We summarize our calculation of the total cross section for top quark production at hadron colliders within the context of perturbative quantum chromodynamics, including resummation of the effects of initial-state soft gluon radiation to all orders in the strong coupling strength

Berger, E. L.

Contopanagos, H.

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. - by a contractor of the U. S. Government under contract No. W-31-104ENG-38. Accordingly. the U. S. Government retains a nonexclusive, royalty-free iscense to publash or reproduce the published form of this contribution. or allow others to do 90, for ANL-HEP-CP-96-83 Theoretical Aspects of Top Quark Production at Hadron Colliders * Edmond L. Berger and Harry Contopanagos High Energy Physics Division, Argonne National Laboratory Argonne, Illinois 60439, USA (October 15, 1996) Abstract We summarize OUT calculation of the total cross section for top quark pro- duction at hadron colliders within the context of perturbative quantum chro- modynamics, including resummation of the effects of initial-state soft gluon radiation to all orders in the strong coupling strength. *Invited paper presented by E, L. Berger at the 28th International Conference on High Energy Physics, Warsaw, Poland, July 25 - 31, 1996. Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. THEORETICAL ASPECTS OF T O P  Q U A R K  P R O D U C T I O N  AT H A D R O N  COLLIDERS EDMOND L. BERGER and HARRY CONTOPANAGOS High E n e m  Physics Division, Argonne National Labomtory, Argonne, IL 60439, USA We nunmarise our calculation of the total cross section for top quark production at hadron colliders within the context of perturbative quantum chromodynamics, includingresummation of the effects of initial-state soft gluon radiation to all  orders in the strong coupling strength. 1 In t roduct ion  and Motivation In hadron interactions at  collider energies, t E  pair production proceeds through partonic hard- scattering processes involving initial-state light quarks q and gluons g .  In lowest-order pertur- bative quantum chromodynamics (QCD), O(CY;)~  the two partonic subprocesses are q + t j  + t + t and g + g + t + f. Calculations of the cross sec- tion through next-to-leading order, O(a:), involve gluonic radiative corrections to these lowest-order subprocesses as well as contributions from the q+g  initial state A complete fixed-order calculation at order O(cry),n 2 4 does not exist. The physical cross section for each production channel is obtained through the convolution (1) The square of the total hadronic center-of-mass energy is S, the square of the partonic center-of- mass energy is s, m denotes the top mass, p is the usual factorization and renormalization scale, and @ij (q ,  p)  is the parton flux. The variable q = - 1 measures the distance from the par- tonic threshold. The indices ij € ( q q , g g )  denote the initial parton channel. The partonic cross sec- tion &j(q,  rn, p) is obtained commonly from fixed- order QCD calculations ', or, as described here, from calculations that go beyond fixed-order per- turbation theory through the inclusion of gluon resummation2T3V4 to all orders in the strong cou- pling strength a,. Unless otherwise specified, we use a z ~ ( p  = m) a.(m)/x. The total physical cross section is obtained after incoherent addition of the contributions from the the qtj and g g  pro- duction channels. Comparison of the partonic cross section at  next-to-leading order with its lowest-order value reveals that the ratio becomes very large in the near-threshold region. Indeed, as q 4 0, the "K- factor" at the partonic level k(q) grows in propor- tion to aln2(q). The large mass of the top quark notwithstanding, the large ratio $(q)  makes it ev- ident that the next-to-leading order result does not necessarily provide a reliable quantitative predic- tion of the top quark production cross section at the energy of the Tevatron collider. The large ra- tio casts doubt on the reliability of simple fixed- order perturbation theory for physical processes for which the near-threshold region in the suben- ergy variable contributes significantly to the phys- ical cross section. Top quark production at the Fermilab Tevatron is one such process, because the top mass is relatively large compared to the energy available. Other examples include the pro- duction of hadronic jets that carry large values of transverse momentum and the production of pairs of supersymmetric particles with large mass. To obtain more reliable theoretical estimates of the cross section in perturbative QCD, it is important first to identify and isolate the terms that provide the large next-to-leading order enhancement and then to resum these effects to all orders in the strong coupling strength. 2 Gluon Radiat ion and Resummation The origin of the large threshold enhancement may be traced to  initial-state gluonic radiative corrections to the lowest-order channels. We re- . mark that we are calculating the inclusive total cross section for the production of a top quark- antiquark pair, i.e., the total cross section for t+i?+anything. The partonic subenergy threshold in question is the threshold for t + f+ any number 1 of gluons. This coincides with the threshold in the invariant mass of the t + i? system for the lowest order subprocesses only. For i + j + t + i? + g, we define the variable z through the invariant (1 - z )  = a r n x  , where k and pt are the four- vector momenta of the gluon and top quark. In the limit that z + 1, the radiated gluon carries zero momentum. After cancellation of soft sin- gularities and factorization of collinear singulari- ties in O(CY:), there is a left-over integrable large logarithmic contribution to the partonic cross sec- tion associated with initial-state gluon radiation. This contribution is proportional to a3 ln2( 1 - z). When integrated over the near-threshold region 1 2 z 2 0, it provides an excellent approximation to the full next-to-leading order physical cross sec- tion as a function of the top mass. At m = 175 GeV, the ratio of the next-to-leading order to the leading order physical cross sections in the leading This ratio shows that the near-threshold logarithm builds up cross section in a worrisome fashion. It suggests that perturbation theory is not converg- ing to a stable prediction of the cross section. The goal of gluon resummation is to sum the series in CYn+2 ln2n (1 - z )  t o  all orders in CY in order to ob- tain a more defensible prediction. Different methods of resummation differ in theoretically and phenomenologically important respects. Formally, if not explicitly in some ap- proaches, an integral over the radiated gluon mo- mentum z must be done over regions in which z + 1. Therefore, one significant distinction among methods has to do with how the inevitable "non- perturbative" region is handled. In the approach of Laenen, Smith, and van Neerven (LSvN) 2, an  undetermined infrared cutoff (IRC) po is intro- duced, with A Q C ~  5 po 5 m. The presence of an extra scale spoils the renormalization group properties of the overall expression. The unfortu- nate dependence of the resummed cross section on this undetermined cutoff is important numerically since it appears in an exponent 2. It is difficult to evaluate theoretical uncertainties in a method that requires an undetermined infrared cutoff. logarithmic approximation is aqq (O+l) /aqq (0)- 1.22. 3 Perturbative Resummation The method of resummation we employ is based on a perturbative truncation of principal-value re- 2 summation5. This method has an important tech- nical advantage in that it does not depend on ar- bitrary infrared cutoffs. Because extra scales are absent, the method permits an evaluation of the perturbative regime of applicability of the method, i.e., the region of the gluon radiation phase space where perturbation theory should be valid. Factorization and evolution lead directly to exponentiation of the set of large threshold loga- rithms in moment (n) space in terms of a finite ex- ponent Epv that includes both perturbative and non-perturbative content. The non-perturbative content is not a prediction of perturbative QCD. In our study of top quark production, we apply the exponent only in the interval in moment space in which the perturbative content dominates. Af- ter inversion of the Mellin transform from moment space to the physically relevant momentum space, the resummed partonic cross sections, including all large threshold corrections, can be written (2) The leading large threshold corrections are con- tained in the exponent Ei j (z ,  a), a calculable polynomial in a. The derivative 3ij(7,  m, z )  = O ( Q ~ )  partonic cross section expressed in terms of inelastic kinematic variables. The upper limit of integration, < 1, is set by the boundary between the perturbative and non-perturbative regimes, well specified within the context of the calculation, and k i n  is fixed by kinematics. Our perturbative resummation probes the threshold down to q 2 70 = (I - zmOz)/2. Be- low this value, perturbation theory, resummed or otherwise, is not to be trusted. For m = 175 GeV, we determine that the perturbative regime is re- stricted to values of the subenergy greater than 1.22 GeV above the threshold (2m) in the qtj chan- nel and 8.64 GeV above threshold in the gg chan- nel. The difference reflects the larger color factor in the gg case. The value 1.22 GeV is comparable to  the decay width of the top quark. d(8i, (0) ( q , m , z ) ) / d z ,  and Sii (0) is the lowest-order 4 Physical cross section Other than the top mass, the only undetermined scale in our approach is the usual QCD factoriza- tion and renormalization scale p. In Fig. 1, we p j+ t f+X. Js= 1.8 TeV / , . .  , . . , , . . I . , .  14U 160 180 2w 220 24c : rn (GeV) Figure 1: Inclusive total cross section for top quark produc- tion in collisions at = 1.8 TeV. The dashed curves show the upper and lower limits while the solid cupye is our central prediction. CDF and DO data an shown. show our total cross section for t i  production as a function of top mass in @ collisions at  = 1.8 TeV. The central value is obtained with the choice p / m  = 1, and the lower and upper limits are the maximum and minimum of the cross section in the range p / m  E {0.5,2). At m = 175 GeV, the full width of the uncertainty band is about 10% . We consider that the variation of the cross section over the range p / m  E {0.5,2} provides a good overall estimate of uncertainty. For comparison, we note that over the same range of p ,  the strong coupling strength a varies by &lo% at m = 175 GeV. In estimating uncertainties, we do not consider ex- plicit variations of our non-perturbative cutoff, ex- pressed through kaz, the upper limit of integra- tion in Eq. (2). This is justified because, for a fixed m and p ,  ha= is obtained by enforcing dominance of the universal leading logarithmic terms over the subleading ones. Therefore, haz is derived and is not a source of uncertainty. At fixed m, the cutoff necessarily varies as p and thus a vary. Our calculation is in agreement with the data '. We find otf(m = 175 GeV,.\/S = 1.8 TeV) = 5.522:::; pb. Our cross section is larger than the next-to-leading order value by about 9%. The top quark cross section increases quickly with the energy of the p@ collider. We provide predictions in Fig. 2 for an upgraded Tevatron operating a t  f i  = 2 TeV. We predict &(m = 175 GeV, 43 = 2 TeV) = 7.562:::; pb. The cen- tral value rises to 22.4 pb at = 3 TeV and 46 pb at f i  = 4 TeV. 5 Other Methods of Resummation Two other groups have published calculations of the total cross section at m = 175 GeV and ,,G = 1.8 TeV: otf(LSvN ') = 4.95ft:ii pb; and &(CMST ') = 4.75+: pb. From the purely numerical point of view, all three agree within their estimates of theoretical uncertainty. How- ever, the resummation methods differ as do the methods for estimating uncertainties. Both the central value and the band of uncertainty of the LSvN predictions are sensitive to their arbitrary infrared cutoffs. To estimate theoretical uncer- tainty, we use the standard p-variation, whereas LSvN obtain theirs primarily from variations of their cutoffs. The group of Catani, Mangano, Nason, and Trentadue (CMNT) calculate a central value of the resummed cross section (also with p / m  = 1) that is less than 1% above the exact next-to- leading order value. There are similarities and differences between our approach to resummation and the method of CMNT. We both use the same universal leading-logarithm expression in moment space, but differences occur after the transfor- mation to momentum space. The differences in the two approaches can be stated more explicitly if we examine the perturbative expansion of the resummed hard kernel X g ( z , a )  in the two a p  proaches. If, instead of restricting the resumma- tion to the universal leading logarithms only, we were to use the full content of 7 f t ( z ,  a), we would arrive at an analytic expression that is equivalent to the numerical inversion of CMNT, (3) [ 1 x,, R N l t 2 a ~ i ,  1n2(1-z)+2.yE ln(1-z) +0(a2). In terms of this expansion, in our, work we retain only the leading term ln2(1 - 2). at order a, but CMNT retain both this term and the subleading term 27~3 In( 1 - 2). Indeed, if the subleading term 2yE In(1 - z )  is discarded in Eq. (3), the residu- als 6 i j / U p o  defined by CMNT increase from 0.18% to 1.3% in the qQ production channel and 3 from 5.4% to 20.2% in the gg channel. After addi- tion of the two channels, the total residual 6/aNLo grows from the negligible value of about 0.8% cited by CMNT to the value 3.5%. While still smaller than the increase of about 9% that we obtain, the increase of 3.5% vs. 0.8% shows the substantial influence of the subleading logarithmic terms re- tained by CMNT. We judge that it is not appropriate to keep the subleading term for several reasons: it is not uni- versal; it is not the same as the subleading term in the exact, O(a3)  calculation; and it can be changed arbitrarily if one elects to keep non-leading terms in moment space. The subleading term is nega- tive and i t  is numerically very significant, if it is integrated throughout the phase space. In the qq channel at  m = 175 GeV and & = 1.8 TeV, its inclusion eliminates more than half of the con- tribution from the leading term. In our view, the presence of numerically significant subleading con- tributions begs the question of consistency. The influence of subleading terms is amplified at higher orders where additional subleading structures oc- cur in the CMNT approach with significant nu- merical coefficients proportional to x' ,  C(3), and so forth. We will present a more detailed discus- sion of these points elsewhere. References 6 Discussion Extending our calculation to much larger values of m in p-ji collisions at  &' = 1.8 TeV, we find that resummation in the principal qq channel produces enhancements over the next-to-leading order cross section of 21%, 26%, and 34%, respectively, for m = 500, 600, and 700 GeV. The reason for the increase of the enhancements with mass at fixed energy is that the threshold region becomes in- creasingly dominant. Since the qq channel also dominates in the production of hadronic jets at very large values of transverse momenta, we sug- gest that on the order of 25% of the excess cross section reported by the CDF collaboration may well be accounted for by resummation. Our theoretical analysis and the stability of our cross sections under p variation provide con- fidence that our perturbative resummation proce- dure yields an accurate calculation of the inclusive top quark cross section at Tevatron energies and exhausts present understanding of the perturba- tive content of the theory. I 140 '60 I80 ZOO 220 ? A 0  260 rn (GrV)  Figure 2: Inclusive total cross section for top quark pro- duction in collisions at 6 = 2.0 TeV. Acknowledgments This work was supported by the U.S. Department of Energy, Division of High Energy Physics, Con- tract No. W-31- 109-ENG-38. 1. P. Nason e t  al, Nucl. Phys. B 303, 607 (1988); 327, 49 (1989); 335, 260 (1990) (E). W. Beenakker et al, Phys. Rev. D 40, 54 (1989); Nucl. Phys. B 351, 507 (1991). 2. E. Laenen, J. Smith, and W.L. van Neerven, Nucl. Phys. B 389, 543 (1992); Phys. Lett. B 321, 254 (1994). 3. E. L. Berger and H. Contopanagos, Phys. Lett. B 381, 115 (1995); Phys. Rev. D 54, 3085 (1996); ANL-HEP-CP-96-51, hep- ph/9606421 (21 June 1996). 4. S. Catani, M. Mangano, P. Nason, and L. Trentadue, Phys. Lett. B 378, 329 (1996); hepph/9604351(18 Apr 1996); M. Mangano and P. Nason, private communication. 5. H. Contopanagos and G. Sterman, NucI. Phys. B 400, 211 (1993); 419, 77 (1994). 6. P. Tipton, plenary session report, these pro- ceedings; A. Castro (CDF collaboration), ibid; J.  Bantly (DO collaboration), ibid. 7. F. Abe e t  a1 , Phys. Rev. Lett. 77, 448 ( 1996). 4 

Theoretical aspects of top quark production at hadron collider

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Theoretical aspects of top quark production at hadron collider

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