def wddpp(E, pvec, lp, mp, kvec, l, m):
if l == m == lp == mp == 0:
k = norm(kvec)
p = norm(pvec)
return 1 / 4 * ((E - omega(k)) * (E - omega(p)) - mydot(kvec, pvec))**2
else:
return 0
def wss(E, pvec, lp, mp, kvec, l, m):
if l == m == lp == mp == 0:
k = LA.norm(kvec)
p = LA.norm(pvec)
return (omega(k) * omega(p) - np.dot(kvec, pvec))**2
else:
return 0
def omega2sum(E, pvec, lp, mp, kvec, l, m, Ytype='r'):
k = LA.norm(kvec)
p = LA.norm(pvec)
if (lp == 0 and mp == 0):
if (l == 0 and m == 0):
S1 = omega(k)**2 + (
(E - omega(k))**2) / 2 + 2 / 3 * (k * qst(E, k) / E2k(E, k))**2
elif l == 2:
#S1 = 4/15*(qst(E,k) / E2k(E,k))**2 * conj(y2(kvec,m,Ytype))
S1 = 4 / 15 * (1 / E2k(E, k))**2 * conj(y2(kvec, m,
Ytype)) # TB, no q
else:
S1 = 0
else:
S1 = 0
if (l == 0 and m == 0):
if (lp == 0 and mp == 0):
S2 = omega(p)**2 + (
(E - omega(p))**2) / 2 + 2 / 3 * (p * qst(E, p) / E2k(E, p))**2
elif lp == 2:
#S2 = 4/15*(qst(E,p) / E2k(E,p))**2 * y2(pvec,mp,Ytype)
S2 = 4 / 15 * (1 / E2k(E, p))**2 * y2(pvec, mp, Ytype) # TB, no q
else:
S2 = 0
else:
S2 = 0
if np.imag(S1 + S2) > 10**-8:
print('Error: imaginary part in omega2sum')
return np.real(S1 + S2)
def K3cubicA(E, pvec, lp, mp, kvec, l, m):
p = norm(pvec)
k = norm(kvec)
Ep = defns.E2k(E, p)
Ek = defns.E2k(E, k)
wp = defns.omega(p)
wk = defns.omega(k)
ap = defns.qst(E, p)
a = defns.qst(E, k)
pterm = E * wp - 3
kterm = E * wk - 3
if lp == l == mp == m == 0:
D3p = Ep**2 - 4
D3 = Ek**2 - 4
out = D3p**3 + D3**3 + 2 * (pterm**3 + kterm**3) + 8 * E**2 * (
pterm * (p * ap / Ep)**2 + kterm * (k * a / Ek)**2)
elif lp == 0 and l == 2:
#out = 16/5 * kterm * (E*a/Ek)**2 * defns.y2real(kvec,m)
out = 16 / 5 * kterm * (E / Ek)**2 * defns.y2real(
kvec, m) # removed a=qk* factor (no q)
elif lp == 2 and l == 0:
#out = 16/5 * pterm * (E*ap/Ep)**2 * defns.y2real(pvec,m)
out = 16 / 5 * pterm * (E / Ep)**2 * defns.y2real(
pvec, mp) # removed ap=qp* factor (no q)
else:
out = 0
#out *= ap**lp * a**l # q factors are NOT included here (no q)
if out.imag > 1e-15:
print('Error: imaginary part in K3cubicA')
return out.real
def p12stark(E, pvec, kvec): # (see notes)
k = LA.norm(kvec)
p = LA.norm(pvec)
beta_k = k / (E - omega(k))
gam_k = sqrt(1 / (1 - beta_k**2))
khat = [ki / k for ki in kvec] if k != 0 else kvec
return [
-pvec[i] + (gam_k * beta_k * (E - omega(p)) -
(gam_k - 1) * np.dot(pvec, khat)) * khat[i]
for i in range(0, 3)
]
def T(E,pvec,kvec): # Cartesian basis
k=LA.norm(kvec); p=LA.norm(pvec)
beta_k = k/(E-omega(k)); gam_k = sqrt(1/(1-beta_k**2))
beta_p = p/(E-omega(p)); gam_p = sqrt(1/(1-beta_p**2))
khat = [ki/k for ki in kvec] if k!=0 else kvec
phat = [pi/p for pi in pvec] if p!=0 else pvec
t = np.zeros((3,3))
for i in range(0,3):
t[i][i] = -1
for j in range(0,3):
t[i][j] += khat[i]*phat[j] * ( gam_k*beta_k*gam_p*beta_p - np.dot(khat,phat)*(gam_k-1)*(gam_p-1) ) - khat[i]*khat[j]*(gam_k-1) - phat[i]*phat[j]*(gam_p-1)
return t
def K2inv(E,kvec,l,m,a0,r0,P0,a2):
k = LA.norm(kvec)
omk = defns.omega(k)
E2star = defns.E2k(E,k)
qk = defns.qst(E,k)
h = sums.hh(E,k)
if h==0:
print('Error: invalid kvec in K2inv')
if l==m==0:
out = 1/(32*pi*omk*E2star) * ( -1/a0 + r0*qk**2/2 + P0*r0**3*qk**4 + abs(qk)*(1-h))
elif l==2 and -2<=m<=2:
# out = 1/(32*pi*omk*E2star*qk**4) * ( -1/a2**5 + abs(qk)**5*(1-h) )
out = 1/(32*pi*omk*E2star) * ( -1/a2**5 + abs(qk)**5*(1-h) ) # TB, no q
else:
return 0
if out.imag > 1e-15:
print('Error in K2inv: imaginary part in output')
else:
out = out.real
return out
def wds(E,pvec,lp,mp,kvec,l,m,Ytype='r'):
if lp==mp==0:
k=LA.norm(kvec); p=LA.norm(pvec)
psk = pstark(E,pvec,kvec)
if l==m==0:
return 1/2*(E*omega(p)-sqrt(wss(E,pvec,0,0,kvec,0,0)))**2 + 2/3*(qst(E,k)*LA.norm(psk))**2
elif l==2:
#return 4/15 * qst(E,k)**2 * conj(y2(psk,m,Ytype))
return 4/15 * conj(y2(psk,m,Ytype)) # TB, no q
else:
return 0
def b2(E,L,nnp,nnk): nnp = np.array(nnp); nnk = np.array(nnk) p = 2*pi/L*LA.norm(nnp); k = 2*pi/L*LA.norm(nnk) omp = defns.omega(p); omk = defns.omega(k) return (E-omp-omk)**2 - (2*pi/L*LA.norm(nnp+nnk))**2
def K3cubicB(E, pvec, lp, mp, kvec, l, m):
#pvec=np.array(pvec); kvec=np.array(kvec)
p = norm(pvec)
k = norm(kvec)
phat = np.array([x / p for x in pvec]) if p != 0 else pvec
khat = np.array([x / k for x in kvec]) if k != 0 else kvec
Ep = defns.E2k(E, p)
Ek = defns.E2k(E, k)
wp = defns.omega(p)
wk = defns.omega(k)
ap = defns.qst(E, p)
a = defns.qst(E, k)
# Bp = p/(E-wp); Bk = k/(E-wk)
# gp = 1/sqrt(1-Bp**2); gk = 1/sqrt(1-Bk**2)
k_p = np.dot(pvec, kvec)
# Denote primed (outgoing) momenta with t
# Denote p_{12}^+ by p12, and p_{12}^- by m12; add t for primed versions
pstark_vec = K3_defns.pstark(
E, pvec, kvec) #gk*Bk*wp*khat + (gk-1)*p*khat_phat*khat + pvec
kstarp_vec = K3_defns.pstark(
E, kvec, pvec) #gp*Bp*wk*phat + (gp-1)*k*khat_phat*phat + kvec
p12t_stark_vec = K3_defns.p12stark(
E, pvec, kvec) #gk*Bk*(E-wp)*khat - (gk-1)*p*khat_phat*khat - pvec
p12_starp_vec = K3_defns.p12stark(
E, kvec, pvec) #gp*Bp*(E-wk)*phat - (gp-1)*k*khat_phat*phat - kvec
pstark = norm(pstark_vec)
kstarp = norm(kstarp_vec)
p12t_stark = norm(p12t_stark_vec)
p12_starp = norm(p12_starp_vec)
p3_p3t = wk * wp - k_p
p12_p12t = (E - wk) * (E - wp) - k_p
p12_p3t = (E - wk) * wp + k_p
p3_p12t = (E - wp) * wk + k_p
t = K3_defns.T(E, pvec, kvec)
u = np.outer(p12t_stark_vec, p12_starp_vec) # tensor needed for full U
if lp == l == mp == m == 0:
out = p3_p3t**3 + 1/4*( (p12_p3t)**3 + (p3_p12t)**3 ) + p12_p3t*pstark**2*a**2 + p3_p12t*kstarp**2*ap**2 \
+ 1/16*p12_p12t**3 + 1/4*p12_p12t*( p12t_stark**2*a**2 + p12_starp**2*ap**2 ) \
+ 1/3*p12_p12t*a**2*ap**2*Sspher(t,0,0,0,0) + 2/3*a**2*ap**2*Uspher(t,u,0,0,0,0)
elif lp == mp == 0 and l == 2:
out = 2/5*p12_p3t*y2real(pstark_vec,m) + 1/10*p12_p12t*y2real(p12t_stark_vec,m) \
+ p12_p12t*ap**2*Sspher(t,0,0,2,m) + 2*ap**2*Uspher(t,u,0,0,2,m) # removed a^2=qk*^2 factor (no q)
# factor of 1/sqrt(15) that Steve has is inside Sspher and Uspher
elif lp == 2 and l == m == 0:
out = 2/5*p3_p12t*y2real(kstarp_vec,mp) + 1/10*p12_p12t*y2real(p12_starp_vec,mp) \
+ p12_p12t*a**2*Sspher(t,2,mp,0,0) + 2*a**2*Uspher(t,u,2,mp,0,0) # removed ap^2=qp*^2 factor (no q)
# factor of 1/sqrt(15) that Steve has is inside Sspher and Uspher
elif lp == l == 2:
out = 3 * p12_p12t * Sspher(t, 2, mp, 2, m) + 6 * Uspher(
t, u, 2, mp, 2, m) # removed (ap*a)^2=(qk*qp*)^2 factor (no q)
# factor of 1/15 that Steve has is inside Sspher and Uspher
else:
out = 0
#out *= ap**lp * a**l # q factors are NOT included here (no q)
if out.imag > 1e-15:
print('Error: imaginary part in K3cubicB')
return out.real
