Contraction Diagram Analysis in Pion-Kaon Scattering
Chaitra Kalmahalli Guruswamy, Ulf-G. Meißner, Chien-Yeah Seng
CContraction Diagram Analysis in Pion-Kaon Scattering
Chaitra Kalmahalli Guruswamy a , Ulf-G. Meißner a,b,c , Chien-Yeah Seng a a Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universit¨at Bonn, D-53115 Bonn, Germany b Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany c Tbilisi State University, 0186 Tbilisi, Georgia
Abstract
We study the contributions from the connected and disconnected contraction diagrams to the pion-kaonscattering amplitude within the framework of SU(4 |
1) partially-quenched chiral perturbation theory.Combining this with a finite-volume analysis, we demonstrate that a lattice calculation of the easier com-putable connected correlation functions is able to provide valuable information of the noisier disconnectedcorrelation functions, and may serve as a theory guidance for the future refinement of the correspondinglattice techniques.
1. Introduction
Pion-kaon ( πK ) scattering is the simplest hadronic scattering process that involves a strange quark,and therefore it plays a crucial role in our understanding of the SU(3) chiral symmetry breaking of theQuantum Chromodynamics (QCD) [1]. The πK scattering amplitude was calculated within the frame-work of Chiral Perturbation Theory (ChPT) at one loop [2, 3] and at two loops [4], with the appearanceof certain low-energy constants (LECs), some of which can be fixed in other processes. Naturally, this alsoprovides motivation for the study of πK scattering using one of the standard first-principle treatmentsof the strong interaction, namely lattice QCD .Furthermore, due to the similar isospin structures, an improved understanding of πK scattering alsoprovides useful insights for πN scattering, which is an important ingredient towards resolving the cur-rent disagreement between the lattice [7, 8, 9, 10, 11] and the dispersion-theoretical [12, 13, 14, 15]determinations of the pion-nucleon sigma term.So far there exists a number of exploratory studies of πK scattering, in both the I = 3 / I = 1 / I = 1 / K ∗ resonance, but it turns out that this channel is much more difficult to handleon the lattice, due to the existence of correlation functions involving the contraction of one or more pairsof quarks at the same temporal point (which are often called “disconnected diagrams”). Such diagramshave low signal-to-noise ratio, and are also the main reason for the increased difficulty in the lattice studyof ππ scattering at lower isospin. Obviously, one cannot claim to have a controlled error analysis in thelattice study of πK scattering without properly understanding the contribution from the disconnecteddiagrams.The recent years have seen a systematic development of a theory analysis of contraction diagrams inhadron-hadron interactions based on Partially-Quenched Chiral Perturbation Theory (PQChPT). Theunderlying principle is rather straightforward: Contraction diagrams that are inseparable in a physicalamplitude would become separable upon the introduction of extra quark flavors. Since this separationis unphysical, it will unavoidably involve new parameters that cannot be fixed by experiment, but canbe determined from lattice simulations. This method was successfully applied in the analysis of ππ For investigations of πK scattering using dispersion relations, see e.g. [5, 6]. a r X i v : . [ h e p - l a t ] J un a) u¯s u¯su¯d u¯d (b) u¯s u¯su¯d u¯d Figure 1: The quark contraction diagrams for I = 3 / πK scattering. The amplitude for diagram (a) and (b) is givenby T a ( s, t, u ) and T b ( s, t, u ) respectively. The thick line indicates the (cid:104) s ¯ s (cid:105) contraction. The time flows in the horizontaldirection. scattering [24, 25, 26] and the parity-odd πN coupling [27]. In this paper we generalize it to πK scatteringin both the finite and infinite volume. We demonstrate that, from the lattice calculation of the two easiercomputable connected diagrams in the I = 3 / I = 1 /
2. Thisprovides a useful theory gauge to the calculation of the latter on lattice. For a discussion of the status ofvarious scattering processes pertinent to chiral dynamics in the continuum and on the lattice, see [28].This work is organized as follows. In Sec. 2 we introduce the different contraction diagrams in πK scattering, and demonstrate how they can be expressed in terms of physical scattering amplitudes in adeformation of QCD with an extended flavor sector. In Sec. 3 we introduce SU(4 |
1) PQChPT in theinfinite volume, and use it to calculate the different contraction diagrams up to one-loop accuracy, O ( p ).In Sec. 4 we discuss the implications of the results above to the actual lattice calculations which arecarried out in a finite volume. The final conclusions are given in Sec. 5.
2. Contraction diagrams in πK scattering Assuming isospin symmetry, the πK scattering amplitude can be categorized into two isospin channels, T / and T / . In particular, T / ( s, t, u ) is given by the scattering amplitude of π + ( k ) K + ( p ) → π + ( k ) K + ( p ), where the Mandelstam variables s, t, u are defined as s = ( k + p ) , t = ( k − k ) , u = ( k − p ) , respectively, subject to the constraint s + t + u = 2( M π + M K ). The I = 1 / T / by appropriate crossing: T / ( s, t, u ) = 32 T / ( u, t, s ) − T / ( s, t, u ) . (1)To construct interpolators of mesons on the lattice, one expresses the meson fields in terms of their“constituent quarks”, for example, π + = u ¯ d and K + = u ¯ s . A lattice study of meson-meson scatteringthen consists of computing correlation functions involving all possible contractions between quark andanti-quark pairs. For instance, the I = 3 / T a ( s, t, u ) and T b ( s, t, u ) depicted in Fig. 1: T / ( s, t, u ) = T a ( s, t, u ) + T b ( s, t, u ) . (2)Both contraction diagrams above are purely connected, as there is no contraction between the quark–anti-quark pair at the same time coordinate. Therefore, they are rather straightforwardly calculable on thelattice. The situation for the I = 1 / T / ( s, t, u ) = T a ( s, t, u ) − T b ( s, t, u ) + 32 T c ( s, t, u ) , (3)2 a) u¯s u¯su¯u u¯u (b) u¯s u¯su¯u u¯u (c) u¯s u¯su¯u u¯u Figure 2: The contraction diagrams in the I = 1 / πK scattering. among which the diagram (c) contains a pair of disconnected contractions and is much noisier on thelattice. However, from the theory point of view, T c is nothing but the s → u crossing of T b and is no morecomplicated than the latter. Therefore, a precise theory description of the individual connected diagramswill automatically provide useful information of the disconnected ones which can be directly contrastedto lattice results.In an ordinary three-flavor QCD the two connected diagrams in Fig. 1 are inseparable in any physicalscattering amplitude, so one cannot study T b ( s, t, u ) by itself. The separation is possible, however, in adeformation of QCD with an extended quark sector. In a generic meson-meson scattering, in order toisolate each contraction diagram one requires a minimum number of four fermionic quarks [24]. But at thesame time one needs also one “bosonic quark”, such that its loop effect cancels with that from the extrafermionic quark, and thus to keep the sea dynamics identical to that of ordinary three-flavor QCD. Thisleads to SU(4 |
1) Partially-Quenched QCD (PQQCD), in which the quark sector reads q = ( u, d, s, j ; ˜ j ),where the first four quarks are fermionic and the last is bosonic. The quark mass matrix is givenby M = diag( ¯ m, ¯ m, m s , ¯ m ; ¯ m ), where m s is the strange quark mass and ¯ m < m s . Notice that thisextended theory is actually simpler than that needed in the analysis of ππ scattering [25, 26]. There,one needs again four fermionic quarks for the diagram separations, but two bosonic quarks in order tokeep the sea dynamics identical to a two-flavor QCD. That leads to an SU(4 |
2) PQQCD which has morepseudo-Nambu-Goldstone (pNG) particles than SU(4 |
1) (see discussions in the next section). The twocontractions T a and T b can now be expressed in terms of physical scattering amplitudes in the extendedtheory: T a ( s, t, u ) = T ( u ¯ s )( d ¯ j ) → ( u ¯ s )( d ¯ j ) ( s, t, u ) ,T b ( s, t, u ) = T ( u ¯ s )( d ¯ j ) → ( d ¯ s )( u ¯ j ) ( s, t, u ) . (4)
3. Analysis in SU(4 |
1) PQChPT
The right-hand side of Eq. (4) can be calculated in the low-energy effective field theory (EFT) ofSU(4 |
1) PQQCD, namely the SU(4 |
1) PQChPT [29, 30, 31, 32, 33, 34, 35]. In this section we summarizethe most important results relevant to this work, while interested readers may refer to the literature citedabove for more details.Firstly, in complete analogy to the ordinary ChPT, the spontaneous chiral symmetry breaking SU(4 | L ⊗ SU(4 | R → SU(4 | V in SU(4 |
1) PQQCD generates pNG particles that are expressed collectively in thefollowing matrix-valued field: Φ = φ η η ˜ φ , (5)3ith: φ = u ¯ u u ¯ d u ¯ s u ¯ jd ¯ u d ¯ d d ¯ s d ¯ js ¯ u s ¯ d s ¯ s s ¯ jj ¯ u j ¯ d j ¯ s j ¯ j , η = u ¯˜ jd ¯˜ js ¯˜ jj ¯˜ j , η = (cid:16) ˜ j ¯ u ˜ j ¯ d ˜ j ¯ s ˜ j ¯ j (cid:17) , ˜ φ = ˜ j ¯˜ j . (6)The supertrace (Str) of Φ is defined as, Str Φ = (cid:88) i =1 φ ii − ˜ φ. (7)Since we know that there are only 5 − Φ (in contrast to the6 − |
2) for ππ scattering), it is more convenient to introduce a supertracelessmatrix Φ (cid:48) = Φ − Str Φ . In particular, the diagonal components in Φ (cid:48) give rise to four independent neutralpNG bosons { π , η, σ a , σ b } by writing:( Φ (cid:48) ) diag = π λ (cid:48) + σ a λ (cid:48) + 12 √ η − σ b ) λ (cid:48) + 12 √ − η + 3 σ b ) λ (cid:48) (8)where λ (cid:48) = 1 √ , − , ,
0; 0) , λ (cid:48) = 1 √ , , , −
2; 0) ,λ (cid:48) = 1 √
12 diag(1 , , − ,
1; 0) , λ (cid:48) = − √
24 diag(1 , , ,
1; 4) . (9)With this we can define the standard non-linear representation of the pNG particles, U = exp (cid:32) √ i Φ (cid:48) F (cid:33) , (10)where F is the pNG boson decay constant in the chiral limit, and proceed to construct the most generaleffective chiral Lagrangian. At O( p ) we get: L (2) = F ∂ µ U † ∂ µ U ] + F χU † + U χ † ] , (11)where χ = 2 B M , with M the quark mass matrix. Expanding Eq. (11) up to the quadratic terms ofpNG fields, we find that there are no mixing terms between different fields, and thus all the 24 pNG fieldsare indeed independent particles, with the leading order (LO) squared masses given by one of the threefollowing mass parameters:˚ M π = 2 B ¯ m, ˚ M K = B ( ¯ m + m s ) , ˚ M η = 23 B ( ¯ m + 2 m s ) , (12)satisfying the Gell-Mann-Okubo formula, 3 ˚ M η = 4 ˚ M K − ˚ M π . In particular, the four neutral parti-cles { π , η, σ a , σ b } have LO masses { ˚ M π , ˚ M η , ˚ M π , ˚ M π } , respectively. One also finds that the pNG fieldpropagators are given by the standard form: S φ ( k ) = ik − M φ + i(cid:15) , (13)except that the σ b propagator acquires an extra negative sign. In short, the diagonalization procedureof the neutral particles in Eq.(8) completely avoids the cumbersome double-pole structures in the usualdiscussions of PQChPT propagators, and greatly simplifies the one-loop analysis.4 able 1: Coefficients of the UV divergence in the SU(4 |
1) PQChPT. i 0 1 2 3 4 5 6 7 8Γ i
116 332 316
18 38 11144 Applying the Lagrangian in Eq. (11) at one loop results in ultraviolet (UV) divergences that areregulated using dimensional regularization (DR) and reabsorbed into the LECs of the most general O( p )chiral Lagrangian without external sources [34, 36]: L (4) = L Str[( ∂ µ U † )( ∂ ν U )( ∂ µ U † )( ∂ ν U )]+ ( L − L )Str[( ∂ µ U † )( ∂ µ U )]Str[( ∂ ν U † )( ∂ ν U )]+ ( L − L )Str[( ∂ µ U † )( ∂ ν U )]Str[( ∂ µ U † )( ∂ ν U )]+ ( L + 2 L )Str[( ∂ µ U † )( ∂ µ U )( ∂ ν U † )( ∂ ν U )]+ L Str[( ∂ µ U † )( ∂ µ U )]Str[ U † χ + χ † U ]+ L Str[( ∂ µ U † )( ∂ µ U )( U † χ + χ † U )]+ L (Str[ U † χ + χ † U ]) + L (Str[ U † χ − χ † U ]) + L Str[ χU † χU † + χ † U χ † U ] . (14)Here it is useful to notice that the LECs { L i } i =1 are identical to those in the ordinary SU(3) ChPT [1],and the only new LEC is L . This can be seen by observing that Eq. (14) is equivalent to the O ( p ) chiralLagrangian of the ordinary SU(3) ChPT at tree level as long as the involved particles are the ordinarySU(3) pNG bosons. The renormalized LECs are defined by L ri = L i − λ Γ i , where λ = − π (cid:18) − d + log(4 π ) − γ E + 1 (cid:19) , (15)with γ E the Euler-Mascheroni constant, and d is the number of space-time dimensions. The divergence( β -function) coefficients { Γ i } are summarized in Tab. 1.Below we quote the analytical results up to O ( p ) needed in this work. First, the physical pion, kaonmasses and the pion decay constant are just the same as in ordinary ChPT [1]: M π = ˚ M π (cid:20) µ π − µ η M K F π (2 L r − L r ) + 8 M π F π (2 L r + 2 L r − L r − L r ) (cid:21) ,M K = ˚ M K (cid:20) µ η M π F π (2 L r − L r ) + 8 M K F π (4 L r + 2 L r − L r − L r ) (cid:21) ,F π = F (cid:20) − µ π − µ K + 8 M K F π L r + 4 M π F π ( L r + L r ) (cid:21) , (16)where µ P = ( M P / π F π ) ln( M P /µ ), with µ the scale of dimensional regularization.The two contraction diagrams T a and T b , expressed as SU(4 |
1) physical scattering amplitudes inEq. (4), are given up to O ( p ) as: T a ( s, t, u ) = µ π F π M π ( M π − M K ) [16 M K M π + 2 M K (14 M π − M π t + t )+ M π (16 M π − M π t − s − su − u )] + µ K F π M K ( M K − M π )[8 M K + 8 M K (2 M π − t ) + M K (8 M π − M π t − s − su − u )5 M π t ] + M π µ η F π M η ( M K − M π ) [ − M K + M K (18 t − M π )+9 M π t ] + t ( M π + t )8 F π ¯ J ππ ( t ) − M π (8 M K − t )72 F π ¯ J ηη ( t ) + t F π ¯ J KK ( t )+ ( M K + M π − s ) F π ¯ J πK ( s ) + ( M K + M π − u ) F π ¯ J πK ( u )+ M π (4 M K − t )12 F π ¯ J πη ( t ) + 8 F π [3 M K + M K (4 M π − t )+3 M π − M π t − s − su − u ] L r + 8 F π ( t − M K )( t − M π ) L r + 4 F π [( s − M π − M K ) + ( u − M π − M K ) ] L r + 8 F π [ M K ( t − M π )+ M π t ] L r + 32 M K M π F π L r + 4 M K M π − t ( M π + t )576 π F π (17)and T b ( s, t, u ) = M K + M π − s F π + µ π F π M π ( M π − M K ) [ − M K (9 M π + 2 t )+ M K (6 M π + 3 M π (5 t + 6 u ) + 8 t + 4 tu ) + M π (24 M π − M π (4 t + 9 u ) − t − tu − u )] + µ K F π M K ( M K − M π ) [30 M K +2 M K (9 M π − t − u ) + M K (3 M π (3 t − u ) − t + 2 tu + 6 u )+ M π t ( − M π + 2 t + u )] + µ η F π ( M K − M π ) [ − M K + M K (15 t +18 u − M π ) − M π ( t − u ) − tu − u ] + 112 F π [2 t + tu + 4 M π + M K (4 M π − t ) − M π (3 t + 2 u )] ¯ J ππ ( t )+ 124 F π [4 M K + M K (4 M π − t − u ) + t ( − M π + 2 t + u )] ¯ J KK ( t )+ 116 F π [ − M K + M K (8 M π − t ) − M π − M π t + tu + 2 u ] ¯ J πK ( u )+ t F π ( M K − M π ) u ¯ J πK ( u ) − M π (4 M K − t )12 F π ¯ J πη ( t ) + 1144 F π [44 M K + M K (56 M π − t − u ) − M π + 6 M π ( t − u ) + 9 tu + 18 u ] ¯ J Kη ( u ) − ( M K − M π ) (16 M K − M π − t )144 F π ¯ J Kη ( u ) u + ( M K − M π ) F π u (9 ¯¯ J πK ( u ) + ¯¯ J Kη ( u )) − F π [ − s − su − u + 3 M K + M K (4 M π − t ) + 3 M π − M π t ] L r + 2 F π [( t − M K )( t − M π ) + ( u − M π − M K ) ] L r − M π ( M K − M π + s ) F π L r + 16 M K M π F π L r + 1384 π F π [11 M K + M K (10 M π − t − u ) + 3 M π + M π (5 t − u ) − t + tu + u ] . (18)6ere, the two-point functions ¯ J P Q and ¯¯ J P Q are defined as [37]:¯ J P Q ( s ) = 132 π (cid:34) (cid:18) ∆ s − Σ∆ (cid:19) ln M Q M P − ν ( s ) s ln [ s + ν ( s )] − ∆ [ s − ν ( s )] − ∆ (cid:35) ¯¯ J P Q ( s ) = ¯ J P Q ( s ) − s π (cid:34) Σ∆ + 2 M P M Q ∆ ln M Q M P (cid:35) , (19)where ∆ = M P − M Q , Σ = M P + M Q , ν ( s ) = (cid:113) [ s − ( M P + M Q ) ][ s − ( M P − M Q ) ] . (20)The third contraction diagram is simply given by T c ( s, t, u ) = T b ( u, t, s ). Finally, we are also interestedin their values at the threshold, s = ( M K + M π ) , t = 0 and u = ( M K − M π ) , which are given by:( T a ) thr = M K M π F π ( M K − M π ) (cid:18) µ π − µ K − M K + M π M η µ η (cid:19) + M K M π F π ( − L r + 32 L r +32 L r − L r + 32 L r + ¯ J πK ( s ) + ¯ J πK ( u )) + M K M π π F π (21)( T b ) thr = − M K M π F π + µ π F π ( M K − M π ) [ M K + 2 M K M π + 5 M K M π − M K M π ]+ µ K F π ( M K − M π ) [2 M K + 2 M K M π + M K M π − M π ] − µ η F π ( M K − M π ) [3 M K + 6 M K M π + 11 M K M π − M K M π ] − F π [( M K − M π ) ( M K + 6 M K M π + M π )] ¯ J πK ( u ) − F π [ M K +4 M K M π − M K M π + 4 M K M π + M π ] ¯ J Kη ( u ) + ( M K + M π ) F π (9 ¯¯ J πK ( u )+ ¯¯ J Kη ( u )) + 8 M K M π F π (cid:18) L r + 2 L r − M K + M π M K L r + 2 L r (cid:19) + ( M K + M π ) (5 M K + M π )384 π F π (22)and ( T c ) thr is obtained by replacing M π → − M π (which also means u → s ) in ( T b ) thr .
4. Finite-Volume Analysis
We now discuss the implications of the results above, which are obtained in a field theory at infinitevolume, to the discrete energies calculated on the lattice in a finite volume. The analysis in this sectionis a straightforward generalization of that in Ref. [26].To do so we construct three effective single-channel scattering amplitudes using T a , T b and T c . First,consider the 2 × | ψ (cid:105) = | u ¯ s (cid:105) | d ¯ j (cid:105) and | ψ (cid:105) = | d ¯ s (cid:105) | u ¯ j (cid:105) .Diagonalizing this matrix gives two single-channel scattering amplitudes: T α ( s, t, u ) = T a ( s, t, u ) + T b ( s, t, u ) , T β ( s, t, u ) = T a ( s, t, u ) − T b ( s, t, u ) . (23)In particular, T α ( s, t, u ) = T / ( s, t, u ). The third single-channel amplitude is simply: T γ ( s, t, u ) = T / ( s, t, u ) = T a ( s, t, u ) − T b ( s, t, u ) + 32 T c ( s, t, u ) . (24)7or each single-channel amplitude one could perform the partial-wave expansion in the center-of-mass(CM) frame: T ( s, t, u ) = ∞ (cid:88) l =0 (2 l + 1) T l ( E ) P l (cos θ ) , (25)where E = √ s is the CM energy, θ is the scattering angle and { P l ( x ) } are the Legendre polynomials.The l = 0 (i.e. S-wave) partial-wave amplitude is parameterized as: T ( E ) = 8 πEp cot δ ( E ) − ip , (26)where p is the CM momentum and δ ( E ) is the S-wave phase shift. At small p one performs the effectiverange expansion: p cot δ ( E ) = − a + 12 r p + . . . , (27)which defines the S-wave scattering length a and effective range r . The S-wave scattering lengths ofthe three single-channel amplitudes above are given by: a α = − π √ s [( T a ) thr + ( T b ) thr ] ,a β = − π √ s [( T a ) thr − ( T b ) thr ] ,a γ = − π √ s (cid:20) ( T a ) thr −
12 ( T b ) thr + 32 ( T c ) thr (cid:21) . (28)In particular, a β is the only one among the three that depends on the unphysical LEC L r . The latterdoes not affect the pNG boson masses and decay constants at O ( p ), but does contribute to the scatteringparameters.The discrete energies E extracted from lattice correlation functions at finite volume can be obtainedby solving the single-channel L¨uscher’s formula [38] (see also Ref. [20, 39] for more discussions): p cot δ ( E ) = 2 πL π − / Z (1; q ) , q = p L π , (29)where L is the lattice size and Z is the L¨uscher zeta-function. This gives the discrete ground-stateenergies of the three channels as known functions of the scattering lengths and lattice size E i = f ( a i , L ),( i = α, β, γ ). Therefore, if we define C i ( τ ) ( i = a, b, c ) as the lattice correlation function corresponding tothe contraction diagram of type i at the Euclidean time τ , then the following combinations of correlationfunctions decay as a single exponential at large τ : C a ( τ ) + C b ( τ ) ∼ A α exp {− E α τ } ,C a ( τ ) − C b ( τ ) ∼ A β exp {− E β τ } ,C a ( τ ) − C b ( τ ) + 32 C c ( τ ) ∼ A γ exp {− E γ τ } . (30)Hence, through a single lattice calculation of the difference between C a and C b which both appear inthe I = 3 / E β and thus fix the unknown LEC L r . After doing so, allthe three discrete energies { E α , E β , E γ } are fully predictable given any set of lattice parameters. Thisis beneficial in multiple ways. For instance, we know that the most difficult disconnected correlationfunction C c ( τ ) depends on three exponents: C c ( τ ) ∼ − A α exp {− E α τ } − A β exp {− E β τ } + 23 A γ exp {− E γ τ } , (31)8nd all the three exponents are known functions of { ¯ m, m s , L } . This provides a useful theory gauge of theaccuracy for lattice studies of C c ( τ ), which directly tests the lattice techniques in handling disconnecteddiagrams. Furthermore, once L r is fixed from lattice data, the SU(4 |
1) chiral Lagrangian (withoutexternal sources) will be completely known to NLO, so it could be applied to the lattice study of otherinteresting hadronic processes such as the ππ → K ¯ K scattering.We end this section by estimating the LEC L r . Integrating out the strange quark from the theory, thegraded algebra SU(4 |
1) reduces to SU(3 | L appearing in this reduced theory should be thesame as in the SU(4 |
2) version and thus is known with a sizeable uncertainty, L r = 1 . . · − , whichcomes from an NNLO analysis of lattice data in Ref. [40]. In our new formalism, L r appears at NLOin the SU(4 |
1) scattering amplitudes, so one could in general expect an order-of-magnitude improvementof its accuracy through the analysis of the lattice πK contraction diagrams, just like what happened tothe combination 3 L r + L r in SU(4 |
2) as demonstrated in Ref. [26]. The actual analysis will appear in afuture work.
5. Conclusions
As a natural generalization of previous works, we perform a PQChPT analysis of the different con-traction diagrams in πK scattering, in both infinite and finite volume. We show that up to O ( p ) thereis only one undetermined LEC in the EFT, which can be fixed by the lattice study of the connectedcontraction diagrams in the I = 3 / τ behavior of the disconnectedcorrelation function in the I = 1 / πN scattering and will be carried out in a follow-up work. Acknowledgements
We thank Feng-Kun Guo for some useful discussion. This work is supported in part by the DFG(Grant No. TRR110) and the NSFC (Grant No. 11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”, and also by the Alexandervon Humboldt Foundation through the Humboldt Research Fellowship (CYS). The work of UGM wasalso supported by the Chinese Academy of Sciences (CAS) through a President’s International FellowshipInitiative (PIFI) (Grant No. 2018DM0034) and by the VolkswagenStiftung (Grant No. 93562).
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