Meta-stable Vacuum in Spontaneously Broken N=2 Supersymmetric Gauge Theory
aa r X i v : . [ h e p - t h ] D ec HIP-2007-41/TH
Meta-stable Vacuum in Spontaneously Broken N = 2 Supersymmetric Gauge Theory
Masato Arai a , Claus Montonen a , Nobuchika Okada b,c andShin Sasaki a a High Energy Physics Division, Department of Physical Sciences, University of Helsinkiand Helsinki Institute of Physics, P.O.Box 64, FIN-00014, Finland b Department of Physics, University of Maryland, College Park, MD 20742, USA c Theory Division, KEK, Tsukuba 305-0801, Japan
Abstract
We consider an N = 2 supersymmetric SU (2) × U (1) gauge theory with N f = 2 masslessflavors and a Fayet-Iliopoulos (FI) term. In the presence of the FI term, supersymmetryis spontaneously broken at tree level (on the Coulomb branch), leaving a pseudo-flatdirection in the classical potential. This vacuum degeneracy is removed once quantumcorrections are taken into account. Due to the SU (2) gauge dynamics, the effectivepotential exhibits a local minimum at the dyon point, where not only supersymmetrybut also U (1) R symmetry is broken, while a supersymmetric vacuum would be realizedtoward infinity with the runaway behavior of the potential. This local minimum is foundto be parametrically long-lived. Interestingly, from a phenomenological point of view, inthis meta-stable vacuum the massive hypermultiplets inherent in the theory play the roleof the messenger fields in the gauge mediation scenario, when the Standard Model gaugegroup is embedded into their flavor symmetry. masato.arai@helsinki.fi claus.montonen@helsinki.fi [email protected] shin.sasaki@helsinki.fi Introduction
Recently, the possibility that a supersymmetry (SUSY) breaking vacuum is not the global min-imum but a local one has been proposed by Intriligator, Seiberg and Shih (ISS) [1]. They haveinvestigated an N = 1 SUSY SU ( N c ) gauge theory (SUSY QCD). The number of flavors istaken to be in the range, N c + 1 ≤ N f < N c , so that this theory is described as the infrared freemagnetic dual theory at low energies and can be analyzed perturbatively. The effective theoryhas the same structure as the O’Raifeartaigh model, and SUSY is broken with the pseudo-flatdirections parameterized by meson fields in the dual theory. The vacuum degeneracy is removedonce one-loop corrections to the K¨ahler potential are taken into account, and a SUSY breakingminimum shows up at the origin in the moduli space. In addition to this minimum, there existSUSY vacua in this model, away from the local minimum. It has been shown in Ref. [1] that thisfalse vacuum can be long-lived and thus meta-stable.The idea of a meta-stable SUSY breaking vacuum opens up a lot of theoretical possibilitiesfor SUSY breaking. For such a false vacuum, the conventional argument using the Witten indexis not applicable, and it is generally possible for a theory to include a local minimum with brokenSUSY even though the Witten index implies the existence of a SUSY vacuum. Similarly, thetheorem [2] that a model with spontaneous SUSY breaking should have an R-symmetry is notapplicable to a model with a SUSY breaking local minimum. This feature is welcome from aphenomenological point of view, because R-symmetry forbids gauginos to obtain masses. TheR-symmetry should be broken spontaneously or explicitly to realize a phenomenologically viablemodel. For example, it has been argued [3] that SUSY breaking in a meta-stable vacuum requiresonly an approximate R-symmetry. Spontaneous R-symmetry breaking in gauged O’Raifeartaighmodels [3] and modified O’Raifeartaigh models [4, 5] with meta-stable SUSY breaking vacuahave been discussed.Since the paper by ISS, there have been lots of explorations of models with meta-stable vacua.A meta-stable SUSY breaking vacuum can simplify the gauge mediation scenario [6] and severalsimple models have been proposed [7, 8, 9, 10, 11, 12]. String theory realizations of the meta-stable SUSY breaking vacuum have been investigated in Refs. [13, 14, 15, 16, 17, 18], where aSUSY breaking scale lower than the string scale can be realized.In N = 1 SUSY models, a meta-stable vacuum can be analyzed only in a weak couplingregime in the (effective) theory by perturbative means. Our lack of knowledge about the non-perturbative K¨ahler potential prevents us from moving away from the weak coupling limit. How-ever, in a class of N = 2 SUSY gauge theories, we can analyze the vacuum structure of a modelbeyond perturbation theory as first demonstrated by Seiberg and Witten [19, 20], using the prop-erties of holomorphy and duality. In Ref. [21, 22], N = 2 SUSY gauge theories perturbed by anappropriate superpotential have been studied beyond the perturbative regime. It has been shownthat such perturbed N = 2 SUSY gauge theories can have meta-stable vacua at generic pointsin the moduli space.In this paper, we revisit the N = 2 SUSY gauge theory with a Fayet-Iliopoulos (FI) terminvestigated in Ref. [23]. The model is based on the gauge group SU (2) × U (1) with N f = 22assless hypermultiplets. At the classical level, this theory has SUSY vacua on the Higgs branch,at the origin of the Coulomb branch. Except for near the origin on the Coulomb branch, theclassical potential possesses SUSY breaking minima along a pseudo-flat direction on the Coulombbranch. These are far away from the Higgs branch and parameterized by moduli parameters,scalars of vector multiplets. In the quantum theory, these pseudo flat directions are removed anda non-trivial local vacuum may arise while the SUSY vacua on the Higgs branch would remain.The effective potential along the pseudo flat direction can be analyzed beyond perturbationtheory by using the exact results in N = 2 SUSY QCD [19, 20]. It is found that the effectivepotential exhibits a local minimum with broken SUSY at the dyon point through the SU (2) gaugedynamics and also that U (1) R is dynamically broken there. The global structure of the potentialis determined in perturbation theory and the effective potential is found to be of the so-calledrunaway type, namely, the potential energy decreases toward infinity where the SUSY vacuumwould be realized. We discuss the vacuum structure of this model in more detail and give a roughestimate of the decay rate of the local minimum to the runaway vacuum and the SUSY vacua onthe Higgs branch. We find that this local minimum is parametrically long-lived and thus meta-stable. Also, we address phenomenological applications of our model. In fact, in this meta-stablevacuum, the massive hypermultiplets inherent in the model play the role of messenger fields inthe gauge mediation scenario when the flavor symmetry among the hypermultiplets is gauged asthe Standard Model gauge group.The organization of this paper is as follows. In §
2, the model is defined and its classicalvacuum structure is studied. In §
3, low energy effective Lagrangian is derived using the exactresults in SUSY QCD. The effective potential is analyzed in §
4, and we show that the effectivepotential exhibits a local minimum at the dyon singular point due to non-perturbative SU (2)effects. In §
5, we give a rough estimate for the decay rate of the local minimum and show thevacuum can be long-lived. Phenomenological applications of the model are addressed in §
6. Thelast section is devoted to our conclusion. Detailed derivations of the effective couplings are givenin an Appendix.
We first define our classical Lagrangian and analyze its classical vacuum . We describe theclassical Lagrangian in terms of N = 1 superfields: Adjoint chiral superfields A i and vectorsuperfields V i in the vector multiplet ( i = 1 , U (1) and the SU (2) gaugesymmetries, respectively), and two chiral superfields Q rI and ˜ Q Ir in the hypermultiplet ( r = 1 , I = 1 , SU (2) color index). The superfield strength is defined by W iα = − D ( e − V i D α e V i ). The classical Lagrangian is given by L = L HM + L VM + L FI , (2.1) L HM = Z d θ (cid:16) Q † r e V +2 V Q r + ˜ Q r e − V − V ˜ Q † r (cid:17) The complete analysis of the classical potential for the one flavor case was originally performed in Ref. [24]. √ (cid:18)Z d θ ˜ Q r ( A + A ) Q r + h.c. (cid:19) , (2.2) L VM = 12 π Im (cid:20) tr (cid:26) τ (cid:18)Z d θA † e V A e − V + 12 Z d θW (cid:19)(cid:27)(cid:21) + 14 π Im (cid:20) τ (cid:18)Z d θA † A + 12 Z d θW (cid:19)(cid:21) , (2.3) L FI = Z d θξV , (2.4)where τ = i πg + θ π and τ = i πe are the gauge couplings of the SU (2) and the U (1) gaugeinteractions, respectively. Here we use the notation, tr( T a T b ) = T ( R ) δ ab = δ ab for the SU (2)generators T a . The common U (1) charge of the hypermultiplets is normalized to one. The lastterm in Eq. (2.1) is the FI term with a coefficient ξ of mass dimension two. In what follows, weassume that ξ >
0. In general, the FI term also appears in F-term, but the SU (2) R symmetryallows us to take a frame so that it appears only in D-term. Because of this, the SU (2) R symmetryis explicitly broken down to its subgroup U (1) R ′ . The global symmetry of the theory turns outto be SU (2) Left × SU (2) Right × U (1) R ′ × U (1) R . From the above Lagrangian, the classical potential is read off as V = 1 g tr[ A , A † ] + g q † r T a q r − ˜ q r T a ˜ q † r ) + q † r [ A , A † ] q r − ˜ q r [ A , A † ]˜ q † r + 2 g | ˜ q r T a q r | + e (cid:16) ξ + q † r q r − ˜ q r ˜ q † r (cid:17) + 2 e | ˜ q r q r | + 2 (cid:16) q † r | A + A | q r + ˜ q r | A + A | ˜ q † r (cid:17) , (2.5)where A , A , q r and ˜ q r are scalar components of the corresponding chiral superfields.There are supersymmetric vacua in this potential. For example, a solution A = A = 0 ,q r = 0 , ˜ q = ( v, , ˜ q = (0 , v ) , ξ − | v | = 0 , (2.6)is a possible supersymmetric vacuum.Let us then investigate the global structure of the vacuum. In order to do that, consider thefollowing field configuration: q r = 0 and ˜ q = ( a, , ˜ q = (0 , b ) , A = diag( a / , − a /
2) and A = diag( a , a ). Here, a and a are complex parameters, and a and b are, for simplicity, takento be real. Then the potential (2.5) is written as V = g a − b ) + e ξ − a − b ) + 2 a (cid:12)(cid:12)(cid:12)(cid:12) a + 12 a (cid:12)(cid:12)(cid:12)(cid:12) + 2 b (cid:12)(cid:12)(cid:12)(cid:12) a − a (cid:12)(cid:12)(cid:12)(cid:12) . (2.7) Without U (1) gauge symmetry (and the FI term), the flavor symmetry SU (2) Left × SU (2) Right is enhancedto O (4) since the representation of Q and ˜ Q are in an isomorphic representation of SU (2) gauge group. V = ( ξ − X ) + X Y , (2.8)where X and Y can be regarded as hypermultiplet and vector multiplet directions, respectively.The plot of the potential is depicted in Fig. 1. The potential has a pseudo flat direction along Y -2 -1 0 1 2X -2 -1 0 1 2Y01020V -2 -1 0 1X -1 -0.5 0 0.5 1X -1 -0.50 0.51Y00.511.5V -1 -0.5 0 0.5 1X Figure 1: Schematic picture of the classical potential. ξ is taken to be 1 in (2.8). For Y > √ ξ (left), the minimum is along X = 0. For Y < √ ξ (right), there are tachyonic directions along X .with X = 0. The vacua on this flat direction are tachyonic along the X -direction for Y < √ ξ ,but there are no longer tachyonic directions for Y > √ ξ . In the case (2.7), a pseudo flat directionis, for instance, parameterized by b = 0 and z = a − a , (2.9)for a + a = 0, which corresponds to X and Y , respectively. Along this direction, the potentialis further minimized with respect to a , whose value at the minimum is given by a = e g + e ξ ! . (2.10)In this example, the gauge symmetry SU (2) × U (1) is broken to a linear combination, U ′ (1). Thepotential energy at this minimum is given by V = ξ e g e + g . (2.11)Therefore, the supersymmetry is broken at the minimum along the pseudo flat direction by thenon-zero FI parameter.We are now interested in what happens to this pseudo flat direction in quantum theory. Since a and a have U (1) R charge +2, the U (1) R symmetry is broken by these non-zero vacuum ex-pectation values (VEVs). We expect that the pseudo-flat direction is lifted up, once quantum5orrections are taken into account, and some non-degenerate vacua would appear after the ef-fective potential is analyzed. This naive expectation seems natural, if we notice that the abovepotential energy is described by the bare gauge couplings, which should be replaced by the effec-tive ones (non-trivial functions of the moduli parameters) in the effective theory. In the followingsections, we will show that quantum corrections actually remove the vacuum degeneracy andleave two vacua on the Coulomb branch , one of which is a local minimum breaking both SUSYand U (1) R symmetry and the other is a runaway vacuum.In addition to this runaway SUSY vacuum in the Coulomb branch, there would be otherSUSY vacua. It is known that there are no quantum corrections on the Higgs branch [25],and we thus expect that the classical SUSY vacua (2.6) survive after quantum corrections aretaken into account. At these SUSY vacua, the hypermultiplets have very small VEVs becauseof the theoretical consistency condition ξ ≪ Λ (see the next section). On the Coulomb branch,except for very near the origin, we have a pseudo flat direction as shown above and no tachyonicdirection towards the Higgs branch. For this reason, we will analyse the quantum theory alongthe Coulomb branch and will see how the effective potential can be modified along the pseudoflat directions. In this section, we describe the low energy Wilsonian effective Lagrangian of our theory. Thedetailed derivation of the effective action is found in Ref. [23]. Here, we briefly summarize theresults for the convenience of the reader.In order to derive the exact low energy effective action L EXACT , which is described by thelight fields, the dynamical scale and the coefficient of the FI term ξ , we need to integrate theaction to zero momentum. However, this is a highly non-trivial task. Without the FI term, thetheory has N = 2 SUSY, which can be utilized to integrate out massive degrees of freedom. Inour model, this is not the case because the SUSY is broken at the classical level. In the followingdiscussion, suppose that the coefficient ξ , the order parameter of SUSY breaking, is much smallerthan the dynamical scale Λ of the SU (2) gauge interaction. Then we can consider the effectiveaction up to the leading order in ξ . The exact effective Lagrangian, if it could be obtained, canbe expanded in the parameter ξ as L EXACT = L SUSY + ξ L + O ( ξ ) . (3.12)Here, the first term L SUSY is the exact effective Lagrangian containing full SUSY quantumcorrections. The second term is the leading term in ξ . Since ξ is a constant and has massdimension 2, L should be a gauge-invariant quantity having mass dimension 2. This simpleconsideration tells us that the second term is nothing but the FI term. Analyzing the effectiveLagrangian up to the leading order in ξ , we obtain the effective potential to order ξ . Thecoefficient of ξ in the effective potential includes the full SUSY quantum corrections. Therefore, The exactness of the FI term is also discussed by using the harmonic superspace formalism in Ref. [26].
6o achieve our aim, what we need to analyze the effective potential is nothing but the effectiveLagrangian L SUSY .Except for the FI term, the classical SU (2) × U (1) gauge theory has a moduli space, whichis parameterized by a and a . On this moduli space except at the origin, the gauge symmetryis broken to U (1) c × U (1). Here U (1) c denotes the gauge symmetry in the Coulomb phaseoriginating from the SU (2) gauge symmetry. Before discussing the effective action of this theory,we should make clear how to treat the U (1) gauge interaction part. In the following analysis,this part is, as usual, discussed as a cut-off theory. Thus, the Landau pole Λ L is inevitablyintroduced in our effective theory, and the defining region of the modulus parameter a is con-strained to lie within the region | a | < Λ L . Because of this constraint, the defining region forthe modulus parameter a is found to be also constrained to be in the same region, since thetwo moduli parameters are related to each other through the hypermultiplets. We take the scaleΛ L to be much larger than the dynamical scale of the SU (2) gauge interaction Λ, so that the U (1) gauge interaction is always weak in the defining region of moduli space. Note that, in ourframework, we implicitly assume that the U (1) gauge interaction has no effect on the SU (2)gauge dynamics. This assumption will be justified in the following discussion concerning themonodromy transformation (see Eq. (3.16)).We first discuss the general formulae for the effective Lagrangian L SUSY , which consists oftwo parts described by light vector multiplets and hypermultiplets, L SUSY = L VM + L HM . Atlow energies, the N = 2 effective Lagrangian of the vector multiplet part, L VM , includes thesuperfield A of the unbroken Abelian subgroup of SU (2) and the Abelian superfield A . Theeffective action consistent with N = 2 SUSY and all the symmetries in our theory is given by L VM = 14 π Im X i,j =1 Z d θ ∂F∂A i A † i + Z d θ τ ij W i W j ! , (3.13)where F ( A , A , Λ , Λ L ) is the prepotential, which is a function of moduli parameters a , a , thedynamical scale Λ, and the Landau pole Λ L . Note that the effective coupling τ (= τ ) appearsthrough the quantum corrections. The effective coupling τ ij is defined as τ ij = ∂ F∂a i ∂a j ( i, j = 1 , . (3.14)The part L HM is described by a light hypermultiplet with appropriate quantum numbers ( n e , n m ) n ,where n e is the electric charge, n m is the magnetic charge, and n is the U (1) charge. This partshould be added to the effective Lagrangian around a singular point in moduli space, since thehypermultiplet is expected to be light there and enjoys correct degrees of freedom in the effective In this paper, we study the Coulomb branch, not the mixed branch like in (2.9) and (2.10), on which it isdifficult to analyze the effective potential. However, at low energies, the SU (2) gauge coupling is much largerthan the U (1) gauge coupling, so that the solution (2.9) and (2.10) is approximately that of the Coulomb branch. There is a possibility that a non-trivial fixed point and a strong coupling phase exist in QED [27]. Thisproblem is non-trivial, and is outside our scope (see also Ref. [26] for related discussions). L HM = Z d θ (cid:16) M † r e n m V D +2 n e V +2 nV M r + ˜ M r e − n m V D − n e V − nV ˜ M r † (cid:17) + √ (cid:18)Z d θ ˜ M r ( n m A D + n e A + nA ) M r + h.c. (cid:19) , (3.15)where M r and ˜ M r denote light quark, light monopole or light dyon hypermultiplet, that is, thelight BPS states, and V D is the dual gauge field of U (1) c . Since the U (1) gauge coupling isweak and does not affect the SU (2) gauge dynamics, the flavor symmetry is effectively that of N = 2 SUSY QCD. Recalling that a plays a role of the hypermultiplet mass if it has vacuumexpectation value, for vanishing VEV of a , the light BPS states belong to a spinor representationof SO (4) ∼ SU (2) − × SU (2) + [28, 20]. A non-zero vacuum expectation value of a breaks thesymmetry down to SU (2) − × U (1) + . At the quantum level, the global U (1) R symmetry isanomalous and the resultant anomaly-free symmetry turns out to be Z ⊂ U (1) R [20].In order to obtain an explicit description of the effective Lagrangian, let us consider themonodromy transformations of our theory. Suppose that the moduli space is parameterized bythe vector multiplet scalars a , a and their duals a D , a D which are defined as a iD = ∂F/∂a i ( i = 1 , Sp (4 , R ),which leaves the effective Lagrangian invariant, and the general formula is found to be [29] a D a a D a → αa D + βa + pa γa D + δa + qa a D + p ( γa D + δa ) − q ( αa D + βa ) − pqa a , (3.16)where α βγ δ ! ∈ SL (2 , Z ) and p, q ∈ Q . Note that this monodromy transformation for thecombination ( a D , a , a ) is exactly the same as that for SUSY QCD with massive quark hyper-multiplets, if we regard a as the common mass of the hypermultiplets such that m = √ a .This fact means that the U (1) gauge interaction part only plays the role of the mass term forthe SU (2) gauge dynamics. This observation is consistent with our assumptions. On the otherhand, the SU (2) dynamics plays an important role for the U (1) gauge interaction part, as can beseen from the transformation law of a D . This monodromy transformation is also used to derivedual variables associated with the BPS states. As a result, the prepotential of our theory turnsout to be essentially the same as the result in Ref. [20] with the additional relation m = √ A , F ( A , A , Λ , Λ L ) = F ( SW ) SU (2) ( A , m, Λ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m = √ A + CA , (3.17)where the first term on the right hand side is the prepotential of N = 2 SUSY QCD with hyper-multiplets having the same mass m , and C is a free parameter. The freedom of the parameter C is used to determine the scale of the Landau pole relative to the scale of the SU (2) dynamics.8he effective potential can be obtained from the action, after eliminating auxiliary fields V = b b ξ + S ( a , a ) n ( | M r | − | ˜ M r | ) + 4 | M r ˜ M r | o + 2 T ( a , a )( | M r | + | ˜ M r | ) − U ( a , a )( | M r | − | ˜ M r | ) , (3.18)where | M r | = M r M † r , | ˜ M r | = ˜ M r ˜ M r † , b ij = (1 / π )Im τ ij is the effective coupling and det b ≡ b b − b . The functions S , T and U are defined as S ( a , a ) = 12 b + ( b − nb ) b det b , (3.19) T ( a , a ) = | a + na | , (3.20) U ( a , a ) = b − nb det b ξ . (3.21)Solving the stationary conditions with respect to the hypermultiplets, we have the following threesolutions: 1 . M = ˜ M = 0; V = b b ξ , (3.22)2 . | M r | = − T − U S , ˜ M = 0; V = b b ξ − S | M r | , (3.23)3 . M = 0 , | ˜ M r | = − T + U S ; V = b b ξ − S | ˜ M r | . (3.24)The solution Eq. (3.23) or Eq. (3.24), in which the light hypermultiplet acquires a vacuumexpectation value, is energetically favored, because det b > S ( a , a ) >
0. Since thehypermultiplet appears in the theory as the light BPS state around the singular point in modulispace, a potential minimum is expected to emerge there. In addition, the points (3.23) and (3.24)are stable in the M, ˜ M directions. This is because they are unique solutions and have lower energythan the point (3.22), and the potential at infinity in M, ˜ M space is dominated by the M , ˜ M terms. On the other hand, the solution Eq. (3.22) describes the potential energy away from thesingular points, which smoothly connects with the solution Eq. (3.23) or Eq. (3.24).It was shown that the effective potential is described by the periods a D , a and the effectivegauge coupling b ij . The periods are the same as that of massive SUSY QCD. Although thereare some different descriptions of the periods it is convenient for our purpose to write them asintegral representations [29] and to write the effective coupling τ ij in terms of the Weierstrassfunctions.We first review how to obtain the periods a D and a . The elliptic curve of N = 2 SUSYQCD with two hypermultiplets having the same mass m was found to be [20] y = x ( x − u ) − Λ
64 ( x − u ) + Λ m x − Λ m , (3.25) We presuppose that the potential is described by the proper variables associated with the light BPS states.For instance, the variable a is understood implicitly as − a D , when we consider the effective potential for themonopole. u = Tr A is identified with the modulus parameter. In this case, the mass formula of theBPS state with the quantum numbers ( n e , n m ) n is given by M BPS = √ | n m a D + n e a + nm/ √ | .If λ is a meromorphic differential on the curve Eq. (3.25) such that ∂λ∂u = √ π dxy , (3.26)the periods are given by the contour integrals a D = I α λ , a = I α λ , (3.27)where the cycles α and α are defined so as to encircle e and e , and e and e , respectively,which will be given explicitly later on (see eq.(3.36)). The meromorphic differential is given by λ SW = − √ π ydxx − Λ = − √ π dxy x − u + m Λ (cid:16) x + Λ (cid:17) . (3.28)The differential has a single pole at x = − Λ and the residue is given byRes λ SW = 12 πi ( − m √ . (3.29)We calculate the periods by using the Weierstrass normal form for later convenience. In thisform, the algebraic curve is rewritten in new variables x = 4 X + u and y = 4 Y , such that Y = 4 X − g X − g = 4( X − e )( X − e )( X − e ) , (3.30) X i =1 e i = 0 , where g and g are explicitly written by g = 116 u + Λ − m Λ ! , (3.31) g = 116 m Λ − u m Λ − u Λ
96 + 2 u ! . (3.32)Converting the Seiberg-Witten differential, Eq. (3.28), into the Weierstrass normal form andsubstituting it into Eq.(3.27), we obtain the integral representations of the periods as follows( a D and a are denoted by a and a , respectively): a i = − √ π − uI ( i )1 + 8 I ( i )2 + m Λ I ( i )3 ( c ) ! , (3.33)where c is the pole of the differential, given by c = − u − Λ . The integrals I ( i )1 , I ( i )2 and I ( i )3 aredefined as I ( i )1 = 12 I α i dXY , I ( i )2 = 12 I α i XdXY , I ( i )3 ( c ) = 12 I α i dXY ( X − c ) . (3.34)10he roots e i of the polynomial defining the cubic are chosen so as to lead to the correct asymptoticbehavior for large | u | , a D ( u ) ∼ i π √ u log u Λ , a ( u ) ∼ √ u , (3.35)A correct choice is the following: e = u − Λ − s u + Λ m s u + Λ − Λ m ,e = u − Λ
64 + 18 s u + Λ m s u + Λ − Λ m , (3.36) e = − u
12 + Λ . Fixing the contours of the cycles relative to the positions of the poles, which is equivalent tofixing the U (1) charges for the BPS states, the final formulae are given by a i = − √ π − uI ( i )1 + 8 I ( i )2 + m Λ I ( i )3 − u − Λ !! − m √ δ i , (3.37)with the integrals I (1) s ( s = 1 , ,
3) explicitly given by I (1)1 = Z e e dXY = iK ( k ′ ) √ e − e , (3.38) I (1)2 = Z e e XdXY = ie √ e − e K ( k ′ ) + i √ e − e E ( k ′ ) , (3.39) I (1)3 = Z e e dXY ( X − c ) = − i ( e − e ) / ( k + ˜ c K ( k ′ ) + 4 k k c − k Π ν, − k k !) , (3.40)where k = e − e e − e , k ′ = 1 − k = e − e e − e , ˜ c = c − e e − e , and ν = − (cid:16) k +˜ ck − ˜ c (cid:17) (cid:16) − k k (cid:17) . The formulae for I (2) s are obtained from I (1) s by exchanging the roots e and e . In Eqs. (3.38)-(3.40), K , E , andΠ are the complete elliptic integrals [30] given by K ( k ) = Z dx [(1 − x )(1 − k x )] / , (3.41) E ( k ) = Z dx − k x − x ! / , Π ( ν, k ) = Z dx [(1 − x )(1 − k x )] / (1 + νx ) . Next let us consider the effective coupling defined in Eq. (3.14). A detailed derivation of theeffective couplings is given in the Appendix. The effective couplings τ and τ are obtained by τ = ∂a D ∂a = ω ω , (3.42) τ = ∂a D ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − τ ∂a ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = − z ω , (3.43)11here ω i is the period of the Abelian differential, ω i = I α i dXY = 2 I ( i )1 ( i = 1 , , (3.44)and z is defined as z = − √ e − e F ( φ, k ); sin φ = e − e c − e . (3.45)Here F ( φ, k ) is the incomplete elliptic integral of the first kind given in (A.3).The effective coupling τ is described in terms of the Weierstrass function. First consider theperiod a D by using the Riemann bilinear relation [31], I α φ I α ω − I α ω I α φ = 2 πi N p X n =1 Res x + n φ Z x + n x − n ω , (3.46)where φ and ω are meromorphic and holomorphic differentials, respectively, N p is the number ofpoles ( N p = 1 in our case), and x ± n are poles of φ on the positive and negative Riemann sheets.Substituting φ = ∂λ SW /∂a and ω = ∂λ SW /∂a into Eq. (3.46), we obtain a D = − N p X n =1 Z x + n x − n λ SW + ˜ C , (3.47)where ˜ C is a constant independent of a . The effective coupling τ is obtained by differentiatingEq. (3.47) with respect to a with a fixed. The integral in Eq. (3.47) after the differentiationcan be evaluated by the uniformization method discussed in the Appendix. After regularizingthe integral by using the freedom of the constant ˜ C , we finally obtain (see also the Appendix fordetails) τ = − πi " log σ (2 z ) + 4 z ω I (1)2 + C, (3.48)where σ is the Weierstrass sigma function, and C is the constant in Eq. (3.17).We now define the Landau pole associated with the U (1) interaction. In the ultraviolet regionfar away from the origin of the moduli space, the effective coupling is dominated by the U (1)gauge interaction since the SU (2) interaction is asymptotic free and small. As we expect, thegauge coupling b is found to be a monotonically decreasing function of the large | a | with fixed u , and vice versa (see, for example, Fig. 2 in the case of fixed a ). The Landau pole is definedas | a | = Λ L at which b = 0. The large Λ L required in our assumption is realized by takingan appropriate value for C . In the following analysis, we fix C = 4 πi , which corresponds toΛ L = 10 − in units of Λ. 12
10 10 201.0251.051.0751.11.1251.151.175 Re( ) u b
Figure 2: The effective gauge coupling b for a = 3 / √ u axis.
32 =1 u u u u u u u u u u u u (-1,1) (-1,1) (0,1) (-1,1) (-1,1) (1,0) -1 1 0 -1 -1 1-1 (1,-1) L = AD point
Figure 3: Flow of the singular points as Re( a ) increases with Im( a ) = 0. In this section, we examine the effective potential minimum numerically. As explained in theprevious section, the minimum is expected to appear at the singular point since it is energeticallyfavored due to the non-zero condensation (see Eq. (3.18)) of the light BPS state such as a quark,monopole or dyon with appropriate quantum number ( n e , n m ) n . Thus, let us first investigate thesingular points, and then analyze the effective potential at the singular point.The singular points on the moduli space is determined by the cubic polynomial [20]. Thesolutions of the cubic polynomial give the positions of the singular points in the u -plane. In the N f = 2 case with the same hypermultiplet masses, the solution is easily obtained as u = − m Λ − Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m = √ a , u = m Λ − Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m = √ a , u = m + Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m = √ a . (4.49)The flow of the singular points with respect to the real hypermultiplet mass is sketched in Fig. 3.For a = 0, the singular points appear at u = u = − Λ / u = Λ /
8. Here, at u = − Λ / uVVVV min V min V min V min V min V min1 2 2 211 Re( )
Figure 4: The effective potential for a = 0 . a = 0 . a = 0 . u axis. Re( ) u V a
Re( ) VV min 2 V min 1 V , min 2 V minmin min Figure 5: The evolution of the potential minima V and V at the singular points u and u , respectively, as a varies on the real u -axis (left) and on the real a axis (right). The solid(dashed) curve shows the plots with(without) dyon condensation.two singular points coincide. For non-zero a > this singular point splits into two singularpoints u and u , which correspond to the BPS states with quantum numbers ( − , − and( − , , respectively. As a is increasing, these singular points, u and u , are moving to theleft and the right on the real u -axis, respectively. The two singular points, u and u , collideand coincide at the so-called Argyres-Douglas (AD) point [32] ( u = ) for a = Λ2 √ , whereit is believed that the theory becomes superconformal. As a increases further, there appeartwo singular points u and u again, and the quantum numbers of the corresponding BPS states,( − , at u and (0 , at u , change into (1 , − − and (1 , , respectively. The singular point u is then moving away to the right faster than u .Now let us examine the effective potential at the singular point. First note that the effective For Im( a ) = 0, it is enough to consider only the case a >
0, since the result for a < u ↔ u , as can be seen from the first two equations in Eq. (4.49). u u u u u u u u a Re( ) = a Im( ) > a Re( ) > a Im( ) > a Re( ) < a Im( ) < Figure 6: Flow of the singular points u and u for general values of a .potential is a function of u and a , V ( a ( u, a ) , a ) (see (3.23) and (3.24)). Furthermore, (4.49)tells us that the singular point is completely determined by the value of a , and therefore thepotential at the singular point is a function of a only. In the following, we investigate theeffective potential at some fixed value of a , and see how the minimum appears at the singularpoint. Then we examine the evolution of the minimum by varying a . In our numerical analysis,we take Λ = 2 √ ξ = 0 . ≤ Re( a ) < Λ2 √ , Im( a ) = 0The effective potentials for several values of a in the range, 0 < a < Λ / √ V and V ,appear at two singular points u and u , respectively, while there is no minimum at the singularpoint u since the monopole condensation is too small for the potential to have a minimum. Inthe middle figure the top and the bottom curves show the effective potential without and withthe dyon condensations, respectively. The cusps are smoothed out in the bottom curve, whichmeans that the correct degrees of freedom in the theory are considered. The two minima in Fig.4 approach each other and their values become smaller as a decreases. Such a behavior can beshown in Figs. 5. The left figure shows the evolution of the potential minima, V and V at the dyon singular points as a changes on the real u axis. The top and the bottom curvesare plots without and with dyon condensates, respectively. From this figure, one sees that thecondensation lowers the potential energy. The right figure shows that the evolution of the po-tential minima along the real a axis. In this plot, the behaviors at the two dyon singular pointscompletely coincide because of the symmetry, a → − a (see (4.49)). From the analysis we findthat the potential is bounded from below, at least along the real u axis, and it is expected thatthere is a (local) minimum at u → − Λ / a → a ) = 0 , Im( a ) = 0Next we examine the effective potential for a complex value of a around a = 0. For ourpurpose, it is sufficient to investigate small values of Im( a ) near Re( a ) = 0 since we want toknow whether the effective potential is bounded from below or not at the point Re( a ) = 0. Onceagain let us go back to the flow of the singular points. Fig. 6 shows the flow of the singular15 V V -2 -1 1 20.004650.00470.00475 0.1 0.2 0.3 0.4 0.50.004650.00470.00475 a V min 2 V minmin 2 V min1 V , minmin
Im( )
Figure 7: The evolution of the minima V and V at u and u , respectively, with varying thepure imaginary part of a along u = − iy axis(left) and the imaginary a axis.points u and u for several complex values of a . The left figure shows the flow as Im( a ) in-creases. The singular points u and u are moving in opposite directions along u = − Λ / − a ) > , Im( a ) > a ) < , Im( a ) <
0) case. The plots of the potential corresponding to these flows are shownin Fig. 7 and 8. The left figure in Fig. 7, corresponding to the left figure in Fig. 6 shows theevolution of the potential minima for two dyons along the u = − a ) axis. Note that in the latter plot, the evolution of the two dyonminima completely coincide as in the case of the right figure in Fig. 5. These two dyon pointsroll down to the point a → u → − a = 0 is bounded from belowin the pure imaginary direction of a . Fig. 8 shows the contour and 3D plots of the effectivepotential at the u -dyon point as a function of Re( a ) > a ) >
0. The dark(light) colorshows lower(higher) value of the effective potential. The effective potential is invariant underRe( a ) → − Re( a ) and/or Im( a ) → − Im( a ), and the plot for other parameter range of a isobtained through this invariance. The plot of the effective potential at u -dyon point is obtainedby exchanging a → − a . In conclusion, the point u → − a → | a | , since the condensationsof the two dyon states are going to overlap with each other (see Fig. 4). Unfortunately, we haveno knowledge about the correct description of the effective theory in this situation. Nevertheless,we conclude that there must appear a local minimum with broken SUSY in the limit a → u = − Λ / −
1, and the value of theeffective potential at the cusp is non-zero, V ≃ . >
0. If we had the correct description ofthe effective theory for a = 0, this cusp might be smoothed out. However, there is no reason forSUSY to be restored at u = −
1, because the correct effective theory must have no singularityin the K¨ahler metric. Therefore, there is the promising possibility of the appearance of a local16 .1 0.2 0.3 0.4 0.50.10.20.30.40.5 a Re( ) a Im( ) a Re( ) V min a Im( ) Figure 8: The contour and the 3D plots of the effective potential at the singular point u as afunction of a .minimum with broken SUSY at u = − a = 0. Note that in this local minimum the global Z ⊂ U (1) R symmetry is broken down to Z .(iii) Re( a ) > Λ2 √ Let us get back to the case of Im( a ) = 0. For a > Λ2 √ , the effective potential has twominima, V and V at two singular points u and u . The dyon condensation is too smallfor the effective potential to have a minimum at u . The plot of the effective potential is shownin Fig. 9. While the evolution of the potential energy with the singular point u is the sameas for 0 < a < Λ2 √ , the potential energy on the quark singular point at u is monotonicallydecreasing, as a is increasing. Thus, there is a runaway direction along the flow of the quarksingular point. We can find the same global structure along the flow of the quark singular pointfor general complex a values.The evolutions of the potential energies according to the flows of the singular points along thereal u -axis are simultaneously plotted in Fig. 10. The global structure of the effective potentialis of the runaway type. However, we found the promising possibility that there exists a localminimum with broken SUSY in the theory. Precisely speaking, since there is no well-definedvacuum in the runaway direction, this minimum with broken SUSY is the unique and promisingcandidate for the vacuum in the theory. Unfortunately, we have no knowledge of the correctdescription about the effective theory around the degenerate dyon point.Finally, we would like to comment on the possible SUSY vacua which are present on the Higgsbranch at the classical level. As we have mentioned in §
2, SUSY vacua at the point a = a = 0would exist even at the quantum level. At the classical level, the SUSY breaking vacua onthe pseudo flat direction roll down to the SUSY vacua on the Higgs branch near the origin | a | , | a | ∼ √ ξ . In our analysis of the Coulomb branch, the effective potential (3.18) does notgive such a picture. Instead, we have found that the effective potential realizes a local minimum17
10 -5 5 10 150.00480.004850.00490.004950.00505
Re( ) u VV min1 V min3 a Re( ) V min3 Figure 9: The evolution of the potential minima at the singular points as a varies on the real u -axis (left) and on the real a axis (right).at u ∼ Λ ≫ ξ , such that the minimum is far away from the Higgs branch. In order to realizethe SUSY vacua on the Higgs branch, which would exist even in the quantum level, appropriatehypermultiplets should be introduced in the effective action, while we, unfortunately, do notknow the correct treatment of such degrees of freedom. In this model, we have found two possible kinds of SUSY vacua: A runaway vacuum at infinity inmoduli space on the Coulomb branch and the vacua on the Higgs branch. Here we will estimatethe decay rate from the local vacuum to these SUSY vacua.First we estimate it for the runaway vacuum. As we discussed earlier, consistency requiresthat we restrict the moduli space to a region bounded by the Landau pole. When the boundary ofmoduli space is located far away from the dynamical scale, the potential energy at the boundaryis almost zero, and the true, almost SUSY, vacuum of the theory lies somewhere in this region.If our world is trapped in the local minimum we found in the previous sections, it will eventuallydecay to this approximately supersymmetric vacuum. The decay rate is expected to be verysmall, as the potential barrier is very wide.As analyzed in the previous section, the effective potential can be described as a function ofthe modulus parameter a . In our calculation, the effective potential is treated in the triangleapproximation [33]. Let us take the path in the direction of Re( a ): climbing up from the localminimum ( a = 0) to the AD point ( a = Λ / √ a = Λ L ). This is similar to the situation in the ISS model.18
10 -5 5 10 150.00470.00480.0049 (Dyon) (Dyon)AD point (Quark) a Re( ) VV Re( ) u AD point (Quark)(Dyons) V min V min1 2 V min3min min V , min1 V min2 V min3 Figure 10: The effective potential energy at each singular point.In the triangle approximation, parameters characterizing the potential are∆ V ± , ∆Φ ± , (5.50)where ∆ V ± and ∆Φ ± are the difference of potential height and the distances between local/Landaupole points and potential barrier (see Fig. 11). Following reference [33], we define λ ± ≡ ∆ V ± ∆Φ ± , c ≡ λ − λ + = ∆ V − ∆ V + ∆Φ + ∆Φ − . (5.51)In our case, ∆Φ + ∼ Λ , ∆Φ − ∼ Λ L , (5.52)and the height of the effective potential is controlled by the SUSY breaking order parameter ξ as V ∼ ξ . (5.53)Through numerical analysis, the ratio ∆ V − / ∆ V + is estimated to be O (10), so that the conditionof Eq. (13) in Ref. [33] can be satisfied, ∆ V − ∆ V + ! ≥ − ∆Φ − − ∆Φ + = 21 − ∆Φ + ∆Φ − ∼ . (5.54)Here we have used ∆Φ + ∆Φ − ∼ ΛΛ L ≪ . (5.55)Since for our choice of parameters, Λ L = 10 − in units of Λ, we can safely use the formula ofthe bounce action [33], B = 32 π c ( √ c − ∆Φ ∆ V + . (5.56)19ecause the parameter c is very small c = λ − λ + = ∆ V − ∆ V + ∆Φ + ∆Φ − ∼
10 ΛΛ L ≪ , (5.57)we find B ∼ Λ L ξ ≫ Λ ξ ≫ . (5.58)Here we used the condition Λ ≫ ξ for our analysis in the previous section to be theoreticallyconsistent. As a result, the decay rate per unit volume Γ /V ∼ e − B from the local minimum tothe Landau pole point is very small, and the vacuum at the local minimum is very long-lived,i.e. meta-stable, as expected. Although the final formula seems to indicate that the decay ratebecomes zero in the limit Λ L → ∞ , it is, in fact, non-zero due to the barrier penetration fromthe local minimum to the runaway direction.Figure 11: Schematic picture of the effective potentialNext we estimate the decay rate from the local minimum to the SUSY vacua present onthe Higgs branch. The most conservative path to such SUSY vacua from the local one is firstclimbing up to the origin a = a = 0 , q = ˜ q = 0 and then rolling down to the SUSY vacuum a = a = 0 , q, ˜ q ∼ √ ξ . In this situation, the potential parameters are estimated from thenumerical analysis to be ∆Φ + = Λ = 2 √ , ∆Φ − = q ξ = √ . , ∆ V + = 0 . , ∆ V − = 0 . . (5.59)These parameters do not satisfy the condition (5.54), so that we can not use the formula (13) inRef. [33]. Instead, we can use another formula (20) in Ref. [33], B = π λ R T h − β + 3 cβ β − + 3 cβ + β − − c β − i . (5.60)20ere β ± ≡ s ± λ ± , R T = 12 β + cβ − cβ − − β + ! . (5.61)The bounce action is estimated to be B ∼ O (10 ) ≫ We have found a meta-stable vacuum with broken SUSY and also broken U (1) R symmetry in theprevious sections. Here we address the application of our model to phenomenology. Supersym-metric extensions of the Standard Model have been considered as one of the most promising waysto solve the gauge hierarchy problem in the Standard Model. Since any supersymmetric partnersof the Standard Model particles have not been observed yet, supersymmetry should be brokenat low energies. The origin of supersymmetry breaking and its mediation to the supersymmetricversion of the Standard Model are still prime questions in particle physics. As mentioned severaltimes before, in order to obtain a realistic model, U (1) R symmetry breaking is necessary as well asbreaking of supersymmetry. This is because U (1) R symmetry forbids gauginos to obtain masses.From this point of view, the meta-stable vacuum we have found is suitable for phenomenology.Furthermore, the model automatically provides the structure necessary in the gauge mediationscenario.First, let us give a brief review on the gauge mediation scenario [6]. The basic structure ofthis scenario is described as the messenger sector superpotential, W = S ˜ΦΦ , (6.62)where S is a gauge singlet chiral superfield, and ˜Φ and Φ are a vector-like pair of chiral superfields,so-called messenger fields, which are charged under the Standard Model gauge group. Supposethat both the scalar component and the F-component of the singlet superfield S develop VEVs sothat SUSY and also U (1) R symmetry are broken. Through quantum corrections with messengerfields, gauginos and scalar partners of the Standard Models particles obtain soft SUSY breakingmasses, M soft ∼ α SM π h F S ih S i , (6.63)where h S i and h F S i are VEVs of the scalar and the F-component of the superfield S , α SM standsfor the Standard Model gauge coupling.Now we return to our model. At the meta-stable vacuum, u → − Λ / a →
0, the modelpossesses flavor symmetry SU (2) − × SU (2) + [20]. The BPS states, dyon hypermultiplet, whichdescribe the low energy effective theory around the meta-stable vacuum, belong to a doublet21nder SU (2) + . Note that the model also includes the other massless hypermultiplets ( ˜ D and D )doublets under SU (2) − at the singular point, u = Λ /
8. However, they can no longer be masslessat the other singular point and they exist as massive states in the meta-stable vacuum. One canunderstand such a structure in the classical theory. In the classical superpotential in Eq. (2.2), W = √ Q Ir ( A + A ) IJ Q Jr , (6.64)considering that the Cartan part of A ∼ diag( A / , − A /
2) is left in the low energy effectivetheory, there are two moduli points, a / a = 0 and − a / a = 0, where the hypermultipletsare massless. But at one of the moduli points, some hypermultiplets are massless, but the othersare massive and are integrated out. The same structure should be realized at the quantum level.Now the massive hypermultiplet would have the following form of the superpotential, W = √ D r ( n m A D + n e A + nA ) D r , (6.65)with certain quantum numbers ( n e , n m ) n as in Eq. (3.15). Since h A D i ∼ Λ at the local minimumwe have found, the massive hypermultiplets are heavy and integrated out from the low energyeffective theory. However, once we take supersymmetry breaking effects into account at theminimum, this superpotential is found to play an important role in phenomenology.Supersymmetry is broken at the local minimum and so the F-component of A D develops aVEV characterized as h F A D i ∼ ξ , so that Eq. (6.65) has the same structure as Eq. (6.62) with A D = h A D i + θ h F A D i ∼ Λ + θ ξ . Therefore, when the SU (2) flavor symmetry is weakly gaugedas, for example, the SU (2) weak gauge group in the Standard Model, the hypermultiplets, ˜ D and D , play the role of messenger fields and the SU (2) gaugino and all doublet scalars in thesupersymmetric Standard Model obtain masses through the gauge mediation such as M soft ∼ α π ξ Λ , (6.66)where α is the SU (2) weak gauge coupling. Suitable choices of model parameters, supersymme-try breaking order parameter ξ and the messenger scale Λ, lead to phenomenologically favoredvalues for soft supersymmetry breaking masses around 1 TeV.In order to obtain a more realistic phenomenological model, it is necessary to extend ourmodel so as to provide a larger flavor symmetry. For example, an ideal choice would be an SU (5) flavor symmetry, whose subgroup can be gauged as the Standard Model gauge group SU (3) C × SU (2) L × U (1) Y ⊂ SU (5) so that all gauginos and scalar partners obtain masses. Toimplement such a large global symmetry into a supersymmetric gauge theory, a model shouldbe based on a general SU ( N ) ( N >
2) gauge group with an appropriate number of flavors. It isnon-trivial to construct such a model more suitable for phenomenology, and we leave this issuefor future works.Our SUSY breaking model is based on N = 2 SUSY gauge theories, while the supersymmetricStandard Model is a chiral theory and should obey N = 1 supersymmetry. It may be somewhatunusual to realize such a setup in four dimensions. As a natural realization, we can considera N = 1 five-dimensional brane world scenario, where the SUSY breaking sector resides in the22ulk while the SUSY Standard Model sector resides on a “3-brane”. The Lagrangian for the bulkfields is described in terms of N = 2 SUSY theory in four-dimensional point of view, while thatfor the brane fields obeys only N = 1 supersymmetry. If we could extend our four-dimensionalmodel to a five-dimensional one, such a natural phenomenological model would be realized. We have investigated an N = 2 supersymmetric gauge theory based on the gauge group SU (2) × U (1) with N f = 2 flavors and the FI term associated with the U (1) gauge group. Thanks to theexact results in N = 2 supersymmetric gauge theories, we can analyze the model beyond pertur-bation theory with respect to the SU (2) gauge coupling, but as a perturbation with respect tothe FI term smaller than the SU (2) dynamical scale. We have found that the effective potentialexhibits a local SUSY breaking minimum at the degenerate dyon point due to the strong SU (2)dynamics. On the other hand, away from the origin of the moduli space, the potential energydecreases as we move toward infinity eventually realizing an almost SUSY vacuum. In additionto this runaway vacuum, there are SUSY vacua on the Higgs branch which survive quantum cor-rections. We have estimated the decay rate of the local minimum in the triangle approximationand found that the false vacuum is parametrically long-lived. In this meta-stable vacuum, notonly SUSY but also the R-symmetry are broken. Interestingly, the basic structure of a messengersector in the gauge mediation scenario is inherent in our model in the meta-stable vacuum. Oncethe flavor symmetry among massive hypermultiplets is gauged as the Standard Model gaugegroup, they play the role of messenger fields and supersymmetry breaking is transmitted intothe SUSY Standard Model sector through the Standard Model gauge interactions. In order toobtain a more realistic phenomenological model, it is necessary to enlarge the gauge group soas to include more flavors. It is an interesting question whether such a model still exhibits ameta-stable vacuum suitable for phenomenology. This direction is worth investigating in thefuture. Acknowledgements
We would like to thank K. Ohta, S. Terashima and N. Yokoi for their useful commentsand in particular for pointing out the existence of SUSY vacua. The work of N. O. is partlysupported by the Grant-in-Aid for Scientific Research in Japan ( ppendix : Derivations of the effective couplings in termsof the Weierstrass functions
In this appendix, we exhibit the derivations of the effective couplings in term of the Weierstrassfunctions. The derivations are applicable for all the case of flavors ( N f = 1 , , N f , corresponding to each flavor case. It is convenient to introducethe uniformization variable z through the map with the Weierstrass ℘ function,( ℘ ( z ) , ℘ ′ ( z )) = ( X, Y ) . (A.1)Using this map, the half period ω i / e i = ℘ ( ω i /
2) ( ω = ω + ω ). Theinverse map is defined as z = Ψ − ( x ) = Z ∞ x dXY = − √ e − e F ( φ, k ) , (A.2)where we changed the integration variable X by t = ( e − e ) / ( X − e ), and F ( φ, k ) is theincomplete elliptic integral given by F ( φ, k ) = Z sin φ dt [(1 − t )(1 − k t )] / ; sin φ = e − e x − e . (A.3)We derive the effective couplings, τ and τ , by using the map of Eq. (A.1). The effectivecoupling τ is described by τ = ∂a D ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = ∂a D ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − τ ∂a ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u . (A.4)The partial derivative of the periods a D and a with respect to a can be calculated usingEqs. (3.27)-(3.28) as ∂a i ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = I α i ∂λ SW ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = Q ( N f ) ( a , Λ N f ) Z e e j dX Y ( X − c ) ( i = j ) , (A.5)where the coefficient Q ( N f ) is given by Q ( N f ) ( a , Λ N f ) = − N f ( √ a ) N f − Λ − N f N f π . (A.6)Using the map of Eq. (A.1), the integral can be described as ∂a i ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = Q ( N f ) Z ω ω j dz ℘ ( z ) − ℘ ( z ))= Q ( N f ) ℘ ′ ( z ) log σ ( z − z ) σ ( z + z ) + 2 zζ ( z ) ! , (A.7)24here ℘ ( z ) = c , ζ ( z ) is the Weierstrass zeta function, and we used the definition of the Weier-strass sigma function, ζ ( z ) = ddz log σ ( z ), and the relation ℘ ′ ( z ) ℘ ( z ) − ℘ ( z ) = ζ ( z − z ) − ζ ( z + z ) + 2 ζ ( z ) . (A.8)Taking into account that Y corresponds to ℘ ′ ( z ) under the map of Eq. (A.1), the pole ℘ ′ ( z ) canbe easily obtained as ℘ ′ ( z ) = − N f ( N f − / Λ − N f N f . (A.9)Using the pseudo periodicity of the Weierstrass sigma function, σ ( z + ω i ) = − σ ( z ) exp (cid:18) ζ (cid:18) ω i (cid:19) (cid:18) z + 12 ω i (cid:19)(cid:19) , (A.10)we obtain ∂a i ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = − πi (cid:20) ω i ζ ( z ) − z ζ (cid:18) ω i (cid:19)(cid:21) . (A.11)The zeta function at half period can be described by integral representations as ζ (cid:18) ω i (cid:19) = − I ( i )2 . (A.12)Substituting Eq. (A.11) into Eq. (A.4) and using the Legendre relation ω ζ (cid:18) ω (cid:19) − ω ζ (cid:18) ω (cid:19) = iπ , (A.13)we finally obtain τ = − z ω . (A.14)Next we derive the effective coupling τ , which is given by differentiating a D of Eq. (3.47)with respect to a with a fixed such as τ = − Z x + n x − n " ∂λ SW ∂u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∂u∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a + ∂λ SW ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + ∂ ˜ C∂a . (A.15)The integral can be evaluated by using the map (A.1). Although the integral contains a di-vergence, it can be regularized by using the freedom of the integration constant ˜ C . Let usdemonstrate this regularization by introducing the regularization parameter ǫ as follows. τ = − Z x +0 + ǫx − + ǫ " ∂λ SW ∂u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − ∂u∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∂a ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ! + ∂λ SW ∂a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + ∂ ˜ C∂a = − Z z + ǫ − z + ǫ dz " − πiω (cid:18) ω ζ ( z ) − z ζ (cid:18) ω (cid:19)(cid:19) + Q ( N f ) ℘ ( z ) − ℘ ( z )) + ∂ ˜ C∂a = − πi log σ (2 z ) − z ω ζ (cid:18) ω (cid:19)! + 1 π log σ ( ǫ ) + 12 + ∂ ˜ C∂a . (A.16)25he divergent part, log σ ( ǫ ), can be subtracted by taking the integration constant such that˜ C = Ca − a − a π log σ ( ǫ ), and we finally obtain Eq. (3.48) with the relation of Eq. (A.12). References [1] K. Intriligator, N. Seiberg and D. Shih, JHEP (2006) 021, arXiv:hep-th/0602239.[2] A. E. Nelson and N. Seiberg, Nucl. Phys. B (1994) 46, arXiv:hep-ph/9309299.[3] K. Intriligator, N. Seiberg and D. 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