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3 (or 4!) loops renormalization constants for lattice QCD

F. Di Renzo, A. Mantovi, V. Miccio and C. Torrero ( 1 ) & L. Scorzato ( 2 ). (1) Università di Parma and INFN, Parma, Italy (2) Humboldt-Universit ä t, Berlin, Germany. 3 (or 4!) loops renormalization constants for lattice QCD. Motivation.

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3 (or 4!) loops renormalization constants for lattice QCD

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  1. F. Di Renzo, A. Mantovi, V. Miccio and C. Torrero(1) &L. Scorzato(2) (1) Università di Parma and INFN, Parma, Italy (2) Humboldt-Universität, Berlin, Germany 3 (or 4!) loops renormalization constants for lattice QCD

  2. Motivation Renormalization constants in (Lattice) Perturbation Theory: is it really a second choice? Can you compare PT and NP? Computational Setup Numerical Stochastic Perturbation Theory (Parma group after Parisi & Wu) Quarks bilinears at 3 (4) loops Z’s in the RI’-MOM scheme. The “perfect” case: Zp/Zs. Treating anomalous dimensions (“tamed” log’s). Perspectives … there is much to do! Outline

  3. Renormalization constants and LPT Despite the fact that there is no theoretical obstacle to computing log-div RC in PT, on the lattice one tries to compute them NP. Popular (intermediate) schemes are RI’-MOM (Rome group) and SF (alpha Coll). >>LPT converges badly and usually computations are 1 LOOP (analytic 2 LOOP on their way). >> Often (large) use is made of Boosted PT (Parisi, Lepage & Mackenzie). >> We can compute to 3 (or even 4) LOOPS! >> We make use of the idea of BPT and we are able to assess convergence properties and truncation errors of the series. >> We want to assess consistency with NP determinations (if available). This is the case: we will focus on Zp/Zs (see Tarantino @LAT05).

  4. Computational tool (NSPT) F. Di Renzo, G. Marchesini, E. Onofri, Nucl.Phys. B457 (1995), 202 F. Direnzo, L. Scorzato, JHEP 0410 (2004), 73 The main assertion is (remember: η is gaussian noise) We now simply implement on a computer the expansion which is the starting point of Stochastic Perturbation Theory NSPT comes as an application of Stochastic Quantization (Parisi & Wu): the field is given an extra degree of freedom, to be thought of as a stochastic time, in which an evolution takes place according to the Langevin equation Both the Langevin equation and the main assertion get translated in a tower of relations ...

  5. We work in the RI’-MOM scheme: compute quark bilinears operators between (off-shell p) quark states and then amputate to get G functions project on the tree level structure Renormalization conditions read where the field renormalization constant is defined via Renormalization scheme (definitions) Martinelli & alNP 445 (1995) 81 One wants to work at zero quark mass in order to get a mass-independent scheme.

  6. We know which form we have to expect for a generic coefficient Renormalization scheme (comments) We compute everything in PT. Usually divergent parts (anomalous dimensions) are “easy”, while fixing finite parts is hard. In our approach it is just the other way around! We actually take the g’s for granted. See J.Gracey (2003): 3 loops! We take small values for (lattice) momentum and look for “hypercubic symmetric” Taylor expansions to fit the finite parts we want to get. RI’-MOM is an infinite-volume scheme, while we have to perform finite V computations! Care will be taken of this (crucial) aspect.

  7. Computational setup Configurations (some hundreds) up to 3 (4...) LOOPs have been generated and stored in order to perform many computations. - Wilson gauge – Wilson fermion (WW) action on 324 and 164 lattices. • Gauge fixed to Landau (no anomalous dimension for the quark field at • 1 loop level). - nf = 0 (both 324 and 164); 2 , 3, 4 (324). We will focus onnf = 2. • Relevant mass countertem (Wilson fermions) plugged in (in order to • stay at zero quark mass).

  8. Think about tree level ... It works pretty well! For a complete list of reference to analytic results we compare to, please refer to Capitani Phys Rep 382(03)113 1Loop example (Zq) Easy example (no log in Landau gauge): what do we expect for the inverse quark propagator?

  9. A first less trivial example would be 1Loop for the scalar current Just be patient for a few minutes: there is something more direct ... The perfect quantity to compute is the ratio Zp/Zs (or Zs/Zp): - quark field renormalization drops out in the ratio; - no anomalous dimension around; • as an extra bonus, from the point of view of the signals the two quantities are “independent”. Therefore, one can verify that the series are inverse of each other.

  10. We now try to resum (@b-1=5.8) using different coupling definitions: • Series are actually inverse of each other and finite V effects are under • control. Irrelevant effects are taken into account by the “hyp-expans”!

  11. Resummation at fixed order (blue=1,green=2,red=3) vs value of the couplings (x-axis): from left to right x0, x1, x2, x3. • Remember, for this quantity we do not need to know an anomalous dimension. It’s tantalizing, so ... go for 4 loops! • Notice: we know the critical mass counterterm!

  12. There is a clean signal! ... and as a byproduct you get the critical mass to 4 loop.

  13. At fixed coupling milder and milder variations with the order. • At fixed order milder and mildervariations changing the coupling. • Resumming at this order the series are almost inverse of each other.

  14. What do quote as a result? This is our sistematic (truncation) error. Take the phenomenologist’s attitude(deviations from previous order): Zp/Zs = .77(1). This is also consistent with sort of “scaling” of deviations from previous order. Compare to NP (see Tarantino @LAT05) Zp/Zs = .75(1).

  15. About “scaling” of deviations from previous order ... (This of course should not be taken too seriously ...)

  16. A caveat on BOOSTED PERTURBATION THEORY! (a trivial one) We now exaggerate the boosting of coupling: x0, x1, x2, x3, …, xi = b/Ph, … The bottom line is obvious: there is no free lunch in BPT ...

  17. Remember that from our master formula (points are the signal, crosses signal minus log) We now go back to Zs (1 LOOP)

  18. Much the same holds for Zp, so apparently there is a common problem.

  19. We plot the signal for the scalar current, the pseudoscalar current and their ratio (guess which is which!) on 324 and 164. Again, 1 LOOP. It is a finite volume effect!

  20. Much the same holds at 2 LOOP ...

  21. What does such a “tamed”-log look like? Compute it in the continuum: this should be a pL effect. Example: look for the “log-signature” for the sunset (result plotted vs log(p2), so it should be a straight line with slope dictated by anomalous dimension). Remember: apparently the log is “tamed” by finite volume.

  22. This is a way of drawing which is closer to what we saw: log (diamonds) and “tamed-log” (circles) on the finite size we are interested in. … so take this signal for the “tamed”-log and plug it into our subtraction!

  23. It works! Here is the signal for Zs

  24. ... and here comes Zp

  25. PRELIMINARY!

  26. Conclusions and perspectives >>NSPT can give you a valuable tool for computation of Z ‘s. >> The effect of Boosted PT can be carefully assessed and convergence properties (which can be not so bad!) can be inspected. >> Care should be taken for finite volume when log’s are in place. >> Configurations are there: many computations are possible (also in other fermionic schemes). Moreover, improvement is on the way ...

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