Color-octet scalars and potentially large CP violation at the LHC

# Color-octet scalars and potentially large Cp violation at the LHC

Xiao-Gang He, German Valencia and Hiroshi Yokoya INPAC, Department of Physics, Shanghai Jiao Tong University, Shanghai, China
Department of Physics, National Taiwan University, Taipei, Taiwan
Department of Physics, Iowa State University, Ames, IA 50011
National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan
July 25, 2019
###### Abstract

We consider the phenomenology of violation in a color-octet extended scalar sector for production and decay at the LHC. In particular we study the effect of the two neutral color-octet scalars and that occur in the model. There are two new sources of violation: a phase in the couplings of to top-quarks; and two phases in the quartic couplings of the scalar potential. In resonant production of a single followed by its decay into pairs through the parton level process , we find large raw asymmetries which can reach 12%. These raw asymmetries are, of course, diluted by standard model (SM) pairs making observation of violation contingent on whether the resonance itself can be extracted from the SM background.

###### pacs:
PACS numbers: 12.15.Ji, 12.15.Mm, 12.60.Cn, 13.20.Eb, 13.20.He, 14.70.Pw
preprint:

## I Introduction

Violation of charge-parity () symmetry beyond the standard model (SM) has yet to be observed but we suspect that it must be there in order to explain the baryon asymmetry of the universe. This gives paramount importance to new searches for violation in high-energy frontier. One tool, proposed many years ago, for searches in collider experiments is the use of triple-product correlations tprods (), which are simple kinematic correlations of the form . These correlations are referred to as “naive-” odd because they reverse sign under the “naive-” operation that reverses the direction of momenta and spin without interchanging initial and final states. In general, correlations of this form can be even or odd, as they are induced by either loop level unitarity phases or by violating phases respectively. In top-quark pair production, these triple-product correlations originate in violating spin correlations with the top-quark (and anti-quark) weak decay acting as spin analyzer.

There exist several recent proposals to search for violation at the LHC using triple-product correlations. Our discussion is based on observables discussed for anomalous top-quark couplings in Ref. cplhc () as well as observables discussed for multi-Higgs models in Ref. cphiggs (). Additional processes that have been discussed recently include and pair production and decay others (). On the other hand, conserving “naive-” odd triple-product correlations have been studied in radiative top-quark decay Hagiwara:2007sz () at one-loop level.

In this paper we consider triple-product correlations in top-quark pair production at the LHC, induced by new violating interactions in a scalar sector extended with a color-octet electroweak-doublet as described in Ref. Manohar:2006ga (). This model incorporates the additional scalars in a manner consistent with minimal flavor violation (MFV) in order to naturally suppress flavor changing neutral currents (FCNC).

Of particular interest to us are the two neutral, color-octet, scalar resonances that occur in the model, . These particles couple at the one-loop level to two gluons and their production at the LHC has been discussed in Ref. Gresham:2007ri () for the conserving case. These particles also couple (dominantly) to top-quark pairs which makes them an ideal candidate to study violation in the process . To this end, we extend the results of Ref. Gresham:2007ri () to include violation, which can occur both in the couplings of to top-quark pairs as well as in certain self-interactions in the scalar potential. We further assume that top-quark decay proceeds as in the SM and serves only to analyze the corresponding spin. Within this framework we find that relatively large raw asymmetries are possible, as large as . These raw asymmetries are diluted by the SM top-quark pairs and their observation is contingent on the resonance itself being observable. For illustration, we present a set of parameters for which the resonance is visible over the SM background and the resulting asymmetry can be as large as a few percent. We also comment on other channels where asymmetries are potentially visible in cases where the single resonance is not.

The LHC has already established new constraints on the new physics beyond the SM, even though it has been operating so far with a reduced energy of 7 TeV. In particular, both ATLAS and CMS have excluded certain color-octet scalars similar to the ones we consider here in a broad mass range noreson () by studying the dijet channel. These exclusion limits, while interesting, do not apply to the models we discuss, where the color-octet resonances decay almost exclusively into top-quark pairs. Their decay modes into dijets occur with branching ratios below for the sets of parameters we use in this study.

Our paper is organized as follows: in Section II we review the relevant aspects of the model for the color-octet scalars with emphasis on the violating phases that have not been studied previously. In Section III we study several benchmark cases that illustrate the generic properties of the raw asymmetries. We also discuss several aspects concerning the observability of the violating signals at the LHC. In Section IV we state our conclusions and, finally, we relegate some analytic formulae to the Appendix.

## Ii Color-octet scalars as a source for Cp violation

We briefly review the case of a scalar sector that has been extended with a color-octet electroweak-doublet scalar with hypercharge , . This particular choice is motivated by the requirement of MFV and has been recently elaborated in Ref. Manohar:2006ga (); Gresham:2007ri (). It was noted in these papers that in a MFV scenario only scalars with the same gauge quantum numbers as the SM Higgs doublet or color-octet scalars with the same weak quantum number as the Higgs doublet can couple to quarks, and this has many interesting consequences for both collider and flavor physics. This color-octet electroweak doublet can be written in the properly normalized component form with the color index as , where is the generator normalized as .

The Yukawa couplings of the color-octet scalars can be parameterized, to the leading order, with the MFV assumption as Manohar:2006ga ()

 L = −√2v~ηU¯URTA^MuULSA0+√2v~ηU¯URTA^MuVKMDLSA+ (1) − √2v~ηD¯DRTA^MdDLSA0†−√2v~ηD¯DRTA^MuV†KMULSA−+h.c.,

where are the diagonalized mass matrices, ; and are the up and down quarks, and ; and  GeV is the Higgs vacuum expectation value (VEV), . The neutral complex field can be further decomposed into a scalar and a pseudo-scalar as . The parameters are expected to be of order one and are in general complex. We will write them as with real, and if there are non-zero phases there is violation beyond the SM.

There is a second possible source of violation in this model in two of the self couplings appearing in the scalar potential. The most general potential with the complex scalar doublets and is given by Manohar:2006ga (),

 V = λ4(H†iHi−v22)2+2m2sTr S†iSi+λ1H†iHiTr S†jSj+λ2H†iHjTr S†jSi (2) + [~λ3H†iH†jTr SiSj+~λ4H†iTr S†jSjSi+~λ5H†iTr S†jSiSj+h.c.] + λ6Tr S†iSiS†jSj+λ7Tr S†iSjS†jSi+λ8Tr S†iSiTr S†jSj + λ9Tr S†iSjTr S†jSi+λ10Tr SiSjS†iS†j+λ11Tr SiSjS†jS†i.

The parameters are in general complex, but without loss of generality, one can choose a convention in which is real. The two other phases cannot be removed and we write them as . Non-zero phases provide the second source of violation beyond the SM present in the model. We note the custodial symmetry requires  Burgess:2009wm (); Carpenter:2011yj (), so that in our parameterization.

If the mass of color-octet scalars is not too large, a tree-level interaction can pair produce them at the LHC through the process . Since the color-octet scalars also couple to quarks, there is an additional contribution from . However this contribution is small because the Yukawa couplings of to quarks are proportional to quark masses. Single production is also possible at tree level from its Yukawa couplings to quarks, but this tree-level contribution is small as it is proportional to the light-quark mass. It has been shown that the single production cross section at the LHC is dominated by a loop induced interaction and can be of order  fb for masses of order a few hundred GeV to a TeV Gresham:2007ri (). If these resonances are produced at the LHC, it will be possible to study their properties, including violation, through their decays to SM particles. The dominant decay mode is although there is also one-loop induced decay into gluon pair:  Manohar:2006ga (); Gresham:2007ri ().

In this paper we wish to study signals of violation at the LHC. This is accomplished by first observing the resonant production of the neutral color-octet scalars through their effective couplings and then studying the correlations that occur in the subsequent decay chain: . The new sources of violation induce triple-product correlations involving the spin of the top and anti-top quarks which are subsequently analyzed by their weak decays. In the dimuon mode chosen above, it is the direction of the muons that analyzes the spin directions. Both the new sources of violation contribute to the resulting asymmetries.

The relevant effective couplings are shown schematically in Figure 1, and they can be written in terms of an effective Lagrangian as

 L(S−t¯t) = ¯t(aR+ibRγ5)TAtSA0R+¯t(aI+ibIγ5)TAtSA0I, L(S−gg) = (FaRGAμνGBμν+FbR~GAμνGBμν)dABCSA0R (3) + (FaIGAμνGBμν+FbI~GAμνGBμν)dABCSA0I,

where is the gluon field strength tensor and .

The couplings to the top-quark occur at tree-level and they contain violation originating in the new phase . They are given by

 SA0R−t¯t:i(aR+ibRγ5)=−iηUmtv(cosαu−isinαuγ5)TA, SA0I−t¯t:i(aI+ibIγ5)=iηUmtv(sinαu+icosαuγ5)TA. (4)

The couplings to gluons occur at one-loop level and can be easily derived following Ref. Gresham:2007ri (). The violation in this case is due to phases in the couplings . We find

 FaR = (√2GF)1/2αs8π[ηUcosαuIq(m2tm2R) − 94v2m2R(λ4cosα4+λ5cosα5){12Is(1)+12Is(m2Im2R)}], FbR = (√2GF)1/2αs8π12m2tm2RηUsinαuf(m2tm2R), FaI = (√2GF)1/2αs8π[−ηUsinαuIq(m2tm2I) + 94v2m2I(λ4sinα4+λ5sinα5){56Is(1)+16Is(m2Rm2I)}], FbI = (√2GF)1/2αs8π12m2tm2IηUcosαuf(m2tm2I). (5)

In these expressions we have assumed that the mass of the charged color-octet scalars is equal to the mass of , , which corresponds to the custodial symmetry conserving case where  Manohar:2006ga (). We have allowed for the mass of to be different, . These two masses are related by . Throughout the calculation, the scalars and (when not in a loop), are taken to be on-shell, in keeping with the narrow width approximation. The loop functions and are defined by:

 Iq(z) = 2z+z(4z−1)f(z),Is(z)=−z(1+2zf(z)), f(z) = 12(ln(1+√1−4z1−√1−4z)−iπ)2  for z<1/4 (6) = −2(arcsin(12√z))2  for z>1/4.

From this it follows that , a factor that generates some suppression in contributions from scalar loops relative to top-quark loops.

## Iii Estimate of CP-odd asymmetries

We will now give numerical estimates for the triple-product correlations that will serve as -odd observables. Following Ref. cplhc (), we know that the best observable for the case of production and decay is the correlation

 ~O1 = ϵμναβpμbpν¯bpαμ+pβμ−b¯b CM−−−−→∝→pb⋅(→pμ+×→pμ−). (7)

The first, covariant, expression is given in terms of the completely antisymmetric Levi-Civita tensor, whereas the second one indicates its reduction to a simple triple-product correlation in the center-of-mass frame. The reasons to select this correlation from the set described in Ref. cplhc () are twofold. First, the dimuon decay of is the cleanest. Second, the fact that the pair is produced from a scalar intermediate state prevents the appearance of correlations involving the beam momentum. Note that although it appears that this correlation requires distinguishing the and jets, it is only necessary to systematically associate one of the jets with one of the muons. For example, the ””-jet could be the one closest to the .

To study the effect of the correlation Eq. (7) we consider the laboratory frame distribution . The violating effects can be isolated by extracting asymmetric terms from this distribution, either by a direct fit or by constructing the integrated counting asymmetry

 A1 ≡ Nevents(~O1>0)−Nevents(~O1<0)Nevents(~O1>0)+Nevents(~O1<0). (8)

To measure this distribution (and its associated integrated asymmetry) we generate events for the process with the aid of MadGraph madgraph (). To generate the signal events we implement the vertices of Figure 1 into the MadGraph code. We also use the default MadGraph SM processes to generate the corresponding events. In all cases we use the default MadGraph cuts requiring the top quark and boson intermediate states to be within 15 widths of their mass shell, the transverse momentum of both muons to be larger than 10 GeV and the pseudo-rapidity of both muons to be . We also use SM parameter values as in MadGraph and the CTEQ-6L1 parton distribution functions Pumplin:2002vw ().

We begin by presenting raw asymmetries (no SM top-quark pair events) using values for the parameters and in the ranges discussed in Ref. Gresham:2007ri (), as well as  TeV. We illustrate the asymmetries utilizing only the dimuon signal, but emphasize that other decay modes can also be used.

The results are summarized in Table 1 where we show for each set of parameters the corresponding resonance () widths (which are completely dominated by the decay mode); production cross-sections at the LHC for  TeV; and a raw asymmetry. This raw asymmetry is estimated separately for each resonance by including only top-quark pair events that result from the decay of that resonance and have an invariant mass within 10 GeV of the resonance,  GeV.

We begin with Case 1 in which we have two well separated resonances,  GeV and  GeV, . The scalar potential parameters , as well as the parameter governing the strength of the coupling, are all chosen to be one for illustration. We also choose violating phases to be , , in order to maximize the violating interaction in the coupling of scalars to . As seen in Table 1, the resulting raw asymmetry can be rather large, about for each resonance. The results also show that the two resonances produce asymmetries that tend to cancel each other out, as can be inferred from Eq. (17). In Fig. 2, we plot the distributions for the two scalars, where is evaluated as in the rest-frame. The solid and dot-dashed lines show the violating case whereas the thin dashed lines illustrate the corresponding conserving case where the phases have been set to zero.

In Case 2 we choose much larger values for the parameters , taking them to be 8 times larger than . With this choice we want to enhance the relative contribution from the scalar loops which are otherwise suppressed by the factor mentioned before. We also introduce non-zero phases . We find that this choice produces a modest increase in the cross-section for but not for . The raw asymmetries remain about the same indicating that the top quark loop is still dominant.

In Case 3 we repeat the parameters of Case 1 except for choosing which has the effect of minimizing the cancellation of asymmetries between the two resonances. This can be seen from the Table 1 where the asymmetry around remains near 12% but the asymmetry near is down to about 8%. The production cross-section for each resonance is also larger with this choice of phase and the widths are affected as well with becoming narrower and wider.

As Eq. (18) indicates, when the two masses are close to each other, the contribution from the top-quark loop tends to cancel out, exposing the effect of . We illustrate this in Case 4 with and phase . The resulting raw asymmetry is smaller but non-vanishing.

Finally, in Case 5, we illustrate the effect of the resonance mass by choosing one of the resonances to be just above the threshold and the other one near the high-end of the LHC reach. With the same phases as in Case 1, the asymmetries are about 25% smaller in this case.

We now turn to the question of observability of the raw asymmetries over the SM background of top-quark pairs. First, there is a matter of statistical sensitivity. At the LHC, the total cross-section for events is approximately  pb. This implies that the expected number of events in the dimuon channel is about for an integrated luminosity  [fb]. Therefore, an optimistic sensitivity for the asymmetry using only the dimuon channel for one year of nominal LHC running, is , where is the branching ratio of the dimuon channel. Of course this can be improved significantly by considering other decay channels. For example, in Ref. cplhc () it is shown how to use the lepton-plus-jets channel and purely hadronic channels to measure asymmetries. A detailed analysis of all channels is beyond the scope of the present paper where we simply seek to establish the possibility of a large raw asymmetry.

Second, as we have seen, the asymmetries due to the two resonances tend to cancel. This indicates the need to isolate a certain window in top-quark pair invariant mass around the resonance to extract a non-zero asymmetry. In the dimuon channel it is not possible to reconstruct so we use the transverse mass for this purpose.111The transverse mass is defined as , where , is missing transverse momentum, is vector-sum of dimuon ( pair) transverse momentum, with the invariant-mass of dimuon ( pair). The correlation between these two variables is shown in Figure 3.

This complication would also be treated differently in other top-quark decay channels, and its resolution will ultimately depend on an observation of a new resonance in events. At the same time, it shows that if one wants to check for asymmetries in events in which a resonance has not been observed, there are good reasons to study separately different small windows in (or the corresponding observable such as ), as small as allowed by statistics.

The resonant cross-section for all the cases discussed in Table 1 is significantly smaller than the cross-section for SM top-quark pairs in a 20 GeV window in which is  fb when the window is centered at  GeV. This number is valid at leading order, including branching ratios for decay into dimuon channel, and including all the kinematic cuts discussed before. The SM events will thus dilute the raw asymmetries by two orders of magnitude making them unobservable for practical purposes. In other words, the large raw asymmetries in Table 1 are not observable at the LHC because the corresponding resonances do not stand out above the SM background. To continue our discussion we thus consider resonances with larger production cross-section. This can be easily achieved by increasing to , which results in an order of magnitude increase in the resonance cross-sections without changing the asymmetries significantly.222The value of in conjunction with masses below one TeV is slightly outside the parameter space allowed by studied in Ref. Gresham:2007ri (). We use it anyway as a simple way to illustrate a resonance that stands above SM background taking into consideration the imprecise nature of indirect constraints on new physics. We illustrate this in Table 2. Notice that we have kept the definition of the raw asymmetry with a 20 GeV window around the resonance as in Table 1. This window does not cover a full width in all cases with , resulting in fractionally smaller asymmetries than would be possible. The precise optimization of the window size, although eventually important for measurement, is beyond the scope of this study.

We now have examples of large raw asymmetries in resonances that may be observable over the SM background as can seen in Figure 4.

In Table 3 we examine the asymmetries that result when all events are included (SM plus new resonances). In the first column we show the asymmetry diluted by SM events. We define this asymmetry using a 20 GeV window around the resonance. As expected, it is roughly one order of magnitude smaller than the raw asymmetries for the resonant cross-sections shown in Table 2, and larger for which has a larger production cross-section. In the second column we show how the asymmetry becomes even smaller when the whole range is used, due to the partial cancellation between the two resonances. Finally, in the last column, we impose a realistic kinematic cut to simulate selecting different windows to separate the two scalar contributions. Note that the numbers in parenthesis in Table 3 give the statistical error in our simulation, which is expected to be the same order as the statistical error for  [fb] at the LHC.

## Iv Results and Conclusion

We have investigated the violating asymmetries that result in events at the LHC from new sources of violation associated with color-octet scalars. We have considered the effect of two new sources of violation: a phase in the couplings of the new resonances to top-quarks; and two phases in the quartic couplings of the scalar potential. We find that the former is responsible for much larger violating effects than the latter. We have shown that these models typically induce large raw asymmetries that can reach 12%.

Observation of the asymmetries is contingent to observation of the new resonance itself and we have presented a rough numerical simulation that illustrates this. An optimistic sensitivity for the asymmetry using only the dimuon channel could be at the LHC with an integrated luminosity of  [fb], when the events in full kinematical region are taken into account. In Table 3 we have shown several examples utilizing kinematical cuts with resulting asymmetries enhanced as large as a few percent by enhancing the scalar resoance contribution and avoiding cancellation between the two scalar contributions, which could be observed at the LHC.

We have presented our analysis for the dimuon channel as this is the cleanest one. However, our study can be easily extended to other top-quark decay channels to increase statistics. We have studied the asymmetries only in the top-quark pair production channel via one new resonance. Other channels, such as pair production, also exhibit violating asymmetries and may be preferable in scenarios where the production cross-section exceeds the single production cross-section.

###### Acknowledgements.
This work was partially supported by NSC, NCTS, SJTU 985 grant, and Excellent Research Projects of National Taiwan University (NTU-98R0526), and in part by DOE under contract number DE-FG02-01ER41155. G.V. thanks the National Taiwan University, Taipei, Taiwan for their hospitality.

## Appendix A CP Violation with a Higgs boson

We review the salient features of violation in production at the LHC induced by a new Higgs-boson as discussed in Ref. cphiggs (). The violation in a suitable extended Higgs sector manifests itself in the form of a neutral Higgs mass eigenstate that has both scalar and pseudo-scalar couplings to the top-quark. In general these couplings can be written as

 L = −mtvH¯t(A+iBγ5)t, (9)

where are real and corresponds to the standard model with one Higgs doublet. Multi-Higgs models achieve maximal violation when , and these couplings reach the Weinberg unitarity bound,  Weinberg:1990me ().

This Higgs-boson is produced at the LHC mostly via gluon fusion. Both the scalar and pseudoscalar cases have been considered in the literature before and these results at leading order can be summarized by the effective couplings

 L = [FaGμνGμν+Fb~GμνGμν]H (10)

The two form factors and can be found, for example in Ref. Spira:1995rr (). For the kinematic regime in which the Higgs boson is heavier that a pair they are given by

 Fa = (√2GF)1/2Aαs12π3Iq(z) Fb = −(√2GF)1/2Bαs8πzf(z) (11)

with , the functions and arise from the one-loop contribution of a top-quark loop and were given in Eq. (6).

The origin of the correlation, Eq. (7) in this model is a violating term in the invariant matrix element squared for of the form

 |M|2 = C1(s,t,u) ϵ(pt,p¯t,pμ+,pμ−)+⋯ (12)

The function can be written in a compact form reflecting two contributions: the channel Higgs amplitude squared; and the interference between the and channels ( in a color singlet state) and the -channel Higgs amplitude. They are given by

 C1(s,t,u) = 24KℓℓAB(|Fa|2+|Fb|2)s2(s−m2H)2+m2HΓ2Hm4tv2 + 8g2sKℓℓ(BRe(Fa)+ARe(Fb))s2(s−m2H)((s−m2H)2+m2HΓ2H)(s2−(t−u)2)m4tv. Kℓℓ ≡ g8(pb⋅pν)(p¯b⋅p¯ν)(πmtΓt)2(πMWΓW)2 (13) × δ(p2t−m2t)δ(p2¯t−m2t)δ(p2W+−M2W)δ(p2W−−M2W).

The delta functions in this expression reflect the use of the narrow-width approximation for all top-quark and -boson propagators. If the Higgs boson is also narrow, the expression simplifies further to

 C1(s,t,u) = 24KℓℓAB(|Fa|2+|Fb|2)s2m4tv2(πmHΓH) δ(s−m2H) (14)

## Appendix B CP Violation with color-octet scalars

In the color-octet model discussed in this paper, the violation in the process takes the same form as Eq. (12) with the form factor of Eq. (14) replaced by the sum of the contributions from the octet neutral scalars and . The overall color factor changes from to , and in the narrow width approximation we have

 C1(s,t,u) = 20Kℓℓ3s2m2t(aRbR(|FaR|2+|FbR|2)(πmRΓR) δ(s−m2R) (15) + aIbI(|FaI|2+|FbI|2)(πmIΓI) δ(s−m2I)) = 20Kℓℓ3s2m4tv2η2Usucu16((πDRmRΓR) δ(s−m2R)+(πDImIΓI) δ(s−m