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Heavy quark expansion parameters from lattice NRQCD
IOLg[m | I_.~ [ i ,a _-k't.'1[ I k l "-!
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear PhysicsB (Proc. Suppl.) 106 (2002) 376-378
www.elscvier.com/locaLe/n~
Heavy quark expansion parameters from lattice NRQCD
*
JLQCD Collaboration: N. Tsutsui a, S. Aoki b, R. Burkhalter c, M. Fukugita d, S. Hashimoto a, K-I. Ishikawa ¢, N. Ishizuka b'~, V. lwasaki b'c, K. Kanaya b,c, T. Kaneko a, Y. Kuramashi a, M. Okawa a, T. Onogi e, S, Tominaga c, A. Ukawa b'c, N. Yamada a, T. Yoshié b'¢ ~High Energy Accelerator Research Organization(KEK),
Tsukuba, Ibaraki 305-0801, Japan
blnstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan CCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan dInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan eyukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Using the lattice NRQCD action for heavy quark, we calculate the heavy quark expansion parameters/~r and B~ for heavy-light mesons and heavy-light-]ight baryons. The results are compared with the mass differences among heavy hadrons to test the validity of HQET relations on the lattice.
1. I n t r o d u c t i o n In the calculation of inclusive decay rates of the heavy hadron, the heavy quark expansion (HQE) technique is widely used. At the order 1/m S of HQE two nonperturbative parameters 1
#~(HQ) -- 2Mù----~(HQ Q(iD)2Q HQ), p~(HQ) =--
1 2MH----~( HO O,ä" BQ "Q},
(1) (2)
appear in the calculation. HQ represents a heavylight meson or heavy-light-light baryon (for b hadrons, Hb = B, B*, Ab, Eb, E~). For instance, the lifetime ratio of b hadrons is given as [1]
defined in the static limit: mQ --+c~. For heavylight meson, #27r,G has been calculated using the lattice version of the Heavy Quark Effective The-
ory [2]. In this work we calculate #2 and #2 on the lattice using the NRQCD action for heavy quark. Although the individual matrix element suffers frorn large perturbative uncertainty due to power divergence in the rnatching calculation, their dir. #.(H~ 2 (1) 2 (2) ferences hke )-,~(H~ ) are free ~ o m the uncertainty of the operator. We calculate both #2,(H~1))- #2(H~2)) and #2(H~1)) - #~(H~2)), and cornpare thern with the corresponding predictions for mass splittings. 2.
T(H~I)) = 1 q- #~(H~I)) -/z~(H~2)) T(H~2)) 2m~ F2 / T./'(1)~
+co
Gk**b
, 2 [ T..l'(2)'~ ] -- I'~G\**b ]
m~
+ 0(1/m3),
The pararneters #5 and #~ can be indirectly obtained from hadron masses, using (3)
with cv - 1.2. While #2 may be evaluated from experimental values of hyperfine splitting, the determination of ~2 requires some theoretical inputs. It should be noted that the parameters are *presented by N. Tsutsui
H Q E T mass f o r m u l a
MHo mQ ----Ä + -
-
--/Zrr --/iG + O
2mQ
1
,
(4)
where Ä is the residual energy difference between MHQ and mQ surviving in the infinite heavy quark limit. ~2 and Iz~ appear in the correction terms of O(1/mo). Therefore, by considering proper mass differences, certain combinations
0920-5632/02/$ - see frontmatter© 2002 ElsevierScienceB.V. All rightsreserved. PI1 S0920-5632(01)01719-4
N. Tsutsui et al./Nuclear Physics B (Proc. Suppl.) 106 (2002) 376-378
0.6
377
0.5
0.5
0.4
0.4 0.3
0.3
C.D
t
0.2
2 2 MBo~-M~ :
0.1
-a~5(B --B)
0
,
0
0.1
i
=
o x p .
0.2 0.3 0.4 t / M ~ [ G e V -1]
:
0.2
0.1
:
--A~5(~'--Z" )
:= : i
0.5
=
exp. 0.6
Figure 1. Hyperfine sp]itting of ground state mesons. Circles is obtained ~om the energy differences, while crosses are from the matrix elements.
0
i
i
0.1
0.2
i
a
0.3 0.4 IIMÆ [ G e V - ' ]
0.5
0.6
Figure 2. Hyperfine splitting of heavy-light-light baryons.
elements/z 2 and #2. of/z~ and ~~ can be extracted. For example, a difference of #~ can be obtained from the mass splitting in a spin multiplet, because Ä and/z2~ have the same value. Also, the spin averaged mass M B = (MB + 3 M ~ ) / 4 does not depend on #2, because/z 2 is proportional to the spin of the light degrees of the freedom and the sum of #2 in the spin multiplet vanishes. 3. L a t t i c e c a l c u l a t i o n s We carry out quenched QCD simulations at /3=6.0 on a 203 x 48 lattice. The NRQCD aetion including all O(1/mQ) terms and the non-perturbatively improved clover action (cs,0=1.769) is adapted for heavy quark and light quark, respectively. Five heavy quark masses amQ=l.3, 2.1, 3.0, 5.0, and 10.0 are used to study the 1/mQ dependence of hadron masses and matrix elements, while three hopping parameters K--0.13331, 0.13384, and 0.13432 are simulated to extrapolate to the chiral limit Kc=0.135284(8). The inverse lattice spacing a-1--1.85(5) GeV is determined with the p meson mass m»--770 MeV. We measure the three-point functions, where OHQ is an interpolating field to create or annihilate the hadron HQ, and O~,a is the operator to be measured, ~ ( i ß ) 2 Q or Q ä . / ~ Q . We divide them by to obtain the desired matrix
(OHQ(tl)OtHQ(O))
4. Hyperfine spUttings From (4) the hyperfine splitting MB. given by -A#2/2mQ, or equivalently
-- MB is
M2. - M 2 = - A ~ 2 -- - ( / z 2 ( B *) - / z 2 ( B ) ) , (5) at the leading order. In Figure 1, we plot our results for - A # ~ together with the measurement of M2. - M 2. We observe that the relation (5) is satisfied very well, while both are significantly lower than the experimental values for B and D mesons. In deriving (5) we used a relation A#~ = #~(B*) - p~(B) = 0,
(6)
which holds in the static limit. However, for the NRQCD action including the spin-magnetic interaction term at O(1/mQ), the operator O~ mixes with Og at order as/mQ. This is the reason why our result for - A # 2 deviates from that of the mass difference in the lighter heavy quark mass region. In other words, the relation (6) may be considered as a renormalization condition for the operator O~.. Similar armlysis can be made for the hyperfine splitting of heavy-light-light baryon, i.e. ~,* ~. splitting. Figure 2 shows the mass difference and the matrix element --A#~. Both are in good agreement.
378
N. Tsutsui et aL /Nuclear Physics B (Proc. Suppl.) 106 (2002) 376-378
0 ° 5
0.1
,
0
0.4
-0.1 0.3
-0.21
-0.2 0.2 -0.3 0.1
0.393(31)--0.21(21)×21
- -
-0.4
exp. 0
0
i
i
0.1
0.2
i
i
0.3 0.4 IlMy [ G e V -1]
-0.5
i
0.5
0.6
0
i
i
0.1
0.2
i
i
0.3 0.4 l / M » [GGV -1]
i
0.5
0.6
Figure 4. - h # 2 measured from the matrix elements is compaxed with the slope of mass difference MAh -- M~, --0.21(21) GeV 2.
Figure 3. I / M B dependence of MAh -- MB.
5. MAh -- MB
The heavy-light meson-baryon mass difference MAh - - MB is given as MAh -- M~
=
Ä(Ab) - Ä(B) 1
[_#2(Aò)+#~(B)].
(7)
The intercept at 1/M»=O yields Ä(Ab) --Ä(B) while the slope is described by - A # 2 = -(~~(Ab) - ~~(B)). In Figure 3 we plot MAh -- M~ as a function of 1~MB. For the intercept we obtain A(Ab) - A(B)=393(31) MeV. in agreement with a previous work by All Khan et al., Ä(Ab) Ä(B)=415(156) MeV. Our result is slightly larger than the experimental values for b and c hadrons. However, to draw a definite conclusion we have to consider several systematic errors, especially the finite volume effect, because our lattice may not be large enough for baryons. The slope obtained from the fit of the mass difference is consistent with zero: -0.21(21) GeV2. Our results of direct measurement of --A# 2 is plotted in Figure 4, which is consistent with the result from mass difference, but have much bettet accuracy. Out result is also compatible with the phenomenological estimate -0.01(3) GeV 2 [1] obtained from a combination (MAh -- MB) -- (MA~ - -
M»).
6. Conclusions We confirm that the lattice measurements of the matrix elements #2 and #2 are consistent with the HQET mass relations. The well-known problem of quenched lattice calculation that the hyperfine splitting is much smaller than the experiments is also reproduced. An important extension of our work is to measure the matrix elements of four-quark operators, which are relevant to the 1 / m ~ corrections to the lifetime ratlos [1]. This work is supported by the Supercomputer Project No.66 (FY2001) of High Energy Accelerator Research Organization (KEK), and also in part by the Grants-in-Aid of the Ministry of Education (Nos. 10640246, 11640294, 12014202, 12640253, 12640279, 12740133, 13640260 and 13740169). K-I.I and N.Y are supported by the JSPS Research Fellowship. R
E
F
E
R
E
N
C
E
S
1. M. Neubert and C. T. Sachrajda, Nucl. Phys. B 483 (1997) 339. 2. V. Gimenez, G. Martinelli and C. T. Sachrajda, Nucl. Phys. B 486 (1997) 227. 3. A. Ali Khan et al., Phys. Rev. D 62 (2000) 054505.
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear PhysicsB (Proc. Suppl.) 106 (2002) 376-378
www.elscvier.com/locaLe/n~
Heavy quark expansion parameters from lattice NRQCD
*
JLQCD Collaboration: N. Tsutsui a, S. Aoki b, R. Burkhalter c, M. Fukugita d, S. Hashimoto a, K-I. Ishikawa ¢, N. Ishizuka b'~, V. lwasaki b'c, K. Kanaya b,c, T. Kaneko a, Y. Kuramashi a, M. Okawa a, T. Onogi e, S, Tominaga c, A. Ukawa b'c, N. Yamada a, T. Yoshié b'¢ ~High Energy Accelerator Research Organization(KEK),
Tsukuba, Ibaraki 305-0801, Japan
blnstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan CCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan dInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan eyukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Using the lattice NRQCD action for heavy quark, we calculate the heavy quark expansion parameters/~r and B~ for heavy-light mesons and heavy-light-]ight baryons. The results are compared with the mass differences among heavy hadrons to test the validity of HQET relations on the lattice.
1. I n t r o d u c t i o n In the calculation of inclusive decay rates of the heavy hadron, the heavy quark expansion (HQE) technique is widely used. At the order 1/m S of HQE two nonperturbative parameters 1
#~(HQ) -- 2Mù----~(HQ Q(iD)2Q HQ), p~(HQ) =--
1 2MH----~( HO O,ä" BQ "Q},
(1) (2)
appear in the calculation. HQ represents a heavylight meson or heavy-light-light baryon (for b hadrons, Hb = B, B*, Ab, Eb, E~). For instance, the lifetime ratio of b hadrons is given as [1]
defined in the static limit: mQ --+c~. For heavylight meson, #27r,G has been calculated using the lattice version of the Heavy Quark Effective The-
ory [2]. In this work we calculate #2 and #2 on the lattice using the NRQCD action for heavy quark. Although the individual matrix element suffers frorn large perturbative uncertainty due to power divergence in the rnatching calculation, their dir. #.(H~ 2 (1) 2 (2) ferences hke )-,~(H~ ) are free ~ o m the uncertainty of the operator. We calculate both #2,(H~1))- #2(H~2)) and #2(H~1)) - #~(H~2)), and cornpare thern with the corresponding predictions for mass splittings. 2.
T(H~I)) = 1 q- #~(H~I)) -/z~(H~2)) T(H~2)) 2m~ F2 / T./'(1)~
+co
Gk**b
, 2 [ T..l'(2)'~ ] -- I'~G\**b ]
m~
+ 0(1/m3),
The pararneters #5 and #~ can be indirectly obtained from hadron masses, using (3)
with cv - 1.2. While #2 may be evaluated from experimental values of hyperfine splitting, the determination of ~2 requires some theoretical inputs. It should be noted that the parameters are *presented by N. Tsutsui
H Q E T mass f o r m u l a
MHo mQ ----Ä + -
-
--/Zrr --/iG + O
2mQ
1
,
(4)
where Ä is the residual energy difference between MHQ and mQ surviving in the infinite heavy quark limit. ~2 and Iz~ appear in the correction terms of O(1/mo). Therefore, by considering proper mass differences, certain combinations
0920-5632/02/$ - see frontmatter© 2002 ElsevierScienceB.V. All rightsreserved. PI1 S0920-5632(01)01719-4
N. Tsutsui et al./Nuclear Physics B (Proc. Suppl.) 106 (2002) 376-378
0.6
377
0.5
0.5
0.4
0.4 0.3
0.3
C.D
t
0.2
2 2 MBo~-M~ :
0.1
-a~5(B --B)
0
,
0
0.1
i
=
o x p .
0.2 0.3 0.4 t / M ~ [ G e V -1]
:
0.2
0.1
:
--A~5(~'--Z" )
:= : i
0.5
=
exp. 0.6
Figure 1. Hyperfine sp]itting of ground state mesons. Circles is obtained ~om the energy differences, while crosses are from the matrix elements.
0
i
i
0.1
0.2
i
a
0.3 0.4 IIMÆ [ G e V - ' ]
0.5
0.6
Figure 2. Hyperfine splitting of heavy-light-light baryons.
elements/z 2 and #2. of/z~ and ~~ can be extracted. For example, a difference of #~ can be obtained from the mass splitting in a spin multiplet, because Ä and/z2~ have the same value. Also, the spin averaged mass M B = (MB + 3 M ~ ) / 4 does not depend on #2, because/z 2 is proportional to the spin of the light degrees of the freedom and the sum of #2 in the spin multiplet vanishes. 3. L a t t i c e c a l c u l a t i o n s We carry out quenched QCD simulations at /3=6.0 on a 203 x 48 lattice. The NRQCD aetion including all O(1/mQ) terms and the non-perturbatively improved clover action (cs,0=1.769) is adapted for heavy quark and light quark, respectively. Five heavy quark masses amQ=l.3, 2.1, 3.0, 5.0, and 10.0 are used to study the 1/mQ dependence of hadron masses and matrix elements, while three hopping parameters K--0.13331, 0.13384, and 0.13432 are simulated to extrapolate to the chiral limit Kc=0.135284(8). The inverse lattice spacing a-1--1.85(5) GeV is determined with the p meson mass m»--770 MeV. We measure the three-point functions
(OHQ(tl)OtHQ(O))
4. Hyperfine spUttings From (4) the hyperfine splitting MB. given by -A#2/2mQ, or equivalently
-- MB is
M2. - M 2 = - A ~ 2 -- - ( / z 2 ( B *) - / z 2 ( B ) ) , (5) at the leading order. In Figure 1, we plot our results for - A # ~ together with the measurement of M2. - M 2. We observe that the relation (5) is satisfied very well, while both are significantly lower than the experimental values for B and D mesons. In deriving (5) we used a relation A#~ = #~(B*) - p~(B) = 0,
(6)
which holds in the static limit. However, for the NRQCD action including the spin-magnetic interaction term at O(1/mQ), the operator O~ mixes with Og at order as/mQ. This is the reason why our result for - A # 2 deviates from that of the mass difference in the lighter heavy quark mass region. In other words, the relation (6) may be considered as a renormalization condition for the operator O~.. Similar armlysis can be made for the hyperfine splitting of heavy-light-light baryon, i.e. ~,* ~. splitting. Figure 2 shows the mass difference and the matrix element --A#~. Both are in good agreement.
378
N. Tsutsui et aL /Nuclear Physics B (Proc. Suppl.) 106 (2002) 376-378
0 ° 5
0.1
,
0
0.4
-0.1 0.3
-0.21
-0.2 0.2 -0.3 0.1
0.393(31)--0.21(21)×21
- -
-0.4
exp. 0
0
i
i
0.1
0.2
i
i
0.3 0.4 IlMy [ G e V -1]
-0.5
i
0.5
0.6
0
i
i
0.1
0.2
i
i
0.3 0.4 l / M » [GGV -1]
i
0.5
0.6
Figure 4. - h # 2 measured from the matrix elements is compaxed with the slope of mass difference MAh -- M~, --0.21(21) GeV 2.
Figure 3. I / M B dependence of MAh -- MB.
5. MAh -- MB
The heavy-light meson-baryon mass difference MAh - - MB is given as MAh -- M~
=
Ä(Ab) - Ä(B) 1
[_#2(Aò)+#~(B)].
(7)
The intercept at 1/M»=O yields Ä(Ab) --Ä(B) while the slope is described by - A # 2 = -(~~(Ab) - ~~(B)). In Figure 3 we plot MAh -- M~ as a function of 1~MB. For the intercept we obtain A(Ab) - A(B)=393(31) MeV. in agreement with a previous work by All Khan et al., Ä(Ab) Ä(B)=415(156) MeV. Our result is slightly larger than the experimental values for b and c hadrons. However, to draw a definite conclusion we have to consider several systematic errors, especially the finite volume effect, because our lattice may not be large enough for baryons. The slope obtained from the fit of the mass difference is consistent with zero: -0.21(21) GeV2. Our results of direct measurement of --A# 2 is plotted in Figure 4, which is consistent with the result from mass difference, but have much bettet accuracy. Out result is also compatible with the phenomenological estimate -0.01(3) GeV 2 [1] obtained from a combination (MAh -- MB) -- (MA~ - -
M»).
6. Conclusions We confirm that the lattice measurements of the matrix elements #2 and #2 are consistent with the HQET mass relations. The well-known problem of quenched lattice calculation that the hyperfine splitting is much smaller than the experiments is also reproduced. An important extension of our work is to measure the matrix elements of four-quark operators, which are relevant to the 1 / m ~ corrections to the lifetime ratlos [1]. This work is supported by the Supercomputer Project No.66 (FY2001) of High Energy Accelerator Research Organization (KEK), and also in part by the Grants-in-Aid of the Ministry of Education (Nos. 10640246, 11640294, 12014202, 12640253, 12640279, 12740133, 13640260 and 13740169). K-I.I and N.Y are supported by the JSPS Research Fellowship. R
E
F
E
R
E
N
C
E
S
1. M. Neubert and C. T. Sachrajda, Nucl. Phys. B 483 (1997) 339. 2. V. Gimenez, G. Martinelli and C. T. Sachrajda, Nucl. Phys. B 486 (1997) 227. 3. A. Ali Khan et al., Phys. Rev. D 62 (2000) 054505.