Non-trivial θ-vacuum Effects in the 2-d O(3) Model

Transcription

1 Non-trivial θ-vacuum Effects in the 2-d O(3) Model Uwe-Jens Wiese Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Bern University CERN, August 2, 2012 Collaborators: Wolfgang Bietenholz (Mexico City) Michael Bögli, Urs Gerber, Ferenc Niedermayer (Bern) Philippe de Forcrand (CERN, ETH Zürich) Michele Pepe (Milano) JHEP 1012 (2010) 020, JHEP 1204 (2012) 117, arxiv:

2 Outline The 2-d O(3) Model at θ = 0 The 2-d O(3) Model at θ 0 Walking near the Conformal Fixed Point at θ = π Conclusions and Outlook

3 Outline The 2-d O(3) Model at θ = 0 The 2-d O(3) Model at θ 0 Walking near the Conformal Fixed Point at θ = π Conclusions and Outlook

4 Euclidean action S[ e] = 1 2g 2 Topological charge Q[ e] = 1 8π Schwarz inequality d 2 x µ e µ e iθq[ e] d 2 x ε µν e ( µ e ν e) Π 2 [S 2 ] = Z S[ e] 4π g 2 Q[ e]

5 Euclidean action S[ e] = 1 2g 2 Topological charge Q[ e] = 1 8π Schwarz inequality Exact mass gap d 2 x µ e µ e iθq[ e] d 2 x ε µν e ( µ e ν e) Π 2 [S 2 ] = Z S[ e] 4π g 2 Q[ e] m = 8 e Λ MS P. Hasenfratz, M. Maggiore, and F. Niedermayer, Phys. Lett. B245 (1990) 522

6 Euclidean action S[ e] = 1 2g 2 Topological charge Q[ e] = 1 8π Schwarz inequality Exact mass gap d 2 x µ e µ e iθq[ e] d 2 x ε µν e ( µ e ν e) Π 2 [S 2 ] = Z S[ e] 4π g 2 Q[ e] m = 8 e Λ MS P. Hasenfratz, M. Maggiore, and F. Niedermayer, Phys. Lett. B245 (1990) 522 Step-scaling function σ(2, u 0 ) = 2Lm(2L), u 0 = Lm(L) = L/ξ(L) M. Lüscher, P. Weisz, and U. Wolff, Nucl. Phys. B359 (1991) 221 J. Balog and A. Hegedus, J. Phys. A: Math. Gen. 37 (2004) 1881

7 Standard lattice action S[ e] = 1 g 2 e x e x+ˆµ. S 2 x,µ

8 Standard lattice action S[ e] = 1 g 2 e x e x+ˆµ. S 2 x,µ Lattice topological charge Q[ e] = 1 A xyz Z 4π t xyz Area of spherical triangle A 123 = 2ϕ [ 2π, 2π], X + iy = r exp(iϕ), X = 1 + e 1 e 2 + e 2 e 3 + e 3 e 1, Y = e 1 ( e 2 e 3 ) B. Berg and M. Lüscher, Nucl. Phys. B190 (1981) 412

9 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields.

10 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields. They do not have the correct classical continuum limit and are thus tree-level impaired.

11 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields. They do not have the correct classical continuum limit and are thus tree-level impaired. Perturbation theory is not applicable to topological lattice actions.

12 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields. They do not have the correct classical continuum limit and are thus tree-level impaired. Perturbation theory is not applicable to topological lattice actions. For δ < π 2 the topological action from above leads to a segmentation of configuration space into distinct topological sectors separated by infinite action barriers.

13 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields. They do not have the correct classical continuum limit and are thus tree-level impaired. Perturbation theory is not applicable to topological lattice actions. For δ < π 2 the topological action from above leads to a segmentation of configuration space into distinct topological sectors separated by infinite action barriers. Instantons as well as any other allowed field configurations have zero action.

14 Topological lattice action with an angle constraint exp( s( e x, e y )) = 1 for e x e y > cos δ, and zero otherwise A. Patrascioiu and E. Seiler, Nucl. Phys. Proc. Suppl. 30 (1993) 184; J. Stat. Phys. 106 (2002) 811 M. Hasenbusch, Phys. Rev. D53 (1996) 3445 W. Bietenholz, U. Gerber, M. Pepe, UJW, JHEP (2010) 1012:020 Topological lattice actions are invariant against (small) deformations of the lattice fields. They do not have the correct classical continuum limit and are thus tree-level impaired. Perturbation theory is not applicable to topological lattice actions. For δ < π 2 the topological action from above leads to a segmentation of configuration space into distinct topological sectors separated by infinite action barriers. Instantons as well as any other allowed field configurations have zero action. The action can be simulated with the Wolff cluster algorithm. U. Wolff, Phys. Rev. Lett. 62 (1989) 361; Nucl. Phys. B334 (1990) 581

15 Correlation function G(x y) = e x e y, G(p) = G(x) exp(ipx) Second moment correlation length ( ) χ F 1/2 ξ 2 (L) = 4F sin 2, χ = G(p = 0), F = G(p = (2π/L, 0)) (π/l) x

16 Correlation function G(x y) = e x e y, G(p) = G(x) exp(ipx) Second moment correlation length ( ) χ F 1/2 ξ 2 (L) = 4F sin 2, χ = G(p = 0), F = G(p = (2π/L, 0)) (π/l) x 100 ξ 2 (L) / a δ / π ξ 2 (L) a = Ag 2 exp ( 2π g 2 ), 1 g 2 = b δ 2 + c

17 Step-scaling approach M. Lüscher, P. Weisz, and U. Wolff, Nucl. Phys. B359 (1991) L xi(2l) L xi(l)

18 Step-scaling approach M. Lüscher, P. Weisz, and U. Wolff, Nucl. Phys. B359 (1991) L xi(2l) 1.8 ξ 2 (2L) / ξ 2 (L) L xi(l) ξ 2 (L) / L S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 75 (1995) 1891

19 Step-scaling function Σ(2, u 0, a/l) = 2Lm(2L), u 0 = Lm(L) = Cut-off effects of the step-scaling function Σ(2, u 0, a/l) = σ(2, u 0 ) + a2 [ ] L 2 B log 3 (L/a) + C log 2 (L/a) +...

20 Step-scaling function Σ(2, u 0, a/l) = 2Lm(2L), u 0 = Lm(L) = Cut-off effects of the step-scaling function Σ(2, u 0, a/l) = σ(2, u 0 ) + a2 [ ] L 2 B log 3 (L/a) + C log 2 (L/a) Standard action O(3) D(1/3) action O(3) D(-1/4) action Topological action Σ(2,u o,a/l) a / L J. Balog, F. Niedermayer, and P. Weisz, Phys. Lett. B676 (2009) 188; Nucl. Phys. B824(2010) 563

21 Topological susceptibility χ t = Q2 V Semiclassical treatment suggests scaling violations due to dislocations ( ) ( ) ( ) 4π 4π c ξ 2 c/2π χ t ξ 2 exp( S d ) exp g 2 = exp g 2 a M. Lüscher, Nucl. Phys. B200 (1982) 61

22 Topological susceptibility χ t = Q2 V Semiclassical treatment suggests scaling violations due to dislocations ( ) ( ) ( ) 4π 4π c ξ 2 c/2π χ t ξ 2 exp( S d ) exp g 2 = exp g 2 a M. Lüscher, Nucl. Phys. B200 (1982) < Q 2 > (L) L / a

23 Optimized constrained action S[ e] = xy s( e x, e y ), s( e x, e y ) = 1 g 2 (1 e x e y ) for e x e y > cos δ and s( e x, e y ) = otherwise

24 Optimized constrained action S[ e] = xy s( e x, e y ), s( e x, e y ) = 1 g 2 (1 e x e y ) for e x e y > cos δ and s( e x, e y ) = otherwise Optimization condition at u 0 = Σ(2, u 0, a/l = 1/10) = σ(2, u 0 ) cos δ = Σ(2,u 0, a/l) Standard action Constraint action Topological action θ =

25 Outline The 2-d O(3) Model at θ = 0 The 2-d O(3) Model at θ 0 Walking near the Conformal Fixed Point at θ = π Conclusions and Outlook

26 Triangulated square lattice with open boundary conditions L Correlation function C(t 1, t 2 ; θ) = t 1 S(t 1, t 2 ) t 2 1 Z(t 1, t 2 ; θ) x S 2 d e x E(t1 ) E(t 2 ) exp( S[ e] + iθq(t 1, t 2 )) exp( m(θ, L)(t 2 t 1 )) Z(t 1, t 2 ; θ) = d e x exp( S[ e] + iθq(t 1, t 2 )) x S 2 Computational procedure M. Hasenbusch, Nucl. Phys. Proc. Suppl. 42 (1995) 764 C(t 1, t 2 ; θ) = C(t 1, t 2 ; θ)z(t 1, t 2 ; θ)/z(0) Z(t 1, t 2 ; θ)/z(0)

27 Optimized constrained action with fixed cos δ = θ = π/ θ = π Σ(2,u 0, a/l) Analytic result σ(π, 2, u 0 = ) = Numerical result σ(π, 2, u 0 = ) = (6) confirms the conjectured exact S-matrix A. B. Zamolodchikov and V. A. Fateev, Sov. Phys. JETP 63 (1986) 913 a/l J. Balog, unpublished

28 θ-dependent massgap Lm(θ, L) at Lm(0, L) =

29 θ-dependent massgap Lm(θ, L) at Lm(0, L) = constrained action, C= m(π,l)/m(0,l) a/l

30 θ-dependent massgap Lm(θ, L) at Lm(0, L) = constrained action, C= m(π,l)/m(0,l) a/l Remaining non-uniform tiny cut-off effects in the per mille range.

31 θ-dependent massgap Lm(θ, L) at Lm(0, L) = constrained action, C= m(π,l)/m(0,l) a/l Remaining non-uniform tiny cut-off effects in the per mille range. Setting the scale at θ = 0 yields analytic result at θ = π.

32 θ-dependent massgap Lm(θ, L) at Lm(0, L) = constrained action, C= m(π,l)/m(0,l) a/l Remaining non-uniform tiny cut-off effects in the per mille range. Setting the scale at θ = 0 yields analytic result at θ = π. θ is a relevant parameter that does not renormalize non-perturbatively.

33 θ-dependent massgap Lm(θ, L) at Lm(0, L) = constrained action, C= m(π,l)/m(0,l) a/l Remaining non-uniform tiny cut-off effects in the per mille range. Setting the scale at θ = 0 yields analytic result at θ = π. θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct theory for each value of θ ] π, π].

34 Outline The 2-d O(3) Model at θ = 0 The 2-d O(3) Model at θ 0 Walking near the Conformal Fixed Point at θ = π Conclusions and Outlook

35 Low-energy effective Wess-Zumino-Novikov-Witten model S[U] = 1 d 2 x Tr[ 2g 2 µ U µ U] 2πikS WZNW [U] Topological WZNW term S WZNW [U] = 1 24π 2 d 2 x dx 3 ε µνρ Tr[U µ UU ν UU ρ U] H 3 Difference between two field extensions S WZNW [U (1) ] S WZNW [U (2) ] = 1 24π 2 d 2 x dx 3 ε µνρ Tr[U µ UU ν UU ρ U] Π 3 [S 3 ] = Z S 3 Quantization of level k exp(2πiks WZNW [U (1) ]) = exp(2πiks WZNW [U (2) ]) k Z

36 Triangular lattice L Meron-cluster algorithm t 1 Q = C Q C, t 2 2Q C Z W. Bietenholz, A. Pochinsky, and UJW, Phys. Rev. Lett. 75 (1995) 4524 Improved estimators for partition and correlation function Z(θ) Z(0) = C cos(θq C ), e x e y exp(iθq) = ( e x r)( e y r) C cos(θq C ) [ δ Cx,C y + (1 δ Cx,C y ) tan(θq Cx ) tan(θq Cy ) ]

37 Walking coupling g(θ, L) = m(θ, L)L, β(θ, g) = L L g(θ, L) m(θ,l) L 9 8 θ=0 7 6 θ=π/ θ=π M L

38 Walking coupling g(θ, L) = m(θ, L)L, β(θ, g) = L L g(θ, L) θ=0-0.5 m(θ,l) L 6 θ=π/ θ=π M L β(α) IRFP -2 θ=0 θ=π/2-2.5 θ=3π/4 θ=0.92 π θ=π α

39 Walking coupling g(θ, L) = m(θ, L)L, β(θ, g) = L L g(θ, L) m(θ,l) L M L θ=0 θ=π/2 θ=π β(α) IRFP -2 θ=0 θ=π/2-2.5 θ=3π/4 θ=0.92 π θ=π Scaling behavior m(θ, L ) θ π 2/3 log( θ π ) 1/2 α m(θ,l) L + [π- m(π,l) L] ML= 2 r 0 ML= 4 r 0 ML= 6 r 0 ML= 8 r 0 ML=12 r 0 ML=16 r 0 ML=24 r 0 ML=32 r M L t 2/3 log(t/t0) -1/2 P. de Forcrand, M. Pepe, UJW, arxiv:

40 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized.

41 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized. Technicolor theories: distance to conformal window controlled by discrete number of techniquark flavors or by the size of technifermion representation, which do affect the perturbative β-function.

42 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized. Technicolor theories: distance to conformal window controlled by discrete number of techniquark flavors or by the size of technifermion representation, which do affect the perturbative β-function. Lessons for simulating technicolor gauge theories Large logarithmic corrections near the edge of the conformal window.

43 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized. Technicolor theories: distance to conformal window controlled by discrete number of techniquark flavors or by the size of technifermion representation, which do affect the perturbative β-function. Lessons for simulating technicolor gauge theories Large logarithmic corrections near the edge of the conformal window. Careful control of cut-off effects is mandatory.

44 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized. Technicolor theories: distance to conformal window controlled by discrete number of techniquark flavors or by the size of technifermion representation, which do affect the perturbative β-function. Lessons for simulating technicolor gauge theories Large logarithmic corrections near the edge of the conformal window. Careful control of cut-off effects is mandatory. Large finite-size effects near the conformal window.

45 Differences between 2-d O(3) model and technicolor theories O(3) model: the parameter θ that determines the distance to the conformal fixed point is continuously varying and does not get renormalized. Technicolor theories: distance to conformal window controlled by discrete number of techniquark flavors or by the size of technifermion representation, which do affect the perturbative β-function. Lessons for simulating technicolor gauge theories Large logarithmic corrections near the edge of the conformal window. Careful control of cut-off effects is mandatory. Large finite-size effects near the conformal window. Particles with a mass much below Λ MS (technidilatons) may arise near the edge of the conformal window.

46 Outline The 2-d O(3) Model at θ = 0 The 2-d O(3) Model at θ 0 Walking near the Conformal Fixed Point at θ = π Conclusions and Outlook

47 Conclusions Topological lattice actions are invariant against small continuous deformations of the lattice field.

48 Conclusions Topological lattice actions are invariant against small continuous deformations of the lattice field. Topological lattice actions are tree-level impaired. Hence they do not have the correct classical continuum limit, one cannot treat them perturbatively, and they may violate a Schwarz inequality.

49 Conclusions Topological lattice actions are invariant against small continuous deformations of the lattice field. Topological lattice actions are tree-level impaired. Hence they do not have the correct classical continuum limit, one cannot treat them perturbatively, and they may violate a Schwarz inequality. Despite these classical deficiencies, topological lattice actions still have the correct quantum continuum limit.

50 Conclusions Topological lattice actions are invariant against small continuous deformations of the lattice field. Topological lattice actions are tree-level impaired. Hence they do not have the correct classical continuum limit, one cannot treat them perturbatively, and they may violate a Schwarz inequality. Despite these classical deficiencies, topological lattice actions still have the correct quantum continuum limit. The topological susceptibility of the 2-d lattice O(3) model does not seem to suffer from dislocation lattice artifacts. Still it is logarithmically ultraviolet divergent, just as in the continuum.

51 Conclusions Topological lattice actions are invariant against small continuous deformations of the lattice field. Topological lattice actions are tree-level impaired. Hence they do not have the correct classical continuum limit, one cannot treat them perturbatively, and they may violate a Schwarz inequality. Despite these classical deficiencies, topological lattice actions still have the correct quantum continuum limit. The topological susceptibility of the 2-d lattice O(3) model does not seem to suffer from dislocation lattice artifacts. Still it is logarithmically ultraviolet divergent, just as in the continuum. The optimized constrained action has extremely tiny cut-off effects, typically in the per mille range.

52 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively.

53 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct physical theory for each value of θ.

54 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct physical theory for each value of θ. Very accurate Monte Carlo simulations confirm the conjectured exact S-matrix for θ = π beyond any reasonable doubt.

55 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct physical theory for each value of θ. Very accurate Monte Carlo simulations confirm the conjectured exact S-matrix for θ = π beyond any reasonable doubt. At θ π the coupling is walking very slowly, thus mimicking the approach to the conformal window in technicolor theories.

56 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct physical theory for each value of θ. Very accurate Monte Carlo simulations confirm the conjectured exact S-matrix for θ = π beyond any reasonable doubt. At θ π the coupling is walking very slowly, thus mimicking the approach to the conformal window in technicolor theories. Outlook Highly optimized ultra-local lattice actions have also been constructed for O(N) models with N > 3 (J. Balog, F. Niedermayer, M. Pepe, P. Weisz, and UJW, to be published) and may even exist for non-abelian lattice gauge theories (M. Bögli, D. Banerjee, F. Niedermayer, M. Pepe, and UJW, work in progress).

57 Conclusions θ is a relevant parameter that does not renormalize non-perturbatively. There is a distinct physical theory for each value of θ. Very accurate Monte Carlo simulations confirm the conjectured exact S-matrix for θ = π beyond any reasonable doubt. At θ π the coupling is walking very slowly, thus mimicking the approach to the conformal window in technicolor theories. Outlook Highly optimized ultra-local lattice actions have also been constructed for O(N) models with N > 3 (J. Balog, F. Niedermayer, M. Pepe, P. Weisz, and UJW, to be published) and may even exist for non-abelian lattice gauge theories (M. Bögli, D. Banerjee, F. Niedermayer, M. Pepe, and UJW, work in progress). Is the 2-d O(3) model perhaps even integrable for 0 < θ < π?