Singlet Operators Research Articles

Renormalization of a Standard Model extension with a Dark Abelian Sector and predictions for the W-boson mass

The described Dark Abelian Sector Model (DASM) extends the Standard Model (SM) by a “dark” sector containing a spontaneously broken U(1)d gauge group. Keeping this dark sector quite generic we only add one additional Higgs boson, one Dirac fermion, and right-handed SM-like neutrinos to the SM. Using the only two singlet operators of the SM with dimension less than 4 (the U(1)Y field-strength tensor and the SM Higgs mass operator |Φ|2) as well as the right-handed neutrino fields we open up three portals to the dark sector. Dark sectors, such as the one of the DASM, that introduce an additional Higgs boson H as well as an additional Z′ gauge boson can have a large influence on the predictions for electroweak precision observables and even accommodate possible dark matter candidates. We consider one of the two Higgs bosons to be the known 125 GeV Higgs boson and parameterize the extension of the scalar sector by the mass of the second Higgs boson, the Higgs mixing angle, and a Higgs self-coupling. We do not assume any mass hierarchy in the gauge sector and use the mass of the additional Z′ boson and a corresponding gauge-boson mixing angle to parameterize the extension of the gauge sector. The fermion sector is parameterized by the mass of the additional fermion and a fermion mixing angle. We describe an on-shell as well as an MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document} renormalization scheme for the DASM sectors and give explicit results for the renormalization constants at the 1-loop level, and, thus, prepare the ground for full NLO predictions for collider observables in the DASM. As a first example, we provide the DASM prediction for the W-boson mass derived from muon decay.

The fate of non-supersymmetric Gross-Neveu-Yukawa fixed point in two dimensions

We investigate the fate of the non-supersymmetric Gross-Neveu-Yukawa fixed point found by Fei et al. in 4 − ϵ dimensions with a two-component Majorana fermion continued to two dimensions. Assuming that it is a fermionic minimal model which possesses a chiral ℤ2 symmetry (in addition to fermion number parity) and just two relevant singlet operators, we can zero in on four candidates. Assuming further that the least relevant deformation leads to the supersymmetric Gross-Neveu-Yukawa fixed point (i.e. fermionic tricritical Ising model), we can rule out two of them by matching the spin contents of the preserved topological defect lines. The final candidates are the fermionic (11, 4) minimal model if it is non-unitary, and the fermionic (E6, A10) minimal model if it is unitary. If we further use a constraint from the double braiding relation proposed by one of the authors, the former scenario is preferable.

Multi-fixed point numerical conformal bootstrap: a case study with structured global symmetry

In large part, the future utility of modern numerical conformal bootstrap depends on its ability to accurately predict the existence of hitherto unknown non-trivial conformal field theories (CFTs). Here we investigate the extent to which this is possible in the case where the global symmetry group has a product structure. We do this by testing for signatures of fixed points using a mixed-correlator bootstrap calculation with a minimal set of input assumptions. This ‘semi-blind’ approach contrasts with other approaches for probing more complicated groups, which ‘target’ known theories with additional spectral assumptions or use the saturation of the single-correlator bootstrap bound as a starting point. As a case study, we select the space of CFTs with product-group symmetry O(15) ⊗ O(3) in d = 3 dimensions. On the assumption that there is only one relevant scalar (ℓ = 0) singlet operator in the theory, we find a single ‘allowed’ region in our chosen space of scaling dimensions. The scaling dimensions corresponding to two known large-N critical theories, the Heisenberg and the chiral ones, lie on or very near the boundary of this region. The large-N antichiral point lies well outside the ‘allowed’ region, which is consistent with the expectation that the antichiral theory is unstable, and thus has an additional relevant scalar singlet operator. We also find a sharp kink in the boundary of the ‘allowed’ region at values of the scaling dimensions that do not correspond to the (N, M ) = (3, 15) instance of any large-N -predicted O(N ) ⊗ O(M ) critical theory.

Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators

In this paper, we present one- and two-loop results for the renormalization of the gluon and quark gauge-invariant operators which appear in the definition of the QCD energy-momentum tensor, in dimensional regularization. To this end, we consider a variety of Green's functions with different incoming momenta. We identify the set of twist-2 symmetric traceless and flavor singlet operators which mix among themselves and we calculate the corresponding mixing coefficients for the nondiagonal components. We also provide results for some appropriate regularization-independent (RI')-like schemes, which address this mixing, and we discuss their application to nonperturbative studies via lattice simulations. Finally, we extract the one- and two-loop expressions of the conversion factors between the proposed RI' and the MSbar schemes. From our results regarding the MSbar-renormalized Green's functions, one can easily derive conversion factors relating numerous variants of RI'-like schemes to MSbar. To make our results easily accessible, we also provide them as Supplemental Material, in the form of a Mathematica input file and, also, an equivalent text file.

Instanton solutions in a model of AdS4/CFT3 correspondence

From the wrapping of the (anti)membranes of 11-dimensional supergravity over , on the internal directions along with an ansatz for its 4-form flux, by solving the original equations and identities, we arrive at scalar differential equations in the Euclidean space; note that the associated bulk solutions and setups break all supersymmeties, parity and scale invariance; the resulting (pseudo) scalar potential, which is Higgs-like with two nearly homogeneous vacua, provides the first-order phase transition and tunneling from the false- to true- vacuum. Here, concentrating on the three (pseudo) scalar modes m2=-2, 4, 10 , which are, in turn, realizable in Wick-rotated and skew-whiffed M2-branes backgrounds, we employ approximate methods and, particularly, Adomian decomposition method to solve the nonlinear second-order partial differential equations, valid in the probe approximation, with the Dirichlet boundary condition or the initial data from a basic exact solution, to get solutions in series expansions near the boundary in different orders of perturbation. Next, making use of the AdS4/CFT3 correspondence rules, after swapping the three fundamental representations of for gravitino, we build the dual singlet operators from the (scalar, fermion and gauge) fields in a 3-dimensional Chern-Simons-matter gauge field theory living on the resultant anti-M2-brnaes; after that, by deforming the corresponding boundary actions with the operators, we get invariant solutions with nonzero finite actions, which ,in turn, are small instantons sitting at the origin of a 3-sphere at infinity, causing instability and mediating false vacuum decay. In other words, the boundary potentials unbounded from below are duals for the collapse of the bulk (thin-wall) vacuum bubbles and big crunch singularities.

Light dark sectors through the Fermion portal

Pairs of Standard Model fermions form dimension-3 singlet operators that can couple to new dark sector states. This “fermion portal” is to be contrasted with the lower-dimensional Higgs, vector and neutrino singlet portals. We characterise its distinct phenomenology and place effective field theory bounds on this framework, focusing on the case of fermion portals to a pair of light dark sector fermions. We obtain current and projected limits on the dimension-6 effective operator scale from a variety of meson decay experiments, missing energy and long-lived particle searches at colliders, as well as astrophysical and cosmological bounds. The DarkEFT public code is made available for recasting these limits, which we illustrate with various examples including an integrated- out heavy dark photon.

Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model

A novel functorial relationship in perturbative quantum field theory is pointed out that associates Feynman diagrams (FD) having no external line in one theory mathbf{Th}_1 with singlet operators in another one mathbf{Th}_2 having an additional U(mathcal{N}) symmetry and is illustrated by the case where mathbf{Th}_1 and mathbf{Th}_2 are respectively the rank r-1 and the rank r complex tensor model. The values of FD in mathbf{Th}_1 agree with the large mathcal{N} limit of the Gaussian average of those operators in mathbf{Th}_2. The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, then into rank 3 tensors and so on. This FD functor can straightforwardly act on the d dimensional tensorial quantum field theory (QFT) counterparts as well. In the case of rank 2-rank 3 correspondence, it can be combined with the geometrical pictures of the dual of the original FD, namely, equilateral triangulations (Grothendieck’s dessins d’enfant) to form a triality which may be regarded as a bulk-boundary correspondence.

The three-loop single mass polarized pure singlet operator matrix element

We calculate the massive polarized three-loop pure singlet operator matrix element AQq(3),PS in the single mass case in the Larin scheme. This operator matrix element contributes to the massive polarized three-loop Wilson coefficient HQq(3),PS in deep-inelastic scattering and constitutes a three-loop transition matrix element in the variable flavor number scheme. We provide analytic results in Mellin N and in x space and study the behaviour of this operator matrix element in the region of small and large values of the Bjorken variable x.

The three-loop polarized pure singlet operator matrix element with two different masses

We present the two-mass QCD contributions to the polarized pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the polarized structure function g1(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the polarized heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals. These integrals depend on the mass ratio through the main argument, and the alphabet includes square–root valued letters.

String corrections to AdS amplitudes and the double-trace spectrum of mathcal{N} = 4 SYM

We consider α′ corrections to four-point correlators of half-BPS operators in mathcal{N} = 4 super Yang-Mills theory in the supergravity limit. By demanding the correct behaviour in the flat space limit, we find that the leading (α′)3 correction to the Mellin amplitude is fixed for arbitrary charges of the external operators. By considering the mixing of double-trace operators we can find the (α′)3 corrections to the double-trace spectrum which we give explicitly for su (4)-singlet operators. We observe striking patterns in the corrections to the spectra which hint at their common ten-dimensional origin. By extending the observed patterns and imposing them at order (α′)5 we are able to reproduce the recently found result for the correction to the Mellin amplitude for leftlangle {mathcal{O}}_2{mathcal{O}}_2{mathcal{O}}_p{mathcal{O}}_prightrangle correlators. By applying a similar logic to the [0, 1, 0] channel of su(4) we are able to deduce new results for the correlators of the form leftlangle {mathcal{O}}_2{mathcal{O}}_3{mathcal{O}}_p-1{mathcal{O}}_prightrangle .

Addition of SU(3) generators and its singlet Hilbert space

We construct the singlet Hilbert space associated with addition of SU(3) generators. This corresponds to the solution of Gauss law in lattice QCD. The normalized basis states are explicitly constructed using Schwinger Bosons. Further, we compute the action of basic singlet operators on these basis states.

Two-loop doubly massive four-point amplitude involving a half-BPS and Konishi operator

The two-loop four-point amplitude of two massless SU(N) colored scalars and two color singlet operators with different virtuality described by a half-BPS and Konishi operators is calculated analytically in maximally supersymmetric Yang-Mills theory. We verify the ultraviolet behaviour of the unprotected composite operator and exponentiation of the infrared divergences with correct universal values of the anomalous dimensions in the modified dimensional reduction scheme. The amplitude is found to contain lower transcendental weight terms in addition to the highest ones and the latter has no similarity with similar amplitudes in QCD.

Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1\u2212-plane

We study the vacuum structure of Nf flavour two-dimensional QED with an arbitrary integer charge k. We find that the axial symmetry is spontaneously broken from {mathbb{Z}}_{k{N}_f} to {mathbb{Z}}_{N_f} due to the non-vanishing condensate of a flavour singlet operator, resulting in k degenerate vacua. An explicit construction of the k vacua is given by using anon-commutative algebra obtained as a central extension of the {mathbb{Z}}_{k{N}_f} discrete axial symmetry and ℤk 1-form (center) symmetry, which represents the mixed ’t Hooft anomaly between them.We then give a string theory realization of such a system with k = 2 and Nf = 8 by putting an anti D-string in the vicinity of an orientifold O1−-plane and study its dynamics using the two-dimensional gauge theory realized on it. We calculate the potential between the anti D-string and the O1−-plane and find repulsion in both weak and strong coupling regimes of the two-dimensional gauge theory, corresponding to long and short distances, respectively. We also calculate the potential for the (Q, −1)-string (the bound state of an anti D-string and Q fundamental strings) located close to the O1−-plane. The result is non-perturbative in the string coupling.

Critical O(N) model to order ϵ4 from analytic bootstrap

We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical model, to fourth order in the -expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson–Fisher model. The anomalous dimensions we obtain for the singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the results for the OPE coefficients, we derive the corrections to the central charges and , which are found to be compatible with the known large expansions. Predictions for the central charge in the strongly coupled 3d model, including the 3d Ising model, are made for various values of , which compare favourably with numerical results and previous predictions.

Aperiodic exchange modulation effect on the specific heat in the Kondo necklace model

In the present work we study the effect of the aperiodic exchange modulation on the spin gap at finite temperature as well as the specific heat of the Kondo necklace model in two and three dimensions. For this purpose, we use a representation for the localized and conduction electrons in terms of local Kondo singlet and triplet operators. A decoupling scheme on the double time Green’s functions is also used to find the dispersion relation for the excitations of the system. The influence of the aperiodic exchange modulation on the spin gap at low temperatures is discussed in the paramagnetic phase. Moreover, we investigate the specific heat as a function of the aperiodic exchange modulation at low temperatures in two cases: above the quantum critical point i.e., along the so-called non-Fermi liquid trajectory and in the Kondo spin liquid state. We have also compared our results with previous bond operator mean-field calculations.

A complex fermionic tensor model in d dimensions

In this note, we study a melonic tensor model in d dimensions based on three-index Dirac fermions with a four-fermion interaction. Summing the melonic diagrams at strong coupling allows one to define a formal large-N saddle point in arbitrary d and calculate the spectrum of scalar bilinear singlet operators. For d = 2 − ϵ the theory is an infrared fixed point, which we find has a purely real spectrum that we determine numerically for arbitrary d < 2, and analytically as a power series in ϵ. The theory appears to be weakly interacting when ϵ is small, suggesting that fermionic tensor models in 1-dimension can be studied in an ϵ expansion. For d > 2, the spectrum can still be calculated using the saddle point equations, which may define a formal large-N ultraviolet fixed point analogous to the Gross-Neveu model in d > 2. For 2 < d < 6, we find that the spectrum contains at least one complex scalar eigenvalue (similar to the complex eigenvalue present in the bosonic tensor model recently studied by Giombi, Klebanov and Tarnopolsky) which indicates that the theory is unstable. We also find that the fixed point is weakly-interacting when d = 6 (or more generally d = 4n + 2) and has a real spectrum for 6 < d < 6.14 which we present as a power series in ϵ in 6 + ϵ dimensions.

Spectra of operators in large N tensor models

We study the operators in the large $N$ tensor models, focusing mostly on the fermionic quantum mechanics with $O(N)^3$ symmetry which may be either global or gauged. In the model with global symmetry we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the $O(N)$ groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the SYK model with real fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest $h=0$ eigenvalue corresponds to the conserved $O(N)$ charges. A class of singlet operators may be constructed from contracted combinations of $m$ symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from $m$ melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical $2k$-particle operators do not receive any ladder corrections and have quantized large $N$ scaling dimensions $k/2$. We construct pictorial representations of various singlet operators with low $k$. For larger $k$ we use available techniques to count the operators and show that their number grows as $2^k k!$. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large $N$ limit. We also study the large $N$ spectrum of low-lying operators in the Gurau-Witten model, which has $O(N)^6$ symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large $N$ tensor integrals with various symmetries.

The two-mass contribution to the three-loop pure singlet operator matrix element

We present the two-mass QCD contributions to the pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the structure function F2(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals that include square root letters in the alphabet, depending also on the mass ratio through the main argument. Numerical results are presented.

Partition function of free conformal fields in 3-plet representation

Simplest examples of AdS/CFT duality correspond to free CFTs in d dimensions with fields in vector or adjoint representation of an internal symmetry group dual in the large N limit to a theory of massless or massless plus massive higher spins in AdSd+1. One may also study generalizations when conformal fields belong to higher dimensional representations, i.e. carry more than two internal symmetry indices. Here we consider the case of the 3-fundamental (“3-plet”) representation. One motivation is a conjectured connection to multiple M5-brane theory: heuristic arguments suggest that it may be related to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N symmetry group that has an AdS7 dual. We compute the singlet partition function Z on S1 × Sd−1 for a free field in 3-plet representation of U(N) and analyse its novel large N behaviour. The large N limit of the low temperature expansion of Z which is convergent in the vector and adjoint cases here is only asymptotic, reflecting the much faster growth of the number of singlet operators with dimension, indicating a phase transition at very low temperature. Indeed, while the critical temperatures in the vector (Tc ∼ Nγ, γ > 0) and adjoint (Tc ∼ 1) cases are finite, we find that in the 3-plet case Tc ∼ (log N)−1, i.e. it approaches zero at large N. We discuss some details of large N solution for the eigenvalue distribution. Similar conclusions apply to higher p-plet representations of U(N) or O(N) and also to the free p-tensor theories invariant under [U(N)]p or [O(N)]p with p ≥ 3.

Flavor structure of Λ baryons from lattice QCD: From strange to charm quarks

We study $\mathrm{\ensuremath{\Lambda}}$ baryons of spin-parity ${\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$ with either a strange or charm valence quark in full $2+1$ flavor lattice QCD. Multiple $SU(3)$ singlet and octet operators are employed to generate the desired single baryon states on the lattice. Via the variational method, the couplings of these states to the different operators provide information about the flavor structure of the $\mathrm{\ensuremath{\Lambda}}$ baryons. We make use of the gauge configurations of the PACS-CS Collaboration and chirally extrapolate the results for the masses and $SU(3)$ flavor components to the physical point. We furthermore gradually change the hopping parameter of the heaviest quark from strange to charm to study how the properties of the $\mathrm{\ensuremath{\Lambda}}$ baryons evolve as a function of the heavy quark mass. It is found that the baryon energy levels increase almost linearly with the quark mass. Meanwhile, the flavor structure of most of the states remains stable, with the exception of the lowest ${\frac{1}{2}}^{\ensuremath{-}}$ state, which changes from a flavor singlet $\mathrm{\ensuremath{\Lambda}}$ to a ${\mathrm{\ensuremath{\Lambda}}}_{c}$ state with singlet and octet components of comparable size. Finally, we discuss whether our findings can be interpreted with the help of a simple quark model and find that the negative-parity ${\mathrm{\ensuremath{\Lambda}}}_{c}$ states can be naturally explained as diquark excitations of the light $u$ and $d$ quarks. On the other hand, the quark-model picture does not appear to be adequate for the negative-parity $\mathrm{\ensuremath{\Lambda}}$ states, suggesting the importance of other degrees of freedom to describe them.