Table Of ContentUB-ECM-PF-06-04; IPPP/06/05
Renormalization Group Improved Sum Rule Analysis for the Bottom Quark Mass
Antonio Pineda1 and Adrian Signer2
1Dept. d’Estructura i Constituents de la Mat`eria, U. Barcelona,
Diagonal 647, E-08028 Barcelona, Catalonia, Spain
2Institute for Particle Physics Phenomenology, Durham, DH1 3LE, England
(Dated: 23 January 2006)
Westudy the effect of resumming large logarithms in thedetermination of thebottom quark mass
through a non-relativistic sum rule analysis. Our result is complete at next-to-leading-logarithmic
accuracy and includes some known contributions at next-to-next-to-leading logarithmic accuracy.
Compared to finite order computations, the reliability of the theoretical evaluation is greatly im-
proved, resulting in a substantially reduced scale dependence and a faster convergent perturbative
series. ThisallowsustosignificantlyimproveoverpreviousdeterminationsoftheMSbottomquark
mass, m , from non-relativistic sum rules. Ourfinal figurereads m (m )=4.19±0.06 GeV.
b b b
6
0
0 Processes involving a b¯b quark pair close to threshold correctionsturnedouttobemuchlargerthananticipated
2
are very sensitive to the bottom quark mass, m , and and,moreover,verystronglyscaledependent. Theuseof
b
n offer a unique opportunity to accurately determine its threshold masses [6, 7, 8], which account for the cancel-
a
value. One of the cleanestobservables where this depen- lation of the pole mass renormalonin the observable,do
J
dence onm showsup is the non-relativisticsum rule [1] not really solve these problems, specially for the strong
3 b
2 12π2e2 d n ∞ ds scale dependence. Overall, this produced a very slowly
1 Mn ≡ n! b (cid:18)dq2(cid:19) Π(q2)(cid:12)q2=0 =Z0 sn+1Rb¯b(s), cthoenvdeertgeernmtisneartieiosn, boefinmgbt.heNdoonm-pienratnutrbsaotuirvceecoofrreercrtoironins
v (cid:12) (1) are known in the limit m /n Λ [9]. Even though
85 where Rb¯b(s) ≡ σ(e+e− → b¯b)/σ(e+e− →µ+µ−), Π(q2) this limit does not hold fobr la≫rge eQnCoDugh n, we can take
1 is the vacuum polarization and eb = −1/3 the elec- it as an orderof magnitude estimate. Numerically, these
1 tric charge of the bottom quark. The typical scale is corrections are very small and can be neglected in com-
0 p 2mb/√n and, provided n is not chosen to be too parison with other sources of errors.
∼
6 large, the left-hand side of Eq. (1) can be reliably com- The situation is very similar in the case of tt¯pair pro-
0 putedusingaweakcouplinganalysis(theright-handside duction near threshold. In this case the use of threshold
/
h can be determined from experiment). To describe such masses results in a well behaved perturbative series for
p processestheoretically,astandardfixed-ordercalculation thepositionofthepeakofthett¯crosssectionand,there-
p- in the strong coupling αs is insufficient due to the pres- fore,mayenableaprecisedeterminationofthetopquark
e ence of terms (αs/v)n 1 at each order in perturba- mass, once experimental data is available. However, the
∼
h tion theory, where v 1/√n 1 is the velocity of the large theoretical uncertainty in the normalization of the
v: heavy quarks. Such te∼rms app≪ear because there are sev- crosssection remained(even if the series is more conver-
i eral scales involved in the problem. There is the hard gent than in the bottom case). It has been claimed [10]
X
scale µh mb, the soft scale µs mbv of the order that the source of this uncertainty is due to potentially
ar of the typ∼ical momentum and, finall∼y, the ultrasoft scale large logv terms, which arise due to the presence of sev-
µus ∼ mbv2 of the order of the typical kinetic energy of eral scales and take the form logµh/µs and logµs/µus.
the heavy quarks. Using effective field theories (for a re- Although it has been shown later [11, 12] that some ex-
view see [2]), the perturbative expansion can be system- pressions used in that paper were incorrect and that the
atically reorganized into an expansion in the two small finalerrorclaimedwassomewhatoveroptimistic[13],the
parameters of the problem, αs and v, and the b¯b cross message that the resummation of logarithms improves
section can be written as the scale dependence has gone through.
α n Giventheimportanceofthelogv termsforthett¯cross
s
Rb¯b(s)=v v (2) section, it is natural to ask whether their inclusion also
Xn (cid:16) (cid:17) improvesthesituationintheb¯bcase. Inourcasewehave
1(LO);α ,v(NLO);α2,vα ,v2(NNLO)... .
× s s s to replace the expansion of Eq. (2) by
(cid:8) (cid:9)
Tcihenetclyoeuffisicniegntthseotfhtrheisshsoelrdieesxcpaannsbieonco[3m].puted most effi- Rb¯b(s)=vXn (cid:16)αvs(cid:17)nXm (αslogv)m (3)
At present, non-relativistic sum rules have been com-
1(LL);α ,v(NLL);α2,vα ,v2(NNLL)... .
puted at next-to-next-to-leading order (NNLO) [4, 5]. × s s s
(cid:8) (cid:9)
This allowed for a precise determination of the bottom Aswewillsee,theselogarithmsareextremelyimportant
quark mass using a well understood perturbative ap- numerically and substantially improve the reliability of
proach. Unfortunately,intheon-shellscheme,theNNLO the theoretical evaluation of the moments.
2
The n-th moment, M , as defined in Eq. (1) is com- the PS/RS mass is set to µ =2 GeV, to ensure it does
n f
puted in the usual way. First we match QCD to non- not exceed the characteristic scale µ . Once the PS/RS
s
relativistic QCD (NRQCD) at the hard scale which we mass is determined, we convert it to m , the MS mass
b
set to µ = m. This theory is then matched to poten- at the renormalization scale m . We use the three-loop
h b
tial NRQCD (pNRQCD) [14]. Solving the correspond- conversion [17] of the pole mass to m and for the PS
b
ing non-relativistic Schr¨odinger equation perturbatively and RS mass a ‘large β ’ [18] and renormalon-based [8]
0
weobtainImG(0,0,E),the imaginarypartofthe Green approximationrespectively for the four-loop term.
function at the origin. Mn can then be written as The moments are evaluated by performing the energy
integration in the complex energy plane using a strictly
∞
dE
M = 48πe2N (4) expanded form as indicated in Eq. (5). The difference
n b cZ−∞ (E+2mb)2n+3 between this evaluation and using Eq. (4) is NNNLO
E and, therefore, beyond the accuracy we are aiming at.
c2 c c ImG(0,0,E),
×(cid:18) 1− 1 23m (cid:19) However, for small values of n, this difference is sizable.
b
In fact, for n = 6 the resulting values for m may
b,PS/RS
where E = √s 2mb, Nc = 3 and c1 and c2 are the differ by up to 45/60MeV depending on how we expand
−
matching coefficients of the currents, normalized to 1 at the prefactor Eq. (5), for n = 8 the difference is up to
leading order. In a strict non-relativistic expansion one 15/25 MeV, whereas for n 10 the values for m agree
b
≥
also expands within 10/15 MeV.
The experimental moments are determined as de-
1 e−nmEb 3E nE2 scribed in Ref. [5]. The moment is split into the con-
= 1 + ...
(E+2m )2n+3 (2m )2n+3 (cid:18) − 2m 4m2 (cid:19) tribution due to the six Υ resonances and the open b¯b
b b b b
(5) continuum. Themainuncertaintycomesfromtherather
treating nE m . We also remark that the loga- poor knowledge of the latter, which we parametrize as
b
rithms involvi∼ng µ always appear in the combination Rcont =0.4 0.2 [19]. Since the continuum contribution
s b¯b ±
log( 4m E/µ2). This confirms that the natural scales issuppressedforlargervaluesofn,resultinginasmaller
− b s
are given by E m /n and µ p 2m /√n. To en- experimental error, we refrain from using n<6.
b s b
∼ ∼ ∼
sure the applicability of perturbation theory, we cannot The main theoretical uncertainty in previous determi-
choose n too large and will restrict ourselves to n 14. nations of the bottom quark mass was due to the huge
≤
The matching coefficients of pNRQCD depend on the scale dependence of the NNLO result, which made it
scales µ = m , µ and µ . In solving the renormal- rather difficult to find a reliable procedure for estimat-
h b s us
ization group equations we have set µ = µ2/m. The ing the theoretical error. It is the main result of this
us s
expressions we use are complete at NLL and NNLO. workto show that the situation improvesconsiderably if
At NNLL they are also complete (in particular we in- a renormalizationgroupimprovedanalysis is performed.
clude the insertions of the renormalization group im- To illustrate this, in Figure 1 we show the dependence
proved potentials to G(0,0,E) up to the desired order of the theoretical value for M (evaluated at LO/LL,
10
inthe MSscheme)exceptfor c . For c we areusing the NLO, NLL, NNLO and NNLL respectively) on µ . For
1 1 s
knownNLL[11]expressionaswellassomepartialNNLL the purpose of illustration we also plot the experimen-
contributions, which include the spin-dependent correc- tal value of the moment including its error. We set the
tions [15], the NNLL ultrasoft corrections to the Green strong coupling to α (M ) = 0.118 and use three-loop
s Z
function, the corrections due to the two-loop beta run- evolution to determine it at lower scales. For the plot
ning, and some contributions coming from the introduc- shown in Figure 1 we set m (2 GeV) = 4.515 GeV
b,PS
tion of partial higher-order terms in the renormalization and m (2 GeV) = 4.370 GeV and vary the soft scale
b,RS
groupimprovedpotentials that appearinthe anomalous aroundits characteristicvalueµ 2m /√10 (indicated
s b
∼
dimension of c . For details we refer to Refs. [5, 16]. In by a dashed vertical line). Note that in this region the
1
particularwe stressthatnotallthe ultrasoftrelatedlog- size of the NNLL corrections (even if large) is consider-
arithmsareincludedinouranalysis. Withthis caveatin ably smaller than the corresponding fixed-order NNLO
mind, we still refer to our full result as NNLL. ones. Moreover, contrary to earlier analyses, the NNLL
WealsoincludeQEDcorrectionsinourresult. Count- result is now more stable with respect scale variations
ing α α2, these corrections enter already at NLO, due than the NLL one (for the range of scales for which the
∼ s
toasingleexchangeofapotentialphoton,butthey have computation is trustworthy). This is of course what one
only a minor numerical impact. They increase the ex- would expect and indicates that the inclusion of the log-
tracted bottom quark mass by less than 10 MeV. arithmssubstantiallyimprovesthereliabilityofthetheo-
The threshold masses we consider in this analysis are retical prediction. Only for scales µ <2 GeV the situa-
s
thepotentialsubtracted(PS)massm (µ )[7]andthe tion gets out ofcontrol,but for these scales the ultrasoft
b,PS f
renormalon subtracted (RS) mass m (µ ) [8]. The scale is below 1 GeV and we can not really rely on our
b,RS f
subtraction scale µ that is needed for the definition of computation. Multiple insertions of corrections to the
f
3
the theoretical error.
5 NNLO mPS=4.515 GeV The experimental error, ∆exp, is simply determined
by extracting the value for m for the two extreme val-
b
4 ues of the experimental moment. The error due to the
D uncertainty in the strong coupling, ∆ , is obtained by
21-0 3 LL(cid:144)LO studying the effect on the extracted boαttom quark mass
1
@
if we vary 0.115 < α (M ) < 0.121. Following common
M10 2 NNLL NLL practise one would esstimZate the theoretical error, ∆ ,
th
by variation of the scale by a factor of two. As is obvi-
NLO
1 ous from Figure 1, for small scales the theoretical result
cannot be trusted. Therefore, in previous analyses, the
0 scale variation was limited to scale choices above a cer-
2 3 4 5 6 7 8
taincutoff, typicallyset to a value around2 GeV. In the
Μs
current analysis we refrain from using such an estimate.
5 NNLO mRS=4.370 GeV There are several reasons. First, such an error estimate
dependscruciallyonthesomewhatarbitrarylowercutoff
4 of the scale variation. Second, it does not take into ac-
D count the fact that the higher-order corrections are siz-
21- 3 LL(cid:144)LO able. Given that the scale dependence is very modest
0
1
@ (for reasonably large scales) compared to the size of the
M10 2 NNLL NNLL corrections, we think that such an error analysis
NLL
wouldconsiderablyunderestimatethetheoreticalerrorin
1 NLO the present case. Finally, the scale variation as depicted
in Figure 1 does not take into account the independent
0 variation of the ultrasoft scale, since in our analysis the
2 3 4 5 6 7 8 latterisdeterminedbythe softscale. Itwouldbeprefer-
Μs
able to be able to vary all scales independently to ob-
tain a better estimate of the uncertainty, in particular
FIG. 1: The moment M10 as a function of µs at LO/LL, since someultrasoftlogarithmsaremissinginourresult.
NLO,NLL,NNLOandNNLLformbPS(2GeV)=4.515GeV We have verified that a naive variation of µ results in
in the PS scheme (upper figure), and for mbRS(2 GeV) = a rather large uncertainty which, however, iussconsistent
4.370GeVintheRSscheme(lowerfigure). Theexperimental
with the final error estimate we propose. Therefore, we
moment with its error is also shown (grey band).
prefer to determine the theoretical error by taking half
thesizeofthehighest-ordercorrectionthatisincludedin
Coulomb potential also seem to be important in this re- our result. More precisely, we determine two values for
gion [20]. mb by equating the experimental and theoretical value
The situation is similar for other values of n. As a (at the scale for which it reaches its maximum) of the
general feature, for increasing n, the scale dependence momentatNNLLandNLLrespectively. Theerrorisde-
increases slightly. This is not surprising since larger n terminedashalfthe differencebetweenthese two values.
induce smaller scales and at some point the applicabil- This procedure assumes a perturbative series where suc-
ity of perturbation theory is questionable. On the other cessive terms become less and less important. For this
hand, as mentioned above, smaller values of n have the to hold we have to use a threshold mass, since for the
disadvantagethatthe non-relativisticapproximationbe- pole mass the NNLL corrections are much larger than
comes less reliable. the NLL ones. In this respect moments with low val-
These findings show that it is possible to improve the ues of n and/or threshold mass definitions with values
accuracyofpreviousdeterminationsofthebottomquark close to the MS mass are better behaved. On the other
mass from non-relativistic sum rules if the renormaliza- hand, the actual size of the correction,and therefore the
tion group improvement is applied. In order to deter- assigned error, increases for such mass definitions.
mine the MS mass we first determine the PS/RS mass We summarize our results in Table I, where we also
with its error, proceeding as follows: we consider M show the combined error, ∆ , which is obtained by
n tot
for n 6,8,10,12,14 and obtain our central value by addingthevariouserrorsinquadrature. Asexpected,the
∈ { }
equating the theoretical and experimental value of the experimental error decreases with increasing n. The re-
moment at the standard scale µ = 2m /√n. For the sultsareallconsistentwitheachother,inparticularifwe
s b
error in the determination of the threshold masses we take into account the additional uncertainty mentioned
consider three sources: the experimental error, the error above for M with n 8, due to the non-relativistic
n
≤
due to the uncertainty in the strong coupling and finally expansion in the energy integration. Related to this,
4
n mb,PS(2 GeV) ∆th ∆exp ∆α ∆tot mb Inconclusion,wehavestudiedtheeffectofresumming
6 4460 40 50 35 70 4135 ± 65 logarithmsfornon-relativisticsumrules. Thelogarithms
8 4505 45 25 30 60 4170 ± 55 turn out to be numerically very important and improve
10 4515 45 15 25 55 4185 ± 50 the reliabilityof the theoreticalcomputation. This man-
12 4520 45 10 20 50 4185 ± 45 ifests itself in a reduced scale dependence and an im-
provement of the convergence of the perturbative series.
14 4520 40 10 15 45 4185 ± 40
It allows us to obtain an accurate value for the MS bot-
n mb,RS(2GeV) ∆th ∆exp ∆α ∆tot mb tom quark mass using a credible error estimate
6 4315 55 50 25 80 4140 ± 70
8 4360 65 30 20 75 4180 ± 65 mb(mb)=4.19±0.06GeV. (8)
10 4370 65 20 10 70 4190 ± 60 At this stage, the main problem appears to be the large
12 4370 65 15 5 65 4190 ± 60 sizeoftheperturbativecorrectionsandtounderstandits
14 4370 65 10 5 65 4185 ± 55 origin. FurtherimprovementsrequirethefullNNLLcom-
putation of the sum rule, especially the potentially large
ultrasoft effects. Obviously, the inclusion of all NNNLO
TABLEI:Extractionofmb,PS/RS(2GeV)witherrorsforvar- effects will also be important and might lead to a better
ious n. All values are given in MeV and rounded to 5 MeV.
controlofthestrongscaledependence forsmallvaluesof
The total error has been obtained by adding the partial errors
µ and the large size of the perturbative corrections.
inquadrature. Thecorresponding valuefortheMS masswith s
Acknowledgements. AP acknowledges discussions
its error is given in the last column.
withA.A.PeninandJ.Soto. ASthanksECMBarcelona
for hospitality during the course of this work.
we note that in computing the moments we do not use
the exact fixed-order coefficient at (α2), since we drop
O s
terms of (α2/(√n)k) with k 1. Again, this neglect is
O s ≥
potentially more of a problem for smaller moments. Let
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