Anomalous chromomagnetic moments of quarks and large transverse energy jets

We consider the jet cross sections for gluons coupling to quarks with an anomalous chromomagnetic moment. We then apply this to the deviation and bounds from QCD found in the CDF and D0 Fermilab data, respectively, to ﬁnd a range of possible values for the anomalous moments. The quadratic and quartic terms in the anomalous moments can ﬁt to the rise of a deviation with transverse energy. Since previous analyses have been done on the top quark total cross section, here we assume the same moment on all quarks except the top quark and ﬁnd the range (cid:117) (cid:107) (cid:56) (cid:117) (cid:91) (cid:117) (cid:107) /(2 m q ) (cid:117) (cid:53) 1.0 (cid:54) 0.3 TeV (cid:50) 1 for the CDF data. Assuming the anomalous moment is present only on a charm or bottom quark which is pair produced results in a range (cid:117) (cid:107) b , c (cid:56) (cid:117) (cid:53) 3.5 (cid:54) 1.0 TeV (cid:50) 1 . The magnitudes here are compared with anomalous magnetic moments that could account for R b and found to be in the same general range, as well as not inconsistent with CERN LEP and SLD bounds on (cid:68)(cid:71) had . (cid:64) S0556-2821 (cid:126) 96 (cid:33) 05821-3 (cid:35)


I. INTRODUCTION
There are several higher-dimensional operators that can be used to evaluate structure from a theory beyond the standard model ͓1͔. In this paper we use the time-honored anomalous magnetic moment as the indicator ͓2-5͔, applied to the quark-gluon vertex, or the anomalous chromomagnetic moment. Several papers have been written analyzing the contribution of such a moment to the top production cross section ͓6-10͔. In this paper, we assume such a moment could apply to any or all of the mainly annihilated, scattered or produced quarks other than the top quark. The results of this will be different from those of the four-Fermion contact interaction operators, and also should be included in evaluations of possible structure. Along with the exchange of a spin 3/2 excited quark state ͓11͔, the E T dependence and angular two-jet behavior could help further limit or establish a composite mechanism, or new interactions from higher mass scales. The same theory that gives an anomalous chromomagnetic moment also may give a four-Fermion contact interaction, but here we analyze only the anomalous moment mechanism in a general context. For a complete analysis of all color current contributions in a particular model, we refer to the recent analysis of the Minimal Supersymmetric Standard Model ͓12͔. The key point in our analysis is that the anomalous moment term Ј q , when compared to the Dirac current ␥ , grows at high E T as O(ЈE T ).
In the top production papers, gluon fusion or quark fusion to a virtual gluon are calculated to produce the t-t pair. Here we include all quark gluon hadronic processes since quarks commonly present in protons and antiprotons could have the small anomalous moments.
Since next-to-leading order QCD corrections have not been calculated for this general set of anomalous moment processes, to compare with the data we follow the Collider Detector at Fermilab Collaboration ͑CDF͒ ͓13,14͔ procedure of calculating the ratio of the theory with structure divided by lowest order QCD and comparing it with the ratio of data divided by next to leading order ͑NLO͒ QCD. We find that an anomalous chromomagnetic coefficient ͉Ј͉ ϭ͉/(2m q )͉ϭ1.0 TeV Ϫ1 fits the CDF rise, and would be roughly upper bounded by 1.3 TeV Ϫ1 , and lower bounded by 0.7 TeV Ϫ1 . The central value for the rise in CDF is not directly supported by the D0 ͓15͔ data, but is within the one sigma systematic energy calibration error curve for D0, which rises to 120% at E T ϭ450 GeV.
We use Ј since in the general case the internal diagram or dynamics might not involve the light external quark, and a new physics model calculation will give Ј directly. The use of the breakup into a vector current (␥ ) and an anomalous chromomagnetic moment term (iЈ ) includes all anomalous moment vertex corrections in Ј including those of QCD. However, at the very large momentum transfers we are considering here, there are form factors on the QCD vertex corrections making up the anomalous chromomagnetic moment either for virtual gluons or for high E T virtual quarks ͑''sidewise form factors'' ͓16͔͒, which will damp like ͓ QCD Ј ϭO(m q /p Ќ 2 )͔ and become irrelevant. At some q 2 , the anomalous moment from new or composite interactions also will evidence a form factor. That is automatically included in the analysis by considering Ј(q 2 ) as a function of q 2 , but in the comparison to the data, we do not need to invoke that dependence yet. We test whether an anomalous magnetic moment equal to the anomalous chromomagnetic moment possibly indicated here would be in conflict with the ⌫ had accuracy at the CERN e ϩ e Ϫ collider LEP, and find it would not be.
Because of discrepancies in the total hadronic cross sections for charm and bottom production at LEP and the SLAC Large Detector ͑SLD͒, R c and R b , we also find a separate range for either the charm or bottom quark only having an anomalous chromomagnetic moment, using the quarkantiquark production cross sections. The range for the CDF data is ͉ b,c Ј ͉ϭ3.5Ϯ1 TeV Ϫ1 . We note that if the R b discrepancy is accounted for by an anomalous magnetic mo- TeV Ϫ1 , and is better than the inclusive jet limits for the b or c quark alone calculated here.

II. ANOMALOUS CHROMOMAGNETIC MOMENT CROSS SECTIONS
The top gluon fusion production ͓7,9,10͔ with anomalous chromomagnetic moments has been calculated already. We use the simplifications of these for the case of zero quark mass in the various channels in which quark-quark-gluongluon processes occur. For completeness, we repeat these formulas here for the massless case along with the massless QCD contribution. We also present here the result with anomalous chromomagnetic moments for quark-antiquark annihilation and scattering to the same quark-antiquark with interference, and its crossed quark-quark scattering. We then present the cross sections for different quarks scattering and annihilating. The cross sections are even in powers of Ј due to the zero quark mass limit being used. This is because at zero mass only gamma matrices occur in the QCD trace and substitution of an anomalous moment term for a ␥ would give an odd number of gamma matrices and resulting in a zero trace, unless accompanied by an additional term. In other terms, at zero quark mass, there is no interference between helicity conserving and helicity flip processes: helicity conserving ones have either no anomalous moment terms or two such terms; helicity flip ones have one anomalous moment for each helicity flipped quark line, and this gets squared in the cross section. Neglect of the mass in a propagator term could give an error of order m b /E T Ϸ2.3% in the amplitude at E T ϭ200 GeV, or 4.5% in the cross section.

A. Gluon processes
The well-known result ͓17͔ for gluon-gluon scattering is For quark-antiquark to gluon-gluon ͓18,17,7,9͔, the cross section is

͑2͒
For gluon-gluon to quark-antiquark, we use the above, replacing the coefficient 16/72 by 16/256, for either the quark or antiquark being observed. For gluon-quark to gluon-quark with the gluon observed ͑or gluon-antiquark to gluon-antiquark͒, the cross section is

͑3͒
For gluon-quark to gluon-quark with the quark observed ͑or gluon-antiquark to gluon-antiquark͒, we interchange t with û in the above equation.

B. Quark processes
For quark antiquark to the same quark antiquark including t-channel exchange as well as s-channel annihilation with their interference, and observing the quark, we have

͑4͒
For the above process observing the antiquark, we interchange t and û in the above formula. For identical quarks scattering to the same identical quarks, we interchange ŝ and û in the above equation, and add a factor of 1/2 for the identical final state quarks. For a quark-antiquark annihilation to a virtual gluon creating a different quark-antiquark pair, as in charm and bottom production, summing over identical cross sections for either quark or antiquark observed ͑which will later be multiplied by 4 for four other light quarks being created͒, we have for the s-channel gluon exchange

͑5͒
For qϩqЈ→qϩqЈ observing the final quark, we have for the t-channel gluon exchange

͑6͒
For observing the final antiquark, we use the above equation with t and û exchanged. For qϩqЈ to qϩqЈ observing the quark, we use the above formula and interchange ŝ and û in the large parentheses. For qϩqЈ to qϩqЈ observing qЈ, we interchange t and û in the large parentheses of the above formula.

III. COMPARISON WITH POSSIBLE ANOMALOUS MAGNETIC MOMENTS IN Z COUPLINGS
The possibility of anomalous magnetic moments of quarks appearing at LEP and SLD at the Z peak and their forward-backward asymmetry has been considered ͓4,8͔.
Here we just get values to indicate the order of magnitude, referring the reader to the more careful and complete analysis by T.G. Rizzo ͓8͔, which gives Ϫ0.012р b e рϪ0.002 at 95% CL, updated to Moriond 1996 data ͓19͔. With an anomalous electric dipole moment as well, it allows positive b e р0.025. Using the integrated cross section ͓4,8͔ for b production at the Z peak, we find with m b ϭ4.5 GeV ͑and Letting the b e terms account for the discrepancy ͓20͔, ⌬R b /R b ϭ0.026 gives two solutions, b e ϭ0.027, and b e ϭϪ0.0069, corresponding to b Ј e ϭ3.0 TeV Ϫ1 and b Ј e ϭϪ0.77 TeV Ϫ1 .
A FB b isolates the cos term in the Z cross section, which can be partly due to the anomalous magnetic moment ͓4,8͔ where the inverse power term has been taken here to match 1ϩ⌬R b /R b . Using the SM value for A b SM the positive b e ϭ0.027 gives a 3.1 discrepancy with ͓20͔ A FB b ϭ0.1002Ϯ0.0028, and is eliminated. The negative b e ϭϪ0.0069 ( b Ј e ϭϪ0.77 TeV Ϫ1 ) value only gives a 0.4 deviation and is consistent. The error on is about 6%, and it is 11% below theory. The correction of the anomalous moment only lowers theory by 0.5%.
Our calculated CDF jet cross sections do not depend on the sign of Јg , and for the case of only the b quark having the anomalous chromomagnetic moment ͉ b Ј g ͉ϭ3.5Ϯ1 TeV Ϫ1 is not inconsistent with the ⌬R b anomalous magnetic moment since there will be a numerical factor depending on how the electric and color charges are distributed among the internal constituents with different masses.
For the case where all quarks except top are given the anomalous chromomagnetic moment, allowing an equal anomalous magnetic moment for each, we find changes in ⌬⌫ had /⌫ had of 0.0017, 0.0005, and Ϫ0.0005, corresponding to the Јϭ1.3, 1.0, and 0.7 TeV Ϫ1 cases, respectively. This is at most a one discrepancy since the fractional error ͓20͔ on ⌬⌫ had /⌫ had is Ϯ0.0019.

IV. RESULTS AND CONCLUSIONS
In calculating the jet transverse energy distributions, we have used the parton distribution functions MRSA Ј ͓21͔.
Since we are taking ratios of anomalous moment contributions to strict QCD, both in leading order, most variance with parton distribution functions ͑PDFs͒ drops out, only being about 2% at E T ϭ450 GeV among the Martin-Roberts-Stirling ͑MRS͒ set of PDFs, or between MRSA Ј and CTEQ3M. Assuming all quarks except the top have the same anomalous chromomagnetic moment, we find in Fig. 1 that the range ͉Ј͉ϭ1.0Ϯ0.3 TeV Ϫ1 will fit the CDF data ͓13,14͔, and be in the allowed systematic error range of the D0 data ͓15͔. The quadratic and quartic terms in Ј give a natural fit to the possible curvature in the data, and indicate FIG. 2. The ratio of the anomalous chromomagnetic moment contribution of either a bottom or charm quark-antiquark pair created to the lowest order QCD jet distribution in transverse energy.
The dashed, solid, and dotted curves are for ͉ b,c Ј ͉ϭ4.5, 3.5, and 2.5 TeV Ϫ1 , respectively. The data are as in Fig. 1.   FIG. 1. The ratio of the anomalous chromomagnetic moment contribution to the lowest order QCD jet distribution in transverse energy. The dashed, solid, and dotted curves are for ͉Ј͉ϭ1.3, 1.0, and 0.7 TeV Ϫ1 , respectively. The crossed data are from CDF run 1a, and the circle data from run 1b ͑preliminary͒.
why the corrections do not show up significantly until E T у200 GeV. It is not inconsistent with an equal anomalous magnetic moment being present for the quarks at the Z peak.
If the charm or bottom quark alone among the lighter quarks possesses a sizable anomalous chromomagnetic moment, we find in Fig. 2 the range ͉ b,c Ј ͉ϭ3.5Ϯ1 TeV Ϫ1 will fit the CDF data, and be in the allowed range of the D0 data. It is not inconsistent with a comparable anomalous magnetic moment explanation ͓8͔ of R b .