# How the axial anomaly controls flavor mixing among mesons

## Abstract

Here, it is well known that, because of the axial anomaly in QCD, mesons with J ^{P} = 0 ^{–} are close to SU(3) _{V} eigenstates; the η'(958) meson is largely a singlet, and the η meson an octet. In contrast, states with J ^{P} = 1 ^{–} are flavor diagonal; e.g., the Φ(1020) is almost pure ¯ss. Using effective Lagrangians, we show how this generalizes to states with higher spin, assuming that they can be classified according to the unbroken chiral symmetry of G _{fl} = SU(3) _{L} × SU(3) _{R}. We construct effective Lagrangians from terms invariant under G _{fl} and introduce the concept of hetero- and homochiral multiplets. Because of the axial anomaly, only terms invariant under the Z(3) _{A} subgroup of the axial U(1) _{A} enter. For heterochiral multiplets, which begin with that including the η and η'(958), there are Z(3) _{A} invariant terms with low mass dimension which cause states to mix according to SU(3) _{V} flavor. For homochiral multiplets, which begin with that including the Φ(1020), there are no Z(3) _{A} invariant terms with low mass dimension, and so states are diagonal in flavor. In this way, we predict the flavor mixing for themore »

- Authors:

- Jan Kochanowski Univ., Kielce (Poland); Johann Wolfgang Goethe Univ., Frankfurt am Main (Germany)
- Johann Wolfgang Goethe Univ., Frankfurt am Main (Germany)
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1438293

- Alternate Identifier(s):
- OSTI ID: 1462430

- Report Number(s):
- BNL-207869-2018-JAAM

Journal ID: ISSN 2470-0010; PRVDAQ; TRN: US1902172

- Grant/Contract Number:
- SC0012704

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 9; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

### Citation Formats

```
Giacosa, Francesco, Koenigstein, Adrian, and Pisarski, Robert D. How the axial anomaly controls flavor mixing among mesons. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.091901.
```

```
Giacosa, Francesco, Koenigstein, Adrian, & Pisarski, Robert D. How the axial anomaly controls flavor mixing among mesons. United States. doi:10.1103/PhysRevD.97.091901.
```

```
Giacosa, Francesco, Koenigstein, Adrian, and Pisarski, Robert D. Tue .
"How the axial anomaly controls flavor mixing among mesons". United States. doi:10.1103/PhysRevD.97.091901.
```

```
@article{osti_1438293,
```

title = {How the axial anomaly controls flavor mixing among mesons},

author = {Giacosa, Francesco and Koenigstein, Adrian and Pisarski, Robert D.},

abstractNote = {Here, it is well known that, because of the axial anomaly in QCD, mesons with JP = 0– are close to SU(3)V eigenstates; the η'(958) meson is largely a singlet, and the η meson an octet. In contrast, states with JP = 1– are flavor diagonal; e.g., the Φ(1020) is almost pure ¯ss. Using effective Lagrangians, we show how this generalizes to states with higher spin, assuming that they can be classified according to the unbroken chiral symmetry of Gfl = SU(3)L × SU(3)R. We construct effective Lagrangians from terms invariant under Gfl and introduce the concept of hetero- and homochiral multiplets. Because of the axial anomaly, only terms invariant under the Z(3)A subgroup of the axial U(1)A enter. For heterochiral multiplets, which begin with that including the η and η'(958), there are Z(3)A invariant terms with low mass dimension which cause states to mix according to SU(3)V flavor. For homochiral multiplets, which begin with that including the Φ(1020), there are no Z(3)A invariant terms with low mass dimension, and so states are diagonal in flavor. In this way, we predict the flavor mixing for the heterochiral multiplet with spin 1 as well as for hetero- and homochiral multiplets with spin 2 and spin 3.},

doi = {10.1103/PhysRevD.97.091901},

journal = {Physical Review D},

number = 9,

volume = 97,

place = {United States},

year = {2018},

month = {5}

}

DOI: 10.1103/PhysRevD.97.091901

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Works referenced in this record:

##
Chiral matrix model of the semi-QGP in QCD

journal, August 2016

- Pisarski, Robert D.; Skokov, Vladimir V.
- Physical Review D, Vol. 94, Issue 3

##
Low lying axial-vector mesons as dynamically generated resonances

journal, July 2005

- Roca, L.; Oset, E.; Singh, J.
- Physical Review D, Vol. 72, Issue 1

##
Phenomenology of scalar and vector mesons in the linear σ model

journal, December 1994

- Ko, Pyungwon; Rudaz, Serge
- Physical Review D, Vol. 50, Issue 11

##
Existence of the critical endpoint in the vector meson extended linear sigma model

journal, June 2016

- Kovács, P.; Szép, Zs.; Wolf, Gy.
- Physical Review D, Vol. 93, Issue 11

##
$B\to {K}_{1}\pi (K)$ decays in the perturbative QCD approach

journal, October 2014

- Zhang, Zhi-Qing; Hou, Zhi-Wei; Yang, Yueling
- Physical Review D, Vol. 90, Issue 7

##
Is ${f}_{0}(1710)$ a glueball?

journal, December 2014

- Janowski, Stanislaus; Giacosa, Francesco; Rischke, Dirk H.
- Physical Review D, Vol. 90, Issue 11

##
Chiral Partners in a Chirally Broken World

journal, January 2009

- Leupold, Stefan; Wagner, Markus
- International Journal of Modern Physics A, Vol. 24, Issue 02n03

##
Mixing and decay constants of pseudoscalar mesons

journal, October 1998

- Feldmann, Th.; Kroll, P.; Stech, B.
- Physical Review D, Vol. 58, Issue 11

##
Regarding the enigmas of $P$ -wave meson spectroscopy

journal, March 1998

- Burakovsky, L.; Goldman, T.
- Physical Review D, Vol. 57, Issue 5

##
How instantons solve the U(1) problem

journal, September 1986

- 't Hooft, G.
- Physics Reports, Vol. 142, Issue 6

##
Nature of Light Scalar Mesons from their Large- ${N}_{c}$ Behavior

journal, March 2004

- Peláez, J. R.
- Physical Review Letters, Vol. 92, Issue 10

##
Phenomenology of pseudotensor mesons and the pseudotensor glueball

journal, December 2016

- Koenigstein, Adrian; Giacosa, Francesco
- The European Physical Journal A, Vol. 52, Issue 12

##
Effective Lagrangians and Field Algebras with Chiral Symmetry

journal, July 1969

- Gasiorowicz, S.; Geffen, D. A.
- Reviews of Modern Physics, Vol. 41, Issue 3

##
${K}_{1}(1270)-{K}_{1}(1400)$ mixing and the fourth generation standard model effects in $B\to {K}_{1}{l}^{+}{l}^{-}$ decays

journal, August 2011

- Ahmed, Aqeel; Ahmed, Ishtiaq; Paracha, M. Ali
- Physical Review D, Vol. 84, Issue 3

##
An effective Lagrangian with broken scale and chiral symmetry II: pion phenomenology

journal, June 1996

- Carter, G.; Ellis, P. J.; Rudaz, S.
- Nuclear Physics A, Vol. 603, Issue 3-4

##
Decays of the vector glueball

journal, June 2017

- Giacosa, Francesco; Sammet, Julia; Janowski, Stanislaus
- Physical Review D, Vol. 95, Issue 11

##
Properties of the vector meson nonet at large ${N}_{c}$ beyond the chiral limit

journal, April 2006

- Leupold, Stefan
- Physical Review D, Vol. 73, Issue 8

##
Toy model for two chiral nonets

journal, August 2005

- Fariborz, Amir H.; Jora, Renata; Schechter, Joseph
- Physical Review D, Vol. 72, Issue 3

##
Physics with the KLOE-2 experiment at the upgraded DAΦNE

journal, June 2010

- Amelino-Camelia, G.; Archilli, F.; Babusci, D.
- The European Physical Journal C, Vol. 68, Issue 3-4

##
A PCAC puzzle: π0→γγ in the σ-model

journal, March 1969

- Bell, J. S.; Jackiw, R.
- Il Nuovo Cimento A, Vol. 60, Issue 1

##
Vector meson-vector meson interaction in a hidden gauge unitary approach

journal, April 2009

- Geng, L. S.; Oset, E.
- Physical Review D, Vol. 79, Issue 7

##
Decay of the pseudoscalar glueball into scalar and pseudoscalar mesons

journal, March 2013

- Eshraim, Walaa I.; Janowski, Stanislaus; Giacosa, Francesco
- Physical Review D, Vol. 87, Issue 5

##
Strong and radiative decays of excited vector mesons and predictions for a new $\varphi (1930)$ resonance

journal, September 2017

- Piotrowska, Milena; Reisinger, Christian; Giacosa, Francesco
- Physical Review D, Vol. 96, Issue 5

##
Meson vacuum phenomenology in a three-flavor linear sigma model with (axial-)vector mesons

journal, January 2013

- Parganlija, D.; Kovács, P.; Wolf, Gy.
- Physical Review D, Vol. 87, Issue 1

##
Masses of the tensor mesons with ${J}^{P}={2}^{-}$

journal, October 2014

- Chen, Wei; Cai, Zi-Xing; Zhu, Shi-Lin
- Nuclear Physics B, Vol. 887

##
Systematization of tensor mesons and the determination of the 2++ glueball

journal, December 2004

- Anisovich, V. V.
- Journal of Experimental and Theoretical Physics Letters, Vol. 80, Issue 12

##
Erratum: Glueball decay rates in the Witten-Sakai-Sugimoto model [Phys. Rev. D **91** , 106002 (2015)]

journal, May 2016

- Brünner, Frederic; Parganlija, Denis; Rebhan, Anton
- Physical Review D, Vol. 93, Issue 10

##
Pseudotensor meson family

journal, January 2015

- Wang, Bo; Pang, Cheng-Qun; Liu, Xiang
- Physical Review D, Vol. 91, Issue 1

##
Glueballs, hybrids, multiquarks

journal, December 2007

- Klempt, Eberhard; Zaitsev, Alexander
- Physics Reports, Vol. 454, Issue 1-4

##
Penguin-dominated $B\to \varphi {K}_{1}(1270)$ and $\varphi {K}_{1}(1400)$ decays in the perturbative QCD approach

journal, November 2014

- Liu, Xin; Zou, Zhi-Tian; Xiao, Zhen-Jun
- Physical Review D, Vol. 90, Issue 9

##
Evidence for a scalar glueball

journal, June 1995

- Amsler, Claude; Close, Frank E.
- Physics Letters B, Vol. 353, Issue 2-3

##
Vacuum properties of mesons in a linear sigma model with vector mesons and global chiral invariance

journal, September 2010

- Parganlija, Denis; Giacosa, Francesco; Rischke, Dirk H.
- Physical Review D, Vol. 82, Issue 5

##
Phenomenology of axial-vector and pseudovector mesons: decays and mixing in the kaonic sector

journal, October 2013

- Divotgey, Florian; Olbrich, Lisa; Giacosa, Francesco
- The European Physical Journal A, Vol. 49, Issue 10

##
Dynamical generation and dynamical reconstruction

journal, October 2009

- Giacosa, Francesco
- Physical Review D, Vol. 80, Issue 7

##
η bound states in nuclei: a probe of flavour-singlet dynamics

journal, March 2006

- Bass, Steven D.; Thomas, Anthony W.
- Physics Letters B, Vol. 634, Issue 4

##
${K}_{1}(1270)-{K}_{1}(1400)$ mixing angle and new-physics effects in $B\to {K}_{1}{l}^{+}{l}^{-}$ decays

journal, October 2008

- Hatanaka, Hisaki; Yang, Kwei-Chou
- Physical Review D, Vol. 78, Issue 7

##
Glueball decay rates in the Witten-Sakai-Sugimoto model

journal, May 2015

- Brünner, Frederic; Parganlija, Denis; Rebhan, Anton
- Physical Review D, Vol. 91, Issue 10

##
Masses and decay constants of the heavy tensor mesons with QCD sum rules

journal, September 2014

- Wang, Zhi-Gang; Di, Zun-Yan
- The European Physical Journal A, Vol. 50, Issue 9

##
Anomalous spin-1-meson decays from the gauged Wess-Zumino term

journal, December 1984

- Gomm, H.; Kaymakcalan, Ö.; Schechter, J.
- Physical Review D, Vol. 30, Issue 11

##
On the gluon content of the η and η′ mesons

journal, May 2007

- Escribano, Rafel; Nadal, Jordi
- Journal of High Energy Physics, Vol. 2007, Issue 05

##
Electromagnetic transition form factors of light vector mesons

journal, August 2010

- Terschlüsen, Carla; Leupold, Stefan
- Physics Letters B, Vol. 691, Issue 4

##
Decays of tensor mesons and the tensor glueball in an effective field approach

journal, December 2005

- Giacosa, F.; Gutsche, Th.; Lyubovitskij, V. E.
- Physical Review D, Vol. 72, Issue 11

##
Axial-Vector Vertex in Spinor Electrodynamics

journal, January 1969

- Adler, Stephen L.
- Physical Review, Vol. 177, Issue 5

##
Vector and axial-vector correlators in a chirally symmetric model

journal, January 2002

- Urban, M.; Buballa, M.; Wambach, J.
- Nuclear Physics A, Vol. 697, Issue 1-2

##
Information on the structure of the ${a}_{1}$ from $\tau $ decay

journal, September 2008

- Wagner, M.; Leupold, S.
- Physical Review D, Vol. 78, Issue 5

##
Is the effective Lagrangian for quantum chromodynamics a $\sigma $ model?

journal, June 1980

- Rosenzweig, C.; Schechter, J.; Trahern, C. G.
- Physical Review D, Vol. 21, Issue 12

##
Chiral dynamics of the two Λ(1405) states

journal, September 2003

- Jido, D.; Oller, J. A.; Oset, E.
- Nuclear Physics A, Vol. 725

##
A fresh look at η 2(1645), η 2(1870), η 2(2030) and f 2(1910) in $\bar{p}p \to \eta \pi ^{0}\pi ^{0}\pi ^{0}$

journal, January 2011

- Anisovich, A. V.; Batty, C. J.; Bugg, D. V.
- The European Physical Journal C, Vol. 71, Issue 1

##
Scalar Glueball in Radiative $J/\psi $ Decay on the Lattice

journal, January 2013

- Gui, Long-Cheng; Chen, Ying; Li, Gang
- Physical Review Letters, Vol. 110, Issue 2

##
The status of glueballs

journal, February 2013

- Ochs, Wolfgang
- Journal of Physics G: Nuclear and Particle Physics, Vol. 40, Issue 4

##
Branching ratios and spectral functions of τ decays: Final ALEPH measurements and physics implications

journal, December 2005

- Schael, S.; Barate, R.; Brunelière, R.
- Physics Reports, Vol. 421, Issue 5-6

##
Light-quark masses and isospin violation

journal, April 1979

- Gross, David J.; Treiman, S. B.; Wilczek, Frank
- Physical Review D, Vol. 19, Issue 7

##
Hadronic η and $ \eta{^\prime}$ decays

journal, December 2005

- Borasoy, B.; Nißler, R.
- The European Physical Journal A, Vol. 26, Issue 3

##
First observations of the rare decays B + → K + π + π − μ + μ − and B +→ ϕK + μ + μ −

journal, October 2014

- Aaij, R.; Adeva, B.; Adinolfi, M.
- Journal of High Energy Physics, Vol. 2014, Issue 10

##
Meson resonances, large *N* _{ c } and chiral symmetry

journal, June 2003

- Cirigliano, Vincenzo; Pich, Antonio; Ecker, Gerhard
- Journal of High Energy Physics, Vol. 2003, Issue 06

##
$\rho \rho $ interaction in the hidden gauge formalism and the ${f}_{0}(1370)$ and ${f}_{2}(1270)$ resonances

journal, December 2008

- Molina, R.; Nicmorus, D.; Oset, E.
- Physical Review D, Vol. 78, Issue 11

##
Chiral model for $\overline{q}q$ and $\overline{qq}qq$ mesons

journal, November 2004

- Napsuciale, Mauro; Rodríguez, Simón
- Physical Review D, Vol. 70, Issue 9