# Long lifetime pair creation in nonrelativistic $v\ll c$ regime?

At the Bottom of page 12 of the notes Quantum Field Theory I by Michael G. Schmidt (U. Heidelberg, 2007), I have seen that

also for very slow massive particles there are quantum ﬁeld theoretical corrections.

I can understand this statement: within an extremely small time period $\Delta t$ the Energy uncertainty

$$\Delta E \geq \frac{\hbar}{2 \Delta t} \approx 2mNc^2$$

leads to creation of $N$ particles with mass $m$. But what I am asking for is:

Can particle-antiparticle pairs live for a longer time period? I guess also yes, because even considering first order Perturbation Theory one has the probability amplitude proportional to

$$\operatorname{sinc}((E_\mathrm{in} - E_\mathrm{out} - E_\mathrm{transferred})t),$$

and if the energy gap $\epsilon := E_\mathrm{in} - E_\mathrm{out} - E_\mathrm{transferred}$ is sufficiently small, the sinc-function tends to the 'classical' delta-distribution (energy conservation law) much slower than for larger energy gaps.

To achieve $E_\mathrm{out} \approx 2Nmc^2 + E_\mathrm{in} - E_\mathrm{transferred}$, the transferred energy must also be high; this can be e.g. the high-frequency part of a photon wave-packet or some other process that does vary very fast in time.

Can long-living pairs be produced in a usually low energetic system with a high frequency collision rate? Should this be treated nonperturbatively, because the perturbative Dyson series expansion may fail for small values of $\epsilon$?

P.S. More Collisions (with environment) $\Rightarrow$ More decoherence $\Rightarrow$ More classical. But if the collisions will be very frequent, will quantum effects like pair production come into play? Will these last longer, when $\epsilon$ is small (resonance)?

## 1 Answer

To generate a particle antiparticle pair, the necessary energy is needed, i.e. at least the sum of the masses, and some left over for the interaction with the field necessary for the production.For example for production of an electron positron pair

a field is necessary for energy momentum conservation in the center of mass of the pair, and here it is provided by a nucleus , Z. The incoming photon is real, but in general it could be virtual, but the energy it carries that will generate the electron positron pair should be larger than twice the mass of an electron.

A heavy slow moving particle will have to lose that much energy to generate a real pair. The virtual interactions that you describe of higher order in the theory, cannot materialize a pair unless the particle becomes even slower by the above algebra.

There are no long lived elementary particles that are heavy, but there are ions. These do produce pairs and the crossections have been calculated under certain conditions.

Lifetime has a meaning for a decay, it is not used for pair creation. If the incoming photon is virtual, there is no way to say when t=0 is, if it is real the whole interaction happens within the electromagnetic characteristic time .