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Predictions of the static quark model on boson decays
Volume 21, number 6
PHYSICS
1 July 1966
LETTERS
References 1. B.A.Arbuzov and A.T.Filippov. Physics Letters 20 (1966) 537. 2. N.Cabibbo, Phys.Rev.Letters 10 (1963) 531. 3. J.H.Christenson et al., Phys.Rev. Letters 13 (1964) 138.
4. N.Cabibbo and E. Ferrari, Nuovo Cim. 18 (1960) 928. 5. U. Camerini et al., Phys. Rev. Letters 13 (1964) 318. 6. J. Bernstein, G. Feinberg and T.D. Lee, Phys.Rev. 139 (1965) B1650.
*****
PREDICTIONS
OF
THE
STATIC
QUARK
MODEL
ON
BOSON
DECAYS
*
H. PIETSCHMANN and W. TRIRRING Institute for Theoretical Physics, University of Vienna, Vienna, Austria Received 12 April 1966
Some decays of the 36 fjq mesons are calculated within the static quark model. Introducing as new parameter the radius of the wave function we calculate the V-y vertex, the P -2y, the P - p+v and the f -21r+y decays.
The static quark model is surprisingly successful in explaining the partial width for decays like B*+B+P,
V+P
+ P’,
i,
or V+P+y
)
P-,V+y.
[l]
In view of this success, it seems desirable to check on whether it is pure coincidence or real support for the model. For reference let us review the decay X0 - p” + y 21. Taking into account the phenomenological X 6 -71mixing of about loo, one predicts a partial width for this decay of about 170 keV. Non-resonating background can be estimated to contribute less than 10 keV. Since empirically only l?(X -p+y)/l?(X +q+27r) e + is known [3], one just gets the prediction PX = i MeV. Further results of the quark model require another parameter, namely the spatial extension of the quark-antiquark wave function cp(x). For this one cannot make a precise prediction, but putting [a(O)]2 = r-3 one can check whether we get a reasonable value for Y. This is a compromise between a (long-ranged) hydrogen-type wave function and a sharply cut-off sphere. Considering first the vector meson-y-vertex one defines a coupling constant f vr by
(Vlj,C@ JO> = (2mV*)-iefvy~peipr where
malization volume. The electric current j,(x) proportional to - fpPCcp - )nPCln - f=,,
is (2)
where (p, n, X) for the quark triplet [4]. Pti is given by rp = ercc + z‘/.lu /lvq V * (3) If one assumes that the quarks in a meson have an effective mass 31Clmeson[5] one has to attribute to them an anomalous magnetic moment CC= CUP - elMmeson in order to retain the “observed” total quark moment nq = clp (tip being the total proton magnetic moment). This causes the al.Lv-term in (3). From eqs. (l)-(3) one derives immediately fW=fwiY=f~iY=3:1:-4~c
c=O.58
(4)
where c is a correction factor for the different anomalous magnetic moment of the quarks in the @i-state due to its rather different mass. Wt and @i are the “idealy mixed” particles with mixing angle arctan 92. With c set equal to unity (4) is of course the SU3 result. Recalling that fv,, is not dimensionless, one adds another mass correction to obtain. (f~ir/~~)
= (f,,y/maW,
’ o’5g
(5)
(1)
lCLis the polarization vector and D the nor-
* The research reported in this document has been partly sponsored by the U . S . Government. 713
Volume 21, number 6
PHYSICS
LETTERS
a value which is in rather good agreement with the fit to isoscalar nucleon form factors [6]. Writing jp out in terms of creation and annihilation operators of quarks and comparing with eq. (1) one finds f w y = (2mJ9?+ i
x 0.8
1 July 1966
V
(6)
1+1e+e-
2
x10-4
8
l/8.2
e+e-
1
x10-4
3
Ml.7
PO
e+e-
0.65 x10-4
3
3b3.6
0.3
9
3/20
l-V - 1+1- = mv(4r ar2/3)(fvr/&2
Transition
(7) Experimental data and the resulting coupling constants are shown in table 1. Relying on ref. 9 we shall use in what follows eq. (4) and f,,
= m;/20
.
&
(9)
where c(vp is the transition moment defined by /.LvP = (P(/.Lz(V,j,
= 0).
(10)
The relevant transition moments as obtained from the quark model [4] are summarized in table 2. The decay widths are again obtained from a standard formula [lo] r(P’2y)
)()2 V ??23 p . (11) 4vP where the index V in (11) refers to the intermediate vector meson. Inserting (4), (8) and the values from table 2, one computes the following widths
= cY(Q2
l?(n” -2~)
= 37 eV
(12a)
r(?l, -2~)
= 1.7 keV
(12b)
lY(Xr -2~)
= 27 keV
(12c)
where the subscript r on X and q denotes ttreal” particles for which 9-X mixing has again been taken into account. Eq. (12a) is by about a factor 714
!J+v-
moments
P
x10-4
Table 2 in units of proton moments.
?
X0
-1
J-i
v5
45
710
:
03
Though this value is too small to give a good fit to the isoscalar form factors [6] it predicts Y= lo-l3 cm, which is a rather reasonable value. Turning to 2~ decays of mesons one observes that in a non-relativistic picture the emission of the first y should just straighten out the two quark spins and then the quark - antiquark system can transform into the second y. Hence the process should go through P - V + y - 2~. The second vertex is given by eq. (1) and the VPyvertex is Lvpy = ~PV~(P@~~V,,$,
fV&J
W
PO
.
Ref.
rv-l+z-/rv
W
where the factor 0.8 is the correction due to the opv-term in eq. (3). The coupling-constant (6) is related to the partial width for decays into leptonpairs by the well-known formula [7] + O(+$)
Table 1 meson decays into lepton-pairs.
Vector
-2J2 __ 3d5
@i
-
-2
0
3J3
3 bigger than the biggest experimental number [ll]. With eq. (12~) we can predict qx
-w)/r(x
-27) = 6.3
(13)
in addition to the total width *. To check eq. (12b) one calculates the decay q -hr+y through the sequence g - py -71~ y. Here, one needs the VPP’-vertex which is given by
LwpT = ifwp, VP(P,P
-PP;)
.
(14)
The relevant coupling constant foa is obtained from the measured p-width to be f3 pn /41r=O.6 [ 11. Straight-forward computation yields = 120 eV rh -b+ 74= b&f& a/24n2)(rn6/m3)l 71 77 (15)
I= 0.26, and hence r(q-'2y)/r(q -2n+y)
=5
(16)
which agrees with the data [2] within the 20% error. * It is interesting to note that from the observed branching ratios of Xo into the various charged modes of the 7727~system [12] and from the known ratio 77 -3770 + no2y,~q+~+n.-8° one can predict the fraction of completely neutral decay products. Subtraction of this fraction from the observed neutral fraction which, obviously, includes X0 -2~ decays, yields xo -+poy/xo -+2y N 3 a2. We are indebted to Dr.B. Buschbeck-Czapp for discussions on this point.
Volume 21, number 6
PHYSICS
Regarding the discrepancy of the r” lifetime one may argue that the v may have an anomalously large r and hence also the virtual p and w in the chain r” - V + y - 2y. This would decrease fv,, and bring the theoretical P(x” - 2~) down. However, if all other predictions of the model, in particular the large q-width (2.6 keV) and the X - py/X - 2~ ratio are experimentally verified one may also question the experimental evidence on the 7r” lifetime. The quark model also provides a means to estimate the absolute rate of the weak decays 71++ p++vcl andK++ p+ + vp (in addition to their ratio !) In analogy with eq. (1) one defines a constant CP by
(P(Ai(x)\O)=
(mp~)-~ cpqneiqX
.
(17)
It is connected to the lifetime by the classical formula l/rp
= (G2c;/47r) m2, mp[l - (mn./“p,2]2.
(18)
The empirical values are cK = c,/3.87 . (19) cll = mr.0.69, Evaluating (17) in the same way as (1) yields [4] c,Jm, = 0.69cos 0.h q(O) (20) cKJ"n = 0.69 sin @*& ~(0) where 0 = (0.06$ is the Cabbibo-angle, which givesr, = (p(O)]-3 = 1.75 X lo-l3 cm, ?-K = = 1.1 X lo- I 3 cm. Here again one faces a rather large value for the quark wave function in the pion. Possible applications of the quark model to more complicated strong decays are limited by the fact that any selection of particular intermediate states well reflect the true situation only very crudely. Thus there remains only one decay, namely @ - p + B. The VVP-vertex can be calculated from an effective quark-meson interaction LI = fi (g/C10)4+(a* v)(xipi)Q
9
(21)
by evaluating (VILI (V). p. is some “mean meson mass” [14], g;/~, is related to the static pionnucleon coupling constant f 2/4 r = 0.08 by &/no = +
.
(22)
Straight-forward computation with the relativistic VVP vertex (4g/~o)EILVuA~~,vp~,Xn yields r(+-+pr)
= (g/clo)2 sin2k{[1x[l-
(mpmr/m&2]X
3. 2 (mp-m,/m-~)2]}2m,m&.
(23)
LETTERS
1 July 1966
where X is the difference between the real and the ideal mixing angles. The factor sin2 A enters because the ideal 9 state is completely decoupled from the p,, -system. The experimental partial width [3] requires A N 4O thus shifting the total mixing angle to about 400 in good agreement with the mass formula. With the VVP-vertex one can estimate the decayXO-4n viathechainX”-+2p+4n tobe of the order eV, and thus to be entirely negligible. We would like to thank J. Kuti for interesting discussions.
References 1. V. V.Anisovich, A.A.AnseIm, Ya. I.Azimov, G. S. Danilov, I.T.Dyatlov, Physics Letters 16 (1965) 194; W. E. Thirring. Phvsics Letters 16 f1965) 335: L. D. Soloview,’ Physics Letters 16 (1965) 345; C. Becchi and G. Morpurgo, Physics Letters 17 (1965) 352; F. Giirsey, A. Pais and L. A. Radicati, Phys.Rev. Letters 13 (1964) 299; M.A.Beg, B.W.LeeandA.Pais, Phys.Rev.Letters 13 (1964) 514; S. Badier and C. Bouchiat, Physics Letters 15 (1965) 96. 2. R. H. Dalitz and D. G. Sutherland, Nuovo Cimento 3’7 (1965) 1777; S. L. Glashow and Socolow, Phys. Rev. Letters 15 (1965) 329. W. H. Barkas, 3. A. H. Rosenfeld, A. Barbaro-Galtieri, P. L.Bastien, J. Kirz and M.Roos, Rev. Mod.Phys. 37 (1965) 633. Acta 4. For details of calculations cf.: W. Thirring, Phys. Austr. Suppl. II (1966) 205; Supp. III, to be published. N. N. Bogolubov et al., Dubna preprint D-1968 (1965). E.B.Hughes, T.A.Griffy, M.R.YearianandR. Hofstadter, Phys. Rev. 139 (1965) B458. Y. Nambu and J. J.Sakurai, Phye. Rev. Letters 8 (1962) 79. D.M.Binnie, A.Duane, M.R.Jane, W.G.Jones, D. C. Mason, J. A. Newth, D. C. Potter, I. V. Rahman, J.Walters, B.Dickinson, R.J.Ellison, A.E. Harckhan, M. Ibbotson, R. Marshall, R. F. Templeman and A. J. Wynroe, Physics Letters 18(1965)348. G. Glass, R. C. 9. J. K. De Pagter, J. I. Friedman, Chase, M.Gettner, E. V. Goeler, R. Weinstein and A.M. Boyarski, Phys. Rev. Letters 16 (1966) 35. 10. L. M. Brown and P. Singer. Phvs. Rev. Letters 8 (1962) 460; Erratum ibid. 16 (l”966) (424). C.Bempora and P.L. Braccini, Phys11. G.BeIIetini, ics Letters 18 (1965) 333. 12. J.Badier et al., Physics Letters 17 (1965) 337. 13. M. Beg and V. Singh, Phys.Rev. Letters 13 (1964) 681.
715
PHYSICS
1 July 1966
LETTERS
References 1. B.A.Arbuzov and A.T.Filippov. Physics Letters 20 (1966) 537. 2. N.Cabibbo, Phys.Rev.Letters 10 (1963) 531. 3. J.H.Christenson et al., Phys.Rev. Letters 13 (1964) 138.
4. N.Cabibbo and E. Ferrari, Nuovo Cim. 18 (1960) 928. 5. U. Camerini et al., Phys. Rev. Letters 13 (1964) 318. 6. J. Bernstein, G. Feinberg and T.D. Lee, Phys.Rev. 139 (1965) B1650.
*****
PREDICTIONS
OF
THE
STATIC
QUARK
MODEL
ON
BOSON
DECAYS
*
H. PIETSCHMANN and W. TRIRRING Institute for Theoretical Physics, University of Vienna, Vienna, Austria Received 12 April 1966
Some decays of the 36 fjq mesons are calculated within the static quark model. Introducing as new parameter the radius of the wave function we calculate the V-y vertex, the P -2y, the P - p+v and the f -21r+y decays.
The static quark model is surprisingly successful in explaining the partial width for decays like B*+B+P,
V+P
+ P’,
i,
or V+P+y
)
P-,V+y.
[l]
In view of this success, it seems desirable to check on whether it is pure coincidence or real support for the model. For reference let us review the decay X0 - p” + y 21. Taking into account the phenomenological X 6 -71mixing of about loo, one predicts a partial width for this decay of about 170 keV. Non-resonating background can be estimated to contribute less than 10 keV. Since empirically only l?(X -p+y)/l?(X +q+27r) e + is known [3], one just gets the prediction PX = i MeV. Further results of the quark model require another parameter, namely the spatial extension of the quark-antiquark wave function cp(x). For this one cannot make a precise prediction, but putting [a(O)]2 = r-3 one can check whether we get a reasonable value for Y. This is a compromise between a (long-ranged) hydrogen-type wave function and a sharply cut-off sphere. Considering first the vector meson-y-vertex one defines a coupling constant f vr by
(Vlj,C@ JO> = (2mV*)-iefvy~peipr where
malization volume. The electric current j,(x) proportional to - fpPCcp - )nPCln - f=,,
is (2)
where (p, n, X) for the quark triplet [4]. Pti is given by rp = ercc + z‘/.lu /lvq V * (3) If one assumes that the quarks in a meson have an effective mass 31Clmeson[5] one has to attribute to them an anomalous magnetic moment CC= CUP - elMmeson in order to retain the “observed” total quark moment nq = clp (tip being the total proton magnetic moment). This causes the al.Lv-term in (3). From eqs. (l)-(3) one derives immediately fW=fwiY=f~iY=3:1:-4~c
c=O.58
(4)
where c is a correction factor for the different anomalous magnetic moment of the quarks in the @i-state due to its rather different mass. Wt and @i are the “idealy mixed” particles with mixing angle arctan 92. With c set equal to unity (4) is of course the SU3 result. Recalling that fv,, is not dimensionless, one adds another mass correction to obtain. (f~ir/~~)
= (f,,y/maW,
’ o’5g
(5)
(1)
lCLis the polarization vector and D the nor-
* The research reported in this document has been partly sponsored by the U . S . Government. 713
Volume 21, number 6
PHYSICS
LETTERS
a value which is in rather good agreement with the fit to isoscalar nucleon form factors [6]. Writing jp out in terms of creation and annihilation operators of quarks and comparing with eq. (1) one finds f w y = (2mJ9?+ i
x 0.8
1 July 1966
V
(6)
1+1e+e-
2
x10-4
8
l/8.2
e+e-
1
x10-4
3
Ml.7
PO
e+e-
0.65 x10-4
3
3b3.6
0.3
9
3/20
l-V - 1+1- = mv(4r ar2/3)(fvr/&2
Transition
(7) Experimental data and the resulting coupling constants are shown in table 1. Relying on ref. 9 we shall use in what follows eq. (4) and f,,
= m;/20
.
&
(9)
where c(vp is the transition moment defined by /.LvP = (P(/.Lz(V,j,
= 0).
(10)
The relevant transition moments as obtained from the quark model [4] are summarized in table 2. The decay widths are again obtained from a standard formula [lo] r(P’2y)
)()2 V ??23 p . (11) 4vP where the index V in (11) refers to the intermediate vector meson. Inserting (4), (8) and the values from table 2, one computes the following widths
= cY(Q2
l?(n” -2~)
= 37 eV
(12a)
r(?l, -2~)
= 1.7 keV
(12b)
lY(Xr -2~)
= 27 keV
(12c)
where the subscript r on X and q denotes ttreal” particles for which 9-X mixing has again been taken into account. Eq. (12a) is by about a factor 714
!J+v-
moments
P
x10-4
Table 2 in units of proton moments.
?
X0
-1
J-i
v5
45
710
:
03
Though this value is too small to give a good fit to the isoscalar form factors [6] it predicts Y= lo-l3 cm, which is a rather reasonable value. Turning to 2~ decays of mesons one observes that in a non-relativistic picture the emission of the first y should just straighten out the two quark spins and then the quark - antiquark system can transform into the second y. Hence the process should go through P - V + y - 2~. The second vertex is given by eq. (1) and the VPyvertex is Lvpy = ~PV~(P@~~V,,$,
fV&J
W
PO
.
Ref.
rv-l+z-/rv
W
where the factor 0.8 is the correction due to the opv-term in eq. (3). The coupling-constant (6) is related to the partial width for decays into leptonpairs by the well-known formula [7] + O(+$)
Table 1 meson decays into lepton-pairs.
Vector
-2J2 __ 3d5
@i
-
-2
0
3J3
3 bigger than the biggest experimental number [ll]. With eq. (12~) we can predict qx
-w)/r(x
-27) = 6.3
(13)
in addition to the total width *. To check eq. (12b) one calculates the decay q -hr+y through the sequence g - py -71~ y. Here, one needs the VPP’-vertex which is given by
LwpT = ifwp, VP(P,P
-PP;)
.
(14)
The relevant coupling constant foa is obtained from the measured p-width to be f3 pn /41r=O.6 [ 11. Straight-forward computation yields = 120 eV rh -b+ 74= b&f& a/24n2)(rn6/m3)l 71 77 (15)
I= 0.26, and hence r(q-'2y)/r(q -2n+y)
=5
(16)
which agrees with the data [2] within the 20% error. * It is interesting to note that from the observed branching ratios of Xo into the various charged modes of the 7727~system [12] and from the known ratio 77 -3770 + no2y,~q+~+n.-8° one can predict the fraction of completely neutral decay products. Subtraction of this fraction from the observed neutral fraction which, obviously, includes X0 -2~ decays, yields xo -+poy/xo -+2y N 3 a2. We are indebted to Dr.B. Buschbeck-Czapp for discussions on this point.
Volume 21, number 6
PHYSICS
Regarding the discrepancy of the r” lifetime one may argue that the v may have an anomalously large r and hence also the virtual p and w in the chain r” - V + y - 2y. This would decrease fv,, and bring the theoretical P(x” - 2~) down. However, if all other predictions of the model, in particular the large q-width (2.6 keV) and the X - py/X - 2~ ratio are experimentally verified one may also question the experimental evidence on the 7r” lifetime. The quark model also provides a means to estimate the absolute rate of the weak decays 71++ p++vcl andK++ p+ + vp (in addition to their ratio !) In analogy with eq. (1) one defines a constant CP by
(P(Ai(x)\O)=
(mp~)-~ cpqneiqX
.
(17)
It is connected to the lifetime by the classical formula l/rp
= (G2c;/47r) m2, mp[l - (mn./“p,2]2.
(18)
The empirical values are cK = c,/3.87 . (19) cll = mr.0.69, Evaluating (17) in the same way as (1) yields [4] c,Jm, = 0.69cos 0.h q(O) (20) cKJ"n = 0.69 sin @*& ~(0) where 0 = (0.06$ is the Cabbibo-angle, which givesr, = (p(O)]-3 = 1.75 X lo-l3 cm, ?-K = = 1.1 X lo- I 3 cm. Here again one faces a rather large value for the quark wave function in the pion. Possible applications of the quark model to more complicated strong decays are limited by the fact that any selection of particular intermediate states well reflect the true situation only very crudely. Thus there remains only one decay, namely @ - p + B. The VVP-vertex can be calculated from an effective quark-meson interaction LI = fi (g/C10)4+(a* v)(xipi)Q
9
(21)
by evaluating (VILI (V). p. is some “mean meson mass” [14], g;/~, is related to the static pionnucleon coupling constant f 2/4 r = 0.08 by &/no = +
.
(22)
Straight-forward computation with the relativistic VVP vertex (4g/~o)EILVuA~~,vp~,Xn yields r(+-+pr)
= (g/clo)2 sin2k{[1x[l-
(mpmr/m&2]X
3. 2 (mp-m,/m-~)2]}2m,m&.
(23)
LETTERS
1 July 1966
where X is the difference between the real and the ideal mixing angles. The factor sin2 A enters because the ideal 9 state is completely decoupled from the p,, -system. The experimental partial width [3] requires A N 4O thus shifting the total mixing angle to about 400 in good agreement with the mass formula. With the VVP-vertex one can estimate the decayXO-4n viathechainX”-+2p+4n tobe of the order eV, and thus to be entirely negligible. We would like to thank J. Kuti for interesting discussions.
References 1. V. V.Anisovich, A.A.AnseIm, Ya. I.Azimov, G. S. Danilov, I.T.Dyatlov, Physics Letters 16 (1965) 194; W. E. Thirring. Phvsics Letters 16 f1965) 335: L. D. Soloview,’ Physics Letters 16 (1965) 345; C. Becchi and G. Morpurgo, Physics Letters 17 (1965) 352; F. Giirsey, A. Pais and L. A. Radicati, Phys.Rev. Letters 13 (1964) 299; M.A.Beg, B.W.LeeandA.Pais, Phys.Rev.Letters 13 (1964) 514; S. Badier and C. Bouchiat, Physics Letters 15 (1965) 96. 2. R. H. Dalitz and D. G. Sutherland, Nuovo Cimento 3’7 (1965) 1777; S. L. Glashow and Socolow, Phys. Rev. Letters 15 (1965) 329. W. H. Barkas, 3. A. H. Rosenfeld, A. Barbaro-Galtieri, P. L.Bastien, J. Kirz and M.Roos, Rev. Mod.Phys. 37 (1965) 633. Acta 4. For details of calculations cf.: W. Thirring, Phys. Austr. Suppl. II (1966) 205; Supp. III, to be published. N. N. Bogolubov et al., Dubna preprint D-1968 (1965). E.B.Hughes, T.A.Griffy, M.R.YearianandR. Hofstadter, Phys. Rev. 139 (1965) B458. Y. Nambu and J. J.Sakurai, Phye. Rev. Letters 8 (1962) 79. D.M.Binnie, A.Duane, M.R.Jane, W.G.Jones, D. C. Mason, J. A. Newth, D. C. Potter, I. V. Rahman, J.Walters, B.Dickinson, R.J.Ellison, A.E. Harckhan, M. Ibbotson, R. Marshall, R. F. Templeman and A. J. Wynroe, Physics Letters 18(1965)348. G. Glass, R. C. 9. J. K. De Pagter, J. I. Friedman, Chase, M.Gettner, E. V. Goeler, R. Weinstein and A.M. Boyarski, Phys. Rev. Letters 16 (1966) 35. 10. L. M. Brown and P. Singer. Phvs. Rev. Letters 8 (1962) 460; Erratum ibid. 16 (l”966) (424). C.Bempora and P.L. Braccini, Phys11. G.BeIIetini, ics Letters 18 (1965) 333. 12. J.Badier et al., Physics Letters 17 (1965) 337. 13. M. Beg and V. Singh, Phys.Rev. Letters 13 (1964) 681.
715