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On the origin of the pion in confinement schemes
Nuclear Physics B134 (1978) 327-338 © North-Holland Publishing Company
ON THE ORIGIN OF THE PION IN CONFINEMENT SCHEMES R. BROUT, F. ENGLERT and J.-M. FRI~RE * Facult( des Sciences, Universit( Libre de Bruxelles, Belgium
Received 21 October 1977
It is argued that the 't Hooft one-dimensional gauge model is a good starting point on how to conceive the pion in confinement schemes. The results of Wu on the quark propagator in this model are analyzed in the light of spontaneously broken chiral symmetry and the existence of the pion is deduced. The corresponding Bethe-Salpeter wave function is exhibited.
1. Introduction Few will contest Nambu's interpretation of PCAC and soft-pion physics in terms o f spontaneously broken chiral symmetry (sbc) as one of the more solid acquisitions in the quest for the foundations o f hadron physics. Nambu's original mechanism for sbc is, however, foreign to our present conception of hadron structure wherein the fundamental fermions, at least in the non-strange uncharmed sector, are taken to be light quarks bound together by confined flux. The mesons are supposed to be linear configurations of q and ~ and the energy scale is set by the confined flux density between them. How then is it possible to generate a zero-mass pion in the limit of zero-mass quarks? What is the mechanism for generating a negative energy which compensates the positive energy of the flux? In hunting for hints for the solution of this problem, our attention was called to the one-dimensional gauge model o f ' t Hooft [1 ] who, so to speak, happened upon a zero-mass bound state without displaying the mechanism of sbc. He found that the quark developed a negative energy correction of tachyonic character to the classical Coulomb energy which in the bound state with the antiquark is precisely what is necessary to cancel the enclosed flux energy. Thus one is confronted with a mechanism completely opposite to that of Nambu wherein the fermions acquire positive mass which is then counterbalanced by the attractive energy necessary to make the pion bound state. In the present paper we shall show that this curious tachyonic component of the
* Charg6 de Recherches du Fonds National de la Recherche Scientifique. 327
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R. Brout et al. / Confinement schemes
quark's self-energy is a consequence of chiral conservation in the context of a linear confinement scheme. The zero-mass bound state arises from a Ward identity generated by the conserved chiral currents. Our development relies on Wu's prescription of integration in the handling of the infrared singularities characteristic of the one-dimensional gauge theory. This prescription, which is different from 't ltooft's and whose validity is yet to be established, gives rise to a propagator structure in which the mechanism of chiral breakdown shows up most clearly. The reader may verify, however, that our proof of the existence of the zero-mass mode in consequence of chiral breakdown (eqs. ( 1 2 ) - ( 1 4 ) ) , is not in any way affected by the choice of this or that prescription. In the present paper we do not discuss which integration procedure is in fact correct. We present the graphical analysis which leads to Wu's result for the quark propagator and thereby to an interpretation of his result in terms of chiral concepts and sbc. It turns out that the theory with massless quarks can only be given sense as a limit. Calculations must be performed in the presence of a mass which at a later stage can be sent to zero. In this respect one is dealing with a phase transition in the usual sense. The propagator structure results in a response function which is singular. As usual the singularity is due to the presence o f a zero-mass bound state. We produce the corresponding Bethe-Salpeter bound-state wave function. What is less conventional is that the response in question is n o t to a mass term (i.e. a u, v mixer where u and v are the chiral components of the Dirac field). Rather it is the response of the v current to an infinitesimal u current source. The rble of the infinitesimal chiral breaker, the mass, is to induce finite communication between u and o in the quark propagator. These results touch at the heart of our problem since they arise from the coupling of the long-range linear Coulomb force, typical of confinement schemes, to massless fermions. This brings us to the question: why look at the one-dimensional 't Hooft model in the first place? Let us recall that this is a one-dimensional gauge model taken in the mathematical limit where the number of internal degrees of freedom of the quark, N, tends to infinity. In this (unphysical) limit the vacuum polarization corrections vanish and the quark propagator is dressed by ladder graphs, which, as is well known, is a sort of generalized Hartree-Fock self-consistent field method consistent with Ward identities. Although it is a comfort to know that the subclass of graphs considered does correspond to the solution of an exact field theory, we do not take this as the principal reason for our study. Rather we take a semi-phenomenological view. The model gives rise to a linear meson structure with flux enclosed between quark and antiquark. Thus the model is conceived as an approximation to the realistic situation in close correspondence with current ideas of confinement. The long-range linear Coulomb field is presumed to have come from a prior stage o f the confinement program. If present thinking is taken as a guide, the condition of the gauge field far from the quarks is a classical problem in which the " p h o t o n " , at least in the center of the flux
329
R. Brout et al. / Confinement schemes
tube has strictly zero mass *. This is not to be tampered with in our one-dimensional model. As is well known, vacuum polarization effects due to massless fields lead to massive photons, hence they are completely foreign to our enterprise. Rather, we address ourselves to the problem of the nature o f the quark's response to this milieu of linear flux, i.e. its local electric field and the concomitant spread of its wave function. It is then most fortunate that 't Hooft's model isolates just these effects. The paper is laid out as follows. The Schwinger-Dyson equation for the quark propagator and Wu's solution are presented and discussed. The bound state BetheSalpeter equation for the pion is solved concomitantly. The detailed analysis of the physics behind these answers is analyzed in mathematical terms using chiral concepts.
2. The spontaneous breakdown of chiral symmetry We shall work in the light-cone gauge (A_ = 0) and in chiral notation u = ~(1 + 0¢3) @,
v = l(1 - ~3) qJ"
(1)
Here a 3 = 3`073 and we use the usual Dirac conventions, with 3'0 hermitian, 3'3 antihermitian. Thus the v(g) particle moves to the left and the u(V) to the right. In the 't Hooft N ~ oo limit the non-Abelian character of the gauge group never becomes manifest and one may just as well work with Abelian notation directly. The action used to generate ladder graphs is then written as S=
d 2 x ( ~ ( 0 _ A + ) 2 + X/2o+(ia+ - g A + ) o
+ X/~u+(iO_.) u - m ( u + o + o+u)} ,
(2)
where for any vector V± = xfi~(Vo -+ V3). We shall focus attention on the vv component o f the quark propagator, Svv. It is convenient to define the self-energy according to Svv =
S~w(l + £vvSvv) •
(3)
Ladder-graph summation gives
zvv-
2g2i
cdkodk 3
J--
-Svv
p + k) .
(4)
Introducing 1 S~vv = X/2(P+ - r n 2 / 2 p - ) - i e p _
* This point of view has been recently enforced by one of us in [3].
(5)
R Brow et al. / Confinement
330
schemes
into (3):
“’ =dT(p+- m2/*pl) - Xv,- iepThis chiral notation Y-
(6)
*
is related to the more familiar Dirac propagator by
SDimcY-=
AY-Y+YSvv= fir+Svv *
As written, eq. (4) is profoundly ambiguous because of the singularity at k = 0 in the integrand. This is a reflection of the long-range linear Coulomb force. From ‘t Hooft’s work, we know that inclusion of the pole at k = 0 gives rise to a divergent term corresponding to the classical Coulomb energy in a box of infinite length. This divergence subsequently cancels in the quark-antiquark bound state, as it must on physical grounds. Thus, though the divergence is important for establishing the fact of confinement, it would be convenient to work in a formalism in which it is precluded at the outset, since it is irrelevant to the bound-state problem. The relevant processes are all the (local) quantum corrections to the classical energy. Wu has invented a prescription which presumably does just that. One first Wick rotates in the p. plane; a perfectly admissible step which introduces no new physics as it is inherent in the specification of the ie factors in S. What is new is the additional instruction to perform integrations symmetrically in the complex k space i.e. the space formed by the complex numbers k3, k. = ik4. Eq. (4) then becomes
+
(P k)” Ip + k I2 + m2 - (p t k)’ Evv ’
(7)
p* sp3 - ip4.
pcp3+ip4,
Wu’s solution which we shall subsequently derive and interpret using chiral concepts, is a remarkable &XT de force: Xvv=$
(Ip12+m2)(
[
(Ip12+m2)2-p
in graphical form,
4gqp I2 112 71
1
(8)
1.
The zero-mass limit is
Zvv=;p[l -[l -$]Lil) .
(9)
As Wu mentions, the solution as Ip 1+ 03(IZvv + g2/rrp *) is ‘t Hooft’s result, the latter being simply that calculated by second-order perturbation theory. Correspond. ing to (8) or (9) the propagator is given by
svv=4g,. g2
(10)
R. Brout et aL / Confinement schemes
331
The Bethe-Salpeter equation in a channel which will be discussed below is
V(p,p*;r,r*)=
--
g2 f s v v ( p - r + k , p * - r * + k * ) 7r 2
X P ( p + k, p* + k*; r, r*)Svv (p + k, p* + k*) dk3dk4
(k')~
(11)
P is the amputated BS wave function between quark and antiquark, of total momentum r, i.e.
r ( p , r) = s ~ ~(p) ¢(p, r) S ~ 4 p - r ) ,
(12)
where ff is the usual BS wave function. The question is whether when m = 0 there is a solution of(11) at r = 0 (and hence a zero-mass bound state). The answer is yes, as is easily seen by differentiation of (4) and (6). Recalling that dp/dp ° = 0 we have
dSvv(P, p ' ) _ d 1 dp" dp* p - ~vv(P, P') S dew = vv-'~Svv dZvv dp" =
_
,
g2 (" d2k dSvv(P + k, p* + k*) ~ J k"7 ap"
g2 t'd 2k
~-iJ~-
dew Svv(P + k ) . Svv(P + k) d(p + k)*
(13)
Thus a solution o f ( 1 1 ) is
r'(p, p*, 0) -
dEw(p, p*) dp" '
(14)
and the existence of the pion is established. Indeed one easily checks that qJ = SvvF(p, 0) Svv is a sum of positive-definite terms in Ip [2. Thus it is a nodeless bound state of q-~; hence pseudoscalar. (In one space dimension only scalar and pseudoscalar mesons exist according to the parity P of the wave function.) On the other hand the charge conjugation of this state is C = - 1 , as quark-antiquark bound states in one space dimension have necessarily PC = +1 in the absence of other degrees of freedom. How this fact may be reconciled with the identification of this zero-mass boson as the precursor of the n(P = - 1 , G = - 1 ) in Minkowski space is discussed in sect. 3. Let us now see what is going on behind this magic. In second order we have for
R. Brout et al. / Confinement schemes
332
the massless case
k .2
_
p +k
rdrdO l J r 2 - e~ 2 / ° IPl eiX + r e w '
g2 f
(15)
where we have changed to polar coordinates and written p as p = IP I e ix. Now change variable to the relative phase 0 - X to give g2 -ix ('dr dO Z(? =-~ -e J~(pl/r+eiO
(16) •
Wu's prescription is to integrate over 0 (the unit circle) first. As promised, the divergence at r -- 0 disappears. Indeed one picks up a pole in the 0 integral only for r > Ipl whereupon Z(vlv) = _eiX .2g2lpl .... j~ dr_ n r3 p
eiXg2 7rlpl
(17)
We have thus established that for massless quarks 1 rr
f dk3dka S~vv(P+ k) = S~vv(p*) = (S~vv(P))* k .2
(I 8)
This is sketched in the "mnemonic" graphs of fig. 1. Exchanging a photon changes a v-propagator into a u-propagator. In this lowest order, this seems to be a hasty conclusion since u-particles have never been brought to bear in the analysis. However when we analyze the full problem we shall see that this interpretation is correct. In order to proceed further, let us suppose that the phenomenom which occurs in lowest order is general: namely every time a photon is exchanged S(v0v)(p)changes into Svv(°)(p) (see fig. 1). The complete ladder summation then reduces to a simple algebraic problem S~ =
1
p - ~/.)
s;~
.
(20)
Combining eq. (20) with its complex conjugate leads to a quadratic equation for Svv, the solution of which is eq. (10) with Zvv given by eq. (9). It is amusing and informative to expand eq. (9) order by order to check the counting of the mnemonic graphs given in fig. 1. The burden of the proof of this method is then one of integration. It must be shown that loops enclosed in loops all integrate out according to the mnemonic pre-
R. Brout et al. / Confinement schemes
333
/
I
/
I
//
,//
\
/
\
\
/ Fig. 1. Equivalence between photon exchanges graphs and u, o zig-zags. Horizontal lines read p - 1 and diagonal lines p * - 1. To each vertex corresponds a factor ig/x/lr. Dashed lines stand for l/k .2.
scription. Clearly the proof b y i n d u c t i o n is established if one proves (see fig. 2) /~ d2k 1 1 _ rr I n = d k.-S p + k l p + kl 2n p*lPl 2n "
(22)
By the same manipulations as in eqs. (15), (16), (17) one easily proves that In is proportional to [p* Ip 12n]-n" the only question is to calculate the numerical factor. For n = 1, I n is ambiguous and for n ~> 2 it is plainly infinite. Thus straightforward
,e'
~
I/"4
,
\~
f
i/
a \1
" " "
~
/" ~ "x\
".
(,a)
#~+~"x
Fig. 2. The numerical equivalence of 2a and b is the content of eq. (22) in the text.
334
R. Brout et al. / Confinement schemes
summation and integration of the perturbation series at m = 0 is mathematical nonsense. The trick that cures the disease is at the same time fraught with physical significance. One switches on a mass term. Integration of the lowest-order term is now more tedious, but still straightforward. One finds (appendix B)
_1 rr
fj
d2k
(p,
l
_
k) + m2/(p ° +
g.) p.
1
(23)
+ m2/p "
Therefore eq. (18) generalizes in the presence of a mass term as well. It is now a straightforward matter to establish (22). Replace m 2 by m 2 + e and expand both sides of (23) in powers of e/(Ip 12 + m2). Equating equal powers of e and t h e n passing to the limit m ~ 0 establishes the required theorem. It is then clear that the whole analysis of Svv at m = 0 is to be understood as a limiting procedure from m ~: O. That is why the ( p * ) - I propagators which occur at intermediate steps are truly unperturbed u-particle propagators. Indeed, one may test for the presence of u by differentiation with respect to p°. Before integration one finds zero for every graph when m -- 0. But as we have emphasized the integrals have meaning only for m ~ 0. Differentiation under the integral sign when m :~ 0 is then permitted and yields a non-vanishing result superficially proportional to m 2. Before integration this factor of m 2 is of course the signal that the u-propagator has been coupled into v. One then integrates and finds that lim m 2 ~ 0 exists, so that the u-particle which was called upon to give the integral existence becomes effectively hidden. In a word, the key theorem is d2k
(p + k) °
m 2.-.~ (p + k)* d2k t.~d lim J d/~" [ I p + k12+ m2] n+l k "~T m2~O
-
d 1 1 clp* p* IPl 2n "
(24)
The proof follows immediately from (23). Thus as m ~ 0 the infinitesimal u,v mixing is built up into a finite effect through the singular (k°) -2 kernel in the integrand i.e. the effect of the long-range Coulomb force. To our knowledge this is a novel effect which has not yet been encountered in phase-transition theory. The branch structure of the propagator which has been emphasized by Wu does not seem to be the relevant point. The phase transition is already present in 't Hooft's work, but is still less transparent. It is therefore instructive to analyze the details o f ' t Hooft's mechanism to obtain the pion, as much of the physics is already present even in this low order of approxi-
R. Brout et al. / Confinement schemes
335
mation. The v-propagator in his treatment reads 1 p _ g2/~p.
(25)
•
(We have omitted the irrelevant divergent contribution to Zvv.) In Minkowski space the pole of S(v~ now Occurs at Po = - P a +g2/~r(Po - P 3 ) . Thus for P3 < < 0, corresponding to particle propagation in the infinite-momentum frame, the v suffers a reduction in energy from IP31 to IP31 -g2/21rlP31. It is precisely this negative shift in energy which car.cels the energy of the confined flux between q and ~, to give the pion. Because of the importance of this result, we present an elementary argument in appendix B which shows clearly that the origin of this effect is chiral conservation. Finally the Bethe-Salpeter wave function (14) is understandable in the same terms as the propagator. Before integration each term in dZvv/dP" vanishes. However, turning on an infinitesimal mass term yields a finite result. The effect is so marked that in response to a u-current source the v current diverges as m --* 0. This is seen from the fact that the graphs contributing to dZ/dp* = U(p, 0) always have at the vertex (the point of differentiation), a pair of u-propagators. The (amputated) legs are v-propagators. Once more this seems a novel circumstance in phase transition theory. The singularity in usual theories would be felt in the u÷v channel. The essential element then is the appearance of p* in the v-propagator, rather than the self-consistent mass as in the usual theories of sbc. It is to be emphasized that this possibility of a p° dependence occurs in consequence of the fact that g has the dimension of mass, hence a reflection of the very fact of confinement, g being related to the flux energy density. The relevant Ward identity is: d S ~ l _ d [S(v~]- l
dp"
-
-
dp"
+ r(p,
0).
(26)
In the chiral limit d [S(v°v)] - l / d p ° vanishes and eq. (13) becomes a homogeneous equation equivalent to the BS equation (11) at r = 0. The Ward identity (26) is the limit of the more general identity which results from v-current conservation r+ r ( p , r) = S~vI (p+ + r+, p _ + r_) - Svvl(p+, p _ ) ,
(27)
taken in the limit r÷ --*0 at fixed infinitesimal r_. The sorting out of the axial and vector currents is reserved for a subsequent paper which will be devoted to questions of gauge and Lorentz invariance.
3. Concluding remarks In view of the above subtleties and the capacity of the one-dimensional gauge theory to produce both a pion and a linear spectrum typical of confinement, it
336
R. Brout et al. / Confinement schemes
seems hard to imagine that within the confinement scheme it is possible to concoct some other mechanism for sbc and the pion. Realistic confinement schemes should be developed with this in mind. How to do this is of course the central problem. Nevertheless on a more modest level, one may now try to incorporate the above lessons into a 3-dimensional semi-phenomenology. To realize this program, one is forced to double the number of chiral components to account for the existence of the quark spin. Thus we now have for the bound states PC = -+1 with the plus sign for symmetric triplets and the minus sign for antisymmetric singlets. Clearly there is now room for the pion as the Nambu-Goldstone particle. Of great interest will be the interrelation between the removal of charge conjugation (and spin) degeneracy and the quantization of Regge intercepts, ttow does the G = +1 spin triplet (p) get its intercept of ~1 ? There remain several technical problems which we have glossed over. The question of gauge invariance must be settled and the meaning of Wu's prescription of symmetric integration be understood. The Lorentz-transformation properties of the solutions must be looked into. Is it true for example that 't Hooft's solution is correct in the infinite-momentum frame, as one might suppose simply on dimensional grounds? If so, the problem simplifies immensely and there will be no need to use all the complexity of Wu's solution. Its function would then be to serve as the guarantee of the gauge and Lorentz invariance of the theory. At all events one may now attack these problems in the light of Wu's work with the optimistic frame of mind that the controversial elements surrounding this problem can now be settled. We are deeply indebted to Giorgio Parisi for emphasizing to us the possible relevance of the 't Hooft model in the quest for the pion and to Claude Truffin for helpful discussions. Appendix A It is instructive to revisit the problem of the quark's self-energy in Coulomb gauge. We carry this out in such a fashion that a conceptual understanding of the negative energy correction emerges clearly and naturally. Placing a point quark in a box of length L gives rise in the first instance to a classical Coulomb energy equal to g2L. We expect the correction to this energy on physical grounds to be negative due to the wave-function spread. The calculation is most transparent in Rayleigh-Schr6dinger perturbation theory. A v-quark moving to the left with momentum - p suffers a change in energy in second order due to its interaction with the Coulomb field equal to AE(p)
g2 f -2-n ~
(~)2
-p+k<0
g2
(~.~2 -p+k
1 (E_p - E_p +k) 1. . . . . . .
(A.I)
K Brout et al. / Confinement schemes
337
The factors ( 2 k ) - 1 / 2 a r e the usual matrix elements for emission and absorption. The peculiar division into positive and negative energy differences results from the fact that the chiral-invariant interaction with the Coulomb field dictates that the recoil particle moving to the right be an anti-v. In other words the virtual process responsible for this term is pair creation to the left of the moving v-particle of momentum - p ; the right-moving intermediate anti-v then annihilates with the left-moving initial V.
In the domains of integration stipulated in (A.1) we have/fp - E _ p + k = k and Ep + E - p + k = k so that
AE(p) = ~
Ikl k
~]
-
p
g2 ?
dk_
27r
k2
p
g2
27rp '
which checks against the answer given in the text in the infinite-momentum frame. Thus it is clear that this tachyonic negative energy arises as a consequence of chiral conservation. Of course that this negative energy exactly cancels the energy of the enclosed flux to make the pion can only be reasoned from the Ward identity arising from the conserved chiral current.
Appendix B We want to calculate I = f dk3dk4 k .2
1
(B.1)
( p + k ) + m 2 / ( p + k) ° '
According to the prescription of symmetric integration, we put k = r e iO, p = IP I e ix to get
I =
dr o
r- 2e i ~
......
m2
( [ P l e ix + r e iO) +
o
IPl e -ix + r e -iO
Changing variables to u = e i(°-x) i dr I = - flp-~r2 ~
unit circle /12
e ix (IPl u + r) _ _ + u ( i p l 2 + r 2 + m2 ) . . . . . . .
Iplr
+ l
udu.
(B.3)
K Brout et al. / Confinement schemes
338
The roots of the denominator are Ul =
IP 12 + r2 + rn2 IPl r IPl 2 + r 2 + m 2
u2
Iplr
] / [ IP 12 + r 2 + m2~ 2
+r~
~ir
I -4,
1//[p12+ r 2 - m2\2 ~ Iplr ) - 4.
(B.4)
Only ui lies inside the unit circle and brings a pole contribution: e Ix rr f dx (IP I: + m2) 2 + x(m 2 -
I=2~p [ a x---i x=r 2 ,
Ip
12) - (Ip 12 + m 2) R
R
R = x / - ~ + 2 x ( m 2 - 1 P l 2 ) + ( I P l Z + m2) 2 .
(B.5)
Straightforward integration then leads to
el× I- R + I p l Z + m 2]~ --- - ~ i=2_~1 1 . x p* + rn2/p
(8.6)
References [1 ] G. 't Hooft, Nucl. Phys. B75 (1975) 461. [2] T.T. Wu, Two-dimensional Yang-Mills theory in the leading I/N expansion, CERN preprint TH 2366 (August 1977). [3] F. Englert, Electric and magnetic confinement schemes, Lectures given at Carg~se Summer School, 1977.
ON THE ORIGIN OF THE PION IN CONFINEMENT SCHEMES R. BROUT, F. ENGLERT and J.-M. FRI~RE * Facult( des Sciences, Universit( Libre de Bruxelles, Belgium
Received 21 October 1977
It is argued that the 't Hooft one-dimensional gauge model is a good starting point on how to conceive the pion in confinement schemes. The results of Wu on the quark propagator in this model are analyzed in the light of spontaneously broken chiral symmetry and the existence of the pion is deduced. The corresponding Bethe-Salpeter wave function is exhibited.
1. Introduction Few will contest Nambu's interpretation of PCAC and soft-pion physics in terms o f spontaneously broken chiral symmetry (sbc) as one of the more solid acquisitions in the quest for the foundations o f hadron physics. Nambu's original mechanism for sbc is, however, foreign to our present conception of hadron structure wherein the fundamental fermions, at least in the non-strange uncharmed sector, are taken to be light quarks bound together by confined flux. The mesons are supposed to be linear configurations of q and ~ and the energy scale is set by the confined flux density between them. How then is it possible to generate a zero-mass pion in the limit of zero-mass quarks? What is the mechanism for generating a negative energy which compensates the positive energy of the flux? In hunting for hints for the solution of this problem, our attention was called to the one-dimensional gauge model o f ' t Hooft [1 ] who, so to speak, happened upon a zero-mass bound state without displaying the mechanism of sbc. He found that the quark developed a negative energy correction of tachyonic character to the classical Coulomb energy which in the bound state with the antiquark is precisely what is necessary to cancel the enclosed flux energy. Thus one is confronted with a mechanism completely opposite to that of Nambu wherein the fermions acquire positive mass which is then counterbalanced by the attractive energy necessary to make the pion bound state. In the present paper we shall show that this curious tachyonic component of the
* Charg6 de Recherches du Fonds National de la Recherche Scientifique. 327
328
R. Brout et al. / Confinement schemes
quark's self-energy is a consequence of chiral conservation in the context of a linear confinement scheme. The zero-mass bound state arises from a Ward identity generated by the conserved chiral currents. Our development relies on Wu's prescription of integration in the handling of the infrared singularities characteristic of the one-dimensional gauge theory. This prescription, which is different from 't ltooft's and whose validity is yet to be established, gives rise to a propagator structure in which the mechanism of chiral breakdown shows up most clearly. The reader may verify, however, that our proof of the existence of the zero-mass mode in consequence of chiral breakdown (eqs. ( 1 2 ) - ( 1 4 ) ) , is not in any way affected by the choice of this or that prescription. In the present paper we do not discuss which integration procedure is in fact correct. We present the graphical analysis which leads to Wu's result for the quark propagator and thereby to an interpretation of his result in terms of chiral concepts and sbc. It turns out that the theory with massless quarks can only be given sense as a limit. Calculations must be performed in the presence of a mass which at a later stage can be sent to zero. In this respect one is dealing with a phase transition in the usual sense. The propagator structure results in a response function which is singular. As usual the singularity is due to the presence o f a zero-mass bound state. We produce the corresponding Bethe-Salpeter bound-state wave function. What is less conventional is that the response in question is n o t to a mass term (i.e. a u, v mixer where u and v are the chiral components of the Dirac field). Rather it is the response of the v current to an infinitesimal u current source. The rble of the infinitesimal chiral breaker, the mass, is to induce finite communication between u and o in the quark propagator. These results touch at the heart of our problem since they arise from the coupling of the long-range linear Coulomb force, typical of confinement schemes, to massless fermions. This brings us to the question: why look at the one-dimensional 't Hooft model in the first place? Let us recall that this is a one-dimensional gauge model taken in the mathematical limit where the number of internal degrees of freedom of the quark, N, tends to infinity. In this (unphysical) limit the vacuum polarization corrections vanish and the quark propagator is dressed by ladder graphs, which, as is well known, is a sort of generalized Hartree-Fock self-consistent field method consistent with Ward identities. Although it is a comfort to know that the subclass of graphs considered does correspond to the solution of an exact field theory, we do not take this as the principal reason for our study. Rather we take a semi-phenomenological view. The model gives rise to a linear meson structure with flux enclosed between quark and antiquark. Thus the model is conceived as an approximation to the realistic situation in close correspondence with current ideas of confinement. The long-range linear Coulomb field is presumed to have come from a prior stage o f the confinement program. If present thinking is taken as a guide, the condition of the gauge field far from the quarks is a classical problem in which the " p h o t o n " , at least in the center of the flux
329
R. Brout et al. / Confinement schemes
tube has strictly zero mass *. This is not to be tampered with in our one-dimensional model. As is well known, vacuum polarization effects due to massless fields lead to massive photons, hence they are completely foreign to our enterprise. Rather, we address ourselves to the problem of the nature o f the quark's response to this milieu of linear flux, i.e. its local electric field and the concomitant spread of its wave function. It is then most fortunate that 't Hooft's model isolates just these effects. The paper is laid out as follows. The Schwinger-Dyson equation for the quark propagator and Wu's solution are presented and discussed. The bound state BetheSalpeter equation for the pion is solved concomitantly. The detailed analysis of the physics behind these answers is analyzed in mathematical terms using chiral concepts.
2. The spontaneous breakdown of chiral symmetry We shall work in the light-cone gauge (A_ = 0) and in chiral notation u = ~(1 + 0¢3) @,
v = l(1 - ~3) qJ"
(1)
Here a 3 = 3`073 and we use the usual Dirac conventions, with 3'0 hermitian, 3'3 antihermitian. Thus the v(g) particle moves to the left and the u(V) to the right. In the 't Hooft N ~ oo limit the non-Abelian character of the gauge group never becomes manifest and one may just as well work with Abelian notation directly. The action used to generate ladder graphs is then written as S=
d 2 x ( ~ ( 0 _ A + ) 2 + X/2o+(ia+ - g A + ) o
+ X/~u+(iO_.) u - m ( u + o + o+u)} ,
(2)
where for any vector V± = xfi~(Vo -+ V3). We shall focus attention on the vv component o f the quark propagator, Svv. It is convenient to define the self-energy according to Svv =
S~w(l + £vvSvv) •
(3)
Ladder-graph summation gives
zvv-
2g2i
cdkodk 3
J--
-Svv
p + k) .
(4)
Introducing 1 S~vv = X/2(P+ - r n 2 / 2 p - ) - i e p _
* This point of view has been recently enforced by one of us in [3].
(5)
R Brow et al. / Confinement
330
schemes
into (3):
“’ =dT(p+- m2/*pl) - Xv,- iepThis chiral notation Y-
(6)
*
is related to the more familiar Dirac propagator by
SDimcY-=
AY-Y+YSvv= fir+Svv *
As written, eq. (4) is profoundly ambiguous because of the singularity at k = 0 in the integrand. This is a reflection of the long-range linear Coulomb force. From ‘t Hooft’s work, we know that inclusion of the pole at k = 0 gives rise to a divergent term corresponding to the classical Coulomb energy in a box of infinite length. This divergence subsequently cancels in the quark-antiquark bound state, as it must on physical grounds. Thus, though the divergence is important for establishing the fact of confinement, it would be convenient to work in a formalism in which it is precluded at the outset, since it is irrelevant to the bound-state problem. The relevant processes are all the (local) quantum corrections to the classical energy. Wu has invented a prescription which presumably does just that. One first Wick rotates in the p. plane; a perfectly admissible step which introduces no new physics as it is inherent in the specification of the ie factors in S. What is new is the additional instruction to perform integrations symmetrically in the complex k space i.e. the space formed by the complex numbers k3, k. = ik4. Eq. (4) then becomes
+
(P k)” Ip + k I2 + m2 - (p t k)’ Evv ’
(7)
p* sp3 - ip4.
pcp3+ip4,
Wu’s solution which we shall subsequently derive and interpret using chiral concepts, is a remarkable &XT de force: Xvv=$
(Ip12+m2)(
[
(Ip12+m2)2-p
in graphical form,
4gqp I2 112 71
1
(8)
1.
The zero-mass limit is
Zvv=;p[l -[l -$]Lil) .
(9)
As Wu mentions, the solution as Ip 1+ 03(IZvv + g2/rrp *) is ‘t Hooft’s result, the latter being simply that calculated by second-order perturbation theory. Correspond. ing to (8) or (9) the propagator is given by
svv=4g,. g2
(10)
R. Brout et aL / Confinement schemes
331
The Bethe-Salpeter equation in a channel which will be discussed below is
V(p,p*;r,r*)=
--
g2 f s v v ( p - r + k , p * - r * + k * ) 7r 2
X P ( p + k, p* + k*; r, r*)Svv (p + k, p* + k*) dk3dk4
(k')~
(11)
P is the amputated BS wave function between quark and antiquark, of total momentum r, i.e.
r ( p , r) = s ~ ~(p) ¢(p, r) S ~ 4 p - r ) ,
(12)
where ff is the usual BS wave function. The question is whether when m = 0 there is a solution of(11) at r = 0 (and hence a zero-mass bound state). The answer is yes, as is easily seen by differentiation of (4) and (6). Recalling that dp/dp ° = 0 we have
dSvv(P, p ' ) _ d 1 dp" dp* p - ~vv(P, P') S dew = vv-'~Svv dZvv dp" =
_
,
g2 (" d2k dSvv(P + k, p* + k*) ~ J k"7 ap"
g2 t'd 2k
~-iJ~-
dew Svv(P + k ) . Svv(P + k) d(p + k)*
(13)
Thus a solution o f ( 1 1 ) is
r'(p, p*, 0) -
dEw(p, p*) dp" '
(14)
and the existence of the pion is established. Indeed one easily checks that qJ = SvvF(p, 0) Svv is a sum of positive-definite terms in Ip [2. Thus it is a nodeless bound state of q-~; hence pseudoscalar. (In one space dimension only scalar and pseudoscalar mesons exist according to the parity P of the wave function.) On the other hand the charge conjugation of this state is C = - 1 , as quark-antiquark bound states in one space dimension have necessarily PC = +1 in the absence of other degrees of freedom. How this fact may be reconciled with the identification of this zero-mass boson as the precursor of the n(P = - 1 , G = - 1 ) in Minkowski space is discussed in sect. 3. Let us now see what is going on behind this magic. In second order we have for
R. Brout et al. / Confinement schemes
332
the massless case
k .2
_
p +k
rdrdO l J r 2 - e~ 2 / ° IPl eiX + r e w '
g2 f
(15)
where we have changed to polar coordinates and written p as p = IP I e ix. Now change variable to the relative phase 0 - X to give g2 -ix ('dr dO Z(? =-~ -e J~(pl/r+eiO
(16) •
Wu's prescription is to integrate over 0 (the unit circle) first. As promised, the divergence at r -- 0 disappears. Indeed one picks up a pole in the 0 integral only for r > Ipl whereupon Z(vlv) = _eiX .2g2lpl .... j~ dr_ n r3 p
eiXg2 7rlpl
(17)
We have thus established that for massless quarks 1 rr
f dk3dka S~vv(P+ k) = S~vv(p*) = (S~vv(P))* k .2
(I 8)
This is sketched in the "mnemonic" graphs of fig. 1. Exchanging a photon changes a v-propagator into a u-propagator. In this lowest order, this seems to be a hasty conclusion since u-particles have never been brought to bear in the analysis. However when we analyze the full problem we shall see that this interpretation is correct. In order to proceed further, let us suppose that the phenomenom which occurs in lowest order is general: namely every time a photon is exchanged S(v0v)(p)changes into Svv(°)(p) (see fig. 1). The complete ladder summation then reduces to a simple algebraic problem S~ =
1
p - ~/.)
s;~
.
(20)
Combining eq. (20) with its complex conjugate leads to a quadratic equation for Svv, the solution of which is eq. (10) with Zvv given by eq. (9). It is amusing and informative to expand eq. (9) order by order to check the counting of the mnemonic graphs given in fig. 1. The burden of the proof of this method is then one of integration. It must be shown that loops enclosed in loops all integrate out according to the mnemonic pre-
R. Brout et al. / Confinement schemes
333
/
I
/
I
//
,//
\
/
\
\
/ Fig. 1. Equivalence between photon exchanges graphs and u, o zig-zags. Horizontal lines read p - 1 and diagonal lines p * - 1. To each vertex corresponds a factor ig/x/lr. Dashed lines stand for l/k .2.
scription. Clearly the proof b y i n d u c t i o n is established if one proves (see fig. 2) /~ d2k 1 1 _ rr I n = d k.-S p + k l p + kl 2n p*lPl 2n "
(22)
By the same manipulations as in eqs. (15), (16), (17) one easily proves that In is proportional to [p* Ip 12n]-n" the only question is to calculate the numerical factor. For n = 1, I n is ambiguous and for n ~> 2 it is plainly infinite. Thus straightforward
,e'
~
I/"4
,
\~
f
i/
a \1
" " "
~
/" ~ "x\
".
(,a)
#~+~"x
Fig. 2. The numerical equivalence of 2a and b is the content of eq. (22) in the text.
334
R. Brout et al. / Confinement schemes
summation and integration of the perturbation series at m = 0 is mathematical nonsense. The trick that cures the disease is at the same time fraught with physical significance. One switches on a mass term. Integration of the lowest-order term is now more tedious, but still straightforward. One finds (appendix B)
_1 rr
fj
d2k
(p,
l
_
k) + m2/(p ° +
g.) p.
1
(23)
+ m2/p "
Therefore eq. (18) generalizes in the presence of a mass term as well. It is now a straightforward matter to establish (22). Replace m 2 by m 2 + e and expand both sides of (23) in powers of e/(Ip 12 + m2). Equating equal powers of e and t h e n passing to the limit m ~ 0 establishes the required theorem. It is then clear that the whole analysis of Svv at m = 0 is to be understood as a limiting procedure from m ~: O. That is why the ( p * ) - I propagators which occur at intermediate steps are truly unperturbed u-particle propagators. Indeed, one may test for the presence of u by differentiation with respect to p°. Before integration one finds zero for every graph when m -- 0. But as we have emphasized the integrals have meaning only for m ~ 0. Differentiation under the integral sign when m :~ 0 is then permitted and yields a non-vanishing result superficially proportional to m 2. Before integration this factor of m 2 is of course the signal that the u-propagator has been coupled into v. One then integrates and finds that lim m 2 ~ 0 exists, so that the u-particle which was called upon to give the integral existence becomes effectively hidden. In a word, the key theorem is d2k
(p + k) °
m 2.-.~ (p + k)* d2k t.~d lim J d/~" [ I p + k12+ m2] n+l k "~T m2~O
-
d 1 1 clp* p* IPl 2n "
(24)
The proof follows immediately from (23). Thus as m ~ 0 the infinitesimal u,v mixing is built up into a finite effect through the singular (k°) -2 kernel in the integrand i.e. the effect of the long-range Coulomb force. To our knowledge this is a novel effect which has not yet been encountered in phase-transition theory. The branch structure of the propagator which has been emphasized by Wu does not seem to be the relevant point. The phase transition is already present in 't Hooft's work, but is still less transparent. It is therefore instructive to analyze the details o f ' t Hooft's mechanism to obtain the pion, as much of the physics is already present even in this low order of approxi-
R. Brout et al. / Confinement schemes
335
mation. The v-propagator in his treatment reads 1 p _ g2/~p.
(25)
•
(We have omitted the irrelevant divergent contribution to Zvv.) In Minkowski space the pole of S(v~ now Occurs at Po = - P a +g2/~r(Po - P 3 ) . Thus for P3 < < 0, corresponding to particle propagation in the infinite-momentum frame, the v suffers a reduction in energy from IP31 to IP31 -g2/21rlP31. It is precisely this negative shift in energy which car.cels the energy of the confined flux between q and ~, to give the pion. Because of the importance of this result, we present an elementary argument in appendix B which shows clearly that the origin of this effect is chiral conservation. Finally the Bethe-Salpeter wave function (14) is understandable in the same terms as the propagator. Before integration each term in dZvv/dP" vanishes. However, turning on an infinitesimal mass term yields a finite result. The effect is so marked that in response to a u-current source the v current diverges as m --* 0. This is seen from the fact that the graphs contributing to dZ/dp* = U(p, 0) always have at the vertex (the point of differentiation), a pair of u-propagators. The (amputated) legs are v-propagators. Once more this seems a novel circumstance in phase transition theory. The singularity in usual theories would be felt in the u÷v channel. The essential element then is the appearance of p* in the v-propagator, rather than the self-consistent mass as in the usual theories of sbc. It is to be emphasized that this possibility of a p° dependence occurs in consequence of the fact that g has the dimension of mass, hence a reflection of the very fact of confinement, g being related to the flux energy density. The relevant Ward identity is: d S ~ l _ d [S(v~]- l
dp"
-
-
dp"
+ r(p,
0).
(26)
In the chiral limit d [S(v°v)] - l / d p ° vanishes and eq. (13) becomes a homogeneous equation equivalent to the BS equation (11) at r = 0. The Ward identity (26) is the limit of the more general identity which results from v-current conservation r+ r ( p , r) = S~vI (p+ + r+, p _ + r_) - Svvl(p+, p _ ) ,
(27)
taken in the limit r÷ --*0 at fixed infinitesimal r_. The sorting out of the axial and vector currents is reserved for a subsequent paper which will be devoted to questions of gauge and Lorentz invariance.
3. Concluding remarks In view of the above subtleties and the capacity of the one-dimensional gauge theory to produce both a pion and a linear spectrum typical of confinement, it
336
R. Brout et al. / Confinement schemes
seems hard to imagine that within the confinement scheme it is possible to concoct some other mechanism for sbc and the pion. Realistic confinement schemes should be developed with this in mind. How to do this is of course the central problem. Nevertheless on a more modest level, one may now try to incorporate the above lessons into a 3-dimensional semi-phenomenology. To realize this program, one is forced to double the number of chiral components to account for the existence of the quark spin. Thus we now have for the bound states PC = -+1 with the plus sign for symmetric triplets and the minus sign for antisymmetric singlets. Clearly there is now room for the pion as the Nambu-Goldstone particle. Of great interest will be the interrelation between the removal of charge conjugation (and spin) degeneracy and the quantization of Regge intercepts, ttow does the G = +1 spin triplet (p) get its intercept of ~1 ? There remain several technical problems which we have glossed over. The question of gauge invariance must be settled and the meaning of Wu's prescription of symmetric integration be understood. The Lorentz-transformation properties of the solutions must be looked into. Is it true for example that 't Hooft's solution is correct in the infinite-momentum frame, as one might suppose simply on dimensional grounds? If so, the problem simplifies immensely and there will be no need to use all the complexity of Wu's solution. Its function would then be to serve as the guarantee of the gauge and Lorentz invariance of the theory. At all events one may now attack these problems in the light of Wu's work with the optimistic frame of mind that the controversial elements surrounding this problem can now be settled. We are deeply indebted to Giorgio Parisi for emphasizing to us the possible relevance of the 't Hooft model in the quest for the pion and to Claude Truffin for helpful discussions. Appendix A It is instructive to revisit the problem of the quark's self-energy in Coulomb gauge. We carry this out in such a fashion that a conceptual understanding of the negative energy correction emerges clearly and naturally. Placing a point quark in a box of length L gives rise in the first instance to a classical Coulomb energy equal to g2L. We expect the correction to this energy on physical grounds to be negative due to the wave-function spread. The calculation is most transparent in Rayleigh-Schr6dinger perturbation theory. A v-quark moving to the left with momentum - p suffers a change in energy in second order due to its interaction with the Coulomb field equal to AE(p)
g2 f -2-n ~
(~)2
-p+k<0
g2
(~.~2 -p+k
1 (E_p - E_p +k) 1. . . . . . .
(A.I)
K Brout et al. / Confinement schemes
337
The factors ( 2 k ) - 1 / 2 a r e the usual matrix elements for emission and absorption. The peculiar division into positive and negative energy differences results from the fact that the chiral-invariant interaction with the Coulomb field dictates that the recoil particle moving to the right be an anti-v. In other words the virtual process responsible for this term is pair creation to the left of the moving v-particle of momentum - p ; the right-moving intermediate anti-v then annihilates with the left-moving initial V.
In the domains of integration stipulated in (A.1) we have/fp - E _ p + k = k and Ep + E - p + k = k so that
AE(p) = ~
Ikl k
~]
-
p
g2 ?
dk_
27r
k2
p
g2
27rp '
which checks against the answer given in the text in the infinite-momentum frame. Thus it is clear that this tachyonic negative energy arises as a consequence of chiral conservation. Of course that this negative energy exactly cancels the energy of the enclosed flux to make the pion can only be reasoned from the Ward identity arising from the conserved chiral current.
Appendix B We want to calculate I = f dk3dk4 k .2
1
(B.1)
( p + k ) + m 2 / ( p + k) ° '
According to the prescription of symmetric integration, we put k = r e iO, p = IP I e ix to get
I =
dr o
r- 2e i ~
......
m2
( [ P l e ix + r e iO) +
o
IPl e -ix + r e -iO
Changing variables to u = e i(°-x) i dr I = - flp-~r2 ~
unit circle /12
e ix (IPl u + r) _ _ + u ( i p l 2 + r 2 + m2 ) . . . . . . .
Iplr
+ l
udu.
(B.3)
K Brout et al. / Confinement schemes
338
The roots of the denominator are Ul =
IP 12 + r2 + rn2 IPl r IPl 2 + r 2 + m 2
u2
Iplr
] / [ IP 12 + r 2 + m2~ 2
+r~
~ir
I -4,
1//[p12+ r 2 - m2\2 ~ Iplr ) - 4.
(B.4)
Only ui lies inside the unit circle and brings a pole contribution: e Ix rr f dx (IP I: + m2) 2 + x(m 2 -
I=2~p [ a x---i x=r 2 ,
Ip
12) - (Ip 12 + m 2) R
R
R = x / - ~ + 2 x ( m 2 - 1 P l 2 ) + ( I P l Z + m2) 2 .
(B.5)
Straightforward integration then leads to
el× I- R + I p l Z + m 2]~ --- - ~ i=2_~1 1 . x p* + rn2/p
(8.6)
References [1 ] G. 't Hooft, Nucl. Phys. B75 (1975) 461. [2] T.T. Wu, Two-dimensional Yang-Mills theory in the leading I/N expansion, CERN preprint TH 2366 (August 1977). [3] F. Englert, Electric and magnetic confinement schemes, Lectures given at Carg~se Summer School, 1977.