- Email: [email protected]
Weak matrix elements on the lattice
Nuclear Physics B (Proc. Suppl .) 16 (1990) 381-384 North-Holland
AT IX ELEMENTS O
WEAK
381
THE LATTICE
F.RAPUANO RVFN - Sezione di Roma - c% Dip . di Fisica, Università "La Sapienza ", Roma, Italy
The lattice formulation is very well suited for a
- INTRODUCTION In this talk I will very briefly summarize the latest results obtained on the lattice for the weak hamiltonian matrix elements . The main points that I will cover are :
numerical treatment but is also subject to limitations intrinsic to the available computing resources. The main outcome of this is in an approximation that we are forced to
1) Meson decay constants fn , fk, fD . . . . .
use to make the calculation possible : the effect of fermion
2) B-parameter for the K0 - K0 transition
loops is neglected (quenched approximation) even though
3) The ®I = 1/2 and ®I = 3/2 amplitudes .
some preliminary calculation for the full QCD has already been performed with limited statistic . In general it is widely
4) The semileptonic meson form factor . The effective hamiltonian for the weak transition is
Hw =
G
Cn(g,Mw)
accepted that this theoretically strong approximation should introduce a systematical error of the order 10 and 20% . There are other sources of systematical error however, that come from the fact that in general one is far away from the continuum limit a-+0 and so O(a) terms
On(p
could be present . Methods heve been proposed to reduce these effects to O(g2a) [6] but in general they are
where Gf is the Fermi constant, Cn (11,M w ) are the
On(p
coefficient of the Wilson expansion and suitable operator, renormalised at a scale g.
is the
Due to asymptotic freedom for IL » AQCD - 100200 MeV the coefficients Cn can be safely computed in perturbation theory . Still the matrix elements of the operator O n must be computed non perturbatively . Different approaches have been proposed to achieve this goal; QCD Sum rule techniques [3] , the 1/N expansion [4]. The lattice method has certain advantages : ®) There are non further assumption apart from QCD and the renormalization group . ®) Very good results have obtained for the hadron spectrum by many groups [5] ®) There are no free parameters once that few physical input from the hadron spectrum are fixed. 0920-5632/90/$3 .50 © Elsevier Science Publishers B .V . Northmllolland
substantially more computer time consumimg.
Furthermore the masses of the quarks are not the (very low) physical ones to keep the correlation lengths small with respect to the lattice size and so the physical
quantities are obtained through an extrapolation to the value of the quark mass for which chiral symmetry, that has been explicitly broken by the Wilson* term, is restored. For the Wilson formulation this critical value of the quark mass
must be found for each lattice as the breaking of chiral symmetry allows an additive mass renormalization . This
step is absent in the Kogut-Susskind formulation where the problem of the fermion doubling is solved at the cost of a four-fold increase in the number of flavor described by the theory, but chiral symmetry is not broken . The ELC and UCLA group use the Wilson formulation while the STAG group uses the Kogut-Susskind formulation [7] .
F. Rapuano/ Weak matrix elements on the lattice
382
. - A1=3/2 and ®S=2 erelevant operator for these transitions are : S2 = ["s
(1- y5) dl [s yM (1- 75) dl d i&'& OLl + [SL
04 = [SL
UL uL 1(tt dLl
- [iL r dL aL Ytt dLl but on the lattice, due to the Wilson term one has mixing with different chirality operators . In this case only dimension 6 operators can contribute. O&S-2 = ÂTr (g,go) [O®S=2
Ci(sI'id)(iI'id)
Hw - C+(IL) 0+(R) + C-(p.) 0-(9) where O+ is responsible for both AI=1/2 and ®I=3/2 transitions while O- gives only a AI=1/2 transition. At short distance one has C- /C+-3 and so an extra enhancement is needed tojustify the experimental value. It has been Proposed [2] that "penguin' operators may be responsible for the extra enhancement.The non perturbative lattice measurements of this amplitudes is now harder due to the mixing of O+ and O- with operators of dimension less than 6 which, on dimensional grounds, have divergent. coefficients in the limit a--)0. This quantities must be nouperturbatively computed and subtracted. [10] making the calculation much more unreliable. Also in this case one obtains the AI=1/2 ko-+i+ic- amplitude using the relationship :
where the ci coefficient can be computed perturbatively [8] .
A(ko-47E+n-) MK
The relevant parameter in ®S=2 KO-Ro transition is
the so called B-parameter defined as: B
_ ~li yA (1- 1f5) d i VL(1- y5) dl&O> %° S/3fKmK
defined in such a way to be 1 in the vacuum saturation approximation and from BkOiO one can obtain the ®I=3/2 k+-+n+nn amplitude using the soft pion relation [9] = GF sin0~os0e K 2
2
mK -m n mK
c4(w)
[`kolOITco> ~
fK MK
2-A1=1/2 For the ®I=1/2 the situation is much more complicate. The general decomposition for the weak hamiltonian is
=Z1
Fcc sinucosu
rnK -mn MK
[C-(R) T + C+(lt) Y1l where the quantities ,y. and y" are related to the matrix elements of O+ and O- between k and n states .
3
S TS Table 1 contains the results come from the new run ELC of the collaboration on the APE computer [12] and from the UCLA group for the meson decay constants . Note that the error for the K,D and D8 form factor is somewhat lower the the pion one . This is due to the fact that one computes fx= (fx/fn)LATT " fn exp in such a way to reduce the systematic error.The B and Bs decay constants are obtained using the relation fB/fD= mD/mB which is certainly correct asymptotically but some doubts may be cast on its use at masses of the order of few GeV. The data for the B-parameter are summarized in table 2. The conclusion from those numbers is that the ®I=3/2 turns out to be too large: ELC obtains (6.5±1)10-8
F. Rapuano/ Weak matrix elements on the lattice
to be compared with the exp. value of 3.7 10-8 . The discrepancy denotes that systematic effect are still present, it must be stressed for example that Electromagnetic contribution (very difficult to measure) have been neglected in these calculation and may completely change the scenario, particularly for the DI=3/2 amplitude .
any case, that the penguin diagram is necessary to obtain the enhancement seen in nature. Table 3 Results for the DI=1/2 amplitude
Table 1 Decay constants n
EL
[11,1 ]
154± 25 140±20±20
A(kS--nV+ir) (Experiment. = 21 .2) A(k+-m+x°)
UCLA-[141
Exp. 131
K
158±13
161+
164
D
180±30
174±52
<290
DS
218±30
234±33
Bd
~ 120
105±34
Bs
~ 150
Results for the
renopamt
383
78
UCLA[14] 11 .6±6.0
from k-nx
ELC[17]
from k-x from k-nx
16±40 35±31
The last point that I want to quote is the measurement ofmeson form factor for semileptonic decay i.e. and «-IJILIDo> where J11= i yP c or a c as only the vector cuiTerits contributes between pseudosclar states . These matrix elements are in general parametrized as
T2 able malization group invariant Bkoko
Notes
ELC[13]
.85±.1
preliminary
STAG[15]
.91±.1
[B(i=2GeV)=.7±.08]
+
2
q
q2
m2K
q)9 f+(q2)
qtL) f°(q2)
where f+ parametrizes the exchange ofa vector particle and PD a scalar particle . Table 4 summarizes the results from the ELC[18] and UCLA[ 19] group.
These values are also reasonably consistent with the Bparameter computed from the 1/Nc expansion B=.7±1 [3] and from the QCD-Sum rule B=.5±.1±.` [16]. Table 3 summarizes the present (obscure) situation for the AI=1/2 amplitude . The UCLA group uses the k-nn matrix element to measure this amplitude. In this case no non-perturbative subtraction is needed but the diagrammatic is more complicate. The ELC group has used both the k-n and k-wic matrix elements also in the new high statistics run but could not give any definite answer due to large fluctuations and the poor quality of the signal. It is clear, in
2 MD -
=(PD + PK 2 2 MID -mK
B-parameter
Notes
Table Results for the meson semileptonic form factors fo(0)
NO)
kn
1 .25 ±10
1 .23 ±19 [191 [exp.1 .07.1 .001
Die
.42 ±11
.63 ±15 [191 .70±20 [181
Dk
.66 ±11
.71 ±23 [191 .74 ±17 [181
F. Rapuano/ Weak matrix elements on the lattice
334
They are in very good agreement with each other and in spectacular agreement with the experimental value for ID(0) from the Tagged Photon Collaboration and the Mark III collaboration. They measure the branching ratio B(Do-4k e v) = (3.8±.5t .6)% and (3.9±.6±.6)°70 . Using those results, IVcsl=.975 and the vector meson assumption, one obtains fb(0)=.73. In conclusion one could say that the results are encouraging enough to improve the calculation both statistically and systematically to shade some light on the most obscure and, at the moment, out of reach problem of the octet enhancement . Furthermore a clear understanding of the penguin diagrams is also necessary to understand the problem ofCP violation in the standard model. eferences 1.
M.K. Gaillard, B.W. Lee : Phys. Rev . Lett. 33, 108 (1974) . G.Altarelli, L.Maiani : Phys. Lett. 52B, 351, (1974) .
2.
M.A. Shifman, A .I. Voinshtein, V.J. Zacharov: Nucl. Phys. 120B, 316 (1977) .
3 . W.A. Bardeen, A.J. Buras, J.M. Jerard: Nucl. Phys. 293B, 787 (1977), Phys. Lett.192B, 369, (1987) . 4.
A Pich, E. De Raphael : Phys. Lett. 158B, 477, (1985), Phys. Lett. 189B, 369, (1987).
5.
See the plenary talk by G. Martinelli at this conferences and refereace therein.
6.
K. Symanzik : in Mathematical Problems in Theoretical Physics, ed. by R. Schrader et al., Lecture Notes in Physics (Springer, Berlin 1983).
7.
See the reports by these groups in the Proceedings of the conference on Lattice Field Theory, FNAL, Nucl.
Phys. Proc . Suppl . 9 (1988) 8.
G. Martinelli: Phys. Lett. 141B, (1984), 395. C. Bernard, T. Draper, A. Soni : Phys. Rev. 36D (1987) 3224.
9.
J.F.Donogue et al: Phys. Lett. 119B, (1982), 412.
10. L. Maiani et. al : Nucl. Phys. 289B, 505, (1987) and C. Bernard, A. Soni in ref 7.
11 . ELC Coll., (1988).
.B. Gavela et al : Phys. Lett. 206B, 113
12. See M.P. Lombardo Talk at the Non Pert. Field Teory par. sess. at this conference. 13. ELC Coll. preliminary, in preparation. 14. C. Bernard, A. Soni in ref 7. 15. G.W..Kilcup, S.R.Sharpe, R.Gupta, A.Patel : preliminary private communication from A.Patel, to be published. 16. L.Bilic et al.: DESY 88-187. 17. L. Maiani in ref 7. 18. M. Crisafulli et al.: Phys. Lett. 233B, 90, (1989). 19. A. El Khadra in ref. 7.
AT IX ELEMENTS O
WEAK
381
THE LATTICE
F.RAPUANO RVFN - Sezione di Roma - c% Dip . di Fisica, Università "La Sapienza ", Roma, Italy
The lattice formulation is very well suited for a
- INTRODUCTION In this talk I will very briefly summarize the latest results obtained on the lattice for the weak hamiltonian matrix elements . The main points that I will cover are :
numerical treatment but is also subject to limitations intrinsic to the available computing resources. The main outcome of this is in an approximation that we are forced to
1) Meson decay constants fn , fk, fD . . . . .
use to make the calculation possible : the effect of fermion
2) B-parameter for the K0 - K0 transition
loops is neglected (quenched approximation) even though
3) The ®I = 1/2 and ®I = 3/2 amplitudes .
some preliminary calculation for the full QCD has already been performed with limited statistic . In general it is widely
4) The semileptonic meson form factor . The effective hamiltonian for the weak transition is
Hw =
G
Cn(g,Mw)
accepted that this theoretically strong approximation should introduce a systematical error of the order 10 and 20% . There are other sources of systematical error however, that come from the fact that in general one is far away from the continuum limit a-+0 and so O(a) terms
On(p
could be present . Methods heve been proposed to reduce these effects to O(g2a) [6] but in general they are
where Gf is the Fermi constant, Cn (11,M w ) are the
On(p
coefficient of the Wilson expansion and suitable operator, renormalised at a scale g.
is the
Due to asymptotic freedom for IL » AQCD - 100200 MeV the coefficients Cn can be safely computed in perturbation theory . Still the matrix elements of the operator O n must be computed non perturbatively . Different approaches have been proposed to achieve this goal; QCD Sum rule techniques [3] , the 1/N expansion [4]. The lattice method has certain advantages : ®) There are non further assumption apart from QCD and the renormalization group . ®) Very good results have obtained for the hadron spectrum by many groups [5] ®) There are no free parameters once that few physical input from the hadron spectrum are fixed. 0920-5632/90/$3 .50 © Elsevier Science Publishers B .V . Northmllolland
substantially more computer time consumimg.
Furthermore the masses of the quarks are not the (very low) physical ones to keep the correlation lengths small with respect to the lattice size and so the physical
quantities are obtained through an extrapolation to the value of the quark mass for which chiral symmetry, that has been explicitly broken by the Wilson* term, is restored. For the Wilson formulation this critical value of the quark mass
must be found for each lattice as the breaking of chiral symmetry allows an additive mass renormalization . This
step is absent in the Kogut-Susskind formulation where the problem of the fermion doubling is solved at the cost of a four-fold increase in the number of flavor described by the theory, but chiral symmetry is not broken . The ELC and UCLA group use the Wilson formulation while the STAG group uses the Kogut-Susskind formulation [7] .
F. Rapuano/ Weak matrix elements on the lattice
382
. - A1=3/2 and ®S=2 erelevant operator for these transitions are : S2 = ["s
(1- y5) dl [s yM (1- 75) dl d i&'& OLl + [SL
04 = [SL
UL uL 1(tt dLl
- [iL r dL aL Ytt dLl but on the lattice, due to the Wilson term one has mixing with different chirality operators . In this case only dimension 6 operators can contribute. O&S-2 = ÂTr (g,go) [O®S=2
Ci(sI'id)(iI'id)
Hw - C+(IL) 0+(R) + C-(p.) 0-(9) where O+ is responsible for both AI=1/2 and ®I=3/2 transitions while O- gives only a AI=1/2 transition. At short distance one has C- /C+-3 and so an extra enhancement is needed tojustify the experimental value. It has been Proposed [2] that "penguin' operators may be responsible for the extra enhancement.The non perturbative lattice measurements of this amplitudes is now harder due to the mixing of O+ and O- with operators of dimension less than 6 which, on dimensional grounds, have divergent. coefficients in the limit a--)0. This quantities must be nouperturbatively computed and subtracted. [10] making the calculation much more unreliable. Also in this case one obtains the AI=1/2 ko-+i+ic- amplitude using the relationship :
where the ci coefficient can be computed perturbatively [8] .
A(ko-47E+n-) MK
The relevant parameter in ®S=2 KO-Ro transition is
the so called B-parameter defined as: B
_ ~li yA (1- 1f5) d i VL(1- y5) dl&O> %° S/3fKmK
defined in such a way to be 1 in the vacuum saturation approximation and from BkOiO one can obtain the ®I=3/2 k+-+n+nn amplitude using the soft pion relation [9]
2
mK -m n mK
c4(w)
[`kolOITco> ~
fK MK
2-A1=1/2 For the ®I=1/2 the situation is much more complicate. The general decomposition for the weak hamiltonian is
=Z1
Fcc sinucosu
rnK -mn MK
[C-(R) T + C+(lt) Y1l where the quantities ,y. and y" are related to the matrix elements of O+ and O- between k and n states .
3
S TS Table 1 contains the results come from the new run ELC of the collaboration on the APE computer [12] and from the UCLA group for the meson decay constants . Note that the error for the K,D and D8 form factor is somewhat lower the the pion one . This is due to the fact that one computes fx= (fx/fn)LATT " fn exp in such a way to reduce the systematic error.The B and Bs decay constants are obtained using the relation fB/fD= mD/mB which is certainly correct asymptotically but some doubts may be cast on its use at masses of the order of few GeV. The data for the B-parameter are summarized in table 2. The conclusion from those numbers is that the ®I=3/2 turns out to be too large: ELC obtains (6.5±1)10-8
F. Rapuano/ Weak matrix elements on the lattice
to be compared with the exp. value of 3.7 10-8 . The discrepancy denotes that systematic effect are still present, it must be stressed for example that Electromagnetic contribution (very difficult to measure) have been neglected in these calculation and may completely change the scenario, particularly for the DI=3/2 amplitude .
any case, that the penguin diagram is necessary to obtain the enhancement seen in nature. Table 3 Results for the DI=1/2 amplitude
Table 1 Decay constants n
EL
[11,1 ]
154± 25 140±20±20
A(kS--nV+ir) (Experiment. = 21 .2) A(k+-m+x°)
UCLA-[141
Exp. 131
K
158±13
161+
164
D
180±30
174±52
<290
DS
218±30
234±33
Bd
~ 120
105±34
Bs
~ 150
Results for the
renopamt
383
78
UCLA[14] 11 .6±6.0
from k-nx
ELC[17]
from k-x from k-nx
16±40 35±31
The last point that I want to quote is the measurement ofmeson form factor for semileptonic decay i.e.
T2 able malization group invariant Bkoko
Notes
ELC[13]
.85±.1
preliminary
STAG[15]
.91±.1
[B(i=2GeV)=.7±.08]
+
2
q
q2
m2K
q)9 f+(q2)
qtL) f°(q2)
where f+ parametrizes the exchange ofa vector particle and PD a scalar particle . Table 4 summarizes the results from the ELC[18] and UCLA[ 19] group.
These values are also reasonably consistent with the Bparameter computed from the 1/Nc expansion B=.7±1 [3] and from the QCD-Sum rule B=.5±.1±.` [16]. Table 3 summarizes the present (obscure) situation for the AI=1/2 amplitude . The UCLA group uses the k-nn matrix element to measure this amplitude. In this case no non-perturbative subtraction is needed but the diagrammatic is more complicate. The ELC group has used both the k-n and k-wic matrix elements also in the new high statistics run but could not give any definite answer due to large fluctuations and the poor quality of the signal. It is clear, in
2 MD -
B-parameter
Notes
Table Results for the meson semileptonic form factors fo(0)
NO)
kn
1 .25 ±10
1 .23 ±19 [191 [exp.1 .07.1 .001
Die
.42 ±11
.63 ±15 [191 .70±20 [181
Dk
.66 ±11
.71 ±23 [191 .74 ±17 [181
F. Rapuano/ Weak matrix elements on the lattice
334
They are in very good agreement with each other and in spectacular agreement with the experimental value for ID(0) from the Tagged Photon Collaboration and the Mark III collaboration. They measure the branching ratio B(Do-4k e v) = (3.8±.5t .6)% and (3.9±.6±.6)°70 . Using those results, IVcsl=.975 and the vector meson assumption, one obtains fb(0)=.73. In conclusion one could say that the results are encouraging enough to improve the calculation both statistically and systematically to shade some light on the most obscure and, at the moment, out of reach problem of the octet enhancement . Furthermore a clear understanding of the penguin diagrams is also necessary to understand the problem ofCP violation in the standard model. eferences 1.
M.K. Gaillard, B.W. Lee : Phys. Rev . Lett. 33, 108 (1974) . G.Altarelli, L.Maiani : Phys. Lett. 52B, 351, (1974) .
2.
M.A. Shifman, A .I. Voinshtein, V.J. Zacharov: Nucl. Phys. 120B, 316 (1977) .
3 . W.A. Bardeen, A.J. Buras, J.M. Jerard: Nucl. Phys. 293B, 787 (1977), Phys. Lett.192B, 369, (1987) . 4.
A Pich, E. De Raphael : Phys. Lett. 158B, 477, (1985), Phys. Lett. 189B, 369, (1987).
5.
See the plenary talk by G. Martinelli at this conferences and refereace therein.
6.
K. Symanzik : in Mathematical Problems in Theoretical Physics, ed. by R. Schrader et al., Lecture Notes in Physics (Springer, Berlin 1983).
7.
See the reports by these groups in the Proceedings of the conference on Lattice Field Theory, FNAL, Nucl.
Phys. Proc . Suppl . 9 (1988) 8.
G. Martinelli: Phys. Lett. 141B, (1984), 395. C. Bernard, T. Draper, A. Soni : Phys. Rev. 36D (1987) 3224.
9.
J.F.Donogue et al: Phys. Lett. 119B, (1982), 412.
10. L. Maiani et. al : Nucl. Phys. 289B, 505, (1987) and C. Bernard, A. Soni in ref 7.
11 . ELC Coll., (1988).
.B. Gavela et al : Phys. Lett. 206B, 113
12. See M.P. Lombardo Talk at the Non Pert. Field Teory par. sess. at this conference. 13. ELC Coll. preliminary, in preparation. 14. C. Bernard, A. Soni in ref 7. 15. G.W..Kilcup, S.R.Sharpe, R.Gupta, A.Patel : preliminary private communication from A.Patel, to be published. 16. L.Bilic et al.: DESY 88-187. 17. L. Maiani in ref 7. 18. M. Crisafulli et al.: Phys. Lett. 233B, 90, (1989). 19. A. El Khadra in ref. 7.