PHYSICAL REVIEW LETTERS week ending PRL 95, 172001 (2005) 21 OCTOBER 2005
Chiral Suppression of Scalar-Glueball Decay
Michael S. Chanowitz* Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, USA (Received 13 June 2005; published 20 October 2005)
Since glueballs are SU 3 Flavor singlets, they should couple equally to u, d, and s quarks, so that equal coupling strengths to and K K are expected. However, we show that chiral symmetry implies the scalar-glueball amplitude for G0 ! qq is proportional to the quark mass, so that mixing with ss mesons is enhanced and decays to K K are favored over . Together with evidence from lattice calculations and experiment, this supports the hypothesis that f0 1710 is the ground state scalar glueball.
DOI: 10.1103/PhysRevLett.95.172001 PACS numbers: 12.39.Mk, 11.30.Rd
1. Introduction.—The existence of gluonic states is a known enhancement of ! relative to ! e .For quintessential prediction of quantum chromodynamics mq 0 chiral symmetry requires the final q and q to have (QCD). The key difference between quantum electrody- equal chirality, hence unequal helicity, so that in the G0 rest namics (QED) and QCD is that gluons carry color charge frame with the z axis in the quark direction of motion, the while photons are electrically neutral. Gluon pairs can then total z component of spin is nonvanishing, jSZj 1. form color singlet bound states, ‘‘glueballs,’’ like mesons Because the ground state G0 gg wave function is isotropic and baryons, which are color singlet bound states of va- (L S 0), the qq final state is pure s wave [4], L 0. lence quarks [1]. Because of formidable experimental and The total angular momentum is zero, and since there is no theoretical difficulties, it is not surprising that this simple, way to cancel the nonvanishing spin contribution, the dramatic prediction has resisted experimental verification amplitude vanishes. With one power of mq Þ 0, the q for more than two decades. Quenched lattice simulations and q have unequal chirality and the amplitude is allowed. predict that the mass of the lightest glueball, G0, a scalar, is The enhancement is substantial, since ms is an order of near ’ 1:65 GeV [2], but the prediction is complicated by magnitude larger than mu and md [5]. But for scalar glue- mixing with qq mesons that require more powerful com- balls of mass ’ 1:5–2 GeV, G0 ! ss is suppressed of putations. Experimentally the outstanding difficulty is that 2 order ms=mG0 , so that it may be smaller than the nomi- glueballs are not easily distinguished from ordinary qq nally higher order G0 ! qqg process, which is SU 3 Flavor mesons, themselves imperfectly understood. This diffi- symmetric. We find that the soft and collinear quark- culty is also exacerbated if mixing is appreciable. gluon singularities of G0 ! qqg vanish for mq 0,as The most robust identification criterion, necessary but they must if G0 ! qq is to vanish at one loop order for not sufficient, is that glueballs are extra states, beyond mq 0. Unsuppressed, flavor-symmetric G0 ! qqg de- those of the qq meson spectrum. This is difficult to apply cays are dominated by configurations in which the gluon in practice, though ultimately essential. In addition, glue- is well separated from the quarks, which hadronize pre- balls are expected to be copiously produced in gluon-rich dominantly to multibody final states. G0 may also decay channels such as radiative J= decay, and to have small flavor symmetrically to multigluon final states (n 3), via two photon decay widths. These two expectations are the three and four gluon components of G2 in Eq. (1), by encapsulated in the quantitative measure ‘‘stickiness,’’ [3] higher dimension operators that arise nonperturbatively, or which characterizes the relative strength of gluonic versus by higher orders in perturbation theory. The IR singular- photonic couplings. ities of G0 ! ggg are canceled by virtual corrections to the Another popular criterion is based on the fact that glue- G0 wave function, while configurations with three (or balls are SU 3 Flavor singlets which should then couple more) well separated gluons hadronize to multihadron final equally to different flavors of quarks. However we show states. The enhancement of ss relative to uu dd is then here that the amplitude for the decay of the ground state most strongly reflected in two body decays: we expect scalar glueball to quark-antiquark is proportional to the K K to be enhanced relative to , while multibody quark mass, M G0 ! qq /mq, so that decays to ss pairs decays are more nearly flavor symmetric. are greatly enhanced over uu dd , and mixing with ss Glueball decay to light quarks and/or gluons cannot mesons is enhanced relative to uu dd . We exhibit the be computed reliably in any fixed order of perturba- result at leading order and show that it holds to all orders in tion theory. However, the predicted ratio, G0 ! ss = standard QCD perturbation theory. G0 ! uu dd 1, is credible since it follows from The result has a simple nonperturbative physical expla- an analysis to all orders in perturbation theory and, in nation, similar, though different in detail, to the well- addition, from a physical argument. The implication that
0031-9007=05=95(17)=172001(4)$23.00 172001-1 © 2005 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 95, 172001 (2005) 21 OCTOBER 2005
G0 ! K K G0 ! is less secure and is From the lowest order Feynman diagrams we obtain best studied on the lattice. Remarkably, it is supported by p an early quenched study of G decay to pseudoscalar 32 2 S 0 X M meson pairs for two ‘‘relatively heavy’’ SU 3 Flavor sym- 3 m metric values of mq, corresponding to mPS ’ 400 and ’ q 2 2 u p3;h3 v p4;h4 ij (4) 630 MeV [6]. Linear dependence on mq implies quadratic 1 cos dependence on mPS [7], which is consistent at 1 with the where u ;v are the q; q spinors for quark and antiquark lattice computations [6]. Chiral suppression could then be 3 4 with center of mass momenta p3;p4, helicities h3;h4, color the physical basis for the unexpected and unexplained indices i; j, and center of mass velocity . Equation (4) lattice result. With subsequent computational and theoreti- includes a colorp factor from the color singlet gg wave cal advances in lattice QCD, it should be possible today to function, C 2=3 verify the earlier study and to extend it to smaller values of ij ij After angular integration the decay amplitude is mPS, nearer to the chiral limit and to the physical pion p mass. If the explanation is indeed chiral suppression, then 16 2 mq 1 M G ! qq f log u v : the couplings of higher spin glueballs should be approxi- 0 0 S 3 1 3 4 ij mately flavor symmetric and independent of m , a pre- PS (5) diction which can also be tested on the lattice. Enhanced strange quark decay changes the expected Squaring Eq. (5), summing helicities and color indices, and experimental signature and supports the hypothesis that integrating over the phase space, the width is f0 1710 is predominantly the ground state scalar glueball. 16 2 2 2 2 1 This identification was advocated by Sexton, Vaccarino, G0 ! qq Sf0mqMG log : (6) and Weingarten [6], and is even more compelling today in 3 1 view of recent results from J= decay obtained by BES Notice that an explicit factor mq appears in the gg ! qq [8,9]—see Ref. [10] for an experimental overview. amplitude, Eq. (4), which is not averaged over the initial In Section 2 we compute M G0 ! qq at leading order gluon direction and which clearly has contributions from for massive quarks, with the expected linear dependence on higher partial waves, J>0. It may then appear that chiral mq. In Section 3 we show that M G0 ! qq /mq to any suppression applies not just to spin-0 glueballs but also to order in S. In Section 4 we describe the infrared singu- glueballs of higher spin. However, when Eq. (4) is squared 2 larities of M G0 ! qqg . We conclude with a brief dis- and the phase space integration is performed, a factor 1=mq cussion, including experimental implications. results from the t and u channel poles, which cancels the 2. G ! qq at leading order.—Consider the decay of 2 0 explicit factor mq in the numerator, so that the total cross a scalar glueball G0 with mass MG to a qq pair [11] section gg ! qq does not vanish in the chiral limit, with quark mass mq. The effective glueball-gluon-gluon because of the J>0 partial waves. coupling is parameterized by 3. G0 ! qq to all orders.—We now show that G ! qq vanishes to all orders in perturbation theory M 0 L eff f0G0Ga Ga (1) for mq 0. Consider the Lorentz invariant amplitude M X p1;p2;p3;p4 X M : (7) where G0 is an interpolating field for the glueball, Ga is the gluon field strength tensor with color index a, and f is 0 The perturbative expansion for MX is a sum of terms an effective coupling constant that depends on the G0 wave arising from Feynman diagrams with arbitrary numbers function. The gg ! qq scattering amplitude is of loops. After evaluation of the loop integrals, regularized as necessary, MX is a sum of terms, M gg ! qq 1 2 M gg ! qq (2) X M X u p3; 3 iu p4; 4 ; (8) i where i pi; i with i 1; 2 are the polarization vectors for massless constituent gluons with four- where u3;u4 are massless fermion, antifermion spinors momentum pi and polarization i. Summing over the [12] of chirality 3; 4. The i are 4 4 matrices, each a polarizations i and averaging over the gluon direction in product of ni momentum-contracted Dirac matrices, the G0 rest frame to project out the s wave, we find 6 6 6 i ki1ki2 ...kini (9) Z f0 where each kia is one of the external four-momenta, M G0 ! qq d X M gg ! qq ; (3) 4 p1;p2;p3;p4. Chiral invariance for mq 0 implies that the number of 2 where X 2p2 p1 MGg projects out the j factors, ni, in Eq. (9) is always odd. Since all external i helicity state that couples to Ga Ga in Eq. (1). momenta vanish and the spinors obey p6 3u3 p6 4u4 0,
172001-2 PHYSICAL REVIEW LETTERS week ending PRL 95, 172001 (2005) 21 OCTOBER 2005 by suitably anticommuting the 6kia, each term in Eq. (8) can Feynman diagrams using the helicity spinor method [12] be reduced to a sum of terms linear in p6 1 and p6 2, which we with numerical evaluation of the 9 dimensional integral: choose to be symmetric and antisymmetric, X Z 2 G0 ! qqg jM G0 ! q 3q4g5 j u p3; 3 iu p4; 4 u p3; 3 Si s; t; u p6 1 p6 2 PS h3;h4; 5 Ai s; t; u p6 1 p6 2 u p4; 4 : 2 X Z Z Z f0 0 2 d 1 d 1 5 X (10) 16 PS h3;h4; 5 A and S are Lorentz invariant functions of the 0 i i M g1g2 ! q 4q3g5 5 X Mandelstam variables s; t; u. Since p p p p , 1 2 3 4 0 0 the symmetric term vanishes and Eq. (8) reduces to M g1g2 ! q 4q3g5 : (13)
M X A s; t; u u p3; 3 p6 1 p6 2 u p4; 4 : (11) We focus here on the infrared singularities, which provide P a consistency check at one loop order that M G0 ! qq where A s; t; u iAi s; t; u . vanishes in the chiral limit. Next consider the integration over the gluon direction, In general there could be soft IR divergences for Eq. (3). In the G rest frame with the z axis chosen along 0 Eq;Eq ;Eg ! 0 and collinear divergences for the quark direction of motion, z^ p^ 3, we integrate over ; ; ! 0. In fact, only the qq collinear divergence 2 qg qg qq d d p^ 1 d 1d cos 1. The Mandelstam variables are occurs: neither dN=dE nor dN=dE diverge at low en- then s M2 and u; t 1 M2 1 cos . Since the g q G 2 G 1 ergy, and only dN= diverges at ! 0. Instead color and helicity components of the G wave function qq qq 0 dN=dE diverges at the maximum energy, E m =2, are symmetric under the interchange of the two gluons, g g G0 and dN=d diverges for ! . Both of these diver- Bose symmetry requires A s; t; u to be odd under p $ qg qg 1 gences are kinematical reflections of the collinear singu- p2. In our chosen coordinate system A is a function only of cos , and must therefore be odd, A cos larity at qq ! 0, for which the qq pair with mqq 0 1 1 recoils with half the available energy against the gluon in A cos 1 . But evaluating u 3 p6 1 p6 2 u4 explicitly [12] we find the opposite hemisphere (paper in preparation [13]). This is precisely the pattern of divergences required if 2 i 1 u 3 p6 1 p6 2 u4 MGe sin 1 (12) G0 ! qq is chirally suppressed to all orders and, in par- ticular, at one loop. For if there were soft divergences in i 1 which is even in cos 1, while the azimuthal factor, e , any of Eq;Eq ;Eg or collinear divergences in qg and qg , i 1 provides the required oddness underR p1 $ p2: e ! then the resulting singularities at mqg;mqg ! 0 would have i 1 i 1 e e . Consequently d cos 1A vanishes, to be canceled by virtual corrections to G0 ! qq , such as and M G0 ! qq 0 to all orders in the chiral limit. gluon self-energy contributions to the quark propagator. Because of our choice of axis, z^ p^ 3, the integral over The absence of these singularities is a consistency check 1 also vanishes. For other choices of z^ the azimuthal and (i.e., a necessary condition) that G0 ! qq is chirally sup- polar integralsR do not vanish separately, but the full angular pressed at one loop order. The collinear divergence for 2 integral, d p^ 1, vanishes in any case. qq ! 0 is canceled by quark loop contributions to the For nonvanishing quark mass, mq Þ 0, chirality-flip gg ! gg amplitude, which in the present context are one amplitudes contribute. With one factor of mq from the loop corrections to the G0 wave function. fermion line connecting the external quark and antiquark, 5. Discussion.—We have shown to all orders in pertur- the i matrices in Eq. (8) include products of even numbers bation theory and with a simple, nonperturbative physical of Dirac matrices, i.e., ni in Eq. (9) may be even. In order argument that the ground state J 0 glueball has a chirally mq there are then contributions to M G0 ! qq , like the suppressed coupling to light quarks, M G0 ! qq /mq, term shown explicitly in Eq. (5). with corrections of higher order in mq=mG. From Eq. (6) The vanishing azimuthal integration for z^ p^ 3 reflects with mu;md;ms varied within 1 limits [5], G0 ! ss the physical argument given in the introduction. The factor dominates G0 ! uu dd by a factor between 20 and i 1 e corresponds to SZ 1 from the aligned spins of the 100. Flavor symmetry is reinstated for G0 ! qqg when the q and q , while the absence of a compensating factorR in A gluon is well separated from the q and q . For sufficiently 2 is due to the projection of the orbital s wave by the d p^ 1 heavy mG one can test this picture by measuring strange- integration and the absence of spin polarization in the ness yield as a function of thrust or sphericity, with en- initial state. hanced strangeness in high thrust or low sphericity events, 4. Infrared singularities of G0 ! qqg .—Although it is but it is unclear if this is feasible for mG ’ 1:7 GeV.Itis 3 of order S, G0 ! qqg is not chirally suppressed and more feasible for the ground state pseudoscalar glueball, may therefore be larger than G0 ! ss , which is of order which is expected to be heavier than the scalar and which 2 2 2 S ms=mG. Setting mq 0 we evaluated the 13 we also expect to be subject to chiral suppression.
172001-3 PHYSICAL REVIEW LETTERS week ending PRL 95, 172001 (2005) 21 OCTOBER 2005
For light scalar glueballs, the best hope to see strange- suppressed two body couplings at zero momentum. I thank ness enhancement is the two body decays, G0 ! A. Soni, H. Murayama, and M. Schwartz for bringing these K K = . Since G0 ! qqg and G0 ! ggg are not works to my attention. chirally suppressed, naive power counting suggests G0 ! qqg ggg G0 ! ss , so that n 3 parton decay amplitudes are probably the dominant mechanism for multihadron production. Then KK will dominate two *Email address: [email protected] body decays while multiparticle final states are approxi- [1] The existence of valence gluons is supported by lattice mately SU 3 Flavor symmetric up to phase space correc- simulations—see K. J. Juge, J. Kuti, and C. J. Morning- tions favoring nonstrange final states. star, Nucl. Phys. B, Proc. Suppl. 63, 543 (1998). Chiral suppression has a major impact on the experi- [2] C. Morningstar and M. J. Peardon, in Scalar Mesons: An mental search for the ground state scalar glueball. Interesting Puzzle for QCD, edited by A. Fariborz, AIP Candidates cannot be ruled out because they decay pref- Conf. Proc. No. 688 (AIP, New York, 2003) p. 220. erentially to strange final states, especially KK , and mix- [3] M. Chanowitz, Proceedings of the Sixth International ing with ss mesons may be enhanced. This picture fits Workshop on Photon-Photon Collisions, Lake Tahoe, 1984 (World Scientific, Singapore, 1985). nicely with the f0 1710 meson: it is copiously pro- [4] Integrating over the gluon direction to project out the duced in radiative decay in the ! KK channel [8] s-wave gg wave function is equivalent to integrating and in the gluon-rich central rapidity region in pp scatter- over the final quark direction with the initial gluon direc- ing [14], has a small coupling [15], mass consistent tion fixed. with the prediction of quenched lattice QCD [2], and a [5] K. Hagiwara et al., (Particle Data Group), Phys. Lett. B strong preference to decay to KK , with B =B KK < 592, 473 (2004). The ratios of u, d and s quark masses are 0:11 at 95% CL [9]. As a rough estimate of the sticki- renormalization-scale invariant for Q>2ms. ness [3], we combine the 95% CL upper limit with [6] J. Sexton, A. Vaccarino, and D. Weingarten, Phys. Rev. central values for radiative decay [8], with the re- Lett. 75, 4563 (1995); Nucl. Phys. B, Proc. Suppl. 47, 128 0 (1996). sult S f0 1710 :S f2 1525 :S f2 1270 ’ >36 :12:1. A more complete discussion of the experimental situation [7] M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968). will be given elsewhere [13]—see also Ref. [10]. [8] J. Z. Bai et al. (BES), Phys. Rev. D 68, 052003 (2003); The interpretation of f0 1710 as the chirally suppressed W. Dunwoodie et al. (Mark-III), in Hadron Spectroscopy scalar glueball can be tested both theoretically and experi- 1997, edited by S.-U. Chung and H. Willutzki, AIP Conf. mentally. Lattice QCD can test the prediction that G0 ! Proc. No. 432 (AIP, New York, 1998), p. 753. KK is enhanced for the ground state J 0 glueball but not [9] M. Ablikim et al. (BES), Phys. Lett. B 603, 138 (2004). for J>0. With an order of magnitude more J= decays [10] M. Chanowitz, Gluonic States: Glueballs & Hybrids, pre- than BES II, experiments at BES III and CESR-C will sented at Topical Seminar on Frontier of Particle Physics study rarer two body decays and multiparticle decays. If, (QCD and Light Hadrons, Beijing, 2004); posted at http:// as seems likely, the rate for multiparticle decays is big, the bes.ihep.ac.cn/conference/2004summersch/p.htm. [11] Elastic scattering, gg ! gg, contributes to the glueball lower bound on B ! f0 1710 will increase beyond its already appreciable value from KK alone. A large wave function, not to the decay amplitude. The dissocia- tion process, gg ! qq qq , in which each gluon makes a inclusive rate B ! f0 , a large ratio B f0 ! transition to a color-octet qq pair, is kinematically forbid- KK =B f0 ! , and approximately flavor-symmetric den for mq Þ 0; an additional gluon exchange is required couplings to multiparticle final states would support the to allow it to proceed on mass shell, which is therefore of identification of f0 1710 as the chirally suppressed, order g4 in the amplitude. ground state scalar glueball. [12] R. Kleiss and W. J. Stirling, Nucl. Phys. B262, 235 (1985). I wish to thank F. Close, M. Golden, and S. Sharpe for [13] M. Chanowitz (to be published). helpful discussions. This work was supported in part by the [14] D. Barberis et al. (WA102), Phys. Lett. B 453, 305 (1999); Director, Office of Science, Office of High Energy and B. R. French et al. (WA76/102), Phys. Lett. B 460, 213 Nuclear Physics, Division of High Energy Physics, of the (1999). U.S. Department of Energy under Contract No. DE-AC03- [15] H. J. Behrend et al. (CELLO), Z. Phys. C 43, 91 (1989). 76SF00098. [16] R. J. Crewther, Phys. Rev. Lett. 28, 1421 (1972); M. S. Chanowitz and J. R. Ellis (SLAC), Phys. Lett. B 40, 397 Note added.—The trace anomaly [16] implies chiral (1972); Phys. Rev. D 7, 2490 (1973); J. C. Collins, suppression of the G0 coupling to and KK at zero A. Duncan, and S. D. Joglekar, Phys. Rev. D 16, 438 four-momentum [17]; however, the extrapolation to the (1977). G0 mass-shell is out of control and could be large. [17] J. M. Cornwall and A. Soni, Phys. Rev. D 32, 764 (1985). Similarly, in the conjectured anti-de Sitter approach to [18] C. Csaki, H. Ooguri, Y. Oz, and J. Terning, J. High Energy QCD the scalar glueball is a dilaton [18], with chirally Phys. 01 (1999) 017.
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