Eur. Phys. J. C (2013) 73:2560DOI 10.1140/epjc/s10052-013-2560-1

Regular Article - Theoretical Physics

Stringent tests of constrained Minimal Flavor Violationthrough �F = 2 transitions

Andrzej J. Buras1, Jennifer Girrbach2,a

1TUM-IAS, Lichtenbergstr. 2a, 85748 Garching, Germany2Physik Department, TUM, 85748 Garching, Germany

Received: 2 July 2013 / Revised: 14 August 2013 / Published online: 6 September 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract New Physics contributions to �F = 2 transi-tions in the simplest extensions of the Standard Model(SM), the models with constrained Minimal Flavor Viola-tion (CMFV), are parametrized by a single variable S(v),the value of the real box diagram function that in CMFVis bounded from below by its SM value S0(xt ). With al-ready very precise experimental values of εK , �Md , �Ms

and precise values of the CP-asymmetry SψKSand of B̂K

entering the evaluation of εK , the future of CMFV in the�F = 2 sector depends crucially on the values of |Vcb|,|Vub|, γ , FBs

√B̂Bs and FBd

√B̂Bd

. The ratio ξ of the lat-ter two non-perturbative parameters, already rather preciselydetermined from lattice calculations, allows then togetherwith �Ms/�Md and SψKS

to determine the range of theangle γ in the unitarity triangle independently of the valueof S(v). Imposing in addition the constraints from |εK | and�Md allows to determine the favorite CMFV values of

|Vcb|, |Vub|, FBs

√B̂Bs and FBd

√B̂Bd

as functions of S(v)

and γ . The |Vcb|4 dependence of εK allows to determine|Vcb| for a given S(v) and γ with a higher precision than itis presently possible using tree-level decays. The same ap-plies to |Vub|, |Vtd | and |Vts | that are automatically deter-mined as functions of S(v) and γ . We derive correlations

between FBs

√B̂Bs and FBd

√B̂Bd

, |Vcb|, |Vub| and γ that

should be tested in the coming years. Typically FBs

√B̂Bs

and FBd

√B̂Bd

have to be lower than their present latticevalues, while |Vcb| has to be higher than its tree-level de-termination, with a significance depending on the evolu-tion of their errors. Within the SM this would imply valuesfor B(K+ → π+νν̄) and B(KL → π0νν̄) that are typicallylarger by (15–20) % than those presently quoted in the lit-erature. The region in the space of these three parametersallowed by CMFV indicates tensions in this class of modelsand hints for the presence of new sources of flavor violationand/or new local operators in �F = 2 data that are strongly

suppressed in these models. As a byproduct we propose toreduce the present uncertainty in the charm contribution toεK by using the experimental value of �MK .

1 Introduction

The simplest class of extensions of the Standard Model(SM) are models with constrained MFV (CMFV) [1–3] thatsimilarly to the SM imply stringent correlations betweenobservables in K , Bd and Bs systems, while allowing stillfor significant departures from SM expectations. In the caseof �F = 2 transitions in the down-quark sector, that is,K0–K̄0 and B0

s,d–B̄0s,d mixings, new physics (NP) in these

models enters only through a single universal real one-loopfunction, the box diagram function S(v), with v standing forthe parameters of a given CMFV model. Moreover it can beshown that in this class of models S(v) is bounded from be-low by its SM value [4]:

S(v) ≥ S0(xt ) ≈ 2.3 (1)

with S0(xt ) given in (9).It is known that the SM experiences some tension in the

correlation SψKS− εK [5–9]. This tension can be naturally

removed in models with CMFV by enhancing the value ofS(v). However, as pointed out in [10, 11] this spoils theagreement of the SM with the data on �Ms,d signaling thetension between �Ms,d and εK in CMFV models.

Now the experimental data on εK , �Md , �Ms are al-ready very precise. Moreover the CP-asymmetry SψKS

isalso measured within the accuracy of a few percent. In allCMFV models we have

SψKS= sin 2β (CMFV), (2)

a e-mail: jennifer.girrbach@gmail.com

Page 2 of 14 Eur. Phys. J. C (2013) 73:2560

and consequently this implies a universal precise value ofthe angle β in the unitarity triangle in this class of mod-els [1].

With |Vus | determined already very precisely the size ofthe tensions mentioned above depends primary on the valuesof

|Vcb|, FBs

√B̂Bs , FBd

√B̂Bd

, B̂K, γ, S(v), (3)

with |Vub| being then a derived quantity from the unitarityof the CKM matrix. The present status of these quantitiescan be found in Table 1. In the case of Bs,d mesons we referalso to [12, 13].

Now the lattice QCD calculations made an impressiveprogress in the determination of B̂K which enters the evalu-ation of εK [14–19]. The most recent world average B̂K =0.767 ± 0.010 [20] is very close to its large N value B̂K =0.75 [21, 22]. Moreover the analyses in [23] and in partic-ular in [24] show that B̂K cannot be larger than its large N

value. Therefore we expect that the final lattice value of B̂K

will be slightly lower than the present one. However, as thelattice precision is already very high, we expect that the fi-nal result will be very close to 0.75 and it is a very goodapproximation to set simply

B̂K = 0.75. (4)

Indeed compared to the present uncertainty from |Vcb| inεK proceeding in this manner is fully justified. Moreoverthis value of B̂K is optimal for CMFV models as with lowervalues it is harder to fit the data on εK and the tensions dis-cussed by us increase.

We are then left with five variables in (3) and we canask the question which values of them would allow to fitall �F = 2 data within the SM and CMFV models. Even ifsuch an exercise is against the spirit that γ and |Vcb| should

be determined from tree-level decays, while FBs

√B̂Bs and

FBd

√B̂Bd

from lattice calculations independently of NPcontributions, in a concrete framework like the SM andCMFV such an exercise makes sense. In fact the determi-nations of some of the non-perturbative parameters fromFCNC data was already performed within the SM with dif-ferent goals in the analyses of the UTfit collaboration [25].See also [9].

The spirit of the present paper differs also from our recentstudies of NP models, in particular in [11, 26–31], where theemphasis has been put on correlations between observables,still to be measured, rather than the correlations between pa-rameters of a given NP model. However, due to the shut-down of the LHC for two years not much progress in test-ing these correlations is expected soon. On the other handthe coming years can be considered as lattice precision era,which will allow to study NP with high precision through

�F = 2 observables for which the data are already very pre-cise. In the field of FCNC processes this is a unique situationuntil now.

In order to reach our goal we need still one input. Ide-ally it should be the angle γ in the unitarity triangle deter-mined in tree-level decays and therefore independently ofS(v). For this reason we present our results as functions ofγ . However, as γ is presently not well known from tree-leveldecays, we will use the range for the ratio

ξ = FBs

√B̂Bs

FBd

√B̂Bd

(5)

from lattice calculations to find which range for γ is consis-tent with experimental value of the ratio �Ms/�Md in thefull class of CMFV models independently of S(v).

In principle, also |Vcb| could be used as input, but thevery strong dependence of εK on |Vcb| as |Vcb|4 invites usin the presence of an accurate value of B̂K to determine |Vcb|from εK .1 As already stated above we determine here |Vcb|to fit FCNC data and this makes sense in this concrete classof models. Comparison with tree-level results for |Vcb| willoffer still another test of CMFV.

Our paper is organized as follows. In Sect. 2 we recall theformulas used in our analysis. In Sect. 3 we calculate

|Vcb|, FBs

√B̂Bs , FBd

√B̂Bd

, |Vub|,|Vtd |, |Vts |, B

(B+ → τ+ντ

) (6)

in CMFV models as functions of S(v) and γ . Among thesemodels a prominent role is played by the SM. These resultswill be compared in the future with more precise values ofthese quantities but already now one can see that CMFVmodels may turn out to have a hard time to give a satis-factory description of them unless appreciable shifts with

respect to the current central values of |Vcb|, FBs

√B̂Bs and

FBd

√B̂Bd

will be found. In Sect. 4 we analyze the uncer-tainty in the determination of |Vcb| induced by the largeuncertainty in the QCD factor ηcc in the charm contribu-tion to εK . We demonstrate how this uncertainty can be re-duced by a factor of three by using the experimental valueof �MK and the large N estimate of long distance contri-butions to the latter. In Sect. 5 we address simplest modelsoutside the CMFV framework, in particular the MFV mod-els at large, where new operators and flavor blind phasescould be present. We also comment on models with U(2)3

flavor symmetry. In Sect. 6 we summarize briefly our results.

1The idea of determining |Vcb| from εK is not new. It was suggestedin [32] with the goal to predict an accurate value of B(K+ → π+νν̄)

within the SM that similarly to εK suffers from |Vcb|4 dependence. Butat that time the precision on B̂K was insufficient so that this strategycould not be used efficiently.

Eur. Phys. J. C (2013) 73:2560 Page 3 of 14

2 Compendium of basic formulas

2.1 �F = 2 observables

Here we collect the formulas we used in our calculations.For the mass differences in the B0

s,d–B̄0s,d systems we have

�Ms = 19.1/ps ·[√

B̂Bs FBs

279 MeV

]2[S(v)

2.31

]

×[ |Vts |

0.040

]2[ηB

0.55

], (7)

�Md = 0.55/ps ·[√

B̂BdFBd

226 MeV

]2[S(v)

2.31

]

×[ |Vtd |

8.5 · 10−3

]2[ηB

0.55

]. (8)

The value 2.31 in the normalization of S(v) is its presentSM value for mt(mt) = 163 GeV:

S0(xt ) = 4xt − 11x2t + x3

t

4(1 − xt )2− 3x2

t logxt

2(1 − xt )3

= 2.31

[mt(mt)

163 GeV

]1.52

. (9)

Concerning |Vtd | and |Vts |, we have first

|Vtd | = |Vus ||Vcb|Rt (10)

with Rt being one of the sides of the unitarity triangle whichgenerally in CMFV can be determined independently ofS(v) [3]:

Rt = ηR

ξ

|Vus |

√�Md

�Ms

√mBs

mBd

,

ηR = 1 − |Vus |ξ√

�Md

�Ms

√mBs

mBd

cosβ + λ2

2+O

(λ4).

(11)

Here ξ is defined in (5) and β is determined also universallythrough (2). We also have

|Vts | = ηR|Vcb| (12)

with |Vcb| determined from εK as discussed below.Now the departure of ηR from unity is very small and

can be calculated by using present lattice input for ξ and thecentral value of β . This gives

Rt = 0.743ξ, ηR = 0.9812. (13)

For the numerics in Sect. 3 we, however, use the relation (11)without the lattice input, except for Eq. (29) below where weuse (13).

The next steps depend on whether ξ or γ is used as aninput. If ξ is used, (13) allows to determine Rt and using therelations

Rb =(

1 − λ2

2

)1

λ

∣∣∣∣Vub

Vcb

∣∣∣∣ =√

1 + R2t − 2Rt cosβ,

cotγ = 1 − Rt cosβ

Rt sinβ

(14)

allows to determine |Vub| and γ .On the other hand if γ from tree-level decays is used as

an input, one could determine Rt and Rb without using thelattice value of ξ [33]:

Rt = sinγ

sin(β + γ ), Rb = sinβ

sin(β + γ ). (15)

In turn ξ could be determined by using (11) and the compar-ison with its lattice value would be another test of CMFV.

Finally, the CP-violating parameter εK is given by

εK = κεeiϕε

√2(�MK)exp

[�(MK

12

)], (16)

where ϕε = (43.51 ± 0.05)◦ and κε = 0.94 ± 0.02 [6, 34]takes into account that ϕε �= π

4 and includes long distanceeffects in �(Γ12) and �(MK

12). Moreover (λi = V ∗isVid )

(MK

12

)∗ = G2F

12π2F 2

KB̂KmKM2W

[λ2

cηccS0(xc)

+ λ2t ηttS(v) + 2λcλtηctS0(xc, xt )

], (17)

where the QCD factors ηij are given in Table 1 andS0(xc, xt ) can be found in [35]. In writing (17) we makea very plausible assumption that in models with CMFV, NPenters only through S(v). In any case the contributions in-volving charm quark are subleading.

We can then expose the main parametric dependence ofεK within the CMFV models as follows [6]:

|εK | = κεCεB̂K |Vcb|2λ2η̄(|Vcb|2(1 − �̄)ηttS(v)

+ ηctS0(xc, xt ) − ηccxc

), (18)

where η̄ and �̄ are the known variables related to the unitar-ity triangle and

Cε = G2F F 2

KmKM2W

6√

2π2(�MK)exp. (19)

As in εK the function S(v) is proportional to |Vcb|4 andnot |Vcb|2 as in �Ms,d it is εK which plays the crucial

Page 4 of 14 Eur. Phys. J. C (2013) 73:2560

role in fixing the favored value of |Vcb|. The prime roles of

�Ms,d is to determine Rt through (11) and then FBs

√B̂Bs

and FBd

√B̂Bd

through (7) and (8), respectively.Finally, eliminating S(v) from all these expressions in

terms of other variables we find the basic formula express-ing the correlation between various quantities discussed inour paper

SψKS= sin 2β = 1

b�Md

[ |εK ||Vcb|2B̂K

− a

], (20)

where

a = rεRt sinβ[ηctS0(xt , xc) − ηccxc

],

b = ηtt

ηB

rε

2rd |Vus |21

F 2Bd

B̂Bd

,(21)

with

rε = κε|Vus |2Cε, rd = G2F

6π2M2

WmBd(22)

and Rt given in (11).In the derivation of (20) the following relations are use-

ful:

sin 2β = 2η̄(1 − �̄)

R2t

, η̄ = Rt sinβ. (23)

2.2 B+ → τ+ν in CMFV

It will also be interesting to consider the tree-level decayB+ → τ+ντ . In the SM B+ → τ+ντ is mediated by theW± exchange with the resulting branching ratio given by

B(B+ → τ+ντ

)SM

= G2F mB+m2

τ

8π

(1 − m2

τ

m2B+

)2

F 2B+|Vub|2τB+

= 6.05|Vub|2(

FB+

185 MeV

)2

. (24)

Evidently this result is subject to significant parametric un-certainties induced in (24) by FB+ and |Vub|. However, re-cently the error on FB+ from lattice QCD decreased signif-icantly [36] so that the dominant uncertainty comes from|Vub|.

To our knowledge B+ → τ+ντ decay has never beenconsidered in CMFV. Here we would like to point out thatin this class of models the branching ratio for this decay isenhanced (suppressed) for the same (opposite) sign of thelepton coupling of the new charged gauge boson relative tothe SM one. Indeed, the only possibility to modify the SMresult up to loop corrections in CMFV is through a tree-level

exchange of a new charged gauge boson, whose flavor inter-actions with quarks are governed by the CKM matrix. Inparticular the operator structure is the same.

Denoting this gauge boson by W ′ and the correspondinggauge coupling by g̃2 one has

B(B+ → τ+ν)

B(B+ → τ+ν)SM=

(1 + r

g̃22

g22

M2W

M2W ′

)2

, (25)

where we introduced a factor r allowing a modification inthe lepton couplings relatively to the SM ones, in particularof its sign. Which sign is required will be known once thedata and SM prediction improve.

If W ′ with these properties is absent, the branching ra-tio in this framework is not modified with respect to the SMup to loop corrections that could involve new particles butare expected to be small. A tree-level H± exchange gener-ates new operators and is outside this framework. In orderto simplify the analysis, in what follows we will neglect di-rect NP contributions to this decay and we will use the SMformula (24). In this manner NP will enter this decay indi-rectly through the value of |Vub| determined in our �F = 2analysis.

3 Numerical analysis

Our analysis uses the inputs of Table 1 where we also col-lect the present values of the quantities in (3) that we willdetermine. In this context let us emphasize that while theresults for weak decay constants from lattice are very new,

the values of FBs

√B̂Bs , FBd

√B̂Bd

, B̂Bs and B̂Bdgiven in

this table are from 2011. But soon these results should beupdated. Still it is tempting to combine the new results forweak decay constants with the values of the B̂i parametersin Table 1 to find

FBs

√B̂Bs = 259(7) MeV,

FBd

√B̂Bd

= 211(10) MeV, ξ = 1.227.

(26)

We observe that ξ is basically unchanged but the values of

FBs

√B̂Bs and FBd

√B̂Bd

are decreased. Consequently as in-vestigated in [10] also �Ms and �Md decrease so that theεK–�Ms,d tension in CMFV identified in [10, 11] would besoftened. But this requires the confirmation of the values in(26) by future lattice calculations.

Before entering our analysis we want to find the range forγ by using the range for ξ in Table 1 together with formulas(11) and (14). We find independently of S(v)2

γ = (66.6 ± 3.7)◦ (CMFV) (27)

2We included the uncertainty from SψKSas well.

Eur. Phys. J. C (2013) 73:2560 Page 5 of 14

Table 1 Values of the experimental and theoretical quantities used asinput parameters

GF = 1.16637(1) × 10−5 GeV−2 [37]

MW = 80.385(15) GeV [37]

sin2 θW = 0.23116(13) [37]

α(MZ) = 1/127.9 [37]

αs(MZ) = 0.1184(7) [37]

mu(2 GeV) = (2.1 ± 0.1) MeV [20]

md(2 GeV) = (4.73 ± 0.12) MeV [20]

ms(2 GeV) = (93.4 ± 1.1) MeV [20]

mc(mc) = (1.279 ± 0.013) GeV [39]

mb(mb) = 4.19+0.18−0.06 GeV [37]

mt(mt ) = 163(1) GeV [20, 40]

Mt = 173.2 ± 0.9 GeV [43]

mK = 497.614(24) MeV [37]

FK = 156.1(11) MeV [20]

B̂K = 0.767(10) [20]

κε = 0.94(2) [6, 34]

ηcc = 1.87(76) [47]

ηtt = 0.5765(65) [41]

ηct = 0.496(47) [48]

�MK = 0.5292(9) × 10−2 ps−1 [37]

|εK | = 2.228(11) × 10−3 [37]

B(B → Xsγ ) = (3.55 ± 0.24 ± 0.09) × 10−4 [37]

B(B+ → τ+ν) = (1.15 ± 0.23) × 10−4 [49]

mBd= mB+ = 5279.2(2) MeV [38]

mBs = 5366.8(2) MeV [38]

FBd= (188 ± 4) MeV [36]

FBs = (225 ± 3) MeV [36]

FB+ = (185 ± 3) MeV [36]

BBs = 1.33(6) [20]

B̂Bs /B̂Bd= 1.05(7) [20]

FBd

√B̂Bd

= 226(13) MeV [20]

FBs

√B̂Bs = 279(13) MeV [20]

ξ = 1.237(32) [20]

ηB = 0.55(1) [41, 42]

�Md = 0.507(4)ps−1 [44]

�Ms = 17.72(4)ps−1 [44]

SψKS= 0.679(20) [37]

Sψφ = −0.01 ± 0.08 [45]

�Γs = 0.116 ± 0.019 [46]

τBs = 1.503(10)ps [44]

τBd= 1.519(7)ps [44]

τB± = (1641 ± 8) × 10−3 ps [44]

|Vus | = 0.2252(9) [44]

|Vcb| = (40.9 ± 1.1) × 10−3 [38]

|V incl.ub | = (4.41 ± 0.31) × 10−3 [38]

|V excl.ub | = (3.23 ± 0.31) × 10−3 [38]

which having much smaller error agrees very well with γ

from tree-level decays obtained by LHCb using 3 fb−1 andneglecting D0 mixing [50]3

γ = (67.2 ± 12)◦, (LHCb). (28)

Similar comment applies to the extraction of γ from U-spinanalysis of Bs → K+K− and Bd → π+π− decays (γ =(68.2 ± 7.1)◦) [52].

As we have seen the values of γ , β and ξ are correlatedwith each other and any pair of these variables can be usedfor phenomenology. We find it useful to use γ,β as fun-damental input parameters as they may turn out to be mostefficient one day in determining the pair (Rt ,Rb) withoutusing the value of ξ . However, in order to be consistent withthe present range for ξ we have to vary β in the range con-sistent with the values for SψKS

and γ in the range in (27).Having this in mind we will proceed in the following steps:4

Step 1 For a given γ and β from SψKSwe will calculate Rb

and Rt by means of (15) and subsequently ξ by using (11).The resulting value for ξ serves here mainly as a test thatwe are consistent with its present lattice values but its futuremore precise value from lattice calculations could narrowthe range for the output values of various quantities evalu-ated in the remaining two steps.

Step 2 Having ξ allows with the help of εK to determine|Vcb| as a function of S(v). Explicitly for central values ofparameters we find (using (13))

|Vcb| = v(ηcc, ηct )√ξS(v)

√√1 + h(ηcc, ηct )S(v) − 1,

v(ηcc, ηct ) = 0.0282, h(ηcc, ηct ) = 24.83.

(29)

The second solution of the quadratic equation for |Vcb|2 isexcluded as it leads to a negative |Vcb|2. The numerical val-ues of h(ηcc, ηct ) and v(ηcc, ηct ) correspond to the centralvalues of ηcc and ηct in Table 1. We will discuss the uncer-tainty due to these parameters in the next section.

With all this information at hand we can determine |Vub|,|Vts | and |Vtd | as functions of S(v).

Step 3 Finally having |Vcb| we can determine√

B̂BdFBd

from (8) and knowing already ξ from Step 1 also√

B̂Bs FBs .

3In [51] both B → DK and B → Dπ decays are used and furthermoreD0 mixing fully included and the combination of results gives as best-fit value γ = 72.6◦ and the confidence interval γ ∈ [55.4,82.3]◦ at68 % CL.4The numerical values in the expressions in this section correspond tothe central values of the remaining input parameters.

Page 6 of 14 Eur. Phys. J. C (2013) 73:2560

As the route to determine all these quantities could ap-pear not transparent we show it in a chart in Fig. 1.

In Fig. 2 we show the dependence of ξ on γ varyingSψKS

in its one σ range. We note that the uncertainty dueto SψKS

is very small. This triple correlation is universal forall CMFV models including the SM and is central for testsof CMFV framework. Measuring γ from tree-level decaysprecisely together with precise values of SψKS

and ξ will bean important test for this class of models as this correlationis independent of S(v). The range for γ in (27) has been ex-tracted from this plot by varying ξ and SψKS

independentlyin their present one σ ranges.

It should be emphasized that this first test is independentof |Vub| and |Vcb| and can become a precision test of CMFVmodels. But in order to find which CMFV model, if any, ischosen by nature we have to consider the quantities whichdepend on S(v). This brings us to Steps 2 and 3.

The S(v) and γ dependence of all quantities of interestis shown in Table 2. We show there also the SM case cor-responding to S(v) = S0(xt ) = 2.31. These results are alsoshown in Fig. 3, where we plot various quantities as func-tions of S(v) for different values of γ . The thickness of thelines shows the uncertainty in SψKS

. Evidently these plotsare correlated with each other:

• Once one of the variables is determined, also S(v) is de-termined and consequently the remaining quantities arepredicted.

• In a given model in which S(v) is known, all quantitiesconsidered are predicted.

Equivalently to Fig. 3 one can find using (20) explicituniversal correlations between various quantities in Table 2.We show few examples below. In all plots we also show the

Fig. 1 The overview of the determination of various quantities dis-cussed in the text with results given in Table 2 (Color figure online)

Fig. 2 γ versus ξ for SψKS∈ [0.659,0.699] (Color figure online)

present best values of all quantities obtained directly fromlattice or tree-level decays.

In order to understand the results in Table 2 let us re-call the problems of CMFV for the present input for |Vcb|,FBs

√B̂Bs and FBd

√B̂Bd

in Table 1 [10, 11]. With the valueof β extracted from SψKS

and γ from �Ms/�Md , εK isin the SM significantly smaller than its experimental value,while �Ms,d are significantly larger than corresponding ex-perimental values. Therefore increasing S(v), while bring-ing εK closer to the data, shifts �Ms,d further from theirexperimental values. Using for example |Vub| = 0.0034 andγ = 68◦ where the SM prediction and the central exper-imental value for SψKS

coincide, the SM prediction for|εK |SM = 0.00186 is below the data. Turning now to CMFV,the value of S(v) at which the central experimental value of|εK | is reproduced turns out to be S(v) = 2.9 [11] to be com-pared with SSM = 2.31. At this value of S(v) with presentinput the central values of �Ms,d read

�Md = 0.69(6) ps−1, �Ms = 23.9(2.1) ps−1. (30)

They both differ from experimental values by 3σ . Using an-other value of |Vub| which worsens the agreement of theSM with the experimental value for SψKS

leads to a differ-ent value of S(v) to reproduce |εK |exp = 0.002228 and thusalso (30) changes.

These problems of CMFV can also be seen when thepresent central values of |Vcb| and of non-perturbative pa-rameters are inserted together with the data on �Ms,d , εK ,|Vcb| and |Vus | into (20). We find then

SψKS= sin 2β = 0.86 ⇒ β = 29.8◦,

Rt = 0.92(31)

and thus

Rb = 0.50, |Vub| = 0.0047, γ = 66.4◦. (32)

Evidently, the fact that SψKSis much larger than the data

requires the presence of new CP-violating phases. This ex-ercise is equivalent to the one performed in [5], where εK

Eur. Phys. J. C (2013) 73:2560 Page 7 of 14

Table 2 CMFV predictions for various quantities as functions of S(v)

and γ . The four elements of the CKM matrix are in units of 10−3,FBs

√B̂Bs and FBd

√B̂Bd

in units of MeV and B(B+ → τ+ν) in units

of 10−4

S(v) γ |Vcb| |Vub| |Vtd | |Vts | FBs

√B̂Bs FBd

√B̂Bd

ξ B(B+ → τ+ν)

2.31 63◦ 43.6 3.69 8.79 42.8 252.7 210.0 1.204 0.822

2.5 63◦ 42.8 3.63 8.64 42.1 247.1 205.3 1.204 0.794

2.7 63◦ 42.1 3.56 8.49 41.4 241.8 200.9 1.204 0.768

2.31 67◦ 42.9 3.62 8.90 42.1 256.8 207.2 1.240 0.791

2.5 67◦ 42.2 3.56 8.75 41.4 251.1 202.6 1.240 0.765

2.7 67◦ 41.5 3.50 8.61 40.7 245.7 198.3 1.240 0.739

2.31 71◦ 42.3 3.57 9.02 41.5 260.8 204.5 1.276 0.770

2.5 71◦ 41.6 3.51 8.87 40.8 255.1 200.0 1.276 0.744

2.7 71◦ 40.9 3.45 8.72 40.1 249.6 195.7 1.276 0.719

Fig. 3 CKM matrix elements versus S(v) for γ = 63◦/67◦/71◦ (green, red, blue). The thickness of the lines corresponds to the 1σ range ofSψKS

∈ [0.659,0.699] (Color figure online)

Page 8 of 14 Eur. Phys. J. C (2013) 73:2560

has been set to its experimental value but sin 2β was pre-dicted. On the other hand setting SψKS

to its experimentalvalue as done in [6] one finds that |εK | is significantly be-low the data.

Thus the only hope for CMFV is that the input on |Vcb|,FBs

√B̂Bs and FBd

√B̂Bd

changes with time. In particular asseen in Table 2:

• The values of FBs

√B̂Bs and FBd

√B̂Bd

have to decreasesignificantly, at least as far as given in (26). But this as-sumes the lowest value of S(v).

• At this value of S(v) in order to obtain agreement of εK

with the data |Vcb| should be larger by roughly 2σ fromits present central tree-level value.

• Increasing S(v) allows to lower the required value of

|Vcb| but simultaneously decreases further FBs

√B̂Bs and

FBd

√B̂Bd

making their values significantly lower thantheir present values without changing their ratio (seeFig. 4). This finding shows that the freedom in choosingS(v) does not necessarily help in solving CMFV prob-lems.

• With increasing γ the value of |Vcb| required by εK

can further be decreased. For a fixed S(v) this increases

FBs

√B̂Bs but decreases FBd

√B̂Bd

as can be read off fromFig. 4.

• For the full range of S(v) and γ considered in Table 2 thevalues of |Vub| and |Vtd | remain in the following ranges:

|Vub| = (3.57 ± 0.16) × 10−3,

|Vtd | = (8.75 ± 0.26) × 10−3,(33)

where in the case of |Vub| we included also the error fromβ , which is irrelevant in the case of |Vtd |. Otherwise theerror in |Vub| would amount to ±0.12. The value of |Vts |follows the one of |Vcb| but is by 1.9 % smaller than

the latter. Also for B(B+ → τ+ν) a narrow range is pre-dicted:

B(B+ → τ+ν

) = (0.77 ± 0.07) × 10−4 , (34)

where the present uncertainty in FB+ has been taken intoaccount.

In Fig. 4 on the left hand side we show the correlation

between FBd

√B̂Bd

and |Vcb| for different value of γ . Anal-

ogous correlation between FBs

√B̂Bs and |Vcb| is shown on

the right hand side. Possibly these two plots showing theallowed ranges for the three parameters in question in theCMFV framework are among the most important results ofour paper. Similarly we show in Fig. 5 the same correlationbut with |Vub|.

Finally, we may ask what all these findings would im-ply for K+ → π+νν̄ and KL → π0νν̄ decays which alsostrongly depend on |Vcb|. As these decays depend withinCMFV models on a different one-loop function X(v), whichcould be larger or smaller than in the SM, nothing definitecan be stated in a general CMFV model. On the other hand,within the SM this function is known. The fact that in thismodel the optimal value of |Vcb| at γ ≈ 67◦ is 0.0429 andnot 0.0409 given in Table 1, automatically implies values forB(K+ → π+νν̄) and B(KL → π0νν̄) that are, respectively,larger by 15 % and 20 % than the ones quoted presentlyin the literature. This difference could matter one day whenother parametric uncertainties decrease.

4 The uncertainty due to ηcc and ηct

It is known that significant uncertainty in the SM predictionfor εK comes from the value of the QCD correction ηcc andto a lesser extent from ηct that even at the NNLO level are

Fig. 4 |Vcb| versus FBd

√B̂Bd

and FBs

√B̂Bs for S(v) ∈ [2.3,2.8]

and γ ∈ [63◦,71◦] (yellow), γ = 63◦/67◦/71◦ (green, red, blue).Cyan: fixed S(v) = 2.4 and γ ∈ [63◦,71◦]. Gray range: 1σ range of

FBd

√B̂Bd

= 226(13) MeV, FBs

√B̂Bs = 279(13) MeV and |Vcb| =

(40.9 ± 1.1) × 10−3 (see Table 1) (Color figure online)

Eur. Phys. J. C (2013) 73:2560 Page 9 of 14

Fig. 5 |Vub| versus FBd

√B̂Bd

and FBs

√B̂Bs for S(v) ∈ [2.3,2.8]

and γ ∈ [63◦,71◦] (yellow), γ = 63◦/67◦/71◦ (green, red, blue).

Cyan: fixed S(v) = 2.4 and γ ∈ [63◦,71◦]. Gray range: 1σ range of

FBd

√B̂Bd

= 226(13) MeV, FBs

√B̂Bs = 279(13) MeV and |V excl

ub | =(3.21±0.31)×10−3 (light gray), |V incl

ub | = (4.41±0.31)×10−3 (darkgray) (see Table 1) (Color figure online)

known only with the accuracy of ±41 % and ±9 %, respec-tively [47, 48]. In our analysis this uncertainty enters the co-efficient h(ηcc, ηct ) of S(v) and v(ηcc, ηct ) in (29). In whatfollows we would like to investigate the impact of these un-certainties on the determination of |Vcb| from εK and pro-pose a method of reducing the uncertainty in ηcc with thehelp of the experimental value of �MK accompanied in par-ticular by future lattice calculations of long distance effectsin �MK .

In Table 3 we show the values of h(ηcc, ηct ) correspond-ing to the range of these QCD corrections calculated in[47, 48]. If we fix γ = 67◦ and S(v) = S0(xt ) = 2.31 andvary ηcc ∈ [1.10,2.64] and ηct ∈ [0.451,0.541] simultane-ously then |Vcb| lies within the range [41.7,44.4] ·10−3 (seealso Fig. 6). Scanning only ηcc(ηct ) and fixing ηct (ηcc) toits central value we find |Vcb| ∈ [42.1,43.9] · 10−3(|Vcb| ∈[42.6,43.4] · 10−3). Translated in the uncertainty in the de-termination of |Vcb| we find an uncertainty of ±2.0 % and±0.9 % due to ηcc and ηct , respectively. The uncertaintydue to ηtt is fully negligible. It is expected that an improvedmatching of short distance calculations of ηcc and ηct tothe lattice calculations of B̂K can significantly reduce thepresent total theoretical error on εK and thus allow a moreaccurate extraction of |Vcb|.

As the large uncertainty in ηcc is disturbing we propose toextract this parameter from the experimental value of �MK .Assuming that in CMFV models �MK is fully dominatedby the SM contributions we decompose it as follows:

�MK = (�MK)cc + (�MK)ct + (�MK)tt + (�MK)LD,

(35)

with the first three short distance contributions obtainedfrom

(�MK)ij = 2 (MK

12

)ij

(36)

Fig. 6 |Vcb| as a function of ηcc and ηct for fixed γ = 67◦ andS(v) = S0(xt ) = 2.31 (Color figure online)

with MK12 given in (17) and i = c, t . For the dominant con-

tribution we have

(�MK)cc =(

λ − λ3

2

)2 G2F

6π2F 2

KB̂KmKηccm2c(mc). (37)

We find then setting mc(mc) at its central value5

ηcc = 2.123

(0.75

B̂K

)w, (38)

5In principle the determination of ηccxc is more useful as this productshould be independent of the scale μc in mc but in order to comparewith the value of ηcc in Table 1 we prefer here to work with ηcc .

Page 10 of 14 Eur. Phys. J. C (2013) 73:2560

Table 3 Coefficient h(ηcc, ηct )

and v(ηcc, ηct ) in Eq. (29) fordifferent values of ηcc,ct

h(ηcc, ηct ) ηcc v(ηcc, ηct ) ηcc

1.10 1.87 2.64 1.10 1.87 2.64

ηct 0.451 18.83 34.93 85.82 ηct 0.451 0.0303 0.0259 0.0207

0.496 15.57 24.83 51.42 0.496 0.0323 0.0282 0.0235

0.541 11.61 18.55 34.21 0.541 0.0341 0.0304 0.0260

where (k = t t, ct,LD)

w = 1 − rtt − rct − rLD, rk = (�MK)k

(�MK)exp. (39)

For rtt and rct we find within the SM

rtt = 0.0021, rct = 0.0030 (40)

confirming the result in [47] that both corrections are below1 % and totally negligible within the SM compared withother uncertainties. As our analysis shows in CMFV mod-els rtt could be increased by 30 % due to the increase ofS(v) but this would still keep these corrections below 1 %.Thus our proposal depends crucially on the estimate of rLD.Yet, as we argue below the error from this contribution issignificantly smaller than the error from the direct NNLOcalculation in [47].

Basically the only results on rLD in QCD that are avail-able are from large N QCD calculations in which at low en-ergies one uses a dual representation of QCD as a theory ofweakly interacting mesons [53–55]. In the case of K0–K̄0

mixing and non-leptonic K-meson decays this approach, de-veloped in [22, 23, 56–58], provided already a quarter ofcentury ago results which are now basically confirmed bythe more sophisticated lattice calculations. This is the caseof the B̂K parameter calculated first in [22, 23] and also thecase of �I = 1/2 rule [58], where the dynamics behind thisrule related predominantly to current–current operators hasbeen identified and the enhancements of �I = 1/2 transi-tions and suppression of �I = 3/2 transitions have beencomputed reaching rough agreement with the data. Preciselythis understanding is presently emerging from lattice calcu-lations [59, 60].

Motivated by the success of this approach, its result forrLD could also be approximately correct. In the large-N ex-pansion, the leading contribution comes from one-loop con-tributions induced by two �S = 1 transitions with virtualππ , πK and KK in the loop. One finds then rLD to bepositive and in the ballpark of 0.3 [61, 62]. The study ofsubleading corrections indicates that similarly to the case ofB̂K these corrections are not large and tend to cancel partlyeach other [22, 34, 63, 64] with some tendency to have op-posite sign to the leading term [64]. While precise calcula-tion of rLD in view of these cancellations is very difficultby analytic methods, based on these studies we expect that

rLD is likely to be positive and in the ballpark of 0.1–0.3.Taking this estimate at face value and using (38) we end upwith

ηcc ≈ 1.70 ± 0.21. (41)

This is consistent with ηcc = 1.87 used by us in the previ-ous section. Moreover, the error is by a factor of at least 3.5smaller than the error obtained by the direct NNLO calcu-lation of ηcc in [47]. Interestingly the authors of the latterpaper would find this result if they varied the scale μc inthe range 1.3 ≤ μc ≤ 1.8 GeV and not in the range 1.0–2.0 GeV.

With the reduction of the error in ηcc by a factor of morethan 3.5 relatively to the one resulting from direct calcula-tion [47], the uncertainty in εK from ηcc alone is reducedfrom roughly 7 % to 2 %. This is lower than the presentuncertainty of 3 % from ηct . Yet, this reduction cannot beappreciated at present as by far the dominant uncertainty inεK comes from |Vcb|. Therefore, presently this reduction hasbasically no impact on the analyses of various extensions ofthe SM reviewed recently in [65], as in such extensions newparameters are present. The situation will change when theerror on |Vcb| will be decreased.

Needless to say, we are aware of the fact that our expecta-tions on the size of rLD and the estimate in (41) require moredetailed investigations and in particular future confirmationfrom lattice simulations. Presently no reliable result on rLD

from lattice is available but important progress towards itsevaluation has been made in [66]. This first result seems toindicate that rLD could be larger than expected by us. We aretherefore looking forward to more precise evaluation of thisimportant quantity from the lattice in order to see whetheralso in this case large N approach passed another test ornot.

In Fig. 7 we show the anatomy of various uncertaintieswith different ranges described in the figure caption. We ob-serve that the reduced error on ηcc corresponding to the cyanregion decreased the allowed region which with future lat-tice calculations could be decreased further. Comparing theblue and cyan regions we note that the reduction in the erroron ηct would be welcomed as well. It should also be stressedthat in a given CMFV model with fixed S(v) the uncertain-ties are reduced further. This is illustrated with the blackrange for the case of the SM. Finally a precise measurement

Eur. Phys. J. C (2013) 73:2560 Page 11 of 14

Fig. 7 |Vcb| versus FBd

√B̂Bd

and FBs

√B̂Bs for γ ∈ [63◦,71◦]. The

yellow region is the same as in Fig. 4: S(v) ∈ [2.31,2.8], ηcc = 1.87,ηct = 0.496. In the purple region we include the errors in ηcc,ct asin Table 3: S(v) ∈ [2.31,2.8], ηcc ∈ [1.10,2.64], ηct ∈ [0.451,0.541].In the cyan region we use instead the reduced error of ηcc as inEq. (41): S(v) ∈ [2.31,2.8], ηcc ∈ [1.49,1.91], ηct ∈ [0.451,0.541].

In the blue region we fix ηct to its central value: S(v) ∈ [2.31,2.8],ηcc ∈ [1.49,1.91], ηct = 0.496. To test the SM we include the black re-gion for fixed S(v) = S0(xt ) = 2.31 and ηcc,ct as in the purple region.The gray line within the black SM region corresponds to ηcc = 1.87and ηct = 0.496. The gray box is the same as in Fig. 4 (Color figureonline)

of γ or equivalently precise lattice determination of ξ willhave impact on Fig. 7. We illustrate this impact in Fig. 8 bysetting in the plots of Fig. 7 γ = (67 ± 1)◦.

5 Going beyond CMFV

It is evident from our analysis and in particular from Fig. 4that in the likely case in which the central values of |Vcb|,FBs

√B̂Bs and FBd

√B̂Bd

will only change by a few percentbut their uncertainties will be significantly reduced, one willhave to look beyond the CMFV framework to understandthe �F = 2 data. Models like Littlest Higgs model with T-parity, various Randall–Sundrum scenarios and general su-persymmetric models having many free parameters will beable to remove all these tensions although improved pre-cision on the quantities in question would imply new con-straints on the parameters of these models. Here we men-

tion simpler models where the discussion can be made moretransparent than in those more complicated models.

First in the case of MFV at large [67–69] in which newoperators can contribute the situation looks a bit better. Herethe presence of left–right scalar operators in a 2HDM orthe MSSM naturally interferes destructively with the SMcontribution suppressing in particular �Ms [70], but onlyslightly �Md and having no impact on εK . Even if inthese models charged Higgs and SUSY particles can en-hance S(v) one can expect that still values of |Vcb| will berequired to be above the ones quoted in Table 1 and the

ones of FBs

√B̂Bs and FBd

√B̂Bd

have to be suppressed.But the problem will be softer than in CMFV. On theother hand left-right operators affect the ratio �Ms/�Md

which in CMFV models agrees well with data. The so-lution then would be left–left scalar operators which af-fect �Ms and �Md by the same factor and keep this ratiofixed.

Fig. 8 |Vcb| versus FBd

√B̂Bd

and FBs

√B̂Bs as in Fig. 7 but for γ = (67 ± 1)◦ (Color figure online)

Page 12 of 14 Eur. Phys. J. C (2013) 73:2560

The latter solution appears to be the best for the 2HDMwith flavor blind phases, the so-called 2HDMMFV [71, 72]as well. But here the presence of new CP-violating phaseschanges the situation radically as now |Vub| can be chosenlarger in order to fit εK . The enhanced value of sin 2β canbe suppressed through these new phases. This implies onthe other hand enhancement of the asymmetry Sψφ to values0.15–0.20 [11] which is 2σ above the central experimentalvalue from LHCb and could be ruled out when the data im-prove.

Possibly the simplest solution to the problems of variousmodels with MFV is to reduce the flavor symmetry U(3)3

to U(2)3 [73–79]. As emphasized in [73, 75, 80] in this caseNP effects in εK and B0

s,d–B̄0s,d are not correlated with each

other so that as pointed out in [26] the enhancement of εK

and suppression of �Ms,d can be achieved, if necessary, in

principle for the values of |Vcb|, FBs

√B̂Bs and FBd

√B̂Bd

inTable 1.

In particular,

• NP effects in εK are of CMFV type and εK can only beenhanced but the size of necessary enhancement dependson the value of |Vub| which similar to 2HDMMFV doesnot have to be low.

• In B0s,d − B̄0

s,d system, the ratio �Ms/�Md is equal to theone in the SM and in good agreement with the data. But inview of new CP-violating phases ϕBd

and ϕBs even in thepresence of only SM operators, �Ms,d can be suppressed.But the U(2)3 symmetry implies ϕBd

= ϕBs and conse-quently a triple SψKS

− Sψφ − |Vub| correlation whichconstitutes and important test of this NP scenario [26].

• The important advantage of U(2)3 models over2HDMMFV is that in the case of Sψφ being very small oreven having opposite sign to SM prediction, this frame-work can survive with concrete prediction for |Vub|.

6 Conclusions

In this paper we have determined FBs

√B̂Bs , FBd

√B̂Bd

,|Vcb|, |Vub|, |Vtd | and |Vts |, necessary to fit very precise dataon εK , �Md , �Ms and SψKS

in CMFV models as functionsof the phase γ and the sole NP free parameter in the �F = 2transitions, the value of the box diagram function S(v). Animportant ingredient of this analysis was a very small erroron B̂K from lattice and the fact that B̂K comes within 1–2 %from its large N value 0.75. The results are shown in Table 2and Figs. 3–5. The chart showing the execution of our strat-egy can be found in Fig. 1.

Our main messages from this analysis are as follows:

• The tension between εK and �Ms,d in CMFV models ac-companied with |εK | being smaller than the data withinthe SM, cannot be removed by varying S(v) when thepresent input parameters in Table 1 are used.

• Rather the value of |Vcb| has to be increased and the val-

ues of FBs

√B̂Bs and FBd

√B̂Bd

decreased relatively to thepresently quoted lattice values. These enhancements andsuppressions are correlated with each other and dependon γ . The allowed regions in the space of these threeparameters, the most important results of our paper, areshown in Figs. 4, 7 and 8.

• We have also pointed out that the removal of the ten-sions in question through the increase of |Vcb| wouldwithin the SM imply values for B(K+ → π+νν̄) andB(KL → π0νν̄) that are respectively larger by 15 % and20 % than the ones quoted presently in the literature. Thisdifference could matter one day when other parametricuncertainties decrease.

• The knowledge of long distance contributions to �MK

accompanied by the very precise experimental value ofthe latter allows a significant reduction of the present un-certainty in the value of the QCD factor ηcc under theplausible assumption that �MK in CMFV models is fullydominated by the SM contribution. This implies the re-duction of the theoretical error in εK and in turn the re-duction of the error in the extraction of the favored valueof |Vcb| in the CMFV framework. Present estimates ofthese long distance contributions using large N QCD al-low for an optimism but more sophisticated lattice calcu-lations are required to fully execute this idea.

We have also discussed simplest extensions of the SMwhich could in principle offer a better description of the datain case CMFV would fail to do so. Models with U(2)3 flavorsymmetry appear to us as most efficient in this respect whilebeing still very simple. In order to see whether the CMFVframework will survive final tests from �F = 2 transitionsfurther progress from lattice calculations and experimentalmeasurements is required. Our wish list includes:

• In particular improved lattice calculations of FBs

√B̂Bs ,

FBd

√B̂Bd

and ξ but also of B̂K , B̂Bs and B̂Bs .• Calculation of long distance contribution to �MK in or-

der to reduce the error on ηcc as proposed by us.• Improved experimental data on |Vcb|, |Vub|, γ , SψKS

andSψφ . In particular a measurement of Sψφ significantly dif-ferent from 0.04 would signal the presence of new phasesbeyond the CMFV framework.

The correlations identified in this paper will allow tomonitor the future developments, likely to indicate that newsources of flavor and CP-violation beyond the CMFV frame-work are present in nature. In this context our main mes-sage is the following one. While until now the search for NPthrough rare decays did not bring any convincing signs ofits presence, it could turn out soon that �F = 2 transitionscombined with the progress made by lattice community willherald the appearance of particular type of NP. This would

Eur. Phys. J. C (2013) 73:2560 Page 13 of 14

not only be exciting news but would also give some direc-tions for searching for this NP in rare decays and even high-energy processes.

Acknowledgements We thank Jean-Marc Gérard for many illumi-nating comments and Chris Sachrajda for useful information on theprogress in lattice calculations. This research was fully financed anddone in the context of the ERC Advanced Grant project “FLAVOUR”(267104).

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