Bound State Calculation in Three-Body Systems with Short Range - PowerPoint PPT Presentation

Building Blocks Doublet S -wave and Bound States Triton Charge Radius Bound State Calculation in Three-Body Systems with Short Range Interactions Jared Vanasse Ohio University May 30, 2016 Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Triton Charge Radius Ingredients of EFT � π ◮ For momenta p < m π pions can be integrated out as degrees of freedom and only nucleons and external currents are left. ◮ For any effective (field) theory write down all terms with degrees of freedom that respect symmetries. ◮ Develop a power counting to organize terms by their relative importance. ◮ Calculate respective observables up to a given order in the power counting. Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Lagrangian Triton Charge Radius The two-body Lagrangian to N 2 LO in EFT � π is  � n +1  � � � 1 � � � ∇ 2 ∇ 2 + γ 2 L 2 = ˆ N † ˆ t †  ∆ t − t  ˆ N + ˆ i ∂ 0 + i ∂ 0 + c nt t i i 2 M N 4 M N M N n =0  � n +1  � 1 � � ∇ 2 + γ 2  ∆ s −  ˆ s † s + ˆ c ns i ∂ 0 + s a a 4 M N M N n =0 � � � � N T ¯ t † i ˆ N T P i ˆ a ˆ P a ˆ s † ˆ + y t N + H . c . + y s ˆ N + H . c . . ◮ c 0 t , c 0 s -range corrections Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

The LO dressed deuteron propagator is given by a bubble sum c (0) c (1) (LO) 0 t 0 t (N 2 LO) (NLO) Im[ p ] 3 S 1 ( S − matrix) ( Z -parametrization) At m π LO coefficients are fit to reproduce the deuteron iγ t ( γ t ≈ 45MeV) pole and at NLO to Re[ p ] reproduce the residue about the deuteron pole (Phillips,Rupak, and Savage (2000 )).

Doublet S-wave and Bound state The three-body Lagrangian is � � �� � ∇ 2 + γ 2 L 3 = ˆ t ˆ ψ † Ω − h 2 (Λ) i ∂ 0 + ψ 6 M N M N � � ∞ � ω ( n ) t i − ω ( n ) t 0 ˆ ψ † σ i ˆ s 0 ˆ ψ † τ a ˆ N ˆ + N ˆ + H . c .. s a n =0 where ψ is an auxiliary triton field. The LO triton vertex function G 0 ( E , p ) is given by following coupled integral equations (Hagen, Hammer, and Platter (2013)) G 0 ( E , p ) = � B 0 + K ℓ =0 ( q , p , E ) ⊗ G 0 ( E , q ) , 0

Building Blocks Doublet S -wave and Bound States Triton Charge Radius The LO kernel in cluster-configuration (c.c) space is � q 2 + p 2 − M N E − i ǫ � � 1 � 0 ( q , p , E ) = − 2 π − 3 K ℓ qp Q 0 − 3 1 qp � D t ( E ,� � q ) 0 . × 0 D s ( E ,� q ) The LO triton vertex function and the inhomogeneous term � B 0 are c.c. space vectors given by � G 0 ,ψ → Nt ( E , p ) � � � 1 , � G 0 ( E , p ) = B 0 = . 1 G 0 ,ψ → Ns ( E , p ) The ⊗ operator is given by � Λ 1 dqq 2 A ( q ) B ( q ) . A ( q ) ⊗ B ( q ) = 2 π 2 0 Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Triton Charge Radius The NLO ( G 1 ( E , p )) and NNLO ( G 2 ( E , p )) triton vertex functions are 1 1 1 1 1 1 2 1 2 2 2 1 2 2 Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Triton Charge Radius The NLO triton vertex function is � � p 2 � + K ℓ =0 G 1 ( E , p ) = G 0 ( E , p ) ◦ R 1 ,� ( q , p , E ) ⊗ G 1 ( E , q ) , E − p 0 2 M N and the NNLO triton vertex function � � � � p 2 � G 2 ( E , p ) = G 1 ( E , p ) − c 1 ◦ G 0 ( E , p ) ◦ R 1 E − ,� p 2 M N + K ℓ =0 ( q , p , E ) ⊗ G 2 ( E , q ) , 0 where  � �  � Z t − 1 1 p 2 − M N p 0 − i ǫ γ t + 4 �   2 γ t  � �  R 1 ( p 0 ,� p ) = �  ,  Z s − 1 p 2 − M N p 0 − i ǫ 1 4 � γ s + 2 γ s and � Z t − 1 � c 1 = . Z s − 1 Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Triton Charge Radius Defining Σ 0 . The dressed triton propagator is given by the sum of diagrams Σ 0 Σ 0 Σ 0 which yields i ∆ 3 ( E ) = i Ω − i Ω H LO Σ 0 ( E ) i Ω + · · · 1 = i 1 − H LO Σ 0 ( E ) , Ω where H LO = − 3 ω 2 π Ω = − 3 ω 2 π Ω = 3 ω t ω s t s π Ω . Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Building Blocks Doublet S -wave and Bound States Triton Charge Radius Defining the functions 1 1 Σ 1 2 2 Σ 2 The NNLO triton propagator is Σ 1 H NLO Σ 0 ( NLO ) Σ 2 H NLO Σ 1 H NNLO Σ 0 Σ 1 Σ 1 2 H NLO Σ 0 Σ 1 ( H NLO ) 2 Σ 0 Σ 0 h 2 ( NNLO ) Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Calculating three-body forces and wavefunction renormalization of triton ◮ Method 1: Fix triton pole position at each order (Fixes three-body forces if binding energy fit to). Calculate residue about triton pole at each order to get triton wavefunction renormalization. ◮ Method 2: Note that in general triton pole and wavefunction renormalization given by perturbative expansion 1 − H Σ( E ) = 1 − ( H 0 + H 1 + · · · )(Σ 0 ( B 0 + B 1 + · · · ) + Σ 1 ( B 0 + B 1 + · · · ) + · · · ) = 0 0 ( E ) − Σ ′ 1 1 1 1 ( E ) Z ψ = Σ ′ ( E ) = 1 ( E ) + · · · = 0 ( E ) + · · · Σ ′ 0 ( E ) + Σ ′ Σ ′ Σ ′

Properly Renormalized Vertex Function Ensuring that triton propagator has pole at triton binding energy gives conditions 1 H LO = , H LO Σ 1 ( B ) + H NLO Σ 0 ( B ) = 0 , Σ 0 ( B ) � � H NNLO + 4 3( M N B + γ 2 t ) � H LO Σ 2 ( B )+ H NLO Σ 1 ( B )+ Σ 0 ( B ) = 0 . H 2 Triton wavefunction renormalization is residue about pole leads to triton vertex functions � � � π Z LO Z LO Γ 0 ( p ) = ψ G 0 ( B , p ) , = ψ Σ ′ 0 ( B ) � � � Σ ′ G 1 ( B , p ) − 1 1 Z LO Γ 1 ( p ) = G 0 ( B , p ) . ψ Σ ′ 2 0 � � Σ ′ G 2 ( B , p ) − 1 1 Z LO Γ 2 ( p ) = G 1 ( B , p ) ψ Σ ′ 2 0 � � � � Σ ′ � 2 Σ ′ Σ 2 3 − 1 − 2 3 M N � 1 2 0 + H 2 G 0 ( B , p ) . Σ ′ Σ ′ Σ ′ 8 2 0 0 0

Triton Charge Form Factor Charge form factor of triton at LO given by three diagrams � � 1 + τ 3 � � ˆ ˆ ˆ N † i ∂ 0 + ie A 0 N 2 (a) (b) (c) NLO and NNLO triton charge form factor 2 2 2 1 1 1 (a) (b) (c) 1 1 1 1 1 1 (a) (b) (c) 1 1 (d) (e) (d) (e) (f) h 2 (g)

Building Blocks Doublet S -wave and Bound States Triton Charge Radius LO triton charge form factor given by � d 4 k � d 4 p � 0 ( E , � p ) χ j ( E , � K , � p ,� Z LO (2 π ) 4 G T P , p 0 ,� P , p 0 , k 0 ,� k ) ψ (2 π ) 4 j = a , b , c × G 0 ( E , � K , k 0 ,� k ) , where G 0 ( E , � K , k 0 ,� k ) is LO triton vertex function in a frame boosted by momentum � K G 0 ( E , � K , k 0 ,� k ) = � B 0 � � � �� � � K · � � k 2 q 2 q , k , 2 k � D (0) + 3 B 0 + k 0 − + ,� R 0 B 0 − q 3 M N 2 M N 2 M N ⊗ G 0 ( B 0 ,� q ) . In the Breit frame we have � Q = � P − � K . Jared Vanasse Bound State Calculation in Three-Body Systems with Short Range

Focusing on diagram (a) we find � � � � µα p − 2 a ( E , � K , � p ,� � � χ ji νβ = ie (2 π ) 4 δ ( k 0 − p 0 ) δ (3) P , p 0 , k 0 ,� k ) k − � Q 3 � 2 � k + 2 i 3 E + k 0 ,� � × i D (0) K ( � k − 1 3 � 3 K ) 2 1 + i ǫ 3 E − k 0 − 2 M N � 1 + τ 3 � µ i δ α β δ ij . × ( � k − 2 3 � Q − 1 3 � 2 P ) 2 1 3 E − k 0 − + i ǫ ν 2 M N Projecting in the doublet S-wave channel gives � � p − 2 χ a ( E , � K , � p ,� k ) = ie (2 π ) 4 δ ( k 0 − p 0 ) δ (3) � � P , p 0 , k 0 ,� k − � Q 3 � 2 � k + 2 i 3 E + k 0 ,� � × i D (0) K 3 ( � k − 1 3 � K ) 2 1 3 E − k 0 − + i ǫ 2 M N � 0 � i 0 × . ( � 3 � 3 � 0 2 / 3 k − 2 Q − 1 P ) 2 1 3 E − k 0 − + i ǫ 2 M N

Triton charge form factor LO charge form-factor contribution from diagram (a) is � T F ( a ) � 0 ( p ) ⊗ A 0 ( p , k , Q ) ⊗ � 0 ( Q 2 ) = Z LO G G 0 ( k ) ψ � T +2 � G 0 ( p ) ⊗ A 0 ( p , Q ) + A 0 ( Q ) , and NLO contribution is � T F ( a ) � 0 ( p ) ⊗ A 1 ( p , k , Q ) ⊗ � 1 ( Q 2 ) = Z LO G G 0 ( k ) ψ T + 2 � 1 ( p ) ⊗ A 0 ( p , k , Q ) ⊗ � G G 0 ( k ) � T T +2 � 0 ( p ) ⊗ A 1 ( p , Q ) + 2 � G G 1 ( p ) ⊗ A 0 ( p , Q ) + A 1 ( Q ) , where � � p 2 � � G n ( p ) = D (0) B 0 − ,� p G n ( B 0 , p ) . 2 M N

Triton charge form factor The vector term is � Λ � 1 1 � A n ( p , Q ) = − M N 1 1 � dqq 2 � � dx 2 π qQx q 2 − 2 3 qQx + 1 0 − 1 9 Q 2 p 0    p 2 + q 2 + 1 9 Q 2 + ( y − 2 3 ) qQx − M N B 0  � × Q 0 q 2 − 2 3 qQx + 1 9 Q 2 p � � 1 � qQx � � � q 2 Q 2 2 × D ( n ) B 0 − − + 2 − y ,� q , s 2 M N 12 M N − 2 / 3 M N and scalar term is � Λ � 1 1 � A n ( Q ) = M N 1 2 � dqq 2 � dx 4 π 2 qQx 3 0 − 1 0 � � 1 � qQx � q 2 Q 2 × D ( n ) B 0 − − + 2 − y ,� q . s 2 M N 12 M N M N