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BRST cohomology and vacuum structure of two-dimensional chromodynamics
__ __ li!!Ei
16 November 1995
&
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 363 (1995) 85-92
BRST cohomology and vacuum structure of two-dimensional chromodynamics E. Abdalla’,
K.D. Rothe
lnstitut ftir Theoretische Physik, Universitiit Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
Received 24 July 1995; revised mannscript received 7 September 1995 Editor: P.V. Landshoff
Abstract Using a formulation of QCDz as a perturbed conformally invariant theory involving fertnions, ghosts, as well as positive and negative level Wess-Zumino-Witten fields, we show that the BRST conditions become restrictions on the conformally invariant sector, as described by a G/G topological theory. By solving the corresponding cohomology problem we are led to a finite set of vacua. For G = SU(2) these vacua are two-fold degenerate.
1. Introduction
Quantum Chromodynamics in 1+ 1 dimensions (:QCD2) has been subject of numerous investigations in the past twenty five years [ 1,2]. However, unlike its abelian counterpart, the exactly soluble Schwinger model [ 31, one had up to recently no hint at its exact integrability. Moreover, traditional topological arguments based on instantons suggest that the vacuum of QCDz is unique, unlike the case of the Schwinger model, where this vacuum is known to be infinitely degenerate. However, arguments have been presented I.41 in favor of the existence of a discrete, but finite set of QCD2 vacua in higher representations of the fermions. The recent formulation [ 51 of QCD2 as a perturbed conformally invariant Wess-Zumino-Witten-type theory [6] turns out to provide an appropriate starting point for a dynamical investigation of the physical
Hilbert-space structure of QCD2. The fundamental framework is provided by the BRST analysis of QCDZ in this formulation, as carried out in Ref. [ 81. In particular it will be our aim to investigate the possible existence of degeneracy of the QCD;! vacuum. In this respect it will be useful to point out the parallelisms with the Schwinger model in the decoupled formulation. We shall show that the conformally invariant sector of QCD2 is described by a level-one G/G topological field theory, thus allowing for a complete classification of the ground states. By explicitly solving the cohomology problem for G = SU( 2)) we find that the vacuum is two-fold degenerate in the left- and rightmoving sector, respectively. In order to provide the necessary framework, we briefly review the essential results of Ref. [ 81. In the light-cone gauge A+ = 0, the QCDz-partition function reads 2 ’ Our conventions are: x = (;i), ‘y+ = 2 (:A), y- = 2 cz),
’ Work supported by Alexander von Humboldt Stiftung. On leave Cram the University of Sao Paula, Brazil. 0370-2693/95/%09.50
A* = A0 f At, & ifabcf
@ 1995 Elsevier Science B.V. All rights reserved
.XSD/0370-2693(95)01193-5
and tr(f#)
= a0 f 81, A, = AgP, etc., with LP.Pl
= Sab, fubcfubd
= fCv&d.
=
E. Abdalla.
K.D. Rothe / Physics Letters B 363 (1995) 85-92
right-moving sector, implying the existence served right- and left-moving BRST currents
J
Db_Dc_ exp( i&F) ,
X
with the corresponding
(1)
gauge-fixed
[ 81
Lagrangian
+ ,yiia+,y, + $~iD-$z
CGF = tr $(a+A_)*
a*.P’7
- ib,{cF,
CT}] ,
= 0,
(8)
where
+ tr( b_ia+c_ ) ,
(2)
,
J(B) F = tr [c+,
of con[ 81
where b- c- are Grassman-valued from the gauge-fixing condition.
ghost fields arising
2. Local decoupled formulation
of QCDz
n_
s
--&D_(v)a+(tia_v-‘)
-(i23)
Ka_v-’ + Xl/y! + b_{c_,c_}
z 0,
Making the change of variable A_ = iva-v-l ti2 = 42
-n+ E (3)
7
one is led to the factorized partition function
[ 5,8]
)
Rh
(4)
where
(9)
&v-l[a:(iea_v-l)lv-
+x2x;
qV-‘ia+V
+ b+{c+,c+)
X0
(10)
are first-class constraints, with sZ_ x 0 playing the role of the Gauss law. It is interesting to note that
a-R+ = -v-la+R_
-i(Cv + l)I[Vl
d2xtr[a+(va_v-‘>I2
1,
and
,
z = z’“‘z’o’z” F
=a_ + [va-v-l,
n(v)
v,
(11)
implying ,
(5) D_d+A_
+ (1 +&);A_
=O.
(12)
and Z(O) F
-
/
Z’O’ = xh xexp
exp (i / d2x,j$,y)
DjV,y
Note that the term proportional in the literature.
,
ZJb*lDc* 3. Non-local decoupled formulation ( 1‘/d2+u[b+ia_c+
+ b_ia+c_])
,
(6)
with I [ g] the WZW functional
r[gl = &
s
d2x tr
tr
of QCDZ
The partition function (5) involves 4th order derivatives. In order to reduce these to 2nd order, we introduce an auxiliary field E via the identity
afig-‘apg exp(~Sfh[a,(Vid_V-‘)12)
I
+ &
to CV has been ignored
JJ dr
d*xs~l”g-‘gg-‘d,gg-‘d,g,
(7)
=/DEexp(-il$tr[cE2
0
where g(l,x) = g(x),g(O,x) = ll. The partition function exhibits a BRST symmetry in the left- and
+
J;;F
La+(i4a_v-l)
I>.
(13)
87
E. Abdalla, K.D. Rothe/ Physics Letters B 363 (1995) 85-92
Making the change of variable [ 581 1<= J;;(y)
-&(/TlkY+fl)
(14)
and making use of the Polyakov-Wiegmann identity [71 Ughl
= l-k1 + F[hl
+ & / d2x tr (g-‘GJ+gha_h-‘)
,
(15)
one arrives at the alternative representation [ $81 (0)z,z,, Z=Z, (0)Z@
(16)
with YDPexp{-i( 1 + C,)r[P]},
Zv =
(17)
s
where V = PV, and
be understood from a unified point of view by first making the change of variables from A+ and A_ to the group-valued fields U and V and only then fixing the gauge. At this stage the gauge-fixing procedure no longer involves a Faddeev-Popov determinant. With this procedure both BRST symmetries result alone from the Jacobians involved in the change of variables, as explained in Ref. [ 81. Note also that the presence of the ghosts in (20), (21) is in fact necessary in order for these constraints to be first class (vanishing central charge). Their presence reflects the fact that in the non-abelian case these ghosts originally appear coupled to the gauge field in the partition function. As we shall see shortly, they are absent in the corresponding constraints (35) of the U( 1) Schwinger model, where they are decoupled from the outset. Finally let us rewrite Zp in ( 18) in terms of an auxiliary field C_ as follows: z, =
vp s
J
where s’ = rip-1+
(18) Note that the WZW action enters with negative level -- ( 1 + Cv) , This will be very important in the following section. There exist [ 81 two BRST currents associated with the partition function ( 16) :
J,-(‘) = tr
[c&
- !jb+{c*,c+}]
,
(19)
+
(2)
lX_eis’ ,
I[i
(22)
tr(a+C_)2
e tr( C-p-‘ia+p)
1.
(23)
By gauging the gh - P - p sector following method of Ref. [ lo], one discovers one further constraint
(T)
e/3C_j3-'
- -$id_p-’
(T)
+ {b_,c_}
i&9-P-’ x 0.
(24)
where fi- E XIX! + {b’o’,c’o’}
As emphasized in Ref. [ 51, this constraint is 2nd class with respect to the constraints (20), (2 1) , and serves to determine the auxiliary field C_.
1+cv-
- ~Via_~-’
z 0, si,
E x2/y; + {b?‘, cy’} M 0,
(20) 1 +Gyi, - T
+
v
4. The Schwinger model revisited (21)
represent first class constraints. These shall play a central role in the characterization of the QCD2 vacuum. Note that we have two BRST charges, and hence two BRST conditions on the physical Hilbert space. As discussed in Ref. [9], their symmetric presence can
In order to gain some feeling for the constraints (9),(10),(20)and(21)itisusefultoseewhatthese constraints correspond to in the U( 1) case. In the U( 1) case, CV = 0 (corresponding to decoupled Faddeev-Popov ghost from the outset). Parametrizing V in (3) by
E. Abdalla, K.D. Rothe/ Physics Letters B 363 (1995) 85-92
88
V=exp(i2&$),
(25)
the WZW functional I[ V] and Maxwell term in (5) reduce to
respectively, so that the partition function (4) reads
(27) Notice that 4 is a negative metric field, corresponding to the fact that the WZW action I[ V J enters in (5) with negative level. From (26) follows the equation of motion q
(q ) +;
(28)
f$=o;
the constraints fI+ (9) and ( 10) take the form
J;;
n*==a,
e2
( ) Cl+;
d-eXypx”O.
where 4 = 4 + &I. The e-dependent term cancels against the anomaly arising from the fermionic integration, so that the gauged partition function coincides with (27). Following Ref. [ lo], the variation of the partition function with respect to W, then leads to the constraints (29). Notice that the Klein-Gordon operator (Cl -t $) projects out the massive mode of 4 satisfying (28), leaving one only with the massless mode. Hence, (29) corresponds to constraints on the massless (conformally invariant) sector of the theory. Indeed, both the curl and divergence of Q, vanish. In the Schwinger model it is clear how to separate the massless (negative metric) field from the massive (physical) excitations. The procedure corresponds to the transition from the local to the non-local formulation of Section 2, and consists in introducing the auxiliary field E via ( 13) with the parametrization (25). This leads to the new effective Lagrangian l=
%i#,y + b+ia_c+ + b-i&_ -
;a,7japq+ $awa,E- gE2,
(33)
where
(29)
In the spirit of Ref. [ lo] these constraints are obtained by the gauging of the effective action in (27) as follows:
where W, is an external field. Parametrizing this field as
~=c$-EE.
(34)
The Lagrangian 2 plays the role of the “non-local” QCDz Lagrangian. We see that q is the negative metric, zero mass field in the usual parametrization of the Schwinger model. In the non-abelian formulation, the fields /?, 6 and V take the role of E, q and 4, respectively, the correspondences being given by p = exp( -2i&E), v = exp(2ifiq) as well as (25). According to our discussion in the non-abelian case we expect two first-class constraints, and one secondclass constraint. The first two one obtains by gauging the fermion-eta sector in a way analogous to the first two equations in (30). This leads to the first-class constraints
w, = +av + a,i, one finds
(32)
which replace the constraints (20) and (21) in the abelian case. Condition (35)) when implemented on the physical states, is just the familiar requirement r111
E. Abdalla. K.D. Rothe / Physics Letters B 363 (1995) 85-92
ac(t40 + rl) Iphys)= 0,
(36)
where p is the “potential” of the free fermionic current (37) Condition (36) characterizes the physical Hilbert space of the Schwinger model, and in particular its ground state structure, which turns out to be infinitely degenerate. In the non-abelian case, this role is taken up by the constraints (20) and (2 1) . In the notation of this section, the partition function ( 22) takes the form Zp = / VOpDC- exp (i / L’> ,
(38)
with L’ = -$(~?~q)~ + $(apE)2
+ C_a+E+
$(apCJ2.
(39) The gauging of L’, following the procedure of Ref. 1:lo] corresponds to the replacement of L’ by Cw=L’fW+
(
a_v+a_E-
-5_ J;;
>
,
(40)
8x1 = EC-Xl, ac-
=ac+
6x2
=o,
=o,
Sb_ = --~,y;,.y, - La-q, 2J;;
6b+ = 0,
(43)
and a similar transformation obtained by the substitutions XI H ~2, c& H C~ and bh ++ b,. These symmetries imply the conservation of the BRST currents, f*B’ = c*d*,
arJ1,B) = 09
(44)
with & given by (35). The condition (36) is seen to be equivalent to the BRST condition (45)
Q&PO) =O,
where Q* is the (nilpotent) charge associated with the currents (44), and ]*o) labels the ground states. The matter and negative metric part of fi* separately satisfy a Kac-Moody algebra with level k = 1 and k = - 1, respectively. As is well known, there exists an infinite set of solutions of (45). This is quite unlike the case of QCD2, where the vacuum degeneracy is finite, as we now demonstrate.
with W+ = a++. Expression (40) can be written as 5. The QCD2 vacuum c”= -~(a,r1')2+~(a~E')2+c_a+E'+~(a,c_)2, (41) with 7’ = 77+ 4, E’ = E + q5. Hence the partition functions associated with L” and Cc’coincide, implying the constraint - +c_-a_v+a_E=o,
(42)
which is just the abelian version of (24). The physical Hilbert space of the Schwinger model factorizes into a massive Fock space and a massless one. The condition (36) only implies a restriction on the massless (conformally invariant) sector which describes the ground state of the theory. This restriction is equivalent to the corresponding BRST condition. Indeed, the action associated with the Lagrangian (33) is invariant under the BRST transformation 2J;;Sq
= -kc_,
From our discussion we conclude that the physical states of QCD:! are obtained by solving the BRST conditions Q:B’lO) = 0,
I*} E
xphys,
(46)
with identification of states differing by BRST exact states. In (46)) Qi”’ are the charges associated with the BRST currents (8) or, equivalently, ( 19). We shall preferably work with the currents (19) of the nonlocal formulation, where the BRST condition (46) becomes a restriction on the conformally invariant sector of tirhys describing the ground states /‘Pa) of QCDz: Q:“’ ]‘Po) = 0.
(47)
The crucial observation now is that the solution of (47) in the conformally invariant F - gh - q sector of ( 16) is identical to the solution of the cohomology problem of a level one G/G coset WZW model,
E. Abdalla, K.D. Rothe / Physics Letters B 363 (I 995) 85-92
90
which corresponds to a topological field theory. In the following we shall restrict ourselves to G = SU( 2). The constraints s2, involve the matter currents
of the Wakimoto realization [ 121 of the (level one) Kac-Moody currents Ja and pa: J,‘=a,,,
.I$ = $?t%x,
(48)
the negative metric-field
currents JnT = nd,, - & c
(t”Vid_P-‘),
Ju = -2
tr
1+cv -0 J+ = -7
tr (t”V’-‘id+V)
(49) where a,
dmdkan-m-k,
m,n,k
c; .
ISO>
J;IJ, J) = JJJ, J), = JIJJ)
and d,
are canonically
]d,, amI = am+n, and [ 4m, Al case of U( 1) (Schwinger model),
Since the two light-cone components can be treated independently, we shall omit the subscript f from here on. The solution of the BRST condition (47) is constructed in terms of the highest weight eigenstates ]J, J) of the charges associated with isospin-3 component of the currents (48) and (49) (zero modes in the corresponding Laurent expansion)
J2I.U)
- c
(56) ,
and the ghost currents JZ”” = f‘,&
&d,-m
n,m
(51)
conjugate
pairs,
= marnfn. In the a, = d, = 0, and
$$,, -+ &a~, where the field p is defined in $ (37). An analogous decomposition in terms of zl,, & and & is made 3 for J$, 2. Expressing the BRST charge in terms of the WaIcimoto variables, it can be decomposed into terms of given degrees by attributing to the fields c, d, d and ~f(h,a,ii,c$-) the degree +1(-l), where df are defined by @ = -& (4 f ib). It turns out that it suffices to study the states which are annihilated by the operator Q(O) of lowest degree, which is nilpotent, as well as quadratic in the fundamental excitations: Q”’ [‘PO)= 0,
(57)
with the highest weight condition cI,a,
Q(O) = c ,;+iziJ, .J) = 0,
J;+‘*]J, J) = 0.
(52)
We define our Fock space be requiring that the state IJ, J) be annihilated by the “positive” frequency parts of the currents (48)-(50): J;>,lJ, Jr) = 0,
?>OlJ, J) = 0,
(53)
c;>&
b;>(-lJ, J) = 0.
(54)
J) = 0,
6$lJ,A
=o.
c?,&
+ c?,&.
(58)
n
The total current, as well as the zeroth mode of the energy momentum-tensor (Virasoro operator .Cu) correspondingly take the form
+
In (53) and (54) the subscript n labels the modes in the Laurent expansion of the corresponding operators. Since bU and cU are canonically conjugate fields, we further require
+ 2&c
n
c [:a-,,d,,
: - : ii_,,&
: -f3&
: cC,b’?_,,:] ,
n
(59) ~0 = ~[J(J+
1) - J(J+
+CCn : a-,,d”
+ ii-,&
I)] + gkbb_,c;]
:
(55)
In order to obtain the states satisfying the BRST condition (47)) in terms of the states I./,4 defined above, we follow closely the work of Ref. [ 131 and make use
J Actually there arc some technical subtleties, the reader to Ref. [ 131.
for which we refer
E. Abdalla. K.D. Rothe / Physics Letters B 363 (1995) 85-92
As shown in Ref. [ 131, the physical states must also be annihilated by these operators: @*\I)
= 0,
Lo/WI)) = 0,
(61)
since dti and b turn out to be BRST exact. This implies I= -J - 1, as well as the absence of non-zeromode excitation, and allows one to write the physical states as linear combinations of
= (~o)““(~o)““(c,f)“‘(c,)“-IJ, Implementation imply [ 131
of condition
I9cl) = CoflJ, -(J
+ 1)).
-J (57)
- 1).
(62)
is then found to
(63)
In order to completely classify the vacua, we still need to know the values which J can take. From the representation theory of Kac-Moody algebras with central charge k [ 141 one learns that the allowed values of J are finite and parametrized by
91
for k = integer (s = 1) reduces to just one condition, 2J+l=r,r=l,..., k+ l.Inourcase, k= 1. In the case G = SU( N) we also expect a discrete, growing number of vacua, in accordance with the results of Ref. [ 15 1. In the case of the Schwinger model [ 31 and its generalization to the Cartan subalgebra of U(N) [ 161, the vacuum is infinitely degenerate. In the U( 1) case this is known since the work of Lowenstein and Swieca [ 111; in the framework of Section 4, these vacua are given by J~\I’o)= 1J, -J), and the infinite degeneracy is a consequence of J taking all integer values. In the non-local formulation of QCD2 we have seen that the massive sector (described by Zp) completely separates from the conformally invariant one (describing the vacuum sector). In Ref. [ 171 the S-matrix has been computed up to pole factors describing the bound states. These factors may differ according to the choice of vacuum.
Acknowledgement
where s = 1,...,qandr= 1, . . ..p - 1, where p and q coprime and defined by k + 2 = p/q. For our case k = 1. We therefore conclude that I = 0, 4, that is, we have a two-fold degeneracy of the ground state.
One of the authors (E.A.) is grateful to 0. Aharony, L. Alvarez Gaumt5, and G. Thompson for some useful discussions and correspondence. He also thanks the Alexander von Humboldt Foundation for financial support making this collaboration possible. We also express our gratitude to M.C.B. Abdalla for a careful reading of the manuscript.
6. Summary
References
25+1=1--(s-l)(k+2), :.J+l
=-r+s(k+2),
(64)
iire
Summarizing, we have shown, the BRST conditions implied restrictions on the conformally invariant ( vacuum) sector of QCD2. By systematically exploring these conditions, we have shown that for a given chirality the vacuum of SU(2) - QCD2 is two-fold degenerate. The same conclusion is suggested by other, quite different considerations based on the equivalence of the level k, G/G mode1 on the space Z to the so-called BF theories, which in turn are equivalent to a ChernSimons theory on C x S’ [ 151. The result obtained in ! 151 for G = SU(2) in particular can be interpreted as the existence of k + 1 states. This agrees with the result (64) obtained from representation theory, which
[ 1I E. Abdalla, M.C.B. AbdaIIa and K.D. Rothe, Nonpemrbative Methods in ‘P.vo-Dimensional Quantum Field Theory, World Scientific 1991. [2] E. Abdalla, M.C.B. Abdalla, hepth 9503002, Phys. Rep., to appear. [ 31 J. Schwinger, Phys. Rev. 128 ( 1962) 2425. [4] E. Witten, Ii Nuovo Cimento A 51 ( 1979) 325. [5] E. Abdalla and M.C.B. Abdalla, Int. J. Mod. Phys. A 10 (1995) 1611; Phys. Len. B 337 (1994) 347. [6] E. Witten, Commun. Math. Phys. 92 (1984) 455. [7] A. Polyakov and P. Wiegmann, Phys. Lett. B 131 (1983) 121. [8] D.C. Cabra, K.D. Rothe and EA. Schaposnik, Heidelberg preprint HD-THEP-95-3 1, heptb 9507043. [9] F. Bastianelli, Nucl. Phys. B 361 ( 1991) 555. [ 101 D. Karabali and H.J. Schnitzer, Nucl. Phys. B 329 (1990) 649.
92
E. Abdalla, K.D. Rothe / Physics Letters B 363 (1995) 85-92
[ II] S. Lowenstein and J.A. Swieca, Ann. Phys. 68 (1971) 172.
[ 151 M. Blau and G. Thompson, Nucl. Phys. B 408 (1993) 345.
[ 121 M. Wakimoto, Commun. Math. Phys. 104 (1989) 605. 1131 0. Aharony and 0. Ganor, J. Sonnenschein, S. Yankielowicz and N. Sochen, Nucl. Phys. B 399 ( 1993) 527. [ 141 V.G. Kac and D.A. Kazhdan, Adv. Math. 34 ( 1979) 97.
1161 L.V. Belvedere, K.D. Rothe, B. Schroer and J.A. Scwieca, Nucl. Phys. B 153 (1979) 112. [ 171 E. Abdalla and M.C.B. Abdalla, CERN-TH/95-81, hepth 9503235.
16 November 1995
&
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 363 (1995) 85-92
BRST cohomology and vacuum structure of two-dimensional chromodynamics E. Abdalla’,
K.D. Rothe
lnstitut ftir Theoretische Physik, Universitiit Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
Received 24 July 1995; revised mannscript received 7 September 1995 Editor: P.V. Landshoff
Abstract Using a formulation of QCDz as a perturbed conformally invariant theory involving fertnions, ghosts, as well as positive and negative level Wess-Zumino-Witten fields, we show that the BRST conditions become restrictions on the conformally invariant sector, as described by a G/G topological theory. By solving the corresponding cohomology problem we are led to a finite set of vacua. For G = SU(2) these vacua are two-fold degenerate.
1. Introduction
Quantum Chromodynamics in 1+ 1 dimensions (:QCD2) has been subject of numerous investigations in the past twenty five years [ 1,2]. However, unlike its abelian counterpart, the exactly soluble Schwinger model [ 31, one had up to recently no hint at its exact integrability. Moreover, traditional topological arguments based on instantons suggest that the vacuum of QCDz is unique, unlike the case of the Schwinger model, where this vacuum is known to be infinitely degenerate. However, arguments have been presented I.41 in favor of the existence of a discrete, but finite set of QCD2 vacua in higher representations of the fermions. The recent formulation [ 51 of QCD2 as a perturbed conformally invariant Wess-Zumino-Witten-type theory [6] turns out to provide an appropriate starting point for a dynamical investigation of the physical
Hilbert-space structure of QCD2. The fundamental framework is provided by the BRST analysis of QCDZ in this formulation, as carried out in Ref. [ 81. In particular it will be our aim to investigate the possible existence of degeneracy of the QCD;! vacuum. In this respect it will be useful to point out the parallelisms with the Schwinger model in the decoupled formulation. We shall show that the conformally invariant sector of QCD2 is described by a level-one G/G topological field theory, thus allowing for a complete classification of the ground states. By explicitly solving the cohomology problem for G = SU( 2)) we find that the vacuum is two-fold degenerate in the left- and rightmoving sector, respectively. In order to provide the necessary framework, we briefly review the essential results of Ref. [ 81. In the light-cone gauge A+ = 0, the QCDz-partition function reads 2 ’ Our conventions are: x = (;i), ‘y+ = 2 (:A), y- = 2 cz),
’ Work supported by Alexander von Humboldt Stiftung. On leave Cram the University of Sao Paula, Brazil. 0370-2693/95/%09.50
A* = A0 f At, & ifabcf
@ 1995 Elsevier Science B.V. All rights reserved
.XSD/0370-2693(95)01193-5
and tr(f#)
= a0 f 81, A, = AgP, etc., with LP.Pl
= Sab, fubcfubd
= fCv&d.
=
E. Abdalla.
K.D. Rothe / Physics Letters B 363 (1995) 85-92
right-moving sector, implying the existence served right- and left-moving BRST currents
J
Db_Dc_ exp( i&F) ,
X
with the corresponding
(1)
gauge-fixed
[ 81
Lagrangian
+ ,yiia+,y, + $~iD-$z
CGF = tr $(a+A_)*
a*.P’7
- ib,{cF,
CT}] ,
= 0,
(8)
where
+ tr( b_ia+c_ ) ,
(2)
,
J(B) F = tr [c+,
of con[ 81
where b- c- are Grassman-valued from the gauge-fixing condition.
ghost fields arising
2. Local decoupled formulation
of QCDz
n_
s
--&D_(v)a+(tia_v-‘)
-(i23)
Ka_v-’ + Xl/y! + b_{c_,c_}
z 0,
Making the change of variable A_ = iva-v-l ti2 = 42
-n+ E (3)
7
one is led to the factorized partition function
[ 5,8]
)
Rh
(4)
where
(9)
&v-l[a:(iea_v-l)lv-
+x2x;
qV-‘ia+V
+ b+{c+,c+)
X0
(10)
are first-class constraints, with sZ_ x 0 playing the role of the Gauss law. It is interesting to note that
a-R+ = -v-la+R_
-i(Cv + l)I[Vl
d2xtr[a+(va_v-‘>I2
1,
and
,
z = z’“‘z’o’z” F
=a_ + [va-v-l,
n(v)
v,
(11)
implying ,
(5) D_d+A_
+ (1 +&);A_
=O.
(12)
and Z(O) F
-
/
Z’O’ = xh xexp
exp (i / d2x,j$,y)
DjV,y
Note that the term proportional in the literature.
,
ZJb*lDc* 3. Non-local decoupled formulation ( 1‘/d2+u[b+ia_c+
+ b_ia+c_])
,
(6)
with I [ g] the WZW functional
r[gl = &
s
d2x tr
tr
of QCDZ
The partition function (5) involves 4th order derivatives. In order to reduce these to 2nd order, we introduce an auxiliary field E via the identity
afig-‘apg exp(~Sfh[a,(Vid_V-‘)12)
I
+ &
to CV has been ignored
JJ dr
d*xs~l”g-‘gg-‘d,gg-‘d,g,
(7)
=/DEexp(-il$tr[cE2
0
where g(l,x) = g(x),g(O,x) = ll. The partition function exhibits a BRST symmetry in the left- and
+
J;;F
La+(i4a_v-l)
I>.
(13)
87
E. Abdalla, K.D. Rothe/ Physics Letters B 363 (1995) 85-92
Making the change of variable [ 581 1<= J;;(y)
-&(/TlkY+fl)
(14)
and making use of the Polyakov-Wiegmann identity [71 Ughl
= l-k1 + F[hl
+ & / d2x tr (g-‘GJ+gha_h-‘)
,
(15)
one arrives at the alternative representation [ $81 (0)z,z,, Z=Z, (0)Z@
(16)
with YDPexp{-i( 1 + C,)r[P]},
Zv =
(17)
s
where V = PV, and
be understood from a unified point of view by first making the change of variables from A+ and A_ to the group-valued fields U and V and only then fixing the gauge. At this stage the gauge-fixing procedure no longer involves a Faddeev-Popov determinant. With this procedure both BRST symmetries result alone from the Jacobians involved in the change of variables, as explained in Ref. [ 81. Note also that the presence of the ghosts in (20), (21) is in fact necessary in order for these constraints to be first class (vanishing central charge). Their presence reflects the fact that in the non-abelian case these ghosts originally appear coupled to the gauge field in the partition function. As we shall see shortly, they are absent in the corresponding constraints (35) of the U( 1) Schwinger model, where they are decoupled from the outset. Finally let us rewrite Zp in ( 18) in terms of an auxiliary field C_ as follows: z, =
vp s
J
where s’ = rip-1+
(18) Note that the WZW action enters with negative level -- ( 1 + Cv) , This will be very important in the following section. There exist [ 81 two BRST currents associated with the partition function ( 16) :
J,-(‘) = tr
[c&
- !jb+{c*,c+}]
,
(19)
+
(2)
lX_eis’ ,
I[i
(22)
tr(a+C_)2
e tr( C-p-‘ia+p)
1.
(23)
By gauging the gh - P - p sector following method of Ref. [ lo], one discovers one further constraint
(T)
e/3C_j3-'
- -$id_p-’
(T)
+ {b_,c_}
i&9-P-’ x 0.
(24)
where fi- E XIX! + {b’o’,c’o’}
As emphasized in Ref. [ 51, this constraint is 2nd class with respect to the constraints (20), (2 1) , and serves to determine the auxiliary field C_.
1+cv-
- ~Via_~-’
z 0, si,
E x2/y; + {b?‘, cy’} M 0,
(20) 1 +Gyi, - T
+
v
4. The Schwinger model revisited (21)
represent first class constraints. These shall play a central role in the characterization of the QCD2 vacuum. Note that we have two BRST charges, and hence two BRST conditions on the physical Hilbert space. As discussed in Ref. [9], their symmetric presence can
In order to gain some feeling for the constraints (9),(10),(20)and(21)itisusefultoseewhatthese constraints correspond to in the U( 1) case. In the U( 1) case, CV = 0 (corresponding to decoupled Faddeev-Popov ghost from the outset). Parametrizing V in (3) by
E. Abdalla, K.D. Rothe/ Physics Letters B 363 (1995) 85-92
88
V=exp(i2&$),
(25)
the WZW functional I[ V] and Maxwell term in (5) reduce to
respectively, so that the partition function (4) reads
(27) Notice that 4 is a negative metric field, corresponding to the fact that the WZW action I[ V J enters in (5) with negative level. From (26) follows the equation of motion q
(q ) +;
(28)
f$=o;
the constraints fI+ (9) and ( 10) take the form
J;;
n*==a,
e2
( ) Cl+;
d-eXypx”O.
where 4 = 4 + &I. The e-dependent term cancels against the anomaly arising from the fermionic integration, so that the gauged partition function coincides with (27). Following Ref. [ lo], the variation of the partition function with respect to W, then leads to the constraints (29). Notice that the Klein-Gordon operator (Cl -t $) projects out the massive mode of 4 satisfying (28), leaving one only with the massless mode. Hence, (29) corresponds to constraints on the massless (conformally invariant) sector of the theory. Indeed, both the curl and divergence of Q, vanish. In the Schwinger model it is clear how to separate the massless (negative metric) field from the massive (physical) excitations. The procedure corresponds to the transition from the local to the non-local formulation of Section 2, and consists in introducing the auxiliary field E via ( 13) with the parametrization (25). This leads to the new effective Lagrangian l=
%i#,y + b+ia_c+ + b-i&_ -
;a,7japq+ $awa,E- gE2,
(33)
where
(29)
In the spirit of Ref. [ lo] these constraints are obtained by the gauging of the effective action in (27) as follows:
where W, is an external field. Parametrizing this field as
~=c$-EE.
(34)
The Lagrangian 2 plays the role of the “non-local” QCDz Lagrangian. We see that q is the negative metric, zero mass field in the usual parametrization of the Schwinger model. In the non-abelian formulation, the fields /?, 6 and V take the role of E, q and 4, respectively, the correspondences being given by p = exp( -2i&E), v = exp(2ifiq) as well as (25). According to our discussion in the non-abelian case we expect two first-class constraints, and one secondclass constraint. The first two one obtains by gauging the fermion-eta sector in a way analogous to the first two equations in (30). This leads to the first-class constraints
w, = +av + a,i, one finds
(32)
which replace the constraints (20) and (21) in the abelian case. Condition (35)) when implemented on the physical states, is just the familiar requirement r111
E. Abdalla. K.D. Rothe / Physics Letters B 363 (1995) 85-92
ac(t40 + rl) Iphys)= 0,
(36)
where p is the “potential” of the free fermionic current (37) Condition (36) characterizes the physical Hilbert space of the Schwinger model, and in particular its ground state structure, which turns out to be infinitely degenerate. In the non-abelian case, this role is taken up by the constraints (20) and (2 1) . In the notation of this section, the partition function ( 22) takes the form Zp = / VOpDC- exp (i / L’> ,
(38)
with L’ = -$(~?~q)~ + $(apE)2
+ C_a+E+
$(apCJ2.
(39) The gauging of L’, following the procedure of Ref. 1:lo] corresponds to the replacement of L’ by Cw=L’fW+
(
a_v+a_E-
-5_ J;;
>
,
(40)
8x1 = EC-Xl, ac-
=ac+
6x2
=o,
=o,
Sb_ = --~,y;,.y, - La-q, 2J;;
6b+ = 0,
(43)
and a similar transformation obtained by the substitutions XI H ~2, c& H C~ and bh ++ b,. These symmetries imply the conservation of the BRST currents, f*B’ = c*d*,
arJ1,B) = 09
(44)
with & given by (35). The condition (36) is seen to be equivalent to the BRST condition (45)
Q&PO) =O,
where Q* is the (nilpotent) charge associated with the currents (44), and ]*o) labels the ground states. The matter and negative metric part of fi* separately satisfy a Kac-Moody algebra with level k = 1 and k = - 1, respectively. As is well known, there exists an infinite set of solutions of (45). This is quite unlike the case of QCD2, where the vacuum degeneracy is finite, as we now demonstrate.
with W+ = a++. Expression (40) can be written as 5. The QCD2 vacuum c”= -~(a,r1')2+~(a~E')2+c_a+E'+~(a,c_)2, (41) with 7’ = 77+ 4, E’ = E + q5. Hence the partition functions associated with L” and Cc’coincide, implying the constraint - +c_-a_v+a_E=o,
(42)
which is just the abelian version of (24). The physical Hilbert space of the Schwinger model factorizes into a massive Fock space and a massless one. The condition (36) only implies a restriction on the massless (conformally invariant) sector which describes the ground state of the theory. This restriction is equivalent to the corresponding BRST condition. Indeed, the action associated with the Lagrangian (33) is invariant under the BRST transformation 2J;;Sq
= -kc_,
From our discussion we conclude that the physical states of QCD:! are obtained by solving the BRST conditions Q:B’lO) = 0,
I*} E
xphys,
(46)
with identification of states differing by BRST exact states. In (46)) Qi”’ are the charges associated with the BRST currents (8) or, equivalently, ( 19). We shall preferably work with the currents (19) of the nonlocal formulation, where the BRST condition (46) becomes a restriction on the conformally invariant sector of tirhys describing the ground states /‘Pa) of QCDz: Q:“’ ]‘Po) = 0.
(47)
The crucial observation now is that the solution of (47) in the conformally invariant F - gh - q sector of ( 16) is identical to the solution of the cohomology problem of a level one G/G coset WZW model,
E. Abdalla, K.D. Rothe / Physics Letters B 363 (I 995) 85-92
90
which corresponds to a topological field theory. In the following we shall restrict ourselves to G = SU( 2). The constraints s2, involve the matter currents
of the Wakimoto realization [ 121 of the (level one) Kac-Moody currents Ja and pa: J,‘=a,,,
.I$ = $?t%x,
(48)
the negative metric-field
currents JnT = nd,, - & c
(t”Vid_P-‘),
Ju = -2
tr
1+cv -0 J+ = -7
tr (t”V’-‘id+V)
(49) where a,
dmdkan-m-k,
m,n,k
c; .
ISO>
J;IJ, J) = JJJ, J), = JIJJ)
and d,
are canonically
]d,, amI = am+n, and [ 4m, Al case of U( 1) (Schwinger model),
Since the two light-cone components can be treated independently, we shall omit the subscript f from here on. The solution of the BRST condition (47) is constructed in terms of the highest weight eigenstates ]J, J) of the charges associated with isospin-3 component of the currents (48) and (49) (zero modes in the corresponding Laurent expansion)
J2I.U)
- c
(56) ,
and the ghost currents JZ”” = f‘,&
&d,-m
n,m
(51)
conjugate
pairs,
= marnfn. In the a, = d, = 0, and
$$,, -+ &a~, where the field p is defined in $ (37). An analogous decomposition in terms of zl,, & and & is made 3 for J$, 2. Expressing the BRST charge in terms of the WaIcimoto variables, it can be decomposed into terms of given degrees by attributing to the fields c, d, d and ~f(h,a,ii,c$-) the degree +1(-l), where df are defined by @ = -& (4 f ib). It turns out that it suffices to study the states which are annihilated by the operator Q(O) of lowest degree, which is nilpotent, as well as quadratic in the fundamental excitations: Q”’ [‘PO)= 0,
(57)
with the highest weight condition cI,a,
Q(O) = c ,;+iziJ, .J) = 0,
J;+‘*]J, J) = 0.
(52)
We define our Fock space be requiring that the state IJ, J) be annihilated by the “positive” frequency parts of the currents (48)-(50): J;>,lJ, Jr) = 0,
?>OlJ, J) = 0,
(53)
c;>&
b;>(-lJ, J) = 0.
(54)
J) = 0,
6$lJ,A
=o.
c?,&
+ c?,&.
(58)
n
The total current, as well as the zeroth mode of the energy momentum-tensor (Virasoro operator .Cu) correspondingly take the form
+
In (53) and (54) the subscript n labels the modes in the Laurent expansion of the corresponding operators. Since bU and cU are canonically conjugate fields, we further require
+ 2&c
n
c [:a-,,d,,
: - : ii_,,&
: -f3&
: cC,b’?_,,:] ,
n
(59) ~0 = ~[J(J+
1) - J(J+
+CCn : a-,,d”
+ ii-,&
I)] + gkbb_,c;]
:
(55)
In order to obtain the states satisfying the BRST condition (47)) in terms of the states I./,4 defined above, we follow closely the work of Ref. [ 131 and make use
J Actually there arc some technical subtleties, the reader to Ref. [ 131.
for which we refer
E. Abdalla. K.D. Rothe / Physics Letters B 363 (1995) 85-92
As shown in Ref. [ 131, the physical states must also be annihilated by these operators: @*\I)
= 0,
Lo/WI)) = 0,
(61)
since dti and b turn out to be BRST exact. This implies I= -J - 1, as well as the absence of non-zeromode excitation, and allows one to write the physical states as linear combinations of
= (~o)““(~o)““(c,f)“‘(c,)“-IJ, Implementation imply [ 131
of condition
I9cl) = CoflJ, -(J
+ 1)).
-J (57)
- 1).
(62)
is then found to
(63)
In order to completely classify the vacua, we still need to know the values which J can take. From the representation theory of Kac-Moody algebras with central charge k [ 141 one learns that the allowed values of J are finite and parametrized by
91
for k = integer (s = 1) reduces to just one condition, 2J+l=r,r=l,..., k+ l.Inourcase, k= 1. In the case G = SU( N) we also expect a discrete, growing number of vacua, in accordance with the results of Ref. [ 15 1. In the case of the Schwinger model [ 31 and its generalization to the Cartan subalgebra of U(N) [ 161, the vacuum is infinitely degenerate. In the U( 1) case this is known since the work of Lowenstein and Swieca [ 111; in the framework of Section 4, these vacua are given by J~\I’o)= 1J, -J), and the infinite degeneracy is a consequence of J taking all integer values. In the non-local formulation of QCD2 we have seen that the massive sector (described by Zp) completely separates from the conformally invariant one (describing the vacuum sector). In Ref. [ 171 the S-matrix has been computed up to pole factors describing the bound states. These factors may differ according to the choice of vacuum.
Acknowledgement
where s = 1,...,qandr= 1, . . ..p - 1, where p and q coprime and defined by k + 2 = p/q. For our case k = 1. We therefore conclude that I = 0, 4, that is, we have a two-fold degeneracy of the ground state.
One of the authors (E.A.) is grateful to 0. Aharony, L. Alvarez Gaumt5, and G. Thompson for some useful discussions and correspondence. He also thanks the Alexander von Humboldt Foundation for financial support making this collaboration possible. We also express our gratitude to M.C.B. Abdalla for a careful reading of the manuscript.
6. Summary
References
25+1=1--(s-l)(k+2), :.J+l
=-r+s(k+2),
(64)
iire
Summarizing, we have shown, the BRST conditions implied restrictions on the conformally invariant ( vacuum) sector of QCD2. By systematically exploring these conditions, we have shown that for a given chirality the vacuum of SU(2) - QCD2 is two-fold degenerate. The same conclusion is suggested by other, quite different considerations based on the equivalence of the level k, G/G mode1 on the space Z to the so-called BF theories, which in turn are equivalent to a ChernSimons theory on C x S’ [ 151. The result obtained in ! 151 for G = SU(2) in particular can be interpreted as the existence of k + 1 states. This agrees with the result (64) obtained from representation theory, which
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