Isosinglet down quark mixing and CP violation experiments

We confront the new physics models with extra isosinglet down quarks in the new CP violation experimental era with sin(2 b ) and e 8 / e measurements, K 1 ! p 1 nn ¯ events, and x s limits. The closeness of the new experimental results to the standard model theory requires us to include full standard model ~ SM ! amplitudes in the analysis. In models allowing mixing to a new isosinglet down quark, as in E 6 , ﬂavor changing neutral currents are induced that allow a Z 0 mediated contribution to B - B¯ mixing and which bring in new phases. In ( r , h ), (cid:132) x s ,sin( g ) (cid:133) , and (cid:132) x s ,sin(2 f s ) (cid:133) plots we still ﬁnd much larger regions in the four down quark model than in the SM, reaching down to h’ 0, 0 < sin( g ) < 1, 2 0.75 < sin(2 a ) < 0.15, and sin(2 f s ) down to zero, all at 1 s . We elucidate the nature of the cancellation in an order l 5 four down quark mixing matrix element which satisﬁes the experiments and reduces the number of independent angles and phases. We also evaluate tests of unitarity for the 3 3 3 Cabibbo-Kobayashi-Maskawa submatrix.


I. INTRODUCTION
The ''new physics'' class of models we use are those with extra isosinglet down quarks, where we take only one new down quark as mixing significantly. An example is E 6 , where there are two down quarks for each generation with only one up quark, and of which we assume only one new isosinglet down quark mixes strongly. This model has shown large possible effects in B-B mixing phases ͓1͔. The new B factory results on sin(2␤) in the standard model ͑SM͒ range, the ⑀Ј/⑀ experimental convergence, the new K ϩ → ϩ result, the ⌬m s limits near the SM prediction, and other new measurements require a finer analysis and a potential challenge to new physics models. In this paper we include the full SM contributions as well as the new physics contributions from the isosinglet down quark model to jointly analyze the constraints from all of these experiments, as well as other flavor changing neutral current ͑FCNC͒ limits and SM Cabibbo-Kobayashi-Maskawa ͑CKM͒ matrix element constraints.
In models allowing mixing to a new isosinglet down quark ͑as in E 6 ) flavor changing neutral currents are induced that allow a Z 0 mediated contribution to B-B mixing and which bring in new phases ͓1-3͔. In (,), "x s ,sin(␥)…, and "x s ,sin(2 s )… plots we still find much larger regions than in the SM, reaching down to Ϸ0, 0рsin(␥)р1, and sin(2 s ) down to zero ͑below the SM range͒, all at 1 limits. The nature of the cancellation in a fourth down quark matrix element V 4d to satisfy the experiments is elucidated. We also establish ranges for the new mixing elements to the new isosinglet down quark, and make a simple estimate of the lower mass limit of the new down quark.
In Sec. II we introduce the scenario with more down quarks as in E 6 , truncate it to one extra down quark, introduce the 4ϫ4 mixing matrix, and apply it to B-B mixing. Section III presents the CP violating B d and B s decay asymmetries, and B s mixing, including the FCNC tree diagram additions. Section IV presents the full SM contributions as well as the four down quark model ͑FDQM͒ amplitudes for the CP violating and FCNC K meson experiments that are used. Section V presents the joint chi-squared analysis and results for the SM and FDQM model for the various plots listed above. Section VI presents the sizes or limits on the matrix elements, mixing angles, phases, FCNC couplings and unitarity quadrangles. Section VII lists the conclusions and projects what the next down quark mass limit might be.

II. ISOSINGLET DOWN QUARK MIXING MODEL
Groups such as E 6 with extra SU(2) L singlet down quarks ͓4͔ give rise to flavor changing neutral currents ͑FCNC͒ through the mixing of four or more down quarks ͓3,5-8͔. The initial quarks of definite weak isospin in E 6 , for each generation are the left handed isodoublet (u iL 0 ,d iL 0 ), their right handed isosinglets u iR 0 and d iR 0 , and the yet to be found isosinglet pairs D iL 0 and D iR 0 . We can take the initial up quark matrix to be the mass eigenstates, so u i 0 ϭu i , giving V u ϭI 3ϫ3 . The down quarks (d i 0 ,D i 0 ), which correspond to the same generations as u i , mix to form mass eigenstates (d i ,D i ) via the matrix V d (6 ϫ6), where d iL 0 ϭV i j d d jL . The weak interaction charged current matrix is then UϭV u † ϫV d , the 3ϫ6 matrix that is the upper three rows of V d . The lower three rows of V d are the three linear combinations of (d i ,D i ) that are the isosinglet D i 0 which cannot couple to up quarks by the weak interactions.
We truncate the V d matrix to the 4ϫ4 matrix using only the D quark that mixes most ͑and dropping the superscript d on V d ), giving the four down quark model ͑FDQM͒. Calling the new down quark mixture D, the weak charged currents of D to u, c, and t quarks are V tD ϭs 34 , V cD ϭs 24 e Ϫi␦ 24 , and V uD ϭs 14 e Ϫi␦ 14 , which are in the fourth column. The fourth row gives the linear combination that is the initial isosinglet D L 0 . The complete 4ϫ4 mixing matrix was given previously ͓9,10͔. The leading terms in the 4ϫ4 down quark mixing matrix with 6 angles and 3 phases are *Email address: djsilver@uci.edu where, to leading order in new angles, The FCNC amplitudes are given in terms of the mixings V 4i to form the isosinglet down quark by ͓5͔ ϪU i j ϵV 4i * V 4 j for i j.
The diagonal neutral current couplings are reduced in strength by the amplitudes into the isosinglet down quarks, becoming The FCNC with tree level Z 0 mediated exchange may contribute part of B d 0 -B d 0 mixing and of B s 0 -B s 0 mixing, and the constraints leave a range of values for the fourth quark's mixing parameters. As shown in Fig. 1 If the FCNC amplitudes are a large contributor to the B d -B d mixing, they introduce three new mixing angles and two new phases over the standard model ͑SM͒ into the CP violating B decay asymmetries. The size of the contribution of the FCNC amplitude U db as one side of the unitarity quadrangle is less than 0. 15 of the unit base ͉V cd V cb ͉ at the 1-level ͑see Sec. VI͒, but we have found ͓3,5,7,8͔ that it can contribute as large an amount to B d -B d mixing as does the standard model. The new phases can appear in this mixing and give total phases different from that of the standard model in CP violating B decay asymmetries ͓7-9,11,12͔.
For B d -B d mixing with the four down quark induced b-d coupling, U db , we have ͓9͔ where with y t ϭm t 2 /m W 2 , unitarity triangle angles. However, they are still sines of the overall phases of the amplitudes in the asymmetries. We analyze the FDQM with the present data, and also show projected results for three different sin(2␣) values of Ϫ1.0, 0, and ϩ1.0, which are allowed under the FDQM, although sin(2␣)ϭϮ1 are not allowed by the SM at 2.

A. sin"2␤… and sin"2␣…
In the four down quark model we use ''sin(2␣)'' and ''sin(2␤)'' to denote results of the appropriate B d decay CP violating asymmetries, but since the mixing amplitudes are superpositions, the experimental results for these asymmetries are not directly related to angles in a triangle. Being imaginary parts of pure complex exponentials, they are sines of phase angles. The asymmetries with FCNC contributions included are ͑for B mixing to B before decay͒ with U stdϪdb 2 defined in Eq. ͑9͒. The same mixing phase occurs in both asymmetries, times the squares of the different decay phases. We take the Moriond results for sin(2␤) from Babar ͓13͔ and Belle ͓14,15͔ to give the weighted average sin(2␤)ϭ0.78Ϯ0.08.

B. sin"␥…
In the four down quark model, what we mean by ''sin(␥)'' is the result of the experiments which would give this variable in the SM ͓16,17͔, as in B s 0 →D s ϩ K Ϫ . Here, the four down quark model involves more complicated amplitudes, and ''sin(␥)'' is not simply sin(␦ 13 ):

C. The ''frequency'' of B s oscillations, x s
In the four down quark model, x s is no longer the simple ratio of two CKM matrix elements, but now involves the Z-mediated annihilations and exchange amplitudes as well.
Here we avoid the full theoretical uncertainty on B B f B 2 , by taking the ratio of x s to x d , which is better calculated theoretically, and in which we have also included the FCNC with Z 0 exchange We now include the amplitude method analysis of LEP with SLD to assign a ⌬ 2 for each ⌬m s calculated in the angular parameter grid ͓18͔.

D. The B s decay asymmetry, sin"2 s …
In the standard model, B s mixing involves (V ts *V tb ) 2 which is almost exactly real, and the leading decay process of b→cc s has no significant phase from the decay which is proportional to V cb 2 . Thus almost no CP violating phase develops in the most likely B s decays. This occurs in the decays B s →J/⌿, B s →D s ϩ D s Ϫ , and B s →J/⌿K S . The near vanishing of this asymmetry is a test of the SM ͓6͔. Below, we will find a strange twist on this, since the FDQM will include a range that includes values smaller than the SM range, and does not exceed it. In the SM the angle s is the small angle in the b-s unitarity triangle, and its nonzero value indicates CP violation.
In the four down quark model, the CP violating B s decay asymmetry is ͑for the mixing to J/ or D s which includes the double FCNC Z 0 exchange proportional to U bs 2 . Because of the additional flavor changing term, in the four down quark model, the angle given by the above asymmetry will not generally be an angle in a triangle.

A. FCNC as an addition to penguin plus box amplitudes
Since CP violation and FCNC experiments with K mesons are approaching the SM range and also limit FCNC amplitudes, we now include the full SM amplitudes with the FCNC Z 0 exchange amplitudes as well. The K meson experiments are ⑀, K ϩ → ϩ , K L →, and we now add the recent and fairly well determined results for Re(⑀/⑀Ј).
We use the amplitudes determined by Buras ͓19,20͔. In order to reconcile the notation between us and Buras and Silvestrini ͓21͔, we relate their Z ds to our U sd by taking

͑17͒
as implied by their definitions in Lagrangians.
In the following formulas, B 0 is the ⌬Sϭ1 box amplitude, S 0 is the ⌬Sϭ2 box amplitude, C 0 is the Z 0 penguin amplitude, D 0 is the off-shell photon penguin ͑with D 0 Ј being the on-shell amplitude͒, and E 0 is the off-shell gluon penguin (E 0 Ј being on-shell͒. Gauge independent combinations are For m t ϭ170 GeV and m c ϭ1.25 GeV, for example, these The FCNCZ 0 exchange with amplitude U ds can be added to the d-s Penguin amplitude with the Z 0 by the substitution to obtain the SM plus FCNC result.

B. Indirect CP violation in epsilon
In K-K mixing, the small indirect CP violation is given through ͉⑀͉ ͓19͔, where we include the substitution from Eq. ͑21͒:

C. Direct CP violation in Re"⑀ЈÕ⑀…
The direct CP violation in K 0 decays, Re(⑀Ј/⑀), has received more accurate measurements that are definitely nonzero. The average of KTeV ͓22͔ and NA48 ͓23͔ gives Re(⑀Ј/⑀)ϭ17.3Ϯ2.4, where the error has been increased by The P's are functions of B 6 The recent detection of two events in K ϩ → ϩ has produced the experimental result ͓24͔ compared to the SM range ͓25͔ of (0.72Ϯ0.21)ϫ10 Ϫ10 . The Poisson probability for the angle parameters is converted to a chi-squared form ͓26͔ which is convolved into the total 2 formula. For this experiment using a logarithmic prior, with 2ϫn obs ϭ4 degrees of freedom ͓26͔, the addition to 2 is The sum of the SM ͓19͔ plus FCNC contributions is obtained by using Eq. ͑21͒:

E. K L \µ ¿ µ À
The short distance weak FCNC contribution to K L → constrains Re(U ds ) and is given from ͓28͔ after including Eq. ͑21͒:

͑29͒
Here, Y (x t )ϭ y Y 0 , and ͓27͔ Y ϭ1.012. The long distance contribution has been analyzed ͓29͔. We make the 1 limit conservatively as the sum of the 1 experimental limit plus the 1 long distance estimate ͓29͔.
From the above K meson formulas, the error formulas were generated using MATHEMATICA.

V. JOINT CHI-SQUARED ANALYSIS OF THE SM AND THE FDQM EXPERIMENTS
FCNC experiments put limits on the new mixing angles and constrain the possibility of new physics contributing to Here we jointly analyze all constraints on the 4ϫ4 mixing matrix obtained by assuming only one of the SU(2) L singlet down quarks mixes appreciably ͓7͔. We use the seven experiments for the 3ϫ3 CKM submatrix elements ͓2͔, which include those on the three matrix elements V us ,V ub ,V cb of the u and c quark rows; ͉⑀͉; B d -B d mixing (x d ); the new limits on ⌬m s , or x s ; and the new measurements for sin(2␤). For studying FCNC, we include V ud and V cd , the bound on B→X s ͑which constrains b→s), the two events in K ϩ → ϩ ͓12,26,30͔, and R b in Z 0 →bb ͓12͔ ͑which directly constrains the V 4b mixing element͒. FCNC experiments will bound the three amplitudes U ds , U sb , and U bd which contain three new mixing angles and three phases. We use the mass of the top quark as m t ϭ174 GeV. We also add FCNC constraints from K L →, now including the large long distance error, and the new and more convergent results for ⑀Ј/⑀ from NA48 ͓23͔ and KTeV ͓22͔.
Related analyses including both SM and FDQM amplitudes in kaon constraints by Barenboim, Botella and Vives ͓31,32͔ precede this work. We have applied a full 2 analysis rather than just 95% C.L. bounds, and have included the new, larger and more exact sin(2␤) results, as well as new K ϩ → ϩ , ⑀Ј/⑀ results, and new and full x s data. We have also included an analysis of the 4ϫ4 mixing matrix parameters and found a crucial cancellation in one of the matrix elements.
We use a method for combining the Bayesian Poisson distribution for the average for the two observed events in K ϩ → ϩ ͓24,26͔ with the chi-squared distribution from the other experiments. This treats the two events with a logarithmic Bayesian prior as four degrees of freedom. This gives a total of ten additional experimental degrees of freedom for the FDQM.
In maximum likelihood correlation plots, we use for axes two output quantities which are dependent on the mixing matrix angles and phases, such as (,), and for each possible bin with given values for these, we search through the nine dimensional angular data set of the 4ϫ4 down quark mixing angles and phases, finding all sets which give results in the bin, and then put into that bin the minimum 2 among them. To present the results, we then draw contours at several 2 in this two dimensional plot corresponding to given confidence levels. In the FDQM analysis including FCNC experiments, there are 17 experimental degrees of freedom, minus 9 parameters, giving 8 remaining degrees of freedom. In contrast to the SM, for the FDQM in Fig. 3, almost the entire region sin(2␤)Ͼ0.48 is allowed at 2, and sin(2␤) can be as low as 0.55 at 1. In the FDQM, all values of sin(2␣) are allowed. In this case, the larger 1 range for sin(2␤) than from the direct experimental measurement is an effect of including so many experiments in the joint fit.

C. Standard model with comparison experiments ",… plot
Here we depart from the analysis of the SM experiments alone to show the effects of the additional three K meson experiments, namely K ϩ → ϩ , K L →, and ⑀Ј/⑀. We see the effects of the x s ϭ(⌬m s /⌫ s )ϭ1.35x d ͉V ts /V td ͉ 2 lower bound in the SM limiting the length of V td ϰͱ(1Ϫ) 2 ϩ 2 and cutting off for Ͻ0.

E. Four down quark model: ",… plots
As in the SM, the plotted and are taken as the coordinates of V ub * , scaling the base of the b-d unitarity quadrangle to unity

͑30͒
The unitaritity quadrangle is given by where the last term has limits ͉U bd /V cb * V cd ͉р0.15, as will be shown later. The near half circles in ␥ϭ␦ϭ␦ 13 (V ub * ϭs 13 e i␦ 13 ) at present are due to ␦ 14 or ␦ 24 ͑which are related͒ becoming some of the source of the observed CP violation in ⑀, so that ␦ 13 is less constrained. Then, ␦ 13 can be closer to zero or 180°so that can also be small or zero. For projected sin(2␣)ϭϩ1, 0, or Ϫ1, we see regions extended beyond the SM regions, which also allow to be small ͑see separately, we find that the K ϩ → ϩ result eliminates the large 1 negative rings from the previous analysis ͓1͔.

F. Fraction of the new FCNC amplitude in ⑀
In order to display how the FCNC Z 0 exchange with the new phases in U ds can account for the CP violation in ⑀ K , we plot the ratio of the FCNC contribution to the experimental result. In Fig. 7, (⑀ FCNC /͉⑀ expt ͉) is shown against the phase of V ub * , which is ␦ 13 . In Fig. 7, while ⑀ FCNC cannot account for the entire ⑀ result, it can account for 60% of it at a 1 confidence level.

G. Standard model: "x s ,sin"␥…… plots
x s is determined in the SM from The largest error arises from the uncertainty in ͉V td ͉, which follows from the present 15% uncertainty in ͱB B f B ϭ230 Ϯ35 MeV from lattice calculations ͓34͔. In the SM, the B factory measurements construct a rigid triangle from the knowledge of ␣ and ␤, and removes this uncertainty in ␥ and x s in the future. From present data for the SM "x s ,sin(␥)… plot in Fig. 8, the limits at 2 are 0.56рsin(␥)р0.99, and 16рx s р48. Because of the approximately linear relation between x s and sin(␥), an exact x s measurement (ϰ1/͉V td ͉ 2 ) can strongly constrain sin(␥) to Ϯ0.07 in the SM.

H. Four down quark model: "x s ,sin"␥…… plots
In the FDQM, the sin(␥) range goes down to zero at 1, or Ϫ0.4 at 2 ͑Fig. 9͒, since now goes down to zero at 1 or to Ϫ0.2 at 2 where ϷϪ0.5. A larger sin(␥) range is thus allowed in the FDQM than in the SM. The x s allowed region in the FDQM is 16 to 60 at 1 or to 80 at 2, which is also larger than the 2 x s range of 48 in the standard model. In the FDQM, there is not an approximately linear relation between sin(␥) and x s as there is in the SM. Thus an accurate measurement of x s still leaves a very large region of sin(␥) available in the FDQM. A subsequent sin(␥) measurement will be needed to distinguish between the two models.

I. The decay asymmetry from B s mixing, sin"2 s …
s is the small angle in the b-s unitarity triangle given by V cb * V cs ϩV tb * V ts ϩV ub * V us ϭ0. In Wolfenstein terms this is Then, sin( s )ϭA 4 /A 2 ϭ 2 . This is small in the standard model where sin(2 s )ϭ2 2 ϭ0.10, or at 2 0.030рsin͑2 s ͒р0.060. ͑35͒ In the FDQM, as seen in Fig. 10, Ϫ0.2рsin(2 s )р0.065 at 2, and down to zero at 1. Here the range continues down to zero since can go down to zero. Hence, a value of sin(2 s ) less than 0. 03 would signify a deviation from the SM.

J. Fourth side of the unitarity quadrangle U bd
Unitarity of the b-d columns has four terms, which may be written as

͑36͒
since ϪU bd ϭV 4b * V 4d . ͑We use U db ϭU bd * .͒ As the unitarity triangle is scaled by ͉V cb * V cd ͉ to make a unit base, the complex plot of U bd is also so scaled. The length of the U bd /͉V cb * V cd ͉ side, as plotted in Fig. 11, is thus less than 0.15, compared to the unit base in the (,) plot, and prefers possibly a more vertical direction. The accuracy of angles and sides of the unitarity triangle must and should reach this accuracy for a good test of the SM.

A. Bound from Z 0 \bb
The weak isovector part of Z 0 →bb is reduced by (1 Ϫ͉V 4b ͉ 2 ) through where C QCD ϭ3(1.0385), and t ϭ0.0094 for m t ϭ174 GeV. Present data and theory give We note that the ͉V 4b ͉ effect is to decrease R b , while the experiment is about 1 above the theory. To lowest order in ͉V 4b ͉ the FDQM effect is The "x s ,sin(2 s )… plot for the B s asymmetry sin(2 s ) in the four down quark model for present data, with contours at 1, 90% C.L., and 2. This gives a contribution to 2 of This 2 is used as a constraint on all angle choices in the fit.

B. 3D matrix element Lego plot
From the 2 surface in the 3D space of the magnitudes of the matrix elements involved in the FCNC, Fig. 12, we can see the limits and ranges of two of the matrix elements. To 5% accuracy, V 4d ϭϪs 14 e i␦ 14 , and its magnitude ranges from 0.035 to 0.085 at 2. To 10% accuracy, V 4b ϭϪs 34 , and its magnitude ranges up to 0.020 at 2. The third FCNC matrix element, ͉V 4s ͉, is bounded by 0.0004. This requires a fine cancellation between its two components in (Ϫs 24 e i␦ 24 Ϫs 12 s 14 e i␦ 14 ), such that s 24 Ϸs 12 s 14 and ␦ 24 ϭ␦ 14 ϩ to get the cancelling minus sign. This means that there is effectively only one new phase, which we may consider as ␦ 14 .
From the cancellation, s 24 ranges from 0.009 to 0.017 at 2. The cancellation is to about 1/20 of the value of s 24 . The third term in V 4s , s 34 (s 23 ϩs 12 s 13 e i␦ 13 ), then contributes р0.0009, which is the same order as the partly canceling terms. The cancellation does not mean fine tuning since one could have parametrized V 4s by a single angle instead. However, the incredibly small size of ͉V 4s ͉р 5 could be considered a fine tuning itself. In comparison to the SM CKM matrix we should note that keeping the leading terms in the real and imaginary parts, V cs ϭ1ϩiA 2 6 , V cd ϭϪ ϪiA 2 5 , and V ts ϭϪA 2 ϪiA 4 . So even in the standard model there are matrix elements whose imaginary parts are as small as O( 4 ), O( 5 ), and O( 6 ).
In the double FCNC Z 0 exchange amplitude in B d -B d mixing, via U db ϭϪV 4d * V 4b Ϸs 14 e Ϫi␦ 14 s 34 ,

͑43͒
it is only the ␦ 14 phase in (U db ) 2 Ϸe Ϫ2i␦ 14 that can add to the SM box diagram term with its phase of (V td * ) 2 .

C. Phases
The cancellation in V 4s to make it small requires ␦ 24 Ϸ␦ 14 ϩ. Thus we can display the phases in a two dimensional plot of ␦ 14 vs ␦ 13 , as in Fig. 13. When ␦ 13 is in its SM range of 40°͓sin(␥)ϭ0.64͔ to 70°͓sin(␥)ϭ0.94͔ the SM terms can be dominant and the small FCNC amplitudes allow each ␦ 14 equally. For certain values of ␦ 14 , near 80°and 270°, the new physics amplitudes can be dominant and ␦ 13 can be large, leading to the enlarged (,) contours that can reach Ϸ0 and extend beyond to ␦ 13 р200°at 2.

D. FCNC phase structure
Using the V 4s cancellation structure with s 24 ϭs 12 s 14 and ␦ 24 ϭ␦ 14 ϩ, we can rewrite the V 4i matrix elements in terms of just one phase in the leading terms the V 4i , and therefore in the U i j and in the FCNC ampli- We note that while s 14 and s 24 are nonzero, the cancellation in V 4s and the ability of s 34 to vanish still allow all U i j to vanish.

E. Variable determination
In general for the three complex matrix elements V 4i , one would expect three magnitudes and three phases. In determining these from the ϪU i j ϭV 4i * V 4 j however, one overall phase would not appear experimentally, due to the V*V structure of the U i j . So we can at best determine three magnitudes and two phases from the U i j . This agrees with the three new angles and two new phases introduced in the 4 ϫ4 unitary matrix where an extra phase has been removed for the definition of the new D 0 down quark. Whereas the three U i j may seem to contain three real and imaginary parts to be determined, they are not independent, since there is one restriction between them, namely that the product is real. So again, we are left with three magnitudes and two phases that can be determined by experiments involving the FCNC amplitudes, which allows us to determine the three new angles and two new phases, just from low energy experiments involving the U i j . With sufficient energy to produce one D quark, the angles s i4 can each be determined separately by the combined weak production of ū D, c D or tD pairs, or from the similar decays of the D quarks.
The cancelation in V 4s has related s 24 ϭs 14 and ␦ 24 ϭ␦ 14 ϩ. Thus there are only effectively two independent new angles and one new phase to be determined from the five independent components of the U i j , leading to an overconstrained system. Finding a consistent solution is then a test of the FDQM. Of course, if more variables are found to be needed, the mixings to five or six down quarks would have to be considered. The present fits have found nonzero values for s 14 and its related s 24 . Yet s 34 may still be small or vanish, and the one new independent phase is still to be determined, although its determination is coupled to that of the CKM ␦ 13 phase.

F. Unitarity tests on the CKM submatrix
Contained in the 4ϫ4 analysis are tests of the unitarity of the 3ϫ3 CKM submatrix contained in the 4ϫ4 FDQM mixing matrix. The FCNC couplings U ds , U sb , and U db measure the deviations from orthogonality of the columns of the CKM submatrix, in d-s, s-b, and d-b projections, respectively. Their sizes will be discussed in Sec. VI G under unitarity quadrangles.
Bounds on the size of the ͉V 4i ͉, iϭ1, 2, 3, bound the deviation from unity of the sum of the squares of the three CKM elements in each column Similarly, for the rows, the ͉s i4 ͉ 2 measure the deviation from unity for the sum of the squares of the CKM row elements. For the d column or u row, since ͉V 4d ͉Хs 14 Ϸ0.035 to 0. 085, unitarity of the CKM three elements of the d column or u row is off by 0.0012р͉V 4d ͉ 2 р0.0072, or ͑52͒ 0.5 4 р͉V 4d ͉ 2 р3 4 . ͑53͒ For the s column, since ͉V 4s ͉р0.40ϫ10 Ϫ3 , the deviation from unitarity of the CKM submatrix is bounded by ͉V 4s ͉ 2 р0.16ϫ10 Ϫ6 ϭ0.6 10 . ͑54͒ For the b column or t row, since ͉V 4b ͉Хs 34 р0.020, the deviation from unitarity of the CKM submatrix is bounded by For the c row, since s 24 Хs 12 s 14 ϭs 14 , the deviation of the CKM from unitarity is a multiple of the u row result from s 14 0.5 6 р͉s 24 ͉ 2 р3 6 . ͑56͒ Finally, the deviation of ͉V 44 ͉ 2 from unity is an overall measure of mixing to the fourth down quark 1Ϫ͉V 44 ͉ 2 ϭs 14 2 ϩs 24 2 ϩs 34 2 . ͑57͒ The right-hand side is dominated by s 14 2 giving ͉V 44 ͉ 2 ϭ1Ϫ͑0.5→3 ͒ 4 . ͑58͒

b-d quadrange
The orthogonality relation between the b and d columns of the 4ϫ4 mixing matrix is

͑59͒
The fourth side of the b-d unitarity quadrangle, scaled to make the base of unit length, is U db * /͉V cd * V cb ͉. From Fig. 11, we see that the length of the FCNC quadrangle side is р0.15 in the vertical or imaginary direction, and р0.06 in the horizontal or real direction. The sides of the b-d unitarity quadrangle can be written in a modified Wolfenstein form as ϵA 4 ͑ ϩi͒, and ͑63͒ where we have introduced ϩi into ϪU db * with a coefficient to make the scaled quadrangle A independent. An example of the scaled b-d quadrangle is shown in Fig. 14. We see that unitarity requires that the length of the FCNC coupling side ϪU db * has to be cancelled by another triangle side to close the triangle, and that occurs in V tb * V td having an addition to the SM formula. The area of the b-d unitarity quadrangle is computed by adding the areas of three subtriangles and a rectangle Area͑bϪd ͒ϭA 2 6 ͓ϩ͑ 1Ϫ ͔͒/2. ͑65͒ We note that if either or both and are nonzero, CP is violated, and the quadrangle has a nonzero area, analogous to the SM unitarity triangle result. However, as we will see below, the area of the b-d quadrangle is different from those of the other unitarity quadrangles by the term above.

s-b quadrangle
The unitarity orthogonality between the s and b columns for the s-b quadrangle is V us * V ub ϩV cs * V cb ϩV ts *V tb ϪU sb ϭ0.

͑66͒
The first term is A 4 (Ϫi), the second term is A 2 , and the third term is ϪA 2 , to leading order. If we scale the base to unit length by dividing by A 2 , then the first term side is of order 0. 02 in length. From Fig. 15, the fourth side of scaled U sb is of order 0. 0001, or 0.5% of the small third side of the triangle. The enclosed angle is then the same as in the SM, s ϭ 2 , and the triangle's or quadrangle's area is A 2 6 /2.

d-s quadrangle
The orthogonality relation between the d and s columns is V ud * V us ϩV cd * V cs ϩV td * V ts ϪU ds ϭ0.

͑67͒
The largest sides of the d-s unitarity quadrangle are of length , being the first and second terms, and the third term is A 2 5 (1Ϫϩi)ϭ0.0004(1Ϫϩi). The fourth side is the FCNC coupling U sd , which is bounded in magnitude by 2.5ϫ10 Ϫ5 ϭ 7 , as seen in Fig. 16. Thus the FCNC fourth FIG. 14. The b-d unitarity quadrangle scaled by A 3 , with sides given as above. side is at most 6% of the small third side. The angle subtended by the small third side is then essentially the same as that by the third and fourth sides, being d ϭA 2 4 . The triangle's or quadrangle's area is also A 2 6 /2.

H. The sum rule for the CP violating B decay asymmetry angles
It was shown before ͓6͔ that as long as the penguin diagrams in the B decays can be neglected, that the sum of the CP violating decay angles, even with new physics contributions, is modulo . This can be seen from Eqs. ͑12͒ and ͑11͒ where in the sum of (2␣ϩ2␤), the B d mixing phase cancels out in general, regardless of its source, and from Eqs. ͑13͒ and ͑16͒, where in the sum of (2␥ϩ4 s ), the phase from B s mixing cancels out. The other tree amplitude decay phases in these equations either cancel or sum to the phase of a product of mixing matrix elements which becomes a product of absolute values squared, with zero phase. This leads to the CP violating B decay angle sum rule ͓6͔ ␣ϩ␤ϩ␥ϩ2 s ϭ, mod . ͑68͒

VII. CONCLUSIONS FOR ISOSINGLET DOWN QUARK MODELS
With much new data, it is still the case that FCNCs can contribute significantly to B d -B d mixing and to B s -B s mixing, and give contributions with new phases. In the FDQM, all sin(2␣) are allowed. In the (,) plane, the FDQM allows large regions for р0 as opposed to the у0 regions in the SM, and in particular, those where goes to zero, both with the present data and with the projected sin(2␣) values.
In new physics models then, the SM phase ␦, or , can be smaller, with the other phases causing much of the presently observed CP violation. In the "x s ,sin(␥)… plots in the FDQM, all of sin(␥)у0 is allowed at present in contrast to sin(␥)у0.55 in the SM, and with no approximately linear relation as in the SM. This will require combining experimental results of x s and sin(␥), to find out if the results correlate to the narrow linear region of the SM analysis. The present range for x s is from 16 to 48 at 2-in the SM, and from 16 to 80 at 2-in the four down quark model. The b-d unitarity triangle, scaled to unit base length, has to be measured to an accuracy of 0.15 or better to begin to limit a fourth side and to verify the SM against the FDQM.
Each E 6 generation also contains an isosinglet ͓4͔ or sterile neutrino, which may provide a connection between the quark and lepton searches for new physics in terms of establishing new particle representations.
The mass of the lightest singlet down quark in E 6 could be roughly related to the mixing angle by 34 2 Ӎm b /m D , and with ͉V 4b ͉Ӎ 34 р0.02 ͑69͒ from combined fits, that gives m D у2500ϫm b ϭ11 TeV. ͑70͒ Using the single R b 90% C.L. limit of ͉ 34 ͉р0.035, which is not as strong as the combined fits, gives m D у4 TeV. The previous analysis ͓1͔ gave a lower limit of 1.2 TeV.