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Does High -Energy Behavior Depend on Quark Masses ?

This research investigates the dependence of high-energy behavior in QCD on quark masses, exploring the possibility of nonzero bound state masses in massless QCD. The study examines the role of renormalization, chiral symmetry breaking, and the behavior of scattering amplitudes. The conclusion suggests that nonzero current quark mass in QCD is necessary for infinitely rising cross-sections.

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Does High -Energy Behavior Depend on Quark Masses ?

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  1. Does High -Energy Behavior Depend on Quark Masses ? V. A. P e t r o v Division of Theoretical Physics Institute for High Energy Physics P r o t v i n o, RF EDS'09: 13th International Conference On Elastic & Diffractive Scattering (Blois Workshop)

  2. Is it possible to get non-zero ( bound state )mass in massless QCD? Coleman & E. Weinberg (1973) “Normal” massless QCD: L0 = - ¼F2 + iψγ∂ψ + g ψγAψ Renormalization: g → g(μ) = F(μ/Λ), Λ = fundamental scale parameter (“dimensional transmutation”) →M ~ Λ Chiral symmetry breaking:‹ψψ› ≠ 0. No new scale. In finite theory (N=4 SYM): g → g. No fundamental scale parameter. No massive particles ( without (super) symmetry breaking).

  3. Physical quantities are invariants of the Renormalization Group М(g, μ) = М(g´, μ´) (g´, μ´) → g(μ) [μ2∂/∂μ 2 + β(g 2)∂/∂g 2]M = 0 М(g, μ) = c μ2exp(- K(g2)) ∂ K(g2)/ ∂g 2 = 1/ β(g 2) μ2∂ g 2 /∂μ 2 = β(g 2)

  4. Scattering Amplitude T( s, t) = F ( s, t; g 2,μ 2) d [T| (g, μ)→ g(μ) ]/d μ = 0 g(μ)~1/log(μ/Λ) at μ→∞ H a d r o n m a s s e s: M2i= Ciμ2exp(- K(g2)) → CiΛ2 ~ exp(- c´/ g2) T(s,0) = H ( s/Λ2 ;{C i}) T(s,0) may not exist if massless particles survive in the spectrum ( pions in the Bogoliubov-Goldstone mode).

  5. Natural condition: g = 0 →T = 0 T forward = H[ (s/μ2) exp (c´/ g2) ], c´ > 0. ! g2→0 ~ s →∞ ! lim T(g2→0) = lim T(s →∞) = 0

  6. Consequences(awful but theoretically admissible) At s →∞: σtot < const/s dσ/d t (t = 0) < const/s2 General lower bounds ( Jin-Martin, 1964) σtot ≥ const / s6(logs)2

  7. If true, the data imply a huge “turn-off scale”: √s turn-off > O ( 10 Te V) ~100000qcd(?!) No intelligible mechanism in QCD

  8. “Rescue”: massive QCD • L0→ L0 + m ψψ • Two RG-invariant mass scales : • Λ1= μ exp (-K (g)) ~ exp(-1/ g2), c´ > 0 • Λ2 = m exp (L (g)) ~ (1/ g2 ) c´´ , c´´ >0 • ∂L (g) ∂g = γ m(g ) / β(g ) • γ m (g ) = - m -1μ∂ m /∂μ • Important: mass scales are namely Lagrangian parameters, not “dynamically generated masses”.

  9. The trick with the free-field limit does not pass through T(s,0,….) = Ф ( s / Λ21 ; Λ22 / Λ21 ) lim T = Ф (∞; ∞) = 0 g →0 lim T = Ф (∞; Λ22 / Λ21 ) = ? s → ∞ Two limits are generally different

  10. An example Im T (s,0) = (s /22)log2 (s /21) Im T (s,0) ∞at s  ∞ Im T (s,0)const(g2)c exp(1/(0 g2)) 0 at g0

  11. Conclusion: Non-zero current quark mass in QCD is a necessary condition to get infinitely rising cross – sections . [?]

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