The Baryon Resonance Spectrum and the 1/Nc Expansion

Richard Lebed

Jefferson Lab

July, 2009

A Tom Cohen (TDC)/ Rich Lebed (RFL) Joint

TDC, RFLPRL 91, 012001 (2003)

TDC, RFLPRD 67, 096008 (2003)

TDC, RFLPRD 68, 056003 (2003)

TDC, RFLPLB 578, 150 (2004)

TDC, Daniel C. Dakin (DCD), Abhinav Nellore (AN), RFLPRD 69, 056001 (2004)

TDC, DCD, AN, RFLPRD 70, 056004 (2004)

TDC, RFLPRD 70, 096015 (2004)

TDC, DCD, Daniel R. Martin, RFLPRD 71, 076010 (2005)

TDC, RFL, PLB 619, 115 (2005) TDC, RFL

PRD 72, 056001 (2005) RFL, PLB 369, 68 (2006) TDC, RFL

PRD 74, 036001 (2006) TDC, RFL

PRD 74, 056006 (2006) Herry J. Kwee (HJK), RFL

PRD 75, 016002 (2007) 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?

π

N

Resonances are poles in complex-valued scattering amplitudes

10−23 s

Resonances are poles in complex-valued scattering amplitudes

10−23 s

Resonances are poles in complex-valued 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 pairs (color structure δαβ)

or from NC quarks (color structure εα1,α2,…, αNc)

Mesons at large NC are:

• free [O(NC0) masses]

• stable [O(1/NC) widths]

• scatter weakly [O(1/NC2) cross sections] [Veneziano (1976)]

qq

“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(NC

0) 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

Chiral Solitons!Quarks!

QUARKS! SOLITONS!

jerk 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 color-fundamental 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 pictureDashen, Jenkins & Manohar; Carone, Georgi & Osofsky; Luty & March-

Russell (1994)

Consider processes that involve the (entangled) interaction of n quarks → Represented by n-body operators:

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]

2-body 3-body

Example: Baryon massesJenkins & 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)

Massdifferencequotient

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

• Breaks both I and J but preserves their vector sum K

,4

2

UUTrf

L

)ˆˆ)(exp()(0 rriFrU

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:

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

2241 ,Tr UUUUL

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 pair creation, using H should be OK

But how to use the 1/NC approach when pair creation is common?

Need a method that treats resonances as poles in meson-baryon scattering amplitudes

• Real part → mass Imaginary part → width

Chiral solitons naturally understand meson-baryon coupling

qq

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 meson-baryon 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 = 1K = 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%) mη+mN=1490 MeV

• N(1650)→ πN (55-90%), ηN (3-10%)

[TDC & RFL (2003)]

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 is about 4-5 times [O(NC)] larger than that to πΣ [Large NC SU(3) selection rule, TDC & RFL (2005)]

NK

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/NC

n [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 3-flavor 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

• Large-NC states that do not occur for NC > 3 are always seen to decouple from observables when NC 3 [TDC & RFL (2006)]

Chiral limit vs. Large NC limit

• 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 3-flavor 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/NC

n: a perturbative expansion

H = c0 NC 1 + c1(8)

NC0 T8 + cJ J2/ NC + …

where T8 = , J2 =

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)?

qqquarks 2

8

qqqq

ii

22

Consistency Conditions[Dashen, Jenkins, Manohar (1994)]

Consider pion-baryon scattering:

Each vertex is O(NC1/2), but the whole process is O(NC

0)!

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) Clesbch-Gordan Coefficients (CGC)

Large NC baryon SU(3) representations are of the form (p,q) = [O(NC

0), O(NC1)]

Meson representations as usual: (p,q) = [O(NC

0), O(NC0)]

Theorem [TDC & RFL (2005)]: SU(3) CGC for (baryon 1) + (meson) → (baryon 2) can be O(NC

0) only ifYmeson = 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

while Λ resonances in an SU(3) 1 (for NC=3), prefer (by a factor NC) decays to πΣ

Empirical example: Coupling constant of Λ(1520) D03 to is about 4-5 times [O(NC)] larger than that to πΣ when threshold p2L+1 taken into account

NK

NK

NK

Electric Multipole Relations

Magnetic Multipole Relations

Example Multipole Relation L = 2 magnetic multipole relation:

Evaluate these amplitudes on resonance:

2

15

)(,m

2

5,2,2

)(,m

2

5,2,2

13

)(,m

2

3,2,2

)(,m

2

3,2,2

1

)1675(D Contains

2

3

)1520(D ContainsC

pnnppnnp

NOMMMM

213

13 106.013.0

GeV102.140

GeV105.82.18

2

1CN

ORHSLHS

RHSLHS

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)