Size and shape of baryons in a large N_c quark model

5 Pages·0.1 MB·English

by  A. J. Buchmann

SIZE AND SHAPE OF BARYONS IN
A LARGE N QUARK MODEL
C
A. J. BUCHMANN
University of Tu¨bingen, Institute for Theoretical Physics, Auf der Morgenstelle 14,
D-72076 Tu¨bingen, Germany
E-mail: [email protected]
3
0
Baryon charge radii and quadrupole moments are calculated in a quark model
0
generalized to an arbitrary number (Nc) of colors. Several relations among the
2
charge radii and quadrupole moments are found. In particular, the relation
n Qp→∆+ = √12rn2 betweentheneutronchargeradiusrn2 andthep→∆+transition
a quadrupole moment Qp ∆+ is shown to hold for physical baryons as well in the
J largeNc limit. →

6

1 Introduction
1
v
1 Charge radii (rB2) and quadrupole moments (QB) are the lowest moments of
3 the chargedensityρinalow-momentumexpansion. Forexample,foramem-
0 ber of the baryon decuplet with unit charge, one has up to q2 contributions
1
0 q2 q2
3 hB|ρ(q)|Bi=1− 6 rB2 − 6 QB+···, (1)
0
/ where q is the photonmomentum transfer to the baryon. The firsttwo terms
h
p arisefromthesphericallysymmetricmonopolepartofρ,whilethe thirdterm
- isobtainedfromthequadrupolepartofthechargedensity. Theycharacterize
p
the total charge, spatial extension, and shape of the system.
e
h Baryon charge radii and quadrupole moments are closely related conse-
: quences of the quark-gluondynamics. They have been calculated in different
v
modelsofbaryonstructure,e.g.,anN =3constituentquarkmodelwithtwo-
i c
X body exchange currents 1. Several new relations among the different r2 and
B
r Q have been abstracted from these model calculations, some of which have
a B
later been shown to be general consequences of the underlying symmetries
and dynamics of quantum chromodynamics 2,3.
2 Large N Quark Model
c
It is instructive to investigate the charge radii and quadrupole moments of
baryonsin a quark model in which the number of colorsN is not fixed to its
c
physical value N = 3 but allowed to take on an arbitrary odd integer value.
c
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c
Among the advantages of treating N as a parameter are:
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(i) an expansion in powers of 1/N of the operator structures contributing to
c
a given observable is obtained, (ii) there is a hierachy in importance of one-,
two-, and three-quark operators.
A multipole expansion of the baryon charge ρ up to quadrupole terms
leads to the following invariants in spin-isospin space
Nc B Nc
ρ = A e e 2σ σ (3σ σ σ σ ) , (2)
i i i j iz jz i j
− N (cid:20) · − − · (cid:21)
Xi c Xi<j
where σ is the z-component of the Pauli spin matrix of quark i, and e is
iz i
the quark charge. The constants A and B in front of the one- and two-quark
operatorsparametrizetheorbitalandcolormatrixelements. Ifthetwo-quark
invariants are generated by gluon exchange, there is a fixed ratio ( 2) of the
−
factors multiplying the monopole (spin-scalar) and quadrupole (spin-tensor)
contributions to ρ.
Three-quark operators are neglected in the present work.
3 Baryon Charge Radii
AccordingtoWitten 4,baryonshavechargeradiiproportionalto N0 inlead-
c
ing orderso that, e.g., nonstrange baryonshave the same size independent of
their spin. Here, we consider the 1/N corrections associated with the spin-
c
dependent two-body operators in Eq. (2). The latter break SU(6) spin-flavor
symmetry and lead to a splitting between octet and decuplet charge radii.
This is analogousto the octet-decuplet mass splittings by spin-dependent in-
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teractions .
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TheN and∆chargeradii arethematrixelementsofthefirsttwoterms
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in Eq. (2) evaluated between large N nucleon and ∆ wave functions :
c
1 4J(J +1)+N (N +2)(2Q 1) Q
r2 = AQ B c c − +6B , , (3)
B Q(cid:26) − N2 N (cid:27)
c c
whereQisthetotalchargeandJ thetotalangularmomentumofthebaryon.
Specializing to J = 1/2 for the nucleon states, one obtains (for neutral
baryons there is no normalization factor 1/Q)
(N 1)(N 3)
r2 = A B c− c− ,
p − N 2
c
(N 1)(N +3)
r2 = B c− c . (4)
n N 2
c
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Similarly, from Eq. (3) we obtain for the ∆ states with J =3/2
3 (N 2 2N +5)
r2 =A B c − c ,
∆++ − 2 N 2
c
(N 2 4N +15)
r2 =A B c − c ,
∆+ − N 2
c
(N 3)(N +5)
r2 = B c− c ,
∆0 N 2
c
2
(N 5)
r∆2− =A−3B cN−2 . (5)
c
From these equations one readily derives that
r2 r2 =r2 r2 (6)
p− ∆+ n− ∆0
isvalidforallN . Eq.(6)containstheSU(6)symmetry-breakingeffectdueto
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the spin-isospindependent (1/N1)correctioninthe baryoncharge. Further
O c
relations and a more complete treatment including the effect of three-body
3
operators can be found in Ref. .
4 Baryon Quadrupole Moments
Quadrupole moments measure the deviation of the baryon’s charge density
from spherical symmetry. For the diagonal quadrupole moment of a non-
strangedecuplet baryonwithchargeQ,totalangularmomentum J, andpro-
jection J one finds
z
B [3J 2 J(J +1)] (4J(J +1)+N (N +2)(2Q 1))
z c c
Q = − − . (7)
B 2(cid:20) J(J +1) (cid:21) N 2
c
Eq. (7) is the matrix element of the quadrupole part of ρ evaluated between
large N quark model wave functions. Specializing to J =J one gets for the
c z
different charge states of the ∆,
2 [15+3N (N +2)]
c c
Q = B ,
∆++ 5 N 2
c
2 [15+N (N +2)]
c c
Q = B ,
∆+ 5 N 2
c
2 [15 N (N +2)]
c c
Q∆0 = 5B − N 2 ,
c
2 [15 3N (N +2)]
c c
Q∆− = 5B − N 2 . (8)
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The ∆ quadrupole moments satisfy the relations
Q∆++ +Q∆− =Q∆+ +Q∆0,
Q∆+ +Q∆− =2Q∆0, (9)
whichholdforallN . Theyarelinearcombinationsofthequadrupolemoment
c
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relations obtained with Morpurgo’s general parametrization method . Only
very weak assumptions, such as invariance of the strong interactions under
isospin rotations, are required to derive them.
The nondiagonal N ∆ transition quadrupole moment in the large N
→
quark model is
1 2(N 1)(N +5)
c c
QN→∆ = 2B p −N , (10)
c
bothforthep ∆+ andthen ∆0 transition. Additionalresultsincluding
→ →
an analysis of strange baryons and three-body operators will be published
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elsewhere .
5 Relations Between Charge Radii and Quadrupole Moments
ComparingEq.(10)andEq.(4),arelationbetweentheneutronchargeradius
rn2 and the p→∆+ transition quadrupole moment Qp→∆+ is obtained
1
Qp→∆+ = √2rn2, (11)
which is exact for N = 3 and N . For intermediate values of N the
c c c
difference between left and right ha→nd∞side is (1/N2). Eq. (11) was origi-
O c
nally derived in an N =3 constituent quark model with two-body exchange
c
currents1. Recentexperiments 9,10 providesome evidencethatitis satisfied
within 20%.
The observables rn2 and Qp→∆+ are closely related because (i) in both
cases one-quark operators do not contribute, (ii) both observables are dom-
inated by the two-quark (B) term, (iii) the relative weight of the monopole
(spin-scalar) and quadrupole (spin-tensor) parts in ρ is uniquely determined
if these arise from gluon exchange.
The generalizationof Eq. (11) to finite momentum transfers is
Gp→∆+(q2)= 3√2Gn (q2). (12)
C2 − q2 C0
TogetherwiththeresultGp→∆(q2)= √2Gn (q2),itcanbeusedtopredict
M1 − M1
the quadrupole (C2) over dipole (M1) ratio in N ∆ electroexcitation from
→
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the elastic neutron form factor data . This agrees very well with C2/M1
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data from N ∆ electroexcitation .
→
Finally, the following relation between the charge radii and quadrupole
moments holds for all N :
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Q∆++ +Q∆+ +Q∆0 +Q∆− =rp2−r∆2+ +rn2 −r∆20. (13)
6 Summary
The size and shape of baryons are different but related aspects of the under-
lying quark-gluon dynamics. A large N quark model shows that there is a
c
number of relations between baryon charge radii and quadrupole moments
that can be put to experimental tests.
Acknowledgments
I thank the organizersfor the invitation. Some financialsupport by the DFG
under title BU 813/3-1 and the Institute for Nuclear Theory is gratefully
acknowledged.
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