Advertisement

Journal of High Energy Physics

, 2018:106 | Cite as

Large-N \( \mathbb{C}{\mathrm{\mathbb{P}}}^{\mathrm{N}-1} \) sigma model on a finite interval and the renormalized string energy

  • Alessandro Betti
  • Stefano Bolognesi
  • Sven Bjarke Gudnason
  • Kenichi Konishi
  • Keisuke Ohashi
Open Access
Regular Article - Theoretical Physics

Abstract

We continue the analysis started in a recent paper of the large-N two-dimensional \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density \( \mathrm{\mathcal{E}} \) (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to \( \mathrm{\mathcal{E}}\left(x,\varLambda,\ L\right)\to \frac{N}{4\pi }{\varLambda}^2 \) at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Lüscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model in two dimensions for LΛ → ∞.

Keywords

1/N Expansion Sigma Models Solitons Monopoles and Instantons 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Supplementary material

13130_2018_7472_MOESM1_ESM.mp4 (369 kb)
ESM 1 (MP4 368 kb)

References

  1. [1]
    A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].CrossRefADSGoogle Scholar
  2. [2]
    E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  3. [3]
    S. Bolognesi, K. Konishi and K. Ohashi, Large-N \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) σ-model on a finite interval, JHEP 10 (2016) 073 [arXiv:1604.05630] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  4. [4]
    R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Non-Abelian monopoles and the vortices that confine them, Nucl. Phys. B 686 (2004) 119 [hep-th/0312233] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  5. [5]
    K. Konishi, A. Michelini and K. Ohashi, Monopole-vortex complex in a θ vacuum, Phys. Rev. D 82 (2010) 125028 [arXiv:1009.2042] [INSPIRE].ADSGoogle Scholar
  6. [6]
    M. Cipriani, D. Dorigoni, S.B. Gudnason, K. Konishi and A. Michelini, Non-Abelian monopole-vortex complex, Phys. Rev. D 84 (2011) 045024 [arXiv:1106.4214] [INSPIRE].ADSGoogle Scholar
  7. [7]
    C. Chatterjee and K. Konishi, Monopole-vortex complex at large distances and non-Abelian duality, JHEP 09 (2014) 039 [arXiv:1406.5639] [INSPIRE].CrossRefMATHADSGoogle Scholar
  8. [8]
    A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  9. [9]
    R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Non-Abelian superconductors: Vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].CrossRefMATHADSGoogle Scholar
  10. [10]
    M. Shifman and A. Yung, Non-Abelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    A. Milekhin, CP (N − 1) model on finite interval in the large-N limit, Phys. Rev. D 86 (2012) 105002 [arXiv:1207.0417] [INSPIRE].ADSGoogle Scholar
  12. [12]
    S. Monin, M. Shifman and A. Yung, Non-Abelian String of a Finite Length, Phys. Rev. D 92 (2015) 025011 [arXiv:1505.07797] [INSPIRE].MathSciNetADSGoogle Scholar
  13. [13]
    A. Milekhin, \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) σ-model on a finite interval revisited, Phys. Rev. D 95 (2017) 085021 [arXiv:1612.02075] [INSPIRE].ADSGoogle Scholar
  14. [14]
    A. Actor, Temperature Dependence Of The \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) Model And The Analogy With Quantum Chromodynamics, Fortsch. Phys. 33 (1985) 333 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  15. [15]
    A. Flachi, M. Nitta, S. Takada and R. Yoshii, Sign Flip in the Casimir Force for Interacting Fermion Systems, Phys. Rev. Lett. 119 (2017) 031601 [arXiv:1704.04918] [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    M. Nitta and R. Yoshii, Self-Consistent Large-N Analytical Solutions of Inhomogneous Condensates in Quantum \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) Model, arXiv:1707.03207 [INSPIRE].
  17. [17]
    A. Flachi, M. Nitta, S. Takada and R. Yoshii, Casimir Force for the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) Model, arXiv:1708.08807 [INSPIRE].
  18. [18]
    D. Pavshinkin, Grassmannian σ-model on a finite interval, arXiv:1708.06399 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Alessandro Betti
    • 1
  • Stefano Bolognesi
    • 2
    • 3
  • Sven Bjarke Gudnason
    • 4
  • Kenichi Konishi
    • 2
    • 3
  • Keisuke Ohashi
    • 5
  1. 1.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheSienaItaly
  2. 2.Department of Physics “E. Fermi”University of PisaPisaItaly
  3. 3.INFN, Sezione di PisaPisaItaly
  4. 4.Institute of Modern PhysicsChinese Academy of SciencesLanzhouChina
  5. 5.Research and Education Center for Natural SciencesKeio UniversityYokohamaJapan

Personalised recommendations