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A mechanism for generating narrow states
Volume 60B, number 1
PHYSICS LETTERS
A MECHANISM FOR GENERATING
22 December 1975
NARROW STATES
F.M. RENARD Ddpartement de Physique Mathdmatique*, 34060 Montpellier Cedex, France Received 4 July 1975 We show how a symmetry property can lead simultaneously to broad states and to narrow states. We discuss then the origin of the symmetry breaking with respect to the properties of the narrow states recently discovered.
In view of the difficulties encountered in the interpretation o f the new meson states we want to draw the attention on a different way for obtaining narrow states. Conversely to the usual procedures which need new quantum numbers (charm, colour, ...) and adequate thresholds (KK, CC, ...) in order to forbid the main decays we invoke a maximal symmetry for the interactions in order to generate stable states but require a small symmetry breaking interaction for allowhag the decays. The basic mechanism is the following; consider two degenerate states V 1 and V 2 having the same couplings to any final state; the mass matrix has the general form: m - imp
A - ira1` 1
A- imp
m 2 - imP]
1" is the common width created by the real intermediate states occurring between V i and V/. (fig. 1); A is a possible self-mass contribution due to virtual intermediate states in the same process. The physically observable states obtained after diagonalization have the eigenvalues zv = m 2 + A - 2 i m 1 "
and
z~ = m 2 - A ,
they correspond to the eigenvectors: IV) =cosO[ V 1) + sin OIV2) =---~1~(I V1) + [V2)),
Fig. 1. One state (V) has the width 2F, the other ( 4 ) is rigorously stable. This procedure can be extended to the case of n degenerate states with the result consisting in one state having the width nP and n - 1 stable states. We used previously [1 ] this mechanism for constructing a vector meson continuum but neglected at that time the decoupled states; if one starts with vector mesons V i having normal couplings, for a mass of 3 GeV one expects a width P of the order of 0.5 GeV and then F V = 21" 1.0 GeV which means that V will effectively constitute a part of the background or vector meson continuum observed in e+e - annihilation in this energy range. We shall now concentrate on the narrow states. In order to have a non zero width we need a small perturbation; writing the 2 × 2 mass matrix in the form m2-imP+611-iTll
A-im1"+612-i3'12
1
A - i m p + 812 - i712 m2 - ira1` + 622 - i3'22 ] we get zV ~ m 2 + A - 2imp zq~m 2-A-i
(711 + 7 2 2 - 2 ~ ' 1 2 )
(2)
[hb) = - s i n 0IV 1 ) + cos 01V 2) = ~ 2 ( - I V 1 ) + IV2)). +(511 -- t522)28mP_--(3"11-- 722)2.1 * Physique math6matique et th4orique, 4quipe de recherche associ6e au CNRS.
with the mixing relations 55
Volume 60B, number 1
PHYSICS LETTERS
22 December 1975
v~
I V) = ~22 [(1 + e)l V 1) + (1 - e)l V2)]
(3)
~
vj Fig. 2.
1
14) = ~ - [ - ( 1
- e)lV1) + (1 + e)lV2)]
where e = "n'/4
Fig. 3. -
0
811 -- 822 -- i('111 -- '122)
2(A -- i m p + 812 -- i'112 ) "
(through fig. 2) and of order ~2 to the decay width
"1ii (through fig. 3) are not large enough in order to exAny amplitude (fl TI 4) can be directly expressed in terms of the (fl Tt Vi) ones (supposed to be known) and of the mixing parameter e calculated from the symmetry breaking corrections. Non zero (fl TI 4) amplitudes may appear either through direct violations of (fl TI V 1) = (fl TI V 2) or through terms proportional to e and constitute effectively the width appearing in zqj. In order to discuss the physical consequences one needs a model answering at least the two following questions: - what are the degenerate states Vi? what is the nature of the symmetry breaking interaction? We don't propose here a well-defined model but just treat an example in order to illustrate the mechanism. With respect to the first question one can imagine the existence of a symmetry (internal or dynamical) of the strong interactions to which the V i pertain with the quantum numbers di; strong interactions would be d-independent (like charge or colour independence or isospin independence inside SU 3 representations) and lead to the maximal symmetry required for the mass matrix. We choose to consider series of
plain P~ as noticed by everybody [2]. Radiative decays (i.e., contributions of order a to the "1i/) Vi -* "1 + hadrons will cancel in the 4 state because vector dominance relates them to the hadronic strong decays V i -~ vector meson + hadrons which cancel exactly for any such final state. On an other hand 4 '3.7 -~ 43.1 + h a d r o n s d e c a y s are known [2] to be isospin conserving semi-strong transitions with coupling constants reduced only by the factor/3 ~ 1/10 with respect to normal strong couplings; a similar factor/3 appears also for the 4N total cross section, i.e., the forward 4N -+ 4N amplitude, with respect to the ordinary vector meson-nucleon cases. Taking this information as input on would conclude that there exist in addition to the symmetric strong interactions a non invariant semi-strong term which occurs in the transitions involving at least two V i. Consider the
vector mesons (V 1, V2), (V'1, V~), (V I, V~) ... and apply the 2 X 2 mechanism to each member of this series;
diagram of fig. 4 which contributes to the self-mass terms 8i/and for V~ to the decay width '1q. The following nice features arise: - the interference terms of order/3 between strong and semi-strong interactions cancel in the 4 like states because of the combinations 811 + 822 -- 2812 and
obviously another possibility would be to consider at least a 3 X 3 mechanism (V 1, V 2, V3) leading to V, 4, 4' states. Weaker interactions would then depend upon d i and create the small asymmetric terms (like electromagnetic interactions distinguish different charges or colour states and SU 3 breaking medium strong interactions distinguish different isospin states). At first sight it seems that normal electromagnetic interactions are not sufficient for our purpose. Photons have to couple differently to Vi(gvl v 4= gV2v like in the p, w, ~ case) in order that gq;• be non zero form eq. (1) or (3) and have a normal intensity required by the ~+~- partial width. But the contributions of order ~ to the symmetry breaking terms 8q 56
Vk
Fig. 4.
'111 + '122 -- 2'112" S o f o r e x a m p l e t h e d e c a y w i d t h o f
the 43.7 is really of order/32 ~ 10 - 2 as required. - no contribution of this type (of order/32) can occur for the lowest 43.1 decay width; the first term (in eq. (2)) is due to the self-mass term and rnP~ (811 -- 822) 2/8mI' is of order ~ ~ 10 -4 which is just the factor required for passing from 500 MeV down to 50 keV. - radiative transitions become now allowed by the vector dominance terms involving the intermediate V i
Volume 60B, number 1
V;,
PttYSICS LETTERS
.
<
Fig. 5. and the semi-strong breaking term (fig. 5); they are however reduced by the factor ~3in the amplitude and by the absence of the important p, w, ~olike intermediate states; one gets therefore only about one keV for modes like 27rT, r/7 .... Let us however come back to the second order electromagnetic transitions between V i. The single photon intermediate state (fig. 2) is too weak [2], giving P~0--*had ~ RP~0 ~u+u - ~ 12 keV, but other selfmass terms like those of fig. 6 are possible. From the
22 December 1975
whether or not one applies also the semi-strong reduction factor/3 because of the occurrence of a V i - V/hadrons transition; this would lead t o / 5 i / ~ ~35m2 7.5 × 10 - 3 GeV 2 and only a negligible contribution of 1 keV to P~0" On the opposite if one keeps this electromagnetic term as it stands, then it is sufficient to explain the ff decay width. One could in this case reconsider the origin of the semi-strong interaction required for the transitions involving two V i (i.e., two ~) and attribute it completely to this kind o f enhanced electromagnetic tadpole diagrams. All the features o f the hadronic and radiative decays of ~b and if' would be due to the peculiar limiting mixing between the initial vector meson states V i slightly perturbed by electromagnetic interactions. However in this case it would be hard to understand a rigorous isospin conservation in all of the purely hadronic decays ~b' ~ + hadrons, a point which need further experimental investigations.
References Fig. 6. mesons electromagnetic mass differences and p - 6o interference one knows [3] that such terms can be relatively large, i.e. 6m 2 ~ 5 X 10 - 3 GeV 2 for tg, w, K* mesons. Applying a scaling factor m 2 / m 2 .~ 15 leads to ~i1 ~ 0.075 GeV 2 and through eq. (2) a width F~ o f the order of 100 keV. But here one has to decide
[1] F.M. Renard, Nucl. Phys. B82 (1974) 1. [2] H. Harari, SLAC-PUB-1514 (1974); CERN Theory Boson Workshop, CERN TH 1964 (1974); J. Ellis, Schladming lectures, CERN TH 1996 (1975); and everybody own calculations. [3] M. Gourdin, F.M. Renard and L. Stodolsky, Phys. Lett. 30B (1969) 347; F.M. Renard, Nucl. Phys. B15 (I 970) 118; Spring. Tracts Mod. Phys. 63 (1972) 98.
57
PHYSICS LETTERS
A MECHANISM FOR GENERATING
22 December 1975
NARROW STATES
F.M. RENARD Ddpartement de Physique Mathdmatique*, 34060 Montpellier Cedex, France Received 4 July 1975 We show how a symmetry property can lead simultaneously to broad states and to narrow states. We discuss then the origin of the symmetry breaking with respect to the properties of the narrow states recently discovered.
In view of the difficulties encountered in the interpretation o f the new meson states we want to draw the attention on a different way for obtaining narrow states. Conversely to the usual procedures which need new quantum numbers (charm, colour, ...) and adequate thresholds (KK, CC, ...) in order to forbid the main decays we invoke a maximal symmetry for the interactions in order to generate stable states but require a small symmetry breaking interaction for allowhag the decays. The basic mechanism is the following; consider two degenerate states V 1 and V 2 having the same couplings to any final state; the mass matrix has the general form: m - imp
A - ira1` 1
A- imp
m 2 - imP]
1" is the common width created by the real intermediate states occurring between V i and V/. (fig. 1); A is a possible self-mass contribution due to virtual intermediate states in the same process. The physically observable states obtained after diagonalization have the eigenvalues zv = m 2 + A - 2 i m 1 "
and
z~ = m 2 - A ,
they correspond to the eigenvectors: IV) =cosO[ V 1) + sin OIV2) =---~1~(I V1) + [V2)),
Fig. 1. One state (V) has the width 2F, the other ( 4 ) is rigorously stable. This procedure can be extended to the case of n degenerate states with the result consisting in one state having the width nP and n - 1 stable states. We used previously [1 ] this mechanism for constructing a vector meson continuum but neglected at that time the decoupled states; if one starts with vector mesons V i having normal couplings, for a mass of 3 GeV one expects a width P of the order of 0.5 GeV and then F V = 21" 1.0 GeV which means that V will effectively constitute a part of the background or vector meson continuum observed in e+e - annihilation in this energy range. We shall now concentrate on the narrow states. In order to have a non zero width we need a small perturbation; writing the 2 × 2 mass matrix in the form m2-imP+611-iTll
A-im1"+612-i3'12
1
A - i m p + 812 - i712 m2 - ira1` + 622 - i3'22 ] we get zV ~ m 2 + A - 2imp zq~m 2-A-i
(711 + 7 2 2 - 2 ~ ' 1 2 )
(2)
[hb) = - s i n 0IV 1 ) + cos 01V 2) = ~ 2 ( - I V 1 ) + IV2)). +(511 -- t522)28mP_--(3"11-- 722)2.1 * Physique math6matique et th4orique, 4quipe de recherche associ6e au CNRS.
with the mixing relations 55
Volume 60B, number 1
PHYSICS LETTERS
22 December 1975
v~
I V) = ~22 [(1 + e)l V 1) + (1 - e)l V2)]
(3)
~
vj Fig. 2.
1
14) = ~ - [ - ( 1
- e)lV1) + (1 + e)lV2)]
where e = "n'/4
Fig. 3. -
0
811 -- 822 -- i('111 -- '122)
2(A -- i m p + 812 -- i'112 ) "
(through fig. 2) and of order ~2 to the decay width
"1ii (through fig. 3) are not large enough in order to exAny amplitude (fl TI 4) can be directly expressed in terms of the (fl Tt Vi) ones (supposed to be known) and of the mixing parameter e calculated from the symmetry breaking corrections. Non zero (fl TI 4) amplitudes may appear either through direct violations of (fl TI V 1) = (fl TI V 2) or through terms proportional to e and constitute effectively the width appearing in zqj. In order to discuss the physical consequences one needs a model answering at least the two following questions: - what are the degenerate states Vi? what is the nature of the symmetry breaking interaction? We don't propose here a well-defined model but just treat an example in order to illustrate the mechanism. With respect to the first question one can imagine the existence of a symmetry (internal or dynamical) of the strong interactions to which the V i pertain with the quantum numbers di; strong interactions would be d-independent (like charge or colour independence or isospin independence inside SU 3 representations) and lead to the maximal symmetry required for the mass matrix. We choose to consider series of
plain P~ as noticed by everybody [2]. Radiative decays (i.e., contributions of order a to the "1i/) Vi -* "1 + hadrons will cancel in the 4 state because vector dominance relates them to the hadronic strong decays V i -~ vector meson + hadrons which cancel exactly for any such final state. On an other hand 4 '3.7 -~ 43.1 + h a d r o n s d e c a y s are known [2] to be isospin conserving semi-strong transitions with coupling constants reduced only by the factor/3 ~ 1/10 with respect to normal strong couplings; a similar factor/3 appears also for the 4N total cross section, i.e., the forward 4N -+ 4N amplitude, with respect to the ordinary vector meson-nucleon cases. Taking this information as input on would conclude that there exist in addition to the symmetric strong interactions a non invariant semi-strong term which occurs in the transitions involving at least two V i. Consider the
vector mesons (V 1, V2), (V'1, V~), (V I, V~) ... and apply the 2 X 2 mechanism to each member of this series;
diagram of fig. 4 which contributes to the self-mass terms 8i/and for V~ to the decay width '1q. The following nice features arise: - the interference terms of order/3 between strong and semi-strong interactions cancel in the 4 like states because of the combinations 811 + 822 -- 2812 and
obviously another possibility would be to consider at least a 3 X 3 mechanism (V 1, V 2, V3) leading to V, 4, 4' states. Weaker interactions would then depend upon d i and create the small asymmetric terms (like electromagnetic interactions distinguish different charges or colour states and SU 3 breaking medium strong interactions distinguish different isospin states). At first sight it seems that normal electromagnetic interactions are not sufficient for our purpose. Photons have to couple differently to Vi(gvl v 4= gV2v like in the p, w, ~ case) in order that gq;• be non zero form eq. (1) or (3) and have a normal intensity required by the ~+~- partial width. But the contributions of order ~ to the symmetry breaking terms 8q 56
Vk
Fig. 4.
'111 + '122 -- 2'112" S o f o r e x a m p l e t h e d e c a y w i d t h o f
the 43.7 is really of order/32 ~ 10 - 2 as required. - no contribution of this type (of order/32) can occur for the lowest 43.1 decay width; the first term (in eq. (2)) is due to the self-mass term and rnP~ (811 -- 822) 2/8mI' is of order ~ ~ 10 -4 which is just the factor required for passing from 500 MeV down to 50 keV. - radiative transitions become now allowed by the vector dominance terms involving the intermediate V i
Volume 60B, number 1
V;,
PttYSICS LETTERS
.
<
Fig. 5. and the semi-strong breaking term (fig. 5); they are however reduced by the factor ~3in the amplitude and by the absence of the important p, w, ~olike intermediate states; one gets therefore only about one keV for modes like 27rT, r/7 .... Let us however come back to the second order electromagnetic transitions between V i. The single photon intermediate state (fig. 2) is too weak [2], giving P~0--*had ~ RP~0 ~u+u - ~ 12 keV, but other selfmass terms like those of fig. 6 are possible. From the
22 December 1975
whether or not one applies also the semi-strong reduction factor/3 because of the occurrence of a V i - V/hadrons transition; this would lead t o / 5 i / ~ ~35m2 7.5 × 10 - 3 GeV 2 and only a negligible contribution of 1 keV to P~0" On the opposite if one keeps this electromagnetic term as it stands, then it is sufficient to explain the ff decay width. One could in this case reconsider the origin of the semi-strong interaction required for the transitions involving two V i (i.e., two ~) and attribute it completely to this kind o f enhanced electromagnetic tadpole diagrams. All the features o f the hadronic and radiative decays of ~b and if' would be due to the peculiar limiting mixing between the initial vector meson states V i slightly perturbed by electromagnetic interactions. However in this case it would be hard to understand a rigorous isospin conservation in all of the purely hadronic decays ~b' ~ + hadrons, a point which need further experimental investigations.
References Fig. 6. mesons electromagnetic mass differences and p - 6o interference one knows [3] that such terms can be relatively large, i.e. 6m 2 ~ 5 X 10 - 3 GeV 2 for tg, w, K* mesons. Applying a scaling factor m 2 / m 2 .~ 15 leads to ~i1 ~ 0.075 GeV 2 and through eq. (2) a width F~ o f the order of 100 keV. But here one has to decide
[1] F.M. Renard, Nucl. Phys. B82 (1974) 1. [2] H. Harari, SLAC-PUB-1514 (1974); CERN Theory Boson Workshop, CERN TH 1964 (1974); J. Ellis, Schladming lectures, CERN TH 1996 (1975); and everybody own calculations. [3] M. Gourdin, F.M. Renard and L. Stodolsky, Phys. Lett. 30B (1969) 347; F.M. Renard, Nucl. Phys. B15 (I 970) 118; Spring. Tracts Mod. Phys. 63 (1972) 98.
57