Delineating the conformal window

Transcription

1 University of Oxford STRONGBSM Kick-off Meeting, CP3-Origins Odense, August 19 th 2010

2 Outline 1 Motivations for Technicolor and New Strong Dynamics BSM 2

3 Outline 1 Motivations for Technicolor and New Strong Dynamics BSM 2

4 In collaboration with: Thomas Pickup and Mike Teper (Oxford U.), ArXiv:

5 What is the world made of?

6 What should the world be made of? See S.Sarkar: Universenet, confronting theory with experiment, Copenhagen 10

7 What should the world be made of?

8 What should the world be made of?

9 Dynamical vs SM-like Higgs sector Natural, v EW F Π dynamical. Fine-tuning, triviality etc.

10 Dynamical vs SM-like Higgs sector Natural, v EW F Π dynamical. Known realizations QCD and Superconductivity Fine-tuning, triviality etc. No known fundamental scalars

11 Dynamical vs SM-like Higgs sector Natural, v EW F Π dynamical. Known realizations QCD and Superconductivity Naturally stable DM. Ω DM,Ω B of same origin? Fine-tuning, triviality etc. No known fundamental scalars Hand-made stability.

12 Dynamical vs SM-like Higgs sector Natural, v EW F Π dynamical. Known realizations QCD and Superconductivity Naturally stable DM. Ω DM,Ω B of same origin? Dynamical flavor sector complicated Fine-tuning, triviality etc. No known fundamental scalars Hand-made stability. Flavor sector simply parametrized

13 Dynamical vs SM-like Higgs sector Natural, v EW F Π dynamical. Known realizations QCD and Superconductivity Naturally stable DM. Ω DM,Ω B of same origin? Dynamical flavor sector complicated Fine-tuning, triviality etc. No known fundamental scalars Hand-made stability. Flavor sector simply parametrized Need to determine the phase diagram of strongly coupled gauge theories

14 Phases of gauge theories - Analytically 3C 2 (R), α 4π = β 0 β 1. (Appelquist, Lane and Muhanta 88; Cohen and Georgi 89; Sannino and 1 Ladder approximation: α c = π Tuominen 04; Dietrich and Sannino 06; Ryttov and Sannino 07) 2 All-orders beta function conjecture(s) (Ryttov and Sannino 08; Antipin and Tuominen 09; Dietrich 09) 3 Dualities (Sannino 09) 4 Compactification approach (Unsal and Poppitz 09; Ogilvie and Myers 09;) 5 Worldline formalism (Armoni 09) 6 Holography (Hong and Yee 06; Alvares, Evans, Gebauer and Weatherill 09) 7 Metric confinenement and causal analytic couplings (Oehme and Zimmerman 80; Nishijima 86; Oehme 1990; Gardi and Grunberg 98; M.T.F, Pickup and Teper 10)

15 Seibergs analysis of SQCD Using holomorphic effective action and dualities Seiberg has argued the following phase diagram of SQCD (Seiberg 94) N f N 1 SQCD, N f Fundamentals 9 8 IR Conformal Confining, chiral symmetry Confining, chiral symmetry broken N C

16 Seibergs analysis of SQCD Using holomorphic effective action and dualities Seiberg has argued the following phase diagram of SQCD (Seiberg 94) N f N 1 SQCD, N f Fundamentals IR Conformal Confining, chiral symmetry Confining, chiral symmetry broken N C Note that in SQCD conformality is lost due to confinement not chiral symmetry breaking (See e.g. Pallante s discussion at this conference)

17 Approaches for Non-Susy theories: Schwinger-Dyson eq. The fermion Schwinger-Dyson gap equation in Euclidean space: is 1 (p) = /p m + g Dkγ µ G µν (p k)s(k)γ ν (p k;p,k) G is gluon propagator, Γ is the 1PI 3-point vertex, Dk = d4 k (2π) 4 and the full fermion propagator, with wave-function renormamlization Z(p) and fermion self-energy Σ(p) is 1 (p) = Z(p)/p Σ(p), (1) (Many authors, see e.g: Cohen and Georgi 88)

18 Rainbow approx to Schwinger-Dyson eq. Rainbow approximation: Γ = igγ and G = (p k) 2 (free propagator in Landau gauge). Then Z(p) = 1,Σ(p) = m + 3C 2 (R)g 2 Dk(p k) 2 Σ(k) Σ(k) 2 + k 2, (2) In the limit of small momentum p Σ(p), the solutions are Σ(p) p γm(µ), Σ(p) p γm(µ) 2, (3) with mass anomalous dimension and critical coupling γ m (µ) = 1 1 α(µ) π,α c α c 3C 2 (R). (4) Solutions correspond to hard and dynamical mass terms. (Simple to analyze the corresponding see (Cohen and Georgi 88))

19 Beyond the Rainbow approximation The rainbow approximation satisfies the Ward identity: g [ Z(p)/p Z(k)/k Σ(p) + Σ(k) ] = (p k)γ(p k;p,k) (5) For Γ = igγ the Ward identity is satisfied by the asymptotic behaviour above, above: Z 1,Σ 0 for p (6) However Z 1 is not true in other gauges: Effectively restricted to work in Landau gauge where coupling and wave-fct renorm vanishes.

20 Conformality lost: Schwinger Dyson analysis of SQCD N f N 1 SQCD, N f Fundamentals 9 8 IR Conformal Confining, chiral symmetry Confining, chiral symmetry broken N C The rainbow approximation gives the lower conformal boundary Nf II = 2.25N c. Chiral symmetry breaking happens much later (Appelquist, Nyffeler and Selipsky 97)

21 Metric Confinement The metric confinement criterion provides a lower bound on the conformal window

22 Metric Confinement The metric confinement criterion provides a lower bound on the conformal window Consider the asymptotic behaviour of the gluon propagator D D µν (k 2 ) dx 4 e ikx A µ (x)a ν (0) D µν (k 2 ) = (g µν k µ k ν /k 2 )D(k 2 ) (7)

23 Metric Confinement The metric confinement criterion provides a lower bound on the conformal window Consider the asymptotic behaviour of the gluon propagator D D µν (k 2 ) dx 4 e ikx A µ (x)a ν (0) D µν (k 2 ) = (g µν k µ k ν /k 2 )D(k 2 ) (7) Define the one-loop β function and gluon anomalous dimension γ of the Callan-Symanzik equations: β(x) dx d ln(q 2 ) = ( β 0 x 2 + β 1 x 3 + ), γ(x,ζ) = (γ 00 + ζγ 01 )x,(κ 2 κ 2 + β x + γ)d = 0 (8)

24 Conformality lost: Metric Confinement Oehme and Zimmermann showed, that asymptotically: ) γ00 lim k k2 D(k 2 ) (ln[ k2 /β 0 µ 2] (9) For β 0 < 0 the asymptotic fall-off of D is faster than k 2 for γ 00 > 0 leading to a superconvergence relation D(p 2 ) = dk 2ImD(k2 ) k 2 p 2 ImD(k 2 ) = 0 (10) Metric confinement, is defined as a phase with no (transverse) gauge field excitations in the state space (defined through the BRST algebra). 0 Oehme and Zimmermann 80

25 Conformality lost: Metric Confinement Oehme and Zimmermann showed, that asymptotically: ) γ00 lim k k2 D(k 2 ) (ln[ k2 /β 0 µ 2] (9) For β 0 < 0 the asymptotic fall-off of D is faster than k 2 for γ 00 > 0 leading to a superconvergence relation D(p 2 ) = dk 2ImD(k2 ) k 2 p 2 ImD(k 2 ) = 0 (10) Metric confinement, is defined as a phase with no (transverse) gauge field excitations in the state space (defined through the BRST algebra). Superconvergence then implies metric confinement! Oehme and Zimmermann 80 0

26 Metric Confinement and SQCD Metric confinement provides a lower bound Nf MC boundary of the conformal window: on the lower

27 Metric Confinement and SQCD Metric confinement provides a lower bound Nf MC on the lower boundary of the conformal window: SQCD saturates the this lower bound (also for SO,Sp gauge groups)! N f N 1 SQCD, N f Fundamentals IR Conformal No metric confinement Confining, chiral symmetry Confining, chiral symmetry broken N C Figure: Phase-diagram of SQCD from holomorphy and the γ 00 = 0 line

28 Metric Confinement and SQCD II If we are interested in non-susy predictions then the question is: How restricting is reproducing the Seiberg phase diagram of SQCD?

29 Metric Confinement and SQCD II If we are interested in non-susy predictions then the question is: How restricting is reproducing the Seiberg phase diagram of SQCD? The ACS conjecture also reproduces SQCD but differs from metric confinement criterion for non-susy theories

30 Metric Confinement and SQCD II If we are interested in non-susy predictions then the question is: How restricting is reproducing the Seiberg phase diagram of SQCD? The ACS conjecture also reproduces SQCD but differs from metric confinement criterion for non-susy theories In fact the ACS confecture predicts no conformal window at all for an SU(2) theory with adjoint fermions (Appelquist, Cohen and Schmaltz 99; Sannino 09)

31 Applying metric confinement to non-susy theories β(x) dx d ln(q 2 ) = ( β 0 x 2 + β 1 x 3 + ), (11) where now x α s /π. The first two coefficients of the expansion are universal, independent of the renormalisation group scheme: 4β 0 = 11 3 C 2(G) 4 3 T(R)N f (12) 16β 1 = 34 3 C2 2 (G) 20 3 C 2(G)T(R)N f 4C 2 (R)T(R)N f. (13) β 0 = 0 determines the upper conformal boundary: N I f = 11 4 C 2 (G) T(R), (14)

32 The condition for metric confinement, in terms of γ 00 is: γ 00 = 1 ( C 2(G) 4 ) 3 T(R)N f < 0. (15)

33 The condition for metric confinement, in terms of γ 00 is: γ 00 = 1 ( C 2(G) 4 ) 3 T(R)N f < 0. (15) From metric confinement the lower bound on the lower boundary of the conformal window is then: N II f N MC f 13C 2 (G)/8T(R). (16)

34 Conformal windows for SU theories with Dirac fermions N f N f Fundamental rep I N f II,MC N f II,SD N f N C 2 index symmetric rep N f N f Adjoint rep N C AS N C (M.T.F, Pickup and Teper 10) N C

35 SD in rainbow approximation vs metric confinement SD rainbow approximation predicts MWT (2 Dirac flavors in the adjoint/2s of SU(2)) to be just below and the NMWT model (2 Dirac flavors in the 2S of SU(3)) well below the conformal window.

36 SD in rainbow approximation vs metric confinement SD rainbow approximation predicts MWT (2 Dirac flavors in the adjoint/2s of SU(2)) to be just below and the NMWT model (2 Dirac flavors in the 2S of SU(3)) well below the conformal window. SD rainbow approximation predicts the 1-family model (8 Dirac flavors in the fundamental of SU(2)) to be just above the lower conformal boundary.

37 SD in rainbow approximation vs metric confinement SD rainbow approximation predicts MWT (2 Dirac flavors in the adjoint/2s of SU(2)) to be just below and the NMWT model (2 Dirac flavors in the 2S of SU(3)) well below the conformal window. SD rainbow approximation predicts the 1-family model (8 Dirac flavors in the fundamental of SU(2)) to be just above the lower conformal boundary. According to MC, MWT and the 1-family model are well above the lower bound of the conformal window with NMWT just (above) it.

38 Perturbation theory and causal analyticity Due to AF, at large momentum transfer Q 2, the coupling constant behaves as x(q 2 ) 1 β 0 ln(q 2 /Λ 2 )

39 Perturbation theory and causal analyticity Due to AF, at large momentum transfer Q 2, the coupling constant behaves as x(q 2 ) 1 β 0 ln(q 2 /Λ 2 ) Physical quantities calculated with the 1-loop running coupling will inherit this Landau singularity.

40 Perturbation theory and causal analyticity In the conformal window the coupling does not diverge: 0 x(q 2 ) x FP for 0 Q 2 <

41 Perturbation theory and causal analyticity In the conformal window the coupling does not diverge: 0 x(q 2 ) x FP for 0 Q 2 < As we approach the upper bound, N f Nf I, the coupling becomes weak on all scales and we may expect perturbation theory to work well.

42 Perturbation theory and causal analyticity In the conformal window the coupling does not diverge: 0 x(q 2 ) x FP for 0 Q 2 < As we approach the upper bound, N f Nf I, the coupling becomes weak on all scales and we may expect perturbation theory to work well. We can see this explicitly by integrating the two-loop β-function β(x) = ( β 0 x 2 + β 1 x 3 + ) x(q 2 ) = 1 c W (z), c = β 1 β 0, z = 1 c e ( Q 2 Λ 2 ) β0 /c,w(z)exp[w (z)] = z.

43 2-loop coupling N f Fundamental rep 10 II,SD N f II,MC N I f 5 N f CA N f II,AO N 0 f N C x Q Fundamental rep SU 3,N f 12 SU 3,N f Q 2 2

44 Causal Analyticity The integrated two-loop coupling x(q 2 ) can be continued to the complex plane Q 2 C

45 Causal Analyticity The integrated two-loop coupling x(q 2 ) can be continued to the complex plane Q 2 C If x(q 2 ) has the analytic structure of a typical physical quantity: a cut for k 2 = Q 2 0 and no other singularities for Q 2 C it is causal analytic maxarg Q x Q SU 2,F,N f 7 SU 3,F,N f 10 SU 3,2S,N f 2 SU 2,Adj,N f Q 2 2

46 Causal Analyticity II There are three singularity structures of the asymptotically free two-loop coupling: causal analytic, branch points into the complex plane, the usual landau singularity maxarg Q x Q SU 2,F,N f 7 SU 3,F,N f 10 SU 3,2S,N f 2 SU 2,Adj,N f Q 2 2

47 Causal analyticity III The 2-loop coupling is causal analytic if: 0 < β 2 0/β 1 < 1 (17)

48 Causal analyticity III The 2-loop coupling is causal analytic if: 0 < β 2 0/β 1 < 1 (17) This is stronger than requiring the two-loop β-function have a fixed-point since, as β 1 0 the bound is violated The coupling is causal analytic all the way down to Nf MC provided C 2 (R) > C 2(G).

49 Causal analyticity III The 2-loop coupling is causal analytic if: 0 < β 2 0/β 1 < 1 (17) This is stronger than requiring the two-loop β-function have a fixed-point since, as β 1 0 the bound is violated The coupling is causal analytic all the way down to Nf MC provided C 2 (R) > C 2(G). True for all two-index reps of SU(N).

50 Causal analyticity III The 2-loop coupling is causal analytic if: 0 < β 2 0/β 1 < 1 (17) This is stronger than requiring the two-loop β-function have a fixed-point since, as β 1 0 the bound is violated The coupling is causal analytic all the way down to Nf MC provided C 2 (R) > C 2(G). True for all two-index reps of SU(N). However, in SQCD causal analyticity breaks down before the lower conformal boundary!

51 Causal analyticity III The 2-loop coupling is causal analytic if: 0 < β 2 0/β 1 < 1 (17) This is stronger than requiring the two-loop β-function have a fixed-point since, as β 1 0 the bound is violated The coupling is causal analytic all the way down to Nf MC provided C 2 (R) > C 2(G). True for all two-index reps of SU(N). However, in SQCD causal analyticity breaks down before the lower conformal boundary! Consistent with the strong-weak duality of SQCD in the IR

52 Conformal window and causal analyticity N f N f Fundamental rep 10 I N f II,MC N f 5 II,SD N f CA N f N C 2 index symmetric rep N f N f Adjoint rep N C AS N C (M.T.F, Pickup and Teper 10) N C

53 Conformal window and analyticity II We should ask how large x(q) is at N MC f?

54 Conformal window and analyticity II We should ask how large x(q) is at Nf MC? We trust(?) 2-loop causal analyticity if N c x is everywhere small for Q 2 C. maxq Nc x Q N f 2 N f 10 N f 7 N f 8 N f 2 N f 12 N f 3 N f N f SU 2,F SU 3,F SU 2,Adj SU 3,2S Figure: N f (N f Nf MC )/(Nf I NMC f )

55 Conformal window and analyticity II We should ask how large x(q) is at Nf MC? We trust(?) 2-loop causal analyticity if N c x is everywhere small for Q 2 C. maxq Nc x Q N f 2 N f 10 N f 7 N f 8 N f 2 N f 12 N f 3 N f N f SU 2,F SU 3,F SU 2,Adj SU 3,2S Figure: N f (N f Nf MC )/(Nf I NMC f ) QCD rule of thumb is to begin to worry about perturbation theory for α s 0.5 N c x s 0.5

56 Summarizing conformal window and analyticity The criterion of metric confinement provides a lower bound Nf MC for the conformal window - saturated in SQCD (SU,SO,Sp). MWT (NMWT) and 1-family model all (just) above Nf MC

57 Summarizing conformal window and analyticity The criterion of metric confinement provides a lower bound Nf MC for the conformal window - saturated in SQCD (SU,SO,Sp). MWT (NMWT) and 1-family model all (just) above Nf MC MWT, NMWT and 1-family are all 2-loop causal analytic down to Nf MC.

58 Summarizing conformal window and analyticity The criterion of metric confinement provides a lower bound Nf MC for the conformal window - saturated in SQCD (SU,SO,Sp). MWT (NMWT) and 1-family model all (just) above Nf MC MWT, NMWT and 1-family are all 2-loop causal analytic down to Nf MC. Causal analyticity implies perturbation theory is consistent - we can interpret this as the theories being perturbative down to Nf MC.

59 Summarizing conformal window and analyticity The criterion of metric confinement provides a lower bound Nf MC for the conformal window - saturated in SQCD (SU,SO,Sp). MWT (NMWT) and 1-family model all (just) above Nf MC MWT, NMWT and 1-family are all 2-loop causal analytic down to N MC f. Causal analyticity implies perturbation theory is consistent - we can interpret this as the theories being perturbative down to N MC f. SQCD is not (2-loop) causal analytic at the lower end of the conformal window - consistent with the above interpretation and strong-weak duality in SQCD.

60 Summarizing conformal window and analyticity The criterion of metric confinement provides a lower bound Nf MC for the conformal window - saturated in SQCD (SU,SO,Sp). MWT (NMWT) and 1-family model all (just) above Nf MC MWT, NMWT and 1-family are all 2-loop causal analytic down to N MC f. Causal analyticity implies perturbation theory is consistent - we can interpret this as the theories being perturbative down to N MC f. SQCD is not (2-loop) causal analytic at the lower end of the conformal window - consistent with the above interpretation and strong-weak duality in SQCD. Even the coupling in 10-flavor QCD was argued to be perturbative by Gardi and Grunberg - though the 2-loop coupling here is rather large.

61 All-orders β-function and anomalous dimensions A different lower bound on the conformal window comes from the conjectured all-orders β-function: Ryttov and Sannino 07 β(x) = β 0 x 21 T(R)N f γ m (x)/(6β 0 ) ( 1 x 2 C 2(G) 1 + 2β 0 β 0 )x, (18) Here, γ m d lnm d ln µ is the fermion mass anomalous dimension

62 All-orders β-function and anomalous dimensions A different lower bound on the conformal window comes from the conjectured all-orders β-function: Ryttov and Sannino 07 β(x) = β 0 x 21 T(R)N f γ m (x)/(6β 0 ) ( 1 x 2 C 2(G) 1 + 2β 0 β 0 )x, (18) Here, γ m d lnm d ln µ is the fermion mass anomalous dimension Since γ m 2 is a rigorous bound from unitarity, this provides a different lower bound on Nf II, N AO f = 11 8 C 2 (G) T(R) < 13 8 C 2 (G) T(R) = NMC f (19)

63 All-orders β-function and anomalous dimensions A different lower bound on the conformal window comes from the conjectured all-orders β-function: Ryttov and Sannino 07 β(x) = β 0 x 21 T(R)N f γ m (x)/(6β 0 ) ( 1 x 2 C 2(G) 1 + 2β 0 β 0 )x, (18) Here, γ m d lnm d ln µ is the fermion mass anomalous dimension Since γ m 2 is a rigorous bound from unitarity, this provides a different lower bound on Nf II, N AO f = 11 8 C 2 (G) T(R) < 13 8 C 2 (G) T(R) = NMC f (19) Moreover, this β-function predicts γ m at the fixed point, β = 0: γ m = 11C 2(G) 4T(R)N f 2T(R)N f

64 Conformal window lower bounds: MC and AO Fundamental rep Adjoint rep N f N f II,SD N f II,MC N I f 5 N f CA N f II,AO N 0 f N C 2 index symmetric rep N C N f N f N C AS N C

65 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 )

66 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 ) With this leading order γ the AOBF can be exactly integrated, yielding the same criterion for causal analyticity as the two-loop analysis!

67 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 ) With this leading order γ the AOBF can be exactly integrated, yielding the same criterion for causal analyticity as the two-loop analysis! Next we can compare γm 1 loop (x FP ) using x FP from two-loop β function to the AOBF prediction γ m = 11C 2(G) 4T(R)N f 2T(R)N f for the (N)MWT models (where x FP is small)

68 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 ) With this leading order γ the AOBF can be exactly integrated, yielding the same criterion for causal analyticity as the two-loop analysis! Next we can compare γm 1 loop (x FP ) using x FP from two-loop β function to the AOBF prediction γ m = 11C 2(G) 4T(R)N f 2T(R)N f for the (N)MWT models (where x FP is small) For (N)MWT we find γm 1 loop (x FP ) = 1.34, 0.6 with x 2 loop FP

69 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 ) With this leading order γ the AOBF can be exactly integrated, yielding the same criterion for causal analyticity as the two-loop analysis! Next we can compare γm 1 loop (x FP ) using x FP from two-loop β function to the AOBF prediction γ m = 11C 2(G) 4T(R)N f 2T(R)N f for the (N)MWT models (where x FP is small) For (N)MWT we find γ 1 loop m (x FP ) = 1.34, 0.6 with x 2 loop FP Going to next order in γ m and β in MS gives O(10)%corrections

70 AOBF and analyticity To leading order γ m (x) = 3 2 C 2(R)x + O(x 2 ) With this leading order γ the AOBF can be exactly integrated, yielding the same criterion for causal analyticity as the two-loop analysis! Next we can compare γm 1 loop (x FP ) using x FP from two-loop β function to the AOBF prediction γ m = 11C 2(G) 4T(R)N f 2T(R)N f for the (N)MWT models (where x FP is small) For (N)MWT we find γm 1 loop Going to next order in γ m and β in MS gives O(10)%corrections The AOBF result is γ AOBF m = 1.3, 0.75 (x FP ) = 1.34, 0.6 with x 2 loop FP

71 Comparing to lattice results (near-) conformality of the (N)MWT models seem consistent with lattice simulations Please see talks at this conference, M.T.F, Pickup and Teper 10 for references

72 Comparing to lattice results (near-) conformality of the (N)MWT models seem consistent with lattice simulations Anomalous dimensions for the MWT model γ m < 1 found in lattice simulations. However at smaller fixed-point couplings than the two-loop prediction. Please see talks at this conference, M.T.F, Pickup and Teper 10 for references

73 Comparing to lattice results (near-) conformality of the (N)MWT models seem consistent with lattice simulations Anomalous dimensions for the MWT model γ m < 1 found in lattice simulations. However at smaller fixed-point couplings than the two-loop prediction. NMWT more uncertain from both the MC/CA and lattice perspective: The 1-loop anomalous dimension is large, due to a large-fixed point coupling and perhaps not trustable. However the next order MS corrections are small and the prediction agrees with the AOBF. Some lattice results point to near-conformal rather than conformal. In that case γ m is scheme dependent. Please see talks at this conference, M.T.F, Pickup and Teper 10 for references

74 Conclusions Metric confinement and causal analyticity provides an interesting approach to studying the conformal window

75 Conclusions Metric confinement and causal analyticity provides an interesting approach to studying the conformal window Applied to SQCD they are in agreement with Seibergs analysis.

76 Conclusions Metric confinement and causal analyticity provides an interesting approach to studying the conformal window Applied to SQCD they are in agreement with Seibergs analysis. Applied to the (N)MWT models they indicate (near-) conformality and (near-) perturbativity.

77 Conclusions Metric confinement and causal analyticity provides an interesting approach to studying the conformal window Applied to SQCD they are in agreement with Seibergs analysis. Applied to the (N)MWT models they indicate (near-) conformality and (near-) perturbativity. In the case of MWT this points to an anomalous dimension γ m < 1 as does the all-orders β-function and recent lattice simulations - caveats apply in the comparison!

78 Conclusions Metric confinement and causal analyticity provides an interesting approach to studying the conformal window Applied to SQCD they are in agreement with Seibergs analysis. Applied to the (N)MWT models they indicate (near-) conformality and (near-) perturbativity. In the case of MWT this points to an anomalous dimension γ m < 1 as does the all-orders β-function and recent lattice simulations - caveats apply in the comparison! Finally the prospects of understanding strong-dynamics and Technicolor models of EWSB with more and more lattice results and LHC data in the future are very exciting!

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