Challenges for the Nelson-Barr Mechanism
PPrepared for submission to JHEP
Challenges for the Nelson-Barr Mechanism
Michael Dine a and Patrick Draper b,a a Santa Cruz Institute for Particle Physics and Department of Physics, University of California, SantaCruz, CA 93106, USA b Department of Physics, University of California, Santa Barbara, CA 93106, USA
E-mail: [email protected] , [email protected] Abstract:
The axion and m u = 0 solutions to the strong CP problem have been subjectto the most careful scrutiny and critique. Basic theoretical issues include hierarchy and fine-tuning problems, quality and genericity of symmetries, and compatibility with solutions tothe electroweak hierarchy problem. We study the similar set of challenges for solutions tostrong CP based on spontaneous CP violation and the Nelson-Barr mechanism. Some of ourobservations have appeared in the literature previously, and others are new; our purpose isto collect and analyze the issues as a whole and provide an assessment of the most plausiblesettings for the Nelson-Barr solution. ArXiv ePrint: a r X i v : . [ h e p - ph ] O c t ontents θ in Non-Supersymmetric Theories 63.3 Models with Strong Dynamics 8 θ at Tree Level 95 SUSY Nelson-Barr Models 12 Z N in SUSY 125.1.1 CP broken by Supersymmetry-Conserving Dynamics 125.1.2 Theories with Flat Directions 135.2 Breaking of Supersymmetry 145.3 Loop Corrections in Supersymmetric Theories 15 There are various naturalness problems of the Standard Model (SM), including the cosmolog-ical constant problem, the hierarchy problem, the hierarchies in the quark and lepton massmatrices, and the strong CP problem. Of these, the last is special. Even modest changes inthe cosmological constant would drastically alter the world around us. Similarly, the valuesof the weak scale and the light quark and lepton masses play critical roles in a range of phe-nomena. But if the CP-violating parameter ¯ θ were, say, 10 − , there would be no appreciablechange in nuclear physics.Theorists may put forward complicated explanations for the smallness of ¯ θ , with manyadditional degrees of freedom, complicated symmetries, and some amount of fine tuning, butthis activity is not particularly satisfying. More compelling would be a theory in which thethe smallness of ¯ θ emerged as an accidental consequence of other structure in a physicaltheory: an explanation of flavor or dark matter, for example. We will refer to this (presentlyhypothetical) phenomenon as incidental CP conservation .Most attention has focussed on three solutions to the strong CP problem: the possibilityof a massless up quark, the Peccei-Quinn (PQ) solution with its associated axion [1, 2], and– 1 –pontaneous CP or P violation with a protection mechanism for ¯ θ [3–13]. The first twosolutions require that the theory possess an approximate U (1) symmetry, the violation ofwhich is primarily due to the QCD anomaly. If the symmetry is not spontaneously brokenat scales above the QCD scale, there must be one or more very light quarks. This is usuallystated as the requirement that the u quark mass vanishes, but the more precise statementis that at scales beyond a few GeV, m u m d < − . Apart from any theoretical issues, thepossibility of a massless quark is strongly disfavored by lattice calculations [14]. If the chiralsymmetry is nonlinearly realized, there is a light axion [1, 2]. The potential for this axiondetermines ¯ θ .The third proposed solution is that CP or P is spontaneously broken and ¯ θ is protectedby extra structure [3–7], the most common example of which is the Nelson-Barr (NB) mech-anism [8–10] in the case of spontaneous CP violation. Since the underlying theory is CP-conserving, the “bare” ¯ θ parameter vanishes. CP must then be spontaneously broken in a waythat ensures a small effective ¯ θ while allowing an order one phase in the CKM matrix (and amechanism for baryogenesis) [3–10]. The NB proposal is striking in that it seeks to solve thestrong CP problem with no low energy consequence, unlike the axion and m u = 0 solutions.On the other hand, in this paper, we will see some relations between these proposals. Setting aside the possibility that m u = 0 leaves the PQ and NB proposals. As currentlyimplemented in an array of models, neither is completely satisfactory from a theoretical pointof view; certainly neither is obviously incidental in the sense defined above. For the PQsolution, the theoretical problems have been extensively discussed, and we will review someof the issues. The primary focus of this paper will be the challenges to obtaining a plausibleimplementation of the NB solution. In both PQ and NB, the inadequacies of current proposalsconcern the structure of the microscopic, ultraviolet theory and particularly the complexityand plausibility of the structures necessary for an effective solution.1. The principal difficulty with the axion mechanism is that the PQ symmetry needs to beof very high quality. If this symmetry is an accident, it must be a remarkably good one.If the symmetry and its breaking are described by a conventional effective field theory,the required quality can be achieved with a Z N symmetry, but requires N ≥
11 or so.This is hardly a compelling explanation for the smallness of an inconsequential param-eter of the Standard Model. In string theory, the situation for light axions appearsbetter, but a solution in this framework requires assumptions about the stabilization ofmoduli which, while perhaps imaginable, at least at present are impossible to verify. Inthe string framework, one must also hypothesize an unconventional cosmology and typ- In the interesting alternative case of spontaneous parity violation, models and their criteria for successwere discussed in [11–13]. Another mechanism in the case of spontaneous CP violation, distinct from NB,involves the introduction particular “shaping symmetries” in the underlying flavor structure [15]. Other solutions [16–18] possess close similarities to the solutions with approximate U(1)s [16, 17] or NB [18]. In [19], the possibility that N is large in order to account for dark matter was considered. It was shownthat dark matter can account for a large value of N , but not large enough to solve the strong CP problem. – 2 –cally some tuning of initial conditions, unless the axion decay constant is surprisinglysmall.2. As we will elaborate in this paper, the NB mechanism is generically on even weaker the-oretical ground. If the implementation is not massively fine-tuned, it requires strong dy-namics or supersymmetry (though not necessarily at scales of order a few TeV). Strongdynamics are insufficient to protect small ¯ θ in the simplest models, and supersymmetricmodels require gauge mediation ( m / (cid:28) splittings in supermultiplets). In addition,new discrete or gauge symmetries and strong coincidences of scales are necessary, aswell as a number of degrees of freedom beyond those required by supersymmetry.Instead of such speculative exercises, one can hope for an experimental resolution. Thediscovery of an axion would, needless to say, answer the question. However, a large part ofthe axion parameter space is currently inaccessible. For the NB solution, there is no similar“smoking gun.” While we will argue that gauge mediation is a requirement, the scale neednot be particularly low.This paper is organized as follows. In Section 2, we review the basic structure of thefermionic sector of NB models. In Section 3, we discuss non-supersymmetric models. Ifsuch models contain fundamental scalars, one would expect the scale of CP violation to behigh in order to limit the fine-tuning. However, constraints imposed by dangerous higher-dimension couplings require a low scale of CP violation, implying enormous fine tuning.Although compositeness can explain the required hierarchy, we argue that the simplest modelstypically fail to retain the necessary NB structure. Setting the fine-tuning issue aside, wediscuss the sorts of symmetries which might ensure vanishing ¯ θ at tree level, and discuss thedangerous radiative corrections to ¯ θ that can arise at one and two loop order. In Section 4we turn to supersymmetry. In theories for which supersymmetry is broken well below some“fundamental” ultraviolet scale (perhaps the Planck, string, or compactification scale), wecan pose more sharply the question of what it means for the bare θ to vanish. We argue thatin practice there is a heavy axion, and thus a sense in which the supersymmetric NB andPQ models can be considered as different limiting cases of axion models. We discuss how theexpectation value of this axion might be fixed and constraints on couplings of the axion topossible CP-violating sectors. We also note that very simple landscape considerations suggestthat vanishing of the “bare θ ” in such frameworks is extremely rare, and these is no obviousanthropic selection effect one might invoke. Finally, we discuss the spontaneous breakingof CP and SUSY and the radiative corrections to ¯ θ in supersymmetric models with gravityand gauge mediation. In gravity mediation, corrections are typically large and spoil the NBsolution. In gauge mediation, the corrections can be smaller, but there are upper bounds onthe ratio of the susy-breaking scale to the scale of CP violation. In Section 6 we summarizeand conclude. – 3 – The Essence of the Nelson-Barr Mechanism
The main challenge in solving the strong CP problem with spontaneous CP violation is tounderstand why Arg det m q < − , (2.1)while there is a large phase in the CKM matrix. Nelson [8] and Barr [9, 10] obtained thefirst simple, phenomenologically viable models which achieve this and elucidated the generalproperties of renormalizable Lagrangians that can exhibit Arg det m q = 0 at tree level.A model with minimal field and symmetry content was obtained by Bento, Branco, andParada (BBP) [20], and serves as a useful starting point for understanding the properties ofthe NB mechanism. The BBP model introduces additional charge ± / SU (2) singlet quarks q , ¯ q , as well as a set of complex fields η a neutral under the SM (we will comment on real fieldslater). The down-type quark mass terms in the BBP model are given by L = µ ¯ qq + a af η a ¯ d ¯ f q + y f ¯ f HQ f ¯ d ¯ f + . . . . (2.2)The η a are assumed to have vevs with relative phases, breaking CP. At tree level, the Lagrangian in (2.2) automatically gives Arg det m q = 0 for the quarkmasses. However, it is not the most general renormalizable Lagrangian allowed by the symme-tries of the SM. Couplings of the form η a q ¯ q and HQ ¯ q must be forbidden. Similarly, we mightlike µ to be the expectation value of a CP-conserving field, which constrains its interactionswith the η a . Discrete symmetries can provide the necessary structure, and we return to thisissue in the next section.The CKM phase in the SM is generated by integrating out the heavy flavor from (2.2).Defining the 4 × M = (cid:32) µ B m d (cid:33) ; m d ≡ yv ; B f = a af η a , (2.3)we need to diagonalize the matrix MM † = (cid:32) µ + BB † Bm Td m d B † m d m Td (cid:33) . (2.4)If the left hand corner of this matrix is larger than the other entries, we can integrate out the In fact, in the original BBP model [20], only a single complex field is introduced with Yukawa couplings( a f η + a (cid:48) f η ∗ ) ¯ d ¯ f q . This structure is sufficient as long as a f and a (cid:48) f are nonzero, a f (cid:54) = a (cid:48) f , and a required discretesymmetry under which η , q , and ¯ q transform is a Z instead of a more general Z N . We consider the form ofEq. (2.2), with multiple η a and vanishing a (cid:48) fa , anticipating possible Z N symmetries as well as the extension ofthe BBP model to supersymmetry. – 4 –eavy state, leaving the 3 × (cid:32) ( m d m Td ) ij − ( m d ) ik B † k B (cid:96) ( m Td ) (cid:96)j µ + B f B † f (cid:33) . (2.5)The diagonalizing matrix is the CKM matrix. Note that this procedure is correct only in thelimit µ + | B f | (cid:29) m d ; otherwise, the CKM matrix is not unitary.Obtaining a large CKM phase strongly constrains the parameters. If there is only onenon-vanishing B f , or if each B f has the same phase, or if µ (cid:29) | B f | , then the CKM matrixis real. However, if there are two distinct, non-vanishing B f of comparable magnitude andwith a large relative phase, and µ (cid:46) | B f | , there is a non-trivial phase. For example, if B = (0 , b, c ), a phase of order Im(b / c) enters the CKM matrix. We see that a rather closecoincidence of scales is required between the real and imaginary parts of different fields. Thesevere challenges for non-susy NB theories will be discussed in the next section. In this section we consider nonsupersymmetric Nelson-Barr models. We begin with a surveyof the basic issues and challenges confronting such models already at tree level, and thenelaborate on two of the issues that arise when radiative corrections are included.
Without supersymmetry, it is a simple matter to construct models of spontaneous CP viola-tion. We can, for example, introduce two real fields, σ and π , the first CP-even and the secondCP-odd, with appropriate NB-type couplings to fermions and a potential that leads to a vevfor each. Likewise with complex fields it is not difficult to spontaneously break CP, if thereis sufficient freedom in the specification of the scalar potential (for a principled discussion ofnecessary and sufficient conditions, see [21].However, NB models, to be viable, must confront several theoretical challenges:1. Further symmetries are necessary to enforce the necessary structure of the mass matrix,even at the renormalizable level. In the BBP model discussed in the previous section,since µ (cid:46) |(cid:104) η a (cid:105)| , it is necessary suppress or forbid dimension-4 couplings of the form η a q ¯ q . Likewise we must suppress HQ ¯ q . One possibility is to allow the new scalarsand fermions to transform under a Z N symmetry (if N >
2, then the scalars must becomplex, as in the model discussed above): η a → e πikN η a , q f → e − πikN q f , ¯ q f → e πikN ¯ q f . (3.1)With other fields neutral, we obtain a Lagrangian of the desired form. It is not difficultto write down models which spontaneously break both CP and the Z N . We will discusspossible gauge symmetries when we consider supersymmetry in the next section.– 5 –. The scale of spontaneous CP breaking m CP should be low compared to the cutoff Λ.Dimension-5 operators such as η ∗ a η b ¯ qq , η a HQ ¯ q (3.2)for example, can induce ¯ θ of order ( m CP / Λ). Note that the Z N symmetry defined inEq. (3.1) (or possible U (1) symmetries) does not help to suppress higher-dimensionoperators like (3.2). Without further symmetries or fine-tuning, even if the cutoff isΛ = M p , suppression of such operators requires m CP (cid:46) GeV . (3.3)3. As in any non-supersymmetric or non-composite model, light scalars are fine-tuned.Here we require at least two such scalars at a scale m CP (cid:28) M p , and the fine-tuning of each of these masses is much worse than just fine-tuning ¯ θ by itself. It is difficult tomake sense of NB models outside of a broader framework in which m CP /M p is naturallysmall.
4. As we have seen in the previous section, to obtain a substantial CKM angle, it is criti-cal that the expectation values of different CP-odd and CP-even fields (times suitablecouplings) coincide to better than an order of magnitude.5. We might want to account for µ dynamically, i.e. through the expectation value ofa fundamental or composite field S . Additional symmetries need to be introduced toavoid inducing phases in S from couplings of S to the η a .6. Even when it vanishes at tree-level, ¯ θ is often generated radiatively at the scale m CP .Loop effects are particularly problematic. They cannot be suppressed simply by addi-tional (bosonic) symmetries or by lowering the scale of CP violation. These corrections willbe the subject of the next section. θ in Non-Supersymmetric Theories Even if one closes one’s eyes to fine tunings, and one is willing to accept a low scale forCP violation, loop corrections are quite problematic in NB models. Threshold correctionsto ¯ θ have to be considered on a model-by-model basis, but certain operators are typicallyproblematic. BBP studied ¯ θ at one loop in [20]. Below, we review and reinterpret their result,and observe further problematic contributions at two loop order. We will see that the oneloop sensitivity of m CP to the UV cutoff requires us to add structure, such as supersymmetryor a dynamical origin for the scalars, and then to consider all of the other issues in that largerframework. In the subsequent section we discuss composite models and see that while thefine-tuning of m CP can be resolved, simple cases will either have difficulty maintaining ¯ θ = 0– 6 – H i h η a ih η b i Q i ¯ d qH η ¯ d j Figure 1 : Example threshold correction to Arg det m d .at tree level, or will have one loop corrections to ¯ θ similar to non-composite models. This willlead us to consider NB in the supersymmetric context.In the BBP model, dangerous contributions to ¯ θ arise at one loop from the Higgs portaloperators ( γ ij η † i η j + λ ij η i η j + cc ) H † H . (3.4) λ ij can be forbidden by a Z N symmetry with N >
2, so we consider the effects of γ ij . Unlessthe γ s are very small, these couplings make a large contribution to the Higgs mass. In thecontext of a solution to the m CP hierarchy problem, there might or might not be a principledreason why the couplings are small, but a priori they indicate only another contribution ofmany to the tuning of m H . At one loop, the diagram of Fig. 1 gives a complex correction tothe SM down-type Yukawa coupling, contributing to a shift in ¯ θ of order∆¯ θ (cid:39) Im Tr y − ∆ y (cid:39) η a a af a bf γ bc η ∗ c π m CP . (3.5)Adequately suppressing ¯ θ requires the a and/or γ couplings to be small.The authors of [20] took the viewpoint that whatever solves the SM hierarchy problemmight suppress the portal couplings. Such suppressions can occur in supersymmetric orcomposite theories (both of which solve the m CP hierarchy problem, but not necessarily thefull m H one). These theories involve significant extra structure beyond the minimal BBPmodel, and the radiative corrections to ¯ θ must be considered in the full theories. Withoutsupersymmetry or extra dynamics, the Higgs mass is simply tuned, and small θ is problematic.At two loop order, there are additional contributions which must be suppressed. In– 7 – q qµ h η ∗ a i h η b i Figure 2 : Example two-loop contribution to the phase of µ .particular, insertions of the operator L η = γ ijkl η i η j η ∗ k η ∗ (cid:96) (3.6)can contribute phases to the operators µ ¯ qq and QH ¯ d . The relevant Feynman diagrams containa loop of gauge bosons and an η loop, with insertions of L η ; an example is given in Fig. 2(this contribution is similar to the “dead duck” graph noted in [8]). The contribution to ¯ θ isof order ∆¯ θ (cid:39) g a af a cf η ∗ b η d γ abcd (16 π ) m CP (3.7)Again, unless the couplings are surprisingly small, the correction is several orders of magnitudeto large. In the supersymmetric case, we will see that these contributions can be suppressed,but new issues will arise. The low scale of CP violation may be protected by strong dynamics. For example, the CP-oddscalars could be pseudogoldstone mesons Π of an SU(N) gauge theory in which condensatesspontaneously break approximate chiral flavor symmetries, (cid:104) ¯ ψ i ψ j (cid:105) = Bf exp( i Π a t a /f Π ) , (3.8)in analogy with the pions of QCD. The Π fields can obtain nonzero vevs naturally from aparticular pattern of chiral symmetry breaking (as in, e.g., Dashen’s model [22]). In this case,BBP-type couplings to the Standard Model and the q ,¯ q messengers (assumed for now to befundamental fermions) might arise from higher-dimensional operators of the form1Λ κ fij ¯ ψ i ψ j ¯ d f q/ Λ → B f Λ Tr (cid:104) κ f e i Π a t a /f Π (cid:105) ¯ d f q + . . . . (3.9)If the hierarchy between the scale of the gauge theory ∼ f Π and the UV cutoff Λ is large, theeffective couplings a af in Eq. (2.2) may be very small, and the effective scale of CP violationmuch smaller than f Π . We can see from the form of Eq. (2.5) that the CKM phase can– 8 –till be large if µ is sufficiently small. Furthermore, the one loop BBP radiative correction– generated here by couplings of the form H † H ¯ ψψ/ Λ – is suppressed when the effective a af couplings are small.Unlike in the fundamental scalar case, however, it is difficult to implement discrete sym-metries needed to keep µ real. Permitting (3.9) while forbidding the similar 4-fermi operator¯ ψψ ¯ qq requires the discrete symmetry to act chirally on ψ , ¯ ψ (and, for example, on q ,¯ q ), butexplicit chiral symmetry breaking is necessary to generate the spontaneous CPV potentialwhen the CP-odd scalars are pseudogoldstones. This breaking might be soft, as in a set ofmasses m for the ψ , ¯ ψ , and thus the coefficient of ¯ ψψ ¯ qq/ Λ might be suppressed by m/ Λ.But if m is not too different from f Π , then f Π / Λ must be less than 10 − , resulting in anunacceptably low value for m CP .It is even more difficult to understand the NB structure and the reality of the effective µ if the messenger fields q ,¯ q are baryons of the gauge theory. In this case the baryon mass is ex-pected to arise principally from spontaneous chiral symmetry breaking, which by constructionbreaks CP.We stress that it is not impossible to build NB-type models with strong dynamics, but itrequires more complicated structures. A minimal example was constructed in Ref. [23], con-sisting of a BBP-type model in which the ¯ ψψ ¯ qq operator is forbidden by a gauged subgroupof the chiral flavor symmetry. This symmetry might also be discrete. The Dashen mass termsare forbidden by the symmetry, but the potential can still break CP with suitable dimension-6operators ( ¯ ψψ ) . Ref. [23] also showed that models with acceptably small radiative correc-tions to ¯ θ could be distinguished by the flavor transformation properties of the CPV spurionspresent in the low-energy theory. BBP-type models with generic couplings possess CPV spu-rions in the infrared in both the fundamental and anti-fundamental representations of SU (3) d ,and as such they fail the criteria of [23]. This is reflected in the large one loop correctionto ¯ θ . However, when the couplings a af are small, as can arise in strongly-coupled modelsas discussed above, the low-energy theory contains only an SU (3) d -fundamental spurion andthe criteria for small corrections to ¯ θ are met. θ at Tree Level Supersymmetry, with SUSY breaking at scales well below the scale of CP violation, can signif-icantly ameliorate the Nelson-Barr fine-tuning problem. In addition, SUSY can forbid some ofthe problematic higher-dimension operators and quantum corrections to ¯ θ encountered in thenon-SUSY case. In this section, we consider supersymmetric Nelson-Barr models and theirsymmetries. We first review some of the problematic aspects of the Peccei-Quinn solution ofthe strong CP problem and their possible resolution. Then we consider more carefully theunderlying premise that CP can naturally be a good symmetry, and as a result that the bare¯ θ vanishes. In both cases the questions are ultraviolet-sensitive and the resolutions dependon the structure of the microscopic theory. In particular, if there is an underlying landscape,small bare ¯ θ is implausible. – 9 –e first review some aspects of the axion solution, with and without supersymmetry. Themost challenging aspect of the Peccei-Quinn solution of the strong CP problem is understand-ing why the global symmetry is so good. Global symmetries should arise only as accidentsof gauge symmetry and the structure of low dimension terms in an effective action. It wasquickly recognized that this is a challenge for the PQ mechanism [24]. From a PQ-violatingpotential V pqv , we can define an axion quality factor, Q a ≡ f a ∂V pqv ( a ) ∂a m π f π . (4.1)Solving the strong CP problem requires Q a < − . (4.2)In a conventional effective field theory analysis (i.e. finite number of degrees of freedom above f a ), small Q a is highly non-generic. If the axion arises as the phase of a field Φ, (cid:104) Φ (cid:105) = f a e ia/f a , (4.3)symmetry violating operators like Φ n +4 M np (4.4)spoil the PQ mechanism even for f a = 10 GeV unless n >
7. Such suppression can beobtained with a discrete Z N symmetry, with N ≥
11, but such a model appears contrived.Witten pointed out early on that string theory provides a possible resolution to theproblem of the quality of the PQ symmetry [25]. This is most easily understood in theframework of supersymmetry. Typically string models possess moduli, Φ, whose imaginarycomponent obeys a discrete shift symmetry:Φ = x + ia ; a → a + 2 π (4.5)This symmetry guarantees that any superpotential is a function of e − Φ at large x . Here x might be π g , for some gauge coupling g .In this setting, the primary question is why the theory sits in an asymptotic region of themoduli space where e − x is very small. It is consistent at least with the fact that the observedgauge couplings are small, but a detailed connection is not possible at present, much lessreliable computations [26].We turn now to theories where CP is a symmetry of the microscopic dynamics. Herewe can make a connection with string axions discussed above. In known string theories, CPis a good symmetry [27–29]. For typical string compactifications, this statement means thatthere is a subspace of the moduli space on which CP is conserved, and CP is spontaneously– 10 –roken on the rest. In supersymmetric theories, the moduli fields include both a CP-evenand a CP-odd scalar, as in Eq. (4.5), and we will refer to them as saxions x i and axions a i ,respectively. We can define a i = 0 as the CP conserving point. CP is spontaneously brokenif some of these axions are stabilized at a i (cid:54) = 0. Generally one or moduli couple to each ofthe gauge groups in the classical theory, providing candidate axions. The question of whetherthere is a non-zero θ is then a question of whether the relevant axions are heavy and fixed atCP conserving points.If the moduli are stabilized supersymmetrically, the CP-even and CP-odd states are fixedtogether. Suppose that we have a single modulus, with W = − αe − Φ /b + W ; K = − log(Φ + Φ † ) . (4.6)with W small, as in the KKLT scenario[30]. ThenΦ ≈ b log( W /α ) . (4.7)Provided W and α are real, Φ is real. If Φ couples to the QCD gauge fields as Φ W α , itgenerates no tree-level contribution to θ . Plausibly, if W is large, CP remains unbroken, andΦ is very heavy.Should W be real? If we assume W results from CP-conserving dynamics, it is automat-ically real. On the other hand, flux landscapes provide a model where complex W appearsmore likely. In such cases W is the sum of many contributions associated with many differentfluxes, of which we expect about half to be CP-even and half to be CP-odd. CP preservationamounts to requiring half of the of the fluxes to vanish. In other words, given 10 states,only 10 conserve CP and have vanishing W , and correspondingly CP-conservation appearsvery non-generic. Moreover, as noted earlier, it is hard to see what might select for small θ .However, absent a sharp UV prediction for W , we can simply take its reality as a requirementof the NB setup.We can ask what may happen when we introduce a sector in which CP is spontaneouslybroken with characteristic scale µ . If this sector does not break supersymmetry, we mightexpect additional, CP-violating terms in the superpotential of order µ e − S . These terms willshift the minimum of the axion field, but their contribution is suppressed if b is large. If, forexample, e − S < − and b = 5, then θ < − . Alternatively, if b = 1, the contributionto θ is suppressed by at least ten order orders of magnitude provided the scale µ is at leastthree orders of magnitude below M p . In non-supersymmetric models (e.g. cases where thescale of SUSY-breaking is (cid:29) µ ) with axions, one would expect the difficulties to be at leastas severe; it is not clear in such contexts that terms violating the Peccei-Quinn symmetrymust be exponentially small.The assumption that W is real constrains a combination of the supersymmetry breakingand CP violating scales. In particular, we might expect CP violation to generation a complexterm in the superpotential, W ∼ µ CP . If there is no suppression of the phase, the requirement– 11 –f cancellation of the cosmological constant yields the constraint: µ CP < M / M p . (4.8) In this section, we assume that any would-be axions are massive and fixed in a CP conservingmanner. We then ask what are the requirements on SUSY NB models required to accountfor a very small ¯ θ . The Lagrangian of (2.2) naturally extends to a superpotential: W = µ ¯ qq + λ af η a q ¯ d f + y f ¯ f H d Q f ¯ d f + . . . . (5.1)For the moment we continue to treat µ as a dimensionful constant. While the absence ofundesirable renormalizable interactions like ηq ¯ q and H d Q ¯ q can be technically natural dueto nonrenormalization theorems, they can be forbidden in a more principled way with, forexample, discrete symmetries like (3.1). Again a coincidence in scales among the η a vevs isrequired, as well as µ (cid:46) | λ af η a | .As emphasized above, putting NB into a larger and more natural framework incurs newchallenges. The prime example in SUSY models is that the η a must be sequestered fromthe supersymmetry breaking sector to avoid, e.g., giving phases to the gluino mass, amongother problems [31]. We might expect the SUSY breaking theory to exhibit either an exact(discrete) R symmetry, or at least approximate accidental one. If there is an identifiableGoldstino field, Z (assumed chiral), then couplings of the η a to Z must be suppressed.Replacing µ by a dynamical field S may be desirable and requires further symmetries.For example, it is critical to forbid renormalizable couplings between S and the η a . Z N in SUSY If CP is violated at or below the scale of supersymmetry breaking, the low-energy theory canbe studied in the non-supersymmetric framework of the previous section. Therefore, we focuson CP violation at scales much higher than those of supersymmetry breaking. We will notattempt to be exhaustive, but we consider models that illustrate some of the challenges. Weconsider two classes of models:1. Models in which the CP violating fields are fixed supersymmetrically. Here there is adiscrete set of vacua and all fields have mass of order the scale of CP violation.2. Models in which the CP violating fields are fixed by SUSY breaking dynamics. We takethe scale of CP violation to be much larger than the scale of SUSY breaking; in thissituation, CP is broken by fields in approximate flat directions.
To write a simple model that breaks CP in isolated vacua, we introduce two fields η and η , odd under a Z symmetry, and fields X and Y that are even. We can also suppose an– 12 – symmetry (for simplicity we will take it to be continuous, but it can also be a discretesubgroup) under which X and Y have R charge 2 and the η i are neutral. Then we can takethe superpotential to have the form, without loss of generality: W = Xµ + X ( aη + bη η + cη ) + Y ( a (cid:48) η + b (cid:48) η η + c (cid:48) η ) . (5.2)This superpotential typically has minima in which η and η have phases, breaking CP. If q , ¯ q are both odd under the Z , with R charge 1, and ¯ d f is even, with R charge 1, then weobtain the NB superpotential at the renormalizable level.There are a number of issues with models of this type. In particular, if supersymmetrybreaking is associated with a Goldstino superfield in a hidden sector, Z , these symmetries willnot forbid Zη η couplings, leading to CP violating phases in ordinary soft breaking terms. Z N symmetries with larger N , while forbidding these couplings, require more structure inorder to obtain a superpotential that is both Z N invariant and spontaneously breaks CP (and Z N ).Another model for spontaneous CP violation has been presented in [32]. In addition toa discrete symmetry, the model relies on a continuous global symmetry to suppress couplingswhich would induce θ at tree level. If the U (1) is replaced by a discrete subgroup, at least a Z × Z symmetry is needed to suppress dangerous renormalizable operators. String theory constructions suggest another possibility which can lead rather naturally to theNB structure. There are two elements. First, string models often possess U (1) symmetriesbeyond those of the Standard Model, as well as additional fields, which can yield the requiredsuperpotential for the NB models. Second, there are often approximate flat directions inwhich CP-odd fields can obtain large expectation values. Under suitable conditions, thesevevs may spontaneously break CP.In particular, the gauge group E , familiar in Calabi-Yau compactifications of the het-erotic string, suggests the possibility of two additional U (1)s at some energy scale as well asseveral additional fields. In terms of O (10) × U (1) ⊂ E , the 27 of E decomposes as27 = 16 − / + 10 + 1 − . (5.3)We will treat the theory as if this symmetry is broken to the Standard Model × U (1) × U (1).Then we can list the fields and their charges under the two U (1)s: Q, ¯ e, ¯ u = ( − / , L, ¯ d = ( − / , − q = (1 , q = (1 , − η = ( − / , H = (1 ,
2) ¯ H = (1 , − S = ( − , . Note that the η is essentially the right-handed neutrino of O (10), while the S is the field in E outside of the 16 or 10. q, ¯ q , and (cid:96), ¯ (cid:96) arise from the 10 of O (10). Anomaly cancellation is– 13 –eadily satisfied by including an additional q , ¯ q , (cid:96) , ¯ (cid:96) , η , S for each generation. In addition,we assume that there is one additional S , ¯ S pair and one additional η, ¯ η pair (and allow thepossibility of other incomplete multiplets, particularly for the Higgs field).With these charge assignments, the most general cubic superpotential involving S, η, q, ¯ q and the ordinary matter fields is precisely that of Eq. (5.1). Moreover, at the renormalizablelevel, the classical theory possesses flat directions with non-zero η i , ¯ η, S i , ¯ S .The flat directions may be lifted by supersymmetry-breaking effects and dimension-5operators. If some of the soft masses in the flat directions are negative, some of the fields willreceive large expectation values. If there are quartic superpotential couplings, e.g. M p η i η j ¯ η and M p S i S j ¯ S , then these expectation values are of the order S , η ∼ m susy M p . (5.5)With several fields, there will typically be CP violating minima of the potential.Many problematic higher-dimension operators are forbidden by holomorphy and theU(1)s. However, a surviving class of dimension-5 operators, S i ¯ Sη j ¯ η , must be forbidden toavoid large phases in S . These couplings can be forbidden by discrete symmetries. One virtueof this type of model is that it is compatible with the existence of a (discrete) R symmetry,which can suppress couplings of the η fields to any would-be supersymmetry-breaking sectorand possible messengers.Another potential difficulty is the large size of the η i expectation values. These aresufficiently large that, depending on the scale of supersymmetry breaking and the suppressionscale, they have the potential to induce ¯ θ through dimension-6 operators. We have already noted that supersymmetry breaking introduces new potential contributionsto ¯ θ . Many of these contributions do not decouple, even as the supersymmetry breakingscale is taken arbitarily large. As a result, a successful supersymmetric solution to strongCP requires suppression of phases in the gluino mass, as well as a high degree of degeneracy,proportionality, and suppression of phases in squark masses and A -terms [31], regardless ofthe scale of supersymmetry breaking.We distinguish two classes of models: those, like gravity-mediated models, where the softbreaking terms of the SM fields are of order m / , and those, like gauge mediated models,where m / is parametrically smaller.Consider first gravity-mediated models. In these models, one general issue is (cid:104) W (cid:105) ∼ m / M p . If (cid:104) W (cid:105) is complex, this feeds into θ through phases, for example, at one loop in thegaugino mass (this is the familiar anomaly-mediated contribution). In Section 4, we raisedgeneral questions about the reality of (cid:104) W (cid:105) , and argued that in flux landscapes, at least, real (cid:104) W (cid:105) is unlikely. More generally, apart from some sort of anthropic selection, no convincingmechanism has been put forward to account for the value of the cosmological constant. Sothe failure of landscape models to account for small phases is troubling.– 14 –n gauge-mediated models, the situation can be significantly better. Comparing theanomaly-mediated to the gauge-mediated gluino mass, we require α s π m / m susy < − . (5.6)This constraint places a loose upper bound on the underlying scale of supersymmetry breakingif W possesses an order one phase.In both gravity and gauge mediation, there may be other strong constraints, dependingon the nature of supersymmetry breaking. If supersymmetry is broken in a hidden sectorthrough a gauge-singlet chiral field, Z , with F Z = f , then any phase in f can feed into softbreaking terms, yielding phases for the gluino, for example, as well as squark mass matrices.These, in turn, contribute to θ . In the models we have studied, these might arise fromcouplings such as W η − Z = λη i η j Z (5.7)at dimension three in W , or even through terms of dimension 2. Such undesirable termscan be forbidden if Z is charged under some symmetry (as in some models of dynamicalsupersymmetry breaking), or by combinations of continuous and discrete symmetries in themodels of CP breaking by pseudomoduli of the sort discussed in the previous section. Forexample, couplings of combinations like η i ¯ η to Z can be forbidden by R symmetries. In themodels with discrete vacua, this problem is more challenging. In gauge-mediated models, itis also necessary to forbid couplings of the η fields to messengers. This can again arise fromthe R symmetries consistent with the flat direction models.If non-renormalizable terms coupling CP-breaking fields to Z are permitted by symme-tries, these will constrain the scale of CP violation. Certain Kahler potential terms are difficultto suppress by symmetries. However, one can contemplate higher scales of CP violation thanin the non-supersymmetric case.Overall, then, both in gravity and gauge mediation, it appears possible to avoid dangerousnew sources of phases at tree level, without large arrays of new fields or excessively complicatednew symmetry structures. Gravity mediation requires stronger constraints on the reality of W . Supersymmetric theories are immunized against many of the types of corrections found innon-supersymmetric theories as a consequence of holomorphy and non-renormalizations. Inparticular, large terms of the form H ∗ Hη ∗ i η j and η i η j η ∗ k η ∗ l need not arise (the correspondingsuperpotential terms can be suppressed by symmetries and the smallness of the µ term).There are, however, new possible sources of corrections to θ . We divide our discussion betweengravity mediated and gauge mediated models. Loop corrections in gravity mediated models,as discussed in [31], are quite problematic. Gauge mediated models are better controlled [18].– 15 –e assume that tree level contributions to phases of gaugino masses are highly sup-pressed. Beyond this, we require, as discussed above, suppression of phases in the underlyingsupersymmetry breaking f term and the superpotential. But there are still potential diffi-culties. As discussed in [31], already at one loop, there are contributions to gaugino massesarising from loops involving heavy fields in the CP violating sector. In the simplest model,the heavy field is a Dirac particle, of mass m D , consisting of a charge 1 / D = (cid:88) B f ¯ d f + µ ¯ q (5.8)and a field of charge − / D = q . There is a soft breaking term, L q ¯ D = A D m D ¯ DD . (5.9)The gluino mass receives contributions proportional to A ∗ D . In general, there is no reasonfor the phase of A to vanish; this requires a very specific alignment of expectation valuesand couplings. It could arise in the presence of an SU (4) symmetry acting on ¯ d and ¯ q –something clearly not present in this structure. The phase must be smaller than 10 − orso. Similarly, there are potential contributions proportional to F η a . In supergravity models,these may naturally be suppressed by ( m / /M p ) / , so they become problematic if the scaleof supersymmetry breaking is greater than 10 GeV or so.As discussed in [31], there are additional contributions arising from phases in soft scalarmass terms. Suppressing these requires a remarkably high degree of degeneracy and propor-tionality. Overall, then, there is a set of issues similar to, but more severe than, the usualflavor problems of supergravity theories.Gauge mediated models are characterized by features which ameliorate the problemsnoted above. First and foremost, new sources of flavor violation are absent, and A terms arehighly suppressed.In addition, insertions of F η a , which also enter in loop corrections to gaugino masses, aresmall if SUSY breaking does not couple to the η a at tree level. SUSY-breaking F -terms forthe η a are generated radiatively from Kahler potential operators such as Z † Zη † a η b /m CP , butin the minimal model they appear only at three loop order. These statements need not holdin theories where messengers mix with other fields so as to gain large A terms, or where thereare “ µ -terms” for some of the η fields.At higher loop order, complex A-terms and flavor-violating soft masses can be generatedin gauge mediation. Such terms can give a weak upper bound on the hierarchy F Z /m CP . Forexample, in minimal gauge mediation, a Kahler potential operator of the form Z † Zq ¯ d f η a /m CP is generated at 3-loop order from loops of the η fields connected to ordinary gauge mediationloops. This operator provides a phase to the gluino mass in a manner similar to a complexA-term of the form A γ ηq ¯ d (although the operator involves heavy fields and cannot be writtenas an A-term at the scale m CP ). Because of the high loop suppression, the bound from ¯ θ is See also the discussion in [32] for the possibility of suppression through alignment. – 16 –eak: F Z /m CP (cid:46) − .Furthermore, all non-minimal flavor violation among the light fields comes from thecoupling a af η a ¯ d f q and the mixing of light right-handed fields with a af (cid:104) η a (cid:105) ¯ d f . If µ (cid:28) a af (cid:104) η a (cid:105) ,the light field is mostly ¯ q , and the mixing is small. Since µ (cid:29) a af (cid:104) η a (cid:105) is in conflict with thelarge CKM phase, and there is no obvious reason for the scales to be coincident, contributionsto ¯ θ in gauge-mediated NB models can be even further suppressed by µ/m CP . We have argued that solving the strong CP problem is not necessarily an arena for modelbuilding cleverness; rather, ideally, the smallness of an inconsequential parameter shouldemerge as a consequence of features of a theory which explains a range of other phenomena.No currently known model for solving strong CP is completely satisfactory from this point ofview.The shortcomings of the axion solution are well-known. Perhaps the most credible re-alization is in string theory, where plausible assumptions about moduli fixing may lead to asolution, albeit with a relatively high-scale axion.In the case of the Nelson-Barr solution, we have argued that non-supersymmetric modelsare at best very complicated, with intricate symmetries required to suppress higher-dimensionoperators. If these operators are simply suppressed by a low scale of CP violation, modelswithout strong dynamics or supersymmetry require a degree of fine-tuning higher than if¯ θ were simply set to zero by hand. Furthermore, we have argued that dynamical modelsbased on vevs for pseudo-Goldstones are nontrivial to construct. Loop corrections in genericnon-SUSY models are even more problematic, making further demands on the theories.Supersymmetric Nelson-Barr fares somewhat better. Coincidences of scales are still re-quired, but light scalars can be technically natural, and holomorphy greatly restricts thehigher-dimension operators that can contribute to ¯ θ . We described a specific structure inwhich the NB mechanism is operative and CP is broken in approximate flat directions by fieldscarrying new gauge symmetries. Additional discrete symmetries can suppress dangerous cou-plings of the CP-violating fields to the hidden sector fields and also couplings to messengers.Loop corrections are known to be highly problematic in generic gravity-mediated models,but in gauge-mediated models, these effects are under control. So supersymmetric modelswith additional symmetries and gauge mediation provide a setting in which the Nelson-Barrmechanism is plausible, at least as viewed at relatively low scales.We have also studied the underlying premise of models that aim to solve the strongCP problem through spontaneous CP violation: that in such theories, the bare θ parameternaturally vanishes. We stressed that this is a question of the nature of the ultraviolet theory.In string theory, the value of θ is generally controlled by the value of an axion field, so thebasic assumption is that there are massive axions whose expectation values conserve CP.Perhaps most problematic for the idea of small θ , however, is the possibility of a landscape.We noted that in flux landscapes, in particular, where the heavy axion expectation value is– 17 –etermined by superpotential parameters, these parameters are likely to be complex in anoverwhelming majority of states.So the current status of the strong CP problem can be described by saying we possessthree solutions, each with significant flaws. The reader is free to develop his or her own viewas to which solution, is any, is most plausible. Unless there are systematic problems withlattice computations which are common to disparate approaches to QCD, the light u quarksolution is ruled out. The axion solution requires either very complicated symmetry struc-tures, or some assumptions about moduli stabilization and an unconventional cosmologicalhistory. The spontaneous CP solution requires supersymmetry, a variety of additional sym-metries, something like gauge mediation, and, perhaps most problematic, an explanation ofwhy moduli are stabilized in a CP-conserving way. Acknowledgements:
This work was supported by the U.S. Department of Energy grantnumber DE-FG02-04ER41286. We appreciate conversations with Nima Arkani-Hamed,Nathaniel Craig, Ravi Kuchimanci, Anson Hook, Nathan Seiberg, Goran Senjanovic, ScottThomas, and Luca Vecchi.
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