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p-Adic Physics

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Year 2018

How to describe family replication phenomenon gauge theoretically?

In TGD framework family replication phenomenon is described topologically (see this). The problem is to modify the gauge theory approach of the standard model to model to describe family replication phenomenon at QFT limit.

1. Identification of elementary particles

1.1 Original picture

The original view about family replication phenomenon assumed that fermions correspond to single boundary component of the space-time surface (liquid bubble is a good analogy) and thus characterized by genus g telling the number of handles attached to the sphere to obtain the bubble topology.

  1. Ordinary bosons would correspond to g=0 (spherical) topology and the absorption/emission of boson would correspond to 2-D topological sum in either time direction. This interpretation conforms with the universality of ordinary ew and color interactions.
  2. The genera of particle and antiparticle would have formally opposite sign and the total genus would be conserved in the reaction vertices. This makes sense if the annihilation of fermion and anti-fermion to g=0 boson means that fermion turns backwards in time emitting boson. The vertex is essentially 2-D topological sum at criticality between two manifold topologies. In the vertex 2-surface would be therefore singular manifold. The analogy to closed string emission in string model is obvious.
1.2 The recent vision

Later the original picture was replaced with a more complex identification.

  1. Fundamental particles - partons - serving as building bricks of elementary particles are partonic 2-surfaces identified as throats of wormhole contacts at which the Euclidian signature of the induced metric of the wormhole contact changes to Minkowskian one. The orbit of partonic 2-surface corresponds to a light-like 3-surface at which the Minkowskian signature of the induced metric changes to Euclidian, and carries fermion lines defining of boundaries of string world sheets. Strings connect different wormhole throats and mean generalization of the notion of point like particle leading to the notion of tensor network (see this).

    Elementary particles are pairs of two wormhole contacts. Both fermions and bosons are pairs of string like flux tubes at parallel space-time sheets and connected at their ends by CP2 sized wormhole contacts having Euclidian signature of induced metric. A non-vanishing monopole flux loop runs around the extrenely flattened rectangle loop connecting wormhole throats at both space-time sheets and traverses through the contacts.

  2. The throats of wormhole contacts are characterized by genus given by the number g of handles attached to sphere to get the topology. If the genera ga,gb of the opposite throats of given wormhole contact are same, one can assign genus to it : g=ga=gb. This can be defended by the fact, that the distance between the throats is given by CP2 length scale and thus extremely short so that ga≠ gb implies strong gradients and by Uncertainty Principle mass of order CP2 mass.

    If the genera of the two wormhole contacts are same: g1=g2, one one can assign genus g to the particle. This assumption is more questionable if the distance between contacts is of order of Compton length of the particle. The most general assumption is that all genera can be different.

  3. There is an argument for why only 3 lowest fermion generations are observed (see this). Assume that the genus g for all 4 throats is same. For g=0,1,2 the partonic 2-surfaces are always hyper-elliptic allowing thus a global conformal Z2 symmetry. Only these 3 2-topologies would be realized as elementary particles whereas higher generations would be either very heavy or analogous to many-particle states with a continuum mass spectrum. For the latter option g=0 and g=1 state could be seen as vacuum and single particle state whereas g=2 state could be regarded as 2-particle bound state. The absence of bound n-particle state with n>2 implies continuous mass spectrum.
  4. Fundamental particles would wave function in the conformal moduli space associated with its genus (Teichmueller space). For fundametal fermions the wave function would be strongly localized to single genus. For ordinary bosons one would have maximal mixing with the same amplitude for the appearance of wormhole throat topology for all genera g=0,1,2. For the two other u(3)g neutral bosons in octet one would have different mixing amplitudes and charge matrices would be orthogonal and universality for the couplings to ordinary fermions would be broken for them. The evidence for the breaking of the universality (see this) is indeed accumulating and exotic u(3)g neutral gauge bosons giving effectively rise to two additional boson families could explain this.
2. Two questions related to bosons and fermions

What about gauge bosons and Higgs, whose quantum numbers are carried by fermion and anti-fermion (or actually a superposition of fermion-anti-fermion pairs). There are two options.

  1. Option I: The fermion and anti-fermion for elementary boson are located at opposite throats of wormhole contact as indeed assumed hitherto. This would explain the point-likeness of elementary bosons. u(3) charged bosons having different genera at opposite throats would have vanishing couplings to ordinary fermions and bosons. Together with large mass of ga≠ gb wormhole contacts this could explain why ga≠ gb bosons and fermions are not observed and would put the Cartan algebra of u(3)g in physically preferred position. Ordinary fermions would effectively behave as u(3)g triplet.
  2. Option II: The fermion and anti-fermion for elementary boson are located at throats of different wormhole contacts making them non-point like string like objects. For hadron like stringy objects, in particular graviton, the quantum numbers would necessarily reside at both ends of the wormhole contact if one assumes that single wormhole throats carries at most one fermion or anti-fermion. For this option also ordinary fermions could couple to (probably very massive) exotic bosons different genera at the second end of the flux tube.
There are also two options concerning the representation of u(3)g assignal to fermions corresponding ot su(3)g triplet 3 and 8⊕ 1.

Option I: Since only the wormhole throat carrying fermionic quantum numbers is active and since fundamental fermions naturally correspond to u(3)g triplets, one can argue that the wormhole throat carrying fermion quantum number determines the fermionic u(3)g representation and should be therefore 3 for fermion and 3bar anti-fermion.

At fundamental level also bosons would in the tensor products of these representations and many-sheeted description would use these representations. Also the description of graviton-like states involving fermions at all 4 wormhole throats would be natural in this framework. At gauge theory limit sheets would be identified and in the most general case one would need U(3)g× U(3)g× U(3)g× U(3)g with factors assignable to the 4 throats.

  1. The description of weak massivation as weak confinement based on the neutralization of weak isospin requires a pair of left and right handed neutrino located with νL and νbarR or their CP conjugates located at opposite throats of the passive wormhole contact associated with fermion. Already this in principle requires 4 throats at fundamental level. Right-handed neutrino however carries vanishing electro-weak quantum numbers so that it is effectively absent at QFT limit.
  2. Why should fermions be localized and su(3)g neutral bosons delocalized with respect to genus? If g labels for states of color triplet 3 the localization of fermions looks natural, and the mixing for bosons occurs only in the Cartan algebra in u(3)g framework: only u(3)g neutral states an mix.
Option II: Also elementary fermions belong to 8+1. The simplest assumption is that both fermions and boson having g1≠ g2 have large mass. In any case, g1≠ g2 fermions would couple only to u(3)g charged bosons. Also for this option ordinary bosons with unit charge matrix for u(3)g would couple in a universal manner.
  1. The model for CKM mixing (see this) would be modified in trivial manner. The mixing of ordinary fermions would correspond to different topological mixings of the three states su(3)-neutral fermionic states for U and D type quarks and charged leptons and neutrinos. One could reduce the model to the original one by assuming that fermions do not correspond to generators Id, Y, and I3 for su(3)g but their linear combinations giving localization to single valued of g in good approximation: they would correspond to diagonal elements eaa, a=1,2,3 corresponding to g=0,1,2.
  2. p-Adic mass calculations (see this) assuming fixed genus for fermion predict an exponential sensitivity on the genus of fermion. In the general case this prediction would be lost since one would have weighted average over the masses of different genera with g=2 dominating exponentially. The above recipe would cure also this problem. Therefore it seems that one cannot distinguish between the two options allowing g1≠ g2. The differences emerge only when all 4 wormhole throats are dynamical and this is the case for graviton-like states (spin 2 requires all 4 throats to be active).
The conclusion seems to be that the two options are more or less equivalent for light fermions. In the case of exotic fermions expected to be extremely heavy the 8+1 option looks more natural. At this limit however QFT limit need not make sense anymore.

3. Reaction vertices

Consider next the reaction vertices for the option in which particles correspond to string like objects identifiable as pairs of flux tubes at opposite space-time sheets and carrying monopole magnetic fluxes and with ends connected by wormhole contacts.

  1. Reaction vertex looks like a simultaneous fusing of two open strings along their ends at given space-time sheets. The string ends correspond to wormhole contacts which fuse together completely. The vertex is a generalization of a Y-shaped 3-vertex of Feynman diagram. Also 3-surfaces assignable to particles meet in the same manner in the vertex. The partonic 2-surface at the vertex would be non-singular manifold whereas the partonic orbit would be singular manifold in analogy with Y shaped portion of Feynman diagram.
  2. In the most general case the genera of all four throats involved can be different. Since the reaction vertex corresponds to a fusion of wormhole contacts characterized in the general case by (g1,g2), one must have (g1,g2)=(g3,g4). The rule would correspond in gauge theory description to the condition that the quark and antiquark su(3)g charges are opposite at both throats in order to guarantee charge conservation as the wormhole contact disappears.
  3. One has effectively pairs of open string fusing along their and and the situation is analogous to that in open string theory and described in terms of Chan-Paton factors. This suggests that gauge theory description makes sense at QFT limit.
    1. If g is same for all 4 throats, one can characterize the particle by its genus. The intuitive idea is that fermions form a triplet representation of u(3)g assignable to the family replication. In the bosonic sector one would have only u(3)g neutral bosons. This approximation is expected to be excellent.
    2. One could allow g1≠ g2 for the wormhole contacts but assume same g for opposite throats. In this case one would have U(3)g× U(3)g as dynamical gauge group with U(3)g associated with different wormhole contacts. String like bosonic objects (hadron like states) could be therefore seen as a nonet for u(3)g. Fermions could be seen as a triplet.

      Apart from topological mixing inducing CKM mixing fermions correspond in good approximation to single genus so that the neutral members of u(3)g nonet, which are superpositions over several genera must mix to produce states for which mixing of genera is small. One might perhaps say that the topological mixing of genera and mixing of u3(g) neutral bosons are anti-dual.

    3. If all throats can have different genus one would have U(3)g× U(3)g× U(3)g× U(3)g as dynamical gauge group U(3)g associated with different wormhole throats. This option is probably rather academic. Also fermions could be seen as nonets.
4. What would the gauge theory description of family replication phenomenon look like?

For the most plausible option bosonic states would involve a pair of fermion and anti-fermion at opposite throats of wormhole contact. Bosons would be characterized by adjoint representation of u(3)g=su(3)g× u(1)g obtained as the tensor product of fermionic triplet representations 3 and 3bar.

  1. u(1)g would correspond to the ordinary gauge bosons bosons coupling to ordinary fermion generations in the same universal manner giving rise to the universality of electroweak and color interactions.
  2. The remaining gauge bosons would belong to the adjoint representation of su(3)g. One indeed expects symmetry breaking: the two neutral gauge bosons would be light whereas charged bosons would be extremely heavy so that it is not clear whether QFT limit makes sense for them.

    Their charge matrices Qgi would be orthogonal with each other (Tr(QgiQgj)=0, i≠ j) and with the unit charge matrix u(1)g charge matrix Q0∝ Id (Tr(Qgi)=0) assignable to the ordinary gauge bosons.These charge matrices act on fermions and correspond to the fundamental representation of su(3)g. They are expressible in terms of the Gell-Mann matrices λi (see this).

How to describe family replication for gauge bosons in gauge theory framework? A minimal extension of the gauge group containing the product of standard model gauge group and U(3)g does not look promising since it would bring in additional generators and additional exotic bosons with no physical interpretation. This extension would be analogous to the extension of the product SU(2)× SU(3) of the spin group SU(2) and Gell-Mann's SU(3) to SU(6)). Same is true about the separate extensions of U(2)ew and SU(3)c.
  1. One could start from an algebra formed as a tensor product of standard model gauge algebra g= su(3)c× u(2)ew and algebraic structure formed somehow from the generators of u(3)g. The generators would be

    Ji,a= Ti ⊗ Ta ,

    where i labels the standard model Lie-algebra generators and a labels the generators of u(3)g.

    This algebra should be Lie-algebra and reduce to the same as associated with standard model gauge group with generators Tb replacing effectively complex numbers as coefficients. Mathematician would probably say, that standard model Lie algebra is extended to a module with coefficients given by u(3)g Lie algebra generators in fermionic representation but with Lie algebra product for u(3)g replaced with a product consistent with the standard model Lie-algebra structure, in particular with the Jacobi-identities.

  2. By writing explicitly commutators and Jacobi identifies one obtains that the product must be symmetric: Ta• Tb= Tb• Ta and must satisfy the conditions Ta• (Tb• Tc)= Tb• (Tc• Ta)= Tc• (Ta• Tb) since these terms appear as coefficients of the double commutators appearing in Jacobi-identities

    [Ji,a,[Jj,b],Jk,c]]+[Jj,b,[Jk,c],Ji,a]] + [Jk,c,[Ji,a],Jj,b]]=0 .

    Commutativity reduces the conditions to associativity condition for the product •. For the sub-algebra u(1)3g these conditions are trivially satisfied.

  3. In order to understand the conditions in the fundamental representation of su(3), one can consider the product the su(3)g product defined by the anti-commutator in the matrix representation provided by Gell-Mann matrices λa (see this and this):

    ab}= 43δa,b Id + 4dabcλc , & Tr(λaλb) =2δab , & dabc= Tr(λaλbc)

    dabc is totally symmetric under exchange of any pair of indices so that the product defined by the anti-commutator is both commutative and associative. The product extends to u(3)g by defining the anti-commutator of Id with λa in terms of matrix product. The product is consistent with su(3)g symmetries so that these dynamical charges are conserved. For complexified generators this means that generator and its conjugate have non-vanishing coefficient of Id.

    Remark: The direct sum u(n)⊕ u(n)s formed by Lie-algebra u(n) and its copy u(n)s endowed with the anti-commutator product • defines super-algebra when one interprets anti-commutator of u(n)s elements as an element of u(n).

  4. Could su(3) associated with 3 fermion families be somehow special? This is not the case. The conditions can be satisfied for all groups SU(n), n≥ 3 in the fundamental representation since they all allow completely symmetric structure constants dabc as also higher completely symmetric higher structure constants dabc... up to n indices. This follows from the associativity of the symmetrized tensor product: ((Adj⊗ Adj)S⊗ Adj)S =(Adj⊗ (Adj⊗ Adj)S)S for the adjoint representation.
To sum up, the QFT description of family replication phenomenon with the extension of the standard model gauge group would bring to the theory the commutative and associative algebra of u(3)g as a new mathematical element. In the case of ordinary fermions and bosons and also in the case of u(3)g neutral bosons the formalism would be however rather trivial modification of the intuitive picture.

See chapter New Particle Physics Predicted by TGD: Part I.



Further evidence for the third generation of weak bosons

Matt Strassler had a blog posting about an interesting finding from old IceCube data revealed at thursday (July 12, 2018) by IceCube team. The conclusion supports the view that so called blasars, thin jets of high energy particles suggested to emerge as matter falls into giant black hole, might be sources of high energy neutrinos. In TGD framework one could also think that blazars originate from cosmic strings containing dark matter and energy. Blazars themselves could be associated with cosmic strings thickened to magnetic flux tubes. The channeling to flux tubes would make possible observation of the particles emerging from the source whatever it might be.

Only the highest energy cosmic neutrinos can enter the IceCube detector located deep under the ice. IceCube has already earlier discovered a new class of cosmic neutrinos with extremely high energy: Matt Strassler has written a posting also about this two years ago (see this): the energies of these neutrinos were around PeV. I have commented this finding from TGD point of view (see this).

Last year one of these blazars flared brightly producing high energy neutrinos and photons: neutrinos and photons came from the same position in the sky and occurred during the same period. IceCube detector detected a collision of one (!) ultrahigh energy neutrino with proton generating muon. The debris produced in the collision contained also photons, which were detected. IceCube team decided to check whether old data could contain earlier neutrino events assignable to the same blasar and found a dramatic burst of neutrinos in 2014-2015 data during period of 150 days associated with the same flare; the number of neutrinos was 20 instead of the expected 6-7. Therefore it seems that the ultrahigh energy neutrinos can be associated with blazars.

By looking the article of IceCube team (see this) one learns that neutrino energies are of order few PeV (Peta electron Volt), which makes 1 million GeV (proton has mass .1 GeV). What kind of mechanism could create these monsters in TGD Universe? TGD suggests scaled variants of both electroweak physics and QCD and the obvious candidate would be decays of weak bosons of a scaled variant of ew physics. I have already earlier considere a possible explanation interms of weak bosons of scaled up variant of weak physics characterizes by Mersenne prime $M_{61}=2^{61}-1}$ (see this).

  1. TGD "almost-predicts" the existence of three families of ew bosons and gluons. Their coupling matrices to fermions must be orthogonal. This breaks the universality of both ew and color interactions. Only the ordinary ew bosons can couple in the same manner to 3 fermion generations. There are indeed indications for the breaking of the universality in both quark and leptons sector coming from several sources such as B meson decays, muon anomalous anomalous (this is not a typo!) magnetic moment, and the the finding that the value of proton radius is different depending on whether ordinary atoms or muonic atoms are used to deduce it (see this).
  2. The scaled variant of W boson could decay to electron and monster neutrino having same energies in excellent approximation. Also Z0 boson could decay to neutrino-antineutrino pair. The essentially mono-chromatic energy spectrum for the neutrinos would serve as a unique signature of the decaying weak boson. One might hope of observing two kinds of monster neutrinos with mass difference of the order of the scaled up W-Z mass difference. Relative mass difference would same as for ordinary W and Z - about 10 per cent - and thus of order .1 PeV.
One can look the situation quantitatively using p-adic length scale hypothesis and assumption that Mersenne primes and Gaussian Mersennes define preferred p-adic length scales assignable to copies of hadron physics and electroweak physics.
  1. Ordinary ew gauge bosons correspond in TGD framework to Mersenne prime Mk= 2k-1, k=89. The mass scale is 90 GeV, roughly 90 proton masses.
  2. Next generation corresponds to Gaussian Mersenne Gaussian Mersenne prime MG,79= (1+i)79-1. There is indeed has evidence for a second generation weak boson corresponding to MG,79 (see this). The predicted mass scale is obtained by scaling the weak boson mass scale of about 100 GeV with the factor 2(89-79/2=32 and is correct.
  3. The next generation would correspond to Mersenne prime M61. The mass scale 90 GeV of ordinary weak physics is now scaled up by a factor 2(89-61)/2= 214 ≈ 64,000. This gives a mass scale 1.5 PeV, which is the observed mass scale for the neutrino mosters detected by Ice-Cube. Also the earlier monster neutrinos have the same mass scale. This suggests that the PeV neutrinos are indeed produced in decays of W(61) or Z(61).
See chapter New Particle Physics Predicted by TGD: Part I.



LSND anomaly is here again!

Sabine Hossenfelder told about the finding of MinibooNe collaboration described in arXiv.org preprint Observation of a Significant Excess of Electron-Like Events in the MiniBooNE Short-Baseline Neutrino Experiment.

The findings give strong support for old and forgotten LSND anomaly - forgotten because it is in so blatant conflict with the standard model wisdom. The significance level of the anomaly is 6.1 sigmas in the new experiment. 5 sigma is regarded as the threshold for a discovery. It is nice to see this fellow again: anomalies are the theoreticians best friends.

To me this seems like a very important event from the point of view of standard model and even theoretical particle physics: this anomaly with other anomalies raises hopes that the patient could leave the sickbed after illness that has lasted for more than four decades after it became a victim of the GUT infection.

LSND as also other experiments are consistent with neutrino mixing model. LSND however produces electron excess as compared to other neutrino experiments. Anomaly means that the parameters of the neutrino mixing matrix (masses, mixing angles, phases) are not enough to explain all experiments.

One manner to explain the anomaly would be fourth "inert" neutrino having no couplings to electroweak bosons. TGD predicts both right and left-handed neutrinos and right-handed ones would not couple electroweakly. In massivation they would however combine to single massive neutrino just like in Higgs massivation Higgs gives components for massive gauge bosons and only neutral Higgs having no coupling to photon remains. Therefore this line of thought does not seem terribly promising in TGD framework.

For many years ago I explained the LSND neutrino anomaly in TGD framework as being due to the fact that neutrinos can correspond to several p-adic mass scales. p-Adic mass scale coming as power of 21/2 would bring in the needed additional parameter. The new particles could be ordinary neutrinos with different p-adic mass scales. The neutrinos used in experiment would have p-adic length scale depending on their origin. Lab, Earth's atmosphere, Sun, ... It is possible that the neutrinos transform during their travel to less massive neutrinos.

What is intriguing that the p-adic length scale range that can be considered as candidates for neutrino Compton lengths is biologically extremely interesting. This range could correspond to the p-adic length scales L(k)∼ 2(k-151)/2L(151), k= 151,157, 163, 167, varying from cell membrane thickness 10 nm to 2.5 μm. These length scales correspond to Gaussian Mersennes MG,k=(1+i)k-1. The appearance of four of 4 Gaussian Mersennes in such a short length scale interval is a number theoretic miracle. Could neutrinos or their dark variants with heff= n× h0 (h= 6× h0 is the most plausible option at this moment, see this and this) together with dark variants weak bosons effectively massless below their Compton length have a fundamental role in quantum biology?

For the TGD based new physics and also for LSND anomaly see chapter New Particle Physics Predicted by TGD: Part I of "p-Adic physics".



Strange spin asymmetry at RHIC

The popular article Surprising result shocks scientists studying spin tells about a peculiar effect in p-p and p-N (N for nucleus) observed at Relativistic Heavy Ion Collider (RHIC). In p-p scattering with polarized incoming proton there is asymmetry in the sense that the protons with vertical polarization with respect to scattering plane give rise to more neutrons slightly deflected to right than to left (see the figure of the article). In p-N scattering of vertically polarized protons the effect is also observed for neutrons but is stronger and has opposite sign for heavier nuclei! The effect came as a total surprise and is not understood. It seems however that the effects for proton and nuclear targets must have different origin since otherwise it is difficult to understand the change of the sign.

The abstract of the original article summarizes what has been observed.

During 2015 the Relativistic Heavy Ion Collider (RHIC) provided collisions of transversely polarized protons with Au and Al nuclei for the first time, enabling the exploration of transverse-single-spin asymmetries with heavy nuclei. Large single-spin asymmetries in very forward neutron production have been previously observed in transversely polarized p+p collisions at RHIC, and the existing theoretical framework that was successful in describing the single-spin asymmetry in p+p collisions predicts only a moderate atomic-mass-number (A) dependence. In contrast, the asymmetries observed at RHIC in p+A collisions showed a surprisingly strong A dependence in inclusive forward neutron production. The observed asymmetry in p+Al collisions is much smaller, while the asymmetry in p+Au collisions is a factor of three larger in absolute value and of opposite sign. The interplay of different neutron production mechanisms is discussed as a possible explanation of the observed A dependence.

Since diffractive effect in forward direction is in question, one can ask whether strong interactions have anything to do with the effect. This effect can take place at the level of nucleons and a quark level and these two effects should have different signs. Could electromagnetic spin orbit coupling cause the effect both at the level of nucleons in p-N collisions and at the level of quarks in p-p collisions?

  1. Spin-orbit interaction effect is relativistic effect: the magnetic field of target nucleus in the reference frame of projectile proton is nonvanishing: B= -γ v× E, γ= 1/(1-v2)1/2. The spin-orbit interaction Hamiltonian is

    HL-S = -μB ,

    where

    μ= gp μNS , μN= e/2mp

    is the magnetic moment of polarized proton proportional to spin S, which no has definite direction due to the polarization of incoming proton beam. The gyromagnetic factor gp equals to gp=2.79284734462(82) holds true for proton.

  2. Only the component of E orthogonal to v is involved and the coordinates in this direction are unaffected by the Lorentz transformations. One can express the transversal component of electric field as gradient

    Er= - ∂rV r/r .

    Velocity v can be expressed as v=p/mp so that the spin-orbit interaction Hamiltonian reads as

    HL-S= γ gp (e/2mp) (1/mp)LS [∂rV/r ] .

    For polarised proton the effect of this interaction could cause the left-right asymmetry. The reason is that the sign of the interaction Hamiltonia is opposite at left and right sides of the target since the sign of L=r× p is opposite at left- and right-hand sides. One can argue as in non-relativistic case that this potential generates a force which is radial and proportional to ∂r[(∂rV(r))/r)].

Consider first the scattering on nucleus.
  1. Inside the target nucleus one can assume that the potential is of the form V= kr2/2: the force vanishes! Hence the effect must indeed come from peripheral collisions. At the periphery responsible for almost forward scattering one as V(r)=Ze/r and one has ∂r(∂rV(r))/r)= 3Ze/r4, r=R, R the nuclear radius. One has R = kA1/3 for a constant density nucleus so that one has ∂r(∂rV(r))/r)= 3k-4eZA-4/3.

    The force decreases with A roughly like A-1/3 but the scattering proton can give its momentum to a larger number of nucleons inside the target nucleus. If all neutrons get their share of the transversal momentum, the effect is proportional to neutron number N=A-Z one would obtain the dependence Z(A-Z)A-4/3 ∼ A2/3. If no other effects are involved one would have for the ratio r of Al and Au asymmetries

    r=Al/Au ∼ Z(Al)N(Al)/Z(Au)A(u) × [A(Au)/A(Al)]4/3 .

    Using (Z,A)=(13,27) for Al and (Z,A)=(79,197) for Au one obtains the prediction r=.28. The actual value is r≈ .3 by estimating from Fig. 4 of the article is not far from this.

  2. This effect takes place only for protons but it deviates proton at either side to the interior of nucleus. One expects that the proton gives its transversal momentum components to other nucleons - also neutrons. This implies that sign of the effect is same as it would be for the spin-orbit coupling when the projectile is neutron. This could be the basic reason for the strange sign of the effect.
Consider next what could happen in p-p scattering.
  1. One must explain why neutrons with R-L asymmetry with respect to the scattering axis are created. This requires quark level consideration.
  2. The first guess is that one must consider spin orbit interaction for the quarks of the polarized proton scattering from the quarks of the unpolarized proton. What comes in mind is that one could in a reasonable approximation treat the unpolarized proton as single coherent entity. In this picture u and d quarks of polarized proton would have asymmetric diffractive scattering tending to go to the opposite sides of the scattering axis.
  3. The effect for d quarks would be opposite to that for u quarks. Since one has n=udd and and p=uud, the side which has more d quarks gives rise to neutron excess in the recombination of quarks to hadrons. This effect would have opposite sign than the effect in the case of nuclear target. This quark level effect would be present also for nuclear targets.
See the chapter New Particle Physics Predicted by TGD: Part II.



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