1. Higgs Mechanism at Finite Chemical Potential with Type-II Nambu-Goldstone Boson Based on arXiv:1102.4145v2 [hep-ph] Yusuke Hama (Univ. Tokyo) Tetsuo Hatsuda (Univ. Tokyo) Shun Uchino (Kyoto Univ.) 4/20 (2011) Dense Strange Nuclei and Compressed Baryonic Matter @ YITP, Kyoto, Japan

2. Contents 1. Introduction 2. Spontaneous Symmetry Breaking and Nambu-Goldstone Theorem 3. Type-II Nambu-Goldstone Spectrum at Finite Chemical Potential 4. Higgs Mechanism with Type-II Nambu-Goldstone Boson 5. Summary and Conclusion ＊ ＊ Our original work

3. Introduction Condensed Matter PhysicsElementary Particle Physics Spontaneous Symmetry Breaking Background: Spontaneous Symmetry Breaking (SSB) Nambu (1960) Cutting Edge Research of SSB Ultracold AtomsColor Superconductivity Extremely similar phenomena Origin of Mass

4. The number of NG bosons and Broken Generators systemSSB pattern G→H Broken generators ( BG) NG boson#NG boson dispersion 2-flavor Massless QCD SU(2) L × SU(2) R → SU(2) V 3 pion 3E(k) ～k Anti- ferromagnet O(3) → O(2)2 magnon 2E(k) ～k FerromagnetO(3) → O(2)2 magnon 1E(k) ～k 2 Kaon condensation in color superonductor U(2) →U(1)3 “kaon” 2E(k) ～k E(k) ～k 2 Chemical potential plays an important role for the number and dispersion of NG bosons One of the most important aspects of SSB The appearance of massless Nambu-Goldstone (NG) bosons Motivation: How many numbers of Nambu-Goldstone (NG) bosons appear? Relations between the dispersions and the number of NG bosons?

5. Nielsen-Chadha Theorem Nielsen and Chadha(1976) analyticity of dispersion of type-II spectral decomposition Classification of NG bosons by dispersions E~p 2n+1 : type-I, E~p 2n : type-II Nielsen-Chadha inequality N I + 2 N II ≧ N BG All previous examples satisfy Nielsen-Chadha inequality

6. Higgs Mechanism Purpose Analyze the Higgs mechanism with type-Ⅱ NG boson at finite chemical potential.  ≠ 0: type-I & type-II N BG ≠N NG = N I +N II  =0: type-I N BG =N NG = N I without gauge bosons ? N NG =(N massive gauge )/3 with gauge bosons N NG =(N massive gauge )/3

7. Type-II Nambu- Goldstone Spectrum at Finite Chemical Potential

8. minimal model to show type-II NG boson Lagrangian SSB Pattern Field parametrization 2 component complex scalar Quadratic Lagrangian mixing by  U(2) Model at Finite Chemical Potential Miransky and Schafer (2002) Hamiltonian Hypercharge

9. Type-II NG boson spectrum Equations of motion (  =0) (  ≠ 0)  ’ 1 massive  ’ 2 type-II  ’ 3 type-I  ’ massive   type-I  massive Nielsen-Chadha inequality: N I =1, N II =1, N I + 2N II = N BG   type-I   type-I dispersions mixing effect

10. Higgs Mechanism with Type-II NG Boson at Finite Chemical Potential

11. Gauged SU(2) Model U(2) Lagrangian field parametrization gauged SU(2) Lagrangian covariant derivative gauge boson mass background charge density to ensure the “charge” neutrality Kapusta (1981)

12. R  Gauge Clear separation between unphysical spectra (A 3   ghost, “NG bosons”) and physical spectra (A 3  i   Higgs) and by taking the  →∞ masses of unphysical particles decouple from physical particles Fujikawa, Lee, and Sanda (1972) Gauge-fixing function  : gauge parameter Landau gauge Feynman gauge Unitary gauge

13. Quadratic Lagrangian coupling new mixing between  1,2   and unphysical modes ( A a  ) What remain as physical modes?

14. Dispersion Relation (p→0, α>>1) diagonaloff-diagonal

15. Field Mass Spectrum and Result total physical degrees of freedom are correctly conserved  ’(Higgs)  ’ 3 (type-I)  ’ 2 (type-II)  ’ 1 (massive) A 1,2,3  T A 1,2,3  T, L Fieldsg=0, μ≠0g≠0, μ≠0 massive21 NG boson1 (Type I), 1(Type II) 0 Gauge boson３×2 T ３×3 T, L Total10

16. Summary We analyzed Higgs Mechanism at finite chemical potential with type-II NG boson with R  gauge Result: ・Total physical degrees of freedom correctly conserved -- Not only the massless NG bosons (type I & II) but also the massive mode induced by the chemical potential became unphysical ・Models: gauged SU(2) model, Glashow-Weinberg-Salam type gauged U(2) model, gauged SU(3) model Future Directions: ・Higgs Mechanism with type-II NG bosons in nonrelativistic systems (ultracold atoms in optical lattice)? -- What is the relation between the Algebraic method (Nambu 2002) and the Nielsen Chadha theorem？ ・Algebraic method: counting NG bosons without deriving dispersions ・Nielsen-Chadha theorem: counting NG bosons from dispersions

17. Back Up Slides

18. Counting NG bosons with Algebraic Method behave canonical conjugate belong to the same dynamical degree of freedom N BG ≠N NG O(3) algebra anti-ferromagnet ferromagnet N BG =N NG N BG ≠N NG Nambu (2002) Q a : broken generators independent broken generators N BG =N NG SU(2) algebra N BG ≠N NG U(2) model Examples

19. The Spectrum of NG Bosons V v vv Future Work

20. Glashow-Weinberg-Salam Model Fieldsg=0  ≠0 g≠0≠0g≠0≠0 Gauge2×43×3+2 NGBType I×1 Type II×1 0 Massive2１

21. Gauged SU(3) Model Fieldsg=0  ≠0 g≠0≠0g≠0≠0 Gauge 2×53×5 NGB 1 (Type I) 2 (Type II) 0 Massive 3１