- Email: [email protected]
Lattice HQET calculation of the Isgur-Wise function
PROCEEDINGS SUPPLEMENTS
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379
Lattice HQET Calculation of the Isgur-Wise Function Joseph Christensen*, Terrence Draper and Craig McNeile t ~' ~Department of Physics and Astronolny, University of Kentucky, Lexington, KY 40506 We calculate the Isgur-Wise function on the lattice, simulating the light quark with the Wilson action and the heavy quark with a direct lattice implementation of the heavy-quark effective theory. Improved smearing functions produced by a variational technique, MOST, are used to reduce the statistical errors and to minimize excited-state contamination of the ground-state signal. Calculating the required matching factors, we obtain (~(1) = -0.64(13) for the slope of the Isgur-Wise function in continuum-HQET in the MS scheme at a scale of 4.0 GeV.
1. T h e T a d p o l e - I m p r o v e d S i m u l a t i o n The Isgur-Wise function is the torm factor of a heavy-light meson in which the heavy quark is taken to be much heavier than the energy scale, >> AQCD. This calculation adds perturbative corrections to the simulation results of Draper & McNeile [1]. The Isgur-Wise function is calculated using the action first suggested by Mandula & Ogilvie [2]:
mQ
3 --ivj [ j=l
where "~j = ~vo and G ( ~ , t = 0) = E1l ( x ) . Better results can be obtained when a smeared source is used in place of a point source. The smearing fimction, f(~), was calculated from a static simulation (vj = 0) via the smearing technique MOST (Maximal Operator Smearing Technique [3]).
2. T h e I s g u r - W i s e F u n c t i o n The Isgur-Wise function was extracted from the lattice simulation as a ratio of three-point functions which was suggested by Mandula & lat Ogilvie [2]. I~,nren(V" v') t 2 is the large At limit
of 'it0
4v0v[i
(vo
This leads to the evolution equation:
u0
[
-- j = l ~
*Presented by J. Christensen at Lattice '97, Edinburgh, Scotland. tCurrently at Department of Physics, University of Utah, Salt Lake City, U T 84112. tThis work is supported in part by the U.S. Department of Energy under grant numbers DE-FG05-84ER40154 and DE-FC02-91ER75661, and by the University of Kentucky (;enter for Computational Sciences. The computations were carried out at NERSC. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V All rights reserved. Pll S0920-5632(97)00775-5
C 3vv' (At)C~v' v (At)
c v(zxt)c
' (zxt)
where At is the time separation between the current operator and each B-meson interpolating field. Draper & McNeile have presented [1] the nontadpole-improved unrenormalized slope of the lattice Isgur-Wise function to demonstrate the efficacy of the computational techniques.
3. Tadpole Improvement for H Q E T Tadpole improvement grew from the observation that lattice links, U, have mean field value, u0 ¢ 1. Therefore, it is better to use an action written as a function of (u/~o). In the Wilson action, each link has a coefficient t~; Uo can be paired with ~; to easily tadpole improve a posteriori any previous non-tadpole-improved calculation.
378
J Christensen et al./Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379
In the HQET, there is no common coefficient (analogous to a) for both Ut and U/. Correspondence between tadpole-improved and nontadpole-improved HQET actions with v 2 = 1 cannot be made via a simple rescaling of parameters, as is done for the Wilson action with a. Fortunately, the evolution equation can be written (as noticed by Mandula & Ogilvie [4]) such that the Uo is grouped with 9j. Thus, tadpole-improved Monte-Caxlo data can be constructed from the non-tadpole-improved data by replacing v nt -+ v tad and by including two overall multiplicative factors: vnt
~ t a d ( t ; ? ~ t a d , v ~ ad)
:
u -0t V ~o Gnt(t ~" .' ~nt, V~t)
(1)
In addition to the multiplicative factors Uo t and "'d).~d, the tadpole-improvement of the simulation requires adjusting the velocity according to ~tad = u0~nt, subject t o (vtad) 2 = 1 and (v't) 2 = 1. Thus, V~ad
~-
vj
=
V~}'t[I Jr" (l -- ?t2)(v~t)2] +
-`/2
(1 -
4. T a d p o l e - I m p r o v e d
Renormalization
UO 3
5vo
=
-i E
3
vj'2X4 - UoVo E
/=1
vjXj
j=l
Uo is the perturbative expansion and 4 v~,tad
~,j =
b,tad rztad ---- Vj Z~vj
: (1 +
UO
'
V~, tad -- v b , t a d ~ t a d 0 ~V0
'
7-vo t a d -- 1-t- 5Vo V0
If one fits to e x p { - t } rather than e x p { - ( t + 1)}, the tadpole-improved wave-function renormalization is reduced to ~Z~ = 5ZQ + (~Mtad -IVo ln(uo))/Vo. Thus the ln(uo) term cancels explicitly and, as in the static case [6], tadpoleimprovement has no effect on 5Z~, to order a. 5. P e r t u r b a t i v e
Renormalizations
We will present our computations of the renorrealization factors elsewhere, but include this comment: Although the tadpole-improved functions include factors of UOlpt [7], these effects axe higher order in a and axe dropped. Only the velocity renormalization is explicitly affected by the perturbative expansion of uo:
By comparing the unrenormalized propagator ztad
- 0 +-
i
+u
0
=
ZQ [v~(ik4) + v:(k~) + Mr] - :
at O(k 2) and using (vr) ~ = (vb) 2 = 1, the perturbative renormalizations can be obtained. Aglietti [5] has done this for a different non-tadpoleimproved action, for the special case ff = Vz2. With momentum shift, p--+p'=(pa+iln(uo),p-) [6], with ~/uoexp(ik4) = exp(i(k4 + iln(uo))) and o E(p)lp=O, the tadpole-improved with X , = Op, perturbative renormalizations are found to be
5M 5ZQ
=
=
- E ( O ) - voln(uo)
ZQ - 1
=
(l+5~g'2CF(ylr2)) v1 167r2
The perturbative renormalizations favor a scale of q*a = 1.9(1) for a, which yields a ~ 0.19(1).
to the renormalized propagator
iH(k,v)
vj
3 = - i v o X 4 - uo E v j X j j=l
6. V e l o c i t y
Renormalization
Mandula & Ogilvie [4] consider the perturbarive velocity renormalization expanded in orders of 9. Our numbers for the velocity renormalization agree with theirs. Another option is to consider, as did both Mandula & Ogilvie and Hashimoto & Matsufuru [8], the non-perturbative renormalization of the velocity. Prom Hashimoto's & Matsufuru's graph, we estimate their --v,npZtad~ 1.05(5). Prom Mannt dula & Ogilvie's result, we notice t h a t Zg,np = / /,,o'vo : \
vj vj \ o' oJ /
J Christensen et al./Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379 U0 X 1.01(1). This is very close to the effect of tadpole-improving, and implies Z t'~d = 1.01(1). We therefore use Zt,.~d _,, ~ 1 as the non-perturbatire velocity renormalization in our calculation to renormalize the slope of the Isgur-Wise function.
7. Renormalization of ('(v. v') We claim that we can convert our Monte-Carlo data into tadpole-improved results and can calculate a renormalized tadpole-improved slope for the Isgur-Wise function. We use the notation Z' = l+SZ~ for the renornlalization of the Isgur-Wise function, with 5Z~: f [2 (1 - v . v % ( v . v ' ) ) l n ( # a ) 2 - . f ' ( v , ¢)] 127r'~ with r(v.v') defined in [9] and primes on Z mid f to indicate the "reduced value." For simplicity, we use the local current, which is not conserved on the lattice; Z~(1) ~ 1. However, the construction in §2 guarantees that the extracted renormalized Isgur-Wise function is properly normalized, ~ren(1) = 1. 8. C o n c l u s i o n s After renormalization of our tadpole-improved results, we obtain ~ n ( 1 ) = -0.64(13) for the slope of the Isgur-Wise function in continuum H Q E T in the MS scheme at a scale of 4.0 GeV. Without renormalization, the slope is • ~unr~n(1) , tad = -0.56(13). Without tadpole-improvement, the slope is ~unren(1) , nt = --0.43(10). We found that the tadpole-improved action (and therefore the tadpole-improved data) cannot be obtained from the non-tadpole-improved act,ion (or data) by a simple rescaling of any parameter. However, the form of the evolution equation allows the construction of the tadpole-improved Monte-Carlo data from the non-tadpole-improved data as described in §3. After tadpole improvement, non-perturbative corrections to the velocity are negligible. Furthermore, tadpole improvement greatly reduces the perturbative corrections to the slope of the Isgur-Wise function.
379
Unrenormalized Isgur-Wise Slope Not Tadpole Improved At 2 3 1
0.154 0.38+I 0.42_+2 0.50+~ _
0.155 0.38+I 0.41_+~ 0.48+~ -,
0.156 0.38+]L 0.41.+~ 0.45+~o _
nc 0.39.+~t 0.41_+~ 0.43_1 +1oo
At 2
0.154 0.49+I
3
0.55+
4
0.65_12I-1° 0.63+11_12 0.59-13+12 0.57_13+13
Tadpole Improved 0.155 0.50_+I
0.156 0.50+I
nc 0.50_+I
0.54+_
o.53_+3
Renormalized Isgur-Wise Slope Tadpole Improved At 0.154 0.155 0.156 n~ 2 0.57+I 0.57.+ I 0.58+~ 0.58_+I 3 0.62+ 3 0.62+3 0.61_+3 0.61+_!J +12 ff-ll (I ¢17 + 1 3 {1 f l A + 1 3 4 0.72-11 0-71-n . . . . . 11 . . . . . la Table 1 The negative of the slope at the normalization point, ~'(1), from both the unrenormalized and the renormalized ratio of three-point functions. This ratio gives the (un)-renormalized Isgur-Wise function ~(v.v') at asymptotically-large times At.
REFERENCES 1. T. Draper and C. McNeile, Nucl. Phys. B (Proc. Suppl. ) 47, 429 (1996). 2. .I.E. Mandula and M.C. Ogilvie, Phys. Rev. D 45, R2183 (1992). 3. T. Draper and C. McNeile, Nucl. Phys. B (Proc. Suppl. ) 34, 453 (1994). 4. J.E. Mandula and M.C. Ogilvie, hep-lat/9602004, hep-lat/9703020. 5. U. Aglietti, Nucl. Phys. B421, 191 (1994). 6. C. Bernard, Nucl. Phys. B (Proc. Suppl. ) 34, 47 (1994). 7. G.P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993). 8. S. Hashimoto and H. Matsufuru, Phys. Rev. D 54, 4578 (1996). 9. A.F. Falk, H. Georgi, B. Grinstein, and M. B. Wise, Nucl. Phys. B343, 1 (1990).
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379
Lattice HQET Calculation of the Isgur-Wise Function Joseph Christensen*, Terrence Draper and Craig McNeile t ~' ~Department of Physics and Astronolny, University of Kentucky, Lexington, KY 40506 We calculate the Isgur-Wise function on the lattice, simulating the light quark with the Wilson action and the heavy quark with a direct lattice implementation of the heavy-quark effective theory. Improved smearing functions produced by a variational technique, MOST, are used to reduce the statistical errors and to minimize excited-state contamination of the ground-state signal. Calculating the required matching factors, we obtain (~(1) = -0.64(13) for the slope of the Isgur-Wise function in continuum-HQET in the MS scheme at a scale of 4.0 GeV.
1. T h e T a d p o l e - I m p r o v e d S i m u l a t i o n The Isgur-Wise function is the torm factor of a heavy-light meson in which the heavy quark is taken to be much heavier than the energy scale, >> AQCD. This calculation adds perturbative corrections to the simulation results of Draper & McNeile [1]. The Isgur-Wise function is calculated using the action first suggested by Mandula & Ogilvie [2]:
mQ
3 --ivj [ j=l
where "~j = ~vo and G ( ~ , t = 0) = E1l ( x ) . Better results can be obtained when a smeared source is used in place of a point source. The smearing fimction, f(~), was calculated from a static simulation (vj = 0) via the smearing technique MOST (Maximal Operator Smearing Technique [3]).
2. T h e I s g u r - W i s e F u n c t i o n The Isgur-Wise function was extracted from the lattice simulation as a ratio of three-point functions which was suggested by Mandula & lat Ogilvie [2]. I~,nren(V" v') t 2 is the large At limit
of 'it0
4v0v[i
(vo
This leads to the evolution equation:
u0
[
-- j = l ~
*Presented by J. Christensen at Lattice '97, Edinburgh, Scotland. tCurrently at Department of Physics, University of Utah, Salt Lake City, U T 84112. tThis work is supported in part by the U.S. Department of Energy under grant numbers DE-FG05-84ER40154 and DE-FC02-91ER75661, and by the University of Kentucky (;enter for Computational Sciences. The computations were carried out at NERSC. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V All rights reserved. Pll S0920-5632(97)00775-5
C 3vv' (At)C~v' v (At)
c v(zxt)c
' (zxt)
where At is the time separation between the current operator and each B-meson interpolating field. Draper & McNeile have presented [1] the nontadpole-improved unrenormalized slope of the lattice Isgur-Wise function to demonstrate the efficacy of the computational techniques.
3. Tadpole Improvement for H Q E T Tadpole improvement grew from the observation that lattice links, U, have mean field value, u0 ¢ 1. Therefore, it is better to use an action written as a function of (u/~o). In the Wilson action, each link has a coefficient t~; Uo can be paired with ~; to easily tadpole improve a posteriori any previous non-tadpole-improved calculation.
378
J Christensen et al./Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379
In the HQET, there is no common coefficient (analogous to a) for both Ut and U/. Correspondence between tadpole-improved and nontadpole-improved HQET actions with v 2 = 1 cannot be made via a simple rescaling of parameters, as is done for the Wilson action with a. Fortunately, the evolution equation can be written (as noticed by Mandula & Ogilvie [4]) such that the Uo is grouped with 9j. Thus, tadpole-improved Monte-Caxlo data can be constructed from the non-tadpole-improved data by replacing v nt -+ v tad and by including two overall multiplicative factors: vnt
~ t a d ( t ; ? ~ t a d , v ~ ad)
:
u -0t V ~o Gnt(t ~" .' ~nt, V~t)
(1)
In addition to the multiplicative factors Uo t and "'d).~d, the tadpole-improvement of the simulation requires adjusting the velocity according to ~tad = u0~nt, subject t o (vtad) 2 = 1 and (v't) 2 = 1. Thus, V~ad
~-
vj
=
V~}'t[I Jr" (l -- ?t2)(v~t)2] +
-`/2
(1 -
4. T a d p o l e - I m p r o v e d
Renormalization
UO 3
5vo
=
-i E
3
vj'2X4 - UoVo E
/=1
vjXj
j=l
Uo is the perturbative expansion and 4 v~,tad
~,j =
b,tad rztad ---- Vj Z~vj
: (1 +
UO
'
V~, tad -- v b , t a d ~ t a d 0 ~V0
'
7-vo t a d -- 1-t- 5Vo V0
If one fits to e x p { - t } rather than e x p { - ( t + 1)}, the tadpole-improved wave-function renormalization is reduced to ~Z~ = 5ZQ + (~Mtad -IVo ln(uo))/Vo. Thus the ln(uo) term cancels explicitly and, as in the static case [6], tadpoleimprovement has no effect on 5Z~, to order a. 5. P e r t u r b a t i v e
Renormalizations
We will present our computations of the renorrealization factors elsewhere, but include this comment: Although the tadpole-improved functions include factors of UOlpt [7], these effects axe higher order in a and axe dropped. Only the velocity renormalization is explicitly affected by the perturbative expansion of uo:
By comparing the unrenormalized propagator ztad
- 0 +-
i
+u
0
=
ZQ [v~(ik4) + v:(k~) + Mr] - :
at O(k 2) and using (vr) ~ = (vb) 2 = 1, the perturbative renormalizations can be obtained. Aglietti [5] has done this for a different non-tadpoleimproved action, for the special case ff = Vz2. With momentum shift, p--+p'=(pa+iln(uo),p-) [6], with ~/uoexp(ik4) = exp(i(k4 + iln(uo))) and o E(p)lp=O, the tadpole-improved with X , = Op, perturbative renormalizations are found to be
5M 5ZQ
=
=
- E ( O ) - voln(uo)
ZQ - 1
=
(l+5~g'2CF(ylr2)) v1 167r2
The perturbative renormalizations favor a scale of q*a = 1.9(1) for a, which yields a ~ 0.19(1).
to the renormalized propagator
iH(k,v)
vj
3 = - i v o X 4 - uo E v j X j j=l
6. V e l o c i t y
Renormalization
Mandula & Ogilvie [4] consider the perturbarive velocity renormalization expanded in orders of 9. Our numbers for the velocity renormalization agree with theirs. Another option is to consider, as did both Mandula & Ogilvie and Hashimoto & Matsufuru [8], the non-perturbative renormalization of the velocity. Prom Hashimoto's & Matsufuru's graph, we estimate their --v,npZtad~ 1.05(5). Prom Mannt dula & Ogilvie's result, we notice t h a t Zg,np = / /,,o'vo : \
vj vj \ o' oJ /
J Christensen et al./Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 377-379 U0 X 1.01(1). This is very close to the effect of tadpole-improving, and implies Z t'~d = 1.01(1). We therefore use Zt,.~d _,, ~ 1 as the non-perturbatire velocity renormalization in our calculation to renormalize the slope of the Isgur-Wise function.
7. Renormalization of ('(v. v') We claim that we can convert our Monte-Carlo data into tadpole-improved results and can calculate a renormalized tadpole-improved slope for the Isgur-Wise function. We use the notation Z' = l+SZ~ for the renornlalization of the Isgur-Wise function, with 5Z~: f [2 (1 - v . v % ( v . v ' ) ) l n ( # a ) 2 - . f ' ( v , ¢)] 127r'~ with r(v.v') defined in [9] and primes on Z mid f to indicate the "reduced value." For simplicity, we use the local current, which is not conserved on the lattice; Z~(1) ~ 1. However, the construction in §2 guarantees that the extracted renormalized Isgur-Wise function is properly normalized, ~ren(1) = 1. 8. C o n c l u s i o n s After renormalization of our tadpole-improved results, we obtain ~ n ( 1 ) = -0.64(13) for the slope of the Isgur-Wise function in continuum H Q E T in the MS scheme at a scale of 4.0 GeV. Without renormalization, the slope is • ~unr~n(1) , tad = -0.56(13). Without tadpole-improvement, the slope is ~unren(1) , nt = --0.43(10). We found that the tadpole-improved action (and therefore the tadpole-improved data) cannot be obtained from the non-tadpole-improved act,ion (or data) by a simple rescaling of any parameter. However, the form of the evolution equation allows the construction of the tadpole-improved Monte-Carlo data from the non-tadpole-improved data as described in §3. After tadpole improvement, non-perturbative corrections to the velocity are negligible. Furthermore, tadpole improvement greatly reduces the perturbative corrections to the slope of the Isgur-Wise function.
379
Unrenormalized Isgur-Wise Slope Not Tadpole Improved At 2 3 1
0.154 0.38+I 0.42_+2 0.50+~ _
0.155 0.38+I 0.41_+~ 0.48+~ -,
0.156 0.38+]L 0.41.+~ 0.45+~o _
nc 0.39.+~t 0.41_+~ 0.43_1 +1oo
At 2
0.154 0.49+I
3
0.55+
4
0.65_12I-1° 0.63+11_12 0.59-13+12 0.57_13+13
Tadpole Improved 0.155 0.50_+I
0.156 0.50+I
nc 0.50_+I
0.54+_
o.53_+3
Renormalized Isgur-Wise Slope Tadpole Improved At 0.154 0.155 0.156 n~ 2 0.57+I 0.57.+ I 0.58+~ 0.58_+I 3 0.62+ 3 0.62+3 0.61_+3 0.61+_!J +12 ff-ll (I ¢17 + 1 3 {1 f l A + 1 3 4 0.72-11 0-71-n . . . . . 11 . . . . . la Table 1 The negative of the slope at the normalization point, ~'(1), from both the unrenormalized and the renormalized ratio of three-point functions. This ratio gives the (un)-renormalized Isgur-Wise function ~(v.v') at asymptotically-large times At.
REFERENCES 1. T. Draper and C. McNeile, Nucl. Phys. B (Proc. Suppl. ) 47, 429 (1996). 2. .I.E. Mandula and M.C. Ogilvie, Phys. Rev. D 45, R2183 (1992). 3. T. Draper and C. McNeile, Nucl. Phys. B (Proc. Suppl. ) 34, 453 (1994). 4. J.E. Mandula and M.C. Ogilvie, hep-lat/9602004, hep-lat/9703020. 5. U. Aglietti, Nucl. Phys. B421, 191 (1994). 6. C. Bernard, Nucl. Phys. B (Proc. Suppl. ) 34, 47 (1994). 7. G.P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993). 8. S. Hashimoto and H. Matsufuru, Phys. Rev. D 54, 4578 (1996). 9. A.F. Falk, H. Georgi, B. Grinstein, and M. B. Wise, Nucl. Phys. B343, 1 (1990).