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Higgs Mechanism Without Spontaneous Symmetry Breaking and Quark Confinement

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Higgs Mechanism Without Spontaneous Symmetry Breaking and Quark Confinement The 10th APCTP-BLTP/JINR-RCNP-RIKEN Joint Workshop on Nuclear and Hadronic Physics Riken, Wako, 17 August 2016 Higgs mechanism without spontaneous symmetry breaking and quark confinement Kei-Ichi Kondo (Chiba Univ., Japan) Based on e-Print: arXiv:1606.06194 [hep-th] Short abstract: We give a manifestly gauge-invariant definition of the Higgs mechanism in the operator level, without relying on spontaneous symmetry breaking which is signaled by a non-vanishing vacuum expectation value of the scalar field. 1 x Introduction: SSB and NG boson Spontaneous symmetry breaking (SSB) is an important concept in physics. SSB occurs when the lowest energy state or the vacuum is degenerate. J SSB of global discrete symmetry G = Z(2) in the real scalar field theory ( ) 1 λ µ2 2 L = @ ϕ∂µϕ − V (ϕ2);V (ϕ2) = ϕ2 − ; ϕ 2 R; λ > 0; (1) rs 2 µ 2 λ The Lagrangian has the Z(2) symmetry: ϕ(x) ! −ϕ(x). V V - unbroken Z(2) symmetry (µ2 ≤ 0) =) SSB of Z(2) symmetry: (µ2 > 0) hϕi = 0 hϕi = v =6 0 2 J SSB of global continuous symmetry G = U(1) in the complex scalar field theory ( ) λ µ2 2 L =@ ϕ∗@µϕ − V (ϕ∗ϕ);V (ϕ∗ϕ) = ϕ∗ϕ − ; ϕ 2 C; λ > 0; (2) cs µ 2 λ The Lagrangian has the U(1) symmetry: ϕ(x) ! eiθϕ(x). V V v θ unbroken U(1) symmetry : =) SSB of U(1) symmetry: hϕ(x)i = 0 hϕ(x)i = v =6 0 For ϕ(x) = jϕ(x)jeiθ(x) 2 C • flat direction: θ(x) ! massless Nambu-Goldstone particle • curved direction: jϕ(x)j ! massive Higgs particle, This is a very beautiful result. 3 J What happens in the gauge field theory with the local continuous symmetry? First, we consider the Abelian-Higgs theory or U(1) gauge-scalar theory: ( ) 1 λ µ2 2 L = − F F µν + (D ϕ)∗(Dµϕ) − V (ϕ∗ϕ);V (ϕ∗ϕ) = ϕ∗ϕ − ; ϕ 2 C; AH 4 µν µ 2 λ µν µ ν ν µ F = @ A − @ A ;Dµ = @µ − iqAµ; (3) • The local U(1) gauge symmetry is completely broken spontaneouslyp by choosing a vacuum with the non-vanishing VEV of the scalar field: hϕ(x)i = v= 2 =6 0. 1 jD ϕj2 = j@ ϕ − iqA ϕj2 ! j − iqA hϕ(x)ij2 + ::: = (qv)2A Aµ + :::; (4) µ µ µ µ 2 µ • The Nambu-Goldstone particle θ (boson) could be generated, but it disappears. It is absorbed into the massless gauge boson Aµ to form the massive vector boson as supplying the longitudinal component. The Brout-Englert-Higgs mechanism or Higgs phenomenon is one of the most well- known mechanisms by which gauge bosons acquire their masses. This is a model for superconductor. The superconductivity is understood from the SSB of U(1) gauge symmetry and the Meisser effect is just the Anderson-Higgs mechanism. 4 J The spontaneous gauge symmetry breaking is a rather misleading terminology. The lattice gauge theory a` la Wilson gives a well-defined gauge theory without gauge fixing. The Elitzur theorem tells us that the local continuous gauge symmetry cannot break spontaneously, if no gauge fixing is introduced. The VEV hϕi of ϕ is rigorously zero regardless of the shape of the scalar potential V : hϕi = 0; (5) • Therefore, we are forced to fix the gauge to cause the non-zero VEV. Even after the gauge fixing, however, we still have the problem. Whether SSB occurs or not depends on the gauge choice. In non-compact U(1) gauge-Higgs model, the SSB occurs hϕi 6= 0 only in the Landau gauge α = 0, and no SSB occur hϕi = 0 in all other covariant gauges with α =6 0, as rigorously shown [KK85,BN86]. In axial gauge, hϕi = 0 for compact models [FMS80]. • A global subgroup H0 remains unbroken after imposing the gauge fixing condition for the original local gauge group G. Such a global symmetry is called the remnant global gauge symmetry. Only a remnant global gauge symmetry of the local gauge symmetry can break spontaneously to cause the Higgs phenomenon. However, such subgroup is not unique and the location of the breaking in the phase diagram depends on the remnant global gauge symmetries in the gauge-Higgs model. 5 The relevant numerical evidences are given on a lattice for different remnant symmetries allowed for various confinement scenarios. Moreover, the transition occurs in the regions where the Fradkin-Shenker-Osterwalder- Seiler theorem assures us that there is no transition in the physical state. • The above observations indicate that the Higgs mechanism should be characterized in a gauge-invariant way without breaking the original gauge symmetry. It is obvious that the non-vanishing VEV of the scalar field is not a gauge-invariant criterion of SSB. We show that a gauge boson can acquire the mass in a gauge-invariant way without assuming spontaneous breakdown of gauge symmetry which is signaled by the non- vanishing VEV of the scalar field. The Higgs phenomenon can be described even without such SSB. The spontaneous symmetry breaking is sufficient but not necessary for the Higgs mechanism to work. • Remember that quark confinement is realized in the unbroken gauge symmetry phase with mass gap. Thus, the gauge-invariant description of the Higgs mechanics can shed new light on the complementarity between confinement phase and Higgs phase. 6 x Conventional Higgs mechanism due to SSB We consider SU(N) Yang-Mills-Higgs theory with the gauge-invariant Lagrangian: 1 1 L = − F µν(x) · F (x) + (D µ[A ]ϕ(x)) · (D [A ]ϕ(x)) − V (ϕ(x) · ϕ(x)): YMH 4 µν 2 µ (1) A We assume that the adjoint scalar field ϕ(x) = ϕ (x)TA has the fixed radial length, ϕ(x) · ϕ(x) ≡ ϕA(x)ϕA(x) = v2: (2) A A A The Yang-Mills field µ(x) = µ (x)TA and ϕ(x) obey the gauge transformation: −1 −1 −1 Aµ(x) ! U(x)Aµ(x)U (x) + ig U(x)@µU (x); ϕ(x) ! U(x)ϕ(x)U −1(x);U(x) 2 G = SU(N): (3) Notice that ϕ(x) · ϕ(x) is a gauge-invariant combination. The covariant derivative −1 Dµ[A ] := @µ − ig[Aµ; ·] transforms Dµ[A ] ! U(x)Dµ[A ]U (x). A A Notation: For the Lie-algebra valued quantities A = A TA and B = B TA, we use A B A A A · B := 2tr(AB) = A B 2tr(TATB) = A B (4) 7 First, we recall the conventional description for the Higgs mechanism. If ϕ(x) acquires a non-vanishing VEV hϕ(x)i = hϕi, then Dµ[A ]ϕ(x) := @µϕ(x) − ig[Aµ(x); ϕ(x)] ! −ig[Aµ(x); hϕi] + :::; (5) and 1 L ! − tr fF µν(x)F (x)g − g2tr f[A µ(x); hϕi][A (x); hϕi]g + ::: YMH 2 G µν G µ 1 = − tr fF µν(x)F (x)g − g2tr f[T ; hϕi][T ; hϕi]gA µA(x)A B(x): (6) 2 G µν G A B µ To break spontaneously the local continuous gauge symmetry G by the non-vanishing VEV hϕi of the scalar field ϕ, we choose the unitary gauge in which ϕ(x) is pointed to a specific direction ϕ(x) ! ϕ1 uniformly over the spacetime. By this procedure the original gauge symmetry G is not completely broken. Indeed, there may exist a subgroup H (of G) which does not change ϕ1. The partial SSB G ! H provides the coset G=H (broken parts) with the mass, while the mass is not supplied for the H-commutative part of Aµ: 1 L ! − tr fF µν(x)F (x)g − (gv)2tr fA µ(x)A (x)g: (7) YMH 2 G µν G=H µ 8 Therefore, the resulting theory is a gauge theory with the residual gauge group H. For G = SU(2), by taking the usual unitary gauge A A3 hϕ1i = vT3; or hϕ1i = vδ ; (8) the kinetic term generates the mass term, 1 − g2v2tr f[T ;T ][T ;T ]gA µAA B = (gv)2(A µ1A 1 + A µ2A 2): (9) G A 3 B 3 µ 2 µ µ • A 1 A 2 The off-diagonal gluons µ ; µ acquire the same mass MW := gv, • A 3 ~ 2 The diagonal gluon µ remains massless. φ(x) ∋ SU(2)/U(1) =S Even after taking the unitary gauge (8), A 3 U(1) gauge symmetry described by µ still remains as the residual gauge symmetry H = U(1), U(1) which leaves ϕ1 invariant x (the rotation around the axis of the scalar field ϕ1). Thus, the SSB is sufficient for the Higgs mechanism to take place. But, it is not clear whether the SSB is necessary or not for the Higgs mechanism to work. 9 x Gauge-invariant Higgs mechanism: SU(2) case Next, we give a gauge-invariant description for the Higgs mechanism. • We construct a composite vector field Wµ(x) from Aµ(x) and ϕ(x) by −1 ^ ^ ^ Wµ(x) := −ig [ϕ(x); Dµ[A ]ϕ(x)]; ϕ(x) := ϕ(x)=v: (1) • We find that the kinetic term of the Yang-Mills-Higgs model is identical to the \mass term" of the vector field Wµ(x): 1 1 D µ[A ]ϕ(x) · D [A ]ϕ(x) = M 2 W µ(x) · W (x);M := gv; (2) 2 µ 2 W µ W as far as the constraint (ϕ · ϕ = v2) is satisfied. This fact is shown explicitly: 2 2 µ −2 µ g v W · Wµ =v 2tr([ϕ; D [A ]ϕ][ϕ; Dµ[A ]ϕ]) −2 µ µ =v f(ϕ · ϕ)(D [A ]ϕ · Dµ[A ]ϕ) − (ϕ · D [A ]ϕ)(ϕ · Dµ[A ]ϕ)g µ =(D [A ]ϕ) · (Dµ[A ]ϕ); (3) 2 where we have used the constraint (ϕ · ϕ ≡ v ) and ϕ · Dµ[A ]ϕ = ϕ · @µϕ + ϕ · (gAµ × ϕ) = gAµ · (ϕ × ϕ) = 0; with ϕ · @µϕ = 0 following from differentiating the constraint.
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