# Missing mass in atomic systems

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• Loganwithallstate
In summary, Logan Knox aspires to understand the physics and nature of the quantum world, but acknowledges that he has a long way to go. He believes that asking the right questions is the first step in finding answers, and he is currently questioning the missing mass in a hydrogen system. He wonders if this missing mass is stored as energy in the form of kinetic or potential energy, and if so, where is it located and in what field. He also wonders if he is asking the wrong questions and is seeking guidance from experts in the field of quantum mechanics and quantum field theory.

#### Loganwithallstate

TL;DR Summary
99% of mass of hydrogen is not in the 3 quarks and the electron, where is it/ how is it expressed?
Hi! My name's Logan Knox and I'm aspiring to eventually understand the physics and nature of the quantum world in its totality, and I have a LONG way to go, but I have to get there by asking the right questions, and I think this is the first step to finding the right question to ask for this problem. The total mass of the up quarks, down quark, and electron in just a non isotope hydrogen system around 9.6 MeV using 1MeV= 1.78266e-30kg
9.611MeV=1.71e-29kg where as the mass of a typical hydrogen system is 1.67e-27kg which is 100x heavier! Since mass is energy I'm assuming (hopefully correctly) that this is energy stored up as either kinetic or potential energy (neither of which do I understand how they manifest on the field level, perhaps excess energy not bound by particles? remember I'm on a learning quest so if that's incorrect an explanation would mean the world) My question is where is the missing mass? Assuming a system with no momentum, where is the excess mass? If it is free energy where is that energy stored, and what field does it occupy, it it energy that has not manifested into particles on the electron field, up quark field, both, neither? I would love to truly understand this problem, but I'm assuming I'm asking the wrong questions, and if so, what is the right one?

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Hello @Loganwithallstate,
!​

Loganwithallstate said:
in terms of QFT what field stores the energy?
I don't speak that language, and I suspect you don't either ?

But in QFTlayman's terms: the quarks are very much bound together by a force that is so strong that we call it the strong force . That's where this energy is sitting.

You could redo the exercise on a proton and an electron versus a hydrogen atom and come out a lot better !
But then, that's QM and not QFT, I suppose !

##\ ##

BvU said:
Hello @Loganwithallstate,
!​

I don't speak that language, and I suspect you don't either ?

But in QFTlayman's terms: the quarks are very much bound together by a force that is so strong that we call it the strong force . That's where this energy is sitting.

You could redo the exercise on a proton and an electron versus a hydrogen atom and come out a lot better !
But then, that's QM and not QFT, I suppose !

##\ ##
Thank you! I was overthinking and jumping down rabbit holes! I looked past the energy of the forces! question though, If I were to measure the energy of the strong force in the atom, where would I find it? For example, is it located in the gluon field? As a practice I am trying to break down just a hydrogen atom's energy/mass into as fundamental blocks as possible.

As I said: don't speak QCD or QFT. I speak a little Google, but so do you.
Better to invoke higher authorities -- of which PF has plenty. You might help them by adding a little context about yourself: it makes a difference if you are close to a PhD in theoretical physics or still in undergrad.

@vanhees71 , @mfb, @Nugatory ?

BvU said:
As I said: don't speak QCD or QFT. I speak a little Google, but so do you.
Better to invoke higher authorities -- of which PF has plenty. You might help them by adding a little context about yourself: it makes a difference if you are close to a PhD in theoretical physics or still in undergrad.

@vanhees71 , @mfb, @Nugatory ?
Thank you for your help and patience! I suppose when it came to google, I could not find any information, I am completely sure it has to do with my phrasing/ not knowing what words to use, every result would be of high school level physics. I understand my phrasing being off, English is not my first language, so I'm having difficulty wording my questions, I appreciate your patience and understanding! Conceptually, I have a decent understanding of QFT and QP, however both my mathematical understanding and my english both need work XD

Well, I 'discovered' my answer is nonsense: if it's a simple binding energy, then the proton mass would be lower than the sum of three quark masses, wouldn't it ?

I go into read-only mode for this thread (better for all) ! Welcome to PF nevertheless

The question, how hadrons (among them particularly the protons and neutrons making up the atomic nuclei of everyday matter) get their masses is a very difficult one and subject of current research.

On the fundamental level the strongly interacting particles in the Standard Model of elementary particle physics are the quarks and gluons. The quarks are spin-1/2 fermions (Dirac particles) carrying electric charges of 2/3 and -1/3 elementary charges and another kind of charge, called the "color charge". Each sort ("flavor") of quarks come in three colors "red", "blue", "green". Then there are the antiquarks with the opposite electric charges of -2/3 and +1/3 elementary charges and the color charges "anti-red", "anti-blue", and "anti-green". The gluons are, similar to the photons in the electromagnetic interactions, massless spin-1 particles, being electrically neutral and carrying color charges of 8 types similar to pairs of color and anti-color of the quarks, but one of the 9 possible combinations of color and anti-color is color-neutral and thus doesn't make up a gluon.

Finally in the Standard Model the matter particles are grouped in 3 families, which are all exact copies of one another except of the fundamental masses these particles have due to the interaction with the famous Higgs-boson field. In each family are 2 sorts of quarks, one with electric charge +2/3 and one with -1/3. In the 1st family you have up- and down-quarks, in the 2nd family, charm- and strang-quarks, and in the 3rd family, top- and bottom-quarks.

Now in contradistinction to the photons the gluons carry themselves color charge and are thus directly interacintg with themselves. The amazing consequence is that the strong interaction becomes weak for reactions between quarks and gluons with large momentum transfer, and only for this high-energetic reactions you can use perturbation theory. In our low-energy everyday-world, however, we don't even see free quarks and gluons but the particles participating in the strong interaction are the hadrons, which turn out to be mainly either a bound state of a quark and an anti-quark (which are bosons and called mesons) or a bound state of 3 quarks (which are fermions and called baryons). All these bound states carry zero net-color charge, i.e., we only observe color-neutral bound states of quarks and gluons.

The proton and the neutron are the lowest baryon states and (together with electrons, which don't participate in the strong interactions) make up the matter around us. Now you cannot make experiments with free quarks and gluons, because there are simply none of them available but you have to make experiments with the hadrons (particularly protons) to find out about the strong interactions and the properties of the quarks and gluons. If you let, e.g., high-energetic electrons collide with protons, you can resolve the quark substructure and figure out how the proton looks like in the sense of a little lump of strongly interacting matter. As it turns out the naive picture of simply binding three quarks together is in fact too naive, and this has to do with the above described confinement property, i.e., that the color-charg-carrying quarks and gluons are always "confined" in color-neutral hadrons.

One important tool to evaluate the quantum field theory of the strong interactions, color chrmodynamics (QCD), is lattice-QCD, which are computer codes evaluating the theory on a discretized space-time lattice (with time being imaginary time, i.e., calculating in the socalled "Euclidean QFT"). With these computer codes you can, e.g., predict how the masses of the hadrons as bound states of quarks and gluons come out and particularly which masses of the various quark flavors you must assume. As it turns out for the light u- and d-quarks making up the protons and neutrons, you get masses of the order of about ##10 \text{MeV}/c^2##. Now the protons and neutrons have a mass of about ##940 \text{MeV}/c^2##.

Thus, e.g., a proton cannot be a simple bound state of the three quarks in the same way as a hydrogen atom is a bound state of a proton and an electron. In this case you'd indeed expect that the bound state has a mass given by the sum of the masses of the constituents minus the binding energy divided by ##c^2##, but in the case of the proton the mass is much larger already than the sum of the above determined quark masses of a few ##\text{MeV}/c^2##. More precisely one calls these quark masses the "current quark masses", and these are the masses appearing in the Standard-Model Lagrangian and are due to the coupling of the quarks to the Higgs field and the Higgs field's vacuum expectation value.

The rest of the mass of the proton thus is dynamically generated by the strong interaction. There is no simple way to understand it from first principles, i.e., from QCD, because in this low-energy regime you cannot use perturbation theory, though lattice QCD tells us that QCD is indeed the right theory to describe strongly interacting particles, including the hadrons as bound states of quarks and gluons and the right "effective degrees of freedom". So we have to rely on effective models to try to understand the mechanism behind this enigmatic "mass generation".

One very intuitive but not really correct model is the socalled MIT bag model. The idea is that around the "constituent quarks" (or "valence quarks") making up the proton (two up + 1 down quark, leading to the right charge of 2/3+2/3-1/3=1 elementary charge of a proton) a kind of "bag" is formed in the interacting QCD vacuum, which is far from being a naive non-interacting vacuum describing simply the absence of particles. The interacting QCD vacuum rather forms both a "quark condensate", i.e., the vacuum expectation value ##\sum_{f} m_f \langle bar{\psi}_f \psi \rangle \neq 0## and a gluon condensate (giving rise to the socalled "trace anomaly"). The MIT bag model now simply assumes that around the valence quarks of the proton a kind of cavity in the interacting QCD vacuum is formed within which the three valence quarks are confined. The diameter of the bag is of the order of 1 fm and the quarks are always in motion within this bag. The corresponding kinetic energy provides most of the mass of the proton, and the three valence quarks are rather "constituent quarks" with an effective mass of 1/3 of the proton mass.

More modern models of the hadrons are more abstract, making use of the socalled "accidental symmetries" of QCD. In the light-quark sector (i.e., looking only at u and d, and also including the somewhat heavier s quarks) the most important symmetry is the socalled chiral symmetry, and indeed chiral models of various kinds lead to a successful description of the strong-interaction properties of hadrons. Of course the strong interaction of hadrons is analogous to the electromagnetic interactions between neutral atoms and molecules, because the hadrons are color neutral, but this "residual forces" hold the atomic nuclei together, which can be described as bound states of protons and neutrons. Because the residual interaction is much weaker than that between the quarks and gluons making up the hadrons, here you can use the usual approximations of many-body quantum theory to understand the properties of the various nuclei in the periodic table of the elements, nowadays even based on ab-initio calculations of the many-body interactions based on chiral symmetry.

One way to experimentally test all these complicated properties of quarks and gluons and the hadrons made up of them are relativistic heavy-ion collisions as done at the LHC at CERN, the Relativistic Heavy Ion Collider (RHIC) at BNL, and GSI here in Germany (with the new FAIR accelerator being under construction right now) etc. Here you let collide (heavy) nuclei at very high energy, and what's formed in these collisions are a lot of quarks and gluons forming a hot and dense medium called the "quark gluon plasma". The reason is that many quarks and gluons are close together in this fireball and colliding at quite high energies. In this circumstances the usually formed hadrons "melt" and the right effective degrees of freedom to describe the fireball are quark-gluon like quasiparticles forming a collectively expanding medium which can pretty well be described by relativistic hydrodynamics. This fireball, however, is rapidly expanding and cooling down, and finally the quarks and gluons are bound together to hadrons again. Because the QGP phase of the fireball lasts just a few ##\text{fm}/c##, we can never observe directly the QGP but must read off the properties of this very "exotic" state of matter from the measured hadron (as well as photon and lepton) spectra which are in fact detected in these experiments.

One of the most interesting questions is the investigation of the phase diagram of strongly interacting matter, with various phase transitions (confinement-deconfinement transition, chiral transition). Here one has to use all kinds of theory (lattice QCD as the most fundamental approach but also effective hadronic models, relativistic hydro dynamics and transport theory etc. etc.) to describe the data from heavy-ion collisions.

Another fascinating way to figure out how the strong interacting matter looks like is now also opened by the detection of gravitational waves. Here the signals coming from "neutron-star mergers" are most important. This is the collision of two neutron stars, which formed a binary star system moving around each other as the planets move around the Sun, but loosing also energy due to the radiation of gravitational waves (which however cannot yet directly detected). Thus they come closer and closer together and finally circling around each other very rapidly and finally merging together and finally forming a new neutron star or a black hole. The gravitational wave signal of this final stage of rapid motion is detectable by gravitational-wave detectors like LIGO/VIRGO and from the detailed gravitational-wave signal one can figure out the properties of the neutron-rich matter making up the neutron stars. In addition, if you are lucky, you can also measure the electromagnetic waves of such an event, also called a "kilo nova". This has been the case for one such neutron-star merger yet, and one could measure not only the gravitational wave signal but also the electromagnetic singals on various wave lengths. From all these signals one can learn a lot about the equation of state of (neutron-rich) strongly interacting matter which is surprisingly close in density and temperature as expected in the collisions of atomic nuclei at intermediate energies as being realized already today in the "beam-energy-scan program" at RHIC and being also one of the main motivations to build the new "Facility for Antiproton and Ion Research" (FAIR) under construction here in Germany.

Paul Colby, etotheipi, BvU and 1 other person
Loganwithallstate said:
Summary:: 99% of mass of hydrogen is not in the 3 quarks and the electron, where is it/ how is it expressed?

Hi! My name's Logan Knox and I'm aspiring to eventually understand the physics and nature of the quantum world in its totality, and I have a LONG way to go, but I have to get there by asking the right questions, and I think this is the first step to finding the right question to ask for this problem.
This is definitely not the right place to start. Like all of us, you have to begin at the beginning.

What's you current level of physics knowledge and mathematical capability?

BvU