Quark Number Fractionalization in N=2 Supersymmetric $SU(2) \times U(1)^{N_f}$ Gauge Theories

11 Pages·0.16 MB·English

by  Giuseppe Carlino

N
Quark Number Fractionalization in = 2 Supersymmetric
SU(2) U(1)Nf Gauge Theories
×
Giuseppe Carlino(1), Kenichi Konishi(1)
8
and
9
9 Haruhiko Terao(2)
1

n
a
J Dipartimento di Fisica – Universita` di Genova(1)

7
Istituto Nazionale di Fisica Nucleare – Sezione di Genova(1)

1 Via Dodecaneso, 33 – 16146 Genova (Italy)
v
Department of Physics, Kanazawa University(2)
7
2 Kanazawa, Japan
0
E-mail: [email protected]; [email protected]
1
0
8
9
/
h
ABSTRACT:
t
-
p
Physicalquark-number chargesofdyons aredetermined, via a formula which generalizesthat of
e
h Wittenfortheelectriccharge,inN =2supersymmetrictheorieswithSU(2) U(1)Nf gaugegroup.
×
:
v The quark numbers of the massless monopole at a nondegenerate singularity of QMS turn out to
i vanish in allcases. A puzzle relatedto CP invariantcases is solved. Generalizationof our results to
X
r SU(Nc) U(1)Nf gauge theories is straightforward.
a ×
GEF-Th-11/1997;KANAZAWA 97-21
December 1997
Introduction
The breakthroughachievedin the celebratedworksofSeiberg andWitten [1,2]has made possi-
ble, for the first time, to go beyond the semiclassical quantization in the study of soliton dynamics
in non Abelian gauge theories in four dimensions.[3] Of particular interest among their properties
is the electric and quark-number fractionalization. In Ref.[4] it was shown that, in the semiclassical
limit, the exact Seiberg-Witten prepotentials and mass formula reproduce these effects correctly,in
accordancewiththe standardsemiclassicalcalculations[5,6]. Inthe caseofthe electricchargefrac-
tionalizationonehashereanexactquantumresultvalidevenwherethesemiclassicalapproximation
breaks down.
The situation for the quark-number fractionalization is somewhat different. As long as the
U(1) symmetries associated with the quark numbers are global, the ”physical quark number of
monopoles” is a somewhat obscure quantity, even though such a quantum number is conservedand
not spontaneouslybroken. In fact, there is no localfield within the theory coupled to the conserved
quark-number currents Jµ. For instance the correlation functions,
i
Πµν(Q)=i d4xe iQx T Jµ(x)Jν(0) (1)
Z − h { i i }i
cannot be easily analyzed at low energies, although at high energies these can be computed pertur-
batively due to asymptotic freedom.
In this paper we consider N = 2 supersymmetric SU(2) U(1)Nf gauge theories (Nf = 1,2,3)
×
where the Abelian factors correspond to the conserved quark numbers; more precisely we consider
these theories in the limit
g 0+, (2)
i
→
gi being the Ui(1) coupling constant. In other words, we introduce the hypothetical weak U(1)Nf
gauge bosons and their N =2 partners in order to probe the strong interaction dynamics, which is
dominated by the SU(2) interactions. This theory has a large vacuum degeneracy parametrized by
N +1 moduli parameters,
f
u= TrΦ2 ; a = A =m /√2, (i=1,2,...N ). (3)
i i i f
h i h i
Thephysicali-thquarknumber(charge)S ofagivenparticleisbydefinitionitslowenergycoupling
i
strength to the U (1) gauge boson, measured in the unit of the coupling constant, g . g is common
i i i
toallparticles(elementaryandsolitonic)anddependsonlyonthescalewhilethechargeS depends
i
on the particle considered. For an elementary particle, say the j-th quark, S = δ . The latter is
i ji
not renormalized, as is well known. Our main aim is to determine the value of S for a given dyon,
i
in each vacuum (u,a ,a ,...,a ).
1 2 Nf
Fractional quark numbers of dyons as boundary effects
The theory is described by the Lagrangian,
1 1
L = Imτ d4θΦ eVΦ+ d2θ WW +
cl †
8π (cid:20)Z Z 2 (cid:21)
Nf 1 1
+ 8πImτi(cid:20)Z d4θA†iAi+Z d2θ 2WiWi(cid:21)+L(quarks), (4)
Xi=1
1
where
L(quarks) = [Z d4θ{Q†ieVQieVi +Q˜ie−VQ˜†ie−Vi}+Z d2θ√2{Q˜iΦQi+AiQ˜iQi}+h.c.], (5)
Xi
where Φ,W and A ,W are N =2 vector supermultiplets containing the gauge bosons. Classi-
i i
{ } { }
cally this theory has the flat directions parametrized by the vevs Eq.(3). In a generic point of such
vacuum moduli space, the gaugesymmetry is brokento U(1) U(1)Nf and the low energyeffective
×
Lagrangian must describe N +1 massless gauge bosons and their superpartners, and eventually,
f
also light quarks or dyons. Its form is restricted by N =2 supersymmetry to be
1 Nf 1 Nf ∂F
L(eff) = Im d2θ τ W W + d4θ A¯ +L(light), (6)
ij i j i
8π Z 2 Z ∂A
iX,j=0 Xi=0 i
where τ = ∂2F , and L(light) describes either light quarks or light dyons. F(a , a ;
ij ∂Ai∂Aj 0 { i}
Λ; Λ ) is the prepotential, holomorphic in its arguments. Also, we introduced a notation
i
{ }
A A, V V, A A , V V , (7)
0 0 D0 D D0 D
≡ ≡ ≡ ≡
to indicate the vector multiplet related to the original SU(2) gauge multiplet. Λ is the position of
i
the Landau pole associated to the i-th U(1) gauge interaction.
The form of L(light) near one of the quark singularities (when u = m2 Λ2), is fixed since the
i ≫
quantum numbers of the light quarks with respect to the U(1) U(1)Nf gauge group are known.
×
On the other hand, the monopoles acquirethe quark numbers dynamically. Semiclassicallythey
arise through the zero modes of the Dirac Hamiltonian in the monopole background [7, 3, 2, 4].
The classical quark numbers of the dyons, which become massless at various singularities of QMS
are given in Table 1. Note that, in contrast to Ref.[4] we have chosen the classical quark number
chargesfor monopoles to start with. 1 The dynamical, fractionalpart of quark number chargescan
be determined as follows.
The structure of the lowenergy effective Lagrangian,when one ofthese monopolesis light(near
one of the singularities in the u plane), is then fixed by their integer quantum numbers, n ,n and
m e
n S(cl), i=1,2,...N . (8)
i ≡ i f
Near the singularity of QMS where a (n ,n , n ) dyon is light L(light) has the form (assuming
m e i
{ }
there is only one such dyon)
L(light) = d4θ[M†enmV0D+neV0+ niViM +M˜†e−nmV0D−neV0− niViM˜]+
Z P P
+ d2θ√2(n A +n A + n A )M˜M +h.c. (9)
m 0D e 0 i i
Z
X
The fact that the monopole is coupled to the weak U(1) gauge fields with the (apparent) integer
charges, does not mean that its physical charges are equal to the classical ones. The point is that
1This is, strictly speaking, unnecessary. One can start with any choice of S and adjust the constant part of aD
proportional to quark masses accordingly, as explained in [4]. The final result for the physical quark number is the
same,whatever initialchoiceforS onemakes,butthefinalformulaslookmostelegantwithourchoice.
2
Table 1: Classical quark number charges and other global quantum numbers of light dyons.
±
denotes the SO(2N ) chirality of the spinor representation. θ gives the value of the effective θ
f eff
parameter where the corresponding dyon becomes massless.
N =1
f
name n n n SO(2) θ
1 m e eff
M 0 1 0 + 0
M 1 1 1 π
′
− −
M 0 1 2 + 2π
′′
−
N =2
f
name n n n n SO(4) SU(2) SU(2) θ
1 2 m e eff
×
M 0 0 1 0 + (2, 1) 0
1
M 1 1 1 0 + (2, 1) 0
2
M 1 0 1 1 (1, 2) π
1′ − −
M 0 1 1 1 (1, 2) π
2′ − −
N =3
f
name n n n n n SO(6) SU(4) θ
1 2 3 m e eff
M 0 0 0 1 0 + 4 0
0
M 1 1 0 1 0 + 4 0
1
M 1 0 1 1 0 + 4 0
2
M 0 1 1 1 0 + 4 0
3
N 0 0 0 2 1 1 π/2
−
3
there are nontrivial boundary effects to be taken into account, just as the Witten’s effect for the
electric charge of the monopole, in the presence of the θ term, (θ/32π2)F F˜µν [8, 9].
µν
In our case, the crucial term is the mixed gauge kinetic term, τ W W of Eq.(6). In fact, this
0i 0 i
term yields a term in the energy
1
Reτ d3xE H (10)
0i i 0
4π Z ·
where E and H standrespectively for the ”electric”field associatedwith the weak,quark number
i 0
U (1) and for the ”magnetic” field associated with the strong U (1) (related to the SU(2)) gauge
i 0
interactions. In the presence of a static monopole,
n
d3xE H d3x( φ (x)) m = 4πn d3xφ (x)δ3(x), (11)
i 0 i m i
Z · ≃Z −∇ ·∇ r − Z
hence Eq.(10) implies that the magnetic monopole, when observed at spatial infinity, possesses an
additional quark number charge,
∂2F
∆S =n Reτ =n Re . (12)
m 0i m
∂a∂a
i
The true, physical i-th quark number charge of such a dyon is therefore given by
S(phys) =n +n Reτ . (13)
i i m 0i
This, which generalizes Witten’s well-known formula [8], is our main result.
Generalization to SU(Nc)
The fractional quark numbers of dyons in N = 2 supersymmetric SU(N ) gauge theories [10]
c
can be found by similar considerations. In the QMS the SU(Nc) gauge group is broken to U(1)Nc.
The dyon carries magnetic and electric charges of each unbroken U(1) and the associated quantum
numbers are denoted by (nr ,nr),(r = 1, ,N ). The gauge couplings in the low energy effective
m e ··· c
action are also generalized to τ , (r,s = 1, ,N ), with the mixed θ terms, (θ /32π2)Fr F˜sµν.
rs ··· c rs µν
Therefore the r-th physical electric charge Qr is found to be
Nc
Qr =nr+ Reτ ns . (14)
e rs m
Xs=1
In order to define the physical quark numbers unambiguously we consider SU(Nc) U(1)Nf
×
gauge theories. The U(1)Nc U(1)Nf effective Lagrangianis given by
×
1 1
L(eff) = Im d2θ τ W W + d4θA A¯ +L(light), (15)
IJ I J DI I
8π Z 2 Z
XI,J XI
whereI,J denotethecombinedsuffix(r,i),(r =1, ,N ,i=1, ,N ). Fromthemixedcouplings
c f
··· ···
betweenthe“color”U(1)andthe“flavor”U(1)fieldstrengthswefindthephysicali-thquarknumber
of a dyon in SU(N ) gauge theories as
c
Nc
S(phys) =n + Reτ nr . (16)
i i ri m
Xr=1
4
Minimal coupling
One might wonder whether, having an ”exact low energy effective Lagrangian” at hand, such
fractional quark number charges should not appear as part of the standard minimal interaction
terms. In fact, it is possible to interpret our result this way. In the case of Witten’s effect this was
pointed out in [11].
There is indeed a large class of arbitrariness in the choice of ”dual” variables A A , V
D 0D D
≡ ≡
V ,correspondingtoashiftofthesevariablesbytermslinearinA, andA ,V, andV , i=1,2,...,N .
0D i i f
These make a subgroup of Sp(2+2N ,R) which leaves Eq.(6) form invariant (see however below).
f
Actually,sincethequark-numberU(1)groupsareonlyweaklygauged,wecanexcludethoseelements
of Sp(2+2N ,R) which transform A (i=1,2,...,N ) to their duals. In such a case, the most
f i f
generalformoftherelevantsubgroupofSp(2+2N ,R)havebeenrecentlyfoundbyAlvarez-Gaum´e
f
et. al. [12]; they have the following general form:
A αA +βA+p A
D D i i
 A   γA +δA+q A 
D i i
, (17)
AiD→AiD+pi(γAD+δA) qi(αAD +βA) piqi
   − − 
 A   A 
 i   i 
α β
where SL(2,R) and p ,q are real.2
i i
(cid:18)γ δ(cid:19)∈
α β 1 β
The transformations relevant to us are the ones with = and p . These
i
(cid:18)γ δ(cid:19) (cid:18)0 1(cid:19)
transformations leave the effective LagrangianEq.(6) invariant, except for the shift,
τ τ +β; τ τ +p . (18)
0i 0i i
→ →
Therefore all of Reτ, Reτ can be eliminated by an appropriate such transformation, i.e., by
0i
choosing β = Reτ, p = Reτ . However N = 2 supersymmetry imposes a simultaneous shift of
i 0i
vector superfields
V V +βV +p V ; V V; V V ; (19)
D D i i i i
→ → →
in the effective Lagrangianinvolving light dyons such as Eq.(9). The net result is that the realpart
in the coefficients of the mixed kinetic terms discussed previously has disappearedand, at the same
time, the dyon with integer quantum numbers (n ,n ,n ) is now coupled minimally to the vector
m e i
fields A and A with charges
µ iµ
Q =n +n Reτ; and S(phys) =n +n Reτ . (20)
e e m i i m 0i
The first is Witten’s effect (Reτ =θ /π), the second is our result.
eff
TheapparentlylocaleffectiveLagrangianEq.(6)(withEq.(9))withanontrivialboundaryeffect,
has been transformed by (19) into an explicitly nonlocal Lagrangian. Such an equivalence is to be
expected after all, in view of the dyonic nature of our monopoles.
2In [12] models with N = 2 dilaton and mass ”spurion” fields are studied and this leads them to consider a
Sp(4+2Nf,R)group. Herewerestrictourselvestorenormalizabletheories: thisleavesonlythequarkmassestobe
replaced by the N = 2 mass ”spurion” fields in their language. The latter is equivalent to gauging the Nf quark-
number U(1) groups, asformulatedhere. Notethat weuseaslightlydifferent notation from[12],ai,aiD instead of
mi,miD,etc. Notethatpi,qi areanyrealnumbers,rationalornot.
5
OurformulaforthephysicalquarknumberEq.(20)canbeexpressedintermsofknownquantities.
Note that in the limit of weak U(1)Nf couplings the low energy effective coupling and θ parameter
of the SU(2) sector as well as the mass formula must be the same as in [2]. It means that the
prepotential is essentially that given in [2]:
Nf
F(a , a ;Λ; Λ )=F(SW)(a , m ;Λ) + C a2, (21)
0 { i} { i} 0 { i} |mi=√2ai i i
Xi=1
where possible terms linear in a and a have been dropped.3 The last term contains the genuine
i
free parameters of the theory, the U (1) coupling constants at a given scale, or the position of the
i
corresponding Landau poles, Λ . Although these affect the U (1) coupling constants g , they do
i i i
{ }
not enter the calculation of the corresponding charges, Eq.(13).
τ can now be expressed as
0i
∂2F ∂a ∂a ∂F ∂F(SW)(a, m ;Λ)
D iD i
τ = = = , where a = = { } . (22)
0i ∂a∂a ∂a a ∂a ai D ∂a ∂a
i i
The partial derivative of a with respect to a can be further rewritten as
D i
∂a ∂a da ∂a ∂a ∂a
D D D D
τ = = = τ . (23)
0i a u u u u
∂a ∂a − da ∂a ∂a − ∂a
i i i i i
The fractional quark charge can now be computed by using the known exact solution for
da da
D
= ω, = ω, a = λ , a= λ , (24)
D SW SW
du I du I I I
α β α β
and their derivatives with respect to a =m /√2. The meromorphic differential λ , related to ω
i i SW
by ω = √2dx = ∂λSW, is given explicitly in [2, 12, 4, 13].
8π y ∂u
Riemann bilinear relation
An equivalent alternative formula for τ can be found by first rewriting the formula (23) as
0i
∂a ∂a ∂a ∂a ∂a
D D
τ = / . (25)
0i (cid:18) ∂a u∂u ai − ∂u ai∂a u(cid:19) ∂u ai
i i
In terms of a meromorphicdifferential φ =∂λ /∂a , and the holomorphic differential ω, this can
i SW i
be written, by using Riemann bilinear relation [14], as
x+
n
τ = φ ω ω φ / ω =2πi Res φ ω/ ω, (26)
0i (cid:18)Iα iIβ −Iα Iβ i(cid:19) Iβ Xn x+n i Zx−n Iβ
wherex+ andx denote the polesofφ inthe firstandthe secondRiemannsheetrespectively. The
n −n i
contour fromx to x+ mustbe takenso as to goaroundthe branchpoint which is not encircledby
−n n
the α-cycle. The positions of the poles x ’s (which are nontrivialfor N =3) are explicitly givenin
n f
[12].
3 Thelowenergyeffective actioninvolvesthesecondorhigher derivativesoftheprepotential. Thetermlinearin
a does affect the constant part of aD which should be fixed by appropriate convention so that the mass formula is
obeyed.
6
Semiclassical limit
As a first check of our result consider the semiclassicallimit, u Λ2, a =m /√2 Λ. In this
i i
≫ ≫
limit Eq.(13) must reduce to the known result [5, 6]:
n a+m /√2
S(phys) n + mArg i . (27)
i ≃ i 2π m /√2 a
i
−
whichisindeedthecaseascanbeverifiedbyadirectcalculation,similartotheoneintheAppendix
C of [4]. 4
Vanishing quark numbers of massless monopoles
An interesting special case is that of the quark number chargesof a masslessdyon, at one of the
singularities of QMS. Since such an object occurs in a theory in which the original SU(2) coupling
constant becomes large in the infrared, the semiclassical method does not apply there.
For simplicity we consider the cases in which the singularity is nondegenerate, with only one
massless monopole. This is the case for N = 1, or for N = 2 or N = 3 with generic and
f f f
nonvanishing masses. The physical quark numbers of these massless monopoles turn out to be zero.
To see how this result comes about, let us consider the one-flavor case and concentrate on the
(n ,n ,n ) = (1,0,0) monopole occurring at the singularity u = u = e iπ/3. (We use the unit,
m e 1 3 −
3 2 8/3Λ2 =1.) The proof in other cases is similar. From Eq.(23) one has
−
·
1 dx dx da dx dx
D
τ = τ ; τ = = / , (28)
01
−2π (cid:18)I xy − I xy(cid:19) da I y I y
α β α β
where use of made of the explicit formulae valid for N = 1, λ = (√2/4π)ydx/x2; y =
f SW
−
x2(x u)+ 2√2a x 1. Near u u , two of the branch points x , x are close to each other
1 3 2 3
− − ≃
(and coalesce at 2u /3 when u = u ), while the third one is near u /3. In the integrations
3 3 3
−
over α cycle (which encircles the nearby branch points x and x ) y can be approximated as y =
2 3
(x x )(x x )(x x ) √x x (x x )(x x ), so that
1 2 3 2 1 2 3
− − − ≃ − − −
p p
dx x2 dx 2π dx 2π
=2 , , (29)
I xy Z xy → x √x x I y → √x x
α x3 2 2− 1 α 2− 1
as u u . On the other hand, the integration over β cycle is also dominated by the region near
3
→
x x , and
2
≃ dx x1 dx 2
=2 I, (30)
I xy Z xy ≃ x √x x ·
β x2 2 2− 1
where the integral I = x1 dx is divergent at u=u . However the dx in the denomi-
Rx2 √(x−x2)(x−x3) 3 Hβ y
nator of τ also diverges as
dx 2
I. (31)
I y ≃ √x x ·
β 2− 1
Therefore
1 1 I
τ =0, S(phys) 0, (32)
01
→−{x √x x − x √x x · I} →
2 2 1 2 2 1
− −
4In [4] the present authors studied the electric and quark number fractionalization in the context of the original
SU(2) gauge theory with Nf quark hypermultiplets. For the quark number charges, it was only possibleto make a
checkthroughthemassformula,whichisknownbothsemiclassicallyandexactly[2].
7
as u u . The same result follows by using the second formula Eq.(26).
3
→
For the massless (1,1,1) dyon at u = u = e+iπ/3 (where the combined α+β cycle vanishes)
1
one finds by a similar analysis that S(phys) =1 1=0.
−
These results are somewhat analogous to the fact that all massless ”dyons” of N = 2 Seiberg-
Witten theories with various N , are actually all pure magnetic monopoles with Q =0. [4]
f e
Some numerical results
It is straightforward to evaluate τ given by the formula (23) or (26) numerically at any point
0i
on the moduli space. In Fig.1 we show Re(τ ), or the physical quark number of (1,0,0) dyon in
01
N=2SQCDwithasinglemasslessflavor. Inthis caseS(phys) approaches 1/2inthe weakcoupling
−
limit, while it rapidly reduced to zero near the singularity where the (1,0,0) BPS state becomes
massless, in accordance with the discussion in the precedent paragraph. The quark number of the
same dyon remains equal to 1/2 in any vacua with real positive u.
−
0.0
θ=−π/3
θ=−π/6
−0.1 θ=0
−0.2
τ)01
e(
R
−0.3
−0.4
−0.5
0.0 1.0 2.0 3.0
|u/u|
3
Fig.1: Re(τ ) in the massless N = 1 theory is shown along the half lines u =
01 f
u expiθ of θ =0, π/6, π/3.
| | − −
CP invariance and quark numbers
It is somewhat surprising that the physical quark number of the monopole takes all possible
fractional values even in a CP invariant theory. An example is the N = 1, m = 0 theory at
f
Argu = π/3; u 1, where θ = 0. Indeed S(phys) of the (1,0) monopole takes all real values
eff
− | | ≥
from0(at u=e iπ/3)to 1/2(at u ). Sucha resultseemsto be atoddwiththe well-known
−
− | |→∞
resultofJackiwandRebbi[7],thatinaCPinvariantSU(2)theorywithafermioninthefundamental
representationthe’tHooft-Polyakovmonopolebecomesadegeneratedoubletwithfermionnumbers
q = 1/2.
± ±
The key for solving this apparent puzzle lies in the vacuum degeneracy. In the argument of Ref
8
[7] the standard monopole 0 M is accompanied by another state
| i≡| −i
M =b M , such that M ψ M =0 (33)
+ † +
| i | −i h −| | i6
where b is the fermion zero mode operator, ψ = bψ (x)+nonzeromodes, b,b = 1. There is a
0 †
{ }
conserved fermion conjugation symmetry , such that
F
[ ,H]=0; b =b†, 2 =1. (34)
F F F F
This lastequation, together with Eq.(33),implies that the monopole states M transformto each
| ±i
other by : M = M . The fermion number operator must be defined as
F F| ±i | ∓i
1 1
S = d4x(ψ ψ ψψ )= (b b bb )+... (35)
† † † †
2Z − 2 −
so that
S = S, [S,ψ]= ψ. (36)
F F − −
One finds q = q from the first of the above. On the other hand, from the M ... M matrix
+ +
− − h −| | i
element of the second of Eq.(36) another relation, q = q +1 follows. Combining these two one
+
−
finds the announced result q = q = 1/2. Of course, the first of Eq.(34) guarantees that the
+
− −
states M are degenerate in mass.
| ±i
In the present theory the role of the fermion conjugation is played by CP symmetry. Under
CP, however, the vacuum is also transformed as u u . What happens in the case of theories
∗
→
at Argu = π/3; u 1, is that the theory is transformed by CP to another theory, related
− | | ≥
to the original one by an exact Z symmetry. This explains the fact that θ = 0 and that
3 eff
the low energy effective monopole theory there has an exact CP invariance in the usual sense [4].
Nonetheless, from the formal point of view the original CP symmetry of the underlying theory
is spontaneously broken in this case, and the Jackiw-Rebbi argument does not apply. In fact,
although the operator relations Eq.(34) and Eq.(36) still hold, the states are now transformed by
M ;u = M ;u : the two states related by CP operation live on two different Hilbert space.
∗
F| ± i | ∓ i
As a result, the first of Eq.(36) yields q = q ; q = q , where q are the quark number of
+∗ − − + − −∗ ±∗
the states M ;u . The second equation, whose matrix elements relates the states in the same
∗
| ± i
vacuum, leads to q =q 1; q =q 1. Note that these four relationsare mutually consistent
− +− −∗ +∗ −
and relates the four charges by q = q = q +1 = q +1. Although the first of Eq.(34) does
+∗ − − − + −∗
imply that the monopoles M ;u and M ;u have the same mass, (and similarly M ;u and
+ ∗ ∗
| i | − i | − i
M ;u ) it does not imply any degeneracy; it rather means that the spectrum of the theories at u
+
| i
and at u are the same, reflecting the Z symmetry of the underlying theory .
∗ 3
Note that along the real positive values of u (for N = 1), where CP is exact and not sponta-
f
neously broken(with a CP invariantvacuum), dyons are found indeed to be doubly degenerate and
have quark numbers 1/2, in accordance with [7].
±
Quark-number current correlation functions
Once the physical quark numbers of light dyons are known, the analogue of the R-ratio as-
sociated with the correlation function Eq.(1) may be computed at low energies by the one loop
contributions of the weakly coupled dyons. By an appropriate normalization one finds that, near
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