Mass, 0 Instantons, the Limit c N and the LargeInstantons, the ·0 Mass, and the Large Nc Limit...
Transcript of Mass, 0 Instantons, the Limit c N and the LargeInstantons, the ·0 Mass, and the Large Nc Limit...
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Instantons, the η′ Mass,
and the Large Nc Limit
Thomas Schaefer
North Carolina State
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Outline
• Introduction: Why Instantons?
U(1)A puzzle, topology, and instantons
• Instantons, the η′ meson, and the Witten-Veneziano relation: large
baryon density
ρ¿ Λ−1QCD: semi-classical approximation under control
• Instantons and the large Nc limit
smooth large Nc limit? Witten-Veneziano relation?
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U(1)A Puzzle
• QCD has a U(1)A symmetry ψL → eiφψL, ψR → e−iφψR
which is spontaneously broken 〈ψLψR〉 = Σ 6= 0
• Goldstone Theorem: massless Goldstone boson mη′ → 0 (mq → 0)
But: η′ is heavy, m2η′ À mqΛQCD
• Resolution: axial anomaly
Aµ
g(
g(
ε
ε
µ
ν
a
a )
)
∂µAµ =
Nf
16π2GaµνG
aµν
But: RHS is a total divergence GG ∼ ∂µKµ
⇒ Topology is important
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Topology in QCD
• classical potential is periodic in variable X
X∆X = +1
X = −1
V
∆
X =
∫
d3xK0(x, t)
∂µKµ =1
32π2GaµνG
aµν
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• classical minima correspond to pure gauge configurations
Ai(x) = iU †(x)∂iU(x)
E2 = B2 = 0
• classical tunneling paths: Instantons
r
X=0 τX=1
E + B2 2
Aaµ(x) = 2ηaµνxνx2 + ρ2
,
GaµνGaµν =
192ρ4
(x2 + ρ2)4.
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• Dirac spectrum in the field of an instanton
t
ε
R
R
L
L R
L
axial charge
violation:
∆QA = 2
• Instanton induced quark interaction (Nf = 2)
u L
d L d R
u R
L = G detf(ψL,fψR,g)
G =
∫
dρn(ρ)
violates U(1)A but
preserves SU(2)L,R
. . . and contributes to
the η′ mass
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• Tunneling rate (barrier penetration factor)
?
ρ
ρ
fluctuations large
n( )
fluctuations small
n(ρ) ∼ exp
[
−8π2
g2(ρ)
]
∼ ρb−5
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QCD at large Nc
• QCD (m = 0) is a parameter free theory. Very beautiful.
But: No expansion parameter
• ’t Hooft: Consider Nc →∞ and use 1/Nc as a small parameter
Nc →∞ ⇒ classical master field
• keep ΛQCD fixed ⇒ g2Nc = const
g
g
Nc
• Could the master field be a multi-instanton?
Witten : No! dn ∼ exp(
− 1g2
)
∼ exp (−Nc)
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U(1)A anomaly at large Nc
• consider θ term L = ig2θ32π2G
aµνG
aµν
• no θ dependence in perturbation theory.
Witten:non-perturbative θ
dependenceχtop =
d2Edθ2
∣
∣
∣
θ=0∼ O(1)
• massless quarks: topological charge screening limm→0 χtop = 0
• How can that happen? Fermion loops are suppressed!
g GG2 ~g GG2 ~
g GG2 ~
g N 8 3
c1
cN
g GG2 ~
g N 14 2
c
Witten:
η′ has to become light
f2πm2η′ = 2Nf χtop(no quarks)
⇒ m2η′ = O(1/Nc)
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Witten-Veneziano relation “works”
χtop ' (200MeV)4 (quenched lattice) ⇒ mη′ ' 900MeV
Where does χtop 6= 0 come from?
Instantons
• large Nc limit?
• Witten-Veneziano relation?
• . . .
???
• topology?
• semi-classical limit?
• charge screening?
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Digression:
Instantons and the mass of the η′:
Large Baryon Density
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QCD at large density (Nf = 2)
• schematic phase diagram
T
µ
<qq> <qq>
diquark condensate breaks
U(1)B and U(1)A
〈qLqL〉 = ρ ei(χ+φ)/2
〈qRqR〉 = ρ ei(χ−φ)/2
• effective lagrangian for U(1)A Goldstone boson
φ
ρ
L = f2
2
[
(∂0φ)2 − v2(∂iφ)
2]
− V (φ+ θ) + L(ρ, χ)
V (φ+ θ) vanishes in perturbation theory12
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η′ mass at large baryon density (Nc = Nf = 2)
• instanton induced effective interaction for quarks with p ∼ pF
u L
d L d R
L/R
u R
−1q
qn(ρ, µ) = n(ρ, 0) exp
[
−Nfρ2µ2
]
ρ ∼ µ−1 ¿ Λ−1QCD
• instanton contribution to vacuum energy
∆
L
L R
R
∆ GI
〈L〉 = A cos(φ+ θ)
A = CNΦ2[
log(
µΛ
)]4(
Λµ
)8
Λ−2
• η′ mass satisfies “Witten-Veneziano” relation f2m2φ = A
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• very dilute instanton gas
RD ρ¿ rIA ¿ RD
ρ ∼ µ−1
rIA = A1/4
RD = m−1φ
• A is the local topological susceptibility
A = χtop(V ) =〈Q2
top〉V
V r4IA ¿ V ¿ R4D
• Global topological susceptibility vanishes
χtop = limV→∞〈Q2
top〉V
V = 0 (m = 0)
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500 1000 1500 2000 2500 µ [MeV]
10
50
100
200
300400500
mP [
MeV
]
mP (inst)
mP (conf) [arb.sc.]
2∆Λ
conf [arb.sc.]
• extrapolate to zero density(
NV
)
∼ 1 fm−4 m′η ∼ 800MeV
• Instantons predict density dependence of mη′
can be checked on the lattice
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Main Course
Instantons and the mass of the η′:
Large Number of Colors
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Instantons at large Nc
• semi-classical ensemble of instantons at large Nc
color
space
coordinate space
instantons are Nc = 2
configurations(
NV
)
= O(Nc)
⇒ εvac = O(bNc) = O(N2c )
• instantons are semi-classical
ρ ' ρ∗ = O(1) Sinst = O(Nc)
• density dn ∼ exp(−Sinst) = O(exp(−Nc))?
NO ! large entropy dn ∼ exp(+Nc)
• topological susceptibility χtop ' (N/V ) = O(Nc)?
NO ! fluctuations suppressed χtop = O(1)
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Digression: N = 4 SUSY Yang Mills
• string/field theory duality (Maldacena)
N = 4 SUSYYM ⇔ IIB strings on AdS5 × S5λ = g2N →∞ ⇔ (ls/R)
4 → 0
(g2 → 0) (gs → 0)
• string theory contains D-instantons characterized by location on
AdS5 × S5 ⇔ field theory instantons∫
d4x∫
dρρ5
∫
dΛab
AdS5 × S5
• k instanton amplitudes
(AdS5 × S5)k → (AdS5 × S5)
k instantons in commuting SU(2)′s
(bound by fermions)
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Instanton ensemble
instanton ensemble described by partition function
Z = 1NI !NA!
∏NI+NA
I
∫
[dΩI n(ρI)]
× exp(−Sint)
n(ρ) = CNc
(
8π2
g2
)2Nc
ρ−5 exp[
− 8π2
g(ρ)2
]
CNc= 0.47 exp(−1.68Nc)
(Nc−1)!(Nc−2)!8π2
g2(ρ) = −b log(ρΛ), b = 113 Nc
Sint = −32π2
g2 |u|2
ρ2Iρ2A
R4IA
(
1− 4 cos2 θ)
+ Score
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• complicated ensemble, size distribution
n(ρ) ∼
exp(−Nc) ρ < ρ∗
const ρ ∼ ρ∗ρ∗ ∼ O(1)
• total density determined by interactions
S(1− body) ∼ S(2− body)
Nc ∼ Nc × 1Nc
×(
NV
)
classical ∼ classical × color overlap × density
• conclude
(
N
V
)
= O(Nc)
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instanton size distribution
0 0.1 0.2 0.3 0.4 0.5
ρ [Λ−1]
0.0
5.0×104
1.0×105
n(ρ)
Nc=3Nc=4Nc=5Nc=6
0 2 4 6 8 10 12ρ
0
1
2
3
D(ρ
)
SU(2)SU(3)SU(4)
B. Lucini, M. Teper
fluctuations in N are 1/Nc suppressed
〈N2〉 − 〈N〉2 = 4b 〈N〉 ∼ O(1) (not O(Nc)!)
also true for topological charge
〈Q2〉 = 4b−r(b−4) 〈N〉 ∼ O(1)
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global observables: (N/V ), 〈qq〉, χtop
0.1 0.2 0.31/Nc
0
1
2
3
4χ to
p , (N
/V)
> [Λ
n ](N/V)<qq>χtop
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meson correlation functions (π, ρ, η′)
0 0.2 0.4 0.6 0.8 1 1.2
x [Λ-1]
0.1
1
10
Π (
x) /Π
0(x)
Nc=3
Nc=4
Nc=5
Nc=6
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meson masses: m2π,m
2ρ ∼ 1, m2
η′ ∼ 1/Nc
2 3 4 5 6 7N
c
0
0.2
0.4
0.6
m2 [
GeV
2 ] mπmρmη’
Note: m2η′ ∼ 1/Nc even though (N/V ) ∼ Nc
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Nucleon Spin
• polarized DIS implies large OZI violation
g Ad
g Au
g Au,d
g0A ' 0.25
g8A ' 0.65
• instanton effects: anomaly, uL → uRdRuL vertex
L
R
R
L
L/R L/R L/R
L/RL/R
I A
I
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Numerical Study
0.2 0.25 0.3 0.35 0.4ρ [fm]
0.6
0.8
1
1.2
1.4
1.6
g A
gA
3
gA
0 (con)
gA
0 (tot)
g3A ' 1.25 agrees with experiment
g0A ' 0.75 too large
Very little OZI violation
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Light Scalars
light scalars can be interpreted as (qq)(qq) states (Jaffe)
explains “inverted” mass spectrum and strong coupling to (ππ), (KK)
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scalar correlation functions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7x [fm]
-0.5
0
0.5
1
1.5
2
2.5
Π(x
)/Π
0(x)
Nc=3
Nc=4
Nc=5
Nc=6
(I=0)
a (I=1)
σ
0
mσ → 1 GeV as Nc →∞
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Tetraquark-meson (qq)(qq)− (qq) mixing
0 0.2 0.4 0.6x [fm]
-60
-40
-20
0
Πm
ix(x
)/Π
0 σ(x)
Nc=3
Nc=4
Nc=5
Nc=6
σ becomes “normal” (qq) scalar as Nc →∞
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Summary
• nice example for semi-classical instanton liquid: QCD at large
baryon density.
can be studied on the lattice
• instanton liquid can have a smooth large Nc limit(
NV
)
= O(Nc), χtop = O(1), m2η′ = O(1/Nc)
• smooth large Nc limit does not imply that Nc = 3 is similar to
Nc =∞ in all regards
η′, scalar mesons, nucleon spin and strangeness
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