1754-0410-2-2 1754-0410 Research article <p>Effects of supersymmetric threshold corrections on high-scale flavour textures</p> Çakir Altan altan.cakir@cern.ch Solmaz Levent lsolmaz@balikesir.edu.tr

Department of Physics, Izmir Institute of Technology, IZTECH, TR35430, Turkey

Balikesir University, Physics Department, TR10300, Balikesir, Turkey

 PMC Physics A 1754-0410 2008 2 1 2 http://www.physmathcentral.com/1754-0410/2/2 10.1186/1754-0410-2-2 hep-ph/0608093 27 6 2007 07 3 2008 07 3 2008 2008 Çakir and Solmaz This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Integration of superpartners out of the spectrum induces potentially large contributions to Yukawa couplings. Supersymmetric threshold corrections therefore influence the CKM matrix prediction in a non-trivial way. We study the effects of threshold corrections on high-scale flavor structures specified at the gauge coupling unification scale in supersymmetry. We first consider high-scale Yukawa textures which qualify as phenomenologically viable at tree level, and find that they are disqualified after incorporating the threshold corrections. Next, we consider Yukawa couplings, such as those with five texture zeroes, which are incapable of explaining flavor-changing processes. Incorporating threshold corrections, however, makes them phenomenologically viable textures. Therefore, supersymmetric threshold corrections are found to have an observable impact on Yukawa couplings of quarks, and any confrontation of high-scale textures with experiments at the weak scale must take into account such corrections.

PACS Codes: 12.60.Jv, 12.15.Hh

1 Introduction and motivation

Supersymmetric theories with general soft breaking terms possess a number of flavor and CP violation sources 1. In general, they are nested in the rigid and soft sectors of the theory, and bear no correlation or selection rules whatsoever. This is the case in supergravity and superstring scenarios where Kähler metric and superpotential couplings are all generic matrices in the space of fermion flavors and depend on the compactification scheme employed. This rather high degree of freedom in sources, textures and structures of the flavor mixings at GUT/string scale needs to be refined by confronting them with experimental data especially on rare processes. In general, testing high-scale flavor structures with experimental data involves three basic ingredients:

1. Specification of flavor textures in rigid and soft sectors at the messenger scale (which we take to be the MSSM gauge coupling unification scale Q = MGUT ~ 1016 GeV).

2. Rescaling of lagrangian parameters to low-scale Q = Mweak ~ TeV via renormalization group flow. This stage is particularly important due to (i) largeness of the logs (log MGUT/Mweak) involved, and (ii) modifications of flavor structures because of mixings with others.

3. Integration out of the superpartners at Mweak to achieve an effective theory which comprises the SM particle spectrum with possible imprints of supersymmety in various couplings. For FCNC phenomenology this step is important as it induces flavor-nonuniversal couplings of gauge and Higgs bosons to fermions.

Any high-scale flavor structure specified in step 1 is classified to be phenomenologically viable if it agrees with experimental data after step 3. The first two steps have been widely discussed in literature by identifying flavor violation sources in general supergravity 2345 and confronting them with experimental data on fermion masses and mixings as well as various observables in kaon and beauty systems 56789.

So far analysis of the third step above has been restricted to TeV-scale supersymmetry where gauge 10 and Higgs 1112 bosons have been found to develop flavor-changing couplings to fermions. In particular, emphasis has been put on the couplings of Z 10 and Higgs 131415161718 to bs¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkgaIjqbdohaZzaaraaaaa@2DCB@ since mixing between second and third generation fermions exhibits a theoretically clean and experimentally wide room for new physics. These analyses have led to conclusion that flavor violation sources in sfermion sector can have a big impact on Higgs phenomenology as well as various rare processes in kaon and beauty systems 1112.

It is thus of prime phenomenological interest to know what impact the integration-out of sparticles can leave on high-scale flavor textures below Mweak. Stating more specifically, can integrating sparticles out of the spectrum render a given high-scale otherwise-viable texture inappropriate or generate CKM matrix from solely the soft sector or modify effects of Yukawa couplings on the soft sector? These are some of the questions which will be addressed in the present work.

In Sec. 2 below we briefly discuss the formalism for determining effects of supersymmetric threshold corrections. We mainly follow results of 1112 therein. In Sec. 3, we first discuss in Sec. 3.1 sensitivities of the GUT-scale CKM-ruled, hierarchic and democratic Yukawa textures to supersymmetric threshold corrections when trilinear couplings are proportional to Yukawas. In Sec. 3.2 we investigate effects of flavor mixings in squark mass-squared matrices on textures analyzed in Sec. 3.1. In Sec. 3.3, we determine effects of threshold corrections on Yukawa textures which would not qualify physical at tree level. In Sec. 4 we conclude.

2 The formalism

The superpotential of the MSSM

W _ = U _ Y u Q _ H _ u + D _ Y d Q _ H _ d + E _ Y e L _ H _ d + μ H _ u H _ d MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaqiaaqaaiabdEfaxbGaayPadaGaeyypa0ZaaecaaeaacqWGvbqvaiaawkWaaGqabiab=LfaznaaBaaaleaacqWF1bqDaeqaaOWaaecaaeaacqWGrbquaiaawkWaamaaHaaabaGaemisaGeacaGLcmaadaWgaaWcbaGaemyDauhabeaakiabgUcaRmaaHaaabaGaemiraqeacaGLcmaacqWFzbqwdaWgaaWcbaGae8hzaqgabeaakmaaHaaabaGaemyuaefacaGLcmaadaqiaaqaaiabdIeaibGaayPadaWaaSbaaSqaaiabdsgaKbqabaGccqGHRaWkdaqiaaqaaiabdweafbGaayPadaGae8xwaK1aaSbaaSqaaiab=vgaLbqabaGcdaqiaaqaaiabdYeambGaayPadaWaaecaaeaacqWGibasaiaawkWaamaaBaaaleaacqWGKbazaeqaaOGaey4kaSIaeqiVd02aaecaaeaacqWGibasaiaawkWaamaaBaaaleaacqWG1bqDaeqaaOWaaecaaeaacqWGibasaiaawkWaamaaBaaaleaacqWGKbazaeqaaaaa@568B@

encodes the rigid parameters μ and Yukawa couplings Yu,d,e (of up quarks, down quarks and of leptons) each being a 3 × 3 non-hermitian matrix in the space of fermion flavors.

The breakdown of supersymmetry is parameterized by a set of soft (i.e. operators of dimension ≤ 3) terms 1

ℒ s o f t = m H u 2 H u † H u + m H d 2 H d † H d + Q ˜ † m Q 2 Q ˜ + U ˜ m U 2 U ˜ † + D ˜ m D 2 D ˜ † + L ˜ † m L 2 L ˜ + E ˜ m E 2 E ˜ † + [ U ˜ Y u A Q ˜ H u + D ˜ Y d A Q ˜ H d + E ˜ Y e A L ˜ H d + μ B H u H d + 1 2 ∑ α M α λ α λ α + h . c . ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@BD33@

where trilinear couplings Yu,d,eA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDcqGGSaalcqWFKbazcqGGSaalcqWFLbqzaeaacqWFbbqqaaaaaa@3337@ like Yukawas themselves are non-hermitian flavor matrices whereas the sfermion mass-squareds mQ,...,E2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWFfbqraeaacqWFYaGmaaaaaa@3418@ are all hermitian. In general, all of the parameters in the second line and off-diagonal entries of the sfermion mass-squared matrices are endowed with CP-odd phases; they serve as sources of CP violation beyond the SM. The Yukawa matrices, trilinear couplings and sfermion mass-squareds facilitate flavor violation in processes mediated by sparticle loops. The MSSM possesses 21 mass parameters, 36 mixing angles and 40 CP-odd phases in addition to ones in the SM 19. Consequently, there is a 97-dimensional parameter space to be scanned in confronting theory with experiments at Mweak. In supergravity or string models the parameters of (1) and (2) are determined by compactification mechanism and structure of the internal manifold 234.

The parameters of (1) and (2) are scale- dependent. They are rescaled to Q = Mweak via the MSSM RGEs 2021222324 with boundary conditions specified at Q = MGUT. The RG running of model parameters is crucial. In fact, various phenomena central to supersymmetry phenomenology e.g. gauge coupling unification, radiative electroweak breaking, induction of flavor structures even for flavor-blind soft terms are pure renormalization effects. The Yukawa couplings, μ parameter and gauge couplings form a coupled closed set of observables 25 in that their scale dependencies are not affected by soft-breaking sector unless some sparticles are decoupled before reaching Mweak. On the other hand, running of the soft masses depend explicitly on rigid parameters of the theory, and they develop, among other things, novel flavor structures thanks to the Yukawa matrices. For instance, evolution of the soft mass-squared of left-handed squarks

d m Q 2 d t = m Q 2 ( Y u † Y u + Y d † Y d ) + ( Y u † Y u + Y d † Y d ) m Q 2 + 2 ( Y u † m U 2 Y u + Y d † m D 2 Y d + Y u A † Y u A + Y d A † Y d A ) + 2 ( m H u 2 Y u † Y u + m H d 2 Y d † Y d ) − 2 ( 16 3 g 3 2 | M 3 | 2 + 3 g 2 2 | M 2 | 2 + 1 15 g 1 2 | M 1 | 2 ) 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeabdaaaaeaajuaGdaWcaaqaaiabdsgaKHqabiab=1gaTnaaDaaabaGae8xuaefabaGae8NmaidaaaqaaiabdsgaKjabdsha0baaaOqaaiabg2da9aqaaiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaGcdaqadaqaaiab=LfaznaaDaaaleaacqWF1bqDaeaaieaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8xDauhabeaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8hzaqgabeaaaOGaayjkaiaawMcaaiabgUcaRmaabmaabaGae8xwaK1aa0baaSqaaiab=vha1bqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWF1bqDaeqaaOGaey4kaSIae8xwaK1aa0baaSqaaiab=rgaKbqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWFKbazaeqaaaGccaGLOaGaayzkaaGae8xBa02aa0baaSqaaiab=ffarbqaaiab=jdaYaaaaOqaaaqaaiabgUcaRaqaaiabikdaYmaabmaabaGae8xwaK1aa0baaSqaaiab=vha1bqaaiab+bcigcaakiab=1gaTnaaDaaaleaacqWFvbqvaeaacqWFYaGmaaGccqWFzbqwdaWgaaWcbaGae8xDauhabeaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqGFGaIHaaGccqWFTbqBdaqhaaWcbaGae8hraqeabaGae8NmaidaaOGae8xwaK1aaSbaaSqaaiab=rgaKbqabaGccqGHRaWkcqWFzbqwdaqhaaWcbaGae8xDauhabaGae8xqaeKae4hiGyiaaOGae8xwaK1aa0baaSqaaiab=vha1bqaaiab=feabbaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqcqGFGaIHaaGccqWFzbqwdaqhaaWcbaGae8hzaqgabaGae8xqaeeaaaGccaGLOaGaayzkaaaabaaabaGaey4kaScabaGaeGOmaiZaaeWaaeaacqWGTbqBdaqhaaWcbaGaemisaG0aaSbaaWqaaiabdwha1bqabaaaleaacqaIYaGmaaGccqWFzbqwdaqhaaWcbaGae8xDauhabaGae4hiGyiaaOGae8xwaK1aaSbaaSqaaiab=vha1bqabaGccqGHRaWkcqWGTbqBdaqhaaWcbaGaemisaG0aaSbaaWqaaiabdsgaKbqabaaaleaacqaIYaGmaaGccqWFzbqwdaqhaaWcbaGae8hzaqgabaGae4hiGyiaaOGae8xwaK1aaSbaaSqaaiab=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@D25A@

with t = (4π)-2 log Q/MGUT, shows explicitly how flavor structure of a given parameter, say mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@, at a given scale Q senses those of the remaining parameters. Indeed, flavor mixings exhibited by mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@ at Q = Mweak can stem from mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@ or Yu,d or Yu,dA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDcqGGSaalcqWFKbazaeaacqWFbbqqaaaaaa@3108@ or all of them. Therefore, a given pattern of flavor mixings in, for instance, kaon system can be sourced by various flavor matrices in rigid as well as soft sectors of the theory.

The flavor structures at Mweak arising from solutions of RGEs are further rehabilitated by taking into account the decoupling of sparticles at the supersymmetric threshold. Indeed, once part of the sparticles are integrated out of the spectrum the effective theory below Mweak can exhibit sizeable non-standard effects in certain scattering channels of the SM particles 101112131415161718. Taking the effective theory below Mweak to be two-Higgs-doublet model (2HDM) one finds

Y d e f f = Y d ( M w e a k ) − γ d + tan ⁡ β   Γ d Y u e f f = Y u ( M w e a k ) − γ u + cot ⁡ β   Γ u MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaacbeGae8xwaK1aa0baaSqaaiab=rgaKbqaaiabdwgaLjabdAgaMjabdAgaMbaakiabg2da9iab=LfaznaaBaaaleaacqWFKbazaeqaaOGaeiikaGIaemyta00aaSbaaSqaaiabdEha3jabdwgaLjabdggaHjabdUgaRbqabaGccqGGPaqkcqGHsislcqaHZoWzdaahaaWcbeqaaiabdsgaKbaakiabgUcaRiGbcsha0jabcggaHjabc6gaUjabek7aIjabbccaGiabfo5ahnaaCaaaleqabaGaemizaqgaaaGcbaGae8xwaK1aa0baaSqaaiab=vha1bqaaiabdwgaLjabdAgaMjabdAgaMbaakiabg2da9iab=LfaznaaBaaaleaacqWF1bqDaeqaaOGaeiikaGIaemyta00aaSbaaSqaaiabdEha3jabdwgaLjabdggaHjabdUgaRbqabaGccqGGPaqkcqGHsislcqaHZoWzdaahaaWcbeqaaiabdwha1baakiabgUcaRiGbcogaJjabc+gaVjabcsha0jabek7aIjabbccaGiabfo5ahnaaCaaaleqabaGaemyDauhaaaaaaaa@6DCD@

where Yd,u(Mweak) are solutions of the corresponding RGEs evaluated at Q = Mweak, and γd,u and Γd,u are flavor matrices arising from squark-gluino and squark-Higgsino loops. Their explicit expressions can be found in 1112.

The physical quark fields are obtained by rotating the original gauge eigenstate fields via the unitary matrices VR,Lu,d MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGsbGucqGGSaalcqWGmbataeaacqWG1bqDcqGGSaalcqWGKbazaaaaaa@332B@ that diagonalize Yu,deff:

( V R d ) † Y d e f f V L d = Y d ¯ , ( V R u ) † Y u e f f V L u = Y u ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5684@

where Yd¯=diag.(hd¯,hs¯,hb¯) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaacbeGae8xwaK1aaSbaaSqaaiab=rgaKbqabaaaaOGaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiOla4YaaeWaaeaadaqdaaqaaiabdIgaOnaaBaaaleaacqWGKbazaeqaaaaakiabcYcaSmaanaaabaGaemiAaG2aaSbaaSqaaiabdohaZbqabaaaaOGaeiilaWYaa0aaaeaacqWGObaAdaWgaaWcbaGaemOyaigabeaaaaaakiaawIcacaGLPaaaaaa@4132@ and Yu¯=diag.(hu¯,hc¯,ht¯) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaacbeGae8xwaK1aaSbaaSqaaiab=vha1bqabaaaaOGaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiOla4YaaeWaaeaadaqdaaqaaiabdIgaOnaaBaaaleaacqWG1bqDaeqaaaaakiabcYcaSmaanaaabaGaemiAaG2aaSbaaSqaaiabdogaJbqabaaaaOGaeiilaWYaa0aaaeaacqWGObaAdaWgaaWcbaGaemiDaqhabeaaaaaakiaawIcacaGLPaaaaaa@417A@ are physical Yukawa matrices whose entries are directly related to running quark masses at Q = Mweak:

h u ¯ = g 2 ( M w e a k ) m u ( M w e a k ) 2 M W sin ⁡ β , h d ¯ = g 2 ( M w e a k ) m d ( M w e a k ) 2 M W cos ⁡ β MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@736F@

with similar expressions for other generations.

In general, whatever flavor textures are adopted at MGUT, the resulting CKM matrix, VCKMcorr≡(VLu)†VLd MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaGccqGHHjIUdaqadaqaaiabdAfawnaaDaaaleaacqWGmbataeaacqWG1bqDaaaakiaawIcacaGLPaaadaahaaWcbeqaaGqaaiab=bcigcaakiabdAfawnaaDaaaleaacqWGmbataeaacqWGKbazaaaaaa@41F5@, must agree with the existing experimental bounds 26. Clearly, in the limit of vanishing threshold corrections Γu,d and γu,d, physical CKM matrix VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ reduces to VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ computed by diagonalizing Yu,d(Mweak). Reiterating, it is with comparison of the predicted CKM matrix, VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@, with experiment that one can tell if a high-scale texture, classified to be viable at tree-level by considering VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ only, is spoiled by the supersymmetric threshold corrections. The experimental bounds on the absolute magnitudes of the CKM entries (at 90% CL) read collectively as:

| V C K M e x p | = ( 0.9739 0.9751 0.2210 0.2270 0.0029 0.0045 0.2210 0.2270 0.9730 0.9744 0.0390 0.0440 0.0048 0.0140 0.0370 0.0430 0.9990 0.9992 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@A39E@

where left (right) window of MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaL4babaaaamaaL4babaaaaaaa@2B85@ in each entry refers to lower (upper) experimental bound on the associated CKM element. Clearly, the largest uncertainity occurs in |Vtd|. These matrix elements are measured at Q = MZ, and for a comparison with predictions of the effective theory below Q = Mweak they have to be scaled from MZ up to Mweak. This can be done without having a detailed knowledge of the particle spectrum of the effective 2HDM at Mweak (as emphasized above, the effective theory may consist of some light superpartners in which case beta functions of certain couplings get modified as exemplified by analyses of b → sγ decay in effective supersymmetry 2728) since RG running of the CKM elements is such that VCKM (1, 1), VCKM (1, 2), VCKM (2, 1), VCKM (2, 2) and VCKM (3, 3) do not evolve with energy scale, to an excellent approximation 293031. Therefore, it is rather safe to confront the CKM matrix predicted by the effective theory at Mweak with the experimental results (7) entry by entry excluding, however, VCKM (1, 3), VCKM (3, 1), VCKM (2, 3) and VCKM (3, 2) for which renormalization effects can be sizeable.

In the next section, we will compute supersymmetric threshold corrections to Yukawa couplings of quarks for certain prototype flavor textures defined at Q = MGUT. In particular, we will evaluate radiatively corrected CKM matrix as well as couplings of the Higgs bosons to quarks to determine the impact of the decoupling of squarks out of the spectrum at Mweak on scattering processes at energies accessible to present and future colliders.

3 High-scale textures and threshold corrections

First of all, for standardization and easy comparison with literature (e.g. with the computer codes ISAJET 32 and SOFTSUSY 33) we take SPS1a' conventions for supersymmetric parameters 34

tan β = 10, m0 = 70 GeV, A0 = -300 GeV, m1/2 = 250 GeV

and completely neglect supersymmetric CP-violating phases, as mentioned before.

Instead of scanning a 97-dimensional parameter space for specifying what high-scale parameter ranges are useful for what low-energy observables, which is actually what has to be done, we simplify the analysis by focussing on certain prototype textures at high scale. In general, for any flavor matrix in any sector of the theory there exist, boldly speaking, three extremes: (i) completely diagonal, (ii) hierarchical, and (iii) democratic textures. There are, of course, a continuous infinity of textures among these extremes; however, for definiteness and clarity in our analysis we will focus on these three structures.

3.1 Flavor violation from Yukawas and trilinear couplings

In this subsection we investigate effects of superymmetric threshold corrections on high-scale textures in which Yukawa couplings exhibit non-trivial flavor mixings and so do the trilinear couplings since we take

Y u , d , e A = A 0 Y u , d , e MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaaieqacqWFzbqwdaqhaaWcbaGae8xDauNaeiilaWIae8hzaqMaeiilaWIae8xzaugabaGae8xqaeeaaOGaeyypa0Jaemyqae0aaSbaaSqaaiabicdaWaqabaGccqWFzbqwdaWgaaWcbaGae8xDauNaeiilaWIae8hzaqMaeiilaWIae8xzaugabeaaaaa@3D03@

at the GUT scale. The soft mass-squareds, on the other hand, are taken entirely flavor conserving i.e. they are strictly diagonal and universal at the GUT scale. It is with direct proportionality of trilinear couplings with Yukawas and certain ansatze for Yukawa textures that, we will study below sensitivities of certain high-scale Yukawa structures to supersymmetric threshold corrections at the TeV scale.

3.1.1 CKM-ruled texture

We take Yukawa couplings of up and down quarks to be

Y u = diag ( 3.5   10 − 6 , 1.3   10 − 3 , 0.4566 ) Y d = ( 6.2368   10 − 5 − 1.4272   10 − 5 5.9315   10 − 7   e 0.3146 i 2.4640   10 − 4 1.07074   10 − 3 − 4.0458   10 − 5 1.6495   10 − 4   e 1.047 i 1.81465   10 − 3 4.8476   10 − 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeGadaaabaacbeGae8xwaK1aaSbaaSqaaiab=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@C49D@

with no flavor violation in the lepton sector: Ye = diag. (1.9 10-5, 4 10-3, 0.071). The flavor violation effects are entirely encoded in Yd which exhibits a CKM-ruled hierarchy in similarity to Yukawa textures analyzed in 5i.e. this choice of boundary values of the Yukawas leads to correct CKM matrix 26 at Mweak upon integration of the RGEs.

At the weak scale the Yukawa matrices, trilinear couplings and squark soft mass-squareds serve as sources of flavor violation. The trilinear couplings, under two-loop RG running 2021222324 with boundary conditions (9), attain the flavor structures

Y u A = ( − 7.2   10 − 3 0 0 1.70   10 − 6   e 0.5641 i − 2.67 2.9   10 − 4 6.24   e 1.047 i   10 − 3 6.8   10 − 2 − 532.7 ) Y d A = ( − 0.204 − 0.191 − 0.138   e − 1.039 i − 0.567 − 3.495 − 1.436 − 0.384   e 1.046 i − 4.19 − 134.24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@C285@

both measured in GeV at Mweak = 1 TeV. Clearly, YuA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqaaaaaa@2EDB@ is essentially diagonal whereas (2, 3), (3, 2) and (2, 2) entries of YdA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqaaaaaa@2EB9@ are of the same size.

Though they start with completely diagonal and universal boundary values, the squark soft squared masses develop flavor-changing entries at Mweak = 1 TeV:

m Q 2 = ( 533.67  GeV ) 2 ( 1.07 0.0 0.0 0.0 1.07 − 2.2   10 − 4 0.0 − 2.2   10 − 4 0.86 ) m D 2 = ( 530.76  GeV ) 2 ( 1.01 0.0 0.0 0.0 1.01 − 1.5   10 − 4 0.0 − 1.5   10 − 5 0.99 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeGadaaabaacbeGae8xBa02aa0baaSqaaiab=ffarbqaaiab=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@9ED9@

with mU2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFvbqvaeaacqWFYaGmaaaaaa@2EA5@ = (497.11 GeV)2 diag. (1.15, 1.15, 0.69). The numerical values of the parameters above exhibit good agreement with well-known codes like ISAJET 32 and SOFTSUSY 33.

The presence of flavor violation in the soft sector of the low-energy theory gives rise to non-trivial corrections to Yukawa couplings and in turn to the CKM matrix. Indeed, use of (11) and (12) in 1112 introduces certain corrections to the tree-level Yukawa matrices Yu,d(Mweak) to generate Yu,deff MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDcqGGSaalcqWFKbazaeaacqWFLbqzcqWFMbGzcqWFMbGzaaaaaa@33F2@ in (4). In fact, VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ (obtained from Yu,d(Mweak)) and VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ (obtained from Yu,deff) compare to exhibit spectacular differences:

| V C K M t r e e | | V C K M c o r r | = ( 0.9746 0.9795 0.2241 0.2015 0.0037 0.0034 0.2240 0.2014 0.9737 0.9788 0.0406 0.0375 0.0079 0.0066 0.0400 0.0371 0.99917 0.9993 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@B408@

where left (right) window of MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaL4babaaaamaaL4babaaaaaaa@2B85@ in (i, j)-th entry refers to |VCKMtree(i,j)|(|VCKMcorr(i,j)|) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabcYha8jabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaGccqGGOaakcqWGPbqAcqGGSaalcqWGQbGAcqGGPaqkcqGG8baFcqGGOaakcqGG8baFcqWGwbGvdaqhaaWcbaGaem4qamKaem4saSKaemyta0eabaGaem4yamMaem4Ba8MaemOCaiNaemOCaihaaOGaeiikaGIaemyAaKMaeiilaWIaemOAaOMaeiykaKIaeiiFaWNaeiykaKcaaa@51CB@. Clearly, |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@| agrees very well with |VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@| in (7) entry by entry. This qualifies (10) to be the correct high-scale texture given experimental FCNC bounds at Q = MZ. However, radiative corrections induced by decoupling of squarks, gluinos and Higgsinos at the supersymmetric threshold Mweak = 1 TeV is seen to leave a rather strong impact on the CKM entries. Consider for instance (1, 1) entries of VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@, VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ and VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@. Present experiments provide a 1.64σ significance to |VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@(1, 1)| around a mean value of 0.745 as is seen from (7). The tree-level prediction, |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@(1, 1)|, takes the value of 0.9746 which is rather close to the center of the experimental interval. However, once supersymmetric threshold corrections are included this tree-level prediction gets modified to |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 1)| = 0.9795. This value is obviously far beyond the existing experimental limits as it is a 13.39σ effect. Similarly, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 2)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(2, 1)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(2, 2)| and |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(3, 3)| are, respectively, 12.36σ, 12.36σ, 11.95σ and 2.30σ effects. Obviously, deviation of |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(i, j)| from |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@(i, j|) (comparison with experiments at Q = MZ is meaningful especially for (i, j) = (1, 1), (1, 2), (2, 1), (3, 3) entries whose scale dependencies are known to be rather mild 293031), when the latter falls well inside the experimentally allowed range, obviously violates existing experimental bounds in (7) by several standard deviations. Consequently, supersymmetric threshold corrections entirely disqualify the high-scale texture (10) being the correct texture to reproduce the FCNC measurements at the weak scale. This case study, based on numerical values for Yukawa entries in (10), manifestly shows the impact of supersymmetric threshold corrections on high-scale textures which qualify viable at tree level.

The physical quark fields, which arise after the unitary rotations (5), acquire the masses

M u ¯ ( M w e a k ) = diag . ( ≃ 0 , 0.545 , 149.45 ) , M d ¯ ( M w e a k ) = diag . ( 3.35   10 − 3 , 5.76   10 − 2 , 2.33 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWaa0aaaeaaieqacqWFnbqtdaWgaaWcbaGae8xDauhabeaaaaGccqGGOaakcqWGnbqtdaWgaaWcbaGaem4DaCNaemyzauMaemyyaeMaem4AaSgabeaakiabcMcaPiabg2da9iabbsgaKjabbMgaPjabbggaHjabbEgaNjabc6caUiabcIcaOiabloKi7iabicdaWiabcYcaSiabicdaWiabc6caUiabiwda1iabisda0iabiwda1iabcYcaSiabigdaXiabisda0iabiMda5iabc6caUiabisda0iabiwda1iabcMcaPiabcYcaSaqaamaanaaabaGae8xta00aaSbaaSqaaiab=rgaKbqabaaaaOGaeiikaGIaemyta00aaSbaaSqaaiabdEha3jabdwgaLjabdggaHjabdUgaRbqabaGccqGGPaqkcqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGUaGldaqadaqaaiabiodaZiabc6caUiabiodaZiabiwda1iabbccaGiabigdaXiabicdaWmaaCaaaleqabaGaeyOeI0IaeG4mamdaaOGaeiilaWIaeGynauJaeiOla4IaeG4naCJaeGOnayJaeeiiaaIaeGymaeJaeGimaaZaaWbaaSqabeaacqGHsislcqaIYaGmaaGccqGGSaalcqaIYaGmcqGGUaGlcqaIZaWmcqaIZaWmaiaawIcacaGLPaaaaaaaaa@7849@

all measured in GeV. In this physical basis for quark fields, VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ governs the strength of charged current vertices for each pair of up and down quarks. These mass predictions are to be evolved down to Q = MZ to make comparisons with experimental results. This evolution depends on the effective theory below Mweak. Speaking conversely, the high-scale texture (10) as to be folded in such a way that resulting mass and mixing patterns for quarks agree with experiments below the sparticle threshold Mweak.

3.1.2 Hierarchical texture

The Yukawa couplings are taken to have the structure (as can be motivated from 35363738)

Y u = ( 2.6463   10 − 4 5.8163   10 − 4 i − 1.0049   10 − 2 − 5.8163   10 − 4 i 2.2587   10 − 3 1.0049   10 − 5 i − 4.8233   10 − 3 − 9.0437   10 − 6 i 0.495 ) Y d = ( 3.9808   10 − 4 8.1167   10 − 4   e 0.734 i − 1.1431   10 − 3 8.1167   10 − 4   e − 0.734 i 2.7997   10 − 3 2.04844   10 − 3 i − 1.1431   10 − 3 − 1.6461   10 − 3 i 4.97   10 − 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaacbeGae8xwaK1aaSbaaSqaaiab=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@0AFD@

with no flavor violation in the lepton sector: Ye = diag. (1.9 10-5, 0.004, 0.071). Here both Yu and Yd exhibit a hierarchically organized pattern of entries. In a sense, the hierarchic nature of Yd in (10) is now extended to Yu so as to form a complete hierarchic pattern for quark Yukawas at the GUT scale.

At the weak scale, the Yukawa matrices above, trilinear couplings, and squark soft mass-squareds serve as sources of flavor violation. The trilinear couplings, under two-loop RG running 2021222324 with boundary conditions (9), obtain the flavor structures

Y u A = ( − 0.4315 − 1.1442 i 10.637 1.1466 i − 4.4531 − 4.8631   10 − 3 i 5.0657 − 0.1046 i − 524.07 ) Y d A = ( − 1.2934 − 2.6494   e 0.734 i 3.1221 2.6428   e − 0.731 i − 9.1395 5.2606 i 3.4532 − 5.6827 i − 135.861 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@C77D@

both measured in GeV at Mweak = 1 TeV. Clearly, in contrast to (11), now both YuA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqaaaaaa@2EDB@ and YdA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqaaaaaa@2EB9@ develop sizeable off-diagonal entries, as expected from (15).

Though they start with completely diagonal and universal boundary values, the squark soft squared masses develop flavor-changing entries at Mweak = 1 TeV:

m Q 2 = ( 533.69  GeV ) 2 ( 1.07 1.9   10 − 5   e 1.144 i 2.14   10 − 3 1.9   10 − 5   e − 1.144 i 1.07 3.17   10 − 4 i 2.14   10 − 3 − 3.17   10 − 4 i 0.86 ) m U 2 = ( 496.76  GeV ) 2 ( 1.16 − 6.66   10 − 6 i 9.6   10 − 3 6.66   10 − 6 i 1.16 − 1.4   10 − 5 i 9.6   10 − 3 1.4   10 − 5 i 0.685 ) m D 2 = ( 531.07  GeV ) 2 ( 1.01 3.3   10 − 5   e 1.06 i 3.75   10 − 4 3.3   10 − 5   e − 1.06 i 1.01 − 6.62   10 − 4 i 3.75   10 − 4 6.62   10 − 4 i 0.99 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeWadaaabaacbeGae8xBa02aa0baaSqaaiab=ffarbqaaiab=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reaebqaaiab=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@48AD@

whose average values show good agreement with (12) but certain off-diagonal entries exhibit significant enhancements when the corresponding entries of Yukawas and trilinear couplings are sizeable.

The flavor-violating entries of Yukawas, trilinear couplings and soft mass-squareds collectively generate radiative contributions γu,d, Γu,d to the Yukawa couplings below Mweak 1112. In fact, VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ (obtained from Yu,d(Mweak)) and (obtained from Yu,deff confront as follows:

| V C K M t r e e | | V C K M c o r r | = ( 0.9745 0.9773 0.2243 0.2118 0.0049 0.0034 0.2240 0.2116 0.9737 0.9766 0.0417 0.0379 0.0109 0.0091 0.0405 0.0370 0.99912 0.99927 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@B502@

where left (right) window of MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaL4babaaaamaaL4babaaaaaaa@2B85@ in (i, j)-th entry refers to |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@(i, j)| (|VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(i, j)|). Clearly, |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@| falls well inside the 1.64σ experimental interval in (7) entry by entry. In this sense, Yukawa matrices in (20) qualify to be the correct high-scale textures given present experimental determination of VCKM at Q = MZ. However, this agreement between experiment and theory gets spoiled strongly by the inclusion of supersymmetric threshold corrections. Indeed, as is shown comparatively by (23), VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ violates the bounds in (7) significantly. More precisely, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 1)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 2)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ (2, 1)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(2, 2)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(3, 3)| turn out to have 7.65σ, 6.83σ, 6.77σ, 6.79σ, 3.28σ significance levels, respectively. These significance levels are far beyond the existing experimental 1.64σ intervals depicted in (7). As a result, supersymmetric threshold corrections are found to entirely disqualify the high-scale texture (15) to be the correct texture to reproduce the FCNC measurements at the weak scale. This case study therefore shows the impact of supersymmetric threshold corrections on high-scale textures which qualify viable at tree level.

The physical quark fields, which arise after the unitary rotations (5), acquire the masses

M u ¯ ( M w e a k ) = diag . ( 0.0065 , 0.98 , 153.82 ) , M d ¯ ( M w e a k ) = diag . ( 0.0071 , 0.155 , 2.37 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWaa0aaaeaaieqacqWFnbqtdaWgaaWcbaGae8xDauhabeaaaaGccqGGOaakcqWGnbqtdaWgaaWcbaGaem4DaCNaemyzauMaemyyaeMaem4AaSgabeaakiabcMcaPiabg2da9iabbsgaKjabbMgaPjabbggaHjabbEgaNjabc6caUiabcIcaOiabicdaWiabc6caUiabicdaWiabicdaWiabiAda2iabiwda1iabcYcaSiabicdaWiabc6caUiabiMda5iabiIda4iabcYcaSiabigdaXiabiwda1iabiodaZiabc6caUiabiIda4iabikdaYiabcMcaPiabcYcaSaqaamaanaaabaGae8xta00aaSbaaSqaaiab=rgaKbqabaaaaOGaeiikaGIaemyta00aaSbaaSqaaiabdEha3jabdwgaLjabdggaHjabdUgaRbqabaGccqGGPaqkcqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGUaGlcqGGOaakcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI3aWncqaIXaqmcqGGSaalcqaIWaamcqGGUaGlcqaIXaqmcqaI1aqncqaI1aqncqGGSaalcqaIYaGmcqGGUaGlcqaIZaWmcqaI3aWncqGGPaqkaaaaaa@7449@

all measured in GeV. In this physical basis for quark fields, VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ is responsible for charged current interactions in the effective theory below Mweak. The morale of the analysis above is that, the high-scale flavor structures (15) are to be modified in such a way that VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ agrees with VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@ with sufficient precision. Aftermath, the question is to predict quark masses appropriately at Q = Mweak so that, depending on the particle spectrum of the effective theory beneath, existing experimental values of quark masses at Q = MZ are reproduced correctly.

3.1.3 Democratic texture

In this subsection, we take Yukawa couplings to be (as can be motivated from relevant works 394041)

Y u = ( 0.1475 0.1443 0.1458 0.1443 0.1475 0.1458 0.1456 0.1458 0.1456 ) Y d = ( 0.01583 0.01452 ( 1 − 10 − 2 i ) 0.01553 ( 1 − 10 − 2 i ) 0.01452 ( 1 + 10 − 2 i ) 0.01944 0.01617 ( 1 + 2   10 − 2 i ) 0.01551 ( 1 + 10 − 2 i ) 0.01617 ( 1 − 2   10 − 2 i ) 0.01604 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaacbeGae8xwaK1aaSbaaSqaaiab=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@DC07@

with no flavor violation in the lepton sector: Ye = diag. (1.9 10-5, 4 10-3, 0.071). Here both Yu and Yd exhibit an approximate democratic structure so that Yu,d(Mweak) generate correctly masses and mixings of the quarks at the weak scale. Clearly, in the exact democratic limit two of the quarks from each sector remain massless, and therefore, a realistic flavor structure is likely to come from small perturbations of the exact democratic texture 394041. Another important feature of exact democratic texture is that all higher powers of Yukawas reduce to Yukawas themselves up to a multiplicative factor, and this gives rise to linearization of and in turn direct solution of Yukawa RGEs in the form of an RG rescaling of the GUT scale texture 25. These properties remain approximately valid for perturbed democratic textures like (20).

At the weak scale, the Yukawa matrices above, trilinear couplings, and squark soft mass-squareds serve as sources of flavor violation. The trilinear couplings, under two-loop RG running 2021222324 with boundary conditions (9), obtain the flavor structures

Y u A = − ( 182.44 175.57 178.81 175.69 182.32 178.81 178.62 178.81 178.67 ) Y d A = − ( 44.41 40.07   e − 0.0117 i 43.39   e 0.0115 i 39.44   e 0.0101 i 55.46   e − 0.0013 i 44.82   e 0.0218 i 43.09   e − 0.099 i 45.17   e − 0.0216 i 44.79   e 0.0016 i ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@E483@

both measured in GeV at Mweak = 1 TeV. Though not shown explicitly, each entry of YuA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqaaaaaa@2EDB@ is complex with a phase around 10-7 - 10-6 in size.

Though they start with completely diagonal and universal boundary values, the squark soft squared masses develop flavor-changing entries at Mweak = 1 TeV:

m Q 2 = ( 533.67  GeV ) 2 ( 1.0 0.0672 0.0670 0.0672 1.0 0.0673 0.0670 0.0673 1.0 ) m U 2 = ( 497.38  GeV ) 2 ( 1.0 0.1526 0.1524 0.1526 1.0 0.1524 0.1524 0.1524 1.0 ) m D 2 = ( 530.59  GeV ) 2 ( 1.0 5.046   10 − 3   e − 0.01 i 4.826   10 − 3   e 0.01 i 5.046   10 − 3   e 0.01 i 1.0 5.289   10 − 3   e 0.02 i 4.826   10 − 3   e − 0.01 i 5.289   10 − 3   e − 0.02 i 1.0 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeWabaaabaqbaeqabeWaaaqaaGqabiab=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@28DD@

whose average values show good agreement with (12) and (17). The off-diagonal entries of each squark soft mass-squared are of similar size due to the democratic structure of the Yukawa couplings. The flavor-mixing entries mU˜2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacuWGvbqvgaacaaqaaiabikdaYaaaaaa@2EBB@ are the largest among all three mass squareds.

The flavor-violating entries of Yukawas, trilinear couplings and soft mass-squareds collectively generate radiative contributions γu,d, Γu,d to the Yukawa couplings below Mweak 1112. In fact, VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ (obtained from Yu,d(Mweak)) and VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ (obtained from Yu,deff) confront as follows:

| V C K M t r e e | | V C K M c o r r | = ( 0.9748 0.9685 0.2229 0.2490 0.0083 0.0085 0.2229 0.2489 0.9739 0.9674 0.0421 0.0463 0.0092 0.0104 0.0419 0.0459 0.99908 0.99889 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@B55E@

where left (right) window of MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaL4babaaaamaaL4babaaaaaaa@2B85@ in (i, j)-th entry refers to |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@(i, j)| (|VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ (i, j)|). Obviously, |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@| agrees very well with |VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@| in (7) entry by entry. This qualifies (20) to be the correct high-scale texture given present experimental determination of VCKM at Q = MZ. The most striking aspect of (23) is the fact that supersymmetric threshold corrections push VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ beyond the experimental bounds. More precisely, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 1)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(1, 2)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(2, 1)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(2, 2)|, |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(3, 3)| turn out to have 17.22σ, 14.21σ, 14.21σ, 15.22σ, 16.40σ significance levels, respectively. These are obviously far beyond the existing experimental 1.64σ significance intervals depicted in (7). As a result, supersymmetric threshold corrections are found to entirely disqualify the high-scale texture (20) to be the correct texture to reproduce the FCNC measurements at the weak scale.

Here, it is worthy of noting that deviation of |VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@(i, j)| from |VCKMtree MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzaaaaaa@352E@ (i, j)| (for i, j = 1, 2) turns out to be similar in size for CKM-ruled (see eq. 13) and democratic (see eq. 23) textures. It is smallest for the hierarchical texture (see eq. 18). Therefore, CKM-ruled texture in (10) and democratic one in (20) exhibit a pronounced sensitivity to supersymmetric threshold corrections in comparison to hierarchical texture in (15).

The physical quark fields, which arise after the unitary rotations (5), acquire the masses

M u ¯ ( M w e a k ) = diag . ( 0.055 , 1.27 , 144.78 ) , M d ¯ ( M w e a k ) = diag . ( 0.099 , 0.27 , 2.4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWaa0aaaeaaieqacqWFnbqtdaWgaaWcbaGae8xDauhabeaaaaGccqGGOaakcqWGnbqtdaWgaaWcbaGaem4DaCNaemyzauMaemyyaeMaem4AaSgabeaakiabcMcaPiabg2da9iabbsgaKjabbMgaPjabbggaHjabbEgaNjabc6caUiabcIcaOiabicdaWiabc6caUiabicdaWiabiwda1iabiwda1iabcYcaSiabigdaXiabc6caUiabikdaYiabiEda3iabcYcaSiabigdaXiabisda0iabisda0iabc6caUiabiEda3iabiIda4iabcMcaPiabcYcaSaqaamaanaaabaGae8xta00aaSbaaSqaaiab=rgaKbqabaaaaOGaeiikaGIaemyta00aaSbaaSqaaiabdEha3jabdwgaLjabdggaHjabdUgaRbqabaGccqGGPaqkcqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGUaGlcqGGOaakcqaIWaamcqGGUaGlcqaIWaamcqaI5aqocqaI5aqocqGGSaalcqaIWaamcqGGUaGlcqaIYaGmcqaI3aWncqGGSaalcqaIYaGmcqGGUaGlcqaI0aancqGGPaqkaaaaaa@708F@

all measured in GeV. In this physical basis for quark fields, VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ is responsible for charged current interactions in the effective theory below Mweak. The morale of the analysis above is that, the high-scale flavor structures (20) are to be modified in such a way that VCKMcorr MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGJbWycqWGVbWBcqWGYbGCcqWGYbGCaaaaaa@353A@ agrees with VCKMexp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAfawnaaDaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeaacqWGLbqzcqWG4baEcqWGWbaCaaaaaa@33DF@ with sufficient precision. Aftermath, the question is to predict quark masses appropriately at Q = Mweak so that, depending on the particle spectrum of the effective theory beneath, existing experimental values of quark masses at Q = MZ are reproduced correctly.

3.2 Inclusion of flavor violation from squark soft masses

In this section we extend GUT-scale flavor structures analyzed in Sec. 3.1 by switching on flavor mixings in certain squark soft mass-squareds. In other words, we maintain Yukawa textures to be one of (10), (15) or (20), and examine what happens to CKM prediction if squared masses of squarks possess non-trivial flavor mixings at the GUT scale.

The effective Yukawa couplings Yu,deff beneath Q = Mweak receive contributions from all entries of mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@(Mweak) via respective mass insertions 1112. Generically, larger the mass insertions larger the flavor violation potential of Yu,deff. Consequently, main problem is to determine the relative strengths of on-diagonal and off-diagonal entries of mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@(Mweak) given that they start with a certain pattern of flavor mixings. Take, for instance, mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@ which evolves with energy scale via (3) at single loop level. Its analytic solution is difficult, if not impossible, given coupled nature of RGEs and further complications brought about by the presence of flavor mixings. However, for an approximate yet instructive analysis, one can consider solving (3) for an infinitesimally small scale change from MGUT down to MGUT - ΔQ:

m Q 2 ( Δ t ) = m Q 2 ( 0 ) − 61 99 Δ t m 1 / 2 2 1 + Δ t { m Q 2 ( 0 ) + 2 m 0 2 1 , Y u † Y u + Y d † Y d } + 2 Δ t ( Y u A † Y u A + Y d A † Y d A ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeGadaaabaacbeGae8xBa02aa0baaSqaaiab=ffarbqaaiab=jdaYaaakiabcIcaOiabfs5aejabdsha0jabcMcaPaqaaiabg2da9aqaaiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaGccqGGOaakcqaIWaamcqGGPaqkcqGHsisljuaGdaWcaaqaaiabiAda2iabigdaXaqaaiabiMda5iabiMda5aaakiabfs5aejabdsha0jabd2gaTnaaDaaaleaacqaIXaqmcqGGVaWlcqaIYaGmaeaacqaIYaGmaaGccqWFXaqmaeaaaeaacqGHRaWkaeaacqqHuoarcqWG0baDdaGadaqaaiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaGccqGGOaakcqaIWaamcqGGPaqkcqGHRaWkcqaIYaGmcqWGTbqBdaqhaaWcbaGaeGimaadabaGaeGOmaidaaOGae8xmaeJaeiilaWIae8xwaK1aa0baaSqaaiab=vha1bqaaGqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWF1bqDaeqaaOGaey4kaSIae8xwaK1aa0baaSqaaiab=rgaKbqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWFKbazaeqaaaGccaGL7bGaayzFaaGaey4kaSIaeGOmaiJaeuiLdqKaemiDaq3aaeWaaeaacqWFzbqwdaqhaaWcbaGae8xDauhabaGae8xqaeKae4hiGyiaaOGae8xwaK1aa0baaSqaaiab=vha1bqaaiab=feabbaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqcqGFGaIHaaGccqWFzbqwdaqhaaWcbaGae8hzaqgabaGae8xqaeeaaaGccaGLOaGaayzkaaaaaaaa@8369@

with the scale variable Δt = (4π)-2 log(1 - ΔQ/MGUT). Here right-handed squark mass-squareds are taken strictly flavor-diagonal, for simplicity. This approximate solution gives enough clue that diagonal entries of mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@ tend to take hierarchically large values at the IR due to the gluino mass contribution, mainly. However, its off-diagonal entries do not have such an enhancement source:

( m Q 2 ) i j ( Δ t ) = ( m Q 2 ) i j ( 0 ) + Δ t [ ( m Q 2 ) i i ( 0 ) + ( m Q 2 ) j j ( 0 ) + 4 m 0 2 ] ( Y u † Y u + Y d † Y d ) i j + Δ t [ ( Y u † Y u + Y d † Y d ) i i + ( Y u † Y u + Y d † Y d ) j j ] ( m Q 2 ) i j ( 0 ) + Δ t ( Y u † Y u + Y d † Y d ) 6 − ( i + j ) j ( m Q 2 ) i 6 − ( i + j ) ( 0 ) + Δ t ( Y u † Y u + Y d † Y d ) i 6 − ( i + j ) ( m Q 2 ) 6 − ( i + j ) j ( 0 ) + 2 Δ t ( Y u A † Y u A + Y d A † Y d A ) i j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeqbdaaaaeaadaqadaqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaakiaawIcacaGLPaaadaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcIcaOiabfs5aejabdsha0jabcMcaPaqaaiabg2da9aqaamaabmaabaGae8xBa02aa0baaSqaaiab=ffarbqaaiab=jdaYaaaaOGaayjkaiaawMcaamaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiikaGIaeGimaaJaeiykaKIaey4kaSIaeuiLdqKaemiDaq3aamWaaeaadaqadaqaaiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaakiaawIcacaGLPaaadaWgaaWcbaGaemyAaKMaemyAaKgabeaakiabcIcaOiabicdaWiabcMcaPiabgUcaRmaabmaabaGae8xBa02aa0baaSqaaiab=ffarbqaaiab=jdaYaaaaOGaayjkaiaawMcaamaaBaaaleaacqWGQbGAcqWGQbGAaeqaaOGaeiikaGIaeGimaaJaeiykaKIaey4kaSIaeGinaqJaemyBa02aa0baaSqaaiabicdaWaqaaiabikdaYaaaaOGaay5waiaaw2faamaabmaabaGae8xwaK1aa0baaSqaaiab=vha1bqaaGqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWF1bqDaeqaaOGaey4kaSIae8xwaK1aa0baaSqaaiab=rgaKbqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWFKbazaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdMgaPjabdQgaQbqabaaakeaaaeaacqGHRaWkaeaacqqHuoarcqWG0baDdaWadaqaamaabmaabaGae8xwaK1aa0baaSqaaiab=vha1bqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWF1bqDaeqaaOGaey4kaSIae8xwaK1aa0baaSqaaiab=rgaKbqaaiab+bcigcaakiab=LfaznaaBaaaleaacqWFKbazaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdMgaPjabdMgaPbqabaGccqGHRaWkdaqadaqaaiab=LfaznaaDaaaleaacqWF1bqDaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8xDauhabeaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8hzaqgabeaaaOGaayjkaiaawMcaamaaBaaaleaacqWGQbGAcqWGQbGAaeqaaaGccaGLBbGaayzxaaWaaeWaaeaacqWFTbqBdaqhaaWcbaGae8xuaefabaGae8NmaidaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGOaakcqaIWaamcqGGPaqkaeaaaeaacqGHRaWkaeaacqqHuoarcqWG0baDdaqadaqaaiab=LfaznaaDaaaleaacqWF1bqDaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8xDauhabeaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8hzaqgabeaaaOGaayjkaiaawMcaamaaBaaaleaacqaI2aGncqGHsislcqGGOaakcqWGPbqAcqGHRaWkcqWGQbGAcqGGPaqkcqWGQbGAaeqaaOWaaeWaaeaacqWFTbqBdaqhaaWcbaGae8xuaefabaGae8NmaidaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdMgaPjabiAda2iabgkHiTiabcIcaOiabdMgaPjabgUcaRiabdQgaQjabcMcaPaqabaGccqGGOaakcqaIWaamcqGGPaqkaeaaaeaacqGHRaWkaeaacqqHuoarcqWG0baDdaqadaqaaiab=LfaznaaDaaaleaacqWF1bqDaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8xDauhabeaakiabgUcaRiab=LfaznaaDaaaleaacqWFKbazaeaacqGFGaIHaaGccqWFzbqwdaWgaaWcbaGae8hzaqgabeaaaOGaayjkaiaawMcaamaaBaaaleaacqWGPbqAcqaI2aGncqGHsislcqGGOaakcqWGPbqAcqGHRaWkcqWGQbGAcqGGPaqkaeqaaOWaaeWaaeaacqWFTbqBdaqhaaWcbaGae8xuaefabaGae8NmaidaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabiAda2iabgkHiTiabcIcaOiabdMgaPjabgUcaRiabdQgaQjabcMcaPiabdQgaQbqabaGccqGGOaakcqaIWaamcqGGPaqkaeaaaeaacqGHRaWkaeaacqaIYaGmcqqHuoarcqWG0baDdaqadaqaaiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqcqGFGaIHaaGccqWFzbqwdaqhaaWcbaGae8xDauhabaGae8xqaeeaaOGaey4kaSIae8xwaK1aa0baaSqaaiab=rgaKbqaaiab=feabjab+bcigcaakiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqaaaakiaawIcacaGLPaaadaWgaaWcbaGaemyAaKMaemOAaOgabeaaaaaaaa@253C@

unless Yukawas or trilinear couplings are given appropriate boundary configurations at the GUT scale. That this is the case can be seen explicitly by considering, for instance, democratic texture for Yukawas (20) together with (9) and strict universality and flavor-diagonality of the soft masses, except

m Q 2 ( 0 ) = m 0 2 ( 1 1 1 1 1 1 1 1 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaaieqacqWFTbqBdaqhaaWcbaGae8xuaefabaGae8NmaidaaOGaeiikaGIaeGimaaJaeiykaKIaeyypa0JaemyBa02aa0baaSqaaiabicdaWaqaaiabikdaYaaakmaabmaabaqbaeqabmWaaaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaqaaiabigdaXaaaaiaawIcacaGLPaaaaaa@3F36@

which contributes maximally to each term of (26). Even with such a democratic pattern for Yukawas, trilinear couplings and mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@(0), however, one obtains at Mweak = 1 TeV

m Q 2 = ( 533.37  GeV ) 2 ( 1.0 − 0.0512 − 0.0510 − 0.0512 1.0 − 0.0513 − 0.0510 − 0.0513 1.0 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6D07@

with similar structures for mU2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFvbqvaeaacqWFYaGmaaaaaa@2EA5@ and mD2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFebaraeaacqWFYaGmaaaaaa@2E83@. Alternatively, if one adopts (10) or (15) setups the off-diagonal entries of squark soft mass-squareds at Mweak are found to remain around m02 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqaIWaamaeaacqaIYaGmaaaaaa@2E67@ which are much smaller than the on-diagonal ones. Therefore, Yukawa textures (and hence those of the trilinear couplings) studied in Sec. 3.1 lead one generically to hierarchic textures for squark soft mass-squareds at Q = Mweak irrespective of how large the flavor mixings in mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@(0) might be. In fact, predictions for CKM matrix remain rather close to those in Sec. 3.1 above. This is actually clear from (26) where off-diagonal entries of mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@ are seen to evolve into new mixing patterns via themselves and those of Yukawas and trilinear couplings. In conclusion, evolution of squark soft masses is fundamentally Yukawa-ruled and when Yukawas at the GUT scale are taken to shoot the measured value of CKM matrix, the mass insertions associated with mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@(Mweak) are too small to give any significant contribution to Yu,deff.

As follows from (26), for generating sizeable off-diagonal entries for mQ,U,D2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbqucqGGSaalcqWFvbqvcqGGSaalcqWFebaraeaacqWFYaGmaaaaaa@3299@(Mweak) it is necessary to abandon either Yukawa textures analyzed in Sec. 3.1. or proportionality of trilinear couplings with Yukawas. Therefore, we take Yukawa couplings at the GUT scale precisely as (20), we maintain (9) for both YdA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWFKbazaeaacqWFbbqqaaaaaa@2EB9@ and YeA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWFLbqzaeaacqWFbbqqaaaaaa@2EBB@, and we take mU2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFvbqvaeaacqWFYaGmaaaaaa@2EA5@(0) and mD2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFebaraeaacqWFYaGmaaaaaa@2E83@(0) strictly flavor-diagonal as in all three case studies carried out in Sec. 3.1. However, we take mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=1gaTnaaDaaaleaacqWFrbquaeaacqWFYaGmaaaaaa@2E9D@(0) as (27) above, and YuA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqaaaaaa@2EDB@ as

Y u A ( 0 ) = − 150  GeV ( 1 1 1 1 1 1 1 1 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaaieqacqWFzbqwdaqhaaWcbaGae8xDauhabaGae8xqaeeaaOGaeiikaGIaeGimaaJaeiykaKIaeyypa0JaeyOeI0IaeGymaeJaeGynauJaeGimaaJaeeiiaaIaee4raCKaeeyzauMaeeOvay1aaeWaaeaafaqabeWadaaabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedabaGaeGymaedaaaGaayjkaiaawMcaaaaa@441D@

which certainly violates (9) that enforces trilinears to be proportional to the corresponding Yukawas. Then two-loop RG running from Q = MGUT down to Q = Mweak gives

Y u A = ( − 262.087 − 259.342 − 260.709 − 259.474 − 261.954 − 260.709 − 260.688 − 260.674 − 260.735 ) Y d A = ( − 41.435 37.091   e − 0.0127 i − 40.408   e 0.0124 i − 36.171   e 0.0102 i 52.230   e − 0.0019 i − 41.558   e 0.0228 i 39.983   e − 0.0300 i 42.075   e − 0.0224 i − 41.683   e 0.0025 i ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@0190@

both measured in GeV at Mweak = 1 TeV. Though not shown explicitly, each entry of YuA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqabiab=LfaznaaDaaaleaacqWF1bqDaeaacqWFbbqqaaaaaa@2EDB@ is complex with a phase around 10-7 - 10-6 in size. On the other hand, squark soft mass-squared at Q = Mweak are given by

m Q 2 = ( 516.58  GeV ) 2 ( 1.0 − 0.13 − 0.13 − 0.13 1.0 − 0.13 − 0.13 − 0.13