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Relativistic effects in pp annihilation into two mesons
222
Nuclear Phystcs B (Proc Suppl ) 8 (1989) 222-22t North-ftolland, Amsterdanl
R E L A T I V I S T I C E F F E C T S IN P P A N N I H I L A T I O N I N T O T W O MESONS
M. MARUYAMA,
T. GUTSCHE,
G.L. STROBEL
* and Amand
FAESSLER
Institut ffir Theoretische Physik, Universitiit Tfibingen, Auf der Morgenstelle 14, 7400 Tfibingen, F.R. Germany The effects of the small components of Dirac spinors used to describe quark and antiquark wave functions in clusters and the Lorentz contraction of cluster wave functions are considered in the proton-antiproton annihilation into two final state mesons. The effects are studied by referring to the nonrelativistic model as a base of comparison.
When studying N 2 ) annihilation mto mesons microscopically in the framework of the constituent quark model, a nonrelativistic hamiltonian approach is usually used to describe the quark dynamics within the initial and final state clusters.
However, m o m e n t a of the
final state mesons are in the relativistic region, even for low energies of the initial protonantiproton system. In p/~ annihilation into two mesons at rest, for example ~rTror pp, the final state mesons are moving quite relativistically in the overall center of mass frame ( ~c ~ 99% and -~ ~ 60%). c We consider the annihilation model into two mesons (referred to as A2 in the literature), as thin model suitably describes the branching ratios of the m a j o r two meson final state channels seen in p/5 annihilation I . Two mesons are produced by annihilation of two q~ paiI~ from the initial pp state and by subsequent creation of one q(~ pair assuming a 3p0 vertex. A relativistic quark annihilation model which agrees with the conventional annihilation model in the nonrelativistic limit is studied. By introducing the effects of the small component, the following Dirac spinor wave function with a Gaussian form to describe a single quark is defined as
=
(1)
where ~7 is the Pauli spin matrix, :g is a two component spinor and N the nonnalizahon factor. The adjustable p a r a m e t e r s R and/~ are the cluster size and the weight of the small component, respectively. The quark m o m e n t u m couples to the spin in the small component, which will change the spin-flavour structure of the annihilation amplitude.
Baryon and
meson cluster wave functions in the initial and final state are described by a product of single quark wave functions with the center of m o m e n t u m motion removed in terms of the * Permanent address: Physics Department, University of Georgia, Athens Ga, USA
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Maruyama et al./ Relativistic effectsin p~ annihilation
223
Peierls-Yoccoz m e t h o d 2. The 3p0 vertex in the relativistic model is described by a scalar interaction for the Dirac spinors of the annihilated/created quark-antiquark pair as
~])~(¢t) ff)q(q-) = COnSt. X~ ~" (¢t __ q~ )~q
(2)
Including the effects of the small components, the transition amplitude of nucleon-antinucleon annihilation into two mesons in the planar A2 model is then simply given by the overlap of the initial and final state clusters, with the appropriate qq pairs being contracted according to the scalar interaction of eq. 2. In table 1 results for the effect of the small component on the transition amplitude in comparison to the nonrelativistic case are shown. TABLE 1 Effect of the small component on the transition amplitude, t is the ratio of relativistic to nonrelativistic transition amplitude for several nucleon and meson radii RN and Rm. ~ is fixed to be 0.36. (2T+]
2S+ILj )
RN = 0.62 Rm = 0.57
t2(~ = 0.36) 0.50 f m 0.57 f m
fm fm
0.62 0.50
fm fm
11S o
pp ww
0.85 0.85
0.86 0.86
0.85 0.85
0.87 0.87
0.88 0.88
0.86 0.86
0.65 0.65
0.65 0.65
0.66 0.66
0.65 0.65 0.65
0.65 0.65 0.65
0.66 0.66 0.66
13S1 pTr w~ 11p 1 p~r 0Jq
31p1 p+ppOq wTr°
Meson branching ratios within the same partial wave are unchanged. The decrease of the absolute value of the transition amplitude when considering the small component can be recovered by readjusting the overall strength of the A2 transition, which is a free parameter. The inclusion of the small component changes the relative weight of the partial waves, i.e., P-wave annihilation decreases with increasing fl more than S-wave annihilation. Within the range of quark cluster radii considered, the result is not sensitive to the size parameters. We take the point of view that the cluster wave functions are described by Gaussians in the rest frame of the cluster. In the frame where the cluster is moving, the wave function of the cluster is contracted in the direction parallel to the cluster motion. We use 'tilded' coordinates for those in a cluster rest frame and 'untilded' ones for the overall center of mass frame. For a cluster moving in the z direction of the overall center of mass frame the 'tilded' and 'untilded' coordinates are related by 3
~.=(E~z, \1vl/
:~=x, ~=y,
(3)
M Maruyama et M./ Relativistic effects in pp annihilatiot~
224
where M is the cluster mass and E is the cluster energy in the overall center of mass frame Furthermore, the cluster wave function is normalized to u m t y in the overall center of mass fran:e. In table 2 the effect of the Lorentz contraction on the transition amplitude is shown at a center of mass energy of 50 MeV. TABLE 2 Effect of the Lorentz contraction on the transition amplitude at Tom = 50 MeV. t is the ratio of relativistic to nonrelativistic transition amplitude. L and S specify orbital angular m o m e n t u m and spin in the p/5 system. mesons
:Tr
prr
pp
1 15(s=l), 1.06(s=o) 1.12
1.16(s=1), 1.15(s=o) 1.17
0.91(s=1), 0.85(s=o) 0.96
1 12(s=1), 1.12(s=o) 1 16
0.92(s=1), 0 85(s=o) 0.88
1.10(s=l), 1.10(s=o) 1.10
[RN "~" 0.62 fro, R,,, = 0 57 fro]
L=0 L=I
t2 = t2 =
2.00 2.04
[RN = 0.50 f r o , Rm = 0.57 f m l L=O L=I
t2 = t2 =
1.36 1.58
[RN = 0.62 f r o , R,n = 0.50 fro] L=O tz = 1.10 L=I f2 = 1.07
Depending quantitatively on the nucleon and meson size, the two-pion product:ons a r e enhanced by a factor up to two, as compared to the channels containing vector mesons. In particular the Lorentz contraction affects the angular dependence of the final state meson channels, thus changing the predictions of the d:fferential cross sections. For further details see also reference 4 Strongly dependent upon input parameters (i.e., cluster sizes and /3 dependence), the p/5 annihilation into two mesons will be sensitively influenced by relativistic corrections. Whereas the effect of the small component on the relevant observables is at the few percent level, the Lorentz contraction will result in sizable modifications of the annihilation amplitudes as originally obtained in the nonrelativistic limit. REFERENCES 1) M. M a r u y a m a , S. Furui and A. Faessler, Nucl. Phys. A472 (1987) 643 2) C. Wong, Phys. Rev D24 (1981) 1416. 3) A.N. Mitra and I. Kumari, Phys. Rev. D15 (1977) 261. 4) M. Maruyama, T. Gutsche, G.L. Strobel and A. Faessler, in preparation.
Nuclear Phystcs B (Proc Suppl ) 8 (1989) 222-22t North-ftolland, Amsterdanl
R E L A T I V I S T I C E F F E C T S IN P P A N N I H I L A T I O N I N T O T W O MESONS
M. MARUYAMA,
T. GUTSCHE,
G.L. STROBEL
* and Amand
FAESSLER
Institut ffir Theoretische Physik, Universitiit Tfibingen, Auf der Morgenstelle 14, 7400 Tfibingen, F.R. Germany The effects of the small components of Dirac spinors used to describe quark and antiquark wave functions in clusters and the Lorentz contraction of cluster wave functions are considered in the proton-antiproton annihilation into two final state mesons. The effects are studied by referring to the nonrelativistic model as a base of comparison.
When studying N 2 ) annihilation mto mesons microscopically in the framework of the constituent quark model, a nonrelativistic hamiltonian approach is usually used to describe the quark dynamics within the initial and final state clusters.
However, m o m e n t a of the
final state mesons are in the relativistic region, even for low energies of the initial protonantiproton system. In p/~ annihilation into two mesons at rest, for example ~rTror pp, the final state mesons are moving quite relativistically in the overall center of mass frame ( ~c ~ 99% and -~ ~ 60%). c We consider the annihilation model into two mesons (referred to as A2 in the literature), as thin model suitably describes the branching ratios of the m a j o r two meson final state channels seen in p/5 annihilation I . Two mesons are produced by annihilation of two q~ paiI~ from the initial pp state and by subsequent creation of one q(~ pair assuming a 3p0 vertex. A relativistic quark annihilation model which agrees with the conventional annihilation model in the nonrelativistic limit is studied. By introducing the effects of the small component, the following Dirac spinor wave function with a Gaussian form to describe a single quark is defined as
=
(1)
where ~7 is the Pauli spin matrix, :g is a two component spinor and N the nonnalizahon factor. The adjustable p a r a m e t e r s R and/~ are the cluster size and the weight of the small component, respectively. The quark m o m e n t u m couples to the spin in the small component, which will change the spin-flavour structure of the annihilation amplitude.
Baryon and
meson cluster wave functions in the initial and final state are described by a product of single quark wave functions with the center of m o m e n t u m motion removed in terms of the * Permanent address: Physics Department, University of Georgia, Athens Ga, USA
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Maruyama et al./ Relativistic effectsin p~ annihilation
223
Peierls-Yoccoz m e t h o d 2. The 3p0 vertex in the relativistic model is described by a scalar interaction for the Dirac spinors of the annihilated/created quark-antiquark pair as
~])~(¢t) ff)q(q-) = COnSt. X~ ~" (¢t __ q~ )~q
(2)
Including the effects of the small components, the transition amplitude of nucleon-antinucleon annihilation into two mesons in the planar A2 model is then simply given by the overlap of the initial and final state clusters, with the appropriate qq pairs being contracted according to the scalar interaction of eq. 2. In table 1 results for the effect of the small component on the transition amplitude in comparison to the nonrelativistic case are shown. TABLE 1 Effect of the small component on the transition amplitude, t is the ratio of relativistic to nonrelativistic transition amplitude for several nucleon and meson radii RN and Rm. ~ is fixed to be 0.36. (2T+]
2S+ILj )
RN = 0.62 Rm = 0.57
t2(~ = 0.36) 0.50 f m 0.57 f m
fm fm
0.62 0.50
fm fm
11S o
pp ww
0.85 0.85
0.86 0.86
0.85 0.85
0.87 0.87
0.88 0.88
0.86 0.86
0.65 0.65
0.65 0.65
0.66 0.66
0.65 0.65 0.65
0.65 0.65 0.65
0.66 0.66 0.66
13S1 pTr w~ 11p 1 p~r 0Jq
31p1 p+ppOq wTr°
Meson branching ratios within the same partial wave are unchanged. The decrease of the absolute value of the transition amplitude when considering the small component can be recovered by readjusting the overall strength of the A2 transition, which is a free parameter. The inclusion of the small component changes the relative weight of the partial waves, i.e., P-wave annihilation decreases with increasing fl more than S-wave annihilation. Within the range of quark cluster radii considered, the result is not sensitive to the size parameters. We take the point of view that the cluster wave functions are described by Gaussians in the rest frame of the cluster. In the frame where the cluster is moving, the wave function of the cluster is contracted in the direction parallel to the cluster motion. We use 'tilded' coordinates for those in a cluster rest frame and 'untilded' ones for the overall center of mass frame. For a cluster moving in the z direction of the overall center of mass frame the 'tilded' and 'untilded' coordinates are related by 3
~.=(E~z, \1vl/
:~=x, ~=y,
(3)
M Maruyama et M./ Relativistic effects in pp annihilatiot~
224
where M is the cluster mass and E is the cluster energy in the overall center of mass frame Furthermore, the cluster wave function is normalized to u m t y in the overall center of mass fran:e. In table 2 the effect of the Lorentz contraction on the transition amplitude is shown at a center of mass energy of 50 MeV. TABLE 2 Effect of the Lorentz contraction on the transition amplitude at Tom = 50 MeV. t is the ratio of relativistic to nonrelativistic transition amplitude. L and S specify orbital angular m o m e n t u m and spin in the p/5 system. mesons
:Tr
prr
pp
1 15(s=l), 1.06(s=o) 1.12
1.16(s=1), 1.15(s=o) 1.17
0.91(s=1), 0.85(s=o) 0.96
1 12(s=1), 1.12(s=o) 1 16
0.92(s=1), 0 85(s=o) 0.88
1.10(s=l), 1.10(s=o) 1.10
[RN "~" 0.62 fro, R,,, = 0 57 fro]
L=0 L=I
t2 = t2 =
2.00 2.04
[RN = 0.50 f r o , Rm = 0.57 f m l L=O L=I
t2 = t2 =
1.36 1.58
[RN = 0.62 f r o , R,n = 0.50 fro] L=O tz = 1.10 L=I f2 = 1.07
Depending quantitatively on the nucleon and meson size, the two-pion product:ons a r e enhanced by a factor up to two, as compared to the channels containing vector mesons. In particular the Lorentz contraction affects the angular dependence of the final state meson channels, thus changing the predictions of the d:fferential cross sections. For further details see also reference 4 Strongly dependent upon input parameters (i.e., cluster sizes and /3 dependence), the p/5 annihilation into two mesons will be sensitively influenced by relativistic corrections. Whereas the effect of the small component on the relevant observables is at the few percent level, the Lorentz contraction will result in sizable modifications of the annihilation amplitudes as originally obtained in the nonrelativistic limit. REFERENCES 1) M. M a r u y a m a , S. Furui and A. Faessler, Nucl. Phys. A472 (1987) 643 2) C. Wong, Phys. Rev D24 (1981) 1416. 3) A.N. Mitra and I. Kumari, Phys. Rev. D15 (1977) 261. 4) M. Maruyama, T. Gutsche, G.L. Strobel and A. Faessler, in preparation.