Chiral symmetry breaking - Effective lagrangian. Abstract In this lectures, I will introduce an effective Lagrangian with determinant interaction [1]

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1 Chiral symmetry breaking - Effective lagrangian Su Houng Lee 1, 1 IPAP and Department of Physics, Yonsei University, Seoul Korea (Dated: October 1, 010) Abstract In this lectures, I will introduce an effective Lagrangian with determinant interaction [1] PACS numbers: Electronic mail:suhoung@phya.yonsei.ac.kr Typeset by REVTEX 1

2 I. A SIMPLE MODEL There are several ways to introduce the following global invariance to effective model The quark and meson field transforms as follows U() L U()R. (1) q L = g L q L q R = g R q R Σ = g L Σg R () A. Linear sigma model In the linear sigma model, one usually introduces the following representation, Σ = σ + iτπ (3) Then, we can write an effective Lagrangian involving the fermion and meson fields, L = q L i /q L + q R i /q R + 1 ( ) 4 Tr µ Σ µ Σ + 1 ) 4 µ Tr (ΣΣ λ [ )] ) 16 µ Tr (ΣΣ g ( q L Σq R + q R Σ q L (4) II. TOY MODEL BY G. T HOOFT Under this transformation, the meson field, which can be thought of being composed of quark-antiquark q R q L, transforms as, The meson field are composed of eight meson fields Now consider the following Lagrangian φ = U L φu R (5) φ = 1 (σ + iη) + 1 (α + iπ) τ, (6) L = Tr[ µ φ µ φ ] V (φ) (7) A potential V 0 that is invariant under eq.(1) is. V 0 = µ Tr[φφ ] + 1 ( ) (λ 1 λ ) Tr[φφ 1 ] + λ Tr[φφ ] = µ (σ + η + α + π ) + λ 1 8 (σ + η + α + π ) + λ ) ((σα + ηπ) + (α π) (8)

3 from Assuming, we get σ = f, σ = f + s, (9) dv (f) df = 0 (10) f = µ /λ 1, (11) and, V 0 = λ ( 1 fs + 1 ) (s + η + α + π λ ) ) + (fα + sα + ηπ) + (α π) from which we read off: (1) m s = λ 1 f = µ, m η = 0, m α = λ f, m π = 0 (13) There are two possible symmetry breaking terms, 1. Chiral symmetry breaking.. U(1) A symmetry breaking. V m = U m + U m; U m = 1 4 meiχ Trφ = 1 meiχ (σ + iη) (14) V a = U a + U a; U a = κe iθ detφ = κe iθ ( (σ + iη) (α + iπ) ) (15) Note under U(1) transformation φ e iω, (σ + iη) e iω (σ + iη), (α + iπ) e iω (α + iπ) (16) A. Explicit chiral symmetry breaking case In this case, we can chose ω = π χ, and Then consequently, we get V = V 0 + V m (17) V m = mσ (18) f = µ /λ 1 + m/(λ 1 f) (19) m s = µ + 3m/f m π = m η = m/f (0) 3

4 B. U A (1) symmetry breaking Consider the other case, where m = 0; we can chose ω = 1 (π θ), and Then consequently, we get V = V 0 + V a (1) V a = κ(σ + π η α ) () f = µ /λ 1 + 8κ/λ 1 (3) m π = 0 m η = 8κ (4) III. VECTOR MESONS We can explicitly introduce vector mesons by two methods. Gauging external symmetry and using hidden local symmetry. A. Massive Yang Mills approach In the massive Yang Mills approach (MYMA), the vector and axial vector mesons are introduced as gauge fields of the external charge. Therefore, the kinetic term in eq. (4) becomes as follows. L = 1 ) 4 f πtr (D µ UD µ Σ + 1 m ρtr (A LµA ) µl + A Rµ A µr, (5) here, we reparametrized Σ = f π U and used the non-linear realization of the pion fields. D µ U = µ U iga L µu + igua R µ, (6) and A L µ = V µ + A µ, A R µ = V µ A µ. This with the vector meson dominance, the external vector field V µ, will be identified with the vector meson fields by V µ = g ρ ρ µ. Then one can obtain g ρππ = g ρ m a = m ρ + fπg ρ (7) 4

5 B. Hidden Local Symmetry The non linear realization that we described above are said to be based on the manifold G/H =SU() L SU()R /SU() V. This is so because the original symmetry U g L Ug R is broken to the vector part only in the vacuum TrU TrgUg. In ref.[], the hidden local symmetry found by noting that can be rewritten as, where the transformation properties for the ξ s are, U(x) = exp[iπ(x)/f π ] (8) U(x) = ξ L ξ R (9) ξ L (x) h(x)ξ L (x)g L, ξ R (x) h(x)ξ R (x)g R (30) Introducing the gauge field V (x) µ for the hidden local symmetry and the covariant derivative, we have the following two lowest order terms, L V L A = f π 4 Tr[D µξ L ξ L + D µξ R ξ R ] = f π 4 Tr[D µξ L ξ L D µξ R ξ R ] (31) Then the most general Lagrangian that reproduces the chiral lagrangian in Eq.(4) is given as, Adding the kinetic term for the vector field we find, L = L A + al V (3) g ρππ = 1 ag m ρ = ag f π (33) [1] G. t Hooft, Phys. Rep (1986). [] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett 54 (1985)