def jac(mg2, v2, mn2, temp, vb2, lam, N):
""" Jacobian for the mg2 gap equation rootfinder """
eps = 1e-12
if mg2 <= 0:
mg2 = eps # regulator for mg=0 case
mg = sc.sqrt(mg2)
return np.array([
1.0 - (lam / (6. * mg)) *
(thermal_tadpole(mg + eps, temp, MU) -
thermal_tadpole(mg, temp, MU)) / eps,
])
def jac(mn2, v2, temp, vb2, lam, N):
""" Jacobian for the mn2 gap equation rootfinder """
eps = 1e-12
if mn2 <= 0:
mn2 = eps # regulator for mg=0 case
mn = sc.sqrt(mn2)
return np.array([
1.0 - (lam * (N + 2.) / (12. * mn)) *
(thermal_tadpole(mn + eps, temp, MU) -
thermal_tadpole(mn, temp, MU)) / eps,
])
def si_3pi_hartree_rhs(v2,
mg2,
mn2,
temp,
vb2,
lam,
N,
symmetric_branch=False):
"""
Right hand side of the equations of motion for the 2PIEA at the
Hartree-Fock level with 3PI symmetry improvement. More precisely,
the vertex Ward identity is enforced in place of the Higgs equation
of motion.
Arguments:
v2 = the Higgs vev squared
mg2 = the Goldstone mass squared
mn2 = the Higgs mass squared
temp = the temperature
vb2 = the Higgs vev squared at zero temperature
lam = the quartic coupling constant at zero temperature
N = the number of field components (i.e., O(N) symmetry)
The units do not matter as long as they are all the same, e.g. GeV.
Returns:
A tuple (v2, mg2, mn2) of updated estimates, suitable for iteration.
"""
assert v2 >= 0
assert mg2 >= 0
assert mn2 >= 0
assert temp >= 0
assert vb2 >= 0
assert lam > 0
assert N >= 1
assert np.floor(N) == N, "N must be an integer"
if temp > 0:
tg = thermal_tadpole(sc.sqrt(mg2), temp, MU)
tn = thermal_tadpole(sc.sqrt(mn2), temp, MU)
else:
tg = 0
tn = 0
if not symmetric_branch:
# broken symmetry branch
return (vb2 - ((N + 1.) * temp**2.) / 12. - tn, 0., lam * v2 / 3.)
else:
# symmetric branch
return (0., (lam / 6.) * (-vb2 + (N + 1) * tg + tn),
(lam / 6.) * (-vb2 + (N + 1) * tg + tn))
def si_2pi_hartree_solution(temp,
vb2,
lam,
N,
maxsteps=20,
tol=1e-3,
update_weight=0.2):
"""
Solution of the equations of motion for the 2PIEA with
symmetry improvement. This is the same procedure used in:
A. Pilaftsis and D. Teresi, Symmetry-Improved CJT Effective Action.
Nucl. Phys. B 874, 594 (2013) http://arxiv.org/abs/1305.3221.
Arguments:
temp = the temperature
vb2 = the Higgs vev squared at zero temperature
lam = the quartic coupling constant at zero temperature
N = the number of field components (i.e., O(N) symmetry)
maxsteps = maximum number of iteration steps (defaults to 20)
tol = the absolute tolerance goal for the solution (component-wise)
symmetric_branch = whether or not to solve for the symmetric or broken
symmetry phase solution (defaults to False)
update_weight = the weighting given to new updates in the iterative
procedure (defaults to 0.2, must be between 0 and 1)
The units do not matter as long as they are all the same, e.g. GeV.
Returns:
A tuple (v2, mg2, mn2) of
v2 = the Higgs vev squared
mg2 = the Goldstone mass squared
mn2 = the Higgs mass squared
"""
assert temp >= 0
assert vb2 >= 0
assert lam > 0
assert N >= 1
assert np.floor(N) == N, "N must be an integer"
assert maxsteps > 1
assert np.floor(maxsteps) == maxsteps, "maxsteps must be an integer"
crit_T2 = 12. * vb2 / (N + 2.)
mn2 = (lam * vb2 / 3.) * (1. - temp**2. / crit_T2)
if temp == 0:
return (vb2, 0., mn2)
elif 0 < temp**2. <= crit_T2:
# analytical solution exists!
tn = thermal_tadpole(sc.sqrt(mn2), temp, MU)
return ((3. * mn2 / lam) + ((temp**2.) / 12.) - tn, 0., mn2)
else:
# symmetric phase - same solution as the unimproved case
return no_si_solution(temp,
vb2,
lam,
N,
maxsteps=maxsteps,
tol=tol,
symmetric_branch=True,
update_weight=update_weight)
def no_si_rhs(v2, mg2, mn2, temp, vb2, lam, N, symmetric_branch=False):
"""
TODO: Currently breaks near the critical temperature!
Right hand side of the equations of motion for the 2PIEA without symmetry
improvement.
Arguments:
v2 = the Higgs vev squared
mg2 = the Goldstone mass squared
mn2 = the Higgs mass squared
temp = the temperature
vb2 = the Higgs vev squared at zero temperature
lam = the quartic coupling constant at zero temperature
N = the number of field components (i.e., O(N) symmetry)
symmetric_branch = Boolean indicating whether to compute for the symmetric
or broken symmetry phase. Defaults to False. Note that this routine
does not do the checking to make sure that the specified phase
exists for the given parameter values.
The units do not matter as long as they are all the same, e.g. GeV.
Returns:
A tuple (v2, mg2, mn2) of updated estimates, suitable for iteration.
"""
assert v2 >= 0
assert mg2 >= 0
assert mn2 >= 0
assert temp >= 0
assert vb2 >= 0
assert lam > 0
assert N >= 1
assert np.floor(N) == N, "N must be an integer"
if temp > 0:
tg = thermal_tadpole(sc.sqrt(mg2), temp, MU)
tn = thermal_tadpole(sc.sqrt(mn2), temp, MU)
else:
tg = 0
tn = 0
if symmetric_branch:
return (0., (lam / 6.) * (-vb2 + (N + 1) * tg + tn),
(lam / 6.) * (-vb2 + (N + 1) * tg + tn)) # mn == mg in sym.
else:
return (vb2 - (N - 1) * tg - 3. * tn, lam * (tg - tn) / 3.,
lam * v2 / 3.)