PHYSICAL REVIEW D VOLUME 53, NUMBER 9 1 MAY 1996 Two loop O N f s corrections to the decay width of the Higgs boson to two massive fermions K Melnikov Institut für Physik, THEP, Johannes Gutenberg Universität, Staudinger Weg 7, Mainz, D 55099, Germany Received 1 December 1995 We present an analytical calculation of additional real or virtual radiation of a light fermion pair in the fermionic decay of the Higgs boson H f 1 f 1 for arbitrary ratios of the Higgs boson mass to the f 1 fermion mass This result gives us a value of the O(N f s ) radiative correction to the inclusive decay rate H f 1 f 1 Using this result in the framework of the Brodsky-Mackenzie-Lepage scheme, we discuss the question of the scale setting in the one-loop QCD correction to the decay width H f 1 f 1 for an arbitrary relation between the Higgs boson and fermion masses PACS numbers: 1480Bn, 138Bx I INTRODUCTION Fermionic decay channels of the standard model SM Higgs boson are important channels for both discovery and investigation of this particle 1, Direct observation of such decays can give important information about Higgs-fermion Yukawa couplings and hence provide still absent check of the symmetry-breaking mechanism in the fermion sector of the SM In this paper we discuss the O(N f s ) correction to the decay of the Higgs boson to the pair of massive fermions H f 1 f 1 for arbitrary relation between the Higgs boson mass and the mass of the fermion f 1 This decay is studied in the literature well enough The one-loop QCD radiative correction to the fermionic partial width of the Higgs boson was calculated long ago 3 Since then the studies of this decay were concentrated on the analyses of the limit m H m 1 from the phenomenological point of view this is definitely a good approximation for the decay H bb ) In this limit the renormalization group methods were applied to this partial decay width 5 and the exact results for the complete O( s ) correction including the power suppressed terms O(m 1 /m H ) were obtained analytically 4,6 Recently in 7 these results were rederived and some new, previously missed, contributions were calculated Our work is motivated by the known fact that the QCD radiative corrections to a number of processes involving Higgs-boson quark interactions appear to be large It is possible to attribute a bulk of them to the running of the Yukawa coupling or equivalently to the running of the fermion mass In this respect it is important to calculate the next-to-leading order QCD radiative corrections in order to reduce the scale ambiguity of the leading order results and check our understanding of the resummation procedures based on the renormalization group equations The complete analyses of the problem requires a full twoloop calculation of the QCD radiative correction to the H f 1 f 1 decay channel for an arbitrary relation between the Higgs boson and the quark masses This task is complicated enough An easier way is provided by the Brodsky-Lepage- Mackenzie BLM method 8 This method gives the possibility to obtain a value of the correct scale in the one-loop QCD correction and hence a good idea of a two-loop contribution 1 by considering the two-loop diagrams, which arise due to the light fermion loop insertions into the gluon propagator in the one-loop QCD correction see Fig 1 Similar arguments and techniques were used for the analyses of the scale setting in the QCD radiative corrections to the parameter and to the top quark decay width 9 In some sense, our results are complementary to the results presented in 4,6,7 we calculate the part of the twoloop QCD radiative correction, which corresponds to the running of the QCD coupling constant, however keeping the relation between the Higgs boson mass and the mass of the fermion arbitrary We mention here that our approach is similar to the one of Ref 10 The subsequent part of the paper is organized as follows: in the next section we discuss real radiation of the light fermion pair in H f 1 f 1; in Sec III we analyze virtual radiation of the fermion pair in two cases m 1 m and m 1 m ;in Sec IV we discuss the O(N f s ) correction to the total decay width H f 1 f 1; finally we present some remarks and conclusions Some comments concerning our notations are in order It is clear that a major part of our discussion applies to the QED case as well Hence, in the first two sections we use the QED terminology While discussing the total decay width H f 1 f 1 in Sec IV, we switch to the QCD notations, explicitly indicate N f dependence of the result, and use appropriate color factors II REAL DECAY RATE H f 1 f 1f f The Higgs boson couples to fermions proportionally to their masses Hence, we consider only diagrams where the Higgs boson is connected with the heavy fermion lines The generic graphs are shown in Fig 1 The decay rate for the process H f 1 f 1f f, normalized to the lowest order width (H f 1 f 1), can be written as the two-dimensional integral 1 Note however that in some cases large higher order corrections cannot be absorbed to the strong coupling constant by choosing the BLM scale 8 0556-81/96/539/5008/$1000 53 500 1996 The American Physical Society
53 TWO LOOP O(N f s ) CORRECTIONS TO THE DECAY WIDTH 501 FIG 1 Generic diagrams for the O(N f s ) correction a Virtual radiation b Real radiation H f 1 f 1f f H f 1 f 1 3 1 F R, F R 1 1r 1y 3 4r1 dy dz1 r 1 4r 4r z z z 1yz 4r 1 14zy16r 1 1yz ln 1yzy1/ 1,y,z 1yzy 1/ 1,y,z The integration variables have the following meaning: m H y is the invariant mass of the pair of heavy fermions f 1 f 1 while m H z gives the invariant mass of the pair of light fermions f f In the limit of interest (m 1 m ) the integration can be split into the soft and hard regions, depending on the energy of the light fermion pair Integrations over these regions are performed separately Our result for the decay rate H f 1 f 1f f can be written as F R f R ln m f 1 m R ln m f 0 H m R H Below we present the functions f () R,f (1) (0) R,f R obtained by the direct integration of Eq 1: f R 1p 1p lnp 1, 5 4 where and 4y1/ 1,y,z14r 1 r 1 z 1yz y 1,y,z, 1 r 1 m 1, r m m, 14r H m 1 H f 1 R 1p 1p 4Li p6li p4 ln1plnp4 ln1plnp 1 ln p 4 ln1p 11p4 4p 3 0p 44p13 61p 1p lnp y1 4r 1 y, 1,y,z1y z yzyz 3 591p 136p 11p, 6 f 0 R 1p 1p 8Li 3 1 1p4Li 3 p 8Li 3 1p10 Li 3 1p 8 Li 3 p53 4 3 ln3 1p 1 6 ln3 p 16 lnpln 1p16 ln1pln1plnp ln pln1p1 Li p ln1plnp 8 Li pln1plnp16 ln1p4 ln1plnp 413pp 3p 3 p 4 1p 1p 58p8p 8p 3 5p 4 31p 1p lnpln1p8 ln1p 591p 136p 31p Li p 17p4 8p 3 4p 15p55 61p 1p 76p14p p 3 17p 4 31p 1p 1 ln1p 361p 1p 575p 4 104p 3 530p 608p133lnp313p 4 160p 3 40p 56p13ln p15p 4 8p 3 8p 8p5lnpln1p Li p 1p 4331p 830p 7 Note, that the double logarithmic form factor f R () is proportional to the infrared divergent part of the partial decay width H f 1 f 1g 3
50 K MELNIKOV 53 The variable p is defined by the equation m H 1p m 1 p Li and Li 3 are di and trilogarithms, defined in accordance with 11 Equations 5 7 give the exact result for the real decay rate H f 1 f 1f f in the limit m H m, m 1 m Let us consider now the limit of the heavy Higgs boson, which is given by the conditions r 1 1, r r 1 In this case we expand the complete formula up to the terms of the order of O(r 1 ) and get F R 1 lnr lnr 1 1lnr 1 ln r 1 13 6 lnr 1 4 1 59 1 6 ln3 r 1 13 1 ln r 1 133 36 lnr 1 53 55 6 r 1 lnr lnr 433 36 3 lnr 1 53 6 3 ln r 1 7 lnr 1 403 18 8 The opposite limit is realized when the mass of the Higgs boson is close to the masses of two heavy fermions: r 1 1 In this case the velocity of the fermion is small The photon couples to a slow fermion proportionally to the velocity of the latter Hence we expect that in the limit r 1 1 the emission of the pair should be suppressed as the square of the velocity Calculating this limit from the complete expression, one finds F R 3 ln r lnr 16 799 ln 3 1 3 3 ln 799 745 ln8 18 43 9 III VIRTUAL RADIATIVE CORRECTION A General formulas In this section we discuss virtual radiation of additional fermion pair First we present some general formulas which are valid for an arbitrary relation between fermion masses m 1 and m Later we analyze two cases of practical importance: m 1 m and m 1 m Additional virtual radiation of a fermion pair corresponds to the insertion of the fermion loop to the gluon line in the one-loop QCD correction Fig 1a The first step in this consideration is to write the contribution of the light fermion pair to the gluon polarization operator through the dispersion integral subtracted at zero-momentum transfer, which corresponds to the QED-like normalization of the coupling constant: 3 1 k d 3 4m 1 m 1 k i 1 4m 10 Because of the vector current conservation, k k part of the polarization operator does not contribute to the physical amplitude To evaluate the O(N f s ) correction to the Yukawa coupling we consider both bare radiative correction to the triangle graph Fig 1a and the counterterms In both we insert a fermion loop into the gluon line After writing the polarization operator through the dispersion integral Eq 10, the integration over the gluon mass factorizes Hence as the first step we evaluate corrections to the Yukawa coupling coming from the massive vector boson exchange between heavy fermions and then integrate this result over the masses of the vector boson with the spectral density given by the fermion contribution to the imaginary part of the gluon polarization operator see Eq 10 It is clear then that the renormalization can already be discussed at the first step of our calculation The counterterm Lagrangian is known from the one-loop QCD radiative correction 3 and can be written as 4 L ct g Y S m, m 1 V m, S m, H 11 Here g Y is the Yukawa coupling, V,S are defined through the quark mass operator, p,ipˆ V p,m 1 S p, 1 and V,S is the derivative of the corresponding quantity with respect to p In accordance with the preceding discussion we explicitly indicate the dependence of the quark mass operator on the gluon mass To proceed further to a more precise discussion let us fix the notations We write the Yukawa coupling of the Higgs boson to the fermions in the following way: ig Y T V H, 13 where T V () is the two-loop form factor The sum of the bare radiative corrections to the vertex and the counterterms gives the following representation for the O(N f s ) correction to the Hf 1 f 1 vertex: 3 As far as we are concerned with the QED-like graphs such subtraction is evidently possible Technically, it is more convenient to change the scale of the coupling constant in the final result than to subtract gluon vacuum polarization at an arbitrary scale 4 This counterterm Lagrangian corresponds to the on-shell subtraction of the quark mass operator Hence m 1 below is the pole mass of the quark
53 TWO LOOP O(N f s ) CORRECTIONS TO THE DECAY WIDTH 503 T V 3 4 d4 k 4 i4m 1 kˆ 4p 1 p k p 1 k m 1 p k m 1 m 1 V m 1, S m 1,, 14 T V 4m d 1 m 1 4m T V 15 We assume here that the matrix element of the matrices in Eq 14 must be evaluated with respect to the on-shell fermion and antifermion spinors This formula is valid for arbitrary relation between the masses m 1 and m Below we consider two special cases m 1 m and m 1 m In Ref 10 it was suggested to calculate first the oneloop integrals with the arbitrary gluon mass and then to integrate this result with the gluon spectral density Here we choose a different way, which, in our opinion, is more suitable for the two special cases we are interested in To demonstrate it, we discuss below the calculation of the contribution of the scalar three-point function term proportional to 4p 1 p in Eq 14 to the two-loop form factor T () V B Contribution of the scalar three-point function In this subsection we indicate all the steps which are necessary to evaluate the integral d I 4m 1 m 1 4m Cm H, 16 Here C(m H, ) is the scalar three-point function: Cm H, d4 k 4 1 k p 1 k m 1 p k m 1 17 Clearly, C(m H, ) is the analytic function in the complex plane of the s variable with the cut going from 4m 1 to along the real axis We write the dispersion representation Cm H, i 1 4 4m1 C I s, ss ln s4m 1 ds ssi C Is,, 18 I i ds 16 4m1 ssi 1 d ss 4m 1 m 1 4m ln s4m 1 It can be seen that in both cases of interest this representation is very convenient for further integration First, in the case m 1 m we have the one-scale integral Integration over provides a simple expression and the subsequent integration over s is cumbersome but trivial The second case m 1 m is more tricky If s4m 1 4m the integration over can be simply performed providing the possibility to make subsequent integration over s The region of integration questioning this opportunity is the region s4m 1 4m It is easy to see however, that the contribution of this region is suppressed as O(m ) Hence it can be completely neglected as long as we are not interested in the light mass power corrections We hope that after this discussion all the steps necessary to evaluate the virtual radiation become clear C Results for the virtual corrections Finally we present the result of our calculation of the virtual radiative corrections In the case m 1 m we write the O(N f s ) correction in the form and T V 1 6 F V F V f V ln m f 1 m V ln m f 0 H m V H The expressions for the quantities f V (i) are f V 1p 1p lnp 1, f 1 V 1p 1p Li p4ln1plnp ln1plnp 1 ln p ln1p 19 0 1 Using this representation in Eq 16 and changing the order of integration we get 16p4p 31p lnp 8 3,
504 K MELNIKOV 53 f V 0 1p 1p Li 3 p4li 3 1p3Li plnp ln1p lnp6 ln1p 4ln1p 4lnpln1pln1pln pln1p ln 1plnpln pln1p 1 6 ln3 p 1 31p 104p10p Li pln1plnp1448p6p 16p4p ln p 4 lnpln1p 4 3 lnp 14p13p ln 1p 16 77 ln1p 3 18 3 In the case when the Higgs boson is much heavier than the fermion f 1, the expression for F V () reads F V 1 lnr lnr 1 1lnr 3 14 77 3 18 1 r lnr lnr 1 ln r 1 3 lnr 14 8 3 1 6 ln3 r 1 1 3 ln r 1 4 9 lnr 1 6lnr 1 4 33ln r 1 16 lnr 1 8 8 9 4 In the opposite case, when the Higgs boson mass is close to the threshold of two heavy fermions, the following result can be obtained: F V m ln m 11 H 3 ln 4m m H 5 6 5 It is instructive to combine this result with the threshold O() correction which can be found in 3 The result for the threshold form factor is then T V 1 1 1 3 ln 4m m H m 3 ln 6 5 m H 11 3 Now it is clear that we can eliminate large logarithms appearing in the next-to-leading order calculation by appropriate choice of the coupling constant The modified minimal subtraction scheme (MS) coupling constant, renormalized at the arbitrary scale, can be related to the on-shell renormalized coupling constant by the equation MS 1 MS 3 ln m 3 O MS Substituting this expression to the equation for the threshold form factor, one finds T V 1 MS 1 3 ln 4 m H 6 5 1 3 ln m H 11 3 6 To eliminate the large logarithms appearing in this expression we have to choose see also 10 two different scales for the MS coupling constant: in the terms exhibiting Coulomb singularity we set m H hence the scale is given by the value of the nonrelativistic three momenta of the corresponding particles while for the part of the correction which does not exhibit Coulomb singularity the reasonable scale is (1/4)m H Hence relevant expression for the threshold form factor reads T V 1 MSm H 4 1 11 9 MSm H /4 1 5 18 7 Our discussion of the threshold region given above is quite similar to the discussion given in Ref 10 for the threshold behavior of the vector current form factors for a more detailed discussion see Ref 1 This similarity is definitely in accord with the universality of the threshold region where dynamics is defined by a long- range Coulomb force Exactly the same technique as above can be used in the equal mass case m 1 m In this case all relevant integrations can be performed very quickly providing the simple results The effective form factor in this case reads f 0 V 1p 1p 1 6 ln3 p lnp 54p5p 16pp 61p 4 ln p 471p4 p4410p44p 91p1p 3 lnp14 4071p5 1389p1p 3 98p 1p 181p1p 4 8 The threshold expansion can be obtained from the previous equation:
53 TWO LOOP O(N f s ) CORRECTIONS TO THE DECAY WIDTH 505 FIG The BLM scale for the one-loop QCD correction to the decay width H f 1 f 1 expressed through the pole quark mass The vertical axes is the ratio BLM /s, the horizontal axes is the ratio m/s f 0 V 137 13094 1 9 6 The high energy expansion in this case is F V 1 6 ln3 r 1 5 6 ln r 1 8 9 lnr 1 14 407 18 r 1 10 lnr 1 4 8 9 30 FIG 3 The BLM scale for the one-loop QCD correction to the decay width H f 1 f 1 expressed through the pole quark mass The vertical axes is the ratio BLM /s, the horizontal axes is the ratio m/s Here 1 is the one-loop QCD radiative correction to the decay width see Ref 3 and f V (0),f R (0) are given by Eqs 3 and 7, respectively 0 is the lowest order value for the decay width H f 1 f 1: 0 3G Fm H m 1 3 4 3 We now apply the BLM scale fixing procedure for the decay width H f 1 f 1 including the full mass dependence of the radiative corrections The general result for the BLM scale is then IV DECAY RATE H f 1 f 1 Let us now discuss the total decay rate H f 1 f 1 which is obtained by summing virtual and real corrections calculated so far 5 In doing so, we find that the double logarithms of the ratio of the square of the mass of the light fermion to the Higgs boson mass cancel, while the single logarithmic term survives The coefficient of this logarithmic term is proportional to the one-loop QCD correction to the total decay rate of H f 1 f 1 3 Hence, we can eliminate this large logarithm by expressing the total decay rate H f 1 f 1 through s ( ) evaluated at the scale m H s After that, the expression for the decay width H f 1 f 1 including the O(N f s ) corrections reads H f 1 f 1 0 1 4 3 ss 1 9 N f ss f 0 V f 0 R 31 5 We remind the reader that in this section we completely switch to the QCD terminology Below s always denotes the QCD coupling constant in the MS scheme BLM sexp f 0 V f 0 R 1 33 The numerical results for the ratio BLM /s as a function of the ratio m 1 /s are shown in Figs and 3 The one-loop radiative correction goes to zero in the vicinity of the point m 1 /s03 Around this point the BLM analyses can not be applied The threshold BLM scale approaches zero, in accordance with the discussion in Sec III Note that the scale is quite low even sufficiently far from the threshold Beyond the point m 1 /s03 the BLM scale for the coupling constant is extremely low being of the order of 00100m H We remind the reader that our discussion was applied to the width expressed through the pole mass of the quark As has been recently pointed out see for instance Ref 9 the low value of the BLM scale which is usually obtained in such cases is connected with the fact that the pole quark mass receives large contributions from the region of the small loop momenta The way to avoid this problem is to express the result for the width in terms of the running quark mass Usually, this transformation is used in the asymptotic regime for the radiative corrections Here we want to check it for the whole mass range For this aim we express the result for the width H f 1 f 1 through the running mass keeping the
506 K MELNIKOV 53 3G Fm H m 1 s 1 17 4 3 3 1 65 3 N f ss ss 3 N f ss 4 m 1 s s 434 313 640 ss 18 36 FIG 4 The BLM scale for the one-loop QCD correction to the decay width H f 1 f 1 expressed through the running quark mass The vertical axes is the ratio BLM /s, the horizontal axes is the ratio m (s)/s terms of the order of O(N f s ) in the O( s ) corrections The expression for the pole mass in terms of the running mass reads m m N f s 1 s 3 48 71 m ln 3 8 1 m 6 ln 13 lnm 18 34 Using expression for the width in terms of the running quark mass, we recalculate the BLM scale The numerical results are presented in Fig 4 We see that the use of the running quark mass in the expression for the width makes the BLM scale for the coupling constant higher for arbitrary relation between the Higgs and the fermion mass excluding the region close to the threshold, where the use of the running mass is artificial This does not make much difference for the mass region not far from the threshold, but for higher energies the difference is huge The curve in Fig 4 can be well approximated by the equation BLM 049m H m m H 35 The important check of our results can be performed by studying the limit m H m 1 As it was mentioned above, the O( s ) corrections to the decay width H f 1 f 1 are known in this limit up to the power suppressed terms O(m 1 /m H ) Expanding expression for the width up to the terms of the order O(m 1 /s) and using expression for the running mass, we get Our result for the width in Eq 36 is in complete agreement with the N f -dependent part of the O( s ) correction given in 4,6,7 V CONCLUSION By no doubts the Higgs interaction with the massive quarks is quite important from the phenomenological point of view A vivid example is provided by the decay mode H b b, which can be used for the detection of the light Higgs boson Even leaving aside the problem of finding this particle, direct measurement of the coupling of the Higgs boson to quarks seems to be necessary The measurement of such a type can test the symmetry-breaking mechanism in the Higgs-fermion sector of the standard model In this paper we present analytical results for the O(N f s ) correction to the decay width of the Higgs boson to the pair of massive fermions for the arbitrary relation between the mass of the Higgs boson and the mass of the fermion We calculate both real and virtual radiation of the light fermion pair in this decay As a byproduct of this analyses, we obtain the formulas see Eq 4 and below for the width of the rare decay: H f 1 f 1f f in the limit m 1 m It seems that the most important phenomenological application of our analyses is connected with the Higgs boson decay to two top quarks In this case the value of the top quark mass and the expected value of the Higgs boson mass suggests that there will be no small or large mass ratios in this problem In this case any results on the next-to-leading order QCD radiative corrections are absent and our results provide the first step in this direction Our analyses of the BLM scale indicates that the use of the running quark mass in the complete expression for the one-loop QCD radiative correction to H f 1 f 1 is definitely a good choice for an arbitrary relation between the Higgs and the fermion masses We hope that if both the running quark mass and the BLM scale for the coupling constant, evaluated in this paper see Eq 35, are used for the description of the one-loop QCD corrected decay width H f 1 f 1, this should substitute quite a reasonable approximation for the description of this process for an arbitrary ratio of the Higgs boson mass to the fermion mass This work was supported by Graduiertenkolleg Teilchenphysik, Universität Mainz
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