Many models, in particular, the brane-world models with two branes, predict the existence of the scalar radion, whose mass can be somewhat smaller than those of all the Kaluza-Klein modes of the graviton and Standard Model (SM) particles. Due to its origin the radion interacts with the trace of the energy-momentum tensor of the SM. The fermion part of the radion interaction Lagrangian is different from that for the SM Higgs boson due to the presence of additional terms playing a role for off-shell fermions. It was shown previously  that for the case of the single radion and single Higgs boson production processes in association with an arbitrary number of SM gauge bosons all the contributions to the perturbative amplitudes appearing due to these additional terms were cancelled out making the processes similar up to a replacement of masses and overall coupling constants. For the case of the associated Higgs boson-radion and the Higgs boson pair production processes involving the SM gauge bosons the similarity property also takes place. However a detailed consideration shows that in this case it is not enough to replace simply the masses and the constants ( and ). One should also rescale the triple Higgs coupling by the factor .
Comparison of associated Higgs boson-radion and Higgs boson pair production processes
E. Boos, S. Keizerov, E. Rakhmetov, K. Svirina
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University
Leninskie Gory, 119991, Moscow, Russia
Faculty of Physics, Lomonosov Moscow State University, Leninskie Gory
119991, Moscow, Russia
One of the characteristic features of brane world models, in particular, of the Randall-Sundrum (RS) model  with a stabilization of the extra space dimension [3, 4], is the existence of the radion [3, 5, 6] – the lowest Kaluza-Klein (KK) mode of the five-dimensional scalar field appearing from the fluctuations of the metric component corresponding to the extra dimension. The radion might be significantly lighter than the other KK modes [7, 8, 9], and therefore it is of a special interest for collider phenomenology (see, e.g.,  - ).
The radion couples to the trace of the energy-momentum tensor of the SM, so the interaction Lagrangian has the following form 
where is a dimensional scale parameter, stands for the radion field and is the trace of the SM energy-momentum tensor. In most of the studies the latter is taken at the lowest order in the SM couplings and the fields are supposed to be on the mass shell. Here we consider the additional terms which come into play for the case of off-shell fermions, so the SM energy-momentum tensor has the following form :
where the first two terms correspond to the conformal anomaly of massless gluon and photon fields, , are the QCD and QED -functions respectively, , and are the SM Higgs, W- and Z-boson fields, is the Standard Model covariant derivative and the summation here is carried out over all the Standard Model fermions.
In the case of on-shell fermions the fermion part of the Lagrangian (1) is the same as for the Higgs boson (with the replacement ), but for off-shell fermions additional terms need to be taken into consideration. These terms in the Lagrangian give additional momentum depending contributions to the fermion-antifermion-radion interaction vertices and new fermion-antifermion-gauge boson-radion vertices which modify the radion production and decay processes making them potentially different from the same processes with the Higgs boson. In paper  it was shown that all the additional contributions as compared to the Higgs boson case are canceled out in the sum of amplitudes for the single radion production in association with an arbitrary number of any SM vector gauge bosons and there remain Higgs-like terms only. This property follows from the structure of any massive fermion current emitting the radion and gauge bosons both for the case of real and/or virtual emmited particles as well as for the case of boson and fermion loops .
In the present paper we consider the associated Higgs boson-radion production in comparison to the Higgs boson pair production processes as a continuation of our previous study . We demonstrate that in the case of the associated Higgs boson-radion production the similarity property is more involved. It is not enough to perform the replacement of two constants ( and ) for getting the amplitude involving the radion from the corresponding amplitude for the Higgs boson. It is explicitly demonstrated that the amplitude of the associated production of the Higgs boson and the radion and an arbitrary number of gauge bosons can be obtained from the corresponding amplitude involving the Higgs boson pair by the replacement of the Higgs and the radion masses, the constant and the Higgs vacuum expectation value and in addition by a rescaling the triple Higgs coupling by a certain factor. As in our previous study we do not consider the well-known differences between the Higgs and the radion processes caused by the conformal anomalies.
The investigation of double Higgs boson production is an important task for experimental measurements of the Higgs field potential profile. This problem is rather tricky even in the high luminosity mode of the LHC, being one of the key arguments for the ILC construction. However, if one of the multidimensional brane world scenarios occurs in nature, the presence of the radion can further complicate the problem of the Higgs potential research due to the similarity of the Higgs boson and the radion properties.
2 Associated Higgs boson-radion production in fermion-antifermion annihiation
Let us first consider the associated Higgs boson-radion production in fermion-antifermion annihilation (Fig.1).
The corresponding contributions to the amplitude simplified using the Dirac equation and the identity have the following form
For clarity one can modify (3) using the simple kinematics relation:
So (3) can be rewritten as follows
Now let us write down the contributions to the amplitude of the double Higgs boson production process (Fig.2) and compare the results:
Notice that in this case there is no contribution like (6).
Thus, the total amplitude of the double Higgs production process yields
Finally one can compare (11) and (15) and see the explicit cancellation of all the contributions that make the difference between the associated Higgs boson-radion and the double Higgs boson production.
i.e, the expressions for the total amplitudes (11) and (15) coincide up to the replacements of the masses and the denominators of the coupling constants and to the renormalization of the triple Higgs coupling by the factor , where
3 Associated Higgs boson-radion production in fusion
As another example let us compare two processes involving gluons: the associated Higgs boson-radion production () and the double Higgs boson production (), the corresponding diagrams are shown below.
One can see that the first four diagrams for the process of the radion production (Fig.3) are similar to those which appear in the SM (Fig.4). But the other three diagrams contain the Higgs boson-fermion-fermion-radion vertex which does not exist in the SM.
One can notice that all the contributions to the amplitudes of these processes have the following similar structure
for , where ;
for , where ;
and , , have the following form
The first line in (21) corresponds to the Higgs boson-fermion-fermion vertex in the process and the second line – to the Higgs boson-radion-fermion-fermion vertex in the process;
Let us notice that the radion-fermion-fermion vertex contains the inverse propagators and . In the expressions , and for the box diagrams this vertex is surrounded by the propagators and . This gives a reduction of a box diagram with the radion to a linear combination of two triangle diagrams and one box diagram with the vertex such as that of the Higgs boson as it is demonstrated in Fig.5.
This can be easily understood with the help of the tree-level illustration of a fermion current with emission of the radion, a gauge boson and the Higgs boson (Fig.6). The product of the radion-fermion-fermion vertex (the black point in Fig.6) and two propagators leads to three terms respectively:
a Higgs-like term with the vertex proportional to the fermion mass (the empty point in Fig.6), a term with the propagator being dropped out, i.e., with the radion and the Higgs boson emission from the same point, and a term with the propagator being dropped out, with the radion and the gauge boson emission from the same point. In this way we get a reduction of each box diagram with the radion to a sum of other contributions.
One can substitute (19)–(21) explicitly into the expressions and open the brackets in order to get a representation of , and as sums of other and contributions with the corresponding numerical factors
The term can be represented as a combination of and contributions as follows
by transforming the Higgs boson-radion vertex in the following way
and multiplying it by the propagator (22)
Finally the sum of all for the process can be written in terms of and the parameter again as it has been done in the previous example (see (16))
which looks very similar to the expression for the process
Thus, once again we see that the amplitudes for these two processes coincide up to the parameter and to the replacements of the masses and the denominators of the coupling constants.
One can notice that we could get the same amplitude if the model contains only the Higgs boson with renormalized parameters:
In other words, the radion contribution can mimic the deviation in the triple Higgs coupling. This fact must be taken into account in the investigation of coupling in the case of the radion detection. However this is valid only for the processes in the first order in the radion coupling constant, in more complicated cases the difference between the radion and the Higgs boson can be more significant and go beyond the simple replacement of the Higgs potential parameters.
4 Cancellations of additional to the Higgs-like contributions in associated Higgs boson-radion production
In paper  it was shown that all the additional contributions as compared to the Higgs boson case are cancelled out in the amplitudes of the single radion production processes. This property follows from the structure of any massive fermion current emitting the radion and an arbitrary number of any SM gauge bosons. Now let us show that the similar general property takes place in the case of the associated Higgs boson-radion production. Above we have already demonstrated the explicit cancellation by the example of the associated Higgs boson-radion production in fermion-antifermion annihilation and in gluon fusion. These were the processes with only two bosons ( or ) in the final state. For the general proof let us consider a fermion current (or a fermion loop) with the emission of an arbitrary number, say , of SM bosons (vector gauge – or Higgs – ) with all possible permutations. stands for the number of gauge bosons and – for the number of Higgs bosons, . Now add another Higgs boson (for ) or another radion (for ) to this current in all possible ways.
There are two possibilities of adding a Higgs boson : a) the one emitted from the fermion line, and b) the one emitted from the boson ( or ) line. For the radion there are the same options plus another one: c) the radion is emitted directly from the four-point vertex with or boson (Fig.7).
Let us first set the notations for all the vertices. For each vertex here all the lines are considered to be incoming. We will use Latin indices () for a parallel consideration of cases with gauge and Higgs bosons. In the case of gauge bosons the Latin indices take values of vector indices () and in the case of Higgs bosons they just reduce to the sign for Higgs boson vertices.
So, stands for the Lorentz part of the fermion-fermion-gauge boson vertex and for the fermion-fermion-Higgs boson vertex respectively
In the same manner we denote the vertex with two gauge bosons and the Higgs boson and the triple Higgs vertex
Now for the radion, the fermion-fermion-radion vertex is a function of the momenta of the incoming fermions, say and . It can be rewritten in terms of the inverse propagators  and takes the following form
The notation unifies the vertices for two gauge bosons and the radion interaction (excluding the anomalies) and for two Higgs bosons and the radion interaction respectively
The term stands for gauge boson-fermion-fermion-radion and Higgs boson-fermion-fermion-radion four-point vertices
Now one can write down the contributions to the amplitudes in the following form
The number runs from to (). In the case of real initial and final fermions one should take into account another amplitude
If the added particle is the Higgs boson, the amplitude takes the form
If the added particle is the radion, one can write the following amplitude
where we used (34) and the equation of motion ( is the outgoing momentum).
Now let us take any number , . In the case of adding a Higgs boson the sum of all amplitudes for the chosen is
For the case of adding a radion the sum has the following form