Table Of ContentMPI/PhT/96–110
October 1996
Three-Loop (α2G M2) Corrections to Hadronic
s F t
O
Higgs Decays
K.G. Chetyrkin, B.A. Kniehl, and M. Steinhauser
7 ∗
9
Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut),
9
1 F¨ohringer Ring 6, 80805 Munich, Germany
n
a
J
3
1
1 Abstract
v
7 Wecalculatethetop-quark-inducedthree-loopcorrectionsof (α2G M2)tothe
7 O s F t
Yukawa couplings of the first five quark flavours in the framework of the minimal
2
1 standard model with an intermediate-mass Higgs boson, with mass M 2M .
H t
0 ≪
The calculation is performed using an effective-Lagrangian approach implemented
7
withthehard-massprocedure. Asanapplication,wederivethe (α2G M2)correc-
9 O s F t
/ tions to the H qq¯partial decay widths, including the case q = b. The couplings
h →
p of the Higgs boson to pairs of leptons and intermediate bosons being known to
- (α2G M2), this completes the knowledge of such corrections in the Higgs sector.
p O s F t
e We express the results both in the MS and on-shell schemes of mass renormaliza-
h
tion. We recover the notion that the QCD perturbation expansions exhibit a worse
:
v convergence behaviour in the on-shell scheme than they do in the MS scheme.
i
X
PACS numbers: 12.38.-t, 12.38.Bx, 14.65.Ha, 14.80.Bn
r
a
∗
Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, 60th October
Anniversary Prospect 7a, Moscow 117312,Russia.
1
1 Introduction
One of the longstanding questions of elementary particle physics is whether nature makes
use of the Higgs mechanism of spontaneous symmetry breaking to endow the particles
with their masses. The Higgs boson, H, is the missing link sought to verify this the-
oretical conjecture in the standard model (SM). The possible range of the Higgs-boson
mass, M , is constrained from below both experimentally and theoretically. The failure
H
of experiments at the CERN Large Electron-Positron Collider (LEP 1) to observe the
¯
decay Z ffH has ruled out the mass range M 65.6 GeV at the 95% confidence
H
→ ≤
level [1]. Depending on the precise value of the top-quark mass, M , the requirement that
t
the vacuum be the true ground state provides an even more stringent theoretical lower
bound [2]. However, this bound may be somewhat relaxed by taking into account the
possibility that the physical minimum of the effective potential might be metastable [3].
Other theoretical arguments bound M from above. The requirements that partial-wave
H
unitarity in intermediate-boson scattering at high energies be satisfied [4] or that pertur-
bation theory in the Higgs sector be meaningful [5] establish an upper bound on M at
H
1/2
about 8π√2/3G 1 TeV, where G denotes Fermi’s constant, in a weakly inter-
F F
≈
acting(cid:16)SM. The tri(cid:17)viality bounds, i.e., MH upper bounds derived through perturbative
[6] or lattice [7] computations by requiring that the running Higgs self-coupling, λ(µ),
stay finite for renormalization scales µ < Λ, where Λ is the cutoff beyond which new
physics operates, are somewhat stronger. In the following, we shall focus our attention
on the lower end of the allowed M range, where M 2M , which will be accessed by
H H t
≪
colliding-beam experiments in the near future.
A Higgs boson with M < 135 GeV decays dominantly to b¯b pairs [8]. This decay
H
mode will be of prime impor∼tance for Higgs-boson searches at LEP 2 [9], the Fermilab
Tevatron [10] or a possible 4-TeV upgrade thereof [11], a next-generation e+e− linear
collider [12], and a future µ+µ− collider [13]. Techniques for the measurement of the
H b¯b branching fraction at a √s = 500 GeV e+e− linear collider have been elaborated
→
in Ref. [14]. The branching ratio of H cc¯ is roughly 30 times smaller than that of
¯ →
H bb [8].
→
Once a novel scalar particle is discovered, it will be crucial to decide if it is the very
Higgs boson of the SM or if it lives in some more extended Higgs sector. To that end,
preciseknowledgeoftheSMpredictionswillbemandatory, i.e.,quantumcorrectionsmust
be taken into account. The present knowledge of quantum corrections to the H qq¯
→
partial decay widths has recently been reviewed in Ref. [15]. At one loop, the electroweak
[16, 17] and quantum-chromodynamical (QCD) [18] corrections are known for arbitrary
masses. The leading high-M term, of (G M2), was first derived by Veltman [19].
H O F H
In the limit M 2M , which we are interested in here, the terms of (X ), where
H t t
≪ O
X = G M2/8π2√2 , tendtobedominant. Theyariseinpartfromtherenormalizations
t F t
of the(cid:16)Higgs wave fun(cid:17)ction and vacuum expectation value, which are independent of the
quark flavour q [20]. In the case of q = b, there is an additional non-universal (X )
t
O
contribution [16, 17, 21], which partly cancels the flavour-independent one. At two loops,
the universal [22] and bottom-specific [23, 24] (α X ) terms are available. Furthermore,
s t
O
2
the first [25] and second [26] terms of the expansion in M2/M2 of the (α2) five-flavour
q H O s
QCD correction have been found. As for the top-quark-induced correction in (α2), the
O s
full M dependence of the non-singlet (double-bubble) contribution [27] as well as the
t
first four terms of the M2/M2 expansion of the singlet (double-triangle) contribution [28]
H t
have been computed. At three loops, the (α3) non-singlet correction is known in the
O s
massless approximation [29].
In this paper, we shall take the next step, to three loops including virtual top-quark
effects. Specifically, we shall evaluate the (α2X ) corrections to the decay widths of
¯ O s t
H qq¯, and in particular of H bb. These corrections may be divided into three
→ →
classes, which are separately finite and gauge independent: (i) The universal correction
originates in the renormalizations of the Higgs wave function and vacuum expectation
value and occurs as a building block in the computation of quantum corrections to any
Higgs-boson production or decay process. It is related to the Higgs- and W-boson self-
energies. In the case of the leptonic decay, H l+l−, this is the only source of (α2X )
→ O s t
corrections [30]. (ii) The quark-specific correction arises from Feynman diagrams where
thetopquarkonlyappearsinaclosedloopwhich isconnected withtheexternalquarkline
by two gluons (and, in some cases, one additional weak neutral boson). It just depends
on the third component of weak isospin of the considered quark. (iii) The bottom-specific
correction emerges from the one-loop seed diagram where one charged boson is emitted
and re-absorbed from the external bottom-quark line, by appropriately adding gluon lines
(and, in some cases, one quark or ghost loop). It is the natural extension of the bottom-
¯
specific (α X ) correction to the H bb decay width [23, 24] by one order of α .
s t s
O →
The next-to-leading-order QCD corrections to the top-quark-induced shifts in the
l+l−H, W+W−H, and ZZH couplings have been found in Ref. [30]. The present paper
completes the knowledge of the (α2X ) corrections to the SM Yukawa couplings, still
O s t
excluding that of the top quark. The Yukawa couplings of the first four quark flavours
receive corrections from the same class of diagrams, while, in the case of the bottom
Yukawa coupling, an additional class of diagrams must be included.
In Ref. [30], it was noticed that the QCD perturbation expansions of the l+l−H,
W+W−H, and ZZH couplings and the electroweak ρ parameter, for which the (α2X )
O s t
correction is also known [31], exhibit striking similarities. If the top-quark mass is renor-
malized according to the on-shell scheme, the coefficients of (α /π)X and (α /π)2X are
s t s t
in each case negative and of increasing magnitude. On the other hand, the corresponding
coefficients in the MS scheme are much smaller and of variant sign. This gave support
to the notion that the use of the pole mass deteriorates the convergence properties of the
QCD perturbation series, which may also be motivated from the study of renormalons
[32]. It is interesting to find out whether this observation is substantiated by the analysis
of the light-quark and bottom Yukawa couplings. We shall return to this issue in Sect. 5.
Our key result for the H b¯b decay width to (α2X ) has recently been presented
→ O s t
without derivation in a brief note [33]. This paper provides the full details of our analysis
and also deals with the H qq¯ decays, where q = u,d,s,c. It is organized as follows.
→
After introducing our notations in Sect. 2, we shall derive, in Sect. 3, a heavy-top-quark
effective Lagrangian with two coefficient functions to be determined by diagrammatic
3
calculation. From this Lagrangian, we shall obtain a generic formula for the H qq¯
→
decay widths valid through (α2X ). In Sect. 4, we shall calculate, at three loops, the
O s t
coefficient functions relevant for the first four quark flavours. In Sect. 5, we shall extend
this analysis to also include the case q = b. Our conclusions will be summarized in Sect. 6.
2 Notations
Inthissection, thenotationisfixed andusefulformulae, whichwillbenecessary tonumer-
ically evaluate our results, are provided. The calculation is performed in the framework of
dimensional regularization with space-time dimension n = 4 2ε. The QCD gauge group
−
is taken to be SU(N ), with N arbitrary. The colour factors corresponding to the Casimir
c c
operators of the fundamental and adjoint representations are C = (N2 1)/(2N ) and
F c − c
C = N , respectively. For the numerical evaluation we set N = 3. The trace normaliza-
A c c
tion of the fundamental representation is T = 1/2. The number of active quark flavours
is denoted by n . Unless otherwise stated, we work in the MS scheme, with µ being the
f
renormalization scale.
The µ dependence of the strong coupling constant, α (µ), and any quark mass, m(µ),
s
is governed by the renormalization-group (RG) equations,
µ2d α (µ) α 2 α k
s s s
=β(α ) = β , (1)
dµ2 π s − π k π
(cid:18) (cid:19) k≥0 (cid:18) (cid:19)
X
µ2d α α k
lnm(µ)=γ (α ) = s γk s , (2)
dµ2 m s − π m π
k≥0 (cid:18) (cid:19)
X
where
1 11 4 1 34 20
β = C Tn , β = C2 4C Tn C Tn , (3)
0 4 3 A − 3 f 1 16 3 A − F f − 3 A f
(cid:20) (cid:21) (cid:20) (cid:21)
1 1 3 97 10
γ0 = 3C , γ1 = C2 + C C C Tn (4)
m 4 F m 16 2 F 6 F A − 3 F f
(cid:20) (cid:21) (cid:20) (cid:21)
are the first few coefficients of the QCD β and γ functions. The next-to-leading-order
(two-loop) solution of Eq. (1) reads
α (µ) 1 β lnln(µ2/Λ2)
s 1
= 1 , (5)
π β0ln(µ2/Λ2) " − β02 ln(µ2/Λ2) #
where Λ is the asymptotic scale parameter.
The relation between the MS mass m(µ) and the on-shell mass M of the top quark
t
is given by
m (µ) 3 µ2 α(nf)(µ) 3 µ2
t s
=1+X 4+ ln + C 1 ln
Mt t 2 Mt2! π F − − 4 Mt2!
4
α(nf)(µ) 2 3 3
s
+ C T ζ(2)
F
π 4 − 2
(cid:26) (cid:18) (cid:19)
7 5 3 21 µ2 9 µ2
+C2 +3ζ(2) +ln2 ζ(3)+ ln + ln2
F "128 (cid:18)−8 (cid:19)− 4 32 Mt2 32 Mt2#
1111 1 3 185 µ2 11 µ2
+C C + ζ(2)(1 3ln2)+ ζ(3) ln ln2
A F "− 384 2 − 8 − 96 Mt2 − 32 Mt2#
71 1 13 µ2 1 µ2
+ C Tn + ζ(2)+ ln + ln2 + (α3,α X ,X2), (6)
F f 96 2 24 Mt2 8 Mt2!) O s s t t
where n = 6, X = G M2/8π2√2 , and ζ is Riemann’s zeta function, with values
f t F t
ζ(2) = π2/6 and ζ(3) (cid:16) 1.202057. Th(cid:17)e (α2) and (X ) corrections were calculated in
≈ O s O t
Refs. [34] and [35, 36], respectively. Iterating Eq. (6), one obtains the scale-independent
MS mass µ = m (µ ) as
t t t
µ α(nf)(M ) α(nf)(M ) 2 3 3
t s t s t
=1+4X C + C T ζ(2)
t F F
M − π π 4 − 2
t (cid:26) (cid:18) (cid:19)
199 5 3 1111
+C2 +3ζ(2) +ln2 ζ(3) +C C
F 128 −8 − 4 A F − 384
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20)
1 3 71 1
+ ζ(2)(1 3ln2)+ ζ(3) +C Tn + ζ(2) + (α3,α X ,X2). (7)
2 − 8 F f 96 2 O s s t t
(cid:21) (cid:18) (cid:19)(cid:27)
In the MS scheme, the relation between the values of α (µ) for n = 5 and n = 6 is given
s f f
by
α(6)(µ) 1 µ2 2 µ2
α(5)(µ) = α(6)(µ) 1+ s T ln +x +ln + (α2) , (8)
s s ( π "−3 m2t t −3 m2t!# O s )
where x (µ) = G m2(µ)/8π2√2 . The constant term of (α2x ) in Eq. (8) represents a
t F t O s t
new result, whihch follows from thie analysis of the next section. Inserting the (X ) term
t
O
of Eq. (6) into Eq. (8), one obtains
α(6)(µ) 1 µ2 µ2
α(5)(µ) = α(6)(µ) 1+ s T ln +X 2+2ln + (α2) . (9)
s s ( π "−3 Mt2 t Mt2!# O s )
3 Effective Lagrangian
In this section, we construct an effective Lagrangian for the interaction between an
intermediate-mass Higgs boson and a quark-antiquark pair. Therefore, in addition to
the pure QCD Lagrangian, the couplings of the quarks to the Higgs boson, H, the neutral
Goldstone boson, χ, and the charged Goldstone bosons, φ±, must be taken into account.
This will produce corrections proportional to X . In this paper, we are not interested in
t
corrections of (X2), and our formulae will not in general be valid in this order.
O t
5
As a starting point, we consider the bare Yukawa Lagrangian,
H0
= J, (10)
LY − v0
where v is the Higgs vacuum-expectation value, the superscript 0 labels bare quantities,
and the operator J is defined as
J = m0q¯0q0 +m0t¯0t0. (11)
q t
q
X
Here, q runs over u, d, s, c, and b. It is easy to see that J is a finite operator, in the
sense that no additional renormalization constant is needed. Our aim is to construct
the equivalent expression for J in the effective theory where the top quark is integrated
out. Because only the leading terms in M are considered, may be written as a linear
t Y
L
combination of three physical operators with mass dimension four, namely,
m0t→∞ eff = H0 C0O′ + C0 O′ +C0 O′ , (12)
LY −→ LY − v0 " 1 1 2q 2q 3q 3q #
Xq (cid:16) (cid:17)
where
2
O′ = G0′ ,
1 aµν
O′ =(cid:16)m0′q¯0′(cid:17)q0′,
2q q
i → ←
O′ =q¯0′ D/0′ D/0′ m0′ q0′ (13)
3q 2 − − q
(cid:20) (cid:18) (cid:19) (cid:21)
are bareoperators inthe n = 5 effective theory and C0 = C0(α0,m0,µ) (i = 1,2q,3q)are
f i i s t
their bare coefficient functions. Here, G is the colour field strength, D is the covariant
aµν µ
derivative, and the parameters and fields of the effective theory are marked by a prime.
Notice that O′ vanishes by the fermionic equation of motion and may be omitted once
3q
C0 and C0 are determined.
1 2q
The relations between the parameters and fields in the full and effective theories read
1/2
q0′ = ζ0 q0,
2q
(cid:16) (cid:17)1/2
G0′ = ζ0 G0 ,
aµ 3 aµ
m0′ =ζ(cid:16)0 (cid:17)m0,
q m,q q
g0′ =ζ0g0, (14)
s g s
where G denotes the gluon field. The renormalization constants ζ0 , ζ0 , and ζ0 play
aµ 2q m,q 3
an important rˆole in the determination of the coefficient functions C0. As may be seen
i
using the method of projectors [37], they may be calculated from the vector and scalar
6
parts of the quark self-energy, Σ (p2) and Σ (p2), and the transverse part of the gluon
V S
self-energy, Π(p2), via
ζ0 =1+Σ0t(0), (15)
2q V
1 Σ0t(0)
ζ0 = − S , (16)
m,q 1+Σ0t(0)
V
ζ0 =1+Π0t(0), (17)
3
where the superscript t indicates that only diagrams containing the top quark have to
be considered and the superscript 0 reminds us of that the parameters are still in their
bare forms. In our convention, the bare quark and gluon propagators are proportional
−1
to p(1+Σ0 (p2)+γ Σ0(p2)) m0(1 Σ0(p2)) and [p2(1+Π0(p2))]−1, respectively.
6 V 5 A − q − S
Nohtice that the axial-vector part of the quark iself-energy, Σ (p2), does not enter our
A
analysis because it leads to corrections of (x2). The renormalization constant ζ0 only
O t g
enters the stage via Eqs. (8) and (9). Through (α2x ), we have ζ0 = (ζ0)−1/2.
O s t g 3
By means of the higher-order formulation [38] of a well-known low-energy theorem
(LET) [39] in connection with the method of projectors [37], one may derive the following
relations which allow one to calculate the coefficient functions:
m0∂
ζ0C0 = t Π0t(0), (18)
3 1 −∂m0
t
m0∂
ζ0 ζ0 C0 C0 =1 Σ0t(0) t Σ0t(0), (19)
2q m,q 2q − 3q − S − ∂m0 S
t
(cid:16) (cid:17)
m0∂
ζ0 C0 = t Σ0t(0). (20)
2q 3q −∂m0 V
t
Here it is understood that the operator (m0∂/∂m0) only acts on those appearances of
t t
m0 which remain after every 1/v0 is saturated by one power of m0 to give m0/v0. As
t t t
may be seen from Eqs. (18)–(20), the calculation of the coefficient functions involves the
quark- and gluon-propagator diagrams containing the top quark, with nullified external
momenta. By means of the LET, the corresponding vertex diagrams may be generated
by attaching an external Higgs-boson line with zero momentum to the quark lines. To the
order of interest here, this requires the evaluation of one-, two-, and three-loop tadpole
diagrams. They are calculated with the help of the program package MATAD, written in
FORM [40], which makes use of integration-by-parts identities developed in Ref. [41].
So far, we have constructed an expression for J in terms of bare operators and coeffi-
cient functions, which both still contain poles in ε. As is well known, different operators of
the same dimension and quantum numbers in general mix under renormalization. Specif-
ically, the renormalized operators, which will be denoted by square brackets, are related
to the unrenormalized ones according to [42]
α ∂ α ∂
[O′]= 1+2 s lnZ O′ 4 s lnZ O′ ,
1 " ∂αs g!# 1 − ∂αs m! q 2q
X
O′ =O′ , (21)
2q 2q
h i
7
where Z and Z are the coupling and mass renormalization constants, respectively, in
g m
pure QCD with n = 5 active flavours. It hence follows that
f
µ2d α ∂ πβ α ∂
[O′]= s [O′]+4 s γ O′ ,
dµ2 1 − ∂αs αs! 1 ∂αs m!Xq h 2qi
µ2d
O′ =0. (22)
dµ2 2q
h i
Note that O′ does not mix with O′ and O′ . On the other hand, the coefficient functions
3q 1 2q
are renormalized according to
1
C = C0,
1 1+2(α ∂/∂α )lnZ 1
s s g
4(α ∂/∂α )lnZ
C = s s m C0 +C0 . (23)
2q 1+2(α ∂/∂α )lnZ 1 2q
s s g
Consequently, the n = 5 effective Lagrangian (12) takes the form
f
H0
eff = C [O′]+ C O′ +C O′ . (24)
LY − v0 " 1 1 2q 2q 3q 3q #
Xq (cid:16) h i h i(cid:17)
The new coefficient functions and the operators are individually finite, but, with the
exception of O′ , they are not separately RG invariant. From now on, O′ will be
2q 3q
omitted becauhse iit vanishes on mass shell and thus does not contribute tohtheiH qq¯
→
decay widths.
It is desirable to arrange for the renormalized coefficient functions and operators to
be separately µ independent up to higher orders. This may be achieved by reshuffling
the terms in Eq. (24). Exploiting the RG invariance of O′ and of the trace of the
2q
energy-momentum tensor, h i
πβ(5)
Θµ = [O′]+ 1 2γ(5) O′ , (25)
µ 2α(5) 1 − m 2q
h i s (cid:16) (cid:17)Xq h i
where all quantities are defined in the n = 5 effective theory, we may construct two new
f
operators which are indeed RG invariant [43], e.g.,
2π πβ(5)
O′ = [O′] 2γ(5) O′ ,
g −β(5) 2α(5) 1 − m 2q !
h i 0 s Xq h i
O′ = O′ . (26)
q 2q
h i h i
Thus, the relevant part of the n = 5 effective Lagrangian may be written as
f
H0
eff = C O′ + C O′ , (27)
LY − v0 gh gi Xq qh qi!
8
where
α(5)β(5)
C = s 0 C ,
g − π2β(5) 1
4α(5)γ(5)
C = s m C +C . (28)
q πβ(5) 1 2q
Asthenew coefficient functionsandoperatorsareseparately RGinvariant, wemay choose
µ = µ for C and C and µ = M for O′ and O′ .
t g q H g q
In order to calculate the H qq¯dehcayiwidthh, itiis more convenient to re-express eff
→ LY
in terms of the operators [O′] and O′ . At the same time, the separation of the scales
1 2q
M and µ is kept, so that Eq. (27)hbeciomes
H t
H0
eff = [O′]+ O′ , (29)
LY − v0 C1 1 q C2q 2q !
X h i
where
α(5)(µ )β(5)(M )
(µ ,M )= s t H C (µ ),
C1 t H α(5)(M )β(5)(µ ) 1 t
s H t
4α(5)(µ )
(µ ,M )= s t γ(5)(µ ) γ(5)(M ) C (µ )+C (µ ). (30)
C2q t H πβ(5)(µ ) m t − m H 1 t 2q t
t
(cid:16) (cid:17)
Notice that, for our purposes, and are needed up to (α x ) and (α2x ), respec-
C1 C2q O s t O s t
tively. It is instructive to expand Eq. (30) in terms ofα(6)(µ ), keeping only terms relevant
s t
for our (α2x ) calculation. Observing that C starts at (α ), while C = 1 to lowest
O s t 1 O s 2q
order, this yields
(µ ,M )=C (µ ),
1 t H 1 t
C
α(6)(µ ) µ2
(µ ,M )= 4γ0 s t C (µ )ln t +C (µ ). (31)
C2q t H − m π 1 t M2 2q t
H
The coefficients C and C must be computed diagrammatically. This will be done
1 2q
for the light-quark flavours in Sect. 4 and for q = b in Sect. 5. In the remainder of this
section, we present a generic formula, derived from Eq. (29), for the H qq¯decay width,
→
appropriate for M 2M , which accommodates all presently known corrections. It
H t
≪
reads
Γ(H qq¯)=ΓBorn 1+δ¯ 2 1+∆QED 1+ ∆weak 1+∆QCD ( )2
→ qq¯ (cid:16) u(cid:17) (cid:20)(cid:16) q (cid:17)(cid:18) q (cid:12)xt=0(cid:19)(cid:16) q (cid:17) C2q
+ ΞQCD . (cid:12) (32)
q C1C2q (cid:12)
i
Here,
N G M m2 4m2 3/2
ΓBorn = c F H q 1 q (33)
qq¯ 4π√2 − M2 !
H
9
is the Born result including the full mass dependence. As is well known [18], we may
avoid the appearance of large logarithms of the type ln(M2/m2) in the QCD correction,
H q
∆QCD, by taking m in Eq. (33) to be the MS mass for n = 5, m(5)(µ), with µ of order
q q f q
M . Consequently, we may put m = 0 in ∆QCD. We may proceed similarly with the
H q q
quantum-electrodynamical (QED) correction, ∆QED, which then takes the form
q
α(µ) 17 3 µ2
∆QED = Q2 + ln , (34)
q π q 4 2 M2 !
H
where α(µ) is the MS fine-structure constant and Q is the fractional quark charge. In
q
turn, m(5)(µ) is then also shifted by a QED correction [35] from the pole mass, M .
q q
∆weak denotes the weak correction with the leading (x ) term stripped off. If we
q xt=0 O t
put m(cid:12)(cid:12)(cid:12)q = 0 and consider the limit MH ≪ 2MW, ∆wqeak xt=0 simplifies to [17]
(cid:12)
(cid:12)
∆weak = GFMZ2 1 3 1 4s2 Q 2(cid:12)+c2 3 lnc2 5 , (35)
q (cid:12)xt=0 8π2√2 "2 − (cid:16) − w| q|(cid:17) w s2w w − !#
(cid:12)
(cid:12)
where c2 = 1 s2 = M2 /M2, M is the Z-boson mass, and M is the W-boson mass.
w − w W Z Z W
∆QCD is the well-known QCD correction in the n = 5 effective theory [25],
q f
α(nf)(µ) 17 3 µ2
∆QCD = s C + ln
q π F 4 2 M2 !
H
α(nf)(µ) 2 691 9 9 105 µ2 9 µ2
+ s C2 ζ(2) ζ(3)+ ln + ln2
π " F 64 − 4 − 4 16 M2 8 M2 !
H H
893 11 31 71 µ2 11 µ2
+C C ζ(2) ζ(3)+ ln + ln2
F A 64 − 8 − 8 12 M2 16 M2 !
H H
65 1 11 µ2 1 µ2
+ C Tn + ζ(2)+ζ(3) ln ln2
F f −16 2 − 6 M2 − 4 M2 !#
H H
α(5)(µ) 17 µ2
= s +2ln
π 3 M2 !
H
α(5)(µ) 2 8851 47 97 263 µ2 47 µ2
+ s ζ(2) ζ(3)+ ln + ln2 . (36)
π ! " 144 − 6 − 6 9 M2 12 M2 #
H H
This correction originates from the class of diagrams where the Higgs boson directly
couples to the final-state qq¯pair, i.e., it does not comprise the double-triangle topologies
of Ref. [28]. The latter will be discussed below. The scale µ in ∆QCD must be identified
q
with that of m(5)(µ) in ΓBorn. It is natural to choose µ = M in order to suppress the
q qq¯ H
logarithms of RG origin.
The corrections of (αnx ), with n = 0,1,2, as well as the n = 6 QCD corrections of
(α ) and (α2) are aOll cosnttained in δ¯ , , and . δ¯ contaifns the universal (αnx )
O s O s u C1 C2q u O s t
10
