Chiral Symmetry Breaking from Monopoles and Duality

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1 Chiral Symmetry Breaking from Monopoles and Duality Thomas Schaefer, North Carolina State University with A. Cherman and M. Unsal, PRL 117 (2016)

2 Motivation Confinement and chiral symmetry breaking are well established [V(r)-V(r 0 )]r Π u Σ g + quenched κ = m ps + m s 2 m ps l, s Continuum N t 16 N t 12 N t 10 N t r/r T MeV Goal: Find defomations of QCD, continuously connected to the full theory, that exhibit χsb and confinement. Bali et al. (2000), Borsanyi et al (2011)

3 Background: Confinement in Weak Coupling Consider SU(2) gauge theory with N ad f = 1 on R 3 S 1 L = 1 4g 2Fa µνf aµν i g 2λa σ D ab λ b + m g 2λa λ a A a µ(0) = A a µ(l) x 4 Large mass limit: Pure YM. Small mass limit: SUSY YM. Small S 1 and m: Confinement can be studied using semi-classical methods, based on monopole-instantons, instantons, and bions. Low energy fields: Holonomy b and dual photon σ

4 Non-perturbative effects Topological classification on R 3 S 1 (GPY) 1. Topological charge Q top = 1 16π 2 d 4 xf F 2. Holonomy (eigenvalues q α of Polyakov line at spatial infinity) [ ] β Ω( x) = Trexp i A 4 dx Magnetic charges Q α M = 1 4π d 2 STr[P α B]

5 Periodic instantons (calorons) Instanton solution in R 4 can be extended to solution on R 3 S E + B r Q top = ±1 X=0 τ X=1 Ω = 1 Q α M = 0 SU(2) solution has = 8 bosonic zero modes dρ ρ 5 d 3 xdx 4 du e 2S 0 2S 0 = 8π2 g 2 4n adj fermionic zero modes d 2 ζd 2 ξ

6 Calorons at finite holonomy: monopole constituents KvBLL (1998) construct calorons with non-trivial holonomy BPS and KK monopole constituents. Fractional topological charge, 1/2 at center symmetric point. 2 (3+1) = 8 bosonic zero modes, 2 2 fermionic ZM. dφ 1 d 3 x 1 d 2 ζe S 1 dφ 2 d 3 x 2 d 2 ξe S 2

7 Topological objects (Q M,Q top ) = ( S 2 B dσ, R 3 S 1 F F) BPS KK BPS KK monopoles (1, 1/2) ( 1, 1/2) ( 1, 1/2) (1, 1/2) instantons (0, 1) (0, 1) bions (0,0) (0,0) Note: BPS/KK topological charges in Z 2 symmetric vacuum. Also have (2,0) (magnetic) bions.

8 Effective potential Instantons and monopoles: Exact solutions, but V(b,σ) = 0. Bions: Approximate solutions V BPS,BPS e 2b e 2S 0 d 3 re S 12(r) S 12 (BPS)(KK) S 12 (r) = 4πL g 2 r (q1 mq 2 m q 1 bq 2 b)+4log(r) (BPS)(BPS) r 12 Saddle point integral after analytic continuation g 2 g 2 (BZJ) V(b,σ) M6 PV L3 e 2S [ ( ) ] 0 8π g 6 cosh ( θ π) cos(2σ) g2 Center symmetric vacuum tr(ω) = 0 preferred Mass gap for dual photon m 2 σ > 0 ( confinement)

9 What about chiral symmetry breaking? Original setup: One adjoint fermion, chiral symmetry is discrete. λλ 0 Z 2Nc Z 2 Light fundamental fermions: Need strong coupling. L Gdet N f ( ψ L ψ R )+h.c. Heavy fundamental fermions: Study explicit breaking of Z N center symmetry.

10 Role of Boundary Conditions Consider flavor twisted boundary conditions ψ(τ +β) = Ω F ψ(τ) Ω F = diag(1,e 2πi/N f,...,e 2πi(N f 1)/N f ) Flavor holonomy Ω F has several interesting properties: 1. N f = N c : Respects Z Nc center symmetry. 2. Large L: Breaks flavor symmetry, but in a controlled fashion. 3. Small L: New semi-classical picture of chiral symmetry breaking: Distributed zero modes and color-flavor transmutation.

11 Large L expectations Flavor holonomy corresponds imaginary flavor (isospin) chemical potential µ F i/l. Can be studied using chiral Lagrangian L = f2 π 4 Tr[ µ U µ U ] BTr[MU +h.c] with µ U = µ U +i[ µ F T F,U]. Consider N f = 2 (isopsin chemical potential) m 2 π 0 = m 2 π m 2 π ± = m 2 π + µ 2 I N f 1 exact Goldstone modes (m=0), others acquire gaps.

12 Small L theory: Perturbation theory Consider center symmetric gauge holonomy (add double trace deformation). For l > N c L theory abelianizes SU(N c ) [U(1)] N c 1 Gapless (Cartan) gluons described by dual photon σ S = g2 8π 2 d 3 x( µ σ) 2 L with F i µν = g2 2πL ǫ µνα α σ i. Remain gapless to all orders in perturbation theory due to emergent shift symmetry σ σ + ǫ.

13 Small L theory: Semiclassical objects Center symmetric background, no fermions: Instanton fractionalize into N c constituents M i e S 0 e i α i σ S 0 = 8π2 g 2 N c α i SU(N c ) root vectors In the ground state these objects proliferate: The monopole-anti-monopole gas. V( σ) m 3 We S 0 cos( α i σ) Mass gap for the dual photon, continuous shift symmetry broken. Massless fermions: Take into account fermion zero modes. i

14 Small L theory: Fermion zero modes Many eigenvalue circles: Polyakov line Flavor holonomy Instanton-monopoles θ flavor singlet twist N c = N f = 4 N c = 4 N f = 3 Zero modes localize on monopoles jumping over flavor eigenvalues Bruckmann, Nogradi, van Baal (2003); Moore et al. (2014)

15 Two basic scenarios (N c = N f ) No flavor twist: Standard t Hooft vertex carried by one monopole M 1 e S 0 e i α 1 σ det F ( ψ f L ψg R ) M i>1 e S 0 e i α i σ Center symmetric flavor holonomy: Single flavor t Hooft vertex carried by each monopole M i e S 0 e i α i σ ( ψ i Lψ i R) trivial flavor holonomy center symmetric holonomy

16 Spontaneous symmetry breaking Unbroken symmetries of flavor twisted theory [U(1) J ] N c 1 [U(1) V ] N f 1 [U(1) A ] N f 1 U(1) Q Shift symmetry Exact flavor symmetry Symmetries of monopole vertex M i e S 0 e i α i σ ( ψ i Lψ i R) Preserves vectorial symmetry [U(1) V ] N f 1 U(1) Q. Breaks axial symmetry [U(1) A ] N f 1 : ( ψ f L ψf R ) eiǫ i ( ψ i Lψ i R)

17 Spontaneous symmetry breaking, continued Monopole vertex is invariant provided [U(1) A ] Nf 1 is combined with [U(1) J ] Nc 1 shift symmetry [Ũ(1) A] Nf 1 ( ψ f L : ψf R ) eiǫ i ( ψ L i ψi R ) e i α i σ e iǫ i e i α i σ Ground state e i α i σ 1. Breaks [U(1) V ] N f 1 [Ũ(1) A] N f 1 [U(1) V ] N f 1 For m = 0 the ground state is degenerate. Massless Goldstone boson { f S σ = L d 3 2 x π 4 Tr[ µ Σ µ Σ ] } BTr[MΣ+h.c.] Microsccopically Σ = e iπ/f π with Π = π a T a and π a = g 2πL σa Color-flavor transmutation

18 Chiral Lagrangian Chiral lagrangian has calculable coefficients { f S σ = L d 3 2 x π 4 Tr[ µ Σ µ Σ ] } BTr[MΣ+h.c.] f 2 π = ( g 6πL ) 2 = N cλm 2 W 24π 2 B = 1 2 ψψ m 3 2 W e 8π λ Also note: VEV of monopole operator can be viewed as effective constituent quark mass m Q m W e 8π2 λ

19 Conclusions and Outlook Calculabe mechanism for chiral symmetry breaking in a compactified version of QCD. Results consistent with continuity between large L (full QCD) and small L theory. Mechanism based on monopole instantons and color flavor transmutation. Study: Extension to N f > N c? Relation between χsb and confinement?