Spontaneous Breaking of Supersymmetry

Spontaneous breaking of supersymmetry

Hiroshi Suzuki

Theoretical Physics Laboratory

Nov. 18, 2009 @ Theoretical science colloquium in RIKEN

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 1 / 29 Electron, proton, neutron, 3He, neutrino, quarks. . . Only one particle in a speciﬁc quantum state

Photon, pion, 4He, W -boson, Z -boson, Higgs particle. . . Any number of particles in a speciﬁc quantum state

Fermi particle (fermion)

Bose particle (boson)

They have quite different statistical properties. . .

SUperSYmmetry (SUSY) postulates invariance under the “exchange” of these bosons and fermions!

Identical particles in microscopic world

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 Photon, pion, 4He, W -boson, Z -boson, Higgs particle. . . Any number of particles in a speciﬁc quantum state

Electron, proton, neutron, 3He, neutrino, quarks. . . Only one particle in a speciﬁc quantum state Bose particle (boson)

They have quite different statistical properties. . .

SUperSYmmetry (SUSY) postulates invariance under the “exchange” of these bosons and fermions!

Identical particles in microscopic world

Fermi particle (fermion)

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 Electron, proton, neutron, 3He, neutrino, quarks. . . Only one particle in a speciﬁc quantum state

Photon, pion, 4He, 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!

Identical particles in microscopic world

Fermi particle (fermion)

Bose particle (boson)

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 Only one particle in a speciﬁc quantum state

Photon, pion, 4He, 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!

Identical particles in microscopic world

Fermi particle (fermion) Electron, proton, neutron, 3He, neutrino, quarks. . .

Bose particle (boson)

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 Only one particle in a speciﬁc quantum state

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!

Identical particles in microscopic world

Fermi particle (fermion) Electron, proton, neutron, 3He, neutrino, quarks. . .

Bose particle (boson) Photon, pion, 4He, W -boson, Z -boson, Higgs particle. . .

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 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!

Identical particles in microscopic world

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 They have quite different statistical properties. . .

SUperSYmmetry (SUSY) postulates invariance under the “exchange” of these bosons and fermions!

Identical particles in microscopic world

Fermi particle (fermion) Electron, proton, neutron, 3He, neutrino, quarks. . . Only one particle in a speciﬁc quantum state Bose particle (boson) Photon, pion, 4He, W -boson, Z -boson, Higgs particle. . . Any number of particles in a speciﬁc quantum state

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 SUperSYmmetry (SUSY) postulates invariance under the “exchange” of these bosons and fermions!

Identical particles in microscopic world

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 Identical particles in microscopic world

SUperSYmmetry (SUSY) postulates invariance under the “exchange” of these bosons and fermions!

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 2 / 29 SUSY⇐⇒

sneutrino? photino?

Higgs particle? ⇐⇒ ? ? ⇐⇒ top quark W , Z bosons ⇐⇒ ? ? ⇐⇒ u, d quarks ? ⇐⇒ electron ? ⇐⇒ neutrino photon ⇐⇒ ?

But our real world is not supersymmetric. . .

Mass spectrum of elementary particles

boson fermion

mass

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 3 / 29 SUSY⇐⇒

sneutrino? photino?

⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

But our real world is not supersymmetric. . .

Mass spectrum of elementary particles

boson fermion

mass Higgs particle? top quark W , Z bosons u, d quarks electron neutrino photon

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 3 / 29 sneutrino? photino?

⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

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, 2009 3 / 29 sneutrino? photino?

? ? ? ? ? ? ?

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, 2009 3 / 29 sneutrino? photino?

? ? ?

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, 2009 3 / 29 sneutrino? photino?

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, 2009 3 / 29 sneutrino? photino?

⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

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, 2009 3 / 29 ⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

But our real world is not supersymmetric. . .

Mass spectrum of elementary particles

boson SUSY⇐⇒ fermion

sneutrino? photino?

mass Higgs particle? top quark W , Z bosons u, d quarks electron neutrino photon

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 3 / 29 ⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

But our real world is not supersymmetric. . .

Mass spectrum of elementary particles

boson SUSY⇐⇒ fermion

sneutrino? photino?

mass Higgs particle? top quark W , Z bosons u, d quarks electron neutrino photon

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 3 / 29 ⇐⇒ ? ? ⇐⇒ ⇐⇒ ? ? ⇐⇒ ? ⇐⇒ ? ⇐⇒ ⇐⇒ ?

But our real world is not supersymmetric. . .

Mass spectrum of elementary particles

boson SUSY⇐⇒ fermion

sneutrino? photino?

mass Higgs particle? top quark W , Z bosons u, d quarks electron neutrino photon

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 3 / 29 ordinarysuper- (continuous) boson(s)fermion(s)

Nambu-Goldstone theorem When symmetry is spontaneously broken, there emerge massless

Spontaneous symmetry breaking The lowest energy (quantum) state of the system is not invariant under the symmetry transformation

SUSY is spontaneously broken?

Chiral symmetry is spontaneously broken! ⇒ pions

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, 2009 4 / 29 ordinarysuper- (continuous) boson(s)fermion(s)

Nambu-Goldstone theorem When symmetry is spontaneously broken, there emerge massless

SUSY is spontaneously broken?

Chiral symmetry is spontaneously broken! ⇒ pions

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, 2009 4 / 29 ordinarysuper- (continuous) boson(s)fermion(s)

Nambu-Goldstone theorem When symmetry is spontaneously broken, there emerge massless

Chiral symmetry is spontaneously broken! ⇒ pions

Spontaneous symmetry breaking

SUSY is spontaneously broken?

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 4 / 29 super- fermion(s)

Spontaneous symmetry breaking

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, 2009 4 / 29 ordinary (continuous) boson(s)

Spontaneous symmetry breaking

SUSY is spontaneously broken? Nambu-Goldstone theorem When supersymmetry is spontaneously broken, there emerge massless fermion(s)

Chiral symmetry is spontaneously broken! ⇒ pions

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 4 / 29 QFT is quantum mechanics of inﬁnitely many variables (ﬁeld)

3 φ(x, t), x ∈ R , 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. . .

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, 2009 5 / 29 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. . .

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)

3 φ(x, t), x ∈ R , t: time

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 5 / 29 SUSY and Quantum Field Theory (QFT)

3 φ(x, t), x ∈ R , 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, 2009 5 / 29 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 (≡ E0), the ground state energy will be very important in what follows

SUSY Quantum Mechanics (QM)

d Hamiltonian of SUSY QM (obtained by p → −i~ dq )

2 d 2 1 1 H = −~ + W 0(q)2 + W 00(q) b†b − , 2 dq2 2 ~ 2

where the function W (q) is called the super-potential

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 6 / 29 Among E, the smallest E (≡ E0), the ground state energy will be very important in what follows

SUSY Quantum Mechanics (QM)

d Hamiltonian of SUSY QM (obtained by p → −i~ dq )

2 d 2 1 1 H = −~ + W 0(q)2 + W 00(q) b†b − , 2 dq2 2 ~ 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, 2009 6 / 29 SUSY Quantum Mechanics (QM)

d Hamiltonian of SUSY QM (obtained by p → −i~ dq )

2 d 2 1 1 H = −~ + W 0(q)2 + W 00(q) b†b − , 2 dq2 2 ~ 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 (≡ E0), the ground state energy will be very important in what follows

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 6 / 29 These can be represented in terms of a “two-story” notation 0 1 |0i ⇔ bosonic, |1i ⇔ fermionic 1 0 and 2 × 2 matrices 0 0 0 1 b ⇔ , b† ⇔ 1 0 0 0

Creation and annihilation operators of a fermion

b† and b obey the relations bb† + b†b = 1, (b†)2 = b2 = 0 and, from these,

† † |0i −→b |1i −→b 0 0 ←−|b 0i ←−b |1i

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 7 / 29 Creation and annihilation operators of a fermion

b† and b obey the relations bb† + b†b = 1, (b†)2 = b2 = 0 and, from these,

† † |0i −→b |1i −→b 0 0 ←−|b 0i ←−b |1i These can be represented in terms of a “two-story” notation 0 1 |0i ⇔ bosonic, |1i ⇔ fermionic 1 0 and 2 × 2 matrices 0 0 0 1 b ⇔ , b† ⇔ 1 0 0 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 7 / 29 Now, the fundamental relation in SUSY QM is H = Q2 where Q is an (hermitian) operator called the super-charge 1 d 0 1 0 0 −i Q = √ −i~ + W (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

SUSY QM

In this two-story notation, 2 d 2 1 1 0 1 0 H = −~ + W 0(q)2 + ~W 00(q) 2 dq2 2 0 1 2 0 −1

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 8 / 29 The supercharge maps the boson into the fermion and vice versa 0 ∗ ∗ 0 Q = , Q = ∗ 0 0 ∗ and generates SUSY transformation

SUSY QM

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 8 / 29 SUSY QM

In this two-story notation, 2 d 2 1 1 0 1 0 H = −~ + W 0(q)2 + ~W 00(q) 2 dq2 2 0 1 2 0 −1 Now, the fundamental relation in SUSY QM is H = Q2 where Q is an (hermitian) operator called the super-charge 1 d 0 1 0 0 −i Q = √ −i~ + W (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, 2009 8 / 29 The zero-energy E = 0 is very special, because

E = 0 → Q2Ψ(q) = 0 → QΨ(q) = 0

Zero-energy state is always annihilated by the supercharge Zero-energy state is always invariant under SUSY transformation

Basic property of SUSY QM (I)

Since H = Q2, the energy eigenvalue E

HΨ(q) = Q2Ψ(q) = EΨ(q)

is positive or zero E ≥ 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 9 / 29 Zero-energy state is always invariant under SUSY transformation

Basic property of SUSY QM (I)

Since H = Q2, the energy eigenvalue E

HΨ(q) = Q2Ψ(q) = EΨ(q)

is positive or zero E ≥ 0 The zero-energy E = 0 is very special, because

E = 0 → Q2Ψ(q) = 0 → QΨ(q) = 0

Zero-energy state is always annihilated by the supercharge

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 9 / 29 Basic property of SUSY QM (I)

Since H = Q2, the energy eigenvalue E

HΨ(q) = Q2Ψ(q) = EΨ(q)

is positive or zero E ≥ 0 The zero-energy E = 0 is very special, because

E = 0 → Q2Ψ(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, 2009 9 / 29 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

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 = Q2)

HΦ(q) = HQΨ(q) = QHΨ(q) = QEΨ(q) = EΦ(q), E > 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 10 / 29 N.B. The pairing is impossible for E = 0 because QΨ(q) = 0

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 = Q2)

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, 2009 10 / 29 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 = Q2)

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, 2009 10 / 29 SUSY is spontaneously broken!

QΨ0(q) = 0 QΨ0(q) = Φ0(q) 6= 0

Possible patterns of the energy spectrum

E0 > 0

E0 = 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 11 / 29 SUSY is spontaneously broken!

QΨ0(q) = Φ0(q) 6= 0

Possible patterns of the energy spectrum

E0 > 0

E0 = 0

QΨ0(q) = 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 11 / 29 SUSY is spontaneously broken!

Possible patterns of the energy spectrum

E0 > 0

E0 = 0

QΨ0(q) = 0 QΨ0(q) = Φ0(q) 6= 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 11 / 29 Possible patterns of the energy spectrum

SUSY is spontaneously broken!

E0 > 0

E0 = 0

QΨ0(q) = 0 QΨ0(q) = Φ0(q) 6= 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 11 / 29 The solution is exp + 1 W (q) 0 Ψ(q) ∝ ~ or Ψ(q) ∝ 0 exp − 1 W (q) ~ The solution, however, must be normalizable Z ∞ dq Ψ(q)†Ψ(q) < ∞ −∞ This requires W (q) → +∞ for q → ±∞ or W (q) → −∞ for q → ±∞

Zero-energy E = 0 state in SUSY QM

Zero-energy E = 0 state satisﬁes the 1st order differential eq. i 0 1 d 1 −1 0 QΨ(q) = −√~ + W 0(q) Ψ(q) = 0 2 1 0 dq ~ 0 1

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 12 / 29 The solution, however, must be normalizable Z ∞ dq Ψ(q)†Ψ(q) < ∞ −∞ This requires W (q) → +∞ for q → ±∞ or W (q) → −∞ for q → ±∞

Zero-energy E = 0 state in SUSY QM

Zero-energy E = 0 state satisﬁes the 1st order differential eq. i 0 1 d 1 −1 0 QΨ(q) = −√~ + W 0(q) Ψ(q) = 0 2 1 0 dq ~ 0 1 The solution is exp + 1 W (q) 0 Ψ(q) ∝ ~ or Ψ(q) ∝ 0 exp − 1 W (q) ~

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 12 / 29 Zero-energy E = 0 state in SUSY QM

Zero-energy E = 0 state satisﬁes the 1st order differential eq. i 0 1 d 1 −1 0 QΨ(q) = −√~ + W 0(q) Ψ(q) = 0 2 1 0 dq ~ 0 1 The solution is exp + 1 W (q) 0 Ψ(q) ∝ ~ or Ψ(q) ∝ 0 exp − 1 W (q) ~ The solution, however, must be normalizable Z ∞ dq Ψ(q)†Ψ(q) < ∞ −∞ This requires W (q) → +∞ for q → ±∞ or W (q) → −∞ for q → ±∞

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 12 / 29 Asymptotic behavior of W (q) determines which. . .

If W (q) behaves as

or There exists E = 0 state and SUSY is not broken Otherwise, for example,

SUSY is spontaneously broken!

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 13 / 29 SUSY will not be spontaneously broken

SUSY will be spontaneously broken

Let us consider two examples

Model I 1 W (q) = ωq2 2

Model II 1 1 W (q) = ωq2 − λq3 2 3

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 14 / 29 SUSY will be spontaneously broken

Let us consider two examples

Model I 1 W (q) = ωq2 2

SUSY will not be spontaneously broken Model II 1 1 W (q) = ωq2 − λq3 2 3

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 14 / 29 Let us consider two examples

Model I 1 W (q) = ωq2 2

SUSY will not be spontaneously broken Model II 1 1 W (q) = ωq2 − λq3 2 3

SUSY will be spontaneously broken

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 14 / 29 The ground state in model I

The Hamiltonian 2 d 2 1 1 0 1 1 0 H = −~ + ω2q2 + ω 2 dq2 2 0 1 2~ 0 −1 represents two independent harmonic oscillators with shifted zero-point energies ±(1/2)~ω: Exactly solvable The unique ground state ! 0 Ψ (q) = 0 ω 1/4 exp − ω q2 π~ 2~ has the energy 1 1 E = ω − ω = 0 0 2~ 2~ and is annihilated by the supercharge

QΨ0(q) = 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 15 / 29 1st excited states in model I

The ﬁrst excited states 0 ! 1/4 Ψ1(q) = 4ω3 ω 2 ∝ QΦ1(q) bosonic 3 q exp − q π~ 2~ and ! ω 1/4 exp − ω q2 Φ (q) = π~ 2~ ∝ QΨ (q) fermionic 1 0 1

have the degenerate energies

3 1 E = ω − ω = ω 1 2~ 2~ ~ 1 1 E = ω + ω = ω 1 2~ 2~ ~

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 16 / 29 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

Spectrum in model I

E4 = 4~ω

E3 = 3~ω

E2 = 2~ω

E1 = ~ω

E0 = 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 17 / 29 The excitation energy from the ground state to the ﬁrst excited fermionic state is ~ω

Spectrum in model I

E4 = 4~ω

E3 = 3~ω

E2 = 2~ω

E1 = ~ω

E0 = 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, 2009 17 / 29 Spectrum in model I

E4 = 4~ω

E3 = 3~ω

E2 = 2~ω

E1 = ~ω

E0 = 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, 2009 17 / 29 Perturbation theory 0 H = H0 + H where 2 d 2 1 1 0 1 1 0 H = −~ + ω2q2 + ω ⇐ Model I 0 2 dq2 2 0 1 2~ 0 −1 1 1 0 1 0 H0 = −λωq3 + λ2q4 − λq 2 0 1 ~ 0 −1

Model II in the perturbation theory

Model II " # 2 d 2 1 λ 2 1 0 H = −~ + ω2q2 1 − q 2 dq2 2 ω 0 1 1 1 0 + ω − λq 2~ ~ 0 −1 is not exactly solvable for λ 6= 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 18 / 29 Model II in the perturbation theory

Model II " # 2 d 2 1 λ 2 1 0 H = −~ + ω2q2 1 − q 2 dq2 2 ω 0 1 1 1 0 + ω − λq 2~ ~ 0 −1 is not exactly solvable for λ 6= 0 Perturbation theory 0 H = H0 + H where 2 d 2 1 1 0 1 1 0 H = −~ + ω2q2 + ω ⇐ Model I 0 2 dq2 2 0 1 2~ 0 −1 1 1 0 1 0 H0 = −λωq3 + λ2q4 − λq 2 0 1 ~ 0 −1

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 18 / 29 No O(λ) term because of the reﬂection symmetry λ → −λ, q → −q At O(λ2):

Z ∞ 3 2 Ψ(0)( )† 0Ψ(0)( ) = ~ λ2 dq 0 q H 0 q 2 −∞ 8ω 2 R ∞ (0) † 0 (0) dq Ψ (q) H Ψ (q) 2 X −∞ 0 n 3 = − ~ λ2 + O(λ4) (0) (0) 8ω2 n>0 E0 − En and 3 2 3 2 E(2) = ~ λ2 − ~ λ2 = 0 0 8ω2 8ω2 Is this just a coincidence?

At O(λ0), 1 1 E(0) = ω − ω = 0 0 2~ 2~

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 19 / 29 At O(λ2):

At O(λ0), 1 1 E(0) = ω − ω = 0 0 2~ 2~ No O(λ) term because of the reﬂection symmetry λ → −λ, q → −q

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 19 / 29 Is this just a coincidence?

At O(λ0), 1 1 E(0) = ω − ω = 0 0 2~ 2~ No O(λ) term because of the reﬂection symmetry λ → −λ, q → −q At O(λ2):

Z ∞ 3 2 Ψ(0)( )† 0Ψ(0)( ) = ~ λ2 dq 0 q H 0 q 2 −∞ 8ω 2 R ∞ (0) † 0 (0) dq Ψ (q) H Ψ (q) 2 X −∞ 0 n 3 = − ~ λ2 + O(λ4) (0) (0) 8ω2 n>0 E0 − En and 3 2 3 2 E(2) = ~ λ2 − ~ λ2 = 0 0 8ω2 8ω2

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 19 / 29 At O(λ0), 1 1 E(0) = ω − ω = 0 0 2~ 2~ No O(λ) term because of the reﬂection symmetry λ → −λ, q → −q At O(λ2):

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 19 / 29 Proof: The would-be E = 0 wave function 1 1 1 1 exp − W (q) = exp − ωq2 − λq3 ~ ~ 2 3 is normalizable to all orders of the power-series expansion to O(λN )

N n 1 ω X 1 λ exp − W (q) ' exp − q2 q3 . 2 n! 3 ~ ~ n=0 ~ Q.E.D. Spontaneous SUSY breaking in this system cannot be seen by perturbation theory!

Non-renormalization theorem!

Non-renormalization theorem

E0 = 0 persists in all orders of power-series expansion w.r.t. λ

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 20 / 29 Spontaneous SUSY breaking in this system cannot be seen by perturbation theory!

Non-renormalization theorem!

Non-renormalization theorem

E0 = 0 persists in all orders of power-series expansion w.r.t. λ

Proof: The would-be E = 0 wave function 1 1 1 1 exp − W (q) = exp − ωq2 − λq3 ~ ~ 2 3 is normalizable to all orders of the power-series expansion to O(λN )

N n 1 ω X 1 λ exp − W (q) ' exp − q2 q3 . 2 n! 3 ~ ~ n=0 ~ Q.E.D.

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 20 / 29 Non-renormalization theorem!

Non-renormalization theorem

E0 = 0 persists in all orders of power-series expansion w.r.t. λ

Proof: The would-be E = 0 wave function 1 1 1 1 exp − W (q) = exp − ωq2 − λq3 ~ ~ 2 3 is normalizable to all orders of the power-series expansion to O(λN )

N n 1 ω X 1 λ exp − W (q) ' exp − q2 q3 . 2 n! 3 ~ ~ n=0 ~ Q.E.D. Spontaneous SUSY breaking in this system cannot be seen by perturbation theory!

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 20 / 29 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

¡ + ¢

Non-renormalization theorem

More generally, for any superpotential W (q), Non-renormalization theorem

If E0 = 0 in the classical theory (i.e., when ~ = 0), then E0 = 0 remains in all orders of perturbation theory

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 21 / 29 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

¡ + ¢

Non-renormalization theorem

More generally, for any superpotential W (q), Non-renormalization theorem

If E0 = 0 in the classical theory (i.e., when ~ = 0), then E0 = 0 remains in all orders of perturbation theory

Proof: Almost the same as above

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 21 / 29 Such stability results from the cancellation between the contribution from bosons and fermions

¡ + ¢

Non-renormalization theorem

More generally, for any superpotential W (q), Non-renormalization theorem

If E0 = 0 in the classical theory (i.e., when ~ = 0), then E0 = 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, 2009 21 / 29 Non-renormalization theorem

More generally, for any superpotential W (q), Non-renormalization theorem

If E0 = 0 in the classical theory (i.e., when ~ = 0), then E0 = 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, 2009 21 / 29 = 0 + 0λ2 + 0λ4 + 0λ6 + ···

Spontaneous SUSY breaking in this system is a purely non-perturbative phenomenon!

Semi-classical approximation

Semi-classical (or WKB, or instanton) approximation yields

3 ~ω 2 ω E0 ' exp − > 0 2π ~ 6λ2

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 22 / 29 Spontaneous SUSY breaking in this system is a purely non-perturbative phenomenon!

Semi-classical approximation

Semi-classical (or WKB, or instanton) approximation yields

3 ~ω 2 ω E0 ' exp − > 0 2π ~ 6λ2 = 0 + 0λ2 + 0λ4 + 0λ6 + ···

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 22 / 29 Semi-classical approximation

Semi-classical (or WKB, or instanton) approximation yields

3 ~ω 2 ω E0 ' exp − > 0 2π ~ 6λ2 = 0 + 0λ2 + 0λ4 + 0λ6 + ···

Spontaneous SUSY breaking in this system is a purely non-perturbative phenomenon!

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 22 / 29 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

Spectrum in model II

E1 ∼ ~ω

E0 > 0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 23 / 29 There is no excitation energy from the ground state to another fermionic ground state! = Nambu-Goldstone theorem

Spectrum in model II

E1 ∼ ~ω

E0 > 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, 2009 23 / 29 Spectrum in model II

E1 ∼ ~ω

E0 > 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, 2009 23 / 29 ground state ⇒ vacuum ground state energy E0 ⇒ vacuum energy density E0 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken excitation energy ⇒ particle mass (E = mc2) Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 ground state energy E0 ⇒ vacuum energy density E0 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken excitation energy ⇒ particle mass (E = mc2) Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

ground state ⇒ vacuum

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken excitation energy ⇒ particle mass (E = mc2) Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

ground state ⇒ vacuum ground state energy E0 ⇒ vacuum energy density E0

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 excitation energy ⇒ particle mass (E = mc2) Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

ground state ⇒ vacuum ground state energy E0 ⇒ vacuum energy density E0 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

ground state ⇒ vacuum ground state energy E0 ⇒ vacuum energy density E0 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken excitation energy ⇒ particle mass (E = mc2)

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 and

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

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

ground state ⇒ vacuum ground state energy E0 ⇒ vacuum energy density E0 if E0 > 0, SUSY broken ⇒ if E0 > 0, SUSY broken excitation energy ⇒ particle mass (E = mc2) Nambu-Goldstone theorem ⇒ Nambu-Goldstone theorem

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 The cancellation between bosons and fermions makes the divergence of the mass of the Higgs particle quite moderate Consistency of string theory requires SUSY

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

and

non-renormalization theorem ⇒ non-renormalization theorem

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 24 / 29 Consistency of string theory requires SUSY

Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

and

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, 2009 24 / 29 Dictionary from SUSY QM to SUSY QFT

Dictionary

SUSY QM SUSY QFT

and

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, 2009 24 / 29 Large Hadron Collider (LHC)

To ﬁnd evidence of SUSY is one of the main objectives. . .

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 25 / 29 No

We do not know the equation, beyond the perturbation theory!

Then can we compute such non-perturbative phenomena in SUSY QFT from ﬁrst principles?

Why? Because the equation is very complicated?

Then why?

We only have perturbation theory + consistency arguments

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, 2009 26 / 29 No

We do not know the equation, beyond the perturbation theory!

No Why? Because the equation is very complicated?

Then why?

We only have perturbation theory + consistency arguments

Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 No

We do not know the equation, beyond the perturbation theory!

Why? Because the equation is very complicated?

Then why?

We only have perturbation theory + consistency arguments

Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 We do not know the equation, beyond the perturbation theory!

No Then why?

We only have perturbation theory + consistency arguments

Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 We do not know the equation, beyond the perturbation theory!

Then why?

We only have perturbation theory + consistency arguments

Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 We do not know the equation, beyond the perturbation theory! We only have perturbation theory + consistency arguments

Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 We only have perturbation theory + consistency arguments

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 Then why? We do not know the equation, beyond the perturbation theory!

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 Non-perturbative study of SUSY QFT

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 26 / 29 Non-perturbative deﬁnition by the lattice

QFT is quantum mechanics of inﬁnitely many variables (ﬁeld)

3 φ(x, t), x ∈ R , t: time ⇓ 3 φ(x, t), x ∈ Z , t ∈ Z

This discretization however breaks SUSY. . .

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 27 / 29 Only quite recently. . . (I. Kanamori, PRD 79 (2009))

Vacuum energy density E0 of a certain SUSY QFT (SUSY Yang-Mills theory) deﬁned in one-dimensional space

2 +0.10 E0/g = 0.09 ± 0.09(sys) −0.08 (stat)

30 -a1 a0(!g) +a2 25 -2 20 2 15 1 10 0 5 1 2 3 4 (energy density) g 0 -5 0 1 2 3 4 5 !g

it appears that the spontaneous SUSY breaking in this system is unlikely. . .

Hiroshi Suzuki (TPL) Spontaneous breaking of supersymmetry Nov. 18, 2009 28 / 29 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, 2009 29 / 29