The two particle inclusive cross section for the reaction $p
+\bar{p}\rightarrow \gamma + c + X$ is studied in perturbative quantum
chromodynamics at order $O(\alpha ^2_s)$. Differential distributions are
provided for various observables, and a comparison is made with data from the
CDF collaboration.Comment: 4 pages, LaTeX, uses stwol.sty, 6 postscript figures. Invited paper
  presented by E. L. Berger at the 28th International Conference on High Energy
  Physics, Warsaw, Poland, July 25 - 31, 199

Bailey, B.

Berger, E. L.

Gordon, L. E.

English

arXiv

The two particle inclusive cross section for the reaction {ital p} + {ital {anti p}} {r_arrow} {gamma} + {ital c} + {ital X} is studied in perturbative quantum chromodynamics at order {ital O}({alpha}{sup 2}{sub s}). Differential distributions are provided for various observables, and a comparison is made from the CDF collaboration

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-4 be submitted manuscript has been authored by a contractor of the U. S. Government under contract No. W-31-109ENG-38. Aecordinglv. the U. S. Government retains a nonexcluiive. rovalty-free license to publish or reproduce the published form of t h i s  contribution. or allow others to do so. for U. S. Government purposes. JAN 1 6 1997 ANL-HEP-CP-96-84 O S T I  hep-ph/9610502 Prompt Photon plus Charm Production at Next-to-Leading Order in QCD * B. Bailey Physics Department, Ecked  College, St. Petersburg, FL 33711, U.S.A. E. L. Berger and L. E. Gordon High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (October 14, 1996) Abstract The two particle inclusive cross section for the reaction p + p --f y + c + X is studied in perturbative quantum chromodynamics at order O ( a t ) .  Differ- ential distributions are provided for various observables, and a comparison is made with data from the CDF collaboration. 'Invited paper presented by E. L. Berger at the 28th International Conference on High Energy Physics, Waxsaw, 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. PROMPT PHOTON PLUS CHARM PRODUCTION AT NEXT-TO-LEADING ORDER IN QCD B. BAILEY Physics Department, Eckerd College, St. Petersburg, FL 33711, U.S.A. E. L. BERGER and L. E. GORDON High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. The two particle inclusive cross section for the reaction p + + y 4- c + X is studied in perturbative quantum chromodynamics at order O(a:). Differential distributions are provided for various obsewables, and a comparison is made with data from the CDF collaboration. 1 Introduction Theoretical study of the reaction p + p  -+ 7 + c + X  is interesting for several reasons. The two particle inclusive cross section probes QCD dynamics in more detail than a single particle inclusive cross section. Specification of the cross section in next- to-leading order in QCD is correspondingly more challenging, and data will permit a study of var- ious correlations including those in the rapidities and transverse momenta of the final photon and charm quark. Second, the data may permit a di- rect measurement of the charm quark density of the proton if it can be shown, as we do 'f2, that the hard scattering cross section is dominated by the contribution from the cg scattering subpro- cess, where the incident c quark is a constituent of one of the initial hadrons. Third, it is valuable to be able to compare data obtained by the CDF Collaboration on prompt photon production in association with charm decay products, such as e*, p* and D*, with next-to-leading order pre- dictions. We have published two next-to-leading order calculations in perturbative QCD of the reaction p + @  -+ y+c+X at collider energies'>2. In most of the data of interest, the charm quark carries a rel- atively large value of transverse momentum, p$ 2 10 GeV. Since this value is large when compared to the mass of the charm quark, we neglect the mass of the quark, and we include hard-scattering subprocesses in our calculation in which the charm quark is a constituent of the initial hadrons. For values of p$ less than 10 GeV or so, it would be LO: c+g -> y t  pT = 15 G e V  + LO Frag. 7 h > CJ R v -4 V x x P \ -0 400 200 0 0.5 1 1.5 2 2 Figure 1: Cross section as a function of e at y7 = 0 in leading order. necessary to retain a finite mass for the quark and to examine a different set of subprocesses in which the massive quark is not a constituent in the ini- tial state but is, instead, produced in the hard scattering '. A calculation with m, # 0 also re- quires resummation of logarithmic terms of the form cr:lnn(p$/m,) for large values of p ~ .  In leading-order, the contributions to associ- ated production include the direct term c + g -+ 7 + c  along with several photon fragmentation pro- cesses typified by the hard scattering subprocess c + q -P c + q followed by long-distance fragmen- tation of the final q into a photon: q -+ y + X. 1 In next-to-leading order, there are gluon-exchange loop corrections to the lowest-order direct process, real gluon emission radiative corrections to the lowest-order direct process, and several new three- body final-state contributions: g + g -+ c + E + 7, q+Q--,c+E+7,c+q--rc+q+7,c+Q--rc+Q+7, c+E+ c+E+7, and c+c --+ c+c+r. We include all of these. In principle, we should also include fragmentation to next-to-leading order. However, in view of the small size of the fragmentation con- tribution and the theoretical problems identified with isolated prompt photon production in next- to-leading order, we treat the fragmentation terms at lowest order. The distinction between our two calcula- tions 't2 resides in the different techniques used in performing the phase-space integrals. In the first paper, we insist on going as far as we can us- ing purely analytical techniques. In the second, we use a combination of analytical and Monte Carlo techniques 6 .  The second approach allows implementation of isolation cuts and other exper- imentally relevant selections. In both cases, we evaluate the gluon exchange and the real gluon emission diagrams analytically, expose the infra- red and collinear singularities, cancel the infra-red singularities, and factor the collinear singularities into incident parton densities or final-state frag- mentation functions. We use dimensional regular- ization to  deal with the singularities. Although somewhat restrictive, the purely analytic calcula- tion provides the size of the two-particle inclu- sive cross section, the relative importance of var- ious contributing subprocesses, and an important check on the more versatile combination2 of ana- lytic and Monte Carlo methods, Some of our analysis is done in terms of the variable t = -$j. $;/(P:)~, and some of the results are presented as histograms in z for fi- nite intervals in z. If z is positive, the photon and charm quark are in opposite hemispheres. If z = 1, the photon and charm quark have equal and opposite values of transverse momentum. This situation occurs for the leading-order direct pro- cess, for gluon exchange loop corrections to the leading-order direct process, whenever an emitted gluon becomes soft, and when a third parton in a three-parton final-state process becomes collinear to an incident parton. The point z = 1 is associ- ated with various soft and collinear poles, exposed when the cross section is expanded as a function 400 2 200 2 a v N -0 h a V \ V z o  -200 z Figure 2: cross section as 8 function of z in next-to-leading order. The dominant contributions are shown. of z. The singularities are expressed in terms of delta functions, 6( 1 - z), and "plus"-distributions, 1/(1 - z)+. The singularities are integrable, and results are expressed as integrals over finite inter- vals in z. 2 Numerical Results Our results are presented at a center-of-mass en- ergy f i  = 1.8 TeV. The renormalization and fac- torization scales are taken as p = p;. We sum over charm and anticharm production throughout. In Fig. 1 the net lowest-order contribution is shown as a function of a at p; = 15 GeV. The lowest order cross section is made up of the lowest order direct term cg * 7c, proportional to 6(1- z ) ,  and fragmentation contributions. In Fig. 1, the direct term provides the peak at z = I, and the various photon fragmentation terms contribute in the re- gion z 2 1. Note the absence of any contribution in the region z < 1, an unrealistic prediction that illustrates the inadequacy of the lowest order ap- proach. The cross section in z resulting from our the next-to-leading order massless-charm calcula- tion is displayed in Fig. 2. The next-to-leading or- der collinear initial-state contributions reduce the peak at a = 1, and the next-to-leading order three- parton final-state contributions broaden the distri- bution. The cross section is finite at  all values of E ,  similar to the situation observed in experiments. 2 c+g -> y+c+x g+g -> y+c+c - c+q -> y+c+q - - - - -  - . - . - . - . ............ E a 10-2 - W \ 3 10-3 - 10-4 - I , ,  , , l , , , , l , , * , \ , , ,  20 40 60 80 100 PT (Gev) Figure 3: The transverse momentum dependence of &/dp;dyYdz, for z integrated over the interval 0.2 < z < 2.0. The upper solid line shows the sum of all subprocesses through next-to-leading order. The dashed line shows the sum of the O(a.)  and O(a3) contributions from the cg initial state. The O(a:) contributions from the 88 and cq initial states are shown as dash-dot and dotted curves. The lower solid line shows the O ( a f )  contribution from the @q, Ec, and cc initial states. . In Fig. 2 we also present the individual contribu- tions from the most important subprocesses. The cg initiated process dominates the cross section, but there are important contributions from the gg and cq initiated subprocesses in the region of small p g .  The predicted distribution in p$ is shown in Fig. 3. As might be expected, the contribution from the qq initial state begins to become impor- tant at large p;. Distributions in p& and in the rapidities of the 7 and c quark may be found in our publi- cations 1$2. In the second paper, we also discuss the influence of photon isolation, and we compare the magnitude and shape of the next-to-leading order and leading order cross sections for various observables. As an example, we present Fig. 4, a distribution in p$ for different selections on z and the isolation cone variable R. Positive correlations in rapidity at collider energies are a dynamical property of the hard- scattering matrix element ?. Indeed, it may be shown that the parton-level matrix element is proportional to cosh-l(Ay), where Ay is the dii- ference in the rapidities of two final-state particles 102 - 101 % 2 - Dl-  100 U a a 5 W 10-1 10-2 E '  ' '  ' " I '  " ' I '  " ' 1  -1  <.ye < 1 - 1  < y ' <  1 .:. . .: - full & a > 0.1 IL > 0.1. R = 0.7 a > 0.1, R = 0.4 - - - - - - - - - _ _  .................... 7 - 1 1 ~ ~ , 1 1 , , , 1 1 , / 1  10 20 30 40 50 PF (Cev) Figure 4: Cross section &/dp% as a function of p;,  with 15 < p; < 45 GeV, and the rapidities of the photon and charm quark restricted to -1.0 < y < 1.0. The solid curve shows the cross section with no further restrictions and the dashed curve the result after the additional selection z > 0.1. The dotted and dash-dot curves display the results d e r  photon isolation restrictions are applied, in addition to the cut on z. in a two-particle inclusive reaction. In Fig. 5,  we present the distribution in the difference of the ra- pidities of the photon and charm quark. In this example, the 7 is selected to have rapidity in the forward region 1 < yy < 2. Under these circum- stances, as may be observed in Fig. 5,  the charm quark is predicted to have positive rapidity also, with Ay = yy - yc peaking a value only slightiy larger than 0. In Fig. 6 we compare our results to CDF data' for photon plus pf production, where the p* is identified as the decay product of a charm quark. The three upper points are obtained8 from the Monte Carlo event-generator Pythia whereas the lower point is our theoretical expectation. The Pythia cross section lies substantially below the data while our cross section exceeds the data some- what. 3 Conclusions We presented the results of next-to-leading or- der QCD calculations of the inclusive production of a prompt photon in association with a charm quark. The two-particle inclusive cross section al- 3 400 100 0 R = 0.7 a > 0.1 h n a a A v W / -  I '  .. AY Figure 5: Cross section du/dAy M a function of the dif- ference Ay = u'f - f l  of the rapidities of the photon and &arm quark, for p + p -+ 7 + c + X at fi = 1.8 TeV. The ratio z is restricted to z > 0.1, and photon isolation is imposed, with R = 0.7. The transverse momentum and rapidity of the photon are selected to be in the intervals 16 < p a  4 45 GeV and 1.0 < y' < 2.0. The dashed curve shows the behavior at leading order. lows more refined tests of the underlying QCD dynamics, revealed, e.g., through rapidity corre- lations. Our cross section shows the expected reduced dependence on the choice of the fac- torization/renormalization/fiagmentation scale in QCD. The sensitivity of the cross section to the. charm quark content of the proton is preserved in next-to-leading order. A comparison with the CDF data shows reasonable agreement. Further work is underway to ioclude effects associated with the finite maas of the charm quark and with charm fragmentation: c 4 DX or c + pX. In other work, we are investigating associated production of photons and charm quarks at H E M  energies and the associated production of photons and bot- tom quarks at collider energies. Acknowledgments The work at Argonne National Laboratory was supported by the US Department of Energy, Di- vision of High Energy Physics, Contract number W-3 1-109-ENG-38. 0 NLOWD . b 9" -F ! lo'' 0 .I.' J F Cross Seaion Measurement Type Figure 6: Ratio of the measured 7cX a o s s  section to that predicted, for various charm decay channels .  References 1. E. L. Berger and L. E. Gordon, Phys. Rev. D 54, 2279 (1996) and references therein. 2. B. Bailey, E. L. Berger and L. E. Gordon, Phys. Rev. D 54,1896 (1996) and references therein. 3. CDF Collaboration, F. Abe et d, Fermilab Report FERMILAB Pub-96/152E1 Submit- ted to Phys. Rev. Lett. 4. P. Nason, S. Dawson, and R.K. Ellis, Nucl. Phys. B 303,607 (1988); B 327 ,49  (1989); B 335, 260 (1990) (E). W. Beenakker, H. Kuijif, W.L. van Neerven, and J. Smith, Phys. Rev. D 40, 54 (1989); W. Beenakker, W.L. van Neerven, R. Meng, G.A. Schuler, and J. Smith, Nucl. PAYS. B 351, 507 5. E. L. Berger, X. Guo, and J. W. Qiu, Phys. Rev. Lett. 76,  2234 (1996); Phys. Rev. D 54, (1996) (1 November, 1996). 6. B. Bailey, J. Ohnemus, and J. F. Owens Phys. Rev. D 46, 2018 (1992). 7. E. L. Berger, Phys. Rev. D 37, 1810 (1988). 8. S. Kuhlmann, CDF Collaboration, private (1991). communication. 4 

Prompt Photon plus Charm Production at Next-to-Leading Order in QCD

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