New Path to Unification of Gravity with Particle Physics
aa r X i v : . [ phy s i c s . g e n - ph ] D ec New Path to Unification of Gravity with Particle Physics
Alexander Burinskii
Laboratory of Theor. Phys. , NSI, Russian Academy of Sciences, B. Tulskaya 52 Moscow 115191 Russia, ∗ The principal new point is that ultra-high spin of the elementary particles makes Einstein’s gravityso strong, that its influence to metric is shifted from Planck to the Compton scale! Compatibilityof the Kerr-Newman (KN) gravity with quantum theory is achieved by implementation of thesupersymmetric Higgs model without modification of the Einstein-Maxwell gravity. We considerthe nonperturbative bag-like solution to supersymmetric generalized LG field model, which createsa flat and supersymmetric vacuum state inside the bag, forming the Compton zone for consistentwork of quantum theory. The bag is deformable, and its shape is controlled by BPS bound, providingcompatibility of the bag boundary with external gravitational and electromagnetic (EM) field. Inparticular, for the spinning KN gravity bag takes the form of oblate disk with a circular string placedon the disk border. Excitations of the KN EM field create circular traveling waves. The super-bagsolution is naturally upgraded to the Wess-Zumino supersymmetric QED model, indicating a bridgefrom the nonperturbative super-bag to perturbative formalism of the conventional QED.
PACS numbers: 11.27.+d, 04.20.Jb, 04.70.Bw
I. INTRODUCTION
Modern physics is based on Quantum theory and Grav-ity. The both theories are confirmed experimentally withgreat precision. Nevertheless, they are contradicting andcannot be combined in a unified theory. One of theprincipal points is the structure of elementary particles,which are considered as pointlike and even structureless(for example electron) in quantum theory, but should bepresented as the extended field configurations for com-patibility with the right hide side of the Einstein equa-tions, G µν = 8 πT µν . Revolutionary step for unification was made in su-perstring theory, however, as mentioned John Schwarz, “...Since 1974 superstring theory stopped to be consideredas particle physics... ” and “... a realistic model of ele-mentary particles still appears to be a distant dream ... ” [1]). One of the reasons of this is that extra dimensionsare compactified with extra tiny radii of order the Plancklength 10 − cm, which does not correlate with charac-teristic lengths of quantum physics and makes impossibleto test extra dimensions with currently available energies.The idea to bring fundamental gravitational scale closeto the weak scale was considered in different approaches,and in particular, in the brane world scenario, where theweakness of the localized 4d gravity is explained by its“leaks” into the higher-dimensional bulk, and the braneworld mechanism allowed to realize ideas of the super-string theory for any numbers of the extra dimensions[2].Alternative ideas were related with nonperturbative4D solutions of the non-linear field models – solitons, inparticular, solitonic solutions to low energy string theory[3–5]. This approach, being akin to the Higgs mechanismof symmetry breaking, is matched with nonperturbative ∗ email: [email protected] approach to electroweak sector of the Standard Model.The most known is the Nielsen-Olesen model of dualstring based on the Landau-Ginzburg (LG) field modelfor a phase transition in superconducting media, and alsothe famous MIT and SLAC bag models [6–8] which aresimilar to solitons, but being soft, deformable and oscil-lating, acquire many properties of the dual string mod-els. Besides, being suggested for confinement of quarks,the bag models assume consistent implementation of theDirac equation. The question on consistency with grav-ity is not discussed usually for the solitonic models, as itis conventionally assumed that gravity is very weak andis not essential on the scale of electroweak interactions.For example, in [9] we read ”... quantum gravity effectsare usually very small, due to the weakness of gravity rel-ative to other forces. Because the effects of gravity areproportional to the mass, or the energy of the particle,they grow at high energies. At energies of the order of E GeV, gravity would have a strength comparable withthat of the other Standard Model interactions.”
Our principal point here is that the assumption on theweakness of gravity is not correct, since it is based onthe underestimation of the role of spin in gravitationalinteractions. Indeed, nobody says that gravity is weak inCosmology where physics is determined by giant masses.Similarly, the giant spin/mass ratio of spinning parti-cles makes influence of gravity very strong in the particlephysics.For the great spin/mass ratio of the elementary parti-cles, about 10 − (in dimensionless units G = c = ~ = 1), the commonly accepted view that gravity is weakand not essential in particle physics up to Planck scale,should be replaced by principally new point of view that GRAVITY IS NOT WEAK , and its influence becomescrucial for the structure of the spinning particles at theCompton scale of the electroweak interactions.We show that spin of the Kerr-Newman (KN) rotat-ing black hole (BH) with parameters of an electron de-forms space-time in the Compton zone so strongly thatthe compatible with gravity structure of spinning particleis determined almost unambiguously as a supersymmet-ric bag model.The KN space-time with ultra-high spin has the nakedsingular ring creating two-sheeted topology. This spacediffers from Minkowski space so strongly that neither theDirac theory nor perturbative QED can be applied, sincethey require the flat space at least in the Compton zone ofthe dressed particle. We show here that this conflict canbe resolved without modification of the Einstein-Maxwellgravity – the space can be cured by a supersymmetric bagmodel , in which the singular region of KN solution is re-placed by the flat internal space of the Compton size. Wefind the corresponding non-perturbative BPS-saturatedsolution in frame of the supersymmetric generalized LGmodel, in which boundary of the bag is formed by thedomain wall (DW) interpolating between the externalKN gravity and the supersymmetric vacuum state in-side the bag. Similar to the usual bag models, the super-bag model is deformable and displays a super-consistencywith the external gravitational and electromagnetic (EM)KN field, in the sense that its shape and dynamics arefully defined by matching its boundary with a special sur-face (which can be called as ”zero gravity surface” (ZG)),where the external gravitational field is compensated bythe EM field. The ZG-surface determines position of theBPS DW-solution, and therefore, it determines shape ofthe bag, as a disk-like configuration with a closed stringplaced at the sharp border of the disk [10–12]. We showthat the supersymmetric LG model can be naturally up-graded to the Wess-Zumino SuperQED model [13], re-vealing connections between the non-perturbative solu-tions of the supersymmetric LG model and the conven-tional perturbative technics used in QED.
II. SUPER-BAG MODEL AS BPS SOLUTIONTO GENERALIZED LG MODELA. Basic features of the ultra-rotatingKerr-Newman solution
It has been recently obtained [14, 15], that the source ofultra-spinning Kerr-Newman (KN) solution can be con-sidered as a superconducting soliton having many fea-tures of the bag model [10, 11, 16], but with the essentialadvantage of compatibility with Einstein-Maxwell grav-ity in four dimensions. As is known, the bag modelstake intermediate position between strings and solitons[17–19]. Although, the bags were initially offered as theextended models of hadrons, [6–8], being based on theAbelian Higgs model of symmetry breaking their indi-cated rather applicability to the Salam-Weinberg modelof leptons, which was one of the reasons to consider thegravitating KN bag as the model for consistent with grav-ity leptons.The spinning KN solution is of particular interest inthis regard, since, as it was obtained by Carter [20, 21], that gyromagnetic ratio of the KN solution is g = 2 , andtherefore corresponds to the external field of the elec-tron. The spin/mass ratio of the electron is about 10 , and structure of source of the KN solution for such ahuge spin should shed the light on origin of the conflictbetween gravity and quantum theory. One can see thatthe KN field with parameters of electron becomes ex-tremely strong on the Compton distances, so that theBH horizons disappear and the Kerr singular ring of theCompton radius a = ~ /m becomes open, which breakstopology of space-time and creates two-sheeted metric.In the Kerr-Schild (KS) approach, metric of the KN so-lutions is [20] g µν = η µν + 2 Hk µ k ν , (1)where η µν is metric of an auxiliary Minkowski space M , (signature ( − + ++)), and H is the scalar function whichfor the KN solution takes the form H KN = mr − e / r + a cos θ , (2)where r and θ are oblate spheroidal coordinates, and k µ is a null vector field k µ k µ = 0 , forming a Kerr congruence– the vortex of polarization of gravitational and electro-magnetic field in the Kerr space-time. The Kerr singularring corresponds to border of the disk r = 0 , in the equa-torial plane cos θ = 0 . Similarly, vector potential of KN solution is alsocollinear with the null direction k µ ,A µ = − er ( r + a cos θ ) k µ (3) −10 −5 0 5 10−10−50510−10−50510 Z FIG. 1: Vortex of the Kerr light-like (null) congruence k µ propagates analytically from negative sheet of Kerr metric, r < , to positive one, r >
0. In the equatorial plane, cos θ =0 , the Kerr congruence is focused on the Kerr singular ring, r = cos θ = 0. The KN metric becomes two-sheeted, since the Kerrcongruence k µ dx µ = dr − dt − a sin θdφ, (4)is out-going at the ‘positive’ sheet of the metric, r > , and passes analytically to ‘negative’ sheet, r < , be-ing extended via ring r = 0 , where it becomes in-going.The two null vector fields k µ ( x ) ± become different at r > r < , leading to two different metrics g ± µν = η µν +2 Hk ± µ k ± ν on the positive and negative sheet ofthe same Minkowski background. Similarly, it leads alsoto two-sheeted vector-potential A ± µ , that makes space in-appropriate for quantum theory, and therefore, conflictbetween quantum theory and gravity is shifted by 22 or-ders earlier then it is usually expected, from the Planckto the Compton scale. As usually, singularity is signalto new physics – theory of more high level. The KNgravitational field is strong near the Kerr singular ringand creates vortex of the space-time polarization in theCompton zone of the dressed electron, which should beflat for work of quantum theory. It is usually assumedthat in vicinity of strong field, gravity should be mod-ified to a new Quantum Gravity. Taking into accountsharp incompatibility of Quantum and Gravity, naturalrequirement for such new theory would be separation oftheir zones of influence: formation of the internal zone (I) – flat core for quantum theory, and external zone (E) – for undisturbed gravitational and electromag-netic fields.There should also be selected intermediate zone (R) – interpolating between (I) and (E) .In the case of strong KN field, these demands becomeso restrictive that determine structure of the new theoryalmost uniquely. It turns out that the flat Compton zonefree from gravity may be achieved without modification ofthe Einstein-Maxwell equations, through SUPERSYM-METRY, which eats up the strong gravitational field inthe core of particle. Expelling gravity from the core ofthe KN spinning particle is similar to expelling the EMfield from superconducting core, and both of these super-phenomena are realized in core of the KN solution by thesupersymmertric LG field model [13, 22–28] in the formof a BPS-saturated Super-Bag solution, for which justthe strong contradiction between Quantum and Gravitydetermines extreme sensitivity of the model to the choiceof the separating surface (R) .The natural choice of this surface for the KN solutionwas suggested by C. L´opez [29]. According (1) and (2) itshould be the surface ”zero gravity” (ZG) at r = R = e m , (5)where function H vanishes H KN ( R ) = 0 , (6)and metric becomes flat, and can be matched with flatMinkowski space for r < R. It turns the L´opez source ofthe KN solution in a shell-like bubble. So far as r is theoblate spheroidal coordinate, [20], related with Cartesiancoordinates by transformations x + iy = ( r + ia ) exp { iφ K } sin θ, z = r cos θ, ρ = r − t, (7) the bubble surface r = R takes the oblate ellipsoidal form– the disk of the thickness R and radius r c = √ R + a , where a = J/m.
For solution without rotation, a = 0 , and bubble turnsinto a sphere of the classical radius r e . Such sphericalshape was suggested by Dirac in [30] as an ”extensibleelectron model” – prototype of the bag models, display-ing one of their basic features of the bags – their deforma-bility.We see that deformations of the KN Super-Bag appearas consequence of the requirement on sharp separation ofthe zones (I),(E),(R) . −15 −10 −5 0 5 10 15 −10 0 10−15−10−505 A. a / R = 0 B. a / R = 3 C. a / R = 7 D. a / R = 10 FIG. 2: (A): Spherical bag without rotation a/R = 0, anddisk-like bags for different values of the rotation parameter:(B)- a/R = 3; (C) - a/R = 7; and (D) - a/R = 10.
B. Spinning bag creates a string
Usually, it is assumed that bags are deformed by ro-tations taking the shape of a string-like flux-tube joiningthe quark-antiquark pair [6].In the KN Super-Bag, the spinning gravitational fieldcontrols disk-like shape of the bag, and string-like struc-ture is formed for a/R >
0, at edge rim of the disk, asshown in Fig.2. In the equatorial plane, this string ap-proaches very close to the Kerr singular ring, see Fig.3A,so, it is really just the singular ring regularized by thebag boundary.Among diverse attempts to use nonperturbative mod-els in the electroweak sector of the Standard Model (SM)[31–35], the central place takes the Nielsen-Olesen (NO)model [36, 37] of the string, which is created as a vortexline in a superconductor.The assumption, that Kerr singular ring is similar toNO model of dual string was done very long ago in[38, 39], where it was noted that excitations of the KN so-lution create traveling waves along the Kerr ring. Later,it was obtained in [5, 40] close connection of the Kerrsingular ring with the Sen fundamental string solution to −4 −3 −2 −1 0 1 2 3 4−3−2−10123 BAG BOUNDARY SINGULAR RING SINGULAR RING BAG BOUNDARY A ) B ) NAKED SINGULAR POINT v = c
FIG. 3: Regularization of the KN string. Boundary of bagfixes cut-off R = r e for the Kerr singular ring. A) The exactKN solution. B) The KN solution is excited by the lowesttraveling mode: emergence of the singular pole. low energy string theory.[44] In the KN bag model thisstring is formed at the sharp boundary of the supercon-ducting disk, as a dual analog of the NO vortex line insuperconductor.In accordance with the condition (6), the KN grav-ity controls position of the bag boundary (R) , and alsomore thin effects, such as excitations of the KN gravitydefine dynamics of the bag and appearance of the trav-eling waves.In particular, it has been shown [11], that the lowestEM excitation of the KN solution creates the travelingwave which has a circulating lightlike node. At this point,surface of the deformed bag touches the Kerr singularring, as it is shown in Fig.3B, which breaks regulariza-tion at this point and creates the lightlike singular pole,which can be considered as emergence of the bare Diracparticle circulating inside the Compton zone of dressedelectron. On the other hand, this pole breaks homogene-ity of the closed circular string, creating the frontal andrear ends turning this string in the open. As usual, theend points of an open string are associated with quarks,and the KN super-bag model turns into a single “bag-string-quark” system, 4D analog of D2-D1-D0-brane sys-tem of the string–M-theory. III. SUPERSYMMETRY ENSURESCONSISTENCY WITH GRAVITYA. Generalized LG field model and domain wall(DW) phase transition
The LG field model of superconductivity is used inmany solitonic models, in particular, in the NO dualstring model, as a field model in the MIT and SLACbag models, and really, it is also the the Higgs modelof symmetry breaking, because the Higgs vacuum itself ”... is analog to a superconducting metal”, [8]. The LGLagrangian used in the NO model (minimal LG model)is L NO = − F µν F µν −
12 ( D µ Φ)( D µ Φ) ∗ − V ( | Φ | ) , (8)where D µ = ∇ µ + ieA µ are the U (1) covariant derivatives,and F µν = A µ,ν − A ν,µ is the corresponding field strength,and potential V has the quartic form V = λ (Φ † Φ − η ) , (9)where η is condensate of the Higgs field Φ , its vacuumexpectation value (vev) η < | Φ | > , [36].The minimal LG model can be used to describe super-conductivity inside the bag – interplay of the KN vector-potential with the Higgs condensate. Since requirements (I),(E),(R) define inside the bag a flat space, the corre-sponding covariant derivatives can be taken as flat, D µ = ∇ µ + ieA µ → D µ = ∂ µ + ieA µ . (10)However, the NO and KN models have opposite spa-cial configurations: the KN bag model should describe asuperconducting disk surrounded by the long-range EMand gravitational field, while the NO model describesvortex of the EM field inside the superconducting Higgscondensate which breaks the external long-range EM andgravitational field. Note, that this is a typical drawbackof the most of soliton models and, in particular, the usualbag models which are formed as a ”cavity in supercon-ductor” [8]. The reason of this disadvantage lies in theuse of the potential (9).The correct opposite configuration – condensation ofthe Higgs field inside the core – requires more complexscalar potential V formed from several complex fieldsΦ i , i = 1 , ,
3, [14]. Kinetic part of the correspondinggeneralized LG model differs from those of the minimalLG model (8) only by summation over the fields Φ i , L GLGkin = − F µν F µν − X i ( D iµ Φ i )( D µi Φ i ) ∗ , (11)while the potential V is changed very essentially, and hasto be formed by analogy with machinery of the N = 1supersymmetric field theory [13] from a superpotentialfunction W (Φ i ) . [45] The scalar potential[46] V ( r ) = X i F i F ∗ i (12)is formed through derivatives of the function W (Φ i ) ,F i = ∂W/∂ Φ i ≡ ∂ i W, (13)where W (Φ i , ¯Φ i ) = Z (Σ ¯Σ − η ) + ( Z + µ ) H ¯ H, (14) µ and η are real constants, and the special notations areintroduced ( H, Z, Σ) ≡ (Φ , Φ , Φ ) , to identify Φ as thecomplex Higgs field H = | H | e iχ , (15)which interacts with the KN vector field A µ as D µ = ∇ µ + ieA µ . The fields Φ and Φ are assumed uncharged,and D iµ = ∇ iµ for i = 2 , . The condition F i = ∂ i W = 0 determines two vacuumstates with V = 0: (I) internal vacua: r < R − δ , where the Higgs field | H | = η, and Z = − µ, Σ = 0 , and (E) external vacuum state: r > R + δ , where the Higgsfield H = 0 , and Z = 0 , Σ = η, separated by spike of the potential V > (R) – a domain wall (DW), interpolating betweenzones (I) and (E) , in the full correspondence with therequirements (I),(E),(R) .Reduction of the corresponding LG equations to Bogo-molny form is performed by minimization of the energydensity per unit area of the DW surface, µ = 12 X i =1 [ X µ =0 |D ( i ) µ Φ i | + | ∂ i W | ] . (16)The four dimensional DW solutions in supersymmet-ric LG model have paid attention in the works [22–28],where it was usually considered the static planar DWspositioned in (x,y) plane, with the transverse to the wallz-direction. However, even in the simplest case of the onefield Φ( z ) and one coordinate z , µ = 12 ( | ∂ z Φ | + | ∂ Φ W | ) , (17)reduction of the LG equation to Bogomolny form turnsout to be nontrivial, since it requires the introduction ofan arbitrary phase factor α, so that (17) can be equiva-lently presented in the form µ = 12 | ∂ z Φ − e iα ∂ ¯Φ ¯ W | + Re e iα ∂ z W, (18)which is saturated by the Bogomolny equation ∂ z Φ = e iα ∂ ¯Φ ¯ W . (19)The DW forming the KN bag is much more compli-cated, since first of all it is not planar, but forms thespheroidal boundary profile of which is shown in Fig.3.Second, it is formed by three chiral fields Φ i , and thirdly,the most important feature is that this DW is not staticand has non-trivial dependence on the phases of the com-plex fields Φ i . The corresponding BPS saturated solutionwas found in [11, 16], where it was shown that the phases α i of the complex fields Φ i should acquire nontrivial de-pendence from time and angular coordinate α = 2 χ ( t, φ ) , α = α = 0 , (20) and the Higgs field becomes oscillating, showing that justin the KN bag model the transformation to Bogomolnyform (18) begins to operate at full power. disk r=0 DW surfaces R=0.9 and R=1 Singular ring zone of ring−string Domain Wall a = 10, R=1 FIG. 4: The domain wall profile (axial section) defined by theoblate spheroidal coordinate r = R. B. Minimal LG model and quantization of theangular momentum
The non-trivial dependence (20) is fixed in zone (I) ,where the generalized LG model is reduced to minimalLG model, and the NO Lagrangian (8) leads to equations (cid:3) A µ = J µ = e | H | ( χ, µ + eA µ ) . (21)One sees that vector potential A µ acquires from theHiggs field the mass term m v = e | H | , and the EM fieldbecomes short-range, with the characteristic parameter λ = 1 / ( e | H | ) corresponding to the penetration depth ofthe EM field in superconductivity. As a consequence,the currents vanish inside the core, J µ = 0 , leading tothe equations (cid:3) A µ = 0 , χ, µ + eA µ = 0 , (22)showing that besides of the massive component A m v µ which falls off receiving the mass m v from the Higgs field,there are also the components of different behavior.Vector-potential of the external KN solution (3) is A µ dx µ = − err + a cos θ ( dr − dt − a sin θdφ ) . (23)It grows near the core and takes maximal value at theboundary of the disk, at r = R = e / m, cos θ = 0 ,A maxµ dx µ = − Re me ( dr − dt − adφ ) . (24)Note, that the component A r is a perfect differential (asit is shown for example in [20]) and can be ignored. Atthe boundary, A maxµ is dragged by the light-like directionof the Kerr singular ring (see Fig.2) and the component A maxφ forms the closed Wilson loop, so that e I A maxφ dφ = 4 πma. (25)The right equation in (22) shows that penetrating insidethe disk vector potential determine oscillating phase ofthe Higgs field as χ = 2 mt + 2 amφ. The condition ofmultiplicity of the periods χ and φ gives 2 am = n, n = i, , , .., which in view of J = ma, leads to quantizationof angular momentum as J = n/ , n = i, , , ... (26)On the other hand (22) shows that phase of Higgs field H = Φ = | H | e i (2 mt +2 amφ ) (27)oscillates with the frequency ω = 2 m which supportsextension of the components A int = me , A inφ = mae inside the disk.[47] At the disk boundary (22) is broken,and according (21) there appear the surface currents J µ . −3−2−1012 −6−5−4−3−2−10123−8−6−4−2024 z − string real slice of complex string z + string φ = const. singular ring FIG. 5: Kerr’s coordinate φ = const. Kerr singular ring dragsthe vector potential, forming a closed Wilson loop along edgeborder of the DW.
IV. SUPERBAG AS NONPERTURBATIVESOLUTION OF THE SUPERQED MODELA. Bosonic sector of the supersymmetric LG model
As we noticed earlier, the generalized LG model basedon the superpotential (14) is not true supersymmetricmodel. The difference is that the true superpotential W is to be a chiral function of the chiral superfields Φ i , whilethe scalar potential V = F i F ∗ i (28)is formed from the chiral part F ∗ i = ∂W/∂ Φ i , (29)but also incudes the antichiral superpotential W + (Φ + i )depending on the antichiral superfields Φ + i F i = ∂W + /∂ Φ + i . (30) These relations are retained in the bosonic sector of thesupersymmetric theory, where the fields Φ i and Φ + i turninto the complex conjugate scalar components of the su-perfields.To get full correspondence with supersymmetric the-ory, the fields Φ i and ¯Φ i in (14), should be considered asindependent chiral fields Φ i and ˜Φ i , and there should alsobe introduced an antichiral superpotential W + (Φ + i , ˜Φ + i ) , which in the bosonic sector turns into complex conju-gated superpotential, built of the complex conjugatedfields ¯ W (Φ ∗ i , ˜Φ ∗ i ) . From the complex point of view, thetransition from (14) to supersymmeric Higgs model maybe considered as complexification of the moduli space –analytical extension from the real section, fixed by condi-tion ¯Φ i = Φ ∗ i , to its complex extension, the manifold withindependent coordinates Φ i and ˜Φ i , supplemented withcomplex conjugate coordinates Φ ∗ i , ˜Φ ∗ i . Therefore, thetransition to bosonic sector of the supersymmetric gen-eralized LG model requires doubling of the chiral field toeliminate their degeneracy on the real slice.Returning to the original work by Morris [42], wherethe potential (14) was suggested for super-generalizationof the Witten’s superconducting string model [43], weshould double the charged chiral fields Σ and Φ , andconsider five chiral superfields Σ ± , Φ ± , and Z, which inWitten’s interpretation of this model as the U ( I ) × U ′ ( I )Higgs field model, acquire the charges ( ± ,
0) for Φ ± , andcharges for the Σ ± fields as (0 , ± . The chiral superpo-tential (14) takes the form W (Φ i , ˜Φ i ) = Z (Σ + Σ − − η ) + ( Z + µ )Φ + Φ − , (31)with identificationΦ i = (Φ + , Φ − , Σ + , Σ − , Z ) . (32)The auxiliary fields F ∗ i = ∂W/∂ Φ i = ( F ∗ + , F ∗− , F ∗ Σ+ , F ∗ Σ − , F ∗ Z ) (33)take the form F ∗± = ( Z + µ )Φ ∓ , (34) F ∗ Σ ± = Z Σ ∓ , (35) F ∗ Z = Σ + Σ − + Φ + Φ − − η , (36)Vacuum expectation values of fields Φ i for which F ∗ i = 0give minima of the potential V = 0 corresponding tosupersymmetric vacuum states. Just as in case (14), weobtain two isolated vacua (I) Φ − Φ + = η , Z = − µ, Σ + = Σ − = 0; (E) Φ − = Φ + = 0 , Z = 0 , Σ + Σ − = η ;separated by the zone (R) of the positive potential V = | Σ + Σ − + Φ + Φ − − η | + | ( Z + µ )Φ + | + | ( Z + µ )Φ − | + | Z | ( | Σ + | + | Σ − | ) . (37) B. Transition to SuperQED model
We note that two oppositely charged superfields Φ + and Φ − give rise to correspondence of the supersymmet-ric LG model to kinetic part of the Wess-Zumino Su-perQED model [13], L SQEDkin = − W a W a +Φ ++ e eV Φ + | θθ ¯ θ ¯ θ +Φ + − e − eV Φ − | θθ ¯ θ ¯ θ , (38)where V is vector superfield, and W a = − ¯ D ¯ DD α V. Inthe same time, the potential part (31) corresponds to themost general renormalizable supersymmetric Lagrangianand gives rise to nonperturbative generalization of theSuperQED model.The chiral superfields Φ ± , are expressed in the compo-nent formΦ ± ( y ) = H ± ( y µ ) + √ θψ ± ( y µ ) + θθF ± ( y µ ) , (39)as functions of the chiral coordinates y µ = x µ + iθσ µ ¯ θ and θ, and the scalar components H ± are independent Higgsfields, splitting of the complex conjugated Higgs field ofthe minimal LG model in (14) and (15). Interplay of theoppositely charged Higgs fields H ± with vector potentialin zone (I) is defined by (22) and yields H ± = | H ± | e ± iχ , ¯ H ± = | H ± | e ∓ iχ , χ = 2 mt + 2 amφ, (40)where the fields ¯ H ± are scalar components of the antichi-ral fieldsΦ + ± ( y + ) = ¯ H ± ( y + µ ) + √ θ ¯ ψ ± ( y + µ ) + ¯ θ ¯ θ ¯ F ± ( y + µ ) , (41)as functions of the antichiral coordinates y + µ = x µ − iθσ µ ¯ θ and ¯ θ. The corresponding nonperturbative solutionwith doubled Higgs fields (40) can be obtained similar to[11].In the Wess-Zumino SuperQED model, the two Weylspinors ψ ± in (39) combine into one massive Dirac spinorof the electron – superpartner of the Higgs doublet H ± , [13].The nonperturbative super-bag solution generates inthe core of spinning particle the flat Compton zone (I) ,which is free from gravity and supersymmetric, represent-ing the conditions for the work of the perturbative Su-perQED model, while the remarkable perturbative prop-erties of the SuprQED – ”miracleous cancellations” ofthe component super-graphs [13] for a link to pertur-bative QED. Note, that in the nonperturbative modelof super-bag, the superpartners cannot be considered asseparate particles, and are integrated as the superfieldcomponents of a single nonperturbative solution. Thesuper-bag model reveals correspondence not only withgravity and electroweak sector of the SM, but also witha nonperturbative version of the SuperQED model. V. OUTLOOK
We have considered principal features of the Kerr-Schild geometry which specify the supersymmetric bagmodel as a new way to particle physics consistent withgravity and electroweak sector of the SM. Two of themare principally new relative to the widespread belief:– the spinning KN gravity is not weak, and becomesvery strong at the Compton scale of the particle physics,– compatibility between Quantum and Gravity can beachieved by means of supersymmetric generalization ofthe matter sector, without modification of the Einstein-Maxwell theory.We considered interplay of the KN gravity with thematter sector based on the supersymmetric generalizedLG field model, which is equivalent to supersymmetricHiggs mechanism of symmetry breaking, and give a non-perturbative solution to generalized LG field model in theform of a super-bag – nonperturbative version of the Su-perQED model. By conception, the 4d super-bag modelhas to be soft and oscillating, similar to the conceptionof the superstring models [7, 17, 18].Due to extreme high spin/mass ratio, impact of thegravitational KN field on the structure of space-time be-comes very strong, and the consistent supersymmetricnonperturbative solutions become very sensible to theexternal Einstein-Maxwell field. As a result,a) the super-bag model creates a free from gravityCompton core of spinning particle, where the supersym-metric vacuum state of the Higgs field provides the flatspace, required for consistent work of quantum theory;b) the super-bag takes the shape of a strongly oblatedisk forming a circular string along its border;c) gravitational and electromagnetic excitations of theKN solution create consistent stringy oscillations of thesuper-bag in the form of traveling waves.Many problems remain to be solved. The closest is theso far unsolved problem of the exact nonstationary (os-cillating or accelerating) generalization of the KN solu-tion, the problem of the consistent solutions of the Diracequation corresponding to confinement of quark insidethe bag, and so on.Nevertheless, the considered here features of the super-bag model are so intriguing that we risk to state that theyreally give the key to solution of the principal problem ofunification of gravity with particle physics.Finally, we should mention very important new as-pect of this study, the direct link to the non-perturbativeWess-Zumino SuperQED model, which provides remark-able cancellations between component diagrams, present-ing a link between the nonperturbative bag-like solutionand the conventional technics of the perturbative QED.
Acknowledgements
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