The θ-model in quantum corrections to classical confinement

Nuclear Physics B131 (1977) 421-428 © North-Holland Publishing Company

T H E 0 - M O D E L IN QUANTUM CORRECTIONS

TO CLASSICAL CONFINEMENT Bo-Sture SKAGERSTAM * Physics Department, Syracuse University, Syracuse, New York 13210, USA ** Received 20 October 1976 (Revised 19 September 1977)

We consider a generalization of the 0-model due to Chodos and show that the property of classical permanent confinement is absent after quantization.

1. I n t r o d u c t i o n

Recently we discussed some questions related to the quantum analogue of classical permanent confinement [1 ] based on arguments due to Chodos and Klein [2]. It was argued that the q u a n t u m fluctuations are so severe that they destroy the original confinement and one is left with a free q u a n t u m field theory or a theory with no ground state. The analysis was based on the fact that a normal ordering is sufficient in order to remove all ultraviolet divergences in two-dimensional polynomial interactions [3]. As was shown in [4] the equation of motions describing the MIT bag Can be obtained from a theory in which the corresponding fields acquire an infinite mass outside the bag. Chodos has now suggested a concrete model, the 0-model [5], which has the same feature of realizing the MIT-bag structure in the limit when the mass of the field outside the bag tends to infinity. The action for a charged scalar field ~b(x) takes the following form S = f f d t d x { O u ~tOu(~ _ B + (B - m2~tq~)O(B

-

ol2~)~q~)} ,

(1)

where 0 (.) is the Heavside step function i.e. 1 if X > 0 O(x) :

(2) 0 if x ~ < 0 .

* Permanent address: Institute of Theoretical Physics, Fack, S-402 20 G6teborg, Sweden. ** Work supported by the Swedish Natural Science Research Council, contract 8244-008. 421

422

B.-S. Skagerstam / Quantum corrections to classical confinement

Compared to the original action of the MIT-bag (1) enjoys the advantage of not imposing complicated constrains on the field ~(x) (the vanishing outside the bag). This fact enables one to consider the quantization in a conventional manner and in [2] it was shown that (1) gives rise to a trivial physical theory. In the present paper we shall consider generalizations of the action (1) and show that the same feature remains i.e. the confinement property of (1), in the limit of large m, does not remain after quantization. This suggest that the result mentioned above may be a very general property of local quantum field theories, at least in two dimensions.

2. The classical action In (1) one considers the function (B - m2q~t~b) O(B - m2~btq~),

(3)

which gives rise to the equation of motion [i]0 = - r n a c~O(B - m2~tqs).

(4)

In the limit m 2 -+ oo this equation reduces to the equation of motion of a free scalar field, which is supposed to describe the physics inside the bag. We would now like to consider a generalization of the function (3) which still contains a reference to the boundary of the bag namely where B = rn2dpt(x) dp(x). The function (3) is continuous at the surface of the bag but has a discontinuous first derivative. This derivative can be made continuous by considering the follow-

(B-m2~t(x)@(x))o(B-m2¢t(x)@(x))

~ide

•

thebag !I

Insidethebag

",3B=m2**(x),(x) Fig. 1.

m2¢t(x)¢(x)

B.-S. Skagerstam / Quantum corrections to classical confinement ing extension of

423

(3)

xOo~(x ) def. xO(x)(1 - e x p ( - a x ) ) ,

(5)

where a is some large positive real number. Physically this corresponds to a situation where the surface of the bag is described not by a step function but a function which tends to the O-model value for large values of the parameter a (see fig. 1). In other words the transition between the inside and outside of the bag is not sharp (compare [2]). We therefore consider the following classical action S = f f d t d x { O u ~bt0uq~ - B + (B - m2¢t~b) O~(B - m2~btq~)}.

(6)

Below we will consider more general regularizations of the function (3). The equations of motion will, of course, be more complicated but inside the bag we still have a massless boson field. In the limit of large massparameter m 2 we therefore do not expect any drastic change of the physics as compared to the 0-model. We are now going to prove that this indeed is the case and that after renormalization only a free field theory can remain.

3. The renormalization We will consider the relativistic quantum fields in one space and one time dimension only and the field ~bis supposed to be a charged scalar one. In unrenormalized parameters the Hamiltonian density, corresponding to the action given by eq. (6), takes the following form H ( x ) : H o ( x ) + B - (B - m2ckt (x) ok(x)) O~(B - m 2 o t (x) ok(x)),

(7)

where

Ho(x) = nt(x) n(x) + o~t(x) a~(x) ~5-

ax

(8)

is the free field part of the Hamiltonian. The ultraviolet divergences in (8) can be removed by a simple normal ordering, defined in the Schr6dinger picture (see [3]). In order to derive a closed expression of the normal ordered Hamiltonian density we consider the spectral resolution of the function (5) 1 ~ exp(-iXx) Oa(x) = - 2-n a f (k + ie)(k + ia) d ~ , where the limit e $ 0 is to be understood after the integration procedure.

(9)

B..S. Skagerstam / Quantum corrections to classicalconfinement

424

Furthermore we notice that by using the exponentiated form of Wick's theorem

[31

exp(i~qi)

=

,n~ A2 ]

exp(i~q~),

(10)

the following expression can easily be derived [ 1,2]" exp(6¢*¢)

(/3/4n)

= 1 -

l n ( A 2 / m 2)

((~/4~)~(A2/m2)]'

Nm e x p 1 -

(1 1)

where 13is a complex number and Nm(') denotes the normal ordering indexed by a parameter m. The interaction term in (7) can therefore be written in the form HI(X) = - ( B - mZ~bt(x) O(x)) Oa(B - m2(b(x) (2(x)) io~ - 2nn - f

e x p ( - i k B ) exp(iXm2q~t(x) ~b(x))

_oo

(

1

1

)

× (X + ie)2(~ + ia) + (~ + ie)(k + ia) 2 d k ,

(12)

where the normal ordering now can be performed explicitly. We obtain the following expression o0

HI(x) = ~

f

exp(-ixe) (

1

(1 - ikm~) \ ( k + ie)z(k + i(x)

_oo

+(A+ie)~)~+io0

N m exp\

1-im~)~

]

'

(13)

where def. m 2 A2 rn~ = ~ - n l n my

(14)

is a renormalized mass parameter. In order to evaluate the contour integrals in (13) it should be remembered that the exponential in (13) should be interpreted as the normal ordering of a power series. The normal ordered Hamiltonian density can therefore be written in the form

H(x) : NmHo(x ) + ~ rt=l

anNm(~fl(x ) 4)(x)) n + ~ rt=l

bnNm(~bt(x) ~b(x))n

B.-S. Skagerstam / Quantum corrections to classical confinement exp(-o.B) ( a m 2 0 " ~ ( x ) c)(x)] ^ + a(o~m2 - 1) Nm exp noa,~ -- 1 / + B R + Y ,

425

(15)

where 3' is a divergent constant which can be put equal to zero and ^ def. B exp(--aRBR) BR = a R -- 1

(16)

is a renormalized bag constant. In eq. (16) we introduced the notation a R = am~ ,

(17)

BR = B / m ~ .

(18)

We notice in passing that MR vanishes in the limit aR -+ oo as it should (compare [2]). It is furthermore not difficult to obtain closed expressions for the coefficients an and bn. In fact we obtain the following expressions

an

_ (_l)n+l (m2)n = e x p ( _ i ~ B a ) ~n 2nnI (m2)n_ 1 aR f (~ + ie)2(~ + iotR)(i + ~j)n+l d~,

(19)

( - 1 ) n+l (m2) n ~ e x p ( - i ~ B R ) ~n 2rrn.~ (rn~)n_ i OeR f (~+ ie)(~ + i0~R)2(i+ ~)n+l d~.

(20)

bn =

In (19) only the pole contributions from ~ = - i e and ~ = - i should be taken into account and in (20) the pole contributions from ~ = - i a R and ~ = - i . The contributions from the other poles have already been included in the Hamiltonian density (15). The first step in the renormalization programme is to consider the coefficients in front of the quadratic interactions in (15) and render it finite by a suitable choice of renormalized parameters. We obtain from the expressions (19) and (20) al=m 2

m20~Rexp(--BR)( ~R ~ I+BR

bl = ap.rn2(exp(-BR) + exp(--0~RBR))

1 ) aR- I '

aR

1) =

(21)

(a R

1) 3 .

(22)

Extracting the mass term of the Gaussian interaction in (15) and collecting all appropriate coefficients, we see that the effective mass can be made finite if we chose c~R and BR as finite parameters. BR will then be divergent and must be subtracted from the Hamiltonian. Eqs. (19) and (20) then show, however, that all higher order A

426

B.-S. Skagerstam / Quantum corrections to classical confinement

(B-m2+t(x)+(x))@reg(B-m2~t(x)#(x)) I I L-_

I I I I I I

B

Inside the bag

I

I I I I

m2#t(x)f(x)

B:m2+t(x)~(x) Fig. 2. interaction terms vanish. This means that after the renormalization mentioned above only a free quantum field theory remains. Let us now consider regularizations of the function (3), XOreg(X),which e.g. have continuous derivatives at x = 0, the surface of the bag (see fig. 2). We notice that (B m2qSt~b) Oreg(B m2q~t~b) is a positive bounded function and that it vanishes for field values such that m24~t~b >~B. In the case of a charged scalar field we can always, in a conventional manner, introduce two independent scalar fields, q5l(x) and ¢2(x), such that [6] -

-

¢(x) -

-

~bl (x) + i~b2(x) x/2

(23)

The expression (B - m 2 ¢ t ¢ ) ®reg(B - m2q~t¢) can therefore regarded as a function of the independent variables ¢1 and ¢2 and quite generally be written as a Fourier transform [7]" (B - m2~bt¢) Oreg(B - rn2¢t~b) = B f

f dP(~l, ~2) exp(i~lqS1 + i~24~2), (24)

where d/a(~l, ~2) = f(~l, ~2) d~l d~2 is a bounded positive measure and normalized to one./2(~1, ~2) can depend on B and m 2 only in the combination B/m 2 due to dimensional reasons. We are going to perform a simple variational computation, in the Schr6dinger picture, i.e. we consider fixed-time n- and q~-fields. We can then make use of (10) and apart from a divergent additive constant we obtain H(x) = Nm(Ho(X ) + B)

(

-

gNmf f

d/a(~ 1,

X exp - ~ - ~ ( ~ l + ~ ) l n ~ m 2 ] ] exp(i~l¢l(X) + i~2q~2(x)) •

(25)

B.-S. Skagerstam / Quantum corrections to classical confinement

427

We now consider two cases: (i) The dependence of the cutoff can be absorbed into the parameters B and rn2 in such a way that the renormalized Hamiltonian density takes the following form (compare e.g. the Gaussian model in [1,2]): H(x) = N m ( g o ( x ) + B R - BR f f

ditR(~l, ~2)

X exp(i ~1~bl(x) + i~2 ¢2 (x))).

(26)

In (26) BR is a renormalized bag constant and the measure ItR(~l, ~2) can depend on both B R and a renormalized mass parameter, mR, through the ratio BR/m~t. Following Coleman [3] we now consider trial states 10, It) such that an(k, It) 10, It) = 0 for n = 1,2 and compute the expectation value of the energy density. By making use of the expression (10) we easily obtain =

-

+ B.

X

-

oRff

ditR(

l,

exp()~l~b I (x) + i~2~b2(x)),

(27)

i.e. 1

(0, It lH(x) l O, It) --- ~ (it: - m 2) + BR

- BR ffdit.(

l,

)expt

In

(28)

Since the measure ItR(81, ~2) is positive and bounded and It a parameter which can be freely varied the Hamiltonian is not bounded from below. The important point here is that the measure ItR(~l, 82) contains points for which 8] + 85 > 87r [8]. On the other hand this follows from the representation (24) since the function X~reg(X) vanishes outside a bounded region of space [7]. We conclude that the renormalization prescription (26) leads to an unphysical theory. (ii) The coefficient in front of Nm (~bt'~b)in a series expansion of (25) should be finite. Here we notice that the measure It(~, 82) can be written in the form dit(~" ~ 2 ) = ~ f ( 7

~1,~2)d~ld~2,

(29)

which follows from the fact that the representation (24) is invariant under a scale transformation ~b~ a¢ and m -+ c~-lm. We can now easily compute the coefficient in front of Nm(q~*0): B _B~ ffd~l d~2f(~B ~1,_~_ ~2) ( ~ + ~ ) exp(_ 1

2

428

B.-S. Skagerstam / Quantum corrections to classical confinement

This factor can be rendered finite by a renormalization procedure where m and B - 8n l n - l ( A 2 / m 2) are kept finite. Higher order terms in ~bt~bin the expansion of (1 1) will therefore vanish when the cutoff A tends to infinity.

4. Conclusions In a simple extension of the technique used in [2] we have proved that q u a n t u m fluctuations in the 0-model destroys the feature of classical permanent confinement of the charged scalar field. Even if the surface of the bag is perfectly smooth we have shown that a non-trivial spectrum cannot arise from a confinement mechanism as suggested in the 0-model. The q u a n t u m fluctuations are so severe that the classical bag-structure completely disappears.

References [1] [21 [3] [4] [5] [6] [7] [8]

B. Skagerstam, Math. Phys. Lett. 1 (1977) 499. A. Chodos and A. Klein, Phys. Rev. D14 (1976) 1663. S. Coleman, Phys. Rev. Dll (1975) 2088. A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471. A. Chodos, Phys. Rev. D12 (1975) 2397. J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill,New York, 1965). M. Reed and B. Simon, Fourier analysis, self-adjointness (Academic Press, New York, 1975). B. Skagerstam, Phys. Rev. D13 (1976) 2827.