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The Wess-Zumino term in the chiral bag model
Volume 182, number 3,4
THE W E S S - Z U M I N O
PHYSICS LETTERS B
25 December 1986
T E R M IN T H E C H I R A L BAG M O D E L ~
M a n n q u e R H O i and Ismail Z A H E D Physics Department, State University of New York at Stony BrOok, Stony Brook, N Y 11794, USA Received 27 September 1986
It is shown that local chiral boundary conditions make consistent an otherwise anomalous skyrmion with defect. A careful analysis of the induced action in the vacuum of the chiral bag model yields the missing part of the Wess-Zumino term. The issues of spin-statistics, hypercharge and strangeness are all consistently described in the chiral bag model.
It is well known that the Wess-Zumino term plays an important role in the spin-statistics assignment of the SU(N)f skyrmion [1 ]. It is also known that this term accounts for most of the anomalous but soft meson decays, and provides a consistency ground for the pertinent low-energy effective theories [1 ]. It is even believed that certain classes of anomalous theories can be made consistent at the quantum level by adding Wess-Zumino terms to the effective action [2]. It is also believed that the Wess-Zumino term acts as a monopole in the meson configuration space, much like the fermion-monopole system in the configuration space of particles [3]. Is there a consistent model for SU(N)f skyrmions with defects? Since the spin-statistics character of the skyrmion stems from the structure of the Wess-Zumino term, the immediate question is how the latter gets modified by the presence of the bag as a defect. The answer to this question will clarify key issues related to spin, isospin, hypercharge, strangeness, etc. in the context of the two-phase models [4]. We show in this letter that the usual local chiral boundary conditions yield naturally a Wess-Zumino term in the fermion-induced action, making consistent an otherwise anomalous model. Our proof will be explicitly carried out in (1 + 1) dimensions and should have a straightforward extension to (3 + 1) dimensions. To appreciate the point, notice that if we were to consider the Wess-Zumino term in the complementary part of a static bag B = V X S 1 in (1 + 1) dimensions W-3 = 2m~3 f d3x eUVaTr(LuLz, L c ) , V/xstx[0,1]
(1)
with L u = g+Ol~g, g = exp ( - 2 i v Y 0 ) and 0 = OaT a, then its variation yields
-- 3 = 67r• 3 f fW
1 d2x e uv Tr((g +fig) LuLv)+ 6/rt~3 f ds f
'VxS 1
0
d oue uvc~Tr((g +6 g ) L v r o ) ,
(2)
b'VxS 1
where we have used the Maurer-Cartan equation
6L u = Ou(g*fg ) + [Lu, (g+fg)] .
(3)
The normalization K3 in (1), (2) will be discussed below. Notice that since the bag constitutes a manifold with Supported in part by the US Department of Energy under Contract No. DE-AC02-76ER13001 with the State University of New York. Permanent address: Service de Physique Thforique, CEN Saclay, F-91191 Gif-sur-Yvette, France.
274
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PHYSICS LETTERS B
Volume 182, number 3,4
25 December 1986
5 Fig. 1. Space-time × [0,1] sheet spanned by a (1 + 1)-dimensional bag.
Ii
V.S I
boundaries (0V¢) as illustrated in fig. 1, the contribution of (2) to the equations of motion depends explicitly on the h o m o t o p y extension via the surface term. This term is unphysical and should be removed. This can be achieved by filhng the bag B = V X S 1 with dynamical quarks, chirally coupled at the surface aB. In euclidean space, this translates into
~k(x, t) = O,
{exp [i 2x/~0 (fl, t) 3'5 ] - ~} ~b(fl, t) = 0 ,
(4)
with the boundary at x =/3. We should point out that in terms of (4), the Sturm-Liouville problem is completely determined and consistent, a point well appreciated in chiral bag phenomenology. This scheme, however, is not unique since we could have used spectral boundary conditions [5] as well. The latter, however, are not local and do not seem to be amenable to a simple physical interpretation. To solve (4), we will use the Weyl representation for the Dirac algebra, 3,O = a 1,
3, l = - a 2,
7 5 = i 7 0 7 1 = o 3.
The general solution to (4) for a time-dependent chiral angle 0 (/3, t) reads ~b(x, t) = exp {-i2x/~[~(1 + 7"5) 0_ + ½(1 - 3'5) 0+]} ~b0(x, t ) ,
(5)
where 0_+ --- O+(x +-t). ~o(X, t) is the usual MIT bag wavefunction. The boundary condition in (4) is fulfilled iff 0 _ 0 3 - t) - 0 + ~ +
t) = 0(/3, t ) .
(6)
There is no other constraint on the two light-cone functions 0_+. To pin down any induced Wess-Zumino term in the fermion vacuum requires a time-dependent boundary condition and thus a time-dependent calculation. This rules out a canonical analysis, since we cannot define a suitable hamiltonian for the time-dependent problem. We can, however, investigate the induced action by computing the fermion determinant of the bagged fields. If we denote by ~0 the euclidean Dirac operator in (4), we have e x p ( - S F) = det(00/~0),
i.e.
1 S F = - t r Ln(~0/~0) = - f ds Tr(Os~ 0 ~01), 0
(7)
which follows from a naive h o m o t o p y extension 0 (x, t) -+ 0 (x, t, s) and such that 0 (x, t, s = 1) = 0 (x, t) and 0 (x, t, s = 0) = 0. From (4), we conclude that 1
S F = --
f a, Tr<~B 0s [exp(i2v~0~5)
- ~] ~ o l ) ,
(8)
0 where A B is the boundary 6-function, and the trace is over spin, isospin, space and time. As it stands (8) is ill-defreed and requires proper regularization. This is naturally achieved by point-splitting in time, SF=--
lim f
dxdtdsABTr(Osexp(i2x/-ffO3,s)So(x,t+e;x,t-e)).
(9)
e~0+ B× [0,1] Using the time-dependent solutions (5), we construct the confined Dirac propagator So in terms of the MIT propagator SO as follows 275
Volume 182, number 3,4 ,
PHYSICS LETTERS B
(exp [-i2x/~0
S o (x, t; x , t') =
( x - t)] -
× S o ( x , t;x', t')
( + 12x/~O÷(x + t')]
0 ) exp [ - i 2X/n0+(x + t)]
exp[
,
0
25 December 1986
+
0
_
,
,
exp[ 12V~0 (x + t ) ]
)
.
(10)
]'he presence of A B in (9) requires that x = x' =/3 G 0B. Due to the confining boundary condition, the short-distance behavior of SO is modified. From the multiple reflection analysis of the propagator [6], we conclude that to leading order
lira
So(X , t + e;x', t - e) = ¼(1 + 7~)(70/4ne)(3 - 7~) + O ( 1 ) ,
(11)
with 7~ = 3"1 • n/3 and n~ = n+ = ±1. Since we are only interested in the induced Wess-Zumino action inside the bag, we will focus on Im SF to leading order in the derivative expansion. Inserting (10), (11) into (9), and using (6)
gives 1
hnSF=4ix,ff f ~ fdxdtTr(OsO~tOOxO)+ .... 0
(12)
B
where ... stands for higher derivative terms. If we impose chiral invariance on (12), we obtain Im S F = ~3
f
Bx[0,1]
d3x e guaTr(LuLvL~) = 27rW3 ,
(13)
with K3 = Nc/247r2; Nc is the number of colored quarks. This term is precisely the complementary part of (1) as defined outside the bag, and t¢3 is the correct normalization factor. Thus, W3 + Im S F = 2m¢ 3 f d3x e~V~Tr(LuLvL~), R×SIX[0,1]
(14)
which is the expected expression for the Wess-Zumino term in (1 + 1) dimensions. Mthough this proof is specific to (1 + 1) dimensions, we believe that our result holds true in (3 + 1) dimensions where the boundary-value problem becomes more involved. Since such concepts as spin, isospin, hypercharge and strangeness are all intricately related to the behaviour of the Wess-Zumino term under adiabatic rotations in either space, isospace, etc., we conclude that they are all accounted for properly in the chiral bag model through the vacuum induced part of the Wess-Zumino term and its complementary part in the meson sector. A similar conclusion has been recently reached by Niemi [7], using a generalized index theorem on open manifolds. An actual construction of a realistic SU(3)f chiral bag remains, however, to be accomplished. We would like to thank Gerry Brown, Hidenaga Yamagishi and Ed Witten for discussions.
References [1] E. Witten, Nucl. Phys. B 223 (1983) 422. [2] L.D. Faddeev and S. Slatashvili, Phys. Lett. B 167 (1985) 255. [3] M.V. Berry, Proc. R. Soc. A 392 (1984) 45; H. Sonoda, Nucl. Phys. B 266 (1986) 410. [4] G.E. Brown and M. Rho, The chiral bag, Comm. Part. Nucl. Phys., to be published. [5] H. Donnelly, Am. J. Math. 99 (1976) 879. [6] J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51 (1983) 1518; T.H. Hansson and R.L. Jaffe, Phys. Rev. D 28 (1983) 882; I. Zahed, Phys. Rev. D 30 (1984) 2221. [7] A.J. Niemi, Phys. Rev. Lett. 54 (1985) 631. 276
THE W E S S - Z U M I N O
PHYSICS LETTERS B
25 December 1986
T E R M IN T H E C H I R A L BAG M O D E L ~
M a n n q u e R H O i and Ismail Z A H E D Physics Department, State University of New York at Stony BrOok, Stony Brook, N Y 11794, USA Received 27 September 1986
It is shown that local chiral boundary conditions make consistent an otherwise anomalous skyrmion with defect. A careful analysis of the induced action in the vacuum of the chiral bag model yields the missing part of the Wess-Zumino term. The issues of spin-statistics, hypercharge and strangeness are all consistently described in the chiral bag model.
It is well known that the Wess-Zumino term plays an important role in the spin-statistics assignment of the SU(N)f skyrmion [1 ]. It is also known that this term accounts for most of the anomalous but soft meson decays, and provides a consistency ground for the pertinent low-energy effective theories [1 ]. It is even believed that certain classes of anomalous theories can be made consistent at the quantum level by adding Wess-Zumino terms to the effective action [2]. It is also believed that the Wess-Zumino term acts as a monopole in the meson configuration space, much like the fermion-monopole system in the configuration space of particles [3]. Is there a consistent model for SU(N)f skyrmions with defects? Since the spin-statistics character of the skyrmion stems from the structure of the Wess-Zumino term, the immediate question is how the latter gets modified by the presence of the bag as a defect. The answer to this question will clarify key issues related to spin, isospin, hypercharge, strangeness, etc. in the context of the two-phase models [4]. We show in this letter that the usual local chiral boundary conditions yield naturally a Wess-Zumino term in the fermion-induced action, making consistent an otherwise anomalous model. Our proof will be explicitly carried out in (1 + 1) dimensions and should have a straightforward extension to (3 + 1) dimensions. To appreciate the point, notice that if we were to consider the Wess-Zumino term in the complementary part of a static bag B = V X S 1 in (1 + 1) dimensions W-3 = 2m~3 f d3x eUVaTr(LuLz, L c ) , V/xstx[0,1]
(1)
with L u = g+Ol~g, g = exp ( - 2 i v Y 0 ) and 0 = OaT a, then its variation yields
-- 3 = 67r• 3 f fW
1 d2x e uv Tr((g +fig) LuLv)+ 6/rt~3 f ds f
'VxS 1
0
d oue uvc~Tr((g +6 g ) L v r o ) ,
(2)
b'VxS 1
where we have used the Maurer-Cartan equation
6L u = Ou(g*fg ) + [Lu, (g+fg)] .
(3)
The normalization K3 in (1), (2) will be discussed below. Notice that since the bag constitutes a manifold with Supported in part by the US Department of Energy under Contract No. DE-AC02-76ER13001 with the State University of New York. Permanent address: Service de Physique Thforique, CEN Saclay, F-91191 Gif-sur-Yvette, France.
274
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PHYSICS LETTERS B
Volume 182, number 3,4
25 December 1986
5 Fig. 1. Space-time × [0,1] sheet spanned by a (1 + 1)-dimensional bag.
Ii
V.S I
boundaries (0V¢) as illustrated in fig. 1, the contribution of (2) to the equations of motion depends explicitly on the h o m o t o p y extension via the surface term. This term is unphysical and should be removed. This can be achieved by filhng the bag B = V X S 1 with dynamical quarks, chirally coupled at the surface aB. In euclidean space, this translates into
~k(x, t) = O,
{exp [i 2x/~0 (fl, t) 3'5 ] - ~} ~b(fl, t) = 0 ,
(4)
with the boundary at x =/3. We should point out that in terms of (4), the Sturm-Liouville problem is completely determined and consistent, a point well appreciated in chiral bag phenomenology. This scheme, however, is not unique since we could have used spectral boundary conditions [5] as well. The latter, however, are not local and do not seem to be amenable to a simple physical interpretation. To solve (4), we will use the Weyl representation for the Dirac algebra, 3,O = a 1,
3, l = - a 2,
7 5 = i 7 0 7 1 = o 3.
The general solution to (4) for a time-dependent chiral angle 0 (/3, t) reads ~b(x, t) = exp {-i2x/~[~(1 + 7"5) 0_ + ½(1 - 3'5) 0+]} ~b0(x, t ) ,
(5)
where 0_+ --- O+(x +-t). ~o(X, t) is the usual MIT bag wavefunction. The boundary condition in (4) is fulfilled iff 0 _ 0 3 - t) - 0 + ~ +
t) = 0(/3, t ) .
(6)
There is no other constraint on the two light-cone functions 0_+. To pin down any induced Wess-Zumino term in the fermion vacuum requires a time-dependent boundary condition and thus a time-dependent calculation. This rules out a canonical analysis, since we cannot define a suitable hamiltonian for the time-dependent problem. We can, however, investigate the induced action by computing the fermion determinant of the bagged fields. If we denote by ~0 the euclidean Dirac operator in (4), we have e x p ( - S F) = det(00/~0),
i.e.
1 S F = - t r Ln(~0/~0) = - f ds Tr(Os~ 0 ~01), 0
(7)
which follows from a naive h o m o t o p y extension 0 (x, t) -+ 0 (x, t, s) and such that 0 (x, t, s = 1) = 0 (x, t) and 0 (x, t, s = 0) = 0. From (4), we conclude that 1
S F = --
f a, Tr<~B 0s [exp(i2v~0~5)
- ~] ~ o l ) ,
(8)
0 where A B is the boundary 6-function, and the trace is over spin, isospin, space and time. As it stands (8) is ill-defreed and requires proper regularization. This is naturally achieved by point-splitting in time, SF=--
lim f
dxdtdsABTr(Osexp(i2x/-ffO3,s)So(x,t+e;x,t-e)).
(9)
e~0+ B× [0,1] Using the time-dependent solutions (5), we construct the confined Dirac propagator So in terms of the MIT propagator SO as follows 275
Volume 182, number 3,4 ,
PHYSICS LETTERS B
(exp [-i2x/~0
S o (x, t; x , t') =
( x - t)] -
× S o ( x , t;x', t')
( + 12x/~O÷(x + t')]
0 ) exp [ - i 2X/n0+(x + t)]
exp[
,
0
25 December 1986
+
0
_
,
,
exp[ 12V~0 (x + t ) ]
)
.
(10)
]'he presence of A B in (9) requires that x = x' =/3 G 0B. Due to the confining boundary condition, the short-distance behavior of SO is modified. From the multiple reflection analysis of the propagator [6], we conclude that to leading order
lira
So(X , t + e;x', t - e) = ¼(1 + 7~)(70/4ne)(3 - 7~) + O ( 1 ) ,
(11)
with 7~ = 3"1 • n/3 and n~ = n+ = ±1. Since we are only interested in the induced Wess-Zumino action inside the bag, we will focus on Im SF to leading order in the derivative expansion. Inserting (10), (11) into (9), and using (6)
gives 1
hnSF=4ix,ff f ~ fdxdtTr(OsO~tOOxO)+ .... 0
(12)
B
where ... stands for higher derivative terms. If we impose chiral invariance on (12), we obtain Im S F = ~3
f
Bx[0,1]
d3x e guaTr(LuLvL~) = 27rW3 ,
(13)
with K3 = Nc/247r2; Nc is the number of colored quarks. This term is precisely the complementary part of (1) as defined outside the bag, and t¢3 is the correct normalization factor. Thus, W3 + Im S F = 2m¢ 3 f d3x e~V~Tr(LuLvL~), R×SIX[0,1]
(14)
which is the expected expression for the Wess-Zumino term in (1 + 1) dimensions. Mthough this proof is specific to (1 + 1) dimensions, we believe that our result holds true in (3 + 1) dimensions where the boundary-value problem becomes more involved. Since such concepts as spin, isospin, hypercharge and strangeness are all intricately related to the behaviour of the Wess-Zumino term under adiabatic rotations in either space, isospace, etc., we conclude that they are all accounted for properly in the chiral bag model through the vacuum induced part of the Wess-Zumino term and its complementary part in the meson sector. A similar conclusion has been recently reached by Niemi [7], using a generalized index theorem on open manifolds. An actual construction of a realistic SU(3)f chiral bag remains, however, to be accomplished. We would like to thank Gerry Brown, Hidenaga Yamagishi and Ed Witten for discussions.
References [1] E. Witten, Nucl. Phys. B 223 (1983) 422. [2] L.D. Faddeev and S. Slatashvili, Phys. Lett. B 167 (1985) 255. [3] M.V. Berry, Proc. R. Soc. A 392 (1984) 45; H. Sonoda, Nucl. Phys. B 266 (1986) 410. [4] G.E. Brown and M. Rho, The chiral bag, Comm. Part. Nucl. Phys., to be published. [5] H. Donnelly, Am. J. Math. 99 (1976) 879. [6] J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51 (1983) 1518; T.H. Hansson and R.L. Jaffe, Phys. Rev. D 28 (1983) 882; I. Zahed, Phys. Rev. D 30 (1984) 2221. [7] A.J. Niemi, Phys. Rev. Lett. 54 (1985) 631. 276