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A lattice study of semileptonic decays of d-mesons
Volume 223, number 1
PHYSICS LETTERS B
1 June 1989
A LATHCE STUDY OF SEMILEPTONIC DECAYS OF D-MESONS M. CRISAFULLI, G. MARTINELLI Dipartimento di Fisica "G. Marconi'; Universit~ degli Studi di Roma "La Sapienza'" 1-00185 Rome, Italy
V.J. HILL and C.T. SACHRAJDA Department of Physics, University of Southampton, Southampton S09 5NH, UK Received 14 March 1989
We present the results of a computation of the matrix elements of the vector current which are relevant for the semileptonic decays of D-mesons. The computations are performed on a 20 X 102× 40 lattice at fl= 6.0 using Wilson fermions in the quenched approximation. From the study of the matrix element ( K - [JulD °) we find that the form factor at zero m o m e n t u m transfer is given by f + ( 0 ) = 0.74_ 0.17, in excellent agreement with the experimental measurement. From a computation of the matrix element ( n - [JulD °) we find that the corresponding form factor at zero m o m e n t u m transfer is given by f + ( 0 ) = 0.70_+ 0.20. Within our statistical errors vector dominance gives a good description of the form factor at the values of the m o m e n t u m transfers accessible on our lattice.
In this letter we present the results of a lattice computation of the matrix elements ( K - IJulD°) ( ( n - I J ~ I D °) ), where Ju =gT~,c (dyuc) (and hence implicitly also of the corresponding matrix elements for charged D-mesons). These matrix elements contain all the information about the hadronic physics present in semileptonic decays of D Ointo light pseudoscalar mesons. Precise knowledge of these matrix elements would allow for a direct determination of the element Vcs of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [ 1 ] from an experimental measurement of semileptonic decay rates. Below we will compare out results with the data from the Tagged Photon Spectrometer Collaboration [2 ] and the MARK III Collaboration [ 3 ]. Using Lorentz and parity invariance the D--, K matrix element ~ can be parametrized in terms of two form factors, which (using the helicity basis) we choose to b e f + (q2) andfO(q2), defined by
#J The following discussion applies equally well to the D - , n matrix element by modifying the appropriate flavour q u a n t u m numbers.
90
/ m 2 ( K - l J , ] D ° ) =~kPD "bpK -q2
+m~-m~ q2 quf°( q2) ,
\
q). f+ (q2) (1)
where q is the momentum transfer, q=PD--PK. Note that we only need to compute matrix elements of the vector current, since the matrix element of the axial component of the V-A weak current between two pseudoscalar states is zero. At zero momentum transf e r f + (0) = f ° ( 0 ) . We have computed these two form factors at several values of q2. Lattice computations of electromagnetic form factors of hadrons containing quarks of equal mass have shown that different choices for the lattice definition of the vector current can lead to results which differ by up to 25% or so (see refs. [4,5], and references therein). These differences are due to corrections of O ( a ) , where a is the lattice spacing. The optimal choice of vector current is the one which is conserved in the lattice theory itself. For this reason where possible we choose to compute the matrix elements ( 1 ) with
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)
V o l u m e 223, n u m b e r 1
PHYSICS LETTERS B
]~ = Pu = ½ [~(x) (7,, - 1 ) VAx)c(x+~)
+$(x+/~) (Tu + 1 ) U ~ ( x ) c ( x ) ] .
(2)
IT'isthe current which, in the case of degenerate quark masses is conserved in the lattice theory defined by the action with Wilson fermions which we are using (for further details see ref. [4] ). We have also computed the form factorsf + a n d f ° using the local current v l°c"
j#= 7~v,oc IZIoc , u - Zz,o~g(x)Tuc(x) .
at large tx. Thus by computing the correlation functions (4) and (7) the form factorsf ÷ a n d f ° can be determined. The two- and three-point correlation functions can be expressed in terms of the quark propagators. For example, in the quenched approximation the correlation function C u for the local current _uv l°c is given by (see fig. 1 )
Cu(tx, ty)= j [dU] exp[--SE(U)]
(3)
In order to extract the physical form factors from those computed on the lattice using the local current, its renormalization constant Zv,o¢ has to be computed non-perturbatively. We do this by using the Ward identity relating the divergence of the conserved vector current to the scalar density (this will be explained in detail below). The matrix elements are calculated by computing the three-point correlation functions
Cu(tx, ty) - ~, exp(iq-y) exp(ip.x) X,y
× (OIT[MK(X)Ju(Y)M~(O) ]10),
1 J u n e 1989
(4)
where MD and MK are interpolating operators for the D- and K-mesons, respectively (throughout this work we use the pseudoscalar densities (75u and aT~s for these operators). Inserting complete sets of states between the operators in ( 4 ), and taking ty and i x - ty to be sufficiently large that only the lightest single-particle state contributes to each of these complete sets, we find (after continuing to euclidean space)
X ~ exp(iq.y) exp(ip.x) x,y
X Tr [TsSc(0, y)7uSs(y, x)ysS~(x, 0) ]
(8)
with the normalization f [ d U] exp[ - S E (U) ] = 1. In eq. ( 8 ) S¢ ( x, y ), Ss ( x, y ) and S~ (x, y) are the charm, strange and light (up or down) quark propagators from x to y in the background gluon field U. SE is the euclidean gluonic action. Throughout this work we have neglected the difference between the masses of the strange and light quarks, i.e. we take S s ( x , y ) =S~(x, y). Then
C~(tx, ty) = j [dU] exp [ --SE(U) ] × ~ exp(iq.y) Yr[TsS~(O,y)7uS,(y, 0; tx, p ) ] , Y
(9)
where the generalized quark propagator Sl is given by
S,(y, 0; & , p ) -
~ exp(ip-x) S~(y, x)75S~(x, O) . (10)
C,(tx, t~)= ~
x ~ K < K - I S z ID°) exp(--EDty) 4EDE K
× exp[ --EK(tx--ty) ] ,
(5)
where EM is the energy of the meson M ( M = D or K), and the ZM'S are defined by ( 0 JMKI K)=X//~K,
(0iMDID) =x/~D.
We have computed the correlation functions on a 1 0 2 × 2 0 × 4 0 lattice at fl=6.0, using the 30 gluon configurations and corresponding propagators S~ (x, 0), S~(x, 0) and S,(x, 0; tx, p) which had been genVI
(6)
The ZM's and the masses of the mesons (and hence their energies) are determined in the standard way by computing the two-point correlation functions:
I
K ( & ) = ~ < O I T [ M ( x ) M * ( O ) ] I0> .1¢
exp ( - m M t~) --~ 2mM
g Z M
(7)
Fig. 1. 91
Volume 223, number 1
PHYSICS LETTERS B
erated for our earlier studies ofhadronic structure and weak matrix elements (for further details see refs. [4,6 ] ). Note that all the propagators have one end at the origin. The S~ propagators were generated with G = 6 and p = 0 for all 30 configurations, and with tz= 15 and p= (zc/10, 0, 0) (the smallest non-zero momentum consistent with our lattice) for 15 of these configurations. The S~'s and S~'s were generated at three different values of the Wilson parameter K=0.1515, 0.1530 and 0.1545 (the chiral limit corresponds to K=-Kc=O.1563(2) [6] ), and the charm quark propagator with Kch= 0.1350 (which corresponds to a mass of the D ° n 1.8 GeV in the limit in which the mass of the light quark goes to zero). In table 1 we present the values of the meson masses and the ZM'Sobtained from all 30 configurations and from the subset of 15 configurations for which we have the S~ propagators with tx= 15 a n d p ¢ 0. The mass of the vector meson containing a charmed quark and a light antiquark, the D*, is also presented for later comparison of our results with the vector dominance model. The results in table 1 can be converted into physical units by using a - ~= 2.2 + 0.1 GeV [ 6 ]. This value of the lattice spacing was obtained by using the rho mass to set the scale; if the proton's mass is used a smaller value for the inverse lattice spacing is obtained (a - l ~- 1.6 GeV). We start by presenting the results for the case in which the correlation function C~ is computed using the "conserved" vector current l?, defined in eq. (2). Since we only have at our disposal propagators which begin or end at the origin and the current is now extended over two lattice spacings, the current itself cannot be put at the origin. The kaon source is fixed to be either at tx=6 or at tx= 15 (typically a separa-
1 June 1989
tion of six lattice spacings or so is needed to isolate the single particle state). Thus in order to be able to evaluate the diagram in fig. 1 we must put the source for the D-meson at the origin. But it now becomes clear that the kaon source cannot be placed at ix= 6 since, it is then not possible to satisfy the inequalities t x - ty >> 0 and ty >> 0. Thus we are restricted to using the 15 configurations with the S~ propagator generated with the kaon source at tx= 15. We place the current at times ty in the range 6-9, and compute the average (weighted by the errors) of the results obtained at these four values of ty. The final state kaon now has three-momentum p = (n/10, 0, 0), and we have computed the form factorsf ÷ a n d f ° at the three different values of momentum transfer for which the D-meson has momentum - p , 0 and p. The results are presented in table 2, together with the corresponding values of q2. The errors are estimated by dividing the 15 configurations into three clusters of five configurations and treating the results from each cluster as being independent. The case in which the D-meson at rest decays into a kaon with momentum (n/10, 0, 0) corresponds to q2,.~ 0 for all three values of the Wilson parameter of the light quarks. Taking q2 to be identically zero for these points, and extrapolating linearly in the light quark mass to the chiral limit, which is relevant for the decay D--,rc£v, we have obtained ~2 f~+ (0) =0.76 + 0.21.
(11)
If we extrapolate to the value of the Wilson parameter appropriate for D--,I~v, i.e. to the value of K for ~z
Notice that the dependence on the mass of the light quark very mild.
is
Table 1 Values of the meson masses and Z's obtained by computing the two-point correlation function (7), using (a) the full set of 30 configurations, and (b) the subset of 15 configurations for which the matrix elements of the conserved current were computed (see text). The errors on the last figure are given in parentheses. K=0.1515
m~a mDa mo.a ZKa 2 Zoa 2
92
K=0.1530
K=0.1545
(a)
(b)
(a)
(b)
(a)
(b)
0.52(2) 0.90(1) 0.93(1) 0.043(5) 0.057(6)
0.50(3) 0.88(2) 0.92(2) 0.027(4) 0.048(5)
0.45(2) 0.87(2) 0.90(1) 0.037(5) 0.051(6)
0.43(3) 0.85(3) 0.89(2) 0.023(5) 0.043(6)
0.35(3) 0.84(2) 0.87(1) 0.030(7) 0.045(6)
0.36(4) 0.83(3) 0.87(3) 0.020(12) 0.039(6)
Volume 223, number 1
PHYSICS LETTERS B
1 June 1989
Table 2 Values of the form factors fo, f÷ and f - obtained by computing the correlation function (4) with the conserved current. PD is the momentum of the initial D-meson and the final state kaon has three-momentum p = (~r/10, 0, 0 ). PD
K
f+
fo
f -
q2a2
-p
0.1515 0.1530 0.1545
0.71 +0.09 0.69-+0.09 0.66+0.12
0.77+0.07 0.75+0.08 0.71 +0.12
-0.11 +0.03 -0.13+0.05 -0.13+0.08
-0.278(3) -0.253(3) -0.226(9)
0
0.1515 0.1530 0.1545
0.87+0.07 0.85+0.08 0.81 -+0.11
0.87+0.06 0.85+0.08 0.80+0.11
-0.06+0.11 -0.10+0.12 -0.09+0.16
-0.016(2) +0.003(2) +0.026(7)
+p
0.1515 0.1530 0.1545
0.98+0.05 0.96+0.07 0.83+0.07
1.07+0.15 1.07+0.19 1.01 +0.24
0.41+0.79 0.41 +0.81 0.62+0.81
+0.117(3) +0.141 (3) +0.169(9)
which the light pseudoscalar meson mass inK= 490 MeV, we obtain f~ (0) =0.78_0.18.
(12)
We have also d e t e r m i n e d f + ( 0 ) by using the entries o f table 2 at all three values o f q2 for each K. In this case we assume the validity o f the vector d o m i n a n c e model ,t3 f + (q2) rn20. f+(0) -q2-m~.
'
(13)
using the masses o f the vector meson found at the corresponding values o f the light quark masses (table 1 ). Eq. ( 1 3 ) is expected to be accurate in the low q2 region a n d is consistent with the observations o f the authors o f refs. [2,3]. The figures in eqs. ( 11 ) a n d (12) are then m o d i f i e d into f~+ (0) = 0 . 7 0 _ + 0 . 2 0 ,
f~ (0) =0.74+0.17.
(14)
The values in eqs. (14) are our best estimates o f f+(O). The results presented above are extremely satisfying. The statistical errors are reasonably small and the f ~ ( 0 ) is in excellent agreement with recent experimental d a t a from the Tagged P h o t o n S p e c t r o m e t e r Collaboration [ 2 ]. This Collaboration measures the relevant branching ratio to be B ( D ° - - , K - e + V e ) = (3.8 _+0.5 + 0.6)%. F r o m this value o f the b r a n c h i n g ratio and the D O lifetime, and assuming that the q 2 b e h a v i o u r o f the form factor is given by ( 13 ), they ~3 Since the D* and the D* are degenerate we do distinguish between them.
obtain the result I Vcs12 I f + ( 0 ) 12=0.50 ± 0.07 + 0.08. A s s u m i n g three generations a n d i m p o s i n g unitarity on the C K M m a t r i x (i.e. taking I V , I = 0 . 9 7 5 ) , this result corresponds t o f + ( 0 ) = 0.73 + 0.05 + 0.07. The M A R K III [ 3 ] C o l l a b o r a t i o n obtain an almost identical result for the branching ratio, B ( D ° - , K - e + v e ) = (3.9 + 0.6 + 0.6) %, and hence the d e d u c e d value o f f + ( 0 ) is also very similar. Both experiments have studied the b e h a v i o u r o f f +(q2) with q2 and found it to be consistent with the vector d o m i n a n c e model. In fig. 2 we plot the results for the form f a c t o r f + as a function o f q 2 / m 20. at K - - 0.1515. F o r c o m p a r i son with the vector d o m i n a n c e m o d e l we also plot the curve o b t a i n e d by using eq. ( 1 3 ) , with roD* as given in table 1 a n d f + ( 0 ) r = o 15~s=0.88. We stress that the increase o f f + (q2) with increasing q2 is m o r e significant than the error bars in fig. 2 indicate, be1. SO
l
f
I
I
I
I
i ¸
i
~
I
I
I
I
I
I
1. Z 5
1. O0
O. 7 5
O. 5 0
8.25
O. 08
-13.4
I
, , I .... -O.Z
I .... G (q/MD-)2
I , G.2
Fig. 2. 93
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PHYSICS LETTERS B
cause o f the correlations in the results at the three different values o f m o m e n t u m transfer. In table 2 we also report our results f o r f ° ( q 2 ) . A single pole model for f O(q2) would predict a behaviour similar to that in ( 13 ) but with the D* mass replaced by the scalar c h a r m e d meson. In this case, however, we do not have a lattice m e a s u r e m e n t o f the scalar meson mass, which we expect would require a much larger statistical sample o f gauge field configurations. It is, however, instructive to report our results f o r f - defined as
f - (q2)= m2-m~< q2 [fO(qZ)_f+(q2)] .
(15)
f - is given in table 2 at the three values o f K and different q2. O u r results f o r f - , which should be zero in the S U ( 4 ) symmetric limit, show that it is different from zero at least for those points for which the statistical errors are reasonably small, i.e. at K = 0.1515 and 0.1530 a n d p D = --p. We have also calculated the form f a c t o r s f o and f + by computing the correlation function ( 4 ) using the local vector current V ~°c. In this case, since the quark and antiquark fields in the vector current are at the same point, we can place the current at the origin, and use the S~ propagators with the kaon source at t--6. The source for the D - m e s o n is then placed at negative times (the results presented below are the weighted averages o b t a i n e d from the values computed by placing D - m e s o n at times between - 9 and -6). The kaon in the final state is now at rest, and we compute the form factors for the three values o f
1 June 1989
m o m e n t u m transfer for which the D-meson has threem o m e n t u m 0, p and 2,0, where p = ( n / 1 0 , 0, 0). The results for f °/Zv~oc and f+/Zvioc, where Zv,oc is the renormalization constant o f the local vector current, are presented in table 3. We expect the points corresponding to PD = 2/' to have a larger systematic error since the m o m e n t u m o f the D - m e s o n is fairly large in this case. F o r this reason we will not consider t h e m further. The most efficient way to c o m p u t e Zv~oc directly is by repeating this computation with the sources for the D - m e s o n and kaon at time zero and 15 respectively, and by c o m p a r i n g the results with those o b t a i n e d using the " c o n s e r v e d current". This c o m p u t a t i o n is in progress. In the m e a n t i m e we estimate Zv,oc using the W a r d identity which relates the divergence o f the vector current to the scalar density:
l(l
(KIV~I~ ]U) = 5
(KlgclO).
(16)
The R H S o f eq. ( 16 ) is not renormalized. Thus, by using the relation between the matrix elements o f the conserved and local currents, one has
Zv'°c(KIVUV~'°~ID)= 5
h
1)
(17) By c o m p a r i n g the matrix elements o f the divergence o f the local current, which can be derived from the matrix elements o f V~°c a n d qu, with those o f the scalar density at the same m o m e n t u m transfer we can derive Z v .... Using the m e t h o d described above we have found
Table 3 Values of the form factorsf°/Zv,o¢ andf+/Zv,oc obtained by computing the correlation function (4) with the local current. PD is the momentum of the initial D-meson and the final state kaon is at rest.
94
PD
K
0
0.1515 0.1530 0.1545
p
2/7
f+ /Zv~oc
f°/Zv,o~
q2aZ
-
1.02_+0.12 1.01_+0.10 1.01 _ + 0 . 1 0
+0.140(10) +0.179(17) +0.244(25)
0.1515 0.1530 0.1545
1.08+0.12 1.09+0.09 1.12_+0.10
1.00_+0.12 0.97-+0.09 0.95+_0.09
+0.084( 11 ) +0.130(18) +0.205(28)
0.1515 0.1530 0.1545
0.84_+0.20 0.74_+ 0.21 0.45-+0.35
0.88_+0.22 0.74 +_0.20 0.51 _ + 0 . 2 6
-0.067(13) - 0.002 (22) +0.099(34)
Volume 223, number 1
Z~oc =0.98 +0.10
PHYSICS LETTERS B
(18)
to be compared with the results obtained from the matrix elements of the vector current between two light pseudoscalars, Zv,oc ~- 0.71 ~4 By assuming Zv~oc= 1, which is compatible with the result given in eq. (18), we see that the results reported in table 3 for PD =P are compatible with the results obtained from the matrix elements of I7". However, a better determination of the renormalization constant of the local current, from a direct comparison of its matrix elements with those of the conserved current at the same q2 would help to reduce the uncertainty. This would also clarify the issue of the dependence of the renormalization constant on the mass of the quark. Recently, some preliminary lattice results for the form factors f + and f o were presented by Bernard, E1-Khadra and Soni [ 7 ]. These authors compute the D--,n and the D ~ K matrix elements on a 163×25 lattice at fl=5.7 and on a 163×33 lattice at fl=6.1, using the local vector current, g ( x ) ~ , s ( x ) . Assuming vector dominance they find f + (0) = 0.71 (23) and f + (0) = 0.63 ( 15 ) for the kaon and pion case respectively to be compared with our results in eq. (14). Since the calculation of ref. [7] uses the local current, the lattice results for its matrix elements must be converted into the continuum ones by multiplying the current by a factor which can be estimated with an accuracy of at most 20%. The preliminary results presented there were not yet sufficiently precise to determine the dependence of the form factor on the momentum transfer. We will not review here the many interesting phenomenological models which have been proposed to study the form factors (see for example refs. [ 8-10 ], and references therein). These have a variety of model assumptions and parameters and consequently a range of predictions (includingf + (0) = 0.6 + 0.1 [ 8 ], 0.76 [9],0.58 [10]). We end this letter with a brief summary of our main results and conclusions. We have evaluated the form factors of the weak vector current relevant for semileptonic decays of D-mesons. Our best results for these
~4 We recall that, up to terms of O(a), Zv,oc is expected to be independent of quark masses.
1 June 1989
form factors at zero momentum transfer are given in eq. ( 14 ). These results have reasonably small (statistical) errors and are in agreement with experimental measurements. The results at non-zero momentum transfer are in general agreement with the vector dominance model (note in particular that the form factors increase with q2 in fig. 2, where q is the momentum transfer). This work adds to our confidence that lattice computations of operator matrix elements are becoming sufficiently precise to be of great phenomenological value. We also wish to stress two technical points. Firstly the use of the "conserved" vector current helps eliminate a significant source of systematic error. Secondly, when studying matrix elements between a heavy and a light particle, it is frequently useful to give the light particle a non-zero three-momentum. In our case when the final state kaon had momentump, wherep= (n/10, 0, 0), it was possible to evaluate the form factors at zero momentum transfer directly (with the D-meson at rest q2= 0). On the other hand when the final state kaon is at rest in order to have q2_~0 we need the D-meson to have a fairly large momentum, 2p, and hence the systematic errors are significant in this case. We are greatly indebted to L. Maiani for many illuminating discussions. Interesting discussions with the authors of ref. [ 7] are gratefully acknowledged. The computations described in this paper were performed using the CRAY-XMPs at the RutherfordAppleton Laboratory and at Cineca. We acknowledge support from the SERC and from INFN which made these computations possible.
References [ 1 ] N. Cabibbo, Phys. Rev. Len. 10 ( 1963 ) 531 ; M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1972) 282. [2] Tagged Photon Spectrometer Collab., J.C. Anjos et al., Fermilab preprint FNAL 88/141-E (1988). [3] MARK 3 Collab., D. Hitlin, Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energies, eds. W. Barrel and R. Ruckl (North-Holland, Amsterdam, 1987); R. Schindler, Proc. XXXIII Intern. Conf. on High energy physics, Vol 1, ed. S.C. Loken (World Scientific, Singapore, 1987) p. 745; D. Coffman, Ph.D. thesis, California Institute of Technology, CALT-68-1415 (1987).
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[ 4 ] G. MartineUi and C.T. Sachrajda, Nucl. Phys. B 306 ( 1988 ) 865. [ 5 ] C.T. Sachrajda, Southampton University preprint SHEP 88/ 89-2, in: Proc. 1988 Lattice Conf., to be published. [6] M.B. Gavela et al., Nucl. Phys. B 306 (1988) 677. [ 7 ] C. Bernard, A. E1-Khadra and A. Soni, presented by A. Soni at the Ringberg Workshop on Hadronic matrix elements and weak decays (April 1988), UCLA preprint U C L A / 8 8 / TEP31; UCLA preprint UCLA/88/TEP/44, in: Proc. 1988 Lattice Conf., to be published.
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[8 ] T.M. Aliev et al., Sov. J. Nucl. Phys. 40 (1984) 527. [9] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637. [ 10 ] B. Grinstein et al., University of Toronto preprint UTPT88-12 (1988).
PHYSICS LETTERS B
1 June 1989
A LATHCE STUDY OF SEMILEPTONIC DECAYS OF D-MESONS M. CRISAFULLI, G. MARTINELLI Dipartimento di Fisica "G. Marconi'; Universit~ degli Studi di Roma "La Sapienza'" 1-00185 Rome, Italy
V.J. HILL and C.T. SACHRAJDA Department of Physics, University of Southampton, Southampton S09 5NH, UK Received 14 March 1989
We present the results of a computation of the matrix elements of the vector current which are relevant for the semileptonic decays of D-mesons. The computations are performed on a 20 X 102× 40 lattice at fl= 6.0 using Wilson fermions in the quenched approximation. From the study of the matrix element ( K - [JulD °) we find that the form factor at zero m o m e n t u m transfer is given by f + ( 0 ) = 0.74_ 0.17, in excellent agreement with the experimental measurement. From a computation of the matrix element ( n - [JulD °) we find that the corresponding form factor at zero m o m e n t u m transfer is given by f + ( 0 ) = 0.70_+ 0.20. Within our statistical errors vector dominance gives a good description of the form factor at the values of the m o m e n t u m transfers accessible on our lattice.
In this letter we present the results of a lattice computation of the matrix elements ( K - IJulD°) ( ( n - I J ~ I D °) ), where Ju =gT~,c (dyuc) (and hence implicitly also of the corresponding matrix elements for charged D-mesons). These matrix elements contain all the information about the hadronic physics present in semileptonic decays of D Ointo light pseudoscalar mesons. Precise knowledge of these matrix elements would allow for a direct determination of the element Vcs of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [ 1 ] from an experimental measurement of semileptonic decay rates. Below we will compare out results with the data from the Tagged Photon Spectrometer Collaboration [2 ] and the MARK III Collaboration [ 3 ]. Using Lorentz and parity invariance the D--, K matrix element ~ can be parametrized in terms of two form factors, which (using the helicity basis) we choose to b e f + (q2) andfO(q2), defined by
#J The following discussion applies equally well to the D - , n matrix element by modifying the appropriate flavour q u a n t u m numbers.
90
/ m 2 ( K - l J , ] D ° ) =~kPD "bpK -q2
+m~-m~ q2 quf°( q2) ,
\
q). f+ (q2) (1)
where q is the momentum transfer, q=PD--PK. Note that we only need to compute matrix elements of the vector current, since the matrix element of the axial component of the V-A weak current between two pseudoscalar states is zero. At zero momentum transf e r f + (0) = f ° ( 0 ) . We have computed these two form factors at several values of q2. Lattice computations of electromagnetic form factors of hadrons containing quarks of equal mass have shown that different choices for the lattice definition of the vector current can lead to results which differ by up to 25% or so (see refs. [4,5], and references therein). These differences are due to corrections of O ( a ) , where a is the lattice spacing. The optimal choice of vector current is the one which is conserved in the lattice theory itself. For this reason where possible we choose to compute the matrix elements ( 1 ) with
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)
V o l u m e 223, n u m b e r 1
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]~ = Pu = ½ [~(x) (7,, - 1 ) VAx)c(x+~)
+$(x+/~) (Tu + 1 ) U ~ ( x ) c ( x ) ] .
(2)
IT'isthe current which, in the case of degenerate quark masses is conserved in the lattice theory defined by the action with Wilson fermions which we are using (for further details see ref. [4] ). We have also computed the form factorsf + a n d f ° using the local current v l°c"
j#= 7~v,oc IZIoc , u - Zz,o~g(x)Tuc(x) .
at large tx. Thus by computing the correlation functions (4) and (7) the form factorsf ÷ a n d f ° can be determined. The two- and three-point correlation functions can be expressed in terms of the quark propagators. For example, in the quenched approximation the correlation function C u for the local current _uv l°c is given by (see fig. 1 )
Cu(tx, ty)= j [dU] exp[--SE(U)]
(3)
In order to extract the physical form factors from those computed on the lattice using the local current, its renormalization constant Zv,o¢ has to be computed non-perturbatively. We do this by using the Ward identity relating the divergence of the conserved vector current to the scalar density (this will be explained in detail below). The matrix elements are calculated by computing the three-point correlation functions
Cu(tx, ty) - ~, exp(iq-y) exp(ip.x) X,y
× (OIT[MK(X)Ju(Y)M~(O) ]10),
1 J u n e 1989
(4)
where MD and MK are interpolating operators for the D- and K-mesons, respectively (throughout this work we use the pseudoscalar densities (75u and aT~s for these operators). Inserting complete sets of states between the operators in ( 4 ), and taking ty and i x - ty to be sufficiently large that only the lightest single-particle state contributes to each of these complete sets, we find (after continuing to euclidean space)
X ~ exp(iq.y) exp(ip.x) x,y
X Tr [TsSc(0, y)7uSs(y, x)ysS~(x, 0) ]
(8)
with the normalization f [ d U] exp[ - S E (U) ] = 1. In eq. ( 8 ) S¢ ( x, y ), Ss ( x, y ) and S~ (x, y) are the charm, strange and light (up or down) quark propagators from x to y in the background gluon field U. SE is the euclidean gluonic action. Throughout this work we have neglected the difference between the masses of the strange and light quarks, i.e. we take S s ( x , y ) =S~(x, y). Then
C~(tx, ty) = j [dU] exp [ --SE(U) ] × ~ exp(iq.y) Yr[TsS~(O,y)7uS,(y, 0; tx, p ) ] , Y
(9)
where the generalized quark propagator Sl is given by
S,(y, 0; & , p ) -
~ exp(ip-x) S~(y, x)75S~(x, O) . (10)
C,(tx, t~)= ~
x ~ K < K - I S z ID°) exp(--EDty) 4EDE K
× exp[ --EK(tx--ty) ] ,
(5)
where EM is the energy of the meson M ( M = D or K), and the ZM'S are defined by ( 0 JMKI K)=X//~K,
(0iMDID) =x/~D.
We have computed the correlation functions on a 1 0 2 × 2 0 × 4 0 lattice at fl=6.0, using the 30 gluon configurations and corresponding propagators S~ (x, 0), S~(x, 0) and S,(x, 0; tx, p) which had been genVI
(6)
The ZM's and the masses of the mesons (and hence their energies) are determined in the standard way by computing the two-point correlation functions:
I
K ( & ) = ~ < O I T [ M ( x ) M * ( O ) ] I0> .1¢
exp ( - m M t~) --~ 2mM
g Z M
(7)
Fig. 1. 91
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erated for our earlier studies ofhadronic structure and weak matrix elements (for further details see refs. [4,6 ] ). Note that all the propagators have one end at the origin. The S~ propagators were generated with G = 6 and p = 0 for all 30 configurations, and with tz= 15 and p= (zc/10, 0, 0) (the smallest non-zero momentum consistent with our lattice) for 15 of these configurations. The S~'s and S~'s were generated at three different values of the Wilson parameter K=0.1515, 0.1530 and 0.1545 (the chiral limit corresponds to K=-Kc=O.1563(2) [6] ), and the charm quark propagator with Kch= 0.1350 (which corresponds to a mass of the D ° n 1.8 GeV in the limit in which the mass of the light quark goes to zero). In table 1 we present the values of the meson masses and the ZM'Sobtained from all 30 configurations and from the subset of 15 configurations for which we have the S~ propagators with tx= 15 a n d p ¢ 0. The mass of the vector meson containing a charmed quark and a light antiquark, the D*, is also presented for later comparison of our results with the vector dominance model. The results in table 1 can be converted into physical units by using a - ~= 2.2 + 0.1 GeV [ 6 ]. This value of the lattice spacing was obtained by using the rho mass to set the scale; if the proton's mass is used a smaller value for the inverse lattice spacing is obtained (a - l ~- 1.6 GeV). We start by presenting the results for the case in which the correlation function C~ is computed using the "conserved" vector current l?, defined in eq. (2). Since we only have at our disposal propagators which begin or end at the origin and the current is now extended over two lattice spacings, the current itself cannot be put at the origin. The kaon source is fixed to be either at tx=6 or at tx= 15 (typically a separa-
1 June 1989
tion of six lattice spacings or so is needed to isolate the single particle state). Thus in order to be able to evaluate the diagram in fig. 1 we must put the source for the D-meson at the origin. But it now becomes clear that the kaon source cannot be placed at ix= 6 since, it is then not possible to satisfy the inequalities t x - ty >> 0 and ty >> 0. Thus we are restricted to using the 15 configurations with the S~ propagator generated with the kaon source at tx= 15. We place the current at times ty in the range 6-9, and compute the average (weighted by the errors) of the results obtained at these four values of ty. The final state kaon now has three-momentum p = (n/10, 0, 0), and we have computed the form factorsf ÷ a n d f ° at the three different values of momentum transfer for which the D-meson has momentum - p , 0 and p. The results are presented in table 2, together with the corresponding values of q2. The errors are estimated by dividing the 15 configurations into three clusters of five configurations and treating the results from each cluster as being independent. The case in which the D-meson at rest decays into a kaon with momentum (n/10, 0, 0) corresponds to q2,.~ 0 for all three values of the Wilson parameter of the light quarks. Taking q2 to be identically zero for these points, and extrapolating linearly in the light quark mass to the chiral limit, which is relevant for the decay D--,rc£v, we have obtained ~2 f~+ (0) =0.76 + 0.21.
(11)
If we extrapolate to the value of the Wilson parameter appropriate for D--,I~v, i.e. to the value of K for ~z
Notice that the dependence on the mass of the light quark very mild.
is
Table 1 Values of the meson masses and Z's obtained by computing the two-point correlation function (7), using (a) the full set of 30 configurations, and (b) the subset of 15 configurations for which the matrix elements of the conserved current were computed (see text). The errors on the last figure are given in parentheses. K=0.1515
m~a mDa mo.a ZKa 2 Zoa 2
92
K=0.1530
K=0.1545
(a)
(b)
(a)
(b)
(a)
(b)
0.52(2) 0.90(1) 0.93(1) 0.043(5) 0.057(6)
0.50(3) 0.88(2) 0.92(2) 0.027(4) 0.048(5)
0.45(2) 0.87(2) 0.90(1) 0.037(5) 0.051(6)
0.43(3) 0.85(3) 0.89(2) 0.023(5) 0.043(6)
0.35(3) 0.84(2) 0.87(1) 0.030(7) 0.045(6)
0.36(4) 0.83(3) 0.87(3) 0.020(12) 0.039(6)
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Table 2 Values of the form factors fo, f÷ and f - obtained by computing the correlation function (4) with the conserved current. PD is the momentum of the initial D-meson and the final state kaon has three-momentum p = (~r/10, 0, 0 ). PD
K
f+
fo
f -
q2a2
-p
0.1515 0.1530 0.1545
0.71 +0.09 0.69-+0.09 0.66+0.12
0.77+0.07 0.75+0.08 0.71 +0.12
-0.11 +0.03 -0.13+0.05 -0.13+0.08
-0.278(3) -0.253(3) -0.226(9)
0
0.1515 0.1530 0.1545
0.87+0.07 0.85+0.08 0.81 -+0.11
0.87+0.06 0.85+0.08 0.80+0.11
-0.06+0.11 -0.10+0.12 -0.09+0.16
-0.016(2) +0.003(2) +0.026(7)
+p
0.1515 0.1530 0.1545
0.98+0.05 0.96+0.07 0.83+0.07
1.07+0.15 1.07+0.19 1.01 +0.24
0.41+0.79 0.41 +0.81 0.62+0.81
+0.117(3) +0.141 (3) +0.169(9)
which the light pseudoscalar meson mass inK= 490 MeV, we obtain f~ (0) =0.78_0.18.
(12)
We have also d e t e r m i n e d f + ( 0 ) by using the entries o f table 2 at all three values o f q2 for each K. In this case we assume the validity o f the vector d o m i n a n c e model ,t3 f + (q2) rn20. f+(0) -q2-m~.
'
(13)
using the masses o f the vector meson found at the corresponding values o f the light quark masses (table 1 ). Eq. ( 1 3 ) is expected to be accurate in the low q2 region a n d is consistent with the observations o f the authors o f refs. [2,3]. The figures in eqs. ( 11 ) a n d (12) are then m o d i f i e d into f~+ (0) = 0 . 7 0 _ + 0 . 2 0 ,
f~ (0) =0.74+0.17.
(14)
The values in eqs. (14) are our best estimates o f f+(O). The results presented above are extremely satisfying. The statistical errors are reasonably small and the f ~ ( 0 ) is in excellent agreement with recent experimental d a t a from the Tagged P h o t o n S p e c t r o m e t e r Collaboration [ 2 ]. This Collaboration measures the relevant branching ratio to be B ( D ° - - , K - e + V e ) = (3.8 _+0.5 + 0.6)%. F r o m this value o f the b r a n c h i n g ratio and the D O lifetime, and assuming that the q 2 b e h a v i o u r o f the form factor is given by ( 13 ), they ~3 Since the D* and the D* are degenerate we do distinguish between them.
obtain the result I Vcs12 I f + ( 0 ) 12=0.50 ± 0.07 + 0.08. A s s u m i n g three generations a n d i m p o s i n g unitarity on the C K M m a t r i x (i.e. taking I V , I = 0 . 9 7 5 ) , this result corresponds t o f + ( 0 ) = 0.73 + 0.05 + 0.07. The M A R K III [ 3 ] C o l l a b o r a t i o n obtain an almost identical result for the branching ratio, B ( D ° - , K - e + v e ) = (3.9 + 0.6 + 0.6) %, and hence the d e d u c e d value o f f + ( 0 ) is also very similar. Both experiments have studied the b e h a v i o u r o f f +(q2) with q2 and found it to be consistent with the vector d o m i n a n c e model. In fig. 2 we plot the results for the form f a c t o r f + as a function o f q 2 / m 20. at K - - 0.1515. F o r c o m p a r i son with the vector d o m i n a n c e m o d e l we also plot the curve o b t a i n e d by using eq. ( 1 3 ) , with roD* as given in table 1 a n d f + ( 0 ) r = o 15~s=0.88. We stress that the increase o f f + (q2) with increasing q2 is m o r e significant than the error bars in fig. 2 indicate, be1. SO
l
f
I
I
I
I
i ¸
i
~
I
I
I
I
I
I
1. Z 5
1. O0
O. 7 5
O. 5 0
8.25
O. 08
-13.4
I
, , I .... -O.Z
I .... G (q/MD-)2
I , G.2
Fig. 2. 93
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cause o f the correlations in the results at the three different values o f m o m e n t u m transfer. In table 2 we also report our results f o r f ° ( q 2 ) . A single pole model for f O(q2) would predict a behaviour similar to that in ( 13 ) but with the D* mass replaced by the scalar c h a r m e d meson. In this case, however, we do not have a lattice m e a s u r e m e n t o f the scalar meson mass, which we expect would require a much larger statistical sample o f gauge field configurations. It is, however, instructive to report our results f o r f - defined as
f - (q2)= m2-m~< q2 [fO(qZ)_f+(q2)] .
(15)
f - is given in table 2 at the three values o f K and different q2. O u r results f o r f - , which should be zero in the S U ( 4 ) symmetric limit, show that it is different from zero at least for those points for which the statistical errors are reasonably small, i.e. at K = 0.1515 and 0.1530 a n d p D = --p. We have also calculated the form f a c t o r s f o and f + by computing the correlation function ( 4 ) using the local vector current V ~°c. In this case, since the quark and antiquark fields in the vector current are at the same point, we can place the current at the origin, and use the S~ propagators with the kaon source at t--6. The source for the D - m e s o n is then placed at negative times (the results presented below are the weighted averages o b t a i n e d from the values computed by placing D - m e s o n at times between - 9 and -6). The kaon in the final state is now at rest, and we compute the form factors for the three values o f
1 June 1989
m o m e n t u m transfer for which the D-meson has threem o m e n t u m 0, p and 2,0, where p = ( n / 1 0 , 0, 0). The results for f °/Zv~oc and f+/Zvioc, where Zv,oc is the renormalization constant o f the local vector current, are presented in table 3. We expect the points corresponding to PD = 2/' to have a larger systematic error since the m o m e n t u m o f the D - m e s o n is fairly large in this case. F o r this reason we will not consider t h e m further. The most efficient way to c o m p u t e Zv~oc directly is by repeating this computation with the sources for the D - m e s o n and kaon at time zero and 15 respectively, and by c o m p a r i n g the results with those o b t a i n e d using the " c o n s e r v e d current". This c o m p u t a t i o n is in progress. In the m e a n t i m e we estimate Zv,oc using the W a r d identity which relates the divergence o f the vector current to the scalar density:
l(l
(KIV~I~ ]U) = 5
(KlgclO).
(16)
The R H S o f eq. ( 16 ) is not renormalized. Thus, by using the relation between the matrix elements o f the conserved and local currents, one has
Zv'°c(KIVUV~'°~ID)= 5
h
1)
(17) By c o m p a r i n g the matrix elements o f the divergence o f the local current, which can be derived from the matrix elements o f V~°c a n d qu, with those o f the scalar density at the same m o m e n t u m transfer we can derive Z v .... Using the m e t h o d described above we have found
Table 3 Values of the form factorsf°/Zv,o¢ andf+/Zv,oc obtained by computing the correlation function (4) with the local current. PD is the momentum of the initial D-meson and the final state kaon is at rest.
94
PD
K
0
0.1515 0.1530 0.1545
p
2/7
f+ /Zv~oc
f°/Zv,o~
q2aZ
-
1.02_+0.12 1.01_+0.10 1.01 _ + 0 . 1 0
+0.140(10) +0.179(17) +0.244(25)
0.1515 0.1530 0.1545
1.08+0.12 1.09+0.09 1.12_+0.10
1.00_+0.12 0.97-+0.09 0.95+_0.09
+0.084( 11 ) +0.130(18) +0.205(28)
0.1515 0.1530 0.1545
0.84_+0.20 0.74_+ 0.21 0.45-+0.35
0.88_+0.22 0.74 +_0.20 0.51 _ + 0 . 2 6
-0.067(13) - 0.002 (22) +0.099(34)
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Z~oc =0.98 +0.10
PHYSICS LETTERS B
(18)
to be compared with the results obtained from the matrix elements of the vector current between two light pseudoscalars, Zv,oc ~- 0.71 ~4 By assuming Zv~oc= 1, which is compatible with the result given in eq. (18), we see that the results reported in table 3 for PD =P are compatible with the results obtained from the matrix elements of I7". However, a better determination of the renormalization constant of the local current, from a direct comparison of its matrix elements with those of the conserved current at the same q2 would help to reduce the uncertainty. This would also clarify the issue of the dependence of the renormalization constant on the mass of the quark. Recently, some preliminary lattice results for the form factors f + and f o were presented by Bernard, E1-Khadra and Soni [ 7 ]. These authors compute the D--,n and the D ~ K matrix elements on a 163×25 lattice at fl=5.7 and on a 163×33 lattice at fl=6.1, using the local vector current, g ( x ) ~ , s ( x ) . Assuming vector dominance they find f + (0) = 0.71 (23) and f + (0) = 0.63 ( 15 ) for the kaon and pion case respectively to be compared with our results in eq. (14). Since the calculation of ref. [7] uses the local current, the lattice results for its matrix elements must be converted into the continuum ones by multiplying the current by a factor which can be estimated with an accuracy of at most 20%. The preliminary results presented there were not yet sufficiently precise to determine the dependence of the form factor on the momentum transfer. We will not review here the many interesting phenomenological models which have been proposed to study the form factors (see for example refs. [ 8-10 ], and references therein). These have a variety of model assumptions and parameters and consequently a range of predictions (includingf + (0) = 0.6 + 0.1 [ 8 ], 0.76 [9],0.58 [10]). We end this letter with a brief summary of our main results and conclusions. We have evaluated the form factors of the weak vector current relevant for semileptonic decays of D-mesons. Our best results for these
~4 We recall that, up to terms of O(a), Zv,oc is expected to be independent of quark masses.
1 June 1989
form factors at zero momentum transfer are given in eq. ( 14 ). These results have reasonably small (statistical) errors and are in agreement with experimental measurements. The results at non-zero momentum transfer are in general agreement with the vector dominance model (note in particular that the form factors increase with q2 in fig. 2, where q is the momentum transfer). This work adds to our confidence that lattice computations of operator matrix elements are becoming sufficiently precise to be of great phenomenological value. We also wish to stress two technical points. Firstly the use of the "conserved" vector current helps eliminate a significant source of systematic error. Secondly, when studying matrix elements between a heavy and a light particle, it is frequently useful to give the light particle a non-zero three-momentum. In our case when the final state kaon had momentump, wherep= (n/10, 0, 0), it was possible to evaluate the form factors at zero momentum transfer directly (with the D-meson at rest q2= 0). On the other hand when the final state kaon is at rest in order to have q2_~0 we need the D-meson to have a fairly large momentum, 2p, and hence the systematic errors are significant in this case. We are greatly indebted to L. Maiani for many illuminating discussions. Interesting discussions with the authors of ref. [ 7] are gratefully acknowledged. The computations described in this paper were performed using the CRAY-XMPs at the RutherfordAppleton Laboratory and at Cineca. We acknowledge support from the SERC and from INFN which made these computations possible.
References [ 1 ] N. Cabibbo, Phys. Rev. Len. 10 ( 1963 ) 531 ; M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1972) 282. [2] Tagged Photon Spectrometer Collab., J.C. Anjos et al., Fermilab preprint FNAL 88/141-E (1988). [3] MARK 3 Collab., D. Hitlin, Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energies, eds. W. Barrel and R. Ruckl (North-Holland, Amsterdam, 1987); R. Schindler, Proc. XXXIII Intern. Conf. on High energy physics, Vol 1, ed. S.C. Loken (World Scientific, Singapore, 1987) p. 745; D. Coffman, Ph.D. thesis, California Institute of Technology, CALT-68-1415 (1987).
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[ 4 ] G. MartineUi and C.T. Sachrajda, Nucl. Phys. B 306 ( 1988 ) 865. [ 5 ] C.T. Sachrajda, Southampton University preprint SHEP 88/ 89-2, in: Proc. 1988 Lattice Conf., to be published. [6] M.B. Gavela et al., Nucl. Phys. B 306 (1988) 677. [ 7 ] C. Bernard, A. E1-Khadra and A. Soni, presented by A. Soni at the Ringberg Workshop on Hadronic matrix elements and weak decays (April 1988), UCLA preprint U C L A / 8 8 / TEP31; UCLA preprint UCLA/88/TEP/44, in: Proc. 1988 Lattice Conf., to be published.
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[8 ] T.M. Aliev et al., Sov. J. Nucl. Phys. 40 (1984) 527. [9] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637. [ 10 ] B. Grinstein et al., University of Toronto preprint UTPT88-12 (1988).