Determination of αS using s-wave quarkonia decays

UCLEAR PHYSIC~ PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 54A (1997) 247-252

ELSEVIER

D e t e r m i n a t i o n of a:, u s i n g s-wave q u a r k o n i a decays J.H.Field D f p a r t e m e n t de Physique Nuclfaire et Corpusculaire, Universit6 de Gen~ve, 24 quai Ernest-Ansermet CH-1211Gen~ve 4. e-mail : [email protected] a , is determined from T, Y/¢ and ~?c decay branching ratios. Relativistic corrections, scale dependent pQCD corrections and non-perturbative corrections derived from the T or J/'¢ inclusive photon spectra are included. The latter, corresponding to 'effective gluon mass' parameters ~_ 1 GeV are essential for a consistent overall description of the data.

1. I n t r o d u c t i o n As first suggested by Applequist and Politzer [1] quarkonium decays into final states containing light hadrons or a photon and light hadrons can, in principle, provide a precise determination of a , at scales of the order of the heavy quark mass. At lowest order in p Q C D the decay V --* light hadrons (V = J/¢,T) is expected to occur via the process shown in Fig.1 Factorisation occurs

Q

g

13Trlq'ITI 1

r,,,1/mQ

r ~ 1 / vmQ

Figure 1. Factorisation (dashed line) between the heavy quark bound state and the hard annihilation amplitude in the process V ~ ggg.

between the long distance physics described by the bound state wave function and the short distance annihilation process caleulable in pQCD. In the static limit the partial decay width is then: 2 - 9)

P(V ~ ggg) =

81M~

I¢(°)1=

(1)

0920-5632/97/$17.00 © 1997 Elsevier Science B.V All rights reserved. PII: S0920-5632(97)00049-2

where ¢(0) is the spatial heavy quark wave function at the origin. A similar factorisation occurs in the QED process V --~ e+e - with the analogous result:

r(v

ee) =

16repot(My) 2 1¢(°)12

(2)

The ratio

Rv

r(v

ao(

-

- 9)

(3)

in which the long distance physics contained in ¢(0) cancels, provides a sensitive estimator for a , . The experimental values of Rv as well as those of other branching ratios R~, Rnc also sensitive to a , [2] are summarised in Table 1. Static limit formulae for the relevant decay widths m a y be found in Ref[3]. As will now be shown, the simple picture of Eqns.(1-3) is strongly modified when relativistic corrections, higher order QCD corrections and non-perturbative effects related to the infra-red cut-off of p Q C D are taken into account. The goal of the present study is to see whether consistent values of o~s can be obtained from the five branching ratios RT, R), Rj/¢, R}/¢ and R,7~. The last systematic study of this type including relativistic corrections and scale dependence was that of Kobel [4] in 1992. 2. R e l a t i v i s t i c C o r r e c t i o n s Relativistic corrections to the static limit formulae (1),(2) arise because the annihilation process does not occur at a fixed point as described

248

JH. Field~Nuclear Physics B (Proc. Suppl.) 54,4 (1997) 247-252

Table 1 Branching ratios used for as determinations [2]. BR Definition Expt. Value

R~ R~

r(ggg)/r(ee) r(999)/r(~) r(~g)/r(ee)

32.6 -4- 0.8 35.7 -4- 1.7 lO.1 + 0.9 5.4 +2.1 -1.6 ( ~ 2 7 ± 0.54) ×~o ~

R,/¢ R)/,~ r(ggg)/r(Tgg) R~o

r(99)/r(,~)

3. O ( a s ) Q C D C o r r e c t i o n s The O ( a , ) correction factor for F(V---* ggg)

is [s] fQCD = 1 + a ' ( # ) B(I.t ) where

S(#) = 3fi02 ln(2-~) - 0.26 - 1.16~s ~o = 11

by

I¢(0)1 ~,

(6)

7r

2n! 3 '

n! = 4 (T),

n! = 3 ( J / C )

but rather over a volume of radius 1 / m q . The main effect as calculated for example by Keung and Muzinieh [5] is a modification of the hard annihilation amphtude. The correction factor for the partial widths is of the form f~z where:

and # is the renormalistation scale.A similar formula applies for r(n -~ gg) [9]. For V -~ ee the corrections are purely virtual and independent of # [10,11]:

f,e, = (1 + C,.,,(v2}) 2

fQcD = ( 1 - ~ , ( m Q ) )

(4)

and

Only transverse contributions from gluon exchange between QQ are included in (6),(7). The validity of this assumption in the presence of relativistic corrections is discussed in the following Section.


C,,,i

6 =

(V ---} ee,

2.16

~l---* gg,

( y ~ ~gg,

r/---*77)

y ~.r~9)

In consequence, relativistic corrections cancel to O(v 2) in R~,, and Rno. Keung and Muzinieh also calculated the O(v 2) correction to the inclusive photon spectrum in V ---+ 7gg. Two recent calculations [6,7] confirmed the results of Ref[5] and evaluated f,,l using wave functions derived from potential models. In the following the results of Ref[7] are used: f;R~frRs/'¢' el

=

0.67:1:0.05

=

0.22 + 0.04

(7)

The errors ate estimated from the spread of (v 2) values for the T and J / ¢ in the literature (mainly from calculations of relativistic corrections to V ---, e + e - ) . As proposed in an ad hoc manner in Ref[8], the relativistic corrections, especially for the J / ¢ , are large.

4. D o u b l e c o u n t i n g o f R e l a t i v i s t i c pQCD Corrections

and

In potential models the bound state properties are determined by a potential of the form 4O~s + kT 3r

v(T) : - - - -

(S)

The first term in Eqn(8) describes perturbative one gluon exchange (OGE) between the Q and the Q, the second the long-range confinement potential. The transverse part of OGE also contributes to fQCD. The question of possible double counting of OGE effects in the wave function and fQCD then arises. Unlike in the analagous QED case (positronium), there is no exact solution to this problem in perturbation theory for e,b quarkonia, since the wave function is largely determined by the confining potential, not by OGE. Two studies of combined relativistic and QCD corrections have been made for the process J / ¢ -~ ee by iterating the OGE potential in

J.H. Field~Nuclear Physics B (Proc. Suppl.) 54A (1997) 247 252 the Bethe-Salpeter equation. In the first [12] it was found that 1¢(0)12 in Eqn(2) is replaced by I¢(r = 1/mc)l 2 and fOOD in Eqn(7) by the much smaller value ( 1 - 0 . 3 6 a , ( m c ) ) . In the second [13] an exphcit formula was given for the combined relativistic/QCD correction factor:

frel+QCD = ( 1 _

(~)

[1

@2)6 C~''((V2))] (9)

This formula reduces to the result of Ref[5] when a , = 0, and to that of Eqn(7) when (v ~) = 0, i.e. 6(0) = 16/3~-. However the actual value of 5 for the J / ¢ with a , = 0.3, /v 2) = 0.28 is such that frel+QCD

=

249

Table 2 Values of the QCD correction coefficient B(m~) for different factorisation ans£tze. Mode Full Hard J / ¢ ~ ggg -3.7 19.0 J / ¢ ---* 7gg -6.7 16.0 J / ¢ ~ ee -5.33 0.0 rlc ~ gg 4.85 9.54 r/c --~ 3'7 -3.38 1.31 Rj/¢ 1.59 19.0 RI 3.0 3.0 Rnc

8.23

8.23

0.84

whereas

ggg, Tgg, but guided by the J / ¢

fret = frel+qCD(O~, = O) = 0.91 and fQCD = het+QC~'( = 0) = 0.49 Naive factorisation, as used in Refs[6,7] for R j / ¢ , R r predicts a correction factor of

f~et x fQcD = 0.45 instead of the actual value of 0.84. The results of Refs[12,13] then indicate that for the case of J / ¢ ~ ee naive factorisation results in strong double counting of the overall correction. A good first approximation since frel+QCD ~-- fret is to set fQCD = 1 if naive factorisation is used.

()

---* ee example, double counting may be avoided by introducing a new factorisation ansatz for the QCD corrections. Instead of factorising into the bound state wave function only the Coulombic part of the O G E contribution, the entire OGE, as well as the quark self-energy contributions, are factorised into the wave function. This defines the ' H a r d ' factorisation ansatz, in contrast to the conventional 'Full' one where these contributions are included in fQCD (see Fig2 ). Values of the coefficient B(mc) defined as in Eqn(6) for different J / ¢ and rk decay widths and branching ratios are shown, for the Full and Hard factorisation ans£tze in Table 2. fQCD is large and positive for Hard factorisation for final states containing t:R J / ,~ gluons, and there is a large difference in SQVD R'

E Full

for the two ans£tze. ~ s/~ ]QCD "R'c however are inJ QCD, dependent of factoristation ansatz to 0 ( ~ ) since the virtual corrections are the same for the numerator and denominator of the branching ratio.

Ha r d 5. T h e d e c a y s T, J / ¢ ---*7 l i g h t h a d r o n s

Figure 2. The Hard and Full factorisation ans£tze for the O G E contribution in V ~ ggg.

No exphcit calculation of combined relativistic and QCD corrections has been made for V

At lowest order in p Q C D these processe are described by the diagram of Figl with one gluon replaced by a photon. Three different calculations of QCD corrections have been made. The first [14] considered modifications to the spectrum in the large z limit

J.H. Field~NuclearPhysicsB (Proc. Suppl.) 54A (1997) 24~252

250

Table 3 Parisi-Petronzio correction factors for Mg = 1.174-0.08 GeV (T), Mg = 0.664-0.08 GeV (J/'¢) BR RT

Rj/¢ R~c

RMV / Rn,

I

dr

fMs 0.71 4- 0.03 0.40 + 0.12 0.89 4- 0.03 0.60 4- 0.18 0.69 4- 0.06

(z =_ 2E.r/Mv ). The leading terms _~ ln(1 - z) were summed to all orders in a~. The second calculation [15] was a parton shower model with a non-perturbative gluon mass cut-off ~_ 0.45 GeV. The last calculation is that of Parisi and Petronzio [16] who introduced a non-perturbative gluon mass _~ 0.8 GeV to explain the soft spect r u m observed in J/¢ decays. The effect of such a non zero gluon mass on decay widths was also calculated. Fits [2] to the experimental inclusive photon spectra [17-19] using this model gave the following values for the non-perturbative 'effective glnon mass' Mg: Mg --

1.17 4- 0.08GeV

(T)

=

0 . 6 6 ± 0.08GeV

(J/e)

The corresponding correction factors fM. for different decay modes from Ref[16] are presented in Table 3. The different theoretical predictions for the photon spectra are compared in Fig.3 The curves B and C show the results of the fits with Mg ~ 0 for the T, J/¢ respectively and give a good representation of the experimental data. Both the calculation of Ref[14] and the QCD Born term with massless gluons are completely excluded by both the charmonium and the b o t t o m o n i u m data. The model of Rei~15] gives a good description of the T spectrum, but predicts a too hard spectrum for the J / ¢ . Relativistic corrections [8] also result in a softening of the spectra, but as shown in Fig.4 for the J / ¢ [17] the effects are modest and cannot explain the discrepencies between experiment and

0.2

o.z.

o.6

o.a ~.

Figure 3. Theoretical predictions for inclusive photon spectra in V --* 7X. A : Born term with massless gluons, dashed line : Ref[14], dotted line : Rei~15], B : Ref[2,16] (T, M9 = 1.17 GeV), C : Ret~2,16] (J/¢, Mg = 0.66 CeV).

Or,,

0.3 N~ot d~

~ dN

0.2

00~

05

0.7

09

1.1

Figure 4. Inclusive photon spectrum in J/¢ ---* 7X. D a t a : Ref[17]. Solid curve : Rei~2,16] ( Mg = 0.66 GeV). Dotted curve: Ref[15]. Dashed curve : Ref[15] with relativistic correction from

Re 5] ( (v 2) : 0.28).

all theoretical calculations except that of Parisi and Petronzio. 6. D e t e r m i n a t i o n

of a,

For each branching ratio and choice of renormalisation scale ~, a , ( # ) is determined by numerical solution of the appropriate equation. For example: Rv = _-

1 0 ( 7r2 - 9 ) ( a ~ )

3

FA

S6cD(

c~ #

(10)

J.H. Field~NuclearPhysics B (Proc. Suppl.) 54A (1997) 247-252

251

Table 4 Confidence levels of a , determinations. T decays. No solution exists for RT, with # = mb/4, Full Fact.

M # "~4

=0

Full

mb 1 × 10 -8 [ 4 × 10 -4 ] 2mb

M Hard 0.001 [ 2 × 10 -s] 0.01 [ 0.0045] 0.008 [ 0.024]

0.16 [ 0.38 ]

#0

Full 1 × 10 -4 [ 2.1 X 10 -4 ] 0.094 [ 0.026]

Hard 0.008 [ 0.0015 ] 0.035 [ 0.082] 0.015 [ 0.039]

Table 5 Confidence levels of as determinations. J / ¢ , ~c decays. No solution exists for R j/e, with # = mc/4, Full Fact.

Mg = O #

Full

4 me 7 × 10 -7 [ 2 × 10 -7 ] 2m~ 1 × 1 0 - 4 [ 2 × 10 -4 ]

Mg # O

Hard 0.09 [ 2.5 x 10 -5 ] 0.02 [ 3.0 x 10 -3 ] 0.012 [ 0.027]

Table 6 Results for a,(mQ). Hard factorisation, # = mQ including Parisi-Petronzio gluon mass correction factors. Errors : experimental, in f M , , in f~el

as(mQ)

BR RT

0.1845 =k 0.0013 + 0.0025 =£ 0.0046 0.215 + 0.010 =k 0.009 0.307 ± 0.007 ± 0.030 ± 0.018 0.19 + 0.07 =t= 0.06 0.236 + 0.039 ± 0.010

Rj/~p Rn¢

Here ' F A ' denotes the factorisation ansatz (Full or Hard). a , ( m Q ) is then calculated from the one-loop evolution equation: 1

1

-

~01n (

(11)

where m q -- M y / 2 . To check the internal consistency of the five different a , determinations the CL (X 2 probability) t h a t the values are consistent with their weighted m e a n is calculated. T h e overall consistency with p Q C D tested by c o m p a r i n g the a , values with those determined in Deep Inelastic Scattering (DIL) [20] evolved, using two loop evolution equations, to the scale mQ:

a,(mb) OIL

~- 0.199 + 0.006 fi= 0.012

as(me) OIL

:

0.303 + 0.017 =k 0.023

Full 0.0024 [ 0.0066 ] 6 X 10 -4 [ 0.0065 ]

Hard 0.71 [0.14] 0.29 [ 0.35 ] 0.15 [ 0.23]

T h e first error is experimental, the second theoretical. Consistency with p Q C D is tested by calculating the CL as described above, including also the DIL values in the weighted mean. In the calculation of the CL the errors on f ~ l and fMg a r e taken into account as well as the exprimental errors on the branching ratios. T h e CL results for the different factorisation ans£tze, different renormalisation scales and inclusion or non-inclusion of the Parisi-Petronzio correction factor fM,, are presented in Table 4 for the T and in Table 5 for the J/¢, ~k. T h e first entry is the CL for consistency of the branching ratios alone (R:r, R ~ in Table 4, R j/e, R)/¢ and Rnc in Table 5), the second, in square brackets, includes also a, (mQ)DIL. Only for Hard factorisation and Mg ¢ 0 is a globally consistent solution with CLb~ x CLce > 10 -2 obtained. Consistent solutions for the T with # = 2rob, Full factorisation and either Mg = 0 or Mg ¢ 0 are also found, but the same p a r a m eter choices for the J/¢, ~k give a CL < 10 -3. T h e r e n o r m a h s a t i o n scale dependence reflects the possible effect of uncalculated higher order corrections. For Mg = 0 these are expected to be similar for c h a r m o n i a and b o t t o m o n i a . T h e a , values for the consistent solution with # = mQ, H a r d factoristion and Mg # 0 are presented in Table 6. T h e corresponding weighted

252

JH. Field~Nuclear Physics B (Proc. Suppl.) 54A (1997) 24~252

averages yield the results: a,(mb)

=

0.1887± 0.0013± 0.0048+ 0.0219 --0.0148

o~,(mc)

=

Acknowledgement I should like to thank my collaborator M.Consoli for many insights into the apphcation of QCD to quarkonia decays.

0.269± 0.007± 0.025+ 0.066 --0.046

The errors are respectively experimental, theoretical, (in f~el and fMg) and due to renormalisation scale dependence. For as (mb) the range of scales considered is rob~2 < # < 2rob and for a , ( m c ) , m ~ / 4 < tt < 2me. 7. C o n c l u s i o n s It has been shown that only when important non-perturbative effects, parameterised by an 'effective gluon mass ' M s "~ 1 GeV for each heavy quark flavour, are taken into account is a consistent description of both branching ratios and inclusive photon spectra possible. It has been previously shown [21,22] that a gluon mass parameter can, quite generally, replace AQCD as the infra red cut-off parameter of pQCD. The basic QCD parameters are then [22] an effective on-shell coupling constant a , ( 0 ) = 0.30 4- 0.05, a gluon mass mg = 1.5 + 1.2 - 0.06 GeV and standard current quark masses at a scale _~ m s. In fact at one loop the relation:

( (nq = (m~ m d m , mcrnb ) ~ yields A(~)CD : 266 MeV, consistent with conventional pQCD phenomenology. At scales < mg, AQGD is not a constant, but a calculable [21] scale dependent parameter, a~ then has no singularity at scales _ AQCD but rather, (as in QED [22]) 'freezes' at a constant value _~ a,(0). See Ref[2] for a discussion of the relation between M s and m s• The main conclusion of this work is that the scale at which conventional asymptotic pQCD breaks down is not (as often conjectured in the literature [23] ) _~ AQCD but rather m s ___ 1.5 GeV.

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