Non-exponential Decay in Quantum Field Theory and in Quantum Mechanics: The Case of Two (or More) Decay Channels

Abstract

We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is exponential) and equals the quantity Γ ₁/Γ ₂, where Γ ₁ and Γ ₂ are the respective tree-level decay widths. However, in the full treatment (both for QFT and QM) it is an oscillating function around the mean value Γ ₁/Γ ₂ and the deviations from this mean value can be sizable. Technically, we study the decay properties in QFT in the context of a superrenormalizable Lagrangian with scalar particles and in QM in the context of Lee Hamiltonians, which deliver formally analogous expressions to the QFT case.

Appendices

Appendix A: The Loop Function

In order to regularize the self-energy diagram of Fig. 1b we introduce at each vertex the vertex-function $\tilde{\phi}(q)$. At the level of the Lagrangian formulation this can be achieved by rendering the Lagrangian nonlocal, see details in Refs. [22, 48–52] and refs. therein. The integral in Eq. (10) takes the form

$$ \varSigma\bigl(p^{2},m^{2}\bigr)=-i\int\frac{d^{4}q}{(2\pi)^{4}} \frac{\tilde{\phi}(q)^{2}}{ [ ( \frac{p+2q}{2} )^{2}-m^{2}+i\varepsilon ] [ ( \frac{p-2q}{2} )^{2}-m^{2}+i\varepsilon ] } . $$

(141)

A general property of Σ(x,m ²) follows from the optical theorem:

$$(\sqrt{2}g)^{2}\operatorname{Im}\bigl[\varSigma\bigl(x,m^{2}\bigr) \bigr]=x\varGamma^{\text{t-l}}(x,m,g) \bigl[ \tilde{\phi}\bigl(q=(0,\mathbf{q}) \bigr) \bigr]^{2}\text{,}$$

where

$$ \mathbf{q}^{2}=\frac{x^{2}}{4}-m^{2} .$$

(142)

Due to spatial isotropy, the function $\tilde{\phi}(q=(0,\mathbf{q}))$ must depend on q ².

For a generic $\tilde{\phi}(q)$, the tree-level decay width is indeed not given by Eq. (2) but takes the following modified form:

$$ \varGamma^{\text{t-l}}(x,m,g)\rightarrow\varGamma^{\text{t-l}}(x,m,g) \bigl[ \tilde{\phi}\bigl(q=(0,\vec{q})\bigr) \bigr]^{2} .$$

(143)

When assuming that $\tilde{\phi}(q)=\tilde{\phi}(\mathbf{q})$ (i.e. when we drop the dependence on q ⁰), we can perform, in the rest frame of the S particle (p=(x,0)), the integral over q ⁰ in Eq. (141), obtaining:

$$ \varSigma\bigl(x^{2},m^{2}\bigr)=\int\frac{d^{3}q}{(2\pi)^{3}} \frac{\tilde{\phi}^{2}(\mathbf{q})}{\sqrt{\mathbf{q}^{2}+m^{2}} ( 4(\mathbf{q}^{2}+m^{2})-x^{2}-i\varepsilon ) } . $$

(144)

In this work we make the following simple choice for numerical evaluations:

$$ \tilde{\phi}(q)=\tilde{\phi}(\mathbf{q})=\theta\bigl(\varLambda^{2}- \mathbf{q}^{2}\bigr) ,$$

(145)

i.e. we work with a sharp cutoff Λ. (In general, the use of smooth functions affects the results only slightly, thus showing a weak dependence on the adopted vertex function. An explicit calculation has been presented in Ref. [21].) In this case, by performing also the spatial integration we obtain the explicit result:

$$ \begin{aligned}[b] \varSigma\bigl(x,m^{2}\bigr)&=\frac{-\sqrt{4m^{2}-x^{2}}}{8\pi^{2}x}\arctan \biggl( \frac{\varLambda x}{\sqrt{\varLambda^{2}+m^{2}}\sqrt{4m^{2}-x^{2}}} \biggr) \\ &\quad {}-\frac{1}{8\pi^{2}}\log \biggl( \frac{m}{\varLambda+\sqrt{\varLambda^{2}+m^{2}}} \biggr) . \end{aligned}$$

(146)

Following comments are in order:

• (i) The imaginary part of the self-energy amplitude $\operatorname{Im}[\varSigma(x^{2},m^{2})]$ is zero for 0<x<2m and nonzero starting at threshold.

• (ii) The real part $(\sqrt{2}g)^{2}\mathrm{Re}[\varSigma(x^{2},m^{2})]$ is nonzero below and above threshold and depends explicitly on the cutoff Λ.

• (iii) For the case $\tilde{\phi}(q)=\theta(\varLambda^{2}-\mathbf{q}^{2})$, the optical theorem takes the form

  $$ (\sqrt{2}g)^{2}\operatorname{Im}\bigl[\varSigma\bigl(x^{2},m^{2} \bigr)\bigr]=x\varGamma^{\text{t-l}}(x,m,g)\theta \biggl( \sqrt{ \varLambda^{2}+m^{2}}-\frac{x}{2} \biggr) \text{,}$$

  (147)

  Besides the additional θ function, it does not depend on the cutoff Λ. Note that Eq. (13) in the text is strictly speaking valid only for $x<2\sqrt{\varLambda^{2}+m_{1}^{2}}$. This condition is numerically met in Figs. 2 and 6.

• (iv) The present choice of $\tilde{\phi}(q)$ breaks Lorentz invariance. This is here not a problem, because we always work in the rest frame of S. It is also not difficult to generalize $\tilde{\phi}(q)$ in order that it is Lorentz invariant and delivers the same results of this paper when the rest frame of S is considered.

As a last step we turn to the issue of the bare and renormalized masses: in this work we started with the bare mass M ₀ entering in the Lagrangian of Eq. (1) and then we derived the renormalized mass M in Eq. (16), arising upon the inclusion and the resummation of the self-energy diagram of Fig. 1b. Alternatively, one could have added a counterterm in Eq. (1):

$$ \mathcal{L\rightarrow L-}\frac{1}{2}CS^{2}\quad \text{with}\ C=R(M_{0}). $$

(148)

In this way the mass equation takes the form

$$M^{2}-M_{0}^{2}-C+R(M)=0\rightarrow M=M_{0} .$$

Clearly, all the physical results of this work would be unaffected by this alternative procedure.

Appendix B: The n-Channel Case

In the n-channel case the interaction Lagrangian takes the form

$$ \mathcal{L}_{int}=\sum_{i=1}^{n}g_{i}S \varphi_{i}^{2} ,$$

(149)

where the particle φ _i has a mass m _i . The spectral function (or mass distribution) d _S (x) can be decomposed as $d_{S}(x)=\sum_{i=1}^{n}d_{S}^{(i)}(x)$ with

$$ \begin{aligned} d_{S}^{(i)}(x)&=\frac{2x}{\pi}\lim_{\varepsilon\rightarrow0} \frac{2g_{i}^{2}\operatorname{Im} [ \varSigma(p^{2},m_{i}^{2}) ] +\varepsilon }{ ( x^{2}-M_{0}^{2}+\operatorname{Re}\varPi(x^{2}) )^{2}+ ( \operatorname{Im}\varPi(x^{2})+\varepsilon )^{2}} , \\ \varPi (x)&=\sum_{i=1}^{n}( \sqrt{2}g_{i})^{2}\varSigma\bigl(x^{2},m_{i}^{2} \bigr) .\end{aligned} $$

(150)

We define a _i (t) as the Fourier transform of $d_{S}^{(i)}(x)$:

$$ a_{i}(t)=\int_{-\infty}^{\infty} dx d_{S}^{(i)}(x)e^{-ixt}. $$

(151)

Out of a _i (t) we define

$$ A_{i}(t)=\bigl \vert a_{i}(t)\bigr \vert ^{2} , \qquad A_{mix,i}(t)=\sum_{j=1,j\neq i}^{n} \frac{a_{i}(t)a_{j}^{\ast}(t)+a_{i}^{\ast}(t)a_{j}(t)}{2} . $$

(152)

The probability that the state |S〉 decays in the time interval (t,t+dt) into the i-th channel reads h _i (t)dt, where

$$ h_{i}(t)=-A_{i}^{\prime}(t)-A_{mix,i}^{\prime}(t) .$$

(153)

It is possible to define the n(n−1)/2 ratio functions

$$ R_{ij}(t)=\frac{h_{i}(t)}{h_{j}(t)}\quad \text{with}\ i<j,\ i=j=1,\ldots, n. $$

(154)

In the BW limit one has R _ij (t)→Γ _i /Γ _j , where Γ _i is the tree-level decay width in the i-th channel. In general, we expect large deviations from the BW limit. Indeed, many unstable particles of the Standard Model decay in more than two channels: the analysis of the temporal behavior can be performed through the formulae derived here. Explicit calculations represent an outlook for the future.