lflav(jmerge)=mergedfl
else
goto 99
endif
enddo
99 continue
c No more merging is possible.
if(.not.dijetflag) then
c Define the initial scale as
c the invariant mass of the remaining system
ptot=0
do j=3,nlegborn
if(lflav(j).ne.onem) then
ptot=ptot+p(:,j)
endif
enddo
if(raisingscales) then
q2merge=max(q2mergeMAX,
$ ptot(0)**2-ptot(1)**2-ptot(2)**2-ptot(3)**2)
else
q2merge=ptot(0)**2-ptot(1)**2-ptot(2)**2-ptot(3)**2
endif
else
c Dijet case: use the scalar sum of the pt of the two partons
q2merge = 0
do j=3,nlegborn
if(lflav(j).ne.onem) then
q2merge=sqrt(p(1,j)**2+p(2,j)**2)+q2merge
endif
enddo
q2merge=q2merge**2
if(raisingscales) then
q2merge=max(q2mergeMAX,q2merge)
endif
endif
if(scales(1).gt.0) then
do j=1,nlegborn
if(abs(lflav(j)).le.st_nlight) then
basicfac=basicfac*
1 sudakov(q2merge0,q2merge,scales(j),lflav(j))
bornfac=bornfac+
1 expsudakov(q2merge0,q2merge,scales(j),lflav(j))
endif
enddo
else
c If scales(1)=0 no merge has taken place: no sudakovs.
mu2=1
muf2=max(q2merge*facfact2,1d0)
inlofac=0
bornfac=0
basicfac=1
nlofac=0
endif
if(st_bornorder.gt.inlofac) then
alphas=pwhg_alphas(max(q2merge*renfac2,1d0),
1 st_lambda5MSB,st_nlight)
c do j=inlofac+1,st_bornorder
c mu2=mu2*max(q2merge*renfac2,1d0)
c nlofac=nlofac+alphas/st_alpha
c basicfac=basicfac*alphas/st_alpha
c enddo
mu2=mu2*max(q2merge*renfac2,1d0)**(st_bornorder-inlofac)
nlofac=nlofac+alphas/st_alpha*(st_bornorder-inlofac)
basicfac=basicfac*(alphas/st_alpha)**(st_bornorder-inlofac)
inlofac=st_bornorder
endif
nlofac=nlofac/inlofac
mu2=mu2**(1d0/inlofac)
b0=(33-2*st_nlight)/(12*pi)
bornfac=1+st_alpha*nlofac*
1 (bornfac+st_bornorder*b0*log(mu2/st_muren2))
st_mufact2=muf2
end
C ------------------------------------------------ C
C - Inputs: - C
C - ******* - C
C - p - Underlying born momenta - C
C - lflav - Flavour list derived from - C
C - flst_born by subjecting it to - C
C - repeated QCD clusterings. - C
C - - C
C - Outputs: - C
C - ******** - C
C - jmerge - Index in lflav of one of the two - C
C - closest partons. - C
C - kmerge - Index in lflav of the - C
C - corresponding parton. - C
C - mergedfl - Flavour of parton resulting from - C
C - combination. - C
C - q2merge - pT^2 scale associated to the - C
C - merging of jmerge and kmerge. - C
C - - C
C - checked 24/03/12 - C
C ------------------------------------------------ C
subroutine findNearestNeighbours(p,lflav,jmerge,kmerge,mergedfl,
$ q2merge)
implicit none
include 'nlegborn.h'
include 'pwhg_st.h'
include 'pwhg_math.h'
C - Input / output:
real * 8 p(0:3,nlegborn)
integer lflav(nlegborn)
integer jmerge,kmerge,mergedfl
real * 8 q2merge
C - Local variables:
real * 8 ycm
integer onem
parameter (onem=1000000)
integer j,k
integer fl1,fl2,fl
integer npartons,nparticles
real * 8 yj,phij,q2j
real * 8 yk,phik,q2k
real * 8 dphi
real * 8 q2
logical dijetflag
common/cdijetflag/dijetflag
q2merge=1d10
ycm=log(p(0,1)/p(0,2))/2
mergedfl=onem
c Count particles and partons in the final state.
c If we have two particles and two partons, it
c is the dijet case, return with no merging.
npartons = 0
nparticles = 0
do j=3,nlegborn
if(abs(lflav(j)).le.st_nlight) npartons = npartons+1
if(lflav(j).ne.mergedfl) nparticles = nparticles+1
enddo
if(npartons.eq.nparticles.and.npartons.eq.2) then
dijetflag = .true.
return
else
dijetflag = .false.
endif
c
do j=3,nlegborn
if(abs(lflav(j)).gt.st_nlight) goto 11
yj=0.5d0*log((p(0,j)+p(3,j))/(p(0,j)-p(3,j)))
if(yj.gt.ycm) then
call validmergeisr(lflav,1,j,fl1)
if(fl1.ne.onem) then
q2j = p(1,j)**2+p(2,j)**2
if(q2j.lt.q2merge) then
q2merge=q2j
jmerge=1
kmerge=j
mergedfl=fl1
endif
endif
else
call validmergeisr(lflav,2,j,fl2)
if(fl2.ne.onem) then
q2j = p(1,j)**2+p(2,j)**2
if(q2j.lt.q2merge) then
q2merge=q2j
jmerge=2
kmerge=j
mergedfl=fl2
endif
endif
endif
do k=j+1,nlegborn
if(abs(lflav(k)).gt.st_nlight) goto 12
call validmergefsr(lflav,j,k,fl)
if(fl.ne.onem) then
yk=0.5d0*log((p(0,k)+p(3,k))/(p(0,k)-p(3,k)))
call phipt2(p(:,k),phik,q2k)
call phipt2(p(:,j),phij,q2j)
dphi=abs(phik-phij)
if(dphi.gt.2*pi) dphi=dphi-2*pi
if(dphi.gt.pi) dphi=2*pi-dphi
q2=((yk-yj)**2+dphi**2)*min(q2k,q2j)
if(q2.lt.q2merge) then
q2merge=q2
jmerge=j
kmerge=k
mergedfl=fl
endif
endif
12 continue
enddo
11 continue
enddo
end
C ------------------------------------------------ C
C - Inputs: - C
C - ******* - C
C - q2h - Upper node scale / bound on Sudakov - C
C - q2l - Lower node scale / bound on Sudakov - C
C - flav - flavour index for the evolving parton - C
C - - C
C - Outputs: - C
C - ******** - C
C - sudakov - The Sudakov form factor. - C
C - - C
C ------------------------------------------------ C
function sudakov(q20,q2h,q2l,flav)
implicit none
real * 8 sudakov,q2h,q2l,q20
integer flav
include 'pwhg_st.h'
include 'pwhg_math.h'
real * 8 lam2
logical isQuark
real * 8 theExponentN,theExponentD
logical ini
data ini/.true./
save ini
if(ini) then
c call sudakov_plotter
ini=.false.
endif
lam2=st_lambda5MSB**2
if(q20.le.lam2.or.q2l.lt.lam2.or.q2h.lt.lam2) then
sudakov=0
goto 999
endif
if(q2l.ge.q2h.or.q2h.le.q20) then
sudakov=1
goto 999
endif
if(flav.eq.0) then
isQuark=.false.
else
isQuark=.true.
endif
if(q2l.le.q20) then
call sudakov_exponent(q20,q2h,q2h,theExponentN,
$ isQuark,2,.true.)
sudakov=exp(theExponentN)
else
call sudakov_exponent(q20,q2h,q2h,theExponentN,
$ isQuark,2,.true.)
call sudakov_exponent(q20,q2l,q2l,theExponentD,
$ isQuark,2,.true.)
sudakov=exp(theExponentN-theExponentD)
endif
999 continue
end
C ------------------------------------------------ C
C - Inputs: - C
C - ******* - C
C - q2h - Upper node scale / bound on Sudakov - C
C - q2l - Lower node scale / bound on Sudakov - C
C - flav - flavour index for the evolving parton - C
C - - C
C - Outputs: - C
C - ******** - C
C - sudakov - The Sudakov form factor. - C
C - Uses 1-loop alpha, no CMW scheme - C
C - etc. - C
C - - C
C ------------------------------------------------ C
function LO_sudakov(q20,q2h,q2l,flav)
implicit none
real * 8 LO_sudakov,q2h,q2l,q20
integer flav
include 'pwhg_st.h'
include 'pwhg_math.h'
real * 8 b0,c,b,lam2
lam2=st_lambda5MSB**2
if(q20.le.lam2.or.q2l.lt.lam2.or.q2h.lt.lam2) then
LO_sudakov=0
goto 999
endif
if(q2l.ge.q2h.or.q2h.le.q20) then
LO_sudakov=1
goto 999
endif
b0=(33-2*st_nlight)/12d0
if(flav.eq.0) then
c=3
b=b0/3
else
c=4d0/3
b=3d0/4
endif
if(q2l.le.q20) then
LO_sudakov= exp(
1 -c/b0*( log(log(q2h/lam2)/log(q20/lam2))
2 *(0.5d0*log(q2h/lam2)-b)
3 -0.5d0*log(q2h/q20)
4 ))
else
LO_sudakov= exp(
1 -c/b0*( log(log(q2h/lam2)/log(q20/lam2))
2 *(0.5d0*log(q2h/lam2)-b)
3 -0.5d0*log(q2h/q20)
4 )
5 +c/b0*( log(log(q2l/lam2)/log(q20/lam2))
6 *(0.5d0*log(q2l/lam2)-b)
7 -0.5d0*log(q2l/q20)
8 ))
endif
999 continue
end
C ------------------------------------------------ C
C - Inputs: - C
C - ******* - C
C - q2h - Upper node scale / bound on Sudakov - C
C - q2l - Lower node scale / bound on Sudakov - C
C - flav - flavour index for the evolving parton - C
C - - C
C - Outputs: - C
C - ******** - C
C - expsudakov - The Sudakov form factor's expon - C
C - -ent MODULO a factor of minus - C
C - alphaS, integrated with alphaS - C
C - fixed. Summed over with the - C
C - relevant alphaS factors this is - C
C - used in compensating the NLO - C
C - correction induced when the - C
C - Sudakov multiplies the Born. - C
C - - C
C ------------------------------------------------ C
function expsudakov(q20,q2h,q2l,flav)
implicit none
real * 8 expsudakov,q2h,q2l,q20
integer flav
include 'pwhg_st.h'
include 'pwhg_math.h'
include 'pwhg_flg.h'
real * 8 b0,c,b,lam2
lam2=st_lambda5MSB**2
if(q20.le.lam2.or.q2l.lt.lam2.or.q2h.lt.lam2) then
c in this case everything is zero, irrelevant
expsudakov=0
return
endif
if(q2l.ge.q2h.or.q2h.le.q20.or.flg_bornonly) then
expsudakov=0
return
endif
b0=(33-2*st_nlight)/12d0
if(flav.eq.0) then
c=3
b=b0/3
else
c=4d0/3
b=3d0/4
endif
if(q2l.le.q20) then
expsudakov=
1 c/pi*(0.25d0*log(q2h/q20)**2 - b*log(q2h/q20))
else
expsudakov=
1 c/pi*(0.25d0*log(q2h/q20)**2 - b*log(q2h/q20))
2 - c/pi*(0.25d0*log(q2l/q20)**2 - b*log(q2l/q20))
endif
end
C ---------------------------------------------- C
C - Inputs: - C
C - ******* - C
C - flav - flavour list derived from flst_born - C
C - by subjecting it to repeated QCD - C
C - compatible clusterings. - C
C - i - index of i-th initial-state partON - C
C - in flav: hence i = 1 or 2 only. - C
C - j - index of j-th final-state partICLE - C
C - particle in flav. - C
C - - C
C - Outputs: - C
C - ******** - C
C - fl - Would-be PDG code of spacelike - C
C - "mother" parton obtained by merging - C
C - (~on-shell) incoming parton i with - C
C - outgoing particle j: - C
C - i -> fl + j - C
C - Note gluons have id=0 in Powheg-Box - C
C - instead of 21. If the splitting is - C
C - not possible in QCD, fl=1000000 ; - C
C - this setting signals to the rest of - C
C - the algorithm that this is not a - C
C - candidate pair for combination. - C
C - - C
C ---------------------------------------------- C
subroutine validmergeisr(flav,i,j,fl)
implicit none
include 'nlegborn.h'
include 'pwhg_flst.h'
include 'pwhg_st.h'
integer onem
parameter (onem=1000000)
integer flav(nlegborn),i,j,fl
integer lflav(nlegborn)
logical validflav
external validflav
if(i.gt.2.or.j.le.2) then ! Remove when development is finished.
write(*,*) 'validmergeisr: fatal error'
write(*,*) 'Routine demands an i.s. and f.s. particle'
write(*,*) 'index for the 2nd and 3rd input values '
write(*,*) 'respectively. Quitting.'
call exit(-1)
endif
if(abs(flav(i)).gt.st_nlight.or.abs(flav(j)).gt.st_nlight) then
fl=onem
return
endif
if(flav(i).eq.flav(j)) then
c g -> g g or q -> g q
fl=0
goto 999
endif
if(flav(j).eq.0) then
c q -> q g
fl=flav(i)
goto 999
endif
if(flav(i).eq.0) then
c g -> qbar q
fl=-flav(j)
goto 999
endif
fl=onem
return
999 continue
C - Check that the flavour list that results from the merging
C - is acceptable e.g. check that for HJJ you don't get back to
C - qqbar->H; if you do then set fl to 1000000, as if the
C - branching were not possible in QCD s.t. it will be neglected
C - as a candidate for clustering.
lflav=flav
lflav(j)=onem
lflav(i)=fl
if(.not.validflav(lflav)) then
fl=onem
endif
end
C ---------------------------------------------- C
C - Inputs: - C
C - ******* - C
C - flav - flavour list derived from flst_born - C
C - by subjecting it to repeated QCD - C
C - compatible clusterings. - C
C - i - index of i-th final-state partICLE - C
C - in flav: hence i = 1 or 2 only. - C
C - j - index of j-th final-state partICLE - C
C - particle in flav. - C
C - - C
C - Outputs: - C
C - ******** - C
C - fl - Would-be PDG code of timelike - C
C - "mother" parton obtained by merging - C
C - outgoing particles i and j: - C
C - fl -> i + j - C
C - Note gluons have id=0 in Powheg-Box - C
C - instead of 21. If the splitting is - C
C - not possible in QCD, fl=1000000 ; - C
C - this setting signals to the rest of - C
C - the algorithm that this is not a - C
C - candidate pair for combination. - C
C - - C
C ---------------------------------------------- C
subroutine validmergefsr(flav,i,j,fl)
implicit none
include 'nlegborn.h'
include 'pwhg_flst.h'
include 'pwhg_st.h'
integer onem
parameter (onem=1000000)
integer flav(nlegborn),i,j,fl
integer lflav(nlegborn)
logical validflav
external validflav
if(i.le.2.or.j.le.2) then ! Remove when development is finished.
write(*,*) 'validmergefsr: fatal error'
write(*,*) 'Routine demands an f.s. and f.s. particle'
write(*,*) 'index for the 2nd and 3rd input values '
write(*,*) 'respectively. Quitting.'
call exit(-1)
endif
if(abs(flav(i)).gt.st_nlight.or.abs(flav(j)).gt.st_nlight) then
fl=onem
return
endif
if(flav(i).eq.-flav(j)) then
c g -> g g or g -> q qbar
fl=0
goto 999
endif
if(flav(j).eq.0) then
c q -> q g
fl=flav(i)
goto 999
endif
if(flav(i).eq.0) then
c q -> g q
fl=flav(j)
goto 999
endif
fl=onem
return
999 continue
C - Check that the flavour list that results from the merging
C - is acceptable e.g. check that for HJJ you don't get back to
C - qqbar->H; if you do then set fl to 1000000, as if the
C - branching were not possible in QCD s.t. it will be neglected
C - as a candidate for clustering.
lflav=flav
lflav(j)=onem
lflav(i)=fl
if(.not.validflav(lflav)) then
fl=onem
endif
end
C ---------------------------------------------- C
C - Inputs: - C
C - ******* - C
C - p - p(0) = Energy, p(3) = p_Z - C
C - - C
C - Outputs: - C
C - ******** - C
C - y - Rapidity - C
C - phi - phi - C
C - q2 - pT^2 w.r.t the beam - C
C - - C
C ---------------------------------------------- C
subroutine phipt2(p,phi,q2)
implicit none
real * 8 p(0:3),phi,q2
q2=p(1)**2+p(2)**2
phi=atan2(p(2),p(1))
end
C ********* DDT / Ellis-Veseli / Nason-Ridolfi Sudakov ************ C
C - - C
C - Output: - C
C - ======== - C
C - The value of the Sudakov exponent defined as the integral, - C
C - from Log [ ql^2/Lambda^2 ] up to Log [ qh^2/Lambda^2 ], of - C
C - - C
C - d Log[ q^2/Lambda^2 ] - C
C - - { - C
C - aSBar*A1*Log[m^2/q^2] + aSBar^2*A2*Log[m^2/q^2] - C
C - + aSBar*B1 + aSBar^2*B2 - C
C - } - C
C - - C
C - where aSBar = aS/2/Pi. - C
C - - C
C - For m2=qh2, except for an overall factor of two this is the - C
C - Sudakov form factor of eq. 32 in the Ellis-Veseli paper - in - C
C - that paper they have two quark lines to consider while here - C
C - we only want to consider one line at a time. The factor of - C
C - two is manifest in the code below as our A1, A2, B1, B2 - C
C - coefficients defined to be HALF of the Ellis-Veseli ones. - C
C - - C
C - The m2 dependence is a relic of the Nason & Ridolfi form of - C
C - the Sudakov form factor, which has the numerator in the large - C
C - log equal to mZZ but the upper bound on the Sudakov integral - C
C - is Q^2. There doesn't seem to be any problem arising when - C
C - you just call the routine with m^2=Q^2, but maybe if this - C
C - gets resolved I can re-do the mathematica integral. - C
C - - C
C - The analytic integral was done in Mathematica assuming no - C
C - flavour thresholds. When q^2 is below the b or c quark - C
C - flavour thresholds a numerical integration is done instead - C
C - using dgauss. The numerical integration and analytic results - C
C - agree very well above these thresholds - try resetting - C
C - debuggingEpsilon below. - C
C - - C
C - For the default values of A1, A2, B1, B2 in the code below - C
C - the Sudakov should correspond to that of Nason and Ridolfi, - C
C - which has an effective B2 term by virtue of the fact that - C
C - the CMW alpha_S is used to multiply the leading & subleading - C
C - term. At least with the calculation done in the way it is we - C
C - can easily play around with the coefficients. - C
C - - C
C - To Use: - C
C - ======= - C
C - q2l = The scale of the lower clustered node. - C
C - q2h = The scale of the higher clustered node. - C
C - m2 = The boson mass squared (argument of the - C
C - log in the exponent of N.R. eq 4.8). - C
C - theExponent - C
C - = The value of the curly brackets in N.R. eq 4.8. - C
C - isQuark = .true. for a quark propagator - C
C - theAccuracy = 0 for 1-loop alphaS and A2=B2=0, - C
C - = 1 for 2-loop alphaS and Powheg A & B coeffs - C
C - = 2 for 2-loop alphaS and NLL A & B coeffs - C
C - - C
C - Notes: - C
C - ====== - C
C - Details for the integration in the sudakov exponent can be - C
C - found in the Mathematica notebook: menlops/DDT_exponent.nb . - C
C - - C
C - The Mathematica notebook shows plots in which the 5-flavour - C
C - and 4-flavour alphaS differ by <2% at pT=3 GeV and 4% at - C
C - pT=2 GeV. Using the C.M.W. alphaS (aS -> aS*(1+aS*K/2*pi)) - C
C - increases these differences but they remain small: 2% at pT=4 - C
C - GeV and 6% at pT=2 GeV. Note well that since the program - C
C - matches alphaS at flavour thresholds, not - C
C - alphaS*(1+alphaS*K/2*pi), since K too actually depends on - C
C - the number of flavours, this means the 3,4 and 5 flavour - C
C - C.M.W. alphaS*(1+alphaS*K/2*pi) DO NOT match at the flavour - C
C - thresholds in pT! Whereas alphaS 4 and 5 flavour couplings - C
C - match at 5 GeV, the nf dependence of K means that the 4 and 5 - C
C - flavour alphaS*(1+alphaS*K/2*pi) actually meet at about 9 GeV - C
C - instead. - C
C - - C
C ***************************************************************** C
subroutine sudakov_exponent(q2l,q2h,m2,theExponent,isQuark,
$ theAccuracy,fixed_nf)
implicit none
include 'nlegborn.h'
include 'pwhg_st.h'
include 'pwhg_flst.h'
include 'pwhg_rad.h'
include 'pwhg_math.h'
real * 8 q2l,q2h,m2,theExponent
logical isQuark
integer theAccuracy
integer nf
real * 8 bnf,bpnf,K
real * 8 A1,B1,A2,B2,zeta3
real * 8 Lq2l,Lq2h,Lm2
real * 8 pwhg_alphas
external pwhg_alphas
real * 8 aSbar
real * 8 theA1coeff,theA2coeff,theB1coeff,theB2coeff
real * 8 eps
logical isQuarkLine
integer accuracy
real * 8 m2_common
common/sudakov_integral/isQuarkLine,accuracy,m2_common
real * 8 dgauss,sudakov_exponent_integrand
external dgauss,sudakov_exponent_integrand
logical pwhg_isfinite
external pwhg_isfinite
real * 8 debuggingEpsilon,tmp
logical fixed_nf ! Assume the number of active flavours in
! is always st_nlight when evaluating the
! Sudakov.
C - Fractional difference (%) between analytic and numerical Sudakov
C - exponent integration which leads to an error on the screen if
C - it is exceeded. Making it negative (recommended) deactivates
C - this debugging.
debuggingEpsilon = -999d0 ! 1d-9
isQuarkLine = isQuark
accuracy = theAccuracy
if(fixed_nf) then
nf=st_nlight
else
if(q2l.lt.rad_charmthr2) then
nf=3
elseif(q2l.lt.rad_bottomthr2) then
nf=4
else
nf=5
endif
endif
bnf = (11d0*CA-2d0*nf)/12/Pi
bpnf = (153 - 19d0*nf) / Pi / 2 / (33 - 2*nf)
K = (67d0/18-Pi**2/6)*CA-5d0/9*nf
if(isQuarkLine) then
A1 = Cf
A2 = Cf*K
B1 = -3d0/2*Cf
C - Powheg spurious B2:
B2 = -3d0/2*Cf*K
C - Davies and Stirling B2 (divided by two):
C - TO DO: adjust by Ellis-Veseli qT-space prescription for B2.
zeta3 = 1.20206
c$$$ B2 = ( Cf*Cf * (Pi*Pi-3d0/4-12*zeta3)
c$$$ $ + Cf*CA * (11*Pi*Pi/9-193d0/12+6*zeta3)
c$$$ $ + Cf*nf*0.5 * (17d0/3-4d0/9*Pi*Pi)
c$$$ $ ) / 2d0
else
A1 = CA
A2 = CA*K
B1 = -2*Pi*bnf
C - Powheg spurious B2:
B2 = -2*Pi*bnf*K
C - Grazzini and De Florian B2 (divided by two):
C - TO DO: adjust by Ellis-Veseli qT-space prescription for B2.
zeta3 = 1.20206
c$$$ B2 = ( CA*CA * (23d0/6+22*Pi*Pi/9-6*zeta3)
c$$$ $ + 4*Cf*nf*0.5d0
c$$$ $ - CA*nf*0.5*(2d0/3+8d0*Pi*Pi/9)
c$$$ $ - 11d0/2*CA*Cf
c$$$ $ ) / 2d0
endif
Lq2l = Log(q2l/st_lambda5MSB/st_lambda5MSB)
Lq2h = Log(q2h/st_lambda5MSB/st_lambda5MSB)
Lm2 = Log(m2/st_lambda5MSB/st_lambda5MSB)
aSbar = pwhg_alphas(q2l,st_lambda5MSB,-1)/2/Pi
c$$$C - We insist the DDT/Ellis-Veseli Sudakov exponent is positive
c$$$C - everywhere. This effectively defines the upper bound of the
c$$$C - Sudakov exponent integral.
c$$$ if( ( (A1+A2*aSbar)*(Lm2-Lq2l) + (B1+B2*aSbar) ) .lt.0) then
c$$$ theExponent=0d0
c$$$ return
c$$$ endif
if(accuracy.eq.0) then
A2 = 0d0 ! NLL coefficient
B2 = 0d0 ! NNLL coefficient
bpnf = 0d0 ! Reduce 2-loop to 1-loop alpha in the calculation
else if(accuracy.eq.2) then
B2 = 0d0 ! NNLL coefficient
else if(accuracy.eq.3) then
B1 = 0d0 ! NLL coefficient
A2 = 0d0 ! NLL coefficient
B2 = 0d0 ! NNLL coefficient
bpnf = 0d0 ! Reduce 2-loop to 1-loop alpha in the calculation
endif
if(q2l.ge.rad_bottomthr2.or.fixed_nf) then
theA1coeff =
$ ( (Lq2h - Lq2l)
$ - Lm2*Log(Lq2h/Lq2l)
$ )/(2*bnf*Pi)
$ + bpnf*( 2*Lm2*(Lq2h - Lq2l)
$ + 2*Lm2*Lq2h*Log(Lq2l)
$ + Lq2l*Lq2h*Log(Lq2l)**2
$ - 2*Lm2*Lq2l*Log(Lq2h)
$ - Lq2l*Lq2h*Log(Lq2h)**2
$ )/(4*bnf**2*Lq2l*Lq2h*Pi)
theA2coeff =
$ ( Lm2*(Lq2l - Lq2h)
$ - Lq2l*Lq2h*Log(Lq2l)
$ + Lq2l*Lq2h*Log(Lq2h)
$ )/(4d0*bnf**2*Lq2l*Lq2h*Pi**2)
$ + bpnf*( 0.5*(Lq2h-Lq2l)*( Lm2*(Lq2l + Lq2h)
$ - 4*Lq2l*Lq2h
$ )
$ + (Lm2 - 2*Lq2l)*Lq2h**2*Log(Lq2l)
$ - Lq2l**2*(Lm2 - 2*Lq2h)*Log(Lq2h)
$ )/(4d0*bnf**3*Lq2l**2*Lq2h**2*Pi**2)
$ + bpnf**2*( (Lq2l - Lq2h)*( 8*Lm2*( Lq2l**2
$ + Lq2l*Lq2h
$ + Lq2h**2
$ )
$ - 27*Lq2l*Lq2h*(Lq2l + Lq2h)
$ )
$ - 6*(4*Lm2 - 9*Lq2l)*Lq2h**3*Log(Lq2l)
$ + 18*(-2*Lm2 + 3*Lq2l)*Lq2h**3*Log(Lq2l)**2
$ + 6*Lq2l**3*(4*Lm2 - 9*Lq2h)*Log(Lq2h)
$ + 18*Lq2l**3*(2*Lm2 - 3*Lq2h)*Log(Lq2h)**2
$ )/(432.*bnf**4*Lq2l**3*Lq2h**3*Pi**2)
theB1coeff =
$ - Log(Lq2h/Lq2l)/(2.*bnf*Pi)
$ - bpnf*( Lq2l - Lq2h - Lq2h*Log(Lq2l)
$ + Lq2l*Log(Lq2h)
$ )/(2.*bnf**2*Lq2l*Lq2h*Pi)
theB2coeff =
$ - (1/Lq2l - 1/Lq2h)/(4.*bnf**2*Pi**2)
$ - bpnf*( Lq2l**2 - Lq2h**2 - 2*Lq2h**2*Log(Lq2l)
$ + 2*Lq2l**2*Log(Lq2h)
$ )/(8.*bnf**3*Lq2l**2*Lq2h**2*Pi**2)
$ - bpnf**2*( 2*Lq2h**3 - 2*Lq2l**3
$ + 3*Lq2h**3*Log(Lq2l)*(2 + 3*Log(Lq2l))
$ - 3*Lq2l**3*Log(Lq2h)*(2 + 3*Log(Lq2h))
$ )/(108.*bnf**4*Lq2l**3*Lq2h**3*Pi**2)
theExponent = A1*theA1coeff + A2*theA2coeff
$ + B1*theB1coeff + B2*theB2coeff
if(debuggingEpsilon.gt.0d0) then
tmp=dgauss(sudakov_exponent_integrand,Lq2l,Lq2h,eps)
tmp=100d0*abs((tmp-theExponent)/(tmp+theExponent))
if(tmp.gt.debuggingEpsilon) then
write(*,*) ''
write(*,*) 'sudakov_exponent: debug mode'
write(*,*) '============================'
write(*,*) 'sqrt(q2l) (GeV) = ',sqrt(q2l)
write(*,*) 'sqrt(q2h) (GeV) = ',sqrt(q2h)
write(*,*) 'sqrt(m2) (GeV) = ',sqrt(m2)
write(*,*) 'analytic - numerical exponent (%) = ',tmp
write(*,*) 'analytic exponent = ',theExponent
write(*,*) 'A1 term = ',A1*theA1coeff
write(*,*) 'A2 term = ',A1*theA2coeff
write(*,*) 'B1 term = ',A1*theB1coeff
write(*,*) 'B2 term = ',A1*theB2coeff
endif
endif
else
eps = 1d-6
m2_common=m2
theExponent=dgauss(sudakov_exponent_integrand,Lq2l,Lq2h,eps)
endif
c$$$ if(theExponent.gt.0d0) theExponent=0d0;
if(.not.pwhg_isfinite(theExponent)) then
write(6,*) ' '
write(6,*) 'Warning: sudakov_exponent is weird.'
write(6,*) 'theExponent = ',theExponent
write(6,*) 'exp(theExponent) = ',exp(theExponent)
write(6,*) 'q_low = ',sqrt(q2l)
write(6,*) 'q_hi = ',sqrt(q2h)
write(6,*) 'm = ',sqrt(m2)
endif
end
C ***************************************************************** C
C - The integrand in Sudakov exponent times q^2 i.e. we effectiv - C
C - ely integrate in log(q2/lambda^2) (Lq2) to better sample - C
C - the Sudakov peak region. This can be used to replace the - C
C - above analytic result in the region below the b- and c-quark - C
C - thresholds when a variable flavour number should come into - C
C - effect (not taken account of in the analytic result!). We see - C
C - excellent agreement above the b-quark threshold between - C
C - numerical and analytic results (basically exact agreement), - C
C - while below pT=5 GeV we see differences of at most 5%, rising - C
C - to 10% and 50% as pT goes below the charm threshold (~2 GeV). - C
C - I suggest using this routine AS AN ENHANCEMENT of the one - C
C - above, i.e. to call the one here with a low value of EPS for - C
C - events in which pT < b-quark threshold. - C
C ***************************************************************** C
function sudakov_exponent_integrand(Lq2)
implicit none
include 'nlegborn.h'
include 'pwhg_st.h'
include 'pwhg_flst.h'