Constraints on light dark sector particles from lifetime difference of heavy neutral mesons

Heavy meson decays with missing energy in the final state offer interesting avenues to search for light invisible new physics such as dark matter (DM). In this context, we show that such new physics (NP) interactions also affect lifetime difference in neutral meson-antimeson mixing. We consider general dimension-six effective quark interactions involving a pair of DM particles and calculate their contributions to lifetime difference in beauty and charm meson systems. We use the latest data on mixing observables to constrain the relevant effective operators. We find that lifetime differences provide novel and complementary flavor constraints compared to those obtained from heavy meson decays.


Introduction
It is a general consensus at present that a large part of our Universe is composed of dark matter (DM) [1,2].The evidence of DM comes primarily from the observation of its gravitational effects, but has been corroborated by several independent methods [3,4].However, other than its existence and weakly interacting nature, little is known about its mass and properties, making the detection a challenging endeavor.There are dedicated experiments searching for DM in distinctive mass windows.In principle, DM does not have to be composed of a single species of particles, giving rise to more complicated dark sectors (DS), where one or a few particles play the role of DM 1 .
If such DS particles are sufficiently light and couple to quarks, they can also be produced in decays of beauty and charm mesons [5,6], presenting new intriguing opportunities for searching DM at low-energy flavor experiments.These decays are characterized by the missing energy E miss in the final state as DS particles produced in the decay escape undetected.In Table 1, we summarize current data on relevant charm and beauty mesons decay modes.
These decays proceed via flavor-changing neutral current transitions, which arise only at the loop level in the Standard Model (SM), with missing energy E miss usually corresponding to a pair of massless neutrinos2 .The SM predictions for these processes are quite small, as can be seen from Table 1.Furthermore, as neutrinos are present in the final state, the corresponding theoretical predictions do not suffer from uncertainties arising due to photon exchange.Precise measurements of these decay modes, therefore, allow for highly sensitive probe of new physics (NP) beyond the SM, including light DM.
Several papers have focused on these decay modes as probes of light invisible NP (see for example, Refs.[5,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]).Here, it is worth mentioning that very recently Belle-II found the evidence for B + → K + E miss , measuring the branching ratio (2.3 ± 0.7) × 10 −5 .The value is higher than the corresponding SM value at a level 2.7σ, and can be explained if there are light invisible NP contributions to the decay, where, for example, E miss is carried away by DM, in addition to SM neutrinos.In this light, there has been reinvigorated interest and motivation to study decays listed in Table 1.
The purpose of this article is to point out that quark-DM interactions relevant to these decays also unavoidably generate a finite NP contribution to the lifetime difference in mesonantimeson mixing, which provides additional constraints on light DM.In particular, we show that lifetime difference can be used to obtain novel bounds on certain quark-DM interactions that remain unconstrained so far.In order to see how the contribution to lifetime difference arises from above-mentioned quark-DM interactions, let us consider the off-diagonal term of the neutral meson M 0 mass matrix [33,34] where M 12 denotes the dispersive part of mixing amplitude, which contributes to the M 0 -M 0 mass difference, whereas Γ 12 is related to absorptive part contributing to lifetime difference in M 0 and M 0 , with M 0 = B 0 , B 0 s , or D 0 mesons.On the right hand side, the first term concerns local |∆F | = 2 interactions and does not have absorptive part.However, the second term, which involves intermediate states n and concerns |∆F | = 1 transitions, generates an absorptive piece, which shows that the lifetime difference is governed by |∆F | = 1 interactions.
The paper is organized as follows.In the next section we define general effective Hamiltonian containing all relevant quark-DM interactions.In Section 3 we calculate DM contribution to the lifetime difference in heavy meson-mixing.In Section 4, we present our results: we constrain the quark-DM interactions using current data on lifetime differences, and compare these with those from c → uE miss , b → {s, d} E miss decays.Finally, we summarize our conclusions in Section 5.

Relevant effective interactions
To keep our analysis general, we work in the framework of effective field theory (EFT) valid at some appropriate low energy scale that is typically set by the mass of decaying meson.Note that the final state in the decay can, in principle, contain a single or more DM or DS fields.The underlying quark-DM interactions will also have one or more DM fields, accordingly.However, operators with a single DM field can only generate an absorptive part of the meson mixing amplitude if they have DM mass close to that of B 0 s , B 0 , or D 0 states and a large width [35].Since, by definition, DM particles should have cosmological-scale lifetimes, we will not consider such operators here.Furthermore, we restrict the scope of our analysis to effective operators of dimension up to six.The general effective Hamiltonian can then be written as where C i (µ) are the Wilson coefficients of effective operators O i evaluated at low energy scale µ.The scale Λ denotes NP scale, which in a specific UV model is typically related to the mass of heavy NP particle(s) mediating interactions between the SM and DM fields.The number of independent operators in Eq. ( 2) depends on the spin of DM particle.The full basis of DM operators have been specified previously also (for example, see Refs.[6,15,36]).
In this paper, we will focus on scenarios where DM is either a scalar or fermion particle.We follow the operator basis provided in Ref. [15], with our notation similar to Ref. [36].Note that our restriction to the operators of dimension six excludes operators with vector dark sector particles such as qi Q j G µν D G Dµν , where G µν D is a strength tensor of the vector DS field V µ D .Note that this restriction, in principle, does not exclude the decays of the heavy quark states into two vector DS particles, In case of scalar DM (denoted as ϕ), there are four operators, where quark indices {q i Q j } = {uc} for c → u, {sb} for b → s, and {db} for b → d quark transitions, and Compared to Ref. [15], we have multiplied operators O q i Q j S,P by quark mass (m Q j ) so that all operators have same dimensions.Note that operators O In case of the Dirac-type fermionic DM (denoted as ψ), the number of effective operators of dimension six is larger.These operators are given by where For the Majorana DM, vector current operators O V V,AV and both tensor current operators vanish.

DM contribution to lifetime difference
As pointed out in the Introduction, the imaginary part of bi-local contributions in Eq. ( 1) arising due to |∆F | = 1 interactions gives contribution to the off-diagonal element Γ 12 of decay width matrix in meson-antimeson mixing.The optical theorem relates the matrix element Γ 12 to the imaginary part of the forward scattering amplitude as Here, T is the transition operator defined as where T is the time-ordering operator and |∆F | = 1 effective Hamiltonian is given in Eq. ( 2).The time-ordered product in Eq. ( 5) is a nonlocal quantity, which in the heavy quark limit can be expanded in powers of 1/m Q as a series of local operators.The leading contribution of this operator product expansion corresponds to the diagram shown in Fig. 1, where each vertex corresponds to insertion of effective operators given in Eqs. ( 3) and (4) for scalar and fermion DM, respectively.Note that since intermediate states are DM particles, there are no contributions from their interference with the SM operators.
Let us first consider contributions of scalar DM operators in Eq. ( 3).The discontinuity of the relevant diagram can be calculated using Cutkosky rules [37].We utilize Package-X [38] to this end and to further simplify the Dirac structures of results.For the operators O and O , we obtain respectively, while the corresponding expressions due to O can be obtained as ) V , respectively, after replacing labels S → P and V → A everywhere therein.Further, , and M 0 denotes B 0 s , B 0 , and D 0 for {q i Q j } = {sb}, {db}, and {uc}, respectively.The quantities ⟨Q , σ µν for a = S, P, V, A, T , respectively, are the matrix elements of ∆F = 2 effective operators, and have been defined in the Appendix A. In our calculation, we take incoming momentumsquared in the loop s = (p Q j + p q i ) 2 ≃ m 2 M 0 .In addition, we also ignore the masses of s and u quarks.
For the fermionic DM operators in Eq. ( 4), we obtain The corresponding contributions of remaining operators are obtained as (Γ M 0 12 ) P S, P P → −(Γ M 0 12 ) SS, SP , (Γ M 0 12 ) AV, AA → (Γ M 0 12 ) V V, V A , after making obvious change of Wilson coefficients and replacing labels S → P and V → A in the matrix elements ⟨Q q i Q j a ⟩.The results for Γ 12 in Eqs. ( 7) and ( 8) are general and applicable to all three heavy meson-antimeson mixing systems: B 0 s -B 0 s , B 0 -B 0 , and D 0 -D 0 .B 0 s -B 0 s and B 0 -B 0 systems: For neutral B systems, the total width difference ∆Γ q (with q = s, d for B 0 s and B 0 systems, respectively) in presence of NP is given by [39] ∆Γ q = 2|Γ q, SM corresponds to the DM contribution, and for neutral B systems is parametrized by (Γ M 12 ) i in Eqs. ( 7) and ( 8).On the other hand, to account for the SM part, we employ the following expression (for a review see Ref. [39]) with where V is the CKM matrix.The numerical values of real coefficients a q , b q , c q (notation originally introduced in Ref. [40]) are taken from Ref. [39].These coefficients exhibit hierarchy |c q | ≫ |a q | ≫ |b q | which together with the smallness of CKM ratio |λ u /λ t | ∼ 10 −2 indicate that the SM value of Γ q 12 /M q 12 is dominated by the c q term.The dispersive part M q 12 in the SM is dominated by the short-distance contribution arising from the box diagram involving top quark and W boson in the loop and given by (see Ref. [41] for the latest review) where G F denotes the Fermi constant [42], m W is the mass of W boson, and Inami-Lim function S 0 (x) [43] parametrizes top quark loop contribution.Note that top mass m t here is in M S-mass scheme.The factor ηB ≃ 0.84 [44] encodes the renormalization group running from heavy scale (m t ) to B meson mass scale, whereas f Bq is the B meson decay constant and B (1 Bq is the bag parameter related to matrix elements of the ∆B = 2 SM effective operator as defined in Appendix A.
On the other hand, the current measurements of ∆Γ q are [45] where Γ d is the total decay width of B 0 meson.D 0 -D 0 system: The width difference in charm system can be parametrized as y D 12 = |Γ D 12 |/Γ D , with Γ D denoting total decay width of D 0 meson.The SM prediction of mass and width difference in neutral charm system is very tiny due to enhanced CKM suppression and more pronounced GIM cancellations between internal light quarks (for a recent review, see Ref. [46]).This allows for charm mixing to be particularly useful for probing indirect NP effects [33,34].On the experimental side, the latest fit results from HFLAV [45] gives In order to obtain conservative bounds, given the highly suppressed Γ D 12 in the SM, in our analysis we will assume that experimental value of y D 12 is saturated by the DM contributions.

Results
We are now in position to present constraints on NP effective interactions in Eqs. ( 3) and ( 4) from lifetime differences in neutral beauty and charm mesons, and confront them with constraints from branching ratios of decay modes listed in Table 1.The details of calculation of decay rate of meson decays and employed hadronic form factors have been relegated to appendix B. To keep our analysis simple, we will work in single operator dominance scenario.

Constraints on scalar DM
In Fig. 2, we show constraints in the Wilson coefficient (C ) vs. DM mass (m ϕ ) plane for all effective operators in the scalar DM scenario.We set Λ = 1 TeV, as the indirect measurements only constraint the combination C Note that unless mentioned otherwise explicitly all constraints in each of plots throughout the paper correspond to exclusion curves at 90% CL, with region above the curves being ruled out.
The first column of Fig. 2 shows constraints on {sb} quark flavor operators from ∆Γ s (solid red line), B(B 0 → K 0 E miss ) (dashed blue line), B(B 0 → K * 0 E miss ) (dash-dotted purple line), B(B + → K * + E miss ) (dotted orange line), and B(B 0 s → E miss ) (dashed brown line).In contrast, the green shaded band corresponds to 90% CL region that can explain the recent Belle II measured value of B(B + → K + E miss ).Concerning branching ratio constraints, note that operators with scalar quark current (sb) are insensitive to decays involving vector meson final state since the corresponding matrix element vanishes, e.g., ⟨K * |sb|B⟩ = 0. Similarly, operators with axial (sγ µ γ 5 b) or pseudoscalar (sγ 5 b) quark currents are not sensitive to decays involving pseudoscalar meson final state as ⟨K|s Γ i b|B⟩ = 0 for Γ i = γ 5 and γ µ γ 5 .Overall, we find that constraints from ∆Γ s are always weaker, by an order or more, compared to those from decay modes.However, note that the region beyond DM mass m ϕ ≥ (m B − m K ( * ) )/2 becomes kinematically inaccessible in three-body decays.This region, except for the operator O sb P , is only constrained from ∆Γ s .In the second column of Fig. 2 we show constraints on { db} quark flavors operators from ∆Γ d (solid red line), B(B 0 → π 0 E miss ) (dashed blue line), B(B + → π + E miss ) (solid green line), B(B 0 → ρ 0 E miss ) (dash-dotted purple line), B(B + → ρ + E miss ) (dotted orange line), and B(B 0 → E miss ) (dashed brown line).The pattern of constraints is similar to those discussed above.But now since allowed phase spaces of B 0 → E miss and B → πE miss decays are relatively larger, their constraints cover the whole region of ∆Γ d bound.In the third column of Fig. 2 we show constraints on operators with {ūc} quark flavors.The bound from y D 12 (i.e., y D 12 ≤ 0.641%) is shown as solid red line, whereas bounds from branching ratios of D 0 → π 0 E miss and D 0 → E miss are shown as dashed blue line and dashdotted green line, respectively.The relevance of including lifetime difference constraint is particularly visible now.Because only a couple of charm decay modes have been measured so far, not all effective operators can be constrained from them.The lifetime difference constraint is the sole constraint on such operators, e.g., (ūγ µ γ 5 c)(ϕ † i ← → ∂ µ ϕ) (see the last plot in Fig. 2).Furthermore, we also note that y D 12 constraint can in general also compete with

Summary
Working in the EFT framework containing a light invisible particle of spin 0 or 1/2 as an explicit degree of freedom, we calculated the contributions of all relevant, effective operators to the lifetime difference in beauty and charm mesons mixings.In our analysis, we considered effective operators of dimension six and focused on operators involving a pair of light NP particles.We also pointed out that the previously considered effective operators containing a single dark sector particle do not contribute to the lifetime difference in neutral heavy mesons.We compared the constraints resulting from the lifetime difference with those from various beauty and charm meson decays.We find that in the beauty sector, the bounds on the Wilson coefficients of relevant, effective operators obtained from the meson decays are typically more robust than those obtained from the lifetime difference.This implies that the contribution of the dark sector particles to the lifetime difference in B mesons is not large.
Contrary to that, for operators relevant to charm physics, we find that lifetime difference in charm mixing is crucial for constraining several operators when the NP particle is massive.We note that the lack of experimental data in charm decays, e.g., D → ρ E miss , implies that vector-axial effective operators are presently not constrained.New constraints on these operators are obtained using the data on lifetime differences in charm mixing.