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Magnetic moments of the S (1535) and S (1650) resonances, and 11 11 low-lying negative parity baryons Neetika Sharma1, A. Mart´ınez Torres2, K. P. Khemchandani2, and Harleen Dahiya1 1Department of Physics, Dr. B.R. Ambedkar National Institute Of Technology, Jalandhar-144011, India. 2 2Instituto de F´ısica, Universidade de São Paulo, 1 0 C.P 66318, 05314-970 São Paulo, SP, Brazil. 2 l (Dated: July 16, 2012) u J Abstract 3 1 The magnetic moments of the negative parity S11(1535) and S11(1650) resonances have been ] h calculated within the framework of the chiral constituent quark model. The explicit contributions p - p coming from the spin and orbital angular momentum, including the effects of the configurations e h mixingbetweenthestates N2P and N4P ,areobtained. Thecalculations havebeenfurther 1/2 1/2 [ | i | i extended to determine the magnetic moment of the low-lying negative parity octet baryons. Since 1 v the chiral quark model incorporates the constituent quarks and Goldstone bosons as effective 1 1 3 degrees of freedom, the effect of the presence of the meson cloud has also been discussed. Further, 3 . when the contributions of the “quark sea” to the spin and orbital angular momentum are added, 7 0 we find interesting results. 2 1 : v i X r a 1 I. INTRODUCTION Scrutinizing the structure and the properties of light hadrons is a key to understanding the mechanism of the strong interactions at low energies. The very feature of the confine- ment of the theory of the strong interactions makes it difficult to access the underlying physics at the low and intermediate energies. However, continuous efforts are being made to investigate this energy region and a lot of important information is being made available through theoretical and experimental studies. One of the most widely accepted phenomena occurring at these energies is that the hadrons become the effective degrees of freedom and the relevant dynamics is governed by the chiral symmetry and its spontaneous breaking. This information has been applied to build a quark model, the chiral quark model [1], which facilitates the calculations of the hadron properties from a theory based on the symmetries of the quantum chromodynamics (QCD). In this work, we use this model to investigate the magnetic moments of the low-lying, spin-parity JP = 1− octet resonances. A lot of 2 work has been done to understand the structure, masses, widths and decay properties of the baryon resonances (see, for example, Refs. [2–10] for different recent approaches studying lightbaryonresonances), however, althoughsomestudieshavebeendedicatedtoobtaintheir electromagnetic properties [11–16], not much attention has been paid to this latter subject. Such an information can be very useful in revealing the quantum mechanical properties of the resonances like their spin structure. This information can be straight forwardly obtained when using a quark model whose reliability is affirmed by including the chiral symmetry while building the formalism and the model in turn can also be useful to trace the quark degrees of freedom at low energies. From the quark model studies, it is known that the magnetic moment of the low-lying spin 1+ baryons receives contributions not only from the valence quarks, but also from many 2 other effects, such as the quark sea, the orbital angular momentum, relativistic effects, the meson cloud effect, etc. [17, 18]. Also they are relatively well-known both experimentally and theoretically [19]. However, as mentioned above, little is known about their low-lying spin 1− counterparts. Negative parity partners of the octet baryons with spin 1 can be 2 2 understood in a quark model as those arising from the excitation of one unit of the orbital angular momentum, and their mass splittings can be traced to the spontaneous breaking of chiral symmetry of QCD [20–22]. The study of the magnetic moment of the nucleon and its 2 excited spectrum including the S11(1535) N∗ resonance, the lightest lying JP = 21− nucleon resonance, provides valuable insight into the nonperturbative aspects of QCD. Thus, in this context too, it would be interesting to examine the QCD predictions for the negative parity states. There are several known baryonic states, but the experimental investigation of the excited baryon spectrum has been long awaited, however the interest in the same seems to grow rapidly in recent times [23]. The magnetic moment of the nucleon resonances can be extracted through the bremsstrahlung processes. The measurement of the magnetic moment of ∆++(1232) was carried out by studying the process π+p γπ+p [19]. Moreover, a → new experiment to study the γp γπ0p reaction has been made recently by the A2/TAPS → collaboration at MAMI [24]. For the S (1535) resonance, it is being proposed that its mag- 11 netic moment can be extracted by investigating the process of γN γηN and π p γηn − → → [13, 14]. Since this resonance strongly couples to the ηN channel, the η meson in the final state can be regarded as a probe of the S (1535) resonance in the intermediate state. 11 Presently, the theoretical studies of the magnetic moments of the spin 1− baryons are 2 based on simple quark models [13], effective Lagrangian approaches [12, 14], chiral constituent quarkmodel[15], andlatticeQCD[16]. Sincethepastdecade, anextensive program exists to collect data on electromagnetic production of one and two mesons at Jefferson Lab, MIT-Bates, LEGS, MAMI, ELSA, and GRAAL [23]. To analyze these data, and thereby refine our knowledge of the baryon spectrum, a variety of physics analysis models have been developed. Theexperimental efforts outlinedabove should becomplemented by high-quality ab initio computations in the phenomenological models which include lattice QCD. In the low energy regime, the chiral constituent quark model (χCQM) [25] successfully explains the “Proton spin crisis” and other related properties [26–29]. The model has been furtherextendedtocalculatetheoctetanddecuplet baryonmagneticmomentsincorporating the sea quark polarizations and their orbital angular momentum contribution through the generalized Cheng-Li mechanism [18]. Recently, the model has been extended to calculate themagneticmoment ofthecharmed baryonswithspin 1+, 3+ andthetransitions 3+ 1+, 2 2 2 → 2 1+ 1+ [30]. Keeping in view the successes of the χCQM, it is instructive to extend the 2 → 2 applicability of the model to study the magnetic moments of the negative parity low-lying nucleon resonances and octet baryons with unit orbital angular momentum. In fact, a calculation of the magnetic moments of the spin 12− and 32− N∗’s has been done in Ref. [15] 3 within the χCQM. As we will discuss in the subsequent sections in detail, we do a more refined calculation for N ’s by including the Cheng-Li mechanism and further extend the ∗ whole calculation for the first excited states of all the 1+ octet baryons. 2 To fulfill the purpose of the present article we first formulate in detail the magnetic mo- mentsofthenegative parityN resonances (S+(1535),S0 (1535), S+(1650),andS0 (1650)). ∗ 11 11 11 11 In Sec. II, the explicit contribution of the spin and orbital angular momentum of the magnetic moment for the above mentioned resonances are calculated in the nonrelativistic quark model. Further, the generalized Cheng-Li mechanism has been incorporated to calculate explicitly the contribution coming from the valence quarks, quark sea polarization and its orbital angular momentum contribution for the spin part and the valence and sea quarks contribution for the orbital angular momentum part in χCQM. The details of these calculations have been presented in Sec. III. Since the χCQM incorporates the constituent quarks and Goldstone bosons as effective degrees of freedom, it is interesting to examine the effects of the presence of the meson cloud. These calculations have also been presented for the case of the negative parity, low-lying octet baryons. In Sec. IV we discuss our numerical results and Sec. V summarizes our results. II. MAGNETIC MOMENT OF BARYONS IN THE QUARK MODEL In the nonrelativistic SU(6) constituent quark model (NCQM), the magnetic moment of the baryon resonances have contribution coming from both quark spin and orbital angular momentum S L µ = µ +µ , (1) B B B with Q µS = µs = i s , (2) B i 2m i Xi Xi i Q µL = µl = i l , (3) B i 2m i Xi Xi i where si and li are the spin and orbital angular momentum of the ith quark and the index i is summed over three quarks. The orbital angular momentum part vanishes for ground state baryons on account of the absence of orbital excitation. If B is the given baryon’s | i 4 SU(6) spin-flavor wavefunction, then we have B µS B = ∆uµ +∆dµ +∆sµ , (4) u d s h | | i B µL B = ∆u(1)µ +∆d(1)µ +∆s(1)µ . (5) u d s h | | i Here, µ = eq (q = u,d,s) is the quark magnetic moment, e and M are the electric q 2Mq q q charge and mass of the q quark, respectively. The polarizations corresponding to the spin and orbital angular momentum, ∆q = q q and ∆q(1) = q(1) q( 1), can be calculated as ↑ ↓ − − − B B B = B + ( 1) B . (6) ↑↓ ± ≡ h |N| i h |N N | i b In Eq. (6), and ( 1) are the number operators given by ↑↓ ± N N = n u +n u +n d +n d +n s +n s , (7) ↑↓ u↑ ↑ u↓ ↓ d↑ ↑ d↓ ↓ s↑ ↑ s↓ ↓ N ( 1) = n u(1) +n u( 1) +n d(1) +n d( 1) +n s(1) +n s( 1), (8) N ± u(1) u(−1) − d(1) d(−1) − s(1) s(−1) − with n (n ) being the number of quarks with spin up (down) and n (n ) is the q↑ q↓ q(1) q(−1) number of quarks with the orbital angular momentum projection m = 1 (m = 1). L L − In this section, we calculate the magnetic moment of the S (1535) and S (1650) nucleon 11 11 resonances in the framework of the NCQM and further extend this to the rest of low-lying negative parity octet baryons. The lowest lying negative parity nucleon resonances are N2P and N4P , where the usual spectroscopic notation 2S+1L is used to indicate 1/2 1/2 J | i | i their total quark spin S = 1, 3 (2S+1 = 2, 4), orbital angular momentum L = 1 (P-wave), 2 2 and total angular momentum J = 1 [31]. The spin angular momentum S = 1 couples with 2 2 the orbital angular momentum L = 1 to give the total angular momentum J = 1 and J = 3. 2 2 The wavefunctions of the N2P and N4P states are explicitly given as 1/2 1/2 | i | i 1 1 1 1 1 N2P = 1 m m ψρ χλ φρ +χρ φλ | 1/2i √2 mXlmsh 2 l s|2 2i(cid:26) 1mlh√2(cid:0) ms ms (cid:1)i 1 +ψλ χρ φρ χλ φλ , (9) 1mlh√2 ms − ms i(cid:27) (cid:0) (cid:1) 1 3 1 1 N4P = 1 m m ψρ χs φρ + ψλ χs φλ , (10) | 1/2i √2 mXlmsh 2 l s| 2 2 ih 1ml ms 1ml ms i where ψ, χ, and φ denote the spatial, spin, and flavor wavefunctions. The superscripts s or ρ (λ) indicate that they are totally symmetric among three quarks, or odd (even) under the exchange of the first two quarks [32]. 5 The physical eigenstates forthe L = 1 negative parity resonances, arelinear combinations of these two and are expressed as S (1535) = cosθ N2P sinθ N4P , (11) 11 1/2 1/2 | i | i− | i S (1650) = sinθ N2P +cosθ N4P . (12) 11 1/2 1/2 | i | i | i Now, the magnetic moments of the S (1535) and S (1650) resonances can be expressed in 11 11 terms of the magnetic moments of the N2P and N4P states and their cross terms 1/2 1/2 | i | i as µ = µ cos2θ +µ sin2θ 2 N2P µ N4P cosθsinθ, (13) S11(1535) N2P1/2 N4P1/2 − h 1/2| z| 1/2i µ = µ sin2θ +µ cos2θ+2 N2P µ N4P cosθsinθ. (14) S11(1650) N2P1/2 N4P1/2 h 1/2| z| 1/2i The value of the mixing angle θ depends on the quark interaction. Assuming a hyper- fine interaction between the quarks, Isgur and Karl [32] predicted a mixing angle θ = tan 1(√5 1)/2 31.7 , which is close to the empirical mixing angle θ 32 found in − ◦ ◦ − ≃ − ≃ − Ref. [33]. Since the magnetic moment has contribution coming from both spin and orbital angular momentum as expressed in Eqs. (2) and (3), we now present the matrix elements for quark spin and orbital angular momentum contributions of the magnetic moments of the states N2P and N4P and the cross terms obtained from mixing the N2P and N4P 1/2 1/2 1/2 1/2 | i | i | i | i states. It is easy to obtain the spin structure of the baryons as expressed in Eq. (6) 8 10 4 5 4 2 N2P+ N2P+ = u + u + d + d + u(1) + d(1), ↑ ↓ ↑ ↓ h 1/2|N| 1/2i 9 9 9 9 9 9 4 5 8 10 2 4 N2P0 N2P0 = u + u + d + d + u(1) + d(1), h 1/2|N| 1/2i 9 ↑ 9 ↓ 9 ↑ 9 ↓ 9 9 14 4 7 2 1 1 1 1 N4P+ N4P+ = u + u + d + d + u(1) + u( 1) + d(1) + d( 1), ↑ ↓ ↑ ↓ − − h 1/2|N| 1/2i 9 9 9 9 18 6 9 3 7 2 14 4 1 1 1 1 N4P0 N4P0 = u + u + d + d + u(1) + u( 1) + d(1) + d( 1), h 1/2|N| 1/2i 9 ↑ 9 ↓ 9 ↑ 9 ↓ 9 3 − 18 6 − 2 2 2 2 N2P+ N4P+ = u u d + d , ↑ ↓ ↑ ↓ h 1/2|N| 1/2i 9 − 9 − 9 9 2 2 2 2 N2P0 N4P0 = u + u + d d . (15) h 1/2|N| 1/2i −9 ↑ 9 ↓ 9 ↑ − 9 ↓ Here, the superscripts + and 0 denote the charge of the resonance state. It is important to mention at this point that there are no cross terms for µL because N2P and N4P z | 1/2i | 1/2i 6 have orthogonal quark spin states which are not affected by µL. Using Eqs. (4) and (5) the z magnetic moment of the N2P+ , N4P+ , and the cross terms can be expressed as | 1/2i | 1/2i 2 1 4 2 µ = µ µ + µ + µ , (16) N2P1+/2 −9 u − 9 d 9 u 9 d 1 2 2 4 µ = µ µ + µ + µ , (17) N2P10/2 −9 u − 9 d 9 u 9 d 10 5 1 2 µ = µ + µ µ µ , (18) N4P1+/2 9 u 9 d − 9 u − 9 d 5 10 2 1 µ = µ + µ µ µ . (19) N4P10/2 9 u 9 d − 9 u − 9 d 4 4 N4P+ µS N2P+ = µ µ , (20) h 1/2| z| 1/2i 9 u − 9 d 4 4 N4P0 µS N2P0 = µ + µ . (21) h 1/2| z| 1/2i −9 u 9 d Using Eqs. (13) and (14), the magnetic moment of the S (1535) and S (1650) states in 11 11 the NCQM are expressed as 2 1 4 2 10 5 µ = µ µ + µ + µ cos2θ+ µ + µ S1+1(1535) (cid:18)−9 u − 9 d 9 u 9 d(cid:19) (cid:18) 9 u 9 d 1 2 4 4 µ µ sin2θ 2 µ µ cosθsinθ, (22) u d u d −9 − 9 (cid:19) − (cid:18)9 − 9 (cid:19) 1 2 2 4 5 10 µ = µ µ + µ + µ cos2θ+ µ + µ S101(1535) (cid:18)−9 u − 9 d 9 u 9 d(cid:19) (cid:18)9 u 9 d 2 1 4 4 µ µ sin2θ 2 µ + µ cosθsinθ, (23) u d u d −9 − 9 (cid:19) − (cid:18)−9 9 (cid:19) 2 1 4 2 10 5 µ = µ µ + µ + µ sin2θ+ µ + µ S1+1(1650) (cid:18)−9 u − 9 d 9 u 9 d(cid:19) (cid:18) 9 u 9 d 1 2 4 4 µ µ cos2θ+2 µ µ cosθsinθ, (24) u d u d −9 − 9 (cid:19) (cid:18)9 − 9 (cid:19) 1 2 2 4 5 10 µ = µ µ + µ + µ sin2θ+ µ + µ S101(1650) (cid:18)−9 u − 9 d 9 u 9 d(cid:19) (cid:18)9 u 9 d 2 1 4 4 µ µ cos2θ+2 µ + µ cosθsinθ. (25) u d u d −9 − 9 (cid:19) (cid:18)−9 9 (cid:19) Using m = m = 1m , we have µ = Q /2m = 2µ and µ = Q /2m = µ . Thus, u d 3 N u u u N d d d − N 7 substituting these values in Eqs. (22)-(25), we obtain 1 5 8 µ = cos2θ+ sin2θ cosθsinθ (26) S1+1(1535) 3 3 − 3 1 8 µ = sin2θ+ cosθsinθ (27) S101(1535) −3 3 5 1 8 µ = cos2θ+ sin2θ + cosθsinθ (28) S1+1(1651) 3 3 3 1 8 µ = cos2θ cosθsinθ. (29) S101(1651) −3 − 3 Using the value of mixing angle θ = 31.7 in Eqs. (26)-(29), one can obtain the magnetic ◦ − momentsoftheN resonancesS+(1535),S0 (1535),S+(1650),andS0 (1650). Themagnetic ∗ 11 11 11 11 moment of the other negative parity low-lying baryon resonances with spin 1 can similarly 2 be calculated using their respective wave functions. The results of these calculations are presented and discussed in Section IV. III. MAGNETIC MOMENT OF BARYONS IN χCQM The key to understanding the magnetic moment of the baryons χCQM formalism [26] is the fluctuation process ′ ′ ′ q GB+q (qq¯)+q , (30) ↑↓ ↓↑ ↓↑ → → ′ ′ where GB represents the emitted Goldstone boson and qq¯ +q constitute the “quark sea” [27, 28]. The effective Lagrangian describing the interaction between quarks and a nonet of GBs can be expressed as = g q¯ Φ+ζ η′ I q = g q¯(Φ)q, where ζ = g /g , g and g L 8 √3 8 ′ 1 8 1 8 (cid:16) (cid:17) are the coupling constants for the singlet and octet GBs, respectively, and I is the 3 3 × identity matrix. The matrix of the GBs can be expressed as ′ π0 +β η +ζ η π+ αK+ u  √2 √6 √3    ′ Φ′ =  π− −√π02 +β√η6 +ζ√η3 αK0 and q =  d  . (31)  αK αK¯0 β 2η +ζ η′   s   − − √6 √3    The parameter a(= g 2) denotes the probability of chiral fluctuation u(d) d(u)+π+( ), 8 − | | → whereas α2a, β2a and ζ2a respectively denote the probabilities of fluctuations u(d) s+ → K (0), u(d,s) u(d,s)+η, and u(d,s) u(d,s)+η′. − → → The spin part µS of the magnetic moment of a given baryon receives contributions from the valence quarks, sea quarks, and orbital angular momentum of the “quark sea” [18] and 8 is expressed as µS = µS +µS +µS , (32) B val sea orbit where µS and µS represent the contributions of the valence quarks and the sea quarks val sea to the magnetic moments due to the spin polarizations. In addition, there is a significant contribution coming from the orbital angular momentum of the “quark sea”, µS , since orbit the GB emitted due to the chiral fluctuation is in the P wave state, l = 1. The details of z h i the valence quark calculations have already been presented in the previous section. We now present the calculations of the sea and orbital angular momentum contributions. The sea quark spin polarizations corresponding to each baryon can be obtained by substituting for each valence quark q ( ) P q ( ) + ψ(q ( )) 2, (33) ↑ ↓ [q, GB] ↑ ↓ ↑ ↓ → − | | when calculating the spin contribution to the magnetic moment. In Eq. (33), P is [q, GB] the probability of emission of GBs from a quark q ( ) and ψ(q ( )) 2 is the probability of ↑ ↓ ↑ ↓ | | transforming a q ( ) quark [28] given by ↑ ↓ a ψ(u ( )) 2 = 3+β2 +2ζ2 u ( ) +ad ( ) +aα2s ( ), ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ | | 6 (cid:0) a (cid:1) ψ(d ( )) 2 = au ( ) + 3+β2 +2ζ2 d ( ) +aα2s ( ), ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ | | 6 (cid:0) a (cid:1) ψ(s ( )) 2 = aα2u ( ) +aα2d ( ) + 2β2 +ζ2 s ( ), (34) ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ | | 3 (cid:0) (cid:1) for the spin up quarks. The orbital angular momentum contribution of each chiral fluctuation is given as [18, 26] e e e µ(q q′ ) = q′ l + q − q′ l , (35) ↑ ↓ q GB → 2M h i 2M h i q GB where l = MGB and l = Mq . The quantities (l , l ) and (M , M ) are h qi Mq+MGB h GBi Mq+MGB q GB q GB the orbital angular momenta and masses of the quarks and GBs, respectively. The orbital moment of each process in Eq. (35) is then multiplied by the probability for such a process to take place to yield the magnetic moment due to all the transitions starting with a given valence quark. For example, 1 β2 ζ2 [µ(u )] = a + + µ(u u )+µ(u d )+α2µ(u s ) , (36) ↑ → (cid:20)(cid:18)2 6 3 (cid:19) ↑ → ↓ ↑ → ↓ ↑ → ↓ (cid:21) 9 1 β2 ζ2 [µ(d )] = a µ(d u )+ + + µ(d d )+α2µ(d s ) , (37) ↑ → (cid:20) ↑ → ↓ (cid:18)2 6 3 (cid:19) ↑ → ↓ ↑ → ↓ (cid:21) 2 ζ2 [µ(s )] = a α2µ(s u )+α2µ(s d )+ β2 + µ(s s ) . (38) ↑ → (cid:20) ↑ → ↓ ↑ → ↓ (cid:18)3 3 (cid:19) ↑ → ↓ (cid:21) The orbital moments of the u, d, and s quarks in terms of the χCQM parameters (a,α,β,ζ), quark masses (M ,M ,M ) and GB masses (M ,M ,M ,M ), are respectively given as u d s π k η η′ 3M2 α2(M2 3M2) β2M ζ2M [µ(u )] = a u K − u + η + η′ µ , u ↑ → (cid:20)2Mπ(Mu +Mπ) − 2MK(Mu +MK) 6(Mu +Mη) 3(Mu +Mη′)(cid:21) (39) 3(M2 2M2) α2M β2M ζ2M [µ(d )] = 2a π − d K η η′ µ , d ↑ → − (cid:20)4Mπ(Md +Mπ) − 2(Md +MK) − 12(Md +Mη) − 6(Md +Mη′)(cid:21) (40) α2(M2 3M2) β2M ζ2M [µ(s )] = 2a K − s η η′ µ . (41) s ↑ → − (cid:20)2MK(Ms +MK) − 3(Ms +Mη) − 6(Ms +Mη′)(cid:21) Using this formalism, we can calculate explicitly the valence, sea, and orbital contributions to the spin angular momentum of the magnetic moments of the baryons. As an example, the µS contribution to the magnetic moment of the S+(1535) is given as 11 2 1 10 5 8 µSval(S1+1(1535))=−(cid:18)9µu+ 9µd(cid:19)cos2θ+(cid:18) 9 µu+ 9µd(cid:19)sin2θ− 9(µu−µd)sinθcosθ, (42) a 2β2 4ζ2 β2 2ζ2 µSsea(S1+1(1535))= 9 (cid:20)(cid:18)5+2α2+ 3 + 3 (cid:19)µu+(cid:18)4+α2+ 3 + 3 (cid:19)µd+3α2µs(cid:21)cos2θ 5a 2β2 4ζ2 β2 2ζ2 5+2α2+ + µu+ 4+α2+ + µd+3α2µs sin2θ − 9 (cid:20)(cid:18) 3 3 (cid:19) (cid:18) 3 3 (cid:19) (cid:21) 8a β2 2ζ2 + 1+α2+ + (µu µd)sinθcosθ, (43) 9 (cid:18) 3 3 (cid:19) − 2 1 10 5 µS (S+(1535))= µ(u )+ µ(d ) cos2θ+ µ(u )+ µ(d ) sin2θ orbit 11 −(cid:18)9 ↑ → 9 ↑ → (cid:19) (cid:18) 9 ↑ → 9 ↑ → (cid:19) 8 (µ(u ) µ(d ))sinθcosθ. (44) ↑ ↑ −9 → − → The orbital angular momentum contribution µL to the magnetic moment of a given baryon receives contributions from the valence and sea quarks as µL = µL +µL , (45) B val sea where µL and µL represent the contributions of the valence and sea quarks to the magnetic val sea moments due to the orbital angular momentum polarizations. The details of the valence quark calculations have already been presented in the previous section, whereas the sea 10