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Calculation of the annihilation rate of P wave quark-antiquark bound states
Volume 60B, number 2
PHYSICS LETTERS
CALCULATION
5 January 1976
OF THE ANNIHILATION
P WAVE QUARK-ANTIQUARK
BOUND
RATE
OF
STATES
R. B A R B I E R I , R. G A T T O * and R. K t 3 G E R L E R CERN, Geneva, Switzerland Received 27 October 1975 In the framework of the gauge theory of strong interactions (in particular for the charm scheme) we calculate the annihilation rates of P wave quark-antiquark bound states of J PC = 0 ++, 2 ++. Applications can be made to the decays of the C = +1 states lying between ~ and ~ ', to their gluonic production, and to f' decay. Annihilations into 23, and the Primakoff productions are also discussed.
A most appealing scheme for strong interaction dynamics is based on colour triplets of fermions (quarks) interacting through colour octets o f massless gauge vector-mesons (gluons). Within such a scheme and under some additional conjectures on the relevant dynamics, the ff decay rate into hadrons, for instance, has been calculated in terms o f the three-gluon annihilation o f a heavy quark-antiquark S-wave b o u n d system [1 ]. We present here the rather m o r e c o m p l e x calculation of the rates for two-gluon annihilation o f P-wave quarkantiquark b o u n d states. A similar calculation does not exist in the positronium literature. Our calculation holds for any scheme where the relevant meson states are described in terms o f heavy fermion pairs anrihilating through massless gluons. The q u a n t u m numbers are, for quark-antiquark P-waves, jPC = 0++, 1 + + 2++, and 1+ - . Our twogluon calculation concerns 0 ++ and 2 ++. Annihilation into t w o gluons of 1++ is indeed forbidden +~ . In particular we shall obtain for the ratio o f the 2 ++ annihilation rate to the 0 ++ annihilation rate, in the a p p r o x i m a t i o n o f zero binding, the result Fam~(Z++)/Fann(0 ++) = 4 / 1 5 ,
(1)
i n d e p e n d e n t o f any parameter. The main purpose o f our w o r k is to apply the results to the newly found states [2] in the mass region b e t w e e n and if', which have been described as P-wave b o u n d states o f a heavy quark (to be called Q) and its antiquark [3]. The two-gluon annihilation rates Farm(2 ++) and I'ann(0 ++) give then directly the measured annihilation rates into hadrons o f such states ( t h r o u g h violation o f the Zweig rule)+2. The same w o u l d apply to annihilation into non-strange hadrons (via Zweig-rule violation) o f f ' ( l 514), if described as a 2 ++ P-wave (s-g)-state, e x c e p t that our a p p r o x i m a t i o n s may badly fail in that case. Whereas the result in eq. (1) for the zero binding a p p r o x i m a t i o n is i n d e p e n d e n t o f dynamical parameters, the * Also Instituto di Fisica dell'Universita, Rome, Italy. .1 Charge conjugation does not forbid a jPC= 1++ state to annihilate into three gluons: it only requires that the colour dependence of the final gluons is described by the antisymmetric fabc tensor. In addition, a jPC = 1++ state can to leading order annihilate into one gluon plus one quark pair. ,2 The dynamical conjectures which underlie our calculation, and the identification of the gluonic widths with the widths for decay into ordinary hadrons of the new particles, are essentially the following, i) The low-lying poles of the QQ four-point Green function are approximate eigenstates of the total Hamiltonian of the theory. The infrared structure of the theory, which gives the poles in the QC~ sector, is not producing any strong connection of them with other sectors of the theory. (The smallness of as(M), the sliding coupling constant at the typical masses of the QQ poles, alone is not enough to guarantee that this is the case). ii) The binding mechanism and the actual parameters of the theory are such that the QQ) bound states are mainly non-relativistic systems, iii) The colour singlet gluon density correlation functions do not show resonant or threshold behaviours in the region of the QC) poles, since this is likely to destroy their computability in perturbation theory even with a small coupling a S. The above conjectures are implicit in the similar calculation for ~5 and nc (ref. [ 1 ]). 183
Volume 60B, number 2
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5 January 1976
formulae we obtain separately for the widths, are 96~2 2 Pann(0++ ) = - ~ [qS,p(0)12, r, (2++) _ 1285M40~s I~b'P(0)12 '
(2,3)
where M is the bound state mass, a S = g2/47r the gluonic fine-structure constant, and ~[,(0) is the derivative at r= 0 of the radial wave function of the bound state. If the lowest-order QED diagrams are employed to calculate annihilation into two photons (which might turn out to be a risky assumption +2), one gets for the ratios of two-gamma to the above two-gluon annihilation 1"(2++ ~ 73,_)P(O++-+ 77)_ 9 (°~) 2 Pann~(2++ ) = Pann(0++) -~ e~ ~ s '
(4)
where eQ is the electric charge of the constituent quark. Similar considerations for the pseudoscalar lowest S-bound state 0 -+ and its first radial excitation (0-+) ' give -+ Fann((O ) ) _ F ( ( 1 - - ) ' - > e + e - ) ['ann(O-+) = 2 (~.) F ( 0 - + - ~ 7 7 ) - 9 4 ~s]t/,2 rann(O -+) r(1--oe+e-) ' r ( 1 - - - > e + e - ) 3e~ ' f'ann(0-+) -~eQ , (5-7) '
2
where 1- - and ( 1 - - ) ' are the corresponding vector states ($ and $ ' for the heavy quark system). The transition amplitude of a JP quark-antiquark bound state of mass M into two gluons of four-momenta klu, k2ta, polarization vectors elue2u , and colour indices a, b is graphically represented in fig. 1. The bubbles denote the Bethe-Salpeter bound-state wave function .t~o~(JP , M; Pts) dependent on the relative four-momentum Pu" In the c.m.s, k l u = (3//2)(1 ,/~), k2u= (3//2)(1, - k ) , elu= (0, el), e2u= (0, ~2), el "/~ = ~2 "]~ = 0. In terms of this Bethe-Salpeter wave function the transition amplitude ~ }b(jp, M; l~)e~t e~] can be written as
eye ;.b (JP,M; 1¢) = - -2~3 6ab : d4p Tr [CNi/(p, kl) qJ(JP,M;p)] ,
(8)
where I K + / ~ - / ( 1 +m
- ½ K + P + / ~ t +m
Ni]= "Yi (½K + P - k 1 ) 2 - r~ ")']+ 7] ( - { K + p + kl)2 - m 27i'
(9) K u = klu+kzu = (M, 0), and the charge conjugation matrix C has been introduced in order to describe our quark-antiquark system by a fermion-fermion equal-mass bound state wave function t~ (JP, M; p). The elementary quark-quark-gluon vertex -
(xa)i/ ig ~ - - 7 u
,
defines the coupling constant g. In the extreme non-relativistic limit the dependence on Pu °fNi/is neglected and the usual expression is obtained for the decay amplitude proportional to the non-relativistic wave function calculated at the origin in the relative coordinate space. This gives the well-known result for the 0 - state, which has zero relative angular momentum. However, to get a non-zero result for the P-waves, because of the vanishing of the non-relativistic wave function at the origin, the terms linear in p in the expansion ofNi/, as well as the terms linear in p coming from the small components of the relativistic wave function, must be retained. In this approximation the relativistic wave function can be expressed in terms of the non-relativistic wave function which has only large components:
$(JP,M;p) - (l+3)q~(JP, M;p)(l+3)/4 , 184
Volume 60B, number 2
p-½K
PHYSICS LETTERS
E2,k2,b
-P ~K
5 January 1976
K-%
£2 kz b
0-. 0"° 2~ (0-') ,
o f--Fig. 1. Feynman graphs for the two-gluon annihilation.
Fig. 2. Two-gluon production mechanism of C-even states in pp collisions.
as
=T " P l . Here m is the constituent mass (m -~ M/2). If this expression for the wave function is inserted in eq. (9) and the expansion in p, one gets
c~ab(JPi/" ' M; k ) ~ i~g2 M2
(10)
of Nif is made up to linear terms
Tr[C(c~O)(Ic)+PlQ~(ifl)(fc))X(JP'M;P)] ,
(11)
where (o) i;
-
,
plc~(ill.l)(lc)=et" p ~ ( f c ) + ~ ( I c ) = ' p -- 2ef'pfif-- 2 T//S.(p) + 2 p ' ] ¢ ( ~ ( / ¢ ) + ieifk~k) , TiSj(q) = Tiq'77/ + 7]q'7"fi ,
(12)
TzA(q) = Tiq'7"Y/--'Y/q'y'Y i •
This is our explicit starting point for the calculation.
Because of Bose statistics for the gluons and parity conservation, the transition amplitude c/ff ~b(o+ ; M;/~) for the
JP= 0 + state can be expressed in terms of two invariant amplitudes
~ab(o+ M; fc) = 8 ab [Afif + Bfcilcj] A =lAg (c'ff~i7 - ]¢i]cjQ'l'(7.1.a) -
2i g2
v~ M2
fd3p Tr [Ca'px(O+,M;p)].
The trace of the 4 X 4.matrix product can be immediately reduced to the trace of a 2 × 2 matrix, since has only the high component 2 × 2 matrix xNR(O+,M;p). One has Tr
(13)
x(O+,M;p)
[Cax] = - T r [CT x] = Tr [o2 oxNR ]NR ,
and then 2i g2
A-
v M2 f d3p Tr [o2a'pxNR(O+,M;p)] .
(14)
To evaluate the right-hand side of eq. (14) the following formulae are useful: 0,+-
xNR(o+,M;p) = ~ ( l l O O [ l m l l ) ~ T ( p ) x ~ , m,l Tr(o2onx~)=(llOOIlnll)(-i)v~,
fd3ppnq~l = ( l l O O l l n l m ) i ~ ' p ( O ) , ~/4n O,+o'p= ~ (-1)nonp -n , n
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where ×~ is the S = 1, S z = l, spin part of the non-relativistic wave function and qST (p) its orbital part in momentum space, whose Fourier transform is q~]n(x) = 4~p(r)y]n (j), normalized to 1. Inserting eqs. (15) into eq. (14) the final expression for the amplitude A is
Correspondingly, for the width, one has
; d~2k craabca+ M'klC'lffab*(o+ M" _ [cj[Cm)= 1 96a2 Iq~[,(O)l2 "~] ,- , , , lm " ' ' k) (8il- fcifCl)(8]m 7 IA[2 = ~ M "
r(°+,M) =~2~S~2
(17)
The decomposition into invariant amplitudes of the transition matrix element for a JP= 2 + state is slightly more complicated. Introducing a traceless symmetric tensor Vii to describe the transformation properties under rotations of the J = 2 state (with definite Jz, e.g. Jz = 0), one has (sum over equal indices is understood): c-~ ~./b(2+ ' M;/~) = 8 ab [BVi/ + CVlm kl~CmSi] + DVlm elniemrfknkr + UVlm lCl~Cm ~ci~.].
(18)
With Vii normalized to 1, using
Vi/VjiVli = 1/v:6 , the coefficients B, ..., E can be projected out by a straightforward calculation. They are found to depend on the quantity
H = -Vrs f d 3 p Tr(PrO2OsXNR(2 + , M; p)) = - Y 2 - ~ ~ ( 0 ) .
(19)
The final expression for H is obtained with the help of eqs. (15), after having made the usual decompositions 0,_+
×NR(2+,M;p) = ~
m,l
< l l 2 0 L l m 1/> ~ n ( p ) x{ ,
(20)
0,±
VrsPrPs = ~ (1120 [ 1 q 1t}pq o t
i
(21)
q,t
Finally, the width of the 2 + state is given by r ( 2 + , M ) _ 128 ~2 LCP(O) L2
(22)
5 M4
As a trivial by-product of eqs. (11) and (12), from the term independent of p, one gets for the two-gluon decay of the S-wave 0 - state 8
r ( 0 - , M) = ~ ~
I~S(0 ) 12 ,
(23)
where ¢s(r) is the non-relativistic wave function for the L = 0 state, normalized to f o r2 drlq~s(r)I 2 = 1. Eq. (17) is the usually quoted formula (see ref. [1]). The calculations that we have presented apply as well, with essentially no modification, to the two-photon decays when computed from the graphs analogous to those in fig. 1. With the substitution rule a 2-+ (9/2)e~c~ 2 one gets from the two-gluon decay widths the corresponding two-photon decay widths [see eqs. (4) and (7) and
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footnote* 3 ]. To apply our results to the p h e n o m e n o l o g y o f the new states X observed in ~ ' ( 3 . 7 ) ~ X + 3' [2], we assume that they are indeed P-wave b o u n d states o f two heavy c o m p o n e n t s (possibly, but not necessarily, charmed quarks). We then get, i n d e p e n d e n t l y o f any parameter Yann(2++)/I'ann(0 ++) = 0 . 2 7 . The absolute rates Farm(2 ++) and l-'ann(0 ++) depend on the not well-known parameter a214~p(0)i 2. We take the value o f this p a r a m e t e r f r o m the masses and widths o f ~ and ~ ' following our previous systematic w o r k [ 4 ] ' 4 We find Farm(2 ++) = 0.64 MeV,
Farm(0 ++) = 2.4 M e V .
F r o m eq. (5) we find
r ~ ( ( 0 - + ) ' )/r~n(0 -+)
-= 0 . 4 5 ,
whereas eq. (6) gives Farm(0 - + ) = 6.7 MeV . For the 23' rates, f r o m eqs. (4) and (7), we get values ['(2 ++ ~ 23') = 0.93 k e V ,
Y(0 ++ -+ 23') = 3.5 k e V ,
F ( 0 - + - + 2 3 ' ) = 9.8 k e V ,
F((0-+)'~
2V) = 4.4 k e V .
We stress that the presence o f parameters (a s and wave f u n c t i o n ) which are not well k n o w n (and which we have taken from a previous w o r k ) makes the numerical predictions not c o m p l e t e l y reliable at this time. We think that t h e y could, h o w e v e r , provide for at least a frame for comparison. The graphs o f fig. 1 also appear in a possible p r o d u c t i o n m e c h a n i s m of the states 0 - + , 0 ++, 2 ++, ( 0 - + ) ' through two-gluons, analogous to the D r e l l - Y a n picture (see fig. 2). Following Einhorn and Ellis [5] , we take a gluon distribution within the p r o t o n o f the form
-} (1- x)S /x , (normalized and averaged over colour and polarizations). Typical integrated p r o d u c t i o n cross-sections obtained through such a m e c h a n i s m are: at s = 5 0 0 G e V 2: o - 0 . 7 ~ b for 0 - + ;
u - 0 . 1 / J b for 0 ++, 2 ++ and ( 0 - + ) ' .
At s = 2800 G e V 2 we find: o ~ 2.2~tb for 0 - + ;
o ~ 0 . 4 p b for 0++, 2++ and ( 0 - + ) ' .
F r o m the t w o - p h o t o n amplitudes o f the type o f fig. 1 we have also o b t a i n e d for the c o h e r e n t P r i m a k o f f production in lead the following typical values for the cross-section * s : at a l a b o r a t o r y p h o t o n m o m e n t u m o f 100 GeV, • s We note that, as is well known, the applicability of the perturbation expansion in a S rests on the absence of infrared divergences in each order. However, when a partial width is computed, logarithms of the mass ratio between light and heavy quarks are likely to appear. This kind of criticism applies also to the "naive" calculation of the q,-transitions between bound states (such as ~ ' ~ P-state + 3', ~ ~ r~c+~', etc.) where the infrared problem is even more severe, since additional infrared divergences appear in the limit of zero binding for the initial and final states involved in the transition. This circumstance may in fact be the main reason for the failure of such a calculation when making a comparison with experimental data. • 4 The values we use are a S = 0.18, Iqsi~(0)l2 = 0.09 (GeV) s for P-states, Iq~S(0)l2 = 0.81 (GeV) a for S-states m c = 1.64 GeV. ,s For r/c there already exists the calculation of Dashen, Muzinich, Lee and Quigg [5 ], who obtain larger values (by factors of 3-4). The reasons for the difference are the much larger 2q, width and slightly higher ~c mass (acting in opposite directions). 187
Volume 60B, number 2 o(7 ~ 0 - + ) = 70 nb,
PHYSICS LETTERS o(q¢ -* 0++) = 4.1 nb,
o ( 7 - + 2++) = 4 nb,
5 January 1976 o(7 ~ ( 0 - + ) ' ) = 3.8 n b ,
at a laboratory photon momentum o f 150 GeV, o(3, ~ 0 - + ) = 1 6 0 n b ,
o(7 ~ 0 + + ) = 1 4 n b ,
o(7 ~ 2++) = 1 5 n b ,
o(7 ~ ( 0 - + ) ' ) = 1 4 n b .
Other cross-sections determined from the 23' amplitudes of the type shown in fig. 1 are the two-photon crosssections e+e - - + e - e + X [7], where in our case X = 0 - + , 0 ++, 2 ++, ( 0 - + ) '. We find at colliding beam energy 2E = 7 GeV: 02~,(0 - + ) = 2 X 1 0 - 2 nb,
a2~,(0++) = 2 X 1 0 - 3 n b ,
02.y(2 ++) = 3 X 10 - 3 nb,
o2.y((0-+) ' ) = 3 × 1 0 - 3 n b ,
whereas for 2 E = 30 GeV, o 2 v ( 0 - + ) is 1 × 1 0 - 1 n b , whereas o27(0++), a2.y(2 ++) and o2.y((0-+) ' ) are all 2 × 10-2nb. As we have already mentioned, application of our results to light mesons is a priori subject to doubt because of large relativistic effects, presumably large light-quark pairs contamination, etc. If we tentatively apply our formulae for I'ann(2 ++) to f'(1514) assumed to consist of sg only, with all parameters [a s at the f' mass, qS~(0), and ms] as suggested from a previous work [ 4 ] ' 6 , we obtain a value Farm(2 ++) = 6 MeV at the f' mass. Assuming that roughly two-thirds of this goes into states consisting o f u, d, fi, d quarks, we find a branching ratio of 10% for decay into such states. This number, although large, is consistent with the present scanty experimental evidence [8]. Better experimental determinations would be of interest. +6 For strange quarks we take m S = 0.54 GeV; at the f' mass we take c~S = 0.21, and for ]qS~(0)l2 we take 0.01 (GeV) 5 ; these values are taken from ref. [4]. References [1] T. Appelquist and H.D. Politzer, Phys. Rev. Letters 34 (1975) 43; A. De Rujula and S.L. Glashow, Phys. Rev. Letters 34 (1975) 46; For general reviews see: M.K. Galliard, B.W. Lee and J.L. Rosner, Rev. Mod. Phys. 47 (1975) 2; H. Harari, W.S. 75/40 Ph, Stanford preprint; H.J. Lipkin, Fermilab Conf. 75/57, THY, Fermilab preprint. [2] W. Braunschweig et al., Phys. Letter 57B (1975) 407; G.J. Feldman et al., Phys. Rev. Letters 35 (1975) 821. [3] E. Eichten et al., Phys. Rev. Letters 34 (1975) 369. [4] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Letters 57B (1975) 455; and Ref. TH. 2036-CERN. [5] M.B. Einhorn and S.D. Ellis, Phys. Rev. Letters 34 (1975) 1190. [6] R.F. Dashen et al., Fermilab Pub/75/18/THY. [7] S. Brodsky, T. Kinoshita, H. Terazawa, Phys. Rev. D4 (1971) 1532. [8] N. Barash-Schmidt et al., Tables of Particle Properties, April 1974 (unpublished).
188
PHYSICS LETTERS
CALCULATION
5 January 1976
OF THE ANNIHILATION
P WAVE QUARK-ANTIQUARK
BOUND
RATE
OF
STATES
R. B A R B I E R I , R. G A T T O * and R. K t 3 G E R L E R CERN, Geneva, Switzerland Received 27 October 1975 In the framework of the gauge theory of strong interactions (in particular for the charm scheme) we calculate the annihilation rates of P wave quark-antiquark bound states of J PC = 0 ++, 2 ++. Applications can be made to the decays of the C = +1 states lying between ~ and ~ ', to their gluonic production, and to f' decay. Annihilations into 23, and the Primakoff productions are also discussed.
A most appealing scheme for strong interaction dynamics is based on colour triplets of fermions (quarks) interacting through colour octets o f massless gauge vector-mesons (gluons). Within such a scheme and under some additional conjectures on the relevant dynamics, the ff decay rate into hadrons, for instance, has been calculated in terms o f the three-gluon annihilation o f a heavy quark-antiquark S-wave b o u n d system [1 ]. We present here the rather m o r e c o m p l e x calculation of the rates for two-gluon annihilation o f P-wave quarkantiquark b o u n d states. A similar calculation does not exist in the positronium literature. Our calculation holds for any scheme where the relevant meson states are described in terms o f heavy fermion pairs anrihilating through massless gluons. The q u a n t u m numbers are, for quark-antiquark P-waves, jPC = 0++, 1 + + 2++, and 1+ - . Our twogluon calculation concerns 0 ++ and 2 ++. Annihilation into t w o gluons of 1++ is indeed forbidden +~ . In particular we shall obtain for the ratio o f the 2 ++ annihilation rate to the 0 ++ annihilation rate, in the a p p r o x i m a t i o n o f zero binding, the result Fam~(Z++)/Fann(0 ++) = 4 / 1 5 ,
(1)
i n d e p e n d e n t o f any parameter. The main purpose o f our w o r k is to apply the results to the newly found states [2] in the mass region b e t w e e n and if', which have been described as P-wave b o u n d states o f a heavy quark (to be called Q) and its antiquark [3]. The two-gluon annihilation rates Farm(2 ++) and I'ann(0 ++) give then directly the measured annihilation rates into hadrons o f such states ( t h r o u g h violation o f the Zweig rule)+2. The same w o u l d apply to annihilation into non-strange hadrons (via Zweig-rule violation) o f f ' ( l 514), if described as a 2 ++ P-wave (s-g)-state, e x c e p t that our a p p r o x i m a t i o n s may badly fail in that case. Whereas the result in eq. (1) for the zero binding a p p r o x i m a t i o n is i n d e p e n d e n t o f dynamical parameters, the * Also Instituto di Fisica dell'Universita, Rome, Italy. .1 Charge conjugation does not forbid a jPC= 1++ state to annihilate into three gluons: it only requires that the colour dependence of the final gluons is described by the antisymmetric fabc tensor. In addition, a jPC = 1++ state can to leading order annihilate into one gluon plus one quark pair. ,2 The dynamical conjectures which underlie our calculation, and the identification of the gluonic widths with the widths for decay into ordinary hadrons of the new particles, are essentially the following, i) The low-lying poles of the QQ four-point Green function are approximate eigenstates of the total Hamiltonian of the theory. The infrared structure of the theory, which gives the poles in the QC~ sector, is not producing any strong connection of them with other sectors of the theory. (The smallness of as(M), the sliding coupling constant at the typical masses of the QQ poles, alone is not enough to guarantee that this is the case). ii) The binding mechanism and the actual parameters of the theory are such that the QQ) bound states are mainly non-relativistic systems, iii) The colour singlet gluon density correlation functions do not show resonant or threshold behaviours in the region of the QC) poles, since this is likely to destroy their computability in perturbation theory even with a small coupling a S. The above conjectures are implicit in the similar calculation for ~5 and nc (ref. [ 1 ]). 183
Volume 60B, number 2
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5 January 1976
formulae we obtain separately for the widths, are 96~2 2 Pann(0++ ) = - ~ [qS,p(0)12, r, (2++) _ 1285M40~s I~b'P(0)12 '
(2,3)
where M is the bound state mass, a S = g2/47r the gluonic fine-structure constant, and ~[,(0) is the derivative at r= 0 of the radial wave function of the bound state. If the lowest-order QED diagrams are employed to calculate annihilation into two photons (which might turn out to be a risky assumption +2), one gets for the ratios of two-gamma to the above two-gluon annihilation 1"(2++ ~ 73,_)P(O++-+ 77)_ 9 (°~) 2 Pann~(2++ ) = Pann(0++) -~ e~ ~ s '
(4)
where eQ is the electric charge of the constituent quark. Similar considerations for the pseudoscalar lowest S-bound state 0 -+ and its first radial excitation (0-+) ' give -+ Fann((O ) ) _ F ( ( 1 - - ) ' - > e + e - ) ['ann(O-+) = 2 (~.) F ( 0 - + - ~ 7 7 ) - 9 4 ~s]t/,2 rann(O -+) r(1--oe+e-) ' r ( 1 - - - > e + e - ) 3e~ ' f'ann(0-+) -~eQ , (5-7) '
2
where 1- - and ( 1 - - ) ' are the corresponding vector states ($ and $ ' for the heavy quark system). The transition amplitude of a JP quark-antiquark bound state of mass M into two gluons of four-momenta klu, k2ta, polarization vectors elue2u , and colour indices a, b is graphically represented in fig. 1. The bubbles denote the Bethe-Salpeter bound-state wave function .t~o~(JP , M; Pts) dependent on the relative four-momentum Pu" In the c.m.s, k l u = (3//2)(1 ,/~), k2u= (3//2)(1, - k ) , elu= (0, el), e2u= (0, ~2), el "/~ = ~2 "]~ = 0. In terms of this Bethe-Salpeter wave function the transition amplitude ~ }b(jp, M; l~)e~t e~] can be written as
eye ;.b (JP,M; 1¢) = - -2~3 6ab : d4p Tr [CNi/(p, kl) qJ(JP,M;p)] ,
(8)
where I K + / ~ - / ( 1 +m
- ½ K + P + / ~ t +m
Ni]= "Yi (½K + P - k 1 ) 2 - r~ ")']+ 7] ( - { K + p + kl)2 - m 27i'
(9) K u = klu+kzu = (M, 0), and the charge conjugation matrix C has been introduced in order to describe our quark-antiquark system by a fermion-fermion equal-mass bound state wave function t~ (JP, M; p). The elementary quark-quark-gluon vertex -
(xa)i/ ig ~ - - 7 u
,
defines the coupling constant g. In the extreme non-relativistic limit the dependence on Pu °fNi/is neglected and the usual expression is obtained for the decay amplitude proportional to the non-relativistic wave function calculated at the origin in the relative coordinate space. This gives the well-known result for the 0 - state, which has zero relative angular momentum. However, to get a non-zero result for the P-waves, because of the vanishing of the non-relativistic wave function at the origin, the terms linear in p in the expansion ofNi/, as well as the terms linear in p coming from the small components of the relativistic wave function, must be retained. In this approximation the relativistic wave function can be expressed in terms of the non-relativistic wave function which has only large components:
$(JP,M;p) - (l+3)q~(JP, M;p)(l+3)/4 , 184
Volume 60B, number 2
p-½K
PHYSICS LETTERS
E2,k2,b
-P ~K
5 January 1976
K-%
£2 kz b
0-. 0"° 2~ (0-') ,
o f--Fig. 1. Feynman graphs for the two-gluon annihilation.
Fig. 2. Two-gluon production mechanism of C-even states in pp collisions.
as
=T " P l . Here m is the constituent mass (m -~ M/2). If this expression for the wave function is inserted in eq. (9) and the expansion in p, one gets
c~ab(JPi/" ' M; k ) ~ i~g2 M2
(10)
of Nif is made up to linear terms
Tr[C(c~O)(Ic)+PlQ~(ifl)(fc))X(JP'M;P)] ,
(11)
where (o) i;
-
,
plc~(ill.l)(lc)=et" p ~ ( f c ) + ~ ( I c ) = ' p -- 2ef'pfif-- 2 T//S.(p) + 2 p ' ] ¢ ( ~ ( / ¢ ) + ieifk~k) , TiSj(q) = Tiq'77/ + 7]q'7"fi ,
(12)
TzA(q) = Tiq'7"Y/--'Y/q'y'Y i •
This is our explicit starting point for the calculation.
Because of Bose statistics for the gluons and parity conservation, the transition amplitude c/ff ~b(o+ ; M;/~) for the
JP= 0 + state can be expressed in terms of two invariant amplitudes
~ab(o+ M; fc) = 8 ab [Afif + Bfcilcj] A =lAg (c'ff~i7 - ]¢i]cjQ'l'(7.1.a) -
2i g2
v~ M2
fd3p Tr [Ca'px(O+,M;p)].
The trace of the 4 X 4.matrix product can be immediately reduced to the trace of a 2 × 2 matrix, since has only the high component 2 × 2 matrix xNR(O+,M;p). One has Tr
(13)
x(O+,M;p)
[Cax] = - T r [CT x] = Tr [o2 oxNR ]NR ,
and then 2i g2
A-
v M2 f d3p Tr [o2a'pxNR(O+,M;p)] .
(14)
To evaluate the right-hand side of eq. (14) the following formulae are useful: 0,+-
xNR(o+,M;p) = ~ ( l l O O [ l m l l ) ~ T ( p ) x ~ , m,l Tr(o2onx~)=(llOOIlnll)(-i)v~,
fd3ppnq~l = ( l l O O l l n l m ) i ~ ' p ( O ) , ~/4n O,+o'p= ~ (-1)nonp -n , n
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where ×~ is the S = 1, S z = l, spin part of the non-relativistic wave function and qST (p) its orbital part in momentum space, whose Fourier transform is q~]n(x) = 4~p(r)y]n (j), normalized to 1. Inserting eqs. (15) into eq. (14) the final expression for the amplitude A is
Correspondingly, for the width, one has
; d~2k craabca+ M'klC'lffab*(o+ M" _ [cj[Cm)= 1 96a2 Iq~[,(O)l2 "~] ,- , , , lm " ' ' k) (8il- fcifCl)(8]m 7 IA[2 = ~ M "
r(°+,M) =~2~S~2
(17)
The decomposition into invariant amplitudes of the transition matrix element for a JP= 2 + state is slightly more complicated. Introducing a traceless symmetric tensor Vii to describe the transformation properties under rotations of the J = 2 state (with definite Jz, e.g. Jz = 0), one has (sum over equal indices is understood): c-~ ~./b(2+ ' M;/~) = 8 ab [BVi/ + CVlm kl~CmSi] + DVlm elniemrfknkr + UVlm lCl~Cm ~ci~.].
(18)
With Vii normalized to 1, using
Vi/VjiVli = 1/v:6 , the coefficients B, ..., E can be projected out by a straightforward calculation. They are found to depend on the quantity
H = -Vrs f d 3 p Tr(PrO2OsXNR(2 + , M; p)) = - Y 2 - ~ ~ ( 0 ) .
(19)
The final expression for H is obtained with the help of eqs. (15), after having made the usual decompositions 0,_+
×NR(2+,M;p) = ~
m,l
< l l 2 0 L l m 1/> ~ n ( p ) x{ ,
(20)
0,±
VrsPrPs = ~ (1120 [ 1 q 1t}pq o t
i
(21)
q,t
Finally, the width of the 2 + state is given by r ( 2 + , M ) _ 128 ~2 LCP(O) L2
(22)
5 M4
As a trivial by-product of eqs. (11) and (12), from the term independent of p, one gets for the two-gluon decay of the S-wave 0 - state 8
r ( 0 - , M) = ~ ~
I~S(0 ) 12 ,
(23)
where ¢s(r) is the non-relativistic wave function for the L = 0 state, normalized to f o r2 drlq~s(r)I 2 = 1. Eq. (17) is the usually quoted formula (see ref. [1]). The calculations that we have presented apply as well, with essentially no modification, to the two-photon decays when computed from the graphs analogous to those in fig. 1. With the substitution rule a 2-+ (9/2)e~c~ 2 one gets from the two-gluon decay widths the corresponding two-photon decay widths [see eqs. (4) and (7) and
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footnote* 3 ]. To apply our results to the p h e n o m e n o l o g y o f the new states X observed in ~ ' ( 3 . 7 ) ~ X + 3' [2], we assume that they are indeed P-wave b o u n d states o f two heavy c o m p o n e n t s (possibly, but not necessarily, charmed quarks). We then get, i n d e p e n d e n t l y o f any parameter Yann(2++)/I'ann(0 ++) = 0 . 2 7 . The absolute rates Farm(2 ++) and l-'ann(0 ++) depend on the not well-known parameter a214~p(0)i 2. We take the value o f this p a r a m e t e r f r o m the masses and widths o f ~ and ~ ' following our previous systematic w o r k [ 4 ] ' 4 We find Farm(2 ++) = 0.64 MeV,
Farm(0 ++) = 2.4 M e V .
F r o m eq. (5) we find
r ~ ( ( 0 - + ) ' )/r~n(0 -+)
-= 0 . 4 5 ,
whereas eq. (6) gives Farm(0 - + ) = 6.7 MeV . For the 23' rates, f r o m eqs. (4) and (7), we get values ['(2 ++ ~ 23') = 0.93 k e V ,
Y(0 ++ -+ 23') = 3.5 k e V ,
F ( 0 - + - + 2 3 ' ) = 9.8 k e V ,
F((0-+)'~
2V) = 4.4 k e V .
We stress that the presence o f parameters (a s and wave f u n c t i o n ) which are not well k n o w n (and which we have taken from a previous w o r k ) makes the numerical predictions not c o m p l e t e l y reliable at this time. We think that t h e y could, h o w e v e r , provide for at least a frame for comparison. The graphs o f fig. 1 also appear in a possible p r o d u c t i o n m e c h a n i s m of the states 0 - + , 0 ++, 2 ++, ( 0 - + ) ' through two-gluons, analogous to the D r e l l - Y a n picture (see fig. 2). Following Einhorn and Ellis [5] , we take a gluon distribution within the p r o t o n o f the form
-} (1- x)S /x , (normalized and averaged over colour and polarizations). Typical integrated p r o d u c t i o n cross-sections obtained through such a m e c h a n i s m are: at s = 5 0 0 G e V 2: o - 0 . 7 ~ b for 0 - + ;
u - 0 . 1 / J b for 0 ++, 2 ++ and ( 0 - + ) ' .
At s = 2800 G e V 2 we find: o ~ 2.2~tb for 0 - + ;
o ~ 0 . 4 p b for 0++, 2++ and ( 0 - + ) ' .
F r o m the t w o - p h o t o n amplitudes o f the type o f fig. 1 we have also o b t a i n e d for the c o h e r e n t P r i m a k o f f production in lead the following typical values for the cross-section * s : at a l a b o r a t o r y p h o t o n m o m e n t u m o f 100 GeV, • s We note that, as is well known, the applicability of the perturbation expansion in a S rests on the absence of infrared divergences in each order. However, when a partial width is computed, logarithms of the mass ratio between light and heavy quarks are likely to appear. This kind of criticism applies also to the "naive" calculation of the q,-transitions between bound states (such as ~ ' ~ P-state + 3', ~ ~ r~c+~', etc.) where the infrared problem is even more severe, since additional infrared divergences appear in the limit of zero binding for the initial and final states involved in the transition. This circumstance may in fact be the main reason for the failure of such a calculation when making a comparison with experimental data. • 4 The values we use are a S = 0.18, Iqsi~(0)l2 = 0.09 (GeV) s for P-states, Iq~S(0)l2 = 0.81 (GeV) a for S-states m c = 1.64 GeV. ,s For r/c there already exists the calculation of Dashen, Muzinich, Lee and Quigg [5 ], who obtain larger values (by factors of 3-4). The reasons for the difference are the much larger 2q, width and slightly higher ~c mass (acting in opposite directions). 187
Volume 60B, number 2 o(7 ~ 0 - + ) = 70 nb,
PHYSICS LETTERS o(q¢ -* 0++) = 4.1 nb,
o ( 7 - + 2++) = 4 nb,
5 January 1976 o(7 ~ ( 0 - + ) ' ) = 3.8 n b ,
at a laboratory photon momentum o f 150 GeV, o(3, ~ 0 - + ) = 1 6 0 n b ,
o(7 ~ 0 + + ) = 1 4 n b ,
o(7 ~ 2++) = 1 5 n b ,
o(7 ~ ( 0 - + ) ' ) = 1 4 n b .
Other cross-sections determined from the 23' amplitudes of the type shown in fig. 1 are the two-photon crosssections e+e - - + e - e + X [7], where in our case X = 0 - + , 0 ++, 2 ++, ( 0 - + ) '. We find at colliding beam energy 2E = 7 GeV: 02~,(0 - + ) = 2 X 1 0 - 2 nb,
a2~,(0++) = 2 X 1 0 - 3 n b ,
02.y(2 ++) = 3 X 10 - 3 nb,
o2.y((0-+) ' ) = 3 × 1 0 - 3 n b ,
whereas for 2 E = 30 GeV, o 2 v ( 0 - + ) is 1 × 1 0 - 1 n b , whereas o27(0++), a2.y(2 ++) and o2.y((0-+) ' ) are all 2 × 10-2nb. As we have already mentioned, application of our results to light mesons is a priori subject to doubt because of large relativistic effects, presumably large light-quark pairs contamination, etc. If we tentatively apply our formulae for I'ann(2 ++) to f'(1514) assumed to consist of sg only, with all parameters [a s at the f' mass, qS~(0), and ms] as suggested from a previous work [ 4 ] ' 6 , we obtain a value Farm(2 ++) = 6 MeV at the f' mass. Assuming that roughly two-thirds of this goes into states consisting o f u, d, fi, d quarks, we find a branching ratio of 10% for decay into such states. This number, although large, is consistent with the present scanty experimental evidence [8]. Better experimental determinations would be of interest. +6 For strange quarks we take m S = 0.54 GeV; at the f' mass we take c~S = 0.21, and for ]qS~(0)l2 we take 0.01 (GeV) 5 ; these values are taken from ref. [4]. References [1] T. Appelquist and H.D. Politzer, Phys. Rev. Letters 34 (1975) 43; A. De Rujula and S.L. Glashow, Phys. Rev. Letters 34 (1975) 46; For general reviews see: M.K. Galliard, B.W. Lee and J.L. Rosner, Rev. Mod. Phys. 47 (1975) 2; H. Harari, W.S. 75/40 Ph, Stanford preprint; H.J. Lipkin, Fermilab Conf. 75/57, THY, Fermilab preprint. [2] W. Braunschweig et al., Phys. Letter 57B (1975) 407; G.J. Feldman et al., Phys. Rev. Letters 35 (1975) 821. [3] E. Eichten et al., Phys. Rev. Letters 34 (1975) 369. [4] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Letters 57B (1975) 455; and Ref. TH. 2036-CERN. [5] M.B. Einhorn and S.D. Ellis, Phys. Rev. Letters 34 (1975) 1190. [6] R.F. Dashen et al., Fermilab Pub/75/18/THY. [7] S. Brodsky, T. Kinoshita, H. Terazawa, Phys. Rev. D4 (1971) 1532. [8] N. Barash-Schmidt et al., Tables of Particle Properties, April 1974 (unpublished).
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