# Proton and Neutron mass

Can we calculate the difference in mass between a Proton and Neutron?
If so, how?

[ Moderator: why did you move this from QM to general physics? ]

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## Answers and Replies

Hi Passion Flower.

neutron is slightly heavier

Mass of proton : 1,6726 x 10^(-27) kg
Mass of neutron: 1,6749 x 10^(-27) kg
Mass of electron: 0,00091x10^(-27) kg

The mass of a neutron is greater than the mass of a
proton because the neutron contains a proton, contains
an electron with some subatomic particles.

neutron = proton + electron + subatomic particles

Hi Passion Flower.

neutron is slightly heavier

Mass of proton : 1,6726 x 10^(-27) kg
Mass of neutron: 1,6749 x 10^(-27) kg
Mass of electron: 0,00091x10^(-27) kg

The mass of a neutron is greater than the mass of a
proton because the neutron contains a proton, contains
an electron with some subatomic particles.

neutron = proton + electron + subatomic particles
So what is the formula?

In energy units (using E = mc^2), the masses are: Proton: 938.272 MeV, neutron:
939.566 MeV, mass difference = 1.293 MeV, electron: 0.511 Mev.

In energy units (using E = mc^2), the masses are: Proton: 938.272 MeV, neutron:
939.566 MeV, mass difference = 1.293 MeV, electron: 0.511 Mev.
I do not think you understand what I am asking. I am asking if we can calculate the masses or at least the difference in mass. And if so, then what is the formula.

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mathman
Science Advisor
From Wikipedia

Outside the nucleus, free neutrons are unstable and have a mean lifetime of 885.7±0.8 s (about 14 minutes, 46 seconds); therefore the half-life for this process (which differs from the mean lifetime by a factor of ln(2) = 0.693) is 613.9±0.8 s (about 10 minutes, 14 seconds). Free neutrons decay by emission of an electron and an electron antineutrino to become a proton, a process known as beta decay:

n0 → p+ + e + νe

From Wikipedia

Outside the nucleus, free neutrons are unstable and have a mean lifetime of 885.7±0.8 s (about 14 minutes, 46 seconds); therefore the half-life for this process (which differs from the mean lifetime by a factor of ln(2) = 0.693) is 613.9±0.8 s (about 10 minutes, 14 seconds). Free neutrons decay by emission of an electron and an electron antineutrino to become a proton, a process known as beta decay:

n0 → p+ + e + νe
I do not want to be difficult but is my question really so hard to understand?

jtbell
Mentor
I am asking if we can calculate the masses or at least the difference in mass. And if so, then what is the formula.

I suspect you are really asking if we can predict the difference in mass from first (or at least deeper) principles.

The mass of a neutron is greater than the mass of a
proton because the neutron contains a proton, contains
an electron with some subatomic particles.

neutron = proton + electron + subatomic particles

This is absolutely not correct. A proton consists of two u quarks, a d quark and a set of virtual quark pairs and gluons. A neutron looks exactly like that with a d quark substituted for one of the u quarks.

I suspect you are really asking if we can predict the difference in mass from first (or at least deeper) principles.
Well if we can predict something that implies we can calculate it right?

At any rate, yes, so if the answer is yes I like to see some formulas.

fzero
Science Advisor
Homework Helper
Gold Member
Hi Passion Flower.

neutron is slightly heavier

Mass of proton : 1,6726 x 10^(-27) kg
Mass of neutron: 1,6749 x 10^(-27) kg
Mass of electron: 0,00091x10^(-27) kg

The mass of a neutron is greater than the mass of a
proton because the neutron contains a proton, contains
an electron with some subatomic particles.

neutron = proton + electron + subatomic particles

This model of the neutron has been known to be inaccurate for the last 45 years, which you might have realized if you kept reading on the web page you took that from.

We now understand that protons and neutrons are composed of quarks. The proton is composed of 2 up quarks and 1 down quark, while the neutron is composed of 2 down quarks and 1 up quark. There's no simple explanation for the exact difference in the mass between the proton and neutron. It's known that the down quark is slightly heavier than the up quark, so the neutron could be predicted to be slightly heavier than the proton. However, the difference between quark masses only explains a small amount of the neutron-proton mass difference.

The remaining mass difference can only be explained by the strong interaction between the constituent quarks of each particle. This interaction within the proton is stronger, since the proton is stable and the neutron is not. So the proton has more binding energy, again leading to a smaller mass than the neutron. However precisely computing the mass difference is not as simple as writing down a formula, since just about all approximation schemes break down when considering the strong interaction at low energies. Lattice QCD is the most promising computational method for answering such questions, and it gives correct results for nucleon masses to within a few percent.

The remaining mass difference can only be explained by the strong interaction between the constituent quarks of each particle. This interaction within the proton is stronger, since the proton is stable and the neutron is not. So the proton has more binding energy, again leading to a smaller mass than the neutron. However precisely computing the mass difference is not as simple as writing down a formula, since just about all approximation schemes break down when considering the strong interaction at low energies. Lattice QCD is the most promising computational method for answering such questions, and it gives correct results for nucleon masses to within a few percent.
Ok Lattice QCD it is then.

Within a few percent is ok.

So what is the formula?

fzero
Science Advisor
Homework Helper
Gold Member
Ok Lattice QCD it is then.

Within a few percent is ok.

So what is the formula?

There is no simple formula to be written down. Lattice QCD starts with the definition of QCD observables via path integrals, discretizing on a lattice (along with techniques that took many years to develop) allows these integrals to be done numerically. The resulting spectrum can be converted to (ratios) of particle masses, but I'm not familiar (and not many people outside of the experts are either) with the details. If you want to see some general ideas about lattice methods, you could take a quick look at http://arxiv.org/abs/hep-lat/0506036 One of the more recent computations of light hadron masses appears in http://arxiv.org/abs/0906.3599 but there aren't many formulas there either.

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Borek
Mentor
From the second paper:

In the work presented here, a full calculation of the light hadron spectrum in QCD,
only three input parameters are required: the light and strange quark masses and the coupling g.

I can be misreading something, but it looks to me like calculated mass of hadron is still a function of something we can't calculate from the first principles - unless we assume quark mass to be the first principle...

PAllen
Science Advisor
From the second paper:

I can be misreading something, but it looks to me like calculated mass of hadron is still a function of something we can't calculate from the first principles - unless we assume quark mass to be the first principle...

My understanding is that the masses of quarks would come from the Higgs mechanism, thus outside QCD. This paper is as fundamental as it gets for QCD.

There is no simple formula to be written down.
There is no need for it to be simple. We passed the time long ago where we had to calculate things by hand.

If you want to see some general ideas about lattice methods, you could take a quick look at http://arxiv.org/abs/hep-lat/0506036
That looks helpful, in Python nevertheless. I would prefer Maple, any worksheets?

Can we do the full calculation in Maple or mathematica (I prefer Maple).
If the answer is no then what software is used to calculate it?

We certainly can do path integrals, Wick rotations and Monte Carlo stuff in Maple. Discretizing on a lattice I am not sure. What is quenched approximation? It seems if we use that then "the Standard Model explains" is not very relevant statement to the method of calculation or do I see that incorrectly?

My understanding is that the masses of quarks would come from the Higgs mechanism, thus outside QCD. This paper is as fundamental as it gets for QCD.
You are probably right that we cannot explain the absolute value of a mass but I am already satisfied if we can calculate the relative masses and according to the second paper "the Standard Model should explain the difference". However rather than asking for an explanation, which is tricky (at least to me) in QM anyway I rather see, how we can actually compute this on our desktop computers.

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fzero
Science Advisor
Homework Helper
Gold Member
From the second paper:

I can be misreading something, but it looks to me like calculated mass of hadron is still a function of something we can't calculate from the first principles - unless we assume quark mass to be the first principle...

There's a computational obstacle that I don't completely understand related to doing computations with light quark masses. Obtaining physical results is done by some sort of scaling procedure (in masses, this is separate from the lattice spacing) and they must require some extra data to complete the definitions of physical results.

My understanding is that the masses of quarks would come from the Higgs mechanism, thus outside QCD. This paper is as fundamental as it gets for QCD.

Almost, but due to renormalization, the quark masses at the electroweak scale are very different from those at GeV scales. There are other papers that compute light quark masses and coupling renormalization at GeV scales, but they also use light pion masses as inputs.

That looks helpful, in Python nevertheless. I would prefer Maple, any worksheets?

Can we do the full calculation in Maple or mathematica (I prefer Maple).
If the answer is no then what software is used to calculate it?

I'd expect many of the lattice codes are still written in Fortran and run on parallel supercomputers. I don't think that Maple or Mathematica are well-suited yet to doing these things. While you could probably put all of the formulas into a notebook, neither program would do the computations in a particularly optimal manner. It's still better to explicitly code hard computations.

We certainly can do path integrals, Wick rotations and Monte Carlo stuff in Maple. Discretizing on a lattice I am not sure. What is quenched approximation?

Quenched means that corrections from fermion loops are being neglected. This is because dealing with Grassmann variables on a lattice is a very hard problem in the first place and the approximation schemes that must be used are computationally intensive.

A side question then would be how computationally intensive is the full calculation. So do I understand correctly that there are two main approaches, e.g. perturbation and QCD lattice and that quenched optimization is a shortcut for the second approach which is a shortcut to the perturbation method?

So how computationally intensive is it? Wasn't this done decades ago? As far as I understand the computing power decades ago can now easily be reproduced on even a laptop.

PAllen
Science Advisor
A side question then would be how computationally intensive is the full calculation. So do I understand correctly that there are two main approaches, e.g. perturbation and QCD lattice and that quenched optimization is a shortcut for the second approach which is a shortcut to the perturbation method?

So how computationally intensive is it? Wasn't this done decades ago? As far as I understand the computing power decades ago can now easily be reproduced on even a laptop.

From the paper on computation of neutron/proton masses (light hadron masses):

Computations were performed on the Blue Gene supercomputers at FZ J¨ulich and IDRIS
and on clusters at Wuppertal and CPT. This work is supported in part by EU grant I3HP,
OTKA grants AT049652, DFG grants FO 502/1-2, SFB-TR 55, EU grants RTN contract
MRTN-CT-2006-035482 (FLAVIAnet), (FP7/2007-2013)/ERC no208740 and the CNRS’s
GDR grant 2921. Useful discussions with J. Charles and M. Knecht are acknowledged

FYI: this paper is considered a landmark culination of 3 decades or work on QCK

PAllen
Science Advisor
A side question then would be how computationally intensive is the full calculation. So do I understand correctly that there are two main approaches, e.g. perturbation and QCD lattice and that quenched optimization is a shortcut for the second approach which is a shortcut to the perturbation method?

So how computationally intensive is it? Wasn't this done decades ago? As far as I understand the computing power decades ago can now easily be reproduced on even a laptop.

From the paper on computation of neutron/proton masses (light hadron masses):

Computations were performed on the Blue Gene supercomputers at FZ J¨ulich and IDRIS
and on clusters at Wuppertal and CPT. This work is supported in part by EU grant I3HP,
OTKA grants AT049652, DFG grants FO 502/1-2, SFB-TR 55, EU grants RTN contract
MRTN-CT-2006-035482 (FLAVIAnet), (FP7/2007-2013)/ERC no208740 and the CNRS’s
GDR grant 2921. Useful discussions with J. Charles and M. Knecht are acknowledged

FYI: this paper is considered a landmark culmination of 3 decades of work on lattice QCD methods.

desktop computers.
Computing the nucleon mass is one of the most expensive calculation done from lattice supercomputers. Not maple on a desktop. Besides, calculating the nucleon mass means that the calculation is not accurate enough to distinguish proton and neutron. There are other calculations dedicated to this difference
http://arxiv.org/abs/hep-lat/0608023

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missed the second page of the discussion before posting

Blue Gene supercomputers at FZ J¨ulich and IDRIS
and on clusters at Wuppertal and CPT.
Am I wrong in thinking we can already have similar speeds with NVIDIA's CUDA technology?

Computing the nucleon mass is one of the most expensive calculation done from lattice supercomputers. Not maple on a desktop. Besides, calculating the nucleon mass means that the calculation is not accurate enough to distinguish proton and neutron. There are other calculations dedicated to this difference
http://arxiv.org/abs/hep-lat/0608023
Interesting article. Do we know what kind of hardware this was computed with?

The apparent lack of computing power makes me wonder, how were those claims that the Standard Model can accurately calculate the relative masses ever verified say 30 years ago? I guess they were not right?

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PAllen
Science Advisor
Am I wrong in thinking we can already have similar speeds with NVIDIA GPU's?

Interesting article. Do we know what kind of hardware this was computed with?

The apparent lack of computing power makes me wonder, how were those claims that the Standard Model can accurately calculate the relative masses ever verified say 30 years ago? I guess they were not right?

Where did you see such claims? I thought the early successes of QCD were all about predicting high energy reactions accurately. Surprisingly, this is much simpler than computing masses. I have a friend who worked on lattice QCD years ago - he likened it to computing weather, only harder.

fzero
Science Advisor
Homework Helper
Gold Member
The apparent lack of computing power makes me wonder, how were those claims that the Standard Model can accurately calculate the relative masses ever verified say 30 years ago? I guess they were not right?

Calculating the relative masses is a huge part of verifying QCD, but there are many other tests. At energies large compared to the mass of the proton, the QCD coupling is much smaller (what's called asymptotic freedom) and perturbation theory is valid. So a huge body of evidence for the correctness of QCD was the agreement between perturbation theory and experiment at those scales.

See also:
http://news.sciencemag.org/sciencenow/2008/11/21-02.html [Broken]

Calculating the relative masses is a huge part of verifying QCD, but there are many other tests. At energies large compared to the mass of the proton, the QCD coupling is much smaller (what's called asymptotic freedom) and perturbation theory is valid. So a huge body of evidence for the correctness of QCD was the agreement between perturbation theory and experiment at those scales.
Got ya.

Looks from the article we at least have the first step:

From the article:

The experimental neutron-proton mass difference:

1.2933317±0.0000005 MeV

The EM contributions is:

em = −0.76±0.30 MeV
(estimated using the Cottingham sum-rule saturated by the Born diagrams),

Difference due to strong isospin breaking:

d−u= 2.05∓0.30 MeV

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