An N/D study of the S_{11} channel πN scattering amplitude
AAn
N/D study of the S channel πN scattering amplitude February 8, 2021
Qu-Zhi Li, † Yao Ma, † Wen-Qi Niu, † Yu-Fei Wang, ‡ Han-Qing Zheng ♥ ,(cid:63) , † Department of Physics and State Key Laboratory of Nuclear Physics and Technology, PekingUniversity, Beijing 100871, P. R. China ‡ Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany ♥ College of Physics, Sichuan University, Chengdu, Sichuan 610065, P. R. China (cid:63)
Collaborative Innovation Center of Quantum Matter, Beijing, Peoples Republic of China
Abstract
Extensive dynamical
N/D calculations are made in the study of S channel low energy π N scatterings, based on various phenomenological model inputs of left cuts at tree level.The subtleties of the singular behavior of the partial wave amplitude at the origin of thecomplex s plane are carefully analysed. It is found that, relying very little on the details ofthe dynamical inputs, the subthreshold resonance N ∗ (890) survives. In a series of recent publications [1] [2] [3], it is suggested that there exists a sub-threshold1 / − nucleon resonance hidden in S channel of πN scatterings, with a pole mass √ s = (0 . ± . − (0 . ± . i GeV. The result is obtained by using the production representation (PKUrepresentation) for partial wave amplitudes [4] [5] [6] [7] [8]. It is found later that the N ∗ (890)pole may also be generated from a conventional and simple K -matrix fit, though the latter suffersfrom the existence of spurious poles on 1st Riemann sheet of complex s plane [9]. Properties of N ∗ (890) are also investigated, such as its coupling to N γ and
N π [10], and
N γ ∗ [11]. It is foundthat its coupling to N π is considerably larger than that of the N ∗ (1535), while its coupling to N γ is comparable to that of the N ∗ (1535). These results on couplings look reasonable and arewithin expectations, hence providing further evidence on the existence of N ∗ (890).However, to firmly establish the existence of such a subthreshold resonance is still a difficulttask. Besides dispersion relations, the most frequently used tools at hand are perturbationchiral amplitudes and their unitarizations (For a recent review, see Ref. [12]), or (unitarized)resonance models. However, these unitarization techniques are far from being perfect when usedin the study of low energy strong interaction physics. Especially, when applying to partial waveamplitudes with unequal mass scatterings, extra difficulties will occur, as will be discussed atsome lengths in this paper. The major difficulties arose at s = 0 point in s plane, where chiralexpansions break down since chiral expansions and partial wave projections do not commutewhen s →
0. The expected decoupling property of heavy resonances when their masses sent toinfinity is also violated in partial wave amplitudes at s = 0, for a purely kinematical reason inpartial wave projections with unequal mass scatterings. The major task of this paper is to showhow the subthreshold resonance persist, irrespective of various difficulties and uncertainties leftin the input quantity – the left part of the scattering amplitude.This paper will provide further evidences on the existence of N ∗ (890), by directly finding apole in the S matrix element calculated from the N/D method. Early studies on low energy πN Corresponding author. a r X i v : . [ nu c l - t h ] F e b catterings via N/D method may be found in Ref. [13] and references therein. Nevertheless, noreport on the possible existence of a subthreshold resonance is known in the literature of
N/D studies, to the best of our knowledge. In our practice of
N/D calculations no spurious poleson first Riemann sheet are found to emerge. Also an
N/D calculation faithfully reproduces allinput dynamical as well as kinematical branch point singularities. We therefore think the
N/D method is rather reliable. However, the calculations in
N/D studies do generate spurious branchcuts and spurious poles on the second sheet, due to the truncation of numerical integrations.Nevertheless, their effects can be evaluated to see that the sum of hazardous contributions benegligible.This paper is organized as follows: section 1 is the introduction. In section 2 a brief intro-duction to the
N/D method is given with a solvable toy model calculation. Also in section 2 weafford a review on the production representation which is found very illuminating in understand-ing the complicated
N/D calculations. Section 3 focuses on the singularity structure of partialwave amplitudes at s = 0, including the discussion on why chiral expansions break down here,and on how high energy contributions enter through an analysis on regge asymptotic behavior of T ( s ) when s →
0. Section 4 devotes to numerical analyses on how can a subthreshold resonanceemerge under various phenomenological inputs.
N/D recipe, a prelude
N/D method
The partial wave T matrix element is expressed as T = N/D , (1)where D contains only the s -channel unitarity cut or the right hand cut R , whereas N onlycontains the left hand cut ( l.h.c. ) or L . See Fig. 1, R = [ s R , + ∞ ) whereas L represents allbranch cuts except R in Fig. 1. In section 3.2 we will briefly review on how to determine thecut structure in Fig. 1 [14].Figure 1: Branch cuts (thick blue lines) of partial wave π N elastic scattering amplitudes, where c L = ( m N − m π ) /m N , c R = m N + 2 m π , s L = ( m N − m π ) , s R = ( m N + m π ) .Partial wave unitarity in single channel approximationIm R T ( s ) = ρ ( s ) | T ( s ) | (2)leads to the following relations Im R [ D ( s )] = − ρ ( s ) N ( s ) , Im L [ N ( s )] = D ( s )Im L [ T ( s )] , (3)2nd subsequential N/D equations: D ( s ) = 1 − s − s π (cid:90) R ρ ( s (cid:48) ) N ( s (cid:48) )( s (cid:48) − s )( s (cid:48) − s ) ds (cid:48) ,N ( s ) = N ( s ) + s − s π (cid:90) L D ( s (cid:48) )Im L [ T ( s (cid:48) )]( s (cid:48) − s )( s (cid:48) − s ) ds (cid:48) , (4)in which a once subtraction to the integrals at s = s below s R are taken. In Eq. (4), the left cutof the partial wave T matrix element, Im L T is an input quantity. Throughout this paper, weonly discuss Im L T extracted from tree level amplitudes. Hence, except in the case with t -channel ρ exchange as discussed in section 4.2, where an arc cut in s plane will be met, L is always onthe real axis. For example, for pure tree level chiral amplitudes, L = ( −∞ , ∪ [ c L , c R ]. We willmake a rather detailed discussion on how to determine l.h.c. s in section 3.2.To solve the integral equations one may substitute D into the second equation of Eq. (4) toget N ( s ) = N ( s ) + ˜ B ( s, s ) + s − s π (cid:90) R ˜ B ( s (cid:48) , s ) ρ ( s (cid:48) ) N ( s (cid:48) )( s (cid:48) − s )( s (cid:48) − s ) ds (cid:48) , (5)with ˜ B ( s (cid:48) , s ) = s (cid:48) − s πi (cid:90) L Disc T (˜ s )(˜ s − s )(˜ s − s (cid:48) ) d ˜ s , (6)and use the inverse matrix method to obtain a numerical solution. Throughout this paper, weset s = 1 GeV , a value a little bit below the elastic threshold s R . Apparently the auxilliaryfunction ˜ B in Eqs. (5) and (6) can be formally written as ˜ B ( s, s ) = T L ( s ) − T L ( s ). Since weonly use tree level input in performing N/D calculations in which the unitarity cut is absent,we actually set ˜ B ( s, s ) = T tree ( s ) − T tree ( s ) (7)throughout the calculations. In this way we avoid the discussions on possible subtractionsencountered at two endpoints of the integral defined in Eq. (6). In this scheme, the subtractionconstant N ( s ) appeared in Eq. (5), the would-be T tree ( s ), serves as a free fit parameter. Theuse of Eq. (7), rather than dealing with the integral with possible truncations defined in Eq. (6),greatly simplifies our calculations.As we will see later in this paper, there exists a subtlety when using Eq. (4) to discussun-equal mass scatterings. The problem comes from a singularity at s = 0 in the partial waveamplitude and in its left cut, which stems from high energy region t → + ∞ through partial waveprojections, and from relativistic spin kinematics as well. But before dealing with this problem,we are more eager in finding a solution of Eq. (4) in a simplified toy model, in the followingsubsection. In N/D recipe the input quantity is Im L T . Let us begin with a simple version of N/D studyby assuming Im L T being simulated by a set of Dirac δ functions:Im L T = (cid:88) i γ i δ ( s − s i ) . (8)The T matrix is analytically solvable for Im L T given in Eq. (8). We (arbitrarily) choose Case
I:one pole at s = 0, and Case
II: one pole at s = − m N , and fit to the “data” obtained from thesolutions of Roy Steiner equations [15] by tuning the parameter γ , and search for poles on the s -plane. Both cases give a good fit to the data, and a sub-threshold pole emerges in each casewith a location listed in Table 1. The phase shift and its PKU decomposition [4] [5] are plotted inFig. 2. On the left of Fig. 2, we see the familiar picture that the background contribution to thephase shift is concave and negative while the subthreshold resonance pole provides a positive andconvex phase shift above threshold to counterbalance the former contribution, and the sum ofthe two reproduces the steadily rising phase shift data. In order to have a better understandingof this phenomenon, a brief introduction to the production representation of partial wave elasticscattering S matrix element is needed. 3 ase I Case II s − m N γ (GeV ) 0.79 1.34 √ s pole (GeV) 0.810 - 0.125i 0.788 - 0.185iTable 1: Subthreshold pole locations using input Eq. (8)..Figure 2: left: fit to the phase shift of Case
II as an example; right: the spectral functionIm L f ( s ) /s of Case
I and II. Notice that the singularity at s = 0 in Case
II is due to thekinematical singularity in ρ ( s ) defined in Eq. (11) rather than dynamical. The “spectral” function drawn on the r.h.s. of Fig. 2 is defined as following: f ( s ) = sπ (cid:90) L ds (cid:48) Im L f ( s (cid:48) ) /s (cid:48) ( s (cid:48) − s ) + sπ (cid:90) R ds (cid:48) Im R f ( s (cid:48) ) /s (cid:48) ( s (cid:48) − s ) , (9)Im L,R f ( s ) = Im L,R (cid:18) ln S ( s )2 iρ ( s ) (cid:19) , (10)where the partial wave S matrix element S = 1 + 2 iρT and ρ is the kinematic factor: ρ ( s ) = (cid:112) s − ( m π + m N ) (cid:112) s − ( m π − m N ) s . (11)Elastic partial wave S matrix elements satisfy a production representation like follows: S = (cid:89) i S i × e iρ ( s ) f ( s ) . (12)For more details about the pole elements S i one is referred to Refs. [4] [5]. The productionrepresentation exhibits some nice features which are very useful in our analyses. One advantageis that the phase shift from different sources are additive. This property is vital in tracingthe physical origin of the phase shift in some situations. As have been stressed repeatedly, thepositive value of Im L f ( s ) /s guarantees that the background contribution to the phase shift benegative and concave, and hence an isolated singularity on the second sheet is needed to takecharge of the steady rise of near threshold phase shift. The circular cut caused by t -channelexchanges and the u -channel cut are not considered yet here, nevertheless they are numericallysmall as comparing with the cut lying on ( −∞ , s L ]. Hence a large and positive Im L f ( s ) /s at s = 0, looks important as suggested in Fig. 2. The strong enhancement of Im L f ( s ) /s at s = 0 isdue to two reasons: one is from the kinematic singularity at s = 0 from Eq. (11), the other is fromthe possible singularity in T ( s ) when s →
0. An example for the former is fit
Case
II in section 2.2where T (0) ∼ constant, while the latter example is provided by fit Case
I where T ( s ) ∼ O ( s − )when s →
0. In general, making use of the property of real analyticity for S matrix elements, Previous examples can be found in Refs. [4] [1] [9]. L f ( s ) as − ln | S ( s ) | / ρ ( s ) = − ln [1 − ρ ( s )Im L T ( s ) + 4 ρ ( s ) | T | ] / ρ ( s ). Hence if T ( s ) does not vanish when s →
0, then ln | S ( s ) | diverges logarithmically, since ρ ( s ) ∝ s − atorigin. It may be worth stressing that the singularities caused by relativistic kinematics trulyexist and bring physical consequences, since they enter the physical equations such as Eq. (2). Agood example to realize this comes from figure 5 and Eq. (55) of Ref. [7]: without the kinematicalsingularity the data curve can simply not be explained.On the other side, the inelastic right hand cut contribution to the phase shift in Eq. (9) shouldalways be positive [3], since in that region | S ( s ) | = η <
1, with η the inelasticity parameter. In N/D calculations, a cutoff to the integral interval has to be adopted, i.e., [ s R , + ∞ ) to be replacedby [ s R , Λ R ]. In this situation, it is not difficult to understand that the truncated N/D integrationactually violates unitarity by introducing a spurious branch cut starting from s = Λ R , in thesense that the effective η parameter exceeds unity: | S ( s ) | = [1 − ρ ( s )Im R T ( s ) + 4 ρ ( s ) | T | ] =[1 + 4 ρ ( s ) | T | ] >
1, when s > Λ R , since a truncation implies actually Im R T = 0. So one hasto be cautious when performing the N/D calculations by monitoring to what extent unitarity isviolated. This may be fulfilled at quantitative level by, for example, calculating the contributionfrom the region above Λ R to the phase shift, through Eq. (9). It is found that, when doingcalculations in this paper, the violation can either be large or small, depending on whether ornot the input quantity T L ( s ) at s = Λ R is too large or small. For the former, an example is the χ PT input which is not valid anymore at Λ R = 1 .
48 GeV , i.e., the value we choose throughoutour calculations. However, it is found in practice that, whenever there appears a large effectfrom the spurious branch cut, there always appears a spurious pole located very close to thecutoff. It provides a contribution to the phase shift with opposite sign and almost cancels thatfrom the spurious cut at low energies numerically. This is reasonable and within expectationsince the net spurious effects generated from the cutoff have to be very small in the low energyregion. If this were not true, the whole calculation scheme would be questioned and be invalid.Encouraged by the discussions made in section 2.2, we plan to make a more realistic N/D calculation at next steps. The first thing needs to be settled down is of course choosing an inputIm L T as much realistic as possible. One may choose the χ PT outputs as an input here, aswhat is adopted in Refs. [1] [2] [3]. A careful analysis reveals, however, that the partial waveprojection of χ PT amplitudes encounters a rather severe problem at s = 0. In the followingsection we dedicate to the study of such a problem. T ( s ) at s = 0 As suggested in the end of the above section, to make a more realistic calculation, one mayuse χ PT amplitudes to extract Im L T [3]. However, the results may not be directly applicable toEq. (4), and should be treated with great care. The integration interval for N ( s ) is L = ( −∞ , O ( p ) level Im L T behaves as O ( s − / ) when s →
0, andat O ( p ) level it behaves as O ( s − / ), while from rather general argument to be discussed insection 3.2 these strong singularities at s = 0 are not physical. Hence one has to find a way toget rid of these artifacts caused by partial wave projections of χ PT amplitudes.In general, singularities of the partial wave T matrix element at s = 0 come from two aspects: • the O ( p n ) ( n ≥
2) level χ PT amplitudes behave as s − n − / when s → n increases when the chiral orderincreases. • the left-hand cut around s = 0 receives singular contribution from high energy region ofcrossed channels, i.e., t, u → ∞ , through partial wave projections;We will address these problems in some details below. It is worth pointing out beforehand, thatthough in N/D approach one has to deal with these problems cautiously, the calculations madein Refs. [3] and [1] are luckily insensitive to the problem. Though the background contribution tothe phase shift will be enhanced incorrectly by the contribution in the vicinity of s = 0, it will belargely compensated by tuning the cutoff parameter Λ L when evaluating the l.h.c. integral. Moreinterestingly even though the integral of f ( s ) is enhanced near s = 0, its derivatives behavesvery differently. For example, df ( s ) ds = − π (cid:90) ds (cid:48) ln | S ( s (cid:48) ) | / ρ ( s (cid:48) )( s (cid:48) − s ) . (13)5ow the integrand near s (cid:48) = 0 behaves like ∼ s (cid:48) ln s (cid:48) and the unwanted singularity is killed. Thisfact may be translated into a more transparent language: in Refs. [3] [1], except the scatteringlength which anyway needs to be fitted by tuning Λ L , the other quantities such as the curvatureof the phase shift curve are quite immune to the disease infected from s = 0. χ PT amplitudes
We notate the process as N ( p, σ ) + π ( q ) → N ( p (cid:48) , σ (cid:48) ) + π ( q (cid:48) ), where p, q, p (cid:48) , q (cid:48) are themomenta and σ, σ (cid:48) are the spins. The Mandelstam variables are s = ( p + q ) = ( p (cid:48) + q (cid:48) ) , t = ( p − p (cid:48) ) = ( q − q (cid:48) ) , u = ( p − q (cid:48) ) = ( p (cid:48) − q ) . (14)The full amplitude can be decomposed as (the following discussions are all for isospin I = 1 / T = ¯ u ( p (cid:48) , σ (cid:48) ) (cid:20) A ( s, t ) + /q + /q (cid:48) B ( s, t ) (cid:21) u ( p, σ ) . (15)The results of the scalar functions A, B are for example listed in Refs. [1,3] (further references arefound in Refs. [16] [17] [18] [19] [20] [21]). From O ( p ) on, the χ PT lagrangian contains 4-point ππN N contact terms, which contributes to the scalar functions as ( C refers to constants), A (cid:2) O ( p ) (cid:3) ⊃ C ( s − u ) , A (cid:2) O ( p ) (cid:3) ⊃ C ( s − u ) . (16)More explicitly, • at O ( p ) (Born and contact diagrams): A = g m N F , B = 1 − g F − m N g F ( s − m N ) − m N g F u − m N ; (17) • at O ( p ) (only contact diagram): A = − c m π F + c ( s − u ) m N F + c F (2 m π − t ) − c ( s − u ) F , B = 4 m N c F ; (18) • at O ( p ) (Born diagram): A B = − m N gF × m π ( d − d ) , B B = 4 m π g ( d − d ) F × su + m N (2 u − m N )( s − m N )( u − m N ) ; (19) • at O ( p ) (contact diagram): A C = − ( d − d )( s − u ) m N F + ( d + d ) m N F ( s − u )(2 m π − t )+ d m N F ( s − u ) + 4 m π d m N F ( s − u ) , B C = ( d − d )( s − u ) F . (20)In the expressions above, g is the axial-vector coupling constant, F is the pion decay constant,and c i , d i are low-energy constants. It seems that A contains O ( p ) constant and B is O ( p − )(and similar for higher order functions), but in fact tree diagrams do not violate the powercounting. This can be seen, by means of Gordon identity: T = ¯ u ( p (cid:48) , σ (cid:48) ) (cid:20) D ( s, t ) + i m N σ µν q µ q (cid:48) ν B ( s, t ) (cid:21) u ( p, σ ) , D = A − s − u m N B . (21)6or instance, D = s − u m N F (cid:20) − g + 2 g m N ( t − m π )( s − m N )( u − m N ) (cid:21) + g m N F ( t − m π ) ( s − m N )( u − m N ) , (22)which is apparently O ( p ). Meanwhile the prefactor of B ( q µ q (cid:48) ν ) is O ( p ), making the wholeamplitude O ( p ).The partial wave projection is done on helicity amplitudes: T ++ = (cid:114) z s (cid:2) m N A ( s, t ) + ( s − m π − m N ) B ( s, t ) (cid:3) , T + − = (cid:114) − z s s (cid:2) ( s − m π + m N ) A ( s, t ) + m N ( s + m π − m N ) B ( s, t ) (cid:3) , (23)where z s = cos θ , and θ is the scattering angle; T ++ stands for that the helicities of the initialand final nucleon are both +1 /
2, while T + − means the final nucleon has helicity − / z s = cos θ ) are: t ( s, z s ) = 2 m π − ( s + m π − m N ) s + [ s − ( m π + m N ) ][ s − ( m π − m N ) ]2 s z s , (24) u ( s, z s ) = m π + m N − s − ( m π − m N ) s − [ s − ( m π + m N ) ][ s − ( m π − m N ) ]2 s z s . (25)The S amplitude is from J = 1 / T J ++ = 132 π (cid:90) − dz s T ++ ( s, t ( s, z s )) d J / , / ( z s ) , T J + − = 132 π (cid:90) − dz s T + − ( s, t ( s, z s )) d J − / , / ( z s ) , (26)where d stands for Wigner small- d matrix. For S channel, T ( S ) = T J =1 / + T J =1 / − . (27)From this formula the singularity at s = 0 is obvious. On the one hand, in Eq. (23) thekinematic effects give an s − / factor, which makes s = 0 a branch point. On the other hand,see for example in Eq. (22), contact terms from χ PT expansions lead to higher and higher orderpolynomials of s − u : T (cid:2) O ( p n ) (cid:3) ⊃ C ( s − u ) n . (28)According to Eq. (25), u ( s → → s − , so when n ≥ T (cid:2) O ( p n ) (cid:3) ( s → ∼ Cs − n − / , (29)where the extra factor − / χ PT calculation is employed, the stronger singularity will occurnear s = 0 and the chiral series breaks down. Apparently this is only an artificial problem causedby chiral expansions since it contradicts the genuine singularity structure expected when s → s →
0. In principle one would not hope Eq. (29) to appear in the expression of Im L T when using N/D . One writes a spectral representation of the partial wave amplitude [14], for t -channel: T ∼ (cid:90) + ∞ σ t dt (cid:48) Σ( s, t (cid:48) ) (cid:90) − d cos θ R (cos θ ) t − t (cid:48) , (30)where Σ is the Mandelstam spectral function, σ t = 4 m π is the threshold of t -channel process,and R is the basis function of the partial wave projection (usually the linear combinations of7igner- d matrices). The function R can be expanded at t = t (cid:48) : R = R + ( t − t (cid:48) ) R + · · · , andonly the leading order causes singularities: T ∝ (cid:90) + ∞ σ t dt (cid:48) Σ( s, t (cid:48) ) β − ln (cid:20) α + βα − β (cid:21) , (31)with α = t (cid:48) − m π + ( s + m π − m N ) s , β = [ s − ( m π + m N ) ][ s − ( m π − m N ) ]2 s . (32)Therefore the left-hand cut is described by the equation α = β ; (33)which gives s ± ( t (cid:48) ) = m π + m N − t (cid:48) ± (cid:113) ( t (cid:48) − m π )( t (cid:48) − m N ) . (34)When t (cid:48) takes the value from σ t to + ∞ , the trajectory of this solution traces out the left-handcut. The following conclusions can be obtained: • when t (cid:48) ∈ [4 m π , m N ], the cut appears as a circle Re s + Im s = ( m N − m π ) , and theendpoint to the right s = m N − m π corresponds to t (cid:48) = σ t = 4 m π ; • when t (cid:48) ∈ (4 m N , + ∞ ), s − generates the cut ( −∞ , m π − m N ), and −∞ corresponds to t (cid:48) → + ∞ ; • when t (cid:48) ∈ (4 m N , + ∞ ), s + generates the cut ( m π − m N , t (cid:48) → + ∞ ;One performs a similar analysis in u -channel and the solution is s ( u (cid:48) ) = ( m π − m N ) u (cid:48) , s ( u (cid:48) ) = 2( m π + m N ) − u (cid:48) . (35)There is a nucleon pole u (cid:48) = m N , giving a segment cut (( m π − m N ) /m N , m π + m N ). When u (cid:48) > σ u = ( m π + m N ) , s gives (0 , ( m N − m π ) ) and s → u (cid:48) → + ∞ ; while s generates ( −∞ , ( m N − m π ) ) with s → −∞ when u (cid:48) → + ∞ .It was pointed out in Ref. [5] that, for meson – meson scatterings, if T ( s, t ) ∼ O ( t n ) when t → ∞ for fixed s , then the partial wave amplitude behaves as T ( s ) ∼ O ( s − n ) when s → | T ( t, cos θ s = 1) | < t ln t , it is expectedthat the proper singularity behavior for s -channel partial wave amplitude near s = 0 is no moresingular than T ( s ) ∼ O ( s − ) (up to some logarithmic corrections). This estimation can evenfurther be improved. It is seen from the above discussions that as s → − , there is a high energycontribution from the t → + ∞ region. In this region the full amplitude is governed by t -channel( ππ → N ¯ N ) reggeon exchanges. The leading Regge trajectory is ∆(1232) with the interceptparameter α ∆ (0) (cid:39) .
19 [22], which leads to a weak singularity T ∼ s − α ∆ (0) , (36)for the partial wave amplitude, when s → − . The s → + limit is the same according to forexample Ref. [22].All the discussions given above in this section are on how to determine l.h.c. s generated dy-namically, or in other words, cuts originated from physical absorptive singularities from crossedchannels. Besides these dynamical l.h.c. s there exists an additional cut ( −∞ ,
0] for pure kinemat-ical reasons: the nucleon spinor wave function provides a √ s branch cut which already showedup in the second equality in Eq. (26). The effect of branch cut singularity from relativistickinematics truly exists, as has already been addressed in section 2.3.8 Numerical analyses of
N/D method
It has been made clear in the above section, that using χ PT inputs of Im L T encounters theproblem that the partial wave projections of χ PT amplitudes lead to a strong but incorrectsingularity at s = 0, by violating what is expected from rather general constraints of quantumfield theory. Nevertheless it is not clear yet, to what extent the use of χ PT results may distort thephysical output. In the following we devote to the study of this problem by invoking O ( p ) and O ( p ) (tree level amplitude only) χ PT results, since in O ( p ) case no free parameter is availablefor a data fit. Nevertheless, the O ( p ) N/D unitarization can still be made and compared withdata, which ends up with a pole location √ s = 1 . − i .
23 GeV and a steadily rise phase shiftlarger than data by roughly 5 degrees at √ s = 1 .
16 GeV.
N/D calculations using pure χ PT inputs
The singularity of O ( p ) Im L T when s → O ( s − − / ). Which, as discussedpreviously, is not physical. We nevertheless still perform the N/D calculation to see whathappens. At O ( p ) level there are four low energy constants (LECs) c i with i = 1 , · · · ,
4. Thereare different sets of c i parameters found in the literature (e.g., Refs. [23], [13], [24], [25], [1]). Forthese LECs, certain bounds, i.e., the positivity constraints [26] are obeyed.A good fit is obtained with the value of fit parameters as c = − . , c = 3 . , c = − . , c = 3 . − ), which are found to be in good agreement with those foundin the literature. The fit parameter N ( s ) = 0 .
51 and the pole is located at √ s = 0 . − . i GeV . (37)The fit can also be done if we do not make a subtraction in the equation for N ( s ) in Eq. (4).In this case we get very similar results and N ( s ) = 0 .
55. In addition, we have also employeddifferent sets of c i , and the results change very little. For instance when we take the centralvalues of c i ’s from Ref. [23] (NLO): c = − . ± . , c = 1 . ± . , c = − . ± . , c =2 . ± .
03 (in units of GeV − ), the pole position is 0 . − . i GeV and the phase shift doesnot differ much from the data (when √ s = 1 .
16 GeV, it is only 1 . ◦ smaller). However, exceptthese nice features, the spurious branch cut becomes annoying here, contributing to the phaseshift at √ s = 1 .
16 GeV roughly -10 ◦ . Nevertheless, this large spurious effect is almost exactlycancelled by a second sheet pole located at √ s = 1 . − . i GeV. It is not totally clear to usyet why and under what situation the spurious branch cut becomes numerically important. Oneconjecture is that the O ( p ) χ PT input itself becomes sick at Λ R = 1 .
48 GeV , hence amplifiesthe contribution of spurious branch cut . Finally, since the net effects generated from the cut at s = Λ R are very small, we still think such type of solutions are acceptable for the evaluation ofphysics at lower energies.As a comparison we plot the “spectral function”, i.e., Im L f ( s ) /s , obtained here together withthose discussed in section 2.2, in Fig. 3. Comparing Eq. (37) with that in table 1, and differentFigure 3: left: N/D fit to the S phase shift with O ( p ) χ PT input, all the spurious contributionsare ignored here; right: a comparison of different Im f ( s ) /s in different situations. l.h.c contributions drawn in Fig. 3, we observe that the O ( p ) calculation overestimates the l.h.c Case
II. As a result, the pole contribution to the phaseshift has to be increased and hence the pole location has to move towards to the right directionin s plane closer to the πN threshold. But of course, such discussions are only meaningful underthe condition that the effects of spurious branch cut and the spurious pole around s = Λ R canceleach other.If all the spurious contributions are ignored, then we may ask a question. The calculationhere and that in Ref. [3] use the same ‘data’ sample and l.h.c. while in Ref. [3] the pole locatesat √ s = 0 . ± . − i (0 . ± .
08) GeV. Comparing with Eq.(37) it is found that there existsa rather large systematic error in determining the pole location. One possible reason to explainthis may come from the fact that in Ref. [3] a truncation of l.h.c. is performed while in herethere is no truncation on the left, see Eqs. (5) and (7). In fact in the calculation made in table 4of Ref. [3], it is found that when sending the cutoff s c to ∞ the pole location moves upwards to √ s = 0 . − i .
21 GeV, i.e., getting closer to Eq. (37).Finally, we have also tested the O ( p ) inputs (at tree level only) and found that the outputsare similar to the situation found in the O ( p ) case, so we no longer discuss the results hereanymore. N/D calculation using phenomenological models
In the above section we have made discussions on the problem encountered when using χ PTresults to estimate the l.h.c . The higher order terms appeared in the chiral lagrangian describing π N interactions are obtained by integrating out heavy degrees of freedom like the ρ meson in t -channel and N ∗ resonances in s and u -channels, etc.. The ill singularities at s = 0 in partialwave chiral amplitudes comes at least partly from integrating out heavy degrees of freedom.To see this more clearly, let us write down an effective interaction lagrangian responsible for t -channel ρ meson exchange, L t = g ρ (cid:126)ρ µ · ( ∂ µ (cid:126)π × (cid:126)π ) + g ρ ¯ N (cid:126)τ · ( γ µ (cid:126)ρ µ + κ m N σ µν ∂ µ (cid:126)ρ ν ) N , (38)where g ρ and κ are resonance coupling constants, (cid:126)τ are Pauli matrices, (cid:126)ρ µ , (cid:126)π and N refer to ρ resonance, pion and nucleon, respectively. For S channel, the ρ exchange contribution to theinvariant amplitude, Eq. (15), can be obtained: A = g ρ κ ( u − s )2 m N ( t − m ρ ) , B = 2 g ρ ( κ + 1) t − m ρ . (39)Now, if a 1 /m ρ expansion is made, at leading order we have A = g ρ κ ( s − u )2 m N m ρ , B = − g ρ ( κ + 1) m ρ . (40)Comparing with Eq. (18), we find that the ρ meson exchange only contributes to c term [29].As we already know from the discussions made in section 3.1, the c term will cause an s − / singularity after partial wave projection. This is avoidable, if we do not make a 1 /m ρ expansionin the beginning. It can be seen that all these resonance exchange amplitudes contain singularityof s − / type at most when s = 0. Therefore partial wave projections and chiral expansions donot commute, which can be checked directly by evaluating t -channel ρ exchange contributionsby making partial wave projections of Eqs. (23) and (26).We further make an asymptotic expansion of the ρ exchange contributions to T ( S ) in thevicinity of s = 0 and find that the first two most singular terms are of type a + bs √ s , (41) Because of two excuses: firstly they cancel each other; secondly, they are from distant places as they areassociated with the cutoff at s = Λ R anyway. It is desirable to know the possible origin of spurious effects sinceit leaves the hope to isolate and to remove them. On the contrary, it is difficult to cure the similar problem inPad´e approximation [27]. In the meson–meson scattering lagrangian, the LECs at O ( p ) level are known to be saturated by heavydegrees of freedom [28]. In meson–baryon system, systematic studies on this point are not known to the authors. O ( s − / ) obtained if the ρ propagator were expanded beforehand. Similarthings happen if we introduce, for example, a u -channel as well as s -channel S resonance ex-change. In this situation one can prove that expanding the N ∗ propagator in the full amplitudesleads contributions to c and c terms in Eq. (18). Not making a chiral expansion beforehand theresonance exchange contribution to the partial wave amplitude can be expanded at s = 0 andsimilar results as Eq. (41) are again obtained, so do the P resonance exchange contributions.Explicit expressions for resonance contributions to parameters, e.g., a and b defined inEq. (41) are obtainable. However, another obscure problem occurs here. These coefficientsdepend on the mass parameters of the exchanged resonances and do not vanish as the resonancemass gets large, which seems to contradict the general expectation from the decoupling theo-rem [30] [31]. Without a deeper understanding on this problem, we point out that the signof contributions from different sources can be different. For example, the t -channel ρ exchangecontribution to parameter a is a ( ρ ) = − g ρ κ ( m N − m π ) πm N ; the
12 + baryon exchange contribution is a ( N ∗ + ) = ( g N ∗ ) ( − m N + m π )( m N +( m N ∗ ) ) πF m N ∗ ; whereas the − resonance exchange contribution is a ( N ∗− ) = ( g N ∗ ) ( m N − m π )( m N +( m N ∗ ) ) πF m N ∗ , which is different in sign as comparing with that of thefirst two contributions. Hence, a certain conspiracy theory of cancellation is assumed to over-come the problem of too large resonance contributions to parameter a , or more accurately, T ( s )near s = 0. In practice, we therefore use the O ( p ) χ PT result plus a polynomial backgroundas the input Im L T , i.e.: Im L T ( s ) = Im L T (1) ( s ) + Im L [ a + bs √ s ] , (42)where a and b are simply two free parameters without relating to resonance parameters anymore.The fit gives with N ( s ) = 0 . a = − .
39 GeV and b = − .
27 GeV − and one second sheetpole is found with √ s = 0 . − . i GeV without sizable spurious branch cut contributions.Since the mass of the ρ meson is fixed we also tried the case of O ( p ) χ PT results plus the ρ meson exchange term and a polynomial. That is,Disc T ( s ) = Disc T (1) ( s ) + Disc T ρ ( s ) + Disc[ a + bs √ s ] . (43)In this case the ρ meson exchange produces an extra arc in s plane [9], see Fig. 4. We getFigure 4: The l.h.c. caused by t -channel ρ meson exchange (circular arc [9]); u -channel exchange(line segment from c L to c R ). The branch point d satisfies | d | = m N − m π [9] N ( s ) = 0 . , a = − .
88 GeV, b = − .
00 GeV − , and one second sheet pole is found located at √ s = 0 . − . i GeV . (44) The reason behind this phenomenon is that as s → N ∗ (890) pole location remains stablehowever, irrespective to the pollution from the spurious effects. It is noticed that the contributionfrom the arc cut generated by t -channel ρ exchange is very small, e.g., it only contributes 1.5 ◦ at √ s = 1 .
16 GeV.The “spectral” function in this case is plotted in Fig. 5. Different contributions to the phaseshift according to the PKU decomposition are plotted in Fig. 6.Figure 5: Comparison among different “spectral” functions. The singular behaviors of T ( s )at s = 0 are O ( s − / ), O ( s − / ) and O ( s ) for O ( p ) χ PT , model Eq. (43) and
Case
II,respectively.Figure 6: Fit results using Eq. (43). Phase shift decomposition: only contributions from physicalingredients are plotted including their summation ‘Total’. It clearly demonstrates that spuriouscontributions cancel each other, otherwise curve ‘Total’ cannot get close to the data.We have made a rather long and exhaustive analyses which has to be stopped somewherewith some regrets. One is that all the calculations made in this paper are performed at treelevel only. At loop level, there are of course dynamical cut contributions, like the circular cut.The latter is estimated in Ref. [1] using the complete O ( p ) χ PT input and it is found that thesign of the circular cut contribution may vary depending on the choice of cutoff parameter, butalways remains small in magnitude: e.g., at √ s = 1 .
16 GeV its contribution to the phase shift is0.2 ◦ when s c = 0 .
32 GeV , and -1.7 ◦ when s c = − .
08 GeV . Since when evaluating the circle12here is no problem like what happens at s = 0, we think this estimation on the smallness of thecircular cut contribution is at least qualitatively reasonable. See Fig. 7, the cutoff parameter s c = 0 .
32 corresponds to evaluating the l.h.c. region covered by the green dashed circle, whichcan be estimated by chiral perturbation theory; whereas s c = − .
08 corresponds to that coveredby the red dotted circle, which is required by best fit. The estimations made in Refs. [1] [3]pointed out that the region where χ PT calculation can be safely used is not enough to generatethe N ∗ (890), i.e., certain help from the contribution in the region s ∈ ( −∞ , .
32) is needed. The singularity in the “spectral” function at s = 0 seems to be helpful. It is realised thatthe rescue task is easily fulfilled by looking at Fig. 5. The fit Case II only contains a weaksingularity at s = 0, i.e., T (0) ∝ const while its contribution in the segment (0 . , s L ] is muchweaker than the O ( p ) ones, but it still affords a pole. The real situation should be much moreoptimistic. It is noticed that the model Eq. (42) behaves quite like O ( p ) χ PT results in theFigure 7: Region of l.h.c. being used in Ref. [1]. The cutoff parameter s c = 0 .
32 corresponds toevaluating the l.h.c. region covered by the green dashed circle, which can be estimated by chiralperturbation theory; whereas s c = − .
08 corresponds to that covered by red dotted circle, whichis required by best fit.region (0 . , s L ) and it is expected continuously to work in the region ( − Λ L , − (cid:15) ) ∪ (+ (cid:15), . L is estimated to be around R = m N − m π for example. We make a test by setting (cid:15) (cid:39) .
05 GeV and cut off the peak around s = 0 in the ‘spectral’ function when s ∈ ( − (cid:15), + (cid:15) ).The N ∗ (890) emerges stubbornly with a location √ s = 0 . − . i GeV.Hence, we conclude that the l.h.c. contributions in total to the phase shift is sizable, based onwhich the N ∗ (890) survives with a rather stable pole location. Considering the level of accuracyof our calculations, we do not try to give statistical error bars here in this paper. It may be somewhat amazing to claim something new in a field under extensive studies formore than half a century. However, according to our studies, a subthreshold broad resonancehas well chance of being existed if the s -wave phase shift steadily rise above the threshold as aconvex curve. The discussion made in the end of the last section suggests that the kinematicalsingularity structure at s = 0 plays a rather important role. This is not surprising. One evenfinds examples in extreme cases that a pole can be generated totally from kinematic reasons.For example, in J = 0 , I = 2 channel of ππ scatterings, there exists a virtual pole which can beunderstood from pure kinematical reasons and it brings important contributions to the phaseshift [6], and can be proved to exist rigorously [32]. Another example is the companionate virtualstate of the nucleon, which can also be explained from pure kinematical reasons [3]. More precisely, the pole position is not stable. For example when taking s c = 0 .
32 GeV the N ∗ (890)degrades into two deep virtual poles. Taking s c = 0 .
32 GeV will cause disasters in other channels as well [1].
13t is also interesting to notice that the N ∗ (890) state may be related to the lowest lying1 / − baryon states suggested by Azimov dated back in 1970 [33] named as N (cid:48) there, havingbeen searched for desperately since then [34]. Contrary to the original proposal that the lowestlying nucleon counterpart lie above the πN threshold, or at least lie above the nucleon mass(on the 1st sheet), the pole position named N ∗ (890) determined in Refs. [1] [2] [3] as well asin this paper, easily escapes of all bounds and limits set up previously [34]. The mass (width)difference may be explained by a familiar mechanism that when a strong coupling gradually beturned on, the pole will move from the real axis above the threshold to the left of s plane off thereal axis. Nevertheless, the calculation made in this paper does not provide any solid evidenceon the relation between the N (cid:48) and N ∗ (890) yet. We plan to investigate this question in future.The N/D calculations discussed in this paper are of rather complicated dynamical ones,however the production representation has been shown to be useful in providing us a simple andpictorial way of understanding the essence of the
N/D calculations: the evidence of the existenceof N ∗ (890) seems to be at least partly from the peculiar singularity structure of the backgroundintegral defined in Eqs. (9). It reads that if T ( s = 0) does not vanish (or does not vanish fastenough), then an s -wave subthreshold resonance exist, in the most attractive channels. Thismay even be a rather universal phenomenon, if the background contributions are universallynegative, as suggested by quantum scattering theory [35] and repeatedly verified by calculationsin quantum field theories. [7] [4] [6]It is apparent that the existence of a light 1 / − nucleon state is crucial for the completionand establishment of the lowest lying 1 / − octet baryons as suggested in Ref. [33], if ever itexists . It will definitely improve our understanding of strong interaction physics as well. Forexample, the N ∗ (890) state, if exists, will definitely force us to rethink the possible physicsbehind f (500) and K (700). Another question it raises is how to interpret spontaneous chiralsymmetry breaking more properly. The textbook explanation on this point is that the axialcharge Q A commutes with the strong interaction hamiltonian, hence if chiral symmetry werenot broken parity doublets would appear in nature. But what was observed as the lowest lying1 / − nucleon is N ∗ (1535), the non-degeneracy of its mass comparing with the nucleon masstherefore indicates that chiral symmetry is spontaneously broken. The emergence of N ∗ (890)may also bring new thinking on the related physics.The authors would like to thank Zhi-hui Guo and De-Liang Yao for helpful discussions atvarious stage of this work, and Ulf-G. Meißner for a careful reading of the manuscript and usefulsuggestions. Especially we thank Igor Strakovsky at George Washington University, for veryinteresting information on the 1 / − octet baryons. This work is support in part by NationalNature Science Foundations of China under contract number 11975028 and 10925522, and by theDeutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 196253076-TRR 110. Reference [1] Y. F. Wang, D. L. Yao, and H. Q. Zheng, Chin. Phys. C , 064110 (2019).[2] Y. F. Wang, D. L. Yao, and H. Q. Zheng, Front. Phys. (Beijing) , 24501 (2019).[3] Y. F. Wang, D. L. Yao, and H. Q. Zheng, Eur. Phys. J. C , 543 (2018).[4] H. Q. Zheng et al. , Nucl. Phys. A , 235 (2004).[5] Z. Y. Zhou and H. Q. Zheng, Nucl. Phys. A , 212 (2006).[6] Z. Y. Zhou et al. , JHEP , 043 (2005).[7] Z. G. Xiao and H. Q. Zheng, Nucl. Phys. A , 273 (2001).[8] J. Y. He, Z. G. Xiao, and H. Q. Zheng, Phys. Lett. B , 59 (2002), [Erratum: Phys.Lett. B 549, 362–363 (2002)].[9] Y. Ma, W. Q. Niu, Y. F. Wang, and H. Q. Zheng, Commun. Theor. Phys. , 105203(2020). For more information on the status of the octet baryons, see the talk of Igor Strakovsky given at
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