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


http://inspirehep.net/record/478677


http://inspirehep.net/author/Lebed%2C%20Richard%20F.?recid=478677&ln=en

http://inspirehep.net/author/Lebed%2C%20Richard%20F.?recid=478677&ln=en



http://arxiv.org/find/hep-ph/1/au:+Manohar_A/0/1/0/all/0/1

http://arxiv.org/find/hep-ph/1/au:+Manohar_A/0/1/0/all/0/1



http://inspirehep.net/author/Jenkins%2C%20Elizabeth%20Ellen?recid=468160&ln=en

http://inspirehep.net/author/Jenkins%2C%20Elizabeth%20Ellen?recid=468160&ln=en

http://books.google.com/books?id=PX2Al8LE9FkC

http://books.google.com/books?id=PX2Al8LE9FkC

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−


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


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


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

· · ·


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


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


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



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



!

!

!

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


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


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


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


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


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


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


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


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


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

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