0.09 nanometres: blow an electron up to Earth size and it's round to within an atom

Here's a question that sounds like a riddle: is the electron round?

An electron has no surface. Nobody has ever found an edge on one, and in every experiment so far it behaves like a point. So "round" can't mean what it means for a ball.

What it can mean is this. Every electron carries an electric charge. Does that charge sit exactly at the centre, or is it nudged, ever so slightly, towards one end?

The answer so far: centred, as far as anyone can measure. 🎯 The best measurement, published in 2023, found that any nudge must be tiny. Give the electron the smallest of its standard sizes, blow it up to the size of the whole Earth, and on this page's arithmetic the nudge would be smaller than a single hydrogen atom.

Getting there took:

a clock built from molecules
two clouds of ions (here, molecules one electron short) in a trap
about 620 hours of patient counting

And the answer reaches well beyond the electron. In the simplest models it points to new particles too heavy for the world's biggest particle collider to make. It also narrows the ideas for one of the biggest open questions there is: why the universe contains any matter at all.

Chapter 1 of 13

A bump on the Earth

On 6 July 2023 the journal Science published a result from JILA, a research institute in Boulder, Colorado, run by the University of Colorado and the US National Institute of Standards and Technology.

The team looked for an off-centre charge on the electron, and found none. What they could say is how big one could be and still have slipped past them: an upper limit. As of October 2026 nobody has beaten it, as far as we can find.

Physicists call this the electron's shape. It is a picture: an off-centre charge would mean the electron's charge sits very slightly more towards one end of its spin axis than the other (every electron has a built-in spin, so it has an axis). Strictly, it is a number that says how the electron's energy changes in an electric field, and that energy is what the experiments measure.

The limit can be read as a distance: how far the charge could sit off-centre. But it's an unimaginably small one, so let's scale it up. Give the electron a size (chapter 2 says which, and why that's a choice) and blow it up to the size of the Earth. The bump grows in the same proportion.

It comes out at most about 0.09 nanometres (0.093, more exactly; this page's arithmetic). A hydrogen atom, the simplest atom, is about 0.1 nanometres across. So on a whole planet, the bump would be a little smaller than one hydrogen atom.

On that chosen size, the electron is round to about one part in 70 million billion.

That's across the atom: from its centre to its edge, hydrogen is smaller than the bump. The JILA team painted roughly the same picture when they announced the result.

Try it, on a ruler of things you know. Guess first.

The bump grows in step with the size you give the electron: the Earth picture is a scaling, not a measurement.

Chapter 2 of 13

How big is an electron?

Here's the catch: to blow an electron up you need a size to start from, and nobody has ever measured one.

At a particle collider at CERN, the European physics laboratory near Geneva, electrons were smashed into their antimatter twins, and they bounced off each other just as points should. In every experiment so far the electron behaves as a point, so any size we give it is a choice.

Physics does offer three standard lengths that go with the electron, built from its mass, its charge and the constants of nature. None is its real size: they are yardsticks, each about 137 times bigger than the last. The smallest is about 2.8 femtometres (a femtometre is a millionth of a billionth of a metre).

Guess first

Start from a size smaller than that, then blow it up to the size of the Earth. The bump comes out…

See the answer

Bigger. Any smaller choice would make the bump bigger, and below about 2.5 femtometres the bump would be bigger than a hydrogen atom.

This page uses that smallest one, the classical electron radius. Of the three it gives the bump the most room, and it gives, roughly, the picture the JILA team described. But it is a convention, not a floor: any smaller choice would make the bump bigger, and below about 2.5 femtometres the bump would be bigger than a hydrogen atom.

On a length about 70 times bigger, the one in the JILA team's own estimate (an estimate we meet near the end), the Earth-sized bump shrinks to about 1.4 picometres; a picometre is a millionth of a millionth of a metre. So "smaller than a hydrogen atom" holds on the smallest standard length and on every bigger one.

None of this changes the physics.

The real result is one number, how far the charge could sit off-centre; the yardstick only picks the everyday picture.

Try the yardstick buttons in the figure above.

Chapter 3 of 13

What “round” means

Push a charge a little off-centre and you get what physicists call an electric dipole moment: the charge times the distance it is pushed. Call it an EDM. For an electron it is measured in e·cm: the electron's charge times a centimetre.

An EDM is not a magnet. The electron certainly is a tiny magnet; the question is whether it also has this electric kind of lopsidedness.

Guess first

An electron spins about an axis. If its charge were lopsided, which of these could it be?

See the answer

Nudged along the spin. For a particle with the electron's kind of spin, the rules of quantum physics allow only one kind of lopsidedness: one end of the spin axis against the other.

An electron's EDM could only point along its spin. An electron has no markings to pick out any other direction, so whatever arrow it carries must lie along the only arrow it has. "Lopsided sideways" is out.

So are an egg and a squashed ball. For a particle with the electron's kind of spin, the rules of quantum physics allow only one kind of lopsidedness: one end of the spin axis against the other. "Pear-shaped" is wrong too (that word belongs to some atomic nuclei), and so is "heavier at one end": an EDM would move charge, not mass.

The language of shape comes from the field itself. In 2011 a team at Imperial College London titled their paper "Improved measurement of the shape of the electron". It's a good picture as long as it stays a picture.

Compare the electron's magnetism, measured to 0.13 parts per trillion. In the same units, any electric lopsidedness is less than about a fifth of a billionth of a billionth of it (this page's arithmetic). So why may an electron carry a magnet but, unless something rare is going on, no EDM? That's a story about mirrors.

Chapter 4 of 13

The mirror test

Picture a spinning electron as a tiny top, with an arrow along its axis for the spin. Its magnet points along the same axis.

Stand a mirror beside it, the axis running along the glass. In the mirror the electron turns the other way, so its spin arrow flips, and its magnet, which comes from the turning, flips too: an ordinary electron, spinning the other way.

Now nudge its charge towards the top end. In the mirror the top is still the top: the nudge stays put while the spin flips. If the laws of physics looked the same in a mirror, this copy would be as good an electron as the original. But there is only one kind of electron, so in a mirror-perfect world the nudge would have to be zero.

Now run the film backwards. Everything turns the other way, so the spin flips. The nudge is about where the charge sits, not how it moves, so it stays put: an EDM fails this test too. Lev Landau pointed this out in 1957. In 1950 Edward Purcell and Norman Ramsey had argued that such things should be tested, not assumed to be zero.

The magnet passes both tests, because it flips with the spin.

Try them.

Failing the mirror test alone would be no surprise. In 1957, testing an idea of Tsung-Dao Lee and Chen-Ning Yang, Chien-Shiung Wu and her colleagues found that the weak force, behind some kinds of radioactivity, fails it. So an EDM would not be the first sign that nature can tell left from right.

The backwards-film test is the rare one: the known rules of particles fail it only very slightly, through the way particles called quarks mix.

An EDM would be a new way for nature to tell a film run backwards from one run forwards.

Chapter 5 of 13

A giant field inside a molecule

How do you look for an EDM? Put the electron in an electric field. If its charge is nudged, its energy would be a little different with its spin along the field than against it. The bigger the field, the bigger the difference.

A free electron just speeds off in an electric field, so put it inside an atom. But in 1963 Leonard Schiff showed that an atom's electrons rearrange themselves to cancel the field, which would hide any EDM completely.

The way out is relativity. Near a heavy nucleus (the tiny, heavy centre of an atom) the inner electrons move fast enough for Einstein's rules to matter, and in 1965 Patrick Sandars showed that the hiding then fails. A heavy atom can even magnify an EDM: in thallium, about 585 times.

Molecules do better still. In a molecule of one heavy atom and one light one, the electrons near the heavy nucleus sit in an enormous electric field, and a small field from the lab lines the molecule up.

JILA used hafnium fluoride ions: a hafnium atom joined to a fluorine atom, one electron short. Inside, an electron feels about 23 billion volts per centimetre, a calculated figure, since nobody can measure it directly. The lab applied only about 58 volts per centimetre, turning round and round so as not to pull the ions out of their trap.

That is about 400 million times the lab's field (this page's arithmetic), and about 100,000 times what any lab could apply directly, by the team's reckoning. Nobody applied 23 billion volts per centimetre: the molecule supplies it, and relativity does the amplifying.

Two fields on one ruler

58 V/cm what the lab applied

23 billion V/cm inside the molecule, calculated

about 400 million times

Chapter 6 of 13

A clock made of molecules

The energy difference is far too small to measure as an energy. So JILA turned it into a clock.

A spinning electron in a magnetic field wobbles round like a tilted top. JILA used a small magnetic field to make the spins wobble about 100 times a second: the clock's tick. (Nº 17 has the same physics: spins in a magnetic field.)

The method is Norman Ramsey's, the idea behind atomic clocks, and it won him a share of the 1989 Nobel Prize.

First pulseA first pulse sets the spins pointing sideways.
WaitThey wobble on their own for up to 3 seconds.
Second pulseA second pulse turns the angle they reached into something that can be counted.

An EDM would make the tick a little faster or slower, depending on which way the molecule faces. At the size of the limit, by under half a part per million. The energy behind that change is what a magnetic field of about 1.6 femtotesla, about a thirty-billionth of the Earth's field, would give a lone electron (this page's arithmetic). A tesla is the unit of magnetic field; a femtotesla is a millionth of a billionth of one.

The experiment doesn't watch electrons spin: it reads the result statistically, from a few hundred ions at a time, over and over. And this clock doesn't tell the time; its job is to compare two ticks.

The figure draws the two clocks of the next chapter as two hands. At true size the EDM's part is invisible, which is the point; the exaggeration slider makes it visible.

After 2.3 seconds, an EDM at the limit would pull the two clocks apart by just 1/4,800 of a turn.

The real experiment had no exaggeration slider. It needed patience.

Chapter 7 of 13

Two clouds and a lot of patience

A change that small would be drowned by anything else that nudges the tick: a drift in the magnetic field, a change in the trap.

So JILA ran two clocks at once, in the same trap: two clouds of ions, with their molecules facing opposite ways. An EDM would push the clocks opposite ways, one faster and one slower. Most noise pushes both the same way, and cancels when you compare them.

The two clouds' magnets also differ by about 1/230, so their ticks drift apart and come back in step about 230 ticks later. The team took data twice: at the start and at that beat.

Try "Jump to the beat" in the figure above.

About 20,000 ions were trapped in each shot. After 3 seconds a few hundred were counted, by breaking them apart with laser light and catching the pieces on a detector.

The data took about 620 hours over two months, about 100 million ions counted in all. The answer stayed hidden from the team until 1 November 2022, so that it could not steer their choices. It was consistent with zero.

Why so long? Halving the uncertainty from counting takes four times as much data: one hour would have left a counting uncertainty bigger than the whole previous best limit (this page's scaling).

The previous bestACME: a beam

That previous best came from ACME, a team that sends a beam of thorium monoxide molecules through its fields. Each molecule flies about 20 centimetres in about a thousandth of a second, inside an even bigger calculated field, about 78 billion volts per centimetre.

2023JILA: a trap

Trapping lets JILA watch each molecule about 3,000 times longer, with far fewer molecules.

Like for like, the 2023 limit is about 2.4 times tighter than ACME's of 2018, and the two results agree.

Chapter 8 of 13

55 years of shrinking bumps

1950The search began with that 1950 suggestion from Purcell and Ramsey.
1959The first electron search, in 1959, used free electrons.
1964Atoms took over in 1964, with caesium.
1968By 1968, a caesium experiment had pushed the limit far enough that, on an Earth-sized electron, the bump could have been at most about the width of a fine hair.
1994, 2002Later, thallium atoms took over, at Berkeley, with results in 1994 and 2002.
2011, 2014Then molecules: ytterbium fluoride at Imperial College in 2011, thorium monoxide from ACME in 2014. By then the Earth-sized bump was down to about the width of DNA.
2017In 2017 JILA's first trapped ions gave a limit that was a confirmation with a new method, not a tighter one.
2018, 2023Then ACME again in 2018, and JILA in 2023: less than a hydrogen atom.

Since 1968 the limit has tightened about 700,000 times (this page's arithmetic). The steps were uneven. One was less than a factor of two, one was about twelve, and one was a deliberate step sideways to try a new technique.

Step through the years, then press Play and keep going down.

The ladder is a selection: other results sit between 1968 and 1994. And the dots stop well above the lines at the bottom. Those lines are the next chapter.

Chapter 9 of 13

What the rules expect

"Round, as far as anyone can measure" is not the same as "round". The Standard Model, today's best theory of particles, does fail the backwards-film test, a little (chapter 4). So it does give the electron an EDM.

But a minute one. The quarks' rule-breaking reaches the electron only in roundabout ways. In 2020 two theorists, Yasuhiro Yamaguchi and Nodoka Yamanaka, worked out how big it should be. On this page's arithmetic, that is billions of times below today's limit (and their estimate is uncertain by about 70%).

Nor is that what these molecules would pick up first. In 2022 three theorists, Yohei Ema, Ting Gao and Maxim Pospelov, showed that here the Standard Model's signal comes mostly from something else: a tiny force between the electron and the nucleus, which mimics an EDM. It is about a thousand times bigger than the estimate before it, from 2014: theory moves too. Yet it is still some 400,000 times below today's limit.

It isn't a nudge on the electron itself. But shown as if it were a bump on an Earth-sized electron, it would be about a quarter of the radius of a proton, the heart of a hydrogen atom (Nº 5 has its size; this page's arithmetic). The electron's own Standard Model EDM would make a bump tens of thousands of times smaller than a proton's radius.

As a bump on an Earth-sized electron

smaller than a hydrogen atomToday's limit: at most about 0.09 nanometres

about a quarter of a proton's radiusThe Standard Model's signal in these molecules, shown as if it were a bump: some 400,000 times below today's limitthis page's arithmetic

tens of thousands of times smaller than a proton's radiusThe electron's own Standard Model EDM: billions of times below today's limitthis page's arithmetic; the estimate is uncertain by about 70%

Signs, not to scale: each row zooms in further.

So the experiments are, in effect, free of any known background (nothing known is big enough to fake the signal): any EDM detected soon would be new physics, something beyond today's best theory.

See how far down the predictions sit with "All the way down" in the figure above.

Chapter 10 of 13

Mountains you can't climb

How can a tabletop experiment say anything about particles too heavy to make? Through what physicists call quantum loops.

A new particle that touches the electron and breaks the backwards-film rule could appear around it for a fleeting moment, as a "virtual" particle, and leave an EDM behind, even if no collider can make it for real. The heavier the particle, the fainter its trace.

So no trace means that any such particles are heavy, or touch the electron only feebly, or break the rule only a little. It's like seeing mountains you cannot climb, with one caution: how high you "see" depends on what you assume.

The view from a tabletop

The foothills: the LHC makes new particles only up to a few TeV.
The far peak: particles heavier than about 40 TeV, the JILA team's estimate for the simplest case.
How high you "see" depends on what you assume.

A sketch, not to scale.

The JILA team's estimate takes the simplest case: particles that touch the electron about as strongly as electromagnetism does, and break the rule as much as they possibly can. Then the limit points to particles heavier than about 40 TeV. A TeV is a million million electronvolts, the unit of energy, and so of mass, that particle physicists use.

That's roughly ten times heavier than anything the Large Hadron Collider can make directly. In its third run, from 2022 to 2026, the LHC collided protons at 13.6 TeV, but that energy is shared among the pieces inside the protons, so it makes new particles only up to a few TeV.

Change the assumptions and the answer moves, from below 1 TeV to several hundred. If the particles reach the electron less directly (through "two loops"), the reach is lower: in one worked example, from a 2026 review by the theorists Maxim Pospelov and Adam Ritz, it is about 10 TeV, and there the 10 TeV sets the strength of an interaction, not the mass of a particle.

Choose the assumptions yourself.

The reach grows slowly: an experiment ten times better sees only about three times heavier. Supersymmetry, the idea that every known particle has a heavier partner, is the classic test case: EDMs don't rule it out, they squeeze it.

The limit is a dial set by assumptions, not a wall.

Chapter 11 of 13

Why is there any matter?

There's one more reason to hunt for new ways of breaking the backwards-film rule: the matter around us.

For every ten billion photons, the particles of light left over from the Big Bang, the universe holds about six protons and neutrons, and almost no antimatter at all.

If the universe began hot and even-handed, matter and antimatter should have been made in equal amounts and wiped each other out almost completely. Something tipped the balance.

In 1967 Andrei Sakharov set out what that something needs.

A process must change the number of protons and neutrons.
Matter and antimatter must behave differently, which takes broken rules of the kind in chapter 4.
Things must at some moment have been thrown out of balance, so that the extra matter was not simply unmade again.

The Standard Model has the ingredients, but its matter–antimatter difference is far too weak. So the matter in the universe very probably needs new rule-breaking.

That's where the electron comes in. Many ideas put the new rule-breaking where it would usually give the electron a measurable EDM, and the results that found nothing squeeze them: in one well-studied model, a 2025 study by two physicists, Biekötter and Olea-Romacho, finds that the predicted EDMs typically come out at least ten times bigger than the limits.

Other ideas survive. One, called leptogenesis, makes the excess at far higher energies and need not give the electron a detectable EDM; some others dodge the limits. So the roundness narrows the options. It does not explain where the matter came from.

Chapter 12 of 13

What's next

Several teams are building experiments that might beat the 2023 limit. Most of what follows is plans and first steps.

JILAThorium fluoride ions

JILA's next experiment uses thorium fluoride ions. In them, the spin state the search needs is the molecule's lowest-energy state, so it cannot decay on its own. The group's 2022 paper predicts that the spins could keep ticking for about 20 seconds if the ions are kept in cold surroundings, at 180 kelvin (about −90 °C); at room temperature, heat radiation from the surroundings would cut that to about 3 seconds. It also plans a conveyor belt of ion traps, to count more ions. No result yet.

ACMEA third version of its beam

ACME is building a third version of its beam. In August 2026 a group working with ACME reported a way to make each thorium monoxide molecule scatter many particles of light, which its authors expect would more than double the statistical sensitivity of an ACME-style search: how much it can learn from its counting, which is only one part of how well it can do.

Other parts are on the way: a paper describes a better source of molecules, and a preprint (a paper posted before other scientists have checked it) describes a new magnetic shield. We found no overall improvement figure, and this page gives none.

Other teamsDifferent molecules

Other teams are working on, or proposing, different molecules: beams of ytterbium fluoride cooled with laser light, beams of barium fluoride, barium fluoride frozen in solid neon, and bigger molecules held in grids of laser light.

As of October 2026, none of them has beaten the 2023 limit, as far as we can find.

Chapter 13 of 13

Round, so far

So what does the 2023 result mean? Five things it does not mean:

Not perfectly round. Round to within the limit. The Standard Model itself predicts a tiny EDM, far below it.
Not a measured shape. An energy shift that an EDM would cause was looked for, and did not show up. "Shape" is a picture of the limit this sets.
Not a size for the electron. The electron behaves as a point. The 0.09 nanometres comes from a chosen yardstick, and a smaller one would make the bump bigger.
Not a wall at 40 TeV. A dial set by assumptions, from below 1 TeV to several hundred.
Not an explanation of matter. It squeezes some ideas for where the matter came from, and leaves others open.

What it does mean is one number, seen several ways. Let's add it all up. 📜

Certificate of roundness

The electron

looked for an off-centre charge, and found none

Bump on an Earth-sized electronthis page's arithmetic: at most 0.093 nm
A hydrogen atom, acrossthe simplest atom: about 0.1 nm
Round toon the smallest standard size: 1 part in 70 million billion
Field inside the moleculecalculated: 23 billion V/cm
Field the lab appliedto line them up: 58 V/cm
Ions trapped per shota few hundred counted: about 20,000
Longest waitbefore reading the clock: 3 seconds
Dataanalysed blind: 620 hours
Tighter since 1968this page's arithmetic: 700,000 ×
Standard Model backgroundbelow today's limit: 400,000 ×
New particles pushed aboveJILA's estimate, simplest models: about 40 TeV

round so far ✦ a clock in a molecule ✦ heavier than the LHC can make

In a trap in Boulder, a few hundred molecules at a time were counted, then a few hundred more.

Inside each one, an electron's spin ticked about a hundred times a second.

Three seconds of waiting, read again and again, for two months.

Nothing extra turned up in the tick.

An electron as big as a planet, and not one hydrogen atom out of round.

Blow an electron up to the size of the Earth, and it is round to within a hydrogen atom.

Part IAn electron the size of the Earth

Chapter 1Nine hundredths of a nanometre

On 6 July 2023 Science published a measurement by Tanya Roussy, Luke Caldwell, Jun Ye, Eric Cornell and colleagues at JILA, the joint institute of the University of Colorado and the National Institute of Standards and Technology in Boulder. They searched for an electric dipole moment (EDM) on the electron, written d_e, and found a value consistent with zero:¹

d_e = (−1.3 ± 2.0_stat ± 0.6_syst) × 10⁻³⁰ e·cm, |d_e| < 4.1 × 10⁻³⁰ e·cm (90%)The first uncertainty is statistical, the second systematic. The 90% upper limit comes from the measured value and its combined uncertainty, 2.1 × 10⁻³⁰ e·cm, treated as a Gaussian folded at zero.

The second number is an upper limit, not a measured dipole moment: the experiment saw nothing, and the limit says how large a dipole moment could be and still have escaped it. As of October 2026 no experiment has beaten it, as far as we can find: the Particle Data Group's 2026 listing uses this result alone,² and a search of arXiv on 25 October 2026 found no tighter electron limit.

Physicists call the dipole moment the electron's shape. It is a picture: an electric dipole moment would mean the electron's charge sits very slightly more towards one end of its spin axis than the other. Strictly it is a number in the equations that says how the electron's energy changes in an electric field, and the experiments measure that energy.

A dipole moment is a charge times a distance, so dividing by the electron's charge e turns the limit into a length: the furthest the electron's charge could sit off-centre. Compare it with the classical electron radius, r_e = 2.818 × 10⁻¹⁵ m (Chapter 2 says why this length),³ and scale the electron up to the Earth's mean radius, 6,371 km:⁴

d_e/e < 4.1 × 10⁻³⁰ cm = 4.1 × 10⁻³² m; (d_e/e)/r_e < 1.455 × 10⁻¹⁷; 1.455 × 10⁻¹⁷ × 6,371 km = 9.27 × 10⁻¹¹ m = 0.093 nmThis page's arithmetic. The fraction is one part in 6.9 × 10¹⁶; the scale factor from r_e to the Earth is 2.26 × 10²¹. The equatorial radius, 6,378 km, gives the same rounding.

Measured against the classical electron radius, the electron is round to about one part in 70 million billion, and an electron the size of the Earth could be lopsided by at most about 0.09 nanometres. The one-standard-error uncertainty, 2.1 × 10⁻³⁰ e·cm, would give 0.047 nm (this page's arithmetic).

The comparison with hydrogen needs care. It fails for the atom's radius: the Bohr radius, a₀ = 0.0529 nm,³ is smaller than 0.093 nm. It holds for the atom's width, twice its radius: 2a₀ = 0.106 nm on the Bohr radius, 0.159 nm on the mean ground-state radius, 1.5a₀ = 0.079 nm, and 0.24 nm on the van der Waals radius chemists use for packing, 0.12 nm.⁵ On every one of these definitions the bump is less than the width of a hydrogen atom: 0.88 of 2a₀, the tightest (this page's arithmetic). It would not be less than the width of every atom: helium is about 0.063 nm across on the measure that gives hydrogen 2a₀, so this page always says "a hydrogen atom".

The JILA team made the same comparison. In the news release that accompanied the paper, Roussy explained that for an Earth-sized electron any asymmetry would be "smaller than the radius of an atom".⁶ This page's 0.093 nm is about an atomic radius, the same picture in order of magnitude; on r_e, though, the press line fails for hydrogen, so this page leaves it as the release's words.

Figure 1 does the scaling for any size, result and yardstick you choose; its inset draws the bump beside a hydrogen atom at one linear scale.

Figure 1🎲 Try it

Blow up an electron

Pick a size for the electron

Guess first: blown up to the size of Neptune's orbit, the bump would be about the size of

Blow it up to the size of

Radius of the blown-up electron

Electron dipole momentWhich result to use

Yardstick

r_e is not a floor: a yardstick below about 2.5 fm would make the Earth-scale bump bigger than a hydrogen atom (this page's arithmetic).

Lopsidedness (d_e/e) ÷ yardstickOff-centre by, as a share of its size

Bump on this electron

Compared with a hydrogen atom (2a₀ = 0.106 nm)Compared with a hydrogen atom

BetweenSomewhere between

The largest charge offset allowed by an electric dipole moment d_e, scaled up with the electron. The offset d_e/e is divided by a yardstick length L and multiplied by the chosen radius R: b = (d_e/e) R/L. The electron has no measured size, so L is a convention (Chapter 2): r_e = 2.818 fm, which gives the JILA team's Earth picture in order of magnitude; ƛ_C/2 = 193 fm, the length in the team's one-loop estimate; or ƛ_C = 386 fm. The proton marker is its charge radius, 0.84 fm (Nº 5). The limit is that of Roussy et al. 2023 (90%); the Standard Model values are those of Chapter 10. This page's arithmetic.Pick how big to blow the electron up, and which result to use. The figure scales up the most the electron's charge could be off-centre and puts it on a ruler beside things you know. The electron has no size anyone has measured, so the yardstick buttons choose one: the smallest standard size gives the bump the most room of the three, and a smaller choice would make it bigger. The middle button is the length in the JILA team's own estimate (an estimate we meet near the end). Two buttons show what the Standard Model, today's best theory of particles, expects (more on that later on).

The bump grows in proportion to the size you give the electron, so the Earth picture is a scaling, not a measurement.

Chapter 2A size for something with no size

The Earth picture needs a size for the electron, and the electron has none we can measure. At CERN's LEP collider, electrons and positrons scattered off each other just as point particles should. Substructure would show up as an extra "contact" interaction between them, and the Particle Data Group's listing, from a combined reanalysis of LEP data, puts the energy scale of any such interaction above 8.3 or 10.3 TeV, depending on the sign assumed, at 95% confidence.⁷ In every experiment so far the electron behaves as a point, so any size we give it is a choice.

Physics offers three standard lengths associated with the electron, built from its mass, its charge and the constants of nature. Each is a factor of the fine-structure constant α ≈ 1/137 away from the next:

r_e = αƛ_C = 2.818 fm < ƛ_C = ħ/m_ec = 386.2 fm < a₀ = ƛ_C/α = 52,918 fmThe classical electron radius, the reduced Compton wavelength and the Bohr radius (CODATA 2022 values, rounded). The ladder of factors α holds for the reduced Compton wavelength only; with h/m_ec the ratios become 2π/α and 1/(2πα).

The reduced Compton wavelength is the length quantum theory builds from the electron's mass with the reduced Planck constant ħ and the speed of light c, and the Bohr radius is the size of hydrogen in Bohr's model. The classical electron radius is the radius at which a sphere of charge e would have an electrostatic energy comparable with the electron's rest energy, and the only one of the three that needs no Planck constant. Nobody thinks it is the electron's size. This page uses it because it is the smallest of the three, so of these it gives the bump the most room. It is a convention, one that gives the JILA team's Earth picture in order of magnitude, and it is not a floor: any shorter length would make the bump bigger. The headline survives only for yardsticks above a threshold:

L* = (d_e/e) × R_⊕ / 2a₀ = 4.1 × 10⁻³² m × 6.371 × 10⁶ m / 1.0584 × 10⁻¹⁰ m = 2.47 fmThis page's arithmetic: the yardstick at which the Earth-scale bump equals hydrogen's width 2a₀. It sits 12% below r_e. On any yardstick below about 2.5 fm the bump would exceed a hydrogen atom.

The JILA team's own theory estimate (Chapter 11) uses half the reduced Compton wavelength, ƛ_C/2 = 193 fm.¹ On that length:

(d_e/e)/(ƛ_C/2) = 4.1 × 10⁻³² m / 1.931 × 10⁻¹³ m = 2.1 × 10⁻¹⁹; × 6,371 km = 1.4 pmThis page's arithmetic. ƛ_C/2 is 68.5 times r_e, so the bump is about 70 times smaller than on the classical radius. On the full reduced Compton wavelength, 386 fm, it is 0.68 pm, 137 times smaller.

"Smaller than a hydrogen atom" holds on r_e and on every larger standard length.

One naming trap: CODATA's "Compton wavelength", λ_C = h/m_ec = 2.426 pm, is 2π times the reduced one; on λ_C/2, 1.213 pm, the Earth-scale bump would be 0.22 pm (this page's arithmetic).³

None of this changes the physics. The physical result is d_e itself, a number in e·cm. The yardstick only decides which everyday picture that number turns into.

Chapter 3What "shape" means here

An electric dipole moment measures how far a charge sits off-centre: charge times distance. For the electron the natural unit is the electron's charge times a centimetre, the e·cm. An electric dipole moment is not a magnet: the electron certainly has a magnetic moment, and the question is whether it also has an electric one.

For a particle with spin, any dipole moment must point along the spin axis. This is a theorem of quantum mechanics (the Wigner–Eckart theorem), and it has a plain meaning: an electron has no features by which to define any other direction, so whatever arrow it carries must lie along the only arrow it has. The JILA paper writes the dipole moment as d_e times the unit vector along the spin.¹ "Lopsided sideways" is not an option.

Nor can the electron be stretched or squashed. A particle of spin ½ can carry static electric moments only up to rank 1, twice its spin: the dipole. So it has no rank-2 electric quadrupole moment: nothing oval. The only asymmetry allowed is one end of the spin axis against the other. Even "pear" is the wrong word: in nuclear physics a pear-shaped nucleus has an octupole deformation, a rank-3 moment, which a spin-½ particle cannot have.⁸ And "heavier at one end" is wrong too: a dipole moment moves charge, not mass.

The language of shape comes from the field itself. In 2011 Jony Hudson, Ed Hinds and colleagues at Imperial College London titled their Nature paper on the molecule ytterbium fluoride "Improved measurement of the shape of the electron".⁹ It is a good picture as long as it stays a picture.

Compare the electron's magnetic moment. In units of the Bohr magneton it has been measured to 0.13 parts per trillion:¹⁰

g/2 = 1.001 159 652 180 59(13)Fan, Myers, Sukra and Gabrielse, 2023: an uncertainty of 0.13 parts per trillion. The departure from 1 is the electron's anomalous magnetic moment.

Divided by the speed of light, the magnetic moment has the units of an electric dipole: μ_B/c = eƛ_C/2 = e × 1.93 × 10⁻¹¹ cm, and the electric dipole moment is below 2.1 × 10⁻¹⁹ of it (this page's arithmetic). The next chapter shows why the electron may carry the one and, unless two symmetries fail, not the other.

Chapter 4The mirror and the film run backwards

Physicists sort arrows by how they behave under two operations. Parity, written P, inverts space through a point: every position r becomes −r. Time reversal, written T, runs the film backwards: every velocity reverses. Arrows come in two kinds, and the two operations treat them differently.¹¹

An electric dipole is a polar vector: it points from one place to another, like a position. Parity flips it; time reversal, which moves nothing, keeps it. Spin, like every angular momentum, is an axial vector: it is defined by a sense of rotation, and the right-hand rule turns that rotation into an arrow. Under parity a rotation keeps its sense, so spin is unchanged; run the film backwards and the rotation reverses, so spin flips. The magnetic moment comes from circulating charge and behaves like spin.

P: s → s, μ → μ, d → −d T: s → −s, μ → −μ, d → dSpin s and magnetic moment μ are axial vectors; the electric dipole moment d is a polar vector (Jackson, Classical Electrodynamics, 3rd edition).

If the electron's dipole moment lies along its spin, as Chapter 3 says it must, parity turns "dipole along spin" into "dipole against spin", and so does time reversal. If the laws were unchanged by P, the electron after P, dipole against spin, would be just as good an electron as the original. But there is only one kind of electron, with one value of d_e, so d_e would have to equal −d_e: zero. The same argument works for T. So a permanent electric dipole moment on a particle with spin violates both P and T, as Lev Landau pointed out in 1957;¹² seven years earlier Edward Purcell and Norman Ramsey had argued that such moments should be tested by experiment, not assumed to be zero.¹³ The magnetic moment passes both tests, because it flips together with the spin.

Figure 2's mirror is a plane containing the spin axis, the mirror you would actually hold up. Reflection in such a plane is parity followed by a half-turn, which does not change the relation between two arrows along the axis, so it gives the same comparison.¹¹ In the drawing the spin flips and the dipole stays.

Figure 2🎲 Try it

The mirror test

Flip it and compare

What the electron carries

Spin points

Moment pointsIts nudge points

Moment relative to spinCompared with the spin

How well knownWhat we know

Spin and the magnetic moment are axial vectors, the electric dipole a polar one (Chapter 4); the mirror is a plane containing the spin axis. Because an EDM can only point along the spin, either operation turns "dipole along spin" into "dipole against spin". A non-zero EDM therefore violates P and T, and, assuming CPT symmetry, CP (Landau 1957). P alone fails in the weak interaction, so the rare part is T. The magnetic moment survives both. Arrows not to scale. This page's model.A spinning electron and a copy of it, mirrored or with time run backwards. Its magnetism flips along with its spin, so the copy is still an ordinary electron. An off-centre charge would not flip, so the copy would point the 'wrong' way. Finding one would mean the laws can tell a film run backwards from one run forwards, which the known rules do only very slightly.

Broken parity alone is nothing new. In 1956 Tsung-Dao Lee and Chen-Ning Yang pointed out that it had never been tested in the weak interaction and proposed experiments;¹⁴ months later Chien-Shiung Wu of Columbia University and colleagues at the National Bureau of Standards found it violated in the beta decay of polarised cobalt-60 nuclei.¹⁵ Landau then proposed CP, parity combined with C, the swap of every particle for its antiparticle, as the symmetry that survives.¹² So an electron EDM would not be the first sign that nature tells left from right. What it would add is a broken time reversal.

That is the rare part. The CPT theorem says that C, P and T combined are an exact symmetry of theories like the Standard Model, so a broken T is a broken CP. CP violation was found in 1964 in the decays of neutral kaons,¹⁶ and in 1973 Makoto Kobayashi and Toshihide Maskawa showed that it could come from a phase in the mixing of three generations of quarks, written in what is now called the Cabibbo–Kobayashi–Maskawa (CKM) matrix (Nobel Prize 2008).¹⁷ The Standard Model breaks CP through quark mixing, and only weakly: the Jarlskog invariant that measures it is J = 3.16 × 10⁻⁵.¹⁸ Broader treatments are in the reviews by Pospelov and Ritz and by Chupp and colleagues, and in the book by Khriplovich and Lamoreaux.^19,20,21 The CKM phase accounts for all the CP violation measured in laboratory experiments so far; the theory has a second CP-breaking parameter, the strong interaction's θ̄ term, which neutron and mercury searches test (Chapter 9).¹⁹

Part IIA clock inside a molecule

Chapter 5A field no laboratory can apply

The way to look for an electric dipole moment is to put the electron in an electric field and see whether its energy depends on which way its spin points. A dipole d_e in a field E has energy −d_e·E, and since d_e lies along the spin, the energy differs between spin along the field and spin against it:

U = −d_e·E, ΔU = U_against − U_along = 2d_eEThe experiment compares the electron's spin aligned and anti-aligned with the field; a non-zero d_e would split the two by 2d_eE, with opposite sign for opposite orientations.

The bigger the field, the bigger the shift. The obvious approach, a bare electron between charged plates, fails: a free electron accelerates out of any electric field. An electron inside a neutral atom seems no better. In 1963 Leonard Schiff proved that in a neutral system of non-relativistic point charges the electrons rearrange to cancel an applied field, so that on average each particle feels none, which would hide any dipole moment completely.²²

The escape is relativity. In 1965 Patrick Sandars showed that the shielding fails for an electron moving relativistically near a heavy nucleus, so that the atom's dipole moment can exceed the electron's own by a factor growing roughly as 10α²Z³, where Z is the atomic number; in thallium it is about −585 times the electron's.^23,24 A rough guide to the speeds involved is Zα, the Bohr-model speed of an inner electron as a fraction of light's: 0.53 for hafnium (Z = 72), 0.51 for ytterbium, 0.59 for thallium, 0.66 for thorium (this page's arithmetic). Nº 11 uses the same rule of thumb for lead and gold, and its caveat holds here too: an orbital has no single speed, and the effect comes from the Dirac equation, not from the electron "getting heavier". By the Z³ rule alone, hafnium beats caesium, the first atom used, by 2.2 times, thallium by 3.2 and thorium by 4.4 (this page's arithmetic).

Molecules add a second amplifier. In a polar molecule made of a heavy atom and a light one, the electrons near the heavy nucleus sit in an intense internal field. A small laboratory field lines the molecule up, and the electron then feels an effective field, E_eff, of tens of billions of volts per centimetre. E_eff cannot be measured directly; it is calculated with relativistic quantum chemistry.

Molecule	E_eff, calculated (GV/cm)
HfF⁺ (hafnium fluoride ion), JILA 2017 and 2023	≈ 23 (no uncertainty given)
ThO (thorium monoxide), ACME	≈ 78
ThF⁺ (thorium fluoride ion), JILA's next experiment	35.2
YbF (ytterbium fluoride), Imperial 2011	26 fully polarised; 14.5 at 10 kV/cm

Sources: HfF⁺, the calculations cited by Roussy et al. 2023; ThO, ACME 2018; ThF⁺, Denis et al. 2015; YbF, Kara et al. 2012, with the 14.5 GV/cm at the 2011 experiment's operating field.^{1,25,26,27,28}

In the JILA experiment the electron sits in one particular electronic state of HfF⁺, labelled ³Δ₁, where E_eff ≈ 23 GV/cm. The laboratory's orienting field is only about 58 V/cm, rotating 375,000 times a second because a static field would pull the ions out of their trap.¹ The effective field is about 4 × 10⁸ times that (this page's arithmetic) and, in the paper's words, "∼10⁵ times larger than what can be directly applied in the lab". Nobody applied 23 GV/cm: the molecule supplies it, and relativity does the amplifying.

Chapter 6A clock that ticks a hundred times a second

The energy shift of Chapter 5 is far too small to measure directly as an energy. The JILA experiment measures it as a frequency, with the method of separated oscillatory fields for which Norman Ramsey shared the 1989 Nobel Prize, the idea behind atomic clocks.²⁹ A first pulse, called a π/2 pulse because it turns the spin through a quarter-turn, puts the electron's spin into an equal superposition of its two orientations, a spin pointing sideways. The spin precesses freely, like a top, for a chosen hold time. A second π/2 pulse turns the angle it has reached into a difference in population between the two states, which is read out. Scan the hold time and the population oscillates: a Ramsey fringe, at the precession frequency.

In HfF⁺ the free precession lasts up to 3 s. A small magnetic field, rotating with the orienting electric field, sets the basic tick through the Zeeman splitting, the energy gap a magnetic field opens between the spin's orientations. Here the gap is 3|g_F|μ_BB_rot, with g_F ≈ −0.0031, and it gives fringes of about 100 Hz.¹ This is the same physics as the spins in a magnetic field of Nº 17, whose energy splitting is 2μB for a moment μ in a field B. An electric dipole moment would add to the tick or take away from it:

f = f₀ ± δ, hf₀ = 3|g_F|μ_BB_rot ≈ h × 100 Hz, δ = 2d_eE_eff/hThe sign of δ depends on whether the molecule, and so E_eff, points along or against the rotating field. The paper's conversion is 2E_eff/h ≈ 1.11 × 10³¹ µHz per e·cm.

How big a change would a dipole moment at the limit make? With E_eff = 23 GV/cm, all of the following is this page's arithmetic:

d_eE_eff = 4.1 × 10⁻³⁰ e·cm × 23 × 10⁹ V/cm = 9.4 × 10⁻²⁰ eV; δ = 45.6 µHz; δ/f₀ = 4.6 × 10⁻⁷; 2πδ × 3 s = 8.6 × 10⁻⁴ radWith h = 4.1357 × 10⁻¹⁵ eV s. The last figure is the extra phase one cloud's spins would gather in a 3 s precession.

So a dipole moment at the limit would change a 100-hertz tick by under half a part per million, and turn the spin by less than a thousandth of a radian in the longest precession; the abstract of the team's preprint speaks of its "sensitivity to 10⁻¹⁹ eV shifts in molecular ions".¹ The energy is what a magnetic field of 1.6 femtotesla would give a lone electron's spin (μ_B = 5.788 × 10⁻⁵ eV/T),³ about 3 × 10⁻¹¹ of the Earth's field of roughly 50 µT (this page's arithmetic).

The experiment does not watch electrons spin: it reads a quantum phase statistically, from a few hundred ions per shot, over many shots. And this clock does not keep time: JILA is the same Boulder institute whose strontium clocks appear in Nº 9's table of records, but here the clock's job is to compare two ticks.

Figure 3 draws the clock: two hands, the two clouds of Chapter 7, in a frame turning a hundred times a second, with warm arcs for the dipole moment's share of each hand's drift. At true size the arcs are invisible; the labelled exaggeration slider makes them visible.

Figure 3🎲 Try it

The clock inside the molecule

Turn up the exaggeration

Hold timeWaiting time

Exaggerate the EDM

Electron EDM (or its Standard Model equivalent)

Upper / lower cloud (EDM aside)Tick rates of the two clouds

EDM shift per cloud δ = 2d_eE_eff/hChange in the tick (true size)

EDM split between the clouds at t (true)How far the EDM pulls the clocks apart

Drawn exaggerated by

How long to average

Hours of data

Statistical error on d_eUncertainty from counting

Ions counted

Ramsey's method in HfF⁺ ions (Roussy et al. 2023), with free precession for up to 3 s (Chapter 6). The fringe frequency is about 100 Hz. Two clouds (doublets) of opposite molecular orientation share the trap; an EDM shifts their frequencies by ±δ (Chapter 6), and their magnetic moments differ by about 1/230 (Chapter 7). Full contrast is drawn; the real fringes are weaker. The exaggeration is labelled; at true size δ/f₀ = 4.6 × 10⁻⁷ at the 2023 limit. The averaging block is Chapter 7's square-root scaling. This page's model and arithmetic.Two clouds of molecules, each with electrons ticking about 100 times a second. An electron EDM would speed one clock up and slow the other down. At true size the difference is far too small to see, so the exaggeration slider makes it bigger; the real effect never changes. Press Run the clock to watch the clouds drift apart and come back in step, then slide the hours to see why it took about 620 hours of counting. The counting uncertainty is in e·cm: the electron's charge times a centimetre, the unit for an off-centre charge.

At the default 2.30 s the clouds' magnetic difference (Chapter 7) has brought the hands back in step. The dipole moment pushes the two clouds opposite ways, so the split it makes between them is twice one cloud's phase: 1.3 × 10⁻³ rad, 1/4,800 of a turn, invisible. An exaggeration of about 580 times makes the split about one radian at 3 s (this page's model). The real experiment has no exaggeration slider, which is why it needed what Chapter 7 describes.

Chapter 7Two clouds, one trap, 620 hours

A shift of 45.6 µHz on 100 Hz would be swamped by anything that nudges the tick: a drift in the magnetic field, a change in the trap. The JILA design cancels most of these by running two clocks at once in the same trap. In the orienting field the ³Δ₁ state of HfF⁺ splits into a series of pairs of levels ("doublets"). The two clouds are the ions in two of them. In those two doublets the molecule is oriented in opposite senses relative to the rotating field, so the electron feels E_eff pointing opposite ways. A dipole moment would therefore shift the two doublets' frequencies by equal amounts in opposite directions, while most sources of noise push both the same way and cancel in the difference.¹

The cancellation is not free. The two doublets' magnetic moments differ by about 1/230, so their ticks differ by about 0.43 Hz, roughly 10⁴ times the dipole moment's shift. The clouds drift out of phase and come back in phase about 230 oscillations later, and the experiment takes its data twice: early, when they are in phase, and again at that beat. In Figure 3 the magnetic-difference button shows this drift: each hand runs at 100.22 or 99.78 Hz, back in step at 2.30 s (this page's model).

The numbers of the run:¹ about 20,000 HfF⁺ ions were trapped in each shot, and typically about 120 per doublet were detected after 3 s. They were counted by state-selective photodissociation: laser light breaks apart molecules in one state, and the fragments land on a microchannel plate detector. The data came in 1,370 blocks over two months, some 620 hours, with about 10⁸ ion detections; 1,329 blocks survived the cuts. The analysis was kept blind until 1 November 2022, so that the choice of cuts could not be steered by the answer. The statistical uncertainty in the dipole-moment channel, the part of the clouds' frequency difference that a dipole moment would shift, was 22.8 µHz, about 30% above the quantum projection noise set by counting a finite number of ions.

The measured frequency in the dipole-moment channel was f_BD = −14.6 ± 22.8_stat ± 6.9_syst µHz. Divided by the conversion of Chapter 6, 1.11 × 10³¹ µHz per e·cm, that is the result of Chapter 1: d_e = (−1.3 ± 2.0 ± 0.6) × 10⁻³⁰ e·cm. The combined one-standard-error uncertainty, 2.1 × 10⁻³⁰ e·cm, improves on ACME's 2018 result, in the paper's words, "by a factor of ∼2".¹ The systematic error budget is set out in a companion paper by Caldwell and colleagues.³⁰

Why 620 hours? Because halving the statistical error takes four times as much data. Figure 3's hours slider scales the published statistical error, 2.0 × 10⁻³⁰ e·cm after 620 hours, by √(620 h/H):

σ(H) = 2.0 × 10⁻³⁰ e·cm × √(620 h / H); σ(1 h) = 5.0 × 10⁻²⁹ e·cmThis page's scaling of the published error, not a published figure; √620 = 24.9. Ions counted scale as about 10⁸ × H/620.

One hour of the same data would have given an error larger than ACME's whole 2018 limit; about twenty hours would have matched it (this page's scaling).

Chapter 8Beam against trap

The previous best limit came from a different design. The ACME collaboration sends a beam of thorium monoxide molecules through a region of controlled fields. Each molecule's electron spin precesses while the molecule flies about 20 cm, roughly 1 ms at a beam speed of 200 m/s, inside an effective field of about 78 GV/cm. In 2018 ACME reported d_e = (4.3 ± 3.1_stat ± 2.6_syst) × 10⁻³⁰ e·cm, so |d_e| < 1.1 × 10⁻²⁹ e·cm.²⁶

ACME II (2018): a beam

Molecule: ThO, neutral
E_eff: about 78 GV/cm (calculated)
Precession: about 1 ms, set by the flight over about 20 cm
Limit: 1.1 × 10⁻²⁹ e·cm

JILA (2023): a trap

Molecule: HfF⁺, an ion
E_eff: about 23 GV/cm (calculated)
Precession: up to 3 s, held in the trap
Limit: 4.1 × 10⁻³⁰ e·cm

The trade is time against numbers: a trap lets JILA watch each molecule about 3,000 times longer (3 s against 1 ms), with far fewer molecules and a weaker field. Per molecule, a dipole moment at the 2023 limit would build 9.7 × 10⁻⁷ rad of phase in ThO (δ = 155 µHz for 1 ms) against 8.6 × 10⁻⁴ rad in HfF⁺, about 900 times more (this page's arithmetic, in Chapter 6's convention; ACME's convention gives half the rate).²⁶

How much tighter is 2023? The two published 90% figures divide to 2.7, but they use different statistical methods. Computed with the same Feldman–Cousins method as ACME's, the JILA limit would be 4.5 × 10⁻³⁰ e·cm, about 2.4 times tighter: the paper's own comparison, which it says holds whichever of the two methods is applied to both.¹ This page's folded-Gaussian bound on ACME's value, 9.5 × 10⁻³⁰ e·cm, gives 2.3. The two central values differ by slightly more than one standard error, which counts as agreement: 2023 confirms ACME's result and tightens it.

Neither molecule measures the electron's dipole moment purely. A heavy molecule is also sensitive to a CP-violating force between the electron and the nucleons (protons and neutrons) of the nucleus, parametrised by a coupling C_S. What HfF⁺ measures is the combination

hf_BD sgn(g_F) = −2E_effd_e + 2W_SC_S, W_S/h = −51 kHzRoussy et al. 2023. If d_e = 0, the same data give C_S = (−1.4 ± 2.2 ± 0.7) × 10⁻¹⁰. W_S, calculated like E_eff, sets how strongly HfF⁺ responds to C_S.

The headline limit assumes the electron's dipole moment is the only source. ThO and HfF⁺ weigh the two terms differently, so together they constrain both without that assumption: |d_e| < 2.1 × 10⁻²⁹ e·cm and |C_S| < 1.9 × 10⁻⁹, from a combined fit to the two results that takes its molecular parameters from Fleig and Jung.^1,31 Chapter 10 shows why C_S matters: the Standard Model's own signal in these molecules comes mostly through it.

Part IIIWhat the roundness tells us

Chapter 9The ladder of limits

After their 1950 paper (Chapter 4),¹³ Purcell and Ramsey set the first limit on the neutron's dipole moment in 1957, with James Smith.³² The first electron search, at the University of Michigan in 1959, used free electrons.³³ Atoms replaced free electrons in 1964, with Sandars and Lipworth's caesium beam,³⁴ and in 1968 Weisskopf and colleagues pushed the caesium limit below 3 × 10⁻²⁴ e·cm.³⁵ Thallium later took over at Berkeley: Eugene Commins and colleagues published (1.8 ± 1.2 ± 1.0) × 10⁻²⁷ e·cm in 1994,³⁶ and Regan, Commins, Schmidt and DeMille set a limit of 1.6 × 10⁻²⁷ e·cm in 2002.³⁷ Then came molecules.

Year	Group · system	Limit (e·cm, 90%)	Earth-sized bump	Tighter than 1968
1968	Weisskopf et al. · caesium	3 × 10⁻²⁴	68 µm	1
1994	Commins et al. · thallium	3.8 × 10⁻²⁷*	86 nm	790
2002	Regan et al. · thallium	1.6 × 10⁻²⁷	36 nm	1,900
2011	Hudson et al. · YbF	1.05 × 10⁻²⁷	24 nm	2,900
2014	ACME I · ThO	8.7 × 10⁻²⁹	2.0 nm	34,000
2017	Cairncross et al. · HfF⁺	1.3 × 10⁻²⁸	2.9 nm	23,000
2018	ACME II · ThO	1.1 × 10⁻²⁹	0.25 nm	270,000
2023	Roussy et al. · HfF⁺	4.1 × 10⁻³⁰	0.093 nm	730,000

A selection: the Particle Data Group lists further results between 1968 and 1994, such as caesium (1989) and thallium (1990).² *The 1994 abstract gives a value, not a limit; 3.8 × 10⁻²⁷ is this page's 90% bound from it, a Gaussian folded at zero with the two errors combined. Earth-sized bumps use the classical electron radius and 6,371 km; the last two columns are this page's arithmetic.

The Imperial group's ytterbium fluoride beam gave 1.05 × 10⁻²⁷ e·cm in 2011, below thallium's.⁹ ACME's first thorium monoxide result, in 2014, reached 8.7 × 10⁻²⁹ e·cm,³⁸ later revised to 9.4 × 10⁻²⁹ with an updated effective field of 78 GV/cm (ACME's 2017 methods paper and 2018 result).^39,26 In 2017 JILA's first trapped-ion experiment, with "spin interrogation times over 700 ms", reached 1.3 × 10⁻²⁸ e·cm:⁴⁰ a confirmation with a new method, not a tighter limit than 2014.

The steps are irregular. One was less than a factor of two (2002 to 2011); the first thorium monoxide result was about twelve times below the ytterbium fluoride one (this page's arithmetic); and one was a deliberate step sideways to a new technique. Over the whole run from 1968 to 2023 the limit has tightened by 3 × 10⁻²⁴/4.1 × 10⁻³⁰ = 7.3 × 10⁵, about 700,000-fold (this page's arithmetic). On an Earth-sized electron, the allowed bump has shrunk from about the width of a fine hair in 1968, to about the width of DNA in 2014, to less than a hydrogen atom in 2023.

Figure 4 steps through the ladder. Its second view, "All the way down", extends the axis to where the Standard Model's predictions sit, the subject of Chapter 10.

Figure 4🎲 Try it

The ladder of limits

Step through the years

Step through the experiments

Limit (90%)Upper limit

On an Earth-sized electron

Tighter than 1968Tighter than 1968 by

Above the molecular Standard Model backgroundAbove the Standard Model's signal in these molecules

Upper limits on the electron EDM at 90% confidence, by publication year, as in the table above; the open ring is this page's 90% bound from the published 1994 thallium value, (1.8 ± 1.2 ± 1.0) × 10⁻²⁷. The band is ACME 2018's range of typical predictions for new physics at 1–100 TeV. Standard Model lines as in Chapter 10: the molecular equivalent, about 1.0 × 10⁻³⁵ e·cm, and the electron's own EDM, about 5.8 × 10⁻⁴⁰ e·cm; dashed, the 2014 benchmark of about 10⁻³⁸ e·cm (Pospelov and Ritz), which the 2022 calculation replaced. This page's arithmetic. As of October 2026 no tighter limit has been published, as far as we can find.Each dot is one experiment's upper limit (the open ring is this page's own reading of a 1994 result): the electron's off-centre charge must be smaller than this. Limits are in e·cm: the electron's charge times a centimetre, the unit for an off-centre charge. Lower is rounder. The shaded band is where, by ACME's 2018 estimate, many ideas for new heavy particles (of 1 to 100 TeV, a unit of mass that chapter 10 explains) would usually put the EDM. Step through the years to watch the allowed bump on an Earth-sized electron shrink, then press Play and keep going down to where the Standard Model's prediction sits, far below.

The electron's searches have sisters, which this page mentions only for context. The neutron's dipole moment is below 1.8 × 10⁻²⁶ e·cm (Paul Scherrer Institute, 2020), and the mercury-199 atom's below 7.4 × 10⁻³⁰ e·cm at 95% confidence (2016).^41,42 Those systems are most sensitive to CP violation among quarks and gluons, including the θ̄ term of Chapter 4, rather than to the electron.

Chapter 10What the Standard Model expects

"Round, as far as anyone can measure" is not the same as "round". The Standard Model breaks CP (Chapter 4), so it gives the electron a dipole moment, but a minute one: the CP violation it is known to have lives in the mixing of quarks and reaches the electron only through loops of virtual particles (Chapter 11 says how loops work). Counted at the level of quarks, every contribution up to three loops vanishes, and a four-loop estimate gives less than 10⁻⁴⁴ e·cm. In 2020 Yasuhiro Yamaguchi and Nodoka Yamanaka found a larger contribution, from hadrons, the composite particles quarks make, acting over longer distances:

d_e(Standard Model) ≈ 5.8 × 10⁻⁴⁰ e·cm (uncertain by about 70%)Yamaguchi and Yamanaka 2020; the 2026 review by Pospelov and Ritz cites it as bringing d_e to about 10⁻⁴⁰ e·cm. The 2023 limit is 4.1 × 10⁻³⁰/5.8 × 10⁻⁴⁰ = 7.1 × 10⁹ times larger (this page's arithmetic).

That is about seven billion times below today's limit.^43,24 But it is not what these experiments would see first. A molecule in a state with unpaired electrons, like the ³Δ₁ states of HfF⁺ and ThO that these experiments use (physicists call it paramagnetic), responds to the combination of Chapter 8: the electron's dipole moment plus the CP-violating electron–nucleus force C_S. In 2022 Yohei Ema, Ting Gao and Maxim Pospelov showed that in the Standard Model the second term wins. They found C_S ≈ 7 × 10⁻¹⁶, which in these molecules mimics an electron dipole moment "three orders of magnitude larger than previously believed", in the authors' words:⁴⁴

d_e^equiv ≈ 1.0 × 10⁻³⁵ e·cmEma, Gao and Pospelov 2022. The 2023 limit is about 4 × 10⁵ times larger (this page's arithmetic).

The earlier benchmark, from Pospelov and Adam Ritz in 2014, had put the Standard Model's molecular background near 10⁻³⁸ e·cm.⁴⁵ The thousandfold revision is a reminder that theory moves too. The 2026 review puts the gap as "roughly five orders of magnitude" of experimental progress before the Standard Model's own CP violation becomes a background.²⁴

Drawn as an equivalent bump on an Earth-sized electron, the molecular signal would be 0.23 fm, about a quarter of a proton's radius (0.84 fm, as in Nº 5), though no charge offset on the electron corresponds to it. The electron's own Standard Model dipole moment would give 1.3 × 10⁻²⁰ m, about 1/64,000 of a proton's radius. Both are this page's arithmetic; Figure 1's Standard Model chips put them on its ruler.

So the experiments are, in effect, background-free: with the known background hundreds of thousands of times below today's sensitivity, any dipole moment detected soon would be new physics.^26,1,24

Chapter 11Seeing mountains you cannot climb

How can a tabletop say anything about particles too heavy to make? Through quantum loops. A new particle that couples to the electron and breaks CP can appear briefly as a virtual particle around the electron and leave a dipole moment behind, even if no collider can make it for real. The heavier the particle, the fainter its trace, so finding no trace sets a lower bound on the mass: seeing mountains you cannot climb, with the caution that how high you "see" depends on assumed couplings and phases.

The JILA paper gives the simplest one-loop estimate: the dipole moment grows with the square of the new particle's coupling to the electron, g, and with the sine of its CP-violating phase, and falls as the square of its mass M:¹

d_e ~ (eƛ_C/2) (g²/2π) sin φ_CP (m_e/M)² ⇔ M ~ g m_ec² √(ƛ_C sin φ_CP / (4π d_e/e))Roussy et al. 2023, eq. (5), written there with a₀α/2 = ƛ_C/2. g is the new particle's coupling to the electron, φ_CP its CP-violating phase, M its mass. The prefactor eƛ_C/2 is μ_B/c (Chapter 3). An order-of-magnitude estimate.

The team's headline is M ≳ (g/α^½) × 40 TeV: for couplings about as strong as electromagnetism, g² ≈ α ≈ 1/137, and a maximal CP phase, the limit points to new particles heavier than about 40 TeV, which the paper calls an order of magnitude beyond the largest-mass particles the LHC can detect directly. The abstract is more cautious, claiming constraints on "broad classes of new physics above 10¹³ electron volts", 10 TeV.¹ The same formula, solved by this page with sin φ = 1 at the 2023 limit, gives:

Coupling (this page's labels)	g²	One-loop M
Order-one coupling	1	about 440 TeV
Electroweak strength	≈ 1/30	about 80 TeV
Electromagnetic strength (the paper's headline)	1/137	38 TeV
Very feeble coupling	10⁻⁶	0.44 TeV

This page's arithmetic (M ≈ 442g TeV, CODATA 2022 constants); the paper quotes 40. The JILA paper's illustrative values for the strong, electromagnetic and weak forces are g² ~ 1, 1/137 and 10⁻⁶; this page names the first and last by size only, since 10⁻⁶ is small because the W boson is heavy, a suppression the formula's (m_e/M)² already counts.

The reach grows only as the square root of the improvement in the limit: the paper's like-for-like 2.4-fold tightening raises it 1.5 times, and on the two published figures the electromagnetic-strength reach went from 23 TeV with ACME's 2018 limit to 38 TeV (this page's arithmetic). Read the other way, a 3 TeV particle with those couplings and a maximal phase would have given a dipole moment about 160 times the 2023 limit (this page's arithmetic; 3 TeV is an example).

If the new particles reach the electron less directly, through two loops, the reach is lower. ACME, with its own formula and couplings of electroweak strength, estimated that its 2018 limit reached about 30 TeV at one loop and 3 TeV at two, for a maximal CP phase, and that theories with new particles at 1–100 TeV "typically predict" dipole moments of 10⁻²⁷ to 10⁻³⁰ e·cm.²⁶ The 2026 review by Pospelov and Ritz works a two-loop example, a CP-violating coupling of the top quark to the Higgs boson: today's limit reaches an operator scale Λ_ht of about 10 TeV, normalised to a phase with sin φ_t = 1/√2 (about 12 TeV at sin φ_t = 1, this page's arithmetic). That is the scale of an effective coupling, not the mass of a particle.²⁴

The review's summary is that dipole-moment sensitivity to CP-odd effective operators "stretches to the O(10 TeV) scale". Supersymmetry, the proposal that every known particle has a heavier partner, is the classic test case. In the review's reading, the Higgs boson's mass already pushes the partners in supersymmetry's generic weak-scale versions above about 10 TeV, and dipole moments now reach those scales: tens of TeV when the partners are alike across all three generations of particle ("flavour-universal"), and 100 TeV and above only in a case the review calls "admittedly optimistic".²⁴ Dipole moments do not rule out supersymmetry; they squeeze it.

For comparison, in its third run (2022–2026) the LHC collided protons at 13.6 TeV, but that energy is shared among the quarks and gluons inside the protons, so it makes new particles directly only up to a few TeV. Run 3 ended on 27 June 2026, and the machine is due to restart as the High-Luminosity LHC in mid-2030.⁴⁶

Models can also dodge the limit with cancellations, or with CP violation confined to heavy quarks; a 2025 paper by Aiko, Endo, Kanemura and Mura builds a model with a second Higgs field (a two-Higgs-doublet model) that could make the universe's matter (Chapter 12) and avoids the limits without cancellations.⁴⁷ Cesarotti and colleagues discuss how to interpret the limit.⁴⁸ Figure 5 lets you set the assumptions.

Figure 5🎲 Try it

Seeing mountains you cannot climb

Choose the assumptions

Coupling to the electronHow strongly they touch the electron

CP-violating phase φHow much they break the backwards-film rule

LimitWhich limit

One-loop mass scale MPushes new particles above

A 3 TeV particle would giveA 3 TeV particle with these couplings would give

Limit used

AssumptionsAssumed

Order-of-magnitude estimates of the mass scale an EDM limit reaches. One loop: the estimate of Roussy et al. 2023 (eq. 5) given above, solved for M. Coupling presets are the labels of the table above; electroweak strength uses g² ≈ 1/30 (ACME 2018 uses α_eff = 4/137). Two loops: the CP-odd top–Higgs operator of Pospelov and Ritz (2026 review chapter), d_e = 4.1 × 10⁻³⁰ e·cm × (10 TeV/Λ)² × sin φ/(1/√2), an operator scale. The shaded foothills are the few TeV the LHC can reach directly. This page's arithmetic; a dial, not a wall.How heavy could new particles be and still leave a mark on the electron? It depends on how strongly they touch the electron and how much they break the backwards-film rule. Choose those and a limit, and the peak shows how far that limit 'sees': a rough estimate, not a wall. The two-loop button shows one example where the particles reach the electron less directly; then the reach is lower. The shaded foothills are what the LHC can make directly.

So the reach is a range, not a number. The 2023 limit reaches from below 1 TeV (one loop, a very feeble coupling such as g² ~ 10⁻⁶, or a small CP phase) to several hundred TeV (one loop, g ~ 1), with two-loop examples near 10 TeV; the JILA team's headline is 40 TeV for couplings of electromagnetic strength.

Chapter 12Why there is matter at all

There is one more reason to look for new CP violation, and it is the matter around us. The universe holds about six protons and neutrons (together, baryons) for every ten billion photons, and essentially no antimatter. In numbers, the baryon-to-photon ratio is

η = n_b/n_γ ≈ 274 × 10⁻¹⁰ × Ω_bh² = 274 × 10⁻¹⁰ × 0.02237 = 6.1 × 10⁻¹⁰Ω_bh² = 0.02237 ± 0.00015 from Planck 2018; the conversion factor, 274 × 10⁻¹⁰, is the one in the Particle Data Group's review of Big-Bang nucleosynthesis, and the product is this page's arithmetic.

That value rests on the Planck satellite's measurement of the density of ordinary matter.⁴⁹ The same ratio comes out of Nº 4's census of the particles around us, with its 411 microwave-background photons in every cubic centimetre, and the same Planck measurement underpins the hot Big Bang of Nº 1. In a universe that began hot and symmetric, matter and antimatter should have been made in equal amounts and annihilated almost completely. Something tipped the balance.

In 1967 Andrei Sakharov set out what that something needs:⁵⁰

Baryon number must change: Some process must turn a state with no excess of protons and neutrons into one with an excess.
C and CP must be violated: Otherwise every process that makes matter has a mirror process, at the same rate, that makes antimatter.
Thermal equilibrium must be broken: In equilibrium, any excess made is unmade at the same rate.

The Standard Model has the ingredients, but not in the amounts needed. Its CP violation is far too weak: in the words of the 2026 Pospelov–Ritz review, it "apparently fails by many orders of magnitude in explaining the magnitude of the observed baryon to photon ratio".²⁴ And with a Higgs boson of 125 GeV (0.125 TeV), the Standard Model's electroweak transition, the epoch in the early universe when the Higgs field took on its non-zero value, is a smooth crossover. Electroweak baryogenesis, making the matter excess at that transition, would need an abrupt, out-of-equilibrium event instead, to meet Sakharov's third condition.^51,52,53 So the asymmetry needs new physics, very probably new CP violation.

That is where the electron comes in. Many proposals put the new CP violation near the electroweak scale, where it would usually give the electron a measurable dipole moment, and the null results squeeze them: in a well-studied CP-violating two-Higgs-doublet model, a 2025 analysis by Biekötter and Olea-Romacho finds the predicted electron dipole moments "typically exceed the experimental limits by at least an order of magnitude".⁵⁴ Electroweak baryogenesis is under pressure; it is not dead.

Other routes survive. Leptogenesis, proposed by Fukugita and Yanagida in 1986, makes the asymmetry first among leptons, the electron's family of particles, at very high energies, and need not give the electron a detectable dipole moment;⁵⁵ and some electroweak scenarios evade the limits, like the two-Higgs example of Chapter 11.⁴⁷ A null result narrows the options; it neither explains the asymmetry nor rules out every idea that could.

Chapter 13The next generation

Several groups are building the experiments that might beat 4.1 × 10⁻³⁰ e·cm. As of October 2026 none has reported a better limit, and what follows is mostly plans and first steps.

JILA's third generation: ThF⁺. JILA's next experiment uses thorium fluoride ions, whose dipole-moment-sensitive ³Δ₁ state is the electronic ground state, "immune to decoherence from spontaneous decay". The group's 2022 paper predicts that, with the ions kept in a 180 K environment, that state would stay coherent for about 20 seconds, which the paper called a tenfold improvement on its HfF⁺ experiment of the time; at room temperature, blackbody radiation would cut its effective lifetime to about 3 s, no longer than the 2023 HfF⁺ measurement's precession. The paper also plans "a conveyor belt of ion traps" to raise the count rate.⁵⁶ The group has since published a high-efficiency way to detect the ions' states,⁵⁷ and ThF⁺'s effective field is calculated at 35.2 GV/cm.²⁷

ACME III: ThO. ACME's third generation is being built. In August 2026 a Michigan State University group led by Xing Wu, an ACME III collaborator, reported the first optical cycling of thorium monoxide, scattering many laser photons from each molecule (Physical Review Research); the authors project that it would more than double the statistical sensitivity of an ACME-style search.⁵⁸ A paper in APS Open Science (May 2026) describes a beam source with about a 40% long-term gain in yield,⁵⁹ and a preprint describes a magnetic shield with three layers rather than five around a precession region five times longer than ACME II's, with magnetic systematic errors estimated to be about 40 times smaller.⁶⁰ We found no published overall improvement factor, and this page quotes none.

Other molecules. At Imperial College, laser-cooled ytterbium fluoride beams have been used for spin interferometry (2026).⁶¹ The NL-eEDM collaboration's supersonic beam of barium fluoride has made a first 34-hour measurement, d_e = (2 ± 3) × 10⁻²⁵ e·cm (Eur. Phys. J. D, 2026).⁶² The EDM³ collaboration and an Italian group are developing barium fluoride frozen in solid neon, with apparatus and spectroscopy papers so far.^63,64 Polyatomic molecules held in optical lattices, grids of laser light, have been projected to reach a statistical error of 10⁻³⁰ e·cm: a projection, not a measurement.⁶⁵

Chapter 14Round, so far

Five things the 2023 result does not mean:

Perfectly round: Round to within the limit. The Standard Model itself predicts a dipole moment, about 6 × 10⁻⁴⁰ e·cm, far below it (Chapter 10).
A measured shape: An energy shift that a dipole moment would cause was looked for and not seen; "shape" is a picture of the resulting limit (Chapter 3).
A size for the electron: The electron behaves as a point; the 0.09 nm comes from a chosen yardstick, and a smaller one would make the bump bigger (Chapter 2).
A wall at 40 TeV: A dial set by assumptions, from below 1 TeV to several hundred TeV (Chapter 11).
An explanation of matter: It squeezes some ways of making the matter in the universe and leaves others open (Chapter 12).

Where things stand, as of 25 October 2026: the best limit is JILA's 4.1 × 10⁻³⁰ e·cm, used alone by the Particle Data Group's 2026 listing;² without assuming the dipole moment is the only source, ThO and HfF⁺ together give 2.1 × 10⁻²⁹ e·cm. JILA's ThF⁺ experiment and ACME III are being built, and no new experiment has reported a better limit, as far as we can find.

The thread through all of it is one number in e·cm and the ways of seeing it: round to one part in 70 million billion on the smallest standard length; a bump of at most 0.093 nm on an Earth-sized electron; and, for any new particles that break CP and couple to the electron, a demand that they be heavier, more feebly coupled or more cleverly arranged than the simplest models allowed. A few hundred ions counted in a trap in Boulder, again and again for two months, showed no extra tick.

An electron the size of the Earth would be round to within a hydrogen atom.

Glossary

Electric dipole moment (EDM), e·cm: A charge times the distance by which it sits off-centre. For the electron it is quoted in e·cm, the electron's charge times a centimetre.
Spin: The electron's intrinsic angular momentum, ħ/2: the only direction an electron has, so any EDM lies along it.
Magnetic moment, Bohr magneton μ_B: The strength of the electron's built-in magnet, very nearly one Bohr magneton, eħ/2m_e.
Axial and polar vector: A polar vector, like a position or an electric dipole, flips under parity; an axial vector, like spin or a magnetic moment, does not. Spin and the magnetic moment flip under time reversal instead.
Three standard lengths: The classical electron radius r_e = 2.818 fm, the reduced Compton wavelength ƛ_C = 386 fm and the Bohr radius a₀ = 0.0529 nm, each a factor of the fine-structure constant α ≈ 1/137 apart. None is a measured size.
P, T and CP (and CPT): Parity inverts space; time reversal runs the film backwards; C swaps particles for antiparticles. The weak force breaks P; the CPT theorem makes a broken T a broken CP.
Effective electric field, E_eff: The field an electron inside a polar molecule responds to, tens of GV/cm; calculated, not measured.
Schiff's theorem: In a neutral system of non-relativistic point charges, the field on each particle averages to zero. Relativity breaks it in heavy atoms.
Zeeman splitting: The energy difference between spin orientations in a magnetic field; in HfF⁺ it sets the experiment's 100 Hz tick.
Ramsey method: Two pulses around a free precession turn the phase the spin gathers into a population of ions that can be counted; JILA ran it on two oppositely oriented clouds, analysed blind.
90% confidence limit: A value the true one is below with 90% confidence, given the data and the statistical method.
C_S (electron–nucleon coupling): A CP-violating force between electrons and nucleons that mimics an EDM in molecules. The Standard Model's molecular signal comes mostly through it.
Standard Model and the CKM matrix: The theory of the known particles. The CKM matrix describes how quarks mix; through it the Standard Model breaks CP, and only weakly (Jarlskog invariant J ≈ 3 × 10⁻⁵).
Quantum loop: A contribution in which virtual particles briefly appear and vanish; it lets particles too heavy to make leave traces.
Operator scale Λ: The energy scale of an effective coupling, not a particle mass.
Supersymmetry: A proposed symmetry pairing every known particle with a heavier partner, whose CP phases would usually give the electron an EDM.
Two-Higgs-doublet model (2HDM): An extension of the Standard Model with a second Higgs field, which can add new CP violation.
Sakharov conditions: Baryon-number violation, C and CP violation, and departure from equilibrium: what it takes to make more baryons (protons, neutrons) than antibaryons. Today η ≈ 6.1 × 10⁻¹⁰ baryons per photon.
Electroweak baryogenesis: Making the matter excess during the electroweak transition; it needs an abrupt transition and new CP violation.
Leptogenesis: Making a lepton excess first, at very high energies, and converting it to baryons; it need not give a detectable electron EDM.

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