Quark-hadron duality for heavy meson mixings in the 't Hooft model

We study local quark-hadron duality and its violation for the $D^0-\bar{D}^0$, $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings in the 't Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-$N_c$ limit. With the 't Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the $D^0-\bar{D}^0$ mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the $D^0-\bar{D}^0$ mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.


Introduction
The theory of heavy quark physics, established since 1980s, has already experienced its mature stage. While its early development is characterized particularly by the heavy quark symmetry, nowadays it is turned into a systematic way to handle non-perturbative aspects of quantum chromodynamics (QCD). Equipped with Wilson's operator product expansion (OPE) [1][2][3] (the ideas were adopted to QCD in Refs. [4][5][6]), certain processes in the deep Euclidean domain are factorized into short and long distance objects. The former is calculated via perturbation theory while the latter is evaluated by non-perturbative methods such as lattice QCD. The OPE formula is then converted into one in the Minkowskian domain, on which physical processes of interest lie, via the analytic continuation. As a result, the observables are expanded by the inverse of heavy quark mass, 1/m Q . This methodology, referred to as the heavy quark expansion (HQE) [7][8][9][10] (see, e.g., Refs. [11,12] for reviews), is quite successful in describing inclusive processes for b quark. The current results for the lifetime ratios of b-hadrons [12][13][14] and the width difference in the B 0 s −B 0 s mixing [15] show an excellent agreement with the Heavy Flavor Averaging Group (HFLAV) data [16].
In contrast to the successful aspects of HQE for b quark, there exists two-fold complexity for treating c quark: (1) charm might be possibly too light for applying HQE and (2) due to Glashow-Iliopoulous-Miani (GIM) mechanism [17], observables undergo severe cancellation unlike the milder one for b quark. Due to the latter, specifically relevant for flavor-changing neutral current (FCNC) processes, observables are subject to the suppressions of SU(3) breaking [18] and/or the tiny product of Cabibbo-Kobayashi-Maskawa (CKM) matrix [19,20] elements, V * cb V ub . One of such notoriously difficult FCNC processes of c quark is the D 0 −D 0 mixing 1 , that proceeds via ∆C = 2 transition (see Refs. [25,26] for reviews). Two possible methods to calculate the D 0 −D 0 mixing exist in the literature: exclusive and inclusive approaches, where the latter is based on HQE. In the exclusive approach [27][28][29][30][31][32][33][34][35][36], the experimental data of hadronic decays are utilized so that the relevant long-distance effect can be properly extracted. The modern analyses [34,36] showed that two-body decays of D 0 meson accommodate roughly a half of the width difference although there lies a difficulty in handling other multi-body modes. Hence, while the order of magnitude of the width difference was reproduced, the quantitative agreement is still not realized in the exclusive approach.
On the other hand, the situation of the inclusive approach to the D 0 −D 0 mixing is somewhat different from that to the exclusive one. Owing to the severe GIM cancellation, the inclusive values of the mass and width differences are considerably suppressed, as can be seen from formulae obtained by the box diagrams in Refs. [37][38][39][40] and also by the heavy quark effective field theory in Refs. [41,42]. The later update including next-toleading order (NLO) corrections, obtainable from proper replacement in the B 0 −B 0 mixing [43][44][45][46] (see also [47]), gives the width difference about four orders of magnitude smaller [48,49] than the HFLAV data [16]. This huge discrepancy is to be contrasted with the exclusive approach, in which the order of magnitude is accommodated. Another point to be mentioned is that the HQE prediction for τ (D + )/τ (D 0 ) in Ref. [13] is in agreement with the HFLAV data [16], albeit the huge uncertainty in the theoretical side, indicating that the HQE for c quark is more or less meaningful in the processes without GIM cancellation.
In order to interpret the aforementioned disagreement, several possibilities are discussed in the literature 2 : first one is attributed to the contributions of higher dimensional operators, potentially leading to an enhancement, as discussed in [32,41,49,51]. For further clarifying this possibility, one should calculate a number of non-perturbative matrix elements for D = 9, 12 operators. Indeed, a new physics contribution is considered a candidate for explaining the gap. See, e.g., Refs. [52][53][54][55] for the studies in the context of new physics. A subtle point discussed in the recent work [56] is that if one adopts µ 1 , a scale at which the bi-local process induced by the ∆C = 1 oparators is calculated, different for individual internal quark contributions, the sufficient enhancement is realized after taking sum over flavors. In this respect, a natural question might be how large the next-to-next-to-leading order (NNLO) QCD corrections [57,58] will be after its completion. Furthermore, another recent study [59] where the dispersion relation is regarded as a 1 For the experimental side, the first evidence was found by Belle [21] and BABAR [22] collaborations in 2007. Subsequent confirmation was made by CDF [23] and LHCb [24] experiments. Currently, the average over large datasets [16] show that the zero values of the mixing parameters are excluded by more than 11.5σ [16], so that the occurrence of the D 0 −D 0 mixing has been firmly verified. See Refs. [16,25] for the detail of the experimental status and references therein. 2 See the status summarized in Ref. [50].
either smearing or 1/N c correction should be taken into account to gain a reasonable result, since local duality is maximally violated even for large energies for this case.
Resonance-based investigation of duality is greatly facilitated with the help of the 't Hooft model [93], 1 + 1 dimensional SU(N c ) gauge theory in the large-N c limit [94][95][96][97], in which case only the planar diagrams give non-vanishing contributions. The Bethe-Salpeter equation [98,99] in the light-cone gauge leads to a relation constraining wave functions and masses of mesons, the so-called 't Hooft equation. Being solvable, the equation unambiguously determines the properties of mesons in this formalism, thereby offering a useful laboratory to examine the non-perturbative dynamics of strong interaction. The (asymptotic) linear Regge trajectory, a key ingredient in (b), can be demonstrated in the model. Supported by such tractable features, discreteness of the mass spectra is shown mathematically [100], as is required by confinement. Posterior to the original work [93], the scattering amplitude, discussion in the axial gauge, chiral symmetry breaking, simulation on the lattice (with finite N c ), generalized parton distribution functions, weak decays of heavy quark, etc., are investigated in Refs. . Particularly noteworthy is that the intermediate meson pole contribution to the heavy-to-light form factor is demonstrated for any heavy quark mass, and the correction to the approximation is also determined, so that QCD dynamics in heavy quark decays can be clarified in 1 + 1 dimensions [115]. Numerical [93,104,106,110,113,118,121,129], semi-analytical [130] and analytical [131][132][133][134][135][136][137][138] methods to obtain solutions to the 't Hooft equation are investigated in the vast literature.
The mentioned tractable features of the 't Hooft model enable us to test quark-hadron duality. In the previous studies, this test is applied for hadronic spectral density functions [72,74,76,82] related to e + e − annihilation and τ decays, deep inelastic scattering [87,88] and heavy meson decays [72, 75-79, 81, 82, 84, 86]. Some of those references analytically gave the oscillating behavior for process rates, which is not captured in the practical OPE, as the energy/heavy quark mass is lowered. Thus, it is broadly considered that the 't Hooft model offers one certain methodology to reliably analyze duality violation while how the result is altered quantitatively in 3 + 1 dimensions remains unclear.
In this work, we study quark-hadron duality and its violation for heavy meson mixings in the 't Hooft model. 5 We first calculate the meson mixings based on the box diagrams in two-dimensions, corresponding to the contributions of four-quark operators in the HQE. Also calculated is the same observable based on the exclusive sum over final states, where the two-body decays are dominant in the large-N c limit since n-mesons' coupling is suppressed by N 1−n/2 c . To perform the exclusive analysis, by following the formalism in Refs. [75,77], we represent the topological amplitude [139][140][141][142] in terms of the overlap integrals for meson wave functions, which can be determined as numerical solutions to the 't Hooft equation. Then, the two calculated quantities are compared, in order to check the validity of the HQE. A non-trivial point in this comparison is that the GIM mechanism potentially affects the order of magnitudes of the observables. The investigation for the D 0 −D 0 mixing, subject to the strong GIM cancellation, is distinguished from ones for 5 Duality violation in the D 0 −D 0 mixing was concerned in Ref. [51], where the matrix element of the higher dimensional operator that linearly depends on strange quark mass avoiding the strong GIM cancellation was mainly discussed. the B 0 q −B 0 q (q = d, s) mixing. We show that the a large correction to the box diagram is realized for D 0 −D 0 when the phase space function is given solely by 4D-like one with certain choices of strange quark mass. As for the B 0 q −B 0 q mixing, the correction is much smaller than that for the D 0 −D 0 mixing, and consistent with the realistic observations in four-dimensions. Furthermore, this work deals with heavy meson decays into light mesons, such as D 0 → π + π − →D 0 , in addition to decays into heavy mesons. Little has been known for duality in the former case while for latter, especiallyB 0 s → B 0 s , an agreement between the partonic rate and the exclusive rate is shown [143] (see also the later study [144]) in the small-velocity limit [145] together with heavy quark and large-N c limits.
This paper is organized as follows: In Sec. 2, the formalism of the meson mixings, including formulae of the width differences, is exhibited. In Sec. 3, we first recapitulate the 't Hooft model to establish the notation. Subsequently calculated is the absorptive part of partonic transitions, cū → uc and bq → qb (q = d, s). Then, by taking the matrix elements, we obtain the formula of the HQE from the four-quark operators. The counterpart in the exclusive approach is also obtained in the large-N c limit. We show the numerical results in regards to violation of local duality in Sec. 4, by first analyzing the width differences from the individual flavors and then showing the results in the presence of the GIM mechanism. Finally, we conclude in Sec. 5.
2 Formalism in the CP conserving limit s mixings For the the D 0 −D 0 mixing, we introduce mass eigenstates denoted by |D 1,2 that diagonalize the Schrödinger equations [146] in the CP-conserving limit, where |D 1 (|D 2 ) coincides with a CP-even (odd) state. The off-diagonal element of the mixing matrix is given by, are associated with the contributions of off-shell and on-shell intermediate states, respectively. The width difference between the two CP states defined by ∆Γ D = Γ can be expressed in terms of the off-diagonal element of the mixing matrix, in the CP-conserving limit. The sign of the above observable is to be determined experimentally in this convention. As for the B 0 q −B 0 q mixing (q = d, s), a commonly adopted convention is based on |B H and |B L , heavier and lighter eigenstates. In the CP conserving limit, one finds that the sign of ∆Γ = Γ H − Γ L depends on that of M 12 unlike in Eq. (2.2), as can be seen in Eq. (2.16) of Ref. [39]. In order to compare the results of the D 0 −D 0 and the B 0 q −B 0 q mixings on the equal footing, the convention similar to that of the D 0 −D 0 mixing is adopted in the B 0 q −B 0 q mixing. That is, we introduce mass eigenstates of |B 1,2 , where |B 1 (|B 2 ) is a CP-even (odd) state, and define ∆Γ Bq = Γ . For this case, the following notation similar to one for the D 0 −D 0 mixing is introduced, Hereafter we exploit Γ 12 , valid in the CP conserving limit, and do not utilize the notation of Γ for brevity: we calculate the D 0 →D 0 transition for the D 0 −D 0 mixing whileB 0 q → B 0 q is computed for the B 0 q −B 0 q mixing, in the common notation of Γ 12 .

Width differences
For the D 0 −D 0 and B 0 q −B 0 q mixings, Γ 12 in Eqs. (2.1, 2.3) are given by the following expressions (α = inc, exc), (2.6) The products of the CKM matrix elements are defined by,  Exploiting the unitarity relation, λ d + λ s + λ b = 0 (λ u(q) + λ c(q) + λ t(q) = 0), one can eliminate λ d (λ u(q) ) in Eq. (2.5) (Eq. (2.6)) and write, where the combinations for individual contributions of flavors are given by, (2.14) One finds that Eqs. (2.11, 2.12) and Eqs. (2.13, 2.14) vanish for s = d and c = u, respectively, so that the first two terms in Eq. (2.9) and Eq. (2.10) are sensitive to flavor symmetry breakings. The characteristic differences between D 0 −D 0 , B 0 d −B 0 d and B 0 s −B 0 s mixings can be found in Eqs. (2.5-2.14). To see this, we exploit the hierarchy of the CKM matrix elements, |λ s | |λ b | for the D 0 −D 0 mixing and |λ u(s) | |λ c(s) | for the B 0 s −B 0 s mixing. If the SU(3) breaking in Eq. (2.11) is larger than the suppression from λ b for D 0 −D 0 mixing, we find that the D 0 −D 0 and B 0 s −B 0 s mixings are approximated by one term, where η(n) is a phase that depends on each intermediate state.
3 Inclusive and exclusive analyses in 1 + 1 dimensions

The 't Hooft model
The QCD Lagrangian in 1 + 1 dimensions has a form apparently similar to one in 3 + 1 dimensions, with the covariant derivative defined by iD µ = i∂ µ + gA µ . For the second term, the sum runs over flavors. m f and g are a bare mass and a bare coupling, respectively, both of which have a unit mass dimension in 1 + 1 spacetime. We introduce the following notation of the QCD coupling, β is require to be a constant in the large-N c limit for the sensible counting for N c , and gives a unit for any dimensional quantities in the model. We adopt the light-cone gauge satisfying A − = 0, in which case the theory becomes ghost-free while the field strength is simplified to be effectively Abelian. With the notations introduced above, the 't Hooft equation is given by, where x and 1 − x represent the light-cone momentum fractions that are carried by q 1 andq 2 , respectively. M n denotes the meson mass while m 1 and m 2 are bare masses of q 1 andq 2 , respectively. φ n is a meson wave function of the n-th (n = 0, 1, · · · ) radial state that satisfies the boundary conditions, φ n (0) = φ n (1) = 0. States labeled by n = even are pseudoscalar mesons with n = 0 being the ground state, the lightest hadron. The other states with n = odd are scalar mesons. As was shown by 't Hooft, Eq. (3.3) is independent of the infrared cut-off. The renormalizations for fermion masses were already taken into account by shifting the bare masses, 3). Furthermore, by introducing a meson decay constant for the n-th radial state consisting of q 1 andq 2 , one writes a matrix element for the axial current, Above we used the mesonic notation, H, for the ground state consisting of q 1q2 , corresponding to n = 0 in Eq. (3.4, 3.5). The matrix element of the pseudoscalar bilinear similar to Eq. (3.6) can be derived by using the equation of motion while one for the scalar bilinear vanishes. As for the matrix element of the vector current, it can be rewritten as one for the axial vector current in Eq. (3.6) by using the relation of the gamma matrix in two-dimensions, as is done in Appendix.

HQE from leading operators
We consider the weak vertex that has a generalized Lorentz structure parameterized as, with V CKM being the CKM matrix element associated with a given process. c V = −c A = 1/2 corresponds to the case where the weak interaction proceeds via the standard modellike V−A current. The W boson propagator given as in 3 + 1 dimensions is, where fixing ξ = 1 leads to the unitary gauge, in which case the contributions of the charged-Goldstone bosons are absent. We keep the contribution that is dominant in the limit of M W → ∞, corresponding to the g µν part in Eq. (3.8). Below, by using these Feynman rules, we give the effective Hamiltonian leading to the absorptive parts of Qq → qQ transition with (Q,q) being (c,ū), (b,d) or (b,s) shown in Fig. 1. The detail of the calculation is given in Appendix. As a result, the absorptive parts of the effective Hamiltonian that contribute to the heavy meson mixing in the considered approximations are given by, The coefficients and the four-quark operators are given by, represent phase space functions that have non-zero values in a physical region, . The leading contribution for large m Q solely comes from the term proportional to F (th) ij . One finds that the coefficients given in Eqs. (3.10, 3.11) are proportional to (c 2 V − c 2 A ). Hence, the observables in meson mixings for c V = ±c A corresponding to the V ± A current vanish, which is not seen in four-dimensions. This is partially attributed to the fact that either vector current or axial current is reducible and can be written by another in two-dimensions. The derivation for Eqs. (3.10, 3.11) by means of the Fiertz rearrangements in two-dimensions is given in Appendix. Furthermore, the non-vanishing result in the limit of m i , m j → 0 for the V × V current (c V = 0, c A = 0) observed via Eqs. (3.10, 3.11) is to be contrasted with Ref. [76], where the contribution of the four-fermion operator in the annihilation-topology, calculated as an absorptive part, is shown to vanish at zeroth order in strong interaction.
The matrix elements in Eq. (3.9) can be taken on the basis of the factorization in the large-N c limit with Eq. (3.6), On r.h.s. of Eqs. (3.18, 3.19), the factor two, arising from two possible ways for taking the currents in inserting vacuum, are considered, and is cancelled out with 1/2 on l.h.s. If we go beyond large-N c limit, an evaluation the non-perturbative matrix elements in Eqs. (3.18, 3.19) should be made, that is beyond our current scope. As long as the four-quark operators are concerned, however, the matrix elements do not give sources of flavor symmetry breaking in Eqs. (2.11-2.14). As a main result in this subsection, one finally obtains the HQE expression of the four-quark operators, , (s, s) for the first case and (u, u), (c, u), (c, c) for the latter two cases. In the limit of m Q → ∞, it is well-know that c const., to be contrasted with the case in 3 + 1 dimensions, Γ (H, inc) ij ∝ m 2 Q , as can be seen from Refs. [37][38][39]. This difference results from the fact that both Fermi constant and decay constant are dimensionless in 1 + 1 spacetime. If we take the massless limit of internal quarks, Eq. (3.20) is recast into, As we shall see later, Eq. (3.21) agrees with the exclusive result in the same limit. The 1/m Q expansion of the contributions of the four-quark operators in Eq. (3.20) can be readily studied in the static limit, m Q = m 1 → ∞, in Eq. (3.3) as was first discussed in Refs. [111,112] Below, we give the final results for the ground state in Ref. [82], where F (n) is a finite object in the static limit, Moreover, it might be useful to introduce δ ≡ R − 1, a quantity power-suppressed by m Q , where the expansion of δ is obtained from Eq. (3.23), One also finds that the phase space functions in Eqs. (3.14-3.17) give corrections of the 1/m Q expansion due to the expansion formulae, ij , H It is possible to numerically obtain the explicit coefficients of each 1/m Q term with a given mass of the spectator quark as in Ref. [82]. However, since the expansion of 1/m Q is not necessary in our current purpose, the numerical results presented in Sec. 4 is a leading term in the limit of m Q → ∞, a certain care must be taken since the inclusion of the 2Dspecific phase space function affects the resultant counting in z s . We also give alternative expressions that take account of F

Topological amplitude in the large-N c limit
In the remaining part of this section, we aim to obtain Γ follow. Even if we take account of resonant contributions for some of the topological amplitudes, that strengthen N c dependence [77], the contribution of T to width is still dominant compared with the others in the large-N c limit. The leading decay amplitudes from two-body pseudoscalar modes in the large-N c limit are given by, where the superscript, (0, 0), represents the ground states in the final particles while the subscripts, (cū) and (i, j) = (d, d), (d, s), (s, d), (s, s), stand for the flavors in an initial and final states, respectively. Similarly, one introduces the decay amplitudes ofB 0 d andB 0 s from the color-allowed tree diagrams as follows, We omitted processes that are given by color-allowed tree diagrams but do not contribute to B 0 q −B 0 q through the most color-favored topology, e.g.,B 0 For additionally including the contributions of scalar and other excited particles in the final states, one generalizes the notations in (3.35) and in (3.36) into, By performing the phase space integral in 1 + 1 dimensions, one writes the partial decay width for H → f 1 f 2 decays, (3.40) where p 12 denotes a momentum of either daughter meson in the rest frame of H. The peculiarity of the phase space, showing that the width looks divergent when M H = M 1 +M 2 or equivalently p 12 = 0, is present in Eq. (3.39). This point is obviously distinct from the case with 3+1 dimensions, Γ ∝ |p 12 | due to the phase space, leading to the vanishing width for p 12 = 0. In the analytical study [76], it was shown that this singularity is cancelled out with an amplitude in the semi-leptonic decay, in which massless particles are involved. As mentioned, we consider the rigorous large-N c limit, where the resonant width associated with strong decays vanishes, in which case the topological amplitude does not develop its imaginary part. One can write the individual internal quark contributions on r.h.s. in Eqs. (2.9, 2.10) by allocating n in Eq. (2.18) on the basis of the relevant quantum numbers. The diagrams for exclusive processes, given by the hadronic degrees of freedom in The momenta denoted by p km is understood as one in Eq. (3.40) with the relevant final state. For the individual (i, j) contribution, the sum over (k, m), representing the tower of kinematically allowed excited particles (in addition to the ground states) for final states, is taken. The prefactor of (−1) k+m comes from η(n) in Eq. (2.18) to account for the parity-odd property of the topological amplitude in the case of k + m = even due to the proportionality to the spatial component of a momentum with overall negative signs. It should be also noted that there exists no orbital angular momentum via the relative motion of particles in the final state in two-dimensions. The explicit formula for color-allowed tree amplitude is obtained in Ref. [75]. Below we denote the momenta of mesons labeled by k, m as q and p, respectively. The kinematical variable defined by ω = q − /p − is determined by, With the generalized Lorentz structure in Eq. (3.7), we can use the formulas [75,78], valid for (Q,q) equal to (c,ū), (b,d) and (b,s) in the limit of M W → ∞, For an on-shell process, q 2 is set to M 2 k in Eqs. (3.43, 3.44). F nm denotes the triple overlap integral while C m and D m are the quark-model type contact terms [78], One can analytically simplify m Q dependence of the width difference in Eqs. (3.41, 3.42) in the massless limit of quarks except for heavy quarks in initial and final states. For this case, the only non-vanishing contribution in Fig. 2 is k = m = 0 due to the vanishing property of the decay constants for excited states, c n = 0 (n = 0), for the massless constituents. Since M k and M m with k = m = 0 vanish in this limit, q 2 → 0 together with ω → 0 follows for both of the two interfering amplitudes in Eqs. (3.41, 3.42), in which case the terms except for C in Eq. (3.44) vanish. Then, the interference of the amplitudes is simplified as, By using c 0 = 1 and φ 0 (x) = 1 (except for the end points) for massless constituents, we find that the width difference in Eqs. (3.41, 3.42) is reduced to, It should be noted that Eq. (3.49) agrees with the HQE result in Eq. (3.21). Therefore, local duality is unambiguously seen in the massless limit of quarks except the heavy decaying one, that is indeed an analogy of the Pauli interference [78,79]. Moreover local duality in the heavy meson mixings is understood as an example of the "exclusive" duality [66], where one exclusive mode approximates the inclusive result. The heavy quark limit is unnecessary to derive duality in this case. Another point to mention is that the twisted sum over exclusive states in Eqs. ( [75] so that whether the topology is twisted affects the asymptotic m Q dependence of the observables.

Local duality for massive flavors
In reality, s and c quarks cannot be regarded as massless particles. Including these masses is crucial in the presence of the GIM mechanism, since otherwise the net observables vanish in the limit where a particular CKM product is neglected. To this end, in this section, we investigate local duality and its violation for those massive quarks by numerically solving the 't Hooft equation. In Sec. 4.1, duality in the contributions from individual flavors is discussed. Subsequently, the result for the GIM combination that appears in the observable is presented in Sec. 4.2.
In numerically solving the 't Hooft equation, standard methods adopted in the literature might be the Multhopp technique (see Ref. [148] and also appendices in Refs. [75,113] for the detail), where the wave function is expanded by the trigonometric basis function. The integral equation is then regarded as an eigenvalue problem, yielding the asymptotically linear Regge trajectory of meson mass spectra. The normalization of the eigenvectors obtained is rescaled so as to satisfy 1 0 dx[φ(x)] 2 = 1. It is often pointed out for the Multhopp technique, however, that the behaviors at the end points, φ n (x) = x β 1 for x ∼ 0 and φ n (x) = (1 − x) β 2 for x ∼ 1 with β 1,2 being πβ 1,2 cot(πβ 1,2 ) = β 2 − m 2 1,2 , are not straightforward to obtain. Then, the BSW-improved Multhopp method [106] is developed, rendering the behavior at the end points better controlled. In this work, we adopt the method in Ref. [82], introducing the following expansion, We then convert the 't Hooft equation into the eigenvalue problem with the recursive formula in Ref. [82], where the accuracy nearby end points are improved by taking large N , and obtain a (n) k and M 2 n . Nonetheless, the endpoint behaviors for x → 0, 1 are still given by square root, so that great care must be taken for the accuracy. As Q that forms the bound state of Qq gets heavier, the meson wave function at the vicinity of x = 1 becomes rather singular. Excited states that are formed by light quark and anti-quark with large n, whose wave functions rapidly oscillate, also cause errors in the presence of the limited precision around the endpoints. In this work, we take N in Eq. (4.1) as 500 and solve the 't Hooft equation, and then truncate heavier (500 − N eff ) excited states, that do not follow the linear Regge trajectory, as well as eigenvectors. N eff is varied to test the stability of the numerical results. Moreover, the numerical analysis requires the evaluation of the overlap integrals for the convolution of meson wave functions in Eqs. (3.45-3.47), distinguished from the simpler one for semi-leptonic decays of heavy mesons in Ref. [82]. In order to guarantee the numerical stability of the result presented below, we neglect the triple overlap integral in Eq. (3.45), that gives a contribution suppressed by at least 1/m 2 Q [78] to the decay amplitudes, relative to the leading terms in Eqs. (3.46, 3.47). For this case, the stability under the variation of N eff is verified. Hereafter we fix N eff = 300. Further improvement in the numerical results entails technical tasks, including the accurate calculation of endpoint behaviors, as well as the precise evaluation of the convolution integral, which are beyond our current scope, while the exclusive results presented below capture the leading behaviors in the 1/m Q expansion. As was obtained in Sec Before proceeding to results, further remarks are addressed: • For the heavy quark decays, spikes of the rate emerge [75,77] when the heavy quark mass gets larger than threshold values for M H = M k + M m due to the hadronic phase space unlike the case in 3 + 1 dimensions. In order to quantify violation of local duality, the middle point between i-th and (i + 1)-th thresholds should be discussed [76]. For the width difference in the heavy meson mixings, the analogous spikes appears for massive final states, as well as decays. The numerical results presented below are based on discrete points for heavy quark mass that are not (exactly) at the thresholds to avoid obvious singularities in Eqs. (3.41, 3.42).  [149] for d = 4. The bare masses for u and d quarks are fixed to zero in what follows. As for the bare coupling, we adopt an ansatz, β = 340 MeV, that is obtained in such a way that the string tension of QCD 4 is fitted [80,127] by (π/2)β 2 = 0.18 GeV 2 . . Due to the vanishing properties of decay constants for the excited states of pions, we find that the sum over pion states in Eq. (3.41) is reduced only to the ground state, as was discussed in Sec. 3.3, so that,

Numerical result for individual flavors
.

(4.2)
By recalling that in the massless limit of u and d quarks, the only surviving contribution in T

Numerical result in the presence of the GIM mechanism
We would like to remind the reader that the inclusive width difference, discussed in Sec. 3.2, has quite different function forms, depending on whether (1) 4D-like phase space term in Eq. (3.14) is only considered or (2) the 2D-specific terms in Eqs. (3.15, 3.16) are additionally included. For the former, the GIM 1 for the D 0 −D 0 mixing defined in Eq. (2.11) behaves like (m s /m c ) 4 while it is (m s /m c ) 2 for the latter in the large m c limit, due to Eq. (3.31) and Eq. (3.33), respectively, meaning that the former is more suppressed. The similar discussion is applied for the B 0 q −B 0 q mixing by replacing m c → m b and m s → m c . Thus, the order of the magnitude of |Γ (exc) /Γ (inc) | strongly depends on whether (1) or (2) is adopted for the inclusive side. Below, we present the results based on both (1) and (2).
In Figs Fig. 6 is of direct relevance in phenomenology, while this is not the case for the B 0 q −B 0 q (q = d, s) mixing, as was discussed in Sec. 2.2. The numerical stabilities under the variation of N eff are confirmed for what are plotted in Figs. 6-8, especially m s /β > 0.14 in Fig. 6.
The ratio of the inclusive observable to the sum of exclusive ones defined in Eqs. (2.2, 2.4) is shown in Figs. 9-12 for the three meson mixings. In obtaining the figures, we included all the three terms in Eqs. (2.9, 2.10). The numerical results are stabilized as the second terms give quite small contributions. One finds that for the D 0 −D 0 mixing, the patterns in Fig. 9 are precisely similar to those in Fig. 6, which are regarded as the cases in the limit of λ b → 0 in Fig. 9. Hence, the net observable for the D 0 −D 0 mixing is enhanced when the phase space is given by one in four-dimensions, as well as Fig. 6. Meanwhile, the patterns in Figs For the B 0 q −B 0 q mixing, further comparison between the four-dimensional observation and two-dimensional results is given in order. For q = d, the HFLAV result for ∆Γ B d [16] is consistent with zero within an error while the four-dimensional HQE result is given by ∆Γ B d = (2.6 ± 0.4) × 10 −3 ps −1 [15]. Due to this situation in four-dimensions, a visible size of the correction to the HQE prediction in the B 0 d −B 0 d mixing is possible, being still consistent with the two-dimensional result in Fig 10. As for q = s, by combining the results of the HFLAV [16] and the HQE [15], one obtains a ratio, ∆Γ

Conclusion
We have studied local quark-hadron duality and its violation in the heavy quark mixings on the basis of one certain dynamical mechanism. For the inclusive analysis, we have obtained the leading HQE expression that arises from the four-quark operators by evaluating the box diagrams in two-dimensions. The resulting width difference scales like a constant for large m Q , with the correction to this starting from 1/m Q , which was clarified in the static limit. Care must be taken for the fact that, in the presence of the GIM mechanism, the order of magnitude for the inclusive observables strongly depends on whether the 4D-like phase space is solely considered or 2D-specific ones are also included.
We have analytically shown that local duality is unambiguously seen in the massless limits for u and d quarks, which might be relevant for D 0 → π + π − →D 0 and B 0 d → π − π + → B 0 d , by comparing the inclusive and exclusive width differences. This is interpreted as an example of the "exclusive" duality. For the massive case, duality violation is numerically investigated for the three meson mixings with the 't Hooft equation being solved. For two massive intermediate contributions, e.g., cū → ss → uc, the spikes for the exclusive width differences appear when the heavy quark mass gets larger than values at each kinematical threshold.
As stressed in the Introduction, the realistic observation in four-dimension indicates that the discrepancy between theory and experiment is of four orders of magnitude for the observable in the D 0 −D 0 mixing when the HQE result is given by the four-quark operators. In an attempt to interpret this observation, we have investigated how the exclusive observable is enhanced, relative to one obtained by the inclusive analysis, in the presence of the GIM mechanism. For the D 0 −D 0 mixing, the enhancement for the exclusive result is shown, confirmed to be larger than 10 3 for 0.14 < m s /β < 0.25, when the phase space function is given by only the 4D-like term, although a huge enhancement is absent when the contributions of the 2D-specific phase space terms are added. As for the B 0 q −B 0 q (q = d, s) mixing, no huge enhancement of the exclusive observable is realized, yet the visible correction to |∆Γ

A Box diagram in 1 + 1 dimensions
For Q(p 1 )q(p 2 ) → q(p 3 )Q(p 4 ), one finds that Fig. 1a with internal quarks being labeled as (i, j) = (d, d), (s, d), (s, s) for cū → uc and (i, j) = (u, u), (c, u), (u, u) for bq → qb (q = d, s) is calculated in d dimensions, The above expression is readily evaluated in the approximation where the momenta of heavy quark (Q) is much larger than ones of the spectator quark (q), i.e., p 1 p 2 , p 4 p 3 . This is done by decomposing the product of the propagators into partial fractions [38] with .

(A.2)
Hereafter we suppress 1/[(1−x i )(1−x j )] in Eq. (A.2), which approaches unity in M W → ∞. By defining an object analogous to the Fermi constant, G F / √ 2 = g 2 2 /8M 2 W , one gets, ij and I (k) ij are loop integrals given by, where some objects analogous to those in 3 + 1 dimensions [38,39] are introduced, There are a few points to be mentioned. First, due to the asymmetric sum of k, terms independent of k vanish in Eq. (A.3). Second, a threshold relevant for two internal quarks is associated with Λ give rises to discontinuity, contributing to the width difference. As we will see later, the discontinuities of G where the functions that have branch cut are introduced by, The discontinuities of Eq. (A.13) in a physical region are (the sign is associated with ones above branch cut), (A.14) where F where in Eq. (A.15) we used γ µ = µν γ ν γ 5 ( 01 = +1) valid in two-dimensions, that yields V µ × V µ = −A µ × A µ . As for the terms proportional toḠ ij andĪ ij , the relevant Fiertz rearrangements are also obtainable straightforwardly, with the equation of motion for heavy quark being implemented. Below, we omit the bilinears that do not contribute to heavy meson mixings for the ground state in the large-N c limit. By substituting Eqs. (A.14-A.16) into Eq. (A.12), we obtain the absorptive part of Fig. 1a, ij + 2H so that (part of) the final result is proportional to the (V ± A) × (V ± A) operator in four-dimensions. This difference is due to the vanishing of γ µ γ α γ µ = (2 − d)γ α , and also to the higher redundancy for products of gamma matrices for d = 2 than that for d = 4. Likewise, one can also calculate the absorptive part of Fig. 1b, which gives the amplitude similar to Eq. (A.17) except that the momentum arrangement for the spinors is different. By combining these results, we finally obtain the effective Hamiltonian in Eq. (3.9) with Eqs. (2.1, 2.3).