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Note on QCD corrections to hadronic Higgs decay
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
N O T E O N Q C D C O R R E C T I O N S T O H A D R O N I C H I G G S DECAY Manuel DREES CERN, CH- 1211 Geneva 23, Switzerland and Ken-ichi H I K A S A National Laboratoryfor High Energy Physics (KEK), Tsukuba-shi, lbaraki 305, Japan Received 9 February 1990
The O(oq) QCD correction to the hadronic decay of a scalar Higgs boson, for which contradicting results have been reported, is calculated. Our result is in agreement with that of Braaten and Leveille but disagrees with that of Janot. The possible origin of the discrepancy is discussed. The total hadronic decay rate and the differential rate to qctgof a pseudoscalar Higgs boson are also presented.
I. Introduction The Higgs boson plays a fundamental role in the SU ( 2 ) × U ( 1 ) s y m m e t r y breaking and the generation o f the masses o f W, Z, quarks, and leptons. The search for the Higgs boson is thus o f prime importance in clarifying the origin o f the Fermi scale. If the Higgs boson mass is below the W pair threshold, it has a large decay branching fraction to hadrons. The lowest-order decay rate, H--,qdl, has been known for a long time ~t. Since quarks have colour interactions, it is necessary to evaluate the Q C D correction to the decay if one needs an accurate estimate o f the rate and a realistic simulation o f the decay. Although the O (a~) correction was calculated a decade ago by Braaten and Leveille [ 2 ], it has remained relatively unknown. Recently we m a d e an independent calculation o f the correction [3], which is in agreement with ref. [2], and extended it to the case o f n o n m i n i m a l and pseudoscalar Higgs bosons. A new calculation by Janot [ 4 ] also appeared recently, who apparently did not know the old result o f ref. [2]. His result, however, does not agree with ref. [2] and ours. Nevertheless, his results seem to have been used in obtaining recent experimental bounds [ 5 ] on Higgs bosons at L E E In this note, we present our calculation o f the correction in some detail and try to clarify the origin o f the difference, because we have listed only the final results in ref. [3 ]. We will also show the differential rate for the decay into qdlg. The pseudoscalar case has not been discussed in the literature. Finally, we discuss the s u m m a t i o n of leading logarithms, which is necessary to have well-behaved corrections in the large Higgs mass limit. We thus provide all relevant formulas for the hadronic decay o f a scalar and a pseudoscalar Higgs particle, which are correct up to both O ( a ~ ) and leading logarithms. In this paper we denote a scalar and a pseudoscalar Higgs boson by H and P, respectively.
w~ The formula for the leptonic decay H--,~-can be found in ref. [ I ]. Following the public recognition of QCD, the hadronic decay rate has been understood to be given the same formula with the suitable replacement of the fermion mass and the inclusion ofa colour factor of 3, provided that m. is large enough for perturbative QCD to be applicable. 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
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2. Scalar Higgs decay
2.1. Lowest order The lowest order amplitude H--+q~l is given by the Feynman diagram of fig. 1, in which our convention for the momenta is also shown:
.#o=fa(p)v(lJ) ,
(2.1)
wherefis the Yukawa coupling defined as
~'=.fCIqH .
( 2.2 )
In the minimal standard model,
f
_
gm _ 2mw
(v/~Gv),nm
(2.3)
where g is the SU(2) gauge coupling, m is the quark mass, mw is the W mass, and Gv is the Fermi coupling constant. Squaring the amplitude (2.1) and summing over spins and colour, we obtain Z I.,#o 1 2 = 2 N c f 2 m [ i ~ , ,
(2.4)
where flo= q
~-~z,
(2.5)
And N o = 3 is a colour factor. The decay is isotropic and integrating over the two-body phase space d ~ 2 - 8flon '
(2.6)
gives the lowest-order decay rate ro(H--*qq)= ~ 1
f
NcGvm2 " mHfl~. dqb2 ~" I~a'ol2 - Ncf2 8n mF,fl 3= 4/7~nx/z
q(p} H(q) .......
-~ q(p)
Fig. 1. Feynman diagram for the lowest-orderhadronie Higgsdecay, H--.qq. The momenta of the panicles are defined in the parentheses. 456
(2.7)
a)
b]
.....<
.....<
d)
el
cl
.....< f)
Fig. 2. Diagrams for the one-loop corrections for Hoq¢l. (a): vertex correction; (b), (c): self-energycorrections; (d)-(f): counterlerms.
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2.2. Virtual one-loop corrections We now proceed to the QCD corrections. The contributing graphs are depicted in fig. 2. Diagram 2a gives the vertex correction, 2b and 2c are quark self-energy corrections and 2d-2f are the counterterm contribution. We will work in the Feynman gauge. Diagram 2a gives the correction to the lowest-order vertex eq. (2.3). Denoting the full vertex function (up to the coupling factor) and A =Ao+A~, where Ao= 1 is the lowest order term, we find for the one-loop correction from diagram 2a A,~2a)
-
{ (~ m 2) Cvoq -log~ 2n 2
+--~o
Li2\l+flo/
1+,8o2 , 8o+1 m2 2flo l°gfl--o~-ll°g2-T fl--~l°gflo----~-r*l°g fl---~ -l+i~
+-~o
°gfl~-l+3
'
(2.8a)
l-fl°21 1+'8° +3} ,8,~-- Ogl_fl,~ '
(2.8b)
for q2<0, and ReA,~2.~- Cvas 2n
{ (~ m 2) l+flt] 1+flo ,n 2 2 -Iog-~- - 2fl~-ol o g ~ l o g ~ 5 -
l+fl~[ . (l-flo) ~floflO l+flo _llog21+flo 3] + ~ Ll2 ~ +log lOgl_,80 ~ +
+
for q2>4m2. Although the physical vertex is at q2=m2, we have also shown the value for negative q2 for the purpose of discussing the renormalization condition later. (in the definition offlo, the factor of m ~ should be replaced by q2 in general.) In eq. (2.8), Li2(x) is a Spence function Li2(x)=-
Tlog(l-t),
(2.9)
o
as =g~/4~ is the QCD coupling constant, and CF=~ is a colour factor. We have regularized the ultra-violet divergence by dimensional regularization with D=4-2~, p is the renormalization scale, and the infra-red singularity is regularized by a gluon mass 2. No problem occurs from the use of a finite gluon mass, since the threegluon coupling does not appear in this order. The self-energy diagrams 2b and 2c are cancelled by the counterterm diagrams 2e and 2f if one uses the onshell renormalization scheme as usually adopted in QED. The mass and wave-function renormalization counterterms in this scheme are given by m5-2 _ 4 ] . 6m_Cva~m 4n [ 3 (~ -log rn_~) +4 ] , Z 2 - 1 = ~ -Cva~[ ~ -n - (~ - l o g ~m 2) +21og)~
(2.10,
Finally, the counterterm for vertex 2d should be included. In arbitrary Yukawa theories, this counterterm is fixed by the renormalization condition for the Yukawa coupling, which can be chosen at will. For the Higgs boson, however, there is no arbitrariness in the choice of the renormalization condition, since the quark mass and the Yukawa coupling are proportional to each other. In fact the mass comes from the vacuum expectation value of the Higgs field:
5g=fdlq(v+H),
(2.11 )
with m= -fv. Since we are concerned with the QCD radiative corrections and the Higgs field carries no coiour, the combination v+H has to be renormalized as a whole. Thus the counterterm for the Yukawa coupling is fixed in terms of the quark mass and wave-function renormalization constants: 457
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= 2[-2($log$)+log$-41.
It should be stressed that this counterterm function at q2= 0:
(2.12)
is not equal to the “naive”
one obtained
by subtracting
the vertex
(2.13) This is in sharp contrast with the case of the vector vertex, for which Ward identity and zero momentum subtraction give identical counterterms. We note that the same counterterm can be used for multi-Higgs cases and also for pseudoscalar Higgs bosons (up to the overall Yukawa coupling factor). The total virtual correction is the sum of eq. (2.8b) and eq. (2.12), which is free from ultra-violet singularities: ReA,=%{-(ylog$-l)log$ 0
+?[I&(+$)
-+h3g2s
+log%logs
and the lowest-order T,,(H+qq)
0
=ro.2
rate eq. (2.7 ) is multiplied
+ 51
+ ylogs
(2.14)
_I},
by a factor 1 + 2 Re (1,:
Re,4, .
(2.15)
2.3. Gluon emission corrections The virtual correction (2.14) still contains an infra-red divergence. In order to obtain a finite result, one has to add corrections from the emission of a real gluon. In our treatment of the singularity, it is convenient to divide the gluon emission into soft and hard parts by a cut-off w. ( < m) in the gluon energy. The contributing diagrams are shown in fig. 3. 2.3.1. Soft gluon corrections The emission of a soft gluon is described factor):
by a universal
factor multiplying
the tree amplitude
(up to a colour
(2.16) Here E is the polarization vector of the gluon, and T” is the SU (3) generator in the fundamental representation. The soft-gluon emission probability integrated over the soft-gluon phase space (defined as Eg < wo) is
H(g)
-------((
_______<;;
al
458
bl
Fig. 3. Diagrams for real gluon emission, H+qqg. of the particles are defined in the parentheses.
The momenta
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\ k---pp k-P)l
polarization, ES < too colour
_ _cF-s
(l+Po logl+Po-1 ) log 4co~
1-flo
( \ 2//0
l+fl°2[ ' ( l - f l ° ) ~ f l fl° o l+fl° _~log21+flo +--~0 El2 ~ +log IOgl_flo l-flo
62 ]
+
1
I+flo ~
log 1- P o )
(2.17)
(The colour factor CF is effective in the sense that it is meaningful only in the rate. ) The soft gluon correction to the decay rate is given by this factor multiplying the lowest-order rate: F~s(H--,qQg) =Fo'd,s.
(2.18)
Adding this and the virtual correction, we find that the 2 dependence disappears, which shows the cancellation of the infra-red singularity explicitly.
2. 3.2. Hard gluon corrections The amplitude for hard gluon emission is calculated from the diagrams in fig. 3:
( P'~l~+2p~' ~Y"+2"O'~']T~v(P)e * . ,lth=fgst2(p)\ 2k'p 2k'p J
(2.19)
In general, there are only two independent kinematic variables in spin-summed three-body decays. (Although the three-body phase space has five variables, three of them reduce to trivial angular variables, on which the squared matrix element does not depend.) As the two independent variables, we may choose two of the following three invariants:
s=(p+P)Z=2p'p+2m 2, t ' = 2 p ' k = t - m 2, u'=2p'k=u-m 2,
(2.20)
with the constraint s+ t'+ u ' = m 2. Another choice is fl and 0, defined as follows:
fl= / 1 - --,4m2 "V
(2.21a)
S
p.k=~ ( m ~ - s ) ( 1 - f l c o s 0 ) .
(2.21b)
In the CM frame of the qQ system, fl is the velocity of the quark, and 0 is the angle between the quark and gluon directions. The three-body phase space in terms of these variables is 1
m 2 (flo2
2
2
(2.22)
d~3= 128n3m 2 d s d t ' = 32n3 (1 ~f12)~ - f l )fl dfldcos0. The spin- and colour-summed matrix element squared is I~Rh I2=4f X
[~7,
2NcCvg 2
t' u'
+--+
= 16f2NcCvg~
2(q2-2m2)(s-4m2) +2+4m2 ( 1 + t'u' (
2sqZfl2 "~
1+ (-~-~_s)2,] 1 f12cos20 _
l ) -2m2(q2-4rn2)(--~+ --~)1
8m2q2fl 2 1 ] ( q 2 - - s ) 2 ( 1 --]~2-C0S2~) 2 "
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Integrating over the three-body phase space with the cut-offEg> OJogives ~2 _r- CFCes { ( l + f l ~ log 1+,6'o _ 1 F i h ( H ~ q { I g ) - , o It k 2,60 1 -rio
+ l+ o . l-/ o T-o [2LI2(I--~o)+2Li2(-4 l -31og 1-fl-~ --41ogflo+ ~
l-
)
log
m2 409o2
o]
l+ o 1+ o log ~ + ~ log 2 1 - flo
1 + flo J - log
l+flo 1 } (3+2f12+3fl~)log l_-~o + ~ o (-3+29fl°2) "
(2.24)
Summing all virtual, soft, and hard corrections, it is readily seen that all singularities cancel and no COodependence remains: F, (H--*qdl+ qdlg) =Fiv +Fts +Fib =Fo
CF~s
d.,
(2.25)
where I ,J.= ~oA(flo)+ ~1
3
,
l+flo
(3+34flo2- 13fl4) log 1--~o + ~
3
(-l+7fl~),
(2.26)
with
A(flo)=(l+'8~)
[
. (l--fl0) (l--flO) 4L12 l-i--~° +2Li2 l+flo
4
--3flo log 1--,8~ --4flo logflo.
-3log
l°gl+fl°-21°gfl°'l-ti{}iog~l+fl°]_] (2.27)
This result agrees with that of Braaten and Leveille [2 ], who employed dimensional regularization to treat both ultra-violet and infra-red divergences. The final result thus does not depend on how the singularities are regularized, as it should. Since we have sued the on-shell renormalization scheme, the quark mass parameter m should be regarded as the position of the pole of the quark propagator, which may be estimated using QCD sum rules [6 ]. We remark that the results shown above hold also for scalar bosons in multi-Higgs models, provided that the proper Yukawa couplingfis used. [Only eq. (2.3) and the last equality of (2.7) should be modified. ] So far we have discussed the QCD correction to H-~qdl, but our result is directly applicable to the O ( a ) QED correction to H-, ~ (f bein'g a quark or a lepton ) with the replacement CF a~-~ aef2 (and No-, 1 for a lepton ).
2.4. Comparison withJanot's result In ref. [4], Janot provides some dctail of this calculation, which makes it possible to pin down the origin of the difference to a certain degree. [ In the following, quote the equations in ref. [4 ] as (Jm, n). ] First, his result of real emission (J5.8) is in agreement with ours, if we correct a misprint: the factor flo2 in the denominator of the infra-red-singular term in (J5.8) (on the first linc) should be/3o. In addition, the cxpression for the second Spence function is somewhat ambiguous; the argument should be squared, not the function itself. Hence the difference should have stemmed from the virtual corrections. His result appears in eq. (J4.7). Besides the same misprint in the infra-red singular term (,82 should be flo), we find two disagreements. In the ~2 The gluon energy can be defined either in the Higgs rest frame or in the q~l CM frame. In the limit O.~o<< m, the result does not depend on this choice. Needless to say, one has to employ larger cut-offs to circumvent the soft and collinear singularities in realistic simulations of the three-jet rate.
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notation o f ref. [ 4 ], we obtained the constant factor of - i instead of - 2, and for the coefficient o f In [ ( 1 -,go) / ( 1 + Bo) ] (the last term ), we found - ( I - fl~ )/rio instead o f - ( 1 + 2flo2 )/2,80. We cannot track down the reason for the discrepancy further, except that Janot seems to have used the four-dimensional y matrix algebra instead of the D-dimensional one [see the sentence above (J4.5) ], which can lead to inconsistent results in dimensional regularization. Moreover, there is no remark about how the renormalization was actually done.
2.5. Comparison with QCD sum rule calculations Calculations closely related to ours have been made in the context of Q C D sum rules. Reinders et al. [7 ] calculated the O ( a 5 ) Q C D correction to the two-point function of the scalar current, which is similar to our calculation. In their calculation, however, there is no principle to fix the renormalization condition, because their scalar current is just an external current which has nothing to do with the quark mass. Therefore, they subtract the vertex function at zero momentum, giving a result which is slightly different from ours. Similar calculations were made for the pseudoscalar current also. The result in ref. [ 7 ] was derived with zero m o m e n t u m subtraction and differs from our result for P decay. However, the result in a review [6] was derived from the divergence of the axial vector current and is in agreement with ours. Finally, a calculation of the Q E D corrections to the decay o f a pseudoscalar boson into a lepton pair has been done by Bergstr6m [ 8 ]. He restricted his calculation to the case mp >> m and obtained a result which is identical to eq. (4.2b), using the correct renormalization condition.
3. Pseudoscalar Higgs decay
Let us now turn to the hadronic decay o f a pseudoscalar Higgs boson P. Because the steps are parallel to that for a scalar H, we will only mention the main differences. The Yukawa interaction is taken as
~=fqiy5 qP,
( 3.1 )
and the lowest-order rate is
Ncf 2
Fo(P--*qq) = ~
mp//o •
(3.2)
Note that the threshold behaviour is S-wave, in contrast to the P-wave behavior of H ~ qct. The virtual correction factor differs slightly: ReAD~2a)(P--,q~I)=ReAt~2a)(H--,q~I)-
log~
(qZ>4m2) ,
(3.3)
i.e., the single logarithm term is absent. The renormalization counterterm is exactly the same as for the scalar case, as is discussed in the last section. The soft-gluon factor is also the same. The hard-gluon emission matrix element squared is
Z ['l/hlz=4f2~'Cvg~
= 16f2NcCvg~
+ u't' + 2q2(s-2m -
I + (q~_s)2 J 1 -fl2cos20
+2-2m2q2
(, ,)] ~7i + ~7i
(q2-s)2 (1 -,82cos20) 2 '
(3.4)
and the integrated rate is in the form of (2.24), with the last line charged to + ~
I
, 1 +/~o ( 3 + 2 P oz + 3 f l g ) l o g ~ + ~ ( 2 9 - 3 f l o 2) .
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Summing all the corrections, we find Fj ( P-* q~l + q()g) = Fo ( P-~q~l )
Cvas 7t
Ap,
(3.6)
where Ap=
l
A(,so) + 1 ~ o (19+2,8g+3,8~)log !-,8o1+,8°+ ~ ( 7 - , 8 g ) .
(3.7)
Note that the functional form of both the squared matrix element (3.4) and the total rate (3.6) differs from the scalar case; in addition, the coupling strength f c a n , of course, be different in the two cases.
4. Limiting behaviours The correction factors d , and dp show interesting behaviours in the limiting cases. Near the threshold m u - - ) 4 m (flo--,0), the corrections are positive and develop a 1/flo singularity: 7Z2
AN ~ ~00 -- 1 ,
(4.1a)
7.(2
Ap--- 2,8o - - 3 .
(4.1b)
It can be seen that these terms come solely from virtual corrections; both soft and hard gluon corrections vanish at the threshold. The 1/,8o singularity can be understood as coming from the colour Coulomb interaction between the quark and antiquark, and signals the formation of bound states ~3. A realistic treatment of this threshold region requires the inclusion of the mixing of the Higgs boson with the quarkonium states, both below and above the open flavour threshold. This phenomenon has been fully discussed in ref. [ 3 ]. In the opposite limit mH >> m (,8o~ 1 ), the corrections change the sign. Moreover, they contain a logarithmic singularity AH -~ _3 log rn~t + 9 ,
(4.2a)
m~ Ap ~_ - S l o g m---5- + 9 .
(4.2b)
The asymptotic equality of the scalar and pseudoscalar corrections indicates the restoration ofchiral symmetry. The appearance of the logarithmic term is related to the ultra-violet behaviour of the mass operator, which we further discuss in the next section. Because of the log factor, the correction becomes large if the quark mass is much smaller than the Higgs mass, rendering the one-loop result inappropriate (the corrected rate eventually becomes negative). In this case, one can sum the leading logarithmic terms as discussed in ref. [2 ]. As far as the leading term is concerned, this is equivalent to replacing the quark mass parameter in the coupling with the running quark mass: F(H-)q~I) =
NcGFffl( mH ) 2
4x/~ 7t
raN,8 3 ,
where rh(mH ) is the running quark mass evaluated at raN: ~3 Fore further discussion, see ref. [7 ].
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(4.3)
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rh(mH):~(2m) (Oq(mH)'] 12,<33-2m) k,a~(2m),/
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{log(4m21A2) ] '2'<33-zN' ) = rh(2m) \
~
~
/
,
(4.4)
A is the QCD scale parameter, and Nf is the effective number of flavours. The running mass at the threshold r?t(2m) may be replaced by m. The expression (4.3), however, only holds at rnH >> m and cannot be used closer to the threshold, because the discarded finite O (cq) piece becomes comparable to the retained logarithmic term. We propose to use the following interpolation formula connecting the full one-loop expression near threshold and the leading-log result at large mH:
(log(4m2/A2))24/(33-2Nf)[
CFOls(
1+
~
m2n']l an + ~ l o g - - ~ - j j .
(4.5)
How does the leading-log summation affect the qclg rate? The answer lies in the origin of the final log factor. The general theorem of the cancellation of mass singularities ensures that all soft and collinear singularities cancel in the rate. The remaining logarithm thus should come from the renormalization [2 ], which suggests that the logarithm belongs to the overall Yukawa coupling factor. Therefore, the quark mass parameter in the coupling f i n the rate for qclg should also be replaced by the running mass.
5. Conclusion
In this note we describe the computation of the O ( ~ ) QCD correction to the hadronic decay of a scalar and pseudoscalar Higgs boson. We gave the virtual and real corrections separately and the differential rate for the real emission, H, P--,qdlg is also given. The distribution for P--,qflg has not appeared elsewhere. A subtlety in the renormalization of the Yukawa coupling is discussed in detail. The difference between our result and that of Janot is shown to arise from the virtual correction which crucially depends on the renormalization condition. The difference of the two results becomes prominent in the large Higgs mass limit mH >> m. If we denote the rate by F=Fo+Fj, where Fo is the lowest order result and F, is the O(~s) correction, Janot [4] obtains a constant correction factor in this limit:
Fl Ja"°'~ 1"o
5 C~a~ 4 rc
(5.1)
On the other hand, Braaten and Leveille [ 2 ] and ourselves found Fl
-- ~ log - - ~ +
--
(5.2)
The appearance of the logarithm may seem strange at first glance: indeed, the QCD correction for a vector current (e.g. for the R ratio in e + e - - , h a d r o n s ) is a constant (]CFa~/n), as is well known. This difference, however, can be understood as follows. The logarithmic term is absent for the R ratio since the vector current is conserved and its anomalous dimension vanishes. The scalar current, or the mass operator, however, has nonvanishing anomalous dimension, which manifests itself in the logarithmic correction. As is mentioned in the previous section, summing over the leading logarithms turns the quark mass parameter in the coupling factor into the running quark mass. This high-mass region rnH >> m has been subsequently studied by Sakai [ 9 ] and Inami and Kubota [ 10 ] in a more rigorous framework, and Russian group even derived the O ( a ~ ) correction in this limit [ 11 ]. Since the nature of the logarithmic factor is so well understood, we believe that the result of ref. [4 ], which does not have the logarithmic term, is incorrect.
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Note added A f t e r s u b m i t t i n g o u r paper, we were i n f o r m e d that the A L E P H c o l l a b o r a t i o n in fact used the c o r r e c t i o n factor o f B r a a t e n - L e v e i l l e a n d us in t h e i r analysis, while the O P A L c o l l a b o r a t i o n c o n s e r v a t i v e l y did not i n c l u d e any Q C D c o r r e c t i o n s w h e n d e r i v i n g t h e i r b o u n d s . T h e size o f the c o r r e c t i o n s is not very large in the mass range r e l e v a n t to these searches. H o w e v e r , the Q C D c o r r e c t i o n s b e c o m e m u c h m o r e i m p o r t a n t at L E P 2 and higherenergy colliders because o f the l o g a r i t h m i c factor.
References [ I ] L. Resnick, M.K. Sundaresan and P.J.S. Watson, Phys. Rev. D 8 (1973) 172. [ 2 ] E. Braaten and J.P. Leveille, Phys. Rev. D 22 (1980) 715. [31 M. Drees and K. Hikasa, Heavy quark thresholds in Higgs physics, KEK TH-224, CERN-TH.5393/89 (May 1989), Phys. Rev. D 41, in press. [41P. Janot, Phys. Lett. B 223 (1989) 110. [ 5 ] ALEPH Collab., D. Decamp et al., Phys. Lett. B 236 ( 1990 ) 233; B 237 (1990) 29 I; OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B 236 (1990) 224. [6] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127 (1985) I. [ 7 ] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B 186 ( 1981 ) 109. [ 8 ] L. Bergstr~Sm,Z. Phys. C 20 ( 1983 ) 135. [9] N. Sakai, Phys. Rev. D 22 (1980) 2220. [10] T. Inami and T. Kubota, Nucl. Phys. B 179 (1981) 171. [ 11 ] S.G. Gorishnii, A.L. Kataev and S.A. Larin, Yad. Fiz. 40 (1984) 517 [Sov. J. Nucl. Phys. 40 (1984) 329].
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N O T E O N Q C D C O R R E C T I O N S T O H A D R O N I C H I G G S DECAY Manuel DREES CERN, CH- 1211 Geneva 23, Switzerland and Ken-ichi H I K A S A National Laboratoryfor High Energy Physics (KEK), Tsukuba-shi, lbaraki 305, Japan Received 9 February 1990
The O(oq) QCD correction to the hadronic decay of a scalar Higgs boson, for which contradicting results have been reported, is calculated. Our result is in agreement with that of Braaten and Leveille but disagrees with that of Janot. The possible origin of the discrepancy is discussed. The total hadronic decay rate and the differential rate to qctgof a pseudoscalar Higgs boson are also presented.
I. Introduction The Higgs boson plays a fundamental role in the SU ( 2 ) × U ( 1 ) s y m m e t r y breaking and the generation o f the masses o f W, Z, quarks, and leptons. The search for the Higgs boson is thus o f prime importance in clarifying the origin o f the Fermi scale. If the Higgs boson mass is below the W pair threshold, it has a large decay branching fraction to hadrons. The lowest-order decay rate, H--,qdl, has been known for a long time ~t. Since quarks have colour interactions, it is necessary to evaluate the Q C D correction to the decay if one needs an accurate estimate o f the rate and a realistic simulation o f the decay. Although the O (a~) correction was calculated a decade ago by Braaten and Leveille [ 2 ], it has remained relatively unknown. Recently we m a d e an independent calculation o f the correction [3], which is in agreement with ref. [2], and extended it to the case o f n o n m i n i m a l and pseudoscalar Higgs bosons. A new calculation by Janot [ 4 ] also appeared recently, who apparently did not know the old result o f ref. [2]. His result, however, does not agree with ref. [2] and ours. Nevertheless, his results seem to have been used in obtaining recent experimental bounds [ 5 ] on Higgs bosons at L E E In this note, we present our calculation o f the correction in some detail and try to clarify the origin o f the difference, because we have listed only the final results in ref. [3 ]. We will also show the differential rate for the decay into qdlg. The pseudoscalar case has not been discussed in the literature. Finally, we discuss the s u m m a t i o n of leading logarithms, which is necessary to have well-behaved corrections in the large Higgs mass limit. We thus provide all relevant formulas for the hadronic decay o f a scalar and a pseudoscalar Higgs particle, which are correct up to both O ( a ~ ) and leading logarithms. In this paper we denote a scalar and a pseudoscalar Higgs boson by H and P, respectively.
w~ The formula for the leptonic decay H--,~-can be found in ref. [ I ]. Following the public recognition of QCD, the hadronic decay rate has been understood to be given the same formula with the suitable replacement of the fermion mass and the inclusion ofa colour factor of 3, provided that m. is large enough for perturbative QCD to be applicable. 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
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2. Scalar Higgs decay
2.1. Lowest order The lowest order amplitude H--+q~l is given by the Feynman diagram of fig. 1, in which our convention for the momenta is also shown:
.#o=fa(p)v(lJ) ,
(2.1)
wherefis the Yukawa coupling defined as
~'=.fCIqH .
( 2.2 )
In the minimal standard model,
f
_
gm _ 2mw
(v/~Gv),nm
(2.3)
where g is the SU(2) gauge coupling, m is the quark mass, mw is the W mass, and Gv is the Fermi coupling constant. Squaring the amplitude (2.1) and summing over spins and colour, we obtain Z I.,#o 1 2 = 2 N c f 2 m [ i ~ , ,
(2.4)
where flo= q
~-~z,
(2.5)
And N o = 3 is a colour factor. The decay is isotropic and integrating over the two-body phase space d ~ 2 - 8flon '
(2.6)
gives the lowest-order decay rate ro(H--*qq)= ~ 1
f
NcGvm2 " mHfl~. dqb2 ~" I~a'ol2 - Ncf2 8n mF,fl 3= 4/7~nx/z
q(p} H(q) .......
-~ q(p)
Fig. 1. Feynman diagram for the lowest-orderhadronie Higgsdecay, H--.qq. The momenta of the panicles are defined in the parentheses. 456
(2.7)
a)
b]
.....<
.....<
d)
el
cl
.....< f)
Fig. 2. Diagrams for the one-loop corrections for Hoq¢l. (a): vertex correction; (b), (c): self-energycorrections; (d)-(f): counterlerms.
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2.2. Virtual one-loop corrections We now proceed to the QCD corrections. The contributing graphs are depicted in fig. 2. Diagram 2a gives the vertex correction, 2b and 2c are quark self-energy corrections and 2d-2f are the counterterm contribution. We will work in the Feynman gauge. Diagram 2a gives the correction to the lowest-order vertex eq. (2.3). Denoting the full vertex function (up to the coupling factor) and A =Ao+A~, where Ao= 1 is the lowest order term, we find for the one-loop correction from diagram 2a A,~2a)
-
{ (~ m 2) Cvoq -log~ 2n 2
+--~o
Li2\l+flo/
1+,8o2 , 8o+1 m2 2flo l°gfl--o~-ll°g2-T fl--~l°gflo----~-r*l°g fl---~ -l+i~
+-~o
°gfl~-l+3
'
(2.8a)
l-fl°21 1+'8° +3} ,8,~-- Ogl_fl,~ '
(2.8b)
for q2<0, and ReA,~2.~- Cvas 2n
{ (~ m 2) l+flt] 1+flo ,n 2 2 -Iog-~- - 2fl~-ol o g ~ l o g ~ 5 -
l+fl~[ . (l-flo) ~floflO l+flo _llog21+flo 3] + ~ Ll2 ~ +log lOgl_,80 ~ +
+
for q2>4m2. Although the physical vertex is at q2=m2, we have also shown the value for negative q2 for the purpose of discussing the renormalization condition later. (in the definition offlo, the factor of m ~ should be replaced by q2 in general.) In eq. (2.8), Li2(x) is a Spence function Li2(x)=-
Tlog(l-t),
(2.9)
o
as =g~/4~ is the QCD coupling constant, and CF=~ is a colour factor. We have regularized the ultra-violet divergence by dimensional regularization with D=4-2~, p is the renormalization scale, and the infra-red singularity is regularized by a gluon mass 2. No problem occurs from the use of a finite gluon mass, since the threegluon coupling does not appear in this order. The self-energy diagrams 2b and 2c are cancelled by the counterterm diagrams 2e and 2f if one uses the onshell renormalization scheme as usually adopted in QED. The mass and wave-function renormalization counterterms in this scheme are given by m5-2 _ 4 ] . 6m_Cva~m 4n [ 3 (~ -log rn_~) +4 ] , Z 2 - 1 = ~ -Cva~[ ~ -n - (~ - l o g ~m 2) +21og)~
(2.10,
Finally, the counterterm for vertex 2d should be included. In arbitrary Yukawa theories, this counterterm is fixed by the renormalization condition for the Yukawa coupling, which can be chosen at will. For the Higgs boson, however, there is no arbitrariness in the choice of the renormalization condition, since the quark mass and the Yukawa coupling are proportional to each other. In fact the mass comes from the vacuum expectation value of the Higgs field:
5g=fdlq(v+H),
(2.11 )
with m= -fv. Since we are concerned with the QCD radiative corrections and the Higgs field carries no coiour, the combination v+H has to be renormalized as a whole. Thus the counterterm for the Yukawa coupling is fixed in terms of the quark mass and wave-function renormalization constants: 457
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= 2[-2($log$)+log$-41.
It should be stressed that this counterterm function at q2= 0:
(2.12)
is not equal to the “naive”
one obtained
by subtracting
the vertex
(2.13) This is in sharp contrast with the case of the vector vertex, for which Ward identity and zero momentum subtraction give identical counterterms. We note that the same counterterm can be used for multi-Higgs cases and also for pseudoscalar Higgs bosons (up to the overall Yukawa coupling factor). The total virtual correction is the sum of eq. (2.8b) and eq. (2.12), which is free from ultra-violet singularities: ReA,=%{-(ylog$-l)log$ 0
+?[I&(+$)
-+h3g2s
+log%logs
and the lowest-order T,,(H+qq)
0
=ro.2
rate eq. (2.7 ) is multiplied
+ 51
+ ylogs
(2.14)
_I},
by a factor 1 + 2 Re (1,:
Re,4, .
(2.15)
2.3. Gluon emission corrections The virtual correction (2.14) still contains an infra-red divergence. In order to obtain a finite result, one has to add corrections from the emission of a real gluon. In our treatment of the singularity, it is convenient to divide the gluon emission into soft and hard parts by a cut-off w. ( < m) in the gluon energy. The contributing diagrams are shown in fig. 3. 2.3.1. Soft gluon corrections The emission of a soft gluon is described factor):
by a universal
factor multiplying
the tree amplitude
(up to a colour
(2.16) Here E is the polarization vector of the gluon, and T” is the SU (3) generator in the fundamental representation. The soft-gluon emission probability integrated over the soft-gluon phase space (defined as Eg < wo) is
H(g)
-------((
_______<;;
al
458
bl
Fig. 3. Diagrams for real gluon emission, H+qqg. of the particles are defined in the parentheses.
The momenta
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\ k---pp k-P)l
polarization, ES < too colour
_ _cF-s
(l+Po logl+Po-1 ) log 4co~
1-flo
( \ 2//0
l+fl°2[ ' ( l - f l ° ) ~ f l fl° o l+fl° _~log21+flo +--~0 El2 ~ +log IOgl_flo l-flo
62 ]
+
1
I+flo ~
log 1- P o )
(2.17)
(The colour factor CF is effective in the sense that it is meaningful only in the rate. ) The soft gluon correction to the decay rate is given by this factor multiplying the lowest-order rate: F~s(H--,qQg) =Fo'd,s.
(2.18)
Adding this and the virtual correction, we find that the 2 dependence disappears, which shows the cancellation of the infra-red singularity explicitly.
2. 3.2. Hard gluon corrections The amplitude for hard gluon emission is calculated from the diagrams in fig. 3:
( P'~l~+2p~' ~Y"+2"O'~']T~v(P)e * . ,lth=fgst2(p)\ 2k'p 2k'p J
(2.19)
In general, there are only two independent kinematic variables in spin-summed three-body decays. (Although the three-body phase space has five variables, three of them reduce to trivial angular variables, on which the squared matrix element does not depend.) As the two independent variables, we may choose two of the following three invariants:
s=(p+P)Z=2p'p+2m 2, t ' = 2 p ' k = t - m 2, u'=2p'k=u-m 2,
(2.20)
with the constraint s+ t'+ u ' = m 2. Another choice is fl and 0, defined as follows:
fl= / 1 - --,4m2 "V
(2.21a)
S
p.k=~ ( m ~ - s ) ( 1 - f l c o s 0 ) .
(2.21b)
In the CM frame of the qQ system, fl is the velocity of the quark, and 0 is the angle between the quark and gluon directions. The three-body phase space in terms of these variables is 1
m 2 (flo2
2
2
(2.22)
d~3= 128n3m 2 d s d t ' = 32n3 (1 ~f12)~ - f l )fl dfldcos0. The spin- and colour-summed matrix element squared is I~Rh I2=4f X
[~7,
2NcCvg 2
t' u'
+--+
= 16f2NcCvg~
2(q2-2m2)(s-4m2) +2+4m2 ( 1 + t'u' (
2sqZfl2 "~
1+ (-~-~_s)2,] 1 f12cos20 _
l ) -2m2(q2-4rn2)(--~+ --~)1
8m2q2fl 2 1 ] ( q 2 - - s ) 2 ( 1 --]~2-C0S2~) 2 "
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Integrating over the three-body phase space with the cut-offEg> OJogives ~2 _r- CFCes { ( l + f l ~ log 1+,6'o _ 1 F i h ( H ~ q { I g ) - , o It k 2,60 1 -rio
+ l+ o . l-/ o T-o [2LI2(I--~o)+2Li2(-4 l -31og 1-fl-~ --41ogflo+ ~
l-
)
log
m2 409o2
o]
l+ o 1+ o log ~ + ~ log 2 1 - flo
1 + flo J - log
l+flo 1 } (3+2f12+3fl~)log l_-~o + ~ o (-3+29fl°2) "
(2.24)
Summing all virtual, soft, and hard corrections, it is readily seen that all singularities cancel and no COodependence remains: F, (H--*qdl+ qdlg) =Fiv +Fts +Fib =Fo
CF~s
d.,
(2.25)
where I ,J.= ~oA(flo)+ ~1
3
,
l+flo
(3+34flo2- 13fl4) log 1--~o + ~
3
(-l+7fl~),
(2.26)
with
A(flo)=(l+'8~)
[
. (l--fl0) (l--flO) 4L12 l-i--~° +2Li2 l+flo
4
--3flo log 1--,8~ --4flo logflo.
-3log
l°gl+fl°-21°gfl°'l-ti{}iog~l+fl°]_] (2.27)
This result agrees with that of Braaten and Leveille [2 ], who employed dimensional regularization to treat both ultra-violet and infra-red divergences. The final result thus does not depend on how the singularities are regularized, as it should. Since we have sued the on-shell renormalization scheme, the quark mass parameter m should be regarded as the position of the pole of the quark propagator, which may be estimated using QCD sum rules [6 ]. We remark that the results shown above hold also for scalar bosons in multi-Higgs models, provided that the proper Yukawa couplingfis used. [Only eq. (2.3) and the last equality of (2.7) should be modified. ] So far we have discussed the QCD correction to H-~qdl, but our result is directly applicable to the O ( a ) QED correction to H-, ~ (f bein'g a quark or a lepton ) with the replacement CF a~-~ aef2 (and No-, 1 for a lepton ).
2.4. Comparison withJanot's result In ref. [4], Janot provides some dctail of this calculation, which makes it possible to pin down the origin of the difference to a certain degree. [ In the following, quote the equations in ref. [4 ] as (Jm, n). ] First, his result of real emission (J5.8) is in agreement with ours, if we correct a misprint: the factor flo2 in the denominator of the infra-red-singular term in (J5.8) (on the first linc) should be/3o. In addition, the cxpression for the second Spence function is somewhat ambiguous; the argument should be squared, not the function itself. Hence the difference should have stemmed from the virtual corrections. His result appears in eq. (J4.7). Besides the same misprint in the infra-red singular term (,82 should be flo), we find two disagreements. In the ~2 The gluon energy can be defined either in the Higgs rest frame or in the q~l CM frame. In the limit O.~o<< m, the result does not depend on this choice. Needless to say, one has to employ larger cut-offs to circumvent the soft and collinear singularities in realistic simulations of the three-jet rate.
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notation o f ref. [ 4 ], we obtained the constant factor of - i instead of - 2, and for the coefficient o f In [ ( 1 -,go) / ( 1 + Bo) ] (the last term ), we found - ( I - fl~ )/rio instead o f - ( 1 + 2flo2 )/2,80. We cannot track down the reason for the discrepancy further, except that Janot seems to have used the four-dimensional y matrix algebra instead of the D-dimensional one [see the sentence above (J4.5) ], which can lead to inconsistent results in dimensional regularization. Moreover, there is no remark about how the renormalization was actually done.
2.5. Comparison with QCD sum rule calculations Calculations closely related to ours have been made in the context of Q C D sum rules. Reinders et al. [7 ] calculated the O ( a 5 ) Q C D correction to the two-point function of the scalar current, which is similar to our calculation. In their calculation, however, there is no principle to fix the renormalization condition, because their scalar current is just an external current which has nothing to do with the quark mass. Therefore, they subtract the vertex function at zero momentum, giving a result which is slightly different from ours. Similar calculations were made for the pseudoscalar current also. The result in ref. [ 7 ] was derived with zero m o m e n t u m subtraction and differs from our result for P decay. However, the result in a review [6] was derived from the divergence of the axial vector current and is in agreement with ours. Finally, a calculation of the Q E D corrections to the decay o f a pseudoscalar boson into a lepton pair has been done by Bergstr6m [ 8 ]. He restricted his calculation to the case mp >> m and obtained a result which is identical to eq. (4.2b), using the correct renormalization condition.
3. Pseudoscalar Higgs decay
Let us now turn to the hadronic decay o f a pseudoscalar Higgs boson P. Because the steps are parallel to that for a scalar H, we will only mention the main differences. The Yukawa interaction is taken as
~=fqiy5 qP,
( 3.1 )
and the lowest-order rate is
Ncf 2
Fo(P--*qq) = ~
mp//o •
(3.2)
Note that the threshold behaviour is S-wave, in contrast to the P-wave behavior of H ~ qct. The virtual correction factor differs slightly: ReAD~2a)(P--,q~I)=ReAt~2a)(H--,q~I)-
log~
(qZ>4m2) ,
(3.3)
i.e., the single logarithm term is absent. The renormalization counterterm is exactly the same as for the scalar case, as is discussed in the last section. The soft-gluon factor is also the same. The hard-gluon emission matrix element squared is
Z ['l/hlz=4f2~'Cvg~
= 16f2NcCvg~
+ u't' + 2q2(s-2m -
I + (q~_s)2 J 1 -fl2cos20
+2-2m2q2
(, ,)] ~7i + ~7i
(q2-s)2 (1 -,82cos20) 2 '
(3.4)
and the integrated rate is in the form of (2.24), with the last line charged to + ~
I
, 1 +/~o ( 3 + 2 P oz + 3 f l g ) l o g ~ + ~ ( 2 9 - 3 f l o 2) .
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Summing all the corrections, we find Fj ( P-* q~l + q()g) = Fo ( P-~q~l )
Cvas 7t
Ap,
(3.6)
where Ap=
l
A(,so) + 1 ~ o (19+2,8g+3,8~)log !-,8o1+,8°+ ~ ( 7 - , 8 g ) .
(3.7)
Note that the functional form of both the squared matrix element (3.4) and the total rate (3.6) differs from the scalar case; in addition, the coupling strength f c a n , of course, be different in the two cases.
4. Limiting behaviours The correction factors d , and dp show interesting behaviours in the limiting cases. Near the threshold m u - - ) 4 m (flo--,0), the corrections are positive and develop a 1/flo singularity: 7Z2
AN ~ ~00 -- 1 ,
(4.1a)
7.(2
Ap--- 2,8o - - 3 .
(4.1b)
It can be seen that these terms come solely from virtual corrections; both soft and hard gluon corrections vanish at the threshold. The 1/,8o singularity can be understood as coming from the colour Coulomb interaction between the quark and antiquark, and signals the formation of bound states ~3. A realistic treatment of this threshold region requires the inclusion of the mixing of the Higgs boson with the quarkonium states, both below and above the open flavour threshold. This phenomenon has been fully discussed in ref. [ 3 ]. In the opposite limit mH >> m (,8o~ 1 ), the corrections change the sign. Moreover, they contain a logarithmic singularity AH -~ _3 log rn~t + 9 ,
(4.2a)
m~ Ap ~_ - S l o g m---5- + 9 .
(4.2b)
The asymptotic equality of the scalar and pseudoscalar corrections indicates the restoration ofchiral symmetry. The appearance of the logarithmic term is related to the ultra-violet behaviour of the mass operator, which we further discuss in the next section. Because of the log factor, the correction becomes large if the quark mass is much smaller than the Higgs mass, rendering the one-loop result inappropriate (the corrected rate eventually becomes negative). In this case, one can sum the leading logarithmic terms as discussed in ref. [2 ]. As far as the leading term is concerned, this is equivalent to replacing the quark mass parameter in the coupling with the running quark mass: F(H-)q~I) =
NcGFffl( mH ) 2
4x/~ 7t
raN,8 3 ,
where rh(mH ) is the running quark mass evaluated at raN: ~3 Fore further discussion, see ref. [7 ].
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(4.3)
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rh(mH):~(2m) (Oq(mH)'] 12,<33-2m) k,a~(2m),/
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{log(4m21A2) ] '2'<33-zN' ) = rh(2m) \
~
~
/
,
(4.4)
A is the QCD scale parameter, and Nf is the effective number of flavours. The running mass at the threshold r?t(2m) may be replaced by m. The expression (4.3), however, only holds at rnH >> m and cannot be used closer to the threshold, because the discarded finite O (cq) piece becomes comparable to the retained logarithmic term. We propose to use the following interpolation formula connecting the full one-loop expression near threshold and the leading-log result at large mH:
(log(4m2/A2))24/(33-2Nf)[
CFOls(
1+
~
m2n']l an + ~ l o g - - ~ - j j .
(4.5)
How does the leading-log summation affect the qclg rate? The answer lies in the origin of the final log factor. The general theorem of the cancellation of mass singularities ensures that all soft and collinear singularities cancel in the rate. The remaining logarithm thus should come from the renormalization [2 ], which suggests that the logarithm belongs to the overall Yukawa coupling factor. Therefore, the quark mass parameter in the coupling f i n the rate for qclg should also be replaced by the running mass.
5. Conclusion
In this note we describe the computation of the O ( ~ ) QCD correction to the hadronic decay of a scalar and pseudoscalar Higgs boson. We gave the virtual and real corrections separately and the differential rate for the real emission, H, P--,qdlg is also given. The distribution for P--,qflg has not appeared elsewhere. A subtlety in the renormalization of the Yukawa coupling is discussed in detail. The difference between our result and that of Janot is shown to arise from the virtual correction which crucially depends on the renormalization condition. The difference of the two results becomes prominent in the large Higgs mass limit mH >> m. If we denote the rate by F=Fo+Fj, where Fo is the lowest order result and F, is the O(~s) correction, Janot [4] obtains a constant correction factor in this limit:
Fl Ja"°'~ 1"o
5 C~a~ 4 rc
(5.1)
On the other hand, Braaten and Leveille [ 2 ] and ourselves found Fl
-- ~ log - - ~ +
--
(5.2)
The appearance of the logarithm may seem strange at first glance: indeed, the QCD correction for a vector current (e.g. for the R ratio in e + e - - , h a d r o n s ) is a constant (]CFa~/n), as is well known. This difference, however, can be understood as follows. The logarithmic term is absent for the R ratio since the vector current is conserved and its anomalous dimension vanishes. The scalar current, or the mass operator, however, has nonvanishing anomalous dimension, which manifests itself in the logarithmic correction. As is mentioned in the previous section, summing over the leading logarithms turns the quark mass parameter in the coupling factor into the running quark mass. This high-mass region rnH >> m has been subsequently studied by Sakai [ 9 ] and Inami and Kubota [ 10 ] in a more rigorous framework, and Russian group even derived the O ( a ~ ) correction in this limit [ 11 ]. Since the nature of the logarithmic factor is so well understood, we believe that the result of ref. [4 ], which does not have the logarithmic term, is incorrect.
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Note added A f t e r s u b m i t t i n g o u r paper, we were i n f o r m e d that the A L E P H c o l l a b o r a t i o n in fact used the c o r r e c t i o n factor o f B r a a t e n - L e v e i l l e a n d us in t h e i r analysis, while the O P A L c o l l a b o r a t i o n c o n s e r v a t i v e l y did not i n c l u d e any Q C D c o r r e c t i o n s w h e n d e r i v i n g t h e i r b o u n d s . T h e size o f the c o r r e c t i o n s is not very large in the mass range r e l e v a n t to these searches. H o w e v e r , the Q C D c o r r e c t i o n s b e c o m e m u c h m o r e i m p o r t a n t at L E P 2 and higherenergy colliders because o f the l o g a r i t h m i c factor.
References [ I ] L. Resnick, M.K. Sundaresan and P.J.S. Watson, Phys. Rev. D 8 (1973) 172. [ 2 ] E. Braaten and J.P. Leveille, Phys. Rev. D 22 (1980) 715. [31 M. Drees and K. Hikasa, Heavy quark thresholds in Higgs physics, KEK TH-224, CERN-TH.5393/89 (May 1989), Phys. Rev. D 41, in press. [41P. Janot, Phys. Lett. B 223 (1989) 110. [ 5 ] ALEPH Collab., D. Decamp et al., Phys. Lett. B 236 ( 1990 ) 233; B 237 (1990) 29 I; OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B 236 (1990) 224. [6] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127 (1985) I. [ 7 ] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B 186 ( 1981 ) 109. [ 8 ] L. Bergstr~Sm,Z. Phys. C 20 ( 1983 ) 135. [9] N. Sakai, Phys. Rev. D 22 (1980) 2220. [10] T. Inami and T. Kubota, Nucl. Phys. B 179 (1981) 171. [ 11 ] S.G. Gorishnii, A.L. Kataev and S.A. Larin, Yad. Fiz. 40 (1984) 517 [Sov. J. Nucl. Phys. 40 (1984) 329].
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