Chiral corrections to the $1^{-+}$ exotic meson mass

We first construct the effective chiral Lagrangians for the $1^{-+}$ exotic mesons. With the infrared regularization scheme, we derive the one-loop infrared singular chiral corrections to the $\pi_1(1600)$ mass explicitly. We investigate the variation of the different chiral corrections with the pion mass under two schemes. Hopefully, the explicit non-analytical chiral structures will be helpful to the chiral extrapolation of the lattice data from the dynamical lattice QCD simulation of either the exotic light hybrid meson or tetraquark state.


I. INTRODUCTION
According to the naive non-relativistic quark model, the meson is composed of a pair of quark and anti-quark. The neutral mesons do not carry the quantum numbers such as J P C = 0 −− , 0 +− , 1 −+ , 2 −+ .... In contrast, the 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 the exotic states in order to emphasize the difference from the mesons within the quark model. In fact, the exotic quantum numbers provide a powerful handle to probe the non-perturbative behavior of QCD [1][2][3]. In this work we focus on the the exotic meson with J P C = 1 −+ , which is a good candidate of the hybrid meson and tetraquark state.
The current status of the π 1 (1400) and π 1 (1600) is a little murky. There exist speculations that the π 1 (1400) might be non-resonant or it may be a tetraquark candidate instead of a hybrid meson. Although there also exist other possible theoretical explanations such as a tetraquark candidate [16,17] or a molecule/four-quark mixture [18], the π 1 (1600) remains a popular candidate of 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's applicable to all possible interpretations of the π 1 mesons.
There are many investigations of the 1 −+ light hybrid meson mass in literature [20][21][22][23][24][25][26][27][28][29][30][31][32][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 rather large pion mass on the lattice. One may wonder whether such a discrepancy may be removed with dynamical lattice QCD simulations using 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 π 1 (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 π 1 (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 Sec. II and present the formalism in Sec. III. In Sec. IV, we present the numerical results and conclude.

II. LAGRANGIANS
In order to calculate the chiral corrections to the π 1 (1600) meson mass up to the one loop order, we first construct the effective chiral Lagrangian [36,37], which can be expressed as follows where L 0 is the free part According to the decay modes of π 1 (1600), we can write down the interaction terms Because of the 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 m 2 π , The π 1 → πππ decay mode may lead to the two-loop self energy diagram π 1 (1600) in Fig. 1. We ignore the contribution from this diagram since we focus on the chiral corrections to the π 1 (1600) mass up to the 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.

III. CHIRAL CORRECTIONS TO THE π1(1600) MASS
With the above preparation, we start to calculate the chiral corrections to the mass of π 1 (1600). The propagator of the π 1 (1600) is defined as where p is the four momenta of π 1 . At the lowest order, the propagator simply reads and its inverse is Here, m 0 denotes the bare mass of π 1 (1600). We separate the self energy Σ µν (p 2 ) into the transversal and longitudinal parts The full propagator reads which can be expressed as Only the transverse part Σ T (p 2 ) will shift the pole position. Therefore we concentrate on the transversal part of the self energy [43] and consider all the Feynman diagrams shown in Fig. 2 and Fig. 3. The π 1 (1600) mass satisfies the relation In order to obtain the quark mass (∼ m 2 π , m 2 η ) dependence of the self energy corrections, it is convenient to adopt the infrared regularization (IR) scheme [44][45][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 doesn't exist a proper power counting rule for the issue we are dealing with. There are a few different mass scales such as the π 1 mass, the π, η meson masses, the masses of other meson resonances, and the chiral symmetry breaking scale. The mass of the π 1 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 still can be used to derive the non-analytic part of an integral. The non-analytical chiral corrections to the self-energy of the π 1 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 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 does not exist consistent power counting. Fortunately, the non-analytical corrections from different approaches are similar if one considers the oneloop 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].
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 as the 'soft poles' and 'hard poles' in Refs. [49,50].
When one expands the loop integral in terms of the small parameters such as m/M or m/µ where µ is the renormalization scale, one notices that 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 π 1 (1600), we set the kinematical region p 2 ∼ M 2 π1 . 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 M 2 π1 ≫ (M +m) 2 and m 2 ≪ M 2 , including those diagrams with the ρπ, ηπ, b 1 (1235)π, f 1 (1285)π and η ′ π as the intermediate states. The second class corresponds to the condition M 2 π1 ∼ M 2 and m 2 ≪ M 2 , where the intermediate states are the π 1 (1600)η and π 1 (1600)η ′ .

A. 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, b 1 , f 1 , η ′ mesons. l and p denote the loop momentum and external momentum respectively. After performing the l-integration, the above integral reads Since we choose the external momentum p near the mass shell of π 1 (1600), we always have ( Obviously we have 0 < x 2 < x 1 < 1. We now divide the integral into three parts according to the integration interval with We first consider I πX . The assumption p 2 ≫ (M + m π ) 2 and m 2 π ≪ M 2 leads to So we can expand x 1,2 in terms of the small parameter a, Then we have Recall that x 1 ∼ O(1) and x 2 ∼ O(a). When x ∈ [0, x 2 ], we can expand the above integral in terms of the parameter After the interchange of summation and integration, we get Clearly I πX is non-analytic in a for noninteger dimension d. We move on to the I (2) πX part. After shifting the integration variable, we get With the replacement x = (x 1 − x 2 )y, one gets .
πX is complex and proportional to (x 1 − x 2 ) d−3 that can be expanded in powers of x 2 . We expand the third integral I πX in terms of x 2 /x, i.e., Obviously I πX only contains the integer powers of a. It is clear that I (3) πX and the real part of I (2) πX 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, I πX 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][45][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 can not 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 L = 1 ǫ − γ E + ln 4π + 1 and we let µ = m π1 . Up to O(m 4 π ) and O(m 4 η ), we collect the one-loop chiral corrections to the self-energy of the π 1 (1600) below

B. η-π loop
Consider the scalar loop integral for η-π loop After performing the l-integration, the above integral reads Similarly, ∆ can be re-expressed as ∆ 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 I  (1). When y ∈ [0, 1 − x 1 ], we can expand the above integral in terms of the parameter y/(1 − x 2 ) Obviously I πX is non-analytic in b for for non-integer dimension d. In other words, I πX also contributes to the 'infrared singular part'. The 'infrared singular part' of I πη with the imaginary part are thus The chiral correction from the ηπ loop diagram reads The η ′ meson mass is dominated by the axial anomaly, which remains large in the chiral limit. The propagators in the η ′ -π 1 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 π 1 η loop diagram with M 2 π1 ∼ M 2 and m 2 ≪ M 2 , 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 The tadpole diagram of the π1(1600) self energy. The O(m 2 π ) and O(m 4 π ) LECs also contribute to the self energy, which are labeled by square and cross respectively.
Within the IR scheme, the 'infrared singular part' of I π1η reads with The chiral corrections from the tadpole diagrams in Fig. 3 are where we have redefined the low energy constants All the divergence can be absorbed by the counter terms in Eq. 13, which also contribute to m π1 Σ tree π1(1600) = e 1 (m 2 π + m 2 η ) + e 2 (m 2 π + m 2 η ) 2 .
Finally we obtain the chiral corrections to the π 1 (1600) mass up to the one loop order, which is the main result of this work One may note that we treat the intermediate states as stable particles in our above calculation, however, the widths of ρ, b 1 , f 1 are not small. The contributions from the widths of the intermediate states to the non-analytic chiral corrections to the π 1 (1600) mass are summarized in Appendix A.

IV. RESULTS AND DISCUSSIONS
We need to deal with the numerous effective coupling constants before the numerical analysis. Actually the experimental information on the π 1 (1600) decays is not rich. From the current experimental data of the π 1 (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.
From Eqs. (3)- (7), the partial decay width of the π 1 (1600) reads where p π is the pion decay momentum. With the total decay width of π 1 (1600) around 300 MeV as input [53], we get For the π 1 π 1 η coupling constant, we use g π1η ∼ 1 1.6Fη GeV −1 ∼ 5.3 GeV −1 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 M 2 π = M 2 η → 0 when m s = m approaches zero simultaneously.
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 π 1 (1600) mass from different loop diagrams with the pion mass in Figs. (4)- (5). The most important chiral correction to the π 1 (1600) mass comes from the π 1 η loop. The chiral corrections from the πρ, πη and πη ′ loops are positive and increase with m π while the corrections from the ηπ 1 , πb 1 and πf 1 loops are negative. On the other hand, the chiral corrections from the ηπ 1 , πb 1 and πf 1 loops are very sensitive to the pion mass.
The coupling constants d i (i = 1, 2), d * j (j = 1, 2) contribute to the tadpole diagram while e k (k = 1, 2) are the 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 of too many unknown coupling constants. According to PDG [4], the π 1 (1600) was observed in the b 1 π, η ′ π and f 1 π modes. The Compass collaboration reported the π 1 (1600) in the ρπ mode [9]. On the other hand, the π 1 (1400) was observed in the ηπ mode. Both the π 1 (1600) and π 1 (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 π 1 (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 π 1 (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 π 1 mesons since our analysis does not rest on the inner structure of the π 1 mesons. Hopefully, the explicit non-analytical chiral structures will be helpful to the chiral extrapolation of the lattice data from the dynamical lattice QCD simulation of either the exotic light hybrid meson or tetraquark state.
Now x 1 , x 2 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's interesting to note that the above expression contains a non-analytic chiral correction to the imaginary part, which is proportional to Γ M 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 π 1 (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, b 1 π, f 1 π loops, we collect the non-analytic chiral corrections to the mass of the π 1 (1600) up to O(m 4 π ),