- Email: [email protected]
Are strange baryons non-spherical?
PHYSICS LETTERS B
Volume 182,number 2
ARE STRANGE Marek JEZABEK aHigh Energy bjnstitute of CDepartment and Service Received
BARYONS
NON-SPHERICAL?
a, Maciej A. NOWAK
18
*
b and Mannque
RHO ’
Physics Laboratory, Institute of Nuclear Physics, ul. Kawiory 26a, PL-30059 Physics, Jagellonian University, ul. Reymonta 4, PI-30059 Cracow, Poland of Physics, SUNY, Stony Brook, NY II 794, USA de Physique VzIPorique, CEN Saclay, F-91 191 Gif-sur- Yvette, France
27 September
December 1986
Cracow, Poland
1986
We calculate the distribution of baryon number for the ground state hyperons. Strange baryons are described as bound kaon-skyrmion systems. A striking consequence of the model is the non-sphericity of baryonic density for strange baryons.
The Skyrme soliton approach to low-energy hadron physics works well in describing non-strange baryons [ I]. Further improvements such as addition of quark core (chiral bag) *I and/or incorporation of vector mesons [3] can lead to even better agreement with the experiment. However, attempts to incorporate strangeness into this model in a standard way [4] run into serious difficulties [5]. Callan and Klebanov [6] have shown that baryons carrying heavy flavors can be described by bound states of the corresponding heavy mesons in the background field of the basic SU, skyrmion. In particular hyperons can be considered as bound kaon-skyrmion systems. The bound states exist in the channels needed to reproduce the quark model quantum numbers of strange baryons. The model encounters some difficulties with the pattern of hyperfine splitting. These problems may be related to higher derivative terms in the effective lagrangian for the Goldstone fields. However, modelindependent mass relations derived within this approach work very well. Thus, it is of considerable interest to work out other predictions of this scheme and compare them with the experiment. In ref. [7] the radial density of baryon number has been calculated. It follows from the model of Callan * Work supported
in part by USDOE ACQ2-76-ER13001. *’ For a recent review see e.g. ref. [ 21.
0370-2693/86/s (North-Holland
under
contract
1 p(r,0,cp) = e.. Tr(LiLjLk), 241.r~ lJk where Li = U’a, U, and r, 0, q denote spherical coordinates in space. Since only spatial derivatives appear in eq. (1) we can substitute a “static” Callan-Klebanov ansatz for U:
(2)
U = NQN, where I N=
0 exp(icur*ri)
,o
DE-
03.50 0 Elsevier Science Publishers Physics Publishing Division)
and Klebanov that the ground states of strange baryons are “larger” than non-strange ones [7]. In the present letter we derive an even more surprising consequence of the bound state approach to strangeness. We shall show that the distributions of baryon number for strange baryons are non-spherical. This result is clearly different from the expectations of many other models of baryons, and it is conceivable that this structure can be seen by experiments. We calculate the density of baryon number
0
0
(3)
1
O>a=-:e>&r,and B.V.
139
Volume 182, number
PHYSICS
2
I8 December
LETTERS B
We shall also use mi.i s a .mi, (IL’-da/& dr. It follows from eq. (5-I that mimi=
L 1
= 1 - 2 sin2 i @+ 0 0 +isinyO
0
0 0
l.
r;+
01
i
0 0 1
(
cos 6
= sin 0 exp(iq)
1 ’
forL = 1,
(8)
(4) (9) where
g,(cr)=eijk
(5)
=O,
1.
p = -(247r2)-l
and 7 is defined by eq. (4), see also ref. [7]. As explained in ref. [6] one can perform a partial wave analysis of kaon modes. In what follows we restrict our discussion to the modes corresponding to K = l/2, where K = Z + L denote generators of combined isospin and spatial rotations. For these modes y = y(r) *’ One can easily derive explicit expressions for .$‘s. For K = l/2 and K3 = l/2 we have:
E=(i),forL
and y’ E d-y/
From eqs. (1) and (2) one derives the following expression for the density:
In eq. (3) ir = r/r and ri are the Pauli matrices. The hedgehog function 0 = e(r) can be obtained in a standard way from the variation of the Skyrme action, c.f. ref. [I]. In eq. (4) A0 denote the Cell-Mann matrices and K, kaonic fields; g is a normalized two-component object, i.e. .!+g = 1,
1986
Tr[N-‘NiN1:‘Nk]
gz(‘Y) = eijk Tr[Q-‘QiQl~‘QkI
~Jw)
= k v$vY)
= - 12 sin20a’,(10a) r2
= 0,
(lob)
-f,(-WK
UOc)
and f,,(a,y)
= eijkTr(QNiN1:lQ-lN-lNk),
(1 la)
. fI(%‘Y)=
eijkTr(QiQjTIN-lNk),
(llb)
f2(OL?7) = eijk Tr(NjNJylQ-‘Qk),
(llc)
(6a)
f3(a,r)
(lld)
t6b)
After a lengthy but straightforward tains:
corresponding to the orbital agular moment L of the kaon. Let us introduce the following shorthand notation: Ni G aiN, Qi E aiQ etc., and define
= eiik Tr(QiNN1:lQ-lN-lNk). calculation
one ob-
hO=--(COSY+2cos2cYcos4:~ p2 + 4s2 sin20 sin4 :r) sin2 o 01’,
(12a)
h 1 = 2(X sin27 - T sin4 $y) QI’
mi = Tr(E[‘r,),
(7a)
Wkr = Tr(G+J+rl),
(7b)
+ i(Z sin2 $7 + IV cos 7) sin 2a sin y y’
(7c)
t b [(T t P - D2)sin4 & t Y sin2y] sin 201, (12b)
x ilk =
Tr[@$Xajt+)~~l.
*2 For K > l/2 y = y(r,o) and non-sphericity of baryonic density is evident. Moreover, there are no bound states in these channels [ 61.
140
2 h, = - ; (W sin2y t 22 sin4 :r) sin2a LY’,
(12c)
Volume 182, number
PHYSICS
2
h3=2[2r(2S.Dr2
V+Z)sin4i7cr’
+ (cos2 i7 + S2 sin2 k7) sin 2a sin 7 7’
(124
+ V sin 201sin4 :7] sin2cY, where D=m. z;i’
S = mini,
P = miimii, 1 I
T = ‘iik elpq nk nq ml.>imp.j, > V = 2Sninj
X = ii eijkXiilnknl,
mi.i,
Y=kie.. yk X ijk -X, W=
ieijkni(IVjk
LETTERS
B
18 December
Eqs. (13a), (13b) contain our main result. It is clear that also for K = l/2 partial waves the distribution of baryon number in strange baryons must depend on the polar angle 0. The above result is surprising and one might be tempted to conclude from this that the Callan-Klebanov ansatz should be rejected. We do not share such an extreme point of view: we see no a priori reason for rejecting the idea that hyperons could be non-spherical. The model of Callan and Klebanov has many attractive features and leads to a few successful results, so, non-sphericity of strange baryons which follows from this model could also be considered as an interesting challenge for both phenomenology and experiment . Integrating p over the solid angle, one derives from eqs. (13) the formula for the radial density:
+ $mj.k), (14)
Z = nimi mi.i -S-D. It is evident now that unlike the Skyrme hedgehog the ansatz of Callan and Klebanov need not lead to a spherical, i.e. independent of 0 and cp,expression for p. For the Skyrme ansatz the only geometrical object which depends on 13and L+J is the normal vector r. However, p does not carry any vector indices and, consequently, it can depend only on (r)2. Thus, p is spherical for the Skyrme ansatz. Callan and Klebanov introduce through their ansatz other objects like mi and its derivatives, so, p need not be spherical for this case. We calculated the density from eq. (9) using the expressions (6a), (6b) for ,$ and obtained for L = 0: p = (27r2r2)-l
1986
and then for the baryon number outside a sphere of radius R :
B(R) = J
da j(r).
(15)
R
The baryon number should not depend on continuous variations of 0(r) and y(r) for e(O) and 0(-) fixed, so, B cannot depend on R explicitly, i.e.; B(R) = @e(R), 7(R)),
(16)
and, consequently,
sin2 :0 (17)
x {e’[l +(I +2c0se)c0s7 + 2 cos e sin4 :7 + 4 sin2 :e sin4:7
-7’sinysine
(cos2~7+sin2~7c0s2e)),
c0s2e]
On the other hand, using dimensional arguments and spherical symmetry of 0 and 7, one can show that (13a)
and for L = 1:
(18)
p = (27&2)-l x {e’(sin2s
~0s~
tr
+ 5
Comparing eqs. (17) and (18) one obtains the following consistency conditions:
sin27 ~0s~ :e
C,=O,
t4sin4~7c0s4~ec0s2e)t7’sinec0s2~e~in7 X (cos2 :7 + sin2 $u c0s2e)}.
(13b)
acpe
(19a) = aqa7.
(19b)
141
PHYSICS LETTERS B
Volume 182, number 2
18 December 1986
We have checked that the above consistency conditions are fulfilled for p given by the formulae (13a), (13b) *3. After elementary integration, using eqs. (13a), (13b), we obtain
in ref. [6] it has been shown that the ground states car respond to L = 1. Thus, we uncover another surprising consequence of the model: the baryonic size of the ground state is larger than the baryonic size of oddparity excited states.
s(e,r)=-;(e
We thank J.P. Blaizot, M. Praszalowicz, V. Vento and K. Zalewski for discussions. We thank also H. Nadeau who rederived some of our formulae. M.J. is indebted to Service de Physique Theorique in Saclay where part of this work was done. M.A.N. is partially supported by the Contract CPBP.Ol.09.
- $
-;sin2e>
sin 6 sin2 &3
- sin2 iy)9(L,
e>
(20)
where 7(L,e)=-sin2:e,
= c0s2:e,
forL=O, for L = 1,
(21)
and -n =G0 < 0. Our result for L = 1 is equivalent to the result given in ref. [7]. It follows immediately from eqs. (20) and (21) that the baryonic size of a strange baryon for L = 1 is larger and for L = 0 smaller than the baryonic size of a non-strange skyrmion. However, *3 Thus, one can calculate the radial density i from B using eq. (17), as it has been done in ref. [ 71. However, despite B can be expressed as B(r) = JJ dp de sin u one cannot identify p and -dw/dr because at! fixed c the consistency conditions analogous to (19a), (19b) are not fulfiied.
the fact that
X w(r,@,lp),
142
References 111 G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 5.52.
[21 G.E. Brown and M. Rho, Comm. Part. Nucl. Phys., to . be published. [31 M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yamagida, Phys. Rev. Lett. 54 (1985) 1215. 141 See e.g.: P.O. Mazur, M.A. Nowak and M. Praszalowicz, Phys. Lett. B 147 (1984) 137. ISI M. Praszalowicz, Phys. Lett. B 158 (1985) 264. [61 C.G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985) 365. [71H. Nadeau, M.A. Nowak, M. Rho and V. Vento, Phys. Rev. Lett., to be published.
Volume 182,number 2
ARE STRANGE Marek JEZABEK aHigh Energy bjnstitute of CDepartment and Service Received
BARYONS
NON-SPHERICAL?
a, Maciej A. NOWAK
18
*
b and Mannque
RHO ’
Physics Laboratory, Institute of Nuclear Physics, ul. Kawiory 26a, PL-30059 Physics, Jagellonian University, ul. Reymonta 4, PI-30059 Cracow, Poland of Physics, SUNY, Stony Brook, NY II 794, USA de Physique VzIPorique, CEN Saclay, F-91 191 Gif-sur- Yvette, France
27 September
December 1986
Cracow, Poland
1986
We calculate the distribution of baryon number for the ground state hyperons. Strange baryons are described as bound kaon-skyrmion systems. A striking consequence of the model is the non-sphericity of baryonic density for strange baryons.
The Skyrme soliton approach to low-energy hadron physics works well in describing non-strange baryons [ I]. Further improvements such as addition of quark core (chiral bag) *I and/or incorporation of vector mesons [3] can lead to even better agreement with the experiment. However, attempts to incorporate strangeness into this model in a standard way [4] run into serious difficulties [5]. Callan and Klebanov [6] have shown that baryons carrying heavy flavors can be described by bound states of the corresponding heavy mesons in the background field of the basic SU, skyrmion. In particular hyperons can be considered as bound kaon-skyrmion systems. The bound states exist in the channels needed to reproduce the quark model quantum numbers of strange baryons. The model encounters some difficulties with the pattern of hyperfine splitting. These problems may be related to higher derivative terms in the effective lagrangian for the Goldstone fields. However, modelindependent mass relations derived within this approach work very well. Thus, it is of considerable interest to work out other predictions of this scheme and compare them with the experiment. In ref. [7] the radial density of baryon number has been calculated. It follows from the model of Callan * Work supported
in part by USDOE ACQ2-76-ER13001. *’ For a recent review see e.g. ref. [ 21.
0370-2693/86/s (North-Holland
under
contract
1 p(r,0,cp) = e.. Tr(LiLjLk), 241.r~ lJk where Li = U’a, U, and r, 0, q denote spherical coordinates in space. Since only spatial derivatives appear in eq. (1) we can substitute a “static” Callan-Klebanov ansatz for U:
(2)
U = NQN, where I N=
0 exp(icur*ri)
,o
DE-
03.50 0 Elsevier Science Publishers Physics Publishing Division)
and Klebanov that the ground states of strange baryons are “larger” than non-strange ones [7]. In the present letter we derive an even more surprising consequence of the bound state approach to strangeness. We shall show that the distributions of baryon number for strange baryons are non-spherical. This result is clearly different from the expectations of many other models of baryons, and it is conceivable that this structure can be seen by experiments. We calculate the density of baryon number
0
0
(3)
1
O>a=-:e>&r,and B.V.
139
Volume 182, number
PHYSICS
2
I8 December
LETTERS B
We shall also use mi.i s a .mi, (IL’-da/& dr. It follows from eq. (5-I that mimi=
L 1
= 1 - 2 sin2 i @+ 0 0 +isinyO
0
0 0
l.
r;+
01
i
0 0 1
(
cos 6
= sin 0 exp(iq)
1 ’
forL = 1,
(8)
(4) (9) where
g,(cr)=eijk
(5)
=O,
1.
p = -(247r2)-l
and 7 is defined by eq. (4), see also ref. [7]. As explained in ref. [6] one can perform a partial wave analysis of kaon modes. In what follows we restrict our discussion to the modes corresponding to K = l/2, where K = Z + L denote generators of combined isospin and spatial rotations. For these modes y = y(r) *’ One can easily derive explicit expressions for .$‘s. For K = l/2 and K3 = l/2 we have:
E=(i),forL
and y’ E d-y/
From eqs. (1) and (2) one derives the following expression for the density:
In eq. (3) ir = r/r and ri are the Pauli matrices. The hedgehog function 0 = e(r) can be obtained in a standard way from the variation of the Skyrme action, c.f. ref. [I]. In eq. (4) A0 denote the Cell-Mann matrices and K, kaonic fields; g is a normalized two-component object, i.e. .!+g = 1,
1986
Tr[N-‘NiN1:‘Nk]
gz(‘Y) = eijk Tr[Q-‘QiQl~‘QkI
~Jw)
= k v$vY)
= - 12 sin20a’,(10a) r2
= 0,
(lob)
-f,(-WK
UOc)
and f,,(a,y)
= eijkTr(QNiN1:lQ-lN-lNk),
(1 la)
. fI(%‘Y)=
eijkTr(QiQjTIN-lNk),
(llb)
f2(OL?7) = eijk Tr(NjNJylQ-‘Qk),
(llc)
(6a)
f3(a,r)
(lld)
t6b)
After a lengthy but straightforward tains:
corresponding to the orbital agular moment L of the kaon. Let us introduce the following shorthand notation: Ni G aiN, Qi E aiQ etc., and define
= eiik Tr(QiNN1:lQ-lN-lNk). calculation
one ob-
hO=--(COSY+2cos2cYcos4:~ p2 + 4s2 sin20 sin4 :r) sin2 o 01’,
(12a)
h 1 = 2(X sin27 - T sin4 $y) QI’
mi = Tr(E[‘r,),
(7a)
Wkr = Tr(G+J+rl),
(7b)
+ i(Z sin2 $7 + IV cos 7) sin 2a sin y y’
(7c)
t b [(T t P - D2)sin4 & t Y sin2y] sin 201, (12b)
x ilk =
Tr[@$Xajt+)~~l.
*2 For K > l/2 y = y(r,o) and non-sphericity of baryonic density is evident. Moreover, there are no bound states in these channels [ 61.
140
2 h, = - ; (W sin2y t 22 sin4 :r) sin2a LY’,
(12c)
Volume 182, number
PHYSICS
2
h3=2[2r(2S.Dr2
V+Z)sin4i7cr’
+ (cos2 i7 + S2 sin2 k7) sin 2a sin 7 7’
(124
+ V sin 201sin4 :7] sin2cY, where D=m. z;i’
S = mini,
P = miimii, 1 I
T = ‘iik elpq nk nq ml.>imp.j, > V = 2Sninj
X = ii eijkXiilnknl,
mi.i,
Y=kie.. yk X ijk -X, W=
ieijkni(IVjk
LETTERS
B
18 December
Eqs. (13a), (13b) contain our main result. It is clear that also for K = l/2 partial waves the distribution of baryon number in strange baryons must depend on the polar angle 0. The above result is surprising and one might be tempted to conclude from this that the Callan-Klebanov ansatz should be rejected. We do not share such an extreme point of view: we see no a priori reason for rejecting the idea that hyperons could be non-spherical. The model of Callan and Klebanov has many attractive features and leads to a few successful results, so, non-sphericity of strange baryons which follows from this model could also be considered as an interesting challenge for both phenomenology and experiment . Integrating p over the solid angle, one derives from eqs. (13) the formula for the radial density:
+ $mj.k), (14)
Z = nimi mi.i -S-D. It is evident now that unlike the Skyrme hedgehog the ansatz of Callan and Klebanov need not lead to a spherical, i.e. independent of 0 and cp,expression for p. For the Skyrme ansatz the only geometrical object which depends on 13and L+J is the normal vector r. However, p does not carry any vector indices and, consequently, it can depend only on (r)2. Thus, p is spherical for the Skyrme ansatz. Callan and Klebanov introduce through their ansatz other objects like mi and its derivatives, so, p need not be spherical for this case. We calculated the density from eq. (9) using the expressions (6a), (6b) for ,$ and obtained for L = 0: p = (27r2r2)-l
1986
and then for the baryon number outside a sphere of radius R :
B(R) = J
da j(r).
(15)
R
The baryon number should not depend on continuous variations of 0(r) and y(r) for e(O) and 0(-) fixed, so, B cannot depend on R explicitly, i.e.; B(R) = @e(R), 7(R)),
(16)
and, consequently,
sin2 :0 (17)
x {e’[l +(I +2c0se)c0s7 + 2 cos e sin4 :7 + 4 sin2 :e sin4:7
-7’sinysine
(cos2~7+sin2~7c0s2e)),
c0s2e]
On the other hand, using dimensional arguments and spherical symmetry of 0 and 7, one can show that (13a)
and for L = 1:
(18)
p = (27&2)-l x {e’(sin2s
~0s~
tr
+ 5
Comparing eqs. (17) and (18) one obtains the following consistency conditions:
sin27 ~0s~ :e
C,=O,
t4sin4~7c0s4~ec0s2e)t7’sinec0s2~e~in7 X (cos2 :7 + sin2 $u c0s2e)}.
(13b)
acpe
(19a) = aqa7.
(19b)
141
PHYSICS LETTERS B
Volume 182, number 2
18 December 1986
We have checked that the above consistency conditions are fulfilled for p given by the formulae (13a), (13b) *3. After elementary integration, using eqs. (13a), (13b), we obtain
in ref. [6] it has been shown that the ground states car respond to L = 1. Thus, we uncover another surprising consequence of the model: the baryonic size of the ground state is larger than the baryonic size of oddparity excited states.
s(e,r)=-;(e
We thank J.P. Blaizot, M. Praszalowicz, V. Vento and K. Zalewski for discussions. We thank also H. Nadeau who rederived some of our formulae. M.J. is indebted to Service de Physique Theorique in Saclay where part of this work was done. M.A.N. is partially supported by the Contract CPBP.Ol.09.
- $
-;sin2e>
sin 6 sin2 &3
- sin2 iy)9(L,
e>
(20)
where 7(L,e)=-sin2:e,
= c0s2:e,
forL=O, for L = 1,
(21)
and -n =G0 < 0. Our result for L = 1 is equivalent to the result given in ref. [7]. It follows immediately from eqs. (20) and (21) that the baryonic size of a strange baryon for L = 1 is larger and for L = 0 smaller than the baryonic size of a non-strange skyrmion. However, *3 Thus, one can calculate the radial density i from B using eq. (17), as it has been done in ref. [ 71. However, despite B can be expressed as B(r) = JJ dp de sin u one cannot identify p and -dw/dr because at! fixed c the consistency conditions analogous to (19a), (19b) are not fulfiied.
the fact that
X w(r,@,lp),
142
References 111 G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 5.52.
[21 G.E. Brown and M. Rho, Comm. Part. Nucl. Phys., to . be published. [31 M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yamagida, Phys. Rev. Lett. 54 (1985) 1215. 141 See e.g.: P.O. Mazur, M.A. Nowak and M. Praszalowicz, Phys. Lett. B 147 (1984) 137. ISI M. Praszalowicz, Phys. Lett. B 158 (1985) 264. [61 C.G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985) 365. [71H. Nadeau, M.A. Nowak, M. Rho and V. Vento, Phys. Rev. Lett., to be published.