## Friday, June 27, 2014

### Up Quark, Down Quark Mass Difference Known With Unprecedented Precision

The News

June 18, 2014 paper estimates that differences in electromagnetic field strength between the proton and neutron account for 1.04 +/- 0.11 MeV of the proton-neutron mass difference using lattice QCD methods, an estimate three or four times more precise that previous state of the art estimates of this component.  The proton gets more of its mass from its electromagnetic field than the neutron does.  (An April 11, 2014 power point description of this paper is also available.)

The same paper estimates, using this calculation, that the difference between the up quark mass and the down quark mass is 2.33 +/- 0.11 MeV.  This is the most precise estimate of the up quark-down quark mass difference to date.  The previous state of the art estimate had a best fit value of 2.5 MeV (within the two sigma confidence interval of the new result), but with an uncertainty of about 0.7 MeV.

Some Details For Experts

The paper assumes that the difference in the strength of the strong force field between the proton and neutron is assumed to be negligible relative to the differences in quark masses and electromagnetic field strengths.  This seems intuitively reasonable, because up quarks and down quarks have identical strong force couplings to each other and both have very small masses relative to the total mass of a nucleon, while having very different electromagnetic couplings.

Put another way, QCD contributions to the mass splitting between the proton and the neutron are almost entirely a function of the difference between the rest masses of the up and down quarks.  This is true even though the gluon field accounts for more than 98% of the mass of protons and neutrons respectively, while the contributions of the rest masses of the quarks themselves is modest.

The dominant source of uncertainty in this estimate arises from lack of clarity over whether the dipole form factor of the inelastic subtraction term in the contribution of the electromagnetic field strength to the mass of the proton and neutron scales as a cubic or quartic polynomial (i.e. whether an exponent in an obscure subpart of the overall equation is 3 or 4).  If it is cubic, then the actual value is higher by 0.09 and the uncertainty drops from +/- 0.11 MeV to +/- 0.04 MeV.  If it is quartic, then the actual value is lower by 0.09 and the uncertainty drops by the same amount.

This paper replicated the result of another paper released in pre-print form by an independent group of lattice QCD investigators just two days earlier on June 16, 2014, using a moderately different approach, so the result can be considered quite reliable.  This paper's result for the difference in mass due to electromagnetic field strength between the proton and neutron is 1.00 MeV +/- 0.16 MeV, and an up quark-down quark mass difference of 2.52 MeV+/- 0.29 MeV.  But, it predicts a total difference between the proton and neutron mass of 1.51 +/- 0.28 MeV (compared to the physical value of 1.2933322 MeV), which is not inconsistent with experiment but has a central value that is high by about 16%.

Incidentally, while both papers have produced highly precise measurements of the difference in nucleon mass attributable to the electromagnetic field energy in these baryons, I am not able to cite any source regarding the absolute value of this contribution of the masses of the proton and neutron, relative to the contribution from the gluon field in these baryons (and to the extent that the two are interrelated this could conceivably be an ill defined quantity).

Both results exploit the Coleman-Glashow relation (which dates to at least 1982 or earlier) which argues that the sum of the mass differences in three different pairs of charged and neutral hadrons (one of which is the proton and neutron pair) with carefully chosen combinations of light scalar differences and other factors that cancel out due to symmetries, should equal zero.  This hypothesis is confirmed experimentally true to the current limits of experimental precision (something that a 2000 paper called a "miracle", but which flows quite naturally from the quark model of QCD and symmetry considerations).

The Masses And Mass Differences Of Exclusively or Predominantly Light Quark Hadrons

Protons and Neutrons

A neutron has a rest mass of 939.565,379(21) MeV.  A proton has a rest mass of 938.272,046(21) MeV. The difference between the rest mass of the neutron and the rest mass of a proton is known to somewhat greater precision; it is 1.293,332,2(4) MeV.

The rest mass of an electron is 0.510,998,928(11) MeV.

The difference between the rest mass of a neutron and the sum of the rest masses of a proton and an electron is 0.782,333,2(4) MeV.  In other words, this is the minimum energy of a photon produced in ordinary beta decay.  About one part in 1201 of the rest mass of a neutron is converted into energy in ordinary beta decay.  The other 1200 out of 1201 parts of the rest mass of a neutron is converted in other kinds of rest mass.

A neutron has one up quark and two down quarks, producing a zero net electric charge.  A proton has two up quarks and one down quark, producing a +1 net electric charge.  Both have total angular momentum equal to 1/2.

Quantum chromodynamics (QCD) estimates of the proton and neutron mass from first principles are accurate to about +/- 1% (i.e. about +/- 1 MeV), although QCD estimates of the proton-neutron mass difference are now significantly more precise than that (while still remaining far less precise than experimental measurements of this quantity).

Due to confinement, no light quark is ever observed at an energy scale of less than that of a pion (about 140 MeV for charged pions and 136 MeV for neutral pions), which is the lightest particle made up of quarks, and the only pseudoscalar mesons (total angular momentum of o and negative parity) made up only of light quarks.

The vector mesons, with total angular momentum 1, that are made up only of up and down quarks are the rho mesons (vector) (cursive p) which has a mass of 775.11 MeV when charged and 775.49 MeV when neutral, and the omega mesons (vector) (cursive lowercase w) which has a mass of 782.65 MeV.

All four of the delta baryons, which are three quark particles made up of combinations of up and/or down quarks with total angular momentum 3/2 (unlike the 1/2 of the proton and neutron) have masses of about 1232 MeV.

The lighest scalar meson (with total angular momentum of 0 and positive parity), the f0(500) has a mass of about 500 MeV and does not have a consensus interpretation of its makeup in a simple quark composite model, although it is sometimes interpreted as primarily consisting of linear combinations of pions, which are in turn made up only of up and down quarks.

Interpreting the Proton-Neutron Mass Difference

Some of the difference in rest mass between a neutron and proton may be attributable to a difference in mass between an up quark and a down quark.  Some of the difference in rest mass between a neutron and a proton may be attributable to a difference in the amount of energy in the strong force field and electromagnetic field in the proton and neutron respectively.

To only slightly oversimplify, using the canonical values for the up and down quark masses (discussed further below), the quarks in a proton (about 9.4 MeV) are 2.5 MeV lighter than in the neutron (about 11.9 MeV), but the proton has combined contributions of the strong force fields and electromagnetic fields that are 1.2 MeV stronger (928.9 MeV in the proton v. 927.7 MeV in the neutron), a difference of about a tenth of a percent in field strength.

The question of how much of this difference is due to differences between up quark and down quark rest masses, and how much of this differences is due to gluon and photon fields in the proton and neutron is highly model dependent.

The particle data group estimate of the mass difference between the up and down quarks is 2.5 MeV, rather than the 1.3 MeV that one would naively expect from the difference between the neutron mass and the proton mass, a value that would struggle to fit in the two sigma error bars of current estimates of absolute masses and mass ratios for these quarks.

According to the Particle Data Group, the up quark has an estimated mass of 2.3 MeV + 0.7/-0.5 MeV, and the down quark has an estimated mass of 4.8 MeV +0.5/-0.3 MeV.  Viewed together, rather than in isolation, the up quark rest masses is estimated to be from 0.38 to 0.58 of the rest mass of the down quark.  The precision with which we know the up and down quark masses is roughly ten million times less than the precision with which we known the difference in rest masses between the proton and neutron.  These estimates are little improved from estimates made at the very dawn of the quark model in the 1970s.

In late March or early April of 2010, a paper in Physical Review Letters by Christine Davies and others (pre-print here and pdf here) made a much more precise estimate of 2.01 +/- 0.14 MeV for the up quark and 4.79 +/- 0.16 MeV for the down quark, which implies a difference between the two rest masses of 2.78 +/- 0.21 MeV (and a low end 0.42 mass ratio), which is just barely consistent with the new results at the two sigma level.  But, the PDG worldwide average analysis has not proclaimed this result to be correct at that level of precision.

Definitional Issues In Light Quark Mass Determinations

Light Quark Pole Masses

Even the definitions of the light quark masses are fraught with problems.  In the Standard Model, the mass of a particle varies with the energy-momentum scale at which it is measured.  In the case of top quarks, bottom quarks, and charm quarks, it is possible and sensible to measure the "pole mass" of a particle - i.e. the mass of a quark measured at an energy scale equal to its rest mass.

In the case of the light quarks, the rest masses of the quarks are customarily evaluated at energy scales of 2 GeV (i.e. at about the mass of two nucleons that collide with each other), at which they are observed (in a confined context), rather than at the rest mass of the quarks themselves.

But, for example, extrapolation of the running of the quark masses suggests that the light quarks should be about 35% heavier at a 1 GeV energy scale.  Naively extrapolating the running of the light quark masses downward from 2 GeV to estimate their "pole mass" produces masses in the hundreds of MeVs, but a more accurately interpretation is that the pole masses of the light quarks are simply ill defined and the extrapolation is being applied beyond the scale where it is valid.  Instead, different definitions of quark masses than pole mass, such as the MS mass scheme are used to generalized the concept of quark masses to the light quarks.  But, it isn't obvious that MS mass is as fundamental a quantity as pole mass.
Koide (1994) calculates the running of the three light quark masses down to their pole masses, even though these values have little practical application, in both a five quark and three quark flavor model. In the five quark flavor model he comes up with pole masses for the up quark of 346.3 MeV, for the down quark of 352.4 MeV and for the strange quark of 489 MeV. In a three quark flavor model he comes up with pole masses of 163.1 MeV for the up quark, 169 MeV for the down quark, and 338 MeV for the strange quark. One could get somewhat lower light quark pole mass values still in a two quark flavor model.
Koide updated these calculations in 1997 and concluded that the pole mass of the up quark was 0.501 GeV, the pole mass of the down quark was 0.517 GeV and the pole mass of the strange quark was 0.687 GeV (based on their measured values at other energy scales), although all sub-1 GeV values were noted with an "*" mark.
A more recent update of the calculations can be found at Xing (2008) does not consider masses running to very low energy scales for light quarks, explaining that "The pole masses of three light quarks are not listed, simply because the perturbative QCD calculation is not reliable in that energy region."
Notably, these perturbative QCD calculations become ill defined at masses higher than the mass of the pion.  So, perturbative QCD calculations break down to the point of being unreliable at energy scales somewhere between 140 MeV and 1000 MeV.

The pole masses in the five quark flavor models are quite similar to the conventional dressed quark masses for the up and down quarks discussed below, however.

Dressed Masses For Light Quarks

Another approach is the look at the "dressed quark" mass which for the up and down quarks is about 0.32 GeV (i.e. about a third of the proton mass, which is the lightest three quark composite particle containing only light quarks), which includes a proportionate share of the gluon field mass of a baryon that contains a light quark in its mass.  This is also sometimes described as a "constituent quark model."  (The link in this paragraph is to a 2013 power point presentation that is quite a good starting point for understanding the state of the effort to determine the quark masses.)

Recent state of the art estimates of dressed quark masses determined with Lattice QCD, however, are closer to 0.25 GeV for up and down quarks, and 0.502 GeV for strange quarks.  This implies fundamental light quark masses of about 8.6 MeV and fundamental strange quark masses of about 227 MeV at an energy scale of about 840 MeV.  These values are about 2.45 times the canonical values for these masses at 2 GeV energy scales.  I may be misinterpreting this analysis, but this seems to me to indicate that the total contribution of the electromagnetic field of a proton to a proton's mass is about 188 MeV.

While "dressed quark" masses for the up and down quark can produce not obviously wrong estimates of hadron masses for most hadrons containing just up and down quarks (and also, for example, for the eta meson which is a mix of up, down and strange quarks with a mass of 547.85 MeV).

But, dressed quark masses for the up and down quarks do seem inconsistent with the masses of the pions.  Charged pions have two light quarks but have a mass less than the dressed mass of even one light quark, and neutral pions which are linear combinations of up-antiup and down-antidown quark pairs are even lighter.  In constituent quark models, the binding energy of the pion implied by the dressed quark masses for the up and down quarks must be approximately negative 360 MeV.  But, it isn't obvious that negative energies are are a physical concept in this context.

An alternate way of understanding the light hadron masses which explains why the pion can be so light is explained in a fairly straightforward way here.  Pions play a special role as "Goldstone" bosons in QCD, i.e. force carriers who arise as a result of a broken symmetry, which give them special properties.

Basically, the right formula for the mass of a meson is equal to a constant scale factor times the positive square root of ((two times the average mass of the quarks in the meson) times (the appropriate value for the QCD scale)).  The QCD scale value, in turn, is a function of the number of quark flavors that are accessible at the energy level involved.  It is about 217 MeV when there is sufficient energy to involve five active quark flavors and abut 350 MeV when there is sufficient energy to involve three active quark flavors.  Arguably, it may be necessary in some cases to invoke a two active quark energy scale, although this involves complex self-referential calculations related to the strange quark mass.

Footnote

Another interesting amateur mass hierarchy paper with a numerological flair can be found here.

UPDATED July 2, 2014 for formatting purposes and to add material to the definitions section of this post.