Lattice QCD (LQCD) is a non-perturbative formulation of Quantum Chromodynamics (QCD), the regnant theory of strong interactions, on a discrete Euclidian space-time grid. Large scale simulations of lattice QCD allow us to calculate the contributions of QCD to the properties, decays and interactions of hadrons composed of quarks and gluons. With the development of highly efficient numerical algorithms and peta-scale computing resources, we are able to calculate, from first principles, many quantities that are being used to both validate QCD and search for new physics beyond the standard model of elementary particles and interactions.

These simulations are extremely computationally intensive and require leadership class computing resources. They are being done at national supercomputer centers at NERSC and Oak Ridge, USQCD clusters at Fermilab and JLab, and institutional computing resources at Los Alamos.

The Los Alamos lattice QCD group has made many pioneering contributions since its inception in 1984. Current staff members include Tanmoy Bhattacharya, Vicenzo Cirigliano, Michael Graesser, Rajan Gupta, Emanuele Mereghetti, and Boram Yoon. They are working with postdoctoral fellows Yong-Chull Jang, Santanu Mondal and Sungwoo Park on five projects:

- Calculations of matrix elements of isovector quark bilinear operators within the neutron and proton states are being carried out to explore signatures of novel scalar and tensor interactions that can arise at the TeV scale. By combining the results for the matrix elements with precision measurements of [ultra]cold neutron decays, one can put constraints on the possible size of these interactions. (see Figure 1) The PNDME (Precision Neutron Decay Matrix Elements) collaboration motivated these calculations and has published a number of state-of-the-art results that have recently been reviewed in the FLAG 2019 report and included in the community averages. Published papers include Probing Novel Scalar and Tensor Interactions from (Ultra)Cold Neutrons to the LHC; Axial, scalar, and tensor charges of the nucleon from 2 + 1 + 1 -flavor Lattice QCD; Isovector charges of the nucleon from 2 + 1 + 1 -flavor lattice QCD
- Calculations of the axial vector, electric and magnetic form factors that are needed to calculate the cross-section of the scattering of neutrinos, electrons and muons off nuclear targets. Our recent work has resolved the violation of the PCAC relation between the form factors. (See Figure 2) Published papers include Axial-vector form factors of the nucleon from lattice QCD; Axial Vector Form Factors from Lattice QCD that Satisfy the PCAC Relation; Nucleon Electromagnetic Form Factors in the Continuum Limit from 2+1+1-flavor Lattice QCD
- Calculations of matrix elements of novel CP violating operators that arise at the TeV scale. Each CP violating interaction makes a contribution to the neutron electric dipole moment (nEDM), and in many BSM theories these are large enough to explain the observed matter-antimatter asymmetry of the universe generated via baryogenesis. Using the tools of effective field theory, these operators are written in terms of quark and gluon fields at the hadronic (~1 GeV) scale. Lattice QCD is then used to calculate their matrix elements within the neutron state. Their contribution to the nEDM is given by dn = ∑i εi gi where εi is the strength (coupling at ~1 GeV scale) of the ith operator and gi is its matrix element. Our goal is to provide high precision results for gi for all operators that make a significant contribution and use future bounds on nEDM to constrain new CP violation, ie, the . (See Figure 3 for current constraints on the split SUSY model.) The operators under investigated include the Q-term, the quark EDM, the quark chromo EDM and Weinberg three gluon operators. Published papers include Neutron Electric Dipole Moment from Beyond the Standard Model; Dimension-5 CP-odd operators: QCD mixing and renormalization; Neutron Electric Dipole Moment and Tensor Charges from Lattice QCD
- Calculations of the matrix elements of flavor diagonal operators with axial, scalar, tensor and vector Lorentz structures. These provide (i) how the spin of the nucleons arises from the spin of the quarks and gluons and their orbital angular momentum; (ii) the nucleon sigma term σπN and its strangeness content σs; (iii) the strength of the interaction of dark matter with nuclear targets; and (iv) the average momentum, transversity and helicity distributions of each quark flavor within nucleons. Published papers include Quark contribution to the proton spin from 2 + 1 + 1 -flavor lattice QCD; Flavor diagonal tensor charges of the nucleon from ( 2 + 1 + 1 )-flavor lattice QCD
- Calculations of matrix elements of light cone operators within the proton state that shed light on the contribution of the angular momentum of quarks to the proton spin. Of particular interest is the extraction of the Sivers and Boer-Mulders distribution functions. (See Figure 5) These calculations are being done in collaboration with Michael Engelhardt (NMSU), and Andreas Schaefer (University of Regensburg, Germany). Published papers include Nucleon transverse momentum-dependent parton distributions in lattice QCD: Renormalization patterns and discretization effects

**Figure 1:** The allowed parameter region of novel scalar and tensor couplings ϵS and ϵT that can arise in extensions of the standard model at the TeV Scale. The PNDME collaboration results are used for the low energy constraints from neutron beta decay and compared with analogous constraints from the Atlas and CMS experiments at the LHC. For details see Isovector charges of the nucleon from 2 + 1 + 1 -flavor lattice QCD

**Figure 2: **Comparison of the tests of the partially Conserved Axial Current (PCAC) and the pion-pole dominance (PPD) hypothesis for the axial form factors. The traditional way of doing the analysis, S2pt, shows large violations, which are resolved by our new analysis strategy, SA4. For perfect validation, the value should be unity up to O(a) corrections at all momenta Q2. (See Axial Vector Form Factors from Lattice QCD that Satisfy the PCAC Relation)

**Figure 3: **Regions in the M2 − μ plane corresponding to various values of dn/de in split SUSY model, obtained by varying gTu,d,s within our estimated uncertainties. Using de ≤ 1.1 × 10−29 e cm and assuming maximal CP violation, the allowed region lies above the solid black line.

(See Flavor diagonal tensor charges of the nucleon from ( 2 + 1 + 1 )-flavor lattice QCD)

**Figure 4: **Our (PNDME) results for the contributions of the up, down, and strange (Δu, Δd, and Δs) quarks to the spin of the proton are compared with other lattice and phenomenological estimates. (See Quark contribution to the proton spin from 2 + 1 + 1 -flavor lattice QCD)

**Figure 5: **Lattice QCD data in the SIDIS limit, η → ∞, as a function of the Collins-Soper parameter ζ=0.83, are compared with the experimental extraction of the SIDIS generalized Sivers shift at ζ=0.83. In addition to lattice data for |bT| ≈ 0.35 fm, results from an earlier DWF-on-Asqtad study are also given. (See Nucleon transverse momentum-dependent parton distributions in lattice QCD: Renormalization patterns and discretization effects)

During 2007—2014, Gupta and Bhattacharya co-led the US wide HotQCD collaboration and investigated the behavior of QCD at finite temperature to elucidate the nature of the chiral transition to the quark-gluon plasma, determine the equation of state, and examine the restoration of U(1) axial symmetry. These results were published in six papers:

(i) The equation of state in (2+1)-flavor QCD;

(ii) QCD Phase Transition with Chiral Quarks and Physical Quark Masses;

(iii) Fluctuations and Correlations of net baryon number, electric charge, and strangeness: A comparison of lattice QCD results with the hadron resonance gas model;

(iv) Chiral transition and U(1)_A symmetry restoration from lattice QCD using domain wall fermions;

(v) Chiral and deconfinement aspects of the QCD transition; and

(vi) Equation of state and QCD transition at finite temperature.

Their work showed that, for physical values of the up, down and strange quark masses, the transition from the hadron phase to the quark-gluon plasma is a crossover, with a crossover temperature T=154(9) MeV (See Chiral and deconfinement aspects of the QCD transition). The equation of state is shown in the Figure 6 below and compared with a similar calculation carried out by the Wuppertal-Budapest collaboration, who used a different discretized lattice action (Stout).

**Figure 6: T**he behavior of (energy – 3pressure), pressure (p) and entropy density (s) of QCD as a function of the temperature calculated by the HotQCD collaboration with the HISQ action and compared with the results from the Wuppertal-Budapest collaboration who used the Stout action (See The equation of state in (2+1)-flavor QCD). The equation of state of QCD, extracted from this data, is used in the analysis of experimental data produced in the collisions of relativistic heavy ions at Brookhaven National Lab and at the Large Hadron Collider.