Correlating $B_q^0 \to \mu^+\mu^-$ and $K_L \to \pi^0\nu\bar\nu$ Decays with Four Generations

The long-awaited $B_s\to \mu^+\mu^-$ mode has finally been observed at rate consistent with Standard Model, albeit lower by 1.2$\sigma$. There is some hint for New Physics in the rarer $B_d^0 \to \mu^+\mu^-$ decay, especially if the currently 2.2$\sigma$-enhanced central value persists with more data. The measurement of $CP$ violating phase $\phi_s$, via both $B_s\to J/\psi K\bar K$ and $J/\psi\pi\pi$ modes, has reached Standard Model sensitivity. These measurements stand major improvement when LHC enters Run 2. Concurrently, the $K_L\to\pi^0\nu\bar\nu$ and $K^+\to\pi^+\nu\bar\nu$ modes are being pursued in a similar time frame. We illustrate the possible correlations between New Physics effects in these four modes, using the fourth generation as example. While correlations may or may not exist in other New Physics models, the four generation model can accommodate enhancements in both $B_d^0 \to \mu^+\mu^-$ and $K_L\to\pi^0\nu\bar\nu$.


I. INTRODUCTION
The 7-and-8 TeV run (Run 1) of the LHC has brought about much progress in particle physics, even though no New Physics (NP) has been uncovered.The hint of possible NP in the forward-backward asymmetry of B → K * ℓ + ℓ − decay [1] from the B factories was eliminated by the measurement of B 0 → K * 0 µ + µ − by LHCb with just 0.37 fb −1 data [2].Although with much anticipation, the possibility or hint of large CP violating (CPV) phase φ s in B 0 s -B 0 s meson mixing at the Tevatron [3,4] was also eliminated by the combined measurements of the B 0 s → J/ψ φ [5] and B 0 s → J/ψ f 0 (980) [6] channels, with 0.37 fb −1 and 0.41 fb −1 data, respectively.Thus, the thread that started with the suggested possible correlations [7] with large direct CPV difference ∆A Kπ ≡ A(B + → K + π 0 ) − A(B 0 → K + π − ) [8], if the source for the latter arises from the electroweak penguin [9], was vanquished.Finally, the hot pursuit for B 0 s → µ + µ − at the Tevatron was overtaken by the LHC, culminating in the observation by the LHCb [10] and CMS [11] experiments [12], albeit again consistent with the Standard Model (SM).This deals another blow to minimal supersymmetric SM (MSSM), where the B 0 s → µ + µ − rate could have been enhanced by three orders of magnitude, the original reason behind the frenzied pursuit!However, LHC Run 1 has brought on its own tantalizing hints.The full combination results of CMS and LHCb for B 0 s → µ + µ − and the even rarer B 0 d → µ + µ − decay have been announced recently [13], B(B 0 d → µ + µ − ) = (3.9+1.6 −1.4 ) × 10 −10 .
(2) At 6.2σ significance, the B 0 s → µ + µ − mode is genuinely observed.The SM expected value, however, would have given 7.6σ, hence a mild "suppression" is suggested.On the other hand, the B 0 d → µ + µ − mode, nonzero at the 3.2σ level, is consistent with SM expectation of (1.06 ± 0.09) × 10 −10 [14] only at 2.2σ level.While 2.2σ should not be taken seriously as a deviation, but if the current central value persists with much more data, it would be more than 3 times enhanced compared with SM! Thus, B 0 d → µ + µ − is the mode to watch at the up and coming LHC Run 2 (13 and 14 TeV).
Compared with initial 7 TeV data, the 1 fb −1 LHCb update for φ s [15] is also providing new implications: which combines both the B 0 s → J/ψ φ and J/ψ π + π − channels, with respective values φ s = 0.07 ± 0.09 ± 0.01, (1 fb −1 J/ψK K, LHCb) (4) φ s = −0.14+0.17 −0.16 ± 0.01, (1 fb −1 J/ψ ππ, LHCb) (5) which are of opposite sign.Eq. ( 5) is improved from an earlier [16] 1 fb −1 result of −0.019 +0.173+0.004−0.174−0.003, mainly due to improvements in tagging of the B 0 s flavor, i.e. particle or anti-particle.With same tagging, Eq. ( 4) is improved from the 0.37 fb −1 result of 0.15 ± 0.18 ± 0.06.Eq. (3) dominates the Heavy Flavor Averaging Group (HFAG) combination [17] of all experiments, φ s = 0.00 ± 0.07, (PDG2014) (6) which is adopted by the Particle Data Group (PDG) [18].The PDG 2014 result, Eq. ( 6), may seem to imply that it would take a long while to probe New Physics via deviations from the SM expectation of φ s ≃ −0.04.This is especially so since the preliminary result [19] of CMS based on 20 fb −1 data, which is mildly negative, is not included in the PDG 2014 combination.However, with 3 fb −1 data at hand, LHCb has already updated its result in the B 0 s → J/ψ π + π − mode to full Run 1 data [20]: This is the single best measurement so far, but comparing Eq. ( 8) with Eq. ( 5), one sees that adding twice more data lead to not only a large reduction of error, but a change in sign of the central value.Again, this is in part due to improved tagging.Unfortunately, LHCb has not updated (see, however, the Note Added ) the B 0 s → J/ψ φ mode to full Run 1 data, in part because the analysis is done simultaneously with ∆Γ s measurement.If the 3 fb −1 update for B 0 s → J/ψ φ keeps the same sign as Eq. ( 4), then φ s would become positive in sign, and would start to deviate obviously from SM expectation!This positive tendency of world average has already been shown in an unofficial combination at a conference [21].In this vein, it is good that the CMS as well as ATLAS experiments, even though not quite competitive with LHCb (see Eq. ( 7)) on φ s , would provide a sanity check.
The emerging prospect that φ s could deviate from SM, even with just Run 1 data, underlies the prospects for LHC Run 2 that would start in 2015.In this respect, given that the central value of Eq. ( 2) for B 0 d → µ + µ − is more than 3 times the SM expectation, it is imperative to follow up with Run 2 data, and the experiments should hone their analyses on B 0 d → µ + µ − , now that the B 0 s → µ + µ − mode is more or less "done" (although it should still be watched).Besides B 0 q → µ + µ − and φ s measurements, a third measurement is the so-called P ′ 5 angular variable in B 0 → K * 0 µ + µ − , where LHCb finds [22] some anomaly.Given that this is one out of many measurables, and especially since the LHCb result is only with 1 fb −1 data, we should first see the 3 fb −1 update.But B 0 → K * 0 µ + µ − is certainly a mode to follow with Run 2 data with sophisticated angular analyses.We also note in passing the so-called R K anomaly [23] in lepton universality violation in B + → K + ℓ + ℓ − decays, which has 2.6σ deviation from SM expectation.It may or may not be related to the P ′ 5 anomaly.Whether P ′ 5 or R K stay interesting as Run 2 data unfolds is unclear at present.But it should be clear that B 0 q → µ + µ − and φ s measurements would remain lampposts for New Physics.What could the situation be in 2016, one year into Run 2? By 2017/18, LHCb expects 8 fb −1 or more data.The data rate is much higher for CMS, but trigger bandwidth and other issues start to cut in, although things would certainly be extremely interesting.As we bring the time frame up to 2017-2018, one should consider the Belle II experiment at the Super KEKB accelerator, now being completed in Japan.However, Belle II is not particularly competitive in φ s and B 0 q → µ + µ − .But given that the two measurables correspond to b ↔ s and b → d transitions, one involving CPV, the other not, there is one particular process that comes to mind: K → πν ν decays, which are s → d transitions.The neutral K 0 L → π 0 νν decay is purely CPV, and is being pursued by the KOTO experiment [24] in Japan, while the charged K + → π + νν mode is pursued by the NA62 experiment [25] at CERN.Both experiments would be operating within a similar time frame.If one has indications for NP in φ s and B 0 q → µ + µ − , likely one would find NP in K → πν ν, and vice versa.
An element of competition between high-and low-energy luminosity frontiers would be quite interesting.
In this paper we study the correlations between the four measurables of B 0 d → µ + µ − , B 0 s → µ + µ − , φ s (CPV phase in B s mixing), and K → πν ν (especially K 0 L → π 0 νν).We shall use the 4th generation (4G) for sake of illustration.Although some may find this extreme, but we challenge anyone to find a way to enhance B 0 d → µ + µ − rate by a factor of three and still survive all constraints.The issue with 4G is the observation of a light Higgs boson, without the anticipated factor of 9 enhancement in cross section.But we have argued [26] that the Higgs boson practically does not enter (i.e. is "orthogonal" to) flavor changing processes at lower energy, and, if one discovers an enhanced B 0 d → µ + µ − decay, it may put some doubt on the Higgs nature of the observed 126 GeV particle.We view the issue as still open [27], to be settled in a similar time frame of 2017-2018.Our 4G study serves to illustrate how New Physics in B 0 q → µ + µ − , φ s , and K → πν ν might be accommodated or correlated.
The paper is organized as follows.We review in Sec.II the formulas and data inputs for our study, and give our numerical results in Sec.III.Given that φ s value for full Run 1 data is still quite volatile, we take the more conservative approach of Eq. ( 6), but do explore the situation for φ s > 0. We offer some discussions and give our conclusion in Sec.IV.

II. FORMULAS AND DATA INPUT
Unless stated otherwise, we use experimental values from PDG 2013 partial update [28].As our purpose is to illustrate correlation between B 0 q → µ + µ − and K → πν ν, updating to PDG2014 does not make major difference.We define the parameters: We adopt the parametrization of Ref. [29] for the 4 × 4 CKM matrix, with convention and treatment of Ref. [30].
In particular, we assume SM-like values for s 12 , s 23 , s 13 and φ ub ≃ φ 3 ≡ γ, with the following input [28] |V us | = 0.2252 ± 0.0009, |V cb | = 0.0409 ± 0.0011, • .This is a simplification, since we try to observe trends, rather than making a full fit.We find taking the "exclusive" measurement value [28] for |V ub | would be overexclusive, with no meaningful solution space.
Having a 4th generation of quarks brings in three new angles and two new phases.In this paper, we take  13), while the narrow green-shaded contours correspond to the 1(2)σ regions of sin 2β/φ1 (Eq.( 14)).Solid-blue lines are labeled 10 10 B(B d → µ + µ − ) contours, where the upper bound of Eq. ( 15) is applied.
for sake of illustration, thereby fixing one of the angles.
A second angle and one of the two phases are fixed by the discussion illustrated below.We are then left with two mixing parameters, for which we take as r ds and φ ds (see Eq. ( 10)), as we are interested in K → πν ν decays.
We have picked m t ′ = 1000 GeV, which is considerably above the experimental bound [18], which is now beyond the nominal [31] unitarity bound (UB).Experiments are now vigorously exploring the so-called vector-like quarks, with signatures beyond b ′ → tW and t ′ → bW (involving tree level flavor changing neutral current decays to Z and Higgs bosons).Even with vector-like (fourth generation) quarks, the experimental bound [18] has reached beyond 700 GeV.In our study, to reduce the number of parameters, we adhere to the sequential fourth generation.
We had suggested, in Ref. [26], that a visibly enhanced B d → µ + µ − could suggest the presence of a 4th generation.Intriguingly, this is supported by current data [13], although more data with targeted study is clearly needed.In Fig. 1, we update Fig. 3(a) of Ref. [26] on the r dbφ db plane defined by for hadronic parameters.We no longer take the ratio with ∆m B d in the formula of the B d → µ + µ − branching ratio.We also update the experimental constraints: from HFAG Winter 2014 [17] and 95% CL limit of LHCb [10], respectively.The latter is softer than the recent CMS and LHCb combination (cf.Eq. ( 2)).
From Fig. 1, we shall consider two scenarios marked as S1/S3 and S2 in Fig. 1, to illustrate respectively, where we have conservatively stayed within 1σ boundaries of both ∆m B d (uncertainty in Eq. ( 13)) and sin 2β/φ 1 .Scenario 1 (and 3) has an enhanced which carries a near maximal 4G CPV phase φ db .For b → s observables, we update both formulas and input parameters of Ref. [33].For the CPV phase φ s ≡ 2Φ Bs in B s -Bs mixing, we use (see e.g.Ref. [34]) with λ bs q ≡ V * qs V qb (q = t, t ′ ).As discussed in Introduction, we adopt the PDG 2014 [18] value of φ s , Eq. ( 6), to be conservative, and impose 1σ or 2σ constraints.
For the B s → µ + µ − branching ratio, because of the sizable width difference ∆Γ s , experimental results have to be compared with the ∆Γ s -corrected [35,36] branching ratio denoted with a bar, which is related to uncorrected one without bar via [37] where with y s = ∆Γ s /(2Γ s ) = 0.069 ± 0.006 [18].The uncorrected branching ratio is given by (see e.g.Ref. [34]) The correction factor η eff = 0.9882±0.0024takes into account NNLO QCD and NLO electroweak corrections [38].
For the nonperturbative parameter R 8 that enters ε ′ /ε, we adopt the value obtained from lattice [53], with the translation by Ref. [51].There is still no reliable result from lattice QCD for R 6 , so we treat [30] it as a parameter, i.e.R 6 = 1.0, 1.5, 2.0, 2.5.We require ε ′ /ε to agree within 1σ error of the experimental value [18] ε for each fixed value of R 6 .on other potentially important observables are in order.We checked that ∆m Bs does not give further constraints in the parameter space of our interest, within hadronic uncertainty from Eq. ( 25).We considered also the ratio ∆m Bs /∆m B d , which is under better theoretical control, as can be read from ξ ≡ f Bs BBs /f B d BB d = 1.268 ± 0.063 [32].Although this ratio provides much stronger constraint than individual ∆m Bs or ∆m B d due to reduced hadronic uncertainty, we find that our parameter space of interest is allowed within uncertainty from ξ at 2σ level.In contrast to ∆m B d,s , ∆m K is polluted by Long-Distance (LD) effects.We have checked that there is no significant change of the SD part from the SM value and (∆m K ) SD is still below the measured value [28].D 0 -D0 mixing is also subject to LD effects.We checked that the SD contribution to the mixing amplitude M D 12 from b ′ (with m b ′ ∼ m t ′ ) could be enhanced up to 3  [left] Allowed region in e. r ds -φ ds ) plane for Scenario 1, i.e. r db e iφ db = 0.0004 e i330 • (enhanced B d → µ + µ − ) and φs = 0.00 ± 0.07 (PDG2014), where the constraint source for each boundary is indicated.The leading constraint is Bs → µ + µ − , where 1(2)σ region -towards larger (smaller) BR in central region (4th-extending-to-1st quadrants) -is (very) light shaded, separated by dashed lines, except: KL → µ + µ − cuts off at upper left, as well as centerright, indicated by light-blue solid lines; 1σ allowed φs (i.e.−0.07) cuts off the 1σ allowed Bs → µ + µ − in right-center, while the K + → π + νν upper bound (solid purple line) cuts off the 2σ allowed Bs → µ + µ − .[right] The allowed region is further overlaid with εK (blue-shaded), ε ′ /ε (narrow green bands corresponding to R6 in increasing order from 1.0, 1.5, 2.0, 2.5) and B(KL → π 0 νν), labeled in 10 −10 units.The illustration is for m t ′ = 1000 GeV (Eq.( 11)), and other parameter choices are discussed in Sec.II.
SM value in the parameter space of our interest, but it is still well below the measured value of ∆m D [28].
B → K ( * ) µ + µ − observables are subject to precise measurements at the LHC and severely constrain NP effects.The 4G t ′ could affect these observables mainly through the Wilson coefficients C 9 and C 10 .We checked that the 4G effects on C 9 is small, within 5% of the SM value (∼ 4.3), in the parameter space of our interest.We find that the 4G effects on C 10 can be as large as unity in some part of the target parameter space we consider.However, adopting the model independent constraint in Ref. [54], we checked that the changes are within 2σ for various modes.

III. RESULTS
We have mentioned in the Introduction that, taking the measurement value of |V ub | extracted from exclusive B decay modes, we could not find any "solution" for New Physics for our purpose.Thus, we use |V ave ub | = 0.00415 ± 0.00049 throughout this work.
To illustrate possible tensions between B d → µ + µ − and φ s measurements, we shall explore three scenarios, with the r db e iφ db values of Eq. ( 16): 1. Scenario 1: r db e iφ db = 0.00040 e i330 • (enhanced B d → µ + µ − ), φ s = 0.00 ± 0.07 (PDG 2014).This scenario treats the possibility that B d → µ + µ − is enhanced to 3 times SM expectation or higher, but takes the more conservative PDG 2014 value for φ s , Eq. ( 6), which is the HFAG Winter 2014 average dominated by the LHCb 1 fb −1 result of Eq. ( 3).As we shall see (the following two scenarios), while the PDG 2014 value could in principle allow for positive φ s values, the positive central value suggested by LHCb's B s → J/ψφ at 1 fb −1 but B s → J/ψππ at 3 fb −1 , leads to a lot of tension.
With formulas and data input given in Sec.II, we first consider Scenario 1 and plot, in Fig. 2[left], the region in the e. r ds -φ ds ) plane allowed by the various constraints.The leading constraint is the now observed B s → µ + µ − mode, Eq. ( 1), illustrated for the 1 and 2σ region by the golden-hued light and very light shaded regions.Various other constraints take precedence in certain regions: the short distance K L → µµ constraint cuts in at the upper-left corner, as well as just right of center; the 1σ φ s = −0.07constraint cuts off the 1σ-allowed B s → µ + µ − near the center of the right-hand  2a, where a sliver of 1σ allowed region remains, but has no overlap with εK, and at 2σ, φs = −0.04still provides lower bound, while KL → µ + µ − and enhanced (suppressed in 4th quadrant) Bs → µ + µ − give upper bound.
The allowed region of Fig. 2[left] is further overlaid, in Fig. 2[right], by the constraints of ε K and ε ′ /ε, as well as K L → π 0 νν contours.The latter, plotted in red-solid lines, are labeled by BR values in 10 −10 units, where the highest "15" on the righthand side is just above the nominal GN bound of Eq. ( 31), while the region of SM strength (or below) is marked by red-dash lines on lefthand side with label "SM".The ε K constraint, plotted in shaded blue with theoretical error (experimental error negligible), prefers small |V * t ′ d V t ′ s | values, except two "chimneys" where the phase of V * t ′ d V t ′ s is small for one near 180 • , and the other is tilted in the fourth quadrant.The ε ′ /ε constraint is more subtle, because of the less known hadronic parameter R 6 (we have fixed R 8 ≃ 0.7, Eq. ( 38)).We illustrate with R 6 = 1.0, 1.5, 2.0, 2.5, with the corresponding (in ascending order) green bands in Fig. 2[right] determined by experimental error of ε ′ /ε.
First, we observe that the ε K and ε ′ /ε constraints disfavor the possible enhancements for ) is in the first quadrant.For R 6 ≃ 1, K L → π 0 νν would be close to SM expectation.However, as R 6 increases, the ε K "chimney" in the 4th quadrant allows for enhanced K L → π 0 νν, up to 1/3 to 1/2 the GN bound, for R 6 reaching 2 or 2.5 in strength.There is a correlation between larger K L → π 0 νν and smaller B s → µµ.If KOTO observes K L → π 0 νν shortly after reaching below the GN bound, a rather large R 6 value would in turn be implied.One argument for larger 25.We see the intricacies and prowess, still, of the various kaon measurements, with K L → π 0 νν as the main frontier, on a par with the ongoing B d , B s → µµ and φ s measurement efforts.
For Scenario 2, where B d → µµ is taken as consistent with SM but φ s is taken as the positive-preferred "LHCb-ICHEP2014" value, namely φ s = 0.070 ± 0.055, (ICHEP2014, LHCb) (40) we skip the intermediate step corresponding to Fig. 2[left], and plot in Fig. 3[left] the results corresponding to Fig. 2[right] (which was for Scenario 1).The regions marked by long dashed lines and very lightly shaded are all beyond 1σ level, indicating more tension, and B s → µµ is no longer the leading constraint.The 2σ lower φ s = −0.04gives bound from below, while on the left it is bounded by 2σ B s → µ + µ − (4.3 × 10 −9 [13]) from above, and on the right, bounded from above by ), a sliver of K + → π + νν (90% CL), and φ s = 0.015 (1σ), respectively.For R 6 = 1, K L → π 0 νν is barely larger than SM value.But as the R 6 value rises, one again has an ε K "chimney" that allows for enhanced K L → π 0 νν practically up to GN bound.Approaching the GN bound for K L → π 0 νν, however, illustrates the tension: For Scenario 3, which has both enhanced B d → µµ and φ s 0 preference of Eq. ( 40), we again skip the intermediate step corresponding to Fig. 2[left], and plot the results in Fig. 3[right], where the allowed region is marked by long dashed lines.There is a sliver of 1σ allowed region in the first quadrant, but there is no overlap with ε K and ε ′ /ε.The 2σ boundaries are: φ s = −0.04(SM value) from below, and K L → µµ and B s → µµ (4.3 × 10 −9 , but the sliver in 4th quadrant is 1.6 × 10 −9 ), respectively.One finds, interestingly, that when ε K and ε ′ /ε (the blue and green shaded bands) are considered, φ s is in fact pushed towards the SM value of −0.04 (hence overlaps with Scenario 1 actually), while only very mild enhancement of K L → π 0 νν is permitted.For R 6 ≃ 1.5, the branching ratio could reach ∼ 2 × 10 −10 , or 1/7 of GN bound, but further enhancement and larger R 6 values are excluded.We infer from this that φ s > 0 together with enhanced B d → µµ would cause severe tension in 4G model, while the Scenario 1 result of Fig. 2[right] suggests that allowing φ s −0.04, i.e.SM value, would allow more space for K L → π 0 νν enhancement.

IV.
DISCUSSION AND CONCLUSION Some discussions were already given in presenting the scenarios of the previous section.Here, we would like to make remarks from a more global view.
If one takes enhanced B d → µ + µ − , but take the conservative PDG 2014 value for φ s (i.e., Scenario 1), Eq. ( 6), then for 1σ range, every measurement other than B d → µ + µ − would be close to SM expectation.A mild enhancement of K L → π 0 νν is possible, but this would take some while for KOTO to reach in sensitivity.We have presented already the intriguing case permitted by the 2σ zone of Fig. 2[right], and the correlation between larger K L → π 0 νν with larger R 6 , and smaller B s → µ + µ − (as well as negative φ s ).But if one takes the more positive central value for φ s concurrent with enhanced B d → µ + µ − , one finds rather strong tension, such that r ds = |V * t ′ d V t ′ s | ≃ 10 −5 is quite small (hence unnatural), together with φ ds = arg(V * t ′ d V t ′ s ) ≃ 0 such that φ s is close to SM expectation, R 6 has to be close to 1, even though K L → π 0 νν can still be mildly enhanced.One sees from combining Fig. 2[right] and Fig. 3[right] that φ s > 0 is hard to sustain (see Note Added ) if B d → µ + µ − turns out enhanced, i.e. there is no solution in case of a fourth generation.
It could happen that B d → µ + µ − ends up SM-like, but φ s moves positive, as indicated by the "LHCb-ICHEP2014" result of Eq. ( 40).This case was illustrated by Scenario 2, where in the 4G framework, we chose a sizable φ db = arg(V * t ′ d V t ′ b ) phase, but still within 1σ of measured sin 2β/φ 1 .The choice of large φ db is in part due to our interest in K L → π 0 νν, which is a purely CPV process.We found that, in the 4G context, the situation is already precarious, i.e. one survives only in 2σ allowed region.Although K L → π 0 νν can indeed be enhanced up to practically the GN bound, the well measured ε K from the kaon system forces φ s 0. There is also the correlation of larger K L → π 0 νν and R 6 , and smaller B s → µ + µ − .If one follows the indication from Eq. ( 40), we find that for φ s > +0.05, no solution space exist for fourth generation model within our setup.
We have used the fourth generation model for illustration, since it provides specific new CKM mixing elements to affect b → s and b → d transitions, and induces correlations with s → d transitions.It is generally viewed that the fourth generation is ruled out by the SM-like Higgs boson production cross section.But we have argued [26] that the Higgs boson does not enter the lower energy processes discussed here, hence these processes are independent flavor checks.Furthermore, loopholes exist for the SM-Higgs interpretation [27].
The correlations and possible tensions in the 4G framework is taken as an illustration of the possible correlation between b → s, b → d, and s → d processes.If one does not accept the 4G model, we stress that B d → µ + µ − might well be found experimentally to be enhanced compared with SM expectation.We know of no other model, except 4G, which could actually achieve this.In general, whatever flavor physics one turns on in some New Physics model, one faces the myriad of constraints discussed in Sec.II.We believe no model can survive intact and "without blemish".Thus, the modes B d → µ + µ − , B s → µ + µ − , φ s and K L → π 0 νν provide "pressure tests" to our understanding of flavor and CP violation.Within 5 to 10 years from now, all the measurements may end up consistent with SM, but we may also face genuine surprises in any one of these forefront measurements.Though differences must exist, we believe there would be correlations between the above four modes in any New Physics model that has a limited set of new parameters.
Turning to an experimental perspective, we stress that the KOTO experiment must first reach below the GN bound for K L → π 0 νν, around 1.4 × 10 −9 , to "start business", as larger values are unlikely.This is the aim for KOTO [24] for the 2015 run.We are reminded, however, that there is another ongoing experiment, NA62, that aims [25] principally to measure K + → π + νν at SM level.We have taken the conservative 90% CL limit from E949.But it is not impossible that NA62 could observe K + → π + νν even above this bound.If so, then things would be even more interesting for KOTO.That is, if NA62 discovers larger value for K + → π + νν, then the GN bound for K L → π 0 νν moves up.This is by itself a volatile situation that ought to be watched.NA62 will start its first physics run in Fall 2014.
Finally, although we are not one hundred percent certain, our study of correlations between the various forefront flavor and CPV probes in the 4G context indicates that φ s > 0 could be a major challenge.We remark that one thread of the paper, which has its origins in Ref. [26], is the mild but long-standing hint of discrepancy between the direct and indirect measurements of sin 2β/φ 1 .Could this thread be the clue that leads to the deconstruction of the tapestry?Let's wait and see in the next 3-4 years!
In conclusion, the possible enhancement of B 0 d → µ + µ − mode could correlate with enhancement of K L → π 0 ν ν decay up to close to the Grossman-Nir bound in the 4th generation model, with the further correlation that B 0 s → µ + µ − is somewhat suppressed while CPV phase φ s 0 and small.Positive values for φ s are disfavored.These measurements would provide "pressure tests" to our understanding of flavor and CP violation for any New Physics model, especially if any deviation is found.They should be followed earnestly in parallel to the final scrutiny of the nature of the 125 GeV boson at LHC Run 2.
at the E949 bound, relatively suppressed B s → µµ, slightly positive φ s .But that would be a great bonanza for all measurements involved, except B(B d → µµ) ∼ SM, including a hadronic physics challenge of R 6 2.5.Here,|V t ′ s | > |V t ′ d | would demand 10 4 × |V * t ′ d V t ′ s | 0.32, in favor of larger K L → π 0 νν.