QCD IN THE LARGE NC LIMIT AND ITS
APPLICATIONS TO NUCLEAR PHYSICS
Alvaro Calle CordonJLab, Theory Center
27th Annual Hampton University Graduate Studies Program (HUGS 2012)
Thursday, June 14, 2012
• QCD in a nutshell
• Hydrogen atom in N dimensions
• Large Nc limit of QCD
• Mesons, baryons and its interactions
LECTURE I: FOUNDATIONS LECTURE II: APPLICATIONS
Bibliography
1. Baryons in the 1/N Expansion. E. Witten. Nucl. Phys. B160 (1979) 57.
2. Phenomenology of large Nc QCD. R. F. Lebed. Czech.J.Phys. 49 (1999) 1273-1306. e-Print: nucl-th/9810080.
3. Large N QCD. A. V. Manohar. Lecture Notes. e-Print: hep-ph/9802419.
4. Large Nc baryons. E. E. Jenkins. Ann. Rev. Nucl. Part. Sci. 48 (1998) 81-119. e-Print: hep-ph/9803349.
5. Aspects of Symmetry (chapter 8). Sidney Coleman (1988). Cambridge University Press.
• NN interaction
• Baryon masses
• Axial current
Thursday, June 14, 2012
Hadron spectrum & Quark ModelGell-Mann & Ne’eman, 1964
3⊗ 3⊗ 3 = 10⊕ 8⊕ 8⊕ 1 3⊗ 3 = 8⊕ 1
SUF(3) Flavor
predicted
Λ(1405)
q =
uds
Approximate symmetry
mu = md = ms
JP =1
2
+
JP =3
2
+
JP =1
2
−
JP = 0−
JP = 1−
Thursday, June 14, 2012
Hadron spectrum & Quark Model
Baryon octect Baryon Wave function
∆++ uuu∆+ (uud+ udu+ duu)/
√3
∆0 (udd+ dud+ ddu)/√3
∆− dddΣ∗+ (uus+ usu+ suu)/
√3
Σ∗0 (uds+ usd+ dus+ dsu+ sud+ sdu)/√6
Σ∗− (dds+ dsd+ sdd)/√3
Ξ∗0 (uss+ sus+ ssu)/√3
Ξ∗− (dss+ sds+ ssd)/√3
Ω− sss
ψ∆++ = ψ(space) ψ(spin) ψ(flavor) ψ(color)
∆++ = u ↑ u ↑ u ↑
S S S
Pauli’s principle
A A
Nc = 3
Particles in Nature are color singletsψ(color) = (rgb− rbg + gbr − grb+ brg − bgr) /√6
Thursday, June 14, 2012
Quantum Chromo Dynamics = gauge theory of the strong interactions
LQCD =
f
qf (iDµγµ −mf ) qf − 1
4F aµνF
a,µν
QCD in a nutshell
SUc(3) gauge group
qf =
qf,rqf,gqf,b
qf,α = Uαβ qf,β
U ≡ exp
−i θa
λa
2
Eight non-commuting generators
λa
2,λb
2
= ifabc λc
2F aµν = ∂µAa
ν − ∂νAaµ − gfabcAb
µAcν
Dµqf ≡ (∂µ + igAµ) qf Aµ = Aaµλa
2
Aµ(x) → Aµ(x)−1
e∂µθ(x)
Aaµ(x) → Aa
µ(x) +1
g∂µθ
a(x) + fabcAcµ(x)θ
b(x)
(QED)
(QCD)
Generators of the adjoint representation
Gluons are in the adjoint representation
g g g2
Fritzsch, Gell-Mann & Leutwyler f = u, d, s, c, b, t
Thursday, June 14, 2012
Asymptotic Freedom
QCD in a nutshell
QCD !!" !!#!$!%&''()!*!%&%%%+s Z
0.1
0.2
0.3
0.4
0.5
s (Q)
1 10 100Q [GeV]
Heavy Quarkoniae+e– AnnihilationDeep Inelastic Scattering
July 2009
David J. Gross, H. David Politzer & Frank Wilczek, 2004
αs ≡ g2/4π
αs(Q2) =
4π
(11− 23Nf ) log
Q2
Λ2
· · ·
Thursday, June 14, 2012
Non-perturbative methods
Chiral Perturbation Theory
Lattice QCD
Large Nc limit of QCD
Jo Dudek lectures next week !
Too much to say about for a two-hours lecture
I’ll try to give you an overview
Thursday, June 14, 2012
Chiral Perturbation Theory
1.0
0.5
!0, !±
K±,K0, K0"
a0, f0"!
#
$,%
n, p
!"0, "±#0, #±, #++
BaryonsMesons
Mass [GeV]
Effective Field Theory of QCD in the chiral limit
mu ∼ 2.0± 1.0 MeV mc ∼ 1.29± 0.11 GeVmd ∼ 4.5± 0.5 MeV 1 GeV mb ∼ 4.19± 0.18 GeVms ∼ 100± 30 MeV mc ∼ 172.9± 1.5 GeV
mu = md = ms = 0
L0QCD =
l=u,d,s
(qL,l i D qL,l + qR,l i D qR,l)
qL =1
2(1− γ5)q qR =
1
2(1 + γ5)q
chiral limit
SUc(3)
uL
uL
uL
dLdLdL
sLsLsL
SUf (3)
SUc(3)
uR
uR
uR
dRdRdR
sRsRsR
SUf (3)
SUL(3)× SUR(3)
Doublets of opposite parity ?
Spontaneous symmetry breaking Nambu-Goldstone bosons
Eight (three) massless pseudo scalar particles !Three (very) light ps mesonsEight light ps mesons
Thursday, June 14, 2012
Hydrogen atom in N-dimensions
+H =
p2
2m− α
r
α = 1/137 = 0.0073
Is perturbation theory applicable? NO !
r → rt, p → p/t, t ≡ 1/(mα2)
H → H
mα2≡ H =
p2
2− 1
r
The expansion parameter disappears, we can absorb the overall energy by redefining the temporal scale
A hidden expansion parameteris the space dimension
Hydrogen atom in N-dimensions with N very large
− 1
2m
∂2
∂r2+
N
r
∂
∂r
− α
r
Ψ = E Ψ
Ψ = r−N/2Ψ , r = N2RRescaling
H =1
N2
− 1
2mN2
∂2
∂R2+
1
8mR2− α
R
Meff = mN2 Veff (R) = 1/(8mR2)− α/R
r0
0.0 0.5 1.0 1.5 2.0!2.0!1.5!1.0!0.50.00.5
R
Veff!R"
To lowest order in 1/N E0 = Veff (r0)/N2 = −2mα2/N2
E0
N=3
= −2/9 mα2
E0
exact
= −1/2 mα2
Thursday, June 14, 2012
Large Nc limit of QCD
G. t’Hooft, 1974
SU(3) → SU(Nc)
t’Hooft double-line notation
Aaµb ∼ qaqb a
b
g
g
g g
g g2
g2
!m m
l l
l
m
k
Nc
g g
" g2Nc
t’Hooft limit
limNc→∞
g2Nc = constant
g ∼ 1/Nc
Thursday, June 14, 2012
Large Nc limit of QCD
!
!
!
Planar diagrams
Quark loops
!
!l
l
l
m m
m
g g
" g2
Non-planar diagrams
!
∼ O(1)
∼ 1/Nc
∼ 1/N2c
Planarity
Leading order Feynman diagrams in the large Nc limit are planar with a minimum number of quark loops
Thursday, June 14, 2012
Mesons in the large Nc limit
Large Nc meson
|1c =1√Nc
qc1 qc1 + · · ·+ qcNc
qcNc
j
j
kk
i
i
Basic assumption: QCD in the large Nc limit is a confining theory and quarks, anti-quarks and gluons must combine in order to give a colorless state.
only one color singlet state is allowed as intermediate stateqjA
jkA
ki q
i ∼ qq
(qkAkl q
l)(AjmAm
j ) ∼ qqg! ! =
n
=
n
f2n
k2 −M2n
O(Nc)
Mn ∼ O(1)
fn ∼ O(
Nc)
Mesons are stable and light
Thursday, June 14, 2012
Baryons in the large Nc limit
E. Witten, 1979
Nc quarks
antisymmetric color singlets
i1i2···iNcqi1qi2 . . . qiNc
current quark mass quark kinetic energy potential energy
HB = Nc m+Nc t+ V = Nc (m+ t+ v) = Nc h
V = N2c
1
Ncv
two-quarks interaction# of quark pairs
Interaction between quarks negligible
Potential felt by any individual quark
baryons are heavy
MB ∼ O(Nc)
∼ O(1/Nc)
∼ O(1)
Hartree picture: each quark move independently in background potential
ψB(x1, . . . , xN ) = ΠNci=1φ(xi)Baryons are heavy in the large Nc limit but
their size and shape are finite
Thursday, June 14, 2012
Meson and baryon interactions
!
"Nc
"Nc
"Nc
1/"Nc
Nc #
Meson decay
Meson-meson interactionΓmeson ∼ O(1/Nc)
Mesons are very narrow
!!
"Nc
"Nc
"Nc
"Nc
1/"Nc
1/"Nc
"Nc
"Nc
"Nc
"Nc
1/Nc
Meson-baryon interaction
Meson-baryon scattering
Mesons are free and non-interacting
Vmeson ∼ O(1/Nc)
BB
m
!
Meson-baryon coupling constant
gANc
Fπ∂iπ
aXia ∼ O(
Nc)
m m
B B !
Meson-baryon scattering
∼ O(1)
mesons are scattered by baryons
Witten’s counting rules
Thursday, June 14, 2012
Spin-Flavor symmetry
Gervais & Sakita, 1984Dashen & Manohar, 1993
consistency conditions
!(q, a) !(q!, b)
N (p) N (p!)
!(q, a) !(q!, b)
N (p) N (p!)
Pion-nucleon scattering amplitude is ∼ O(1)
∼ O(
Nc)but the pion-nucleon vertex is
cancellation between diagrams
+ ∝ N2c g
2A
F 2π
Xia, Xjb
= O(Nc0)
spin-flavor symmetrySU(2Nf )
u u Spin
u d Flavor
u d Spin-Flavor
J = I =1
2,3
2, . . . ,
Nc
2
B =
N∆
∆5/2...
∆Nc/2
Thursday, June 14, 2012
Baryon-baryon interaction
· · · · · ·
B B
BB
m ! ∼ O(Nc)
Spin-flavor structure of the NN potential
Kaplan, Savage & Manohar, 1996
V (r) = VC(r) + (σ1 · σ2)(τ1 · τ2)WS(r) + (τ1 · τ2)WT (r)S12 ∼ Nc
Box and cross box diagram cancellations
Banerjee, Cohen & Gelman, 2002
OBE potential at leading Nc ACC & E. Ruiz Arriola, 2010
Thursday, June 14, 2012
1/Nc - Chiral Perturbation Theory
1.11.21.31.41.51.61.71.81.9
100 200 300 400 500 600 700 800m
[GeV
]M [MeV]
PACS-CS Lattice1/Nc-ChPT (Fit I)
1/Nc-ChPT (Fit II)
Nucleon and Delta masses
0.91
1.11.21.31.41.51.6
100 200 300 400 500 600 700 800
mN
[GeV
]
M [MeV]
PACS-CS Lattice1/Nc-ChPT (Fit I)
1/Nc-ChPT (Fit II)
ACC & Goity, 2012
Thursday, June 14, 2012
1/Nc - Chiral Perturbation TheoryAxial current gA
ACC & Goity, 2012
+ +
Exact cancellation in the large Nc limit and almost exact for Nc=3
All the pion mass dependence of gA is dominated by the tad-
pole diagram and and the counter-terms
!
!
!
!!
!!
!
!
!! ! ! !
!
"
""
"
""
"
"
"
""
"
"
"
""
"
200 300 400 500 600 7001.0
1.1
1.2
1.3
1.4
M!!MeV"
g A
The result is a mild dependence of gA with the pion mass
Thursday, June 14, 2012
Summary
Large Nc limit is a fundamental feature of QCD
It is not restricted to low or high energies, so it may be considered a non-perturbative method to solve QCD at low-energies
There are a lot of situations where quantities in the large Nc limit are not far from the real world Nc=3
The combination of the large Nc limit & ChPT looks promising to explain recent lattice QCD results for hadronic quantities
Thursday, June 14, 2012
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