Spontaneous breaking of supersymmetry

Transcription

1 Spontaneous breaking of supersymmetry Hiroshi Suzuki Theoretical Physics Laboratory Nov. 18, Theoretical science colloquium in RIKEN Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

2 Identical particles in microscopic world Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

3 Identical particles in microscopic world Fermi particle (fermion) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

4 Identical particles in microscopic world Fermi particle (fermion) Bose particle (boson) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

5 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Bose particle (boson) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

6 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Bose particle (boson) Photon, pion, 4 He, W -boson, Z -boson, Higgs particle... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

7 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Only one particle in a specific quantum state Bose particle (boson) Photon, pion, 4 He, W -boson, Z -boson, Higgs particle... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

8 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Only one particle in a specific quantum state Bose particle (boson) Photon, pion, 4 He, W -boson, Z -boson, Higgs particle... Any number of particles in a specific quantum state Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

9 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Only one particle in a specific quantum state Bose particle (boson) Photon, pion, 4 He, W -boson, Z -boson, Higgs particle... Any number of particles in a specific quantum state They have quite different statistical properties... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

10 Identical particles in microscopic world Fermi particle (fermion) Electron, proton, neutron, 3 He, neutrino, quarks... Only one particle in a specific quantum state Bose particle (boson) Photon, pion, 4 He, W -boson, Z -boson, Higgs particle... Any number of particles in a specific quantum state They have quite different statistical properties... SUperSYmmetry (SUSY) postulates invariance under the exchange of these bosons and fermions! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

11 But our real world is not supersymmetric... Mass spectrum of elementary particles boson fermion mass Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

12 But our real world is not supersymmetric... Mass spectrum of elementary particles boson fermion mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

13 But our real world is not supersymmetric... Mass spectrum of elementary particles boson SUSY fermion mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

14 But our real world is not supersymmetric... Mass spectrum of elementary particles boson SUSY fermion mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

15 But our real world is not supersymmetric... Mass spectrum of elementary particles boson SUSY fermion mass Higgs particle?? top quark W, Z bosons? u, d quarks electron neutrino photon? Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

16 But our real world is not supersymmetric... Mass spectrum of elementary particles boson SUSY fermion mass Higgs particle??? top quark W, Z bosons?? u, d quarks? electron? neutrino photon? Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

17 But our real world is not supersymmetric... Mass spectrum of elementary particles boson SUSY fermion mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

18 But our real world is not supersymmetric... Mass spectrum of elementary particles boson sneutrino? SUSY fermion photino? mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

19 But our real world is not supersymmetric... Mass spectrum of elementary particles boson sneutrino? SUSY fermion photino? mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

20 But our real world is not supersymmetric... Mass spectrum of elementary particles boson sneutrino? SUSY fermion photino? mass Higgs particle? W, Z bosons photon top quark u, d quarks electron neutrino Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

21 Spontaneous symmetry breaking The situation s.t. the mass spectrum does not reflect a symmetry, if the symmetry is spontaneously broken Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

22 Spontaneous symmetry breaking The situation s.t. the mass spectrum does not reflect a symmetry, if the symmetry is spontaneously broken Spontaneous symmetry breaking The lowest energy (quantum) state of the system is not invariant under the symmetry transformation Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

23 Spontaneous symmetry breaking The situation s.t. the mass spectrum does not reflect a symmetry, if the symmetry is spontaneously broken Spontaneous symmetry breaking The lowest energy (quantum) state of the system is not invariant under the symmetry transformation SUSY is spontaneously broken? Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

24 Spontaneous symmetry breaking The situation s.t. the mass spectrum does not reflect a symmetry, if the symmetry is spontaneously broken Spontaneous symmetry breaking The lowest energy (quantum) state of the system is not invariant under the symmetry transformation SUSY is spontaneously broken? Nambu-Goldstone theorem When ordinary (continuous) symmetry is spontaneously broken, there emerge massless boson(s) Chiral symmetry is spontaneously broken! pions Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

25 Spontaneous symmetry breaking The situation s.t. the mass spectrum does not reflect a symmetry, if the symmetry is spontaneously broken Spontaneous symmetry breaking The lowest energy (quantum) state of the system is not invariant under the symmetry transformation SUSY is spontaneously broken? Nambu-Goldstone theorem When super- symmetry is spontaneously broken, there emerge massless fermion(s) Chiral symmetry is spontaneously broken! pions Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

26 SUSY and Quantum Field Theory (QFT) Since SUSY changes the number of fermions (and of bosons) in the system, it is naturally discussed only in the 2nd quantization framework (= the number representation) Quantum Field Theory (QFT) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

27 SUSY and Quantum Field Theory (QFT) Since SUSY changes the number of fermions (and of bosons) in the system, it is naturally discussed only in the 2nd quantization framework (= the number representation) Quantum Field Theory (QFT) QFT is quantum mechanics of infinitely many variables (field) φ(x, t), x R 3, t: time Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

28 SUSY and Quantum Field Theory (QFT) Since SUSY changes the number of fermions (and of bosons) in the system, it is naturally discussed only in the 2nd quantization framework (= the number representation) Quantum Field Theory (QFT) QFT is quantum mechanics of infinitely many variables (field) φ(x, t), x R 3, t: time This is technically complicated, so let us consider QFT in an artificial space, consisting of a single point φ(, t) q(t), t: time This is quantum mechanics of a single degree of freedom... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

29 SUSY Quantum Mechanics (QM) Hamiltonian of SUSY QM (obtained by p i d dq ) H = 2 2 d 2 dq ( 2 W (q) 2 + W (q) b b 1 ), 2 where the function W (q) is called the super-potential Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

30 SUSY Quantum Mechanics (QM) Hamiltonian of SUSY QM (obtained by p i d dq ) H = 2 2 d 2 dq ( 2 W (q) 2 + W (q) b b 1 ), 2 where the function W (q) is called the super-potential Schrödinger equation for the wave function (or state) Ψ(q) HΨ(q) = EΨ(q), where E is called the energy eigenvalue and Ψ(q) is the eigenfunction Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

31 SUSY Quantum Mechanics (QM) Hamiltonian of SUSY QM (obtained by p i d dq ) H = 2 2 d 2 dq ( 2 W (q) 2 + W (q) b b 1 ), 2 where the function W (q) is called the super-potential Schrödinger equation for the wave function (or state) Ψ(q) HΨ(q) = EΨ(q), where E is called the energy eigenvalue and Ψ(q) is the eigenfunction Among E, the smallest E ( E 0 ), the ground state energy will be very important in what follows Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

32 Creation and annihilation operators of a fermion b and b obey the relations and, from these, bb + b b = 1, (b ) 2 = b 2 = 0 0 b 1 b b b 1 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

33 Creation and annihilation operators of a fermion b and b obey the relations and, from these, bb + b b = 1, (b ) 2 = b 2 = 0 0 b 1 b b b 1 These can be represented in terms of a two-story notation ( ) ( ) bosonic, 1 fermionic 1 0 and 2 2 matrices b ( ) 0 0, b 1 0 ( ) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

34 SUSY QM In this two-story notation, H = ( 2 d 2 2 dq ) ( 2 W (q) ) + ( ) 2 W 1 0 (q) 0 1 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

35 SUSY QM In this two-story notation, H = ( 2 d 2 2 dq ) ( 2 W (q) Now, the fundamental relation in SUSY QM is H = Q 2 ) + ( ) 2 W 1 0 (q) 0 1 where Q is an (hermitian) operator called the super-charge Q = 1 [ i d ( ) ( )] W 0 i (q) 2 dq 1 0 i 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

36 SUSY QM In this two-story notation, H = ( 2 d 2 2 dq ) ( 2 W (q) Now, the fundamental relation in SUSY QM is H = Q 2 ) + ( ) 2 W 1 0 (q) 0 1 where Q is an (hermitian) operator called the super-charge Q = 1 [ i d ( ) ( )] W 0 i (q) 2 dq 1 0 i 0 The supercharge maps the boson into the fermion and vice versa ( ) ( ) ( ) ( ) 0 0 Q =, Q = 0 0 and generates SUSY transformation Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

37 Basic property of SUSY QM (I) Since H = Q 2, the energy eigenvalue E HΨ(q) = Q 2 Ψ(q) = EΨ(q) is positive or zero E 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

38 Basic property of SUSY QM (I) Since H = Q 2, the energy eigenvalue E HΨ(q) = Q 2 Ψ(q) = EΨ(q) is positive or zero E 0 The zero-energy E = 0 is very special, because E = 0 Q 2 Ψ(q) = 0 QΨ(q) = 0 Zero-energy state is always annihilated by the supercharge Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

39 Basic property of SUSY QM (I) Since H = Q 2, the energy eigenvalue E HΨ(q) = Q 2 Ψ(q) = EΨ(q) is positive or zero E 0 The zero-energy E = 0 is very special, because E = 0 Q 2 Ψ(q) = 0 QΨ(q) = 0 Zero-energy state is always annihilated by the supercharge Zero-energy state is always invariant under SUSY transformation Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

40 Basic property of SUSY QM (II) For any positive energy eigenvalue E > 0 HΨ(q) = EΨ(q), E > 0, one may always apply Q to produce another state Φ(q) Φ(q) = QΨ(q) that has the identical energy E (recall H = Q 2 ) HΦ(q) = HQΨ(q) = QHΨ(q) = QEΨ(q) = EΦ(q), E > 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

41 Basic property of SUSY QM (II) For any positive energy eigenvalue E > 0 HΨ(q) = EΨ(q), E > 0, one may always apply Q to produce another state Φ(q) Φ(q) = QΨ(q) that has the identical energy E (recall H = Q 2 ) HΦ(q) = HQΨ(q) = QHΨ(q) = QEΨ(q) = EΦ(q), E > 0 All E > 0 states consist of pairs of bosonic and fermionic states! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

42 Basic property of SUSY QM (II) For any positive energy eigenvalue E > 0 HΨ(q) = EΨ(q), E > 0, one may always apply Q to produce another state Φ(q) Φ(q) = QΨ(q) that has the identical energy E (recall H = Q 2 ) HΦ(q) = HQΨ(q) = QHΨ(q) = QEΨ(q) = EΦ(q), E > 0 All E > 0 states consist of pairs of bosonic and fermionic states! N.B. The pairing is impossible for E = 0 because QΨ(q) = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

43 Possible patterns of the energy spectrum E 0 = 0 E 0 > 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

44 Possible patterns of the energy spectrum E 0 = 0 E 0 > 0 QΨ 0 (q) = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

45 Possible patterns of the energy spectrum E 0 = 0 E 0 > 0 QΨ 0 (q) = 0 QΨ 0 (q) = Φ 0 (q) 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

46 Possible patterns of the energy spectrum SUSY is spontaneously broken! E 0 = 0 E 0 > 0 QΨ 0 (q) = 0 QΨ 0 (q) = Φ 0 (q) 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

47 Zero-energy E = 0 state in SUSY QM Zero-energy E = 0 state satisfies the 1st order differential eq. QΨ(q) = i ( ) [ 0 1 d dq + 1 ( )] W 1 0 (q) Ψ(q) = Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

48 Zero-energy E = 0 state in SUSY QM Zero-energy E = 0 state satisfies the 1st order differential eq. QΨ(q) = i ( ) [ 0 1 d dq + 1 ( )] W 1 0 (q) Ψ(q) = The solution is ( ( exp + 1 Ψ(q) W (q)) ) 0 or ( ) 0 Ψ(q) exp ( 1 W (q)) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

49 Zero-energy E = 0 state in SUSY QM Zero-energy E = 0 state satisfies the 1st order differential eq. QΨ(q) = i ( ) [ 0 1 d dq + 1 ( )] W 1 0 (q) Ψ(q) = The solution is ( ( exp + 1 Ψ(q) W (q)) ) 0 The solution, however, must be normalizable This requires or or dq Ψ(q) Ψ(q) < W (q) + for q ± W (q) for q ± ( ) 0 Ψ(q) exp ( 1 W (q)) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

50 Asymptotic behavior of W (q) determines which... If W (q) behaves as There exists E = 0 state and SUSY is not broken Otherwise, for example, or SUSY is spontaneously broken! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

51 Let us consider two examples Model I W (q) = 1 2 ωq2 Model II W (q) = 1 2 ωq2 1 3 λq3 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

52 Let us consider two examples Model I W (q) = 1 2 ωq2 SUSY will not be spontaneously broken Model II W (q) = 1 2 ωq2 1 3 λq3 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

53 Let us consider two examples Model I W (q) = 1 2 ωq2 SUSY will not be spontaneously broken Model II W (q) = 1 2 ωq2 1 3 λq3 SUSY will be spontaneously broken Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

54 The ground state in model I The Hamiltonian H = ( 2 2 d 2 dq ω2 q 2 ) ( ) ( ) ω 0 1 represents two independent harmonic oscillators with shifted zero-point energies ±(1/2) ω: Exactly solvable The unique ground state ( ) 0 Ψ 0 (q) = ( ω ) 1/4 ( π exp ω 2 q2) has the energy E 0 = 1 2 ω 1 2 ω = 0 and is annihilated by the supercharge QΨ 0 (q) = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

55 1st excited states in model I The first excited states ( ) 0 Ψ 1 (q) = ( ) 4ω 3 1/4 ( π q exp ω 3 2 q2) QΦ 1 (q) and Φ 1 (q) = ( ( ω ) 1/4 ( ) π exp ω 2 q2) QΨ 1 (q) 0 bosonic fermionic have the degenerate energies E 1 = 3 2 ω 1 ω = ω 2 E 1 = 1 2 ω + 1 ω = ω 2 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

56 Spectrum in model I E 4 = 4 ω E 3 = 3 ω E 2 = 2 ω E 1 = ω E 0 = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

57 Spectrum in model I E 4 = 4 ω E 3 = 3 ω E 2 = 2 ω E 1 = ω E 0 = 0 The excitation energy from the ground state to the first excited bosonic state is ω Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

58 Spectrum in model I E 4 = 4 ω E 3 = 3 ω E 2 = 2 ω E 1 = ω E 0 = 0 The excitation energy from the ground state to the first excited bosonic state is ω The excitation energy from the ground state to the first excited fermionic state is ω Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

59 Model II in the perturbation theory Model II H = [ 2 2 is not exactly solvable for λ 0 d 2 dq ( 2 ω2 q 2 1 λ ) ] 2 (1 0 ω q 0 1 ) ( ) ( ω λq ) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

60 Model II in the perturbation theory Model II H = [ 2 2 d 2 dq ( 2 ω2 q 2 1 λ ) ] 2 (1 0 ω q 0 1 is not exactly solvable for λ 0 Perturbation theory H = H 0 + H ) ( ) ( ω λq where H 0 = ( 2 d 2 2 dq ) ( ) 2 ω2 q ( ) ω 0 1 ( H = λωq ) ( ) ( ) 2 λ2 q λq ) Model I Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

61 At O(λ 0 ), E (0) 0 = 1 2 ω 1 2 ω = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

62 At O(λ 0 ), E (0) 0 = 1 2 ω 1 2 ω = 0 No O(λ) term because of the reflection symmetry λ λ, q q Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

63 At O(λ 0 ), E (0) 0 = 1 2 ω 1 2 ω = 0 No O(λ) term because of the reflection symmetry λ λ, q q At O(λ 2 ): dq Ψ (0) 0 (q) H Ψ (0) (q) = dq Ψ(0) 0 (q) H Ψ (0) n (q) 2 n>0 E (0) 0 E (0) n 8ω 2 λ2 = 3 2 8ω 2 λ2 + O(λ 4 ) and E (2) 0 = 3 2 8ω 2 λ ω 2 λ2 = 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

64 At O(λ 0 ), E (0) 0 = 1 2 ω 1 2 ω = 0 No O(λ) term because of the reflection symmetry λ λ, q q At O(λ 2 ): and dq Ψ (0) 0 (q) H Ψ (0) (q) = dq Ψ(0) 0 (q) H Ψ (0) n (q) 2 n>0 E (0) 0 E (0) n 8ω 2 λ2 E (2) 0 = 3 2 8ω 2 λ ω 2 λ2 = 0 Is this just a coincidence? = 3 2 8ω 2 λ2 + O(λ 4 ) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

65 Non-renormalization theorem! Non-renormalization theorem E 0 = 0 persists in all orders of power-series expansion w.r.t. λ Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

66 Non-renormalization theorem! Non-renormalization theorem E 0 = 0 persists in all orders of power-series expansion w.r.t. λ Proof: The would-be E = 0 wave function exp ( 1 ) [ W (q) = exp 1 ( 1 2 ωq2 1 )] 3 λq3 is normalizable to all orders of the power-series expansion to O(λ N ) exp ( 1 ) W (q) exp ( ω N 2 q2) ( ) 1 λ n n! 3 q3. n=0 Q.E.D. Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

67 Non-renormalization theorem! Non-renormalization theorem E 0 = 0 persists in all orders of power-series expansion w.r.t. λ Proof: The would-be E = 0 wave function exp ( 1 ) [ W (q) = exp 1 ( 1 2 ωq2 1 )] 3 λq3 is normalizable to all orders of the power-series expansion to O(λ N ) exp ( 1 ) W (q) exp ( ω N 2 q2) ( ) 1 λ n n! 3 q3. Q.E.D. Spontaneous SUSY breaking in this system cannot be seen by perturbation theory! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29 n=0

68 Non-renormalization theorem More generally, for any superpotential W (q), Non-renormalization theorem If E 0 = 0 in the classical theory (i.e., when = 0), then E 0 = 0 remains in all orders of perturbation theory Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

69 Non-renormalization theorem More generally, for any superpotential W (q), Non-renormalization theorem If E 0 = 0 in the classical theory (i.e., when = 0), then E 0 = 0 remains in all orders of perturbation theory Proof: Almost the same as above Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

70 Non-renormalization theorem More generally, for any superpotential W (q), Non-renormalization theorem If E 0 = 0 in the classical theory (i.e., when = 0), then E 0 = 0 remains in all orders of perturbation theory Proof: Almost the same as above SUSY theories typically exhibit this sort of remarkable stability under perturbative (or radiative) corrections Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

71 Non-renormalization theorem More generally, for any superpotential W (q), Non-renormalization theorem If E 0 = 0 in the classical theory (i.e., when = 0), then E 0 = 0 remains in all orders of perturbation theory Proof: Almost the same as above SUSY theories typically exhibit this sort of remarkable stability under perturbative (or radiative) corrections Such stability results from the cancellation between the contribution from bosons and fermions + Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

72 Semi-classical approximation Semi-classical (or WKB, or instanton) approximation yields E 0 ω ( 2π exp 2 ω 3 ) 6λ 2 > 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

73 Semi-classical approximation Semi-classical (or WKB, or instanton) approximation yields E 0 ω ( 2π exp 2 ω 3 ) 6λ 2 > 0 = 0 + 0λ 2 + 0λ 4 + 0λ 6 + Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

74 Semi-classical approximation Semi-classical (or WKB, or instanton) approximation yields E 0 ω ( 2π exp 2 ω 3 ) 6λ 2 > 0 = 0 + 0λ 2 + 0λ 4 + 0λ 6 + Spontaneous SUSY breaking in this system is a purely non-perturbative phenomenon! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

75 Spectrum in model II E 1 ω E 0 > 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

76 Spectrum in model II E 1 ω E 0 > 0 The excitation energy from the ground state to the first excited bosonic state is ω Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

77 Spectrum in model II E 1 ω E 0 > 0 The excitation energy from the ground state to the first excited bosonic state is ω There is no excitation energy from the ground state to another fermionic ground state! = Nambu-Goldstone theorem Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

78 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

79 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT ground state vacuum Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

80 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT ground state vacuum ground state energy E 0 vacuum energy density E 0 Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

81 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

82 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken excitation energy particle mass (E = mc 2 ) Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

83 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken excitation energy particle mass (E = mc 2 ) Nambu-Goldstone theorem Nambu-Goldstone theorem Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

84 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT and ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken excitation energy particle mass (E = mc 2 ) Nambu-Goldstone theorem Nambu-Goldstone theorem non-renormalization theorem non-renormalization theorem Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

85 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT and ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken excitation energy particle mass (E = mc 2 ) Nambu-Goldstone theorem Nambu-Goldstone theorem non-renormalization theorem non-renormalization theorem The cancellation between bosons and fermions makes the divergence of the mass of the Higgs particle quite moderate Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

86 Dictionary from SUSY QM to SUSY QFT Dictionary SUSY QM SUSY QFT and ground state vacuum ground state energy E 0 vacuum energy density E 0 if E 0 > 0, SUSY broken if E 0 > 0, SUSY broken excitation energy particle mass (E = mc 2 ) Nambu-Goldstone theorem Nambu-Goldstone theorem non-renormalization theorem non-renormalization theorem The cancellation between bosons and fermions makes the divergence of the mass of the Higgs particle quite moderate Consistency of string theory requires SUSY Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

87 Large Hadron Collider (LHC) To find evidence of SUSY is one of the main objectives... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

88 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

89 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

90 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

91 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Why? Because the equation is very complicated? Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

92 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Why? Because the equation is very complicated? No Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

93 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Why? Because the equation is very complicated? Then why? No Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

94 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Why? Because the equation is very complicated? Then why? No We do not know the equation, beyond the perturbation theory! Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

95 Non-perturbative study of SUSY QFT Just as in SUSY QM, the spontaneous SUSY breaking can occur in SUSY QFT and it can be a non-perturbative phenomenon Then can we compute such non-perturbative phenomena in SUSY QFT from first principles? No Why? Because the equation is very complicated? Then why? No We do not know the equation, beyond the perturbation theory! We only have perturbation theory + consistency arguments Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

96 Non-perturbative definition by the lattice QFT is quantum mechanics of infinitely many variables (field) φ(x, t), x R 3, t: time φ(x, t), x Z 3, t Z This discretization however breaks SUSY... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

97 Only quite recently... (I. Kanamori, PRD 79 (2009)) Vacuum energy density E 0 of a certain SUSY QFT (SUSY Yang-Mills theory) defined in one-dimensional space E 0 /g 2 = 0.09 ± 0.09(sys) (stat) (energy density) g a 0 (!g) -a 1 +a !g it appears that the spontaneous SUSY breaking in this system is unlikely... Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29

98 Summary SUSY is a very interesting possibility, but it must be spontaneously broken to be true in the real world To study nonperturbative spontaneous breaking of SUSY from first principles, we need a non-perturbative formulation of SUSY QFT This is not yet available, although we had recently a promising success at least partially Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, / 29