Anomalous magnetic moment of electron

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]

• pictures from Gyromagnetic Ratio and Anomalous Magnetic Moment

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

• http://www.wolframalpha.com/input/?i=1%2F%28137*2pi%29
• The event is represented in a Feynman diagram, where space extends along the horizontal axis and time moves up the vertical axis.

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

• http://docs.google.com/viewer?a=v&q=cache:5hOX9DCrL7sJ:www.physics.ohio-state.edu/~kass/P780_L3_sp03.ppt+anamalous+magnetic+moment+electron+feynman+diagram&hl=ko&gl=us&pid=bl&srcid=ADGEEShmuOjISGxcCejbd6l7kWuRiTY7AtBHwKpZ_Zec4dSTPlJ8kqZSA80srABAl8PEFKnJJVfrawIHlkI0Z9S5wA1ArJMpmMERZp3I3ppK4BN5drRWx4mJi8VTW_wf8xjrs3v1VOqX&sig=AHIEtbTIAIEubf5ZHYPXPv4aE6ImvmxEVw

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

• Numerical calculation of electron g-2 at 4 loops in QED S. Laporta
  

 

 

five-loop Feynman diagrams

• There are 12,672
• http://www.strings.ph.qmul.ac.uk/~bigdraw/feynman/slide3.html[1]

anaomalous muon magnetic dipole moment

• anaomalous muon magnetic dipole moment is still unknown
• http://eskesthai.blogspot.com/2010/12/muon.html

 

memo

• Calculation of the Anomalous Magnetic Moment of the Electron from the Evans-Unified Field Theory

 

 

 

 

computational resource

• number of Feynman diagrams http://oeis.org/A005413

encyclopedia

• http://en.wikipedia.org/wiki/G-factor_%28physics%29
• http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment
• http://en.wikipedia.org/wiki/Bohr_magneton

expositions

• Xiaoyi, anomalous magnetic moment
• Brian Hayes, “g-OLOGY,” American Scientist 92, no. 3 (2004): 212. http://www.americanscientist.org/issues/num2/g-ology/1

articles

• Broadhurst, D. J, and D. Kreimer. 1996. Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. hep-th/9609128 (September 16). doi:doi:10.1016/S0370-2693(96)01623-1. http://arxiv.org/abs/hep-th/9609128.
• 1948 슈와잉거 (J. Schwinger, Phys.Rev. 73(1948) 416L)

books

• Mandl, Franz, and Graham Shaw. 2010. Quantum Field Theory. John Wiley & Sons. http://books.google.de/books/about/Quantum_Field_Theory.html?id=Ef4zDW1V2LkC&redir_esc=y
  • chapter 9