The low energy limit of the 3-flavor extended Linear Sigma Model. Jonas Schneitzer. Johann Wolfgang Goethe Universität Frankfurt am Main

Transcript

1 The low energy limit of the 3-flavor extended Linear Sigma Model Jonas Schneitzer Johann Wolfgang Goethe Universität Frankfurt am Main Fachbereich Physik Institut für Theoretische Physik

2 Outline 1 LECs The extended Linear Sigma Model 3 The low energy limit 4 Outlook

3 L χp T = 1 ( µ φ) 1 M φ φ ) + (c δ 0 δ a,bδ c,d + c d 0 d abid icd φ aφ b φ cφ d ) + (c δ δ a,b δ c,d + c δ,b δa,cδ b,d + c d d abi d icd µφ aφ b µ φ cφ d ) + (c δ 4 δ a,bδ c,d + c δ 4,b δa,cδ b,d + c d 4 d abid icd µφ a νφ b µ φ c ν φ d

4 The extended Linear Sigma Model

5 The basic Lagrangian L elsm = D µ Φ] D µφ m 0 Φ Φ λ 1 ( Φ Φ) ] λ Φ Φ 1 4 ( ) L µν L µν + R µν m R µν + 1 L + µ L µ + R µ ] ] R µ + H Φ + Φ + c 1 detφ detφ ] g + i L µν L µ, L ν] + R µν R µ, R ν] ] + h1 Φ Φ L µ L µ + R µ R µ + h L µ Φ + Φr µ + h 3 ΦR µ Φ L µ + g 3 L µ L ν L µl ν + R µ R ν ] R µr ν + g4 L µ L µl ν L ν + R µ R µr ν ] R ν + g 5 L µ L µ R µ R µ + g6 L µ L µ L µ L µ + R µ R µ R µ R µ ] D µφ = µφ ig 1 L µ Φ ΦR µ ] Φ = S + ip L µ = V µ + A µ R µ = V µ A µ Ψ = ψ i λ i, λ i = U(3)-generators i = 0, 1,..., 8

6 The basic Lagrangian L elsm = D µ Φ] D µφ m 0 Φ Φ λ 1 ( Φ Φ) ] λ Φ Φ 1 4 ( ) L µν L µν + R µν m R µν + 1 L + µ L µ + R µ ] ] R µ + H Φ + Φ + c 1 detφ detφ ] g + i L µν L µ, L ν] + R µν R µ, R ν] ] + h1 Φ Φ L µ L µ + R µ R µ + h L µ Φ + Φr µ + h 3 ΦR µ Φ L µ + g 3 L µ L ν L µl ν + R µ R ν ] R µr ν + g4 L µ L µl ν L ν + R µ R µr ν ] R ν + g 5 L µ L µ R µ R µ + g6 L µ L µ L µ L µ + R µ R µ R µ R µ ] D µφ = µφ ig 1 L µ Φ ΦR µ ] Φ = S + ip L µ = V µ + A µ R µ = V µ A µ Ψ = ψ i λ i, λ i = U(3)-generators i = 0, 1,..., 8

7 The basic Lagrangian L elsm = D µ Φ] D µφ m 0 Φ Φ λ 1 ( Φ Φ) ] λ Φ Φ 1 4 ( ) L µν L µν + R µν m R µν + 1 L + µ L µ + R µ ] ] R µ + H Φ + Φ + c 1 detφ detφ ] g + i L µν L µ, L ν] + R µν R µ, R ν] ] + h1 Φ Φ L µ L µ + R µ R µ + h L µ Φ + Φr µ + h 3 ΦR µ Φ L µ + g 3 L µ L ν L µl ν + R µ R ν ] R µr ν + g4 L µ L µl ν L ν + R µ R µr ν ] R ν + g 5 L µ L µ R µ R µ + g6 L µ L µ L µ L µ + R µ R µ R µ R µ ] D µφ = µφ ig 1 L µ Φ ΦR µ ] Φ = S + ip L µ = V µ + A µ R µ = V µ A µ Ψ = ψ i λ i, λ i = U(3)-generators i = 0, 1,..., 8

8 The basic Lagrangian L elsm = D µ Φ] D µφ m 0 Φ Φ λ 1 ( Φ Φ) ] λ Φ Φ 1 4 ( ) L µν L µν + R µν m R µν + 1 L + µ L µ + R µ ] ] R µ + H Φ + Φ + c 1 detφ detφ ] g + i L µν L µ, L ν] + R µν R µ, R ν] ] + h1 Φ Φ L µ L µ + R µ R µ + h L µ Φ + Φr µ + h 3 ΦR µ Φ L µ + g 3 L µ L ν L µl ν + R µ R ν ] R µr ν + g4 L µ L µl ν L ν + R µ R µr ν ] R ν + g 5 L µ L µ R µ R µ + g6 L µ L µ L µ L µ + R µ R µ R µ R µ ] D µφ = µφ ig 1 L µ Φ ΦR µ ] Φ = S + ip L µ = V µ + A µ R µ = V µ A µ Ψ = ψ i λ i, λ i = U(3)-generators i = 0, 1,..., 8

9 Basic Assumptions The following 4-field interaction terms will be can be neglected: Interactions with 0 or 1 P i -fields Interactions with exactly 3 P i -fields Interactions with P i -fields and different heavy fields

10 Spontaneous chiral symmetry breaking m 0 m 0, m 0 > 0 Vacuum expectation value of σ N and σ S become non-zero: σ N σ N + φ N, σ S σ S + φ S This leads to mixing terms of the skalar and vector mesons and the pseudoskalar and axial-vector mesons: D µφ µφ ig 1 L µ Φ ΦR µ ] ig 1 L µ S φ S φ R µ ], S φ = S φ (φ N, φ S ) Resolved by shifting the vector and axial-vector fields: V µ,a V µ,i + µ Sa, A µ,a A µ,a + µ Pa µ Sa = ω ai µs i, µ Pa = ω ai µp i Nearly every term in the Lagrangian is greatly expanded by this!

11 Example of the new terms h 1 Φ Φ L µ L µ + R µ R µ h 1 Φ Φ L µ L µ + R µ R µ + h 1 (S + S φ ) + P (V µ,a µ Sa + A µa µ Pa) + h 1 (S + S φ ) + P ( ( µ Sa) + ( µ Pa) ) + h 1 ((Vµ,a) SS φ + Sφ + (A µ,a) ) Other terms generate up to 13 new terms!

12 Terms classified by types The terms in the Lagrangian can be sorted into multiple types: L A = L A,kin,mass + L AAP P + L AV P + L ASP L V = L V,kin,mass + L V V P P + L V P P L S = L S,kin,mass + L SSP P + L SP P L P = L P,kin,mass + L P P P P

13 The low energy limit

14 The functional integral f, f, = N DP DSDV µda µ exp i d 4 xl elsm + i d 4 xl GF L GF = ξ A ( µaµ a ) ξ V ( µv µ a ) f = P a, S a, V µ,a, A µ,a

15 The functional integral The heavy field elsm integrals integrals: 1 I A = N A DA µ exp d 4 xd 4 y A µ,i(x)o µν A,ij (x, y)aν,j(y) + i 1 I V = N V DV µ exp d 4 xd 4 y V µ,i(x)o µν V,ij (x, y)vν,j(y) + i I S = N S DS exp i d 4 xd 4 y S i(x)o S,ij (x, y)s j(y) + i I η = N η Dη exp i d 4 xd 4 y η (x)o η (x, y)η (y) + i d 4 xj µ A,i (x)aµ,i(x) d 4 xj µ V,i (x)vµ,i(x) d 4 xj S,i (x)s i(x) d 4 xj η (x)η (x) Or, in euklidian coordinates: I ψ = N Dψ µ exp 1 d 4 x E d 4 y E ψ µ,i(x E )O µν ψ,e,ij (x E, y E )ψ ν,j(y E ) d 4 x E J µ ψ,i (x E)ψ µ,i(x E ) I φ = N Dφ exp 1 + d 4 x E d 4 y E φ i(x E )O φ,e,ij (x E, y E )φ j(y E ) d 4 x E J φ,i (x E )φ i(x E )

16 The functional integral The Euclidian Integrals are Gauß-integrals with analythic solutions: ] 1/ ψ,e,ij (x E, y E ) 1 exp I ψ = N ψ deto µν d 4 x E d 4 y E J ψ,µ,i (x E )O µν 1 ψ,e,ij (x E, y E )J ψ,µ,j (y E ) deto µν ψ,e,ij (x E, y E )] 1/ generates loop-order terms 1 J ψ,µ,i(x E )O µν 1 ψ,e,ij (x E, y E )J ψ,µ,j (y E ) generates the 4P i interaction terms

17 The functional integral ] O φ,e,ij = xe δ ij + m φ,ij δ(x E y E ) + +c 1,ijkl P k P l + c,ijkl γp k γ P l + c 3,ijkl P k P l xe +... ] δ(x E y E ) ] O µν ψ,e,ij = ( xe δ ij + m ψ,ij )gµν E + (1 ξ ψ)δ ij x µ E x ν E δ(x E y E ) + c 1,ijkl µ P k ν P l + c,ijkl γp k γ P l g µν ] E δ(xe y E ) m ij is non-diagonal in i, j = 0, 8 we seperate the φ 0/ψ µ 0 integration from the rest and rotate these fields into f µ 1N /f µ 1S, ωµ N /ωµ S and σ/σ For an easier calculation we choose the Feynman gauge ξ = 1 Now the first part is easy to invert: ( ) O 1 φ,e,ij = δ ij xe + m δ(x E y E ) = 1 n φ,i m ( 1) n x φ,i m δ(x E y E ) n=0 φ,i O µν 1 ψ,e,ij = δ ij g µν ( ) E xe + m δ(x E y E ) = 1 n ψ,i m ( 1) n x ψ,i m δ(x E y E ) n=0 ψ,i

18 The functional integral Since the source J φ,i /J µ ψ,i is generally of order P, we only keep terms of order P 0 from O 1 φ,e,ij /Oµν 1 ψ,e,ij only the terms we have inverted contribute! The axial-vector source J µ A (VµP, SP ) would only lead to terms in the form of V µ P and S P which are again neglected

19 J µ V,k J µ V,k = g 1 µ P ap b f abk ( ) + g ν ν Pa µ Pb f abk + S φ,a µ Pb P c (g 1 + h h 3 )d abi f ick + (g1 h 3)d aci f ibk + (h + h 3 )d aki f ] icb d abc = d abc + 3 (δ abδ c0 + δ acδ b0 + δ bc δ a0 δ a0 δ b0 δ c0 ) f abc = f abc

20 J S,k J S,k = c 1 c 10 P 0 + c 18 P 8 ] P aβ ak λ 1 S φ,a P b P cδ ak δ bc ) λ S φ,a P b P c ( 3 d aki d ibc d aci d ibk g 1 µ Pa µ Pb d abk + S φ,a µ Pb µ h1 Pc δ akδ bc + (g1 h 3 )d abi d ick + h ] + h 3 d aki d ibc g 1 µ( µ PaP b )d abk S φ,a µ( µ Pb P c) hf aci d ibl + h 3(d ali f icb + f ali d icb )] ω lk ω lk = iω K (δ l4δk5 δ l5 δk4 + δ l6 δk7 δ l7 δk6)

21 Approximations S φ,8 S φ, = 0 + O(10 ) φs φ N 1.08 = 1 + O(10 ) m V,a = m V + O(10 1 ) m S,a = m S + O(10 1 ) ω = ω a1 = ω f1n = ω f1s + O(10 1 ) = ω K1 + O(10 1 ) P η + O(η ), P η + O(η )

22 Approximated J µ V,k and J S,k J µ V,k = g ω ν ( ν P a µ P b ) + ( g 1 + ) ] 3 S φ,0ω(g1 h 3 ) µ P ap b f abk J S,k = c 1 3 S3 φ,0 P 0P a (δ ak 3δ a0 δ k0 δ a8 δ k8 ) + λ 1 P b P c + h 1ω ] µp b µ P c S φ,a δ ak δ bc ( + 3 S λ φ,0 PaP b + 3 S φ,0ω g1 + h ) h 3 µp a µ P b g 1 ω µ ( µ P ap b )] d abk

23 L 0 4P L4P 0 = c1 (c 10 δ a0 + c 18 δ a8 ) β bcd λ 1 4 δ abδ cd λ ] 8 d abi d icd P ap b P cp d + ω h1 δ abδ cd + h ] h 3 d abi d icd + (g 1 h 3 )d aci d ibd µp ap b µ P cp d + ω4 g5 + g 6 δ ab δ cd + g ] 3 + g 4 d abi d icd + g 3d aci d ibd µp a νp b µ P c ν P d

24 Results L elsm = 1 ( µpa) 1 m a P a ) + (c δ 0 δ a,b δ c,d + c d 0 d abi d icd P ap b P cp d ) + (c δ δ a,bδ c,d + c δ,b δa,cδ b,d + c d d abid icd µp ap b µ P cp d ) + (c δ 4,I δ a,bδ c,d + c δ 4,I,b δa,cδ b,d + c d 4,I d abid icd µp a νp b µ P c ν P d ) + (c δ 4,II δ a,bδ c,d + c δ 4,II,b δa,cδ b,d + c d 4,II d abid icd µ νp a µ νp b P cp d ) + (c δ 4,III δ a,bδ c,d + c δ 4,III,b δa,cδ b,d + c d 4,III d abid icd µ ν P a µp b P c ν P d Takes the same Form as the χp T -results for K and π interactions except for additional terms in O( 4 µ ) The c1-term and the differences of d ab0 and d ab8 lead to different LEC s whenever η is involved (c δ/d n c δ/d n,0, cδ/d n )!

25 Outlook

26 Outlook Check the χp T calculations again Compare the numerical results Look for a way to include η without introducing more than 10 new constants