arXiv:1606.06963v3 [physics.gen-ph] 31 Aug 2016 2 farida [2] [email protected] [1] h ietuicto fgaiywt toginteraction strong with inhibits gravity also of and unification principle, gauge direct theory the the on quantum based a gravity constructing of quantization. from resist invariance physicists but coordinate prevents manifold), (the space-time a symmetry in similar interactions have Gravitational also (QYMT). theory Yang-Mills etna constant Newtonian otyter fsrn neatos osse gauge possesses group the interactions, the on QCD, strong based that symmetry of indicate theory experiments worthy related other and funda- quantum a of [1–6]. play physics (QCD) strong-interaction to chromodynamics the identified in role were mental fields tensor proposal, constant gravitational strong equations a field with Einstein-type theory by two-tensor governed strong interac- are the a fields where strong tensor in interactions, gravitational of formulated and was was strong field of idea field non-abelian This spin-2 a nonlinear tions. describe a to suc- of used the -interaction which in self gravity, strong cessive of concept the proposed l h acltosdn ntenmrclltieQCD lattice numerical the in done calculations the All co-workers his and Salam Abdus seventies early the In neada h ert fisdevelopments its of secrets sci- centuries?” future our the to during of at advances hidden; next and the lies ence at future glance the a cast which behind veil an ie n re horizon” freer and and outlooks, wider new a steel; and his methods gains of new temper finds the he tests investigator the (1900). ibr (1900). 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O. 38 ie the times This Bern-Carrasco-Johansson SU r fgaiya ag itne[] ntesm way same the in Glashow-Weinberg-Salam, distances[7]; of large theory the- gauge at Einstein’s to the gravity leads of which action ory Weyl derived but Ein- GR be stein nonrenormalizable cannot quantizing gravity) by straightforwardly induced mechanically tum for- gravity strong the mulation. within problems, matter fteguegaiydaiy n h ouinto R solution on the existence Yang-Mills and quantum duality; the gauge/gravity the of h isenHletato o uegaiyis gravity pure for action Einstein-Hilbert the C em ntanbe oee,ta snttecase: this the disprove not to attempt is valiant that The however, unattainable; seems QCD nonrenormalizable eta.()Enti eea eaiiya neetv lon theory effective the an of as limit relativity distance general Einstein po- (2) (QED) potential, electrodynamic tential. QCD quantum non-relativistic non-relativistic the and perturbative/short of (1) component induces: behaviorrange theory) distance gravity short strong the of describes (which action follow- Weyl the In Becchi- ing, P.953). and ([8], fields transformations gravitational nonlinear Rouet-Stora the by non- of resulting absorbed renormalizations the are Here, divergences P.967). renor- gauge-invariant & be In P.963 to shown ([8], QYMT. been malizable has with tensor curvature merged the quadratic in possesses be terms which action gravitational to a gravity case, this the scheme allows unification the problem. that such the existing is solve an formulation to gravity is used Strong there be whether can of that question scheme unification the to turn us let community. important physics apparently upon and have special may ef- very solution inspiring a the such of which example fect outstanding an offers sibility C cini cl nainl udai ntefield the in quadratic invariantly scale strengths is action QCD [7]. h ac eei htqatmgaiy(..aquan- a (i.e. gravity quantum that is here catch The aignwrcle omn h rgno h problem, the of origin the mind to recalled now Having h rgno h iclisi o la ous: to clear now is difficulties the of origin The (2) 4 h yaia raigo h cl naineof invariance scale the of breaking dynamical the wt t hrceitcms gap), mass characteristic its (with L otesltoso the of solutions the to nue eea eaiiy(R san as (GR) relativity general induces e × F U urs oeta,wihdescribes which potential, quarks) t orsodn edeutoso the of equations field corresponding µν n udai ntecraue is curvature, the in quadratic and e i (1) D hsieial ed othe to leads inevitably This CD. hooy saaa,Pakistan Islamabad, chnology, (i.e. Y , eue to reduces non-unitary hs h nfiaino rvt with gravity of unification the Thus, . BJ obecp and double-copy (BCJ) . − U etiomass neutrino hsi the is This and ) (1) Q fe h spontaneous the after renormalizable, to of ation rm facie prima ose origo et fons unitary 4 n dark and G impos- EW while and = g - 2 symmetry breakdown[9, 10]. The paper is organized as follows. In section II, we QCD possesses four remarkable properties that strong briefly review the BCJ double-copy construction of grav- gravity must have for it to be called a complete theory ity scattering amplitudes. Section III is devoted to the of strong interactions. The first is asymptotic freedom review of strong gravity theory. Most importantly, we (i.e., the logarithmic decrease of the QCD coupling con- prove that BCJ double-copy construction exists within stant α (Q2) 1/(ln Q2) at large momentum transfers, the strong gravity formulation. The calculation of the di- s 0 ∼ 0 or equivalently the decrease of αs at small distances, mensionless strong coupling constant is done in the sec- αs(r) 1/(ln r)) which permits one to perform con- tion IV. The theoretically obtained value is tested ex- sistent∼ theoretical computations of hard processes using perimentally in the section V. We present strong gravity perturbation theory. This property also implies an in- as a massive spin-two theory in the section VI. Here, crease of the running coupling constant at small momen- we show that the dynamics of strong gravity theory is tum transfer, that is, at large distances. The second fully symmetric, but its vacuum state is asymmetric. We important property is the confinement, in which quarks also show in this section that electroweak and custodial and gluons are confined within the domain of their strong symmetries can be induced dynamically. Critical tem- interaction and hence cannot be observed as real physical perature, fundamental mass and mass gap of the QCD objects. The physical objects observed experimentally, at vacuum are obtained in the section VII. This leads to large distances, are hadrons (mesons and baryons). The the derivation of the effective pure Yang-Mills potential. third characteristic property is the dynamical breakdown The gauge-gravity duality property of strong gravity the- of chiral symmetry, wherein the vector gauge theories ory is studied in the section VIII. We also show that with massless Dirac fermion fields ψ are perfectly chiral strong gravity possesses UV regularity and dynamical symmetric. However, this symmetry is broken dynami- chiral symmetry breaking in this same section. Confine- cally when the vector gauge theory is subjected to chiral ment and asymptotic freedom properties of the strong SU(2) rotations. This is the primary reason why chiral gravity is studied in the section IX. In this section, we symmetry is not realized in the spectrum of hadrons and calculate the energy density of QCD vacuum. The exis- their low energy interactions[11, 12]. The fourth prop- tence of quantum Yang-Mills theory on R4 is established erty is the mass gap( ∆). Here, every excitation of the in the section X. The vacuum stabilizing property of QCD vacuum has minimum positive energy (i.e. ∆ > 0); Higgs boson with mass mH = 129GeV is studied in sec- in other words, there are no massless particles in the tion XI. The solutions to the neutrino mass, dark energy theory[9, 10]. Additionally, and dark matter problems are presented in the sections strong gravity must also be able to reproduce the two XII, XIII and XIV respectively. The physics of the re- fundamental parameters of QCD (i.e., coupling αs and pulsive gravity and cosmic inflation is presented in the fundamental quark mass mq [13], P.178). section XV. Conclusion is given in the section XVI. Thus, the three demands that must be met by strong gravity theory for it to be called a unification scheme for QYMT-GR are: II. THEORETICAL PRELIMINARIES (1) It must admit the four QCD properties afore-listed. (2) It must be able recover the fundamental parame- Research in strong gravity has always had a rather ters of QCD (i.e., αs and mq). unique flavor, due to conceptual difficulty of the field, (3) It must be able to reproduce Einstein’s general rel- and remoteness from experiment. We argue, in this pa- ativity as the limiting case of its long-distance behavior. per, that if the conceptual misconception namely, that Any theory that fulfills these three demands can be gravity is bedeviled with many untamable− infinities termed ”a unified theory of nature”. that beclouds the field could be circumvented, then the− In the present paper, we study the structure of a dy- complexity enshrined in the field would become highly namically broken scale-invariant quantum theory (Weyl’s trivialize. action) within the context of strong gravity formulation, The most powerful tool for removing this conceptual and its general properties. The major problem which has difficulty is encoded in a long-known formalism: that to be faced immediately is the unresolved question of uni- the asymptotic states of gravity can be obtained as ten- tarity of pure gravity: Weyl’s action is non-unitary while sor products of two gauge theory states (i.e. gravity = the Einstein-Hilbert action for pure gravity is unitary. gauge gauge). This idea was extended to certain in- This problem is circumvented within the framework of teracting⊗ theories, in 1986, by Kawai, Lewellen and Tye strong gravity: where the unitary Einstein-Hilbert term [14]; and to strong-gravitational theory by A. Salam and is induced after the breakdown of the scale invariance C. Sivaram in 1992 [6]. The modern understanding of of Weyl’s action ([6], P.324). To put it in a proper and this double-copy formalism is largely due to the work of succinct context, Einstein GR emerges from the Weyl’s Bern, Carrasco and Johansson (BCJ). Formally, double- action after the dynamical breakdown of its scale invari- copy construction (also known as BCJ construction) is ance. Hence Einstein’s theory of gravity is not a funda- used to construct a gravitational scattering amplitude by mental theory of nature but the classical output of the using modern unitarity method, and the scattering am- more fundamental gluon-dependent Weyl’s action. plitudes of two gauge theory as building blocks [15, 16]. 3

This pathbreaking technique of computing perturbative loops, a valid gravity scattering amplitude is obtained by scattering amplitudes, which led to a deeper understand- replacing color factors with kinematic numerators in a ing of quantum field theory, gravity, and to powerful new gauge-theory scattering amplitude. The resulting gauge- tools for calculating QCD processes, was awarded the coupling doubling is called BCJ/double-copy property 2014 J.J. Sakurai Prize for Theoretical Particle Physics [15, 16]. [17]. The gluon’s scattering amplitudes, (in terms of cubic BCJ construction has overturned the long-accepted graphs) at L loops and in D dimensions, are given by dogma on Einstein’s GR, which posits that GR is non- ([15, 16, 18, 19], and the references therein): renormalizable. This new approach breaths new life LD into the search for a fundamental unified theory of na- (L) L−1 m−2+2L d ℓ 1 cini Am = i gα LD (1) ture based on the ”supergravity” approach. Supergrav- (2π) Si Di i ∈ cubic ity tries to tame the infinities encountered in the Ein- X Z stein’s theory of gravity by adding ”supersymmetries” to where m is the number of points, gα is the dimen- it. In a variant of the theory called N = 8 supergrav- sionless gauge coupling, Si are the standard symmetry ity, which has eight new ”mirror-image” particles (grav- factors and Di are denominators encoding the structure itinos) allow physicists to tame the infinities present in of propagator in the cubic graphs. ci are the color factors the Einstein’s theory of gravity: other variants of super- and ni are the kinematic numerators. BCJ construction gravity are N = 2, 4 Yang-Mills-Einstein-Supergravity posits that within the gauge freedom of individual cubic (YMESG) and N = 0 Yang-Mills-Einstein (YME) theo- graphs, there exist unique amplitude representations that ries ([18, 19], and the references therein) Supergravity − make kinematic factors ni obey the same general alge- is like a ”young twig, which thrives and bears fruit braic identities as color factors. Hence, color/kinematics only when it is grafted carefully and in accordance duality holds: ni ci [15, 16]. with strict horticultural rules upon the old stem”. The double-copy⇐⇒ principle then states that once the As to the N = 0 YME theory (where N = 0 means color/kinematics duality is satisfied (i.e., ni ci), that there are no supersymmetries in the theory), we the L-loop scattering amplitudes of a supergravity⇐⇒ theory claim that this theory is by no means different from the (with N 4) are given by broken-scale-invariant Weyl’s action. This assertion can ≥ only be true if this action naturally possesses BCJ and k m−2+2L dLDℓ 1 n2 M (L) = iL−1 α i guage-gravity duality properties. The BCJ property is m 2 (2π)LD S D ∈ i i established in the next subsection, and we show that the   i Xcubic Z potential, carried by the broken-scale-invariant Weyl’s (2) action, possesses this property in the subsection D of where dimensionless kα is the gravity coupling; and it section III of this paper. The gauge-gravity duality is assumed that the two involved gauge fields are from property of strong gravity is established in section VIII: the same Yang-Mills theory. From Eqs. (1) and (2), we this is our ”guide post on the mazy paths to the have hidden truths” of neutrino mass and dark energy prob- A(L) = M (L) k =2g (3) lems. The discovery made here is that both problems are m m ⇐⇒ α α connected by the effective vacuum energy (or effective Eq.(3), which is valid for all variants of supergravity Weyl Lagrangian). with N 4, is the expected gauge-coupling doubling or BCJ property.≥ This property shows that gravitons and gluons should be part of a fundamental unified theory of A. Perturbative Quantum Gravity and Color/ nature. Kinematics Duality: A Review However, the devil is in the detail: the color-kinematics duality (ni ci) is more or less a conjecture; and QCD (one of the variants of Yang-Mills theory) is the the scattering-amplitude⇐⇒ method of probing the quan- current well-established theory of the strong interactions. tum nature of gravity is full of many mathematical land- Due to its asymptotic-free nature, perturbation theory mines. Nevertheless, the conclusions of N = 8 super- is usually applied at short distances; and the ensuing gravity theory are indisputable. For we are convinced predictions have achieved an astonishing success in ex- that the gauge-coupling doubling and gauge-gravity du- plaining a wide range of phenomena in the domain of ality should exist in the correct theory of quantum grav- large momentum transfers. Upon closer consideration ity without appealing to supersymmetries. This is where the question arises: Can perturbation theory be used to strong gravity theory (or point-like gravity) kicks in. Our explore the quantum behavior of gravity at short dis- present knowledge of the theory of strong gravity puts us tances as well? The answer to that question is a re- in a position to attack successfully the problem of quan- sounding yes! The discovery of BCJ principle is now our tum gravity/point-like gravity by using powerful- window into the quantum world of gravity with tamable mathematical tools (formula operators from differential infinities at short distances. This principle states that, geometry with their duality and supersymmetry-like prop- regardless of the number of spacetime dimensions and erties) bequeathed to us by antiquity. 4

We conclude this section with a great quote from one from QCD gluon field as a sum ([4], P.572): of the greatest revolutionary mathematicians the world a b a b c has ever known (David Hilbert) [20]: ”If we do not suc- GµGν ηab + GµGν Gσdabc + ... (4) ceed in solving a mathematical problem, the reason fre- quently consists in our failure to recognize the more gen- where ηab is the SU(3)C color-metric, dabc is the totally eral standpoint from which the problem before us appears symmetric 8 8 8 1 coefficient and Ga is the dressed ⊗ ⊗ → µ only as a single link in a chain of related problems. Af- gluon field. The curvature would be generated by the a ter finding this standpoint, not only is this problem fre- derivatives of Gµ ([6],P.323). The 2-gluon configuration quently more accessible to our investigation, but at the can then be written from Eq.(4) as same time we come into possession of a method which a b is applicable also to related problems” The ”stand- gµν (x)= GµGν ηab (5) point” discovered in this paper is the strong− gravity theory. with

g = det(gµν (x)) (6)

III. STRONG GRAVITY THEORY: A REVIEW Eq.(5) is taken as the dominating configuration in the excitation systematics. In this picture, the metric is con- structed from a gluon-gluon interaction, and the gluon- We briefly review the standard formulation of strong gluon effective gravity-like potential (effective Rieman- gravity theory in this section: (for more details see [1– nian metric, gµν ) would act as a metric field passively 6, 8] and the references therein). Beginning with the gauging the effective diffeomorphisms (general coordi- two-gluon phenomenological fields (i.e. double-copy con- nate transformations), just as is done by the Einstein struction), we re-establish strong gravity as a renormal- metric field for the general coordinate transformations of izable four-dimensional quantum gauge field theory by the covariance group ([5], P.174). varying Weyl action with respect to the spacetime met- It is crystal-clear that Eq.(5), as put forward by the ric constructed out of the two-gluon configuration. In proponents of strong gravity, is by no means different this case, the two-point configuration (which leads to the from the double-copy structure of gauge fields in the quantization of space-time itself) naturally introduces BCJ construction (gravity = gauge gauge); as such a minimum length 2rg (i.e. ”intergluonic distance”); we should be able to arrive at the⊗ same conclusions. where rg is the ”gluonic radius”. It should be empha- The BCJ formalism (double-copy construction) is formu- sized here that this way of quantizing space-time be- lated by using scattering-amplitude method. Similarly, gins from the trajectories of two 2-gluons,i.e., curves or we show that double-copy construction can be obtained paths of the geometry used. This method of construct- by using formula operators from the differential geom- ing spacetime geometry from 2-gluon phenomenology has etry. Our approach puts BCJ formalism on a proper been shown to be compatible with nature: The visualiza- mathematical footing: it puts flesh on the bones of BCJ tion of the QCD vacuum (i.e.visualization of action formalism. density of the Euclidean-space QCD vacuum in three-dimensional slices of a 243 36 spacetime lat- × tice), by D. B. Leinweber, has shown that empty space A. Scale-Invariant-Confining Action for Strong is not empty; rather it contains quantum fluctuations in Gravity Theory the gluon field at all scales (this is famously referred to as ”gluon activity in a vacuum”) [21]. This can only In analogy with the scale-invariant QCD action which mean one thing: that gluon field is the fundamental field i is quadratic in the field strengths Fµν (with dimension- of nature, and the spacetime metric/gravity is emergent less coupling), we have the corresponding Weyl action from 2-gluon configuration. This is the main argument for gravity ([6], P.322): of BCJ/double-copy construction. Simpliciter! By taking the vacuum states of hadron to be color- I = α d4x√ gC Cαβγδ (7) less (i.e. color-singlet), the approximation of an external W − s − αβγδ QCD potential (the hadron spectrum above these levels) Z can be generated by color-singlet quanta. Based on the where αs is purely dimensionless and can be made into 2 fully relativistic QCD theory, these contributions have to a running coupling constant αs(Q0). It’s worth noting come from the summations of suitable Feynman diagrams that Eq.(7) is not only generally covariant but also lo- in which dressed n-gluon configurations are exchanged cally scale invariant ([7], P.6). The Weyl’s tensor (Cαβγδ) between several ”flavors” of massless quarks. Thus, the is constructed out of the corresponding Riemann curva- simplest such system (with contributions from n-gluon ir- ture tensor, i.e., the covariant derivatives involving gauge reducible parts n =2, 3, ..., and with the same Lorentz fields, characterized with the generators of the conformal quantum numbers) will have∞ the quantum numbers of 2- group. In the following, the metric is generated by Eq.(5) gluon. The color singlet external field is then constructed ([6], P.323). 5

The Weyl curvature tensor is defined as the traceless Eq.(15) leads to the field equations [24]: part of the Riemann curvature [22]:

1 δIW 1 Cαβγδ = Rαβγδ (Rαγ ηβδ Rαδηβγ √ ggµαgνβ = Tµν (17) − n 2 − − δgαβ −2 − Rβγ ηαδ + Rβδηαγ ) − Eq.(17) would be of fourth-order in the form ([6], 1 + R(η η η η ) (8) P.323): (n 1)(n 2) αγ βδ − αδ βγ − − Eq.(8) is constructed by using the trace-free property 1 1 g (Rγ );δ + R ;δ Rδ Rδ 2R Rδ + g R Rγδ of Weyl tensor: 2 µν γ ;δ µν ;δ − µ;ν;δ − ν;µ;δ − µδ ν 2 µν γδ αγ α 1 γ ;δ γ γ 1 γ 2 η C = C = 0 (9) [2gµν (R ) 2(R );µ;ν 2R Rµν + gµν (R ) ] αβγδ βαδ − 3 γ ;δ − γ − γ 2 γ 1 By contracting Eq.(8) with itself, we get = Tµν (18) 4αs 4 C Cαβγδ = R Rαβγδ R Rβδ αβγδ αβγδ − (n 2) βδ The corresponding fourth-order Poisson equation and − its linearized solution are given as([6], P.323 & 325): 2 + R2 (10) (n 2)(n 1) δ 4V = km δ3(r) − − s∇ 0 In four-dimension (n = 4), Eq.(10) reduces to; V (r)= αr (19)

2 αβγδ αβγδ βδ 1 2 It is clear from Eq.(18) that its left-hand side van- C CαβγδC = RαβγδR 2RβδR + R ≡ − 3 ishes whenever Rµν is zero (the vanishing of a tensor is (11) an invariant statement ([23],P.146)), so that any vacuum Thus, Eq.(7) becomes, solution of Einstein equations would also satisfy the ones from the quadratic action. A complete exact solution of 4 αβγδ βδ 1 2 the field Eq.(18) (with metric signature + ) for a IW = αs d x√ g(RαβγδR 2RβδR + R ) − − − 3 general spherical symmetric vacuum metric− is −− given as Z (12) ([6], P.323-324):

ds2 = αdt2 βdr2 r2dθ2 r2 sin2 θdφ2 (20) B. Gauss-Bonnet Invariant Theorem − − − where For space-time manifold topologically equivalent to λ1 2 flat space, the Gauss-Bonnet theorem relates the various α =1 λ2r λ3r (21) quadratic terms in the curvature as [7]: − r − −

I = α d4x√ g(R Rαβγδ 4R Rαβ+R2)=0 GB − s − αβγδ − αβ β = [α]−1 (22) Z (13) Using this property, we can rewrite Eq.(12) as λ1, λ2, and λ3 in Eq.(21) are suitable constants, related to the coupling constant. Dimensional analysis and nat- ural unit formalism then tell us that coupling constant IW IW GB = IW IGB = IW (14) (α) would remain dimensionless provided that λ1 carries −→ − −1 the dimension of distance ([L]GeV ), λ2 the dimension of mass ([M]GeV ), and λ3 the dimension of squared mass ([M]2, GeV 2). If we take the mass to be the mass of the 4 βδ 1 γ 2 IW = 2αs d x√ g RβδR (Rγ ) (15) quark (m ), then we can rewrite Eq.(21) as − − − 3 q Z  

where Rβδ is the Ricci tensor, which is a symmetric λ1 2 2 tensor due to the Bianchi identities of the first kind, αs =1 mqr mq r (23) γ − r − − and its trace defines the scalar curvature Rγ = R ([23], P.153). By using Eqs.(7) and (15), we have For the pure Yang-Mills theory (i.e. QCD without quarks), mq 0 and Eq.(23) reduces to 4 βδ 1 2 1 4 αβγδ → d x√ g RβδR R = d x√ gCαβγδC − − 3 2 − λ Z   Z α =1 1 (24) (16) s − r 6

Based on the strong gravity theory and the formalism The corresponding solution of the Eq.(30) for a point of the vacuum solution of Einstein field equations [3, 23, mass source is given as ([26], P. 3): 25], λ1 = Gf m. C C C With this value, Eq.(24) reduces to V (r)= 1 2 e−β1/r + 3 e−β2/r (31) r − r r G m α = g =1 f (25) where C = k2M/8πα , C = k2M/6πα , C = s 00 − r 1 3 2 3 3 k2M/42πα β = α1/2(α k2)−1/2 G3/2, and β = 3, 1 3 1 × f 2 and Eq.(20) becomes −1/2 α1/2 2 (3α α ) kh2 G3/2.Mi is unknown invari- 3 2 − 1 × f −1 ant mass (but we identified it to be the invariant mass of Gf m Gf m   ds2 = 1 dt2 1 dr2 the final hadronic state of the theory, M m (because − r − − r ≡     final observable particle state must be color singlet)). r2dθ2 r2 sin2 θdφ2 (26) By using Eqs.(28) and (29), β = and thus Eq.(31) − − 2 reduces to ∞ where mass m is the only allowed mass in the the- C C ory, and is due to the self-interaction of the two gluons V (r)= 1 2 e−β1/r (32) (glueball). Eq.(26) is the well celebrated Schwarzschild r − r vacuum metric except that instead of normal Newtonian −19 −1 gravitational constant (GN 10 GeV ), we have 2 ≈ −1 k m strong-gravitational constant (Gf 1GeV ). C = ≈ 1 16π k2m C2 = C. Broken Scale Invariance and 12π Perturbative/Short Distance Behavior −1 3/2 β1 = k Gf (33)

−1 As expected, the resulting infinity β2 = is tamed by Once we have ΛQCD Gf 1GeV , the scale invari- ∞ ance would be broken.≡ An additional≈ Einstein-Hilbert the nonlinear nature of the Weyl’s action. term linear in the curvature would be induced, but the From Eqs.(32) and (33), we have full action would still preserve its general coordinate in- k2 m 4 variance ([6],P.324): V (r)= 1 e−β1/r (34) 16πr − 3   Eq.(32) is the exact equation obtained for the broken I = d4x√ g α R Rµν α R2 + k−2α R eff − − 1 µν − 2 3 scale invariance and perturbative behavior of strong grav- Z (27)
ity in ([6], P.325). Here the induced Einstein-Hilbert term incorporates the phenomenological term 1/k2 = 1 ([8], P.954 32πGN D. Double-copy Construction in Strong Gravity & 967): this term is called graviton propagator/ ”pure Yang-Mills” propagator . By comparing Eq.(27) with Eq.(15), we have From Eq.(34), we can write 2 kα α1 = α3 =2 V (r)= C (35) 16πr 2 α2 = (28) 2 2 3 where the dimensionless gravity coupling kα k m = 4 −β1/r ≡ 32πGN m and C 1 3 e is the ”group-theoretic Using natural units formalism, we can write constant”× of strong≡ gravity− theory. It is to be recalled that the interaction energy, to the 1 k−2 = 1 1017GeV (29) leading order, of two static (i.e., symmetric) color sources 32πGN ≈ × of QCD without quarks (pure Yang-Mills theory) is given by [27–30]: where G 10−19GeV −1 (in natural units) ([2], P. N ≈ 2668). g2 E(r)= α C (36) Eq.(27) gives rise to the mixture of fourth-order and 4πr second-order field equations([6], P.324), whose solutions for the field of a localized mass involves Yukawa and Where dimensionless gauge coupling g2 g2(r) m , α ≡ × rg the normal 1/r potential terms. and mrg is an arbitrary renormalization group scale for- mally invoked, in quantum field theory, to keep the scale- α 4V + β 2V km δ3(r) (30) dependent gauge coupling (g2(r)) dimensionless. Since ∇ ∇ ≈ 0 7

Eq.(35) is also the energy of two interacting gluons, we and 115); can write (from Eqs.(35) and (36)) g N = r (39) eσ1+σ2∈r 1 V (r)= E(r) kα =2gα (37) r ⇐⇒ X − Eq.(37) is the required BCJ property. We have there- The summation sign in Eq.(39) can be converted into fore proved the existence of double-copy construction in an integral, because for a particle in a box, the states of strong gravity. It is remarkable to note that despite dif- the system have been found to be very close. Using the ferent approaches taken by supergravity (scattering am- density of single-particle states function, Eq.(39) reduces plitude method) and strong gravity (effective potential to; method), we still arrive at the same conclusion (see Eqs. ∞ (3) and (37)). D( )d N = ∈ ∈ (40) eσ1+σ2∈ 1 Z0 − IV. QCD EVOLUTION where D( )d is the number of allowed states in the energy range∈ ∈to +d and is the energy of the The body of experimental data describing the single-particle∈ state.∈ Using∈ the∈ density of states as a strong interaction between nucleons (which is the non- function of energy, we have ([33], P. 290); perturbative aspect of QCD for r ) is consistent −→ ∞ with a strong coupling constant behaving as αs 1 4πV m [31]: obviously this aspect of QCD is consistent with≈ the D( )d = 2m d ∈ ∈ h3 ∈ p ∈ Eq.(25) for r .   −→ ∞ One of the discoveries about strong force is that it with diminishes inside the nucleons, which leads to the free movement of gluons and quarks within the hadrons. The p = √2m implication for the strong coupling is that it drops off at ∈ very small distances. This phenomenon is called ”asymp- totic freedom” or perturbative aspect of QCD, be- 2m 3/2 D( )d =2πV 1/2 d (41) cause gluons and massless quarks approach a state where ∈ ∈ h2 ∈ ∈ they can move without resistance in the tiny volume of   the hadron [32]. Hence for the strong gravity to describe where p is the momentum of particle, m its mass and h the perturbative aspect of QCD correctly, it must repro- is the Planck constant. By putting Eq.(41) into Eq.(40), duce the value of strong coupling constant αs (by us- we have ing the observed properties of gluons: the mediators of ∞ strong force) that is compatible with the experimental 2m 3/2 1/2 d data. This is what we set out to do in this section. N =2πV ∈ ∈ (42) h2 eσ1+σ2∈ 1   Z0 −

A. Gluon Density but σ1 = σ2 µeff and σ2 =1/kT. µeff is the effective potential, k is× the Boltzmann constant and T denotes temperature ([33], P.116). Since there is no restriction The first thing to note here is that gluon, being a on the total number of bosons (gluons), the effective po- bosonic particle, obeys Bose-Einstein statistics. The tential is always equals to zero (µeff = 0) (this is true Fermi-Dirac and Bose-Einstein distribution functions are for the case where the minimum of the effective poten- given as ([33], P. 115); tial continuously goes to zero as temperature grows[34]). g Thus, Eq.(42) reduces to; = r (38) r σ1+σ2∈ ℵ e r 1 ∞ ± 2m 3/2 1/2 d N =2πV ∈ ∈ (43) where the positive sign applies to fermions and the h2 e∈/kT 1 negative to bosons. is the number of particles in the   Z0 − ℵr single-particle states, gr is the degenerate parameter, σ1 is the coefficient of expansion of a gas of weakly cou- By using the standard integral(where ς(z) is the Rie- pled particles (an ideal configuration for describ- mann zeta function and Γ(z) is the gamma function) ing the asymptotic freedom/perturbative regime ∞ z−1 of QCD) inside the volume V . σ2 is the Lagrange un- x dx = ς(z)Γ(z) (44) determined multiplier and is energy of the r-th state. x ∈r e 1 The value of ”σ1” for boson gas at a given temperature is Z0 − determined by the normalization condition ([33], P. 112 8

Eq.(43) becomes Thus Eq.(52) becomes

3/2 8Ξ(E )4 2πmkT ρ = vac (54) N =2.61V (45) vac 3 h2 27k   Eq.(54) is the energy density of a single gluon. But 2 Using m = E/c and the average kinetic energy of based on double-copy construction (see section II, Eqs.(3) boson gas in three-dimensional space E = 3kT/2, Eq. and Eq.(37)), Eq.(54) is multiplied by 2, and thus, (45) reduces to; 4 16Ξ(∆εvac) 3/2 3 2ρ = (55) N (2.61)(3π) k vac 3 = T 3 (46) 27k V (hc)3   Eq.(55) now represents two-point correlator-vacuum 3 2 3 N (2.61)(3π) / k energy density. By comparing Eq.(50) with Eq.(55), we Define n and Ξ 3 = 2.522 g ≡ V ≡ (hc) × have 7 −3 10 (mK) . Hence the gluonh density (ng)i can be ex- 16Ξ 19 −3 pressed as; αs = g00 = =2.336 10 (meV ) 27k3 × 3 15 −1 ng = ΞT (47) As 1m =5.070 10 GeV , the above equation leads to × Eq.(47) is the required result for the finite temperature and density relation for gluon. α = g = C C =0.1797 (56) s 00 QCD × grav. B. Strong-gravity Coupling Constant Eq.(56) is the required strong (-gravity) coupling con- stant at the starting point of QCD evolution. In the next section, we show the compatibility of Eq.(56) with The principle of general covariance tells us that the the perturbative QCD, which is the theory that describes energy-momentum tensor in the vacuum (with zero mat- asymptotic freedom regime analytically. ter and radiation) must take the form;

T00 = K ρ (48) h i V. PERTURBATIVE QUANTUM Here ρ has the dimension of energy density and K CHROMODYNAMICS describesh ai real (strong-) gravitational field [35]. Hence Eq.(48) reduces to; Computations in perturbative QCD are formally based on three conditions: (1) that hadronic interactions be- 4 T00 = K(Evac) (49) come weak at small invariant separation r Λ−1 ; ≪ QCD (2) that the perturbative expansion in α (Q2) is well- and K = g00 = CQCD Cgrav(strong gravity cou- s 0 × − defined mathematically; (3) factorization dictates that pling). CQCD is a dimensionless coefficient which is en- tirely of QCD origin and is related to the definition of all effects of collinear singularities, confinement, non- QCD on a specific finite compact manifold. Similarly, perturbative interactions, and the dynamics of bound state can be separated constituently at large momentum Cgrav is a dimensionless coefficient which is entirely of gravitational origin [35–38]. Therefore Eq.(49) becomes transfer in terms of (process independent) structure func- tions Gi/H (x, Q), hadronization functions DH/i(z,Q), or 4 T00 = g00(Evac) (50) in the case of exclusive processes, distribution amplitudes φH (xi,Q) [39, 40]. The asymptotic freedom property of Recall that energy density (ρvac) can also be written as perturbative QCD(β0 = 11 (2/3)nf ) is given as ([41], P. 1): − Evac −1 ρvac = = V Evac (51) V × 2 4π 2 2 αs(Q0)= 2 < 0.2 for Q0 > 20GeV (57) Q0 Eq.(51) is justified by the standard box-quantization β0 ln( Λ2 ) procedure [35]. Hence we have In the framework of perturbative QCD, computations ρ = n E (52) of observables are expressed in terms of the renormalized vac g vac 2 × coupling αs(µR). When one takes µR close to the scale −1 where ng V (number density). of the momentum transfer Q0 in a given process, then From the≡ average kinetic energy for gas in three- α (µ2 Q2) is indicative of the effective strength of the s R ∼ 0 dimensional space, we have T = 2Evac/3k. With this strong interaction in that process. Eq.(57) satisfies the value, Eq.(47) reduces to following renormalization group equation (RGE) [42]:

3 dα 8Ξ(Evac) µ2 s = β(α )= (b α2 + b α3 + b α4 + O(α5)) (58) ng = (53) R 2 s 0 s 1 s 2 s s 27k3 dµR − 9

with the gravitational field generated by a spherically sym- metric mass m, on the assumption that the electric b = (33 2n )/12π (59) 0 − f charge, and orbital angular momentum (L) of the mass are all zero [25]. b = (153 19n )/24π2 (60) 1 − f It turns out that the Schwarzschild vacuum solution of the Einstein field equations can be understood in terms 5033 325 2 3 b2 = (2857 nf + nf )/128π (61) of the Pauli-Fierz relativistic wave equations for massive − 9 27 spin-2 particles which would mediate a short-range ten- where Eqs.(59-61) are referred to as the 1-loop, 2-loop sor force ([3], P. 117). It follows that the two interacting a b and 3-loop beta-function coefficients respectively. The gluon fields (Gµ and Gν ) are considered to be dressed minus sign in Eq.(58) is the origin of asymptotic free- gluon fields of the gravitational field, i.e., the col- dom, i.e., the fact that the strong coupling becomes weak ors of the gluon fields are covered or hidden within the for hard processes. Eq.(58) shows that RGE is dependent spacetime base-manifold (ηab) of the color SU(3) prin- on the correct value of a purely dimensionless strong cou- cipal bundle ([4], P. 572), thereby making the− observable pling constant ( αs). Thus the precise calculation of its asymptotic states of gravity to be color-singlet/color- value (without appealing to the choice of renormalization neutral. Hence the resulting glueball (massive particle 2 scheme and scale choice Q0) would be the holy grail of formed as a result of the self-interaction of two gluons) perturbative QCD. of the theory (with spherically symmetric mass m and quantum numbers J PC =2−+) would still have the total angular momentum of 2. The validity of this statement A. Experimental Test is proved by using the well-known Pauli-Fierz relativis- tic wave equations for massive particles of spin-2([3], P. We begin by reviewing the systematic study of QCD 124): coupling constant from deep inelastic measurements in 2 ([43] and the references therein), where many experi- φµν + m φµν = 0 (63) mental data were collected and analyzed at the next-to- leading order of perturbative QCD (see Tables 2,3 and 6 of [43]) by using deep inelastic scattering (DIS) struc- µν ∂µφ = 0 (coordinate gauge condition) (64) 2 ture functions F2(x, Q ).In these experimental results, we 2 are more interested in the αs(90GeV )=0.1797 (in the Table 6 of [43]) obtained when the number of points is µ 613. This is the exact value we obtained theoretically in φµ = 0 (conformal gauge condition) (65) Eq.(56). Hence, we have not only demonstrated that the perturbative expansion for hard scattering amplitudes 2 converges perturbatively at αs = αs(90GeV )=0.1797 φµν = φνµ (symmetric condition) (66) but also able to prove that QCD is a strong-gravity- derived theory: an astonishing discovery! We have For the symmetric condition (Eq.(66)), the coordinate also validated the asymptotic freedom property of per- gauge condition given in Eq.(64) eliminates four out of turbative QCD given in Eq.(57): namely, that the start- the ten components of the wave function φµν of the 2 2 ing point of QCD evolution is Q0 = 90GeV for αs = Eq.(63); and the condition given in Eq.(65) eliminates 0.1797 < 0.2. one more, leaving 5 degrees of freedom: Having tested Eq.(56) experimentally, we therefore proceed to rewrite the renormalization group equation (Eq.(58) ) as: 2S +1= D =5= S = 2 (67) ⇒ As a result of the Eq.(67), the following is true: strong 2 3 4 β(αs)= b0(0.1797) + b1(0.1797) + b2(0.1797) + ... gravity, as a massive spin-2 theory, has five degrees of − (62) freedom ( D =5).   Eq.(62) is an echo of ”composition independence or Recall that the parity (P) and charge (C) quantum universality property” of the coupling αs to all orders numbers can be expressed by in the perturbative expansion for hard scattering ampli- tudes. P = ( 1)J+1 (68) −

VI. STRONG GRAVITY AS A MASSIVE C = ( 1)J (69) SPIN-TWO THEORY − and In the Einstein’s GR, the Schwarzschild vacuum is the solution to the Einstein field equations that describes J = L + S (70) 10

where J is the total angular momentum, L is the or- B. Groups of Motions in Strong Gravity Admitting bital angular momentum and S is the spin. Custodial and Electroweak Symmetries Thus, for the Schwarzschild vacuum solution (i.e., L = 0), we have The fundamental theorem in the theory of strong grav- ity (as a massive spin-2 theory) contains two statements, − J PC =2 + (71) namely: (1) Strong gravity is a pseudo-gravity ([5], P.173). Requiring instead that φµν = φνµ (2) Strong gravity, as a massive spin-2 field theory, 6 (antisymmetric condition), we would have has five degrees of freedom. The first statement means obtained 2S +1= D =1= S =0 and J PC =0−+, that the strong gravity must have a fundamental group ⇒ which is a pseudoscalar state. An important SO(n1,n2). The group SO(n1,n2) is the special real consequence of this discovery is that the un- pseudo-orthogonal group in n1 + n2 dimensions. This derlying dynamics of the strong gravity theory group has a non-compact group that is isomorphic to a is fully symmetric (i.e. φµν = φνµ = S = 2 ) generalized rotation group (involving spherical (with pos- ⇒ but its ground/vacuum state is asymmetric (i.e. itive curvature) and hyperbolic (with negative curvature) n1,n2 φµν = φνµ = S = 0; meaning that the vacuum rotations) in R . Its maximal compact subgroup is 6 ⇒ state must have massive spin-zero particle(s) given as SO(n ) SO(n ).The second statement forces − 1 × 2 glueball/meson with mass m): this is a formal us to write n1 + n2 = 5. a description of spontaneous symmetry-breaking From the Eq.(5), the dressed gluon field Gµ can be a phenomenon. separated into asymptotic-flat connection (Nµ ), i.e. the constant curvature (zero-mode) of the field and the a a a a normal gluon field (Aµ): Gµ = Nµ + Aµ ([4], P.572 & A. Effective Lagrangian of a Massive Spin-2 [5], P.174). By using the de Sitter group formalism for Theory the spacetime of constant curvature, the non-compact groups (de Sitter groups) for strong gravity are SO(4, 1) By using effective field theory (EFT) and the property and SO(3, 2). The group SO(4, 1) is associated with the of strong gravity (as a massive spin-2 theory, D = 5), spacetime manifold of constant positive curvature (de- the effective Lagrangian of the theory is characterized by noted by S(+)), representing spherical rotations, and [44]: SO(3, 2) is associated with the manifold of constant neg- ative curvature (denoted by S( )), representing hyper- − O bolic rotations. The two spaces are embedded in the L = i (72) di−4 manifold with signature (+ ). The maximal compact i MX −− X subgroups for the two non-compact groups are ([3], P. 132): where Oi are operators constructed from the light fields (with light mass), and information on any SO(4) SO(1) SO(4) SU(2) SU(2) (75) heavy degrees of freedom ( with heavy mass MX ) × ≈ ≈ × 1 −4 is encoded in the coupling di . For i = 1, we have MX

O SO(3) SO(2) SU(2) U(1) (76) L = 1 (73) × ≈ × d1−4 MX Eqs.(75) and (76) can be used to label left-right and isospin-hypercharge symmetries respectively: Using D = d1 = 5 means that the operator O1 must carry the dimension of squared energy (O E2) for the 1 SU(2)L SU(2)R (77) effective Lagrangian to carry the dimension∼ of energy: ×

E2 SU(2) U(1) (78) L = (74) L × Y MX Eq.(77) is called custodial symmetry of the Higgs sec- Eq.(74) is the effective Lagrangian of the strong grav- tor. This symmetry is spontaneously broken to the di- ity theory. The invariant mass/energy operator E2 = agonal/vector subgroup after the Higgs doublet acquires p pµ = m2 is called a flat space/Poincar˙einvariant. This a nonzero vacuum expectation value (VEV): SU(2) µ L × is characterized by an irreducible representation of the SU(2)R SU(2)V [45]. Eq.(78) is the electroweak Poincar˙egroup (with spin J ), and can be used to de- gauge symmetry−→ of the Standard Model (SM) of par- scribe a composite field ([3], P. 133-137) with five intrinsic ticle physics. degrees of freedom (i.e. D = d1 = 5). The importance of To break the electroweak symmetry at the weak this statement will be made manifest in the next subsec- scale and give mass to quarks and leptons, Higgs dou- tion. blets (that can sit in either 5H or 5H ) are needed. The 11 extra 3 states are color triplet Higgs scalars. The cou- composite field with five independent components, which plings of these color triplets violate lepton and baryon occurs naturally out of the strong gravity formulation, number, and also allows the decay of nucleons through is identified as the Higgs field H transforming in five- the exchange of a single color triplet Higgs scalar. In or- dimensional representations (5H ). As we shall soon der not to violently disagree with the non-observation of show, Eq.(79) connects the solution of the dark nucleon (e.g. proton) decay, the mass of the single color energy problem to the neutrino mass problem. triplet must be greater than 1011GeV [46]. It is to be The chain of symmetry-breakings in the Eq.(80) has remarked here that this heavy∼ mass would not disallow varying energy scales but the Lagrangian L of the whole the violation of lepton and baryon number: this is the system remains invariant: the physics of vacuum seems key to unlocking the mystery of neutrino mass problem. to obey effective field theory rather than quantum field We shall return to this a little later. theory. If the composite light field (with its five indepen- dent components) in the subsection A of section VI is taken to be the Higgs field, transforming in five- VII. SOME CONSEQUENCES OF STRONG dimensional representation (i.e. 5H ), then nature would GRAVITY AND THEIR PHYSICAL be permanently cured of its vacuum catastrophe dis- INTERPRETATIONS ease. In this case the invariant mass/energy operator of the light field would now be taken to be the VEV of the This section is entirely devoted to the consequences of Higgs doublets (i.e. E υ = 246GeV ), and the heavy strong gravity. In this case, we show the hitherto un- mass of color triplet Higgs≡ scalar would be encoded in the known connection between hadronic size, physical lat- d1−4 coupling 1/MX =1/MX . Here MX is the heavy mass tice size and gluonic radius (rg ). From this, we calcu- characteristic of the symmetry-breaking scale of the high- late the second-order phase transition/critical tempera- energy unified theory [47]. Once the high-energy unified ture Tc, and the fundamental hadron mass of QCD. theory that is compatible with nature is found, the value of MX will show up automatically. This is where pure Yang-Mills propagator kicks in. A. Calculation of the Gluonic Radius and Second-order Phase Transition Temperature

C. Type-A 331 Model The configuration at T >Tc for mass of the glueball for pure SU(3)C is shown in the Fig.1 [49]. Where 2rg One of the beyond-SM’s of particle physics is the is the intergluonic invariant separation. S and P repre- SU(3)C SU(3)L U(1)X or 331 model, in which sent scalar and pseudoscalar glueball / gauge fields re- the three× fundamental× interactions (i.e. electromagnetic, spectively. This figure is a perfect representation of 2- weak and strong interactions) of nature are unified at gluon phenomenological field. It is interesting to note a particular energy scale MU . This model is formu- that Fig.1 has exactly the same structure with one-loop lated by extending the electroweak sector of the SM graviton self-energy diagram ([8], P. 955). This is not gauge symmetry. The unification of the three interac- a mere coincidence, it only shows the compatibility of 17 tions occurs at the energy scale MU 1 10 GeV Eq.(5) with the tetrad formulation of GR, and the exis- in the type-A variant of this model.≈ In this× variant tence of double-copy construction in all the variants of of the model, the 331 symmetry is broken to repro- quantum gravity theory. In what follows, we will heavily duce the SM electroweak sector at the energy scale of rely on the correctness of the Fig.1 as the valid geom- 16 MX = 1.63 10 GeV [48]. It is apparent from the etry for strong gravity theory from the point of view of Eq.(29) that×k−2 = M = 1 1017GeV this is not 2-gluon phenomenology (double-copy construction). U × − surprising because electromagnetic, strong and weak We can therefore rewrite Eq.(25) for Tc and gluonic nuclear interactions are all variants of Yang-Mills inter- radius rg as action Hence the type-A 331 model is compatible with − the nature and MX (which is identified as the mass of the single color triplet Higgs scalar) = 1.63 1016GeV . [1 αs] rg × Tc = − (81) Thus Eq.(74) becomes Gf

2 2 By using Eq.(56), Eq.(81) becomes E (246GeV ) −3 L = = 16 =3.7 10 eV (79) MX 1.63 10 GeV × × 0.8203rg Tc = (82) and the symmetry-breaking pattern is Gf

2 MX E We now calculate the value of r by using the value of SU(3)L U(1)X SU(2)L U(1)Y U(1)Q (80) g × −→ × −→ the momentum transfer, at which αs converges pertur- 2 2 It is to be emphasized that the calculated value in the batively (i.e., Q0 = 90GeV ): see subsection A of section Eq.(79) is purely based on the principle of naturalness: a V. 12

Recall that the energy-wavelength relation is given as hold simultaneously [34]. Interestingly, we have a pri- ori claimed, during the calculation of gluon density, that hc µeff = 0: an assertion that is justified by the fact glue- Q0 = (83) λ ball, a self-conjugated particle with neutral color and zero electric charge, has a vanishing effective/chemical poten- Based on the geometry of Fig.1, we can write its as- tial (i.e. µ = 0) ([47], P.565). Thus the second-order sociated wavelength as; eff chiral phase transition temperature is calculated by using gluonic radius (Eq.(86)) , and thus Eq.(82) becomes λ =2πrg (84)

Tc =0.129GeV = 129MeV (89) Hence Eq.(83) reduces to; 38 27 3 −1 −2 where Gf = 10 GN =6.674 10 m kg s and hc 1GeV =1.78 10−27×kg. × Q0 = × 2πrg Hence, the chiral second-order phase transition in the strong gravity theory occurs when Tc = 129MeV and µ = 0. Exactly the same values were obtained in ℏ eff c [34, 51] for second-order chiral phase transition in QCD rg = (85) Q0 vacuum. We have thus established that strong gravity theory exhibits second-order chiral symmetry in the limit But Q2 = 90GeV 2 = Q = 9.487GeV and ℏ = 0 0 of vanishing quark masses ( mq 0). It is worth not- 6.582 10−16eV s. Thus Eq.(85)⇒ reduces to, → × ing here that the pure SU(3)C vacuum metric (Eq.(26)), obtained in the limit mq 0, is compatible with the r =2.08 10−17m (86) → g × glueball mass configuration given in the Fig.1, because Fig.1 was obtained in the limit of vanishing quark masses [49].

B. Charmed Final Hadronic State of Strong Gravity

Since we have shown that strong gravity theory pos- sesses SU(2) gauge field (i.e. isospin symmetry, SU(2)V ) in the subsection B of section VI, it is pertinent to inves- tigate the structure of the fundamental mass formula of the theory. In lattice QCD theory, the lattice spacing plays the role FIG. 1: Diagram for the contribution to the glueball (two- of ultraviolet cutoff, since distances shorter than ”a” is gluon) mass. not accessible. In the limit of vanishing of quark masses (mq 0), this is the only dimensional parameter and therefore→ all dimensionful quantities e.g. hadron and Eq.(86) is the required gluonic radius. Clearly Eq.(86) quark masses will have to be given in units of the lat- is related to the radius of hadron (r ) [1, 3–6]: h tice spacing ([12], P. 271):

rh = 10 rg (87) 1 × m = f(α(1/a),a) (90) a From the lattice QCD simulation performed at the ini- tial run β = 2.2 on a L3T = 243 48 lattice gives the It is clear from Eq.(90) that the unknown function f ×−15 physical lattice size (La)of2.08 10 m [50]. By using is dependent on the strong coupling and lattice spacing. Eq.(86), we can write × This equation is by no means different from Eq.(25):

2 rh La = 10 rg (88) m = (1 g00) (91) × Gf −

Hence Eqs.(86-88) show the connection between the It is evident from Eqs.(90) and (91) that 1 rh and gluonic radius, radius of hadron and the physical lattice a ≡ Gf f(α(1/a),a) (1 α ) = (1 g ). By using Eqs.(56) size. ≡ − s − 00 It is generally believed that at sufficiently high tem- and (87), Eq.(91) becomes perature / density, the QCD vacuum undergoes a phase m =1.29GeV = 1290MeV (92) transition into a chirally symmetric phase. Here, the chirally symmetric phase transition will be second-order Eq.(92) is the fundamental, color-singlet mass scale of phase transition iff the conditions T = 0 and µ =0 QCD vacuum. c 6 eff 13

The η(1295) pseudoscalar state/ η meson state to the number of charge states: with J PC multiplets of J PC = 0−+ has− mass value + of mη = 1294 4MeV ([46], P.32). Similarly, the N p ± 2 N = 0 = (94) charm-quark (with charge 3 ) has the mass value (mc) N n of 1.275 0.025GeV ([46], P.23). In terms of the resum-     ± ming threshold logarithms in the QCD form factor for The isospin symmetry (SU(2)V ) then demands that both the B-meson decays to next-to-leading logarithmic accu- charge states should have the same energy in order to racy, the mass formula for the charm-quark is given as preserve the invariance of the Hamiltonian (H) of the mc = mb mB + mD 1.29GeV [52]. Where mb, mB system. This means that isospin symmetry is a state- − ≈ and mD denote bottom-quark, B- and D-mesons respec- ment of the invariance of H of the strong interactions tively. under the action of the Lie group SU(2). However, The correctness of the strong gravity theory in describ- the near mass-degeneracy of the neutron and proton ing reality/nature is clear from the above-quoted values. points to an approximate symmetry of the Hamiltonian For we have shown in the section VI of this paper that describing the strong interactions [54, 55]. The mass even though the underlying dynamics of the strong grav- gap (m ) which is responsible for the approximate gap − ity theory is fully symmetric (φµν = φνµ), its vacuum symmetry of strong interaction in this case must be − state is nonetheless asymmetric (φµν = φνµ) with the the energy difference between the proton state and neu- PC 6 −+ pseudoscalar quantum numbers J =0 . In combin- tron state of the proton-neutron SU(2) doublet funda- ing this fact with the Eq.(92), the existence of the pseu- mental representation (with gauged isospin symmetry): doscalar η-meson state with J PC = 0−+ and mass m m m 1.29MeV . Where m and m are the − gap ≡ n − p ≈ p n mη = 1290MeV in the QCD vacuum is established. masses of proton and neutron respectively ([47],P.152). It − If we take the dynamically induced coupling constant is to be noted here that mn mp is the transition (ex- 2 − in the second part of the Eq.(28) (i.e. α2 = 3 ) as the citation) energy needed to transform neutron into pro- fundamental charge of QCD vacuum attributed to the ton ([23], P.548). In this picture, the mass gap is noth- − charm-quark (and taking into consideration Eq.(92)), ing but the energy difference between these two states in − then we can say that charm-quark also exist in the QCD the isospin space. From the foregoing, the approximate vacuum. Thus, the fundamental quantities of the QCD SU(2)V isospin symmetry of the strong nuclear force is vacuum are η meson (one of the examples of hadrons) dependent on the non-vanishing of m , and hence the − gap and charm quark. Based on this understanding, we color-singlet mass spectrum of the QCD matter must de- − posit that the final hadronic state of strong gravity theory pend on it. is charmed (i.e. m = mη = mc = 1290MeV ). Thus Eq.(93) becomes In the next subsection, we establish the exis- 3 tence of mass gap within the formulation of strong m = 10 (mn mp) = 1290MeV (95) gravity (by using the vector sugroup (i.e. isospin × − symmetry SU(2)V ) of the custodial symmetry in and the Eq.(77)); and also justify the validity of us- ing the dynamically induced coupling constant m = m m 1.29MeV (96) 2 gap n − p ≈ (α2 = 3 ) as the fundamental charge of the QCD vacuum. It is to be recalled that the fundamental charge (of U(1) and SU(2) gauge fields) is related to the elec- troweak coupling constants via the Weinberg-Salam geo- C. Mass Gap metric relations: e = g1 cos θw = g2 sin θw and cos θw = mW /mZ [56]. Where g1 and g2 are the gauge cou- QCD is widely accepted as a dynamical quantum gauge plings of U(1) and SU(2) gauge fields respectively. θw theory of strong interactions not only at the fundamental is the mixing angle, e is the fundamental charge, mW quark-gluon level, but also at the hadronic level. In this is the mass of W -boson and mZ is the mass of Z- picture, any color-singlet mass scale parameter must be boson. By using mW = 80.385GeV,mZ = 91.1876GeV 0 expressed in terms of the mass gap [53]: [57],e = α2 = 2/3, we have θw 28.17 and g2 = 0.6666666667/0.4720892507 = 1.4121623522≈ . This is the nucleon coupling constant for the two-flavor (i.e. pro- m = const mgap ton and neutron) SU(2) representation. The value of g × 2 m = const mgap = 1290MeV (93) (= 1.4121623522) is to be compared with the nucleon × axial coupling constant computed from two-flavor SU(2) where const. denotes arbitrary constant. lattice QCD: gA =1.412(18) [58]. In particle physics, particles that are affected equally In the next subsection, we demonstrate that the values by the strong force but having different charges, such as of mgap and Tc do not only play a very important role protons and neutrons, are treated as being different states in the Big Bang nucleosynthesis but are also part of the of the same nucleon-particle with isospin values related primordial constituents of the QCD vacuum. 14

D. Big Bang Nucleosynthesis (BBN) By using the values of αs and m, we proceed to solve Eqs.(19) and (34) completely. From Eq.(34), we have BBN refers to the production of relatively heavy nuclei k2 m 32πG m from the lightest pre-existing nuclei (i.e., neutrons and F = N × ≡ 16π 16π protons with mgap = 1.29MeV ) during the early stages −19 of the Universe. Cosmologists believe that the necessary F =2GN m =2.580 10 (102) × and sufficient condition for nucleosynthesis to have oc- curred during the early stages of the universe is that the Eq.(102) is to be compared with the ratio of the proton mass to the Planck mass scale ( Mproton 10−19). value of equilibrium neutron fraction (Xn) or the neu- MP lanck ≈ −1 tron abundance must be close to the optimum value, i.e., By using Eq.(29) and the value of Gf ( 1GeV [2], ≈ X 50% ([23], P.550). In fact, the value of X at the P. 2668), the last part of Eq.(33) becomes n ≈ n time t = 0 was calculated to be Xn = 0.496 = 49.6% β =3.162 108GeV −1 (103) ([23], P.549). 1 × The equilibrium neutron fraction for temperature T & One of the properties of the confining force is the 3 1010K is given as ([23], P.550): × notion of ”dimensional reduction” which suggests that −1 the calculation of a large planar Wilson loop in D = 4 E/kT Xn 1+ e (97) dimensions reduces to the corresponding calculation in ≈ h i D = 2 dimensions. In this case, the leading term for where E = mgap = 1.29MeV. By using natural unit the string tension is derived from the two-dimensional approach (i.e., setting the Boltzmann constant k = 1) strong-coupling expansion ([59], P.49-50). and using the value of critical temperature (T = Tc = Following this line of reasoning, αs is made into a di- 129MeV ), Eq.(97) reduces to mensionful coupling (dimensional transmutation) as fol- lows: 0.01 −1 Xn 1+ e = 49.75% (98) ≈ σ α [m]4−D =0.1797 (1.29GeV )2 ≡ s × The value in the Eq.(98) is compatible with the value σ =0.299GeV 2 (104) obtained at the time t = 0 (i.e., Xn = 49.6%) , and is approximately equal to the optimum value (X 50%). Note that αs is dimensionless (as expected) only in four n ≈ This can only mean two things: (i) mgap and Tc existed dimensions, but here we use D = 2 in order to obtain at time t = 0 of BBN processes. (ii) These two quanti- the Wilson-like string tension (which represents the ties are the fundamental quantities of QCD / quantum geometry of the Weyl’s action because it is rotationally vacuum. symmetric). Eq.(104), which is called string tension, is According to the detailed calculations of Peebles and to be compared with the value σ =0.27GeV 2 [60]. With Weinberg, the abundance by weight of cosmologically these values, the confinning potential (Vconf )/linearly ris- produced helium is given as ([23], P.554): ing potential in the Eq.(19) reduces to

XH4e =2Xn (99) Vconf (r)= σr (105) By combining Eqs.(98) and (99), we have

XH4e = 99.5% (100) and the perturbative aspect (Vpert) of strong gravity (Eq.(34)) becomes Eq.(100) confirms the validity of Eq.(99), namely, that the total amount of neutrons before nucleosynthesis must F 4 (Fe−β1/r) be equal to total amount of helium abundance after the Vpert(r)= (106) nucleosynthesis. r − 3 r + The threshold for the reaction p + νe n + e is at → Where the color factor (CF )/Casimir invariant associ- me + mgap = 1.8MeV ([23], P.544). Thus the mass of ated with gluon emission from a fundamental quark electron (me) is me =0.51MeV. present in the Eq.(106) for SU(3) gauge group (with− The invariance of the mass gap is supported by the N = 3) is given as − following transitions ([23], P.548): 1 1 4 E E = m for n + ν p + e− CF = N = (107) e − ν gap ←→ 2 − N 3 +   Eν Ee = mgap for n + e p + ν − ←→ and E + E = m for n p + e− + ν (101) ν e gap ←→ n ∞ β1 r Eq.(101) clearly shows that mass gap is invariant under e−β1/r = ( 1)n (108) −  n! crossing-symmetry. n=0 X 15

eff Hence the effective pure Yang-Mills potential (VY M accounts for the self-interaction between the gluons ( the (r)) of strong gravity theory (from Eqs.(105) and (106)) fons et origo of nonlinearity in the Yang-Mills theory). is Thus, strong gravity theory is a gauge theory: we men- tion in passing that Eq.(112) is also obtainable from the eff VY M (r)= Vpert(r)+ Vconf (r) Eq.(109). −β1/r In the next subsection, we prove that the Einstein’s eff F 4 (Fe ) V (r)= + σr (109) theory of gravity can also be derived from the same Y M r − 3 r equation (Eq.(27)) that gave rise to the Eq.(106).

VIII. GAUGE-GRAVITY DUALITY B. Effective Einstein General Relativity In this section, we show that strong gravity theory pos- sesses gauge-gravity duality property. So far, we have been dealing with the short-range be- havior of the strong gravity theory. In this subsection, A. NRQED and NRQCD Potentials we take a giant step towards deriving the Einstein GR entirely from the strong gravity formulation. To set the The perturbative non-relativistic quantum electro- stage, we rewrite Eq.(27) as: dynamics (NRQED) that gives rise to a repulsive 4 µν 2 Coulomb potential between an electron-electron Ieff = d x√ g α1Rµν R α2R − − − − pair is due to one photon exchange, and this repulsive Z
Coulomb potential is given by [61]: d4x√ gk−2α R − 3 α Z V (r)= e (110) QED r By using Eq.(28), the above equation becomes

α(0) 4 µν 2 2 where the QED running coupling αe = 2 . I = d x√ g 2R R R 1− (Q ) eff − − µν − 3 − α(0) 1/137 and (Q2) are the vacuum polarization Z   ≈ insertions [62]. Similarly, the perturbative componentQ of 2 d4x√ gk−2R (114) Q − the NRQCD potential between two gluons or between a Z quark and antiquark is given as [61]:

4 α (r) 1. The Matter Action V (r)= s (111) QCD −3 r Without using any rigorous mathematics, we would where the strong running coupling αs(r) must expo- like to show that the part of the Eq.(114) containing the nentiate in order to account for the nonlinearity of the quadratic terms is in fact the matter action (IM ). From gluon self-interactions. Eq.(16), we have The total color-singlet NRQCD potential is ([61], P.273 & [63], P.39): 4 µν 2 2 IM d x√ g 2Rµν R R = 4 α (r) ≡ − − 3 V (r)= s + kr (112) Z   QCD −3 r d4x√ gC Cµναβ − µναβ Obviously, Eq.(106) contains both the NRQED po- Z tential (Eq.(110)) and NRQCD potential (Eq.(111)). I = d4x√ gC Cµναβ (115) M − µναβ Hence the perturbative/short-range aspect of the strong Z gravity theory (derived completely entirely from the The fact that Weyl Lagrangian density (C Cµναβ ) broken-scale-invariant Weyl’s action in the Eq.(27)) uni- µναβ is a conserved quantity due to its general covariance prop- fies NRQED and NRQCD with one single coupling con- erty means that we can write stant F : µναβ 1 4 e−β1/r δ Cµναβ C =0 (116) Vpert(r)= F (113) r − 3 r
  This ensures the conservation of energy-momentum. By using the principle of stationary action on the It is important to note that the QCD part (second Eq.(115) and taking Eq.(116) into consideration, we have term) of the Eq.(113) is QED-like (first term) apart from the color factor 4/3 which shows that there is more 1 − δI = d4x√ g gµν C Cµναβ δg (117) than one gluon and the exponential function which M 2 − µναβ µν − − Z
16

Recall that the energy-momentum tensor is defined as [64]: δ √ gR = √ gR δgµν + Rδ√ g + √ ggµν δR − − µν − − µν 2δ (√ g M ) 2δ M (126) Tµν − − L = − L + gµν M (118)
≡ √ g δgµν L − λ λ where M is the matter conserved Lagrangian density. δRµν = (δΓµλ);ν (δΓµν );λ (127) Using theL Weyl conserved Lagrangian density, we have −

µναβ 2δ Cµναβ C T = − + g C Cµναβ (119) ∂ ∂ µν δgµν µν µναβ √ ggµν δR = (√ ggµν δΓλ ) (√ ggµν δΓλ )
− µν ∂xν − µλ −∂xλ − µν
(128) By using Eq.(116), Eq.(119) reduces to

µναβ Tµν = gµν Cµναβ C or 1 µν µν µν µναβ δ√ g = √ gg δgµν (129) T = g Cµναβ C
(120) − 2 − Clearly Eq.(120) is a conserved (due
to Eq.(116)) sym- metric (due to the presence of gµν ) tensor ([23], P. 360); µν µρ νσ δg = g g δgρσ (130) and its nonlinearity represents the effect of gravitation − on itself. To deal with this nonlinear effect, Princi- Eq.(128) vanishes when we integrate over all space ple of Equivalence is normally invoked, in which any ([23], P. 364). Thus, for the pure gravitational part,we point X in an arbitrarily strong gravitational field is the have same as a locally inertial coordinate system such that 1 gαβ(X)= ηαβ ([23], P. 151). δIG = √ g Hence Eq.(117) becomes 16πGN − × Z µρ νσ 1 µν 4 1 Rµν g g δgρσ g R δgµν d x δI = d4x√ gT µν δg (121) − 2 M 2 − µν   Z 1 µν 1 µν 4 δIG = √ g R g R δgµν d x (131) Eq.(121) is the equation of energy-momentum tensor 16πGN − − 2 for a material system described by matter action [23]. Z   From Eqs. (121), (125) and (131), we have

2. Pure Gravitational Action 1 µν 1 µν δIG = δIM = R g R = − ⇒ 16πGN − 2 −2 1   By using the value of k (= ) from Eq.(29), the 1 µν 32πGN T linear term part of the Eq.(114) is written as − 2 µν 1 µν µν δIG + δIM = R g R +8πGN T =0 (132) I = 2 d4x√ gk−2R − 2 G − − Z By using 1 4 IG = d x√ gR (122) −16πGN − γδ Z gαγgβδA = Aαβ (133) We can therefore write and redefining the resulting indices as µ and ν, we get 1 I = I + I (123) R g R = 8πG T (134) eff M G µν − 2 µν − N µν By using the general covariance property of Weyl’s ac- It should be noted that all terms in the Eq.(134) are tion, we can write already present in the Eqs.(15) and (18), as such the un- derlying symmetry (general coordinate invariance) of the δI = δI + δI =0 (124) eff M G Eq.(7) is still preserved in a covariant manner. Eq.(132) However, this can only be true iff ensures the conservation of energy-momentum (which is a statement of general covariance [23], P. 361). Thus,

δIM + δIG =0 δIG = δIM (125) the Weyl’s action given in the Eq.(123) would be sta- ⇐⇒ − tionary / invariant with respect to the variation in gµν , µν The curvature scalar R can be defined as g Rµν , and iff Eq.(132) holds. Interestingly, it holds because the following standard equations are valid ([23], P.364): Eq.(132) is the Einstein field equations, and hence the 17 full Weyl’s action is stationary with respect to the varia- D. Breaking of Chiral Symmetry in Strong Gravity tion in gµν .This is precisely what we expect: that the in- Theory variance of Weyl’s action is maintained by inducing gen- eral relativity. Hence the general covariance property of QCD admits a chiral symmetry in the advent of Eq.(7) has been revealed because the statement that δIeff vanishing quark masses. This symmetry is broken should vanish is ”generally covariant”, and this leads to spontaneously by dynamical chiral symmetry; and the energy-momentum conservation ([23], P. 361). broken explicitly by quark masses. The nonpertur- Conclusively, the perturbative aspect of strong grav- bative scale of dynamical chiral symmetry breaking is ity theory (i.e. Eq.(27)) possesses quantum gauge theory around Λx 1GeV [65]. Apparently, the chiral sym- ≈ (Eq.(106)) and gravity theory (Eq.(134)); thus proving metry in the strong gravity is broken spontaneously by −1 the existence of gauge-gravity duality in the strong grav- its inherent dynamical chiral symmetry breaking Gf = ity formulation. ΛQCD = Λx 1GeV . In much the same spirit, the calculated value≈ of mass scale of the theory reverberates the existence of the approximate symmetry in the strong interaction: m =1.29GeV and G−1 =Λ 1GeV . f QCD ≈ C. Ultraviolet Finiteness IX. CONFINEMENT AND ASYMPTOTIC The strong gravity program adopts the Wilsonian FREEDOM viewpoint on quantum field theory. Here the basic in- put data to be fixed ab initio are the kind of quantum In the past few decades it became a common knowl- fields (i.e., gluon fields) carrying the theory’s degrees of edge that confinement is due to a linearly rising potential freedom (one graviton equals two gluons: BCJ construc- between static test quarks / gluons in the 4-dimensional tion), and the underlying symmetry (spherical/rotational pure Yang-Mills theory (see Eq.(105)). The fact that con- symmetry). The fact that two gluons are used to con- finement (i.e. non-perturbative aspect of QCD) is a sim- struct spacetime metric means that the resulting gravity ple consequence of the strong coupling expansion means must be point-like. This fact is encoded in the three- that an infinitely rising linear potential becomes highly dimensional Dirac delta functions in the first part of the non-trivial in the weak coupling limit of the theory. This Eqs.(19), and (30). The point-like nature of gravity in short-scale weak coupling limit is called asymptotic free- this picture is the origin of ultraviolet (UV) divergence. dom [66, 67]. By all standards, these two properties of The question here is: Is Eq.(106) (the effective poten- QCD contradict all previous experience in physics with tial carried by Eqs.(27)) UV finite, or perturbatively strong force decreasing with distance. The asymptotic renormalizable? This question can be answered by us- freedom part of the paradox has been correctly resolved ing Eqs.(106) and (108): [27, 28], leaving out the hitherto unresolved color confine- ment property of the non-perturbative QCD regime. As we have remarked previously, a complete theory of strong interaction should be able to explain these two properties F 4F β β2 β3 β4 V (r)= 1 1 + 1 1 + 1 ... of QCD simultaneously (i.e., the dominance of asymp- pert r − 3r − r 2r2 − 6r3 24r4 − totic freedom at the small scale distances (quark-gluon   (135) regime) and the emergence of infrared slavery (confine- It is to be noted, from subsection C of section III, ment) at long scale distances (hadronic regime)). These −1 dual properties of QCD are succinctly depicted in the that the expression for β1( with dimension of GeV E−1) contains inverse of boson fields dimension (E−−→1), Eq.(109). The linearly rising potential means that the potential and fermion fields dimension (G3/2 E−3/2). So it f between a static gluon-gluon pair keeps rising linearly as suffices to posit that β contains both boson−→ and fermion 1 one tries to pull the two constituents apart (see Eq.(105)). fields: A perfect replica of supersymmetric fermion-boson Thus they are confined in a strongly bound state [66]. field duality. Let us now test for the UV behavior of the Based on the dynamics of Eq.(105), an infinite amount Eq. (106): of energy would be required to pull the two constituents of bound glueball/meson state apart. Vpert(r) = [1 + + ...]=0 The resulting force of strong gravity theory is called −→ ∞ − ∞ − ∞ ∞ − ∞ ∞− r 0 Yang-Mills-Gravity force (FYMG(r)), because Eq.(7) (136) which gives rise to the confining potential is the Clearly Eq.(136) is a host of infinities, but they all −Weyl’s action for gravity ([6], P.322), and the− action in cancel out, thus rendering Eq.(106) UV finite. Hence, the Eq.(27) which gives rise to the perturbative QYMT strong gravity theory has UV regularity. Interestingly, also contains− Einstein-Hilbert action for gravity. To this is the main conclusion of the theories of supergravity explain− the behavior of this force at both small and large (”enhanced cancellations”). distance scales, we differentiate Eq.(109) with respect to 18 the gluon-gluon separating distance r (and taking into and F (r) = 2.6 1038GeV 2. The negative sign YMG − × consideration Eq.(107)): of FYMG(r) is the hallmark of the asymptotic freedom and the weakness of gravitational field (FYMG(r) < 0) −β1/r −β1/r F 1 CF e F C β e at the Planck scale! This would also disallow the forma- F (r)= − F 1 + σ YMG − r2 − r3 tion of singularity at the centre of a blackhole (see Fig.3
(137) for more details). Based on the foregoing, we therefore The summing graphs of strong gravitational gluody- assert that strong gravity theory is consistent and well- behaved down to Planck distance scale ( 10−19GeV −1) namics are shown in the Fig.2. The blue graphs are ∼ the graphs of the effective pure Yang-Mills potential . (Eq.(109)), while the red plots are the graphs of the Yang-Mills-Gravity force (Eq.(137)). It is easy to show that these equations possess UV asymptotic freedom (al- beit with tamable infinities) and infrared (IR) slavery behaviors of the QCD. For us to see these behaviors, the following facts are in order: (i) If the radial derivative of potential is positive, then the force is attractive. (ii) If the radial derivative of potential is negative, then the force becomes repulsive [68]. (iii) Since only color singlet states (hadrons)/ or dressed glueball can exist as free ob- servable particles, we multiplied the gluon-distance scale (in the Figs.2 and 3) by factor of 10 in order to convert gluon radius to the more observable hadronic radius (in line with the Eq.(87)). (iv) The graphs in the Figs.2 and 3 are plotted by using the highly interactive plotting software [69]. The strong interaction is observable in two areas: (i) on a shorter distance scale ( for 10−19GeV −1 r −1 ≤ ≤ 3.0277GeV ), FYMG(r) is repulsive (i.e., negative force) and reducing in strength as we probe shorter and shorter distances (up to Planck length (10−19GeV −1)). This makes Eq.(137) to be compatible with the asymp- totic freedom property of QCD, where the force that holds the quark-antiquark or gluon-gluon together de- creases as the distance between them decreases. Being FIG. 2: Summing graphs of strong gravitational gluodynam- a repulsive force (within the range 10−19GeV −1 r ics. 3.0277GeV −1 ), it would disallow the formation of quark-≤ ≤ antiquark / gluon-gluon singularity because the con- stituents can only come close up to a minimum distance scale at which the repulsive force would be strong enough to prevent further reduction in their separating distance. (ii) On a longer distance scale (r 3.0278GeV −1), A. Energy density of QCD vacuum ≥ FYMG(r) becomes attractive (i.e., positive force). Here F (r) does not diminish with increasing distance. Af- YMG The scale invariance of the strong gravity is broken ter a limiting distance (r = 104GeV −1) has been reached, at Λ 1GeV ([6]. P. 324). Hence the associated it remains constant at a strength of 0.299GeV 2 (no mat- QCD distance scale≈ would be given as r = G =1GeV −1. In ter how much farther the separating distance between the g f terms of the observable hadronic radius (see Eq.(87)), we quarks /gluons). Meanwhile, the linearly rising potential −1 −15 have rh = 10GeV =1.972 10 m =1.972 fm. The keeps on increasing ad infinitum (see the blue curve in × eff QCD potential at this distance scale is given as V = the Fig.3). This phenomenon is called color confinement Y M in QCD. The explanation is that the amount of work- 2.495761GeV from the Fig.3, and the energy density (ε) done against a force of 0.299GeV 2 (= 2.449 105N) is of the QCD vacuum is calculated as: enough to create particle-antiparticle pairs within× a short distance r = 104GeV −1 = 1.972 10−12m than to keep eff × VY M 2.495761GeV 3 on increasing the color force indefinitely. ε = 3 = 3 3 =0.325GeV / fm (138) (rh) (1.972) fm By using [69], we demonstrate that Eqs.(109) and (137) are consistent and well-behaved down to the Planck scale: (i) At r = 10−19GeV −1 = 1.972 10−35m (Planck Eq.(138) is to be compared with the value calculated × length), V eff (r) = 2.58 1019GeV (Planck energy) from the Lattice QCD (ε 0.33GeV /fm3) ([70], P.54). Y M × ≈ 19

the quantum level of the theory because Yang-Mills field shows quantum behavior that is very similar to its clas- sical behavior at short distance scales ([71], P.1-2). How- ever, the Maxwell’s theory must be replaced by its quan- tum version (i.e. QED; photon-electron interaction), and the nonlinear part (AΛA) must now describe the self- interaction of gluons (which is the source of nonlinear- ity of the theory). The fact that the physics of strong interaction is described by a non-abelian gauge group G = SU(3) (i.e. QCD), suggests immediately that the potentials of the four-dimensional quantum Yang-Mills field must be the sum of the linear QED (dA) and nonlin- ear QCD (AΛA) potentials at quantum level. Thus the first composite hurdle for any would-be solution of the problem to cross is to: (1) obtain QED +QCD potential at short distances with a single unified coupling constant. (2) The two potentials must perfectly explain the indi- vidual physics of QED and QCD at the quantum scale. (3) The two potentials must be obtained from a four- dimensional quantum gauge theory. To surmount this composite hurdle, one must first of all establish the existence of four-dimensional quantum gauge theory with FIG. 3: Graphs of pure Yang-Mills potential (in blue) and gauge group G = SU(N), and then every other thing will Yang-Mills-Gravity force (in red). follow naturally.

X. EXISTENCE OF QUANTUM YANG-MILLS 1. Jaffe-Witten Existence Theorem ([71],P.6) THEORY ON R4 The official description of this (i.e. Yang-Mills exis- The existence of quantum Yang-Mills theory on R4 tence and mass gap) problem was put forward by Arthur (with its characteristic mass gap) is one of the seven (now Jaffe and Edward Witten. Their existence theorem is six) Millennium prize problems in mathematics that was briefly paraphrased as follows: The existence of four- put forward by Clay Mathematics Institute in 2000 [71]. dimensional quantum gauge theory (with gauge group The problem is stated as follows: SU(N)) can be established mathematically, by defining Prove that for any compact simple gauge group G = a quantum field theory with local quantum field operators SU(N), a fully renormalized quantum Yang-Mills theory in connection with the local gauge-invariant polynomials, exists on R4 and has a non-vanishing mass gap. in the curvature F and its covariant derivatives, such as T rFij Fkl(x). In this case, the correlation functions of the quantum field operators should be in agreement A. Solution-plan with the predictions of perturbative renormalization (i.e. the theory must have UV regularity) and The first thing to note here is that Yang-Mills theory asymptotic freedom (i.e. the weakness of strong is a non-abelian gauge theory, and the idea of a gauge force at extremely short-distance scale); and there theory emerged from the work of Hermann Weyl [72] (the must exist a stress tensor and an operator product expan- same Weyl that formulated the Weyl’s action that was sion, admitting well-defined local singularities predicted used in the formulation of strong gravity theory, based by asymptotic freedom. on the Weyl-Salam-Sivaram’s approach [6]). By using the eye of differential geometry, we observed The Maxwell’s theory of electromagnetism is one of that the solution to the problem is concealed in the math- the classical examples of gauge theory. In this case, the ematical structures rooted in the differential geometry . gauge symmetry group of the theory is the abelian group In other words, the above-stated existence theorem is the U(1). If A designates the U(1) gauge connection (lo- mathematical description of the strong gravity formula- cally a one-form on spacetime), then the potential of the tion. field is the linear two-form F = dA. To formulate the classical version of the Yang-Mills theory, we must re- 4 place the gauge group U(1) of electromagnetism by a 2. R −Weyl-Salam-Sivaram Theorem[6] compact gauge group SU(N), and the potential aris- ing from the field would be a generalized form of the The Weyl-Salam-Sivaram theorem is in fact the geo- Maxwell’s: F = dA + AΛA. This formula still holds at metrical interpretation of the Jaffe-Witten existence the- 20 orem. In the following, the local quantum field operators but the energy difference between the two sub-states of a b are the two strong tensor fields (Gµ(x) and Gν (x); two the proton-neutron configuration: mgap = mn mp gluons forming double-copy construction) used to con- 1.29MeV. Hence the mass formula of QCD (Eq.(93))− and≈ struct the spacetime metric in the section III of this the stable Higgs boson mass (see next section) must be paper. These local quantum fields have a direct connec- expressed in terms of this mass gap. a b tion (via g = det(GµGν ηab)) with the gauge-invariant Conclusively, the two gauge groups that are needed local polynomials in the curvature C and its covariant to accurately describe the solution to this Millennium αβγδ derivatives: √ gCαβγδC (x). Note that ”T r” in prize problem are SU(3) for the establishment of the the Jaffe-Witten− existence theorem denotes an invariant existence theorem and SU− (2) for describing the mass quadratic form on the Lie algebra of group G. Simi- gap of the solution.−Hence, the− Weyl-Salam-Sivaram larly, √ g in the Weyl-Salam-Sivaram theorem denotes existence theorem of strong gravity puts quantum an invariant− quadratic form on the gauge group SU(3). gauge field theory (QFT) on a solid mathematical The correlation function in this case is nothing but the footing of the differential geometry; in this sense, spacetime metric (gµν (x)) constructed out of the two lo- QFT is a full-fledged part of mathematics. a b cal quantum fields (Gµ(x) and Gν (x)), and used as a function of the spatial cum temporal distance between these two random variables (gluons). We have painstak- XI. STABILITY OF VACUUM: A HINT FOR ingly demonstrated that this spacetime metric agrees, at PLANCK SCALE PHYSICS FROM mH = 126GeV short distance scales, with the predictions of asymptotic freedom (i.e. the weakness of strong force at extremely The 126GeV Higgs mass seems to be a rather special short distance scales (see section IX)) and perturba- value, from all the a priori possible values, because it just tive renormalization (i.e. the existence of UV regular- at the edge of the mass range implying the stability of ity of the theory at short distances; the theory should Minkowski vacuum all the way down to the Planck scale be able to regularize its own divergences at extremely −1 19 [74]. If one uses the Planck energy (GN 10 GeV ) as short distance scales, say, r = 0 (see subsection C the cutoff scale, then the vacuum stability≈ bound on the of section VIII)). There also exist a stress energy- mass of the Higgs boson is found to be 129GeV. That momentum tensor (Eq.(17)), and field product expan- is, vacuum stability requires the Higgs boson mass to sion (Eq.(18)), having local singularities encoded in the be mH = 129GeV [75] . A new physics beyond SM is three-dimensional Dirac delta functions (Eqs. (19) and thus needed to reconcile the discrepancy between 126GeV (30)) predicted by asymptotic freedom. Overall the and 129GeV mass of Higgs boson. The first thing to broken-scale-invariant Weyl action (Eq.(27)) is observe here is that the vacuum stability bound on the the required perturbative four-dimensional quan- mass of Higgs boson (mH = 129GeV ) has exactly the tum gauge field theory with its inherent gauge same ”number-structure” with the values that we have group SU(3) that gives rise to color/Casimir fac- been working with in this paper. tor 4/3 (Eq.(107)). However, for this statement to be By using Eq.(93), we can write valid the theory must possess both QED and QCD po- tentials (i.e. F = dA + AΛA). Happily, the theory does mH = const. m (139) possess these potentials with a single coupling constant × (see Eqs. (106), (110), (111) and (113)). Comparing the energy scale of the pure Yang-Mill propagator in the Eq.(29) (k−2 =1 1017GeV ) with the The fact that the scale invariance of Weyl action is −1 19 × −1 Planck scale ( G 10 GeV ) shows a magnitude broken at the strong scale Λ = G 1GeV ([6], N QCD f ≈ difference of 10≈2. By using≈ this value as our constant (i.e. −1 P.324) which is equal to its dynamical chiral sym- G − N metry breaking scale [65] is a clear indication of the const. = k−2 ), we get exactly mH = 129GeV : existence of proton as the− fundamental hadron of the −1 theory. In this case, one must therefore investigate the G m = m N = 129GeV (140) ground state (neutron state) of the proton state us- H k−2 ing isospin symmetry. But for this to be possible, the   gauge group that describes isospin symmetry must ex- Eq.(140) is very important because: (1) it shows the ist within the framework of the theory. This is where coupling of Higgs mass (mH ) to the fundamental mass, custodial symmetry (Eq.(77)) kicks in. The vector sub- and mass gap of the QCD vacuum (m = 1290MeV = group of custodial symmetry is in fact the isospin sym- 103 m ). (2) It connects Higgs mass to the Planck × gap metry: SU(2)L SU(2)R SU(2)V [73]. This isospin energy scale. To show the vacuum stability property of symmetry then× demands−→ that the Hamiltonian (H) of the Eq.(140), we eliminate the fundamental mass of the proton-neutron state must be zero. However, the near QCD vacuum by using the value of critical temperature mass-degeneracy of the neutron and proton in the SU(2) from Eq.(89) (T m = 10Tc): doublet representation points to an approximate isospin ≡ symmetry of the Hamiltonian describing the strong inter- G−1 m = T N = 129GeV (141) action [54, 55]. The mass gap in this picture is nothing H k−2   21

Obviously, T >Tc (see subsection A of section VII). are Dirac particles (with particles and antiparticles being This is the well-known vacuum stability condition in different objects thereby conserving the lepton number) the second-order phase transition theory; while the or Majorana particles (with particles and antiparticles condition for vacuum instability is T

D> 4) ([47], P. 216): where

′ ′ m 2 − 2 4 D T † T (1 2r ) ζ d = MX gαβ(LαLτ2Φ)C (Φ τ2LβL)+H.c. (144) g = − + (153) L 00 (1 + m )2 r(1 + m )2 Xαβ 2r 2r

where MX is a heavy mass ( of a single color triplet Higgs scalar ) characteristic of the symmetry-breaking ζ(1 + m ) g = g = 2r (154) scale of the high-energy unified theory, D is called a 01 10 − r1/2 dimension-D operator and its value in this case is D =5. gαβ is a yet-unknown symmetric 3 3 matrix of coupling constants. With D =5, Eq.(144) becomes× m g =(1+ )4 (155) 11 2r ′ ′ 1 T † T 5 = gαβ(LαLτ2Φ)C (Φ τ2LβL)+ H.c. (145) L MX αβ X 2 g22 = g11r (156) The electroweak symmetry breaking VEV (= υ = 246GeV [91]) of the Higgs field leads to the Majorana neutrino mass term([47], P. 216); 2 2 2 g33 = g22 sin θ = g11r sin θ (157) 2 ′ ′ M 1 υ T † mass = gαβ ναLC νβL + H.c. (146) L 2 MX The quantity m represents an effective gravitational αβ X mass, and ζ is an electric-charge dependent parameter From Eq.(146), the Majorana mass matrix has ele- [92]. Since neutrinos are electrically neutral, we set ζ to ments ([47], P. 216) zero: ζ =0. Hence Eq.(152) reduces to

2 g00 0 0 0 L υ Mαβ = gαβ (147) 0 g11 0 0 MX gµν = (158)  0 0 g22 0  with ([47], P. 208)  0 0 0 g33    L L and Mαβ = Mβα (148)

Eq.(148) is the reason why the gαβ matrix must be m 2 symmetric. With α = β =0, 1, 2, Eq.(147) reduces to (1 2r ) g00 = − m 2 (159) (1 + 2r ) 2 L υ M00 = g00 (149) MX g01 = g10 =0

2 L υ This matrix (Eq.158) has Euclidean space signature M11 = g11 (150) MX ++++ . It’s worth noting that for us to impose Lorentz signature on the above matrix, we must invoke the Levi- Civita indicator on the matrix to account for the special 2 relativity in the limiting case, and to also transform the L υ M22 = g22 (151) metric from 4 dimensions to 3+1 dimensions. It doesn’t MX matter whether we insert the Lorentz signature before or (It is worth noting that if all the diagonal elements of after solving the Eq.(158), due to the fact that it is a gαβ are all 1’s, then the Eqs. (147) and 149-151 reduce diagonalized matrix [93]. to Eq.(74).) The fact that Majorana neutrino Lagrangian preserves The gravitational potential (gµν ) which is capable of CP symmetry means that it possesses symmetry axis representing a combined gravitational and electromag- (θ = 0). The reason why Majorana neutrino Lagrangian netic field outside a spherically symmetric material preserves CP symmetry is that Majorana particles are in- distribution is given as [92]; variant to CP transformation (because Majorana particle = Majorana antiparticle) ([47], P. 203-205). g00 g01 0 0 Consequently (by setting θ = 0), Eqs.(157-158) reduce g10 g11 0 0 to gµν = (152)  0 0 g22 0  g =0 (160)  0 0 0 g33  33   23

we further impose the condition that the common ori- gin is centred at zero (i.e., x0 C0 = 0), then Eq.(168) g00 0 00 reduces to − 0 g11 0 0 gµν = (161)  0 0 g22 0  3 2 2 2 2 2 0 0 00 r = (xi Ci) = (x1 C1) +(x2 C2) +(x3 C3)   − − − −   i=1 Hence, Eq.(147) becomes X (169) where x C , x C and x C are the radii of the 1 − 1 2 − 2 3 − 3 g00 0 00 spheres. By using unit sphere formalism individually on 2 L υ 0 g11 0 0 the three sphere, Eq.(169) reduces to Mαβ = (162) MX  0 0 g22 0  0 0 00 3   2 2   r = (xi Ci) =3 (170) − By solving Eq.(159) completely for mass m, we have i=1 X 1/2 Thus, Eq.(167) becomes 2r(1 g00 ) m = − 1/2 (163) (1 + g00 ) g22 = 11.6766 (171)

m 2 (1+ 2r ) Multiplying Eq.(159) by m 2 and solve the resulting and Eq.(161) reduces to (1+ 2r ) equation completely for mass m;

1/2 1/2 0.1797 0 0 0 m = 2r[1 (g11g00) ] (164) ± − 0 3.8922 0 0 gµν = (172) where sign in Eq.(164) leads to the same result. By  0 0 11.6766 0  ± comparing Eq.(163) with Eq.(164), we get  0 0 00    2 16 1/2 2 With MX =1.63 10 GeV (see subsection C of sec- 1 (1 g ) 2 00 × υ g11 = 1 − 1/2 (165) tion VI) and υ = 246GeV [91], = 3.7meV (see g00 " − (1 + g )2 # MX 00 Eq.(79)). Hence Eqs.(149-151) reduce to Since our calculated value for g00 is g00 =0.1797, thus Eqs.(156) and (165) reduce to m0 =0.665meV (173)

g11 =3.8922 (166)

m1 = 14.401meV (174) 2 g22 =3.8922r (167)

We now look for an ingenious way to eliminate r2 in Eq.(167). It is tempting to straightforwardly use unit m2 = 43.203meV (175) sphere formalism but this direct approach will not work L L L L because M is a linear superposition of three different where m0 M , m1 M ,m2 M and m3 0. αβ ≡ 00 ≡ 11 ≡ 22 ≡ neutrino masses, albeit from the same source. The best And Eq.(162) reduces to mathematical approach that we can use to circumvent this problem is the 3-sphere formulation (note that this 0.1797 0 0 0 0 3.8922 0 0 approach is anchored on the fact that 3-sphere is a sphere M L =3.7meV (176) in 4-dimensional Euclidean space) [94, 95]: αβ  0 0 11.6766 0   0 0 00  3   2 2 2 2 r = (xi Ci) = (x0 C0) + (x1 C1) + For the purpose of book-keeping, we set m m , − − − 0 ≡ 1 i=0 m1 m2 and m2 m3. It is evident from Eqs.(173-175) X 2 2 ≡ ≡ (x2 C2) + (x3 C3) (168) that m1 < m2 < m3, which is clearly a Normal Mass − − Hierarchy signature. The validity of which can also be We turn Eq.(168) on its head by using it to represent confirmed by considering the approach of M. Kadastik et three spheres (representing three types of neutrino) with al [96]. common origin. This reduces Eq.(168) to ordinary linear 2 2 2 superposition of three spheres (in two-dimension, they m1 + m2 +3m3 N1 = − 2 2 (177) reduce to circles) with common origin / source. Suppose 2m1 + m2 24

with 0.058269eV. To obtain an accurate result, we must con- vert the calculated value of sum of the neutrino into two N1 > 1 normal mass hierarchy decimal places in conformity with the denorminator of → N < 1 inverted mass hierarchy Eq.(183). Thus 1 → N 1 degeneratemasses (178) 1 ≈ → 3 Taking the values of m1,m2, and m3 from Eqs.(173- m 0.06eV (184) 175), Eq.(177) gives the value N 28, which satisfies i ≈ 1 i=1 the criterion of normal mass hierarchy≈ in Eq.(178). X The mass-squared difference is defined mathematically Consequently, as 0 2 Ων h 0.00064 (185) ∆m2 = m2 m2 (179) ≈ ij i − j Eqs.(184) and (185) are the fiducial parameter values where i > j. Based on the Eq.(179) (and taking into that have been taken to be valid for the background Cos- account Eqs.(173-175)), we have the following equations: mology to be consistent with the most recent cosmologi- cal measurements [99]. Here, it turns out that the South ∆m2 = m2 m2 =2.06 10−4eV 2 (180) 21 2 − 1 × Pole Telescope (SPT) cluster abundance is lower than preferred by either the WMAP9 or Planck+WMAP9 po- larization data for the Planck base ΛCDM model; but 2 2 2 −3 2 assuming a normal mass hierarchy for the sum of of the ∆m31 = m3 m1 =1.87 10 eV (181) − × neutrino masses with mν 0.06eV ([46], P.237 & 239) the data sets are found to be≈ consistent at the 1.0σ level for WMAP9 and 1.5σPlevel for Planck+WMAP9 [100]. 2 2 2 −3 2 ∆m32 = m3 m2 =1.57 10 eV (182) Obviously, our calculations confirm that the Planck base − × ΛCDM model’s prediction of sum of the neutrino masses is correct. 1. Experimental Test

(1) The combined results of all solar experiments with XIII. DARK ENERGY Super-Kamiokande-I zenith spectrum and KamLAND data give ∆m2 = ∆m2 = 2 10−4eV 2 at 99.73% C. sol 21 × For the strong gravity theory to be a complete theory L.[97]. This experimental value is compatible with our of QCD and gravity, it must be tested okay at both Eq.(180): Thus confirming the validity of our m1 and m2 small and large scale distances. The large distance ,here, values. is the cosmological scale where ”dark energy” is domi- (2) From the atmospheric neutrino oscillation exper- nant Dark energy (ρ) is an unknown form of energy, iments, the bound on the mass of the heaviest neutrino which− was invented to account for the acceleration of the is m3 & 40meV [98]. This value experimentally confirms expanding universe. The observed value (upper limit) our value in the Eq.(175). We therefore assert that the of ρ is ρobserved (2.42 10−3eV )4 [101] A major values of our m1,m2 and m3 conform with the experi- outstanding problem≈ is that× most quantum− field theo- mental data. ries naively predict a huge value for the dark energy: the prediction is wrong by a factor of 10120 [102]. The origin of the problem is now clear to us: Eqs.(118) and B. Observational Test (120) clearly show that the energy-momentum tensor is related to the invariant (Weyl) Lagrangian density, but The energy density of light massive neutrinos is given not to the total energy density of a vacuum, which is not as ([47], P. 590-591): operationally measurable, due to quantum fluctuations! The energy density of any given system, such as the 3 universe, is categorized into two parts: one is due to mi the true vacuum (ρ) and the other to the matter and Ω0 h2 = i=1 (183) ν 94P.14eV radiation (pressure (p)) present in the system. These two types of energy density are related by the energy- 0 2 where Ων h is the neutrino energy density (which is momentum tensor Tµν [23]: also known as the Gershtein-Zeldovich limt or Cowsik- 3 McClelland limit) and mi is the sum of the three ρ 0 0 0 i=1 0 p 0 0 T = (186) P 3 µν  0− 0 p 0  active neutrino masses. From Eqs.(173-175), m = i 00− 0 p i=1  −  P   25

( Note that by putting Eq.(186) into Eq.(134), two We understand, of course, that the energy of vacuum things happen: (1) The energy density of the true vac- is extremely large (due to quantum fluctuations) but the uum becomes negative, meaning repulsive gravity and (2) strong gravity and Majorana-neutrino Lagrangian the energy density of matter and radiation becomes pos- (a conserved quantity that encodes the informa- itive, meaning attractive gravity. These two results are tion about the dynamics of the universe) tell us compatible with the observations. The gravity of ordinary that it is only the effective Lagrangian of the universe 2 matter/energy is always attractive, while the gravity of that is physically measurable (i.e. υ /MX =3.7meV ). true vacuum (i.e., dark energy) is always repulsive.) Since it has been observationally confirmed that the ac- celeration of the expanding universe is controlled by the XIV. DARK MATTER OR LEFTOVER energy density of true vacuum (ρ),but not by the mat- YANG-MILLS-GRAVITY FORCE? ter/energy content of the universe, we can write (from Eq.(186)) A. The Galaxy Rotation Problem (GRP)

T00 = ρ (187) The GRP is the inconsistency between the theoretical By combining Eq.(120) with Eq.(187), we get Eq.(50): prediction and the observed galaxy rotation curves, as- 4 suming a centrally dominated mass associated with the ρ = g00[Evac] (188) observed luminous material. The direct computation of Note that the Weyl Lagrangian density scales as mass profiles of galaxies from the distribution of stars αβγδ 4 and gas in spirals and mass-to-light ratios in the stellar CαβγδC [Evac] (where Evac denotes the effective (Weyl) Lagrangian∼ of the vacuum), due to the scale in- disks, utterly disagree with the masses derived from the variance of the Weyl’s action in the Eq.(7) ([47], P.206). observed rotation curves using Newtonian force law of To see the repulsive nature of the dark energy, we com- gravity. Based on the Newtonian dynamics, most of the bine Eqs.(134) and (188) to get mass of the galaxy had to be in the galactic bulge near the center, and that stars and gas in the disk portion should ρ 0 0 0 orbit the center at decreasing velocities with increasing 0000 radial distance, away from the galactic center [103–105]: G = 8πG (189) µν − N  0000  This is achieved by equating the centripetal force experi- enced by the orbiting gas/stars to the Newton force law  0000    ([46], P.241): where G R 1 g R is the Einstein’s tensor. µν ≡ µν − 2 µν The negative sign in the Eq.(189) is the hallmark of the Fc = FN , repulsive nature of dark energy. It is to be noted that we G M have not invoked the presence of the famous cosmological v = N = v(r) 1/√r (191) constant (Λ) in the Eq.(134) because it is not needed r r ⇒ ∝ for the expanding or contracting universe ([46], P.232). where v is the speed of the orbiting star, M is the All what is needed for the accelerating expansion of the centrally dominated mass of the galaxy and r is the radial universe, as currently observed, is the Eq.(189); while the distance from the center of the galaxy. combination of the Eqs.(134) and (186) tell us that the However, the actual observations of the rotation curve universe will either expand (if the right-hand side of the of spirals completely disagree with the Eq.(191): the Eq.(134) is negative (ρ)) or contract (if the right-hand curves do not decrease in the expected inverse square side of the Eq.(134) is positive (p)). Albert Einstein root relationship. Rather, in most galaxies observed, one was right after all: the introduction of the fudged factor finds that v becomes approximately constant out to the (Λ) was his greatest blunder. You cannot out-einstein largest values of radial distance (r) where the rotation Einstein! curve can be measured ([46], P.241). A solution to this Using Eqs.(56) and (79), Eq.(188) becomes problem was to hypothesize the existence of a substan- ρ = (2.41 10−3eV )4 (190) tial invisible amount of matter to account for this inex- × plicable extra mass/gravity force that keeps the speed of Obviously, Eq.(190) compares favorably with the up- orbiting stars/gas approximately constant for extremely per bound value of the observed ρ (ρobserved = (2.42 large values of r. This extra mass/gravity was dubbled 10−3eV )4) [101]. × ”dark matter” [106]. It has long been suggested that the nonrelativistic mas- Though dark matter is by far the most accepted expla- sive neutrinos may give a significant contribution to the nation of the rotation problem, other alternatives have energy density (i.e. the so-called dark energy) of the uni- been proposed with varying degrees of success. The verse ([47], P.590). This statement has been confirmed to most notable of these alternatives is the Modified Newto- 2 be true via Eqs.(176) and (188): with Evac = υ /MX = nian Dynamics (MOND), which involves modifying the 3.7meV . Newton force law by phenomenologically adding a small 26 fudged factor α0: As a result of the Eqs.(137) and (194), the follow- ing facts emerge: (1) empty space/vacuum is permeated GN Mm with constant-attractive-gravitational force (dark mat- FMOND = 2 + α0 (192) r ter) with mass Mg = 546.809MeV . (2) The dark matter is stable on cosmological time scales due to the flat curve Within the central bulge of galaxy, the first term of the property of M for r (see red curve in the Fig.3). Eq.(192) dominates, and to the largest value of r where g (3) Newtonian dynamics−→ and ∞ Einstein’s GR need no mod- the rotation curve can be measured (the domain of dark ifications. (4) MOND is phenomenologically correct and matter), the second term dominates. MOND has had a happens to be compatible with YMG force law. remarkable amount of success in predicting the flat rota- tion curves of low-surface-brightness galaxies, matching the Tully-Fisher relation of the baryonic distribution, and the velocity dispersions of the small orbiting galaxies of XV. REPULSIVE GRAVITY AND COSMIC the local group [107]. INFLATION The ensuing fundamental question here is: Do we re- ally need to modify Newtonian dynamics and Ein- It is true (from the Eq.(137)) that FYMG can only get stein’s GR before we could account for this extra grav- more repulsive as we probe shorter and shorter distances. itational force with no ”origin”? The answer is a big No! As such, the separating distance between two gluons can- The two theories are fantastically accurate in their re- not taper to zero. This means that the theory ”realizes spective domains of validity. But the core of the problem asymptotic freedom” because two gluons cannot sit on is that we assumed that both theories should be valid top of each other (i.e. separating distance r = 0 is for- at all distance scales (from particle physics scale (say, bidden), hence they are almost free to move around due −1 Planck scale) to the edge of the Universe); but the irony to the non-existent of attractve force at r Λ . ≪ QCD is that they are not. It turns out that in order to solve This explanation is then carried by analogy into the con- GRP, one needs a force law that is valid for all distance struction of spacetime geometry. The fact that the space- scales. This is where BCJ construction kicks in. As we time metric (gµν ) is ab initio constructed out of the two have painstakingly demonstrated (by obtaining Eqs.(106) entangled gluons (BCJ construction) means that space- and (134) from Eq.(27)) , the major conclusion of double time cannot realize singularity (i.e. r = 0). From the 6 copy construction is the existence of gauge/gravity dual- foregoing, one is therefore forced to ask a fundamentally ity. This duality property led to the formulation of the disturbing question: How did our universe ”begin”, or Yang-Mills-Gravity (YM) force in the Eq.(137). A close what existed ”before” the Big Bang? perusal of Eqs.(137) and (192) shows that both equations A. C. Doyle famously claimed that ”once you eliminate are essentially the same; and that Eq.(137) explains the the impossible, whatever remain, no matter how improb- universal rotation curve perfectly, by producing able, must be the truth.” In line with this quote, we posit a flatly stable curve at large values of r (see red that the behavior of the universe during the first fraction curve of the Fig.3) (a universal rotation curve can of a second (t < 10−44s) after the Big Bang can only be expressed as the sum− of exponential distributions of be a matter for conjecture but we are certain that t =0 6 visible matter that reduce to zero with large values of r and r = 0 due to the ever-increasing repulsive nature of 6 away from the center of galaxy, and spherical dark matter FYMG as we probe short-distance scales. Hence, perhaps halo (just like σ in the Eq.(137)) that tapers to flat ro- our universe had its origin in the ever-recurring interplay tation curve with constant speed and gravitational force between expanding and contracting universe. We are [108]) Hence, the deep significance of structure of the sure of the former but the latter is highly unlikely, given Eq.(137)− to cosmology is not accidental but fundamental the present behavior of dark energy and the ever-constant 2 to the evolutionary histories of our universe. effective Lagrangian of the vacuum υ /MX =3.7meV . As pointed out by V. de Sabbata and C. Sivaram, a The fact that most of the calculations done using deviation from the Newton’s inverse square law can arise Planck epoch parameters (i.e., Planck time, Planck en- naturally from R + R2 theory (such as strong gravity ergy and Planck length) conform to what is obtainable in theory), whose solution gives a Newtonian/Coulomb po- nature strongly suggests that any epoch less than Planck tential and a Yukawa term ([26], P.4). Thus it is natural epoch is operationally meaningless. Thus, a plausible to investigate the behavior of the Eq.(137) on a cosmo- theory can be constructed (starting from the Planck time logical scale. Of course, Eq.(137) tapers to σ on the t = 10−44s) by bringing the calculations done in the cosmological scale: section IX of this paper to bear: After about 10−44s (with Planck length 10−19GeV −1) the repulsive grav- 38 2 44 ity was FYMG = 2.6 10 GeV (= 2.129 10 N). F =0.299GeV 2 =2.449 105N (193) This caused the universe− × to undergo an− exponential× ex- YMG × pansion (due to the exponential nature of FYMG). The −44 with mass (Mg) exponential expansion lasted from 10 s after the Big Bang/Bounce to time 104GeV −1(= 6.6 10−21s) : this × Mg = FY M = 546.809MeV (194) is the time that produced the stable-flat curve (red curve p 27 in the Fig.(3)), signaling the demise of the exponential (i) the exponential expansion (governed by FYMG = era. Following this cosmic inflationary (exponential ex- 2.129 1044N) and the normal accelerating expansion pansion) epoch, the dark matter which is nothing (governed− × by Eq.(189)). − but the remnant of Yang-Mills-Gravity force, FYMG = 2.449 105N dominates and the universe continues to expand× but at− a less rapid rate. The battle of supremacy between dark energy (repulsive gravity) and dark mat- XVI. CONCLUSION ter (attractive gravity) was won by dark energy during the time t =6.6 10−21s: since dark energy is intimately We have shown, in this paper, that the point-like the- connected to the× spacetime metric itself (see Eq.(189)), it ory of quantum gravity (strong gravity theory) is geo- would have increased tremendously during the exponen- metrically equivalent to the four-dimensional, nonlinear tial expansion period, when the space increased in size quantum gauge field theory (i.e. QYMT), and the Ein- by factor of 1023(= 104GeV −1/10−19GeV −1) in a small stein General Relativity. The inherent UV regularity, fraction of a second! The victory of dark energy over dark BCJ and gauge-gravity duality properties of this renor- matter means that our universe will continue to expand malizable theory allowed us to solve four of the most ad infinitum: we are living in a runaway universe. difficult problems in the history of physics: namely, dark Another formal way of explaining the inflationary matter, existence of quantum Yang-Mills theory on R4, epoch (i.e. the vacuum-dominated universe approach) neutrino mass and dark energy problems. of the universe which makes the pre-existing universe In any geometric field theory, all physical quantities scenario conceivable− can be effected by using Fig.3. and fields should be induced from one geometric entity In this approach, it is− believed that the universe passed (Weyl’s action) and the building blocks of the geometry through an early epoch of vacuum dominance (i.e. infla- used (2-gluon configuration/double-copy construction). tion), presided over by the varying potential energy (i.e. This principle has been inspired by Einstein’s statement Eq.(113)) of the scalar field, called inflaton ([47], P.564). ”a theory in which the gravitational field and elec- When the scalar field reached the minimum of the poten- tromagnetic− field do not enter as logically distinct struc- tial (which corresponds to the minimum of the potential tures, would be much preferable” and established in curve (blue curve) in the Fig.3) exponential expansion this paper. As we have demonstrated,− this principle im- ended. Based on the law of conservation of energy, the plies that the Weyl Lagrangian density used to construct reduction in the potential energy (due to the rolling down the field equations of the strong gravity theory is com- of the inflaton from the top of the potential curve, i.e. posed of the building blocks (two gluons) of the geome- the decaying of the inflaton field) generated hot (quark- try and their derivatives (in which the curvature arised gluon) plasma epoch (which later generated the matter in terms of derivatives of the dressed gluon field). In and radiation epoch). So from then on, the Big Bang other words, Weyl Lagrangian is not constructed, a pri- evolved according to the Standard Cosmological Model ori, from different parts (each corresponding to a cer- ([47], P.564); and governed by the Eqs.(105), (134), (189) tain field) as usually done. This makes strong gravity and (193). theory to pass the test of unification principle. We conclude this section with the following facts: (1) The initial conditions of our universe are the Planck Acknowledgement epoch parameters (i.e., Planck time, Planck energy and Mr. O. F. Akinto is indebted to the Department of Planck length). (2) Inflationary theory is correct. (3) Physics, CIIT, Islamabad and the National Mathemati- The expansion epoch of the universe consists two phases: cal Center Abuja, Nigeria for their financial support.

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