# Physicists adding 3 decimals to the fine structure constant is a big accomplishment. Why?

Yesterday, a team of physicists from France announced a breakthrough in nailing down a "magic number" by adding three decimals to the the fine-structure constant (news article; technical paper)

$$\alpha^{-1}\approx 137.035\,999\,206(11)$$

To the layman's eyes, 3 more decimals does not seem so spectacular. Why is this such a big deal when it is about the fine-structure constant?

• Comments are not for extended discussion; this conversation has been moved to chat. – tpg2114 Dec 12 '20 at 16:58

The fine structure constant tells us the strength of the electromagnetic interaction.

There are some misleading statements in the news story. The big one is how to read the result,

\begin{align} \alpha_\text{new}^{-1} & = 137.035\,999\,206(11) \\ &= 137.035\,999\,206\pm0.000\,000\,011 \end{align}

The digits in parentheses give the uncertainty in the final digits; you can see that the traditional $$\pm$$ notation is both harder to write and harder to read for such a high-precision measurement. The new high-precision experiment is better than the average of all measurements as of 2018, which was

$$\alpha_\text{2018}^{-1} = 137.035\,999\,084(21)$$

You can see that the new uncertainty is smaller than the old uncertainty by a factor of about two. But even more interesting is that the two values do not agree: the new result $$\cdots206\pm11$$ is different from the previous average $$\cdots086\pm 22$$ by about five error bars. A "five sigma" effect is a big deal in physics, because it is overwhelmingly more likely to be a real physical difference (or a real mistake, ahem) than to be a random statistical fluctuation. This kind of result suggests very strongly that there is physics we misunderstand in the chain of analysis. This is where discoveries come from.

This level of detail becomes important as you try to decide whether the explanations for other puzzles in physics are mundane or exciting. The abstract of the technical paper refers to two puzzles which are impacted by this change: the possibility that a new interaction has been observed in beryllium decays, and the tension between predictions and measurements of the muon’s magnetic moment, which is sensitive to hypothetical new interactions in a sneakier way.

• "This kind of result suggests very strongly that there is physics we misunderstand in the chain of analysis.": I wouldn't put it this way. Such kind of discrepancies are not rare in the history of the determination of fundamental constants ($\alpha$ and $G$ are two prominent examples), and are most likely due to missing or underestimating a source of error. It's not that we misunderstand the physics but those are complex experiments and it's easy to overlook subtle phenomena. – Massimo Ortolano Dec 4 '20 at 10:04
• Following on from @MassimoOrtolano's point, while not directly about measuring fundamental constants, the tale of the superluminal neutrinos that weren't indicates the sort of thing that can cause errors and be very hard to track down – Chris H Dec 4 '20 at 12:20
• @MassimoOrtolano I'm not sure what Rob had in mind but to my mind, misunderstanding your experiment in such a way that your underestimate your error by a factor of 5 is missing physics. – jacob1729 Dec 4 '20 at 15:09
• @jacob1729 In my experience (I work in the field and I have contacts with colleagues who work in a number of experiments related to fundamental constants), such errors are not caused by missing physics. For instance, in reference to ChrisH example, I spoke with someone who made further analysis of the experiment with the aim of duplicating it and the consensus was that there was an error in the operation of the timing system. And I know of other similar examples, where the issue is rather in technical details. – Massimo Ortolano Dec 4 '20 at 15:33
• @MassimoOrtolano I suggested the possibility of a mistake in the sentence preceding the one you quoted. And the sentence you quoted was phrased carefully: there is a big difference between “misunderstood physics” and its much more exciting subset, “new physics.” But your callout of the need for caution is appreciated. – rob Dec 4 '20 at 16:29

The fine structure constant $$\alpha$$ pops up all over the place in physics, and its presence in equations that physicists have to solve makes it important to measure its true value, so more accurate predictions can be obtained from those equations.

More than that, in the case of $$\alpha$$ it is actually the ratio of other fundamental constants of nature whose values are very accurately known. If the calculated value of alpha differed from its measured value, it would hint at something missing from our understanding of the underlying physics- some subtle effect that no one had ever seen before, and which now must be explained. This is the stuff of which Nobel prizes are made, which makes adding significant digits to a measurement like that worthwhile.

• Great for-the-layman explanation. – e2-e4 Dec 4 '20 at 4:04
• Since the 2019 SI redefinition, it's no longer the case that $\alpha^{-1} = 4\pi\epsilon_0 \hbar c / e^2$ is a ratio of independently-known constants: all of $\hbar, c, e$ now have exactly-defined values. Electromagnetism is now a one-parameter theory, and whether you call that parameter $\alpha$ or $\epsilon_0$ or $\mu_0$ is a matter of algebra and personal taste. – rob Dec 4 '20 at 5:07
• @rob it was always a one-parameter theory. ε, μ, & c were exactly specified. Now ε & μ (instead of $e$) have an uncertain factor proportional to α. (ℏ's uncertainty was an unrelated artifact of the standard kilogram's uncertainty.) – alexchandel Dec 4 '20 at 21:50
• What is the difference between calculated and measured values? – Acccumulation Dec 6 '20 at 2:33

I am adding a few thoughts after two very good answers are already there, so I won't repeat material already covered, except to note the element of misreporting in the news item; see answer by rob.

What I want to do is say a little about what it is like to do this kind of physics. I have never myself reported a new high-precision measurement of a fundamental constant, but I have worked with people who have. The classic example of this type of work is the measurement by Lamb and Retherford in 1947 of a tiny effect in hydrogen that is now known as the Lamb shift. It was important because although it only contributes minutely to the energy, it shows that there is an effect in hydrogen that is not predicted by a Dirac equation model of hydrogen; this both stimulated the development of quantum electrodynamics and gave to the theoreticians a definite number to aim at in checking their work.

Ever since then experimental physicists of a certain type have been painstakingly seeking higher and higher precision as a way to probe for new physical effects in the fundamental particle interactions. It is a sort of quiet companion to the particle collision experiments. The latter are a lot more expensive and yield a lot more data, but just occasionally the quiet sort of experiment, working at low energy and ultra-high precision, lends a little nudge. It is very exciting when this happens.

It is not yet clear whether this has happened with this new measurement. The 5 sigma difference means either someone made a mistake, or there is new physics. The mistake could be, for example, that the uncertainty in the old value was underestimated by a factor 2 or 3, and the values just happened to disagree by a few sigma. It is the sort of tantalizing situation that in cosmology would be called a "tension" between differing measurements of ostensibly the same quantity.

• I never got my head around how confident these experimentalists are with their systematic errors in the first place. – user224659 Dec 5 '20 at 0:37
• @TheoreticalMinimum - a big part of learning (experimental) physics is to learn how to have a confident and correct error estimate. I remember it being much harder than handling the physical formulas. It's a whole science by itself. – Aganju Dec 5 '20 at 5:41
• @TheoreticalMinimum if you're good at it, you do a blinded measurement (so you don't know whether your result agrees with previous measurements until you're ready to publish) and you check literally every possible parameter you can adjust in order to check whether it alters the result when it shouldn't, you check whether the residuals of fitting your data to a lineshape have structure, etc etc. Of course we're not perfect but it's an extremely painstaking field – llama Dec 5 '20 at 8:19

(Comment was deleted, so I'm turning it into an answer.)

This 2018 blog post at Résonaances explains some of the physical implications. In brief: the precision with which $$\alpha$$ is known is a limiting factor in the precision of standard model calculations of the magnetic moments of the electron and muon. The level of precision that both experiment and theory have achieved in determining $$g_e$$ is such that non-standard-model processes in higher order Feynman diagrams could affect the agreement between experiment and theory.