I would lay down my life for two brothers or eight cousins.- J.B.S. Haldane, from here.

# Dispatches From Turtle Island

Observations That Transcend Law and Politics

## Saturday, February 28, 2015

## Tuesday, February 24, 2015

### A Theoretical Case For The Higgs Boson As A Composite Particle Made Of Gauge Bosons

The case for the Higgs boson as a scalar combination of electroweak gauge bosons in a gauge invariant fashion is made in a preprint by F.J. Himpsel (of the University of Wisconsin at Madison Physics Department who primarily works in condensed matter physics; another of his interesting ideas about fundamental physics and additional explanation of the one described in the preprint is found here). The preprint makes a theoretical case for a Higgs boson as a composite object made of gauge bosons whose mass is half of the Higgs vacuum expectation value (vev) at the tree level before adjustment for higher order loops that bridge the gap between the tree level estimate and the experimentally observed value.

The paper is notable because it takes these related notions from being mere numerology, to something with some plausible theoretical foundation and can close the gaps between rough first order theoretical estimates and reality in a precise, calculable, falsifiable way. There is a lot of suggestive evidence that he is barking up the right tree. And, there is much to like about the idea of a Higgs boson as a composite of the Standard Model gauge bosons as discussed in the final section of this post.

The preprint discusses that the Higgs boson mass is close to half of the Higgs vev, 123 GeV (actually closer to 123.11 GeV), a 2% difference. This gap is material, on the order of a five sigma difference from the experimentally measured value, but the explanation for the discrepancy is interesting and plausible (some mathematical symbols translated into words):

The combined margin of error weighted experimental value of the

The experimentally measured value of the

The other fundamental particle masses in the Standard Model are as follows:

* The

* The

* The

* The

* The

* The

* The

* The

* Each of the

* The

* The

*

The only other experimentally determined Standard Model constants not set forth above are:

* The strong force coupling constant (about 0.1185 +/- 0.0006 at the Z boson mass energy scale per the Particle Data Group).

* The U(1) electroweak coupling constant g', which is known with exquisite precision. I believe that the value of g is about 0.65293 and that the value of g' is about 0.34969 (both of which are known with much greater precision).

* The four parameters of the CKM matrix, which are known with considerable precision. In the Wolfenstein parameterization, they are λ = 0.22537 ± 0.00061 , A = 0.814+0.023 −0.024, ρ¯ = 0.117 ± 0.021 , η¯ = 0.353 ± 0.013.

* The four parameters of the PMNS matrix; three of which are known with moderate accuracy. These three parameters are theta12=33.36 +0.81/-0.78 degrees, theta23=40.0+2.1/-1/5 degrees or 50.4+1.3/-1.3 degrees, and theta12=8.66+0,44/-0.46 degrees.

None of these are pertinent to the issues discussed in this post. All of these except the CP violating phase of the PMNS matrix and the quadrant of one of the PMNS matrix parameters has been measured with reasonable precision.

General relativity involves two constants (Newton's constant which is 6.67384(80)×10

In addition, Planck's constant (6.62606957(29)×10

A small number of additional experimentally determined constants may be necessary to describe dark matter phenomena and cosmological inflation.

The intermediate equations in the tree-level analysis in the paper suggests how the 125.98 +/- 0.03 GeV/c^2 value that the simple 2H=2W+Z formula suggests could be close to the actual result. If this similar formula were true, the combined electroweak fits of the W boson mass, top quark mass and Higgs boson mass favor a value at the low end of that range, perhaps 125.95 GeV/c^2.

* * *

An alternative possibility, in which one half of the Higgs boson mass equals exactly the sum of the squares of the massive fundamental bosons of the Standard Model (and the sum of the squares of the masses of the fundamental fermions has the same value as the sum of the squares of the boson masses), i.e., for Higgs vev=V, Higgs boson mass=H, W boson mass=W, and Z boson mass=Z:

H^2=(V^2)/2-W^2-Z^2, implies a Higgs boson mass of 124.648 +/- 0.010 GeV, with combined combined electroweak fits favoring a value at the high end of this range (perhaps 124.658 GeV). (This would imply a top quark mass of about 174.1 GeV which is consistent with the current best estimates of the top quark mass; a slightly lower top quark mass of 173.1 GeV would be implied if the Higgs boson mass were, for instance, 125.95 GeV; both of these top quark masses are within 1 standard deviation of the experimentally measured value of the top quark mass).

This could also have roots in the analysis in the paper, which includes half of the square of the Higgs vev, the W boson mass, and the Z boson mass in its analysis (and has terms for the photon mass that drop out because the photon has a zero mass).

* * *

Thus, both alternative possibilities (2H=2W+Z and sum of F^2= sum of B^2= 1/2 Higgs vev) are roughly consistent with the experimental evidence, although they are clearly inconsistent with each other using pole masses for each boson.

It is also possible, however, that the correct scale at which to evaluate the masses in these formulas might, for example, be closer to the Higgs vev energy scale of about 246 GeV, and that at that scale, both formulas might be true simultaneously. Renormalization of masses as energy scales increase shrink the fundamental gauge boson masses more rapidly than they shrink the fundamental fermion masses, and an energy scale at which the sums of the squares of the fermion masses equal the sum of the squares of the boson masses is less than 1,000 GeV (i.e. 1 TeV) and not less than the value determined using the pole masses of the respective fundamental particles.

In a naive calculation using the best known values of the fundamental fermion and boson masses, the sum of the square of the fermion masses are not quite equal to the sum of the square of the boson masses (including the Higgs boson mass). But, the uncertainties in the top quark mass, the Higgs boson mass, and the W boson mass (in that order) are sufficient to make it unclear how the sum of the square of the fermion masses relate to the sum of the square of the boson masses either at pole masses or at any other energy scale. The uncertainties in these three squared masses, predominate over any uncertainties in the masses of the other 11 fermions, the Z boson (whose mass is known with seven times more precision than the W boson despite having a higher absolute value), and uncertainty in the value of the Higgs vev.

If these top quark and Higgs boson mass measurements could be made about three times more precise than the current state of the art experimental measurments, most of the current uncertainty regarding the nature of the Higgs boson mass and the relationship of the fundamental particle masses to the Higgs vev could be eliminated. The remainder of the LHC experiments will almost surely improve the accuracy of both of these measurements, but may be hard pressed to improve them by that much before these experiments are completed.

It is always a plus to be able to derive any experimentally determined Standard Model parameter from other Standard Model parameters, reducing the number of degrees of freedom in the theory. This would make electroweak unification even more elegant.

More deeply, if the Higgs boson is merely a composite of the Standard Model electroweak gauge bosons, then:

(1) The hierarchy problem as conventionally posed evaporates, because the Higgs boson mass itself is no longer fine tuned. The profound fine tuning of the Higgs boson mass in the Standard Model is the gist of the hierarchy problem. The demise of the hierarchy problem removes an important motivation for SUSY theories.

(2) Some of the issues associated with the hierarchy problem migrate to the question of why the W and Z boson mass scales are what they are, but if the Higgs boson mass works out to be such that the fermionic and bosonic contributions to the Higgs vev are identical, then the W and Z boson masses are a function of the fermion masses and an electroweak mixing angle, or equivalently, the fermion masses are a function of the electroweak boson masses and the texture of the fermion mass matrix.

(3) The spotlight in the mystery of the nature of the fermion mass matrix would cease to be arbitrary coupling constants with the Higgs boson and return squarely to W boson interactions which are the only means in the Standard Model by which fermions of one flavor can transform into fermions of another flavor.

(4) There are no longer any fundamental Standard Model bosons that are not spin-1 (the hypothetical spin-2 graviton is not part of the Standard Model). All fundamental Standard Model fermions are spin-1/2. Eliminating a fundamental spin-0 boson from the Standard Model changes the Lie groups and Lie algebras which can generate the Standard Model fundamental particles without excessive or missing particles in a grand unified theory.

(5) If the Higgs boson couplings are derivative of the W and Z boson couplings, then neutrinos, which couple with the W and Z boson, although only via the weak force, should derive their masses via the composite Higgs boson mechanism, just as other fermions in the Standard Model do. This implies that neutrinos should have Dirac mass just like other particles that derive their masses from the same interactions.

(6) A composite Higgs boson makes models with additional Higgs doublets less plausible, except to the extent that different combinations of fundamental Standard Model gauge bosons can generate these exotic Higgs bosons, which if they can, would then have masses and other properties that could be determined from first principles and would be divorced from supersymmetry theories.

(7) A composite Higgs boson might slightly tweak the running of the Standard Model coupling constants, influencing gauge unification at high energies (as could quantum gravity effects). A slight tweak of 1-2% to the running of one or more of the coupling constants over from the electroweak scale to the Planck or GUT scale (more than a dozen orders of magnitude), is all that would be necessary for the Standard Model coupling constants to unify at some extremely high energy. It may also have implications for the unstable v. metastable v. stable nature of the vacuum.

The paper is notable because it takes these related notions from being mere numerology, to something with some plausible theoretical foundation and can close the gaps between rough first order theoretical estimates and reality in a precise, calculable, falsifiable way. There is a lot of suggestive evidence that he is barking up the right tree. And, there is much to like about the idea of a Higgs boson as a composite of the Standard Model gauge bosons as discussed in the final section of this post.

The preprint discusses that the Higgs boson mass is close to half of the Higgs vev, 123 GeV (actually closer to 123.11 GeV), a 2% difference. This gap is material, on the order of a five sigma difference from the experimentally measured value, but the explanation for the discrepancy is interesting and plausible (some mathematical symbols translated into words):

The resulting value MH = ½v = 123 GeV matches the observed Higgs mass of 126 GeV to about 2%. A comparable agreement exists between the tree-level mass of the W gauge boson MW = ½gv = 78.9 GeV in (8) and its observed mass of 80.4 GeV. Such an accuracy is typical of the tree-level approximation, which neglects loop corrections of the order the weak force coupling constant = g^2/4pi which is approximately equal to 3%. It is reassuring to see the Higgs mass emerging directly from the concept of a Higgs boson composed of gauge bosons.The summary at the end of the paper notes that:

In summary, a new concept is proposed for electroweak symmetry breaking, wherethe Higgs boson is identified with a scalar combination of gauge bosons in gauge invariant fashion.That explains the mass of the Higgs boson with 2% accuracy. In order to replace the standard Higgs scalar, the Brout-Englert-Higgs mechanism of symmetry breaking is generalized from scalars to vectors.The ad-hoc Higgs potential of the standard model is replaced by self-interactions of the SU(2) gauge bosons which can be calculated without adjustable parameters. This leads to finite VEVs of the transverse gauge bosons, which in turn generate gauge boson masses and self-interactions.Since gauge bosons and their interactions are connected directly to the symmetry group of a theory via the adjoint representation and gauge-invariant derivatives, the proposed mechanism of dynamical symmetry breaking is applicable to any non-abelian gauge theory, including grand unified theories and supersymmetry.

In order to test this model, the gauge boson self-interactions need to be worked out. These are the self-energies [of the W+/- and Z bosons] and the four-fold vertex corrections [of the WW,WZ, and ZZ boson combinations]. The VEV of the standard Higgs boson which generates masses for the gauge bosons and for the Higgs itself is now replaced by the VEVs acquired by the W+/- and Z gauge bosons via dynamical symmetry breaking. Since the standard Higgs boson interacts with most of the fundamental particles, its replacement implies rewriting a large portion of the standard model. Approximate results may be obtained by calculating gauge boson self-interactions within the standard model, assuming that the contribution of the standard Higgs boson is small for low-energy phenomena. The upcoming high-energy run of the LHC offers a great opportunity to test the characteristic couplings of the composite Higgs boson, as well as the new gauge boson couplings introduced by their VEVs. If confirmed, the concept of a Higgs boson composed of gauge bosons would open the door to escape the confine of the standard model and calculate previously inaccessible masses and couplings, such as the Higgs mass and its couplings.

**Background: The Standard Model Constants**The combined margin of error weighted experimental value of the

**Higgs boson mass**as of the latest updated results from the LHC in September of 2014 is 125.17 GeV with a one sigma margin of error in the vicinity of about 0.3 GeV-0.5 GeV. In other words, there is at least a 95% probability that the true Higgs boson mass is between 124.17 GeV and 126.17 GeV, and the 95% probability range is probably closer to 124.37 GeV and 125.97 GeV.The experimentally measured value of the

**Higgs vev**is 246.2279579 +/- 0.0000010 GeV. Conceptually, this is a function of the SU(2) electroweak force coupling constant g and the W boson mass. In practice, it is determined using precision measurements of muon decays.The other fundamental particle masses in the Standard Model are as follows:

* The

**top quark mass**is 173.34 +/- 0.76 GeV (determined based upon the data from the CDF and D0 experiments at the now closed Tevatron collider and the ATLAS And CMS experiments at the Large Hadron Collider as of March 20, 2014).* The

**bottom quark mass**is 4.18 +/- 0.03 GeV (per the Particle Data Group). A recent QCD study has claimed, however, that the bottom quark mass is actually 4.169 +/- 0.008 GeV.* The

**charm quark mass**is 1.275 +/- 0.025 GeV (per the Particle Data Group). A recent QCD study has claimed, however, that the charm quark mass is actually 1.273 +/- 0.006 GeV.* The

**strange quark mass**is 0.095 +/- 0.005 GeV (per the Particle Data Group).* The

**up quark mass**and**down quark mass**, are each less than 0.01 GeV with more than 95% confidence, although the up quark mass and down quark pole masses are ill defined and instead are usually reported at an energy scale of 2 GeV.* The

**tau charged lepton mass**is 1.77682 +/- 0.00016 GeV (per the Particle Data Group).* The

**muon mass**is 0.1056583715 +/- 0.0000000035 GeV (per the Particle Data Group).* The

**electron mass**is 0.000510998928 +/- 0.000000000011 GeV (per the Particle Data Group).* Each of the

**three Standard Model neutrino mass eigenstates**(regardless of the neutrino mass hierarchy that proves to be correct) is less than 0.000000001 GeV.* The

**W boson mass**is 80.365 +/- 0.015 GeV.* The

**Z boson mass**is 91.1876 +/- 0.021 GeV.*

**Photons**and**gluons**have an exactly zero rest mass (as does the hypothetical**graviton**).The only other experimentally determined Standard Model constants not set forth above are:

* The strong force coupling constant (about 0.1185 +/- 0.0006 at the Z boson mass energy scale per the Particle Data Group).

* The U(1) electroweak coupling constant g', which is known with exquisite precision. I believe that the value of g is about 0.65293 and that the value of g' is about 0.34969 (both of which are known with much greater precision).

* The four parameters of the CKM matrix, which are known with considerable precision. In the Wolfenstein parameterization, they are λ = 0.22537 ± 0.00061 , A = 0.814+0.023 −0.024, ρ¯ = 0.117 ± 0.021 , η¯ = 0.353 ± 0.013.

* The four parameters of the PMNS matrix; three of which are known with moderate accuracy. These three parameters are theta12=33.36 +0.81/-0.78 degrees, theta23=40.0+2.1/-1/5 degrees or 50.4+1.3/-1.3 degrees, and theta12=8.66+0,44/-0.46 degrees.

None of these are pertinent to the issues discussed in this post. All of these except the CP violating phase of the PMNS matrix and the quadrant of one of the PMNS matrix parameters has been measured with reasonable precision.

General relativity involves two constants (Newton's constant which is 6.67384(80)×10

^{−11}m^{3}*kg^{−1}*s^{−2}and the cosmological constant which is approximately 10^{-52}m^{-2}).In addition, Planck's constant (6.62606957(29)×10

^{−34}J*s) and the speed of light (299792458 m*s^{−1}) are experimentally measured constants (even though the speed of light's value is now part of the definition of the meter), that are known with great precision and which must be known to do fundamental physics in the Standard Model and/or General Relativity.A small number of additional experimentally determined constants may be necessary to describe dark matter phenomena and cosmological inflation.

**A Higgs Boson Mass Numerology Recap**The intermediate equations in the tree-level analysis in the paper suggests how the 125.98 +/- 0.03 GeV/c^2 value that the simple 2H=2W+Z formula suggests could be close to the actual result. If this similar formula were true, the combined electroweak fits of the W boson mass, top quark mass and Higgs boson mass favor a value at the low end of that range, perhaps 125.95 GeV/c^2.

* * *

An alternative possibility, in which one half of the Higgs boson mass equals exactly the sum of the squares of the massive fundamental bosons of the Standard Model (and the sum of the squares of the masses of the fundamental fermions has the same value as the sum of the squares of the boson masses), i.e., for Higgs vev=V, Higgs boson mass=H, W boson mass=W, and Z boson mass=Z:

H^2=(V^2)/2-W^2-Z^2, implies a Higgs boson mass of 124.648 +/- 0.010 GeV, with combined combined electroweak fits favoring a value at the high end of this range (perhaps 124.658 GeV). (This would imply a top quark mass of about 174.1 GeV which is consistent with the current best estimates of the top quark mass; a slightly lower top quark mass of 173.1 GeV would be implied if the Higgs boson mass were, for instance, 125.95 GeV; both of these top quark masses are within 1 standard deviation of the experimentally measured value of the top quark mass).

This could also have roots in the analysis in the paper, which includes half of the square of the Higgs vev, the W boson mass, and the Z boson mass in its analysis (and has terms for the photon mass that drop out because the photon has a zero mass).

* * *

Thus, both alternative possibilities (2H=2W+Z and sum of F^2= sum of B^2= 1/2 Higgs vev) are roughly consistent with the experimental evidence, although they are clearly inconsistent with each other using pole masses for each boson.

It is also possible, however, that the correct scale at which to evaluate the masses in these formulas might, for example, be closer to the Higgs vev energy scale of about 246 GeV, and that at that scale, both formulas might be true simultaneously. Renormalization of masses as energy scales increase shrink the fundamental gauge boson masses more rapidly than they shrink the fundamental fermion masses, and an energy scale at which the sums of the squares of the fermion masses equal the sum of the squares of the boson masses is less than 1,000 GeV (i.e. 1 TeV) and not less than the value determined using the pole masses of the respective fundamental particles.

In a naive calculation using the best known values of the fundamental fermion and boson masses, the sum of the square of the fermion masses are not quite equal to the sum of the square of the boson masses (including the Higgs boson mass). But, the uncertainties in the top quark mass, the Higgs boson mass, and the W boson mass (in that order) are sufficient to make it unclear how the sum of the square of the fermion masses relate to the sum of the square of the boson masses either at pole masses or at any other energy scale. The uncertainties in these three squared masses, predominate over any uncertainties in the masses of the other 11 fermions, the Z boson (whose mass is known with seven times more precision than the W boson despite having a higher absolute value), and uncertainty in the value of the Higgs vev.

If these top quark and Higgs boson mass measurements could be made about three times more precise than the current state of the art experimental measurments, most of the current uncertainty regarding the nature of the Higgs boson mass and the relationship of the fundamental particle masses to the Higgs vev could be eliminated. The remainder of the LHC experiments will almost surely improve the accuracy of both of these measurements, but may be hard pressed to improve them by that much before these experiments are completed.

**What Does A Composite Higgs Boson Hypothesis Imply?**It is always a plus to be able to derive any experimentally determined Standard Model parameter from other Standard Model parameters, reducing the number of degrees of freedom in the theory. This would make electroweak unification even more elegant.

More deeply, if the Higgs boson is merely a composite of the Standard Model electroweak gauge bosons, then:

(1) The hierarchy problem as conventionally posed evaporates, because the Higgs boson mass itself is no longer fine tuned. The profound fine tuning of the Higgs boson mass in the Standard Model is the gist of the hierarchy problem. The demise of the hierarchy problem removes an important motivation for SUSY theories.

(2) Some of the issues associated with the hierarchy problem migrate to the question of why the W and Z boson mass scales are what they are, but if the Higgs boson mass works out to be such that the fermionic and bosonic contributions to the Higgs vev are identical, then the W and Z boson masses are a function of the fermion masses and an electroweak mixing angle, or equivalently, the fermion masses are a function of the electroweak boson masses and the texture of the fermion mass matrix.

(3) The spotlight in the mystery of the nature of the fermion mass matrix would cease to be arbitrary coupling constants with the Higgs boson and return squarely to W boson interactions which are the only means in the Standard Model by which fermions of one flavor can transform into fermions of another flavor.

(4) There are no longer any fundamental Standard Model bosons that are not spin-1 (the hypothetical spin-2 graviton is not part of the Standard Model). All fundamental Standard Model fermions are spin-1/2. Eliminating a fundamental spin-0 boson from the Standard Model changes the Lie groups and Lie algebras which can generate the Standard Model fundamental particles without excessive or missing particles in a grand unified theory.

(5) If the Higgs boson couplings are derivative of the W and Z boson couplings, then neutrinos, which couple with the W and Z boson, although only via the weak force, should derive their masses via the composite Higgs boson mechanism, just as other fermions in the Standard Model do. This implies that neutrinos should have Dirac mass just like other particles that derive their masses from the same interactions.

(6) A composite Higgs boson makes models with additional Higgs doublets less plausible, except to the extent that different combinations of fundamental Standard Model gauge bosons can generate these exotic Higgs bosons, which if they can, would then have masses and other properties that could be determined from first principles and would be divorced from supersymmetry theories.

(7) A composite Higgs boson might slightly tweak the running of the Standard Model coupling constants, influencing gauge unification at high energies (as could quantum gravity effects). A slight tweak of 1-2% to the running of one or more of the coupling constants over from the electroweak scale to the Planck or GUT scale (more than a dozen orders of magnitude), is all that would be necessary for the Standard Model coupling constants to unify at some extremely high energy. It may also have implications for the unstable v. metastable v. stable nature of the vacuum.

### The Proca Model and Podolsky Generalized Electrodynamics

**Background and Motivation**

The Standard Model, and in particular, the quantum electrodynamics (QED) component of the Standard Model, assumes that the photon does not have mass (although photons do, of course, have energy, and hence are subject to gravity in general relativity in which gravity acts upon both matter and energy).

Now, almost nobody seriously thinks that the assumption of QED that the photon is massless is wrong, because the predictions of QED are more precise, and are indeed more precisely experimentally tested than any other part of the Standard Model, or for that matter almost anything in experimental physics whatsoever. There is no meaningful experimental or theoretical impetus to make the assumption that the photon is massless.

But, generalizations of Standard Model physics that parameterize deviations from the Standard Model expectation, provide a useful tool for devising experimental tests to confirm or contradict the Standard Model and to quantify how much deviation from the Standard Model has been experimentally excluded.

Also, any time one can demonstrate that it is possible to have a new kind of force with a massive carrier boson that behaves in a manner very much like QED, but not exactly like QED, that is theoretically rigorous, these theories, once their implications are understood, can be considered as possible explanations for explaining unsolved problems in physics.

For example, many investigators have considered a massive "dark photon" as a means by which dark matter fermions could be self-interacting very similar to the models discussed below, because self-interacting dark matter models seem to be better at reproducing the dark matter phenomena that astronomers observe, than dark matter models in which dark matter fermions interact solely via gravity and Fermi contact forces (i.e. the physical effects of the fact that two fermions can't be in the same place at the same time) with other particles and with each other.

**The Proca Model and Podolsky Generalized Electrodynamics**

A paper published in 2011 and just posted to arVix today for some reason evaluates the experimental limitations on this assumption.

The Proca model of Romanian physicist Alexandru Proca (who became a naturalized French citizen later in life when he married a French woman) was developed on the eve of World War II, mostly from 1936-1938 considers a modification of QED in which the photon has a tiny, but non-zero mass. Proca's equations still have utility because they describe the motion of vector mesons and the weak force bosons, both of which are massive spin-1 particles that operate only at short ranges as a result of their mass and short mean lifetimes.

Experimental evidence excludes the possibility that photons have Proca mass (according to the 2011 paper linked above and cited below) down to masses of about 10

^{-39}grams (which is roughly equivalent to 10

^{-6}eV/c

^{2}). This is on the order of 10,000 lighter than the average of the three neutrino masses (an average which varies by about a factor of ten between normal, inverted and degenerate mass hierarchies). The exclusion (assuming that no mass is discovered) would be about 100 times more stringent if an experiment proposed in 2007 is carried out. This exclusion for Proca mass is roughly equal to the energy of a photon with a 3 GHz frequency (the frequency of UHF electromagnetic waves used to broadcast television transmissions); visible light has more energy and a roughly 300 THz frequency (10,000 times more energetic).

The Particle Data Group's best estimate of the maximum mass of the photon is much smaller than the limit cited in the 2011 article, with a mass of less than 10

^{-18}eV/c

^{2}from a 2007 paper (twelve orders of magnitude more strict). A 2006 study cited by not relied upon by PDG claimed a limit ten times as strong. A footnote at the PDG entry based on some other 2007 papers notes that a much stronger limit can be imposed if a photon acquires mass at all scales by a means other than the Higgs mechanism (with formatting conventions for small numbers adjusted to be consistent with this post):

When trying to measure m one must distinguish between measurements performed on large and small scales. If the photon acquires mass by the Higgs mechanism, the large-scale behavior of the photon might be effectively Maxwellian. If, on the other hand, one postulates the Proca regime for all scales, the very existence of the galactic field implies m < 10Ordinarily a massive photon would break the gauge symmetry of QED, which would be inconsistent with all sorts of experimentally confirmed theoretical predictions that rely upon the fact that the gauge symmetry of QED is unbroken.^{-26}eV/c^{2}, as correctly calculated by YAMAGUCHI 1959 and CHIBISOV 1976.

But, it is possible to find a loophole in the assumption that a massless photon would break the gauge symmetry of QED. Specifically, the Podolsky Generalized Electrodynamics model, proposed in the 1940s by Podolsky, incorporates a massive photon in a manner that does not break gauge symmetry. In this model, the photon has both a massless and massive mode, with the former interpreted as a photon and the later tentatively associated with the neutrino by the Podolsky when the model was formulated (an interpretation that has since been abandoned for a variety of reasons). In the Podolsky Generalized Electrodynamics model, Coulomb's inverse square law describing the electric force of a point charge is slightly modified. Podolosky Generalized Electrodynamics is equivalent to QED in the limit as Podolsky's constant "a" approaches zero.

Podolosky Generalized Electrodynamics is also notable because it can be derived as an alternative solution to one that uses a set of very basic assumptions to derive Maxwell's Equations from first principles, and does so in a manner that prevents the infinities found in QED because it has a point source (which Feynman and others solved for practical purposes with the technique of renormalization) from arising.

In the Podolsky Generalized Electrodynamics model, there is a constant "a" with units of length associated with the massive mode of the photon must have a value that is experimentally required to be smaller than the current sensitivity of any current experiments. But, "a" is required as a consequence of the "value of the ground state energy of the Hydrogen atom . . . to be smaller than 5.6 fm, or in energy scales larger than 35.51 MeV."

In practice (for reasons that are not obvious without reading the full paper), this means that deviations from QED due to a non-zero value of "a" could be observed only at high energy particle accelerators.

It isn't inconceivable, however, to imagine that Podolsky's constant had a value on the order of the Planck length (i.e. 1.6 * 10

^{-35}meters), which would be manifest only at energies approaching the Planck energy which is far beyond the capacity of any man made experiment to create, a value which could be correct without violating any current experimental constraints that have been rigorously analyzed to date.

**Selected References**

* B. Podolsky, 62 Phys. Rev. 68 (1942).

* B. Podolsky, C. Kikuchi, 65 Phys. Rev. 228 (1944).

* B. Podolsky, P. Schwed, 20 Rev. Mod. Phys. 40 (1948).

* R. R. Cuzinatto, C. A. M. de Melo, L. G. Medeiros, P. J. Pompeia, "How can one probe Podolsky Electrodynamics?", 26 International Journal of Modern Physics A 3641-3651 (2011) (arVix preprint linked to in post).

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