Why can't a single photon produce an electron-positron pair?

In reading through old course material, I found the assignment (my translation):

Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.

My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.

How to solve this task?

• Related: physics.stackexchange.com/q/12488/2451 , physics.stackexchange.com/q/13513/2451 and links therein. – Qmechanic Mar 28 '12 at 12:10
• @Qmechanic: I don't know if this counts as homework as it's part of a course that was given several years ago? The tag wiki isn't very helpful. On the other hand, I'm grateful for hints as opposed to answers. – Andrey Mar 28 '12 at 12:41
• I didn't add the homework tag. If you think it doesn't apply, please edit it out again. – Qmechanic Mar 28 '12 at 12:56
• Yes, the homework tag does apply here. The tag wiki is meant to explain that the tag doesn't just apply to actual homework questions, but rather any question of an educational nature - that is, any question where the real goal is to learn something more general than just the answer to the problem. – David Z Mar 28 '12 at 22:14

Another way of solving such problems is to go to another reference frame, where you obviously don't have enough energy.

For example you've got a $5 MeV$ photon, so you think that there is plenty of energy to make $e^-e^+$ pair. Now you make a boost along the direction of the photon momentum with $v=0.99\,c$ and you get a $0.35 MeV$ photon. That is not enough even for one electron.

• what a boost? why do I need a boost here? – Ben Aug 16 at 12:44
• @Ben Don't ask extra questions in comments. Make a new question. – Kostya Aug 19 at 16:28

Other way to see why this is impossible is to look at inverse process: why annihilating positron and electron can't give up only one photon? Imagine these two particles at rest near each other (or look at center-of-mass system). They will annihilate giving 1MeV of energy, but single photon can't pick this energy up by itself because it would also have E/c of momentum and starting setup, the two charged particles, had non. You need two photons that move in opposite directions.

Check if momentum can be conserved. That ought to do the trick.

• Is it necessary to have an equal number of particles/momenta in a CM-frame? – Ben Aug 16 at 12:46

This has a very simple answer that also works the other way around (see my answer to Why during annihilation of an electron and positron 2 gamma rays are produced instead of 1?). The idea is that this process cannot satisfy both momentum and energy conservation simultaneously. Let's prove it.

Let us consider that the one photon in question is moving in the $z$ direction with energy $\hbar\omega$ and momentum $\hbar\omega/c$. That is, the $4$-vector describing the photon is $k=(\hbar\omega,0,0,\hbar\omega)$ in this frame (let's assume $c=1$). Now, the electron-positron system has $4$-vectors $p_1$ and $p_2$ describing their movement. Momentum energy conservation implies

$$k=p_1+p_2$$

If we square this equation (that is, take the inner product under the Lorenz signature), we have, noting $k^2=0$, $p_1^2=m^2$, and $p_2^2=m^2$, that

$$m^2+p_1\cdot p_2=0$$

Now, this equation is entirely frame independent. Thus, if we pick a frame in which the total momentum is zero, we have that $p_1=(m\gamma,m\beta\gamma\cos{\theta},0,m\beta\gamma\sin{\theta})$ and $p_2=(m\gamma,-m\beta\gamma\cos{\theta},0,-m\beta\gamma\sin{\theta})$ (where $\gamma$ is the Lorenz factor $1/\sqrt{1-\beta^2}$ and $\beta$ is the velocity in natural units). This gives

$$p_1\cdot p_2=m^2\gamma^2\left(1-\beta^2\right)=m^2$$

That is, the kinematic equation requires $2m^2=0$, which is not possible for an electron.

This is a lot of math to give not much intuition. The real intuition lies in the fact that there cannot exist a frame in which the photon has zero momentum, but there does exist a frame in which the electron-positron system has zero momentum. This is incompatible with relativity, and so this process is not kinematically possible.

This applies for the reverse process (where this argument becomes slightly more inherent). Like I said above, it may be helpful to check out my answer to a related question.

I hope this helped!

• Why then $\pi^0\rightarrow2\gamma$ and $\pi^0\rightarrow e^{+}e^{-}$ are possible? Because the neutral pion is composite? – Alexander Cska Sep 22 at 20:46
• This answer is about single photon decay into (massive) antiparticle pairs. The first decay is digamma production, which is perfectly allowed (go to the center of mass frame), and the second is allowed because the pion has more than twice the electron mass. Nothing about the composite-ness of the neutral pion needs to be applied to explain these decays. – Bob Knighton Sep 23 at 2:34
• Why in the zero momentum frame the 3 -momentum has x and z components. Why not x and y? – Alexander Cska Oct 5 at 14:38
• @AlexanderCska It's not strictly important. You could write the momentum in general in terms of Euler angles, but rotational invariance (in particular, the freedom to choose my spatial coordinate system) gives me the freedom to pick the electron-positron pair to be moving strictly in the $x,z$ plane. Strictly speaking, I could have even chosen them to be moving only in the $z$-direction (in retrospect, I'm not exactly sure why I didn't...) – Bob Knighton Oct 5 at 14:43
• Anyway, I really like your answer. Back at school we used the fact that $p<\frac{E}{c}$ – Alexander Cska Oct 5 at 14:50

Very clear explanation by Griffiths:

The Feynman rules enforce conservation of energy and momentum at each vertex, and hence for the diagram as a whole. It follows that the primitive QED vertex (two electrons or a positron and an electron, coupled to a photon)by itself does not represent a possible physical process. We can draw the diagram, but calculation would assign to it the number zero. The reason is purely kinematical: $e^- \rightarrow e^- + \gamma$ would violate conservation of energy. (In the center-of-mass frame the electron is initially at rest, so its energy is $mc^2$. It cannot decay into a photon plus a recoiling electron because the latter alone would require an energy greater than $mc^2$.) Nor, for instance, is $e^+ +e^-\rightarrow \gamma$ possible, although it's easy enough to draw the diagram. In the center-of-mass system the electron and positron enter symmetrically with equal and opposite velocities, so the total momentum before the collision is obviously zero. But the final momentum cannot be zero, since photons always travel at the speed of light; an electron-positron pair can annihilate to make two photons, but not one.

Within a larger diagram, however, these figures are perfectly acceptable, because, although energy and momentum must be conserved at each vertex, a virtual particle does not carry the same mass as the corresponding free particle.

In fact, a virtual particle can have any mass, whatever the conservation laws require. The virtual particles do not lie on their mass shell. External lines, by contrast, represent real particles, and these do carry the “correct” mass.

no. a single photon can decay to an electron-positron pair. It must do so however near a nucleus to conserve momentum. 2 gamma-ray reactions may be rare but pair production process dominates gamma-ray reactions as energy increases and also as nearby nucleus increases in mass.

If $e^+$ $e^+$ give photon From consevation of momuntum $$P_{before} = P_{after}$$ So, $$p_e + p_{e^+} = p_{photon} \\ p_{photon} = 0$$ And we know the momentum of photon can not be zero So , there must be two outgoing photons in opposite direction .

• The question asks about the inverse process. – jinawee Jan 8 '16 at 12:18

protected by ACuriousMind♦Jul 25 '17 at 0:17

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