On a possible effective four-boson interaction and its implications at the upgraded LHC

We consider a possibility of a spontaneous generation of fourfold effective interactions of electroweak gauge bosons $W^a$ and $B$. The conditions for the spontaneous generation are shown to lead to a set of compensation equations for parameters of the interaction. In case of a realization of a non-trivial solution of the set, important electro-weak parameter $\sin^2\theta_W$ is defined. The existence of two non-trivial solutions is demonstrated, which provide a satisfactory value for the electromagnetic fine structure constant $\alpha$ at scale $M_Z$: $\alpha(M_Z) = 0.007756$. There is a solution with the high effective cut-off being close to the Planck mass by the order of magnitude. The most interesting solution corresponds to effective cut-off $\simeq 10^2\,TeV$. This solution gives quite definite prediction for non-perturbative effects in processes $p+p\to\bar t t (W^\pm, Z)$, which could be observed at the upgraded LHC.


Introduction
The Standard Model of particles' interactions is fairly considered as quite successful theory. It explains the phenomena in high energy physics experiments, and give us consecutive interpretation of their totality structure. But till now we cannot tell that SM is all-sufficient theory. First of all, by the reason of that it cannot explain at least lowest energy gravitational interaction in coupling with other ones and in the same way as others. Just it cause theorists to make attempts of different SM extensions building.
However, secondly, even if we shall leave such problems as scale hierarchy aside, we immediately face another, so to say, more prosaic problem. We cannot admit the Standard Model as accomplished theory simply by the reason of that there are too many external parameters we have to bring into it for provide it's expository power. The number of these parameters (such as coupling constants and matter fields' masses ratios) reaches up to 29. Of course, we can hope that determination of their value will be supplied with the mentioned wouldbe extension of the SM. But we see striking contrast between existing theory's insularity and it's wide experimental validation just in this insularity, on the one hand, -and further developed constructions' not fixed status with the absence of any data confirmation at present day and indeterminate perspectives in the foreseeable future, on the other hand.
Provided the wouldbe extended theory will be really able to reduce the totality of all data to one general principle, we have to make extremely great progress in experiment technology for making this theory as such validated as SM is validated just now. This problem prompts us to make efforts in other direction. Particularly, we can attempt to lead out necessary evaluations just within the existing theory structure. And those minimal extensions which we'll have to build anyway, must rather be non-structural and deal not with new fields and particles, but with new type of just known ones' effective interactions. And the searching region for such interactions may be, of course, indicated by the fact that SM and general quantum field theory conception are of the less successfulness in describing of lower energy processes. We can ask ourselves: may be, there is some deep correlation between the fact of failing of perturbation theory in such phenomena description and the presence of special low energy quantum effects, which are to be taken into account in some way? This situation patently corresponds with such effects as superconductivity and superfluidity, where classical local theory was powerless and consecutive analysis in terms of fundamental quantum theory equations was inaccessible also, but where the solution was found in the framework of non-perturbative contributions. We have no perturbative source of a "force", which bind electrons into Cooper's pair, but we can describe their behavior in this pairing.
The method of effective non-local interactions building, which we shall try to apply to the mentioned above problem in this work, was grown up from N.N. Bogolyubov's compensation conception 1,2 developed and successfully applied just in the superconductivity theory. Although in field theory it acquire some new specialities, stated above analogy seems to be quite encouraging for us. And on the other hand, we've just also quite successfully applied this approach to the range of particle physics low energy processes. The compensation approach was applied 3,4 to the problem of the spontaneous generation of effective interactions in quantum field theories. Most impressive effectiveness of the method was demonstrated in light meson physics, where spontaneously generated Nambu -Jona-Lasinio interaction 5,6 building allowed us 7 to predict main particles' properties with good precision using only fundamental QCD parameters without external parameters bringing in. Also applications to the composite Higgs particle problem 8 and to the spontaneous generation of the wouldbe anomalous three-boson interaction 9,10 , to be discussed below, can be mentioned. Our aim in this work is to demonstrate principal possibility of finding solution for fundamental SM parameters problem in terms of effective interactions. Correspondingly, we build a simple model, being guided by our previous experience in similar, but more advanced models. In case of success of this attempt it would be really important step, hopefully opening a road to the more sophisticated and more close to reality theorizing upon this important subject. But, in the same time, just with this simplified approach we shall present some predictions, suitable for upgraded LHC experiments studies.
In works 4,5,6,7,8,9,10,11 N.N. Bogoliubov compensation principle 1,2 was applied to studies of a spontaneous generation of effective non-local interactions in renormalizable gauge theories. The method and applications are also described in full in the book 12 .
In particular, papers 9,10 deal with an application of the approach to the electroweak interaction and a possibility of spontaneous generation of effective anomalous three-boson interaction of the form with uniquely defined form-factor F (p i ), which guarantees effective interaction (1) acting in a limited region of the momentum space. It was done in the framework of an approximate scheme, which accuracy was estimated to be ≃ (10 − 15)% 3 . Would-be existence of effective interaction (1) leads to important non-perturbative effects in the electro-weak interaction. It is usually called anomalous three-boson interaction and it is considered for long time on phenomenological grounds 13,14 . Our interaction constant G is connected with conventional definitions in the following way where g ≃ 0.65 is the electro-weak coupling. The best limitations for parameter λ read 15 where subscript denote a neutral boson being involved in the experimental definition of λ.
Here z 0 is a dimensionless parameter, which is connected with value of a boundary momentum, that is with effective cut-off Λ 0 according to the following definition 9,? Let us note, that the solution of the analogous compensation procedure in QCD correspond to g(z 0 ) = 3.817 11 , that gives satisfactory description of the lowmomentum behavior of the running strong coupling. It is instructive to present in Fig. 1 the behavior of form-factor F (p, −p, 0) in dependence on momentum p, where and F (z) = 0 for z > z 0 . As a rule the existence of a non-trivial solution of a compensation equation impose essential restrictions on parameters of a problem. Just the example of these restrictions is the definition of coupling constant g(z 0 ) in (4). It is advisable to consider other possibilities for spontaneous generation of effective interactions and to find out, which restrictions on physical parameters may be imposed by an existence of non-trivial solutions. In the present work we consider possibilities of definition of links between important physical parameters, first of all with relation to the fine structure constant α.

Weinberg mixing angle and the fine structure constant
Let us demonstrate a simple model, which illustrates how the well-known Weinberg mixing angle could be defined. Let us consider a possibility of a spontaneous generation of the following effective interaction of electroweak gauge bosons where we maintain the residual gauge invariance for the electromagnetic field. Here indices a, d correspond to charged W -s, that is they take values 1, 2, while index b corresponds to three components of W defined by the initial formulation of the electro-weak interaction. Definition (7) corresponds to convenient rule for Feynman rules for corresponding vertices, e.g. for the first term in (7) ı where components of W a have indices µ, ν and incoming momenta and indices (p, ρ) and (q, σ) refer to fields W b . Let us remind the relation, which connect fields W 0 , B with physical fields of the Z boson and of the photon Thus in terms of the physical states (W + W − Z A ) wouldbe effective interaction (7) has the following form Interactions of type (10) were earlier introduced on phenomenological grounds in works 16,17 and are subjects for experimental studies. Let us introduce an effective cut-off Λ and consider a possibility of a spontaneous generation of interaction (7). In doing this we proceed with the add-subtract procedure, which was used throughout works 3,4,5,6,7,8,9,10 . Now we start with usual form of the Lagrangian, which describes electro-weak gauge fields W a and B L = L 0 + L int ; and W a µν is the well-known non-linear Yang-Mills field of W -bosons. Then we perform the add-subtract procedure of expression (7) Now let us formulate compensation equations for wouldbe interaction (7). We are to demand, that considering the theory with Lagrangian L ′ 0 (13), all contributions to four-boson connected vertices, corresponding to interaction (7) are summed up to zero. That is the undesirable interaction part in the would-be free Lagrangian (13) is compensated. Then we are rested with interaction (7) only in the proper place (14) We would formulate these compensation equations using experience acquired in the course of application of the method to the Nambu -Jona-Lasinio interaction and the triple weak boson interaction (1). As is demonstrated in book 12 (Section 3.3), the first approximation for the problem of spontaneous generation of the Nambu -Jona-Lasinio interaction takes form-factor F (p) to be a step function Θ(Λ 2 − p 2 ) and only horizontal diagrams of the type presented in Fig. 2 are taken into account. The next approximation, described in detail in 4 and in 12 (Chapter 5) includes also vertical diagrams and form-factor F (p) is uniquely defined as a solution of the set of the compensation conditions in terms of standard Meijer functions. We have demonstrated, that the first approximation gives satisfactory results and the next ones serves for its specification. In the present work we just use the first approximation.
So let us introduce effective cut-off Λ, which is a subject for definition by solutions of the problem and use just a step function Θ(Λ 2 − p 2 ) for the effective form-factor.
In this way we have the following set of compensation equations, which corresponds to diagrams being presented in Fig. 2 Factor 2 in equations here corresponds to sum by weak isotopic index δ a a = 2, a = 1, 2.
We have the following solutions of set (15) in addition to the evident trivial one: Note, that absence of some x i in a solution means that this x i is arbitrary. Then, following the reasoning of the approach, we assume, that the Higgs scalar corresponds to a bound state consisting of a complete set of fundamental particles.
Here we study the wouldbe effective interaction (7, 10) of the electroweak bosons, so we take into account just these bosons as constituents of the Higgs scalar. There are two Bethe-Salpeter equations for this bound state, because constituents are either W a W a or Z Z. These equations are presented in the two rows of Fig. 3. In approximation of very large cut-off Λ these equations have the following form with notations of (15) Now we look for solutions of set (15,25,26,27) for variables x 2 , x 3 , x 4 , x 5 , Λ, which gives appropriate value for α(M Z ) = 0.007756, according to relation (27).
From definition of parameters in experimental work 19 and from (7) we have Results (31, 32) lead to the following prediction for parameters a W 0 , k W 0 for the two solutions Comparing the two last expressions and taking from experimental work 19 the following limitations There is also solution (30) with very large cut-off Λ. It is remarkable, that this solution correspond to the cut-off being of the order of magnitude of the Planck mass M P l = 1.22 × 10 16 T eV . Of course effective coupling constants G i in this case are extremely small. This possibility in case of its realization may serve as an explanation of hierarchy problem 21 . Indeed, with this solution the actual values for the masses of W, Z, H and value α(M Z ) may be reconciled with the effective cut-off being defined by the gravitational Planck mass. So the actual relation between the electro-weak scale and the gravity scale may acquire at least a qualitative interpretation.
We would draw attention to the low cut-off case also. Value of Λ (29) is close to boundary value of the momentum in the problem of a spontaneous generation of anomalous triple W interaction (1). Indeed, value of the electro-weak gauge constant g at this boundary (4) Then the following relation is to be fulfilled where α ew is defined in (27). Then this relation is an equation for parameter Λ. The solution of this equation gives Λ = 7.91413 · 10 2 T eV .
We see, that this value is of the same order of magnitude as value 5.2262 · 10 2 T eV in solution (29). Now we could formulate results in a rather different manner. We have two interesting values for possible cut-off Λ. The low value (44), which is compatible with previous results 9,10 by the order of magnitude, and the Planck mass. Let us consider set of equations (16,25,26) for these values of the cut-off. Earlier we have fixed actual value for electromagnetic constant α(M Z ) and calculated values for the cutoff (29,30). Now we fix Λ and calculate α(M Z ). In this way for values (44) and the Planck mass we obtain respectively

Experimental implications
Effective interaction (10) leads to effects in inclusive reactions Unfortunately with values for effective coupling constants G 2 , G 4 for preferable solution (29, 31) one could hardly hope for achieving the necessary precision even at the upgraded LHC. However there is a possibility for an enhancement of the effect in processes involving t-quarks due to large m t . Let us consider the wouldbe contribution of interaction (10,29) for √ s = 8 T eV the following estimate For the same process with the negative W we have For process p + p →t + t + Z we have the following contribution For process pp → tt W with both charge signs we have in new data results, which agree both the SM value ≃ 232 f b and the predicted one ≃ 363 f b . However one sees, that data of both collaborations slightly favor the last predicted value. Let us note, that additional contributions ∆σ(ttW, Z) increase with the energy and for the updated energy of the LHC √ s = 14 T eV they become ∆σt tZ (14 T eV ) = 578 f b.
b Results for √ s = 7 T eV see in 24 . Our predictions are to be compared with the SM calculations 25,26,27 in Table 1 c . We have already noted, that results for √ s = 8 T eV do not contradict the current data (54, 55, 56). As for √ s = 14 T eV , we see from the Table, that the most promising process for testing of the present results at the upgraded LHC is p + p →t t W ± . Indeed, the total additional contribution to production of charged W with top pair is around 1.6 pb, that more than twice exceeds the corresponding total SM value. Note, that we do not include in the Table process p + p →t t γ, because the effect here is significantly less pronounced. Namely, for √ s = 13 T eV we have σ SM = 1.744 ± 0.005 pb 27 , whereas the effect of interaction (47) is calculated to be ∆σ = 0.125 pb. Let us also note, that our estimations of the effect might have accuracy around 10% according to the experience of applications of the approach to several examples (see book 12 ).
Provided the prediction being confirmed, the first non-perturbative effect in the electroweak interaction would be established.

Conclusion
To conclude let us draw attention to the the results in view of the compensation approach to the problem of a spontaneous generation of an effective interaction. First of all, the results are obtained exclusively due to application of this approach. We would emphasize that the existence of a non-trivial solution of compensation conditions always impose strong restrictions on parameters of the problem. We see such restrictions in both problems of the spontaneous generation of the Nambu -Jona-Lasinio interaction 7 and the triple anomalous weak boson interaction 9,10 . In the present work such conditions for existence of interaction (7) are shown to define the Weinberg mixing angle, that leads to results (45) for the electromagnetic coupling constant.
It is also worth mentioning, that the wouldbe effective interaction under consideration leads to significant experimental effects being shown in Section 3, which likely may be either proved or disproved in forthcoming studies at the LHC.

Acknowledgements
The work is supported in part by grant NSh-7989.2016.2 of the President of Russian Federation.