The $$\eta _c$$ decays into light hadrons using the principle of maximum conformality
Eur. Phys. J. C
The ηc decays into light hadrons using the principle of maximum conformality
BoLun Du 0
XingGang Wu 0
Jun Zeng 0
Shi Bu 0
JianMing Shen 0
0 Department of Physics, Chongqing University , Chongqing 401331 , People's Republic of China
In the paper, we analyze the ηc decays into light hadrons at the nexttoleading order QCD corrections by applying the principle of maximum conformality (PMC). The relativistic correction at the O(αs v2)order level has been included in the discussion, which gives about 10% contribution to the ratio R. The PMC, which satisfies the renormalization group invariance, is designed to obtain a scalefixed and schemeindependent prediction at any fixed order. To avoid the confusion of treating nf terms, we transform the usual MS pQCD series into the one under the minimal momentum space subtraction scheme. To compare with the prediction under conventional scale setting, RConv,mMOM−r = 4.12−+00..2380 × 103, after applying the

PMC, we obtain RPMC,mMOM−r = 6.09−+00..5652 ×103, where
the errors are squared averages of the ones caused by mc and
mMOM. The PMC prediction agrees with the recent PDG
value within errors, i.e. Rexp = (6.3 ± 0.5) × 103. Thus we
think the mismatching of the prediction under conventional
scalesetting with the data is due to improper choice of scale,
which however can be solved by using the PMC.
The heavy quark mass provides a natural hard scale for
the heavy quarkonium decays into light hadrons or photons.
Calculations of their decay rates are considered as one of the
earliest applications of pQCD. The charmonium has become
a popular field since the discovery of J /ψ resonance at SLAC
and Brookhaven in 1974. There are lots of successful
experimental studies about charmonium, including the precise
measurements of spectrum, lifetimes and branch ratios, cf. a
comprehensive review given in the PDG [
1
]. At the same time,
many theoretical efforts have been tried for an appropriate
description of charmonium. As an important breakthrough,
a systematic pQCD analysis of the heavy quarkonium
inclusive annihilation and production has been given within the
nonrelativistic QCD theory (NRQCD) in 1995 [
2
].
According to the NRQCD framework, the quarkonium
decay rate can be factored into a sum of products of the
shortdistance coefficients and the longdistance matrix elements
(LDMEs). The shortdistance coefficients are perturbatively
calculable in a power series of αs . The LDMEs can be
estimated by means of the velocity power counting rule, i.e.
the LDMEs can be classified in terms of the relative
velocity between the constituent quarks of the heavy quarkonium.
Especially, the colorsinglet ones can be directly related to the
wavefunction (derivative of the wavefunction) at the origin,
which then can be calculated via proper potential models.
The decay rates of the pseudoscalar quarkonium into light
hadrons and photons have been calculated at the
nexttoleading order (NLO) level [
3,4
]. The relativistic corrections
at the O(αs v2)order have been given in Refs. [
5,6
]. Within
the NRQCD factorization framework, the decay rate of the
ηc into light hadrons or photons can be expressed as
(ηc → LH) =
ηcO1(1 S0)ηc
and
(ηc → γ γ ) =
Fγ γ (1 S0)
mc2
+
Gγ γ (1 S0)
mc4
ηcP1(1 S0)ηc + · · ·
ηcO1(1 S0)ηc
ηcP1(1 S0)ηc + · · · ,
(1)
(2)
where F1, G1, Fγ γ and Gγ γ are short distance coefficients.
The symbol · · · stands for the contributions from
highdimensional LDMEs which are at least at the level of O(v4 ).
mc is the cquark pole mass.1 v2 is the squared heavy quark
or antiquark velocity in the meson rest frame. For the case of
ηc, it can be calculated by
1 The choice of pole mass avoids the ambiguity of using MSmass for
separating the renormalization group involved βterms [
7
].
2 ηcP1(1 S0)ηc
v ηc = mc2 ηcO1(1 S0)ηc
.
R =
(ηc → LH)
(ηc → γ γ )
= R0(μ) 1 +
To suppress the uncertainty from the LDMEs, one usually
calculates the ratio
81π2CF a2(μ), μ is
where a(μ) = αs (μ)/(4π ), R0(μ) = 2α2 NC
2
an arbitrary renormalization scale, and β0 = 11 − 3 n f (n f
being the active flavor number) is the leading βterm of the
renormalization group function. It is noted that the
factorization scale dependence is missing at this level, which is
the case even at the NNLO level [
8
], we are thus free of the
factorization scalesetting problem.
It is conventional to take the renormalization scale as the
typical momentum flow of the process or the one to eliminate
the large logs of the pQCD series, we call this conventional
scalesetting approach. As will be shown later, such a
simple treatment on scale introduces large scale uncertainty and
makes the lowerorder prediction unreliable. At present, the
ηc decays into light hadrons or photons have been
calculated up to NNLO level, which however still shows a poor
pQCD convergence [
8–10
]. Thus by simply pursuing
higherandhigher order terms may not be the solution for those
highenergy processes. In fact, even if we obtain a small
scale uncertainty for global quantities such as the total
crosssection or the decay rate at a certain fixed order, it is due
to cancelations among different orders; the scale uncertainty
for each order is still uncertain and could be very large. Two
such examples for Higgs boson decay and the hadronic
production of Higgs boson can be found in Refs. [
11,12
]. When
one applies conventional scalesetting, the renormalization
scheme and initial renormalization scale dependence are
introduced at any fixed order. Thus, a proper scalesetting
approach is important for the fixedorder predictions.
Such large scale uncertainty has long been observed, and
to improve the accuracy of R, Ref. [
13
] suggested to resum
the finalstate chains of the vacuumpolarization bubbles and
got RNNA = (3.01 ± 0.30 ± 0.34) × 103 for the naive
nonAbelianization resummation [
14
] and RBFG = (3.26 ±
0.31±0.47)×103 for the backgroundfieldgauge
resummation [
15
], respectively. Both predictions are consistent with
the world average given by Particle Data Group (PDG) in
year 2000 [
16
], which gives Rexp = (3.3 ± 1.3) × 103. As
an attempt, the authors of Ref. [
13
] also presented a
prediction by using the Brodsky–Lepage–Mackenzie (BLM)
scale(3)
setting approach [
17
] and got a much larger R value, i.e.
RBLM = 9.9 × 103.
We should point out that those predictions are different
from the value derived from the new experimental
measurements, which shows Rexp = (6.3 ± 0.5)×103 [
1
]. As will be
shown later, the BLM prediction given in Ref. [
13
] is
questionable. Thus it is interesting to show whether an improved
pQCD analysis could be done and could explain the new
Rexp, as is the purpose of this paper. Especially, it is
important to show whether the mismatching of the data and the
pQCD prediction is caused by improper choice of scale or
by some other reasons.
A novel scalesetting approach, the Principle of
Maximum Conformality (PMC) [
18–21
], has been developed in
recent years. The PMC satisfies renormalization group
invariance [22] and it reduces in NC → 0 Abelian limit [
23
] to the
standard Gell–Mann–Low method [
24
]. A more convergent
pQCD series without factorial renormalon divergence can be
obtained. The PMC scales are physical in the sense that they
reflect the virtuality of the gluon propagators at a given order,
as well as setting the effective number (n f ) of active flavors.
The resulting resummed pQCD expression thus determines
the relevant “physical” scales for any physical observable,
thereby providing a solution to the renormalization
scalesetting problem. Because all the schemedependent {βi
}terms in pQCD series have been resummed into the
running couplings with the help of renormalization group
equation, the PMC predictions are renormalization scheme
independent at every order. Such scheme independence can be
demonstrated by using commensurate scale relations [
25
]
among different observables. A number of PMC applications
have been summarized in the review [
26–28
]. The PMC
provides the underlying principal for the BLM, and in the
following, we shall adopt the PMC to set the renormalization scale.
Up to NLO level, the expression of R can be rewritten as
R = r1,0a2(μ) + r2,0 + 2β0r2,1 a3(μ) + O(a4),
where the MScoefficients ri, j can be read from Eq. (4), in
which ri,0 are conformal ones. Following the standard PMC
procedures, we get
R = r1,0a2(Q1) + r2,0a3(Q1),
where ln Q21/μ2 = −r2,1/r1,0. Here, we have set the
unknown PMC scale Q2 = Q1 such that to ensure
the schemeindependence of R under any
renormalization schemes via proper commensurate scale relations [
25
],
whose exact value can be determined by the NNLO term
which is not available at present.
If directly using the MSscheme expression (4), we shall
obtain a small PMC scale Q1 = 0.86 or 0.78 GeV for the
prediction with or without relativistic correction. It is already
close to the lowenergy region, this explains why a large
RBLM is obtained in Ref. [
13
]. [At the NLO level, the BLM
(5)
(6)
prediction is the same as the PMC prediction if all n f terms
are pertained to αs running.] For this case, a reliable
prediction can only be obtained by using certain lowenergy
αs model, which however will introduce extra model
dependence for the prediction.
Following the idea of PMC, only those {βi }terms that
are pertained to the renormalization of running coupling
should be absorbed into the running coupling. For the
processes involving threegluon or fourgluon vertex, the
scalesetting problem is more involved [
29
]. The MOM scheme
is a physical scheme which is based on the
renormalization of the triplegluon vertex at some symmetric offshell
momentum. The MOM scheme carries information about
the vertex at a specific momentum configuration. This
external momentum configuration is nonexceptional and there
are no infrared issues, thus avoiding the confusion of
distinguishing {βi }terms. Thus to avoid the ambiguity of
applying the PMC on R, similar to the case of QCD BFKL
Pomeron [
30–32
], we shall first transform the results from
the MSscheme to the momentum space subtraction scheme
(MOMscheme) [
33,34
] and then apply the PMC. Another
reasons for choosing the MOM scheme lie in that a
better pQCD convergence can be obtained by using the MOM
scheme than using the MSscheme, and a more reasonable
PMC scale in perturbative region can be achieved.
For the purpose, we adopt the perturbative relation
between the MSscheme running coupling and the
mMOMscheme one as [
35
]
aMS(μ) = amMOM(μ) 1 − 4D1amMOM + · · · ,
where for the Landau gauge, D1 = d1,0 + d1,1n f , d1,0 =
114649 NC , and d1,1 = − 158 . We then obtain
RmMOM(μ)
13π 2
− 2
+ 1 ,
= RmMOM(μ) amMOM 2β0 ln μ2
0 4mc2 + 1
165
+ 2 +
7π 2
3
0 81π2CF amMOM(μ) 2. After applying
where RmMOM(μ) = 2α2 NC
the PMC, we obtain a new PMC scale Q1 = exp(−3d1,1)Q1,
which equals to 1.99 or 1.80 GeV for the prediction with or
without relativistic correction. Such a larger PMC scale
indicates a reliable pQCD prediction can be achieved by using
the mMOM scheme.
To do the numerical calculation, we adopt the cquark
and bquark running masses as the MSscheme ones [
1
]:
mc(mc) = (1.27 ± 0.03) GeV and mb(mb) = (4.18+−00..0043)
GeV. By using the relation between the pole mass m Q and
the MSscheme running mass m¯ Q [
36–39
]:
5 6
µ (GeV)
2
3
4
7
8
9
10
Fig. 1 The ratio R at the NLO level versus the initial choice of μ under
the mMOM scheme. mc = 1.49 GeV. The symbol “−r ” stands for
relativistic corrections. For conventional scale setting, the sensitivity of
μ is very large. After applying the PMC, R is independent to the choice
of μ
we obtain mc = 1.49 ± 0.03 GeV. To be consistent, we adopt
the twoloop αs running, whose behavior is fixed by using
the reference point αs (m Z ) = 0.1181 ± 0.0011 [1]. And we
adopt v2 ηc = 0.430 GeV2/mc2 [
40,41
].
Numerical results of the QCD asymptotic scales MS and
mMOM under Landau gauge are listed in Table 1, where the
errors are dominantly caused by the uncertainty αs (m Z ) =
±0.0011. The asymptotic scales for different schemes satisfy
the relation [
11,35
], mMOM/ MS = exp(2D1/β0).
As a crosscheck, by using the same input parameters, we
obtain the same MSscheme prediction on R under
conventional scalesetting as that of Ref. [
13
]. Due to the reasons
listed above, we shall adopt the mMOMscheme to do our
following discussions.
We present the PMC prediction on R at the NLO level
versus the initial choice of μ in Fig. 1, which is under the
mMOM scheme and both the results before and after
applying the PMC are presented. Under conventional scale setting,
R shows a strong scale dependence which decreases with the
increment of μ. More explicitly, by varying μ from mc to
4mc, the ratio R will change from ∼ 9 × 103 to ∼ 3 × 103.
(7)
(8)
2 x 104
1.8
1.6
1.4
1.2
n f = 5
0.228+0.014
−0.014
0.397+0.025
−0.024
(9)
Conv.
PMC
Conv−r
PMC−r
Conv.
Conv.r
PMC
PMCr
After applying the PMC, the PMC scale Q1 is the same for
any choice of μ, leading to scale independent prediction. The
relativistic correction brings an extra ∼ 2% contribution to
the conventional prediction and ∼ 14% contribution to the
PMC prediction. Thus the relativistic correction is
important, especially for the PMC predictions. Figure 1 shows that
if choosing μ = Q1, the values of R under conventional
scale setting shall be equal to the PMC ones.
After applying the PMC, due to the elimination of
divern
gent renormalon terms as n!β0 αsn, the pQCD series shall be
more convergent. We present the LO and NLO terms of R
before and after applying the PMC in Table 2. We define a
parameter κ = RNLO/RLO to show the relative importance of
the NLOterm and the LOterm. Table 2 confirms that a better
pQCD convergence can be achieved by applying the PMC. A
larger κ and a larger scale uncertainty for each term indicate
that one cannot get the exact value for each term by using a
guessed scale suggested by conventional scalesetting.
Analyzing the pQCD series in detail, we observe that the
scale errors for conventional scalesetting are rather large for
each term, and a possible net small scale error for a pQCD
approximant is due to correlations/cancelations among
different orders. On the other hand, due to the fact that the
running of αs at each order has its own {βi }series governed
by the renormalization group equation, the βpattern for the
pQCD series at each order is a superposition of all the {βi
}terms which govern the evolution of the lowerorder αs
contributions at this particular order. Thus, inversely, the PMC
scale at each order is determined by the known βpattern,
and the individual terms of R at each order shall be well
determined.
We present the theoretical uncertainties for the
conventional and the PMC scale settings in Fig. 2, in which the errors
are squared averages of the ones from the choices of the
cquark pole mass mc and the asymptotic scale mMOM. As a
comparison, the experimental prediction of Ref. [
1
] is also
presented. Under conventional scalesetting, Fig. 2 shows
that the errors caused by mc and mMOM is smaller than the
case of PMC scalesetting, which is however diluted by the
quite large scale uncertainty. For example, the value of R with
[or without] relativistic corrections shall be varied within the
large region of 4.12−+14..5609 × 103 [or 4.21−+15..5066 × 103] for
μ ∈ [mc, 4mc]. Under PMC scalesetting, the scale
uncertainty is greatly suppressed, and the R uncertainty is
dominated by the choices of two parameters mc and mMOM,
which give about 10% contribution to R. The value of R
decreases with the increment of mc, and increases with the
increment of mMOM. More explicitly, we have
where the first error is for mc ∈ [1.46, 1.52] GeV and the
second one is caused by taking mMOM to be the values
listed in Table 1.
Figure 2 shows that the conventional prediction of R with
or without relativistic correction is about 3.6σ deviation from
the data. This discrepancy becomes even larger by including
the NNLO term [
8
], thus the authors there even doubt the
validity of NRQCD theory for this particular observable.2
However, Fig. 2 shows that after applying the PMC, the
pQCD prediction and the data are consistent with each other
within reasonable errors even at the NLO level. The
condition of the branching ratio 1/R is similar. This indicates that
the large discrepancy between the data and the pQCD
predic2 The NNLO results given in Ref. [
8
] cannot be conveniently adopted
for setting the PMC scales. We need to confirm which n f terms are
conformal and which are not, thus the scalesetting procedures are much
more involved. And, we think such a complex NNLO calculation need
to be confirmed by other groups.
tion is caused by improper choice of renormalization scale,
and a simple guessed scale may lead to false prediction or
false conclusion. Thus a proper setting of the renormalization
scale is important for lowerorder predictions.
As a final remark, one may also calculate R by using the
determined predictions on the decay widths (1) and (2)
separately. If taking all input parameters as the central value of our
present choices, we obtain RConv.,mMOM−r ∼ 2.57 × 10−3
and RPMC,mMOM−r ∼ 2.64 × 10−3, both of which are quite
different from our above predictions (11,13). Thus there are
large differences for those two treatments on R, which starts
at αs4order level. Such large differences can be explained by
the weaker pQCD convergence as can be seen from Table 2,
which shall be suppressed by including moreandmore loop
terms. We prefer the usually adopted way of using Eq. (4)
to calculate R, in which the uncertainty from the LDME is
suppressed and there is no factorization scale dependence up
to the NNLO level.
As a summary, in this paper, we have studied the ratio of
the ηc(1S) decay rate into hadrons over its decay rate into
photons by applying the PMC. The PMC provides a
systematic way to set the optimal renormalization scale for high
energy process, whose prediction is free of initial
renormalization scale dependence at any fixed order. A more
convergent pQCD series can be achieved and the residual scale
dependence due to unknown highorder terms are highly
suppressed. Figure 2 shows that the large discrepancy between
the data and the pQCD prediction by using a guessed scale
suggested by conventional scalesetting can be cured by
applying the PMC. The PMC, with its solid physical and
theoretical background, greatly improves the precision of
standard model tests, and it can be applied to a wide variety of
perturbatively calculable processes.
Acknowledgements This work was supported in part by the National
Natural Science Foundation of China under Grant No. 11625520.
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