Why is Hadronic Vacuum Polarization a Matrix Element of Current Products?

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SUMMARY

The discussion centers on the concept of Hadronic Vacuum Polarization (HVP) as a matrix element of current products, specifically in the context of quantum field theory as presented in Peskin & Schroeder's textbook. The formula σ(e^{+}e^{-})=(1/2s)ImM(e^{+}e^{-}→e^{+}e^{-}) is derived, where the matrix element iM is expressed in terms of the electromagnetic current J^{\mu} of quarks. The key point of confusion is the representation of the hadronic vacuum polarization tensor i∏^{\mu\nu}_{h}(q) as an integral involving the time-ordered product of currents, which reflects the vacuum fluctuations induced by virtual photons.

PREREQUISITES
  • Quantum Field Theory (QFT) fundamentals
  • Understanding of matrix elements in particle physics
  • Familiarity with Peskin & Schroeder's QFT textbook
  • Knowledge of electromagnetic current and its role in particle interactions
NEXT STEPS
  • Study the derivation of the hadronic vacuum polarization tensor in QFT
  • Explore the implications of vacuum fluctuations in quantum electrodynamics (QED)
  • Learn about the time-ordered product of operators in quantum field theory
  • Investigate the role of virtual particles in particle interactions
USEFUL FOR

Physicists, graduate students in theoretical physics, and anyone studying quantum field theory and particle interactions, particularly those interested in hadronic processes and vacuum polarization effects.

ndung200790
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Please teach me this:
In chapter 18.4 Peskin&Schoeder(QFT) they consider the annihilation of electron and positron to hadron.Ignoring the mass of the electron,we have:

σ(e[itex]^{+}[/itex]e[itex]^{-}[/itex])=(1/2s)ImM(e[itex]^{+}[/itex]e[itex]^{-}[/itex]→e[itex]^{+}[/itex]e[itex]^{-}[/itex]).

We have:

iM=(-ie)[itex]^{2}[/itex]u[itex]^{-}[/itex](k)[itex]\gamma[/itex][itex]_{\mu}[/itex]v(k[itex]_{+}[/itex](-i/s)(i[itex]\Pi[/itex][itex]^{\mu\nu}_{h}[/itex](q))(-i/s)v[itex]^{-}[/itex](k[itex]_{+}[/itex][itex]\gamma[/itex][itex]_{\nu}[/itex]u(k).

I do not understand why they can write:

i∏[itex]^{\mu\nu}_{h}[/itex](q)=-e[itex]^{2}[/itex][itex]\int[/itex]d[itex]^{4}[/itex]xe[itex]^{iqx}[/itex]<T{J[itex]^{\mu}[/itex](x)J[itex]^{\nu}[/itex](0)>.

Where J[itex]^{\mu}[/itex] is the electromagnetic current of quarks.
Thank you very much for your kind helping.
 
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Now,I think that it is because the fluctuation of vacum(that is excited by virtue photon) making a loop that is product of two currents(not more complex including many hadronics).Is that correct?
 

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