QCD IN THE LARGE NC LIMIT AND ITS APPLICATIONS TO …C LIMIT AND ITS APPLICATIONS TO NUCLEAR PHYSICS...
Transcript of QCD IN THE LARGE NC LIMIT AND ITS APPLICATIONS TO …C LIMIT AND ITS APPLICATIONS TO NUCLEAR PHYSICS...
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