Study of Electromagnetic Corrections to QCD

**Principal Investigator:**

Hinnerk Stüben

**Affiliation:**

Regionales Rechenzentrum, Universität Hamburg (Germany)

**Local Project ID:**

hhh43

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

*The fundamental constituents of the strong nuclear force are quarks and gluons, which themselves bind together to form the familiar building blocks of nuclear physics, protons and neutrons. The two most common forms of quarks are the up quark and the down quark. The quarks carry electric charges +2/3 (up) and −1/3 (down). A proton is composed of two up quarks and one down quark (it has charge +1), whereas the neutron has two down and one up quark (it is charge-neutral). The understanding of the strong nuclear force has now matured to the level where quantitative statements can be made about the role of electric charges on the quark-gluon structure of matter.*

The elementary constituents of the *strong nuclear force* are *quarks* and *gluons*, which themselves bind together to form the familiar building blocks of nuclear physics, *protons* and *neutrons*. The interactions between the elementary quarks and gluons are described by the fundamental relativistic quantum field theory of Quantum Chromodynamics (QCD).

Electromagnetism is understood in terms of Quantum Electrodynamics (QED) which describes the interactions between *electrically charged* particles and *photons*. The frontiers of experimental measurement and theoretical calculation have successfully tested this theory to better than *parts-per-billio*n accuracy.

The two most common forms of quarks are the *up* quark and the *down* quark. The electric charge of the up quark is +2/3, whereas the down quark carries charge −1/3. A proton is composed of two up quarks and one down quark, adding to a net charge of +1, whereas the neutron has two down and one up and hence producing a chargeneutral object.

Owing to the difficulty of performing QCD calculations (neglecting electromagnetism), the strong force of QCD has not been tested to anywhere near the same degree as QED. Nevertheless, recent progress has seen the value for the proton mass from Lattice QCD agree with experiment at an unprecedented accuracy of 2%, i.e. Lattice QCD simulations have now reached a precision where electromagnetic corrections from QED become important.

In the present work, simulations in *dynamically-coupled* QCD+QED were performed. In addition to the up and down quarks the (next heavier) strange quark (with charge −1/3) was treated dynamically as well. Essential for the dynamics are the gluon and photon fields which are able to communicate through the dynamical sea quarks popping in and out of existence within the vacuum. Figure 1 shows an example of the interplay between the QED magnetic field and the QCD topological charge density of the vacuum from a simulation on a 24³×48 lattice. Note that it is not straightforward to visualise QCD because the gluon fields describe eight charges (and are represented in the computer by arrays of complex 3×3 matrices).

*Isospin breaking effects* (due to the differences between up and down quarks) are crucial for the existence of our Universe. Our Universe would not exist in the present form if the neutron-proton mass difference would only be slightly different. If it would be larger than the binding energy of the deuteron, no fusion would take place. If it would be a little smaller, all hydrogen would have been burned to helium. Hence, dynamically-coupled QCD+QED simulations were employed to conduct an investigation into the isospin splittings of the low-lying hadron spectrum, as shown in Figure 2. Given the long-range nature of QED (the photon is massless), it is expected that the lattice simulations will be affected by the fact that it is performed on a finite volume, hence in all cases, results are presented for independent simulations on two different volumes.

Taking the results from Figure 2, a total neutron-proton mass splitting of 1.27(75)(50)MeV was found, where the central value and statistical error are taken from the larger (48³×96) volume and the difference between the two volumes is taken as a systematic error. This is in excellent agreement with the experimentally observed mass splitting of 1.30MeV.

An important feature of these simulations is the clean separation of electromagnetic (QED) and strong (QCD) contributions to this splitting where it was found that the contribution from QED is −1.53(25)(50)MeV and the contribution from QCD is 2.79(67)(40)MeV.

Hence, it is observed that the stability of the universe is due to a finely tuned cancellation between the effects of the electromagnetic and strong forces of nature.

One of the few places where there exists tension between experiment and standard model predictions is in the anomalous magnetic moment of the muon. Currently the experimental and theoretical values have similar errors, and the discrepancy is about 3 standard deviations. The planned Muon g–2 Experiment at Fermilab aims to reduce the experimental uncertainty to 140 parts-per-billion. Thus it is essential to get the theoretical uncertainties down to a comparable precision – this will require the hadronic vacuum polarisation (HVP) contributions to be known to better than 0.5%. Reaching this target demands the inclusion of QCD+QED effects to properly understand how the behaviour of quarks are modified when their electric charges are turned on. Having the timely advantage of state-of-the-art ensembles of fully dynamical QCD+QED gauge field configurations an initial investigation of the electromagnetic contributions to the hadronic vacuum polarisation tensor was performed.

In Figure 3, the electromagnetic effects in the hadronic vacuum polarisation tensor Π is isolated by considering the ratio of the up or down quark contributions to Π with a fictitious n quark with zero electrical charge. The masses of all three quarks are tuned to be the same so that the observed signal is a purely electromagnetic effect. As expected, we see a larger electromagnetic effect for the up quark than for the down quark.

These results show that the electromagnetic effect must be taken into account to get a calculation of the vacuum polarization tensor to better than 1%.

**Project Team**

Taylor Haar, Roger Horsley, Waseem Kamleh, Zachary Koumi, Yoshifumi Nakamura, Holger Perlt, Dirk Pleiter, Paul Rakow, Gerrit Schierholz, Arwed Schiller, Hinnerk Stüben, Alex Westin, Ross Young, James Zanotti

**References**

R. Horsley, Z. Koumi, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. Stüben, R. D. Young, J. M. Zanotti, *Isospin splittings in the decuplet baryon spectrum from dynamical QCD+QED*, arXiv:1904.02304 [hep-lat]

A. Westin, R. Horsley, W. Kamleh, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, H. Stüben, R. D. Young, J. M. Zanotti, *Anomalous magnetic moment of the muon with dynamical QCD+QED*, arXiv:1902.01518 [hep-lat]

**Scientific Contact:**

Dr. Hinnerk Stüben

Universität Hamburg

Regionales Rechenzentrum

Schlüterstraße 70, D-20146 Hamburg (Germany)

e-mail: hinnerk.stueben [at] uni-hamburg.de

*JSC Project ID: hhh43*

*April 2019*