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Geometric interpretation of a new hadronization model
Volume 195, number 3
PHYSICS LETTERSB
10 September 1987
G E O M E T R I C I N T E R P R E T A T I O N OF A N E W H A D R O N I Z A T I O N M O D E L M. GASPERINI Dipartimento eli Fisica Teorica dell'Universit~, Corso M. D'Azeglio 46, and INFN, Sezione di Torino, 1-10125 Turin, Italy
Received 22 April 1987
It is shown that the phenomenologicalpotential, used in the context of a recent successful hadronization scheme to realize an effective quark confinement, can be reproduced by coupling, to first order, the quark field to the geometry of an anti-de Sitter vacuum. This seems to suggest a geometricinterpretation whith could explain and justify the manifest Lorentz noninvariance of the original effectivehadronization model.
A new interesting hadronization model has recently been proposed [ 1-3]; this model, based on a quark wave function which satisfies a generalized Dirac equation, (i~,o0, +i~,-x/x2o - m) ~t =0,
(1)
is in excellent agreement with the present data on the D Oand F + lifetimes, thus explaining a longstanding problem in the phYSiCSof charm decays [ 2 ], and gives also predictions for the decays of D O and D + into two pseudoscalar mesons which compare quite well with the available data [ 3]. An unsatisfactory formal aspect of this model, as explicitly recognized by t h e authors [ 1-3], is the manifest Lorentz noninvariance ofeq. (1), which, as stressed in ref. [2], puts some limitations on its applications. According to the proponents of the model, the phenomenological term added to the usual Dirac equation is physically justified as the term responsible for the confinement of the quarks, which cannot appear asymptotically and are to be considered as free particles within a region with an effective radius Xo-- 0.2-0.3 fm [2]. From a geometrical point of view, the phenomenon of confinement can be represented by embedding the confined particle in a c u r v e d manifold of given radius, for example a de Sitter "microuniverse" [4,5] instead of the usual Minkowski space. The aim of this paper is to point out then a possible
geometric interpretation of eq. (1), by showing that the nonhermitian interaction t e r m i(uy.x~u can indeed be obtained by coupling the quark to a de Sitter geometry, and that, choosing a "cosmological" constant [ A [ = l / x ~ , the effective equation (1) appears just as an approximation (in the limit of very small distances, X/Xo << 1 ) of the exact theory in a de Sitter space. In this geometric realization of the hadronization model the nonconservation of the Dirac current is naturally interpreted as a curvature effect (in a curved geometry the current satisfies a generally covariant continuity equation). The most interesting aspect of a geometric interpretation, however, is that in a curved spacetime the global Lorentz symmetry is broken: we can then understand in this way and justify the lack of Lorentz invariance ofeq. (1), noting, however, that, if the interaction with the geometry is included to all orders, in the exact equation (see below) the local Lorentz symmetry is restored. We start recalling that, in order to write the Dirac equation in a curved manifold, the Dirac operator is to be generalized by introducing covariant derivatives, that is i~'aOa--,iTUDu, where [6] D u=Ou-klt.ogaby[a~ b] .
(2)
Here COu~b is the Lorentz connection, 7u= V~,a~'a, where 7a are the usual flat-space Dirac matrices and VS is the vierbein field, that relates the world metric tensor gu, to the Minkowski metric ~/~baccording to
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
453
Volume 195, number 3
PHYSICS LETTERS B
g/ly= VflaV,‘Q, (Notations: Greek letters denote holonomic world indices, Latin letters a, b, c, ... are tangent space Lorentz indices. For all the other conventions regarding the Dirac matrices and the deflnition of covariant derivatives we follow ref. [ 61.) Considering in particular a riemannian (i.e., torsionless) geometry, the Lorentz connection can be expressed simply in terms of the Ricci rotation coefficients Cahcas follows: c0fiba= V&‘(Cahc-&a
-
Crab),
-
dij+
X,Xj
&
dX’ dXj, >
(4)
where x2=Stixixj (conventions: i, j,, k, ... run from 1 to 3) and ,JIis the constant which sets the scale of the space. The components of the vierbein field {Vpa} are then
Kj=a,j+
v44i=o=
x$
(a-
vj4,
1)
(5)
f= (1 -Ax”) -‘I, and the inverse field {vfla} is given by [where
$3
p’,=p,-
(&1).
(6)
With this choice of the vierbeins, the only nonvanishing components of the Ricci rotation coefficients are (i, j, k= 1, 2, 3) Ct44
=
and 454
=
-
C4i4
-&,Jf+ 2Jf
A straightforward computation of the Dirac operator $‘D, (using also the property of the Dirac matrices y”~t~~‘] =Y[“Y~Y’]+2~“tbyc1) leads finally to the following expression for the Dirac equation (iy@D,- m) w = 0 in a de Sitter space
(&z. +
y4d4+iy.V
$ (dcz-
l)(x.y)(x*V
>
dt2
(
va4= l/J,
Cijk= - Cj,
(3)
where c,bc=~p,~,~[pVvlc and P, is the inverse vierbein field, such that vMav’b= dab. Working in Cartesian rectangular coordinates, xl”= (x, t), the line-element of the de Sitter geometry takes the form ds2 = (1 -/Ix*)
10 September 1987
(r$ _A>&&Jf 2x2 f
(7)
-m
>
y=O.
(9)
For Ax2 CK1, eq. (9) becomes [iY”a, + +ilx2y4~, - fLl(x-y)(x-V -i&.x--m]v=O.
)
(10)
In the limit x-0 we get then eq. (1)) provided we identify rl= - l/x;. Therefore this geometric interpretation of the hadronization model seems to work only embedding the quark in an anti-de Sitter space (/i < 0). This result is suggestive and very much reminiscent of the results of ref. [ 41, where an effective confinement is realized by representing geometrically the hadronic bags as microuniverses with just the anti-de Sitter metric. The validity of this interpretation is to be verified, however, by considering also higher-order contributions of the geometry to the effective potential. This point, as well as many other points left open in the present investigation, in particular those related to the general questions raised by the phenomenology of the meson decays, will be analysed in forthcoming publications. As a conclusive remark, it is interesting to recall that the anti-de Sitter vacuum appears naturally in the compactification of higher-dimensional unified models (see for example ref. [ 71). It is a pleasure to thank I. Bediaga and E. Predazzi
Volume 195, number 3
PHYSICS LETTERS B
for t h e i r k i n d i n t e r e s t a n d v e r y useful c o m m e n t s t h a t led to i m p r o v e the original m a n u s c r i p t . I a m also v e r y grateful to A. G i o v a n n i n i for m a n y fruitful discussions o n t h e general i d e a s u n d e r l y i n g this paper.
References [ 1] I. Bediaga, E. Predazzi and J. Tiomno, Phys. Lett. B 181 (1986) 395.
10 September 1987
[ 2 ] J.L. Basdevant, I. Bediaga and E. Predazzi, A new hadronization model: implicit charm decay, Torino University preprint DFTT 86/30 (December 1986), to be published. [ 3 ] J.L. Basdevant, I. Bediaga, E. Predazzi and J. Tiomno, A new hadronization scheme: the case of explicit charm decay, Torino University preprint DFTT 87/1 (March 1987), to be published. [4] A. Salam and J. Strathdee, Phys. Rev. D 18 (1978) 4596. [5] Z. Haba, Phys. Lett. B 78 (1978) 421. [6] F.W. Hehl and B.K. Datta, J. Math. Phys. 12 (1971) 1334. [7] Y. Tosa, Phys. Rev. D 30 (1984) 339.
455
PHYSICS LETTERSB
10 September 1987
G E O M E T R I C I N T E R P R E T A T I O N OF A N E W H A D R O N I Z A T I O N M O D E L M. GASPERINI Dipartimento eli Fisica Teorica dell'Universit~, Corso M. D'Azeglio 46, and INFN, Sezione di Torino, 1-10125 Turin, Italy
Received 22 April 1987
It is shown that the phenomenologicalpotential, used in the context of a recent successful hadronization scheme to realize an effective quark confinement, can be reproduced by coupling, to first order, the quark field to the geometry of an anti-de Sitter vacuum. This seems to suggest a geometricinterpretation whith could explain and justify the manifest Lorentz noninvariance of the original effectivehadronization model.
A new interesting hadronization model has recently been proposed [ 1-3]; this model, based on a quark wave function which satisfies a generalized Dirac equation, (i~,o0, +i~,-x/x2o - m) ~t =0,
(1)
is in excellent agreement with the present data on the D Oand F + lifetimes, thus explaining a longstanding problem in the phYSiCSof charm decays [ 2 ], and gives also predictions for the decays of D O and D + into two pseudoscalar mesons which compare quite well with the available data [ 3]. An unsatisfactory formal aspect of this model, as explicitly recognized by t h e authors [ 1-3], is the manifest Lorentz noninvariance ofeq. (1), which, as stressed in ref. [2], puts some limitations on its applications. According to the proponents of the model, the phenomenological term added to the usual Dirac equation is physically justified as the term responsible for the confinement of the quarks, which cannot appear asymptotically and are to be considered as free particles within a region with an effective radius Xo-- 0.2-0.3 fm [2]. From a geometrical point of view, the phenomenon of confinement can be represented by embedding the confined particle in a c u r v e d manifold of given radius, for example a de Sitter "microuniverse" [4,5] instead of the usual Minkowski space. The aim of this paper is to point out then a possible
geometric interpretation of eq. (1), by showing that the nonhermitian interaction t e r m i(uy.x~u can indeed be obtained by coupling the quark to a de Sitter geometry, and that, choosing a "cosmological" constant [ A [ = l / x ~ , the effective equation (1) appears just as an approximation (in the limit of very small distances, X/Xo << 1 ) of the exact theory in a de Sitter space. In this geometric realization of the hadronization model the nonconservation of the Dirac current is naturally interpreted as a curvature effect (in a curved geometry the current satisfies a generally covariant continuity equation). The most interesting aspect of a geometric interpretation, however, is that in a curved spacetime the global Lorentz symmetry is broken: we can then understand in this way and justify the lack of Lorentz invariance ofeq. (1), noting, however, that, if the interaction with the geometry is included to all orders, in the exact equation (see below) the local Lorentz symmetry is restored. We start recalling that, in order to write the Dirac equation in a curved manifold, the Dirac operator is to be generalized by introducing covariant derivatives, that is i~'aOa--,iTUDu, where [6] D u=Ou-klt.ogaby[a~ b] .
(2)
Here COu~b is the Lorentz connection, 7u= V~,a~'a, where 7a are the usual flat-space Dirac matrices and VS is the vierbein field, that relates the world metric tensor gu, to the Minkowski metric ~/~baccording to
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
453
Volume 195, number 3
PHYSICS LETTERS B
g/ly= VflaV,‘Q, (Notations: Greek letters denote holonomic world indices, Latin letters a, b, c, ... are tangent space Lorentz indices. For all the other conventions regarding the Dirac matrices and the deflnition of covariant derivatives we follow ref. [ 61.) Considering in particular a riemannian (i.e., torsionless) geometry, the Lorentz connection can be expressed simply in terms of the Ricci rotation coefficients Cahcas follows: c0fiba= V&‘(Cahc-&a
-
Crab),
-
dij+
X,Xj
&
dX’ dXj, >
(4)
where x2=Stixixj (conventions: i, j,, k, ... run from 1 to 3) and ,JIis the constant which sets the scale of the space. The components of the vierbein field {Vpa} are then
Kj=a,j+
v44i=o=
x$
(a-
vj4,
1)
(5)
f= (1 -Ax”) -‘I, and the inverse field {vfla} is given by [where
$3
p’,=p,-
(&1).
(6)
With this choice of the vierbeins, the only nonvanishing components of the Ricci rotation coefficients are (i, j, k= 1, 2, 3) Ct44
=
and 454
=
-
C4i4
-&,Jf+ 2Jf
A straightforward computation of the Dirac operator $‘D, (using also the property of the Dirac matrices y”~t~~‘] =Y[“Y~Y’]+2~“tbyc1) leads finally to the following expression for the Dirac equation (iy@D,- m) w = 0 in a de Sitter space
(&z. +
y4d4+iy.V
$ (dcz-
l)(x.y)(x*V
>
dt2
(
va4= l/J,
Cijk= - Cj,
(3)
where c,bc=~p,~,~[pVvlc and P, is the inverse vierbein field, such that vMav’b= dab. Working in Cartesian rectangular coordinates, xl”= (x, t), the line-element of the de Sitter geometry takes the form ds2 = (1 -/Ix*)
10 September 1987
(r$ _A>&&Jf 2x2 f
(7)
-m
>
y=O.
(9)
For Ax2 CK1, eq. (9) becomes [iY”a, + +ilx2y4~, - fLl(x-y)(x-V -i&.x--m]v=O.
)
(10)
In the limit x-0 we get then eq. (1)) provided we identify rl= - l/x;. Therefore this geometric interpretation of the hadronization model seems to work only embedding the quark in an anti-de Sitter space (/i < 0). This result is suggestive and very much reminiscent of the results of ref. [ 41, where an effective confinement is realized by representing geometrically the hadronic bags as microuniverses with just the anti-de Sitter metric. The validity of this interpretation is to be verified, however, by considering also higher-order contributions of the geometry to the effective potential. This point, as well as many other points left open in the present investigation, in particular those related to the general questions raised by the phenomenology of the meson decays, will be analysed in forthcoming publications. As a conclusive remark, it is interesting to recall that the anti-de Sitter vacuum appears naturally in the compactification of higher-dimensional unified models (see for example ref. [ 71). It is a pleasure to thank I. Bediaga and E. Predazzi
Volume 195, number 3
PHYSICS LETTERS B
for t h e i r k i n d i n t e r e s t a n d v e r y useful c o m m e n t s t h a t led to i m p r o v e the original m a n u s c r i p t . I a m also v e r y grateful to A. G i o v a n n i n i for m a n y fruitful discussions o n t h e general i d e a s u n d e r l y i n g this paper.
References [ 1] I. Bediaga, E. Predazzi and J. Tiomno, Phys. Lett. B 181 (1986) 395.
10 September 1987
[ 2 ] J.L. Basdevant, I. Bediaga and E. Predazzi, A new hadronization model: implicit charm decay, Torino University preprint DFTT 86/30 (December 1986), to be published. [ 3 ] J.L. Basdevant, I. Bediaga, E. Predazzi and J. Tiomno, A new hadronization scheme: the case of explicit charm decay, Torino University preprint DFTT 87/1 (March 1987), to be published. [4] A. Salam and J. Strathdee, Phys. Rev. D 18 (1978) 4596. [5] Z. Haba, Phys. Lett. B 78 (1978) 421. [6] F.W. Hehl and B.K. Datta, J. Math. Phys. 12 (1971) 1334. [7] Y. Tosa, Phys. Rev. D 30 (1984) 339.
455