Supersymmetry Breaking Triggered by Monopoles
Csaba Csaki, David Curtin, Vikram Rentala, Yuri Shirman, John Terning
SSupersymmetry Breaking Triggeredby Monopoles
Csaba Cs´aki , David Curtin , , Vikram Rentala , ,Yuri Shirman , John Terning Department of Physics, LEPP, Cornell University, Ithaca, NY 14853. C. N. Yang Institute for Theoretical PhysicsStony Brook University, Stony Brook, NY 11794. Department of Physics, University of Arizona, Tucson, AZ 85721. Department of Physics, University of California, Irvine, CA 92697. Department of Physics, University of California, Davis, CA 95616. [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract
We investigate N = 1 supersymmetric gauge theories where monopole condensationtriggers supersymmetry breaking in a metastable vacuum. The low-energy effectivetheory is an O’Raifeartaigh-like model of the kind investigated recently by Shih wherethe R -symmetry can be spontaneously broken. We examine several implementationswith varying degrees of phenomenological interest. a r X i v : . [ h e p - t h ] J a n Introduction
Magnetic monopoles are fascinating for many reasons. Decades before Grand Unified Theo-ries were considered, Dirac [1] realized that the existence of monopoles implies charge quan-tization. It turns out that monopoles and GUTs are intrinsically connected, and monopolescan arise dynamically as topologically stable gauge field configurations from spontaneousgauge symmetry breaking [2]. Their dynamics appear quite distinct from other kinds ofobjects in quantum field theory, and actions that incorporate monopoles into a theory withelectric charges have to be either Lorentz-violating [3] or non-local [4]. A magneticallycharged condensate leads to a magnetic dual Meissner effect and represents one possibleexplanation for confinement [5]. (For an excellent review of these ideas the reader is directedto [6].)In 1994 Seiberg and Witten [7] were able to use elliptic curves to find the low-energyeffective action of N = 2 supersymmetric SU (2) gauge theories. As the gauge symmetryis broken to U (1) we would expect to find heavy topological monopoles and dyons. Theydiscovered that some of those topological states become massless weakly coupled particlesat certain singular points on the moduli space, where the electric gauge coupling diverges.Furthermore, softly breaking these theories to N = 1 supersymmetry (SUSY) lifted the mod-uli space and induced the massless monopole to condense, leading to electric confinementand providing an illuminating perspective on the well-known N = 1 phenomenon of gauginocondensation. (A pedagogical introduction can be found in [8].) These results were soon gen-eralized to higher gauge groups [9]. The higher-dimensional moduli spaces of these theoriescontain singular submanifolds where both electric and magnetic charges of the same U (1)gauge group become simultaneously massless, providing the first example of a self-consistentquantum field theory where this particle content arises dynamically. It is also possible toapply these methods to the analysis of minimally supersymmetric N = 1 theories in theCoulomb phase and extract the holomorphic parts of the low-energy effective action [10].SUSY is, of course, extremely interesting for phenomenological reasons, the most impor-tant one being the stabilization of the weak scale. While there are several possible mech-anisms for breaking supersymmetry [11] and mediating its breaking to the supersymmetricstandard model, no clear favorite has emerged. It is therefore prudent to continue lookingfor new ways of breaking SUSY. The unique properties of monopoles, and the fact that theyarise as light states dynamically and calculably in some theories, motivate the constructionof SUSY-breaking models that rely on monopole dynamics. The hope is that eventually somenew mechanisms with desirable, novel features might be found. In this paper, we initiatethe study of such constructions .Models involving meta-stable SUSY-breaking [13] circumvent the restrictive Witten indexconstraint [14]. The topic enjoyed a fresh surge of interest since Intriligator, Seiberg, and Shihshowed how theories as simple as SQCD in the free magnetic phase can feature metastableSUSY-breaking vacua [15], and since then many models have been proposed to incorporate Theories with light monopoles and dyons are also worth studying since they might yield a possiblemechanism of electroweak symmetry breaking [12]. N = 2 SYM might plausibly feature such SUSY-breaking vacua, and their intuitionturned out to be correct. Deformed N = 2 theories can generate SUSY-breaking localminima at generic points of their moduli spaces [17], but the metastable vacua do not lieon the singularity and the monopoles of the theory play no role. Ref. [18] considers an N = 2 model in the Coulomb phase softly broken to N = 1 which classically breaks SUSYin a metastable vacuum via a Fayet-Illiopoulos term. They find that the SUSY-breakingsurvives the nonperturbative quantum effects, and that light dyons undergo condensation atthe meta-stable SUSY-breaking minimum.Here, we want to construct a dynamical model of supersymmetry-breaking where SUSYwould be restored in the absence of a monopole condensate. Our starting point will be the N = 1 SU (2) model [19], where the gauge symmetry is broken to U (1) on the moduli spaceand there are singular submanifolds on which monopoles or dyons become massless. We thendeform the theory to obtain monopole condensation near a point on the singular submanifoldof the moduli space. In the low-energy limit we find an effective O’Raifeartaigh-type modelof the form recently investigated by Shih [20] which features a metastable SUSY-breakingminimum.The paper is structured as follows. In Section 2 we briefly review the N = 1 SU (2) modelwith massless monopoles and parameterize our ignorance of the K¨ahler potential to writedown an effective theory near a point on the singular submanifold. In Section 3 we reviewthe Shih-O’Raifeartaigh model and derive the scaling behavior of some important quantitiesdetermined at 1-loop. We then deform the SU (2) model to let monopole condensationtrigger supersymmetry breaking in Section 4. Section 5 explores some variations of thismodel, and we conclude with Section 6. SU (2) Model
The basis for our model of SUSY-breaking is the SU (2) model [19]. After briefly reviewingits main features, we will expand the theory around a particular point in moduli space inorder to explicitly parameterize our ignorance of the incalculable K¨ahler potential. Our starting point is an N = 1 SUSY model [19] with a SU (2) × SU (2) × SU (2) gaugesymmetry that is is broken down to a diagonal U (1) at low energies. This makes it possible toapply Seiberg-Witten methods [7,10] to obtain information about the holomorphic quantities(the superpotential and gauge kinetic term) of the model. The particle content of theunderlying electric theory is SU (2) SU (2) SU (2) Q Q Q SU (2) i gauge groups become strongly coupled below scales Λ i . For simplicity welet Λ i = Λ. The moduli space is spanned by four gauge invariants M i = det Q i = 12 ( Q i ) αβ ( Q i ) γδ (cid:15) αγ (cid:15) βδ ,T = 12 ( Q ) β α ( Q ) β α ( Q ) β α (cid:15) α β (cid:15) α β (cid:15) α β , (2.2)and at generic points in the moduli space the SU (2) gauge group is broken down to thediagonal U (1), so the theory is in the Coulomb Phase. The holomorphic quantities of thelow-energy theory are described by an elliptic curve y = (cid:2) x − (cid:0) Λ M + Λ M + Λ M − M M M + T (cid:1)(cid:3) − . (2.3)Rescaling the variables by defining u SW = 2 (cid:0) Λ M + Λ M + Λ M − M M M + T (cid:1) , Λ SW = 2 Λ , we see that Eq. (2.3) is identical to the N = 2 SU (2) SYM curve [7], y = (cid:18) x − u SW (cid:19) − Λ SW . (2.4)The elliptic curve represents a torus with complex structure, and the low-energy U (1) eff holomorphic gauge coupling is given as the ratio of the two periods of the torus. The torus canbe transformed by an SL (2 , Z ) transformation, which corresponds to transforming the low-energy effective U (1) gauge theory into a dyonic dual description. In the electric descriptionthe electric gauge coupling approaches zero as u → ∞ . The roots of the N = 2 SU (2)SYM elliptic curve are degenerate for u = ± , meaning that the torus becomes singularon the corresponding submanifolds of the full moduli space. This causes the electric gaugecoupling to diverge on these submanifolds, whereas the magnetic/dyonic gauge coupling goesto zero. Therefore certain monopoles or dyons, which are large, massive and strongly coupledtopological objects in the weakly coupled electric regime u → ∞ , become elementary, light,and weakly coupled (the magnetic coupling goes to zero) near the respective singularities.The monopoles and dyons of the SU (2) model become massless whenΛ M + Λ M + Λ M − M M M + T ± = 0 . (2.5)Near these two points in moduli space, the effective potential can be approximated as W eff = 1Λ (cid:2) − Λ M − Λ M − Λ M + M M M − T ± (cid:3) E ± ˜ E ± + { HOT } , where E ± , ˜ E ± are monopoles/dyons, which are light, elementary and weakly coupled nearthe singularity. The higher-order terms { HOT } only contain higher powers of the term3n square brackets and cannot change the location of the singularity. Higher powers ofmonopoles/dyons in the superpotential are forbidden by global symmetries (including ananomalous U (1) R ) and holomorphy. Rescaling the moduli to have mass dimension 1, thisbecomes W eff = (cid:20) − M − M − M + M M M Λ − T Λ ± (cid:21) E ± ˜ E ± + { HOT } . (2.6) We will deform the SU (2) model in a way that will lead to SUSY-breaking triggered bymonopole condensation. To that end we want to find the effective theory near a singularpoint on the moduli space, defined by M = 2Λ , M , = 0 , T = 0 . (2.7)The existence and stability of a SUSY-breaking minimum (after perturbations are included)in the vicinity of this point will depend on the exact form of the K¨ahler potential, whichcannot be calculated using Seiberg-Witten techniques in an N = 1 theory. Instead we willexpand the effective Lagrangian in small fluctuations around the supersymmetric state (2.7)and restrict the form of the Kahler potential using unbroken global symmetries. Expanding M = 2Λ + δM , Eq. (2.6) becomes W eff = (cid:20) − T Λ − δM − M − M (cid:21) E ± ˜ E ± + { HOT } . (2.8)where { HOT } now includes terms like M E ± ˜ E ± / Λ and T M E ± ˜ E ± / Λ . (We explictly keepthe T / Λ term because it gives the lowest-order contribution to the potential for T .) Whilethe K¨ahler potential is not determined by holomorphy, the weakly coupled degrees of freedomnear the singular point are the monopoles and the moduli, and the K¨ahler potential is non-singular in terms of these fields with an expansion in inverse powers of Λ. The globalsymmetries are then used to constrain the K¨ahler potential. There is an S symmetry whichswitches the M i ’s and SU (2) i ’s around, as well as a slightly less obvious Z which acts oneach of the electric quarks as Q i → e inπ/ Q i . (2.9)This is an anomaly-free Z subgroup [21] of anomalous U (1) global symmetry under whicheach of the electric quarks has charge 1. Under this symmetry, the moduli transform as M i → e inπ M i and T → e inπ/ T , meaning that M i and T both have charge 2 under the Z .The x and y coordinates of the elliptic curve, Eq. (2.3), have charge 2 and 0 respectively,while Λ has charge 0.Around the point in moduli space (2.7), the global symmetries of the model are brokenfrom S to S , which exchanges M and M , and Z is broken to Z , under which T → − T and the M i are singlets. Defining a field basis ϕ i = ( δM , M , M , T, E + , ˜ E + ), we write the4¨ahler potential as an expansion in the small fluctuations K = ϕ † j K ji ϕ i + O (cid:18) ϕ Λ (cid:19) , (2.10)where K ji is a hermitian positive-definite matrix. The symmetries then restrict K ji to be ofthe form K ji = α δe iθ δe iθ δe − iθ β γ δe − iθ γ β κ η
00 0 0 0 0 η , (2.11)where all parameters are real and positive definiteness requires κ > η > β > γ , α ( β + γ ) > δ . The precise values of these parameters are unknown but presumably O (1).We can now define new degrees of freedom ˜ M i to diagonalize the upper 3 × K ji . Upon rescaling, all degrees of freedom can then be made canonical to quadratic order inthe K¨ahler potential, giving a effective superpotential valid in the neighborhood of Eq. (2.7), W eff = (cid:20) a ˜ M + b ˜ M + c ˜ M − d T Λ (cid:21) E + ˜ E + , (2.12)where the coefficients a, b, c, d are unknown complex O (1) numbers into which we have ab-sorbed the canonical rescaling of the monopole fields. As long as the S symmetry is unbrokenone can show that c = 0, but we include this coefficient for generality since it will be inducedperturbatively by explicit S breaking effects, as discussed in Section 4. Triggering SUSY-breaking via monopole condensation can be achieved by deforming the SU (2) model to resemble the Shih-O’Raifeartaigh model [20] in the low-energy limit (near asingular point of moduli space). In this section we will briefly review the Shih-O’Raifeartaighmodel and then derive some scaling behavior which will be important in ensuring the stabilityof our SUSY-breaking local minimum against incalculable corrections. In [20], Shih wrote down an O’Raifeartaigh model with a single pseudomodulus and R -charges other than 0 or 2 which can break R -symmetry spontaneously without tuning. Thesuperpotential is W = λXφ φ + m φ φ + 12 m φ + f X . (3.1) See [22] for some studies of spontaneous R -breaking in models with multiple pseudomoduli. R -charges are R X = 2 , R φ = − , R φ = 1 , R φ = 3. The tree-level scalar potential is V = | λφ φ + f | + | λXφ + m φ | + | λXφ + m φ | + | m φ | . (3.2)Via field redefinitions we can let all the parameters be real and positive. It is useful to definethe two dimensionless parameters y = λfm m , r = m m . (3.3)For y <
1, there exists a pseudoflat direction that breaks SUSY: φ i = 0 , X arbitrary ⇒ (cid:104) V (cid:105) = f . (3.4)The field X is a pseudomodulus, meaning it does not receive a potential at tree-level butdoes get one at 1-loop. The pseudomoduli space is stable in a neighborhood of the origin | X | < X max , where X max ≡ m λ − y y . (3.5)The 1-loop Coleman-Weinberg potential [23] can stabilize X at the origin for r < ∼ R -symmetry and induce a non-zero (cid:104) X (cid:105) for r > ∼
2, see Fig. 1. There is also aSUSY-runaway: φ = (cid:114) f m λ X , φ = − (cid:114) f Xm , φ = (cid:115) λ f X m m , X → ∞ (3.6)Along this runaway direction, V = f m m / ( λ | X | ), so the value of | X | at which the potentialenergy becomes equal to the false vacuum energy f is | X | = X cross ≡ m λy = (cid:115) fλry . (3.7)For smaller values of | X | , the potential energy is larger than f along the runaway direction.The width of the potential barrier that separates the false vacuum from the runaway scaleswith negative powers of y and λ , so if either is small the parametric longevity of the SUSY-breaking minimum can be guaranteed. We will eventually construct a model which reduces to the Shih-O’Raifeartaigh model ina low-energy limit, up to incalculable K¨ahler corrections and higher-order terms in the su-perpotential. To ensure that these incalculable contributions to the scalar potential do notdestabilize the false vacuum we will have to understand the scaling behavior of the pseudo-modulus mass, VEV and the gradient of the potential barrier.6 x V cw r x V cw r Figure 1: Two examples of the 1-loop Coleman-Weinberg potential V CW (shifted by aconstant) that generate zero and nonzero VEV for X . Note that | X | = (cid:113) fλ ˜ x and V CW = λ f ˜ V CW .The first step is to separate out the f /λ scaling from the r, y scaling by redefining adimensionless version of the 1-loop Coleman-Weinberg potential V CW ( | X | ) = 164 π Tr( − F M log M Λ (3.8)in the following fashion:˜ x = √ λ | X |√ f , ˜ V CW (˜ x ) ≡ λ f V CW (cid:18) ˜ x √ λ (cid:112) f (cid:19) . (3.9)Then ˜ V CW (˜ x ) depends only on ˜ x, y, r (up to an additive constant). In these units, X max fromEq. (3.5) becomes ˜ x max = 1 − y y / r / . (3.10)We can now easily explore the r, y scaling of ˜ V CW (˜ x ) numerically.There are two regimes of interest. For the first numerical scan we let r ∈ (0 ,
10) toexplore the interesting r ∼ O (1) behavior of V CW and make the plots of (cid:104) X (cid:105) , the gradientof the potential barrier and the mass of the pseudomodulus shown in Figures 2, 3 and 4.The second scan looked at log ( r ) ∈ (1 ,
8) to extract scaling behaviors with r and y varying over many orders of magnitude. These extracted scalings turned out to givereasonable order-of-magnitude estimates for r ∼ O (1) as well. The results can be summarizedas follows: • For any r , there exists a y max < ≤ (cid:104)| X |(cid:105) 1. From the scan, we are able to extract the followingscaling behavior: y max ≈ r (3.11)where the error is ∼ ∼ < 1% for r ∼ ∼ 10, and > 100 respectively.Hence we can extract an interesting constraint for large r that must be satisfied toguarantee the existence of a SUSY-breaking local minimum: λf < ∼ m for r > ∼ . (3.12) • The pseudomodulus VEV (cid:104) X (cid:105) is 0 for r < ∼ 2. For r > ∼ (cid:104) X (cid:105) ≈ ry X max = (1 − y ) m λ (3.13)with errors O (10%), O (1%) when r ∼ O (1) and r > 10 respectively. • The maximum (positive) gradient between (cid:104) X (cid:105) and X max is roughly given by (cid:20) ∆ V ∆ | X | (cid:21) max ∼ π (cid:114) yr × (cid:112) λ f = 18 π λ f m , (3.14)as long as y is not very close to y max , in which case the gradient approaches 0. • As shown in Fig. 4, the behavior of the pseudomodulus mass mostly depends on y withthe exception of the dip near r ≈ 2. Ignoring the dip, the mass scales as m X ∼ y × (cid:112) λ f = 13 (cid:112) λ f m m (3.15)Let us reexamine the lifetime of the SUSY-breaking minimum in light of these scalings.The pseudomodulus VEV is at (cid:104) X (cid:105) = 0 or (cid:104) X (cid:105) ≈ (1 − y ) m (4 λ ) − . The smallest value of | X | at which the SUSY-runaway has a lower potential energy than the potential of the pseudoflatdirection is X cross = m ( λy ) − . If (cid:104) X (cid:105) is not zero, one can show using y < y max ∼ /r that (cid:104) X (cid:105) /X cross ≈ ry (1 − y ) / < ∼ (1 − y ) / < / 2, so regardless of r the barrier widthscales as X cross , or ∼ O ( y − λ − ). Therefore, if either y or λ is small, the longevity of thesupersymmetry breaking vacuum is guaranteed. Since y − > ∼ r for a SUSY-vacuum, thepresence of such vacuum when r is large also guarantees its longevity. Now we want to deform the SU (2) model such that, near the monopole singularity, themonopoles condense and the low-energy effective theory (below the condensation scale)resembles the Shih-O’Raifeartaigh model of metastable SUSY-breaking. This mechanismshould be dynamical in the sense that SUSY is restored in the weak-coupling limit Λ → (cid:104)| X |(cid:105) in units of (a) (cid:112) f /λ and (b) X max . White areasindicate that the 1-loop Coleman-Weinberg potential slopes away from the origin withoutany local minima. Notice that for r > ∼ R -symmetry is spontaneously broken.Figure 3: The maximum value of the gradient dVdX in the interval | X | ∈ ( (cid:104)| X |(cid:105) , X max ), inunits of (cid:112) λ f . 9igure 4: The pseudomodulus mass, m X , generated by V CW in units of (cid:112) λ f .To achieve this, we introduce SU (2) -singlet fields φ , , , Z, Y and the following tree-levelsuperpotential to the electric theory: W tree = ˜ m ( QQ ) A + ˜ λ Λ UV ( QQ ) B φ φ + 12 m φ + m φ φ + a Z Λ UV Q Q Q Z + a Y ( QQ ) C Y. (4.1)Here, ( QQ ) A,B,C are linear combinations of Q , Q , Q and Λ that become the canonical˜ M , , perturbations around the point, Eq. (2.7), in the IR. The electric quark mass ˜ m andthe UV-physics scale Λ UV must be much smaller and larger, respectively, than Λ to protectthe nonperturbative SU (2) dynamics. For the same reason the Yukawa coupling a Y mustbe much less than unity. The couplings ˜ λ and a Z are perturbative. These deformationsexplicitly break the S symmetry, while the Z symmetry is reduced to Z , with T and Z both having charge 1. Crucially, this is still sufficient to prevent any quadratic K¨ahler mixingin the low-energy theory except amongst the M i perturbations, but in Eq. (2.12) a nonzero c ∼ O (max { π ˜ λ ΛΛ UV , π a Z ΛΛ UV , π a Y } ) will be generated.In the magnetic theory near the monopole singularity W tree is mapped to δW = − µ ˜ M + λ ˜ M φ φ + m φ + m φ φ + m Z ZT + m Y ˜ M Y, (4.2)where µ ∼ m Λ (cid:28) Λ , Λ ∼ ˜ λ Λ / Λ UV (cid:28) m Z ∼ a Z Λ / Λ UV (cid:28) Λ, and m Y ∼ a Y Λ (cid:28) Λ.10y absorbing phases into the fields appropriately, all the parameters can be made real andpositive. The rationale behind choosing the particular form of the deformations Eq. (4.2) is the fol-lowing: the mass terms for ZT and ˜ M Y stabilize the respective moduli at the origin, while F ˜ M = aE + ˜ E + − µ forces the monopoles to condense, which creates an effective tadpolefor ˜ M . This generates an effective Shih-O’Raifeartaigh model, where the pseudomodulus isa mixture of the composite degrees of freedom ˜ M , ˜ M and the tadpole is generated by themonopole condensate.Let us examine this more carefully. Setting F ˜ M ,T,Y,Z = 0 gives (cid:104) ˜ M (cid:105) = (cid:104) T (cid:105) = (cid:104) Z (cid:105) = 0 , (cid:104) Y (cid:105) = − c (cid:104) E + ˜ E + (cid:105) m Y . (4.3)To ensure that we have massless monopoles, set F ˜ E + ,E + = 0 by enforcing (cid:104) ˜ M (cid:105) = − ba (cid:104) ˜ M (cid:105) . (4.4)If the monopoles condense the remainder of the theory looks exactly like the Shih-O’Raifeartaighmodel. Minimizing | F ˜ M | + | F ˜ M | under the assumption that φ i = 0 gives the monopoleVEV (cid:104) E + ˜ E + (cid:105) = aa + b µ . (4.5)The ˜ M tadpole ensures that monopole condensation is energetically preferable. The tree-level vacuum energy is (cid:104) V (cid:105) = b a + b µ , (4.6)and receives contributions from both nonzero F ˜ M , .It is now clear that in the low-energy limit this theory resembles the Shih-O’Raifeartaighmodel, with the pseudomodulus X corresponding to a mixture of the composite ˜ M and ˜ M while the tadpole f X is generated by monopole condensation, with f ∼ µ . (In fact we maysimply set X = ˜ M and f = abµ / ( a + b ) ∼ m Λ. The ˜ M content of the pseudomodulus hasno effect other than to rescale its mass by an O (1) factor with respect to the correspondingexpression for the Shih-O’Raifeartaigh model.)To summarize, we have shown that the point (cid:104) ˜ M (cid:105) = (cid:104) T (cid:105) = (cid:104) Z (cid:105) = (cid:104) φ i (cid:105) = 0 , (cid:104) Y (cid:105) = − c (cid:104) E + ˜ E + (cid:105) m Y , (cid:104) E + ˜ E + (cid:105) = aa + b µ , (cid:104) ˜ M (cid:105) = − ba X, (cid:104) ˜ M (cid:105) = X, (4.7)constitutes a tree-level stable pseudomoduli space parameterized by the value of ˜ M . Thetree-level spectrum can be divided into four groups:11 ˜ M , ˜ M , ˜ M : These three chiral superfields have a supersymmetric spectrum. Thereare two massive modes with masses O ( m Y , µ ) and one zero mode chiral superfield. Thefermion component of the zero mode is the Goldstino. The complex scalar componentof the zero mode multiplet is X . | X | is the pseudomodulus and receives a VEV at 1-looplevel, whereas the phase of X is the Goldstone boson of the global U (1) R under which X has charge +2. This is not a global symmetry of the electric superpotential, butis an accidental symmetry in the IR when irrelevant (nonrenormalizable) interactionsare neglected. • T, Z : Their spectrum is also supersymmetric and massive with masses m Z + O ( µ (cid:112) m Z / Λ). • Y, E + , ˜ E + : Two massive chiral superfields have the same mass as the non-zero modesin the ˜ M i group. The other superfield is eaten by the magnetic gauge superfield sincethe U (1) mag is broken by the monopole VEV. • φ , φ , φ : The scalar and fermion masses of these fields are identical to the corre-sponding masses from the Shih-O’Raifeartaigh model (with the substitution f → abµ / ( a + b )), and are furthermore the only masses that depend on the pseudo-modulus.The 1-loop Coleman-Weinberg potential for X is generated exclusively by the φ i masses, giv-ing us an effective low-energy Shih-O’Raifeartaigh model below the monopole condensationscale and a corresponding SUSY-breaking vacuum. All the results from Section 3 carry overand apply near the origin of our field perturbations. We will now check what constraints the various scales in the theory must satisfy to ensurethat the Shih-O’Raifeartaigh metastable SUSY-breaking vacuum of the deformed SU (2) model is not wiped out by 1 / Λ suppressed corrections which we have so far neglected underthe assumption that they would be small in the neighborhood of the monopole singularity.There are two sources for these corrections: (a) irrelevant terms in the superpotential,Eqns. (2.12) & (4.2), and (b) cubic and higher order terms in the fields in the K¨ahlerpotential. We can ignore the higher-order corrections in evaluating the Coleman-Weinbergpotential, since their contributions are subdominant to the tree-level mass-dependence onthe pseudomodulus | X | (this will be demonstrated below). That means we must check twothings: that the higher order corrections do not destabilize any field directions that were flatprior to taking those corrections into account, and that those corrections do not overpowerthe Coleman-Weinberg potential and destabilize the | X | VEV.Since the flat direction corresponding to the Goldstone boson of the broken U (1) is pro-tected, and assuming that the tree-level masses of M , T , Z , Y , φ , and φ are sufficientlylarge, all we need to worry about are the 1 / Λ suppressed corrections involving the pseudomodulus X . The dangerous ones are K¨ahler terms cubic in X and non-renormalizable su-perpotential terms. These terms are allowed because of the spontaneous breaking (4.4) and12xplicit breaking (4.1) of the Z global symmetry to Z . Both types contribute terms of theform δV ∼ V X + X † Λ ∼ m Λ( X + X † ) . (4.8)The potential for X generated by V CW looks like a mexican hat in the X -complex-plane.The phase is undetermined, but | X | receives a VEV from V CW . Adding terms like Eq. (4.8),i.e. linear terms in X , will generate a definite VEV for θ X while shifting the VEV of | X | . Soto ensure stability, we must check that the linear tilt due to a deformation like δV ∼ m Λ X does not overpower the stabilizing effect of V CW ( | X | ). (We now drop the absolute value signsand use X to describe the component of that field along the tilt direction.) The potentialfor X including higher order corrections looks schematically like this: V ( X ) ∼ V CW ( X ) + m Λ X (4.9)(Note that we have replaced µ by m Λ, which is sufficient for the required order-of-magnitudeestimates.) To make sure that the local minimum of V ( X ) is not destroyed by the tilt, werequire that the rough scale of the gradient of the potential barrier is much larger than thescale of the gradient of the tilt. Using the result of our numerical scan in Eq. (3.14), weobtain the following inequality which must be satisfied to ensure that our mechanism ofSUSY-breaking survives the effect of higher-order corrections: (cid:18) ∆ V ∆ X (cid:19) max (cid:29) m Λ ⇒ O (10 − ) (cid:18) ΛΛ UV (cid:19) m Λ m (cid:29) m Λ , (4.10)which can be simplified to m Λ (cid:28) O (10 − ) × (cid:18) ΛΛ UV (cid:19) . (4.11)There is also another constraint on the scales from SUSY-breaking: λm Λ m m ∼ y < y max ∼ r , (4.12)which becomes ΛΛ UV < ∼ (cid:16) m Λ (cid:17) (cid:18) Λ m (cid:19) (4.13)To illustrate these constraints, define the powers c UV , c m , c , c such thatΛΛ UV ∼ − c UV , m Λ ∼ − c m , m Λ ∼ − c , m Λ ∼ − c . (4.14)Then equations (4.11), (4.13) imply c > c UV , c < ∼ 12 ( c m + c UV ) , (4.15)13n addition to c UV , c m ≥ / Λ UV ∼ . 01 and m ∼ m , i.e. c UV = 2 , c = c . Then c m ≥ 14 and c , = 8 whenthe inequality is saturated. (For c m > c , can be somewhat larger.) In this case thehierarchies of the model are m (cid:28) m , (cid:28) Λ (cid:28) Λ UV . (4.16)Finally, to ensure that ˜ M is not destabilized by K¨ahler corrections m Y ∼ a Y Λ (cid:28) Λcannot be too small. The lower bound is a Y (cid:29) m Λ . (4.17)We emphasize that these constraints, while restricting us to certain areas of the model’sparameter space, do not represent tuning. There is no particular balancing of parametersinvolved in stabilizing the false vacuum. The above hierarchies merely guarantee that certainpotentially destabilizing contributions are subdominant. By inspection of Eq. (4.1) it is clear that in the weak coupling or classical limit (Λ → Q i = φ i = Z = Y = 0. This means that supersymmetry breaking depends on the nonperturbative SU (2) dynamics.In the Λ (cid:54) = 0 case this model has two runaways which both resemble the runaway in theShih-O’Raifeartaigh model. The first runaway takes F ˜ M , F φ i → F ˜ M (cid:54) = 0. The other runawayis the only supersymmetric runaway in this model. Increasing the monopole VEV from (cid:104) E + ˜ E + (cid:105) = aµ / ( a + b ) to µ /a sets F ˜ M = 0. F ˜ M , F φ i are identical to the F -terms inthe Shih-O’Raifeartaigh model with the replacement X → ˜ M and f → bµ a , and are takento zero via the Shih-O’Raifeartaigh runaway, Eq. (3.6). In both cases F Y,Z,T, ˜ M = 0 via theVEVs in Eq. (4.3) just like in the SUSY-breaking minimum, and ˜ M is free to move howeverit has to in order to set F E, ˜ E = 0. The trajectory of ˜ M depends on the other fields andimplicitly includes all the corrections to the monopole mass in Eq. (2.12) that we can ignoreclose to the origin of our perturbations.Assuming we can trust our approximately canonical K¨ahler potential, the potential en-ergy along both runaways V (cid:24)(cid:24)(cid:24) SUSY run = b µ ( a + b ) + m m µ λ | ˜ M | aba + b , V SUSY run = m m µ λ | ˜ M | ba (4.18)becomes equal to the potential energy of the SUSY-breaking pseudomoduli space V P MS = b a + b µ . (4.19) Using the word runaway implies that there is a SUSY-breaking minimum along this runaway at X = ∞ ,but it is in fact more likely that the fields would eventually roll into the SUSY-runaway. M = m m λ µ a + b ab . (4.20)For the explicit examples of scales considered below Eq. (4.15), this is much less than Λ,making the calculation trustworthy, but even if noncanonical K¨ahler corrections becomeimportant it would not significantly change the result that the width of the potential barrierbetween the SUSY-breaking pseudomoduli space and both runaways is of roughly the samesize along the X direction. (The same can be said for the distance along the φ i directions,since along both runaways φ i scales with (cid:104) E + ˜ E + (cid:105) / ∼ µ .) However, since the SUSY-runawayis separated from the SUSY-breaking local minimum by an O ( µ ) potential barrier, the decaypath of the false vacuum is much more likely to be across the tiny barrier created by V CW to either the SUSY-breaking or supersymmetric runaway. Therefore, since λ is small, thesame arguments that ensure longevity of the false vacuum in the Shih-O’Raifeartaigh modelapply here as well. There is one possible source of tuning in this model, which is the alignment of the defor-mations in the electric theory. If the coefficients of Q i and Λ in the linear combinations( QQ ) A,B,C are not properly chosen in the electric superpotential Eq. (4.1) then they do notcorrespond to the canonical IR degrees of freedom ˜ M i and the effective IR superpotentialwill not exactly resemble Eq. (4.2).We can get a feeling for the required level of alignment by considering the following moregeneral superpotential, δW = − µ ( ˜ M + (cid:15) ˜ M + (cid:15) ˜ M ) + λ ( ˜ M + (cid:15) ˜ M + (cid:15) ˜ M ) φ φ + m φ + m φ φ + m Z ZT + m Y ( ˜ M + (cid:15) ˜ M + (cid:15) ˜ M ) Y, (4.21)Most of these misalignments are harmless, shifting tree-level VEVs or inducing tree-leveltadpoles for the pseudomodulus. However, some can destabilize the SUSY-breaking vacuum.The (cid:15) term, apart from inducing a tree-level pseudomodulus tadpole like (cid:15) , shifts theeffective λ -coupling in the fermion contribution to V CW . For r < 10, the maximum allowedvalues of (cid:15) that do not destroy the local minimum lie in the range (cid:15) max ∼ − − − depending on r and y , with (cid:15) max decreasing for larger r and smaller y . This represents therequired level of tuning in the ( QQ ) A,B linear combinations.The (cid:15) and (cid:15) terms give a mass to the pseudomodulus m X = (cid:15) Y m Y ∼ (cid:15) Y a Y Λ, where (cid:15) Y = ( (cid:15) − ba (cid:15) ). Adding a pseudomodulus mass to the Shih-O’Raifeartaigh model createsa SUSY minimum at X = f /m X and tilts the pseudomoduli space away from the origin.We have to make sure that V CW is not overwhelmed by this gradient, which at a position (cid:104) X (cid:105) on the pseudomoduli space is given by (cid:18) ∆ V ∆ X (cid:19) mX = m X (cid:104) X (cid:105) + f m X . (4.22)15ranslating this to our effective Shih-O’Raifeartaigh model and using Equations (3.13),(3.14), we obtain the following upper bounds: (cid:15) Y a Y (cid:28) π (cid:16) ΛΛ UV (cid:17) mm when r < ∼ 2, i.e. (cid:104) X (cid:105) = 0 √ π (cid:16) ΛΛ UV (cid:17) mm when r > ∼ 2, i.e. (cid:104) X (cid:105) (cid:54) = 0 , (4.23)where we have used the fact that for small λ , (cid:104) X (cid:105) (cid:29) f when r > ∼ a Y isalready a small number satisfying 10 − (cid:28) a Y (cid:28) 1. The constraint Eq. (4.23) gives (cid:15) Y a Y (cid:28) (cid:26) ∼ − when r < ∼ 2, i.e. (cid:104) X (cid:105) = 0 ∼ − when r > ∼ 2, i.e. (cid:104) X (cid:105) (cid:54) = 0 . (4.24)If we take a Y ∼ − , (cid:15) Y could be as big as 10 − for (cid:104) X (cid:105) = 0 and unity for (cid:104) X (cid:105) (cid:54) = 0,representing the required level of tuning in the ( QQ ) C linear combination. SU (2) model In Section 4 we constructed a theory of monopole-triggered SUSY-breaking based on the SU (2) model. This method of deforming models with massless monopoles to induce SUSY-breaking seems fairly general, and it is instructive to try and embed the Shih-model differ-ently into the monopole sector.The first alternative possibility is to make φ composite instead of the pseudomodulus X . Starting from Eq. (2.12), this would lead us to add the deformations and singlet fields δW = − f ˜ M + X ( λT φ − µ ) + m φ φ + m Z Z ˜ M + m Y Y ˜ M . (5.1)These couplings preserve the Z global symmetry as long as φ and φ are also charged. The˜ M tadpole induces a monopole VEV which generates a mass for T , completing the Shih-O’Raifeartaigh sector. The coupling λ ∼ (Λ / Λ UV ) comes from an operator that is higherdimensional than in the model of Section 4. The stability of X against incalculable K¨ahlercorrections as well as the existence of a SUSY-breaking minimum requires the hierarchy m (cid:28) µ , m (cid:28) Λ (cid:28) Λ UV . (5.2)Turning off the SU (2) gauge interactions (Λ → 0) does not destroy the SUSY-breakingpseudomoduli space at tree-level, meaning the gauge interactions deform a classically existingSUSY-breaking minimum, similar to [18]. Since the goal of this paper is to find a model ofdynamical monopole-triggered SUSY-breaking we will not pursue this possibility further.The ‘most dynamical’ embedding of the Shih-O’Raifeartaigh sector into the SU (2) modelis to make both the pseudomodulus X and φ composite. Since we are using ‘more’ of themonopole sector to break SUSY this requires fewer deformations: δW = − f ˜ M + λ ˜ M T φ + m φ φ + m Y Y M . (5.3)16gain the ˜ M tadpole induces a monopole VEV, which now provides a tadpole for ˜ M aswell as a mass for T , which act as the pseudomodulus and φ respectively. Stability of X against K¨ahler corrections and existence of the SUSY-breaking vacuum requires m (cid:28) Λ dueto the smallness of λ ∼ (Λ / Λ UV ) : (cid:16) m Λ (cid:17) (cid:28) (cid:18) ΛΛ UV (cid:19) , m Λ > ∼ (cid:18) ΛΛ UV (cid:19) . (5.4)As desired, SUSY is restored in the Λ → M adjusting to keep thetrajectory on the singularity.This latter model appears more elegant than the model of Section 4, but it suffers theunfortunate drawback that K¨ahler corrections render the T -mass incalculable, a result ofthe monopole VEV doing double duty. While this also does not fulfill our requirementsfor a calculable monopole-triggered dynamical SUSY-breaking theory, these two alternativedeformations of the SU (2) model demonstrate how one might produce many more modelsof SUSY-breaking that include monopole dynamics. Monopoles have many unique characteristics that make them very interesting. Their unusualdynamics might hold the key to constructing novel models of supersymmetry (or perhapselectroweak) breaking. Topological monopoles, traditionally treated as nonperturbative ob-jects, can be calculationally controlled using Seiberg-Witten methods. This opens up newavenues for model building.In our model, supersymmetry breaking is triggered by monopole condensation. A suitablydeformed SU (2) theory with massless monopoles takes on the form of an effective Shih-O’Raifeartaigh model with a meta-stable SUSY-breaking local minimum. In constructingsuch a model within the limitations of N = 1 supersymmetry it is important to check thatincalculable K¨ahler corrections do not destabilize the false vacuum. We have shown thatthese contributions can be controlled, and through appropriate choices of scales can be madesubdominant. Additional deformations of the SU (2) model, with varying characteristics,demonstrate more generally how models with massless monopoles might be deformed toinduce SUSY-breaking. It is our hope that this will open up new investigations which mighteventually yield elegant mechanisms of breaking supersymmetry that circumvent some ofthe problems encountered by other approaches. It would be interesting to explore SUSY-breaking in theories that have both massless electrically and magnetically charged particles. Acknowledgements We would like to thank Matt Baumgart, Luis ´Alvarez-Gaum´e, Mario Martone and ZoharKomargodski for valuable conversations. C.C., Y.S. and J.T thank the Aspen Center for17hysics and C.C. and J.T. also the Kavli Institute for Theoretical Physics where part of thiswork was completed.The work of C.C. and D.C. was supported in part by the National Science Foundationunder grant PHY-0757868. V.R. was supported by Department of Energy under grantDE-FG02-04ER-41298. Y.S. was supported by National Science Foundation under grantPHY-0970173. J.T. was supported by the Department of Energy under grant DE-FG02-91ER406746. References [1] P. A. M. Dirac, “Quantised Singularities in the Electromagnetic Field, Proc. R. Soc. A 133 (1931) 60.[2] G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B (1974)276; A. M. Polyakov, “Particle spectrum in quantum field theory,” JETP Lett. (1974) 194 [Pisma Zh. Eksp. Teor. Fiz. , (1974) 430].[3] P. A. M. Dirac, “The Theory of magnetic poles,” Phys. Rev. (1948) 817.[4] D. Zwanziger, “Quantum field theory of particles with both electric and magneticcharges,” Phys. Rev. (1968) 1489.[5] G. ‘t Hooft, in Proceedings of the European Physical Society International on HighEnergy Physics, Palermo, 1975, edited by A. Zichichi (Editrice Compositori, Bologna,1976), p. 1225; S. Mandelstam, “Vortices And Quark Confinement In Nonabelian GaugeTheories,” Phys. Rept. (1976) 245.[6] J. Preskill, “Magnetic Monopoles,” Ann. Rev. Nucl. Part. Sci. (1984) 461;[7] N. Seiberg and E. Witten, “Electric–Magnetic Duality, Monopole Condensation, andConfinement in N = 2 supersymmetric Yang-Mills theory,” Nucl. Phys. B 426 (1994)19, hep-th/9407087 ; “Monopoles, Duality and Chiral Symmetry Breaking in N = 2Supersymmetric QCD,” Nucl. Phys. B 431 (1994) 484, hep-th/ 9408099 .[8] L. Alvarez-Gaume, S. F. Hassan, “Introduction to S duality in N = 2 supersymmetricgauge theories: A Pedagogical review of the work of Seiberg and Witten,” Fortsch.Phys. , (1997) 159-236, hep-th/9701069 .[9] A. Hanany and Y. Oz, “On the Quantum Moduli Space of Vacua of N = 2 Super-symmetric SU ( N c ) Gauge Theories,” Nucl. Phys. B 452 (1995) 283, hep-th/9505075 ;P. C. Argyres, M. R. Douglas, Nucl. Phys. B448 (1995) 93-126, hep-th/9505062 .[10] K. A. Intriligator, N. Seiberg, “Phases of N = 1 supersymmetric gauge theories infour-dimensions,” Nucl. Phys. B431 (1994) 551-568, hep-th/9408155 .1811] E. Poppitz, S. P. Trivedi, “Dynamical supersymmetry breaking,” Ann. Rev. Nucl. Part.Sci. , 307-350 (1998). [hep-th/9803107]; Y. Shadmi and Y. Shirman, “DynamicalSupersymmetry Breaking,” Rev. Mod. Phys. 72 (2000) 25, hep-th/9907225 ; K. A. In-triligator, N. Seiberg, “Lectures on Supersymmetry Breaking,” Class. Quant. Grav. ,S741-S772 (2007). [hep-ph/0702069]; M. Dine, J. D. Mason, “Supersymmetry and ItsDynamical Breaking,” Rept. Prog. Phys. , 056201 (2011). [arXiv:1012.2836 [hep-th]].[12] C. Csaki, Y. Shirman and J. Terning, “Electroweak Symmetry Breaking From MonopoleCondensation,” Phys. Rev. Lett. (2011) 041802, hep-ph:1003.1718 .[13] S. Dimopoulos, G. R. Dvali, R. Rattazzi, “A Simple complete model of gauge mediatedSUSY breaking and dynamical relaxation mechanism for solving the µ problem,” Phys.Lett. B413 (1997) 336-341, hep-ph/9707537 .[14] E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B 202 (1982) 253.[15] K. Intriligator, N. Seiberg and D. Shih, “Dynamical SUSY breaking in meta-stablevacua,” JHEP (2006) 021, hep-th/0602239 [16] M. Dine, J. L. Feng, E. Silverstein, “Retrofitting O’Raifeartaigh models with dynami-cal scales,” Phys. Rev. D74 (2006) 095012. hep-th/0608159 ; R. Kitano, H. Ooguri,Y. Ookouchi, “Direct Mediation of Meta-Stable Supersymmetry Breaking,” Phys. Rev.