Using $e^+e^-\rightarrow\gamma + X$ as an example, we show that the
conventional factorization theorem in perturbative quantum chromodynamics
breaks down for isolated photon cross sections in a well defined part of phase
space. Implications and physical consequences are discussed.Comment: 11 pages, RevTex, 1 figure in postscrip

Berger, Edmond L.

Guo, Xiaofeng

Qiu, Jianwei

English

arXiv

Using {ital e{sup +}e{sup -} {r_arrow} {gamma} + X} as an example, we show that the conventional factorization theorem of perturbative QCD breaks down for isolated photon cross sections in a specific part of phase space. Implications are discussed

Berger, E. L.

UNT Digital Library

*. . , *  by a contractor of the U. S. Government udder contract No. W-31-104ENG-38. Accordingly, tha U. S. Government retains a nonexclusive, royaltyfrw license to publish BREAKDOWN OF CONVENTIONAL FACTORIZATION FOR ISOLATED PHOTON CROSS SECTIONS EDMOND L. BERGER", XIAOFENG GUO' and JIANWEI QIU' " High Energy Physics Division, Argonne National Laboratory, Argonne, iL 60439, U S A  ' Department of Physics and Astronomy, Iowa State Universaty, rlmes, I A  50011, USA Using e+e- -+ y + X as an example, we show that the conventional factorization theorem of perturbative QCD breaks down for isolated photon cross sections in a specific part of phase space. Implications are discussed. Much of the predictive power of perturbative QCD derives from factoriza- tion theorems Conventional factorization expresses a physical quantity as the convolution of a partonic part with a nonperturbative matrix element, and it requires that the perturbatively calculated partonic part be infrared safe or- der by order in ad. Predictions then follow when processes with different hard scattering but the same matrix elements are compared. Using e+e- --+ y X  as an example, we demonstrate that the perturbatively calculated partonic part for the isolated photon cross section is not infrared safe in a well-defined phase space. The essence of isolation is that a cone of half-angle 6 is drawn about the direction of the photon's momentum, and the isolated cross section is defined for photons accompanied by less than a specified amount of hadronic energy in the cone, e.g., E r n e  5 E,,. At high energy, photons can result from long-distance fragmentation of quarks and gluons, themselves produced in short-distance hard collisions. In such fragmentation contributions, hadronic energy in the isolation cone has two sources: a) energy from parton fragmentation, Etrag , and b) energy from non-fragmenting final-state partons, E;z3&ns , that enter the cone. When the maximum hadronic energy allowed in the isolation cone is saturated by the fragmentation energy, E,,, = Efrag, there is no allowance for energy in the cone from other final-state partons. In particular, if there is a gluon in the final state, the phase space for this gluon becomes restricted. By contrast, isolation does not affect the virtual gluon exchange contribution. Therefore, for isolated photons, there is a possibility that the infrared singularity from the virtual contribution may not be cancelled completely by the restricted real contribu- tion. We showed 293 that such incomplete cancellation of infrared singularities appears first at  next-to-leading order (NLO) in the quark fragmentation con- 1 Paper presented at the XIV International Conference on Particles and Nuclei, May 23-28, 1996, Williamsburg, Virginia. ~~~~~~ OF THIS BOeLfM DXSCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spc- cifie commercial product, process, or service by trade name, trademark, manufac- turer. or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. P \ b W % 10-7 10-8 - - 0 b L x7 Figure 1: Comparison of the one-loop quark fragmentation contributions to the isolated cross section and the inclusive cross section in e+e- - - /X as a function of x-, = 2 E - 1 6 .  tributions. If conventional factorization were true, the fragmentation contributions to the cross section for isolated photons would be expressed in the factorized form (1) zy = 2Ey/&, x, = 2Ec/&, Eh = Emax/Er ,  and the sum extends over c = q,  ij and g. DC-?(z, 6) is the nonperturbative function that describes fragmentation of parton "c" into a photon. The lower limit of integration results from the isolation requirement with the assumption that all fragmentation energy is in the isolation cone3. Because of the isolation condition, the phase space constraints are different in three regions: a) z-, < l/(l-+q,), b) z7 = l / ( l + c b ) ,  c) xy > 1/(1 + ~ h ) .  We show below that the next-to-leading order partonic hard part for quark fragmentation, Eqdu~,0 , - -q , /d3pq ,  is infrared sensitive. When zy < 1/(1 + ~ h ) ,  subprocesses with two-body final states do not contribute. Therefore, there is no leading-order quark (or antiquark) fragmen- tation contribution, and one-loop virtual diagrams do not contribute. The well-known 1/(1- zg) infrared singularity of the real gluon emission diagrams, as zq = xy/z + 1, will remain in x. After convolution with Dq-.7(r), this inverse power infrared sensitivity yields a logarithmic divergence propor- tional to!n(l/z,-(l+q,)). Asshown in Fig. 1, as ty -+ l / ( l + ~ h ) ,  the isolated cross section becomes larger than the inclusive cross section, which is certainly not physical. This infrared sensitivity in &2+oee--qx signals a breakdown of conventional perturbative factorization. When zr = 1/(1+ fh), x q  = z7/z = 1 is possible. Therefore, the one-loop virtual diagrams, which are proportional to 6(1 - z q )  will contribute. However, 2 isolation constraints limit the phase space of real gluon emission in the real subprocess, e+e- -+ qqg. Consequently, the infrared divergences in the real and virtual contributions do not cancel completely in the isolated case, the inclusive case. In n = 4 - 2.5 dimensions, we find unlike At x, = 1, corresponding to  z7 = 1/(1+ q,), the partonic part for quark fragrdentation in Eq. (2) i s  infrared divergent, and the perturbative calculation is not well-defined. Conventional perturbative factorization again breaks down. When x7 > 1/(1+ ~ h ) ,  due to the finite cone size, there is a very small region of phase space where the isolated cross section is not exactly equal to the inclusive cross section. This region of phase space is proportional to Ehb2/4. In this narrow region above x7 = 1/(1 + Eh), dl one-loop contributions to the isolated cross section have a logarithmic divergence Ln( 1 + Eh - 1/z7) as x7 --t 1/(1+ ~ h )  from above3. In summary, the NLO partonic part for the quark fragmentation contribu- tion to the isolated cross section, calculated in perturbative QCD, is infrared sensitive when x7 5 1/( 1 + ~ h ) .  Conventional perturbative factorization for the cross section of isolated photons in e f e -  annihilation breaks down for x7 in the neighborhood of 1/(1+ ~ h ) .  Breakdown of factorization means that the isolated cross section cannot be calculated reliably in perturbative QCD near x7 = 1/( I + E ~ ) .  This result challenges theorists to find a modified factorization scheme. References 1. J. C. Collins, D. E. Soper and G. Sterman, in Perturbdive Qvuntum Chmmody~mics,  ed. A. H. Mueller (World Scientific, Singapore, 1989). 2. E. L. Berger, X. Guo and J. Qiu, Phys. Rev. Lett. 76, 2234 (1996). 3. E. L. Berger, X. Guo and J. Qiu, Argonne report ANL-HEP-PR-96-37, heegph/9605324, Phys.Rev. D, in press. Acknowlegdment This work was supported in part by the U. S. Department of Energy, Division of High Energy Physics, Contract No. W-31-109-ENG38. 3 

Breakdown of conventional factorization for isolated photon cross sections

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Breakdown of Conventional Factorization for Isolated Photon Cross Sections

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