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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)
77
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)
78
CHAPTER VIII: Symmetries of QCD
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.
80
• 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)
82
• 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
• 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
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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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
dτ
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
2ψ
– 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