# Leptonic widths of heavy quarkonia: S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O({alpha}{sub s}v{sup 2})

## Abstract

We construct the S-wave part of the electromagnetic vector annihilation current to O({alpha}{sub s}v{sup 2}) on the lattice for heavy quarks whose dynamics are described by the NRQCD action, and v is the nonrelativistic quark velocity inside the meson. The lattice vector current for QQ annihilation is expressed as a linear combination of lattice operators with quantum numbers L=0, J{sup P}=1{sup -}, and the coefficients are determined by matching this lattice current to the corresponding continuum current in QCD to O(v{sup 2}) at one-loop. The annihilation channel gives a complex amplitude and a proper choice for the contours of integration is needed; a simple Wick rotation is not possible. In this way, and with a careful choice of subtraction functions in the numerical integration, the Coulomb-exchange and infrared singularities appearing in the amplitudes are successfully treated. The matching coefficients are given as a function of the heavy quark mass Ma in lattice units. An automated vertex generation program written in Python is employed, allowing us to use a realistic NRQCD action and an improved gluon lattice action. A change in the definition of either action is easily accommodated in this procedure. The final result, when combined with lattice simulation results,more »

- Authors:

- SUPA, School of Physics, University of Edinburgh, Edinburgh EH9 3JZ (United Kingdom)
- Department of Physics, University of Regina, Regina, Saskatchewan S4S 0A2 (Canada)
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Cambridge CB3 0WA (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20933223

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevD.75.014008; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANNIHILATION; B QUARKS; C QUARKS; COMPUTERIZED SIMULATION; GLUONS; LATTICE FIELD THEORY; MASS; MESONS; PARTICLE DECAY; PARTICLE WIDTHS; QUANTUM CHROMODYNAMICS; QUANTUM NUMBERS; QUARK-QUARK INTERACTIONS; QUARKONIUM; S WAVES; SINGULARITY; T QUARKS; VECTOR CURRENTS

### Citation Formats

```
Hart, A., Hippel, G. M. von, and Horgan, R. R.
```*Leptonic widths of heavy quarkonia: S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O({alpha}{sub s}v{sup 2})*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.014008.

```
Hart, A., Hippel, G. M. von, & Horgan, R. R.
```*Leptonic widths of heavy quarkonia: S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O({alpha}{sub s}v{sup 2})*. United States. doi:10.1103/PHYSREVD.75.014008.

```
Hart, A., Hippel, G. M. von, and Horgan, R. R. Mon .
"Leptonic widths of heavy quarkonia: S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O({alpha}{sub s}v{sup 2})". United States.
doi:10.1103/PHYSREVD.75.014008.
```

```
@article{osti_20933223,
```

title = {Leptonic widths of heavy quarkonia: S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O({alpha}{sub s}v{sup 2})},

author = {Hart, A. and Hippel, G. M. von and Horgan, R. R.},

abstractNote = {We construct the S-wave part of the electromagnetic vector annihilation current to O({alpha}{sub s}v{sup 2}) on the lattice for heavy quarks whose dynamics are described by the NRQCD action, and v is the nonrelativistic quark velocity inside the meson. The lattice vector current for QQ annihilation is expressed as a linear combination of lattice operators with quantum numbers L=0, J{sup P}=1{sup -}, and the coefficients are determined by matching this lattice current to the corresponding continuum current in QCD to O(v{sup 2}) at one-loop. The annihilation channel gives a complex amplitude and a proper choice for the contours of integration is needed; a simple Wick rotation is not possible. In this way, and with a careful choice of subtraction functions in the numerical integration, the Coulomb-exchange and infrared singularities appearing in the amplitudes are successfully treated. The matching coefficients are given as a function of the heavy quark mass Ma in lattice units. An automated vertex generation program written in Python is employed, allowing us to use a realistic NRQCD action and an improved gluon lattice action. A change in the definition of either action is easily accommodated in this procedure. The final result, when combined with lattice simulation results, describes the electromagnetic decays of heavy quarkonia, notably the {upsilon} meson.},

doi = {10.1103/PHYSREVD.75.014008},

journal = {Physical Review. D, Particles Fields},

number = 1,

volume = 75,

place = {United States},

year = {Mon Jan 01 00:00:00 EST 2007},

month = {Mon Jan 01 00:00:00 EST 2007}

}