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The Skyrmions and quarks in nuclei
639~
Nuclear Physics A434 (1985) 639c-650~ North-Holland, Amsterdam
THE SKYRMIONSAND QUARKS IN NUCLEI *
Mannque RHO Service de Physique Theorique, France
CEN Saclay, 91191 Gif-sur-Yvette
- cedex,
It is proposed that the quark-bag description and the Skyrmion description of baryons are related to each other by quantized parameters. Topology (through a chiral anomaly) plays an important role in bridging the fundamental theory of the strong interactions (QCO) to effective theories. Some consequences on the efforts to see quark degrees of freedom in nuclear matter are discussed. It is suggested that at low energies there will be no "smoking gun" evidences for quark presence in nuclei.
1. THE SKYRMIONS
AND THE "CHESHIRE CAT" MODEL
As long as quantum chromodynamics is required
to resort to effective
the presumed properties
(QCD) is intractable
one
theories that embody as fully as possible
of QCD. Bag models,
models etc are all such effective
at low energies,
non-relativistic
quark potential
theories. Recent developments
suggest that
when one goes from a fundamental
theory (such as QED or QCO) to an effective
theory, some physical parameters
such as coupling constants, masses, etc are
no longer arbitrary, zation phen~enon, eliminate triguing
but constrained
some of the ambiguities phenomenal
to have quantized
valuesI. This quanti-
which seems to be closely tied to topology, serves to inherent in effective
as the quantized
theories. Such in-
Hall effect, charge fractionization
field theory and condensed matter physics and others are beleived nected to this property The Skyrmion
of effective
description
theories.
of the baryons 2'3 is an effective
and presents a case where there appears a quantized one to identify baryons from a non-linear throws a whole new light on the structure connection
the role of confinement,
theory of QCO
parameter which enables
u-model Lagrangian.
This feature
of the nucleon, with possibly
to the old meson theoretic approaches
understanding
in
to be con-
and promises
chiral sy~try
a deep
to be crucial in
and asymptotic
freedom
in nuclear environment. Arguments
based on large-NC QCD4 suggest strongly that an effective
gian with a quLzntieed parameter, will describe
namely the Wess-Zumino
low-energy QCD dynamics
sufficently
accurately5.
*Invited talk given at 10th International Heidelberg, July 30-August 3, 1984
Conference
0375-9474/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Lagran-
term, exists which Nobody has. yet
on Particles
and Nuclei,
640~
M. Rho / The skyrmions and quarks in nuclei
found a truly satisfactory
one. However surprisingly
gian in its truncated form describes
enough, the Skyrme Lagran-
already many of the baryon dynamics as
equally well as the models based on explicit account of quarks, such as bag and potential models6y7. effective
Lagrangians
of the fundamental
The Skyrme Lagrangian
is thus a sumple version of QCD
in large-NC limit. The Wess-Zumino
term carries a vestige
theory and knows about the constituents,
It appears to be possible
quarks4.
to "derive" the Skyrmion heuristically
starting
from a quark bag*. Imagine we start with a big bag of radius R, in which NC quarks will be inserted to create a baryon. Now if the bag is large enough, then the effect of Goldstone bosons on the quark spectrum - through the bouni7.?Y$ll,,o b in'Yp$ = e eing the chiral angle related to the Gold-
dary condition
stone pion field on the bag surface - will be weak. The chiral angle will be nearly zero. In this case, almost all the physical quantities-the ge, axial charge and energy density-will
be.lodged
Suppose now the bag is squeezed adiabatically,
so that the chiral angle
turns away from zero. For a baryon of the baryon charge B = n, 8 rnr. The boundary
baryon char-
in the quarks inside the bag. 8
goes up to
condition breaks the CP symmetry for 0 # 0, n/Z, IT. Because
of this CP asymmetry,
(a bag without NC valence quarks) acqui6,9,10,11 res a non-zero barvon charse. The calculations show that
B - + vat = 71/2+e < e <
an "empty" bag
[e -: sin 2el for 0 2 e 2 a/2-e, =l- 1 [e - i.sin 201 for TI. The discontinuity at 8 = IT/~ 2 E,E+O, is due to the fact that
a positive-Dirac
level for K = O+ (K = $, + ;C) plunges into the negative energy
Dirac sea at B = IT/~. Thus although there is a CP symmetry at this angle, 16 ; this corresponds to the zero-mode result of charge fractionizavat = *7 12 tion found in other areas of physics .
B
With N, quarks inserted into the bag, the net baryon charge within the bag is
e=oto
B(Nc) = 1 - $
~=TI,
re - $ sin zel,
for 0 < 0 2 IT. Thus as the angle turns from
, the leakage being complete when
the baryon charge "leaks
e = 71. This leakage phenomnenon can be easily understood in terms of an anomaly. Instead of the CP asymmetric
e-dependent
boundary
condition, one can work with
a CP- symmetric boundary condition by making a chiral rotation $ -f $' = V$ with V(r) = exp(i7.P $(r)u,),
e E e(R). The price one has to pay is then that the
Dirac equation will have a coupling on the surface to an axial "gauge field" (not a real gauge field, but an effective
one) Cu = (V-l y6 aU V)"s
i y" Dp$'= 0, Dp = ap + yS CI1 with 6, defined so that it is non-vanishing surface, and vanishing everywhere
on an infinitesimal
else. Now it is well-known
coupled gauge field produces a vector anomaly
strip on the
that an axially
(in contrast to a vector-coupled
M. Rho / The skyrmions and quarks in nuclei
gauge field which produces an axial anomaly).
641~
Thus the bag isolated from the
rest of the system (i.e. the exterior pion sector) does not conserve the baryon charge and hence the leakage.* Since the charge must be conserved found in the exterior
globally,
the missing
sector. In fact, Skyrme' proposed
charge must be
in 1961 that an entire
baryon charge can be lodged in the meson sector. His proposal turns
out to be
correct. To see this, we have to elevate 8 to a dynamical variable n(r) 0 E o(R). To make the bag model consistant with chiral invariance, be identified with the Goldstone bosons live outside-of
with
e(r) must
boson field. We assume that the Goldstone
the bag. In terms of the guaternion,
restricting
for the
moment to the chiral SU(2) x SU(2) case, U(r) = +- [o(r) + i?*il(r)]
,
U+U= 1
TI where IS is the scalar meson field, 71' the Goldstone for the case), the dynamics processes, stabilizing
in the exterior sector is, for long-wave-length
governed by the non-linear term (with L
IJ
dsK
d-model Lagrangian
= U-I
au u)
=-$
Tr [Lzl + $
where F, is the pion decray constant,E which will be determined higher derivative
boson field (IT+,n-, IT'
supplemented
Tr [Ln, LV12 +
by a
. ..
an arbitrary dimensionless
constant
later, and . .. stands for other quartic terms and
terms. Skyrme model corresponds
to this Lagrangian without
the extra . .. terms. Skyrme proposed* - and arguments based on large NC QCD 495 - that this Lagrangian should describe not only meson dynamics but support also low-energy
properties
of baryons.
Thus the first term in the Lagrangian
should provide, e.g., soft-pion This has been explicitly sible, baryons
theorems on both meson and baryon targets. 14 demonstrated to be the case . For this to be pos-
(fermions) must emerge from mesons
(bosons). Indeed, it is now
firmly established that through non-trivial topology, fermions do emerge from 15,5 the chiral field ,the fermions are baryons5 (called Skyrmions) and when suitably quantized,
rotating Skyrmions
correspond
to low-lying baryons, N and
A7. A deep connection
to the quantization
of physical parameters mentioned
is seen when one considers SU(3) x SU(3). In the has a Lagrangian
*
This argument
case
of the Skyrme type, with U now a 3 x 3 matrix
is based on H.B. Nielsen's
before
of SU(3) x SU(3),one still involving octet
"infinite hotel" model
13
M. Rho / The skyrmions and quarks in nuclei
642c
of Goldstone
bosons.
In addition,
term, required by perturbative rnr contributing
there is an extra term, called Wess-Zumino
and non-oerturbative
to the action wherer
disc5. The Wess-Zumino of the effective
anomalies.
It has a form
is an integral over a .five dimensional
term has a quantized
value required by the consistency
theory, the condition being that m = NC = the number of color.
It is this term that signals that baryons are made up of N,quarks when NC is odd, they satisfy fermion statistics.
Although
it can be used for the same purpose by suitably embedding
and that
r = 0 for SU(2)xSU(2) an SU(2) mapoing
into
an SU(3) mapping. The Wess-Zumino predicts
term, with its parameter
the processes
involving
fixed by the constraint,
transitions
uniquely
of even number of bosons-todd num-
ber of bosons. Thus once one assumes F
= FK = F = Fn,, then NC = 3 fixes unitn- + + a+a-y. quely such transitions as K'K- -L 7r+TI-?TB , K+ +nnev,n+n'a-vandn' 16 Agreement with experiments is in general good and, in some cases, excellent . We will now argue that the Skyrmion described radius of the chiral bag is ween
0:0(R) and R. Thus
bag with N
quarks
shrunk
above is obtained when the
to zero* .There is a direct relation bet-
13 = 0, 71 correspond
respectively
to R = - , 0 for a
(Fig. 1).
0
0.5
1.0Rifml
FIGURE 1 Relation between R and B(R), from Ref.8. (A kink at R M 0.44fm, an artifact of the sharo surface,is smoothed).
M. Rho / The skymions
The first quantity
and quarks in nuclei
643~
to look at is the baryon number 2y5. Consider
From the action I = ISk + mr where
the limit R = 0.
ISk is the Skyrme action, one obtains the
baryon current an = -?-2 cuvaB 24~ This comes entirely
Tr [L"LaLBl.
from the Wess-Zuminoterm.
[The argument is made here for
SU(3) x SU(3), but the same cesult
holds for SU(2) x SU(2).1
that
, choose the hedgehog configuration
8 # 0 [where % = /d3x Bo(x)l
uo(r) = ,iT*fe(r)
In order to see
,iQ.Pe(r)
o
0
1>
or (
with o(0) = TI, o(m) = 0 so as to assure finite energy. For this configuration ?r
B = -I$[-e(r) + + sin 28(r)]:
= 1
.
Thus when R is zero, the entire baryon charge is lodged in the meson sector. This confirms
the leakage of a baryon due to the quantum anomaly.
If R # 0, then the charge is shared. Inside the bag, it was found to be 1 -
he
- + sin 281; outside, due to a defect, the integration goes from R
to ,", so we have $8 of R. This result
- i sin 201. The sum is 1, as it should be, independently
makes it physically
plausible
that the Skyrmion
is a baryon.
Along with the baryon charge, other things leak out too Bygylo. Consider the 10 vacuum energy inside the bag, in particular nvac = Evac R ,
Qvac(e) n(e,t)
= ;$
m!e,t)
= - +
R,, = w,, R
- n(o,t)i Q, e -tlnnl
1n siw(Q,) .
Both 'vat and 'total = 'vat + $a1 ($a1 is the contribution of each valence quark) are plotted in Fig. 2. At 0 = 0, Rtot= 2.04 ... . the M.I.T. value, and as e is dialled to
TI, Qtot leaks out, the leakage complete at 8 = a.Closely
ted to the leakage of energy is the axial flux
O(e)
= f d2S
o(e)
nn na r=R
rela-
644c
M. Rho / The skyrmions and quarks in nuclei
where the expectation
value of the axial current is taken with respect to the
bag vaccum or the combined system of the vaccum plus the valence quarks. The relation
is
subtraction
To arrive at this expression,One
is needed to render the quantities
finitelO. The result is given in Fig. 3. Again completely
at 0 = T.Unlike
+ Rval I "leaks" out
&[C$ac
_
_.
the baryon charge which is a topological
there is no simple relation for these other quantities titioned into the interior and exterior
invariant
,
as to how they are par-
regions. Nevertheless,
physical obser-
vables in general depend little on the size of R.
0
n/4
n/2
3n/4
-to
n
n/l
Q
n/2
3n/C
n
El FIGURE 3
FIGURE 2 g
R vs.8 from Ref. 10
(related to RO) vs.8 from
Ref. 10.
It is sometimes claimed
in the literature
that one can constrain
the bag size
by looking at, say, the axial form factor of the nucleon gA(q2). This claim is not supported when the Dirac polarization account. The axial coupling
is dictated by the axial flux Fig. 4, 0 is fairly independent the controversy.
(chiral anomaly)
- and in general pion couplings
is suitably taken into to the nucleon -
Q(e) in the quark sector. As one can see in of the radius within the range of R relevant to
This should come as no great surprise.
M. Rho / The skyunions
64Sc
and quarks in nuclei
t 1.01 0.5
I 0.6
I 0.7
I 1 0.8 R(fmf 0.9
FIGURE 4 The axial flux per Nc vs.R, from Refs. 8 and 10 It is perhaps a more sensible thing to do to proceed in the reverse sense by requiring
that the axial charge gA be independent
consequences
from such a requirement.
Skyrme's quartic
of R and extracting
the
This allows us "to derive" heuristically
term. To do this, we use the axial-current
conservation
a"A: = 0. This requires the axial flux in the quark sector @ that of the Skyrmion sector ~skyrmion. the number of colorequal
/dQvac\ skyrmion = WcjalQ_*
side is a known quantity
its contribution
$2
to be equal to quark 0 2 TI, this is (for
to NC). R@
The right-hand
In the range
from the quadratic
*
for given 8. The left-hand side receives
and quartic terms of the Skyrme Lagrangian.
Explicitly
ip
skyrmion
= 4riR2Fi ($)r=R[l
+ E' ~1
, 7l
We propose8 to determine E2 and hence the Skyrme quartic term as
9 -+ 7~ from
the axial flux balance. The result is given in Fig. 5 Physically freedom
what this means is that one "integrates"
out the quark-bag
degrees of
to obtain the quartic term. [This is similar to integratina out quark
fields in an effective theory in which quarks are coupled to chiral fields 17 (e.g. o-model) . In such a process, one obtains the Wess-Zumino term and quartic
terms (which need not give precisely
naturally quartic
the Skyrme type).] We are thus
invited to look at the quark sector to understandtherole
term I*_ The physics of the quark sector is determined
Dirac levels as the angle e is dialled toward
of the
by the flow of
7~. Figure 6 has the relevant
0 0
0.1
0.2
0.3
n/2
n
3n/2 Q 2n
0.4 0.5R(fmI
FIGURE 5
FIGURE 6
The strength of the Skyrme quartic term ($) resulting from the shrinking bag. From Ref.8.
Flow of KF = O+,O-,lf levels inside the bag. The arrow indicates the crossing point.
level flow pattern. We find that the Skvrme quartic
let-m is generated
crosses from above the oscillating
K
q
as, at e = ec, a K = O- level
1+ level. The O- level remains the first
excited
level after this chiral angle nc and plunges into the negative-energy
sea at
6 = 3a/2. Thus the Skyrme quartic
term can be viewed as a quantum effect
associated with the f+ - O- (in K') level crossing*. It seems plausible tation of an effective
that this level crossing is closely related to the excidegree of freedom
nally associated with the short-range
corresponding
repulsion
to the w meson traditio-
in nucleon-nucleon
interactions.
There are two compelling
arguments why this notion is plausible. One is the re.7c who replaced the Skyrme quartic term by Adkins and Nappi
sult of a calculation
by w mesons coupled to the baryon current,obtaining Skyrme model by the quartic
a stability provided
in the
term. The results for all nucleon and A static pro-
perties come out to be quite simiiar. The other is the result of a calculation by Jackson, Jackson and Pasquier a Born-Oppenhei~r
of the nucleon-nucleon
treatment of the Sky~ion-Sky~ion
that at small relative distances between the nucleons, spin-independent
D-wave
term in its particular
IW scattering
data at threshold18.
based on
It is found
there emerges a spin-iso-
repulsion which can be well reprensented
*The Skyrme quartic
interaction
interactions.
by the exchange of an
form is found to be consistent with (Another coincidence?).
w meson. The level crossing argument set in roughly at R M 0.4-0.5fm
suggests that the w degrees of freedom
for a 6 = 1 system and at R * 0.7-0.8fm
for a
B = 2 system. Let us suppose that there is, as conjectured, the level crossing and the Skyrme quartic
an intimate relation between
term on the one hand and the level
crossing and the w exchange on the other hand. Then an important consequence is that we would have a quark-based distancesI*. quartic
A second equally
description
of an "w" exchange at short
important consequence
is that although the Skyrme
term is not unique, 1‘t could contain many of the requisite
ingredients
for short-distance
dynamics not directly visible and hence there must be some
strong constraints
on what further terms can appear in the effective
(For instance, an o field cannot be naive@ the quartic
added to the Lagrangian
term). Finally one notes a striking
the quantities mathematical
Bvac (6) and R@(8)vac.
qualitative
Lagrangian. in addition to
resemblance
between
The former is related to a quantity of 20 , a topolon invariant
interest, namely the Atiyah-Patodi-Singer
gical invariant,so
although it may be purely coincidental
the latter has a similar topological conjecture*
that the P
invariant.
in the quartic
If this
it could also be that
is true, then Skyrme's
term is quantized may have a deep mathe-
matical meaning. What we have arrived at is a "Cheshire Cat" model nucleon as a bag surroundedbya
8,22*
of the baryons. The
small pion cloud (Skyrmion cloud) and the nu-
cleon as a Skyrmion with a point "bag" may very well represent two sides of a same coin. The so-called bag radius R can be big or small depending upon the kind of approximations
that are made, the quartic term replacing the part of
the quark bag that is increasing or decreasing (bag), the expansion Skyrmion
parametgr
sector, it is l/N,
higher-order
corrections
in size. In the quark sector
is the color fine structure
in c1S and l/N, are best controlled.
was that at low energies and/or low momentum Q, 0.44fm. What happens at higher energies
transfers,
and mo~ntum
be done on these questions.
Our suggestion8
the optimal radius is transfers(qz)?
happens when nucleons are jammed into higher densities(P)? needs to
constant cts; in the
, The optimal radius must be the one at which
What
More quantitative
work
But it is obvious what can happen: the
chiral angle e will turn toward - and relax at - a smaller value as q2 or density is increased.
Thus seen by deeply inelastic electrons,
clear matter could look bigger relative that the transition
a nucleon in nu-
to the free space. We further expect
from the region in which nucleon degrees of freedom
are
* H.B. Nielsen coined this name in analogy to the Cheshire cat in Lewis Caroll's "Alice in Wonderland".
648~
M. Rho / The skymions
appropriate
and quarks in nuclei
to the region in which quark degrees of freedom are appropriate
not be abrupt. Thus the results on the electrodisintegration ("small" bag) and the quenching
of the axial-vector
("big" bag) may be not incompatible mately
will
of the deuteron
coupling constant g,,,
in the framework of QCDz2. One would ulti-
like to "see" quarks in nuclei. ?ur oicture strongly suggests that
"seeing" quarks is going to be just as hard, painstaking "seeing" mesons
and uneventful
as
in nuclei, boring perhaps but a clever solution nevertheless.
2. OTHER DEVELOPMENTS Here is a list of what has been going on recently elsewhere. 1. One interesting at a description
problem
is whether or not one cannot arrive
of the nucleon equivalent
account of confinement yes for topological quantities
theoretical
to the Skyrmion without an explicit 23 . I would think the answer is 17 the baryon current) , no for dynamical
and asymptotic
quantities
(e.g. stability)
freedom
(e.g.
unless miracles
occur through radiative correctionsf
Now in the Skyrmion model: 2. Current algebra and soft-meson low naturally
theorems on meson and baryon targets fol-
from the Skvrme Laqranoian 14 . demonstrated
plus the Wess-Zumino
term. This has
been explicitly
3. Upon quantizing
the Skyrmions,
one obtains the N and A as rotational
vels for the case of SU(2) x SU(2)7 and the spin - i octets and spin - G
ledecu-
plets for the case of SU(3) x SU(3)24. 4. Embedding SO(3) [instead of SU(2) as above] into SU(3), one obtains dibaryon resonances
(B=2)25. In particular,
a O+ SU(3)-singlet
state called H is
predicted
at as low energy as 2.2GeV. This is quite similar to the doubly stran26 ge six-quark state predicted in the M.I.T. bag model . 5. The N*'s and A*'s can be described 27
pling of the Skyrmion 6. The electroweak formulated
in terms of rotation-vibration
interactions
in a consistent
with Skyrmions
way, free of anomalies
(in place of quarks) can be 28
.
7. The monopole
catalysis of the proton decay an be described 29 Skyrmion whose topological knot is unwound by a monopole . 8. The nuclear matter-quark of Skyrmions
9. The N-N interaction
*
plasma phase transition
19,31
an example being chiral anomalies.
in terms
resembles
obtained from the Skytmion-Skyrmion
the Paris potential.
They do sometimes,
in terms of a
can be described
alone. The infinite density limit of the Skyrmions
quark gas3'. resembles
cou-
.
a
interactions
M. Rho / The skyrrnions and quarks in nuclei
In conclusion,
it is quite remarkable
at least qualitatively,
649C
that such a simple Lagrangian
so much of QCD. It would be interesting
embodies,
to see whether
and where it breaks down. ACKNOWLEDGEMENTS I am very grateful A.S. Goldhaber,
for collaborations
and/or discussions
with 3G.E. Brown,
A.D. Jackson, V. Vento, G. Ripka and I. Zahed.
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Phys. Lett. 1408 (1984) 280
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Phys. Lett. -139B (1984) 217. and J. Rubinstein, 3. Math. Phys. _!j(1968) 1762; J. Math. Phys. -11 (1970) 2611.
650~
M. Rho / The skyrmions and quarks in nuclei
16) G.
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17)
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18)
A.D. Jackson, H.K. Lee and M. Rho, paper in preparation.
191 A. Jackson, A.D. Jackson
and V. Pasqieur,
Nucl. Phys. A (in press).
T. Eguchi, P. Gilkey and A.Hanson,.Phys.
Reports 66 (1980) 215.
20)
See,e.g.,
21)
H.B. Nielsen and I. Zahed, unpublished.
22)
See, e.g. M. Rho, Pion Interactions Science 2, in nress.
23)
S. Kahana and G. Ripka, Nucl. Phys. A., to be published; G. Ripka, Trieste Lecture (1984).
24)
E. Guadagnini, Nucl. Phys. 8236 (1984) 35; P.O. Mazur, N.A. Nowak and l%-&-aszalowicz, model", unpublished (1984).
25)
A.P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers and A. Stern, Phys. Rev. Lett. -52 (1984) 887.
26)
R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195; V.A. Matveev and P. Sorba, NuGo Cimento -45A (1978) 257.
27)
C. Hajduk and B. Schwesinger, Phys. Lett. 1408 (1984) 172; A. Hayashi and G. Holzwarth, Phys. Lett. laW(1984) 175; skyrmion breathing mode", J.D. Breit and C.R. Nappi, "Phase shifts nhe Princeton preprint (1984); I. Zahed, U.G. Meissner and U.B. Kaulfuss, "Low-lying resonances in the Skyrme model in semi-classical approximation, Niels Bohr Institute preprint, to be published (1984).
28)
E. D'Hoker and E. Farhi, Phys. Lett. 1348 (1984) 86.
29)
C.G. Callan and E. Witten, Nucl. Phys. 8239 (1984) 161.
30)
M. Kutschera, C.J. Pethick and D.G. Ravenhall, soliton model", Illinois preorint (1984).
31)
Z. Hlousek, "Interbaryonic force as a soliton-soliton University preprint (1984). me model" ,&own
within Nuclei, in Ann. Rev. NucL. Part.
"SU(3) extension
of the Skyrme
"Dense matter in the chiral interaction
in Skyr-
Nuclear Physics A434 (1985) 639c-650~ North-Holland, Amsterdam
THE SKYRMIONSAND QUARKS IN NUCLEI *
Mannque RHO Service de Physique Theorique, France
CEN Saclay, 91191 Gif-sur-Yvette
- cedex,
It is proposed that the quark-bag description and the Skyrmion description of baryons are related to each other by quantized parameters. Topology (through a chiral anomaly) plays an important role in bridging the fundamental theory of the strong interactions (QCO) to effective theories. Some consequences on the efforts to see quark degrees of freedom in nuclear matter are discussed. It is suggested that at low energies there will be no "smoking gun" evidences for quark presence in nuclei.
1. THE SKYRMIONS
AND THE "CHESHIRE CAT" MODEL
As long as quantum chromodynamics is required
to resort to effective
the presumed properties
(QCD) is intractable
one
theories that embody as fully as possible
of QCD. Bag models,
models etc are all such effective
at low energies,
non-relativistic
quark potential
theories. Recent developments
suggest that
when one goes from a fundamental
theory (such as QED or QCO) to an effective
theory, some physical parameters
such as coupling constants, masses, etc are
no longer arbitrary, zation phen~enon, eliminate triguing
but constrained
some of the ambiguities phenomenal
to have quantized
valuesI. This quanti-
which seems to be closely tied to topology, serves to inherent in effective
as the quantized
theories. Such in-
Hall effect, charge fractionization
field theory and condensed matter physics and others are beleived nected to this property The Skyrmion
of effective
description
theories.
of the baryons 2'3 is an effective
and presents a case where there appears a quantized one to identify baryons from a non-linear throws a whole new light on the structure connection
the role of confinement,
theory of QCO
parameter which enables
u-model Lagrangian.
This feature
of the nucleon, with possibly
to the old meson theoretic approaches
understanding
in
to be con-
and promises
chiral sy~try
a deep
to be crucial in
and asymptotic
freedom
in nuclear environment. Arguments
based on large-NC QCD4 suggest strongly that an effective
gian with a quLzntieed parameter, will describe
namely the Wess-Zumino
low-energy QCD dynamics
sufficently
accurately5.
*Invited talk given at 10th International Heidelberg, July 30-August 3, 1984
Conference
0375-9474/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Lagran-
term, exists which Nobody has. yet
on Particles
and Nuclei,
640~
M. Rho / The skyrmions and quarks in nuclei
found a truly satisfactory
one. However surprisingly
gian in its truncated form describes
enough, the Skyrme Lagran-
already many of the baryon dynamics as
equally well as the models based on explicit account of quarks, such as bag and potential models6y7. effective
Lagrangians
of the fundamental
The Skyrme Lagrangian
is thus a sumple version of QCD
in large-NC limit. The Wess-Zumino
term carries a vestige
theory and knows about the constituents,
It appears to be possible
quarks4.
to "derive" the Skyrmion heuristically
starting
from a quark bag*. Imagine we start with a big bag of radius R, in which NC quarks will be inserted to create a baryon. Now if the bag is large enough, then the effect of Goldstone bosons on the quark spectrum - through the bouni7.?Y$ll,,o b in'Yp$ = e eing the chiral angle related to the Gold-
dary condition
stone pion field on the bag surface - will be weak. The chiral angle will be nearly zero. In this case, almost all the physical quantities-the ge, axial charge and energy density-will
be.lodged
Suppose now the bag is squeezed adiabatically,
so that the chiral angle
turns away from zero. For a baryon of the baryon charge B = n, 8 rnr. The boundary
baryon char-
in the quarks inside the bag. 8
goes up to
condition breaks the CP symmetry for 0 # 0, n/Z, IT. Because
of this CP asymmetry,
(a bag without NC valence quarks) acqui6,9,10,11 res a non-zero barvon charse. The calculations show that
B - + vat = 71/2+e < e <
an "empty" bag
[e -: sin 2el for 0 2 e 2 a/2-e, =l- 1 [e - i.sin 201 for TI. The discontinuity at 8 = IT/~ 2 E,E+O, is due to the fact that
a positive-Dirac
level for K = O+ (K = $, + ;C) plunges into the negative energy
Dirac sea at B = IT/~. Thus although there is a CP symmetry at this angle, 16 ; this corresponds to the zero-mode result of charge fractionizavat = *7 12 tion found in other areas of physics .
B
With N, quarks inserted into the bag, the net baryon charge within the bag is
e=oto
B(Nc) = 1 - $
~=TI,
re - $ sin zel,
for 0 < 0 2 IT. Thus as the angle turns from
, the leakage being complete when
the baryon charge "leaks
e = 71. This leakage phenomnenon can be easily understood in terms of an anomaly. Instead of the CP asymmetric
e-dependent
boundary
condition, one can work with
a CP- symmetric boundary condition by making a chiral rotation $ -f $' = V$ with V(r) = exp(i7.P $(r)u,),
e E e(R). The price one has to pay is then that the
Dirac equation will have a coupling on the surface to an axial "gauge field" (not a real gauge field, but an effective
one) Cu = (V-l y6 aU V)"s
i y" Dp$'= 0, Dp = ap + yS CI1 with 6, defined so that it is non-vanishing surface, and vanishing everywhere
on an infinitesimal
else. Now it is well-known
coupled gauge field produces a vector anomaly
strip on the
that an axially
(in contrast to a vector-coupled
M. Rho / The skyrmions and quarks in nuclei
gauge field which produces an axial anomaly).
641~
Thus the bag isolated from the
rest of the system (i.e. the exterior pion sector) does not conserve the baryon charge and hence the leakage.* Since the charge must be conserved found in the exterior
globally,
the missing
sector. In fact, Skyrme' proposed
charge must be
in 1961 that an entire
baryon charge can be lodged in the meson sector. His proposal turns
out to be
correct. To see this, we have to elevate 8 to a dynamical variable n(r) 0 E o(R). To make the bag model consistant with chiral invariance, be identified with the Goldstone bosons live outside-of
with
e(r) must
boson field. We assume that the Goldstone
the bag. In terms of the guaternion,
restricting
for the
moment to the chiral SU(2) x SU(2) case, U(r) = +- [o(r) + i?*il(r)]
,
U+U= 1
TI where IS is the scalar meson field, 71' the Goldstone for the case), the dynamics processes, stabilizing
in the exterior sector is, for long-wave-length
governed by the non-linear term (with L
IJ
dsK
d-model Lagrangian
= U-I
au u)
=-$
Tr [Lzl + $
where F, is the pion decray constant,E which will be determined higher derivative
boson field (IT+,n-, IT'
supplemented
Tr [Ln, LV12 +
by a
. ..
an arbitrary dimensionless
constant
later, and . .. stands for other quartic terms and
terms. Skyrme model corresponds
to this Lagrangian without
the extra . .. terms. Skyrme proposed* - and arguments based on large NC QCD 495 - that this Lagrangian should describe not only meson dynamics but support also low-energy
properties
of baryons.
Thus the first term in the Lagrangian
should provide, e.g., soft-pion This has been explicitly sible, baryons
theorems on both meson and baryon targets. 14 demonstrated to be the case . For this to be pos-
(fermions) must emerge from mesons
(bosons). Indeed, it is now
firmly established that through non-trivial topology, fermions do emerge from 15,5 the chiral field ,the fermions are baryons5 (called Skyrmions) and when suitably quantized,
rotating Skyrmions
correspond
to low-lying baryons, N and
A7. A deep connection
to the quantization
of physical parameters mentioned
is seen when one considers SU(3) x SU(3). In the has a Lagrangian
*
This argument
case
of the Skyrme type, with U now a 3 x 3 matrix
is based on H.B. Nielsen's
before
of SU(3) x SU(3),one still involving octet
"infinite hotel" model
13
M. Rho / The skyrmions and quarks in nuclei
642c
of Goldstone
bosons.
In addition,
term, required by perturbative rnr contributing
there is an extra term, called Wess-Zumino
and non-oerturbative
to the action wherer
disc5. The Wess-Zumino of the effective
anomalies.
It has a form
is an integral over a .five dimensional
term has a quantized
value required by the consistency
theory, the condition being that m = NC = the number of color.
It is this term that signals that baryons are made up of N,quarks when NC is odd, they satisfy fermion statistics.
Although
it can be used for the same purpose by suitably embedding
and that
r = 0 for SU(2)xSU(2) an SU(2) mapoing
into
an SU(3) mapping. The Wess-Zumino predicts
term, with its parameter
the processes
involving
fixed by the constraint,
transitions
uniquely
of even number of bosons-todd num-
ber of bosons. Thus once one assumes F
= FK = F = Fn,, then NC = 3 fixes unitn- + + a+a-y. quely such transitions as K'K- -L 7r+TI-?TB , K+ +nnev,n+n'a-vandn' 16 Agreement with experiments is in general good and, in some cases, excellent . We will now argue that the Skyrmion described radius of the chiral bag is ween
0:0(R) and R. Thus
bag with N
quarks
shrunk
above is obtained when the
to zero* .There is a direct relation bet-
13 = 0, 71 correspond
respectively
to R = - , 0 for a
(Fig. 1).
0
0.5
1.0Rifml
FIGURE 1 Relation between R and B(R), from Ref.8. (A kink at R M 0.44fm, an artifact of the sharo surface,is smoothed).
M. Rho / The skymions
The first quantity
and quarks in nuclei
643~
to look at is the baryon number 2y5. Consider
From the action I = ISk + mr where
the limit R = 0.
ISk is the Skyrme action, one obtains the
baryon current an = -?-2 cuvaB 24~ This comes entirely
Tr [L"LaLBl.
from the Wess-Zuminoterm.
[The argument is made here for
SU(3) x SU(3), but the same cesult
holds for SU(2) x SU(2).1
that
, choose the hedgehog configuration
8 # 0 [where % = /d3x Bo(x)l
uo(r) = ,iT*fe(r)
In order to see
,iQ.Pe(r)
o
0
1>
or (
with o(0) = TI, o(m) = 0 so as to assure finite energy. For this configuration ?r
B = -I$[-e(r) + + sin 28(r)]:
= 1
.
Thus when R is zero, the entire baryon charge is lodged in the meson sector. This confirms
the leakage of a baryon due to the quantum anomaly.
If R # 0, then the charge is shared. Inside the bag, it was found to be 1 -
he
- + sin 281; outside, due to a defect, the integration goes from R
to ,", so we have $8 of R. This result
- i sin 201. The sum is 1, as it should be, independently
makes it physically
plausible
that the Skyrmion
is a baryon.
Along with the baryon charge, other things leak out too Bygylo. Consider the 10 vacuum energy inside the bag, in particular nvac = Evac R ,
Qvac(e) n(e,t)
= ;$
m!e,t)
= - +
R,, = w,, R
- n(o,t)i Q, e -tlnnl
1n siw(Q,) .
Both 'vat and 'total = 'vat + $a1 ($a1 is the contribution of each valence quark) are plotted in Fig. 2. At 0 = 0, Rtot= 2.04 ... . the M.I.T. value, and as e is dialled to
TI, Qtot leaks out, the leakage complete at 8 = a.Closely
ted to the leakage of energy is the axial flux
O(e)
= f d2S
o(e)
nn na
rela-
644c
M. Rho / The skyrmions and quarks in nuclei
where the expectation
value of the axial current is taken with respect to the
bag vaccum or the combined system of the vaccum plus the valence quarks. The relation
is
subtraction
To arrive at this expression,One
is needed to render the quantities
finitelO. The result is given in Fig. 3. Again completely
at 0 = T.Unlike
+ Rval I "leaks" out
&[C$ac
_
_.
the baryon charge which is a topological
there is no simple relation for these other quantities titioned into the interior and exterior
invariant
,
as to how they are par-
regions. Nevertheless,
physical obser-
vables in general depend little on the size of R.
0
n/4
n/2
3n/4
-to
n
n/l
Q
n/2
3n/C
n
El FIGURE 3
FIGURE 2 g
R vs.8 from Ref. 10
(related to RO) vs.8 from
Ref. 10.
It is sometimes claimed
in the literature
that one can constrain
the bag size
by looking at, say, the axial form factor of the nucleon gA(q2). This claim is not supported when the Dirac polarization account. The axial coupling
is dictated by the axial flux Fig. 4, 0 is fairly independent the controversy.
(chiral anomaly)
- and in general pion couplings
is suitably taken into to the nucleon -
Q(e) in the quark sector. As one can see in of the radius within the range of R relevant to
This should come as no great surprise.
M. Rho / The skyunions
64Sc
and quarks in nuclei
t 1.01 0.5
I 0.6
I 0.7
I 1 0.8 R(fmf 0.9
FIGURE 4 The axial flux per Nc vs.R, from Refs. 8 and 10 It is perhaps a more sensible thing to do to proceed in the reverse sense by requiring
that the axial charge gA be independent
consequences
from such a requirement.
Skyrme's quartic
of R and extracting
the
This allows us "to derive" heuristically
term. To do this, we use the axial-current
conservation
a"A: = 0. This requires the axial flux in the quark sector @ that of the Skyrmion sector ~skyrmion. the number of colorequal
/dQvac\ skyrmion = WcjalQ_*
side is a known quantity
its contribution
$2
to be equal to quark 0 2 TI, this is (for
to NC). R@
The right-hand
In the range
from the quadratic
*
for given 8. The left-hand side receives
and quartic terms of the Skyrme Lagrangian.
Explicitly
ip
skyrmion
= 4riR2Fi ($)r=R[l
+ E' ~1
, 7l
We propose8 to determine E2 and hence the Skyrme quartic term as
9 -+ 7~ from
the axial flux balance. The result is given in Fig. 5 Physically freedom
what this means is that one "integrates"
out the quark-bag
degrees of
to obtain the quartic term. [This is similar to integratina out quark
fields in an effective theory in which quarks are coupled to chiral fields 17 (e.g. o-model) . In such a process, one obtains the Wess-Zumino term and quartic
terms (which need not give precisely
naturally quartic
the Skyrme type).] We are thus
invited to look at the quark sector to understandtherole
term I*_ The physics of the quark sector is determined
Dirac levels as the angle e is dialled toward
of the
by the flow of
7~. Figure 6 has the relevant
0 0
0.1
0.2
0.3
n/2
n
3n/2 Q 2n
0.4 0.5R(fmI
FIGURE 5
FIGURE 6
The strength of the Skyrme quartic term ($) resulting from the shrinking bag. From Ref.8.
Flow of KF = O+,O-,lf levels inside the bag. The arrow indicates the crossing point.
level flow pattern. We find that the Skvrme quartic
let-m is generated
crosses from above the oscillating
K
q
as, at e = ec, a K = O- level
1+ level. The O- level remains the first
excited
level after this chiral angle nc and plunges into the negative-energy
sea at
6 = 3a/2. Thus the Skyrme quartic
term can be viewed as a quantum effect
associated with the f+ - O- (in K') level crossing*. It seems plausible tation of an effective
that this level crossing is closely related to the excidegree of freedom
nally associated with the short-range
corresponding
repulsion
to the w meson traditio-
in nucleon-nucleon
interactions.
There are two compelling
arguments why this notion is plausible. One is the re.7c who replaced the Skyrme quartic term by Adkins and Nappi
sult of a calculation
by w mesons coupled to the baryon current,obtaining Skyrme model by the quartic
a stability provided
in the
term. The results for all nucleon and A static pro-
perties come out to be quite simiiar. The other is the result of a calculation by Jackson, Jackson and Pasquier a Born-Oppenhei~r
of the nucleon-nucleon
treatment of the Sky~ion-Sky~ion
that at small relative distances between the nucleons, spin-independent
D-wave
term in its particular
IW scattering
data at threshold18.
based on
It is found
there emerges a spin-iso-
repulsion which can be well reprensented
*The Skyrme quartic
interaction
interactions.
by the exchange of an
form is found to be consistent with (Another coincidence?).
w meson. The level crossing argument set in roughly at R M 0.4-0.5fm
suggests that the w degrees of freedom
for a 6 = 1 system and at R * 0.7-0.8fm
for a
B = 2 system. Let us suppose that there is, as conjectured, the level crossing and the Skyrme quartic
an intimate relation between
term on the one hand and the level
crossing and the w exchange on the other hand. Then an important consequence is that we would have a quark-based distancesI*. quartic
A second equally
description
of an "w" exchange at short
important consequence
is that although the Skyrme
term is not unique, 1‘t could contain many of the requisite
ingredients
for short-distance
dynamics not directly visible and hence there must be some
strong constraints
on what further terms can appear in the effective
(For instance, an o field cannot be naive@ the quartic
added to the Lagrangian
term). Finally one notes a striking
the quantities mathematical
Bvac (6) and R@(8)vac.
qualitative
Lagrangian. in addition to
resemblance
between
The former is related to a quantity of 20 , a topolon invariant
interest, namely the Atiyah-Patodi-Singer
gical invariant,so
although it may be purely coincidental
the latter has a similar topological conjecture*
that the P
invariant.
in the quartic
If this
it could also be that
is true, then Skyrme's
term is quantized may have a deep mathe-
matical meaning. What we have arrived at is a "Cheshire Cat" model nucleon as a bag surroundedbya
8,22*
of the baryons. The
small pion cloud (Skyrmion cloud) and the nu-
cleon as a Skyrmion with a point "bag" may very well represent two sides of a same coin. The so-called bag radius R can be big or small depending upon the kind of approximations
that are made, the quartic term replacing the part of
the quark bag that is increasing or decreasing (bag), the expansion Skyrmion
parametgr
sector, it is l/N,
higher-order
corrections
in size. In the quark sector
is the color fine structure
in c1S and l/N, are best controlled.
was that at low energies and/or low momentum Q, 0.44fm. What happens at higher energies
transfers,
and mo~ntum
be done on these questions.
Our suggestion8
the optimal radius is transfers(qz)?
happens when nucleons are jammed into higher densities(P)? needs to
constant cts; in the
, The optimal radius must be the one at which
What
More quantitative
work
But it is obvious what can happen: the
chiral angle e will turn toward - and relax at - a smaller value as q2 or density is increased.
Thus seen by deeply inelastic electrons,
clear matter could look bigger relative that the transition
a nucleon in nu-
to the free space. We further expect
from the region in which nucleon degrees of freedom
are
* H.B. Nielsen coined this name in analogy to the Cheshire cat in Lewis Caroll's "Alice in Wonderland".
648~
M. Rho / The skymions
appropriate
and quarks in nuclei
to the region in which quark degrees of freedom are appropriate
not be abrupt. Thus the results on the electrodisintegration ("small" bag) and the quenching
of the axial-vector
("big" bag) may be not incompatible mately
will
of the deuteron
coupling constant g,,,
in the framework of QCDz2. One would ulti-
like to "see" quarks in nuclei. ?ur oicture strongly suggests that
"seeing" quarks is going to be just as hard, painstaking "seeing" mesons
and uneventful
as
in nuclei, boring perhaps but a clever solution nevertheless.
2. OTHER DEVELOPMENTS Here is a list of what has been going on recently elsewhere. 1. One interesting at a description
problem
is whether or not one cannot arrive
of the nucleon equivalent
account of confinement yes for topological quantities
theoretical
to the Skyrmion without an explicit 23 . I would think the answer is 17 the baryon current) , no for dynamical
and asymptotic
quantities
(e.g. stability)
freedom
(e.g.
unless miracles
occur through radiative correctionsf
Now in the Skyrmion model: 2. Current algebra and soft-meson low naturally
theorems on meson and baryon targets fol-
from the Skvrme Laqranoian 14 . demonstrated
plus the Wess-Zumino
term. This has
been explicitly
3. Upon quantizing
the Skyrmions,
one obtains the N and A as rotational
vels for the case of SU(2) x SU(2)7 and the spin - i octets and spin - G
ledecu-
plets for the case of SU(3) x SU(3)24. 4. Embedding SO(3) [instead of SU(2) as above] into SU(3), one obtains dibaryon resonances
(B=2)25. In particular,
a O+ SU(3)-singlet
state called H is
predicted
at as low energy as 2.2GeV. This is quite similar to the doubly stran26 ge six-quark state predicted in the M.I.T. bag model . 5. The N*'s and A*'s can be described 27
pling of the Skyrmion 6. The electroweak formulated
in terms of rotation-vibration
interactions
in a consistent
with Skyrmions
way, free of anomalies
(in place of quarks) can be 28
.
7. The monopole
catalysis of the proton decay an be described 29 Skyrmion whose topological knot is unwound by a monopole . 8. The nuclear matter-quark of Skyrmions
9. The N-N interaction
*
plasma phase transition
19,31
an example being chiral anomalies.
in terms
resembles
obtained from the Skytmion-Skyrmion
the Paris potential.
They do sometimes,
in terms of a
can be described
alone. The infinite density limit of the Skyrmions
quark gas3'. resembles
cou-
.
a
interactions
M. Rho / The skyrrnions and quarks in nuclei
In conclusion,
it is quite remarkable
at least qualitatively,
649C
that such a simple Lagrangian
so much of QCD. It would be interesting
embodies,
to see whether
and where it breaks down. ACKNOWLEDGEMENTS I am very grateful A.S. Goldhaber,
for collaborations
and/or discussions
with 3G.E. Brown,
A.D. Jackson, V. Vento, G. Ripka and I. Zahed.
REFERENCES 1) R. Jackiw , Comments in Nucl. and Part. Phys., to appear (1984); lectures at Les Houches Summer School, Les Houches, France (1983). 2) T.H.R. Skyrme, Proc. Roy. Sot. London Ser. A -260 (1961) 127; Nucl. Phys. -31 (1962) 556. 3) N.K. Pak and H.C. Tze, Ann. Phys. (N.Y.) 117 (1979) 164; J. Gibson and H.C. Tze, Nucl. Phys. B183 m81) 524; A.P. Balachandran, V.P. Nair, S.G. Rzv and A. Stern, Phys. Rev. Lett. -49 (1982) 1124; Phys. Rev. _D27 (1983) 1153. 4) G. 'tHooft, Nucl. Phys. 872 (1974) 461; 875 (1974) 461; E. Witten, Nucl. Phys. B'IbtT(1979) 57 5) E. Witten, Nucl. Phys. -B223 (1983) 423, 433. 6) M. Rho, A.S. Goldhaber and G.E. Brown, Phys. Rev. Lett. 51 (1983) 747; A.D. Jackson and M. Rho, Phys. Rev. Lett. -51 (1983) 751.7) a. G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552; b. G.S. Adkins and C.R. Nappi, Nucl. Phys. B233 (1984)m; c. G.S. Adkins and C.R. Nappi, Phys. Lett. 'I3TB (1984) 251 8) G.E. Brown, A.D. Jackson, M. Rho and V. Vento, Phys. Lett. _1408 (1984) 285. 9) J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51 (1983) 1518; I. Zahed, U.G. Meissner and A. Wirzba, Phys. Lxt. B, in press 1984; P.J. Mulders, "Theoretical aspects of hybrid chiral bag models", M.I.T. preprint (1984). 10) L. Vepstas,
A.D. Jackson and A.S. Goldhaber,
Phys. Lett. 1408 (1984) 280
11) M. Jezabek and K. Zalewski, "Direct evaluation of baryon numbers of empty chiral bags", Zeit. fiir Phys. in press (1984). 12) R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398; R. Jackiw and J.R. Schrieffer, NuclT-Phys. B190 [FS31 (1981) 253. 13) H.B. Nielsen,unpublished. 14) H.J. Schnitzer, 15) R. Finkelstein J.G. Williams,
Phys. Lett. -139B (1984) 217. and J. Rubinstein, 3. Math. Phys. _!j(1968) 1762; J. Math. Phys. -11 (1970) 2611.
650~
M. Rho / The skyrmions and quarks in nuclei
16) G.
Kramer, W.F. Palmer and S.S. Pinsky, Phys. Rev. D, in press.
17)
A. Dhar and S.R. Wadia, Phys. Rev. Lett. 52 (1984) 959; classique couple aux M. Gaudin, "Lagranaien effectif du champ %iral fermions", Saclay preprint (1984); R.I. Nepomechie, "Evaluating fermion determinants through the chiral anomaly", Brandeis preprint (1984); A. Manohar and G. Moore, "Anomalous inequivalence of phenomenological theories", Harvard preprint HUTP-84/A007 (1984).
18)
A.D. Jackson, H.K. Lee and M. Rho, paper in preparation.
191 A. Jackson, A.D. Jackson
and V. Pasqieur,
Nucl. Phys. A (in press).
T. Eguchi, P. Gilkey and A.Hanson,.Phys.
Reports 66 (1980) 215.
20)
See,e.g.,
21)
H.B. Nielsen and I. Zahed, unpublished.
22)
See, e.g. M. Rho, Pion Interactions Science 2, in nress.
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