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Eur. Phys. J. Plus (2017) 132: 118
DOI 10.1140/epjp/i2017-11374-3
Superconductivity due to soft super-symmetry breaking Balwant S. Rajput
Eur. Phys. J. Plus (2017) 132: 118 DOI 10.1140/epjp/i2017-11374-3
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Superconductivity due to soft super-symmetry breaking Balwant S. Rajputa Kumaun University, Nainital, India Received: 16 December 2016 c Societ` Published online: 8 March 2017 – a Italiana di Fisica / Springer-Verlag 2017 Abstract. Revisiting the super-symmetric dyons in N = 2 super-symmetric theory and analyzing the possible soft breaking of N = 2 super-symmetric Yang Mills theory to N = 0 by making the dynamically generated mass scale ∧ a function of dilation spurion, it has been demonstrated that the scalar and auxiliary components of pre-potential, constructed in terms of dilation, are frozen to be constant to generate soft breaking of N = 2 theory and it has been shown that, as soon as these soft breaking terms are turned on, monopole condensation appears and we get a unique ground state and the superconducting phase. It is also shown that in this soft breaking of N = 2 super-symmetry, the superconductivity phase occurs due to condensation of monopoles only and the dyons do not condensate near the real u-plane.
1 Introduction Monopoles and dyons have become the intrinsic part of all current grand unified theories [1] and super-symmetric models [2–9] but the most important role of monopoles and dyons in physics is their participation in the mechanism of confinement through their condensation [10], leading to efficient microscopic theories of superconductivity [11–14], dual-superconductivity [15, 16] and color superconductivity [17]. In super-symmetric GUT’s, the monopole ground state, in general, carries fractional electric charge as well as color hypercharge [18]. This is manifestation of fermion fractionalization [19] with the axial anomaly effect [20] properly taken into account. Super-symmetry provides the first realistic testing ground for the idea of fermion fractionalization and induced fractional charge on a monopole. Thus the occurrence of fractionally charged dyons along with their super-partners for each charge state leads to the possibility of observation of dyons. All super-symmetric theories with holomorphic super-potentials have moduli space as the result of flat directions in such potentials [4, 5, 21–24]. All points located on these flat directions represent degenerate but physically in-equivalent ground states. A classical moduli space cannot be lifted quantum mechanically due to tight constraints imposed by N = 2 super-symmetry. Such a moduli space incorporates Higgs, Coulomb and confinement phases in the theory where superconductivity, dual superconductivity and color superconductivity occur in confinement phase and several interesting phenomena, like appearance of massless solitonic states, normalization of electric-magnetic dualities and discontinuities in the BPS states of dyons may occur in Coulomb phase. Super-symmetric quantum field theories are easier to analyze and are much more tractable than non-super-symmetric theories due to the constraints which follow from super-symmetry. In particular, the homomorphicity of the super-potential when combined with global symmetries enables one to find many exact results [25, 26]. In all these exact solutions, the singularities of quantum moduli space of the theory correspond to the appearance of massless monopoles and dyons. Consequently, the microscopic super-potential, explicitly breaking N = 2 to N = 1 super-symmetry, has been introduced [8, 22] to explore physics near N = 2 singularities and it has been found that the generic N = 2 vacuum is lifted leaving only a singular loci of moduli space at the N = 1 vacua where monopole and dyons can condensate leading to confinement and superconductivity. Constructing the effective Lagrangian near a singularity u = ∧2 in moduli space for N = 2 super-symmetric theory with SU (2) gauge group, it has been shown in our recent papers [27, 28] that when a mass term is added to this Lagrangian, the N = 2 super-symmetry is reduced to N = 1 super-symmetry yielding the dyonic condensation which leads to confinement and superconductivity as the consequence of generalized Meissner effect. Constructing the total effective Lagrangian of N = 2 SU (2) gauge theory with Nf = 2 quark multiplets and quark chemical potential at classical and quantum levels, it has been demonstrated in our another recent paper [29] that baryon number symmetry is spontaneously broken as a consequence of the SU (2) strong gauge dynamics and the color a
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superconductivity dynamically takes space at the non-SUSY vacuum. Revisiting the super-symmetric dyons in N = 2 theory in present paper, we have analyzed the possible soft breaking of N = 2 super-symmetric Yang Mills theory to N = 0 by making the dynamically generated mass scale ∧ a function of dilation spurion and it has been shown that monopole couplings are well behaved near the singularity u = ∧2 while the dyon couplings are favored at the singularity u = −∧2 . It has been demonstrated that the scalar and auxiliary components of pre-potential, constructed in terms of dilation, are frozen to be constant to generate soft breaking of N = 2 theory and as soon as these soft breaking terms are turned on, monopole condensation appears and we get a unique ground state and the superconducting phase but dyons do not condense near the real u- plane. It has also been shown that this condensation of monopoles leads to confinement and the superconductivity. Thus in soft breaking of N = 2 super-symmetry to N = 0, the superconductivity phase occurs due to condensation of monopoles only.
2 Super-symmetric dyons in N = 2 theory The simplest four dimensional super-symmetric model in which boundary terms enter as central charge [21] may be constructed in terms of the following N = 2 super-symmetric dyonic Lagrangian [26] in SU (2) gauge theory with the generalized gauge field strength Gµν : 1 1 1 a a µ L = − Gaµv Gµv a + (Dµ φ) (Dµ φ)a + (Dµ P ) (D P )a 4 2 2 ¯ µ (Dµ ψ)a − |q|ǫabc ψ¯a γ5 ψ c P b + iψaγ − i|q|ǫabc ψ¯a ψ b P c − V (φ, P ),
(1)
where
1 1 1 (|q|ǫabc φb φc )2 + [|q|ǫabc P b P c ]2 − (|q|ǫabc φb P c )2 (2) 4 4 2 The dyonic charge q in this equation is the complex quantity with its electric and magnetic charges as its real and imaginary parts q = e − ig, (2a) V (φ, P ) =
and ψ is Dirac spinor, φ is scalar field and P is the pseudo-scalar field (all these fields are in the adjoint representation of the gauge group). This Lagrangian of SU (2) Wess-Zumino model [30] obviously describes the dyon-iso-vector-fermion system. In this model the vacuum energy V (φ, P ) is independent of values of φ and P in certain directions in field space. As long as φ and P commute, the vacuum energy is classically zero. φ and/or P may have non-zero vacuum expectation values, spontaneously breaking some of gauge symmetries. Since the potential for Higg’s fields has flat directions corresponding to [φ, P ] = 0, by performing a chiral rotation we can assume that P = 0. Then the potential term V (φ, P ) in eq. (1) preserves chiral-symmetry but breaks the super-symmetry yielding the dyonic solutions for 0|φa |0 = φ = v (3)
and
0|P a |0 = P = 0.
Thus the original massless gauge multiplet splits into two different super-symmetric multiplets, one of them is still massless and contains the fields Vµ3 , φ3 , P 3 and Ψ 3 , while the other one gets non-zero mass and contains the fields Vµ± , φ± and ψ ± . In the general case, for Lagrangian (1) the fields associated with dyons may be written in the following form: 1 (Ei φ) =φa Gaoi + Pa ǫijk Gjka , 2 1 (Bi φ) = φa ǫijk Gjka + Pa Gaoi . 2
(4)
The super-symmetric current associated with Lagrangian (1) may be readily derived in the following form Sµ = σ αβ Gaαβ γµ ψ a + (Dα φ)a γ α γµ ψ a + (Dα P )a γ α ψ¯a − |q|γµ γ5 ∈abc φb Pc ψ a ,
(5)
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which leads to the following expressions for electric and magnetic charges: QE = d3 x∂i (Ei φ) QM = d3 x∂i (Bi φ) = k,
(6)
with (Ei φ) and (Bi φ) given by eqs. (4). These central charges are non-vanishing only if the vacuum expectation value φ or P is non-zero. The presence of these central charges in the algebra implies that [30] M 2 ≥ Q2E + Q2M ,
(7)
where M is the dyonic mass. In the special case given by eq. (3), the non-zero expectation value φ = v = 0 is responsible for spontaneous breaking of SU (2) to U (1) and eqs. (6) give QE = ve QM = vg.
(8)
Then, eq. (7) reduces to M ≥ v|q|,
(9)
¯ Q} = 2γµ P µ − 2QE − 2iγ5 QM , {Q,
(10)
which is saturated at classical level for all states in the theory. The bound on mass in eqs. (7) and (9) is the consequence of the presence of central charges. In the presence of central charges QE and QM , given by eqs. (6), the super-symmetric algebra is modified in to following form [31]:
¯ are super-symmetric generators and the bracket {} denotes anti-commutation. This modification where Q and Q in algebra leads to partial breaking of super-symmetry [30, 32] and the unbroken super-symmetry pairs the bosonic zero-modes with fermionic zero-modes (which can be interpreted as Goldstone modes of the broken super-symmetry). Generically, the solutions satisfying Bogomol’ny bound [33], v ≥ k,
(11)
breaks only half of the super-symmetry in N = 2 theory and the unbroken half super-symmetry pairs bosonic zero modes with fermonic zero modes [31]. The nature of dyons is strongly perturbed by fermionic sector which couples with them since the zero energy solutions for dyons continue to exist for both iso-spinor and iso-vector fermions. The simplest four-dimensional model in which boundary terms enter as external electric and magnetic charges in N = 2 super-symmetric theory may be obtained [7] in terms of the Lagrangian density given by eq. (1) where all fields are in adjoint representation of SU (2) and the classical potential V (φ, P ) is given by V (φ, P ) =
1 tr (φ, P )2 . |q|2
(12)
In the Prasad-Sommerfield limit [34], we have but 0|φ|0 = v = 0, where
V (φ, P ) = 0,
(13)
φ¯a = Aφa + BP a ,
(14)
with A and B as constants satisfying the condition A2 + B 2 = 1. Then the Lagrangian density (1) yields the following expressions for electric and magnetic fields as zero-order solution of eq. (13): Eia = (Di φ)a sin α Bia = (Di φ)a cos α, where
α = tan−1 e/g.
(15)
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For α = 0, the Lagrangian density (1) is invariant under the super-symmetric transformations, which breaks only half of super-symmetry. This partial breaking of super-symmetry is the consequence of the fact that the algebra of super-symmetric charges in N = 2 super-symmetric theory is modified in the form given by eq. (10) by the topological charges of dyons. The presence of these central charges in the algebra implies the inequality (7) for dyonic mass. The states saturating this inequality are BPS-saturated states. In the classical potential (12) of N = 2 theory without hypermultiplets (i.e. without quarks) let us set P = φ† , such that 1 V (φ, P ) = V |φ| = ′ 2 tr(φ, φ† )2 , (16) (g ) where g ′ is the gauge coupling constant of the underlying microscopic theory. As long as φ and φ† commute, the scalar potential V (φ) = 0 even for non-vanishing expectation value of φ, given by eq. (13), which spontaneously breaks SU (2) to U (1) showing that the theory has a continuum of gauge in-equivalent vacua called the classical moduli space parameterized by [24, 25] 1 (17) u = trφ2 = v 2 , 2 where for SU (2) gauge group we have set φ= with 3
σ =
1 3 vσ , 2
1 0 0 −1
.
(18)
This parameter u is a good local coordinate on the classical moduli space. Such a classical moduli space is the consequence of existence of flat directions along which the scalar potential (16) vanishes. All points located on these flat directions represent degenerate but physically in-equivalent ground states. For generic φ the low-energy effective Lagrangian, containing a single N = 2 vector multiplet V , is expressed in terms of a holomorphic function F (v) =
1 τ (v)v 2 , 2
(19)
where τ (v) is effective coupling in the vacuum parameterized by v. The metric is dS 2 = Im
∂ 2 F (v) dvd¯ v = Im τ (v)dvd¯ v, ∂v 2
where
(20)
∂2F . (21) ∂v 2 and nm , respectively, as follows in the classical moduli space: i2 g g √ , and nm = (22) 8π 2 τ (v) =
Defining electric and magnetic charge numbers ne e ne = √ 2 the dyonic charge may be written as
q = e − ig =
√
8πinm 2 ne − , g i2
(23)
and the dyonic mass, given by the upper limit of inequality (9), may be written as M=
√ √ 8πinm 2v|ne − | = 2v|ne − τcl nm |, i2 g
(24)
8πi . g i2
(25)
where τcl =
Identifying this τcl as the effective coupling constant τ (v) of eq. (19), we get F (v) =
4πiv 2 . g i2
(26)
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In terms of τcl , the dyonic charge of eq. (23) may be written as √ q = 2(ne − τcl nm ) = (ne , nm ).
(27)
Setting v=a and
∂F = aD . ∂a
(28)
we may write eq. (20) for metric of classical moduli space in the following form: ∂aD dad¯ a = Im daD d¯ a ∂a i a − dad¯ aD ), = − (daD d¯ 2
dS 2 = Im
(29)
which is symmetrical in a and aD . In general a and aD are the functions of local holomorphic coordinate u and hence the metric of eq. (29) may be written as daD d¯ a · dud¯ u du d¯ u i daD d¯ a aD da d¯ − =− dud¯ u, 2 du d¯ u du d¯ u
dS 2 = Im
which is invariant under adding a constant to aD or a. Under the duality transformation, τcl → τcl + 1,
(30)
(31)
we have (ne , nm ) → (ne − nm , nm ),
(32)
which incorporates the transformation of a monopole (0, nm ) to a dyon (−nm , nm ) and the transformation of a dyon (1, 1) to a monopole (0, 1). In Coulomb gauge, the theory has a massless photon and hence it is subject to the standard electric-magnetic duality, −1 q = (ne , nm ) → (−nm , ne ) = q, (33) τcl which incorporates the inversion of τcl . The transformation (32) and (33) generate an infinite duality group SL(2, Z). Thus there is a natural family of parameters related by SL(2, Z) and in classical moduli space the singular points are associated with extra massless particles. If a general matrix ⌈∈ SL(2, Z) is taken as αβ ⌈= = M, (34) γ δ then the transformation,
aD a
→
a′D a′
=M
aD a
,
(35)
gives a′D = αaD + βa a′ = γaD + δa,
(36)
which give the following transformation of the periodic matrix τ11 : ′ τ11 =
β + ατ11 . δ + rτ11
(37)
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It gives the following modular transformation of the holomorphic function F (a): F ′ (a) = Fτ (a′ ) =
1 1 βδa2 + αγa2D + βγaD a + F (a). 2 2
(38)
Thus, for M =S=
0 1 −1 0
,
(39)
we get the transformation (33) yielding a′D = a; a′ = −aD ; 1 ′ S τ11 = τ11 =− τ11
(40) (41)
and F ′ (a) = Fs (as ) = −aaD + F (a).
(42)
Similarly, for M =T =
1 −1 0 1
,
(43)
we have the transformation (31) which gives a′D = aD − a = aTD ; a′ = a = aT ;
′ τ11 = 1 + τ11
and F ′ (a) = FT (aT ) =
1 2 a + F (a). 2
(44)
(45)
The transformation (33) incorporates the transformation of an electric charge (1, 0) to a monopole (0, 1), i.e. it leads to the monopole region. On the other hand, the transformation (32) transforms a monopole (0, 1) to a dyon (−1, 1), i.e. it leads to the dyon region The mass of the BPS-state in quantum moduli space M may be written, in terms of a and aD by using relations (30) along with relation (22) which yield ∂aD τ (a) = . (46) ∂a Then relation (24) for mass of BPS state may be written as follows: √ M = 2|Z|,
(47)
where Z = ane − aD nm
(48)
is the central charge of super-symmetric algebra. Equations (40) and (44) demonstrate that in the monopole and dyon regions the natural independent variables to be used are a(m) = −aD
(49)
a(d) = aD − a,
(50)
and respectively. Equations (42) and (45) demonstrate that in these regions the holomorphic function F (a) transforms to the following pre-potentials, respectively, F ′ (a′ ) = Fs (aS ) = −aaD + F (a) and F (a) = Fτ (aT ) =
1 2 a + F (a). 2
(51)
(52)
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3 Super-conductivity due to soft breaking of N = 2 super-symmetry to N = 0 Let us break N = 2 super-symmetry to analyze the superconductivity arising due to resulting condensation of monopoles and dyons. To achieve this end, let us make the dynamical scale ∧ of N = 2 theory a function of N = 2 vector multiplet which is then frozen to become a spurion, whose F and D components [35, 36] softly breaks N = 2 theory down to N = 0. As soon as these soft breaking terms are turned on, monopole condensation appears and we get a unique ground state and the super-conducting phase. Let us introduce N = 2 vector multiplets in the pre-potential, F =
1 2 τa , 2
(53)
∂F ∂s
(53a)
to make F (a, s) in such a way that sD =
is monodromy invariant. Then scalar and auxiliary components of this super-field are frozen to be constants to generate soft breaking of N = 2 theory. Let us introduce dilation field s-such that Λ = e−is , (54) with real and imaginary part of s given as Im s ∼
1 g2
and Re S ∼ θ, where g is the coupling constant. Then in each region of the moduli space, we can write a pre-potential in the form F = a2 f (a/∧).
(55)
For this function the transformation (49) in the monopole region implies that 1 F − aaD 2
(56)
is modular invariant. For the dynamically generated mass scale ∧, the explicit expressions of a(u) and aD (u) may be written as [27, 28, 30]
∧2 − u 1 1 2 F − , , 1; , 2 2 2 1 − u/∧2 1 1 1 − u/∧2 u − ∧2 F , , 2; aD (u) = i , 2 2 2 2 a(u) =
(57) (58)
where F (α, β, γ; x) is the usual hyper geometric function defined as 1 F (α, β, γ, x) = B(β, γ − β)
1
0
(1 − t)γ−β−1 tβ−1 (1 − xt)−α dt,
(59)
with B(β, γ − β) as the usual β-function. From these relations we get iF
τ11 =
1 1 u−∧2 2 , 2 , 1; u+∧2
∂aD . = 2∧2 ∂a F 21 , 21 , 1; u+∧ 2
(60)
Using the differential properties of periods a(u) and aD (u) given by eqs. (57) and (58), it may readily be shown that i 1 F − aaD = − u. 2 π
(61)
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After introducing the spurion field s through eq. (54), we have two vector multiplets, a0 = s and a1 = a, and the 2 × 2 matrices of couplings, τ11 =
∂2F ∂aD ; = 2 ∂a ∂a
τ01 =
∂2F ∂s∂a
τ00 =
∂2F . ∂s2
(62)
Using eqs. (54) and (55) along with eqs. (51) and (62), we have ∂ ∂ =i∧ ; ∂s ∂∧ aD = τ11 =
a2 ∂F = 2af + f ′ ∂a ∧
(63)
4a ∂aD a2 = 2f + f ′ + 2 f ′′ ∂a ∧ ∧
τ01 = −3 τ00 = −
(64)
ia2 ′ ia3 ′′ f − 2f ∧ ∧
(65)
a3 ′ a4 ′′ f − 2f , ∧ ∧
(66)
where’ denotes the derivative. The explicit expression of τ11 has been obtained in the form of eq. (60) in terms of hypergeometric function. The other coupling matrices may be written, from eqs. (63)–(66), as follows in terms of τ11 : τ01 = i(aD − aτ11 ) ∂τ11 ∂τ01 = iτ01 − a2 ∂a ∂a and
(67)
∂τ00 ∂τ11 = iτ01 − a2 . ∂a ∂a
We may also write ∂F 1 2 = 2i F − aaD = u = sD ∂s 2 π and τ01 =
2 ∂u 2i ; τ00 = π ∂a π
2u − a
∂u ∂a
(68)
,
(69)
which show that sD is modular invariant. In the region where the coordinate describing light field is a⌈ with ⌈, given by eq. (34), as an element of SL(2, Z), the prepotential may be written as F⌈ = F⌈ (a⌈ , s) = a2⌈ f⌈ (a⌈ /∧), (70) with the coupling given by ⌈
τij =
∂ 2 F⌈ ∂ai⌈ ∂aj⌈
.
(71)
Using relations (36), the transformation rules for aΓ = aΓ (a, s) and aΓD = aΓD (a, s) may be written in the following form: a⌈ = a⌈ (a, s) = γaD (a, s) + δa ⌈
aD (a, s) = αaD (a, s) + βa.
(72)
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Using these relations, we also have
∂ ∂ 1 · = ∂a⌈ ⌈ γτ11 + δ ∂a basis ∂ γτ01 ∂ ∂ = − . ∂s ⌈basis ∂s rτ11 + δ ∂a
(73)
The transformation rule (37) then gives ⌈
τ11 =
β + ατ11 . δ + γτ11
(74)
The transformation rules (38) and (73) lead to ⌈
τ01 ; γτ11 + δ 2 γτ01 = τ00 − γτ11 + δ
τ01 = ⌈
τ00 and also to
∂F⌈ ∂s
= ⌈basis
∂F , ∂s
(75)
(76)
which shows the modular invariance of ∂F ∂s . The full Lagrangian in the region u = ∧2 , in this case, may be readily written in the following form [25, 28, 29]: (∧2 ) 1 ∂2F 1 4 ∂F ¯i i α 2 Im d θ i W W = d θ iA + L 4π ∂A 2 ∂A ∂Ai α i
¯ + d2 θ (nm AD + ne A)M M ¯ † e(−2nm VD −2ne V ) M ˜ + H.C. , + d4 θ M † e(2nm VD +2ne V ) M + M (77) with i = 0, 1, A0 = S and A1 = A, where VD is the dual of vector super-field V and AD is the dual of chiral super-field A, and the effective coupling τ in the vacuum parameterized by a is given by eq. (25) which may also be written as eq. (21) with v = a. The variable θ in these equations denotes the vacuum deformation due to SU (2) strong gauge dynamics such that the relations (25) become τ=
∂2F 8πi θ ∂2F . = = ′2 + 2 2 ∂a ∂A g 2π
(78)
To determine the vacuum structure in this region, S is frozen to be a constant. Its lowest component fixes the scale ∧ but we also freeze its auxiliaries F0 and D0 . Eliminating the auxiliary fields Fm , Fm ¯ and Fa , the potential is obtained as V =
1 [|m|2 + |m ˜ 2 | + 2|a|2 (|m|2 + |m ˜ 2 |) 2b11 1 ¯˜ + b01 D0 (|m|2 − |m| ˜ + F0 m ¯ m) ˜ 2 )] + [b01 (F¯0 mm b det bij 1 |D0 |2 + |F0 |2 , + b11 2
where
(79)
∂ 2 F (a, s) 1 1 τm τij = Im , (80) 4π 4π ∂ai ∂aj ¯ . For small values of F0 and D0 with respect to ∧, the right-hand and m and m ¯ are the scalar components of M and M side of eq. (79) is an exact expression which incorporates super-symmetry breaking. The expression (79) also contains bij is invariant under the generators the contribution of the metric coming from the Kahler potential. The quantity det b11 S and T , given by (39) and (43), of the modular group SL(2, Z). Minimizing the potential V of eq. (79) in the monopole region specified by eq. (49), we get √ ∂V 2 1 2 2 2 ¯˜ = 0 = (|m | + |m| ˜ )m + 2|a| m + b01 F0 m (81) ∂m ˜ b11 b11 bij =
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and
√ 2 1 ∂V 2 2 2 (|m | + |m| ˜ )m ˜ + 2|a| m ˜ + b01 F0 m ˜ = 0, = ¯ b11 b11 ∂m ˜ where we have set D0 = 0. These equations give √ 2 b01 F0 (|m|2 − |m| ˜ 2 ) = 0, b11
which yields
|m|2 = |m| ˜ 2.
Setting m = ρ,
m ˜ = ǫρ
(82)
(83)
(84)
and F0 = f0 ,
where ǫ = ±1, we get the following result from eq. (81): 1 b01 ǫf0 2 2 ρ ρ + b11 |a| + √ = 0, b11 2
(85)
which gives ρ = 0, b01 f0 ρ2 = −b11 |a|2 ± √ > 0, 2
(86)
since
1 Im τ11 > 0. (87) 4π Equation (85) determines the region in the u-plane, where the monopole acquires a vacuum expectation value. Substituting eq. (87) into eq. (79), we obtain b11 =
V =−
2 4 det b 2 ρ − f , b11 b11 0
(88)
which implies that the monopole region, where monopole acquires a vacuum expectation value, is favored and we have confinement and the resulting super-conductivity. In the monopole region the effective potential (88) may be written as follows in terms of magnetic variables: V (m) =
−2
ρ4 − (m)
det b(m)
b11
(m)
b11
f02 ,
(89)
where b(m) is given by (80) and F (a, s) is given by eq. (51). In the Higgs’s region, ρ2 = 0 and the potential (88) maps to V (h) = −
det b(h) (h) b11
(2)
f0 ,
(90)
where b(h) is obtained from eq. (80) with F (a) = 12 τ a2 . This potential has no minimum out side the monopole region near u = ∧2 and hence the fields roll towards the monopole region. Let us now turn to the dyon region specified by eq. (43) leading to the transformation (44) for parameters a and aD and coupling constant τ11 and the transformation (45) for the prepotential F (a). Using these relations the transformations of other couplings may be readily written as follows: ′ τ01 = iτ01
and
′ τ00 = τ00 .
(91)
Comparing these transformations with those given by eqs. (39)–(42) in monopole regions, we get the following relations between dyon and monopole variables: (d)
(m)
a(d) (u) = ia(m) (1 − u); aD (u) = i[aD (−u) − a(m) (−u)], (d)
(m)
(d)
(m)
τ11 (u) = τ11 (−u) − 1; τ01 (u) = τ01 (−u)
(92)
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and (d)
(m)
τ00 (u) = −τ0i (−u).
(93)
These transformation rules of τij coupling have been derived under the residual ZR ∈ U (1)R symmetry whose generator acts on the u-plane as u → −u. (94) Equations (49), (51), (57) and (58) in monopole region show that monopole couplings are well behaved near u = ∧2 , while it follows from relations (44) and (45) in the dyon region that the dyon couplings are well behaved near u = −∧2 . It is obvious, from eqs. (49), (51) and (57), that near the monopole region, au(m) ∼ i(u − ∧2 )
(95)
and (m)
τ01 ∼ i. These equations lead to the result that au(m) → 0 as
u → ∧2 .
(96)
Thus eqs. (92) and (93) lead to (d)
a(u) → u + ∧2 and (d)
τ01 = 0
at u = −∧2 ,
and hence eq. (86) shows that dyon does not condensate along the real u-axis. On the other hand, it follows, from eqs. (92), that: (m)
Im τ0
> 0,
(97)
and hence condition (87) is satisfied and thus as soon as f = 0, the monopole will condensate leading to confinement and super-conductivity. Thus in the soft breaking of N = 2 super-symmetry to N = 0, the super-conducting phase occurs due to condensation of monopoles only.
4 Discussion The scalar and auxiliary components of the pre-potential, given by eq. (53) in terms of dilation introduced by eq. (54), are frozen to be constants to generate soft breaking of N = 2 theory. As soon as these soft breaking terms are turned on, monopole condensation appears and we get a unique ground state and the super-conducting phase. Matrices of coupling are given by eqs. (64)–(66) in terms of spurion s and the vector multiplet a. Other coupling matrices related to these coupling matrices are given by eqs. (67) and (69). Equation (77) gives the full Lagrangian in the region near u = ∧2 in moduli space of N = 2 super-symmetric theory with the dynamical scale ∧ as a function of an N = 2 vector multiplet which is frozen to become a spurion whose F and D components softly breaks N = 2 theory down to N = 0 theory. Minimizing the corresponding potential V , given by eq. (79) in the monopole region specified by eq. (39), eqs. (81) and (82) have been obtained by setting D0 = 0. These equations lead to relation (85) which determines the region in the u-plane, where monopole acquires a vacuum expectation value. Equation (88) implies that such a region is favored and it contains the confinement and the resulting superconductivity. Equation (89) gives the effective potential in terms of magnetic variables in monopole region. The potential (88) has no minimum outside the monopole region near u = ∧2 and it makes the fields roll towards the monopole region. Equations (92) give relationships between dyon and monopole variables. It is obvious from these relations and eqs. (40), (41) and (42) in the monopole region that monopole couplings are well behaved near u = ∧2 , while dyon couplings are well behaved near u = −∧2 . In view of eqs. (95) and (97), it is demonstrated by eq. (87), that as soon as f0 = 0, the monopoles will condense but dyons do not condense near the real u-plane. This condensation of monopoles leads to confinement and the superconductivity. Thus in soft breaking of N = 2 super-symmetry to N = 0, the super-conductivity phase occurs due to condensation of monopoles only.
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