Scalar scattering amplitude in Gaussian wave-packet formalism

We compute an $s$-channel $2\to2$ scalar scattering $\phi\phi\to\Phi\to\phi\phi$ in the Gaussian wave-packet formalism at the tree-level. We find that wave-packet effects, including shifts of the pole and width of the propagator of $\Phi$, persist even when we do not take into account the time-boundary effect for $2\to2$, proposed earlier. The result can be interpreted that a heavy scalar $1\to2$ decay $\Phi\to\phi\phi$, taking into account the production of $\Phi$, does not exhibit the in-state time-boundary effect unless we further take into account in-boundary effects for the $2\to2$ scattering. We also show various plane-wave limits.


Introduction and summary
It is well-known that a plane-wave S-matrix is ill-defined when taken literally because its matrix element is proportional to the energy-momentum delta function, which always gives either zero or infinity when squared to compute a probability. On the other hand, we may define an S-matrix in the Gaussian wave-packet basis without such an infinity [1,2].
It has been claimed that the Gaussian formalism gives a deviation from the Fermi's golden rule [3,4], in which the probability is suppressed only by a power of the deviation from the energy-momentum conservation rather than the conventional exponential suppression; 1 see also Refs. [6,7,8].
In Ref. [2], a scalar decay Φ → φφ has been computed in the Gaussian formalism, and the previously-claimed power-law deviation from the Fermi's golden rule has been identified to come from the configuration in which the decay interaction is placed near a time-boundary. As we will see, this configuration is realized, even if the in/out states are at a distance. To examine the in-boundary effect for 1 → 2 more in detail, it is desirable to take into account the production process of the decaying Φ.
In this paper, we compute a tree-level s-cannel scalar scattering φφ → Φ → φφ in the Gaussian formalism. We find that wave-packet effects, including shifts of the pole and width of the propagator of Φ, persists even when we do not take into account the time-boundary effect, proposed earlier. The result can be interpreted that a heavy scalar decay Φ → φφ, taking into account the production of Φ, does not exhibit the in-state 1 → 2 time-boundary effect unless we do not take into account the in-state 2 → 2 time boundary. This paper is organized as follows: In Sec. 2, we present basic setup of the Gaussian formalism, and compute the Gaussian S-matrix for the s-channel 2 → 2 scattering: φφ → Φ → φφ. In Sec. 3, we discuss the possible time-boundary effects. In Sec. 4, we focus on the bulk contribution and show that wave effects exist even when we neglect the boundary contributions. In Sec. 5, we present several plane-wave limits of the obtained result. In Sec. 6, we present summary and discussion. In Appendix A, we compare with the φφ → φφ scattering in the φ 4 theory.

Gaussian S-matrix
Here we first review the Gaussian formalism, and obtain the S-matrix for the s-channel 2 → 2 scalar scattering: φφ → Φ → φφ.

Gaussian basis
We review the Gaussian formalism, following Ref. [2], to clarify the notation in this paper. A free scalar field operatorφ at x = x 0 , x (in the interaction picture) can be expanded by the plane basis:φ (x) = d 3 p 2p 0 (2π) 3/2 e ip·xâ ϕ (p) + h.c.
We may also expandφ by the creation and annihilation operators of the free Gaussian wave packets:φ where X = X 0 , X is the center of the wave packet; P is the central momentum of the wave packet; σ and X 0 are fixed (and can differ) for each field participating in the scattering; and the coefficient function becomes .
By e.g. sandwiching between p| and |p , we can show the completeness of the Gaussian basis in the one-particle subspace: Namely, the Gaussian basis can expand any one-particle wave function ψ(x) = x | ψ as where we used the short-hand notation |Π = |ϕ, σ; Π etc. and x | Π is given in Eq. (8). Note the following relation: In the large-σ expansion, we get where in which

In and out states
We consider the s-channel scalar scattering φφ → Φ → φφ. Since both the in and out states are of φ, we omit the label φ hereafter. Generically, one particle in the in-and out-states can be asymptotic to an arbitrary free wave function Ψ(x) = x | Ψ , which can be expanded by the Gaussian basis as Therefore without loss of generality, we may assume that the asymptotic free states are Gaussian, and we will do so hereafter. We prepare the in and out Heisenberg states |in; σ 1 , Π 1 ; σ 2 , Π 2 and |out; σ 3 , Π 3 ; σ 4 , Π 4 , respectively, by where we have defined the free states etc., and take See Sec. 3 for further discussion.

Gaussian two-point function
In this subsection, we omit the labels ϕ and σ as they are all equal, except for the mass m ϕ . In the later application, ϕ will be the intermediate heavy scalar Φ. We want to put the expansion (19), into the time-ordered two-point function: Now we can check that . (32) Putting this into the two-point function (31), We have recovered the ordinary plane-wave propagator as we should, since we integrate over the complete set. 2 As usual, using with being an arbitrary positive infinitesimal, we may rewrite it into more familiar form:

Gaussian S-matrix
Now we compute the probability amplitude under the assumption (28): whereĤ I int (t) = e iĤ free t Ĥ −Ĥ free e −iĤ free t is the interaction Hamiltonian in the interaction picture. In the plane-wave S-matrix, one subtracts the first term in the Dyson series (36), writeŜ =1 + iT , and concentrate on the transition amplitude fromT . In the Gaussian formalism, we do not need such regularization of dropping the first term1 because the inner product of the free states would remain finite even for identical momenta. 3 When we integrate over the final state momenta P 3 and P 4 , the contribution from1 would automatically drop out even if we take the plane-wave limit after all the computations. Hereafter, we omit the trivial term σ 3 , Π 3 ; σ 4 , Π 4 | σ 1 , Π 1 ; σ 2 , Π 2 from S when we call it "transition amplitude".
In this paper, we consider the following simplest interaction Hamiltonian: whereφ andΦ are given in Eq. (1). The tree-level transition amplitude is given by where T x,x is the time ordering with respect to x and x only. Hereafter, we concentrate on the s-channel process because it is dominant in the near on-shell process of our interest.
For example, a part of the s-channel process is The Wick contraction with the external line gives, for example, where the propagator of Φ becomes the same as the plane-wave one, as we have seen in the previous sub-section. Then the contribution (39) becomes In total there will be factor 8 from the other Wick contractions. To summarize, where t := x 0 and t := x 0 are the production and decay times of Φ, and M := m Φ is the heavy scalar mass. This is the starting equation for our computation.
Hereafter, we consider the leading approximation in the plane-wave limit (24): where for a = 1, . . . , 4, in which X a is the center of wave packet at a reference time t = 0 and V a is its central velocity: with m := m φ . We perform the Gaussian integral over the positions of interaction to get where we have dropped a phase factor that cancels out in the square |S| 2 and have defined the following: • Energies and momanta for in and out states: • The averaged space-like width-squared of the in-and out-states, respectively: • For any three vector Q, and • The time-like width-squared of the overlap of the in-and out-states: (54) • The interaction time for the in-and out-states: • The overlap exponent for the in-and out-states: We can show the non-negativity of R in and R out as in Sec. 3.1 in Ref. [2]; our case corresponds to the σ 0 → ∞ limit in its Appendix C.1.
We see from Eq. (47) that a configuration that has large R in or R out of initial and final-state phase space (Π 1 , . . . , Π 4 ) and of the internal momentum p gives an exponentially suppressed wave-function overlap and the corresponding amplitude is also suppressed exponentially.

Separation of bulk and time boundaries
After integrating over t and t , we get where we have defined the window functions as in Ref. [2] G in (T ) := and Physically, the complex variable T in (T out ), or especially its real part T in = T in ( T out = T out ), corresponds to an "interaction time" at which the interaction occurs between the initial (final) φφ and the internal Φ. In terms of the Gauss error function the above two functions are represented as follows: For convenience, we distinguish the bulk effects from the in-and out-boundary ones as where for the interaction between the initial φφ state and the intermediate Φ, and for the interaction between the final φφ state and the intermediate Φ, Here, the following sign function for a complex variable has been defined: More explicitly, where we define the step function for a real variable as Under the above classification of the in-and out-window functions, we divide the probability amplitude S into two parts: where S bulk contains the pure bulk contributions from G bulk in-int (T in ) and G bulk out-int (T out ), while every term of S boundary includes at least one boundary window function.   o y S z s z P f z u x j 1 r p r m X 7 A 2 N n S 8 o W V i 5 c u r y b W r l y 9 d n 1 9 Y / N G w X f a n s H z h m M 5 X l H X f G 6 Z N s 8 H Z m D x o u t x r a V b / E A / f C H 6 D z r c 8 0 3 H f h n 0 X F 5 p a Q 3 b r J u G F q A q V 6 w + q m 6 k W J r R k 5 w V M l J I g X z 2 Figure 1: Schematic diagram in position space. Each of blue and red lines denotes the trajectory of the center of wave-packet for in and out states φ, respectively. The thick dashed line denotes the trajectory of internal particle Φ, while the black dots of its ends indicate that the interactions occur in a finite range with the spatial and time-like widths ∼ √ σ in and √ ς in (∼ √ σ out and √ ς out ) around the point Ξ(T in ) in at time T in (point Ξ(T out ) out at time T out ), respectively. Circles are a reminder that each packet is given with a finite width, namely with the widths ∼ √ σ 1 and √ σ 2 (∼ √ σ 3 and √ σ 4 ) at times T 1 and T 2 (T 3 and T 4 ) for the initial (final) wave packets. In the perturbation theory, we consider time evolution of the in-state from T in to T out in the interaction picture, which are chosen near T 1 , T 2 and T 3 , T 4 , respectively, and the S-matrix element is taken with the out-state at T out . The left figure shows an s-channel scattering without a backward propagation in the sense of the oldfashioned perturbation theory. The right figure explains that there always exists a final state configuration that realizes, e.g. T 1 T out no matter how large we take a cluster-decomposition limit:

Interpretation of boundary effect
We present and clarify two different interpretations of the result (57). We consider a finite time interval T out −T in . Without loss of generality, we focus on the initial time boundary at T in unless otherwise stated. First we stress that when we integrate over the final-state phase space Π 3 and Π 4 with varying interaction time T out (= T out ) accordingly to Eq. (55), there always exists a final-state configuration that gives a significant in-boundary effect at T in , no matter what initial configuration we take, even a cluster-decomposition limit |Ξ 1 (T in ) − Ξ 2 (T in )| → ∞ and/or take T in → −∞; see Fig. 1.
To illustrate qualitative behavior, let us tentatively focus on the expressions in the following limit [2]: 4 which results in 5 Note that the illustrative limit (71) implies that near the boundary, ( T out − T in ) 2 2ς out , the deviation from the "energy conservation" is large: From Eq. (72), we see that the boundary effect may become significant when T is near the in-boundary, namely when ( T − T in ) 2 2ς out with ( T ) 2ς out as said above: Note that the apparent exponential growth for the energy non-conserving limit ( T ) 2 2ς out is cancelled out by the existing energy conservation factor coming from That is, the exponential suppression factor for a deviation from the energy conservation, e −( Tout) 2 /2ςout , is cencelled and replaced by the power suppression factor 1/ T in the boundary effect. Recall that the boundary contribution from the configuration ( T out − T in ) 2 2ς out arises even if X 3 and X 4 are at a distance. 6 The existence of boundary effect crucially depends on the relation (28). The key question is the following: Can we well approximate the real physical setup in experiment, namely the Schrödinger-picture in-state e −iĤt |in; Π 1 Π 2 , by the "free Schrödinger-picture" state e −iĤ free t |Π 1 Π 2 , evolving in a virtual free world without any interaction, at t = T in when interactions are not negligible? 7 If not, what state should we prepare for e −iĤt |in; Π 1 Π 2 at t = T in ? Here we introduce two different constructions: "free" and "dressed", which say yes and no for the first question, respectively.

Quantum mechanics basics
For the discussion below, let us recall the basics of quantum mechanics and spell out our notation. We identify the Schrödinger, Heisenberg, and interaction pictures at an arbitrary 5 In Eq. (72), we cannot take |T out −T in | √ 2ς out → 0 limit because of the assumption (71). When correctly taken, this limit is finite; see Ref. [2]. 6 Suppose we consider the probability from the amplitude (57), P = |S| 2 , for a special case T1 = T2 = Tin and T3 = T4 = Tout: P (TinΠ1Π2 → ToutΠ3Π4). It satisfies P (TinΠ1Π2 → ToutΠ3Π4) → 0 in the limits Tout → Tin and |Xi − Xj| → ∞ for all i = 1, 2 and j = 3, 4. We also have P (TinΠ1Π2 → TinΠ1Π2) = 1. Here, P (TinΠ1Π2 → ToutΠ3Π4) represents a transition probability for not only short distance interactions but also long distance ones such as the Coulomb potential; see also the discussion below Eq. (36). 7 In this section, we omit to show the trivial dependence on σ1, σ2, etc.
reference time t r : For an arbitrary operatorÔ in the Schrödinger picture, we relate them by 8 and for a time-independent state |Ψ in the Heisenberg picture by where we have usedÛ If an eigenbasis |Φ exist in the Schrödinger picture,Ô |Φ = o |Φ , the corresponding operators in the interaction and Heisenberg pictures have the following eigenbases, respectively: The time dependence of these eigenbases is different from that of the states (78) and (79). Typically in our computation,Ô stands forĤ free .

"Free" construction
So far, we have chosen an arbitrary initial (final) time T in (T out ) anywhere near T 1 (T 3 ) and/or T 2 (T 4 ). In the "free" construction we identify the in and out Schrödinger-picture states at times T in and T out , respectively, with a "free Schrödinger picture" state that evolves in a virtual free world governed by the free Hamiltonian no matter how significant interactions are at these times: where we have defined the "free Schrödinger" state that evolves in the virtual free world: 8 Recall that in the interaction picture, we separate an expectation value as Ψ| e iĤ (t−tr) e −iĤ free (t−tr) e iĤ free (t−tr)Ô e −iĤ free (t−tr) e iĤ free (t−tr) e −iĤ (t−tr) |Ψ .
In other words, the in and out states are given in the Heisenberg picture as in the Schrödinger picture as and in the interaction picture as One can trivially check the following: We also see that the Heisenberg-picture relation (85) reads in the Schrödinger picture, and in the interaction picture, The "free" construction puts more emphasis on the interaction picture, in which the identification (90) appears most natural. We can also rewrite the probability amplitude as an inner product of the interaction-picture states at an arbitrary time t: which becomes Eq. (36) when we set the arbitrary reference time t r = 0 as before. 9 Note that the t dependence drops out of the expression, and hence the probability does not depend on t.
We may say that the boundary effects remain even if the interaction is taken into account in the following sense [4] (see also Ref. [10]): Suppose that we transform the free states by a unitary operatorV (κ) withV † (κ)V (κ) =1 in Eq. (91): Then the S-matrix becomes and Accordingly the order κ 2 contribution of the transition amplitudes are invariant under the unitary change of the free states.

"Dressed" construction
To repeat, we have chosen an arbitrary initial time T in anywhere near T 1 and/or T 2 . One might feel it strange to identify the initial state as in Eq. (83) for a wave-packet configuration (Π 1 , . . . , Π 4 ) that gives a significant overlap of the final-state wave-packets at T out T in so that interactions are not negligible at T in as in the right panel in Fig. 1. In particular, the boundary interaction (72) crucially depends on the arbitrarily chosen T in : For a given fixed initial and final state configuration (Π 1 , . . . , Π 4 ), the boundary contribution drops off exponentially as we shift the arbitrarily chosen T in backwards in time.
The boundary effect is a consequence of the above-mentioned identification of the Heisenberg state |in; Π 1 Π 2 and |out; Π 3 Π 4 at T in and T out , respectively. What if we identify different states at T in and T out ? Suppose that we take into account the interactions from T in (< T in ) to T in and from T out (> T out ) to T out (backward in time as T out < T out ) in addition to the "free" construction above: 9 Or else, we may rewrite and redefine all the free states e iĤ free tr |Φ , each being anĤ free -eigenstate, to be |Φ .
where we have replaced |Π 1 Π 2 and |Π 3 Π 4 in the "free" construction (85) by We note that the free basis |Π 1 Π 2 and the state T e −i T in T inĤ int (t −tr) dt |Π 1 Π 2 are different from each other; the same note applies for the out ones. Note also that we can rewrite the Heisenberg-picture states (97) as In the Schrödinger picture, these are equivalent to and in the interaction picture, Just as in the free construction (91), we may write the S-matrix as an inner product of the interaction-picture state at an arbitrary time t: from which the t-dependence drops out. Hereafter, we come back to the choice t r = 0. We note that S and S are physically different. If we could take the limits T in → −∞ and T out → ∞, we would be able to write 10 However, the limits do not commute with the final-state integral of infinite volume over Π 3 and Π 4 as we will see below.

Comparison of two constructions
The in-boundary effect for the fixed configuration (Π 1 , . . . , Π 4 ) disappears from S , which includes the interaction from the time T in (or sufficiently earlier time than T in − √ 2ς out for the given final state configuration) to T in in Eq. (98). In the original S in the "free" construction, interactions at t < T in does not appear. If we start from S for the configuration (Π 1 , . . . , Π 4 ), we recover the boundary effect of S by sharply switching off interactions at t < T in .
Here in S , although the free wave packets in |Π 1 Π 2 are given experimentally at T 1 and T 2 , we identify |Π 1 Π 2 with the Heisenberg state at much earlier time T in , not at somewhere T in near them. Namely, the Schrödinger-picture state e −iĤt |in; Π 1 Π 2 at t → T in is identified with the "free Schrödinger-picture" state e −iĤ free t |Π 1 Π 2 that is time-evolved backward t → T in in a virtual free world governed byĤ free , even for the case where interactions are not negligible for t < T in . In |in; Π 1 Π 2 , interactions are put at times much earlier than T in at which the supposedly free in-state is to be defined.
For the particular in and out-state configuration (Π 1 , . . . , Π 4 ) with (T out − T in ) 2 2ς out , we may always choose T in T in − √ 2ς out , and the in-boundary effect for this configuration drops out of S , but there always exist other configuration (Π 3 , Π 4 ) that has the in-boundary effect at T out T in accordingly to Eq. (55). Therefore, the probability summed over (Π 3 , Π 4 ) has the in-boundary effect for any fixed T in .
Let us rephrase the above discussion in a slightly different way. As we move T in backwards, the bulk region expands, and the effective in-boundary at T in goes back in time. For a given T in , the in-boundary contribution arises from the out state that has overlap of out wave packets at T in . Therefore, the T in → −∞ limit is not uniform because the region of in-boundary effect in Π 3 Π 4 moves along with T in . For these out states for given T in , the boundary effect persists. If such an out state is not included, the boundary effect disappears.
To summarize so far, for any configuration of Π 3 and Π 4 , there always exists a T in that removes the boundary effect, while for any T in , there always exists a configuration of Π 3 and Π 4 that yields an in-boundary effect. Therefore it is subject to debate whether or not the limit (105) can be taken to remove all the time boundary effects.
The expression for boundary effect in the second term in Eq. (72) vanishes exponentially in the limit T in → −∞. In the "dressed" construction, this is natural because this limit corresponds to taking into account all the interactions from −∞, for the fixed initial and 10 The "dressed" construction corresponds to the ordinary plane-wave computation of taking the T → final state configurations. In the "free" construction, one emphasizes the fact that no matter how much we take the limit T in → −∞, there always exists a final state configuration with ( T out − T in ) 2 2ς out for a given T in . The difference of two constructions is the order of procedures: taking the limit T in → ∞ first vs integrating over the infinite volume of (Π 3 , Π 4 ) first.
So far, both constructions have pros and cons, subject to one's theoretical prejudice. Ultimately, experiment should determine which (or else) is right. Currently, an experiment is on-going [11] based on the "free" construction [12]. In this paper, we will leave the choice of constructions open, and concentrate on the wave effect that persists even when we only take into account the bulk effects. See Sec. 4.2 for related discussion on the in-boundary effect for 1 → 2 decay of Φ → φφ. and the typical "average energy" for the 2 → 2 process By the saddle-point approximation, we get Here, the p dependence of the exponent e F is of the form where w := σ + P − ς δE δV + i δX + T ς δV , in which 11 δT and we have defined the "average momentum" for the 2 → 2 process P := σ in P in + σ out P out σ in + σ out (119) and the "interaction time" for the 2 → 2 process Note that the last term in Eq. (115) (in its second line) can be dropped out since it is a pure imaginary constant. The saddle point ∂ F ∂p i = 0 is at 12 that is, Now we can rewrite F without any approximation as where Let us separate two terms corresponding to the momentum and energy conservation from F * : where we have defined δP := P out − P in , and the "average velocity" for the 2 → 2 process and have used the identity We see from the first term in the parentheses in Eq. (130) that the suppression is weaker when the "impact parameter" δX is parallel to the "momentum transfer" δV . This combination δV 2 δX 2 − δV · δX 2 is always non-negative due to the Cauchy-Schwarz inequality.
Also from the second term, the suppression is weaker when the difference of the average position of in and out states is close at the "2 → 2 interaction time" T ς , namely when δX + T ς δV is small. For the integrating over p, the Gaussian factor is (2π) 3 Finally we get the differential amplitude for a fixed configuration of initial and final states (Π 1 , . . . , Π 4 ): where we have defined the dimensionless amplitude M; cf. Eq. (189): Several comments are in order: • All the terms in F * are negative or zero, and hence F * gives always a suppression factor.
• In the amplitude (134), the plane-wave limit σ → ∞ gives a delta function for the momentum conservation: • Likewise, the limit ςσ + σ + +ς(δV ) 2 → ∞ gives a delta function for the energy conservation: • In the squared amplitude |S| 2 , the factor e −σ(δP ) 2 gives the momentum conservation in the limit σ → ∞: We note that the infinity δ 3 (0) from δ 3 δP 2 that appears in the plane-wave computation, using the right-hand side in Eq. (136), is tamed in the current wave-packet one: The would-be delta function squared becomes another would-be delta function again.
• Likewise, the factor in |S| 2 gives the energy conservation in the limit ςσ + σ + +ς(δV ) 2 → ∞: Note that the energy conservation is deformed by the wave-packet effect V σ · δP , which goes to zero in the momentum conserving limit: δP → 0.
• It is remarkable that the wave effect persists even without the time-boundary effect. Namely, the real and imaginary parts of the pole of propagator are shifted as in Eq. (135). Even when p * P in and Ω(p * ) E in , the pole position of the propagator is shifted such that the mass-squared M 2 and decay width Γ are shifted by (δT/ς + ) 2 and −2E in δT/ς + M , respectively.

In-boundary effect for decay
Here we discuss how our result for the 2 → 2 scattering φφ → Φ → φφ can be applied to the 1 → 2 decay process Φ → φφ. In Sec. 3, we have presented two different constructions regarding the boundary effect. For the 1 → 2 decay Φ → φφ [2], the key question for its inboundary effect is how we can better take into account the production process of Φ. Which approximates an experimentally prepared state of Φ better at an initial time T decay in ? Is it the Heisenberg state in the free construction, or in the dressed construction? 13 In our result for the 2 → 2 s-channel scattering of φφ → Φ → φφ, the interaction time T in would correspond to T decay in for the Φ → φφ decay. Here we note that the in-boundary effect of the decay becomes significant when the decay-interaction point around T out is near the center of the in-state wave packet at T decay in T in , namely when Therefore, one might interpret that the limit δT → 0, which necessarily arises when we integrate over the final state phase space Π 3 and Π 4 , corresponds to the in-boundary for the 1 → 2 decay. By taking δT → 0 in Eq. (135), we obtain We see that there is no 1 → 2 in-boundary effect in the 2 → 2 bulk amplitude. If the inboundary effect of 1 → 2 decay exists, it can only emerge from the in-boundary effect of 2 → 2 scattering.

Various limits
Here, we take several limits where σ in and/or σ out goes to infinity.

Plane-wave limit for initial state
First we take the plane-wave limit for the initial state σ in → ∞ for fixed σ out : 13 See the discussion in Secs. 3.3 and 3.4 for subtleties on taking T → −∞ limit.
where, since σ and ςσ + σ + +ς(δV ) 2 stay finite, both of the momentum and energy conservations are violated. The above limited values lead to where we used the result of Eq. (148) in the last steps of Eqs. (149) and (150). From the above information, we get the limit of propagator To summarize, We see that the momentum conservation is broken by ∼ √ σ out , and the energy conservation by ∼ √ ς out , along with the shift −V out · δP in the plane-wave limit for the initial state.

Plane-wave limit for final state
Similarly, we may take the plane-wave limit for the final state σ out → ∞ for fixed σ in : Ω(p * ) → ω out (P out ) = E out , The limit of propagator becomes 1 − (Ω(p * )) 2 − δT To summarize, We see that the momentum conservation is broken by ∼ √ σ in , and the energy conservation by ∼ √ ς in , along with the shift −V in · δP in the plane-wave limit for final state.

Plane-wave limit for both
Finally, we take the double-scaling limit σ in , σ out → ∞ for fixed σ out /σ in : ςσ + σ + + ς δV 2 = σ σ in σ + ∆V 2 out + σout σ + ∆V 2 in + σ σ + δV The limits (168) and (170) lead to the momentum and energy conserving delta functions δ 3 (P out − P in ) and δ(E out − E in ) as in Eqs. (136) and (137), respectively. Then we obtain δE ≈ −δV · P, Ω(p * ) = E in − V in · (P in − p * ) + ςout ς in E out − V out · (P out − p * ) 1 + ςout where ≈ denotes that we have used the energy and momentum conservation from the above mentioned delta functions. Based on the above information, we derive the plane-wave limit of the propagator: We see that the propagator is reduced to the plane-wave form. To summarize, where δ 4 (P out − P in ) = δ(E out − E in ) δ 3 (P out − P in ).

Discussion
In this paper, we have computed the Gaussian S-matrix for the s-channel 2 → 2 scalar scattering: φφ → Φ → φφ. We have found that the wave effects persist even without the time-boundary effect. As a future work, it would be interesting to study the integrated probability after performing the final state integral over the positions X 3 and X 4 : Then we may read off how the ordinary plane-wave differential cross section arises, and see the derivation from it due to the wave effects. It would also be interesting to study the factorization in the limit E 2 in − P 2 in → M 2 .
After integrating over x and t (neglecting the time-boundaries), we get the expression for the probability amplitude, namely the dimensionless S-matrix: We may compare this result with the relation between the dimensionful plane-wave S-matrix element S plane and the dimensionless plane-wave amplitude M plane : We see that gives the proper normalization, where M plane = −λ for the current case. That is,