$B^0-\bar B^0$ entanglement for an ideal experiment on the direct CP violation $\phi_3/\gamma$ phase

$B^0$--$\bar B^0$ Entanglement offers a conceptual alternative to the single charged B-decay asymmetry for the measurement of the direct CP violating $\gamma/\phi_3$ phase. With $f=J/\Psi K_L,J/\Psi K_S$ and $g=(\pi\pi)^0,(\rho_L\rho_L)^0$ the 16 time-ordered double decay rate Intensities to $(f, g)$ depend on the relative phase between the the $f$- and $g$-decay amplitudes given by $\gamma$ at tree-level. Several constraining consistencies appear. An intrinsic accuracy of the method at the level of $\pm 1^\circ$ could be achievable at Belle-II with an improved determination of the penguin amplitude to $g$-channels from existing facilities.

B 0 -B0 Entanglement offers a conceptual alternative to the single charged B-decay asymmetry for the measurement of the direct CP violating γ/φ3 phase.With f = J/ΨKL, J/ΨKS and g = (ππ) 0 , (ρLρL) 0 the 16 time-ordered double decay rate Intensities to (f, g) depend on the relative phase between the the f -and g-decay amplitudes given by γ at tree-level.Several constraining consistencies appear.An intrinsic accuracy of the method at the level of ±1 • could be achievable at Belle-II with an improved determination of the penguin amplitude to g-channels from existing facilities.
There is considerable interest in improving the precision for the direct CP violation φ 3 /γ phase γ = arg(−V ud V * ub /V cd V * cb ) in the b-d unitarity triangle of the Cabibbo-Kobayashi-Maskawa (CKM) flavour mixing matrix of quarks [1,2].This angle connects the sides for decay amplitudes of the B system dominated by tree diagrams, so that its measurement is a bona-fide determination of the Standard Model (SM) parameters.This is important in order to search for New Physics in the loop contributions of penguin -rare decays-and box -mixing-diagrams.
The most precise result from a single analysis at the LHCb experiment is [3] γ = (65.4+3. 8  −2.2 ) • .It uses the GGSZ method [4] for B ± → DK ± with the choice of D → K S π + π − , D → K S K + K − 3-body decays.In charged B decays, the observation of CP violation (CPV) needs [5] the interference of two amplitudes with different weak phases -changing sign from particles to antiparticles-and strong phases -invariant under CP-.This mismatch is what originates a CP-violating asymmetry in the corresponding decay rates for B + and B − .In this case D represents a D 0 or D0 meson reconstructed from a final state that is common to both, D 0 and D0 being produced respectively by b → cūs and b → ucs tree level diagrams.The parameters of their mixing have been simultaneously determined in the analysis of [3].The amplitude of the decay where m 2 ± are the squared invariant masses of the K S h ± particle combinations, that define the position of the decay in the Dalitz plot.The parameter r B is the ratio of the magnitudes of the B − → D0 K − and B − → D 0 K − amplitudes, δ B being their strong relative phase.Neglecting CP violation in charm decays, the chargeconjugated amplitudes for B + decay satisfy ĀD = A D .The sensitivity to γ is obtained by comparing the distributions in the Dalitz plots of D decays from B + and B − mesons.As a consequence, the variation of the strong phase within the Dalitz plot is needed.Complementary information from measurements performed by CLEO [6] and BESIII [7][8][9] is available.Alternative methods [10][11][12] correspond to different choices for the decay channels of the D's.One understands the complexity of the analyses.
In this paper we discuss an ideal conceptual experiment for γ by exploiting the B 0 − B0 Einstein-Podolsky-Rosen (EPR) Entanglement [13].The use of the EPR correlation was proposed in Refs.[14][15][16][17] for several decay channels in the B factories.Entanglement has been instrumental in the past for the observation of Time-Reversal-Violation by the BABAR Collaboration [18] using the concept and method given in Refs.[19][20][21] and with far reaching information [22][23][24].The method for γ consists in the observation of the coherent double decay to flavor-non-specific products.In it, the extraction of the γ phase is free from the essential strong phases contamination needed in charged B decays.The necessary interference between amplitudes containing the V cd V * cb and V ud V * ub sides of the unitarity triangle is automatic from the two terms of the entangled B 0 − B0 system.The double rate intensity to the (f, g) and (g, f ) pairs of CPeigenstate decay products, with f = J/ψK S , J/ψK L and g = π + π − , π 0 π 0 , ρ + L ρ − L , ρ 0 L ρ 0 L , will do the job from CPconserving and CP-violating transitions, as we demonstrate below.The measurement of the time-ordered intensities for these 16 = 2(f ) × 4(g) × 2(time ordering) combined processes is rich in physics and consistencies, leading to the relative phase γ responsible of direct CP violation.The Belle-II experiment at the upgraded KEK facility would have the opportunity to perform the analysis presented here if enough integrated luminosity is accumulated in the coming years.
The choice of the ρ L ρ L channels together with ππ channels is motivated by their common CP properties, as seen in the change of basis [25] from the two-particle states with definite helicity to L-S coupling

arXiv:2204.05567v1 [hep-ph] 12 Apr 2022
In particular, for helicities λ 1 = λ 2 = 0, the ρρ system from B decay is in states with L = S = 0, 2, so it has definite symmetry properties under C = +, P = + and CP = +.Therefore, we may use a unified theoretical framework for the discussion of the time-ordered intensities associated to the double decays (f, g) and (g, f ) with decay times t 1 , t 2 such that ∆t ≡ t 2 − t 1 > 0. For any g decay products, the choice of f = J/ψK L defines for the ∆t living partner a CP-forbidden transition, whereas f = J/ψK S corresponds to a CP-allowed transition.
Taking the transition amplitude from the C = − entangled B 0 − B0 state to the time-ordered decay products f and g, its square and integration over the initial decay time at fixed ∆t ≡ t, leads to the double-decay intensity [24] (3) with Γ the common decay width of the eigenstates B H = pB 0 + q B0 , B L = pB 0 − q B0 with definite time evolution, ∆M = M H − M L their mass difference and A f,g H,L = f, g| T |B H,L their decay amplitudes.In the absence of CP violation in the mixing for this system, |p/q| = 1.As anticipated, this intensity presents interference terms between f and g, either direct or through mixing.With ∆Γ = 0 for B d decays, there are time-independent and oscillatory terms in t with different physics.Due to the definite (anti)symmetry of the C = − entangled state, Eq.( 3) satisfies the following expected symmetry property: the combined transformation t → −t and f ↔ g is the identity.Hence the interest in the separate measurements of I(f, g; t) + I(g, f ; t) and I(f, g; t) − I(g, f ; t) in order to separate even and odd terms in t.As a consequence, we find it convenient to express Eq.(3) in the basis {cos 2 (∆M t/2), sin 2 (∆M t/2), sin(∆M t)} of time dependencies as where Γ f is the average decay probability to f from B 0 and B0 .Î(f, g; t) is a reduced intensity, with m , and I f g od = −I gf od the "intensity parameters" for each decay pair.The I d parameter shows up since the t = 0 separation between the two decays for each (f, g) pair, so it is the signal for a direct correlation between the decay amplitudes.
We introduce the usual Mixing×Decay interference λ ≡ q p Ā A from B 0 and B0 relevant at any time, for each decay amplitude -either λ f or λ g -.In terms of the complex λ, we have the three combinations with the constraint The calculation of the intensity parameters in Eq.( 4) for each double-decay rate (f, g) is then obtained from the combinations (5) as Whereas I d and I m contain real number terms and then select the real part of the time evolution, I od contains imaginary terms selecting the imaginary part of the time evolution.In addition, we observe that the time-even parameters are symmetric under the f ↔ g exchange, and the odd parameter is antisymmetric, as anticipated.
The decay channels f = J/ψK L , J/ψK S are known to be well described by their tree level amplitudes, which satisfy |λ f | = 1, i.e.C f = 0.As an important consequence of Eq.( 6), a non-vanishing I od intensity parameter is trapping penguin amplitudes through their modulus contribution C g = 0 (|λ g | = 1) for any decay channel g.The phase of λ f is to a great accuracy the mixing phase q/p = e −2iφ M , with φ M = β in the SM.Assuming also ∆F = ∆Q, that is no wrong sign decays, we have imposed by the two opposite CP-eigenvalues for the decay products J/ψK L and J/ψK S .The surviving terms in Eq.( 6) are linear in λ f , implying consistency relations for the absolute and relative normalizations of the intensity parameters, leading in turn to consistencies for the double-decay timedependent reduced intensities Î(L, g; t) + Î(S, g; t) = 1, ∀g, ∀t .
Using the exchange symmetry properties, Eqs.(8,9) are also valid for the time-ordered (g; L, S) decays.
They provide a controlled connection between the CPforbidden and CP-allowed time-dependent transitions for any of the four decay products g.The λ g amplitudes (g = ππ, ρ L ρ L ) can be parameterized as where φ g is a weak phase in the decay b → uūd, and both ρ g = 1 and φ g = γ are due to the penguin contributions.At tree level, all g states considered here would have φ g = γ.Notice that the (R f R g +S f S g ) combination appearing in the I f g d intensity parameter is blind to the phase of q/p and it directly probes where the ± corresponds to f = L, S, respectively.As anticipated, no mediation of the mixing is present in the I d parameter of the intensity.Thus the determination of this direct correlation between the two decay products in Eq.( 4) for these processes becomes where clearly the mixing is not present.If penguin contributions were not relevant, we would have at tree level I Lg d = sin 2 γ for all CP-forbidden transitions, With the expected g-dependent penguin contributions through both ρ g and γ − φ g ≡ g , to be discussed below, Eq.( 12) provides a powerful consistency from the four g's and the two f 's for the extraction of the CPV γ-phase.
Let us focus now on the different information to be accessed by the measurement of the other intensity parameters.In the case of I m , the combination in Eq.( 6) involves the λ f λ g product, which connects the f, g decay amplitudes through the mixing.The use of Eqs.(7,10) leads to The result (14) depends on the phase 2φ M + φ g , indicating explicitly that the I m parameter denotes a correlation between the two f and g decay channels induced through the mixing.As already advertised, I Lg m + I Sg m = 1 ∀g, as for the other term even in time.
The two (f, g) time-even intensity parameters combine in the observable sum of intensities for the time-ordered exchange of decay products f ↔ g.We obtain the result (15) for f = L, S correspondingly.As seen, the contributions of the direct CPV phase φ g and the mixing-induced CPV phase 2φ M + φ g separate in two different time-dependent behaviors, the second naturally needing a time slice to become apparent.For any of the two f channels and the four g channels, these two terms are separately apparent when with n = 0, 1, 2...
The third intensity parameter I od can be separated out from the difference of the two time-ordered intensities, where I Lg od + I Sg od = 0 ∀g in this case.It is worth remarking that this intensity parameter would vanish iff the penguin contribution were absent in the g decay channels.As the CPV mixing sin 2φ M (sin 2β in the SM) is the best measured parameter in this field, Eq.( 17) can be used to measure the deviation of ρ g in each of the four g-channels from 1, induced by the penguin amplitude, and check its prediction from the isospin analysis given below.Consistently, the measurement of observable (17) for both f = L and f = S has to reproduce a change of sign, providing in particular the relative normalization of events in these two decay channels.
Besides the factor depending on ρ g , the observables are also affected by the penguin amplitudes in a departure of the phase φ g from a common γ/φ 3 through to be extracted from a dedicated isospin analysis.The procedure follows the original ideas of Gronau and London along the path described in Refs.[26,27].The neutral and charged B-meson decays differ in the presence versus absence, respectively, of the penguin contribution to the amplitudes for each final h = π, ρ L system.The charged decay amplitudes A +0 = A(B + → h + h 0 ) and Ā+0 = A(B − → h − h 0 ) have a final (h ± h 0 ) isospin 2 state and, therefore, only the ∆I = 3/2 tree-level amplitude contributes with the weak phase γ.It is convenient to define, with the same notation for both neutral decay channels ππ and ρ L ρ L and using g = ± or 00 for the corresponding decay charges, in such a way that the double ratio gives The isospin triangular relations with these complex ratios are Eqs. ( 21) allow to get a g and a g by using all the branching ratios of the processes B ± → h ± h 0 ; B 0 , B0 → h + h − , h 0 h 0 .In Table I we give the summary of our isospin analysis with the present PDG data [28].Taking into account that the ρ + L ρ − L channel is the one with larger branching ratio, we must conclude that the error in  Benchmark ρg gives us an estimate of the uncertainty due to the present knowledge of the penguin pollution in the determination of γ/φ 3 .An important improvement in the branching ratios entering in the isopin analysis is expected as an outcome of Belle-II and LHC experiments that will reduce this error significantly.
The intrinsic accuracy of the method proposed in this paper is controlled by our ability to extract φ g .In order to estimate the expected uncertainty in that extraction, we proceed as follows (further details are provided in the Supplementary Material).First, we fix input values of φ M and γ.For each decay channel g = ρ + L ρ − L , ρ 0 L ρ 0 L , π + π − , π 0 π 0 , we also fix input values of ρ g and g , which fix φ g = γ − g , following the three different benchmark cases in Table II.Next, considering the decay channels f = L, S, we compute the six coefficients I Sg d,m,od , I Lg d,m,od , which control the four time-dependent decay channels (f, g), (g, f ), for each g.Then, for each g, a given number of events is generated according to the four double-decay intensities.The procedure is repeated in order to produce our simulated data, from which I Sg d,m,od are extracted including uncertainties, I Lg d,m,od are given by eq.( 8).Finally ρ g , φ g , φ M are obtained with a simple fit.Notice that the intensity parameters I d,m,od depend, respectively, on φ g , φ M + φ g and φ M phases.Therefore, the inclusion of the I m term together with I d in the fit allows to avoid the discrete degeneracy φ g → φ g + π, with the information of the quadrant for φ M .
We show the results of our analysis in two scenarios A and B taking into account the Belle-II projected luminosity [29][30][31] and the corresponding branching ratios: scenario A assumes 1000 ρ + L ρ − L events of type B ρρ , 50 ρ 0 L ρ 0 L events of type B ρρ , 200 π + π − events of type B − ππ and 50 π 0 π 0 events of type B − ππ .Scenario B assumes 500 ρ + L ρ − L events of type B ρρ and 100 π + π − events of type B + ππ .The results of the fit to the generated I f g d,m,od in both scenarios are given in Table III.From the results in Scenario A we conclude that, since γ = φ g + g , the error δφ ρ + , hence the importance of its improvement, as already mentioned.Even before these improvements we can do better and fit the three I f g d,m,od for all channels in terms of γ, g and ρ g including all the information of the isospin analysis.In this case the result is γ = 1.222 ± 0.080 = (70.0± 4.6) • .Note that the error on γ is smaller than the error in ρ + L ρ − L due to a unique γ in all channels, which presents a quantitative conclusion: the present proposal could provide a measurement of γ below the 1 • error if the errors in the isospin analysis can be reduced to the level of δφ ρ For the more conservative scenario B, we get an intrinsic error δφ ρ + L ρ − L = 1.8 • .Again, using all the information used in the isospin analysis and φ M , we estimate γ = 1.221 ± 0.085 = (70.0± 4.9) • , which reinforces the idea that, with this method, it could be statistically possible to go below 1 • of precision in the determination of γ, thanks to the expected improvements in the data entering in the isospin analysis.
To conclude, with B 0 -B0 Entanglement, we consider the double decay rate Intensity to flavor-non-specific channels governed by the c-and u-quarks.It offers a conceptual alternative to the decay of single B ± mesons for the extraction of the direct CPV γ/φ 3 phase.The needed interference between two decay amplitudes is provided by the exchanged terms of the entangled state and no strong phases appear as essential ingredients.The 8 timesymmetric (f, g) Intensities with f = J/ΨK L , J/ΨK S , g = (ππ) 0 , (ρ L ρ L ) 0 have a tree-level common γ phase, g = ρ + L ρ − L being the benchmark channel.Several constraining consistencies among the different intensities appear.We find that an intrinsic accuracy of the order of 1 degree could be achievable for the relative phase of the f -and g-amplitudes.The present limitation of ±5 • , to be improved by the existing experimental facilities, comes from the phase of the penguin contribution in the g-amplitude, extracted from an isospin analysis to neutral and charged B decays.

SUPPLEMENTARY MATERIAL
As discussed in the main text, the intrinsic limitation of the method is controlled by our ability to extract φ g .The procedure to estimate the expected uncertainty in that extraction is the following.
1. We fix input values of φ M = β = 0.384, γ = 1.222, and, for each decay channel g = ρ + L ρ − L , ρ 0 L ρ 0 L , π + π − , π 0 π 0 , we also fix input values of ρ g and g , which fix φ g = γ − g .We consider the three different benchmark cases in Table II.2. Considering the decay channels f = L, S, the six coefficients I Sg d,m,od , I Lg d,m,od are computed: they control the four time-dependent combinations (f, g), (g, f ), for each g.

3.
For each g, we generate values of t, the events, distributed according to the four double-decay intensities.In order to incorporate the effect of experimental time resolution, each t is randomly displaced following a normal distribution with zero mean and σ = 1 ps.Additional experimental effects such as efficiencies are not included.Generation proceeds until a chosen number of events N g with |t| ≤ 5 τ B 0 has been obtained with the four (f, g), (g, f ) combinations altogether.These N g events are binned.by eq.( 8), and similarly for decay channels ρ 0 L ρ 0 L , π + π − , π 0 π 0 according to the different benchmarks B ρρ , B ± ππ in Table II.
6. Finally we extract ρ g , φ g , φ M , with a simple fit to the I Sg d,m,od .
Concerning the number of events, with the Belle-II design luminosity [29] and the branching ratios BR(g), BR(f ), we assume that it would be possible to collect 1000 events for g = ρ + L ρ − L , 200 events for g = π + π − and 50 events for both g = ρ 0 L ρ 0 L and g = π 0 π 0 channels.We show the results of our analyses for two scenarios.
• In scenario B we assume to have 500 ρ + L ρ − L events of type B ρρ and 100 π + π − events of type B + ππ .
I(S, ρ + ρ − ; t) 020 = 1.1 • gives an idea of the intrinsic statistical limiting error we would expect in the determination of γ for the assumed number of events.Combining φ ρ + L ρ − L with ρ + L ρ − L = 0.008 ± 0.091 would bring the error in γ to the present error in ρ + L ρ − L

4 .
The procedure is repeated in order to obtain mean values and standard deviations in each bin: these constitute our simulated data, as illustrated in Figure1, which corresponds to g = ρ + L ρ − L (benchmark B ρρ in TableII), N g = 1000 events and 20 bins in [0; 5 τ B 0 ].The black dots with bars are the mean values and uncertainties, the red curves are the extracted double-decay intensities, and the blue curves correspond to the I f g d term in each intensity.There are no significant differences if one considers, for example, 15 or 10 bins.5.From the simulated data, one can obtain I

IFIG. 1 :
FIG. 1: Simulated data, 1000 events, benchmark B ρρ .Black dots with bars indicate mean values and associated uncertainties; the red curves are the extracted double-decay intensities, while the blue curves correspond to the I f g d term in each intensity.

TABLE I :
Summary of isospin analyses results.

TABLE II :
Benchmark cases used in the numerical simulations.

TABLE III :
Results of the fit