- Email: [email protected]
Quantum Chromodynamics on the lattice
i
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A623 (1997) 81c-89c
Quantum Chromodynamics on the Lattice F. Rapuano Dipartimento di Fisica, Uuiversit£ di Roma 'La Sapienza' and INFN, Sezione di Roma, P.le A. Moro 2, 1-00185 Roma, Italy. In this talk I give a brief overview of Quantum Chromodynamics on the Lattice and report on some of the most recent results obtained in the field. 1. I N T R O D U C T I O N The theoretical knowledge of the fundamental interactions that we have reached today is quite deep. In particular it is widely believed that strong interactions are described by Quantum Chromodynamics (QCD). QCD has the property of asymptotic freedom that opens the possibility of using perturbative methods in the short distance region where the running coupling constant a,(q2), q~ is the scale relevant to the process under consideration, is small. In the perturbative regime QCD accurately reproduces experimental data. The situation is much more complicated for all processes that involve long distance interactions. One must find reliable non-perturbative methods. The lattice formulation of QCD has proven to be a first principle method which has no a-priori assumptions, can be systematically improved and is well suited to numerical calculations. My talk is organized as follows: in section 2 I give a brief introduction to the lattice method. In section 3 I discuss the main sources of errors and in section 4 I describe some recent results. 2. T H E L A T T I C E M E T H O D The Lattice formulation of QCD (LQCD) [1] is based on the functional formulation of Quantum Field Theory. The expectation value of a generic operator (9 is defined as:
-_ ½ f II dtv.]
(1)
where U,,, ~bq and Cq are the fields of the theory, respectively gluons, quarks and antiquarks of flavour q and SQCDis the QCD action. Z is the same integral without (9 in such a way that (1) = 1. Wilson [2] proposed, as a prescription to define the theory and the integral in eq.(1), to discretize space-time on a finite 4-dimensional mesh of sites with spacing a. Quarks on the sites exchange gluons on the links in such a way that local SU(3) gauge invariance is preserved. We have then a regularized theory with the introduction of an UV cutoff due to the finiteness of the lattice spacing a, and an IR cutoff due to the finite space-time volume. The introduction of a finite and discrete space-time makes by itself the number 0375-9474/97/$17.00 © 1997 - Elsevier Science B.V. All rights reserved. PII: S0375-9474(97)00425-9
E Rapuano /Nuclear Physics A623 (1997) 81c~89c
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of degrees of freedom in eq.(1) finite but still the integral is analytically unmanageable even for a moderate number of lattice sites. Going to Euclidean time via a Wick rotation allows the use of statistical techinques to evaluate the integral in eq.(1) as we now have a real positive exponential factor that can be interpreted as a probability distribution of field configurations. The technique that is used to numerically evaluate (1) consists in the generation of a set of independent field configurations with probability ezp(-SQvD) with a Montecarlo method. On each of these field configurations one measures the value of O. The average value is then: 1 ~v,
=
Z o, +
(2)
{=I
where Arc is the number of independent field configurations, Oi is the value of O on the i-th field configuration and 6 represents both the statistical error introduced by the evaluation the integral and the systematic error due the discretization of space-time. The QCD Lattice action does not contain dimensionful parameters. AU dimensionful quantities are expressed in terms of powers of the lattice spacing a. The continuum limit is reached, in QCD, in the limit of vanishing coupling constant 9. The relation between # and the lattice spacing a is obtained by perturbatively solving the renormalization group equation. In turn one can replace the cut-off with a dimensionful scale parameter ALat which is not fixed by the theory. For a lattice measure to have a meaning in the continuum one should check cut-off independency: physical quantities expressed in terms of ALat should not depend on the value of the coupling constant g used in the simulation when the value of g is small enough (or a -1 is large enough). Then if one physical quantity (e.g. the string tension or a hadron mass) is used to fix AL~t, all other physical quantities will be lattice predictions. 3. T H E P R O B L E M
OF ERRORS
The discretization of the theory is not a trivial issue. On one hand one may have lost symmetries whose restoration in the continuum limit should explicitly be verified (rotational invariance is an obvious example. See [1] for a discussion of the question of symmetry breaking on the lattice). On the other hand, as stated before, we are introducing various sources of errors that should be carefully analized to be sure of the reliability of the measures. 3.1. S t a t i s t i c a l error In some sense this is the simplest to estimate. If the set of N~ configurations on which the measure is performed axe independent we have that 6,t,t is of order 1/v/N~. At present all reliable lattice simulations are done on samples of hundreds of configurations that reasonably represent the configuration space, and the independence of the sample of configurations can be easily verified. Statisticalerror is then quite under control and does not deserve any further discussion. 3.2. Systematic error This is a m u c h more delicate problem in lattice Q C D . The sources of systematic errors are m a n y and each one poses different problems. I will analyse the most rclewnt ones
E Rapuano/Nuclear Physics A623 (1997) 81c~9c
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and the possible solutions. • Finite lattice spacing. The value of a in recent simulations ranges between 0.05 and 0.1 fm. This corresponds to an UV cut-off lr/a ~ 6 + 12 GeV. It is very difficult to estimate apriori whether these values can be considered large enough. Many simulations have checked that an approximate scaling behaviour is observed at 92 < I for pure gauge quantities and light quark hadronic spectrum. Other quantities like heavy quark systems show quite larger scaling violations. An important point is that the discretized gauge part SG of SLQCD = So + SF has scaling violations O(a 2) while the fermionic part SF, discretized according to the Wilson procedure, has substantially larger O(a) violation. In principle one can remove these terms by a proper definition of the lattice action and operators, as proposed by Symanzik [3]. Many groups have proposed different methods of "improvement" of the lattice action [4]. The most promising one is an all-order non-perturbative improvement that reduces discretization errors to O(a 2) [5]. I cannot describe these techniques for lack of time. • Finite Volume. This error is, in today's simulation, quite small. Finite volumes leads to the use of unphysically large quark masses in such a way that for the typical correlation length involved in the simulation (e.g. ~ ~ l i m a where m x is a hadron mass) one has << L, L being the linear spacial extention of the lattice. One then is forced to work with light quark masses mq ~ 100 MeV or larger and then extrapolate to the physical quark masses with an error that, in any case, is not overwhelmingly large. • The Quenched Approximation. So far we have described errors that are intrinsic to the numerical simulation of the discretized theory. Actually to cope with the available computer power, we shall now make an approximation. The fermionic part of the lattice action is:
SF = ~bq(J~+ mq)~bq
(3)
where ~ is the discretized version of the QCD covariant derivative and mq is the bare mass of the quark of flavour q. The fermionic degrees of freedom can be analitically integrated:
f
= a t( 9 +
(4)
and one can introduce an effective action:
e-s"' = II aet( q
+ mq) -So
(5)
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E Rapuano /Nuclear Physics A623 (1997) 81c-89c
as a weight distribution. One can show that the discretized gluon action Sa is a local quantity (i.e. depending only on neighouring links) which is easy to evaluate numerically. The determinant instead is a highly non-local quantity which represents a formidable problem from the computational point of view. The quenched approximation consists in setting the fermionic determinant to one. This amounts to neglect the effects of virtual quark loops. Quarks propagate then as valence quarks in the background gauge fields. While arguments like the approximate validity of the Zweig rule and the small amount of momentum carried by sea-quarks in hadrons support the quenched approximation, it is difficult to quantify its effects, as different observables will be affected by fermion loops in a different fashion. As an example we have that a p meson will become stable (i.e. F = 0) and we can naively estimate that its mass will be affected by a shift ~m ~ gF = F. So the error on the width is of course 100 % while the error on the mass is much smaller. The pion-nucleon a-term also is a quantity that should be very much influenced by quark loops. One of the latest determinations, [6] gives a~N --~ 45 MeV. It is well known that the calculation of the a-term in absence of sea-quarks gives a much smaller result [7]:
2[
_ 3
m,~
m= - mN +
m mA]
~ 26MeV
(6)
Lattice simulations in the quenched case [8] agree quite well with this result while preliminary unquenched simulations [9] seem to confirm that the sea-quark contibution is substantial. In conclusion the systematic error coming from the quenched approximation may of the order 10 + 15% on hadron masses but can be larger for other observables. At present many groups are making simulations taking into account quark loops. The results have quite limited statistics and may suffer from finite volume effects. The removal of the quenched approximation is a step that should be achieved with the next generation of computer with speeds of the order of one to ten TFlops. 4. R E S U L T S It is practically impossible to summarize all results obtained in LQCD in the last years. I will concentrate on those that I consider more relevant to confirm the lattice as a reliable non-perturbative method and onto some phenomenologically important predictions. For a full review of the state of the art of lattice simulation the proceedings of the annual Lattice QCD conference is the best reference. 4.1. T h e h a d r o n s p e c t r u m Hadron masses are computing from the so called two-point correlation functions:
(7)
n
where H(t) is an interpolating operator with the quantum number of the particle under study (e.g. qTsq for a pion, (/Tiq for a p meson etc.), E,, is the energy of the n-th radial
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Table 1 Predicted meson masses in GeV for various lattices
Exper. C60 W60 C62a W62a C62b W62b W64 C64
Mp M,.. 0.770 0.686 (see [10]) 0.809(7) 0.6849(3) 0.808(3) 0.6849(1) 0.81(1) 0.6849(5) 0.803(6) 0.6851(2) 0.79(1) 0.6856(5) 0.797(7) 0.6853(3) 0.796(4) 0.6853(2) 0.792(4) 0.6855(2)
M~ 1.019 0.977(7) 0.978(3) 0.98(1) 0.984(6) 1.00(1) 0.989(7) 0.990(4) 0.994(4)
Table 2 As in table 1 for baryon masses
MN
MA~
M~
Ma
Mo
Exper.
0.9389
1.135
1.3181
1.232
1.6724
C60 W60 C62a W62a C62b W62b W64 C64
1.09(5) 1.19(5) 1.1(1) 1.17(7)
1.21(4) 1.29(4) 1.22(8) 1.28(6)
1.32(4) 1.3(1) 1.60(9) 1.40(4) 1.46(7) 1.71(4) 1.34(7) 1.39(5)
1.1(2)
1.2(2)
1.4(1)
1.6(3)
1.9(2)
1.2(1) 1.3(1) 1.40(9) 1.50(9) 1.72(5) 1.21(9) 1.32(8) 1.43(6) 1.4(2) 1.72(9) 1.2(1) 1.29(8)1.41(7) 1.3(2) 1.7(1)
excited state. At large time separation and zero momentum, a single particle exponential behaviour will set in. From a two paramer tlt one can tllen extract the mass of the lowestlying state. The leptonic decay constant of the pion, f~ can be similarly obtained from the correlation of the fourth component of the axial current (A0(t)A*0(0)). I will report the results obtained in a large statistics study (N~ > 200) at values of a ranging from 0.05 to 0.1 fm (a -1 ranging between 2.0 and 4.0 GeV) and lattices sizes from 183 > 64 to 24 a × 64 [10]. The data show a quite good scaling behaviour for all quantities under study except for the vector meson decay constant. Table 1-3 summarize the results obtained by fixing the value of a from the mass of the K* meson. Lattices axe labeled with a W or a C according to the fermionic action used, ordinary Wilson [2] (W) or tree-level improved [4] (C), and by the value of/~ = 6 / f . Masses and pseudoscalar decay constants agree within 5 to 15% to the experimental data. The quenching error can be more relevant for the nucleon mass. Vector mesons decay constants are, on the contrary, quite far from the physical values. Preliminary studies seem to indicate that the non-perturbative O(a ~) [5] removal is particularly important for this quantity.
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Table 3 As in table 1 for the leptonic decay constants f.(GeV) Exper. C60 W60 C62a W62a C62b W62b W64 C64
0.1307 0.134(9) 0.161(7) 0.124(9) 0.140(6) 0.14(2) 0.140(8) 0.15(1) 0.144(9)
1
0.28 0.35(3) 0.41(2) 0.33(3) 0.38(2) 0.26(4) 0.39(2) 0.32(1) 0.25(1)
fr(GeV) 0.1598 0.149(8) 0.173(6) 0.143(8) 0.157(5) 0.16(2) 0.157(7) 0.166(8) 0.158(8)
I
1
fK*
f, 0.23 0.30(1) 0.357(8) 0.281(7) 0.335(6) 0.25(2) 0.34(1) 0.296(6) 0.243(8)
0.33(2) 0.38(1) 0.30(2) 0.36(1) 0.25(3) 0.37(2) 0.31(1) 0.25(1)
A particularly interesting result comes from the UKQCD collaboration for the spectroscopy of heavy baryons [11] on a lattice of size 243 × 48 at a value of a -1 of 2.9 ± 0.2 GeV. The simulation is performed at a heavy quark mass close to the charm mass and an extrapolation is performed to the mass of the beauty quark. Their results are shown in table 4. The lattice data agree rather well, within the errors, with the experimental data, when available. 4.2. Heavy-light mesons leptonic decay constants.
Conceptually the technique is the same as for the fight quark systems even though various methods have been developed to increase the reliability of the prediction as far as the extraction of the low-lying state is concerned. The decay constant of the B meson is experimentally not measured yet and its knowledge is extremely important in the Standard Model because it would allow a determination, via the measurement of the leptonic decay width, of IV~I which is known with a quite large error. Also it is connected to the amount of CP violation in the process B ~ ~bK, [12] There are many results on this subject and I have reported some of them in table 5 which is taken from ref. [13] with some recent data added and various modifications. It should be considered that these simulations have different statistics and systematics; see ref. [13] for a detailed description of the data and the original references. The leptonic decay constant of the D, meson, fD,, has been measured by various experiments and the experimental world average for gives a value of 241 + 21 :k 30 MeV [14] which is very good agreement with the lattice ones. 4.3. Other predictions.
I will conclude with a short list of other predictions that are quite important. There are recent analysis about quark masses that give ~ = r~,+,,,~ = 3.1+0.3 MeV, [15] for the fight 2 quak mass and m, = 100+21+10 MeV [15] and ra, = 100+21+10 MeV [10] for the strange quark mass. A large amount of work is currently underway to simulate semileptonic decays of heavy quarks [16]. The calculation involves now three point functions and are quite more computing intensive than the two point functions needed for the spectrum. The
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Table 4 Heavy baryons masses from ref. [11]. Results are quoted with a statistical error (first) and a systematic error (second) arising from the uncertainty in the calibration of the lattice spacing. Where available, the experimental data are reported. h =
Baryon
Quark Content
Ah ~h ~ Eh =' --h --h=* ~h
(ud)h (uu)h (uu)h (us)h (us)h (u~)h (ss)h
Exp. (MeV) 2285(1)
charm
Latt. (GeV) 2.27 +4_s +s 3-
h =
Exp. (MeV)
5641(50) 5814(60) 5870(60)
beauty
Latt. (GeV) 5.64 +5 +s -5 -2
2453(1)
2.46 +~ +s -3 -S
2530(7) 2468(4) 2560 t
2.44 +~- -5+4 2.41 +a +4 -3 -4 2.57 +s +6 -3 -6
2643(2)
2.55 +s +s -4 -5
5.90 +4 +4 -6 -5
2.68 -4+s +s_s
5.99 +5 +s -5 -5
2.66 +~_ +s_~
6.00
2704(20)
5.77 +s +4 -6 -4
5.78 +5 +4 -6 -3 5.76 +3 +4 -5 --3 5.90 +s +4 -6 -4
+4 +5 -5 -5
results for the decays D --* K and D -+ K* give, for the form factors that parametrize the amplitudes, agree quite well with the experimental ones. As for the decay constants, this supports the predictions for the processes B --+ ~r and B --* p and the determination of the Isgur and Wise function. Other important results for the study of CP violation in the B system is the mixing parameter BB(#) = ( B- I O I B ) / Js ~ 2M ~~. where O is the AB = 2 operator b,/,(1-Ts)ql~7,(1-75)q at a scale #. Ref. [13] again is a good starting point for a study of the available results. Open problems like the value of the CP violation parameters e'/e and the A I = 1/2 rule are the goal of the lattice theoretical developements. These problems are particularly hard ones because of the mixing of the relevant operators with operators, which are absent in the continuum, and that being divergent, should be non perturbatively subtracted on the lattice. 5. C O N C L U S I O N S I believe that the lattice can be considered a reliable tool for non-perturbative calculations. The good agreement between simulated and measured quantities makes us confident that the predictions for free parameters of the Standard Model and are quite reliable. The error sources are under control and further improvements are possible. The quenched approximation is only due to the available computer power but we will be able to simulate full QCD with the next generation of computers. One word is worth saying about the role that special purpose computers have had in the developement of Lattice Quantum Chromodynamics. Many projects have been realized. From the first generation projects designed at Columbia [17], at IBM [18], at INFN-Italy [19] and in Japan [20] with speed in the range 1 + 10 GFlops to the most advanced one like the APE-mille project at INFN that will reach a speed of 1 TFlops, all have given a tremendous boost to these calculations.
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Table 5 Summary of results for values and ratios of leptonic decay constants of pseudoscalar mesons. Results with no quoted values for fl or a -1 come from more than one lattice Group
fl
a -1
fB
lB,
(GeV) (MeV) (MeV) 6.0 1.9 6.0 197(18) 6.O 2.O 176(~I) 6.0 2.33 6.1 2 6.2 2.7 160(~) 194(~) 6.3 3.0 187(38) 207(41) 6.4 3.3
DeG-L APE UKQCD LANL BDHS UKQCD BLS ELC ELC ELC PSI-WUP LANL 6.2 APE
205(40)
2.6
180(32) 205(35)
fBo/fS
fD
(MeV) 190(33) 218(9) 1.17(12) 199(~) 229(~) 174(53) 1.22(~) 185(4~) 212(4~) 1.11(6) 208(38) 210(40) 194(15) 1.08(6)
1.14(8)
fD.
fD,/fD
(MeV) 222(16) 1.17(22) 240(9) 1.11(1) 1.13(9) 260(I~) 1.14(2) 234(72) 1.35(22) 1.18(2) 230(36) 1.11(6) 230(50)
198(17) 200(18) 186(29) 218(15) 221(17) 237(16)
1.07(4)
REFERENCES
1. For a detailed discussion of all aspects of LQCD see: H.J. Rothe, Lattice Gauge Theory, World Scientific, Singapore, 1992. I. Montvay, G. Muenster, Quantum Fields on a Lattice, Cambridge University Press, Cambridge, 1994. 2. K. Wilson, Phys. Rev D10 (1974) 2445. 3. K. Simanzik, Nucl. Phys. B226 (1983) 187, 205. 4. B. Sheikoleslami and R. Wohlert, Nucl. Phys. B259 (1985) 572. G. Heatly et al. Nucl. Phys. B413 (1994) 461, and refs. therein. 5. M. Lfischer et al., 14th International Symposium on Lattice Field Theory, St. Louis, 4-8 June 1996,hep-lat/9608049 and refs. therein. 6, J. Gasser, H. Leutwyler, M.E. Sainio, Phys. Left. B253 (1991) 252. 7. T.P. Cheng, Phys. Rev. D13, (1976) 2161. R.L. Jaffe, Phys. Rev. D21, (1980) 3215. 8. Ape Collaboration Phys. Left. B258 (1991) 195. 9. M. Fukugita, Y. Kuramashi, M. Okawa, A. Ukawa, Phys.Rev. D51 (1995) 5319. 10. C. Allton, V. Gimenez, L. Giusti, F. Rapuano, hep-lat/9611021. 11. UKQCD Collaboration (K.C. Bowler et al.), Phys.Rev. D54 (1996) 3619. 12. M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, Nucl.Phys. B369 (1992) 139. J.L. Rosner, Rev. Mod. Phys. 64 (1992), 1151. 13. J. Flynn, 14th International Symposium on Lattice Field Theory, St. Louis, 4-8 June 1996, hep-lat/9610010. 14. J. D. Richman, XXVIII International Conference on High Energy Physics, Warsaw
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1996, hep-ex/9701014. 15. R. Gupta, T. Bhattacharya, hep-lat/9605039. 16. C. AUton et al., Phys Lett. B 345 (1995) 513. UKQCD Collaboration, Nucl. Phys. B447 (1995) 425. 17. N. Christ, in Lattice Gauge Theory using Parallel Processors, ed. X. Li, A. Qiu and H. Ren, Gordon and Breach, New York, 1987. 18. J. Beetem, M. Denneau and D. Weingarten, IEEE proceedings of the 12th Annual International symposium on Computer Architecture, IEEE Computer Society (1985). 19. Ape Collaboration (C. Battista et al.), Int. J. of High Speed Computing 5 (1993) 637. 20. Y. Iwasaki et. al, Comp. Phys. Comm. 49 (1994) 449.
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A623 (1997) 81c-89c
Quantum Chromodynamics on the Lattice F. Rapuano Dipartimento di Fisica, Uuiversit£ di Roma 'La Sapienza' and INFN, Sezione di Roma, P.le A. Moro 2, 1-00185 Roma, Italy. In this talk I give a brief overview of Quantum Chromodynamics on the Lattice and report on some of the most recent results obtained in the field. 1. I N T R O D U C T I O N The theoretical knowledge of the fundamental interactions that we have reached today is quite deep. In particular it is widely believed that strong interactions are described by Quantum Chromodynamics (QCD). QCD has the property of asymptotic freedom that opens the possibility of using perturbative methods in the short distance region where the running coupling constant a,(q2), q~ is the scale relevant to the process under consideration, is small. In the perturbative regime QCD accurately reproduces experimental data. The situation is much more complicated for all processes that involve long distance interactions. One must find reliable non-perturbative methods. The lattice formulation of QCD has proven to be a first principle method which has no a-priori assumptions, can be systematically improved and is well suited to numerical calculations. My talk is organized as follows: in section 2 I give a brief introduction to the lattice method. In section 3 I discuss the main sources of errors and in section 4 I describe some recent results. 2. T H E L A T T I C E M E T H O D The Lattice formulation of QCD (LQCD) [1] is based on the functional formulation of Quantum Field Theory. The expectation value of a generic operator (9 is defined as:
(1)
where U,,, ~bq and Cq are the fields of the theory, respectively gluons, quarks and antiquarks of flavour q and SQCDis the QCD action. Z is the same integral without (9 in such a way that (1) = 1. Wilson [2] proposed, as a prescription to define the theory and the integral in eq.(1), to discretize space-time on a finite 4-dimensional mesh of sites with spacing a. Quarks on the sites exchange gluons on the links in such a way that local SU(3) gauge invariance is preserved. We have then a regularized theory with the introduction of an UV cutoff due to the finiteness of the lattice spacing a, and an IR cutoff due to the finite space-time volume. The introduction of a finite and discrete space-time makes by itself the number 0375-9474/97/$17.00 © 1997 - Elsevier Science B.V. All rights reserved. PII: S0375-9474(97)00425-9
E Rapuano /Nuclear Physics A623 (1997) 81c~89c
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of degrees of freedom in eq.(1) finite but still the integral is analytically unmanageable even for a moderate number of lattice sites. Going to Euclidean time via a Wick rotation allows the use of statistical techinques to evaluate the integral in eq.(1) as we now have a real positive exponential factor that can be interpreted as a probability distribution of field configurations. The technique that is used to numerically evaluate (1) consists in the generation of a set of independent field configurations with probability ezp(-SQvD) with a Montecarlo method. On each of these field configurations one measures the value of O. The average value is then: 1 ~v,
Z o, +
(2)
{=I
where Arc is the number of independent field configurations, Oi is the value of O on the i-th field configuration and 6 represents both the statistical error introduced by the evaluation the integral and the systematic error due the discretization of space-time. The QCD Lattice action does not contain dimensionful parameters. AU dimensionful quantities are expressed in terms of powers of the lattice spacing a. The continuum limit is reached, in QCD, in the limit of vanishing coupling constant 9. The relation between # and the lattice spacing a is obtained by perturbatively solving the renormalization group equation. In turn one can replace the cut-off with a dimensionful scale parameter ALat which is not fixed by the theory. For a lattice measure to have a meaning in the continuum one should check cut-off independency: physical quantities expressed in terms of ALat should not depend on the value of the coupling constant g used in the simulation when the value of g is small enough (or a -1 is large enough). Then if one physical quantity (e.g. the string tension or a hadron mass) is used to fix AL~t, all other physical quantities will be lattice predictions. 3. T H E P R O B L E M
OF ERRORS
The discretization of the theory is not a trivial issue. On one hand one may have lost symmetries whose restoration in the continuum limit should explicitly be verified (rotational invariance is an obvious example. See [1] for a discussion of the question of symmetry breaking on the lattice). On the other hand, as stated before, we are introducing various sources of errors that should be carefully analized to be sure of the reliability of the measures. 3.1. S t a t i s t i c a l error In some sense this is the simplest to estimate. If the set of N~ configurations on which the measure is performed axe independent we have that 6,t,t is of order 1/v/N~. At present all reliable lattice simulations are done on samples of hundreds of configurations that reasonably represent the configuration space, and the independence of the sample of configurations can be easily verified. Statisticalerror is then quite under control and does not deserve any further discussion. 3.2. Systematic error This is a m u c h more delicate problem in lattice Q C D . The sources of systematic errors are m a n y and each one poses different problems. I will analyse the most rclewnt ones
E Rapuano/Nuclear Physics A623 (1997) 81c~9c
83c
and the possible solutions. • Finite lattice spacing. The value of a in recent simulations ranges between 0.05 and 0.1 fm. This corresponds to an UV cut-off lr/a ~ 6 + 12 GeV. It is very difficult to estimate apriori whether these values can be considered large enough. Many simulations have checked that an approximate scaling behaviour is observed at 92 < I for pure gauge quantities and light quark hadronic spectrum. Other quantities like heavy quark systems show quite larger scaling violations. An important point is that the discretized gauge part SG of SLQCD = So + SF has scaling violations O(a 2) while the fermionic part SF, discretized according to the Wilson procedure, has substantially larger O(a) violation. In principle one can remove these terms by a proper definition of the lattice action and operators, as proposed by Symanzik [3]. Many groups have proposed different methods of "improvement" of the lattice action [4]. The most promising one is an all-order non-perturbative improvement that reduces discretization errors to O(a 2) [5]. I cannot describe these techniques for lack of time. • Finite Volume. This error is, in today's simulation, quite small. Finite volumes leads to the use of unphysically large quark masses in such a way that for the typical correlation length involved in the simulation (e.g. ~ ~ l i m a where m x is a hadron mass) one has << L, L being the linear spacial extention of the lattice. One then is forced to work with light quark masses mq ~ 100 MeV or larger and then extrapolate to the physical quark masses with an error that, in any case, is not overwhelmingly large. • The Quenched Approximation. So far we have described errors that are intrinsic to the numerical simulation of the discretized theory. Actually to cope with the available computer power, we shall now make an approximation. The fermionic part of the lattice action is:
SF = ~bq(J~+ mq)~bq
(3)
where ~ is the discretized version of the QCD covariant derivative and mq is the bare mass of the quark of flavour q. The fermionic degrees of freedom can be analitically integrated:
f
= a t( 9 +
(4)
and one can introduce an effective action:
e-s"' = II aet( q
+ mq) -So
(5)
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E Rapuano /Nuclear Physics A623 (1997) 81c-89c
as a weight distribution. One can show that the discretized gluon action Sa is a local quantity (i.e. depending only on neighouring links) which is easy to evaluate numerically. The determinant instead is a highly non-local quantity which represents a formidable problem from the computational point of view. The quenched approximation consists in setting the fermionic determinant to one. This amounts to neglect the effects of virtual quark loops. Quarks propagate then as valence quarks in the background gauge fields. While arguments like the approximate validity of the Zweig rule and the small amount of momentum carried by sea-quarks in hadrons support the quenched approximation, it is difficult to quantify its effects, as different observables will be affected by fermion loops in a different fashion. As an example we have that a p meson will become stable (i.e. F = 0) and we can naively estimate that its mass will be affected by a shift ~m ~ gF = F. So the error on the width is of course 100 % while the error on the mass is much smaller. The pion-nucleon a-term also is a quantity that should be very much influenced by quark loops. One of the latest determinations, [6] gives a~N --~ 45 MeV. It is well known that the calculation of the a-term in absence of sea-quarks gives a much smaller result [7]:
2[
_ 3
m,~
m= - mN +
m mA]
~ 26MeV
(6)
Lattice simulations in the quenched case [8] agree quite well with this result while preliminary unquenched simulations [9] seem to confirm that the sea-quark contibution is substantial. In conclusion the systematic error coming from the quenched approximation may of the order 10 + 15% on hadron masses but can be larger for other observables. At present many groups are making simulations taking into account quark loops. The results have quite limited statistics and may suffer from finite volume effects. The removal of the quenched approximation is a step that should be achieved with the next generation of computer with speeds of the order of one to ten TFlops. 4. R E S U L T S It is practically impossible to summarize all results obtained in LQCD in the last years. I will concentrate on those that I consider more relevant to confirm the lattice as a reliable non-perturbative method and onto some phenomenologically important predictions. For a full review of the state of the art of lattice simulation the proceedings of the annual Lattice QCD conference is the best reference. 4.1. T h e h a d r o n s p e c t r u m Hadron masses are computing from the so called two-point correlation functions:
(7)
n
where H(t) is an interpolating operator with the quantum number of the particle under study (e.g. qTsq for a pion, (/Tiq for a p meson etc.), E,, is the energy of the n-th radial
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Table 1 Predicted meson masses in GeV for various lattices
Exper. C60 W60 C62a W62a C62b W62b W64 C64
Mp M,.. 0.770 0.686 (see [10]) 0.809(7) 0.6849(3) 0.808(3) 0.6849(1) 0.81(1) 0.6849(5) 0.803(6) 0.6851(2) 0.79(1) 0.6856(5) 0.797(7) 0.6853(3) 0.796(4) 0.6853(2) 0.792(4) 0.6855(2)
M~ 1.019 0.977(7) 0.978(3) 0.98(1) 0.984(6) 1.00(1) 0.989(7) 0.990(4) 0.994(4)
Table 2 As in table 1 for baryon masses
MN
MA~
M~
Ma
Mo
Exper.
0.9389
1.135
1.3181
1.232
1.6724
C60 W60 C62a W62a C62b W62b W64 C64
1.09(5) 1.19(5) 1.1(1) 1.17(7)
1.21(4) 1.29(4) 1.22(8) 1.28(6)
1.32(4) 1.3(1) 1.60(9) 1.40(4) 1.46(7) 1.71(4) 1.34(7) 1.39(5)
1.1(2)
1.2(2)
1.4(1)
1.6(3)
1.9(2)
1.2(1) 1.3(1) 1.40(9) 1.50(9) 1.72(5) 1.21(9) 1.32(8) 1.43(6) 1.4(2) 1.72(9) 1.2(1) 1.29(8)1.41(7) 1.3(2) 1.7(1)
excited state. At large time separation and zero momentum, a single particle exponential behaviour will set in. From a two paramer tlt one can tllen extract the mass of the lowestlying state. The leptonic decay constant of the pion, f~ can be similarly obtained from the correlation of the fourth component of the axial current (A0(t)A*0(0)). I will report the results obtained in a large statistics study (N~ > 200) at values of a ranging from 0.05 to 0.1 fm (a -1 ranging between 2.0 and 4.0 GeV) and lattices sizes from 183 > 64 to 24 a × 64 [10]. The data show a quite good scaling behaviour for all quantities under study except for the vector meson decay constant. Table 1-3 summarize the results obtained by fixing the value of a from the mass of the K* meson. Lattices axe labeled with a W or a C according to the fermionic action used, ordinary Wilson [2] (W) or tree-level improved [4] (C), and by the value of/~ = 6 / f . Masses and pseudoscalar decay constants agree within 5 to 15% to the experimental data. The quenching error can be more relevant for the nucleon mass. Vector mesons decay constants are, on the contrary, quite far from the physical values. Preliminary studies seem to indicate that the non-perturbative O(a ~) [5] removal is particularly important for this quantity.
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Table 3 As in table 1 for the leptonic decay constants f.(GeV) Exper. C60 W60 C62a W62a C62b W62b W64 C64
0.1307 0.134(9) 0.161(7) 0.124(9) 0.140(6) 0.14(2) 0.140(8) 0.15(1) 0.144(9)
1
0.28 0.35(3) 0.41(2) 0.33(3) 0.38(2) 0.26(4) 0.39(2) 0.32(1) 0.25(1)
fr(GeV) 0.1598 0.149(8) 0.173(6) 0.143(8) 0.157(5) 0.16(2) 0.157(7) 0.166(8) 0.158(8)
I
1
fK*
f, 0.23 0.30(1) 0.357(8) 0.281(7) 0.335(6) 0.25(2) 0.34(1) 0.296(6) 0.243(8)
0.33(2) 0.38(1) 0.30(2) 0.36(1) 0.25(3) 0.37(2) 0.31(1) 0.25(1)
A particularly interesting result comes from the UKQCD collaboration for the spectroscopy of heavy baryons [11] on a lattice of size 243 × 48 at a value of a -1 of 2.9 ± 0.2 GeV. The simulation is performed at a heavy quark mass close to the charm mass and an extrapolation is performed to the mass of the beauty quark. Their results are shown in table 4. The lattice data agree rather well, within the errors, with the experimental data, when available. 4.2. Heavy-light mesons leptonic decay constants.
Conceptually the technique is the same as for the fight quark systems even though various methods have been developed to increase the reliability of the prediction as far as the extraction of the low-lying state is concerned. The decay constant of the B meson is experimentally not measured yet and its knowledge is extremely important in the Standard Model because it would allow a determination, via the measurement of the leptonic decay width, of IV~I which is known with a quite large error. Also it is connected to the amount of CP violation in the process B ~ ~bK, [12] There are many results on this subject and I have reported some of them in table 5 which is taken from ref. [13] with some recent data added and various modifications. It should be considered that these simulations have different statistics and systematics; see ref. [13] for a detailed description of the data and the original references. The leptonic decay constant of the D, meson, fD,, has been measured by various experiments and the experimental world average for gives a value of 241 + 21 :k 30 MeV [14] which is very good agreement with the lattice ones. 4.3. Other predictions.
I will conclude with a short list of other predictions that are quite important. There are recent analysis about quark masses that give ~ = r~,+,,,~ = 3.1+0.3 MeV, [15] for the fight 2 quak mass and m, = 100+21+10 MeV [15] and ra, = 100+21+10 MeV [10] for the strange quark mass. A large amount of work is currently underway to simulate semileptonic decays of heavy quarks [16]. The calculation involves now three point functions and are quite more computing intensive than the two point functions needed for the spectrum. The
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Table 4 Heavy baryons masses from ref. [11]. Results are quoted with a statistical error (first) and a systematic error (second) arising from the uncertainty in the calibration of the lattice spacing. Where available, the experimental data are reported. h =
Baryon
Quark Content
Ah ~h ~ Eh =' --h --h=* ~h
(ud)h (uu)h (uu)h (us)h (us)h (u~)h (ss)h
Exp. (MeV) 2285(1)
charm
Latt. (GeV) 2.27 +4_s +s 3-
h =
Exp. (MeV)
5641(50) 5814(60) 5870(60)
beauty
Latt. (GeV) 5.64 +5 +s -5 -2
2453(1)
2.46 +~ +s -3 -S
2530(7) 2468(4) 2560 t
2.44 +~- -5+4 2.41 +a +4 -3 -4 2.57 +s +6 -3 -6
2643(2)
2.55 +s +s -4 -5
5.90 +4 +4 -6 -5
2.68 -4+s +s_s
5.99 +5 +s -5 -5
2.66 +~_ +s_~
6.00
2704(20)
5.77 +s +4 -6 -4
5.78 +5 +4 -6 -3 5.76 +3 +4 -5 --3 5.90 +s +4 -6 -4
+4 +5 -5 -5
results for the decays D --* K and D -+ K* give, for the form factors that parametrize the amplitudes, agree quite well with the experimental ones. As for the decay constants, this supports the predictions for the processes B --+ ~r and B --* p and the determination of the Isgur and Wise function. Other important results for the study of CP violation in the B system is the mixing parameter BB(#) = ( B- I O I B ) / Js ~ 2M ~~. where O is the AB = 2 operator b,/,(1-Ts)ql~7,(1-75)q at a scale #. Ref. [13] again is a good starting point for a study of the available results. Open problems like the value of the CP violation parameters e'/e and the A I = 1/2 rule are the goal of the lattice theoretical developements. These problems are particularly hard ones because of the mixing of the relevant operators with operators, which are absent in the continuum, and that being divergent, should be non perturbatively subtracted on the lattice. 5. C O N C L U S I O N S I believe that the lattice can be considered a reliable tool for non-perturbative calculations. The good agreement between simulated and measured quantities makes us confident that the predictions for free parameters of the Standard Model and are quite reliable. The error sources are under control and further improvements are possible. The quenched approximation is only due to the available computer power but we will be able to simulate full QCD with the next generation of computers. One word is worth saying about the role that special purpose computers have had in the developement of Lattice Quantum Chromodynamics. Many projects have been realized. From the first generation projects designed at Columbia [17], at IBM [18], at INFN-Italy [19] and in Japan [20] with speed in the range 1 + 10 GFlops to the most advanced one like the APE-mille project at INFN that will reach a speed of 1 TFlops, all have given a tremendous boost to these calculations.
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Table 5 Summary of results for values and ratios of leptonic decay constants of pseudoscalar mesons. Results with no quoted values for fl or a -1 come from more than one lattice Group
fl
a -1
fB
lB,
(GeV) (MeV) (MeV) 6.0 1.9 6.0 197(18) 6.O 2.O 176(~I) 6.0 2.33 6.1 2 6.2 2.7 160(~) 194(~) 6.3 3.0 187(38) 207(41) 6.4 3.3
DeG-L APE UKQCD LANL BDHS UKQCD BLS ELC ELC ELC PSI-WUP LANL 6.2 APE
205(40)
2.6
180(32) 205(35)
fBo/fS
fD
(MeV) 190(33) 218(9) 1.17(12) 199(~) 229(~) 174(53) 1.22(~) 185(4~) 212(4~) 1.11(6) 208(38) 210(40) 194(15) 1.08(6)
1.14(8)
fD.
fD,/fD
(MeV) 222(16) 1.17(22) 240(9) 1.11(1) 1.13(9) 260(I~) 1.14(2) 234(72) 1.35(22) 1.18(2) 230(36) 1.11(6) 230(50)
198(17) 200(18) 186(29) 218(15) 221(17) 237(16)
1.07(4)
REFERENCES
1. For a detailed discussion of all aspects of LQCD see: H.J. Rothe, Lattice Gauge Theory, World Scientific, Singapore, 1992. I. Montvay, G. Muenster, Quantum Fields on a Lattice, Cambridge University Press, Cambridge, 1994. 2. K. Wilson, Phys. Rev D10 (1974) 2445. 3. K. Simanzik, Nucl. Phys. B226 (1983) 187, 205. 4. B. Sheikoleslami and R. Wohlert, Nucl. Phys. B259 (1985) 572. G. Heatly et al. Nucl. Phys. B413 (1994) 461, and refs. therein. 5. M. Lfischer et al., 14th International Symposium on Lattice Field Theory, St. Louis, 4-8 June 1996,hep-lat/9608049 and refs. therein. 6, J. Gasser, H. Leutwyler, M.E. Sainio, Phys. Left. B253 (1991) 252. 7. T.P. Cheng, Phys. Rev. D13, (1976) 2161. R.L. Jaffe, Phys. Rev. D21, (1980) 3215. 8. Ape Collaboration Phys. Left. B258 (1991) 195. 9. M. Fukugita, Y. Kuramashi, M. Okawa, A. Ukawa, Phys.Rev. D51 (1995) 5319. 10. C. Allton, V. Gimenez, L. Giusti, F. Rapuano, hep-lat/9611021. 11. UKQCD Collaboration (K.C. Bowler et al.), Phys.Rev. D54 (1996) 3619. 12. M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, Nucl.Phys. B369 (1992) 139. J.L. Rosner, Rev. Mod. Phys. 64 (1992), 1151. 13. J. Flynn, 14th International Symposium on Lattice Field Theory, St. Louis, 4-8 June 1996, hep-lat/9610010. 14. J. D. Richman, XXVIII International Conference on High Energy Physics, Warsaw
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1996, hep-ex/9701014. 15. R. Gupta, T. Bhattacharya, hep-lat/9605039. 16. C. AUton et al., Phys Lett. B 345 (1995) 513. UKQCD Collaboration, Nucl. Phys. B447 (1995) 425. 17. N. Christ, in Lattice Gauge Theory using Parallel Processors, ed. X. Li, A. Qiu and H. Ren, Gordon and Breach, New York, 1987. 18. J. Beetem, M. Denneau and D. Weingarten, IEEE proceedings of the 12th Annual International symposium on Computer Architecture, IEEE Computer Society (1985). 19. Ape Collaboration (C. Battista et al.), Int. J. of High Speed Computing 5 (1993) 637. 20. Y. Iwasaki et. al, Comp. Phys. Comm. 49 (1994) 449.