Decay Constants of B and D Mesons from Non-perturbatively Improved Lattice QCD

a r X i v :h e p -l a t /0007020v 3 10 O c t 2001

Edinburgh 2000/14IFUP-TH/2000-17JLAB-THY-00-25

SHEP 0008

Decay Constants of B and D Mesons from Non-perturbatively Improved Lattice QCD

UKQCD Collaboration

K.C.Bowler a ,L.Del Debbio b ,1,J.M.Flynn b ,https://www.360docs.net/doc/822679864.html,cagnina a ,V.I.Lesk b ,2,C.M.Maynard a ,

D.G.Richards a ,c ,d

a Department of Physics &Astronomy,University of Edinburgh,Edinburgh EH9

3JZ,Scotland,UK b Department of Physics &Astronomy,University of Southampton,Southampton,

SO171BJ,UK

c Je?erson

Laboratory,MS 12H2,12000Je?erson Avenue,Newport News,VA

23606,USA.d Department

of Physics,Old Dominion University,Norfolk,VA 23529,USA.

PACS numbers:12.38Gc,13.20Fc,13.20Hc,14.40Lb,14.40Nd

1Introduction

The accurate determination of the B and D meson decay constants is of pro-

√

found importance in phenomenology.The combination f B

f B=195(6)+24

?23

MeV

f D=206(4)+17

?10

MeV

f B

S =220(6)+23

?28

MeV

f D

S =229(3)+23

?12

MeV

f B

s

/f B=1.13(1)+1?5

f D

s

/f D=1.11(1)+1?3

f K=150(3)+12

?8MeV

f B?=28(1)+3?4

f D?=8.6(3)+5?9

f B?

s

=25(1)+2?3

f D?

s

=8.3(2)+5?6

f B?/f B?

s

=1.10(2)+2?6

f D?/f D?

s

=1.04(1)+2?4

The?rst error quoted is statistical,the second systematic.Systematic errors are discussed in Section4,with the main contributions itemized in Table11. 2Details of the calculation

2.1Improved action and operators

In the Wilson formulation of lattice QCD,the fermionic part of the action has lattice artifacts of O(a)(where a is the lattice spacing),while the gauge action di?ers from the continuum Yang-Mills action by terms of O(a2).To leading order in a the Symanzik improvement program for on-shell quantities involves adding the SW term to the fermionic Wilson action,

S SW=S W?c SW iκ

and??μis the symmetric lattice derivative.The generic current renormalisation is as follows(J=A,V):

J R=Z J(1+b J am q)J I(3) where Z J is calculated in a mass-independent renormalisation scheme.

The bare quark mass,am q,is

am q=1

κ?

1

κe?=

1

κ1

+

1

2

+θ(?t)e m Q t

1?γ4

2 ←D k+←D?k

ψQ(9)

so that

(A stat

R )0=Z stat

A

(1+b stat

A

am q)(A stat

I

)0(10)

The c stat

A term is not implemented in this calculation,but its value is negative

which will be signi?cant when the static point is compared to the in?nite mass extrapolation.

2.3De?nitions of mesonic decay constants

The pseudoscalar and vector meson decay constants,f P and f V,are de?ned by

0|A Rμ(0)|P =ipμf P,(11) 0|V Rμ(0)|V =i?μM2V

Input and derived parameters.The lattice spacing is set by fπ.

β=6.2β=6.0

Improvement coe?cients from various https://www.360docs.net/doc/822679864.html,NL refers to Bhattacharya et al.and BPT is boosted perturbation theory.

β=6.2

β=6.0

LANL

ALPHA BPT LANL ALPHA BPT Z A 0.818(2)(5)0.807(8)(2)

0.81630.807(2)(8)0.7906(94)

0.8038b A 1.32(3)(4)?1.241.28(3)(4)?1.26c A ?0.032(3)(6)?0.038(4)?0.012?0.037(4)(8)?0.083(5)?0.013Z V 0.7874(4)0.7922(4)(9)0.79590.770(1)0.7809(6)0.7820b V 1.42(1)(1)1.41(2)1.241.52(1)1.54(2)1.26c V

?0.09(2)(1)

?0.21(7)

?0.026

?0.107(17)(4)

?0.32(7)

?0.028

β?0.10

?0.08

?0.06

?0.04

?0.02

0.00

c A

ALPHA

Bhattacharya

et al.

β

?0.4

?0.3

?0.2

?0.1

c V

PT Fig.1.The improvement coe?cients c A (left)and c V (right)as functions of the coupling β.Note that the vertical scales in the two plots are di?erent.The values of the coe?cients are shown for the two values of the coupling used in this work.The dashed line through the ALPHA c A points is their interpolating function.

Changes in c A and c V can have a particularly large e?ect on the extracted values of the decay constants when the pseudoscalar and vector meson masses are not small (in lattice units),since the improved current matrix elements are given by,

0|A I 4|P = 0|A 4|P +c A sinh (aM P ) 0|P |P 0|V I i |V,? = 0|V i |V,? +c V sinh (aM V ) 0|T i 4|V,?

(13)

5

7

9

11

13

15

17

19

21

23

timeslice

0.011

0.0120.0130.0140.0150.016

0.0170.0180.019R

4

Fig.2.The mixing of the axial current with the pseudoscalar density.β=6.0,κH =0.1123,κL =0.13344.The ratio plotted is de?ned in equation (19).

when the ground state is isolated.In particular at β=6.0the heavy-light meson mass at the heaviest kappa is somewhat bigger than one such that sinh aM ～1.3.This is illustrated for the pseudoscalar decay constant in Fig-ure 2.The ?gure shows the ratio R 4(between PA and PP correlators,de?ned in equation (19)and proportional to f P ),for several di?erent values of c A at β=6.0.It can be seen that using the NP value of c A from the ALPHA col-laboration decreases R 4by ～20%relative to the case c A =0.The plot also shows the ratio determined using the NP value of c A from Bhattacharya et al.At β=6.0the disagreement between ALPHA and Bhattacharya et al.in the mixing coe?cients is striking.Bhattacharya et al.try to estimate some of the cut-o?e?ects in their determination of c A by looking at the di?erence between two-and three-point derivatives.They also note that since a 2only halves between β=6.0and β=6.2,where disagreement is substantially decreased,even higher order cut-o?e?ects are playing a dominant r?o le.Collins and Davies [27]have examined the e?ect of using higher-order derivatives in determining c A on the data set used in this work and point out that the di?erence in the value of c A from di?erent orders of derivative disappears once the chiral limit is taken.There is clearly a large O (a )ambiguity in the value of c A .

Bhattacharya et al.have computed all the coe?cients needed for this calcu-lation in a consistent fashion and their determinations are used here for that reason.The values of the mixing coe?cients c A and c V are an order of mag-nitude larger in the ALPHA determinations than in BPT at β=6.0.This has a large impact on the HQS relation between,and scaling of,the decay

constants.When the values of the mixing coe?cients from Bhattacharya et al.are used the HQS relation is satis?ed and there is good scaling.

The ALPHA collaboration compute b V in the SF scheme at very small quark masses and in Bhattacharya et al.the heaviest quark mass is am q～0.13. One may question whether these coe?cients can be used around the charm mass:atβ=6.0,am c～0.75and atβ=6.2,am c～0.5.However,the e?ective normalisation of the vector[28,29]and axial[30]currents has been measured at these quark masses and found to be in good agreement with the determinations of Z V and b V3.

A value for b m is required to compute the rescaled quark mass used in the chiral extrapolations.Whilst there is a non-perturbative determination[31], it is only at one value of the coupling,and so this work follows[15]and uses the BPT value with which the non-perturbative value is consistent.

Expressions for the static improvement and renormalisation coe?cients c stat

A ,

b stat A and Z stat

A

,have been derived in perturbation theory[11],

Z stat

A

=1.0+ ln(aμ)

2?0.056(7)g20+O(g40)

c stat

A

=?

1

MS scheme.The

coupling g is then evaluated in the

MS coupling.

Table3

Pseudoscalar and vector masses in lattice units atβ=6.2andβ=6.0.Fit ranges are12?22atβ=6.2and10?22(P),10?21(V)atβ=6.0.

β=6.2

κHκL aM P aM V

0.13460.739+1?10.775+2?2 0.12330.13510.721+2?10.759+2?2

0.13530.714+2?10.752+3?2

0.13460.505+1?10.563+2?2 0.12990.13510.484+1?10.546+2?2

0.13530.476+1?10.540+2?2

β=6.0

κHκL aM P aM V

0.133441.006+2?11.056+2?2 0.11730.134170.981+2?21.034+3?2

0.134550.969+2?21.026+4?4

0.133440.675+1?10.759+2?2 0.12730.134170.646+2?10.736+3?2

0.134550.631+2?10.727+4?3

2.6Extraction of masses and decay constants of heavy-light mesons

The pseudoscalar meson masses are extracted from the asymptotic behaviour of the two-point correlation functions,

C SS P P(t, p)= x e?i p· x ?S P(t, x)?S?P(0, 0) (16)

t→∞

→

Z S P( p) 2

The decay constants are extracted from the large time behaviour of di?er-ent two-point correlation functions at zero momentum.The PA correlation function is used for the pseudoscalar decay constant:

C SL P A(t, 0)= x A I4(t, x)?S?P(0, 0) (17)

t→∞

→Z L A Z S P

cosh(aM V(t?T/2))e?aM V T/2

2aM V

where V I j is a spatial component of the improved local vector current operator. Again,S denotes a smeared or spatially extended interpolating?eld operator and L a local operator.The factor Z L V is the overlap of the local vector operator with the vector state,Z L V= r?r k 0|V I k|V( 0, ?) ,from which the vector decay constant can be extracted via equation(12).

Matrix elements proportional to the decay constants can be extracted from ratios of correlation functions.The ratio

tanh(aM P(T/2?t))(19) R4≡C SL P A(t)Z S

P

is used for the pseudoscalar case and is shown in Figure2for various values of c A.The vector decay constant is determined using

C SL V V(t)

(20)

Z S V

Results for the decay constants are shown in Table4.

An alternative method is to?t several correlators simultaneously allowing an estimation of the contamination of the ground state signal by excited states.

Table4

Pseudoscalar and vector decay constants in lattice units.Atβ=6.2the?t ranges are14?21(P)and15?23(V).Atβ=6.0the?t ranges are14?21(P)and 16?23(V).

β=6.2

κHκL af P f V

0.13460.0873+7?87.98+8

?8

0.12330.13510.0829+7?88.17+10

?9

0.13530.0814+8?98.23+11

?10

0.13460.0804+6?75.76+5

?5

0.12990.13510.0765+6?85.79+6

?5

0.13530.0751+6?95.79+7

?6

β=6.0

κHκL af P f V

0.133440.1217+10

?10

7.80+9

?8

0.11730.134170.1162+10

?11

7.89+11

?10

0.134550.1142+12

?14

7.84+16

?11

0.133440.1102+6

?9

5.62+6

?5

0.12730.134170.1053+7

?9

5.59+7

?6

0.134550.1032+8

?11

5.51+9

?7

An eight-parameter?t is made to C SS P P,C SL P A and C SL P P,allowing for ground and ?rst excited state contributions.Results for the pseudoscalar decay constant are consistent with the single ratio?t.The ratio method is used for central values in the following but the di?erence from the multi-exponential?ts is quoted in Table14below as one measure of systematic error.

2.6.1The static-light axial current

The static-light axial current can be extracted from axial-axial correlation functions.The local-local and local-smeared correlation functions were used in this work,with large-time behaviour given by

C LL AA(t, 0)= x A L4(t, x)A L?4(0, 0) t→∞→(Z L A)2e?a?Et(21) C LS AA(t, 0)= x A S4(t, x)A L?4(0, 0) t→∞→Z S A Z L A e?a?Et

2

4

6

8

10

12

14

t

0.20.40.60.811.21.4a ?E

e f f

Fig.3.The e?ective mass plot for the LL and LS static-light axial correlation function.κL =0.13344.The lines on the plot show the value of the ?t,and the ?t ranges.

where ?E is the unphysical di?erence between the mass of the meson and the mass of the bare heavy quark.The static-light amplitude,Z L ,is

Z L =af stat

P

2

(22)

The smearing function is again the one described in [35].

The static-light correlation functions were generated at only one value of the coupling,β=6.0,and without the covariant derivative operators necessary to improve the current.The assumption that the matrix element of the improve-ment term is of the same order of magnitude as the primary term leads to O (10%)error in the static point associated with the absence of improvement.The signal of the static-axial current is very quickly overwhelmed by statistical noise,making the ?t di?cult.Simultaneously ?tting to both the local-local and local-smeared correlator gives a better estimate of the local current.This is shown in Figure 3.The renormalised values of Z L (without the current improvement term)are shown in Table 5.

2.7Extraction of masses and decay constants of light-light mesons

The masses and decay constants of the light-light pseudoscalar mesons are

extracted from a simultaneous ?t to three correlation functions:C SS P P ,C SL

P P

and C SL

P A .The smearing function used for the light-lights is the ‘fuzzing’of

Values of a?E and Z L from simultaneous?ts to the LL and LS correlation functions. The?t ranges are8-12for the LL and4-12for the LS.

κL Z L a?E

0.134600.134600.0640+7?7 0.135100.134600.0610+6?7 0.135300.134600.0599+6?7 0.135100.135100.0587+5?6 0.135300.135100.0575+6?6 0.135300.135300.0559+6?7

β=6.0

κ1κ2af P

The inverse lattice spacing,a?1(GeV)

β=6.2β=6.0

= fπ

aM P

00.01

0.020.03

am q2

0.05

0.0550.06

0.065

a f P

Fig.4.Dependence of the light pseudoscalar decay constant f P on quark masses.The lines give f P as a function of one of the quark masses,a ?m q 2,for ?xed values of the other quark mass,a ?m q 1.Filled symbols are the lattice data points.The lowest line shows the extrapolation of a ?m q 1to the normal quark mass,with the open triangle giving f K from the position of the kaon on this line.

by the charged pion according to

a 2(m 2π)expt =2Ba m

n (26)

and that for the strange quark mass by

a 2(m 2K )I =B (a m

n +a m s )(27)

where

(m 2K )I =

1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

am q

0.8

0.82

0.84

a M P

00.0050.010.0150.020.0250.030.035

0.0750.080.0850.09a f

P

Fig.5.The chiral extrapolation of the pseudoscalar decay constant (top)and pseu-doscalar mass (bottom)against rescaled light quark mass.β=6.2and κH =0.1200.The vertical lines show the strange and normal quark masses.

3.2Chiral extrapolations of heavy-light masses and decay constants A linear dependence of heavy-light masses and decay constants on the light quark mass is assumed:

aK i =αi +βi a m

q (30)

where K i is f P ,f V /a ,m P or m V .Some example extrapolations are shown

in Figure 5.The results for the extrapolated quantities are shown in Ta-bles 8and 9.

3.3Heavy quark extrapolations

Heavy Quark Symmetry (HQS)implies asymptotic scaling laws [39]for the decay constants in the in?nite heavy quark mass limit.Away from this limit,heavy quark e?ective theory ideas motivate the following ans¨a tze for the de-pendence on the heavy meson masses:

ΦP (M P )≡Θ(M B ,M P )f P

√M P

+

ηP f V √

M V

+ηV

Table8

Masses and decay constants at physical light quark masses.β=6.2.

κHκL aM P af P aM V f V

κn0.696(2)0.077(1)0.736(3)8.4(1)

0.1233

κs0.726(1)0.0857(8)0.764(2)8.05(9)

κn0.455(2)0.071(1)0.523(3)5.8(1)

0.1299

κs0.491(1)0.0789(7)0.551(1)5.77(6)

Table9

Masses and decay constants at physical light quark masses.β=6.0.

κHκL aM P af P aM V f V

κn0.945(3)0.109(2)1.005(5)7.9(2)

0.1173

κs0.985(2)0.119(1)1.039(3)7.8(1)

κn0.603(2)0.099(1)0.704(5)5.5(1)

0.1273

κs0.651(2)0.1080(9)0.740(3)5.60(7)

whereΘdenotes logarithmic corrections given at leading order by[40],

Θ(M B,M)= α(M)

MS

=295MeV[41].

HQS also relates the pseudoscalar and vector decay constants as follows[40];

U(

3αs(

4π

+O(1/

where

M)is de?ned to eliminate the radiative corrections in U(

M)≡U(

3αs(

4π (35)

Calculated values of U(

M)=ω0+ω1

M

+

ω2

M2

(36)

HQS implies thatω0=1.However,ω0can also be left as a free parameter to test the applicability of HQS.Likewise,HQS can be applied to setγP=γV in?ts using equations(31)and(32).However,higher order QCD corrections would modify this in a similar way to U(

γV

= 1+8M)

00.010.02

0.030.04

(a/r 0)

2

0.36

0.38

0.4

0.42

0.44

r 0f K

Fig.6.The dependence of f K on the lattice spacing.

a continuum extrapolation of r 0f K against a 2/r 2

0.To examine the scaling be-haviour of the decay constant in this work,a comparison with the ALPHA results is shown in Figure 6.The line shows the linear extrapolation to the continuum limit (CL)that was performed by ALPHA,excluding the point at the coarsest lattice spacing,β=6.0.There are two main di?erences in the calculations.First,ALPHA use degenerate light quark masses,whereas this work speci?cally takes into account the non-degeneracy of the quarks.How-ever,ALPHA have checked on the same dataset analysised in this work that using degenerate light quarks has negligible e?ect.A more obvious di?erence is in the values of the improvement coe?cients,c A and b A .As noted earlier,improvement coe?cients may di?er by O (a )terms,depending on the improve-ment condition used.A particular choice of conditions used to determine the coe?cients may result,for a particular quantity,in smaller discretisation ef-fects at ?nite lattice spacing.This seems to be the case for the decay constants when the improvement coe?cients of Bhattacharya et al.are used rather than those determined by ALPHA,especially at β=6.0.However,as the con-tinuum limit is approached the values of f K from the two studies converge.

The major source of uncertainty in the heavy-light decay constants comes from the determination of the lattice spacing.Additional systematic errors arise from discretisation e?ects and from the heavy quark extrapolations.Since these latter errors are related to the size of the heavy quark masses,the heavy masses used in this work are listed in Table 10.All these errors will now be discussed.

The ambiguity in the decay constants from the lattice spacing is quite large:there is an overall 16%variation in the value of f B when the scale is set by m ρ,