Hadrons and lattice QCD

ELSEVIER

Nuclear Physics A721 (2003) 40~49~ www.elsevier.com/locate/npe

Hadrons and Lattice QCD Akira Ukawa Center for Computational Physics and Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan We review recent progress of lattice &CD. 1.

Overview

QCD is a beautiful theory. Given a single coupling constant a#(,~) and six quark masses m&L Q(P), %(PU), mc(ll), wh), ml(P) a t some scale 1-1,the theory promises to explain the entire phenomena of the strong interactions. To bring this promise to reality, thus fulfilling the vision which Hideki Yukawa conceived in 1935, we need to be able to integrate the Feynman path integral of &CD. Lattice QCD[l] is a framework which provides a concrete mean to carry out this task with the help of computers. In a quarter of a century since the ground-breaking papers [2,3], lattice QCD and numerical simulations have progressed significantly both in depth and width of the subjects. In this talk, we try to review recent progress in lattice &CD. We start with calculations of hadron spectrum and fundamental constants of lattice QCD (Sec. 2). The key point here is a quantitative understanding of the limitation of the quenched approximation, and subsequent progress in dynamical quark simulations of &CD. We next turn to the issue of chiral symmetry(Sec. 3). With increased precision of full QCD data, mismatches with predictions of chiral perturbat,ion theory have been observed. We discuss this issue, and review recent applications of chirally symmetric fermion formalisms to this problem. In Sec. 4 we discuss recent results on the K meson weak decay amplitudes relevant for the AI = l/2 rule and the direct CP violation parameter ~‘16. In the last few years there has been substantial progress in the long-standing problem of simulating lattice QCD at finite quark chemical potential [4]. Space limitations force us to skip this interesting subject in this proceedings, and we refer to recent reviews[5,6] for details. 2. Hadron 2.1.

Spectrum

Quenched

light

and Fundamental hadron

Constants

of QCD

spectrum

The light hadron mass spectrum provides a benchmark to verify both the validity of QCD and the methodological effectiveness of numerical approach. This calculation is also essential for determining the physical scale of QCD and quark masses. For precision calculations of the hadron spectrum, it is essential to control various systematic errors within a few% level. This requires a lattice size of L x 3 fm or more to 0375-9474/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved. doi:lO.l016/S0375-9474(03)01015-7

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1.8 1.6 1.4 50) 1.2 9 E 1.0 0.8

-

0.6

0 $ input experiment

0.85 1 I’ 0.0

0.4

Figure 1. Quenched light hadron mass spectrum as compared with experiment (horizontal bars) [8]. Statistical errors and sum of statistical and systematic errors are indicated. Results with K input and 0 input are shown.

02

,A 0.4

0.6 a [G&i-‘]

0.8

1.0

12

Figure 2. Continuum extrapolation of 4 and K* meson masses in quenched (open circles) and iVf = 2 full (filled symbols) QCD [16]. Diamonds at a = 0 show experimental values.

suppress finite-size effects, good control over the chiral extrapolation to the limit of zero quark masses mg -+ 0, and that over the continuum extrapolation to vanishing lattice spacing a -+ 0. Within the quenched approximation which ignores sea quark effects, the GFll Collaboration [7] initiated a systematic effort toward this goal, which has been completed by the recent work of the CP-PACS Collaboration[8]. Their result for the ground state spectrum of meson octet and baryon octet and decuplet is shown in Fig. 1. Here 7r and p meson masses are used to fix the u and d quark masses (assumed degenerate) and lattice spacing, and either the K meson mass (filled symbols) or 4 meson mass (open symbols) to fix the s quark mass. We observe an overall agreement of the calculated spectrum with experiment at a 10% level. However, there is a systematic discrepancy between them at finer resolution, which exceeds the estimated errors. Furthermore, the quenched results, and hence the pattern of disagreement, differ depending on the choice of input to fix quark masses. This uncertainty is substantial; for the strange quark mass, we find rnp(p = 2 GeV) = 142!i8 MeV with 4 input, and 114’8, MeV with K input, which differ by 25%. This illustrates the limitation of quenched &CD, and demonstrates the necessity of full QCD with dynamical quarks to carry out precision predictions from lattice &CD. 2.2.

Full

QCD

simulation

and sea quark

effects

in the meson

sector

Full QCD simulations with dynamical quarks have a rather long history [9]. It is. however, only recently that the computer power has reached the level of 1 Tflops which has turned out to be needed to allow systematic exploration of sea quark effects.

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Over the past several years, simulations dynamically incorporating degenerate 1~and d quarks (Nf = 2 full QCD) have been carried out by the SESAM-TxL [lo-121, UKQCD[lS, 141, CP-PACS[15,16], QCDSF[17], and JLQCD[18] Collaborations using Wilson-type quark actions. The MILC Collaboration, on the other hand, has used the staggered (Kogut-Susskind) action in their Nf = 2 simulations [19]. They have recently started Nf = 3 simulations [20] in which s quark is treated dynamically as well as ‘11and d quarks. A similar attempt using Wilson-type action is also being started in Ref. [21]. One of the questions addressed in these simulations is whether dynamical sea quark effects account for the discrepancy observed in the quenched calculations. In Fig. 2 we plot the continuum extrapolation of the K’ and 4 meson masses using K meson mass as input to fix the strange quark mass [15,16]. The quenched results (open symbols) converge to values smaller than experiment (diamonds) by about 5%, as already observed in Fig. 1. In Nf = 2 full QCD (filled points) the discrepancy is sizably reduced after the continuum extrapolation. A subtle point with the Nf = 2 result above is the need of continuum extrapolation. The linear extrapolation employed in Fig. 2 is based on the fact that the tadpole-improved clover quark action used in the simulation has errors of O(g’a). Recent work [18] using fully O(e) improved action (i. e., with errors of O(a’)) shows evidence that an increase of hyperfine splitting by inclusion of dynamical quarks is a physical effect of continuum &CD. 2.3. Quark masses Quark masses are one of the fundamental constants of nature, on a par with lepton masses such as electron mass. They can only be determined theoretically by working out the functional relation between hadron massesand quark massessince quarks are confined within hadrons. The masses of quarks have been repeatedly calculated in lattice QCD as a part of hadron spectrum calculations. Recent progress consists in precision determinations in quenched QCD and systematic evaluations in full QCD with dynamical quarks. In Fig. 3 we plot a recent result[l5] for the average u-d quark mass mud and strange quark mass m, at p = 2 GeV in the MS scheme, calculated in quenched QCD and in Nf = 2 full QCD and extrapolated to the continuum limi:. In quenched QCD there is a large uncertainty of about 25% in the strange quark mass depending on the input hadron mass; this is an inherent uncertainty which cannot be shrunk within quenched &CD. This problem almost disappears in Nf = 2 full &CD. At the same time the values of quark masses are also reduced by about 20%, lying close to the lower edge of the band given by the Particle Data Group as covering the existing lattice and phenomenological estimates. One may expect that incorporating s quark further reduces the light quark masses. Recent work based on the MILC Nf = 3 configurations [20] indicate that this is the case[23]. Let us briefly touch on the values of charm and bottom quark masses. For the charm quark, the most advanced attempt using fully non-perturbative renormalization factor for conversion of lattice values to the continuum ones yield myS(m,) = 1.301(34) GeV [24] in quenched &CD.

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quenched N,=O

.

,“I, am N,=2

.

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Particle Data Group evaluation

mue(2GeV) x 10

I

L

Figure 3. Light quark massesin quenched and Nf = 2 full QCD [15]. The u and d quark masses (assumed degenerate) are multiplied by 10 for ease of drawing (filled circles). The s quark mass is computed using 4 meson mass (down triangles) or K meson mass (up triangles) as input. Particle Data Group evaluation of lattice and phenomenological estimates is also shown (vertical bars)[22].

Figure 4. Strong coupling constant a6N’=5’(h/iz) in the MS scheme converted to Nf = 5 flavors and calculated at the 2 boson mass. Experimental determinations are taken from Ref. [22]. Lattice results are from Davies et al ‘97[28], SESAM[SO], QCDSF-UKQCD[29], and Davies et al ‘02[31].

The problem for bottom quark is the control of systematic errors associated with large values of bottom quark mass mb x 4GeV at currently accessible lattice spacings a-l x 2 GeV, i.e., mba > 1. Thus effective field theory interpretation of lattice heavy quark action has to be invoked for analyses. We quote mrs(mb) = 4.27(9) GeV [25] from a recent analysis. 2.4. Strong coupling constant The strong coupling constant cyI(p) has been estimated in a variety of high-energy scattering and decay experiments [22]. Lattice QCD attempts to determine this quantity from low energy calculations. One of the methods employed is to evaluate cam at the cutoff scale p= l/a using short-distance quantities such as the gluon condensate, and fix the scale l/u from hadron masses [26]. Another, more rigorous, approach is to determine the renormalization-group evolution of the running coupling using the Schrodinger functional finite-size technique[27]. Both methods are being used and expanded to full QCD recently. In Fig. 4 we compile recent lattice results for a, [2831] and compare them with phenomenological determinations [22]. With lattice data, inclusion of sea quark effects is not yet complete; Davies et al ‘97 [28] made an extrapolation from the quenched (Nf = 0) and Nf = 2 results linearly in Nf to estimate the dynamica! clfects of 5 rqu.~rlr Th<>

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0.1

0.2

‘4

0.3 r, mtwI

0.5

Figure 5. Chiral behavior of mz/mp in Nf = 2 full QCD at a z 0.1 fm[18]. Lines show chiral logarithm fits leaving the decay constant f free or fixed to the experimental value.

Figure 6. Chiral behavior of mz/m, in quenched Nf = 0 QCD obtained with the overlap formalism[38].

QCDSF-UKQCD value is based on Nf = 2 full &CD, and the effects of strange sea quark are not incorporated. Finally the recent value of Davies et al ‘02 [31] based on MILC Nf = 3 simulations is taken at a single lattice spacing a M 0.13 fm. The scatter of values seen in Fig. 4 may partly originate from the deficiencies described here. We consider that progress expected in the next several years toward Nf = 3 simulations would clarify the situation. 3. Issues and Progress with

Chiral

Symmetry

3.1. Matching with chiral perturbation theory Ideally one would like to carry out simulations as close as possible to the physical u and d quark mass corresponding to the meson mass ratio mrr/mP x 0.18. This is presently not feasible because of the increase of the condition number of the quark Dirac operator toward light quarks c(D) cx l/m,. Simulations so far have been limited to mn/mp M 0.4 in quenched &CD, and to M 0.6 in full QCD with Wilson-type quark actions (with the Kogut-Susskind action, the ratio has been reduced to M 0.4). Hence, to reach the physical point a rather long extrapolation is needed. Traditionally, polynomial extrapolations have been often employed for this purpose in the literature. Close to mp = 0, however, one expects logarithmic infrared singularities arising from the vanishing of pion mass. Chiral perturbation theory makes explicit predictions on the functional forms of physical quantities as functions of bare quark masses. Hence one may employ these forms for the extrapolation. In Fig. 5 we show recent high statistics data for the pseudo scalar meson mass squared as a function of bare quark mass m, for N, = 2 full QCD[lS]. Chiral perturbation theory

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4%

predicts that

The curvature we expect from this formula is indicated by the concave curve for which we assumed f = fX = 93 MeV. On the other hand, lattice data (circles) are quite straight, not showing signals for the logarithm in the range of quark mass explored in Fig. 5 (ma/mp M 0.8 - 0.6). This example illustrates the recent observation by several studies that chiral behavior with current lattice data often fails to conform with the expectation from chiral perturbation theory. One possible reason behind this phenomena is the large values of quark massesexplored SO far with Wilson-type quark actions. In the case of Fig. 5 the lowest point corresponds to m, M 500 MeV. Smaller quark masses may be needed before the singular behavior becomes manifest [32]. Model studies have been made trying to incorporate effects of large quark masses[33]. They indicate that a linear behavior as found in Fig. 5 may change closer to mq = 0, leading to uncertainties of 10 - 20% at the physical point. 3.2. Lattice

fermions

with

exact

chiral

symmetry

A very important recent development with chiral symmetry is the realization that one can write down chirally symmetric lattice fermion actions. More than one way to achieve this is known, and they are called domain-wall fermion [34], overlap fermion [35] and fixedpoint action [36]. These actions avoid the Nielsen-Ninomiya theorem by using infinitely many fields or its equivalent, and are all based on the Ginsparg-Wilson relation[37] for the lattice Dirac operator given by Dy5 + ~sD = 2aDRysD

(2)

where R is a local operator. This formula states that chiral symmetry in the sense of Ward identities among Green’s functions is violated only by local terms which vanish as the lattice spacing a + 0. A number of tests of these formalisms has been made, mostly within quenched &CD, checking the chiral Ward identities and chiral properties of observables such as pseudo scalar meson mass, chiral order parameter and weak matrix elements such as the K” - Ks mixing matrix. They show quite encouraging results, exhibiting both good chiral behavior and small scaling violation. In Fig. 6 we show a recent result obtained with the overlap fermion action for the pseudo scalar meson mass squared in quenched QCD [38]. Quenched chiral perturbation theory predicts that m~ocmo(l-blogmo+bmo+~~~)

(3)

where m. denotes bare quark mass. The CP-PACS Collaboration explored the light quark mass region down to m?r/mp z 0.4 using the conventional fermion action, and found that their data indicates the presence of the logarithmic singularity wit,h 6 M 0.1[8]. In Fig. 6. the calculation was pushed down to a much smaller mass of m,/m, E 0.2 which almost reaches the physical point (ma/m,, M 0.18). We observe quite strong evidence for the

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presence of logarithmic singularity, with an estimated value of the parameter 6 = 0.26(3). We consider this to be the success of both the overlap formalism itself and the prediction of quenched chiral perturbation theory. Similar results have been obtained with the fixed point fermion action [39]. 4. Weak Amplitudes

of Hadrons

Calculating QCD corrections to the weak interacbion matrix elements is an important subject of lattice &CD. Quantities such as the B meson decay constant fs and weak form factors are crucial theoretical ingredients for constraining the Cabibbo-KobayashiMaskawa quark mixing matrix [40]. Another challenging problem, which we wish to address here, is understanding of K -+ ~YTamplitudes. Physically they are relevant for the long-standing issue of the N = l/2 rule, and the direct CP violation parameter cl/e; ReAo(K ReAz(K

E’ -E =

+ a~(1 = 0)) x 22 -+ mr(I = 2))

(20.7 f 2.8) x 1O-4 (15.3 f 2.6) x 1O-4

(4

KTeV (FNAL) NA48 (CERN)

(5)

Quantitatively deriving these numbers is crucial for verifying our understanding of CP violation within the Standard Model. There are two difficulties with lattice QCD calculations of these amplitudes. One is the presence of two particles in the final state; extracting the state with correct relative momenta to satisfy energy conservation from the Km Green’s function is non-trivial[41]. Another problem is chiral symmetry; the behavior of these amplitudes for small pion mass are controlled by this symmetry. If one can keep chiral symmetry, it is possible to bypass the first problem by invoking the relation between K + 7rr amplitudes and the K --t x and K -+ vacuum amplitudes which follow from chiral symmetry [42], and calculating the latter two-body Green’s functions. Recently, exploiting chiral symmetry of the domain wall fermion, the RIKEN-BNLColumbia Collaboration [43] and the CP-PAGS Collaboration [44] carried out extensive quenched simulations of the latter amplitudes, with which they attempted to extract the physical K + 7r~ amplitudes. In Figs. 7-8 we show results of the CP-PACS Collaboration for the real part of the amplitudes which are relevant for the AI = l/2 rule. These results are obtained at the lattice spacing o-l M 2 GeV where studies with the K meson B parameter indicate that scaling violation is, not large. The physical prediction is obtained by taking the meson mass rnk to the chiral limit m M 2 - 0 . We observe that, while the I = 2 amplitude is consistent with experiment, the I = 0 amplitude is a factor two too small compared to the experimental value. Consequently, the AI = l/2 rule is explained only partially. The result for t’/c is shown in Fig. 9; the lattice value is too small and opposite in sign compared to experiment. The RIKEN-BNL-Columbia Collaboration analyzed data in a different way, and obtained the AI = l/2 rule consistent with experiment. Their result for t’/~ is similar to those of the CP-PACS.

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2.5

experiment (7 16%32 .24%32 chiral log. quadratic

-

-L-LA

0.1

0.2

0.3 0.4 mM2[Geti]

0.5

0.6

0.7

Figure 7. Chiral behavior of the I = 2 K + 7~ amplitude calculated with the reduction method[44].

0

0.1

0.2

0.3 0.4 mu2 [G&l

L

1

0.5

0.6

I

L-~~L--A

0.7

0.6

Figure 8. Chiral behavior of the I =, 0 K + 7i-x amplitude calculated with the reduction method[44].

The origin of these unsatisfactory results has not been clarified yet. One possible reason is remnant of chiral symmetry breaking due to a finite fifth dimensional size of the domain wall fermion. It may induce a large mixing of the dimension six four-quark weak operators with dimension three operators. Other possibilities are quenching and higher order corrections in chiral perturbation theory. These points need to be examined in further analyses. It is possible that one or more of these reasons conspire to invalidate the use of K + z amplitude for calculating the K + ~7r amplitude. If this is the case, we need to deal directly with the amplitudes with two pions in the final state. Lellouch and Liischer derived a formula for the physical K + XX amplitude in this case[45]. It would be an interesting and challenging problem to attempt a direct calculation of the K + 7~ amplitude using their formula. A first step in this direction has been taken in a recent work[46]. 5. Concluding

Remarks

Lattice QCD has witnessed significant progress over the last several years. On the computational side, Tflops-scale dedicated parallel computers such as CP-PACS have enabled a precision calculation of the quenched light hadron spectrum, which have exposed the deficiency of the quenched approximation quantitatively. Emphasis has shifted to full QCD since then, and important effects of sea quarks such as consistency of predictions independent of inputs and a decrease of quark masses have been uncovered in N/ = 2 dynamical simulations. Clearly time is ripe for making serious attempts to treat the s quark dynamically in addition to u and d quark to realize full QCD simulations with the realistic quark spectrum. The MILC Collaboration has started such an attempt with the

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25,

NA46 10 5

0 16%32 .24$x32 t

i 0

0.1

0.2

0.3

0.4 0.5 mu2 [GeV’I

0.6

0.7

0.6

Figure 9. Chiral behavior of t’/e calculated with the reduction method[44].

Kogut-Susskind quark action[20]. Another attempt using the non-perturbatively O(a)improved Wilson-type quark action is being pursued [21]. Another major development is lattice fermion formalisms with exact chiral symmetry. They have been shown to possess the expected chiral properties, and they have come to be used in practical simulations aiming-to calculate a variety of observables. These fermion actions take much more computer time than the conventional ones. With steady progress of computer power, notably with the next dedicated computers QCDOC and apeNEXT which are scheduled for completion late next year, we expect feasibility to use them for large-scale full QCD simulations to increase. We then hope that the next PaNiC conference will see substantial progress of lattice QCD beyond those achieved up to now. Acknowledgements This work is supported by Grant-in-aid of the Ministry of Education (No. 12304011 and 13640259). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

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