Abstract
This paper explores long-range interactions between magnetically charged excitations of the vacuum of the dual Landau–Ginzburg theory (DLGT) and the dual Abrikosov vortices present in the same vacuum. We show that, in the London limit of DLGT, the corresponding Aharonov–Bohm-type interactions possess such a coupling that the interactions reduce to a trivial factor of e2πi×(integer). The same analysis is done in the SU(N c)-inspired [U(1)]\(^{N_{\mathrm{c}}-1}\)-invariant DLGT, as well as in DLGT extended by a Chern–Simons term. It is furthermore explicitly shown that the Chern–Simons term leads to the appearance of knotted dual Abrikosov vortices.
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Acknowledgements
This work was supported by the Portuguese Foundation for Science and Technology (FCT, program Ciência-2008) and by the Center for Physics of Fundamental Interactions (CFIF) at Instituto Superior Técnico (IST), Lisbon. The author is grateful to the whole staff of the Department of Physics of IST for their cordial hospitality.
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Appendices
Appendix A: Some details of the derivation of Eq. (14)
With the use of Eq. (12), and owing to the conservation of j μ , one has
We furthermore assume the standard normalization 〈1〉=1 of the functional average, which implies a division by the functional integral \(\int{ \mathcal{D}}J_{\mu}\mathrm{e}^{-(2\pi\eta)^{2}\int_{x,y}J_{\mu}^{x}J_{\mu}^{y}D_{\mathsf{m}}^{xy}}\) corresponding to the first term on the right-hand side of Eq. (A.1). Thus, we always imply that the measure \({\mathcal{D}}J_{\mu}\) is normalized by a division by this integral.
The last term in Eq. (A.1) can be represented, through the integration by parts, as \((\mu\eta)^{2}\int_{x,y} D_{\mathsf{m}}^{xy}j_{\mu}^{x}\int_{u} D_{0}^{yu}j_{\mu}^{u}\). The y-integration in this expression is straightforward:
where the equality \(\frac{1}{\mathbf{p}^{2} (\mathbf{p}^{2}+\mathsf{m}^{2})}=\frac{1}{\mathsf{m}^{2}} (\frac{1}{\mathbf{p}^{2}}-\frac{1}{\mathbf{p}^{2}+\mathsf{m}^{2}} )\) has been used at the last step. Using further the explicit form of m, Eq. (13), we can represent the last term in Eq. (A.1) as \(\frac{1}{2\nu}\int_{x,y}j_{\mu}^{x} j_{\mu}^{y} (D_{0}^{xy}-D_{\mathsf{m}}^{xy} )\).
In the second term on the right-hand side of Eq. (A.1), one can use the equality \(\partial_{\nu}^{y}\int_{z} D_{0}^{yz}j_{\lambda}^{z}=\int_{z} D_{0}^{yz}\partial_{\nu}^{z} j_{\lambda}^{z}\), which yields the same y-integration as in Eq. (A.2): \(\int_{y} D_{\mathsf{m}}^{xy}D_{0}^{yz}= \frac{1}{\mathsf{m}^{2}} (D_{0}^{xz}-D_{\mathsf{m}}^{xz} )\). Upon the subsequent integration by parts, we obtain for this term the following expression: \(\frac{2\pi i}{\mu\nu}\varepsilon_{\mu\nu\lambda} \int_{x,y}J_{\mu}^{x} j_{\nu}^{y}\partial_{\lambda}^{x} (D_{0}^{xy}-D_{\mathsf{m}}^{xy} )\). Noticing also the definition of the Gauss’ linking number, \(\hat{L}(j,J)= \varepsilon_{\mu\nu\lambda}\int_{x,y}J_{\mu}^{x}j_{\nu}^{y}\partial_{\lambda}^{x}D_{0}^{xy}\), we arrive at Eq. (14).
Appendix B: Some details of the derivation of Eq. (26)
For Θ’s obeying condition (24), one readily obtains the inequality
Owing to this inequality, the term \(\frac{\mu^{2}}{8\varTheta^{2}\nu}(\partial_{\mu}\varphi_{\mu})^{2}\) in Eq. (25) can be neglected in comparison with the absolute value of the term \(-\frac{1}{8\eta^{2}}\varPhi_{\mu\nu}^{2}\). The resulting Gaussian φ μ -integration can be performed by seeking the saddle-point function in the form \(\varphi_{\mu}=\varphi_{\mu}^{(1)}+i\varphi_{\mu}^{(2)}\), and solving the so-emerging system of equations for \(\varphi_{\mu}^{(1)}\) and \(\varphi_{\mu}^{(2)}\). The result can be written as
where \(R_{\mu}\equiv2\pi J_{\mu}+\frac{\mu}{m}j_{\mu}\) and \(S_{\mu}\equiv\frac{\mu}{m^{2}} \varepsilon_{\mu\nu\lambda}\partial_{\nu}j_{\lambda}\) are, respectively, the real and the imaginary parts of the current which couples to φ μ in Eq. (25). The obtained real and imaginary parts of the saddle-point function φ μ entering Eq. (B.2) read
and
with the new mass parameter \({\mathcal{M}}\equiv\frac{\mu^{2}\eta^{2}}{\varTheta}\). Furthermore, in the limit (B.1) at issue, the \({\mathcal{O}}({\mathcal{M}}/m)\)-terms in Eqs. (B.3) and (B.4) should be neglected compared to 1. That yields the following saddle-point expressions for \(\varphi_{\mu}^{(1)}\) and \(\varphi_{\mu}^{(2)}\):
Substituting them into Eq. (B.2), we obtain for the Wilson loop in the limit (24) expression (26).
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Antonov, D. Triviality of the Aharonov–Bohm interaction in a spatially confining vacuum. Eur. Phys. J. C 72, 2015 (2012). https://doi.org/10.1140/epjc/s10052-012-2015-0
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DOI: https://doi.org/10.1140/epjc/s10052-012-2015-0