2000 character limit reached
Reference frame dependence of the periodically oscillating Coulomb field in the Proca theory
Published 26 Feb 2024 in hep-ph and hep-th | (2402.17789v1)
Abstract: The Proca theory of the real massive vector field admits non-equilibrium solutions, where the asymptotic dynamics of the electric field is dominated by the periodically oscillating Coulomb component. We discuss how such field configurations are seen in different reference frames, where we find an intriguing spatial pattern of the vector field and the electromagnetic field associated with it. Our studies are carried out in the framework of the classical Proca theory.
- These statements can be justified by doing the angular integration in (I) and (4). Writing d3ksuperscript𝑑3𝑘d^{3}kitalic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_k as dωkωk2dΩ(𝒌^)𝑑subscript𝜔𝑘superscriptsubscript𝜔𝑘2𝑑Ω^𝒌d\omega_{k}\omega_{k}^{2}d\Omega(\hat{{\bm{k}}})italic_d italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d roman_Ω ( over^ start_ARG bold_italic_k end_ARG ), this amounts to the evaluation of ∫𝑑Ω(𝒌^)differential-dΩ^𝒌\int d\Omega(\hat{{\bm{k}}})∫ italic_d roman_Ω ( over^ start_ARG bold_italic_k end_ARG ) in the above-mentioned expressions.
- Positive even β𝛽\betaitalic_β’s are preferable because such a choice facilitates analytical evaluation of the discussed fields. The chosen values of β𝛽\betaitalic_β guarantee differentiability of the vector field in the sense specified below (I). These technical remarks follow from the discussion presented in Damski (2023a).
- The mean is defined as ∑i=1n|δ𝑬i|/|𝑬i|superscriptsubscript𝑖1𝑛𝛿subscript𝑬𝑖subscript𝑬𝑖\sum_{i=1}^{n}|\delta{\bm{E}}_{i}|/|{\bm{E}}_{i}|∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | italic_δ bold_italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | / | bold_italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | divided by n𝑛nitalic_n, where 𝑬isubscript𝑬𝑖{\bm{E}}_{i}bold_italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the i𝑖iitalic_i-th vector displayed in Fig. 1 for a given mR𝑚𝑅mRitalic_m italic_R and δ𝑬i𝛿subscript𝑬𝑖\delta{\bm{E}}_{i}italic_δ bold_italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the subleading contribution to such 𝑬isubscript𝑬𝑖{\bm{E}}_{i}bold_italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (e.g. n=18𝑛18n=18italic_n = 18 for mR=6𝑚𝑅6mR=6italic_m italic_R = 6).
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