Does High -Energy Behavior Depend on Quark Masses ?

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 . [?]