- Email: [email protected]
A dynamical treatment of radiative transitions in the charmonium picture
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
A DYNAMICAL TREATMENT OF RADIATIVE TRANSITIONS IN THE CHARMONIUM PICTURE M. CHAICHIAN and R. KOGERLER CERN, Geneva, Switzerland
Received 29 March 1976 A dynamical scheme is proposed to resolve the problem of radiative transitions among the new particles within the quark-antiquark bound state picture. Contrary to the conventional treatments, here the photon is not coupled to the free quarks but is taken to be dominated by the spectrum of mesons which enter the off-mass shell hadronic vertices. The hadronic couplings are, in turn, calculated through the overlaps of their wave functions. The over-all normalization for the strength of hadronic couplings is obtained by requiring the elastic form factor at zero momentum to be one. The resulting values for the radiative widths are considerably smaller than the conventionally calculated ones and are in agreement with experiment. It is unnecessary to explain nowadays how much the discovery of the new particles has pushed forward physical understanding of hadrodynamics. It is especially the quark picture into which one hopes to get more insight by applying it to the new phenomena. A promising version of this model describes the hadrons as being constructed out of (at least) four fractionally charged quarks - the three ordinary (p, n, X) and a fourth heavy quark which we shall call charmed quark (c) without necessarily identifying it with the charm model. Then the new resonances are understood as weakly bound c~ states, their small width resulting from the action of (an improved) Zweig rule. Though a lot of experimental facts can be used as arguments in favour of this picture, there are still some problems, one of the most serious being the radiative decays of some of these states: the E1 transition between the ~'s and the even-C-states (0 ++, 1+÷, 2 ÷÷ .... ) around 3.5 GeV [1], and the M1 transitions between ff's (n3S1) and r~c'S (n 1S0). Calculations with the conventional quark model (where the photon couples directly to the free quarks and the binding is accounted for only by an overlap integral of the two wave functions) have led [2] to results which lie high above the experimental limits. Even after taking into account charmed meson decay corrections, the values are in some cases by a factor of 2 - 3 too high [3]. On the other hand, it is by now known that quark model calculations of this type give wrong results, with similar discrepancies, even for some radiative decays of light mesons [4]. There exist other calculations which base on different grounds like broken SU(4) sum rules [5] but yield, at least for heavy c~ mesons, an order of magnitude too high widths. This unsatisfactory situation has led us to attempt a calculation within a scheme, where in ~iddition to other correcting effects, we avoid problems like point coupling of the photon to free quarks and/or Dirac magnetic moment of them as used in conventional quark calculations. There are four types of decays which we shall consider: a) J + ÷ ~ $ + 7 ,
b) ~ ' - + S + + + 7 ,
t
c) ~O(~0')-+~c(nc)+7,
t
--)"
d)~/c
4+7.
(1)
.
Here J++ denotes the even-C-states 0 ++, 1++, 2 ÷+ which we consider as P wave c~ bound states. We start from a SU(4) quark model where the forces between the quarks emerge from the exchange of coloured [the coloured group being SU(3)] giuons and the coupling is given by the standard non-Abelian gauge field theory. Thus the forces have a small distance part, coming from one gluon exchange and behaving like a Coulomb force, and a long range force which hopefully results from such a field theory and is responsible for quark confinement. Asymptotic freedom determines the energy dependence of the quark-giuon coupling gs (g 214.r = cq). In a previous paper [6] a detailed and systematic analysis of such a model was given. The non-relativistic potential V(r) = - 4 a s / 3 r + V 0 + •r ,
(2) 75
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
was used to determine the wave functions and the zero-order energy eigenvalues, whereas the next order (relativistic) effects (both spin dependent and spin independent) are taken into account by the expectation values of the FermiBreit Hamiltonian. In this way, it was not only possible to get very reasonable values for masses and leptonic widths of both the heavy and the light mesons, but also to estimate the importance of the relativistic corrections. The corresponding result was that for the mesons higher than 1 GeV the non-relativistic description is not too bad and it is particularly good for the c~ bound states. Encouraged by this apparent non-relativistic nature of the c~ bound states and by the fact that in almost all decays (1) the centre-of-mass momentum is relatively small (0.15-0.3 5 GeV), we shall attack the problem of the radiative decays (1) within a non-relativistic picture. Let us consider the radiative transition A -~ B + 3',
(3)
where A and B are c~ bound states. Let the photon be dominated by those vector mesons V i which can couple to the particles A and B without violating Zweig rule (i.e., by ~b, ~', ~", etc.). As we are working in a non-relativistic picture (With SchrOdinger wave functions), we have to take into account all time-ordered diagrams, i.e., the ""direct graph" (fig. 1a) and the "Z graph" (fig. 1b) ,i. The hadronic couplings we calculate according to a model where the incoming quarks of particle A combine in a SU(4) ® SU(2) symmetric way with a qq pair created out of vacuum [7] (see fig. 2). In this way the coupling strengths are determined only up to a unique normalization constant 3'. The value of 3' we fix by computing in the same way the electro-magnetic form factors of the charmed mesons F+(0 - + ) and F÷'(0 - + ) and requiring them to be one at q2 = 0. In this way, having the wave functions, one obtains a parameter-free prediction for all the interesting radiative decay widths. Here we only quote the main ingredients and calculational steps and we shall present the detailed analysis including the results and discussions on the pure hadronic vertices in a future communication [8]. The transition matrix element for process (3) is given by (BIJ~mIA) = ~ e ~ k Vi
m'~i 1 ^,~(k) T(k) m2vi_t6ABVi ~ ,
(4)
where f/-1 are couplings of the vector mesons to the (real) photon, ~ABVi ,(k) are the corresponding hadronic couplings for the off-mass shell V i (but without the normalization factor 3') and the tensors T(k) stem from the tensor decomposition of the corresponding hadronic amplitudes. In the following we restrict ourselves to the first three mesons t~, t~', ~". The neglected terms give less than 5% corrections. The vector meson-photon couplings in this model are connected with the value of the vector meson wave function at the origin by *2
4 Iffv(0)N/l-(16/3~r)as(m?~)'
(5)
which together with the parameters of ref. [6] give
.f~ = 11.48,
f~, = 16.27,
f~,, = 18.23,
(6)
which reproduce the experimental leptonic widths fairly well. The values (6), however, represent the on-mass shell coupling (i.e., q 2 = m y2 ) and we do not know how to continue them to q 2 = 0. So we make the most straightforward assumption that their off-mass-shell continuation is obtained by a c o m m o n factor ~ *s, i.e.,f~0 i =
f~i(q2=O) = rlf~i(m¢;i). 2 41 We are indebted to J.S. Bell for enlightening discussion On this point. ,5 The square-root factor stems from the first order relativistic correction (i.e., one gluon vertex correction) which has a considerable effect here. +a To be more precise, one should take different r/i for different ~0i'sand find them by utilizing the form factor conditions for further particles like 0+*(c~.), etc. 76
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
8
A
u)
=
b)
Fig. 1. Schematic description of the radiative transition A~B+7.
Fig. 2. The hadronic coupling in the non-relativistic quark-pair creation model.
For the evaluation of the hadronic coupling constants, we start from all the possible invariant amplitudes but restrict ourselves to those which preserve gauge invariance when coupled to a massless particle. The number of resulting amplitudes are: one couplingg for
0 ÷+ ~ VV,
two couplings h 1 and h 2 for
1++ ~ VV,
three couplings cx,/3and 3' for
2++ ~ VV,
one coupling f for
0 -+ -~ VV.
(7)
The coupling constants we get by calculating the matrix elements in the quark model described above. The matrix element for A ~ B+V is given there by (e.g., for A =J+÷ state): (B, VITIAO,)) = 7 ~
m', l'
C(lmlm'lOO)C(llll'lJA, k)(ebB~V [ ~ eh l~' vm' a
) IA,BV(kB) , m , l
(8)
where the ¢'s are the SU(8) parts of the wave functions, evac is the same part for the SU(4) singlet cE pair which is created out of the vacuum in a spin triplet P wave state 0 +÷. The overlap functions I m, l contain the spatial parts of the wave functions and (in the rest frame of A) are given by
I m'IA,BC(kB)= 8(kB' +kc) (2rt) -9/2 fY~(k)exp({ikB(2x-y+z) exp (~1k0'1"- z - x ) ) × Y[(l&l)~A(IXJ)~(ly[)~(Izl) elk d x d y dz.
(9)
We perform the SU(8)® 0(3) part of the overlaps and get the various invariant tensor amplitudes (in the nonrelativistic limit, of course). Their weights are then identified with the hadronic coupling constants. We have evaluated the necessary overlap integrals numerically using the wave functions obtained in ref. [6]. The values of these couplings will be quoted elsewhere [8]. Here we only mention some of their general features: i) the couplings decrease with increasing order of radial excitation of the vector mesons Vi which couple to the photon. This fact is easily traced back to the increasing number of nodes in the corresponding wave functions; ii) for a specific process, the coupling constants in general change sign if a particle is replaced by its next radial excitation ,4 All these features, which can be well understood by inspecting the contributing spatial wave functions in full detail, reduce the effective couplings g" which actually enter the formula for the widths and which are given by -
gABs0 [1÷ f¢ gAB¢/ + -f -~ gABS" 1
gAB~ = 7 - - ~ - ¢
(I0)
Here each coupling constant should be understood as the appropriate combination of one part originating from *4 For a similar conclusion, but in a different context, see ref. [9].
77
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
the direct diagram (fig. la) and one from the Z diagram (fig. lb). Without presenting all numerical values for the various coupling constants (they will be given in ref. [8]), we quote here only the fact that the suppression factor in brackets lies in most cases around 0.4-0.6. There is an additional effect; for some decay configurations which include P states (namely, for ~ ' ~ 0 ÷÷ + V ( 1 - - ) , qJ'-+ 2 +÷ + V and 1++-~ if+V) one of the coupling constants in the invariant amplitude decompositions, originating from the direct graphs only, vanishes identically, whereas for the corresponding "crossed" reactions (i.e., 0 +÷ -+ if+V, 2 ÷÷ ~ ff+V and 4' ~ I+++V) there is no such effect. It is the Z diagram which gives a non-vanishing contribution to these couplings and in this way restores a sort of "crossing symmetry" between corresponding channels. We also want to mention another feature of the model: it, is not only non-relativistic but it also uses a static version of the spin projection operator (whereas the momentum dependence of the spatial part is fully taken into account in the overlap irltegral). One, of course, can correct [10,11 ] for this by applying Lorentz boosts to the SU(8) ® 0(3) wave functions of the mesons. The main outcome of these are Wigner rotations. Because we know in the model the average internal momenta of the quarks, we can estimate these effects and find them to change the results by maximally 20%, with the exception of q / ~ %(2.8)+ 7 (see below). We shall discuss this in full detail elsewhere [8]. We are now in the position to calculate the ratios of all the couplings and the widths without any unknown constant entering. In order to have the absolute magnitude of each width, the only remaining problem is to determine the over-all normalization constant 7- As a normalization criterion we use the fact that the electric form factor of a charged pseudoscalar meson is equal to unity at zero momentum transfer. Thus we start by calculating the form factor of the charmed meson F ÷ (considered as a cX bound state) *s in our model. The relevant formula is
eFF+(q2) = 7x
~i
em2 i 1 em~i f~i(q2 ) m2i_q2 gFF4~i(q2) + 7c ~i f~i(q 2) m~i-1 q2 gFF~i (q2)"
(1 1)
For q2 = 0 we set fe~.(q2 =0) ~ reg.(m2.) and f~i(O) = 77fq~.(m~i),(see footnote *a). Furthermore, we take into t t { t account the possibility that the normalization constant "Ix for the vertex FF~b (corresponding to the creation of a X~ pair from the vacuum) can be different from that of FF~b (connected to the creation of a c~ pair) which we denote by 7c. The resulting equation then becomes 3'X
~i gFF~i "Yc ~gFFqJi ~ + - .. 77 i f~i
=1,
(12)
•
which together with the analogous equation for the F '+ form factor enable us to determine two unknowns 7x and 7c/77 uniquely ,6. The results are 7x = 10.32,
7c/77 = 4.02.
(13)
It is amusing to mention that if we set 7c = Yx * 7 by hand, the corresponding results would be 7x = 7c = 10.32 and 77= 2.56. This value of 77, i.e., the suppression factor stemming from off-mass-shell extrapolation of f~o, is almost the same as obtained from various other considerations [12]. Using the resulting value for 7c/77 *6, the widths of all the radiative decays can be determined without any free parameter. The results are given in tables 1 and 2. Included in the tables are also the values for the final CM momenta kCM and the relevant kinematical factors, since there is still an uncertainty concerning experimental mass spectrum. Besides the well-established values of ~(3.095) and ~'(3.685) We have used the following masses:
*s We consider the F meson and not the D meson since the former is more non-relativistic than the latter and since at the same time we also avoid the appearance of particles like O and w where the non-relativistic description is definitely poor. ,6 For the calculation of the radiative widths, we need only -rc/r/and not ~c itself. ,7 In general one expects ~'c < "rh. For detailed estimates, see ref. [8].
78
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
Table 1 Transition
0++-~
1++~
+~.
Table 2
kCM (GeV)
Main kinematical factor
0.314
--J--1ka rn~÷
Calculated width (keV) 39.3
~0
~*/c
+3'
kCM (GeV)
Main kinematical factor
Calculated width (keV)
0.281
---!-I k a
18.7
3m~v
+~,
0.355
1---~k3 3m~+
2++--,~0 +,y
0.384
1 [10 1 + i l k 3 5rn~+~-f~'~ mS]
$' ~ 0+++,y
0.246
1 ka 3m~,
¢' ~ 1+++7
0.212
1---~k3 3m~,
~' -_. 2+*+7
0.177a) 0.132 b)
1 /10 1 . ~ i c 3m~, \ ~ " k-'2+ m ~ ]
a) with m(2 *+) = 3.510 GeV;
Transition
1.6
67.3
1.3
0.05
a
~' ~r/c +3'
~0'
0.779
0.171 c) ' +~, --'r~c 0"255d)
~'c ~* e/ +'r
0.390 c) 0.309 d)
lk3 3m~, 1 __
49.6(?) e)
k3
3rn~,
1 - - k3 m 2, 17c
4.8 c) 16.3 d) 11.6 c) 7 3 d)
c) with rn(r/c) = 3.510 GeV; d) with m(~c) = 3.420 GeV; e) see text.
21.1a) 16.3 b)
b) with m(2 *+) = 3.550 GeV;
- for 0 ++, 1++, 2 ++, the calculated values from ref. [6], i.e., mow = 3.425, m l ~ = 3.466, m2+. = 3.503. For illustrative purposes, the mass of 2 ++ was alternatively chosen as 3.55; - for the paracharmonium states we chose m~ = 2.8 and m~, = 3.51 (and alternatively 3.42). qC qC The corresponding calculations with the harmonic oscillator potential will be given elsewhere [8]. Preliminary estimates show that just the overlap integrals ('9) themselves do not change much between the two potentials mentioned. However, the relative importance of different terms in (10) which determines the magnitude of the effective couplings gABT, as well as the values for 7x and 7c/r/, depend more on the choice of the potential. As one can see from the tables, the calculated widths are in agreement with the experimental limits: P ( 4 ~ ~7c + 7) < 11 keV [ 13] and 1-'(4' ~ (C =+) + 7) < 11 -+5(?) keV [13] for each C-even state between 3.4 and 3.6 GeV. The only exception might be the 2 ++ if it lies lower than 3.55 GeV. There are rumours that the branching ratio of 4'(3.685) into a state at 3.41 plus 7 is about 7+3% [implying P ( 4 ' --' 3.41 +3') ~ 15 keV]. If this value is verified, we could only interpret the 3.41 as being not only the 0 ++ state. It could be understood then either as the '7'c (lying very near to the 0 ++) and/or the 2 ++. In view of the non-conclusive experimental situation, we consider further speculation too early. There is one further remark necessary: in the decay 4 ' ~ r/c(2.8)+ 7 the CM m o m e n t u m k is very high (= 0.779 GeV) which would put under doubt the non-relativistic approximation used in our scheme (in this case, e.g., the Wigner rotations could bring up to 100% corrections). Therefore the given value of 49.6 k e V i n table 2 is tentative and could be changed by a factor of 2 - 3 . Finally, for comparison, let us mention that in the conventional quark model calculations, the expression for the width of, say, 4 ~ r/c+ 3' is given by [2] 79
Volume 63B, number 1
PHYSICS LETTERS 2
P c ° n v e n t ' ( ~ ~ ~c+T) = 4 °~
1) 2
5 July 1976
k3
mcc l+k/x/m~c+k21II2'
1 = f~c(X) cos~- ~(x) dx.
(14a,b)
In our scheme
where ~ is given by eq. (10) and gAB ~i are proportional to the overlap integrals of kind (9). Comparing (14a) with (15), one gets formally I~convent.I2 = (8/3) 2 Ill 2 ~ (8/3) 2 = 7.11,
(16)
while in our case I~12 = 0.2(3,c/n) 2 = 3.23,
(17)
which gives P(ff ~ %+),) = 18.7 keV, for mnc = 2.8 and -3/x/12 c~ quark content oft/c. It is our pleasure to thank R. Barbieri, Y. Frishman, D. Horn, H.B. Nilsen, B. Richter and J. Weyers for discussions. We are especially grateful to J.S. Bell and J. Prentki for critically reading the manuscript and for valuable discussions.
References [1] W. Braunschweig et al., Phys. Letters 57B (1975) 407; G.J. Feldman et al., Phys. Rev. Letters 35 (1975) 821. [2] E. Eichten et al., Phys. Rev. Letters 34 (1975) 369; J. Borenstein and R. Shankar, Phys. Rev. Letters 34 (1975) 619. [3] E. Eichlen, K. Gottfried, T. Kinoshita, K. Lane and T.-M. Yan, Cornell University preprint CLNS-323 (1975). [4] D.H. Boal, R.H. Graham and J.W. Moffat, Toronto University preprint (1975); P.J. O'Donnell, Toronto University preprint (1975). [5] E. Takasugi and S. Oneda, Ohio State University preprint COO-1545-176 (1976). [6] R. Barbieri, R. Gatto, R. Kbgerler and Z. Kunszt, Nuclear Phys. B 105 (1976). [7] R. Van Royen and V.F. Weisskopf, Nuovo Cimento 50 (1967) 617; R. Carlitz and M. Kislinger, Phys. Rev. D2 (1970) 336; A. Le Yaouanc, L. Olivier, O. P~ne and J.C. Raynal, Phys. Rev. D7 (1973) 2223. [8] M. Chaichian and R. KiSgerler, to be published. [9] R. Aviv, Y. Goren, D. Horn and S. Nussinov, Tel-Aviv University preprint 471-75 (1975); M. Chaichian and M. Hayashi, CERN preprint TH.2082 (1975) - to appear in Phys, Letters B. [10] A. Krzywicki and A. Le Yaouanc, Nuclear Phys. B14 (1969) 246; H. Lipkin, Phys. Rev. 183 (1969) 1221. [11 ] A. Le Yaouanc, L. Olivier, O. P~ne and J.C. Raynal, Phys. Rev. D9 (1974) 2636. [12] D. Sivers, J. Townsend and G. West, SLAC preprint SLAC-PUB-1636 (1975); Chan Hong-Mo, Ken-ichi Konishi, J. Kwiecinski and R.G. Roberts, Phys. Letters 60B (1976) 469. [13] J.W. Simpson et al., Phys. Rev. Letters 35 (1975) 699.
80
PHYSICS LETTERS
5 July 1976
A DYNAMICAL TREATMENT OF RADIATIVE TRANSITIONS IN THE CHARMONIUM PICTURE M. CHAICHIAN and R. KOGERLER CERN, Geneva, Switzerland
Received 29 March 1976 A dynamical scheme is proposed to resolve the problem of radiative transitions among the new particles within the quark-antiquark bound state picture. Contrary to the conventional treatments, here the photon is not coupled to the free quarks but is taken to be dominated by the spectrum of mesons which enter the off-mass shell hadronic vertices. The hadronic couplings are, in turn, calculated through the overlaps of their wave functions. The over-all normalization for the strength of hadronic couplings is obtained by requiring the elastic form factor at zero momentum to be one. The resulting values for the radiative widths are considerably smaller than the conventionally calculated ones and are in agreement with experiment. It is unnecessary to explain nowadays how much the discovery of the new particles has pushed forward physical understanding of hadrodynamics. It is especially the quark picture into which one hopes to get more insight by applying it to the new phenomena. A promising version of this model describes the hadrons as being constructed out of (at least) four fractionally charged quarks - the three ordinary (p, n, X) and a fourth heavy quark which we shall call charmed quark (c) without necessarily identifying it with the charm model. Then the new resonances are understood as weakly bound c~ states, their small width resulting from the action of (an improved) Zweig rule. Though a lot of experimental facts can be used as arguments in favour of this picture, there are still some problems, one of the most serious being the radiative decays of some of these states: the E1 transition between the ~'s and the even-C-states (0 ++, 1+÷, 2 ÷÷ .... ) around 3.5 GeV [1], and the M1 transitions between ff's (n3S1) and r~c'S (n 1S0). Calculations with the conventional quark model (where the photon couples directly to the free quarks and the binding is accounted for only by an overlap integral of the two wave functions) have led [2] to results which lie high above the experimental limits. Even after taking into account charmed meson decay corrections, the values are in some cases by a factor of 2 - 3 too high [3]. On the other hand, it is by now known that quark model calculations of this type give wrong results, with similar discrepancies, even for some radiative decays of light mesons [4]. There exist other calculations which base on different grounds like broken SU(4) sum rules [5] but yield, at least for heavy c~ mesons, an order of magnitude too high widths. This unsatisfactory situation has led us to attempt a calculation within a scheme, where in ~iddition to other correcting effects, we avoid problems like point coupling of the photon to free quarks and/or Dirac magnetic moment of them as used in conventional quark calculations. There are four types of decays which we shall consider: a) J + ÷ ~ $ + 7 ,
b) ~ ' - + S + + + 7 ,
t
c) ~O(~0')-+~c(nc)+7,
t
--)"
d)~/c
4+7.
(1)
.
Here J++ denotes the even-C-states 0 ++, 1++, 2 ÷+ which we consider as P wave c~ bound states. We start from a SU(4) quark model where the forces between the quarks emerge from the exchange of coloured [the coloured group being SU(3)] giuons and the coupling is given by the standard non-Abelian gauge field theory. Thus the forces have a small distance part, coming from one gluon exchange and behaving like a Coulomb force, and a long range force which hopefully results from such a field theory and is responsible for quark confinement. Asymptotic freedom determines the energy dependence of the quark-giuon coupling gs (g 214.r = cq). In a previous paper [6] a detailed and systematic analysis of such a model was given. The non-relativistic potential V(r) = - 4 a s / 3 r + V 0 + •r ,
(2) 75
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
was used to determine the wave functions and the zero-order energy eigenvalues, whereas the next order (relativistic) effects (both spin dependent and spin independent) are taken into account by the expectation values of the FermiBreit Hamiltonian. In this way, it was not only possible to get very reasonable values for masses and leptonic widths of both the heavy and the light mesons, but also to estimate the importance of the relativistic corrections. The corresponding result was that for the mesons higher than 1 GeV the non-relativistic description is not too bad and it is particularly good for the c~ bound states. Encouraged by this apparent non-relativistic nature of the c~ bound states and by the fact that in almost all decays (1) the centre-of-mass momentum is relatively small (0.15-0.3 5 GeV), we shall attack the problem of the radiative decays (1) within a non-relativistic picture. Let us consider the radiative transition A -~ B + 3',
(3)
where A and B are c~ bound states. Let the photon be dominated by those vector mesons V i which can couple to the particles A and B without violating Zweig rule (i.e., by ~b, ~', ~", etc.). As we are working in a non-relativistic picture (With SchrOdinger wave functions), we have to take into account all time-ordered diagrams, i.e., the ""direct graph" (fig. 1a) and the "Z graph" (fig. 1b) ,i. The hadronic couplings we calculate according to a model where the incoming quarks of particle A combine in a SU(4) ® SU(2) symmetric way with a qq pair created out of vacuum [7] (see fig. 2). In this way the coupling strengths are determined only up to a unique normalization constant 3'. The value of 3' we fix by computing in the same way the electro-magnetic form factors of the charmed mesons F+(0 - + ) and F÷'(0 - + ) and requiring them to be one at q2 = 0. In this way, having the wave functions, one obtains a parameter-free prediction for all the interesting radiative decay widths. Here we only quote the main ingredients and calculational steps and we shall present the detailed analysis including the results and discussions on the pure hadronic vertices in a future communication [8]. The transition matrix element for process (3) is given by (BIJ~mIA) = ~ e ~ k Vi
m'~i 1 ^,~(k) T(k) m2vi_t6ABVi ~ ,
(4)
where f/-1 are couplings of the vector mesons to the (real) photon, ~ABVi ,(k) are the corresponding hadronic couplings for the off-mass shell V i (but without the normalization factor 3') and the tensors T(k) stem from the tensor decomposition of the corresponding hadronic amplitudes. In the following we restrict ourselves to the first three mesons t~, t~', ~". The neglected terms give less than 5% corrections. The vector meson-photon couplings in this model are connected with the value of the vector meson wave function at the origin by *2
4 Iffv(0)N/l-(16/3~r)as(m?~)'
(5)
which together with the parameters of ref. [6] give
.f~ = 11.48,
f~, = 16.27,
f~,, = 18.23,
(6)
which reproduce the experimental leptonic widths fairly well. The values (6), however, represent the on-mass shell coupling (i.e., q 2 = m y2 ) and we do not know how to continue them to q 2 = 0. So we make the most straightforward assumption that their off-mass-shell continuation is obtained by a c o m m o n factor ~ *s, i.e.,f~0 i =
f~i(q2=O) = rlf~i(m¢;i). 2 41 We are indebted to J.S. Bell for enlightening discussion On this point. ,5 The square-root factor stems from the first order relativistic correction (i.e., one gluon vertex correction) which has a considerable effect here. +a To be more precise, one should take different r/i for different ~0i'sand find them by utilizing the form factor conditions for further particles like 0+*(c~.), etc. 76
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
8
A
u)
=
b)
Fig. 1. Schematic description of the radiative transition A~B+7.
Fig. 2. The hadronic coupling in the non-relativistic quark-pair creation model.
For the evaluation of the hadronic coupling constants, we start from all the possible invariant amplitudes but restrict ourselves to those which preserve gauge invariance when coupled to a massless particle. The number of resulting amplitudes are: one couplingg for
0 ÷+ ~ VV,
two couplings h 1 and h 2 for
1++ ~ VV,
three couplings cx,/3and 3' for
2++ ~ VV,
one coupling f for
0 -+ -~ VV.
(7)
The coupling constants we get by calculating the matrix elements in the quark model described above. The matrix element for A ~ B+V is given there by (e.g., for A =J+÷ state): (B, VITIAO,)) = 7 ~
m', l'
C(lmlm'lOO)C(llll'lJA, k)(ebB~V [ ~ eh l~' vm' a
) IA,BV(kB) , m , l
(8)
where the ¢'s are the SU(8) parts of the wave functions, evac is the same part for the SU(4) singlet cE pair which is created out of the vacuum in a spin triplet P wave state 0 +÷. The overlap functions I m, l contain the spatial parts of the wave functions and (in the rest frame of A) are given by
I m'IA,BC(kB)= 8(kB' +kc) (2rt) -9/2 fY~(k)exp({ikB(2x-y+z) exp (~1k0'1"- z - x ) ) × Y[(l&l)~A(IXJ)~(ly[)~(Izl) elk d x d y dz.
(9)
We perform the SU(8)® 0(3) part of the overlaps and get the various invariant tensor amplitudes (in the nonrelativistic limit, of course). Their weights are then identified with the hadronic coupling constants. We have evaluated the necessary overlap integrals numerically using the wave functions obtained in ref. [6]. The values of these couplings will be quoted elsewhere [8]. Here we only mention some of their general features: i) the couplings decrease with increasing order of radial excitation of the vector mesons Vi which couple to the photon. This fact is easily traced back to the increasing number of nodes in the corresponding wave functions; ii) for a specific process, the coupling constants in general change sign if a particle is replaced by its next radial excitation ,4 All these features, which can be well understood by inspecting the contributing spatial wave functions in full detail, reduce the effective couplings g" which actually enter the formula for the widths and which are given by -
gABs0 [1÷ f¢ gAB¢/ + -f -~ gABS" 1
gAB~ = 7 - - ~ - ¢
(I0)
Here each coupling constant should be understood as the appropriate combination of one part originating from *4 For a similar conclusion, but in a different context, see ref. [9].
77
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
the direct diagram (fig. la) and one from the Z diagram (fig. lb). Without presenting all numerical values for the various coupling constants (they will be given in ref. [8]), we quote here only the fact that the suppression factor in brackets lies in most cases around 0.4-0.6. There is an additional effect; for some decay configurations which include P states (namely, for ~ ' ~ 0 ÷÷ + V ( 1 - - ) , qJ'-+ 2 +÷ + V and 1++-~ if+V) one of the coupling constants in the invariant amplitude decompositions, originating from the direct graphs only, vanishes identically, whereas for the corresponding "crossed" reactions (i.e., 0 +÷ -+ if+V, 2 ÷÷ ~ ff+V and 4' ~ I+++V) there is no such effect. It is the Z diagram which gives a non-vanishing contribution to these couplings and in this way restores a sort of "crossing symmetry" between corresponding channels. We also want to mention another feature of the model: it, is not only non-relativistic but it also uses a static version of the spin projection operator (whereas the momentum dependence of the spatial part is fully taken into account in the overlap irltegral). One, of course, can correct [10,11 ] for this by applying Lorentz boosts to the SU(8) ® 0(3) wave functions of the mesons. The main outcome of these are Wigner rotations. Because we know in the model the average internal momenta of the quarks, we can estimate these effects and find them to change the results by maximally 20%, with the exception of q / ~ %(2.8)+ 7 (see below). We shall discuss this in full detail elsewhere [8]. We are now in the position to calculate the ratios of all the couplings and the widths without any unknown constant entering. In order to have the absolute magnitude of each width, the only remaining problem is to determine the over-all normalization constant 7- As a normalization criterion we use the fact that the electric form factor of a charged pseudoscalar meson is equal to unity at zero momentum transfer. Thus we start by calculating the form factor of the charmed meson F ÷ (considered as a cX bound state) *s in our model. The relevant formula is
eFF+(q2) = 7x
~i
em2 i 1 em~i f~i(q2 ) m2i_q2 gFF4~i(q2) + 7c ~i f~i(q 2) m~i-1 q2 gFF~i (q2)"
(1 1)
For q2 = 0 we set fe~.(q2 =0) ~ reg.(m2.) and f~i(O) = 77fq~.(m~i),(see footnote *a). Furthermore, we take into t t { t account the possibility that the normalization constant "Ix for the vertex FF~b (corresponding to the creation of a X~ pair from the vacuum) can be different from that of FF~b (connected to the creation of a c~ pair) which we denote by 7c. The resulting equation then becomes 3'X
~i gFF~i "Yc ~gFFqJi ~ + - .. 77 i f~i
=1,
(12)
•
which together with the analogous equation for the F '+ form factor enable us to determine two unknowns 7x and 7c/77 uniquely ,6. The results are 7x = 10.32,
7c/77 = 4.02.
(13)
It is amusing to mention that if we set 7c = Yx * 7 by hand, the corresponding results would be 7x = 7c = 10.32 and 77= 2.56. This value of 77, i.e., the suppression factor stemming from off-mass-shell extrapolation of f~o, is almost the same as obtained from various other considerations [12]. Using the resulting value for 7c/77 *6, the widths of all the radiative decays can be determined without any free parameter. The results are given in tables 1 and 2. Included in the tables are also the values for the final CM momenta kCM and the relevant kinematical factors, since there is still an uncertainty concerning experimental mass spectrum. Besides the well-established values of ~(3.095) and ~'(3.685) We have used the following masses:
*s We consider the F meson and not the D meson since the former is more non-relativistic than the latter and since at the same time we also avoid the appearance of particles like O and w where the non-relativistic description is definitely poor. ,6 For the calculation of the radiative widths, we need only -rc/r/and not ~c itself. ,7 In general one expects ~'c < "rh. For detailed estimates, see ref. [8].
78
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
Table 1 Transition
0++-~
1++~
+~.
Table 2
kCM (GeV)
Main kinematical factor
0.314
--J--1ka rn~÷
Calculated width (keV) 39.3
~0
~*/c
+3'
kCM (GeV)
Main kinematical factor
Calculated width (keV)
0.281
---!-I k a
18.7
3m~v
+~,
0.355
1---~k3 3m~+
2++--,~0 +,y
0.384
1 [10 1 + i l k 3 5rn~+~-f~'~ mS]
$' ~ 0+++,y
0.246
1 ka 3m~,
¢' ~ 1+++7
0.212
1---~k3 3m~,
~' -_. 2+*+7
0.177a) 0.132 b)
1 /10 1 . ~ i c 3m~, \ ~ " k-'2+ m ~ ]
a) with m(2 *+) = 3.510 GeV;
Transition
1.6
67.3
1.3
0.05
a
~' ~r/c +3'
~0'
0.779
0.171 c) ' +~, --'r~c 0"255d)
~'c ~* e/ +'r
0.390 c) 0.309 d)
lk3 3m~, 1 __
49.6(?) e)
k3
3rn~,
1 - - k3 m 2, 17c
4.8 c) 16.3 d) 11.6 c) 7 3 d)
c) with rn(r/c) = 3.510 GeV; d) with m(~c) = 3.420 GeV; e) see text.
21.1a) 16.3 b)
b) with m(2 *+) = 3.550 GeV;
- for 0 ++, 1++, 2 ++, the calculated values from ref. [6], i.e., mow = 3.425, m l ~ = 3.466, m2+. = 3.503. For illustrative purposes, the mass of 2 ++ was alternatively chosen as 3.55; - for the paracharmonium states we chose m~ = 2.8 and m~, = 3.51 (and alternatively 3.42). qC qC The corresponding calculations with the harmonic oscillator potential will be given elsewhere [8]. Preliminary estimates show that just the overlap integrals ('9) themselves do not change much between the two potentials mentioned. However, the relative importance of different terms in (10) which determines the magnitude of the effective couplings gABT, as well as the values for 7x and 7c/r/, depend more on the choice of the potential. As one can see from the tables, the calculated widths are in agreement with the experimental limits: P ( 4 ~ ~7c + 7) < 11 keV [ 13] and 1-'(4' ~ (C =+) + 7) < 11 -+5(?) keV [13] for each C-even state between 3.4 and 3.6 GeV. The only exception might be the 2 ++ if it lies lower than 3.55 GeV. There are rumours that the branching ratio of 4'(3.685) into a state at 3.41 plus 7 is about 7+3% [implying P ( 4 ' --' 3.41 +3') ~ 15 keV]. If this value is verified, we could only interpret the 3.41 as being not only the 0 ++ state. It could be understood then either as the '7'c (lying very near to the 0 ++) and/or the 2 ++. In view of the non-conclusive experimental situation, we consider further speculation too early. There is one further remark necessary: in the decay 4 ' ~ r/c(2.8)+ 7 the CM m o m e n t u m k is very high (= 0.779 GeV) which would put under doubt the non-relativistic approximation used in our scheme (in this case, e.g., the Wigner rotations could bring up to 100% corrections). Therefore the given value of 49.6 k e V i n table 2 is tentative and could be changed by a factor of 2 - 3 . Finally, for comparison, let us mention that in the conventional quark model calculations, the expression for the width of, say, 4 ~ r/c+ 3' is given by [2] 79
Volume 63B, number 1
PHYSICS LETTERS 2
P c ° n v e n t ' ( ~ ~ ~c+T) = 4 °~
1) 2
5 July 1976
k3
mcc l+k/x/m~c+k21II2'
1 = f~c(X) cos~- ~(x) dx.
(14a,b)
In our scheme
where ~ is given by eq. (10) and gAB ~i are proportional to the overlap integrals of kind (9). Comparing (14a) with (15), one gets formally I~convent.I2 = (8/3) 2 Ill 2 ~ (8/3) 2 = 7.11,
(16)
while in our case I~12 = 0.2(3,c/n) 2 = 3.23,
(17)
which gives P(ff ~ %+),) = 18.7 keV, for mnc = 2.8 and -3/x/12 c~ quark content oft/c. It is our pleasure to thank R. Barbieri, Y. Frishman, D. Horn, H.B. Nilsen, B. Richter and J. Weyers for discussions. We are especially grateful to J.S. Bell and J. Prentki for critically reading the manuscript and for valuable discussions.
References [1] W. Braunschweig et al., Phys. Letters 57B (1975) 407; G.J. Feldman et al., Phys. Rev. Letters 35 (1975) 821. [2] E. Eichten et al., Phys. Rev. Letters 34 (1975) 369; J. Borenstein and R. Shankar, Phys. Rev. Letters 34 (1975) 619. [3] E. Eichlen, K. Gottfried, T. Kinoshita, K. Lane and T.-M. Yan, Cornell University preprint CLNS-323 (1975). [4] D.H. Boal, R.H. Graham and J.W. Moffat, Toronto University preprint (1975); P.J. O'Donnell, Toronto University preprint (1975). [5] E. Takasugi and S. Oneda, Ohio State University preprint COO-1545-176 (1976). [6] R. Barbieri, R. Gatto, R. Kbgerler and Z. Kunszt, Nuclear Phys. B 105 (1976). [7] R. Van Royen and V.F. Weisskopf, Nuovo Cimento 50 (1967) 617; R. Carlitz and M. Kislinger, Phys. Rev. D2 (1970) 336; A. Le Yaouanc, L. Olivier, O. P~ne and J.C. Raynal, Phys. Rev. D7 (1973) 2223. [8] M. Chaichian and R. KiSgerler, to be published. [9] R. Aviv, Y. Goren, D. Horn and S. Nussinov, Tel-Aviv University preprint 471-75 (1975); M. Chaichian and M. Hayashi, CERN preprint TH.2082 (1975) - to appear in Phys, Letters B. [10] A. Krzywicki and A. Le Yaouanc, Nuclear Phys. B14 (1969) 246; H. Lipkin, Phys. Rev. 183 (1969) 1221. [11 ] A. Le Yaouanc, L. Olivier, O. P~ne and J.C. Raynal, Phys. Rev. D9 (1974) 2636. [12] D. Sivers, J. Townsend and G. West, SLAC preprint SLAC-PUB-1636 (1975); Chan Hong-Mo, Ken-ichi Konishi, J. Kwiecinski and R.G. Roberts, Phys. Letters 60B (1976) 469. [13] J.W. Simpson et al., Phys. Rev. Letters 35 (1975) 699.
80