- Email: [email protected]
Vector meson decays and the algebra of currents
PHYSICS
Volume 19, number 6 PP - PP - 2.5 . pp.+fi7 Assuming that T44/TI4
is near this value, then
T12 2.56 --+2_ l/TI4+3.96 ’ T13 which 1s estimated at 1.4, and is not very sensitive, the important point bemg that 6 >> FIi’. This compares favourably with the latest experiments, which give [4]
1 December 1965
where T 1s the reaction matrlx assuming degenerate masses and where relations between amplitudes have been established using group theory. Also i,j and n are channel labels, n being the below threshold particle combination. I wish to thank Professor P. T. Matthews for suggesting this problem and for many profitable discussions. References 1 F. Hussam and P. Rotelll, Physics Letters 16 (1965)
183. Imperial College preprmt ICTP/65/28; P. Wintermtz et al., Dubna preprint 1966; L.Schfflke, Heidelberg preprmt 1965, Chai S.Lai, Phys.Rev.Letters 15 (1965) 563. 2. P.T.Matthews and A.SaIam, Nuovo Cimento 13 (1959) 381. n a. E.Segre in Nobel lectures 1942-1962 (Elsevier 1964) p. 513. 4. BaItay, Barash et al., Phys.Rev. Letters 15 (1965) 532.
T12= 1.34. I.13 In general the corrected amplitude assummg one mtermediate state below threshold can be written as
TiJ= ‘ij
LETTERS
6Tin Tjn - 1+ 6 *****
VECTOR
MESON
DECAYS
AND
THE
ALGEBRA
OF
CURRENTS
*
V. S. MATHUR ** and L. K. PANDIT *** Department of Physzcs and Astronomy, UnzversEty
of Rochester,
Rochester,
New
York
Received 15 November 1965
Fubini and Furlan [l] have recently developed a powerful method for extracting physical mformatlon from the algebra of currents. This method, along with the hypothesis of partially conserved axial vector currents [2], has been used by Adler [3] and Weisberger [4] to 0bta.m the renormalization of the pdecay axial vector coupling constant. Many authors [5] have used the same method to study the renormalization of the strangeness - changmg axial vector coupling constant m the leptonic decays of the hyperons. In a recent paper [6] the present authors have shown how the Fubml-Furlan-Adler-Weisberger technique may also be used to obtain information on the low energy interactions of hadrons; and in particular have studied the ti-system. The various successful applications of the Fubmi-Furlan technique are particularly significant, since no higher symmetries of the HamIltonian are assumed in the analysis of the problems considered. In the present note, we study the decay of vector mesons. We shall follow the notation of Pandit and Schechter [5]. Starting with the commutator of the strangeness-preserving axial “charges?
and taking the matrix elements betweenp+ states, we obtain: * Supported m part by the U S Atomic Energy Commission ** On leave of absence from the Centre of Advanced Studies m Theoretical Physics and Astrophysics of Delhi, Delhi, In&a *** On leave of absence from the Tata Institute of Fundamental Research, Bombay, India
Umverslty
523
Volume 19, number 6
PHYSICS
C {tp+]Bi]n)(n n
LETTERS
1
December 1965
]$I P+) -(nJ+I o+)} = 2(P+l r3l P+) ,
(2)
where momenta and spin variables have been suppressed. Conservation of parity, angular momentum and Gconlugation, imply that the only single-particle states occurring m the sum are those of a no and an w. To evaluate these single-particle contributions, we use the followmg effective pnn- and wps-mteractions:
The contribution of the many particle intermediate states can be expressed, followmg Adler [3] and Weisberger [4] in terms of the total np cross sections, continued to zero-mass pion. The sum rule of eq. (2) becomes *: + 6l;j
(5)
,
(‘5) We may estimate the integral lp” by taking mto account the low lymg resonances in the pn-system The p-meson (mass 1020 MeV) is known experimentally to be very weakly coupled to pn and so may be neglected. There is some indication that the np-resonance Ai (mass 1090 MeV, width 125 MeV) [7] is an axial vector meson [8]. The resonance [7] A2 at 1310 MeV will be much less important since it occurs at a higher energy. The value of 1; due to Al is ln = 3.6 GeV-2 ,
(7)
P
and gives only about 5% contribution to our sum rule (5). Takmg (G&, /4n = 2 correspondmg to the p width l?(p) w 100 MeV, and using the experimental valb ues \$I = 1.18, (GNNn/48) = 14.6, we obtain from the sum rule (5): (Gzpv/C)
= 14.8/M;
.
(8)
Usmg now the model of Gell-Mann, Wagner and Sharp [9], this value of Gwpn may be used to evaluate the width for the w - 3n decay. We find, I+(w) = 10.8 NLeV ,
(9)
to be compared with the experimental value [8] 9.4 f 1.7 MeV. Had we neglected the contmuum contribution I$ we would have obtained the value r(w) = 11.5 MeV. The result obtained above is in remarkable agreement with experimental value. The original result of Gell-Mann et al. [9] gave a result for the w width too small by a factor of about 20. An improved result was found only more recently using SU(6) and related higher symmetry schemes [lo]. Our result is of particular sigmficance, since it has been obtained without the use of any higher symmetry. It is the * We neglect the “off-mass-shell”
effects
Czero
mass”
plon) which IS 3ustlfled to the extent that for the plonlc form
factors q&n for the smgle-particle
terms
(0) /K&(O)
= KCpn (0) /K&*(O)
O”(W,O) “a”(w)
524
0) represents
t
and
&-7 where $(W,
= 1
np cross
section
p
for “zero-mass”
I
p plon.
(See m this connection
Adler [6]).
Volume 19, number 6
PHYSICS
LETTERS
1 December 1965
sum-rule (5) that, through its dynamical content relates the couplmg constants Gpan and G,on One can, if one lrkes, turn around the argument, and conclude from our sum rule that q~meson is indeed weakly coupled to the pn system agam a result of some of the recent higher symmetry schemes [lo]. A similar analysis can now be made for the K* decays. We take now the matrix elements of the commutator (1) between single K*+ states, and the matrix element of the commutator of strangeness-changing axial Tzharges”:
LB;, Bfl =Y + Q
,
between single p+ states.
In the former case the smgle particle intermediate states contrlbutmg are K” and K*O and in the latter case r(o and Tf*O. Thus we now have the coupling constants GK*~, GK*K%,
GpKK and GpK*K entering through the effective
interactions:
(11) AS a fairly good approximation, we can again neglect the multrparticle contributions and proceed 111the standard manner. From the two sum rules we can determme GK*pv and GpK*K once we fix GK*& from the K* width and assume GpKK as given by the universal coupling of p to the conserved current
[ll]
(taking,
of course,
isospin
(G&x /4n N 2). Our results are:
02) GiKeK/”
= 3/M&.
(13)
A comparison can now be made with the estimate of Bardakci et al. [lOkfor the partial widt$ of K* + Kxn according to the model that this decay is dominated by the processes K - pK - ~r?rKand K - K*n tin. For our values (12) and (13) of the coupling constants, we find r(K*KTT) z 25 keV to be compared with the experimental upper limit of 100 keV. We urlsh to thank Professor
R. E. Marshak and Dr. G. Guralnik for their interest in this work.
1. S.Fubmi and G.Furlan, Physics 1 (1965) 229; G.Furlan, F.Lannoy, C.Rosett.1 and G.Segr6, Nuovo Cimento 38 (1965) 1747, and preprmt (Trleste) (1965); S.Fubim, G.Furlsn and C.Roseti, preprmt (Trieste) (1965). 2. M.Gell-Mann and M.L&y, Nuovo Cimento 16 (1960) ‘705; Y.Nambu. Phvs.Rev.Letters 4 (1960) 380. 3. S.L.Adler, Phys.Rev.Letters 14 (1965) 1051. 4. W. I. Wetsberger, Phvs. Rev. Letters 14 (1965) 1047. 5. L.K.Pandtt &d J.Schechter, Physms Letters 19 (1965) 56; D. Amati, C. Bouchlat and J.Nuyts, Physics Letters 19 (1965) 59; A.Sato and S.Sasakl, preprint (Osaka) (1965); C.A.Levmson and I. J.Muzimch, Phys.Rev.Letters 15 (1965) 715. 6. V.S.Mathur and L.K.Pandit, preprmt (Rochester) (1965). For application to 711interaction, see S.L.Adler, preprint (Harvard) (1965) ; and also I. J.Muzinich and S.Nussmov, Physics Letters 19 (1965) 248. 7. A.H.Rosenfeld, A.Barbaro-Galtlerl, W.H.Barkas, P.L.Bastien, J.Kirz and M.Roos, Rev.Mod.Phys.36 (1964) 977. 8. See for example S.L.Glashow and R.H.Socolow, Phys.Rev.Letters 15 (1965) 329. 9. M.Gell-Mann, D.Sharp and W. Wagner, Phys.Rev.Letters 8 (1962) 261. IO. B.&k&a and K.C.Wall, Phys.Rev.Bl39 (1965) 1355; K.Bardakcl, J.M.Cornwall, P.G.O.Freund and B.W.Lee, Phys.Rev.Leiters 14 (1965) 264, J.S.Gerstem, Phys.Rev.Letters 14 (1965) 453: F.Hussam, Physics Letters 15 (1965) 78;. Fayyazuddiu, Rlazuddm and M. S. Razmi, preprint (1965)) S. Badier and C. Bouchiat, preprmt (1965); H. J. Schnitzer, preprint (1965). 11. J.J.Sakural, Ann.Phys.11 (1960) 1.
525
Volume 19, number 6 PP - PP - 2.5 . pp.+fi7 Assuming that T44/TI4
is near this value, then
T12 2.56 --+2_ l/TI4+3.96 ’ T13 which 1s estimated at 1.4, and is not very sensitive, the important point bemg that 6 >> FIi’. This compares favourably with the latest experiments, which give [4]
1 December 1965
where T 1s the reaction matrlx assuming degenerate masses and where relations between amplitudes have been established using group theory. Also i,j and n are channel labels, n being the below threshold particle combination. I wish to thank Professor P. T. Matthews for suggesting this problem and for many profitable discussions. References 1 F. Hussam and P. Rotelll, Physics Letters 16 (1965)
183. Imperial College preprmt ICTP/65/28; P. Wintermtz et al., Dubna preprint 1966; L.Schfflke, Heidelberg preprmt 1965, Chai S.Lai, Phys.Rev.Letters 15 (1965) 563. 2. P.T.Matthews and A.SaIam, Nuovo Cimento 13 (1959) 381. n a. E.Segre in Nobel lectures 1942-1962 (Elsevier 1964) p. 513. 4. BaItay, Barash et al., Phys.Rev. Letters 15 (1965) 532.
T12= 1.34. I.13 In general the corrected amplitude assummg one mtermediate state below threshold can be written as
TiJ= ‘ij
LETTERS
6Tin Tjn - 1+ 6 *****
VECTOR
MESON
DECAYS
AND
THE
ALGEBRA
OF
CURRENTS
*
V. S. MATHUR ** and L. K. PANDIT *** Department of Physzcs and Astronomy, UnzversEty
of Rochester,
Rochester,
New
York
Received 15 November 1965
Fubini and Furlan [l] have recently developed a powerful method for extracting physical mformatlon from the algebra of currents. This method, along with the hypothesis of partially conserved axial vector currents [2], has been used by Adler [3] and Weisberger [4] to 0bta.m the renormalization of the pdecay axial vector coupling constant. Many authors [5] have used the same method to study the renormalization of the strangeness - changmg axial vector coupling constant m the leptonic decays of the hyperons. In a recent paper [6] the present authors have shown how the Fubml-Furlan-Adler-Weisberger technique may also be used to obtain information on the low energy interactions of hadrons; and in particular have studied the ti-system. The various successful applications of the Fubmi-Furlan technique are particularly significant, since no higher symmetries of the HamIltonian are assumed in the analysis of the problems considered. In the present note, we study the decay of vector mesons. We shall follow the notation of Pandit and Schechter [5]. Starting with the commutator of the strangeness-preserving axial “charges?
and taking the matrix elements betweenp+ states, we obtain: * Supported m part by the U S Atomic Energy Commission ** On leave of absence from the Centre of Advanced Studies m Theoretical Physics and Astrophysics of Delhi, Delhi, In&a *** On leave of absence from the Tata Institute of Fundamental Research, Bombay, India
Umverslty
523
Volume 19, number 6
PHYSICS
C {tp+]Bi]n)(n n
LETTERS
1
December 1965
]$I P+) -
(2)
where momenta and spin variables have been suppressed. Conservation of parity, angular momentum and Gconlugation, imply that the only single-particle states occurring m the sum are those of a no and an w. To evaluate these single-particle contributions, we use the followmg effective pnn- and wps-mteractions:
The contribution of the many particle intermediate states can be expressed, followmg Adler [3] and Weisberger [4] in terms of the total np cross sections, continued to zero-mass pion. The sum rule of eq. (2) becomes *: + 6l;j
(5)
,
(‘5) We may estimate the integral lp” by taking mto account the low lymg resonances in the pn-system The p-meson (mass 1020 MeV) is known experimentally to be very weakly coupled to pn and so may be neglected. There is some indication that the np-resonance Ai (mass 1090 MeV, width 125 MeV) [7] is an axial vector meson [8]. The resonance [7] A2 at 1310 MeV will be much less important since it occurs at a higher energy. The value of 1; due to Al is ln = 3.6 GeV-2 ,
(7)
P
and gives only about 5% contribution to our sum rule (5). Takmg (G&, /4n = 2 correspondmg to the p width l?(p) w 100 MeV, and using the experimental valb ues \$I = 1.18, (GNNn/48) = 14.6, we obtain from the sum rule (5): (Gzpv/C)
= 14.8/M;
.
(8)
Usmg now the model of Gell-Mann, Wagner and Sharp [9], this value of Gwpn may be used to evaluate the width for the w - 3n decay. We find, I+(w) = 10.8 NLeV ,
(9)
to be compared with the experimental value [8] 9.4 f 1.7 MeV. Had we neglected the contmuum contribution I$ we would have obtained the value r(w) = 11.5 MeV. The result obtained above is in remarkable agreement with experimental value. The original result of Gell-Mann et al. [9] gave a result for the w width too small by a factor of about 20. An improved result was found only more recently using SU(6) and related higher symmetry schemes [lo]. Our result is of particular sigmficance, since it has been obtained without the use of any higher symmetry. It is the * We neglect the “off-mass-shell”
effects
Czero
mass”
plon) which IS 3ustlfled to the extent that for the plonlc form
factors q&n for the smgle-particle
terms
(0) /K&(O)
= KCpn (0) /K&*(O)
O”(W,O) “a”(w)
524
0) represents
t
and
&-7 where $(W,
= 1
np cross
section
p
for “zero-mass”
I
p plon.
(See m this connection
Adler [6]).
Volume 19, number 6
PHYSICS
LETTERS
1 December 1965
sum-rule (5) that, through its dynamical content relates the couplmg constants Gpan and G,on One can, if one lrkes, turn around the argument, and conclude from our sum rule that q~meson is indeed weakly coupled to the pn system agam a result of some of the recent higher symmetry schemes [lo]. A similar analysis can now be made for the K* decays. We take now the matrix elements of the commutator (1) between single K*+ states, and the matrix element of the commutator of strangeness-changing axial Tzharges”:
LB;, Bfl =Y + Q
,
between single p+ states.
In the former case the smgle particle intermediate states contrlbutmg are K” and K*O and in the latter case r(o and Tf*O. Thus we now have the coupling constants GK*~, GK*K%,
GpKK and GpK*K entering through the effective
interactions:
(11) AS a fairly good approximation, we can again neglect the multrparticle contributions and proceed 111the standard manner. From the two sum rules we can determme GK*pv and GpK*K once we fix GK*& from the K* width and assume GpKK as given by the universal coupling of p to the conserved current
[ll]
(taking,
of course,
isospin
(G&x /4n N 2). Our results are:
02) GiKeK/”
= 3/M&.
(13)
A comparison can now be made with the estimate of Bardakci et al. [lOkfor the partial widt$ of K* + Kxn according to the model that this decay is dominated by the processes K - pK - ~r?rKand K - K*n tin. For our values (12) and (13) of the coupling constants, we find r(K*KTT) z 25 keV to be compared with the experimental upper limit of 100 keV. We urlsh to thank Professor
R. E. Marshak and Dr. G. Guralnik for their interest in this work.
1. S.Fubmi and G.Furlan, Physics 1 (1965) 229; G.Furlan, F.Lannoy, C.Rosett.1 and G.Segr6, Nuovo Cimento 38 (1965) 1747, and preprmt (Trleste) (1965); S.Fubim, G.Furlsn and C.Roseti, preprmt (Trieste) (1965). 2. M.Gell-Mann and M.L&y, Nuovo Cimento 16 (1960) ‘705; Y.Nambu. Phvs.Rev.Letters 4 (1960) 380. 3. S.L.Adler, Phys.Rev.Letters 14 (1965) 1051. 4. W. I. Wetsberger, Phvs. Rev. Letters 14 (1965) 1047. 5. L.K.Pandtt &d J.Schechter, Physms Letters 19 (1965) 56; D. Amati, C. Bouchlat and J.Nuyts, Physics Letters 19 (1965) 59; A.Sato and S.Sasakl, preprint (Osaka) (1965); C.A.Levmson and I. J.Muzimch, Phys.Rev.Letters 15 (1965) 715. 6. V.S.Mathur and L.K.Pandit, preprmt (Rochester) (1965). For application to 711interaction, see S.L.Adler, preprint (Harvard) (1965) ; and also I. J.Muzinich and S.Nussmov, Physics Letters 19 (1965) 248. 7. A.H.Rosenfeld, A.Barbaro-Galtlerl, W.H.Barkas, P.L.Bastien, J.Kirz and M.Roos, Rev.Mod.Phys.36 (1964) 977. 8. See for example S.L.Glashow and R.H.Socolow, Phys.Rev.Letters 15 (1965) 329. 9. M.Gell-Mann, D.Sharp and W. Wagner, Phys.Rev.Letters 8 (1962) 261. IO. B.&k&a and K.C.Wall, Phys.Rev.Bl39 (1965) 1355; K.Bardakcl, J.M.Cornwall, P.G.O.Freund and B.W.Lee, Phys.Rev.Leiters 14 (1965) 264, J.S.Gerstem, Phys.Rev.Letters 14 (1965) 453: F.Hussam, Physics Letters 15 (1965) 78;. Fayyazuddiu, Rlazuddm and M. S. Razmi, preprint (1965)) S. Badier and C. Bouchiat, preprmt (1965); H. J. Schnitzer, preprint (1965). 11. J.J.Sakural, Ann.Phys.11 (1960) 1.
525