Theory of the Mass Gap in Qcd and Its Vacuum
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THEORY OF THE MASS GAP IN QCD AND ITS VACUUM V. Gogokhia WIGNER RESEARCH CENTER FOR PHYSICS, HAS, BUDAPEST, HUNGARY [email protected] GGSWBS'12 (August 13-17) Batumi, Georgia 5-th "Georgian-German School and Workshop in Basic Science 0-0 Lagrangian of QCD LQCD = LYM + Lqg 1 L = ¡ F a F a¹º + L + L YM 4 ¹º g:f: gh: a a a abc b c 2 F¹º = @¹Aº ¡ @ºA¹ + gf A¹Aº; a = Nc ¡ 1;Nc = 3 j j j j j Lqg = iq¹®D®¯q¯ +q ¹®m0q¯; ®; ¯ = 1; 2; 3; j = 1; 2; 3; :::Nf j a a j D®¯q¯ = (±®¯@¹ ¡ ig(1=2)¸®¯A¹)γ¹q¯ (cov: der:) 0-1 Properties of the QCD Lagrangian QCD or, equivalently Quantum ChromoDynamics 1). Unit coupling constant g 2). g » 1 while in QED (Quantum ElecroDynmics) it is g ¿ 1. 3). No mass scale parameter to which can be assigned a physical meaning. Current quark mass m0 cannot be used, since the quark is a colored object QCD without quarks is called Yang-Mills, YM 0-2 The Ja®e-Witten (JW) theorem: Yang-Mills Existence And Mass Gap: Prove that for any compact simple gauge group G, quantum Yang- Mills theory on R4 exists and has a mass gap ¢ > 0. (i). It must have a "mass gap". Every excitation of the vacuum has energy at least ¢ (to explain why the nuclear force is strong but short-range). (ii). It must have "quark con¯nement" (why the physical particles are SU(3)-invariant). (iii). It must have "chiral symmetry breaking" (to account for the "current algebra" theory of soft pions). We need Mass Gap responsible for the NP dynamics 0-3 Gluon SD equation 0 0 D¹º(q) = D¹º(q) + D¹½(q)i¦ρσ(q; D)Dσº(q) n o 1 D (q) = i T (q)d(q2) + »L (q) ¹º ¹º ¹º q2 1 D0 (q) = i fT (q) + »L (q)g ¹º ¹º ¹º q2 2 T¹º(q) = ±¹º ¡ (q¹qº=q ) = ±¹º ¡ L¹º(q) 0-4 S D = + Γ D S G D + GDµ + G D D D + T3 D + D T3 D D D T D D 3 + D T4 D D q gh t (1) 2 ¦ρσ(q; D) = ¦ρσ(q) + ¦ρσ(q) + ¦ρσ(D) + ¦ρσ (q; D ) (2) 4 (20) 3 +¦ρσ (q; D ) + ¦ρσ (q; D ): 0-5 The Mass Gap s 2 ¦ρσ(q; D) = ¦ρσ(q; D) ¡ ¦ρσ(0; D) = ¦ρσ(q; D) ¡ ±ρσ¢ (D) ¢2(D) ´ ¢2(¸; ®; D) Problems A. The ¯rst problem is how to satisfy the ST identity 2 q¹qºD¹º(q) = i» but without going to ¢ (D) = 0 limit. B. The second problem is how to make the relevant gluon propagator purely transversal, since the ghosts will fail to do this when ¢2(D) will be explicitly present. 0-6 Methods Theory of matrices Subtractions at the gluon propagator level Theory of generalized functions (distributions) Dimensional regularization method Theory of functions of complex variable The Weierstrass-Sokhotsky-Casorati theorem which describes the behavior of the Laurent expansions near their essential (zero) singularities in order to get ¯nite results (Renormalization program) ¸ ! 1; ® ! 0 0-7 Solution 2 INP 2 PT D¹º(q; ¢ ) = D¹º (q; ¢R) + D¹º (q) 1 ¢2 DINP (q; ¢2 ) = iT (q)dINP (q2; ¢2 ) ; dINP (q2; ¢2 ) = R ¹º R ¹º R q2 R q2 n o PT PT 2 2 D¹º (q) = i T¹º(q)d (q ) + »L¹º(q) (1=q ) PT 2 ®s 1 2 d (q ) = 2 2 = 2 2 ; q ! 1; AF 1 + ®sb ln(q =¤QCD) b ln(q =¤QCD) 2 2 ®s(mZ ) = 0:1184; ¤QCD = 0:09 GeV ; b = (11=4¼) for Y M 0-8 Some remarks 2 1Ã®s(¸) 2 ®s(¸)!0 2 ¤INP Ã¡1Ã¸ ¢ (¸; ®s(¸)) ¸!1 ¡! ¤PT ; 2 2 2 2 ¤INP ´ ¢R; ¤PT ´ ¤QCD INP QCD (= QCD =) P T QCD; Intrinsically non-perturbative QCD (INP QCD) con- ¯nes gluons (no free gluons, dressed gluons are sup- pressed in incoming and outgoing states) Perturbative QCD (PT QCD) is asymptotically free 0-9 VACUUM ENERGY DENSITY (VED) IN THE QUANTUM YM THEORY The vacuum of QCD is a very complicated con- ¯ning medium. Its dynamical and topological com- plexity means that its structure can be organized at various levels: classical and quantum, dynamical and topological. It is mainly NP by origin, character and magnitude, since the corresponding coupling constant is large. However, the virtual gluon ¯eld con¯gura- tions and excitations of the PT origin, character and magnitude, due to AF, are also present there. 0-10 The bag constant I B = V EDPT ¡ V ED B = V EDPT ¡V ED = V EDPT ¡[V ED¡V EDPT +V EDPT ] = V EDPT ¡ [V EDINP + V EDPT ] = ¡V EDINP > 0 The formalism which makes it possible to calculate the VED from ¯rst principles is the e®ective potential approach for composite operators J.M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D, 10 (1974) 2428. 0-11 The VED S D + + + +... D T3 D T3 + D T4 D D D S S = + + + ... D i Z d4q n o V (D) = T r ln(D¡1D) ¡ (D¡1D) + 1 ;V (D ) = 0 2 (2¼)4 0 0 0 0-12 The bag constant II BYM = ¡²YM Z q2 · ¸ 1 eff 3 = dq2 q2 ln[1 + 3®INP (q2)] ¡ ®INP (q2) ; ¼2 0 4 2 2 INP 2 INP zc q ¢R d (q ) ´ ® (z) = ; z = 2 zc = 2 z qeff qeff 4 BYM = qeff £ YM ; Z 1 · ¸ 1 INP 3 INP YM = dz z ln[1 + 3®s (z)] ¡ ®s (z) : ¼2 0 4 0-13 0.04 max zc =0.45 0.02 eff 4 0 zc =1.3786 ) / q c 0 (z YM B -0.02 Physical Region Non-Physical Region -0.04 0 0.5 1 1.5 2 2.5 0-14 zc 4 Figure 1: The BYM =qeff e®ective potential vs. zc. The non-physical 0 region is zc ¸ zc , since BYM should be always positive. At zc = 0 the e®ective potential is also zero. 0.04 0.02 Physical Region Non-Physical Region eff 4 0 zc =1.3786 ) / q c 0 (z YM ε -0.02 min zc =0.45 -0.04 0 0.5 1 1.5 2 2.5 0-15 zc 4 Figure 2: The ²YM =qeff e®ective potential vs. zc. The non-physical 0 region is zc ¸ zc , since ²YM should be always negative. At zc = 0 the e®ective potential is also zero. YM (zc)=@zc = 0 2 max ¡1 2 4 qeff = (zc ) ¢R = 2:2¢R 4 4 BYM = ¡²YM = 0:1273 £ ¢R = 0:0263 GeV General properties of the bag constant being: ² colorless (color-singlet); ² electrically neutral; ² transversal, i.e., depending only on "physical" de- grees of freedom of gauge bosons; 0-16 ² the explicit gauge invariance; ² uniqueness, i.e., it is free of all the types of the PT contributions now; ² ¯niteness; ² positiveness; ² no imaginary part (stable vacuum); ² existence of the stationary state for the corre- sponding YM energy density (negative pressure); ² the ¯nal dependence on the mass gap only; ² a good numerical agreement with phenomenology (gluon condensate). 0-17 Contribution of BYM to the dark energy problem Bag constant may also contribute to the so-called dark energy density. At least, from the qualitative point of view it satis¯es almost all the criteria neces- sary for the dark energy/matter candidate. From the quantitative numerical point of view it is also much better than the estimate from the Higgs ¯eld's contribution to the VED by S. Weinberg, arXiv:astroph/9610044 8 4 %H » 10 GeV : ¡2 4 %our » 10 GeV 0-18 exp ¡46 4 %vac » 10 GeV WMAP Collaboration, E. Komatsu et al., Astro- phys. J. Suppl. 180 (2009) 330 So relatively to the value inferred from the cosm. const. (experimental value exp 54 %H =%vac » 10 ; exp 44 %our=%vac » 10 0-19 Numerical value in di®erent units 4 BYM = 0:0263 GeV = 3:4 GeV=fm3 = 3:4 £ 1039 GeV=cm3: 1 GeV = 1:6 £ 1010J = 4:45 £ 10¡23GW h; 1W = 10¡3kW = 10¡6MW = 10¡9GW 17 3 BYM » 10 GW h=cm 0-20 3 17 EYM = BYM cm » 10 GW h Total production of primary energy of the 25 EU countries in 2004 was 7 Et » 10 GW h: The number is taken from "Energy FOR THE FU- TURE", a position paper of the EPS, www.eps.org Approximately 1=3 of this energy was produced by nuclear power plants 0-21 Energy from the QCD vacuum The bag constant is the energy density of the purely transversal severely infrared singular virtual gluon ¯eld con¯gurations which are not only stable (no imagi- nary part), but are being in the stationary state as well, i.e., in the state with the minimum of energy. That is why it makes sense to discuss the "releasing" of the bag energy EYM from the vacuum. V E = ¡B V = ¡E »¡¸3; ¸ ! 1 vac YM YM cm3 Let us imagine now that we can release the ¯nite portion EYM from the vacuum in k di®erent places (di®erent "vacuum energy releasing facilities" (VERF)). 0-22 It can be done by nm times in each place, where m = 1; 2; 3:::k. Then the releasing energy Er becomes Xk Er = EYM nm = EYM (n1 + n2 + n3 + ::: + nk) m=1 Xk 2 Er = EYM £ lim nm » ¸ ; ¸ ! 1 (k;n )!1 m m=1 ER = Evac ¡ Er = Evac[1 + O(1=¸)]; ¸ ! 1; i.e., the QCD vacuum is in¯nite and permanent reservoir of energy.