# A Lattice QCD Calculation of Vector Meson Decay Constants

**Principal Investigator:** Eric B. Gregory, Institut für Theoretische Physik, FB-C, Universität Wuppertal (Germany)

**HPC Platform:** JUQUEEN of JSC

**Date Published: **January 2015

**JSC Project ID: **hwu13

Researchers from Bergische Universität Wuppertal (BUW) are using lattice quantumchromodynamics (QCD) to calculate vector meson decay constants f_{V} , where V represents a vector meson such as ρ, ω, Φ, etc.

A meson is a composite particle composed of quarks *q* and antiquarks *q ^{-}* bound together by the strong force which is carried by gluons. In a simplified picture, we might describe a meson as a bound quark-antiquark pair exchanging gluons. Quarks have intrinsic spin of ½. In a vector meson the quark and antiquark are arranged such that the meson has a total spin of

*J*= 1, and the quantum field describing the vector meson is related to its mirror image in the same way that a vector is.

The vector meson decay constant measures the overlap of the wave function of the quark and antiquark. If the meson is a flavor-singlet (e.g., ρ^{0} , ω, *Φ*, *J/ψ*) then the decay constant is related to the rate of leptonic decay such as ρ → e^{+} e^{−} . In the case of non-singlet vector mesons (such as ρ^{±}, *K*^{∗}) decay constants are related to the coupling to the weak vector boson *W*^{±}. This is depicted diagramatically in Figure 1.

*Fig. 1.: A vector meson coupled to a photon. The decay constant f*

_{V}is related to the strength of the coupling at the vertexIn practice, the description of a meson as just a quark and antiquark pair is incomplete. Quantum field theory tells us that to fully account for the properties of the meson, we must consider that in the vicinity of the meson there exists a fluctuating cloud of quark-antiquark pairs. These pairs are created and then annihilate each other on extremely short time scales. Within each pair the antiquark has exactly the opposite spin and charge and color as the quark, so the fluctuations do not change the net quantum numbers of the meson. Nevertheless, these fluctuations have great impact on the overall properties of the composite particle. Figure 2 is an example of how this cloud of fluctuations contributes to the decay constants *f _{V}* .

*Fig. 2.: The same interaction as Figure 1, but showing some of the quantum fluctuations contributing to f*

_{V}.These “sea quarks” interact not only with the original “valence” quark and antiquark, but also with each other. A calculation of *f _{V}* must include these fluctuations and we do this non-perturbatively with the tools of lattice QCD. The calculation has been performed through large-scale simulation initially involving more than 10 million hours of CPU time on the JUQUEEN HPC system at JSC. We simulate the sea of quark and gluon fluctuations in a four-dimensional discretized box of space-time. On stored configurations representing snapshots of these fluctuations, we invert a large, sparse matrix representing the interactions of the quantum fields.

The process is repeated thousands of times to generate an expectation value of a correlation function representing the meson propagating across the lattice, as depicted in Figure 3. Offline, we fit this data to extract both the mass of the vector mesons and their decay constants as fit parameters.

*Fig. 3.: Diagram representing the propagation of a ρ meson from x to y.*

We can vary the input parameters such as quark mass and the discretization scale. Figure 4 shows results as a function of quark mass.

*Fig. 4.: Rho meson decay constants f*

_{ρ}as a function of the pion mass squared, which serves as a proxy for the quark mass. Larger β values equate to finer lattice scales.*© for all images: Bergische Universität Wuppertal, Theoretische Physik, Fachbereich C*

**Scientific Contact:**

Dr. Eric B. Gregory

Bergische Universität Wuppertal

Fachbereich C - Mathematik und Naturwissenschaften

Theoretische Physik

Gaussstrasse 20

D-42097 Wuppertal/Germany

e-mail: gregory@uni-wuppertal.de

*January 2015*