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085_tions to the partial decay width _(H ! bb) in the order s

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085_tions to the partial decay width _(H ! bb) in the order s

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     The large top quark mass expansion for Higgs boson decays into bottom quarks and into gluons

     PostScript? processed by the SLAC/DESY Libraries on 3 Jul 1995.

     NIKHEF-H/95-027 June 1995

     S.A. Larinab , T. van Ritbergenb, J.A.M. Vermaserenb

     a b

     Institute for Nuclear Research of the Russian Academy of Sciences 60th October Anniversary Prospect 7a, Moscow 117312, Russia NIKHEF-H, P.O. Box 41882, 1009 DB, Amsterdam, The Netherlands

     We calculate the large top quark mass expansions for the H ! bb decay rate in the order 2 and for the H ! gluons decay rate in the order 3 . The obtained s s expansions rapidly converge in the region of their validity, MH < 2mt, i.e. below the threshold of tt production.

     Abstract

     1 Introduction

     With the discovery of the top quark 1], the Higgs boson remains as the only fundamental particle of the Standard Model which is not found experimentally. Considerable e orts were devoted to calculate perturbative corrections within the Standard Model to di erent Higgs boson decay channels, for a review see 2]. In the present paper we calculate top quark mass corrections to Higgs boson decays into bottom quark-antiquark pairs and into gluons, i.e. to the Higgs boson partial widths ?(H ! bb) and ?(H !gluons). The decay process H ! bb is the dominant decay channel for the Higgs boson with an intermediate mass MH < 2MW and will be of prime importance in the future experimental searches of the Higgs boson at colliders. The decay channel H ! gluons is interesting since heavy quarks that mediate this process contribute to the decay rate without being suppressed by their large mass which eventually may provide a way to count the number of heavy quarks beyond the Standard Model. The purpose of the present paper is to examine the convergence properties of the series of top quark mass suppressed corrections to the Higgs boson decay rate in the region 1

     HEP-PH-9506465

     MH < 2mt, where the large top quark mass expansion can be applied. These corrections a priori can be sizable but they turn out to be small for the processes considered so far. In ref. 3] we considered the top quark mass corrections to the Z boson and lepton decays into hadrons. In this letter we continue this program to the case of the Higgs decays into hadrons. Throughout this paper in our calculations we use dimensional regularization 4] and the standard modi cation of the

minimal subtraction scheme 5], the MS scheme 6].

     2 corrections to ?(H bb) s We use the optical theorem to calculate the top mass suppressed corrections to the partial decay width ?(H ! bb) in the order s . To this end we should calculate the imaginary parts of the 3-loop Higgs boson propagator diagrams of non-singlet and singlet types with top quark loops listed in g.1a and g.1b correspondingly.

     2 The order

     !

     2

     decay width ?(H ! bb). Thin lines indicate bottom quark propagators, thick lines indicate top quark propagators and spiral lines indicate gluons. The symbol indicates a Yukawa type Higgs{quark vertex. Diagrams 1a are of the non-singlet type, Diagram 1b is of the singlet type.

     Figure 1. Diagrams contributing to top mass suppressed corrections to the partial

     To extract the contribution to this channel from the singlet "double triangle" diagram in g.1b we should subtract from the imaginary part of this diagram the contribution to ?(H ! gluons) associated with two gluon cut in this singlet diagram. The contribution from this two gluon cut can be straightforwardly calculated 8] by the use of Feynman parameters. For the calculation of the propagator diagrams in g.1 we applied the diagrammatic large top quark mass expansion to each of the diagrams separately, using the technique which was previously used in 3] and which is based on the method developed in 7]. Although the formal parameter of this expansion is not small (the mass ratio MH =mt can be larger than 1) this expansion is applicable below the threshold for the production of tt pairs, i.e. MH < 2mt. (One can say that the real expansion parameter is MH =2mt.) For the actual calculations we relied on the symbolic manipulation program FORM 9] and 2

     we used the package MINCER 10] together with additional FORM routines to perform massive integrals. The Yukawa type Higgs{quark couplings contribute a factor mb for the non-singlet diagrams and a factor mbmt for the singlet diagram. We work in the leading order in small b-quark mass, mb << MH , and nullify other light quark masses. This technically means that for the non-singlet diagrams the mass in the b-quark propagators is nulli ed from the start but for the singlet diagram one power of the b-quark mass from the propagators must be kept to prevent the trace over the Dirac matrices in the b-triangle from vanishing. The higher small mb -corrections can be found in 11] for non-singlet contributions and in 12] for the full case. After addition of the massless (mt independent) non-singlet contributions calculated in 13] we obtain the following result for the large top quark mass expansion of the Higgs decay rate into bb in the leading order of the b-quark mass

     2 2

     (H ! bb) = ?NS (H ! bb)+?S (H ! bb); ?NS (H (m ! bb) = GF MHp2 b )

    n 1 + 4

     (6) 2 (6)

     "

     (6)

     s

     !

     f NS; +

     1 0 1

     s

     !2

     f NS; + f NS;

     2 0 2

     f NS; = CF ( ),

     1 0 17 4

     MH + f NS; MH + f NS; MH + O( MH ) 5 ; mt mt mt mt (1)

     2 2 2 2 4 4 2 3 6 8 6 8 2

     !3

     f NS; = CF +CACF +TF CF Nf +TF CF

     2 0 2 2 1 2

     691 64

     893 64

     337 72

     ? ? ln( MH ) + ln ( MH ) ? ? ? ln( MH ) + ln ( MH ) ? + + + ln( MH ) ?

    ln ( MH ) ? ? ? ln( MH ) + ln ( MH ) , mt mt

     9 4 3 3 8 2 105 16

     2

     2

     9 8

     2

     2

     31 8 3

     11 48

     2

     71 12

     2

     2

     11 16

     2

     2

     2

     65 16

     3

     1 12

     2

     11 6

     2

     2

     1 4

     2

     2

     2

     3

     1 12

     2

     11 6

     2

     1 4

     2

     2

     2

     f NS; = TF CF

     2

     107 450

     ln( MH ) , mt

     1 15

     2

     2

     f NS; = TF CF ? f NS; = TF CF

     2 3

     529 58800

     +

     1 280

     ln( MH ) , m2 t

     2

     2719 3572100

     1 2835

     ln( MH ) , m2 t

     2

     3

     S (H

     2 0

     (m ! bb) = GF MHp2 b ) n 4 4

     (6) 2

     2

     (6)

     s

     !2

     f2S;0 + f2S;1 MH m2 t

     2 2 3 1 6 2

     + f2S;2 MH m4 t

     4

     + f2S;3 MH m6 t

     6

     f S; = TF CF ? ln( MH ) + + ? m2 t

     2

     14 3

     1 6

     2

     mb ? + ln ( MH ) ,

     2 2

     7 720 2

     + O ( MH ) 5 ; mt (2)

     8 8

     !3

     f S; = TF CF ?

     2 1

     41 1080

     ln( MH ) + m2 t

     2 2 2

     2011 16200

     + ?

     7 180 2

     +

     7 720

     mb ln ( MH ) , 2

     2

     2

     f S; = TF CF ?

     2 2

     47 15120

     ln( MH ) + m2 t

     28307 3175200

     + ?

     1 1008

     1 252 2

     +

     1 1008

     mb ln ( MH ) , 2

     2

     2

     f S; = TF CF ?

     2 3

     59 168000

     ln( MH ) + m2 t

     100381 105840000

     + ?

     13 100800

     13 25200

     +

     13 100800

     where we presented separately the contributions from the non-singlet diagrams, ?NS (H ! bb), and from the singlet diagram, ?S (H ! bb). The contribution of the two gluon cut is subtracted from the singlet part (the corresponding terms are indicated in curly brackets). GF is the Fermi constant. CF = and CA = 3 are the Casimir operators of the fundamental and adjoint representation of the colour group SU (n), n = 3 is the number of quark colours, TF = is the trace normalization of the fundamental representation. The result is for 6- avour QCD, Nf = 6, s is the strong coupling constant and mb ( ), mt( ) are running MS bottom and top quark masses. The e ects of using the pole mb mass instead of the running mb mass can be found in 14]. The singlet type coe cients f S; and f S; agree with the previous calculations 15] and 12] correspondingly. The non-singlet type coe cients agree with the previous calculation 16]. The singlet coe cients f S; , f S; are new results. To obtain the result in e ective 5- avour QCD where the non-singlet top quark contributions decouple 17], one should substitute the b-quark mass mb and the coupling constant s in terms of their values in the e ective 5- avour QCD by appling the known decoupling relations 18, 19, 3] :

     4 3 1 2 (6) (6) 2 0 2 1 2 2 2 3 (6) (6) (6)

     mb ln ( MH ) , 2

     2

     2

     s

     ( )=

     (5)

     s

     ( )+

     2 ) 41 +

     (5)

     s

     ( )

     (5)

     !2

     TF ln( 3 mt ( ) ) + O ( s )

     2 3 2

     (3)

     ( ) T C ? 1 ln ( mb ( ) = mb ( F F 8 mt ( ) ) ! # 89 + O( ) 5 ln( (4)

    + 24 m ( ) ) ? 288 s t Substitution of (3) and (4) in (1)+(2) and putting

    = MH gives the following result for

     (6) (5)

     s

     !2

     2

     2

     2

     2

     3

     2

     4

     the Higgs decay rate into bb in e ective 5- avour QCD

     p ?(H ! bb) = 3GF MH (mb ) 41 + 4 2

     5 2

     2

     (5)

     s

     !

     f +

     1

     (5)

     s

     !2

     f +O

     2

     !3 3 s 5;

     (5)

     f =

     1 2

     17 3

     5:66667,

     17 12 2 2 113 1620 2 3 2 1 9 2 7 1080 1 1512

     (5)

     f =h ? ? ? ln(x) + ln (y) i +h ? ? ln(x) + ln (y) xi ? + ln(x) + ln (y) x i + h? ? ? ln(x) + ln (y) x + O(x ) +

     9235 97 144 6 3 5233 7 24300 1080 1837 680400 3411287 4286520000 1 2 1512 13 151200 1 3240 2 2 2 3193 6804000 13 151200 2 3 4

     (y) i + h0:15138 ? 0:06975 ln(x) + 0:0064815 ln (y) x i + h?0:009227

    + 0:00030864 ln(x) + 0:00066137 ln (y) i x + ?0:00005276 ? 0:0004693

ln(x) + 0:00008598 ln (y) x + O(x )

     2 2 2 2 2 3 4

     30 h :71675 ? 0:66667 ln(x) + 0:11111 ln

     where, x = MH =mt , y = mb =MH . Please note in the leading order of the large top quark mass expansion the presence of ln(MH =mt ) coming from the singlet contribution which is not a ected by the decoupling procedure (since the singlet type contributions are renormalized independently of the nonsinglet ones). It shows the violation of decoupling for the considered process due to the singlet contribution, which was rst found in 15, 12]. We conclude this section with stressing the fast convergence of the obtained large top quark mass expansion for ?(H ! bb) in the region below the threshold of tt production, MH < 2mt as can be seen from the fast decrease of the coe cients of the expansion. Even in the region close to this threshold (where x 4) one observes a fast convergence of the large top quark mass expansion.

     2 2 2 2 2 2

     3 corrections to ?(H gluons) s Higgs boson decay into gluons is possible since both the Higgs particle and gluons couple to massive quark loops. An important property of this partial decay rate is that contributions from heavy quarks in these loops are not suppressed by their mass such that measurement of this decay rate counts the number of heavy quarks. The contributions to ?(H ! gluons) start from the order s (and this process is therefore not as prominent phenomenologically as the Higgs decay rate into bottom quarks which receives tree level contributions).

     3 The order

     !

     2

     5

     (H ! gluons) is known 20, 21] in the order s in the leading order of the large top quark mass expansion and the s correction turns out to be large (about 2/3 of the leading s contribution). In this section we present the higher orders in the large top quark mass expansion of the Higgs decay rate into (two and three) gluons H ! gg(g) mediated by top quark loops. The gqq nal states (with light quarks q=u,d,s,c,b) generated in the chain H ! gg ! gqq are also included. For this process we work in the approximation in which the rst ve quark avours are massless. By applying the optical theorem, the necessary contributions are given by the imaginary part of the singlet 4-loop diagrams of the Higgs propagator in which each of the external Higgs vertices is located in a di erent top quark loop. Clearly, for a Higgs mass below the top threshold, MH < 2mt, the only physical cuts that can be drawn in these propagator type diagrams are through gluons and light quarks. All contributing diagrams are of the singlet type and were encountered

previously in the calculation of ?(Z ! hadrons) 3] where a complete

    list of the necessary 4-loop singlet diagrams was given. Applying the

    diagrammatic large quark mass expansion to all contributing diagrams

    yields the result in six avour QCD, Nf = 6,

     3 3 2

     p ?(H ! gluons) = GF MH 72 2

     3

     (6)

     s

     (6)

     !2

     +

     s

     !"

     TF D 1 + h MH + h MH + O MH mt mt mt

     2 1 2 2 2 2 2 4 6 4 6 1 3 2 2 2 3 4 6

     (

     !

     h =

     1 2

     h + h MH + h MH + O MH mt mt mt

     0 3 4 6

     !# )

     ;

     (6)

     7 60

     , ,

     11 6

     h =

     2 2 0 3

     1543 100800 103 12

     h = CA ? ln( M2H ) + CF ? 2 2 H +TF Nf ? + ln( M2 ) + TF ? ln( MH ) ,

    m2 t

     2

     3 2 7 3 2 3 7 3 2 3

     h = CA ? +TF Nf ?

     1 3 71 80 2 3

     77 360 29 120

     ln( M2H ) + CF ? ln( m22t ) 2 2 H + ln( M2 ) + TF ? ln( MH ) , m2

    t

     2

     13 720 7 40 7 90 29 120 7 90 16973 604800

     h

     i

     ln( MH ) + CF ? h = CA 2 2 +TF Nf ? + ln( M2H ) + TF

     47459 432000

     2

     h

     83 10080

     1543 33600

     89533 3024000

     1543 151200

     89533 3024000

     ln( m22 ) t

     i

     1543 151200

     ln( MH ) , m2 t

     2

     where D = n ? 1 is the number of generators of the colour group SU (n) (D = 8 for

     2

     6

     QCD). In the leading order of the large top quark mass expansion our s result agrees with 21]. Explicit checks show that the coe cients of the logarithms in eq. (6) are in agreement with the required renormalization group invariance of the physical quantity ?(H ! gluons). Furthermore, the results that are presented in this paper were obtained in an arbitrary covariant gauge for the gluon elds i.e. keeping the gauge parameter as a free parameter in the calculations. The explicit cancellation of the gauge dependence in the physical quantities gives a good check of the results. Applying the decoupling relation (3), putting = MH and substituting the QCD colour factors gives the following result for e ective 5 avour QCD

     3

     p ?(H ! gluons) = GF MH 36 2

     3 2 7 60 1543 100800 2

     (5)

     s

     !2 "

     h +

     2 3

     (5)

     s

     !

     h + O( s ) ;

     3 2

     #

     h =1+ x+ x + O(x ) 1 + 0:116667x + 0:015308x + O(x ),

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