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Meta‐Stable Supersymmetry Breaking in an N = 1 Perturbed Seiberg‐Witten Theory

S. Sasaki – University of Helsinki and Helsinki Institute of Physics M. Arai – Sogang University C. Montonen – University of Helsinki N. Okada – KEK

Deposited 06/06/2019

Citation of published version:

Sasaki, S., Arai, M., Montonen, C., Okada, N. (2008): Meta‐Stable Supersymmetry Breaking in an N = 1 Perturbed Seiberg‐Witten Theory. AIP Conference Proceedings, 1078(1). DOI: https://doi.org/10.1063/1.3052003

Cite as: AIP Conference Proceedings 1078, 486 (2008); https://doi.org/10.1063/1.3052003 Published Online: 26 January 2009

Shin Sasaki, Masato Arai, Claus Montonen, and Nobuchika Okada

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AIP Conference Proceedings 1078, 486 (2008); https://doi.org/10.1063/1.3052003 1078, 486

* University ofHelsinki and Helsinki Institute ofPhysics, PO.Box 64, FIN-00014, Finland tCQUeST, Sogang University, Shinsu-dong 1, Mapo-gu, Seoul 121-742, Korea University ofHelsinki, PO.Box 64, FIN-00014, Finland trheory Division, KEK, Tsukuba 305-0801, Japan

Abstract. In this contribution, we discuss the possibility of meta-stable supersymmetry (SUSY) breaking vacua in a perturbed Seiberg-Witten theory with Fayet-Iliopoulos (FI) term. We found meta-stable SUSY breaking vacua at the degenerated dyon and monopole singular points in the moduli space at the nonperturbative level. Keywords: Supersymmetry breaking PACS: 11.30.Pb

THE MODEL 1 4 2 + 47rim [ru (! deA!A1 + ~ Jd 0Wt)] SUSY breaking at meta-stable vacua in various SQCD + d40 [Qte2V2+2V1Qr +Qre-2V2-2V1Qrt] models has been intensively studied since the proposal J of the ISS model [l]. The other interest is meta-stable 2 SUSY breaking in perturbed Seiberg-Witten theories [2, +h [! d 0 Qr(A2+A1)Qr +h.c.], (2) 3, 4]. In the following, we focus on this possibility. We consider four-dimensional Jf/ = 2 SU(Nc) x U(l ), where V2,A2 and Vi,A1 are vector and chiral super Ni flavors SQCD with FI term. Supersymmetry in the fields belonging to the SU ( 2) and U ( 1) vector multi model is partially broken down to Jf/ = 1 due to the pres plets respectively. The chiral superfields Q1 and (Ir are ence of adjoint mass terms. The extra U( 1) part is nec hypermultiplets that are in the fundamental and anti essary for the FI term and treated as cut-off theory with fundamental representations of the SU(2) gauge group Landau pole A£. With the help of the Seiberg-Witten so (r = 1,2 is the flavor index, and I= 1,2 is the SU(2) lution, we can analyze the theory in exact way provided color index). W is the Jf/ = 1 superfield strength and TiJ the Landau pole is very far away and the perturbation are complex gauge couplings. terms are very much smaller than the SU(Nc) dynami The second term Xsoft is the soft SUSY breaking term cal scale A. In the following, we focus on Ne = Ni = 2 given by case and show that there are SUSY breaking meta-stable minima in the full quantum level. (3)

JV = I SUSY preserving deformation of In Xsoft, µ 1, µ2 are masses corresponding to U ( 1) and JV =2 SQCD SU(2) part of the adjoint scalars and A is the FI param eter. In the absence of Xsoft, the gauge symmetry is bro Let us consider a tree-level Lagrangian ken as SU(2) x U(l)---+ U(l)c x U(l) on the Coulomb branch ("LJ ,LJfi=2 ("LJ (1) 2,, = 2,,SQCD + 2,,soft· Here x'sici5 is the Lagrangian for Jf/ = 2 SU(2) x U(l) (4) super Yang-Mills with Ni = 2 massless fundamental hypermultiplets where q, q are hypermultiplet scalars. Once we tum on Xsoft, there are SUSY vacua on the Coulomb and Higgs 4 2 2 \/m [Tr{ r22 (f d eA!e v2A2e- v2 branches. We are going to investigate the quantum effec 2 tive action on the Coulomb branch. 2 +~ j d e w])}]

CP1078, Supersymmetry and the Unification ofFundamental Interactions, Proceedings of the 111" International Conference (SUSY08), edited by D.-K. Hong and P. Ko © 2008 American Institute of Physics 978-0-7354-0609-4/08/$23.00

486 QUANTUM THEORY where U = U(a1,a2),S = S(a1,a2) > 0 are functions of a1,a2 and 4Z = IMI = IMI [6]. The stationary point The exact low energy effective Lagrangian is described (10) where the light hypermultiplet acquires a vacuum by light fields, the SU(2) dynamical scale A, the Landau expectation value by the condensation of the BPS states pole AL, the masses µi (i = 1,2) and the FI parameter is energetically favored because S > 0. A. If the perturbation terms are much smaller than the Due to the abovementioned reason, we focus on the dynamical scale A, the effective Lagrangian Xexact is singularity points in the moduli space. To find the explicit given by potential, we need the moduli space metric, and hence the prepotential. The monodromy transformation around Xexact = Xsusy+Xpert + tl(µf,A). (5) the singular points in the moduli space dictates us that Here the first term xsusy describes an Jf/ = 2 SUSY La the U ( 1) modulus a1 can be interpreted as the common grangian containing full quantum corrections. The sec hypermultiplet mass min the SU(2) gauge theory. This ond term Xpert. includes the masses and the FI terms in fact implies that the prepotential in our model is given by the leading order. First we consider the general formulas for the effective §(a2,a1,A,AL) = g;jf;_d)(a2,m,A)I +cay, (11) Lagrangian XsusY- The Lagrangian Xsusy is given by m=v12a1 two parts, vector multiplet part xvM and hypermultiplet partxHM, where g;jf:(_J) is the prepotential for SU(2) massive xsusy = xvM + XHM- (6) SQCD with common mass m = vf2a1. The constant C is a free parameter which is used to fix the Landau pole The xvM part consists of U(l)c and U(l) vector mul AL to the appropriate value1 . tiplets. The effective Lagrangian for these vector multi The singular points on the moduli space are deter plets is mined by a cubic polynomial [7]. The solutions of the cubic polynomial give the positions of the singular points xvM _!__Im ±[fd 4 e ~§ A; in the u-plane. In the case Ne = NJ = 2 with a common 47r i,j=l uAi hypermultiplet mass m, which is regarded as the modulus vf2a1 here, the solution is obtained as +~ j d2e riJwiwJ] , (7) A2 A2 A2 2 where§= §(A2,A1,A,AL) is a prepotential as will be u1 = -mA- 8, u2 = mA- 8' U3 = m + s· (12) discussed below. The effective gauge coupling TiJ is de- The singular points correspond to dyons, a monopole and fined by TiJ = with moduli ai- The hypermultiplet a~J;,1 a quark. The behavior of the singularity flow along a1 partXHM is direction can be found in [6]. At the singular points in the moduli space, a2 and a1 are related to each other and the XHM = d4e [M;e2nmVw+2neV2+2nV1Mr J potential is a function of a1 only. To find the stationary +Mre-2nmVw-2neV2-2nV1MrtJ points of the potential along the a1 direction is a difficult task and we need the help of numerical analysis. +hJ d 20 [Mr(nmAw +neA2 +nA1)Mr Let us start from the µ 1 = A = 0 case. Fig. 1 shows the global structure of the potential along the Re( a1) +h.c.], (8) direction. As a result, we found the global SUSY minima at a1 = 0 in the degenerated dyon and monopole singular whereMr,Mr are chiral superfields and Vw,Aw are dual points. variables of V2,A2. These hypermultiplets correspond Next, let us tum on µ1 and A. In the presence of the to the light BPS dyons, monopoles and quarks. which soft term, the gauge dynamics favors the monopole and are specified through the appropriate quantum numbers the dyon points at a1 = 0 as SUSY vacua besides the (ne, nm)n- Here ne and nm are the electric and magnetic runaway vacua. It implies that if we add µ1 0, 0 charges of U(l)c, respectively, whereas n is the U(l) -=/- A -=/- terms which produce a vacuum at a point different from charge. The potential is a function of M,M,a1,a2. We a1 = 0 at the classical level, SUSY is dynamically broken found stationary points alongM,M directions at (l)M = as a consequence of the discrepancy of SUSY conditions M = 0 and (2)M-=/- 0,M-=/- 0. The potential value at each between the classical and the quantum theories. Actually, stationary points are evaluated as (1) V(a2,a1) = U, (9) (2) V(a2,a1) = U -4S4t4, (10) 1 We fix C = 4ni which implies AL ~ I 017-is A.

487 Vmin the location is shifted to negative values of Re( a1) as is expected from the discussion in the previous paragraph. AD point 0.0004 Furthermore, the potential energy has a non-zero value 0.0003 and therefore SUSY is dynamically broken. We find that

0.0002 / the potential energies at the left and right dyon singular I points also have the same structure. A qualitative picture 0.0001 / / of the evolutions of the potential minima is depicted in -i--~....,...... ~~2...... ~¼ 3 ~~~...... -5 Re(a1) Fig. 4.

leftdyon FIGURE 1. Global structure of vacuum. Solid and dashed quark curves show the evolutions of the potential energies at the rightdyon monopole and left(right) dyon points for O <:: Re(a1) < A/(2,/2). The potential energies at the right dyon(dotted) and quark(dash-dotted) po ints for Re(a1) > A/(2,/2) are also plot ted. We have fixed A = 2,/2. ~:".______A

turning on the mass µ1 and the FI parameter A realizes R,(u) such a situation. In this case, the classical vacuum is at a1 = - A/ µ1, different from the point a1 = 0 which the FIGURE 4. Qualitative picture of the evolutions of the po dynamics favors. A resultant SUSY breaking vacuum is tential minima. realized at non-zero value of a1. This is very similar to In addition to these local minima, there are supersym the SUSY breaking mechanism discussed in the Izawa metric vacua on the Higgs branch which survive from Yanagida-Int:riligator-Thomas model in Jf/ I SUSY = the quantum corrections. We estimated the decay rate gauge theory [8, 9]. We show a schematic picture of our from our local minima to the SUSY vacua on the Higgs situation in Fig. 2. branch and found that the decay rate can be taken to be fu 11 effective very small. This means our local minima are nothing but potential meta-stable SUSY breaking vacua. ' Vt =faO, A =fa O \ t•1 = A= 0 quantum theory tree level \ ...... , ACKNOWLEDGMENTS

The work of S. S is supported by the bilateral pro gram of Japan Society for the Promotion of Science and FIGURE 2. Schematic picture of SUSY breaking mecha Academy of Finland, "Scientist Exchanges". msm v.s. mm >. = 0.19 REFERENCES 0. 0008

>. = 0.17 1. K. Intriligator, N. Seiberg and D. Shih, ffiEP 0604 (2006) >. = 0.15 021 [arXiv:hep-th/0602239]. I 2. H. Ooguri, Y Ookouchi and C. S. Park, arXiv:0704.3613 [hep-th]. 3. J. Marsano, H. Ooguri, Y Ookouchi and C. S. Park, Nucl. -2 . 2 -2 . 1 Phys. B 798 (2008) 17 [arXiv:0712.3305 [hep-th]]. 4. G. Pastras, arXiv:0705.0505 [hep-th]. FIGURE 3. Local SUSY breaking minimum at the 5. M. Arai, C. Montonen, N. Okada and S. Sasaki, Phys. Rev. monopole singular point for µ1 = µ2 = 0.1 and A D 76 (2007) 125009 [arXiv:0708.0668 [hep-th]]. 0.15,0.17,0.19 from bottom to top. Here O <:: Re(a1) < 6. M. Arai, C. Montonen, N. Okada and S. Sasaki, ffiEP 0803 A/2,/2. (2008) 004 [arXiv:0712.4252 [hep-th]]. 7. N. Seiberg and E. Witten, Nucl. Phys. B 431 (1994) 484 Let us see in detail how this works for non-zero values [arXiv:hep-th/9408099]. of µ1, µ2 and A. Fig. 3 shows the evolution of the poten 8. K. I. Izawa and T. Yanagida, Prog. Theor. Phys. 95 (1996) tial energies at the monopole point V~in for several val 829 [arXiv:hep-th/9602180]. ues of A as a function ofRe(a1) with µ1 = µ2 = 0.1. The 9. K. A. Intriligator and S. D. Thomas, Nucl. Phys. B 473 (1996) 121 [arXiv:hep-th/9603158]. potential minimum is no longer realized at a1 = 0, but

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