# How come “smaller, weaker” particles are more massive (have higher energies)?

Something has always struck me as counter-intuitive: when reading about high-energy experiments such as the LHC, they are always looking for stuff on a really small scale with MASSIVE energies.

I guess this quote illustrates it (Wikipedia: Gravitons):

Attempts to extend the Standard Model or other quantum field theories by adding gravitons run into serious theoretical difficulties at high energies...

Shouldn't something really small in size be really low in energy?

Take the Higgs boson for instance. I just can't grasp it. If this particle is mediating a fundamental interaction that isn't very strong, how come it isn't much easier to see? Wouldn't we be swimming in a soup of them?

Finally, with gravitons (Wikipedia: Gravitons again):

For example, a detector with the mass of Jupiter and 100% efficiency, placed in close orbit around a neutron star, would only be expected to observe one graviton every 10 years, even under the most favorable conditions.

But, I mean, doesn't gravity exist everywhere?

I run into this apparent paradox in regard to all kinds of particles, when reading about physics. I'm sure it's just some fundamental point I'm not getting.

In high energy physics the energy scale is very important. As you said matter is probed at smaller and smaller distances, and that requires more energy. Why is that?

Well in natural units ($c = \hbar = 1$) we have some quantities that mix with each other, i.e there is very little difference between them (mainly just a proportionality constant) In particular:

$$[Velocity] = number$$ $$[Energy] = [Mass] = [Momentum]$$ and $$[Mass] = [Length]^{-1}$$

From this it follows that $[Energy]$ is actually just inverse $[Length]$ hence the smaller the distances probed, the higher the energy scales.

If these above relations seem strange think of them like this. The highest achievable velocity is the speed of light $c$ and we already set that to one by our choice of natural units. This means that any other velocity will range from $0 \leq v \leq 1$ thus its a scalar.

Also, from $E^2 = (pc)^2 + (mc^2)^2$ it follows that $E^2 = p^2 + m^2$. The last one, which is the core of your question follows from the fact that $\hbar/(mc)$ has units of length and in natural units it becomes $m^{-1}$.

To finish up, your last point about gravity follows from the fact that gravitons interact very weakly at the energy scales we are probing because gravity only becomes relevant at extremely small distances of the order of the Planck Length,

$$\ell _{{\text{P}}}={\sqrt {\frac {\hbar G}{c^{3}}}}\approx 1.616\;199(97)\times 10^{{-35}}{\mbox{ m}}$$

This equates to huge energies that we have no access to currently. All this done above is called dimensional analysis.

Don't consider the Higgs boson as a fundamental interaction because it is not. The reason we need high energies to produce the Higgs is for a different reason. As others pointed out, the Higgs is an excitation of the Higgs field. The boson itself is very massive. Remember mass = energy. To produce a massive boson you need to supply at least enough energy to produce its mass. This kind of energy is not available in our everyday lives. Only the LHC has enough power to produce energy scales that high. But that doesn't affect other particles interacting with the Higgs field to gain mass.

For a subatomic particle, the gravitational effects are extremely small due to their tiny masses. For gravity to become relevant for individual particles we need to investigate them at the planck length scales. But gravity in general is relevant in the universe, and that is because astronomical objects are very massive and their combined mass produces gravitational fields that have observable effects.

• I'm surprised that my question was understood so easily – thanks for the quick and relevant answer. I'm afraid, though, that the part about gravitation once again violates my intuition: I've always read that gravity only becomes relevant at large scales, such as astronomical scales and doesn't play any role on small scales because it's a very weak interaction? – dalgard Jan 6 '15 at 14:00
• I understand about gravity now: in simple terms, the distance a single particle may be swerved by gravity is really small, but put them all together and it adds up. Thanks! – dalgard Jan 6 '15 at 14:08
• @dalgard Thats right! – Constandinos Damalas Jan 6 '15 at 14:12
• @dalgard The reason gravity dominates on large scales is that it's always additive - while for example the electromagnetism of a magnet is just the residual electromagnetic force of its particles (most of them negate each other), there's no "antigravity" to balance gravity. It's still extremely weak (think about how easy it is for you to overcome the gravity of the whole planet while throwing a ball, for example), but it will always increase in force with increasing mass/energy. When talking about individual particles, gravity loses this advantage, of course, so it's just a very weak force. – Luaan Jan 7 '15 at 8:12
• Why does mass scale with the inverse of length? – 0 0 Jan 7 '15 at 9:38

One should add a word of caution with regards to gravity. Gravity does not behave like any of the other forces. It is therefor not even clear that gravity can be quantized, which is well documented in over half a century of utter failure to produce a self-consistent quantum theory of gravity. The clues that gravity may not be a force like all others are actually pretty obvious in the thermodynamics of black holes, which look a lot more like classical phase transition phenomena than macroscopic quantum states. It is therefor perfectly possible that gravity is a thermodynamics remnant of another force which we haven't seen, yet.

• Good point. I didn't mean to assume, just used it as an example. – dalgard Jan 6 '15 at 15:46
• Nobody knows for sure. There may be a QFT of gravity, after all, or there may not be... it's all up in the air and nature doesn't want to give this one up easily. – CuriousOne Jan 6 '15 at 16:01

in the case of the Higgs boson, very high energies are needed to create it because it is quite massive. The Higgs boson is best understood as an excitation of the Higgs field, and a good way to excite this field (since it couples to mass) is to "pluck" it with an interaction involving other pretty massive particles, such as the top quark. It is still difficult to excite this field though, and this difficulty is in some way a measure of the mass of the Higgs boson (the energy of this excited state of the Higgs field). Perhaps somebody with a better background in QFT can give a better picture of this than I can, as I am certainly no expert here.

For other particles, such as the graviton, they can be very hard to detect because they don't interact in easily detected ways. Neutrinos are a great example of a particle that is all over the place (millions are passing through you every second), but only interact via the weak force (ignoring their low mass gravitational interaction). It's hard to detect neutrinos because they mostly just pass through out detectors, and we have to work hard to create favorable conditions for their detection (look up Super-K or Ice Cube for an idea of what neutrino detection entails). Gravitons, as far as I know, should only interact gravitationally, making it even harder to detect them, since Gravity is such a weak force compared to the other fundamental forces. You would also have to be very careful to distinguish graviton events from other events (like neutrino events). This is also no easy task. Thus it would be hard to detect individual gravitons because of the way they interact.

• Interesting about the Higgs boson. So I should look to quantum field theory in order to understand how it works? – dalgard Jan 6 '15 at 16:14
• But if the boson mediates a force and the force is there all the time, wouldn't the particle – or rather, the excitation – be there all the time, too, for anyone to see? – dalgard Jan 6 '15 at 16:14
• You should look to QFT in order to understand the Higgs field, and then understand the Higgs boson as an excitation of that field. Note that the Higgs boson does not mediate any force though. – JotThisDown Jan 6 '15 at 17:36
• Gravitons, on the other hand, are theorized to mediate the gravitational force. Two particles would interact gravitationally by exchanging virtual gravitons in much the same way that two electrons interact electromagnetically by exchanging virtual photons. These virtual gravitons cannot be detected (even in theory) for the same reason that virtual photons cannot be detected. Thus just because there is gravity around us does not mean that the virtual gravitons that are mediating it can be easily detected. – JotThisDown Jan 6 '15 at 17:38
• If (non-virtual) gravitons are generated by some physical process, then they could in principle be detected, but doing so would be infeasible thanks to the fact that they only interact via the gravitational force. This is again akin to the fact that we can detect photons that are generated by physical processes (an electron changing energy levels for example), although in the case of these photons, detection is much easier because they interact electromagnetically. – JotThisDown Jan 6 '15 at 17:40

by the uncertainty principle , if uncertainty in position is very small for probing smaller distances then uncertainty in momentum is very large so in energy .and energy is length inverse in natural units.so probing smaller distances requires very high energy.

strength of fundamental forces comes from their coupling constants.if the interaction is long range like electromagnetism then exchange particle is photon which is very light. similarily for weak forces, exchange particle W-Z bosons are more massive. it is again given by uncertainty principle.

yes we are swimming in the higgs-field ,it is background field that is all over the universe but the mass of higgs bososn is very high(125.6 gev) ,so it requires very high energy to create them in the lab.

yes , gravity is everywhere but Unambiguous detection of individual gravitons, though not prohibited by any fundamental law, is impossible with any physically reasonable detector. The reason is the extremely low cross section for the interaction of gravitons with matter.it is very weak.