The width and decay modes of charmed tensor mesons

Volume 91B, number 3,4

PHYSICS LETTERS

21 April 1980

THE WIDTH AND DECAY MODES OF CHARMED TENSOR MESONS Vladimir PRIVMAN and Paul SINGER Department of Physics, Technion, Israel Institute of Technology, Haifa, Israel Received 22 October 1979

We calculate the principal strong and electromagnetic decays of the charmed tensor mesons D** and F** assuming their masses lie between 2.2 and 2.7 GeV/c2 . Their total width is found to vary strongly with the expected mass. The three-body decay D** ~ D*~rTrwill prevail over the two-body modes if mD** > 2.6 GeV/c2. D **° ~ D*°3, emerges as the major electromagnetic decay. These modes derive from the TVV vertex for which various alternative forms are considered. It is shown that production of TcV c pairs in e+e- annihilation can pinpoint the correct TVV form.

The discovery o f low-spin ( 0 - , 1 - ) charmed mesons has generally confirmed the theoretical predictions on their masses and a remarkable experimental and theoretical effort is now under way to study and explain their various decay modes. Charmed mesons with spin 2 +, to complement the well-studied uncharmed multiplet (A 2, f . . . . ), are also expected, however, none were discovered so far [1]. In this paper we calculate the possible decay modes of the still elusive charmed tensor mesons D**(C = 1, S = 0, / = ½) and F**(C = 1, S = 1, I = 0) and we show that the possible dominance o f the decays by particular channels is strongly dependent on the actual (yet unknown) mass o f these particles. We consider the main two- and three-body strong decays as well as some electromagnetic ones and our analysis shows that for mD** > 2.6 GeV/c 2 the three-body decay channel D** ~ D*Trlr becomes the dominant one. Most theoretical estimates for the masses o f D** and F** fall between 2.25 and 2.7 GeV/c 2, with F** being heavier than D** b y several tens o f MeV [2]. We show that for this range of masses, the expected total width of D** varies between ~ 1 MeV and ~ 600 MeV, and that o f F** between ~ 10 -4 MeV and ~ 90 MeV. The production of charmed tensor mesons in e l e c t r o n - p o s i t r o n annihilation is also estimated and we expect it to be maximal around 5 - 6 GeV. This process is shown to be the appropriate tool for distinguishing among the various 436

possible forms of the interactions responsible for the main three-body decays. As the charmed tensor mesons (Tc) are expected to lie above the m D + rn;r ~ 2 GeV/c 2 threshold, their decays are allowed to proceed via strong and electromagnetic interactions. Our calculations are performed using the method o f effective lagrangians to lowest order, which is known to give sufficiently accurate resuits for estimating the decays describable by tree diagrams. The strength of the vertices is determined from the observed decays in the non-charmed sector and related to the charmed ones via SU(4) symmetry as described below. The two-body decays T c ~ VcP , Pc V, Pc P, Pc7 have already been considered with a similar approach and using a restricted SU(4) symmetry by Randa and Donnachie [3], assuming mD**+ = 2.62 GeV/c 2, m F , , = 2.66 GeV/c 2. For these two-body channels we carry over their results, extending their calculations to cover the range of possible T c masses we consider. Let us turn to the three-body decays. A scan o f the possible final states and o f the interaction vertices leading to them indicates that the most likely threebody decays are T c -+ PcTr~ and T e -->Vc~rlr, with the pion pair dominated b y an intermediate p meson. The strong vertices responsible for these modes are TVP and TVV, followed b y V ~ PP. To estimate the T c PcTrTr modes we use an SU(4)-symmetric effective lagrangian

Volume 91B, number 3,4

Table 1 Partial decay widths of D** and F** for mass values mD** = 2.62 GeV/c 2 , mF** = 2.66 GeV/c2 . Decay mode

Width (MeV)

+ D%r ~ D%rTr D** ~ D1r D** ~ D ~ r D** -~ FK D **° + D*07 D **° ~ D07 D **+~ D*+'r F** ~ D*K F** ~ DK F** ~ F*7 F** ~ F3' F** ~ Fr7

94 83 17 1.4 0.17 12 0.33 0.30 20 7.4 0.20 0.089 0.060

D**

D**

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PHYSICS LETTERS

LTV P =gTVPfiikeuupa~UT~a~PV~c~Pk ,

aLS = P l "P2/M2

,

bLs = 3

,

cLS

=

--1

_

?_2

~,,-2

_pl/m

(3)

,

dLS = --1 -- 2p2/M 2 , eLS = - - 4 , where P l , P2 are the four-momenta o f the vector mesons. (b) The tensor-meson-dominance model o f Renner [5], which was used recently by Gampp and Genz to calculate appropriate decays in the ¢ / J family [7], which gives aR=Pl'P2/M2,

bR = 1 ,

eR=d R=-I,

(4) eR = 0 . Both models are gauge invariant when used with protons. As these models have identical a amplitudes (V1, V2 in s wave) which gives the main contribution to the rates, we have also devised an additional ad hoe model (c) having a different form-factor dependence in the a amplitude, namely (1) aps = P l "P2/M2 +p2p2/M2M1M2 ,

where #, u . . . . are Lorentz indices, T, V, P denote tensor, vector and pseudoscalar fields and i, ], k = 1. . . . . 15. gTVP is determined from A 2 ~ p~ to be IgTVP [ = (19.6 + 1.5) GeV 2 and we allow the intermediate p to decay into a pion pair, using IgpTr,r I = 6.09 -+ 0.07. Calculating the decay width we find that these modes contribute only a few percent o f the three-body final states (e.g. for mD** = 2.62 GeV/c 2 one has P(D** -* D~lr) = 1.4 -+ 0.3 MeV, while other three-body modes amount to nearly 100 MeV - see table 1). In order to calculate the T c -~ VcTrTr decay, the knowledge o f the TVV vertex is needed. The most general effective lagrangian is [4,5]

+ (e/M2)Og~V~ofluaaV~~ ) + h . c . , where a . . . . . e are five dimensionless independent amplitudes and M is the tensor-meson mass. For the present calculations we adopt the values of two models which are typical o f the approaches used: (a) The dual model o f Levy and Singer [4] which was successfully applied to the calculation of rare decays o f non-charmed tensor mesons [6] and which gives

bps = 1, (5)

Cps = d p s = - 1 ,

eps = 0 ,

where M1, M 2 are the vector-meson masses. Although this choice for a is somewhat arbitrary, it is in fact inspired bY an effective lagrangian LTVI~¢a cc TuvOu v with Ouu being the e n e r g y - m o m e n t u m tensor o f the free vector field V1, from which eq. (5) obtains for V23- V 1 . For processes with photons, gauge invariance is assured when (5) is used in conjunction with vectormeson dominance. The SU(4) symmetry is implemented in the form (Lorentz indices omitted) 1

LTW = ~gTwdiikriViV k ,

i , ] , k = O , 1 . . . . . 152a )'"

whereby the symmetry is enlarged to encompass V0 and TO, as suggested b y Okubo [8]. When L T W [eqs. (2) and (6)] is rewritten in terms o f the physical fields, we use the physical masses in a, ..., e. This accounts for additional SU(4) breaking, beyond their use in the phase-space integrals. With ( 2 ) - ( 6 ) and 0-meson dominance for the pion pair we calculate D** ~ D*lrrr, after determining the value o f g y v v from A 2 -+ coTr+Tr- (in this mode the pion pair is purely I = 1, making it especially suited for calibrating gTVV). We find that the three models considered give very similar results for the widths and the decay distribution dP/ds, where s = (qTh + q~r2)2, although the value o f g T v v varies somewhat ([ g ~ v [ 437

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PHYSICS LETTERS

PS = 49 -+ 6 G e V - 1, IgTRw I = 46 +- 6 G e V - 1 , IgTVV I = 30 -+ 4 GeV-1). In the following, we use for definiteness model (c) when quoting numerical results. The calculation shows that for T c masses above 2.5 GeV/c 2 the three-body decay D** -+ D*nn contributes substantially to the width, while if roD** exceeds 2.6 GeV/c 2 it even becomes the major decay mode. Our model also implies the following relations among the various charge modes within the Pc~r~r or within the Vcnn channels (neglecting electromagnetic mass differences):

o

X X X X X ~

< ~ ~

t

X X X X X X X X X X X X

I I I I X ~

F [ D **+ ~ D*0(D0)~z+n 0 ] = F[D**0 ~ D*+(D+)~-n0] = 2P[D **+ --> D*+(D+) ~z+~- ]

(7a)

= 2P[D**0 -> D*0(D0)n+~r- ] , r IF** -+ F*(F)~r~] = 0 .

(Tb)

The contribution of the major decay modes to the tensor-meson widths is given in table 1 for the mass values rnF** = 2.66 GeV/c 2 , roD** = 2.62 GeV/c 2, which used in ref. [3]. For m D . , m F . , m F and m D the reported experimental values are used. The T c Vc~, decays are calculated from eqs. ( 2 ) - ( 6 ) and vector-meson dominance o f the electromagnetic current, with the gv'r couplings determined from the measured V ~ £+£- widths. In table 2 we present the variation of the main partial decay widths of D** and F** as a function of their physical masses. It ic clear from the tables that strong (D** ~ D ' m r ) and electromagnetic (D** ~ D*7) decays related to the TVV vertex may play a very important part in the decays o f charmed tensor mesons. In fig. 1 we given the calculated dI'/ds distribution, compared to the one expected from phase space alone. We have also estimated the production o f TcP c and TcV c pairs in e+e - collisions, using the TVP and TVV couplings and vector-meson dominance o f the em current. To ensure correct behaviour at high energies, the TVV amplitude is modified by a multiplying form factor, which we take as (mv/Ecm) n for Ecru > m v , and unity for Ecru < m V [9], where m V is the mass of the exchanged vector meson. In the processes we consider the main contribution to production comes from the 7 ~ (virtual if/J) state, which is then the only one kept, thus m V = mo/j. In order to determine the minimal possible n we require o(e+e - ~ VcV c + VcP c + PcPc + TeV c +TcPc) < t~(e+e - ~ hadrons), and we 438

I

I

%

k~

i t

2

Z~

.fi

2 ¢'q

. ~o

%

I

X X~ ~ ~

Volume 91B, number 3,4

PHYSICS LETTERS

0.24

21 April 1980

0.2

~'~>' 0.16

(by

0.0

~I~ o o~

A

=E

-0.2

u hi o.o(

0.4

- 0.4

Fig. 1. The differential decay rate dF/ds for the decay D** D%r~r, s = (qTr1 + q~r2)2 with mD** = 2.62 GeV/c 2. Continuous curve: present model; dotted curve: phase space.

-0.6

o.1

02 s(GeV 2 )

0.3

use the theoretical calculation o f Boal and Wright [9] for the channels not containing tensor mesons. One finds nLS = 5, n R = 4, nps = 3. Our calculation then shows that for all three models o(e+e - ~ T c V c + T c P c ) reaches a m a x i m u m at energies o f 5 - 6 GeV, where it might c o n t r i b u t e as m u c h as 20% o f a(e+e - -> hadrons). The angular distribution o f the p r o d u c t s in e+e -~ TcP c is 1 + cos20, where 0 is the center-of-mass angle with the e+e - axis. In e+e - -+ TcVc, the angular distrib u t i o n F(O) is

F(O) =

1 + f ( E c m ) cos20 ,

(8)

where f ( E c m ) is a f u n c t i o n sensitive to the m o d e l used for the T V V vertex. In fig. 2 we present f ( E c m ) for the amplitudes o f eqs. ( 2 ) - ( 4 ) and it appears that e+e - -> T c V c w o u l d be a very good l a b o r a t o r y for distinguishing a m o n g the various models for TVV. In concluding, we stress that the e x p e c t e d w i d t h o f charmed tensor mesons varies considerably over the range o f their possible masses, extending for D** from a narrow w i d t h o f ~ 3 MeV if mD** ~ 2.25 G e V / c 2 to a w i d t h o f several h u n d r e d MeV for roD** ~ 2.7 G e V / c 2. This has serious implications on the searches for these mesons. The F** is generally one order o f magnitude narrower. As to the e x p e c t e d modes, we have shown that the previously neglected [3] threeb o d y decay D** -~ D*nTris the d o m i n a n t feature for " h e a v y " tensor mesons. The D** -> DTnr on the other side is only a m i n o r c o n t r i b u t i o n to the width. In this respect, we remark that configuratiops not derivable from the T W vertex m a y also occur, with the pion

-0.8

4

I

5

I

6

I

7

8

ECM (GeV)

Fig. 2. The function f(Ecm) of eq. (8) for the three versions of TVV vertex considered: (a) eq. (3); (b) eq. (4); (c) eq. (5). pair in an I = 0 state. Thus D** ~ D(rrTr)/= 2 w o u l d c o m e from a TTP vertex w i t h (T) ~ 7rTrand D** D*(lrlr)/= 0 from a TVS one with (S) ~ ~rTr. V e r y little is k n o w n about such vertices and from the noncharmed tensor-meson decays there is good reason to believe that t h e y do not contribute significantly [6]. In any case, the mere detection o f F** -~ F*(F)Trrr or the violation o f relations (7a) w o u l d be a direct check on our assumed TVV d o m i n a n c e in these decays. We also call a t t e n t i o n to the significant value o f D **0 ~ D * 0 7 which could contribute as m u c h as 2 0 30% to the total width for low roD** and 5 - 1 0 % in the upper part o f the mass range ,1. This is a direct %1

The charged mode D **÷ --*/)*+7 is down by a factor of 40 when using experimental "r-V vertices, or by a factor of 16 if we use SU(4) values, with our form for LTVV [eq. (6)]. X 2 (3555) ~ ~/J + 7 is also derivable from the TVV vertex and we get for this width 2.1 -+ 0.8 MeV. With a branching ratio of 16 -+ 3% this implies F(X2 ~ all) = 13 +- 7 MeV, somewhat on the higher side. We caution that the vector dominance model which we use for estimating the T ~ V7 decays has not been verified yet for the TVV vertex, not even in the non-charmed sector. Moreover, a large extrapolation from q2 = m~r to q2 = 0 is used for qvT, which could require corrections by factor of ~- 2 (see, e.g., ref. [11]). 439

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manifestation of the strength of the TVV vertex, as determined from the non-charmed sector. For the similar radiative decays of non-charmed tensor mesons we obtain F(A 0 -+ 6o~/) = 7.1 MeV, p ( f 0 ~ p07) = 6.0 MeV, F(A 2 ~/93') = 0.55 MeV. The possibility of having T ~ V7 radiative decays of several MeV strength and the very exciting implications for 7 - 7 physics have indeed been already mentioned in the literature [6,10]. One of us (P.S.) would like to express his thanks to the Theory Groups at SLAC and University of Washington, Seattle for their hospitality during the summer of 1979, when this work was completed. He also acknowledges informative discussions on the topic of this paper with G. Feldman, F. Gilman, G. Goldhaber and H. Harari. This research was supported in part by the Israel Commission for Basic Research.

References [ 1] For a review of charmed meson production and properties see: T. Appelquist, R.M. Barnett and K. Lane, Ann. Rev. Nucl. Part. Sci. 28 (1978) 387.

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[2] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147; P. Becher and M. Bohm, Phys. Lett. 60B (1976) 189; J. Finkelstein and S.S. Pinsky, Phys. Rev. D15 (1977) 360; J.T. Kiehl, D.B. Lichtenberg and J.G. Wills, Lett. Nuovo Cimento 18 (1977) 283; E. Eichten et al., CLNS-425 (1979). [3] J. Randa and A. Donnachie, Nucl. Phys. B129 (1977) 528. [4] N. Levy and P. Singer, Phys. Rev. D4 (1971) 2177. [5] B. Renner, Nucl. Phys. B30 (1971) 634. [6] N. Levy, P. Singer and S. Toaff, Phys. Rev. D13 (1976) 2662; D15 (1977) 1403. [7] W. Gampp and H. Genz, Phys. Lett. 76B (1978) 319; 79B (1978) 276. [8] S. Okubo, Phys. Rev. D13 (1976) 1994. [9] See, e.g., J. Layssac and F.M. Renard, Nuovo Cimento 6A (1971) 134; D.C. Boal and A.C.D. Wright, Phys. Rev. D16 (1977) 1505. [10] J. Babcock and J.L. Rosner, Phys. Rev. D14 (1976) 1286; J.W. Alcock, W.N. Cottingham and I.H. Dunbar, Nucl. Phys. B145 (1978) 85. [11] R.M. Egloff et al., Phys. Rev. Lett. 43 (1979) 657.