物理の本

詳細pdf版

---

Peskin 1

Peskin 2

Peskin 3

Peskin 4

Peskin 5

Peskin 6

Peskin 7

Peskin 8

---

Peskin & Schroeder. An Introduction to Quantum Field Theory

ghostのLagranigan (21.52)を計算する

import sympy

量の定義

g, tw, m_W = sympy.symbols("g, \\theta_w, m_W")
h = sympy.symbols("h")

W_p, W_m, Z_0, A = sympy.symbols("W^+, W^-, Z^0, A")

phi_p, phi_m, phi_3 = sympy.symbols("\\phi^+, \\phi^-, \\phi^3")

c_p, c_m, c_Z, c_A = sympy.symbols("c^+, c^-, c^Z, c^A")
_c_p, _c_m, _c_Z, _c_A = sympy.symbols("\\bar{c}^+, \\bar{c}^-, \\bar{c}^Z, \\bar{c}^A")

v = 2*m_W / g
gp = g * sympy.tan(tw)

A_1 = (W_p + W_m) / sympy.sqrt(2)
A_2 = sympy.I * (W_p - W_m) / sympy.sqrt(2)
A_3 = A * sympy.sin(tw) + Z_0 * sympy.cos(tw)
A_Y = A * sympy.cos(tw) - Z_0 * sympy.sin(tw) # = B

phi_1 = (phi_p+phi_m) / sympy.sqrt(2)
phi_2 = -sympy.I * (phi_p-phi_m) / sympy.sqrt(2)

c_1 = (c_p+c_m) / sympy.sqrt(2)
c_2 = -sympy.I * (c_p-c_m) / sympy.sqrt(2)
_c_1 = (_c_p+_c_m) / sympy.sqrt(2)
_c_2 = sympy.I * (_c_p-_c_m) / sympy.sqrt(2)

c_3 = c_Z * sympy.cos(tw) + c_A * sympy.sin(tw)
c_Y = c_A * sympy.cos(tw) - c_Z * sympy.sin(tw)
_c_3 = _c_Z * sympy.cos(tw) + _c_A * sympy.sin(tw)
_c_Y = _c_A * sympy.cos(tw) - _c_Z * sympy.sin(tw)

phi_0 = sympy.Matrix([0, 0, 0, v])
phi = sympy.Matrix([phi_1, phi_2, phi_3, v+h])
c = sympy.Matrix([c_1, c_2, c_3, c_Y])
_c = sympy.Matrix([_c_1, _c_2, _c_3, _c_Y])

T_1 = 1/2 * sympy.Matrix([
[0, 0, 0, 1],
[0, 0, -1, 0],
[0, 1, 0, 0],
[-1, 0, 0, 0]])

T_2 = 1/2 * sympy.Matrix([
[0, 0, 1, 0],
[0, 0, 0, 1],
[-1, 0, 0, 0],
[0, -1, 0, 0]])

T_3 = 1/2 * sympy.Matrix([
[0, -1, 0, 0],
[1, 0, 0, 0],
[0, 0, 0, -1],
[0, 0, 1, 0]])

T_Y = 1/2 * sympy.Matrix([
[0, -1, 0, 0],
[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, -1, 0]])

T = [T_1, T_2, T_3, T_Y]

Gauge boson

特に構造定数$[T^a, T^b] = f^{abc} T^c$は$f^{12Y} = 1$のみ． Lagrangianは$g^b f^{abc} (\partial^\mu\bar{c}^a) A^b_\mu c^c$

L_ghost_gauge = - (g * _c_1 * A_2 * c_Y) - (gp * _c_2 * A_Y * c_1) - (g * _c_Y * A_1 * c_2) + (g * _c_Y * A_2 * c_1) + (g * _c_2 * A_1 * c_Y) + (gp * _c_1 * A_Y * c_2)
L_ghost_gauge = sympy.expand(L_ghost_gauge)

Lpp = L_ghost_gauge.coeff(_c_p*c_p) * _c_p * c_p
Lmm = L_ghost_gauge.coeff(_c_m*c_m) * _c_m * c_m
LZZ = L_ghost_gauge.coeff(_c_Z*c_Z) * _c_Z * c_Z
LAA = L_ghost_gauge.coeff(_c_A*c_A) * _c_A * c_A

display(sympy.expand(Lpp.simplify()))
display(sympy.expand(Lmm.simplify()))
display(sympy.expand(LZZ.simplify()))
display(sympy.expand(LAA.simplify()))

$\displaystyle - i A \bar{c}^+ c^{+} g \cos{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)} + i Z^{0} \bar{c}^+ c^{+} g \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}$

$\displaystyle i A \bar{c}^- c^{-} g \cos{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)} - i Z^{0} \bar{c}^- c^{-} g \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}$

$\displaystyle 0$

$\displaystyle 0$

Lpm = L_ghost_gauge.coeff(_c_p*c_m) * _c_p * c_m
LpA = L_ghost_gauge.coeff(_c_p*c_A) * _c_p * c_A
LpZ = L_ghost_gauge.coeff(_c_p*c_Z) * _c_p * c_Z

display(sympy.expand(Lpm.simplify()))
display(sympy.expand(LpA.simplify()))
display(sympy.expand(LpZ.simplify()))

$\displaystyle 0$

$\displaystyle i W^{-} \bar{c}^+ c^{A} g \cos{\left(\theta_{w} \right)}$

$\displaystyle - i W^{-} \bar{c}^+ c^{Z} g \sin{\left(\theta_{w} \right)}$

Lmp = L_ghost_gauge.coeff(_c_m*c_p) * _c_m * c_p
LmA = L_ghost_gauge.coeff(_c_m*c_A) * _c_m * c_A
LmZ = L_ghost_gauge.coeff(_c_m*c_Z) * _c_m * c_Z

display(sympy.expand(Lmp.simplify()))
display(sympy.expand(LmA.simplify()))
display(sympy.expand(LmZ.simplify()))

$\displaystyle 0$

$\displaystyle - i W^{+} \bar{c}^- c^{A} g \cos{\left(\theta_{w} \right)}$

$\displaystyle i W^{+} \bar{c}^- c^{Z} g \sin{\left(\theta_{w} \right)}$

LZp = L_ghost_gauge.coeff(_c_Z*c_p) * _c_Z * c_p
LZm = L_ghost_gauge.coeff(_c_Z*c_m) * _c_Z * c_m
LZA = L_ghost_gauge.coeff(_c_Z*c_A) * _c_Z * c_A

display(sympy.expand(LZp.simplify()))
display(sympy.expand(LZm.simplify()))
display(sympy.expand(LZA.simplify()))

$\displaystyle - i W^{+} \bar{c}^Z c^{+} g \sin{\left(\theta_{w} \right)}$

$\displaystyle i W^{-} \bar{c}^Z c^{-} g \sin{\left(\theta_{w} \right)}$

$\displaystyle 0$

LAp = L_ghost_gauge.coeff(_c_A*c_p) * _c_A * c_p
LAm = L_ghost_gauge.coeff(_c_A*c_m) * _c_A * c_m
LAZ = L_ghost_gauge.coeff(_c_A*c_Z) * _c_A * c_Z

display(sympy.expand(LAp.simplify()))
display(sympy.expand(LAm.simplify()))
display(sympy.expand(LAZ.simplify()))

$\displaystyle i W^{+} \bar{c}^A c^{+} g \cos{\left(\theta_{w} \right)}$

$\displaystyle - i W^{-} \bar{c}^A c^{-} g \cos{\left(\theta_{w} \right)}$

$\displaystyle 0$

Goldstone boson

M = sympy.Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]])

for a in range(3):
    for b in range(3):
        M[a, b] = -g**2 * (T[a]*phi_0).dot(T[b]*phi)
    M[a, 3] = -g * gp * (T[a]*phi_0).dot(T[3]*phi)

for b in range(3):
    M[3, b] = -g * gp * (T[3]*phi_0).dot(T[b]*phi)

M[3, 3] = -gp**2 * (T[3]*phi_0).dot(T[3]*phi)

L_ghost_goldstone = (_c.T * M * c)[0, 0]
L_ghost_goldstone = L_ghost_goldstone.expand()

Lpp = L_ghost_goldstone.coeff(_c_p*c_p) * _c_p * c_p
Lmm = L_ghost_goldstone.coeff(_c_m*c_m) * _c_m * c_m
LZZ = L_ghost_goldstone.coeff(_c_Z*c_Z) * _c_Z * c_Z
LAA = L_ghost_goldstone.coeff(_c_A*c_A) * _c_A * c_A

display(sympy.expand(Lpp.nsimplify()))
display(sympy.expand(Lmm.nsimplify()))
display(sympy.expand(sympy.simplify(LZZ.nsimplify())))
display(sympy.expand(sympy.simplify(LAA.nsimplify())))

$\displaystyle \frac{i \bar{c}^+ \phi^{3} c^{+} g m_{W}}{2} - \frac{\bar{c}^+ c^{+} g h m_{W}}{2} - \bar{c}^+ c^{+} m_{W}^{2}$

$\displaystyle - \frac{i \bar{c}^- \phi^{3} c^{-} g m_{W}}{2} - \frac{\bar{c}^- c^{-} g h m_{W}}{2} - \bar{c}^- c^{-} m_{W}^{2}$

$\displaystyle - \frac{\bar{c}^Z c^{Z} g h m_{W}}{2 \cos^{2}{\left(\theta_{w} \right)}} - \frac{\bar{c}^Z c^{Z} m_{W}^{2}}{\cos^{2}{\left(\theta_{w} \right)}}$

$\displaystyle 0$

Lpm = L_ghost_goldstone.coeff(_c_p*c_m) * _c_p * c_m
LpA = L_ghost_goldstone.coeff(_c_p*c_A) * _c_p * c_A
LpZ = L_ghost_goldstone.coeff(_c_p*c_Z) * _c_p * c_Z

display(sympy.expand(Lpm.simplify()))
display(sympy.expand(LpA.nsimplify()))
display(sympy.expand(LpZ.nsimplify()))

$\displaystyle 0$

$\displaystyle - \frac{i \bar{c}^+ \phi^{+} c^{A} g m_{W} \sin{\left(\theta_{w} \right)}}{2} - \frac{i \bar{c}^+ \phi^{+} c^{A} g m_{W} \cos{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2}$

$\displaystyle \frac{i \bar{c}^+ \phi^{+} c^{Z} g m_{W} \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2} - \frac{i \bar{c}^+ \phi^{+} c^{Z} g m_{W} \cos{\left(\theta_{w} \right)}}{2}$

Lmp = L_ghost_goldstone.coeff(_c_m*c_p) * _c_m * c_p
LmA = L_ghost_goldstone.coeff(_c_m*c_A) * _c_m * c_A
LmZ = L_ghost_goldstone.coeff(_c_m*c_Z) * _c_m * c_Z

display(sympy.expand(Lmp.simplify()))
display(sympy.expand(LmA.nsimplify()))
display(sympy.expand(LmZ.nsimplify()))

$\displaystyle 0$

$\displaystyle \frac{i \bar{c}^- \phi^{-} c^{A} g m_{W} \sin{\left(\theta_{w} \right)}}{2} + \frac{i \bar{c}^- \phi^{-} c^{A} g m_{W} \cos{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2}$

$\displaystyle - \frac{i \bar{c}^- \phi^{-} c^{Z} g m_{W} \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2} + \frac{i \bar{c}^- \phi^{-} c^{Z} g m_{W} \cos{\left(\theta_{w} \right)}}{2}$

LZp = L_ghost_goldstone.coeff(_c_Z*c_p) * _c_Z * c_p
LZm = L_ghost_goldstone.coeff(_c_Z*c_m) * _c_Z * c_m
LZA = L_ghost_goldstone.coeff(_c_Z*c_A) * _c_Z * c_A

display(sympy.expand(LZp.nsimplify()))
display(sympy.expand(LZm.nsimplify()))
display(sympy.expand(LZA.simplify()))

$\displaystyle \frac{i \bar{c}^Z \phi^{-} c^{+} g m_{W} \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2} + \frac{i \bar{c}^Z \phi^{-} c^{+} g m_{W} \cos{\left(\theta_{w} \right)}}{2}$

$\displaystyle - \frac{i \bar{c}^Z \phi^{+} c^{-} g m_{W} \sin{\left(\theta_{w} \right)} \tan{\left(\theta_{w} \right)}}{2} - \frac{i \bar{c}^Z \phi^{+} c^{-} g m_{W} \cos{\left(\theta_{w} \right)}}{2}$

$\displaystyle 0$

LAp = L_ghost_goldstone.coeff(_c_A*c_p) * _c_A * c_p
LAm = L_ghost_goldstone.coeff(_c_A*c_m) * _c_A * c_m
LAZ = L_ghost_goldstone.coeff(_c_A*c_Z) * _c_A * c_Z

display(sympy.expand(LAp.simplify()))
display(sympy.expand(LAm.simplify()))
display(sympy.expand(LAZ.simplify()))

$\displaystyle 0$

$\displaystyle 0$

$\displaystyle 0$

sympy.simplify(_c.T * c)

$\displaystyle \left[\begin{matrix}\bar{c}^+ c^{+} + \bar{c}^- c^{-} + \bar{c}^A c^{A} + \bar{c}^Z c^{Z}\end{matrix}\right]$