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Strong couplings and magnetic moments in soliton models
Volume 146B, number .3,4
PHYSICS LETTERS
11 October 1984
STRONG COUPLINGS AND MAGNETIC MOMENTS IN SOLITON MODELS E. GUADAGN1NI Dipartimento di Fisica, Universitd di Pisa, Piazza Torricelli, 2, 56100 Pisa, Italy Received 10 May 1984
We consider the large-N chiral models of strong dynamics. The model independent properties of the meson-baryon strong couplings and some aspects concerning the magnetic moments are discussed.
Many properties of meson phenomenology emerge in the large-N limit of QCD [1]. We do not know how to produce a systematic 1/N expansion of QCD. However, it is plausible to assume that in this limit the physics is described by an effective current-algebra lagrangian [2], where the chiral group SU(3) X SU(3) is realized in a non-linear way [3]. In fact, under reasonable assumptions the correct pattern of dynamical chiral symmetry breaking occurs [4]. A minimal model of this type is described in terms o f a field U ( x ) E SU(3), which is related to the octet of the pseudo-scalar mesons by
U(x) = 1 + (2i/F~r) xa~ra(x) + ....
(1)
The small fluctuation expansion describes the low energy physics of the mesons. Quite surprisingly, it turns out that the baryons also can be described in these models [5]. They appear as solitons of the current-algebra lagrangian [6]. This fact confirms the Skyrme idea that at low energy the nucleons can be considered as solitons "made" of pions [5], and agrees with largeN arguments [7], which suggest that the baryons may emerge as solitons in the 1/N expansion. First of all there are topological reasons for the stability of non-trivial static solutions of the equations of motion [8] ; the topological charge corresponds to the baryonic number. Then, when the classical soliton is quantized, the right quantum numbers of the "unitary spin" SU(3) and of the spin are forced by the presence o f the Wess-Zumino term [9] in the effective lagrangian [6, I 0 ]. In order to prevent the size o f the soliton to shrink 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
tO zero it is necessary to add to the usual lagrangian with the term (F2/16) Tri~uU+OuUI
(2)
other terms ,x ; the simplest one is the Skyrme term
[5] (1/32e 2) Tr[U+~uU, U+~vU] 2
(3)
The static properties of the nucleons in the Skyrme model are in agreement with the data within ~30% [12]. Clearly, there is some arbitrariness in the choice of the terms to be added in the lagrangian. Therefore a quantitative analysis of a particular model does not represent a conclusive test. In fact, only the constraints imposed by symmetry considerations have a predictive value in an effective lagrangian. In this note we consider the model independent properties o f the strong couplings between the pseudoscalar mesons and the baryons, and some qualitative features of the magnetic moments are pointed out. In the case of two flavours, a discussion of these arguments has been given in ref. [12]. Here, within SU(3), we want to find the value of the parameter a, which gives the relative strength of the F and D couplings. In addition, consistency with the results obtained for SU(2) requires that the strong decays of the baryons of the decouplet should be correctly predicted from the knowledge ofgTrNN. We use the same method of ref. [12] ;the asymptotic behaviour of the soliton so,1 Recently, an interesting alternative has been proposed through the inclusion of vector mesons [ 11]. 237
Volume 146B, number 3,4
1,1i =
~ Tr[QAoiA-1] + ....
PHYSICS LETTERS
(14)
where the constant/3 is model dependent. The operator Tr[QAoiA -1] is certainly present in the expression of/~i, but we have also other terms. Now, large-N arguments suggest that the terms containing time derivatives of the variable A are suppressed [12]. Without time derivatives of A, the only operator with the right transformation properties is Tr [QA oiA - 1 ]. However, it is a too restrictive assumption to consider this operator only. For instance, with two flavours we would get ~n//.tp = --1. To obtain a reasonable value o f Lln//./p, part of the anomalous contribution to the electromagnetic current was taken into account in ref. [12]. We find here a delicate point; it is difficult to make model independent predictions on the magnetic moments without the introduction of new free parameters. We can use anyway an handwaving argument to get a rough idea of the situation. We assume that the relevant contribution of the missing terms in the expression (14) is given simply by 2u0qS,
(15)
where q is the electric charge of the baryons (in units of e) and S is the spin operator. This is what a complete relativistic quantization suggests. So, we take/~0 equal to eti/2mpc. We then compute the expectation value of the operator
{3Tr[QAo3A-1] + 2/a0qS 3
(16)
Table 1 Magnetic moments. Baryon
Predicted
Experiment
Quark model
n p A ~+ ~Z° ---
-1.32 2.76 -0.66 2.76 -1.44 -1.32 -1.44
-1.913 2.793 -0.614 2.33 -1.41 -1.20 -1.85
-1.86 2.79 -0.58 2.68 -1.05 -1.40 -0.47
References [11 See e.g.S. Coleman, I21 [31
[41
[51 [61
z+ ~- 0 - - - g:zj 3 ,
+u0, z-
"-- = ~/3 -/a 0 .
(17)
With t3 = - 3 . 3 n.m. we obtain table 1. The values of the magnetic moments predicted in a quark model [ 14], are also shown. It is remarkable that with this simple argument one obtains the magnetic moments of the baryons with nearly the same accuracy of the quark model. In fact, using the expression (16) we have ignored SU(3) breaking due to the quark masses. Summarizing, the model independent properties of the strong couplings between the pseudo-scalar mesons and the baryons are in good agreement with experiment. This supports the soliton picture of the baryons
+- 0.005 -2_0.13 _+0.25 +_0.06 -2_0.75
at low energy. As for the magnetic moments, a more detailed analysis of an explicit model is needed to confirm the encouraging results obtained with general arguments.
on the baryon states with spin up. We obtain
p=-
11 October 1984
l / N , Lectures Intern. School of Subnuclear Physics "Ettore Majorana" (Erice, 1979), and references quoted therein. E. Witten, Ann. Phys. (NY) 128 (1980) 363. S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C.G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. S. Coleman and E. Witten, Phys. Rev. Lett. 45 (1980) 100; E. Guadagnini and K. Konishi, Phys. Scripta 26 (1982) 67. T.H.R. Skyrme, Proc. Roy. Soc. A260 (1961) 127; NucL Phys. 31 (1962) 556; J. Mat. Phys. 12 (1971) 1735. N.K. Pak and H.C. Tze, Ann. Phys. 117 (1979) 164; J. Gibson and H.C. Tze, Nucl. Phys. B183 (1981) 524; D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762; J.G. Williams, J. Math. Phys. 11 (1970) 2611; J. Goldstone and F. Wilcek, Phys. Rev. Lett. 47 (1981) 986; A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49 (1982) 1124; Phys. Rev. D27 (1983) 1153; A. Chodos and C.B. Thorn, Phys. Rev. D 12 (1975) 2733; M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Lett. 51 (1983) 747; A.D. J acson and M. Rho, Phys. Rev. Lett. 51 (1983) 751; E. Witten, Nucl. Phys. B223 (1983) 433; G. Adkins and C.R. Nappi, Nucl. Phys: B233 (1984) 109; J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51 (1983) 1518; F. Wilcek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250; E. D'Hoker and E. Farhi, Phys. Lett. 134B (1984) 86; P.O. Mazur, M.A. Novak and M. Praszalowicz, Cracow preprint TPJU 4/84.
239
Volume 146B, number 3,4 [7] [81 [9] [101 [11]
240
PHYSICS LETTERS
E. Witten, Nucl. Phys. B160 (1979) 57. E. Witten, Nucl. Phys. B223 (1983) 422. J. Wess and B, Zumino, Phys. Lett. 37B (1971) 95. E. Guadagnini, Nucl. Phys. B236 (1984) 35. G.S. Adkins and C.R. Nappi, Phys. Lett. 137B (1984) 251.
11 October 1984
[121 G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [ 13] S. Gasiorowicz, Elementary particle physics (Wiley, New York, 1967). [ 14] D.H. Perkins, Introduction to high energy physics (Addison-Wesley,Reading, MA, 1983).
PHYSICS LETTERS
11 October 1984
STRONG COUPLINGS AND MAGNETIC MOMENTS IN SOLITON MODELS E. GUADAGN1NI Dipartimento di Fisica, Universitd di Pisa, Piazza Torricelli, 2, 56100 Pisa, Italy Received 10 May 1984
We consider the large-N chiral models of strong dynamics. The model independent properties of the meson-baryon strong couplings and some aspects concerning the magnetic moments are discussed.
Many properties of meson phenomenology emerge in the large-N limit of QCD [1]. We do not know how to produce a systematic 1/N expansion of QCD. However, it is plausible to assume that in this limit the physics is described by an effective current-algebra lagrangian [2], where the chiral group SU(3) X SU(3) is realized in a non-linear way [3]. In fact, under reasonable assumptions the correct pattern of dynamical chiral symmetry breaking occurs [4]. A minimal model of this type is described in terms o f a field U ( x ) E SU(3), which is related to the octet of the pseudo-scalar mesons by
U(x) = 1 + (2i/F~r) xa~ra(x) + ....
(1)
The small fluctuation expansion describes the low energy physics of the mesons. Quite surprisingly, it turns out that the baryons also can be described in these models [5]. They appear as solitons of the current-algebra lagrangian [6]. This fact confirms the Skyrme idea that at low energy the nucleons can be considered as solitons "made" of pions [5], and agrees with largeN arguments [7], which suggest that the baryons may emerge as solitons in the 1/N expansion. First of all there are topological reasons for the stability of non-trivial static solutions of the equations of motion [8] ; the topological charge corresponds to the baryonic number. Then, when the classical soliton is quantized, the right quantum numbers of the "unitary spin" SU(3) and of the spin are forced by the presence o f the Wess-Zumino term [9] in the effective lagrangian [6, I 0 ]. In order to prevent the size o f the soliton to shrink 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
tO zero it is necessary to add to the usual lagrangian with the term (F2/16) Tri~uU+OuUI
(2)
other terms ,x ; the simplest one is the Skyrme term
[5] (1/32e 2) Tr[U+~uU, U+~vU] 2
(3)
The static properties of the nucleons in the Skyrme model are in agreement with the data within ~30% [12]. Clearly, there is some arbitrariness in the choice of the terms to be added in the lagrangian. Therefore a quantitative analysis of a particular model does not represent a conclusive test. In fact, only the constraints imposed by symmetry considerations have a predictive value in an effective lagrangian. In this note we consider the model independent properties o f the strong couplings between the pseudoscalar mesons and the baryons, and some qualitative features of the magnetic moments are pointed out. In the case of two flavours, a discussion of these arguments has been given in ref. [12]. Here, within SU(3), we want to find the value of the parameter a, which gives the relative strength of the F and D couplings. In addition, consistency with the results obtained for SU(2) requires that the strong decays of the baryons of the decouplet should be correctly predicted from the knowledge ofgTrNN. We use the same method of ref. [12] ;the asymptotic behaviour of the soliton so,1 Recently, an interesting alternative has been proposed through the inclusion of vector mesons [ 11]. 237
Volume 146B, number 3,4
1,1i =
~ Tr[QAoiA-1] + ....
PHYSICS LETTERS
(14)
where the constant/3 is model dependent. The operator Tr[QAoiA -1] is certainly present in the expression of/~i, but we have also other terms. Now, large-N arguments suggest that the terms containing time derivatives of the variable A are suppressed [12]. Without time derivatives of A, the only operator with the right transformation properties is Tr [QA oiA - 1 ]. However, it is a too restrictive assumption to consider this operator only. For instance, with two flavours we would get ~n//.tp = --1. To obtain a reasonable value o f Lln//./p, part of the anomalous contribution to the electromagnetic current was taken into account in ref. [12]. We find here a delicate point; it is difficult to make model independent predictions on the magnetic moments without the introduction of new free parameters. We can use anyway an handwaving argument to get a rough idea of the situation. We assume that the relevant contribution of the missing terms in the expression (14) is given simply by 2u0qS,
(15)
where q is the electric charge of the baryons (in units of e) and S is the spin operator. This is what a complete relativistic quantization suggests. So, we take/~0 equal to eti/2mpc. We then compute the expectation value of the operator
{3Tr[QAo3A-1] + 2/a0qS 3
(16)
Table 1 Magnetic moments. Baryon
Predicted
Experiment
Quark model
n p A ~+ ~Z° ---
-1.32 2.76 -0.66 2.76 -1.44 -1.32 -1.44
-1.913 2.793 -0.614 2.33 -1.41 -1.20 -1.85
-1.86 2.79 -0.58 2.68 -1.05 -1.40 -0.47
References [11 See e.g.S. Coleman, I21 [31
[41
[51 [61
z+ ~- 0 - - - g:zj 3 ,
+u0, z-
"-- = ~/3 -/a 0 .
(17)
With t3 = - 3 . 3 n.m. we obtain table 1. The values of the magnetic moments predicted in a quark model [ 14], are also shown. It is remarkable that with this simple argument one obtains the magnetic moments of the baryons with nearly the same accuracy of the quark model. In fact, using the expression (16) we have ignored SU(3) breaking due to the quark masses. Summarizing, the model independent properties of the strong couplings between the pseudo-scalar mesons and the baryons are in good agreement with experiment. This supports the soliton picture of the baryons
+- 0.005 -2_0.13 _+0.25 +_0.06 -2_0.75
at low energy. As for the magnetic moments, a more detailed analysis of an explicit model is needed to confirm the encouraging results obtained with general arguments.
on the baryon states with spin up. We obtain
p=-
11 October 1984
l / N , Lectures Intern. School of Subnuclear Physics "Ettore Majorana" (Erice, 1979), and references quoted therein. E. Witten, Ann. Phys. (NY) 128 (1980) 363. S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C.G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. S. Coleman and E. Witten, Phys. Rev. Lett. 45 (1980) 100; E. Guadagnini and K. Konishi, Phys. Scripta 26 (1982) 67. T.H.R. Skyrme, Proc. Roy. Soc. A260 (1961) 127; NucL Phys. 31 (1962) 556; J. Mat. Phys. 12 (1971) 1735. N.K. Pak and H.C. Tze, Ann. Phys. 117 (1979) 164; J. Gibson and H.C. Tze, Nucl. Phys. B183 (1981) 524; D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762; J.G. Williams, J. Math. Phys. 11 (1970) 2611; J. Goldstone and F. Wilcek, Phys. Rev. Lett. 47 (1981) 986; A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49 (1982) 1124; Phys. Rev. D27 (1983) 1153; A. Chodos and C.B. Thorn, Phys. Rev. D 12 (1975) 2733; M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Lett. 51 (1983) 747; A.D. J acson and M. Rho, Phys. Rev. Lett. 51 (1983) 751; E. Witten, Nucl. Phys. B223 (1983) 433; G. Adkins and C.R. Nappi, Nucl. Phys: B233 (1984) 109; J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51 (1983) 1518; F. Wilcek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250; E. D'Hoker and E. Farhi, Phys. Lett. 134B (1984) 86; P.O. Mazur, M.A. Novak and M. Praszalowicz, Cracow preprint TPJU 4/84.
239
Volume 146B, number 3,4 [7] [81 [9] [101 [11]
240
PHYSICS LETTERS
E. Witten, Nucl. Phys. B160 (1979) 57. E. Witten, Nucl. Phys. B223 (1983) 422. J. Wess and B, Zumino, Phys. Lett. 37B (1971) 95. E. Guadagnini, Nucl. Phys. B236 (1984) 35. G.S. Adkins and C.R. Nappi, Phys. Lett. 137B (1984) 251.
11 October 1984
[121 G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [ 13] S. Gasiorowicz, Elementary particle physics (Wiley, New York, 1967). [ 14] D.H. Perkins, Introduction to high energy physics (Addison-Wesley,Reading, MA, 1983).