NNLO QCD corrections to the B -> X_s gamma matrix elements using interpolation in m_c

0.3 MB·English

by  Mikolaj Misiak

TTP06-24
SFB/CCP-06-37
IFT-16/2006
hep-ph/0609241
7
0 ¯ →
NNLO QCD Corrections to the B X γ
0 s
2
Matrix Elements Using Interpolation in m

n c
a
J

8

2
v
Miko laj Misiak1,2 and Matthias Steinhauser3
1
4
2
1 Institute of Theoretical Physics, Warsaw University,
9
0 Hoz˙a 69, PL-00-681 Warsaw, Poland.
6
0
/ 2 Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland.
h
p
- 3 Institut fu¨r Theoretische Teilchenphysik, Universit¨at Karlsruhe (TH),
p
e D-76128 Karlsruhe, Germany.
h
:
v
i
X
r
a
Abstract
¯
One of the most troublesome contributions to the NNLO QCD corrections to B X γ
s
→
originates from three-loop matrix elements of four-quark operators. A part of this contribution
that is proportional to the QCD beta-function coefficient β was found in 2003 as an expansion
0
in m /m . In the present paper, we evaluate the asymptotic behaviour of the complete contri-
c b
bution for m m /2. The asymptotic form of the β -part matches the small-m expansion
c b 0 c
≫
very well at the threshold m = m /2. For the remaining part, we perform an interpola-
c b
tion down to the measured value of m , assuming that the β -part is a good approximation
c 0
at m = 0. Combining our results with other contributions to the NNLO QCD corrections,
c
we find (B¯ X γ) = (3.15 0.23) 10−4 for E > 1.6 GeV in the B¯-meson rest frame.
s γ
B → ± ×
The indicated error has been obtained by adding in quadrature the following uncertainties:
non-perturbative (5%), parametric (3%), higher-order perturbative (3%), and the interpolation
ambiguity (3%).
1 Introduction
The decay B¯ X γ is a well-known probe of new physics at the electroweak scale. The
s
current world a→verage for its branching ratio with a cut E > 1.6 GeV in the B¯-meson rest
γ
frame reads [1]
(B¯ X γ)exp = 3.55 0.24 +0.09 0.03 10−4, (1.1)
B → s Eγ>1.6GeV ± −0.10 ± ×
where the first error is combine(cid:16)d statistical and system(cid:17)atic. The second one is due to the theory
input on the shape function. The third one is caused by the b dγ contamination.
→
The totalerror inEq.(1.1)amounts to around7.4%, i.e. it isof thesamesize asthe expected
(α2) corrections to the perturbative transition b Xpartonγ. On the other hand, the relation
O s → s
Γ(B¯ X γ) Γ(b Xpartonγ) (1.2)
→ s ≃ → s
holds up to non-perturbative corrections that turn out to be smaller (see Section 7).
Consequently, evaluating the Next-to-Next-to-Leading Order (NNLO) QCD corrections to
b Xpartonγ is of crucial importance for deriving constraints on new physics from the mea-
su→remesnts of B¯ X γ.
s
→
In the calculation of b Xpartonγ, resummation of large logarithms (α lnM2 /m2)n is
→ s s W b
necessary at each order in α , which is most conveniently performed in the framework of an
s
effective theory that arises from the Standard Model (SM) after decoupling the heavy elec-
troweak bosons and the top quark. The explicit form of the relevant effective Lagrangian is
given in the next section. The Wilson coefficients C (µ) play the role of coupling constants at
i
the flavour-changing vertices (operators) Q .
i
The perturbative calculations are performed in three steps:
(i) Matching: Evaluating C (µ ) at the renormalization scale µ M ,m by requiring
i 0 0 W t
∼
equality of the SM and effective theory Green’s functions at the leading order in
(external momenta)/(M ,m ).
W t
(ii) Mixing: Calculating the operator mixing under renormalization, deriving the effective
theory Renormalization Group Equations (RGE) and evolving C (µ) from µ down to the
i 0
low-energy scale µ m .
b b
∼
(iii) Matrix elements: Evaluating the on-shell b Xpartonγ amplitudes at µ m .
→ s b ∼ b
In the NNLO analysis of the considered decay, the four-quark operators Q ,...,Q and the
1 6
dipole operators Q and Q must be matched at the two- and three-loop level, respectively.
7 8
Three-point amplitudes with four-quark vertices need to be renormalized up to the four-loop
level, while “only” three-loop mixing is necessary in the remaining cases. The matrix elements
are needed up to two loops for the dipole operators, and up to three loops for the four-quark
operators.
The NNLO matching was calculated in Refs. [2,3]. The three-loop renormalization in the
Q ,...,Q and Q ,Q sectors was found in Refs. [4,5]. The results from Ref. [6] on the
1 6 7 8
{ } { }
four-loop mixing of Q ,...,Q into Q will be used in our numerical analysis.1
1 6 7
1 The small effect ( 0.35% in the branching ratio) of the four-loop mixing [6] of Q ,...,Q into Q is
1 6 8
−
neglected here. It was not yet known in September 2006 when the current paper was being completed.
1
As far as the matrix elements are concerned, contributions to the decay rate that are propor-
tional to C (µ ) 2 are completely known at the NNLO thanks to the calculations in Refs. [7,8].
7 b
| |
These two-loop results have recently been confirmed by an independent group [9,10]. Two- and
three-loop matrix elements in the so-called large-β approximation were found in Ref. [11] as
0
expansions in the quark mass ratio m /m . Such expansions are adequate when m < m /2,
c b c b
which is satisfied by the measured quark masses. Finding all the remaining (“beyond-β ”) con-
0
tributions to the matrix elements is a very difficult task because hundreds of massive three-loop
on-shell vertex diagrams need to be calculated.
In the present work, we evaluate the asymptotic form of the m -dependent NNLO matrix
c
elements in the limit m m /2 using the same decoupling technique as in our three-loop
c b
≫
Wilson coefficient calculation [3]. We find that the asymptotic form of the β -part matches
0
the small-m expansion very well at the cc¯ production threshold m = m /2. The same is
c c b
true for the Next-to-Leading Order (NLO) matrix elements. Motivated by this observation,
we interpolate the beyond-β part to smaller values of m assuming that the β -part is a good
0 c 0
approximation at m = 0. Combining our results with other contributions to the NNLO QCD
c
corrections, we find an estimate for the branching ratio at (α2).
O s
Our paper is organized as follows. In Section 2, we introduce the effective theory and
collect the relevant formulae for the B¯ X γ branching ratio. The contributions that are
s
→
known exactly in m are described in Section 3. Expressions for the NNLO matrix elements
c
in the large-β approximation and in the m m /2 limit are presented in Sections 4 and 5,
0 c b
≫
respectively. Section 6 is devoted to discussing the interpolation in m . Section 7 contains
c
the analysis of uncertainties. We conclude in Section 8. Our numerical input parameters are
collected in Appendix A. Appendix B contains a discussion of the cc¯ production treatment in
the interpolation.
2 The effective theory
Following Section 3 of Ref. [12], the B¯ X γ branching ratio can be expressed as follows:
s
→
V∗V 2 6α
[B¯ X γ] = [B¯ X eν¯] ts tb em [P(E )+N(E )], (2.1)
B → s Eγ>E0 B → c exp V π C 0 0
(cid:12) cb (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where α = α (0) 1/137.036 an(cid:12)d N(E(cid:12) ) denotes the non-perturbative correction.2 The
em em 0
m -dependence of B¯ ≃X eν¯ is accounted for by
c c
→
V 2 Γ[B¯ X eν¯]
ub c
C = → , (2.2)
V Γ[B¯ X eν¯]
(cid:12) cb(cid:12) u
(cid:12) (cid:12) →
(cid:12) (cid:12)
with neglec(cid:12)ted(cid:12)spectator annihilation. P(E ) is given by the perturbative ratio
0
Γ[b X γ] V∗V 2 6α
→ s Eγ>E0 = ts tb em P(E ). (2.3)
V /V 2 Γ[b X eν¯] V π 0
cb ub u (cid:12) cb (cid:12)
| | → (cid:12) (cid:12)
(cid:12) (cid:12)
2 See Eqs.(3.10)and(4.7) of Ref.(cid:12)[12]. Th(cid:12)e correctionsfound in Eqs.(3.9) and(3.14) ofRef. [13]as wellas
Eq. (28) of Ref. [14] should be included in N(E ), too.
0
2
Our goal is to calculate the NNLO QCD corrections to the quantity P(E ). The denominator
0
on the l.h.s. of Eq. (2.3) is already known at the NNLO level from Refs. [15,16].
The relevant effective Lagrangian reads
4G 8 2
= (u,d,s,c,b)+ F V∗V C Q +V∗V Cc(Qu Q ) , (2.4)
Leff LQCD×QED √2 " ts tb i i us ub i i − i #
i=1 i=1
X X
where
Qu = (s¯ γ Tau )(u¯ γµTab ),
1 L µ L L L
Qu = (s¯ γ u )(u¯ γµb ),
2 L µ L L L
Q = (s¯ γ Tac )(c¯ γµTab ),
1 L µ L L L
Q = (s¯ γ c )(c¯ γµb ),
2 L µ L L L
Q = (s¯ γ b ) (q¯γµq),
3 L µ L q
(2.5)
Q = (s¯ γ TabP) (q¯γµTaq),
4 L µ L q
Q = (s¯ γ γ γ Pb ) (q¯γµ1γµ2γµ3q),
5 L µ1 µ2 µ3 L q
Q = (s¯ γ γ γ TabP) (q¯γµ1γµ2γµ3Taq),
6 L µ1 µ2 µ3 L q
Q = e m (s¯ σµνb )FP,
7 16π2 b L R µν
Q = g m (s¯ σµνTab )Ga .
8 16π2 b L R µν
The last term in the square bracket of Eq. (2.4) gives no contribution at the Leading Order
(LO) and only a small contribution at the NLO (around +1% in the branching ratio — see
Eq. (3.7) of Ref. [12]). Consequently, we shall neglect its effect on the NNLO QCD correction
and omit terms proportional to V in the analytical formulae below. However, our numerical
ub
results will include the V terms at the NLO. The same refers to the electroweak corrections
ub
that amount to around 3.7% in P(E ) [12,17].
0
−
The quantity P(E ) depends quadratically on the Wilson coefficients3
0
8
P(E ) = Ceff(µ ) Ceff(µ ) K (E ,µ ), (2.6)
0 i b j b ij 0 b
i,j=1
X
where the “effective coefficients” are defined by
C (µ), for i = 1,...,6,
i
Ceff(µ) =  C (µ)+ 6 y C (µ), for i = 7, (2.7)
i 7 j=1 j j
 C (µ)+ 6 z C (µ), for i = 8.
8 Pj=1 j j
The numbers yand z arPe defined so that the leading-order b sγ and b sg matrix
j j
→ →
elements of the effective Hamiltonian are proportional to the leading-order terms in Ceff and
7
Ceff, respectively [18]. This means, in particular, that K = δ δ + (α ). In the MS scheme
8 ij i7 j7 O s
with fully anticommuting γ , ~y = (0,0, 1, 4, 20, 80) and ~z = (0,0,1, 1,20, 10) [19].
5 −3 −9 − 3 − 9 −6 − 3
3 In Eq. (30) of Ref. [7], K was denoted by G /G .
ij ij u
e
3
In Eq. (2.6), we have assumed that all the Wilson coefficients are real, as it is the case in
the SM. Consequently, K is a real symmetric matrix.
ij
Once the MS-renormalized4 coefficients Ceff(µ) are perturbatively expanded
i
Ceff(µ) = C(0)eff(µ)+α (µ)C(1)eff(µ)+α2(µ)C(2)eff(µ)+ α3(µ) , (2.8)
i i s i s i O s
(cid:16) (cid:17)
where
e e e
α(5)(µ )
α (µ ) s b , (2.9)
s b
≡ 4π
the exepression for P(E ) can be cast in the following form:
0
P(E ) = P(0)(µ )+α (µ ) P(1)(µ )+P(1)(E ,µ )
0 b s b 1 b 2 0 b
+ α2(µ ) P(2)(µ )+hP(2)(E ,µ )+P(2)(E ,iµ ) + α3(µ ) . (2.10)
s b 1 e b 2 0 b 3 0 b O s b
h i (cid:16) (cid:17)
Here, P(0) and P(ek) originate from the tree-level matrix element of Qe
1 7
P(0)(µ ) = C(0)eff(µ ) 2,
b 7 b
(1) (cid:16) (0)eff (cid:17) (1)eff
P (µ ) = 2C (µ )C (µ ),
1 b 7 b 7 b
(2) (1)eff 2 (0)eff (2)eff
P (µ ) = C (µ ) +2C (µ )C (µ ), (2.11)
1 b 7 b 7 b 7 b
(cid:16) (cid:17)
(k) (0)eff (2)
while P depend only on the LO Wilson coefficients C . The NNLO correction P is
2 i 3
defined by requiring that it is proportional to products of the LO and NLO Wilson coefficients
(0)eff (1)eff
only (C C ).
i j
(1) (2)
3 The corrections P and P
2 3
(1) (2)
The corrections P and P are known exactly in m . In order to describe their content, we
2 3 c
expand K (E ,µ ) in α (µ )
ij 0 b s b
K = δ δ +α (eµ )K(1) +α2(µ )K(2) + α3(µ ) . (3.1)
ij i7 j7 s b ij s b ij O s b
(cid:16) (cid:17)
The coefficients K(1)eare easily deerived from the kneown NLO results
ij
1
(1) (1) (0)eff (1)
K = Rer γ L +2φ (δ), for i 6, (3.2)
i7 i − 2 i7 b i7 ≤
182 8
K(1) = + π2 γ(0)effL +4φ(1)(δ), (3.3)
77 − 9 9 − 77 b 77
44 8 1
K(1) = π2 γ(0)effL +2φ(1)(δ), (3.4)
78 9 − 27 − 2 87 b 78
(1) (1)
K = 2(1+δ )φ (δ), for i,j = 7, (3.5)
ij ij ij 6
4 The evanescent operators are as in Eqs. (23)–(25) of Ref. [4].
4
where
2
µ
b
L = ln . (3.6)
b m1S!
b
The matrix γˆ(0)eff and the quantities r(1) as functions of
i
2
m (µ )
c c
z = (3.7)
m1S !
b
can be found respectively in Eqs. (6.3) and (3.1) of Ref. [20]. The bottom mass is renormalized
in the 1S scheme [21] throughout the paper. The charm mass MS renormalization scale µ is
c
(1)
chosen to be independent from µ . For future convenience, we quote r :
b 1,2
1666 80
(1) (1)
r (z) = 6r (z) = +2[a(z)+b(z)] iπ. (3.8)
2 − 1 − 243 − 81
The exact expressions for a(z) and b(z) in terms of Feynman parameter integrals can be found
in Eqs. (3.3) and (3.4) of Ref. [20]. Their small-m expansions up to (z4) read [22,23]
c
O
16 5 π2 5 3π2 1 1 7 2π2 π2
a(z) = 3ζ(3)+ L + L2 + L3 z + + L
9 ("2 − 3 − 2 − 4 ! z 4 z 12 z# 4 3 − 2 z
1 1 7 π2 3 457 5π2 1 5
L2 + L3 z2 + +2L L2 z3 + L L2 z4
− 4 z 12 z(cid:19) "−6 − 4 z − 4 z# 216 − 18 − 72 z − 6 z!
π2 z 1 π2 1 5
+ iπ 4 +L +L2 + L + L2 z2 +z3 + z4 + (z5L2),(3.9)
" − 3 z z! 2 2 − 6 − z 2 z! 9 #) O z
8 π2 2π2 1 1
b(z) = 3+ L z z3/2 + +π2 2L L2 z2
−9 ( − 6 − z! − 3 (cid:18)2 − z − 2 z(cid:19)
25 1 19 1376 137 2π2
+ π2 L +2L2 z3 + + L +2L2 + z4
(cid:18)−12 − 9 − 18 z z(cid:19) − 225 30 z z 3 !
10 4
+ iπ z +(1 2L )z2 + + L z3 +z4 + (z5L2), (3.10)
− − z − 9 3 z O z
(cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
where
L = lnz. (3.11)
z
The functions φ(1)(δ 1 2E /m1S) with i,j 1,2,7,8 can be found in Appendix E of
ij ≡ − 0 b ∈ { }
Ref. [12]. The remaining ones (that affect P(1.6GeV) by 0.1% only) can be read out from
∼
the results of Ref. [24]. In particular,
1 1 1
φ(1)(δ) = δ 1 δ + δ2 + lim φ(1)(δ), (3.12)
47 −54 (cid:18) − 3 (cid:19) 12 mc→mb 27
1
(1) (1)
φ (δ) = φ (δ). (3.13)
48 −3 47
5
(1) (1) (2)
Once all the ingredients of K have been specified, P and P are evaluated by simple
ij 2 3
substitutions to Eq. (2.6)
8
(1) (0)eff (0)eff (1)
P = C C K , (3.14)
2 i j ij
i,j=1
X
8
(2) (0)eff (1)eff (1)
P = 2 C C K . (3.15)
3 i j ij
i,j=1
X
(2)
4 The β -part of P
0 2
(2)
The only NNLO correction to P(E ) in Eq. (2.10) that has not yet been given is P . For this
0 2
contribution, we shall neglect the tiny LO Wilson coefficients of Q ,...,Q . The NLO matrix
3 6
elements of these operators affect the branching ratio by only around 1% [20]. Thus, neglecting
the corresponding NNLO ones has practically no influence on the final accuracy.
Let us split K(2) into the β -parts K(2)β0 and the remaining parts K(2)rem
ij 0 ij ij
K(2) = A n +B = K(2)β0 +K(2)rem, (4.1)
ij ij f ij ij ij
where n stands for the number of massless flavours in the effective theory, and
f
3 3 2 33
K(2)β0 β A = 11 n A , K(2)rem A +B . (4.2)
ij ≡ −2 0 ij −2 − 3 f ij ij ≡ 2 ij ij
(cid:18) (cid:19)
Following Ref. [11], we shall take n = 5. Effects related to the absence of real cc¯ production
f
in b Xpartonγ and to non-zero masses in quark loops on gluon propagators are relegated
→ s
to K(2)rem. Thus, the only m -dependent contributions to K(2)β0 originate from charm loops
ij c ij
containing the four-quark vertices Q and Q .
1 2
The explicit K(2)β0 that we derive from the results of Refs. [7,8,11,15,20] read
ij
3 290 100
K(2)β0 = β Re r(2)(z)+2 a(z)+b(z) L L2 +2φ(2)β0(δ), (4.3)
27 0 −2 2 − 81 b − 81 b 27
(cid:26) (cid:20) (cid:21) (cid:27)
1
K(2)β0 = K(2)β0, (4.4)
17 −6 27
3803 46 80 8 98 16
K(2)β0 = β π2 + ζ(3)+ π2 L L2 +4φ(2)β0(δ), (4.5)
77 0 − 54 − 27 3 9 − 3 b − 3 b 77
(cid:26) (cid:18) (cid:19) (cid:27)
1256 64 32 188 8 8
K(2)β0 = β π2 ζ(3)+ π2 L + L2 +2φ(2)β0(δ), (4.6)
78 0 81 − 81 − 9 27 − 27 b 9 b 78
(cid:26) (cid:18) (cid:19) (cid:27)
K(2)β0 = 2(1+δ )φ(2)β0(δ), for i,j = 7. (4.7)
ij ij ij 6
The small-m expansion of Rer(2)(z) up to (z4) was calculated by Bieri et al. [11]
c 2 O
67454 124π2 4
Rer(2)(z) = 11280 1520π2 171π4 5760ζ(3)+6840L
2 6561 − 729 − 1215 − − − z
(cid:16)
6
1440π2L 2520ζ(3)L +120L2 +100L3 30L4 z
− z − z z z − z
64π2 2 (cid:17)
(43 12ln2 3L )z3/2 11475 380π2 +96π4 +7200ζ(3)
z
− 243 − − − 1215 −
(cid:16)
1110L 1560π2L +1440ζ(3)L +990L2 +260L3 60L4 z2
− z − z z z z − z
2240π2 2 (cid:17)
+ z5/2 62471 2424π2 33264ζ(3) 19494L 504π2L
z z
243 − 2187 − − − −
(cid:16) 2464 15103841 7912 2368
5184L2 +2160L3 z3 π2z7/2 + + π2 + ζ(3)
− z z − 6075 − 546750 3645 81
(cid:18)
(cid:17)
147038 352 88 512
+ L + π2L + L2 L3 z4 + (z9/2L4). (4.8)
6075 z 243 z 243 z − 243 z O z
(cid:19)
The function φ(2)β0(δ) reads
77
1−δ
φ(2)β0(δ) = β φ(1)(δ)L +4 dx F(2,nf) , (4.9)
77 0 77 b
" Z0 #
where F(2,nf) as a function of x = 2E /m is given in Eq. (9) of Ref. [8].5
γ b
The remaining functions φ(2)β0(δ) will be neglected in our numerical analysis. It should
ij
not cause any significant uncertainty for E = 1.6GeV. For this particular cut, the NLO
0
(1)
functions φ (δ) affect the branching ratio by around 4% only, which is partly due to a
ij −
(1) (1)
certain convention in their definitions (φ (0) = 0 for (ij) = (77), and φ (1) = 0). An
ij 6 77
analogous convention is used for φ(2)β0(δ). The known φ(2)β0(δ) affects the branching ratio by
ij 77
around 0.4% only. If the NLO pattern is repeated, an effect of similar magnitude is expected
from the−other φ(2)β0(δ).
ij
As in Eq. (4.1), we can split P(2) = P(2)β0+P(2)rem and express P(2)β0 in terms of K(2)β0, by
2 2 2 2 ij
analogy to Eq. (3.14)
P(2)β0 C(0)eff C(0)eff K(2)β0. (4.10)
2 ≃ i j ij
i,j=1,2,7,8
X
The “ ” sign is used above only because we skip i,j = 3,4,5,6 in the sum.
≃
(2) 2
5 The full correction P in the limit m m /
2 c b
≫
The present section contains the main new result of our paper, namely the asymptotic form
(2)
of P in the limit m m /2. It has been evaluated by performing a formal three-loop
2 c ≫ b
decouplingofthecharmquarkintheeffectivetheory, usingthemethodthathasbeenpreviously
applied by us to the calculation of the three-loopmatching at the electroweak scale [3,25]. Once
the charm decoupling scale is set equal to µ , one recovers the asymptotic form of the matrix
b
elements in the large m limit. Details of this calculation will be presented elsewhere [26].
c
5 The originalcalculation of F(2,nf) and severalother contributions to the photon spectrum was performed
in Ref. [27].
7
9
4 Re(a+b) Re(r( 2 ))
2
8 n=1
n=0 n=2
3 7
6
2
5
n=0
4
1 n=1
n=2
3
mc/mb mc/mb
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
(2)
Figure 1: Re(a+b) (left plot) and Rer (right plot) as functions of m /m = √z. See the text
2 c b
for explanation of the curves.
The m -dependence of P(E ) at the NLO is dominated by Re[a(z)+b(z)]. At the two-loop
c 0
level, we find the following asymptotic form of the functions a(z) and b(z)
4 34 1 5 101 1 1 1393 1
Rea(z) = L + + L + + L + + , (5.1)
3 z 9 z 27 z 486 z2 15 z 24300 O z3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
4 8 1 2 76 1 4 487 1
Reb(z) = L + L + L + + . (5.2)
−81 z 81 − z 45 z 2025 − z2 189 z 33075 O z3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Ima(z) = 4iπ and Imb(z) = 4 iπ are exactly constant for z > 1.
9 81 4
(2)
For the real part of the three-loop function r (z) introduced in Eq. (4.3), we find
2
8 112 27650 1 38 572 10427 8
Rer(2)(z) = L2 + L + + L2 L + π2
2 9 z 243 z 6561 z 405 z − 18225 z 30375 − 135
(cid:18) (cid:19)
1 86 1628 19899293 8 1
+ L2 L + π2 + . (5.3)
z2 2835 z − 893025 z 125023500 − 405 O z3
(cid:18) (cid:19) (cid:18) (cid:19)
The small-m expansion of this function has been given in Eq. (4.8).
c
The dotted line in the left plot of Fig. 1 corresponds to the exact expression for Re(a+b).
The solid line presents its small-m expansion up to (z4). The dashed lines are found from
c
the large-m expansion including terms up to (z−n)Owith n = 0,1,2, which is indicated by
c
O
labels at the curves. The solid and dashed lines in the right plot of Fig. 1 present the same
(2)
expansions of Rer . No exact expression is known in this case.
2
The plots in Fig. 1 clearly demonstrate that a combination of the small-m expansion for
c
m < m /2 and the large-m expansion for m > m /2 (even in the n = 0 case) leads to a
c b c c b
(2)
reasonable approximation to Re(a+b) and Rer for any m . Moreover, no large cc¯threshold
2 c
effectsareseenatm = m /2. Theseobservationsmotivateustocalculatethen = 0terminthe
c b
(2)rem
large-m expansion of P and use it in Section 6 to estimate this quantity for m < m /2.
c 2 c b
(2)rem
Theexpressions thatwehave foundfortheleadingtermsinthelarge-m expansion ofK
c ij
(1)
are presented below. The necessary leading terms of K are easily derived from Eqs. (5.1)
ij
8
(1)
and (5.2), taking into account that only φ with i,j > 2 do not vanish at large z, and that
ij
(1)
φ are z-independent for i,j = 4,7,8.
ij
(2)rem (2)rem 1 (2)rem 1 (1) 2 1
K = 36K + = 6K + = K +
22 11 O z − 12 O z 27 O z
(cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19)
218 208 2 1
= L + , (5.4)
D
243 − 81 O z
(cid:20) (cid:21) (cid:18) (cid:19)
127 35 2
(2)rem (1) (1) (1) (1)rem
K = K K + L K + (1 L )K
27 27 77 324 − 27 D 78 3 − D 47
(cid:18) (cid:19)
4736 1150 1617980 20060 1664 1
L2 + L + ζ(3)+ L + , (5.5)
− 729 D 729 D − 19683 243 81 c O z
(cid:18) (cid:19)
127 35 2 1
(2)rem (1) (1) (1) (1)
K = K K + L K + (1 L )K + , (5.6)
28 27 78 324 − 27 D 88 3 − D 48 O z
(cid:18) (cid:19) (cid:18) (cid:19)
1 5 3 1237 232 70 20 1
K(2)rem = K(2)rem + L K(1) + ζ(3)+ L2 L + ,(5.7)
17 −6 27 16 − 4 D 78 − 729 27 27 D − 27 D O z
(cid:18) (cid:19) (cid:18) (cid:19)
1 5 3 1
(2)rem (2)rem (1)
K = K + L K + , (5.8)
18 −6 28 16 − 4 D 88 O z
(cid:18) (cid:19) (cid:18) (cid:19)
2 32 224 628487 628
K(2)rem = K(1) 4φ(1)(δ)+ L K(1) L2 + L π4
77 77 − 77 3 z 77 − 9 D 27 D − 729 − 405
(cid:18) (cid:19)
31823 428 26590 160 2720 256
+ π2 + π2ln2+ ζ(3) L2 L + π2L
729 27 81 − 3 b − 9 b 27 b
512 1
(2)rem
+ πα + 4φ (δ) + , (5.9)
27 Υ 77 O z
(cid:18) (cid:19)
50 8 2 16 112 364 1
K(2)rem = + π2 L K(1) + L2 L + +X(2)rem + , (5.10)
78 − 3 3 − 3 D 78 27 D − 81 D 243 78 O z
(cid:18) (cid:19) (cid:18) (cid:19)
50 8 2 1
K(2)rem = + π2 L K(1) +X(2)rem + , (5.11)
88 − 3 3 − 3 D 88 88 O z
(cid:18) (cid:19) (cid:18) (cid:19)
where
26 4
(1)rem (1)
K = K β L , (5.12)
47 47 − 0 81 − 27 b
(cid:18) (cid:19)
2
µ
c
L = ln , (5.13)
c
mc(µc)!
and the “decoupling logarithm”
2
µ
b
L L L = ln . (5.14)
D b z
≡ − mc(µc)!
9