Chiral perturbation theory for pentaquark baryons and its applications

We construct a chiral lagrangian for pentaquark baryons assuming that the recently found Theta^+ (1540) state belongs to an antidecuplet of SU(3) flavor symmetry with J^P = (1/2)^(+-). We derive the Gell-Mann-Okubo formulae for the antidecuplet baryon masses, and a possible mixing between the antidecuplet and the pentaquark octet. Then we calculate the cross sections for pi^- p ->K^- Theta^+ and gamma n ->K^- Theta^+ using our chiral lagrangian. The resulting amplitudes respect the underlying chiral symmetry of QCD correctly. We also describe how to include the light vector mesons in the chiral lagrangian.


1.
Introduction -Recently, five independent experiments reported observations of a new baryonic state Θ + (1540) with a very narrow width < 5 MeV [1,2,3,4,5,6], which is likely to be a pentaquark state (uudds) [7]. Arguments based on quark models suggest that this state is a member of SU(3) antidecuplet with spin J = 1 2 or 3 2 . The hadro/photo production cross section would depend on the spin J and parity P of the Θ + , and it is important to have reliable predictions for these cross sections. The most proper way to address these issues will be chiral perturbation theory.
In this paper, we construct a chiral lagrangian for pentaquark baryons assuming they are SU(3) antidecuplet with J = 1 2 and P = +1 or −1. (The case for J = 3 2 can be discussed in a similar manner, except that antidecuplets are described by Rarita-Schwinger fields.) Then we calculate the mass spectra of antidecuplets, their possible mixings with pentaquark octets, the decay rates of antidecuplets, and cross sections for π − p → K − Θ + and γn → K − Θ + . Finally we describe how to include light vector mesons in our framework, and how the low energy theorem is recovered in the soft pion limit. 2. Chiral lagrangian for a pentaquark baryon decuplet -Let us denote the Goldstone boson field by pion octet π, baryon octet including nucleons by B, and antidecuplet including Θ + by P.
MeV is the pion decay constant, transforms as It is convenient to define another field ξ(x) by Σ(x) ≡ ξ 2 (x), which transforms as The 3 × 3 matrix field U (x) depends on Goldstone fields π(x) as well as the SU(3) transformation matrices L and R. It is convenient to define two vector fields with following properties under chiral transformations: Note that V µ transforms like a gauge field. The transformation of the baryon octet and pentaquark antidecuplet P including Θ + (I = 0) can be chosen as where all the indices are for SU(3) flavor. The pentaquark baryons are related to P abc = P (abc) by, for example, P 333 = Θ + , P 133 = 1 Then, one can define a covariant derivative D µ , which transforms as D µ B → U D µ BU † , by Chiral symmetry is explicitly broken by non-vanishing current-quark masses and electromagnetic interactions. The former can be included by regarding the quark-mass matrix m = diag (m u , m d , m s ) as a spurion with transformation property m → LmR † = RmL † . It is more convenient to use ξmξ + ξ † mξ † , which transforms as an SU (3) octet. Electromagnetic interactions can be included by introducing photon field A µ and its field strength tensor where Q ≡ diag (2/3, −1/3, −1/3) is the electric-charge matrix for light quarks (q = u, d, s). Now it is straightforward to construct a chiral lagrangian with lowest order in derivative expansion.
The parity and charge-conjugation symmetric chiral lagrangian is given by where where P is the parity of Θ + , Γ + = γ 5 , and Γ − = 1, and m P is the average of the pentaquark decuplet mass. The Gell-Mann-Okubo formulae for pentaquark baryons will be obtained from Expanding this, we get the mass splittings ∆m i ≡ m i − m P within the antidecuplet: wherem = m u = m d ignoring small isospin-breaking effects. If the newly observed state at a mass 1862 ± 2 MeV is identified as Ξ 3/2 , we find Recently Jaffe and Wilczek suggested there could be an ideal mixing between pentaquark antidecuplet P abc and pentaquark octet O a b with the same parities [9]. This idea has been generalized further by other groups [10,11]. In chiral lagrangian approach, such a general mixing arises from Expanding this leads to where B m = 2β m (m s −m)/ √ 3 and we borrowed baryonoctet notation for pentaquark octet states in Eq. (9). Note that the relative sign between n N 0 and p N + (and also Σ 0 Σ 0 ) is negative, unlike the case in Ref. [10]. This is due to the 10 nature of the P. One could write down the same mixing between pentaquark antidecuplet P abc and the ordinary baryon octet B, but such terms will be highly suppressed compared to the above term, since it is a mixing between qqq and qqqqq.
Finally the baryon decuplet can only couple to pentaquark octet O, but not to pentaquark antidecuplet P, since 10 ⊗ 8 ⊗ 10 does not contain SU (3) singlet. This implies that N (1440) or N (1710) cannot be pure pentaquark antidecuplets, because they have substantial branching ratios into ∆π final states. They could be mixed states of pentaquark octet and pentaquark antidecuplet, and their productions and decays will be more complicated than pure antidecuplet case. Since the current data on baryon sectors are not enough to study such mixings in details, we do not pursue the mixing further in the following.
Parameters in the above lagrangian are taken to have the following numerical values: m B ≈ 940 MeV is the nucleon mass, D ≈ −0.81 and F ≈ −0.47 at tree level, and we assumem = 0 and m 2 η = (4/3) m 2 K . The coupling C PN is determined from the decay width Γ Θ of the Θ + , which is dominated by K + n and K 0 p modes as Γ Θ /2 = Γ Θ + →K + n = Γ Θ + →K 0 p : where p * is the kaon momentum in the Θ + rest frame and the signs are for P (Θ + ) = ±1, respectively. Then the C PN is determined as Cahn and Trilling [12] argues that Γ Θ = (0.9 ± 0.3)MeV using the DIANA results [2]. We present our results proportional to the C 2 PN rescaled by the factor 1 MeV/Γ Θ . Such a small C PN can be understood as following: this coupling is related to the matrix element of hadronic axial vector current operator (with zero baryon number) between a pentaquark baryon and an ordinary baryon. Since two states have different number of valence quarks, this matrix element should be highly suppressed compared to the ordinary axial vector coupling (D + F ) or H PN .
The coupling H PN is also unknown, and determines transition rates between pentaquark antidecuplets with pion or kaon emission. Unfortunately, such decays are all kinematically forbidden, and cannot be used to fix H PN . However, we expect that H PN = O(1), without any suppression as in C PN . With this remark in mind, we will assume H PN can vary between −4 and 4 in the numerical analysis to be as general as possible.
3. π − p → K − Θ + -Let us first consider π − p → K − Θ + as applications of our chiral lagrangian for pentaquark baryons. In Figs. 1 (a) and (b), we show the relevant Feynman diagrams. Note that only Fig. 1 (a) was considered in the literature. However, there is an s−channel N 0 (1647) exchange diagram [ Fig. 1 (b) ] in our chiral lagrangian, since Θ + is not an SU(3) singlet, but belongs to the antidecuplet. Therefore one has to The amplitude for π − p → K − Θ + is given by We show the total cross section, rescaled by the factor Γ Θ /MeV, as functions of pion energy E π at the proton rest frame in Fig. 2 depending on the Θ + parity, and varied −4 ≤ H PN ≤ 4. Note that the sign of H PN is very important. If H PN > 0 (< 0), two contributions will have constructive (destructive) interference. Thus our results differ from the previous results where only the n contribution was included. Also the cross section is sensitive to H PN , and may be useful to fix H PN . Following the spirit of chiral perturbation theory at lowest order, we did not include model-dependent form factors, keeping in mind that our results get unreliable as the E π gets as large as ∼ 2.5 GeV.
Note that the even parity and the odd parity cases can be distinguished from the cross section for the parity-odd case is smaller than that for the parity-even case.
4. Photo-production of Θ + -In order to study photoproduction of Θ + on nucleons, we need to know the magnetic dipole interaction terms. For the nucleon octet, The anomalous magnetic moments of nucleons are at tree-level chiral lagrangian. Using κ p = 1.79 and κ n = −1.91, we get κ D = 2.87 and κ F = 0.836. For the pentaquark baryon P, the relevant term is We expect that |κ Θ (≡ κ P )| ≈ |κ D | ≈ |κ F |. On the other hand, a calculation in soliton picture predicts that κ Θ ≈ 0.3, which is rather small [13]. We vary κ Θ between −1 and 1. We ignore transition magnetic moments between nucleon octet and pentaquark antidecuplet, since this transition involves qqq and qqqqq.
The relevant Feynman diagrams for γn → K − Θ + are shown in Fig. 3. One salient feature of our approach based on chiral perturbation theory is the existence of a contact term for γK − nΘ + vertex [ Fig. 3 (d)] that arises from the C PN term in Eq. (4c) with Eq. (2c), which is necessary to recover U(1) em gauge invariance within spontaneously broken global chiral symmetries. The resulting amplitude is The cross sections and the angular distributions in the center of momentum frame are shown in Fig. 4 (a) and (b). Note that the parity-even case has larger cross section, and has a sharp rise near the threshold. The angular distribution shows that the forward/backward scattering is suppressed in the negative parity case, whereas the forward peak is present in the positive parity case. Therefore the angular distribution could be another useful tool to determine the parity of Θ + . Therefore, once C 2 PN is determined from Γ Θ , one could determine the parity of Θ + , and make a rough estimate of κ Θ from the photoproduction cross section.
5. Including light vector mesons -One can also introduce light vector mesons ρ µ , which transforms as under global chiral transformations [14]. Then ρ µ (x) transforms as a gauge field under local SU(3)'s defined by Eq. (1), as V µ does. The covariant derivative D µ can be defined using ρ µ instead of V µ . Note that (ρ µ − V µ ) has a simple transformation property under chiral transformation: and it is straightforward to construct chiral invariant lagrangian using this new field. In terms of a field strength tensor ρ µν , It is important to notice that N Θ + K * coupling should be highly suppressed, since it can appear only in combination of (ρ µ − V µ ), which vanishes in the low-energy limit. In other words, the low-energy theorem is violated if one includes only nΘ + K * diagram, without including the nΘ + Kπ contact term arising from the V / term. Therefore, one should be cautious about claiming that the K * exchange is important in π − p → K − Θ + . Detailed numerical analysis of vector-meson exchange is straightforward, but beyond the scope of the present work and will be pursued elsewhere [15].
6. Conclusion -In conclusion, we constructed a chiral lagrangian involving pentaquark baryon antidecuplet and octet, the ordinary nucleon octet and Goldstone bosons. Using this lagrangian, we derived the Gell-Mann-Okubo formula and the mixing between the pentaquark antidecuplet and pentaquark octet. We also calculated the cross sections for π − p → K − Θ + and γn → K − Θ + for J P = 1 2 ± . In particular, we emphasized that it is very important to respect chiral symmetry properly in order to get correct amplitudes for these processes. All these observables depend on parameters in our chiral lagrangian, which have relations with underlying QCD, but are uncalculable from QCD at present. Photo-production data will be particularly useful in identifying the parity of Θ + , because the threshold behavior of the cross section and its angular distributions strongly depend on the parity. Once the coupling C PN is determined from the decay width of Θ + , then the parity and other couplings H PN and κ Θ could be determined from the hadro/photoproduction cross sections for Θ + . It is straightforward to apply our approach to other related processes such as γp → K 0 Θ + or K + p → π + Θ + , or other pentaquark baryons. Finally we have outlined how to incorporate the vector meson degrees of freedom in our scheme, the details of which will be discussed in the separate publication [15].
This work is supported in part by KOSEF through CHEP at Kyungpook National University and by the BK21 program. The research of JL in the High Energy Physics Division at Argonne National Laboratory is supported by the U. S. Department of Energy, Division of High Energy Physics, under Contract W-31-109-ENG-38. JL thanks KAIST and KIAS for their hospitality during this work.