� Outlines Public Vices and Private Virtues of Future High Precision Physics Giampiero PASSARINO Dipartimento di Fisica Teorica, Universit` a di Torino, Italy INFN, Sezione di Torino, Italy
✁ Outlines A personal (and technical) perspective
✂ Outlines Outlines (1, 2,) The present of two loop calculus 1 A probable decision about its usefulness is possible inductively by studying its success (verifiable consequences) The future of two loop calculus 2 A prospective case study, per aspera ad astra
✄ Outlines Outlines (1, 2,) The present of two loop calculus 1 A probable decision about its usefulness is possible inductively by studying its success (verifiable consequences) The future of two loop calculus 2 A prospective case study, per aspera ad astra
☎ Outlines Outlines (1, 2,) The present of two loop calculus 1 A probable decision about its usefulness is possible inductively by studying its success (verifiable consequences) The future of two loop calculus 2 A prospective case study, per aspera ad astra
✆ The Tree Part I The loop tree: embedded case study
✝ The Tree flow-chart Feynman Feynman Rules Diagrams Green (Pseudo) Functions Observables UV IPS Counterterms Ren. Eq.
✞ The Tree Loop calculus in a nutshell Theorem Definition Any algorithm aimed at An algorithm is optimal when reducing the analytical there is a minimal complexity of a (multi - loop) number of terms, Feynman diagram is zeros of denominators generally bound to correspond to solutions replace the original of Landau equations integral with a sum of the nature of the many simpler diagrams, singularities is not badly introducing overestimated. denominators that show zeros.
✟ ✠ ✠ ✠ ✠ The Tree Sunny-side up Progress Achievements In the past years an basic 2 L integrals have been evaluated enormous progress in the field of 2 L integrals for e.g. analytic expressions massless 2 2 scattering; for the two loop planar gg gg ✡ qg qg and and non-planar box qQ qQ as well as Bhabha master integrals scattering. connected with the tensor integrals have been determined.
☛ ✠ ✠ ✠ ✠ The Tree Sunny-side up Progress Achievements In the past years an basic 2 L integrals have been evaluated enormous progress in the field of 2 L integrals for e.g. analytic expressions massless 2 2 scattering; for the two loop planar gg gg ✡ qg qg and and non-planar box qQ qQ as well as Bhabha master integrals scattering. connected with the tensor integrals have been determined.
☞ ✏ ✌ ✍ ✎ The Tree Status of HO loop calculations zero or one Impressive calculations (up to four loops) for zero or one kinematical variable, e.g. g 2, R , -function 1 Computations involving more than one kin. var. is a new art Example We would like to have n 4 Green functions to all loop orders, from maximally supersymmetric YM amplitudes to real life it’s a long way
✑ ✏ ✌ ✍ ✎ The Tree Status of HO loop calculations zero or one Impressive calculations (up to four loops) for zero or one kinematical variable, e.g. g 2, R , -function 1 Computations involving more than one kin. var. is a new art Example We would like to have n 4 Green functions to all loop orders, from maximally supersymmetric YM amplitudes to real life it’s a long way
✒ ✏ ✌ ✍ ✎ The Tree Status of HO loop calculations zero or one Impressive calculations (up to four loops) for zero or one kinematical variable, e.g. g 2, R , -function 1 Computations involving more than one kin. var. is a new art Example We would like to have n 4 Green functions to all loop orders, from maximally supersymmetric YM amplitudes to real life it’s a long way
✓ The Tree Main road Step 1 reduce reducible integrals Step 2 construct systems of IBP or Lorentz invariance identities Step 3 reduce irreducible integrals to generalized scalar integrals Step 4 solve systems of eqns in terms of MI Step 5 evaluate MI, e.g. differential eqns, MB representations, nested sums, etc.
✔ The Tree Main road Step 1 reduce reducible integrals Step 2 construct systems of IBP or Lorentz invariance identities Step 3 reduce irreducible integrals to generalized scalar integrals Step 4 solve systems of eqns in terms of MI Step 5 evaluate MI, e.g. differential eqns, MB representations, nested sums, etc.
✕ The Tree Main road Step 1 reduce reducible integrals Step 2 construct systems of IBP or Lorentz invariance identities Step 3 reduce irreducible integrals to generalized scalar integrals Step 4 solve systems of eqns in terms of MI Step 5 evaluate MI, e.g. differential eqns, MB representations, nested sums, etc.
✖ The Tree Main road Step 1 reduce reducible integrals Step 2 construct systems of IBP or Lorentz invariance identities Step 3 reduce irreducible integrals to generalized scalar integrals Step 4 solve systems of eqns in terms of MI Step 5 evaluate MI, e.g. differential eqns, MB representations, nested sums, etc.
✗ The Tree Main road Step 1 reduce reducible integrals Step 2 construct systems of IBP or Lorentz invariance identities Step 3 reduce irreducible integrals to generalized scalar integrals Step 4 solve systems of eqns in terms of MI Step 5 evaluate MI, e.g. differential eqns, MB representations, nested sums, etc.
✘ ✙ ✙ ✙ ✙ The Tree But, for the real problem Loop integrals are not enough assemblage of scattering amplitudes infrared divergenges collinear divergenges numerical programs
✚ The Tree IBP and LI Tools Let us point out one A popular and quite drawback of this solution. successful tool in dealing Consider, for instance, the with multi-loop diagrams is following result, represented by the IBPI and LII. Arbitrary integrals can be reduced to an handful of Master Integrals (MI)
✛ ✣ ✣ ✥ ✧ ✧ ✌ ✣ ✣ ✌ ✧ ✣ ✌ ✌ ✥ ✌ ✡ ✏ ✜ ✤ ✣ The Tree IBP example Example 1 B 0 ✜ 1 ✡ 2 ✢ p ✡ m 1 ✡ m 2 ✌ p 2 ✡ m 2 ✡ m 2 1 2 ✣✦✜ m 2 m 2 p 2 ✜ n 3 ✣ B 0 ✜ p ✡ m 1 ✡ m 2 1 2 p 2 m 2 m 2 1 2 ✜ n 2 A 0 ✜ m 1 2 m 2 2 A 0 ✜ m 2
★ The Tree IBP example Around threshold We know that at the normal threshold the leading behavior of ✤✪✩ 1 ✫ 2 , B 0 ✜ 1 ✡ 2 ✣ is Conclusion: reduction to MI apparently overestimates the singular behavior; of course one can derive the right expansion at threshold, but the result is again a source of cancellations/instabilities.
✬ The Tree Two-loop conceptual problems Unstable internal WSTI vs LSZ Unphysical behaviors Two loop ` a la LSZ induced by self-energy The LSZ formalism is insertions into 1 L diagrams; unambiguously defined they signal the presence of only for stable particles, an unstable particle and are and it requires some the consequence of a care when external misleading organization of unstable particles PT. appear Around thresholds These regions are not accessible with approximations, e.g. expansions.
✭ The Tree Two-loop conceptual problems Unstable internal WSTI vs LSZ Unphysical behaviors Two loop ` a la LSZ induced by self-energy The LSZ formalism is insertions into 1 L diagrams; unambiguously defined they signal the presence of only for stable particles, an unstable particle and are and it requires some the consequence of a care when external misleading organization of unstable particles PT. appear Around thresholds These regions are not accessible with approximations, e.g. expansions.
✮ The Tree Two-loop conceptual problems Unstable internal WSTI vs LSZ Unphysical behaviors Two loop ` a la LSZ induced by self-energy The LSZ formalism is insertions into 1 L diagrams; unambiguously defined they signal the presence of only for stable particles, an unstable particle and are and it requires some the consequence of a care when external misleading organization of unstable particles PT. appear Around thresholds These regions are not accessible with approximations, e.g. expansions.
✯ The Tree Technical problems I Reduction to MI It remains Algebraic problem, to generalize to more than few scales Buchberger algorithm to construct Gr¨ obner bases to compute the MI seems to be inefficient New bases?
✰ The Tree Technical problems I Reduction to MI It remains Algebraic problem, to generalize to more than few scales Buchberger algorithm to construct Gr¨ obner bases to compute the MI seems to be inefficient New bases?
✱ The Tree Technical problems I Reduction to MI It remains Algebraic problem, to generalize to more than few scales Buchberger algorithm to construct Gr¨ obner bases to compute the MI seems to be inefficient New bases?
✲ ✏ ✳ ✏ The Tree Technical problems II Although HTF (usually) have nice properties, expansions are often available with good properties of convergence the expansion parameter has the same cut of the function where is the limit? One - loop, Nielsen - Goncharov ✡ m 2 cuts) harmonic Two - loop, one scale ( s 0 polylogarithms 4 m 2 cuts) generalized Two - loop, two scales ( s harmonic polylogarithms next? New higher transcendental functions?
✴ Reduction Part II Future of 2 L calc: exploratory case study
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