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FCNC IN SUSY THEORIES C. A. Savoy Service de Physique Théorique, C.E. de Saclay, 91191 Gif-s/Yvette, France Recent work on flavour changing neutral current effects in supersymmetric models is reviewed. The emphasis is put on new issues related to solutions to the flavour problem through new symmetries: GUTs, horizontal symmetries, modular invariances. Invited Talk at theHEP95 Euroconference, Brussels, July 95. 6 1 Introduction angle, λ = .22: Y : Y : Y = λ8 : λ4 : 1,Y : Y : Y = t c u b s d 9 λ4 : λ2 : 1,Y : Y : Y = λ4 : λ2 : 1,V = λ,V ∼ 9 τ µ e us cb Arichliterature is availableaboutFCNC restrictionson λ2,V ∼λ3. 1 ub supersymmetric extensions of the standard model. Nev- n At the levelof the effective theory, belowthe Planck ertheless, both the LEP (and Tevatron) constraints on a scale, the supersymmetry breaking effects reduce to supersymmetrictheoriesandsomefreshinsightonspon- J gauginomassesandthesoftinteractionsinthescalarpo- 8 taneously broken supergravities from superstrings have tential. Thescalar(mass)2 matrixdependontheKähler encouraged a recent revival of this subject. potential and on the supersymmetry breaking auxiliary 1 The basic supersymmetry induced FCNC (SFCNC) fields. Theuniversalityorflavourindependencehypothe- v effects are produced by the analogues of the Standard 5 sisassumesequalmassesforallsquarksattheunification. Model loop diagrams for neutral current processes, with 2 At lower energies, radiative corrections from Yukawa in- quarksandvectorbosonsreplacedbysquarksandgaugi- 2 teractions split this degeneracy with flavour dependent 1 nos. If quark(lepton)andsquark(slepton)mass matrices shifts. The triscalar couplings are basically proportional 0 arenotdiagonalinthesamebasis,eventhethecouplings to couplings in the superpotential. Again, if universality 6 toneutralgauginostofermionsandsfermionswillnotbe 9 is assumed for the proportionality factors, referred to as diagonal and will induce FCNC effects. There are sev- / A-parameters, their equality is spoilt at lower energies h eral sources of flavour mixing in gaugino couplings that by the calculable radiative corrections. p wenowturntodiscuss. However,Iwanttokeepinmind p- that supersymmetry must be a local symmetry, namely, UniversalityofsofttermsisoftenassumedinSFCNC e asupergravitytheory,atthefondamentallevel. Thishas studies. Then, the most striking effects of radiative cor- h implications on the structure of the low energy effective rections are of two kinds. Gauge corrections are almost v: theory (and vice-versa, which is even more important!) universal and attenuate loop effects by an overall rise in i Withinthegeneralframeworkofsupergravity,athe- thesquarkmassesifgluinosarerelativelyheavy. Yukawa X oryisdefinedbythegaugeandmattersuperfields,andby corrections dominated by the top coupling, Yt, tend to ar their couplings encoded in the Kähler potential and the alignthe downsquarkmasseigenstatesto the upquarks superpotential. The low-energy theory is then fixed by (if tanβ is not too large). This reverses the pattern of the values of the auxiliary fields that provide supersym- gaugino couplings in comparison with the gauge boson metry breaking and their couplings to the light fields. ones. Charginocouplingstodownsquarksandupquarks The supersymmetric part of this effective theory con- areapproximatelyflavourdiagonalwhilegluinoandneu- tains the supersymmetrized gauge couplings and the su- tralino couplings become proportional to the CKM ma- persymmetrized Yukawa couplings, encoded in an effec- trix. However, the expected physical effects are either tive superpotential W =P[YUQiUjH +YDQiDiH + consistent with the present overallbounds on supersym- ij 2 ij 1 YELiEjH ], where H and H are the two Higgs super- metricparticlesortheydependonunknownmixingsand ij 1 1 2 phases,buttheb→sγ transitionprovidesinterestingin- fields, and the matter superfields are as follows: Q,L, formation. contain the SU(2) doublets of quarks and leptons, weak and U,D,E contain the right-handed quarks and lep- Thus, universality naturally suppresses SFCNC ef- tons. The physical content of the three Yukawa cou- fects as it amounts to postulate the largestpossible hor- pling 3×3 matrices is given by their eigenvalues Y (i = izontal symmetry, U(3)5, for each of the 5 irreps of the i u,c,t;d,s,b;e,µ,τ) as well as the CKM matrix V. The Standard Model in the 3 fermion families, as an acci- observed quark masses and mixings and lepton masses dentalsymmetry,i.e.,asymmetryofthescalarpotential revealastronghierarchyconvenientlydisplayedinterms in the limit where all Yukawa couplings vanish. This is ofasmallparameterwhichwechooseto be the Cabibbo justified if supergravityy couplings to the supersymme- trybreakingareflavourindependent. Aswenowturnto couplings. This interesting idea is discussed in more de- discuss, they are not necessarily so. tail in the contributions1 of E. Dudas and F. Zwirner. For this reason, it is not developped here. 2 Flavour theories and supersymmetry Motivated by superstrings, as well as symmetries proposed to explain the structure of Yukawa couplings, ThefermionunitintheStandardModelisafamilyof15 new analyses2,9 have been performed on FCNC transi- fermions that provide a non-trivial anomalous-free rep- tions produced by non-universality in supergravitycou- resentation of the gauge group. GUTs are attempts to plings. Of course, the results are model dependent, one understand the fermion pattern by (vertical )unification variablebeingtheamountoftheflavourindependentsu- of the elements of the family within a representation of persymmetry breaking (in the dilaton direction) respon- a larger gauge theory at very high energies. The tripli- sible for gaugino masses, that attenuates SFCNC. With cation of families is a puzzle. But these fermion repli- this provisothe moreimportantconstraintsinthe quark cas do not look as clones since they quite differ by the sector are coming from K-physics. The lepton sector is strong hierarchy in their Yukawa couplings. The natu- lesssensitivetogauginomasses,andleptonflavourviola- ral explanation of this situation is to hypothetize that tions put severe constraints on the parameters,but only quarks/leptons of the same charge have different quan- as functions of unknown lepton mixing angles. tum numbers of some new symmetries at high energies Nevertheless, in this talk I would like to focus on (symmetries that commute with the Standard Model- the SFCNC problem from the stand-point of different symmetries have been called horizontal). attemptstoexplaintheoriginofflavour,henceoffermion As in many particle physics issues, hints come from masses and mixings. superstrings models, where one finds examples of com- pactifications with fermion families and neither vertical 3 SFCNC effects from SUSY unification nor horizontal unification. Instead, there are in general additionalabelianU(1)symmetriesthatdifferentiatebe- Recently, the question of FCNC effects arising from tween fermions. Moreover, the superstring theory parti- SUSY GUTs has been analysed in detail in a series of clemassesandcouplingsarefielddependentdynamically papers3 . This possibility was pointed out already some determined quantities. time ago, but the fact that the top Yukawa coupling is Aconspicuousresultofsuperstringstudiesisthatthe solargeconsiderablyenhancesthe effects. The ideais to three families of quark superfields may couple to super- estimatetherenormalizationcorrectionfromtherunning gravityaccordingto differentterms inthe Kählerpoten- of the soft parameters in the theory from the supergrav- tial. The relevantlowenergylimit of superstringmodels ityscale (M ) down to the GUT scale (M ) in Planck GUT are described by a N =1 suoergravities. The zero-mass presence of very large Yukawa couplings, which is cer- string spectrum contain an universal dilaton S, moduli tainly the case for Y . In a GUT, above M , the fol- t GUT fields, related to the compactification of six superfluous lowing part of the superpotential give also rise to loop dimensions, denoted by Tα(α = 1..m), and matter chi- diagramsP[YiUjEiUjH3+YiDj QiLjH3′ +YiDj DiEjH3′]in- ral fields Ai. A crucial role is played by the target-space volvingtheHiggstripletpartners. ThecouplingY isall- t modular symmetries SL(2,Z) , transformations on the wayslarge,while Y =Y is largeinO(10)unificationor b τ Tα that are invariances of the effective supergravity the- evenforSU(5)withlargetanβ.Theeffectoftherunning ory. In string models of the orbifold type, the matter from M to M can be very important: the τ Planck GUT R fields Ai transform under SL(2,Z) according to a set of is roughly reduced by a factor (1−Y2/2Y2 ), defineed t max numbers, n(α), called the modular weights of the fields at M , where Y2 is the value of Y for a Landau i GUT max t Ai with respect to the modulus Tα. pole at M . The mass splitting with respect to e Planck R e The dilaton superfield in these theories does have and µ will remain at low energies and produce lepton R e universal supergravity couplings to matter superfields. flavourviolatingprocesses. Ofcoursethe resultsalsode- But the moduli couplings are fixed by modular invari- pendontheanglesdefinedbythediagonalizattionofthe ances. Thus, the Kähler potential and the superpoten- lepton and slepton masses. Assuming naive GUT rela- tial can have different dependences on the moduli for tions for the lepton mixings - cum grano salis in view of each flavour. On the other hand, these moduli corre- the bad naive GUT predictions for the two light families spond to flat directions of the scalar potential so that - one gets sizeable FCNC effects in large regions of the their vev’s are fixed by quantum corrections. Assuming parameter space. For large Y the effects are even big- b thattherelevantonescomefromthelightsector,namely ger. The results can be illustrated by assuming univer- by the coupling of moduli to quarks and leptons in the sal boundary conditions at M , so that the slepton Planck low energy theory, it has been suggested that modular splitting is only due to the Higgs triplet. In this case, invariances can also provide a theory of flavour, by pre- it is possible to present detailed predictions for the vari- dicting the hierarchies in the moduli dependent Yukawa ous lepton flavour violating processes (for quark FCNC, those are concealed by the analogouscontributions from byFroggattandNielsen6tounderstandsuchahierarchi- the MSSM superpotential). cal pattern goes as follows: (i) The key assumption is a Of course if one attempts a real theory of fermion gaugedhorizontalU(1) symmetryviolatedbythesmall X masses based on GUTs, and O(10) has been preferred quark masses so that small Yukawa couplings are pro- in this respect4, for instance, by the introduction of tectedbythissymmetry. TheeffectiveU(1) symmetric X non-renormalizableinteractionsanddiscretesymmetries, theory below some scale M is supposed to be natural to therewillbe correspondingconstraintsonthe softscalar theextentthatallparametersareofO(1). ThescaleM is masses and couplings. The framework will be similar to thelimitofvalidityoftheeffectivetheory,ofO(M ) Planck what is discussed herebelow in the case of abelian hori- ifone adopts a superstringpointof view. The X-charges zontal symmetries. of quarks,leptons and Higgses are free parameters to be fixedaposterioriandsimplydenotedq ,u ,d ,l ,e ,h ,h , i i i i i 1 2 forthedifferentflavours,wherei=1,2,3isthefamilyin- 4 The pseudo-Goldstone approach dex. (ii) One (or more) Froggatt-Nielsenfield Φ, a Stan- dardModelgauge singletis introduced, and we normalize Dimopoulos and Giudice5 invoke the pseudo-Goldstone the U(1) so that its charge is X = −1. The effec- phenomenon to enforce FCNC suppression. They as- X Φ tive (non-renormalizable) U(1) allowed couplings are sume a large Π U(3)’accidental’ symmetry X of the scalar potAe=nQti,aUl,,D,iLn,cElu)ding the scalar masses, in then of the form giUj(Φ/M)qi+uj+h2QiUjH2, with anal- ogous expressions for the H couplings to down quarks the limit of vanishing Yukawa couplings. They intro- 1 and leptons. The coefficients gU, etc, are taken to be duceon-purposemultiplets,sayintheAdj(U(3)5),whose ij vev’s break U(3)5 → U(1)15 or → [U(2) × U(1)]5. natural, i.e., of O(1), unless they are required to vanish by the U(1) symmetry. (iii) The small parameter λ The remaining symmetries entail the following form X is identified with the ratio (<Φ>) as the U(1) symme- for each one of the sfermion mass matrices: m2 = M X eA try is broken by the Φ v.e.v.. Below the scale < Φ >= e−iθAdiag(me2A1, me2A2 me2A3)eiθA, where θA are matrices, λM, one recovers the Standard Model with the effective each one containing five Goldstone fields living in the coset U(3)/U(1)3 ( the extension to [U(2) × U(1)] is Yukawa coupling matrices given by YiUj = λ|qi+uj+h2|, obvious). These are massless states as the potential is YiDj = λ|qi+dj+h1|, YiEj = λ|li+ej+h1|. The Yukawa ma- trixentriescorrespondingtonegativetotalchargeshould flat along the θ directions. Actually, they are ’pseudo- A vanish but these zeroes are filled by the diagonalization Goldstone’ states since the flavour symmetries are ex- of the λ -dependent metrics. plicitly broken by the Yukawa couplings. The latter are taken a priori as given by the quark masses and The X-charges are now chosen to fit the hierarchy in the mass eigenvalues and mixing angles. The experi- CKM mixings. Then, at the quantum level, the hid- mentalmasses(atO(M ))ofthethirdfamiliesgive: den flavour symmetry is broken by loops with quarks Planck h +q +u =0andx=h +q +d =h +l +e ,where that spoil the flatness along the θ directions. By mini- 2 3 3 1 3 3 1 3 3 A theparameterxdependsontheassumedvaluefortanβ. mization one obtains the θ vev’s (and masses)in terms A of the Yukawa couplings Y , such that the m2’s are all With this restriction the Yukawa couplings depend only alignedtotheYukawacoupAlingsY butm2 toetAhematrix on the charge differences qi−q3, ui−u3,..., ei−e3 and A eQ x. Y2 +K+Y2K. The quark squark alignment is as good asUpossibleD, still the m2 disalignment could induce too Recently, there has been an intensive investigation muchKK¯ mixing. TheisQis avoidedif the remaining acci- of this model7,8, including a classification of the possible charge assignments8. But the question I would like dentalsymmetryisU(2)×U(1)sothatm2 =m2 .This eQ1 eQ2 to discuss here was first investigated by Leurer, Nir and can be implemented5 by enlarging the accidental U(3)5 Seiberg9 in the Froggatt-Nielsen framework. Just like symmetry to O(8), spontaneously broken into O(7). the Yukawa couplings, the soft supersymmetry break- In spite of its formal elegance, this approach does ing terms contain powers of the Φ -field to implement not address the flavour problem as far as the expected the U(1) symmetry. The scalar mass matrices have a X dependence of the Yukawa couplings onnew fields is not corresponding hierarchy among their elements, so that envisaged while it might provide a prediction for quark masses as well. Also, the necessarily large number of ad (me2A)i¯= fAijλ|qi−qj|, A=Q,U,D,L,E, where, in the ab- sence of any other symmetry principle, the coefficients hoc Goldstone fields could mitigate one’s enthusiasm. f are all of the order of the supersymmetry break- Aij ing parameter m2 , where m is the gravitino mass. 3/2 3/2 5 The supersymmetric Froggatt-Nielsen ap- Even in the flavour basis that diagonalizes quark mass proach matrices, the squark mass matrices will still be of the same non-diagonal form. Therefore large FCNC effects The smalness of the mass ratios and mixing angles faces might be induced from loopdiagrams with the exchange uswithaproblemofnaturalness. Thedirectioninitiated of neutral sfermions (gluino, photino,...) in possible dis- agreement with experiments. Indeed, with only one Φ me2i¯∼3(Xi−Xj)m23/2nΦcos2θ/λ|Xi−Xj|.Similarresults -field , the acceptable U(1) charge asignements yield also follow for triscalar couplings. X (m2 ) (m2) ∝ md,whichimplymuchtoolargeFCNC The consequences for SFCNC are an improvement eD 12 eQ 12 ms effects in K-physics. One solution9 is to double the with respect to those in the previous section. For in- Froggatt-Nielsen, with another abelian symmetry and a stance, the contribution to KK¯ mixing can be reduced smaller scale. In this case it is possible to strongly sup- by choosingmodels8 withchargesd1 =d2, andthe same press (m2 ) . Interestingly enough, the model predicts trick is possible to avoid too much lepton flavour viola- large (me2D)12 leading to sizeable DD¯ mixing that could tion. eU 12 be experimentally tested. Another solution8 is to assume only one more sin- 7 Conclusion glet Φ′ and an appropriate charge asignment so that (m2 ) (m2) ∝ m2d, which is just enough. Remark- Supersymmetry is the highway connection between eD 12 eQ 12 m2s flavour physics at low energies and flavour theories at ably,inthismodelallanomaliesrelatedtoU(1) canbe X the Planck scale. SFCNC phenomenology provide very cancelled,whileintheothermodelsonehastorelyupon selective constraints in this adventure. the Green-Schwarz mechanism7,8. 8 References 6 Horizontal symmetries in supergravity 1. E. Dudas, these Proceedings; F. Zwirner,these Pro- On one hand, horizontal symmetries are a natural way ceedings. to solve the family puzzle and the fermion mass hierar- 2. A. Brignole, L. E. Ibań˜ez and C. Munõz, Nucl. chy,andgive some restrictionsonsquarkmassesas well. Phys. B422 (1994) 235; D. Choudhury, F. Eber- On the other hand, in string inspired supergravity, the lein, A. König, J. Louis and S. Pokorski Phys. sfermionmassesdependonthemodularpropertiesofthe Lett.B342 (1995) 180; P. Brax and M. Chemtob, matter fields and their modular dependence might well Phys. Rev. D51 (1995) 6550; J. Louis and Y. Nir, be related to the origin of flavour. What if one imposes Nucl. Phys. B447 (1995) 18. P. Brax and C. A. both symmetries on a broken supergravity model? This Savoy, Nucl. Phys. B447 (1995) 227. A. Brignole, has been recently investigated10. For definiteness, let us L. E. Ibań˜ez, C. Munõz, and C. Scheich, Madrid define the modular properties by some set of modular preprint FTUAM-95-26, hep-ph/9508258. weights n(iα) associated to each of the matter fields, and 3. R.BarbieriandL.J.Hall,Phys. Lett. B338 (1994) their transformation under an abelian U(1)X symmetry 212; R. Barbieri, L. J. Hall, and A. Strumia, hep- implementing the Froggatt-Nielsen mechanism, by their ph/9501334. chargesXi. Analogously,n(Φα) andXΦ areintroducedfor 4. S. Dimopoulos, L. Hall and S. Raby, Phys. Rev. the singlet field Φ. Now, let us require the supergravity Lett., 68(1992) 1984;Phys. Rev., D45(1992) 4192; theory to be invariant under these SL(2,Z) and U(1)X M. Carena, S. Dimopoulos, C.E. Wagner and S. transformations. Then, one shows the very interesting Raby, CERN preprint TH95-053, hep-ph/9503488. relation: (q −q )n(α) = X (n(α)−n(α)) betweencharge 5. S. Dimopoulos and G. F. 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