Abstract
In recent years it has been learned that scalar superpartner masses and trilinear couplings should both generically be larger than about 20 TeV at the short-distance string scale if our world is described by a compactified string or M-theory with supersymmetry breaking and stabilized moduli (Acharya et al. in arXiv:1006.3272 [hep-ph], 2010). Here we study implications of this, somewhat generally and also in detail for a particular realization (compactification of M-theory on a G 2 manifold) where there is significant knowledge of the superpotential and gauge kinetic function, and a light gluino. In a certain sense this yields an ultraviolet completion of minimal flavour violation. Flavour violation stems from off-diagonal and non-universal diagonal elements of scalar mass matrices and trilinear couplings, and from renormalization group running. We also examine stability bounds on the scalar potential. While heavy scalars alone do not guarantee the absence of flavour problems, our studies show that models with heavy scalars and light gluinos can be free from such problems.
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Notes
If this inequality is violated, there arises too large an off-diagonal term in the super-CKM basis.
The values used here are those from [28]. The functions S(x i ,x j ) are the well-known Inami–Lim functions [29] entering in the SM box contributions, S(x) is listed also in Appendix A.2. \(G_{\tilde{g}}(x_{\tilde{g}})\) is the loop function of the box diagram involving internal gluinos and squarks and it is defined in Appendix A.2.
That is, \(Y^{f}_{\mathrm{diag}}=\hat{Y}^{f}=U^{f}_{\mathrm{R}}Y^{f} U^{f\dagger}_{\mathrm{L}}\) and consequently trilinear terms are rotated as \(\hat{a}^{f}=U^{f}_{\mathrm{R}}a^{f} U^{f\dagger}_{\mathrm{L}}\) and soft mass-squared matrices as \({\hat{m}}^{2}_{\tilde{f}\mathrm{LL}}=U^{f}_{\mathrm{L}} m^{2}_{\tilde{Q}} U^{f\dagger}_{\mathrm{L}}\) and \({\hat{m}}^{2}_{\tilde{f}\mathrm{RR}}=U^{f}_{\mathrm{R}} m^{2}_{\tilde{f}} U^{f\dagger}_{\mathrm{R}}\).
ϵ′ being the parameter measuring the direct CP violation in the decay amplitude of K→2π, \(\epsilon^{\prime i(\delta_{2}-\delta_{0})} \mathrm{Re}[A_{2}](\mathrm{Re} [A_{0}] \mathop {\mathrm {Im}}[A_{2}]/\mathrm{Re}[A_{2}]-\mathop {\mathrm {Im}}[A_{0}])/(\sqrt{2} \mathrm{Re}[A_{0}])\), \(A_{I} e^{i\delta_{I}}=\langle\pi\pi(I)| H^{\Delta S=1}_{\mathrm{eff}}| K^{0} \rangle\), I=0,2.
The simplified expression (16) is derived considering the \(\mathcal{D}\)-flat direction \(\alpha^{2}=|H_{d}^{0}|^{2}+|\tilde{\nu}_{m}|^{2}=|\tilde{e}_{L_{i}}|^{2}=|\tilde{e}_{R_{j}}|^{2}\) (m≠i,j) in the limit \(\alpha\gg[m^{2}_{H_{d}}+|\mu|^{2}-(\hat{m}^{2}_{\tilde{\nu}})_{mm}]/[(\hat{Y}_{ii}^{e})^{2} +(\hat{Y}_{jj}^{e})^{2}]\) with \(\alpha^{2}>|H_{d}^{0}|^{2}\) [42].
Neither program is completely suited for precisely the calculation required here.
We express ϵ as \(\epsilon= \epsilon^{\mathrm{SM}} + \delta\epsilon^{\mathrm{SUSY}}, \quad\delta\epsilon^{\mathrm{SUSY}}= \delta\epsilon^{H^{\pm}} + \delta \epsilon^{\tilde{\chi}^{\pm}}+ \delta\epsilon^{\tilde{\chi}^{0}} + \delta\epsilon^{\tilde{\chi}^{0} \tilde{g}} + \delta\epsilon^{\tilde{g}}\), where δϵ SUSY is the total SUSY contribution and the individual terms refer to the charged Higgs, the chargino, the neutralino, the neutralino–gluino, and the gluino contribution, respectively.
For the uncertainties in the hadronic matrix element calculations, we have used the bag parameters of [50].
One could start working in the basis where Y d is diagonal and Y u is not. Recall that it is not possible to work in a basis where both are diagonal precisely due to the CKM matrix.
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Acknowledgements
We particularly thank Bobby Acharya for encouragement and participating in the initial stages of the analysis, and we also thank Francesca Borzumati for clarifications on issues regarding b→sγ. We thank Luca Silvestrini for clarifications on the loop functions entering into the Wilson coefficients of the ΔS=2 Hamiltonian. We thank Brent Nelson and Werner Porod for useful discussions. J.K. would like to thank CINVESTAV in Mexico City for hospitality during stages of this work. This work was supported by the Michigan Center for Theoretical Physics, and by the German Science Foundation (DFG) via the Junior Research Group “SUSY Phenomenology” within the Collaborative Research Centre 676 “Particles, Strings and the Early Universe”.
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Appendices
Appendix A: Notation
1.1 A.1 Wilson coefficients
We follow various references [30, 31, 66] for the extraction of the effective Hamiltonians. The ΔS=2 operators involved in (5) are
where P L and P R are the left- and right-handed projection operators, respectively, \(\tilde{O}_{i}= O_{i} (L\leftrightarrow R)\), and \(\langle O_{i}\rangle =\langle\tilde{O}_{i}\rangle \).
1.2 A.2 Loop functions
We collect in this appendix the loop functions that we have used in our analysis.
Appendix B: Details of the running from M G down to \(\protect \mu_{\tilde{f}}\)
We take the scalar soft-squared mass matrices to be proportional to the unit matrix at M G. Their running to \(\mu_{\tilde{f}} \sim m_{3/2} \), the scale at which the scalars decouple, will produce off-diagonal entries. We require these off-diagonal elements to be significantly smaller than the diagonal elements, since otherwise the mass-insertion approximation would not be justified. To be concrete and conservative, let us consider m 3/2=m 0=20 TeV and demand
at \(\mu_{\tilde{f}}\). As we consider small values of tanβ, we can neglect the contributions of Y d and a d to the running. Furthermore, we consider CKM-like matrices diagonalizing Y u, which implies \(Y^{u\dagger} Y^{u} \sim Y^{u} Y^{u\dagger} \sim y_{t}^{2} \mathop {\mathrm {diag}}(0,0,1)\). Consequently, Y u does not affect the running of the off-diagonal elements of \(m^{2}_{\tilde{f}}\). Of course, the same is true of the terms in the renormalization group equation (RGE) of \(m^{2}_{\tilde{f}}\) that involve gauge couplings. Thus, the only relevant terms in the RGE are those proportional to a u† a u and a u a u†. Approximating the right-hand side of the RGE by a constant value (leading-log approximation), we then obtain at \(\mu_{\tilde{f}}\)
Using (2, 23), \(U^{u}_{\mathrm{L}} \sim U^{u}_{\mathrm{R}} \sim V_{\mathrm{CKM}}\) as well as \(Y^{u}_{\mathrm{diag}} \sim \mathop {\mathrm {diag}}(\lambda^{8},\lambda^{4},1)\), where λ≈0.23 is the sine of the Cabibbo angle, and assuming no accidental cancellations, we obtain
where x max is the maximum value of \(x^{f}_{ij}\). Thus, the strongest constraint stems from \(\tilde{f} = \tilde{u}\) and ij=23 in (35),
With \(A_{\tilde{f}} = 1.5 \frac{m_{0}^{2}}{m_{3/2}} \approx30~\mbox {TeV}\), this yields x max∼1.8. To be conservative, we have chosen \(x_{\mathrm{max}} = \sqrt{2}\) for our numerical analysis.
Appendix C: Comments on MFV
Trilinear terms
The term MFV [17] refers to scenarios where all higher-dimensional operators, constructed from SM and fields with Yukawa interactions, are invariant under CP and under the flavour group G F . Here \(G_{F} = \mathrm {SU}(3)_{q_{L}}\ {\otimes}\ \mathrm {SU}(3)_{u_{R}}\ {\otimes}\ \mathrm {SU}(3)_{d_{R}} \ {\otimes}\ \mathrm {SU}(3)_{l_{L}}\ {\otimes}\ \mathrm {SU}(3)_{e_{R}}\ {\otimes} \mathrm {U}(1)_{B} \ {\otimes}\ \mathrm {U}(1)_{L} \ {\otimes}\ \mathrm {U}(1)_{Y}\ {\otimes}\ \mathrm {U}(1)_{PQ} \ {\otimes}\ \mathrm {U}(1)_{e_{R}}\), and the Yukawa couplings are formally regarded as auxiliary fields that transform under G F . As a consequence, MFV requires that the dynamics of flavour violation is completely determined by the structure of the ordinary Yukawa couplings and in particular, all CP violation originates from the CKM phase.
Because of the running of all couplings of a theory, this scenario can only be realized at one particular scale, usually a low energy scale. Starting with parameters defined at M G, MFV can only be a good approximation at M EW, if
where A f is a universal mass parameter for all families and kind of fermions that is small in comparison with other soft masses of the theory and if Yukawa couplings are small. This can be analyzed by studying the dependence of the RGEs of Yukawa and trilinear couplings on Y f [16, 67].Footnote 9 If just the third-family Yukawa couplings are evolved, of course the size of A f does not matter because no off-diagonal terms are produced. With a full RG evolution of complex 3×3 Yukawa and trilinear matrices with small off-diagonal values, MFV can be emulated, albeit never reproduced, for sufficiently small values of A f [16].
Soft-squared masses
The one-loop running of the soft-squared parameters \((m^{2}_{\tilde{f}})_{ij}\) in the SCKM basis is governed by the β functions
where f∈{u,d} and the functions \(G_{m^{2}_{\tilde{f}}}\) contain flavour-diagonal contributions to the running involving gauge couplings and gaugino masses. Note that at an arbitrary scale μ≠M G, the terms which contain
are not diagonal because of the different running of the diagonal elements in \(m^{2}_{\tilde{Q}}\) and \(m^{2}_{\tilde{f}}\). Therefore, off-diagonal terms will necessarily be induced.
Recall that even if we consider only the running of the Yukawa couplings of the third family, this will produce a split in the masses of \(m^{2}_{\tilde{f}}\). We can always choose to go to the basis where one of the Yukawa couplings is diagonal at M G, but this does not guarantee diagonal soft mass-squared matrices in the SCKM basis because the fact that
necessarily implies that not all of the matrices \(\hat{m}^{2}_{\tilde{f}\mathrm{LL}} = U^{f}_{L}m^{2}_{\tilde{Q}}U^{f\dagger}_{L}\) and \(\hat{m}^{2}_{\tilde{f}\mathrm{RR}} = U^{f}_{R}m^{2}_{\tilde{f}}U^{f\dagger}_{R}\) are diagonal. As is known ϵ is very sensitive to this [68]. If the coupling of the particles beyond the SM was of the same order as that of the SM particles, this would push the limit on the scale of new physics entering into the ΔS=2 processes up to
In the lepton sector, we assume heavy right-handed neutrinos that decouple close to the GUT scale. Therefore only the superpartners of right- and left-handed charged leptons as well as left-handed neutrinos can induce flavour violation. Considering the structure of fermion masses we are using, see Sect. 5.4 and references therein, the Yukawa coupling matrix for neutrinos is the same as that for the up-quark sector at M G, therefore the flavour violation induced in this scenario is relatively small. Table 4 shows the MI parameters relevant for the observables ℓ i →ℓ j γ.
In the G 2-MSSM case, where all the examples that are known [6] correspond to the case that trilinear couplings are proportional to Yukawa couplings, we can have a theory, depending on the choice of Yukawa couplings, for which at low energy, all flavour violation present is below the experimental bounds. It is only in this sense that we can say that we have an ultraviolet version of MFV but not in the sense in which MFV is defined. For the case of the relation (2) with \(c^{f}_{ij}\) as in (22), that is, O(1) real random numbers between 0 and \(\sqrt{2}\), at low energies CP violating phases in addition to the CKM phase appear but also flavour violation is below experimental bounds. With \(c^{f}_{ij}\) as in (23), i.e., with random numbers between 0 and \(\sqrt{2}\) and explicit CP phases at M G, flavour violation is more difficult to neglect but still below the experimental bounds. This means that even with more CP phases than in the SM and large mixing present due to the choice of Yukawa couplings, after the running to low energy we obtain a theory which satisfies flavour-violation constraints. Not surprisingly, the reason are the large scalar masses. What is not a trivial result of the analysis is that still bounds on the size of trilinear terms can be obtained.
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Kadota, K., Kane, G., Kersten, J. et al. Flavour issues for string-motivated heavy scalar spectra with a low gluino mass: the G 2-MSSM case. Eur. Phys. J. C 72, 2004 (2012). https://doi.org/10.1140/epjc/s10052-012-2004-3
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DOI: https://doi.org/10.1140/epjc/s10052-012-2004-3