Chiral Extrapolation of the Nucleon-Nucleon S-wave scattering lengths

Transcription

1 Chiral Extrapolation of the Nucleon-Nucleon S-wave scattering lengths Jaume Tarrús Castellà Departament d Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos Universitat de Barcelona Work with: Joan Soto tarrus@ecm.ub.es August 01 J. Tarrús (UB) August 01 1 / 17

2 Aim Objective: To study the quark mass dependence of the NN S-wave scattering lengths in NNEFT with dibaryon fields. To use lattice data to determine the relevant (combinations) of low energy constants. NNEFT with dibaryon fields Soto, JTC (08); Soto, JTC (10) Two dibaryon fields an isovector ( 1 S 0) and an isoscalar ( 3 S 1), coupled to pions and nucleons according to Chiral Symmetry. All low energy constants of natural size. Perturbative pions. Dimensional regularization and MS, so all scales are explicit. Integrating out the dibaryons we obtain a theory with enhanced contact interactions similar to KSW. Bedaque, Grießhammer (99) J. Tarrús (UB) August 01 / 17

3 NNEFT with Dibaryons NNEFT Lagrangian The pion and one nucleon sectors as well as the nucleon pion interaction are the usual ones: L χpt = L () +L (4) +... Gasser, Leutwyler (84) N f = L () = F [ ] 0 4 Tr D µ U D µu +χ U +U χ, χ = B 0 ˆm1, m π = B 0 ˆm, ˆm = mu +m d ( ) U = e it π/f 0, π π α τ α = π α T α π 0 π + = π π 0 L πn = L (1) πn +L() πn L (1) πn = N ( id 0 g A σ u ) N, ( D L () πn = N + ig A { σ D,u 0 }+c 1 Tr[χ + ]+c u 0 m N 4m c 3 u u N +i c 4 ǫijk σ k u i u j +c 5 χ + c 6 ǫ ijk F + ij 8m σ k c [ ) 7 ǫ ijk Tr F + ij ]σ k N. N 8m N J. Tarrús (UB) August 01 3 / 17

4 NNEFT with Dibaryons The Dibaryon-Pion interaction The LO Lagrangian: L (1) D = 1 ] [D Tr s ( id 0 +δ ms )D s + D v ( i 0 +δ m ) ( Dv v +ic sv D v Tr[ ud s] h.c.), with D s = Dsτ a a, and d 0D s = 0D s + 1 [[u, 0u],Ds]. The NLO Lagrangian ] ] L () D [D =s1tr s(um u+u Mu )D s +s Tr [D s(um u+u Mu )D s + +v 1D v D [ ] vtr u Mu +um u +... M = ˆm1 J. Tarrús (UB) August 01 4 / 17

5 NNEFT with Dibaryons The Dibaryon-Nucleon interaction The LO Lagrangian: the NLO Lagrangian: L (1) DN =As (N σ τ a τ N )D s,a + As (N σ τ τ a N)D s,a+ + Av (N τ σσ N ) D v + Av (N τ σ σn) D v, L () DN =Bs (N σ τ a τ D N )D s,a + Bs (N σ τ τ a D N)D s,a+ + Bv (N τ σσ D N ) D v + Bv (N τ σ σ D N) D v + B v (D in τ σ i σ D jn )Dv j + B v (D in τ σ σ i D jn)dv j. J. Tarrús (UB) August 01 5 / 17

6 NNEFT with Dibaryons The Dibaryon-Nucleon interaction The LO Lagrangian: the NLO Lagrangian: L (1) DN =As (N σ τ a τ N )D s,a + As (N σ τ τ a N)D s,a+ + Av (N τ σσ N ) D v + Av (N τ σ σn) D v, L () DN =Bs (N σ τ a τ D N )D s,a + Bs (N σ τ τ a D N)D s,a+ + Bv (N τ σσ D N ) D v + Bv (N τ σ σ D N) D v + B v (D in τ σ i σ D jn )Dv j + B v (D in τ σ σ i D jn)dv j. All terms must have natural size!: A s,v Λ 1/ χ, B s,v Λ 5/ χ, B v Λ 5/ χ J. Tarrús (UB) August 01 5 / 17

7 NNEFT with Dibaryons Dibaryon Propagator Dibaryon Propagator Tree level i E+δ m i One loop i E+δ m i +i A i m N p π = E is always smaller than the self-energy term: E p Λ χ. J. Tarrús (UB) August 01 6 / 17

8 NNEFT with Dibaryons Dibaryon Propagator Dibaryon Propagator Tree level i E+δ m i One loop i E+δ m i +i A i m N p π = E is always smaller than the self-energy term: E p Λ χ. Size of δ m i Matching the one loop propagator to the ERE: δ m i 1 πa i m π Λ χ i = s,v. a 3 S 1 1 = 5.38fm( 36.7MeV ) S 0 1 a1 = 3.7fm( 8.3MeV ) J. Tarrús (UB) August 01 6 / 17

9 NNEFT with Dibaryons Counting Counting There are two momentum regions with different counting: pnneft, p m π, E p /Λ χ = π A i m Np 1 m π = i( E +δ mi ) i = s,v. Potential pions q 0 q /m N Dibaryon-Nucleon interactions J. Tarrús (UB) August 01 7 / 17

10 NNEFT with Dibaryons Counting Counting There are two momentum regions with different counting: pnneft, p m π, E p /Λ χ = π A i m Np 1 m π = i( E +δ mi ) i = s,v. Potential pions q 0 q /m N Dibaryon-Nucleon interactions /πnneft, p m π Λ χ, no dynamic pions = = ie i δ mi +i A i m N p π Λχ m π Dibaryon-Nucleon interactions i = s,v. J. Tarrús (UB) August 01 7 / 17

11 Chiral Extrapolation of the scattering lengths Chiral extrapolation of the scattering lengths In /πnneft the scattering lengths up to NLO are given by a 1 i = πδm i m NA i πδlo m i m NA i with NLO = A i,nlo/a i 1. δm LO i,δm NLO i (1 NLO)+ πδnlo m i, i = s( 1 S 0),v( 3 S 1). m NA i, NLO are obtained matching NNEFT pnneft /πnneft. J. Tarrús (UB) August 01 8 / 17

12 Chiral Extrapolation of the scattering lengths NNEFT pnneft /πnneft Matching NNEFT to pnneft: Radiation pions pnneft is obtained by integrating out nucleons of and pions E m π. Among the later there are the radiation pions q 0 q m π. ( mπ/λ ) Parametric suppression of O χ, but numerically O(1). However they add up to zero, due to Wigner Symmetry. Fleming, Mehen, Stewart (00) J. Tarrús (UB) August 01 9 / 17

13 Chiral Extrapolation of the scattering lengths NNEFT pnneft /πnneft Further contributions are obtained introducing potential pions, q 0 q /m N, inside the radiation pion loops. p m πm N p 1 p +m π As a consequence n potential pion exchanges can be simplified Each bubble should add a parametric suppression of m π/λ χ but in fact π m π/λ χ 1.!. The resumation can be carried out in the 1 S 0 channel ( 3 S 1 channel in the whole diagram). J. Tarrús (UB) August / 17

14 Chiral Extrapolation of the scattering lengths NNEFT pnneft /πnneft Summing up all this class of radiation pion diagrams for the 3 S 1 channels adds up to zero! Reason: The contracted pion exchange, 4 nucleon vertex can be removed through field redefinitions. 1 S 0 channel the resummation cannot be carried out, but the argument still applies to one potential pion insertion. Potentially large unknown contributions in the 1 S 0 channel. J. Tarrús (UB) August / 17

15 Chiral Extrapolation of the scattering lengths NNEFT pnneft /πnneft Matching NNEFT to pnneft Residual Mass (δ m) up to NLO δ mi = δ m i +δm N+, i=s,v. δm N are contributions to the matching to the nucleon mass reshuffled into δ mi through field redefinitions. δm N = 4c 1m π 3g A 64πF0 m 3 π. J. Tarrús (UB) August 01 1 / 17

16 Chiral Extrapolation of the scattering lengths NNEFT pnneft /πnneft Matching pnneft /πnneft p m π Λ χ Non-local potentials become local and can be organized in powers of p /m π Residual masses (δ mi ) up to NLO Contribution to the residual mass and the time derivative. Contributions to the dibaryon time derivative can be reabsorbed by field redefinitions. Dibaryon-Nucleon Interactions (A i) up to NLO The dibaryon-nucleon vertex gets a contribution from this vertex. J. Tarrús (UB) August / 17

17 Chiral Extrapolation of the scattering lengths Fit to lattice data a 1 i ( =ζ i1 1 g A m [ N 16πF0 m π )+ ( g A m N πF 0 ) ( m m 3 ) π ln π µ ζ i g A m N 3πF0 ln ( m ) ] π µ m π +ζ i3m 3 π, i = s( 1 S 0 ),v( 3 S 1 ). 1 S 0 πδ ms m N A s 3 S 1 πδ mv m N A v ( ) 1 g S A 0 3m N F 0 1 A + csv g v A AsAv 3 A ( s 3 g S A 1 3m N F 0 1 A + csv g s A AsAv 3 A v ζ 1 ζ π((s 1 +s )/B 0 8c 1 ) m N A s π(v 1 /B 0 8c 1 ) m N A s ζ 3 g A (s 1 +s )/B 0 8c 1 8F 0 A s ) g A (v 1 /B 0 8c 1 ) 8F 0 A + 5 v ( ) g + A m N log() F 0 18π ( ) g A m N 6+13log() 56π F 0 J. Tarrús (UB) August / 17

18 Chiral Extrapolation of the scattering lengths Fit to lattice data Lattice data from NPLQCD collaboration, Beane et al. (06) a fm 4 3 a 1 S0 a fm 6 4 a 3 S mπ mπ,phys 1 3 Solid(dashed) line correspond to LO(NLO) mπ mπ,phys LO χ d.o.f ζ 1 (MeV ) ζ (MeV 1 ) 1 S S NLO χ d.o.f ζ 1 (MeV ) ζ (MeV 1 ) ζ 3 (MeV ) 1 S S J. Tarrús (UB) August / 17

19 Conclusions Conclusions Contributions to the 3 S 1 scattering length with one radiation pion and n potential pion are of O(1), but add up at to zero. In the 1 S 0 channel it is not possible to compute such diagrams for an arbitrary number of potential pions, there are possible large contributions missing. We have given chiral extrapolation formulas for the scattering lengths up to O ( m 3 π/λ χ ) Lattice data favors an unbounded deuteron in the chiral limit. J. Tarrús (UB) August / 17

20 Conclusions Thank you for your attention J. Tarrús (UB) August / 17