Particle decay in Gaussian wave-packet formalism revisited

We derive the Fermi's golden rule in the Gaussian wave-packet formalism of quantum field theory, proposed by Ishikawa, Shimomura, and Tobita, for the particle decay within a finite time interval. We present a systematic procedure to separate the bulk contribution from those of time boundaries, while manifestly maintaining the unitarity of the $S$-matrix unlike the proposal by Stueckelberg in 1951. We also revisit the suggested deviation from the golden rule and clarify that it indeed corresponds to the boundary contributions, though their physical significance is yet to be confirmed.


Introduction
Strictly speaking, the S-matrix in quantum field theory is defined only by using wave packets; see any textbook, e.g., Refs. [1,2]. The derivation of a physical quantity, such as a decay rate, in terms of plane waves is "actually more a mnemonic than a derivation" [2]. Ishikawa and Shimomura have proposed a formulation of a free Gaussian wave packet in relativistic quantum field theory [3]; see also Refs. [4][5][6][7] for earlier related works. Ishikawa and Tobita have developed a systematic method to approximate the S-matrix in various limits in the Gaussian wavepacket formalism [8][9][10]; further development has been made by themselves and Tajima to include the photon state [11]. The authors have claimed that there can be a deviation from Fermi's golden rule if we consider an S-matrix with finite time interval [8][9][10][11].
Stueckelberg correctly pointed out in 1951 that the plane-wave S-matrix with finite time interval exhibits an extra ultraviolet (UV) divergence coming from the interaction point at the boundary in time [12]. In order to remove it within the plane-wave formalism, a phenomenological factor has been introduced so that the uncertainty of the initial and final times of the process can be taken into account. This has led to the violation of unitarity, and the necessary modification of the S-matrix to cure the pathology has become complicated and rather intractable.
In this paper we revisit the Gaussian wave-packet formalism to derive Fermi's golden rule. We separate the bulk effect from the boundary ones, while manifestly maintaining the unitarity. We further show that the possible deviation from Fermi's golden rule claimed in Refs. [8][9][10][11] indeed corresponds to the decay at the boundary in time.
For clarity, in Sects. 2-4 we will first spell out our results using an example of the tree-level decay process of a heavy scalar into a pair of light scalars φφ due to the super-renormalizable interaction φφ. In order to show how to generalize our results to include the momentum-dependent factors in PTEP 2018, 123B01 K. Ishikawa and K.-y. Oda the interaction and in the wave functions, in Sect. 5 we will then turn to the tree-level decay process of a pseudoscalar ϕ into a pair of photons due to the non-renormalizable interaction ϕF μνF μν . More generalization will be presented in Appendix A. The paper is organized as follows: In Sect. 2 we review the Gaussian wave-packet formalism for the scalar field. In Sect. 3, we reformulate the Gaussian S-matrix and present a systematic procedure to separate the bulk contribution from the boundary ones. In Sect. 4 we obtain the decay probability and derive Fermi's golden rule. We briefly discuss the boundary effect too. In Sect. 5 we generalize our result to the decay into the diphoton final state. In Sect. 6 we summarize our results. In Appendix A we review the Gaussian wave-packet formalism for the scalar, spinor, and vector. In Appendix B, we show the saddle-point approximation of the Gaussian wave packet in the large-width (plane-wave) expansion. In Appendix C, we show the expressions for the plane-wave and particle limits of the decaying particle and for the decay at rest. In Appendix D we present possible expressions for the boundary limit.

Gaussian formalism
We review the Gaussian formalism. As stated above, we consider the decay of a heavy real scalar into a pair of light real scalars φφ by the following interaction: where κ is a coupling constant of mass dimension unity. The interaction Hamiltonian density is H int = −L int . We write the initial and final momenta as p 0 and p 1 , p 2 , respectively. In this section we will let stand for either or φ. We write their masses as m and m φ and consider the case m > 2m φ .

Plane-wave S-matrix
First we briefly review the plane-wave computation of the S-matrix. We can expand the free field operatorˆ (I) (x) at x = x 0 , x = (t, x) in the interaction picture in terms of the annihilation and creation operators of planes waves: where we work in the (−, +, +, +) metric convention and write the kinetic energy Throughout this paper we use both x 0 and t interchangeably (as well as X 0 and T that appear below). We define the following free one-and two-particle states: 1 where (SB) refers to the time-independent basis state in the Schrödinger picture (see Appendix A.1), which are the eigenstates of the free Hamiltonian: In terms of these states, the free field operator in Eq.
(2) can also be written aŝ where |x (IB) = e iĤ free t |x (SB) is the position basis state in the interaction picture; see Appendix A.2.
Usually, the time-independent in and out states in the Heisenberg picture are defined as the eigenstates of the total Hamiltonian that become close to the free states in Eq. (4) at sufficiently remote past and future in the following sense: 2 e −iĤ t out; p 1 , p 2 whereĤ =Ĥ free +Ĥ int is the total Hamiltonian. To be more precise, Eqs. (7) and (8) are meaningless in themselves and should rather be understood as follows (see any textbook, e.g., Refs. [1,2]): The in and out states are really defined by wave packets such that, for arbitrary smooth and sufficiently fast-decaying functions g in p 0 and g out p 1 , p 2 , they satisfy 1 The two-particle state is normalized to where1 is the identity operator in the two-particle subspace. 2 This can be formally rewritten as the interaction-picture state becoming close to the time-independent Schrödinger basis state as for t → T out (→ ∞).

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as t → T in (→ −∞) and t → T out (→ ∞), respectively. 3 The S-matrix is defined by For T in and T out in the sufficiently remote past and future, respectively, one obtains whereÛ in which T denotes the time ordering andĤ (I) int (t) = e iĤ free tĤ int e −iĤ free t is the interaction Hamiltonian in the interaction picture.
As is well known, the expression in Eq. (12) is badly divergent when squared, being proportional to the momentum-space delta function δ 4 (0). Also, one needs to insert an infinitesimal imaginary part for the interaction Hamiltonian by hand in order to make the perturbation in Eq. (13) convergent. This is because the overlap between plane waves can never be suppressed no matter how remote a past and future one moves on, which is the reason one needs the wave packets in Eqs. (9) and (10) for complete treatment of the S-matrix. The cluster decomposition never occurs for the infinitely spread plane waves, while it does for properly defined wave packets.

Gaussian basis
Now we switch from the plane-wave basis to the Gaussian basis. Detailed notations for this subsection can be found in Appendix A.
Instead of the plane-wave expansion of Eq.
(2), one may also expand the free field in terms of the annihilation and creation operators of the free Gaussian wave: where √ σ is the width of the wave packet; X is the location of the center at time T (and we write collectively X = X 0 , X = (T , X ), as stated above); and P is its central momentum. We also use the shorthand notation := (X , P) , := (X , P) , PTEP 2018, 123B01 K. Ishikawa and K.-y. Oda so thatˆ The explicit form of the coefficient function f ,σ ; (= f ,σ ;X ,P ) is obtained as 4 Throughout the main text, we abbreviate e.g. |σ ; to | , in which it is understood that the σ can be different from each other among the in-and out-state particles.
In the large-σ expansion, the leading saddle-point approximation gives where is the location of the center of the wave packet at time t, in which V (P) := P/E (P); see Appendix B. 5 Within this leading-order approximation, the width of the wave pack remains constant in time.

Free Gaussian wave-packet states
Now we can explicitly prepare the free wave-packet states, employed in the right-hand sides of Eqs. (9) and (10), 4 Note that the two "interaction basis" states are the ones at different times: where t = x 0 and T = X 0 in = (X , P) as always. 5 (t) has implicit dependence on m , P, and X (= (T , X )).

Gaussian S-matrix
Suppose that the interaction in Eq. (1) is negligible at some initial and final times T in and T out . Then we may define the corresponding in and out states, following Eqs. (9) and (10), by 7 Now the Gaussian S-matrix is the inner product between these physical states: Note that these in and out states become close, in the sense of Eq. (22), to the free states of Eq. (21), which are square-integrable and of finite norm. 8 This is in contrast to the plane-wave S-matrix of Eq. (11), which is the inner product between the states that become close to the plane waves of Eq. (4), which are not square-integrable, not elements of the Hilbert space, and hence not the physical states. 9 Due to this finiteness of the Gaussian S-matrix, the probability for the transition φφ is simply its square: |S| 2 . 10 There is no need of the hand-waving argument of the momentum delta function δ 4 (0) becoming spacetime volume, etc. 6 Explicitly, g in (g out ) is a (multiple of independent) free Gaussian wave function(s): where each "interaction basis" state is the one at a different time: . Note also that we have written the states in Eq. (21) as the time-independent Schrödinger basis states, rather than the interaction basis ones, in the sense that they are independent of the time coordinate t that will appear later in H (I) (t), the interaction Hamiltonian in the interaction picture. (Otherwise the two-particle state would have two reference times T 1 and T 2 meaninglessly.) 7 If we took T 0 = T in and T 1 = T 2 = T out , we would obtain respectively. 8 Note, however, the issue in footnote 3. 9 One can extend the notion of Hilbert space to include distributions (such as the Dirac delta "function") by using the rigged Hilbert space, namely the Gelfand triple. In the end, from a given plane-wave S-matrix, one can obtain a physically measurable probability only by convoluting it with wave packets. 10 So far, we have not considered any boundary effect as we assume here that the interactions are negligible at T in and T out ; see also footnote 3.
At the first order in the Dyson series of Eq. (13), the S-matrix becomes

Gaussian S-matrix: separation of bulk and boundary effects
Now we compute the Gaussian S-matrix. In Sec. 3.1, we obtain the S-matrix in the leading saddlepoint approximation (18)

Saddle-point approximation in plane-wave limit
With the leading saddle-point approximation of Eq. (18) in the large-width expansion for all the in and out wave packets, we obtain the S-matrix for a given configuration ( 0 , 1 , 2 ): 11 where the symbols indicate the following: • E A are the on-shell energies: with m 0 := m and m a := m φ (a = 1, 2) being their masses-this is merely a rephrasing of Eq. (3). • V A are the corresponding group velocities: We may freely choose either variable P A or V A , which are in one-to-one correspondence. • √ σ s is the spatial size of the interaction region: Hereafter, we abbreviate e.g. 2 A=0 to A . (We also let the lower-case letters a, b, . . . run for the final states 1 and 2 such that a := 2 a=1 , etc.) • The overline denotes the following weighted sum (and not the complex conjugate): For arbitrary scalar and three-vector quantities C A and Q A , respectively, we define We further define, for any Q A , where σ t is the time-like size of the interaction region: • T is what we call the intersection time, around which the interaction occurs: where X A = A (0) is the location of the center of each wave packet at our reference time t = 0: • R is what we will call the overlap exponent that gives the suppression factor accounting for the non-overlap of the wave packets at the intersection point: • We define the momentum and energy shifts, etc., as: • "i [· · · ]" denotes the irrelevant pure imaginary terms that are independent of x. We will neglect them hereafter as they disappear when we take the absolute square of S.
Note that each quantity defined in the above list is a fixed real number for a given configuration of the wave packets ( 0 , 1 , 2 ). Later we will treat X a (a = 1, 2) as variables of six degrees of freedom; T, X a , and R are dependent ones. (If we vary the final state momenta then P a (a = 1, 2) also become variables; V a , σ t , δP, δE, and δω become dependent ones accordingly.) For any pair of three-vectors Q A and Q A (A = 0, 1, 2), we get where we define, for any Q A , 12 Note that we always have δQ 1 − δQ 2 = Q 1 − Q 2 . In particular, or more concretely, Then we get Note that for a parent particle at rest, P 0 = 0, we may simply replace δV a → V a . Expressions in various limits are shown in Appendix C.
Let us prove the non-negativity of R. In general, the weighted average for any real vector Q satisfies From this, one can deduce the non-negativity of R as follows: At time t, the center of each wave packet is located at The square completion of Z 2 = Z 2 − Z 2 with respect to t shows that Z 2 takes its minimum value σ s R at t = T: As Z 2 ≥ 0 for any t, we obtain σ s R ≥ 0, hence the non-negativity of R.
In particular, if the center of all three wave packets coincide at Z A = x at some time t, then Z = x and Z 2 = 0. Equation (48) shows that this can be the case when and only when t = T (for V 2 > 0), and that we get no suppression in such a case, R = 0. Let us see the physical meaning of T. Suppose that we recklessly take the particle limit σ t , σ s → 0 in the second line in Eq. (27) even though the expression itself is obtained in the contrary plane-wave expansion. Then we see that the interaction indeed occurs around the spacetime point which we call the intersection point.
One can show (without taking the particle limit) that the intersection point in Eq. (49) is transformed properly by spacetime translation: By a constant spacetime translation PTEP 2018, 123B01 K. Ishikawa and K.-y. Oda the center of each wave packet (at t = 0) and its average transform as and hence One can also check that the overlap exponent R is translationally invariant (as it should physically be): In particular, we may choose such that the center of the initial wave packet at t = 0, X 0 = X 0 − V 0 T 0 , is kept invariant. Then the center of each final-state wave packet, X a , is shifted as 13 Later, this translation will correspond to the zero mode, Eq. (91).

Spacetime integral over the position of the interaction point
One can exactly perform the Gaussian integrals over the interaction point x = (t, x) in Eq. (27) to get where we have defined the window function, in which is the Gauss error function. In the small-and large-|z| limits, its (asymptotic) expansion reads, respectively, From Eq. (58), we see that the S-matrix is exponentially suppressed unless the momentum is nearly conserved, δP ∼ 0. This is also the case for the energy conservation δω ∼ 0 except in the boundary regions, at which the translational invariance is explicitly broken; see Sect. 3.4 below. As stated above, the overlap exponent R gives another suppression when the wave packets do not overlap.

Separation of bulk and boundary effects
It is convenient to separate the window function in Eq. (59) into the bulk part and the in-and out-boundary ones: where One can rewrite the boundary parts: where G bdry (z) := erf (z) − sgn(z) .
More explicitly, the bulk part reads  where is the step function. 14 We note that G bdry (z) is discontinuous at z = 0 but the combination e z 2 G bdry (z) 2 is continuous and finite everywhere on the complex z plane (except at the origin z = 0); see Fig. 2. In particular, in the limit |z| → ∞, we obtain 15 The explicit formula in the boundary limit |T − T in/out | σ t δω is that is, 16 where 14 As we see in Eq. (70), this step function appears only at T = T in/out and hence does not contribute when summed with G bdry and integrated over T. That is, it appears only at z = 0 and does not contribute when integrated over z in Fig. 2. This might be non-vanishing for a more realistic non-Gaussian wave packet. 15 In terms of the relevant combination, we get  is the Dawson function, whose (asymptotic) expansions read More explicitly, the large-√ σ t δω expansion gives In Fig. 3, we plot G bdry right at either boundary T = T in/out .

Limit of large argument
In the limit the possible leading contributions are We see that the range of T in this limit can be separated into the following regions: • In the bulk region where the intersection time T is well separated from both the boundary times T in and T out , we obtain hence the name "window function." • In the in and out boundary regions T ∼ T in and T out (namely |T − T in | σ t δω and |T − T out | σ t δω), the contribution from the second and third lines, respectively, in Eq. (80) becomes sizable: We see that the exponential suppression e −σ t (δω) 2 for σ t (δω) 2 1 becomes absent in the boundary region.
In Refs. [8][9][10][11], the authors have claimed that contributions from the boundary region can become non-negligible and that there can be physical consequences. 17 In this paper we leave this issue open and proceed by taking into account only the bulk region contribution of Eq. (82); we will briefly comment on the boundary effects in Sect. 4.3.
Equation (83) appears singular in the simultaneous limit |T − T in/out | → 0 and σ t δω → 0. This apparent singularity is an artifact of first taking the limit in Eq. (80): If we take the limit |T − T in/out + iσ t δω| √ σ t in the original expression, Eq. (59), we obtain It is manifest that we have no singularity.

Decay probability: derivation of Fermi's golden rule
Recalling the (over-)completeness of the Gaussian basis in Eq. (A.22), we see that the decay probability into an infinitesimal phase-space range [X a , X a + dX a ] and [P a , P a + dP a ] (a = 1, 2) is (85) 17 One might need a justification for placing the interaction around T in/out , which are defined to be the times at which the very interactions are negligible; see Eq. (22). Note that this contradiction in identifying the in-state with the free state at the remote past, as in Eq. (9), already exists in the ordinary plane-wave computation of the decay rate because the interaction for the decay never becomes negligible even in the infinite-past limit, and one needs to dump the interaction by hand by introducing the ±i term. A better treatment would be to take into account the production process of the parent particle, namely, to compute the φφ → φφ scattering and the one-loop correction to it in the wave-packet formalism, which will be presented elsewhere. We note that this expression is exact up to the leading saddle-point approximation, Eq. (18).

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In Sect. 4.1 we show how to diagonalize the overlap exponent R. In Sect. 4.2 we focus on the bulk contribution G bulk and derive Fermi's golden rule. In Sect. 4.3, we briefly comment on the boundary contribution G bdry .

Diagonalization of the overlap exponent
Now we want to perform the Gaussian integral over the central positions of the wave packets X a . We may rewrite R, in matrix notation, as follows: where the superscript "t" denotes transposition; as defined in Eq. (39), for a = 1, 2; and M is the following real symmetric 6 × 6 matrix: in which Hereafter, we employ the shifted δX a = X a − (V a T a + X 0 − V 0 T 0 ) as six integration variables. One can check that M has a zero eigenvector: where we have normalized − → X 0 as − → X 0 t − → X 0 = 1. This is a direct consequence of the translational invariance under Eq. (57). This zero-mode will eventually give the factor T out − T in , which is the characteristic of Fermi's golden rule.

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where we have assumed V 2 > 0 in deriving the former, which is two-fold degenerate per each ± sign, providing four of the five. 19 After some computation, we obtain We define new integration variables (y 0 , y 1 , . . . , y 5 ) by ⎡ ⎢ ⎣ y 0 . . .
In particular, we get As stated above, the integral over y 0 does not have a Gaussian suppression and will yield the factor ∝ (T out − T in ). Note that as O is a special orthogonal matrix.

Bulk contribution: derivation of Fermi's golden rule
Now we concentrate on the bulk contribution, Eq. (65). Physically, this takes into account the bulk region of Eq. (81), in which the window function takes the particularly simple form of Eq. (82), by which the spatial d 6 y integral is confined within the range that satisfies where the explicit form of T is given in Eq. (44). We note that T is linear in δX a , and hence in y I . In a typical non-singular configuration of (P 1 , P 2 ) with V 2 > 0, the integrals over all the other five variables y 1 , . . . , y 5 are confined by the Gaussian factor within the range of the order of √ σ s ; see Eq. (93). By definition, the interaction point of the bulk region is well separated from the boundaries, 19 The eigenvector for the latter is proportional to ⎡ ⎣ (δV1) 2 +(δV 2) which can be explicitly checked to be orthogonal to the zero-eigenvector δV 1 δV 2 . and hence the window function can be regarded as unity for the integral over y 1 , . . . , y 5 . 20 That is, we may safely perform each integral over these five variables as simply Gaussian:

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in which we used the product of eigenvalues given in Eq. (94).
Rewriting δX a in Eq. (44) by y 0 , . . . , y 5 using the latter of Eq. (95), can read off the coefficient of y 0 in T. After some computation, we obtain where the dots denote the terms linear in y 1 , . . . , y 5 , which are fixed to be of the order of √ σ s by the above Gaussian integrals and are neglected hereafter. Now the region of the window function T in < T < T out corresponds to and the y 0 integral yields To summarize, the integral over (X 1 , X 2 ) results in 21 In the wave limit σ s , σ t → ∞, we obtain 20 Though we have taken the leading saddle-point approximation in the large-σ A expansion in obtaining Eq. (27), we still consider that the wave packets are well localized compared to the whole spacetime volume in which the decay occurs, say |T − T in/out | √ σ s . This is consistent with the treatment of the current work restricted within the bulk region. 21 When the expression for the probability in Eq. (103) grows to the order of unity as one increases T out −T in , one should, e.g., include the phenomenological factor introduced by Weisskopf and Wigner [13,14]. This is nothing but Fermi's golden rule: the decay probability per time interval T out − T in . The resultant total decay rate reads 22

Comments on the boundary contribution
We examine the contributions in Eq. (66), which come from either the in or out boundary region |T − T in/out | σ t δω (tentatively closing our eyes to the point discussed in footnote 17). Formulae for the boundary contributions in the boundary limit are summarized in Appendix D.
Let us estimate the effect of the d 6 y integral over the Gaussian peak e −(T−T in/out ) 2 /σ t in Eq (83), which results from the limit (T−T in/out ) 2 σ t + σ t (δω) 2 1: As discussed in the paragraph containing Eq. (84), this expression is valid only when σ t (δω) 2 1 at T = T in/out ; see Appendix D for possible generalization.
Naively, the integral over the above-mentioned Gaussian peak would be estimated by taking the formal limit σ t → 0, 23 and by regarding the integral d 5 y e −R as Gaussian, Eq. (99); the remaining y 0 integral would again give the factor (δV 1 ) 2 + (δV 2 ) 2 : (108) 22 Let us review the textbook computation: One can use d 3 Pa 2Ea = d 4 P a δ (P a ) 2 + m 2 φ θ P 0 a and integrate over d 4 P 2 to get where p 1 = |P 1 |, E 1 = p 2 1 + m 2 φ , and p 0 = |P 0 | = E 2 0 − m 2 . One may perform the integral in the last line by p 1 dp 1 = E 1 dE 1 to obtain Eq. (105). 23 It should be understood that the limit σ t → 0 is taken with fixed √ σ t δω. We may further take the plane-wave limit σ s → ∞, which renders the factor in the square brackets into the delta function δ 3 (δP): where we have also replaced δω by δE; see Eq. (37). We see that the ultraviolet behavior of the momentum integral is which is convergent. This convergence itself is independent of the limits that we have taken.
There is no ultraviolet divergence from the boundary regions if the decay is due to the superrenormalizable interaction of Eq. (1). In contrast, if the decay of scalar were due to a marginal operator of dimension four, we would have got a linearly divergent integral instead of Eq. (110). 24 We comment on the possible ultraviolet divergence at the boundary. First, one might want to take into account the "uncertainty" of T in/out that is defined in our treatment to be the time (at which the interacting state can well be identified with the free state), by "diffusing the boundary" à la Stueckelberg [12]. This would provide an additional UV suppression factor on the momentum integral, but the necessary unitarity violation requires the change of the very definition of the Smatrix. Second, as stated in footnote 17, the identification of the interacting state with the free state at T in/out cannot be justified for the boundary contribution. Third, in realistic (particle physics) situations, there is no ideally sharp time boundary but some production and detection mechanisms that are extended in spacetime. The phenomenology on the boundary region could strongly depend on the microscopic physics of the boundary. Thus, the boundary contribution depends on the situation or might not be valid when it is ultraviolet divergent. Further discussion and the implications will be presented elsewhere.

Diphoton decay
In order to exhibit how to generalize the simplest scalar decay by the interaction in Eq. (1) to more realistic cases, we consider the decay of a pseudoscalar into a diphoton pair: where ε λμνρ is the totally antisymmetric tensor and g is a coupling constant of mass dimension −1.
For the pion decay, we set g = √ 2α π f π , where α 1/137 and f π 130 MeV are the fine structure and pion decay constants, respectively.
It is actually straightforward to generalize the previous analysis to the diphoton decay. The photon field operator can be expanded in terms of the creation/annihilation operators of the plane and Gaussian waves aŝ In obtaining the S-matrix, all we have to do is to replace κ by in Eq. (58). The spin-summed decay probability is then, from Eq. (85), where we have used s 1 ,s 2 ε λμνρ P λ 1 μ (s 1 , P 1 ) P ν 2 ρ (s 2 , P 2 ) 2 = 2 (P 1 · P 2 ) 2 .
After taking the plane-wave limit, the final expression for Fermi's golden rule, Eq. (104), becomes dP T out − T in = g 2 m 4 8E 0 where we used, under the momentum delta function and the on-shell condition, The total decay rate is That is, the replacement in the final expression reads κ 2 → g 2 m 4 /2 (and of course m φ → 0). 25 One can explicitly check that the next-to-leading-order terms in the expansion in Eq. (B.19) cancel out in the final expression of the probability dP. For example, the saddle-point momentum remains massless at the next-to-leading order: P ± i x− (t) σ 2 = P 2 + O 1 σ 2 .

Summary
We have reformulated the Gaussian S-matrix within a finite time interval in the Gaussian wave-packet formalism. The normalizable Gaussian basis allows the computation of the decay probability without the momentum-space δ 4 (0) singularity that necessarily appears in the one involving the plane-wave basis. We have performed the exact four-dimensional integration over the interaction point x for the decay probability. The unitarity is manifestly maintained throughout the whole computation.
We have proposed a separation of the obtained result into the bulk and boundary parts. This separation corresponds to whether the interaction point is near the time boundary or not and hence is rather intuitive and easy to envisage. Fermi's golden rule is derived from the bulk contribution. As a byproduct, we have also shown that the ultraviolet divergence in the boundary contribution is absent for the decay of a scalar into a pair of light scalars by the superrenormalizable interaction, though its physical significance is yet to be confirmed. We have generalized our results to the case of diphoton decay and to more general initial and final state particles.  In the large-σ limit, we may iteratively solve the saddle-point condition of Eq. (B.6) by P s = P + 1 P + 2 P + · · · with n P = O σ −n . The result is where (t) := X + V (t − T ) with V := v(P), corresponding to Eq. (19). The zeroth and second derivatives read 35 where we used In particular, the necessary conditions in Eqs. (B.12) and (B.13) read, at the leading order, Appendix C. Wave and particle limits for decaying particle C. 1