Meson hyperfine splittings and leptonic decays

Volume 57B, number 5

PHYSICS LETTERS

MESON

HYPERFINE

SPLITTINGS

AND LEPTONIC

4 Augustus 1975

DECAYS

R. B A R B I E R I *, R. G A T T O * * , R. K ( ) G E R L E R * and Z. K U N S Z T * * CERN-Geneva-Switzerland • *Instituto di Fisica and Instituto Nazionale di Fisica Nucleare, Roma, Italy Received 3 May 1975 The 3S-I S mass separations of the bound quark-antiquark systems, as arising from one-gluon exchange, in the charm color quartet scheme, are discussed for the ground mesons and their first radial excitations expecially in connection to leptonic decay rates. Overall consistency requires large modifications for the positronium-like formulae for annihilation and the possible role in this direction of an outstanding gluon radiative correction to the leptonic width is pointed out.

The m o s t recent data on S ( 3 . 1 ) [1], S '(3.7) [2], and e+e - -+ hadrons at higher energy [2] seem, to this m o m e n t , to be in general a g r e e m e n t w i t h the charm m o d e l based o n four quarks u, d, s, c, each in three colors, interacting through an S U ( 3 ) - c o l o r o c t e t o f gluons, according to the picture o f e+e - - a n n i h i l a t i o n suggested b y A p p e l q u i s t and Politzer [3] and De Rujula and Glashow [4] , 1 , 2 . A n essential role in such a picture is played by the e x p e c t e d suppression o f the amplitudes for 9 ~ n o n - c h a r m e d hadrons .3'4, S ' ~ non, l An interesting check of these ideas may come from measurement of the modes ~0 ~ old hadrons + 3"4, ¢, ~ old hadrons + 3,. By comparing the three gluon annihilation (O(asa) where a s is the gluon coupling constant) to the annihilation into 2 gluons + photon (O(a2sa) one finds a ratio for r(~0 + hadrons + V) to F(~0 ~ hadrons) of (16/15)(a/as). This is about 4% for as= 0.18. *2 Charm might play in weak interactions the role suggested by Glashow et al. [5]. This is not relevant however to the considerations developed in the present note. The modifications to our conclusions which would result from a different model containing heavy quarks and colored gluons are straightforward. ,3 Experimental knowledge of the rough magnitude of the branching ratios for decays which are twice forbidden by Zweig rule would be of great interest. For ¢ ~ ~ + 7r + *r, ~ ~o+ f, (but not for ¢ --, ¢ + f* or ~ ~ ~o+ W), ¢ --, .f* + 3*r, etc, and similarly for ~', one expects partial widths of about 0.5 - 0.15% of the corresponding widths for ~0 (or ~0') into old hadrons (i.e. not-charm-containing). Such decays proceed through 5 gluons whereas ~0 (or ¢ ' ) decay into old hadrons proceed through 3 gluons. We note that channels such as ~, or ~V'~ c c ~ + 2 gluons ~ ~ + n + n, etc. could modify this picture. For additional remarks on lightquark pair-production see also the last sentences of footnote*S.

charmed-hadrons *4's and for S ' ~ S + n o n - c h a r m e d hadrons. A m o r e recent d e v e l o p m e n t has b e e n p r o p o s e d b y De Rujula et a l . [7] w h o postulate a picture o f quark binding consisting o f a universal interaction (which they take S U ( 4 ) symmetric, b u t we shall here m o d i f y this point), responsible for the i m p r i s o n m e n t , to which one-gluon exchange effects are added. We shall here consider such a picture limiting our a t t e n t i o n to the effect o f the spin d e p e n d e n t one-gluon term (responsible for hyperfine-splitting) and treat it directly in conj u n c t i o n with any o t h e r available information. We shall treat on this basis the splitting b e t w e e n 3S and IS (ssO-states and (ccO-states. F o r the latter states we assume that the additional i n f o r m a t i o n f r o m the S - w i d t h s into leptons and hadrons m a y safely be i n t r o d u c e d into this description. This leads to a picture o f the pseudoscalar states W, W', 7/c consistent w i t h data so far. Predictions for the mass differences (D* - D) and (F * - F) also follow directly. We t h e n apply the description to 2S states, n a m e l y a ps n o n e t (rt', K ' , E 1 , E 2 ( 1 4 2 0 ) , •/'c) and a vector n o n e t ( p ' , K * ' , co', ~', S ' ) with rough estimates o f the e x p e c t e d masses. An essential p o i n t ,4 The important decays ~0 ~ to + ~r++ n-, ~0 --, ~ + K + + K-, XO~ co + K + + K - are expected to proceed through three gluon~ Assuming that the final pair is in s-state, the ratios of the corresponding phase spaces are: 1 : 0.35" 0.51. Similarly, for ~ ' ~ t o + rr++ ~r-, ~ ' ~ + K++ K-, ~0'~ to + K++ K-, the ratios are 1 : 0.50:0.65 (for ¢ ' ~ to + ~r++ *rto ¢ ~ to + 7r++ *r- such ratio would be 1.3). Deviations from such figures are evidence for differences in couplings and for structure. ,s See next page. 455

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of our description is the incorporation of an exceptionally large gluon radiative correction (similar to a corresponding one for positronium) in the calculation of the leptonic widths. In absence of such a correctior, the relative success of the theory in describing hyperfine splittings would violently conflict with the data on leptonic decays. For the non-relativistic spin-spin interaction we assume the form given by one gluon exchange

H'= ~Cts~(Xl - X2) ~

.

(1)

In eq. ( 1 ) a s is the gluon coupling constant and m 1 , m 2 are quark masses. We recall that De Rujula et al. [7] estimate the mass of the non-strange quarks, u, d, from the assumption that the anomalous magnetic moment of the quark is negligible. The mass m u is then

m u = mproton/Uproton = 938 MeV/2.79 = 336 MeV .(2)

4 A u g u s t u s 1975

The mass of the strange quark, s, is then estimated from the assumption that the hyperfine splitting calculated from (1) is the only contribution to the massdifferences K * - K and p - lr, with the same values of the coupling constant t~s and of the meson wavefunction. This gives ( K * - K)/(p - 7r) ~ mu/m s = 0 . 6 3 ,

(3)

and from (2) m s = 0.54 .

(4)

We shall now proceed by defining the two unmixed (s~-states of spin 1 and 0, 3S and 1S to be called ~o0 and r/~ respectively. On the same assumptions one has 90 - r/~ ~. ( K * - K) 2/(p - rr) ~ 250 MeV.

(5)

We estimate the mass of~o0 as 90 ~ 2 K * - p

= 1.014GeV,

(6)

from which one obtains t

*s If the ~0'-width turns o u t to be m u c h larger than the sum of F(~0'~ ~0 + 2rt) + I'(~0' ~ P-states-(co-)+ 3') + P(~O(3.1)--, hadrons) a real decay into DD might account for the discrepancy, provided 2 m D is sufficiently lower than m~/(for estimates see ref. [6]). Possible.additional m o d e s are ~k'~ nc + to, to ~ 3rt and same with ~o in place o f to (both to and ~o off-mass shell). With a coupling.

(f/m~

D one obtains, treating nc p~to)p(rlc)e~)__ __ e(to)

as stable

.

(m~o' -mrs) 2

x f9 m~

1 [~___~mtoFto [m~o,+mrlc~ :~

[(m~0'- mnc)2 - sl w2

ds-

(s - m 2 ) 2

With m n ~ 3.1 this gives I" ~ ( / ~ / 4 n ) (6 keV). It is n o t possible '{o predict the value o f f a t this time. One can only compare with similar interactions n o t Zweig forbidden, such as p --* tort, obtaining the suggestion F < 60 keV. Virtual production of light-quark pairs, ~0'~ cc-uu, with subseq u e n t annihilation into ordinary h a d r o n s c ~ u ~ ~ u~--, ordinary hadrons, could also account for a fraction o f the ~0'-width. With the mass values chosen in this work the thresholds for light-quark pair production are at 4 GeV for e+e - ~ cc uu, and at 4.4 GeV for e+e - ~ t e s s . Note that a process such as, e+e - -+ c ~ u ~ ~ ordinary hadrons, can entirely proceed within the strong infrared sector. This pro.~ rides a m e c h a n i s m for annihilation into ordinary h a d r o n s which is n o t (at least directly) affected by the smallness o f a s for large s and it is even independent o f the existence of charmed hadrons.

456

r/0 ~ 0.765 G e V .

(7)

The value in eq. (7) is substantially lower than the value 2K - Ir = 0.854 GeV. We consider the value in eq. (7) as more reliable since it is based on the estimate (6), which cannot be much in error for the little mixed ~o-meson, and on the direct use of (1) as the only source of hyperfine splitting. It therefore presumably correctly incorporates the perturbation on the (s-ff)-wavefunction from kinetic terms and from the additional gluon exchange terms other than that in eq. (1). For the w 0 and 770 states, of quark composition (u6 + dd)/x/2, this approach leads to masses 6o0 = P0 and r/0 = 7r. The 1.6% p - ~ mass difference effect is neglected and arises from mixing and higher orders. To calculate the hyperfine splitting $0 - (r/c)0 for the 3S and 1S (~)-states, e for~e+e -. we use as additional imput the width, I'¢, The non-relativistic description of the electro-magnetic decay into e+e - of a quark-antiquark bound state leads to the approximate expression for the width, for small ots

i, e ~ 3e~ot2 16rr l~o(0)l 2 ( l - 16 ) 3 m2 ~ as '

(8)

where m is the bound state mass, ~o(0) its wavefunction at the origin, eQe is the quark-charge, and the factor 3 in front comes from color. The last factor in eq. (8) is a comparatively larger gluon radiative correction due

Volume 57B, number 3

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to exchange of a transverse gluon between the two quarks, whose propagator has a non-infrared part

/3(s) -1 = ~ ( s i

6tw/K2 --/~o6vo/K 2 , not taken into account in the building-up of the meson wave-function. The radiative correction in eq. (8) is similar to a corresponding term in the theory of positronium [8]. Its role is decisive in reducing the conflict between the description of hyperfine splitting and lep, tonic width in the ( ~ ) .system, although a full quantitative picture, already for that mass region, will require a more complete theoretical formulation .6 . Both F e and:the hyperfine splitting, ~00-(r/e)0, are proportional to the squared wavefunction ig(O) l2 of the (c~) system. By eliminating I~(0)12 one finds

1

~0-(rlc)0

= _ _

as

(m¢12

2a2i-16t~s/3n\m

el

e

I'~v '

- m~0)2) -1 ,

(11)

where the sum is extended over all quark-antiquark pairs, i, of mass m! 0) The square of the c m energy is denoted by s. For Zthe masses m(.0) we take from the precedin~ discussion: rn~0)= m~6)= rr, m~0)= r/~= 765 MeV, m(40) = (r/c) 0 = 3.05 GeV. From eq. (11) one has at the physical masses, s = ,12 = (549 MeV) and s = 7/'2 = (958 MeV), (.12) 13(s = r/2) = (540 MeV) 2 , /3.(s= 7/'2) = (440 MeV) 2 , and the decompositions r/= (0.8 2) ~22 (ufi + dd) - (0.5 8 ) (s'g) + (0.02) (c~)! 13 )

(9)

7?'= (0.46)~22 (ufi + dd) + (0.88)(sg) + (0.03)(68)!14)

where m c is the mass of the c-quark. We can estimate t~s, following ref. [3] and [4], from the ratio of F (~ hadrons) to 1-'($ ~ e+e - ) _ 18no~ 2 P ( ~ - ~ h a d r o n s ) 1 - 16Ots/3Zr 5 ( r r 2 _ 9 ) P(~0 ~ e + e - ) '

4 Augustus 1975

°ts3

(10)

neglecting possible gluon radiative correction to F b, which should not contribute very much to the determination o f a s because of the high power, a 3 , in eq. (10). With [1] P~0 = (67 + 20)keV, F~ = (5.2 + 1.3) keV, eq. (10) gives a s = 0.18 + 0.03. From eq. (9), assuming m c = 1.65 GeV, one then finds if0 - (r/c)0'= (~5 -+20) MeV. Neglecting the mass difference ff - ~00, (r/c) 0 is expected between 3.03 and 3.07 GeV. Also, from eqs. (8) and the value o f a s one finds for the bound cE system 1¢(0)12 = (0.06 + 0.02) GeV 3. We have assumed that the value of a s to be used in computing the hyperfine splitting ~00 - (r~c)0 is essentially the same one obtains from eq. (10). This seems to be a sensible assumption but we cannot give any convincing justification. Following De Rujula et al. [4, 7] we assume an annihilation singlet contribution,/3, energy dependent, and connecting with each other the quark-antiquark pairs i = ufi, dd, ~, cEin the 1S state. In an inverse propagator formalism for the mass matrix we can write ,6 After specifying a constant rule for separating the infrared terms included in the binding potential, higher radioactive contributions in ~ts must be included. These points will be treated in a future publication.

The decomposition in eqs. (13) is not much different from those obtained by Lee and Quigg [9] for one solution of the SU(4) mass formula, whereas that in eq. (14) differs from the corresponding solution by Lee and Quigg in that it leads to slightly larger strangequark versus non-strange-quark content for rl'. The small (c~) contents are to be considered as approximate within factors of two. In the lack of experimental information on the mass of the physical r/c we extrapolate from the current asymtotic freedom assumptions the hypothesis/3(s = r/2) ~ 0, in which case rio = 3.05 GeV, and is pure (c~). The solution is made rather unstable however by the vicinity of a pole not very distant from the zero of/3, and rests heavily on the assumption/3(s = 772) = 0. The rate for 8 ~ ( m n c _ 3 1 1 5 ,) F(~k ~ ~c + 3') ~ - ~ m2 (m~ - mnc)3 5 mqj is only of 0.2 keV well compatible with the data .7 . The decay modes ~ ~ r/~,, ~ ~ r/~, through the (c~) contamination in eqs. (13) and (14) are difficult to estimate because of the large photon m o m e n t u m involved. Phase-space alone gives upper limits of the order of 5 keV for the widths but we expect the actual values to be much smaller. ,7 Eq. (15) is valid only if the overlap between the aS and IS wavefunction is complete [ 10]. 457

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The hyperfine splittings D* - D, F* - F; ff - (r/c)0, can be calculated from eqs. (1): D._D_msltPD(O)

2'

F*------F m u ~--(0)

D*-D

' ~O-(rlc)0

mcl~OD(0) 12 mu ~%(0----~](1'6)_

where ~0(0) is the value of the wavefunction at the origin. In our realistic description we shall assume an approximate linear potential V = V 0 + K r (Eichten et al. [15]) with a universal value of K in which case [~0(0)12 is proportional to the reduced mass of the system. This gives D*-D ms'+mc F*-F-mu+m c'

D*-D 2me ~-(r/c)0-mu+me"

(17)

Eqs. (17) give ( D * - D ) / ( F * - F) = 1.1 and ( D * - D)/ (~ - (r/c)0) = 1.7 for m c = 1.65 and with the previous values-of m u and m sWith ~b - (r/c)O = (45 -+20) MeV, as determined before, one obtains D * - D = (75 + 30) MeV and F* - F = (70 + 30 MeV). For such small splittings the only decays modes expected are D* ~ D3', F* ~ F3'. Since D* and F* production from e÷e - is expected to be relatively much more abundant, as soon as kinematicaUy possible, in comparison to D and F production, one would simultaneously expect a larger 3,-ray yield. For the radial 2S excitations " " ~ ' (3.7) and 7/c' one can make a straight comparison of the hyperfine splitting with the corresponding one for 1S states, and write , ¢/-(nc)0 ~0 - 07c)0

2 e ~s(~0~) m,,r~,

- as(qO

(18)

m~ F~ '

where Ots(~' ) and t~s(~) are the values of ors at the ifand ~'-masses. We have all informations to evaluate (18) except for the coupling constant ratio. Using for this tentatively the formula [3,4] as(qj)

1 + 1~

m r ) -1' as(~k) log \ rn~ !

(17)

and with the experimental values I'~0 = (5.2 + 1.3) keV, e t i I'~0, = (2.2 -+0.5) keV [1,2], one has ~k - (r/c) 0 = (0.6 + 0.2) (ff - (r/e)0) = (30 -+10) MeV if (~ - (r/c)0) = 45 MeV, or r/e ~ 3.67 GeV. A similar approach can be attempted to describe the splittings between a higher vector meson 23S nonet, (p', co', ~0', K*'), and a higher pseudoscalar 21S nonet Or', E 1, E 2 -- E, K ' ) where we have identified E(1.42) 458

4 Augustus 1975

with an almost pure (ss-) state, E2, companion of~0'. From eq. (1) and from the assumption that the wavefunction stays roughly constant in going from IS to 2S one obtains from the (ss-) 21S state at ~ 1.42 GeV the corresponding 2 3S state (~p')at ~ 1.6 GeV. After identifying p' with p'(1.6), one obtains for K' and It' masses of 1.2+ 1.3 GeV, for K', and 1.1 + 1.2 GeV for 7r', from use of

~o,_E 2

\ ~ ]

,

(18)

and of E 2 ~ 2 K ' - 7 r ' . For El, of quark composition 1/X/~(ufi + dd) the mass prediction depends heavily on the mixing. Finally, from p' ~ 1.6 GeV and ~0' ~. 1.6 GeV one also has K*' ~ 1.6 GeV. Besides all the uncertainties, implicit in the approach, relativistic corrections, neglected here, appear to be non-negligible for the 2S states [11 ]. One main difference between our approach and that by De Rujula et al. [7] lies in the fact, mentioned at the beginning, that we do not rely on any SU(4) symmetry. Instead, for the (c~) system we rely on the additional experimental information from ~b~ hadrons and ~k e+e - ; whereas for the light mesons we essentially adopt the assumption implicit in the work of De Rujula et al. [7], that the product t~sl~O(O) 12 for the different light mesons (ps and vector), in the expression for the onegluon-exchange hyperfine separation, can be treated as (or replaced by) an effective SU 3-invariant quantity. Now, what if we extend down to the (s~)-system our introduction of independent imputs, i.e. of P(~0 ~ 3rr) and I~(~o~ e+e-)? To do this we have to rely heavily on the use of the approximate expression (8) for the leptonic width of~p, and on the even more questionable use of the expression corresponding to eq. (10) °t3 = 97ra 2 I'(~0-~ 3r 0 1 - 16as/3~r 100r2_9)i,(~o_~e+e)

(19)

From eq. (19) and the experimental data we estimate at the ~o-mass as = 0.34 -+ 0.01 [ 1 2 ] * a , a n d , w i t h the use of eq. (8), as I~o(0)12 = (0.035 + 0.005) GeV 3. This is to be compared with the value 0.065 GeV 3 that one calculates (for instance for K * - K) for the ,a From the asymptotic freedom equation similar to eq. (17) relating as(~O)to as(qJ) one would obtain a s = 0.25 + 0.06 for as(~0) 0.18 -+0.03. =

Volume 57B, number 5

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above SU 3-invariant quantity (which is, or replaces, a s l ¢ ( 0 ) 12 in the hyperfine splitting). The agreement is not at all e~cellent. This was expected at least for three reasons (related among themselves): larger value o f a s, dependence o f F ( ¢ ~ 3rr) on the wavefunction also outside the origin, failures o f the non-relativistic approximation. Nevertheless it is indicative of a possible overall validity of the model and shows the essential role o f the introduction o f non-infrared radiative corrections in the amplitudes for one-photon (and possibly also • for 3 gluon-) annihilation of the bound states, o f which the factor (1 - 16as/3rr ) continues perhaps to be a most important term also in the low mass region of the ~o. In the absence o f such factor one would have found a much larger discrepancy (by about a factor of 1.6) in the comparison between the quantities ashO(O) 12 from the ~o-width and from the hyperfine splitting. One o f us (R.G.) would like to thank Prof. G. Barbiellini for useful conversations and information on the experimental data.

4 Augustus 1975

References [ 1] J.E. Augustin et al., presented by B. Jean Marie at the Paris Conference on Neutrino physics, March 1975. [2] G.S. Abrams et al., L.B.L. 3687, Berkeley, 1975. [3] T. Appelquist and H.D. Politzer, Phys. Rev. Lett. 34 (1975) 43. [4] A. De Rujula and S.L. Glashow, Phys. Rev. Lett. 34 (1975). [5] S.L. Glashow, I. Ilioupolous and L. Maiani, Phys. Rev. D2 (1970) 1285. [6] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, to appear in Phys. Lett. B. [7] A. De Rujula, H. Georgi and S.L. Glashow, Harvard preprint (to be published). [8] R. Karplus and A. Klein, Phys. Rev. 87 (1952) 848. [9] B.W. Lee and C. Quigg, Fermi Lab. Physics Notes 74/1, December 74. We thank Dr. M. Gaillard for showing to us this work. [10] J. Borenstein and R. Sharkar, Phys. Rev. Letters 34 (1975) 619. [ 11] E. £ichten, K. Gottfried, T. Kinoshita, J. Kogut, K.D. Lane and T.M. Yan, Phys. Rev. Letters 34 (1975) 369. [12] R. Barbieri, R. Gatto, R. KOgerler and Z. Kunszt (to be published).

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