Sensitivity to new supersymmetric thresholds through flavor and CP violating physics

Maxim Pospelov,1,2 Adam Ritz,1 and Yudi Santoso1

1Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1 Canada2Perimeter Institute for Theoretical Physics, Waterloo, ON, N2J 2W9, Canada

(Received 29 August 2006; published 18 October 2006)

Treating the MSSM as an effective theory below a threshold scale �, we study the consequences ofhaving dimension-five operators in the superpotential for flavor and CP-violating processes. Below thesupersymmetric threshold such terms generate flavor changing and/or CP-odd effective operators ofdimension six composed from the standard model fermions, that have the interesting property ofdecoupling linearly with the threshold scale, i.e. as 1=��msoft�, where msoft is the scale of softsupersymmetry breaking. The assumption of weak-scale supersymmetry, together with the stringentlimits on electric dipole moments and lepton flavor-violating processes, then provides sensitivity to � ashigh as 107 � 109 GeV. We discuss the varying sensitivity to these scales within several MSSMbenchmark scenarios and also outline the classes of UV physics which could generate these operators.

DOI: 10.1103/PhysRevD.74.075006 PACS numbers: 12.60.Jv, 11.30.Er, 11.30.Hv, 12.10.Kt

I. INTRODUCTION

Weak-scale supersymmetry (SUSY) is a theoreticalframework that helps to soften the so-called gauge hier-archy problem by removing the powerlike sensitivity of thedimensionful parameters in the Higgs potential to thesquare of the ultraviolet cutoff �. This feature, amongothers, has stimulated a large body of theoretical work onweak-scale supersymmetry, supplemented by continuingexperimental searches, which now spans almost three dec-ades. Yet the supersymmetrized version of the standardmodel (SM), the minimal supersymmetric standard model(MSSM), suffers from well-known problems such as thelarge array of allowed free parameters responsible for softSUSY breaking, and the consequent possibility of largeflavor and CP violating amplitudes. The absence ofCP-violation at the O�1� level in the soft-breaking sectorof the MSSM, as suggested by the null results of electricdipole moment (EDM) searches and the perfect accord ofthe observed K and B meson mixing and decay with thepredictions of the SM, implies that the soft-breaking sectorof the MSSM somehow conserves CP and does not sourcenew flavor-changing processes. Whether or not such apattern of soft-breaking masses is theoretically feasible isthe subject of on-going studies addressing the mechanismof SUSY breaking and mediation (see, e.g. [1]). In thiswork, we will make the assumption that an (approxi-mately) flavor-universal and CP-conserving soft-breakingsector is realized, and study the consequences of the pres-ence of SUSY-preserving higher-dimensional operators onflavor and CP-violating observables.

These operators may be thought to emerge from newphysics at some high-energy scale �, which is larger thanthe electroweak scale. Even though the field content of theMSSM may be perfectly ‘‘complete’’ at the electroweakscale, it is clear that almost by construction the MSSMcannot be a fundamental theory because of the requiredhigh-energy physics responsible for SUSY breaking and

mediation. In recent years there is also a more phenome-nological motivation for a new threshold, namely, the newphysics responsible for neutrino masses (assuming they areMajorana) and mixings. Beyond these primary concerns,the possibility of new thresholds, intermediate between theweak and the GUT scales, is also suggested by the axionsolution to the strong CP problem, by the SUSY lepto-genesis scenarios [2] and, more entertainingly, by thepossibility of a lowered GUT/string scale arising fromthe large radius compactification of extra dimensions [3].In summary, given the assumed existence of weak-scalesupersymmetry, there seems ample motivation to expectadditional new physics thresholds above the electroweakscale and possibly below the GUT scale. The presence ofsuch thresholds will generically be manifest not justthrough corrections to relevant and marginal operators,but also through the presence of higher-dimensionaloperators.

As is easy to see, both Kahler terms and the super-potential can receive additional nonrenormalizable termsat the leading dimension-five level [4,5]. Some of theseoperators are well known and were studied in connectionwith baryon-number violating processes and also the see-saw mechanism for neutrino masses. However, to the bestof our knowledge, an analysis of the full set of dimension-five operators with respect to flavor and CP-violatingobservables is still lacking. The purpose of this paper isthus to consider all possible dimension-five extensions ofthe MSSM superpotential and Kahler terms, concentratingon those that conserve lepton and baryon number and areR-parity symmetric. We initiated such a study recently [6],and will provide further details and extensions in thepresent work. As we shall see, such operators can inducelarge corrections to flavor-changing and/or CP-violatingamplitudes and therefore can be efficiently probed withexisting experiments and future searches.

There is a clear parametric distinction between theeffects induced by nonuniversal soft-breaking terms and

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by the higher-dimensional extensions of the superpotential.Whereas the former typically scale as m�2

soft times one ortwo powers of the flavor-mixing angle �ij in the squark(-slepton) sector, the latter decouple as ��msoft�

�1�0ij, where�0ij parametrizes flavor violation in the dimension-fiveoperators. When � is relatively large, and thus the thresh-old corrections to the soft-terms are small, we may havescenarios where �ij ’ 0 while �0ij are significant, and thecorrections to the superpotential can be the dominantmechanism for SUSY flavor and CP violation, providingconsiderable sensitivity to �. At the same time, the addi-tional CP and flavor violation introduced in this way canbe rendered harmless by simply increasing �.

The layout of the paper is as follows. In the next sectionwe list the possible operators in the MSSM superpotentialand the Kahler terms at dimension-five level, including forcompleteness those that violate R-parity. The relevantsupersymmetric renormalization group equations for theoperators of interest are included in an Appendix. InSec. III, we perform the required calculations at theSUSY threshold to connect this extension of the super-potential with the resulting Wilson coefficients in front ofvarious effective SM operators of phenomenological inter-est. Section IVaddresses the consequent predictions for themost sensitive CP-odd and flavor-violating amplitudes andinfers the characteristic sensitivity to � in each channel. InSec. V, we perform this analysis within four SPS bench-mark scenarios [7] (see also [8]) in order to infer thedependence of this sensitivity on the features of theSUSY spectrum. Section VI contains a discussion andalso a brief analysis of the general classes of new physicswhich could be responsible for these operators, while ourconclusions are summarized in Sec. VII.

II. DIMENSION-5 OPERATORS IN THE MSSM

In this section, we will enumerate all the allowed struc-tures in the superpotential and Kahler potential up todimension 5 according to the standard symmetries of theMSSM (see, e.g. [4,5]). We begin by recalling in Table Ithe chiral superfields of the MSSM [9] along with theirgauge quantum numbers.

The matter parity, PM, is defined in the usual way:

PM � ��1�3�B�L�; (1)

where B is the baryon number and L the lepton number.This can be restated as R-parity, defined as

PR � ��1�3�B�L��2s; (2)

where s is the spin of the component field. All knownstandard model particles have PR � �1, while their super-partners have PR � �1. However, when using the super-field formalism it is often more convenient to use matterparity in which all fields belonging to the same superfieldhave the same value of PM.

The superpotential of the MSSM contains a number ofdimensionless parameters, and one dimensionful parame-ter � or, in equivalent language, is composed from onedimension three and several dimension-four operators:

W �3� � ��HdHu; (3)

W �4� � YuUQHu � YdDQHd � YeELHd; (4)

where gauge and generation indices are suppressed. Allthese terms conserve R-parity. In counting the dimensions,one should keep in mind that we are implicitly includingdim�d2�� � 1. At the renormalizable level, there are addi-tional terms that are forbidden by matter/R parity butallowed by gauge invariance,

W �3�6R � ��

0LHu; (5)

W �4�6R � �LLE� �0LQD� �00UDD: (6)

Going beyond the renormalizable level, at dimension-five there are a number of operators allowed by symme-tries. It is worth recalling that in the standard model, abovethe electroweak scale, there is only a single class ofdimension-five operators allowed by symmetries-the see-saw operator-which can naturally provide a smallMajorana mass for the active neutrinos. Within theMSSM, the list is only slightly longer. Suppressing avariety of gauge and generational indices, the collectionof dimension-five operators can be presented in the follow-ing schematic form:

W �5� � cqqQUQD� cqeQULE� chHuHdHuHd

� c�HuLHuL� cp1UUDE� cp2QQQL: (7)

The final two terms in this list violate baryon and leptonnumber by one unit, and therefore induce proton decay.Detailed studies of these operators induced by triplet Higgsexchange have been conducted over the years in the con-text of SUSY GUT models [4,5] (for a recent assessment,see e.g. [10]). The HuLHuL operator is the superfieldgeneralization of the SM seesaw operator and can beresponsible for generating the neutrino masses and mix-

TABLE I. Representations and quantum numbers for chiralfields in the MSSM.

Superfield SU�3�C SU�2�L U�1�Y PM

Q 3 2 1=6 �1U �3 1 �2=3 �1D �3 1 1=3 �1L 1 2 �1=2 �1E 1 1 1 �1Hu 1 2 1=2 �1Hd 1 2 �1=2 �1

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ings. Assuming neutrinos are Majorana, the flavor structureof c� is currently being determined in neutrino physicsexperiments (see e.g. [11]).

Going over to R-parity violating terms (see e.g. [12]),one finds additional dimension-five operators,

W �5�6R � c 6R1QUHdE� c

6R2HuHdHuL� c

6R3QQQHd; (8)

that can be obtained from (7) upon the simple substitutionL$ Hd.

If we now consider the Kahler potential, it is easy to seethat at dimension-four one has the standard �yeV� opera-tors, where � represents a generic MSSM chiral superfield,and the additional dimension-four operators LeVHyd thatviolate R parity. In all cases, V should be chosen as thecorrect linear combination of individual vector superfieldsto insure gauge invariance. At dimension-five level, wehave three additional structures that are allowed by allgauge symmetries and R-parity,

K �5� � cuQUHyd � cdQDH

yu � ceLEH

yu ; (9)

and several further operators that violate R-parity,

K �5�6R � c 6RK1EHdH

yu � c

6RK2QUL

y � c 6RK3UEDy

� c 6RK4QQDy: (10)

At this point, it is important to recall that the equationsof motion can be utilized within the effective Lagrangian toremove various redundancies in the full set of higher-dimensional operators listed above. We will work withtree-level matching at the �-threshold and thus, if oneleaves aside corrections from SUSY breaking, all thestructures in K�5� can be reduced on the superfield equa-tions of motion and absorbed into W �4� and W �5�. Indeed,in the limit of exact SUSY, the superfield equation ofmotion for e.g. Hyu reads

�D 2Hyu / ��Hd � YuQU; (11)

where �D is the spinorial derivative. Substituting this intothe expression for K�5�, we observe that the operatorLEHyu , for example, reduces to a linear combination ofthe usual Yukawa structure with �Ye � �ce and thedimension-five superpotential term:

Zd4�ceLEH

yu /

Zd2�ceLE �D2Hyu

/Zd2���ce�LEHd � ceYuQULE�:

(12)

The inclusion of soft SUSY breaking terms in the equationof motion would change this analysis only slightly; newsoft-breaking structures such as dimension-four four-sfermion interactions ~Q ~U ~L ~E and new trilinear termssuch as ~L ~EHyu [13] would appear. Since the analysis of

higher-dimensional soft-breaking terms goes beyond thescope of the present paper, we choose to eliminate allKahler higher-dimensional terms via the equations of mo-tion and analyze only the corrections to superpotential.

Comparing theHuLHuL-induced neutrino masses to thecharacteristic mass splitting�0:01� 0:1� eV observed inneutrino oscillations, we deduce the corresponding rangeof the energy scales ��:

�0:01�0:1� eVc�hH2ui)��c

�1� �1014�1016�GeV:

(13)

The actual mass scale of the new states responsible forgenerating the effective term HuLHuL can be lower than��. Indeed, in the seesaw scheme c� � Y2

�M�1R , and the

mass of the right-handed neutrinos MR can be smaller than�� if Y� is small. A considerably smaller energy scale forMR than 1014 GeV is indeed suggested by SUSY lepto-genesis scenarios [2].

The mediation of proton decay by the QQQL andDUUE operators has been extensively studied over morethan two decades in the context of SUSY GUT models.Typically, such operators are induced by the exchange of acolor-triplet Higgs superfield, and therefore the operatorsare proportional to the square of the Yukawa couplings. Forthis study, we will not go into the details of how such termswere generated, and simply deduce the sensitivity to cp1

and cp2. The absence of proton-decay at the level of ��1 >1032 yr implies a rather stringent upper bound on thebaryon and lepton number violating couplings cp,

�p c�1p > 1024 GeV; (14)

which is well above the scale of quantum gravity,1019 GeV. The discrepancy in the scales �p and �� issomewhat problematic for SUSY GUTs, and is part of thedoublet-triplet splitting problem. In any event, the disparitybetween �p and �� clearly illustrates the fact that theenergy scales associated with the effective operators in(7) could be widely different, and thus motivates a dedi-cated study to determine the sensitivity to cqq, cqe and ch.

III. INDUCED OPERATORS AT THE SUSYTHRESHOLD

We begin our analysis by making explicit the color andflavor structure of the new dimension-five operators. It iseasy to see that the SU�2� indices can be contracted in onlyone way, via the antisymmetric tensor �ij. Therefore, wesuppress these indices in the expression below:

W �WMSSM�yh�hHdHuHdHu�

Yqeijkl�qe�UiQj�EkLl

�Yqqijkl�qq�UiQj��DkQl��

~Yqqijkl�qq�UitAQj��DktAQl�: (15)

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Here yh, Yqe, Yqq, and ~Yqq are dimensionless coefficientswith the latter three also being tensors in flavor space,while the �’s are the corresponding energy scales. Theparentheses (. . .) in (15) denote the contraction of colorindices. Note, that for the case of one generation there isonly one way of arranging the SU�3� indices, as �QtAU��QtAD� reduces to �QU��QD� upon the use of the com-pleteness relation for the generators of SU�3�.

From the superfield formulation of Eq. (15), one caneasily move to the component form using the standardrules of supersymmetric field theories. However, the fullinteraction Lagrangian resulting from (15) is quite cum-bersome, and we will quote only those terms that are1=� and of potential phenomenological importance,namely, the terms in the Lagrangian that involve two SMfermions and two sfermions. As an example, the QULEoperator in the superpotential generates the followingsemileptonic two fermion-two boson interaction terms:

Zd2�QULE � �UQ ~L ~E� � �EL ~Q ~U� � �UEc ~Q ~L� �UL ~Q ~E�

� �QcL ~U� ~E� � �EQ ~L ~U�: (16)

In this expression, letters with a tilde atop denote sfer-mions, and four-dimensional spinors are used for fermionswith Q being the left-handed quark doublet and Qc itscharge conjugate, etc. The generalization of (16) to therest of the operators in (15) is straightforward.

At the next step, we integrate out the squarks andsleptons to obtain operators composed from SM fieldsalone, or to be more precise, from the fields of a type IItwo-Higgs doublet model. This procedure is facilitated bythe observation that the first two terms in (16) have a closeresemblance to the LR squark and slepton mixing terms,with the only difference being that instead of the usual

me�u��tan 1� and/or Au�e�mue mixing coefficients one hasdimension three fermion bilinear insertions �UQ and �EL. Itis then clear that ~L ~E� and ~Q ~U� can be integrated outstraightforwardly encountering loop integrals that are com-mon in the MSSM literature. Notice that only in the firsttwo terms in (16) can the sfermions be integrated out at oneloop, as the remaining terms contain a slepton and asquark, and so integrating them out requires at least twoloops.

A. Corrections to the SM fermion masses

The SM operators of lowest dimension that are of phe-nomenological interest are the fermion masses. In Fig. 1,we show the one-loop diagrams that lead to the logarithmicrenormalization of the fermion masses. Cutting the ultra-violet divergence at the corresponding threshold �, wearrive at the following expression for fermion massescorrected by the dimension-five operators:

�Me�ij � �M�0�e �ij � Y

qeklij�M

�0�u ��kl

3 ln��qe=msq�

8�2�qe�A�u �� cot��;

�Md�ij � �M�0�d �ij � K

qqklij�M

�0�u ��kl

ln��qq=msq�

4�2�qq�A�u �� cot��;

�Mu�ij � �M�0�u �ij � Y

qeijkl�M

�0�e �kl

ln��qe=msl�

8�2�qe�Ae �� tan�� � Kqq

ijkl�M�0�u ��kl

ln��qq=msq�

4�2�qq�A�u �� cot��;

(17)

with implicit summation over the repeated flavor indices,and we have also defined the combination,

Kqq � Yqq �2

3~Yqq; (18)

that will reappear again below. M�0�e;d;u denote the unper-turbed mass matrices arising from dimension-four terms inthe superpotential.

Some of the mass corrections in (17) correspond to new‘‘nonholomorphic’’ operators such as �UQHyd , which breaksupersymmetry, and scale as �m=m �A=�� log�. Theother set of corrections survive in the limit of unbrokenSUSY, scaling as �m=m ��=�� log�. This is a cor-rection to the standard mass term in the superpotential,UQHu generated by the dimension-five operator. Given thenonrenormalization theorem for the superpotential [14], itmay look surprising that such corrections could arise at all.

FIG. 1. A one-loop correction to the masses of SM fermionsgenerated by the dimension-5 operators in the superpotential.Here and below crossed vertices stand for the two-fermion-two-boson operators generated by the dimension-five operators.

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A more careful look at the diagram in Fig. 1 reveals that itis the dimension 5 Kahler term QUHyd that receives alogarithmic loop correction, leading to the quark masscorrection in (17) upon the use of the equation of motion(12).

B. Dipole operators

Dimension-five dipole operators first arise at two-looporder via integrating out the heavy-flavor squarks from�EL ~U� ~Q, as in Fig. 2. The results for these diagrams can

be deduced from the calculations of the two-loop Barr-Zee-type supersymmetric diagrams in the limit of largepseudoscalar mass [15]. In the charged lepton sector theyresult in

Le �Au �� cot�

�qem2sq

e

12�3 �Mu��klY

qeklij

�Ei�F�PLEj

� �h:c:�; (19)

where we treated LR squark mixing as a mass insertion,and used PL �

1��5

2 and �F� � F����. In the quark

sector the corresponding results are more cumbersome towrite down due to a large number of possible diagrams.

C. Semileptonic four-fermion operators

Going up another dimension, we now considerdimension-six four-fermion operators composed from theSM fermion fields generated by the operators (15). Tworepresentatives of the relevant one-loop diagrams areshown in Fig. 3. The loop functions entering these calcu-lations are identical to those found in the calculation of thecorrections to the SM fermion masses arising from theSUSY threshold [16]. We will generalize the results of[6] by working with the full loop function [16],

I�x; y; z� � �xy ln�x=y� � yz ln�y=z� � zx ln�z=x�

�x� y��y� z��z� x�; (20)

which satisfies

I�z; z; z� �1

2z; (21)

allowing us to consider several benchmark SUSY spectralater on. All the SUSY masses, msq, msl, Mi and the �parameter are considered to be somewhat larger than MW ,so that the effects of gaugino-Higgsino mixing in thechargino and neutralino sector are not particularly impor-tant for the values of the loop integrals.

Integrating out gauginos and sfermions at one-looplevel, we find the following semileptonic operators,sourced by the QULE term in the superpotential,

Lqe �1

�qe

�2s3�

M�3I�m2~u1; m2

~u2; jM3j

2�

�1

4�M�1I�m

2~e1; m2

~e2; jM1j

2�

�Yqeijkl �UiQj

�EkLl

� �h:c:�: (22)

In this expression, we retained the gluino-squark contribu-tion as the largest in the squark sector and the sfermion-bino contribution in the lepton sector. If all SUSY massesare approximately the same, then the second term in thesquare bracket of Eq. (22) is subdominant, but this may notbe the case if the masses in the slepton-bino sector aresignificantly lighter than in the squark-gluino sector.Notice the absence of contributions from SU�2� gauginos,that turn out to be suppressed by additional power(s) ofMW=msoft. Finally, as expected the overall coefficient infront of the semileptonic operator (22) scales as��msoft�

�1.

D. Four-quark operators

Purely hadronic operators in (15) give rise to the follow-ing four-quark effective operators upon integrating outgluinos and squarks:

Lqq �1

�qq

s6�

M�3I�m2~q1; m2

~q2; jM3j

2�Kqq�

8

3� �UQ�� �DQ�

� � �UtAQ�� �DtAQ��� �h:c:�; (23)

FIG. 2. A representative of the two-loop SUSY threshold dia-grams that generate dipole amplitudes and contribute to EDMs,�! e�, the anomalous magnetic moment of the muon, etc.

FIG. 3. One-loop SUSY threshold diagrams that generatedimension-six four-fermion operators composed from the SMfields. Diagram (a) is a squark-gluino loop giving rise to asemileptonic operator, and diagram (b) is a squark-Higgsinoloop leading to a four-fermion operator in the down-quark sector.

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where the summation over flavor is carried out exactly as inEq. (15).

It is well known that the strongest constraints on FCNCin the quark sector often arise from �F � 2 amplitudes inthe down-squark sector that contribute to the mixing ofneutral K and B mesons. It is easy to see that such ampli-tudes are not present in Eq. (23) where two of the quarksare always of the up-type. Of course, they can be convertedto down-type quarks at the expense of an additional loopwith W bosons, but this introduces an additional numericalsuppression. In any event, the conversion of the right-handed quark field U into a D field would necessarilyrequire additional Yukawa suppression. There is, however,a more-direct one-loop SUSY threshold diagram that cangive rise to �F � 2 amplitudes in the down-quark sector.As shown in Fig. 3(b), it consists of a Higgsino-up-squarkloop. The result for this diagram,

Ldd �1

�qe

1

8�2 ��I�m2

~u1; m2

~u2; j�j2��Y�u�im�Y

�d�njK

qqijkl

�1

3� �QmDn�� �DkQl� � � �Qmt

ADn�� �DktAQl�

�

� �h:c:�; (24)

inevitably contains additional suppression by the Yukawacouplings of the up and down-type quarks originating fromthe Higgsino-fermion-sfermion vertices.

Notice that in the limit �� msq the overall coefficientin Eq. (24) scales as �����1 log��=msq� and thus has onlymild dependence on the soft-breaking scale. In this case,the operator (24) must have an explicit superfield general-ization. Indeed, it is easy to see that in this limit (24)corresponds to a dimension 6 Kahler term: QyDDyQ.

E. Modifications to the Higgs sector and sparticlespectrum

Thus far, we have not considered the consequences ofthe presence of the first operator in (15), which consistsentirely of Higgs superfields. Its most obvious implicationis a modification of the Higgs potential and the sparticlespectrum. The addition to the Higgs potential, linear in yh,has a simple form:

�Vh � �2��yh

�h��HyuHu� � �H

ydHd���HdHu� � �h:c:�:

(25)

If ��yh has a cumulative phase, this would create mixingbetween A and the h, H bosons that violate CP symmetry.However, its most important consequence for our studyhere will be an induced complex shift of the bilinear softparameter m2

12, which enters one-loop contributions forfermion EDMs.

The mixing of left- and right-handed sfermions is alsoaffected by this term. In addition to the usual � orA-proportional mixing, we have the following contribution

to the mixing matrix element of ~uL and ~uR,

��M2~u�LR � �mu

yhv2SM

�hcos2�; (26)

and analogous formulae for ~e and ~d with the cos�! sin�substitution. In this expression, v2

SM � 4M2W=g

2W corre-

sponds to the SM Higgs vacuum expectation value (v.e.v.)The neutralino mass matrix also receives two new (com-

plex) entries, i.e. Majorana masses for the neutral compo-nents of ~Hu and ~Hd, proportional to yh��1

h v2SMcos2� and

yh��1h v2

SMsin2� respectively.

IV. PHENOMENOLOGICAL CONSEQUENCESAND SENSITIVITY TO �

In this section, we estimate the sensitivity of variousexperimental searches to the energy scales �qe and �qq. Ofcourse, one of the most important issues is then the as-sumed flavor structure of the new couplings Yqe, Yqq, and~Yqq. Since we are thinking of � as an intermediate scaleand wish to explore the full reach of precision measure-ments, we will make the generous assumption that thesecoefficients are complex, of order one, and do not factorizeinto products of Yukawa matrices in the superpotential:Yqe � YuYe. It is clear that a much more restrictive as-sumption, e.g. minimal flavor violation, would dramati-cally reduce the sensitivity to these operators, but we willnot explore this option here.

A. Naturalness bounds-fermion masses and the �-term

With the above assumption on flavor structure, weshould first investigate the requirement that the correctionsto masses of the SM fermions do not exceed their measuredvalues, as otherwise we will face a new fine-tuning prob-lem in the flavor sector. Taking �MuAu�kl � �MuAu�33 mtAt 175 GeV 300 GeV and using the expression for�me in (17), we arrive at the following estimate,

�me 3mtAt

8�2�qe ln�

�qe

msq

� 1 MeV

107 GeV

�qe ; (27)

which clearly suggests that the ‘‘naturalness’’ scale for thenew physics encoded in semileptonic dimension-five op-erators in the superpotential is on the order of 107 GeV,while the analogous sensitivity in the squark sector isslightly lower, �qq 106 GeV. This high naturalnessscale is simply a restatement of the small Yukawa cou-plings for the light SM fermions. However, perhaps sur-prisingly, we will see below that this sensitivity is not thedominant constraint on the threshold scale.

Before we proceed to estimate the effects induced byfour-fermion operators, we would like to consider theeffective shift of the QCD �-angle due to the mass correc-tions (17). Assuming an arbitrary overall phase for the Yqq

matrices relative to the phases of the eigenvalues of Yu andYd, one typically finds the following shift of the �� parame-

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ter,

� ��Im�md�

md

Im�YqqmtAt�

4�2md�qq ln��qe

msq

�

10�10 1017 GeV

�qq : (28)

This is a remarkable sensitivity of �� to new sources ofCPand flavor violation, and can translate into a strong boundon �qq depending on how the strong CP problem isaddressed. If it is solved by an axion, there are no con-sequences of (28). However, in other possible approaches�� ’ 0 is engineered by hand, using e.g. discrete symmetriesat high energies [17]. In this case, dimension-five operatorscan pose a potential threat up to the energies �qq 1017 GeV, which by itself is very remarkable. Futureprogress in measuring the electric dipole moments(EDMs) of neutrons and heavy atoms [18], can clearlybring this scale up to the Planck scale and beyond.

B. Electric dipole moments from four-fermionoperators

Electric dipole moments (EDMs) of the neutron [19] andheavy atoms and molecules [20–26] are primary observ-ables in probing for sources of flavor-neutral CP violation.The high degree of precision with which various experi-ments have put limits on possible EDMs translates intostringent constraints on a variety of extensions of thestandard model at and above the electroweak scale (see,e.g. [18,27]). Currently, the strongest constraints onCP-violating parameters arise from the atomic EDMs ofthallium [20] and mercury [21], and that of the neutron[19]:

jdTlj< 9 10�25e cm;

jdHgj< 2 10�28e cm;

jdnj< 3 10�26e cm:

(29)

When �� is removed by an appropriate symmetry, theEDMs are mediated by higher-dimensional operators. Both(22) and (23) are capable of inducing the atomic/nuclearEDMs if the overall coefficients contain an extra phaserelative to the quark masses. Restricting Eq. (15) to the firstgeneration and dropping the U�1� contribution, we find thefollowing CP-odd operator:

LCP�odd � �1

�qe

s3�jM3YuejI�m2

~u1; m2

~u2; jM3j

2� sin�

�� �uu�� �ei�5e� � � �ui�5u�� �ee��; (30)

with the CP-violating phase � � arg�M�3Yqe� in a basis

with real me andmu. Taking into account the QCD runningfrom the superpartner mass scale to 1 GeV, and upon theuse of hadronic matrix elements over nucleon states,hNj �uujNi and hNj �ui�5ujNi, we can make a connection

to the CS and CP coefficients in the effective CP-oddelectron-nucleon Lagrangian,

L � CS �NN �ei�5e� CP �Ni�5N �ee: (31)

The isospin singlet part of the CS coefficient is given by

CS � �4s

3��qe Im�M�3Yuuee�I�m2

~u1; m2

~u2; jM3j

2�

2 10�4�1 GeV�qe��1; (32)

where in the latter equality we also assumed maximalviolation of CP, jYqej�MZ�

sin�O�1�, and chose thesuperpartner masses degenerate at 300 GeV. The quarkmatrix element, hNj� �uu� �dd�=2jNi ’ 4, is in accordwith standard values for the quark masses and the nucleon-term.

Using the same assumptions, and the pseudoscalar ma-trix element over the neutron, hnj �ui�5ujni ’�0:4�mN=mu� �ni�5n, we obtain a similar expression forthe neutron CP coefficient,

CP �s

6��qe

�0:4

mn

mu

�Im�M�3Y

uuee�I�m2~u1; m2

~u2; jM3j

2�

4 10�3�1 GeV�qe��1: (33)

Comparing (32) and (33) to the limits on CS and CPdeduced from the bounds on the EDMs of Tl and Hg [28],we obtain the following sensitivity to the energy scale �qe,

�qe * 3 108 GeV from TI EDM; (34)

�qe * 1:5 108 GeV from Hg EDM: (35)

These are remarkably large scales, and indeed not far fromthe intermediate scale suggested by neutrino physics. Infact, the next generation of atomic/molecular EDM experi-ments have the chance of increasing this scale by 2–3orders of magnitude which would put it close to the scalesoften suggested for right-handed neutrino masses.

Going over to purely hadronic CP-violating operators,e.g. Cud� �di�5d�� �uu�, we note that these would induce theEDMs of neutrons, and EDMs of diamagnetic atoms me-diated by the Schiff nuclear moment S� �g�NN�. In particu-lar, we have for the CP-odd isovector pion-nucleoncoupling,

�g �1��NN � �4 10�2 Cudmd

; (36)

with

Cud � �s

9��qq Im�M�3Yuudd�I�m2

~u1; m2

~u2; jM3j

2�; (37)

obtained as for the semileptonic operators above. Thetypical sensitivity to �qq in this case is somewhat lowerthan in the case of semileptonic operators,

�qq * 3 107 GeV from Hg EDM:

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Semileptonic operators involving heavy quark super-fields are also tightly constrained by experiment, via thetwo-loop diagrams of Fig. 2. Assuming no additional CPviolation in the soft-breaking sector and taking into ac-count only the stop loops, we obtain the following resultfor the EDM of the electron:

de � e

12�3

Im�Yttee��qe

mtjAt ��� cot�j

jm2~t1�m2

~t2j

ln�m2

~t1

m2~t2

�; (38)

wherem~t1 andm~t2 are the stop masses in the physical basis.Assuming maximal CP-odd phases and large stop mixing,we arrive at the following estimate for de,

de 108 GeV

�qe 10�27e cm; (39)

which together with the sensitivity to the electron EDMinferred from the constraint on dTl, jdej & 1:610�27e cm, translates to

�qe * 6 107 GeV:

Expressions similar to (38) can be obtained for the quarkEDMs and color EDMs, furnishing similar sensitivity to�qq.

C. Lepton flavor violation

Searches for lepton-flavor violation, such as the decay�! e� and �! e conversion on nuclei have resulted instringent upper bounds on the corresponding branchingratio [29] and the rate of conversion normalized on capturerate [30]1:

Br ��! e��< 1:2 10�11; (40)

R��! e�on Ti�< 4:3 10�12: (41)

Focussing first on�! e conversion, one can deduce thesensitivity of these searches to the energy scale of thesemileptonic operators (15) as the conversion is mediatedby the ( �uu) ( �ei�5�) and ( �uu) ( �e�) operators, and thusinvolves the same matrix elements as does CS. Indeed, thecharacteristic amplitude for the scalar operator has theform

GF���2p �e� � �

4s3��qe Im�M�3Y

uue��I�m2~u1; m2

~u2; jM3j

2�:

(42)

Using the bounds on such scalar operators derived else-where (see e.g. [32]), we conclude that �! e conversioncurrently probes energy scales as high as

�qe * 1 108 GeV from �� ! e�on Ti: (43)

However, it is important to note that the bound on theconversion rate is necessarily proportional to ��e��2 andthus these effects decouple as ��qe��2 in contrast to thelinear decoupling of the EDMs.

A sensitivity to slightly lower scales arises from the two-loop-mediated �! e� process. We have

Br ��! e�� � 384�2�2e�

4G2Fm

2�; (44)

with the transition amplitude generated in the same manneras de,

�e� �

12�3

Re�Yttee��qe

mtjAt ��� cot�j

jm2~t1�m2

~t2j

ln�m2

~t1

m2~t2

�; (45)

where once again the sensitivity in the branching fraction isweakened relative to the EDMs by quadratic decouplingwith the threshold scale.

Future progress in lepton flavor violation searchesshould be able to extend the reach of these probes by 1–2 orders of magnitude. Disregarding a factor of a fewbetween (34) and (43), we conclude that currently theEDMs and searches for lepton flavor violation probe theseextensions of the MSSM up to similar energy scales of108 GeV.

It is also worth noting that sensitivity to �qe, that issomewhat more robust to changing assumptions on theflavor structure of Yue, can also be achieved through com-parison of the two modes of charged pion decay into firstand second generation leptons. The typical sensitivity to�qe in this case could be as large as

�qe * 104 GeV from �� e universality in� decay:

Finally, the two-loop amplitudes in Fig. 3 would alsogive corrections to the anomalous magnetic moments of eand �. In the latter case, one can estimate the sensitivity to�qe as no higher than about 1 TeV.

D. K and B meson mass-difference

Often, the most constraining piece of experimental in-formation comes from the contributions of new physics tothe mixing of neutral mesons, K and B. In the case ofgeneric couplings Yqq and ~Yqq, the four-fermion operators(24) will contain ( �sRdL) ( �sLdR) and ( �bRdL) ( �bLdR) terms.Using a simple vacuum factorization ansatz, we find

hK0j� �dRsL�� �dLsR�j �K0i �

�1

24�

1

4

�mK

ms �md

�2�mKf2

K

’ 2mKf2K;

hK0j� �dRtAsL�� �dLt

AsR�j �K0i �1

18mKf

2K � 0:055mKf

2K;

(46)

and therefore can neglect ( �dRtAsL) ( �dLtAsR) due to its smallmatrix element. Taking into account the one-loop QCD

1A recent announcement from the SINDRUM II collaborationsuggests a slightly stronger constraint, R��! e�onAu�< 710�13 [31].

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evolution of the ( �dRsL) ( �dLsR) operators from the SUSYthreshold scale down to 1 GeV,

� �dRsL�� �dLsR�1 GeV ’ 5� �dRsL�� �dLsR�MZ; (47)

we can estimate the contribution of additional four-fermionoperators to the mass splitting of K mesons:

�mK � �2 RehK0jLddj �K0i

’ �5mKf

2K�I�m

2~u1; m2

~u2; j�j2�

6�2�qq Yds; (48)

where we took � to be real, and introduced the followingnotation for the relevant combination of Yukawa cou-plings:

Y ds � Re�Y�u�i1�Y�d�2jKqqij12 � Re�Yu�i2�Yd�1jK

qq�ij21: (49)

It is easy to see that the presence of Yu and Yd in (49)results in a significant numerical suppression of the corre-sponding Wilson coefficients even with Yqq O�1�. Theminimal suppression is realized with an intermediate stop-loop, in which case the first term in (49) becomes of orderVtdytys, while the second term is Vtsytyd. Numerically,this corresponds to a suppression factor,

Y ds 3 10�4 tan�50

; (50)

which is also tan�-dependent.Putting all the factors together, with the same assump-

tion of degenerate soft mass parameters at 300 GeV, wecome to a disappointingly weak result:

�mK 3 10�6 eVtan�50

200 GeV

�qq ; (51)

with the actual measured value of the mass splitting being3:5 10�6 eV. The calculation of �mB results in a similarsensitivity level, prompting the conclusion that neither�mK nor �mB can probe the flavor structure of additionaldimension-five operators beyond the SUSY threshold. TheCP-violating observable �K will obviously be more sensi-tive by almost 3 orders of magnitude, resulting in

�qq * 105 GeVtan�50

from �K; (52)

which is still clearly inferior to the sensitivity of EDMs andlepton flavor violation. Moreover, it is easy to see that if thecomplete theory at scales � also provides new dimension-six operators in the Kahler potential, the possible conse-quences of those for �mK and �mB would be considerablymore serious that of the dimension-five operators. We givean explicit example of this in the discussion section.

E. Constraining the Higgs operator

The strength of the constraints on QULE and QUQDcomes primarily from the fact that such operators flip thechirality of light fermions without paying the usual price of

small Yukawa couplings. This was a consequence of ourassumption on the arbitrary flavor structure of thedimension-five operators. Should all transitions from uRto uL and eR to eL be suppressed by mf=v, the constraints(34), (43), and (52) would be relaxed all the way to theweak scale and below. Therefore, it would come as nosurprise if the effective operator in the Higgs potential wereto have very weak implications for CP-violating physicsand no consequences at all for the flavor-changingtransitions.

We note first of all that the mixing of the left- and right-handed d-squarks is affected by the Higgs operator (26).This feeds into the one-loop d-quark EDM diagram, wherethis parameter behaves similarly to the insertion of thecomplex A-term, Aeff

d yhv2SM��1

h and leads to a contri-bution to dd that does not grow with tan�. The typicalsensitivity of the neutron and mercury EDMs to the imagi-nary part of the Ad parameter [33], with the superpartnermasses in the ballpark of 300 GeV, then implies

�h * 1 TeV maximal CP violation, neutron EDM:

(53)

Of course, a mere increase of the superpartner masses toaround 1 TeV would completely erase this sensitivity.

Another possibly interesting CP-violating effect wouldcome from the admixture of the pseudoscalar Higgs A tothe scalars h and H at tree level. Subsequent Higgs ex-change would then induce the CS operator [34,35], orcontribute to the two-loop EDM of quarks and electrons[36]. The latter results in contributions to observableEDMs that are tan�-dependent and furnish a sensitivityto �h up to a few TeV.

These are relatively minor effects. However, it turns outthat significant sensitivity to this operator can indeed arisethrough its shift of the Higgs potential (25), and morespecifically the effective shift of the m2

12 parameter,

m212HuHd !

�m2

12 ��yhv2

SM

�h

�HuHd; (54)

assuming the reality of �. The quantity in the parenthesesis an effectivem2

12 parameter, which is complex on accountof Im�yh�. Moreover, its complex phase is enhanced in thelarge tan� limit because m2

12 ’ m2Atan�1�. The resulting

phase affects the one-loop SUSY EDM diagrams.Assuming for simplicity a common mass scale msoft forsleptons, gauginos, and �, we have [33]

de eme tan�

16�2m2soft

�5g2

2

24�g2

1

24

�sin�

arg�M2

�m212�eff�

�: (55)

Expanding to leading order in 1=�h, and imposing thepresent limit on de, we find the sensitivity

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�h * 2 107 GeV�tan�50

�2�300 GeV

MSUSY

��300 GeV

mA

�2;

(56)

which reaches impressively high scales for maximal tan�.

V. CONSTRAINTS WITHIN MSSM BENCHMARKSCENARIOS

A summary of the characteristic sensitivity to the thresh-old scale � in different channels is given in Table II,assuming generic and degenerate SUSY spectra. In thissection, we will go somewhat further and examine thevariation in sensitivity among a few benchmark SUSYscenarios, with different spectra chosen in order to satisfyother constraints, including requiring the correct relic LSPdensity to form dark matter. We have also included theleading one-loop SUSY evolution of the dimension-fiveoperators from the �-scale to the soft threshold. Theseeffects are generally on the order of 20–40%, and therelevant RG equations are included in the appendix.

The benchmark spectra we have chosen are the follow-ing representative SPS points [7]:

(i) SPS1a-msugra(ii) SPS2-focus point

(iii) SPS4-large tan� in the funnel region(iv) SPS8-gauge mediationThe results are shown in Figs. (4–6), with the observ-

ables normalized to their current experimental bound plot-ted against the threshold scale �. All of the scenariosexhibit broadly similar sensitivity to the degenerate spec-trum utilized previously. However, there is still significantvariation in terms of the level of sensitivity exhibitedwithin different benchmark spectra.

If we focus first on Fig. 4, the constraints on CS and�g�NN are most stringent within SPS1a, while SPS2 exhibitsleast sensitivity simply through having a generically heavySUSY spectrum.

Figure 5 shows the constraints imposed by lepton flavor-violating observables, and the quadratic sensitivity to � isclearly evident in comparison to the EDM bounds. Thestrongest constraints again arise within SPS1a and SPS4

TABLE II. Sensitivity to the threshold scale. Note that the naturalness bound on Im�Yqq� doesnot apply to the axionic solution of the strong CP problem, the best sensitivity to Im�yh� isachieved at maximal tan�, and the Hg EDM constraint on Im�Yqq� applies when at least one pairof quarks belongs to the 1st generation.

Operator Sensitivity to � (GeV) Source

Yqe3311 107 Naturalness of me

Im�Yqq3311� 1017 Naturalness of ��, dnIm�Yqeii11� 107 � 109 Tl, Hg EDMsYqe1112, Yqe1121 107 � 108 �! e conversionIm�Yqq� 107 � 108 Hg EDMIm�yh� 103 � 108 de from Tl EDM

1054321

5

4

3

2

1

05020 765

5

4

3

2

1

010050201098

Λ [107 GeV]

dT l (CS )

Λ [105 GeV]

dHg (gπ )NN

FIG. 4 (color online). Plots of dTl�Cs� and dHg� �g�, normalized to the current experimental bound, are shown versus the thresholdscale � in four benchmark scenarios; the corresponding CP-odd coupling is set to unity at the threshold. In these plots and those below,SPS1a � red�solid�, SPS2 � blue�dashed�, SPS4 � green�dotted�, SPS8 � brown�dot� dashed�. The shaded region is above thecurrent experimental bound.

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due to having a generically lighter SUSY spectrum. Thedifferences in the constraints from�! e� are particularlymarked, with SPS2 and SPS8 exhibiting rather minimalsensitivity. This can be understood from the two-loopamplitude as due to the relatively small stop mixing inthese cases.

Figure 6 exhibits the constraints from the electron EDM.The constraints from the two-loop amplitude on the leftnaturally exhibit very similar features to the constraintsfrom �! e�, given the similarity between the two dipoleamplitudes. The constraints on the Higgs operator on theright of Fig. 6 are most pronounced in SPS4 as one wouldexpect due to tan�-enhancement. The weak constraintfrom SPS2 is primarily because of the large value of m2

12which acts to suppress the effect of the additive complexshift.

VI. DISCUSSION

So far we have kept our discussion rather general withinthe context of effective field theory, without concerningourselves with the details of particular renormalizable UVmodels. In the case of the seesaw operator and proton-decay operators, such models are well studied. We wouldnow like to briefly provide an example of how the effectiveterms in the superpotential studied in this paper can begenerated by renormalizable interactions.

We will limit our discussion here to scenarios in whichthese operators can be generated by tree level exchange ofadditional heavy states. As a basic example, consider theMSSM with an expanded Higgs sector-an additional heavysinglet S and heavy pair of doublets H0u and H0d. This issufficient to generate all the operators we have consideredassuming renormalizable interactions of the form:

10543

5

21

4

3

2

1

05020 76

5

5

4

3

2

1

010050201098

Λ [107 GeV]

R (µ → e)

Λ [105 GeV]

Br (µ → e )γ

FIG. 5 (color online). Plots of �! e conversion and �! e�, normalized to the current experimental bound, are shown versus thethreshold scale � in four benchmark scenarios. The labelling conventions for the curves are as in Fig. 4.

100502010543

5

4

3

2

1

0

321

5

4

3

2

1

0

10050201054

Λ [107 GeV]

dT l (de(2l))

Λ [106 GeV]

dT l (de(yh ))

FIG. 6 (color online). Plots of two contributions to dTl�de�, normalized to the current experimental bound, are shown versus thethreshold scale � in four benchmark scenarios. The labelling conventions for the curves are as in Fig. 4.

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W � 12MS

2 � SHuHd ��HdHu ��0H0dH0u

�UQ�YuHu � Y0uH0u� �DQ�YdHd � Y0dH0d�

� EL�YeHd � Y0eH0d�: (57)

Integrating out the singlet S will clearly generate theoperator �HuHd�

2. The complex parameters � and �0 arethe eigenvalues of a 2 2 complex matrix of� parametersthat can always be reduced to diagonal form by bi-unitarytransformations in the �Hu;H

0u� and �Hd;H

0d� spaces.

Assuming the hierarchy, �0 � �, we can integrate outheavy Higgs superfields, producing a set of dimension-five operators

W �5� � 2

MHuHdHuHd �

Y0eY0u�0

ELUQ

�Y0uY

0d

�0�UQ��DQ�: (58)

Comparing (58) with (15), we can make the identificationyh=�h � 2=M, ~Yqq � 0, Yqe=�qe � Y0eY0u=�0, andYqq=�qq � Y0uY

0d=�

0 and translate the sensitivity to the�’s into a sensitivity to the extra Higgs fields. Since apriori there is no correlation or dependence between thetwo sets of Yukawa matrices, one can expect novel flavorand CP violating effects induced by (58). More specificpredictions could be made in models that predict or con-strain the Yukawa couplings Y and Y0, due e.g. to horizon-tal flavor symmetries, Yukawa unification, or discretesymmetries such as parity or CP.

All the effects which decouple as 1=�, when put in thelanguage of the model (57), probe the exchange of heavyDirac fermions, namely, Higgsino particles composed from~H0u and ~H0d. It is then natural to ask the question of whetherthe dimension-six operators induced by the exchange of theheavy scalar Higgses could provide better sensitivity to�0.It is easy to see that in the case of arbitrary Y0u;d O�1�, thecontribution of dimension-six operators to the K mesonmass splitting is

�mK 0:25 GeV3

�02) �0 * 8 106 GeV; (59)

while �K is sensitive to scales 1 108 GeV. The reasonwhy this dimension-six contribution dominates so dramati-cally over (51) and (52) is the suppression of thedimension-five effects by loop and Yukawa factors (50).

We conclude that �F � 2 processes mediated bydimension-six operators in the MSSM extended by anadditional pair of Higgses comes very close in sensitivityto the estimates (34) and (38), with the latter being some-what more stringent. This statement does of course dependon the SUSY mass spectrum, and having heavier squarksand gluinos would reduce the EDM sensitivity. In contrastwe should also note that unlike the previous limits (34) and(43), the constraint (59) is essentially ‘‘static,’’ i.e. difficultto improve upon, as there is a limited extent to which new

physics contributions to �mK and �K can be isolated fromSM uncertainties.

We will end this section with a few additional remarkson issues that we touched on in this work:

(1) Thus far, we have studied the subset of all possibledimension-five operators neglecting, for example,R-parity violation. It is easy to see, however, thatno strong constraints on the R-parity violating termsin (8) would arise at dimension-five level. Indeed,limits on R-parity violation usually come from SMprocesses which have to be bilinear in R-parityviolating parameters. Thus, only a combination oftwo dimension-five terms, or a dimension-five termwith a dimension-four term, would induce four-fermion operators, for example. Since thedimension-four terms are tightly constrained (see,e.g. [12]), one would not expect the limits ondimension-five operators to be competitive with(34).

(2) A primary goal of any theory of CP violation is toprovide a solution to the strong CP problem. Wehave shown that the effective shift in � can be quitesignificant even if higher-dimensional operators aresuppressed by 1017 GeV. This has implications forsolutions to the strong CP problem that do notemploy the dynamical relaxation of ��. For example,if �� � 0 is achieved due to a new global symmetrythat forcesmu � 0 at the dimension-four level but isbroken e.g. by quantum gravity effects, one couldexpect the emergence of Planck-scale suppressedoperators in the superpotential and, remarkablyenough, progress in neutron EDM measurementsby just one or 2 orders of magnitude would directlyprobe such a scenario. Similarly, supersymmetricmodels that construct a small �� using discrete sym-metries can also be affected by these operators, withpossible observable consequences for the neutronEDM.

(3) Since the CP-odd effective interaction CS �NN �ei�5eprovides the leading sensitivity to the energy scaleof new physics encoded in the semileptonicdimension-five operator in the superpotential, it isprudent to recall that the best constraint on CScomes from the EDM of the Tl atom, which isalso used for extracting a constraint on de. Tomake both bounds independent of the possibilityfor mutual cancellations, one should use experimen-tal information from other atomic EDM measure-ments. In this respect, the interpretation ofpromising new molecular EDM experiments thataim to improve the sensitivity to de [25,26] willrequire additional theoretical input on the exactdependence on CS.

(4) Finally, we would like to emphasize that the mainresult of this paper, namely, the sensitivity to thehigh-energy scale in Eqs. (34) and (43), is quite

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robust in the sense that it has a mild dependence onthe SUSY threshold as exhibited in the precedingsection. For example, an increase of the averagesuperpartner mass to 3 TeV would reduce the sensi-tivity to �qe and �qq by only a factor of 10, stillprobing scales of a few 107 GeV. Contrary tothis, the dependence of the electron EDM on theHiggs operator is highly dependent on the details ofthe SUSY spectrum, as taking tan� 5 and mA MSUSY 1 TeV would reduce the sensitivity to �hto a few TeV.

VII. CONCLUSIONS

Continuing progress in precision experiments searchingfor CP- and flavor-violation provides an increasingly strin-gent test for models of new physics beyond the electroweakscale, and supersymmetric theories, in particular. In thispaper, we have presented an analysis of flavor and CPviolating effects in a two-stage theoretical framework:assuming first that the SM becomes supersymmetric at ornear the weak scale, and then that the MSSM gives way atsome higher scale � to a theory with additional degrees offreedom. If nature indeed chooses supersymmetry, bothsteps can clearly be justified. The first one follows fromthe solution to the gauge hierarchy problem offered bySUSY and the evidence for the second (third, etc.) energyscale comes from rather intrinsic features that are requiredby, but not contained within, the MSSM: mediation ofSUSY breaking, neutrino masses, not to mention problemswhich require other solutions, i.e. baryogenesis, the strongCP problem, etc.

We have examined new flavor and CP-violating effectsmediated by dimension-five operators in the superpotentialto show that sensitivity to these operators extends farbeyond the weak scale, and indeed probes very high en-ergies. The semileptonic operators that mediate flavor-violation in the leptonic sector and/or break CP could bedetectable even if the scale of new physics is as high as109 GeV. Since the effects studied here decouple linearly,an increase of sensitivity by just 2 orders of magnitudewould already start probing the scales that are relevant forMajorana neutrino physics. It is also important to note thattheoretically, should a major breakthrough in the precisionof EDM measurements take place, there is ample room forthe EDMs of paramagnetic atoms to probe CP-violatingoperators suppressed by the GUT or string scale withoutfacing the SM background from the Kobayashi-Maskawaphase, which is known to induce tiny EDMs in the leptonsector [37].

The MSSM can contain a variety of new sources offlavor- and CP-violation related to the soft-breaking sector.This plethora of sources appears highly excessive given therather minimalist pattern of CP and flavor violation ob-served experimentally. A number of model-building sce-narios have addressed this issue, often successfully,

especially if supersymmetry is broken at a relatively low-energy scale. Supposing that the wish of many theorists isgranted, and a CP-symmetric, and flavor-conserving, pat-tern of soft SUSY breaking is achieved in a compellingmanner, we may ask the following question: is there anynew information about such a SUSY theory that could beprovided by the continuation of the low-energy precisionexperimental program? This paper provides a clear affir-mative answer to this question.

ACKNOWLEDGMENTS

This work was supported in part by NSERC, Canada.Research at the Perimeter Institute is supported in part bythe Government of Canada through NSERC and by theProvince of Ontario through MEDT.

APPENDIX

In this appendix, we summarize the 1-loop renormaliza-tion group equations used to evolve the dimension-fiveoperators down to the soft threshold. This evolution ofcourse arises purely from the Kahler terms.

In general, we have:

ddtyh � yh

�1

16�2

�Tr�6yuyyu � 6ydyyd � 2yeyye �

� 6g22 �

6

5g2

1

�� . . .

�; (A1)

ddtYqeijkl �

1

16�2

�Yqeijkm�y

ye ye�ml � Y

qeijml�2y

ye ye�mk

� Yqeimkl�yyuyu � y

ydyd�mj � Y

qemjkl�2y

yuyu�mi

� Yqeijkl

�16

3g2

3 � 3g22 �

31

15g2

1

��� . . . ; (A2)

ddtYqqijkl �

1

16�2

�Yqeijkm�y

yuyu � y

ydyd�ml � Y

qqijml�2y

ydyd�mk

� Yqqimkl�yyuyu � y

ydyd�mj � Y

qqmjkl�2y

yuyu�mi

� Yqqijkl

�32

3g2

3 � 3g22 �

11

15g2

1

��� . . . ; (A3)

and similarly for ~Yqqijkl, where yu, yd, ye are 3 3 Yukawamatrices, and the dots represent higher order terms.

In practice, since only the third generation Yukawacouplings are significant, we can make use of the simpli-fied RGEs,

ddtyh ’

yh16�2

�6yty�t � 6yby�b � 2y�y�� � 6g2

2 �6

5g2

1

�;

(A4)

ddtYqeuuee ’

Yqeuuee16�2

��

�16

3g2

3 � 3g22 �

31

15g2

1

��; (A5)

SENSITIVITY TO NEW SUPERSYMMETRIC THRESHOLDS . . . PHYSICAL REVIEW D 74, 075006 (2006)

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ddtYqettee ’

Yqettee16�2

�3y�t yt � y

�byb

�

�16

3g2

3 � 3g22 �

31

15g2

1

��; (A6)

ddtYqeuue� ’

Yqeuue�16�2

��

�16

3g2

3 � 3g22 �

31

15g2

1

��; (A7)

ddtYqette� ’

Yqette�16�2

�3y�t yt � y

�byb

�

�16

3g2

3 � 3g22 �

31

15g2

1

��; (A8)

ddtYqquudd ’

Yqquudd16�2

��

�32

3g2

3 � 3g22 �

11

15g2

1

��: (A9)

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