1. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics

2. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 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-Universität, Berlin, Germany

3. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics OutlineMotivation 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) Perspectives … there is much to do! Quarks bilinears at 3 (4) loops Z’s in the RI’-MOM scheme. The “perfect” case: Z p /Z s. Treating anomalous dimensions (“tamed” log’s).

4. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 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). >> Often (large) use is made of Boosted PT (Parisi, Lepage & Mackenzie). >> LPT converges badly and usually computations are 1 LOOP (analytic 2 LOOP on their way). (analytic 2 LOOP on their way). >> 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. 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 Z p /Z s (see Tarantino @LAT05). This is the case: we will focus on Z p /Z s (see Tarantino @LAT05).

5. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics Computational tool (NSPT) F. Di Renzo, G. Marchesini, E. Onofri, Nucl.Phys. B457 (1995), 202 F. Direnzo, L. Scorzato, JHEP 0410 (2004), 73 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... 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

6. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics Renormalization scheme (definitions) Martinelli & al NP 445 (1995) 81 One wants to work at zero quark mass in order to get a mass- independent scheme. We work in the RI’-MOM scheme: compute quark bilinears operators between (off-shell p) quark states and then amputate to get  functions project on the tree level structure where the field renormalization constant is defined via Renormalization conditions read

7. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 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  ’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. We know which form we have to expect for a generic coefficient

8. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 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 32 4 and 16 4 lattices. - Gauge fixed to Landau (no anomalous dimension for the quark field at 1 loop level). 1 loop level). - n f = 0 (both 32 4 and 16 4 ); 2, 3, 4 (32 4 ). We will focus on n f = 2. - Relevant mass countertem (Wilson fermions) plugged in (in order to stay at zero quark mass). stay at zero quark mass).

9. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 1Loop example (Z q ) Easy example (no log in Landau gauge): what do we expect for the inverse quark propagator? 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

10. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics The perfect quantity to compute is the ratio Z p /Z s (or Z s /Z p ): A first less trivial example would be 1Loop for the scalar current Just be patient for a few minutes: there is something more direct... - 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.

11. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics - Series are actually inverse of each other and finite V effects are under control. Irrelevant effects are taken into account by the “hyp-expans”! control. Irrelevant effects are taken into account by the “hyp-expans”! - We now try to resum (@  -1 =5.8) using different coupling definitions:

12. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics - 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! Resummation at fixed order (blue=1,green=2,red=3) vs value of the couplings (x-axis): from left to right x 0, x 1, x 2, x 3.

13. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics There is a clean signal!... and as a byproduct you get the critical mass to 4 loop. critical mass to 4 loop.

14. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics - At fixed coupling milder and milder variations with the order. - At fixed order milder and milder variations changing the coupling. - Resumming at this order the series are almost inverse of each other.

15. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics What do quote as a result? This is our sistematic (truncation) error. Take the phenomenologist’s attitude (deviations from previous order) : Z p /Z s =.77(1). This is also consistent with sort of “scaling” of deviations from previous order. Compare to NP (see Tarantino @LAT05) Z p /Z s =.75(1).

16. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics About “scaling” of deviations from previous order... (This of course should not be taken too seriously...)

17. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics A caveat on BOOSTED PERTURBATION THEORY! (a trivial one) We now exaggerate the boosting of coupling: x 0, x 1, x 2, x 3, …, x i =  /P h, … The bottom line is obvious: there is no free lunch in BPT...

18. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics We now go back to Z s (1 LOOP) Remember that from our master formula (points are the signal, crosses signal minus log)

19. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics Much the same holds for Z p, so apparently there is a common problem.

20. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics It is a finite volume effect! We plot the signal for the scalar current, the pseudoscalar current and their ratio (guess which is which!) on 32 4 and 16 4. Again, 1 LOOP.

21. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics Much the same holds at 2 LOOP...

22. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics Remember: apparently the log is “tamed” by finite volume. 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(p 2 ), so it should be a straight line with slope dictated by anomalous dimension).

23. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics 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!

24. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics It works! Here is the signal for Z s

25. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics... and here comes Z p

26. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics PRELIMINARY!

27. 3 (or 4!) loops renormalization constants for lattice QCD Francesco Di Renzo Nicosia - September 14, 2005 Workshop on Computational Hadron Physics >> The effect of Boosted PT can be carefully assessed and convergence properties (which can be not so bad!) can be inspected. >> NSPT can give you a valuable tool for computation of Z ‘s. >> 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... Conclusions and perspectives