We describe the hexagon function bootstrap for solving for six-gluon scattering amplitudes in the large N_c limit of N=4 super-Yang-Mills theory. In this method, an ansatz for the finite part of these amplitudes is constrained at the level of amplitudes, not integrands, using boundary information. In the near-collinear limit, the dual picture of the amplitudes as Wilson loops leads to an operator product expansion which has been solved using integrability by Basso, Sever and Vieira. Factorization of the amplitudes in the multi-Regge limit provides additional boundary data. This bootstrap has been applied successfully through four loops for the maximally helicity violating (MHV) configuration of gluon helicities, and through three loops for the non-MHV case.

@article{Dixon:2014xca,
      author         = "Dixon, Lance J. and Drummond, James M. and Duhr, Claude
                        and von Hippel, Matt and Pennington, Jeffrey",
      title          = "{Bootstrapping six-gluon scattering in planar ${\cal
                        N}=4$ super-Yang-Mills theory}",
      year           = "2014",
      eprint         = "1407.4724",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      reportNumber   = "SLAC-PUB-16008",
}

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar N=4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at NNNLLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.

@article{Dixon:2014voa,
      author         = "Dixon, Lance J. and Drummond, James M. and Duhr, Claude
                        and Pennington, Jeffrey",
      title          = "{The four-loop remainder function and multi-Regge
                        behavior at NNLLA in planar N = 4 super-Yang-Mills
                        theory}",
      journal        = "JHEP",
      volume         = "1406",
      pages          = "116",
      doi            = "10.1007/JHEP06(2014)116",
      year           = "2014",
      eprint         = "1402.3300",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}

• Full function
• Coproduct
• Symbol
• Near-collinear limit
• Multi-Regge limit

For some additional expansions, see this page.

We introduce a generating function for the coefficients of the leading logarithmic BFKL Green's function in transverse-momentum space, order by order in alpha_s, in terms of single-valued harmonic polylogarithms. As an application, we exhibit fully analytic azimuthal-angle and transverse-momentum distributions for Mueller-Navelet jet cross sections at each order in alpha_s. We also provide a generating function for the total cross section valid to any number of loops.

@article{DelDuca:2013lma,
      author         = "Del Duca, Vittorio and Dixon, Lance J. and Duhr, Claude
                        and Pennington, Jeffrey",
      title          = "{The BFKL equation, Mueller-Navelet jets and
                        single-valued harmonic polylogarithms}",
      journal        = "JHEP",
      volume         = "1402",
      pages          = "086",
      doi            = "10.1007/JHEP02(2014)086",
      year           = "2014",
      eprint         = "1309.6647",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-ph",
      reportNumber   = "SLAC-PUB-15706, IPPP-13-76, DCPT-13-152",
}

This thesis examines the mathematical structure of scattering amplitudes and correlation functions in perturbative quantum field theory. The focus is on maximally supersymmetric N=4 super Yang-Mills theory, though the techniques presented here should be applicable in other contexts as well. Multi-loop Feynman integrals typically do not evaluate to classical functions, but in many cases they can be expressed in terms of certain types of iterated integrals that generalize classical polylogarithms. I show throughout this thesis how certain analytic properties of these functions can be used in conjunction with physical constraints to produce highly efficient methods of calculation. In the first half of this work, I focus on quantities in N=4 super Yang-Mills theory that can be expressed in terms of polylogarithmic functions of two variables, presenting results for the multi-Regge kinematical limit of six-point amplitudes and for the four-point correlation function of stress-tensor multiplets. In the second half of this work, I extend the analysis to functions of three variables in order to calculate the six-point remainder function, which describes the scattering of six gluons in planar N=4 super Yang-Mills theory. I present analytical results for the remainder function at three and four loops.

@phdthesis{Pennington:2013th,
      author       = {Jeffrey Pennington},
      title        = {Generalized polylogarithms in perturbative quantum field theory},
      school       = {Stanford University},
      year         = 2013,
}

• Chapter 1 (MRK)
• Chapter 4 (R63)
• Chapter 5 (R64)

We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N=4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of "hexagon functions" which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematics limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann-zeta-valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to -7.

@article{Dixon:2013eka,
      author         = "Dixon, Lance J. and Drummond, James M. and von Hippel,
                        Matt and Pennington, Jeffrey",
      title          = "{Hexagon functions and the three-loop remainder
                        function}",
      journal        = "JHEP",
      volume         = "1312",
      pages          = "049",
      doi            = "10.1007/JHEP12(2013)049",
      year           = "2013",
      eprint         = "1308.2276",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}

The three-loop four-point function of stress-tensor multiplets in super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus obtaining the three-loop correlation function analytically. The integrals have the generic structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of leading singularities to obtain the rational coefficients, the symbol — with an appropriate ansatz for its structure — as a means of characterising multiple polylogarithms, and the technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then lift to find the corresponding polylogarithmic functions. The final formulae are numerically confirmed. The techniques we develop can be applied more generally, and we illustrate this by analytically evaluating one of the integrals contributing to the same four-point function at four loops. This example shows a connection between the leading singularities and the entries of the symbol.

@article{Drummond:2013nda,
      author         = "Drummond, James and Duhr, Claude and Eden, Burkhard and
                        Heslop, Paul and Pennington, Jeffrey and others",
      title          = "{Leading singularities and off-shell conformal
                        integrals}",
      journal        = "JHEP",
      volume         = "1308",
      pages          = "133",
      doi            = "10.1007/JHEP08(2013)133",
      year           = "2013",
      eprint         = "1303.6909",
      archivePrefix  = "arXiv",
      reportNumber   = "HU-EP-13-15, IPPP-13-09, DCPT-13-18, --SLAC-PUB-15409,
                        LAPTH-016-13, CERN-PH-TH-2013-058",
}

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.

@article{Pennington:2012zj,
      author         = "Pennington, Jeffrey",
      title          = "{The six-point remainder function to all loop orders in
                        the multi-Regge limit}",
      journal        = "JHEP",
      volume         = "1301",
      pages          = "059",
      doi            = "10.1007/JHEP01(2013)059",
      year           = "2013",
      eprint         = "1209.5357",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      reportNumber   = "SLAC-PUB-15251",
}

We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w,w*). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (\nu,n) which are related to (w,w*) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (\nu,n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to NNNLLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

@article{Dixon:2012yy,
      author         = "Dixon, Lance J. and Duhr, Claude and Pennington, Jeffrey",
      title          = "{Single-valued harmonic polylogarithms and the
                        multi-Regge limit}",
      journal        = "JHEP",
      volume         = "1210",
      pages          = "074",
      doi            = "10.1007/JHEP10(2012)074",
      year           = "2012",
      eprint         = "1207.0186",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      reportNumber   = "SLAC-PUB-15132",
}

• Full function
• Coproduct
• Symbol
• Near-collinear limit
• Multi-Regge limit

For some additional expansions, see this page.

We use a combination of a 't Hooft limit and numerical methods to find non-perturbative solutions of exactly solvable string theories, showing that perturbative solutions in different asymptotic regimes are connected by smooth interpolating functions. Our earlier perturbative work showed that a large class of minimal string theories arise as special limits of a Painlevé IV hierarchy of string equations that can be derived by a similarity reduction of the dispersive water wave hierarchy of differential equations. The hierarchy of string equations contains new perturbative solutions, some of which were conjectured to be the type IIA and IIB string theories coupled to (4, 4k − 2) superconformal minimal models of type (A, D). Our present paper shows that these new theories have smooth non-perturbative extensions. We also find evidence for putative new string theories that were not apparent in the perturbative analysis.

@article{Iyer:2010ex,
      author         = "Iyer, Ramakrishnan and Johnson, Clifford V. and
                        Pennington, Jeffrey S.",
      title          = "{Non-perturbative String Theory from Water Waves}",
      journal        = "J.Phys.",
      volume         = "A44",
      pages          = "375401",
      doi            = "10.1088/1751-8113/44/37/375401",
      year           = "2011",
      eprint         = "1011.6354",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}

We uncover a remarkable role that an infinite hierarchy of non-linear differential equations plays in organizing and connecting certain ĉ<1 string theories non-perturbatively. We are able to embed the type 0A and 0B (A,A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We observe that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A,D) minimal string backgrounds. We explain how these and several string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painleve IV equation plays a key role in organizing the string theory physics, joining its siblings, Painleve I and II, whose roles have previously been identified in this minimal string context.

@article{Iyer:2010ss,
      author         = "Iyer, Ramakrishnan and Johnson, Clifford V. and
                        Pennington, Jeffrey S.",
      title          = "{String Theory and Water Waves}",
      journal        = "J.Phys.",
      volume         = "A44",
      pages          = "015403",
      doi            = "10.1088/1751-8113/44/1/015403",
      year           = "2011",
      eprint         = "1002.1120",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}

Plasma can be used as a convenient medium for manipulating intense charged particle beams, e.g., for ballistic focusing and steering, because the plasma can effectively reduce the self-space charge potential and self- magnetic field of the beam pulse. We previously developed a reduced analytical model of beam charge and current neutralization for an ion beam pulse propagating in a cold background plasma. The reduced-fluid description provides an important benchmark for numerical codes and yields useful scaling relations for different beam and plasma parameters. This model has been extended to include the additional effects of an applied solenoidal magnetic field. Simulations show that the self-magnetic field structure of the ion beam pulse propagating through background plasma can be complex and non-stationary. The linear system of Maxwell’s equations for the self-electromagnetic fields can be solved analytically in Fourier space. For a strong enough applied magnetic field, poles emerge in Fourier space. These poles are an indication that whistler and low-hybrid waves can be excited by the beam pulse.

@article{Davidson:2007zzb,
      author         = "Davidson, R.C. and Kaganovich, I. and Sefkow, A.B. and
                        Startsev, E. and Pennington, J.S.",
      title          = "{Charge and Current Neutralization of an Ion Beam Pulse
                        by Background Plasma in Presence of Applied Magnetic Field
                        and Gas Ionization}",
      journal        = "Conf.Proc.",
      volume         = "C070625",
      pages          = "3675",
      year           = "2007",
      reportNumber   = "PAC07-THPAS083",
}

Type 0A string theory in the (2,4k) superconformal minimal model backgrounds, with background ZZ D-branes or R-R fluxes can be formulated non-perturbatively. The branes and fluxes have a description as threshold bound states in an associated one-dimensional quantum mechanics which has a supersymmetric structure, familiar from studies of the generalized KdV system. The relevant bound state wavefunctions in this problem have unusual asymptotics (they are not normalizable in general, and break supersymmetry) which are consistent with the underlying description in terms of open and closed string sectors. The overall organization of the physics is very pleasing: The physics of the closed strings in the background of branes or fluxes is captured by the generalized KdV system and non-perturbative string equations obtained by reduction of that system (the hierarchy of equations found by Dalley, Johnson, Morris and Watterstam). Meanwhile, the bound states wavefunctions, which describe the physics of the ZZ D-brane (or flux) background in interaction with probe FZZT D-branes, are captured by the generalized mKdV system, and non-perturbative string equations obtained by reduction of that system (the Painleve II hierachy found by Periwal and Shevitz in this context).

@article{Carlisle:2005wa,
      author         = "Carlisle, James E. and Johnson, Clifford V. and
                        Pennington, Jeffrey S.",
      title          = "{D-branes and fluxes in supersymmetric quantum
                        mechanics}",
      year           = "2005",
      eprint         = "hep-th/0511002",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}

We study the Type 0A string theory in the (2,4k) superconformal minimal model backgrounds, focusing on the fully non-perturbative string equations which define the partition function of the model. The equations admit a parameter, Gamma, which in the spacetime interpretation controls the number of background D-branes, or R-R flux units, depending upon which weak coupling regime is taken. We study the properties of the string equations (often focusing on the (2,4) model in particular) and their physical solutions. The solutions are the potential for an associated Schrodinger problem whose wavefunction is that of an extended D-brane probe. We perform a numerical study of the spectrum of this system for varying Gamma and establish that when Gamma is a positive integer the equations' solutions have special properties consistent with the spacetime interpretation. We also show that a natural solution-generating transformation (that changes Gamma by an integer) is the Bäcklund transformation of the KdV hierarchy specialized to (scale invariant) solitons at zero velocity. Our results suggest that the localized D-branes of the minimal string theories are directly related to the solitons of the KdV hierarchy. Further, we observe an interesting transition when Gamma=-1.

@article{Carlisle:2005mk,
      author         = "Carlisle, James E. and Johnson, Clifford V. and
                        Pennington, Jeffrey S.",
      title          = "{Backlund transformations, D-branes, and fluxes in
                        minimal type 0 strings}",
      journal        = "J.Phys.",
      volume         = "A40",
      pages          = "12451-12462",
      doi            = "10.1088/1751-8113/40/41/013",
      year           = "2007",
      eprint         = "hep-th/0501006",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
}