Renormalized electric charge(coupling to photon) of fermions are universal?
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1-loop correction(figure below) to a vertex where fermions couples to a photon is evaluated in Matthew Schwarz's textbook eqn (19.57)
iM=iF2(p2)σμν2m+iγμ(1+f(p2))iM=iF2(p2)σμν2m+iγμ(1+f(p2))
where
f(p2)=e2R8π2∫10dxdydzδ(x+y+z−1)⋅⋅[lnzΛ2(1−z)2m2R−xyp2+zm2γ+p2(1−y)(1−x)+m2R(1−4z+z2)(1−z)2m2R−xyp2+zm2γ]f(p2)=eR28π2∫01dxdydzδ(x+y+z−1)⋅⋅[lnzΛ2(1−z)2mR2−xyp2+zmγ2+p2(1−y)(1−x)+mR2(1−4z+z2)(1−z)2mR2−xyp2+zmγ2]
where Λ2,mγ,mRΛ2,mγ,mR are cut-off scale and dressed photon mass (to deal with IR divergence) and mass of fermion in the loop. Although I didn't complete the integral yet, it appears that the result can depend on m2RmR2. In such cases, will fermions with different mass have different (renormalized) coupling to photon?
If that is true, what will happen if mRmR is so heavy or so light?
iM=iF2(p2)σμν2m+iγμ(1+f(p2))iM=iF2(p2)σμν2m+iγμ(1+f(p2))
where
f(p2)=e2R8π2∫10dxdydzδ(x+y+z−1)⋅⋅[lnzΛ2(1−z)2m2R−xyp2+zm2γ+p2(1−y)(1−x)+m2R(1−4z+z2)(1−z)2m2R−xyp2+zm2γ]f(p2)=eR28π2∫01dxdydzδ(x+y+z−1)⋅⋅[lnzΛ2(1−z)2mR2−xyp2+zmγ2+p2(1−y)(1−x)+mR2(1−4z+z2)(1−z)2mR2−xyp2+zmγ2]
where Λ2,mγ,mRΛ2,mγ,mR are cut-off scale and dressed photon mass (to deal with IR divergence) and mass of fermion in the loop. Although I didn't complete the integral yet, it appears that the result can depend on m2RmR2. In such cases, will fermions with different mass have different (renormalized) coupling to photon?
If that is true, what will happen if mRmR is so heavy or so light?
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Explanation:
Standard Model of Particle Physics. The diagram shows the elementary particles of the Standard Model (the Higgs boson, the three generations of quarks and leptons, and the gauge bosons), including their names, masses, spins, charges, chiralities, and interactions with the strong, weak and electromagnetic forces. It also depicts the crucial role of the Higgs boson in electroweak symmetry breaking, and shows how the properties of the various particles differ in the (high-energy) symmetric phase (top) and the (low-energy) broken-symmetry phase (bottom).
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