Nucleon Observables as Probes for Physics Beyond the Standard Model

**Principal Investigator:**

Dr. Karl Jansen

**Affiliation:**

DESY Zeuthen

**Local Project ID:**

pr74yo

**HPC Platform used:**

SuperMUC-NG at LRZ

**Date published:**

In high-precision low energy particle physics experiments, small but significant discrepancies have been found when compared to expectations from theory. This has substantially increased the interest in precision nucleon structure measurements. This so-called *precision-frontier *of particle physics is serving a complementary role to new physics searches in the *high-energy frontier*, such as those pursued by experiments at the Large-Hadron Collider (LHC) at CERN. A major challenge in such searches is determining the precise contribution from the strong force component of the Standard Model. Experimentally, the proton, which is stable and abundantly available, is an optimal probe for studying strong interaction phenomena and is thus the target of several ongoing experiments such as at MAMI in Mainz, Jefferson Lab, and Fermi Lab as well as at the planned Electron-Ion Collider (EIC) in the US, where an extensive program to map out the proton’s rich sea structure is foreseen. Theoretically, strong interaction phenomena are governed by Quantum Chromodynamics (QCD), and at energy scales relevant to the proton structure, the only known way of studying QCD from first principles is via large scale simulations using the lattice formulation.

Our project on SuperMUC-NG targets such simulations, using the so-called twisted mass fermion formulation of lattice QCD, which has the advantage that observables, such as nucleon structure observables we are targeting here, converge faster to the continuum limit. Using the resources provided at LRZ, our Extended Twisted Mass Collaboration (ETMC) [1] has simulated QCD gluon configurations which include dynamical mass degenerate up and down, as well as the strange, and charm quark flavors (N_{f}=2+1+1) with masses tuned to their physical values and at three values of the lattice spacing parameter. These state-of-the-art ensembles are being used as building blocks for a wide range of lattice QCD studies in addition to nucleon structure which we highlight here.

**Physical point N _{f}=2+1+1 ensembles**

Lattice QCD refers to the discretization of the Lagrangian of QCD on a 4-dimensional Euclidean grid that allows to generate representative configurations via a Markov chain Monte Carlo process. The generated gauge configuration ensembles are then used to calculate hadronic quantities of interest. Continuum physics requires extrapolating using ensembles with multiple values of the lattice spacing and volume to the continuum and infinite physical box length limit. Several algorithmic improvements we have implemented during our allocations on SuperMUC include better integration schemes that scale more favorably with the lattice volume and advanced linear solvers [2, 3]. These algorithmic improvements complement the increased availability of computational resources to allow for the simulation of ensembles with larger volumes and with finer values of the lattice spacing, as we document in terms of SuperMUC core-hours over the years in Figure 1.

**Nucleon axial form-factors**

The ensembles generated on SuperMUC have been used to obtain the axial form factors of the nucleon [4], and in particular to obtain the individual up-, down-, strange-, and charm-quark contributions [5]. The axial structure of the nucleon is important for both understanding strong interaction dynamics and in revealing new physics. For example, neutrino elastic scattering on protons is sensitive to the strange axial form factor of the proton which at zero momentum transfer determines the strange quark contribution to the proton spin. Contributions from the strange quark, such as the so-called strange σ-term, also enter in the determination of cross sections for a class of popular cold dark matter candidates.

In Figure 2 we show our calculation of the strange axial form factor computed for the first time using N_{f}=2+1+1 configurations at the physical point. Our result confirms a negative strange contribution to the nucleon axial form factor and the value in the forward limit (Q^{2}=0) that is compatible with results from phenomenology.

Combining the individual up-, down-, and strange-quark contributions, we construct the SU(3) flavor octet combination, i.e. the combination u+d-2s, as shown in Figure 3. Of interest in particular are the so-called disconnected quark loop contributions to this combination: The strange-quark component of the SU(3) octet combination is disconnected, while the u+d component has both connected and disconnected contributions. A non-zero disconnected u+d-2s combination signals deviation from the SU(3) flavor symmetric point. This is shown in Figure 4, where we see that the deviation from zero is up to 10% for the octet axial form factor at low values of Q^{2} and decreasing at larger values. The significance of this result, beyond that it is the first such calculation at the physical point, is also reflected by the fact that in many phenomenological studies the SU(3) flavor symmetric point is assumed to extract such quantities.

Our analysis for the extraction of the full set of nucleon structure observables on the ensembles generated on SuperMUC is ongoing. In particular, the axial form factors presented here have been carried out using one N_{f}=2+1+1 ensemble at the physical point. While this removes the most significant systematic, namely that introduced by the extrapolation to physical quark masses that was required when simulating at heavier-than-physical quark masses, the full analysis on the ensembles shown in Figure 1 will allow us to extrapolate to the continuum limit thus also eliminating any systematics introduced by the finite lattice spacing. Of particular significance is the planned simulation shown in Figure 1 with the open circle, which will allow us to break new ground in terms of the lattice spacing, which will be the smallest ever simulated with this action, as well as in terms of the lattice volume, which will be the largest and for which large-scale resources such as SuperMUC are crucial.

[1] ETMC: https://github.com/etmc; http://www-zeuthen.desy.de/~kjansen/etmc/

[2] C. Alexandrou, S. Bacchio and J. Finkenrath, *"Multigrid approach in shifted linear systems for the non-degenerated twisted mass operator"*, Comput. Phys. Commun., 51-64 (2019), doi:10.1016/j.cpc.2018.10.013 [arXiv:1805.09584 [hep-lat]].

[3] C. Alexandrou, S. Bacchio, P. Charalambous, P. Dimopoulos, J. Finkenrath, R. Frezzotti, K. Hadjiyiannakou, K. Jansen, G. Koutsou and B. Kostrzewa, *et al.*, *"Simulating twisted mass fermions at physical light, strange and charm quark masses"*, Phys. Rev. **D98**, no.5, 054518 (2018) doi: 10.1103/ PhysRevD.98.054518 [arXiv:1807.00495 [hep-lat]].

[4] C. Alexandrou, S. Bacchio, M. Constantinou, P. Dimopoulos, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, B. Kostrzewa and T. Leontiou, et al., *"Nucleon axial and pseudoscalar form factors from lattice QCD at the physical point"*, Phys. Rev. **D103**, no.3, 034509 (2021) doi:10.1103/PhysRevD.103.034509 [arXiv:2011.13342 [hep-lat]].

[5] C. Alexandrou, S. Bacchio, M. Constantinou, K. Hadjiyiannakou, K. Jansen and G. Koutsou, *“Quark flavor decomposition of the nucleon axial form factors”*, Phys. Rev. **D104**, 074503 (2021), doi:10.1103/ PhysRevD.104.074503 [arXiv:2106.13468 [hep-lat]].

Scientific Contact

Prof. Dr. Karl Jansen

NIC , DESY Zeuthen

Platanenallee 6

D-15738 Zeuthen

+49 33762 / 77-286

Karl.Jansen@desy.de