Top condensation as a motivated explanation of the top forward-backward asymmetry

# Top condensation as a motivated explanation of the top forward-backward asymmetry

Yanou Cui, Zhenyu Han, and Matthew D. Schwartz Center for the Fundamental Laws of Nature
Harvard University,
Cambridge, MA 02138, USA
###### Abstract

Models of top condensation can provide both a compelling solution to the hierarchy problem as well as an explanation of why the top-quark mass is large. The spectrum of such models, in particular topcolor-assisted technicolor, includes top-pions, top-rhos and the top-Higgs, all of which can easily have large top-charm or top-up couplings. Large top-up couplings in particular would lead to a top forward-backward asymmetry through -channel exchange, easily consistent with the Tevatron measurements. Intriguingly, there is destructive interference between the top-mesons and the standard model which conspire to make the overall top pair production rate consistent with the standard model. The rate for same-sign top production is also small due to destructive interference between the neutral top-pion and the top-Higgs. Flavor physics is under control because new physics is mostly confined to the top quark. In this way, top condensation can explain the asymmetry and be consistent with all experimental bounds. There are many additional signatures of topcolor with large mixing, such as top(s)+jet(s) events, in which a top and a jet reconstruct a resonance mass, which make these models easily testable at the LHC.

## 1 Introduction

The CDF collaboration recently reported the measurement of a large forward-backward asymmetry () in top pair production with 5 fb of data in both semi-leptonic cdf-1 and dileptonic cdf-2 channels. The observed asymmetry, for GeV deviates from the standard model by more than 3. The new CDF results are consistent with earlier observation of large top asymmetries by both CDF and D0 based on smaller data sets d0-2007 ; cdf-08 ; cdf-09 , and moreover the significance of the discrepancy has grown over time. Constraints from other measurements, such as the total cross section and the dijet bound, make model building curiously difficult.

There have been basically two classes of models proposed to explain the top quark anomaly, -channel and -channel models, both of which must have non-universal flavor structure. The first class of models provide a tree-level contribution to through the exchange of a new particle in the -channel. This contribution cannot be too large without affecting the overall rate, so the new particle should be heavy ( 1 TeV). Even then, one expects the cross section to grow as a function of the invariant mass, which is not seen. The contribution to can be enhanced by making the new particle colored and parity-violating, which leads to axigluon models Frampton:2009rk . However, to get the right contribution, the axigluon has to have unusual opposite sign couplings to and . Moreover, dijet bound constraints force these models to couple more weakly to ups than tops Bai:2011ed .

The -channel models can explain if some new particle has couplings to up and top Jung:2009jz . -channel models are not constrained by the dijet data, but there are constraints from the total rate, as well as same-sign tops at the Tevatron, and single top production. In these models, the large flavor-violating coupling appears initially to be ad-hoc and unnatural.

In order to make -channel models more appealing theoretically, there have been attempts to embed them in flavor-conserving models by introducing new horizontal symmetries Grinstein:2011yv ; Ligeti:2011vt ; Jung:2011zv ; Nelson:2011us ; Babu:2011yw . In the existing models, the top asymmetry anomaly is explained by new physics uncorrelated to the profound puzzle of electroweak symmetry breaking (EWSB) which necessitates new physics interpretation on its own. The models would be much more compelling if the anomaly corresponded to a previously existing mechanism of EWSB. In this paper, we demonstrate that there is indeed such a possibility in the framework of top condensation Miransky:1988xi ; Miransky:1989ds ; Bardeen:1989ds ; Hill:1991at . Top condensation models can provide particles which naturally have large top-up couplings, and can be exchanged in the -channel to produce the observed .

In top condensation, electroweak symmetry is dynamically broken, as in technicolor (and in QCD), through the formation of bound states. The difference from technicolor is that the condensate is made up of the top quark itself, rather than new techniquarks. A realistic and viable framework for top condensation is topcolor assisted technicolor (TC2) Hill:1994hp . In this framework, there are two sources of EWSB, or three if QCD is included. The majority of EWSB and the majority of the the and masses are due to technicolor, with a small contribution from topcolor; the majority of the top quark mass comes from the topcolor condensate. The light quarks can get mass from the extended technicolor or a scalar Higgs. In this way, TC2 solves one of the difficulties of technicolor, namely how to explain large top Yukawa without flavor problem. We will review the TC2 setup in Section 2.

One generic consequence of having two sources of electroweak symmetry breaking is that there are two sets of Goldstone bosons. One set is eaten by the and to give them their masses, the other set are an triplet of top-pions. These top-pions are similar to the charged Higgses and pseudoscalar in two-Higgs doublet models, but couple only to the third generation, at leading order. Topcolor also generically has a top-Higgs, which is the scalar bound state. In addition, there should be angular and radial excitations. The lightest of these, in analogy to QCD, is expected to be a vector, the top-rho. In the unbroken phase, all of these particles couple predominantly to , where here is the top-color eigenstate.

After electroweak symmetry, the right-handed top can have large mixing with the right-handed charm and up-quarks. That only the right-handed top quark has large mixing is a consequence of having only a condensate, not , which is in turn a consequence of top-hypercharge being attractive in the channel but repulsive in . While there are strong constraints on left-handed mixing, there is substantial freedom in mixing without violating flavor bounds, an appealing natural feature of this model. If there is substantial mixing between and , there will be large flavor changing couplings in the mass basis. Thus the top-pions, top-Higgs, and top-rho can all contribute to production, with exactly the desired features of the -channel models to explain .

One critical feature for the viability of this model is the large interference effects among the new physics particles, and between the new physics and the standard model. The interference between the top-mesons and the standard model in production is destructive, making it possible to be consistent with the measured total cross section and at the same time maintain a large enough . Large couplings generically predict the production of abundant same-sign top quarks. This is also true in our model. However, due to destructive interference between the top-Higgs and the top-pion, as well as that between the top-Higgs and the top-rho, there exists large parameter space within current experimental bounds.

The organization of this paper is as follows. We begin in Section 2 with a review of the TC2 setup. This review is based on the Hill and Simmons technicolor review Hill+Simmons , with a slightly different emphasis and a few more relevant details. The relevant low energy theory after mixing is expanded in Section 2.3. In Section 3, we isolate the important relevant couplings in the low energy theory and discuss their phenomenological implications. In particular, we discuss in Section 3.1, the rate in Section 3.2 and the same-sign top rate in Section 3.3. The bounds are combined in Section 3.4. Section 4 discusses additional constraints and signatures. Section 5 presents our conclusions.

## 2 Topcolor assisted technicolor

In top condensation, the condensate which breaks electroweak symmetry is made up of the top quark itself, rather than new techniquarks as in technicolor. For this to work, the top quark must not be confined in the condensate. Such behavior is different from QCD, but not unreasonable. Indeed, exactly such a situation happens in a non-relativistic situation with the formation of Cooper pairs and breaking of in the BCS theory of superconductivity. The behavior is consistent with calculations in the Nambu-Jona-Lasinio (NJL) model, although these calculations are very approximate. Realistic natural models of topcolor can only provide around GeV of the EWSB vev  GeV, so they must be supplemented by another EWSB sector, leading to topcolor-assisted technicolor (TC2). In Section 2.1 we give an overview of top-condensation. Section 2.2 specifies to TC2. The low energy effective theory of TC2 including the relevant mixing is discussed in Section 2.3.

### 2.1 Overview

Top condensation is based on the observation that strong interactions among fermions, such as with a 4-Fermi operator with large coefficient, can lead to bound states. These bound states can pick up a negative mass squared through RG evolution leading to spontaneous symmetry breaking. Top condensation observes that the top quark is already expected to have strong interactions, since its Yukawa coupling, , so it is natural to have strong dynamics associated with the third generation alone. In the NJL model, where one includes only a single dimension 6 operator and only Fermion loops, the qualitative features of top-condensation can be shown to be reasonable. In this way the proximity of the top mass to the electroweak vev is required, and explained, in contrast to in technicolor theories, where the large top mass is extremely challenging to explain.

The typical picture in top condensation is that the top first forms a bound state, at a scale . In topcolor models, the scale is associated with a new force, topcolor, which acts on the third generation quarks. If topcolor is strong, and itself Higgsed so that the topcolor gauge bosons are massive, then the leading operator generated by integrating out the topcolor gauge bosons is

 −g2¯tt1p2−M2¯tt→g2M2¯tt¯tt∼g¯ttHt−12m2HH2t, (1)

where the top-Higgs, has been integrated-in in the last step and at the scale . At this point, is just an auxiliary field, with no kinetic term. It picks up a kinetic term from renormalization-group evolution and becomes dynamical below the scale . The vanishing of the kinetic term at is called a compositeness boundary condition.

At tree level, the mass-squared of is positive, but it also gets radiative corrections. By dimensional analysis,

 m2H=(1−γ)M2g2 (2)

where is an anomalous dimension (in the NJL model, ). For sufficiently large , the top-Higgs mass squared can be negative, signalling spontaneous symmetry breaking. In order for the top-Higgs to get an expectation value lower than , the anomalous dimension should be close to its critical value. That some tuning is required is not surprising, since the Higgs mass is getting quadratic corrections. Thus it is expected  TeV. If is a little larger, there is a little hierarchy problem. For much larger , the full hierarchy problem is reintroduced.

Since the top-Higgs vev and the top mass are generated from the same dynamics, they are related. Explicitly,

 v2=γm2tlnM2g2m2t+⋯ (3)

where represent the contribution from additional operators generated at the scale where topcolor is integrated out (typically these are small corrections). Since TeV by naturalness, this log is small so we expect . Typically GeV. Thus, topcolor as an explanation of electroweak symmetry breaking is incomplete.

Realistic models of top condensation have two sources of electroweak symmetry breaking: the first, topcolor, generates a EWSB vev around 60-100 GeV. In topcolor-assisted technicolor, this is supplemented by another source of EWSB providing the remaining contribution to , such as technicolor. Technicolor itself is a beautiful explanation of the and masses, but does not explain fermion masses. Extended technicolor (ETC) can explain the fermion masses through a new force, but has flavor problems. These flavor problems are ameliorated by assuming non-QCD like dynamics (walking technicolor) for the ETC sector in which operators involving techniquarks run faster than operators involving standard model quarks. However, even these models have difficulty explaining the large top quark mass. In this way, topcolor “assists” technicolor by handling the top mass. The main ingredient of top-color assisted technicolor relevant to the top forward-backward asymmetry is simply that there are two EWSB sectors, and one of them couples only to the top quark at leading order. Since the details of the technicolor side are irrelevant, we will abbreviate that sector with a simple scalar Higgs doublet . This approximation is called Bosonic topcolor Samuel:1990dq .

### 2.2 The model

In this section, we briefly review the relevant ingredients in a realistic topcolor-assisted technicolor model. As explained in Section 2.1, these models have two sources of electroweak symmetry breaking: one from the topcolor sector, generating a vev GeV, and one from another sector. For simplicity, we will take this other sector to be a single Higgs doublet model. This can be replaced with an extended technicolor sector with little consequence for the top asymmetry and the model bounds. Our presentation will closely follow the Hill-Simmons review Hill+Simmons , with a few added details and simplifications.

In minimal topcolor, the third generation couples to one copy of gauge group and the first two generations to another copy of gauge group. The standard model is the diagonal combination of these two groups. The group coupling to the 3rd generation (topcolor) must be much stronger than the other , to generate the top condensate. The two ’s are necessary so that only a forms and not . In this setup, there is one which couples to everything. Variations are possible, with 3 copies of , one for each generation (which mimic extra-dimensional models), or the top-triangle moose Chivukula:2009ck (inspired by Higgsless models), which has two ’s, but we stick to this minimal construction for simplicity.

In addition to the standard model and the new gauge bosons for the extra and , we need a field which spontaneously breaks topcolor, giving the topgluons a mass of order TeV. When gets a vev,

 SU(3)1×SU(3)2×U(1)1×U(1)2⟨Φ⟩→SU(3)QCD×U(1)Y. (4)

So should be charged under all these groups. In particular, if is a bifundamental, as shown in the moose diagram in Figure 1, it will automatically break the group down to the diagonal. We also include a Higgs which couples to everything to parameterize the technicolor contribution to EWSB. There are 4 couplings and associated with the two ’s and two ’s. The field content is given in Table 1.

In top-condensation, when gets a vev electroweak symmetry will be broken. Since the bottom quark is charged also under , one must be careful not to have as well, which would generate a large bottom quark mass. The hypercharge conveniently can achieve this. Indeed, with the standard model hypercharge assignments, this is attractive in the channel and repulsive in the channel. If the coupling is large (but not too large or else ), this can allow for only a condensate Hill+Simmons .

In TC2, electroweak symmetry is broken by both top condensation, through and by technicolor through . With two sources of EWSB, there will be two sets of Goldstone bosons. One set are eaten by the and , and the other set are called top-pions.

### 2.3 Effective Theory

The easiest way to study the phenomenology of this model is through an effective linear sigma model. To do this we restore the full gauge symmetries by introducing a sigma field , where are the Pauli matrices. The top-Higgs doublet in the linear sigma model can be represented by

 Ht=Σ(fπ+1√2ht0)=(fπ+1√2(ht+iπ0)iπ+)+⋯ (5)

With this normalization a kinetic term gives the proper normalization to the top-pions and the top-Higgs. In the chiral Lagrangian for the top-pions, the kinetic term is normalized to . In TC2, there is a second Higgs, , which contains the pions from technicolor with decay constant , with similar kinetic terms. The electroweak vev gets contributions from both . The expectation from the NJL model is that GeV, although can really be anything. Constraints from suggests that GeV Burdman:1997pf .

Including the the Higgs fields and and the link field restores the full gauge symmetry. Then, including the operators allowed by the full symmetries, we can construct the effective Lagrangian. First, the top-pion interactions are generated from the top mass term via

 LΣ =λt¯¯¯¯TLHttR+h.c. (6) =λtfπ√2t¯t+λt√2(iπ0¯tγ5t+htt¯t)+λtπ+¯tRbL+λtπ−¯¯bLtR+⋯ (7)

Here we can identify as the topcolor contribution to the top mass, and the top-pion couplings, before diagonalization, as and . The relationship between the top-Higgs and top-pion couplings here is particular to the linear-sigma model and not constrained by symmetries. 111In some sense, a linear multiplet is more natural in topcolor than in technicolor. Since the topcolor gauge coupling must be close to its critical value, the spontaneous breaking of chiral symmetry is a relatively small effect, . Close to criticality, the phase transition should be smooth, and the linear representation of bound state should be a fairly good transition even after chiral symmetry is broken Chivukula:1998uf .

The (technicolor) Higgs field has normal Yukawa interactions with the first two generations

 LH =Yu22CLHcR+Yu12CLHuR+Yu21ULHcR+Yu11ULHuR +Yd22CLH†sR+Yd12CLHdR+Yd21ULHsR+Yd11ULHdR+h.c.. (8)

Note that light quark masses can be generated in technicolor with much less walking than heavy quark masses, so this model alleviates a lot of the tension in ETC models.

Finally, there are terms allowed by symmetry which mix the third generation with the other two. These must involve both and to be invariant under the ’s. These operators must be at least dimension 5, and we characterize them with a scale . The leading operators relevant to the subsequent discussion are

 LHΦ=c32¯¯¯¯TLHcRΦΛ+c31¯¯¯¯TLHuRΦΛ+d32¯¯¯¯TLH†sRΦΛ+d31¯¯¯¯TLH†dRΦΛ+c23¯¯¯¯CLHtRΦ†Λdet Φ†Λ3+c13¯¯¯¯ULHtRΦ†Λdet Φ†Λ3+d23¯¯¯¯CLH†bRΦΛdet ΦΛ3+d13¯¯¯¯ULH†bRΦΛdet ΦΛ3+c33¯¯¯¯TLtRHdet ΦΛ3+d33¯¯¯¯TLH†det Φ†Λ3. (9)

The factors are necessary to maintain the symmetry.

The scale should be larger than the vev. That is, , where is the topgluon mass. Since the topcolor interactions are strong, we expect . Once topcolor and electroweak symmetry are broken, by and , the effective low energy quark mass matrices are

 MU=fT⎛⎜ ⎜ ⎜⎝Yu11Yu12c13ε4Yu21Yu22c23ε4c31εc32ε   m0fT+c33ε3,⎞⎟ ⎟ ⎟⎠MD=fT⎛⎜ ⎜⎝Yd11Yd12d13ε4Yd21Yd22d23ε4d31εd32εd33ε3.⎞⎟ ⎟⎠ (10)

For the top quark mass to be generated by topcolor and the bottom quark mass not to be too small, we expect He:1998ie ; Burdman:1999sr .

Actually, can be much smaller Burdman:2000in , since there is another contribution to the bottom mass due to non-perturbative effects. Scaling arguments predict that GeV, which can be interpreted as an instanton effect Hill:1994hp . Alternatively, can even be , if and are very small. For there are many more relevant higher dimension operators and the theory is not predictive. We will suppose that and so that there will be mixing between the right-handed top and the first two generations. It is this mixing which generates a large .

To proceed, we will assume that terms of order or higher can be neglected, but the other terms are free parameters. Considering the possible instanton contribution, we allow for the -quark mass to be arbitrary as well. Then the effective mass matrices reduce to

 MU=fT⎛⎜ ⎜⎝Yu11Yu120Yu21Yu220Yu31Yu32mtfT⎞⎟ ⎟⎠,MD=fT⎛⎜ ⎜ ⎜⎝Yd11Yd120Yd21Yd220Yd31Yd32mbfT.⎞⎟ ⎟ ⎟⎠ (11)

For the working point GeV we have GeV using . So . Thus there can easily be mixing between the right-handed top and first two generation up-type quarks.

The one prediction of topcolor here is the zeros in the 3rd column. The masses are diagonalized with rotations and , constrained by the CKM matrix . As discussed in Ref. Buchalla:1995dp , the zeros in Eq. (11) imply and are almost vanishing. On the other hand, and can be large when the couplings and are large. This means the right-handed top can have large mixing with the first two generations. Going to the mass basis rotates the right-handed top quark as

 tR→UR11tR+UR12cR+UR13uR. (12)

This generates large flavor changing top couplings from the couplings in Eq. (7). We find

 Lπ =igttπ(π0¯tγ5t)+igtuπ(π0¯tLuR)+igtcπ(π0¯tLcR) +gttht(ht¯tt)+gtuh(ht¯tLuR)+gtcht(ht¯tLcR)+h.c., (13)

where . In the linear sigma model, the top-Higgs couplings are the same as the neutral top-pion couplings. Indeed the two fields form a complex scalar. This relation may not hold in a more complete theory, but it is a reasonable starting point and simplifies the parameter space.

The charged pion couplings are related to the neutral pion couplings by symmetry, as in Eq. (7). For example, the interaction has coupling .

Topcolor models in general should also contain another doublet, the bottom-Higgs . This bound state should be present in the effective Lagrangian Chivukula:1998uf and may be light (of order the top-higgs mass). We will not discuss the bottom-Higgs further simply because it is unrelated to .

Finally, there are the couplings of various excitations of the condensate. The lightest excitation is expected to be the top-rho. The top-rho couplings can be modeled, such as using hidden local symmetry models or vector meson dominance. We simply assume that before mixing the top-rho couples to with a strength that is a free parameter. After mixing, it picks up couplings to right-handed up and charm quarks, just like the top-pions and the top-Higgses. The top-rho is expected to be heavier than the top-pion and the top-Higgs Chivukula:1990bc , but have to be moderately light ( TeV) to give sizable . For completeness, we write its Lagrangian as

 Lρ=gttρρμ¯tγμt+gtcρρμ¯tRγμcR+gtuρρμ¯tRγμuR. (14)

We will only discuss the top-Higgs, top-pions and top-rho in the phenomenology section.

There are additional fields in TC2, such as the bottom-Higgs, the charged rho’s, the topgluon, the excitations of the link field , and all the regular ETC particles. For constraints and signatures of these other particles, we refer the reader to the review Hill+Simmons . None of these fields are relevant to , so we will ignore them.

### 2.4 Flavor and electroweak constraints

Before moving on to phenomenology of topcolor model with large flavor-violating associated with , we would like to briefly discuss related flavor constraints and how such model could be compatible with them.

There is no GIM mechanism in TC2 and hence there is a real danger of generating too large FCNCs, for example, through top-pion exchange. However, the top-pions only couple to the third generation before mixing. Large flavor-changing couplings in the first two generations are only produced when both the right-handed and the left-handed tops have large mixings with the first two generations. This is avoided because only the right-handed top has large mixings as we have discussed. Flavor constraints involving the top are very weak, so the only concern left is flavor problems coming from the -sector. In fact, there are in strong -physics constraints on FCNC operators involving and and the because these produce tree-level FCNCs involving after EWSB Fox:2007in . In our case, these bounds do not apply because the top-mesons decay hadronically and there are no electroweak penguins involving the neutral top-mesons alone. Flavor constraints in next-to-minimal flavor violating models Agashe:2005hk , in which new physics couples only to the third generation, are in general fairly weak.

There are additional constraints from the down-type sector, which is not directly related to the flavor-violation necessary for . Even in the down sector, the constraints will be absent if the and elements are small. These terms originate from the and terms in Eq. (9) which involve and . It is easy to believe that these terms might be small for a reason related to the smallness of the down and strange quark masses. In any case, that and are small is a standard assumption in TC2 and we have no further insight into its dynamical origin. Our observation here is simply that or do not have to be small as well.

Based on the reasonable assumption that down-sector RH and LH mixings are both small, it is easy to see that potential dangerous mixing induced via e.g. tree-level exchanging an s-channel or via box diagram involving can be efficiently suppressed. Meanwhile, there is concern from mixing which involves the up-sector only, and may get large contribution from exchanging if both and are sizable. One way to relieve this while keeping possible large flavor violation is to have or small, but not both. For our later discussion we will assume a large .

The model may also receive important electroweak constraint from since it directly involves the unsuppressed vertex in the loop Burdman:1997pf ; Braam:2007pm . However, this constraint is subject to large uncertainties in subleading calculations Hill+Simmons . Moreover, according to related studies in Hill:1994di ; Burdman:1997pf ; Hill+Simmons vector states such as top-rhos may cancel the contribution from the top-pion if they are moderately light (), which is also what we need to generate significant , as discussed later. The charged top-pions could also be heavy, which would further alleviate .

## 3 Phenomenology

In the previous section we reviewed some features of topcolor-assisted technicolor. This explanation of electroweak symmetry naturally explains both the hierarchy and the large top Yukawa with fewer flavor problems than technicolor has without topcolor’s help. Unfortunately, very little of this model’s spectrum is currently calculable. However, some general features were noted

• Electroweak symmetry is broken by both technicolor and topcolor. So in addition to the Goldstone bosons eaten by the and , there is another set, the top-pions, which are bound states and predominantly couple to the top quark.

• In addition, there is expected to be a radial excitation of , the top-Higgs, and angular excitations, such as a top-rho.

• Power counting in the low-energy effective theory implies that there can be large mixing of with . This generates large flavor-changing interactions mediated by the top-mesons.

There are other particles in the theory, such as the charged top-pions and top-rhos, the top-gluon and all the technicolor excitations, such as the techni-rhos. Some of these particles were discussed briefly in the previous section, and their constraints have been discussed elsewhere. They do not contribute directly to interesting top-quark signatures, so we will not discuss them any further.

In this section, we will discuss the effect of having large flavor changing interactions, coming from

 L=gπ(i¯tLuRπ0+¯tLuRht)+gρ(¯tRγμuRρμ)+h.c., (15)

where we have set the top-Higgs and top-pion couplings equal to a constant we call . This does not have to be true, but it is motivated by a linear-sigma model embedding, in which the top-pion and top-Higgs form a complex scalar , and simplifies the parameter space. There are of course plenty of other interactions of these fields. But it is these specific terms which are relevant to the top forward-backward asymmetry.

The top-pion and top-Higgs masses are not expected to be the same, however, with the top-Higgs typically having while the top-pions, being pseudogoldstone bosons can be lighter, . The top-rho is expected to be heavier than both the top-pion and the top-Higgs, based on extrapolations form QCD. None of these masses are calculable, although the pion mass is often estimated using the a fermion loop approximation and the Pagels-Stokar formula Miransky:1988xi ; Miransky:1989ds ; Bardeen:1989ds ; Hill:1991at . In this paper, we simply take them to be free parameters.

In this section, we describe how the top-particles can produce the measured while maintaining an acceptable production rate. We will also discuss the closely related direct measurements of same-sign tops.

### 3.1 Top forward-backward asymmetry AtFB

The idea that the top forward-backward asymmetry could be evidence of a new particle, , being exchanged in the -channel was proposed in Jung:2009jz . The -channel exchange works by exploiting the Rutherford singularity which enhances scattering in the forward region. A vector boson with moderate mass, such as a Jung:2009jz (or a top-rho) can generate a sizable asymmetry. Alternatively could be a scalar Shu:2009xf ; Nelson:2011us (such as a top-Higgs or top-pion). In the case of a scalar there is a competition between Rutherford singularity and spin-conservation, which reduces the efficiency of generating a large . This pushes the scalar mass towards lower end as in order to generate . Both the vector and scalar by themselves will generate an excess of events. However, if both are included, there can be destructive interference, allowing a large but a small effect on the .

The experimental measurement of is presented in a number of ways: in the rest frame, or in the lab frame, and folded or unfolded, if an attempt is made to correct back to the parton level. Unfortunately the unfolding depends on the angular distribution of the tops, which is model dependent, and unfolding using the standard model, which has a tiny may give an unreliable result. In this paper, we follow the approach of Nelson:2011us . We impose rapidity cuts on the tops, , and demand that and compare the after these cuts with reconstructed asymmetry measured by CDF (for ): . We require the asymmetry to be within of the central value, which is about . Our calculations are performed with Calchep v3.0 calchep and checked with Madgraph v4.4.26 Alwall:2007st and by hand. We use CTEQ6l PDF and choose the factorization scale and renormalization scale to be .

In Figure 2 we show the contributions of the top-pion, top-Higgs and top-rho to as a function of mass, with . The single particle contributions are shown as dotted lines, with the top-pion and top-Higgs having an identical effect. The dashed green line shows the effect of having a top-pion and a top-Higgs with . The solid black curve shows the effect of adding a 500 GeV top-rho to the mix. Many other combined curves are possible. This illustrates that even with a heavy vector, there can be a positive contribution to due to interference with the scalars. To get there is a large region of parameter space. The space become somewhat more restricted with the cross-section and the same-sign top bounds are also imposed.

One could also consider the charged top-pion contribution to . The charged pions couple initially to . Due to the smallness of the -quark PDF, the -channel exchange of through this interaction has a negligible contribution to . mixing will only weaken the effect. One could imagine that there might be large mixing, which would allow for to be created through with a -channel . However, mixing is strongly constrained by CKM unitarity bounds Hill+Simmons , so this contribution cannot generate a sizable asymmetry either.

### 3.2 t¯t cross section

Introducing new processes involving new light states with sizable coupling raises concerns about large deviations in the cross section, . As pointed out in previous work e.g.Shu:2009xf , contributions from the -channel exchange amplitude squared, which is positive, could be cancelled by a negative contribution from interference with QCD, rendering small the net deviation from the measured . Indeed there can easily be interference between different new physics contributions as well and one has to do a careful analysis. In our model, we see that the interferences between QCD and all three new particles are negative and those among the new particles are all positive. This is easily seen by going to the limit of the threshold and examine different contributions to the matrix element squared, as shown in Table 2. For reference, the full cross section is given in Appendix A.

The cross section has been measured at the Tevatron and the LHC and is in good agreement with the standard model. There is also data on the differential cross section as a function of the invariant mass, but the error bars are still large. Thus, we compare only to the inclusive cross-section. The cross section is enhanced at NLO at both the Tevatron and the LHC. The -factor can depend on the experimental cuts used, so it is difficult to compare to the measured cross section directly. Instead, we will simply require that the effect of the tree-level new physics contributions to produce less than a change from the standard model , also computed at tree level. This roughly corresponds to band by measurement cdf-ttbar .

Figure 3 shows the percent change in the total cross section as a function of particle masses. We see that the top-pions (or top-Higgs) have negative interference with the standard model and pull the cross section down. The top-rho tends to increase the cross section at low mass. The effect is so large that in fact it is basically impossible for the top-rho to be less than 300 GeV. For large top-rho mass, the contribution is negative. Note that even for a 500 GeV top-rho and light top-pion and top-Higgs (black curve), even though both the scalars and the vector separately want to lower the cross section, their combined effect is a small positive contribution, at the level.

### 3.3 Same-sign tops

One of the most pressing problems for -channel explanations of the top asymmetry is that these models generically produce an abundance of same-sign tops. Indeed the same coupling explaining automatically allows for . Even at the Tevatron, this could have been seen. At the LHC, although no same-sign top bound has been published yet, any same-sign top excess would have lit up the inclusive search for same-sign lepton pairs. Thus it is impossible to have a large contribution to same-sign tops.

To get a bound, we use the published Tevatron bounds. There are two CDF results. One tt-cdf1 , which is model dependent, has a new particle which produces through -channel exchange. This particle is assumed to be heavier than so there are also contributions from and . The combined bound comes out to 800 fb. The second CDF result tt-cdf2 uses more data but looks exclusively for states. The bound here is 500 fb. We will use this second bound since it is stronger.

For the cross-section, we saw that interference between new physics and the standard model, and interference among the new physics particles themselves conspired to have a small effect on the total cross section. For production, in contrast, the standard model contribution is tiny and comes only from weak processes. So the interference between new physics and the standard model cannot help. However, interference between different topcolor processes producing could be negative. This is illustrated in Table 3, where we list the contribution from interferences to the matrix element squared. In fact, there is an exact cancellation between the top-Higgs and top-pion in the limit that their masses are degenerate and they couple with the same strength.

There are various ways to understand the top-pion/top-Higgs interference effect. The cancellation can be seen by direct computation: top-pion exchange produces a amplitude with the same magnitude but opposite sign as top-Higgs exchange, due to the extra in the pseudoscalar interaction. There is also a symmetry interpretation. If we write as a complex scalar, then can be thought of as carrying a chiral charge whose conservation forbids . This mechanism was employed in Nelson:2011us where a complex scalar was introduced directly for this purpose. Here we are observing that the top-pion and top-Higgs automatically combine into this complex scalar. The symmetry is broken by the top-pion/top-Higgs mass splitting and any difference in couplings. But there is still a large cancellation even if the particles are separated by hundreds of GeV.

Figure 3 shows the same-sign top cross section for the top-pion alone, which is the same as the cross section for the top Higgs alone, and also the cross section when both the top-pion and top-Higgs are present. The interference substantially suppresses the same-sign top rate. The left panel shows the rate for just and production. There is also a contribution to same-sign tops from the top-rho and also from on-shell top-pion/top-Higgs pair production, or production in association with a top, with . These contributions are included in the right panel. We have not included interference effects for these extra channels, although when the top-pion and top-Higgs are close in mass, there will be destructive interference in these channels as well.

### 3.4 Combined constraints

Having discussed all the experimental constraints separately, we will now consider combining all the measurements together. Estimates in topcolor based on the NJL model and scaled-up QCD give us some sense of what parameter range is natural. Following these estimates, we consider , , and couplings between 0 and 1.5.

First, suppose we just have the top-pion. Looking at Figure 2, we can see that we would need a coupling larger than and a pion mass GeV to explain the asymmetry. In this regime from Figure 3 we see that there would be too little produced, with around a decrease from the standard model as well as a marginally unallowed production of same-sign tops, as we see in Figure 4. Note that for , the channels which include the top-pion decaying to do not contribute.

Next, let us include the top-Higgs. This helps with the cross section and with same-sign tops. However, it is still hard to get enough asymmetry. Finally, we add the top-rho. This additional contribution can add to the asymmetry without producing much more or same-sign tops. Moreover, due to interference between the top-rho and the top-scalars, the combination of the 3 particles is more than the sum of their parts. There is a large parameter space, including regions with light top-pions as well as regions where the top-pions are heavier than .

In Figure 5 we show the allowed parameter space for the couplings and (see Eq. (15)) for two representative mass points. On the left, we take , and GeV. This has a light top-pion, below the top-mass. The top-pion can also be heavier than , as shown on the right, where , and GeV. Note that for the second mass point, we have inverted the hierarchy between and , introducing destructive interferences that helps ameliorate the same sigh top bounds.

In Figure 6, we show the allowed range of pion masses. To generate these curves, we allowed the top-Higgs and top-rho masses to vary over and and . The region with no top-rho () is shown on the left side. Without a top-rho, the top-pion mass has to be very light. A light top-pion is probably ruled out by direct searches, although the precise bound depends on how the top-pion decays, which is somewhat model dependent. If we look for the region of top-pion and top-rho masses which are allowed by the and same-sign top bounds, practically any values of and are allowed for some coupling strength and mass. In fact, taking removes the same-sign bound completely and the bound is controlled by destructive interference between the new physics and the standard model, as we have discussed..

## 4 Other signatures

We have shown the neutral top-pion, the top-Higgs and the neutral top-rho can combine to give a large top forward-backward asymmetry with small effects on the total cross section and a not-yet-detectable number of same-sign tops produced. These new particles do not generate any new strong flavor constraints. Particles closely related to these, such as the charged top-pions or the charged top-rhos, can contribute to observables like . However, the charged top-pions could be heavy and the constraints would be weak. From the model-independent point of view the main point is that there are no strong flavor constraints directly involving only the particles related to the explanation.

The top-mesons will have interesting signatures at the LHC depending on how they are produced and decay. Generically, we know there should be a sizable coupling of these particles to and . If the top-pion is lighter than the top, it can decay to through an off-shell-top. Particles heavier than the top could decay directly to . If the top-rho is heavy enough, it will decay to or rather than . The top mesons can be singly produced through gluon fusion through a top-loop. They can also be produced in association with a top, or a or or pair produced from through the exchange of a -channel top. There are many possibilities.

A generic consequence of having particles coupling strongly to is that there should be resonances visable at the LHC. For example, top-pion pair production would give a large number of events. A single top-pion could be produced in association with a top through . Thus there would also be events. In general, looking at events with tops and jets and looking at top-jet invariance mass distributions should discover or rule out this model fairly early on at the LHC.

One more consequence of the couplings we have discussed is that if there is a large coupling and the top-pion is lighter than the top, then we could have . Indeed, for with a coupling , the branching ratio to this channel would be 7%, with the top decaying 93% of the time to . What this looks like depends on how the top-pion decays. The top-pion might decay to . This coupling is difficult to calculate if the -quark mass comes from non-perturbative effects (instantons) in the top-color sector. It does get a calculable contribution from top/top-pion loops which make some branching fraction to inevitable. If the coupling is small enough then the decay mode or might dominate. Alternatively, the decay through an off-shell top could dominate as well, depending on the couplings. In any case, the observed rate for the cross section is based on a leptonic branching ratio taken from the standard model. So it is possible that that rate might be slightly higher, with a slightly lower branching ratio to making the top production seem more consistent with the standard model than it actually is.

Let us consider the situation in which . Then we would have 7% of the time. This excess of hadronically decaying tops could help explain an excess seen by CDF in events with two massive fat jets pekka . It might also explain the recently observed excess in multijets near the top-mass seen by a different CDF group Aaltonen:2011sg . In fact, this decay mode could also explain the excess in +jets events seen by CDF Aaltonen:2011mk . For example, this excees could be produced by a top-rho decaying to , with . If decays to dominantly, then it is hard to explain the CDF excess since the peak does not seem to be rich in ’s. It is unclear without futher model details and more calculations whether the rate for production of these modes is consistent with the size of the excess.

If the top-pions are heavier than , they cannot be the explanation of the CDF +jets bump. However, it is still possible that the bump might be due to particles in the technicolor sector, as proposed in Eichten:2011sh . These particles are also present in topcolor-assisted technicolor, but we have nothing new to say about them from previous work.

## 5 Conclusion

The measurements of the top-quark forward-backward asymmetry at the Tevatron suggest new physics may be showing up strongly coupled to the top. That new physics would show up here is not surprising to the many theorists who have had to contend with the large mass hierarchy between the top mass and the other fermions. Indeed, top-condensation and its realization in topcolor Hill:1991at was suggested more than 2 decades ago as a natural combination of dynamical symmetry breaking with an explanation of the large top quark mass. Topcolor even predates the discovery of the top quark itself. When topcolor is combined with technicolor, the resulting framework, topcolor-assisted technicolor (TC2), provides a solution of the hierarchy problem and the large top quark mass without some of the strongest flavor-changing neutral current problems of extended technicolor itself. The combined framework explains the origin of mass as a team effort: the top mass comes from top-color, the and masses come mostly from technicolor, the -quark mass comes from instantons, and the light quark masses come from extended technicolor. Taking this one step further, most of the visible matter in the universe is baryons, which get their masses from QCD, neutrino masses can come from the seesaw mechanism associated with additional new physics at a high scale, and dark matter might come from a totally decoupled sector, such as from axions.

In this paper, we have shown that TC2 also contains all ingredients for producing the large anomaly without violating other constraints. The spectrum contains three relevant particles, the neutral top-pion, the top-Higgs and the neutral top-rho. All of these can have large top-up couplings due to mixing of the right-handed top with the lighter up-type right-handed quarks. This mixing can be naturally large in TC2. Thus, there is automatically an excess in the top-quark forward-backward asymmetry over the standard model due to -channel exchange of these particles. Intriguingly there is constructive interference among the particles in producing a large , but destructive interference affecting the overall cross section. Same-sign top production is a signature of generic points in parameter space of this model. If the top-Higgs and neutral top-pion are close in mass, the current experimental bound on same-sign top production can be avoided without much tuning. Nevertheless, without additional symmetries, it seems impossible to avoid a same-sign top-signal, particularly from on-shell pion or top-Higgs decays, which should be visible at the LHC early on. This would be a clear signature of this type of explanation of the excess.

There are no devastating flavor problems in TC2, even with large mixing. Of course, TC2 does have difficulty with precision electroweak, but there is still no reliable way to calculate and and new particles can also contribute. Thus it is impossible to rule out the model based on precision electroweak alone. Flavor-changing neutral currents are not bad in TC2 because topcolor only couples to top, at leading order, so it naturally falls into next-to-minimal flavor violation Agashe:2005hk . The light quark masses are generated with extended technicolor. However, because ETC is no longer responsible for the bottom or top masses, the tuning required is not very severe. Constraints such as the rate, , are dangerous, but not directly related to the particles responsible for explaining . To the extent that flavor physics is under control in previous incarnations of TC2, they are under control here.

Top-color assisted technicolor has a number of signals that will show up early on at the LHC. The top-pions and other top-mesons should be produced in abundance. They can be singly produced through gluon fusion, as well as pair produced through -channel top-exchange or produced in association with tops through initial states. Top-pions can also be produced from decays of heavier top-particles. With large couplings necessary to explain , there should be an abundance of events with multiple tops and multiple jets. Looking for a resonance peak in the top-jet invariant mass would be a clean test for this model. Thus, forthcoming results from the LHC should soon reveal whether top-condensation can be the explanation to the large top forward-backward asymmetry observed at the Tevatron.

Note Added: As this manuscript was being finalized, the CMS collaboration released a same sign top bound from early LHC running Collaboration:2011dk . This bound is somewhat stronger than the Tevatron bounds. Topcolor-assisted technicolor can still explain and be consistent with this observation, but the parameter space is more constrained: the top-Higgs and neutral top-pion should be closer in mass (if they are degenerate with the same coupling strength, same-sign top production is absent) and the top-rho should also be on the heavier side, which is anyway consistent with some expectations from topcolor Chivukula:1990bc .

## Acknowledgments

We thank Sekhar Chivukula, Howard Georgi, Sunghoon Jung, Yasunori Nomura and James Wells for helpful discussions and Daniel Whiteson for helping us understand the CDF same-sign top results. This work is supported by the NSF under grants PHY-0855591 and PHY-0804450 and the DOE under grant DE-SC003916.

## Appendix A t¯t interference effects

In this appendix, we give formulas for the partonic matrix elements for pair production. These formulas are useful for seeing the signs and strengths of various interference effects.

 MM∗=Mgg+Mππ+Mh