arXiv:1011.2659v2 [hep-th] 18 Feb 2011 cl .Udrti ult ueptnilms term mass superpotential a strong-coupling duality their this q µ infrared-free Under below to energies Λ. dual scale at are theories content, gauge QCD matter supersymmetric suitable ISS notably with most the of theories, asymptotically of mechanism Certain gauge ingredient ISS free [4]: essential duality classic Seiberg An is now mechanism [3]. the breaking on SUSY relying models many scale. be fundamental naturally the can below which magnitude Λ, of breaking involve orders SUSY then The will breakdown. scale SUSY to leading additional generate terms in- can as condensation superpoten- gaugino such or tree-level effects stantons the non-perturbative SUSY, While at preserves coupled Λ. tial con- strongly scale sector becomes infrared which hidden some theory breaking gauge SUSY a 2]. the [1, tains (DSB) models, breaking DSB SUSY In dynamical of models study raigi eatbevcu at vacuum metastable SUSY a triggers in eventually term breaking linear this of presence The ierterm linear a u loteoii fteheacywtotapaigto physics. stability, UV appealing the without the hierarchy just of the details not of fundamental the explain origin the to the also hope of typ- but may units are one so in These scale, small effects. exponentially non-perturbative most ically at by can broken ra- it tree- theorem against be renormalization at unbroken the stabilized by is thus then SUSY level, is if Second, scale, corrections. grand-unified Planck diative the the e.g. or scale, scale scale fundamental electroweak high the some between di- and hierarchy quadratic The from Model vergences. of senses. potential Standard Higgs distinct supersymmetric the the protects two SUSY broken in softly First, problem hierarchy electroweak ∗ 1 lcrncades [email protected] address: Electronic iul h rseto nesadn h rgno h hier the theory. se- of field take origin however effective the shall within understanding u we purely of the Letter prospect this to the In riously explanation completion. an magnitude di- UV of relegating known whose orders scale, breaking many fundamental SUSY be an below tree-level to altogether of happen point parameters model second mensionful a the with disregard content course be of might One h oesw ilb nlsn nti oeaeDSB are Note this in analysing be will we models The to motivation major a as serves point second This o-nrysprymty(UY a leit the alleviate can (SUSY) supersymmetry Low-energy q ˜ o h lmnaymte superfields matter elementary the for ffcieter hc rassprymtydnmclyin dynamically supersymmetry This breaks th which in sector. group theory gauge gauge o effective free coupled asymptotically Some strongly an rely SQCD. a and superpotential of models massive composites Our in as breaking arise dynamically. supersymmetry generated of are mechanism scales all which ecntutnweape fmdl fmetastable of models of examples new construct We µ Λ .INTRODUCTION I. M o h opst nrrdfield infrared composite the for eatbesprymtybekn ihu scales without breaking supersymmetry Metastable 1 oksrß 5 -20 abr,Germany Hamburg, D-22607 85, Notkestraße M etce lkrnnSnhornDESY Elektronen-Synchrotron Deutsches 0. = q ˜ , q un into turns ei Br¨ummerFelix archy M n- d . etr[–] eew aeteie oehtfurther somewhat idea the whose take models we constructing Here by [5–7]. sector q nteU smpe rtt udai n hnt a SU( to group then gauge the and theory. quadratic IR a effective to the first in mapped operator linear is term UV superpotential cubic the tran- originally in strong-coupling an two which sec- of by use gauge sitions, making coupled thus are strongly We a tor. of composites themselves epciey hsalw o agnloperator marginal a for allows This respectively. gener- situation, models this Several remedy to ating coupled). constructed been strongly have become can theory (for ul yaia xlnto fwytesaeo SUSY of scale since the small, why is of breaking explanation dynamical fully where eoe togyculda cl Λ scale a at coupled strongly becomes eet ekyculdda hoyivligacomposite dif- a field a involving of “meson” theory means dual by coupled described weakly ferent, be can dynamics infrared its eta ilcnanaterm a contain will tential hc orsod oamass a to corresponds which cl Λ scale ntefrifae eedu ihtedsrdlna term linear desired the with composite up the end for we infrared far the in h coefficient The nuht guarantee to enough raig tde o nov n udmna aspa- dynamically. parameter truly marginal mass broken The fundamental is any supersymmetry involve so not rameters, does it breaking; (with hie hc ontsffrfo ntblte unotto out only turn The instabilities from groups. suffer gauge require not the do matter of which ranks remaining choices the the on on and constraints bifundamental the content strong single is a are (which with there version with), Φ, simplest concerned are its we In one detail. in struction e fSU( of ber ˜ ,and Φ, , ceaial,w ilpoeda olw.Consider follows. as proceed will we Schematically, nasrc es h S ehns osntoe a offer not does mechanism ISS the sense strict a In ti h i fti etrt ehotteaoecon- above the out flesh to Letter this of aim the is It D µ ∗ 4 = N µ lrvoe,adedu iha infrared an with up end and ultraviolet, e λ & n N rmsrn ag yaiso oeauxiliary some of dynamics gauge strong from losu osatwt ipecubic simple a with start to us allows = h SU( the sadmninesculn.TeSU( The coupling. dimensionless a is eatbestate. metastable a h atrfilswuddcul eoethe before decouple would fields matter the Λ N = N Q nSiegdaiyado h ISS the on and duality Seiberg on n n SU( and ) F rnfrigas transforming uesmer raigin breaking supersymmetry 1 = loptnilyalwdbtuncalculable). but allowed potentially also q h lcrcqaksuperfields quark electric the f λ = ∼ Λ W n f atrbcmssrnl ope,and coupled, strongly becomes factor ) W N M eff Q where , Λ eff Φ n λ ∼ n = n W ) / Λ = aor epciey and respectively, flavours ) vnulyst h cl fSUSY of scale the sets eventually Λ q × λ N µ λ q/ = ˜ N Λ λ SU( ≪ < hnteeetv superpo- effective the Then . uthwvrb hsnsmall chosen be however must Λ N F Λ Q λ N Λ n Λ µ utb u nb hand by in put be must Λ and , N n n q Φ˜ q = q . ˜ M n atrsuperfields matter and ) ⊗ , q + ere ffedmare freedom of degrees λ f 1 + Λ . . . , r h vrl num- overall the are N . . . N for ncrancases certain In . ⊗ q EY10-195 DESY and , n ˜ and N factor ) q n > N ta At . 1 ⊗ (1) (2) (3) 2
SU(N) SU(n) SU(F ) SU(F − n) SU(f) SU(f − N) I tion of λj , bringing it into the form Φ 1 1 1 1 Q 1 1 1 1 λ1 q˜ 1 1 1 1 . .. Pe 1 1 1 1 λ1 p 1 1 1 1 (λI )= λf or (λI )= .. 0 j j . λF TABLE I: Gauge symmetries and non-abelian flavour sym- 0 metries for vanishing superpotential. The symmetries in the four rightmost columns are global, while SU(N) and SU(n) are gauged. (6)
(depending on whether f < F or f F ). For simplicity ≥ we will from now on consider the case where all λi are equal, λ1 = λ2 = ... λ; it is straightforward to extend II. GENERAL FRAMEWORK ≡ the analysis to the more general case where all λi are nonzero and of similar magnitude. Situations where there The basic ingredients of the models we are investigat- are large hierarchies among the λi, or where some of the ing are the gauge group SU(n) SU(N), where we take λi vanish, are less interesting as will become clear below. n N without loss of generality,× and some matter fields ≤ Both gauge factors are required to be asymptotically allowing for a superpotential resembling Eq. (1). We thus free, so 3N > F and 3n>f. They run to strong cou- introduce a bifundamental field Φ along with f copies of pling in the infrared. We now distinguish two cases, SU(n) antiquarksq ˜ and F copies of SU(N) quarks Q. namely SU(N) becoming strongly coupled at higher en- We also allow for SU(n) fundamentals p and SU(N) an- ergies than SU(n) and vice versa. tifundamentals P . The matter content is summarized in Table I. It is restricted by the absence of gauge anoma- lies: Anomaly cancellatione for SU(n) requires f N, A. The case ΛN > Λn with f N spectator fields p present if f >N.≥ Anal- − ogous statements hold for SU(N). One could include Suppose first that SU(N) becomes strongly coupled at more SU(N) SU(n) bifundamentals, or fields in larger × a scale ΛN , where SU(n) gauge couplings are still neg- representations, but we will refrain from that for now. ligible. We are then dealing with SU(N) SQCD with F A superpotential will explicitly break some of the flavours of quarks Q and antiquarks Φ, P . SQCD with flavour symmetries of Table I. The most general renor- N colours and F flavours has a known and calculable malisable superpotential respecting the gauge symme- 3 weakly coupled infrared description if N e F < 2 N in tries is terms of its Seiberg dual theory [4], so we≤ restrict our- selves to the case where N and F are in this range. Some I j I A b j W = λj QI Φ˜q + µA QI P + mj pbq˜ (4) of the mesonic degrees of freedom q in the dual theory can be identified with the QΦ composites, qI QI Φ/ΛN . ∼ I I e Here we have absorbed a factor ΛN in the definition of where λj is an F f matrix, µA is an F (F n) ma- the fields q so that they have canonical dimension. Be- b × × − trix, mj is an f (f N) matrix, and where we have low the scale ΛN the degrees of freedom are those of the × − 2 suppressed gauge indices. In keeping with the principle magnetic dual of SU(N), and the SU(N) singlet fields we that all dimensionful parameters should be of the order started with. In the infrared, where the magnetic gauge of the fundamental scale, the last two terms in Eq. (4) dynamics becomes negligible, they constitute an SU(n) I will only lead to the decoupling of rank (µA) pairs of Q, P gauge theory with F + f N flavours. The effective b − and rank (mj ) pairs of p, q˜ in the UV. We will therefore superpotential contains a term omit them (and redefine the fields and parameters ac- e I j cordingly). Tree-level masses for the remaining fields can Weff = λj ΛN qI q˜ + ... (7) be forbidden by a Z3 or R-symmetry, or by imposing that SU(F n) and SU(f N) should be classically preserved. descending from Eq. (5). It gives a supersymmetric mass − − I The superpotential we are working with is then λΛN to rank (λj ) pairs of quarks and antiquarks. The infrared structure of this theory does not appear to be very interesting at first sight, since for λ of order W = λI Q Φ˜qj . (5) j I one the massive SU(n) flavours should be integrated out, and one is left with an SU(n) gauge theory with massless Flavour rotations permit a singular value decomposi- matter. Instead we now keep all quarks light by dialling λ 1. The SU(n) gauge coupling will become strong at ≪ some lower scale Λn; if λ is chosen such that λΛN < Λn, all flavours will remain dynamical to below that scale. 2 For N = 2, 3 or n = 2, 3 one could also include renormalisable Supposing that SU(n) is in the free magnetic range as baryonic operators, without affecting our conclusions. well, we end up with the ISS model of metastable SUSY 3 breaking. After another Seiberg duality on the SU(n) fac- SU(n) SU(N) SU(N) SU(N − n) tor we obtain a low-energy effective theory which breaks Φ 1 1 SUSY in a metastable vacuum. q˜ 1 1 There are two possible issues here. First, choosing λ Q 1 1 small may be regarded as problematic in view of natu- Pe 1 1 ralness. We stress however that λ is merely a marginal parameter, as opposed to a relevant one. Furthermore, the tuning is used to overcome the discrepancy between TABLE II: Matter content and symmetries for F = f = N > the two dynamically generated scales Λn and ΛN , which n. The first two symmetries SU(n) and SU(N) are gauge sym- could in principle be close together. The naturalness metries. The SU(N) and SU(N − n) in the last two columns problem we eventually aim to solve with dynamical su- represent classically unbroken global symmetries. The flavour persymmetry breaking, by comparison, involves the po- SU(N), in particular, corresponds to the diagonal subgroup tentially much larger hierarchy between the SUSY break- of SU(F ) × SU(f) in Table I. ing scale and the UV completion scale. Second, for generic choices of F , f, N and n one ends SU(n) SU(N) SU(N − n) up with both massive and massless SU(n) flavours. This q 1 is because the superpotential Eq. (7) gives masses to only q˜ 1 F or f of the F + f N quarks and antiquarks (and to e − p 1 even less if some of the λi in Eq. (6) are zero). An effec- B 1 1 1 tive ISS model with both massive and massless flavours Be 1 1 1 still gives rise to supersymmetry breaking at tree-level. However, some of the pseudo-moduli will no longer be stabilized at one-loop [10]. Indeed it was found in [11] TABLE III: Matter content and symmetries below the SU(N) that along the directions where this happens, the two- confinement scale. SU(n) is gauged, while SU(N) and SU(N− loop contribution to the effective potential causes a run- n) are global. away towards the supersymmetric vacuum. A runaway may also appear if there are large hierarchies between the quark masses, in which case the tachyonic two-loop We thus obtain an even stricter condition than before. contributions from the heavier quarks may overwhelm The resulting model is discussed in Sect. IV. the one-loop contributions from the lighter ones. It has subsequently been argued [12] that higher-dimensional operators in the superpotential may stabilize these run- III. MODELS IN THE STABLE RANGE aways away from the point of maximal unbroken symme- try, and that phenomenologically promising metastable Postponing the case n = N until later, we now in- vacua may appear as a consequence (for some recent de- vestigate models with ΛN > Λn and F = f = N >n in velopments see e.g. [13]). While such ideas may lead to detail. More precisely, to have SU(n) in the free magnetic 3 interesting generalizations when applied to our model, range we demand n
SU(N − n) SU(N) SU(N − n) is mapped into Mf 1 ⊗ 1 √N! 1 1 − p 1 α1...αN−n j1 N n jN−n ˜ N−2n ǫ (˜pj1 χ˜α1 ) (˜pjN−n χ˜αN−n ) (N n)! (Λn) ··· B 1 1 1 − Be 1 1 1 √N! 1 = N−2n det(˜pχ˜) . χ 1 (N n)! (Λn) − χ˜ 1 (16) On the baryonic branch the constraint Eq. (11) can 2 TABLE IV: Field content after a Seiberg duality transfor- be satisfied by setting BB = (ΛN ) andp ˜ = 0. Then − mation on SU(n). Only the SU(N − n) listed on the left is Weff in Eq. (13) becomes precisely the magnetic super- gauged; it is the dual magnetic gauge group of SU(n). potential of the ISS model,e which is well-known to break supersymmetry in a metastable state. The vacuum en- ergy is T is a Lagrange multiplier to enforce the constraint 2 V = n λΛnΛN . (17) Eq. (11). Note that here and in the following we are h i | | omitting uncalculable prefactors of order one. We will All scales are generated by dimensional transmutation. consider low-energy fluctuations on the baryonic branch The only small parameter, λ, is dimensionless. of this theory, where the constraint is satisfied with 2 The need for a small marginal parameter is common in BB = (ΛN ) and det M = 0. The K¨ahler potential − models which use the ISS mechanism in SUSY-breaking for such fluctuations is approximately canonical [9]. models without scales [7, 8]. To gain some intuition on Fore λ of order one, q andq ˜ will decouple around the a realistic upper bound on λ, let us construct a limiting scale ΛN and the gauge degrees of freedom will form case for which our approximations can still be considered a pure super-Yang-Mills theory which does not break reliable. Take for instance n = 8, N = 10, and assume SUSY. We instead choose λ sufficiently small, such that that the SU(n) coupling is gn 1 at the scale ΛN ; with λΛN Λn, so q andq ˜ remain as light flavours of SU(n). ≈ ≪ this value SU(n) is still perturbative but on the brink The SU(n) gauge coupling will now become strong at the of strong coupling. With gn(ΛN )=1.1 and the naive lower scale Λn. Since we chose n and N in the range 3 one-loop estimate n 1 i1...in j1...jN−n c1 cn 1 N−n We are now in a position to couple our model to the det M = ǫ ǫc1...cn q q p˜ p˜ n! i1 ··· in j1 ··· jN−n visible sector. There are many models of direct gauge- (15) mediated or messenger gauge-mediated SUSY breaking, 5 SU(N − n) SU(N) SU(N) SU(N − n) SU(N − n) SU(N − n) SU(N) SU(N − n) Qe 1 1 M 1 1 ⊗ 1 Q 1 1 η˜ 1 1 χ 1 1 η 1 1 χ˜ 1 1 ζ 1 1 Pe 1 1 χ 1 1 ρ 1 1 ρ˜ 1 1 TABLE V: The field content for Λn > ΛN , after a Seiberg duality transformation on SU(n). As before, the first two σ 1 1 columns represent gauge symmetries and the last two columns σ˜ 1 1 represent global flavour symmetries. TABLE VI: Fields and symmetries after a second Seiberg relying on ISS-like metastable vacua (for some examples, duality transformation, now on SU(N). Only the first two see [14]). It would be especially interesting to see if the SU(N −n) factors are gauged (but weakly coupled in the IR). common problems of direct gauge mediation (such as the gaugino mass problem and the Landau pole problem) 3 can somehow be overcome in an extension of our model, X Z possibly even while preserving its nice UV properties. For now we leave this issue for future work. 1 2 It is instructive to see how the model fails to give rise Y to a metastable vacuum in the case Λn > ΛN . Below the scale Λn we should use the Seiberg dual of SU(n), which is a SU(N n) magnetic gauge theory. It contains FIG. 1: The F = f = N = n quiver. Circles with label i − represent SU(N)i symmetries, and arrows represent bifunda- a meson Q = Φ˜q/Λn and two dual quarks which we call mental chiral superfields. Node 3 may or may not be gauged. again χ andχ ˜. The full field content is listed in Table V. The superpotentiale becomes, up to Affleck-Dine-Seiberg terms generated by the dual gauge dynamics, the linear term in the meson) does not have full rank in flavour space, however, since there is no linear term Weff = χQχ˜ + λΛn QQ. (19) for ζ. As mentioned in Sect. II, this will destabilize the SUSY-breaking point at two-loop level [11]. If λ is of order one, thene Q and Q decouplee supersym- metrically at the scale Λn. The remaining fields form an SU(N n) SU(N) gauge theorye with massless matter IV. THE UNCALCULABLE CASE F = f = N = n and no− superpotential,× which does not break SUSY. Considering instead again the case that λ 1, so ≪ We next turn to the case F = f = N = n, adapt- that Q and Q are kept light up to scales < Λ , the N ing our notation slightly for convenience. The non- theory becomes effectively SQCD with N colours and abelian symmetry group is SU(N)1 SU(N)2 SU(N)3. 2N n flavourse of quarks Q,χ ˜ and antiquarks Q, × × − The third factor arises as the diagonal subgroup of the P . It is in the free magnetic range, which is eas- 3 SU(F ) SU(f) in Table I which is preserved by W . Since ily checked to follow from n ˜ 2 b = b =0, M =Λ2 ½, BB = Λ1. (24) − Acknowledgements Even if the conjecturedf metastablee vacuum exists, it should somehow be prevented from decaying into the true vacuum too quickly for the model to be viable. I would like to thank Mark Goodsell for useful dis- It is amusing to note that one may recover the model cussions, and Joerg Jaeckel for helpful comments on the of Sect. III from the model of this Section, by breaking manuscript. 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