Ward Identities and the Pion Decay Constant

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Ward identities and the pion decay constant

Sajid Ali

Fakult¨atf¨urPhysik

Journal Club, 18.12.2020, Universit¨atBielefeld Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Noether’s theorem Symmetries and conservatin laws QCD Ward identities The fermionic action The Axial WI Gell–Mann–Oakes–Renner N =1 SYM theory SUSY WIs Renormalization Numerical results WIs analysis Methods Chiral limit Continuum limit The conservation of momentum is related to the homogeneity • of space. Invariance under translation in time means that the law of • conservation of energy is valid. The conservation of angular momentum results from the • isotropy of space. Gauge symmetry leads to conservation of electric charge. • and so on • What is the quantum counterpart of Noether theorem? ans: Ward-Takahashi identities John Clive Ward and Yasushi Takahashi

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Noether’s theorem Every symmetry of the action has corresponding conservation law if the motion is governed by the principle of least action. (classical physics) What is the quantum counterpart of Noether theorem? ans: Ward-Takahashi identities John Clive Ward and Yasushi Takahashi

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Noether’s theorem Every symmetry of the action has corresponding conservation law if the motion is governed by the principle of least action. (classical physics)

The conservation of momentum is related to the homogeneity • of space. Invariance under translation in time means that the law of • conservation of energy is valid. The conservation of angular momentum results from the • isotropy of space. Gauge symmetry leads to conservation of electric charge. • and so on • Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Noether’s theorem Every symmetry of the action has corresponding conservation law if the motion is governed by the principle of least action. (classical physics)

The conservation of momentum is related to the homogeneity • of space. Invariance under translation in time means that the law of • conservation of energy is valid. The conservation of angular momentum results from the • isotropy of space. Gauge symmetry leads to conservation of electric charge. • and so on • What is the quantum counterpart of Noether theorem? ans: Ward-Takahashi identities John Clive Ward and Yasushi Takahashi Taken altogether, the massless action has the symmetry

SU(Nf )LxSU (Nf )R xU (1)V xU (1)A.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Symmetries in QCD

Vector and axial vector unitary transformations

iαTi iαTi ψ0 = e ψ, ψ¯0 = ψ¯ e− , (nonsinglet vector) iα1 iα1 ψ0 = e ψ, ψ¯0 = ψ¯ e− , (ﬂavour singlet vector)

iαγ5Ti iαγ5Ti ψ0 = e ψ, ψ¯0 = ψ¯ e , (nonsinglet axial)

iαγ51 iαγ51 ψ0 = e ψ, ψ¯0 = ψ¯ e , (ﬂavour singlet axial) iαλ iαˆλ ψ0 = e ψ, ψ¯0 = ψ¯ e , (short hand) Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Symmetries in QCD

Vector and axial vector unitary transformations

iαTi iαTi ψ0 = e ψ, ψ¯0 = ψ¯ e− , (nonsinglet vector) iα1 iα1 ψ0 = e ψ, ψ¯0 = ψ¯ e− , (ﬂavour singlet vector)

iαγ5Ti iαγ5Ti ψ0 = e ψ, ψ¯0 = ψ¯ e , (nonsinglet axial)

iαγ51 iαγ51 ψ0 = e ψ, ψ¯0 = ψ¯ e , (ﬂavour singlet axial) iαλ iαˆλ ψ0 = e ψ, ψ¯0 = ψ¯ e , (short hand)

Taken altogether, the massless action has the symmetry

SU(Nf )LxSU (Nf )R xU (1)V xU (1)A. The expectation value of arbitrary O Z 1 S[ψ,ψ¯,U] 0 O 0 = D[ψ,ψ¯,U] O[ψ,ψ¯,U] e− . h | | i Z Local, unitary, non-anomalous and inﬁnitesimal symmetry transformations

ψ ψ + δψ, ψ¯ ψ¯ + δψ¯. → →

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The partition function

Z Z ¯ S[ψ,ψ¯,U] ¯ S[ψ0,ψ0,U0] Z = D[ψ,ψ¯,U] e− = D[ψ0,ψ0,U0] e− . Local, unitary, non-anomalous and inﬁnitesimal symmetry transformations

ψ ψ + δψ, ψ¯ ψ¯ + δψ¯. → →

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The partition function

Z Z ¯ S[ψ,ψ¯,U] ¯ S[ψ0,ψ0,U0] Z = D[ψ,ψ¯,U] e− = D[ψ0,ψ0,U0] e− .

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The partition function

Z Z ¯ S[ψ,ψ¯,U] ¯ S[ψ0,ψ0,U0] Z = D[ψ,ψ¯,U] e− = D[ψ0,ψ0,U0] e− .

The expectation value of arbitrary O Z 1 S[ψ,ψ¯,U] 0 O 0 = D[ψ,ψ¯,U] O[ψ,ψ¯,U] e− . h | | i Z Local, unitary, non-anomalous and inﬁnitesimal symmetry transformations

ψ ψ + δψ, ψ¯ ψ¯ + δψ¯. → → Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The partition function

Z Z ¯ S[ψ,ψ¯,U] ¯ S[ψ0,ψ0,U0] Z = D[ψ,ψ¯,U] e− = D[ψ0,ψ0,U0] e− .

The expectation value of arbitrary O Z 1 S[ψ,ψ¯,U] 0 O 0 = D[ψ,ψ¯,U] O[ψ,ψ¯,U] e− . h | | i Z Local, unitary, non-anomalous and inﬁnitesimal symmetry transformations

ψ ψ + δψ, ψ¯ ψ¯ + δψ¯. → →

0 δO 0 0 OδS 0 = 0. h | | i − h | | i The simplest case O = 1, 0 δO 0 vanishes h | | i 0 δS 0 = 0. h | | i Linear inﬁnitesimal transformation corresponding to unitary transformations ˆ ψ ψ0 = 1 + iε(x)λ ψ, ψ¯ ψ¯ = ψ¯ 1 + iε(x)λ . → → 0

a a ˆ a a λ = 1,τ ,γ5,γ5τ and λ = 1, τ ,γ5,γ5τ (Dirac,ﬂavor) − − Z 4 ˆ ˆ ˆ δS = i d xψ¯ ελγµ ∂µ +γµ λ∂µ ε +iεAµ λγµ +γµ λ +ε λM +M λ ψ

ˆ λγµ + γµ λ = 0,

∂µ (εψ) = ∂µ (ε)ψ + ε∂µ (ψ).

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The fermionic action

Z 4 S = d x ψ¯ γµ Dµ + M ψ, Dµ = ∂µ + iAµ a a ˆ a a λ = 1,τ ,γ5,γ5τ and λ = 1, τ ,γ5,γ5τ (Dirac,ﬂavor) − − Z 4 ˆ ˆ ˆ δS = i d xψ¯ ελγµ ∂µ +γµ λ∂µ ε +iεAµ λγµ +γµ λ +ε λM +M λ ψ

ˆ λγµ + γµ λ = 0,

∂µ (εψ) = ∂µ (ε)ψ + ε∂µ (ψ).

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The fermionic action

Z 4 S = d x ψ¯ γµ Dµ + M ψ, Dµ = ∂µ + iAµ

Linear inﬁnitesimal transformation corresponding to unitary transformations ˆ ψ ψ0 = 1 + iε(x)λ ψ, ψ¯ ψ¯ = ψ¯ 1 + iε(x)λ . → → 0 ˆ λγµ + γµ λ = 0,

∂µ (εψ) = ∂µ (ε)ψ + ε∂µ (ψ).

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The fermionic action

Z 4 S = d x ψ¯ γµ Dµ + M ψ, Dµ = ∂µ + iAµ

Linear inﬁnitesimal transformation corresponding to unitary transformations ˆ ψ ψ0 = 1 + iε(x)λ ψ, ψ¯ ψ¯ = ψ¯ 1 + iε(x)λ . → → 0

a a ˆ a a λ = 1,τ ,γ5,γ5τ and λ = 1, τ ,γ5,γ5τ (Dirac,ﬂavor) − − Z 4 ˆ ˆ ˆ δS = i d xψ¯ ελγµ ∂µ +γµ λ∂µ ε +iεAµ λγµ +γµ λ +ε λM +M λ ψ Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The fermionic action

Z 4 S = d x ψ¯ γµ Dµ + M ψ, Dµ = ∂µ + iAµ

Linear inﬁnitesimal transformation corresponding to unitary transformations ˆ ψ ψ0 = 1 + iε(x)λ ψ, ψ¯ ψ¯ = ψ¯ 1 + iε(x)λ . → → 0

a a ˆ a a λ = 1,τ ,γ5,γ5τ and λ = 1, τ ,γ5,γ5τ (Dirac,ﬂavor) − − Z 4 ˆ ˆ ˆ δS = i d xψ¯ ελγµ ∂µ +γµ λ∂µ ε +iεAµ λγµ +γµ λ +ε λM +M λ ψ

ˆ λγµ + γµ λ = 0,

∂µ (εψ) = ∂µ (ε)ψ + ε∂µ (ψ). ˆ ∂µ (ψγ¯ µ λψ) = ψ¯ λM + M λ ψ.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Z 4 ˆ δS = i d x ε ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ − ˆ 1 λ = 1,λ = 1, ∂µ Vµ = 0, Vµ = ψγ¯ µ ψ, − 2 a ˆ a a a a 1 a λ = τ ,λ = τ , ∂µ V = 2ψ¯[M ,τ ]ψ, V = ψγ¯ µ τ ψ, − µ µ 2 ˆ 1 λ = γ5,λ = γ5, ∂ A = 4ψ¯M γ5ψ (+?), A = ψγ¯ γ5ψ, µ µ µ 2 µ a ˆ a a a a 1 a λ = γ5τ ,λ = γ5τ , ∂µ Aµ = 2ψ¯ M ,τ γ5ψ, Aµ = ψγ¯ µ γ5τ ψ. | { {z } } 2 nonsinglet AWI These identities hold as expectation values. O can be non-zero inside space-time region and zero outside. may be considered to be valid locally. also valid for renormalized quantities.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Z 4 ˆ δS = i d x ε ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ − ˆ ∂µ (ψγ¯ µ λψ) = ψ¯ λM + M λ ψ. These identities hold as expectation values. O can be non-zero inside space-time region and zero outside. may be considered to be valid locally. also valid for renormalized quantities.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Z 4 ˆ δS = i d x ε ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ − ˆ ∂µ (ψγ¯ µ λψ) = ψ¯ λM + M λ ψ.

ˆ 1 λ = 1,λ = 1, ∂µ Vµ = 0, Vµ = ψγ¯ µ ψ, − 2 a ˆ a a a a 1 a λ = τ ,λ = τ , ∂µ V = 2ψ¯[M ,τ ]ψ, V = ψγ¯ µ τ ψ, − µ µ 2 ˆ 1 λ = γ5,λ = γ5, ∂ A = 4ψ¯M γ5ψ (+?), A = ψγ¯ γ5ψ, µ µ µ 2 µ a ˆ a a a a 1 a λ = γ5τ ,λ = γ5τ , ∂µ Aµ = 2ψ¯ M ,τ γ5ψ, Aµ = ψγ¯ µ γ5τ ψ. | { {z } } 2 nonsinglet AWI These identities hold as expectation values. O can be non-zero inside space-time region and zero outside. may be considered to be valid locally. also valid for renormalized quantities. + 1 2 1 1 2 P = P iP = ψ¯[τ iτ )γ5ψ = d¯γ5u, − 2 − 1 P = P1 + iP2 = ψ¯[τ1 + iτ2)γ ψ =u ¯γ d, − 2 5 5 + 1 2 1 1 2 A = A iA = ψγ¯ µ (τ iτ )γ5ψ, µ µ − µ 2 − 1 A = A1 + iA2 = ψγ¯ (τ1 + iτ2)γ ψ. −µ µ µ 2 µ 5

For degenerate masses the AWI reads

(r)a 1 ∂ A = 2m(r)P(r)a, where P(r)a = ψτ¯ aγ ψ µ µ 2 5

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

To compare with experimental results, the quantities need to renormalized. (r)a a (r)a a (r)a a P = ZP P , V = ZV V , A = ZA A , (r) (r) m = Zm m, ψψ¯ = ZS ψψ¯ , m = mbare + mres . h i h i For degenerate masses the AWI reads

(r)a 1 ∂ A = 2m(r)P(r)a, where P(r)a = ψτ¯ aγ ψ µ µ 2 5

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

To compare with experimental results, the quantities need to renormalized. (r)a a (r)a a (r)a a P = ZP P , V = ZV V , A = ZA A , (r) (r) m = Zm m, ψψ¯ = ZS ψψ¯ , m = mbare + mres . h i h i

+ 1 2 1 1 2 P = P iP = ψ¯[τ iτ )γ5ψ = d¯γ5u, − 2 − 1 P = P1 + iP2 = ψ¯[τ1 + iτ2)γ ψ =u ¯γ d, − 2 5 5 + 1 2 1 1 2 A = A iA = ψγ¯ µ (τ iτ )γ5ψ, µ µ − µ 2 − 1 A = A1 + iA2 = ψγ¯ (τ1 + iτ2)γ ψ. −µ µ µ 2 µ 5 Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

To compare with experimental results, the quantities need to renormalized. (r)a a (r)a a (r)a a P = ZP P , V = ZV V , A = ZA A , (r) (r) m = Zm m, ψψ¯ = ZS ψψ¯ , m = mbare + mres . h i h i

+ 1 2 1 1 2 P = P iP = ψ¯[τ iτ )γ5ψ = d¯γ5u, − 2 − 1 P = P1 + iP2 = ψ¯[τ1 + iτ2)γ ψ =u ¯γ d, − 2 5 5 + 1 2 1 1 2 A = A iA = ψγ¯ µ (τ iτ )γ5ψ, µ µ − µ 2 − 1 A = A1 + iA2 = ψγ¯ (τ1 + iτ2)γ ψ. −µ µ µ 2 µ 5

For degenerate masses the AWI reads

(r)a 1 ∂ A = 2m(r)P(r)a, where P(r)a = ψτ¯ aγ ψ µ µ 2 5 The propagator 1 (r)a (r)b Mπ t 0 φ (p = 0,t)φ (0) 0 = δab e− , h | | i 2Mπ √ω

The decay π eνe is described by a Lagrangian, it has the coupling term→

GF cosθc (¯uγµ (1 γ5)d)(¯eγµ (1 γ5)νe ), √2 − − this is an examlple of operator product expansion. This leads to the axial vector current between π and vacuum. The relation to physical pion deﬁnes Fπ as

(r)a 2 (r)a ∂µ Aµ = Mπ Fπ φ

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The matrix element with single pion at rest

(r)a b Mπ t 0 φ (x) π (p = 0) = δab e− , h | | i The decay π eνe is described by a Lagrangian, it has the coupling term→

(r)a 2 (r)a ∂µ Aµ = Mπ Fπ φ

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The matrix element with single pion at rest

(r)a b Mπ t 0 φ (x) π (p = 0) = δab e− , h | | i The propagator 1 (r)a (r)b Mπ t 0 φ (p = 0,t)φ (0) 0 = δab e− , h | | i 2Mπ √ω Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The matrix element with single pion at rest

(r)a b Mπ t 0 φ (x) π (p = 0) = δab e− , h | | i The propagator 1 (r)a (r)b Mπ t 0 φ (p = 0,t)φ (0) 0 = δab e− , h | | i 2Mπ √ω

The decay π eνe is described by a Lagrangian, it has the coupling term→

(r)a 2 (r)a ∂µ Aµ = Mπ Fπ φ compare with nonsinglet AWI

2 (r)a (r)a (r) (r)a M Fπ 0 φ π = 0 ∂µ A π = 2m 0 P π , π h | | i h | µ | i h | | i

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The divergence of the axial vector current acts as interpolating ﬁeld operator of the pion

(r)a b 2 Mπ t ∂µ 0 A (x)π (p = 0) = M Fπ δab e− , h | µ i π this is partially conserved axial vector current or PCAC relation. This allows also to compute Fπ from the pair correlators between A4 and ∂t A4 M F 2 (r)+ (r) π π Mπ t A (p = 0,t)A −(0) e− , h 4 4 i ∼ √ω Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The divergence of the axial vector current acts as interpolating ﬁeld operator of the pion

2 (r)a (r)a (r) (r)a M Fπ 0 φ π = 0 ∂µ A π = 2m 0 P π , π h | | i h | µ | i h | | i 1 λ = λˆ = γ (τ1 + iτ2), 5 2 + + O = P (0) = d¯(0)γ5 u(0), 0 δP 0 = u¯(0)u(0) + d¯(0)d(0) , h | | i h i

Z 4 (r) + ¯ d x ∂µ A −(x)P (0) = u¯(0)u(0) + d(0)d(0) , − h µ i h i 2 4 Z Fπ Mπ 4 + (r) d x φ −(x)φ (0) = Nf Σ , − 2mr h i | {z } 2 2Mπ− at p=0 2 2 (r) (r) F M = m Nf Σ . π π − Gell–Mann–Oakes–Renner (GMOR) relation

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

0 δO 0 0 OδS 0 = 0. h | | i − h | | i Z 4 ˆ δS = d x ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ . − | {z } =0 Z 4 (r) + ¯ d x ∂µ A −(x)P (0) = u¯(0)u(0) + d(0)d(0) , − h µ i h i 2 4 Z Fπ Mπ 4 + (r) d x φ −(x)φ (0) = Nf Σ , − 2mr h i | {z } 2 2Mπ− at p=0 2 2 (r) (r) F M = m Nf Σ . π π − Gell–Mann–Oakes–Renner (GMOR) relation

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

0 δO 0 0 OδS 0 = 0. h | | i − h | | i Z 4 ˆ δS = d x ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ . − | {z } =0 1 λ = λˆ = γ (τ1 + iτ2), 5 2 + + O = P (0) = d¯(0)γ5 u(0), 0 δP 0 = u¯(0)u(0) + d¯(0)d(0) , h | | i h i Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

0 δO 0 0 OδS 0 = 0. h | | i − h | | i Z 4 ˆ δS = d x ∂µ (ψγ¯ µ λψ) + ψ¯ λM + M λ ψ . − | {z } =0 1 λ = λˆ = γ (τ1 + iτ2), 5 2 + + O = P (0) = d¯(0)γ5 u(0), 0 δP 0 = u¯(0)u(0) + d¯(0)d(0) , h | | i h i

Z 4 (r) + ¯ d x ∂µ A −(x)P (0) = u¯(0)u(0) + d(0)d(0) , − h µ i h i 2 4 Z Fπ Mπ 4 + (r) d x φ −(x)φ (0) = Nf Σ , − 2mr h i | {z } 2 2Mπ− at p=0 2 2 (r) (r) F M = m Nf Σ . π π − Gell–Mann–Oakes–Renner (GMOR) relation Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Numerical analysis of SUSY Ward identities and continuum extrapolation. Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

N =1 supersymmetric Yang-Mills theory Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Spectrum

Figure: Qualitative: Formation of bound states at low energy. Covariant derivative in adjoint representation • a a a b c 2 (Dµ λ) = ∂µ λ + g f A λ , a = 1,...,N 1 bc µ c − fermion ﬁeld λ a (gluino) Gluino mass term 1 m λ¯ λ a breaksSUSY softly • 2 g˜ a In contrast toQCD: • 1) adjoint representation of SU(Nc ) ¯ T 2) Majorana condition, λa = λa C a 1 3) λ is Majorana spinor ﬁeld, “Nf = 2 ”

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Supersymmetric Yang-Mills theory

Z 4 n 1 µν a i ¯ a 1 ¯ ao SSYM = d x F F + λaγµ (Dµ λ) mg˜λaλ − 4 a µν 2 − 2

Field strength tensor • a a a a b c a F = ∂µ A ∂ν A ig f A A , Gauge ﬁeld A (gluon) µν ν − µ − bc µ ν µ Gluino mass term 1 m λ¯ λ a breaksSUSY softly • 2 g˜ a In contrast toQCD: • 1) adjoint representation of SU(Nc ) ¯ T 2) Majorana condition, λa = λa C a 1 3) λ is Majorana spinor ﬁeld, “Nf = 2 ”

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Supersymmetric Yang-Mills theory

Z 4 n 1 µν a i ¯ a 1 ¯ ao SSYM = d x F F + λaγµ (Dµ λ) mg˜λaλ − 4 a µν 2 − 2

Supersymmetric Yang-Mills theory

Z 4 n 1 µν a i ¯ a 1 ¯ ao SSYM = d x F F + λaγµ (Dµ λ) mg˜λaλ − 4 a µν 2 − 2

Field strength tensor • a a a a b c a F = ∂µ A ∂ν A ig f A A , Gauge field A (gluon) µν ν − µ − bc µ ν µ Covariant derivative in adjoint representation • a a a b c 2 (Dµ λ) = ∂µ λ + g f A λ , a = 1,...,N 1 bc µ c − fermion field λ a (gluino) Gluino mass term 1 m λ¯ λ a breaksSUSY softly • 2 g˜ a In contrast toQCD: • 1) adjoint representation of SU(Nc ) ¯ T 2) Majorana condition, λa = λa C a 1 3) λ is Majorana spinor field, “Nf = 2 ” µ δQ(y) ∂ Sµ (x)Q(y) = . − δ¯ε(x)

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY Ward identities

Noether’s Theorem in classical theory WIs in quantum theory → S (x) is super current • µ Q(y) is local insertion operator • ε¯(x) is fermionic parameter • RHS of equation is contact term, which is zero if Q is • localised at space-time points diﬀerent from x

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY Ward identities

Noether’s Theorem in classical theory WIs in quantum theory →

µ δQ(y) ∂ Sµ (x)Q(y) = . − δ¯ε(x) Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY Ward identities

Noether’s Theorem in classical theory WIs in quantum theory →

µ δQ(y) ∂ Sµ (x)Q(y) = . − δ¯ε(x)

S (x) is super current • µ Q(y) is local insertion operator • ε¯(x) is fermionic parameter • RHS of equation is contact term, which is zero if Q is • localised at space-time points diﬀerent from x Above transformations result in following Ward identities: δQ(y) ∇µ Sµ (x)Q(y) =mg˜ χ(x)Q(y) + XS (x)Q(y) , − δ¯ε(x)

2i h (cl) i 2i h (cl) i Sµ (x) = tr P (x)σρν γµ λ(x) , χ(x) = tr P (x)σµν λ(x) . − g ρν g µν χ(x) : due to bare gluino mass (m ) g˜ BreakSUSY XS (x) : due to lattice regularization (a)

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY transformations on the lattice with P, T, Majorana nature, and gauge invariance:

iga δUµ (x)= ε¯(x)γµ Uµ (x)λ(x) + ε¯(x + µˆ)γµ λ(x + µˆ)Uµ (x) , − 2 1 (cl) δλ(x)= + P (x)σ ε(x). 2 µν µν χ(x) : due to bare gluino mass (m ) g˜ BreakSUSY XS (x) : due to lattice regularization (a)

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY transformations on the lattice with P, T, Majorana nature, and gauge invariance:

iga δUµ (x)= ε¯(x)γµ Uµ (x)λ(x) + ε¯(x + µˆ)γµ λ(x + µˆ)Uµ (x) , − 2 1 (cl) δλ(x)= + P (x)σ ε(x). 2 µν µν Above transformations result in following Ward identities: δQ(y) ∇µ Sµ (x)Q(y) =mg˜ χ(x)Q(y) + XS (x)Q(y) , − δ¯ε(x)

2i h (cl) i 2i h (cl) i Sµ (x) = tr P (x)σρν γµ λ(x) , χ(x) = tr P (x)σµν λ(x) . − g ρν g µν Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

SUSY transformations on the lattice with P, T, Majorana nature, and gauge invariance:

2i h (cl) i 2i h (cl) i Sµ (x) = tr P (x)σρν γµ λ(x) , χ(x) = tr P (x)σµν λ(x) . − g ρν g µν χ(x) : due to bare gluino mass (m ) g˜ BreakSUSY XS (x) : due to lattice regularization (a) Zero spatial momentum WIs and expansion in a basis of 16 Dirac matrices; the surviving contributions form a set of two non-trivial independent equations: (S,O) (T ,O) (χ,O) 1C1,t + AC1,t = BC1,t , (S,O) (T ,O) (χ,O) 1Cγ4,t + ACγ4,t = BCγ4,t . 1 1 A = ZT ZS− , B = amS ZS− ,

1xb t + Ayb t Bzb t = 0, b = 1,2 , , − ,

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Renormalization

ZS ∇µ Sµ (x)Q(y) + ZT ∇µ Tµ (x)Q(y) = mS χ(x)Q(y) + O(a).

ZS and ZT are renormalization coeﬃcients, mS = mg˜ m¯. − 2i h (cl) i 2i h (cl) i Tµ (x) = tr Pµν (x)γν λ(x) ,Q(y) = ∑ tr Pij (y)σij λ(y) . g g i

Renormalization

ZS ∇µ Sµ (x)Q(y) + ZT ∇µ Tµ (x)Q(y) = mS χ(x)Q(y) + O(a).

ZS and ZT are renormalization coeﬃcients, mS = mg˜ m¯. − 2i h (cl) i 2i h (cl) i Tµ (x) = tr Pµν (x)γν λ(x) ,Q(y) = ∑ tr Pij (y)σij λ(y) . g g i

1xb t + Ayb t Bzb t = 0, b = 1,2 , , − , Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Numerical results of correlation functions

0.06 0.06 0.04 0.04 0.02 0.02 ) ) O O ,t S, ,t 0 T, 0 ( ( 1 1 C C 0.02 0.02 − − 0.04 0.04 − − 0.06 0.06 − 0 4 8 12162024283236404448 − 0 4 8 12162024283236404448 t t

0.06 0.04 0.02 ) O ,t χ, 0 ( 1 C 0.02 − 0.04 − 0.06 − 0 4 8 12162024283236404448 t

0.06 0.06 0.04 0.04 0.02 0.02 ) ) O O ,t ,t 4 4 S, 0 T, 0 ( ( γ γ C C 0.02 0.02 − − 0.04 0.04 − − 0.06 0.06 − 0 4 8 12162024283236404448 − 0 4 8 12162024283236404448 t t

0.06 0.04 0.02 ) O ,t 4 χ, 0 ( γ C 0.02 − 0.04 − 0.06 − 0 4 8 12162024283236404448 t

Figure: β = 5.6 and κ = 0.1655. 1.2 Global 0.9 κ = 0.1655

1 0.6 − S Z

S 0.3

am 0 0.3 − 0.6 − 0 2 4 6 8 1012141618202224 tmin

0.14 Global 0.12 β = 5.6 0.10

Extrapolation to the Chiral 1 − S 0.08 Z S limit using the subtracted 0.06 1 am gluino mass (amS ZS− ) 0.04 0.02

0.00 3.001 3.009 3.017 3.024 3.032 3.040 1/2κ

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The Global method

CalculateA andB by minimizing the quantity:

2 tmax 2 (xi,t + Ayi,t Bzi,t ) ∑ ∑ 2− i=1 t=tmin σt 0.14 Global 0.12 β = 5.6 0.10

Extrapolation to the Chiral 1 − S 0.08 Z S limit using the subtracted 0.06 1 am gluino mass (amS ZS− ) 0.04 0.02

0.00 3.001 3.009 3.017 3.024 3.032 3.040 1/2κ

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The Global method 1.2 Global 0.9 CalculateA andB by κ = 0.1655

1 0.6 − minimizing the quantity: S Z

S 0.3

2 tmax 2 am 0 (xi,t + Ayi,t Bzi,t ) ∑ ∑ 2− 0.3 − i=1 t=tmin σt 0.6 − 0 2 4 6 8 1012141618202224 tmin Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The Global method 1.2 Global 0.9 CalculateA andB by κ = 0.1655

1 0.6 − minimizing the quantity: S Z

S 0.3

2 tmax 2 am 0 (xi,t + Ayi,t Bzi,t ) ∑ ∑ 2− 0.3 − i=1 t=tmin σt 0.6 − 0 2 4 6 8 1012141618202224 tmin

0.14 Global 0.12 β = 5.6 0.10

Extrapolation to the Chiral 1 − S 0.08 Z S limit using the subtracted 0.06 1 am gluino mass (amS ZS− ) 0.04 0.02

0.00 3.001 3.009 3.017 3.024 3.032 3.040 1/2κ ∑α Aα xˆiα = 0, i = (b,t),α = 1,2,3 We employ the method of maximum likelihood 1 L = (A xˆ )(D 1) (A xˆ ), D = A A ( x x xˆ xˆ ) 2 ∑ α iα − ij β jβ ij ∑ α β iα jβ iα jβ i,α,j,β α,β h i −

ﬁnd Aα such that L is minimum

0.14 GLS 0.12 β = 5.6 0.5 0.10

1 0.08 −

0.4 S Z L 0.3 S 0.06 am 0.2 0.04 0.3 0.2 0.02 -0.1 0 B 0.1 0.1 0.00 0.2 A 0.3 0 0.4 3.003 3.010 3.017 3.025 3.032 3.040 1/2κ

(a) Determination of A and B. (b) Chiral extrapolation.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The generalised least squared (GLS) method 1xb t + Ayb t Bzb t = 0, b = 1,2 , , − , We employ the method of maximum likelihood 1 L = (A xˆ )(D 1) (A xˆ ), D = A A ( x x xˆ xˆ ) 2 ∑ α iα − ij β jβ ij ∑ α β iα jβ iα jβ i,α,j,β α,β h i −

ﬁnd Aα such that L is minimum

0.14 GLS 0.12 β = 5.6 0.5 0.10

1 0.08 −

0.4 S Z L 0.3 S 0.06 am 0.2 0.04 0.3 0.2 0.02 -0.1 0 B 0.1 0.1 0.00 0.2 A 0.3 0 0.4 3.003 3.010 3.017 3.025 3.032 3.040 1/2κ

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The generalised least squared (GLS) method 1xb t + Ayb t Bzb t = 0, b = 1,2 , , − , ∑α Aα xˆiα = 0, i = (b,t),α = 1,2,3 0.14 GLS 0.12 β = 5.6 0.5 0.10

1 0.08 −

0.4 S Z L 0.3 S 0.06 am 0.2 0.04 0.3 0.2 0.02 -0.1 0 B 0.1 0.1 0.00 0.2 A 0.3 0 0.4 3.003 3.010 3.017 3.025 3.032 3.040 1/2κ

(e) Determination of A and B. (f) Chiral extrapolation.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The generalised least squared (GLS) method 1xb t + Ayb t Bzb t = 0, b = 1,2 , , − , ∑α Aα xˆiα = 0, i = (b,t),α = 1,2,3 We employ the method of maximum likelihood 1 L = (A xˆ )(D 1) (A xˆ ), D = A A ( x x xˆ xˆ ) 2 ∑ α iα − ij β jβ ij ∑ α β iα jβ iα jβ i,α,j,β α,β h i −

ﬁnd Aα such that L is minimum 0.14 GLS 0.12 β = 5.6 0.10

1 0.08 − S Z S 0.06 am 0.04

0.02

0.00 3.003 3.010 3.017 3.025 3.032 3.040 1/2κ

(h) Chiral extrapolation.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

ﬁnd Aα such that L is minimum

0.5

0.4 L 0.3

0.2 0.3 0.2 -0.1 0 B 0.1 0.1 0.2 A 0.3 0.4 0

(g) Determination of A and B. Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

ﬁnd Aα such that L is minimum

0.14 GLS 0.12 β = 5.6 0.5 0.10

1 0.08 −

0.4 S Z L 0.3 S 0.06 am 0.2 0.04 0.3 0.2 0.02 -0.1 0 B 0.1 0.1 0.00 0.2 A 0.3 0 0.4 3.003 3.010 3.017 3.025 3.032 3.040 1/2κ

(i) Determination of A and B. (j) Chiral extrapolation. 0.265 Pion 0.221 β = 5.6

2 0.176 ) π − a 0.132 am ( 0.088

0.044

0.000 3.005 3.010 3.015 3.020 3.025 3.030 3.035 3.040 2 1 1/2κ (ama π ) ∝ amS ZS− − (analogous to GOR relation) Figure: Extrapolation to the chiral 2 limit using ma π . −

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The adjoint Pion

The m is obtained • a π numerically− in simulations of N =1 SYM theory It is being used for the • tuning of chiral limit 2 1 (ama π ) ∝ amS ZS− − (analogous to GOR relation)

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The adjoint Pion

0.265 The m is obtained Pion • a π 0.221 numerically− in simulations β = 5.6

2 0.176 ) π − of N =1 SYM theory a 0.132 am It is being used for the ( 0.088 • tuning of chiral limit 0.044 0.000 3.005 3.010 3.015 3.020 3.025 3.030 3.035 3.040 1/2κ

Figure: Extrapolation to the chiral 2 limit using ma π . − Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

The adjoint Pion

0.265 The m is obtained Pion • a π 0.221 numerically− in simulations β = 5.6

2 0.176 ) π − of N =1 SYM theory a 0.132 am It is being used for the ( 0.088 • tuning of chiral limit 0.044 0.000 3.005 3.010 3.015 3.020 3.025 3.030 3.035 3.040 2 1 1/2κ (ama π ) ∝ amS ZS− − (analogous to GOR relation) Figure: Extrapolation to the chiral 2 limit using ma π . − Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Chiral limit

0.265 0.144 GLS β = 5.6 0.221 Global 0.120 Pion 0.176 0.096 1 − S

β = 5.6 Z

0.132 S 0.072 am

0.088 am 0.048

0.044 0.024

0.000 0.000 3.001 3.009 3.017 3.024 3.032 3.040 0.0334 0.0266 0.0866 0.1466 0.2066 − 2 1/2κ (ama π) − 1 (a) Comparison. (b) ∆(amS ZS− ).

1 2 Figure: 4(a): Comparison of amS ZS− from two methods with ma π . 1 2 1 − 4(b): ∆(amS ZS− ) using ma π and amS ZS− . − Handling of discretisation eﬀects

What about O(a)? in

ZS ∇µ Sµ (x)Q(y) + ZT ∇µ Tµ (x)Q(y) = mS χ(x)Q(y) + O(a).

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Continuum limit

0.09

0.07 ) 1 − S

Z 0.05

S Continuum limit in physical scale m ,χ 0 0.03 using tmin=3 w ∆( 0.01

0.01 − 0.00 0.04 0.08 0.12 0.16 0.20 0.24 2 (a/w0,χ) Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Continuum limit

0.09

0.07 ) 1 − S

Z 0.05

S Continuum limit in physical scale m ,χ 0 0.03 using tmin=3 w ∆( 0.01

0.01 − 0.00 0.04 0.08 0.12 0.16 0.20 0.24 2 (a/w0,χ)

Handling of discretisation eﬀects

What about O(a)? in

ZS ∇µ Sµ (x)Q(y) + ZT ∇µ Tµ (x)Q(y) = mS χ(x)Q(y) + O(a). Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Suﬃciently large tmin, β = 5.6

0.3 0.25 GLS GLS 0.24 0.2 κ = 0.1645 κ = 0.1650 1 1 − − S 0.18 S 0.15 Z Z S S 0.12 0.1 am am

0.06 0.05

0 0 2 4 6 8 10 12 2 4 6 8 10 12 tmin tmin 0.2 0.15 GLS GLS 0.16 0.12 κ = 0.1655 κ = 0.1660 1 1 − − S 0.12 S 0.09 Z Z S S 0.08 0.06 am am

0.04 0.03

0 0 2 4 6 8 10 12 2 4 6 8 10 12 tmin tmin β tmin from mgg˜ tmin from w0 5.4 4 4 5.45 5 5 5.5 5 6 5.6 7 7

Table: The choice of tmin at a ﬁxed physical time slice distance from the gluino-glue mass (mgg˜ ) and from the physical scale w0.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Fixed physical time slice distance

mgg˜,β0 w0,βi tmin,βi = tmin,β0 , tmin,βi = tmin,β0 , mgg˜,βi w0,β0

where i = 1,2,3, β0 = 5.4 and βi = 5.45,5.5,5.6. Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Fixed physical time slice distance

mgg˜,β0 w0,βi tmin,βi = tmin,β0 , tmin,βi = tmin,β0 , mgg˜,βi w0,β0

where i = 1,2,3, β0 = 5.4 and βi = 5.45,5.5,5.6.

β tmin from mgg˜ tmin from w0 5.4 4 4 5.45 5 5 5.5 5 6 5.6 7 7

Table: The choice of tmin at a ﬁxed physical time slice distance from the gluino-glue mass (mgg˜ ) and from the physical scale w0. 0.08

) 0.06 1 − S Z

S 0.04 m ,χ 0

w 0.02 ∆(

0.00

0.00 0.05 0.10 0.15 0.20 0.25 2 (a/w0,χ)

(b) Fit by one step procedure.

Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Continuum limit

0.13 0.11 ) 1 0.09 − S Z

S 0.07 m ,χ

0 0.05 w

∆( 0.03 0.01 0.01 − 0.00 0.04 0.08 0.12 0.16 0.20 0.24 2 (a/w0,χ)

(a) Fit by two step procedure. Outline Noether’s theorem Ward identities N =1 SYM theory SUSY WIs WIs analysis

Continuum limit

0.13 0.08 0.11 ) ) 0.06 1 0.09 1 − S − S Z Z S 0.07 S 0.04 m m ,χ ,χ 0 0.05 0 w w 0.02

∆( 0.03 ∆( 0.01 0.00 0.01 − 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.00 0.05 0.10 0.15 0.20 0.25 2 (a/w ) 2 0,χ (a/w0,χ)

Figure: Spectrum predicted by the VY and its generalisation form two distinct supermultiplets. The degenracy of masses is shifted when mg˜ is switched on. Non-singlet case:

iw aT a µ a Vector trans. ψ0 = e ψ ∂ Vµ (x)O(y) = 0 a a → h i iw T γ5 µ a a Axial trans. ψ0 = e ψ ∂ A (x)O(y) = 2m P (x)O(y) → h µ i h i a a ˜ a constants cA,cV in O(a) improved currents, AI ,µ = Aµ + acA∂µ P decay constants etc. Singlet→ case: The matrix element of divergence of axial ﬂavour singlet current between two on-shell states

µ 5 h ˜ i ZA α ∂ J β = 2 α ψγ¯ 5m¯ψ β + ZF α tr Fµν Fµν β h | µ | i h | | i h | | i 1). mass term 2). Adler-Bell-Jackiw anomaly Topologial charge →

Chiral Ward identity

SU(Nf )L SU(Nf )R UA(1) UV (1) × × × a a ˜ a constants cA,cV in O(a) improved currents, AI ,µ = Aµ + acA∂µ P decay constants etc. Singlet→ case: The matrix element of divergence of axial ﬂavour singlet current between two on-shell states

µ 5 h ˜ i ZA α ∂ J β = 2 α ψγ¯ 5m¯ψ β + ZF α tr Fµν Fµν β h | µ | i h | | i h | | i 1). mass term 2). Adler-Bell-Jackiw anomaly Topologial charge →

Chiral Ward identity

SU(Nf )L SU(Nf )R UA(1) UV (1) × × × Non-singlet case:

iw aT a µ a Vector trans. ψ0 = e ψ ∂ Vµ (x)O(y) = 0 a a → h i iw T γ5 µ a a Axial trans. ψ0 = e ψ ∂ A (x)O(y) = 2m P (x)O(y) → h µ i h i Singlet case: The matrix element of divergence of axial ﬂavour singlet current between two on-shell states

µ 5 h ˜ i ZA α ∂ J β = 2 α ψγ¯ 5m¯ψ β + ZF α tr Fµν Fµν β h | µ | i h | | i h | | i 1). mass term 2). Adler-Bell-Jackiw anomaly Topologial charge →

Chiral Ward identity

SU(Nf )L SU(Nf )R UA(1) UV (1) × × × Non-singlet case:

iw aT a µ a Vector trans. ψ0 = e ψ ∂ Vµ (x)O(y) = 0 a a → h i iw T γ5 µ a a Axial trans. ψ0 = e ψ ∂ A (x)O(y) = 2m P (x)O(y) → h µ i h i a a ˜ a constants cA,cV in O(a) improved currents, AI ,µ = Aµ + acA∂µ P decay constants etc. → 1). mass term 2). Adler-Bell-Jackiw anomaly Topologial charge →

Chiral Ward identity

SU(Nf )L SU(Nf )R UA(1) UV (1) × × × Non-singlet case:

µ 5 h ˜ i ZA α ∂ J β = 2 α ψγ¯ 5m¯ψ β + ZF α tr Fµν Fµν β h | µ | i h | | i h | | i Chiral Ward identity

SU(Nf )L SU(Nf )R UA(1) UV (1) × × × Non-singlet case:

µ 5 h ˜ i ZA α ∂ J β = 2 α ψγ¯ 5m¯ψ β + ZF α tr Fµν Fµν β h | µ | i h | | i h | | i 1). mass term 2). Adler-Bell-Jackiw anomaly Topologial charge →