Flavor Symmetric Sectors and Collider Physics

Flavor Symmetric Sectors and Collider Physics

Benjamín Grinstein bgrinstein@ucsd.edu Department of Physics, University of California at San Diego, La Jolla, CA 92093    Alexander L. Kagan kaganalexander@gmail.com Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA    Michael Trott mtrott@perimeterinstitute.ca Perimeter Institute for Theoretical Physics, Waterloo, ON N2J-2W9, Canada    Jure Zupan111On leave of absence from University of Ljubljana, Depart. of Mathematics and Physics, Jadranska 19, 1000 Ljubljana, Slovenia and Josef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.
jure.zupan@cern.ch Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA
July 11, 2019

We discuss the phenomenology of effective field theories with new scalar or vector representations of the Standard Model quark flavor symmetry group, allowing for large flavor breaking involving the third generation. Such field content can have a relatively low mass scale and couplings to quarks, while being naturally consistent with both flavor violating and flavor diagonal constraints. These theories therefore have the potential for early discovery at LHC, and provide a flavor safe “tool box” for addressing anomalies at colliders and low energy experiments. We catalogue the possible flavor symmetric representations, and consider applications to the anomalous Tevatron forward backward asymmetry and mixing measurements, individually or concurrently. Collider signatures and constraints on flavor symmetric models are also studied more generally. In our examination of the forward backward asymmetry we determine model independent acceptance corrections appropriate for comparing against CDF data that can be applied to any model seeking to explain the forward backward asymmetry.

I Introduction

In recent years, the study of New Physics (NP) that lies close to the electroweak (EW) energy scale has been motivated primarily by the hierarchy problem. However, it is possible that the correct solution to this problem or the detailed nature of EW symmetry breaking remain to be proposed. Experimental input, as expected from the LHC, is crucial. Furthermore, hints for new physics (NP) may have already emerged from the Tevatron. In this paper we are motivated by recent experimental anomalies at the Tevatron and the strong discovery potential at LHC to explore collider signatures of new physics (NP) sectors that are flavor symmetric. They will be taken to be invariant under the global flavor symmetry group , or its subgroup (where the quarks of the first two families are in doublets of the corresponding factors). The group is the global symmetry of the Standard Model (SM) in the limit where one can neglect the Yukawa interactions


where and are the up and down quark Yukawa matrices, respectively.

The NP sectors will contain scalar or vector fields that have masses and couplings to quarks. At the same time they will be consistent with flavor changing neutral current (FCNC) constraints precisely because of their flavor structure, as long as the breaking of the flavor symmetries is sufficiently small. In the SM the top and bottom Yukawa couplings break the flavor group to its subgroup. We take this breaking into account in our analysis of NP sectors that are initially symmetric. The existence of the symmetry at low energies protects these theories against dangerously large FCNC’s, e.g., in neutral meson mixing. This protection satisfies the naturalness criteria of Glashow and Weinberg Glashow:1976nt () and is not the result of simply tuning parameters. Note that NP models that have an approximate symmetry are sometimes referred to as models of next to minimal flavor violation Agashe:2005hk (). The breaking of in the SM is due to the other quark masses and the CKM mixing angles, and is thus small. The precise mechanism by which is broken in the NP sector will not be important when we explore flavor diagonal collider signatures. However, the nature of breaking will be relevant to our discussion of low energy FCNC’s, see below.

Scalar and vector fields with dimension four invariant direct couplings to quarks are limited in their allowed charge assignments by flavor and the SM gauge symmetry. There are only 14 different nontrivial flavor representations allowed in each case. In the case of symmetric models the possible NP fields are conveniently classified in terms of these representations, with the understanding that they need not come in complete multiplets. A systematic exploration of new flavor symmetric sectors is therefore feasible, either in general, or with the aim of explaining a particular anomaly.222Color symmetric fermion content that mixes with the SM fermion fields is not as constrained in its allowed representations. Initial studies of vector-like fermions have also been undertaken in Refs. Grossman:2007bd (); Arnold:2010vs (). Of the models studied only two were natural in the Glashow-Weinberg sense Arnold:2010vs ().

New flavor symmetric sectors that are perturbatively coupled to quarks are particularly interesting to consider as candidate explanations for Tevatron anomalies. In the first part of this paper, we focus our attention on two anomalies: (i) the CDF measurement of the forward backward asymmetry, , for Aaltonen:2011kc () is away from the NLO SM prediction. (A recent DØ  analysis Abazov:2011rq () does not observe a significant dependence in the “folded” detector level asymmetries, but it appears to be consistent with the CDF detector level measurements within errors.) The inclusive forward backward asymmetry, averaged over the CDF semileptonic Aaltonen:2011kc () and hadronic CDF-dilepton () decay samples and the recent DØ measurement Abazov:2011rq (), is from the NLO SM prediction; (ii) the like sign dimuon asymmetry measured by DØ is away from the SM expectation Abazov:2010hv (); Williams:2011nc (). Each of these anomalies, if confirmed, points to a relatively low scale of NP with a significant coupling to quarks. We identify flavor symmetric models that have the potential to explain them either individually or simultaneously, and study related constraints. In the case of symmetric models, under the assumption that the NP only couples to quarks, some hierarchies among these couplings would be required in order to consistently explain the anomaly, e.g., due to the absence of dijet or resonances at the Tevatron and LHC. Thus, breaking of the symmetry to would be necessary. Alternatively, one could consider symmetric models where the more constrained quark couplings would simply be absent.

It is possible that the above anomalies could be due to statistical fluctuations or underestimates of theoretical or experimental errors. Even if this turns out to be the case, the models we explore in this paper are interesting in their own right, as they have strong discovery prospects at LHC. Again, this is because their flavor symmetric structures allow for sub-TeV NP mass scales. In the second part of this paper, we address the phenomenology of flavor symmetric sectors more generally. The global flavor symmetries we consider could be accidental, or they could be a remnant of the underlying mechanism generating the SM flavor structure (such as non-Abelian horizontal symmetries). We do not concern ourselves with the UV origin of these symmetries, but instead focus on the collider and low energy phenomenology of the new sectors. This approach is inspired by effective field theories (EFT), where one generally constructs all possible interactions consistent with the symmetries of interest. The analysis of flavor diagonal collider constraints and signatures can then be kept quite general, i.e., independent of the way is broken, as already mentioned. For simplicity, in attempts to explain a measured deviation from the SM we will only consider the phenomenology of single multiplets (or the corresponding multiplets), effectively assuming that there is a significant mass gap with other possible representations. Moreover, we will only consider their couplings to quarks. Note that more generally, these fields could couple to additional states transforming under or , possibly providing them with additional decay channels.

The determination of low energy flavor physics constraints on flavor symmetric models generally requires the breaking of to be specified. When determining these constraints we assume the Minimal Flavor Violation (MFV) hypothesis Chivukula:1987py (); D'Ambrosio:2002ex (), i.e., that all breaking of is due to the SM Yukawas. This enforces maximal consistency with FCNC constraints through a symmetry principle, and allows us to explore how low the NP mass scale can be. In all the models we consider, the new states can have EW scale masses. In MFV models that lead to class-2 operators (those that involve right handed fields) in the language of Kagan:2009bn (), the breaking of can actually be orders of magnitude larger then assumed in MFV, while still obeying the FCNC bounds.

The paper is organized as follows. In Section II we list all vector and scalar representations of the form we have motivated, and write down in detail the vector field Lagrangians for two examples. In Section III we systematically discuss the potential of models of this form to explain the anomaly, the DØ dimuon anomaly, and related phenomenology. In Section IV we explore existing bounds on these models from LEP, electroweak precision data (EWPD), FCNCs and dijet studies at the LHC and Tevatron. In Section V we discuss additional LHC phenomenology. Finally, in Section VI we give our conclusions. Many details have been relegated to the Appendices. In Appendix A we list the details of flavor symmetric vector Lagrangians, in Appendix B we gives the details of scattering calculations and phenomenology, and in Appendix C we give a detailed discussion of constraints from meson mixing amplitudes.

Ii Symmetric Representations

Case Couples to
1,8 1 0 (1,1,1)
1,8 1 0 (1,1,1)
1,8 1 0 (1,1,1)
1,8 3 0 (1,1,1)
1,8 1 0 (1,8,1)
1,8 1 0 (8,1,1)
1,8 1 -1 (,3,1)
1,8 1 0 (1,1,8)
1,8 3 0 (1,1,8)
,6 2 -1/6 (1,3,3)
,6 2 5/6 (3,1,3)
Table 1: The flavor and gauge representations for vector fields that can couple directly to quarks through symmetric dimension four interactions without the insertion of a Yukawa matrix. denotes the right handed conjugate representation of the left handed SM doublet.

We are interested in scalars and vectors that couple directly to quarks through dimension four interactions. The scalar fields of this form are renormalizable models, while the vector fields are nonrenormalizable. In this section we list all the possible representations of and the SM gauge group that such fields can have when is unbroken. The vector field representations are listed in Table 1 and the scalar field representations are listed in Table 2 (the latter have been studied and classified in Arnold:2009ay (); Manohar:2006ga ()). This completes the program initiated in Arnold:2009ay (). The complete set of symmetric representations which can couple directly to quarks through dimension four interactions appear in these tables as submultiplets of the representations. For example, in model , the corresponding symmetric vector representations would consist of a triplet, a complex doublet and a singlet of , which are color singlets (octets), or a subset of these.

Case Couples to
1 2 1/2 (3,1,)
8 2 1/2 (3,1,)
1 2 -1/2 (1,3,)
8 2 -1/2 (1,3,)
3 1 -4/3 (3,1,1)
1 -4/3 (,1,1)
3 1 2/3 (1,3,1)
1 2/3 (1,,1)
3 1 -1/3 (,,1)
1 -1/3 (,,1)
3 1 -1/3 (1,1,)
1 -1/3 (1,1,3)
3 3 -1/3 (1,1,3)
3 -1/3 (1,1,)
2 1/2 (1,1,1) ,   
Table 2: Different scalar representations that are not singlets under the flavor group that are symmetric Arnold:2009ay () (the upper rows). The two flavor singlet representations are in the last row and were discussed in Manohar:2006ga ().

Several remarks are in order before we construct the Lagrangians.

  • , and carry the same quantum numbers and are thus a sub-classification of interactions of a single field. For instance, in the case of a color singlet vector with the same couplings to this is just the baryonic . We found it useful to split the interactions into three subgroups. At colliders there is no interference among these interactions up to effects suppressed by light quark masses (but if there are relations between their couplings this can have important consequences for the predicted cross section; for example, for a purely axial gluon the NP interference with the SM amplitude does not contribute to the top pair production cross section Cao:2010zb (); Blum:2011up (); Tavares:2011zg ()). In the treatment of FCNCs the interference effects are trivial to include in the analysis.

  • Many of the scalar and vector fields do not lead to proton decay at any order in perturbation theory due to the SM gauge symmetry and . The vectors X–XI and scalars carry baryon number and may lead to proton decay if they also couple to lepto-quark bilinears, e.g., and for fields X and XI, respectively. This type of coupling is not possible for scalars or vectors in the color representation and can be forbidden for the color representation fields by extending the flavor group to the lepton sector of the SM Arnold:2009ay () .

  • We assume that the new quanta have weak scale masses and that the cut-off of the theory is well above the weak scale so that we only need to focus on dimension four interactions for most of our discussion. Other dimension four couplings such as , with the hypercharge field strength and the vector fields, are not directly relevant to the phenomenology of interest in this paper. We leave the exploration of these interactions to a future publication.

  • The kinetic terms (with flavor breaking insertions) can always be made universal through field redefinitions. Below, we only write down the interaction terms.

  • Tree-level exchanges of fields in a single representation of cannot explain both of the Tevatron anomalies simultaneously for any of the models considered. Models and do, however, lead to enhanced , while not modifying the differential spectrum. At the same time they give new CP violating contributions to and mixing of the right order of magnitude to yield the observed like-sign dimuon asymmetry.

The interaction Lagrangians for the color triplet and sextet scalar fields are given in Arnold:2009ay (). For vector fields the symmetric interactions are given by , where represents a product of generators of color, flavor, and weak SU(2), or some subset thereof, while are the , or family triplets, as appropriate 333In symmetric models the Lagrangians are trivially obtained from the corresponding symmetric Lagrangians, allowing for the possibility that only particular submultiplets of the symmetric representations are present.. To write down the breaking interactions it is useful to (initially) adopt some of the formalism of MFV and promote to spurions that formally transform as bi-fundamentals of


Here are elements of , respectively. Assuming full MFV breaking of , all interactions are then formally invariant under even for nonzero Yukawa couplings. We will mostly work to the first nontrivial order in top Yukawa insertions (the resummation to all orders can be done using a nonlinear representation of , see Feldmann:2008ja (); Kagan:2009bn (); Feldmann:2009dc (); Barbieri:2011ci ()). The explicit forms of the interactions for all the vector models are given in Appendix A. Here we show two examples, models and .

Fields are flavor octets, color singlets (octets). The individual field components are , where the color label and flavor label both run over . To compress the expressions we introduce


with flavor (color) Gell-Mann matrices () normalized to . The renormalizable interactions between quarks and fields are


There are also terms that break ,


We kept only the breaking due to insertions. These are diagonal in the up-quark basis and change the coupling of third generation quarks to the vector fields (explicitly written out in Appendix A, Eqs. (33), (34)). Note that can be complex and possibly a source of beyond the SM CP violation, of interest when considering mixing, while is real. Insertions of are also possible to break the symmetry further and are almost diagonal in the up quark basis, while the off-diagonal elements lead to FCNC’s. We postpone the discussion of these until Section IV.4.

The breaking also splits the vector mass spectrum. The flavor invariant mass terms are


where the color and indices are suppressed, and there is no summation over (the Kronecker delta insures the proper normalization for the color octet fields). Note that is defined when rotating to the mass eigenstate basis; see Appendix A. Adding the breaking terms and , the mass spectrum of the vector states is (suppressing the labels)


Note that are all real. The vectors and are degenerate since is only broken by light quark Yukawas, not by .

Fields are weak doublets in the bi-fundamental representation of the flavor group. The color anti-triplets have field components , and color sextets the field components , with the weak index, the color indices, while and are the indices of the and representations respectively. The tree level quark coupling Lagrangian terms are (suppressing all the indices apart from color, see also Eqs. (58), (59))


Note that the fields transform as , where and . The mass terms are


where we have suppressed all the traced over flavor, color and weak indices (except in the last two terms where we show explicitly the weak contractions). We use , and similarly for the sextet. Note that the last two terms break the mass degeneracy between charge and charge components of weak doublets. The flavor is broken through Yukawa insertions


where we do not write down the terms with more than two Yukawa insertions or the similar terms with Higgs fields. The resulting symmetric spectrum for charge vectors is


The interactions of mass eigenstates with mass eigenstate quarks (denoted with primes) are given by (showing explicitly only color contractions, see also Eqs. (60), (60))


where and are the mass eigenstate vector fields of the doublet. The residual flavor universality of these interactions can be broken by insertions of the spurions and . In MFV this is the only form of further flavor breaking. The rest of the Lagrangian constructions are collected in the Appendix A.

Iii Phenomenology of Tevatron anomalies

We now discuss two recent experimental anomalies observed at the Tevatron: the large forward-backward asymmetry , and the like-sign dimuon anomaly in decays. In this section, we systematically address the following questions:

  • Is it possible to explain either of the two anomalies assuming symmetric models? By which charge and flavor assignments?

  • Are closely related experimental constraints simultaneously obeyed?

  • Is it possible to explain both anomalies using just a single symmetric field?

A common feature of models put forward to explain the anomaly Gresham:2011fx (); Nelson:2011us (); Jung:2011id (); Ferrario:2009bz (); Arhrib:2009hu (); Arnold:2009ay (); AguilarSaavedra:2011vw (); Kamenik:2011wt (); Cheung:1995nt (); Antipin:2008zx (); Gupta:2009wu (); Hioki:2009hm (); Choudhury:2009wd (); Hioki:2010zu (); HIOKI:2011xx (); Blum:2011up (); Jung:2009pi (); Zhang:2010dr (); Delaunay:2011gv (); Jung:2011zv (); Jung:2009jz (); Cheung:2009ch (); Shu:2009xf (); Dorsner:2009mq (); Cao:2009uz (); Barger:2010mw (); Cao:2010zb (); Xiao:2010hm (); Cheung:2011qa (); Cao:2011ew (); Shelton:2011hq (); Berger:2011ua (); Barger:2011ih (); Bhattacherjee:2011nr (); Patel:2011eh (); Barreto:2011au (); Craig:2011an (); Buckley:2011vc (); Shu:2011au (); Jung:2011ua (); Fox:2011qd (); Cui:2011xy (); Duraisamy:2011pt (); AguilarSaavedra:2011ug (); Dorsner:2010cu (); Blum:2011fa (); Gresham:2011dg (); Grinstein:2011yv (); Delaunay:2011vv (); Babu:2011yw (); Ligeti:2011vt (); Tavares:2011zg (); Isidori:2011dp (); Frampton:2009rk (); Chivukula:2010fk (); Bai:2011ed (); Xiao:2010ph (); Ferrario:2009ee (); Martynov:2010ed (); Bauer:2010iq (); Chen:2010hm (); Burdman:2010gr (); Degrande:2010kt (); Choudhury:2010cd (); Cao:2010nw (); Foot:2011xu (); Haisch:2011up () is that they have NP couplings to the up quark. They fall roughly into two classes, those with -channel exchange above a TeV444For a recent exception with a sub-TeV axigluon, see Tavares:2011zg (). Frampton:2009rk (); Ferrario:2009bz (); Ferrario:2009ee (), in which case the axial vector NP couplings to the top quarks and up quarks must be of opposite sign Ferrario:2009bz (); Frampton:2009rk (), and those with sub-TeV -channel exchange Jung:2009jz (); Cheung:2009ch (); Frampton:2009rk (); Shu:2009xf (); Arhrib:2009hu (); Dorsner:2009mq (); Cao:2009uz (), in which case large inter-generational couplings are required (for additional possibilities, see Kamenik:2011wt ()). The couplings in either class could arise from large flavor violation in the underlying theory, which may lead to violations of FCNC constraints in mixing, , or , unless the couplings are carefully aligned (see, e.g., Shelton:2011hq (); Jung:2011zv (); Shu:2011au ()). Moreover, the -channel models can lead to excessive (flavor violating) single top or same sign top pair production at the Tevatron and LHC Jung:2009jz ().

However, flavor violation is not necessary for large Grinstein:2011yv (). Note that in production no net top quark flavor charge is generated. Furthermore, models with an unbroken subgroup do not generate FCNCs in processes with light quarks. The exact size of FCNCs then depends on the size of breaking. If this breaking is MFV-like the FCNCs are generically suppressed below present bounds. The flavor symmetries also eliminate single top and same sign top production.

Many new models have also been put forward to explain the dimuon anomaly Dobrescu:2010rh (); Buras:2010mh (); Jung:2010ik (); Chen:2010aq (); Blum:2010mj (); Buras:2010pz (); Buras:2010zm (); Trott:2010iz (). Together with possible indications for deviations from the SM in decays and decays, it may point to a NP phase in mixing. Intriguingly, MFV suffices to explain the dimuon anomaly Ligeti:2010ia (). After discussing flavor symmetric fields and the anomaly, we will examine whether these fields can also give large enough contributions to mixing, under the assumption of MFV breaking of .

iii.1 General analysis of the forward backward asymmetry

We entertain the possibility that is enhanced above SM levels via tree level exchanges of flavor symmetric scalars or vectors. The experimental evidence for such enhancement is as follows. Using of data CDF measured an inclusive asymmetry in the rest frame (fixing Aaltonen:2011kc (). In a channel with both and decaying semileptonically an even larger asymmetry was found, CDF-dilepton (). Similarly, a recent DØ  analysis finds using of data  Abazov:2011rq (). Combining in quadrature the statistical and systematic errors of the three measurements gives . This is to be compared to the SM prediction from an approximate NNLO QCD calculation Ahrens:2011uf () with and using the MSTW2008 set of PDFs Martin:2009iq (). Inclusion of electroweak corrections leads to an enhancement of the asymmetry, with recently obtained in Ref. Hollik:2011ps (). In the frame, a recent approximate NNLO calculation Kidonakis:2011zn () gives with , to be compared with the CDF value of  Aaltonen:2011kc (). The approximate NNLO SM predictions use the known NLO results Antunano:2007da (); Bowen:2005ap (); Kuhn:1998kw () and build on recent progress in NNLO calculations Moch:2008ai (); Czakon:2009zw (); Beneke:2009ye (); Kidonakis:2008mu (); Cacciari:2008zb (); Kidonakis:2010dk (). DØ  also reports a leptonic asymmetry to be compared to the MC@NLO prediction of  Abazov:2011rq ().

CDF reported evidence that the anomalously large asymmetry rises with the invariant mass of the system, with , while  Aaltonen:2011kc (). A similar rise of the asymmetry with respect to the absolute top vs. anti-top rapidity difference was also reported by CDF with and  Aaltonen:2011kc (). The recent DØ  analysis Abazov:2011rq () does not observe a significant rise of the “folded” detector level asymmetry with respect to and . However, until these results are unfolded they can not be directly compared to the CDF measurements, although at the detector level they appear to be consistent within errors. We collect the above results in Table 3.

Observable Measurement SM predict.
GeV) Aaltonen:2011kc ()
GeV) Aaltonen:2011kc () Ahrens:2011uf ()
) Aaltonen:2011kc () Ahrens:2011uf ()
) Aaltonen:2011kc () Ahrens:2011uf ()
pb Aaltonen:2009iz ()
Table 3: Measurements and predictions for observables in production at the Tevatron. We quote the approximate NNLO QCD prediction of from Ahrens:2011uf () using MSTW2008 PDFs Martin:2009iq (). The two other choices for PDFs give results in agreement with these Ahrens:2011uf (). Among the cross section predictions obtained in Ahrens:2011mw () we quote the one.

Any NP enhancement of must not spoil the agreement between the measured production cross section, , and the SM predictions. At NLO with NNLL summation of threshold logarithms, the SM prediction is pb Ahrens:2011mw () (using MSTW2008 pdf sets and choice of kinematic variables and resummations – the other choices give consistent results but with larger error bars). This is somewhat smaller than the approximate NNLO result (for ), pb Kidonakis:2011jg () (see also Cacciari:2008zb (); Kidonakis:2008mu (); Moch:2008ai ()). Both of these results agree well, within errors, with the measured CDF result based on of data Aaltonen:2009iz () . Thus the NP contribution to the cross section, , is tightly constrained.

Good agreement between experiment and SM predictions is also seen in the differential cross section . This has important implications for the viability of different NP models. For instance, comparing the measured and predicted cross sections together with the measured and predicted , for GeV, one finds that the NP contributions need to reduce the backward -scattering cross section (a statement valid at ). This can only happen if NP interferes with the SM Grinstein:2011yv (). NP in the -channel which interferes with the single gluon exchange amplitude must therefore be due to color octet fields. In general -channel resonances lead to significant effects in . However, this may be avoided for a purely axial gluon that is broad Tavares:2011zg (), in particular regions of parameter space. There are no such clear requirements on the charge assignments of possible -channel NP contributions. However, a characteristic high mass tail in the spectrum could lead to tension with the Tevatron and future LHC cross section measurements at large .

We collect expressions to be used in our analysis below. The total cross section , forward-backward asymmetry , and the cross sections for forward and backward scattering are defined as


where is the angle between incoming proton and outgoing top quark. We use NLO SM predictions for and , and LO predictions for the NP corrections (including interference with the SM). To obtain we define a partonic level asymmetry,


which is to be compared against the binned unfolded partonic level results of Aaltonen:2011kc (). We use the NLO + NNLL SM predictions for the forward, backward and total cross sections Ahrens:2010zv (), the spectrum Ahrens:2010zv () and Ahrens:2011uf (). For concreteness, for GeV we take the central values pb and , while for GeV we take pb and (MSTW08 pdfs); for the inclusive asymmetry (in the rest frame) we take .

iii.1.1 Acceptance effects

As pointed out in Gresham:2011pa (); Jung:2011zv (), care is needed when comparing NP predictions to the experimental parton level and differential spectrum deduced by CDF Aaltonen:2009iz (); Aaltonen:2011kc (), since the deconvolution was done assuming the SM. The acceptance corrections are especially important if NP enhances top production in the very forward region This is because the SM event distribution is more central. We take into account the CDF experimental cuts using correction factors . For the th bin in one needs to multiply the calculated partonic cross section by in order to compare with the CDF measured partonic cross section


There is no summation over in this equation. Since CDF is using SM acceptances and no bins in in the deconvolution of the measurement in Aaltonen:2011kc (), the are given by the ratio of acceptances for the NP model and the SM


where are calculated by splitting each -th bin into bins in


and the sum is over the bins in . Here is the acceptance for each bin, and is the corresponding cross section integrated over the bin. The above expressions are approximate in so far as the bins have finite sizes, and the spill-over of events between different bins is not taken into account. The acceptances are calculated by simulating the partonic cross section using MadGraph4.4.30 Alwall:2007st (), decaying the top quarks in Pythia6.4 Sjostrand:2006za (), which also simulates the LO showering and hadronization, and using PGS for detector simulation. The events were read into Mathematica, where the cuts from Aaltonen:2008hc (); Aaltonen:2011kc () were implemented. The resulting values for the acceptances are collected in Table 4.

Table 4: Acceptances (in %) to be used for emulating the CDF deconvolution to parton level measurements (cf. eqs. (15) - (17)). Note that the do not depend on the NP model and are symmetric in .
Figure 1: The correction factors for -channel dominated vector models (left) and (right), where ()=()[dashed red]; ()[solid blue].

In Fig. 1 the correction factors are shown as a function of the bin for two benchmarks points (corresponding to illustrative couplings and mass valuers) in models which exhibit substantial departures from the SM acceptances. Results for and for the two model benchmark points in Fig. 1 were shown previously without acceptance corrections Grinstein:2011yv (). The differential distributions with acceptance corrections are compared to those in  Grinstein:2011yv () in Fig. 2. We see that the corrections bring the predicted spectrum for the light vector example into good agreement with experiment. For the remaining representations discussed below, the correction factors are not as important. For models , and the are below , and for the others they are below , for all bins and all benchmarks points considered.

The pattern of acceptance corrections can be understood from the angular dependence of the NP contribution to the differential cross section in each model. For instance, the -channel exchange of a vector with mass leads to a Rutherford scattering peak in the forward direction for . Specifically, the expressions for the NP cross sections contain characteristic -channel factors in the denominators, whose angular dependence is reinforced by factors in the numerators, where is the top quark scattering angle in the center of mass frame. Thus, models with light vector masses favor forward top-quark production at large , yielding that are substantially less than 1 in the high bins. In models (-channel) and , (-channel), the angular dependence introduced by the characteristic factors in the denominators is offset by factors in the numerators, which leads to central NP top-quark production, as in the SM. The result is that the in these models are actually slightly larger than 1.

To apply the acceptance corrections to we follow CDF, where was obtained using four bins in and : above or below  GeV and positive or negative. The correction factor for each of the four bins is given as in Eqs. (15)-(17), except that the sum in (17) now runs over all with either or , and over the appropriate values of with either  GeV or  GeV. We find that the corrections are small for all of the models we consider. For instance, for the light vector color octet example in Fig. 1 the shift is from an uncorrected to a corrected , where the first and second numbers are the low and high mass bins in . The small shifts in are due to the coarse binning in Gresham:2011pa (). In particular, the high mass bin is dominated by events with near 450 GeV, which are more central, as in the SM.

Figure 2: The effect of acceptance corrections on the vector model discussed in Grinstein:2011yv (). Before acceptance corrections (left), after corrections applied (right). For the parameters (): solid red (); dashed blue ().

iii.1.2 The phenomenology of symmetric scalar fields

The flavor symmetric models introduced in Section II and collected in Tables 1, 2 can couple to light and heavy quark bilinears with unsuppressed couplings. They are thus interesting candidates to explain the anomaly, as noted in Arnold:2009ay (); Grinstein:2011yv (); Ligeti:2011vt (). In the case of scalars, singlet color triplets or color sextets and doublet color singlets have previously been identified as being promising for explaining the anomaly Shu:2009xf (); Grinstein:2011yv (); Ligeti:2011vt (); Nelson:2011us (); Babu:2011yw (); AguilarSaavedra:2011hz (); Blum:2011fa (). Here, we will focus on some of the flavored versions listed in Table 2. Our results overlap with past studies, but we also include color triplet and sextet scalars that couple to initial state down quarks, that have not been studied as extensively. We find that these models may also be viable, although they generically require a coupling that is a factor of larger than if the up quark is in the initial state. This rule of thumb also holds for the vector models that we will study in next subsection.

Figure 3: Upper two rows: predictions for the inclusive cross sections and inclusive forward-backward asymmetry as a function of scalar mass for models and couplings (solid line), (dotted), (dot-dashed), (dashed) compared to and experimental (shaded) bands. Predictions for with below and above 450 GeV and for are compared to the experimental data from Aaltonen:2009iz (); Aaltonen:2011kc () in the last row for benchmark points labeled with a in the inclusive predictions ( GeV for