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Magnetic moments of baryons in a constituent quark model with relativistic and pionic corrections
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
M A G N E T I C M O M E N T S O F B A R Y O N S IN A C O N S T I T U E N T Q U A R K M O D E L W I T H R E L A T I V I S T I C A N D P I O N I C C O R R E C T I O N S :: Joseph C O H E N ~ a n d H.J. W E B E R Department of Physics and Jesse W. Beams Laboratory, University of Virginia, Charlottesville, VA 22901, USA Received 30 May 1985
The magnetic moments of the baryon octet are calculated in time-ordered perturbation theory in the harmonic oscillator quark model, which is supplemented by a small Dirac quark wave function to include the pion-quark coupling at long distances. Relativistic, pionic and quark confinement corrections are found to be substantial. Contributions from the pionic quark pair and exchange currents nearly cancel each other. A naive o-meson quark-pair current seems to be ruled out by the data.
The successful description o f the baryon magnetic moments in the nonrelativistic harmonic oscillator quark model (NQM) [1] rests on assuming Dirac magnetic moments for constituent quarks o f mass m u = m d = mq for the non-strange quarks and m s for the strange quark. The proton's magnetic moment agrees ,~1 with experiment provided mq ~ mN/3; and m s -GeV/c 2 is required for the hyperons, thus breaking the SU(6) symmetry. These light quark masses in conjunction with a scalar, harmonic confmement potential and the color hyperfine interaction also give a good description of the low lying hadron masses. But the model fails in the multi-quark sector to account for the nuclear forces at low energy, because quark-pair creation, i.e. mesonic degrees of freedom, are missing from the dynamics. In S states, moreover, the quark energy is comparable to the constituent quark mass so that such a nonrelativistic theory is only qualitative. In the MIT bag model [2], single-quark matrix elements are calculated covariantly, massless but confined quarks have finite Dirac magnetic moments, but the magnetic moments of the nucleon come out much smaller than in the NQM. For a better description the pion--quark coupling o f chiral bag models (CBM) [3] ~'~ Work supported in part by the US National Science Foundation. i Present address: Department of Physics, Indiana University, Bloomington, IN 47405, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Hoiland Physics Publishing Division)
is required, which also supplies the nuclear forces at long distances. However, the models suffer from additional recoil and center-of-mass (CM) corrections which are ambiguous and originate from the use of approximate relativistic multi-quark wave functions, while the NQM is translation invariant. Relativistic corrections in the constituent quark model (CQM) [4,5] derive from a two-quark Dirac equation obtained from the Bethe-Salpeter equation without retardation, i~ ~(r 1 ,r2, O/at = ndd(rl ,r 2, t),
2 H = ~ (-ict] .~. + {3]rn]) - ~K ~l~2kl .k2(r 1 - r 2 ) 2 ]=1 + G275 • (1)'75~ (2) e x p ( - m n Ir 1 - r 2 I)/4rt Ir 1 - r 2 1 .
(1) The kj are the color generators in the scalar two-body confinement potential, and G is the p i o n - q u a r k coupling constant. Here we include relativistic corrections to lowest order O(mq 1) in the CQM by the explicit use of a small Dirac quark wave function [4]
q(r) = X? (g(r), -if(r),-~:) = X? (1, - ( i n - V )/2 m q)g(r), f(r) = (1/2mq) dg/dr = - ( a r / 2 m q)g(r), ~ = r/r.
(2)
The large S-state component 229
Volume 165B, number 4,5,6
g(r)= N.exp[ - ~a 1 (r2 _R2)]
PHYSICS LETTERS
(3)
takes into account the confinement potential in eq. (1) with the harmonic oscillator constant a so that r -lp/2= 1/x/~ is the root-mean-square (RMS) threPe. quark proton radius. Note that 1/x/~is related to, but not identical with, the bag radius R. The value ofx/~ = 200 MeV used here corresponds to a proton RMS (three-quark core) radius of 1 fm, while a bag radius of that size would correspond to vC~ = 247 MeV. The three-quark baryon wave functions are written in terms of the usual Jacobi-quark coordinates [1] p, k and CM distance R. Relativistic corrections to lowest order O(mq 1) in the three-quark wave function may be incorporated by writing it in spinor form as (e.g. for the nucleon)
= ×*N(g0,--%03 %), fO = -(ar3/2mq)go' 3
gO(p, X) = __lSI 1 g(r/) =N(a/rr) 3/2 exp [ - {a(p 2 + X2)], /= (4) where 3 /=1
(52 - R 2 ) =p 2 +
(7)
The well-known axial-vector coupling constant gA = s in the nonrelativistic NQM is substantially modified in the CQM by relativistic corrections [4] to become gA = -~(1 -- ~/12m2)(1 + a/am2) -1 .
(8)
Agreement with experiment requiresg A = ¼ and yields the constraint (for non-strange quarks) 12 2 a = i~mq, (9) which we use in the following so that there is only one independent parameter in the non-strange sector of the CQM, viz. rp = 1/X/'dlike the bag radius in the CBM, as was emphasized above. Also, from eqs. (5) and (8), relativistic corrections are typically 25% to lowest order. Pion cloud corrections to gA in potential models with direct pion-quark coupling are known to be negligible [6]. The axial-vector current is taken to stay unrenormalized in the transition from QCD-current quarks to the constituent quarks of the CQM. Finally, in addition to the effective, space-like (Yukawa) meson-quark couplings, the electromagnetic baryon currents can be calculated by minimal substitution in eq. (1) from the Dirac quark current e(~ + lr3)7 z as follows [4]:
"r~q
- [FO,T --(2mN/q)PT] (i/2mN)~Nvqv}(*N)TUN(P)'
N = (1 + c~/4m~) -1/2.
(5)
Along with relativistic corrections the small Dirac quark wave function allows one to include the usual pionquark coupling at long distances with its coupling constant G determined from the resulting rrNN vertex at R=0,
(1 +a/4m~) -1 ~(G/3mq)exp(--q2/6oOi*N'q~N ,
(6) so that at q2 = O,
(10) for isospin T = 0, I, q2 = _q2, and to lowest order in p, p' for plane-wave nucleon spinors UN, where P0,T =N2"31-T[m + (~/4m2)(1 --q2]9a)] exp(--q2/6a),
I"T = N2(q/3m q)(s) T exp(-q2/6a),
(11)
with N from eq. (5). The spin currents in eq. (10) yield the nucleon magnetic moments (in nuclear magnetons, e/2mN, nan.)
G f d3p d3X ~Ni75~ 0N exp(--iq'k ~/~--~)
230
= 14.4.
= (I - q 2 / a m 2 ) - I ~N(p'){ [P0,T + (q/2mN)r'r]
d3 p d3X(g2 + f~) = 1,
i.e.,
-
g2NN/4rr = (G2/4rr)(~ 2mN/3mq)2(1 + c~/4mq2)-2
<[3] N(p')ITu(,)TI [3] N(p))
and the normalization is so that
f d3pd3X 't#tN~N = f
26 December 1985
2 lap = (2mN/3mq)(1 + ~/4m2) -1, /an = -- ~lap. (12)
They both differ by the factor ~(1 * ~/4m2) -1 = ~N 2 = 0.5417 (using eq. (9)) from the nonrelativistic
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
IJ~+ = (2mN/27ms)(1 +ax/4m2s )-1 B
B
C'DB
(a)
C
D
+ (_~)(8mN/27mq)(a + a/4m2) -1 ,
B
and for the cascade
(b)
/axo_ = - -~ [1 + (ol/4m2)(ms/mq) 1/2] -12mN/3m
Fig. 1. (a) Photon coupling to the three-quark core of the baryon B from the octet. (b) Pionic vertex renormalization of (a); C, D are the allowed baryons from the octet and decuplet.
× (2mN/3mq)(_2).
q2/4m2)-I
X UA [(q/2mA)F1Tu
+(2mA/q)Fl(i/2mA)OUVqv]
1 F 1 = - ~(1 + ax/4m~)-l(1/3ms)exp(-q2/6ax)
UA'
(13)
yields the magnetic m o m e n t of the A-hyperon in nuclear magnetons as /aA = " ~ ( 1 + ax/ams2) -1
2mN/3ms,
s
-- ~ (1 +(a/am2q)[3ms/(2ms+mq) ] 1/2)-1
NQM results, of w h i c h N 2 comes from the small Dirac quark wave function via eq. (5) and ] from the boundstate nature o f the large component (confmement). Since the measured magnetic m o m e n t o f the A-hyperon requires a strange quark mass m s > mq, we use the uds basis [1] for the hyperons [i.e. take the s quark to be distinguishable from the u, d quarks and the harmonic oscillator constant a x = ot(mx/mq)l/2 with mx = 3mqms/(2m q + ms) in the S = - i sector] rather than SU(6) symmetric wave functions. The A-current (AITUeslm) = (1 -
(15)
(14)
for the Y.-+hyperons one finds similarly
(16)
Note that these values again differ from the NQM re2 sults by ~ and the wave function renormalization factor. The magnetic m o m e n t contributions from the p h o t o n - q u a r k coupling o f fig. l a in table 1 with relativistic corrections to lowest order are comparable to those of the bare bag model. Thus, only an unrealistically large three-quark proton core 1]Vr~ = rp = 1.64 fm corresponding to mq = 125 MeV/c 2 would reproduce the experimental nucleon magnetic moments. Color hyperfine effects are also much too small to improve the situation [1]. However, pionic corrections at long distances are significant and necessary to understand better (within ~ 1 0 % of the data) the magnetic moments o f the baryon octet. The pion does not couple to the strange quark and, in the uds basis, one finds that the p i o n baryon couplings are those o f the SU(6) symmetry. Their common gaussian form factor and strength ocGN2/mq appear in the rtNN vertex of eq. (6) and are characteristic of the CQM. The parameter-free evaluation of the pionic contributions to the baryon magnetic moments in time-ordered perturbation theo-
Table 1 Baryon magnetic moments t~(CQM for rp = 1/x/r~= 1 fm, rnq = 208 MeV/c z , mx = 297 MeV]c2). a)
b)
Baryon
/z3,_Bq
/zn
p n A E+ Z~o ~-
2.443 -1.629 -0.597 2.372 -0.887 -1.363 -0.557
0.420 -0.420 0 0.250 -0.250 0.014 -0.014
/~q,-3q+lr 2.696 -1.991 -0.614 2.475 -1.088 -1.365 -0.552
c)
d)
, (ex)
tzrr-pair
"rr
tZo-pair
-0.345 0.345 0 0 0 0 0
0.267 -0.267 0 0 0 0 0
0.267 -0.178 0 0.118 -0.059 0 0
d)
/~total e)
NQM f)
Experiment g)
2.618 -1.913 -0.614 2.475 -1.088 -1.365 -0.552
2.70 -1.80 -0.60 2.59 -1.01 -1.36 -0.46
2.793 -1.913 -0.613 2.379 -1.10 -1.250 -0.69 -1.85
-+ 0.004 -+ 0.020 -+ 0.05 -+ 0.014 -+ 0.04 h) -+ 0.75
a) Fig. la. b) Fig. 2. c) Figs. la, lb, 2. d) Fig. 3. e) Figs. 1-3. f) Ref. [9]. g) Ref. [7]. h) Ref. [8]. 231
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
Their contribution to the nucleon magnetic moments is isovector and given by
B
C
,'
',
B
C
÷
B
;, C
B
÷
B
- "j{ C
mrr(2) + m(eX) = 3(gTrNN/5rr)2(1 + ~/4m2) 2 (r3N/mN)
B
Fig. 2. Pion current contributions to the baryon magnetic moment in time-ordered perturbation theory.
X f
ry of figs. lb and fig. 2 gives the results listed in table 1, which resemble those of the CBM's [3]. Note that the magnetic moment of the A-hyperon has only contributions.from fig. lb but not from fig. 2. The strange quark mass m s = 297 MeV/c 2 follows from the experimental value [7] /~A = --0.613 -+ 0.004 n.m. The magnetic moment contributions from one-pionexchanges (OPE) are neither complete nor gauge invariant without the pionic quark-pair current operator (cf. fig. 3) /'~2)(r; r l , r2) =
(G2/9m2) [-~(1)8 (r - r
+~ (2)8 (r --r 2)~(1 ) • V1 ] (, (1)
1)~(2)" V2
X • (2)) 3
X exp(-mr~lr 1 -r21)/4rrlr1 - r 2 l
x
-
,
(17)
X , (2))3o(1)- Vln(2)'V 2
r)"o0' - '21)},
X (~(1) X
V1)exp(-molr 1 -r2[)/47rlr1 - r 2 l
+ (1 ~- 2)
satisfies gauge invariance in conjunction with the relativistic spin-orbit correction of the quark charge operator
-
q).
X [1I(1) X P l ' VlS(r - r l ) ] +"
,(2) = m(2). N
(18)
G2mN fd3o d 3 Xgo(g 2 I 4m 2
× [r 1 × (o N ×
+B
Fig. 3. q~-pair currents. 232
j
B
(21)
The magnetic moment contribution from jo(2) in nuclear magnetons is obtained from
-
B ~ B
(20)
(17')
The pionic pair and exchange currents at the location r are connected via gauge invariance to the pion exchange potential VopE and the quark charge operator in lowest order O(mql), + ~ r 3 ( / ) ] 6(r
and 0 for the A, I¢+ and _---0hyperons. The exchange current contribution m (ex) is easy to calculate in momentum space [5]. For the nucleon there is considerable cancellation between the ~r-pair and exchange current contribution. This is obvious from eq. (19) and the numerical values shown in table 1. Since the attractive two-pion-exchange potential plays an important role in the nucleon-nucleon interaction at medium range, which is ofter parametrized by an effective scalar-isoscalar a-meson exchange, it is reasonable to consider also the correspotading ameson quark-pair current for the magnetic moments. A naive a-pair current
p(S~(r) = ( 1/8mq)[~ 2 1 +17.3(i) ]
Io(r) = exp(-m rr)/4rrr.
O(1)(r) = . ~ [ t 1
(19)
/'(2)(r;rl,r 2) = -('G2"4/mq)la2"'a + }r3(1)18( r _ r l )
which has recently been studied [5], and the pion-exchange current
/~ex) = _(G/3mq)2 (,(1)
dqq4 exp(-q2/2a) [ - 1 / 2 a + 1/(m2+q2)]
m~~r +q2
o
s~.(3)h + g-N J
Vl)exp(-molrl-r2l)]4rdr 1-r21
+ (1 ~ 2)1, which yields for the nucleon
(22)
Volume 165B, number 4,5,6
PHYSICS LETTERS
rn(2) _- (G 2mN/81r 2 -9o~rnq)(1 2 (3))(1 + ~/4mq) 2 -1 + 5r N o,N X f 0
dq q4 . e x p ( - q 2 / 2 ~ ) , m 2 + q2 o
(23)
26 December 1985
boosting the large quark wave function as in eq. (2). In other words, the two-body quark confinement potential in eq. (1) must be accompanied by a one-body potential.
References with G from eq. (7) and m o = -~ GeV/c 2 from ref. [41, m(2) = 0 for the A and ~ 0 hyperons, ~m (2) for the E'~ and ½m(2)n for the I~- hyperon. WherePas m (2) and r n ~ x) are purely isovect3or, m(2.)N gives the correct proton-to-neutron ratio - 3" However, the numerical values in table 1 show that such a naive o-pair current would affect the nucleon and ~-+ magnetic moments adversely, had we included it in the total magnetic moments. The data does not appear to support it. In conclusion, we have calculated the baryon octet magnetic moments using harmonic oscillator wave functions and constituent quark masses with relativistic and pionic contributions. With two constituent quark masses mq = 208 MeV/c 2 and m s = 297 MeV/c 2 (and harmonic oscillator parameter a = ~ r n 2 from gA -5 g ) , w e reproduce the magnetic moments of the baryon octet with ~ 1 0 % accuracy. This is comparable to the NQM, but the CQM improves gA via relativistic corrections and becomes consistent with the nuclear forces at long distances b y including a p i o n - q u a r k coupling. To obtain better agreement with the magnetic moments at a smaller and more realistic three-quark core radius than 1/x/if = 1 fm, an independent small Dirac quark wave function is needed, instead o f simply -
[ 1] See, e.g., F. Close, An Introduction to quarks and partons (Academic Press, New York, 1979), and references therein; A. DeRtijula, H. Georgi and S. Glashow, Phys. Rev. D12 (1975) 147; N. Isgur and G. Karl, Phys. Rev. D21 (1980) 3175 ; H.J. Lipkin, Quark model spectroscopy (Steamboat Springs, CO, 1984), AIP Conf. Proc. Vol. 123 (AIP, New York, 1984) p. 346; N. Isgur in: Proc. 2nd Kaon factory physics Workshop (Vancouver, 1981), TRIUMF Report TRI-81-4 (1981). [2] T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [3] S. Theberge and A.W. Thomas, Nucl. Phys. A393 (1983) 252; F. Myhrer, Phys. Lett. 125B (1983) 359. [4] M. Bozoian and H.J. Weber, Phys. Rev. C28 (1983) 811. [5] H.J. Weber and M. Weyrauch, University of Virginia preprint (1985). [6] R. Tegen, M. Schedl and W. Weise, Phys. Lett. 125B (1983) 9; A.S. Rinat, Phys. Lett. 134B (1984) 11. [7] Particle Data Group, Rev. Mod. Phys. 56 (1984) No. 2 (!l). [8] R. Rameika, A. Beretvas, L. Deck, T. Devlin, K.B. Luk, R. Whitman, P.T. Cox, C. Dukes, J. Dworkin, O.E. Overseth, R. Handier, B. Lundberg, L. Pondrom, M. Sheaff, C. Wilkinson and K. Heller, Phys. Rev. kett. 52 (1984) 581. [9] J. Franklin, Phys. Rev. D30 (1984) 1542.
233
PHYSICS LETTERS
26 December 1985
M A G N E T I C M O M E N T S O F B A R Y O N S IN A C O N S T I T U E N T Q U A R K M O D E L W I T H R E L A T I V I S T I C A N D P I O N I C C O R R E C T I O N S :: Joseph C O H E N ~ a n d H.J. W E B E R Department of Physics and Jesse W. Beams Laboratory, University of Virginia, Charlottesville, VA 22901, USA Received 30 May 1985
The magnetic moments of the baryon octet are calculated in time-ordered perturbation theory in the harmonic oscillator quark model, which is supplemented by a small Dirac quark wave function to include the pion-quark coupling at long distances. Relativistic, pionic and quark confinement corrections are found to be substantial. Contributions from the pionic quark pair and exchange currents nearly cancel each other. A naive o-meson quark-pair current seems to be ruled out by the data.
The successful description o f the baryon magnetic moments in the nonrelativistic harmonic oscillator quark model (NQM) [1] rests on assuming Dirac magnetic moments for constituent quarks o f mass m u = m d = mq for the non-strange quarks and m s for the strange quark. The proton's magnetic moment agrees ,~1 with experiment provided mq ~ mN/3; and m s -GeV/c 2 is required for the hyperons, thus breaking the SU(6) symmetry. These light quark masses in conjunction with a scalar, harmonic confmement potential and the color hyperfine interaction also give a good description of the low lying hadron masses. But the model fails in the multi-quark sector to account for the nuclear forces at low energy, because quark-pair creation, i.e. mesonic degrees of freedom, are missing from the dynamics. In S states, moreover, the quark energy is comparable to the constituent quark mass so that such a nonrelativistic theory is only qualitative. In the MIT bag model [2], single-quark matrix elements are calculated covariantly, massless but confined quarks have finite Dirac magnetic moments, but the magnetic moments of the nucleon come out much smaller than in the NQM. For a better description the pion--quark coupling o f chiral bag models (CBM) [3] ~'~ Work supported in part by the US National Science Foundation. i Present address: Department of Physics, Indiana University, Bloomington, IN 47405, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Hoiland Physics Publishing Division)
is required, which also supplies the nuclear forces at long distances. However, the models suffer from additional recoil and center-of-mass (CM) corrections which are ambiguous and originate from the use of approximate relativistic multi-quark wave functions, while the NQM is translation invariant. Relativistic corrections in the constituent quark model (CQM) [4,5] derive from a two-quark Dirac equation obtained from the Bethe-Salpeter equation without retardation, i~ ~(r 1 ,r2, O/at = ndd(rl ,r 2, t),
2 H = ~ (-ict] .~. + {3]rn]) - ~K ~l~2kl .k2(r 1 - r 2 ) 2 ]=1 + G275 • (1)'75~ (2) e x p ( - m n Ir 1 - r 2 I)/4rt Ir 1 - r 2 1 .
(1) The kj are the color generators in the scalar two-body confinement potential, and G is the p i o n - q u a r k coupling constant. Here we include relativistic corrections to lowest order O(mq 1) in the CQM by the explicit use of a small Dirac quark wave function [4]
q(r) = X? (g(r), -if(r),-~:) = X? (1, - ( i n - V )/2 m q)g(r), f(r) = (1/2mq) dg/dr = - ( a r / 2 m q)g(r), ~ = r/r.
(2)
The large S-state component 229
Volume 165B, number 4,5,6
g(r)= N.exp[ - ~a 1 (r2 _R2)]
PHYSICS LETTERS
(3)
takes into account the confinement potential in eq. (1) with the harmonic oscillator constant a so that r -
= ×*N(g0,--%03 %), fO = -(ar3/2mq)go' 3
gO(p, X) = __lSI 1 g(r/) =N(a/rr) 3/2 exp [ - {a(p 2 + X2)], /= (4) where 3 /=1
(52 - R 2 ) =p 2 +
(7)
The well-known axial-vector coupling constant gA = s in the nonrelativistic NQM is substantially modified in the CQM by relativistic corrections [4] to become gA = -~(1 -- ~/12m2)(1 + a/am2) -1 .
(8)
Agreement with experiment requiresg A = ¼ and yields the constraint (for non-strange quarks) 12 2 a = i~mq, (9) which we use in the following so that there is only one independent parameter in the non-strange sector of the CQM, viz. rp = 1/X/'dlike the bag radius in the CBM, as was emphasized above. Also, from eqs. (5) and (8), relativistic corrections are typically 25% to lowest order. Pion cloud corrections to gA in potential models with direct pion-quark coupling are known to be negligible [6]. The axial-vector current is taken to stay unrenormalized in the transition from QCD-current quarks to the constituent quarks of the CQM. Finally, in addition to the effective, space-like (Yukawa) meson-quark couplings, the electromagnetic baryon currents can be calculated by minimal substitution in eq. (1) from the Dirac quark current e(~ + lr3)7 z as follows [4]:
"r~q
- [FO,T --(2mN/q)PT] (i/2mN)~Nvqv}(*N)TUN(P)'
N = (1 + c~/4m~) -1/2.
(5)
Along with relativistic corrections the small Dirac quark wave function allows one to include the usual pionquark coupling at long distances with its coupling constant G determined from the resulting rrNN vertex at R=0,
(1 +a/4m~) -1 ~(G/3mq)exp(--q2/6oOi*N'q~N ,
(6) so that at q2 = O,
(10) for isospin T = 0, I, q2 = _q2, and to lowest order in p, p' for plane-wave nucleon spinors UN, where P0,T =N2"31-T[m + (~/4m2)(1 --q2]9a)] exp(--q2/6a),
I"T = N2(q/3m q)(s) T exp(-q2/6a),
(11)
with N from eq. (5). The spin currents in eq. (10) yield the nucleon magnetic moments (in nuclear magnetons, e/2mN, nan.)
G f d3p d3X ~Ni75~ 0N exp(--iq'k ~/~--~)
230
= 14.4.
= (I - q 2 / a m 2 ) - I ~N(p'){ [P0,T + (q/2mN)r'r]
d3 p d3X(g2 + f~) = 1,
i.e.,
-
g2NN/4rr = (G2/4rr)(~ 2mN/3mq)2(1 + c~/4mq2)-2
<[3] N(p')ITu(,)TI [3] N(p))
and the normalization is so that
f d3pd3X 't#tN~N = f
26 December 1985
2 lap = (2mN/3mq)(1 + ~/4m2) -1, /an = -- ~lap. (12)
They both differ by the factor ~(1 * ~/4m2) -1 = ~N 2 = 0.5417 (using eq. (9)) from the nonrelativistic
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
IJ~+ = (2mN/27ms)(1 +ax/4m2s )-1 B
B
C'DB
(a)
C
D
+ (_~)(8mN/27mq)(a + a/4m2) -1 ,
B
and for the cascade
(b)
/axo_ = - -~ [1 + (ol/4m2)(ms/mq) 1/2] -12mN/3m
Fig. 1. (a) Photon coupling to the three-quark core of the baryon B from the octet. (b) Pionic vertex renormalization of (a); C, D are the allowed baryons from the octet and decuplet.
× (2mN/3mq)(_2).
q2/4m2)-I
X UA [(q/2mA)F1Tu
+(2mA/q)Fl(i/2mA)OUVqv]
1 F 1 = - ~(1 + ax/4m~)-l(1/3ms)exp(-q2/6ax)
UA'
(13)
yields the magnetic m o m e n t of the A-hyperon in nuclear magnetons as /aA = " ~ ( 1 + ax/ams2) -1
2mN/3ms,
s
-- ~ (1 +(a/am2q)[3ms/(2ms+mq) ] 1/2)-1
NQM results, of w h i c h N 2 comes from the small Dirac quark wave function via eq. (5) and ] from the boundstate nature o f the large component (confmement). Since the measured magnetic m o m e n t o f the A-hyperon requires a strange quark mass m s > mq, we use the uds basis [1] for the hyperons [i.e. take the s quark to be distinguishable from the u, d quarks and the harmonic oscillator constant a x = ot(mx/mq)l/2 with mx = 3mqms/(2m q + ms) in the S = - i sector] rather than SU(6) symmetric wave functions. The A-current (AITUeslm) = (1 -
(15)
(14)
for the Y.-+hyperons one finds similarly
(16)
Note that these values again differ from the NQM re2 sults by ~ and the wave function renormalization factor. The magnetic m o m e n t contributions from the p h o t o n - q u a r k coupling o f fig. l a in table 1 with relativistic corrections to lowest order are comparable to those of the bare bag model. Thus, only an unrealistically large three-quark proton core 1]Vr~ = rp = 1.64 fm corresponding to mq = 125 MeV/c 2 would reproduce the experimental nucleon magnetic moments. Color hyperfine effects are also much too small to improve the situation [1]. However, pionic corrections at long distances are significant and necessary to understand better (within ~ 1 0 % of the data) the magnetic moments o f the baryon octet. The pion does not couple to the strange quark and, in the uds basis, one finds that the p i o n baryon couplings are those o f the SU(6) symmetry. Their common gaussian form factor and strength ocGN2/mq appear in the rtNN vertex of eq. (6) and are characteristic of the CQM. The parameter-free evaluation of the pionic contributions to the baryon magnetic moments in time-ordered perturbation theo-
Table 1 Baryon magnetic moments t~(CQM for rp = 1/x/r~= 1 fm, rnq = 208 MeV/c z , mx = 297 MeV]c2). a)
b)
Baryon
/z3,_Bq
/zn
p n A E+ Z~o ~-
2.443 -1.629 -0.597 2.372 -0.887 -1.363 -0.557
0.420 -0.420 0 0.250 -0.250 0.014 -0.014
/~q,-3q+lr 2.696 -1.991 -0.614 2.475 -1.088 -1.365 -0.552
c)
d)
, (ex)
tzrr-pair
"rr
tZo-pair
-0.345 0.345 0 0 0 0 0
0.267 -0.267 0 0 0 0 0
0.267 -0.178 0 0.118 -0.059 0 0
d)
/~total e)
NQM f)
Experiment g)
2.618 -1.913 -0.614 2.475 -1.088 -1.365 -0.552
2.70 -1.80 -0.60 2.59 -1.01 -1.36 -0.46
2.793 -1.913 -0.613 2.379 -1.10 -1.250 -0.69 -1.85
-+ 0.004 -+ 0.020 -+ 0.05 -+ 0.014 -+ 0.04 h) -+ 0.75
a) Fig. la. b) Fig. 2. c) Figs. la, lb, 2. d) Fig. 3. e) Figs. 1-3. f) Ref. [9]. g) Ref. [7]. h) Ref. [8]. 231
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
Their contribution to the nucleon magnetic moments is isovector and given by
B
C
,'
',
B
C
÷
B
;, C
B
÷
B
- "j{ C
mrr(2) + m(eX) = 3(gTrNN/5rr)2(1 + ~/4m2) 2 (r3N/mN)
B
Fig. 2. Pion current contributions to the baryon magnetic moment in time-ordered perturbation theory.
X f
ry of figs. lb and fig. 2 gives the results listed in table 1, which resemble those of the CBM's [3]. Note that the magnetic moment of the A-hyperon has only contributions.from fig. lb but not from fig. 2. The strange quark mass m s = 297 MeV/c 2 follows from the experimental value [7] /~A = --0.613 -+ 0.004 n.m. The magnetic moment contributions from one-pionexchanges (OPE) are neither complete nor gauge invariant without the pionic quark-pair current operator (cf. fig. 3) /'~2)(r; r l , r2) =
(G2/9m2) [-~(1)8 (r - r
+~ (2)8 (r --r 2)~(1 ) • V1 ] (, (1)
1)~(2)" V2
X • (2)) 3
X exp(-mr~lr 1 -r21)/4rrlr1 - r 2 l
x
-
,
(17)
X , (2))3o(1)- Vln(2)'V 2
r)"o0' - '21)},
X (~(1) X
V1)exp(-molr 1 -r2[)/47rlr1 - r 2 l
+ (1 ~- 2)
satisfies gauge invariance in conjunction with the relativistic spin-orbit correction of the quark charge operator
-
q).
X [1I(1) X P l ' VlS(r - r l ) ] +"
,(2) = m(2). N
(18)
G2mN fd3o d 3 Xgo(g 2 I 4m 2
× [r 1 × (o N ×
+B
Fig. 3. q~-pair currents. 232
j
B
(21)
The magnetic moment contribution from jo(2) in nuclear magnetons is obtained from
-
B ~ B
(20)
(17')
The pionic pair and exchange currents at the location r are connected via gauge invariance to the pion exchange potential VopE and the quark charge operator in lowest order O(mql), + ~ r 3 ( / ) ] 6(r
and 0 for the A, I¢+ and _---0hyperons. The exchange current contribution m (ex) is easy to calculate in momentum space [5]. For the nucleon there is considerable cancellation between the ~r-pair and exchange current contribution. This is obvious from eq. (19) and the numerical values shown in table 1. Since the attractive two-pion-exchange potential plays an important role in the nucleon-nucleon interaction at medium range, which is ofter parametrized by an effective scalar-isoscalar a-meson exchange, it is reasonable to consider also the correspotading ameson quark-pair current for the magnetic moments. A naive a-pair current
p(S~(r) = ( 1/8mq)[~ 2 1 +17.3(i) ]
Io(r) = exp(-m rr)/4rrr.
O(1)(r) = . ~ [ t 1
(19)
/'(2)(r;rl,r 2) = -('G2"4/mq)la2"'a + }r3(1)18( r _ r l )
which has recently been studied [5], and the pion-exchange current
/~ex) = _(G/3mq)2 (,(1)
dqq4 exp(-q2/2a) [ - 1 / 2 a + 1/(m2+q2)]
m~~r +q2
o
s~.(3)h + g-N J
Vl)exp(-molrl-r2l)]4rdr 1-r21
+ (1 ~ 2)1, which yields for the nucleon
(22)
Volume 165B, number 4,5,6
PHYSICS LETTERS
rn(2) _- (G 2mN/81r 2 -9o~rnq)(1 2 (3))(1 + ~/4mq) 2 -1 + 5r N o,N X f 0
dq q4 . e x p ( - q 2 / 2 ~ ) , m 2 + q2 o
(23)
26 December 1985
boosting the large quark wave function as in eq. (2). In other words, the two-body quark confinement potential in eq. (1) must be accompanied by a one-body potential.
References with G from eq. (7) and m o = -~ GeV/c 2 from ref. [41, m(2) = 0 for the A and ~ 0 hyperons, ~m (2) for the E'~ and ½m(2)n for the I~- hyperon. WherePas m (2) and r n ~ x) are purely isovect3or, m(2.)N gives the correct proton-to-neutron ratio - 3" However, the numerical values in table 1 show that such a naive o-pair current would affect the nucleon and ~-+ magnetic moments adversely, had we included it in the total magnetic moments. The data does not appear to support it. In conclusion, we have calculated the baryon octet magnetic moments using harmonic oscillator wave functions and constituent quark masses with relativistic and pionic contributions. With two constituent quark masses mq = 208 MeV/c 2 and m s = 297 MeV/c 2 (and harmonic oscillator parameter a = ~ r n 2 from gA -5 g ) , w e reproduce the magnetic moments of the baryon octet with ~ 1 0 % accuracy. This is comparable to the NQM, but the CQM improves gA via relativistic corrections and becomes consistent with the nuclear forces at long distances b y including a p i o n - q u a r k coupling. To obtain better agreement with the magnetic moments at a smaller and more realistic three-quark core radius than 1/x/if = 1 fm, an independent small Dirac quark wave function is needed, instead o f simply -
[ 1] See, e.g., F. Close, An Introduction to quarks and partons (Academic Press, New York, 1979), and references therein; A. DeRtijula, H. Georgi and S. Glashow, Phys. Rev. D12 (1975) 147; N. Isgur and G. Karl, Phys. Rev. D21 (1980) 3175 ; H.J. Lipkin, Quark model spectroscopy (Steamboat Springs, CO, 1984), AIP Conf. Proc. Vol. 123 (AIP, New York, 1984) p. 346; N. Isgur in: Proc. 2nd Kaon factory physics Workshop (Vancouver, 1981), TRIUMF Report TRI-81-4 (1981). [2] T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [3] S. Theberge and A.W. Thomas, Nucl. Phys. A393 (1983) 252; F. Myhrer, Phys. Lett. 125B (1983) 359. [4] M. Bozoian and H.J. Weber, Phys. Rev. C28 (1983) 811. [5] H.J. Weber and M. Weyrauch, University of Virginia preprint (1985). [6] R. Tegen, M. Schedl and W. Weise, Phys. Lett. 125B (1983) 9; A.S. Rinat, Phys. Lett. 134B (1984) 11. [7] Particle Data Group, Rev. Mod. Phys. 56 (1984) No. 2 (!l). [8] R. Rameika, A. Beretvas, L. Deck, T. Devlin, K.B. Luk, R. Whitman, P.T. Cox, C. Dukes, J. Dworkin, O.E. Overseth, R. Handier, B. Lundberg, L. Pondrom, M. Sheaff, C. Wilkinson and K. Heller, Phys. Rev. kett. 52 (1984) 581. [9] J. Franklin, Phys. Rev. D30 (1984) 1542.
233