Standard Model with Four Generations and Multiple Higgs Doublets

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1 RGE,SDE,SM4 p. 1/2 Standard Model with Four Generations and Multiple Higgs Doublets 1st IAS-CERN School Jan 26th, 2012, Singapore Chi Xiong Institute of Advanced Studies Nanyang Technological University

2 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4)

3 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop)

4 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop) Strong Yukawa couplings and Quasi-fixed point

5 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop) Strong Yukawa couplings and Quasi-fixed point Bound states/condensates of the 4th generation

6 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop) Strong Yukawa couplings and Quasi-fixed point Bound states/condensates of the 4th generation Schwinger-Dyson equation(sde)

7 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop) Strong Yukawa couplings and Quasi-fixed point Bound states/condensates of the 4th generation Schwinger-Dyson equation(sde) Implications of RGE+SDE

8 RGE,SDE,SM4 p. 2/2 Outlines Standard Model with four generations (SM4) Renormalization group equation (RGE) (2-loop) Strong Yukawa couplings and Quasi-fixed point Bound states/condensates of the 4th generation Schwinger-Dyson equation(sde) Implications of RGE+SDE arxiv: (NPB 847), (PLB 694), (NPB 848) (P. Q. Hung, CX)

9 RGE,SDE,SM4 p. 3/2 Why SM4? A sequential family of heavy quarks and leptons has not been ruled out yet G. D. Kribs, T. Plehn, M. Spannowsky and T. Tait, PRD 76, (2007); P. Q. Hung and M. Sher, PRD 77, (2008); H. He, N. Polonsky and S. Su, PRD 64, (2001); M. Chanowitz, PRL 87, (2001) V. A. Novikov, L. B. Okun, A. N. Rozanov and M. I. Vysotsky, PLB 529 (2002);... B. Holdom, PLB 686(2010); J. Erler, P. Langacker, PRL 105, (2010),...

10 RGE,SDE,SM4 p. 3/2 Why SM4? A sequential family of heavy quarks and leptons has not been ruled out yet G. D. Kribs, T. Plehn, M. Spannowsky and T. Tait, PRD 76, (2007); P. Q. Hung and M. Sher, PRD 77, (2008); H. He, N. Polonsky and S. Su, PRD 64, (2001); M. Chanowitz, PRL 87, (2001) V. A. Novikov, L. B. Okun, A. N. Rozanov and M. I. Vysotsky, PLB 529 (2002);... B. Holdom, PLB 686(2010); J. Erler, P. Langacker, PRL 105, (2010),... A heavy 4th generation can alleviate the naturalness (hierarchy) problem of SM3

11 RGE,SDE,SM4 p. 3/2 Why SM4? A sequential family of heavy quarks and leptons has not been ruled out yet G. D. Kribs, T. Plehn, M. Spannowsky and T. Tait, PRD 76, (2007); P. Q. Hung and M. Sher, PRD 77, (2008); H. He, N. Polonsky and S. Su, PRD 64, (2001); M. Chanowitz, PRL 87, (2001) V. A. Novikov, L. B. Okun, A. N. Rozanov and M. I. Vysotsky, PLB 529 (2002);... B. Holdom, PLB 686(2010); J. Erler, P. Langacker, PRL 105, (2010),... A heavy 4th generation can alleviate the naturalness (hierarchy) problem of SM3 A heavy 4th generation can trigger the dynamical electroweak symmetry breaking.

12 RGE,SDE,SM4 p. 3/2 Why SM4? A sequential family of heavy quarks and leptons has not been ruled out yet G. D. Kribs, T. Plehn, M. Spannowsky and T. Tait, PRD 76, (2007); P. Q. Hung and M. Sher, PRD 77, (2008); H. He, N. Polonsky and S. Su, PRD 64, (2001); M. Chanowitz, PRL 87, (2001) V. A. Novikov, L. B. Okun, A. N. Rozanov and M. I. Vysotsky, PLB 529 (2002);... B. Holdom, PLB 686(2010); J. Erler, P. Langacker, PRL 105, (2010),... A heavy 4th generation can alleviate the naturalness (hierarchy) problem of SM3 A heavy 4th generation can trigger the dynamical electroweak symmetry breaking. Standard Model with 4 Generations is probably one of the simplest extensions to SM3

13 RGE,SDE,SM4 p. 4/2 Why SM4? From our study with RGE+SDE, a heavy 4th generation drives the Yukawa couplings to the strong region binding force for condensates

14 RGE,SDE,SM4 p. 4/2 Why SM4? From our study with RGE+SDE, a heavy 4th generation drives the Yukawa couplings to the strong region binding force for condensates brings the cutoff scale down to Λ TeV hierarchy problem

15 RGE,SDE,SM4 p. 4/2 Why SM4? From our study with RGE+SDE, a heavy 4th generation drives the Yukawa couplings to the strong region binding force for condensates brings the cutoff scale down to Λ TeV hierarchy problem leads to a quasi-fixed point around Λ at two-loop level Landau pole, triviality

16 RGE,SDE,SM4 p. 4/2 Why SM4? From our study with RGE+SDE, a heavy 4th generation drives the Yukawa couplings to the strong region binding force for condensates brings the cutoff scale down to Λ TeV hierarchy problem leads to a quasi-fixed point around Λ at two-loop level Landau pole, triviality suggests the restoration of scale invariance above Λ new conformal theories?

17 RGE,SDE,SM4 p. 5/2 RG running of couplings in SM3 We begin with the RGE approach. The RG running of gauge couplings and Higgs couplings (quartic and Yukawa) in SM3 (M h = 120GeV 180GeV ) g 2 3 g 2 2 g t log E 91.2Gev t log E 91.2GeV

18 RGE,SDE,SM4 p. 6/2 RGE in SM4 We study the evolutions of gauge and Higgs couplings (quartic and Yukawa) in SM4

19 RGE,SDE,SM4 p. 6/2 RGE in SM4 We study the evolutions of gauge and Higgs couplings (quartic and Yukawa) in SM4 At two-loop level (MS scheme), 16π 2 dy i dt = β yi

20 RGE,SDE,SM4 p. 6/2 RGE in SM4 We study the evolutions of gauge and Higgs couplings (quartic and Yukawa) in SM4 At two-loop level (MS scheme), 16π 2 dy i dt = β yi e.g. the Higgs quartic coupling β λ = 24λ 2 + 4λ(3g 2 t + 6g 2 q + 2g 2 l 2.25g g 2 1) 12(3g 4 t + 6g 4 q + 2g 4 l ) + (16π 2 ) 1 [180g 6 t +288g 6 q + 96g 6 l (3g 4 t + 6g 4 q + 2g 4 l 80g 2 3(g 2 t +2g 2 q))λ 6λ 2 (24g 2 t + 48g 2 q + 16g 2 l ) 312λ 3 192g 2 3(g 4 t + 2g 4 q)] +...

21 RGE,SDE,SM4 p. 6/2 RGE in SM4 We study the evolutions of gauge and Higgs couplings (quartic and Yukawa) in SM4 At two-loop level (MS scheme), 16π 2 dy i dt = β yi e.g. the Higgs quartic coupling β λ = 24λ 2 + 4λ(3g 2 t + 6g 2 q + 2g 2 l 2.25g g 2 1) 12(3g 4 t + 6g 4 q + 2g 4 l ) + (16π 2 ) 1 [180g 6 t +288g 6 q + 96g 6 l (3g 4 t + 6g 4 q + 2g 4 l 80g 2 3(g 2 t +2g 2 q))λ 6λ 2 (24g 2 t + 48g 2 q + 16g 2 l ) 312λ 3 192g 2 3(g 4 t + 2g 4 q)] +... other couplings β yq, β yl, β yt, β gi,i=1,2,3... (Machacek and Vaughn, 1983)

22 RGE,SDE,SM4 p. 7/2 Zeros of β-functions These RGEs can be integrated numerically, but first we search for roots of β yi = 0 with fixed gauge couplings.

23 RGE,SDE,SM4 p. 7/2 Zeros of β-functions These RGEs can be integrated numerically, but first we search for roots of β yi = 0 with fixed gauge couplings. g3 2 g2 2 g1 2 λ gt 2 gq 2 gl

24 RGE,SDE,SM4 p. 8/2 Zeros of β-functions The fixed point values of the quartic and Yukawa couplings are approximately λ /(4π) , g 2 t /(4π) 2 0.2, g 2 q /(4π) , g 2 l /(4π) 2 0.

25 RGE,SDE,SM4 p. 8/2 Zeros of β-functions The fixed point values of the quartic and Yukawa couplings are approximately λ /(4π) , g 2 t /(4π) 2 0.2, g 2 q /(4π) , g 2 l /(4π) 2 0. The gauge couplings contribute only small fluctuations. These correspond to naive MS masses (using m H = v 2λ, m f = vg f / 2, v = 246 GeV ) m H = 1.44 TeV, m t = 0.97 TeV, m q = 1.26 TeV, m l = 1.28 TeV.

26 RGE,SDE,SM4 p. 8/2 Zeros of β-functions The fixed point values of the quartic and Yukawa couplings are approximately λ /(4π) , g 2 t /(4π) 2 0.2, g 2 q /(4π) , g 2 l /(4π) 2 0. The gauge couplings contribute only small fluctuations. These correspond to naive MS masses (using m H = v 2λ, m f = vg f / 2, v = 246 GeV ) m H = 1.44 TeV, m t = 0.97 TeV, m q = 1.26 TeV, m l Questions: Can this (quasi)fixed point be reached? If yes, at what energy scale? = 1.28 TeV.

27 RGE,SDE,SM4 p. 9/2 RG running of Higgs couplings The RG running of Higgs couplings (quartic and Yukawa), light mass cases (M q = 120GeV 250GeV ) M l 100 GeV g 2 l g 2 q M q 250 GeV M q 160 GeV M q 120 GeV g 2 t Λ t log E 91.2GeV

28 RGE,SDE,SM4 p. 10/2 RG running of Higgs couplings The RG running of Higgs couplings (quartic and Yukawa), heavy mass case (M q = 300GeV 500GeV ) M q 350 GeV M l 250 GeV g 2 l g 2 q 40 M q 500 GeV M q 300 GeV 30 M l 400 GeV M l 200 GeV g 2 t 20 Λ t log E 91.2GeV

29 RGE,SDE,SM4 p. 11/2 Landau Pole vs. Fixed Point Compare 1-loop and 2-loop results Landau Pole vs. Fixed Point 4 2 M q 500 GeV M l 400 GeV Α l Α q Α t M q 120 GeV M l 100 GeV t log E 91.2GeV

30 RGE,SDE,SM4 p. 12/2 RG running of Higgs couplings From the numerical calculations, we see that The evolution of Higgs couplings run into a quasi-fixed point at some scale Λ FP Λ FP decreases when the mass the 4th generation increases

31 RGE,SDE,SM4 p. 12/2 RG running of Higgs couplings From the numerical calculations, we see that The evolution of Higgs couplings run into a quasi-fixed point at some scale Λ FP Λ FP decreases when the mass the 4th generation increases An interesting thing is, by increasing their masses just about 3 times, the 4th generation brings Λ FP from GeV down to few TeV.

32 RGE,SDE,SM4 p. 12/2 RG running of Higgs couplings From the numerical calculations, we see that The evolution of Higgs couplings run into a quasi-fixed point at some scale Λ FP Λ FP decreases when the mass the 4th generation increases An interesting thing is, by increasing their masses just about 3 times, the 4th generation brings Λ FP from GeV down to few TeV. The existence of a quasi-fixed point the triviality problem; The physical consequences of shifting the scale Λ FP down to TeV level provide an alternative solution to the hierarchy problem.

33 RGE,SDE,SM4 p. 13/2 Comments For the RGEs, the expansion parameters are g 2 t /16π 2 0.2, g 2 q /16π , g 2 q /16π , λ /16π

34 RGE,SDE,SM4 p. 13/2 Comments For the RGEs, the expansion parameters are g 2 t /16π 2 0.2, g 2 q /16π , g 2 q /16π , λ /16π These represent strong quartic and Yukawa couplings. The exact location of Λ FP and the values of Higgs couplings at Λ FP should be studied non-perturbatively, e.g. put on lattice

35 RGE,SDE,SM4 p. 13/2 Comments For the RGEs, the expansion parameters are g 2 t /16π 2 0.2, g 2 q /16π , g 2 q /16π , λ /16π These represent strong quartic and Yukawa couplings. The exact location of Λ FP and the values of Higgs couplings at Λ FP should be studied non-perturbatively, e.g. put on lattice β 2 loop (g ) = 0 may give a clue in finding β(g ) = 0 where the scale invariance is restored at some energy

36 α π or α 4π? An order of one expansion parameter? We have seen similar situations before, e.g., the Wilson-Fisher ɛ-expansion, g 4 = 16π 2 ɛ/3 for the physical value ɛ = 1, ɛ = 4 d. RGE,SDE,SM4 p. 13/2 Comments For the RGEs, the expansion parameters are g 2 t /16π 2 0.2, g 2 q /16π , g 2 q /16π , λ /16π These represent strong quartic and Yukawa couplings. The exact location of Λ FP and the values of Higgs couplings at Λ FP should be studied non-perturbatively, e.g. put on lattice β 2 loop (g ) = 0 may give a clue in finding β(g ) = 0 where the scale invariance is restored at some energy

37 RGE,SDE,SM4 p. 14/2 Wilson-Fisher ɛ-expansion µ d dµ g 4(µ) = ɛ g 4 (µ) + 3g2 4(µ) 16π 2 + O(g3 4(µ)) µ d dµ g 2(µ) = g 2 (µ)[ 2 + g 4(µ) 16π 2 + O(g 4(µ))] where ɛ = 4 d and g 2 and g 4 come from terms g 2 φ 2 /2, g 4 φ 4 /4! respectively.

38 RGE,SDE,SM4 p. 14/2 Wilson-Fisher ɛ-expansion µ d dµ g 4(µ) = ɛ g 4 (µ) + 3g2 4(µ) 16π 2 + O(g3 4(µ)) µ d dµ g 2(µ) = g 2 (µ)[ 2 + g 4(µ) 16π 2 + O(g 4(µ))] where ɛ = 4 d and g 2 and g 4 come from terms g 2 φ 2 /2, g 4 φ 4 /4! respectively. Solving β(g 4 ), β(g 2 ) = 0 equations, one finds a non-trivial fixed point at g 4 = 16π2 ɛ 3, g 2 = 0.

39 RGE,SDE,SM4 p. 14/2 Wilson-Fisher ɛ-expansion µ d dµ g 4(µ) = ɛ g 4 (µ) + 3g2 4(µ) 16π 2 + O(g3 4(µ)) µ d dµ g 2(µ) = g 2 (µ)[ 2 + g 4(µ) 16π 2 + O(g 4(µ))] where ɛ = 4 d and g 2 and g 4 come from terms g 2 φ 2 /2, g 4 φ 4 /4! respectively. Solving β(g 4 ), β(g 2 ) = 0 equations, one finds a non-trivial fixed point at g 4 = 16π2 ɛ 3, g 2 = 0. For d=3, it corresponds to g 4 = 16π 2 / or g 4/16π 2 = 1/3

40 RGE,SDE,SM4 p. 15/2 Wilson-Fisher ɛ-expansion The critical exponent ν is then given by the ɛ-expansion ν = 1/2 + ɛ/12 + 7ɛ 2 / ɛ 3 + O(ɛ 4 )

41 RGE,SDE,SM4 p. 15/2 Wilson-Fisher ɛ-expansion The critical exponent ν is then given by the ɛ-expansion ν = 1/2 + ɛ/12 + 7ɛ 2 / ɛ 3 + O(ɛ 4 ) At 1-loop level, ν = 0.58, at 2-loop level, ν = 0.63, at 3-loop level, ν = 0.61 while the experimental value is ν = 0.63

42 RGE,SDE,SM4 p. 15/2 Wilson-Fisher ɛ-expansion The critical exponent ν is then given by the ɛ-expansion ν = 1/2 + ɛ/12 + 7ɛ 2 / ɛ 3 + O(ɛ 4 ) At 1-loop level, ν = 0.58, at 2-loop level, ν = 0.63, at 3-loop level, ν = 0.61 while the experimental value is ν = 0.63 It is fortunate though still somewhat mysterious that an expansion in powers of 1 should work so well. (Weinberg, QFT II)

43 RGE,SDE,SM4 p. 15/2 Wilson-Fisher ɛ-expansion The critical exponent ν is then given by the ɛ-expansion ν = 1/2 + ɛ/12 + 7ɛ 2 / ɛ 3 + O(ɛ 4 ) At 1-loop level, ν = 0.58, at 2-loop level, ν = 0.63, at 3-loop level, ν = 0.61 while the experimental value is ν = 0.63 It is fortunate though still somewhat mysterious that an expansion in powers of 1 should work so well. (Weinberg, QFT II) We do not expect such a precise calculation, but hopefully inclusion of higher order terms will not shift the location and the values of the quasi-fixed point by an order of magnitude.

44 RGE,SDE,SM4 p. 16/2 Bound States/Condensates The Yukawa couplings become strong around Λ FP. Can the 4th generation fermions form bound states or condensates by the Yukawa coupling?

45 RGE,SDE,SM4 p. 16/2 Bound States/Condensates The Yukawa couplings become strong around Λ FP. Can the 4th generation fermions form bound states or condensates by the Yukawa coupling? We perform a non-relativistic analysis at quantum mechanics level, with a Higgs-exchange potential where α Y = m 1 m 2 /4πv 2. V (r) = α Y (r) e m H(r)r r

46 RGE,SDE,SM4 p. 16/2 Bound States/Condensates The Yukawa couplings become strong around Λ FP. Can the 4th generation fermions form bound states or condensates by the Yukawa coupling? We perform a non-relativistic analysis at quantum mechanics level, with a Higgs-exchange potential V (r) = α Y (r) e m H(r)r where α Y = m 1 m 2 /4πv 2. The possibility of forming bound states is characterized by r K f = g3 f 16π λ.

47 RGE,SDE,SM4 p. 17/2 Bound States/Condensates The criteria is K f > 2 (variational method) K f > 1.68 (numerical method)

48 RGE,SDE,SM4 p. 17/2 Bound States/Condensates The criteria is K f > 2 (variational method) K f > 1.68 (numerical method) If we use the fixed-point values of the quartic and Yukawa couplings, we find that the 4th generation may form loosely bound state, while the top quark cannot. K q = 1.82, K l = 1.92, K t = 0.82

49 RGE,SDE,SM4 p. 17/2 Bound States/Condensates The criteria is K f > 2 (variational method) K f > 1.68 (numerical method) If we use the fixed-point values of the quartic and Yukawa couplings, we find that the 4th generation may form loosely bound state, while the top quark cannot. K q = 1.82, K l = 1.92, K t = 0.82 An interesting region is around the dip, i.e. λ 0, where the Yukawa potential becomes a strong Coulomb-like potential. formation of condensates (Rafelski, Fulcher and Klein, 1978).

50 RGE,SDE,SM4 p. 18/2 Bound States/Condensates 2.5 Α l Region I Region II Α q Λ 4Π 1 Α t 0.5 FP t log E 91.2GeV Region I: Condensates v.s. Region II: Fixed Point Note: Neither technicolor nor other unkown interactions are introduced for condensates.

51 RGE,SDE,SM4 p. 19/2 Bound States/Condensates Kf K0 4 2 K NR K K q K 0 K l K 0 Kt K t log E 91.2GeV (m q = 450 GeV and m l = 350 GeV) K f K 0 with K f = gf 3/16π λ and K 0 = The horizontal dotted line indicates an estimate of K f where the non-relativistic method is still applicable and the vertical dotted lines enclose the region where a fully relativistic approach is needed.

52 RGE,SDE,SM4 p. 20/2 Schwinger-Dyson Equation In SM4 the Yukawa couplings become strong around TeV

53 RGE,SDE,SM4 p. 20/2 Schwinger-Dyson Equation In SM4 the Yukawa couplings become strong around TeV The perturbative approach becomes less reliable when approaching Λ FP

54 RGE,SDE,SM4 p. 20/2 Schwinger-Dyson Equation In SM4 the Yukawa couplings become strong around TeV The perturbative approach becomes less reliable when approaching Λ FP Lattice? Nielson-Ninomiya no-go theorem (chiral gauge theory) mirror fermion

55 RGE,SDE,SM4 p. 20/2 Schwinger-Dyson Equation In SM4 the Yukawa couplings become strong around TeV The perturbative approach becomes less reliable when approaching Λ FP Lattice? Nielson-Ninomiya no-go theorem (chiral gauge theory) mirror fermion dispersion relation? π N system

56 RGE,SDE,SM4 p. 20/2 Schwinger-Dyson Equation In SM4 the Yukawa couplings become strong around TeV The perturbative approach becomes less reliable when approaching Λ FP Lattice? Nielson-Ninomiya no-go theorem (chiral gauge theory) mirror fermion dispersion relation? π N system We use Schwinger-Dyson approach (Gap Equation, mean field theory, Hartree-Fock approximation,...)

57 RGE,SDE,SM4 p. 21/2 Schwinger-Dyson Equation Consider Yukawa couplings in SM4 (truncated to only the 4th generation) L Y = g b q L Φ b R g t q L Φ t R + h.c. Φ = iτ 2 Φ, q L = (t, b ) L as usually defined in the SM. φ 0 = + t Figure 1: Graphic representation of the Schwinger-Dyson equation for the quark self-energy (quenched approximation)

58 RGE,SDE,SM4 p. 22/2 Gap Equation For simplicity we only consider the 4th generation quarks. From the SDE the quark self energy satisfies Σ 4Q (p) = +2g2 4Q (2π) 4 d 4 q 1 Σ 4Q (q) (p q) 2 q 2 + Σ 2 4Q (q)

59 RGE,SDE,SM4 p. 22/2 Gap Equation For simplicity we only consider the 4th generation quarks. From the SDE the quark self energy satisfies Σ 4Q (p) = +2g2 4Q (2π) 4 d 4 q 1 Σ 4Q (q) (p q) 2 q 2 + Σ 2 4Q (q) which can be converted to a differential equation Σ 4Q (p) = α α c where α c = π/2, α = g 2 4Q /4π Σ 4Q (q) q 2 + Σ 2 4Q (q)

60 RGE,SDE,SM4 p. 22/2 Gap Equation For simplicity we only consider the 4th generation quarks. From the SDE the quark self energy satisfies Σ 4Q (p) = +2g2 4Q (2π) 4 d 4 q 1 Σ 4Q (q) (p q) 2 q 2 + Σ 2 4Q (q) which can be converted to a differential equation Σ 4Q (p) = α α c where α c = π/2, α = g 2 4Q /4π Σ 4Q (q) q 2 + Σ 2 4Q (q) compared with α c = π/3 in strong QED (Fukuda & Kugo, Bardeen, Leung & Love)

61 RGE,SDE,SM4 p. 23/2 Gap Equation Early work: K. Johnson, M. Baker and R. Willey. PR136(1964), 163(1967)

62 RGE,SDE,SM4 p. 23/2 Gap Equation Early work: K. Johnson, M. Baker and R. Willey. PR136(1964), 163(1967) Numerical analysis: Fukuda and Kugo, Nucl. Phys. B117(1976)

63 RGE,SDE,SM4 p. 23/2 Gap Equation Early work: K. Johnson, M. Baker and R. Willey. PR136(1964), 163(1967) Numerical analysis: Fukuda and Kugo, Nucl. Phys. B117(1976) Analytic analysis: Leung, Love and Bardeen, Nucl. Phys. B273(1986)

64 RGE,SDE,SM4 p. 23/2 Gap Equation Early work: K. Johnson, M. Baker and R. Willey. PR136(1964), 163(1967) Numerical analysis: Fukuda and Kugo, Nucl. Phys. B117(1976) Analytic analysis: Leung, Love and Bardeen, Nucl. Phys. B273(1986) The SDEs are similar, but the boundary conditions are different dσ lim p4 p 0 dp = 0 2 dσ lim p2 p Λ dp 2 + Σ(p) = 0

65 RGE,SDE,SM4 p. 24/2 solutions asymptotic solutions in the weak and strong coupling regions: Σ(p) p 1+ 1 α αc, for α αc α Σ(p) p 1 sin[ 1(lnp + δ)], for α > α c α c

66 RGE,SDE,SM4 p. 24/2 solutions asymptotic solutions in the weak and strong coupling regions: Σ(p) p 1+ 1 α αc, for α αc α Σ(p) p 1 sin[ 1(lnp + δ)], for α > α c α c Numerical Solutions u t u t t t

67 RGE,SDE,SM4 p. 25/2 Condensates One can compute the condensates t L t R = b L b R = 3 4π 4 d 4 q Σ 4Q (q) q 2 + Σ 2 4Q (q)

68 RGE,SDE,SM4 p. 25/2 Condensates One can compute the condensates t L t R = b L b R = 3 4π 4 d 4 q Σ 4Q (q) q 2 + Σ 2 4Q (q) Self energy, condensates and induced scalar mass depend on the cutoff and the Yukawa couplings as π Σ(0) Λ e +1 α/α c 1, t t Λ 3 e α/α c 1, δm 2 φ Λ 2 e 2π π α/α c

69 RGE,SDE,SM4 p. 25/2 Condensates One can compute the condensates t L t R = b L b R = 3 4π 4 d 4 q Σ 4Q (q) q 2 + Σ 2 4Q (q) Self energy, condensates and induced scalar mass depend on the cutoff and the Yukawa couplings as π Σ(0) Λ e +1 α/α c 1, t t Λ 3 e α/α c 1, δm 2 φ Λ 2 e 2π π α/α c Miransky fixed-point? In our case the exponential factors cannot suppress them simultaneously. To avoid fine-tuning, one has to choose a cutoff at TeV scale.

70 RGE,SDE,SM4 p. 26/2 Cutoff vs. Yukawa couplings α α c Λ Σ(0) Λ (GeV) Table 1: The relation between the cutoff scale and the Yukawa coupling.

71 Cutoff vs. Yukawa couplings α α c Λ Σ(0) Λ (GeV) Table 2: The relation between the cutoff scale and the Yukawa coupling. Α c Α FP FP cutoff Cutoff scale cutoff vs Yukawa couplingα 0 0 Α c Α Α g 2 4Π RGE,SDE,SM4 p. 26/2

72 RGE vs. SDE two figures from RGE and SDE respectively 5 Α l 4 Region I Region II Α q 3 2 Λ 4Π 1 FP t log E 91.2GeV 4 3 Region I Region II Α q Α l RGE,SDE,SM4 p. 27/2

73 RGE,SDE,SM4 p. 28/2 Multiple Higgs doublets We might have three Higgs doublets: One fundamental, two composite H 1 = (π +, π, π 0, σ) H 2 = ( b t, t b, t t b b, t t + b b ) H 3 = ( τ ν τ, ν ττ, ν τν τ τ τ, ν τν τ + τ τ )

74 RGE,SDE,SM4 p. 28/2 Multiple Higgs doublets We might have three Higgs doublets: One fundamental, two composite H 1 = (π +, π, π 0, σ) H 2 = ( b t, t b, t t b b, t t + b b ) H 3 = ( τ ν τ, ν ττ, ν τν τ τ τ, ν τν τ + τ τ ) Their effective mass terms are described by (H 1, H 2, H 3 ) M (H 1, H 2, H 3 ) T

75 RGE,SDE,SM4 p. 28/2 Multiple Higgs doublets We might have three Higgs doublets: One fundamental, two composite H 1 = (π +, π, π 0, σ) H 2 = ( b t, t b, t t b b, t t + b b ) H 3 = ( τ ν τ, ν ττ, ν τν τ τ τ, ν τν τ + τ τ ) Their effective mass terms are described by (H 1, H 2, H 3 ) M (H 1, H 2, H 3 ) T The existence of the Nambu-Goldstone bosons leads to detm = 0 modified gap equation

76 RGE,SDE,SM4 p. 29/2 Summary At 2-loop level, the Yukawa sector of SM4 has a non-trivial quasi-fixed point

77 RGE,SDE,SM4 p. 29/2 Summary At 2-loop level, the Yukawa sector of SM4 has a non-trivial quasi-fixed point A heavy 4th generation (m q > 400 GeV) can set Λ FP to be order of TeV

78 RGE,SDE,SM4 p. 29/2 Summary At 2-loop level, the Yukawa sector of SM4 has a non-trivial quasi-fixed point A heavy 4th generation (m q > 400 GeV) can set Λ FP to be order of TeV A heavy 4th generation also drives Yukawa couplings become strong at TeV scale

79 RGE,SDE,SM4 p. 29/2 Summary At 2-loop level, the Yukawa sector of SM4 has a non-trivial quasi-fixed point A heavy 4th generation (m q > 400 GeV) can set Λ FP to be order of TeV A heavy 4th generation also drives Yukawa couplings become strong at TeV scale Bound states/condensates of the 4th generation can be formed by exchanging Higgs bosons

80 RGE,SDE,SM4 p. 29/2 Summary At 2-loop level, the Yukawa sector of SM4 has a non-trivial quasi-fixed point A heavy 4th generation (m q > 400 GeV) can set Λ FP to be order of TeV A heavy 4th generation also drives Yukawa couplings become strong at TeV scale Bound states/condensates of the 4th generation can be formed by exchanging Higgs bosons The perturbative (RGE) and non-perturbative (SDE) approaches lead to consistent results.

arxiv: v3 [hep-ph] 23 Feb 2011

arxiv: v3 [hep-ph] 23 Feb 2011 Renormalization Group Fixed Point with a Fourth Generation: Higgs-induced Bound States and Condensates arxiv:0911.3890v3 [hep-ph] 23 Feb 2011 P.Q. Hung and Chi Xiong pqh@virginia.edu, xiong@virginia.edu