We critically re-examine our earlier derivation of the effective low energy action for QCD in 4 dimensions with chiral fields transforming non-trivially under both color and flavor, using the method of anomaly integration. We find several changes with respect to our previous results, leading to much more compact expressions, and making it easier to compare with results of other approaches to the same problem. With the amended effective action, we find that there are no stable soliton solutions. In the context of the quark soliton program, we interpret this as an indication that the full low-energy effective action must include additional terms, reflecting possible modifications at short distances and/or the non-trivial structure of the gauge fields in the vacuum, such as  \neq 0. Such terms are absent in the formalism based on anomaly integration

Frishman, Yitzhak

Hanany, A

Karliner, M M

arXiv

CERN Document Server

WIS-95/25/June-PH
TAUP-2264-95
hep-ph/9507206
On the Stability of Quark Solitons in QCD
?
Yitzhak Frishman
y
and Amihay Hanany
z
Department of Particle Physics
Weizmann Institute of Science
76100 Rehovot Israel
Marek Karliner
x
School of Physics and Astronomy
Beverly and Raymond Sackler
Faculty of Exact Sciences
Ramat Aviv Tel-Aviv, 69987, Israel
ABSTRACT
We critically re-examine our earlier derivation of the eective low energy action for QCD in
4 dimensions with chiral elds transforming non-trivially under both color and flavor, using the
method of anomaly integration. We nd several changes with respect to our previous results,
leading to much more compact expressions, and making it easier to compare with results of other
approaches to the same problem. With the amended eective action, we nd that there are
no stable soliton solutions. In the context of the quark soliton program, we interpret this as
an indication that the full low-energy eective action must include additional terms, reflecting
possible modications at short distances and/or the non-trivial structure of the gauge elds in
the vacuum, such as
〈
F 2

6= 0. Such terms are absent in the formalism based on anomaly
integration.
? Research supported by the Israel Science Foundation administered by the Israel Academy
of Sciences and Humanities.
y e-mail: fnfrishm@wicc.weizmann.ac.il
z e-mail: ftami@wicc.weizmann.ac.il
x e-mail: marek@vm.tau.ac.il
1
1. Introduction and Motivation
In the recent years some progress have been made towards establishing the
connection between the phenomenologically successful non-relativistic constituent-
quark model (NRQM), and QCD’s fundamental degrees of freedom. Kaplan
[1]
proposed a physical picture combining some of the features of the chiral quark
model
[2]
and the skyrmion
[3−5]
. It was postulated that at distances smaller than
the connement scale but large enough to allow for nonperturbative phenomena
the eective dynamics of QCD is described by chiral dynamics of a bosonic eld
which takes values in U(NcNf). This eective theory admits classical soliton
solutions. Assuming that they are stable and may be quantized semiclassically,
one then nds that these solitons are extended objects with spin 1=2, and that
they belong to the fundamental representation of color and flavor. Their mass is of
order QCD and radius of order 1=QCD .
[6]
It is very tempting to identify them as
the constituent quarks. Thus the constituent quarks in this model are \skyrmions"
in color space.
It turns out that in 2 space-time dimensions this picture is exact
[7]
. Thanks
to exact non-abelian bosonization one can rewrite the action of QCD2 in terms of
purely bosonic variables, which are chiral elds 2 U(NcNf ). It is then straight-
forward to demonstrate that the only non-trivial static solutions of the classical
equations of motion are those which contain either a soliton and an anti-soliton or
Nc solitons. The solitons transform under both flavor and color, yet their bound
states are color singlets and have the quantum numbers of baryons and mesons.
In four space-time dimensions there is no exact bosonization, and therefore
any attempt at derivation of a similar picture in four dimensions must rely on
2
certain approximations. In our previous work
[8]
we have derived the approximate
low-energy eective chiral lagrangian with target space in U(NcNf ) using the
approach based on integration of the anomaly equations
[9−12]
. Equivalent results
were independently obtained in ref. 13, using an a priori dierent approach.
The purpose of the current work is to critically re-examine the results of ref. 8,
with particular emphasis on the question whether the action we have derived can
support stable, time independent classical solutions. The existence of such solu-
tions appears to us to be a necessary condition for establishing the physical picture
in which constituent quarks are solitons of a low-energy eective action of QCD in
four dimensions.
We nd several changes with respect to our previous results, leading to much
more compact expressions, and making it easier to compare with results of other
approaches to the same problem, and in particular with the action proposed by
Kaplan
[1]
.
With the amended eective action, we nd that there are no stable soliton
solutions. In the context of the quark soliton program, we interpret this as an
indication that the full low-energy eective action must include additional terms,
reflecting possible modications at short distances and/or the non-trivial structure
of the gauge elds in the vacuum, such as
〈
F 2

6= 0. Such terms are absent in the
formalism based on anomaly integration.
The layout of the paper is as follows. In Section 2 we rederive the low energy
classical action, resulting in much more compact nal expressions. In Section 3 we
examine the necessary conditions for existence of stable, time independent classical
solutions with nite energy, and conclude that in order for such solutions to exist,
3
the eective action must contain additional terms which are not present in our
derivation. Section 3 is devoted to discussion and interpretation of the results.
2. Derivation of the eective action
We follow the conventions and notation of ref. 8.
The variation of the determinant under the axial transformation of the external
elds, including terms up to zeroth power of the cuto , is given by
−i
 logZ

=
1
42



1
4
FF +
1
12
HH −
2i
3
(AAF +AFA
+ FAA)−
8
3
AAAA

+ 162DA

+
2i
3
[DF; A
] +
i
3
[F ; D
A]
+
1
3
fDA; AA
g − 2ADA
A−
−
2
3
fDA; fA; Agg+
1
3
(DDD
A)

+O(−2)
(1)
where F = @V −@V + i [V; V ] + i [A; A] = 12(L +R); DA = @A +
i [V; A] and H =
1
2 (R − L) = (DA −DA). (cf. eq. (4.1) of ref. 8).
V is the external source coupled to the vector current of the fermions, while A
is the external source coupled to the axial current of the fermions.
An action which has equation (1) as its variation is given by
−i logZ1 =−
22
2
Z
Tr(AA
) + S5CS [R]− S
5
CS [L]
−
i
482
Z
 Tr

i
2
(R + L) fL; Rg+ (LL −
1
2
LR +RR)LR

−
1
122
Z
Tr

1
4
(F)
2 − iF  [A; A]−
1
2
(DA
)2 − (AA)
2

(2)
4
where S5CS is the ve dimensional Chern-Simons action
5
S5CS [R] =
1
242
Z
 Tr(R@R@R +
3
2
iRRR@R −
3
5
RRRRR):
(3)
To derive equation (2), it is useful to employ
S5CS [R] =
1
82
Z
d5x Tr [R (@R@R + i fRR; @Rg −RRRR)]
+
1
242
Z
d4x Tr

R

fR ; @Rg+
3
2
iRRR

:
(4)
For the special case where R = @! + i [R; !] we have
S5CS [R] =
1
242
Z
d4x Tr

!

@R@R +
1
2
i fRR ; @Rg −
1
2
iR@RR

:
(5)
The resulting eective action takes the form
Se =
2
22
Z
Tr

DUD
U−1 − (L −R)
2

−
1
2402
Z
 Tr(JJJJJ)
+
i
482
Z
 Tr

i
2
R

R; J
}
+RRR J
+
1
2
R JR J − J J JR

−
i
482
Z
 Tr

i
2
(R + L) fL; Rg+ (LL −
1
2
LR +RR)LR

+
i
482
Z
 Tr

i
2
(U−1RU + L)
n
L; R
U

o
+ (LL −
1
2
LR
U
 +R
U
R
U
 )LR
U


+
1
1922
Z
Tr

[2(D(U
−1DU))2 − (DU
−1DU)
2
− 4i(LD
U−1DU +RD
UDU−1) + 2(U−1RUL
 − LR
)]
+ 16[iF  [A; A] +
1
2
(DA
)2 + (AA)
2)]

(6)
6
where DU = @U + iRU − iUL, DU−1 = @U−1 + iLU−1 − iU−1R, J =
U−1i@U and J = −UJU−1.
In order to make contact with the usual formulation of QCD, we write down
the eective action for L = R = G in eq. (6), to obtain
Se =
2
22
Z
Tr(DUD
U−1)−
1
2402
Z
 Tr(JJJJJ)
+
i
482
Z
 Tr

i
2
G

G; J
}
+ (GG +
1
2
G J + J J)G J

+
i
482
Z
 Tr

i
2
(U−1GU +G)
n
G; G
U

o
+ (GG −
1
2
GG
U
 +G
U
G
U
 )GG
U


+
1
1922
Z
Tr
n
2

D(U
−1DU)
2
− (DU
−1DU)
2
−4iG

DU−1; DU

+ 2(U−1GUG
 −GG
)
}
;
(7)
where GU = U
−1GU − U−1i@U = U−1(G + J)U .
3. Stability analysis of classical solutions
Our initial hope was to nd nontrivial stable minima of the action (7). In
the process of looking for such solutions, we observed some numerical instabilities,
which caused us to suspect that the action (7) might be unbounded from below.
In order to demonstrate that this is indeed the case, it is sucient to show
this for one trial set of functions, Utrial(x) and Gtrial(x). In order to simplify the
stability analysis, we will therefore begin with classical action without gauge elds,
Gtrial(x) = 0.
Had such an action lead to stable soliton solutions, it would amount to an
(approximate) bosonization of free quarks in four dimensions, which would have
7
been an interesting result in its own right. As the following shows, this has not
been attained in the present formalism. We shall comment later on what we believe
might be the possible reasons for this.
We therefore set the gauge eld to zero, resulting in
Se [0; 0; U ] =
2
22
Z
Tr(JJ
)−
1
2402
Z
 Tr(JJJJJ)
−
1
1922
Z
Tr[2(@J
)2 + (JJ)
2]
(8)
Motivated by the Skyrme model, we are looking for a radially symmetric hedge-
hog solution. We choose the classical solution to be a eld of the form
Uc = e
if(r)~ r^
where ~ are pauli matrices, the generators of some SU(2) subgroup of U(NcNf) and
f(r) is a radial shape function with boundary conditions f(0) = , f(1) = 0. The
choice of the embedding will become relevant only if stable solutions exists, and
then it should be discussed together with quantization of the collective coordinates.
The Wess-Zumino term vanishes and the rest of the terms are given by
−(Jc)
2 = (f 0)2 +
2 sin2 f
r2
(JcJ
c
)
2 = (f 0)4 −
4(f 0)2 sin2 f
r2
(9)
(@J
c)2 =

f 00 +
2f 0
r
−
sin2 f
r2
2
Next, we dene an eective potential Ve as minus the action divided by NcNf
and integrated over space only,
8
Ve (f) =
22

1Z
0
dr [r2(f 0)2 + 2 sin2 f ]
+
1
48
1Z
0
dr
"
2

rf 00 + 2f 0 −
sin(2f)
r
2
+ [r2(f 0)4 − 4(f 0)2 sin2 f ]
# (10)
Finding stable solutions of the action (8) is now reduced to functional mini-
mization of the eective potential (10).
Consider a family of trial functions
f(r) = 2 tan−1(
a
r
) (11)
where a is a variational parameter which also determines the soliton size. The
functions (11) satisfy the boundary conditions at both r = 0 and r!1:
f(r = 0) = 
f(r !1) = 0
(12)
The rst condition is needed to ensure that the solution carries one unit of
winding number, which in our normalization corresponds to one quark.
For this family of trial functions,
Ve (a) = 6
2a−
1
96a
(13)
Hence for a! 0 the potential is unbounded below. Thus, when attempting to solve
the equations of motion, we will nd that the soliton prole will be \squeezed" to
9
zero width around the origin. This shrinking of the classical solution to zero size
occurs despite the presence of both two- and four-derivative terms in the action
(8).
A similar eect occurs already in the  model
[14]
. Also there, the approximate
action with up to four derivatives on the  and ~ elds, constrained to 2+~2 = f2
has its classical solutions \squeezed" to zero width, and with energy tending to
−1.
In two dimensions, stabilization of the soliton is provided by the mass term.
In four dimensions, one may add a mass term
2m4Q [1− cos(f)] (14)
where mQ is some scale related to the original quark mass in the QCD Lagrangian,
(not quite the bare mass itself, as there is normal ordering to be performed; see
Appendix of ref. [7]). Such a mass term does not provide the desired stabilization,
however. For the trial function above, this term will have a divergent contribution
to the potential coming from the integration over large r. Since the stabilization
problem occurs at small distances, this large-r divergence due to the mass term
cannot cure the problem. In order to isolate the large-r divergence, we will treat
the problem of large r by putting a cut-o R. We expect that eventually such a
cut-o will actually be provided by the connement in QCD.
Now the contribution of the mass term to the potential will be
16m4Qa
3

R
a
− tan−1(
R
a
)

(15)
which tends to zero as \a" tends to zero, thus not changing the fact that the
10
potential is unbounded below for small scales a.
4. Discussion and Interpretations
There are various options to overcome these diculties. The rst is by choosing
a dierent regularization scheme. This may change the coecients of the dimen-
sion four operators which appear in the Lagrangian. In particular it may change
the terms in such a way that we will have a commutator squared as in the Skyrme
model. This probably is the only known action which produces a positive Hamilto-
nian. A second way out of this problem is to refer to non-perturbative corrections
which will change the form of the coecients in such a way as to get a Skyrme like
action.
We should remark, however, that in general we would not expect a scheme
change to influence physical results, like the emergence of constituent quarks. It
may happen, however, that due to the approximations made, we may be able to
derive certain quantities in one scheme and not in another.
Recall that in two dimensions, the scheme was completely xed by requiring
vector conservation and that the axial be the dual of the vector. The latter re-
quirement was a result of our wish to have the bosonic version correspond to the
fermionic one, and in the latter the axial is indeed the dual of the vector (see
our work, ref. [8] for details). We do not have an analogous requirement in four
dimensions as yet.
Let us also remark that our classical congurations tend to be \squeezed" to
zero size, and with energy tending to −1. The troublesome part is at short
11
distances. But this is precisely the regime of high momenta, where our approxi-
mations are inadequate, as we have neglected terms with six derivatives or more.
So we either have to nd a better approximation, or maybe exclude some short
distance region.
A nal comment. We expect the eective action (7), after integrating out the
gauge elds and taking trace over color, to yield an eective action in flavor space.
But due to the non-positive nature of the potential that we discovered above, we
do not expect, within the present approximation, to get the Skyrme model with
the assumed standard positive-denite stabilizing four-derivative term
[3;4]
.
Acknowledgements: This research was supported by the Israel Science Foundation
administered by the Israel Academy of Sciences and Humanities. The research of
M.K. was supported in part by the Weizmann Center at the Weizmann Institute
and by a Grant from the G.I.F., the German-Israeli Foundation for Scientic Re-
search and Development. Y.F. would like to thank I. Klebanov, for pointing out
the existence of ref. 14.
12
REFERENCES
1. D. B. Kaplan, Phys. Lett.B235(1990)163; Nucl. Phys.B351(1991)137.
2. A. Manohar and H. Georgi, Nucl. Phys. B234(1984)189.
3. T. H. R. Skyrme, Proc. Roy. Soc. London B260(1961)127.
4. E. Witten, Nucl. Phys. B223(1983)422 and 433; G. Adkins, C. Nappi, and
E. Witten, Nucl. Phys. B228(1983)552; SU(3)f extension in E. Guadagnini,
Nucl. Phys. B236(1984)35; P. O. Mazur, M. A. Nowak, and M. Prasza lowicz,
Phys. Lett. B147(1984)137.
5. E. Witten in Lewes Workshop Proc.; A. Chodos et al., Eds; Singapore, World
Scientic, 1984.
6. G. Gomelski, M. Karliner and S. B. Selipsky, Phys. Lett. B323(1994)182.
7. J. Ellis, Y. Frishman, A. Hanany and M. Karliner, Nucl. Phys. B382(1992)189.
8. Y. Frishman, A. Hanany and M. Karliner Nucl. Phys. B424(1994)3.
9. A. A. Andrianov and L. Bonora, Nucl. Phys. B233(1984)232.
10. J. Balog, Phys. Lett. B149(1984)197.
11. J. L. Manes, Nucl. Phys. B250(1985)369.
12. W. A. Bardeen and B. Zumino, Nucl. Phys. B244(1984)421.
13. P.H. Damgaard and R. Sollacher, Phys. Lett. B322(1994)131.
14. I. Aitchison, C. Fraser, E. Tudor and J. Zuk, Phys. Lett. 165B(1985)162.
13


On the stability of quark solitons in QCD

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