P r a m a n a - J. Phys., Vol. 37, No. 1, July 1991, pp. 71-81. © Printed in India.

Geometric interpretation of hadronization model

S BISWAS and L DAS Department of Physics, University of Kalyani, Kalyani 741 235, India

MS received 13 August 1990; revised 10 December 1990

Abstract. A hadronization model termed as geometric dielectric confinement model is described. The model describes the charmed meson decays quite successfully. In the model we assume that the non-abelian gauge field describing the colour force simulates the effect of a medium having space-dependent dielectric constant. The quarks produced in weak decays move in the dielectric medium such that they are free in limited region of space (r ~- 0) and cannot appear as asymptotic states resulting in hadronization. It is found that the dielectric medium resembles anti-desitter microuniverse and the quarks behave essentially as free particles damped by gaussian distribution. The model reproduces from a single Lagrangian the quark motion as well as the form of dielectric function.

Keywnrds. Geometrization; weak interaction; charm decays.

PACS Nos 12.35; 14-80

1. Introduction

Recently we proposed models for confining photon, quark and gluon (Biswas and Kumar 1989; Kumar and Biswas 1989). In these studies we observed that a strong gravitational background confines particles that satisfy Maxwell-like equations. The strong gravity formalism to strong interaction is well known (Biswas and Mukherjee 1983; Biswas et al 1989; Sivaram and Sinha 1979; Salam et al 1971). It has been shown that when one writes down massless quark and gluon field equations in the strong gravity background describing the strong interaction, the equations take the form of Maxwell-like equations in dielectric medium having the properties of particle confinement. The dielectric function is then identified as strong background. In this paper we adopt a similar approach to treat the quarks taking part in weak interaction for describing the charmed meson decays. We call it geometric colour dielectric model (GCDM). We show that the non-abelian gauge field describing the colour force simulates the effect of a medium that can be described by a gravitation-like background and is found to resemble an anti-deSitter vacuum. The quarks produced in the weak interaction then move in this background as if they are moving in a dielectric medium. The medium thus provides a trap resulting in hadronization. A new interesting model has been recently proposed to explain the present data on the D ° and F ÷ lifetimes, thus explaining a longstanding problem in the physics of charmed decays. The model known as hadronization model (Basdevant et al 1987) starts from a Dirac equation with an extra gaussian damping factor for the wave function. As a result the generalized Dirac equation for the model is given by

(i?udu + i ' l .x /x 2 - m)~, = 0, (1)

71

72 S Biswas and L Das

thus introducing a non-hermitian term -~i't 'x/x2o~, in the hamiltonian. As noted by the proponents (Basdevant et a11986, 1987), the unsatisfactory aspect of this model is the explicit Lorentz noninvariance of eq. (1) putting some limitations on its applications. The physical aspects of eq. (1) relating to charmed meson decays may be put as follows.

In all conventional models of charmed meson decay one usually assumes that the quarks produced in weak decay behave as if they were free particles. This assumption amounts to saying that the confining aspect as well as asymptotic freedom must be taken into account when it comes to comparing hadronic and leptonic decays of charmed mesons. As the quarks are confined but not the leptons which appear as asymptotic states, we should have some mechanism so that the quarks initially free for very short separation make a transition to some hadronic final state i.e., hadronization occurs as separation increases. The solution of (1) is

~,(x, t) ,-~ e x p ( - x2/2Xo2)~q .... (2)

where ~kfrce is the solution of free Dirac equation. The physical aspects of charmed meson decays are now contained in (1) or (2). At a very short distance, assuming that the quarks are produced at x = 0; the quarks are free but as the separation increases the non-hermitian term or the gaussian damping factor makes

~t <qJl qJ> - -1. <qJ IH - H~JqJ> ~ 0, i

resulting in hadronization. In view of the excellent agreement of the model with the data, it is necessary to obtain a first principle derivation of (1). Gasperini (1987) tried for a geometric interpretation of (1) embedding the confined particle in a de-Sitter microuniverse and showed that the nonhermitian interaction term i~ ' t 'x~ can be obtained by coupling the quark to a de-Sitter geometry. Equation (1) emerges when one adopts a cosmological constant A = 1/X2o and assumes that X/Xo << 1. Although the interpretation of Gasperini seems encouraging, it does not pay any attention to QCD that is barely needed to explain confinement and asymptotic freedom whose ignorance is the root of all "anomalies" encountered in explaining the charmed meson decays. Moreover the use of negative cosmological constant is not suitably explained in the interpretation of Gasperini (1987).

In this paper we try to get an answer to these problems from our recently proposed model (Biswas et al 1990; Biswas and Kumar 1989; Biswas and Das 1989). We call this model as GCDM. In G C D M we assume that the non-abelian gauge field describing the colour force simulates the effect of a medium having space-dependent dielectric constant. The simulated medium resembles anti-deSitter microuniverse and the quarks behave essentially as free particles damped by a Gaussian distribution. It is shown elsewhere (Kumar 1990) that G C D M resembles with the colour dielectric model of Nielsen and Patkos (1982) derived from QCD Lagrangian to explain the non-perturbative sector of QCD.

The data of hadronic and semileptonic decays of pseudoscalar charmed mesons (D, D °, F +) have proven very hard to explain theoretically. It is believed that in charmed meson decays the charmed quark is the only relevant issue whereas the noncharmed quark remains as a spectator. In implicit charmed meson decays (i.e., dealing with total decay widths) W-radiationis the dominant part through the process

Geometr ic in terpre ta t ion o f hadroniza t ion model 73

C ~ W + + s, W + ~ u + d so that C--*uds, the other quark remains as a spectator (Averill et al 1989). The other contributors like W-exchange (WE) and W-annihilation (WA) are hoped to be negligible. But the result z (D+)>2t(Do) poses a problem suggesting that WE contributes to D ° and not to D +. Furthermore the result BR (D + ~ X I v ) / B R ( D ° --* X l ÷ v) ~ 2"5 necessitates estimating the actual magnitude of WE in D o decay. In recent theoretical approaches there are several models that describe the two-body decay of charmed mesons (Ruckl 1983; Buras et al 1986; Kamal 1986b; Baur et al 1986). In §2 we describe G C D M where the space dependent dielectric function is obtained from a single Lagrangian. The quark motion is considered in a region where colour electric and magnetic field are zero but the potential A t # 0. The resulting generalized Dirac equation can be obtained by the replacement P---, P - A or from our model where the quarks are considered to be moving in a colour dielectric medium. The space dependent dielectric function simulates a background with deSitter geometry with negative cosmological constant. The conclusions drawn are given at the end.

2. Model for hadronization

The Lagrangian describing a colour field is

l 1 i#v L , = - ~ F ~ v F . (3)

The field equation can be east in a form (Biswas et ai 1990) in low momentum approximation as

O, f~v = - J~, (4)

where f~ , is the abelian part of F~, and all the non-linear terms like A.A.A., A A etc. is absorbed in J~, in the r.h.s, of (4). Equation (4) when compared with QED can be considered as polarization current arising out of closed gluon loops. Equation (4) can be cast in Maxwell-like form in a dielectric medium with space-dependent dielectric function. In this description non-abelian gluon behaves as abelian boson moving in a space dependent dielectric medium. Like Dicke (Dicke 1957; Biswas and Kumar 1989) we assume that the dielectric function modifies the space time such that

ds 2 = (1 /e(r) )d t 2 - e(r)(dx 2 + dy 2 + dz2), (5)

where e(r) is now determined self consistently from a Lagrangian describing the field e(r), gauge field A/~, as well as, the quark field ~(r). The metric (5) would lead to bending of meter sticks depending upon their location in space-time. If such meter sticks are defined as unchanged in length, the variation is then interpreted as due to curved space-time. We have then an Einstein-like theory for the field e(r). Another approach (Dicke 1957; Biswas et al 1989) is to scale down the length and time at every space-time points so that we are still in flat space-time. To do so we introduce the scaling

L = Lo /e ½ (6)

o9 = to o/e ½ (7)

74 S Biswas and L Das

so that we are still in the co-ordinate system given by

ds 2 = dt 2 _ (dx 2 + dy 2 + dz 2)

= ~l#~dx~dx" (8)

with r/oo = --: r/i t = - r/22 = - 1/33 = 1, t/o = 0, i -¢:j. To work with co-ordinate system (8), the parameter m has to be scaled down as

m = mo/~ ½. We treat e(r) as scalar field and take the Lagrangian as

L f -~- ½fay ~ e d v ~, (9) with

foo = l/e, A t = f z2 = f 3 3 = --e,

fo = 0 for i # j , (10)

corresponding to (6). The Lagrangian for the gauge field A. (we omit the super script 'i' henceforth) is taken as

Lg = 16rtF~,.fU~F~afBr, (11)

and the Lagrangian for the quark field is taken as

L , = -- ½(- iffffud,~b + iduff'~u~b) - m~ff~b (12) where

~ ' + ~ ' ~ = 2f~L (13)

m = mo e½,

the ~u are usual flat space gamma matrices. Field equations now follow from the variational principle

_ I L ( - r/)½d'x = 0. (14)

Here L is the sum of (9), (11) and (12). Variations w.r.t, e, ff and Au give the field equations. On the assumption that e is time independent and spherically symmetric, we get for ~ using

dL dL Oude,~ o~=O'

the equation

2028 1[-(¢~t~ 2 (VS) 2 ] 8~( _n~)

1 i - o " i -~ mff~b'~. -~(~¢,~ 0 o ~ - ~ , ' o # - i -.-o ) (15)

Geometric interpretation of hadronization model

As e is time independent we write (15) as

V2 ~ - - I ( V ~ ) 2 = 1 f i - ,or 3 1 -~

75

h.c. - mff~b + 4-~(eE2 + B2/g)}. (15a)

The variations w.r.t. ~b and ~ give

i~°Oo~b + i~'c31~b + tfl ~ ~k - moe½~, = 0, (16)

and,

iff°do~ + i~iOi~ + i f l ( ~ ) ~ + moe½~ =O, (17)

where # = ½&-UOr. (18)

From (16) and (17) we get

i ~ ° Oo~O - iOo~°O = - ( i ~ O,~k - i d , ~ k ) + 2m~k. . . (19)

We use (19) in (15) to eliminate ~ / a i ~ like terms so that

V2e - l ( v e ) 2 = - ~{i~ff~ ° Oo~O-iO~ 60o i f - 2mlff~k} -- I ( e E 2 + BZ/e).

(19a)

Assuming that ~, ~ e x p ( - iwt), eq. (19a) reduces to

- ~--~(Ve)2 = -{ogff~b~7 ° - mff~b + 8~(~E2 + B2/e)}. V2e

The asymptotic freedom demands the quarks to be free at a short distance. This is achieved when we take

e - - * l r-'*O

ff~b ---, constant. (20) r'-*O

Equation (19a) reduces to

V2e (Ve)2 - e½(~°og- mo)~d/-l(eEZ+B2/e,) . (21) 2e

Let us consider a situation E = 0 and B = 0 but A~, ~ 0 so that

Au = (0, V2(x)). (22)

Let us write in view of (20)

76 S Biswas and L Das

as ~o = 7oe+ and m = mo e+. W e have then f rom (21)

2e½V2e$ = _ 22%½, or,

V 2 e $ = - - ~a. (23)

As e is spherical ly symmet r ic we write (23) as

r 2 dr r2 et (24)

The solut ion of (24) is given by

e t = ( a b 1 , 2 \ + r - ~2 r ) . (25)

The b o u n d a r y condi t ion e -~ 1 as r ~ 0 al lows us to put a = 1 and b = 0, so tha t ( taking 2"/6 = a)

e = (1 - at2) 2, (r < a-t) .

This is our required solution. I t should be ment ioned tha t the solut ion (25) is valid within a region r < R where R = a -+ is the distance at which e ~ 0 . As ~, -- ,0 because of gauss ian d a m p i n g factor, the r.h.s o f (21) becomes pract ical ly zero at r ~ ~ so that R ~ ~ , leading to perfect confinement . As we are interested in small r behav iour of quarks , we take air << 1 and evaluate (16) in this approx imat ion . We a r range (16) as

(i,°e3o + iy'd, + ~-f2(7"r)-moe)~ =O,

1 de - +/t~r R 2 = (26)

2 e - + r

N o w for air << 1

~a-+/dr - - "~ 2ar £-½

so tha t (26) is a p p r o x i m a t e d for very small r as

( i,° t~o + i,i t~, + R---T(¥'r) - mo )~b = O

where 1

- - - - - - - a . R 2

(27)

(28)

Equa t i on (27) is exact ly the general ized Di rac equa t ion of Basdevan t et al (1987) deal ing with cha rmed meson decays.

The s imulated backg round is now given by

1 ds2 --- (1 - ar2) 2 dr2 - (1 - ar2)2(dx2 + dy 2 + dz 2) (29)

Geometric interpretation of hadronization model 77

which is nothing but an anti-deSitter geometry. The cosmological constant is now 2a = 2/R 2, the sign of a is now determined by co$7° ~b - m~b which is basically energy density of quark around the centre of confinement (massless quark approximation). So as r becomes large co~,), ° - m~/, ~ 0 lending R large. This is the most welcome feature of our model i.e., free quarks are not asymptotic states and disappear in the free particle limit R --* oo (i.e., no hadronization or lepton case). The reader is referred to Basdevant for detailed discussion in this respect.

To understand the physical origin of hadronization in our model we approximate the r.h.s of (18) as - 2e so that 2 now corresponds to every density around the centre of confinement. To clarify our standpoint let us evaluate the energy momentum tensor from the relation

c~L¢ T~ = ~,,~-~,~ - 6~ L¢. (30)

For ~ = e, A~ and ~, L¢ will be equal to Lf , L o L , of (9), (11) and (12) respectively. Using (30), (9) and (11) we get

so that T, - f e.~e.,- 6,,½f ~,,~,p

is always positive definite. Similarly

(31)

(32)

Hence !

T ° ( A . ) = 8~(~E2 + B2/~).

In deriving (33) we have used

(33)

F~v = A~,v - Av,~,

0A E = - o t - v~o

B = V x A. (34)

Remembering that L , = 0 by virtue of (16) and (17)

A

and hence (18) can be written as

VZe = - To°(~) + m ~ + Tee(e)- T°(A~,). (35)

The source of the background is basically an effective energy density term given by the r.h.s of (35)just like G o = - k T ° equation of general relativity. Because of opposite

78 S Biswas and L Das

sign of T°(e) and T°(At); at a very short distance there is a possibility of cancellation between these two terms even if both terms being large. This allows us to have a confined system held together by its own gravitational force (Dicke 1957; Biswas and Kumar 1990). The simulation of anti-deSitter like geometry is the reason for having confinement in our model. The approximation of r.h.s of (18) or (19) might seem too adhoc. But to justify our approximation, let us solve (18) with (static approx.)

~b ,,~ const, exp ( - iwt),

B = 0 ,

E = constant (36)

considered to be valid at small r region. In such a case we get from (16) and (17), since i~Tic~i~b like terms are zero

ffi~°(9o~ -- idoff~°~ = 2rnoe½ff~b.

Hence eq. (19a) turns out to be

V2e ( ve )2 E 2 2e - e8--n" (37)

The solution is

e,,~(sina_____rr) z

which in the limit r--, 0 gives

e "~ (1 - - t ir2) 2 (38)

where a = E2/8R. Thus we see that constant energy density approximation around the centre of confinement (i.e., r ~ 0) is quite a probable situation.

Having thus explained the origin of hadronization we turn back to the QCD content of our model. Some days back Nielsen and Patkos (1982) obtained from QCD Lagrangian an effective theory which describes the long distance (i.e., the confinement-scale) behaviour of QCD by defining a phase averaging of the gauge fields over small four dimensional space-time hypercubes. The effective Lagrangian density is

where

- 1 2 2 1 4 , L = i~ ,~ ' (~ , - iBj,/X)~k - m~,d./+ ~a.(cg.X ) - U(X) - 2--~Z Tr(F~.F ~ ),

(39)

X = ~-~Trk

k(Xo) = av. Pexp [i~,,oA,dx,]. (40)

For notations and other details the reader is referred to Williams and Thomas (1986) and Aoki and Hyuga (1989) for numerical solution. This model is very much similar

Geometric interpretat ion o f hadronizat ion model 79

to our G C D M where e = X 4 acts like dielectric constant. By rescaling of ~O as well as B, terms we can bring (39) very close to our GCDM. This is discussed elsewhere.

Some comments may be in order regarding the formal Lorentz non-invariance of Basdevant-hamiltonian. It may be noted that our Lagrangian is locally Lorentz invariant and hence the emergence of non-hermitian term in the hamiltonian needs clarification.

In Gasperini's discussion one needs a negative cosmological constant as an input; however, in our approach as the energy densities corresponding to quark and colour are always positive and approximated as 22, it occurs in the generalized Dirac equation with the chosen sign, as if it behaves as negative cosmological constant. One may be skeptical about the definition ~ in (13) leading to (27) instead of adopting the Vierbien formalism as is done in writing down Dirac equation in curved space-time. However, it is observed that starting with deSitter geometry, Dirac equation

(iTUDu - m)ff = 0 (41)

reduces to the form (28) again for a½r << 1. This lends support also to the correctness of our approach. Here (Gasperini 1987)

Du = 0u + ¼(.Oual,~'ta~ 'b],

CO~, b = Lorentz connection,

V~ = Vierbein field. (42)

Thus the geometrization of the basic interaction in the sense of dielectric function advocated by us earlier is the root of hadronization proposed by us and others (Basdevant et al 1987). The formal non-covariance of (27) is a basic defect in Basdevant's model. The Hamiltonian corresponding to (27) is

H = - •(i¥.a + i(¥.r)a - m)~, (43)

so the probability

~<¢lqJ> = 1 , 7<~blH - H I~k> (44)

is not conserved in time and the non-conservation of the Dirac current is naturally interpreted as curvature effects. In our approach though we started from a local Lorentz invariant Lagrangian, it is the co-ordinate system (5) and consequent definition of ~ that have broken the local Lorentz symmetry. However, if the interaction with the geometry is properly included to all order, as in (41) and (42), the local Lorentz symmetry will be restored (Gasperini 1987). To complete the discussion we show that e(r) in the model is due to the effect of an imaginary part of a gauge field related to the weak interaction. Equation (43) can be written as

H = ff(7"P + m)~

provided we make the replacement

P --* P = - iO - ira.

80 S Biswas and L Das

The vector A that now goes as minimum coupling i.e.,

P - . P - A

has the form

A = VA(r), 2(r) = iar2/2.

Though we have restricted ourselves to the approximate form (27), the same argument will also a pp ly to the exact equation (26) in a slightly modified way. Unlike usual gauge theories, 2(r) is pure imaginary leading to the non-conservation of probability as mentioned earlier. The non-hermitian term in (43) finds its justification in our formalism which is basically a curved space-time description. The way the hadroniz- ation takes place may be understood as follows. The solution of generalized Dirac equation is of the form

~b(r, t) '-~ exp (i2(r))~b(r, t), (45)

~b(r, t) obeys free Dirac equation. So if at t = 0, ~b(r, 0) is peaked at a value (r(0)> > 0, the wave packet having velocity v o will move towards the centre r = 0 and the probability become maximum i.e., hadrons coming towards r = 0 will create a quark at r = 0, which while moving towards r > 0 will then hadronize as the probability decreases due to the term exp (i2(r)). So this type of hadronization is a basic ingredient of our GCDM.

3. Conclusion

In retrospect we provide an alternative formulation of QCD based on space-time description. The model has the following features.

(i) It describes the long distance behaviour of QCD in the sense that the confinement is reproduced in the model. The asymptotic freedom is incorporated through boundary condition. (ii) The hadronization model of Basdevant is reproduced using a first principle derivation. (iii) Gasperini's (1987) interpretation is inherent in our formalism. (iv) The model does not require to assume a negative cosmological constant. It emerges naturally. (v) Colour field is confined within a region 1/x o and quarks are almost free in this region (Xo = R). (vi) Outside Xo, the free quark wave function is damped by a gaussian factor and in this region hadronization takes place. As r increases harmonic oscillator like terms are also there. The net result is that the further the particle, the more it is absorbed. For sufficient large r (these coming from e2 term in Dirac equation), ff~b and E practical go to zero as if the effective energy density is very small implying thereby that Xo -" (energy density)- 1 becomes infinite i.e., we have perfect confinement. (vii) The application of the model to charmed meson decays has been carried out (Basdevant et al 1987; Kamal and Sinha 1987; Kamal and Verma 1987; Kobayashi and Maskawas 1973; Blok and Shifman 1987; Chau and Cheng 1987). They obtained the results

z(D+)-~ 9"2 x 10-13s,

z(D°) --- 4"7 x 10-t3s,

BR(D ° - , ! + X) ~- ½BR(D + - , l + x),

~- 9.4~o

Geometric interpretat ion o f hadronizat ion model

B R ( D ° ~ e + X)~_ 7.1%,

z ( F + ) ~ 2 . 7 6 x 1 0 - 1 a s .

T h e r e a d e r is r e f e r r e d to B a s d e v a n t et al (1987) f o r de ta i l s .

81

Acknowledgement

I a m t h a n k f u l to Prof . P D a s g u p t a fo r usefu l d i s c u s s i o n s .

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