CHAPTER VIII: Symmetries of QCD - T39 Physics Group · CHAPTER 8. SYMMETRIES OF QCD • QCD is...

7
QCD is based on local SU (3) c gauge symmetry In addition: global symmetries 8. 1. N¨other’sTheorem L QCD (x)= ¯ ψ(x)[ iγ μ D μ m ] ψ(x)+ 1 4 G a μν (x)G μν a (x) (a =1, ··· , 8) where D μ = μ igA μ (x), A μ (x)= A μ a (x) λ a 2 and ψ(x)= u d s . Let L QCD be invariant under a global transformation of the quark fields: ψ(x) ψ (x) = exp [iΓ a Θ a ] ψ(x)=1+ iΓ a Θ a ψ ±··· with Γ a : generators of U (N ) or SU (N ), Θ a independent of x. Define: N¨other current J a μ (x)= L QCD (μ ψ) ∂ψ Θ a = ¯ ψ(x)γ μ Γ a ψ(x) If L QCD is invariant under the global transformation, then N¨other current J J a μ is con served: μ J μ a (x)=0 If current is localized in space, then N¨other current has conserved charge. Q a = d 3 xJ 0 a (x)= d 3 x ψ (x)Γ a ψ(x) ˙ Q a = dQ a dt =0 because V d 3 x · J a = V d f · J a =0 8. 2. Baryon Number and Flavor Currents a ) Global U (1) symmetry: ψ(x) e iθ ψ(x) J μ B (x)= ¯ ψ(x)γ μ ψ(x) conserved charge: B = d 3 x ψ (x)ψ(x) Baryon number b ) Isospin current: ψ(x)= u(x) d(x) Isospin doublet (N f = 2) - Assume: equal masses m u = m d . - SU (2) f transformation: ψ(x) ψ (x) = exp i τ i 2 θ i - τ i 2 : SU (2) generators (i =1, 2, 3) Pauli matrices - L QCD with m q m u = m d is invariant under SU (2) f conserved current: V μ i (x)= ¯ ψ(x)γ μ τ i 2 ψ(x) conserved isospin charge: Q i = d 3 xV 0 i (x)= d 3 x ψ (x) τ i 2 ψ(x) ˙ Q i =0 [ H, Q i ]=0 c ) Flavor current in SU (3) f : ψ = u d s (N f = 3) - Assume: m u = m d = m s - SU (3) f transformation: ψ(x) ψ (x) = exp i λ j 2 θ j (j =1, ··· , 8) conserved current: V μ i (x)= ¯ ψ(x)γ μ λ i 2 ψ(x) conserved charge: Q i = d 3 x ψ (x) λ i 2 ψ(x) d ) Symmetry breaking: m s = m u,d = m L mass = ¯ ψ(x) m 0 0 0 m 0 0 0 m s ψ(x) μ V μ i (m s m) CHAPTER VIII: Symmetries of QCD

Transcript of CHAPTER VIII: Symmetries of QCD - T39 Physics Group · CHAPTER 8. SYMMETRIES OF QCD • QCD is...

Page 1: CHAPTER VIII: Symmetries of QCD - T39 Physics Group · CHAPTER 8. SYMMETRIES OF QCD • QCD is based on local SU(3) c gauge symmetry • In addition: global symmetries 8.1. N¨other’s

CHAPTER 8. SYMMETRIES OF QCD

• QCD is based on local SU(3)c gauge symmetry

• In addition: global symmetries

8. 1. Nother’s Theorem

LQCD(x) = ψ(x) [ iγµDµ −m ] ψ(x) +

1

4Ga

µν(x)Gµνa (x) (a = 1, · · · , 8) (8.1)

where Dµ = ∂µ − igAµ(x), Aµ(x) = Aµa(x)

λa

2and ψ(x) =

u

d

s

.

• Let LQCD be invariant under a global transformation of the quark fields:

ψ(x)→ ψ(x) = exp [iΓaΘa] ψ(x) = 1 + iΓaΘaψ ± · · · (8.2)

with Γa: generators of U(N) or SU(N), Θa independent of x.

• Define: Nother current

Jaµ(x) = − ∂LQCD

∂(∂µψ)

∂ψ

∂Θa= ψ(x)γµΓaψ(x) (8.3)

• If LQCD is invariant under the global transformation, then Nother current Jaµ is con-

served:

∂µJµa (x) = 0 (8.4)

• If current is localized in space, then Nother current has conserved charge.

Qa =

d3x J0

a(x) =

d3xψ†(x)Γaψ(x)

Qa =dQa

dt= 0

because

V

d3x ∇ · J a =

∂V

df · J a = 0

(8.5)

77

CHAPTER 8. SYMMETRIES OF QCD

• QCD is based on local SU(3)c gauge symmetry

• In addition: global symmetries

8. 1. Nother’s Theorem

LQCD(x) = ψ(x) [ iγµDµ −m ] ψ(x) +

1

4Ga

µν(x)Gµνa (x) (a = 1, · · · , 8) (8.1)

where Dµ = ∂µ − igAµ(x), Aµ(x) = Aµa(x)

λa

2and ψ(x) =

u

d

s

.

• Let LQCD be invariant under a global transformation of the quark fields:

ψ(x)→ ψ(x) = exp [iΓaΘa] ψ(x) = 1 + iΓaΘaψ ± · · · (8.2)

with Γa: generators of U(N) or SU(N), Θa independent of x.

• Define: Nother current

Jaµ(x) = − ∂LQCD

∂(∂µψ)

∂ψ

∂Θa= ψ(x)γµΓaψ(x) (8.3)

• If LQCD is invariant under the global transformation, then Nother current Jaµ is con-

served:

∂µJµa (x) = 0 (8.4)

• If current is localized in space, then Nother current has conserved charge.

Qa =

d3x J0

a(x) =

d3xψ†(x)Γaψ(x)

Qa =dQa

dt= 0

because

V

d3x ∇ · J a =

∂V

df · J a = 0

(8.5)

77

CHAPTER 8. SYMMETRIES OF QCD

• QCD is based on local SU(3)c gauge symmetry

• In addition: global symmetries

8. 1. Nother’s Theorem

LQCD(x) = ψ(x) [ iγµDµ −m ] ψ(x) +

1

4Ga

µν(x)Gµνa (x) (a = 1, · · · , 8) (8.1)

where Dµ = ∂µ − igAµ(x), Aµ(x) = Aµa(x)

λa

2and ψ(x) =

u

d

s

.

• Let LQCD be invariant under a global transformation of the quark fields:

ψ(x)→ ψ(x) = exp [iΓaΘa] ψ(x) = 1 + iΓaΘaψ ± · · · (8.2)

with Γa: generators of U(N) or SU(N), Θa independent of x.

• Define: Nother current

Jaµ(x) = − ∂LQCD

∂(∂µψ)

∂ψ

∂Θa= ψ(x)γµΓaψ(x) (8.3)

• If LQCD is invariant under the global transformation, then Nother current Jaµ is con-

served:

∂µJµa (x) = 0 (8.4)

• If current is localized in space, then Nother current has conserved charge.

Qa =

d3x J0

a(x) =

d3xψ†(x)Γaψ(x)

Qa =dQa

dt= 0

because

V

d3x ∇ · J a =

∂V

df · J a = 0

(8.5)

77

CHAPTER 8. SYMMETRIES OF QCD

• QCD is based on local SU(3)c gauge symmetry

• In addition: global symmetries

8. 1. Nother’s Theorem

LQCD(x) = ψ(x) [ iγµDµ −m ] ψ(x) +

1

4Ga

µν(x)Gµνa (x) (a = 1, · · · , 8) (8.1)

where Dµ = ∂µ − igAµ(x), Aµ(x) = Aµa(x)

λa

2and ψ(x) =

u

d

s

.

• Let LQCD be invariant under a global transformation of the quark fields:

ψ(x)→ ψ(x) = exp [iΓaΘa] ψ(x) = 1 + iΓaΘaψ ± · · · (8.2)

with Γa: generators of U(N) or SU(N), Θa independent of x.

• Define: Nother current

Jaµ(x) = − ∂LQCD

∂(∂µψ)

∂ψ

∂Θa= ψ(x)γµΓaψ(x) (8.3)

• If LQCD is invariant under the global transformation, then Nother current Jaµ is con-

served:

∂µJµa (x) = 0 (8.4)

• If current is localized in space, then Nother current has conserved charge.

Qa =

d3x J0

a(x) =

d3xψ†(x)Γaψ(x)

Qa =dQa

dt= 0

because

V

d3x ∇ · J a =

∂V

df · J a = 0

(8.5)

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8. 2. Baryon Number and Flavor Currents

a ) Global U(1) symmetry:

ψ(x)→ eiθψ(x) ⇒ J

µB(x) = ψ(x)γµψ(x) (8.6)

conserved charge: B =

d3

xψ†(x)ψ(x) ⇔ Baryon number

b ) Isospin current: ψ(x) =

u(x)

d(x)

Isospin doublet (Nf = 2)

- Assume: equal masses mu = md.

- SU(2)f transformation: ψ(x)→ ψ(x) = expiτi

2θi

-τi

2: SU(2) generators (i = 1, 2, 3)→ Pauli matrices

- LQCD with mq ≡ mu = md is invariant under SU(2)f

⇒ conserved current: Vµi (x) = ψ(x)γµ τi

2ψ(x)

⇒ conserved isospin charge: Qi =

d3

x V0i (x) =

d3

x ψ†(x)τi

2ψ(x)

Qi = 0 ⇔ [ H, Qi ] = 0

c ) Flavor current in SU(3)f : ψ =

u

d

s

(Nf = 3)

- Assume: mu = md = ms

- SU(3)f transformation: ψ(x)→ ψ(x) = exp

iλj

2θj

(j = 1, · · · , 8)

⇒ conserved current: Vµi (x) = ψ(x)γµ λi

2ψ(x)

⇒ conserved charge: Qi =

d3

x ψ†(x)λi

2ψ(x)

d ) Symmetry breaking: ms = mu,d = m

Lmass = ψ(x)

m 0 0

0 m 0

0 0 ms

ψ(x) ⇒ ∂µVµi ∝ (ms −m)

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CHAPTER VIII: Symmetries of QCD

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8. 3. QCD with Massless Quarks: Chiral Symmetry

• Start with Nf = 2:

LQCD(x) = ψ(x)iγµDµψ(x)− 1

4Ga

µν(x)Gµνa (x)

L0

QCD

+Lmass (8.7)

where Lmass = ψ(x)

mu 0

0 md

ψ(x)

• QCD in the limit of massless quarks: L0QCD = ψ(x)iγµ∂

µψ(x) + Lquark-gluon + Lglue

• Left- and right-handed quark fields: ψ = ψR + ψL

ψR =1

2(1 + γ5)ψ

ψL =1

2(1− γ5)ψ

(8.8)

where γ5 = γ5 = iγ0γ1γ2γ3 =

0

0

with =

1 0

0 1

γ5, γµ = 0, γ25 =

0

0

• Quark field:

ψ(x) =

s

d3p

(2π)32Ep

bs(p)us(p)e−ip·x + d†

s(p)vs(p)eip·x (8.9)

us(p) = N

χs

σ·pEp+m χs

Ep=|p | , m=0−−−−−−−−→ N

χs

σ·p|p | χs

(8.10)

σ · p|p | = h = ±1

“right”-

“left”-handed.

79

8. 3. QCD with Massless Quarks: Chiral Symmetry

• Start with Nf = 2:

LQCD(x) = ψ(x)iγµDµψ(x)− 1

4Ga

µν(x)Gµνa (x)

L0

QCD

+Lmass (8.7)

where Lmass = ψ(x)

mu 0

0 md

ψ(x)

• QCD in the limit of massless quarks: L0QCD = ψ(x)iγµ∂

µψ(x) + Lquark-gluon + Lglue

• Left- and right-handed quark fields: ψ = ψR + ψL

ψR =1

2(1 + γ5)ψ

ψL =1

2(1− γ5)ψ

(8.8)

where γ5 = γ5 = iγ0γ1γ2γ3 =

0

0

with =

1 0

0 1

γ5, γµ = 0, γ25 =

0

0

• Quark field:

ψ(x) =

s

d3p

(2π)32Ep

bs(p)us(p)e−ip·x + d†

s(p)vs(p)eip·x (8.9)

us(p) = N

χs

σ·pEp+m χs

Ep=|p | , m=0−−−−−−−−→ N

χs

σ·p|p | χs

(8.10)

σ · p|p | = h = ±1

“right”-

“left”-handed.

79

(1± γ5)us(p) = N

χs

hχs

±

hχs

χs

= N

(1± h)χs

±(1± h)χs

(8.11)

⇒ 12(1 + γ5) projects on h = +1 (right handed).

⇒ 12(1− γ5) projects on h = −1 (left handed).

• Massless QCD:

L0QCD = ψL(x)iγµD

µψL(x) + ψR(x)iγµDµψR(x) + Lglue (8.12)

where

ψLiγµDµψL =

1

4ψ†(1− γ5)γ0γµD

µ(1− γ5)ψ

=1

4ψ(1 + γ5)iγµD

µ(1− γµ)ψ

=1

2

ψiγµD

µψ − ψiγµγ5Dµψ

ψRiγµDµψR =

1

4ψ†(1 + γ5)γ0γµD

µ(1 + γ5)ψ

=1

2

ψiγµD

µψ + ψiγµγ5Dµψ

(8.13)

• Global transformation: chiral SU(2)R × SU(2)L symmetry

ψR(x)→ expiτj

2θj

R

ψR(x)

ψL(x)→ expiτk

2θk

L

ψL(x)

(8.14)

with τi: Pauli matrices (i = 1, 2, 3)

• Mass term breaks this symmetry explicitly.

Lmass = ψ(x)

mu 0

0 md

ψ(x) = ψ mψ

= ψR mψL + ψL mψR

(8.15)

⇒ Quark mass term mixes left- and right-handed quarks.

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• In the limit mu,d → 0: conserved currents:

JµR,i(x) = ψR(x)γµ τi

2ψR(x)

JµL,i(x) = ψL(x)γµ τi

2ψL(x)

(8.16)

• Convenient to introduce vector and axial vector current:

Vµi (x) = J

µR,i(x) + J

µL,i(x) = ψ(x)γµ τi

2ψ(x)

Aµi (x) = J

µR,i(x)− J

µL,i(x) = ψ(x)γµγ5

τi

2ψ(x)

(8.17)

( ∂µVµi = 0 , ∂µA

µi = 0 )

• Conserved charge:

QVi (t) =

d3

x V0i (x) =

d3

x ψ†(x)τi

2ψ(x) (Vector charge)

QAi (t) =

d3

x A0i (x) =

d3

x ψ†(x)γ5τi

2ψ(x) (Axial charge)

(8.18)

d

dtQ

Vi (t) = i

H, Q

Vi

= 0 ,

d

dtQ

Ai (t) = i

H, Q

Ai

= 0 (8.19)

• Generalization to 3 flavor (Nf = 3) ⇒ SU(3)R × SU(3)L symmetry

replace τi → λi: Gell-Mann matrices (i = 1, · · · , 8)

• Lie algebra of the vector and axial charges:

Q

Vi (t), Q

Vj (t)

= ifijk Q

Vk (t)

Q

Vi (t), Q

Aj (t)

= ifijk Q

Ak (t)

Q

Ai (t), Q

Aj (t)

= ifijk Q

Vk (t)

(8.20)

with fijk: structure constant of SU(3).

8. 4. Realizations of Chiral Symmetry

• Wigner-Weyl realization:

Ground state (“vacuum”): QVi |0 = 0, Q

Ai |0 = 0

⇒ Total symmetry between positive and negative parity.

81

!

"

0.5

1.0

!

"

#, $

"!

%

K

K"

N

!

", #

···

···

··

···

Mas

s[G

eV]

! "

# $

PseudoscalarMesons

(Jp = 0#)

! "

# $

“Gap”" ! 1GeV

• Spectrum of states in Wigner-Weyl realization

⇒ Parity doublets: for each state of positive parity, there must be a state of equal

mass with negative parity. But:

a ) For nucleon with Jp = 12

+, there is no equal mass partner with Jp = 1

2

−.

b ) For pseudoscalar mesons with Jp = 0−, there is no chiral partner with Jp = 0+.

c ) Vector- and Axialvector-mesons:

– Vector mesons: Jp = 1−

– Axial vector mesons: Jp = 1+

• Current correlation function:

ΠµνV (q) = i

d4x eiq·x0|T [V µ(x)V ν(0)]|0

ΠµνA (q) = i

d4x eiq·x0|T [Aµ(x)Aν(0)]|0

(8.21)

ΠµνV,A(q) =

qµqν − q2gµν

ΠV,A(q2) (8.22)

• In Wigner-Weyl realization:

QV |0 = 0 , QA |0 = 0 ⇒ ΠV (q2) ≡ ΠA(q2) (8.23)

• Spectral functions: ηV,A(s) = 4π ImΠV,A(q2 = s)

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• But empirically:

0.0 0.5 1.0 1.5 2.0s !GeV2"

!

"#meson

a1#meson

!A

!V

• These observations lead to the Nambu-Goldstone realization of chiral symmetry:

Ground state does not have all the symmetries of Lagrangian density.

QVi |0 = 0

Isospin symmetry

, QAi |0 = 0 (8.24)

⇒ Axial symmetry is spontaneously broken.

8. 5. Goldstone’s Theorem

For every spontaneously broken global symmetry, there exists a massless state that

carries the quantum numbers of the corresponding symmetry charge.

QAi |0 = 0 , H |0 = E0 |0

Define |Φi ≡ QAi |0

then: H |Φi = HQAi |0 = Q

Ai H |0 = Q

Ai E0 |0 = E0 |Φi

|Φi energetically degenerate with ground state (vacuum) ⇒ Massless Goldstone Boson.

|Φi are states with spin/parity Jp = 0− “Pseudoscalar”.

For Nf = 2; i = 1, 2, 3; Isospin I = 1 ⇒ Pions (π+, π0

, π−)

• Goldstone’s theorem:

In the Nambu-Goldstone realization of (spontaneously broken) chiral symmetry, the

Goldstone bosons are weakly interacting at low energies.

83

Proof: Consider a state of n Goldstone bosons |(Φ)n = (QA

)n |0.

H |(Φ)n = H (Qi · · ·Qk)

n-times

|0 = (Qi · · ·Qk)H |0 = E0 |(Φ)n

⇒ Each Goldstone boson has energy-momentum relation ε = |q |. Since n (massless)

Goldstone bosons are degenerate with vacuum, it follows that

⇒ Goldstone bosons do not interact in the limit |q |→ 0.

• Low energy QCD is realized in the form of an effective field theory of weakly interacting

Goldstone bosons.

(Pions for Nf = 2; Pseudoscalar meson octet (π, K, K, η) for Nf = 3)

8. 6. Spontaneous Symmetry Breaking

• Another standard example of spontaneous symmetry breaking: Ferro-magnet

Spin system: Hamiltonian H = H0 +

i<j

Gij σi · σj

Invariant under rotational symmetry in3

(O(3) symmetry)

• Low temperature: Magnetization has non-zero expectation value

M = 0 , T = 0

preferred direction in space ⇒ O(3) symmetry is spontaneously broken (Nambu-

Goldstone realization).

Order parameter:

TcT

! M"

At high temperature T > Tc: O(3) symmetry restored in Wigner-Weyl realization.

84

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• Goldstone boson: Magnon “Spin wave”

in QCD: M ↔ Chiral (quark) condensation qq

• Chiral condensate qq is the order parameter of spontaneously broken chiral symmetry

in QCD.

8. 7. Chiral Condensate (Quark Condensate)

• “Perturbative” and “non-perturbative” vacuum

• Quark field operator:

ψ(x) =

d3p

(2π)32Ep

bpu(p)e−ip·x + d†

pv(p)eip·x

= ψ(+)(x) + ψ(−)(x)

(8.25)

ψ+(x) =

d3p

(2π)32Ep

b†pu

†(p)eip·x + dpv†(p)e−ip·x

(8.26)

⇒ Perturbative vacuum: bp |0 = 0, dp |0 = 0

⇒ Non-perturbative vacuum: |Ω: bp |Ω = 0, dp |Ω = 0

ψ(+)(x) |Ω = 0 , ψ(−)(x) |Ω = 0

• Wick’s theorem: T ψ(x)ψ(y) = : ψ(x)ψ(y) : “normal product”

+ 0|T ψ(x)ψ(y)|0 iSF (x, y)

• Definition of normal product:

: bpd†q :≡ −d†

qbp etc. (8.27)

In the perturbative vacuum:

0| : ψ(x)ψ(y) : |0 = 0

and the standard Feynman propagator is SF (x, y) = −i0|T ψ(x)ψ(y)|0.

In the non-perturbative vacuum:

Ω|T ψ(x)ψ(y)|Ω iSF (x, y)

= Ω| : ψ(x)ψ(y) : |Ω+ iSF (x, y)

85

• Definition of quark condensate:

ψψ = itr limy→x+

SF (x, y)− SF (x, y)

= −tr limy→x+

Ω| : ψ(x)ψ(y) : |Ω(8.28)

• For Nf = 2, flavor with ψ =

u

d

ψψ = uu+ dd ; qq with q = u, d

8. 8. Quark Condensate and Spontaneously Broken Chiral Symmetry

• Spontaneous breaking of chiral symmetry (Nambu-Goldstone realization) implies non-

trivial vacuum characterized by non-vanishing chiral condensate:

QAj |0 = 0 ⇔ ψψ = 0

• Sketch of proof: introduce Pj(x) = ψ(x)iγ5τj

2ψ(x)

Relation:

QA

j (t), Pk(x, t)

= − i

2δjkψ(x)ψ(x) (8.29)

Use: QAi (t) =

d

3x A0i (x, t);

ψα(x, t), ψ†

β(y, t)

= δαβδ3(x− y )

ψα(x, t), ψβ(y, t)

= 0

ψ†

α(x, t), ψβ(y, t)

= 0

(8.30)

• Take expectation value of (8.29):

0|QAj Pk − PkQ

Aj |0 = − i

2δjkψψ (8.31)

⇒ If QAj |0 = 0 ⇔ ψψ = 0

• Chiral condensate ψψ = ψRψL + ψLψR:

Order parameter of spontaneously broken chiral symmetry.

86

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8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

Page 7: CHAPTER VIII: Symmetries of QCD - T39 Physics Group · CHAPTER 8. SYMMETRIES OF QCD • QCD is based on local SU(3) c gauge symmetry • In addition: global symmetries 8.1. N¨other’s

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

ψψT,V = −tr limy→x+

DADψDψ ψ(x)ψ(y)e−SE(T,V )

DADψDψ e−SE(T,V )

(8.32)

• Euclidean action: SE(T, V ) =

β

0

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of ψψ

Critical temperature Tc 190 MeV ∼ ΛQCD

TcT

!! " "#!

mq$0 mq%0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq = 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p) quantum state of pion,

Normalization πi(p)|πj(p) = 2Epδij(2π)3δ3(p− p ) where Ep =

p 2 + m2.

0|Aµj (x)|πk(p) = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d = 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

– Combine with (8.29):

QA

1 (t), ψ(x, t)iγ5τ1

2ψ(x, t)

= − i

2ψ(x)ψ(x)

= − i

2

uu + dd

(8.35)

– Take expectation value:

0|QA

1 , ∂µA1µ

|0 = − i

2(mu + md) ψψ (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

=

3

j=1

d

3p

(2π)32Ep|πj(p)πj(p)| (8.37)

0|QAj (t = 0)|πk(p) = iδjk fπ Ep (2π)

3δ3(p )

and 0|∂µAjµ(x)|πk(p) = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf2

π = −1

2(mu + md) ψψ (8.38)

88

For mu + md 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

ψψ −(0.3 GeV)3

uu dd −(0.24 GeV)3 −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89

For mu + md 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

ψψ −(0.3 GeV)3

uu dd −(0.24 GeV)3 −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89

For mu + md 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

ψψ −(0.3 GeV)3

uu dd −(0.24 GeV)3 −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89

For mu + md 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

ψψ −(0.3 GeV)3

uu dd −(0.24 GeV)3 −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89

For mu + md 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

ψψ −(0.3 GeV)3

uu dd −(0.24 GeV)3 −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89