Parton Distribution Functions and Tensor Gluons

We further consider a possibility that inside the proton and, more generally, inside the hadrons, there are additional partons - tensor gluons, which can carry a part of the proton momentum. The tensorgluons have zero electric charge, like gluons, but have a larger spin. Inside the proton a nonzero density of the tensorgluons can be generated by the emission of tensorgluons by gluons. The last mechanism is typical for non-Abelian tensor gauge theories. The process of gluon splitting suggests that part of the proton momentum that was carried by neutral partons is shared between vector and tensor gluons. We derive the regularised evolution equations for the parton distribution functions that take into account these new processes. In particular, this will allow to solve numerically the extended DIGLAP equations and to find out the ratio of densities between gluons and tensorgluons.


Introduction
It was predicted that the Bjorken scaling should be broken by logarithms of a transverse momentum Q 2 and that these deviations from the scaling law can be computed for the deep inelastic structure functions [1,2,3,4,5,6,7,8,9,10,11,12]. In the leading logarithmic approximation the results can be formulated in the parton language [13] by assigning the well determined Q 2 dependence to the parton densities [1,2,14,15,3].
In this article we shall further consider a possibility that inside the proton and, more generally, inside the hadrons there are additional partons -tensorgluons, which can carry a part of the proton momentum [16,17,18,19]. It was proposed in [22] that a possible emission of tensorgluons inside proton will produce a tensorgluon "cloud" of neutral tensorgluon partons in addition to the quark and gluon "clouds". Tensorgluons have zero electric charge, like gluons, but have a larger spin. Inside the proton a nonzero density of the tensorgluons can be generated by the emission of tensorgluons by gluons [16,17,18,19].
The last mechanism is typical for non-Abelian tensor gauge theories, in which there exists a gluon-tensor-tensor vertex of order g. The number of gluons changes not only because a quark may radiate a gluon or because a gluon may split into a quark-antiquark pair or into two gluons [14,15,3], but also because a gluon can split into two tensorgluons [16,17,18,19,20,21,23]. The process of gluon splitting into tensorgluons suggests that part of the proton momentum which was carried by neutral partons is shared between vector and tensorgluons.
The proposed model was formulated in terms of a field theory Lagrangian, which describes the interaction of the gluons with their massless partners of higher spin, the tensorgluons [16,17,18,19]. The characteristic property of the model is that all interaction vertices between gluons and tensorgluons have dimensionless coupling constants in four-dimensional space-time. That is, the cubic interaction vertices have only first order derivatives and the quartic vertices have no derivatives at all. These are familiar properties of the standard Yang-Mills field theory. In order to understand the physical properties of the model it was important to study the tree-level scattering amplitudes. A very powerful spinor helicity technique [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] was used to calculate high-order tree-level diagrams with the participation of tensorgluons in [20].
Here we shall further develop the regularisation technique proposed earlier for the splitting amplitudes. This will allow to solve numerically the generalised DGLAP evolution equations for the parton distribution functions that take into account the processes of emission of tensorgluons by gluons. The present paper is organised as follows. In section two the basic formulae for scattering amplitude and splitting functions are recalled, definitions and notations are specified, the details of the regularisation scheme are presented. In section three we derive the regularised evolution equations for the parton distribution functions that take into account the creation of tensorgluons. Section four contains concluding remarks and summarises the physical consequences of the tensorgluons hypothesis.

Splitting Functions
It was proposed in [22] that a possible emission of tensorgluons inside proton will produce a tensorgluon "cloud" of neutral tensorgluon partons in addition to the quark and gluon "clouds". Our goal is to specify the regularisation of the generalised DGLAP equations [3,5,6,7,8,9,10,4] that was introduced in [22].
In the generalized Yang-Mills theory [16,17,18,19] all interaction vertices between gluons and tensorgluons have dimensionless coupling constant. Using these vertices one can compute the scattering amplitudes of gluons and tensorgluons. The color-ordered scattering amplitudes involving two tensorgluons of helicities h = ±s, one negative helicity gluon and (n−3) gluons of positive helicity were found in [20]. These scattering amplitudes have been used to extract splitting amplitudes of gluons and tensorgluons [21]. Since the collinear limits of the scattering amplitudes are responsible for parton evolution, one can extract from these expressions the splitting probabilities for tensorgluons [22]: The invariant operator C 2 for the representation R is defined by the equation t a t a = C 2 (R) 1 and tr(t a t b ) = T (R)δ ab . These functions satisfy the relations The kernel P T G (z) has a meaning of variation per unit time of the probability density of finding a tensorgluon inside the gluon, P GT (z) -of finding gluon inside the tensorgluon and P T T (z) -of finding tensorgluon inside the tensorgluon. One should define the regularisation procedure for the singular factors (1 − z) −2s+1 and z −2s+1 reinterpreting them as the . The regularisation has been defined in the following way [22]: where f (z) is any test function that is sufficiently regular at the end points and, as one can see, the defined substraction guarantees the convergence of the integrals. Using the same arguments as in the standard case [3] we should add the delta function terms into the definition of the diagonal kernels so that they will completely determine the behaviour of P qq (z) , P GG (z) and P T T (z) functions.
One should add δ(z − 1) to the diagonal kernels P qq (z), P GG (z) and P T T (z) with the coefficients that have been determined by using the momentum sum rule: Thus we completed the definition of the kernels appearing in the evolution equations (3.6).
For completeness we shall present also quark and gluon splitting functions [3]: 2N , T (R) = 1 2 for the SU(N) groups. At the end of this section we shall discuss, what type of processes could also be included into the evolution equations (3.6). In (3.6) we ignore contribution of the highspin fermionsq i of spin s + 1/2, which are the partners of the standard quarks [16,17,18,19], supposing that they are even heavier than the top quark. That is, all kernels P qq , P Gq , P Tq , Pqq, Pq q , Pq G , Pq T and P T q with the emission ofq i are taken to be zero. These terms can be included in the case of very high energies, but at modern energies it seems safe to ignore these contributions. In the evolution equation for tensorgluons in (3.6) one could also include the kernels P T T ′ that describe the emission of tensorgluons by tensorgluons, the T T ′ T ′′ vertex [16,17,18,19]. That also can be done, and the number of evolution equations in that case will tend to infinity. In this article we shall limit ourselves by considering only emissions that always involve the standard gluons and spin-2 tensors ignoring infinite "stairs" of transitions between tensorgluons.

Regularisation of Generalised DGLAP Equations
The deep inelastic structure functions can be expressed in terms of parton densities [3,5,6,7,8,9,10,4]. If q i (x, Q 2 ) is the density of quarks of type i (summed over colors) inside a nucleon target with fraction x of the proton longitudinal momentum in the infinite momentum frame, then the unpolarised structure functions can be represented in the following form: The Q 2 dependence of the parton densities is described by the integro-differential equations for quark q i (x, t) and gluon densities G(x, t), where t = ln(Q 2 /Q 2 0 ) [3,5,6,7,8,9,10,4]. If there is an additional emission of tensorgluons in the proton, then one should introduce the corresponding density T (x, t) of tensorgluons and the integro-differential equations that describe the Q 2 dependence of parton densities in this general case has the following form [22]: The α(t) is the running coupling constant (α = g 2 /4π) is the one-loop Callan-Simanzik coefficient. In particular, the presence of the spin-two tensorgluons in the proton will give The density of tensorgluons T (x, t) changes when a gluon splits into two tensorgluons or when a tensorgluon radiates a gluon. This evolution is described by the last equation (3.6).
The tensorgluon kernels (2.1) are singular at the boundary values similar to the case of the standard kernels (2.5), though the singularities are of higher order compared to the standard case. The '+' prescription in is defined as and so Considering the splitting probabilities for spin two tensorgluons we have to define '+++' prescription as 13) and so (3.14) For spin 2 we shall have and the regularsation of these kernels can be represented in the following form:

Conclusion
Let us summarise the physical consequences of the tensorgluons hypothesis. Among all parton distributions, the gluon density G(x, t) is one of the least constrained functions since it does not couple directly to the photon in deep-inelastic scattering measurements of the proton F 2 structure function. Therefore it is only indirectly constrained by scaling violations and by the momentum sum rule which resulted in the fact that only half of the proton momentum is carried by charged constituents -the quarks -and that the other part is ascribed to the neutral constituents. As it was suggested in [22], the process of gluon splitting leads to the emission of tensorgluons and therefore a part of the proton momentum that is carried by the neutral constituents here is shared between gluons and tensorgluons. The density of neutral partons in the proton is therefore given by the sum of two functions: G(x, t) + T (x, t), where T (x, t) is the density of the tensorgluons.
To disentangle these contributions and to decide which piece of the neutral partons is the contribution of gluons and which one is of the tensorgluons one should measure the helicities of the neutral components, which seems to be a difficult task. The other test of the proposed model will be the consistency of the mild Q 2 behaviour of the moments of the structure functions with the experimental data.
In supersymmetric extensions of the Standard Model [46,47] the gluons and quarks have natural partners -gluinos of spin s=1/2 and squarks of spin s=0. If the gluinos appear as elementary constituents of the hadrons, then the theory predicts the existence of new hadronic states, the R-hadrons [48,49]. These new hadronic states can be produced in ordinary hadronic collisions, and they decay into ordinary hadrons with the radiation of massless photino -the massless partner of the photon, which takes out a conserved Rparity. Depending on the model the gluinos may be massless or massive, depending on the remaining unbroken symmetries and the representation content of the theory. The existing experimental data give evidence that most probably they have to be very heavy [50,51].
It seems that phenomenological limitation on the existence of the R-hadrons is much stronger that the limitation on the existence of the tensorgluons inside the ordinary hadrons.
This is because gluinos change the statistics of the ordinary hadrons: the proton has to have a partner, the R-proton, which is a boson and is indeed a new hadron.
The existence of tensorgluon partons inside the proton does not predict a new hadronic state, a proton remains a proton. The tensorgluons will alternate the parton distribution functions of a proton. The question is to which extent the tensorgluons will change the parton distribution functions. The regularisation of the splitting amplitudes developed above (3.16)-(3.18) will allow a numerical solution of the generalised DGLAP evolution equations (3.6) for the parton distribution functions that takes into account the processes of emission of tensorgluons by gluons. The integration can be performed using the algorithms developed in [42,43,44,45] and will allow to find out the ratio of densities between gluons and tensorgluons.