- Email: [email protected]
Nc expansion of QCD
Nuclear Physics A 790 (2007) 489c–492c
Excited baryon decays in 1/Nc expansion of QCD∗ J.L. Goity
a
and N.N. Scoccola
b
a
Department of Physics, Hampton University, Hampton, VA 23668, USA. Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA.
b
Physics Department, CNEA, (1429) Buenos Aires, Argentina. Universidad Favaloro, Sol´ıs 453, (1078) Buenos Aires, Argentina. CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina.
The decays of excited baryons are analyzed using the 1/Nc expansion of QCD. These studies are performed up to order 1/Nc , and include positive and negative parity states. 1. INTRODUCTION The 1/Nc expansion of QCD [1] has proven to be a useful tool for analyzing the baryon properties. This is mostly due to the existence of a contracted spin-flavor symmetry in the large Nc limit[2,3]. Although SU(2Nf ) is not an exact symmetry in the excited baryon sector[4], its Nc0 breaking is small as several analyses of the baryon spectrum have shown[5–7]. We apply here this approach to the study of the decays of excited non-strange baryons belonging to the 70-plet of negative parity and the 56-plets of positive parity. 2. STRONG DECAYS The strong P partial wave decay width of a resonance with total angular momentum, spin and isospin J ∗ , S ∗ , I ∗ , respectively, into a ground state baryon with quantum numbers J, I and a pseudo-scalar meson with isospin IP can be expressed as Γ[P ,IP ] =
kP MB |B(P , IP , J, I, J ∗ , I ∗ , S ∗ )|2 , 8π 2 MB∗ (2J ∗ + 1)(2I ∗ + 1)
(1)
where B (P , IP , J, I, J ∗, I ∗ , S ∗ ) are the reduced matrix elements of the baryonic operator. Such operator admits an expansion in 1/Nc that has the general form[8,9] [ ,I ]
P P B[μ,α]
=
kP Λ
P
q,j
Cq[P ,IP ] ξ () Gq[j,IP ]
[P ,IP ] [μ,α]
(2)
The factor (kP /Λ)P is included to take into account the chief meson momentum dependence of the partial wave, where the scale Λ is taken to be equal to 200 MeV. The operator ξ () drives the transition from the (2 + 1)-plet to the singlet O(3) state. The operators ∗ We acknowledge support by the NSF (USA) through grant # PHY-9733343 and by the CONICET and ANPCYT (Argentina) through grants PIP 02368 and PICT 03-08580.
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.083
490c
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
G give transitions within the spin-flavor representations in which the excited and GS baryons reside. They can be written as products of the generators of the spin-flavor algebra acting on excited quark state λ = si , ta , gia and on the core Λc = (Sc )i , (Tc )a , (Gc )ia . The dynamics of the decays is encoded in the effective dimensionless coefficients Cq[P ,IP ] . Here, they are obtained by fitting the available empirical decay widths[10]. In the case of the negative parity baryons we have, in addition, two extra unknown parameters: the mixing angles θ2J between the two sets of excited nucleon states NJ∗ , where J = 1/2, 3/2. The corresponding operators and results of the fits are shown in Table 1. We observe Table 1 Fit parameters. LO: There is a two-fold ambiguity for the angle θ1 . For θ3 there is an [2,1] almost two-fold ambiguity which leads to two slightly different values of C6 . NLO: Due to lack of empirical data, three-body operators and the subleading operator for η emission are not included. No degeneracy in θ1 but almost two-fold ambiguity in θ3 . Values of coefficients which differ in the corresponding fits are indicated in parenthesis. Operator 1 Nc 1 Nc 1 Nc
1 Nc 1 Nc 1 Nc 1 Nc 1 Nc
ξ
ξ
(1)
(1)
(s Tc )
ξ (1) (t Sc )
ξ
(1)
ξ
ξ (1) (s ξ (1) (t
ξ
(1)
(g
ξ
(1)
(g
ξ (1) (s
ξ
[1,1] [0,1]
[1,1] [0,1]
(g Sc )
(1)
g
[0,1]
(1)
g
[1,1] [0,1]
[2,1]
[i,a] [2,1] Tc )[1,1] [2,1] Sc )[1,1] [2,1] Sc )[1,1] [2,1] Sc )[2,1] [2,1] Gc )[2,1] [0,0]
s
θ1 θ3 χ2dof
Fit parameters LO
NLO
31 ± 3
23 ± 3
-
(7.4, 32.5) ± (27, 41)
-
(20.7, 26.8) ± (12, 14)
-
(−26.3, −66.8) ± (39, 65)
4.6 ± 0.5
3.4 ± 0.3
-
−4.5 ± 2.4
-
(−0.01, 0.08) ± 2
-
5.7 ± 4.0
-
3.0 ± 2.2
(−1.86, −2.25) ± 0.4
−1.73 ± 0.26
11 ± 4 1.56 ± 0.15 0.35 ± 0.14 (3.00, 2.44) ± 0.07 1.5
17 ± 4 0.39 ± 0.11 (2.82, 2.38) ± 0.11 0.9
that the LO operators already provide a reasonable description of the decay widths. The NLO corrections play some role in improving the results, although the corresponding coefficients are not well determined due to the large error bars in the empirical widths. In any event, a clear dominance of the one-body operators is observed. The operators and results of the fits corresponding to the pion P wave decays of = 0 excited (Roper multiplet) baryons are given in Table 2. Note that for positive parity resonances only the total operators Λ = λ + Λc are needed. We observe that in this case
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
491c
Table 2 Operators and fits corresponding to the pion P wave decays of = 0 excited baryons Operator
1 Nc2
1 Nc
ξ
(0)
ξ (0) G
Fit parameters LO NLO
[1,1]
([S , G]) χ2dof
[1,1] [1,1]
18.7 ± 2.4
17.0 ± 1.6
4.05
24.4 ± 6.3 0.1
the LO fit leads to rather poor results. In fact, the NLO operator is essential to obtain a good description of these decays. It should be noted, however, that the coefficient of the LO order operator remains stable, and that the one of the NLO operator is of natural size. The operators and fits for the decays of the = 2 baryons are given in Table 3. For the P-wave decays, the LO analysis already provides an excellent fit. In the case of the F wave decays, the LO analysis gives a reasonable fit provided the empirical errors are taken to be 30% or more, which is the expected accuracy of such an analysis. If in the NLO fit the errors are taken at its empirical estimates the χ2dof turns out to be rather large. The main contribution to this quantity comes from the πΔ decay of the N(1680) for which only an upper bound is given in Ref.[10]. As shown in Table 3, if such a decay is removed from the analysis a quite good fit is obtained. Another interesting observation Table 3 Operators and fits corresponding to the pion P and F wave decays of = 2 excited baryons. In the LO fit of the F wave decays the empirical errors are taken to be ≥ 30%. In the corresponding NLO fit, the decay N(1680) → π Δ has not been considered. Operator
1 Nc2 1 Nc2
1 Nc2 1 Nc2
ξ (2) G
[1,1]
6.83 ± 0.77
4.09 ± 0.47
-
0.11 ± 2.41
({S , G}) χ2dof
0.15
0.43 ± 6.09 0.44
ξ (2) G
1.01 ± 0.09
0.84 ± 0.04
-
−1.30 ± 0.21
1.73
−0.26 ± 0.47 0.38
ξ (2) ([S , G])[1,1]
ξ
1 Nc
(2)
1 Nc
ξ
[1,1]
[2,1] [1,1]
[3,1]
ξ (2) ([S , G]) (2)
Fit parameters LO NLO
[1,1] [3,1] [2,1] [3,1]
({S , G}) χ2dof
that points to the consistency of the framework based on the approximate O(3)×SU(2Nf ) symmetry is that the predicted suppression of the η channels in the decays positive parity baryons is clearly displayed by the observed decays. This represents a strong experimental confirmation that those baryons do belong primarily into a symmetric representation of SU(2Nf ). In the case of the negative parity baryons the η channel is not suppressed and is indeed very important. This implies that these states belong primarily into the
492c
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
mixed-symmetric representation of SU(2Nf ). Thus, η channels serve as a selector of the spin-flavor structure of excited baryons. 3. RADIATIVE DECAYS The situation concerning the radiative decays is still in a preliminary stage. In Ref.[11] a LO analysis of the decays of the (1− , 70)-plet states has been made. Perhaps the most important conclusion of that study is that, although there are two-body operators already contributing to this order, the amplitudes are dominated by one-body operators. Of course, to have a clearer understanding a NLO analysis is required. For the positive parity resonances, an study including up to NLO corrections is currently under way[12]. Preliminary results show that, as for the strong decays, the LO analysis seems to be problematic for Roper baryons. On the other hand, for the = 2 states at least some LO model independent relations seem to be well satisfied. For example, for the Δ(1950) N decays one obtains AN 3/2 /A1/2 = 1.29 to be compared with the empirical value 1.27 ± 0.24. 4. CONCLUSIONS To conclude, the 1/Nc expansion of QCD provides a systematic approach to the properties of the excited baryons. The analysis of the masses shows that the Nc0 breaking of the spin-flavor symmetry is small. For strong decays one finds, in general, a dominance of the one-body LO operators. In some cases, as e.g. the D wave decays of the negative parity excited baryons, the 1/Nc corrections are not well established due to the rather large uncertainties of the empirical data. In the particular case of the Roper baryons, two-body NLO operators seem to be required to obtain a good description of the empirical data. In the case of the radiative decays, there is work in progress. Preliminary results indicate that the situation is similar to that of strong decays. REFERENCES 1. G. ’t Hooft, Nucl.Phys. B 72 (1974) 461; E. Witten, Nucl.Phys. B 160 (1979) 57. 2. J.L. Gervais and B. Sakita, Phys.Rev.Lett. 52 (1984) 87; Phys.Rev. D30 (1984) 1795. 3. R. Dashen and A.V. Manohar, Phys.Lett. B 315 (1993) 425; Phys.Lett. B 315 (1993) 438; E. Jenkins, Phys.Lett. B 315 (1993) 441. 4. J.L. Goity, Phys.Lett. B 414 (1997) 140. 5. C.E. Carlson, C.D. Carone, J.L. Goity and R.F. Lebed, Phys.Lett. B 438 (1998) 327; Phys.Rev. D 59 (1999) 114008. 6. C.L. Schat, J.L. Goity and N.N. Scoccola, Phys.Rev.Lett. 88 (2002) 102002; Phys.Rev. D 66 (2002) 114014; Phys.Lett. B 564 (2003) 83. 7. N. Matagne and F. Stancu, Phys.Rev. D 71 (2005) 014010; Phys.Lett. B 631 (2005) 7; Phys.Rev. D 73 (2006) 114025. 8. J.L. Goity, C.L. Schat and N.N. Scoccola, Phys.Rev. D 71 (2005) 034016. 9. J.L. Goity and N.N. Scoccola, Phys.Rev. D72 (2005) 034024. 10. Particle Data Group, S. Eidelman et al., Phys.Lett. B 592 (2004) 1. 11. C.E. Carlson and C.D. Carone, Phys.Rev. D58 (1998) 053005. 12. J.L. Goity and N.N. Scoccola, in preparation.
Excited baryon decays in 1/Nc expansion of QCD∗ J.L. Goity
a
and N.N. Scoccola
b
a
Department of Physics, Hampton University, Hampton, VA 23668, USA. Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA.
b
Physics Department, CNEA, (1429) Buenos Aires, Argentina. Universidad Favaloro, Sol´ıs 453, (1078) Buenos Aires, Argentina. CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina.
The decays of excited baryons are analyzed using the 1/Nc expansion of QCD. These studies are performed up to order 1/Nc , and include positive and negative parity states. 1. INTRODUCTION The 1/Nc expansion of QCD [1] has proven to be a useful tool for analyzing the baryon properties. This is mostly due to the existence of a contracted spin-flavor symmetry in the large Nc limit[2,3]. Although SU(2Nf ) is not an exact symmetry in the excited baryon sector[4], its Nc0 breaking is small as several analyses of the baryon spectrum have shown[5–7]. We apply here this approach to the study of the decays of excited non-strange baryons belonging to the 70-plet of negative parity and the 56-plets of positive parity. 2. STRONG DECAYS The strong P partial wave decay width of a resonance with total angular momentum, spin and isospin J ∗ , S ∗ , I ∗ , respectively, into a ground state baryon with quantum numbers J, I and a pseudo-scalar meson with isospin IP can be expressed as Γ[P ,IP ] =
kP MB |B(P , IP , J, I, J ∗ , I ∗ , S ∗ )|2 , 8π 2 MB∗ (2J ∗ + 1)(2I ∗ + 1)
(1)
where B (P , IP , J, I, J ∗, I ∗ , S ∗ ) are the reduced matrix elements of the baryonic operator. Such operator admits an expansion in 1/Nc that has the general form[8,9] [ ,I ]
P P B[μ,α]
=
kP Λ
P
q,j
Cq[P ,IP ] ξ () Gq[j,IP ]
[P ,IP ] [μ,α]
(2)
The factor (kP /Λ)P is included to take into account the chief meson momentum dependence of the partial wave, where the scale Λ is taken to be equal to 200 MeV. The operator ξ () drives the transition from the (2 + 1)-plet to the singlet O(3) state. The operators ∗ We acknowledge support by the NSF (USA) through grant # PHY-9733343 and by the CONICET and ANPCYT (Argentina) through grants PIP 02368 and PICT 03-08580.
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.083
490c
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
G give transitions within the spin-flavor representations in which the excited and GS baryons reside. They can be written as products of the generators of the spin-flavor algebra acting on excited quark state λ = si , ta , gia and on the core Λc = (Sc )i , (Tc )a , (Gc )ia . The dynamics of the decays is encoded in the effective dimensionless coefficients Cq[P ,IP ] . Here, they are obtained by fitting the available empirical decay widths[10]. In the case of the negative parity baryons we have, in addition, two extra unknown parameters: the mixing angles θ2J between the two sets of excited nucleon states NJ∗ , where J = 1/2, 3/2. The corresponding operators and results of the fits are shown in Table 1. We observe Table 1 Fit parameters. LO: There is a two-fold ambiguity for the angle θ1 . For θ3 there is an [2,1] almost two-fold ambiguity which leads to two slightly different values of C6 . NLO: Due to lack of empirical data, three-body operators and the subleading operator for η emission are not included. No degeneracy in θ1 but almost two-fold ambiguity in θ3 . Values of coefficients which differ in the corresponding fits are indicated in parenthesis. Operator 1 Nc 1 Nc 1 Nc
1 Nc 1 Nc 1 Nc 1 Nc 1 Nc
ξ
ξ
(1)
(1)
(s Tc )
ξ (1) (t Sc )
ξ
(1)
ξ
ξ (1) (s ξ (1) (t
ξ
(1)
(g
ξ
(1)
(g
ξ (1) (s
ξ
[1,1] [0,1]
[1,1] [0,1]
(g Sc )
(1)
g
[0,1]
(1)
g
[1,1] [0,1]
[2,1]
[i,a] [2,1] Tc )[1,1] [2,1] Sc )[1,1] [2,1] Sc )[1,1] [2,1] Sc )[2,1] [2,1] Gc )[2,1] [0,0]
s
θ1 θ3 χ2dof
Fit parameters LO
NLO
31 ± 3
23 ± 3
-
(7.4, 32.5) ± (27, 41)
-
(20.7, 26.8) ± (12, 14)
-
(−26.3, −66.8) ± (39, 65)
4.6 ± 0.5
3.4 ± 0.3
-
−4.5 ± 2.4
-
(−0.01, 0.08) ± 2
-
5.7 ± 4.0
-
3.0 ± 2.2
(−1.86, −2.25) ± 0.4
−1.73 ± 0.26
11 ± 4 1.56 ± 0.15 0.35 ± 0.14 (3.00, 2.44) ± 0.07 1.5
17 ± 4 0.39 ± 0.11 (2.82, 2.38) ± 0.11 0.9
that the LO operators already provide a reasonable description of the decay widths. The NLO corrections play some role in improving the results, although the corresponding coefficients are not well determined due to the large error bars in the empirical widths. In any event, a clear dominance of the one-body operators is observed. The operators and results of the fits corresponding to the pion P wave decays of = 0 excited (Roper multiplet) baryons are given in Table 2. Note that for positive parity resonances only the total operators Λ = λ + Λc are needed. We observe that in this case
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
491c
Table 2 Operators and fits corresponding to the pion P wave decays of = 0 excited baryons Operator
1 Nc2
1 Nc
ξ
(0)
ξ (0) G
Fit parameters LO NLO
[1,1]
([S , G]) χ2dof
[1,1] [1,1]
18.7 ± 2.4
17.0 ± 1.6
4.05
24.4 ± 6.3 0.1
the LO fit leads to rather poor results. In fact, the NLO operator is essential to obtain a good description of these decays. It should be noted, however, that the coefficient of the LO order operator remains stable, and that the one of the NLO operator is of natural size. The operators and fits for the decays of the = 2 baryons are given in Table 3. For the P-wave decays, the LO analysis already provides an excellent fit. In the case of the F wave decays, the LO analysis gives a reasonable fit provided the empirical errors are taken to be 30% or more, which is the expected accuracy of such an analysis. If in the NLO fit the errors are taken at its empirical estimates the χ2dof turns out to be rather large. The main contribution to this quantity comes from the πΔ decay of the N(1680) for which only an upper bound is given in Ref.[10]. As shown in Table 3, if such a decay is removed from the analysis a quite good fit is obtained. Another interesting observation Table 3 Operators and fits corresponding to the pion P and F wave decays of = 2 excited baryons. In the LO fit of the F wave decays the empirical errors are taken to be ≥ 30%. In the corresponding NLO fit, the decay N(1680) → π Δ has not been considered. Operator
1 Nc2 1 Nc2
1 Nc2 1 Nc2
ξ (2) G
[1,1]
6.83 ± 0.77
4.09 ± 0.47
-
0.11 ± 2.41
({S , G}) χ2dof
0.15
0.43 ± 6.09 0.44
ξ (2) G
1.01 ± 0.09
0.84 ± 0.04
-
−1.30 ± 0.21
1.73
−0.26 ± 0.47 0.38
ξ (2) ([S , G])[1,1]
ξ
1 Nc
(2)
1 Nc
ξ
[1,1]
[2,1] [1,1]
[3,1]
ξ (2) ([S , G]) (2)
Fit parameters LO NLO
[1,1] [3,1] [2,1] [3,1]
({S , G}) χ2dof
that points to the consistency of the framework based on the approximate O(3)×SU(2Nf ) symmetry is that the predicted suppression of the η channels in the decays positive parity baryons is clearly displayed by the observed decays. This represents a strong experimental confirmation that those baryons do belong primarily into a symmetric representation of SU(2Nf ). In the case of the negative parity baryons the η channel is not suppressed and is indeed very important. This implies that these states belong primarily into the
492c
J.L. Goity, N.N. Scoccola / Nuclear Physics A 790 (2007) 489c–492c
mixed-symmetric representation of SU(2Nf ). Thus, η channels serve as a selector of the spin-flavor structure of excited baryons. 3. RADIATIVE DECAYS The situation concerning the radiative decays is still in a preliminary stage. In Ref.[11] a LO analysis of the decays of the (1− , 70)-plet states has been made. Perhaps the most important conclusion of that study is that, although there are two-body operators already contributing to this order, the amplitudes are dominated by one-body operators. Of course, to have a clearer understanding a NLO analysis is required. For the positive parity resonances, an study including up to NLO corrections is currently under way[12]. Preliminary results show that, as for the strong decays, the LO analysis seems to be problematic for Roper baryons. On the other hand, for the = 2 states at least some LO model independent relations seem to be well satisfied. For example, for the Δ(1950) N decays one obtains AN 3/2 /A1/2 = 1.29 to be compared with the empirical value 1.27 ± 0.24. 4. CONCLUSIONS To conclude, the 1/Nc expansion of QCD provides a systematic approach to the properties of the excited baryons. The analysis of the masses shows that the Nc0 breaking of the spin-flavor symmetry is small. For strong decays one finds, in general, a dominance of the one-body LO operators. In some cases, as e.g. the D wave decays of the negative parity excited baryons, the 1/Nc corrections are not well established due to the rather large uncertainties of the empirical data. In the particular case of the Roper baryons, two-body NLO operators seem to be required to obtain a good description of the empirical data. In the case of the radiative decays, there is work in progress. Preliminary results indicate that the situation is similar to that of strong decays. REFERENCES 1. G. ’t Hooft, Nucl.Phys. B 72 (1974) 461; E. Witten, Nucl.Phys. B 160 (1979) 57. 2. J.L. Gervais and B. Sakita, Phys.Rev.Lett. 52 (1984) 87; Phys.Rev. D30 (1984) 1795. 3. R. Dashen and A.V. Manohar, Phys.Lett. B 315 (1993) 425; Phys.Lett. B 315 (1993) 438; E. Jenkins, Phys.Lett. B 315 (1993) 441. 4. J.L. Goity, Phys.Lett. B 414 (1997) 140. 5. C.E. Carlson, C.D. Carone, J.L. Goity and R.F. Lebed, Phys.Lett. B 438 (1998) 327; Phys.Rev. D 59 (1999) 114008. 6. C.L. Schat, J.L. Goity and N.N. Scoccola, Phys.Rev.Lett. 88 (2002) 102002; Phys.Rev. D 66 (2002) 114014; Phys.Lett. B 564 (2003) 83. 7. N. Matagne and F. Stancu, Phys.Rev. D 71 (2005) 014010; Phys.Lett. B 631 (2005) 7; Phys.Rev. D 73 (2006) 114025. 8. J.L. Goity, C.L. Schat and N.N. Scoccola, Phys.Rev. D 71 (2005) 034016. 9. J.L. Goity and N.N. Scoccola, Phys.Rev. D72 (2005) 034024. 10. Particle Data Group, S. Eidelman et al., Phys.Lett. B 592 (2004) 1. 11. C.E. Carlson and C.D. Carone, Phys.Rev. D58 (1998) 053005. 12. J.L. Goity and N.N. Scoccola, in preparation.