The Wilson-Fisher Fixed Point via ERG Hidenori SONODA Physics Department, Kobe University 24 February 2010 at INT-10-45W RG workshop Abstract I construct the RG flows of the Wilson action for the real scalar field theory in D = 3 using the exact renormalization group. The construction results in a systematic and analytical way of computing the critical exponents at the Wilson-Fisher fixed point. Typeset by FoilTEX
Introduction 1 1. We consider a real scalar theory in D dimensional Euclid space: S = p 1 K (p/λ) 2 φ(p)φ( p) p 2 d D x ( m 2 0 2 φ2 + λ ) 0 4! φ4 where K is a cutoff function with K(p) { = 1 (p 2 1) 0 (p 2 ) 1 K(p) Λ plays the role of a momentum cutoff. 0 1 We study the correlation functions S = [dφ] e S / [dφ]e S. p 2
2. Critical behavior The critical value m 2 0,cr(λ 0 ) depends on λ 0. m 2 0 > m 2 0,cr (unbroken phase) φ = 0 m 2 0 < m 2 0,cr (broken phase) φ 0. The critical behavior is characterized by 2 critical exponents 2 y E = 1 ν > 0, η > 0 For m 2 m 2 0,cr the scaling formula holds: φ(r)φ(0) = 1 r D 2+ηF ±(r/ξ) { + if m 2 > m 2 0,cr if m 2 < m 2 0,cr where ξ is the correlation length: ξ m 2 m 2 0,cr(λ) 1 y E = m 2 m 2 0,cr(λ) ν
3. We can understand the critical behavior from the RG flows in the space of Wilson actions with a fixed momentum cutoff Λ. 3 critical trajectory λ 1 dim renormalized trajectory 2 m 0 WF fixed point 2 m (λ ) 0,cr 0 2 dim renormalized trajectories 0 Gaussian fixed point m 2 m 2 and λ parametrize the renormalized trajectories out of the GFP.
4. The RG flows are given in the infinite dimensional space of Wilson actions. 4 5. The flows out of the gaussian fixed point make a two-dimensional subspace, where the RG flows are given by { d dt m2 = 2m 2 + β m (m 2, λ) d dt λ = λ + β(m2, λ) with the gaussian fixed point at m 2 = λ = 0. 6. The Wilson-Fisher fixed point lies on this subspace, and it is given by (m 2, λ ) satisfying { 2m 2 + β m (m 2, λ ) = 0 λ + β(m 2, λ ) = 0
7. The goal is a concrete realization of Wilson s picture for D = 3 using the exact renormalization group. Especially, construction of the renormalized trajectories parametrized by m 2, λ analytical calculation of the critical exponents y E, η (Cf. Parisi s method with the Callan-Symanzik equation) 5 8. The ǫ expansions for the critical exponents with ǫ 4 D [Fisher & Wilson] { ye = 2 ǫ 3 + η = ǫ2 54 + high precision results from high order calculations [Brézin, Le Gillou, Zinn-Justin] Drawback The expansions do not work for theories that can be defined only for specific dimensions such as chiral theories or supersymmetric theories.
9. Outline 6 (a) introduce ERG for the Wilson action (b) construction of 2-dim renormalized trajectories out of the GFP (c) analytical formulas for β m, β in terms of the Wilson action (d) perturbative calculation of the exponents y E, η
Brief introduction to ERG 7 1. The initial action S Λ0 = 1 2 p p 2 ( m K(p/Λ 0 ) φ(p)φ( p) d D 2 x 0 2 φ2 + λ ) 0 4! φ4 }{{} =S I,Λ0 We define the Wilson action by S Λ = 1 2 p p 2 K (p/λ) φ(p)φ( p) + S I,Λ where exp [S I,Λ [φ]] Z [dφ ] exp " 1 2 Z p # p 2 K(p/Λ 0 ) K (p/λ) φ (p)φ ( p) + S I,Λ0 [φ + φ ]
2. A consequence of the gaussian functional integration: exp [S I,Λ [φ]] = Z [dφ ] exp " 1 2 Z p # p 2 K(p/Λ ) K (p/λ) φ (p)φ ( p) + S I,Λ [φ + φ ] 8 3. Equivalently, the Λ dependence is given by the Polchinski equation: Λ Λ S Λ = Z p (p/λ) p 2 " p 2 K (p/λ) φ(p) δs Λ δφ(p) + 1 ( δsλ δs Λ 2 δφ( p) δφ(p) + δ 2 S Λ δφ( p)δφ(p) )# where (p) (p) 2p 2 d dp 2K(p) 0 1 p 2
4. S Λ have the same correlation functions as S Λ0 : 9 { 1 φ(p)φ( p) φ(p)φ( p) K(p/Λ) 2 SΛ + 1 1/K(p/Λ) p 2 φ(p 1 ) φ(p n ) N i=1 1 K(p i /Λ) φ(p 1) φ(p N ) SΛ are independent of Λ.
Renormalization for D = 3 10 1. The solution S Λ of the Polchinski equation is determined uniquely by the initial condition at Λ = Λ 0. 2. Renormalization: for lim Λ0 S Λ to exist, m 2 0 must be given an appropriate Λ 0 dependence. 3. Notation: S Λ = n=1 p 1,,p 2n δ(p 1 + p 2n ) u 2n (Λ; p 1,,p 2n )φ(p 1 ) φ(p 2n )
4. Equivalently, S Λ (m 2, λ; µ) for the renormalized theory can be constructed by the following conditions: (a) Conditions at Λ = µ: { u2 (Λ = µ; p, p) = m 2 p 2 + O(p 4 ) u 4 (Λ = µ; p i = 0) = λ (b) Asymtptotic conditions for Λ : u 2 (Λ; p, p) Λ linear in p 2 u 4 (Λ; p 1, p 2, p 3, p 4 ) Λ independent of p i u 2n 6 (Λ; p 1,,p 2n ) Λ 0 5. S Λ is now parametrized by m 2, λ, and a renormalization scale µ. 11
6. The µ dependence of S Λ is given by the RG equation: 12 8 >< >: µ S Λ µ = βo λ + β m O m + γn O m m 2S Λ O λ λ S Λ N R» j φ(p) δs Λ p δφ(p) + K(p/Λ)(1 K(p/Λ)) p 2 δs Λ δs Λ δφ(p) δφ( p) + δ2 S Λ δφ(p)δφ( p) β, β m, η are functions of m 2, λ,µ. N counts the number of φ insertions: Nφ(p 1 ) φ(p n ) = n φ(p 1 ) φ(p n ) ff 7. This is equivalent to the RG equation: ( µ ) µ + β λ + β m m 2 φ(p 1 ) φ(p n ) = nγ φ(p 1 ) φ(p n )
Universality 13 1. How does S Λ depend on the arbitrary choice of K(p)? 2. An infinitesimal change δk(p) of the cutoff function changes S Λ by δs Λ = δz Z» Λ δk (p/λ) 2 N p K (p/λ) φ(p) δs Λ δφ(p) + 1 p 2δK (p/λ) 1 ( δsλ δs Λ 2 δφ(p) δφ( p) + δ 2 S Λ δφ(p)δφ( p) where δz Λ is determined so that δs Λ satisfies the normalization condition p 2δu 2(Λ = µ; p, p) = 0 p2 =0 )
3. Equivalence: 14 1 K(p) φ(p)φ( p) 2 S(t) + 1 1/K(p) p 2» 1 = (1 δz(t)) (K + δk)(p) φ(p)φ( p) 2 (S+δS)(t) ny 1 K(p i ) φ(p 1) φ(p n ) S(t) i=1 = 1 n 2 δz(t) «n Y i=1 1 (K + δk)(p i ) φ(p 1) φ(p n ) (S+δS)(t) 4. δk is absorbed by the change of m 2, λ, and normalization of φ. { δm 2 = δu 2 (Λ = µ; p = 0) δλ = δu 4 (Λ = µ; p i = 0) 1 1/(K + δk)(p) + p 2
Construction of Wilson s RG flows 15 It takes 3 steps to construct Wilson s RG flows. Step 1: Combine Polchinski s equation and the RG equation: Λ Λ µ Z» «(p/λ) = K (p/λ) γ p µ + β λ + β m m 2 «φ(p) δs Λ δφ(p) S Λ + 1 p 2 ( (p/λ) 2γK (p/λ) (1 K (p/λ))) 1 2 ( δsλ δs Λ δφ(p) δφ( p) + δ 2 S Λ δφ(p)δφ( p) )
Step 2: rescaling by dimensional analysis m 2 has dimension 2, λ has 1, and φ(p) has D+2 2 : Λ Λ + µ «µ + 2m2 m 2 + λ λ S Λ = Z p p µ φ(p) + D + 2 p µ 2 This amounts to rescaling that restores the original Λ. Combining Step 1 & 2, (λ + β) λ + (2m 2 + β m ) m 2 S Λ Z» j φ(p) D + 2 = p µ + γ + (p/λ) «p p µ 2 K (p/λ) ( (p/λ) 2γK(1 K) 1 δsλ δs Λ + p 2 2 δφ(p) δφ( p) + ff δsλ φ(p) δφ(p) 16 «δsλ φ(p) δφ(p) δ 2 S Λ δφ(p)δφ( p) )
Step 3: Set Λ = µ (fixed) and write S Λ = S(m 2, λ). Finally, we obtain (λ + β) λ + (2m 2 + β m ) m 2 S(m 2, λ) (1) Z» j φ(p) D + 2 = p ν + γ + (p/µ) «ff δs φ(p) p p ν 2 K(p/µ) δφ(p) + 1 p ( (p/µ) 2γK(1 K))) 1 ( ) δs δs 2 2 δφ(p) δφ( p) + δ 2 S δφ(p)δφ( p) From now on we can set µ = 1. 17
The RG flow is given by 18 and we obtain the scaling formula { d dt m2 = 2m 2 + β m (m 2, λ) d dt λ = λ + β(m2, λ) φ(p1 e t ) φ(p n e t ) m 2 e 2 t (1+ t β m ), λe t (1+ t β) = e t {D+n( D+2 2 +γ )} φ(p1 ) φ(p n ) m 2,λ
Determination of β, β m in terms of S 19 1. Expansion S(m 2, λ)[φ] = n=1 p 1,,p 2n δ(p 1 + +p 2n ) u 2n (m 2, λ;p 1,,p 2n ) φ(p 1 ) φ(p 2n ) where u 2, u 4 are normalized by { u2 (m 2, λ; p, p) = m 2 p 2 + u 4 (m 2, λ; p i = 0) = λ (2) 2. Substituting the normalization conditions (2) into (1), we obtain
20 β m = 1 1 2 q q 2 ( (q) ηk(q)(1 K(q))) u 4(m 2, λ;q, q, 0,0) 2γ = 1 1 2 p 2 q q 2 ( (q) ηk(q)(1 K(q)))u 4(m 2, λ; q, q, p, p) β 4λγ = 1 1 2 q 2 ( (q) ηk(q)(1 K(q))) u 6(m 2, λ;q, q, 0,0, 0,0) q These determine β m, β, γ in terms of S(m 2, λ). p 2 =0
Perturbative calculations 21 1. In practice, a mass independent scheme is more convenient: u m 2 2 (m 2, λ; p = 0) u p 2 2 (m 2, λ; p, p) u 4 (m 2, λ; p i = 0) m 2 =0 m 2 =p 2 =0 m 2 =0 = 1 = 1 = λ This gives { βm = C(λ) + β m (λ)m 2 β = β(λ)
2. RG flow equations 22 { d dt m2 = (2 + β m (λ))m 2 + C(λ) d dt λ = λ + β(λ) 3. Wilson-Fisher fixed point (m 2, λ ) { (2 + βm (λ ))m 2 + C(λ ) = 0 λ + β(λ ) = 0 4. Critical exponents: { ye = 2 + β m (λ ) η = 2γ(λ )
5. Lowest non-trivial loop expansions β m = λ 2 β q (q) ( = 3λ 2 q 2γ = λ2 2 p 2 q,r ) 1 q 2 m2 q 2 (q)(1 K(q)) q 4 (q) 1 K(r) q 2 r 2 1 K(q+r+p) (q+r+p) 2 p 2 =0 23 6. Fixed point gives λ = 3 R q 1 (1 K) q 4, m 2 = y E = 2 + β m (λ ) = 2 1 2 λ q η = 1 2 λ 2 p 2 q,r (q) 1 K(r) q 2 r 2 λ 2 R q 2 λ R q q 4 q 2 q 4 1 K(q+r+p) (q+r+p) 2 p 2 =0
7. Wegner-Houghton limit K(p) = θ(1 p 2 ) 24 η = { ye = 2 1 3 = 5 3 2 ln 2 1 54 = 0.38629... 54 = 0.007154... η comes out too small.
Application to the WZ model in D = 3 25 1. The bare action S Λ0 = 1 1 ( p 2 φ(p)φ( p) + χ( p) σ i pχ(p) + F(p)F( p) ) 2 p K(p/Λ 0 ) ( g d 3 ( x φ 2 if + χχφ ) ) gv2 2 2 if where χ χ T σ y. 2. Classical analysis: scalar potential (φ 2 v 2 ) 2 (a) v 2 > 0 Z 2 broken, SUSY exact (b) v 2 < 0 Z 2 unbroken, SUSY broken Z 2 : φ(x,y, z) φ(x, y, z), χ(x, y, z) σ y χ(x, y, z)
3. Critical exponents: 26 (a) Scale dimension of g (v 2 v 2 ): 1 + 1 3 q 23 q (1 K) 2 q 4 + q (1 K) 2 q 4 + 3 2 q (1 K) q 4 (1 K) q 4 WH = 1 + 3 7 = 1.428... (b) Anomalous dimension η = 1 23 q (1 K) q q 4 (1 K) 2 + 3 q 4 2 q (1 K) q 4 WH = 1 7 = 0.142...
Conclusions 27 1. Wilson s RG flows can be constructed concretely using ERG. 2. Fixed points and critical exponents can be calculated by loop expansions. 3. ERG is applicable to dimension specific theories such as chiral and supersymmetric theories. 4. The nature of expansions is unclear due to the absence of an obvious expansion parameter. 5. Optimization should help for better numerical accuracy.
Appendix: Redefinition of β m, β, γ 28 8 < : β m = C 1 λ + C 2 λ 2 + + m 2 (B 1 λ + B 2 λ 2 + ) β = A 1 λ 2 + A 2 λ 3 + γ = D 2 λ 2 + D 3 λ 3 + By redefining 8 >< >: m 2 c 1 λ + c 2 λ 2 + + m 2 (1 + b 1 λ + b 2 λ 2 + ) λ λ + a 1 λ 2 + a 2 λ 3 + φ (1 + d 1 λ + d 2 λ 2 + )φ we can make λ 2 d dt m2 = 2m 2 1 24(2π) 2 d dt λ = λ γ = 0