doi:10.5281/zenodo.2566644 -al [email protected] E-mail: livac-research.org LIVAC, Italy; Rome, hsi la o ntnei h omlgclconstant, cosmological the itself, in vacuum instance for physical Λ clear the is with this coincide can energy, is which [1], energy Dark Introduction 1 fundamental spin; mechanism; interactions. Higgs matter; dark ergy; Keywords— in- fundamental hydrody- to exert correspond also teractions. may vortices which Doped forces namic reproduces spin. dynamics possible whose any vac- vortices, of trig- quantized zone by particles gering Brillouin to mass the give they of polarized and quasi-lattice uum’s edge circularly the Here are around phonons bosons mechanism. hydrody- Higgs Nambu-Goldstone pro- a the the to via of able interactions reinterpretation matter, and namic dark particles particle massive with duce en- dark doped with and fun- coinciding ergy superfluid, field, a scalar as quantum treated damental is vacuum physical the Here Abstract = * iitr elItuin,dell’Universit dell’Istruzione, Ministero κρ vac rmNmuGlsoebsn omsieparticles massive to bosons Nambu-Goldstone From nwhich in , unu aum iaatvcu;dr en- dark vacuum; dilatant vacuum; quantum i h yrdnmc fadpdvacuum. doped a of hydrodynamics the via κ sEnti’ osatand constant Einstein’s is ig,dr etradtevacuum: the and sector dark Higgs, ∼ 9 fteuies’ mass- universe’s the of 69% el iec (MIUR), Ricerca della e a ` 2 eray1,2019 16, February v2, ac Fedi Marco ρ vac 1 eusv ocs hc ih orsodt fundamen- to and correspond attractive might exert which to them forces, to allows repulsive according also and, it a heavier, chirality, their vortices as the matter, making dark mass-acquisition the process, framework, in participates this vacuum, the In of dopant spin. inter- and structures, actions mass particles’ produces and also breaking and acquisition, symmetry spontaneous the of drives hydrodynamics processes whose field scalar quantum that a is of picture and resulting sector The dark themselves. the vacuum unique with quantum coincides is field process Higgs the the that that vacuum, suppose and to doped allowed mass-acquisition superfluid, then Higgs-like a are a in we If described 7] be can 6, process 5, 4, [3, the vacuum? for described (BEC) the condensate condensate tachyonic Bose-Einstein Higgs’ special is And field. viscosity Higgs apparent the possible of Dilatant the to thing? are refer same could we the [22] to vacuum vacuum refer or quantum to different and names different sector something using dark really the field with [22]. Higgs along in the proven is and a described Hence, as dilatancy, rea- the vacuum acts be for could which that son and [2], vacuum superfluid matter the of interaction dark dopant the particle to diffused due with process, vortices mass-acquisition quantized the trigger and mechanism, Nambu- here the which Higgs are bosons, vacuum the Goldstone superfluid quanta, the to in vacuum’s analogous phonons quantum of where mechanism scalar, pairs a vortex-antivortex fundamental, via as a the vac- – as describe quantum field considered can in One annihilating – and uum forming energy. pairs dark virtual of corre- action units whose pulsive (J vacuum, pressure the to of spond density energy the is * / m 3 = a,hnejsiyn h re- the justifying hence Pa), doi:10.5281/zenodo.2566644 iyaptnilterm, potential a sity φ a aumsaewt h ab-odtn oo state boson Nambu-Goldstone the with state vacuum nal density Lagrangian The reads occurs. breaking symmetry the state vacuum the serve of shift Nambu- energy a an to case corresponds this in boson, boson, spin-0 Goldstone a field to correspond scalar can real a as By of field circle a states). with (vacuum potential minima sombrero Goldstone a observe Let breaking symmetry Spontaneous 2 physical of hydrodynamics vacuum. the by Driven interactions. tal oanwdgnrt tt,ltu a ifrn vacuum different a say us and let state state, ground state degenerate the new both charge, a shifts (chiral) to and topological current the a from case our In acquisition mass is conserved the symmetry-induced starts bo- The then Higgs process. which (massless and boson mode) Goldstone son’s the with the coincides of the which in formation event phonon the circular as a vacuum considering superfluid then quasi- are vacuum’s We of a zone lattice. consider Brillouin first us the Let around [9]. current emerge and modes fluids phonon relaxation where that Frenkel time, within know quasi-lattice We a mech- form presented. Higgs superfluids being the is to approach that emerg- the anism for the fundamental treat is to fact allowed boson, are Nambu-Goldstone we ing account, into taken ∗ ic h uefli edo h hsclvcu is vacuum physical the of field superfluid the Since φ φ = sahdoyai process hydrodynamic a as h θ eacmlxsaa edoeigteconstraint the obeying field scalar complex a be L v i 2 ( = = = = constant − − − − ε 2 2 1 2 1 1 hnti urn oncsteorigi- the connects current this Then . v ( ( ive ∂ 2 v ( θ ) µ ∂ − φ The . n nldn nisLgaga den- Lagrangian its in including and µ J i ∗ θ µ θ λ ) | ∂ 0 ∂ )( = ( i µ µ φ hc ste eeeaeonce degenerate then is which , ∂ U θ φ − ∗ θ µ φ )( ( + θ v 1 ihn osritadthis and constraint no with 2 − ive + ) ) m ∂ v φ ymtytransformation symmetry µ θ v ∂θ 2 i = θ θ 2 φ sapoo n this and phonon a as , m ) ∂ . ∗ 2 2 µ = ve φ with , v θ 2 i = ε θ . + ) n osntpre- not does and erdfiethe redefine we m λ U 2 Q v ( → 2 emerges , 1 = ) ∞ current We . (1) θ 2 h ie oino atce sdsrbdhr.Mroe in Moreover here. described as particle, a of motion tized of interpretation the zit- (and spin relation to the constant finding terbewegung), Planck link per- [12] moment Salesi each and cami in fluctuations, is vortical vacuum hydrodynamic by The turbed turn. complete discussed 2 a also just to and the constant account Planck into of meaning taking That, break- symmetry occurs). (spontaneous ing sombrero the of “state top “rest the symmetric on su- the the to in respect with state vacuum energy perfluid favorable more a vor- as quantized appearing tices are fluctuations vacuum’s that confirming reads It creation pairs process. continuous electron-positron annihilation a and in virtual vacuum the quantum the of populating vacuum of hydro- e.g. definition the the of fluctuations, chiral from validity a comes the with approach to agrees dynamic hint this Another and theory. antivortex gauge an 2, and Eq. vortex in a Also, principle vacuum. the superfluid respects the n vortex in quantized action a least of of action turn of the quantum boson The by Goldstone the massless. fact be the must with accords phonon the of we Finally units. use same Both the phonon. have the and of time-independent path are the defining Lagrangian, action the (quantum) for abbreviated the as constant action Planck abbreviated sider a the is in Lagrangian that turn the know for complete We a vortex. expresses quantized action) of quantum (the turn, single a sidering where a in circulation the vortex indicating quantum relation Sommerfeld [11]. semimetals Weyl are in phonons been instance, polarized for have known, circularly phonons phonon. other and chiral circular [10] observed by a driven as states hypothesis, Topological this within resented, 0 sue ohpstv n eaievle representing values negative and positive both assumes | h eosrto a tr rmteBohr- the from start may demonstration The J p 0 ( = 0 h ) h ¯ | sPac osatand constant Planck is θ k i orfrt h hnn h pseudo-momentum The phonon. the to refer to 6= .Tecretms etplgclyrep- topologically be must current The 0. h S I n ∆ 0 safnaetl nenl quan- internal, fundamental, a as C E = = p ∆ ,w e htPac constant Planck that see we 1, dx R t = p = d 2 p x h nh π ecnteeoecon- therefore can We . smmnu.B con- By momentum. is | s | = π h ¯ nE.() refers (3), Eq. in / :ti supports this 2: h represented ∆ E ∆ t Re- . (3) (2) doi:10.5281/zenodo.2566644 osat enn htmse r rdce yteHiggs the by predicted are masses that meaning constant, in invariant is which Higgs lep- the of with Lagrangian Let tons interaction coupling. the Yukawa consider the instance for via us know, for we holds as also but, this fermions and mechanism; Higgs bosons the massive completing the produce they know Yang-Mills we the theory, of bosons gauge the massless the of join these phonons bottom Once energy. the kinetic has around only vacuum. sombrero Goldstone’s moving the phonon in circular excitation This hydrodynamic quantized corresponds and a spon- massless produced to via is phonon boson) breaking circular Nambu-Goldstone symmetry the taneous (the cannot Thus, vacuum we the (chiral) and gap. in topological annihilated mass a is a vacuum to have the refers Here and time charge. in change action not abbreviated the in S parametrization time of ab- the sence as exactly states, vacuum zero-frequency implying time of independent the operator is in charge that, (topological) see the We argument, volume. Goldstone the in quanta vacuum’s of ber arises, where healing the also tube vortex is 3D which arising the phonon of the length circular define the can we of Now radius spacetime. all over integrating by the to Referred reads action. it standard (1), parametrization density the the Lagrangian to include arrive we we time if by action, reduced the back poloidal to Coming and vortex-particle. torus-shaped toroidal cir- a quantized the of combined in directions the quanta vacuum’s as of described culation is spin 4 Sect. 0 = h V ned h iclto aho h hnndoes phonon the of path circulation the Indeed, . a stesatrn eghand length scattering the is stevlm ntevcu nwihtevortex the which in vacuum the in volume the is = R R L − int S dt 2 1 d v = = 2 Q ( SU R ξ ∂ = g µ e ≡ R ( θ dt d 2 L ¯ L r )( ) φ Z ∂ n where and e x 8 d µ R − J π 3 θ V 0 x aN + + ) (
x φ dt = ) † N m e ¯ = 2 R − omlzdnum- normalized a (6) 0 v L g 2
e d 3 sa arbitrary an is dt x W ± , (7) (5) (4) Z Q , 3 hr h aso h lcrnis electron the of mass the where a eepesdi h form the in vacuum expressed superfluid be the can of par- spectrum free Bogoliubov massive The a of ticle. formation the sub- toward the stage represents sequent polaron interac- matter impurity-boson phonon-dark weak interac- The for resulting tion. even the strong be BEC, can the tion in Bo- mode a sound If with goliubov resonant vacuum. is the momentum-energy matter throughout transferred diffused dark the also but the be filaments, of can cosmic profile particles flat in the and for galac- curve, responsible the rotation is in it It concentrated where is discs, polaron. matter tic Bose dark a that believed originates is dopant vacuum, a superfluid as the acting of particles, is boson matter’s it Goldstone dark the with process, of (phonon) mass-acquisition interaction the the in that hypothesized stage further a As Bose to bosons Goldstone From 3 doped the Higgs of the excitation of vacuum. hydrodynamic excitation a an probably as field, results itself boson the Higgs Ultimately, modes. excitation Bogoliubov different on tube (following vortex vortices evolution: canonical and the bosons the Goldstone in of involved generation quasi-lattice different different vacuum’s on The of depend zones might seq.). Brillouin Model and Standard 3 the funda- of Sect. is masses in vacuum discussed superfluid (as the mental case in this (in matter) dopant dark a of particle presence the but phonon) (the bo- son Nambu-Goldstone the by vacuum’s triggered of vortex) energy (the quanta kinetic the from arises observable, an With measured. be to have but mechanism L int interaction matter vacuum-dark via polarons = √ g e 2 v e ¯ L − ε e k R − φ = + → p e ¯ R − E √ e 1 k 2 L − ( E
 k + v + + √ 0 g 2 g √ H e n e 2 2 v 0  g → = e ¯ c L − ) m , otxtrs and torus) vortex e e R − . + eems,as mass, Here e ¯ R − e L −
H (10) , (9) (8) doi:10.5281/zenodo.2566644 E fvcu’ una hs aitna is Hamiltonian a whose in quanta, vacuum’s particles) of matter BEC (dark impurities Let mobile interaction. consider Yukawa to the us polaron via Bose occurs the fermion massive from a passage final The quan- vacuum. for tum spectrum in excitation help Bogoliubov may the This calculating propagation. frequen- light high with compatible very cies, at al- vacuum propagation dilatant transverse-wave all, lows through travel after can and, solidification sound quasi-lattice vacuum only vacuum’s as stress, shear light: to due of occurs, speed cor- the vacuum dilatant to the responds in clearly sound [22] of speed in the computations that suggest successful the free-particle that a notice to and superfluidity, for for regime speed critical and aypr ftesl-nryi ae noacutadthe imagi- and account small into taken a is since self-energy the well-defined, of part is nary polaron a where via found is polaron the of (respec- energy self-energy the tively), the and function Green’s interacting where k ao and laron [13] function Green’s G the and (10) Eq. Consid- ering contribution. matter’s dark to Nambu- thanks real The momentum, the a below. into in to changes discussed pseudo-momentum starts as particle, boson’s vortex, Goldstone phonon massive torus circular a free the of a way form this becoming In mass acquire dynam- 2). phonon the (Sect. in ics Bose partecipating a those which are polarons in in formation involved case quanta vacuum’s matter, The dark vac- arises. of of polaron particle quantum a a and between uum and quanta vacuum’s tween and with particles, ter and momentum volume, with respectively, (impurity) ticle quantum), vacuum’s and a length of mass the is where ξ 0 (  p g ξ H , k 1 z epcieyrpeetteitrcinsrnt be- strength interaction the represent respectively b n ) = , k † 0 + − = ε ε g 1 k k Q k and G ∑ c ∑ − E 2 = k = stekntceeg favcu’ quantum vacuum’s a of energy kinetic the is k / 0 k k ξ ∑ 2 ( ξ hck ¯ ε = p m d 4 k  k Q ( , k d π † p h b z ¯ − ε with , k † h ¯ , ) 2 k † raeabsnadadr atrpar- matter dark a and boson a create 1 p d µ z − 2 k b ˆ , µ k ) = a 2 1 k where , hmclptnil nE.(11) Eq. In potential. chemical + / / nwhich in + = stekntceeg fdr mat- dark of energy kinetic the is ξ m 2 c p V m g Q 2 g z + pe fsudi h superfluid the in sound of speed V Q ehv hnnrgm for regime phonon a have We . ∑ − steceia oeta ( potential chemical the is ∑ k Re , ξ p k k p 0 , ∑ , k stemmnu fapo- a of momentum the is q ε 0 and , k b q ( k † = b p + k † , q ∑ + ε E d p q k ( k † a ) b ti interesting is It . p 0 − k † , stescattering the is 0 q z − ) d q k b r h non- the are G 0 k b 0 ( k b k p , k , , + z V ) − (12) (11) sa is 1 m g = Q Q 4 tr.Fo q 4w banteYkw neato in interaction Yukawa the by obtain given we space, temper- 14 real zero Eq. a From at BEC ature. the of correlation density-density where eiu ftepolaron, the of residue ewe w oaoswt momenta with polarons two between in and order residue second unitary with considering To still length, concentration. scattering the matter dark on vac- pend matrix particle; matter scattering quantum-dark zero-energy uum’s the is mass) in reduced order first is To polaron the unity. to close is 3,oecnsatt ecieishdoyai behavior [14] hydrodynamic (GPE) its equation BEC Gross-Pitaevskii describe the doped to from a start as can considered one is [3], vacuum physical here dynamics As vortex from Spin 4 galactic the in as where such discs. concentrated, matter is dark matter as of baryon concentration also scale, the macroscopic justify concerned; could is on this interaction also matter matter-dark true baryon as a be far must (e.g. vortex-particle this par- matter’s a ticles, dark if diffused that, attracts fact fermion) interest- the also fundamental on is It reflect dynamics below. to the discussed ing which as in form, torus, takes spin a of of doped form the the the in assumes theorem, tube tube second Helmholtz’s vortex obeying (phonon) a Then, boson vacuum. becomes Goldstone and initial mass the acquires polaron, mat- the dark con- in particle the with ter of interaction length the coherence Through (and densate). (5) length healing the with ε r p h itnebtendr atrprilsand particles matter dark between distance the = χ f i ( ( ξ h ¯ p p Z k ∂ψ , ∂ 1 n ban adusefcieinteraction effective Landau’s obtains one , p z , t = ) p f = T ( 2 = r = ) n = ) 1 Q − where , − n ± 2 h 0 − m ∂ 2 p T 2 T ∑ ∇ / 2 2 χ 2 ( ψ m n p ( T 0 p Q , + m z 1 π ) − Q g = z / r ψ e 2 p ∂ − − 2 2 | z a √ ψ , π
ξ E p 2 − a h nrysitof shift energy the | T p r 1 2 p 2 / / 1 1 ,

ξ −

m − p n z , = Q 2 r ψµ ξ ε as p (with p osntde- not does ensthe defines 2 0 ) , m r (14) (13) (16) (15) the ξ doi:10.5281/zenodo.2566644 eivdi h urn hoeia rmwr,btwith as radius with but point-particles, quanta, framework, vacuum’s with of theoretical deal vortices toroidal current not approach the do this we in in believed that solved fact is the particles by fundamental the of torus. vortex a say should will we it or and ring, loop vortex closed a a form form indeed does must it boundary, a if on line, end vortex not the theorem, where second vortex, Helmholtz’s the to of origin the at ψ 2) (Sect. Gold- the phonon of stone path circular the to corresponding line, vortex which in eai[6,wosget[7 htteqatmpotential particle quantum a the of and that [17] Salesi suggest by who proposed [16], Recami was spin, includes Barut-Zanghi different which the theory, a of On reformulation hydrodynamic a numbers. basis, spin all represent satisfactorily v with integral line ( circulation tized contin- 2 is phase modulo its condensate uous space, the in function Since continuous a be must potential. quantum the represents where equation Euler the of analogue continuity the the and i.e. equation equations, hydrodynamic the write can repre- phase the In sentation quanta. between length scattering the quantum, a g of mass the where 1 m = h rbe futailtdvrec n fteradius the of and divergence ultraviolet of problem The = and  4 I ξ n h uefli’ est aihs According vanishes. density superfluid’s the and 0 ∂ρ π C ∂ v ψ endi q 5.Hr h ai ftevelocities the of ratio the Here (5). Eq. in defined a v t d S 2 h ¯ x + = ψ C 2 fvcu’ unai h otx(i.1 can 1) (Fig. vortex the in quanta vacuum’s of stecnest aefnto,with function, wave condensate the is · / v v = m sacoelo nsaeta utecrl a encircle must that space in loop close a is m h ¯ S S ∇ · = √ o-nryprmtr where parameter, low-energy a π ∇ ϕ ρ e steeoedfieaantequan- the again define therefore us Let . Q  2 Γ stesprudvlct and velocity superfluid the is e m π ∂ρ ∂ i v = ,aaoosyt q 2,truhthe through (2), Eq. to analogously ), ϕ h ¯ S t , n − = ρ + ≡ 2 h 2 1 ¯ svcu est.Fo 1)we (16) From density. vacuum is ∇ ∇ m 2 m Γ µ  · ∇ v ~ 0 ( 2 µ √ ρ 2 S h hmclptniland potential chemical the ( 0 √ d ρ n − v − ρ S = 1 2 = ) g ρ ∇ 0 , + · ± (17) 0 ~ v 1 S 2 h ¯ , m 2 ± ∇ 2 √ ,... 2 √ ρ a ) ρ sagain is r  = m (20) (19) (18) (21) 2 ξ as , 5 otxtb.O h ih:avre ig(otxtrs where torus) (vortex ring vortex a velocity, right: toroidal the the On tube. vortex a 1: Figure fe ahqatmfrigtevre a oe along moved has vortex 720°, M of the a rotation forming toroidal quantum a system each after the after state same i.e. the (fermion), in returns spin-½ then have direction, would poloidal after vortex the position the in same turn in the one turns in completed two return having complete to to direction needs toroidal vortex the the time same the bo- massive resulting fermion). the or (by son vortex quantum the the by by then kept is triggered chirality spin is whose the (phonon) vortex boson for The Nambu-Goldstone account particles. well fundamental to of seems geometry topol- non-trivial Their of superfluids ogy. emergence in the tori demonstrate vortex and into [18] Krstulovic evolving Villois, tubes vortex Also lyze (Fig. vortices considered. torus-shaped are work, be 2) this to however In has explained. spin now different Any vortex-particle of immediate. concept quite the is with link the motion, ternal being motion ternal units, natural (using vortex. the of velocity h pnaglrmmnu ( (respectively), momentum directions angular spin cited ratio the the the Defining for as velocities spin rotations. angular of poloidal explain type to can toroidal other spin of any bi-component terms a mechanical such in that notice to ing favcu’ unu oigi h ou-otxneeded torus-vortex the in flowing quantum vacuum’s a If bu-ti ah(i.2o h et.I sinterest- is It left). the on 2 (Fig. path obius-strip ¨ ~ s h ieto fsi.B okn tsi sin- as spin at looking By spin. of direction the ntelf:avre iewt eln length healing with line vortex a left: the On ~ ~ v v S S × = v ~ 2 s where , 2 h oodlvlct and velocity poloidal the 1 h ¯ m = ρ − )myttlyaiefo t in- its from arise totally may 1) 1 ∇ ρ S sdtrie yteratio the by determined is ) = 2 1 m ∇ R R 2 2 v , 3 ω h translational the 1 tal. et , ω ξ 2 actually , sthe as ana- (22) v 1 is doi:10.5281/zenodo.2566644 unu ntetrsvre a eepesda follows as expressed be can torus-vortex a spheroidal the of a in position quantum the into defining equation horn-torus parametric the The determined vortex. be of may evolution spin-0 further of case by The spin-½. corre- to rotations sponds toroidal two each revolution poloidal One ω vacuum’s 12 with simulation a right quanta. the On instance spin-½. for fermions are of characteristics position These typical the particle. starting the to its of rotation velocity in 720° a angular returns after quantum poloidal each the velocity, of angular 1/2 toroidal ratio M the a Given along the flows of trajectory. vortex quantum torus Each the particle. forming a vacuum of superfluid spin the determine may one poloidal 2: Figure hsscinaotsi,i sas neetn oconsider to interesting also is it conclude spin, To about section approaches? this different vortex and reconcile characteristics a vacuum’s then dynamics between Could similarity string. certain a and a line time, notice same also the vacuum’s can At also de- geometry. we that loop-like to a excluded scale, have not higher may is quanta a it on but however, Gravity particles case, Quantum scribe our Loop In in 20]. loops [19, re- also theorized may the particles sembles fundamental of shape toroidal The 2 and torus, φ the of that and tube where 0 1      / and ω x y z π 2 = ( = ( = r . ( = θ ξ ≥ 0 r r otxtrs h ai ftetria nua eoiyt the to velocity angular toroidal the of ratio The torus. Vortex sin n + + r hsshvn rirr ausbten0 between values arbitrary having phases are 2 π ξ ξ ξ ( / ω dt cos cos stedsac ewe h etro the of center the between distance the is 1 ) dt / ( ( ω ω + ( 2 1 1 π φ ω ω dt dt 0 / 1 2 ) dt + + = ω ) φ φ 0 0 1 n fe h cancellations the after and n 2 )) )) = = sin cos d S φ . ( ( / ω ω dt 2 2 dt dt , ω + + 2 θ θ = 0 0 ) ) d θ obius-strip ¨ / dt (23) (24) and 6 dpnigo hi hrlt)i rvnb h Bernoulli the by driven is chirality) their vortices on between repulsion (depending or attraction Huang that be- as confirms forces well [7], of as dynamics Pshenichnyuk, the vortices. to quantum back tween dilatant come of us force Let viscous vacuum. the the to due also actually is relativity, cial this [23], as the in regime for shown (relativistic) reason As dilatant increases. a stress to shear passes vac- which quantum bi-regime uum, a Pioneer but [22]. then the superfluid in a precession simply of Not perihelion solutions Mercury’s vacuum, of exact the and the of anomaly behavior via dilatant demonstrated probably the is as for vacuum the reason in superfluid the dopant greater, also a bi-component the of a presence is The energy of part. dark picture which energy. the of dark to of vacuum, comes is probable that therefore most vacuum, the One superfluid we as the course, matter of Of dark dopant vacuum. of doped think a we immediately in when vortices clue Model quantum Standard important the as very of particles a fundamental is the This describe Gauss’s of gravity. or for the law, law match Coulomb’s vortices of two behavior mathematical between is superfluid interactions the the once doped, Interestingly, via in- occur The effect. and Bernoulli forces the repulsive interact. or superfluid attractive way the are the teractions in change forming particles vortices doping the superfluid which a in [21] In considered matter. is dark in to done resorting indeed by in- as work, fundamental this dopant, justify a to including able without be teractions, to expect, behave would not we vor- probably as quantum would vacuum that superfluid fact the the in tices on a hy- reflect in the to vortices superfluid, studied doped quantum who between [21], interactions Pshenichnyuk drodynamic to refer us Let interactions Hydrodynamic 5 parity. matter-antimatter represent vor- to chiral able that also noted are tices be super- under- should of for it Eventually, foundations also versa. symmetry. mechanical important vice quantum be or the may fermion standing ratio a varying into a boson Thus, but a unvaried mass transform particle’s would the leave would vortex revolutions/rotations the ratio in the in change opportune an that apparent eaiitcms nraei spe- in increase mass relativistic doi:10.5281/zenodo.2566644 atce,cue ytevlct ed ftesuperfluid the [21] of as dopant’s written fields the be velocity can on vortices the acting by density caused energy particles, kinetic Coulomb’s The in charges ex- same law. chirality, electric the positive/negative different possess have as they vortices actly if the attractive if or repulsive chirality is force. to force Bernoulli equivalent to The is also mathematically, divergence, and, of law Gauss’s definition the via which, oia hre eoigrplino attraction, or repulsion denoting charges logical where charges, body the point of elements of ume amount arbitrary spatially an ∑ a of to density fields due charge electric force. force with Bernoulli the body the law consider extended to Gauss’s we also when to hence equivalent Indeed, field, electric is the law for Coulomb’s also and gravitation universal of law which Newton’s gravity, to equivalent for is law Gauss’s Eq. equals that mathematically notice We 25 effect. Bernoulli gener- pressure-related weaker, a them ate makes and interaction other their each where reinforce 1 points fields a the obeying where su- Points fields a velocity function. as vortices’ superfluid the a of in perposition arises force surface Bernoulli the calculated. to to and drops normal distance, superfluid tor healing the vor- the of within the density zero on the dominates while which surface, tex energy, kinetic of density where reads This force. K K k ++ + = − 1 , n = = e + + K ( 2 2 Q ( h m ¯ h m ¯ r 2 i 2 = ) Q ) Q / eoe nitga vrifiieia vol- infinitesimal over integral an becomes , ( + ( R R − 2 ρ 2 e + + v = F ψ in ee osm rdfeettopo- different or same to refer signs F 2 ξ / ξ b ∞ g 2 4 yrdnmclyepessthe expresses hydrodynamically 2 2 2 = = πε )( )( 1 4 Z R Z R R 0 S S 2 2 2 Z K g S + + V + ( vrwihteitga is integral the which over , ( r | r ψ d d d r ) ) 2 2 ∞ n 2 − 2 n − ρ + ( + d ( r r r 2 0 4 ξ ) ξ | ) dS 2 Rd dS 2 2 r ˆ − − ρ dv n cos h u fthe of sum the , ( 2 2 r Rd Rd ) α saui vec- unit a is cos
cos α α d ) ) , , sthe is (27) (29) (26) (25) (28) / r 7 particle, vortices, the between distance [26] at be- online distance available short version to Animated due them. vortices tween the vacuum’s of fields the of velocity from the exchange arises of directly the interaction which particles as matter represented dark with is coupled flow quanta gluon The config- spin uration. (non-destructive) mutual in (quarks) vortex-tori interacting 3: Figure gopdi h lo)wihpoedfo n vortex one from particles proceed continu- matter which a gluon) dark as the and flow in quanta vacuum’s (grouped gluon of the flow represents ous principle. also exclusion 3 Pauli Figure and repulsion electrostatic avoided by however matter-matter be would but vortices, to of -½ hydrodynamics according the or possible ½ albeit spin annihilation, they with (vortex-vortex) if both annihilate contact would in fundamen- electrons, got two two that e.g. suppose fermions, ex- us Pauli tal let with a agrees and in also principle That clusion ½). i.e. -½, toroidal ½, spin, of (spin mutual sense rotations the with regards pro- vortices as topology, a torus non-destructive in quarks three respectively) three charge, as shows red ton, 3 and an- green Fig. also blue, to (with interaction. distance, strong short hydrodynamics very the the a alyze on at reflect put torus-vortices to of interesting is it Finally elusive have discussed. for distance been a need at no vortices between with Interactions vacuum, gravitons. hydrody- doped a a demon- as in were defined force that namic be would If gravity quantum cor- field. strated, and gravitational exists the flow to incoming real responds it 25] a [24, that in hypothesized but de- flux, is law incoming Gauss’s virtual that a reflect as gravity to scribes important is dop- It the particle. with ing associated is which coordinates cylindrical ξ ipie ersnaino rtna ytmo three of system a as proton a of representation Simplified h eln eghand length healing the R α h aiso doping a of radius the h zmta nl in angle azimuthal the doi:10.5281/zenodo.2566644 odtn oos a rge h asaqiiinpro- mass-acquisition the Nambu- trigger the can of role bosons, the Goldstone playing vacuum, phonon superfluid polarized the circularly in re- probably mechanism, As approach Higgs attention. the an renewed gards Such and greater action). therefore vor- deserves of quantum a quantum of (a turn tex one Planck to corresponds phenomena. itself interactions hydrodynamic constant their quantum as particles, spin simplest massive and the explain vortex- are razor, vacuum to doped Ockham solution superfluid, the a to in according particles see, can we As Conclusion edi i.3cudas sueLsae geometry. group tains group L-shaped the the for assume of though also sented, generators could nine the 3 all fields; There, colour Fig. the in of field structure the dynamics investigate field which color the gauge with Cardoso’s agrees of also and by to presented [27] quarks similar Jehle interacting respects for 26]). some model [28, topological in the (see is flows representation broader visual much 3 This simplicity actually greater Fig. for are thin In they here but particle. evident: are the is tubes of –– flux mass the another the to for vortex responsible a again by from driven flowing flow gluon quanta the vacuum’s in momentum its matter dark another specifically interac- –– discussed, previously to strong what to quark of According in a mechanism tion. circulate from hydrodynamic only flow the not also within i.e. vacuum, but quark, doped representing torus-vortex a the the spin of of (with constituents quanta the fluid the since but static, flow) be object, internal not solid the would a quark for a as of structure the Therefore, are quarks, wavefunctions the the Here of another. color to the one changes from passing flow the gluon to this and that quantum fact quarks, of between exchange gluon occurring the chromodynamics, to according another, to ψ 7 = N √ 2 1 − 2 | 1 r ψ ψ ψ r ¯ = i 1 5 3 − = = = eeaos iigtegunoctet. gluon the giving generators, 8

| | | b r g b b ¯ g r r ¯ ¯ ¯ SU

i i i , , , , ( 3 ψ ) 8 C atc iuain [28], simulations lattice QCD = SU √ 1 6 ( N ψ ψ ψ | r 4 2 6 r ¯ = i = = = + 3

| )

g b r U b b b g n nta ob- instead one ¯ ¯ ¯ b ¯

i (

3 , , , − ) r repre- are 2 | g g ¯ i (30) . 8 ue nmto fsrn neato codn oFg 3 Fig. to according http://www.8128.info/strong-interaction at interaction strong of D Erasmo animation Wolfgang vortices, puter and toroidal comments in for spin Recami sim- of computer figures for and Dominici ulations Lorenzo thanks author The Acknowledgements universe. the remaining in the mass-energy [22], to- vacuum which dilatant dopant), are (the (the gether matter energy prod- dark dark particle joint superfluid and the of vacuum) but hydrodynamics nothing that the be of and would uct i.e. universe, observe the of, we in part that ter are reality we the which approach, this Within the between exchange matter the dark vortices. particle of and light quanta the vacuum’s in electrostatic of revisited virtual the be in of cases) may exchange both interaction the in Also bosons (gauge by) photons driven high. (or is with quanta grouped coincide vacuum’s interaction gluons particles that matter’s strong possibility dark the the with to flux, dopant, gluon model, a the as this to matter and in dark come, particle we of repul- when role and attractive the chi- both About their therefore sive. on be depending can Forces vortices, topo- rality. the the be- from of acting arise charges charges force hy- logical electric apparent Bernoulli where them, as the tween but via are fundamental forces) (which as funda- drodynamic Model not the Standard read produce the also therefore of can interactions vortices revolutions The mental of fully ratio rotations. hydrodynamic which to the torus-vortices, parti- as finally spin with and represent coupling matter, by dark polarons cle Bose becoming cess, References BROE G., BERTONE, [2] S., TSUJIKAWA, L., AMENDOLA, [1] abig (2010) Cambridge searches. and models (2010) Cambridge Press, Observations. and Theory atceDr atr Observations, Matter. Dark Particle abig nvriyPress, University Cambridge ∼ abig University Cambridge %o aynmat- baryon of 5% ulrfrcom- for aumler ¨ MFedi.html akEnergy. Dark ∼ 5 of 95% doi:10.5281/zenodo.2566644 RCM,E,SLS,G.: SALESI, E., 075301 119, RECAMI, Lett. [12] Rev. Phys. J., SHI, D., LIU, [11] al., et L. Zhang, J., Xiao, M., Li, J., Yi, H., ZHU, [10] EPST,S.: ESPOSITO, [15] L.P., PITAEVSKII, S., GIORGINI, , F. DALFOVO, [14] M. Georg BRUUN, A., CAMACHO-GUARDIAN, [13] VLVK G.E.: VOLOVIK, [5] V.I.: SBITNEV, [4] K.: Rajat BHADURI, and S. DAS, [3] VLVK G.E.: VOLOVIK, [6] HAG K.: HUANG, [7] HAG K.: HUANG, [8] BLAO,D,BAHI,V . TRA- V., V. BRAZHKIN, D., BOLMATOV, [9] c,ITc,Rjk (2015) Mechan- Rijeka InTech, Quantum ics, of Applications in Topics lected Medium perfluid condensate Bose-Einstein from arXiv:1411.0753[gr-qc]. preprint: energy dark and n.Sr oor Phys. Monogr. Ser. Int. oor Phys. Monogr. ic igpr (2016) Singapore tific, (2017) (2018). 579-582 pp. Science, 6375, phonons. chiral of Observation thermo- liquid of dynamics. theory phonon The K., CHENKO, universe. perfluid TIGR,A e.Md Phys. Mod. Rev. : A. STRINGARI, Condensate 8 Bose-Einstein a Quasiparti- in between cles Interaction Effective Landau Some (1996) ph/9607025v1 considerations particles: spinning simple of hydrodynamics and chanics 302(2018) 031042 , ri:un-h90091(1999) arXiv:quant-ph/9902019v1 , c.Rep. Sci. akeeg n akmte nasu- a in matter dark and energy Dark uefli Universe Superfluid A 156 nteRl fSi nQatmMe- Quantum in Spin of Role the On n alvn,MR e.:Se- (ed.): M.R. Pahlavani, In: . hsclVcu saSeilSu- Special a is Vacuum Physical rpit ri:3950 (2013) arXiv:1309.5707 Preprint: h nvrei eimdroplet helium a in Universe The h uefli Universe Superfluid The o.1 7-1 (2013) 570-618 1, Vol. , 2 2 (2012). 421 , rpit arXiv:quant- Preprint: . 117 (2003) bu kinematics About 71 hs e.X Rev. Phys. , ol Scien- World . 6 (1999) 463 , akmatter Dark 359 n.Ser. Int. , Issue , , , 9 JHE H.: JEHLE, [27] D [26] M.: FEDI, [25] R.T.: CAHILL, [24] dilatant to due mass Relativistic 2019, M. FEDI, [23] dilatant a as vacuum Physical 2018, M. FEDI, [22] PSHENICHNYUK: I.A. [21] F.: VIDOTTO, C., ROVELLI, [20] J.: PULLIN, A., ASHTEKAR, [19] D., PROMENT, G., KRSTULOVIC, 98, A., VILLOIS, A57 Rev. [18] Phys. G., SALESI, E., RECAMI, [17] E.: RECAMI, G., SALESI, [16] (1981) hadrons and quarks fFg ,UL http://www.8128.info/strong- URL: 3, interaction Fig. of Gravity Quantum http://hal.archives-ouvertes.fr/hal-01423134 France, Superfluid CCSD, HAL-Archives, III. fluid: arxiv.org/abs/physics/0307003 https:// at: Preprint kinetic doi:10.5281/zenodo.2566526 energy, relativistic for formula quantum and vacuum https://zenodo.org/record/2566589) reposi- tory: anomaly zenodo on Pioneer Phys., (and J. doi:10.1139/cjp-2018-0744 to Can. precession, solutions perihelion Mercury’s and exact yields fluid fluids quantum (2017) in vortices 2014 Press, University to Theory Spinfoam Introduction and Gravity Elementary Quantum An Gravity: Quantum ity Superfluid a of Model (2016) arXiv:1604.03595v2 Gross-Pitaevskii the for H.: SALMAN, (1998) Barut-Zanghi (1997) the 6 of limit Theory. quantum and mulation ULR . nmto fsrn interaction strong of animation W.: AUMLER, ¨ ol cetfi,2017 Scientific, World , onain fPyisLtes o.1,No. 10, Vol. Letters, Physics of Foundations MFedi.html oooia hrceiaino leptons, of characterization Topological yrdnmc fteDr Super- Dark the of Hydrodynamics otxFlmn rcigMethod Tracking Filament Vortex A rvt sqatmfa in-flow. foam quantum as Gravity hs et B Lett. Phys. , aritrcin fheavy of interactions Pair yrdnmclrefor- Hydrodynamical arXiv:1705.10072v1 , opQatmGrav- Quantum Loop oain Loop Covariant 104 ,207-211 3, , Cambridge , 2017, , , , doi:10.5281/zenodo.2566644 FME,RO,VNALN . UGS,E.: BURGESS, J., ALLEN, VAN R.O., FIMMEL, [29] BICUDO, N., CARDOSO, M., CARDOSO, [28] ntnDC,UA(1980) USA Wash- D.C., Office, ington Information Technical and beyond Scientific and Saturn, Jupiter, to first Pioneer: scaling D Casimir Rev. the of study system, microscopic quark-gluon-antiquark and hybrid static the for P.: atc C optto fteclu fields colour the of computation QCD Lattice 81 354(2010) 034504 , NASA . Phys. , 10