A NEW MODEL FOR W,Z, HIGGS BOSONS MASSES CALCULATION AND VALIDATION TESTS BASED ON THE DUAL GINZBURG-LANDAU THEORY(revised-2015)

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1 A NEW MODEL FOR W,Z, HIGGS BOSONS MASSES CALCULATION AND VALIDATION TESTS BASED ON THE DUAL GINZBURG-LANDAU THEORY(rvisd-5) Stfan Mhdintanu CITON Cntr of Thnology and Enginring for Nular Projts, Str. Atomistilor No. 9, BOP-5-MG-, Oras Magurl, Ilfov, Romania Formrly Snior Rsarhr PACS numbrs:..jb,.6.n,.5.ha,.38.g,.38.mh, 953.Sf Ky words: Partils physis, W,Z, H bosons, G-L thory, bta day mhanism Abstrat In this papr was r-visitd th dual Ginzburg-Landau modl for th alulation of Lornz for, olor magnti gluons urrnt, and th nrgy of vortx lins for a vortx triangular latti typ Abrikosov within a nulon, to find thir maning. For now, it was found that ths nrgis would orrspond to th subatomi partils, W, Ζ, Higgs bosons, pion π, and of nulon itslf. Also, it was dtrmind th fusion tmpratur of two nulons. Th modl prmits to xplain th bta day mhanism of radioisotops to b th sam as th dark ounts in th as of suprondutors, and to think in this ontxt to th nhanmnt of bta day for nular transmutation of radioativ wast. A link with gravity is disussd. In this modl to a suprondutor analogu, w us th Higgs fild ( v.. v 7GV ), and hn a Higgs boson to gnrat bu Shwingr fft ± W pairs for bta day alulation.. Introdution Usually, th masss of W, Z, ar alulatd by taking into aount a priori a Higgs fild, and th dfault using th Higgs mhanism and Higgs boson. Soon aftr th advnt of QCD, t Hooft and Mandlstam [] proposd th dual suprondutor snario of onfinmnt; th QCD vauum is thought to bhav analogously to an ltrodynami suprondutor but with th rols of ltri and magnti filds bing intrhangd: a ondnsat of olor magnti gluons xpls ltri filds from th vauum. If on now puts ltri harg and anti-harg into this mdium,

2 th ltri flux that forms btwn thm will b squzd into a thin, vntually stringlik, Abrikosov-Nilsn-Olson (ANO) vortx whih rsults in linar onfinmnt. Th dual suprondutor mhanism [] is an altrnativ that dos not rquir th ad ho introdution of a Higgs fild but instad uss dynamially gnratd topologial xitations to provid th srning suprurrnts. For xampl, U() latti gaug thory ontains Dira olor magnti gluons in addition to photons. Th dual suprondutor hypothsis postulats that ths olor magnti gluons provid th irulating olor magnti urrnts that onstrain th olor ltri flux lins into narrow flux tubs. thooft has shown that objts similar to th Dira monopols in U() gaug thory an also b found in non-ablian SU(N) modls. Th rsults ar onsistnt with a dual vrsion of th Ginzburg-Landau modl of suprondutivity. Important in undrstanding fild (magnti) dpndn was Abrikosov s fild thortial approahs basd on Ginsburg-Landau thory [] for typ I suprondutors ( κ, κ λ ξ, λ is th pntration dpth, ξ th ohrn lngth ) and typ ons ( κ ) II suprondutors, whih allows magnti flux Φ to pntrat th suprondutor in a rgular array quantizd in units of lmntary flux quantum Φ π.important was th quantization in a ring, flux Φ n, ( n, ±, ) In th prsnt papr w rvisitd G-L modl [],[],[5],[6],[7],[8],[9], in ordr to alulat th valus of th Lornz for, th urrnt, and th nrgis of th Abrikosov vortx lins insid of th nulon, in natural units, in viw to sarh for its rlvan; for th tim bing, it was found that this would orrspond to nrgis for subatomi partils, suh as that of W, Z, Higgs bosons, and of pion π. In stion is applid this modl to th mhanism of bta day. Also th onntion with gravity to b analyzd. In this modl to a suprondutor ± analogu, w us Higgs fild, and hn a Higgs boson, to gnrat W pairs for bta day alulation.. Th dsription of th analogu modl of nulon to a suprondutor Th normal ors that xist in typ-ii suprondutors in th mixd stat ar not sharply dlinatd. Th valu of numbr dnsity of suprltrons of n s is zro at th ntrs of th ors and riss ovr a haratristi distan ξ, th ohrn lngth. Th magnti fild assoiatd with ah normal or is sprad ovr a rgion with a diamtr of λ, and ah normal or is surroundd by a vortx of irulating urrnt. Th QCD vauum an b viwd as a dual suprondutor haratrizd by a olor magnti gluons ondnsat [],[8],[9],[], whn mbdding a stati sa quarks q q pair into th vauum. Th or of th flux tub is just a normal onduting vortx whih is stabilizd by solnoidal magnti suprurrnts, j s, in th surrounding vauum.

3 In ordr to alulat distintly th nrgy stats (masss) in natural units, firstly w rdriv th fild quations of olor magnti gluons urrnt and of th ltri flux. Thrfor, hr is adoptd a basi dual form of Ginzburg-Landau (G-L) thory [],[], [5],[6],[7], whih gnralizs th London thory to allow th magnitud of th ondnsat dnsity to vary in spa. As bfor, th supronduting ordr paramtr is a omplx funtion (x ), whr (x ) is th ondnsat dnsity n s. Also is dfind th wav funtion ( x) ns xp( iϕ ( x)), whr n s is th London (bulk) ondnsat dnsity, and ϕ ar ral funtions dsribing th spatial variation of th ondnsat. Th haratristi sal ovr whih th ondnsat dnsity varis is ξ, th G-L ohrn lngth or th vortx or dimnsion. Th x dnot th radial distan of points from th z -axis, th suprondutor oupying th half spa x. Outsid of th suprondutor in th half spa x, on has B E H H, whr, th xtrnal vtor H is paralll to th surfa. Th Ψ thory of suprondutivity [] is an appliation of th Landau thory of phas transitions to suprondutivity. In this as, som salar omplx funtion fulfils th rol of th ordr paramtr. First of all, w writ th magnti indution B urla A, whr A is th ltromagnti fild potntial. To obtain th full systm of quations w must inorporat th Maxwll quation π B j j () and th divrgn B () Th xtndd Maxwll s quations (in gs ) whih allow for th possibility of magnti hargs analog with ltri hargs ( olor magnti gluons ondnsat), th Gauss law for magntism is divb π ρ m,and th Faraday s law of indution ontains a π B π nw trm or, in SI, µ j m, whr µ ; E m j, also, th t E π Ampr law is idntial to th on without monopols: B j t Th Ampr's law, xprssd as th intgral ovr any arbitrary loop, whr J s is th urrnt nlosd by this loop, is: B dl µ J s (3) A hargd partil moving in a B fild xprins a sidways for that is proportional to th strngth of th magnti fild, th omponnt of vloity that is prpndiular to th magnti fild and th harg of th partil. This for is known as Lorntz for and is givn by : qm FL q( E vb) ( B v( E )) () µ 3

4 , whr, qm -th magnti harg, B in [Tslas], FL in [N] ) In absn of a magnti fild. on gts for fr nrgy of th suprondutor, J.Pitavski []: f b f a dv n m (5) Hr, f n is th fr nrgy at, i.. f n is th fr nrgy of th normal stat. Lt us onsidr th bhavior in prsn of a magnti fild. Th dnsity of th magnti fild is B 8π must b addd to th intgrand (5). But this is insuffiint in th gradint trm in (5) is not invariant with rspt of gaug transformations: A A γ (6) And for phas transformation ϕ ϕ γ (7) Th gradint of phas ϕ dfins th vloity of th suprondutiv pairs (in our as of th olor magnti gluons ondnsat!) v s ϕ (8) m Equation (8) is not invariant undr a suh transformation. To rstor th rquird invarian, on must inlud a furthr trm ontaining th vtor potntial v s ϕ A (9) m Finally, on gts for th supronduting urrnt dnsity js nsvs ns ϕ A () m, and th magnti indution is B A () Applying th url oprator to both sids of () and using (), w obtain th London quation ns js B B () m λ Thrfor, to rstor th invarian in (5), on substitut for [ i( ) A] nrgy thn taks th form th ombination, whih is obviously gaug invariant. Th final xprssion for th fr

5 dv B b a A i m f f n π 8 (3) Hr, th magnti indution must b xprssd as in (). On an obtain th basi quations of Ginzburg-Landau thory by varying this funtional with rspt to A and. Carrying first variation with rspt to A, w find aftr a simpl alulation: ( ) ) ( π δ δ π ϕ δ dv B A div AdV urlb A m m i f () Th sond intgral an b transformd into an intgral ovr rmot surfa and disappars. To minimiz th fr nrgy, th xprssion in th brakts must b qual to zro. This rsults in th Maxwll quation ) _ ( SI in j j urlb s s π (.),or s j s j A. π (5), providd that th urrnt dnsity is givn by ( ) A m m i j s.. * * * (5.) Aording to th dfinition of s n w an substitut ) xp( ϕ i n s. Thn (5.) boms (6) Equation (6) oinids with (). This justifis our idntifiation of with s n.variation of (3) with rspt givs, aftr simpl intgration by parts, ds A i m dv b a A i m f δ ϕ δ δ (7) Th sond intgral is ovr th surfa of th sampl. Th volum intgral vanishs whn b a A i m (8) 5 Ψ A m j s ϕ

6 Equations (5) and (8) form th omplt systm of th Ginzburg-Landau(G-L) thory.. m. In quation (6), to mphasiz: λ.7 6[ m] ns., I did a lot of π multipliations, and I usd th quantizd flux: Φ, and Ψ ns ; 3 3 _ monopols V *. 5m, V 3π r 3, r.8[ fm] n s 8.8 [ C. N. ]. m Sin, th magnti harg of olor magnti gluons bing [6] π 37 g d π 68. 5, and assuming that th lassial ltron radius b qual to th lassial olor magnti gluons radius from whih on has th olor magnti gluons mass mm g d m 7m, th valu of λ rmains unmodifid. Thus, w obtain / j s. ns ϕ m n m s A (9) j ns m,or j s π ϕ A π λ λ,or j s Φ ϕ A π λ λ () W an assum that th indution vtor B is dirtd along th z -axis. Thn th vtor potntial A an b hosn along th y -axis and B da dx () W must solv th G-L quations (5) and (8) for this on-dimnsional problm subjt to th s n boundary onditions: x,, B, () x, ( a / b) H, B s s ϕ m n A Th quantity i is gaug invariant, J.Pitavski [], whn A A ϕ. If w transforming th quation dimnsionlss by: 6

7 x x, λ A A, H λ B B A (3) H Substituting ths variabls into G-L quations (8) and (5). Th G-L quations for our on-dimnsionally problm tak th form (hr ar omittd th hats): κ A 3 (), and A A 3 (5) Th boundary onditions () ar:, E A for x, A for x Not that th boundary ondition A Equations () and (5) giv A ( A ) onstnt (6) κ This xprssion is an nrgy, and as follows from boundary onditions that this nrgy must qual unity. A ( A ) (7) κ For κ λ ξ whn λ ξ th ltrial fild pntrats only slightly into supronduting phas, and th pntration is of ordr κ, th wav funtion is small in this rgion and givs only a small ontribution. Lt us onsidr th distan x κ and κ A. Thn on an nglt th right-hand sid (r.h.s) of () and th solution mathd to (9) bllow is κ x. Substituting this in (5), w find A κ x. Th main ontribution ariss from th rgion whr hangs rapidly, whih is of th ordr of κ. Thr is not ltri fild in this rgion and on an put A in (7). Solving this quation for, w hav κ ( ) (8) This quation hav a simpl solution tanh( κx ) (9) Th suprondutors of sond kind ar thos with κ, and λ ξ. 7

8 W now onsidr th phas transition in suprondutors of th sond kind. For this w an omit th non-linar ( ) trm in (8), w hav i A a (3) m This quation oinids with th Shrodingr quation for a partil of mass m and harg ( in th as of dual, th fator for th harg, whih is spifi to th pairs, it is atually ) in a magnti fild H (in our as th hromo-ltrial flux E () ). Th quantity a plays th rol of nrgy ( E ) of that quation. Th minimum nrgy for a suh partil in a uniform ltro-magnti fild is H C Kgm ( ) ω B J. s [ J ], H -an xtrnal ltro-magnti 8m Cs Kg m s fild of a dipol ratd by th pair u u (th hromoltrial olors fild) d N H E r C (3.) π,whr r.5[ fm] -is th ltrial flux tub radius, d.8[ fm] -th distan btwn J th two quarks hargs, usually H [ A m], but hr is usd as B µ H Am p H d Z u ± π W ± d n Fig.a. Abrikosov s triangular latti for a nulon (proposal) 8

9 Fig.b. Th Giant-Vortx typ arrangmnt for th nulon Hn, quation (3) has a solution only if a H 8m, whn following powrlaw onformal map is applid for omplx numbr of th r.h.s of (3), or quivalntly if th ltro-magnti fild is lss than an uppr ritial fild, s fig.a. m a Φ H H π ξ (3) π N 8.33 π ξ C, and in trms of π J B H.76 π ξ Am Th partil nrgy is ( ) 6.8 8[ J ] 6.8 5[ Kg] 39[ GV ] 8mξ, with λ ξ ; λ.7 ξ.[ fm], or κ. 5 (of typ II-suprondutor). κ.5 On of th haratristi lngths for th dsription of suprondutors is alld th ohrn lngth. It is rlatd to th Frmi vloity for th matrial and th nrgy gap ( k T B ) assoiatd with th ondnsation to th supronduting stat. It has to do with th fat that th supronduting ltron dnsity annot hang quikly-thr is a minimum lngth ovr whih a givn hang an b mad, lst it dstroy th supronduting stat. For xampl, a transition from th supronduting stat to a normal stat will hav a transition layr of finit thiknss whih is rlatd to th ohrn lngth. Howvr, suprfluids possss som proprtis that do not appar in ordinary mattr. For instan, thy an flow at low vloitis without dissipating any nrgy i.. zro visosity. At highr vloitis, nrgy is dissipatd by th formation of quantizd vortis, whih at as "hols" in th mdium whr suprfluidity braks down. 9

10 Mor xatly, this quantity is alld th orrlation or haling lngth [], and is dfind T as ξ ( T ) ξ T T ξ, whr ξ a E F g (3.) is for T, a π from [7], E g -gap nrgy, k B Boltzmann onstant; at onfinmnt T 75[ MV ] [ K], and th Frmi vloity of ltrons (olor magnti gluons) is * 7m E F υ F 7m (3.), whr as th Frmi nrgy w hav for olor magnti gluons ondnsat viwd as 3 ns boson ondnsat EbosonCond 3.3 Tk B. 7E F 7m, whr E ( 3π n ) 3 F * 7m s (3.3), numrially, w hav: 9.3 [ J ] 55[ MV ] E F 3,whr V.5[ fm], and th vloity of olor magnti gluons is υ F [ m s],and ξ. 6[ ] at T (3.) m Not that, if w us only th mass of ltrons (as in th as of suprondutors), th vloity obtaind is gratr than th spd of light, so this strngthns th us olor magnti gluons ondnsat. In th following w will onsidr th strutur of th mixd stat. Th main problm is to undrstand how th ltri fild pntrat in th bulk of th suprondutor. Lt us again onsidr a supronduting ylindr in th ltri fild. It is natural to xpt that th normal rgions, with thir aompanying ltri fild, ar ylindrial tubs paralll to th fild. Th ltrial flux insid suh tub must b intgral multipl n of th flux quantum π Φ π usually.7 5[ Tm ] (3) Th ltrial fild is onntratd insid th tub. At larg distans from th tub it is shildd by annular supronduting flowing around th tub. This urrnt is analog of th suprfluid vloity fild surrounding th vortx lins in th suprfluid liquid. W an thn pitur th mixd stat as an array of quantizd vortx lins. Suh vortx lins wr prditd by A.A. Abrikosov in 957. Thir xistn is ruial for xplaining th propritis of typ II suprondutors (dual in our as).

11 Th prsn of a vortx lin in th ntr of th tub inrass th fr nrgy of th supronduting mdia. Th G-L quations ar solvd analytially only for λ ξ (nar T this mans κ ). Thus, whn th ltrial flux is applid paralll to th supronduting ylindr, th first flux pntrating should b loatd along th axis of th ylindr. Substituting js from Maxwll quation, w an rwrit () as: ns ϕ m π π A n m s ϕ Φ π A λ ϕ Φ π A j s From Maxwll quation (in SI ): urlb j s Φ λ ϕ A js π λ urlb, or A λ B Φ ϕ / π (33) Th phas ϕ in prsn of vortx lin is not a singl-valud funtion of th oordinats. For a vortx lin with minimum flux Φ, th phas inras by π on travrsing a losd ontour that nlos th lin. Thus th intgral along suh a ontour is ϕ dl π (3) Intgrating (33) w find ( A λ B) dl Φ (35) It is not diffiult to hk that in th rang λ x ξ (36) Th sond trm from l.h.s of (35) givs th main ontribution. W tak th ontour of intgration in (35) a irl of radius x. For this gomtry th vtor ( B) has only on omponnt ( B) ϕ along th ontour. Th intgration is thn simpl and w hav db Φ ( B) ϕ (37) dx π xλ To not (in gs ):

12 π ( B) n v ϕ s sϕ Equation (37) thn givs v s ϕ mx for th suprfluid vloity as it must b for a vortx lin in a suprfluid of partils with mass m. Intgrating of (37) for B givs Φ λ B( x) log (38) π λ x This quation is valid in th intrval (36) with logarithmi auray. Noti also that vry vortx arris th flux Φ and hn th man valu of B ovr th ross-stion of th ylindr is B ν Φ (39), whr ν is th numbr of lins pr unit ara. This rsult is invalid nar th uppr ritial flux H whr th ors of th vortx lins bgin to ovrlap. To alulat this numbr w hav to tak into aount th intration btwn vortx lins. As th first stp w hav to find th ltrial fild trough a loop of arbitrary radius surrounding th lin without th rstrition (36). Lt us alulat th url of th both sids of (33). Not that url ϕ π nz δ ( x) (), and urla B whr δ (x) -th Dira funtion Whr r is th two-dimnsional radius-vtor in th x y plan and n z is a unit vtor along axis z (W assum that th axis of th vortx lin oinids with z ). Indd, intgrating ϕ along th ontour nirling th lin and transforming th intgral by Stoks thorm into an intgral ovr a surfa spanning th ontour, w hav aording to (3) url ϕ ds ϕ dl π () Sin this quation must b satisfid for any suh ontour of intgration, w hav (). Finally, w obtain B λ urlurlb n z Φ δ ( x) () Using th vtor idntity urlurlb divb B B, w obtain

13 B λ B Φ δ ( x) (3) This quation is valid only at all distans x ξ () Throughout all th spa xpt th lin x quation (3) oinids with th London quation () Th δ (x) funtion on r.h.s dfins th haratr of th solution at x. Atualy this singularity has alrady bn dfind in (38), whih is valid at small x. Th solution of this quation at x is B( r) onst Κ ( x λ ), whr Κ is th Hankl funtion of imaginary argumnt. Th offiint must b dfind by mathing with th solution of (38). Using th asymptoti formula Κ ( x) log( γ ) for x, C whr γ. 78 (C is Eulr s onstant), w finally hav x Φ B( x) Κ ( x λ ) (5) π λ Using quation (5) w an rwrit (38) as: Φ λ B( x) log, x λ (6) π λ γ x In opposit limit of larg distans on an us th asymptoti xprssion x Κ x x ( ) ( π ) for x. Thus, at larg distans from th axis of th vortx lin th fild drass aording to x λ B x Φ ( ) 3, x λ (7) (8π xλ ) Aordingly th suprondutiv urrnt dnsity drass (in SI ): j ϕ π db dx x λ ( π ) 3 5 8(π Φ xλ ) (8) W an now alulat th nrgy of th vortx lin. Th magnti part of fr nrgy orrsponding to London quation is givn by th intgral. F B [ B λ ( urlb) ] dv (8.) 8π Indd, by varying th xprssion with rspt to B, w immdiatly obtain th London quation (). Th main ontribution to th intgral is du to th sond trm, whih ontains a logarithmi divrgn. Substituting (37) in (8.), and intgrating in th rang (36), w obtain for th nrgy pr unit lngth of vortx lin. Φ π λ λ log ξ (9) 3

14 Equation (9), xplains why only vortx lins with th minimum flux Φ ar th most favorabl. Th nrgy of a lin is proportional to th squar of its magnti flux. Thus, th fragmntation of on lin with th flux n Φ into n lins with flux Φ rsults in an n- fold gain in nrgy. A disussion of th physial bakground of this nrgy an b found,.g. in th books [3], [], [5], as rlatd to Dirihlt s nrgy and harmoni maps. Thus, in [3], whn is indud a magnti stray fild h whih has a rtain nrgy, aording to th stati Maxwll quation, th stray fild satisfis url ( h) ; div ( u h), whr u, is xtndd by outsid Ω. Th first quation implis that h an b writtn as th gradint of funtion U. By th sond quation, this U is a solution of U div(u) in th distribution sns (sin, url ( U ), and div( U ) U ). Thr xists xatly on solution suh that th intgral U dx 3 R u Udx Ω (9.) is finit, and for this hoi of U, this intgral givs th main ontribution to th miromagnti nrgy. It is alld th magntostai nrgy [3]. In our trms, B u onst, U urlb B U div( u) u onst. B, sin x Φ Substituting u B from (6) with x λ log(...) on th x λ π λ boundary, or th dual gaug omponnt of th total ltrial fild J, whn B monopols.656 (9.) Am,and U from (37), on hav Φ Φ λ π dx Ω 8π π λ π λ x Ω Φ π λ dx x Φ π λ log( x) Φ λ log π λ ξ Hr, th fator π is usd to onvrt from ( gs) ( SI). Baus th magnti indution of th olor magnti gluons urrnt whih is powrd monopols by ltri fild givn by a pair of quarks ( H ), B H H, as rsulting from th omparison (9.) with (3.) and (3), it has th raw flow onsquns squzing this romoltrial flux into a vortx lin, followd by foring an organization into a triangular Abrikosov latti, s figur. λ ξ (9.3)

15 Th or of vry vortx an b onsidrd to ontain a vortx lin, and vry partil in th vortx an b onsidrd to b irulating around th vortx lin. Vortx lins an start and nd at th boundary of th fluid or form losd loops. Th prsn of vortx lin whih inrass th fr nrgy of th supronduting mdia with L, it is thrmodynamially favorabl if th ontribution is ngativ; i.. if L Φ H L π µ, and B H µ, µ,or π µ H H (9.) Φ Substituting (9.3) in (9.), w find th lowr ritial fild Φ λ π J B H log log( κ ). 5 (9.5) π λ ξ π λ Am, whr ξ., and whn nar th axis, for x. 6 ξ, whn th indution is B ξ ). 5 H (9.6) ( Lt us th rsults obtaind to th alulation of th nrgy of intration of vortx lins. It is important that quation (3), whih dfins th distribution of th fild, is linar on. It mans that undr ondition () th fild produd by diffrnt vortx lins is additiv. Lt us onsidr two vortx lins plad at x and x sparatd by a distan d from ah othr. Thn, B B B. Th nrgy of th lins is givn by (8.). Lt us transform th first trm in intgrand by mans of () (to multiply with B ), whih givs B λ Φ λ ( urlb) [ B urlurlb ( urlb) ] B ( x) z [ δ ( x x) δ ( x x )] (5) Th first trm in th r.h.s an b transformd into th form B urlurlb ( urlb) div( B urlb) Th volum intgration of this trm in (8.) an b rdud to an intgration ovr a rmot surfa. This intgral disappars, baus of th fast dras of th fild. Baus w ar intrstd hr in th nrgy of intration of th lins, w must taks into aount only th mixd trms of th typ B z ( x) δ ( x x ). (Trms liks B z ( x) δ ( x x ) ontribut to th slf-nrgy of th vortx lins (9). Now th intgration in (8.) is trivial. W hav for th intration nrgy LΦ L int ( B ( x) B ( x)) (5) 8π Both trms on th right ontribut qually and using (5) w hav (5) 5

16 Φ Φ d int ( d) Bd ( x) Κ (53) π 8π λ λ On an also us th asymptoti xprssion for int (s (7)) Φ λ x λ int, x λ (5) 7 3 π λ d Whn th distan d λ ξ th ors of vortx lins ovrlap []. Th quation () is no longr valid. Howvr, (39) is still valid. Lt us onsidr a losd ontour nar th surfa of th ylindr. Th hang of wav funtion on passing round th ontour is π ν S, whr S is th ross-stion ara of th ylindr and ν -th numbr of vortx lins. On obtain from (6) that th ltri flux is m js Φ Φ ν S dl (55) ns Lt us rall that a similarly rlationship [], [], it was introdud for th first tim by London, alld fluxoid quation. Eah fluixoid, or vortx, is assoiatd with a singl quantum of flux rprsntd as Φ,and is surroundd by a irulating supprurrnt, j, of spatial xtnt, λ. As th s applid fild inrass, th fluxoids bgin to intrat and as th onsqun nsmbls thmslvs into a latti. A simpl gomtrial argumnt for th spaing, d of a triangular latti thn givs th flux quantization ondition [3], Bd Φ (56) 3, whr B, is th indution. Th solution of Ginzburg-Landau phnomnologial fr nrgy (3) is usful for undrstanding th Abrikosov flux latti. Th oordinat-dpndnt ordr paramtr ϕ dsribs th flux vortis of priodiity of a triangular latti. Flutuations from ϕ hang th stat to, th minimization of fr nrgy with rspt to, givs th ground stat ϕ (r ). Th fr nrgy is givn by, ( B H ) b f f n i A a dv (57) m 8π, th avrag magnti indution is B( y,,). Th fr nrgy has solutions of vortis of triangular form. Th oordinats of th thr vrtis of a triangular vortx ar givn by (,), ( l,), and 3, l. Th flutuation from ground stat orrsponding to that of triangular latti is that for small flutuations. Th dviation of th fr nrgy from th man-fild valu F FFM with rspt to th thrmal nrgy, k B T, an b usd to obtain th physial proprtis of th flutuations whih ar usful for undrstanding th mltd vortx lattis. Th dviation from th triangular Abrikosov latti is dfind as 6

17 ( r ) a D a ϕ (58) / whih uss th spatial and thrmal avrags alulatd with th probability xp ( F k T ). Classially, B k BT D (59) F FMF masurs th flutuations from th triangular vortx stat. Th flutuation in th distan btwn vortxs boms: as, ( T T ) B (6) FM 5 -as, TFM T B ; (6) -as 3, a vortx transition blow th transition tmpratur s [], whr, TFM -th flux-latti mlting tmpratur, and. from Lindmann ritrion of latti mlting whn d l, and th flux quantization ondition l Φ B, B π n κ. For numrial valus T 75[ MV ], in as of symmtry braking, th as, rsults TFM T, and in as, rsults T FM [ kv ] or.5 9 K, by using (56) in pla of Φ B d with d.398[ fm] (a vry prisly valu), and κ, whih is th tmpratur of fusion (mlting!) of two nulons. This triangular latti orrsponds to th arrangmnt of th quarks pairs u u, uu, dd in th fram of a nulon, s fig.a, fig.b. A dirt numrial analysis allows to obtain th following valus for th urrnt, for and nrgy. Thus, from (8 ) th urrnt for x λ is givn by: j ϕ ( λ ).57[ A / fm ] (6) For x λ, th urrnt dnsity drass at j F 3. 5[ A fm ] Not that vloity υ F, morovr, if on onsidrs th olor magnti gluons urrnt givn by quation (), as j ϕ nsvf g D, whr th magnti harg is: π 37 g d π (63) If w us th rang x. λ, thn th urrnt is obtaind by drivation of (6): 7

18 π jϕ x λ x λ υ Fi.38[ m s] 9.89 g D n V s υ Fi g D (6) E E8 ohrn[m]; ns*.-6 E-6.E5 B; I*.59; Vfi*.8;VEF*.8; *.8 E-7 x ohrnef lambda nss Cohrn_Bx I B/Th/Bx VEF Vfi.E In ordr to mak a orrt hoi of ohrn lngth, wr plottd in figur. xprssions: (6), (3.3) (6), (8) and n s, all assoiatd with th rang of x. Th rsult indiats that for x.[ fm], th vloity boms largr than of th light, and th ohrn grows fastr, rahing valus largr than th radius of th nulon. Also, small valus with thos of th ξ < < λ, it follows that B > B.76, whn B is givn by (3), as a funtion of th ξ, or by (6) as B (x). Thrfor, th bst hoi is to onsidr ξ λ, whn υ < F, but stritly λ ξ, as. ξ. (6.) That orrspond with th valu alulatd abov (3.) as a Frmi vloity. This rsult marks th ssntial proof of this modl, namly th onsidration of th olor magnti gluons ondnsat. From ( ) and (7), th Lorntz for is: FL qvfi B.5. [ N] (65),whn B is givn by (6) and x λ, for th uppr limit: B( λ ).75[ J Am ] (66) With B from (7) and for x λ, w hav 8

19 J (67) B(7λ ) Am, thn, th for boms F L.8 6[ N], or in trms of nrgy barrir F L x 95[ MV ] (68),or th nulon ovrall. In as of x ξ (), with (38) J B ().35 Am (69), whih rspt (9.6). Th magnti nrgy rsults from (9), and (9.3), and for ( λ x ξ ) from (36): [ J fm].66[ GV ].9 fm (7), th for on th flux tub (string tnsion). Now, from (5) and d ( 6)λ ξ, w hav int.3 [ J fm] [ MV ] (7) What would b th valu of th mass of th pion π, omposd of a pair of quarks u d intrating at a distan d λ.66[ fm] of th radius of th nulus. Now, othrs important valus of nrgy: ( ) int ( d x λ ; x.)*.7[ fm]. 9[ J ] (7.), and from (69) with x ξ.7[ fm] ; (7.) h V (H ) 8π 5. [ J ] (7.3) Now, th vortx nrgy is: vortx V H 8π.6 8[ J ] (7.), whr V -is th volum, s fig., aordingly, th orrsponding quivalntly masss ar M vortx 73[ GV ] ±, whih sms to b qual to th mass of W boson pair as bn ratd insid nulons du of Higgs fild v.. v 7GV, for xampl by th Shwingr fft [3]. Th nrgy of th nutral boson Ζ is assimilatd with th vortx-vortx thr pairs intration nrgy [], Z 3* int pair 9[ GV ], whn from (7) int pair int ( d.7.6; x.λ ).85 9[ J ] 3.33[ GV ], (7.) is th nrgy of ah of thr pairs of vortx outrmost ( d ) vortis lins whih intrating (rpl) at th ntr of th triangl situatd at x.λ, thus, bing gnratd a nutral urrnt in th zon of Z during th triangular arrangmnt of th latti, s fig.a, or fig.b. Now, is possibl that th vortis start to oals into a giant vortx (GV) [6], s, fig. b., 9

20 Thus, from (7), rsults an anothr nrgy stat-maximum possibl ( d ), probabl that of Higgs boson (H): H 3* int ( d.7.6; x λ ).7 8[ J ] 35[ GV ] (7.) Hr, a fator of was introdud to orrt on for pairs in th G-L modl. 3. Bta day halftim alulation- an ssntial tst of modl validation Blow w will dmonstrat that th mhanism for bta day of radioisotops is th sam as th dark ounts in th as of suprondutors []. In [3], ar disussd thr typs of possibl flutuations in supronduting strip (assimilatd in our as with th nulon) whih rsult in dissipation. Eah on auss transition to th normal stat from th mtastabl supronduting stat whn urrnts ar los to th ritial valu I : (a) Spontanous nulation of a normal-stat blt aross th strip with π -ϕ phas slip as in thin wirs (a vry phas slip maning Φ nrgy rlasd). (b) Spontanous nulation of a singl vortx nar th dg of th strip and its motion aross to th opposit dg aompanid by a voltag puls. () Spontanous nulation of vortx-antivortx pairs and thir unbinding as thy mov aross th strip to opposit dgs du to th Lorntz for, as wll as th opposit pross of nulation of vortis and antivortis at th opposit dgs and thir annihilation in th strip middl. In [3] ar drivd th nrgy barrirs for thr dissipativ prosss mntiond within th GL thory. Considr a thin-film strip (on of thr vortxs of th nulus) of width w r * λ, s fig a, fig.b. W hoos th oordinats so that x w. Sin w ar intrstd in bias urrnts whih may approah dpairing valus, th supprssion of th supronduting ordr paramtr ( ) must b takn into aount. Also in [3], is usd th standard GL funtional, givn abov in (3). W will us th as (a), a vortx rossing from on strip dg to th opposit on indus a phas slip without rating a normal rgion aross th strip (on of thr vortxs of nulus) width. Whn, is trating th vortx as a partil moving in th nrgy potntial formd by th supronduting urrnts around vortx ntr insid th strip and by th Lorntz for indud by th bias urrnt. In [3], it was drivd th nrgy potntial and is found th vortx rossings rat (phas slips and orrsponding voltag pulss) in th framwork of Langvin quation for visous vortx motion by invoking th known solution of th orrsponding Fokkr-Plank quation. Finally, from[3] th asymptoti stimat for th dark ounts rat, rsults as: 3 ν h k BT RΩ π ν h π ξ I R ν ( I, ν h ) Φ Y (73) π w µ I, and whr th bias urrnt is : w I I ( κ κ ) (7.) π ξ

21 I (7.) Hr, hz λ -th axial ( z ) hight of th olor magnti gluons ondnsat. Hr, th ritial urrnt at whih th nrgy barrir vanishs for a singl vortx rossing: µ wi I ; (7.3).7π ξ And th thrmodynami ritial fild is : Φ H (7.) π ξ λ, whr µ κ, and (7.5), whr υ h τ GL ( vortx Qbind ) -is th nrgy of th vortx during rossing th barrir of hight by quantum tunnling in pla of th thrmal ativation usd in [3], and vortx Q bind ovrpassing an ohmi rsistan along a transvrs path way of th nulid: Rq Rnulid RΩ (75) R π ( ξ w) R q nulid Hr, vortx M W from (7.), and Qbind is th bta day nrgy as obtaind from th data of ah radionulid of bta day typ (Nulids hart ). In th as of bta disintgration n p ν, or d Φ λ ; Λ 8π Λ u W ν,or and th bias urrnt is: d( 3) u( 3) ( 3 3) In β day, nrgy is usd to onvrt a proton into a nutron, whil mitting a positron ( ) and an ltron nutrino ( ν ): Enrgy p n ν So unlik h z ( ν ) ( z ) xp[ z tan ( z )] Y ( z) ν β ; β day annot our in isolation baus it rquirs nrgy du to th mass of th nutron bing gratr than th mass of th proton. β day an only happn insid nuli whn th valu of th binding nrgy of th mothr nulus is lss than that of th daughtr nulus. Th diffrn btwn ths nrgis gos into th ration of onvrting a proton into a nutron, a positron and a nutrino and into th kinti nrgy of ths partils. In all th ass whr β day is allowd nrgtially (and th proton is a part of a nulus with ltron shlls) it is aompanid by th ltron aptur (EC) pross, whn an atomi ltron is apturd by a nulus with th mission of a nutrino: Enrgy p n ν

22 Howvr, in proton-rih nuli whr th nrgy diffrn btwn initial and final stats is lss than m. 56MV, thn β day is not nrgtially possibl, and ltron aptur is th sol day mod. This day is also alld K-aptur baus th innr most ltron of an atom blongs to th K-shll of th ltroni onfiguration of th atom, and this has th highst probability to intrat with th nulus. Thrfor, th ad-ho bias urrnt ratd during vortx rossing through th nrgy barrir is: u ( 3) nrgy d( 3) ( 3 3) At th first sight, th ohmi rsistan of this ad-ho ltrial iruit ratd by th bias urrnt I ( ) du of quarks transformation ( d u), or ( EC( u d), is givn as: Qbind Rnulid (76) τ V GL vortx, and th supronduting quantum rsistan is: R q () 6. 5kΩ,whr th vortx potntial is V vortx H ξ, H from (3.) Giordano [9] has suggstd that phas slips du to marosopi quantum tunnling may b th aus of th low tmpratur rsistan tail in th D wirs h studid in zro fild. On possibl mhanism for our low tmpratur rsistivity tail ould b quantum tunnling of vortis through th nrgy barrir []. On xpts a rossovr from thrmal ativation to quantum tunnling to our whn [], in (73) in pla of thrmal ativation w us th quantum tunnling: k T τ B GL A vortx moving from x w, during th tim τ GL. W stimat th total intration nrgy intration with th nighborhood vortxs or with th on giant-vortxs, fig.b, of othrs nulons from th nulid nulus, during th tim τ GL along th vortx path by mathing (7.), (7.) and (7.) as: Φ w Φ hz Qbind ( Φ I ) κ ( κ ) π ξ 8π λ π ξ h z H I I I, whr th ratio z was hosn as a variabl in (73), through (7.5); I µ This is, in fat, th work don by th Lorntz for on th vortx path of th lngth w. Now, w prod to appliation to som radionulids whih day bta, and bginning with th nutron. Thus, th liftim of th fr nutron is a basi physial quantity, whih is rlvant in a varity of diffrnt filds of partil and astrophysis. Bing dirtly rlatd to th wak intration haratristis it plays a vital rol in th dtrmination of th basi paramtrs lik oupling onstants or quark mixing angls as wll as for all ross stions rlatd to (77) wak p n intration. From th most pris masurmnt within this lass of xprimnts, rsults τ 886.3[ s]. n From Nulid hart-, rsult: 99 Mo ( β, T 65h, Q. 356 MV );

23 85 Kr, Q. 687 MV β and T 8h, γ ray 5KV (.6%) T. 756yr ; γ ray 5KV (95%) T. 8h ; 55 Cs 56 Ba ν, Qβ. 75MV, γ ray 66KV (85%) T 3. 8yr t Numrially, with ths data rsult: τ GL.5 [ s], Y 56, ν h 57, and with w rλ as variabl, th volution of dark ount rat R v, and of R Ω, for diffrnt isotops ar givn in fig. 5. Hr, th fration of th bias urrnt to ritial urrnt I I as usd in Y (z) from (7.5) : 37 was ddud sparatly for 55 Cs as that of th platau zon in fig.6, rsptivly of I µ I.5, by using th ondition R ν T. W an obsrv that this valu orrsponds with th xptd valu from quarks transformation of ± ( ) * vvortx *.6 9r [ Am] [ A], whr th vortx rossing vloity is vvortx λ τ GL, and r -K shll radius. Thrfor, th bias urrnt, whih is prpndiularly on th olor magnti gluons urrnt, is I.5µ I 5[ A] ; whr, µ., and th olor magnti gluons urrnt is: I 3. 5[ A] as givn by (7.), and I.6[ A] from (7.3). From (77), rsults Q bind.9 3[ J ] or nar qually with Q β of th almost of nulids, for xampl, for Mo, Q.356MV.7 3[ J ] β. Now, in as of quantum tunnling th transmission offiint [8] is τ GL T. 66, whih is too small, so, a vortx rossing from on strip dg to th opposit on indus a phas slip, as a), is th only viabil mhanism for dark ounts (bta day). Thus, it was stablishd a logarithmi quation of th β day rat whih rsulting a straight lin as a funtion of th barrir width ( w rλ ) for vry nulid, fig.5, it 6 drasing in as of long livd nulids, lik F. Thrfor, this volution is a disiv validation tst of ntirly modl. Th fator r dsrib th intrations of th nulons insid of th nulus, bing a omplx funtion of mass, Z, Q. bind 3

24 Rdark[/s],Qbind[J].E E-3 85Kr 5.E-7.E-6 n-886s 8h.E-9.E-.E-5.E-8.E- Mo 65h vortx 37Cs 3.8yr quantum tunnling 6F.56yr 6.E-7.E-7 3.E-7.E-7 Fig.5. Th volution of dark ount ( β day) rat as funtion of barrir width..e-.e-7 barrir Ibias.E-7.E-3 w-barrir widthr*lambda R-dark Qbind R_Ohm Rdark.E.E97.E87.E77.E67.E57 Rdark[/s].E7.E37.E7.E7.E7.....E-3.E-3 I/I Fig.6 Th volution of th dark ount rat ( β ) day as a funtion of th bias urrnt.. Conntion with gravity Th for for flux tub squzing, as qually to th Lorntz for, is also th for of gravity.

25 Blow w giv a possibl brakthrough possibl xplanation for a suh gravity for. Thus, if w look at a vry simplifid (salar form) of Einstin's quation aftr multiplying with urvatur radius ζ, th radius (objt radius) R of urvatur of spatim is givn as: (78) ζ 8π G. pζ If th prssur p on th surfa of th tub is onsidrd to b R gnratd by th gravitation for qual with th ontra- Lornz for F L applid on th urvatur of spa-tim ζ situatd in th ntr of vortx, its rol bing to ountratd th dstrution of suprondutivity. π ζ p FL Κ (79) G With Lorntz for alulatd abov (65) ζ nulon GM FL Κ Κ Κ.5[ N] R R (8).5 Κ ζ nulon GM * 6.67 *.6 7, whr.8 39 (8) R R. 5 *. 7,and Κ - h vauum lastiity,or from (79) Κ. (8) G Now, if w onsidr B K 53 So, to hk this rational, firstly, w onsidr th attration of a nulon-earth whn th spatim urvatur of th Earth is hosn littl ovr th Shwarzshild radius GMm p GMm p Κ R GM Gm p FG Earth nulon REarth 6 R R G (83) Earth nulon Earth.5 9 * FG.6 6 or.3 7 K. Thrfor, in th as of a nulon, if w us in pla of th urvatur ζ, whih is too smallr.7 5[ m] than of Plank lngth, w us just it as th lowr limit, thn is obtaind an invariant, a surprising rsult: G l P 3 3 8π G FL 8π G π (8) R π l π G π λ λ P 5

26 6.7 *.55 R. 6[ ] λ Curv nulon m R Curv nulon 5. Conlusions In this papr was r-visitd a modl G-L, in ordr to alulat th Lornz for, th urrnt, and th nrgis of th Abrikosov vortx lins insid of th nulon. Thus, it was found that ths nrgis orrspond of subatomi partils, W, Ζ, H bosons, and of mson π. So, th nulon an b sn as a triangular latti with thr pairs of quarksantiquarks in th tips of th triangl or as giant vortx du of th oalsn of vortis lins of thr vortxs (W). Ths axial vortxs (filamnts) intrating in th latral plan. A onntion with gravity as a ountratd for to th suprondutivity dstrution, is disussd. All of th kys of this modl, as bing analogous to a suprondutor linging vry wll, starting with th us of Maxwll's quation with olor magnti gluons, furthr mass, harg, and th numbr of olor magnti gluons (dnsity), whih dfin th pntration dpth and th ohrn lngth, and finally to th onntion with gravity. Also, w an say that, baus no fr quarks wr dttd, th sam is tru for olor magnti gluons, both of whih ar onfind togthr and in full in nulons, whn th tmpratur of th univrs has rahd * K. Th modl prmits to xplain th bta day mhanism of radioisotops to b th sam as th dark ounts in th as of suprondutors, and to think in this ontxt to th nhanmnt of bta day for nular transmutation of radioativ wast.. REFERENCES. G. t Hooft, in High Enrgy Physis, d. A. Zihii (Editri Compositori, Bologna, 976); S. Mandlstam, Phys. Rpt. C 3, 5 (976); V. Singh, D.A. Brown and R.W. Haymakr, Strutur of Abrikosov Vortis in SU() Latti Gaug Thory, PHYS.LETT.B 36 5 (993), arxiv:/hp-lat/93.. K.H.Bnnmann, J.B.Kttrson, Suprondutivity onvntional and unonvntional Suprondutors, vol., Springr, L.N. Bulavskii, M.J. Graf, C.D. Batista, V.G. Kogan, Phys. Rv. B 83, 56 ().. M. Tinkham, Introdution to Suprondutivity, (MGraw-Hill, Nw York, 975). 5. Rudolf Ptr Hubnr, N. Shopohl, G. E. Volovik, Vortis in unonvntional suprondutors and suprfluids, Springr,. 6. Rudolf Ptr Hubnr, Magnti flux struturs in suprondutors, Springr, 6

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