ERG for a Yukawa Theory in 3 Dimensions Hidenori SONODA Physics Department, Kobe University, Japan 14 September 2010 at ERG2010 Abstract We consider a theory of N Majorana fermions and a real scalar in the 3 dimensional Euclid space. We construct the RG flow of three parameters using ERG. We look for a non-trivial fixed point by loop expansions. For N = 1, we compare the critical exponents with those of the Wess-Zumino model at the fixed point. We need to improve the numerical accuracy, perhaps, by going to 2-loop. Typeset by FoilTEX
Plan of the talk 1 1. Spinors in D = 3 2. A Yukawa model in R 3 3. ERG formalism (a quick review) 4. RG flow for the Yukawa model 5. 1-loop results 6. Comparison with the Wess-Zumino model 7. Summary and conclusions
Spinors in D = 3 2 1. Minkowski space (a) We can choose the gamma matrices pure imaginary: γ 0 = σ y, γ 1 = iσ x, γ 2 = iσ z satisfy {γ µ, γ ν } = 2η µν 1 2 (b) A Lorentz transformation transforms a two-component spinor ψ as ψ Aψ where A SL(2, R) real! Its complex conjugate ψ ψ γ 0 transforms as ψ ψa 1
(c) Given ψ = 1 2 (u + iv), where u, v are Majorana (real) spinors, 3 ψ (iγ µ µ m) ψ = 1 ( ) ũ(iγ }{{} µ µ m)u + ṽ(iγ µ µ m)v 2 real where ũ u T σ y transforms as ũ ũa 1. (σ y A T σ y = A 1 ) (d) Parity { ψ(x, y, t) iγ 1 ψ( x, y, t) = σ x ψ( x, y, t) ψ(x, y, t) ψ( x, y, t)iγ 1 = ψ( x, y, t)( )σ x forbids the mass term ψψ.
2. Euclid space 4 (a) A spatial rotation transforms a two-component spinor ψ as ψ Uψ where U SU(2) complex! and ψ ψ T σ y as ψ ψu 1 (b) Though a Majorana spinor ψ is not real, the action d 3 x 1 2 ψ ( σ + m) ψ makes sense for the anticommuting ψ.
5 (c) Parity { ψ(x, y, z) σy ψ(x, y, z) ψ(x, y, z) ψ(x, y, z)( )σ y forbids the mass term ψψ. (d) Parity Parity & π-rotation around the y-axis gives { ψ(x, y, z) iψ( x, y, z) ψ(x, y, z) i ψ( x, y, z)
A Yukawa model in R 3 6 1. The classical lagrangian L = 1 2 ( φ)2 + 1 2 N I=1 χ I σ χ I + 1 2 m2 φ 2 + λ 4! φ4 + g φ 1 2 N I=1 χ I χ I where φ is a real scalar, and χ I (I = 1,, N) are Majorana spinors. 2. three dimensionful parameters [m 2 ] = 2, [λ] = 1, [g] = 1/2 3. A fermion mass term is forbidden by the Z 2 invariance (parity): { φ( x) φ( x) χ I ( x) iχ I ( x)
4. Classical phase analysis 7 { m 2 > 0 Z 2 exact φ = 0, m χ = 0 m 2 < 0 Z 2 broken φ = 0, m χ 0 5. Related models with the same Z 2 invariance (a) N = 1 Wess-Zumino model (1 SUSY), classically λ = 3g 2 L W Z = 1 2 ( φ)2 + 1 2 χ σ χ + g φ1 2 χχ + 1 8 g2 ( φ 2 v 2) 2 Classical phase analysis Z 2 SUSY φ masses v 2 < 0 exact broken 0 m χ = 0, m 2 φ = g2 2 ( v2 ) v 2 > 0 broken exact ±v m χ = m φ = g v
(b) N 2 Gross-Neveu model with O(N) invariance 8 L GN = 1 2 I χ I σ χ I g 0 2N ( 1 2 I χ I χ I ) 2 Expectation Z 2 I 1 2 χ Iχ I m χ g 0 > g 0,cr exact 0 = 0 g 0 < g 0,cr broken 0 0 O(N) unbroken
6. We wish to do two things: 9 (a) construct the RG flow of m 2, λ, g for the Yukawa model, (b) find a non-trivial fixed point and obtain the critical exponents there. 7. The non-trivial fixed point is accessible from the gaussian fixed point. Perturbation theory is applicable. 8. Motivated by works by Wipf, Synatschke-Czerwonka, Braun,... on SUSY models in 1 & 2 dimensions. 9. Talks on related subjects by Wipf, Synatschke-Czerwonka, Scherer.
Gross-Neveu N>1 g g 0 0,cr RG flows for the Yukawa theory with N Majorna fermions λ Z broken 2 Z exact 2 Wilson-Fisher f.p. (m 2*, λ *, g * ) 10 g m 2 Gaussian f.p. 2 m 2 m cr (g) Wess-Zumino g fixed N=1
ERG formalism (a quick review) 11 1. Wilson action with a momentum cutoff Λ S Λ = 1 2 p p 2 K ( p)φ(p)φ( p) + S I,Λ Λ 1 K(x) 0 1 x 2 2. the ERG differential equation by J. Polchinski (1983) Λ Λ S I,Λ = 1 2 p (p/λ) p 2 { δsi,λ δs I,Λ δφ( p) δφ(p) + δ 2 } S I,Λ δφ( p)δφ(p)
where (p/λ) Λ K ( p Λ) Λ. 12 3. Λ independence of the correlation functions 4. Vertices defined by 1 φ(p)φ( p) φ(p)φ( p) K( Λ) p 2 SΛ + 1 1/K ( Λ) p p 2 φ(p 1 ) φ(p n ) N 1 i=1 K( p i Λ ) φ(p 1) φ(p N ) SΛ S Λ = n=1 1 (2n)! p 1,,p 2n δ(p 1 + p 2n ) u 2n (Λ; p 1,, p 2n )φ(p 1 ) φ(p 2n )
5. S Λ determined uniquely by the ERG diff eq & two sets of conditions: one at Λ = µ, another at Λ. (a) conditions at Λ = µ introduce m 2 and λ: u 2 (µ; 0, 0) = m 2 u p 2 2 (µ; p, p) = 1 p 2 =0 u 4 (µ; 0,, 0) = λ (b) asymptotic conditions for Λ for renormalizability u 2 (Λ; p, p) Λ linear in p 2 u 4 (Λ; p 1,, p 4 ) Λ u 2n 6 (Λ; p 1,, p 2n ) Λ 0 independent of p i This guarantees that the flow starts from the gaussian fixed point. 13
6. µ dependence where O m O λ N µ S Λ µ = βo λ + β m O m + γn m 2S Λ λ S Λ [ φ(p) δs Λ p δφ(p) + K ( p Λ ) (1 K( p Λ ) ) { δs Λ δs Λ p 2 δφ(p) δφ( p) + δ2 S Λ δφ(p)δφ( p) }] 14 N counts φ: RG equation N φ(p 1 ) φ(p n ) = n φ(p 1 ) φ(p n ) ( µ ) µ + β λ + β m m 2 φ(p 1 ) φ(p n ) = nγ φ(p 1 ) φ(p n )
7. dimensional analysis ( Λ Λ + µ µ + 2m2 m 2 + λ λ ) S Λ = p ( p i φ(p) p i + 5 ) 2 φ(p) δsλ δφ(p) 15 8. Combining the ERG diff eq, µ dependence, and dimensional analysis, we obtain { (λ + β) λ + (2m 2 } + β m ) m 2 S(m 2, λ) = + p [ { p i φ(p) p i + (p/µ) 2γK(1 K) 1 p 2 2 ( 5 2 γ + (p/µ) ) } δs φ(p) K(p/µ) δφ(p) { δs δs δφ(p) δφ( p) + δ 2 } ] S δφ(p)δφ( p) where we have set Λ = µ (fixed).
9. RG flow in the parameter space 16 { d dt m2 = 2m 2 + β m (m 2, λ) d dt λ = λ + β(m2, λ) 10. Scaling formula 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 {3+n( 5 2 +γ )} φ(p1 ) φ(p n ) m 2,λ 11. Universality The dependence on the choice of K is absorbed into the parameters m 2, λ. Hence, the critical exponents are universal.
RG flow for the Yukawa model 17 1. RG flow d dt m2 = 2m 2 + β m (m 2, λ, g) d dt λ = λ + β λ(λ, g) d dt g = 1 2 g + β g(λ, g) 2. A fixed point (m 2, λ, g ) is determined by 2m 2 + β m (m 2, λ, g ) = 0 λ + β λ (λ, g ) = 0 1 2 g + β g (λ, g ) = 0
3. y E = 1 ν determined by the linearized RG equation at the fixed point: 18 d dt ( m 2 m 2 ) = y E ( m 2 m 2 ) Hence, y E = 2 + m 2β m m 2,λ,g 4. The anomalous dimensions γ φ, γ χ determined by γ φ = γ φ (λ, g ), γ χ = γ χ (λ, g ) 5. parametrization and normalization determined through the low momentum expansions of the vertices
m 2 p 2 φ I I p p p p J J p 2 =m 2 =0 p 2 =m 2 =0 p 2 =m 2 =0 m 2 =0 pi =m 2 =0 = 0 = 1 = 1 = i p σδ IJ + O(p 2 ) = g N δ IJ 19 pi =m 2 =0 = λ
6. The beta functions are computed from the RG equation 20 µ µ S Λ = β λ ( λ S Λ ) + β g ( g S Λ ) + β m O m + γ φ N φ + γ χ N χ where O m m 2S Λ since we have incorporated m 2 in the scalar propagator. O m = m 2S Λ { K(1 K) 1 δsλ δs Λ (q 2 + m 2 ) 2 2 δφ(q) δφ( q) + q δ 2 } S Λ δφ(q)δφ( q)
1-loop results 21 1. 1-loop calculations give β m = 1 2 λµ q q 2 2g2 N µ (1 K) q q 2 [ { 1 m 2 λ 2 µ q q + 2 g2 1 4 Nµ q q 4 3 (1 K) + 1 } ] ( + ) 6 β λ = 3 λ2 (1 K) + 24 g4 (1 K) 3 µ q q 4 Nµ q q 4 { 4λ g2 1 Nµ q q 4 3 (1 k) + 1 } ( + ) 6 [ β g = g g2 (1 K) 2 (1 K) 3 + Nµ q 4 q 4 q q
+ q q 4 where (q) 2q 2 d dq 2 (q) = The anomalous dimensions are given by { 1 3 (1 K) + 1 } ] ( + ) 6 ( 2q 2 d dq 2 ) 2 K(q). ( γ φ = g2 1 5 µ q q4(1 K) 6 + 1 6 γ χ = g2 1 1 Nµ 2 q4 (1 K) q ) 22
2. Choice of K with one parameter a for optimization 23 K(q) = 1 if 0 < q 2 < a 2 1 q 2 1 a 2 if a 2 < q 2 < 1 0 if 1 < q 2 1 0 K(q) a 2 1 q 2 Two limits: { Litim a 2 = 0 Wegner-Houghton a 2 = 1
3. a dependence of y E for N = 1 24 1.4 N=1 1.35 1.3 1.25 y E 1.2 1.15 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 a Optimized at a 0.646, giving y E 1.36.
4. γ φ = γ χ is expected from SUSY, but 25 0.5 0.45 gamma-phi gamma-chi 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 a γ φ 0.170, γ χ 0.056 at a = 0.646
5. N dependence 26 y E N 1 γ φ N 1 2 γ χ N 0 1.4 1.35 N=1 N=10 N=100 0.5 0.45 N=1 N=10 N=100 0.07 0.06 N=1 N=10 N=100 y E 1.3 1.25 1.2 1.15 1.1 1.05 gamma-phi 0.4 0.35 0.3 0.25 0.2 gamma-chi 0.05 0.04 0.03 0.02 1 0.15 0.01 0.95 0 0.2 0.4 0.6 0.8 1 a 0.1 0 0.2 0.4 0.6 0.8 1 a 0 0 0.2 0.4 0.6 0.8 1 a The large N limits are independent of a. The above results agree with the large N limit of the Gross-Neveu model.
Comparison with the Wess-Zumino model 27 1. Classical action with an auxiliary field L = 1 2 ( φ)2 + 1 2 χ σ χ + 1 2 F 2 + g { if 1 2 ( φ 2 v 2) + φ 1 2 χχ } where v 2 = 2 g 2 m 2. 2. Linear SUSY transformation (arbitrary constant spinor ξ) δφ δχ δif = χξ = ( σ φ if ) ξ = χ σξ
28 3. The supersymmetric φif + 1 2 χχ is forbidden by the Z 2 φ( x) φ( x) χ( x) iχ( x) F ( x) F ( x) 4. R-invariance g g, φ φ, F F 5. ERG is consistent with the linear SUSY. 6. RG equation µ µ S Λ = β g ( g S Λ ) + β v 2 ( v 2S Λ ) + γn
29 7. RG flow { d dt g = 1 2 g + β g d dt v2 = v 2 + β v 2 8. 1-loop results γ = 1 g 2 (1 K) 2 µ [ q q 4 = g3 µ 3 q = µ + v 2 g 2 q q 2 µ β g β v 2 (1 K) 2 + 1 q 4 2 [ 3 q ] (1 K) q 4 q (1 K) 2 q 4 + q ] (1 K) q 4 9. y E = 1 ν determined by y E = 1 + v 2β v 2 v 2,g
10. At 1-loop, we have reproduced the relation found by Wipf,... 30 γ + y E = 3 2 11. No optimization available, but γ depends little on a. 0.076 γ = 5(2a2 + 3a + 1) 6(26a 2 + 33a + 11) Comparable with γ χ 0.056 for the Yukawa model. gamma-star 0.0755 0.075 0.0745 0.074 0.0735 0.073 0.0725 0.072 0.0715 0.071 0 0.2 0.4 0.6 0.8 1 a
Summary and conclusions 31 WZ Y (N = 1) Y (large N) GN (large N) y E 1.429-1.424 1.36 1 1 γφ 0.071-0.076 0.17 0.5 0.5 γχ = γφ 0.056 0 0 1. ERG gives an exact formulation of the RG flows for the Yukawa model in D = 3. 2. Perturbative approximations give reasonable numerical results. 3. Hopefully two-loop calculations (in progress) will improve the numerical accuracy.