Table Of Content

Rome1-1442/06 IFIC/06-31 RM3-TH/06-24 IFUM-880-FT hep-ph/0611266 Analytic Results for Virtual QCD Corrections to Higgs Production and Decay 7 0 0 2 U. Aglietti∗, n a Dipartimento di Fisica, Università di Roma “La Sapienza” and INFN, Sezione di Roma, J P.le Aldo Moro 2, I-00185 Rome, Italy 5 R. Bonciani , 3 † v Departament de F´ısica Teòrica, IFIC, CSIC – Universitat de València, 6 6 E-46071 València, Spain 2 1 G. Degrassi , ‡ 1 6 Dipartimento di Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre, 0 Via della Vasca Navale 84, I-00146 Rome, Italy / h p A. Vicini § - p Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, e h Via Celoria 16, I–20133 Milano, Italy : v i X Abstract r a We consider the production of a Higgs boson via gluon-fusion and its decay into two photons. We compute the NLO virtual QCD corrections to these processes in a general framework in which the coupling of the Higgs boson to the external particles is mediated by a colored fermion and a colored scalar. We present compact analytic results for these two-loop corrections that are expressed in terms of Harmonic Polylogarithms. The expansion of these corrections in the low and high Higgs mass regimes, as well as the expression of the new Master Integrals which appear in the reduction of the two-loop amplitudes, are also provided. For the fermionic contribution, we provide an independent check of the results already present in the literature concerning the Higgs boson and the production and decay of a pseudoscalar particle. Key words: Feynman diagrams, Multi-loop calculations, Higgs physics PACS: 11.15.Bt; 12.38.Bx; 13.85.Lg; 14.80.Bn; 14.80.Cp. ∗Email: [email protected] †Email: Roberto.Bonciani@ific.uv.es ‡Email: degrassi@fis.uniroma3.it §Email: [email protected] 1 Introduction The Higgs searches program at the TEVATRON and at the LHC requires from the theo- retical side the highest possible level of accuracy in the prediction of the production cross- sections and of all the decay channels. Over the years a lot of effort has been devoted to the study of the QCD, and also EW, corrections to the various production mechanisms and decays in the Standard Model and beyond (for a recent review see Ref.[1]). The gluon-fusion process gg H +X [2] is the dominant production mechanism. Its → present knowledge includes the NLO [3, 4, 5] and NNLO QCD corrections [6] and the two- loop EW corrections [7, 8, 9]. The QCD corrections to Higgs production at finite transverse momentum have also been discussed [10]. While the NLO QCD corrections and the two- loop EW light fermion contribution are known completely, namely for arbitrary value of the Higgs mass and of the other relevant particles in the loops, the NNLO QCD corrections are only known in the heavy top limit while the result for the two-loop EW top contribution is valid only for intermediate Higgs mass, i.e. m 2m . H W ≤ The Higgs decay H γγ [11] is, for light values of the boson mass, a very promising → channel. It has been studied in great detail including the NLO QCD [12, 13] and the two- loop EW corrections [14, 8, 15, 16]. The NLO QCD corrections are now known in a closed analytic form [13, 5], while for the EW corrections their knowledge is similar to that of the gluon fusion process. Given the importance of the Higgs physics program, it is highly desirable to have the radiative corrections to the various reactions expressed in analytic form that can be easily implemented in computer codes. With respect to this, it should be recalled that the complete result concerning the NLO QCD corrections to the gluon fusion process has been reported in Ref.[4] via a rather lengthy formula expressed in terms of a one-dimensional in- tegralrepresentation. Actuallythecalculationofthetwo-looplight-fermionEWcorrections to the Higgs production and decay [8] has shown that corrections of this kind can be calcu- lated analytically, expressing the results in terms of Harmonic Polylogarithms (HPL) [17], a generalization of Nielsen’s polylogarithms, and an extension of the HPL, the so-called Generalized Harmonic Polylogarithms (GHPL) [18]. The idea lying behind the introduction of (G)HPLs is to express a given integral coming from the calculation of a Feynman diagram in a unique and non-redundant way as a linear combination of a minimal set of independent transcendental functions. These functions are expressed as repeated integra- tions over a starting set of basis functions and this set depends strongly on the problem one has to solve, being connected directly to the threshold structure of the diagrams under consideration. An inspection of the threshold structure of the NLO QCD corrections to the gluon- fusion process and to H γγ decay shows that these corrections can be fully expressed → in terms of the original set of HPLs introduced in [17]. A FORTRAN program [19] and a Mathematica package [20] that efficiently evaluate these functions are available. The aim of this paper is to provide analytic expressions, in terms of HPLs, for the NLO QCD corrections to the Higgs production cross section via gluon fusion, i.e. gg H, in a → general form that can be applied both to the SM and to models beyond it, and, moreover, to provide an independent check for the formulas already present in the literature. The production mechanism is assumed to be mediated by colored fermion and scalar particles. 1 As a byproduct we also present the NLO QCD corrections to the Higgs decay into two photons, i.e. H γγ. A similar project has been carried out in Ref.[5]. There, the authors → started from the result of Ref.[4]1 expressed as a one-dimensional integral representation. Expanding this result in a power series, employing the theorem that two analytic functions are the same if their Taylor series are the same, they were able to rewrite it in terms of HPLs. In our case we explicitly compute all the relevant Feynman diagrams, expressing the result in term of HPLs. The calculational techniques we employed are the Laporta algorithm [21] for the reduction to Master Integrals (MIs) and the differential equation method [22] for their calculation (the calculation is implemented in FORM [23] codes). To complete an independent check of the results presented in Ref.[4] we also computed the NLO QCD corrections to the pseudoscalar production and decay, i.e. gg A, A γγ. → → The paper is organized as follows: in Section 2 we discuss the QCD corrections to the decay width H γγ. Section 3 is devoted to the study of the Higgs production via the → gluon fusion mechanism. The following section contains the analytic expressions for the virtual QCD corrections to the fermionic contribution in gg A, A γγ. Finally we → → present our conclusions. We include also two Appendices. In the first one we collect the expansions of the relevant functions in the two regimes: for Higgs mass much lighter than the particles mediating the Higgs interaction with the vector bosons and in the opposite case. In the second Appendix we collect the MIs not already present in the literature, that enters the calculation of the NLO QCD corrections. 2 The H γγ Decay Width → We begin by considering the decay width H γγ. Being the Higgs boson electrically → neutral its coupling to the photon is mediated at the loop-level by charged particles. For the latters we assume a vector boson neutral under SU(N ), a fermion and a scalar particle c in a generic R , R SU(N ) representation, respectively, whose coupling’s strengths to the 1/2 0 c Higgs are: m A2 1/2 HVV = gλ m , HFF = gλ , HSS = gλ , (1) 1 W 1/2 0 2m m W W where g is the SU(2) coupling, m is the W mass, m is the fermion mass, A is a generic W 1/2 coupling with the dimension of mass and λ are numerical coefficients2. i The partial decay width for the reaction H γγ can be written as: → G α2m3 Γ(H γγ) = µ H 2 , (2) → 128√2π3 |F| where the function can be organized with respect to the lowest order term and its QCD F corrections as: A2 = λ Q2N +λ Q2 N +λ Q2N , (3) F 1 1 1F1 1/2 1/2 1/2F1/2 0 0 0m2 F0 0 1In Ref.[4] the QCD corrections were considered only for the fermion contribution. 2The SM is recovered with λ1 =λ1/2 =1, λ0 =0, Nc =3 and R1/2 =3. 2 where m is the mass of the scalar particle, while Q and N , i = 0,1/2,1, are the electric 0 i i charges and the representation numbers under SU(N ) of the scalar, fermion and vector c boson particles, respectively. Writing: (1l) (2l) = + +... (4) Fi Fi Fi we have at the one-loop level (1l) = 2(1+6y ) 12y (1 2y )H(0,0,x ), (5) F1 1 − 1 − 1 1 (1l) = 4y 2 1 4y H(0,0,x ) , (6) F1/2 − 1/2 − − 1/2 1/2 (1l) = 4y [1+(cid:2)2y (cid:0)H(0,0,x )(cid:1)] . (cid:3) (7) F0 0 0 0 In Eqs.(5-7) m2 √1 4y 1 y i , x − i − , (8) i i ≡ m2 ≡ √1 4y +1 H − i with m the mass of the vector particle and, employing the standard notation for the HPLs, 1 H(0,0,z) labels a HPL of weight 2 that results to be3 1 H(0,0,z) = log2(z) . (9) 2 The QCD corrections to the lowest order result can be written as α (2l) s (2l) = C(R ) , (10) FQCD π i Fi i=(0,1/2) X whereC(R)istheCasimirfactoroftheR representation(inparticular, forthefundamental i and the adjoint representations of SU(N ) we have C = (N2 1)/(2N ) and C = N , c F c − c A c respectively). We consider first the fermion contribution (the relevant Feynman diagrams are shown in Fig. 1 (a)–(d)). (2l) The expression for depends upon the renormalized mass parameter employed. In F1/2 the case of MS quark masses we have m2 (2l) (2l,a) (2l,b) 1/2 = (x )+ (x )ln , (11) F1/2 F1/2 1/2 F1/2 1/2 µ2 ! where µ is the ’t Hooft mass and 36x 4x(1 14x+x2) 4x(1+x) (2l,a) (x) = − ζ H(0,x) F1/2 (x 1)2 − (x 1)4 3 − (x 1)3 − − − 8x(1+9x+x2) 2x(3+25x 7x2 +3x3) H(0,0,x)+ − H(0,0,0,x) − (x 1)4 (x 1)5 − − 4x(1+2x+x2) + [ζ H(0,x)+4H(0, 1,0,x) H(0,1,0,x)] (x 1)4 2 − − − 4x(5 6x+5x2) 8x(1+x+x2 +x3) + − H(1,0,0,x) (x), (12) (x 1)4 − (x 1)5 H1 − − 12x 6x(1+x) 6x(1+6x+x2) (2l,b) (x) = H(0,x)+ H(0,0,x), (13) F1/2 −(x 1)2 − (x 1)3 (x 1)4 − − − 3All the analytic continuations are obtained with the replacement m2 m2 iǫ − H →− H− 3 (cid:13) H;A g f (cid:13) (a) (b) ( ) (d) (cid:13) H g s (cid:13) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Figure 1: The Feynman diagrams for the decay process H,A γγ. Diagrams (a)–(d) have → a fermion, labeled by “f”, running in the loop, while diagrams (e)–(n) have a scalar, labeled by “s”. with 9 1 7 (x) = ζ 2 +2ζ H(0,x)+ζ H(0,0,x)+ H(0,0,0,0,x)+ H(0,1,0,0,x) 1 2 3 2 H 10 4 2 2H(0, 1,0,0,x)+4H(0,0, 1,0,x) H(0,0,1,0,x). (14) − − − − In Eqs.(12,14) ζ ζ(n) are the Riemann’s zeta functions. n ≡ (2l) The expression for in case the one-loop result is given in terms of on-shell fermion F1/2 masses is given instead by: 4 (2l) (2l,a) (2l,b) = (x )+ (x ) . (15) F1/2 F1/2 1/2 3F1/2 1/2 Eq.(15) is in agreement with the results presented in [13, 5]. (2l) Wenowpresent thescalarcontribution, , assuming thatboththemassofthescalar, F0 m , and thecoupling Aarerenormalized inthe MSscheme (therelevant Feynman diagrams 0 are shown in Fig. 1 (e)–(n)). We find m2 (2l) = (2l,a)(x )+ (2l,b)(x )+ (2l,c)(x ) ln 0 , (16) F0 F0 0 F0 0 F0 0 µ2 (cid:18) (cid:19) (cid:16) (cid:17) where 14x 24x2 x(3 8x+3x2) 34x2 (2l,a) (x) = ζ + − H(0,x)+ H(0,0,x) F0 −(x 1)2 − (x 1)4 3 (x 1)3(x+1) (x 1)4 − − − − 4 8x2 [ζ H(0,x)+4H(0, 1,0,x) H(0,1,0,x)+H(1,0,0,x)] −(x 1)4 2 − − − 2x2(5 11x) 16x2(1+x2) − H(0,0,0,x)+ (x), (17) − (x 1)5 (x 1)5(x+1) H1 − − 6x2 6x2 (2l,b) (x) = H(0,x) H(0,0,x), (18) F0 (x 1)3(x+1) − (x 1)4 − − 3 (2l,c) (1l) (x) = . (19) F0 −4 F0 (2l) We provide also assuming that the mass of the scalar is renormalized on-shell while F0 the A coupling is still given as an MS one. It reads 7 m2 (2l) = (2l,a)(x )+ (2l,b)(x )+ (2l,c)(x )ln 0 . (20) F0 F0 0 3 F0 0 F0 0 µ2 (cid:18) (cid:19) 3 Virtual Corrections to gg H Production Mecha- → nism In this section we present the analytic expressions for the virtual two-loop QCD corrections for Higgs boson production via the gluon fusion mechanism. Being the Higgs boson neutral under SU(N ), its coupling to the gluons is mediated by a loop of colored particles. As in c Section 2, we consider a fermion and a scalar particle, that run in the loops. The Feynman diagrams relevant for the NLO corrections to the production cross section are shown in Fig. 2. The hadronic cross section can be written as: 1 σ(h +h H +X) = dx dx f (x ,µ2)f (x ,µ2) 1 2 → 1 2 a,h1 1 F b,h2 2 F × a,b Z0 X 1 τ H dz δ z σˆ (z), (21) ab × − x x Z0 (cid:18) 1 2(cid:19) where τ = m2/s, µ is the factorization scale, f (x,µ2), the parton density of the H H F a,hi F colliding hadron h for the parton of type a, (a = g,q,q¯) and σˆ the cross section for the i ab partonic subprocess ab H + X at the center-of-mass energy sˆ = x x s = m2/z. The → 1 2 H latter can be written as: σˆ (z) = σ(0)zG (z), (22) ab ab where 2 G α2(µ2) A2 1−2i σ(0) = µ s R λ T(R ) (1l) (23) 128√2π (cid:12) i m2 i Gi (cid:12) (cid:12)i=0,1/2 (cid:18) 0(cid:19) (cid:12) (cid:12) X (cid:12) (cid:12) (cid:12) is the Born-level contribution with (cid:12)(1l) = (1l) and T(R ) are the(cid:12)matrix normalization G(cid:12)i Fi i (cid:12) factors of the R representation (T(R) = 1/2 for the fundamental representation ofSU(N ), i c T(R) = N for the adjoint one). c 5 g H;A f g g (a) (b) ( ) (d) (e) (f) g H s g g (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) (x) Figure 2: The Feynman diagrams for the production mechanism gg H,A. Diagrams → (a)–(h) have a fermion, labeled by “f”, running in the loop, while diagrams (i)–(x) have a scalar, labeled by “s”. Up to NLO contributions we can write α (µ2) G (z) = G(0)(z)+ s R G(1)(z), (24) ab ab π a,b with (0) G (z) = δ(1 z)δ δ , (25) ab − ag bg π2 µ2 G(1)(z) = δ(1 z) C +β ln R + (2l) +... (26) gg −  A 3 0 µ2 Gi  (cid:18) F(cid:19) i=0,1/2 X   The dots in Eq.(26) represent the contribution from the real emission that we have not (1) (1) (1) written as well as the other NLO factors G (z), G (z), G (z). In Eq.(26) β = (11C gq qq qq¯ 0 A − 4n T(R ) n T(R )/6 with n (n ) the number of active fermion (scalar) flavor in the f f s s f s − representation R (R ). f s (2l) The function can be cast in the following form: Gi 6 A2 1−2i (2l) = λ T(R ) C(R ) (2l,CR)(x )+C (2l,CA)(x ) Gi i m2 i i Gi i AGi i (cid:18) 0(cid:19) (cid:18) (cid:19) −1 A2 1−2j (1l) λ T(R ) +h.c. (27) × j m2 j Gj  j=0,1/2 (cid:18) 0(cid:19) X   with (2l,CR) = (2l). The infrared regularized functions (2l,CA), after subtraction of the Gi Fi Gi infrared poles, are found to be: 4x x(1+8x+3x2) 2(1+x)2 (2l,CA)(x) = 3+ H(0,0,0,x) (x) G1/2 (x 1)2 (x 1)3 − (x 1)2 H2 − (cid:20) − − +ζ H(1,0,0,x) , (28) 3 − (cid:21) 4x 3 x(1 7x) 4x (2l,CA)(x) = + − H(0,0,0,x)+ (x) , (29) G0 (x 1)2 −2 (x 1)3 (x 1)2H2 − (cid:20) − − (cid:21) with 4 3ζ 1 (x) = ζ 2 +2ζ + 3 H(0,x)+3ζ H(1,x)+ζ H(1,0,x)+ (1+2ζ ) H(0,0,x) 2 2 3 3 2 2 H 5 2 4 1 2H(1,0,0,x)+H(0,0, 1,0,x)+ H(0,0,0,0,x)+2H(1,0, 1,0,x) − − 4 − H(1,0,0,0,x). (30) − (2l) The analytic expression of is in agreement with that reported in Ref.[5] based on the G1/2 results presented in Ref.[4]. 4 Pseudoscalar Higgs: A γγ and gg A → → To complete an independent check of the results of Ref.[4] in this section we consider the virtual NLO QCD corrections to the decay width of a pseudo-scalar particle A in two photons, Γ(A γγ), and to its production cross section via gluon fusion, σˆ(gg A). → → As in Ref.[4] we assume the interaction of the A particle with gluons mediated only by the top quark. Because the NLO QCD corrections to these two processes are calcu- lated in Dimensional Regularization, a prescription for the γ matrix is needed. We use 5 the same prescription of Ref.[4], i.e. ’t Hooft–Veltman one [24], that, as it is well known, breaks manifestly Ward Identities. The latters need to be restored explicitly, with a finite ¯ renormalization. If Z and Z are the renormalization constants of the vertex Hff, Hff¯ Aff¯ ¯ scalar Higgs-fermion-antifermion, and Aff, pseudo-scalar Higgs-fermion-antifermion, respectively, the contribution of the finite renormalization can be found using [4, 25]: α S Z = Z +2C . (31) Aff¯ Hff¯ F π 7 4.1 Decay Width A γγ → In analogy with Eq.(2), we write: G α2m3 Γ(A γγ) = µ A 2 . (32) → 128√2π3 |E| Assuming the strength of the coupling of the pseudoscalar to the top quark equal to Att = gη m /(2m ), with m the top mass and η a numerical coefficient, the function can be t t W t t E written as (Q = 2/3): t α = η Q2N (1l) + SC (2l) +... . (33) E t t c Et π FEt h i The leading order term is (1l) = 4y H(0,0,x ), (34) Et t t where y , x are given by Eq.(8) with i = t. At the NLO, assuming an MS top mass we t t have: m2 (2l) = (2l,a)(x )+ (2l,b)(x )ln t , (35) Et Et t Et t µ2 (cid:18) (cid:19) where: 4x (2l,a) (x) = [ζ 4H(0, 1,0,x)+H(0,1,0,x) 5H(1,0,0,x)] Et −(x 1)2 3 − − − − 4x[2(1 x)2 ζ (1 x2)] 8x(1 x2) 2 + − − − H(0,x)+ − H(0,0,x) (x 1)3(1+x) (x 1)3(1+x) − − 6x(1+x) 8x(1+x2) + H(0,0,0,x) , (36) (x 1)3 − (x 1)3(1+x)H1 − − 6x 6x (2l,b) (x) = H(0,x)+ H(0,0,x). (37) Et −(x 1)(1+x) (x 1)2 − − The corresponding expression for an OS top mass is given by: 4 (2l) (2l,a) (2l,b) = (x )+ (x ) . (38) Et Et t 3Et t 4.2 Production Cross Section gg A → The expressions for the relevant quantities in the gg A production cross section can be → easily obtained from those in Section 3, with the substitutions: T(R ) 1/2, C(R ) i i → → C , , . In particular, the Born-level partonic cross section (Eq.(23)) is: F F → E G → K G α2(µ2) 1 2 σ(0) = µ s R η (1l) , (39) 128√2π 2 tKt (cid:12) (cid:12) (cid:12) (cid:12) (1l) (1l) (cid:12) (cid:12) with = . The NLO virtual contributio(cid:12)n to the (cid:12)gluon fusion subprocess (Eq.(26)) Kt Et is: π2 µ2 G(1)(z) = δ(1 z) C +β ln R + (2l) , (40) gg − A 3 0 µ2 Kt (cid:20) (cid:18) F(cid:19) (cid:21) 8 where β = (11C 2n )/6, with n the number of active flavor, and 0 A f f − −1 (2l) = (1l) C (2l,CF)(x )+C (2l,CA)(x ) +h.c. (41) Kt Kt F Kt t AKt t (cid:18) (cid:19) (cid:16) (cid:17) with (2l,CF) = (2l) and Kt Et 4x 12x2 (2l,CA)(x) = [ζ H(1,0,0,x) 2 (x)]+ H(0,0,0,x). (42) Kt (x 1)2 3 − − H2 (x 1)3 − − Eqs.(38,41) are in agreement with the corresponding expressions presented in Ref.[5]. 5 Conclusions In this paper, we considered the virtual NLO QCD corrections to the processes H γγ → and gg H. We assumed the coupling of the Higgs boson to the photons and gluons → to be mediated by fermionic and scalar loops. We provided analytic formulas for these corrections that are valid for arbitrary mass of the fermion or scalar particle running in the loops and of the Higgs boson. They are given in a very compact form as a combination of HPLs. The calculation here presented was done using the Laporta algorithm for the reduction of the scalar integrals to the MIs and the differential equations method for the evaluation of the latters. A part of the MIs needed for the calculation was already known in the literature. We explicitly give the analytic results for the MIs that were not known. Wechecked ourresults forthedecaywidth oftheHiggsbosonintwo photonsandforthe partonic cross section of the gluon fusion by performing an independent calculation in the region of small Higgs mass via an asymptotic expansion in the variable r m2/m2 1, ≡ H ≪ with m the mass of the fermion or scalar particle, up to the first 4-5 orders. We considered also the NLO virtual QCD corrections to A γγ and gg A assuming → → the coupling of the pseudoscalar boson to the external particles mediated by a fermion. We find complete agreement with the results previously known in the literature concerning the production and decay of a (pseudo)scalar Higgs boson mediated by fermionic loops. This provides an independent check of the formulas given in Refs.[4, 5] and extends them to the case of a scalar particle running in the loops. Acknowledgments The authors want to thank M. Spira and A. Djouadi for useful communications regarding Ref.[4]. R. B. wishes to thank the Department of Physics of the University of Florence and INFN Section of Florence for kind hospitality, and in particular S. Catani for useful discussions during a large part of this work. Discussions with G. Rodrigo are gratefully acknowledged. The work of R. B. was partially supported by Ministerio de Educación y Ciencia(MEC)undergrantFPA2004-00996,GeneralitatValencianaundergrantGV05-015, and MEC-INFN agreement. Note added After our work was completed a paper on a similar subject has appeared on the Web [27]. 9