Internal particle width effects on the the triangle singularity mechanism in the study of the and puzzle
Abstract
In this article, the analyticity of triangle loop integral with complex masses of internal particles is discussed in a new perspective, base on which we obtain the explicit width dependence of the absorptive part of the triangle amplitude. We reanalyze the decay pattern of with the width effects included in the triangle singularity (TS) mechanism. Based on the present experimental information, we provide a selfconsistent description of the , , and decay channels for . Our results confirm the claim that the TS mechanism plays a decisive role in the understanding of the and puzzle. Namely, the observed differences of resonances within the mass region of GeV are originated from the same state. For the isospin violated process , we identify an additional contribution to the mixing via the TS mechanism.
I Introduction
It has been a longstanding question on the existence of glueball in hadron spectroscopy. This exotic object as the bound state of gluons predicted by QCD has been a crucial piece of information for our understanding of strong interaction theory in the nonperturbative regime. The corresponding theoretical study and experimental search for glueball states have been the topical subjects in hadron physics. However, although tremendous efforts have been made, the indisputable evidence for their existence is still lacking. In the glueball spectrum the lowlying states include scalar (), tensor () and pseudoscalar (). So far, the lattice QCD (LQCD) simulations Morningstar:1999rf (); Bali:1993fb (); Chen:2005mg (); Chowdhury:2014mra (); Richards:2010ck (); Sun:2017ipk () suggest that their typical masses are about , , and GeV, respectively. The mass hierarchy has been a stable feature from LQCD. While there have been topical reviews on the glueball spectrum in the literature for the scalar and tensor, our focus in this work is on the pseudoscalar glueball. We will discuss the longstanding controversial issues involved in the identification of the pseudoscalar glueball candidate, and stress that a selfconsistent picture can only be obtained with a special kinematic effect, i.e. the triangle singularity (TS) or Landau singularity mechanism Landau:1959fi (); Cutkosky:1960sp (); bonnevay:1961aa (); Peierls:1961zz (), implemented.
In the literature the most promising candidate for the pseudoscalar glueball has been assigned to the since it was introduced as an additional state to the nearby and around early 1990s Bai:1990hs (); Bolton:1992kb (); Augustin:1989zf (); Augustin:1990ki (); Bertin:1995fx (); Bertin:1997zu (); Cicalo:1999sn (); Bai:2004qj (). These three states of isospin 0 and similar masses cannot fit the pattern arising from the SU(3) flavor symmetry of quark model in the light quark sector. A broadly accepted classification has been that the and belong to the isospin0 radial excitation states in the SU(3) flavor multiplet due to the mixing between the flavor singlet and octet. The as an outnumbered state was then proposed to be the ground state pseudoscalar glueball candidate. Such an assignment was based on phenomenological studies which predicted the mass of the pseudoscalar glueball around 1.4 GeV Faddeev:2003aw (). This proposal seemed to be able to accommodate the experimental observations with the theoretical prediction, and had attracted a lot of efforts to further explore the structure and production mechanism of the as the pseudoscalar glueball candidate Donoghue:1980hw (); Close:1980rv (); Barnes:1981kp (); Close:1987er (); Amsler:2004ps (); Masoni:2006rz (); Klempt:2007cp ().
Notice that the mass of is far below the expected value from LQCD Morningstar:1999rf (); Bali:1993fb (); Chen:2005mg (); Chowdhury:2014mra (); Richards:2010ck (); Sun:2017ipk (). In the literature a lot of theoretical studies have focused on the consequence of the quark and glueball mixings by assuming the to be the pseudoscalar glueball candidate. Investigations of the pseudoscalar glueball mixings with the light and its mass positions can be categorized into the following classes: (i) Quantify mixings among the ground state pseudoscalar mesons and , and the pseudoscalar glueball which is assigned to Rosenzweig:1981cu (); Cheng:2008ss (); Close:1996yc (); Li:2007ky (); Gutsche:2009jh (); Li:2009rk (); Eshraim:2012jv (); (ii) Identify mechanisms that cause the low mass of pseudoscalar glueball Faddeev:2003aw (); Cheng:2008ss () compared with the lattice QCD (LQCD) calculations Morningstar:1999rf (); Bali:1993fb (); Chen:2005mg (); Chowdhury:2014mra (); Richards:2010ck (); Sun:2017ipk (). However, because of modeldependence it has been very difficult to make progress on establishing unambiguously the glueball nature of .
It is a challenge to bring down the pure gauge glueball mass from GeV to GeV. The very relevant issue is that whatever the mechanism could be it requires an abnormally strong coupling between the light quark states and pure gauge glueball. It has been quoted broadly that the QCD sum rules for pseudoscalar glueball led to relatively low masses. However, it should be noted that the pseudoscalar glueball sum rules are very sensitive to assumptions made in the calculations. As noted explicitly in Ref. Senba:1981iy () the pseudoscalar glueball mass in QCD sum rules has large uncertainties and is very sensitive to the gluon condensation. The question about the pseudoscalar glueball mass is inevitably correlated with the  mixing because of the axial anomaly Witten:1979vv (); Veneziano:1979ec (); Novikov:1979uy (). The deviation of the  mixing angle from the ideal one between the flavor octet and singlet indicates the crucial role played by the anomaly. Meanwhile, the  mixing angle does not determine the mass as a usual mixing scheme would suggest. On the contrary, its dependence of the topological charge density has to be taken into account.
In Ref. Cheng:2008ss () a dynamical approach for the glueball mixing was explored by implementing the mixing into the equations of motion for the anomalous Ward identity and a low mass about 1.4 GeV for the physical pseudoscalar glueball. This approach was extended to accommodate the in Ref. Tsai:2011dp () and a similar result was extracted. However, an analysis of Ref. Mathieu:2009sg () based on the same dynamics yields a lower bound of about 2 GeV for the pseudoscalar glueball mass. In Ref. Qin:2017qes () a revisit of the mixing scheme of Refs. Cheng:2008ss (); Tsai:2011dp () was carried out, and the numerical results of Refs. Cheng:2008ss (); Tsai:2011dp () were confirmed except that the approximation for extracting the pseudoscalar glueball mass appeared to be problematic ^{3}^{3}3A detailed deduction can be found in Ref. Qin:2017qes ().. After curing this problem, it shows that the physical glueball mass will favor to be higher than 2 GeV which is remarkably consistent with the conclusion of Ref. Mathieu:2009sg () and matches the LQCD simulations. It is interesting to note that the analysis of Ref. Gabadadze:1997zc () also suggests a pseudoscalar glueball mass above 2 GeV, although the physical state should be lighter than the quenched state from the pure YangMills gauge theory Gabadadze:1997zc (). The mass difference between the quenched pure gauge state and the QCD state is of the order of , which means that a low mass state around 1.4 GeV is unfavored. In fact, a lot of puzzling questions arised from not only conflicts between the early experimental observation and the first principle LQCD simulations, but also between the early phenomenological studies and the LQCD results.
The change of situation was triggered by the highstatistics experimental data from BESIII. There have been a number of the exclusive decay channels measured with high precision, where contributions from can be clearly identified BESIII:2012aa (); Ablikim:2011pu (); Ablikim:2010au (). It shows that in the vicinity of 1.4 GeV there is only one BreitWigner peak structure in the invariant mass spectrum for . Similar feature can be found in the radiative decays of . In the hadronic production channels, such as , , , etc., there is also only one BreitWigner peak present in the invariant spectrum. The interesting observation is that the peak positions somehow are slightly different in exclusive channels. Further insights into this puzzling problem was gained from the measurement of isospin breaking effects into with at BESIII BESIII:2012aa (), where the isospin breaking effects are found to be unexpectedly large, i.e. . This value is nearly one order of magnitude larger than that produced by the mixing. It was then discovered by the authors of Ref. Wu:2011yx () that the significantly enhanced isospin breaking effects are caused by the socalled “triangle singularity” (TS) mechanism. It was demonstrated in Ref. Wu:2011yx () and later a detailed analysis Wu:2012pg () that at the mass of the nonvanishing coupling of the initial to the intermediate and then their rescatterings into by the exchange of a Kaon or antiKaon allow a perfect satisfaction of the TS condition. While the detailed analytical properties of the triangle diagrams will be discussed later, a simple way to picture the TS mechanism is that it corresponds to such a kinematic condition that all the internal states of the triangle loop can approach their onshell condition simultaneously. As a consequence of such a leading singularity within the loop function, it will provide significant interferences in exclusive decays of and produce the shift of peak positions of a single state in different channels and unexpectedly large isospin breaking effects in its decays into Wu:2011yx (); Wu:2012pg (). The dominance of the TS mechanism in was later confirmed by Ref. Aceti:2012dj () in a chiral unitary approach.
Although the TS mechanism seems to be promising for understanding sofar all the existing puzzles about the and signals, later according to Ref. Achasov:2015uua (), the nonzero width of in the triangle loop integral can lead to significant suppressions of the decay rate. Therefore, the dominance of triangle diagrams in the isospin violated channel may become questionable. Implications of such a possibility suggests that the width effects due to the internal states should not be neglected. In order to clarify the role played by the TS mechanism, we carry out a coherent and quantitative investigation of the decays of into , and including the width effects in the TS mechanism.
As follows, in Sect. II we first introduce the TS conditions and discuss in detail the analytical properties of the triangle loop amplitude when nonzero width is considered. Then we will explore the decay patterns for , and , and clarify the role played by the TS mechanism in Sect. III. In particular, we will show that an additional transition process which enhances the mixing via the TS mechanism should contribute to . The calculation results and discussions will be given in Sect. IV, and a conclusion will be given in Sect. V.
Ii Analytic properties of the triangle loop integral with nonzero widths
To understand the impact of unstable internal meson on the triangle loop integral of (denoted by in the following), we consider a typical triangle amplitude shown in Fig. 1,
(1) 
where the , and are the masses for , and , respectively. Since meson is unstable, a finite width has been introduced in its propagator. Due to the wave vertex and the polarization of meson, the amplitude is actually a tensor integral. However, this tensor integral can be reduced to a sum of scalar 3point integral and some 2point integrals Achasov:2015uua (). By studying the analytical property of the 3point and 2point integrals, we can learn the property of the physical amplitude . The TS condition applies to the physical amplitude where all the internal particles approach their onshell condition simultaneously. In such a sense the reduction of into 3point and 2point loop integrals means that the manifestation of the physical TS contributions is given by the sum of the reduced loop integrals although some contributions are from the 2point loops. Such a clarification is essential for the reason that we actually deal with the physical process instead of single loop integrals which are only part of the dynamics of the physical process. The kinematic condition that all the internal particles are onshell determines the kinematics for all the reduced loops which have to be considered simultaneously. To be more specific, within the TS kinematics the twobody on shell condition in the 2point integrals has been contained in the 3body onshell condition. With the above clarification, the TS contribution in this work is referred to the overall contributions from the physical integral instead of a reduced 3point integral, and the influence of the finite width effects are also referred to its impact on the overall loop function.
Some more features about the TS loops should be pointed out before we proceed to the detailed analysis:

The presence of the TS kinematics implies that the main contributions of the triangle loops come from the kinematic region near the onshell condition for the internal particles. For physical processes where the internal motions of the internal particles can be treated nonrelativistically the scalar triangle loop can be directly integrated out, and the leading logarithmic singularity can be explicitly extracted. In particular, for nonrelativistic heavy meson loop transitions where the TS mechanism is present, the loop amplitudes can be analyzed in the nonrelativistic effective field theory (NREFT) framework and a powercounting scheme can be established.

For loop transitions involving only light hadrons sometimes the nonrelativistic approximation can hardly be justified. In such a case analysis of the triangle loop in the Mandelstam representation should be more appropriate. For most cases of the physical loop integrals, an empirical form factor has to be included to cut off ultraviolet divergence when the internal particles go off shell, which will inevitably introduce some modeldependence, although in general such an uncertainty can be under control. Even for convergent physical loops it is often checked that unphysical contributions from relatively large momentum transfers are reliably estimated and then removed Xue:2017xpu (). For the physical triangle loop of Eq. (1), it converges with the choice of the vector propagator for the . But in order to examine the sensitivities of the loop integrals to unphysical contributions from the ultraviolet region, we include a form factor and compare the numerical results for different cutoff parameters. The detailed discussions will be given later in Section III and IV.
ii.1 Analytical expression
We first consider a typical scalar loop integral
(2) 
where the notation is the same as that in Fig. 1. Taking the Feynman parameterization, it can be expressed as
(3) 
where is a homogeneous polynomial of , i.e.
(4) 
In the case where the internal masses are real, the leading singularity of is determined by the Landau equation
(5) 
This condition means that the extremum of touches 0. Physically, it means that the internal particles are stable and simultaneously become onshell. The Landau equation gives . If this holds for , then there exist solutions for the Landau equation within the physical region which satisfies . This leads to the kinematic bounds for the triangle singularity.
In general, for the fixed external fourmomentum squares and there are two solutions for which satisfy the Landau equation, i.e.
(6)  
(7) 
where . Likewise, when and are fixed, we obtain two solutions for , i.e.
(8)  
(9) 
However, the conditions that with the fixed and or with the fixed and do not cause divergence because the corresponding solution is out of the region . Only when () meets (), the triangle singularity occurs within the physical region and can possibly produce detectable effects in experimental observables.
With the fixed (since it is an external particle), for any given , there is a lying within , where and are labeled as critical values for and , beyond which the triangle singularity no longer exists. Taking the same notation as Ref. Liu:2015taa (), these critical values are given by
(10)  
(11) 
A caveat arising from the above discussion is that the internal particles are stable ones, i.e. they do not have a width in the propagator. In reality the nonvanishing coupling for to pion () and kaon () demonds that the propagator for must contain an imaginary part. Therefore, a detailed investigation of the width effects in the TS mechanism is necessary and useful for a better understanding of the underlying dynamics.
To accommodate the width effects in the triangle loops, we consider complex masses for the internal particles. With the help of Spence function:
(12) 
which is also called dilogarithm function , as a special case for polylogarithm function when , an analytic expression of the transition matrix has been worked out by G.’t Hooft and M. Veltman tHooft:1978jhc ():
(14)  
where functions have the following expressions:
(15)  
(16)  
(17) 
and , and are kinematic variables:
(18)  
(19)  
(20) 
This analytic expression is valid for both real and complex internal masses. In Eq. (14) are functions of the integration variable :
(21)  
(22)  
(23) 
and denotes the value of when .
The function in Eq. (14) can be written in terms of the Spence function with characteristic structures:
(24)  
where
(25) 
with
(26) 
The function arises from
(27) 
with the argument in limited in .
The following features with the TS kinematics will help understand better the analytical properties of the triangle loop amplitude:

In the vicinity of the TS kinematics the main contributions of the transition amplitude are given by the absorptive part. In particular, when the internal masses are real, the logarithmic divergence of the TS manifests itself in the absorptive part of scalar integral. So we will focus on the absorptive part in the analysis.

For the case that the internal masses are all real, the absorptive part can be derived analytically according to the Cutkosky rule. This allows us to examine the width effects on the triangle loop amplitude by comparing them with the Cutkosky rule result. In particular, the width dependence can be highlighted in the absorptive part of the amplitude.

For the physical case, namely the isospinviolating transition , it should be noted that the dispersive part becomes negligible due to the cancellation between the charged and neutral triangle loop amplitudes. This actually leads to a rather modelindependent behavior of the TS contributions to the isospin violations in ^{4}^{4}4Note that even though form factors are often introduced to cut off divergence in the loop integral, in the TS kinematics the dependence of the form factors is relatively small due to the small virtuality for the coupling vertices. For the isospinviolating decay of the cancellation between the charged and neutral loop amplitudes will further reduce the model dependence. Detailed discussion and demonstration of such a consequence has been provided in Refs. Wu:2011yx (); Wu:2012pg (). . As stressed at the beginning, the only thing left behind is the width effect that should be quantified by explicit and selfconsistent calculations.
In the following sections we focus on the derivation of absorptive part of the scalar integral under the influence of the finite width of the intermediate states. We will provide detail analysis of and , for the reason that the physical widths of the by weak decays are much smaller than that of the meson. However, it is checked in the end that the same analytic expression is still valid when any of the internal state has a complex mass.
ii.2 The motion of singularities
For convenience, we express the amplitude of Eq. (14) in a concise form
(28)  
(29)  
(30)  
(31) 
with
(32)  
(33)  
(34)  
(35) 
Our task in this subsection is to extract the imaginary part of the loop amplitude and investigate the movement of the singular kinematics manifested by the location of in the complex plane. The general condition for the TS requires that the following kinematic constraints are satisfied, i.e. , , and . Moreover, the maximum allowed value for or is generally very close to the normal threshold. It allow us to make a substitution of by in some steps as a reasonable approximation. Therefore, in the following discussion of the finite width effects on the imaginary part of the amplitude, we can apply this approximation to simplify the deduction without loss of accuracy. The numerical result of this approximation compared with the exact one will be discussed in the end of this section.
To proceed, we will start with of which the imaginary part depends on the positions of on the complex plane. The latter will then rely on the locations of and as given by Eq. (31). Therefore, the motion of the TS can be illustrated by tracing the locations of , where are defined as the roots of , and we define the is the larger one of the two roots. In the case of finite , may deviate slightly from real axis which will be our focus in this work.
According to the definition of (Eq. 24), the divergence of occurs when , i.e. the denominator of . Taking into account that and are complex functions of and , and for sufficiently small width all the dependent terms only contribute to the imaginary parts of and , the TS condition actually corresponds to and the imaginary part of the complex mass will push and away from the real axis.
In order to discuss how moves when the meson has a small but finite width, we need to first write down the explicit expression of as the roots of function and then expand the with respect to (the mass square of the ) to the first order of .
Since has nothing to do with the complex mass , is not affected by . Thus, their locations on the real axis are
(36)  
(37) 
The only difference between and is the sign in front of their square roots. For simplicity we just need to show the expressions for . The locations of and as the larger root of and , respectively, are
(38)  
(39) 
Hence the variations of the locations of and in association with the presence of the finite width for the can be examined by
(40)  
(41) 
Similarly, we have
(42)  
(43) 
where the kinematic variables are defined as follows:
(44) 
It should be noted that within the region , the value of is always negative, which means that actually moves upward away from the real axis and moves downward away from the real axis with the increasing . These expansions give the sign for the imaginary parts, according to which we can conclude that both and move upward and that both and moves downward under the influence of the increasing .
Speaking of the real part of , for , and , it can be proved that . For and , it can be easily verified that . For and , it can also be proved that . Hence the locations of are quite clear, and we are to take a look at . The real part of is slightly complicated. They are
(45)  
(46)  
(47)  