Few-Body in EFT Gautam Rupak Mississippi State University EMMI: - PowerPoint PPT Presentation

Few-Body in EFT Gautam Rupak Mississippi State University EMMI: The Systematic Treatment of the Coulomb Interaction in Few-Body Systems Darmstadt, May 30 - June 3, 2016

Outline • N-d scattering • Field-redefinition • p-p fusion

A little detour on the lattice • Consider: ; a ( b, γ ) c a ( b, c ) d • Need effective “cluster” Hamiltonian -- acts in cluster coordinates, spins,etc. • Calculate reaction with cluster Hamiltonian. Many possibilities --- traditional methods, continuum EFT, lattice method

Adiabatic Projection Method Initial state | ~ R i Evolved state | ~ R i τ = e − τ H | ~ R i τ h ~ R 0 | H | ~ R i τ Energy measurements in cluster basis. Divide by the norm matrix as these are R 0 = ⌧ h ~ R | ~ not orthogonal basis R 0 i ⌧ [ N ⌧ ] ~ R, ~ Microscopic Hamiltonian L 3( A − 1) Cluster Hamiltonian smaller matrices in practice!! L 3 -- acts on the cluster CM and spins

Neutron-Deuteron System Convergence: L =7, b =1/100 MeV − 1 25 Pine, Lee, Rupak, EPJA 2013 20 15 E (MeV) 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 τ (MeV − 1 ) - grouping R found efficient ∼ 30 × 30

Something still missing ... long range Coulomb · · · ✓ α ✓ α ◆ ◆ α µ O O p 2 p 2 p

Spherical-wall method ψ short ( r ) ∝ j 0 ( kr ) cot δ s − n 0 ( kr ) , ψ Coulomb ( r ) ∝ F 0 ( kr ) cot δ sc + G 0 ( kr ) R w Adjust from free theory: j 0 ( k 0 R w ) = 0 IR scale setting − L/ 2 L/ 2 Hard spherical wall boundary conditions, Borasoy et al. 2007 Carlson et al. 1984 Even older ?

p-p Coulomb Subtracted Phase Shift 60 3% error in fits Analytic 50 b=1/100 MeV − 1 b=1/200 MeV − 1 δ sc (degree) 40 T = T c + T sc 30 e 2 i σ − 1 T c ≈ 2 π µ 2 ip 20 e 2 i ( σ + δ sc ) − 1 T ≈ 2 π 10 Rupak, Ravi PLB 2014 µ 2 ip 0 0 5 10 15 20 25 30 p (MeV)

Improvement 0 0 0 0 0 0 n − d -1 b =1/100 MeV n − d (EFT) − 15 − 15 − 15 − 15 − 15 -1 p − d Breakup Breakup Breakup Breakup Breakup b =1/150 MeV -20 breakup p − d (EFT) -1 b =1/200 MeV δ 0 (degrees) − 30 − 30 − 30 − 30 − 30 δ 0 (degree) STM -40 − 45 − 45 − 45 − 45 − 45 Quartet − S Quartet − S Quartet − S Quartet − S Quartet − S − 60 − 60 − 60 − 60 − 60 -60 − 75 − 75 − 75 − 75 − 75 -80 0 50 100 p (MeV) − 90 − 90 − 90 − 90 − 90 0 0 0 0 0 20 20 20 20 20 40 40 40 40 40 60 60 60 60 60 80 80 80 80 80 100 100 100 100 100 120 120 120 120 120 140 140 140 140 140 p (MeV) Elhatisari, Lee, Meißner, Rupak (2016) Pine, Lee, Rupak ( 2013 )

n-d doublet channel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � - � - � ( � / � ) ��� � � � - � p cot δ = − 1 /a + rp 2 / 2 � � 1 + p 2 /p 2 - � 0 � a ∼ 0 . 65 fm , - � � � � � � � � r ∼ − 150 fm , � � � � � - � � � � � � � � � � � � � � � p 0 ∼ 13 MeV � � � � � � � � � � � � � � � � � � � � � � - � ��� ��� ��� ��� ��� ��� ��� � / � ERE form van Oers & Seagrave (1967) -what EFT for modified ERE Virtual state at 0.5 MeV Girard & Fuda (1979) - Efimov physics

Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state lattice QCD with B field, even with heavy pions?

Phillips-Girard-Fuda �� � �� � � ( ��� ) � �� Preliminary � � � � ��� ��� ��� ��� ��� ��� ��� � � ( ��� ) 3-body correlation

Adhikari-Torreao ��� Preliminary ��� � � ( ��� ) ��� � ��� Adhikari - Torreao � � ��� � Virtual � � � ��� � � � � � ��� � � - �� - �� - �� - �� � � � ( �� )

Field Redefinition 2 → r → → → → 2 M ) N + d † L = N † ( i D t ) d t + d † D t ) d s + t † ( i ∂ 0 + t ( � i s ( � i D Ω ) t † NP i N + h.c ) � g s ( d a † N ¯ � g t ( d i P a N + h.c ) t s � w t ( t † σ i Nd i t + h.c. ) � w s ( t † τ a Nd a s + h.c. ) Start here and write in terms of only nucleon fields 2 → r → → D t = ( � i 4 M + ∆ t ) , � i ∂ 0 � alternatives possible 2 → → r → D s = ( � i 4 M + ∆ s ) , � i ∂ 0 � 2 → r → → D Ω = ( i ∂ 0 + 6 M + ∆ Ω ) . i

Integrate out fields ∂ L ∂ t † = 0 → D Ω ) − 1 ( w t σ i Nd i t + w s τ a Nd a ⇒ t = ( i s ) → t † ( i D Ω ) t You see that it generates dimer-nucleon interaction ... more to come ∂ L † = 0 ∂ d i t j i → → ⇒ d j t ) − 1 [ g t NP i N + w t w s N † σ i τ a ( i D Ω ) − 1 Nd a t = ( − i s ] D i j → → → t N † σ i σ j ( i D Ω ) − 1 N D t δ ij − w 2 t = − i − i D Now things get interesting

Finally integrate the last dimer field and write 2 → 2 M ) N � g 2 i j → r → L = N † ( i 2 [( NP i N ) † ( � i t t ) − 1 NP j N + h.c. ] ∂ 0 + D � 1 k l a b → → → P a N ) † + g t w t w s ( NP k N ) † ( � i 2[ g s ( N ¯ t ) − 1 N † ( i D Ω ) − 1 σ l τ a N ]( � i ) − 1 D D → i j → ⇥ [ g s N ¯ P b N + g t w t w s N † ( i D Ω ) − 1 σ i τ b N ( � i t ) − 1 NP j N ] + h.c. , D a b → → → s N † τ a τ b ( i D Ω ) − 1 N D s δ a b − w 2 ( − i ) = [ − i D i j → → → s N † ( i D Ω ) − 1 σ i τ a N ( − i t ) − 1 N † σ j τ b ( i D Ω ) − 1 N ] − w 2 t w 2 D -- generates higher-body terms -- need to remove time-derivatives

Keep upto 3-body contact interactions 2 → D t ) − 1 δ i j + w 2 w 2 i j → → → r t ) − 1 = ( � i t N † σ i σ j ( i D Ω ) − 1 N + t 4 M ) N † σ i σ j N , ( � i ( i ∂ 0 + D ∆ 2 ∆ 3 t ∆ Ω t 2 → D s ) − 1 δ a b + w 2 w 2 a b → → a b → r → ) − 1 = ( � i s ) − 1 , s N † τ a τ b ( i D Ω ) − 1 N + s 4 M ) N † τ a τ b N ⌘ ( � i ( � i ( i ∂ 0 + D D ∆ 2 ∆ 3 s ∆ Ω s Almost home, and write 2 → 2 M ) N � g 2 i j → → r L = N † ( i 2 [( NP i N ) † ( � i t t ) − 1 NP j N + h.c. ] ∂ 0 + D � g 2 a b → 2 [( N ¯ s ) − 1 N ¯ s P a N ) † ( � i P b N + h.c. ] D → → → � g t g s w t w s [( N ¯ P a N ) † ( � i D s ) − 1 N † ( i D Ω ) − 1 σ i τ a N ( � i D t ) − 1 NP i N + h.c. ] ⌘ L 1 + L 2 + L 3 +...

Field Redefinition To remove time-derivative from two-body try i N † ( NP i N ) + b 1 ¯ a N † ( N ¯ N → N + a 1 P † P † P a N ) Bedaque, Grießhammer (2000) Bedaque, Rupak, Grießhammer, Hammer (2003) After a good amount of elbow grease L 2 = � g 2 ( NP i N ) † NP i N � g 2 ( N ¯ P a N ) † N ¯ t s P a N ∆ t ∆ s g 2 g 2 2 2 ↔ ↔ t [( NP i N ) † ( N s [( NP a N ) † ( N P i N ) + h.c. ] � P a N ) + h.c. ] � r r 8 M ∆ 2 8 M ∆ 2 t s using a 1 = g 2 t / ∆ 2 t b 1 = g 2 s / ∆ 2 s

Removing time-derivatives from three-body contact interaction requires, another field redefinition. However, the leading momentum independent term is simple  g 2  g 2 t w 2 + g 4 s w 2 + g 4 � � ( N ¯ P a N ) † N † τ a τ b N ( N ¯ t t ( NP i N ) † N † σ i σ j N ( NP j N ) − s s P b N ) − ∆ 2 6 ∆ 3 ∆ 2 6 ∆ 3 t ∆ Ω s ∆ Ω t s g 2 t g 2 g 2 t g 2  g t g s w t w s � [( N ¯ s s P a N ) † N † τ a σ i N ( NP i N ) + h.c. ] − − − 4 ∆ 2 4 ∆ t ∆ 2 ∆ t ∆ s ∆ Ω t ∆ s s Bedaque, Rupak, Grießhammer, Hammer (2003) Need to pull out the SU(4) symmetric piece

p-p fusion p 1 Z i † h | NN ( s ; ~ N ( ~ p ) P ( s ) N ( ~ p k, p ) i = d Ω ˆ k/ 2 + ~ k/ 2 � ~ p ) | 0 i (2 ⇡ ) 3 p 4 ⇡ Projector cm, relative momentum h NN ( s 0 ; ~ k 0 , p 0 ) | NN ( s ; ~ k, p ) i = � (3) ( ~ k � ~ k 0 ) � ( p � p 0 ) � ss 0 with projectors = 1 P ( s ) P ( s 0 ) † i h X Tr 2 δ ss 0 Ave . pol Chen, Rupak, Savage (1999)

k, p ) i = 1  † � Z d cos ✓ P ( s 0 ) P ( s 0 ) h NN ( s 0 ; ~ b O ab ; cd N c N d | NN ( s ; ~ k 0 , p 0 ) | N ⇤ a N ⇤ ab O ab ; cd ab 2 cd Chen, Rupak, Savage (1999) Fleming, Mehen, Stewart (2000) (+) d 3 q ( ~ q ) Z ~ p p p y ( − 2 µ ) 2 Z d hg A ( ~ ✏ ∗ ✏ ∗ ) x ˆ d × ~ ∼ q 2 + � 2 (2 ⇡ ) 3 project the appropriate p-wave channels r 32 π s-wave comparison | h d ; x | A − y | pp i | = g A C η γ 3 Λ ( p ) δ xy indices contracted

Thank you