PHYSICAL REVIEW D VOLUME 27, NUMBER 10 15 MAY 1983

Confinement in a Bethe-Salpeter model with logarithmic potential

S. N. Biswas, S. R. Choudhury, K. Datta, and Ashok GoyalDepartment of Physics and Astrophysics, Unlverst'ty ofDelhi, Delhi 110-007, India

(Received 14 June 1982)

We show that the covariant Bethe-Salpeter equation for two scalar quarks confined

through a kernel of the form E—[(p —p')~] '~ possesses a discrete spectrum: this kernel is

a simple covariant generalization of the potential in configuration space proportional to lnr.

To investigate the spectrum, we adopt a new approximation scheme to convert the covariant

Bethe-Salpeter equation into a three-dimensional one, the consistency of the scheme being

checked by applying it to the Wick-Cutkosky equation for the covariant Coulomb problem

and obtaining a Balmer-type formula. The spectrum as revealed by investigating the S-wave

part of the resulting three-dimensional equation by semiclassical methods is essentially ofthe type obtained from the Schrodinger equation with a lnr potential; relativistic effects

depress the energy levels.

I. INTRODUCTION

In an earlier paper,' we discussed the confinement

spectrum for a system of two scalar quarks in thecontext of the relativistic Bethe-Salpeter (BS) equa-tion. We considered the two-quark system interact-ing through a confining kernel of the formK(p —p')= —(2n) P5' '(p p'); in c—onfigurationspace this kernel is of the form it3R whereR =—(x —xo )'r and p is the effective interactionstrength. We found that the covariant confined sys-tem, considered in the framework of the BS equa-tion, possesses no discrete spectrum and concludedthat as the configuration-space kernel PR is notpositive definite over the entire range, the confinedstates "leak" away. This behavior is to be contrast-ed with that of the same system interacting througha three-dimensional oscillator potential. Indeed, wedo recover the desired discrete confinement spec-trum from the BS equation if we adopt an instan-taneous approximation for the momentum-spacekernel. This consists in replacing E(p p') by—E(0,p —p '), which in configuration space leads to apotential of the form 5(xo)V(

~

x~

). The resultingBS equation is a three-dimensional one and obvious-

ly lacks covariance; however, it represents a schemefor examining relativistic corrections to the energylevels of the confinement spectrum obtained fromthe Schrodinger equation with a potential V(

~

x~).

Apart from the harmonic-oscillator potentialV(

~

x~

)-r, there exist in the literature discus-sions of the confinement spectrum for potentials ofthe form lnr and r" (v&0) in the context ofSchrodinger theory. In this note, we investigate thenature of the spectrum one obtains from the BSequation with a kernel of the formE(p —p')=A, [(p —p') ] ~ which is the covariant

generalization (in momentum space) of the three-dimensional logarithmic potential. In Sec. II wefind that the covariant BS equation possesses adiscrete spectrum. To examine the nature of thespectrum, we adopt in Sec. III a new scheme for thereduction of the covariant equation to three-dimensional form: the soundness of the scheme ischecked by using it ro reduce the Wick-Cutkosky BSequation for the relativistic Coulomb problem tothree-dimensional form and obtaining the Balmerspectrum. In the present case, the resulting three-dimensional equation is of the Schrodinger typewith a momentum-dependent potential. In Sec. IV,we treat the S-wave part of this equation by semi-

classical methods and show that the energy levels ofthe confined system are -inn, where n is the excita-tion quantum number. In conclusion, we make afew remarks about the spectrum of the relativisticgeneralization of the linear potential for which

&(p —p') -~l(p —p')'1 '.

II. THE BS EQUATION:EXISTENCE OF THE DISCRETE SPECTRUM

Consider the Wick-rotated covariant BS equationfor scalar quarks of mass m written in the form

2 2

p+ — +m p —— +m P(p)2

2 f &(p p')f(p')d p' — (I)

Here P(p) is the two-body BS amplitude in momen-

tum space, P=(0,iE), E being the total energy ofthe system, and p is the relative four-momentumvariable of the two-body system. K stands for thekernel which we choose to be of the form

27 2439

BISWAS, CHOUDHURY, DATTA, AND GOYAL

A, [(p —p') ] / as the relativistic generalization ofthe three-dimensional configuration-space logarith-mic potential. With this choice of the kernel, wefirst show that the iterated BS equation (1}does pos-sess a kernel of the Hilbert-Schmidt type. For this,we first rewrite Eq. (1) in the following form:

'2 —1

f(P) =2 P+— +m

P

For Eq. (3) to be consistent, we put, as in Ref. 2,

2

p+ — +mP2

2

p —— +m' P(p)=S(p) .2

(4)

Substitution of (4) in (3) easily yields a three-dimensional integral equation for S(p). Further, ifwe write

4,f(p') [(p' —P /2) +m ][ s)2]a (2) S(p }

p+m —E/4

III. REDUCTION TO THREE-DIMENSIONALFORM

In order to examine the nature of this discretespectrum and to consider relativistic effects on theenergy levels of the system confinixl by the three-dimensional logarithmic potential, we study the BSequation in its reduced three-dimensional form.There exist several approaches for reducing the BSequation to a Schrodinger-type equation. We firstnote that the use of the simple instantaneous approx-imation yields in the present case a nonlocal in-tegrodifferential equation in configuration space.Equation (1) then becomes

2 2

p+ — +mP2 p —— +m P(p)

2

d4, P(P') (3)~2 . [(~ ~ i)2]3 2/

where 1 &a &2. The kernel E(p —p') is thus sym-metrized; in particular, for our case, a= —,. Withthe equation written in this form, it is straightfor-ward to note that the second iterated kernelEC2(p p') has—a singularity at p=p' of the form

[(p —p') ] . Thus the iterated kernel K2 for3a= —, behaves as (p —p') near p=p'. We next

consider the integral equation with K2(p,p') as thekernel for a = —, and easily find that

[IC2(p,p', E)—E2(p,p', 0}]

X [E2(p,p', E)—E2(p,p', 0)]

is a Hilbert-Schmidt operator: instead of consider-ing K(E) we need to consider E2(E)—E2(0) in or-der to cancel the large momentum contribution.Thus E2(E)—K2(0) is a completely continuous ker-nel. Hence, following Tiktopoulos, the operator[1—A%2(E)] ' is meromorphic in )(,. We thus con-clude that our equation possesses a discrete spec-trum.

we find that 1((p) satisfies the equation

Q2

f(p ')d p'277 [(p p ')2]&/2($2+m 2)1/2

In coordinate space, Eq. (5) assumes the form

( —V +co )g(r)

A, m Ei(m~

r —r '~

)d3r '

&V(r) g(r ')

4m.ir —r'/

(6)

where we have put ~ =m —E /4.The function Ki(m

~

r —r'~ ) is the modifiedBessel function of the second kind and V(r) is theFourier transform of [(p —p ') ] /, i.e., it is thelogarithmic potential in configuration space. To en-sure that this Schrodinger-type equation which hasemerged from the BS equation leads to mass-independent energy-level splittings, we should startwith an effective BS coupling A, which is of the form

g m where g has the dimensions of energy anddenotes the depth parameter of the resulting loga-rithmic potential.

Equation (6), as it stands, is nonlocal in character.We recover the usual local form with logarithmicpotential if we consider it in the extreme nonrela-tivistic limit. This can be easily seen by chang&ng toa new variable z—:m

~

r —r '~

and then consideringthe limit of large m. The integrodifferential equa-tion (6) does not, however, lend itself easily to prac-tical calculations.

Instead, we show that there exists yet another ef-fective way to convert the covariant BS equation to

27 CONFINEMENT IN A BETHE-SALPETER MODEL WITH. . . 2441

T 2

p+ — +mP2

T

p —— +m P(p)

~ I 0(p')d"p'

(p —p')'

a three-dimensional equation which exhibits rela-tivistic corrections in an interesting way for thepresent problem. To exhibit this method, we firstconsider the reduction of the well-known Wick-Cutkosky equation embodying the co variantCoulomb problem for two scalar particles of equalmass interacting through the exchange of a masslessscalar particle. The relevant Wick-Cutkosky equa-tion is

p ~——p2sin8 cosP,

p2 ——p, sin8 sing,

p3 =pgcosO,

p, =co sinP/(cosha —cosP),

p& ——co sinha /(cosha —cosP ),—m E/4

The new variables vary in the following domains:

0&/&2~, 0&8&m, 0&P&~,and

—oo (0,' ( oo ~

We first transform the equation by using the map-ping'

In the new variables, the Wick-Cutkosky equationtakes the form

da 'df), 4H (a 'P'O'P ')H(a, P, 8,$)=

cosh u —a' —cos0 co cosh u'+4 E sinh a' (9)

where 8 is the angle between the vectors (P', 8', P') and (P,8,$ ) and dQ4 is the four-dimensional solid angle.For going over to the reduced three-dimensional equation in these new variables we first rewrite Eq. (9) by

substituting

4(a,P, 8,$ ) =H(a, P, 8,$ )(co cosh a + —,E sinh a )

to obtain

(co cosh a+ 4E sinh a)4(a, p, 8,$)=2

dQ4da'4(a', P', O', P')

Sm cosh a —a' —cose(10)

The reduction to three-dimensional form now resultsfrom putting a =a' in the right-hand side of thisequation. Consistency then demands, as in Ref. 2,that the left-hand side of the above equation be in-dependent of a, i.e.,

I

values are given by'

A,a

8a~2 1/2

(13)

(co cosh a+ —,E sinh a)4(a,p, 8,$)=p(p, 8,$} .

We therefore obtain for f(p, 8,$) the equation

f(P8$)= 2a I dQq ' '~, (11)g~ (1—cosO~)

where

a—:I da'(co cosh a'+ —,E sinh a') ' . (12)

Equation (11) can be readily compared to theSchrodinger equation for the Coulomb problemtransformed on to the surface of a four-dimensionalsphere displaying the well-known O(4) symmetry.Equation (11) indeed admits O(4) spherical harmon-ics as eigenfunctions and the corresponding eigen-

where C„'" is a Gegenbauer polynomial. Using Eq.(12) and noting that ~ ——me, e being the bindingenergy, we obtain, in the nonrelativistic limit,

1

(n +1)'This is the well-known Balmer formula for the hy-drogen atom whose connection with the O(4) har-monics was pointed out long ago by Fock." Wetherefore conclude that the present prescription forthe reduction of the covariant BS equation to three-dimensional form leads to a consistent nonrelativis-tic limit.

Turning now to the equation of present interest,viz. , (3), under the set of transformations defined in(8) we obtain the analog of Eq. (9),

BISWAS, CHOUDHURY, DATTA, AND GOYAI. 27

da 'dQ 4 ( cosha ' —cosP')H(a ',P ', 8', P'}

H(a, P, 9,$)=[cosh(a —a'}—cos8] (co cosh a'+ —,E sinh2a')

(14)

Under the procedure outlined above for the Wick-Cutkosky equation we obtain from (14) the follow-

ing three-dimensional equation:

P(P, 8,$ ) = (a bco—sP)817 2N

I

ous approximation. To study the confinement spec-trum that results from such a momentum-dependentpotential, we consider the equation for the reducedS-wave radial function, i.e., we put

f(r) = &oo(&)r

d4(1—cosO")

(15) and obtain for R (r) the equation

where a is given by (12) and b is given by00

2 I & 2 ~

b = da'cosha'(co cosha'+ 4E sinh a')

(16)

d2

dr

A,P)1 — V(r) R (r) +co R (r)

2~2

A,cxy

2V(r)R (r);

27T2

We next transform Eq. (15) to three-dimensionalmomentum space by using the variables

p =m tanp/2, p ~——p sin8 cosp, p2 ——p sin9 sing, and

p 3 —p cos8. Further, multiplying both sides of Eq.(15) by sec (P/2) and replacing 1(|(P) byP(p)/cos (P/2) we obtain for P the followingthree-dimensional equation in momentum space:

(p'+co')P(p)=2 (ai+Pip')

2772

further, defining

A,P,X(r)= 1 — V(r) R(r)2772

we obtain for X(r) an equation in which the energydependence of the potential is explicitly exhibited:

d2 g (a ) N p) ) V(r), +co2 X(r)=, , X(r) .

dr 2m 1 —(A,Pi/2~ ) V(r)

where

(t (p ')P [(- -.)2]3n '

CO 1a, =(a b)—,13, =—(a+b)2 2co

(17)(19)

Consider the energy dependence of the parametersappearing in Eq. (19). Since our equation is of theSchrodinger type, it is reasonable to put

The corresponding Schrodinger-type equation inconfiguration space becomes

E=2m+@, e/m «1so that

(20)

A,Q)( —V +m')g(r)= V(r)g(r)

2~2

2V [V(r)g(r)], (18)

IV. THE S-WAVE SPECTRUM

Equation (18) is a Schrodinger-type equation witha momentum-dependent potential. This is to becontrasted with the nonlocal equation (6) which re-sulted from the same BS equation in the instantane-

where V(r) = 4nln(r/ro) —is th. e logarithmic poten-tial obtained from the Fourier transform of[(p —p ') ] !and ro is an arbitrary scale parame-ter. Equation (18) is the resultant Schrodinger-typethree-dimensional equation which emerges from thecovariant BS eqaution in the present approximationscheme and forms the basis of our detailed discus-sion of the confinement spectrum;

co =m E /4= —me —.2 — 2 2 (21)

b=~ (~2+E2/4)1/2

Thus in the domain where (20) and (21) are valid,

a=2/m, b=m —mE

and using Eq. (17),

Further, Eqs. (12), (16), and (17) show that theparameters a

~and P~ are energy dependent. The in-

tegrals in Eqs. (12) and (16) defining a and b may beevaluated exactly and the results are

1 2~(2m —E ) —Aa= — ln,

—,(2m —E )+A1/2

(2m —E ) E44.

27 CONFINEMENT IN A BETHE-SALPETER MODEL WITH. . . 2443

at- —n. /2m, Pt- n—./2m e2 (22) where r„ is given by

Consider next the sign of A, required for confine-ment. From (22) we find that for large e (thoughstill small compared to m) at «Pt. Thus in thislimit we have, from Eq. (19),

1.e.,

1+(g /2e)ln(r„/ro) =0,

r~ =roexp( —2e/g ), E))'g

(25)

d A V(r)2 + X=mEL ~dr' 4am

Since V(r) = 4n. ln—(r/ro), confinement requiresthat A, &0. Recalling that A, is of the form g m, weput

A, /2n = gm—/4n, g &0,having the dimensions of mass, typically, 1 GeV.Using (20) and (22) we obtain from (19) the approxi-mate equation for the S-wave levels in the limit ofsmall e/m:

d'X (g'm/4~) V(r)dr' 1 —(g'/8m. e) V(r)

(23)

We note that even though we are considering the en-

ergy levels of the system for small e lm, theSchrodinger equation with the logarithmic potentialyields e-klnn, n being the excitation quantumnumber; thus for sufficiently large n, g /e is asymp-totically small. To perform a WKB analysis of Eq.(23) we write it in the form

Similarly, r, is given by the equation

g min(r, /ro)=mE' .

1+(g Ie)ln(r, lro)(26)

The WKB quantization condition now yields

roexp(2e/g ) (g /2) in(r/ro)2 , drv'2m

roexp( —2E/g ) 1+(g /2e)ln(r/ro)

1/2

1.e.,

0 3/2 2z' exp[( —2e/g )z]

e exp(2e/g ) dz, /2g 0 (2 z )1/2

=nb .

nh2

e' exp(2e/g ) (g /2e) I'( —,),

Extending the upper limit of the integral withoutbound, we have

+ V(r)X =meX,dp

(24)1.e.,

e=—,g inn+1

r &r&r, ,

]~

V (r)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

FlG. l. Classical turning points for the potentials P(r).

where the new "potential" V(r) has the structureshown in Fig. 1, V(0)=2m@ &0. The classicalmotion for given e is then confined between

2v m rog exp(e, /g )1"(—, ) =nh, (29)

1.e.,

To examine whether any relativistic effects are to befound in the spectrum (27), we examine;, theSchrodinger equation with a logarithmic potentialwhich results from Eq. (23) in the limit of small

g /a. The corresponding equation then is

+s gm V(r)X, =meX, , (28)dr 2 4~

where we have labeled the Schrodinger wave func-tion X, and where, as before, V(r) = —4m ln(r Iro) asis shown in Fig. 2.

Performing an analogous WKB analysis, we have

rt'=roexp(e, lg )2

and the WKB quantization condition now givesTIS

2 I v'2m [ , e, ——,g ln—(rlro)]'/dr=nb .

The integral is evaluated by elementary means toyield

2AAA HIS%'AS, CHOUDHURY, DATTA, AND GOYAL

g mln-r

FIG. 2. Classical turning points for the potentialln(r /ro).

s=g inn+

Comparing (27) and (29), we find that relativistic ef-fects depress the energy levels while retaining theirgeneral character; this has been noted in similar situ-ations elsewhere. '

V. CONCLUSIONS

We have shown that the BS equation for twoscalar particles of equal mass interacting through a

kernel which in momentum space behaves like

[(p —p') ] ~ and is the covariant generalization ofthe logarithmic confinement potential possesses adiscrete spectrum. A new procedure for the reduc-tion of the equation to three-dimensional formwhich, in the case of the Wick-Cutkosky equationfor the relativistic Coulomb problem leads to thecorrect Balmer formula, yields a Schrodinger-typeequation with a momentum-dependent potential. A%KB analysis of the equation shows that the bind-ing energy e-g inn, g being the dimensional cou-pling with the dimensions of energy; relativistic ef-fects depress the energy levels in relation to those inthe Schrodinger theory, though in our approxima-tion scheme their general character remains unal-tered. Finally, we observe that the discrete nature ofthe spectrum which appears in the present case can-not be immediately generalized to the case of thelinear confining potential. The regulated form ofthe kernel for such a potential is, as given byPagels, ' of the form

&(P P') -'9 (P P') +"—

with g ultimately tending to zero. If we iterate thiskernel, we see that the number of iterations requiredto reach a Hilbert-Schmidt kernel is proportional toI/ri. Since rl tends to zero, we can no longer con-clude that the kernel in question possesses a discretespectrum.

S. N. Biswas, S. R. Choudhury, K. Datta, and AshokGoyal, Phys. Rev. D 26, 1983 (1982).

S. N. Biswas, K. Datta, and Ashok Goyal, Phys. Rev. D25, 2199 (1982).

3See, for instance, C. Quigg, Report No. Fermilab Conf-

81/78, 1981 (unpublished).~N. I. Akhiezer and I. M. Glazman, Theory of Linear

Operators in Hilbert Space (Ungar, New York, 1961),p.57. See also R. Courant and D. Hilbert, Methods ofMathematica/ Physics (Wiley, New York, 1966), Vol. 1,p. 112.

5G. Tiktopoulos, Phys. Rev. 133, B1231 (1964).G. Tiktopoulos, J. Math. Phys. 6, 573 (1965);J. R. Hen-

ley, Phys. Rev. D 20, 2532 (1979);J. S. Goldstein, Phys.Rev. 91, 1516 (1953). See Ref. 2 for other references.

7See Ref. 2.G. C. Wick, Phys. Rev. 96, 1124 (1954); R. E. Cutkosky,

ibid. 96, 1135 (1954).9H. S. Green, Nuovo Cimento 5, 866 (1957).DA. Erdelyi, Math. Ann. 115,456 (1938).

H. Pagels, Phys. Rev. D 15, 2991 (1977)~