Anomalous interactions between mesons with nonzero spin and glueballs

Topologically nontrivial fluctuations control the anomalous interactions for the $\eta$ and and $\eta \prime$ pseudoscalar mesons. We consider the anomalous interactions for mesons with higher spin, the heterochiral nonets with $J^{P C} = 1^{+ -}$ and $2^{- +}$. Under the approximation of a dilute gas of instantons, the mixing angle between non-strange and strange mesons decreases strongly as $J$ increases, and oscillates in sign. Anomalous interactions also open up new, rare decay channels. For glueballs, anomalous interactions indicate that the $X(2600)$ state is primarily gluonic.

Introduction: Quantum Chromodynamics (QCD) is close to the chiral limit, where the up, down and strange quarks (u, d, and s) are very light.Consequently, when the global chiral symmetry is spontaneously broken in the vacuum, from SU (3) L × SU (3) R × U A (1) → SU (3) V , nine pseudo-Goldstone bosons should appear.Instead, there are only eight: the usual octet of pions, kaons, and the η meson, while the η ′ is much heavier than expected.This occurs because the axial U A (1) symmetry of the classical theory is broken by quantum effects, through the anomaly of Adler, Bell, and Jackiw [1,2].This splits the singlet η ′ meson from the octet mesons, and gives it a mass through fluctuations which are topologically nontrivial [3][4][5][6][7].The most familiar example are instantons: classical solutions of the gluon field equations in Euclidean spacetime [3], whose effects can be computed semiclassically [4,5].While instantons dominate at high temperature, in vacuum truly quantum fluctuations also contribute [7].
While anomalous interactions are especially dramatic for the pseudoscalar multiplet, it is natural to ask how the axial anomaly affects other mesons, such as conventional mesons with higher spin, or unconventional ones, such as glueballs.As both mesons with nonzero spin and glueballs are massive, the effects of the axial anomaly are more subtle, affecting the mass splittings, mixing, and decays of some fields in these multiplets.
The anomalous interactions between heterochiral and homochiral mesons are very different.Heterochiral mesons have anomalous interactions with no derivatives, which directly affect their mass spectrum, and with few derivatives, which affect their decays.In contrast, homochiral mesons only have anomalous interactions with many derivatives, through the Wess-Zumino-Witten term [9].
In this Letter we construct the anomalous interactions for the underlying quark operators, and their counterparts for heterochiral mesons and for the pseudoscalar glueball, in a dilute gas of instantons (DGI).After reviewing the well known case of J = 0, the extension to heterochiral mesons with spin J = 1 and J = 2, and then with a glueball, is direct.Because of the axial anomaly, massless quarks have exact zero modes, so that instanton contributions to anomalous operators can be computed by saturating these operators with these zero modes [4,5].The only change with nonzero spin is that the vertices which tie the zero modes differs.
At the outset we acknowledge that the topological structure of the vacuum is surely more complicated than a dilute gas of instantons [7].Nevertheless, the anomalous operators which we compute in this work are novel, and we expect a DGI to give a first estimate of their magnitude.Indeed, a recent analysis of the chiral phase transition near the chiral limit suggests that a DGI may well underestimate the effects of topologically nontrivial fluctuations [10].
The present analysis is meant to motivate further study from numerical simulations on the lattice, and especially from experiment.Thus we concentrate on phenomenology, notably the splitting and mixing between mesons in a given multiplet, and on new decay channels which open up for mesons and glueballs.
Heterochiral multiplets: Mesons are classified according to their quantum numbers under spin, parity, and charge conjugation, J P C .The total spin J = L + S is the sum of angular momentum L and the spin S, with P = (−1) L+1 and C = (−1) L+S .With massless quarks, classically left and right handed quarks are invariant the symmetry group of Here q L,R = P L,R q, where P L,R = 1 2 (1 ∓ γ 5 ).U L and U R are flavor rotations in SU L (3) and SU R (3), respectively, while exp(∓iα/2) is a rotation for axial U A (1).This transformation relates nonets with the same spin and opposite parity.
A heterochiral meson with spin zero is proportional to the quark operator q L q R ; those of higher spin are given just by inserting powers of the covariant color derivative, ← → D µ , between the quark fields.For J = 0, 1, and 2, these are Φ, Φ µ and Φ µν , as shown in Table (I).Because ← → D µ only acts upon color and not flavor, these mesons all transform identically under chiral rotations [8].Typically bosonic fields in an effective Lagrangian have dimensions of mass.To ensure this it is necessary to introduce the dimensionful constants M 0 , M 1 , and M 2 for J = 0, 1, and 2 in Table (I).Since the spin is increased by inserting more powers of ← → D µ , the power of M increases with J, ∼ 1/M J+2 J .A major concern in the phenomenological analysis below is the relative magnitude of these mass scales.
We begin by reviewing the experimental evidence for heterochiral multiplets.
(i) Heterochiral mesons with J = 0: Besides the usual pions and kaons, there are the flavor eigenstates, η N ≡ 1/2 (ūu + dd) and η S ≡ ss.Because of the axial anomaly, Eq. ( 6), these mix to form the physical η and η ′ states: The mixing angle, 11], is large and negative.This demonstrates that the axial anomaly ensures that the physical states are closer to the octet and singlet configurations, respectively [12,13].In all they form a pseudoscalar nonet, P ij = 1 2 qj iγ 5 q i .The assignment for the scalar mesons, with J P C = 0 ++ , is still under debate [14-21] [22].In all, Φ = S + iP , Table (I).

I).
Instanton induced interactions: It is well known that instantons generate the interaction [4,5,13] Anticipating later results, we introduce the J-dependent coupling This is a weighted average over the instanton density n(ρ), which for three massless quarks and three colors is given by [29][30][31]: The expression for the running coupling constant g(ρΛ MS ) is given in Eq. ( 13) ( of Supplementary Material) to two loop order, while the instanton density n(ρΛ MS ) is illustrated in Fig. (1).See the Supplementary Material (SM) for further details.Taking the renormalization mass scale Λ MS = 300 MeV [24], for J = 0 we obtain k 0 ≈ 2.57 • 10 6 GeV −5 [32].
Assuming that the effective bosonic field Φ is proportional to the quark bilinear [13,33,34], The bosonic coupling a 0 depends on k 0 and the constant M 0 in Table I, a 0 = k 0 M 6 0 /48 > 0. The mixing angle of Eq. ( 2) is then [35] where the chiral condensate of non-strange quarks ϕ N can be expressed in terms of the pion decay constant A dilute gas of instantons gives negative β 0 , in agreement with phenomenology.Imposing the phenomenological value β 0 = −43.6 • and using the parameters of Refs.[36,37], so that the value of M 0 is close to that for ϕ N .The generic anomalous interaction for three flavors is illustrated in the left part of Fig. (2).The only change with higher spin is that as J increases, powers of ← → D µ are inserted between the zero modes.This is responsible for the factor of ρ 2J in the anomalous interactions, k J in Eq. ( 4).
For spin one, the simplest anomalous interaction is quadratic in Φ µ and linear in Φ: where we introduce the symbol [38] ϵ Given the transformation properties of Φ and Φ µ in Table (I), Eqs. ( 6) and ( 9) are manifestly invariant under SU L (3)×SU R (3).Similarly, as the product of three heterochiral fields, these terms are not invariant under U A (1), but Z(3).These anomalous interactions were first obtained in Ref. [8] entirely from considerations of symmetry.In this paper we now compute their magnitude, as well as anomalous glueball interactions, in a DGI.
To relate the k J to physical processes, we need the values for the constants of proportionality M J between quark and mesonic operators.For spin one, we find k 1 = 9.91 • 10 6 GeV −7 , which for M 1 = M 0 gives The corresponding mixing angle is approximately: For a DGI this mixing angle is positive.Using the value for a strange quark condensate ϕ S = ⟨0| η S |0⟩ ≈ 130 MeV, and assuming M 1 = M 0 = 170 MeV, we obtain a small value of β 1 ≃ 0.75 • ; for a larger value of M 1 = 270 MeV, the mixing angle increases to As illustrated in Fig. (3) [40], experimental results favor a positive value [41][42][43][44], as do numerical simulations on the lattice [45].The anomalous interactions in Eq. ( 9) also open up new decay modes.For example, Γ(ρ(1700) → h 1 (1415) π) = 0.027(M 1 /M 0 ) 6 MeV, which if M 1 = M 0 is rather small.Other anomalous decays are discussed in the Supplementary Material.Measuring such processes can be used to fix the value of M 1 .FIG. 3. β1 in a DGI compared to the experiment [41][42][43][44] and the lattice (LQCD) [45], for M1 = M0 = 170 MeV and M1 = 270 MeV.
An interaction term with one J = 0 meson and two J = 2 heterochiral mesons is We find k 2 = 4.05 • 10 7 GeV −9 , so when M 2 = M 0 , The mixing angle for the pseudotensor multiplet is negative [46], Assuming that M 2 = M 0 , the DGI gives a small mixing angle, β 2 ≈ −0.05 • .This agrees with lattice QCD [47], but not with the large value of β 2 ≃ −42 • extracted in Ref. [26] from the decay rates.To fit such a large mixing angle requires M 2 = 2.4 M 0 .We see that anomalous terms generate mixings between the octet and singlet for all (pseudo-)heterochiral mesons.These mixing angles do decrease strongly with J, for two reasons.First, comparing the values in Eqs. ( 8), (36), and (14), each a J decreases by about ≈ 1/10 as J increases by one (assuming that M 0 = M 1 = M 2 ).This is because anomalous coupling k J in Eq. ( 4) involves ρ 2J , and a DGI peaks at small ρΛ M S ∼ 0.5, Fig. (1).Second, tan β J is proportional to the inverse of the mass squared of the mesons, Eqs. ( 7), (12) and (15).For J = 0, the η and η ′ are pseudo-Goldstone bosons, and so much lighter than ordinary mesons, with J = 1 and 2. The former may be an artifact of a dilute gas of instantons; the latter is not.Further, that the sign of β J flips as J changes is dynamical, and does not follow just from the chiral symmetry.This is a nontrivial test of our model, and appears to agree with experiment.
Besides mixing terms, there are also anomalous terms which involve derivatives of the spin zero field Φ, and so exclusively affect decays.For example, a term which couples heterochiral mesons with J = 0, 1, and 2 is In a DGI An anomalous interaction coupling two heterochiral J = 0 mesons to a J = 2 meson is For a DGI, Again, numerous anomalous decay channels open up.For example, Γ(η 2 (1870 MeV. Measuring such processes will significantly constrain the values of M 1 and M 2 , and test the consistency of our approach.
Besides anomalous mesonic interactions, those involving glueballs follow immediately, and are illustrated in the right part of Fig. 2.An anomalous interaction between a pseudoscalar glueball and heterochiral mesons is given by the term In a DGI c g ≈ 11.Then, by using Ref. [48], we obtain Γ( G0 → K Kπ) ≈ 0.24 GeV and Γ( G0 → ππη ′ ) ≈ 0.05 GeV.In contrast to the anomalous decays between heterochiral mesons, these are large values.Notably, the BE-SIII collaboration has recently seen a pseudoscalar resonance, denoted as X(2600), in the ππη ′ channel [49].
Our results support the interpretation of this resonance as mostly gluonic, with a decay enhanced by the chiral anomaly [50].Further anomalous decays involving heterochiral mesons with higher spin follow directly, and include interactions such as G0 ϵ[Φ Φ µ Φ µ ] − c.c. .We conclude by noting that there are many other anomalous interactions which can be computed with our techniques.
The authors thank Fabian Rennecke and Adrian Königstein for useful discussions.R.D.P. was supported by the U.S. Department of Energy under contract DE-SC0012704, and by the Alexander von Humboldt Foundation.F.G. acknowledges support from the Polish National Science Centre (NCN) through the OPUS project 2019/33/B/ST2/00613.S.J. acknowledges financial support through the project Development Accelerator of the Jan Kochanowski University of Kielce, co-financed by the European Union under the European Social Fund, with no.POWR.03.05.00-00-Z212 / 18 and is thankful to the Nuclear Theory divison of the Brookhaven National Labaratory for warm hospitality during his visit in which the current project was initiated.

SUPPLEMENTAL MATERIAL DETAILS ON THE INSTANTON SOLUTION
Instantons are self-dual solutions of the classical Euclidean equation of motion of QCD.The representation of a singular gauge instanton solution involves three parameters: the instanton location z, instanton size ρ, and gauge group orientation U : where the t'Hooft symbol is defined as and for the anti-self-dual solution η → η The corresponding fermionic zero-modes of the Dirac operator are where a ∈ {1, • • • , N c } is the color index, i, j, k ∈ {1, 2} are the spinor indices, the anti-symmetric tensor ε is included and three Pauli ⃗ σ matrices are used to represent the gamma matrices in the chiral representation, To obtain the quark zero-modes in the case of an anti-instanton, we perform the substitution φ The quark zero-mode is approximated using a free fermionic propagator at a large distance from the instanton centre z: where The zero-mode determinant for the diagonal source which reduces to the N f -point Green's functions in which two fermion lines connect the source to the origin, as shown in Fig. 2.

FROM INSTANTONS TO EFFECTIVE LAGRANGIANS
An ansatz for the effective Lagrangian in terms of the quark zero-modes and the isospin half Dirac spinor ω j given in Ref. [5] is written as: which leads to the following form of the amplitude with the source J (x f ) : Considering the limit in Eq.( 23) within Eq. ( 25), we obtain where the factor ρ −1 is included because of the dimensional difference between the quark zero mode and the quark propagator.We have also neglected the averaging over gauge rotation.This is possible because one can show that the integration over the gauge group element always contains the singlet part, which is identical to the determinant term [71].A comparison with the previous equation implies that The generating function in the semi-classical limit (for N f = 3) reads where k 0 is defined in Eq. ( 4).The numerical value 7.07534 in Eq. ( 5) is taken from Ref. [31] (upon settings N f = N c = 3 within the modified subtraction scheme.Moreover, the running coupling at the two-loop order for small x-values is: One can proceed similarly for higher spins and estimate various couplings within the DGI model.For instance, the Lagrangian that includes spin-1 heterochiral nonet in terms of quark zero-modes reads: where the coupling k 1 is defined in Eq. ( 4).In the last line, the property of the Fermi-statistics is used for the quark fields.An analogous expression for the case of spin-2 holds: The interaction Lagrangian for the pseudoscalar glueball in Ref. (18) is, in terms of the quark and gluonic fields: The pseudoscalar glueball is linked to the topological charge density via: where λ g = ⟨0| g 2 tr(G µν Gµν ) |0 −+ ⟩ is expected to be larger [72] than the scalar glueball coupling strength 15 GeV 3 calculated within the DGI model in Ref. [73].Considering the modification due to the gluonic fields in Eq. ( 44), we obtain the following numerical value for the decay constant of the pseudoscalar glueball for a DGI:

EXTENDED FORMS OF THE LAGRANGIANS
In this section, we present the extended forms of the Lagrangians used in the main part of the paper.The interaction with the scalar mesons has been disregarded, as their precise nonet identification is not yet known.
The extended form of Eq. ( 9) describes the anomalous decay of the orbitally excited vector mesons: An analogous Lagrangian for spin-2 fields in Eq. ( 13) is, explicitly: The Lagrangian in Eq. ( 16) is, in detail: The decay of a spin-2 heterochiral meson into two pseudoscalars is given by Eq. ( 17).In terms of the component where Z π = Z η N = 1.709,Z η S = 1.539 and Z K = 1.604 are from Ref. [48].Other interaction Lagrangians for the pseudoscalar glueball can be obtained, such as

DECAY FORMULAS
The Lagrangian in Eq. ( 9) gives the decay rate where the Heaviside step-function is denoted by Θ(x), κ i 's are given in Table II, and the modulus of the outgoing particle momentum is  The decay rate from the Lagrangian in Eq. ( 16) is For J = 2 mesons, while experimentally the situation is unclear for the excited tensor mesons [74], we assume it to be f 2s ≡ f 2 (2300), and the expression of its decay into two pions reads Decays for heterochiral mesons with spin-1 are given in Table III, and those with spin-2 mesons in Table IV.

FIG. 2 .
FIG.2.Anomalous processes induced by instantons: to left, cubic couplings between the heterochiral-type, Φ's, and to the right, their coupling to a glueball field, G.

TABLE II .
Decay coefficients for heterochiral mesons.

TABLE IV .
Widths for the anomalous decays of spin-2 mesons in a DGI.