Non-supersymmetric Yang-Mills Theory Research Articles

Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles — the Tr(ϕ3) theory — based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the “stringy” Tr(ϕ3) amplitudes secretly contains the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr(ϕ3) field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. These zeros are not manifest in Feynman diagrams but are made obvious by the connection of these amplitudes to the new understanding of associahedra arising from “causal diamonds” in kinematic space. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties for Tr(ϕ3) amplitudes hold and further generalize to the “stringy” Tr(ϕ3) amplitudes. The “kinematic causal diamond” picture suggests a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

Symmetry properties of Wilson loops with a Lagrangian insertion

Null Wilson loops in mathcal{N} = 4 super Yang-Mills are dual to planar scattering amplitudes. This duality implies hidden symmetries for both objects. We consider closely related infrared finite observables, defined as the Wilson loop with a Lagrangian insertion, normalized by the Wilson loop itself. Unlike ratio and remainder functions studied in the literature, this observable is non-trivial already for four scattered particles and bears close resemblance to (finite parts of) scattering processes in non-supersymmetric Yang-Mills theory. Moreover, by integrating over the insertion point, one can recover information on the amplitude, as was recently done to compute the full four-loop cusp anomalous dimension. We study the general structure of the Wilson loop with a Lagrangian insertion, focusing in particular on its leading singularities and their (hidden) symmetry properties. Thanks to the close connection of the observable to integrands of MHV amplitudes, it is natural to expect that its leading singularities can be written as certain Grassmannian integrals. The latter are manifestly dual conformal. They also have a conformal symmetry, up to total derivatives. We find that, surprisingly, the conformal symmetry becomes an invariance in the frame where the Lagrangian insertion point is sent to infinity. Furthermore, we use integrability methods to study how higher Yangian charges act on the Grassmannian integral. We evaluate the n-particle observable both at tree- and at one-loop level, finding compact analytic formulas. These results are explicitly written in the form of conformal leading singularities, multiplied by transcendental functions. We then compare these formulas to known expressions for all-plus amplitudes in pure Yang-Mills theory. We find a remarkable new connection: the Wilson loop with Lagrangian insertion in mathcal{N} = 4 super Yang-Mills appears to predict the maximal weight terms of the planar pure Yang-Mills all-plus amplitude. We test this relationship for the two-loop n-point Yang-Mills amplitude, as well as for the three-loop four-point amplitude.

Higgs Amplitudes from N=4 Supersymmetric Yang-Mills Theory.

Higgs plus multigluon amplitudes in QCD can be computed in an effective Lagrangian description. In the infinite top-mass limit, an amplitude with a Higgs boson and n gluons is computed by the form factor of the operator TrF^{2}. Up to two loops and for three gluons, its maximally transcendental part is captured entirely by the form factor of the protected stress tensor multiplet operator T_{2} in N=4 supersymmetric Yang-Mills theory. The next order correction involves the calculation of the form factor of the higher-dimensional, trilinear operator TrF^{3}. We present explicit results at two loops for three gluons, including the subleading transcendental terms derived from a particular descendant of the Konishi operator that contains TrF^{3}. These are expressed in terms of a few universal building blocks already identified in earlier calculations. We show that the maximally transcendental part of this quantity, computed in nonsupersymmetric Yang-Mills theory, is identical to the form factor of another protected operator, T_{3}, in the maximally supersymmetric theory. Our results suggest that the maximally transcendental part of Higgs amplitudes in QCD can be entirely computed through N=4 super Yang-Mills theory.

More on soft theorems: Trees, loops, and strings

We study soft theorems in a broader context, addressing their fate at loop level and their universality in effective field theories and string theory. We argue that for gauge theories in the planar limit, loop-level soft gluon theorems can be made manifest already at the integrand level. In particular, we show that the planar integrand for N=4 SYM satisfies the tree-level soft theorem to all orders in perturbation theory and provide strong evidence to this effect for integrands in N<4 SYM. We consider soft theorems for non-supersymmetric Yang-Mills theories and gravity, and show the validity of integrand soft theorem, while loop corrections to the integrated soft theorems are intimately tied to the presence of conformal anomalies. We then address the question of universality of the soft theorems for various theories. In effective field theories with F^3 and R^3 interactions, the soft theorems are not modified. However for gravity theories with R^2 phi interactions, the sub-sub-leading order soft graviton theorem, which is beyond what is implied by the extended BMS symmetry, requires modifications at tree level for non-supersymmetric theories, and at loop level for N<5 supergravity due to anomalies. Finally, for superstring amplitudes at finite alpha', via explicit calculation for lower-point examples as well as world-sheet OPE analysis for arbitrary multiplicity, we show that the superstring amplitudes satisfy the same soft theorem as its field-theory counterpart. This is no longer true for bosonic closed strings due to the presence of R^2 phi interactions.

Glueball decay rates in the Witten-Sakai-Sugimoto model

We revisit and extend previous calculations of glueball decay rates in the Sakai-Sugimoto model, a holographic top-down approach for QCD with chiral quarks based on D8 probe branes in Witten's holographic model of nonsupersymmetric Yang-Mills theory. The rates for decays into two pions, two vector mesons, four pions, and the strongly suppressed decay into four pi0 are worked out quantitatively, using a range of the 't Hooft coupling which closely reproduces the decay rate of rho and omega mesons and also leads to a gluon condensate consistent with QCD sum rule calculations. The lowest holographic glueball, which arises from a rather exotic polarization of gravitons in the supergravity background, turns out to have a significantly lower mass and larger width than the two widely discussed glueball candidates f0(1500) and f0(1710). The lowest nonexotic and predominantly dilatonic scalar mode, which has a mass of 1487 MeV in the Witten-Sakai-Sugimoto model, instead provides a narrow glueball state, and we conjecture that only this nonexotic mode should be identified with a scalar glueball component of f0(1500) or f0(1710). Moreover the decay pattern of the tensor glueball is determined, which is found to have a comparatively broad total width when its mass is adjusted to around or above 2 GeV.

Deep inelastic scattering structure functions of holographic spin-1 hadrons with N f ≥ 1

Two-point current correlation functions of the large $N$ limit of supersymmetric and non-supersymmetric Yang-Mills theories at strong coupling are investigated in terms of their string theory dual models with quenched flavors. We consider non-Abelian global symmetry currents, which allow one to investigate vector mesons with $N_f > 1$. From the correlation functions we construct the deep inelastic scattering hadronic tensor of spin-one mesons, obtaining the corresponding eight structure functions for polarized vector mesons. We obtain several relations among the structure functions. Relations among some of their moments are also derived. Aspects of the sub-leading contributions in the $1/N$ and $N_f/N$ expansions are discussed. At leading order we find a universal behavior of the hadronic structure functions.

Ultraviolet structure of half-maximal supergravity with matter multiplets at two and three loops

Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in $D=4$ and $D=5$. In a recent paper, Bossard, Howe and Stelle conjecture the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in $D=4$ at three loops and in $D=5$ at two loops. By assuming the required superspace exists in $D=5$, they argue that no new divergences should occur at two loops even with the addition of Abelian-vector matter multiplets. Up to possible issues with the $SL(2,\mathbb{R})$ global anomaly of the theory, they reach a similar conclusion in $D=4$ for two and three loops. The divergences we find contradict these predictions based on the existence of the desired off-shell superspaces. Furthermore, our $D=4$ results are incompatible with the new divergences being due to the anomaly. We find that the two-loop divergences of half-maximal supergravity are directly controlled by the divergences appearing in ordinary nonsupersymmetric Yang-Mills theory coupled to scalars, explaining why half-maximal supergravity develops new divergences when matter multiplets are added. We also provide a list of one- and two-loop counterterms that should be helpful for constraining any future potential explanations of the observed vanishings of divergences in pure half-maximal supergravity.

Ultraviolet cancellations in half-maximal supergravity as a consequence of the double-copy structure

We show that the double-copy structure of gravity forbids divergences in pure half-maximal (16 supercharge) supergravity at four and five points at one loop in D<8 and at two loops in D<6. We link the cancellations that render these supergravity amplitudes finite to corresponding ones that eliminate forbidden color factors from the divergences of pure nonsupersymmetric Yang-Mills theory. The vanishing of the two-loop four-point divergence in D=5 half-maximal supergravity is an example where a valid counterterm satisfying the known symmetries exists, yet is not present. We also give explicit forms of divergences in half-maximal supergravity at one loop in D=8 and at two loops in D=6.

Deep inelastic scattering from holographic spin-one hadrons

We study deep inelastic scattering structure functions from hadrons using different holographic dual models which describe the strongly coupled regime of gauge theories in the large $N$ limit. Particularly, we consider scalar and vector mesons obtained from holographic descriptions with fundamental degrees of freedom, corresponding to ${\cal {N}}=2$ supersymmetric and non-supersymmetric Yang-Mills theories. We explicitly obtain analytic expressions for the full set of eight structure functions, i.e., $F_1$, $F_2$, $g_1$, $g_2$, $b_1$, $b_2$, $b_3$, $b_4$, arising from the standard decomposition of the hadronic tensor of spin-one hadrons. We obtain the relations $2 F_1 = F_2$ and $2 b_1 = b_2$. In addition, we find $b_1 \sim {\cal {O}}(F_1)$ as suggested by Hoodbhoy, Jaffe and Manohar for vector mesons. Also, we find new relations among some of these structure functions.

High energy behavior of a six-point R-current correlator in $$ \mathcal{N} = 4 $$ supersymmetric Yang-Mills theory

We study the high energy limit of a six-point R-current correlator in N=4 supersymmetric Yang-Mills theory for finite N_c. We make use of the framework of perturbative resummation of large logarithms of the energy. More specifically, we apply the (extended) generalized leading logarithmic approximation. We find that the same conformally invariant two-to-four gluon vertex occurs as in non-supersymmetric Yang-Mills theory. As a new feature we find a direct coupling of the four-gluon t-channel state to the R-current impact factor.

Resizing the conformal window: Aβ-function ansatz

We propose an ansatz for the nonperturbative beta function of a generic non-supersymmetric Yang-Mills theory with or without fermions in an arbitrary representation of the gauge group. While our construction is similar to the recently proposed Ryttov-Sannino all order beta function, the essential difference is that it allows for the existence of an unstable ultraviolet fixed point in addition to the predicted Bank-Zaks -like infrared stable fixed point. Our beta function preserves all of the tested features with respect to the non-supersymmetric Yang-Mills theories. We predict the conformal window identifying the lower end of it as a merger of the infrared and ultraviolet fixed points.

Conformality or confinement: (IR)relevance of topological excitations

We study aspects of the conformality to confinement transition for non-supersymmetric Yang-Mills theories with fermions in arbitrary chiral or vectorlike representations. We use the presence or absence of mass gap for gauge fluctuations as an identifier of the infrared behavior. Present-day understanding does not allow the mass gap for gauge fluctuations to be computed on R*4. However, recent progress allows its non-perturbative computation on R*3xS*1 by using either the twisted partition function or deformation theory, for a range of S*1 sizes depending on the theory. For small number of fermions, Nf, we show that the mass gap increases with increasing radius, due to the non-dilution of monopoles and bions, the topological excitations relevant for confinement on R*3xS*1. For sufficiently large Nf, we show that the mass gap decreases with increasing radius. In a class of theories, we claim that the decompactification limit can be taken while remaining within the region of validity of semi-classical techniques, giving the first examples of semiclassically solvable Yang-Mills theories at any size S*1. For general non-supersymmetric vectorlike or chiral theories, we conjecture that the change in the behavior of the mass gap on R*3xS*1 as a function of the radius occurs near the lower boundary of the conformal window and give non-perturbative estimates of its value. For vectorlike theories, we compare our estimates of the conformal window with existing lattice results, truncations of the Schwinger-Dyson equations, NSVZ beta function-inspired estimates, and degree of freedom counting criteria. For multi-generation chiral gauge theories, to the best of our knowledge, our estimates of the conformal window are the only known ones.

MHV one-loop amplitudes in Yang-Mills from generalised unitarity

In this letter, we exploit generalised unitarity in order to calculate the cut-constructible part of one-loop amplitudes in non-supersymmetric Yang-Mills theory. In particular, we rederive the n-gluon MHV amplitudes for both the adjacent and non-adjacent gluon helicity configurations from 3- and 4-particle cuts alone.

Holographic confinement/deconfinement phase transitions of AdS/QCD in curved spaces

Recently Herzog has shown that deconfinement of AdS/QCD can be realized, in the hard-wall model where the small radius region is removed in the asymptotically AdS space, via a first order Hawking-Page phase transition between a low temperature phase given by a pure AdS geometry and a high temperature phase given by the AdS black hole in Poincare coordinates. In this paper we first extend Herzog's work to the hard wall AdS/QCD model in curved spaces by studying the thermodynamics of AdS black holes with spherical or negative constant curvature horizon, dual to a non-supersymmetric Yang-Mills theory on a sphere or hyperboloid respectively. For the spherical horizon case, we find that the temperature of the phase transition increases by introducing an infrared cutoff, compared to the case without the cutoff; For the hyperbolic horizon case, there is a gap for the infrared cutoff, below which the Hawking-Page phase transition does not occur. We also discuss charged AdS black holes in the grand canonical ensemble, corresponding to a Yang-Mills theory at finite chemical potential, and find that there is always a gap for the infrared cutoff due to the existence of a minimal horizon for the charged AdS black holes with any horizon topology.

Proof of ultraviolet finiteness for a planar non-supersymmetric Yang–Mills theory

This paper focuses on a three-parameter deformation of N = 4 Yang–Mills that breaks all the supersymmetry in the theory. We show that the resulting non-supersymmetric gauge theory is scale invariant, in the planar approximation, by proving that its Green functions are ultraviolet finite to all orders in light-cone perturbation theory.

Amplitudes in pure Yang-Mills and MHV Diagrams

We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Bäcklund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.

Loop amplitudes in pure Yang-Mills from generalised unitarity

We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.

Quantum integrability in (super) Yang–Mills theory on the light-cone

We employ the light-cone formalism to construct in the (super) Yang–Mills theories in the multi-color limit the one-loop dilatation operator acting on single trace products of chiral superfields separated by light-like distances. In the N=4 Yang–Mills theory it exhausts all Wilson operators of the maximal Lorentz spin while in nonsupersymmetric Yang–Mills theory it is restricted to the sector of maximal helicity gluonic operators. We show that the dilatation operator in all N-extended super-Yang–Mills theories is given by the same integral operator which acts on the (N+1)-dimensional superspace and is invariant under the SL(2|N) superconformal transformations. We construct the R-matrix on this space and identify the dilatation operator as the Hamiltonian of the Heisenberg SL(2|N) spin chain.

Scaling violations in Yang–Mills theories and strings in AdS5

String solitons in AdS 5 contain information of N=4 SUSY Yang–Mills theories on the boundary. Recent proposals for rotating string solitons reproduce the spectrum for anomalous dimensions of Wilson operators for the boundary theory. There are possible extensions of this duality for lower supersymmetric and even for non-supersymmetric Yang–Mills theories. We explicitly demonstrate that the supersymmetric anomalous dimensions of Wilson operators in N=0,1 Yang–Mills theories behave, for large spin J, at the two-loop level in perturbation theory, like log J. We compile the analytic one- and two-loop results for the N=0 case which is known in the literature, as well as for the N=1 case which seems to be missing.

Stable supergravity dual of nonsupersymmetric glue

We study non-supersymmetric fermion mass and condensate deformations of the AdS/CFT Correspondence. The 5 dimensional supergravity flows are lifted to a complete and remarkably simple 10 dimensional background. A brane probe analysis shows that when all the fermions have an equal mass a positive mass is generated for all six scalar fields leaving non-supersymmetric Yang Mills theory in the deep infra-red. We numerically determine the potential, produced by the background, in the Schroedinger equation relevant to the study of O^++ glueballs. The potential is a bounded well, providing evidence of stability and for a discrete, confined spectrum. The geometry can also describe the supergravity background around an (unstable) fuzzy 5-brane.