Search

We consider the Sudakov form factor in planar N=4 supersymmetric Yang-Mills theory in the off shell kinematical regime, which can be achieved by considering the theory on its Coulomb branch. We demonstrate that for up to three loops both the infrared-divergent as well as the finite terms do exponentiate, with the coefficient accompanying log2(m2) determined by the octagon anomalous dimension Γoct. This behavior is in stark contrast to previous conjectural accounts in the literature. Together with the finite terms we observe that for up to three loops the logarithm of the Sudakov form factor is identical to twice the logarithm of the null octagon O0, which was recently introduced within the context of integrability-based approaches to four point correlation functions with infinitely large R charges. The null octagon O0 is known in a closed form for all values of the ’t Hooft coupling constant and kinematical parameters. We conjecture that the relation between O0 and the off shell Sudakov form factor will hold to all loop orders.

We present the technique for resummation of flux tube excitations series arising in pentagon operator expansion program for polygonal Wilson loops and amplitudes in $$N = 4$$ SYM. Here we restrict ourselves with contributions of one-particle effective states and illustrate our technique on the example of one loop MHV6 amplitude.

We present the technique for resummation of flux tube excitations series arising in pentagon operator expansion program for polygonal Wilson loops in $\mathcal{N}=4$ SYM. Here we restrict ourselves with contributions of one-particle effective states and consider as a particular example ${\mathrm{NMHV}}_{6}$ amplitude at one-loop. The presented technique is also applicable at higher loops for one effective particle contributions and has the potential for generalization to contributions with more effective particles.

In this article, we continue the investigation of hep-th 1611.02179 regarding iterative properties of dual conformal integrals in higher dimensions. In d=4, iterative properties of four and five point dual conformal integrals manifest themselves in the famous BDS ansatz conjecture. In hep-th 1611.02179 it was also conjectured that a similar structure of integrals may reappear in d=6. We show that one can systematically, order by order in the number of loops, construct combinations of d=6 integrals with 1/(p^2)^2 propagators with an iterative structure similar to the d=4 case. Such combinations as a whole also respect dual conformal invariance but individual integrals may not., Comment: 23 pages, 7 figures

In this paper we present the all-loop conjecture for the integrands of Wilson line form factors, also known as the reggeon amplitudes, in $$\mathcal{N} = 4$$ SYM. The key role is played by a new gluing operation in momentum twistor space, which is used to derive the BCFW recursion for the loop integrands of the reggeon amplitudes.

We consider an exactly solvable inhomogeneous Dicke model which describes an interaction between a disordered ensemble of two-level systems with single mode boson field. The existing method for evaluation of Richardson-Gaudin equations in the thermodynamical limit is extended to the case of Bethe equations in Dicke model. Using this extension, we present expressions both for the ground state and lowest excited states energies as well as leading-order finite-size corrections to these quantities for an arbitrary distribution of individual spin energies. We then evaluate these quantities for an equally-spaced distribution (constant density of states). In particular, we study evolution of the spectral gap and other related quantities. We also reveal regions on the phase diagram, where finite-size corrections are of particular importance., Comment: 19 pages, extended version, accepted for publication in Nuclear Physics B

We discuss the universal soft behavior of form factors in the N=4 maximally supersymmetric Yang–Mills theory in the limit where the momentum of one of the particles tends to zero. We present details of how the tree-level form factors of this theory are related to eigenfunctions of a gl(4|4) integrable spin chain.

In this note we consider tree level form factors of operators from stress tensor supermultiplet with light like operator momentum q 2 = 0. The presentation of form factors in terms of the regulated integral over Grassmannian is given. The conjectured formula is verified by successfully reproducing known answers in the MHV and N k−2MHV, k ≥ 3 sectors as well as appropriate soft limit behavior.

The Grassmannian representation for gauge-invariant amplitudes for arbitrary number of legs with one of them being off-shell is derived for the case of N = 4 SYM. The obtained formula are successfully checked against known BCFW results for MHV n , NMHV4 and NMHV5 amplitudes.

In this paper we present the all-loop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in N=4 $$ \mathcal{N}=4 $$ SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon tree-level amplitudes and loop integrands starting from corresponding expressions for on-shell amplitudes. The introduced gluing procedure is used to derive the BCFW recursions both for tree-level reggeon amplitudes and their loop integrands. In addition we provide predictions for the reggeon loop integrands written in the basis of local integrals. As a check of the correctness of the gluing operation at loop level we derive the expression for LO BFKL kernel in N=4 $$ \mathcal{N}=4 $$ SYM.

We consider the description of reggeon amplitudes (Wilson lines form factors) in N=4 SYM within the framework of four dimensional ambitwistor string theory. The latter is used to derive scattering equations representation for reggeon amplitudes with multiple reggeized gluons present. It is shown, that corresponding tree-level string correlation function correctly reproduces previously obtained Grassmannian integral representation of reggeon amplitudes in N=4 SYM., Comment: 18 pages

We consider the description of form factors of local and Wilson line operators (reggeon amplitudes) in N=4 SYM within the framework of four dimensional ambitwistor string theory. We present the explicit expressions for string composite operators corresponding to stress-tensor operator supermultiplet and Wilson line operator insertion. It is shown, that corresponding tree-level string correlation functions correctly reproduce previously obtained Grassmannian integral representations. As by product we derive four dimensional tree-level scattering equations representations for the mentioned form factors and formulate a simple gluing procedure used to relate operator form factors with on-shell amplitudes., Comment: 46 pages, 3 figures; v3: multiple changes in the text, typos corrected, references added

Maximally supersymmetric Yang–Mills theories have several remarkable properties, among which are the cancellation of UV divergences, factorization of higher loop corrections and possible integrability. Much attention has been attracted to the N = 4 D = 4 SYM theory. The N = ( 1 , 1 ) D = 6 SYM theory possesses similar properties but is nonrenormalizable and serves as a toy model for supergravity. We consider the on-shell four point scattering amplitude and analyze its perturbative expansion within the spin-helicity and superspace formalism. The integrands of the resulting diagrams coincide with those of the N = 4 D = 4 SYM and obey the dual conformal invariance. Contrary to 4 dimensions, no IR divergences on mass shell appear. We calculate analytically the leading logarithmic asymptotics in all loops. Their summation leads to a Regge trajectory which is calculated exactly. The leading powers of s are calculated up to six loops. Their summation is performed numerically and leads to a smooth function of s . The leading UV divergences are calculated up to 5 loops. The result suggests the geometrical progression which ends up in a finite expression. This leads us to a radical point of view on nonrenormalizable theories.

We study the problem of the condensate (stochastic average) origination for an auxiliary field in the Kardar-Parisi-Zhang equation and its matrix generalization. We cannot reliably conclude that there is a condensate for the Kardar-Parisi-Zhang equation in the framework of the one-loop approximation improved by the renormalization group method. The matrix generalization of the Kardar-Parisi-Zhang equation permits a positive answer to the question of whether there is a nonzero condensate, and the problem can be solved exactly in the large-N limit.

In this paper we consider tree level form factors of operators from stress tensor operator supermultiplet with light-like operator momentum q 2 = 0. We present a conjecture for the Grassmannian integral representation both for these tree level form factors as well as for leading singularities of their loop counterparts. The presented conjecture was successfully checked by reproducing several known answers in MHV and N k−2MHV, k ≥ 3 sectors together with appropriate soft limits. We also discuss the cancellation of spurious poles and relations between different BCFW representations for such form factors on simple examples.

Soft theorems for the form factors of 1/2-BPS and Konishi operator super-multiplets are derived at tree level in N = 4 $$ \mathcal{N}=4 $$ SYM theory. They have a form identical to the one in the amplitude case. For MHV sectors of stress tensor and Konishi operator supermultiplets loop corrections to soft theorems are considered at one loop level. They also appear to have universal form in the soft limit. Possible generalization of the on-shell diagrams to the form factors based on leading soft behavior is suggested. Finally, we give some comments on inverse soft limit and integrability of form factors in the limit q 2 → 0.

We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable. We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations., 15 pages, 2 figures

The main aim of this paper is to study the scattering amplitudes in gauge field theories with maximal supersymmetry in dimensions D=6,8 and 10. We perform a systematic study of the leading ultraviolet divergences using the spinor helicity and on-shell momentum superspace framework. In D=6 the first divergences start at 3 loops and we calculate them up to 5 loops, in D=8,10 the first divergences start at 1 loop and we calculate them up to 4 loops. The leading divergences in a given order are the polynomials of Mandelstam variables. To be on the safe side, we check our analytical calculations by numerical ones applying the alpha-representation and the dedicated routines. Then we derive an analog of the RG equations for the leading pole that allows us to get the recursive relations and construct the generating procedure to obtain the polynomials at any order of (perturbation theory) PT. At last, we make an attempt to sum the PT series and derive the differential equation for the infinite sum. This equation possesses a fixed point which might be stable or unstable depending on the kinematics. Some consequences of these fixed points are discussed., Comment: 43 pages, 13 figures, pdf LaTex, v2 minor changes and references added

In this paper we discuss different recursion relations (BCFW and all-line shift) for the form factors of the operators from the $$ \mathcal{N}=4 $$ SYM stress-tensor current supermultiplet T AB in momentum twistor space. We show that cancelations of spurious poles and the equivalence between different types of recursion relations can be naturally understood using geometrical interpretation of the form factors as a special limit of the volumes of polytopes in ℂℙ4 in close analogy with the amplitude case. We also show how different relations for the IR pole coefficients can be easily derived using the momentum twistor representation. This raises an intriguing question — which of powerful on-shell methods and ideas can survive off-shell?

We consider the on-shell amplitudes in $ \mathcal{N}=\left( {1,1} \right) $ SYM in D = 6 dimensions within the spinor helicity and on-shell superspace formalism. This leads to an effective and straightforward technique reducing the calculation to a set of scalar master integrals. As an example, the simplest four point amplitude is calculated in one and two loops in the planar limit. All answers are UV and IR finite and expressed in terms of logs and Polylogs of transcendentality level 2 at one loop, and 4 and 3 at two loops. The all loop leading logarithmical asymptotics at high energy is obtained which exhibits the Regge type behaviour. The intercept is calculated in the planar limit and is equal to $ \alpha (t)=1+\sqrt{{\frac{{g_{{Y\;M}}^2{N_c}\left| t \right|}}{{32{\pi^3}}}}} $ .

In this paper a supersymmetric version of a generalized unitarity cut method in application to MHV and NMHV for form factors of operators from the N=4 SYM stress-tensor current supermultiplet $T^{AB}$ at one loop is discussed. The explicit answers for 3 and 4 point NMHV form factors at tree and one loop level are obtained. The general structure of n-point NMHV form factor at one loop is discussed as well as the relation between form factor with super momentum equal to zero and the logarithmic derivative of the superamplitude with respect to the coupling constant.

In this paper we develop a supersymmetric version of unitarity cut method for form factors of operators from the chiral truncation of the the $\mathcal{N}=4$ stress-tensor current supermultiplet $T^{AB}$. The relation between the superform factor with supermomentum equals to zero and the logarithmic derivative of the superamplitude with respect to the coupling constant is discussed and verified at tree- and one-loop level for any MHV $n$-point ($n \geq 4$) superform factor involving operators from chiral truncation of the stress-tensor energy supermultiplet. The explicit $\mathcal{N}=4$ covariant expressions for n-point tree- and one-loop MHV form factors are obtained. As well, the ansatz for the two-loop three-point MHV superform factor is suggested in the planar limit, based on the reduction procedure for the scalar integrals suggested in our previous work. The different soft and collinear limits in the MHV sector at tree- and one-loop level are discussed., Latex, 35 pages, 6 figures; v2: typos corrected; v3: published version

In this paper we study the form factors for the half-BPS operators $\mathcal{O}^{(n)}_I$ and the $\mathcal{N}=4$ stress tensor supermultiplet current $W^{AB}$ up to the second order of perturbation theory and for the Konishi operator $\mathcal{K}$ at first order of perturbation theory in $\mathcal{N}=4$ 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 $W^{AB}$ and $\mathcal{K}$ 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.

We discuss the latest progress in understanding the infrared behavior of inclusive cross sections and the form factors for the so-called half-BPS operators in the N=4 SYM theory. In both cases, we observe the exponentiation of the infrared divergences while the finite parts do not have such a simple behavior. We show how the infrared divergences cancel in inclusive cross sections when the emission of soft and collinear quanta is taken into account. We calculate the finite parts of the form factors in the two-loop approximation. They are rather complicated and are expressed in terms of generalized polylogarithms of several variables. But the principle of maximal transcendentality is nevertheless still satisfied.

Using the algorithm of constructing the IR finite observables suggested and discussed in details in our previous publications, we consider construction of such observables in N=8 SUGRA in NLO of PT. In general, contrary to the amplitudes defined in the presence of some IR regulator, such observables do not reveal any simple structure., Comment: 11 pages, LATEX

In this paper we give all the details of the calculation that we presented in our previous paper ArXiv:0908.0387 where the infrared structure of the MHV gluon amplitudes in the planar limit for ${\cal N}=4$ super Yang-Mills theory was considered in the next-to-leading order of perturbation theory. Explicit cancellation of the infrared divergencies in properly defined inclusive cross-sections is demonstrated first in a toy model example of "conformal QED" and then in the real ${\cal N}=4$ SYM theory. We give the full-length details both for the calculation of the real emission and for the diagrams with splitting in initial and final states. The finite parts for some inclusive differential cross-sections are presented in an analytical form. In general, contrary to the virtual corrections, they do not reveal any simple structure. An example of the finite part containing just the log functions is presented. The dependence of inclusive cross-section on the external scale related to the definition of asymptotic states is discussed., 49 pages, LATEX, 6 eps figures; Minor changes, Refs added

The infrared structure of MHV gluon amplitudes in ${\cal N}=4$ super Yang-Mills theory is considered in the next-to-leading order of PT. Explicit cancelation of the infrared divergencies in properly defined cross-sections is demonstrated. The remaining finite parts for some inclusive differential cross-sections are calculated analytically. In general, contrary to the virtual corrections, they do not reveal any simple structure., Comment: 13 pages, LaTex; Minor changes, typos corrected, references added

We consider a full Leigh-Strassler deformation of the ${\cal N}=4$ SYM theory and look for conditions under which the theory would be conformally invariant and finite. Applying the algorithm of perturbative adjustments of the couplings we construct a family of theories which are conformal up to 3 loops in the non-planar case and up to 4 loops in the planar one. We found particular solutions in the planar case when the conformal condition seems to be exhausted in the one loop order. Some of them happen to be unitary equivalent to the real beta-deformed ${\cal N}=4$ SYM theory, while others are genuine. We present the arguments that these solutions might be valid in any loop order., 19 pages, 3 axodraw figures

We claim that if by a choice of the couplings the theory can be made conformally invariant (vanishing of the beta functions) it is automatically finite and vice versa. This is demonstrated by explicit example in supersymmetric gauge theory. The formalism is then applied to the beta deformed ${\cal N}=4$ SYM theory and it is shown that the requirement of conformal invariance = finiteness can be achieved for any complex parameter of deformations., Comment: 15 pages, Latex, 1 figure axodraw style