unquenching the quark model

BRAG meeting@Bonn 9/2/07-1Simon Capstick, Florida State University


Unquenching the Quark Model

• Motivation• Required ingredients• Self-consistent baryon masses in

presence of continuum (baryon-meson) states

• SU(6) (flavor-spin) symmetry– Minimum set of intermediate states

• Prior and ongoing work• Lessons from covariant calculation of - • Collaborators: Danielle Morel (PhD@FSU, JMU);

Mike Pichowsky (Chicago mercantile exchange), Sameer Walawalkar (PhD@FSU, CMU);Eric Swanson (Pittsburgh);Ted Barnes (ORNL/U Tennesssee, Knoxville)


Motivation

E. Teller:

“From mesons all manner of forces you get; the infinite part, you simply forget”


Motivation…• In QCD qqq(qq) configurations possible

in baryons: effect on CQM?– Expand in basis of baryon-meson

intermediate states– B’M loops self energies ~O(), mixing – requires calculation of form factors– Many thresholds

close in energy

(k)


Required ingredients

To calculate self energies and mixing:• Need model of spectrum (including states

not seen in experiment) wave functions vertices

• Need model of B(0) B’(-k)M(k) vertices and their momentum dependence

– Focus on baryons• Many calculations of effects in mesons

exist, not surveyed here


Self-consistent mass calculation

• Assume your qqq (bare) quark model baryon masses M0

B depend on:– Parameters: strong coupling 0, quark mass

m0, string tension b0…, 3P0 (pair creation) coupling strength

– Hamiltonians: Hqqq, Hpc

• Parameters determined by a fit to the spectrum and decays in the absence of higher Fock-space componentsEB = M0

B (0,m0,b0,…) = physical mass


Self-consistent quark model masses…

• Correction due to B ! B’C is

where

– Imaginary part of loop integral is ½B! B’C

• Perform sum over B’C intermediate states• Adjust parameters 0,m0,b0… for self-

consistent solution with EB = physical mass – In principle should solve a similar equation for

meson masses EC


Self-consistent mass calculation

• Equivalent to second order perturbation theory in decay Hamiltonian Hpc

– Can calculate the (momentum space) continuum (B’C) component of physical baryon states

– Can calculate the mixing between different baryon states due to continuum states (B! B’C! B’’)


Intermediate (continuum) states

• Baryon self energy due to individual B’M loops are comparable to widths – convergence?

• Calculations applied to ground and L=1 states;

• Intermediate states B’M– Ground states baryons B’{*} – Mesons:

• Blask, Huber and Metsch (’87) used ground state (pseudoscalar) octet mesons M{}

• Zenczykowski (’86), Silvestre-Brac and Gignoux (’91), Y. Fujiwara (’93) used pseudoscalar and vector mesons M{*}


Intermediate states…

• Essential problem: there are lots of B’M thresholds nearby in energy– N, similar thresholds to N(1535), (1116)K, etc.

– Cannot study effects on spectrum by restricting M to (or even all pseudoscalars) or B’ to N, (or even all octet and decuplet ground states)


SU(6) (flavor-spin) symmetry• Zenczykowski: gedanken

experiment• assume SU(3)fSU(2)spin, only ground

state B, B’ and M exist• assume all octet and decuplet ground

state baryons have same mass M0B &

same wvfn.• assume all pseudoscalar and vector

ground state mesons have mass M0M &

same wvfn.

– All loop integrals now the same, apart from SU(6) factor at vertices: = N ?


SU(6)flavor-spin weights

N K K * K N N0 0

N 25/6 32/6 1/6 9/6 8/6 (c-2½s)2

/6(2½c+s)2

/6

0 0

8/6 25/6 8/6 0 10/6 0 0 5(c-2½s)2

/65(2½c+s)2

/6

N K* K* *K* N SUM

N 59/6 64/6 11/6 27/6 16/6 33/6 0 48

16/6 95/6 16/6 0 38/6 0 57/6 48


SU(6) flavor symmetry limit

Sum of loops for N and same only if include all B’M combinations (non-strange, strange, or both) consistent with quantum numbers– Need both pseudoscalar and vector mesons

• This is true for any baryon in the [56,0+] ground states (octet and decuplet baryons), sum is 48– Also true in 3P

0 model (reduces to SU(6)W in this limit)• If all the thresholds and all wave functions are

the same, there are no mass splittings between ground state baryons

• Requires:– SU(3)f breaking turned off (ms = mu,d)– Spin-dependent splittings (-N, -, etc.) turned off


Away from SU(3)f limit• Relax assumption of SU(3)f symmetry in meson &

baryon masses (Tornqvist & Zenczykowski, ’84)

– Use – Assume:

1. Spectral function(s,mB,mM) broad, on scale of baryon mass splittings (ultimately calculate using 3P0)Assume form (s,mB,mM) = (s½ -[mB+mM])

2. Assume: m0N = m0

´ m00

m0 = m0

= m0* = m0

0 + 0 ´ m01

m0 = m0

* = m00 + 20 ´ m0

2 m0

= m00 + 30 ´ m0

3

3. Similar pattern of SU(3)f breaking for meson masses (m= m, ideal mixing,…)


Away from SU(3)f limit…

• Only two thresholds for each external baryon with k strange quarks:– Non-strange quark pair production Ek

0, strange quark pair production Ek

0+2• Call sums over relevant SU(6) weights wk

0, and wk

2

• Adopt similar notation for output baryon masses: mN = m ´ m0

m = m = m* ´ m1 m = m* ´ m2 m ´ m3

• Mass shifts:



• Ignores factor

– [Close to 1 if mk small c.f. hxi +Ek0, but

following works without this assumption]

• From SU(6)/3P0: wk+12 - wk

2 = 4 for k = 0,1,2– Subtract integral relation for k from the

same one for k+1

– Work to first order in mk+1-mk, 0,



• Independent of k:– Output baryon masses show same pattern

as input masses– The difference in pole positions in the r.h.s.

of the (self-consistent) integral equation leads to in denominator


Away from SU(6)fs limit

• Now assume:– Ground-state baryon masses exhibit this simple equal-

spacing pattern– PS and V mesons have their experimental values

• Now output masses exhibit SU(6)fs breaking effects– E.g. m- mN 0– In first order in meson mass differences,

c´ cos(P), s´ sin(P) (perfect mixing P=10o)

– Constant C / d(-N)/dEthreshold > 0 (as threshold becomes more distant, mass shift decreases) [but 0.18 + 0.82 0]


Away from SU(6)fs limit

• Similar formulae can be derived for *-, -, *-,...

• If intermediate baryon masses already split (by e.g. OGE, with similar pattern)– If meson masses satisfy Gell-Mann Okubo

SU(6) mass formula• Get 2N + 2= 3+ (Gell-Mann Okubo, octet)• Get * - = * - * = - * (equal spacing rule for

decuplet)• Get * - = * - (SU(6) relation)

– Recover SU(6) relations for baryon masses!


Both mechanisms necessary

• Result is very interesting– Can interpret SU(6)

breaking effects in baryon spectrum as partly due to spin-flavor dependent exchanges (e.g. OGE)

– Partly due to loop effects

• (not same as OBE)


Results from prior work…• Effects on spectrum are substantial

– Zenczykowski finds many mass splittings close to analyses without qqq residual interactions

– Other calculations show splittings in bare P-wave baryon masses which resemble spin-orbit effects (Blask, Huber and Metsch, Silvestre-Brac and Gignoux, Fujiwara)

• Solution to spin-orbit problem in baryons?

• Lack self-consistent treatment of external and intermediate states—converged? – Such convergence slower in 3P0 NRQM:

mesons, Geiger and Isgur– Faster in covariant model based on

Schwinger-Dyson Bethe-Salpeter approach: M. Pichowsky, S. Walawalkar, SC


Baryon self energies in relativized 3P0 model

• PhD thesis of Danielle Morel (PhD@FSU, now at JMU); study ground and L=1 excited external states

• Calculate vertices as a function of loop momentum using 3P0 model (analytic, Maple)– Use mixed relativized-model wavefunctions

(expanded up to N=7 band)– Include intermediate states BM with

• Mesons M{*} • Baryons B{*}, including all

excitations up to N=3 band• Roughly 300 to 500 intermediate states!


Baryon self energies in relativized 3P0 model…

• Usual 3P0 model gives vertices that are too hard, loops get large contributions from high momenta– Soften with pair-creation operator form factor

exp(-f2[pq-pq]2)

• Currently revisiting calculation to allow self-consistent renormalization of quark model parameters (hard work!)

• T. Barnes and E. Swanson looking at shifts in charmonium spectrum due to D,D*,Ds,Ds

* meson pairs


splitting in a covariant model

• Are large self energies from hadron loops an artifact of a non-relativistic approach, or of the 3P0 model?– M. Pichowsky, S. Walawalkar, S.C.

[PRD60, 054030 (1999)]• Examine and self energies in covariant

model based on Schwinger-Dyson approach• Calculated pseudoscalar-pseudoscalar and

vector-pseudoscalar loops


Loop calculation

• Vector-pseudoscalar-pseudoscalar (VPP) form factors from quark loop integrations (similarly for VVP)– Requires:

• knowledge of the u-, d- and s-quark propagators• P and V meson Bethe-Salpeter amplitudes

– taken from phenomenological studies of EM form factors and strong and weak decays of P and V mesons


Form factors

• Form factors are evaluated numerically and fit to simple exponential form (VPP)– fVPP(p1,p2)

• Note additional suppression when mP > mV/2


Results

• Self energies rapidly decrease with increasing mass of the intermediate state mesons– Form factors

significantly softer than in NRQM/3P0 model calculations

– Additional suppression with mass


Conclusions/Outlook

• The next Fock-space component is likely more important than differences among qqq models– calculating its effects requires:

• Use of full SU(6)-related set of intermediate states, spatially-excited intermediate baryons

• Careful treatment of mixing

• Renormalization of parameters in quark model needs to be carried out– Renormalize s, quark masses, string tension– This requires examining mass shifts of more than

just N, , and their negative-parity excitations

• Need additional suppression when mB >> mB

’+ mM

unquenching the quark model

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