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

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

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

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?

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

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

Type-II Nambu- Goldstone Spectrum at Finite Chemical Potential

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

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

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

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)

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

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

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

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

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

Back Up Slides

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

The Spectrum of NG Bosons V v vv Future Work

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

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