Chiral corrections to the 1⁻⁺ exotic meson mass^*

According to the naive non-relativistic quark model, mesons are composed of a quark and an anti-quark. The neutral mesons do not carry quantum numbers such as J^PC = 0⁻⁻, 0⁺⁻, 1⁻⁺, 2⁻⁺.... In contrast, non-conventional mesons such as the hybrid meson, tetraquark states and glueballs are allowed in quantum chromodynamics (QCD) and can have these quantum numbers. Sometimes these states are denoted as exotic states in order to emphasize the difference from mesons within the quark model. In fact, the exotic quantum numbers provide a powerful handle to probe the non-perturbative behavior of QCD [1–3]. In this work we focus on the exotic meson with J^PC = 1⁻⁺, which is a good candidate for both hybrid meson and tetraquark state.

There are three candidates with J^PC = 1⁻⁺: π₁(1400), π₁(1600) and π₁(2000). Their masses and widths are (1376±17, 300±40) MeV, and (2014 ± 20 ± 16, 230 ± 21 ± 73) MeV [4] respectively. π₁(1600) was first observed in the reaction π⁻p → π⁻π⁻π⁺p in 1998 [5, 6]. Later the π₁(1600) was confirmed in the η'π [7], f₁(1285)π [8, 9] and b₁(1235)π channels [10, 11]. Some experiments also indicated the possible existence of π₁(1400) [12–14] and π₁(2000) [8]. The existence of π₁(2000) awaits further experimental confirmation. This state was not included in the PDG since 2010 [15].

The current status of the π₁(1400) and π₁(1600) is a little murky. There is speculation that the π₁(1400) might be non-resonant or it may be a tetraquark candidate instead of a hybrid meson. Although there are other possible theoretical explanations such as a tetraquark candidate [16, 17] or a molecule/four-quark mixture [18], the π₁(1600) remains a popular candidate for the light hybrid meson [19]. The present calculation is based on the following three facts: the 1⁻⁺ exotic quantum number, the SU(3) flavor structure and the current available decay modes. In other words, it is applicable to all possible interpretations of the π₁ mesons.

There are many investigations of the 1⁻⁺ light hybrid meson mass in the literature [20–33]. The 1⁻⁺ mass extracted from the quenched lattice QCD simulation ranges from 1.74 GeV [34] and 1.8 GeV [35] to 2 GeV [25], which is significantly larger than the experimental value. This apparent discrepancy is slightly disturbing. One possible reason may be due to the fact that all these lattice QCD simulations were performed with quenched configurations and a rather large pion mass on the lattice. One may wonder whether such a discrepancy may be removed with dynamical lattice QCD simulations using the physical pion mass. Then one may make chiral extrapolations to extract the physical mass of the hybrid meson.

In this work we shall derive the explicit expressions of the non-analytical chiral corrections to the π₁(1600) mass up to one-loop order, which may be used to make the chiral extrapolations if the dynamical lattice QCD simulations are available. Throughout our analysis, we focus on the variation of the π₁(1600) meson mass with m_u,d or m_π. In the SU_F(3) chiral limit m_u,d,s → 0, m_π,η → 0. The SU_F(2) chiral limit is adopted where m_u,d → 0 and m_s remains finite. Then, the eta meson mass does not vanish due to the large strange quark mass.

This paper is organized as follows. We construct the effective chiral Lagrangians in Section 2 and present the formalism in Section 3. In Section 4, we present the numerical results and conclude.

In order to calculate the chiral corrections to the π₁(1600) meson mass up to one-loop order, we first construct the effective chiral Lagrangian [36, 37], which can be expressed as follows

where 𝓛₀ is the free part

According to the decay modes of π₁(1600), we can write down the interaction terms

Because of chiral symmetry and its spontaneous breaking, all the pionic coupling constants should vanish when either the pion momentum or its mass goes to zero. The S-wave coupling constants g_b1π and g_f1π arise from the finite current quark mass correction. Therefore, these coupling constants are proportional to ,

The π₁ → πππ decay mode may lead to the two-loop self energy diagram π₁(1600) in Fig. 1. We ignore the contribution from this diagram since we focus on the chiral corrections to the π₁(1600) mass up to one-loop order in this work. Moreover, some contribution of this two-loop diagram may have been partly included in the one-loop diagram with the intermediate ρ and π meson because the ρ meson is the two-pion resonance.

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Furthermore, we need the chiral interaction between the π₁(1600) and the pseudo scalar mesons, which is similar to the chiral Lagrangians of the vector mesons [38–42]. It should be stressed that the π₁π₁π interaction is forbidden by G-parity conservation. We have

For the π₁π₁ππ and π₁π₁ηη interaction, we have

In order to absorb the divergence in the one-loop chiral corrections, we need the following counter terms

𝓛_counter is similar to the chiral Lagrangians of the vector mesons in the form of 〈χ₊〉〈V_μV^μ〉 and 〈χ₊〉²〈V_μV^μ〉, where V_μ is the vector meson and the notation χ₊ is related to the current quark mass.

With the above preparation, we start to calculate the chiral corrections to the mass of π₁(1600). The propagator of the π₁(1600) is defined as

where p is the four momentum of π₁. At the lowest order, the propagator simply reads

and its inverse is

Here, m₀ denotes the bare mass of π₁(1600).

We separate the self energy Σ_μν(p²) into the transversal and longitudinal parts

The full propagator reads

which can be expressed as

Only the transverse part Σ_T(p²) will shift the pole position. Therefore we concentrate on the transverse part of the self energy [43] and consider all the Feynman diagrams shown in Fig. 2 and Fig. 3. The π₁(1600) mass satisfies the relation

In order to obtain the quark mass dependence of the self energy corrections, it is convenient to adopt the infrared regularization (IR) scheme [44–46] to calculate the loop integrals. Usually, the IR method is used in order not to break the power counting while dealing with the integral. Unfortunately, there is no proper power counting rule for the issue we are dealing with. There are a few different mass scales such as the π₁ mass, the π, η meson masses, the masses of other meson resonances, and the chiral symmetry breaking scale. The mass of the π₁ is so high that the π, η and other light mesons can take large momenta, and thus the convergence of a chiral expansion is not ensured. However, for our purpose, the IR method can still be used to derive the non-analytic part of an integral. The non-analytical chiral corrections to the self-energy of the π₁ are inherent and intrinsic due to the presence of the chiral fields, and the non-analytical chiral structures are universal and model independent to a large extent. One may derive them using very different theoretical approaches such as the chiral quark model, effective chiral Lagrangians at the hadronic level or rigorous chiral perturbation theory (ChPT). With ChPT, one can include both analytical and non-analytical corrections order by order with consistent power counting. In contrast, with the effective chiral Lagrangians at the hadronic level as employed in this work, there is no consistent power counting. Fortunately, the non-analytical corrections from different approaches are similar if one considers the one-loop diagrams. The non-analytical structures may play an important role in the chiral extrapolation of the dynamical lattice QCD simulation of the 1⁻⁺ exotic meson mass, which is sensitive to the pion mass on the lattice. Within the IR scheme, the so-called 'infrared singular part' turns out to be the main contribution of the loop integral in the chiral limit. However, one can also find the full expressions of these loop integrals by performing the standard Lorentz invariant calculation in Refs. [47, 48].

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For a certain diagram, there are three mass scales, M_π1 and the masses of the two intermediate states m, M. We assume M > m. The main contribution of a loop integral comes from the poles of the propagators, which are called the 'soft poles' and 'hard poles' in Refs. [49, 50].

When one expands the loop integral in terms of small parameters such as m/M or m/μ, where μ is the renormalization scale, the 'soft part' contribution contains all the terms which are non-analytic in the expansion parameter. In contrast, the 'hard part' is a local polynomial in these parameters which can be absorbed by the low energy constants of higher order Lagrangians [46].

Since we are interested in the small chiral fluctuations around the mass shell of π₁(1600), we set the kinematical region . In particular, we set the the regularization scale to be M_π1. These self-energy diagrams can be divided into two categories. The first class of diagrams fulfills the condition and m² ≪ M², including those diagrams with the ρπ, ηπ, b₁(1235)π,f₁(1285)π and η'π as the intermediate states. The second class corresponds to the condition and m² ≪ M², where the intermediate states are the π₁(1600)η and π₁(1600)η'.

3.1. The light meson pion loop

Now we deal with the light meson pion loop integration corresponding to diagrams (a)–(d) in Fig. 2. Consider the scalar loop integrals

where X represents the ρ, b₁, f₁, η' mesons. l and p denote the loop momentum and external momentum respectively. After performing the l-integration, the above integral reads

with

Since we choose the external momentum p near the mass shell of π₁(1600), we always have . Δ can be re-expressed as Δ = b(x−x₁)(x−x₂), with

Obviously we have 0 < x₂ < x₁ < 1. We now divide the integral into three parts according to the integration interval

with

We first consider . The assumption p² ≫ (M + m_π)² and leads to

So we can expand x_1,2 in terms of the small parameter a,

Then we have

Recall that and . When x ∈ [0,x₂], we can expand the above integral in terms of the parameter x/x₁

After the interchange of summation and integration, we get

Clearly is non-analytic in a for non-integer dimension d.

We move on to the part. After shifting the integration variable, we get

With the replacement x = (x₁ − x₂)y, one gets

is complex and proportional to (x₁ – x₂)^d–3 that can be expanded in powers of x₂.

We expand the third integral in terms of x₂/x, i.e.,

Obviously only contains the integer powers of a.

It is clear that and the real part of are regular in a and will not produce any infrared singular terms for an arbitrary value of the dimension d. Thus these parts can be absorbed into the low energy constants of the effective Lagrangian. On the other hand, develops an infrared singularity as a → 0 for negative enough dimension d. This part is the so-called 'infrared singular part' of I_πX in the IR method of Refs. [44–46]. The 'infrared singular part' contains all the terms which are non-analytic in a as the typical chiral log terms ln a. Such terms cannot be absorbed into the low energy constants of the effective Lagrangian. Furthermore, the contribution of the 'infrared singular part' dominates the I_πX as a → 0.

Finally we obtain the 'infrared singular part' in I_πX with the imaginary part,

where and we let μ = m_π1.

Up to and , we collect the one-loop chiral corrections to the self-energy of the π₁(1600) below:

3.2. η-π loop

Consider the scalar loop integral for the η-π loop:

After performing the l-integration, the above integral reads

with

Similarly, Δ can be re-expressed as Δ = b(x−x₁)(x−x₂), with

Obviously we have

So we can expand x_1,2 in terms of a and b

With the same method, we divide the integral into three parts

with

The and are similar for the case in the previous section, where belongs to the 'infrared singular part' of I_πη and contains an imaginary part. However, the is quite different. To calculate the , we first shift the integration variable

Since and , when y ∈ [0,1 − x₁], we can expand the above integral in terms of the parameter y/(1 − x₂)

Obviously is non-analytic in b for for non-integer dimension d. In other words, also contributes to the 'infrared singular part'. The 'infrared singular part' of I_πη with the imaginary part is thus

The chiral correction from the ηπ loop diagram reads

3.3. η(η')-π₁ loop

The η' meson mass is dominated by the axial anomaly, which remains large in the chiral limit. The propagators in the η'-π₁ loop do not produce a 'soft pole' contribution. In other words, the loop integral does not contain the 'infrared singular part'.

Now we consider the π₁η loop diagram with and m² ≪ M², which is similar to the nucleon self energy diagram. We can use the standard IR method in Ref. [46] to obtain the 'infrared singular part'. First we define the dimensionless variables

The corresponding scalar loop integral is

where

Within the IR scheme, the 'infrared singular part' of I_π1η reads

with

and the regularization scale μ = m_π1. The chiral correction from the π₁η loop diagram reads

3.4. Tadpole diagrams

The chiral corrections from the tadpole diagrams in Fig. 3 are

where we have redefined the low energy constants

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All the divergence can be absorbed by the counter terms in Eq. (13), which also contribute to m_π1

Finally we obtain the chiral corrections to the π₁(1600) mass up to one-loop order, which is the main result of this work

Note that we treat the intermediate states as stable particles in our above calculation. However, the widths of ρ, b₁, f₁ are not small. The contributions from the widths of the intermediate states to the non-analytic chiral corrections to the π₁(1600) mass are summarized in Appendix.

We need to deal with the numerous effective coupling constants before the numerical analysis. Actually the experimental information on the π₁(1600) decays is not rich. From the current experimental data of the π₁(1600) decays, we can make a very rough estimate of the values of g_ρπ, g_ηπ, g_η'π, g_f1π and g_b1π. The others still remain unknown.

A partial wave analysis in Ref. [51] gives the branching ratio

An analysis based on the VES experiment leads to [52]

The E852 collaboration reported [8]

In order to make a very rough estimate of these coupling constants, we combine the above measurements and set the branching ratio to be

From Eqs. (3)–(7), the partial decay width of the π₁(1600) reads

where is the pion decay momentum.

With the total decay width of π₁(1600) around 300 MeV as input [53], we get

For the π₁π₁η coupling constant, we use where the F_η ≈ 0.1 GeV is the decay constant of η. This ad hoc value was estimated with the very naive dimensional argument, which might be too large.

From the tree-level Lagrangian of chiral perturbation theory,

We consider two cases in the numerical analysis. Case 1 corresponds to the SU_F(3) chiral limit where when m_s = m approaches zero simultaneously.

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Since the strange quark is sometimes treated as a heavy degree of freedom in the lattice QCD simulation, we also consider Case 2, which corresponds to the SU_F(2) chiral limit. Now we fix the strange quark mass and let the up and down quark mass approach zero. In the SU_F(2) chiral limit, the η meson mass remains finite. We have

We collect the variation of the chiral corrections to the π₁(1600) mass from different loop diagrams with the pion mass in Figs. (4)–(5). The most important chiral correction to the π₁(1600) mass comes from the π₁η loop. The chiral corrections from the πρ, πη and πη' loops are positive and increase with m_π while the corrections from the ηπ₁, πb₁ and πf₁ loops are negative. Furthermore, the chiral corrections from the ηπ₁, πb₁ and πf₁ loops are very sensitive to the pion mass.

The coupling constants d_i (i = 1,2), contribute to the tadpole diagram while e_k (k = 1,2) are low energy constants. They are unknown at present. Although this kind of contribution may be significant, we do not present their variations with the pion mass because there are too many unknown coupling constants.

According to PDG [4], the π₁(1600) was observed in the b₁π, η'π and f₁π modes. The Compass collaboration reported the π₁(1600) in the ρπ mode [9]. The π₁(1400) was observed in the ηπ mode. Both the π₁(1600) and π₁(1400) signals are very broad with a decay width of 241±40 MeV and 330±35 MeV respectively [4]. These two signals overlap with each other. In this work, we have taken into account all the above possible decay modes and calculated the one-loop chiral corrections to the π₁(1600) mass. We have employed two different methods to deal with the loop integrals and derived all the infrared singular chiral corrections explicitly.

From the available experimental measurement of the partial decay width of the π₁(1600) meson, we extract the coupling constants. We investigate the variation of the different chiral corrections with the pion mass under two schemes. The present calculation is applicable to all possible interpretations of the π₁ mesons since our analysis does not rest on the inner structure of the π₁ mesons. Hopefully, the explicit non-analytical chiral structures will be helpful to the chiral extrapolation of lattice data from the dynamical lattice QCD simulation of either the exotic light hybrid meson or the tetraquark state.

Appendix. Contributions generated by the finite widths of the intermediate states

In this Appendix we deal with the scalar loop integrals when the intermediate states have a finite decay width Γ.

with

where the X represents ρ,b₁, f₁, M and Γ are the corresponding mass and width, and

We expand x_1,2 in terms of a

In our case, Γ_X∼m_π. We treat the as and get

The original integral can be re-expressed as

Now x₁, x₂ are complex while the integration variable x is real, which renders the evaluation of the integral straightforward. We have

After extracting the non-analytic chiral corrections from the above expression, we get

It is interesting to note that the above expression contains a non-analytic chiral correction to the imaginary part, which is proportional to and vanishes when Γ → 0. In comparison, when we treat the intermediate states as stable particles, the imaginary parts of the chiral corrections to the self-energy of the π₁(1600) are analytic in the pseudo-scalar meson mass. In the limit of Γ = 0, we recover the results in the previous sections in the text.

For the ρπ, b₁π, f₁π loops, we collect the non-analytic chiral corrections to the mass of the π₁(1600) up to ,