import sympy
from equation_database.util.doc import bib, equation
[docs]
@bib()
def bibtex():
bibtex: str = r"""
@article{Jezo:2025jyu,
author = "Je{\v{z}}o, Tom{\'a}{\v{s}} and Klasen, Michael and Puck Neuwirth, Alexander",
title = "{Conversion of photons to dileptons in the Kroll-Wada and parton shower approaches}",
eprint = "2506.23162",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "MS-TP-25-18",
month = "6",
year = "2025"
}"""
return bibtex
[docs]
@equation(
latex=r"\rho = \frac{W_{\text{pair}}}{W_{\gamma}} = \frac{2 \alpha}{3 \pi} \int_{2m_e}^E \,\mathrm{d}{M_{ee}} \left(\frac{k'}{k}\right) \frac{(E+M_R)^2 + M_R^2 -M_{ee}^2}{(E+M_R)^2 + M_R^2} \\\cdot \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \left[ \frac{R_T}{M_{ee}} + \frac{2(E+M_R)^2 M_{ee} }{(2EM_R +E^2 + M_{ee}^2)^2}R_L\right]"
)
def equation_2_1(
rho=sympy.Symbol("rho"),
W_pair=sympy.Symbol("W_pair"),
W_gamma=sympy.Symbol("W_gamma"),
alpha=sympy.Symbol("alpha"),
m_e=sympy.Symbol("m_e"),
E=sympy.Symbol("E"),
M_ee=sympy.Symbol("M_ee"),
k_prime=sympy.Symbol("k_prime"),
k=sympy.Symbol("k"),
M_R=sympy.Symbol("M_R"),
R_T=sympy.Symbol("R_T"),
R_L=sympy.Symbol("R_L"),
):
ratio = W_pair / W_gamma
integrand = (
(k_prime / k)
* ((E + M_R) ** 2 + M_R**2 - M_ee**2)
/ ((E + M_R) ** 2 + M_R**2)
* sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
* (1 + 2 * m_e**2 / M_ee**2)
* (
R_T / M_ee
+ 2 * (E + M_R) ** 2 * M_ee / ((2 * E * M_R + E**2 + M_ee**2) ** 2) * R_L
)
)
integral_form = (
2 * alpha / (3 * sympy.pi) * sympy.Integral(integrand, (M_ee, 2 * m_e, E))
)
return (sympy.Eq(rho, ratio), sympy.Eq(ratio, integral_form))
[docs]
@equation(
latex=r"\frac{1}{N_{\gamma^*}} \frac{\,\mathrm{d} N_{ee}}{\,\mathrm{d} M_{ee}} = \frac{2 \alpha}{3 \pi} \frac{1}{M_{ee}} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right)S\approx \frac{1}{M_{ee}}"
)
def equation_2_2(
N_gamma_star=sympy.Symbol("N_gamma_star"),
dN_ee=sympy.Symbol("dN_ee"),
dM_ee=sympy.Symbol("dM_ee"),
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
S=sympy.Symbol("S"),
):
lhs = (1 / N_gamma_star) * (dN_ee / dM_ee)
middle = (
2
* alpha
/ (3 * sympy.pi)
* (1 / M_ee)
* sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
* (1 + 2 * m_e**2 / M_ee**2)
* S
)
rhs = 1 / M_ee
return (sympy.Eq(lhs, middle), sympy.Eq(middle, rhs))
[docs]
@equation(
latex=r"\frac{\,\mathrm{d}^2 N_{ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{3 \pi} \frac{1}{M_{ee}^2} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \,\mathrm{d} N_{\gamma^*}"
)
def equation_2_3(
d2N_ee=sympy.Symbol("d2N_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
dN_gamma_star=sympy.Symbol("dN_gamma_star"),
):
lhs = d2N_ee / dM_ee_sq
rhs = (
alpha
/ (3 * sympy.pi)
* (1 / M_ee**2)
* sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
* (1 + 2 * m_e**2 / M_ee**2)
* dN_gamma_star
)
return sympy.Eq(lhs, rhs)
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - \frac{M_{ee}^2}{s}\right)^3"
)
def equation_2_31(
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
alpha=sympy.Symbol("alpha"),
e_e=sympy.Symbol("e_e"),
M_ee=sympy.Symbol("M_ee"),
s=sympy.Symbol("s"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = alpha * e_e**2 / (3 * sympy.pi) * (1 / M_ee**2) * (1 - M_ee**2 / s) ** 3
return sympy.Eq(lhs, rhs)
[docs]
@equation(
latex=r"\approx \frac{\alpha}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - 6 \frac{m_e^4}{M_{ee}^4} - 8 \frac{m_e^6}{M_{ee}^6} \right) \,\mathrm{d} N_{\gamma^*}"
)
def equation_2_4(
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
dN_gamma_star=sympy.Symbol("dN_gamma_star"),
):
rhs = (
alpha
/ (3 * sympy.pi)
* (1 / M_ee**2)
* (1 - 6 * m_e**4 / M_ee**4 - 8 * m_e**6 / M_ee**6)
* dN_gamma_star
)
return rhs
[docs]
@equation(latex=r"P_{g \to qq}(z) = T_F ( z^2 + (1-z)^2)")
def equation_2_5(
P_g_to_qq=sympy.Function("P_g_to_qq"),
T_F=sympy.Symbol("T_F"),
z=sympy.Symbol("z"),
):
rhs = T_F * (z**2 + (1 - z) ** 2)
return sympy.Eq(P_g_to_qq(z), rhs)
[docs]
@equation(latex=r"P_{\gamma \to ee}(z) = e_e^2 ( z^2 + (1-z)^2)")
def equation_2_6(
P_gamma_to_ee=sympy.Function("P_gamma_to_ee"),
e_e=sympy.Symbol("e_e"),
z=sympy.Symbol("z"),
):
rhs = e_e**2 * (z**2 + (1 - z) ** 2)
return sympy.Eq(P_gamma_to_ee(z), rhs)
[docs]
@equation(
latex=r"\,\mathrm{d} \mathcal P_{\gamma \to ee} = \frac{\alpha}{2\pi} \frac{\,\mathrm{d} Q^2}{Q^2} P_{\gamma \to ee}(z) \,\mathrm{d} z"
)
def equation_2_7(
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
alpha=sympy.Symbol("alpha"),
dQ_sq=sympy.Symbol("dQ_sq"),
Q=sympy.Symbol("Q"),
P_gamma_to_ee=sympy.Function("P_gamma_to_ee"),
z=sympy.Symbol("z"),
dz=sympy.Symbol("dz"),
):
rhs = alpha / (2 * sympy.pi) * (dQ_sq / Q**2) * P_gamma_to_ee(z) * dz
return sympy.Eq(dP_gamma_to_ee, rhs)
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} P_{\gamma \to ee}(z) \,\mathrm{d} z"
)
def equation_2_8(
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
P_gamma_to_ee=sympy.Function("P_gamma_to_ee"),
z=sympy.Symbol("z"),
dz=sympy.Symbol("dz"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = alpha / (2 * sympy.pi) * (1 / M_ee**2) * P_gamma_to_ee(z) * dz
return sympy.Eq(lhs, rhs)
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P_{\gamma \to ee}(z) \,\mathrm{d} z"
)
def equation_2_9(
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
y_minus=sympy.Symbol("y_minus"),
y_plus=sympy.Symbol("y_plus"),
P_gamma_to_ee=sympy.Function("P_gamma_to_ee"),
z=sympy.Symbol("z"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = (
alpha
/ (2 * sympy.pi)
* (1 / M_ee**2)
* sympy.Integral(P_gamma_to_ee(z), (z, y_minus, y_plus))
)
return sympy.Eq(lhs, rhs)
[docs]
@equation(latex=r"p = p_1 + p_2")
def equation_2_10(
p_1=sympy.Symbol("p_1"),
p_2=sympy.Symbol("p_2"),
):
return p_1 + p_2
[docs]
@equation(latex=r"p_1 = z p + k_T")
def equation_2_11(
k_T=sympy.Symbol("k_T"),
p=sympy.Symbol("p"),
z=sympy.Symbol("z"),
):
return k_T + p * z
[docs]
@equation(latex=r"p_2 = (1-z) p - k_T")
def equation_2_12(
k_T=sympy.Symbol("k_T"),
p=sympy.Symbol("p"),
z=sympy.Symbol("z"),
):
return -k_T + p * (1 - z)
[docs]
@equation(
latex=r"p^2 = M_{ee}^2 = 2m_e^2 + 2 p_1\cdot{}p_2 = 2m_e^2 + 2 ( z (1-z) p^2 - k_T^2 - (2z-1) p\cdot{}k_T)"
)
def equation_2_13(
p=sympy.Symbol("p"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
p_dot_p2=sympy.Symbol("p_dot_p2"), # p·p_2
p_dot_kT=sympy.Symbol("p_dot_kT"), # p·k_T
z=sympy.Symbol("z"),
k_T=sympy.Symbol("k_T"),
):
lhs = p**2
middle = 2 * m_e**2 + 2 * p_dot_p2
rhs = 2 * m_e**2 + 2 * (z * (1 - z) * p**2 - k_T**2 - (2 * z - 1) * p_dot_kT)
return (sympy.Eq(lhs, M_ee**2), sympy.Eq(M_ee**2, middle), sympy.Eq(middle, rhs))
[docs]
@equation(latex=r"p_1^2 = m_e^2 = z^2 p^2 + k_T^2 + 2 z p \cdot{} k_T")
def equation_2_14(
p_1=sympy.Symbol("p_1"),
m_e=sympy.Symbol("m_e"),
z=sympy.Symbol("z"),
p=sympy.Symbol("p"),
p_dot_kT=sympy.Symbol("p_dot_kT"), # p·k_T
k_T=sympy.Symbol("k_T"),
):
return sympy.Eq(p_1**2, m_e**2), sympy.Eq(
m_e**2, z**2 * p**2 + k_T**2 + 2 * z * p_dot_kT
)
[docs]
@equation(
latex=r"M_{ee}^2 = 2m_e^2 + 2 ( z (1-z) M_{ee}^2 - k_T^2 ) - \frac{2z-1}{z} (m_e^2 - z^2 M_{ee}^2 - k_T^2 )"
)
def equation_2_15(
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
z=sympy.Symbol("z"),
k_T=sympy.Symbol("k_T"),
):
rhs = (
2 * m_e**2
+ 2 * (z * (1 - z) * M_ee**2 - k_T**2)
- (2 * z - 1) / z * (m_e**2 - z**2 * M_ee**2 - k_T**2)
)
return sympy.Eq(M_ee**2, rhs)
[docs]
@equation(
latex=r"z = \frac 1 2 \pm \frac 1 2 \sqrt{ 1 - 4 \frac{m_e^2}{M_{ee}^2} + 4k_T^2 }"
)
def equation_2_16(
z=sympy.Symbol("z"),
m_e=sympy.Symbol("m_e"),
M_ee=sympy.Symbol("M_ee"),
k_T=sympy.Symbol("k_T"),
):
sqrt_term = sympy.sqrt(1 - 4 * m_e**2 / M_ee**2 + 4 * k_T**2)
z_plus = sympy.Rational(1, 2) + sympy.Rational(1, 2) * sqrt_term
z_minus = sympy.Rational(1, 2) - sympy.Rational(1, 2) * sqrt_term
return (sympy.Eq(z, z_plus), sympy.Eq(z, z_minus))
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + \frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}"
)
def equation_2_17(
alpha=sympy.Symbol("alpha"),
e_e=sympy.Symbol("e_e"),
m_e=sympy.Symbol("m_e"),
M_ee=sympy.Symbol("M_ee"),
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = (
alpha
* e_e**2
/ (3 * sympy.pi)
* (1 / M_ee**2)
* (1 + m_e**2 / M_ee**2)
* sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
)
return sympy.Eq(lhs, rhs)
[docs]
@equation(
latex=r"\approx \frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \,\mathrm{d} N_{\gamma^*} \left( 1 - \frac{m_e^2}{M_{ee}^2} - 4 \frac{m_e^4}{M_{ee}^4} \right)"
)
def equation_2_18(
alpha=sympy.Symbol("alpha"),
e_e=sympy.Symbol("e_e"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
dN_gamma_star=sympy.Symbol("dN_gamma_star"),
):
rhs = (
alpha
* e_e**2
/ (3 * sympy.pi)
* (1 / M_ee**2)
* dN_gamma_star
* (1 - m_e**2 / M_ee**2 - 4 * m_e**4 / M_ee**4)
)
return rhs
[docs]
@equation(
latex=r"P_{\gamma\to ee}^m = e_e^2 \left( 1- 2z+ 2z^2 + 2\frac{m_e^2}{M_{ee}^2}\right)"
)
def equation_2_19(
P_gamma_to_ee=sympy.Symbol("P_gamma_to_ee"),
e_e=sympy.Symbol("e_e"),
z=sympy.Symbol("z"),
m_e=sympy.Symbol("m_e"),
M_ee=sympy.Symbol("M_ee"),
):
rhs = e_e**2 * (1 - 2 * z + 2 * z**2 + 2 * m_e**2 / M_ee**2)
return sympy.Eq(P_gamma_to_ee, rhs)
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P^m_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P^m_{\gamma \to ee}(z) \,\mathrm{d} z"
)
def equation_2_20(
alpha=sympy.Symbol("alpha"),
M_ee=sympy.Symbol("M_ee"),
y_minus=sympy.Symbol("y_minus"),
y_plus=sympy.Symbol("y_plus"),
z=sympy.Symbol("z"),
P_gamma_to_ee=sympy.Function("P_gamma_to_ee"),
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = (
alpha
/ (2 * sympy.pi)
* (1 / M_ee**2)
* sympy.Integral(P_gamma_to_ee(z), (z, y_minus, y_plus))
)
return sympy.Eq(lhs, rhs)
[docs]
@equation(
latex=r"=\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + 2\frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}"
)
def equation_2_21(
alpha=sympy.Symbol("alpha"),
e_e=sympy.Symbol("e_e"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
):
rhs = (
alpha
* e_e**2
/ (3 * sympy.pi)
* (1 / M_ee**2)
* (1 + 2 * m_e**2 / M_ee**2)
* sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
)
return rhs
[docs]
@equation(latex=r"M_{ee}^2 = Q^2 = \frac{p_T^2}{z(1-z)}")
def equation_2_22(
M_ee=sympy.Symbol("M_ee"),
Q=sympy.Symbol("Q"),
p_T=sympy.Symbol("p_T"),
z=sympy.Symbol("z"),
):
return (sympy.Eq(M_ee**2, Q**2), sympy.Eq(Q**2, p_T**2 / (z * (1 - z))))
[docs]
@equation(
latex=r"\,\mathrm{d} \Phi^{\text{FF}}_\text{ant} = \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} s_{IK} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{ij} \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}"
)
def equation_2_23(
dPhi_FF_ant=sympy.Symbol("dPhi_FF_ant"),
f_FF_Kallen=sympy.Symbol("f_FF_Kallen"),
s_IK=sympy.Symbol("s_IK"),
Gamma_ijk=sympy.Symbol("Gamma_ijk"),
dy_ij=sympy.Symbol("dy_ij"),
dy_jk=sympy.Symbol("dy_jk"),
dphi=sympy.Symbol("dphi"),
):
rhs = (
(1 / (16 * sympy.pi**2))
* f_FF_Kallen
* s_IK
* sympy.Heaviside(Gamma_ijk, 1)
* dy_ij
* dy_jk
* (dphi / (2 * sympy.pi))
)
return sympy.Eq(dPhi_FF_ant, rhs)
[docs]
@equation(
latex=r"\bar{a}_{e/\gamma}^{\text{FF},\gamma} = \frac{1}{s_{IK}} \frac 1 2 \frac{1}{y_{ij} + 2 \mu_e^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 \mu_e^2}{y_{ij} + 2 \mu_e^2}\right]"
)
def equation_2_24(
a_bar_e_gamma=sympy.Symbol("a_bar_e_gamma"),
s_IK=sympy.Symbol("s_IK"),
y_ij=sympy.Symbol("y_ij"),
y_ik=sympy.Symbol("y_ik"),
y_jk=sympy.Symbol("y_jk"),
mu_e=sympy.Symbol("mu_e"),
):
rhs = (
(1 / s_IK)
* sympy.Rational(1, 2)
* (1 / (y_ij + 2 * mu_e**2))
* (y_ik**2 + y_jk**2 + 2 * mu_e**2 / (y_ij + 2 * mu_e**2))
)
return sympy.Eq(a_bar_e_gamma, rhs)
[docs]
@equation(
latex=r"\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}^m}{\,\mathrm{d} M_{ee}^2} \propto 4 \pi \alpha e_e^2 \frac 1 2 \frac{1}{M_{ee}^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 m_e^2}{M_{ee}^2}\right] \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}"
)
def equation_2_25(
dP_gamma_to_ee=sympy.Symbol("dP_gamma_to_ee"),
dM_ee_sq=sympy.Symbol("dM_ee_sq"),
alpha=sympy.Symbol("alpha"),
e_e=sympy.Symbol("e_e"),
M_ee=sympy.Symbol("M_ee"),
y_ik=sympy.Symbol("y_ik"),
y_jk=sympy.Symbol("y_jk"),
m_e=sympy.Symbol("m_e"),
f_FF_Kallen=sympy.Symbol("f_FF_Kallen"),
Gamma_ijk=sympy.Symbol("Gamma_ijk"),
dy_jk=sympy.Symbol("dy_jk"),
dphi=sympy.Symbol("dphi"),
):
lhs = dP_gamma_to_ee / dM_ee_sq
rhs = (
4
* sympy.pi
* alpha
* e_e**2
* sympy.Rational(1, 2)
* (1 / M_ee**2)
* (y_ik**2 + y_jk**2 + 2 * m_e**2 / M_ee**2)
* (1 / (16 * sympy.pi**2))
* f_FF_Kallen
* sympy.Heaviside(Gamma_ijk, 1)
* dy_jk
* (dphi / (2 * sympy.pi))
)
return sympy.Eq(lhs, rhs) # Note: using Eq instead of proportional
[docs]
@equation(
latex=r"0< \Gamma_{ijk} = y_{ij} y_{jk} y_{ik} - y_{jk} \mu_i^2 - y_{ik} \mu_j^2"
)
def equation_2_26(
Gamma_ijk=sympy.Symbol("Gamma_ijk"),
y_ij=sympy.Symbol("y_ij"),
y_jk=sympy.Symbol("y_jk"),
y_ik=sympy.Symbol("y_ik"),
mu_i=sympy.Symbol("mu_i"),
mu_j=sympy.Symbol("mu_j"),
):
rhs = y_ij * y_jk * y_ik - y_jk * mu_i**2 - y_ik * mu_j**2
return (sympy.Gt(0, Gamma_ijk), sympy.Eq(Gamma_ijk, rhs))
[docs]
@equation(latex=r"z = \frac 1 2 \pm \frac 1 2 \sqrt{1-\frac{4m_e^2}{M_{ee}^2}}")
def equation_2_27(
z=sympy.Symbol("z"),
m_e=sympy.Symbol("m_e"),
M_ee=sympy.Symbol("M_ee"),
):
sqrt_term = sympy.sqrt(1 - 4 * m_e**2 / M_ee**2)
z_plus = sympy.Rational(1, 2) + sympy.Rational(1, 2) * sqrt_term
z_minus = sympy.Rational(1, 2) - sympy.Rational(1, 2) * sqrt_term
return (sympy.Eq(z, z_plus), sympy.Eq(z, z_minus))
[docs]
@equation(latex=r"1 = y_{ij} + 2 \mu_e^2 + y_{jk} + y_{ik}")
def equation_2_28(
y_ij=sympy.Symbol("y_ij"),
mu_e=sympy.Symbol("mu_e"),
y_jk=sympy.Symbol("y_jk"),
y_ik=sympy.Symbol("y_ik"),
):
return sympy.Eq(1, y_ij + 2 * mu_e**2 + y_jk + y_ik)
[docs]
@equation(
latex=r"y_\pm = \frac{\pm\sqrt{\left(M_{ee}^{2} - 2 m_{e}^{2}\right)^{-1} \left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)} + \left(- M_{ee}^{2} + s\right) }{2 s }"
)
def equation_2_29(
y_plus=sympy.Symbol("y_plus"),
y_minus=sympy.Symbol("y_minus"),
M_ee=sympy.Symbol("M_ee"),
m_e=sympy.Symbol("m_e"),
s=sympy.Symbol("s"),
):
discriminant = (
(M_ee**2 - 2 * m_e**2) ** (-1)
* (M_ee**2 - s)
* (M_ee**4 - 2 * M_ee**2 * m_e**2 - M_ee**2 * s + 6 * m_e**2 * s)
)
sqrt_term = sympy.sqrt(discriminant)
common_term = (-(M_ee**2) + s) / (2 * s)
y_plus_val = sqrt_term + common_term
y_minus_val = -sqrt_term + common_term
return (sympy.Eq(y_plus, y_plus_val), sympy.Eq(y_minus, y_minus_val))