Forward-backward asymmetry of top quark in unparticle physics

The updated CDF measurement of the forward-backward asymmetry (FBA) in the top quark production p{bar p} ->t{bar t} at Tevatron (with the CMS energy 1.96 TeV) shows a deviation of 2*sigma from the value predicted by the Standard QCD Model. We present calculation of this quantity in the scenario where colored unparticle physics contributes to the s-channel of the process, and obtain the regions in the plane of the unparticle parameters lambda and dU which give the values of the FBA and of the total t{bar t} production cross section compatible with the present measurements.

Due to C-parity invariance, it is known that the forward-backward asymmetry (FBA) of top quark pair production at Tevatron vanishes at leading order (LO) [1] in the standard QCD model (SM). The inclusive non-zero charge asymmetry can be induced by (i) radiative corrections to quark-antiquark annihilation and (ii) interference between different amplitudes contributing to gluon-quark scattering qg → ttq andqg → ttq [1,2]. Within the SM, this leads, at the Tevatron with √ s = 1.96 TeV, to the nonzero but relatively low prediction [3], Measurement of any significant deviation from this SM prediction could be attributed to the new physics effects.
When D0 Collaboration published the first measurement on the FBA in top-quark pair production in the pp laboratory frame with 0.9 fb −1 of data, an unexpectedly larger FBA value was indicated [4]. By using 1.9 fb −1 [5], the CDF Collaboration observed the asymmetry (at √ s = 1.96 TeV) to be A pp FB = 0.17 ± 0.08 , A tt FB = 0.24 ± 0.14 , in the pp frame and tt frame, respectively. The updated CDF result with luminosity of 3.2 fb −1 , in the pp (lab) frame, is [6] A pp FB (exp) = 0.193 ± 0.065(stat) ± 0.024(syst) .
As seen in Eq. (3), the large value of the FBA of top-quark is not smeared by the statistics.
In general, the top-pair production by the new physics could be through s-, t-and uchannel and the situation depends on the property of the new particle. No matter to which channel they contribute, the extensions of the SM in the framework of particle physics, such as axigluon [8,11], Z ′ [9], W ′ [10], diquarks [13], etc., have some drawbacks. For instance, in order to explain the observed top-quark FBA value, one has to introduce unimaginably large flavor-changing couplings in the t-and u-channels. The couplings in the s-channel could be as large as the strong gauge coupling of the SM. However, beside the serious constraint from the invisible production of a new resonance, the sign of the top-quark couplings has to be chosen opposite to that of the light quarks in order to get the correct sign of the FBA value.
In order to avoid the aforementioned problems, we study in this work the top quark FBA in the framework of unparticle physics which is dictated by the scale of conformal invariance.
An exact scale-invariant "stuff" cannot have a definite mass unless it is zero. Therefore, in order to distinguish it from the conventional particles, Georgi named the "stuff" unparticle [24,25]. It was found that the unparticle has a noninteger scaling dimension d U and behaves as an invisible particle [24] (see also [26]). Further implications of the unparticle to collider and low energy physics are discussed in Refs. [27][28][29]. We will adopt three aspects of unparticle physics in order to present a possible explanation of the aforementioned large value of the FBA. Firstly, if we take the protecting symmetry to be exact, then, due to the unique character of indefinite mass, no visible resonant unparticle will be produced in the pp collisions. Secondly, by utilizing the noninteger scale dimension, the differential cross section for tt production could be enhanced without fine-tuning the large couplings of unparticle and quarks. Finally, to match the interaction structure of the SM, the considered scale invariant stuff (unparticle) is a vector boson and carries color charges [30]; it has chiral couplings to quarks and its representation in SU(3) c belongs to color-octet.
Since there is no well established approach to give a full theory for unparticle interactions, we study instead the topic from the phenomenological viewpoint. In order to escape the large couplings from flavor changing neutral currents (FCNCs), the couplings of unparticle to quarks are chosen to be flavor conserving. Hence, we write the interactions of colored unparticle with quarks as where g χ = l q χ /Λ d U −1 U and χ = V and A. Here, l q χ is the dimensionless coupling and the index q denotes the quark flavor, Λ U is the scale at which the unparticle is formed, and is taken as d U . By following the scheme shown in Ref. [31], the propagator of the colored vector unparticle is written as with After introducing the interactions of unparticle with quarks and the virtual unparticle propagator, we can now calculate the tt pair production at the quark level. Using Eqs. (4) and (5), the scattering amplitude for qq → tt by unparticle exchange in the s-channel is where flavor q denotes the light u and d quark, and p = p q + pq = p t + pt. The t-channel does not contribute, due to flavor-conserving vertices Eq. (4). The equations of motion implȳ q/ pq = 0 andq/ pγ 5 q = −2m qq γ 5 q. Thus, the factor 2(d U − 2)/(d U − 1) in the propagator is associated with the light quark mass and is negligible. Consequently, the scattering amplitude combined with the SM contributions is given by with g s being the strong coupling of the QCD SM andŝ = (p q + pq) 2 = (p t + pt) 2 . For explicitly showing the differential cross section in tt invariant mass frame, we choose the relevant coordinates of particle momenta as with β 2 t = 1−4m 2 t /ŝ. The polar angleθ is the relative angle between outgoing top-quark and the incoming q-quark. The spin and color averaged amplitude-square is straightforwardly obtained as As a consequence, the differential cross section for qq → tt process as a function ofθ in tt frame is found to be In the s-channel, only the terms linear in cosθ will contribute directly to the forwardbackward asymmetry. From Eq. (11), the relevant effects are associated with g q A g t A and in which the former is from the interference between unparticle and SM while the latter is from the contribution of unparticle itself. In both terms, we see clearly that the axial-vector couplings are the essential to generate the FBA.
To obtain the hadronic cross section from the parton level, we have to consider the convolution with the parton distribution functions. Thus, the differential cross section at the hadronic level is where f i (f j ) is the parton distribution function of the parton q i (q j ) in the proton (antiproton), the angle θ represents the angle between the three-momentum of the produced t quark and the three-momentum of the proton p (⇔ of the quark q) in the lab system (center of mass system of pp). The sum over (i, j) in Eq. (12) is over all parton pair combinations qq = q iqj for the scattering process q iqj → tt (q i , q j = u, d, s).
In the following, all the unprimed kinematic quantities are in the lab system, and all the "hatted" kinematic quantities are in the center of mass system (CMS) of qq (⇔ CMS of tt).
Taking into account the relations p q = x 1 p p and pq = x 2 pp in the lab system, considering the four-momentum conservation in the scattering qq → tt in the qq CMS, and relating the lab and qq CMS quantities via the corresponding boost relations, the following relation can be obtained between the angle θ and its qq CMS analogθ =θ(θ, x 1 , x 2 ): where β t is the aforementioned quantity involvingŝ = (p q + pq The relevant independent kinematic quantities in the integration (12) are all lab-related: x 1 , x 2 , and θ. On the other hand, Eq. (13) shows that the qq CMS-related angleθ is a function of the aforementioned three independent quantities x 1 , x 2 , and θ. The quantity ∂σ qq→tt /∂ cos θ appearing as integrand in Eq. (12) is obtained directly from Eqs. (11) and (13) by applying the derivatives at fixed x 1 and x 2 ∂σ qq→tt (θ, where the partial derivatives ∂/∂ cos θ are at fixed x 1 and x 2 . The total hadronic cross section σ(pp → tt) and the corresponding forward-backward asymmetry are then obtained by the corresponding integrations of the expression (12) in the lab frame Another physical observable of experimental interest is the invariant mass distribution dσ/dM tt [32] where M 2 tt = (p t + pt) 2 The integrations in Eqs. (16)- (18) are performed across the kinematically allowed regions, i.e., such that β t and cosθ are real.
After having presented formulas for the three physical observables, we can now numerically investigate the unparticle contributions to top-quark pair production. At first, in order to reduce the number of free parameters, we assume that the colored vector unparticle is flavor blind, i.e. g t V = g t A = g q V = g q A = g. Then, the remaining unknown parameters appearing in the physical quantities are: g = λ/Λ d U −1 U , and the scale dimension d U . This means that in such a case we have only two independent parameters λ and d U , both dimensionless quantities, and we can fix the scale Λ U formally to an arbitrary value. We will set it equal to Λ U = 1 TeV. We will see that, unlike the situation in the axigluon model [8,11] in which g q A = −g t A is necessary to get the positive sign in FBA, in unparticle physics the flavor-blind and chirality-independent couplings are enough to fit the data.
Further, it is necessary to consider the measurements of at least two of the aforementioned three observables, namely σ(pp → tt) and A pp FB , Eqs. (16)- (17), in order to restrict the area of the parameters λ and d U . The value of the tt production cross section σ(pp → tt) was measured by the CDF Collaboration [33] σ(pp → tt) exp = 7.50 ± 0.31 (stat) ± 0.34 (syst) ± 0.15 (th) pb = 7.50 ± 0.48 pb .
On the other hand, the SM prediction is σ(pp → tt) SM = 6.73 +0.71 −0.79 pb [34], which includes the contributions from the tree-level, the next-to-leading order in α s , and the next-to-leading in threshold logarithms (LO+NLO+NLL). In the specific case of using the CTEQ6.6 parton distribution functions [35] (which we use), the central value for the SM prediction goes slightly down to σ(pp → tt) SM = 6.61 pb, Ref. SM amplitude is the tree-level amplitude [rescaled accordingly in order to obtain σ(pp → tt) SM = 6.6 pb, see below], which gives A pp FB = 0. Therefore, we will regard the A pp FB as calculated according to Eq. (17) [using Eqs. (15), (12) and (11)] to be responsible for the deviation of the experimental from the SM FBA value where the uncertainties (±0.065, ±0.024, ±0.015) were added in quadrature.
In our calculations we use the CTEQ6.6 parton distribution functions, the value m t = 175 GeV for the t quark (pole) mass, and for the QCD coupling the value α s ≈ α s (m t ) ≈ 0.11.
With such values, we obtain the tree-level σ(SM; tree) ≈ 4.85 pb. We use for the SM amplitude the rescaling factor It shows the region in the d U -λ parameter plane which simultaneously fulfills the experimental constraints (19) and (20). This region lies between the two dotted and simultaneously between the two dashed lines. The central measured values σ ≈ 7.5 pb and 14 are achieved at λ = 2.05 and d U = 1.28. In Fig. 3, we scanned over the free parameter space in finite steps ∆λ = 0.1 and ∆d U = 0.01.
In Fig. 4 we present the average values of dσ/dM tt in eight different M tt -intervals ("bins") as used by the CDF measurement [32]. The presented results are for: (a) the QCD SM case results (including the QCD SM) are above the CDF measurements, with the exception of the first two bins M tt ≤ 450 GeV. The deviations could plausibly be ascribed to two main uncertainties [36]: • (i) The chosen scales of renormalization (µ R ) and factorization (µ F ) for which the usual possible values could be taken between m t /2 and 2m t . Here we adopted µ R = µ F = m t .
• (ii) The M tt -dependent NLO effects which include the NLO parton distribution fuction (PDF). Here for simplicity we just use a M tt -independent scale factor value of K=1.36 (i.e., the factor √ K = √ 1.36 for the tree-level SM amplitude A SM ) to fit the tt SM production cross section with LO calculations.
For a detailed analysis of the various uncertainties see Ref. [36]. Nonetheless, most of our results for dσ/dM tt are at least marginally compatible with the CDF results, within 2σ.
In order to make a more detailed inspection, in with the CDF measured values [37] subtracted by the (LO+NLO+NLL) SM values [38].
This subtraction is needed for comparison with our results, for the same reason as in the We see that the experimental uncertainties are very large, especially for A t,high tron; the latter measured value shows 2σ deviation from the QCD SM value. With a natural assumption of quark flavor-blind and chirality-independent interactions to unparticle, our calculations indicate that the aforementioned unparticle contributions can explain this deviation. We found an area of the (two-)parameter space of the unparticle physics which gives the results compatible with the measurements of the forward-backward asymmetry and of the total cross section for the tt production at the Tevatron. The resulting values of the differential dσ/dM tt cross section and the M tt -restricted forward-backward asymmetries are only marginally compatible with the measured values.