Abstract

New compactification mechanism of String theory is proposed.

Applied to Superstring theory it explains quarks fractional

charges and quark/lepton generations of Standard Model.

Theory also explains Regge slope values, quark mixing and neutrino oscillations.

Also it offers explanation for "Dark matter" puzzle of modern astrophysics.

New compactification mechanism of String theory is proposed.

Applied to Superstring theory it explains quarks fractional

charges and quark/lepton generations of Standard Model.

Theory also explains Regge slope values, quark mixing and neutrino oscillations.

Also it offers explanation for "Dark matter" puzzle of modern astrophysics.

Introduction.

Superstring theory [1] is considered by many High-Energy physicists as a good candidate for Theory of Everything since it allows for formulation of all fundamental forces of nature (Electromagnetic, Strong, Weak and Gravitational) as modes of excitations of strings.

However, it turns out that String theory can only be consistently formulated either in 10 (Superstring) or 26 (Bosonic and Heterotic strings) dimensions [2, 1].

The original idea of formulating physical theory in higher than 4 dimensional space-time goes back to 1914 when physicist from Helsingfors Gunnar Nordström introduced extra space dimension to include electromagnetism in his theory of gravitation [3].

In 1919 Theodor Kaluza realized that solving Einstein's equations (published year after Nordström's theory) for general relativity using five dimensions automatically produces Maxwell's equations for electromagnetism. He published his discoveries in 1921 after discussions with Einstein [4].

At the same time *most of observational material* tells us that we live in 4 dimensional space-time. To solve this "compactification problem" two different approaches were suggested: Kaluza-Klein and Calabi-Yau.

In 1926 O.Klein proposed [5] that extra spacial dimension in Kaluza theory got curled up in a circle of small radius (aka "compactified"). Same idea applied to Superstring theory means that *some* spacial dimensions (6) are compactified, while others (3) are not, which leaves unanswered question why some dimensions are better then others.

Also it implies the existence of standing waves in compactified dimensions (so called "Kaluza-Klein mass tower") which are not observed experimentally.

Calabi-Yau solutions suggests that 6 spatial extra dimensions are compactified as complex 3 dimensional manifolds (with 6 real dimensions) [1,6,7]. These are rather reach and sophisticated mathematical models based on existence of compact Kähler manifolds with vanishing Ricci curvature [6]. There are two alternatives here to chose from: one is that "hidden" spacial dimensions are so small that they can not be observed today. Another (popular in D-brane theories) is exactly the opposite - extra dimensions are large, but we are confined to a small subset on which it intersects with D-brane [8, 9].

To avoid all those inconsistencies present in both approaches to compactification we propose "Motley" string model, which treats all spacial dimensions equally and complies with all known experimental material.

True colors of Strings.

First we formulate two postulates:

Postulate 1: Every spacial dimension of String has a unique intrinsic property which we call "color".

Postulate 2: There is force between spacial dimensions of string such that it makes string dimensions of complementary colors (Red_{i}, Green_{i}, Blue_{i}) interact and unite in a *colorless threads* perceived as *observable dimensions*.

"Color" property of String's spacial dimensions is somewhat similar to 3 "color charges" of quarks in Quantum Chromo Dynamics [10 - 13], but has different meaning, since it is viewed here as **intrinsic characteristic of spacial dimensions** in Motley String theory corresponding to different values of **string tension** tensor *T _{i}* in different dimensions.

String state at very high energies (early universe, Planck length about 10^{-33}cm)
is such that all String spacial dimensions are in a free state similar to quark-gluon plasma of Quantum Chromo Dynamics.

9 spacial dimensions at Planck scale (cross section)

At lower energies (modern cold universe) strong "color" force becomes dominant and makes String's complimentary (or using classical optics term "additive") spacial dimensions (Red_{i}, Green_{i}, Blue_{i}) interact to form 3 threads (in case of 9+1 dimensional Superstring) which appear to be colorless from distances larger than size of baryons (proton, neutron, etc). Spacial dimensions of additive "colors" are "glued" together due to Color Force. As a result humans perceive the original (9+1) dimensional String as 3+1 dimensional space time! See Figure 2 below:

Modern Cold Universe: 3 observable colorless dimensions after compactification

Outside of Planck energy scale strings spacial dimensions are *confined* in 3 colorless observable threads.

Observable by humans Gray spacial dimensions *D(i)* are essentially compactified threads of Red(i), Green(i) and Blue(i) string spacial dimensions

**D(i) = R(i) + G(i) + B(i)**, where i =1,2,3

Original 9 spacial dimensions of our Universe, created during Big Bang some 15 Billion years ago, are compactified, or "glued" together and thus not directly seen by human eye, or any currently available experimental devices.

Only *indirect* results coming from mesuring Regge slope values from QCD and meson spectroscopy give us an idea about the deepest level of Universe structure.

Since in our model *all* spacial dimensions are treated uniformly we avoid questions like "why some spacial dimensions are compactified while others are not".

Also there are no standing waves in curved dimensions of Klein compactification and therefore no extra mass values (aka Kaluza-Klein mass tower).

Equally important, there is no need for Calabi-Yau and somewhat artificial "large extra dimensions" models invented to explain "unseen" spacial dimensions [8, 9].

String tension *T*, on one hand, is related to the Fundamental length of
the String *l* via

*l*^{2} = 1/(*pi***T*), where *pi* is Mathematical constant.

On the other hand, String tension *T* is also related to Regge slope *alpha* of
QCD via

*T* = 1/(2*pi***alpha*)

Major textbook on String theory "Superstring theory" is quite vague on string Tension *T*, *the key element* of string theory. On page one it is mentioned that Regge slope is about 1 GeV^{-2} [1].

But closer look at Regge slope reveals something *VERY interesting and ALL important*: leading Regge trajectories of (quark - anti quark) mesons are quite *different* for mesons built of different quark-antiquark pairs!

Regge slope *alpha* can be experimentally measured and is related to particle spin *J* and mass *M* via

*alpha* = *J*/(*M*^{2})

Figure 3 below illustrates (very schematically) how Regge slopes look like for mesons built of Up and Down (blue line), Charm and Strange (green line) and Top and Bottom (red line) quarks and anti-quarks:

Regge slopes for mesons made of Up and Down, Charm and Strange, Top and Bottom quark-antiquark pairs, based on papers [18-22].

Regge *intercept* is common for All mesons made up of quark-antiquark pairs and is generally considered to be about 0.47~0.5 (depending on mesons chosen for numerical analysis and accuracy of linear trajectory).

At the same time, numerous experimental and computational results, collected from mid 1970s, show that Regge *slope* is around 0.88 ∼ 0.9 GeV^{−2} for mesons build of light *u* and *d* quarks and is slowly decreasing to about 0.40 GeV^{-2} for mesons build of heavy *t* and *b* quarks, depending on quark content of hadron [18-22].

Which is exactly what one would expect given *different composition* of strings (Red_{i}, Green_{i}, Blue_{i}) of different tensions *T _{i}* for different quark-antiquark pairs making up mesons.

Nambu-Goto string with massive spinning quarks model shows that Regge slopes for different (quark-anti quark) mesons are quite different!

First analytical models for Regge slopes for different quark-antiquark mesons (and generalizations for baryons) were suggested as early as in 1976.

See [18-22] for details on various Regge models available and experimental and computational results in hadron spectroscopy.

Therefore, this *well established fact* can be seen as the first and the most
*direct confirmation* of Motley String theory. Both experimentally and by
different computational models of (quark-anti quark) mesons!

Applications to Standard Model and Cosmology

Excitation (e.g. soliton) of one *color* dimension of a colorless threads could be considered a quark of charge 1/3, while excitation of 2 color dimensions of a colorless thread could be perceived as a quark of charge 2/3. Total number of different modes of excitations on 3 color dimensions on each thread equals 6 (3 single + 3 pairs) and thus we arrive at 6 flavors of quarks (up, down, charm, strange, top, bottom) known from Standard Model [14].

In a similar way, different type of excitations not related to "color" which live on 3 *colorless* observable threads result in 6 known types of leptons [14].

Since color force is much stronger than Electromagnetic and Weak forces, energy required to produce excitation on String color components/dimensions is much higher than one required for an excitation on observable colorless threads. And that explains why quark masses (which form baryons and mesons) are generally much higher than masses of leptons (electron, muon, tau, etc).

Same Motley String model could be easily applied to Bosonic String formulated in 26 dimensions (26 = 1 + 5^{2}) and Heterotic String (combination of left-moving excitations of Bosonic string and right-moving modes of Superstring).

Interestingly Motley String theory also offers plausible explanation for "Dark Matter" puzzle present in modern cosmology, which basically states that about 95% of total matter of the Universe is made of unseen "dark" matter/energy [16].

According to Particle Data Group web site we have for Up (+2/3) and Down(-1/3) quark masses:

Mass of U quark: ~ 2.3 MeV

Mass of D quark: ~ 4.8 MeV

While at the same time masses of Neutron and Proton are:

Mass of Neutron (U+D+D): ~ 1008 MeV

Mass of Proton (U+U+D): ~940 MeV

Which means that most (~95%) of hadron masses come from *"binding energy"* (or quark-gluon plasma), not from masses of quarks making up hadron.

Much the same like hadrons masses, most of Universe's observable mass comes from energy related to Motley String spacial dimensions, compactified because of Color Force, and therefore not seen (aka "Dark matter").

Which means that "Dark matter problem" in fact may be seen as *experimental confirmation* of Motley String theory.

Energy "frozen" (aka *confined*) in spacial dimensions after compactification can not be observed directly by available particle accelerators and could well be the so called "dark matter" (aka energy) astrophysicists were looking for since 1930s [16].

Careful re-calculations (e.g. using Lattice QCD methods) of quark/hadron and lepton masses based on our Motley String model could possibly shed some light on "dark matter" mystery of modern astrophysics.

Motley String theory and its Color Force also offers *alternative to Higgs mechanism of mass generation* for particles and therefore explains how neutrinos get their mass from modes of excitations of compactified spacial dimensions of Motley String.

This in turn makes hunting of hypothetical Right-handed neutrinos (required by Higgs mechanism) not necessary because neutrinos are known to be Left-handed (to take part in Weak interactions), and their mass origins and oscillations are now explained by Motley String theory.

Motley String and Quarks mixing and Neutrinos Oscillations

For quarks of charge -1/3 (d,s,b) there are following three combinations of colors/flavors on
1. (Red_{1}, Green_{1}, Blue_{1})

2. (Red_{2}, Green_{2}, Blue_{2})

3. (Red_{3}, Green_{3}, Blue_{3})

And for quarks of charge +2/3 (u,c,t) we have another 3 possible combinations of colors/flavors on *two spacial colorless* threads of Motley String:

1. (Red_{1}, Green_{1}, Blue_{1}) + (Red_{2}, Green_{2}, Blue_{2})

2. (Red_{1}, Green_{1}, Blue_{1}) + (Red_{3}, Green_{3}, Blue_{3})

3. (Red_{2}, Green_{2}, Blue_{2}) + (Red_{3}, Green_{3}, Blue_{3})

On the other hand, Gluons are known to carry and transmit "color charge" between quarks.

Gluon may therefore be considered combination of excitations (Red_{i}, Green_{j}, Blue_{k}) of different *colorful spacial dimensions* with indexes (i,j,k) having different values.

Therefore one might think that "quark mixing" between six known types of quarks above is related to exchange of Gluons between string's 3 colorless threads and "quark mixing angle" introduced by Nicola Cabibbo in 1963 in an attempt to explain quark mixing [13].

Weak Force bosons (W and Z) may be considered excitations (Red_{i}, Green_{i}, Blue_{i}) of different *colorless spacial threads* of compactifed string dimensions.

In a similar fashion we have 3 different types of excitations on *TWO colorless spacial threads* for electron, muon and tau leptons, and 3 more modes of excitations on *SINGLE spacial colorless threads* for three different types of neutrinos (electron, muon and tau neutrinos).

Now if we replace QCD bosons (Gluons) with Weak force bosons (W and Z) we get exactly the same mechanism working and mixing (aka oscillating) both quarks and neutrinos!

Exchange of excitations on *compactified colorless spacial threads* requires far less energy
and is therefore happening more frequently than exchange of excitations on *colorful un-compactified dimensions* of Super string.

Which explains why Cabibbo matrix (using abbreviated terms) for *quark* mixing has *small* non-diagonal entries compared with Pontecorvo matrix for *neutrino* mixing, where non-diagonal entries are *considerably larger*! (see complete CKM and PMNS matrices on page 4 in [18] for details).

Therefore we have to conclude that Motley String theory provides explanation for both "quark mixing" and neutrino oscillations detected by 3 major experiments (South Dacota, Sudbury and Super-Kamiokade) [17].

Conclusion

Motley String model and idea of "colorful" spacial dimensions introduced in this article offers consistent and uniform approach to compactification problem present in all String models (Superstring, Bosonic, Heterotic).

It eliminates inconsistencies of compactification mechanisms proposed earlier (Kaluza-Klein, Calabi-Yau manifolds, etc).

Also it solves several major problems present in Standard Model:

1. explains number of particle generations (6 quarks and 6 leptons) of Standard Model.

2. explains fractional charges of quarks and quark/gluon confinement.

3. establishes the link between Multi-dimensional String theories and observable 4-dimensional world.

4. offers alternative to Higgs mechanism for particles mass generation and thus explains neutrino's mass and experimentally observed neutrino oscillations.

5. offers single mechanisms for both quark mixing and neutrino oscillations, which explains sharp difference between values in Cabibbo (CKM) and Pontecorvo (PMNS) mixing matrices for quarks and neutrinos.

6. offers solution for "dark" matter/energy problem of modern astrophysics.

**References:**

[1] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press, 1987.

[2] Michael B. Green, John H. Schwarz, "Anomaly Cancellations in Supersymmetric D=10 Gauge Theory and Superstring Theory", Physics Letters B149 (1984) pp. 117-122.

[3] G.Nordström, "Über die Möglichkeit, das Elektromagnetische Feld und das Gravitationsfeld zu vereiningen", Physikalische Zeitschrift, 1914.

[4] T.Kaluza, "Zum Unitätsproblem der Physik", Sitzungsberichte Preußische Akademie der Wissenschaften 96, 69, 1921.

[5] O.Klein, "Quantentheorie und fünfdimensionale Relativitätstheorie". Zeitschrift für Physik A 37 (12): 895–906, 1926.

[6] E.Calabi, "The space of Kähler metrics", Proc. Internat. Congress Math. Amsterdam, 2, pp. 206–207, 1954.

[7] B.R.Greene, "String Theory On Calabi–Yau Manifolds", arXiv:hep-th/9702155v1.

[8] J.Polchinski, TASI Lectures on D-branes, arXiv:hep-th/9611050.

[9] R.Brandenberger and C.Vafa, "Superstrings in the early universe". Nuclear Physics B 316 (2): 391–410, 1989.

[10] W.Greiner, A.Schäfer, Quantum Chromodynamics. Springer, 1994.

[11] B.V.Struminsky, "Magnetic moments of barions in the quark model". JINR-Preprint P-1939, Dubna, Russia. Submitted on January 7, 1965.

[12] F.Tkachov, "A contribution to the history of quarks: Boris Struminsky's 1965 JINR publication". arXiv:0904.0343

[13] V.A.Matveev and A.N.Tavkhelidze, "THE QUANTUM NUMBER COLOR, COLORED QUARKS AND QCD", Report presented at the 99th Session of the JINR Scientific Council, Dubna, 19-20 January 2006.

[14] F.Halzen, A.Martin, "Quarks & Leptons: An Introductory Course in Modern Particle Physics", John Wiley & Sons, 1984.

[15] "Dark matter" article on Wikipedia and refs there.

[16] Neutrino oscillations experiments:

1. Homestake, South Dacota, USA: https://en.wikipedia.org/wiki/Homestake_experiment

2. Sudbury Neutrino Observatory, Ontario, Canada: https://en.wikipedia.org/wiki/Sudbury_Neutrino_Observatory

3. Super-Kamiokande experiment, Japan: https://en.wikipedia.org/wiki/Super-Kamiokande

[17] Luis Ibanez and Angel Urugana, "String Theory and Particle Physics", Cambridge University Press, 2012.

[18] J. Pasupathy, Phys. Rev. Lett. 37 (1976) 1336.

[19] K. Igi, Phys. Lett. B 66 (1977) 276; Phys. Rev. D 16 (1977) 196.

[20] S. Filipponi and Y. Srivastava, Hadronic Masses and Regge Trajectories, HUTP- 97/A093 [hep-ph/9712204]

[21] L. Burakovsky and T. Goldman, On the Regge Slopes Intramultiplet Relation, https://arxiv.org/abs/hep-ph/9802247v1

[22] L.D. Soloviev "Relativistic quantum model of confinement and the current quark masses": https://arxiv.org/abs/hep-ph/9803483v1

Historical Notes:

The idea of String's colorful spacial dimensions and compactification due to "color" force first came to me in the second half of 1990 when after graduation from Department of Physics of Leningrad State University (LGU) I joined Physical Technical Institute of Academy of Sciences of USSR (Department of Plasma Physics and Astrophysics, Laboratory of Plasma-gaso dynamics). During my study years at LGU I was quite interested in High Energy Physics and first two years attended so called "theory track" (advanced math/phys courses by professors from LOMI designed for future theorists) and seminars organized at Institute of Physics (NIIF) by members of Math Physics and Field Theory cathedras. First draft of this paper was created in late August of 1991 (one night in my wife's parents apartment in Old Peterhoff when knee pain kept me up late).

In June 1992 I got a tourist visa for USA and in late August 1992 left Russian Federation for New York, USA. First days of my American life I spent on Princeton University campus. At that time I contacted Alexander M. Polyakov (one of his students gave me his phone number) who kindly agreed to have a look in my article and suggested that I leave it under the door of his office. But since article was hand written on poor quality soviet paper and I had more pressing issues to solve it was not meant to be so.

Article is published here largely in its original form with some new references (thanks to Internet) and possible application of Motley String theory to cosmology ("dark" matter/energy).

Version 2.0 added section on recently observed neutrinos oscillations as well as quark mixing, both explained by Motley String.

Version 3.0 added reference to experimental and computational material on Regge slope values for different (quark-anti quark) mesons, essentially confirming main ideas of Motley String regarding different values of T_{i} string tension in different spacial dimensions. This is the most direct confirmation of Motley String theory so far.

Version 3.5 added explanation for quite different non-diagonal mixing matrix entries for quarks (Cabibbo) and neutrinos (Pontecorvo) matrices.

Version 4.0 added more results on various Regge slope models and values from hadron spectroscopy.

Version 5.0, released 28 February 2019, added Figure 3 with Regge slopes for mesons made of different quark-antiquark pairs, illustrating (schematically) how different string tension *T _{i}* values in different spacial dimensions affect the QCD hadronic spectroscopy.

Those Regge slope results collected from mid 1970s provide *first direct evidence in support of Motley String theory*.

If you find the ideas outlined in this article worthy of discussion and/or publication or would like to provide comments/feedback, please do not hesitate and get in touch via email found on my Resume page.

In the second part of 2017 I was (a bit unexpectedly) invited to several major international conferences on High Energy Physics and finally presented my version of String theory to scientific community on 11th of December 2017 in Rome.

Number 26 seems to have VERY special meaning not only in High Energy Physics, but also in my life!

Lord works in mysterious ways, as they say in "Blues Brothers" :-).

Any form of financial assistance (e.g. research grant, scholarship, employment contract, etc.) would be much appreciated.

Copyright © 1991, 2014, 2018, 2019 George Yury Matveev