Geometric Solution of the Hierarchy Problem by Means of Einstein-Cartan Torsion

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1 Geomeic Soluion of he Hieachy Poblem by Means of Einsein-Caan Tosion Cal F. Diehe III and Joy Chisian Einsein Cene fo Local-Realisic Physics, 15 Thackley End, Oxfod OX LB, Unied Kingdom Two of he majo open quesions in paicle physics ae: (1 Why ae hee no elemenay femionic paicles obseved in he mass-enegy ange beween he elecoweak scale and he Planck scale? And (, wha mechanical enegy may be counebalancing he divegen elecosaic and song foce enegies of poin-like chaged femions in he viciniy of he Planck scale? In his pape, using a hiheo unecognized mechanism deived fom he non-linea amelioaion of Diac equaion known as he Hehl-Daa equaion wihin Einsein-Caan-Sciama-Kibble exension of geneal elaiviy, we pesen deailed numeical esimaes suggesing ha he mechanical enegy aising fom he gaviyinduced fou-femion self-ineacion in his heoy can addess boh of hese quesions in andem. I. INTRODUCTION Fo ove a cenuy Einsein s heoy of gaviy has povided emakably accuae and pecise pedicions fo he behaviou of macoscopic bodies wihin ou cosmos. Fo he elemenay paicles in he quanum ealm, howeve, Einsein-Caan heoy of gaviy may be moe appopiae, because i incopoaes spinos and associaed osion wihin a covaian descipion [1[. Fo his eason hee has been consideable inees in Einsein-Caan heoy, in he ligh of he field equaions poposed by Sciama [3 and Kibble [. Fo example, in a seies of papes Poplawski has agued ha Einsein-Caan-Sciama-Kibble (ECSK heoy of gaviy [5 solves many longsanding poblems in physics [[7[[9. His concen has been o avoid singulaiies endemic in geneal elaiviy by poposing ha ou obseved univese is pehaps a black hole wihin a lage univese [7. Ou concen, on he ohe hand, is o poin ou using numeical esimaes ha ECSK heoy also offes soluions o wo longsanding poblems in paicle physics. The fis of hese poblems can be aced back o he fac ha gaviy is a consideably weake foce compaed o he ohe foces. When Newon s gaviaional consan is combined wih he speed of ligh and Planck s consan, one aives a he enegy scale of GeV, which is some 17 odes of magniude lage han he heavies known elemenay femion (he op quak obseved a he mass-enegy of 175 GeV. Thus hee is a diffeence of some 17 odes of magniude beween he elecoweak scale and he Planck scale. Thee have been many aemps o explain his diffeence, bu none is as simple as ou explanaion based on he osion conibuions wihin he ECSK heoy. The second poblem we addess hee concens he well known fac ha as we appoach he Planck lengh, m, he elecosaic and song foce self-enegies of poin-like femions become divegen. We will show, howeve, ha osion conibuions wihin he ECSK heoy esolves his difficuly as well, a leas numeically, by counebalancing he divegen elecosaic and song foce enegy densiies nea he Planck scale. In fac, he negaive osion enegy associaed wih he spin angula momenum of elemenay femions may well be he long sough afe mechanical enegy ha couneacs he divegen posiive enegies semming fom hei elecosaic and song nuclea chages. II. STATIC COUNTERPART OF THE HEHL-DATTA EQUATION The ECSK heoy of gaviy is an exension of geneal elaiviy allowing spaceime o have osion in addiion o cuvaue, whee osion is deemined by he densiy of ininsic angula momenum, eminiscen of he quanummechanical spin [1[[3[[5[[7[[9[10[11[1[13[1[15[1. As in geneal elaiviy, he gaviaional Lagangian densiy in he ECSK heoy is popoional o he cuvaue scala. Bu unlike in geneal elaiviy, he affine connecion Γi k j is no esiced o be symmeic. Insead, he anisymmeic pa of he connecion, Sk ij = Γ [i k j (i.e., Eleconic addess: fdiehe@mailaps.og Eleconic addess: jjc@alum.bu.edu

2 he osion enso, is egaded as a dynamical vaiable simila o he meic enso g ij in geneal elaiviy. Then, vaiaion of he oal acion fo he gaviaional field and mae wih espec o he meic enso gives Einsein-ype field equaions ha elae he cuvaue o he dynamical enegy-momenum enso T ij = (/ gδl/δg ij, whee L is he mae Lagangian densiy. On he ohe hand, vaiaion of he oal acion wih espec o he osion enso gives he Caan equaions fo he spin enso of mae [5: s ijk = 1 κ S[ijk, whee κ = πg c. (1 Thus ECSK heoy of gaviy exends geneal elaiviy o include ininsic spin of mae, wih femionic fields such as hose of quaks and lepons poviding naual souces of osion. Tosion, in un, modifies he Diac equaion fo elemenay femions by adding o i a cubic em in he spino fields, as obseved by Kibble, Hehl and Daa [1[[5. I is his nonlinea Hehl-Daa equaion ha povides he heoeical backgound fo ou poposal. The cubic em in his equaion coesponds o an axial-axial fou-femion self-ineacion in he mae Lagangian, which, among ohe hings, geneaes a spino-dependen vacuum-enegy em in he enegy-momenum enso (see, fo example, Ref. [13. The osion enso S k ij appeas in he mae Lagangian via covaian deivaive of a Diac spino wih espec o he affine connecion. The spin enso fo he Diac spino ψ hen uns ou o be oally anisymmeic: s ijk = i c ψγ [i γ j γ k ψ, ( whee ψ ψ γ 0 := (ψ 1, ψ, ψ 3, ψ is he Diac adjoin of ψ and γ i ae he Diac maices: γ (i γ j = g ij. The Caan equaions (1 ende he osion enso o be quadaic in spino fields. Subsiuing i ino he Diac equaion in he Riemann-Caan spaceime wih meic signaue (+,,, gives he cubic Hehl-Daa equaion [1[[5: i γ k ψ :k = mc ψ + 3κ c ( ψγ 5 γ k ψ γ 5 γ k ψ, (3 whee he colon denoes a geneal-elaivisic covaian deivaive wih espec o he Chisoffel symbols, and m is he mass of he spino. The Hehl-Daa equaion (3 and is adjoin can be obained by vaying he following acion wih espec o ψ and ψ (especively, wihou vaying i wih espec o he meic enso o he osion enso [13: S = d x g { 1 κ R i c ( ψγ k ψ :k ψ :k γ k ψ mc 3κ ψψ c 1 } ( ψγ 5 γ k ψ( ψγ 5 γ k ψ. ( The las em in his acion coesponds o he effecive axial-axial, fou-femion self-ineacion menioned above: L AA = g 3κ c ( 1 ψγ 5 γ k ψ( ψγ 5 γ k ψ. (5 This self-ineacion em is no enomalizable. Bu i is an effecive Lagangian densiy in which only he meic and spino fields ae dynamical vaiables. The oiginal Lagangian densiy fo a Diac field in which he osion enso is also a dynamical vaiable (giving he Hehl-Daa equaion, is enomalizable, since i is quadaic in spino fields. Bu as we will see enomalizaion may no be equied, if ECSK gaviy uns ou o be wha is ealized in Naue. Befoe poceeding fuhe we noe ha he above acion is no he mos geneal possible acion wihin he pesen conex. In addiion o he axial-axial em, an axial-veco and a veco-veco ems can be added o he acion, albei as non-minimal couplings (see, fo example, Ref. [15. Howeve, i has been agued in Ref. [13 ha minimal coupling is he mos naual coupling of femions o gaviy because non-minimal couplings ae souced by componens of he osion ha do no appea naually in he models of spinning mae. Fo his eason we will confine ou eamen o he minimal coupling of femions o gaviy and he coesponding Hehl-Daa equaion, while ecognizing ha sicly speaking ou neglec of non-minimal couplings amouns o an appoximaion, albei a ahe good one. Moving fowad o ou goal of numeical esimaes, if we equie he acion ( o be invaian unde local U(1 phase ansfomaions, hen ψ :k ansfoms o ψ :k + iqa k ψ/ fo a chage q and a gauge field A k, and eq. (3 genealizes o i γ k ψ :k + q γ k A k ψ = mc ψ + 3κ c ( ψγ 5 γ k ψ γ 5 γ k ψ. (

3 In he es fame of he paicle and ani-paicle, wih he meic signaue (+,,,, his equaion simplifies o which can be fuhe simplified o + ψ1 +ψ 1 +ψ 1 + ψ +ψ i +ψ + qca 0 = mc ψ3 ψ 3 +ψ 3 ψ +ψ ψ whee we have used γ 0 = i γ 0 ψ + qca 0 γ 0 ψ = mc ψ + 3κ c ( ψγ 5 γ 0 ψ γ 5 γ 0 ψ, (7 3κ c and γ 5 = ψ 3 ψ {ψ1ψ 3 + ψψ + ψ 1 ψ3 + ψ ψ}, ( +ψ 1 +ψ ( If we now epesen he paicles and ani-paicles wih wo-componen spinos ψ a and ψ b, especively [17, whee ( ( ψ1 ψ3 ψ a := and ψ ψ b := (10 ψ ae he wo-componen spinos consiuing he fou-componen Diac spino, hen he above equaion can be wien as wo coupled paial diffeenial equaions: i ψ a + i ψ b + qca 0 ψ a = mc ψ a + 3κ c {ψ 1ψ 3 + ψψ + ψ 1 ψ3 + ψ ψ} ψ b (11 qca 0 ψ b = mc ψ b 3κ c {ψ 1ψ 3 + ψψ + ψ 1 ψ3 + ψ ψ} ψ a. (1 Unlike he case in Diac equaion, hese equaions fo he spinos ψ a and ψ b ae coupled equaions even in he es fame. They decouple in he limi when he osion-induced axial-axial fou-femion ineacion is negligible. On he ohe hand, a low enegies i is easonable o assume ha, in analogy wih he Diac spinos in fla spaceime, he above wo-componen spinos fo fee paicles decouple in he es fame, admiing plane wave soluions of he fom ogehe wih so ha iniially hey ae equal o each ohe [17: ψ a ( = e i(mc / ψ a (0 and ψ b ( = e +i(mc / ψ b (0, (13 ψ b ( = e +i(mc / ψ a (, (1 ψ b (0 = ψ a (0. (15 Subsiuion of his fom of soluions ino eqs. (11 and (1 educes hese equaions o he following wo equaions: mc ψ a ( + qca 0 ψ a ( = mc ψ a ( + 3κ c { ( } mc cos ψ a (0 ψ b ( (1 and mc ψ b ( qca 0 ψ b ( = mc ψ b ( 3κ c { ( } mc cos ψ b (0 ψ a (. (17 Now we ae only concened abou he saic scenaio, fo which (i.e., a ime = 0 hese pai of equaions educe o + qca 0 ψ a(0 = mc ψ a (0 + 3κ c ψ a (0 ψ b (0 (1 and qca 0 ψ b (0 = mc ψ b (0 3κ c ψ b (0 ψ a (0. (19 3

4 In ode o obain scala counepas of hese (now essenially decoupled equaions, we muliply hem hough fom he lef by ψ a (0 = (ψ 1(0, ψ (0 and ψ b (0 = (ψ 3(0, ψ (0, especively, and in he ligh of eq. (15 aive a + qca 0 and qca 0 ψ a (0 3κ c ψ a (0 = mc ψ a (0 (0 ψ b (0 + 3κ c ψ b (0 = mc ψ b (0. (1 Subsiuing in SI unis fo he scala field A 0 = V/c = qc πε 0 in he Loenz gauge (whee V is he elecic poenial, and fo κ = πg c, ogehe wih he pobabiliies ψ b (0 = ψ a (0 1/ 3 of finding he paicles in he volume 3 [cf. eq. (15, we finally aive a ou cenal equaions, which hold a leas fo = 0, fo any elecoweak femion of chage q and mass m and is ani-paicle in he Riemann-Caan spaceime: and q πε 0 3πG c = mc 3 ( 3πG c q πε 0 = mc 3, (3 whee is he adial disance fom q and he wo equaions coespond o he paicle and ani-paicle, especively. Noe ha, as one would expec, if we muliply hough each of hese wo equaions by 3, hen fo femion ani-femion pai annihilaion he ems on he LHS of he equaions cancel ou, leaving mc in enegy o fly off as phoons. I is also woh noing ha wihou amelioaing he Diac equaion wih a cubic em, eq. ( would educe fo an elecon o α / e = m e c, giving e = (α /m e c m, whee α = e πε 0 c is he fine sucue consan. This is one half of he classical elecon adius. Expeimenal evidence, howeve, suggess ha elecon adius is much smalle. As we shall see, ou calculaions wih he cubic em included pedics he elecon adius o be of he ode of 10 3 m, which is close o he Planck lengh. This may un ou o be he coec value of he elecon adius. Needless o say, wha we have pesened above is a deivaion of eq. ( wihin a heoy ha may be viewed as a semi-classical heoy of Diac fields in a Riemann-Caan spaceime [[5. I can be inepeed also as a heoy of gaviy-induced fou-femion self-ineacion wihin sandad geneal elaiviy [1[5. A possible second-quanized genealizaion of his heoy is beyond he scope of ou pape. Howeve, any such genealizaion mus necessaily epoduce he Hehl-Daa equaion (3 fo single femions even a easonably high enegies, jus as Diac equaion emains valid fo single femions a high enegies [1. I is heefoe no uneasonable o base ou numeical esimaes below on eq. ( deived above. We shall soon see ha his equaion is boh necessay and sufficien fo ou puposes. Finally, i is impoan o noe hee ha, despie he appeaance of fou spinos in he ineacion em of eq. (, i descibes he self-ineacion of a single femion, of ange m, no muual ineacions among he spins of fou disinc femions. Tha is o say, i does no descibe a spin field of some so as a caie of a new ineacion [5. If, howeve, one insiss on inepeing he ineacion em in eq. ( as descibing ineacions among fou disinc femions, hen he mass of he coesponding exchange boson would have o exceed 10 1 GeV, which is evidenly quie uneasonable. Wha is moe, as we shall soon see, wihin ou scheme any coecions due o vacuum polaizaion ae auomaically compensaed fo in he poducion of elecoweak mass-enegy, dicaed by eqs. ( and (3 above. III. PARTICLE MASSES VIA TORSION ENERGY CONTRIBUTION Fo ou numeical analysis i is insucive o bing ou he physical conen of eq. ( in puely classical ems. To his end, ecall ha when he self-enegy of an elecon is aibued solely o is elecosaic conen, i is found ha ha enegy is divegen, povided we assume he elecon o be a poin paicle. This enegy is called self-enegy, because i aises fom he ineacion of he chage of he elecon wih he elecosaic field ha i iself is ceaing. Wih he help ou esul ( above we can avoid his fundamenal difficully as follows. Muliplying hough eq. ( wih 3/π, i can be wien as ( q πε 0 3 π 3 ( 3πG c 3 3 π 3 = ( mc 3 π 3. (

5 Wih π 3 /3 ecognized as a volume of a sphee of adius, i is now easy o ecognize each quaniy in he paenhesis in his equaion as enegy, and each em as he coesponding enegy densiy. Now he fis em on he lef of he equaion is hee imes he elecosaic enegy densiy a a disance fom he chage, wih he lae given by [1 u saic = q 3π ε 0. (5 And he quaniy in he fis paenhesis is he oal enegy of a hin spheical shell of chage q a a disance [1: U sphee = q πε 0. ( Unfounaely he fis em in eq. ( diveges as 0. Bu if we cu-off a he Planck lengh, l P = G /c 3, hen by seing q = e fo an elecon we obain 3 e 3π ε GeV m 3. (7 Alhough finie, his is sill an exemely lage enegy densiy. Bu such a lage enegy densiy fo chaged lepons is neve ealized in Naue. A naual quesion hen is: Is hee a negaive mechanical enegy densiy ha cancels ou mos of his enegy o poduce he obseved es mass-enegy of lepons? We believe he answe lies in he second em of eq. (, which as we saw above aises fom he non-linea amelioaion of he Diac equaion wihin he ECSK heoy. Indeed, if we again se he Planck lengh cu-off fo in he second em of eq. (, hen we obain 9 G c lp GeV m 3. ( Compaing his value wih he elecosaic enegy densiy a he Planck lengh cu-off esimaed in eq. (7 we see a once ha he osion-induced mechanical enegy ( can indeed counebalance he huge elecosaic enegy. This is a supising obsevaion, consideing he widespead belief ha he numeical diffeences which aise [beween GR and ECSK heoies ae nomally vey small, so ha he advanages of including osion ae eniely heoeical [1. Moving fowad o ou goal of numeical esimaes, le us noe ha wheneve ems quadaic in spin happen o be negligible, hen he ECSK heoy is obsevaionally indisinguishable fom geneal elaiviy. Theefoe, fo posgeneal-elaivisic effecs, he densiy of spin-squaed has o be compaable o he densiy of mass. The coesponding chaaceisic lengh scale, say fo a nucleon, is efeed o as he Caan o Einsein-Caan adius, defined as [[1 Ca (l P λ C 1 3, (9 whee λ C is he Compon wavelengh of he nucleon. Now i has been noed by Poplawski [[7[[9 ha quanum field heoy based on he Hehl-Daa equaion may ave divegen inegals nomally encouneed in calculaing adiaive coecions, by self-egulaing popagaos. Moeove, he mulipole expansion applied o Diac fields wihin he ECSK heoy shows ha such fields canno fom singula, poin-like configuaions because hese configuaions would violae he consevaion law fo he spin densiy, and hus he Bianchi ideniies. These fields in fac descibe non-singula paicles whose spaial dimensions ae a leas of he ode of hei Caan adii, defined by he condiion ɛ κs, (30 whee s c ψ is he spin densiy, ɛ mc ψ is he es enegy densiy, and ψ 1/ 3 is he pobabiliy densiy, giving he adius (9. Consequenly, a he leas he de Boglie enegy associaed wih he Caan adius of a femion (which is appoximaely 10 7 m fo an elecon may inoduce an effecive ulaviole cuoff fo i in quanum field heoy in he ECKS spaceime. The avoidance of divegences in adiaive coecions in quanum fields may hus come fom spaceime osion oiginaing fom ininsic spin. Poplawski and ohes, howeve, ook ɛ o be he mass-enegy densiy of he femion o aive a he Caan adius (9. Bu i is easy o wok ou fom he fis em of ou eq. ( ha a he Caan adius he elecosaic enegy densiy fo an elecon is sill exemely lage: 3 π (10 7 m GeV m 3. (31 5

6 Fo his eason i is no coec o idenify ɛ wih he es mass-enegy densiy, which is GeV m 3 fo an elecon a he Caan adius. The elecosaic enegy densiy of an elecon is hus eleven odes of magniude highe. Theefoe ɛ is bee idenified wih he elecosaic enegy densiy (31, povided mos of i is canceled ou. If in eq. ( we se he elecosaic enegy densiy appeaing in is fis em o be equal o he spin enegy densiy induced by he fou-femion self-ineacion appeaing in is second em and solve fo, hen we obain π = α l P m. (3 Which is abou fify one imes lage han he Planck lengh, and is a emakably simple consan in ems of he Planck lengh and he fine sucue consan. Accoding o eq. (, his is he effecive adius a which enegy densiy due o spin densiy should compleely compensae he huge elecosaic enegy seen in (7. In ou view his is he coec Caan adius, a leas fo he chaged lepons, ha may sill povide a plausible mechanism fo aveing singulaiies, since i is close o he Planck lengh. I is impoan o noe, howeve, ha hese huge enegy densiies neve acually occu in Naue, because accoding o ou eq. ( hey ae auomaically compensaed. The physical mechanism descibed above is simply o enable exacion of he adius fo diffeen chaged femions. In ode o obain an obseved mass-enegy fo he elemenay femions, we now posi ha hee is a vey small diffeence beween he adii of hei elecosaic enegy densiy and hei spin enegy densiy: := x. We do no popose a specific eason fo his diffeence, bu one possible eason may be ou neglec o include axial-veco and veco-veco ineacions in he acion ( fo he deivaion of Hehl-Daa equaion, as we discussed ealie [13[15. While exclusion of such non-minimal couplings may be jusified on physical gounds, hei inclusion in he deivaion of eq. ( would no have allowed us o decouple he equaions of moion (1 and (17 fo paicles and ani-paicles a = 0. Consequenly, in ha case he pobabiliy densiies such as ψ a (0 1/ 3 could no be assumed o be exacly he same fo he elecosaic em and he spin densiy em in hose equaions. Pehaps a second-quanized genealizaion of he ECSK-Diac heoy would evenually lead us o a bee undesanding of he oigin(s of. In ode o appoximae he diffeence, we hold he adius fo he spin enegy densiy o ha of he cancellaion adius, because his adius is consan fo a given chaged lepon, and because we expec spin enegy densiy o be he same fo all chaged lepons. We hen vay he adius fo he elecosaic and he es mass-enegy densiies, which we ake o be he same. Using eq. (, his leads us o he following fomula fo ou numeical esimaes: x 3 πg c = m xc x 3. (33 As shown in he Appendix below, we wee able o find soluions fo x fo he chaged lepons using abiay pecision in Mahemaica. The fis in ou esuls lised below is he soluion fo up o 0 significan figues. Then, using he same pecision fo compaison, we lis he esuls fo e fo an elecon, µ fo a muon, and τ fo a auon, along wih he ani-femions using eq. (3: = m 0.0 MeV, e = m MeV, µ = m 10 MeV, τ = m 1777 MeV, e+ = m MeV, µ+ = m 10 MeV, τ+ = m 1777 MeV. Evidenly, vey minue changes in he adii ae seen o cause lage changes in he obseved es mass-enegies of he femions. Bu as he diffeences in he adii go lage, he esulan mass-enegies go highe, as one would expec. Needless o say, fo he ani-lepons he esuls fo x will be lage han ahe han smalle, by he same amoun. I seems exaodinay ha Naue would subscibe o such iny diffeences esuling fom lage numbe of significan figues, bu ha migh explain why he undelying elaionship beween he obseved values of he masses of he

7 elemenay paicles has emained elusive so fa. In addiion o he possible easons fo his menioned above, i is no inconceivable ha he diffeence beween he spin enegy densiy and he elecosaic enegy densiy adii aises due o puely geomeical facos. We also suspec ha hee may possibly be some kind of symmey beaking mechanism a wok simila o he Higgs mechanism, and his symmey beaking esuls in he obseved mass-enegy geneaion. As a consisency check, le us veify ha he iny lengh diffeences seen above vanish, 0, as he coesponding es mass-enegy diffeences end o zeo: E 0. To his end, we ecas eq. (33 fo abiay x in a fom involving only es mass-enegy on he RHS as: If we now se 3πG x c x 3 = m x c. (3 A and B 3πG c, (35 hen, wih E = m x c and seing x = as he cancellaion adius fo which E = 0, we obain This allows us o deive a geneal expession fo x when E 0: Fom his expession i is now easy o see ha and convesely, using (3, lim E 0 { A x A3 B 3 x = E { A lim B x x = B/A. (3 A x A3 B 3 x = E. (37 } } x 3 = E = x = B/A =, (3 = E = 0. (39 Consequenly, wih x, we see fom he above limis ha 0 as E 0, and vice vesa. As a ough esimae he calculaion fo he adius q of elemenay quaks can be pefomed in a simila manne as ha fo elecons, since a such sho disances he song foce educes o a Coulomb-like foce. One mus also faco-in he elecosaic enegy, so ha a elaionship like he following mus be calculaed, say, fo he op quak: 9 qx + α s c qx 3πG c ( c = m c. (0 Hee α s is he appopiae song foce coupling (we use 0.1, and we have used a geneic expession fo he oal enegy densiy of he song field as a spheical shell less spheical symmey o mach he ohe ems. Needless o say, a cancellaion adius diffeen fom ha of he chaged lepons has o be calculaed fis, by seing 9 q + α s c q 3 q = 3πG c 3 qx ( c. (1 A calculaion of he adius fo he op quak based on eq. (0 can be found in he Appendix. We expec i o be only a vey ough esimae of he acual value of he adius. Since only one spin densiy is involved, he above calculaion migh be able o appoximae he behaviou of he quaks. Bu hee may be slighly diffeen adial diffeences fo he elecosaic pa and he song foce pa, which would make solving fo hose diffeences vey difficul. The calculaion of he adii q fo he up and down quak will pobably be poblemaic as well, since hei masses ae no well known. Bu if an undelying elaionship is discoveed, hen ha may help o know hose masses bee. In a simila ough manne we appoximae he adius n fo neuinos by eplacing α wih he coupling fo he weak foce, α w 1/9.5, in eq. (33 and using he mass-enegy uppe limi fom Ref. [0. The esuls ae consisen wih adius of 10 3 m. I appeas ha ou appoximaions ae of he same ode fo all lepons and quaks, and ha accoding o ou ough esimaes he size of all elemenay femions could be vey close o he Planck lengh. 3 q 7

8 IV. POSSIBLE SOLUTION OF THE HIERARCHY PROBLEM As alluded o in he inoducion, he Hieachy Poblem efes o he fac ha gaviaional ineacion is exemely weak compaed o he ohe known ineacions in Naue. One way o appeciae his diffeence is by combining he Newon s gaviaional consan G wih he educed Planck s consan and he speed of ligh c. The esuling mass scale is he Planck mass, m P, which some have speculaed o be associaed wih he exisence of smalles possible black holes [7. If we compae he Plank mass wih he mass of he op quak (he heavies known elemenay paicle, c m P = G kg, GeV m = c kg, hen we see ha hee is some 17 odes of magniude diffeence beween hem. This illusaes he enomous diffeence beween he Planck scale and he elecoweak scale. Many soluions have been poposed o explain his diffeence, such as supesymmey and lage exa dimensions, bu none has been univesally acceped, fo one eason o anohe. Fuhemoe, ecen expeimens pefomed wih he Lage Hadon Collide ae gadually uling ou some of hese poposals. Bu egadless of he naue of any specific poposal, i is clea fom he above values ha pedicions of numbes wih a leas 17 significan figues ae necessay o successfully explain he diffeence beween m P and m. We saw fom ou numeical demonsaion in he pevious secion ha wihin he ECSK heoy minue changes in lengh can induce sizable changes in he obseved masses of elemenay paicles, and ha we do have numbes a ou disposal wih moe han 17 significan figues fo poducing hose masses. Moeove, all lengh changes occuing in ou demonsaion ae aking place close o he Planck lengh. Thus, since we ae canceling ou nea he Planck lengh o obain masses down o he elecoweak scale, ous is clealy a possible mechanism fo esolving he Hieachy Poblem. Wihin he ECSK heoy, which exends geneal elaiviy o include spin-induced osion, gaviaional effecs nea mico scales ae no necessaily weak. On he ohe hand, since osion is poduced in he ECSK heoy by he spin densiy of mae, i is mosly confined o ha mae, and hus is a vey sho ange effec, unlike he infinie ange effec of Einsein s gaviy poduced by mass-enegy. In fac he osion field falls off as 1/, as shown in he calculaions of Sec. III, since i is poduced by spin densiy squaed, confined o he mae disibuion [9. To compae he senghs of gaviaional and osion effecs a vaious scales, we may define a mass-dependen dimensionless gaviaional coupling consan, Gm c, and evaluae i fo he elecon, op quak, and Planck masses: α Ge α G α GP α e = = Gm e c = Gm c = Gm P c , , = 1, e πε 0 c Hee α e is he elecomagneic coupling consan, o he fine sucue consan. Fom hese values we see ha nea he Planck scale he gaviaional coupling is vey song compaed o he elecomagneic coupling. Howeve, as we noed above and in Sec. III, nea he Planck scale osion effecs due o spin densiy ae also vey song, albei wih opposie polaiy compaed o ha of Einsein s gaviy, akin o a kind of ani-gaviy effec of a vey sho ange. Fo ou demonsaion above we have used elecosaic enegy densiy and spin densiy fo mae in a saic appoximaion, fo which he field equaion wihin he ECSK heoy educes o G 00 = T 00. A numeical esimae fo G 00 fom he conibuions of he elecosaic enegy and spin densiy pas of T 00 a ou cancellaion adius gives G 00 sa = πg c m ( and G 00 spin = πg 3 πg c c m. (3

9 Evidenly, hese field senghs a he cancellaion adius ae quie lage even fo a single elecon. Founaely hey ae neve ealized in Naue, because, as we can see, hey cancel each ohe ou o poduce G 00 ne = 0. On he ohe hand, if we use only he mass-enegy densiy fo elecon a he cancellaion adius, hen we obain G 00 mass m, which is again some 1 odes of magniude off he mak. Wha is moe, he lae field sengh does no fall off as fas as ha due o he spin-induced osion field. Thus i is easonable o conclude ha wihou he cancellaion of divegen enegies due o he fou-femion ineacion we have exploed hee, ou univese would be highly impobable. 9 V. CONCLUDING REMARKS In his pape we have addessed wo longsanding quesions in paicle physics: (1 Why ae hee no elemenay femionic paicles obseved in he mass-enegy ange beween he eleco-weak scale and he Planck scale? And (, wha mechanical enegy may be counebalancing he divegen elecosaic and song foce enegies of poin-like chaged femions in he viciniy of he Planck scale? Using a hiheo unecognized mechanism exaced fom he well known Hehl-Daa equaion, we have pesened numeical esimaes suggesing ha he osion conibuions wihin he Einsein-Caan-Sciama-Kibble exension of geneal elaiviy can addess boh of hese quesions in conjuncion. The fis of hese poblems, he Hieachy Poblem, can be aced back o he exeme weakness of gaviy compaed o he ohe foces, inducing a diffeence of some 17 odes of magniude beween he elecoweak scale and he Planck scale. Thee have been many aemps o explain his huge diffeence, bu none is simple han ou explanaion based on he spin induced osion conibuions wihin he ECSK heoy of gaviy. The second poblem we addessed hee concens he well known divegences of he elecosaic and song foce self-enegies of poin-like femions a sho disances. We have demonsaed above, numeically, ha osion conibuions wihin he ECSK heoy esolves his difficuly as well, by counebalancing he divegen elecosaic and song foce enegies close o he Planck scale. I is widely acceped ha in he sandad model of paicle physics chaged elemenay femions acquie masses via he Higgs mechanism. Wihin his mechanism, howeve, hee is no saisfacoy explanaion fo how he diffeen couplings equied fo he femions ae poduced o give he coec values of hei masses. While he Higgs mechanism does besow masses coecly o he heavy gauge bosons and a massless phoon, and while ou demonsaion above does no funish a fundamenal explanaion fo he femion masses eihe, we believe ha wha we have poposed in his pape is wohy of fuhe eseach, since ou poposal also offes a possible esoluion of he Hieachy Poblem. Needless o say, he geomeical cancellaion mechanism fo divegen enegies we have poposed hee also dispels he need fo mass-enomalizaion, wih ou cancellaion adius acing as a naual cuoff adius aming he infiniies. Thus boh classical and quanum elecodynamics appeas o be moe complee wih osion conibuions included. Appendix: Calculaions of Cancellaion Radii using Wolfam Mahemaica In his appendix we explain how we used he abiay-pecision in Mahemaica o solve he numeical equaions ou o significan figues. Each equaion displayed below deived fom ou cenal equaion ( is simplified so ha only he numeical facos have o be used, since he dimensional unis cancel ou, leaving lenghs in mees. Fo decimal facos, he numbes mus be padded ou o digis wih zeos. Then he numeical pa of elecosaic enegy densiy is defined as A and he numeical pa of spin enegy densiy is defined as B, jus as in eq. (35 above. These ae hen used houghou o pefom he calculaions. Fo he values of vaious physical consans involved in he calculaions we have used he 01 CODATA values, Ref. [19, and values fom he Paicle Daa Goup, Ref. [0. Calculaion of he Cancellaion Radius fo Chaged Lepons using Fomula (: 3 πg c = 0 (A.1 A:=N[( ( ( /, ; B:=N[(3π( ( /

10 10 (( , ; N[Solve[A B == 0,, //Las { } Calculaion of Radius e of Elecon and Posion e 3πG c = m e c 3 e 3πG e c m e c e = 0 (A. e+ + 3πG c = m e+c e πG e+ c m e+ c e+ = 0 (A.3 C1:=N[( (( , ; N [ Solve [ A B e C1 e == 0, e, //Las N [ Solve [ A + B e+ C1 e+ == 0, e+, //Las { e } { e } Calculaion of Radius µ of Muon and Ani-Muon µ 3πG c = m µ c 3 µ 3πG µ c m µ c µ = 0 (A. µ+ + 3πG c = m µ+c µ πG µ+ c m µ+ c µ+ = 0 (A.5 C:=N[( (( , ; N [ Solve [ A B µ C µ == 0, µ, //Las N [ Solve [ A + B µ+ C µ+ == 0, µ+, //Las { µ } { µ } Calculaion of Radius τ of Tauon and Ani-Tauon τ 3πG c = m τ c 3 τ 3 πg τ c m τ c τ = 0 (A. τ+ + 3πG c = m τ+c τ πg τ+ c m τ+ c τ+ = 0 (A.7 C3:=N[( (( , ; N [ Solve [ A B τ C3 τ == 0, τ, //Las N [ Solve [ A + B τ+ C3 τ+ == 0, τ+, //Las { τ } { τ } Calculaion of he Cancellaion Radius fo Quaks using Fomula (1: 9 + α s c 3πG c q ( = + 3α s c q (93πG c = 0 (A. D:=N[(3(1/10( ( , ; N [ Solve[((A + D q (9B == 0, q, //Las

11 11 { q } Calculaion of Radius q of Top Quak: 9 q + α s c q 3πG c q = m c 3 q + 3α s c (93πG q c q 9 m c q = 0 (A.9 E:=N[9 ( (( , ; N[Solve[(A + D (9B q q E q == 0, q, //Las { q } [1 F. W. Hehl, P. von de Heyde, G. D. Kelick, and J. M. Nese, Geneal elaiviy wih spin and osion: Foundaions and pospecs, Rev. Mod. Phys., 393 (197. [ A. Tauman, in Encyclopedia of Mahemaical Physics, edied by J. -P. Fancoise, G. L. Nabe, and Tsou S. T. (Elsevie, Oxfod, 00, vol., p 19. [3 D. W. Sciama, The physical sucue of geneal elaiviy, Rev. Mod. Phys. 3, 3 (19. [ T. W. B. Kibble, Loenz invaiance and he gaviaional field, J. Mah. Phys., 1 (191. [5 F. W. Hehl and B. K. Daa, Nonlinea spino equaion and asymmeic connecion in geneal elaiviy, J. Mah. Phys. 1, 133 (1971. [ N. J. Poplawski, Mae-animae symmey and dak mae fom osion, Phys. Rev. D 3, 0033 (011. [7 N. J. Poplawski, Nonsingula, big-bounce cosmology fom spino-osion coupling, Phys. Rev. D 5, (01. [ N. J. Poplawski, Cosmological consequences of gaviy wih spin and osion, Ason. Rev., 10 (013. [9 N. J. Poplawski, Univese in a black hole wih spin and osion, axiv: (01. [10 T. Oin, Gaviy and Sings: Cambidge Monogaphs on Mahemaical Physics (Cambidge Univesiy Pess, 00. [11 F. Rohlich, Classical Chaged Paicles, 3d ediion, (Wold Scienific, Singapoe, 007. [1 M. Blagojević and F. W. Hehl (edios, Gauge Theoies of Gaviaion (Impeial College Pess, London, 013. [13 G. de Beedo-Peixoo, L. Feidel, I.L. Shapio, and C.A. de Souza, Diac fields, osion and Babeo-Immizi paamee in cosmology, Jounal of Cosmology and Asopaicle Physics 0, 017 (01. [1 J. Magueijo, T. G. Zlosnik, and T. W. B. Kibble, Cosmology wih a Spin, Phys. Rev. D 7, 0350 (013. [15 A. S. Rudenko and I. B. Khiplovich, Gaviaional fou-femion ineacion in he ealy Univese, Phys.Uspekhi 57, 17 (01. [1 J. Boos and F. W. Hehl, Gaviy-induced fou-femion conac ineacion: libeaing he inemediae W and Z ype gauge bosons, axiv: (01. [17 D. Giffihs, Inoducion o Elemenay Paicles, nd Revised Ed. (Wiley-VCH, Hoboken, New Jesey, 00, chap. 7. [1 R. P. Feynman, R. B. Leighon, and M. Sands, The Feynman Lecues on Physics (Addison-Wesley, Reading, Massachuses, 19, vol. II, chap., p 1. [19 P. J. Moh, D. B. Newell, and B. N. Taylo, CODATA Recommended Values of he Fundamenal Physical Consans: 01, Zenodo (015. [0 K.A. Olive e al. (Paicle Daa Goup, Chinese Physics C 3, (01; see also 015 updae.