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
(9)
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
(10)
with no flavor violation in the lepton sector: Y
e
= diag. (1.9 10-5, 4 10-3, 0.071). The flavor violation effects are entirely encoded in Y
d
which exhibits a CKM-ruled hierarchy in similarity to Yukawa textures analyzed in [5]i.e. this choice of boundary values of the Yukawas leads to correct CKM matrix [26] at M
weak
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 [20–24] with boundary conditions (9), attain the flavor structures
(11)
both measured in GeV at M
weak
= 1 TeV. Clearly, is essentially diagonal whereas (2, 3), (3, 2) and (2, 2) entries of 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 M
weak
= 1 TeV:
(12)
with = (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 [11, 12] introduces certain corrections to the tree-level Yukawa matrices Yu,d(M
weak
) to generate in (4). In fact, (obtained from Yu,d(M
weak
)) and (obtained from Yu,deff) compare to exhibit spectacular differences:
where left (right) window of
in (i, j)-th entry refers to . Clearly, || agrees very well with || in (7) entry by entry. This qualifies (10) to be the correct high-scale texture given experimental FCNC bounds at Q = M
Z
. However, radiative corrections induced by decoupling of squarks, gluinos and Higgsinos at the supersymmetric threshold M
weak
= 1 TeV is seen to leave a rather strong impact on the CKM entries. Consider for instance (1, 1) entries of , and . Present experiments provide a 1.64σ significance to |(1, 1)| around a mean value of 0.745 as is seen from (7). The tree-level prediction, |(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 |(1, 1)| = 0.9795. This value is obviously far beyond the existing experimental limits as it is a 13.39σ effect. Similarly, |(1, 2)|, |(2, 1)|, |(2, 2)| and |(3, 3)| are, respectively, 12.36σ, 12.36σ, 11.95σ and 2.30σ effects. Obviously, deviation of |(i, j)| from |(i, j|) (comparison with experiments at Q = M
Z
is meaningful especially for (i, j) = (1, 1), (1, 2), (2, 1), (3, 3) entries whose scale dependencies are known to be rather mild [29–31]), 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
(14)
all measured in GeV. In this physical basis for quark fields, 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 = M
Z
to make comparisons with experimental results. This evolution depends on the effective theory below M
weak
. 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 M
weak
.
3.1.2 Hierarchical texture
The Yukawa couplings are taken to have the structure (as can be motivated from [35–38])
(15)
with no flavor violation in the lepton sector: Y
e
= diag. (1.9 10-5, 0.004, 0.071). Here both Y
u
and Y
d
exhibit a hierarchically organized pattern of entries. In a sense, the hierarchic nature of Y
d
in (10) is now extended to Y
u
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 [20–24] with boundary conditions (9), obtain the flavor structures
(16)
both measured in GeV at M
weak
= 1 TeV. Clearly, in contrast to (11), now both and 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 M
weak
= 1 TeV:
(17)
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,dto the Yukawa couplings below M
weak
[11, 12]. In fact, (obtained from Yu,d(M
weak
)) and (obtained from Yu,deffconfront as follows:
where left (right) window of
in (i, j)-th entry refers to |(i, j)| (|(i, j)|). Clearly, || 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 V
CKM
at Q = M
Z
. However, this agreement between experiment and theory gets spoiled strongly by the inclusion of supersymmetric threshold corrections. Indeed, as is shown comparatively by (23), violates the bounds in (7) significantly. More precisely, |(1, 1)|, |(1, 2)|, | (2, 1)|, |(2, 2)|, |(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
(19)
all measured in GeV. In this physical basis for quark fields, is responsible for charged current interactions in the effective theory below M
weak
. The morale of the analysis above is that, the high-scale flavor structures (15) are to be modified in such a way that agrees with with sufficient precision. Aftermath, the question is to predict quark masses appropriately at Q = M
weak
so that, depending on the particle spectrum of the effective theory beneath, existing experimental values of quark masses at Q = M
Z
are reproduced correctly.
3.1.3 Democratic texture
In this subsection, we take Yukawa couplings to be (as can be motivated from relevant works [39–41])
(20)
with no flavor violation in the lepton sector: Y
e
= diag. (1.9 10-5, 4 10-3, 0.071). Here both Y
u
and Y
d
exhibit an approximate democratic structure so that Yu,d(M
weak
) 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 [39–41]. 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 [20–24] with boundary conditions (9), obtain the flavor structures
(21)
both measured in GeV at M
weak
= 1 TeV. Though not shown explicitly, each entry of 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 M
weak
= 1 TeV:
(22)
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 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,dto the Yukawa couplings below M
weak
[11, 12]. In fact, (obtained from Yu,d(M
weak
)) and (obtained from Yu,deff) confront as follows:
where left (right) window of
in (i, j)-th entry refers to |(i, j)| (| (i, j)|). Obviously, || agrees very well with || in (7) entry by entry. This qualifies (20) to be the correct high-scale texture given present experimental determination of V
CKM
at Q = M
Z
. The most striking aspect of (23) is the fact that supersymmetric threshold corrections push beyond the experimental bounds. More precisely, |(1, 1)|, |(1, 2)|, |(2, 1)|, |(2, 2)|, |(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 |(i, j)| from | (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
(24)
all measured in GeV. In this physical basis for quark fields, is responsible for charged current interactions in the effective theory below M
weak
. The morale of the analysis above is that, the high-scale flavor structures (20) are to be modified in such a way that agrees with with sufficient precision. Aftermath, the question is to predict quark masses appropriately at Q = M
weak
so that, depending on the particle spectrum of the effective theory beneath, existing experimental values of quark masses at Q = M
Z
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,deffbeneath Q = M
weak
receive contributions from all entries of (M
weak
) via respective mass insertions [11, 12]. 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 (M
weak
) given that they start with a certain pattern of flavor mixings. Take, for instance, 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 M
GUT
down to M
GUT
- ΔQ:
(25)
with the scale variable Δt = (4π)-2 log(1 - ΔQ/M
GUT
). Here right-handed squark mass-squareds are taken strictly flavor-diagonal, for simplicity. This approximate solution gives enough clue that diagonal entries of 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:
(26)
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
(27)
which contributes maximally to each term of (26). Even with such a democratic pattern for Yukawas, trilinear couplings and (0), however, one obtains at M
weak
= 1 TeV
(28)
with similar structures for and . Alternatively, if one adopts (10) or (15) setups the off-diagonal entries of squark soft mass-squareds at M
weak
are found to remain around 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 = M
weak
irrespective of how large the flavor mixings in (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 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 (M
weak
) are too small to give any significant contribution to Yu,deff.
As follows from (26), for generating sizeable off-diagonal entries for (M
weak
) 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 and , and we take (0) and (0) strictly flavor-diagonal as in all three case studies carried out in Sec. 3.1. However, we take (0) as (27) above, and as
(29)
which certainly violates (9) that enforces trilinears to be proportional to the corresponding Yukawas. Then two-loop RG running from Q = M
GUT
down to Q = M
weak
gives
(30)
both measured in GeV at M
weak
= 1 TeV. Though not shown explicitly, each entry of is complex with a phase around 10-7 - 10-6 in size. On the other hand, squark soft mass-squared at Q = M
weak
are given by
(31)
where small phases in off-diagonal entries of and are neglected. A comparison with (22) reveals spectacular enhancements in mass insertions pertaining and .
The trilinear couplings (30) and squark mass-squareds (31) give rise to non-trivial changes in flavor structures of Yu,d(M
weak
) by generating effective Yukawas Yu,deffbeneath Q = M
weak
. Then the CKM matrix obtained from Yu,d(M
weak
) and obtained from Yu,deffcompare as:
where left (right) window of
in (i, j)-th entry refers to |(i, j)| (|(i, j)|). Obviously, || agrees very well with || as was the case in (23). This qualifies (20) to be the correct high-scale texture given present experimental determination of V
CKM
at Q = M
Z
. However, implementation of supersymmetric threshold corrections is seen to leave a big impact on certain entries of the physical CKM matrix. Indeed, |(1, 1)|, |(1, 2)|, |(2, 1)|, |(2, 2)|, |(3, 3)| turn out to have 6.06σ, 23.99σ, 23.89σ, 26.52σ, 4.35σ significance levels, respectively. These are to be contrasted with standard deviations computed for (23) in Sec. 3.1.3 above. Needless to say, these deviations are far beyond the experimental sensitivities and thus supersymmetric threshold corrections completely disqualify the flavor textures (20) in a way different than (23) due to new structures (27) and (29).
Finally, physical quark fields, which arise after the unitary rotations (5), acquire the masses
(33)
all measured in GeV. These mass predictions are close to those obtained within democratic texture. As in all cases discussed in Sec. 3.1. especially light quark masses fall outside the existing experimental bounds, and choice of the correct high-scale texture must reproduce both and quark masses in sufficient agreement with experiment.
3.3 A purely soft CKM ?
In Sec. 3.1 and 3.2 we have discussed how prediction for the CKM matrix depends crucially on the inclusion of the supersymmetric threshold corrections. This we did by negation i.e. we have taken certain Yukawa textures which are known to generate CKM matrix correctly at tree level, and then included threshold corrections to demonstrate how those the would-be viable flavor structures get disqualified.
In this section we will do the opposite i.e. we will take a Yukawa texture which is known not to work at all, and incorporate supersymmetric threshold corrections to show how it can become a viable one, at least approximately. For sure, a highly interesting limit would be to start with exactly diagonal Yukawas at the GUT scale and generate CKM matrix beneath M
weak
via purely soft flavor violation i.e. flavor violation from sfermion soft mass-squareds and trilinear couplings, alone. However, this limit seems difficult to realize, at least for SPS1a' parameter values, since it may require tuning of various parameters, in particular, soft mass-squareds of Higgs and quark sectors [11, 12]. Even if this is done by a fine-grained scan of the parameter space, it will possibly cost a great deal of fine-tuning. Indeed, threshold corrections depend on ratios of the soft masses [11], and generating a specific entry of the CKM matrix can require a judiciously arranged hierarchy among various soft mass parameters – a parameter region certainly away from the SPS1a' point.
Therefore, we relax the constraint of strict diagonality and consider instead GUT-scale Yukawa matrices with five texture zeroes which are known to be completely unphysical as they cannot induce the CKM matrix [42]. In fact, this kind of textures has recently been found to arise from heterotic string [43] when the low-energy theory is constrained to be minimal supersymmetric model [44, 45]. Consequently, we take Yukawas at Q = M
GUT
to be
(34)
with no flavor violation in the lepton sector: Y
e
= diag. (1.9 10-5, 0.004, 0.071). Both Y
u
and Y
d
are endowed with five texture zeroes, and they precisely conform to the structures found in effective theories coming from the heterotic string [43].
Besides, though left unspecified in [43], we take sfermion mass-squareds strictly flavor-diagonal as in Sec. 3.1, and let obey (9). For trilinear couplings pertaining to squark sector we take
(35)
both measured in GeV. These trilinear couplings do not obey (9); they are given completely independent flavor structures, in particular, they exhibit (1) mixing between second and third generations. The first generation of squarks is decoupled from the rest completely.
Two-loop RG running down to Q = M
weak
modifies GUT-scale textures (35) to give
(36)
both measured in GeV. The texture zeroes in (35) are seen to elevated to small yet nonzero values via RG running. The squark soft mass-squareds, on the other hand, exhibit the following flavor structures at M
weak
= 1 TeV:
(37)
where off-diagonal entries are seen to be hierarchically small so that contributions to Yu,defffrom squark soft mass-squareds are expected to be rather small.
The use of Yukawas, trilinear couplings and squark mass-squareds, all rescaled to M
weak
= 1 TeV via RG running, give rise to modifications in Yukawa couplings after squarks being integrated out. In fact, the CKM matrix obtained from Yu,d(M
weak
) and obtained from Yu,deffcompare as:
where left (right) window of
in (i, j)-th entry refers to |(i, j)| (|(i, j)|).
It is clear that by no means qualifies to be a realistic CKM matrix: |(i, j)| = 0 for (i, j) = (1, 3), (3, 1), (2, 3), (3, 2); moreover, Cabibbo angle is predicted to be one order of magnitude smaller. In addition, its diagonal elements turn out to be well outside the experimental limits. However, once supersymmetric threshold corrections are included certain entries are found to attain their experimentally preferred ranges. Indeed, |(1, 1)| and |(3, 1)| fall right at their upper bounds, and |(1, 3)| far exceeds the experimental bound. The predictions for these entries are not good enough; they need to be correctly predicted by further arrangements of the GUT-scale textures. Nevertheless, for the main purpose of illustrating how threshold corrections influence flavor structures at the IR end, the results above are good enough for what has to be shown since all other entries turn out to be in rather good agreement with experimental bounds. The case study illustrated here shows that, even unphysical Yukawa textures with five texture zeroes, can lead to acceptable CKM matrix predictions once supersymmetric threshold corrections are incorporated into Yukawa couplings.
The corrected Yukawa couplings lead to the following quark mass spectrum:
(39)
all measured in GeV. These predictions are not violatively outside the experimental limits, except for the up quark mass. A rehabilitated choice for the GUT-scale textures (34) should lead to a fully consistent prediction for CKM matrix (with much better precision than in, especially the (1, 3), (3, 1) entries of (38) above) together with precise predictions for quark masses (modulo sizeable QCD corrections while running from Q = M
weak
down to hadronic scale).