Transcript The Baryon Resonance Spectrum and the 1/Nc Expansion Richard Lebed Jefferson Lab July, 2009

The Baryon Resonance Spectrum and
the 1/Nc Expansion
Richard Lebed
Jefferson Lab
July, 2009
A Tom Cohen (TDC)/ Rich Lebed (RFL) Joint

TDC, RFL
PRL 91, 012001 (2003)

TDC, DCD, Daniel R. Martin, RFL
PRD 71, 076010 (2005)

TDC, RFL
PRD 67, 096008 (2003)

TDC, RFL, PLB 619, 115 (2005)

TDC, RFL
PRD 72, 056001 (2005)

RFL, PLB 369, 68 (2006)

TDC, RFL
PRD 68, 056003 (2003)

TDC, RFL
PLB 578, 150 (2004)

TDC, RFL
PRD 74, 036001 (2006)

TDC, Daniel C. Dakin (DCD),
Abhinav Nellore (AN), RFL
PRD 69, 056001 (2004)

TDC, RFL
PRD 74, 056006 (2006)

TDC, DCD, AN, RFL
PRD 70, 056004 (2004)

Herry J. Kwee (HJK), RFL
PRD 75, 016002 (2007)

TDC, RFL
PRD 70, 096015 (2004)

HJK, RFL, JHEP 0710, 046 (2007)

HJK, RFL, J. Phys. A41,015206
(2008)
What do we really know about baryon
resonances?
Most authoritative book available?
Thousands of pages,
but not a single mention
of baryon resonances!
Most authoritative book available
Baryon resonances fill
(as opposed to lightest
N, Λ, Σ, Ξ, Ω, or heavy-quark
baryons)
116 out of 1389 pages
Are they orbital and radial
excitations of bags of quarks?
Ground state
Excited state
Are they nucleon/meson bound
states?
N
π
Resonances are poles in complexvalued scattering amplitudes
10−23 s
Resonances are poles in complexvalued scattering amplitudes
10−23 s
Resonances are poles in complexvalued scattering amplitudes
10−23 s
Outline
1) Hadrons in large NC QCD
2) Inspiration: Chiral solitons
3) The K quantum number
4) Degenerate resonance multiplets
5) Evidence from decay modes
6) 1/NC corrections
Just the Facts, Ma’am



Large NC QCD means:
•
•
QCD gauge group is enlarged from SU(3) to SU(NC)
Fundamental representation (under which quarks transform) has NC
states, labeled by values of the color quantum number
It is a well-defined gauge theory limit when αs ~ 1/NC [’t Hooft (1974)]
•
•
Meaningful to perform a 1/NC expansion
Color singlets formed from qq pairs (color structure δαβ)
or from NC quarks (color structure εα1,α2,…, αNc)
Mesons at large NC are:
•
•
•
free
stable
scatter weakly
[O(NC0) masses]
[O(1/NC) widths]
[O(1/NC2) cross sections] [Veneziano (1976)]
“Classical” Large NC Baryons
[Witten (1979)]


Baryon masses in large NC are O(NC1) (NC quarks!)
•
Much heavier than mesons  Treat semiclassically
Ground-state band assumed to have totally symmetric spin ×
flavor wave function, each quark in an orbital s wave
•
•
•
•
Supported by phenomenology: For NC = 3 these states fill a single
symmetric 56 of spin × flavor SU(6): N, Δ, Σ, Ω, etc.
Meson-baryon trilinear coupling scales as NC1/2
Meson-baryon scattering amplitude scales as NC0
Combinatorics of the NC quarks plus ’t Hooft scaling (αs ~ 1/NC)
gives O(NC0) potential energy per quark
Hartree-Fock approximation holds; size of baryon scales as O(NC0)
Summary of Hadrons in 1/NC

If NC is considered large, mesons exist and live long
enough to be detected
•


Fact: Unflavored mesons seen all the way up to 2 GeV and
beyond; if their lifetimes were sufficiently short, only
collections of π’s would be seen
Baryons have NC quarks but don’t grow in size
with NC
Meson-baryon scattering amplitudes don’t grow or
shrink with NC
Dialogue Concerning the Two
Chief Large NC Baryon Systems
Quarks!
QUARKS!
jerk
Chiral Solitons!
SOLITONS!
jerk
Quark vs. Soliton Picture

Quark picture [Dashen, Jenkins, Manohar; Carone, Georgi,
Osofsky; Luty, March-Russell (1994)]:
•
•
Does not assume any particular quark model
Baryon wave function composed entirely of NC colorfundamental representation interpolating fields (well-defined
“constituent quarks” subsuming gluons, sea quarks)—
“quarks”, for short
A.J. Buchmann and RFL,
PRD 62, 096005 (2000)
Studying baryons: Quark picture
Dashen, Jenkins & Manohar; Carone, Georgi & Osofsky; Luty & MarchRussell (1994)



Consider processes that involve the (entangled) interaction of n
quarks → Represented by n-body operators:
2-body
3-body
Generic n-body operators are suppressed by (1/NC)n
From the operators can construct a Hamiltonian for baryon that
is perturbative in powers of 1/NC [Effective theory]
Example: Baryon masses
Jenkins & RFL [1995]




For baryons, there are known perturbative parameters in
addition to 1/NC: e.g., the s quark mass is small compared to
typical gluon binding energies, leading to “strangeness”
expansion parameter ε ≈ 0.3
Consider baryon masses, taking into account the 1/NC and ε
dependence of each n-body operator in the Hamiltonian
To each such operator exists a unique combination of baryon
masses, all of which are measured
Ask: Does the operator whose coefficient is, e.g., ε/(NC)3 lead to
a mass combination of relative size (0.3)/33 ≈ 0.01?
Jenkins & RFL (1995)
Mass
difference
quotient
Success!

The mass spectrum gives direct and strong evidence that the
1/NC expansion works for baryons, even for NC as small as 3!

Similar analysis shows that this success persists for magnetic
moments, charge radii, axial-vector couplings, etc. for the
ground-state baryons (p, n, Λ, etc.). These baryons are either
stable or tend to have long (> 10–10 sec) lifetimes

But what about the baryon resonances, the N*’s, which have
the shortest lifetimes allowed by the Heisenberg uncertainty
principle (≈10–23 sec)?
Excited Baryons in Quark Picture

May try this method and assume that the first excited band of
baryons is the large NC analogue of the NC = 3 SU(6) 70: A
single l = 1 quark and a symmetrized “core” of NC –1 quarks
[Carlson, Carone, Goity, RFL (1998-9); Goity, Schat, Scoccola (2002)]

In practice, one finds that, as expected, the largest operator
coefficients are all of expected size, but most are much smaller
•

The 1/NC expansion holds, but additional dynamical suppressions
are at work
But is this an adequate physical picture for baryon resonances?
Quark Picture vs. Soliton
Picture

Soliton Picture [Adkins, Nappi, Witten; Hayashi, Eckart,
Holzwarth, Walliser (1983-4)]:
•
Treats baryons as solitons (i.e., non-dissipating “lumps” of
energy) in chiral Lagrangians
•
•
•
Automatically allows correct couplings to chiral π, K, η
Known to be compatible with large NC (esp. Skyrme model)
Basic field configuration: hedgehog

f2
L
Tr  U  U   , U 0 (r )  exp(iF (r )rˆ ˆ)
4
Breaks both I and J but preserves their vector sum K

•

Calculating in Soliton Models


Accomplished by choosing particular “profile function” F(r), and
which higher-derivative terms to retain in chiral Lagrangian
Example: In the original Skyrme model (1961-2!) corresponds
to keeping just one particular 4-derivative term:
L   TrU  U , U  U 
1
4



2


2
Choosing F(0) = –π, F(∞) = 0, gives the Skyrmion a topological
charge, and is a fermion for NC odd [Witten (1983)]
Chiral soliton model calculations became a cottage industry in
the mid-1980s [Review: Zahed & Brown (1986)]
Choice of F(r), higher-order terms in L are model dependent
Why Should Quark and Soliton
Pictures Have Anything in Common?

Quark Picture: Assumes ground-state band of baryons in spin
× flavor symmetric multiplet (“56”)

Soliton Picture: Assumes ground-state has maximal-symmetry
hedgehog configuration

In the strict large NC limit, purely group-theoretical matrix
elements in either picture for ground-state baryons are equal
[Manohar (1984)]
Which Picture Works Better for
Baryon Resonances?

Resonances by definition are unstable against strong decay

A Hamiltonian H (quark picture) suggests free asymptotic states

In limit of suppressed

But how to use the 1/NC approach when pair creation is
common?

Need a method that treats resonances as poles in mesonbaryon scattering amplitudes
•

Real part → mass
qq pair creation, using H should be OK
Imaginary part → width
Chiral solitons naturally understand meson-baryon coupling
The K Quantum Number

Chiral soliton models, which treat baryons as heavy,
semiclassical objects, are consistent with large NC
(Adkins, Nappi, Witten, 1983-4)

The K quantum number indicates the underlying soliton state,
but by itself does not give a single eigenstate of I or J
Take appropriate linear combinations to form baryon state
Clebsch Clebsch Clebsch Clebsch

Mesons with well-defined I and J are coupled to baryon state to
form scattering amplitude (Hayashi et al., Mattis et al., 1980’s)
Clebsch Clebsch Clebsch Clebsch
What Does K Have to Do with
Large NC?

Scattering with underlying K conservation (s channel) is
equivalent to the t-channel rule It = Jt (Mattis & Mukerjee, 1986)
Clesbch Clebsch Clebsch Clebsch

The It = Jt rule follows directly from large NC “consistency
conditions” (Kaplan & Manohar, 1997)
•

The consistency conditions (order-by-order in NC unitarity in
meson-baryon scattering) follow from ’t Hooft-Witten NC power
counting (Dashen, Jenkins, Manohar, 1993-4)
K-amplitudes form a correct large NC description of mesonbaryon scattering (TDC & RFL, 2003) ☺
Master Scattering Expression
(Two Flavors)




Meson (spin s, isospin i, in Lth partial wave) scattering on
baryon (spin = isospin R)
Unprimed = initial state, primed = final state
[X] ≡ 2X+1
τ: “Reduced amplitude” [Mattis & Peskin (1985), Mattis (1986)]
Degenerate Resonance Multiplets







A resonance of given I, J appears as a pole in some particular τ
But τ depends only upon K and L
Resonances have no memory of the L used to create them
(except by restricting total J)
Resonances in large NC are labeled solely by K
Particular reduced K amplitudes τ appear in multiple partial
waves labeled by I, J, L
The same resonant pole, same mass and width, appears in
multiple partial waves: Degenerate resonance multiplets
Linear relations among partial waves hold at all energies
Sample Amplitude Results
K=1
K=0
Phenomenological Evidence

Evidence from the baryon resonance spectrum is ambiguous
since 1/NC corrections so large—almost all light-quark
resonances lie in a range of only a few hundred MeV

Look at decay modes instead: e.g., negative-parity N1/2 states
•
•

K=0 amplitude couples to ηN and not πN
K=1 amplitude couples to πN and not ηN
Particle Data Book: One finds two such prominent resonances,
•
•
N(1535)→ ηN (30-55%), πN (35-55%)
N(1650)→ πN (55-90%), ηN (3-10%)
[TDC & RFL (2003)]
mη+mN=1490 MeV
3-Flavor Results

Amplitude relations also occur for 3 flavors, at the price of more
intricate group theory
•
•

First derivation: Mattis & Mukerjee (1988);
Corrected and expanded: TDC & RFL (2004)
Examples:
•
•
Λ(1670) only 5 MeV above ηΛ threshold but has 10-25%
branching ratio to this channel [partner to N(1535)]
Coupling of Λ(1520) D03 to K N is about 4-5 times [O(NC)] larger
than that to πΣ [Large NC SU(3) selection rule, TDC & RFL (2005)]
Synthesis of Quark and Soliton
Pictures

Fact: At large NC, every multiplet of the old-fashioned
SU(2NF)×O(3) quark model (NF = # of light flavors) is a
collection of complete K-multiplets of states: Compatibility
[TDC & RFL (2003, 2005)]

e.g., the multiplet that for NC=3 is (70,1–) contains all states with
a pole in K=0,¹⁄2 1, ³⁄2, 2 → 5 distinct multiplets

May explain why the old quark model works somewhat well for
explaining resonance spectroscopy but also demands lots of
experimentally absent states

Nothing in large NC prevents states in different quark model
multiplets from mixing: Configuration Mixing
[TDC, DCD, AN, RFL (2004)]
“The Story So Far:”

“In the beginning the
Universe was created
[with only three color
charges].”

“This has made a lot of
people angry and been
widely regarded as a
bad move.”
1/NC Corrections: Theory

Just as the leading 2-flavor meson-baryon scattering
amplitudes have It = Jt , those with | It – Jt | = n are suppressed
by 1/NCn [TDC, CDC, AN, RFL (2004); see also Dashen, Jenkins, Manohar
(1995), Kaplan & Savage (1996), Kaplan & Manohar (1997)]

The same statements (as well as Yt = 0) hold true for the 3flavor case [RFL (2006)]

How to use this in practice:
•
•
•
•
Write the master scattering expression in the t channel
Identify It and Jt entries, which are equal at leading order
Copy scattering expression, allowing all terms with | It – Jt | = 1,
suppressed by 1/NC
Obtain amplitude relations holding at leading and first subleading
order
1/NC Corrections: Applications

Amplitude relations among πN→πΔ mixed partial waves
•
•
Not enough πN→ πN amplitudes to eliminate all first-order
corrections!)
Results: Amplitude relations with 1/NC2 corrections really are about
a factor 3 better than those with 1/NC corrections
[TDC, CDC, AN, RFL (2004)]

Similar amplitude relations for γN→ πN
[TDC, CDC, RFL, DRM (2005)]

Similar amplitude relations for eN → eπN
[RFL, Lang Yu (arXiv:0905.0122)]
E2− multipole amplitudes
A few more for the road


Spurious states
•
Chiral limit vs. Large NC limit
•
•

Large-NC states that do not occur for NC > 3 are always seen to
decouple from observables when NC  3 [TDC & RFL (2006)]
mπ = 0 in the chiral limit, mΔ – mN = O(1/NC), so instability of Δ
shows NC = 3 universe closer to chiral than large NC limit
But even intrinsically chiral-limit observables like πN scattering
lengths show evidence of a 1/NC expansion [TDC & RFL (2006)]
πN→ππN
•
Experimental evidence exists for the large NC expectation that
πN→ππN amplitudes are dominated by πN→πΔ, ρN, ωN at
leading NC order [Kwee & RFL (2007)]
Summary and Prospects (I)
1)
Baryons are tractable in the 1/NC QCD expansion because
they contain NC quarks, so that the independent operators
that act upon them form a hierarchy in powers of 1/NC
2)
Experimental evidence for ground-state baryons strongly
supports—indeed, requires—the 1/NC expansion
3)
Large NC QCD provides a clear method to incorporate even
the most evanescent of baryon resonances
4)
They appear in multiplets degenerate in mass and width,
labeled by a quantum number K originally introduced in chiral
soliton models
Summary and Prospects (II)
5)
NC = 3 data provides strong evidence for this soliton-inspired
approach through decay branching ratios and amplitude
relations
6)
It is now known how to include 1/NC corrections, study 3flavor processes, remove NC>3 spurious states, include chiral
physics, and study multi-meson decays and other processes
Now one has effective field theory methods for analyzing
processes with both stable (under strong decay) baryons and
baryon resonances
So What Next?
Crunch time: Global numerical analysis of multiple processes,
including 1/NC corrections
Convince the Baryon Resonance Analysis Group to try out some of
this analysis!
With sufficient time and effort, we can come to understand this
mysterious 116 pages of the Particle Data Book, inspired by old
ideas used in a new way
Building the Quark Picture
A.J. Buchmann and RFL, PRD 62, 096005 (2000)
Calculating in the Quark Picture


Baryons are classified according to spin × flavor of quark wave
functions → can be analyzed using Hamiltonian with operators
carrying particular spin × flavor transformation properties
An operator acting upon n quarks (n-body operator) generically
suppressed by 1/NCn: a perturbative expansion
H = c0 NC 1 + c1(8) NC0 T8 + cJ J2/ NC + …
where
T8
=
 q

quarks



8
2
q
,
J2
=
  i   i 
 q
q   q
q 

2
2
  


ck: dimensionless coefficients (× ΛQCD), should be of order unity
Easy to include SU(3), isospin breaking: e.g., c1(8)  εc1, ε ≈ 0.3
Most incisive test: Form linear combinations of observables
proportional to a single ck /c0; do they really turn out to be O(1)?
Consistency Conditions
[Dashen, Jenkins, Manohar (1994)]

Consider pion-baryon scattering:

Each vertex is O(NC1/2), but the whole process is O(NC0)!

Necessary cancellations only possible if intermediate baryon
can have spin 1/2, 3/2, …, NC/2 and is degenerate in mass at
leading order  “Contracted symmetry” that generates the
whole SU(6) “56”
3-Flavor Master Scattering
Expression

First derivation, Mattis & Mukerjee (1988); corrected and
expanded by TDC & RFL (2004)
A Property of SU(3) ClesbchGordan Coefficients (CGC)

Large NC baryon SU(3)
representations are of the form (p,q)
= [O(NC0), O(NC1)]

Meson representations as usual:
(p,q) = [O(NC0), O(NC0)]

Theorem [TDC & RFL (2005)]:
SU(3) CGC for
(baryon 1) + (meson) → (baryon 2)
can be O(NC0) only if
Ymeson = YB2(max) – YB1(max)
3-Flavor Phenomenology

Extending to SU(3) multiplets provides new decay
selection rules [TDC & RFL (2005)]

e.g., the N(1535) [K=0] pole fills an SU(3) 8 (for NC=3) that is
“η-philic” and “π-phobic”

while the N(1650) [K=1] pole fills an SU(3) 8 (for NC=3) that is
“π-philic” and “η-phobic”

Empirical example: Λ(1670) only 5 MeV above ηΛ threshold but
has 10-25% branching ratio to this channel
3-Flavor Phenomenology

The 3-flavor theorem provides new decay selection rules
[TDC & RFL (2005)]

e.g., Λ resonances in an SU(3) 8 (for NC=3), prefer (by a factor
NC) πΣ decays to K N

while Λ resonances in an SU(3) 1 (for NC=3), prefer (by a factor
NC) K N decays to πΣ

Empirical example: Coupling constant of Λ(1520) D03 to K N is
about 4-5 times [O(NC)] larger than that to πΣ when threshold
p2L+1 taken into account
Electric Multipole Relations
Magnetic Multipole Relations
Example Multipole Relation

L = 2 magnetic multipole relation:
 1
3  m , p (  ) n
m , n (  ) p 
M
M
 M 5
M 5
  O 2
2
,
2
,
2
,
2
,
2
NC

2
2




ContainsD13 (1520)
ContainsD15 (1675)
m , p (  ) n
3
2, 2,
2

m , n (  ) p
3
2, 2,
2



Evaluate these amplitudes on resonance:
LHS  RHS
1
LHS  RHS
2

18.2  8.5103 GeV 1

140.2 103 GeV 1
 1
 0.13  0.06  O 2
 NC



Decoupling Spurious States
Which states are artifacts of NC>3? [TDC & RFL (2006)]

The old quark model multiplets do not give the answer
(configuration mixing is possible in general)

States may be considered “spurious” in meson-baryon
scattering only if:
•
•
•
They appear in an SU(3) multiplet not accessible through
scattering of NC=3 mesons and baryons
They appear in a part of an SU(3) multiplet not accessible through
scattering of NC=3 mesons and baryons (large strangeness)
They require an initial ground-state baryon with spin > ³⁄2 (not
available for NC=3)
Decoupling Spurious States

For the first two categories: The SU(3) CGC are all proportional
to 1–3/NC
[TDC & RFL (2006)]

Can be shown to work even if SU(3) completely broken to
SU(2)×U(1)
The Chiral and 1/Nc Limits

mπ = 0 in the chiral limit, Δ ≡ mΔ – mN = O(1/NC)

Then is δ ≡ mπ /Δ large or small?
•
•

[TDC & RFL (2006)]
Expt: δ ≈ 1/2
Noncommutative limits
Look at baryon quantities that are more sensitive to chiral limit:
Scattering lengths
•
One of the first chiral Lagrangian calculations: Peccei (1968)

Convergence of chiral expansion much more rapid than 1/NC
limit, but 1/NC corrections also discernable

Similarly for other chiral results, e.g., Adler-Weisberger sum rule
πN→ππN



NC power counting: Amplitudes with the smallest number of
hadrons dominate
πN→ππN amplitudes given by πN→πΔ, ρN, ωN at leading NC
order [Kwee & RFL (2007)]
For such channels, the PDG gives large amounts of data with
large uncertainties
•


Tabulated branching fractions such as “10-50%” are not uncommon
Predicted multiplets of resonant poles in general agreement with
data, but analysis performed only at leading NC order (1/NC
corrections not yet computed)
Most incisive test of pole structure: Mixed partial-wave
amplitudes (e.g., πN→πΔ SD11 is pure K=1)