物理の本

詳細pdf版

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Peskin 1

Peskin 2

Peskin 3

Peskin 4

Peskin 5

Peskin 6

Peskin 7

Peskin 8

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Peskin & Schroeder. An Introduction to Quantum Field Theory

Problem 21.2 (LL)

$e^-e^+ \to W^-W^+$を計算する．RRはこれ

CM frameで考える．電子・陽電子はmasslessとする． $W^\pm$の偏極$\epsilon_1$, $\epsilon_2$には縦波成分が存在する． 陽電子のhelicityは右なので，spinor $v$のhelicityは左．

import sympy

量の定義

t, tw, e, m_W, E = sympy.symbols("\\theta, \\theta_w, e, m_W, E", real=True)
p = sympy.sqrt(E**2 - m_W**2)
s = 4 * E**2
m_Z = m_W / sympy.cos(tw)

gamma0 = sympy.Matrix([
[0, 0, 1, 0],
[0, 0, 0, 1],
[1, 0, 0, 0],
[0, 1, 0, 0]
])

gamma1 = sympy.Matrix([
[0, 0, 0, 1],
[0, 0, 1, 0],
[0, -1, 0, 0],
[-1, 0, 0, 0]
])

gamma2 = sympy.Matrix([
[0, 0, 0, -1j],
[0, 0, 1j, 0],
[0, 1j, 0, 0],
[-1j, 0, 0, 0]
])

gamma3 = sympy.Matrix([
[0, 0, 1, 0],
[0, 0, 0, -1],
[-1, 0, 0, 0],
[0, 1, 0, 0]
])

def slashed(vector):
    # vector^mu
    sl = vector[0] * gamma0 - vector[1] * gamma1 - vector[2] * gamma2 - vector[3] * gamma3
    return sl
def prod(vec1, vec2):
    ip = vec1[0] * vec2[0] - vec1[1] * vec2[1] - vec1[2] * vec2[2] - vec1[3] * vec2[3]
    return ip

electron, positronのspinor

u_L = sympy.sqrt(2*E) * sympy.Matrix([0, 1, 0, 0])
v_L = sympy.sqrt(2*E) * sympy.Matrix([1, 0, 0, 0])

$e^-$の運動量$p_1^\mu$，$e^+$の運動量$p_2^\mu$，$W^-$の運動量$k_1^\mu=k_-^\mu$，$W^+$の運動量$k_2^\mu=k_+^\mu$

p_1 = sympy.Matrix([E, 0, 0, E])
p_2 = sympy.Matrix([E, 0, 0, -E])
k_1 = sympy.Matrix([E, p * sympy.sin(t), 0, p * sympy.cos(t)])
k_2 = sympy.Matrix([E, -p * sympy.sin(t), 0, -p * sympy.cos(t)])

$W^-$の偏極ベクトル$\epsilon^{\mu\ast}_{1\bullet}$

e_1Rc = sympy.Matrix([0, sympy.cos(t), -1j, -sympy.sin(t)]) / sympy.sqrt(2)
e_1Lc = sympy.Matrix([0, sympy.cos(t), 1j, -sympy.sin(t)]) / sympy.sqrt(2)
e_1longc = sympy.Matrix([p, E*sympy.sin(t), 0, E*sympy.cos(t)]) / m_W

$W^+$の偏極ベクトル$\epsilon^{\mu\ast}_{2\bullet}$

e_2Rc = sympy.Matrix([0, -sympy.cos(t), -1j, sympy.sin(t)]) / sympy.sqrt(2)
e_2Lc = sympy.Matrix([0, -sympy.cos(t), 1j, sympy.sin(t)]) / sympy.sqrt(2)
e_2longc = sympy.Matrix([p, -E*sympy.sin(t), 0, -E*sympy.cos(t)]) / m_W

def iM_12a(f, af, pol1, pol2):
    coef = sympy.I * e**2 * m_Z**2 / (s*(s-m_Z**2))
    mat = af.T * gamma0 * (prod(pol1, pol2) * (slashed(k_1 - k_2)) - slashed(pol1) * prod(2*k_1+k_2, pol2) + slashed(pol2) * prod(k_1+2*k_2, pol1)) * f
    #
    return coef * mat[0, 0]

def iM_12b(f, af, pol1, pol2):
    coef = - sympy.I * e**2 / (2*sympy.sin(tw)**2 * (s-m_Z**2))
    mat = af.T * gamma0 * (prod(pol1, pol2) * (slashed(k_1 - k_2)) - slashed(pol1) * prod(2*k_1+k_2, pol2) + slashed(pol2) * prod(k_1+2*k_2, pol1))* f
    #
    return coef * mat[0, 0]

def iM_3(f, af, pol1, pol2):
    coef = - sympy.I * e**2 / (2*sympy.sin(tw)**2 * prod(p_1-k_1, p_1-k_1))
    mat = af.T * gamma0 * slashed(pol2) * slashed(p_1-k_1) * slashed(pol1) * f
    return coef * mat[0, 0]

def dsdc(f, af, pol1, pol2):
    iM = iM_12a(f, af, pol1, pol2) + iM_12b(f, af, pol1, pol2) + iM_3(f, af, pol1, pol2)
    R = e**4 / (48*sympy.pi * E**2)
    dsdc =  sympy.Abs(iM) ** 2 / (64*sympy.pi*E**2 * R)
    dsdc = dsdc.subs([(E, 500), (m_W, 80.433), (tw, sympy.asin(sympy.sqrt(0.22290)))])
    dsdc = dsdc.simplify()
    return dsdc

def plot_dsdc(y, label):
    return sympy.plotting.plot_parametric(sympy.cos(t), y, (t, 0.1, 3), xlabel=r"$\cos\theta$", ylabel=r"$d\sigma/d\cos\theta$", yscale="log", ylim=(0.01, 100), axis_center=(-1, 0.01), legend=True, label=label, show=False)

$e^-_Le^+_R \to W^-_RW^+_R$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Rc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Rc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Rc, e_2Rc))))
pp = plot_dsdc(dsdc(u_L, v_L, e_1Rc, e_2Rc), r"$(+, +)$")

$\displaystyle \frac{i e^{2} m_{W}^{2} \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{E \left(4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}\right)}$

$\displaystyle - \frac{2 i E e^{2} \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{\left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}$

$\displaystyle \frac{i E e^{2} \left(- E \cos{\left(\theta \right)} + \sqrt{E^{2} - m_{W}^{2}}\right) \sin{\left(\theta \right)}}{\left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_LW^+_L$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Lc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Lc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Lc, e_2Lc))))
mm = plot_dsdc(dsdc(u_L, v_L, e_1Lc, e_2Lc), r"$(-, -)$")

$\displaystyle \frac{i e^{2} m_{W}^{2} \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{E \left(4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}\right)}$

$\displaystyle - \frac{2 i E e^{2} \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{\left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}$

$\displaystyle \frac{i E e^{2} \left(- E \cos{\left(\theta \right)} + \sqrt{E^{2} - m_{W}^{2}}\right) \sin{\left(\theta \right)}}{\left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_RW^+_L$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Rc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Rc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Rc, e_2Lc))))
pm = plot_dsdc(dsdc(u_L, v_L, e_1Rc, e_2Lc), r"$(+, -)$")

$\displaystyle 0$

$\displaystyle 0$

$\displaystyle \frac{i E^{2} e^{2} \cdot \left(1 - \cos{\left(\theta \right)}\right) \sin{\left(\theta \right)}}{\left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_LW^+_R$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Lc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Lc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Lc, e_2Rc))))
mp = plot_dsdc(dsdc(u_L, v_L, e_1Lc, e_2Rc), r"$(-, +)$")

$\displaystyle 0$

$\displaystyle 0$

$\displaystyle \frac{i E^{2} e^{2} \left(\cos{\left(\theta \right)} + 1\right) \sin{\left(\theta \right)}}{\left(- 2 E^{2} + 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} + m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_LW^+_\text{ln} + W^-_\text{ln}W^+_R$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Lc, e_2longc) + iM_12a(u_L, v_L, e_1longc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Lc, e_2longc) + iM_12b(u_L, v_L, e_1longc, e_2Rc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Lc, e_2longc) + iM_3(u_L, v_L, e_1longc, e_2Rc))))
mzzp = plot_dsdc(dsdc(u_L, v_L, e_1Lc, e_2longc) + dsdc(u_L, v_L, e_1longc, e_2Rc), r"$(-, 0)+(0, +)$")

$\displaystyle - \frac{2 \sqrt{2} i e^{2} m_{W} \sqrt{E^{2} - m_{W}^{2}} \left(\cos{\left(\theta \right)} + 1\right)}{4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}}$

$\displaystyle \frac{4 \sqrt{2} i E^{2} e^{2} \sqrt{E^{2} - m_{W}^{2}} \left(\cos{\left(\theta \right)} + 1\right)}{m_{W} \left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}$

$\displaystyle \frac{\sqrt{2} i E e^{2} \cdot \left(2 E^{2} \cos^{2}{\left(\theta \right)} + 2 E^{2} \cos{\left(\theta \right)} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - 2 E \sqrt{E^{2} - m_{W}^{2}} - m_{W}^{2} \cos{\left(\theta \right)} - m_{W}^{2}\right)}{m_{W} \left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_RW^+_\text{ln} + W^-_\text{ln}W^+_L $

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1Rc, e_2longc) + iM_12a(u_L, v_L, e_1longc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1Rc, e_2longc) + iM_12b(u_L, v_L, e_1longc, e_2Lc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1Rc, e_2longc) + iM_3(u_L, v_L, e_1longc, e_2Lc))))
zmpz = plot_dsdc(dsdc(u_L, v_L, e_1Rc, e_2longc) + dsdc(u_L, v_L, e_1longc, e_2Lc), r"$(+, 0)+(0, -)$")

$\displaystyle - \frac{2 \sqrt{2} i e^{2} m_{W} \sqrt{E^{2} - m_{W}^{2}} \left(\cos{\left(\theta \right)} - 1\right)}{4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}}$

$\displaystyle \frac{4 \sqrt{2} i E^{2} e^{2} \sqrt{E^{2} - m_{W}^{2}} \left(\cos{\left(\theta \right)} - 1\right)}{m_{W} \left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}$

$\displaystyle \frac{\sqrt{2} i E e^{2} \cdot \left(4 E^{2} \cos^{2}{\left(\theta \right)} - 4 E^{2} \cos{\left(\theta \right)} - 4 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} + 4 E \sqrt{E^{2} - m_{W}^{2}} + 2 m_{W}^{2} \cos{\left(\theta \right)} - 2 m_{W}^{2}\right)}{2 m_{W} \left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

$e^-_Le^+_R \to W^-_\text{ln}W^+_\text{ln}$

display(sympy.nsimplify(sympy.simplify(iM_12a(u_L, v_L, e_1longc, e_2longc))))
display(sympy.nsimplify(sympy.simplify(iM_12b(u_L, v_L, e_1longc, e_2longc))))
display(sympy.nsimplify(sympy.simplify(iM_3(u_L, v_L, e_1longc, e_2longc))))
zz = plot_dsdc(dsdc(u_L, v_L, e_1longc, e_2longc), r"$(0, 0)$")

$\displaystyle \frac{i e^{2} \left(- 2 E^{2} - m_{W}^{2}\right) \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{E \left(4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}\right)}$

$\displaystyle \frac{2 i E e^{2} \sqrt{E^{2} - m_{W}^{2}} \cdot \left(2 E^{2} + m_{W}^{2}\right) \sin{\left(\theta \right)}}{m_{W}^{2} \cdot \left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}$

$\displaystyle \frac{i E e^{2} \cdot \left(2 E^{3} \cos{\left(\theta \right)} - 2 E^{2} \sqrt{E^{2} - m_{W}^{2}} - m_{W}^{2} \sqrt{E^{2} - m_{W}^{2}}\right) \sin{\left(\theta \right)}}{m_{W}^{2} \cdot \left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}$

zz.extend(pm)
zz.extend(mp)
zz.extend(mzzp)
zz.extend(zmpz)
zz.show()

(21.108)

display((sympy.I * e**2 * v_L.T * gamma0 * slashed(k_2-k_1) * u_L)[0, 0])
print(sympy.latex(sympy.simplify(iM_12a(u_L, v_L, e_1longc, e_2longc))))
print(sympy.latex(sympy.simplify(iM_12b(u_L, v_L, e_1longc, e_2longc))))
print(sympy.latex(sympy.simplify(iM_3(u_L, v_L, e_1longc, e_2longc))))

$\displaystyle - 4 i E e^{2} \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}$

\frac{i e^{2} \left(- 2 E^{2} - m_{W}^{2}\right) \sqrt{E^{2} - m_{W}^{2}} \sin{\left(\theta \right)}}{E \left(4 E^{2} \cos^{2}{\left(\theta_{w} \right)} - m_{W}^{2}\right)}
\frac{2 i E e^{2} \sqrt{E^{2} - m_{W}^{2}} \cdot \left(2 E^{2} + m_{W}^{2}\right) \sin{\left(\theta \right)}}{m_{W}^{2} \cdot \left(2 E^{2} \cos{\left(2 \theta_{w} \right)} + 2 E^{2} - m_{W}^{2}\right) \tan^{2}{\left(\theta_{w} \right)}}
\frac{i E e^{2} \cdot \left(2 E^{3} \cos{\left(\theta \right)} - 2 E^{2} \sqrt{E^{2} - m_{W}^{2}} - m_{W}^{2} \sqrt{E^{2} - m_{W}^{2}}\right) \sin{\left(\theta \right)}}{m_{W}^{2} \cdot \left(2 E^{2} - 2 E \sqrt{E^{2} - m_{W}^{2}} \cos{\left(\theta \right)} - m_{W}^{2}\right) \sin^{2}{\left(\theta_{w} \right)}}