The Electrostrong Relation Nikola Perkovic
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Nikola Perkovic. The Electrostrong Relation. 2017. hal-01645382
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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Electrostrong Relation
by Nikola Perkovic
Institute of Physics and Mathematics, University of Novi Sad
Abstract
In an attempt to eliminate the Landau Pole from QED by “borrowing” asymptotic freedom from QCD, I was successful in uniting the coupling constants of the two, respectively, in the “Electrostrong relation”. The Landau pole, estimated at around , leads to a value of in the Electrostrong Relation, instead of infinity as was the case before. In addition,− the Strong CP problem is also solved proving that there is no eVneed for fine tuning in Q = . QCD. The first step however, was improving the measurement for the running of the strong coupling constant that can be tested experimentally for value of for the energy scale, QCD integration parameter offering such a prediction for the first time. Q (ΛQ ) ΛQ
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I Introduction
The two dimensionless running constants and will be united in a single equation, the Electrostrong Relation, allowing us to both provide a solution for the Strong CP problem and remove the problem arising due to infinities inQ known Q as the Landau pole [1] when on a very large value of the fine-structure constant therefore we will effectively make QED a mathematically complete theory. The Landau pole Q will be eliminated since QCD enjoys Q → ∞ asymptotic freedom [2], which will be “borrowed” offering us exactly what we need since - expansion (see Appendix 1 for a short introduction to -expansion) describes the asymptotic behavior of a denominator of convergents of continued fractions. The paper will also provide the first prediction for the value and improve the accuracy for the measurement of the running values of the strong coupling constant. (ΛQ ) In QED vacuum polarization corrections to processes involving the exchange of virtual photons result in a dependence, of the effective fine structure constant , which is known as the running of a coupling constant which is, for the running of the fine structure constant, parameterizedQ as: Q
(1.1) − − where at the scale of the Landau pole, approximately Q = −around ∆ Q , the problem of infinities arises. This problem does not exist within QCD since in the running of the strong coupling constant there is asymptotic freedom. eV
The strong interactions Q do not violate CP symmetry in the manner of the weak interactions since the gauge interactions in QCD involve vector currents instead of chiral currents. Experiments too do not indicate CP violation in the QCD sector. However, if one writes the additional term in the Lagrangian density as:
(1.2) A A μ ℒ = Fμ F̃ it contains the topological term with the QCD vacuumπ angle , which violates CP symmetry and it can, if non-zero, lead to experimentally detectable CP violating effects. In short this is reffered to as the Strong CP problem.
II Running of and
The QCD Lagrangian is 훂퐒 퐐 훂 퐐 μ μ A A μ ℒQ = ∑ Ψ̅ , (i ∂μ − gs t Aμ − m )Ψ , − Fμ F
2 where are the Dirac -matrices, the are quark-field spinors for a quark of flavor and mass μ, with a color-index a that runs from to and is the strong Ψq, f gauge coupling. Let’s remember that Lebesgue measure was used in the introduction to - m a = N = gs = π expansion. A generalization of the Lebesgue measure for any locally compact√ group is known as the Haar measure [4]. We know that the simple compact Lie group we need is (see the paper by G. Nagy [5]). From the invariant metric with a unitarity, provided SU N that all eigenvalues different. For matrixes and we get and in a ds = g dx dx given set of coordinates we attain the invariant measure:† † † W W WdW + dWW =
(2.1) d x = √det g x ∏ dx we parameterize the matrix as therefore we conclude that:
W W = exp it T (2.2) det g ~ det Q ∏| − | the distribution of eigenvalues on the unitary group > is given by the invariant measure:
(2.3) d = ∏|exp i − exp(i )| ∏ d which is valid for groups. If we > impose the constraint mod and implement it -function, the density distribution of the eigenvalues in is given by the formula by a δ above as well. ForU the N group this would be of the form:Σ = π SU N SU
q q ∑ e ∑ e (2.4) | | H = ∏|exp i − exp(i )| = ∑ e̅q ∑ eq > | | q q We must first describe the running of and then proceed∑ e̅ to ∑connect e̅ it to the running of . Using -functions for QCD, a group theory we have for Q Q SU i= , , ,…,n (2.5) b b b = − − − ⋯ π π π 3 where the QCD -function has a negative sign due to gluons carrying color charge which leads to self-interactions. At one-loop order we determine:
(2.6) = − b ln Now we introduce the -expansion for a real number :
q
∞ (2.7) − x = ∑ c q where for all we have where = doesn’t have to be an integer and is a floor function [6]. We connect equations 2.6 and 2.7 which yield: n c q c q
(2.8) − Q ΛQ q = + 풪 meaning that from equation 2.7 equals π which isQ the correction due to QCD effects and is the integration constant which coresponds to the scale where the x x = Q perturbatively defined coupling would deverge and we use a value of but other values ΛQ = can be used as well. Knowing the values MeV
c = , c = . − . n , c = − . − where the coefficient of the -dependent piece in the term is yet to be determined and is . n − . n − . , c = − . + . n − . n + . n + C the number of flavors, allows us to establish that is consistent with the C n Vandermonde determinant of the Haar measure for SU(3). We introduce the = (∑ eq) ⁄ ∑ eq vacuum expectation value of the Higgs field [7]. When a local rather than global symmetry is spontaneously broken the gauge field of the broken H symmetry becomes massive,v leading= toGeV what should be a Goldstone scalar to become a longitudinal mode of the massive vector, which works for both abelian and non-abelian local symmetries. In the non-abelian case, for each spontaneously broken generator of the local symmetry the corresponding gauge field becomes massive. The Higgs mechanism푎 gives quarks mass and leaves photons and gluons푎 T A휇 x massless therefore we measure for the first time, being that no other theory predicted the running value of for the parameter: (ΛQ ) ΛQ
v (2.9) Q ) ln (Λ (ΛQ ) = π 4
This allows experimental tests to determine if the predicted value of is accurate and therefore test the -expansion method. The equation for the running value of is then: (ΛQ ) Q
(2.10) (ΛQ ) Q = Q ln ( Q ) where it is evident that QCD enjoys asymptotic freedom.Λ For we add squared values for and . The numerical results are provided in the table 1 bellow. All the values are in Q great agreement with experimental values such as provided by [8], [9], [10], [11]. Particle Data v, ΛQ Q Group [12] provides the world average . The method offered by - expansion offers much desired [13] accuracy and simplicity. Mz = . Using the Lyapunov exponent for the rate of exponential divergence from the initial perturbed conditions, we have where the constant has the form and − ′ where we used the Housdorff →∞ dimension [14]. See ℒC. xGong, ≔ lim Lyapunovn Exponentlog| T ofx | Classical SU(3) Gauge Theory [15] for a more = τdetailed log explanation− log on theτ relationship ≔ dim { ofx ∈the[ , gauge : ℒ couplingx = } and the SU(3) Lyapunov exponent. From this we can deduce that in eq. (1.1) for the running of the fine-structure constant:
(2.11) π ∆ Q = Allowing us to eliminate the Landau pole in the running Q of by establishing a link between and which eliminates the infinities since QCD enjoys asymptotic freedom by establishing: Q Q Q
(2.12) − − π Q = − The results are in good agreement with experimental values Q [16], [17].
Energy [GeV] − Q Q Q Λ = . . . . . . . z M = . . . M = . . M = . . 5
Table 1: Measurments for the electromagnetic and strong nuclear couplings for the values of .
In order to prove the lack of Landau pole훬푄 in my= equations 푀푒푉 I provide the measurements albeit the Landau pole is not relevant for particle physics but is purely of academic interest. The value of the Landau pole is estimated roughly to be around , for this value and where the uncertainties are high since we are dealing eV Q = with such a high energy− level. . Q = . When it comes to the Strong CP problem we will look at the -dependence on the vacuum energy. If we take to be the phase of the overall Lagrangian where is the up quark mass and is the down quark mass and we can treat the ℒ = (m u̅u + m d̅d) cos ⁄ + Lagrangian term as a perturbation. The only degrees of freedom which can readily be excited in h. c. m m QCD are pions and as M. Dine shows in his paper TASI Lectures on The Strong CP Problem [18], the relation of this dependence is where is the rest energy of a pion, however we will only be working with charged pions and and as such we obtain: π π π E = E 푓 cos+ − E π π
(2.13) E nqq̅ Ry Eπ = = π where is the number of constituent푓 quarks/anti-quarksπ cos Q and = M forπ charged pions and the Ry is the Rydberg unit of energy that is measured at 13.605693009(84) eV. Since the rest energy nqq̅ nqq̅ ≡ of charged pions is measured to be MeV we can use it to measure the running of the fine structure constant effectively solving the Eπ = . Strong CP problem. For other Electrostrong− relations, such as for electrons, muons and their π relationship with the Koide formula [19] see Q =Appendix M = 2. .
III Conclusion and Debate
In the Standard Model, constants in the one-loop -functions are given as we can summarize these results and insert SU × SU × U the number of fermion generations and Higgs doublets obtaining us b = T R d(R )d R + T S d(S )d S − C G , and . N = N = b = / Theseb three= − intersections/ b =point − towards a range from to and correspond to a coupling ranging from to where we see an agreement M = GeV M = GeV for in the table where − − . This means that, regardless = ≈ of unification, the equations measure the running− of− the couplings in excellent agreement with bothM the experimental≈ GeV values and the theoretical = predictions = . of the SM.
Appendix 1: β-expansion 6
Let be a real number and be the -transformation such that where is the largest integer that does not exceed . Every can be > T i: [ , → [ , T i x = uniquely expanded into a finite or infinite series: x − [ x] [x] x x ∈ [ ,
(A.1) x x x x x = + + + ⋯ + + ⋯ where and for all . For any Borel set where C is a constant that depends on such that where the measure of the x = [ x] + x = (T i x) n A ⊆ [ , length/energy scale is -invariant meaning− that does not implicitly depend on and if is C A A A C Lebesgue measurable [3], then where is compact. T i A If then any irrational number A = supcan {be kwritten : k ⊂ as A} k where and for any is a convergent of the continued x ∈ [ , p x ⁄q x = [c x , c x , … , c x ,… ] fraction expansion− of where and are relatively prime which allows us to study the + asymptoticc x behavior= [x ] of thec denominator x = a T x n x p x q x k (A.2) k x log log →∞lim = For any there exists two positive constantsn andπ which depend on thus for all : > A n (A.3) k log log − {x ∈ [ , : | − | } A ∙ e If then for almostn all π exists a positive integer that depends on , such that for all : log > π ⁄ log x ∈ [ , N x n N
(A.4) p x |x − | > x − x In the opposite case when q xwe have an integer and for all : log < π ⁄ log N n N
(A.5) p x |x − | < x − x where the convergents of the -expansion ofq xare . x x = x ⁄ + x ⁄ + x ⁄ + ⋯ + x ⁄ 7
Appendix 2: Electrostrong Relations for Electrons, Muons and the Higgs Boson
The relationship between the fine-structure constant and the electron rest energy/mass is well known and established but its relationship with the rest energy of muons, taus and the Higgs boson is not detailed enough. For the time being it has been well established that at the energy level of Z bosons, the fine-structure constant is approximately 1/128 and the trong coupling is approximately 0.1184 and its relationship with electron, muon and Higgs boson rest energy/mass is not explained at all. For the rest energy of electrons, we have an equation:
(2) Q Ry Ry E = Q = where Ry is the Rydberg unit of energy, s is the fines structure constant that equals , value provided by NIST, is the strong coupling constant at M zero momentum transfer and evidently is part of the Koide formula and shouldn’t be . s confused with the momentum scale . For muons, we have≡ a similar equation where the Q Q ≡ rest energy equals: Q Q = Mμ
(3) Q Ry Ry Eμ = Q = where we have to account for the runnings(Mμ) of the M μcoupling s(M constants.μ) M μFor the energy level of muons the fine-structure constant is approximated to be around 1/136 in which case the strong coupling constant and evidently the second part of the Koide formula . The Koide formula relates the rest energies/masses of three leptons, namely electrons, s(Mμ) = . muons and taus . It s been discovered by dr. Yoshio Koide in 1981. it states that in the ’ ’ Qformula:= Q = /
(4) Q E + Eμ + Eτ = (√E + √Eμ + √Eτ) = (√E + √Eμ + √Eτ) where the parameters and Q are the addition of this paper. Additionally we have a formula for rest energy of the Higgs boson that has been measured to be approximately 125.36(41) GeV [20]. As such, we have:Q Q
(5) Ry E퐻푖푔푔푠 = s(M퐻푖푔푔푠) M퐻푖푔푔푠 8 where the running of the two coupling has been measured with experimental values available at [21]. The results are in great agreement with the experimental result provided by ATLAS in ref. [20].
IV References
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[6] D. Hilbert and S. Cohn-Vossen. Geometry and the Imaginaion, 1999.
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[8] L3 Collaboration, Phys. Lett. B284, 1992.
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[11] OPAL Collaboration, Z. Phys. C55, 1992.
[12] J. Beringer et al. (PDG), PR D86, 010001, 2012.
[13] G. Dissertori, The Determination of the Strong Coupling Constant, Institute for Particle Physics, ETH Zurich, 2015.
[14] J. W. Harris and H. Stocker, Hausdorff dimension, Handbook of Mathematics and Computational Science, 1998.
[15] C. Gong, Lyapunov Exponent of Classical SU(3) Gauge Theory, arXiv:hep-lat/9209018.
[16] S. Bethke, The 2009 World Average of αS , Eur. Phys. J. C64, 689, 2009.
[17] K. Nakamura et al. (PDG), J. Phys. G37, 075021, 2010.
[18] M. Dine, TASI Lectures on The Strong CP Problem, arXiv:hep-ph/0011376.
[19] A, Rivero, A. Gsponer, The strange formula of Dr. Koide. 2005 arXiv:hep-ph/0505220v1.
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[20] The ATLAS collaboration, Measurement of the Higgs boson mass in the H→ZZ →4ℓH→ZZ →4ℓ and H→ H→ channels with s√s=13TeV pppp collisions using the ATLAS detector 2017. http://cds.cern.ch/record/2273853. ∗ ∗ [21] G. Dissertori, The Determination of the Strong Coupling Constant 2015. arXiv:1506.05407v1.
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