def fermion_propagator():
""" Calculate expression for first-order corrected fermion propagator. """
latex = Latex()
# 1. Write out amplitude TODO include regulator terms
m = []
ind = "\sigma_1"
sum_terms = [Sub("p", ind), C("m")]
m.append(Frac(Prod([Sum(sum_terms), Gamma(ind)]), Term("p^2 - m^2 + i\\epsilon")))
m.append(Integral("k", norm=True))
m.append(I())
m.append(C("e"))
m.append(Term("\\gamma^\\mu"))
ind = "\sigma_2"
sum_terms = [Sub("p", ind), Sub("(-k)", ind), C("m")]
m.append(Frac(Prod([Sum(sum_terms), Gamma(ind)]), Term("(p - k)^2 - m^2 + i\\epsilon")))
m.append(I())
m.append(C("e"))
m.append(Term("\\gamma^\\nu"))
m.append(Term("D_{\\mu \\nu}(k)"))
ind = "\sigma_3"
sum_terms = [Sub("p", ind), C("m")]
m.append(Frac(Prod([Sum(sum_terms), Gamma(ind)]), Term("p^2 - m^2 + i\\epsilon")))
latex.add(" ".join([t.latex() for t in m]))
# 2. Use Feynman parameterization to rewrite propagators
m_ = []
ds = []
for term in m:
if term.is_fraction():
# TODO
m_.append(term.numer)
ds.append(term.denom)
else:
m_.append(term)
# Integrals
for i, d in enumerate(ds):
a = Term("0")
b = Term("1" + "".join([" - z_{0}".format(j + 1) for j in range(i)]))
m_.append(Integral("z_{0}".format(i+1), norm=True, limits=(a, b)))
# Frac
denom = ""
denom += "\\left["
ds_ = []
for i, d in enumerate(ds):
ds_.append("\\left(" + d.latex() + "\\right) " + "z_{0}".format(i + 1))
denom += " + ".join(ds_)
denom += "\\right]"
denom += "^{0}".format(len(ds))
frac = Frac(Term("1"), Term(denom))
m_.append(frac)
m = m_
latex.add(" ".join([t.latex() for t in m]))
# 3. Distribute numerators (p + m) ish terms
# 4. Solve each internal momenta integral
# 5. Move terms independent of z's to left
# 6. Groan. Solve.
# 7. Solve remaining integrals, contract metrics, etc.
# 8. (OPTIONAL) Evaluate traces for non-Abelian theories.
latex.render()
def calculate(config_str, internal_momenta):
""" Calculate expression for configuration. """
latex = Latex()
################################################
######## CONSTRUCT AMPLITUDE ##########
################################################
try:
internal_momenta = internal_momenta.split()
config, amp = make_amplitude(config_str, internal_momenta)
except:
raise ParseException("Error while parsing.")
# Render
latex.add_text("\\section*{Raw amplitude}")
amp.latex_add(latex)
if RENDER_ALL:
latex.render()
################################################
######## SIMPLIFY NUMERATOR ##########
################################################
amp.numer = amp.numer.expand()
# Render
latex.add_text("\\section*{Simplified numerator}")
amp.latex_add(latex)
if RENDER_ALL:
latex.render()
################################################
######## FEYNMAN'S TRICK ##########
################################################
denom_ = []
for arg in amp.denom.args:
if type(arg) == sy.Pow:
base, power = arg.args
for _ in range(power):
denom_.append(base)
else:
denom_.append(arg)
n = len(denom_)
amp.const *= gamma(n)
zs = [sy.Symbol("{{ z_{{ {0} }} }}".format(i + 1)) for i in range(n)]
amp.denom = sum([d * z for (d, z) in zip(denom_, zs)]).expand()**n
for i, z in enumerate(zs):
a = 0
b = 1 - sum(zs[:i])
amp.integrals_zs.append((z, a, b))
# Render
latex.add_text("\\section*{Feynman parameterization}")
latex.add_text("Here, we perform the following expansion:")
latex.add_text("""$$
\\frac{1}{A_1} \\cdots \\frac{1}{A_n} = (n-1)! \\int\\limits_0^1 dz_1
\\int\\limits_0^{1-z_1} dz_2
\\cdots
\\int\\limits_0^{1-z_1-\\cdots-z_{n-1}} dz_n
\\frac{1}{(z_1 A_1 + \\cdots + z_n A_n)^n}
$$""")
latex.add_text(
"We use this form because a single denominator raised to a power can be simplified with the Golden Integral."
)
amp.latex_add(latex)
if RENDER_ALL:
latex.render()
################################################
######## SPLIT NUMERATOR INTO TERMS ##########
################################################
amps = []
for numer_ in sy.Add.make_args(amp.numer):
amp_ = amp.copy()
amp_.numer = numer_
amps.append(amp_)
# Render
latex.add_text("\\section*{Expanded numerator}")
latex.add_text(
"We split the numerator into additive terms, to process individually. The following is a list of such terms:"
)
for amp_ in amps:
amp_.latex_add(latex)
if RENDER_ALL:
latex.render()
################################################
######## EVAL. INTERNAL MOMENTA ##########
################################################
# TODO evaluate internal momenta integrals
# At this point we stop with numer and denom and combine them into
# one expression, `inner`, which is a sum of fractions.
# TODO update this comment
# Render an explanation
latex.add_text("\\section*{{Golden Integral}}")
latex.add_text("We resolve internal momentas with this transformation:")
latex.add_text("""
$$\\int \\frac{d^d q}{(2 \pi)^d} \\frac{(q^2)^a}{(q^2 + D)^b} = i \\frac{\\Gamma (b-a-\\frac{1}{2}d) \\Gamma (a + \\frac{1}{2} d)}{(4 \\pi)^{d/2} \\Gamma(b) \\Gamma(\\frac{1}{2}d)} D^{-(b-a-d/2)}$$
""")
latex.add_text(
"After this section, all internal momenta should disappear. We will now resolve each term in a queue. Each term may produce additional terms, which are pushed to the back of the queue and resolved later."
)
integrated_amps = []
#for i, amp_ in enumerate(amps):
i = 0
while len(amps) > 0:
# Pop off one amplitude
amp_ = amps[0]
amps = amps[1:]
i += 1
latex.add_text(
"\\section*{{Evaluating internal momenta in this term ({0} terms left)}}"
.format(len(amps)))
amp_.latex_add(latex)
if RENDER_ALL:
latex.render()
# Find an internal momenta
if len(amp_.integrals_internal) > 0:
(k, _, _) = amp_.integrals_internal[0]
latex.add_text("Integrating over ${0}$\\\\".format(k))
# TODO cleanup weird namespacing
k_down_dummy = Momentum(k, "DUMMY", 0)
k_up_dummy = Momentum(k, "DUMMY", 1)
k2_dummy = k_down_dummy * k_up_dummy
# Decompose denominator
# denom = denom_nopow ^ b
denom_nopow, b = amp_.denom.args[0], amp_.denom.args[1]
# Completing the square
# Denominator is always quadratic in momenta
G = denom_nopow # aliasing for convenience
latex.add_text("Completing the square\\")
A = G.collect(k2_dummy).coeff(k2_dummy)
G = sy.simplify(G - A * k2_dummy)
B_up = G.collect(k_down_dummy).coeff(k_down_dummy)
B_down = G.collect(k_up_dummy).coeff(k_up_dummy)
B = B_up + Amplitude.flip_variant(B_down)
G = sy.simplify(G - B_up * k_down_dummy - B_down * k_up_dummy)
C = G
D = -(B_up * Amplitude.flip_variant(B_up)) / (4 * A) + C
latex.add("A = " + latex.get(A))
latex.add("B = " + latex.get(B))
latex.add("C = " + latex.get(C))
"""
The denominator is in the form:
A k^2 + Bk + C
We define a new variable, q, such that
q = A^(1/2) k + B / (2 A^(1/2))
and replace k:
k = q / A^(1/2) - B / (2A)
d^d k = (1 / A^(1/2)) d^d q
The substitution k -> q yields:
A k^2 + Bk + C |-> q^2 + D
where we define D = C - B^2 / (4A)
"""
# Prepare to replace numerator
k_name = k
q_name = "q_{0}".format(len(amp_.qs) + 1) # TODO sloppy af
amp_.qs.append(q_name)
q_up = Momentum(q_name, "DUMMY", 1)
q_down = Momentum(q_name, "DUMMY", 0)
any_name = sy.Wild("a")
any_ind = sy.Wild("b")
any_variant = sy.Wild("c")
any_B_up = B_up.replace(Momentum(any_name, "DUMMY", any_variant),
Momentum(any_name, any_ind, any_variant))
any_B_down = Amplitude.flip_variant(any_B_up)
# Actually replace numerator
amp_.numer = amp_.numer.replace(
Momentum(k_name, any_ind, 1),
Momentum(q_name, any_ind, 1) / (A**0.5) - any_B_up / (2 * A))
amp_.numer = amp_.numer.replace(
Momentum(k_name, any_ind, 0),
Momentum(q_name, any_ind, 0) / (A**0.5) - any_B_down / (2 * A))
amp_.numer = sy.simplify(amp_.numer)
# Replace denominator
amp_.denom = (q_down * q_up + D)**b
# Replace integral
# TODO replace integral
amp_.integrals_internal[0] = (q_name, _, _)
amp_.numer /= A**0.5
latex.add_text("After ${0} \\to {1}$ substitutions".format(
k_name, q_name))
amp_.latex_add(latex)
if RENDER_ALL:
latex.render()
# Expand the numerator into different amplitudes and multiply them
# back in the end
amps__ = []
for numer in sy.Add.make_args(amp_.numer.expand()):
new_amp = amp_.copy()
new_amp.numer = numer
amps__.append(new_amp)
# Render
latex.add_text(
"\\section*{{Expanding numerator into {0} term(s)}}".format(
len(amps__)))
for amp__ in amps__:
amp__.latex_add(latex)
if RENDER_ALL:
latex.render()
# Finish q_name integration for each amplitude separately
for amp__ in amps__:
prod = sy.Mul.make_args(amp__.numer)
# Get k-vectors
# TODO Figure out a way to collect qs nicely
qs = [
q for q in prod
if isinstance(q, Momentum) and q.args[0].name == q_name
]
latex.add_text("\\subsection*{Integrating this term:}")
amp__.latex_add(latex)
latex.add_text(
"Found {0} q-vector terms in the numerator.\\\\".format(
len(qs)))
# Simplify q vectors
# Ward identity for odd tensors
if len(qs) % 2 == 1:
# TODO integral evaluates to zero
latex.add_text("Term vanishes due to Ward identity\\\\")
amp__.const = 0
continue
if len(qs) > 0:
# TODO convert higher-order even tensor integral to
# scalar integral
pass
else:
# TODO Assume a = 0 for now
# This is obviously wrong in general but will be easier
# to fix with a good test case
a = 0
# Golden integral
#c, a = term.as_coeff_exponent(sy.Symbol(k2))
c_ = sy.I * gamma(b - a -
(4 - EPS) / 2) * gamma(a + (4 + EPS) / 2)
c_ /= gamma(b)
c_ /= (4 * sy.pi)**2
# Part of the Golden integral d^q factor
c_ *= (2 * sy.pi)**4
amp__.const *= c_
amp__.denom = D**(b - a - 2
) # TODO generalize to d-dimensions
# with 2 -> D / 2
# Add to amps if nonzero
amps.append(amp__)
# Remove internal integral
amp__.integrals_internal = amp__.integrals_internal[1:]
# Render
latex.add_text("Apply golden integral")
amp__.latex_add(latex)
if RENDER_ALL:
latex.render()
else:
integrated_amps.append(amp_)
## Compress inners by term
#inners_dict = {}
#for (c_, expr_) in amp.inners:
# if expr_ not in inners_dict:
# inners_dict[expr_] = 0
# inners_dict[expr_] += c_
#amp.inners = [(v, k) for (k, v) in inners_dict.items()]
#amp.latex_add2(latex)
amps = integrated_amps
latex.add_text("\\section*{Final amplitudes after momenta integration}")
for amp_ in amps:
amp_.latex_add(latex)
if RENDER_ALL:
latex.render()
##########################################
######## CUTOFF INTEGRATIONS ##########
##########################################
latex.add_text("\\section*{Integrating cutoffs}")
latex.add_text(
"Here we integrate all $t$-variables, which represent the upper and lower cutoffs."
)
uv = sy.Symbol(config["Lamb"])
integrated_amps = []
for amp_ in amps:
latex.add_text("\\subsection*{Integrating this term}")
amp_.latex_add(latex)
expr_ = 1 / amp_.denom
latex.add_text("Denominator only")
latex.add(latex.get(expr_))
if RENDER_ALL:
latex.render()
# Integrate cutoffs
for (t, a, b) in amp_.integrals_cutoffs:
# Integrate w.r.t. cutoff
expr_ = sy.integrate(expr_, (t, a, b))
latex.add_text("Integrating wrt ${0}$...".format(t))
latex.add(latex.get(expr_))
if RENDER_ALL:
latex.render()
# Collecting highest order term
old_expr = expr_
while True:
old_expr = expr_
expr_ = sy.expand_log(expr_, force=True)
expr_ = get_highest_log_term(expr_, uv)
expr_ = sy.simplify(expr_)
if old_expr == expr_:
# No more changes
break
latex.add_text("Keeping only highest order term...")
latex.add(latex.get(expr_))
if RENDER_ALL:
latex.render()
amp_.numer *= expr_
amp_.denom = 1
amp_.integrals_cutoffs = []
integrated_amps.append(amp_)
amps = integrated_amps
######################################
######## Z INTEGRATIONS ##########
######################################
latex.add_text("\\section*{Integrating $z$-variables}")
latex.add_text(
"Here we integrate all $z$-variables, the Feynman parameters.")
integrated_amps = []
for amp_ in amps:
latex.add_text("\\subsection*{Integrating this term}")
amp_.latex_add(latex)
# Integrate cutoffs
expr_ = amp_.numer
for (z, a, b) in amp_.integrals_zs[::-1]:
# Integrate w.r.t. cutoff
# Rationalize decimal powers first, or sympy breaks
expr_ = sy.nsimplify(expr_, tolerance=0.001, rational=True)
expr_ = sy.integrate(expr_, (z, a, b))
latex.add_text("Integrating wrt ${0}$...".format(z))
latex.add(latex.get(expr_))
if RENDER_ALL:
latex.render()
amp_.numer = expr_
amp_.integrals_zs = []
integrated_amps.append(amp_)
amps = integrated_amps
latex.add_text(
"\\section*{Final amplitudes after $z$-variable integration}")
for amp_ in amps:
amp_.latex_add(latex)
if RENDER_ALL:
latex.render()
################################################
######## EVALUATE SPINS AND GAMMAS ##########
################################################
latex.add_text("\\section*{Evaluating spins and gamma matrices}")
#latex.add_text("TODO jk do it yourself you slags, here's the sum, have fun. Don't forget to take traces/multiply by -1 for internal fermion loops.")
latex.add_text("TODO. Here's the sum for now:")
latex.add("\\; + \\;".join([amp_.get_latex(latex) for amp_ in amps]))
if RENDER_ALL:
latex.render()
################################################
######## RENDER ##########
################################################
#amp.latex_add2(latex)
latex.render()
return "\\; + \\;".join([amp_.get_latex(latex) for amp_ in amps])