

 7 Aug 1996

DISPERSIVE THEORY OF CHARMONIUM ON THE LATTICE

A. BOCHKAREV a Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455

Physical contribution of the high-energy part of hadronic spectrum is incorporated to describe the short-distance part of the correlator of heavy-quark currents obtained by quenched Monte-Carlo on a 83 \Lambda 16- and 163 \Lambda 32-lattices for fi = f6; 6:3g. The lattice artifacts in the short-distance behavior of that correlator are isolated. The physical short-distance part of the correlator is fitted by the relevant expressions of perturbative QCD, which allows one to obtain the renormalized charmed quark mass with rather high accuracy m

_MS

c (mc) = 1:22(5) GeV .

1 Correlator of the heavy-quark currents The correlator of the vector currents of charmed quarks:

\Pi (q2)_* = i Z dxeiqx ! 0jT fj_(x)j*(0)g j0 ? (1) where j_ = _c fl_ c can be used to calculate the parameters of perturbative QCD, such as the renormalized charmed-quark mass mc and the strong coupling constant ffs(4m2c) 1. On one hand this correlator satisfies the standard dispersion relation:

\Pi (q2)_* = (q_q* \Gamma g_* q2) q2 Z ds ae(s)s2(s + q2) + d1g_* + d2(q_q* \Gamma g_*q2)

(2) with the singular subtraction constants d1;2. The spectral density ae(s) is proportional to the observable inclusive cross-section of charm-anticharm production in the e+e

\Gamma -annihilation. It is well described by the following simple

ansatz 1:

aephen(s) = s ` f m2resffi(s \Gamma m2res) + 14ss2 `(s \Gamma so) ' (3) with the low-lying resonance of mass mres = mJ=, residue f proportional to the electromagnetic width of the J=-meson, and a smooth continuum spectrum with some effective threshold so ? m2res. The experimental values of the spectrum parameters are:

4ss2f ' 0:6 ; mres ' 3:1 GeV ; so ' (4 GeV )2 (4) aTalk at the workshop "Continuous Advances in QCD '96", Minneapolis, March 1996.

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On the other hand this correlator is computable in perturbative QCD in the vicinity of vanishing momentum q2 = 0 as that point is far away from the nearest threshold 4m2c due to charmed quark-antiquark pair. The parameter of the perturbative loop expansion here is ffs(4m2c) ! 1. Following 1 we consider the ratios rn = Mn+1=Mn of moments of the correlator (1). The moments are defined as

Mn = 1n! ae` \Gamma ddq2 '

n

\Pi __(q2)oe

q2=0 (5)

The applicability of perturbative QCD near q2 = 0 implies the following expansion for the ratios rn:

rn = 14m2

c a

n + bn ffs(4m2c) \Gamma cn G

(2)

(4m2c)2 ! (6)

where fan; bn; cng are known numbers 1. The term , an comes from one loop of free charmed quarks. The term , bn comes from the two-loop diagrams corresponding to one-gluon exchange. The gluon condensate G(2) j ! 0j(ffs=ss) Ga_* Ga_*j0 ? is the vacuum average of the first nontrivial operator in the Wilson operator-product expansion for the correlator of heavy-quark currents. The coefficient cn originating from the Wilson coefficient function starts with one loop of the heavy-quark propagators.

The lower ratios r2;3;4, originating from short distances, are well reproduced in perturbative chromodynamics. The typical virtuality corresponding to the nth moment is , 4m2c=n. The coefficient bn grows with n. At large n ? 8 the perturbation-theory based expansion (6) is irrelevant. The gluon condensate shows up in the intermediate ratios r5;6;7 1 sensitive to larger distances and, hence, nonperturbative fluctuations. The lower moments receive significant contribution from the high-energy (continuum) part of hadronic spectrum, whereas the higher moments correspond to long distances and are saturated by the lowest resonance J=. The moment M0 is quadratically divergent and renormalizes the photon mass, whereas the moment M1 is logarithmically divergent and renormalizes the photon wave function. All the other moments are physical quantities: their ultraviolet singularities are absorbed by the renormalization of the QCD Lagrangian parameters, the charmed-quark mass and the strong coupling constant.

One can obtain a good estimate 1 of the renormalized QCD Lagrangian parameters by comparing the perturbative expressions for the lower ratios (6) to the same quantities obtained phenomenologically on the basis of dispersion relation (2). The principal uncertainty of this estimate is that the continuum

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contribution, significant in the lower ratios, is not known well enough from experiment. It is also only an approximation to identify the spectral density of the correlator of charmed-quark currents with the inclusive cross-section of the charm-anticharm production in e+e

\Gamma -annihilation because of the contribution

of light-quark currents to that inclusive cross-section. To avoid this problem and obtain an accurate way of calculating the renormalized charmed-quark mass and the strong coupling constant normalized on the charmed threshold it was suggested in 2, 3 to calculate the correlator (1) by Monte-Carlo in Lattice QCD. Then the only experimental information needed is the low-lying resonance mass (mJ=). It is used to fix the scale. This idea can only work if one can isolate physical contribution of the high-energy part of hadronic spectrum in Monte-Carlo data on the background of short-distance lattice artifacts. We demonstrate below that this is indeed the case.

Figure 1: Ratios of the neighboring moments of the single resonance approximation to the correlator, on three lattices of different sizes: 84; 164; 324. The resonance mass (mres = 0:5) is given in the units of the inverse lattice spacing. Solid lines are to guide the eye. Dashed line shows the moments of the subtracted correlator. Horizontal dotted line is a prediction

of the continuum theory.

2 Lattice artifacts of the moments We plan to analyse Monte-Carlo data obtained on relatively small and coarse lattices. Hence the lattice artifacts is the number-one problem to deal with. As usual we distinguish finite-volume from the cut-off effects.

3

Rather than doing Fourier transform of the correlator (1) and evaluating derivatives of (5) on a discrete lattice we calculate the moments directly in the X-space:

Mn = 122n n!(n + 1)! Z d4x x2n \Pi (x) (7)

We consider the correlator (1) in the infinitly-narrow-resonance approximation to be the propagator of a free field of mass mres:

\Pi res(q) = 1m2

res + 4a2 P

4 _ sin

2(q_a=2) (8)

The moments of a single resonance (8) computed with the help of eqn.(7) on lattices of different size are shown on Fig. 1. The straight horizontal line is expected in the continuum theory rresn = 1=m2res. One can see strong finitevolume effects on small lattices 84, 164 which persist even on the big (in a computational sense) 324-lattice in the high moments. The higher moments originate in the long-distance part of the correlator. One can see the horizontal plateau of the continuum theory show up in the lower moments on the big 324- lattice. On bigger lattices that plateau will expand to higher moments, but it will always remain somewhat above the prediction of the continuum theory due to cut-off O(a2)-effects. Those cut-off effects in the moments of a single resonance are negligible for a \Delta mres ! 1 3.

The lower moments of the correlator (1) are expected to be well reproduced in the loop expansion of perturbative QCD, where the dominant term is the one loop of free quarks with the renormalized mass _mc. We need to know therefore the lattice artifacts of one loop of free Wilson fermions (as the MonteCarlo data analysed was obtained for Wilson fermions). The moments of the correlator (1) in the one-loop (free quarks) approximation are shown in Fig. (2) for lattices of various sizes. Again one finds finite-volume effects which are particularly strong in high moments. Even on the big 324-lattice the moments of one loop of Wilson quarks deviate from the continuum theory prediction (diamonds) due to the cut-off O(a)-effects. One can reconcile nicely the lattice moments (for L * 32) with the expectations of the continuum theory by using only a simple relation between the quark mass of the continuum theory _mc and the mass parameter of the Wilson propagator mc j 1=2^ \Gamma 4 3 :

_m2c = m2c = ( 1 + mc ) (9) All the masses in (9) are in the units of a. The mass _mc is defined from the behavior of the quark propagator near vanishing momentum. It has nothing to do with the pole mass.

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Figure 2: Ratios of the neighboring moments of the subtracted (dashed lines) and unsubtracted (solid lines) correlator of pseudoscalar currents, saturated by one loop of free Wilson fermions with ^ = 0:1 (amc = 1) and periodic boundary conditions, computed numerically on lattices of indicated sizes. Lines connect the data to guide the eye. The diamonds is a

prediction of the continuum theory with the quark mass _mc j mc.

Note that the relation (9) describing O(a)-fermionic cut-off effects is valid in the infinite volume. Corrections to this formula due to finite volume were studied in 3.

Strong finite-volume effects imply that one has to fit Monte-Carlo data not with the continuum-theory expressions for the moments but with the moments of a single resonance or free quarks, calculated in the same box where the Monte-Carlo simulation was done, which is what we do.

3 Dispersion relations on the Lattice Our aim is to fit the short-distance part of the correlator of heavy-quark currents with the expressions of perturbative QCD. The short-distance part of a two-point correlator is determined to a great extent by the high-energy part of the spectrum. The problem therefore is to distinguish between the physical contribution of the high-energy part of spectrum and the pure lattice artifacts which will eventually dominate as one moves into the domain of very short distances. We claim to see a window of intermediate distances where the contribution of the high-energy part of sectrum is significant while the lattice

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artifacts do not dominate. Incorporating the high-energy part of the spectrum into the fit of Monte-Carlo data for the two-point correlators amounts to generalization of the dispersion relation to the lattice theory. The problem, in other words, is to construct the correlator of the lattice theory corresponding to a given spectral density.

We explore the following extention of the dispersion relation for the twopoint correlator \Pi (q2) to the lattice theory:

\Pi (q2) = ~d1 + ~d2 4a2

4X

_

sin2(q_a=2) (10)

+ 4a2

4X

_

sin2 i q_a2 j!

4 Z

ds ae(s)s2 (s + 4

a2 P

4 _ sin

2(q_a=2))(11)

The factor , sin2 is the inverse of the discrete Laplacian:Z

d4q (2ss)4 e

iqx 4

a2

4X

_

sin2 i q_a2 j = 1a2

4X

_

(2ffi(x) \Gamma ffi(x \Gamma ^a_) \Gamma ffi(x + ^a_))

(12) Therefore the coefficient ~d2 in (10) is precisely the renormalization of the photon wave function in the case of vector currents of charmed quarks. The renormalized correlator would be given by eqn.(10), (11) with ~d1 = ~d2 = 0, considered as a function of the renormalized Lagrangian parameters (we also call it "subtracted correlator"). The subtraction constants d1;2 are ultraviolet singular. In contrast to the continuum theory the term against d2 in (10) is not a polynomial in q2, so it contributes to all the moments, not only to M1. We find that its contribution to Mn decreases very fast with n. If the quark or resonance mass in units of a is less than 1, the ratio r2 is affected significantly by the term , d2 (in contrast to the continuum theory, where this ratio is physical). The ratio r3 is affected very little, whereas the contamination of all the other ratios rn?3 is negligible. One can see the difference between the subtracted and unsubtracted correlators on Figs. (1) and (2). The short-distance lattice artifacts are confined to the ratios r1;2 with rather small effect on r3 as soon as the relevant masses in units of a are less than 1. We therefore choose to fit the ratios r3 and r4 with the expressions of perturbative QCD.

4 Monte-Carlo data The ratios of the moments of the correlator of vector currents of the continuum theory 1 are shown on Fig. 3. Higher moments of the correlator, obtained on

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Figure 3: Ratios of the neighboring moments in the continuum theory for the correlator of interpolating currents of the J=-meson. Burst symbols correspond to the phenomenological evaluation of the correlator via dispersion relation with the smooth hadronic continuum spectrum modelled as in [2]. Squares show the contribution of a single J=-resonance.

Diamonds are moments of one loop of free quarks of mass mc = 1:26 GeV .

the basis of the dispersion relation and experimental information about the spectral density, approach the moments of a single resonance (J=-meson), which always lie on a straight horisontal line. The higher moments therefore come from long distances. The lower moments (r2;3) are seen to be fitted well by the moments of one loop of free charmed quarks, they are short-distance quantities. The two-loop correction due to one-gluon exchange will improve agreement between perturbative predictions and experimental predictions of the lower moments of the correlator. Incorporation of the gluon condensate will improve fit in somewhat higher moments r5;6;7 1.

Compare Fig. 3 with Fig. 4, which shows the ratios of the successive moments of the same correlator computed by Monte-Carlo on 163 ? 32-lattice for fi = 6, ^ = 0:1100 with the tadpole-improved Wilson Clover action 4. One can see strong finite-volume effects in Monte-Carlo data in the domain of higher moments. The moments of a single-resonance (solid line) exhibit the same finite-volume effects and fit Monte-Carlo very well in a wide range of higher moments. This allows us to fix the scale of the lattice theory.

The lower moments frn; n ! 6g are seen to deviate strongly from the single-resonance curve. All Monte-Carlo moments are fitted remarkably well

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Figure 4: Monte-Carlo data on the 163 \Theta 32; fi = 6 lattice for vector currents. ^ = 0:1100. The solid line is a single resonance approximation. The dashed line has been obtained with

the phenomenological charmonium spectrum incorporated via the dispersion relation.

by the dashed line. This line shows the moments of the lattice correlator corresponding to the spectral density which incorporates a smooth continuum spectrum in addition to the low-lying resonance. That lattice correlator is obtained using the disperion relation (11). The parameters of the spectral density are chosen to have experimental values as in (4). Although the lattice artifacts in Monte-Carlo data are strong, they are well described by the natural modification of the denominator in (11), which describes propagation of a state of a given invariant mass s on a finite lattice. We find evidence that it is a good approximation to ignore possible lattice artifacts in the spectral density, as it is done in (11). Incorporation of the high-energy (continuum) part of hadronic spectrum changes the line of ratios rn dramatically in small n. As expected the high-energy part of spectrum is important to descibe the shortdistance behaviour of the correlator. It is less trivial, that one can identify the physical contibution of the continuum spectrum and distinguish it from the short-distance lattice artifacts, which are confined to the lowest ratios r1;2, on the lattices of reasonable size.

5 How to fit Monte-Carlo data at short times The successfull incorporation of the continuum spectrum helps to fit MonteCarlo data on shorter distances and hence fix the scale on small lattices. Fig.

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Figure 5: Resonance mass from the conventional fit of the zero-momentum component of

the correlator of vector currents to cosh[(t \Gamma L=2)mres] on the smaller 83 \Theta 16 lattice.

5 shows the conventional way of extracting the lowest-resonance mass from the two-point correlator of local currents on the small 83 ? 16-lattice: MonteCarlo data are fitted by the expression for the correlator in the signle-resonance approximation. Horizontal plateau is expected at large times which height is given by the resonance mass. That plateau is not seen on Fig. 5 because the 83 ? 16 lattice is too small. The same type of picture holds (see Fig. 6) in terms of the moments: the resonance mass exptracted from the single-resonance fit to different moments should exhibit a horizontal plateau. On the 83 ? 16 lattice that plateau only starts to form at higher moments. At lower moments the resonance mass deviates from the horizontal line. This deviation is due to significant contribution of the high-energy part of hadronic spectrum at short distances.

We now include the continuum spectrum as in eqn. (4) via the dispersion relation (11) keeping the ratio of the effective continuum threshold to the resonance mass fixed to have the experimental value so=m2res ss 1:7 and use the resonance mass as the only fit parameter. The result of such a fit is shown on Fig. 6. One can see that the incorporation of the continuum spectrum via the dispersion relation (11) allows one to fix the resonance mass unambiguously from the data on relatively short distances.

Proceed now to a bigger 163 ? 32 lattice, where one can determine the

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Figure 6: Effective resonance mass from fitting the moment ratios of Monte-Carlo data on the 83\Theta 16; fi = 6 lattice for vector currents. ^ = 0:1060. The upper line is a single-resonance

approximation.

resonance mass by the conventional fit anumbiguously since the time extent is large enough for the high-energy excitations to decay. The ground state dominates and constitutes horizontal plateau at t * 10. The resulting resonance masses are shown in Tables 2, 3. One can compare them to the resonance masses extracted from the small lattice by means of the dispersion relation (incorporating the continuum spectrum), shown in Table 1. The agreement is very good. The corresponding spectrum of charmonium bound states is shown on Fig. 8.

6 The renormalized charmed-quark mass We fit the lower moment ratios r3;4 of Monte-Carlo data with the free-quarks approximation to the correlator (1):

r3(free Wilson quarks) = r3(Monte-Carlo) (13) The Wilson parameter ^ corresponding to those free quarks determines the the renormalized quark mass in accordance with (9) if Monte-Carlo is done on sufficiently large lattice, such as 163 ? 32. On smaller lattices the relation (9) needs to be corrected due to finite-volume effects. This is done by means of the dispersion relation 3.

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Figure 7: Resonance mass from the conventional fit of the zero-momentum component of

the correlator of vector currents to cosh[(t \Gamma L=2)mres] on the bigger 163 \Theta 32 lattice.

In order to fix the subtraction scheme one should be sensitive to the ffscorrection in the ratios rn. The study of the lattice artifacts on the two-loop level is not done yet. However, since the term bn ffs of eqn. (6) is small in the lower ratios r3;4, its value may be taken from the continuum theory:

rn = an4m2

c ,

n

,n j 1 + ffs bn=an (14)

We take the value of the strong coupling constant ffs(mc) ss 0:3 and the coefficients bn=an - from the two-loop calculations of the continuum theory 1 for the vector, pseudoscalar and scalar channels:

,psc3 = 1:02 ; ,vec3 = 0:95 ; ,sca3 = 0:97 ,psc4 = 0:96 ; ,vec4 = 0:92 ; ,sca4 = 0:91 (15)

The values of ,n in (15) correspond to our choice of the coefficients bn in the_ M S-subtraction scheme with the normalization point _ = mc 5. After the fit of Monte-Carlo data with free fermions (one-loop approximation) the mass of those fermions must corrected by the coefficients , in accordance with (15).

The results for the renormalized charmed-quark mass are shown in Tables 1, 2 , 3. The mass is extracted from the ratios r3 and r4 independently in three different channels: pseudoscalar, vector and scalar; on two lattices of size 83 \Lambda 16 and 163 \Lambda 32, for a few values of the hopping parameter ^ in the vicinity of ^charm of which ^ = 0:1060 (for fi = 6) and ^ = 0:1150 (for fi = 6:3)

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Figure 8: Charmonium spectrum obtained on the 163 \Theta 32 lattice at fi = 6 (squares) and fi = 6:3 (diamonds) and on the small lattice 83 \Theta 16 at fi = 6 (crosses) with the clover-andtapole-improved action. The pseudoscalar mass mjc = 2:979GeV is used for normalization.

The wide horizontal lines are experimental data.

are shown. The values of ^ correspond to the tadpole-improved Wilson Clover action 4. The data at fi = 6:3 checks the scaling properties of the renormalized heavy-quark mass. By definition this physical mass should stay fixed and finite as a ! 0 (fi ! 1).

One can see a rather stable value of the renormalized charmed-quark mass m _MSc (mc) = 1:22(5)GeV , which is well in agreement with the estimates of the continuum theory 6: m _MSc (mc) ss 1:23GeV . The previously reported lattice result is 7 : m _MSc (mc) = 1:5(3)GeV .

7 Conclusions Study of the short-distance behavior of the correlator of heavy-quark currents on the lattice appeares as a very promissing way to accurately calculate the parameters of perturbative QCD.

Acknowledgments This report is based on work done in collaboration with Ph. de Forcrand. Computer time for this project was provided by the Minnesota Supercomputer Institute and by the Pittsburgh Supercomputer Center.

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Table 1: The charm-quark mass obtained from fitting the ratios r3, r4 of the subtracted correlator on the small lattice 83 \Theta 16 for fi = 6 , ^ = 0:1060. The masses mres and _mc are in units of the lattice spacing. The masses m

_MS

c [aJ=] and m

_MS c [ajc] are in GeV . Theyare obtained assuming that the lattice spacing is fixed from the vector channel and from the

pseudoscalar channel respectively (a

\Gamma 1 ss 1:9GeV ). The indicated errors are statistical.

fit to r3 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c]j

c 1.57(1) .596(2) .381(2) 1.16(1) 1.15(1) J= 1.61(2) .651(4) .405(3) 1.22(1) 1.21(1)

O/o 1.87(10) .609(9) .329(10) 1.15(3) 1.14(3)

fit to r4 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c]j

c 1.57(1) .623(3) .398(2) 1.17(1) 1.16(1) J= 1.61(2) .666(4) .415(3) 1.23(1) 1.22(1)

O/o 1.87(10) .649(10) .350(10) 1.19(3) 1.18(3)

Table 2: Same as in Table 1 for the big lattice 163 ? 32, fi = 6, ^ = 0:1060. fit to r3 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c ]j

c 1.562(5) .618(2) .396(2) 1.208(6) 1.190(6) J= 1.600(6) .660(1) .413(2) 1.246(6) 1.228(6)

O/o 1.86(4) .645(2) .347(2) 1.230(8) 1.212(8)

fit to r4 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c ]j

c 1.562(5) .645(1) .413(2) 1.224(6) 1.206(6) J= 1.600(6) .677(1) .424(2) 1.258(6) 1.240(6)

O/o 1.86(4) .686(3) .369(7) 1.27(1) 1.25(1)

References

1. M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B 147, 385, 448

(1979). 2. A. Bochkarev, Nucl. Phys. (Proc. Suppl.) 42, 219 (1995) (1995). 3. A. Bochkarev, Ph. de Forcrand, Determination of the renormalized heavy-quark mass in Lattice QCD To appear in Nuclear Physics B, Preprint TPI-MINN-95-17/T, NUC-MINN-95-15/T, HEP-UMN-TH1348, IPS Report 95-15, . 4. B. Sheikholeslami, R. Wohlert, Nucl. Phys. B 259, 572 (1985); G.

Martinelli, C. Sachrajda, A. Vladikas, Nucl. Phys. B 358, 212 (1991); P. Lepage, P. Mackenzie, Phys. Rev. D 48, 2250 (1993).

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Table 3: Same as in Table 2 at fi = 6:3 , ^ = 0:1150. The lattice spacing : a

\Gamma 1 ss 3:7GeV

fit to r3 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c ]j

c 0.805(6) .307(2) .381(2) 1.149(6) 1.148(6) J= 0.836(7) .336(2) .401(3) 1.214(9) 1.212(9)

O/o 0.89(6) .338(5) .38(2) 1.23(3) 1.23(3)

fit to r4 mres _mc _mc=mres m _MSc [aJ=] m _MSc [aj

c ]j

c 0.805(6) .321(2) .399(2) 1.165(6) 1.164(6) J= 0.836(7) .344(2) .411(2) 1.222(6) 1.222(6)

O/o 0.89(6) .359(8) .40(2) 1.27(4) 1.27(4)

5. S. Narison, Phys. Lett. B 197, 405 (1987). 6. For a review see: S. Narison, A fresh look into the heavy-quark mass

values, preprint CERN-TH 7405/94. 7. A. Gonzalez Arroyo, F. Yndurain, G. Martinelli, Phys. Lett. B 117, 437

(1982); C. Allton et. al., Nucl. Phys. B 431, 667 (1994).

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