Table Of ContentThe π2 terms in the s–channel QCD observables
D.V. Shirkov
Bogoliubov Laboratory, JINR, 141980 Dubna, Russia
e-address: [email protected]
Abstract
We analyze the effect of π2–terms in the QCD perturbative expansions for the
1
0 s–channel effective coupling and observables, the effect known from the 80s. We
0 remindthatthesetermscanbecollected intospecificfunctions—strongs–channel
2
coupling α˜(s) and its effective powers A (s) free of ghost singularities. Further on,
k
n
westudythestructureofperturbationtheoryforobservablesanditsreformulation
a
J in terms of nonpower perturbation expansion over the set A (s) .
k
{ }
3 Then we discuss the influence of this effect on the numerical values of α¯ as
s
2
extracted from experiments. The main result is that the common two-loop (NLO,
2 NLLA) approximation widely used in the five-quark (10 GeV . √s . 170 GeV)
v region for a shape analysis contains a systematic negative error of a 1–2 per cent
6 order of magnitude for the extracted α¯(2). Our physical conclusion is that the
0 s
1 α¯ (M2) value averaged over the f = 5 data
s Z
9
0 < α¯ (M2) > 0.124
s Z f=5
0 ≃
0
appreciably differs from the currently accepted “world average” (= 0.118).
/
h
p
-
p
1 Preamble
e
h
:
v Usually, physical quantities in the time-like channel, like the cross-section
Xi ratio of the inclusive e+e− hadron annihilation or the τ–decay process,
→
r are presented in the form of two- or three-term perturbation expansion
a
R(s)
= 1+r(s); r(s) = c α¯ (s)+c α¯2 +c α¯3 +... (1)
R 1 s 2 s 3 s
0
(our coefficients c = C π−k are normalized differently from the commonly
k k
adopted, like inRefs.[1,2,3]) over powers of effective QCDcoupling α¯ which
s
is supposed ad hoc to be of the same form as in the Euclidean domain, e.g.,
1 b lnL 1
α¯(3)(s) = 1 + b2(ln2L lnL 1)+b ;
s β L − β2 L2 β3L3 1 − − 2
0 0 0 (cid:2) (cid:3)
1 5 1 b
+ b3 ln3L+ ln2L+2lnL 3b b lnL+ 3 .
β4L4 (cid:20) 1(cid:18)− 2 − 2(cid:19)− 1 2 2 (cid:21)
0
Here, L = ln(s/Λ2) and for the beta-function we use normalization
β(α) = β α2 β α3 β α4 +... = β α2 1+b α+b α2 +... ,
0 1 2 0 1 2
− − − −
(cid:0) (cid:1)
1
that is also free of π powers. Numerically,
33 2f 153 19f
β (f) = − ; b (f) = − ; b (4 1) = 0.490−0.089.
0 12π 1 2π(33 2f) 1 ± +0.076
−
Coefficients c = d δ include “π2 structures” δ proportional to
k≥3 k k k
−
lower c :
k
(πβ (f))2c 5
δ = 0 1 , δ = (πβ )2(c + b c ); π2β2(4 1) = 4.340−.666. (2)
3 3 4 0 2 6 1 1 0 ± +.723
These structures δ arise[4, 5, 6, 7] in the course of analytic continuation
k
from the Euclidean to Minkowskian region. Coefficients d should be treated
k
as a genuine kth–order ones. Just they have to be calculated with the help
of relevant Feynman diagrams.
To illustrate, consider the three–flavor case for τ–decay, f = 4,5 cases
for e+e− hadron annihilation and Z decay (with f = 5) — see Table 1 in
0
→
which we also give values for the π2–terms.
Table 1
Process f c c = d c d = c δ δ δ
1 2 2 3 3 3 3 3 4
−
τ decay 3 1/π .526 0.852 1.389 0.537 5.01
e+e− 4 .318 .155 -0.351 0.111 0.462 2.451
e+e− 5 .318 .143 -0.413 -0.023 0.390 1.752
Z decay 5 .318 .095 -0.483 -0.094 0.390 1.576
0
Here, all coefficients c , d and δ , due to normalization (1), are of an
k k k
order of unity. One can see that, in the high energy region, contribution of
δ prevails in c .
3 3
2 Preliminary quantitative estimate
In practice, the π2–terms often dominate in higher expansion coefficients.
This effect is especially strong in the f = 5 region. Meanwhile, just in
this region people often use the so-called NLLA approximation, that is the
two-term representation
O(s) = C (α¯ /π)+C (α¯ /π)2 (3)
1 s 2 s
foranobservableO(s)whennext, thethree-loop, coefficientC isnotknown.
3
This is the case, e.g., with event–shape[8] analysis.
On the basis of the numerical estimates of Table 1, in such a case, we
recommend to use the three-term expression
π2β2
O3∆(s) = d1(cid:26)α¯s − 3 0α¯s3(cid:27)+d2α¯s2 = c1α¯s +c2α¯s2 −δ3α¯s3 (4)
2
i.e., to take into account the known predominant π2 part of the next coeffi-
cient c . As it follows from the comparison of the last expression with the
3
previous, two–term one, the α¯ numerical value extracted from eq.(4), for
s
the same measured value O , will differ by a positive quantity (e.g., in the
obs
f = 5 region with α¯ 0.12 0.15)
s
≃ ÷
πδ α¯3 f=5 1.225α¯3
( α¯ ) = 3 s = s 0.002 0.003
s 3
△ 1+2πd α¯ (cid:12) 1+0.90α¯ ≃ ÷
2 s(cid:12)20÷100GeV s
(cid:12)
(cid:12)
that turns to be numerically important.
Moreover, in the f = 4 region, where the three-loop approximation is
commonly used in the data analysis, the π2 term δ of the next order turns
4
out also to be essential. Hence, we propose to use the four-term expression
O4∆(s) = d1α¯s +d2α¯s2 +c3α¯s3 −δ4α¯s4; c3 = d3 −δ3 (5)
(instead of the three-term one (1)) that is equivalent to
π2β2 5
O∆(s) = d α¯ 0α¯3 b π2β2α¯4 +d α¯2 π2β2α¯4 +d α¯3 (6)
4 1(cid:26) s− 3 s − 16 0 s(cid:27) 2 s − 0 s 3 s
(cid:8) (cid:9)
with δ and δ defined[4, 7] in eq.(2).
3 4
The three– and two–term structures in curly brackets are related to spe-
cific expansion functions α˜ and A defined below (10) and entering into the
non-power expansion (11).
Toestimateroughlythenumericaleffectofusingthislastmodifiedexpres-
sion (5), we take the case of e+e− inclusive annihilation. For √s 3 5GeV
≃ ÷
with α¯ 0.28 0.22 one has
s
≃ ÷
πδ α¯4 f=4 1.07α¯4
( α¯ ) = 4 s = s 0.005 0.002
s 4
△ 1+2πd α¯ (cid:12) 1+0.974α¯ ≃ ÷
2 s (cid:12)3÷5GeV s
(cid:12)
(cid:12)
— an important effect on the level of ca 1 2%.
÷
Moreover, the ( α¯ ) correction turns out to be noticeable even in the
s 4
△
lower part of the f = 5 region! Indeed, at √s 10 40 GeV with α¯
s
≃ ÷ ≃
0.20 0.15 we have
÷
( α¯ ) f=5 0.71α¯4 (1.1 0.3) 10−3 (. 0.5%).
△ s 4|10÷40GeV ≃ s ≃ ÷ ·
3 Non-power expansion in the Minkowskian region
The so–called π2 terms in the s–channel perturbative expansions for the
invariant coupling and observables have a simple origin.
3
As it is well known, the usual invariant coupling originally defined [9]
in terms of real constants z , counter-terms of finite Dyson renormalization
i
transformation, can be expressed via a product of dressed symmetric vertex
and propagator amplitudes taken at space-like values of their arguments.
α¯(Q2,α) = αΓ2(Q2,α) d (Q2,α).
i
Yi
Hence, by construction, it is a real function defined in the Euclidean region.
Transition to the time-like region, with logs branching lnQ2 lns iπ
→ −
transforms all relevant amplitudes into complex functions Γ(s,α),d (s,α).
i
Here, the problem of appropriate defining of effective coupling in the time-
like domain arises.
Forthisgoal,weshallfollowtheideadevisedintheearly80sbyRadyushkin
[4]andKrasnikov–Pivovarov [5]. There, anintegraltransformation Rreverse
to the dipole representation for the Adler function has been used.
We propose to treat this representation as an integral operation
∞ ds
R(s) D(z) = Q2 R(s) D R(s) (7)
→ Z (s+z)2 ≡ { }
0
transforming a function R(s) of a real positive (time-like) argument into a
function D(z) given in the cut complex plane with analytic properties equiv-
alent to those following fromtheK¨allen–Lehmann integralrepresentation. In
particular, the function D(Q2) is real on the positive (space-like) real axis
at z = Q2 +i0;Q2 0.
≥
The reverse operation is expressible in the form of a contour integral
i s+iε dz
R(s) = D ( z) R D(Q2) .
pt
2π Z z − ≡
s−iε (cid:2) (cid:3)
With the help of the latter, one can define[11, 12] an effective invariant
time-like coupling α˜(s) = R[α¯ (Q2)] . Omitting some technical details, we
s
give a few resulting[4, 5, 12] expressions.
E.g., starting with one–loop α¯(1) = [β ln(Q2/Λ2)]−1 one has R α¯(1) —
s 0 s
h i
1 1 1 L 1 π s
α˜(1)(s) = arctan = arctan ; L = ln . (8)
β (cid:20)2 − π π(cid:21) β π L Λ2
0 L>0 0
2 3
At the same time, to α¯(1)(Q2) and α¯(1)(Q2) there correspond
s s
(cid:16) (cid:17) (cid:16) (cid:17)
1 L
A(1)(s) R α¯(1) 2 = and A(1)(s) = .
2 ≡ h(cid:0) s (cid:1) i β02[L2 +π2] 3 β03[L2 +π2]2
4
In the two–loop case, for a “popular” expression
1 lnl Q2
β α¯(2) (Q2) = b (f) ; l = ln
0 s,pop l − 1 l2 Λ2
one obtains[4] the two-loop “pop” effective s–channel coupling
b L b ln √L2 +π2 +1
α˜(2)(s) = 1+ 1 α˜(1)(s) 1 . (9)
pop (cid:18) L2 +π2(cid:19) − β (cid:2) L2 +π2(cid:3)
0
Both the expressions (8) and (9) are monotonically decreasing with a finite
IR α˜(0) = 1/β (f = 3) 1.4 value. Meanwhile, higher functions go to the
0
≃
zero A (0) = 0 at the IR limit.
k
In the case L π, it is possible to expand α˜ and A in powers of π2/L2.
k
≫
Thenfunctionsα˜ andA canbepresentedasexpansionsinpowersofcommon
2
α¯ 1/L. They correspond to curly brackets in (6).
s
≃
In [4, 5], as a starting point for observables in the Euclidean, i.e., space-
like domain Q2 > 0, the perturbation series
D (Q2) = 1+ d α¯k(Q2)
pt k s
Xk≥1
has been assumed. It contains powers of usual, RG summed, invariant cou-
pling α¯ (Q2) that obeys unphysical singularities in the infrared (IR) region
s
around Q2 Λ2.
≃ 3
By using the R transformation, we obtain in the Minkowskian region the
“transformed” expansion over a non-power set of functions
R (s) R D (Q2) = 1+ d A (s); A (s) = R α¯k(Q2) (10)
π ≡ pt k k k s
(cid:2) (cid:3) Xk≥1 (cid:2) (cid:3)
free of the mentioned singularities. Properties of these functions have been
analyzed in detail in our previous paper[13] — see also Ref. [14]. For a more
detailed numerical information on the functions α˜ , A and A see Ref.[15].
2 3
Here, we give condensed information that will be enough for a few illus-
trations.
Table 2
Three-loop APT results for Λ(5) = 290GeV; α¯ (M2) = 0.125
MS s z
√s/GeV 5 10 15 20 30 50 60 90 150
α¯ (s) .235 .195 .177 .165 .153 .137 .133 .125 .115
s
α˜(s) .221 .186 .170 .160 .148 .136 .132 .123 .114
10A .456 .330 .275 .246 .214 .180 .169 .149 .129
2
100A .871 .555 .436 .357 .299 .232 .213 .177 .143
3
5
Both in the Figure 1 and in Table 2, we give 3-loop solutions for α¯ as
s
well as for the modified, so–called global (for detail, see paper [13]) functions
α˜ = A , A and A calculated within the MS scheme for the cases Λ =
1 2 3 (5)
215GeV, α¯ (M2) = 0.118 and Λ = 290GeV, α¯ (M2) = 0.125.
s Z (5) s Z
(cid:19)(cid:15)(cid:23)
a (cid:11)4(cid:21)(cid:12)
V
(cid:19)(cid:15)(cid:22)
a
(cid:11)V(cid:12)
a (cid:21)
(cid:11)4 (cid:12)
DQ
(cid:19)(cid:15)(cid:21)
(cid:20)(cid:18)(cid:21)
V 4(cid:3)(cid:11)*H9(cid:12)
(cid:19)(cid:15)(cid:20)
(cid:20) (cid:20)(cid:19) (cid:20)(cid:19)(cid:19)
Figure 1: Effective global Minkowskian, α˜ , and Euclidean, α expansion
an
functions, as compared with the standard one α¯ (at Λ = 350MeV and
s (5)
α¯ (M2) = 0.118).
s Z
Wehavechosenthesetwocasesaslimitingonesasfarasinmanypractical
cases real figures lie between these limits.
In the first figure we give three curves α¯ , α˜ and α related to the same
s an
physical case for Λ = 350MeV and α¯ (M2) = 0.118. The curves α˜ and α
3 s Z an
on the figure go a bit slanting than usual, the α¯ , dotted curve. This is quite
s
natural, as they both are regular in the vicinity of the Λ singularity.
Meanwhile, only two first, α˜ and α have direct physical meaning (com-
an
pare with conclusion of [13]). Just their values have to be determined from
any given experiment. Nevertheless, in the four- and five–flavour regions one
can still refer to α¯ and α¯ (M2) as to traditional theoretical objects.
s s Z
Now, instead of (1), with due account to (10), we have
α˜(s)
r(s) = +d A (s)+d A (s) (11)
2 2 3 3
π
6
with beautifully decreasing coefficients d . Just this nonpower expansion,
k
strictly speaking, should be used instead of its approximations, eqs.(4) and
(6), for data analysis in the time-like region.
At the same time, in the Euclidean, we have also non-power expansion
α (Q2)
d(Q2) = an +d (Q2)+d (Q2) (12)
2 2 3 3
π A A
that can be related to (11) by transformation (7) in the framework of Invari-
ant Analytic Approach (refs.[16, 17]).
These non-power expansions, free of unphysical singularities, jointly form
a correlated system. The latter has been studied in detail in Refs.[13] and
[18]. We call it Analytic Perturbation Theory (APT).
4 Numerical illustrations
To illustrate, let us start with a few cases in the f = 5 region.
To begin with, consider the Υ decay. According to the Particle Data
Group (PDG) overview (see their Fig.9.1 on page 88 of Ref.[1]), this is (with
α¯ (M ) 0.170 and α¯ (M2) = 0.114) one of the most “annoying” points
s Υ ≃ s Z
of their summary of α¯ (M2) values. It is also singled out theoretically. The
s Z
expression for the ratio of decay widths starts with the cubic term
R(Υ) = R α¯3(M )(1+e α¯ ) with e 1. (13)
0 s Υ 1 s 1 ≃
Due to this, the π2 correction1 is rather big here
A α¯3 1 2(πβ )2α¯2 . (14)
3 ≃ s − 0 s
(cid:0) (cid:1)
Accordingly,
2
∆α¯ (M ) = (πβ )2α¯3(M ) 0.0123,
s Υ 3 0 s Υ ≃
that corresponds to
∆α¯ (M ) = 0.006 with α¯ (M ) = 0.120. (15)
s Z s Z
Now, let us turn to a few cases analyzed by the three-term expansion
formula (1). For the first example, take e+e− hadron annihilation at √s =
42GeV and 11GeV.
A common form (see, e.g., Eq.(15) in Ref.[2]) of theoretical presenting of
the QCD correction in our normalization looks like
re+e−(s) = 0.318α¯s(s)+0.143α¯s2−0.413α¯s3. (16)
1
First proposalof taking into account this effect in the Υ decay was discussed[5] more
than a quarter of century ago. Nevertheless, in current practice it is neglected.
7
Starting with re+e−(42) 0.0476, one has α¯s(42) = 0.144. Along with our
≃
new philosophy, one should use instead
re+e−(s) = 0.318α˜(s)+0.143A2(s) 0.023A3(s) (17)
−
that yields α˜(42) = 0.142 with α¯ (42) = 0.145 and α¯ (M2) = 0.127 to be
s s Z
compared with α¯ (M2) = 0.126 under a usual analysis.
s Z
Quite analogously, for re+e−(11) 0.0661; α¯s(11) = 0.200, we obtain
≃
α˜(10) = 0.190 that corresponds to α¯ (M2) = 0.129 instead of 0.130.
s Z
For the next example, we take the Z inclusive decay. Experimental ra-
0
tio R = Γ(Z hadrons)/Γ(Z leptons) = 20.783 .029 is usually
Z 0 0
→ → ±
presented as follows: R = R (1+r (M2)) with R = 19.93. A common
Z 0 Z Z 0
form (see, e.g., Eq.(15) in Ref.[2]) of presenting of the QCD correction in our
normalization looks like
r (M2) = 0.3326α¯ +0.0952α¯2 0.483α¯3.
Z Z s s − s
To [r ] = 0.04184 there corresponds α¯ (M2) = 0.1241 with Λ(5) =
Z obs s Z MS
292MeV. In the APT case,from
r (M2) = 0.3326α˜(M2)+0.0952A (M2) 0.094A (M2) (18)
Z Z Z 2 Z − 3 Z
we obtain α˜(M2) = 0.122 and α¯ (M2) = 0.124 that relates to Λ(5) =
Z s Z
290MeV. Note that here the three-term approximation of (6) gives the same
relation between the α¯ (M2) and α˜(M2) values.
s Z Z
Nevertheless, inaccordancewithourpreliminary estimate forthe ( α¯ )
s 4
△
role, even the so-called NNLO theory needs some π2 correction in the W =
√s . 50GeV region.
Now, turn to the experiments in the HE Minkowskian (mainly with a
shape analysis) that usually are confronted with two-term expression (3). As
it has been shown below, the main theoretical error in the f = 5 region can
be expressed in the form
( α¯ (s) f=5 1.225α¯3(s) 0.002 0.003. (19)
△ s |20÷100GeV ≃ s ≃ ÷
An adequate expression for the shift of an equivalent α¯ (M2) value is
s Z
[ α¯ (M2)] = 1.225α¯ (s)α¯ (M2)2. (20)
△ s Z 3 s s Z
We give results of our approximate APT calculations, mainly by Eqs.(19)
and(20), inthe formof Table3 andFigure2. At thelast column ofthe Table
3 in brackets we indicate difference between the APT and usual analysis. By
bold figures the results of the three–loop analysis are singled out.
8
Table 3
a
The APT revised part (f = 5) of Bethke’s[2] Table 6
√s loops α¯ (s) α¯ (m2) α¯ (s) α¯ (m2)
s s z s s z
Process GeV No ref.[2] ref.[2] APT APT
b
Υ-decay 9.5 2 .170 .114 .182 .120 (+6)
e+e−[σ ] 10.5 3 .200 .130 .198 .129(-1)
had
e+e−[j&sh] 22.0 2 .161 .124 .166 .127(+3)
e+e−[j&sh] 35.0 2 .145 .123 .149 .126(+3)
e+e−[σ ] 42.4 3 .144 .126 .145 .127(+1)
had
e+e−[j&sh] 44.0 2 .139 .123 .142 .126(+3)
e+e−[j&sh] 58 2 .132 .123 .135 .125(+2)
Z had. 91.2 3 .124 .124 .124 .124 (0)
0
→
e+e−[j&sh] 91.2 2 .121 .121 .123 .123(+2)
e+e−[j&sh] 133 2 .113 .120 .115 .122(+2)
e+e−[j&sh] 161 2 .109 .118 .111 .120(+2)
e+e−[j&sh] 172 2 .104 .114 .105 .116(+2)
e+e−[j&sh] 183 2 .109 .121 .111 .123(+2)
e+e−[j&sh] 189 2 .110 .123 .112 .125(+2)
Averaged < α¯ (M2) > values 0.121; 0.124;
s z f=5
a
“j & sh” = jets and shapes; Figures in brackets in the last column give the
difference ∆α¯s(MZ2) between common and APT values.
b
Taken from Ref.[1].
Let us note that our average over events from Table 6 of Bethke’s review
[2] nicely correlates with recent data of the same author (see Summary of
Ref.[19]). The best χ2 fit yields α¯ (M2) = 0.1214 and α¯ (M2) =
s Z [2] s Z APT
0.1235. This gives minimum χ2 = 0.197 and χ2 = 0.144 with impressive
[2] APT
ratio ( 0.73) illustrating the effectiveness of the APT procedure.
≃
On the Fig.2 by open circles and bullets ( , ) we give two– and three–
◦ •
loops data mainly fromFig.10 of paper [2]. The only exclusion is the Υ decay
taken from the Table 6 of the same paper. By crosses we marked the new
“APT values” calculated approximately mainly with help of Eq.(19).
For clearness of the π2 effect, we skipped the error bars. They are the
same as in the mentioned Bethke’s figure and we used them for calculating
χ2.
9
Figure2: ThenewAPTanalysisforα¯ inthefive-flavourtime–likeregion. Crosses
s
(+) differ from circles ( , ) by π2 correction (19). Solid APT curve relates to
◦ •
Λ(5) = 270MeV and α¯ (M2) = 0.124. To compare, we give also the standard
MS s Z
(dot-and-dash curve) α¯ (at Λ(5) = 213MeV and α¯ (M2) = 0.118) taken from
s s Z
Fig.10 of paper [2].
5 Conclusion
We have established a few qualitative effects:
1. Effective positive shift ∆α¯ = +0.002 in the upper half ( 50GeV) of
s
≥
the f = 5 region for all time-like events that have been analyzed up to now
in the NLO mode.
2. Effective shift ∆α¯ +0.003 in the lower half (10 50GeV) of the
s
≃ ÷
f = 5 region for all time-like events that have been analyzed in the NLO
modes.
3. The new value
α¯ (M2) = 0.124 (21)
s Z
by averaging over the f = 5 region.
These results are based on a plausible hypothesis on the “π2– terms”
prevalence in expansion coefficients for observable in the Minkowskian do-
10
