3
VOLUME 26, NUMBER 21 PHYSI CA, L REVIEW LETTERS 24 Mz~ 1971 ular, if a, is equal to zero, then 2m, '&m, [m, +(m, '-a, ')'"]. Using an idea of Penrose, " Chris- todolou' has shown that one can get arbitrarily near this limit. One can interpret this as ex- tracting the rotational energy of the black hole. J. Weber, Phys. Rev. Lett. 22, 1320 (1969). J. Weber, Phys. Rev. Lett. 24, 276 (1970) . J. Weber, Phys. Rev. Lett. 25, 180 (1970). G. W. Gibbons and S. W. Hawking, "The Detection of Short Bursts of Gravitational Radiation" (to be pub- lished) . 5G. B. Field, M. J. Bees, and D. W. Sciama, Com- ments Astrophys. Space Phys. 1, 187 (1969). B. Carter, Phys. Rev. Lett. 26, 331 (1971). W. Israel, Phys. Rev. 164, 1776 {1967). 8 R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967). R. Penrose, private communication. ' R. Penrose, in Battelle Rencontxes 2967, edited by C. M. de Witt and J. A. Wheeler (Benjamin, New York, 1968) . "R. Penrose and S. W. Hawking, Proc. Roy Soc. Ser. A 314, 529 (1970). S. W. Hawking, Proc. Roy. Soc. , Ser. A 300, 187 (1967). R. Penrose, Riv. Nuovo Cimento 1, Num. spec. , 252 (1969) . H. Bondi, M. G. J. van der Burg, and A. W. K. Metz- uer, Proc. Roy. Soc. , Ser. A 269, 21 (1962). R. Penrose, Phys. Rev. Lett. 10, 66 (1963). D. Christodolou, Phys. Rev. Lett. 25, 1596 (1970). Heavy Mesons and Neutral Axial-Vector Currents* P. Carruthers Laboratory of Nuclear Studies, Cornell University, Ithaca, New Yorh 14850 (Received 1 April 1971) It is suggested that the eighth and ninth axial-vector currents are dominated by the J+ =0 mesous q(549), q'(958), aud the (presumed) J =1+ mesons D(1285) aud E(1 242). Pre- dictions are made concerning the decay modes of D and 8 as well as for the DNA and ENN couplings. The existence of an octet of axial-vector cur- rents may be regarded as well established. How- ever the status of a possible ninth current Fo„', an "axial baryon current, " has remained obscure, with some authors doubting its existence and oth- ers ascribing special virtues to it. In the present note we suggest that the g'(960) and E(1422) are manifestations of this current and suggest exper- imental means of illuminating this question. At the same time we present a parallel interpreta- tion of the eighth axial current 5, „' in terms of the properties of the q(550) and D(1285) mesons, The decay modes of D and E are rather simi- lar, with rprm and KKm [with KK coming from the 0" state m„(1016)] observed for both mesons. " (The E meson also decays through KK*, which is kinematically forbidden for D. ) Apart from the K*I7 mode, we interpret the gem and EZm final states as due to the sequence D(E) - eg or D(E) -~„7l with subsequent decays ~-27t, m„-vtq, and v„-KZ. As in an earlier paper' we consider e and 7T„ to be members of a scalar nonet. In the following me shall relate the axial-scalar-pseu- doscalar couplings to the scalar-pseudoscalar- pseudoscalar couplings by a "universality" argu- ment based on partial conservation of axial-vec- tor current (PCAC) for all nine axial-vector cur- rents. We consider the nine axial-vector currents $+' (i =0, ~ ~ ~, 8). The matrix elements of the isospin current F„' may be regarded as dominated by the n (0 ) and the A, (1'). Since there is a. nonet of pseudoscalars [the ninth being q'(958)], it is rea- sonable to ask about a companion nonet of axial- vector mesons and further to speculate that the (0, 1') nonets dominate the corresponding cur- rents. The experimental situation is unclear, but it appears that the A, (1070), D(1285), K„(1240), and E(1422) are good candidates for a nonet. " Comparison with the pseudoscalar octet suggests that D is mostly octet and E the singlet. With this assignment the assumed A, and K~ masses imply that the mixing is negligible. Since qg' mixing is small (sl0'), we shall ignore mixing here and associate the pairs (q, D) and (g', E) with the unmixed currents 5, „, ' and 7, &', respec- tively. The coupling constants g, „D, g, „, D, etc. , may be related to g, «, g„„„by a generalization of a universality argument proposed in Ref. 3 for the (n, A, ) complex. The matrix element (v~F„'~ e) is given in terms of the form factors F, and F,

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VOLUME 26, NUMBER 21 PHYSI CA, L REVIEW LETTERS 24 Mz~ 1971

ular, if a, is equal to zero, then 2m, '&m, [m,+(m, '-a, ')'"]. Using an idea of Penrose, "Chris-todolou' has shown that one can get arbitrarilynear this limit. One can interpret this as ex-tracting the rotational energy of the black hole.

J. Weber, Phys. Rev. Lett. 22, 1320 (1969).J.Weber, Phys. Rev. Lett. 24, 276 (1970) .J. Weber, Phys. Rev. Lett. 25, 180 (1970).G. W. Gibbons and S. W. Hawking, "The Detection

of Short Bursts of Gravitational Radiation" (to be pub-lished) .

5G. B. Field, M. J. Bees, and D. W. Sciama, Com-ments Astrophys. Space Phys. 1, 187 (1969).

B. Carter, Phys. Rev. Lett. 26, 331 (1971).

W. Israel, Phys. Rev. 164, 1776 {1967).8R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8,

265 (1967).R. Penrose, private communication.

' R. Penrose, in Battelle Rencontxes 2967, edited byC. M. de Witt and J. A. Wheeler (Benjamin, New York,1968) ."R. Penrose and S. W. Hawking, Proc. Roy Soc. Ser.

A 314, 529 (1970).S. W. Hawking, Proc. Roy. Soc., Ser. A 300, 187

(1967).R. Penrose, Riv. Nuovo Cimento 1, Num. spec. ,

252 (1969) .H. Bondi, M. G. J. van der Burg, and A. W. K. Metz-

uer, Proc. Roy. Soc., Ser. A 269, 21 (1962).R. Penrose, Phys. Rev. Lett. 10, 66 (1963).D. Christodolou, Phys. Rev. Lett. 25, 1596 (1970).

Heavy Mesons and Neutral Axial-Vector Currents*

P. CarruthersLaboratory of Nuclear Studies, Cornell University, Ithaca, New Yorh 14850

(Received 1 April 1971)

It is suggested that the eighth and ninth axial-vector currents are dominated by the J+=0 mesous q(549), q'(958), aud the (presumed) J =1+ mesons D(1285) aud E(1 242). Pre-dictions are made concerning the decay modes of D and 8 as well as for the DNA andENN couplings.

The existence of an octet of axial-vector cur-rents may be regarded as well established. How-ever the status of a possible ninth current Fo„',an "axial baryon current, " has remained obscure,with some authors doubting its existence and oth-ers ascribing special virtues to it. In the presentnote we suggest that the g'(960) and E(1422) aremanifestations of this current and suggest exper-imental means of illuminating this question. Atthe same time we present a parallel interpreta-tion of the eighth axial current 5, „' in terms ofthe properties of the q(550) and D(1285) mesons,

The decay modes of D and E are rather simi-lar, with rprm and KKm [with KK coming from the0" state m„(1016)]observed for both mesons. "(The E meson also decays through KK*, which iskinematically forbidden for D.) Apart from theK*I7 mode, we interpret the gem and EZm finalstates as due to the sequence D(E) - eg or D(E)-~„7l with subsequent decays ~-27t, m„-vtq, andv„-KZ. As in an earlier paper' we consider e

and 7T„ to be members of a scalar nonet. In thefollowing me shall relate the axial-scalar-pseu-doscalar couplings to the scalar-pseudoscalar-pseudoscalar couplings by a "universality" argu-ment based on partial conservation of axial-vec-

tor current (PCAC) for all nine axial-vector cur-rents.

We consider the nine axial-vector currents $+'(i =0, ~ ~ ~, 8). The matrix elements of the isospincurrent F„' may be regarded as dominated by then (0 ) and the A, (1'). Since there is a. nonet ofpseudoscalars [the ninth being q'(958)], it is rea-sonable to ask about a companion nonet of axial-vector mesons and further to speculate that the(0, 1') nonets dominate the corresponding cur-rents. The experimental situation is unclear, butit appears that the A, (1070), D(1285), K„(1240),and E(1422) are good candidates for a nonet. "Comparison with the pseudoscalar octet suggeststhat D is mostly octet and E the singlet. Withthis assignment the assumed A, and K~ massesimply that the mixing is negligible. Since qg'mixing is small (sl0'), we shall ignore mixinghere and associate the pairs (q, D) and (g', E)with the unmixed currents 5, „,

' and 7, &', respec-tively.

The coupling constants g, „D, g,„,D, etc. , maybe related to g, «, g„„„bya generalization of auniversality argument proposed in Ref. 3 for the(n, A, ) complex. The matrix element (v~F„'~ e)is given in terms of the form factors F, and F,

VOLUME 26, NUMBER 21 PHYSICAL REVIEW LE Y'r ERS 24 Mwv 1971

2 (3)

Experimental determinations of the ~mA and

A,NN coupling mould provide a direct test of theproposed universal pattern relating the A, and 7).

couplings.If we now apply the same arguments to the ma-

trix elements (g ( F»' ( e) and (w ( 5, &' ( n ~), we find,instead of (2),"

g Dg, „~= 2m~'F, G, „„/(m „'-m, '),

g~g, „,~=2mD'F „G,„,„/(m„'-m, ').(4)

(5)

&.(p )l~„'l~(p))=- ((p+P')„F,(t)+(P'-P)„F,(t)], (I)

where t =(p'-p)'. We now make two assumptions:(1) F, and F, are dominated by the m- and A, -polecontributions [this includes the assumption F,(~)=F,(~) =0]; (2) PCAC holds, i.e., 9 "F„'obeys anunsubtracted dispersion relation for all nine axi-al currents.

The validity of the PCAC assumption presum-ably depends on the existence of a limit in whichthe currents are conserved and in which the pseu-doscalar nonet mesons are massless Goldstone-Nambu particles, rather than on the smallnessof the pseudoscalar mass. ' We identify this limitwith that of (spontaneously broken) scale invari-ance, assuming as previously' that this limit ischaracterized by U(3) SU(3) symmetry of the en-ergy-density. The large mass of the g' makes itimplausible-that the-ninth axial current 5„,' isconserved in Bie SU(3) SSU(3) limit, ' which pointsto the existence of an SU(3) 8SU(3)-invariantscale-breaking operator. It is of great interestto check, even qualitatively, the predictionscharacteristic of this class of theories. "

As shown in Ref. 3, the assumptions followingEq. (1) lead to the following relation between the

EmA, and ewe coupling constants":

(2)

This type of relation is easily extended to arbi-trary external particles and provides a principlesimilar to that used to "derive" universal p cou-ling. " In the present case the universal patternis a relation (enforced by PCAC) between thecouplings of two particles (A, and w). It was al-ready noted' that Eq. (2) leads to a reasonableprediction for A, —~w and also leads simply tothe Kawarabayashi-Suzuki- Fayyazuddin-Riazud-din relation. " A similar idea was applied byMich" to the nucleon, giving"

If the current F»' is changed to F,„', Eqs. (4)and (5) hold with D replaced by F.. It is also pos-sible to change external labels from & to e' and

q to g'.The constant gn =g„by SU(3), and g„may be

estimated to be m&~/yz (y~'/4n = 2) by Weinberg'ssecond sum rule. " SU(3) symmetry also gives

The latter "equality" holds to about20%%uo, while there is little evidence bearing onthe equality of I' „and I,.

The relation (4) is not very useful because the

&gq coupling is difficult to obtain. '" The properinterpretation of Eq. (4) is that G, „„is small, aresult in accord with the numerical results ofRef. 3. Hence we turn to Eq. (5) which is usefulsince the decay 7tN- gw can be directly observed.If we attribute the full width"' of 80 MeV tothis mode, the coupling constant is G, , „/4w

N—= 0.05 GeV . Equation (5) then predicts g, ,~'/4m=7. 4. From the decay formula 1(D-w„w)= (g„„,~'/24w)p'/m~', we find I"(D - w~n) = 10.6MeV for m, =0.98 GeV. This prediction is asatisfactory fraction of the total D width (30+ 5

MeV). The v„m pair decays into both KZm and

gm7t, the latter mode also receiving contributionsfrom D —&g-2ng, which we found difficult to es-timate. Some experimental light can be shed onthis process by subtracting off the mNw part ofthe spry mode and attributing the remainder to &

The relations (2) and (5) deal with the octetproperties of the axial currents and axial me-sons. The best way to test these relations is(1) to separate the A, —em component of the A,-3m decay; (2) to make a study of the D- v„vmode [and incidentally to clarify the relation be-tween n„(980) and w„(1020)].

Finally we consider the predictions for the(singlet) meson E. The interesting relationsare"

gag, „s=2m 'F „.G,„„,/(~„'-~, '),gsg~~~E =2m@ F „,G~ ~ „,/(y+~ m~ ). (7)

Equation (7) is of little use since it is very diffi-cult to estimate G„,z, . Equation (8) is more

Npromising since the E - ~g mode can be deter-mined experimentally (an upper bound gs„,~/4w= 15.9 is obtained if we attribute the entire 2nqmode, I'= 50 MeV, to this process), and a roughestimate of G,«. can be obtained from a polemodel for g'- ~g-2m'. From these numbers oneobtains the value (valid to within a factor of 2)g~/F „.=2.9 GeV, not far from g„/F, .

The knowledge of gs/F „.allows the prediction

1347

VOLUME 26, NUMBER 21 PHYSlCAL REVIEW LETTERS 24 Mxv 1971

of the ratio gs~~/g„». For the D and F. mesonsthe formulas analogous to Eq. (3) are

2gDgDEÃ D F'qg7lNN/ Nr

2gEgx~~™xF.g, ~~/2~~

(8)

(9)

Equation (8) may be used to estimate g»„. Sincethe gNN coupling is very smali2'23 (with g v»'/4m of order 10 ' to 10 '), Eq. (8) predicts thatg»„'/4v is also small, say 10 ' to 10 '. If weuse the crude value for gx/F ~. obtained in thepreceding paragraph we find g»~/g„. » = 0.4.The numerical uncertainties are so great that we

only can state that g»N should be roughly equal

*Work supported in part by the National Science Foun-dation.

M. Boos et al. , Phys. Lett. BBB, 1 (1970).B. Lorstad, Ch. d'Andlau, A. Astier, J. Cohen-Ga-

nouna, M. Della-Negra, M. Aguilar-Benitez, J. Bar-low, L. D. Jacobs, P. Malecki, and L. Montanet, Nucl.Phys. 814, 68 (1969). This paper summarizes much ofthe evidence on the D and E mesons.

P. Carruthers, Phys. Rev. D 8, 959 (1971).Although e (700) seems reasonably well established,

the situation with regard to ~z is obscure. We have re-garded 6 (962), v~(980), and v„(1016) as manifestationsof the same state (see Hef. 8).

We are disregarding (conflicting) evidence for a sec-ond K&, as well as the controversial H(990). A pre-ferred value 4 =0 for E(1422) is listed in Ref. 1, al-though inspection of the original sources convinced theauthor that 1' is equally acceptable.

J. L. Rosner and E. %. Colglazier, Phys. Bev. Lett.26, 988 (1971), have proposed a nonet of axial-vectormesons in which the SU(8) singlet is identified with arecently discovered resonance at 955 MeV. In our pro-posal the assumed similarity of the pseudoscalar and

axial-vector nonets suggests that the I+ SU(8) singletshould be more massive than the 1+ octet. The A& tra-jectory is often considered to be degenerate with thepion trajectory. With our assignments, this roughlygeneralizes {iftrajectories are parallel), as indicated

by m&& -m„= 1.14, m& -m z——1.36, m& -m zi = 1.10,2 2 2 2 2 2

m„~ —mr = 1.29. Interchanging D and E would com-pletely ruin the rough equality of the above squared-mass differences.

VB. Dashen, Phys. Bev. 188, 1245 (1969).BP. Carruthers, Phys. Bev. D 2, 2265 (1970).S. L. Glashow, in HaChons and Their Interactions,

edited by A. Zichichi (Academic, New York, 1968).' This point of view is not necessarily in contradic-

tion with "nonlinear realizations" of chiral symmetryin which the quantities Po&~ and q' do not appear.Such models may be only approximations to a morecomplete dynamics.

f1The constants E~ and gA occuring in Eq. (2) are de-fi~ed by (ol&;„'I~; (p)& = ip„z~, &Olt;&'I&;(P ~)&= gee&(P, A, ), i = 1,2, B. The effective coupling constantsare defined by the Lagrangians Z~« = g&«eA" && and

g«~ ——2t",„~ex . For further details see Bef. 8.~2

~2M. Gell-Mann and F. Zachariasen, Phys. Rev. 124,958 (1961).

13K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett.16, 255 (1966); Riazuddin and Fayyazuddin, Phys. Bev.147, 1071 (1966).

H. T. Nieh, Phys. Bev. 164, 1780 (1968). Equation(8) amounts to reinterpreting t"z in terms of the A&NN

coupling constant g& zz.' The coupling conventions used here correspond to

effective interactions g~z = g~&N y&vN A&+ " and

71' = ig~ gPNVy5N ~ X.$6gz and I'

& are defined in analogy to g~ and I"~ in Ref.11. gDgq and gappy are defined by g~ ~=gD q6D 8~'g

pg ~7j +pf D 8p+ e

~~S. Weinberg, Phys. Bev. Lett. 18, 507 (1967).~An. additional problem for D-eq is the kinematic

sensitivity to the e "mass" which is near threshold.A proper treatment would begin with the amplitude(2v O',„'I n&.~~B.Ammar, W. Kropoc, H. Yarger, R. Davis, M. Mott,

H. Werner, M. Derrick, T. Fields, F. Schweingruber,D. Hodge, and D. D. Reeder, Phys. Bev. Lett. 21,1882 (1968), and Phys. Rev. D 2, 480 (1970).

OM. A. Abolins, R. Graven, G. A. Smith, L. H. Smith,A. B. Wicklund, R. L. Lander, and D. E. Pellet, Phys.Hev. Lett. 25, 469 (1970).

There are three other relations involving externalpairs (e,q'), (d, g), (d, q'). These all correspond tokinematically forbidden reactions.

S. B. Deans and %. Holladay, Phys. Rev. 165, 1886{1969).

R. C. Chase, E. Coleman, H. W. J. Courant, E. Mar-quit, E. W. Petraske, H. F. Romer, and K. Ruddick,Phys. Lett. SOB, 659 (1969). These authors obtain alarger value for g&z~ than that given in Ref. 22, usingdifferent assumptions. In either case g~» /4v issmall.

i348