# Etaprime and Eta Mesons with Connection to Anomalous Glue

###### Abstract

We review the present understanding of and meson physics and these mesons as a probe of gluon dynamics in low-energy QCD. Recent highlights include the production mechanism of and mesons in proton-nucleon collisions from threshold to high-energy, the effective mass shift in the nuclear medium, searches for possible and bound states in nuclei as well as precision measurements of decays as a probe of light-quark masses. We discuss recent experimental data, theoretical interpretation of the different measurements and the open questions and challenges for future investigation.

###### Contents

## I Introduction

The meson is special in Quantum Chromodynamics, the theory of quarks and gluons (QCD), because of its strong affinity to gluons. Hadrons, their properties and interactions, are emergent from more fundamental QCD quark and gluon degrees of freedom. QCD has the property of asymptotic freedom. The coupling which describes the strength of quark-gluon and gluon-gluon interactions decreases logarithmically with increasing (large) four-momentum transfer squared, . In the infrared, at low , quark-gluon interactions become strong. Quarks become confined inside hadron bound states and the vacuum is not empty but characterized by the formation of quark and gluon condensates. The physical degrees of freedom are emergent hadrons (protons, mesons …) as bound states of quarks and gluons. Baryons like the proton are bound states of three valence quarks. Mesons are bound states of a quark and antiquark.

Glue is manifest in the confinement potential which binds the quarks. This confinement potential corresponds to a restoring force of 10 tonnes regardless of separation. Quarks are bound by a string of glue which can break into two colorless hadron objects involving the creation of a quark-antiquark pair corresponding to the newly created ends of two confining strings formed from the original single string of confining glue. There are no isolated quarks. The QCD confinement radius is of order 1fm=m. This physics at large coupling is beyond QCD perturbation theory and described either using QCD inspired models of hadrons which build in key symmetries of the underlying theory or through computational lattice methods. About 99% of the mass of the hydrogen atom, 938.8 MeV, is associated with the confinement potential with the masses of the electron 0.5 MeV and the proton 938.3 MeV. Inside the proton the masses of the proton’s constituent two up quarks and one down quark are about 2.2 MeV for each up quark and 4.7 MeV for the down quark.

Besides generating the QCD confinement potential, glue plays a special role in the light hadron spectrum through the physics of the isoscalar and mesons including their interactions. The QCD Lagrangian with massless quarks is symmetric between left- and right-handed quarks (which are fermions) or between positive and negative helicity quarks. However, this symmetry is missing in the ground state hadron spectrum. The lightest mass hadrons, pions and kaons, are pseudoscalar mesons called Goldstone bosons associated with the spontaneous breaking of chiral symmetry between left- and right-handed quarks. These mesons are special in that the square of their masses are proportional to the masses of their constituent valence quark-antiquark pair. (In contrast, the leading term in the masses of the proton and spin-one vector mesons is determined by the confining gluonic potential with contributions from the light quark masses treated as small perturbations.) The lightest mass pions, the neutral with mass 135 MeV and charged with mass 140 MeV, play an important role in nuclear physics and the nucleon-nucleon interaction. The isosinglet partners of the pions and kaons, the pseudoscalar and mesons are too massive by about 300-400 MeV for them to be pure Goldstone states. They receive extra mass from non-perturbative gluon dynamics through a quantum effect called the axial anomaly. This glue comes with non-trivial topology. The physics of Goldstone bosons and the axial anomaly are explained in Section II below. Gluon topology is an effect beyond the simplest quark models and involves non-local and long range properties of the gluon fields. Theoretical understanding of the and involves subtle interplay of local symmetries and non-local properties of QCD. Examples of topology in other branches of physics include the Bohm-Aharanov effect and topological phase transitions and phases of matter in condensed matter physics, the 2016 Nobel Prize for Physics.

The and mesons come with rich phenomenology. The is predominantly a flavor-singlet state. This means that its wavefunction is approximately symmetric in the three lightest quark types (up, down and strange) that build up light hadron spectroscopy. These different species of quarks couple to gluons with equal strength. The meson has strong coupling to gluonic intermediate states in hadronic reactions from low through to high energies. An example from high energy reactions is the decay . The is made of a heavy charm-anticharm quark pair with mass 3686 MeV. Its decay to the light quark meson plus a photon involves the annihilation of the charm-anticharm quark pair into a gluonic intermediate state which then forms the meson made of a near symmetric superposition of light quark-antiquark pairs (up-antiup, down-antidown and strange-antistrange).

In this article we will discuss the broad spectrum of processes involving the that are mediated by gluonic intermediate states. The last 20 years has seen a dedicated programme of and meson production experiments from nucleons and nuclei close to threshold as well as in high energy collisions. Studies of and meson production and decay processes combine to teach us about the interface of glue and chiral dynamics, the physics of Goldstone bosons, in QCD. Measurements of and production in nuclear media are sensitive to behavior of fundamental QCD symmetries at finite density and temperature. In finite density nuclear media, for example in nuclei and neutron stars, hadrons propagate in the presence of long range mean fields that are created by nuclear many body dynamics. Interaction with the mean fields in the nucleus can change the hadrons’ observed properties, e.g., their effective masses, magnetic moments and axial charges. Symmetries between left- and right-handed quarks, which are spontaneously broken in the ground state, are partially restored in nuclear media with a reduced size of the quark condensate. At (large) finite temperature there is an effective renormalization of the QCD coupling which becomes reduced relative to the zero temperature theory for the same four-momentum transfer squared. One expects changes in hadron properties in the interaction region of finite temperature heavy-ion collisions. This article surveys and meson physics as a probe of QCD dynamics emphasizing recent advances from experiments and theory.

In addition to the topics discussed here, the physics of glue in QCD features in many frontline areas of QCD hadron physics research. The planned electron-ion-collider (EIC) has an exciting programme to study the role of glue in nucleons and nuclei over a broad range of high energy kinematics Accardi et al. (2016); Deshpande (2017). The search for hadrons containing explicit gluon degrees of freedom in their bound state wavefunctions is a hot topic in QCD spectroscopy, e.g., possible glueball states built of two or three valence gluons and hybrids built of a quark-antiquark pair and a gluon Klempt and Zaitsev (2007). Gluons in the proton play an essential role in understanding the proton’s internal spin structure Aidala et al. (2013). Studies of the QCD phase diagram Braun-Munzinger and Wambach (2009) from high density neutron stars Lattimer and Prakash (2016) to high temperature quark-gluon plasma and a color-glass condensate postulated to explain high density gluon matter in high energy collisions Gyulassy and McLerran (2005) are hot topics at the interface of nuclear and particle physics research. On the theoretical side, much effort is invested in trying to understand the detailed dynamics which leads to the QCD confinement potential Greensite (2011).

The plan of this paper is as follows.

In Section II we introduce the key theoretical issues with the and mesons and their unique place at the interface of chiral and non-perturbative gluon dynamics. Here we explain the different gluonic effects at work in and meson physics and how they are incorporated in theoretical calculations.

Section III discusses the strong CP puzzle. The observed matter antimatter asymmetry in the Universe requires some extra source of CP violation beyond the quark mixing described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the electroweak Standard Model. The non-perturbative glue which generates the large mass also has the potential to break CP symmetry in the strong interactions. This effect would be manifest as a finite neutron electric dipole moment proportional to a new QCD parameter, , which is experimentally constrained to be very small, less than . One possible explanation for the absence of CP violation here involves a new light-mass pseudoscalar particle called the axion. The axion is also a possible dark matter candidate to explain the “missing mass” in the Universe. While no axion particle has so far been observed, these ideas have inspired a vigorous program of ongoing experimental investigation to look for them.

Sections IV-VII focus on and phenomenology. In Section IV we discuss the information about QCD which follows from and decay processes. The amplitude for the meson to three pions decay depends on the difference between the lightest up and down quark masses and provides valuable information about the ratio of light quark masses. Studies of and decays tell us about their internal quark-gluon and spatial structure. In addition, searches for rare decay processes provide valuable tests of fundamental symmetries.

Section V discusses and production in near-threshold proton-nucleon collisions. The experimental program on and nucleon interactions has focused on near-threshold meson production in proton-nucleon collisions and photoproduction from proton and deuteron targets Wilkin (2017); Krusche and Wilkin (2014); Metag et al. (2017); Moskal et al. (2002). Recent highlights include the use of polarization observables in photoproduction experiments to search for new excited nucleon resonances Anisovich et al. (2017), measurement of the nucleon scattering length through the final state interaction in proton-proton collisions Czerwinski et al. (2014b) and measurement of the spin analyzing power to probe the partial waves associated with production dynamics in proton-proton collisions Adlarson et al. (2018b).

Section VI deals with the and in QCD nuclear media and the formation of possible meson-nucleus bound states. Recent photoproduction experiments in Bonn have revealed an effective mass shift in nuclear medium, which is about -40 MeV at nuclear matter density Nanova et al. (2013). Studies of the transparency of the nuclear medium to the propagating allow one to make a first (indirect) measurement of the -nucleus optical potential. One finds a small width of the in medium Nanova et al. (2012) compared to the depth of the optical potential meaning that the may be a good candidate for possible bound state searches in finite nuclei.

Mesic nuclei, if discovered in experiments, are a new exotic state of matter involving the meson being bound inside the nucleus purely by the strong interaction, without electromagnetic Coulomb effects playing a role. Strong attractive interactions between the meson and nucleons mean that both the and are prime targets for mesic nuclei searches, with a vigorous ongoing program of experiments in both Europe and Japan Metag et al. (2017). Searches for possible mesic nuclei are focused on helium while searches for bound states are focused on carbon and copper.

The effective mass shift in nuclei of about -40 MeV at nuclear matter density is in excellent agreement with the prediction of the Quark Meson Coupling model Bass and Thomas (2006) which works through coupling of the light up and down quarks in the meson to the (correlated two pion) mean field inside the nucleus. Here, the experiences an effective mass shift in nuclei which is catalyzed by its gluonic component Bass and Thomas (2014). Without this glue, the would be a strange quark state after SU(3) breaking with small interaction with the mean field inside the nucleus.

Shifting from finite density to finite temperature, there are also hints in data from RHIC (the Relativistic Heavy-Ion Collider) for possible mass suppression at finite temperature, with claims of at least -200 MeV mass shift Csorgo et al. (2010); Vertesi et al. (2011).

Section VII discusses and production in high-energy hadronic scattering processes from light-quark hadrons. The ratio of to meson production at high transverse momentum, , in high-energy proton-nucleus and nucleus-nucleus collisions is observed to be independent of the target nucleus in relativistic heavy-ion collision data from RHIC at Brookhaven National Laboratory and the ALICE experiment at the Large Hadron Collider at CERN, indicating a common propagation through the nuclear medium in these kinematics. Interesting effects are also observed in high-energy production. The COMPASS experiment at CERN found that odd exotic partial waves are strongly enhanced in relative to exclusive production in collisions of 191 GeV negatively charged pions from hydrogen Adolph et al. (2015), consistent with expectations Bass and Marco (2002) based on gluon-mediated couplings of the .

In Section VIII we give conclusions and an outlook to possible future experiments which could shed new light on the structure and interactions of the and .

Earlier reviews on and meson physics, each with a different emphasis, are given in the volume edited by Bijnens et al. (2002). The lecture notes of Shore (2008) provide a theoretical overview of gluonic effects in physics. Axion physics is reviewed in Kawasaki and Nakayama (2013). Leutwyler (2013) discusses light-quark physics with focus on the meson and Kupsc (2009) gives an overview of the analysis of and meson decays. Meson production in proton-proton collisions close-to-threshold is discussed in detail in the reviews by Moskal et al. (2002), Krusche and Wilkin (2014) and Wilkin (2017). The present status of meson-nucleus interaction studies is reviewed in Metag et al. (2017).

## Ii QCD symmetries and the and

Symmetries are important in hadron physics. Protons and neutrons with spin are related through isospin SU(2), which is expanded to SU(3) to include and hyperons. Likewise, one finds SU(2) multiplets of spin-zero and spin-one mesons, e.g., the charged and neutral spin-zero pions are isospin partners and reside inside SU(3) multiplets together with kaons. This spectroscopy suggests that these hadronic particles are built from simpler constituents. These are spin quarks labeled up, down and strange (their flavor denoted , and ). These quarks carry electric charges and where, e.g., a proton is built from two up quarks and a down quark, and a neutron is built of two down quarks and an up quark. The spin-zero and spin-one mesons are built of a quark-antiquark combination. The hadron wavefunctions are symmetric in flavor-spin and spatial degrees of freedom. The Pauli principle is ensured with the quarks and antiquarks being antisymmetric in a new label called color SU(3), red, green and blue.

High energy deep inelastic scattering experiments probe the deep structure of hadrons by scattering high energy electron or muon beams off hadronic targets. Deeply virtual photon exchange acts like a microscope which allows us to look deep inside the proton. One measures the inclusive cross section. These experiments reveal a proton built of nearly free fermion constituents, called partons.

The deep inelastic results and spectroscopy come together when color is made dynamical in the theory of Quantum Chromodynamics, QCD. Quarks carry a color charge and interact through colored gluon exchange, like electrons interacting through photon exchange in Quantum Electrodynamics, QED. QCD differs from QED in that gluons also carry color charge whereas photons are electrically neutral. (The dynamics is governed by the gauge group of color SU(3) instead of U(1) for the photon.) This means that the Feynman diagrams for QCD include 3 gluon and 4 gluon vertices (as well as the quark gluon vertices) and that gluons self-interact. For excellent textbook discussions of QCD and its application to hadrons see Close (1979); Thomas and Weise (2001).

Gluon-gluon interactions induce asymptotic freedom: the QCD version of the fine structure constant for quark-gluon and gluon-gluon interactions, , decreases logarithmically with increasing resolution . Gluon bremsstrahlung results in gluon induced jets of hadronic particles which were first discovered in high energy collisions at DESY Ellis (2014). Quark and gluon partons play a vital role in high energy hadronic collisions, e.g., at the Large Hadron Collider at CERN Altarelli (2013). Deep inelastic scattering experiments also tell us that about 50% of the proton’s momentum perceived at high is carried by gluons, consistent with the QCD prediction for the deepest structure of the proton. QCD theory also predicts that about 50% of the proton’s angular momentum budget is contributed by gluon spin and orbital angular momentum Bass (2005); Aidala et al. (2013).

Glue in low energy QCD is manifest through the confinement potential which binds quarks inside hadrons. Color-singlet glueball excitations (bound states of gluons) as well as hybrid bound states of a quark and antiquark plus gluon are predicted by theory but still awaiting decisive experimental confirmation.

The decay amplitude for and the ratio of cross-sections for hadron to muon-pair production in high energy electron-positron collisions, , are proportional to the number of dynamical colors , giving an experimental confirmation of .

This dynamics is encoded in the QCD Lagrangian. We first write the quark field as the sum of left- and right-handed quark components where and project out different states of quark helicity. The vector gluon field is denoted . For massless quarks, the QCD Lagrangian reads

(1) |

Here describes the quark-gluon interaction; is the gluon field tensor with the last term here generating the 3-gluon and 4-gluon interactions. The quark-gluon dynamics is determined by requiring invariance under the gauge transformations

(2) |

where describes rotating the local color phase of the quark fields.

For massless quarks the left- and right- handed quarks transform independently under chiral rotations which rotate between up, down and strange flavored quarks. Finite quark masses through the Lagrangian term explicitly breaks the chiral symmetry by connecting left- and right-handed quarks,

(3) |

Quark chirality (-1 for a left-handed quark and +1 for a right-handed quark) and helicity are conserved in perturbative QCD with massless quarks.

Low energy QCD is characterized by confinement and dynamical chiral symmetry breaking. There is an absence of parity doublets in the light-hadron spectrum. For example, the proton and the lowest mass N*(1535) nucleon resonance (that one would normally take as chiral partners) are separated in mass by 597 MeV. This tells us that the chiral symmetry for light and (and ) quarks is spontaneously broken.

Spontaneous symmetry breaking means that the symmetry of the Lagrangian is broken in the vacuum. One finds a non-vanishing chiral condensate connecting left- and right-handed quarks

(4) |

This spontaneous symmetry breaking induces an octet of
light-mass
pseudoscalar Goldstone bosons associated with SU(3)
including the
pions and kaons
which are listed in Table I
and also
– see below –
(before extra gluonic effects in the singlet channel)
a flavor-singlet Goldstone state.
^{1}^{1}1
Goldstone’s theorem tells us that
there is
one massless pseudoscalar boson for
each symmetry generator that does not annihilate the vacuum.

The Goldstone bosons couple to the axial-vector currents which play the role of Noether currents through

(5) |

with the corresponding decay constants (which determine the strength for, e.g., ) and satisfy the Gell-Mann-Oakes-Renner (GMOR) relation Gell-Mann et al. (1968)

(6) |

with MeV. The mass squared of the Goldstone bosons is in first order proportional to the mass of their valence quarks, Eqs. (5,6). This picture is the starting point of successful pion and kaon phenomenology.

A scalar confinement potential implies dynamical chiral symmetry breaking. For example, in the Bag model of quark confinement is modeled by an infinite square well scalar potential. When quarks collide with the Bag wall, their helicity is flipped. The Bag wall thus connects left and right handed quarks leading to quark-pion coupling and the pion cloud of the nucleon Thomas (1984). Quark-pion coupling connected to chiral symmetry plays an important role in the proton’s dynamics and phenomenology, e.g., transferring net quark spin into pion cloud orbital angular momentum and thus playing an important role in the nucleon’s spin structure Bass and Thomas (2010).

The light mass pion is especially important in nuclear physics, also with strong coupling to the lightest mass p-wave nucleon resonance.

Meson | wavefunction | mass (MeV) |
---|---|---|

135 | ||

140 | ||

140 | ||

498 | ||

498 | ||

494 | ||

494 | ||

The QCD Hamiltonian is linear in the quark masses. For small quark masses this allows one to perform a rigorous expansion perturbing in , called the chiral expansion Gasser and Leutwyler (1982). The proton mass in the chiral limit of massless quarks is determined by gluonic binding energy and set by , which sets the scale for the running of the QCD coupling , MeV for QCD with 3 flavors Patrignani et al. (2016).

The lightest up and down quark masses are determined from detailed studies of chiral dynamics. One finds MeV and MeV whereas the strange quark mass is slightly heavier at MeV (with all values here quoted at the scale GeV according to the Particle Data Group Patrignani et al. (2016)).

When electromagnetic interactions are also included, the leading order mass relations (6) become Georgi (1984)

(7) |

where is the electromagnetic contribution Dashen (1969) and . Substituting the pion and kaon masses gives the leading-order quark mass ratios

(8) |

The leading order GMOR formula, Eq. (6), gives the Gell-Mann Okubo formula Gell-Mann (1961); Okubo (1962) for the octet state

(9) |

Numerically . The meson mass and this mass contribution agree within 4% accuracy.

However, this is not the full story. The quark condensate in Eq. (6) also spontaneously breaks axial U(1) symmetry meaning that one might also expect a flavor-singlet Goldstone state which mixes with the octet state to generate the isosinglet bosons. However, without extra input, the resultant bosons do not correspond to states in the physical spectrum. The lightest mass isosinglet bosons, the and , are about 300-400 MeV too heavy to be pure Goldstone states, with masses MeV and MeV. One needs extra mass in the flavor-singlet channel to connect to the physical and mesons. This mass is associated with non-perturbative gluon dynamics.

The flavor-singlet channel is sensitive to processes involving violation of the Okubo-Zweig-Iizuka (OZI) rule, where the quark-antiquark pair (with quark chirality equal two) propagates with coupling to gluonic intermediate states (with zero net chirality); see Fig. 1. The OZI rule Okubo (1963); Zweig (1964); Iizuka (1966) is the phenomenological observation that hadronic processes involving Feynman graphs mediated by gluons (without continuous quark lines connecting the initial and final states) tend to be strongly suppressed.

To see the effect of the gluonic mass contribution consider the - mass matrix for free mesons with rows and columns in the octet-singlet basis

(10) |

At leading order in the chiral expansion (taking terms proportional to the quark masses ) this reads

(11) |

Here is the flavor-singlet gluonic mass term.

In the notation of Eq.(7) these singlet and mixing terms are

(12) |

The masses of the physical and mesons are found by diagonalizing this matrix, viz.

(13) |

One obtains values for the and masses:

(14) | |||||

Here the lightest mass state is the and heavier state is the . Summing over the two eigenvalues in Eq.(14) gives the Witten-Veneziano mass formula Witten (1979a); Veneziano (1979)

(15) |

The gluonic mass term is obtained by substituting the physical values of , and to give GeV. Without the gluonic mass term the would be approximately an isosinglet light-quark state () with mass degenerate with the pion and the would be a strange-quark state with mass — mirroring the isoscalar vector and mesons.

When interpreted in terms of the leading order mixing scheme, Eq. (13), phenomenological studies of various decay processes give a value for the - mixing angle between and Gilman and Kauffman (1987); Ball et al. (1996); Ambrosino et al. (2009). The has a large flavor-singlet component with strong affinity to couple to gluonic degrees of freedom. Mixing means that non-perturbative glue through axial U(1) dynamics plays an important role in both the and and their interactions.

The gluonic mass term is associated with the QCD axial anomaly in the divergence of the flavor-singlet axial-vector current. While the non-singlet axial-vector currents are partially conserved (they have just mass terms in the divergence), the singlet current satisfies the divergence equation Adler (1969); Bell and Jackiw (1969)

(16) |

where
is called the topological charge density.
The anomalous gluonic term is induced by QCD quantum
effects associated with renormalization of the singlet
axial-vector current
^{2}^{2}2
In QCD the flavor-singlet axial-vector
current can couple through gluon intermediate states;
see Fig. 2.
Here the triangle Feynman diagram is essential
with the axial-vector current
and two gluon couplings and
as the three vertices.
When we regularize the ultraviolet behavior of momenta
in the triangle loop, we find that we can preserve current conservation at the quark-gluon-vertices
(necessary for gauge invariance)
or partial conservation of the axial-vector current but
not both simultaneously.
Current conservation wins and induces
the gluonic anomaly term in the singlet divergence equation,
Eq.(16), from the ultraviolet point-like part of the triangle loop.
.
Here is the gluon field tensor and
.
For reviews of anomaly physics
see Shifman (1991) and Ioffe (2006).
Since gluons couple equally to each flavor of quark
the anomaly term cancels in the divergence equations
for non-singlet currents like
and
.

The QCD anomaly means that the singlet current is not conserved for massless quarks. Non-perturbative gluon processes act to connect left- and right-handed quarks, whereas left- and right-handed massless quarks propagate independently in perturbative QCD with helicity conserved for massless quarks.

The integral over space
is quantized
with either integer or fractional values
and measures a property called
the topological winding number.
This winding number vanishes in perturbative QCD and
in QED
but is finite with non-perturbative glue,
e.g.,
it is an integer for instantons
(tunneling processes in the QCD vacuum that
flip quark chirality)
Crewther (1978)
^{3}^{3}3
For a gluon field with gauge transformation
,
.
Finite action requires that
should tend to a pure gauge configuration
when with finite surface term
integral
which takes
quantized values, the topological winding number.
.
The gluonic mass term is generated by
glue associated with this non-trivial topology,
related perhaps to confinement or to instantons
Fritzsch and Minkowski (1975); Kogut and Susskind (1975); Witten (1979b); ’t Hooft (1976b, a).
The exact details of this gluon dynamics are still debated.

It is interesting to consider QCD in the limit of a large number of colors, . There are two well defined theoretical limits taking and either (the number of flavors) or held fixed. The gluonic mass term has a rigorous interpretation as the leading term when one makes an expansion in in terms of a quantity called the Yang-Mills topological susceptibility,

(17) |

– for extended discussion see Shore (1998, 2008). Here

(18) |

is calculated in the pure glue theory (without quarks). If we assume that the topological winding number remains finite independent of the value of then as Witten (1979a). In recent computational QCD lattice calculations Cichy et al. (2015) have computed both the pure gluonic term on the right-hand side of Eq.(18) and the meson mass contributions with dynamical quarks in the Witten-Veneziano formula Eq.(15) and find excellent agreement at the 10% percent level. This calculation gives MeV, very close to the phenomenological value 180 MeV which follows from taking GeV in the Witten-Veneziano formula Eq.(15).

Independent of the detailed QCD dynamics one can construct low-energy effective chiral Lagrangians which include the effect of the anomaly and axial U(1) symmetry Di Vecchia and Veneziano (1980); Rosenzweig et al. (1980); Witten (1980); Nath and Arnowitt (1981); Kawarabayashi and Ohta (1980); Leutwyler (1998) and use these Lagrangians to study low-energy processes involving the and . We define as the unitary meson matrix where denotes the octet of would-be Goldstone bosons associated with spontaneous chiral symmetry breaking with the Gell-Mann matrices (SU(3) generalisations of the isospin SU(2) Pauli matrices that couple to pions), is the singlet boson and is the singlet decay constant (at leading order taken to be equal to =92 MeV). With this notation the kinetic energy and mass terms in the chiral Lagrangian are

(19) |

with the meson mass matrix. The gluonic mass term is introduced via a flavor-singlet potential involving the topological charge density which is constructed so that the Lagrangian also reproduces the axial anomaly. This potential reads

(20) |

where is eliminated through its equation of motion to give the gluonic mass term for the . The Lagrangian contains no kinetic energy term for , meaning that the gluonic potential does not correspond to a physical state; is therefore distinct from mixing with a pseudoscalar glueball state. The coupling in Eq.(20) reproduces the picture of the as a mixture of chirality quark-antiquark and chirality-zero gluonic contributions; see Fig. 1.

Higher-order terms in become important when we consider scattering processes involving more than one or Di Vecchia et al. (1981), e.g., the term gives an OZI-violating tree-level contribution to the decay . For the in a nuclear medium at finite density, the medium dependence of may be introduced through coupling to the mean field in the nucleus through the interaction term . Here denotes coupling to the field. Again eliminating through its equation of motion, one finds the gluonic mass term decreases in-medium independent of the sign of and the medium acts to partially neutralize axial U(1) symmetry breaking by gluonic effects Bass and Thomas (2006). We return to this physics in Section VI below. In general, couplings involving give OZI-violation in physical observables.

Recent QCD lattice calculations suggest (partial) restoration of axial U(1) symmetry at finite temperature Bazavov et al. (2012); Cossu et al. (2013); Tomiya et al. (2017).

There are several places that glue enters and meson physics: the gluon topology potential which generates the large mass, possible small mixing with a lightest mass pseudoscalar glueball state (which comes with a kinetic energy term in its Lagrangian) and, in high momentum transfer processes, radiatively generated glue associated with perturbative QCD. Possible candidates for the pseudoscalar glueball state are predicted by lattice QCD calculations with a mass above 2 GeV Morningstar and Peardon (1999); Gregory et al. (2012a); Sun et al. (2017). These different gluonic contributions are distinct physics.

We have so far discussed the and at leading order in the chiral expansion. Going beyond leading order, one becomes sensitive to extra SU(3) breaking through the difference in the pion and kaon decay constants, , as well as new OZI-violating couplings. One finds strong mixing also in the decay constants. Two mixing angles enter the system when one extends the theory to in the meson momentum Leutwyler (1998), viz.

(21) |

These mixing angles follow because the eigenstates of the mass matrix involve linear combinations of the different decay constants separated by SU(3) breaking multiplying the meson states. In the SU(3) symmetric world one would have , with both vanishing for massless quarks. One finds a systematic expansion, large chiral perturbation theory, in , and , where are the light quark masses and .

Phenomenological fits have been made to production and decay processes within this two mixing angle scheme. Best fit values quoted in Feldmann (2000) are

(22) |

with the fits assuming that any extra OZI-violation beyond can be turned off in first approximation. Similar numbers are obtained in Escribano and Frere (2005) and Shore (2006) and in recent QCD lattice calculations Bali et al. (2018); Ottnad and Urbach (2018). To good approximation, this scheme reduces to one mixing angle if we change to the quark flavor basis and , viz. Feldmann (2000). These numbers correspond to a mixing angle about -15 in the leading order formula Eq.(13) Feldmann et al. (1998).

Recent QCD lattice calculations give values for the mixing angles: Gregory et al. (2012b) and Michael et al. (2013); Urbach (2017) in the quark flavor basis and in the (leading order) octet-singlet basis Christ et al. (2010).

Before discussing phenomenology, we first mention two key issues connected to the QCD anomaly which need to be kept in mind when understanding the . Observables do not depend on renormalization scales and are gauge invariant; that is, they do not depend on how a theoretician has set up a calculation.

First, the current picks up a dependence on the renormalization scale through the two-loop Feynman diagram in Fig. 2 Kodaira (1980); Crewther (1978). This means that the singlet decay constant in QCD is sensitive to renormalization scale dependence. This is in contrast to which is measured by the anomaly-free current . A renormalization group (RG) scale invariant version of suitable for phenomenology can be defined by factoring out the scale dependence or, equivalently, taking the RG scale dependent quantity evaluated at . Numerically, the RG factor is about 0.84 if we take as typical of the infrared region of QCD and evolve to infinity working to in perturbative QCD Bass (2005).

Second, the topological charge density is a total divergence . Here is the anomalous Chern-Simons current

(23) |

with the gluon field and is the QCD coupling. The current is gauge dependent. Gauge dependence issues arise immediately if one tries to separate a “ contribution” from matrix elements of the singlet current . This means that isolating the gluonic leading Fock component from the involves subtle issues of gauge invariance and only makes sense with respect to a particular renormalization scheme like the gauge invariant scheme Bass (2009).

## Iii The strong CP problem and axions

The gluonic topology term (20) which generates the gluonic contribution to the mass also has the potential to induce strong CP violation in QCD. One finds an extra term, , in the effective Lagrangian for axial U(1) physics which ensures that the potential

(24) |

is invariant under axial U(1) transformations with acting on the quark fields being compensated by .

The term is odd under CP symmetry. If it has non-zero value, induces a non zero neutron electric dipole moment Crewther et al. (1979)

(25) |

Experiments constrain .cm at 90% confidence limit or Pendlebury et al. (2015). New and ongoing experiments aim for an order of magnitude improvement in precision within the next five years or so Schmidt-Wellenburg (2016).

Why is the strong CP violation parameter so small? QCD alone offers no answer to this question. QCD symmetries allow for a possible term but do not constrain its size. The value of is an external parameter in the theory just like the quark masses are.

Non-perturbative QCD arguments tell us that if the lightest quark had zero mass, then there would be no net CP violation connected to the term Weinberg (1996). However, chiral dynamics including the decay discussed below tells us that the lightest up and down flavor quarks have small but finite masses. In the full Standard Model the parameter which determines the size of strong CP violation is , where is the quark mass matrix. Possible strong CP violation then links QCD and the Higgs sector in the Standard Model that determines the quark masses.

A possible resolution of this strong CP puzzle is to postulate the existence of a new very-light mass pseudoscalar called the axion Weinberg (1978); Wilczek (1978) which couples through the Lagrangian term

(26) | |||||

Here the term in denotes possible fermion couplings to the axion . The mass scale plays the role of the axion decay constant and sets the scale for this new physics. The axion transforms under a new global U(1) symmetry, called Peccei-Quinn symmetry Peccei and Quinn (1977), to cancel the term, with strong CP violation replaced by the axion coupling to gluons and photons. The axion here develops a vacuum expectation value with the potential minimized at . The mass of the QCD axion is given by Weinberg (1996)

(27) |

Axions are possible dark matter candidates. Constraints from experiments tells us that must be very large. Laboratory based experiments based on the two-photon anomalous couplings of the axion Ringwald (2015), ultracold neutron experiments to probe axion to gluon couplings Abel et al. (2017), together with astrophysics and cosmology constraints suggest a favored QCD axion mass between eV and 3 meV Baudis (2018); Kawasaki and Nakayama (2013), which is the sensitivity range of the ADMX experiment in Seattle Rosenberg (2015), corresponding to between about and GeV. The small axion interaction strength, , means that the small axion mass corresponds to a long lifetime and stable dark matter candidate, e.g., lifetime longer than about the present age of the Universe. If the axions were too heavy they would carry too much energy out of supernova explosions, thereby observably shortening the neutrino arrival pulse length recorded on Earth in contradiction to Sn 1987a data Kawasaki and Nakayama (2013). Possible axion candidates would also need to be distinguished from other possible 5th force light mass scalar bosons Mantry et al. (2014).

## Iv and decays

For the and mesons there are two main decay types: hadronic decays to 3 pseudoscalar mesons and electromagnetic decays to two photons. The hadronic decays are sensitive to the details of chiral dynamics and, for decays into 3 pions, the difference in the light up and down quark masses. The two photon decays tell us about the spatial and quark/gluon structure of the mesons with extra (more model dependent) information coming from decays to final states Rosner (1983). Searches for rare and forbidden decays of the and mesons constrain tests of fundamental symmetries.

The total widths quoted by the Particle Data Group are keV for the meson and MeV for the Patrignani et al. (2016) with the result including the total width value determined directly from the mass distribution measured in proton-proton collisions, MeV Czerwinski et al. (2010). The main branching ratios for the decays are at , at , and for the two photon decay . For the the main decays are at and at Patrignani et al. (2016).

### iv.1 Hadronic decays

The decay is of key interest. This process is driven by isospin violation in the QCD Lagrangian, the difference in light-quark up and down quark masses . In the absence of small (few percent) electromagnetic contributions Baur et al. (1996)), the decay amplitude is proportional to which is usually expressed in terms of the ratio

(28) |

where and is the strange quark mass. Expansion in chiral perturbation theory (in the light-quark masses) converges slowly due to final state pion rescattering effects. Fortunately, these can be resummed using dispersive techniques allowing one to make a precise determination of the ratio of light quark masses from experiments, for a review see Leutwyler (2013).

Recent accurate measurements of the decay to charged pions, , have been performed by the WASA-at-COSY experiment at FZ-Jülich Adlarson et al. (2014b), the KLOE-2 Collaboration at LN-Frascati Anastasi et al. (2016) and at BES in Beijing Ablikim et al. (2017). The neutral 3 pion decay has most recently been measured by WASA Adolph et al. (2009), KLOE Ambrosino et al. (2011), the Mainz A2 Collaboration Prakhov et al. (2018) and at BES Ablikim et al. (2015a).

Taking the precise data on from KLOE-2 as input, Colangelo et al. (2017) find . Combining this result with quoted in the lattice Ref. Aoki et al. (2017), they obtain the light quark mass ratio . Similar results have been obtained by Guo et al. (2017) who include both KLOE-2 and WASA data for this decay and get . Similar values for were found using earlier data by Kampf et al. (2011); Kambor et al. (1996). These numbers compare with which follows from the simple leading-order calculation in Eq. (8).

The decay is also driven by isospin violation. In addition to the QCD processes involved in the decay, here there are also important contributions from the sub-processes plus mixing to give the 3 pion final state and with .

These decays contrast with the process which is the dominant decay with leading QCD term not driven by the difference in and . Here the singlet component in both the initial and final state isoscalar mesons and through mixing means that the reaction is potentially sensitive also to OZI-violating couplings, e.g., from the term at next-to-leading order in in the chiral Lagrangian. The leading order amplitude for this decay is proportional to and vanishes in the chiral limit. The large branching ratios for this decay tell us that non-leading terms play a vital role.

### iv.2 Two-photon interactions

The two photon decays of the , and mesons are driven by the QED axial anomaly.

For the , in the chiral limit

(29) |

where is the two-photon coupling, is the number of colors (=3) and is the electromagnetic coupling. Without the QED anomaly the decay amplitude would be proportional to and vanish for massless quarks.

For the isoscalar mesons one also has to consider the QCD gluon axial anomaly. In the chiral limit one finds the relation Shore and Veneziano (1992)

(30) |

Here and denote the two photon couplings of the physical and topological charge density term. Chiral corrections are discussed in Shore (2006) within the context of the two mixing angle scheme. The observed decay rates for the and suggest small gluonic coupling, , with the gluonic term contributing at most 10% of the decay Shore (2006). Most accurate measurements of the and decays come from KLOE-2 Babusci et al. (2013a) and BELLE Adachi et al. (2008) respectively.

When one or both of the photons becomes virtual, the pseudoscalar meson coupling to two photon amplitudes involve transition form factors associated with the spatial structure of the mesons.

There are measurements in both space-like, ,
and time-like, , kinematics
where is the four-momentum transfer in the reaction
^{4}^{4}4Here denotes the squared four-momentum
transfer of the virtual photon
and should not be confused with “”
in our previous discussion where it denoted the topological
charge density.
For consistency with the literature we here keep
for both cases.
.
The space-like region can be studied
through fusion processes in
electron-positron collisions,
with and
production data from
CELLO Behrend et al. (1991),
CLEO Gronberg et al. (1998),
BABAR del Amo Sanchez et al. (2011)
and KLOE-2 Babusci et al. (2013a).
The time-like region is studied in meson decays
, ,
e.g.,
Dalitz decays to lepton pairs in the final state
with positive equal to the invariant mass of
the final state lepton pair .
Single and double Dalitz decays can be studied.
Recent measurements for the
come from the A2 Collaboration at Mainz
Adlarson et al. (2017a),
WASA-at-COSY
Adlarson et al. (2016),
and
NA60 at CERN Arnaldi et al. (2009),
with data from
BES-III Ablikim et al. (2015b) for the .

Production of a pseudocalar meson through fusion of a real and deeply virtual photon, , are described by perturbative QCD in terms of light-front wavefunctions Lepage and Brodsky (1980); Feldmann and Kroll (1998). In the asymptotic large limit, the transition form-factors for

(31) |

Here, mixing is encoded in the decay constants and are the quark charge factors. The light-cone wavefunctions describe the amplitude for finding a quark-antiquark pair carrying light-cone momentum fraction and and transverse momentum . These amplitudes are normalized via

(32) |

As we explained in Section II, one cannot separate an anomalous contribution from when working with gauge invariant observables, e.g., using renormalization. The small OZI-violation in is consistent with RG effects and with the quark-antiquark leading Fock component moving in a topological gluon potential. Glue may be (strongly) excited in the intermediate states of hadronic reactions.

The low region is described using form-factors

(33) |

The slope parameter

(34) |

is often quoted for the decays. Values extracted for the from timelike decays are GeV and GeV from BES-III Ablikim et al. (2015b), with close to the and masses which appear with vector meson dominance of the virtual photon. In the space-like region the CELLO Collaboration found GeV Behrend et al. (1991). Note that the width term is important here for the because of its large mass and short life time. For the slope measured in time-like decays, the most precise measurement of is GeV from the A2 Collaboration at Mainz Adlarson et al. (2017a).

Extending the final states from charged leptons to charged pions, the process includes contributions from both the transition form-factor and also the box anomaly shown in Fig. 3. Recent measurements are from WASA Adlarson et al. (2012) and KLOE-2 Babusci et al. (2013b). For a recent theoretical discussion see Kubis and Plenter (2015).

The transition form factor for deeply virtual was interpreted in Kroll and Passek-Kumericki (2013) to give a quite large (radiatively generated) two gluon Fock component in the wavefunction. In this calculation the glue enters at next-to-leading order. Exclusive central production of the in high-energy proton-proton collisions at the LHC has been suggested as a cleaner probe since here the glue enters at leading-order Harland-Lang et al. (2013).

In lower energy experiments, quark model inspired fits including a “gluonium admixture” Rosner (1983) have been performed to various low energy processes including the decay by the KLOE Collaboration Ambrosino et al. (2009, 2007); Gauzzi (2012) suggesting a phenomenological “gluonium fraction” of . Various theoretical groups’ analyses of the same data suggest values between zero and about 10% depending on form-factors that are used in the fits Thomas (2007); Escribano and Nadal (2007); Di Donato et al. (2012). When trying to extract a “gluonic content” from experiments it is important to be careful what assumptions about glue have gone into the analyses. Photon coupling decay processes are theoretically cleaner with less model dependence in their interpretation.

At high energies, heavy-quark meson decays to light-quark states including the proceed through OZI-violating gluonic intermediate states, e.g., to and giving experimental constraints on the flavor-singlet components in these mesons. In high energy processes large branching ratios for and meson decays to final states have been observed and are believed to be driven in part by coupling to gluonic intermediate states Browder et al. (1998); Aubert et al. (2001); Behrens et al. (1998); Ball et al. (1996); Fritzsch (1997); Atwood and Soni (1997); Hou and Tseng (1998); Bali et al. (2015); Dighe et al. (1996, 1997).

### iv.3 Precision tests of fundamental symmetries

Precision measurements of the muon’s anomalous magnetic moment are an important test of the Standard Model. The anomalous magnetic moment is induced by quantum radiative corrections to the magnetic moment with the proportionality constant between the particle’s magnetic moment and its spin. The present experimental value from BNL Bennett et al. (2006)

(35) |

differs from the present best theoretical expectation by

(36) |

– a 4.1 deviation Jegerlehner (2017). This result is a puzzle also since possible new physics contributions which might have resolved the discrepancy are now seriously challenged by LHC data which are, so far, consistent with the Standard Model and no extra new particles in the mass range of the experiments. New experiments at Fermilab and J-PARC plan to check this result with the Fermilab experiment improving the present statistical error on from 540 to 140 ppb or Hertzog (2016).

One key issue is the size of low-energy QCD hadronic contributions to the muon . These are the biggest source of theoretical uncertainty in the Standard Model prediction with one important ingredient being the hadronic contributions to virtual photon-photon scattering with meson intermediate states. These are sensitive to the , and transition form-factors. Various calculations appear in the literature; see Table 5.13, page 474, in Jegerlehner (2017). Contributions to from the and are typically about with pion contributions between about 5 and 8 . The total hadronic contribution to including vacuum polarization effects is about 690 with a net light-by-light contribution of about 10 after summing over terms with positive and negative signs.

Studies of meson decays also provide new precision tests of discrete symmetries: charge conjugation, , and charge-parity, Jarlskog and Shabalin (2002). The and mesons are eigenstates of parity , charge conjugation and combined parity with eigenvalues = -1, = +1 and = -1. tests include searches for forbidden decays to an odd number of photons. e.g., Nefkens et al. (2005a), (which is also forbidden by angular momentum conservation) Adlarson et al. (2018a), and Nefkens et al. (2005b). Charge conjugation invariance has also been tested in the decay. Here violation can manifest itself as an asymmetry in the energy distributions for and mesons in the rest frame of the meson. The results were found consistent with zero Adlarson et al. (2014b). A possible violating asymmetry in the decay was determined consistent with zero Adlarson et al. (2016).

## V - and -nucleon interactions

Close-to-threshold and production is studied in photon-nucleon and proton-nucleon collisions. Photon induced reactions are important for studies of nucleon resonance excitations; for a recent review see Krusche and Wilkin (2014). meson production is characterized by the strong role of the s-wave N*(1535) resonance. For studies of higher mass excited resonances, recent advances with double polarization observables are playing a vital role. Recent measurements for the come from Mainz Witthauer et al. (2017, 2016), Jefferson Laboratory Al Ghoul et al. (2017); Senderovich et al. (2016) and GrAAL Levi Sandri et al. (2015), with partial wave analysis studies reported in Anisovich et al. (2015).

For the , (quasi-free) photoproduction from proton and deuteron targets has been studied at ELSA Crede et al. (2009); Jaegle et al. (2011); Krusche (2012), MAMI Kashevarov et al. (2017) and by the CLAS experiment at Jefferson Laboratory Dugger et al. (2006); Williams et al. (2009) with new double polarization observables reported in Collins et al. (2017). The production cross-section is isospin independent for incident photon energies greater than 2 GeV, where -channel exchanges are important. At lower energies, particularly between 1.6 and 1.9 GeV where the proton cross-section peaks, the proton and quasi-free neutron cross-sections show different behavior. These data have recently been used in partial wave analysis revealing strong indications of four excited nucleon resonances contributing to the production process: , , , and . Details including the branching ratios for coupling to the are given in Anisovich et al. (2017).

In proton-nucleon collisions the and production processes proceed through exchange of a complete set of virtual meson hadronic states, which in models is usually truncated to single virtual meson-exchange, e.g., , , , and (correlated two-pion) exchanges Faldt and Wilkin (2001); Nakayama et al. (2003); Pena et al. (2001); Deloff (2004); Shyam (2007). For the OZI-violating production is also possible through excitation of non-perturbative glue in the interaction region Bass (1999). The exchange process can also induce nucleon resonance excitation, especially the N*(1535) with production, before final emission of the or meson. The production mechanism is studied through measurements of the total and differential cross-sections, varying the isospin of the second nucleon and polarization observables with one of the incident protons transversely polarized Moskal (2004). The interpretation of these processes is sensitive to the choice of exchanged mesons and nucleon resonances included in the models and the truncation of the virtual exchange contributions which affects, e.g., the meson nucleon form-factors in the calculations.

The near-threshold meson production in nucleon-nucleon collisions has been investigated extensively in the CELSIUS, COSY and SATURNE facilities. The results determined by different experiments for the total Bergdolt et al. (1993); Hibou et al. (1998); Chiavassa et al. (1994); Calen et al. (1996); Smyrski et al. (2000); Moskal et al. (2004, 2010) and differential Moskal et al. (2004); Abdel-Bary et al. (2003); Moskal et al. (2010); Petren et al. (2010) cross-sections for the and for the quasi-free reactions Calen et al. (1996, 1998); Moskal et al. (2009) are consistent within the estimated uncertainties. In the different experiments mesons could be produced up to excess energy of 92 MeV at CELSIUS, 502 MeV at COSY and 593 MeV at SATURNE.

production has been measured in proton-proton collisions close-to-threshold (excess energy between 0.76 and MeV) by the COSY-11 collaboration at FZ-Jülich Moskal et al. (1998, 2000a, 2000b); Khoukaz et al. (2004); Czerwinski et al. (2014a); Klaja et al. (2010b) and at MeV and 8.3 MeV by SPESIII Hibou et al. (1998) and 144 MeV by the DISTO Collaboration at SATURNE Balestra et al. (2000).

For near-threshold meson production, the cross-section is reduced by initial state interaction between the incident nucleons and enhanced by final state interactions between the outgoing hadrons. For comparing production dynamics a natural variable is the volume of available phase space which is approximately independent of the meson mass. Making this comparison for the neutral pseudoscalar mesons, it was found that production of the meson is about six times enhanced compared to the which is six times further enhanced compared to the . The production amplitudes for the and have the same (nearly constant) dependence on the phase space volume in the measured kinematics close-to-threshold, whereas the production amplitude for the exhibits possible growth with decreasing phase space volume due to strong -proton attractive interaction Moskal et al. (2000b). The large production cross-section is driven by strong coupling to the N*(1535).