A
bstract
Starting from the ACV approach to transplanckian scattering, we present a development of the reduced-action model in which the (improved) eikonal representation is able to describe ...particles’ motion at large scattering angle and, furthermore, UV-safe (regular) rescattering solutions are found and incorporated in the metric. The resulting particles’ shock-waves undergo calculable trajectory shifts and time delays during the scattering process — which turns out to be consistently described by both action and metric, up to relative order
R
2
/b
2
in the gravitational radius over impact parameter expansion. Some suggestions about the role and the (re)scattering properties of irregular solutions — not fully investigated here — are also presented.
It is widely believed that at small x, the BFKL resummed gluon splitting function should grow as a power of 1/x. But in several recent calculations it has been found to decrease for moderately ...small-x before eventually rising. We show that this ‘dip’ structure is a rigorous feature of the Pgg splitting function for sufficiently small αs, the minimum occurring formally at log(1/x)∼1/αs. We calculate the properties of the dip, including corrections of relative order αs, and discuss how this expansion in powers of αs, which is poorly convergent, can be qualitatively matched to the fully resummed result of a recent calculation, for realistic values of αs. Finally, we note that the dip position, as a function of αs, provides a lower bound in x below which the NNLO fixed-order expansion of the splitting function breaks down and the resummation of small-x terms is mandatory.
On the basis of a renormalization group analysis of the kernel and of the solutions of the BFKL equation with subleading corrections, we propose and calculate a novel expansion of a properly defined ...effective eigenvalue function. We argue that in this formulation the collinear properties of the kernel are taken into account to all orders, and that the ensuing next-to-leading truncation provides a much more stable estimate of hard Pomeron and of resummed anomalous dimensions.
We propose a matrix evolution equation in (x,kt)-space for flavour singlet, unintegrated quark and gluon densities, which generalizes DGLAP and BFKL equations in the relevant limits. The matrix ...evolution kernel is constructed so as to satisfy renormalization group constraints in both the ordered and antiordered regions of exchanged momenta kt, and incorporates the known NLO anomalous dimensions in the MSbar scheme as well as the NLx BFKL kernel. We provide a hard Pomeron exponent and effective eigenvalue functions that include the n_f-dependence, and give also the matrix of resummed DGLAP splitting functions. The results connect smoothly with those of the single-channel approach. The novel P_{qa} splitting functions show resummation effects delayed down to x=0.0001, while both P_{ga} entries show a shallow dip around x=0.001, similarly to the gluon-gluon single-channel results. We remark that the matrix formulation poses further constraints on the consistency of a BFKL framework with the MSbar scheme, which are satisfied at NLO, but marginally violated by small n_f/N_c^2-suppressed terms at NNLO.
Based on the ACV approach to transplanckian energies, the reduced-action model for the gravitational
S
-matrix predicts a critical impact parameter
such that
S
-matrix unitarity is satisfied in the ...perturbative region
b
>
b
c
, while it is exponentially suppressed with respect to
s
in the region
b
<
b
c
that we think corresponds to gravitational collapse. Here we definitely confirm this statement by a detailed analysis of both the critical region
b
≃
b
c
and of further possible contributions due to quantum transitions for
b
<
b
c
. We point out, however, that the subcritical unitarity suppression is basically due to the boundary condition which insures that the solutions of the model be ultraviolet-safe. As an alternative, relaxing such condition leads to solutions which carry short-distance singularities presumably regularized by the string. We suggest that through such solutions — depending on the detailed dynamics at the string scale — the lost probability may be recovered.
We present a full study of the 3-body problem in gravity in a flat (2+1)-dimensional spacetime, and in the nonrelativistic limit of small velocities. We provide an explicit form of the ...Arnowitt?Deser?Misner Hamiltonian in a regular coordinate system and we set up all the ingredients for canonical quantization. We emphasize the r?le of a U(2) symmetry under which the Hamiltonian is invariant and which should generalize to a U(N ? 1) symmetry for N bodies. This symmetry seems to stem from a braid group structure in the operations of looping of particles around each other, and guarantees the single valuedness of the Hamiltonian. Its r?le for the construction of single-valued energy eigenfunctions is also discussed.
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