Linking U(2) U(2) to O(4) via decoupling

Transcription

1 Linking U(2) U(2) to O(4) via decoupling Norikazu Yamada (KEK, GUAS) in collaboration with Tomomi Sato (KEK, GUAS) Based on Linking U(2) U(2) to O(4) model via decoupling, PRD91(2015) [arxiv: [hep-lat]] RG flow of linear sigma model with U A (1) anomaly, PoS LATTICE 2014, 191 (2015)[arXiv: [hep-lat]] More about vacuum structure of Linear Sigma Model, PoS LATTICE 2013, 430 (2014)[arXiv: [hep-lat]]

2 Outline Goal: Study chiral phase transition of massless two-flavor QCD at finite T and in vanishing μ in terms of ε expansion Assumption: Non-zero breaking of U A (1) symmetry remains at T c Focusing on: i) Whether 2 nd order phase transition is possible? ii) What is the universality class? ( critical exponents) How: Analyze RG-flow of the 3-d Ginsburg-Landau-Wilson model. IRFP 2 nd order possible Conclusions: i) No IRFP, but still 2 nd order phase transition is possible. ii) Exponent differs from those in O(4) due to non-decoupling effects.

3 Effective theory approach Pisarski and Wilczek, PRD29, 338 (1984) Look at RG flow of 3-d linear σ model (LσM) The nature in PT 2-f QCD depends on fate of UA(1) at Tc. (Nf 3 1 st order) i) Largely broken SU(2) SU(2) O(4) LσM Wilson-Fisher FP 2 nd with O(4) scaling is possible ii) Fully, effectively restored U(2) U(2) [or O(2) O(4)] LσM IRFP? 1 st or 2 nd with U(2) U(2) scaling iii) If breaking is small, U(2) U(2) LσM with U A (1) [U A (1) broken LσM]???

4 UA(1) broken LσM Φ = 2(φ 0 iχ 0 )t 0 + 2(χ i + iφ i )t i e 2iθ A L ΦR (L SU L (2), R SU R (2), θ A Re). L U(2) U(2) = 1 2 tr [ µ Φ µ Φ ] m2 0 tr [ Φ Φ ] + π2 3 g 1 ( tr[φ Φ] ) 2 + π 2 3 g 2tr [ (Φ Φ) 2] Rewriting in terms of components L total = O(4) LσM Setting T=Tc m 2 φ=0 m 2 χ= ca > 0 (Thus χ is massive) At Tc, calculate β-functions in d=4-ε dims with ε=1.

5 β functions 1-loop calc. with dim reg., a mass-dep. scheme and w=0 yields O(4) LSM βˆλ = ϵˆλ +2ˆλ ( ) 6 f(ˆµ) 4ˆλ 2 +6ˆλĝ 2 +3ĝ2 2 8ˆλẑ 6ĝ 2 ẑ +4ẑ 2, βĝ2 = ϵĝ ˆλĝ ) 3 f(ˆµ)ĝ 2 (ˆλ 2ˆx + 1 ( ) 3 h(ˆµ)ĝ 2 4ˆλ +ĝ 2 4ẑ, βˆx = ϵˆx +4f(ˆµ) (ˆλˆx ˆx 2) + 1 ( ) 12 (1 f(ˆµ)) 8ˆλ 2 6ˆλĝ 2 3ĝ2 2 +8ˆλẑ +6ĝ 2 ẑ 4ẑ 2, βẑ = ϵẑ + 1 ( 2ˆλ 2 2 ˆλĝ ) 2 +2ˆλẑ 1 ( 6 h(ˆµ) 4 ˆλ 2 +3ĝ 22 8 ˆλ ) ẑ +4ẑ ( ) 6 f(ˆµ) 2ˆλ 2 +3ˆλĝ 2 +3ĝ2 2 2ˆλẑ 6ĝ 2 ẑ +12ˆλˆx +6ĝ 2 x 12ˆxẑ +4ẑ 2, μ 0 (IR limit) with mχ fixed??? (mχ with μ fixed O(4) LσM)

6 RG-flow (1) 1 μ 2 /m χ 2 µ 2 /c A =0.01,x^=0,z^=0 1 μ 2 /m χ 2 µ 2 /c A =100,x^=0,z^= g^2 0 g^ λ^ λ^ No IRFP

7 RG-flow (2) IR 4 3 IR g^2 2 1 m χ n=10 n= ^ g^2 2 1 n=10 n=0 m χ ^ RG flows can be classified by its IR behavior into two types: 1. All couplings diverge 1 st order 2. ^λ approaches ε/2 and others diverge 2 nd order?

8 How to interpret no IRFP but ^λ ^λfp Usually, no IRFP 1 st order But, in the present case, we infer that the system undergoes 2 nd order. Reason: U A (1) broken LσM = O(4) LσM in IR limit If, in IR limit, massive χ decouples and arbitrary n-point functions of φ i agree btw two theories: Confirmed for arbitrary 4-point functions to 1-loop. O(4) LσM has Wilson-Fisher FP in IR limit. IR limit of U A (1) broken LσM should have it, too.

9 Critical exponents Critical exponents Universality class Powerful tool to analyze various critical phenomena ν, η, α, β, γ, δ, ω ν : correlation length ~ t ν (t: reduced temperature) η : φ(x) φ(0) ~ x d+2 η ω : scaling dim. of the leading irrelevant op.

10 Critical exponents in O(4) model Model ν η ω O(4) (a few % error.) O(4) ε-exp (at leading order) 2/(4-ε) 0 ε Hasenbusch and Vicari, PRB84, (2011)

11 Critical exponents in UA(1) model Model ν η ω O(4) (a few % error.) O(4) ε-exp (at leading order) 2/(4-ε) 0 ε Hasenbusch and Vicari, PRB84, (2011) U A (1) ε-exp (at leading order) 2/(4-ε) 0 2 5ε/3 This work At least, one of the critical exponents, ω, is different from O(4). Two-loop calculation is underway.

12 Reason for different ω ω = dβˆλ dˆλ ˆλ=ˆλIRFP ω : determined by RG dimension of leading irrelevant op. In O(4) LσM, ω O(4) = ϵ

13 Reason for different ω In UA(1) broken LσM, L total = IR limit Leading irrelevant op. ω UA (1)broken =2 5ϵ/3, Non-decoupling causes non-universality.

14 Attractive Basin (large UA(1) breaking) c A / 2 =1,x^( )=-1,z^( )=1 ca / 2 =1,x^( )=0,z^( )=1 ca / 2 =1,x^( )=1,z^( )=1 mχ 2 /Λ 2 =

15 Attractive Basin (small UA(1) breaking) c A / 2 =0.01,x^( )=-0.3,z^( )=0.3 ca / 2 =0.01,x^( )=0,z^( )=0.3 ca / 2 =0.01,x^( )=0.3,z^( )=0.3 mχ 2 /Λ 2 = For smaller m χ2, attractive basin shrinks in vertical (g 2 ) direction. In order to realize 2 nd order transition, g 2 has to be tuned

16 Impact on the nature of SχSB So far, three possibilities have been discussed for the nature of transition: i) 1st order ii) 2nd order with O(4) scaling iii)2nd order with U(2) U(2) scaling Our study suggests fourth possibility: iv) 2nd order with O(4)-like scaling

17 Summary U(2) U(2) LSM with a finite UA(1) breaking is studied in ε-expansion. A novel possibility for the nature of chiral phase transition of massless two-flavor QCD, 2nd order with a scaling different from O(4). Difference from O(4) comes from non-decoupling. Non-decoupling effects induced non-universality.