^{3}

We derive Fermi’s golden rule in the Gaussian wave-packet formalism of quantum field theory, proposed by Ishikawa, Shimomura, and Tobita, for 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

Strictly speaking, the

Ishikawa and Shimomura have proposed a formulation of a free Gaussian wave packet in relativistic quantum field theory [

Stueckelberg correctly pointed out in 1951 that the plane-wave

In this paper we revisit the Gaussian wave-packet formalism to

For clarity, in

The paper is organized as follows: In

We review the Gaussian formalism. As stated above, we consider the decay of a heavy real scalar

First we briefly review the plane-wave computation of the

Throughout this paper we use both

We define the following free one- and two-particle states:^{1}

In terms of these states, the free field operator in Eq. (

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. (^{2}^{3}

The

For

As is well known, the expression in Eq. (

Now we switch from the plane-wave basis to the Gaussian basis. Detailed notations for this subsection can be found in

Instead of the plane-wave expansion of Eq. (

The explicit form of the coefficient function ^{4}

Throughout the main text, we abbreviate e.g.

In the large-^{5}

Now we can explicitly prepare the free wave-packet states, employed in the right-hand sides of Eqs. (^{6}

As stated above,

Suppose that the interaction in Eq. (^{7}

Now the Gaussian

Note that these in and out states become close, in the sense of Eq. (^{8}^{9}^{10}

Using Eq. (

At the first order in the Dyson series of Eq. (

Now we compute the Gaussian

Schematic figure for a configuration with fixed

With the leading saddle-point approximation of Eq. (^{11}

◦

◦

We may freely choose either variable

◦

Hereafter, we abbreviate e.g.

◦ The overline denotes the following weighted sum (and not the complex conjugate): For arbitrary scalar and three-vector quantities

We further define, for any

◦

◦

◦

◦ We define the momentum and energy shifts, etc., as:

◦ “

Note that each quantity defined in the above list is a fixed real number for a given configuration of the wave packets

For any pair of three-vectors ^{12}

Note that we always have

Then we get

Note that for a parent particle at rest,

Let us prove the non-negativity of

From this, one can deduce the non-negativity of

The square completion of

As

In particular, if the center of all three wave packets coincide at

Let us see the physical meaning of

One can show (without taking the particle limit) that the intersection point in Eq. (

One can also check that the overlap exponent

In particular, we may choose
^{13}

Later, this translation will correspond to the zero mode, Eq. (

One can exactly perform the Gaussian integrals over the interaction point

From Eq. (

It is convenient to separate the window function in Eq. (

One can rewrite the boundary parts:

More explicitly, the bulk part reads
^{14}

We note that ^{15}

Normalized boundary function

The explicit formula in the boundary limit ^{16}

More explicitly, the large-

In

In the limit

We see that the range of

◦ In the

◦ In the

We see that the exponential suppression

In Refs. [^{17}

Equation (

It is manifest that we have no singularity.

Recalling the (over-)completeness of the Gaussian basis in Eq. (

We note that this expression is exact up to the leading saddle-point approximation, Eq. (

In

Now we want to perform the Gaussian integral over the central positions of the wave packets

Hereafter, we employ the shifted

One can check that

Writing the other five normalized eigenvectors as ^{18}^{19}

We define new integration variables

In particular, we get

As stated above, the integral over

Now we concentrate on the bulk contribution, Eq. (

In a typical non-singular configuration of ^{20}

Rewriting

To summarize, the integral over ^{21}

In the wave limit

This is nothing but Fermi’s golden rule: the decay probability per time interval ^{22}

We examine the contributions in Eq. (

Let us estimate the effect of the

As discussed in the paragraph containing Eq. (

^{23}

We may further take the plane-wave limit

There is no ultraviolet divergence from the boundary regions if the decay is due to the superrenormalizable interaction of Eq. (^{24}

We comment on the possible ultraviolet divergence at the boundary. First, one might want to take into account the “uncertainty” of

In order to exhibit how to generalize the simplest scalar decay by the interaction in Eq. (

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 as
^{25}

In obtaining the

After taking the plane-wave limit, the final expression for Fermi’s golden rule, Eq. (

The total decay rate is

That is, the replacement in the final expression reads

We have reformulated the Gaussian

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.

We thank Hiromasa Nakatsuka for valuable contributions at the early stages of this work and for reading the manuscript. We are grateful to Osamu Jinnouchi, Arisa Kubota, Terence Sloan, and Risa Ushioda for stimulating discussion. We thank Akio Hosoya, Izumi Ojima, and Masaharu Tanabashi for useful comments. The work of K.I. and K.O. are in part supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Nos. 24340043 (K.I.) and 15K05053 (K.O.).

Open Access funding: SCOAP

We may always separate the total Lagrangian density

Correspondingly, we may separate the Hamiltonian (density)

We list the time dependence of the physical state, operator, and eigenbasis in the Heisenberg, Schrödinger, and interaction pictures in the ^{26}

Time dependences in the different pictures.

Picture | State | Operator | Basis |
---|---|---|---|

Heisenberg | |||

Schrödinger | |||

Interaction |

Throughout this paper,

Let us spell out the ordinary plane-wave basis as a preparation for the Gaussian basis.

A free field operator ^{27}

Here and hereafter, the annihilation operators

A free massless (massive) one-particle state with a definite helicity (spin)

As in the ordinary quantum mechanics, the one-particle position eigenbasis

We may call the position eigenbasis in the interaction picture at time

Concretely, we get

The completeness still holds,

Now we may rewrite:

We define a free Gaussian wave-packet state

Analogously to the plane-wave basis in Eq. (

Concretely, we obtain
^{28}^{29}

Now we define the creation operator of the free wave packet ^{30}^{31}

Note that

To obtain the explicit form of the expansion in terms of the Gaussian basis, one may put Eq. (

Using Eqs. (

Note that

In the wave limit of the initial state,

(Recall that ^{36}

In the limit, the eigenvalues in Eq. (

In the particle limit of the initial state

(Recall that

More concretely,

Without loss of generality, we may set

Finally, we list the corresponding expression to Eqs. (

An experimentalist-friendly parametrization for the decay at rest might be

Without loss of generality, we may set

To cultivate intuition, we present the results for a simple configuration

^{1}The two-particle state is normalized to

^{2}This can be formally rewritten as the interaction-picture state becoming close to the time-independent Schrödinger basis state as

^{3}Strictly speaking, this cannot apply for a decay process. No matter how remote a past we move on to,

^{4}Note that the two “interaction basis” states are the ones at different times:

^{5}

^{6}Explicitly,

^{7}If we took

^{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

^{10}So far, we have not considered any boundary effect as we assume here that the interactions are negligible at

^{11}Recall that we abbreviate

^{12}The abuse of notation for

^{13}The average over the initial and final states is shifted as

^{14}As we see in Eq. (

^{15}In terms of the relevant combination, we get

^{16}We have assumed

^{17}One might need a justification for placing the interaction around

^{18}Recall that the zero eigenvector

^{19}The eigenvector for the latter is proportional to

^{20}Though we have taken the leading saddle-point approximation in the large-

^{21}When the expression for the probability in Eq. (

^{22}Let us review the textbook computation: One can use

^{23}It should be understood that the limit

^{24}Naively, dimensional analysis tells that the tree-level two-body decay of a scalar due to a dimension-

^{25}One can explicitly check that the next-to-leading-order terms in the expansion in Eq. (

^{26}We choose our reference time to identify these three pictures to be

^{27}The dependence of

^{28}Though not quite useful, we may also write down the time-shifted Gaussian wave function in an integral form:

^{29}One can explicitly show that

^{30}We note that, in the Gaussian formulation, the postulation (c) in Ref. [

^{31}When we expand

^{32}In taking the large-

^{33}Explicitly, one may, e.g., take

^{34}We have taken up to the

^{35}We may also rewrite

^{36}Note, however, that the