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TTP92–32 Nov. 1992 QCD CORRECTIONS TO TOPONIUM PRODUCTION AT HADRON COLLIDERS1 J.H. Ku¨hn and E. Mirkes 3 9 Institut fu¨r Theoretische Teilchenphysik 9 1 Universität Karlsruhe n a Kaiserstr. 12, Postfach 6980 J 4 7500 Karlsruhe 1, Germany 1 v 4 0 2 1 0 3 9 / h p - p Abstract e h : v Toponium production at future hadron colliders is investigated. Per- i X turbative QCD corrections to the production cross section for gluon r fusionarecalculatedaswellasthecontributionsfromgluon-quarkand a quark-antiquark collisions to the total cross section. The dependence on the renormalization and factorization scales and on the choice of the parton distribution functions is explored. QCD corrections to the branching ratio of η into γγ are included and the two-loop QCD po- t tential is used to predict the wave function at the origin. The branching ratio of η into γZ, ZZ, HZ and WW is compared with the γγ t channel. 1Supported by BMFT Contract 055KA94P 1 Introduction The production of heavy quarks at hadron colliders has received a lot of attention from the experimental as well as from the theoretical side. Higher order QCD cal- culations for open top quark production [1, 2, 3] provide the theoretical basis for the current experimental limits [4] on the mass of the top quark. Given the current range for the top mass of 90 to 180 GeV deduced from electroweak radiative corrections [5] ongoing experimental studies at the TEVATRON should be able to discover the top quark in the near future. Planned experiments at LHC or SSC will lead to a huge number of tt¯events and may pin down the top quark mass with a remaining uncertainty of perhaps 5 GeV [6]. Aside from the intrinsic interest in the precise top mass determination, the aim to fix the input in one-loop electroweak corrections [7] constitutes one of the important motivations for the drive for future increased precision in the top mass. In [8] it has been proposed to decrease the experimental error significantly throughthestudyofη initsγγ decaychannel.Making useofthelargeluminosity t of future hadron colliders, one should be able to overcome the tiny cross section multiplied by the small branching ratio into two photons. The excellent gamma energy resolution, originally developed for the search for a light Higgs boson, would in principle allow the measurement of the mass of the bound state at about 100 MeV accuracy. This would result in a determination of the top quark mass limited only by theoretical uncertainties. It was shown that the toponium signal could be extracted from the γγ background with a signal to noise ratio decreasing from 2/3 down to 1/8 for m between 90 and 120 GeV. t The estimates of [8] for the production and decay rate were based on the Born approximation and on fairly crude assumptions on the bound state wave function which was derived from a Coulomb potential, neglecting QCD corrections. The signal to noise ratio and the statistical significance of the signal as estimated in [8] is just sufficient for the discovery. Therefore a more precise evaluation of the production rate is mandatory. In lowest order the reaction is induced by gluon-gluon fusion (fig. 1a). QCD corrections will be calculated in this work. They affect and indeed increase the rate in this channel by about 50%. Quark-gluon scattering and quark-antiquark annihilation into η ‘ enter at order α3. As we shall see the qg process increases t s the production cross section by about 8% whereas the effect of qq¯annihilation is negligible. The production cross section for η with nonvanishing transverse momentum t is infrared finite in the order considered here. The total cross section, however, exhibitssingularitiesthathavetobeabsorbedinappropriatelychosenanddefined 1 parton distribution functions. The dependence on the factorization as well as renormalization scales will be studied as well as the dependence on the choice on differentparametrizationsofpartondistributionfunctions.AfullNLOcalculation requires the corresponding corrections for the physically interesting branching ratio. These are available and will be implemented. The bound state production cross section and the branching ratio of η into t γγ depend (markedly!) on R(0)2, the square of the bound state wave function at the origin and hence on the potential. The range of R(0) compatible with the current knowledge of the perturbative two-loop QCD potential and the present experimental results for Λ will be explored. QCD As we shall see in the following, fairly optimistic assumptions are necessary to open even a narrow window for toponium discovery in the mass range of m = 90 110 GeV. This is the consequence of the dominance single quark t − decays (SQD) and the small branching ratio into two photons. The situation improves dramatically if new hypothetical quarks are considered with suppressed single quark decay. Examples are b′ or isosinglet quarks with small mixing with ordinary d,s or b quarks. These would dominantly decay into gauge and Higgs bosons, leading to spectacular signatures. Theoutlayofthispaperisasfollows:Insection2theQCDcorrectionstogluon gluonfusion into η andthe cross section forthe qq¯andqG initiated processes will t be derived. Insection 3the formulae forthe decay ratesofη will belisted andthe t dependence of the wave function on the choice of the potential studied. Section 4 contains the numerical evaluation of the production cross section and the study of scale dependencies. Section 5 contains our conclusions. A brief account of this work has been presented in [9]. 2 QCD corrections to hadronic η production t For the complete evaluation of NLO corrections for η production in gluon gluon t fusion virtual corrections (fig. 1b) are required as well as corrections from real gluon radiation (fig. 1c,d). In addition one needs the cross section for quark- antiquark annihilation into η +gluon (fig. 2a) and the (anti-)quark-gluon (fig. 2b) t initiated reaction into η +quark. Technically it is quite convenient to employ t dimensional regularization for both ultraviolet and infrared singularities. The calculation of amplitudes for bound state production (or decay) can be performed in two different ways. One may either evaluate directly the amplitude for the ¯ production of a (QQ) bound state with the desired spin and orbital angular momentum configuration and arrive at an amplitude proportional to the wave function at the origin or its derivative [10, 11]; or alternatively one may evaluate the production rate for open QQ¯ and subsequently identify the rate at threshold 2 with the rate for bound state production, substituting R(0)2 dPS (k +k ;k ,...,k ,k ,k ) dPS (k +k ;k ,...,k ,q) (1) n 1 2 3 n Q Q¯ n−1 1 2 3 n −→ 2πM where M = 2m + E and k = k = q/2. For the one particle final state t Bind Q Q¯ relevant for resonant η production in gluon gluon fusion this implies (in 4 2ǫ t − dimensions): R(0)2 dPS (k +k ;k ,k ) 2πδ(s M2) (2) 2 1 2 Q Q¯ ⇒ 2πM − and for the η + parton configuration that will be of main concern in the following t R(0)2 1 4πs ǫ 1 dPS (k +k ;k ,k ,k ) dt (3) 3 1 2 3 Q Q¯ ⇒ 2πM 8πs ut Γ(1 ǫ) (cid:18) (cid:19) − where the Mandelstam variables are defined by s = (k +k )2 t = (k q)2 u = (k q)2 = M2 t s (4) 1 2 1 2 − − − − Inthelatterapproachitisimportanttomakesurethat—inthelimitofvanishing relativevelocity—thecrosssectionreceivessolelycontributionsfromtheQQ¯ spin parity configuration corresponding to the bound state. It is therefore mandatory to restrict the evaluation of the amplitude to QQ¯ color singlet configurations and in the reaction under consideration to those three gluon states that are totally antisymmetric with respect to color as well as to their Lorentz structure. The former approach leads typically to more compact expressions2. However, for states with JPC = 0−+ the formula for the bound state amplitude involves the γ5 matrix whose formulation in the context of dimensional regularization introduces complications. This purely formal difficulty is absent in the second approach, which for this reason has been adopted in this calculation. 2.1 The hadronic cross section In this section we define the general structure of the cross sections for η produc- t tion in hadronic collisions h (P )+h (P ) η +X γ +γ +X (5) 1 1 2 2 t → → within the framework of perturbative QCD. Here h ,h are unpolarized hadrons 1 2 with momenta P . The hadronic cross section in NLO is thus given by i σH(S) = dx dx fh1(x ,Q2)fh2(x ,Q2)σâb(s = x x S,α (µ2),µ2,Q2) (6) 1 2 a 1 F b 2 F 1 2 s F Z 2 See for instance [13] for particularly impressive examples. 3 where one sums over a,b = q,q¯,g. fh(x,Q2) is the probability density to find a F parton a with fraction x in hadron h if it is probed at scale Q2 and σâb denotes F the parton cross section for the process a(k = x P )+b(k = x P ) η +X (7) 1 1 1 2 2 2 t → from which collinear initial state singularities have been factorized out at a scale Q2 and implicitly included in the scale-dependent partondensities fh(x,Q2). We F a F next specify the possible partonic subprocesses that contribute to η production t in LO and NLO. 2.2 Gluon fusion In Born approximation the cross section forgg η (fig. 1a) is given by (N = 3) t C → 1 1 1 R(0) σˆgg = 2 2πδ(s M2) (8) Born 2s644(1 ǫ)2 |MBorn| 2πM − − X with N2 1 2 = C − g416(1 2ǫ)(1 ǫ) (9) |MBorn| 4N s − − C TheQQ¯ statehasbeenprojectedontothecolorsingletconfiguration.Thedivision by 64 4 (1 ǫ)2 is implied by the average over 82 gluon colors and (n 2)2 gluon · · − − helicities in n dimensions. g will be replaced by α through g2 = 4πα µ2ǫ. The s s s s Born cross section can therefore be cast into the form π2α2R(0)2 (1 2ǫ) σˆgg = s µ4ǫ − δ(1 z) (10) Born 3s M3 (1 ǫ) − − where z = M2/s. Virtual corrections to the rate for QQ¯ production close to threshold (fig. 1b) can be combined to yield3 [12] 4πµ2 ǫ 2 = 2 Γ(1+ǫ) virtual Born Born |M ∗M | |M | M2 ! X X α π2 1 1 1 s C +b N b +A (11) × π F v 0ǫUV − Cǫ2IR − 0ǫIR ! where 4 π2 5 A T ln2+C 5 +N π2 +1 F F C ≡ − 3 4 − ! (cid:18)12 (cid:19) 11 2 b N n T (12) 0 C f F ≡ 6 − 3 3 Here CF =4/3; NC =3; and TF=1/2. 4 1/ǫ and 1/ǫ represent poles arising from ultraviolet and infrared divergent UV IR integrals respectively. The mass singularity 2ln(m /M) in [12] has been replaced i by 1/ǫ . Since we are considering bound state production the 1/v term (v de- IR notes the relative velocity of the two heavy quarks) in eq. (11) is absorbed in the instantaneous potential and thus effectively dropped. The QCD coupling is renormalized in the MS scheme and is given by α α α 1 s = MS(µ2) 1 MSb +ln4π γ (13) 0 E π π − π ǫ − (cid:20) (cid:18) UV (cid:19)(cid:21) Combining Born result and virtual corrections one thus obtains for resonant production α 4πµ2 ǫ µ2 σˆgg = σ (ǫ) 1+ s Γ(1+ǫ) b ln Born+virtual 0 " π M2 ! 0 M2 1 1 N b +A δ(1 z) (14) − Cǫ2IR − 0ǫIR !# − where the normalization factor 1π2R2(0) (1 2ǫ) σ (ǫ) = α2 (µ2)µ4ǫ − (15) 0 s 3 M3 MS (1 ǫ) − has been split off for convenience. In a second step the cross section for gg gQQ¯ (16) → has to be calculated (fig. 1c,d). Only the kinematical situation will be treated ¯ where the relative momentum of Q and Q vanishes. In an intermediate step the matrix element for ggg QQ¯ (17) → will be considered (fig. 1e,f) and reaction (16) can be subsequently obtained through crossing. The QQ¯ state will be projected onto a color singlet configuration through δ /√3. The overall wave function of the gluons is of course totally ij symmetric. Requiring a totally antisymmetric configuration in color space implies, in turn, a totally antisymmetric wave function in Minkowski space. This latter condition secures JPC = 0−+ for the QQ¯ bound state. (Conversely, requiring symmetric wave functions in both color and Minkowski space would lead to JPC = 1−−.) The amplitude arising from the diagram fig. 1e is given by: ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) = 1 2 3 1 2 3 A i[( k + k + k )/2+m] 1 2 3 u¯((k +k +k )/2)( ig ǫ ) − 6 6 6 ( ig ǫ ) 1 2 3 − 6 1 [( k +k +k )2/4 m2] − 6 2 1 2 3 − − i[( k k + k )/2+m] 1 2 3 − 6 − 6 6 ( ig ǫ )v((k +k +k )/2) (18) × [( k k +k )2/4 m2] − 6 3 1 2 3 1 2 3 − − − 5 to be multiplied the color factor λa1 λa2 λa3 δ 1 1 ij = (if +d ) (19) 2 2 2 ! √3 √34 a1a2a3 a1a2a3 ij As discussed above, only the f term will be retained. The amplitude is obtained from the six permutations of the diagram without triple boson coupling i 1 ˆ = f ǫ (k ,k ,k ;ǫ ,ǫ ,ǫ ) (20) A √34 a1a2a3 ijmA i j m i j m i,j,m X The second type of amplitues, represented by fig. 1f, involve the triple gluon vertex. Their sum reads as follows ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) = 1 2 3 1 2 3 B i[( k k + k )/2+m] 1 2 3 u¯((k +k +k )/2) ( igγ ) − 6 − 6 6 ( ig ǫ ) 1 2 3 " − α [( k1 k2 +k3)2/4 m2] − 6 3 − − − i[( k + k k )/2+m] 1 2 3 +( ig ǫ ) 6 6 − 6 ( igγ ) v((k +k +k )/2) (21) − 6 3 [(k1 +k2 k3)2/4 m2] − α # 1 2 3 − − i − ( g) (k k )αǫ ǫ +(k +2k )ǫ ǫα +( 2k k )ǫαǫ ×(k +k )2 − 1 − 2 1 2 1 2 1 2 − 1 − 2 1 2 1 2 n o with a color factor λbλa3 δ 1 1 ij f = f (22) 2 2 ! √3 a1a2b √32 a1a2a3 ij Three cyclic permutations are combined in this class of diagrams i 1 2 = f ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) (23) B √34 a1a2a3 iB i j m i j m i,j,mX(cyclic) The ghost amplitude is easily obtained from eq. (21) through the replace- C ment of the curly bracket ... kβ, where k ,k and k denote the momenta { } → 2 1 2 3 of the incoming ghost, antighost and gluon respectively. The squared matrix element is finally obtained in n = 4 2ǫ dimensions − 1 2 = + 2 2 2 = f2 g6128(1 ǫ) (24) |M|R |A B| − |C| 48 a1a2a3 − a1,a2,a3 X X X X st+tu+us 2 M8 +s4 +t4 +u4 (1 2ǫ)+4ǫM2 × (s M2)(t M2)(u M2)! " stu − # − − − where s = (k +k )2 t = (k +k )2 u = (k +k )2; M = 2m+E (25) 1 2 1 3 1 3 Bind 6 After crossing (k k ) this result is applicable to the reaction of interest 3 3 → − (16). In the QQ¯ threshold region (with the color restriction discussed above) one obtains 1 1 1 dσ(gg gQQ¯) = 2 dPS (26) → 2s 644(1 ǫ)2 |M|R 3 − X :=g6|M|2 R | {z } where gluon helicities and colors were averaged as usual. Replacing the three- by the two-particle phase space as indicated in eqs. (1) and (3) one obtains α 3M2 s dσ(gg gη ) = σ (ǫ) (27) t 0 → π 2 s st+tu+us 2 M8 +s4 +t4 +u4 4πsµ2 ǫ dt × (s M2)(t M2)(u M2)! stu ut ! Γ(1 ǫ) − − − − The term proportional 4ǫM2 in eq. (24) without singular denominator will not contribute in the limit ǫ 0 and has therefore been dropped. In the limit ǫ 0 → → the result coincides with eq. (8.45) of [13]. For the subsequent discussion it will be convenient to follow closely the treatment of NLO corrections for the Drell-Yan process presented in the classical paper by G.Altarelli et al. [14]. The invariants s,t and u are expressed in terms of M2,y and z by M2 M2 M2 s = ; t = (1 z)(1 y); u = (1 z)y (28) z − z − − − z − and one arrives at Gluon α 4πµ2 ǫ σ(gg g(QQ¯)) = σ (ǫ) sΓ(1+ǫ) → 0 π M2 ! 6 1 fggzǫ(1 z)1−2ǫy−ǫ(1 y)−ǫdy (29) ×Γ(1+ǫ)Γ(1 ǫ) − − − Z0 The normalization factors have been chosen to allow for easy combination with the virtual corrections and fgg is given by (1 ǫ) fgg = − M2 2 (1 2ǫ) |M|R − 2 z (1 z)(1 y)y 1 = − − − 4 (1 z)[1 y(1 z)][(1 z)y +z]! − − − − 1 1 z4 +1+(1 z)4((1 y)4 +y4) + (30) × − − y 1 y! h i − As a consequence of the symmetry of fgg with respect to y (1 y) it suffices ↔ − to evaluate the 1/y term. The y integration can be performed with the help of 7 the distribution identities 1 1 lnx x−1−ǫ = δ(x)+ ǫ −ǫ (cid:18)x(cid:19)+ − x !+ 1 1 lnz ln(1 z) zǫ(1 z)−1−2ǫ = δ(1 z)+ +ǫ 2ǫ − − −2ǫ − (cid:18)1−z(cid:19)+ 1−z − (1−z) !+ 1 1 ln(1 y) lny y−1−ǫ(1 y)−ǫ = δ(y)+ ǫ − ǫ (31) − −ǫ y! − y − y ! + + and 1 1 π2 = 1 ǫ2 +... (32) Γ(1 ǫ)Γ(1+ǫ) − 6 − Combining the Born result with real radiation and virtual corrections one obtains α 4πµ2 ǫ s σ(gg η +X) = σ (ǫ) δ(1 z)+ Γ(1+ǫ) → t 0 ( − π M2 ! µ2 π2 1 b ln N +A δ(1 z) P (z)+N F(z) (33) × " 0 M2 − C 3 ! − − ǫ gg C #) where b and A are given in eqs. (12) and 0 11z5 +11z4 +13z3 +19z2 +6z 12 F(z) = Θ(1 z) − − " 6z(1+z)2 1 ln(1 z) + 4 +z(1 z) 2 ln(1 z)+ 4 − (34) (cid:18)z − − (cid:19) − 1−z !+ 2(z3 2z2 3z 2)(z3 z +2)z ln(z) 1 + − − − − 3 (1+z)3(1 z) − ! 1 z # − − and 1 1 P = 2N + +z(1 z) 2 +b δ(1 z); (35) gg C 0 z (cid:18)1−z(cid:19)+ − − ! − The 1/ǫ term can be absorbed in the gluon densities. If these are defined at a factorization scale Q2, one effectively subtracts from the above result the term F α 4πµ2 ǫ 1 s σ (ǫ) Γ(1+ǫ) P (z)+C (z) (36) 0 2π Q2F ! (cid:18)−ǫ gg gg (cid:19) In the MS scheme C = 0, in the DIS scheme gg 1 z C (z) = n z2 +(1 z)2 ln − +8z(1 z) 1 (37) gg f − − z − − (cid:20)(cid:16) (cid:17) (cid:18) (cid:19) (cid:21) 8 The partonic cross section is then given by 1π2R(0)2 σˆgg = α2 (µ2) δ(1 z) s 3 M3 MS ( − α (µ2) µ2 π2 π2 4 + MS δ(1 z) b ln +N 1+ +C 5 T ln(2) 0 C F f π * − M2 12! 4 − !− 3 ! Q2 ln F P (z) C (z)+N F(z) . (38) gg gg C − M2 − +) 2.3 Gluon quark scattering Gluonquarkscatteringg(k )+q(k ) q(k )+η (P)proceedsevidently inleading 1 2 3 t → order α3 through the diagrams in fig. 2b. Collinear singularities have to be ab- s sorbedinthe structure functions.The (dimensionally regularized) squared matrix ¯ element for the production of QQ at threshold is given by s2 +u2 1 ǫ gq 2(s,t,u) = 32 (1 2ǫ)+ (39) |M | "−(s+u)2 t − t# and has to be supplemented by the color factor 2 1 λaλb λb 1 δ = (40) ij (cid:12)(cid:12)√3 2 2 !ij 2 !kl(cid:12)(cid:12) 3 X(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and the coupling constant g(cid:12)6. After spin and color av(cid:12)eraging the cross section can s then be cast into a form identical to eq. (29) with fgg replaced by 1 z 1+y2(1 z)2 ǫ fgq = − (41) 9(1 z)(1 y) "(1 y(1 z))2 − 1 2ǫ# − − − − − The y integration is performed as before and only a single pole remains α 4πµ2 ǫ 1 σgq = σ (ǫ) sΓ(1+ǫ) P (z) 0 2π M2 ! −ǫ gq (cid:26) (1 z)2 2(1 z) + ln − P (z)+C − (1 lnz)+C z (42) gq F F z ! z − ) with 1+(1 z)2 P (z) = C − (43) gq F z The 1/ǫ term can again be absorbed by the (quark) structure function. This amounts effecively to the subtraction of α 4πµ2 ǫ 1 s σ (ǫ) Γ(1+ǫ) P (z)+C (z) (44) 0 2π Q2F ! (cid:18)−ǫ gq gq (cid:19) 9