Vertex function

<h2 id="definition">Definition</h2>
<p>The vertex function 
  
    
      
        
          Γ
          
            μ
          
        
      
    
    {\displaystyle \Gamma ^{\mu }}
  
 can be defined in terms of a <a href="/facts/Functional_derivative/urNSIWYc">functional derivative</a> of the <a href="/facts/Effective_action/slyeRgDh">effective action</a> Seff as
</p>

Γ
          
            μ
          
        
        =
        −
        
          
            1
            e
          
        
        
          
            
              
                δ
                
                  3
                
              
              
                S
                
                  
                    e
                    f
                    f
                  
                
              
            
            
              δ
              
                
                  
                    ψ
                    ¯
                  
                
              
              δ
              ψ
              δ
              
                A
                
                  μ
                
              
            
          
        
      
    
    {\displaystyle \Gamma ^{\mu }=-{1 \over e}{\delta ^{3}S_{\mathrm {eff} } \over \delta {\bar {\psi }}\delta \psi \delta A_{\mu }}}

<p>The dominant (and classical) contribution to 
  
    
      
        
          Γ
          
            μ
          
        
      
    
    {\displaystyle \Gamma ^{\mu }}
  
 is the <a href="/facts/Gamma_matrix/C2y9Aumy">gamma matrix</a> 
  
    
      
        
          γ
          
            μ
          
        
      
    
    {\displaystyle \gamma ^{\mu }}
  
, which explains the choice of the letter.  The vertex function is constrained by the symmetries of quantum electrodynamics — <a href="/facts/Lorentz_invariance/K5AEuGKg">Lorentz invariance</a>; <a href="/facts/Gauge_invariance/fSW3gggv">gauge invariance</a> or the <a href="/facts/Photon_polarization/fh6oqC5E">transversality</a> of the photon, as expressed by the <a href="/facts/Ward_identity/8Fksdh1f">Ward identity</a>; and invariance under <a href="/facts/Parity_(physics)/emSbJ4hD">parity</a> — to take the following form:
</p>

Γ
          
            μ
          
        
        =
        
          γ
          
            μ
          
        
        
          F
          
            1
          
        
        (
        
          q
          
            2
          
        
        )
        +
        
          
            
              i
              
                σ
                
                  μ
                  ν
                
              
              
                q
                
                  ν
                
              
            
            
              2
              m
            
          
        
        
          F
          
            2
          
        
        (
        
          q
          
            2
          
        
        )
      
    
    {\displaystyle \Gamma ^{\mu }=\gamma ^{\mu }F_{1}(q^{2})+{\frac {i\sigma ^{\mu \nu }q_{\nu }}{2m}}F_{2}(q^{2})}

<p>where 
  
    
      
        
          σ
          
            μ
            ν
          
        
        =
        (
        i
        
          /
        
        2
        )
        [
        
          γ
          
            μ
          
        
        ,
        
          γ
          
            ν
          
        
        ]
      
    
    {\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]}
  
, 
  
    
      
        
          q
          
            ν
          
        
      
    
    {\displaystyle q_{\nu }}
  
 is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F1(q2) and F2(q2) are <i><a href="/facts/Form_factor_(quantum_field_theory)/A499wcHH">form factors</a></i> that depend only on the momentum transfer q2.  At tree level (or leading order), F1(q2) = 1 and F2(q2) = 0.  Beyond leading order, the corrections to F1(0) are exactly canceled by the field strength renormalization.  The form factor F2(0) corresponds to the <a href="/facts/Anomalous_magnetic_moment/s8fVr0An">anomalous magnetic moment</a> <i>a</i> of the fermion, defined in terms of the <a href="/facts/Land%25C3%25A9_g-factor/LD8uHVGW">Landé g-factor</a> as:
</p>

a
        =
        
          
            
              g
              −
              2
            
            2
          
        
        =
        
          F
          
            2
          
        
        (
        0
        )
      
    
    {\displaystyle a={\frac {g-2}{2}}=F_{2}(0)}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Nonoblique_correction/Qq45yLue">Nonoblique correction</a></li></ul>

<ul><li>Gross, F. (1993). <i>Relativistic Quantum Mechanics and Field Theory</i> (1st ed.). <a href="/facts/Wiley-VCH/4oQWIG4d">Wiley-VCH</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0471591139.</li>
<li><a href="/facts/Michael_Peskin/f8mVDAP0">Peskin, Michael E.</a>; Schroeder, Daniel V. (1995). <a href="https://archive.org/details/introductiontoqu0000pesk"><i>An Introduction to Quantum Field Theory</i></a>. Reading: Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-50397-2.</li>
<li><a href="/facts/Steven_Weinberg/G61D6kJw">Weinberg, S.</a> (2002), <a href="https://archive.org/details/quantumtheoryoff00stev"><i>Foundations</i></a>, The Quantum Theory of Fields, vol. I, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-55001-7</li></ul>
<h2 id="external-links">External links</h2>
<ul><li> Media related to Vertex function at Wikimedia Commons</li></ul>

Vertex function open-in-new

Vertex function