Anomalous magnetic moment of electron

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imported>Pythagoras0님의 2015년 3월 17일 (화) 18:31 판
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introduction

  • amplitude = sum of integrals = \(\sum_{n\text{ loops}}\) sum of integrals
  • anomalous electron magnetic dipole moment 1.00115965219
  • theoretical computation matches 11 digits with experiments
  • as n grows, number of Feynman diagrams grows exponentially
  • integrals are becoming difficult

Lande's question

  • Bohr magneton \(\mu_0=e\hbar /2mc\)
  • spin magnetic dipole moment \(\mu_s\)
  • Q. \(\mu_s=\mu_0\) ? (Back and Lande 1925)
  • We define $g$ to be the gyromagnetic ratio

\[\mu_s=\frac{g\mu_0}{2}\]


classical magnetic moment

  • read spin system first
  • gyromagnetic ratio is defined as "magnetic dipole moment"/"angular momentum"

\[\gamma = \mu/L=-e/2m_e\] [/pages/7141159/attachments/4562863 I15-62-g20.jpg]


orbital

  • Let $e$, $m_e$, $v$, and $r$ be the electron's charge, mass, velocity, and radius, respectively.
  • A classical electron moving around a nucleus in a circular orbit
    • orbital angular momentum, \(L=m_evr\)
    • magnetic dipole moment, \(\mu= -evr/2\)
  • we get $\gamma=\mu/L=-e/2m_e$

spin

  • A classical electron of homogeneous mass and charge density rotating about a symmetry axis
    • spin angular momentum, \(L=(3/5)m_eR^2\Omega\)
    • magnetic dipole moment, \(\mu= -(3/10)eR^2\Omega\), where $R$ and $\Omega$ are the electron's classical radius and rotating frequency
  • we get \(\gamma = \mu/L=-e/2m_e\)


anamalous electron magnetic dipole moment

Dirac theory

  • In Dirac’s theory a point like spin 1/2 object of electric charge $q$ and mass $m$ has a magnetic moment

\[\mathbf{\mu}=q\mathbf{S}/m\]

  • so the Bohr magneton of the electron becomes

\[\mu_\mathrm{B} = {{e \hbar} \over {2 m_\mathrm{e}}}\] since the spin of the electron is \(S=\frac{\hbar}{2}\)


g-factor

  • there are correction terms to the spin magnetic moment of the electron as shown by experiments
  • actual spin magnetic moment of the electron involves the spin g-factor (gyromagnetic ratio)

\[\vec{\mu}_S \ = g_e \mu_\mathrm{B} \frac{\vec{S}}{\hbar}=g\frac{e}{2 m_{e}} \ \vec{S}\]

  • The g factor sets the strength of an electron’s interaction with a magnetic field.


classical vs quantum

  • In classical physics (left) magnetic lines of force (perpendicular to the page) induce a curvature in the electron’s path.
  • In quantum electrodynamics (right) the electron interacts with the field by emitting or absorbing a photon.

[/pages/3589069/attachments/4562673 2004329152457_150.gif]

  • vertex correction

$$g = 2 ( 1 + \alpha/2\pi)$$ where $$\alpha= e^2/(4\pi \hbar c) \sim 1/137.036$$ is the fine structure constant

$$g/2=1+c_1\frac{\alpha}{2\pi}+c_2(\frac{\alpha}{2\pi})^2+c_3(\frac{\alpha}{2\pi})^3+\cdots=1.00115965219+\cdots$$



Feynmann diagrams

 

tree level and one-loop diagrams

  • 1 one-loop diagram
    [/pages/7141159/attachments/4563145 2004329152921_150.gif]
  • Feynman, Julian Schwinger, Sin-Itiro Tomonaga and Freeman Dyson
  • Schwinger showed that the one-loop contribution to the "anomalous magnetic moment" of the electron is \(\alpha/{2\pi}=0.00116\cdots\)
  • Schwinger, Julian. 1948. On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Physical Review 73, no. 4 (February 15): 416. doi:10.1103/PhysRev.73.416. 
     
  • http://www.wolframalpha.com/input/?i=fine+structure+constant%2F%282pi%29

 

 

two-loop diagrams

  • 7 two-loop diagrams
    [/pages/3589069/attachments/4562669 2004329153354_150.gif]
    [/pages/7141159/attachments/4562733 I15-62-g2c.jpg]

 

 

three-loop diagrams

  • 72 three-loop diagrams
  • [/pages/3589069/attachments/4562671 200432915395_150.gif]
  • Kinoshita, Toichiro. 1995. New Value of the alpha^{3} Electron Anomalous Magnetic Moment. Physical Review Letters 75, no. 26 (December 25): 4728. doi:10.1103/PhysRevLett.75.4728. 
     


four-loop diagrams

  •  891 diagrams

 

 

five-loop Feynman diagrams


anaomalous muon magnetic dipole moment


 

memo

 

 

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