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97/1490 CLNS IASSNS-HEP 97-60 SNUTP 97-073 hep-ph/9708355 Chiral Perturbation Theory for Tensor Mesons 1 Chi-Keung Chowa and Soo-Jong Reyb;c Newman Laboratory for Nuclear Studies Cornell University, Ithaca NY 14853 USA School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton NJ 08540 USA Physics Department, Seoul National University, Seoul 151-742 KOREA a b c ckchow@lnsth.lns.cornell.edu, sjrey@ias.edu abstract Interactions of a2 ; K2 ; f2 and f20 tensor-mesons with low-energy ; K; ; 0 pseudo-scalar mesons are constrained by chiral symmetry. We derive a chiral Lagrangian of tensor mesons in which the tensor mesons are treated as heavy non-relativistic matter elds. Using 1=Nc counting, we derive relations among unknown couplings of the chiral Lagrangian. Chiral perturbation theory is applied to the tensor-meson mass matrix. At one-loop there are large corrections to the individual tensor meson masses, but the singlet-octet mixing angle remains almost unchanged. We argue that all heavy mesons of spin 1 share common feature of chiral dynamics. Submitted to Physics Letters B 1 Work supported in part by NSF Grant, NSF-KOSEF Bilateral Grant, KOSEF Purpose-Oriented Research Grant 94-1400-04-01-3 and SRC-Program, Ministry of Education Grant BSRI 97-2410, the Monell Foundation and the Seoam Foundation Fellowships. An important application of chiral perturbation theory is to describe the interaction of matter elds (such as nucleons 1, 2] or hadrons containing a heavy quark 3, 4, 5, 6]) with lowmomentum pseudo-Goldstone bosons { the pions, kaons and eta. In Ref. 7] chiral perturbation theory was used to describe the interactions of the , K , and ! vector mesons with lowmomentum pseudo-Goldstone bosons. In this article, we will extend the formalism to study the lowest-lying tensor meson nonet. This nonet is expected to have quantum numbers J P = 2+, and contains the isotriplet a2 (1320) and the S = 1 isodoublets K2 (1430). The two isosinglet states are not as well established, but experimental evidences suggest that they are probably f2 (1275) and f20 (1525). The mass di erence between these nine lowest-lying tensor mesons are small compared to the chiral symmetry breaking scale 4 f 1 GeV. Hence, chiral perturbation theory should be applicable as a systematic expansion procedure for a class of processes involving tensor mesons and soft Goldstone bosons. In the past, chiral perturbation theory has been used extensively to study processes which do not have a tensor meson in the nal state. In such decays, the nal state pions are not soft enough that the application of the chiral Lagrangian to such processes is not justi ed a priori. At the best, they serve as a phenomenological model. The pseudo-Goldstone boson elds can be written as a 3 3 special unitary matrix (1) = exp 2i ; f where 0p +p + K+ 1 2 6 B C p2 + p6 K 0 C : (2) =B A @ 0 2 p K K 6 y , where L 2 SU(3)L and R 2 SU(3)R . At leading Under chiral SU(3)L SU(3)R , ! L R order in chiral perturbation theory, f can be identi ed with the pion or kaon decay constant (f 132 MeV, fK 160 MeV). It is convenient, when describing the interactions of the pseudo-Goldstone bosons with other elds to introduce p = exp if = : (3) Under chiral SU(3)L SU(3)R , ! L U y = U Ry ; (4) where in general U is a complicated function of L, R and the meson elds . For transformations V = L = R in the unbroken SU(3)V subgroup, U = V . 0 0 1 The tensor meson elds are introduced as a 3 3 octet matrix 0a f B p2 + p6 ~ B O = B a2 B @ K2 0 2 (8) 2 a+ 2 a p + fp 2 6 0 K2 0 2 (8) 2 K2 + K2 0 2f p 6 (8) 2 1 C C C; C A (5) (6) (We have deliberately made our notations for the tensor mesons identical to that for the vector mesons in Ref. 7] except for an additional tilde.) By de nition these tensor mesons are symmetric and traceless Lorentz tensors, ~ O ~ S ~ =O ; ~ =S ; ~ O = 0; ~ S = 0: (7) (8) (9) (10) and as a singlet ~ S = f2(0) : Moreover, the polarizations of the tensor mesons are necessarily orthogonal to the momentum, ~ ~ p O = p S = 0: Under chiral SU(3)L SU(3)R , ~ ~ O ! U O U y; and under charge conjugation, ~ ~ C O C 1 = OT ; ~ ~ CS C 1 = S ; ~ ~ S !S ; C C 1 = T: We construct a chiral lagrangian for tensor mesons by treating the tensor mesons as heavy static elds 8, 9] with xed four-velocity v , with v2 = 1. Eq. (8) becomes ~ ~ v O = v S = 0: (11) The chiral lagrangian density which describes the interactions of the tensor mesons with the low-momentum , K and mesons has the general structure L = Lkin + Lint + Lmass: At the leading order in the derivative and quark mass expansions, i~ i~ ~ ~ Lkin = 2 S y (v @)S 2 Tr Oy (v D)O ; 2 (12) (13) and ~ ~~ + h:c: Lint = 2i g1 S y Tr (O A )v i ~~ + 2 g2 Tr (fOy ; O gA )v ~ ; where and ~ ~ ~ D O = @ O + V ; O ]; (14) (15) (16) where M is the quark mass matrix M = diag(mu; md ; ms), and i V = 1 ( @ y + y@ ); A = 2 ( @ y y@ ): 2 Finally, to linear order in quark mass expansion, 8 0 ~~ ~~ Lmass = ~0 + ~2 Tr M S y S + ~8 + ~2 Tr M Tr Oy O ~ ~ ~ ~ ~~ + 1 Tr (Oy M )S + h:c: + 2 Tr (fOy ; O gM ); 2 2 (17) (18) M = 1 ( M + yM y): 2 As mentioned in Ref. 7], one of the mass parameters can be removed by a simultaneous phase rede nition of O and S, and only the singlet-octet mass di erence ~ ~0 ~8 is physically relevant. It is yet unclear which f2 state corresponds to the lowest-lying isosinglet tensor meson, but for all reasonable choices ~ 300 MeV and can be regarded as a quantity of order mq . Note that ~ is of order Nc 1 and vanishes in the large Nc limit. To x couplings in the chiral Lagrangian, we analyze the spectrum of tensor mesons given at leading order in chiral perturbation theory. The analysis is essentially identical to the SU(3) analysis. In the approximation with exact isospin symmetry mu = md = m, we nd that a2 ^ and K2 tensor mesons are interaction and energy eigenstates simultaneously and have masses as: ^ Ma = 8 + 2 ~ 2 m ^ (19) MK = 8 + ~2(m + ms) 2 2 while the f (0) and f (8) mesons have mass matrix 0 1 2 + 2 ~2(m + 2ms) p6 ~1(m ms) A ^ ^ 3 M (8 0) = @ 8 p ~ (m m ) : 2 s 0 61^ We have abbreviated combinations of parameters as 8 (20) (21) = ~8 + ~8 Tr M 3 0 = ~0 + ~0Tr M: Using the above relations, we nd that the singlet-octet mass matrix elements can be expressed in terms of the experimentally measured tensor-meson masses: 4 (8 M11 0) = 3 MK 1 Ma 3 4 (8 M22 0) = Mf 0 + Mf 3 MK + 1 Ma 3 h4 1 M M 0 M 4 M + 1 M i: (8 (8 M12 0) = M21 0) = (22) f f 3 MK 3 a 3K 3a The mass eigenstates of octet-singlet mixing tensor mesons are parametrized by a mixing angle T: ! ! (8) ! jf2 i : jf2i = cos T + sin T (23) 0i (0) jf sin cos 2 2 2 2 2 2 2 2 2 2 2 2 v uM 0 4M + 1M u 1 p: tan T = t 4 f 3 1 K 3 a = 0:556 (24) Mf 3 3 MK 3 Ma Here, we have identi ed Mf = 1270 MeV and Mf 0 = 1525 MeV respectively. This translates 2 2 2 2 2 2 2 2 The above relation of octet-singlet mass matrix suggests the usual SUV (3) prediction for the tangent of the mixing angle 2 T T jf2 i into a mixing angle T = 29:1 , which is close but not exactly the value lin = 26 as quoted in Particle Data Group 10]. For comparison T = 35 for an ideal mixing, and experimentally 1:1) 11]. T is measured to be (25:3 In the large Nc limit, quark loops are suppressed. Thus the leading diagrams in the meson sector contain a single quark loop. As a result, the octet and the singlet tensor mesons can be combined into a single nonet matrix ~ ~ 1~ (25) N = O + p3 S : ~ The chiral Lagrangian in the large Nc limit is expressed exclusively in terms of N tensor eld. At leading order in 1=Nc, the kinetic and interaction parts are given by ~ ~ Lkin = 2i TrN y v DN g ~~ (26) Lint = i~2 Tr(fN y ; N gA )v 2 while the mass matrix part is given by ~ ~~~ ~~ Lmass = 2 TrN y N + 22 Tr fN y ; N gM : (27) Comparison with the chiral we Lagrangian have constructed above indicates that in the Nc ! 1 limit ~ ! 0; ~0 ! 0; ~8 ! 0; (28) 4 g 1 p g1 ! 2~2 ; ~1 ! 2 2 tan T ! p : ~p 3 3 2 This means that jf2i state becomes `pure' (ss) state and the nonet matrix is given by 1 0a f p +p a+ K2 + C B 2 2 a2f p + p K2 0 C : C N = B a2 B @ A 2 2 0 K2 f20 K2 0 2 2 0 2 2 (29) (30) In leading order one has the ideal mixing angle ( T )N !1 = 35 which is close but o by 20% from the actual value T = 29:1 obtained from Eq. (24). Note, however, that the mixing angle is quite sensitive to the f2 masses, and Eq. (24) can be brought into consistency with ideal mixing by just shifting f20 (1525) up by 40 MeV. This discrepancy may be due to contamination of f2 f20 tensor mesons with exotic states such as tensor glueball states and/or exotic mesons such as , ! ! bound-state resonances 12]. The coupling constants g1;2 are free parameters in chiral perturbation theory. To estimate ~ these coupling constants, we decompose the quark-model wave functions of vector and tensor mesons in their rest frames (v = (1; ~ )) into their isospin, spin and spatial parts. 0 jV a( i )i = j a i jS = 1; Sz = ii j1s; L = 0; Lz = 0i; jT a( ij i = j a i jS = 1; Sz = ii j2p; L = 1; Lz = ji + (i $ j) trace in (ij): (31) Note that the vector and tensor mesons are di erent only in their spatial wave functions. Current algebra 13, 14] suggests that the axial current coupling is given by Aai V a SiV y; where (32) V = exp( i (S L)z ); (33) 1 with a small parameter (sin2 8 in the baryon sector, and is expected to be of similar magnitude for mesons). In the zeroth order of an expansion in , the axial current coupling Aai = a S i does not couple to the spatial wave functions at all. As a result, g1;2 = g1;2, i.e., the ~ same parameters govern the axial couplings of the vector and tensor mesons. As discussed in p ~ 7], under the assumption of ideal mixing, g1 = g1 = 2= 3 and g2 = g2 = 1 in the nonrelativistic ~ constituent quark model. If we employ the non-relativistic chiral quark model 15], these factors are further reduced by a factor of 3=4. In chiral perturbation theory, the leading order corrections to the tensor meson masses are 3=2 of order mq and arise from one-loop self-energy diagrams due to virtual pseudoscalar meson 5 exchange. A straightforward calculation gives i h22 2 3 Ma = 8 1f 2 g2 3 m3 + 2mK + 3 m3 + g1 m3 ~ ~2 h23 i 1 ~ ~2 K MK = 8 1f 2 g2 2 m3 + 5 m3 + 6 m3 + g1 m3 3K (08) M11 = 8 1f 2 g1 3m3 + 4m3 + m3 ~2 K h2 i 2 2 (08) ~ ~2 M22 = 8 1f 2 g2 2m3 + 3 m3 + 3 m3 + g1 m3 K 2 2 (08) (08) ~~ (34) M12 = M21 = + 8 1f 2 2 g1g2( 3m3 + 2m3 + m3 ): K 3 The mass corrections are quite substantial. For example, ma 450 MeV. On the other hand, the singlet-octet mixing angle T after leading order chiral perturbation one-loop correction is taken into account remains very small. The mixing angle is 2 s v uM 0 4M + 1M u M tan T = t 4 f 3 1 K 3 a MK 3 Ma Mf + M ; 3 2 2 2 2 2 2 (35) where 4 M +1 M 3K3a = 8 1f 2 g1 + 3 g2 ~2 2 ~2 Using the large Nc relation between g1 and g2, we ~ ~ M= 2 2 (08) + M22 1 m3 4 m3 + m3 : 3 3K nd that (36) 2~2 1 m3 4 m3 + m3 : g2 (37) 8 f2 3 3K For a reasonable range g2 0:75 1, the mass correction M ~ 6 MeV. The corresponding shift in mixing angle is about 1 . Its smallness (when contrasted with the huge corrections to individual masses) is presumably because the combination M has to vanish in the large Nc limit, i.e., M = O(Nc 1). M A remark is in order. Various heuristic arguments indicate that J = 2 tensor mesons might exhibit moderate mixing with other 2++ meson states. The rst is tensor glueball. In the past, there has been various phenomenological model for the mixing. It has been argued that, once the glueball mixing is taken into account, the f2 meson is in ideal mixing 11]. The second are tetra-quark states. There has been no systematic analysis of their in uence to the mixing. Within chiral perturbation theory the tetraquark states might be treatable systematically as a perturbation of (8L ; 8R) (8L; 8R) (8L ; 8R) irreducible state. Electromagnetic correction 6 to the mass matrix is of theoretical interest. In this case one has to bear in mind that both short-distance and long-distance e ects contributions have to be taken into account. It is straightforward to generalize the chiral perturbation theory to (qq) mesons of higher spin. The spin-s meson elds are introduced in terms of 3 3 octet matrix O s and a singlet S s . The spin-s mesons are irreducible representations of the Lorentz group, viz. totally symmetric and traceless components. The polarization of these mesons are necessarily orthogonal to the momentum. For heavy static elds the four-velocity also satis es vO s = 0. Chiral Lagrangian of heavy spin-s tensor mesons is exactly the s=v S same as Eqs.(13)-(18): i i Lkin = s! S y s (v @)S s s! Tr Oy s (v D)O s 8 0; s = 0; > > 12 12 1 1 2 1 1 1 1 1 1 Lint > > <i + (h:c:) = > s! g1S y Tr (O A )v > > + i g Tr(fOy ; O gA )v : ; otherwise, s! 2 1 1 1 1 1 Lmass = s!0 S y s S 1 1 s + s! TrO 1 8 y 1 s O 1 s + s!1 Tr(Oy s M )S s 2 + (h:c:) + s! Tr(fOy 1 s ;O 1 s gM ): (38) Since the interaction Lagrangian structure is exactly the same as 2++ tensor mesons, the leading order and the one-loop correction to mass spectra and mixing angles follow exactly the same pattern. Hence the singlet-octet mixing for these higher spin mesons are also related to their masses through Eq. (24), and be ideally mixed in the large Nc limit. For example, by plugging in the masses of the spin-3 mesons one obtains the mixing angle = 28 , in qualitative agreement with the large Nc prediction. Moreover, the chiral one-loop correction will take the form of Eq. (35) where M will be given by Eq. (37) up of spin-dependent multiplicative constants. Note that, however, these higher spin mesons are heavier in mass and there are many other possible sources of contamination, which cannot be treated within the framework of chiral perturbation theory. In this letter, we have studied chiral perturbation theory of heavy tensor mesons of spin 2. We have shown that the octet - singlet mixing angle is quite close to `ideal mixing' and oneloop correction to the mixing angle is negligible. On the other hand, mass spectra themselves receive quite a sizable corrections. We conclude that the chiral perturbation theory `explains' naturally why the singlet-octet mixing is sign cantly o the ideal mixing for pseudo-scalar 7 Goldstone bosons while close to the ideal mixing for all higher spin 1 mesons. We thank S. von Dombrowski, F. Wilczek, M.B. Wise and T.M. Yan for useful discussions. References 1] E. Jenkins and A.V. Manohar, Phys. Lett. B255 558 (1991). 2] E. Jenkins and A.V. Manohar, Phys. Lett. B259 353 (1991). 3] M.B. Wise, Phys. Rev. D45 2188 (1992). 4] G. Burdman and J.F. Donoghue, Phys. Lett. B280 287 (1992). 5] T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin and H.L. Yu, Phys. Rev. D46 1148 (1992); erratum D55 5851 (1997). 6] P. Cho, Nucl. Phys. B396 183 (1993). 7] E. Jenkins, A.V. Manohar and M.B. Wise, Phys. Rev. Lett. 75 2272 (1995). 8] E. Eichten and B. Hill, Phys. Lett. B234 511 (1990). 9] H. Georgi, Phys. Lett. B240 447 (1990). 10] Particle Data Group, Phys. Rev. D54 98 (1996) 11] Crystal Ball Collaboration, Phys. Lett. B385 425 (1996). 12] J.L. Rosner, Phys. Rev. D241347 (1981); J.L. Rosner and S.F. Tuan, Phys. Rev. D27 1544 (1983); S.S. Eremyan and A.E. Nazaryan, Sov. J. Nucl. Phys. 45 1088 (1987) ; F. Caruso and E. Predazzi, Z. Phys. C33 (1987) 569. 13] F. Buccella, H. Kleinert, C.A. Savoy, E. Celeghini and E. Sorace, Nuovo Cimento A69 133 (1970). 14] F.E. Close, An Introduction to Quarks and Partons, Academic Press (1979), Sect. 6.4.1. 15] A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189. 8

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