The finding is a problem of QED or to the standard view about what proton is. Lamb shift is the effect distinguishing between the states hydrogen atom having otherwise the same energy but different angular momentum. The effect is due to the quantum fluctuations of the electromagnetic field. The energy shift factorizes to a product of two expressions. The first one describes the effect of these zero point fluctuations on the position of electron or muon and the second one characterizes the average of nuclear charge density as "seen" by electron or muon. The latter one should be same as in the case of ordinary hydrogen atom but it is not. Does this mean that the presence of muon reduces the charge radius of proton as determined from muon wave function? This of course looks implausible since the radius of proton is so small. Note that the compression of the muon's wave function has the same effect.
Before continuing it is good to recall that QED and quantum field theories in general have difficulties with the description of bound states: something which has not received too much attention. For instance, van der Waals force at molecular scales is a problem. A possible TGD based explanation and a possible solution of difficulties proposed for two decades ago is that for bound states the two charged particles (say nucleus and electron or two atoms) correspond to two 3-D surfaces glued by flux tubes rather than being idealized to points of Minkowski space. This would make the non-relativistic description based on Schrödinger amplitude natural and replace the description based on Bethe-Salpeter equation having horrible mathematical properties.
Addition: This posting has been subject to continual modifications as I have been fighting with the model armed with my miserable calculational skills. Therefore I made a bigger updating which hopefully provides a clearer representation. The calculations are represented in a little article at my homepage.
1. Basic facts and notions
Can one say anything interesting about the possible mechanism behind the anomaly if one accepts TGD framework? How the presence of muon could reduce the charge radius of proton? Let us first list the basic facts.
- One can say that the size of muonic hydrogen characterized by Bohr radius is by factor me/mμ=211.4 smaller than for hydrogen atom and equals to 250 fm. Hydrogen atom Bohr radius is .53 Angstroms.
- Proton contains 2 quarks with charge 2e/3 and one d quark which charge -e/. These quarks are light. The last determination of quark masses gives masses, which are mu=2 MeV and md=5 MeV (I leave out the error bars). The standard view is that the contribution of quarks to proton mass is of same order of magnitude. This would mean that quarks are not too relativistic meaning that one can assign to them a size of order Compton wave length of order 4×re≈600 fm in the case of u quark (roughly twice the Bohr radius of muonic hydrogen) and 10×re≈24 fm in the case of d quark. These wavelengths are much longer than the proton charge radius and for u quark more than twice longer than the Bohr radius of the muonic hydrogen. That parts of proton would be hundreds of times larger than proton itself sounds a rather weird idea. One could of course argue that the scales in question do not correspond to anything geometric.
In TGD framework this is not the way out since quantum classical correspondence requires this geometric correlate.
- There is also the notion of classical radius of electron and quark. It is given by r= α hbar/m and is in the case of electron this radius is 2.8 fm whereas proton charge radius is .877 fm and smaller. The dependence on Planck constant is only apparent as it should be since classical radius is in question. For u quark the classical radius is .52 fm and smaller than proton charge radius. The constraint that the classical radii of quarks are smaller than proton charge radius gives a lower bound of quark masses: p-adic scaling of u quark mass by 2-1/2 would give classical radius .73 fm which still satisfies the bound. TGD framework the proper generalization would be r= αKhbar/m, where αK is Kähler coupling strength defining the fundamental coupling constant of the theory and quantized from quantum criticality. Its value is very near or equal to fine structure constant in electron length scale.
- The intuitive picture is that light-like 3-surfaces assignable to quarks describe random motion of partonic 2-surfaces with light-velocity. This is analogous to zitterbewegung assigned classically to the ordinary Dirac equation. The interpretation of zitterbewegung radius as classical radius looks rather natural. The notion of braid emerging from Chern-Simons Dirac equation via periodic boundary conditions means that the orbits of partonic 2-surface effectively reduces to braids carrying fermionic quantum numbers. These braids in turn define higher level braids which would move inside a structure characterizing the particle geometrically. Internal consistency suggests that the classical radius r=&alphaKhbar/m characterizes the size scale of the zitterbewegung orbits of quarks.
I cannot resist the temptation to emphasize the fact that Bohr orbitology is now reasonably well understood. The solutions of field equations with higher than 3-D CP2 projection describing radiation fields allow only generalizations of plane waves but not their superpositions in accordance with the fact it is these modes that are observed. For massless extremals with 2-D CP2 projection superposition is possible only for parallel light-like wave vectors. Furthermore, the restriction of the solutions of the Chern-Simons Dirac equation at light-like 3-surfaces to braid strands gives the analogs of Bohr orbits. Wave functions of -say electron in atom- are wave functions for the position of wormhole throat and thus for braid strands so that Bohr's theory becomes part of quantum theory.
- In TGD framework quantum classical correspondence requires -or at least strongly suggests- that also the p-adic length scales assignable to u and d quarks have geometrical correlates. That quarks would have sizes much larger than proton itself how sounds rather paradoxical and could be used as an objection against p-adic length scale hypothesis. Topological field quantization however leads to the notion of field body as a structure consisting of flux tubes and and the identification of this geometric correlate would be in terms of Kähler (or color-, or electro-) magnetic body of proton consisting of color flux tubes beginning from space-time sheets of valence quarks and having length scale of order Compton wavelength much longer than the size of proton itself. Magnetic loops and electric flux tubes would be in question. Also secondary p-adic length cale characterizes field body. For instance, in the case of electron the causal diamond assigned to electron would correspond to the time scale of .1 seconds defining an important bio-rhythm.
2. A general formula for Lamb shift in terms of proton charge radius The charge radius of proton is determined from the Lamb shift between 2S- and 2P states of muonic hydrogen. Without this effect resulting from vacuum polarization of photon Dirac equation for hydrogoen would predict identical energies for these states. The calculation reduces to the calculation of vacuum polarization of photon inducing to the Coulomb potential and an additional vacuum polarization term. Besides this effect one must also take into account the finite size of the proton which can be coded in terms of the form factor deducible from scattering data. It is just this correction which makes it possible to determine the charge radius of proton from the Lamb shift.
- In the article The Lamb shift Experiment in Muonic Hydrogen the basic theoretical results related to the Lamb shift in terms of the vacuum polarization of photon are discussed. Proton's charge density is in this representation is expressed in terms of proton form factor in principle deducible from the scattering data. Two special cases can be distinguished corresponding to the point like proton for which Lamb shift is non-vanishing only for S wave states and non-point like proton for which energy shift is present also for other states. The theoretical expression for the Lamb shift involves very refined calculations. Between 2P and 2S states the expression for the Lamb shift is of form
Δ E(2P3/2F=2 − 2S1/2F=1)=a-brp2 +crp3= 209.968(5) − 5.2248 × r2p + 0.0347 × r3p meV .
where the charge radius rp=.8750 is expressed in femtometers and energy in meVs.
- The general expression of Lamb shift is given in terms of the form factor by
E(2P-2S)=∫ (d3q/(2π)3)× (-4π α ) (F(q2)/q2) × (Π(q2)/q2)× ∫ (| Ψ2P(r)|2-|Ψ2S(r)|2)exp(−iq• r) dV .
Here Π is is a scalar representing vacuum polarization due to decay of photon to virtual pairs.
The modification of the formula is due to the normalization of the 2P and 2S states. These are in general different. The normalization factor 1/N is same for all terms in the expression of Lamb shift for a given state but in general different for 2S and 2P states. Since the lowest order term dominates by a factor of ≈ 40 over the second one, one one can conclude that the modification should affect the lowest order term by about 4 per cent. Since the second term is negative and the modification of the first term is interpreted as a modification of the second term when rp is estimated from the standard formula, the first term must increase by about 4 per cent. This is achieved if this state is orthogonalized with respect to the flux tube state. For states Ψ0 and Ψtube with unit norm this means the modification
Ψ0→ (1/(1-| C|2)× (Ψi -CΨtube) ,
C=〈 Ψtube| Ψ0〉 .
In the lowest order approximation one obtains
a-br2p+crp3→ (1+|C|2)a-brp2+crp3 .
Using instead of this expression the standard formula gives a wrong estimate rp from the condition
a-b r2p,1+crp,13→ (1+| C|2)a-brp2+crp3 .
This gives the equvalent conditions
rp,12= rp2- | C|2a/b ,
Ptube≡| C|2≈ (2b/a)× rp2 × (rp-rp,1)/rp) .
The resulting estimate for the leakage probability is Ptube≈ .0015. The model should be able to reproduce this probability.
3. Could the notion of field body explain the anomaly?
The large Compton radii of quarks and the notion of field body encourage the attempt to imagine a mechanism affecting the charge radius of proton as determined from electron's or muon's wave function.
- Muon's wave function is compressed to a volume which is about 8 million times smaller than the corresponding volume in the case of electron. The Compton radius of u quark more that twice larger than the Bohr radius of muonic hydrogen so that muon should interact directly with the field bodies of u quarks. The field body of d quark would have size 24 fm which is about ten times smaller than the Bohr radius so that one can say that the volume in which muons sees the field body of d quark is only one thousandth of the total volume. The main effect would be therefore due to the two u quarks having total charge of 4e/3.
One can say that muon begins to "see" the field bodies of u quarks and interacts directly with u quarks rather than with proton via its elecromagnetic field body. With d quarks it would still interact via protons field body to which d quark should feed its electromagnetic flux. This could be quite enough to explain why the charge radius of proton determined from the expectation value defined by its wave function wave function is smaller than for electron. One must of course notice that this brings in also direct magnetic interactions with u quarks.
- What could be the basic mechanism for the reduction of charge radius? Could it be that the electron is caught with some probability into the flux tubes of u quarks and that Schrödinger amplitude for this kind state vanishes near the origin? The original idea was based on classical probability: the flux tube portion of state would not contribute to the charge radius and since the portion of the ordinary state would bbe smaller, and effective reduction of the charge radius would be implied. Unfortunately just the opposite occurs and this is due to
the fact that the expression for the Lamb shift involves also constant term besides powers of charge radius. What happens is that the normalization factor of the standard contribution increases by the orthogonalization with the flux tube state. The effect is therefore genuinely quantum mechanical having no classical counterpart.
- I have blundering with this model for a week and it seems that no bad misunderstandings are present anymore. Precise numerical factors are of course dangerous at this age. By the earlier general argument one should have Ptube= .0015. This value of leakage probability is obtained for z=1 and N=2 corresponding to single flux tube per u quark. If the flux tubes are in opposite directions, the leakage into 2P state vanishes by parity. Note that the leakage does not affect the value of the coefficient a in the general formula for the Lamb shift.
The radius of the flux tube is by a factor 1/4 smaller than the classical radius of electron and one could argue that this makes it impossible for electron to topologically condense at the flux tube. For z=4 one would have Ptube= .015, which is 10 times too large a value. Numerical errors are possible. Note that the nucleus possess a wave function for the orientation of the flux tube. If this corresponds to S-wave state then only the leakage beween S-wave states and standard states is possible.
- This effect would be of course present also in the case of electron but in this case the u quarks correspond to a volume which million times smaller than the volume defined by Bohr radius so that electron does not in practice "see" the quark sub-structure of proton. The probability P for getting caught would be in a good approximation proportional to the value of |Ψ(r_u)|2 and in the first approximation one would have
Pe/Pμ ≈ (aμ/ae)3 =(me/mμ)3≈ 10-7
from the proportionality &Psii propto 1/a_i3/2, i=e,μ.