38 research outputs found
An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM
In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry
constrains multi-loop n-edged Wilson loops to be basically given in terms of
the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a
function of conformally invariant cross ratios. We identify a class of
kinematics for which the Wilson loop exhibits exact Regge factorisation and
which leave invariant the analytic form of the multi-loop n-edged Wilson loop.
In those kinematics, the analytic result for the Wilson loop is the same as in
general kinematics, although the computation is remarkably simplified with
respect to general kinematics. Using the simplest of those kinematics, we have
performed the first analytic computation of the two-loop six-edged Wilson loop
in general kinematics.Comment: 17 pages. Extended discussion on how the QMRK limit is taken. Version
accepted by JHEP. A text file containing the Mathematica code with the
analytic expression for the 6-point remainder function is include
The Quark Beam Function at NNLL
In hard collisions at a hadron collider the most appropriate description of
the initial state depends on what is measured in the final state. Parton
distribution functions (PDFs) evolved to the hard collision scale Q are
appropriate for inclusive observables, but not for measurements with a specific
number of hard jets, leptons, and photons. Here the incoming protons are probed
and lose their identity to an incoming jet at a scale \mu_B << Q, and the
initial state is described by universal beam functions. We discuss the
field-theoretic treatment of beam functions, and show that the beam function
has the same RG evolution as the jet function to all orders in perturbation
theory. In contrast to PDF evolution, the beam function evolution does not mix
quarks and gluons and changes the virtuality of the colliding parton at fixed
momentum fraction. At \mu_B, the incoming jet can be described perturbatively,
and we give a detailed derivation of the one-loop matching of the quark beam
function onto quark and gluon PDFs. We compute the associated NLO Wilson
coefficients and explicitly verify the cancellation of IR singularities. As an
application, we give an expression for the next-to-next-to-leading logarithmic
order (NNLL) resummed Drell-Yan beam thrust cross section.Comment: 54 pages, 9 figures; v2: notation simplified in a few places, typos
fixed; v3: journal versio
Form Factors in N=4 Super Yang-Mills and Periodic Wilson Loops
We calculate form factors of half-BPS operators in N=4 super Yang-Mills
theory at tree level and one loop using novel applications of recursion
relations and unitarity. In particular, we determine the expression of the
one-loop form factors with two scalars and an arbitrary number of
positive-helicity gluons. These quantities resemble closely the MHV scattering
amplitudes, including holomorphicity of the tree-level form factor, and the
expansion in terms of two-mass easy box functions of the one-loop result. Next,
we compare our result for these form factors to the calculation of a particular
periodic Wilson loop at one loop, finding agreement. This suggests a novel
duality relating form factors to periodic Wilson loops.Comment: 26 pages, 10 figures. v2: typos fixed, comments adde
Non-global Structure of the O({\alpha}_s^2) Dijet Soft Function
High energy scattering processes involving jets generically involve matrix
elements of light- like Wilson lines, known as soft functions. These describe
the structure of soft contributions to observables and encode color and
kinematic correlations between jets. We compute the dijet soft function to
O({\alpha}_s^2) as a function of the two jet invariant masses, focusing on
terms not determined by its renormalization group evolution that have a
non-separable dependence on these masses. Our results include non-global single
and double logarithms, and analytic results for the full set of non-logarithmic
contributions as well. Using a recent result for the thrust constant, we
present the complete O({\alpha}_s^2) soft function for dijet production in both
position and momentum space.Comment: 55 pages, 8 figures. v2: extended discussion of double logs in the
hard regime. v3: minor typos corrected, version published in JHEP. v4: typos
in Eq. (3.33), (3.39), (3.43) corrected; this does not affect the main
result, numerical results, or conclusion
From correlation functions to Wilson loops
We start with an n-point correlation function in a conformal gauge theory. We
show that a special limit produces a polygonal Wilson loop with sides. The
limit takes the points towards the vertices of a null polygonal Wilson loop
such that successive distances . This produces a fast moving
particle that generates a "frame" for the Wilson loop. We explain in detail how
the limit is approached, including some subtle effects from the propagation of
a fast moving particle in the full interacting theory. We perform perturbative
checks by doing explicit computations in N=4 super-Yang-Mills.Comment: 37 pages, 10 figures; typos corrected, references adde
Correlation function of null polygonal Wilson loops with local operators
We consider the correlator of a light-like polygonal Wilson loop
with n cusps with a local operator (like the dilaton or the chiral primary
scalar) in planar N =4 super Yang-Mills theory. As a consequence of conformal
symmetry, the main part of such correlator is a function F of 3n-11 conformal
ratios. The first non-trivial case is n=4 when F depends on just one conformal
ratio \zeta. This makes the corresponding correlator one of the simplest
non-trivial observables that one would like to compute for generic values of
the `t Hooft coupling \lambda. We compute F(\zeta,\lambda) at leading order in
both the strong coupling regime (using semiclassical AdS5 x S5 string theory)
and the weak coupling regime (using perturbative gauge theory). Some results
are also obtained for polygonal Wilson loops with more than four edges.
Furthermore, we also discuss a connection to the relation between a correlator
of local operators at null-separated positions and cusped Wilson loop suggested
in arXiv:1007.3243.Comment: 36 pages, 2 figure
On form factors in N=4 sym
In this paper we study the form factors for the half-BPS operators
and the stress tensor supermultiplet
current up to the second order of perturbation theory and for the
Konishi operator at first order of perturbation theory in
SYM theory at weak coupling. For all the objects we observe the
exponentiation of the IR divergences with two anomalous dimensions: the cusp
anomalous dimension and the collinear anomalous dimension. For the IR finite
parts we obtain a similar situation as for the gluon scattering amplitudes,
namely, apart from the case of and the finite part has
some remainder function which we calculate up to the second order. It involves
the generalized Goncharov polylogarithms of several variables. All the answers
are expressed through the integrals related to the dual conformal invariant
ones which might be a signal of integrable structure standing behind the form
factors.Comment: 35 pages, 7 figures, LATEX2
On soft singularities at three loops and beyond
We report on further progress in understanding soft singularities of massless
gauge theory scattering amplitudes. Recently, a set of equations was derived
based on Sudakov factorization, constraining the soft anomalous dimension
matrix of multi-leg scattering amplitudes to any loop order, and relating it to
the cusp anomalous dimension. The minimal solution to these equations was shown
to be a sum over color dipoles. Here we explore potential contributions to the
soft anomalous dimension that go beyond the sum-over-dipoles formula. Such
contributions are constrained by factorization and invariance under rescaling
of parton momenta to be functions of conformally invariant cross ratios.
Therefore, they must correlate the color and kinematic degrees of freedom of at
least four hard partons, corresponding to gluon webs that connect four eikonal
lines, which first appear at three loops. We analyze potential contributions,
combining all available constraints, including Bose symmetry, the expected
degree of transcendentality, and the singularity structure in the limit where
two hard partons become collinear. We find that if the kinematic dependence is
solely through products of logarithms of cross ratios, then at three loops
there is a unique function that is consistent with all available constraints.
If polylogarithms are allowed to appear as well, then at least two additional
structures are consistent with the available constraints.Comment: v2: revised version published in JHEP (minor corrections in Sec. 4;
added discussion in Sec. 5.3; refs. added); v3: minor corrections (eqs. 5.11,
5.12 and 5.29); 38 pages, 3 figure
On the renormalization of multiparton webs
We consider the recently developed diagrammatic approach to soft-gluon
exponentiation in multiparton scattering amplitudes, where the exponent is
written as a sum of webs - closed sets of diagrams whose colour and kinematic
parts are entangled via mixing matrices. A complementary approach to
exponentiation is based on the multiplicative renormalizability of intersecting
Wilson lines, and their subsequent finite anomalous dimension. Relating this
framework to that of webs, we derive renormalization constraints expressing all
multiple poles of any given web in terms of lower-order webs. We examine these
constraints explicitly up to four loops, and find that they are realised
through the action of the web mixing matrices in conjunction with the fact that
multiple pole terms in each diagram reduce to sums of products of lower-loop
integrals. Relevant singularities of multi-eikonal amplitudes up to three loops
are calculated in dimensional regularization using an exponential infrared
regulator. Finally, we formulate a new conjecture for web mixing matrices,
involving a weighted sum over column entries. Our results form an important
step in understanding non-Abelian exponentiation in multiparton amplitudes, and
pave the way for higher-loop computations of the soft anomalous dimension.Comment: 60 pages, 15 figure