Final Report Summary - LOWXGLUE (Low-x gluon distribution from the discretised BFKL equation)
The objective of the project is to investigate the properties of gluon density with the BFKL equation. The BFKL equation describes the dynamics of gluon-gluon interactions in high energy particle reactions. Such self-interacting gluonic system was observed at HERA in the inclusive and diffractive virtual photon-proton scattering at the cms energies between 30 and 240 GeV.
Gluons are the carriers of strong interactions. Unlike photons, which are the carriers of electromagnetic interactions, these gluons interact with each other as well as with quarks. At short distances, the quantum chromodynamic (QCD) Lagragian describes the interactions of gluons with quarks and gluons among themselves. The BFKL equation collects the effects of gluon self-interactions (including the interactions with quarks) and provides, today, the most complete description of these effects at short distances.
In the first year of the grant, 2011, the participants have concentrated on the evaluation of an unexpected property of the discrete solution of the BFKL formalism - namely that the solution to the BFKL equation connects the low and extreme high-energy behaviour in a way which contradicts the so called decoupling theorem. The decoupling theorem states that the low- and high-energy behaviour decouple from each other at experimental scales which are below the scales characteristic for new physics. The multigluon interactions, which are essential for the BFKL dynamics, overcome this limitation due to emergence of discrete eigenstates which are highly non-local in transverse momentum space.
The question investigated in 2011 was whether the eigenstates which connect the low and extremely high energy behaviour substantially contribute to the description of data and whether their contribution could not be absorbed into the some free parameters of the theory. These parameters are the phases of the infrared boundary, which crucially determines the data description and are today still only poorly known. To answer this question the numerical procedures, which were developed for the 2010 evaluation, were substantially improved and the data was reanalysed taking into account the effects of supersymmetry (SUSY). The result indicated that, within the model of the infrared boundary condition developed for the 2010 evaluation, the data show a clear sensitivity to SUSY with the threshold at a scale of 10 TeV, i.e. well above the HERA energies of up to about 200 GeV in spite of the purely known infrared boundary condition.
This connection between the low and extremely high energy behaviour of the scattering amplitudes provided by the green function solution of the BFKL equation was further investigated in 2012 in order to understand exactly where the decoupling theorem could be violated and also to understand how the BFKL formalism matches the DGLAP formalism which involves only renormalisation group improved perturbation theory in which no such violation of the decoupling theorem is predicted.
The essence to the resolution of this apparent puzzle is the fact that the discrete BFKL equation with running coupling is solved in terms a universal green function, which does not require any cut-off on the BFKL integral. The solution assumes, in a very general way, that the non-perturbative infrared region of QCD imposes a certain phase on the oscillatory parts of the eigenfunctions at some small transverse momentum. This treatment of the infrared boundary leads to a discrete set of eigenvalues, ?, of the BFKL kernel, since only certain values permit the construction of eigenfunctions which simultaneously obey these phase conditions at low transverse momentum and the large transverse momentum boundary conditions imposed by the asymptotic freedom.
This is in contrast to the 'usual' treatment - see e.g. the paper Phys. Rev. D 49 (1994) 4402, in which a lower transverse momentum cut-off is imposed on the amplitude, i.e. the amplitude is assumed to vanish below the cut-off.
In our approach, the amplitudes are particularly sensitive to the exact values of the discrete ? which are related to the non-perturbative phases, ?, at the cut-off. These phases are determined, in turn, by the gluon-gluon interactions of the non-pertubative QCD, which lead to rich structures below this cut-off.
A rigid cut-off (either UV or IR) destroys the scale invariance of the BFKL kernel - and hence the validity of scale covariant solutions. As pointed out in the paper, the imposition of such cut-offs has no effect on the position of the leading singularity but does affect the pre-factor, which is controlled by the form of this singularity. In terms of the discrete BFKL pomeron, this means that the position of the leading pole is unaffected, but the sub-leading poles are indeed sensitive to an infrared cut-off. We found that the sub-leading singularities are essential in order to obtain a good fit to HERA data. In the green function solution, the scale and conformal invariance (which is central to the BFKL formalism for fixed coupling) is broken in a controlled way, namely only through the running of the coupling. This running, however, generates a large transverse momentum boundary which is, even for relatively large sub-leading eigenvalues like ? < 0.1 beyond all the scales of BSM physics which are presently under discussion. It was important to understand how the running of the coupling constant can have such far-reaching consequences. For this purpose we derive analytically the main properties of the discrete pomeron solution using the LO BFKL equation. This derivation provides a qualitative physical explanation of the mechanism by which the BSM effects modify the discrete pomeron structures and lead to a genuine change of the eigenvalues and eigenfunctions. It also elucidates the role of the infrared phases which define the boundary condition and which can be indirectly determined from data. In LO BFKL, it was shown analytically that the BSM effects cannot be absorbed into the (at present) unknown infrared phases.
Thus our main conclusion is that within a treatment of the infrared behaviour of QCD in terms of specific phases for the oscillation of the amplitude, which is considerably preferably to the simple imposition of an infrared cut-off, the locations of the discrete Regge poles and their residues are determined by both infrared and ultraviolet boundary conditions and are sensitive to the details of the running of the coupling. This can be exploited to use the quality of the fit to HERA data as an indicator of the presence (or otherwise) of new physics beyond the Standard Model.
Gluons are the carriers of strong interactions. Unlike photons, which are the carriers of electromagnetic interactions, these gluons interact with each other as well as with quarks. At short distances, the quantum chromodynamic (QCD) Lagragian describes the interactions of gluons with quarks and gluons among themselves. The BFKL equation collects the effects of gluon self-interactions (including the interactions with quarks) and provides, today, the most complete description of these effects at short distances.
In the first year of the grant, 2011, the participants have concentrated on the evaluation of an unexpected property of the discrete solution of the BFKL formalism - namely that the solution to the BFKL equation connects the low and extreme high-energy behaviour in a way which contradicts the so called decoupling theorem. The decoupling theorem states that the low- and high-energy behaviour decouple from each other at experimental scales which are below the scales characteristic for new physics. The multigluon interactions, which are essential for the BFKL dynamics, overcome this limitation due to emergence of discrete eigenstates which are highly non-local in transverse momentum space.
The question investigated in 2011 was whether the eigenstates which connect the low and extremely high energy behaviour substantially contribute to the description of data and whether their contribution could not be absorbed into the some free parameters of the theory. These parameters are the phases of the infrared boundary, which crucially determines the data description and are today still only poorly known. To answer this question the numerical procedures, which were developed for the 2010 evaluation, were substantially improved and the data was reanalysed taking into account the effects of supersymmetry (SUSY). The result indicated that, within the model of the infrared boundary condition developed for the 2010 evaluation, the data show a clear sensitivity to SUSY with the threshold at a scale of 10 TeV, i.e. well above the HERA energies of up to about 200 GeV in spite of the purely known infrared boundary condition.
This connection between the low and extremely high energy behaviour of the scattering amplitudes provided by the green function solution of the BFKL equation was further investigated in 2012 in order to understand exactly where the decoupling theorem could be violated and also to understand how the BFKL formalism matches the DGLAP formalism which involves only renormalisation group improved perturbation theory in which no such violation of the decoupling theorem is predicted.
The essence to the resolution of this apparent puzzle is the fact that the discrete BFKL equation with running coupling is solved in terms a universal green function, which does not require any cut-off on the BFKL integral. The solution assumes, in a very general way, that the non-perturbative infrared region of QCD imposes a certain phase on the oscillatory parts of the eigenfunctions at some small transverse momentum. This treatment of the infrared boundary leads to a discrete set of eigenvalues, ?, of the BFKL kernel, since only certain values permit the construction of eigenfunctions which simultaneously obey these phase conditions at low transverse momentum and the large transverse momentum boundary conditions imposed by the asymptotic freedom.
This is in contrast to the 'usual' treatment - see e.g. the paper Phys. Rev. D 49 (1994) 4402, in which a lower transverse momentum cut-off is imposed on the amplitude, i.e. the amplitude is assumed to vanish below the cut-off.
In our approach, the amplitudes are particularly sensitive to the exact values of the discrete ? which are related to the non-perturbative phases, ?, at the cut-off. These phases are determined, in turn, by the gluon-gluon interactions of the non-pertubative QCD, which lead to rich structures below this cut-off.
A rigid cut-off (either UV or IR) destroys the scale invariance of the BFKL kernel - and hence the validity of scale covariant solutions. As pointed out in the paper, the imposition of such cut-offs has no effect on the position of the leading singularity but does affect the pre-factor, which is controlled by the form of this singularity. In terms of the discrete BFKL pomeron, this means that the position of the leading pole is unaffected, but the sub-leading poles are indeed sensitive to an infrared cut-off. We found that the sub-leading singularities are essential in order to obtain a good fit to HERA data. In the green function solution, the scale and conformal invariance (which is central to the BFKL formalism for fixed coupling) is broken in a controlled way, namely only through the running of the coupling. This running, however, generates a large transverse momentum boundary which is, even for relatively large sub-leading eigenvalues like ? < 0.1 beyond all the scales of BSM physics which are presently under discussion. It was important to understand how the running of the coupling constant can have such far-reaching consequences. For this purpose we derive analytically the main properties of the discrete pomeron solution using the LO BFKL equation. This derivation provides a qualitative physical explanation of the mechanism by which the BSM effects modify the discrete pomeron structures and lead to a genuine change of the eigenvalues and eigenfunctions. It also elucidates the role of the infrared phases which define the boundary condition and which can be indirectly determined from data. In LO BFKL, it was shown analytically that the BSM effects cannot be absorbed into the (at present) unknown infrared phases.
Thus our main conclusion is that within a treatment of the infrared behaviour of QCD in terms of specific phases for the oscillation of the amplitude, which is considerably preferably to the simple imposition of an infrared cut-off, the locations of the discrete Regge poles and their residues are determined by both infrared and ultraviolet boundary conditions and are sensitive to the details of the running of the coupling. This can be exploited to use the quality of the fit to HERA data as an indicator of the presence (or otherwise) of new physics beyond the Standard Model.