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 QCD3 (or 4!) loops renormalization constants for lattice QCDFrancesco Di RenzoFrancesco Di Renzo

Nicosia - September 14, 2005Nicosia - September 14, 2005Workshop on Computational Hadron PhysicsWorkshop on Computational Hadron Physics



3 (or 4!) loops 3 (or 4!) loops renormalization constants renormalization constants

for lattice QCDfor lattice QCDF. Di Renzo, A. Mantovi, V. Miccio and C. TorreroF. Di Renzo, A. Mantovi, V. Miccio and C. Torrero ((11))

&& L. ScorzatoL. Scorzato ((22))

(1) (1) Università di Parma Università di Parma andand INFN, INFN, Parma, ItalyParma, Italy

(2) (2) Humboldt-UniversitHumboldt-Universitäät,t,Berlin, GermanyBerlin, Germany



OutlineOutline

MotivationMotivationRenormalization constantsRenormalization constants in ( in (LatticeLattice) ) Perturbation Perturbation TheoryTheory::is it really a second choice? Can you compare PT and is it really a second choice? Can you compare PT and NP?NP?Computational SetupComputational SetupNumerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory ((Parma Parma group after group after Parisi & WuParisi & Wu))

PerspectivesPerspectives… … there is much to do!there is much to do!

Quarks bilinears at 3 (4) loopsQuarks bilinears at 3 (4) loopsZZ’s in the ’s in the RI’-MOMRI’-MOM scheme. The “perfect” case: scheme. The “perfect” case: ZZpp/Z/Zss. . Treating anomalous dimensions (Treating anomalous dimensions (“tamed” log’s“tamed” log’s).).



Renormalization constantsRenormalization constants and and LPTLPTDespite the fact that there is no theoretical obstacle to computing 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. log-div RC in PT, on the lattice one tries to compute them NP. Popular (intermediate) schemes are Popular (intermediate) schemes are RI’-MOMRI’-MOM ( (Rome groupRome group) and ) and SFSF ( (alpha Collalpha Coll).).

>>>> Often (large) use is made of Often (large) use is made of Boosted PTBoosted PT ( (ParisiParisi, , Lepage & Lepage & MackenzieMackenzie).).

>>>> LPT converges badlyLPT converges badly and usually computations are and usually computations are 1 LOOP1 LOOP (analytic 2 LOOP on their way).(analytic 2 LOOP on their way).

>>>> We can compute to We can compute to 33 (or even (or even 44) ) LOOPSLOOPS!!

>>>> We make use of the idea of BPT and we are able to assess We make use of the idea of BPT and we are able to assess convergenceconvergence properties and truncation errorsproperties and truncation errors of the series. of the series. >>>> We want to assess We want to assess consistency with NP determinations consistency with NP determinations (if (if available). available). This is the case: we will focus on This is the case: we will focus on ZZpp/Z/Zss (see (see TarantinoTarantino @LAT05). @LAT05).



Computational tool (Computational tool (NSPTNSPT))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 NSPT comes as an application of Stochastic QuantizationStochastic Quantization ( (Parisi & WuParisi & Wu): ): the field is given an extra degree of freedom, to be thought of as a 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 stochastic time, in which an evolution takes place according to the LangevinLangevin equation equation

Both the Langevin equation and the main assertion get translated in Both the Langevin equation and the main assertion get translated in a a tower of relationstower of relations ... ...

The main assertion is (remember: The main assertion is (remember: ηη is is gaussian noisegaussian noise))

We now simply implement on a computer the We now simply implement on a computer the expansionexpansion which is the which is the starting point of Stochastic Perturbation Theorystarting point of Stochastic Perturbation Theory



Renormalization scheme Renormalization scheme (definitions) (definitions) Martinelli & alMartinelli & al NP NP 445 (1995) 81445 (1995) 81

One wants to work at One wants to work at zero quark masszero quark mass in order to get a in order to get a mass-mass-independent schemeindependent scheme..

We work in the We work in the RI’-MOMRI’-MOM scheme: compute quark bilinears scheme: compute quark bilinears operators between (off-shell p) quark states and then amputate operators between (off-shell p) quark states and then amputate to get to get functions functions

project on the tree level structureproject on the tree level structure

where the field renormalization constant is defined viawhere the field renormalization constant is defined via

Renormalization conditions read Renormalization conditions read



Renormalization scheme Renormalization scheme (comments)(comments)We compute everything in PT. Usually divergent parts We compute everything in PT. Usually divergent parts (anomalous dimensions) are “easy”, while fixing finite parts is (anomalous dimensions) are “easy”, while fixing finite parts is hard. In our approach it is just the other way around! hard. In our approach it is just the other way around!

We actually take the We actually take the ’s for granted. See ’s for granted. See J.GraceyJ.Gracey ( (20032003): 3 ): 3 loops!loops!

We take small values for (lattice) momentum and look for We take small values for (lattice) momentum and look for “hypercubic symmetric” Taylor expansions“hypercubic symmetric” Taylor expansions to fit the finite parts to fit the finite parts we want to get.we want to get.RI’-MOMRI’-MOM is an is an infinite-volume schemeinfinite-volume scheme, while we have to , while we have to perform perform finite V computationsfinite V computations! Care will be taken of this ! Care will be taken of this (crucial) aspect.(crucial) aspect.

We know which form we have to expect for a generic We know which form we have to expect for a generic coefficientcoefficient



Computational setupComputational setupConfigurations (some hundreds) up to Configurations (some hundreds) up to 33 ( (44...) ...) LOOPsLOOPs have been have been generated and stored in order to perform many computations.generated and stored in order to perform many computations.

- Wilson gauge – Wilson fermion (- Wilson gauge – Wilson fermion (WWWW) action on ) action on 323244 and and 161644 lattices.lattices.- Gauge fixed toGauge fixed to Landau Landau (no anomalous dimension for the quark (no anomalous dimension for the quark field atfield at 1 loop level).1 loop level).- - nnff = 0 = 0 (both (both 323244 and and 161644); ); 22 , , 33,, 4 4 ( (323244). ). We will focus onWe will focus on nnf f = 2= 2..

- Relevant Relevant mass countertemmass countertem (Wilson fermions) plugged in (in (Wilson fermions) plugged in (in order toorder to stay at zero quark mass).stay at zero quark mass).



1Loop 1Loop example example ((ZZqq))Easy example (no log in Landau gauge): what do we expect for Easy example (no log in Landau gauge): what do we expect for the inverse quark propagator?the inverse quark propagator?

Think about tree level Think about tree level ......

It works pretty well!It works pretty well!

For a complete list of reference to analytic results For a complete list of reference to analytic results we compare to, please refer to we compare to, please refer to Capitani Phys Rep Capitani Phys Rep 382(03)113382(03)113

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

(pa)2

Z q(1)



The perfect quantity to compute is the ratio The perfect quantity to compute is the ratio ZZpp/Z/Zss (or (or ZZss/Z/Zpp): ):

A first less trivial example would be A first less trivial example would be 1Loop1Loop for the for the scalar currentscalar current

Just be patient for a few minutes: there is something more Just be patient for a few minutes: there is something more direct ...direct ...

- quark field renormalization drops out in the ratio;- quark field renormalization drops out in the ratio;

- - no anomalous dimensionno anomalous dimension around; around;

- as an extra bonus, from the point of view of the signals the two as an extra bonus, from the point of view of the signals the two quantities are “independent”. Therefore, one can verify that the quantities are “independent”. Therefore, one can verify that the series are inverse of each other.series are inverse of each other.



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

0 0.2 0.4 0.6 0.8 1-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

(pa)2

(Zp/Z

s)(1)

0 0.2 0.4 0.6 0.8 1-1.5

-1.45

-1.4

-1.35

-1.3

-1.25

(pa)2

(Zp/Z

s)(2)

0 0.2 0.4 0.6 0.8 1-5.4

-5.3

-5.2

-5.1

-5

-4.9

-4.8

-4.7

-4.6

(pa)2

(Zp/Z

s)(3)



- Remember, for this quantity we do not need to know an Remember, for this quantity we do not need to know an anomalous dimension. It’s tantalizing, so ... anomalous dimension. It’s tantalizing, so ... go for 4 loops!go for 4 loops!

Notice: we know the critical mass counterterm!Notice: we know the critical mass counterterm!

Resummation at fixed order (Resummation at fixed order (blueblue=1,=1,greengreen=2,=2,redred=3) vs value of the couplings =3) vs value of the couplings (x-axis):(x-axis):from left to right xfrom left to right x00, x, x11, x, x22, x, x33..

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.30.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92ZP/ZS

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.31.08

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26ZS/ZP



There is a clean signal!There is a clean signal!

... and as a byproduct you get the ... and as a byproduct you get the critical mass to 4 loopcritical mass to 4 loop..

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-21.5

-21

-20.5

-20

-19.5

-19

(pa)2

(Zp/Z

s)(4)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.830

31

32

33

34

35

36

37

38

39

40

(pa)2

mcr(4

)



- At fixed coupling milder and milder variations with the order.At fixed coupling milder and milder variations with the order.

- At fixed order milder and milderAt fixed order milder and milder variations changing the variations changing the coupling.coupling.- Resumming at this order the series are almost inverse of each Resumming at this order the series are almost inverse of each other.other.

0.15 0.2 0.25 0.30.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92Zp/Zs

0.15 0.2 0.25 0.31.08

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26

1.28Zs/Zp



What do quote as a result? This is our sistematic (truncation) What do quote as a result? This is our sistematic (truncation) error.error.Take the phenomenologist’s attitudeTake the phenomenologist’s attitude (deviations from previous order)(deviations from previous order): : ZZpp/Z/Zss

= .77(1)= .77(1)..This is also consistent with sort of “scaling” of deviations from This is also consistent with sort of “scaling” of deviations from previous order. previous order.

0.15 0.2 0.25 0.30.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92Zp/Zs

0.15 0.2 0.25 0.31.08

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26

1.28Zs/Zp

Compare to NP (see Compare to NP (see TarantinoTarantino @LAT05) @LAT05) ZZpp/Z/Zss = .75(1) = .75(1)..



About “scaling” of deviations from previous order ...About “scaling” of deviations from previous order ...

(This of course should not be taken too seriously ...)(This of course should not be taken too seriously ...)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

N-1.2

(N

)



A caveat on BOOSTED PERTURBATION THEORY! A caveat on BOOSTED PERTURBATION THEORY! (a trivial one)(a trivial one)

We now exaggerate the boosting of coupling: We now exaggerate the boosting of coupling: xx00, , xx11, , xx22, , xx33, …, , …, xxii = = /P/Phh, …, …

The bottom line is obvious: there is no free lunch in BPT ... The bottom line is obvious: there is no free lunch in BPT ...

0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.7

0.8

0.9

1

1.1

1.2

1.3Zp/Zs



We now go back to We now go back to ZZs s (1 LOOP)(1 LOOP)

Remember that from our master formula Remember that from our master formula (points are the signal, crosses signal (points are the signal, crosses signal minus log)minus log)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

(pa)2

Z s(1)



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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

(pa)2

Z p(1)



It is a It is a finite volume effectfinite volume effect!!

We plot the signal for the We plot the signal for the scalarscalar current, the current, the pseudoscalarpseudoscalar current and current and their their ratioratio (guess which is which!) on (guess which is which!) on 323244 and and 161644. Again, 1 . Again, 1 LOOP.LOOP.

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(pa)2



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

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

(pa)2



Remember: apparently the Remember: apparently the log is “tamed” by finite volumelog is “tamed” by finite volume..What does such a What does such a “tamed”-log“tamed”-log look like? Compute it in the look like? Compute it in the continuum: this should be a continuum: this should be a pLpL effect. Example: look for the effect. Example: look for the “log-signature” for the sunset (result plotted vs log(p“log-signature” for the sunset (result plotted vs log(p22), so it ), so it should be a straight line with slope dictated by anomalous should be a straight line with slope dictated by anomalous dimension).dimension).

0 0.5 1 1.5 2 2.5 3 3.50.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

log(p2)

(1)



This is a way of drawing which is closer to what we saw: log 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 (diamonds) and “tamed-log” (circles) on the finite size we are interested in.interested in.

… … so take this signal for the “tamed”-log and plug it into our so take this signal for the “tamed”-log and plug it into our subtraction!subtraction!

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.005

0.01

0.015

0.02

0.025

(pa)2



It works! Here is the signal for It works! Here is the signal for ZZss

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(pa)2

Z s(1)



... and here comes ... and here comes ZZpp

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

(pa)2

Z p(1)



0.2 0.25 0.3 0.350.74

0.76

0.78

0.8

0.82

0.84

0.86

0.22 0.24 0.26 0.28 0.30.74

0.76

0.78

0.8

0.82

Zq

0.2 0.25 0.3 0.350.65

0.7

0.75

0.8

0.85

0.9

0.22 0.24 0.26 0.28 0.30.68

0.7

0.72

0.74

0.76

0.78

ZV

0.2 0.25 0.3 0.350.76

0.78

0.8

0.82

0.84

0.86

0.88

0.22 0.24 0.26 0.28 0.30.76

0.78

0.8

0.82

0.84

ZA

PRELIMINARY!PRELIMINARY!



>>>> The effect of The effect of Boosted PTBoosted PT can be carefully assessed and can be carefully assessed and convergence properties (which can be not so bad!) can be convergence properties (which can be not so bad!) can be inspected.inspected.

>>>> NSPT NSPT can give you acan give you a valuable tool valuable tool for computation of for computation of Z Z ‘s.‘s.

>>>> Care should be taken for Care should be taken for finite volumefinite volume when when loglog’s are in place.’s are in place.

>>>> Configurations are there: many computations are possible Configurations are there: many computations are possible (also in other fermionic schemes). Moreover, improvement is on (also in other fermionic schemes). Moreover, improvement is on the way ...the way ...

Conclusions and perspectivesConclusions and perspectives

Documents

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