Particle creation in colour-electric fields

Volume 113B, number 4

PHYSICS LETTERS

24 June 1982

PARTICLE CREATION IN COLOUR-ELECTRIC FIELDS ¢r Jan AMBJORN and Richard J. HUGHES California Institute o f Technology, Pasadena, CA 91125, USA

Received 23 February 1982

The decay of the Yang-Mills vacuum in a uniform colour-electric field is calculated using the method of Bogoliubov transformations. The result does not agree with that obtained by summation of the corresponding perturbation series.

1. lntroductior~ Recently, it has become popular

to study Yang-Mills theory in the presence of a classical background field in' the hope of obtaining information about the ground state. The first investigation of this type in a uniform, "abelian-like" background field was performed by Saviddy et al. [1], who calculated the one-loop effective action. They found that as a consequence of the asymptotic freedom of the theory, this quantity developed a non-trivial minimum in an external colour-magnetic field. Since then the constant field configurations, and modifications of these, have beeff extensively studied in a number of contexts [ 2 - 6 ] , but the imaginary part of the effective action has been a continual source of controversy. In the Schwinger proper-time [7] representation of the causal propagator in the external field the differences between the results obtained may be expressed in terms of the different paths used to perform the proper-time integration. However, when it is possible to define asymptotic in- and out-states there is a unique relation between the choice of such states and the causal propagator, Therefore if one can fred a "physical" criterion for the choice of asymptotic states, the causal propagator, and hence the path to choose in its proper-time representation are determined. For scalar and spinor QED such a criterion is well known in the presence of a uniform external electric

field. The asymptotic in- (respectively out-) states are defined as the ones behaving semi-classically (JWKBlike) in the infinite past (respectively future). We show that this definition may be immediately generalized to gluons propagating in a constant colour-electric field. With this definition of in- and out-states we are able to calculate the decay rate of the vacuum using the technique of Bogoliubov transformations. The field theory thereby constructed is a non-trivial example of (a linearized version of) Yang-MiUs theory canonically quantized in a covariant gauge [8]. A feature of this construction is that physical states are required to satisfy a subsidiary condition analogous to that of covariant gauge QED [9]. Our method is consistent with this condition. 2. Vacuum decay in scalar QED. In this section we show how to define appropriate in- and out-states for a charged scalar field in a constant external electric field. This was first done by Nikishov [7] (for spinor QED) and subsequently generalized by Rumpf and others [10,12]. We reproduce this example here as it will facilitate the generalization to charged spin-one fields. The equation of motion for a charged scalar field, ~, in an external electromagnetic field is

W2 +

= o,

(aa)

D u = 3u - igA u.

(lb)

where Work supported in part by the US Department of Energy under Contract No. DE-AC-03-81ER40050.

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If we choose the potential

A u = -Su3Et ,

(2)

as representing the constant electric field, then we may choose the solutions of eq. (1) to be

24 June 1982

wavefunctions at t ~ _0% +9p, contain both positive and negative "frequencies" at t ~ +~. The two bases are related by Bogoliubov coefficients [14], +

9p

=Ap+gp +Bp ~ p ,

-+gp(t, x) = Np {exp0 p.x)/[(2n) 3 ] 1/2 }

× D_ (1/2T.ia)(exp(+-irr/4)~),

Oa)

- % = Cp +gp +Dp _9p, (7a, b)

which are given by Ap = Dp = [1 + exp(2na)] 1/2

or, ±gp(t, x) = Np {exp(ip'x)/[(2n) 3 ] 1/2}

× exp (-i[¼n + arg P(71 - ia)] },

× D_ (1/2-+ia) (-exp(T-iTr/4)~),

(3b)

where a = - ( p 2 +m2)/2gE,

~ = (2gE)l/2(t +p3/gE) '

Oc, d) Np ; exP(¼m)/(2gE) 1/4, p ~- (Pl'P2'P3)"

Oe' f)

The solutions ±9p will be referred to as outgoing positive- or negative-"frequency" wavefunctions [I0, 13], respectively, because they behave semiclassically (JWKB-like) when t -+ 0% i.e. the asymptotic behaviour of ±9p is given by

lira -gp + ~ PO(t) exp[i Sp(±)(t)].

(4)

(8a)

Bp = Cp = i exp(rta).

(Sb)

The expansion coefficients of the field operator, • (t, x), in terms of the in- (respectively out-) bases become in- (respectively out-) creation or annihilation operators satisfying the usual canonical commutation relations. In- and out-vacuum states can be defined as in the free field case, and the unitary operator, U, connecting the in- and out-states can be constructed:

p

{ P 9p + b ~ p - % } = ~ (O~p+9p P

[Oou t ) = U - 1 ]Oin),

ap = U - 1ap U.

_p

(9a) (9b, c)

t - - + oo

where S(p+-)(t)is the classical action (or equivalently, the Hamilton-Jacobi function) of a massive charged particle, t S(p~)(t)=xt • f at' H(p, t'), (5) the hamfltonian is

H(p, t) = [(p -

gA)2 +m21 1/2,

(6)

and Po(t) is a slowly varying function. Similar statements may be made for the incoming wavefunctions ±~p, in the limit t ~ ,oo. In this way we obtain two sets of wavefunctions which are complete and orthonormal with respect to the (indefinite) scalar product of the Klein-Gordon equation (1). The solutions +9 and +9 have norm = +1, while - 9 and _ 9 have norm = - 1 . A field of the type that we are considering produces pairs, and this process manifests itself in the time evolution of the wavefunctions. The positive "frequency" 306

and similarly for the other creation and annihilation operators. (The explicit form of U is given in ref. [15] .) For the vacuum-to-vacuum amplitude we find

(OoutlOin)=(Oin[U]Oin)=exp(~lnDp). ,,p

(9d)

Insertion of the expressions (8) for the Bogoliubov coefficients into eq. (9d) gives the well-known vacuum persistence amplitude in an external electric field. Our knowledge of the in- and out-states makes it trivial to write down the causal propagator in the Feynman representation:

iG(x, x') = O(t - t') ~

p

(+9p(X)Ap I +~p(X')}

----

F

+ @(t' - t) ~ ( ~p(X)Dp 1 9p(X )}(10) P This propagator has the property that it propagates out-going "positive frequency" modes to the future, and in-coming "negative frequency" modes to the past.

PHYSICS LETTERS

V o l u m e 113B, n u m b e r 4

It is simple to show that [ 11 ]

(OoutlT(q,(x)rb t (x ,))lOin>

ic(x,x')=--

(ll)

<0 out IOin)

By using the explicit form of the wave-functions and Bogoliubov coefficients we may write this propagator as iG(x, x') =

[exp(iTr/4)/(47rgE) 1/2]

.. d3p XJ ~ exp [ip "(x -x')l Ja(~, ~'),

(12a)

where J a ( ~, ~') = I~(~ - i a ) [ 0 (~ -

~')D(1/2_ia)(ebr[4~)

X D_(1/2_ia)(-ei~/4~ ') + ®(~' - ~) X D_ (1/2- ia) (-ei~rl4~)o-

(12b)

(1/2-ia)( ei~rl4~')]"

J ( ~ , ~') has the following integral representation [10]

f ds (si 2s)-1/2 e~0+ X exp {i {2as

0 1 - -g [(~ + ~')2tanh s

+ [(~ _ ~')2 _ ie] coth s} }.

(13)

When We use the representation (13) in eq. (12a), the integral can be performed, and we get the usual Schwinger proper-time representation of the Feymnan propagator [7]. We wish to emphasize that if we de-

Jm S

:-ReS

COMPLEX "PROPER -TIME" PLANE

24 J u n e

1982

form the integration contour in eq. (13) at s -~ °°we only get a representation OfJa(~, ~') when Isl ~ ~oin the fourth quadrant of the complex plane. In the next section we will use eq. (13) for a's having Im a = - 1 . We are then forced to choose a contour, C, of the form shown in fig. 1, for small Re a,

3. Yang-Mills fields. In this section we will study Yang-MiUs theory in a constant, external colour-electric field, for which it is convenient to choose a ("covariant") background gauge fixing condition [16]. A particular feature of this gauge is that it forces the Yang-MiUs theory to look like the electrodynamics of charged, massless, spin-one fields [17,18], and therefore encourages comparison with the scalar QED problem of section 2. However, another consequence of this choice is that the Fock space constructed by the canonical quantization procedure contains ("unphysical") states of negative norm. In order for the theory to be physically sensible one must show that it is possible to project out a Hilbert space of physical states by the imposition of a suitable subsidiary condition. Recently, it has been shown how to do this for the case of Yang-Mills theory quantized in the usual covariant gauges [8], and this procedure may be readily generalized to background gauges [15]. Once we have acquired control over the Hilbert space of states in this way, we may generalize the method of section 2 to the Yang-MiUs problem. In- and out-states can be defined in the Fock space, and the U-operator which connects the in- and out-states becomes a unitary operator on the physical Hilbert space. Consequently, I(Oout JOin) I ~< 1, and in fact, we fred the decay probability to be twice that of a single massless, charged scalar field. When we work in the external field approximation, the Yang-Mills lagrangian in the presence of an external, "abelian-like", colour-electric field pointing in the three-direction of the Lie algebra (say), is identical with that of a massless, charged spin-one field, with a dipole moment corresponding to a gyromagnetic ratio of two [18]. If we make the change of variable At~ + au ~ A 3 ' 4]g

=2-1/2(A1 - kt _ i A 2 ) ,

(14a)

W*_~2-1/2(A1 /z - g +iA2), /z(14b, c)

Fig. 1.

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where A u is the potential representing the external electric field of eq. (2), the quantum fluctuations a, play no part in the approximation to which we work. The lagrangian for the W, fields is then L =-

ID w - D

12 - i g F y * W

The basis functions may be expressed in terms of a basis of the Klein-Gordon equation: - t h e in- or outbasis of section 2, for example. This is trivial for the ghost fields, ~ and 7- For the charged vector field, Wu, we may choose f.",,; = q'~p,

+ i[(D#r~)tDU~ + D r~(D~)*],

6Wu = XDu~,

8wut= X(Dt,~)t,

(17a)

87 = XDuWU,

8rl? = X(DuWU)?,

(17b)

where X? = -X, is all that remains of the BRS invariance [19] of the full lagrangian +1. The corresponding conserved BRS-charge is QB = f d3x [(Dn Wit)? 80 ~ - ~? ~0(Ou WU)]"

(18)

This charge is nilpotent, Q2 = 0, and the physical Hilbert space is projected out by the subsidiary condition QB [physical> = 0.

(t9)

We may now perform the standard canonical quantization by defining a conserved scalar product, and expanding our fields in terms of a set of basis functions with operator-valued expansion coefficients. As a scalar product we take

(r;).

(fn,g~) = i f d3x f;~o(g~).

(20a) (20b)

Here f~o, fn' f~ are solutions of eqs. (16a, b, c), respectively. 4:1 Details willbe published elsewhere [15]. 308

(21a)

f~,p = Ne~(D 3 ++-D0)~0p,

(21b)

where (e+)u = (-+l/x/~, 0, 0, 1/x/~), Cp are the in- or out-basis-functions of section 2 with rn 2 = 0, and the orthogonality properties are


i= 1,2,

= 0,

< f + , f ) = +1.

(22a)

D2~=0.

(16a, b, c) The invariance of L (up to a total divergence)under the infinitesimal transformations

>=

i = 1, 2,

(15)

where Fur and Dr, are the field strength and covariant derivative corresponding to the abelian field of eq. (2), and ~, ~ are anticommuting ghost fields with opposite ghost number. The equations of motion are ( D 2 ~ - 2 i g F ~ ) W v = 0 , D2~=0,

24 June 1982

(22b, c)

The choice of ~p automatically defines an in- and out-basis for the ghost and vector fields, and it is clear that the f ~ have JWKB-like behaviour in the infinite past and future. The Fock space constructed by acting on the vacuum with the various creation operators contains negative and zero norm states. However, the subspace satisfying the subsidiary condition (19) (the "physical" Fock space) is the direct sum of a positive norm subspace, H, (the physical Hilbert space) and a subspace consisting of zero norm states orthogonal to any state satisfying (19). This zero norm subspace can therefore be factored out in the usual way. The U-operator which connects the in- and out-states can be constructed as in the scalar case (the Bogoliubov coefficients are the same), and we get [15] [(Oout (scalar)l Oin(scalar ))14 I
.

(23)

I[2 In this equation the power of four on the right-hand side arises with the four polarization states which the vector field, Wu, has in the covariant gauge, while the power of (-2) comes from the two ghost fields, which have Fermi statistics. Thus we find a decay rate twice as large as that of a massless scalar field. This is physicall3/reasonable as the subsidiary condition of eq. (19) defines a Hilbert space which is essentially that of two positive metric, massless charged particles. To see this we may quantize the vector field in a new basis, h~, a = 1, 2, +, the h~ being linear combinations of the f~. They satisfy

Volume 113B, number 4

(ha, h #) = (fa,f/~),

h~=D/~p,

PHYSICS LETTERS D# h"a = 8 a _ ¢ p . (24a, b, c)

Ttfis basis is the direct analogue of the free photon one, with h~+ playing the role of the longitudinal polarization state, e~ ~ kU. The BRS-charge, (18), becomes a simple generalization of that of covariant gauge QED, where the ghost fields are free [15] iQB = ~

p

{[a~_p(~)ap(h_) -ap(~)at_p(h)]

+ [b~_p(~)bp(h) - bp(~)b?_p(hll }.

(25)

Here, at (~), b ? (/l _) create a ~-ghost particle and an h_ vector antiparticle, respectively. Similar definitions apply to the other operators in this equation. In covariant gauge QED one can restrict oneself to the zero-ghost sector [8]. However, in our case ghosts and h e particles are created (the total ghost-number is conserved), but because ~Bn°ut= ~rr-1 ~Bt3in/-rvt[= QiBn, as can easily be checked) the physical in-Fock space is mapped onto the physical out-Fock space, and S-matrix elements depend only on the h~, 2 particles. Eq. (23) shows this in the simplest case. The last question which we wish to address is the relation of our method to the Schwinger proper-time representation. The causal propagator can be constructed, as in section 2, in terms of the f ~ or h~. Because

(+-)fu+_~ e~ D_(1/2_ia+)(eeiTr/4~) e ip'x,

(26)

where

a± = +-i + a,

a = - ( p ~ + p~)/2gE,

(27a, b)

we see that our causal propagator may be written in the Schwinger proper-time representation provided we choose the path shown in fig. 1 in the complex propertime plane. This path is the one chosen in ref. [3], and the result of that calculation is in agreement with ours.

4. Discussion and conclusions. In this letter we have derived the vacuum persistence amplitude of Y a n g Mills theory in a constant, external colour-electric field by defining particle and anti-particle modes in the presence of the field, and canonically quantizing the resulting field theory. The result obtained is in agreement with that of ref. [3], but disagrees with our previous calculation [4].

24 June 1982

We believe that this discrepancy arises with the different "physical" situations underlying the two methods of calculation. In the present case we have defined our field theory in the presence of the electric field. Conversely, in ref. [4], we implicitly assumed that the effective lagrangian was obtained by summation of the perturbation series for the effective lagrangian with an adiabatically switched coupling constant ("ie"-prescription). Contrary to the analogous situation in scalar QED, these two methods give different results. One may well ask which of our two results is "correct". However, one should bear in mind that neither approach addresses the physical nature of the sources which support the external field. Indeed, as remarked in ref. [4], this type of calculation ignores the coupling of quantum fluctuations to these sources and thereby introduces a violation of gauge invariance. In that case we found a negative decay rate. In the present case we have been careful to maintain BRS-invarlance throughout the calculation, and thereby we have ensured that we find a positive ("physical") decay rate. It is a pleasure to thank K. Johnson for several helpful conversations.

R eferen oes [1] G.K. Sawidy, Phys. Lett. 71B (1977) 133; S.G. Matinyan and G.K. Sawidy, Nuel. Phys. B134 (1978) 539; I.A. Batalin, S.G. Matinyan and G.K. Sawidy, Yad. Fiz. 26 (1977) 407 [Soy. J. Nuel. Phys. 26 (1977) 214] ; H. Pagels and E. Tomboulis, Nucl. Phys. B143 (1978) 485. [2] J. Ambjern and P. Olesen, Nucl. Phys. B170 (1980) 60, and references therein; H. Leutwyler, Nucl. Phys. B179 (1981) 129; LS. Brown and W.I. Weisberger, Nucl. Phys. B157 (1979) 285. [3] A. Yfldiz and P.H. Cox, Phys. Rev. D21 (1980) 1095. [4] J. Ambjern and R.J. Hughes, K~llen's theorem and the unphysical nature of abe/Jan external fields in YangMills theory, Caltech preprint CALT-68-849 (1981), to be published in Nucl. Phys. B. [5] B.J. Harrington and C.H. Tabb, Phys. Rev. D22 (1980) 3049. [6] V. Sehanbacher, Gluon propagator and effective lagrangian in quantum chromodynamics, Tubingen preprint (1980). [7] J. Schwinger, Phys. Rev. 82 (1951) 664. [8] T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66 (1979), and references therein.

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[9] S.N. Gupta, P~oc. Phys. Soe. A63 (1950) 681 ; K. Bleuler, Helv. Phys. Acta 23 (1950) 567; B. Lautrup, Kgl. Dan. Vid. Sel. Mat.-Fys. Medd. 35 no. 11 (1967) 1; N. Nakanishi, Prog. Theor. Phys. 35 (1966) 1111. [10] A.I. Nikishov~ Soy. Phys. JETP 30 (1970) 660. [11 ] H. Rumpf, Phys. Lett. 61B (1976) 272; Nuovo Cimento 35B (1976) 321; H. Rumpf and H.K. Urbantke, Ann. Phys. 114 (1978) 332. [12] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Soy. Phys. JETP 41 (1975) 191 ; D.M. Gitman, J. Phys. A10 (1977) 2007.

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[13] N.M.J. Woodhouse, Phys. Rev. Lett. 36 (1976) 999; D.J. Simms and N.M.J. Woodhouse, Lectures on geometric quantization (Springer, Berlin). [14] N.N. Bogolinbov, Soy. Phys. JETP 7 (1958) 51. [15] J. Ambj#rn and R.J. Hughes, Canonical quantization in non-abelian background fields, Calteeh preprint, in preparation. [16] J. Honerkamp, Nucl. Phys. B48 (1972) 269. [17] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376. [18] R.J. Hughes, Phys. Lett. 97B (1980) 246; Nucl. Phys. B186 (1981) 376. [19] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287.