B and B_s decay constants from QCD Duality at three loops

0.17 MB·English

by  J. Bordes

October 2004
B B
and decay constants from QCD Duality at
s
three loops1
5
0
0
2
J. Bordesa, J. Pen˜arrochaa and K. Schilcherb

n
a
J aDepartamento de F´ısica Teo´rica-IFIC, Universitat de Valencia
4 E-46100 Burjassot-Valencia,Spain
bInstitut fu¨r Physik, Johannes-Gutenberg-Universit¨at

2 D-55099 Mainz, Germany
v
8
2
3
0
1
4 Abstract
0
/
h Using special linear combinations of finite energy sum rules which
p
minimize the contributionofthe unknowncontinuumspectralfunc-
-
p tion, we compute the decayconstantsof the pseudoscalarmesonsB
e and B . In the computation, we employ the recent three loop cal-
s
h
culation of the pseudoscalartwo-point function expanded in powers
:
v of the running bottom quark mass. The sum rules show remarkable
Xi stability over a wide range of the upper limit of the finite energy
integration. Weobtainthefollowingresultsforthepseudoscalarde-
r
a cay constants: f =178±14 MeV and f =200±14 MeV. The
B Bs
results are somewhat lower than recent predictions based on Borel
transform, lattice computations or HQET. Our sum rule approach
of exploiting QCD quark hadron duality differs significantly from
the usual ones, and we believe that the errors due to theoretical
uncertainties are smaller.
PACS: 12.38.Bx,12.38.Lg.
1Supported by MCYT-FEDER under contract FPA2002-00612, and partnership Mainz-
ValenciaUniversities.
1
1 Introduction
The decay constant of a pseudoscalar meson B consisting of a heavy b-quark
q
and a light quark q, with q = u,d,s, is defined through the matrix element of
the pseudoscalar current:
<Ω|(m +m )qiγ b)(0)|B >= f M2 .
b q 5 q Bq Bq
Thesedecayconstantsareofgreatphenomenologicalinterestsincetheyenter
intheinputtonon-leptonicB-decays,inthehadronicmatrixelementsofB−B¯
mixingandintheextractionof|V |fromtheleptonicdecaywidthsofB-Mesons.
cb
Knowledge of the decay constants allows to estimate the so-called hadronic B
parameter which is directly related to the deviation of the vacuum saturation
hypothesis. The decayconstantsaretherefore ofcentralinterestto the ongoing
experiments carried out in B-factories. Unfortunately these matrix elements
could not be measured directly so far, so that we have to rely on theoretical
calculations. As the calculations mustbe non-perturbativethere areessentially
two approaches,QCD sum rules and lattice simulations.
Themethodofthesumruleshasbeensuccessfullyappliedsincethepioneer-
ingworkofShifman,VainshteinandZacharov[1]tocalculatevariouslowenergy
parametersinQCD.ParticularsumrulesarebasedonBorel,Hilberttransforms,
positive moments or inverse moments. Sum rule calculations of the decay con-
stants have been performed since the eighties using Boreltransform techniques
with results within the range f = 160−210 MeV and f /f = 1.09−1.22,
B Bs B
[2,4,5,7,6,8]. AmorerecentsumrulecalculationsusingthenewO(α2)corre-
s
lationfunctionwithoneheavyandonelightcurrent[3]yieldshighercentralval-
uesf =210±19MeVandf /f =1.16±0.04. LatticeQCDdeterminations
B Bs B
alsogive results in a wide range f =161−218MeVand f /f =1.11−1.16
B Bs B
[11, 12] (for a review and a collection of the results, see [9]). A recent HQET
calculation [10] yields f = 206±20MeV. The large variation of the quoted
B
values indicates that there is still room for improvement.
In general, the sum rule technique assumes duality. In our analysis duality
means that weighted integrals over experimental measured amplitudes should
agree with the same integrals evaluated in QCD perturbation theory.
We use a method based on finite energy sum rules (FESR) which equates
positive moments of data and QCD theory to evaluate of the decay constants
ofthe pseudoscalarbottommesons (f andf ). The method weproposehere
B Bs
employs a particular combination of positive moments of FESR in order to op-
timize the effect of the available experimental information. On the theoretical
2
side it uses the asymptotic (large momentum expansion) of QCD, i.e. an ex-
pansion in m2/s, where m is the mass of the bottom quark and s the square
b b
of the CM energy. Such an expansion makes sense as long as s is far enough
from the continuum threshold. We will consider the perturbative expansion up
to second order in the strong coupling constant and up to fourteen powers in
the expansion of the b-mass over the energy, using the results of the reference
[13]. On the phenomenological side of the sum rule we will consider the lowest
lying pseudoscalarB -meson. In our method we circumventthe problem of the
q
unknown continuum data by a judicious use of quark-hadronduality. We use a
linear combination of finite energy sum rules of positive moments, designed in
suchawaythat the contributionofthe data inthe regionabovethe resonances
turns out to be practically negligible [14].
The present paper differs from our earlier pilot work [15] in that O(α )2
s
corrections to the perturbative piece and the O(α ) corrections to the leading
s
non-perturbativetermareincluded. Inthiswaythestabilityofthepredictionof
f is improvedand the errors are reduced. The stability analysis is also placed
B
on more solid footing.
The plan of this note is the following: in the next section we briefly review
the theoreticalmethod proposed,in the third sectionwe discuss the theoretical
and experimental inputs used in the calculation and in the fourth one we write
up the conclusions. Finally in the appendix we discuss briefly the main issues
of the polynomials used in the finite energy sum rule.
2 The method
The two-point function relevant to our problem is:
Π(s=q2) = i dxeiqx <Ω|T(j (x)j (0))|Ω>,
5 5
Z
where < Ω| is the physical vacuum and the current j (x) is the divergence of
5
the axial-vector current:
j (x) = (M +m ):q(x)iγ Q(x):
5 Q q 5
M is the mass of the heavy bottom quark Q(x), and m stands for the mass
Q q
of the light quarks q(x), up, down or strange. The starting point of our sum
rules is Cauchy’s theorem applied to the two-point correlation function Π(s),
weighted with a polynomial P(s)
1
P(s) Π(s) ds = 0. (1)
2πi
IΓ
The integrationpathextends overacircle ofradius|s|=s ,andalongboth
0
sides of the physical cut s ∈ [s ,s ] where s is the physical threshold.
phys. 0 phys.
Neither the polynomial P(s) nor the power of the integration variable change
3
the analytical properties of Π(s), so that we obtain the following sum rule:
1 s0 1
P(s)ImΠ(s)=− P(s) Π(s) ds, (2)
π 2πi
Zsphys. I|s|=s0
Duality now means that on the left hand side we can use experimental in-
formationstartingfromthephysicalthresholds tos ,whereasontheright
phys. 0
hand side, i. e. on a circle of radius s , we can use the theoretical input given
0
by QCD and the operator product expansion. The integration radius s has
0
to be chosen large enough so that the asymptotic expansion ΠQCD(s) of QCD,
whichincludesperturbativeandnon-perturbativeterms,constitutesagoodap-
proximation to the two-point correlator on the circle.
Onthelefthandsideofequation(2),wecanparametrizetheabsorptivepart
of the two-point function by means of a single resonance B and the hadronic
q
continuum of the bq channel from sphys. with sphys. > s . Therefore, we
cont. cont. phys.
write the representation of the hadronic experimental data in the form:
1 1 1
ImΠ(s) = ImΠres.(s) + ImΠcont. θ(s−sphys.), (3)
π π π phys. cont.
where sphys. stands for the physicalthresholdof the continuum physicalregion,
cont.
and therefore it is an input in our calculation. It is worth mentioning that it
must not be confused with the duality parameter (s ) that is determined by
0
stability requirements in most versions of finite energy QCD sum rules.
On the right hand side of equation (2) the theoretical input necessary to
write down the two-point function relevant to the integration over the circle of
radius s has both perturbative and non-perturbative contributions,
0
ΠQCD(s) = Πpert.(s) + Πnonpert.(s), (4)
For the perturbative piece, we take the two-point correlationfunction Πpert.(s)
which has been calculated in [13] for one massless and one heavy quark as an
expansion up to second order (three loops) in the strong coupling constant α
s
and as a power series in the pole mass of the heavy quark (M2/s) up to the
b
seventh order. We have checked that for the employed range of values of the
radiuss oftheintegrationcontour,thepowerseriesconvergeswellanddoesnot
0
introduce any appreciable error in the calculation. In the case of one and two
loops the known complete analytic expressions of the two-point function could
be used, but we also use the mass expansion in this case because the results of
theintegrationcanbegiventheanalytically. Numericallyitmakesnodifference
whether one uses the complete analytic expressions or the expansions.
The authorsof[13]obtainthe followingcompactexpansionofthe two-point
function in terms of the pole mass M ,
b
α (M ) α (M ) 2
Πpert.(s)=Π(0)(s)+ s b Π(1)(s)+ s b Π(2)(s), (5)
π π
(cid:18) (cid:19) (cid:18) (cid:19)
4
where the different terms of the expansion in α have the form:
s
6 3 −s k M2 j
Π(i)(s)=(M +m )2M2 A(i) ln b (i=0,1,2). (6)
b q b j,k M2 s
j=−1k=0 (cid:18) b (cid:19) (cid:18) (cid:19)
X X
andthe coefficients A(i) ofthe QCDperturbativeexpansionaregivenin[13]to
j,k
order (M2/s)7. For instance, the first term of the expansion is given by:
b
3 −s M2 −s
Π(0)(s) = (M +m )2s 3−2ln +4 b ln +
16π2 b q M2 s M2
(cid:26) (cid:18) b (cid:19) (cid:18) b (cid:19)
−s M2 2 2 M2 3 1 M2 4 1 M2 5
−3−2ln b + b + b + b +
M2 s 3 s 6 s 15 s
(cid:20) (cid:18) b (cid:19)(cid:21)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 M2 6 2 M2 7
b + b +....
30 s 105 s
(cid:18) (cid:19) (cid:18) (cid:19) )
In the asymptotic expansion (4) there are also non-perturbative terms due
to the quark and gluon condensates. In our calculations we will include these
terms up to dimension six [2]:
1 α
Πnonpert.(s)=(M +m )2 M hq¯qi 1+2 s (7)
b q b s−M2 π
(cid:26) (cid:20) b (cid:16) (cid:17)
α M2
+2 s ln b +O(m /M )
π −s+M2 q b
b (cid:21)
1 1 α 1 1 M2
− sG2 − M + b hqσGqi
12s−M2 π 2 b (s−M2)2 (s−M2)3
b D E (cid:20) b b (cid:21)
8 2 M2 M4
− π + b − b α hqqi2
27 (s−M2)2 (s−M2)3 (s−M2)4 s
(cid:20) b b b (cid:21) (cid:27)
The α correction to the quark condensate [3] turns out to be small but
s
non-negligible.
Before going on, a comment on the asymptotic expansion is in order. It is
knownthattheconvergenceoftheperturbativeexpansionofthetwo-pointfunc-
tion, when written in terms of the pole mass, is rather poor. In fact, in many
calculations involving heavy quarks, the first and second order loop contribu-
tions are typically of the same order of magnitude making difficult to achieved
convergence in the results. On the other hand, the expansion in terms of the
running mass converges much faster over a wide range of the renormalization
scale. Henceforward, for calculational purposes, we will consider the relations
among the pole and the running mass in the appropriate order in the coupling
constant[16,17,18]inordertorewritethe perturbativepieceoforder(α )i (6)
s
in the form
6 3 −s k m2(µ) j
Π(i)(s)=m2(µ)(m (µ)+m (µ))2 A˜(i) ln b (i=0,1,2).
b b q j,k µ2 s
j=−1k=0 (cid:18) (cid:19) (cid:18) (cid:19)
X X
(8)
5
and similarly for the non-perturbative piece. The coefficients A˜(i) depend on
j,k
the mass logarithmsln(m2/µ2) up to the third power. As Π(s) is not knownto
b
all orders in α , the results of our analysis will depend to some extend on the
s
choiceofthe renormalizationpointµ. Inthe sumruleconsideredherethereare
two obvious choices, µ=m and µ=s , the radius of the integration contour.
b 0
The former choice will sum up the mass logs of the form ln(m2/µ2) and the
b
latter choice the ln(−s/µ2) terms. We find that the for the choice µ = m the
b
convergence of the perturbative terms is significantly better, so we will adopt
this value in the presentation here.
For the experimental side, we can split up the absorptive part into the es-
tablished lowest lying pseudoscalar bq resonance B and an unknown hadronic
q
continuum :
1 1
ImΠ(s) = M4 f2 δ(s−M2 ) + ImΠcont. θ(s−sphys.) (9)
π Bq Bq Bq π phys. cont.
where M and f are, respectively, the mass and the decay constants of the
Bq Bq
pseudoscalar meson B .
q
Looking back to equation (2) and taking all the theoretical parameters in-
cluding the mass of the B -meson as our inputs in the calculation, we see that
q
the decay constant could be computed if we would have accurate information
on the hadronic continuum contribution, which is, however, not the case.
To cope with this problem we make use of the freedom of choosing the
polynomial in equation (2). We take for P(s) a polynomial of the form:
P (s)=a +a s+a s2+a s3+...+a sn, (10)
n 0 1 2 3 n
wherethecoefficientsarefixedbyimposinganormalizationconditionatthresh-
old
P s =M2 = 1, (11)
n ph. Bq
(cid:16) (cid:17)
and requiring that the polynomial P (s) should minimize the contribution of
n
the continuum in the range sphys.,s in a least square sense, i.e.,
cont. 0
h i
s0
skP (s) ds=0 for k =0,...n−1, (12)
n
Zspcohnyts..
The polynomials obtained in this way are closely related to the Legendre poly-
nomials. In the appendix the explicit form of the set of polynomials use in this
work is given.
Thiswayofintroducingthepolynomialweightinthesumrulemakesnegligi-
ble the integrationof 1 ImΠcont. θ(s−sphys.), so thatinthe phenomenological
π phys. cont.
side of the sum rule only the contributionof the B resonanceremains. On the
q
other hand, as we will see in the results, it increases the value of the duality
parameter s and, therefore, the asymptotic expansion of QCD can be trusted
0
safely in the integration contour.
To the extend that Im Πcont. can be approximated by an n degree polyno-
phys.
mialtheseconditionsleadtoanexactcancellationofthecontinuumcontribution
6
to the left hand side of equation (2). As a welcome side effect, this choice of
polynomial will enhance the role of the B resonance. Notice however that
q
increasingthedegreeofthepolynomialP (s),willrequiretheknowledgeoffur-
n
ther terms in the mass expansionandin the non-perturbativeseries. Therefore
the polynomial degree cannot be chosen arbitrarily high.
Tochecktheconsistencyofthemethod,wehaveconsideredasecondtofifth
degreepolynomialsandtheresultsarecompatiblewithintherangeoftheerrors
introducedbytheinputsofthecalculation. Wealsohavecheckedexplicitlythat
a smooth continuum contribution had no influence on the result.
The sum rule method above was previously used in the calculation of the
charm mass from the cc experimental data. The continuum data in this case
wereknownfromtheBESIIcollaboration[19]andwereshowntohavenoinflu-
enceonthepredictedmass[21]. Employingthesametechnique,averyaccurate
prediction of the bottom quark mass was also obtained using the experimental
information of the upsilon system [22].
After these considerations we proceed with the analytical calculation. The
integralsthat we have to evaluate onthe righthand side of the sum rule, equa-
tion (2), are
k
1 −s
J(p,k)= sp ln ds, (13)
2πi µ2
I|s|=s0 (cid:18) (cid:19)
for k = 0,1,2,3 and p = −6,−5,..,n+1. These integrals can be found e.g. in
reference [20]. After integration, equation (2) yields the sum rule
M4 f2 P(M2 )= m2(µ)(m (µ)+m (µ))2 (14)
Bq Bq Bq b b q
n 2 6 3 i
α (µ)
× a s A˜(i) m2j(µ) J(q−j,k)
q π j,k b
q=0i=0j=−1k=0 (cid:18) (cid:19)
XX X X
+ non-perturbative terms
where,forbrevity,wehavenotwrittendownthenon-perturbativetermsexplic-
itly. The contribution of the continuum is neglected.
Pluggingthetheoreticalandexperimentalinputs(physicalthreshold,quarks
and meson masses, condensates and strong coupling constant) into the sum
rule, we obtain the decay constant f for various values of the degree n of
Bq
the polynomial and various values of s . Given the correct QCD asymptotic
0
correlator and the correct hadronic continuum, the calculation of the decay
constant should, of course, not depend either on s or on the degree n of the
0
polynomialinthesumrule(2). Accordingly,foragivennwechoosetheflattest
region in the curve f (s ) to extract our prediction for the decay constant. To
B 0
bespecificwechoosethepointofminimalslope. Ontheotherhand,fordifferent
polynomials,the valueoff ,extractedinthisway,coulddiffer fromeachother
B
as the cancellationof the continuum may be incomplete or the QCD expansion
not accurate enough. We find, however, practically the same results for all our
polynomials. This additional stability is truly remarkable as the coefficients
of the polynomials of order n = 2,3,4 and 5 are completely different and the
7
respective predictions are based on completely different superpositions of finite
energy moment sum rules. This extended consistency leads us to attach great
confidence in our numbers and associated errors.
3 Results
We calculate the decay constants for the B and B heavy mesons. In the first
s
case we take m =0 everywhere. In the second case we retain m =m 6=0 in
q q s
the factor (M +m )2 in front of the correlation function only. Further terms
b q
in the power series in m2/s in (5) are completely negligible for the integration
s
radii s we use in the calculation.
0
Theexperimentalandtheoreticalinputsareasfollows. Thephysicalthresh-
olds isthe squaredmassofthe lowestlyingresonanceinthe bq channel. For
ph.
q being the light quark u, we have:
s =M2 =5.2792 GeV2 =27.87,GeV2 (15)
ph. B
whereasthecontinuumthresholdsphys.istakenfromthenextintermediatestate
cont.
Bππ in an s-wave I = 1, i. e.
2
sphys. =(M +2m )2 =30.90 GeV2.
cont. B π
For q being the strange quark we take:
s =M2 =5.3692 GeV2 =28.83,GeV2 (16)
ph. Bs
The continuum threshold starts in this case at the value:
sphys. =(M +2m )2 =31.92 GeV2.
cont. Bs π
In the theoretical side of the sum rule we take the following inputs. The
strong coupling constant at the scale of the electroweak Z boson mass [24]
α (M )=0.118±0.003 (17)
s Z
is run down to the computation scale using the four loop formulas of reference
[23]. For the quark and gluon condensates (see for example [3]) and the mass
of the strange quark [25] we take:
<qq >(2GeV) = (−267 ± 17 MeV)3,
α
< s GG>= 0.024 ± 0.012 GeV4,
π
<qσGq >= m2 <qq >, with m2 = 0.8 ±0.2GeV,
0 0
m (2GeV) = 120 ±50MeV,
s
<ss>= (0.8±0.3)<qq >. (18)
As discussed, above we fix the renormalizationscale µ in the theoretical side of
thesumtobeµ=m (m ),astheperturbativeseriesinα iswellundercontrol,
b b s
8
i.e. thefirstandsecondordertermsareonlyafewpercentofthedominantzero
order one and the second order one is much smaller than the first order one.
We use a reasonable variation of µ to estimate the corresponding error in our
final result.
Finally, for the bottom quark, a value m (m ) ≈ 4.20GeV is generally ac-
b b
cepted. For consistencywe takethe resultm (m ) = 4.19 ±0.05GeVofwhich
b b
has been obtained by us [22] with a similar sum rule method.
In order to calculate the decay constant for the pseudoscalar meson B, we
proceedinthewaydescribedabove. Wecomputef asafunctionofs withthe
B 0
four different sum rules (2) corresponding to n=2,3,4,5. The results, plotted
in Fig. 1 show remarkable stability properties. We define the optimal value of
s as the center of the stability region (represented by a cross in Fig.1) where
0
the first and/or second derivative of f (s ) vanishes. At these values of s we
B 0 0
obtain the following consistent results:
f =175 MeV for n=2
B
f =177 MeV for n=3
B
f =178 MeV for n=4
B
f =178 MeV for n=5
B
Notice from Fig. 1 that for the fourth degree polynomial (n=4) there is an
stability region of about 50 GeV2 around s . In this region the decay constant
0
changesby less than 1 percent. This stability regionis pushedup to 100 GeV2
inthecaseofn=5. Withtheseconsiderationsweestimateaconservativeerror
inherent to the method of ±3MeV.
Other sources of errors arising in the calculation of f are the quark con-
B
densateswhichaffectthe resultby ±3MeV,andthe bottommasswhich,inthe
rangegivenabove,producesavariationinthedecayconstantof−13,+10MeV.
This last one is the main source of uncertainty in the final result. Finally we
have changed the renormalization scale in the range µ ∈ [3,6] GeV. We esti-
mate an error of ±6MeV associated to this change in µ which is roghly related
to the convergence of the asymptotic expansion
Other errors due to the QCD side of the sum rule (higher order terms in
m2/s and the error on α (m ) are negligible in comparison).
b s Z
Adding quadratically all the errors, we finally quote the following result for
the decay constant of the light meson B:
f = 178 ± 14(inp.) ± 3(meth.) MeV. (19)
B
The first errorcomes fromthe inputs of the computationand the truncated
QCD theory whereas the second is due to the method itself.
Proceeding in the same fashion, but keeping the mass of the strange quark
in the overallfactor and the order m /m in the one loop contribution, we find
s b
the decay constant for the B meson (f ),
s Bs
f = 200 ± 14(inp.) ± 3(meth.) MeV.
Bs
9
f (Gev )
B
0,19
0,18
n=5
n=4
n=3
0,17
n=2
0,16
50 100 150 2
S ( Gev )
0
Figure 1: Decay constant f as a function of the integration radius s for
B 0
m (m )=4.19GeV. The lines represent the sum rule (2) for several choices of
b b
the polynomial P (s).
n
(compare Fig. 2)
Intheanalysisoftheoreticalerrorstheonlynewingredientistheuncertainty
coming from the strange quark mass which turns out to be negligible.
Theratioofthedecayconstantsf and f (whichwouldbe1inthechiral
Bs B
limit) is of special interest. We find:
f
Bs = 1.12 ± 0.01(inp.) ± 0.03(meth.). (20)
f
B
Inthecalculation,theuncertaintiesduetothemethodandtothetheoretical
inputs (mainlyto thebottomquarkmass)arecorrelated,sothatthe finalerror
is very small.
Finally we should mention that in the past the stability region was often
determined in a rather ad hoc fashion by considering the intercept of sum rule
predictionsformomentsdifferingbyonepower. Wefindthat,suitablymodified,
this prescription also works in our sum rule method with similar results but
larger errors.
10