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Strong Gravity Approach to QCD and General Relativity

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Strong Gravity Approach to QCD and General Relativity Strong Gravity Approach to QCD and General Relativity O. F. Akinto, Farida Tahir Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan A systematic study of a Weyl type of action, which is scale free and quadratic in the curvature, is undertaken. The dynamical breaking of this scale invariance induces general relativity (GR) as an effective long distance limit of the theory. We prove that the corresponding field equations of the theory possess an effective pure Yang-Mills (i.e. QCD without quarks) potential, which describes the asymptotic freedom and color confinement properties of QCD. This inevitably leads to the solutions of quantum Yang-Mills existence on R4 (with its characteristic mass gap), and dark matter problems. The inherent Bern-Carrasco-Johansson (BCJ) double-copy and gauge-gravity duality properties of this formulation lead to the solutions of the neutrino mass and dark energy problems. This approach provides a strong gravity basis for the unification of quantum Yang-Mills theory (QYMT) with Einstein GR. Keywords: Weyl action, BCJ double-copy, gauge-gravity duality. I. INTRODUCTION [7]. The origin of the difficulties is now clear to us: ”Who of us would not be glad to lift the QCD action is scale invariantly quadratic in the field i veil behind which the future lies hidden; to strengths Fµν (i.e.non-unitary) and renormalizable, while cast a glance at the next advances of our sci- the Einstein-Hilbert action for pure gravity is unitary and ence and at the secrets of its developments nonrenormalizable. Thus, the unification of gravity with during future centuries?” David Hilbert QCD seems unattainable; however, that is not the case: (1900). − The valiant attempt to disprove this prima facie impos- ”It is by the solution of problems that sibility offers an outstanding example of the inspiring ef- the investigator tests the temper of his steel; fect which such a very special and apparently important he finds new methods and new outlooks, and solution may have upon physics community. gains a wider and freer horizon” David Hilbert (1900). − Having now recalled to mind the origin of the problem, let us turn to the question of whether there is an existing In the early seventies Abdus Salam and his co-workers unification scheme that can be used to solve the problem. proposed the concept of strong gravity, in which the suc- Strong gravity formulation is such the unification scheme cessive self -interaction of a nonlinear spin-2 field was that allows the gravity to be merged with QYMT. In used to describe a non-abelian field of strong interac- this case, a gravitational action which possesses quadratic tions. This idea was formulated in a two-tensor theory terms in the curvature tensor has been shown to be renor- of strong and gravitational interactions, where the strong malizable ([8], P.963 & P.967). Here, the resulting non- tensor fields are governed by Einstein-type field equations gauge-invariant divergences are absorbed by nonlinear 38 renormalizations of the gravitational fields and Becchi- with a strong gravitational constant Gf 10 times the ≈ Rouet-Stora transformations ([8], P.953). In the follow- Newtonian constant GN . Within the framework of this proposal, tensor fields were identified to play a funda- ing, the dynamical breaking of the scale invariance of mental role in the strong-interaction physics of quantum Weyl action (which describes the short distance behavior chromodynamics (QCD) [1–6]. of strong gravity theory) induces: (1) perturbative/short- All the calculations done in the numerical lattice QCD range component of the non-relativistic QCD potential, and other related experiments indicate that QCD, the and non-relativistic quantum electrodynamic (QED) po- arXiv:1606.06963v3 [physics.gen-ph] 31 Aug 2016 worthy theory of strong interactions, possesses gauge tential. (2) Einstein general relativity as an effective long symmetry based on the group SU(3) color of quantum distance limit of the theory This is the fons et origo − of the gauge/gravity duality;− and the solution to Yang-Mills theory (QYMT). Gravitational interactions 4 also have similar symmetry (the coordinate invariance the quantum Yang-Mills existence on R and dark in a space-time manifold), but resist quantization. This matter problems, within the strong gravity for- prevents physicists from constructing a quantum theory mulation. of gravity based on the gauge principle, and also inhibits The catch here is that quantum gravity (i.e. a quan- the direct unification of gravity with strong interaction tum mechanically induced gravity) cannot be derived straightforwardly by quantizing nonrenormalizable Ein- stein GR but Weyl action which leads to Einstein’s the- ory of gravity at large distances[7]; in the same way [1] [email protected] the gauge theory of Glashow-Weinberg-Salam, GEW = [2] farida [email protected] SU(2) U(1) , reduces to U(1) after the spontaneous L × Y Q 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.
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