The quark spectral functions and the Hadron Vacuum Polarization from application of DSEs in Minkowski space
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V. Šauli
The quark spectral functions and the Hadron Vacuum Polarization from application of DSEs in Minkowski space
Received: 5 December 2019 / Accepted: 16 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The hadronic vacuum polarization function h for two light flavors is computed on the entire domain of spacelike and timelike momenta using a framework of Dyson–Schwinger equations. The analytical continuation of the function h is based on the utilization of the Gauge Technique with the entry of QCD Green’s functions determined from generalized quark spectral functions. For the first time, the light quark spectral functions are extracted from the solution of the gap equation for the quark propagator. The scale is set up by the phenomena of dynamical chiral symmetry breaking, which is a striking feature of low energy QCD.
1 Introduction The hadron vacuum polarization function h (x) is conventionally defined through the vacuum expectation a b of current-current correlator such that ab (x) = q 0| jq (0) jq (x)|0 where the sum runs over the all quark h flavors. It is also an alternative name for the part of the photon self-energy (x) due to the quark loops. Using the continuous functional formalism it can be precisely defined as double differentiation of generating functional [φ S M ] with respect to the photon fields A: (x − y)μν =
δ 2 [φ S M ] |φ − · · · , δ Aμ (x)δ Aν (y) S M
(1.1)
where φ S M stand for whole known ensemble of Standard Model fields and where the dots stands for the inverse of the free photon propagator. Using a standard routine [1,2] one can derive for the hadronic part of the Fourier transform of (1.1) a well known expression μν
h (s) = −ie2 Nc
q
eq2 T r
d 4k μ (k − q, k)Sq (k)γ ν Sq (k − q), (2π)4 q
(1.2)
where the photon momentum satisfies s = q 2 , eq is the quark charge in units of electron charge e and the trace μ is taken in Dirac space and q is the dressed quark-photon proper vertex, Sq is the dressed quark propagator, both functions satisfy their own Dyson–Schwinger equations, solutions of them in Minkowski space will be the subject of presented paper. Together with the leptonic polarization function and loops containing gauge bosons W, Z , the function h completes the (inverse) photon propagator (1.1). In the spacelike domain of momenta the polarization function is responsible for a smooth and slow increase of the running QED charge. However, for positive s the V. Šauli (B) Department of Theoretical Physics, NPI Rez near Prague, Czech Academy of Sciences, Prague, Czech Republic E-mail: [email protected]
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V. Šauli
complexity of hadronic polarization h causes measurable interference effect in the fine structure constant αQ E D . Photon polarization function offers a great amount of physical information and in the timelike domain of momenta, it is measured with continuously improved accuracy for many reasons. It is an experimental fact, that heavier quark q is, larger quantum fluctuati
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