4. Three-body properties Φ(s,m 2 ) = M(s)+

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1 . Three-body propertie Φ(,m 2 ) = M() 2M() (m 1) 2 1 (,m 2,)Φ(,m 2 ) Intereted in m 2 behaviour.1 Three-body unitarity Γ (m 1) 2 = Γ (m 1) 2 dic m 2 =9 Φ(,m 2 ) = Φ(,m 2 ) Φ(,m 2 ) = 2M( ) (m 1) 2 1 [Φ(,m2 )] Φ(λ2,m2 )] Φ Φ = (Φ Φ )(Φ Φ ) = dic m 2 =9 Φ(λ2,m2 ) dic = Φ(λ2,m 2 ) Will get integral eqn for dic m 2 =9 Φ We know dic = Φ(λ2,m 2 ) from ubenergy unitarity 1

2 λ 2 = 2i F(,t(,x),m 2 ) dic = Φ(λ2,m 2 ) = 2iρ( )M(λ2 )F0 (,m2 )θ(λ2 ) F 0 (,m 2 ) = F(,t(,x),m 2 )dx Integral eqn: dic m 2 =9 Φ(,m 2 ) = 2M( ) (m 1) 2 d 1 2iρ( )M(λ2 )F 0 (,m2 ) 2M( ) (m 1) 2 d 1 dic m 2 =9 Φ(λ2,m 2 ) Solve by iteration 2

3 Firt iterate i the inhomogeneou term: 2M( ) (m 1) 2 d 1 2iρ( )M(λ2 )F 0 (,m2 ) M( ) 1(,m 2,) σ(,m 2 ) M() Ψ(1) (,m 2,) B(,m 2,) J = 0 λ2 σ(,m 2 ) So firt iteration give dic (1) m 2 =9 Φ(,m 2 ) = i (m 1) 2 d { F 0 (,m2 )ρ(λ2 )σ(λ2,m2 )Ψ(1) (,m2, )} m 2 1 = 2i Note: phae pace = dλ2 dt m 2 = σ(,m 2 )ρ( )d dx 3

4 All order: Ψ (1) (,m 2,) Ψ(,m 2,) where Ψ(,m 2,) = B(,m 2,) 2M() (m 1) 2 dµ 2 1 (µ 2,m 2,)Ψ(,m 2,µ 2 )... dic m 2 =9 Φ(,m 2 ) = i (m 1) 2 d { F 0 (,m2 )ρ(λ2 )σ(λ2,m2 )Ψ(λ2,m2, )} = 2i The model atifie THREE body unitarity!

5 But note: (a) Ψ hould be ymmetric in ; not the cae for 0 bit cut-off at = 0 (b) recattering erie not ame a Feynman graph (apart from triangle) In practice, epecially within the context of an iobar-like approach, we don t focu on threebody unitarity, but rather on quai two-body unitarity i.e. particle reonance cattering 5

6 .2 Particle-reonance cattering Φ(,m 2 ) = M()φ(,m 2 ): M() ha cut pole in heet II at II = c. φ(,m 2 ) ha ame cut, but not the econd heet pole. Define particle-reonance production amplitude by φ( c,m 2 ) = lim II c ( c II ) g 2 Φ( II,m 2 ) we are intereted in m 2 tructure of φ( c,m 2 ), pecifically identifying particle reonance branch point at m 2 = ( c 1) 2 and aociated dic = i 6

7 Φ(,m 2 ) = M() 2 (m 1) 2 d 1 (,m 2,)M( )φ(,m 2 ) 1. Branch point at m 2 = ( c 1) 2 m 2 9 ( C 1) 2 C Γ Γ (m 1) 2 A we continue down in m 2 through the m 2 9 cut, the end-point of the -contour at = (m 1) 2 goe into the econd heet acro the cut, and will hit the pole at II = c. Singularity at (m 1) 2 = c i.e. m 2 = ( c 1) 2. Will ee it i q root branch point ( woolly cut ) 7

8 2. Dicontinuity acro woolly cut = difference between two m 2 -continuation which leave the pole on RHS of contour and on LHS of contour. Thi difference i Φ(,m 2 ) Φ(,m 2 ) = 2M( ) (m 1) 2 { d 1 [M(II )φ(λ2ii,m 2 ) M(λ2II )φ(λ2ii,m 2 )]} [...] = M(II )[φ(λ2ii,m 2 ) φ(λ2ii,m 2 )] φ(ii,m 2 )[M(λ2II ) M(λ2II )] Lat difference i 2πiφ(II,m 2 )δ(λ2ii c ). Firt difference i Φ(II,m2 ) Φ(λ2II,m2 ) So dic wc Φ(,m 2 ) = πig 2 M( )φ( c,m 2 ) 1 ( c,m 2, ) 2M( ) (m 1) 2 d 1 (,m 2, )dic wc Φ(II,m2 ) Another integral eqn for the dic, but thi time the inhomogeneou term i not an integral! 8

9 Firt iteration i inhomogeneou term which i [dic wc Φ(,m 2 )] (1) = πig 2 M( )φ( c,m 2 ) 1( c,m 2, ) Keeping pole contribution on both ide, [dic wc φ( c,m 2 )] (1) = πig 2 φ( c,m 2 ) 1( c,m 2, c ) Now recall that 1 /σ i proportional to J = 0 1 c projection of RPE proce c Carrying out iteration to all order, [dic wc φ( c,m 2 )] = πiφ( c,m 2 )σ( c,m 2 )R( c,m 2, c) 1 = i σ( c,m 2 ) = {[m 2 ( c 1) 2 ][m 2 ( c 1) 2 ]} 1/2 /m 2 9

10 Here R i the particle-reonance cattering R amplitude R atifie quai two-body dicontinuity = i Subtantial m 2 -dependence can be generated in ome cae, uing truncated integration. Could impact the extraction of reonance pole poition. Implementation of thee woolly dic relation: ue effective K matrix/p-vector formalim, with phae pace σ intead of ρ 10

11 Summary two-body unitarity analyticity croing linear ingle-variable integral equation for iobar correction function minimal et of contraint need only two-body amplitude employ tandard angular momentum decompoition of IM urpriing (?) three-body tructure included now: include in exptal fit and compare with other approache (effective Hamiltonian, relativitic cattering formalim) 11