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|
module parameters
! User input:
! next = number of external gluons (initial+final)
! npoints = number of PS points to compute (negative to keep going)
! npermtry = number of colour flow permutations to include per PS
! point (needs to be greater than 2 and smaller than the
! maximal, (next-1)!)
! alphas = the value of the strong coupling
! sqrtshat = collision energy
integer, parameter :: next=5
integer, parameter :: npermtry=24
integer*8, parameter :: npoints=10000
real*8, parameter :: alphas=0.12d0
real*8, parameter :: sqrtshat=1000d0
! constants:
integer, parameter :: ninc=2,nfin=next-ninc
real*8, parameter :: pi=3.14159265358979323846d0
! External momenta (generated by 'call get_momenta()'):
real*8, dimension(0:3,next) :: p
! helicity (polarisation) and permutation used for amplitude
! computation:
integer, dimension(next) :: ip
! final or initial state particle:
integer, dimension(next) :: ifinal
data ifinal(1:next) / -1,-1,nfin*1 /
complex*16, dimension(npermtry,0:maskr(next)) :: amp
end module parameters
program matrix
use parameters
implicit none
integer :: ih,iperm,jperm,i,ib1,ib2,ib,ih1,ih2,towrite=1
integer*8 :: iden,ipoint=0,inonzero=0,nperm
integer, dimension(next,npermtry) :: ips
real*8, dimension(next) :: pmass=0d0
real*8 :: amp2,integral=0d0,uncertainty=0d0,flux,matrix_element,wgt, &
perm_vol
real*8, parameter :: conv=389379660d0
! number of colour permutations
nperm=factorial(next-1)
! colour, polarisation incoming gluons: 8*8, 2*2
! identical final state particle factor: nfin!
iden=8*8*2*2 * factorial(nfin)
! flux factor
flux=1d0/(2d0*sqrtshat**2)
do
if (npoints.gt.0 .and. inonzero.eq.npoints) exit
ipoint=ipoint+1
call get_momenta(sqrtshat,pmass,wgt)
do i=1,2
p(0,i)=-p(0,i) ! treat all momenta as outgoing
enddo
! weight from Ramdo is logarithmic and does not
! include 1/(2pi)^(3n-4). Compensate...
wgt=exp(wgt)/((2*pi)**(3*nfin-4))
! Check if PS point passes generation cuts
if (fail_cuts()) cycle
inonzero=inonzero+1
amp2=0d0
! Determine which npermtry colour ordered amplitudes to
! compute. Do this randomly.
do iperm=1,npermtry
if (npermtry.eq.nperm) then
! Summing over all permutations.
call set_iperm(iperm,ips(1,iperm))
elseif (npermtry.lt.nperm) then
if (iperm.eq.1) then
call get_iperm_rn(ips(1,iperm))
else
111 call get_iperm_rn(ips(1,iperm))
! second (or third, etc.) should always be different
! from previous ones. Simply regenerate. This is not
! optimal, but sufficient if npermtry<<nperm, which
! should be the case for large number of gluons
do i=1,iperm-1
if (all(ips(1:next,iperm).eq.ips(1:next,i))) goto 111
enddo
endif
else
write (*,*) 'npermtry should be smaller than',nperm
stop 1
endif
enddo
do iperm=1,npermtry
ip(1:next)=ips(1:next,iperm)
! Compute the matrix colour ordered amplitude, first computing
! the wavefunctions correspoding to the sum of the first
! next-1 particles, dotting that into the gluon polarisation
! for the next particle.
call compute_amplitude(iperm)
enddo
!!$ do iperm=1,1
!!$ write (*,*) ips(1:next,iperm) ,'=>',iperm
!!$ do i=0,maskr(next)
!!$ write (*,*) i,amp(iperm,i)
!!$ enddo
!!$ enddo
! Loop over all colour ordered amplitudes computed, and multiply
! by the colour factor to get the matrix element squared. If
! npermtry<nperm, also include the weight from the MC-ing over
! the colour-ordered amplitudes. The colour factor for the
! off-diagonal contributions includes a factor 2, because only
! upper triangle of colour matrix is included.
do ih=0,maskr(next)
do iperm=1,npermtry
! do some bit moving to align the helicity with the
! permutation iperm
ih1=0
do ib=1,next
call mvbits(ih,ips(ib,iperm)-1,1,ih1,ib-1)
enddo
do jperm=iperm,npermtry
! do some bit moving to align the helicity with the
! permutation jperm
ih2=0
do ib=1,next
call mvbits(ih,ips(ib,jperm)-1,1,ih2,ib-1)
enddo
! Compute the weight from the MC-ing over permutations
if (iperm.eq.jperm) then
perm_vol=dble(nperm)/dble(npermtry)
else
perm_vol=dble(nperm*(nperm-1))/dble(npermtry*(npermtry-1))
endif
! Compute the amplitude squared
amp2=amp2+dble(amp(iperm,ih1)*dconjg(amp(jperm,ih2)))* &
color_factor_list(ips(1,iperm),ips(1,jperm))*perm_vol
enddo
enddo
enddo
! Include the strong coupling and the identical particle factor.
amp2=amp2*(4*pi*alphas)**nfin/dble(iden)
! Include the flux factor, the phase-space weight, and the
! conversion from GeV->pb.
matrix_element=amp2*flux*wgt*conv
! Add the current matrix element to the total cross section
integral=integral+matrix_element
uncertainty=uncertainty+matrix_element**2
if (inonzero.gt.10*towrite .and. towrite.lt.10000) towrite=towrite*10
if (mod(inonzero,towrite).eq.0) then
write (88,*) ipoint,inonzero,matrix_element,integral/dble(ipoint), &
sqrt(abs(uncertainty/dble(ipoint)-(integral/dble(ipoint))**2)/dble(ipoint)), &
100d0*sqrt(abs(uncertainty/dble(ipoint)-(integral/dble(ipoint))**2)/dble(ipoint))/(integral/dble(ipoint))
write (*,*) ipoint,inonzero,matrix_element,integral/dble(ipoint), &
sqrt(abs(uncertainty/dble(ipoint)-(integral/dble(ipoint))**2)/dble(ipoint)), &
100d0*sqrt(abs(uncertainty/dble(ipoint)-(integral/dble(ipoint))**2)/dble(ipoint))/(integral/dble(ipoint))
call flush(88)
endif
enddo
! Print the final result to the screen
integral=integral/dble(npoints)
uncertainty=uncertainty/dble(npoints)
uncertainty=sqrt(abs(uncertainty-integral**2)/dble(npoints))
if (npoints.gt.10) write (*,*) 'sigma:',integral,'+/-',uncertainty,'(',100d0*uncertainty/integral,'%)'
contains
logical function fail_cuts()
! Cuts on the phase-space point.
use parameters
implicit none
integer i,j
fail_cuts=.false.
do i=3,next
if (pt(p(0,i)).lt.60d0) then
fail_cuts=.true.
return
endif
if (abs(eta(p(0,i))).gt.2d0) then
fail_cuts=.true.
return
endif
if (i.ne.next) then
do j=i+1,next
if (DeltaR(p(0,i),p(0,j)).lt.0.7d0) then
fail_cuts=.true.
return
endif
enddo
endif
enddo
end function fail_cuts
real*8 function pt(p)
! transverse momentum of 'p'
implicit none
real*8, dimension(0:3) :: p
pt=sqrt(p(1)**2+p(2)**2)
end function pt
real*8 function eta(p)
! pseudo-rapidity of 'p'
implicit none
real*8, dimension(0:3) :: p
real*8 :: theta
theta=acos(p(3)/sqrt(p(1)**2+p(2)**2+p(3)**2))
eta=-log(dtan(theta/2d0))
end function eta
real*8 function delta_phi(p1,p2)
! azimuthal difference of 'p1' and 'p2'
implicit none
real*8, dimension(0:3) :: p1,p2
real*8 :: denom
denom=pt(p1)*pt(p2)
delta_phi=acos((p1(1)*p2(1)+p1(2)*p2(2))/denom)
end function delta_phi
real*8 function deltaR(p1,p2)
! Distance (Delta-R) between 'p1' and 'p2'
implicit none
real*8, dimension(0:3) :: p1,p2
deltaR=sqrt(delta_phi(p1,p2)**2+(eta(p1)-eta(p2))**2)
end function deltaR
real*8 function color_factor(iperm,jperm)
! Compute colour factor for permutation numbers 'iperm' and
! 'jperm'
use parameters
implicit none
integer :: iperm,jperm
integer, dimension(next) :: iper,jper,ips
call set_iperm(iperm,ips)
iper(1:next)=ips(:)
call set_iperm(jperm,ips)
jper(1:next)=ips(:)
color_factor=color_factor_list(iper,jper)
end function color_factor
real*8 function color_factor_list(iper,jper)
! Given two permutations (iper and jper) compute corresponding
! colour factor. If the two configurations are identical, include
! an additional factor 2.
use parameters
implicit none
integer :: i,j,k,id0,jd0,id1,jd1,colfac,nperms
integer, dimension(next) :: iper,jper
integer, dimension(0:1,next) :: idel,jdel
logical, dimension(next) :: lper
! add the factor 2 if need be
if (all(iper.eq.jper)) then
colfac=1
else
colfac=2
endif
! Determine the delta's to contract
do i=1,next
! delta's of the amplitude
idel(0,i)=iper(i)
jdel(0,i)=iper(mod(i,next)+1)
! delta's of the conjugate amplitude
idel(1,i)=jper(i)
jdel(1,i)=jper(mod(i,next)+1)
enddo
! No closed string of delta's found so far: set all the labels to
! false
lper(1:next)=.false.
! Loop over the delta's.
do k=1,next
! If this delta already belonged to a previously closed
! string. Skip it
if (lper(k)) cycle
! The two labels of the delta in the amplitude
id0=idel(0,k)
jd0=jdel(0,k)
! Go searching to see to which delta's it is connected
do
do j=1,next
if (jdel(1,j).eq.jd0) then
! Found the delta in the conjugate amplitude to which
! the delta in the amplitude is connected to
id1=idel(1,j)
exit
endif
enddo
if (id0.eq.id1) then
! The delta in the conjugate amplitude, links back to the
! starting delta. Hence, one closed string found: add a
! factor 3 to the colour factor and start with the next
! delta in the amplitude.
colfac=colfac*3
exit
endif
do i=1,next
if (idel(0,i).eq.id1) then
! Found delta in the amplitude to which the delta in
! the conjugate amplitude connects to. Hence, it's
! part of a closed string, so set corresponding lper
! to true.
jd0=jdel(0,i)
lper(i)=.true.
exit
endif
enddo
enddo
enddo
color_factor_list=dble(colfac)
end function color_factor_list
integer*8 function factorial(ifact)
! computes the factorial of 'ifact'. Uses long integers to deal
! with large numbers
implicit none
integer, value :: ifact
factorial=1
do while (ifact.gt.1)
factorial=factorial*ifact
ifact=ifact-1
enddo
end function factorial
end program matrix
subroutine set_iperm(iperm,ipss)
! Given permutation number 'iperm', fills the corresponding
! permutation. This is slow when the number of permutations is
! large, and easily becomes the most time-consuming part of the
! calculation if used too often.
use parameters
implicit none
integer :: iperm,i
integer, dimension(next-1) :: ips
integer, dimension(next) :: ipss
integer :: iflag
do i=1,next-1
ips(i)=i
enddo
do i=1,iperm-1
call ipnext(ips,next-1,iflag)
enddo
ipss(1:next-1)=ips(:)
ipss(next)=next
end subroutine set_iperm
subroutine get_iperm_rn(ips)
! Computes a random permutation 'ips'.
use parameters
implicit none
integer :: i,j,k,itemp
integer,dimension(1:next-1) :: ileft
integer,dimension(next) :: ips
real*8, external :: rn
! Create a pool of numbers to pick from
do i=1,next
ileft(i)=i
enddo
! Randomly remove a number from the pool 'ileft' and add it to the
! 'ips' permutation list.
do i=1,next-1
k=0
itemp=int(rn(1)*(next-i))+1
do j=1,itemp
k=k+1
do while (ileft(k).eq.0)
k=k+1
enddo
enddo
ips(i)=ileft(k)
ileft(k)=0
enddo
! Last number in permutation is always 'next': this removes all cyclic
! permutations.
ips(next)=next
end subroutine get_iperm_rn
subroutine compute_amplitude(iperm)
use parameters
implicit none
integer,parameter :: mw=maskr(next-1),mm=maskr(next-1)+1,zero=0
integer :: imom,i,ih,level,j,isplit,jsplit,iwf,nnext,iperm, &
a3,b3,a4,b4,c4,ia,ib,ic,ext,nh,icount
complex*16, dimension(4,mm,mm) :: wf
integer,dimension(mm) :: nwf
real*8, dimension(0:3,mm) :: pp
complex*16 :: propagator,contribution
complex*16,dimension(4) :: wfout
complex*16,parameter :: ci=(0d0,1d0)
complex*16,dimension(4,0:1) :: wffinal
nwf=0
nnext=next-1
do i=1,next
ext=0
ext=ibset(ext,i-1)
pp(0:3,ext)=p(0:3,ip(i))
do ih=0,1
nwf(ext)=nwf(ext)+1
call v_ext(pp(0,ext),ih,1,wf(1,nwf(ext),ext))
enddo
enddo
do level=1,nnext-1
do i=1,nnext-level
ext=0
nh=1
do j=0,level
ext=ibset(ext,i+j-1)
nh=nh*2
enddo
nwf(ext)=nh
wf(1:4,1:nwf(ext),ext)=(0d0,0d0)
pp(0:3,ext)=0d0
do j=1,nnext
if (btest(ext,j-1)) pp(0:3,ext)=pp(0:3,ext)+pp(0:3,ibset(zero,j-1))
enddo
do isplit=1+trailz(ext),popcnt(ext)-1+trailz(ext)
a3=merge_bits(ext,zero,maskr(isplit))
b3=merge_bits(zero,ext,maskr(isplit))
do ib=1,nwf(b3)
do ia=1,nwf(a3)
call gluon3(wf(1,ia,a3),pp(0,a3), &
wf(1,ib,b3),pp(0,b3), &
wfout)
icount=(ib-1)*nwf(a3)+ia
wf(1:4,icount,ext)=wf(1:4,icount,ext)+wfout(1:4)
enddo
enddo
do jsplit=isplit+1,popcnt(ext)-1+trailz(ext)
a4=a3
b4=merge_bits(b3,zero,maskr(jsplit))
c4=merge_bits(zero,b3,maskr(jsplit))
do ic=1,nwf(c4)
do ib=1,nwf(b4)
do ia=1,nwf(a4)
call gluon4(wf(1,ia,a4),wf(1,ib,b4),wf(1,ic,c4),wfout)
icount=(ic-1)*nwf(a4)*nwf(b4)+(ib-1)*nwf(a4)+ia
wf(1:4,icount,ext)=wf(1:4,icount,ext)+wfout(1:4)
enddo
enddo
enddo
enddo
enddo
if (level.ne.nnext-1) then
propagator=-ci/(pp(0,ext)**2-pp(1,ext)**2-pp(2,ext)**2-pp(3,ext)**2)
wf(1:4,1:nwf(ext),ext)=wf(1:4,1:nwf(ext),ext)*propagator
endif
enddo
enddo
amp(iperm,:)=0d0
do iwf=1,nwf(ext)
do i=1,2
contribution= &
(wf(1,iwf,ext)*wf(1,i,ibset(zero,nnext))- &
wf(2,iwf,ext)*wf(2,i,ibset(zero,nnext))- &
wf(3,iwf,ext)*wf(3,i,ibset(zero,nnext))- &
wf(4,iwf,ext)*wf(4,i,ibset(zero,nnext)))
ih=iwf-1
if (i.eq.2) ih=ibset(ih,next-1)
amp(iperm,ih)=amp(iperm,ih)+contribution
enddo
enddo
end subroutine compute_amplitude
subroutine v_ext(p,ihel,ifinal,wf)
! External gluon wavefunction. From HELAS.
implicit none
integer :: ihel,ifinal
real*8, dimension(0:3) :: p
complex*16, dimension(4) :: wf
real*8, parameter :: rzero=0d0,rhalf=0.5d0,sqh=sqrt(0.5d0)
real*8 :: hel,pt2,pp,pt,pzpt
hel = dble(2*ihel-1)
pt2 = p(1)**2+p(2)**2
pp = min(p(0),sqrt(pt2+p(3)**2))
pt = min(pp,sqrt(pt2))
pp = p(0)
pt = sqrt(p(1)**2+p(2)**2)
wf(1) = dcmplx( rZero )
wf(4) = dcmplx( hel*pt/pp*sqh )
if ( pt.ne.rZero ) then
pzpt = p(3)/(pp*pt)*sqh*hel
wf(2) = dcmplx( -p(1)*pzpt , -ifinal*p(2)/pt*sqh )
wf(3) = dcmplx( -p(2)*pzpt , ifinal*p(1)/pt*sqh )
else
wf(2) = dcmplx( -hel*sqh )
wf(3) = dcmplx( rZero , ifinal*sign(sqh,p(3)) )
endif
end subroutine v_ext
subroutine gluon3(wf1,pwf1,wf2,pwf2,wf)
! Colour-ordered three gluon interaction
implicit none
complex*16,dimension(4) :: wf1,wf2,wf
real*8,dimension(0:3) :: pwf1,pwf2
complex*16, parameter :: prefact=(0d0,1d0)/sqrt(2d0)
complex*16 :: TMP1,TMP2,TMP3
TMP1 = (wf1(1)*wf2(1)-wf1(2)*wf2(2)-wf1(3)*wf2(3)-wf1(4)*wf2(4))
TMP2 = (wf1(1)*pwf2(0)-wf1(2)*pwf2(1)-wf1(3)*pwf2(2)-wf1(4)*pwf2(3))
TMP3 = (wf2(1)*pwf1(0)-wf2(2)*pwf1(1)-wf2(3)*pwf1(2)-wf2(4)*pwf1(3))
wf(1) = prefact*(TMP1*(pwf1(0)-pwf2(0))+2d0*TMP2*wf2(1)-2d0*TMP3*wf1(1))
wf(2) = prefact*(TMP1*(pwf1(1)-pwf2(1))+2d0*TMP2*wf2(2)-2d0*TMP3*wf1(2))
wf(3) = prefact*(TMP1*(pwf1(2)-pwf2(2))+2d0*TMP2*wf2(3)-2d0*TMP3*wf1(3))
wf(4) = prefact*(TMP1*(pwf1(3)-pwf2(3))+2d0*TMP2*wf2(4)-2d0*TMP3*wf1(4))
end subroutine gluon3
subroutine gluon4(wf1,wf2,wf3,wf)
! Colour-ordered four gluon interaction
implicit none
complex*16,dimension(4) :: wf1,wf2,wf3,wf
complex*16, parameter :: prefact=(0d0,0.5d0)
complex*16 :: TMP1,TMP2,TMP3
TMP1 = (wf1(1)*wf2(1)-wf1(2)*wf2(2)-wf1(3)*wf2(3)-wf1(4)*wf2(4))
TMP2 = (wf1(1)*wf3(1)-wf1(2)*wf3(2)-wf1(3)*wf3(3)-wf1(4)*wf3(4))
TMP3 = (wf2(1)*wf3(1)-wf2(2)*wf3(2)-wf2(3)*wf3(3)-wf2(4)*wf3(4))
wf(1) = prefact*(2d0*wf2(1)*TMP2-wf1(1)*TMP3-wf3(1)*TMP1)
wf(2) = prefact*(2d0*wf2(2)*TMP2-wf1(2)*TMP3-wf3(2)*TMP1)
wf(3) = prefact*(2d0*wf2(3)*TMP2-wf1(3)*TMP3-wf3(3)*TMP1)
wf(4) = prefact*(2d0*wf2(4)*TMP2-wf1(4)*TMP3-wf3(4)*TMP1)
end subroutine gluon4
subroutine ipnext(ia, n, flag)
! Compute next permutation starting from the list 'ia' (of length
! 'n').
implicit none
integer ::n,flag,i,j,itemp
integer, dimension(*) :: ia
i=n-1
do while (i.ge.1 .and. ia(i).gt.ia(i+1))
i=i-1
enddo
if (i.lt.1) then
flag = -1
return
endif
j = n
do while (ia(i).gt.ia(j))
j = j-1
enddo
itemp = ia(i)
ia(i) = ia(j)
ia(j) = itemp
i = i+1
j = n
do while (i.lt.j)
itemp = ia(i)
ia(i) = ia(j)
ia(j) = itemp
i = i+1
j = j-1
enddo
flag = 1
return
end subroutine ipnext
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