# Top-pions and single top production at HERA and THERA

###### Abstract

We study single top quark production at the HERA and THERA colliders as coming from the FCNC vertices tc and tcZ that appear at one loop level in Topcolor-assisted technicolor models (TC2). In contrast with previous expectations, we find that the production cross section is of order pb for HERA and pb for THERA (even lower that the SM prediction). Therefore, none of the two colliders are capable to probe the pseudoscalar top-pion or the scalar top-higgs predicted in TC2 models via single top production.

###### pacs:

14.80.-j;14.65.Ha;12.60.Nz1. Introduction

There has been an increased interest in studying forbidden or highly suppresed processes as they become an ideal site to search for new physics lying beyond the Standard Model (SM). In particular, it is expected that if there is any new physics associated with the mas generation mechanism, it may be more apparent in the top quark interactions than in the all the lighter fermionstopreview . Along this line, it has been suggested that t-channel single top production could be rather sensitive to the non-SM flavor changing neutral coupling (FCNC) vertices (Z)belyaev ; peccei . While in general, vector and scalar FCNC vertices are very suppresed by the GIM mechanism in the SM, some extensions of the SM can generate some of them at the tree levelrosado . This is the case of top-color assisted technicolor (TC2), where a composite scalar (and a pseudo-scalar) appears as a result of a strong interaction and can give rise to sizeable FCNC () couplingshill . Then, at the one loop level, these couplings can give rise to effective couplings ().

In particular, the pseudo-scalar flavor diagonal and the flavor changing couplings that are relevant for the TC2 prediction of single top quark production arehill ; yuan :

(1) |

where GeV is the top-pion decay constant, , and the terms come from the diagonalization of the up and down quark matrices. Their numerical values are and , with a free parameter. There are also couplings involving the scalar top-higgs , and their effect on the production of single top at HERA is similar to that of the top-pion. In order to simplify our discussion we will not consider them in this work. Notice that in this model the flavor changing scalar coupling only involves the right handed component of the charm quarkyuan .

Recently, it was suggested that the neutral pseudo-scalar top-pion (or the neutral scalar top-higgs ) contribution to the single top quark production cross section could be significant, of order 1-6 pb, at the HERA and THERA colliders for a top-pion mass between 200 and 400 GeVyue .

In the present letter, we re-examine this possibility and we find that the cross section due to the top-pion is rather of order much less than 1 fb. In fact, the contribution of the top-pion (or the top-higgs) is much smaller the SM contribution which is of order less than 1 fb and is not observable at HERA baur ; fritz . Unfortunately, none of these two colliders HERA and THERA are thus capable to probe the TC2 scalar particles through single top production.

2. The FCNC single top production at HERA

We consider the tree level FCNC process

(2) |

where momentum conservation dictates , and the Mandelstam variables are defined as , and . In this work, we take the top quark mass GeV and the charm quark as massless. We also neglect terms proportional to the electron mass in the differential cross section. However, we will use MeV for the upper and lower limits of the Mandelstam variable t in the phase space integral. This is done in order to avoid the divergence that appears when the exchanged photon goes on-shell ()fritz . The lower and upper limits for are . is the C.M. energy of the collider: GeV for HERA and TeV for THERA.

Through a series of loop diagrams the FCNC tc vertex gives rise to effective tcV vertices, with V=,Z. These diagrams have been calculated and their results given in terms of the following general effective tcV couplingsyue :

(3) |

(In the notation of yue and .) The coefficients are given in Ref.yue . We have computed them and have found agreement, except for the fact that the left handed component of the c quark cannot participate in the tc vertex. The reason for this is that it is only the right handed component of the c quark that appears in the tc vertex, which gives rise to tc. Nevertheless, this correction does not change in any way the fact that the cross section for single top production turns out to be negligible for our TC2 model.

3. General tc vertex with current conservation

For this t-channel process it is well known that HERA will get the bulk of the production cross section from the region where the momentum transfer is very small and the exchanged photon is quasi-realfritz . The contribution from Z exchange is several orders of magnitude smaller. In fact, the experimental situation is such that the scattered positron usually escapes through the rear beam pipe, and events with greater than 1GeV are rejectedzeus . The reason for this is because the photon propagator tends to infinity in the limit , and the phase space integration must be done carefuly. In particular, the effective couplings of Eq. (3) are not the most convenient to work with. It turns out that the coefficient is different from zero in the on-shell limit (see Eq. (5) below). This means that if taken isolated (disregarding the interference with and ) its contribution to the cross section diverges very rapidly. Indeed, the contribution of the other coefficients and will also diverge, but in such a way as to cancel out the contribution from . Unlike the strategy followed by Ref. yue , we will not use the coupling in the form of Eq. (3). Instead, we will perform a Gordon decomposition to transform it into a more suitable form; one that has no such big cancellations.

With the bulk of the contribution coming from a quasi real photon we should bear in mind that electromagnetic vector current conservation implies that the coupling must vanish when the photon goes on shelldeshpande . In order to make this apparent, we take the coupling of Eq. (3) and change it from the (, , ) basis to the (, , ) basisdeshpande . We re-write Eq. (3) through a Gordon decomposition:

(4) |

where is the momentum of the exchange photon (or Z), and with the new coefficients given by

Based on the results of Ref.yue we can make a numerical calculation of the coefficients defined above. First, let us see the values of for a top-pion mass GeV and for :

(5) | |||||

Here, we have separated the factor that gives the dependence on . The cancellation in Eq. (4) can be easily verified. For small but not zero we have that varies linearly with . Nevertheless, all the other form factors remain nearly constant for small and even greater (than 1 GeV) values of . As mentioned before, the fact that is directly proportional to is expected from electromagnetic current conservationdeshpande . To illustrate this point and to see that the dominant contribuion comes from the magnetic transition coefficient , let us now note the numerical values of , and for a top-pion mass GeV and for GeV:

(6) | |||||

Because is proportional to its contribution to the cross section will not diverge in the limit. This is not quite the case for and ; for them there is still a divergent behaviour, although this time it is only a logarithmic one. On the other hand, the coupling contribution is proportional to the electron mass, and it is therefore much smaller than that of .

The coefficients for tcZ are of similar values, even though their contribution is very small we have included them in our numerical results shown in Figs. (1) and (2).

4. Discussion and results

The differential cross section contribution from photon exchange, disregarding terms proportional to the electron mass, is given bykidonakis :

(7) | |||||

As mentioned in Ref.kidonakis the bulk of the cross section is given by the photon exchange diagram (). This is because of its logarithmic divergent behavior that is taken under control with the electron mass. However, our numerical results include the (negligible) contribution from Z exchange.

The total cross section is given by:

with the charm quark PDF and the minimum value of . The C.M. energy of the collider is GeV for HERA and TeV for THERA.

The limits for are:

(8) | |||||

We have evaluated the total cross section with the CTEQ6M PDFcteq , running at a fixed scale . We have also run with the energy scale and have seen no significant change.

In Figure (1) we show the production cross section for the process coming from the FCNC vertices tc and tcZ. For a GeV and we obtain a tiny cross section pb, which is even smaller than the SM contribution to single top production. As mentioned before, the dominant contribution comes from the photon diagram so that is proportional to . We can compare with Ref kidonakis , when they take for at HERA they obtain a very small cross section of order pb including NNLO QCD corrections. They instead prefer to discuss because of the small sea charm quark pdf density. In our model, the tu coupling is very small and it will not generate a sizeable tu effective vertex. Also, notice that in our notation we have . In Fig (1) we see that for and GeV the cross section is pb. This is four orders of magnitude smaller than the cross section of Ref. kidonakis because in our case which corresponds to two orders of magnitude smaller. This is only a rough comparison, as we do not include NNLO QCD corrections.

In figure (2) we show the production cross section for the THERA collider. It is 4-5 orders of magnitude greater than the one obtained at HERA, but still too small for an experiment with estimated luminosity of 500 pb.

As a final remark, let us compare with the results of Ref. yue . There, the cross section values are 6 orders of magnitude greater. As mentioned before, the the cross section in Ref. yue was calculated from an effective coupling in the form of Eq. (3). This means that a very large cancellation must take place in the small region, and any standard numerical integration may not be able to handle this situation properly. Rather, the numerical integration is likely to become unstable for small . Perhaps this is the reason why a (very large) cut in (GeV) had to be applied in their calculation. In this work, we have instead used the effective coupling in the form of Eq. (4). This new coupling does not induce the large cancellations of the previous one. The numerical integration is stable, even in the low region. We use the limits given in Eq. (8), for which can reach small values of order GeV.

In conclusion, we have shown that neither HERA nor THERA can probe the effects of a top-pion (or top-higgs) of TC2 models via single top quark production.

###### Acknowledgements.

We want to thank C.-P. Yuan for helpful discussions. We thank Conacyt for support.## References

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