New Loop-Regularization and Intrinsic Mass Scales in QFTs

Transcription

1 New Loop-Regularization and Intrinsic Mass Scales in QFTs Yue-Liang Wu ITP, Beijing, CAS (CAS) References: USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 1

2 New Loop-Regularization and Intrinsic Mass Scales in QFTs Yue-Liang Wu ITP, Beijing, CAS (CAS) References: Y.L.Wu, Int. J. Mod. Phys. A18 (2003) ; USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 1

3 New Loop-Regularization and Intrinsic Mass Scales in QFTs Yue-Liang Wu ITP, Beijing, CAS (CAS) References: Y.L.Wu, Int. J. Mod. Phys. A18 (2003) ; Y.B. Dai & Y.L.Wu, Eur.Phys. J. C direct 2004 (DOI) /epjcd/s , hep-ph/ USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 1

4 New Loop-Regularization and Intrinsic Mass Scales in QFTs Yue-Liang Wu ITP, Beijing, CAS (CAS) References: Y.L.Wu, Int. J. Mod. Phys. A18 (2003) ; Y.B. Dai & Y.L.Wu, Eur.Phys. J. C direct 2004 (DOI) /epjcd/s , hep-ph/ Y.L.Wu, Mod. Phys. Lett. A19 (2004) 2191, hep-th/ USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 1

5 Contents INTRODUCTION OVERLAPPING DIVERGENCES AND IMPORTANT THEOREMS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 2

6 Contents INTRODUCTION CONCEPT OF IRREDUCIBLE LOOP INTEGRALS OVERLAPPING DIVERGENCES AND IMPORTANT THEOREMS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 2

7 Contents INTRODUCTION CONCEPT OF IRREDUCIBLE LOOP INTEGRALS CONSISTENCY CONDITIONS OF GAUGE INVARIANCE OVERLAPPING DIVERGENCES AND IMPORTANT THEOREMS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 2

8 Contents INTRODUCTION CONCEPT OF IRREDUCIBLE LOOP INTEGRALS CONSISTENCY CONDITIONS OF GAUGE INVARIANCE LOOP-REGULARIZATION PRESCRIPTION OVERLAPPING DIVERGENCES AND IMPORTANT THEOREMS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 2

9 Contents INTRODUCTION CONCEPT OF IRREDUCIBLE LOOP INTEGRALS CONSISTENCY CONDITIONS OF GAUGE INVARIANCE LOOP-REGULARIZATION PRESCRIPTION INTRINSIC MASS SCALES AND REGULATING DISCTRIBUTION FUNCTIONS OVERLAPPING DIVERGENCES AND IMPORTANT THEOREMS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 2

10 APPLICATION TO LOW ENERGY DYNAMICS OF QCD USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 3

11 APPLICATION TO LOW ENERGY DYNAMICS OF QCD DYNAMICALLY SPONTANEOUS SYMMETRY BREAKING USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 3

12 APPLICATION TO LOW ENERGY DYNAMICS OF QCD DYNAMICALLY SPONTANEOUS SYMMETRY BREAKING SCALARS AS LIGHTEST COMPOSITE HIGGS BOSONS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 3

13 APPLICATION TO LOW ENERGY DYNAMICS OF QCD DYNAMICALLY SPONTANEOUS SYMMETRY BREAKING SCALARS AS LIGHTEST COMPOSITE HIGGS BOSONS 31 PREDICTIONS FOR MASS SPECTRUM AND SCALES USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 3

14 APPLICATION TO LOW ENERGY DYNAMICS OF QCD DYNAMICALLY SPONTANEOUS SYMMETRY BREAKING SCALARS AS LIGHTEST COMPOSITE HIGGS BOSONS 31 PREDICTIONS FOR MASS SPECTRUM AND SCALES CONCLUSIONS & REMARKS USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 3

15 Introduction Symmetry and QFTs are the Basic Foundation in Physics All known basic forces of nature, i.e., gravitational, electromagnetic, weak and strong forces, are governed by the symmetries: U(1) Y SU(2) L SU(3) c SO(1, 3) GL(4, R) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 4

16 Introduction Symmetry and QFTs are the Basic Foundation in Physics All known basic forces of nature, i.e., gravitational, electromagnetic, weak and strong forces, are governed by the symmetries: U(1) Y SU(2) L SU(3) c SO(1, 3) GL(4, R) Quantum field theories (QFTs) have successfully been applied to treat the underlying theories of elementary particles and also deal with effective theories for composite particles at low energies as well as critical phenomena (or phase transitions) in statistical mechanics and condensed matter physics. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 4

17 Introduction Symmetry and QFTs are the Basic Foundation in Physics All known basic forces of nature, i.e., gravitational, electromagnetic, weak and strong forces, are governed by the symmetries: U(1) Y SU(2) L SU(3) c SO(1, 3) GL(4, R) Quantum field theories (QFTs) have successfully been applied to treat the underlying theories of elementary particles and also deal with effective theories for composite particles at low energies as well as critical phenomena (or phase transitions) in statistical mechanics and condensed matter physics. An important issue for making QFTs to be physically meaningful is the elimination of ultraviolet (UV) divergences without spoiling symmetries of the original theory. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 4

18 Real World Prefers the Symmetry Breaking Phase In nature: C (Charge), P (parity), T (time), CP all are violated USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 5

19 Real World Prefers the Symmetry Breaking Phase In nature: C (Charge), P (parity), T (time), CP all are violated In Universe: Matter-antimatter is asymmetric, or Baryogenesis USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 5

20 Real World Prefers the Symmetry Breaking Phase In nature: C (Charge), P (parity), T (time), CP all are violated In Universe: Matter-antimatter is asymmetric, or Baryogenesis In SM: U(1) Y SU(2) L U(1) em USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 5

21 Real World Prefers the Symmetry Breaking Phase In nature: C (Charge), P (parity), T (time), CP all are violated In Universe: Matter-antimatter is asymmetric, or Baryogenesis In SM: U(1) Y SU(2) L U(1) em In GUTs/SUSY/String: E 8 E 8, SO(32), SO(10) U(1) em SU(3) c USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 5

22 Real World Prefers the Symmetry Breaking Phase In nature: C (Charge), P (parity), T (time), CP all are violated In Universe: Matter-antimatter is asymmetric, or Baryogenesis In SM: U(1) Y SU(2) L U(1) em In GUTs/SUSY/String: E 8 E 8, SO(32), SO(10) U(1) em SU(3) c In Strong QCD, U(3) L U(3) R U(3) V U(1) V USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 5

23 Question 1: How to Avoid the Infinity Problem in QFTs Underlying theory might not be a quantum theory of fields, it could be something else, for example, string or superstring USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 6

24 Question 1: How to Avoid the Infinity Problem in QFTs Underlying theory might not be a quantum theory of fields, it could be something else, for example, string or superstring One may modify the behavior of field theory at very large/small momentum. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 6

25 Question 1: How to Avoid the Infinity Problem in QFTs Underlying theory might not be a quantum theory of fields, it could be something else, for example, string or superstring One may modify the behavior of field theory at very large/small momentum. Regularization Schemes Cut-off regularization USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 6

26 Question 1: How to Avoid the Infinity Problem in QFTs Underlying theory might not be a quantum theory of fields, it could be something else, for example, string or superstring One may modify the behavior of field theory at very large/small momentum. Regularization Schemes Cut-off regularization Pauli-Villars regularization USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 6

27 Question 1: How to Avoid the Infinity Problem in QFTs Underlying theory might not be a quantum theory of fields, it could be something else, for example, string or superstring One may modify the behavior of field theory at very large/small momentum. Regularization Schemes Cut-off regularization Pauli-Villars regularization Lattice regularization USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 6

28 Differential Regularization USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

29 Differential Regularization Dimensional regularization USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

30 Differential Regularization Dimensional regularization γ 5 PROBLEM USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

31 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

32 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY WRONG GAP EQUATION USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

33 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY WRONG GAP EQUATION 1 = GN c 8π 2 1 = GN c 8π 2 ) (Λ 2 m 2t ln Λ2 m 2 t ( m 2t m 2t ln µ2 m 2 t ) Cut-off Reg. Dim. Reg. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

34 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY WRONG GAP EQUATION 1 = GN c 8π 2 1 = GN c 8π 2 ) (Λ 2 m 2t ln Λ2 m 2 t ( m 2t m 2t ln µ2 m 2 t ) Cut-off Reg. Dim. Reg. This motivates us to a new regularization Scheme USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

35 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY WRONG GAP EQUATION 1 = GN c 8π 2 1 = GN c 8π 2 ) (Λ 2 m 2t ln Λ2 m 2 t ( m 2t m 2t ln µ2 m 2 t ) Cut-off Reg. Dim. Reg. This motivates us to a new regularization Scheme Symmetry-Preserving Loop Regularization (LR) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

36 Differential Regularization Dimensional regularization γ 5 PROBLEM PROBLEM FOR SUPERSYMMETRY WRONG GAP EQUATION 1 = GN c 8π 2 1 = GN c 8π 2 ) (Λ 2 m 2t ln Λ2 m 2 t ( m 2t m 2t ln µ2 m 2 t ) Cut-off Reg. Dim. Reg. This motivates us to a new regularization Scheme Symmetry-Preserving Loop Regularization (LR) Infinity Free Intrinsic Mass Scales USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 7

37 Question 2: How the symmetry broken down? USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

38 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

39 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? Dynamically via a given Gap equation? USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

40 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? Dynamically via a given Gap equation? Geometrically via a specific compactification? USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

41 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? Dynamically via a given Gap equation? Geometrically via a specific compactification? This motivates us to a combined symmetry breaking mechanism: USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

42 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? Dynamically via a given Gap equation? Geometrically via a specific compactification? This motivates us to a combined symmetry breaking mechanism: Dynamically Spontaneous Symmetry Breaking (DSSB) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

43 Question 2: How the symmetry broken down? Spontaneously via a given Higgs potential? Dynamically via a given Gap equation? Geometrically via a specific compactification? This motivates us to a combined symmetry breaking mechanism: Dynamically Spontaneous Symmetry Breaking (DSSB) Gap Equation Minimal Condition USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 8

44 Symmetry-Preserving Loop Regularization (SPLR) Weinberg s folk theorem: Why Quantum Field Theory USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 9

45 Symmetry-Preserving Loop Regularization (SPLR) Weinberg s folk theorem: Why Quantum Field Theory Any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 9

46 Symmetry-Preserving Loop Regularization (SPLR) Weinberg s folk theorem: Why Quantum Field Theory Any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. Wilson-Kadanoff and Gell-Mann-Low RG Flow USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 9

47 Symmetry-Preserving Loop Regularization (SPLR) Weinberg s folk theorem: Why Quantum Field Theory Any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. Wilson-Kadanoff and Gell-Mann-Low RG Flow Physical phenomena at any interesting renormalization energy scale can be described by integrating out the physics at higher energy scales. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 9

48 Implications There must exist in any case a characteristic energy scale (CES) M c. One can always make an QFT description at a sufficiently low energy scale in comparison with the CES M c. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 10

49 Implications There must exist in any case a characteristic energy scale (CES) M c. One can always make an QFT description at a sufficiently low energy scale in comparison with the CES M c. There must exist the so-called sliding energy scale (SES) µ s to describe the low energy dynamics in the infrared regime of QFTs. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 10

50 Implications There must exist in any case a characteristic energy scale (CES) M c. One can always make an QFT description at a sufficiently low energy scale in comparison with the CES M c. There must exist the so-called sliding energy scale (SES) µ s to describe the low energy dynamics in the infrared regime of QFTs. The explicit regularization method is governed by a physically meaningful CES M c and a physically interesting SES µ s. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 10

51 Implications There must exist in any case a characteristic energy scale (CES) M c. One can always make an QFT description at a sufficiently low energy scale in comparison with the CES M c. There must exist the so-called sliding energy scale (SES) µ s to describe the low energy dynamics in the infrared regime of QFTs. The explicit regularization method is governed by a physically meaningful CES M c and a physically interesting SES µ s. There should be no doubt of existing a new symmetry-preserving and infinity-free regularization scheme. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 10

52 Irreducible Loop Integrals (ILIs) Introduction of the concept of irreducible loop integrals (ILIs): Any n-fold ILIs that are evaluated from n-loop overlapping Feynman integrals of loop momenta k i (i = 1,, n) are defined as the loop integrals in which there are no longer the overlapping factors (k i k j + p ij ) 2 (i j) that appear in the original overlapping Feynman integrals. 1-fold ILIs for one loop integrals d 4 k 1 I 2α = (2π) 4 (k 2 M 2 ) 2+α, I 2α µν = d 4 k I 2α µνρσ = (2π) 4 d 4 k (2π) 4 k µ k ν k ρ k σ, α = 1, 0, 1, (k 2 M 2 ) 4+α I 2, I 2µν Quadratically div.; I 0, I 0µν Logarithmically div. k µ k ν (k 2 M 2 ) 3+α, USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 11

53 Symmetry-Preserving Consistency Conditions For Regularized ILIs I R 2µν = 1 2 g µν I R 2, I R 2µνρσ = 1 8 g {µνg ρσ} I R 2, I R 0µν = 1 4 g µν I R 0, I R 0µνρσ = 1 24 g {µνg ρσ} I R 0 g {µν g ρσ} g µν g ρσ + g µρ g νσ + g µσ g ρν USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 12

54 Symmetry-Preserving Consistency Conditions For Regularized ILIs I R 2µν = 1 2 g µν I R 2, I R 2µνρσ = 1 8 g {µνg ρσ} I R 2, I R 0µν = 1 4 g µν I R 0, I R 0µνρσ = 1 24 g {µνg ρσ} I R 0 g {µν g ρσ} g µν g ρσ + g µρ g νσ + g µσ g ρν Divergence Cancellation USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 12

55 Symmetry-Preserving Consistency Conditions For Regularized ILIs I R 2µν = 1 2 g µν I R 2, I R 2µνρσ = 1 8 g {µνg ρσ} I R 2, I R 0µν = 1 4 g µν I R 0, I R 0µνρσ = 1 24 g {µνg ρσ} I R 0 g {µν g ρσ} g µν g ρσ + g µρ g νσ + g µσ g ρν Divergence Cancellation The regularized quadratic divergences cancel each other for Underlying gauge theories due to Gauge Invariance. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 12

56 Regulator-Free Scheme Consistency conditions lead to a regulator-free scheme for underlying QFTs. General Theorem: The convergent integrations can safely be carried out, only the divergent integrations destroy the gauge invariance. Demonstration I 2µν = a g µν I 2, a = I 2µν g µν /(4I 2 ) = 1/4 I 2 00 = a g 00 I 2, a = 1/2 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 13

57 General Regularization Prescription Replacing in the ILIs the loop integrating variable k 2 and the loop integrating measure d 4 k by the regularized ones [k 2 ] l and [d 4 k] l k 2 [k 2 ] l k 2 + M 2 l, d 4 k [d 4 k] l lim N,M 2 l N l=0 c N l d 4 k with lim N,M 2 l N c N l (M 2 l ) n = 0 c N 0 = 1 (n = 0, 1, ) l=0 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 14

58 Explicit and Simple Solution M 2 l = µ 2 s + lm 2 R, c N l = ( 1) l N! (N l)! l! Explicit Forms for the Regularized ILIs I R 0 and I R 2 I R 2 = I R 0 = y 0 (x) = i 16π { 2 M2 c µ 2 [ ln M2 c µ γ 2 w y 2 ( µ2 ) ] } M 2 c i 16π [ ln M2 c 2 µ γ 2 w + y 0 ( µ2 x 0 dσ 1 e σ σ M 2 c ) ], y 1 (x) = e x 1 + x, y 2 (x) = y 0 (x) y 1 (x), x M 2 c = lim N,M R M 2 R/ ln N, µ 2 = µ 2 s + M 2, γ w = γ E = USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 15

59 Two Intrinsic Mass Scales M c - UV Cut-off scale & Characteristic Energy Scale (CES) µ s - IR Cut-off scale & Sliding Energy Scale (SES) Regulating Distribution Functions The Regularized ILIs in Proper-Time Formalism I R 2α = i( 1)α Γ(α + 2) lim 0 N d 4 k (2π) 4 dτ W N (τ; M c, µ s ) τ α+1 e τ(k2 +M 2 ) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 16

60 Conditions for the Regulating Distribution Function W N (τ; M c, µ s ) W N (τ = 0; M c, µ s ) = 0 so as to eliminate the singularity at τ = 0 which corresponds to the UV divergence for the momentum integration; W N (τ = ; M c, µ s = 0) = 1 for ensuring the regulating distribution function not to modify the behavior of original theory in the IR regime; W N (τ; M c, µ s = 0) = 1 which ensures the proper-time formalism to recover the original ILIs in the physical limits; lim N W N (τ; M c, µ s = 0) = 1 for τ 1/M 2 c lim N W N (τ; M c, µ s = 0) = 0 for τ < 1/M 2 c, so that M c acts as the UV cutoff scale. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 17

61 The Simple Form of W N (τ; M c, µ s ) ( ) N N W N (τ; M c, µ s ) = e τµ2 s 1 e τm2 R = ( 1) l N! (N l)! l! e τ(µ l=0 M 2 R = M 2 ch w (N) ln N, h w (N)1 h w (N ) = 1 The Regularized ILIs after integrating over τ I R 2α = lim N,M 2 l d 4 k (2π) 4 N l=0 c N l i( 1) α (k 2 + M 2 + M 2 l )α+2 = i( 1) α [d 4 k] l (2π) 4 1 ([k 2 ] l + M 2 ) α+2 which exactly reproduces the prescription for the new SPLR method. 2 s +lm2 R ) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 18

62 Generalization to More Closed Loops UV-divergence preserving parameter method 1 Γ(α + β) a α = bβ Γ(α)Γ(β) 0 u β 1 du [a + bu] α+β Here u is the UV-divergence preserving integral variable. As a consequence, the UV divergence for the momentum integration transfers into the one for u integration. General Form for n-fold ILIs After safely performing (n-1) convergent integrations over the momentum USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 19

63 k i (i = 1,, n 1), I (n) with Overall 1-fold ILIs n 1 = i=1 n 1 I (n) µν = i=1 I (1) n (µ 2 n) = 0 I (1) n µν (µ2 n) = 0 du i F is (x lm ) (u i + ρ i ) is I(1) n (µ 2 n) du i F is (x lm ) (u i + ρ i ) is I(1) n µν (µ2 n) d 4 1 k n (k 2 n + M 2 + µ 2 n) n d 4 k nµ k nν k n (k 2 n + M 2 + µ 2 n) n+1 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 20

64 Important Features The functions ρ i and µ 2 n have the limits ρ i 0, µ 2 n 0 for u 1,, u n 1 The integral over the n-th loop momentum k n describes the overall divergent property of n-loop diagrams The sub-integrals over the variables u i (i = 1, 2,, n 1) characterize the UV divergent properties for the one-loop, two-loop,, (n-1)-loop sub-diagrams respectively USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 21

65 Important Theorems The Key Theorem In the Feynman loop integrals the overlapping divergences which contain overall divergences and also divergences of sub-integrals will become factorizable in the corresponding ILIs. The Main Theorem All the overlapping divergent integrals can be made to be harmless via appropriate subtractions. USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 22

66 General Prescription of Loop Regularization Universally replace in the n-fold ILIs the n-th loop momentum square k 2 n and the integrating measure d 4 k n as well as the UV-divergence preserving integral variables u i (i = 1,, n 1) and du i by the regularizing ones [k 2 n] l and [d 4 k n ] l as well as [u i ] l and [du i ] l with n being arbitrary: k 2 n [k 2 n] l k 2 n + M 2 l = k 2 n + µ 2 s + lm 2 R, N d 4 k n [d 4 k n ] l lim c N N,M 2 l d 4 k n R l=0 u i [u i ] l u i + M 2 l /µ 2 s = u i lm 2 R/µ 2 s N du i [du i ] l lim c N l du i N,M R l=0 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 23

67 Low Energy Dynamics of QCD Well-Known Phenomena in Real Would Hadrons are considered to be the bound states of quarks and gluons Lightest pseudoscalar mesons are successfully described by current algebra with PCAC The chiral symmetry U(3) L U(3) R is strongly broken down QCD was motivated from the studies of hadrons at low energy Low energy dynamics of QCD remains unsolved due to nonperturbative effects of strong interactions USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 24

68 Important Issues on Low Energy Dynamics of QCD How the chiral symmetry is dynamically broken down How the instanton plays the role as a quantum topological solution of nonperturbative QCD Whether the effective meson theory should be realized as a linear σ model or a non-linear σ model Whether the lowest lying U(3) V nonet scalar mesons corresponds to the chiral partners of the lowest lying nonet pseudoscalar mesons Whether there exist other lighter isospinor scalar mesons κ 0 (900MeV) as the lowest lying scalar mesons USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 25

69 Does the singlet scalar meson f 0 ( ) (or the σ) truly exist, why it is so light. Answers to the Questions Based on Three Basic Realistic Assumptions The (approximate) chiral symmetry of QCD Lagrangian Bound State Solutions of Nonperturbative QCD Important Instanton Effects USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 26

70 Applications of Loop Regularization Method Quantum Chiral Dynamics is realized nonlinearly SU(3) L SU(3) R Symmetry is Dynamically spontaneous broken down Gap Equations as Minimal Conditions of Effective Potential Nonet scalar mesons play the role of composite Higgs Bosons Nonet scalar mesons are the chiral partners of the lowest lying nonet pseudoscalar mesons Chiral Effective Field Theory has the same parameters as QCD (5 parameters: v o (Λ QCD ), µ s (g s (µ)), m u, m d, m s ) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 27

71 Consistent Predictions 18 mass spectrum for nonet scalar and pseudoscalar mesons 2 (6) mixing angles among the neutral mesons 2 Intrinsic Mass Scales M c 1 GeV and µ s Λ QCD Quark condensate < qq > or vacuum expectation value v o κ 0 (900MeV) is the lowest lying isospinor scalar mesons Lightest σ and heaviest η all are duo to Instanton Effects USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 28

72 Effective Chiral Lagrangian QCD Lagrangian for Light Quarks L QCD = qγ µ (i µ + g s G a µt a )q qmq 1 2 trg µνg µν q = (u, d, s), M = diag.(m 1, m 2, m 3 ) diag.(m u, m d, m s ) Approximate Global Chiral Symmetry U(3) L U(3) R, m i << Λ QCD (i = 1, 2, 3) Instanton Effects via t Hooft Determination L inst = κ inst e iθ inst det( q R q L ) + h.c., κ inst e 8π2 /g 2 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 29

73 U(1) L U(1) R U(1) V Effective Four Quark Interactions-NJL at low energy L 4q = 1 µ 2 f ( q Li q Rj )( q Rj q Li ) + h.c. Effective Lagrangian for Quarks and Bound States Integrating over the gluon field and considering the bound state solution L eff (q, q, Φ) = qγ µ i µ q + q L γ µ A µ L q L + q R γ µ A µ R q R [ q L (Φ M)q R + h.c. ] + 2µ 2 f tr ( ΦM + MΦ ) µ 2 f trφφ + µ inst (det Φ + h.c.) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 30

74 Φ ij the effective meson fields Φ ij 1 µ 2 f q Rj q Li + 2M ij, < φ > µ inst << µ 2 f Φ(x) = ξ L (x)φ(x)ξ R (x), φ (x) = φ(x) = a=9 a=0 U = ξ L(x)ξ R (x) = ξ2 L(x) = e i2π(x) f φ a (x)t a, Π (x) = Π(x) = a=9 a=0 Π a (x)t a Generating Functionals for effective chiral Lagrangian of mesons 1 Z DG µ DqD qe i d 4 xl QCD = 1 Z 1 = Z eff DΦDqD qe i d 4 xl eff (q, q,φ) DΦe i d 4 xl eff (Φ) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 31

75 Chiral Effective Lagrangian Applying the Schwinger s proper time technique to the determinant of Dirac operator and adopting the Loop Regularization method for momentum integrals with N c L eff (Φ) = 1 216π 2trT 0 [ D µ ˆΦD µ ˆΦ + D µ ˆΦ D µ ˆΦ ( ˆΦ ˆΦ M 2 ) 2 ( ˆΦ ˆΦ M 2 ) 2 ] + N c 16π 2M2 c trt 2 [ ( ˆΦ ˆΦ M 2 ) + ( ˆΦ ˆΦ M 2 ) ] + µ 2 mtr ( ΦM + MΦ ) µ 2 f trφφ + µ inst (det Φ + h.c.) ˆΦ = Φ M, M = V M = diag.( m 1, m 2, m 3 ), USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 32

76 V = diag.(v 1, v 2, v 3 ), m i = v i m i T 0 = diag.(t (1) 0, T(2) 0, T(3) 0 ), T 2 = diag.(t (1) 2, T(2) 2, T(3) 2 ) and T (i) 0 ( µ2 i ) = ln M2 c M 2 c µ 2 i γ w + y 0 ( µ2 i ), µ 2 M 2 i = µ 2 s + m 2 i c T (i) 2 ( µ2 i ) = 1 µ2 i [ ln M2 c M 2 c M 2 c µ 2 i γ w y 2 ( µ2 i ) ] M 2 c USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 33

77 Dynamically Spontaneous Symmetry Breaking Dynamically Generated Effective Potential V eff (Φ) = trˆµ 2 m ( ΦM + MΦ ) trˆµ2 f (ΦΦ + Φ Φ) trλ[ ( ˆΦ ˆΦ ) 2 + ( ˆΦ ˆΦ) 2 ] µ inst (det Φ + h.c.) with ˆµ 2 f, ˆµ2 m and λ the three diagonal matrices ˆµ 2 f = µ 2 f N c 8π 2 ( M 2 c T 2 + M 2 T 0 ) ˆµ 2 m = µ 2 m N c 8π 2 ( M 2 c T 2 + M 2 T 0 ), λ = N c 16π 2T 0 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 34

78 Spontaneous Symmetry Breaking Vacuum Expectation Values (VEVs) Φ(x) = ξ L (x)φ(x)ξ R (x), φ(x) = V + ϕ(x), V =< φ >= diag.(v 1, v 2, v 3 ) Minimal Conditions/Generalized Gap Equations (ˆµ ) 2 f v i i + (ˆµ ) 2 m m i i 2λ i m 3 i + µ inst v 3 /v i = 0, i = 1, 2, 3, v 3 = v 1 v 2 v 3 Expanding in terms of the quark masses m i ) 2 (ˆµ = f i µ2 f + 2µ fo m i [ 1 + ( ) k mi α k ] µ o k=1 ) 2 (ˆµ = m i µ2 m + 2µ fo m i [ 1 + ( ) k mi α k ] µ o k=1 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 35

79 λ i = λ λ o k=1 ( ) k mi β k, λ o = N c µ o 16π 2 µ 2 o = µ 2 s + v 2 o, m i = m i [1 + m i /(2v o )] The Unknown Parameters µ 2 f, µ inst, v o, µ s, λ, m u, m d, m s Three Constraints 3v o (1 ɛ o )/4 v inst 2 λ(v inst v 3 v 2 o) µ 2 f 10 λv o 2 + 2λ o vo(1 2 2v2 o ) + 4 µ 2 µ 2 f µ 2 m o USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 36

80 with ɛ o = λ o λ [ ( 2v 2 o µ 2 o ) ( v 2 ) o 3 µ 2 o 1 2v o α 1 (1 r) + r + 3 µ o ( 2v 2 o µ 2 o 1 3 λ λ o ) ms v o ] µ fo 2λ o v o (1 r), µ inst 2 λv inst Here r and α 1 are given by r = µ2 s µ 2 o µ2 o [1 + µ2 s M 2 c µ 2 o + O( µ2 o )] M 2 c α 1 (1 r) = 2v o µ o [ µ 2 s 2µ 2 o + O( µ2 o ) ] M 2 c USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 37

81 Parameter Definitions µ 2 f = µ 2 f N c 8π 2 µ 2 m = µ 2 m N c 8π 2 ( M 2 ct 2 ( µ2 o M 2 c ) ) + vot 2 0 ( µ2 o ) M 2 c ) ( M 2 ct 2 ( µ2 o ) + v M ot ( µ2 o ) c M 2 c λ = N c 16π 2T 0( µ2 o ), T M 2 0 ( µ2 o ) = ln M2 c c M 2 c µ 2 o γ w + y 0 ( µ2 o ) M 2 c Gap Equation without Instanton (v inst = 0) N c 8π 2 µ 2 f [ M 2 c µ 2 o ( ln M2 c µ 2 o ) γ w y 2 ( µ2 o ) M 2 c ] = 1 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 38

82 Lightest Composite Higgs Bosons & Mass Formulations The Composite Higgs Bosons - Scalar Mesons and Would-be Goldstone Bosons-Pseudoscalar Mesons 2ϕ = a f f s a + 0 κ f s κ 0 0 a 0 a f 8 + κ 0 κ f f s, 2Π = π η 8 + π 1 3 η 0 π + K + π η η 0 K 0 K K0 2 6 η η 0, USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 39

83 Mass Formulations: Pseudoscalar Mesons at Leading Order m 2 π ± 2µ3 P f 2 (m u + m d ) m 2 K ± 2µ3 P f 2 (m u + m s ) m 2 K 0 2µ3 P f 2 (m d + m s ) m 2 η 8 2µ3 P f [ (m u + m d ) m s ] = 1 3 (4m2 K m 2 π) m 2 η 8 η 0 2µ3 P 2 f 2 3 [ 2m s (m u + m d ) ] = (m2 K m 2 π) m 2 η 0 2µ3 P 2 f 2 3 (m u + m d + m s ) + 12 v3 f µ 2 inst = 1 3 (2m2 K + m 2 π) + 24 v3 f λv 2 inst USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 40

84 µ 3 P = ( µ 2 m + 2 λv 2 o)v o 12 λv 3 o 3v o f 2 Scalar Mesons - Lightest Composite Higgs Bosons m 2 a ± 0 m 2 a 0 0 2(2 m u + m d ) m u + 2v inst v 3 8v 2 o m 2 k ± 0 2(2 m u + m s ) m u + 2v inst v 2 8v 2 o m 2 k 0 0 2(2 m d + m s ) m d + 2v inst v 1 8v 2 o m 2 f 8 m 2 u + m 2 d + 4 m 2 s v inst(2v 1 + 2v 2 v 3 ) 8v 2 o m 2 f s 2( m 2 u + m 2 d + m 2 s) 4 3 v inst(v 1 + v 2 + v 3 ) 2v 2 o m 2 f s f 8 2(2 m 2 s m 2 u m 2 d) 2 3 v inst(2v 3 v 1 v 2 ) 0 USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 41

85 Mixing Angles tan 2θ P = 2 2[1 9v instv 3 m 2 K m π 2 ] 1 tan 2θ S = 2m2 f s f 8 m 2 f s m 2 f 8 Two More Constraints Normalization Condition from Kinetic Term λv 2 o = f 2 /4, T 0 v 2 o = (4πf)2 4N c Λ 2 f (340MeV) 2 Trace of Mass Matrix v inst v 3 = 1 6 (m2 η 8 + m 2 η 0 2m 2 K) = 1 6 (m2 η + m 2 η 2m 2 K) USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 42

86 Numerical Results Input Parameters f π = 94MeV v o = 340MeV m u 3.8MeV m d 5.7MeV m s /m d 20.5 Output Predictions µ f 144MeV, µ inst 8.0MeV M c 922MeV, µ s 333MeV < ūu > < dd > < ss >= (242MeV) 3 m π 139MeV, m π exp 139MeV m K 0 500MeV, m K 0 exp 500MeV USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 43

87 m K ± 496MeV m η 503MeV, m η 986MeV, m a0 978 MeV, m κ0 970 MeV, m f MeV, m σ 677 MeV, θ P 18 o, η 8 = cos θ P η + sin θ P η η 0 = cos θ P η sin θ P η f 8 = cos θ S f 0 + sin θ S σ f s = cos θ S σ sin θ S f 0 m K ± exp 496MeV m η exp 548MeV m η exp 958MeV m exp. a 0 = ± 1.4 MeV PDG m exp. κ 0 = 797 ± 19 ± 43 MeV E7912 m epx. f 0 = 980 ± 10 MeV PDG m exp. σ = ( ) MeV PDG θ S 18 o USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 44

88 Conclusions & Remarks Symmetry-Preserving Loop Regularization is a Useful Tool for Underlying and Effective QFTs USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 45

89 Conclusions & Remarks Symmetry-Preserving Loop Regularization is a Useful Tool for Underlying and Effective QFTs Low Energy Dynamics of QCD Can Well be Described by Nonlinearly Realized Effective Chiral Theory USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 45

90 Conclusions & Remarks Symmetry-Preserving Loop Regularization is a Useful Tool for Underlying and Effective QFTs Low Energy Dynamics of QCD Can Well be Described by Nonlinearly Realized Effective Chiral Theory Dynamically Spontaneous Symmetry Breaking Mechanism Can well be Established in Strong Interactions USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 45

91 Conclusions & Remarks Symmetry-Preserving Loop Regularization is a Useful Tool for Underlying and Effective QFTs Low Energy Dynamics of QCD Can Well be Described by Nonlinearly Realized Effective Chiral Theory Dynamically Spontaneous Symmetry Breaking Mechanism Can well be Established in Strong Interactions Lightest Scalar Mesons are Truly the Composite Higgs Bosons that Have First been Observed in Nature USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 45

92 Instanton Effects are Important and Significant USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 46

93 Instanton Effects are Important and Significant All Higher Order Corrections can be Systematically Evaluated without Involving any Additional Parameters USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 46

94 Instanton Effects are Important and Significant All Higher Order Corrections can be Systematically Evaluated without Involving any Additional Parameters More Precise Measurement and Confirmation of the σ and κ Scalar Mesons will be a Direct Test to the Quantum Chiral Dynamics of Low Energy QCD for Mesons USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 46

95 Instanton Effects are Important and Significant All Higher Order Corrections can be Systematically Evaluated without Involving any Additional Parameters More Precise Measurement and Confirmation of the σ and κ Scalar Mesons will be a Direct Test to the Quantum Chiral Dynamics of Low Energy QCD for Mesons The Symmetry Breaking Mechanism may be applicable to Other Theories, Such as: Electroweak symmetry breaking, Symmetry breaking in GUTs / SUSY USTC-Shanghai Workshop/ Talk by Yue-Liang Wu / October back to start 46