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Decoupling and Destabilizing in Spontaneously Broken August 2007 Decoupling and Destabilizing in Spontaneously Broken Supersymmetry Jonathan A. Bagger and Adam F. Falk Department of Physics and Astronomy, The Johns Hopkins University 3400 North Charles Street, Baltimore, Maryland 21218 The supersymmetric analog of the Goldberger-Treiman relation plays a critical role in the low energy effective theory of models in which supersymmetry is spontaneously broken in a hidden sector. The interactions that connect the hidden and visible sectors break a global symmetry, which implies that the low energy theory must be constructed consistently in inverse powers of the messenger scale. The Goldberger-Treiman relation determines the couplings of the Goldstino to the visible sector fields. These couplings are fixed by the soft supersymmetry breaking terms within a power counting scheme that is stable under radiative corrections. We describe the power counting of the low energy effective theory, first for a toy model of extended technicolor and then for the supersymmetric standard model. One implication of this work for supersymmetry phenomenology is the observation that Goldstino loops can destabilize the weak scale if the low energy theory is not constructed consistently. Another is that Goldstino loops induce all visible sector operators not forbidden by symmetries. The magnitudes of these operators are determined by the consistent power counting of the low energy effective theory. PACS numbers: 11.10.-z, 11.30.Pb, 12.60.Jv I. INTRODUCTION generally the low energy effective theory of the visible sector in models with hidden sector symmetry breaking. The primary purpose of this paper is to make an ele- We assume that the symmetry breaking at a high scale f mentary but essential observation about the qualitative is communicated to the visible sector through nonrenor- features of models with spontaneously broken supersym- malizable interactions suppressed by an even higher scale metry in a hidden sector. We will argue that the visible M. We will argue that because the global symmetry of the (visible + hidden) theory is enhanced as M , sector soft supersymmetry breaking terms must obey a → ∞ self-consistent power counting if the weak scale is not to one can consistently count powers of 1/M in the visible be destabilized. This power counting is the one that fol- sector, even below the scale f at which dynamics in the lows from the fact that an explicit symmetry-breaking hidden sector may become strongly coupled. As a conse- parameter controls all communication between visible quence, in the low energy theory of the visible sector the and hidden sector fields. Because of the Goldberger- ultraviolet cutoff may be taken as high as M, while the Treiman relation [1], the same power counting simultane- coefficients of nonrenormalizable operators, suppressed ously governs both the soft symmetry breaking terms and by powers of 1/M, may depend on nonperturbative hid- the Goldstino interactions. We will show that Goldstino den sector dynamics or logarithms of f/M. These terms loops enforce the correct power counting by introducing look, from a visible sector point of view, like match- destabilizing divergences in an incorrectly constructed ing terms generated at M, but they actually depend on physics at the lower (but still high) scale f M. low energy theory. We will also see that Goldstino loops ≪ induce all operators in the low energy effective theory not Because these more general features are not particular forbidden by symmetries, with coefficients determined by to supersymmetry, we will first discuss them in the some- the consistent power counting. what simpler context of a toy model of extended techni- arXiv:0708.3364v1 [hep-ph] 24 Aug 2007 An immediate consequence of our work concerns the color. We will then apply the lessons learned to super- minimal supersymmetric standard model (MSSM). In symmetry, and conclude with an analysis of the MSSM. Refs. [2, 3] it was noted that nonholomorphic scalar tri- linears can be added to the soft supersymmetry breaking lagrangian with weak scale coefficients, II. EFFECTIVE FIELD THEORY OF AN EXTENDED TECHNICOLOR MODEL ∗ ¯ ∗ mW quh¯ d + mW qdhu , (1) where hu and hd are Higgs fields, and q, d¯ andu ¯ are This paper focuses on power counting in models with squarks. According to the usual power counting, these hidden sector supersymmetry breaking. Nevertheless, terms are soft, so they should not destabilize the hierar- almost all of the relevant physics can be understood chy. In this paper we show that this argument is correct, by studying a simple toy model of extended technicolor but incomplete. When Goldstinos are included, the coef- (ETC). We will develop this example in considerable de- ficients of the nonholomorphic trilinears must be of order tail and map the conclusions of our discussion onto su- 2 mW /M (where M is the messenger scale), or Goldstino persymmetry at the end. loops will destabilize the hierarchy. The ETC model has a “visible sector” consisting of A secondary purpose of this paper is to explore more quark fields QL and QR, transforming under a global 2 SU(2)LQ SU(2)RQ: in the techniquark sector, communicated by a nonrenor- × malizable interaction. Note that breaking the full sym- QL LQQL , QR RQQR . (2) metry SU(2) SU(2) SU(2) SU(2) → → LQ × RQ × LT × RT → SU(2)L SU(2)R SU(2) requires both symmetry It also has a “hidden sector” with techniquarks TL and × → breaking parameters: the condensate T LTR and the TR, transforming under a separate global chiral symme- spurion α/M 2. h i try SU(2) SU(2) , LT × RT Expanding the interaction (6) to linear order in Π, T L LT T L , T R RT T R . (3) m → → mQ Q + i Q ΠQ + ..., (7) L R f L R The two sectors are coupled by a nonrenormalizable con- tact term that arises from extended technicolor at a high we see that the chiral symmetry breaking mass term is scale M, related to the Goldstone boson coupling via m = gπQQf. α This is the famous Goldberger-Treiman relation. It fol- Q T T Q . (4) M 2 L L R R lows solely from the fact that the chiral symmetry is bro- ken spontaneously. Note that the Goldstone-quark cou- The parameter α/M 2 may be viewed as a “spu- pling is suppressed by m/f f 2/M 2. rion,” in the sense that it is a constant that carries It is instructive to consider∼ the power counting in SU(2)4 quantum numbers. The ETC interaction (4) this theory from a top-down, “Wilsonian” point of view. breaks SU(2) SU(2) SU(2) and SU(2) From this perspective, one starts with a theory defined LQ × LT → L RQ × SU(2)RT SU(2)R, leaving the SU(2)L SU(2)R sub- at short distances and then progressively integrates out group intact.→ We put scare quotes around× “spurion” be- modes of longer and longer wavelength to arrive at an cause we do not necessarily view α as representing the effective theory at lower energies. At each energy scale, vacuum expectation value of a field. This spurion (ac- the relevant symmetries are those that are unbroken at knowledging this possible abuse of terminology, we now that scale. The natural sizes of the operator coefficients drop the quotes) carries a power counting dimension of are just the magnitudes that are generated by this proce- 1/M 2; most importantly, it alone controls the SU(2) dure. Delicate cancellations between coefficients present LQ × SU(2)RQ SU(2)LT SU(2)RT SU(2)L SU(2)R at short distances and corrections generated by Wilso- symmetry× breaking in× the low energy→ effective× theory. nian running are considered to be unnatural. At some scale f, the technicolor interaction becomes Above the ETC scale M, we assume that we have a 3 strong and the techniquarks condense, T LTR = f . (In theory in which the techniquark and quark fields inter- this paper we are interested only in theh generali question act with each other freely. (The theory could be non- of power counting the mass scales present in the theory, renormalizable, but any nonrenormalizable operators at so we drop all factors of 4π. Of course, the correct count- that scale give even smaller contributions at low energies ing of such loop factors may be phenomenologically im- than the ones we are interested in.) When we integrate portant in a particular theory.) This condensate breaks out modes at and above the scale M, including the ETC the SU(2)LT SU(2)RT symmetry to its diagonal sub- sector of the theory, we generate a tower of operators sup- group, and produces× three Goldstone bosons in the ef- pressed by powers of 1/M. Below M, these nonrenormal- fective theory below f. The product T LTR is replaced izable operators provide the only interactions between by a composite field containing the Goldstone multiplet the quark and techniquark sectors. Π= πaT a, We now integrate out momenta between M and the lower scale f. All modes have energies below M, so no T T f 2Σ= f 3eiΠ/f . (5) L R → powers of M are induced in the numerators of the op- erator coefficients. Thus the organization of operators This is in accord with the general construction of Callan, according to power counting in 1/M is preserved. Of Coleman, Wess and Zumino [4, 5], which follows sim- course, this is the standard behavior of an effective field ply from the structure of the spontaneously broken sym- † theory. metry. The field Σ transforms as Σ LT ΣRT under The final step is to run the theory from f to m SU(2) SU(2) .
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