Wave-packet effects: a solution for isospin anomalies in vector-meson decay

There is a long-standing anomaly in the ratio of the decay width for ψ(3770)→D0D0¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (3770)\rightarrow D^0\overline{D^0}$$\end{document} to that for ψ(3770)→D+D-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (3770)\rightarrow D^+D^-$$\end{document} at the level of 9.5σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$9.5\,\sigma $$\end{document}. A similar anomaly exists for the ratio of ϕ(1020)→KL0KS0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (1020)\rightarrow K_\text {L}^0K_\text {S}^0$$\end{document} to ϕ(1020)→K+K-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (1020)\rightarrow K^+K^-$$\end{document} at 2.1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.1\,\sigma $$\end{document}. In this study, we reassess the anomaly through the lens of a Gaussian wave-packet formalism. Our comprehensive calculations include the localization of the overlap of the wave packets near the mass thresholds and the composite nature of the initial-state vector mesons. The results align within a ∼1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 1 \sigma $$\end{document} confidence level with the Particle Data Group’s central values for a physically reasonable value of the form-factor parameter, indicating a resolution to these anomalies. We also check the deviation of a wave-packet resonance from the Briet–Wigner shape and find that wide ranges of the wave-packet size are consistent with the experimental data.


Introduction
There is a long-standing anomaly (discrepancy between experimental and theoretical results) in the ratio of the decay width for ψ(3770) → D 0 D 0 to that for ψ(3770) → D + D − .A similar but weaker anomaly exists for the ratio of ϕ(1020) → K 0 L K 0 S to ϕ(1020) → K + K − .On the other hand, the ratio of Υ(4S) → B + B − to Υ(4S) → B 0 B 0 is consistent with the standard theoretical predictions.
At the quark level, these processes are1 ϕ(ss) → K + (us) K − (sū) , ψ(cc) → D + cd D − (dc) , Υ bb → B + ub B − (bū) , ϕ(ss) → K 0 (ds) K 0 s d → K 0 L K 0 S , ψ(cc) → D 0 (cu) D 0 (uc) , Υ bb → B 0 db B 0 bd , which can be summarized as V QQ → P (Qq) + P qQ , where V and P are vector and pseudo-scalar mesons, respectively, and Q and q are heavy (s, c, b) and light (u, d) quarks, respectively.This ratio of decay widths is theoretically clean because most of the quantumchromodynamics (QCD) corrections cancel out between the numerator and the denominator.These decay processes are via strong interaction, and hence in the limit of exact isospin symmetry u ↔ d, the ratio becomes unity.The isospin violation makes a deviation from unity.
We name the ratio of the widths as2 The experimental results are combined by the Particle Data Group (PDG) [ The theoretical prediction of the decay rates for V → P + P − and V → P 0 P 0 is based on the plane-wave formalism so far.The tree-level result of the chiral perturbation theory reads where g V + (g V 0 ) is the coupling between V and P + P − (P 0 P 0 ) and m V , m P + , and m P 0 are the masses of V , P + , and P 0 , respectively.Even if we assume an isospin-symmetric coupling g V + = g V 0 , the difference in the pseudo-scalar-meson masses m P + ̸ = m P 0 results in a deviation in R V from unity: Putting the mass values in Ref. [1], 4 we obtain Comparing Eqs. ( 2) and ( 4), we see that the isospin-symmetric limit for the coupling g V + = g V 0 results in the anomaly at the level of 2.1 σ, 9.5 σ, and 0.32 σ for ϕ, ψ, and Υ, respectively.We briefly review the theoretical accounts for the anomaly within plane-wave formalism.For R ϕ , it turned out that radiative corrections make the anomaly more significant [2]: The standard quantum-electrodynamics (QED) corrections make the theoretical prediction of the ratio 4 % larger, and isospin-breaking corrections to the ratio g 2 ϕ+ /g 2 ϕ0 further make it "some 2 %" [2] larger, leading to a larger anomaly of roughly 5.2 σ assuming that the error is dominated by that in Eq. (2).In Ref. [3] the authors introduce a smeared decay rate that is a function of the energy difference between the initial and final plane-wave states; this smearing is by the Lorentzian distribution due to the inclusion of the width as well as by a phenomenological form factor put by hand to regularize an ultraviolet (UV) divergence; the anomaly for ϕ can be explained with a mass parameter M ≃ 1.5 GeV in the phenomenological form factor.In Ref. [4], the authors have estimated the effects of the electromagnetic structure of kaons and other model-dependent contributions to the radiative corrections, and the resultant corrections have turned out to be tiny.In Ref. [5], two (a Breit-Wigner and a non-relativistic Lorentzian) types of averaged decay widths over the initial-state energy are introduced with two phenomenologically chosen energy intervals 1.010-1.060GeV and 1.000-1.100GeV, to relax the anomaly.
For R ψ , another type of averaged decay width is introduced in Ref. [6], and the resultant anomaly has become even more significant.There is no explanation for this 9.5 σ anomaly so far.
The above smearing/averaging over the energy provides significant effects because the decay V → P P is near the threshold m V ≃ 2m P .In situations near the threshold, it is desirable to treat the decay more rigorously by using wave packets for the initial and final states.Recall that the S-matrix in the plane-wave formalism contains the energy-momentum-conserving delta function and is theoretically ill-defined when computing the probability rather than the rate.A well-defined decay probability can be calculated only as a transition from a wave packet to a pair of wave packets.This is theoretically more reliable.
In the previous analyses [3,4,5], it has been assumed that the transition processes are described by the (plane-wave) rates alone. 5In this paper, we present an analysis based on the transition probability of the normalized states, wave packets, without the divergence of the delta-function squared.Concretely, we compute the decay V → P P in the Gaussian wave-packet formalism [9,10,11,12,13]; see also Refs.[14,15,16]. 6In particular, we include a wave packet effect, called the in-time-boundary effect for the decay, by simply limiting the time-integral of the decay-interaction point to t > T in [10].Here, T in is the time from which the interaction is switched on.This procedure is proven to provide approximate modeling of the full production process of V in the corresponding two-to-two wave-packet scattering, say, e + e − → V → P P [13]; see also Refs.[20,21,11,12,22,23,24] for related discussions.The organization of this paper is as follows: In Section 2, we will introduce the minimum basics of calculating the (generalized) S-matrix that describes wave-packet-to-wave-packet transitions considering the initial state's decaying nature when wave packets take the Gaussian form.In Section 3, we will review significant properties of the Gaussian wave-packet S-matrix.In Section 4, we will compare the theoretical predictions for the ratios of R ϕ , R ψ , and R Υ in the wave-packet and the plane-wave formalisms taking into account the form factor of the vector mesons.In Section 5, we will discuss the constraint from the resonant shape in the electron-positron-collider experiments for ϕ and ψ.In Section 6, we will provide a summary and further discussions.In Appendix A, we will review the form-factor details for vector mesons used for our analysis.In Appendix B, we will provide the details on how to derive the total probability of V → P P under non-relativistic approximations.In Appendix C, a brief review on how to derive the plane-wave decay rate for V → P P will be provided.In Appendix D, we will comment on a specific formal limit where the wave-packet decay rate coincides with the plane-wave decay rate.In Appendix E, we will briefly consider the isospin violation on the ρ system.

Basics of Gaussian wave-packet formalism
For the near-threshold decay, the velocities in the final state are small, and the overlap of the wave packets becomes more significant in general.Therefore, it is important to take them into account.
Here, we spell out how to compute the probability for the V → P P decay in the Gaussian wave-packet formalism.Throughout this paper, we work in the natural units ℏ = c = 1.Readers who are more interested in analyses of experimental results rather than detailed theoretical formulation may skim through this section.

Wave-packet S-matrix
In the Gaussian wave-packet formalism, a transition from an initial wave-packet state |WP 0 ⟩ to a two-body final wave-packet state |WP 1 , WP 2 ⟩ is characterized by the following generalized S-matrix [9]: where U describes the unitary time evolution from the initial time T in to the final time T out in which T denotes the time-ordering and H (I) int is the interaction Hamiltonian density in the interaction picture.The local interaction point (t, x) is integrated in the four-dimensional spacetime.It is noteworthy that the wave-packet states |WP 0 ⟩ and |WP 1 , WP 2 ⟩ are normalizable and hence the transition amplitude (5) is finite, unlike in the ordinary plane-wave formalism. 7Through the Dyson series expansion of U (T out , T in ), a perturbative S-matrix can be systematically constructed at any order of perturbation using Wick's theorem, as in the plane wave case [11].Throughout this paper, the subscripts 0, 1, and 2 denote V , P , and P , respectively.
A free Gaussian wave packet is characterized by a set of parameters m, σ, X 0 , X, P , where m is the mass; σ is the width-squared; and X 0 is a reference time at which the wave packet takes the Gaussian form with the central values of the peak position X and momentum P .
Within the chiral perturbation theory, the effective-interaction-Hamiltonian density is where V, P ± , P 0 , and P 0 are the fields representing the vector meson, the charged pseudoscalar mesons, the neutral pseudo-scalar meson, and its antiparticle, respectively, and g V + and g V 0 are the vector-meson effective couplings to the charged pseudo-scalars and to the neutral pseudo-scalars, respectively.In this paper, we take the isospin-symmetric limit, with which the effective coupling g eff takes the form in the momentum space g eff (λ 0 , P 0 , P 1 , P 2 ) := g V ε µ (P 0 , λ 0 ) (P µ 1 − P µ 2 ) , where P 0 , P 1 , and P 2 are the four-momenta of the vector meson V , the pseudo-scalar meson P , and its antiparticle P , respectively, and ε µ is the polarization vector of the vector meson with λ 0 being its helicity. 8n this paper, we investigate the transition from an off-shell initial state for V to an on-shell final state for P P , having an off-shell energy E 0 and on-shell ones E 1 , E 2 , respectively: where m V and m P are the masses of V and P , respectively; E 0 := m 2 V + P 2 0 is the onshell energy of V ; and Γ V is the "decay width" of V , or more precisely, the imaginary part of its plane-wave propagator divided by m V ; see Ref. [13] for detailed discussion; see also footnote 13.Throughout this paper, we take the narrow-width approximation for Γ V as in Eq. (10). 10 Here, the off-shell V should eventually be regarded as an intermediate state for a scattering process that includes the production of V , which necessarily introduces the in-time-boundary effect appearing below.
Their wave functions take the form where N V is a wave-function (field) renormalization factor for V due to its offshellness 11 and describes the location of the center of the wave packet at a time t with the central velocity Now, it is straightforward to compute the S-matrix from Eq. ( 5) at the leading order in the Dyson series (6) with the effective Hamiltonian (7) from the wave functions (11) [11]: S V →P P = ig eff (λ 0 , P 0 , P 1 , P 2 ) where the notation follows Eq. ( 27) of Ref. [11] (see also below for a short summary). 12Differences from the previous calculation [11] are the following four points: First, the coupling is changed to κ/ √ 2 → g eff .Second, the "decay width" of V is included as the phenomenological factor e − Γ V 2 (t−T 0 ) , 13 where T 0 := X 0 0 is the initial time from which V starts to exist. 14Third, 10 Theoretically, ΓV is obtained as the imaginary part of the plane-wave V propagator at the loop level.See footnote 13.  11 NV shows a factor that accounts for the possible extra decrease of the norm of the initial state due to the off-shellness ΓV > 0. Anyway, NV will drop out of the final ratio of the decay probabilities. 12In Eq. ( 14), we have dropped the overall phase factor which is irrelevant to the calculation of the probability, while properly taking into account the real damping factor e − Γ V 2 (t−T 0 ) coming from the imaginary part of E0; see Eq. (11). 13When one includes the production process in the amplitude, e.g., as e + e − → V → P P , the result does not change whether we expand the complete set of intermediate states of V by the Gaussian-wave-packet or planewave bases; see Sec. 2.3 in Ref. [12].The imaginary part mV ΓV of the plane-wave propagator of V appears through loop corrections, and when translated to the decay process V → P P , its effect can be expressed as the phenomenological factor e −Γ V (t−T 0 )/2 in the plane-wave formalism.Here, we also phenomenologically take into account the exponentially decaying nature of the initial wave packet of V through the channel that is common to the plane-wave decay, namely, through the bulk effect that appears below.See also footnote 9.
14 T0 is indeed irrelevant in the sense that the time-translational invariance results in the dependence of the < l a t e x i t s h a 1 _ b a s e 6 4 = " J V 9 P 9 C w h T G m W q p q 6 c u c 9 4 L 6 v i H S Y V l e c M p V p 9 x 1 U D X G t z V 8 K 2 1 A K 9 p h t o g r 4 O a 0 j c 6 V P J s p T w 7 I 8 + e S 1 4 k s C / 6 n + y L B P Z F c 9 h X / T 0 + k 5 3 e T E y W e e f e 7 5 F 0 6 9 e P q l l 1 9 5 9 c z Z c 6 e T E y W e e f e 7 5 F 0 6 9 e P q l l 1 9 5 9 c z Z c 6   T out (! 1)

Bulk region
< l a t e x i t s h a 1 _ b a s e 6 4 = " s L F    with an infinitesimal ϵ > 0 is conventionally put by hand for the future and past infinite times t → ±∞, which is depicted by the damping of the opacity of the orange region.This factor eventually results in the propagator ∝ p 2 + m 2 − iϵ −1 in the conventional Feynman diagram calculation.
N V in Eq. ( 11) is introduced.Fourth, we have included a phenomenological form factor F due to the composite nature of V : where R 0 describes a typical length scale of the compositeness of V ; see Appendix A. The normalization is such that F becomes unity for V 1 = V 2 .Now we provide a brief introduction to other variables in the first two lines of ( 14) (see Section 3.1 of [11] for more details):

•
√ σ s is a typical spacial size of the region of interaction: final result only on the difference Tin − T0.Furthermore, this dependence on Tin − T0 cancels out between the numerator and denominator of the final ratio of the decay probabilities, as we will see.(Physically, we would expect Tin ≃ T0.) • √ σ t is a typical temporal size of the interaction region: 15 • T is the time of intersection of the three wave packets: where is the location of the center of each wave packet at our reference time t = 0.As mentioned above, each wave packet takes the Gaussian form centered at X A at its reference time X 0 A .• R is called the overlap exponent, which provides the exponential suppression when wave packets are separated from each other: • We write the deviation of energy-momentum from the conserved values (for their central values of wave packets) where ω A := E A − V • P A is the "shifted energy" of each packet.
A schematic figure is shown in the left panel of Fig. 1, compared with the plane-wave counterpart in the right.
After the square completion of t and the analytic Gaussian integration over x in (14), as made in [11], we represent the S-matrix as follows: where the window function G(T) is defined as16 15 We adopt the following notation for arbitrary scalar and vector variables C and C, respectively: being the Gauss error function.The window function G(T) becomes unity for T in ≪ T ≪ T out and zero for T ≪ T in and for T out ≪ T. For a given configuration of in and out states, which fixes the value of σ t , the time regions are called the bulk, in-time-boundary, and out-time-boundary regions, respectively.In the phenomenological analysis below, we will neglect the out-time-boundary contributions as we will discuss.

Differential decay probability
From the S-matrix (22), the differential decay probability can be derived as where we have taken the average over the helicity λ 0 , which results in the helicity-averaged effective coupling: Here, the last equality further assumes the vanishing initial momentum P 0 = 0. We will compute the integrated decay probability under this assumption in Sec.2.3 and in Appendix B.
Hereafter, we assume both the following conditions Physically, each of these conditions is satisfied when at least one of the following three conditions is met: when the interaction time T is apart enough from T in (or T out ) compared to the temporal width of the overlap √ σ t , which typically corresponds to the "bulk-like" case (Fig. 2, left); V when the "mean lifetime through bulk effect" Γ −1 V is much shorter than the temporal width of the overlap region √ σ t , which typically corresponds to the so-to-say "decay within wave-packet overlap" case (Fig. 2, right);

Overlap region
< l a t e x i t s h a 1 _ b a s e 6 4 = " U s x Z 2 P + 7 n 6 H f l J i p x w b L b P q j o Z E i g X e 4 9 r 7 N B O 0 / M 3 5 q U Y d 3 l H y K h 7 e y H B 3 f F q 9 H 6 a D a y P 7 q S b 7 o B P 0 h 1 r O Z v o p e e j c s 0 4 a c 9 3 c s y q 9 2 f y d s / z z f C 8 e 6 1 k u 4 e e n i x r s X L 1 + / v H L V 5 u 9 P n y S v k z d g 5 t s g 7 5 G r s A r Z I j d A n k e + J t + Q b 9 c f r P + y / n D 9 N 9 b 1 i R M c 8 w r J / a z / / h + S + O 5 S < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = "

Overlap region
< l a t e x i t s h a 1 _ b a s e 6 4 = " P z C u Z p 3 0 X B P S n 1 J J Z U 5 I l 7 y L s n o g e 4 g 9 V I t O S j 9 + + M X j + n s 7 q 5 M 3 j G + M P 0 x T z t a 9 c v n q U c W z d P + 1 l 8 6 O e q m M f 8 b K 3 C w 1 L y m J + z 1 f Q w l X r I M 7 e a R i p u s v w t c 1 L n c J j 9 j p D n c H o e y q / 6 u S z D r y + F b y n x r S X l N 5 X y m 0 t q o O J g a X H Q 8 b K + j 6 v 4 V t + z V T y q 7 8 8 q f q z i x S r e q + K 7 In state In state "Bulk-like" "Decay within wave-packet overlap" is hard to draw in the position space and is not shown here.The overlap region is determined by both the initial and final states as in Fig. 1.
when the deviation from the conservation of the shifted-energy, δω, is much larger than the inverse of the temporal width of the overlap 1/ √ σ t , namely the "violation of shifted-energy" case.
This assumption ( 27) is made for simplicity, and there is no obstacle to using the full form (23) in the numerical computation in principle, but the result would remain the same approximately because this is anyway satisfied in the ordinary bulk-like case as well as when anything interesting happens around the (in-)time-boundary.Under the assumption (27), the following asymptotic form is obtained [11]: where we have defined the "bulk window function" in which the sign function for a complex variable is +1 for ℜz > 0 or (ℜz = 0 and ℑz > 0), −1 for ℜz < 0 or (ℜz = 0 and ℑz < 0), Here and hereafter, ℜ and ℑ denote the real and imaginary parts, respectively.Equation ( 29) describes the ordinary "bulk contribution" for the quantum transition from the in to out states in the time period [T in , T out ].The second and third terms of Eq. ( 28) show the contributions near the in and out time boundaries T in and T out , respectively.More explicitly, 18 Because the contribution from the out-time-boundary T ≃ T out is suppressed by the extra dumping factor e −Γ V (T−T 0 ) in the differential probability (25), 19 it is safe to neglect the outtime-boundary contribution, and we can take with which the second line in Eq. ( 28) goes down to zero.With this limit, |G(T)| 2 reads where [GG] bdry (T) : [GG] intf (T) : The three functions [GG] bulk , [GG] bdry and [GG] intf describe the square of the bulk term, the square of the in-time-boundary term, and the interference between the bulk and in-boundary terms, respectively.With the approximation ( 27) and the limit (32), the differential probability (25) takes the simpler form and can be classified into the following three parts: with dP "type" where the argument "type" discriminates the three types of contributions. 18Precisely speaking, Eq. ( 31) is given except right at the boundary T = Tin + Γ V σ t 2 or T = Tout + Γ V σ t 2 , which is rather a peculiarity of how to define a boundary value and is out of our current interest. 19Here, we physically assume Tout − Tin ≫ Γ −1 V with T0 ≃ Tin; see also footnote 14.
where I bdry is written as an integration of a function of V − : in which The integral (46) will be evaluated numerically.

Interference contribution
Integrating the interference contribution in Eq. ( 37), we obtain where the definition of new parameters is as follows: The tilde denotes that the values are evaluated at the saddle point for the interference contribution.In Fig. 5, we have also plotted the P-factors for ρ 0 → π + π − and ρ + → π + π 0 by adopting the same formulas (41), ( 45) and (48) for a purpose of qualitative comparison between the decays with narrow phase spaces (ϕ, ψ, and Υ) and that with broad phase spaces (ρ), knowing that it is speculative whether we can still use the non-relativistic approximation. 20As expected, the difference between ρ 0 → π + π − and ρ + → π + π 0 is small since the magnitude of the isospin breaking is much smaller even for a smaller √ σ π .
4 Analysis of ratio of decay probabilities R V In this section, we discuss the ratio R V of decay probabilities for three vector mesons ϕ, ψ, and Υ in the wave-packet formalism.We compare each with the PDG result and find an agreement around a reasonable value of R 0 of the form factor (for the compositeness of V ).
In particular, the 9.5 σ discrepancy for ψ is dramatically ameliorated.We find that the effect of the form factor is significant in both the wave-packet and plane-wave formalisms.
Figure 6: The ratio comparing the decay rates of ϕ → K + K − to ϕ → K 0 K 0 is drawn as a function of R 0 for two fixed wave-packet sizes of the Kaons of √ σ K = 1 MeV −1 (left panel) and √ σ K = 0.1 MeV −1 (right panel).The experimental result is provided by the PDG [1] [shown in Eq. ( 2)].[shown in Eq. ( 2)].
• For the ψ decay in Fig. 7, the full wave-packet result in the red solid line can fit the PDG result around the form-factor size R 0 ≃ 2 × 10 −3 MeV −1 and 3 × 10 −3 MeV −1 for the wave-packet size of the decay product √ σ D = 1 MeV −1 and 0.01 MeV −1 , respectively.
• For the Υ decay in Fig. 8, the full wave-packet result in the red solid line can fit the PDG result around the form-factor size R 0 ≃ 2 × 10 −3 MeV −1 and 2 × 10 −3 MeV −1 for the wave-packet size of the decay product √ σ B = 1 MeV −1 and 0.01 MeV −1 , respectively.
• As discussed in Fig. 3, if √ σ P is sufficiently large, the bulk contribution becomes exponentially suppressed compared to the boundary one.In this regime, we may still formally evaluate the ratio between the (exponentially small) bulk contributions of P 0 and P − : where the exponent is from Eq. ( 43).This ratio becomes either exponentially large or small due to the mass difference between P + and P 0 , where the magnitude of the exponents is much greater than O(1).For example, we obtain σ P ≳ 10 MeV −2 and 100 MeV −2 if we estimate √ σ P to be larger than the smallest radius of an electron in atoms that interact with decay products of P , namely, the Bohr radius divided by a typical atomic number of the detector atoms, say, a B /Z ≃ 3 MeV −1 and 10 MeV −1 for lead and iron with Z = 82 and 26, respectively.As introduced, the experimental results of R ϕ , R ψ , and R Υ are around unity, and they disagree with R bulk V , both for ϕ and ψ.So, the R bulk V curves are completely out of the depicted ranges of the left panels of Figs. 6, 7, and 8.

Planewave analysis
For a comparison with the wave-packet results, we also show results with the plane-wave decay rate Γ plane (see Eq. (168) in Appendix C), taking into account the relativistic form factor (103).The resultant plane-wave ratio becomes where the parton-level contribution to the ratio is and the other factor is from the relativistic form factor (103) written in terms of the masses and R 0 .
Figure 9: The plane-wave ratio comparing the decay rates of ϕ → K + K − to ϕ → K 0 K 0 is drawn as a function of R 0 , where the captions "rel" and "Non-rel" mean the relativistic and non-relativistic results shown in Eqs. ( 61) and (63), respectively.For comparison, we also show the wave-packet results for two fixed wave-packet sizes of the Kaons of √ σ K = 1 MeV −1 (left panel) and √ σ K = 0.1 MeV −1 (right panel).In both panels, the plane-wave results are the same since they are independent of σ K .The experimental result, shown in Eq. ( 2), is provided by the PDG [1].
For another comparison, we will also show analyses using its non-relativistic approximated form: with where "⇒" represents the operation of taking the non-relativistic approximation and ≈ denotes equality under the non-relativistic approximation.The contributions from the form factor are not canceled out in R plane

V
. 21 Note that the ratio (63) can be obtained from the wave-packet counterpart by taking the limits Γ V → 0 and σ P → ∞ in Γ V P V →P P ; see Appendix D for details.

Plane-wave results
We provide comments on the plane-wave results shown in Figs. 9, 10, and 11 below: • For all of the vector mesons, ϕ, ψ, and Υ, the parton-level ratios (4) (under the isospinsymmetric limit for the couplings, 22 without taking into account the form factor) are disfavored with the PDG's central values at the level of 2.1 σ, 9.5 σ, and 0.32 σ, respectively.(right panel).The other conventions are the same as in Fig. 9.
• On the other hand, when the form-factor effect is included, which is compulsory since the vector mesons are composite particles, we can see agreements with the PDG's results.It suggests the importance of the form factor in addressing the ratio, where its effect is not fully canceled.Also, we can confirm that the non-relativistic results approximate their relativistic counterparts well for the current system.
• We can find appropriate ranges of the form-factor parameter R 0 , where theoretical predictions agree with the PDG's results for both the full wave-packet and plane-wave curves.For each vector meson, the favored regions of R 0 for the wave packet and the plane wave are close to each other but different.
The calculation based on the plane wave works successfully, even though the presumption of free plane waves characterizing initial and final states is, at most, a viable approximation.The wave packet-based calculation provides a comprehensive approach, accounting for all aspects of the quantum nature inherent in the initial and final states, thereby enhancing its reliability.
It would be important to precisely discuss the theoretically valid region of R 0 , which depends on many details on the strong interaction.We leave this point for future research.Note that we also briefly consider the isospin violation on the ρ system, where the result is separately available in Appendix E since it might be out of our main interest.

Constraint from the shape of vector-meson resonances
In general, it is expected that the resonance shape of V is modified by the inclusion of the wave packet effects.That is, vector mesons, produced as resonances in electron-positron colliders, are subjected to a shape-fitting process.This section addresses the constraint on the wave-packet size of pseudoscalar mesons through the resonance shape of the process e − e + → V → P P .Sufficiently precise resonance data from experiments is available for ϕ and ψ, facilitating this purpose.However, detailed resonance data for Υ is currently unavailable.Consequently, our focus is maintained on the instances of ϕ and ψ.Our analysis in this section is meant to be a brief consistency check, assuming the factorization of the production and decay processes of V in both the wave-packet and plane-wave formalisms, and hence is confined to data around the peak.

Invariant mass distribution of decaying vector-meson wave packet
First, we summarize the invariant mass distribution of the decaying vector-meson when the Gaussian wave packet describes the decaying state.
We define a Lorentz-invariant mass squared M 2 for the pair of pseudo-scalars in the final state: where V − is the magnitude of V − := V 1 − V 2 with V a = P a /E a for a = 1, 2 (see Eq. ( 108) in Appendix B).We will use which results in where we have approximated that V decays at rest in the last step.
It is straightforward to derive the following forms after integrating Eq. ( 38) over the final state phase space, except for V − , under the current non-relativistic approximation, which is easily rewritten as the invariant mass distribution by use of Eq. (67): where dP bulk dP bdry Here, we consider the distribution of M instead of M 2 due to the convenience of comparing the wave-packet shape with the non-relativistic Breit-Wigner (BW) shape, which is the wellknown resonant shape for the decaying plane wave with the decay rate Γ V ; see the next subsection. 23 We note that, under the current setup T out → ∞, the factor N 2 V in C V →P P (recall Eq. ( 40)) can be determined by the normalization that is obtained after integrating over M ; see Eqs. ( 41), (45), and (48). 23Experimentally, one may perform a precision experiment by measuring the ratio per each bin ∆M near the resonance, in principle:

Breit-Wigner shape
For the plane-wave calculation, it is well-known that the non-relativistic Breit-Wigner distribution nicely describes the shape of a narrow resonance, 24 where m res , E, and Γ are the resonant mass, the total energy in the center-of-the-mass frame, and the total width of an intermediate resonant particle, respectively.Note that Since we used the non-relativistic approximation, we are adopting the non-relativistic Breit-Wigner shape (73) for comparison.

Method of analyzing resonant shape
We assume the following factorization for the resonant production, where the cross-section of the resonant production of V and its subsequent decay into P and P , σ e − e + →V →P P , can be factorized in the wave-packet (WP) and plane-wave (PW) formalisms, respectively as σ PW-FF e − e + →V →P P (M ) = N PW-FF e − e + →V f NR-BW (M ) where we consider the two cases for PW with and without the form factor (FF); the two cases are discriminated by the short-hand notations "PW-FF" and "PW-Parton".For the form-factor part of (77), we used the relation in Eq. (66) to convert V − to M .
N WP e − e + →V , N PW-Parton e − e + →V , and N PW-FF e − e + →V possess the mass dimension of minus one and describe the factorized production part e − e + → V via the e − e + collision at the center-of-themass energy M .Here, we take these three factors to be independent of M since the primal structure of the resonance is in dP V →P P /dM or f NR-BW , and we use only the data points near the peak of a resonance. 25We will take m res and Γ for f NR-BW (M ) in Eq. ( 73) as m V 24 The relativistic Breit-Wigner distribution takes which is not used for the calculation.Mandelstam's variable s is equal to E 2 . 25As is widely known, under the narrow-width approximation in the plane-wave calculation at the resonant peak M = mres, we can derive the factorized form explicitly: σ PW e − e + →V →P P (M ) ≃ σ PW e − e + →V Br V → P P ; refer to, e.g., Chapter 16 of [28].And at this point, N PW e − e + →V is determined as Also, we note that the width-to-mass ratios of the vector mesons take Γ ϕ /m ϕ ≃ 0.42% and Γ ψ /m ψ ≃ 0.72%, where the adaptation of the narrow-width approximation is justified.
and Γ V , respectively; see also Eq. ( 74).The actual analysis for e − e + → V → P P will be done in the following manner: • We focus on the values of the experimentally-given cross sections only around the resonant peak, namely in [m V − Γ V /2, m V + Γ V /2] since the factorized forms in Eqs. ( 75), (76), and (77) may work only around the peak.Here, we will adopt the PDG values for m V and Γ V [1].
• In the analysis, we fix the values of Γ V and m P as confirmed by the PDG group [1], while we treat m V as an unfixed parameter and will determine it through our statistical fit.The isospin-symmetric coupling g V and the wave-function renormalization factor for V , N V are taken as unity since it can be absorbed into the factor N e − e + →V .Furthermore, for simplicity, we focus on T in = T 0 , where the exponential decay factor in C V →P P in Eq. ( 40) does not work.
• Under the current scheme, σ WP e − e + →V →P P has six parameters {N WP e − e + →V , m V , Γ V , m P , R 0 , σ P }, σ PW-FF e − e + →V →P P has five parameters {N PW-FF e − e + →V , m V , Γ V , m P , R 0 }, and σ PW-Parton e − e + →V →P P has three parameters {N PW-Parton e − e + →V , m V , Γ V }, respectively.We will determine them through statistical analysis.We remind ourselves that the vector-meson wavepacket size σ V does not appear in σ WP e − e + →V →P P under the saddle-point approximation.

Result of ϕ
In Ref. [29], the latest result of the resonant shape of ϕ through e − e + → ϕ → K + K − measured with the CMD-3 detector in the center-of-mass energy range 1010-1060 MeV was reported, where the Born cross sections of e − e + → ϕ → K + K − around the resonance are available in Table I of [29].According to our guideline, we adopt the seven data points from 1018.0 MeV to 1021.3 MeV and adopt the χ 2 functions: where i discriminates the seven points of M where experimental data is available; σ exp As examples, we show the fitted distributions for the two sets of the fixed parameters √ σ K = 10 MeV −1 and R 0 = 0.0015 MeV −1 for the left panel of Fig. 12,26 √ σ K = 10 MeV −1 and R 0 = 0.01 MeV −1 for the right panel of Fig. 12, where the two remaining parameters {N e − e + →ϕ , m ϕ } take the best-fit values, and the values of the χ 2 over the degrees of freedoms (DOFs), which is currently five, at the best-fit points are calculated as We comment on the difference between the plane-wave resonant shapes with and without the form factor.Without the factor, the shape obeys the Breit-Wigner distribution (73) and is symmetric under the reflection around the peak (M = m ϕ ), while taking into account it, the resonant shape becomes asymmetric under the reflection around the peak.The magnitude of the asymmetry is governed by the part R 2 0 M 2 − 4m 2 K + of the form factor. So, for a greater R 0 , a more significant asymmetry will be realized, as observed in Fig. 12.
Here, we comment on the origin of the "over-5σ" values of χ 2 /(DOFs): this is because the resolution of the experimental results near the peak is very high, and the current simple scheme for σ e − e + →V →P P in Eqs.(75), (76) and (77) is not enough for discussing statistical significance precisely.On the other hand, however, we are able to discuss the relative significance between the wave-packet and plane-wave results.According to Eq. (80), the shape of the wave-packet resonant distribution is at least as good as that of the plane-wave resonant distribution at the focused parameter point, where we conclude that the wave-packet result at the first parameter point (for the left panel of Fig. 12) is consistent with the experiment.Note that at the first parameter point, R 0 is taken as a typical value in the current scheme of the form factor (see Appendix A), and a wave packet with a greater size looks similar to the plane wave.
We also see the significance of the wave-packet results over a broad range of √ σ K under R 0 = 0.0015 MeV −1 .In Fig. 13, we plot the "minimized" χ 2 /(DOFs) defined by which measures the statistical significance for √ σ K .We do not consider the PW-FF case since no sizable difference is generated when R 0 = 0.0015 MeV −1 , as shown in the left panel of Fig. 12, and the form-factor part does not depend on √ σ K .Under the simple guideline that a wave-packet result is at least as good as the ordinary plane-wave one, from Fig. 13, we can put the lower bound on √ σ K as From Fig. 14, when R 0 is large as ∼ 10 −2 MeV −1 , the resonant distribution of the wave packet becomes identical with that of the plane-wave without taking into account the form factor.Meanwhile, when R 0 ∼ 10 −3 MeV −1 , where this size is favored with the agreement in R ψ , we observe the deviation from the BW shape in the wave-packet distribution.Note that all three kinds of distributions agree with the experimental data for the larger and smaller R 0 .We comment on the large asymmetry observed in the right panel of Fig. 14, namely the large deviation of "BW with FF" in the low M range.As mentioned in the previous subsection, the asymmetry under the reflection around the peak originates from the parts R 2 0 M 2 − 4m 2 D + and R 2 0 M 2 − 4m 2 D 0 of the form factors.The realized asymmetry in R 0 = 0.01 MeV −1 becomes extensive when M is less than the range used for the statistical fit, so this case is considered to be disfavored even though the limited part near the resonant peak is fitted to the experimental results well.
To clarify the experimentally-valid range for √ σ D , we see the curve of the "minimized" even though the wave-packet shape does not exceed the BW shape in the goodness of fit.To summarize, within the current scheme for the production cross section, no significant bounds on √ σ D are imposed.This is because, as recognized from Fig. 14, the experimental results still have sizable errors for the ψ's resonant shape.

Summary and discussion
In this paper, we have discussed the long-standing anomaly in the ratio of the decay rates of the vector mesons ϕ and ψ, namely, R ϕ = Γ(ϕ → K + K − ) /Γ ϕ → K 0 L K 0 S and R ψ = Γ(ψ → D + D − ) /Γ ψ → D 0 D 0 , where the strong interaction causes the decay channels, and they measure isospin breakings.If we estimate their theoretical values in the plane-wave formalism without considering the effects originating from the composite nature of the initialstate vector mesons, they are disfavored with the PDG's central values at the level of 2.1 σ and 9.5 σ.In particular, there has been no explanation for the latter 9.5 σ anomaly so far.
The decay channels that we focus on are near the mass thresholds, where the velocities in the final state are small, and hence the localization of the overlap of the wave packets is more significant.Here, we fully take into account such effects in the Gaussian-wave-packet formalism.We carefully clarified the properties of one-to-two-body non-relativistic quantum transitions between normalizable physical states described by Gaussian wave packets under the presence of the decaying nature of the initial state, which is a full-fledged calculation taking into account the essences that are missing in the plane-wave calculations.
The result shows agreement with the PDG's combined results within ∼ 1 σ confidence level.We conclude that the long-standing anomalies in R ϕ and R ψ are resolved.
In the calculation, the above-mentioned compositeness has been described by the form factor.The agreement is achieved when we appropriately take the form-factor parameter at around the physically reasonable value R 0 ∼ (500 MeV) −1 .
We also analyzed and made a comment on the bb-vector-meson counterpart Υ, namely R Υ = Γ(Υ → B + B − ) /Γ Υ → B 0 B 0 , where the plane-wave calculation without considering the above-mentioned composite nature already agrees at the 0.32 σ level with the corresponding PDG result due to the smallness of the mass difference between B ± and B 0 .The wave-packet result agrees well with the PDG result around the same value of R 0 .
We mention that the same form factors can be formally multiplied on the ratio of the plane-wave decay rates in order to partially take into account the wave-packet effects, though the wave-packet approach is more comprehensive in describing quantum transitions.By doing so, around the same value of R 0 , the plane-wave results can also be made to agree with the PDG ones.
In general, the shape of a wave-packet resonance deviates from the Briet-Wigner shape, where the magnitude of the deviation depends on the size of wave packets.For ϕ and ψ, experimental data is available, and we put constraints on the size.We found that when the size of the wave packets is small, the derivation from the Briet-Wigner shape tends to be sizable.Both for ϕ and ψ, wide ranges of the wave-packet size are consistent with the experimental data.
In the decay channels of the vector mesons, the non-relativistic approximation works fine, which simplifies the integrations in the S-matrices and the final-state phase spaces in the wavepacket formalism.Many other quantum transitions in high-energy physics are relativistic, and it is worthwhile to establish the general method to perform such integrations without relying on the non-relativistic approximation.Also, analyzing resonant productions precisely requires the full transition probabilities, including production parts.Doing more dedicated analyses on resonant shapes will be another important task.
Here, a vector meson V QQ decays into two light pseudo-scalar mesons P (Qq) and P qQ .We approximate the mass and momentum for each psuedoscalar P (P ) by those of the constituent quark Q (Q): m P ≃ m Q and p P ≃ p 1 (p P ≃ p 2 ), respectively.In this paper, we focus on the situation where the masses of the two pseudo-scalar mesons are almost the same (due to the approximated flavor-isospin SU (2) symmetry), and the mass relation is near the decay threshold, m V ≈ 2 m P . (101) Therefore, we can treat the process as a non-relativistic one, and thus, we conclude that where V 1 and V 2 are the (non-relativistic) velocities of P and P , respectively.Finally, we reach the spin-independent dimensionless function suitable for our purpose, This is the form factor shown in Eq. ( 15) for the matrix elements of the meson decays (with m V ≈ 2 m P ) in the rest system.Now we estimate a typical value of the parameter R 0 in Eq. ( 98).The quarkonium potential can be approximated by a sum of the confining linear potential and the QCD Coulomb potential where a s is known as a s = 1.95 GeV −1 [40] and α s is the QCD fine structure constant.For the domain where r ≲ r c := a s √ α s ≃ 1.5 GeV −1 ,30 the wave function can be approximated by the Coulomb form (98).One can estimate r by equating the potential and kinetic energies, V (r) ∼ K Q , where K Q ∼ m V − 2m Q = O(10) MeV.Since K Q is much smaller than the typical energy scale a −1 s = 0.5 GeV of the potential, the typical QQ distance can be estimated by equating two terms in the right-hand side of Eq. (106): r ∼ r c .The use of Coulomb wave function (98) is marginally justified, which suffices for our current consideration.See e.g.Ref. [42] for further refinement.

B.2 Bulk contribution
We compute the bulk contribution in Eq. (37).First, we perform the position integrals +∞ −∞ d 3 X 1 d 3 X 2 .As in Ref. [11], we obtain where y 0 becomes a flat direction under the (unphysical) no-decay limit Γ V → 0 (as considered in [11]), while the other five directions are not flat directions irrespective of Γ V : where we also used Eq.(31) and took the limit T out → ∞.The range of the integration is given by the bulk window function W (T) in Eq. (31).After the position integrations, we obtain where the damping factors σt π e −σt(δω) 2 and σs π 3/2 e −σs(δP ) 2 provide the approximate conservation for the mean energy and momentum, respectively. 31So far the expression (123) does not assume P 0 = 0 nor the non-relativistic approximation given in Sec.B.1.Next, we perform the momentum integrals under the saddle-point approximation with P 0 = 0 in the non-relativistic approximation given in Sec.B.1: where the factor 1/8 = 1/2 3 is from the Jacobian and the exponent is We see that Thereby, The stationary point (V s + , V s − ) that satisfies ∂F bulk (V s + , V s − ) ∂V + = 0 and ∂F bulk (V s + , V s − ) ∂V − = 0 (131) is found to be There is no V − in the exponent, and we will perform the numerical computation for the V − integral.On the other hand, the saddle point of V + is located at V + = 0.With it in mind, since the other part can be obtained by taking complex conjugation.At first, we perform the square completion of the T part: where in the last fine, we changed the variable to T ′ := T − Γ V σ t /2 and took the limit T out → ∞.
In order to use the analytic formula for σ t > 0 and α ∈ C, 32 where Ei(z) is the exponential integral function defined by the principal value of we add an extra term in the denominator of the integrand of I intf such that which would underestimate the integral to some extent.Now, we reach the following analytic form with 32 One of the necessary conditions for this relation is "(Tin + α ̸ ∈ R) & (ℑ(Tin) ̸ = ℑ(α)) & (ℜ(Tin) ≥ ℜ(α))".In our case, the set of these conditions is manifestly fulfilled since α corresponds to "Tin − ΓV σt/2 + iσtδω".
Stationary points (V s + , V s − ) are defined by and we find one stationary point in the kinetic region (V + ≃ 0 and V − > 0), Note that
t e x i t s h a 1 _ b a s e 6 4 = " d E i 9 S R w S q G d A I 2 s Q g I S p + w d 5/ B Q = " > A A A u O n i c t V r L b x t F G J + W V y m P p i A h J C 4 O U e h D S d l U p S C k S m V 5 Z Q + o a e y 2 k e L U s r 3 r 2 M r a u + y u 3 a R u D l w 4 8 A 9 w 4 A Q S B w R X O H H j w j / A o V d u V Y + t x I U D 3 3 w 7 s + t 9 z G N t k S j x 7 u z 8 f t 9 7 Z j y z H d 8 d h J F h P D x x 8 p l n n 3 v + h V M v n n 7 p 5 V d e P b N 0 9 r X b o T c O u s 6 t r u d 6 w U 6 n H T r u Y O T c i g a R 6 + z 4 g d M e d l z n T u f g Y / r 8 z s Q J w o E 3 a k R H v r M 3 b O + P B r 1 B t x 1 B U 2 u p 1 u x 4 r h 0 e D e G j u T N o G c 3 l p u v 0 o v N R M x j s 9 6 M L r a U V 4 5 K B P 7 W Z i 6 u G 8 d 7 G R m 2 D t a w Q 9 r P l n X 3 j K 9 I k N v F I l 4 z J k D h k R C K 4 d k m b h P C 7 S z a I Q X x o 2 y N T a A v g a o D P H X J M T g N 2 D L 0 c 6 N G G1 g P 4 v w 9 3 u 6 x 1 B P e U M 0 R 0 F 6 S 4 8 B c A s k Z W j b + M n 4 w n x p / G z 8 Y j 4 1 8 h 1 a e y 0 k e L U s r 3 r 2 M r a u + y u 3 a R u D l w 4 8 A 9 w 4 A Q S B w R X O H H j w j / A o V d u V Y + t x I U D 3 3 w 7 s + t 9 z G N t 4 S j e 2 d n 5 f t / 7 m / H M d n x 3 E E a G 8 f D U 6 W e e f e 7 5 F 8 6 8 e P a l l 1 9 5 9 d z S + d d u h 9 4 4 6 D o 7 a e y 2 k e L U s r 3 r 2 M r a u + y u 3 a R u D l w 4 8 A 9 w 4 A Q S B w R X O H H j w j / A o V d u V Y + t x I U D 3 3 w 7 s + t 9 z G N t 4 S j e 2 d n 5 f t / 7 m / H M d n x 3 E E a G 8 f D U 6 W e e f e 7 5 F 8 6 8 e P a l l 1 9 5 9 d z S + d d u h 9 4 4 6 D q 3 u p 7 r B b u d d u i 4 g 5 F z K x p E r r P r B 0 5 7 2 H G d O 5 3 D j u H D g m 8 8 z 9 t r e e X h X J E r W H s / v 9 7 1 n Z m f c C r x e F B v G o 1 O n n 3 j y q a e f O f P s 2 e e e f + H F c + c v v H Q r 8 o d h 2 9 1 p + 5 4 f 7 r b s y P V 6 A 3 c n 7 s W e u x u E r t 1 v e e 7 t 1 t E H 4 y I c J Z O r m P m N 2 u o p k T b 2 o T P y P t 7 T E e o Q Z 6 o x P u u x C l y A H k 2 M g V N 4 I v d F M o K p J V y W y L i s l E K j k 4 + T v P + A j S / 0 y s + N z U 3 4 l W N j H E 0 n o 6 T y c c A y b 5 D z g 4 e j e E z e Z H a H a G 0 X W t 5 S a B C k 2 k 8 y y / P j M 5 w p p 8 n f K 8

s g 5 I
s i a n u w L 6 d S Z m r c / E a i l Y r R l 1 N R W 6 m j N q K + e 1 p L w 6 z O 1 K 3 H e w v w 4 / 9 4 h T y S e X K n l F z D 1 N 9 0 u a m i c j D / f 5 k j S b a y R Z R 2 W e 1 x u R x O z 1 u d g t B b s 1 p + 5 m y i 4 b w z

s 9 S
N b J a L 6 8 a w S x y p R r B K 9 K r G r M j 5 n e S E f n 8 t Z o T 8 + y 2 T I q 1 N / f J b J k F e o / v j M Z e i v r J Y E E n X G Z 5 m 0 6 e O z T J p O J u j O 0 7 P l g O 4 M P V v 0 d e f m 2 e I + 2 6 x c 1 R K 5 L P G c L J I l 3 8 n k s z / f y d T r S N z p U 8 h y l P C c n z 5 l L X i y w L / 6 f 7 I s F 9 s V z 2 F f 9 P T 6 T n f 4 m n / I c a K S V 0 G D 1 x E 8 A d a q + M Z G b R f x k N c r H L 7 q H m J 0 g 0 h P Y C L D U S n 6 a m P S j 9 i e j e v m U m J 8 I n x y c X + C v R t f E F 7 f W V l b X V 6 7 c v L J w 3 W T v T 5 8 h r 5 H X Y e Z b J e + S 6 7 A K 2 S I 7 q O 2 X 5 B v y 7 f L D 5 V + W f 1 3 + L e l 6 + h T D v E x y P 8 t / / A c J T v K 4 < / l a t e x i t > T < l a t e x i t s h a 1 _ b a s e 6 4 = " P z C u Z p 3 0 K r D 5 u M 7 m O z f b o B 1 0 8 a X b b R s q m 0 e 7 a m 1 3 F u 3 Z t 7 z b t N g e u / A M c e g L R A + q f w Y U r B y T K l R P i W C Q u H P j m 8 4 y 9 f s z D u y J R s v Z 4 f r / v P T M 7 4 7 b v 9 s P I M J 6 e O P n c 8 y + 8 + N K p l 0 + / 8 u p r r 5 8 5 e + 6 N W 6 E 3 C j r O z Y 7 n e s F O u x U 6 b n / o 3 I z 6 k e v s + I H T G r R d 5 3 b 7 8 F P 6 / P b Y C c K + N 2 x E 9 3 1 n b 9 A 6 G P a 7 / U 4 r g q b 9 s 2 e a 4 d 0 g m j T D / s G g t R 8 d 7 5 9 d M F a u b G y s r 1 + p G S v r a 6 u r l + m F 8 e H a h r F e W 1 0 x 8 G e B s J 8 t 7 9 x b X 5 M m s 5 D 5 / t G O v 4 + 2 e + e b v + j x I 1 g M + I 9 F K U V O e I d M k G 6 t o H 3 X 1 s o V Z 0 Y v z 4 w b f P 6 h 9 v L 0 7 e M 7 4 3 / g L 9 v z O e G j + B B c P x 3 5 3 H N 5 z t R 8 A u 1 i g E 1 g G 0 U t / d Q y 8 e o g 8 X h R r Z Z A z 9 f O a / A / B c C + 5 6 i D 3 S x N p g 0 w B 6 i x h E 2 l b p O x 1 Z c a 8 2 M h 7 C d a R k G y S 5 K L O S 9 u 1 j j G S 5 2 A Y + D + 4 H m L + H 0 O c q Y A L o 7 Z C l k q d D r I Q 2 P A 3 g z 8 7 0 D l F 7 n 3 H T i D 6 A z 6 v g 4 x Z 4 g D 6 l f u 5 B r v q I p w x d 1 E 5 s L 8 X S H L u P u h w p e 3 v o l x b k h 0 s n G S 9 x + y 8 Y V e e Z m x u Q m / c m y E o + l 0 l F Q + 9 l n m D T N + c H E U j 8 j 7 z O 4 A r e 1 B y 0 W F B n 6 i / T S z P D / u 4 k x Z J n + 3 I H 9 P G d N 2 K d P D A t N D B Z N I p y b c 5 b m a y i z o 5 j x 8 X s P L M r 6 t 0 r i 1 k

4 e 5 U
4 n 7 D v b X 4 e c e s S v 5 5 E I l r 4 i 5 y 3 S / o K l 5 P P J w n y 9 J s 7 l G 4 n V U 6 n m 9 E U n M X p + L 3 V K w W 3 P q b i b s s j E s 5 a / N a I V c j i W U k 5 e h M 2 r 1 h F m U H b d 6 F c c t M W 9 9 J l 5 L y W v N q K + p 1 N e c U W M V c 7 5 O e 9 r j F 6 9 S z q 1 b p 7 0 k P / R r V S y j v F q r y L C U M s p r t p o d p t I O e e V W s 0 r 4 O a 0 j c 6 V P J s p T w 7 I 8 + e S 1 4 k s C / 6 n + y L B P Z F c 9 h X / T 0 + k 5 3+ x p / y H G g k l d B g 9 c R P A H W q v j G V m 3 n 8 d D X K x y + 6 h 5 i e I N I T 2 B C w 1 E p + m h j 3 o / b H o 3 r x l J i f C N O 3 p v m r 0 T X x x a 2 1 l d X 1 l c s 3 L i 9 c M 9 n 7 0 6 f I O + R d m P l W y U f k G q x C t s h N Q t + j f k R + I I + X n y z / s v z b 8 u 9 x 1 5 M n G O Z N k v l Z / u M / E 9 f 1 U Q = = < / l a t e x i t >p t < l a t e x i t s h a 1 _ b a s e 6 4 = " x d w J l j x a o N b L B 7 D 4 e x t 4 8 c Y z a X b b R s q m 0 e 7 a m 1 3 F u 3 Z t 7 z b t N g e u / A M c e g L R A + q f w Y U r B y T K l R P i W C Q u H P j m 8 4 y 9 f s z D u y J R s v Z 4 f r / v P T M 7 4 7 b v 9 s P I M J 6 e O P n c 8 y + 8 + N K p l 0 + / 8 u p r r 5 8 5 e + 6 N W 6 E 3 C j r O z Y 7 n e s F O u x U 6 b n / o 3 I z 6 k e v s + I H T G r R d 5 3 b 7 8 F P 6 / P b Y C c K + N 2 x E 9 3 1 n b 9 A 6 G P a 7 / U 4 r g q b 9 s 2 e a 4 d 0 g m j T D / s G g t R 8 e 7 5 9 d M F a u b G y s r 1 + p G S v r a 6 u r l + m F 8 e H a h r F e W 1 0 x 8 G e B s J 8 t 7 9 x b X 5 M m s 5 D 5 / t G O v 4 + 2 e + e b v + j x I 1 g M + I 9 F K U V O e I d M k G 6 t o H 3 X 1 s o V Z 0 Y v z 4 w b f P 6 h 9 v L 0 7 e M 7 4 3 / g L 9 v z O e G j + B B c P x 3 5 3 H N 5 z t R 8 A u 1 i g E 1 g G 0 U t / d Q y 8 e o g 8 X h R r Z Z A z 9 f O a / A / B c C + 5 6 i D 3 S x N p g 0 w B 6 i x h E 2 l b p O x 1 Z c a 8 2 M h 7 C d a R k G y S 5 K L O S 9 u 1 j j G S 5 2 A Y + D + 4 H m L + H 0 O c q Y A L o 7 Z C l k q d D r I Q 2 P A 3 g z 8 7 0 D l F 7 n 3 H T i D 6 A z 6 v g 4 x Z 4 g D 6 l f u 5 B r v q I p w x d 1 E 5 s L 8 X S H L u P u h w p e 3 v o l x b k h 0 s n G S 9 x + y 8 Y V e e Z m x u Q m / c m y E o + l 0 l F Q + 9 l n m D T N + c H E U j 8 j 7 z O 4 A r e 1 B y 0 W F B n 6 i / T S z P D / u 4 k x Z J n + 3 I H 9 P G d N 2 K d P D A t N D B Z N I p y b c 5 b m a y i z o 5 j x 8 X s P L M r 6 t 0 r i 1 k

4 e 5 U
4 n 7 D v b X 4 e c e s S v 5 5 E I l r 4 i 5 y 3 S / o K l 5 P P J w n y 9 J s 7 l G 4 n V U 6 n m 9 E U n M X p + L 3 V K w W 3 P q b i b s s j E s 5 a / N a I V c j i W U k 5 e h M 2 r 1 h F m U H b d 6 F c c t M W 9 9 J l 5 L y W v N q K + p 1 N e c U W M V c 7 5 O e 9 r j F 6 9 S z q 1 b p 7 0 k P / R r V S y j v F q r y L C U M s p r t p o d p t I O e e V W s 0 7 j w 9 W w 7 o z t C z R V 9 3 b p 4 t 7 r P N y l U t k c s S z 8 k i W f K d T D 7 7 8 5 1 M n d 3 c c e 4 d A i 5 V d R 4 1 K O z F D / A 8 n + / C 6 5 w a 8 7 1 T f u f i f q 5 6 R 3 6 U I E c M m + 6 z y k 6 G O M r G 3 s M K O 7 T j 5 P w t 9 l K I u 7 w D Z F S 9 v R D h 7 n g 5 + i D J h r i P 6 q S b 7 o C P k h 1 r M R v v p e a j c v U 4 a c 9 3 M 8 y y 9 2 e y d k / z T f C 8 e 6 l g u 2 p 3 M z j p 6 t R P n 7 R P c T 0 B J G e w I a A p V b y 0 8 S 4 H 7 U / H t W L p 8 T 8 R P j k 4 P w C f z W 6 J r 6 4 t b a y + t 7 K + v b 6 w n W T v T 9 9 h r x B 3 o S Z b 5 V s k O u w C t k i N 0 G e Q 7 4 m 3 5 B v l x 8 u / 7 L 8 a P m 3 u O v p U w z z K s n 8 L P / + H 1 g e 7 p Q = < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = " X M B j e B L 1 F 7 C d r 2 y a H r i P Z p l z j 9 0 a i e P y X m J 8 L H j b l 5 / m p 0 R X x x e 3 V 5 5 e 3 l t V t r 8 z d M 9 v 7 0 G f I 6 e Q N m v h W y T m 7 A K m S T b I O 8 z 8 h D 8 j 3 5 Y e n H p V + X f l v 6 P e p 6 + h T D v E x S P 0 t / / Q d 9 c v j 3 < / l a t e x i t > r d t p G w a 7 a 6 9 2 V W 8 a 2 N 7 0 7 T b H L h y 4 4 R E T 0 X i g P g z O M A J c e H Q P w F x L B I X D n z z z Y y 9 t n c e 3 h W J k r X H 8 / t 9 7 5 n Z G b d D v x 8 n l v X 4 1 F N P P / P s c 8 + f f u H M i y + 9 / M r Z c + d f v R U H o 6 j j 3 e w E f h D t t F u x 5 / e H 3 s 2 k n / j e T h h 5 r U H b 9 2 6 3 D z + m z 2 8 f e V H c D 4 a N 5 F 7 o 7 Q 1 a B 8 N + t 9 9 p J d B 0 u 9 k K w y g 4 3 j+ 3 Y K 1 Y 6 1 e u r K 7 V r J X L G x 9 s W P T C s t Y 3 V i / X V u G C / i w Q / r M Vn H / 9 K 9 I k L g l I h 4 z I g H h k S B K 4 9 k m L x P C 7 S 1 a J R U J o 2 y N j a I v g q o / P P X J C z g B 2 B L 0 8 6 N G C 1 k P 4 f w B 3 u 7 x 1 C P e U M 0 Z 0 B 6 T 4 8 B c B s k Y W r T + s H 6 0 n 1 q / W T 9 a f 1 r 9 S r j F y U F 3 u w W e b Y b 1 w / + z X b 9 T / 0 a I G 8 J m Q X o Z S 6 p y Q L r m C u v Z B 9 x B b q B U d h j + 6 / + 2 T + o f b i + O 3 r e + t v 0 D / R 9 Z j 6 2 e w Y H j 0 d + e H G 9 7 2 Q 2 C X a x Q D 6 w B a q e / u o h c P 0 Y e L U o 1 c c g T 9 Q u 6 / A / B c C + 5 6 i D 0 2 x L p g 0 w B 6 y x h k 2 l b p O x l Z e a 8 2 M h 7 C d a J l G 6 S 5 q L K S 9 u 1 j j F S 5 2 A a + A O 4 H m L + H 0 O c q Y C L o 7 Z G l K U + H W A l t e B r B n 5 v r H a P 2 I e e m E b 0

7 4 9 P
R + 2 k 2 s D 6 6 k 2 6 6 A z 5 K d 6 z l b K K X n o / K N e O k P d / L M a v e n 8 n b P c k 3 x v P u p Z L l J o y T m q o 4 M y 1 N 9 T R l L n p B H / 8 4 r b A 8 b z z B G n K u G u h a g 7 s a v r U W 4 T X N U B f k d V B T + k a H T p 6 r l e f m 5 L l z y U s k 9 i X / k 3 2 J x L 5 k D v u q v 8 d n 8 9 N f 9 q n r d t p G w a 7 a 6 9 2 V W 8 a 2 N 7 0 7 T b H L h y 4 4 R E T 0 X i g P g z O M A J c e H Q P w F x L B I X D n z z z Y y 9 t n c e 3 h W J k r X H 8 / t 9 7 5 n Z G b d D v x 8 n l v X 4 1 F N P P / P s c 8 + f f u HM i y + 9 / M r Z c + d f v R U H o 6 j j 3 e w E f h D t t F u x 5 / e H 3 s 2 k n / j e T h h 5 r U H b 9 2 6 3 D z + m z 2 8 f e V H c D 4 a N 5 F 7 o 7 Q 1 a B 8 N + t 9 9 p J d B 0 u 9 k K w y g 4 3 j + 3 Y K 1 Y 6 1 e u r K 7 V r J X L G x 9 s W P T C s t Y 3 V i / X V u G C / i w Q / r M V n H / 9 K 9 I k L g l I h 4 z I g H h k S B K 4 9 k m L x P C 7 S 1 a J R U J o 2 y N j a I v g q o / P P X J C z g B 2 B L 0 8 6 N G C 1 k P 4 f w B 3 u 7 x 1 C P e U M 0 Z 0 B 6 T 4 8 B c B s k Y W r T + s H 6 0 n 1 q / W T 9 a f 1 r 9 S r j F y U F 3 u w W e b Y b 1 w / + z X b 9 T / 0 a I G 8 J m Q X o Z S 6 p y Q L r m C u v Z B 9 x B b q B U d h j + 6 / + 2 T + o f b i + O 3 r e + t v 0 D / R9 Z j 6 2 e w Y H j 0 d + e H G 9 7 2 Q 2 C X a x Q D 6 w B a q e / u o h c P 0 Y e L U o 1 c c g T 9 Q u 6 / A / B c C + 5 6 i D 0 2 x L p g 0 w B 6 y x h k 2 l b p O x l Z e a 8 2 M h 7 C d a J l G 6 S 5 q L K S 9 u 1 j j F S 5 2 A a + A O 4 H m L + H 0 O c q Y C L o 7 Z G l K U + H W A l t e B r B n 5 v r H a P 2 I e e m E b 0

Figure 1 :
Figure 1: Schematic figure for the finite wave-packet process (left) and the infinite plane-wave process (right), without taking into account the decay width Γ V .In the left, we have shown the time of intersection T; the spatial and temporal sizes of the overlap √ σ s and √ σ t , respectively; the center of wave packets Ξ A (A = 0, 1, 2); and the initial and final times of the scattering T in and T out , respectively.Also, the bulk T in ≪ t ≪ T out , in-time-boundary (|t − T in | ≲ √ σ t ), and out-time-boundary (|T out − t| ≲ √ σ t ) regions are shown.(This panel corresponds to the bulk-like case |T − T in | ≫ √ σ t ; see Fig. 2.) In the right, the spatial overlap of the plane waves never decreases in time, and hence the interaction would be never switched off, and the scattering would be never completed; therefore the extra damping factor e ∓ϵt d 3 x H (I) int (t,x) S F w 4 8 M 0 3 M 7 v e x z z W F o k S 7 8 7 O 7 / e 9 Z 8 Y z 2 w 3 9 Y Z x Y 1 q M T T z z 5 1 N P P P H v y u V P P v / D i S 6 f P n H 3 5 Z h x M o p 5 3 o x f 4 Q b T T 7 c S e P x x 7 N 5 J h 4 n s 7Y e R 1 R l 3 f u 9 U 9 / I g + v z X 1 o n g Y j F v J 3 d D b G 3 U O x s P + s N d J o O l 6 s n 9 m x b p o 4 U + j f L H B L 1 Y I / 9 k K z r 7 6 J W k T l w S k R y Z k R D w y J g l c + 6 R D Y v j d J R v E I i G 0 7 Z E Z t E V w N c T n H j k m p w A 7 g V 4 e 9 O h A 6 y H 8 P 4 C 7 X d 4 6 h n v K G S O 6 B 1 J 8 + I s A 2 S C r 1 h / W T 9 Z j 6 6 H 1 s / W n 9 a + U a 4 Y c V J e 7 8 N l l W C / c P / 3 V a 8 1 / t K g R f C Z k k K G U O i e k T 95 H X Y e g e 4 g t 1 I o e w 0 / v f f e 4 + c H 2 6 u x N 6 w f r L 9 D / e + u R 9 S t Y 1 b y O k t e E u V e L + z b 2 N + E X H n F r + e R 8 L a / I u a t 0 P 2 + o O R t 5 h M / X l N n c I G w d l X n e b E S S s z e X Y n c 0 7 M 6 S u t s p u 2 o M y / g b C 1 q h l u N I 5 R R l m I x a A 2 k W 5 c e t Q c 1 x S 8 7 b X I j X 0 f I 6 C + p r a / W 1 F 9 R Y x 1 y s d W 1 l 7 l 9 2 1 m 9 b N g S t 3 x A E J B F I P i D + D C x I 3 B I c e + A M Q x y J x 4 c A 3 3 8 z s e h / z W F s k S r w 7 O 7 / f 9 5 4 Z z 2 w 3 9 I d x Y l k P T z z 2 + B N P P v X 0 y W d O P f v c 8 y + c P n P 2 x Z t x M I l 6 3 o 1 e 4 A f R T r c T e / 5 w 7 N 1 I h o n v 7 Y S R 1 x l 1 f e 9 W 9 / A D + v z W 1 I v i Y T B u J X d D b 2 / U O R g P + 8 N e J 4 G m 3 d Z + O / G O k t l w f L x / Z s W 6 a O F P o 3 y x w S 9 W C P / Z C s 6 + / D l p E 5 c E h n a b n b 8 x L M e 7 y j p B R 9 / Z C g r v j 1 e j 9 N B t Y H 9 1 J N 9 0 B n 6 Q 7 1 n I 2 0 U v P R + W a c d K e b + W Y V e / P 5 O 2 e 5 5 v h e f d a y X I T x n l N R H I / 6 3 / 8 B x e u 8 u U = < / l a t e x i t > T in < l a t e x i t s h a 1 _ b a s e 6 4 = " J V 9 P 9 C w h 4 2 M 3 6 C + b m E 7 8 z l O O u H D g m 8 8 z 9 t r e e X h X J E r W H s / v 9 7 1 n Z m f c C r x e F B v G o 1 O n n 3 j y q a e f O f P s 2 e e e f + H F c + c v v H Q r 8 o d h 2 9 1 p + 5 4 f 7 r b s y P V 6 A 3 c n 7 s W e u x u E r t 1 v e e 7 t 1 t E H 9 P n tk R t G P X / Q i O 8 F 7 n 7 f P h z 0 O r 2 2 H U P T X r N v x 9 1 O a B / V G g f n F 4 y V q x s b 6 + t X a 8 b K + t r q 6 h V 6 Y b y z t m G s 1 1 Z X D P x Z I O x n y 7 / w y u e k S R z i k z Y Z k j 5 x y Y D E c O 0 R m 0 T w u 0 d W i U E C a N s n Y 2 g L 4 a q H z 1 1 y Q s 4 C d g i 9 X O h h Q + s R / D + E u z 3 W O o B 7 y h k h u g 1 S P P g L A V k j i 8 b v x g / G Y + N n 4 0 f j T + N f I d c Y O a g u 9 + C z l W D d 4 O D c F 6 / W / 1 G i + v A Z k 2 6 G k u o c k w 7 Z Q F 1 7 o H u A L d S K d o I f 3 f / q c f 2 9 7 c X x G 8 b 3 x l + g / 3 f G I + M n s G A w + r v 9 8 K a 7 / T W w i z W K g L U P r d R 3 d 9 G L R + j D R a F G D h l B v 4 D 5 7 x A 8 Z 8 N d F 7 H H m l g H b O p D b x G D S N s q f S c j K + 7 V Q s Y j u I 6 V b P 0 0 F 2 V W 0 r 4 9 j J E s F 1 v A 5 8 N 9 H / P 3 C P p c A 0 w I v V 2 y N O X p A C u h B U 9 D + H N y v S P U P m D c N K L 3 4 f M a + N g G D 9 C n 1 M 9 d y N U A 8 Z S h g 9 q J 7 a V Y m m P 3 U J d j Z W 8 f / W J D D l E N f P R U Y v 8 A M 4 v 7 j 7 I 6 g G g C J o I e 9 I r 6 6 h j + J 6 0 2 q S N T S G 7 A 8 0 Q u 5 f k Q e i V j A N X / U 1 Z D N 5 h N 9 I 7 G m n J E o N U A u f l V N T y 9 7 y C e X 1 X D N x D b q I x z I H L U o x S d X V e 1 n t Y z Z b D w s x o 2 x K i N y X Y a P X 3 s N k a e o y O m + a I k A x q l + D d Y x B N N R L i 4 h I t z m T I 9 R 1 1 W Q 1 x v d w I h k j R C n h N 2 n f g g w A r z M O N P F H g / z X E f u U J W q / L K G K a o I b b l c T L k R w X P R J C 9 N W y V4 y I c J Z O r m P m N 2 u o p k T b 2 o T P y P t 7 T E e o Q Z 6 o x P u u x C l y A H k 2 M g V N 4 I v d F M o K p J V y W y L i s l E K j k 4 + T v P + A j S / 0 y s + N z U 3 4 l W N j H E 0 n o 6 T y c c A y b 5 D z g 4 e j e E z e Z H a H a G 0 X W t 5 S a B C k 2 k 8 y y / P j M 5 w p p 8 n f K 8 H y 1 l M w s M a 5 i 1 a u I T 6 D 9 F k M u 5 a L N n 8 i l N o T 4 x g R e x n C j M C 7 l 8 6Y G z / X m A 7 G E L t O v y 6 I p X 4 9 4 h f W I h 9 F W f a c K 0 n V c g J r x b 2 E y D G 8 Z 4 s o v Z t + Y 9 8 j a 1 H l + C f 6 v T Z n t Z R L u p i t 2 u s P g 4 E q W 5 5 R 6 B U 0 r J e + L T L a N d z 4 g V T y b a W b S 9 d c Y 4 r e J E V P N o E V U X Q N l l V C W l i y z J M t M c S r b e N w v w v 8 T c q d k J 6 0 L t a 1 i l r o m i y V l s b R 1 M a W 6 m N r a y H g s b Z 5 N K c 9 m g U c V q 6 S 2 8 y x N G G O y W F 3 U i J W Y p a 7 J Y k l Z L G 1 d T K k u p r Y 2 M h 5 L m 2 d T y r N Z 4 F H X 1 Z a is g 5 I s i a n u w L 6 d S Z m r c / E a i l Y r R l 1 N R W 6 m j N q K + e 1 p L w 6 z O 1 K 3 H e w v w 4 / 9 4 h T y S e X K n l F z D 1 N 9 0 u a m i c j D / f 5 k j S b a y R Z R 2 W e 1 x u R x O z 1 u d g t B b s 1 p + 5 m y i 4 b w z

s 9 S
N b J a L 6 8 a w S x y p R r B K 9 K r G r M j 5 n e S E f n 8 t Z o T 8 + y 2 T I q 1 N / f J b J k F e o / v j M Z e i v r J Y E E n X G Z 5 m 0 6 e O z T J p O J u j O 0 7 P l g O 4 M P V v 0 d e f m 2 e I + 2 6 x c 1 R K 5 L P G c L J I l 3 8 n k s z / f y d TZ z R 0 V 3 i H g U l X n U f 3 S X n w f z / P 5 L r z O q T H f O + V 3 H u 7 n q n f k h y l y y L D Z P q v s Z I i j H O w 9 q L B D O 0 r P 3 x I v R b j L 2 0 d G 1 d s L M e 6 O T 0 c f p N m Q 9 F G d d N M d 8 G G 6 Y y 1 m 4 7 3 U f F S u H i f t + X a O W f b + T N 7 u S b 4 x n n c v l S z X Y Z z U V M a Z a a m r p y 5 z 0 Q v q + E d p h e V 5 o w n W g H H V Q N c a 3 N X w r b U Q r 2 m G O i C v j Z r S N z p U 8 h y l P C c n z 5 l L X i y w L / 6 f 7 I s F 9 s V z 2 F f 9 P T 6 T n f 4 m n / I c a K S V 0 G D 1 x E 8 A d a q + M Z G b R f x k N c r H L 7 q H m J 0 g 0 h P Y C L D U S n6 a m P S j 9 i e j e v m U m J 8 I n x y c X + C v R t f E F 7 f W V l b X V 6 7 c v L J w 3 W T v T 5 8 h r 5 H X Y e Z b J e + S 6 7 A K 2 S I 7 q O 2 X 5 B v y 7 f L D 5 V + W f 1 3 + L e l 6 + h T D v E x y P 8 t / / A c J T v K 4 < / l a t e x i t > T < l a t e x i t s h a 1 _ b a s e 6 4 = " P z C u Z p 3 0 K r D 5 u M 7 m O z f b o B 1 0 8 a X b b R s q m 0 e 7 a m 1 3 F u 3 Z t 7 z b t N g e u / A M c e g L R A + q f w Y U r B y T K l R P i W C Q u H P j m 8 4 y 9 f s z D u y J R s v Z 4 f r / v P T M 7 4 7 b v 9 s P I M J 6 e O P n c 8 y + 8 + N K p l 0 + / 8 u p r r 5 8 5 e + 6 N W 6 E 3 C j r O z Y 7 n e s F O u x U 6 b n / o 3 I z 6 k e v s + I H T G r R d 5 3 b 7 8 F P 6 / P b Y C c K + N 2 x E 9 3 1 n b 9 A 6 G P a 7 / U 4 r g q b 9 s 2 e a 4 d 0 g m j T D / s G g t R 8 d 7 5 9 d M F a u b G y s r 1 + p G S v r a 6 u r l + m F 8 e H a h r F e W 1 0 x 8 G e B s J 8 t 7 9 x b X 5 h 0 s n G S 9 x + y 8 Y V e e Z m x u Q m / c m y E o + l 0 l F Q + 9 l n m D T N + c H E U j 8 j 7 z O 4 A r e 1 B y 0 W F B n 6 i / T S z P D / u 4 k x Z J n + 3 I H 9 P G d N 2 K d P D A t N D B Z N I p y b c 5 b m a y i z o 5 j x 8 X s P L M r 6 t 0 r i 1 k

4 e 5 U
4 n 7 D v b X 4 e c e s S v 5 5 E I l r 4 i 5 y 3 S / o K l 5 P P J w n y 9 J s 7 l G 4 n V U 6 n m 9 E U n M X p + L 3 V K w W 3 P q b i b s s j E s 5 a / N a I V c j i W U k 5 e h M 2 r 1 h F m U H b d 6 F c c t M W 9 9 J l 5 L y W v N q K + p 1 N e c U W M V c 7 5 O e 9 r j F 6 9 S z q 1 b p 7 0 k P / R r V S y j v F q r y L C U M s p r t p o d p t I O e e V W s 0 r 4 O a 0 j c 6 V P J s p T w 7 I 8 + e S 1 4 k s C / 6 n + y L B P Z F c 9 h X / T 0 + k 5 3 + x p / y H G g k l d B g 9 c R P A H W q v j G V m 3 n 8 d D X K x y + 6 h 5 i e I N I T 2 B C w 1 E p + m h j 3 o / b H o 3 r x l J i f C N O 3 p v m r 0 T X x x a 2 1 l d X 1 l c s 3 L i 9 c M 9 n 7 0 6 a X b b R s q m 0 e 7 a m 1 3 F u 3 Z t 7 z b t N g e u / A M c e g L R A + q f w Y U r B y T K l R P i W C Q u H P j m 8 4 y 9 f s z D u y J R s v Z 4 f r / v P T M 7 4 7 b v 9 s P I M J 6 e O P n c 8 y + 8 + N K p l 0 + / 8 u p r r 5 8 5 e + 6 N W 6 E 3 C j r O z Y 7 n e s F O u x U 6 b n / o 3 I z 6 k e v s + I H T G r R d 5 3 b 7 8 F P 6 / P b Y C c K + N 2 x E 9 3 1 n b 9 A 6 G P a 7 / U 4 r g q b 9 s 2 e a 4 d 0 g m j T D / s G g t R 8 d 7 5 9 d M F a u b G y s r 1 + p G S v r a 6 u r l + m F 8 e H a h r F e W 1 0 x 8 G e B s J 8 t 7 9 x b X 5 h 0 s n G S 9 x + y 8 Y V e e Z m x u Q m / c m y E o + l 0 l F Q + 9 l n m D T N + c H E U j 8 j 7 z O 4 A r e 1 B y 0 W F B n 6 i / T S z P D / u 4 k x Z J n + 3 I H 9 P G d N 2 K d P D A t N D B Z N I p y b c 5 b m a y i z o 5 j x 8 X s P L M r 6 t 0 r i 1 k

4 e 5 U
4 n 7 D v b X 4 e c e s S v 5 5 E I l r 4 i 5 y 3 S / o K l 5 P P J w n y 9 J s 7 l G 4 n V U 6 n m 9 E U n M X p + L 3 V K w W 3 P q b i b s s j E s 5 a / N a I V c j i W U k 5 e h M 2 r 1 h F m U H b d 6 F c c t M W 9 9 J l 5 L y W v N q K + p 1 N e c U W M V c 7 5 O e 9 r j F 6 9 S z q 1 b p 7 0 k P / R r V S y j v F q r y L C U M s p r t p o d p t I O e e V W s 0 r 4 O a 0 j c 6 V P J s p T w 7 I 8 + e S 1 4 k s C / 6 n + y L B P Z F c 9 h X / T 0 + k 5 3 + x p / y H G g k l d B g 9 c R P A H W q v j G V m 3 n 8 d D X K x y + 6 h 5 i e I N I T 2 B C w 1 E p + m h j 3 o / b H o 3 r x l J i f C N O 3 p v m r 0 T X x x a 2 1 l d X 1 l c s 3 L i 9 c M 9 n 7 0 6 H H n 3 s 8 S d O P X n 6 q a e f e f b M 0 t n n t i N / H H a 8 m x 1 / 4 I c 7 7 V b k D f o j 7 2 b c j w f e T h B 6 r W F 7 4 N 1 q H 7 5 L n 9 + a e G H U 9 0 e N + G 7 g 7 Q 1 b B 6 N + t 9 9 p x d C 0 3 Y x b 4 3 1 r f 2 n Z u m j h T 2 3 m 4 o p lv b G + X l v n L c u E / 2 z 6 Z 1 / 4 l D S J S 3 z S I W M y J B 4 Z k R i u B 6 R F I v j d J e v E I g G 0 7 Z E p t I V w 1 c f n H j k m p w E 7 h l 4 e 9 G h B 6 y H 8 P 4 C 7 X d 4 6 g n v K G S G 6 A 1 I G 8 B c C s k Z W r N + t H 6y H 1 i / W j 9 a f 1 r 9 S r i l y U F 3 u w m e b Y b 1 g / 8 x n L 9 b / 0 a K G 8 B m T X o p S 6 h y T L n k L d e 2 D 7 g G 2 U C s 6 D D + 5 9 8 X D + t t b K 9 N X r O + s v 0 D / b 6 0 H 1 k 9 g w W j y d + f 7 G 9 7 W 1 8 A u 1 y g C 1 i G 0 U t / d Q S 8 e o g 9 X p B q 5 Z A L 9 A u 6 / A / B c C + 5 6 i D 0 y x L p g 0 x B 6 y x h k 2 l b p O x t Z e a 8 2 M h 7 C d a x l G y a 5 q L K S 9 u 1 j j F S 5 2 A Y + H + 6 H m L + H 0 O c q Y E L o 7 Z H V k q c j r I Q 2 P A 3 h z 8 3 0 j l D 7 g H P T i N 6 D z 6 v g 4 x Z 4 g D 6 l f u 5 B r g a I p w x d 1 E 5 u L 8 X S H L u L u h x p e / v o l x b k

m 9 q 8 N
x s P p B L 6 H H 9 e j y a 6 v X I I L c e G W C 0 d d + p g m Q d F 6 B m 4 l u Y C i N a x r j y i / k 3 5 l 1 y q X S e X 4 X / l 0 p m e 5 W E O 8 m K n e 4 w u L i S F T m l X 0 H T S s n 6 I p X d w j s f k D q e j S Q z 6 f p r C v H b w I j p Z t A 8 q m 6 A c g o o x 0 i W X Z B l J z i d b S L u 5 + D / M b l d s S b j s 0 p a + f i s k m a S C a b z 9 H w 5 Y D p D z x d 9 0 7 l 5 v r j P N y t X t U Q t S z 4 n y 2 S p d z L F 7 C 9 2 M k 1 2 c y e 5 d w iE V N 1 5 1 L C w F z / E 8 3 y x C 2 9 y a i z 2 T s X d A P d z 9 T v y 4 w Q 5 5 t h 0 n 1 V 1 M i R Q L v Y e V d ih n S T n b 8 x L E e 7 y D p F R 9 / Z C j L v j 5 e j 9 J B t Y H 9 1 J N 9 0 B H y c 7 1 n I 2 0 U v P R + W a c d K e r 2 e Y V e / P Z O 2 e 5 Z v i e f d q w 5 c u r l + 5 e P n G 5 e V r N n 9 / + h R 5 i b w M M 9 8 6 e Z N c g 1 X I J r k J 8 j 4 m n 5 M v y V d r 3 6 z 9 v P b r 2 m + s 6 8 k T H P M 8 y f y s / f E f r r / w U g = = < / l a t e x i t > ⌧ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " M 6 k G e d 1 d f 3 E P 3 j B 0 / 5 u F E 7 G o 3 9 n h + v / 9 7 Z j L j b u A N o 9 g w H h 4 7 / t j j T z z 5 1 I m n T z 7 z 7 H P P n z p 9 5 o W b k T 8 J e + 6 N n u / 5 4 U 7 X j l x v O H Z v x M P Y c 3 e C 0 L V H X c + 9 1 T 1 4 j z 6 / N X X D a O i P W / G 9 w N 0 b 2 f v j Y X / Y s 2 N o 6 p w + 2 4 7 t S c d o e 1 4 7 u h P G s 3 Y 0 3 B / Z n f i o c 3 r F u G j g T 6 N 8 s c E u V g j 7 2 f L P s 3 J d S + S y x H O y S J Z 8 J 5 P P / n w n U 2 c 3 d 1 p 4 h 4 B L V Z 1 H j U p 7 8 S M 8 z + e 7 8 D q n x n z v l N 9 5 u J + r 3 p G f p M g J w 2 b 7 r L K T I Y 5 y s P e 4 x g 7 t N D 1 / S 7 w U 4 S 7 v C B l V b y / E u D t e j e 6 k 2 Z D 0 U Z 1 0 0 x 3 w S b p j L W b j v d R 8 V K 4 e J + 3 5 R o 5 Z 9 v 5 M 3 u 5 5 v h m e d 6 + V L N d h n N d U x p l p q a u n L n

Figure 3 :
Figure 3: The P-factors (54) are shown for ϕ → K + K − and ϕ → K 0 K 0 (in the top row), ψ → D + D − and ψ → D 0 D 0 (in the middle row), and Υ → B + B − and Υ → B 0 B 0 (in the bottom row), where we take a typical value 0.0015 MeV −1 = (0.67 GeV) −1 for R 0 ; see Appendix A. The wave-packet treatment breaks down when the wave-packet size of the decay product √ σ P is smaller than the de-Broglie wavelength of P , which is depicted by the hatched region.

Figure 4 :
Figure 4: The distributions of the P-factors (54) representing ψ → D + D − are shown as functions of (σ D ) 1/2 and R 0 , where the bulk, boundary, and interference ones are shown by the orange, magenta, and green color, respectively.

Figure 7 :
Figure 7: The ratio comparing the decay rates of ψ → D + D − to ψ → D 0 D 0 is drawn as a function of R 0 for two fixed wave-packet sizes of the D-mesons of √ σ D = 1 MeV −1 (left panel) and √ σ D = 0.01 MeV −1 (right panel).The experimental result is provided by the PDG [1][shown in Eq. (2)].

Figure 8 :
Figure 8: The ratio comparing the decay rates of Υ → B + B − to Υ → B 0 B 0 is drawn as a function of R 0 for two fixed wave-packet sizes of the B-mesons of √ σ B = 1 MeV −1 (left panel) and √ σ B = 0.01 MeV −1 (right panel).The experimental result is provided by the PDG [1]

Figure 10 : 1 (
Figure 10: The plane-wave ratio comparing the decay rates of ψ → D + D − to ψ → D 0 D 0 is drawn as a function of R 0 .For comparison, we also show the wave-packet results for two fixed wave-packet sizes of the D-mesons of √ σ D = 1 MeV −1 (left panel) and √ σ D = 0.01 MeV −1 (right panel).The other conventions are the same as in Fig. 9.

Figure 11 :
Figure 11: The plane-wave ratio comparing the decay rates of Υ → B + B − to Υ → B 0 B 0 is drawn as a function of R 0 .For comparison, we also show the wave-packet results for two fixed wave-packet sizes of the B-mesons of √ σ B = 1 MeV −1 (left panel) and √ σ B = 0.01 MeV −1

σ
PW-Parton e − e + →V →P P (M ) = N PW-Parton e − e + →V f NR-BW (M ) , and error of the experimentally-determined cross section at the point i, respectively.σ WP i , σ PW-Parton i and σ PW-FF i represent the theoretical values of the corresponding cross sections at the energy point identified by i.

Figure 13 :
Figure 13: We plot the variable χ 2 ϕ /(DOFs) min defined in Eq. (83) to compare the significance of the wave-packet calculation with the plane-wave one for various √ σ K .Here, R 0 is fixed as 0.0015 MeV −1 and for each √ σ K , m ϕ and N e − e + →ϕ are determined to (locally) minimize the corresponding χ 2 function in Eq. (78).The black curve and blue dashed horizontal line describe the values in the wave-packet and plane-wave calculations without form factor, where the latter is manifestly independent of √ σ K .

Figure 14 :
Figure 14: The fitted resonance distributions of ϕ in e − e + → ψ → 2D are drawn for the fixed parameters √ σ D = 1 MeV −1 and R 0 = 0.0015 MeV −1 (for the left panel), and √ σ D = 1 MeV −1 and R 0 = 0.01 MeV −1 (for the right panel), where the mass of ψ and the normalization factor N e − e + →ψ are determined through our statistical analysis based on the χ 2 function defined in Eq. (89).The best-fit parameters and the χ 2 functions for the left/right panel are shown in Eqs.(90) and (91) / in Eqs.(92) and (93), respectively.The other conventions are the same as those of Fig. 12.

Figure 15 :
Figure 15: We plot the variable χ 2 ψ /(DOFs) min defined in Eq. (94) to compare the significance of the wave-packet calculation with the plane-wave one for various √ σ D .Here, R 0 is fixed as 0.0015 MeV −1 and for each √ σ D , m ψ and N e − e + →ψ are determined to (locally) minimize the corresponding χ 2 function in Eq. (89).The black curve and the blue dashed horizontal line describe the values in the wave-packet and plane-wave calculations, respectively, where the latter is manifestly independent of √ σ D .