Lessons from the LHCb measurement of CP violation in $B_s\to K^+K^-$

The LHCb experiment measured the time-dependent CP asymmetries $C_{KK}$ and $S_{KK}$ in $B_s\to K^+K^-$ decay. Combining with the corresponding CP asymmetries $C_{\pi\pi}$ and $S_{\pi\pi}$ in $B\to \pi^+\pi^-$ decay, we find that the size of $U$-spin breaking in this system is of order $20\%$. Moreover, the data suggest that these effects are dominated by factorizable contributions. We further study the constraints on new physics contributions to $b\to u\bar uq$ ($q=s,d$). New physics that is minimally flavor violating (MFV) cannot be distinguished from the Standard Model (SM) in these decays. However, new physics that is not MFV can mimic large $U$-spin breaking. Requiring that the $U$-spin breaking parameters remain below the size implied by the data leads to a lower bound of $5-10$ TeV on the scale of generic new physics. If the new physics is subject to the selection rules that follow from the Froggatt-Nielsen (FN) mechanism or from General Minimal Flavor Violation (GMFV), the bound is relaxed to 2 TeV.


I. INTRODUCTION
CP violation in neutral meson decays has provided stringent tests of the Standard Model (SM) and has been a very effective probe of new physics, with examples such as the measurement of CP violation in K L → ππ decays [1] which led to the prediction that there is a third generation of fermions [2], and the measurement of time-dependent CP violation in B → J/ψK S decays [3,4] which proved that the Kobayashi-Maskawa mechanism is the dominant source of the observed CP violation, and excluded alternatives such as the superweak CP violation model [5] and approximate CP [6].For a recent review, see [7].
Recently, the LHCb collaboration provided the first observation of time-dependent CP violation in B s decays [8].The time dependent CP asymmetry in B s → K + K − decay is given by = −C KK cos(∆m s t) + S KK sin(∆m s t) cosh(∆Γ s t/2) + A ∆Γ KK sinh(∆Γ s t/2) .
CP symmetry would imply C KK = S KK = 0 and A ∆Γ KK = 1.Combining the measurement reported in Ref. [8] with the one reported in Ref. [9], the ranges of the relevant parameters are C KK = +0.172± 0.031, S KK = +0.139± 0.032, A ∆Γ KK = −0.897± 0.087.(1.2) In this work we present the theoretical interpretation of this measurement, and explore what can (or cannot) be learned from it.
Given that there are hadronic parameters playing a role, one can go in two directions: • Assume the SM, and extract the hadronic parameters.Then we can learn about U -spin breaking by comparing to the U -spin related parameters extracted from B d → π + π − .
• Use the consistency of the various measurements with approximate U -spin symmetry to obtain constraints on new physics contributions to the b → uūs and b → uūd decays.
For the latter study, we assume that, for processes that get contributions from SM treelevel diagrams that are not CKM suppressed, the contributions from new physics can be neglected.We allow, however, contributions of a-priori arbitrary size and phase to flavor changing neutral current (FCNC) processes and to CKM suppressed tree level processes.
Concretely for B s physics, we assume that new physics contributions to B s → J/ψφ are negligible, while the size and the phase of new physics contributions to B s − B s mixing and to B s → K + K − are only constrained by experimental data.
As concerns the assumption that b → ccs decays are dominated by the SM, note that Much -if not all -of our ability to extract lessons on the SM and on new physics from we analyze side-by-side the two processes.In Section II we introduce the necessary formalism and notations.In Section III we present model independent considerations that make our analysis as generic and as data-driven as possible.In Section IV we use the experimental data to extract the relevant hadronic parameters, and in Section V we find the size of U -spin breaking effects.In Section VI we put bounds on the size of new physics contributions to the B s → K + K − and to B d → π + π − decays.

II. FORMALISM AND NOTATIONS
We follow the formalism and notations presented in Ref. [10].
The neutral mass eigenstates of the B s − B s system are given by with the normalization In terms of the dispersive and absorptive parts of the ∆B = 2 transition amplitudes, we have We define the decay amplitudes, and the parameter λ s f : We denote A KK ≡ A s K + K − and λ KK ≡ λ s KK .The parameters of Eq. (1.2) are given by Note that a consistency check is provided by (C KK ) 2 + (S KK ) 2 + (A ∆Γ KK ) 2 = 1.The B s → K + K − decay goes via the b → uūs quark transition.It depends on the following CKM combinations: Within the SM, one can write the decay amplitudes as follows: (2.7) A few comments are in place regarding Eqs.(2.7): • Within the SM, the only source of CP violation is the CKM matrix.Hence, only the λ j bs factors are complex conjugated between A SM KK and A SM KK .
• The r s factor is, in general, complex.Its phase is a so-called strong phase, which is the same in A KK and A KK .
• P s , T s (and r s ) include in them not only QCD matrix elements, but also electroweak parameters which are neither flavor dependent nor CP violating, such as G F .
• The b → uūs transition has SM tree and penguin contributions.We define T s as the tree contribution, and P s j as the penguin contribution with an intermediate j-quark. (2.9) The B d → π + π − decay goes via the b → uūd quark transition.It depends on the following CKM combinations: (2.10) Within the SM, one can write the decay amplitudes as follows: where r d is, in general, complex.The b → uūd transition has SM tree and penguin contributions.We define T d as the tree contribution, and P d j as the penguin contribution with intermediate j-quark.Then, using CKM unitarity, we have (2.12) Note again that P d , T d (and r d ) include in them not only QCD matrix elements, but also flavor-universal CP conserving electroweak parameters.

III. MODEL-INDEPENDENT CONSIDERATIONS
The CP asymmetry in wrong-sign semileptonic B s decays is given by The experimental world average of A s SL is given by [11] A implying For our purposes we can then approximate |q s /p s | = 1 and use The time dependent CP asymmetry in B s → J/ψφ decay is given by .
The C ψφ , S ψφ and A ∆Γ ψφ parameters depend on λ ψφ in a way similar to Eq. (2.5).Within the SM, the b → ccs transition has tree and penguin contributions.In a way similar to our analysis of B s → K + K − , we can write Here, however, the second term in parenthesis is both CKM and loop suppressed, and can thus be neglected.We further assume that new physics effects on decay processes with SM tree level contributions that are not CKM suppressed are negligible.We thus obtain: Eqs. (3.8) and (3.5) imply that λ ψφ is a pure phase: Measurements of various CP asymmetries in B s decays via b → ccs lead to the following world average for φ s [11]: Eqs. (3.9) and (3.5) allow us to express λ KK as follows: Using Eq. (1.2), we obtain The experimental world average of A d SL is given by [10] implying For our purposes we can then approximate |q d /p d | = 1 and use Similarly to B s → J/ψφ, the B → J/ψK S decay is dominated by a single CKM combination and, furthermore, new physics contributions can be safely assumed to be negligible.

IV. THE STANDARD MODEL
The values of the CKM parameters that play a role in the B s → K + K − and B d → π + π − decays are known from tree level decays [10,12]: Note that, using CKM unitarity relations, we obtain Neglecting the O(λ 4 ) correction,we can rewrite the decay amplitudes, and the λ f parameters, Defining we obtain, for the CP asymmetries, and for the decay rates (averaged over B q and B q ), Using the experimental values of the five observables C KK , S KK , C ππ , S ππ and we can obtain the values of the five hadronic parameters.Solving for the central values of the experimental observables, we obtain:  the sign of δ s and δ d are determined from the sign of C KK and C ππ , respectively, as they are the only observables explicitly sensitive to sin δ q .The 1σ allowed ranges in the |r q | − cos δ q plane are shown in Fig. 1.
A lesson to be drawn from these results is that, while in the Factorizable contributions cancel to a good approximation (roughly, m s /m b ) in the double ratio, so the deviation from unity is affected mainly by non-factorizable contributions [18][19][20][21].The data support the assumption that the non-factorizable contributions are small.
To incorporate first-order U -spin breaking, we write where we assume small breaking, i.e. |p/P | 1 and |t/T | 1.Without loss of generality, we can choose P to be real.Then there are seven hadronic parameters.Given the five observables that we use, C KK , S KK , C ππ , S ππ and R Γ , we can extract five of these seven parameters.In principle, we can extract P by considering the individual decay rates, rather than their ratio, but this has no significance for our analysis.
When we consider the U -spin breaking effects to first order only, the list of five parameters The fourth U -spin breaking parameter, Σi ≡ Im(t/T + p/P ), does not play a role at first order in these observables.We learn that the U -spin breaking effects are of order 0.1 − 0.2, so that U -spin is a good approximate symmetry of this system.
(5.6)This is another way of observing that the U -spin breaking effects in P s / P d and much larger than those in r s /r d , consistent with the assumption that the leading correction comes from the f K /f π factor in the factorizable contributions, and that the non-factorizable contributions are small.
A particularly interesting combination of parameters is the following: It has been noted by Gronau, that in the U -spin limit, the following relation holds [22,23]: Somewhat surprisingly, the experimental data show a strong violation of this relation: [R CΓ ] exp = −2.8± 0.5. (5.9) With our parametrization, and to first order in the U -spin breaking parameters (5.10) We learn that the large deviation of R CΓ from the U -spin limit prediction is a consequence of (twice) three breaking parameters that add up in the same direction.

VI. BEYOND THE SM
It is not a simple task to establish, or to constrain, new physics contributions to CP asymmetries (and even more so to decay rates) in decay modes where there are two SM contributions that differ in their weak and strong phases.Without any extra information, a SM interpretation of the measurements when new physics actually plays a role is (almost) always possible.
In the case of B s → K + K − , there is, however, extra information, which is provided by its (approximate) U -spin relation with B d → π + π − [13][14][15][16].In the presence of new physics, a SM analysis would lead to wrong values of P q and T q .In particular, the new physics contributions might mimic U -spin breaking effects.Thus, the fact that a SM analysis of the experimental data led to small U -spin breaking effects suggests that we can constrain the new physics by demanding that it does not lead to spurious U -spin breaking that is larger than observed.Such constraints assume that there is no cancellation between new physics effects and genuine U -spin breaking effects.
To see how new physics can mimic U -spin breaking, we now present our formalism for including new physics.Any contributions to the four decay amplitudes of interest can be written as follows: A KK = P s λ c bs + T s λ u bs = P s λ c bs (1 + r s λ u bs /λ c bs ), Distinguishing the SM and new physics contributions, we write: A KK = P s SM λ c bs (1 + r s SM R bs uc e +iγ + r s NP e +iθs ), where, in A KK and A ππ , e iγ and e iθq are complex-conjugated.Without loss of generality, we can rewrite these amplitudes as follows: where For our purposes, We assume that |r q NP | 1, |r q SM |, and expand the following two ratios to first order in r q NP : P s ) Before proceeding, we note that by factoring out of the SM contributions the CKM parameters λ c bq and R bq uc , we leave in P q SM and r q SM only QCD matrix elements and flavoruniversal electroweak parameters, such as G F .Thus, in the U -spin limit, we have P s SM = P d SM and r s SM = r d SM .In contrast, r q NP does include in it, in general, flavor-dependent factors, and thus U -spin does not imply r s NP = r d NP .In the U -spin limit, we thus obtain from Eqs. (6.6) ) We In what follows we require that the new physics contributions do not lead to much larger deviations from unity.
We translate the upper bounds on the new physics contributions to lower bounds on the scale of new physics Λ NP .To do so, we assume that Λ NP m W , so that the new physics can be presented by dimension-six terms, and we include in X bq only flavor dependent factors.We will compare g 2 X bq /Λ 2 NP to the size of the SM tree level contribution, which we take to be g 2 λ u bq /m 2 W , with g the weak coupling constant.Concretely, we study five classes of models: • Flavor anarchy: X bq = O(1).

A. Flavor anarchy
We refer to new physics as being anarchic when it has neither flavor suppression nor phase alignment.In other words, it is suppressed by its high scale (and possibly a loop factor) but by no other small parameters, X bq = O(1).In this case, we expect that where the CKM factors compensate for the λ c bs and λ c bd factors that are pulled out of the parenthesis in Eqs.(6.2), and r s AN and r d AN are of the same order of magnitude but not equal.Examining Eqs.(6.7) and (6.4), we learn that the largest modification will be due to the b s term, which is enhanced by (R bs uc ) −1 .The physics behind this result is simple: the SM contributions to P s , P d , T s and T d are proportional to λ c bs , λ c bd , λ u bs and λ u bd , respectively.Anarchic new physics will modify most strongly the term that within the SM is the most strongly CKM-suppressed, which is T s .In the U -spin limit, we obtain: Ts , where we use the notation s γ ≡ sin γ, and similarly to all phases.The ratio between the corrections to r s /r d and Ps / Pd can be estimated as follows: Since the corrections are comparable, the cancellation between the contributions to T s / T d and P s / P d could be accidental, so we require that (b s r s AN )/(r SM |λ c bs |) ∼ < 0.30.Given that we consider anarchic new physics, we further assume that sin θ s / sin γ = O(1) and that , where Λ AN is the high scale of the anarchic new physics.We obtain the following bound:

B. Flavor anarchy with phase alignment
The largest spurious U -spin breaking due to new physics occurs when the flavor structure is anarchic, as in Eq. (6.10), but the new physics phase is set at a special value.Concretely, we consider the case where the new physics phases assume the values of The consequences of this scenario can be straightforwardly read from Eqs. (6.11):In the FN framework [24,25], there is a U (1) FN symmetry that is broken by a small spurion λ FN .The small flavor parameters -mass ratios and CKM angles -are accounted for by different powers of λ FN , depending on the charges of the relevant fields.The small parameter λ FN is commonly taken to be of the order of the Cabibbo angle, λ FN ≈ 0.2, and conventionally taken to carry charge Q FN (λ FN ) = −1 under the U (1) FN symmetry.For our purpose, however, where we aim to find the parametric suppression of flavor-changing dimension-six terms, one can relate this suppression directly to the suppression of the CKM angles (and, in some case, quark masses).Thus, if the new physics is subject to the FN selection rules, then the leading contributions to b → uūq are suppressed by λ

D. General MFV
In the GMFV framework [27], the only sources of flavor [U (3)] 5 breaking are the Yukawa matrices of the SM, but there could be new sources of (flavor-universal) CP violation, X bq = e iθ λ t bq .(For a study in a related framework, see ref. [28].)Matching to Eqs. (6.2), GMFV implies: r s NP e +iθs = −r s GMFV e +iθ (1 + R bs uc e iγ ), U -spin implies r s GMFV = r d GMFV .With these replacements, and using U -spin, we obtain: and  We find that the measured values are consistent with U -spin breaking at the level of 30% or smaller.Furthermore, the U -spin breaking is a factor ∼ 4 smaller in the double ratio of matrix elements, ( Ts / Td )/( Ps / Pd ), than in each of these ratios separately.This result is consistent with the assumption that the leading effect is coming from f K /f π which cancels in the factorizable contributions to the double ratio, with the remaining non-factorizable contributions and contributions of O(m s /m b ) much smaller.
With regard to new physics, the main tool that we use to probe it is the observation that, in general, when interpreting the experimental results assuming the SM, the new physics New physics subject to Minimal Flavor Violation (MFV) does not mimic U -spin breaking.
In fact, analyzing the results in the MFV framework is identical to carrying out the analysis in the SM framework.Yet, if the scale of MFV new physics is below 160 GeV, it will bring the situation closer to a single phase dominance, suppressing the CP asymmetries to below their observed values, and is thus excluded.
Our bounds on the scale of new physics assume that there are no cancellations between the spurious and the genuine U -spin breaking effects.Such cancellations can relax our bounds by factors of O(1).We further did not include loop factors for the new physics contributions.Note, however, that the relevant decay processes, b → uūq (q = s, d), are not flavor changing neutral current (FCNC) processes, and have tree level contributions already in the SM.Finally, while S KK and S ππ depend on, respectively, B s −B s and B d −B d mixing, which are FCNC processes, we used purely experimental data to include the neutral meson mixing parameters, and thus our analysis is independent of the mixing mechanism.
All B and B s decay amplitudes (and, in particular, the leading semileptonic decays) are suppressed by at least such a factor, or even stronger.If new physics contributes significantly to this decay, then the whole consistency of the data with the CKM picture seems accidental, which is very unlikely.Moreover, for new physics to be comparable to the SM contribution, it needs to be lighter than O(TeV) and have tree level flavor changing couplings to quarks.Thus, very likely it should have been directly observed at the LHC, and affect other flavor observables significantly.In contrast, the b → uūs decay can get significant contributions from new physics at the 10 TeV scale, and it is Cabibbosuppressed compared to the charmless semileptonic b decays.The b → uūd decay is an intermediate case, as it has the same suppression as the charmless semileptonic b decays, and can get significant contributions from new physics lighter than O(5 TeV).

Figure 1 .
Figure1.The 1σ allowed ranges for |r q | vs. cos δ q : (purple) q = s, from S KK and C KK ; (pink) q = s, from A ∆Γ KK and C KK ; (yellow) q = d, from S ππ and C ππ , for negative A ∆Γ ππ ; (orange) q = d, from S ππ and C ππ , for positive A ∆Γ ππ .

. 4 )
Matching to the parametrization of Eqs.(4.3), we haveP s = P s SM (1 + a s r s NP ), r s = r s SM + b s r s NP 1 + a s r s NP , P d = P d SM (1 + a d r d NP ), r d = r d SM + b d r d NP 1 + a d r d NP .(6.5) learn that new physics contributions can lead to | P d / P d | = 1, and/or to |r s /r d | = 1 even with no U -spin breaking.We recall that Eq. (4.10) gives, for the experimental central values, | P s / P d | exp − 1 ≈ 0.26, |r s /r d | exp − 1 ≈ 0.07.(6.8)

. 26 )
E. Minimal flavor violation (MFV)In the MFV framework[26], the only sources of flavor [U (3)] 5 breaking and of CP violation are the Yukawa matrices of the SM, X bq = λ t bq .MFV implies:r s NP e iθs = −r MFV (1 + R bs uc e iγ ),r d NP e iθ d = −r MFV (1 + R bd uc e iγ ) .(6.27) contributions will mimic U -spin breaking effects.If the new physics contributions are large, they can generate large spurious U -spin breaking.Assuming that there are no cancellations between the new physics effects and genuine U -spin breaking effects, we found lower bounds on the scale of classes of new physics with various flavor structures: for flavor anarchy Λ AN ∼ > 5 TeV, for Froggatt-Nielsen (FN) selection rules Λ FN ∼ > 2 TeV, and for General Minimal Flavor Violation (GMFV) Λ GMFB ∼ > 2 TeV.We further demonstrated that new physics at a scale as high as O(10 TeV) can generate significant spurious U -spin breaking.
We define q d and p d for the B d − B d system in a similar way to Eq. (2.1), andA d f , A We denote A ππ ≡ A d π + π − and λ ππ ≡ λ d ππ .The parameters of the time-dependent CP asymmetry in B → π + π − are given by 80 , cos δ s = −0.49± 0.14 (sin δ s < 0) , |r d | = 4.64 ± 0.45 , cos δ d = −0.84± 0.04 (sin δ d < 0) , Note that, since A ∆Γ ππ is not measured, there is a discrete ambiguity in |r d | and cos δ d .We present the solution that corresponds to a negative A ∆Γ ππ , which gives |r d | close to |r s |.This is not the case if A ∆Γ ππ is positive (|r d | = 0.43, cos δ d = +0.10).Once |r q | and cos δ q are fixed, the T d and P d contributions are comparable, |r d |R bd uc ≈ 1.9, the B s → K + K − decay is dominated by the P s contribution, |r s |R bs uc ≈ 0.10.The B s → K + K − and B d → π + π − are related by U -spin, the SU (2) symmetry under which s and d form a doublet.Concretely, U -spin requires P ≡ P s = P d , T ≡ T s = T d .(5.1) U -spin breaking effects are expected to be of O(m s /Λ QCD ) ∼ 0.3.Indeed, the best fit values that we obtained, | P s / P d | = 1.26 and |r s /r d | = 1.07 are consistent with this expectation.It is interesting to note that the breaking effect in | P s / P d | is much larger than the one in |r s /r d |.The latter is, in fact, a double ratio: The fact that |r s /r d | and | P s / P d | are close to one, suggests that U -spin breaking effects are small.This result leads us to investigate in detail the U -spin breaking in this system.V. U -SPIN BREAKING .15) In this scenario, there is no deviation from the U -spin relation for | P s / P d |, but the U -spin relation for |r s /r d | is violated.Thus, we require that the contribution to |r s /r d | − 1 does not exceed the experimental value, for which Eq. (4.10) gives |r s /r d | exp − 1 ≈ 0.07.Estimating for the flavor-anarchic phase-aligned case r PA /r SM ∼ m 2 resulting in X bq = e iθq × O(|λ c bq |).Matching to Eqs. (6.2), FN implies: Thus, we require that the FN contribution to|r s /r d |−1 does not exceed |r s /r d | exp −1 ≈ 0.07.Taking s θs /s γ = O(1), and estimating, for FN models, r FN /r SM ∼ m 2 W /Λ 2 FN , we require: 19))Assuming that s θq = O(1), we can obtain a lower bound on the scale of new physics.The strongest bound can be obtained by noticing that FN predicts a much larger deviation from the U -spin relations in |r s /r d | than in | Ps / Pd |, in contrast to the experimental data (and to the expectations from naive factorization).The former is enhanced over the latter by a