Multi-meson systemsin lattice QCD

William Detmold[NPLQCD Collaboration]

University of Washington

Lattice 2008

Phys Rev D76:074507, 2007 Phys. Rev. Lett. 100:082004, 2008

Phys Rev D77:057502, 2008 arXiv:0803.2728

arXiv:0807.1856 + ...

• n<14 pion and kaon systems in QCD

• Two and three body interactions

• Investigate pion and kaon condensates

• Poor man’s nuclear physics

• Many body physics

• Exponentially bad signal/noise ameliorated

• Testing ground

• Meson condensates

• EOS in neutron stars (kaons [Kaplan&Nelson])

Why?

Scattering states• Maiani-Testa: extracting multi-hadron S-matrix

elements from Euclidean lattice calculations of corresponding Green functions is impossible

• Lüscher: volume dependence of two-particle energy levels ⇒ scattering phase-shift up to inelastic threshold

• Exact relation provided r«L

• Used for ππ, KK, NN, ΛN

p cot !(p) =1

"LS

!"pL

2"

#2$

S [x] =|!j|<!!

!j

1|!j|2 ! x

! 4"!

Multi-boson energies

• Large volume expansion of GS energy of n meson system to 1/L7

• 2 & 3 body interactions (N body: L-3(N-1))

• Derived in low energy EFT, but relativistic

• n=2: reproduces expansion of Lüscher

• Can include higher PW, higher body, excited states

[Beane, WD & Savage; WD & Savage ]

[See talk of T Luu for fermion systems]

Multi-boson energies• 1/L7: Relativistic effects

[WD+Savage arXiv:0801.0763]

!En =4! a

M L3nC2

!1!

" a

! L

#I +

" a

! L

#2 $I2 + (2n! 5)J

%

!! a

! L

"3 #I3 + (2n! 7)IJ +

$5n2 ! 41n + 63

%K

&

+! a

! L

"4 #I4 ! 6I2J + (4 + n! n2)J 2 + 4(27! 15n + n2)I K

+(14n3 ! 227n2 + 919n! 1043)L+ 16(n! 2) (T0 + nT1)!"

+nC31L6

!̂L3 + nC3

6"a3

M3L7(n + 3) I + O

!L!8

"

a = a +2!

L3a3 r !̂L

3 = !L3

!1! 6a

" LI"

+72"a4r

MLI I,J , ...: geometric

constants

Many mesons correlators• Eg: n π+ correlator (mu=md)

• n!2 Wick contractions: (13!)~1019

• Maximal isospin states: one quark propagator

Cn(t) =!0"""# $

x

d!5u(x, t)u!5d(0, 0)%n"""0

&

C3(t) = tr [!]3 ! 3 tr [!] tr!!2

"+ 2 tr

!!3

"

! =!

x !5S(x, t; 0)!5S†(x, t; 0) i,αj,β

! Ae!Ent

Lattice details• Calculations use MILC gauge configurations

• L=2.5 fm, a=0.12 fm, rooted staggered

• also L=3.5 fm and a=0.09 fm

• NPLQCD: domain-wall quark propagators

• mπ ~ 291, 318, 352, 358, 491, 591 MeV

• 24 propagators / lattice in best case

• Iz=n=1,...,12 pion and (S=n) kaon systems

n-meson energies• Effective energy plots: log[Cn(t)/Cn(t+1)]

5 10 15 200.2180.2200.2220.2240.2260.228

t!b

E 1!"

5 10 15 200.4400.4450.4500.4550.4600.465

t!b

E 2!"

5 10 15 200.660.670.680.690.700.710.72

t!b

E 3!"

5 10 15 200.880.900.920.940.960.981.00

t!b

E 4!"

5 10 15 201.10

1.15

1.20

1.25

t!b

E 5"

5 10 15 201.351.401.451.501.55

t!b

E 6!"

5 10 15 201.601.651.701.751.801.851.90

t!b

E 7!"

5 10 15 20

1.9

2.0

2.1

2.2

t!b

E 8!"

5 10 15 20

2.2

2.3

2.4

2.5

t!bE 9!"

5 10 15 202.42.52.62.72.82.9

t!b

E 10!"

5 10 15 202.72.82.93.03.13.23.3

t!b

E 10!"

5 10 15 20

3.03.23.43.63.8

t!b

E 12!"

mπ = 352 MeV

n-meson energies• Effective energy plots: log[Cn(t)/Cn(t+1)]

mπ = 352 MeV

5 10 15 200.2180.2200.2220.2240.2260.228

t!b

E 1!"

5 10 15 200.4400.4450.4500.4550.4600.465

t!b

E 2!"

5 10 15 200.660.670.680.690.700.710.72

t!b

E 3!"

5 10 15 200.880.900.920.940.960.981.00

t!b

E 4!"

5 10 15 201.10

1.15

1.20

1.25

t!bE 5"

5 10 15 201.351.401.451.501.55

t!b

E 6!"

5 10 15 201.601.651.701.751.801.851.90

t!b

E 7!"

5 10 15 20

1.9

2.0

2.1

2.2

t!b

E 8!"

5 10 15 20

2.2

2.3

2.4

2.5

t!b

E 9!"

5 10 15 202.42.52.62.72.82.9

t!b

E 10!"

5 10 15 202.72.82.93.03.13.23.3

t!b

E 10!"

5 10 15 20

3.03.23.43.63.8

t!bE 12!"

3.3 GeV

n-meson energies• Clean signals for n=1,...,12

!

! !!

!

!

!

!

! !

!

""

"" "

" " " " "

"

##

## # # #

#

##

#

$$

$$ $

$ $ $

$ $$

24681012!1.0!0.50.00.51.0

m" # 591 MeVm" # 491 MeVm" # 352 MeVm" # 291 MeV

2 4 6 8 10 120.000

0.005

0.010

0.015

0.020

0.025

0.030

n

$E n!E n

~ 8 GeV !

n correlations

• Fit effective energies to extract two parameters: a, η3 from 1/L expansion

• Use 12 eff. energies in n-t-correlated analysis

• Large correlation matrix: correlated χ2

• Reduces uncertainties as n pion correlators “explore more of the lattice”

n correlations!!

n ! 3

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"

m "f "4%#### 3L !!

n ! 5

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"m "f "4%#### 3L !!

n ! 7

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"

m "f "4%#### 3L

!!

n ! 9

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"

m "f "4%#### 3L

!!

n ! 11

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"

m "f "4%#### 3L

!!

n ! 12

0.14 0.15 0.16 0.17 0.18

0.2

0.4

0.6

0.8

1.0

m"a##""!I$2"m "f "4%#### 3L

n≤3 n≤5 n≤7

n≤9 n≤11 n≤12

Thr

ee-b

ody

Two body

2π+ and 2K- interaction

!

!

!

!

!

!

"

#

1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

m!! f!

m!a"" !!"I#2#

!

!

!!

!

!

"

#

3.0 3.2 3.4 3.6 3.8 4.00.2

0.3

0.4

0.5

0.6

0.7

0.8

mK ! fK

mKa!! K

K"I"1#

curves: Weinberg

[See talk of M Savage]

a=0.09

m!/f! mK/fK

3π+ and 3K- interaction

!

!

!

!

!"

1.0 1.5 2.0 2.5 3.0 3.5 4.0!0.5

0.0

0.5

1.0

1.5

m"! f"

m"f "4#$$$$ 3L

!

! !

!

!

"

3.0 3.2 3.4 3.6 3.8 4.0!1.0

!0.5

0.0

0.5

1.0

1.5

mK ! fK

mKf K4"#### 3$

K

L

m!/f! mK/fK

Naïve dimension analysis: 1

3π+ and 3K- interaction

!

!

!

!

!"

1.0 1.5 2.0 2.5 3.0 3.5 4.0!0.5

0.0

0.5

1.0

1.5

m"! f"

m"f "4#$$$$ 3L

!

! !

!

!

"

3.0 3.2 3.4 3.6 3.8 4.0!1.0

!0.5

0.0

0.5

1.0

1.5

mK ! fK

mKf K4"#### 3$

K

L

a=0.09

m!/f! mK/fK

Naïve dimension analysis: 1

• Chemical potential

• Analytic form: EOS

• Numerically using finite difference

Chemical Potentials

µ =d E

d n

!!!!V const

Isospin Chemical PotentialmK ! 580 MeVm" ! 291 MeV

2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

!2.5 fm"3#"$

%%%%%%%%%%%%&"$

m"$1

mK ! 597 MeVm" ! 352 MeV

L ! 2.5 fmL ! 3.5 fm

2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

!2.5 fm"3#"$

%%%%%%%%%%%%&"$

m"$1

mK ! 640 MeVm" ! 491 MeV

2 4 6 8 10 120.0

0.1

0.2

0.3

0.4

!2.5 fm"3#"$

%%%%%%%%%%%%&"$

m"$1

mK ! 678 MeVm" ! 591 MeV

2 4 6 8 10 120.0

0.1

0.2

0.3

0.4

!2.5 fm"3#"$

%%%%%%%%%%%%&"$

m"$1

2+3 body fit No 3 body LOχPT

Kaon Chemical PotentialmK ! 580 MeVm" ! 291 MeV

2 4 6 8 10 120.000.050.100.150.200.250.30

!2.5 fm"3#K$

%%%%%%%%%%%%%&K$

mK$1

mK ! 597 MeVm" ! 352 MeV

L ! 2.5 fmL ! 3.5 fm

2 4 6 8 10 120.000.050.100.150.200.250.30

!2.5 fm"3#K$

%%%%%%%%%%%%%&K$

mK$1

mK ! 640 MeVm" ! 491 MeV

2 4 6 8 10 120.000.050.100.150.200.250.30

!2.5 fm"3#K$

%%%%%%%%%%%%%&K$

mK$1

mK ! 678 MeVm" ! 591 MeV

2 4 6 8 10 120.000.050.100.150.200.250.30

!2.5 fm"3#K$

%%%%%%%%%%%%%&K$

mK$1

2+3 body fit No 3 body LOχPT

Summary

• Explored n<14 meson systems

• Clean signals for all n

• Two and three-body interactions

• π- and K- chemical potentials

• Important contribution from π π π

• Results consistent with LOχPT: Kaplan/Nelson analysis

Numerical precision• Double precision is not enough: not gauge

invariant

• Need arbitrary precision contraction code

• Propagator precision??

Double precision

64!digit precision

5 10 15 20

2.0

2.2

2.4

2.6

2.8

t!b

E 9

5 10 15 202.2

2.4

2.6

2.8

3.0

3.2

3.4

t!b

E 10

5 10 15 201.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

t!b

E 11

5 10 15 20

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

t!b

E 12

Numerical precision• Double precision is not enough: not gauge

invariant

• Need arbitrary precision contraction code

• Propagator precision??

Double precision

64!digit precision

5 10 15 20

2.0

2.2

2.4

2.6

2.8

t!b

E 9

5 10 15 202.2

2.4

2.6

2.8

3.0

3.2

3.4

t!b

E 10

5 10 15 201.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

t!b

E 11

5 10 15 20

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

t!b

E 12

Numerical precision• Double precision is not enough: not gauge

invariant

• Need arbitrary precision contraction code

• Propagator precision??

Double precision

64!digit precision

5 10 15 20

2.0

2.2

2.4

2.6

2.8

t!bE 9

5 10 15 202.2

2.4

2.6

2.8

3.0

3.2

3.4

t!b

E 10

5 10 15 201.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

t!b

E 11

5 10 15 20

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

t!b

E 12

!!

! !

!

!

!

!

!

!

!

0 20 40 60 80 100 120 140!200

!180

!160

!140

!120

!100

!80

!60

Digits of precision

Log 10! " t#C 13$t

%#&n=13 & matrix identites• Pauli principle: C13(t) correlator vanishes

• Matrix Identity: vanishes ∀ 12x12 matrices

• Numerically

C12(t) = T 121 ! 66T2T

101 + 440T3T

91 + 1485T 2

2 T 81 ! 2970T4T

81 ! 15840T2T3T

71 + 19008T5T

71 ! 13860T 3

2 T 61

+36960T 23 T 6

1 + 83160T2T4T61 ! 110880T6T

61 + 166320T 2

2 T3T51 ! 332640T3T4T

51 ! 399168T2T5T

51

+570240T7T51 + 51975T 4

2 T 41 ! 554400T2T

23 T 4

1 + 623700T 24 T 4

1 ! 623700T 22 T4T

41 + 1330560T3T5T

41

+1663200T2T6T41 ! 2494800T8T

41 + 492800T 3

3 T 31 ! 554400T 3

2 T3T31 + 3326400T2T3T4T

31

+1995840T 22 T5T

31 ! 3991680T4T5T

31 ! 4435200T3T6T

31 ! 5702400T2T7T

31 + 8870400T9T

31

!62370T 52 T 2

1 + 1663200T 22 T 2

3 T 21 ! 3742200T2T

24 T 2

1 + 4790016T 25 T 2

1 + 1247400T 32 T4T

21

!3326400T 23 T4T

21 ! 7983360T2T3T5T

21 ! 4989600T 2

2 T6T21 + 9979200T4T6T

21 + 11404800T3T7T

21

+14968800T2T8T21 ! 23950080T10T

21 ! 1478400T2T

33 T1 + 4989600T3T

24 T1 + 415800T 4

2 T3T1

!4989600T 22 T3T4T1 ! 1995840T 3

2 T5T1 + 5322240T 23 T5T1 + 11975040T2T4T5T1 + 13305600T2T3T6T1

!15966720T5T6T1 + 8553600T 22 T7T1 ! 17107200T4T7T1 ! 19958400T3T8T1 ! 26611200T2T9T1

+43545600T11T1 + 10395T 62 + 246400T 4

3 ! 1247400T 34 ! 554400T 3

2 T 23 + 1871100T 2

2 T 24

!4790016T2T25 + 6652800T 2

6 ! 311850T 42 T4 + 3326400T2T

23 T4 + 3991680T 2

2 T3T5 ! 7983360T3T4T5

+1663200T 32 T6 ! 4435200T 2

3 T6 ! 9979200T2T4T6 ! 11404800T2T3T7 + 13685760T5T7

!7484400T 22 T8 + 14968800T4T8 + 17740800T3T9 + 23950080T2T10 ! 39916800T12 (A12)

C13(t) = T 131 ! 78T2T

111 + 572T3T

101 + 2145T 2

2 T 91 ! 4290T4T

91 ! 25740T2T3T

81 + 30888T5T

81

!25740T 32 T 7

1 + 68640T 23 T 7

1 + 154440T2T4T71 ! 205920T6T

71 + 360360T 2

2 T3T61

!720720T3T4T61 ! 864864T2T5T

61 + 1235520T7T

61 + 135135T 4

2 T 51 ! 1441440T2T

23 T 5

1

+1621620T 24 T 5

1 ! 1621620T 22 T4T

51 + 3459456T3T5T

51 + 4324320T2T6T

51 ! 6486480T8T

51

+1601600T 33 T 4

1 ! 1801800T 32 T3T

41 + 10810800T2T3T4T

41 + 6486480T 2

2 T5T41 ! 12972960T4T5T

41

!14414400T3T6T41 ! 18532800T2T7T

41 + 28828800T9T

41 ! 270270T 5

2 T 31 + 7207200T 2

2 T 23 T 3

1

!16216200T2T24 T 3

1 + 20756736T 25 T 3

1 + 5405400T 32 T4T

31 ! 14414400T 2

3 T4T31 ! 34594560T2T3T5T

31

!21621600T 22 T6T

31 + 43243200T4T6T

31 + 49420800T3T7T

31 + 64864800T2T8T

31 ! 103783680T10T

31

!9609600T2T33 T 2

1 + 32432400T3T24 T 2

1 + 2702700T 42 T3T

21 ! 32432400T 2

2 T3T4T21

!12972960T 32 T5T

21 + 34594560T 2

3 T5T21 + 77837760T2T4T5T

21 + 86486400T2T3T6T

21

!103783680T5T6T21 + 55598400T 2

2 T7T21 ! 111196800T4T7T

21 ! 129729600T3T8T

21

!172972800T2T9T21 + 283046400T11T

21 + 135135T 6

2 T1 + 3203200T 43 T1 ! 16216200T 3

4 T1

!7207200T 32 T 2

3 T1 + 24324300T 22 T 2

4 T1 ! 62270208T2T25 T1 + 86486400T 2

6 T1

!4054050T 42 T4T1 + 43243200T2T

23 T4T1 + 51891840T 2

2 T3T5T1 ! 103783680T3T4T5T1

+21621600T 32 T6T1 ! 57657600T 2

3 T6T1 ! 129729600T2T4T6T1 ! 148262400T2T3T7T1

+177914880T5T7T1 ! 97297200T 22 T8T1 + 194594400T4T8T1 + 230630400T3T9T1

+311351040T2T10T1 ! 518918400T12T1 + 4804800T 22 T 3

3 ! 32432400T2T3T24

+41513472T3T25 ! 540540T 5

2 T3 ! 9609600T 33 T4 + 10810800T 3

2 T3T4

+3243240T 42 T5 ! 34594560T2T

23 T5 + 38918880T 2

4 T5 ! 38918880T 22 T4T5

!43243200T 22 T3T6 + 86486400T3T4T6 + 103783680T2T5T6 ! 18532800T 3

2 T7

+49420800T 23 T7 + 111196800T2T4T7 ! 148262400T6T7 + 129729600T2T3T8

!155675520T5T8 + 86486400T 22 T9 ! 172972800T4T9 ! 207567360T3T10

!283046400T2T11 + 479001600T13 (A13)

12

T ji = tr

!Xi

"j

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 291 MeV

2 4 6 8 10 12

0.14

0.16

0.18

0.20

0.22

0.24

n

m!a!!!I"2"

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 352 MeV

2 4 6 8 10 12

0.20

0.25

0.30

0.35

n

m!a!!!I"2"

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 491 MeV

2 4 6 8 10 120.30

0.35

0.40

0.45

0.50

n

m!a!!!I"2" ! ! ! ! ! ! !

! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #

! ! ! ! ! ! !! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 591 MeV

2 4 6 8 10 12

0.45

0.50

0.55

0.60

0.65

n

m!a!!!I"2"

Pion scattering

!!

!!

!!

!!

!!

!

" ""

""

""

""

" "

# # # # # # ## # # #

!!

!!

!!

!!

!!

!

" ""

""

""

""

" "

# # # # # # ## # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

2 4 6 8 10 12

0.20

0.25

0.30

0.35

n

m!a !!!I"2" LO

NLO

NNLO

Exact two-bodyN3LO

m=3,...,8

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 291 MeV

2 4 6 8 10 12

0.14

0.16

0.18

0.20

0.22

0.24

n

m!a!!!I"2"

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 352 MeV

2 4 6 8 10 12

0.20

0.25

0.30

0.35

n

m!a!!!I"2"

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 491 MeV

2 4 6 8 10 120.30

0.35

0.40

0.45

0.50

n

m!a!!!I"2" ! ! ! ! ! ! !

! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #

! ! ! ! ! ! !! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 591 MeV

2 4 6 8 10 12

0.45

0.50

0.55

0.60

0.65

n

m!a!!!I"2"

Pion scattering

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

!!

!!

! !!

!!

! !

""

" "" " " "

"" "

##

# # # # # ## # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 291 MeV

2 4 6 8 10 12

0.14

0.16

0.18

0.20

0.22

0.24

n

m!a!!!I"2"

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

!!

!!

!!

!!

!!

!

" " "" "

""

""

" "

# # # # # # ## # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 352 MeV

2 4 6 8 10 12

0.20

0.25

0.30

0.35

n

m!a!!!I"2"

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

! !! ! ! ! !

!!

!!

" "" " " " " " " "

"

# # # # # # # # # # #

$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 491 MeV

2 4 6 8 10 120.30

0.35

0.40

0.45

0.50

n

m!a!!!I"2" ! ! ! ! ! ! !

! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #

! ! ! ! ! ! !! ! ! !

" " " " " " " " " " "

# # # # # # # # # # #$

$$$$$$$$

$$

$$$$$$$

$$$

$$$$$$

$$$$

$$$$$

$$$$$

$$$$

$$$$$$

$$$

$$$$$$$

$$

$$$$$$$$

$

$$$$$$$$$

m! " 591 MeV

2 4 6 8 10 12

0.45

0.50

0.55

0.60

0.65

n

m!a!!!I"2"

Pion scattering

Expansion shows no sign of breakdown

Pion scattering• Extractions of mπa from four orders in L

n!2

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!3

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!4

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!5

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!6

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$NNL

O% n!7

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!8

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!9

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!10

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!11

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!12

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!2

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!3

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!4

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

n!5

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!6

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$NLO

% n!7

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

n!8

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!9

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!10

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%n!11

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!12

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

n!2

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!3

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!4

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!5

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!6

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$LO%

n!7

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!8

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!9

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!10

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!11

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

n!12

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$LO%

Lüscher exact

two-body

Pion scattering• Extractions of mπa from four orders in L

n!2

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!3

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!4

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!5

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!6

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$NNL

O% n!7

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!8

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!9

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!10

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!11

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!12

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!2

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!3

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!4

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

n!5

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!6

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$NLO

% n!7

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

n!8

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!9

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!10

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%n!11

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

% n!12

5 10 15 20

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NLO

%

Lüscher exact

two-body

Pion scattering• Extractions of mπa from four orders in L

n!2

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!3

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!4

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!5

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!6

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!bm"a"""I!2#$NNL

O% n!7

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!8

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!9

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!10

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

n!11

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O% n!12

5 10 15 200.10

0.15

0.20

0.25

0.30

0.35

0.40

t!b

m"a"""I!2#$NNL

O%

Lüscher exact

two-body

N3LO

!3,4" !3,5" !3,6" !3,7" !3,8" !3,9" !3,10" !3,11" !3,12"

!4,5" !4,6" !4,7" !4,8" !4,9" !4,10" !4,11" !4,12"

!5,6" !5,7" !5,8" !5,9" !5,10" !5,11" !5,12"

!6,7" !6,8" !6,9" !6,10" !6,11" !6,12"

!7,8" !7,9" !7,10" !7,11" !7,12"

!n,m" ! !8,5"

5 10 15 20

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

t#b

m"a ""$I!2%

!8,9" !8,10" !8,11" !8,12"

!9,10" !9,11" !9,12"

!10,11" !10,12"

!11,12"

• Two energies to cancel 3-body: 45 combinations

Lüscher exact

two-body

{n,m}={5,8}

Three meson interactions

• At 1/L6, point-like three-boson interaction must occur [Braaten, Nieto ‘95]

• RGI 3BI: physically meaningful

• Depends logarithmically on L

• Naive dimensional-analysis

• Combinations of energy shifts isolatesthe RGI interaction

!(L)3

m!f4! !(L)

3 ! 1

π+π+π+ interaction

mπ = 352 MeV

n ! 3

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 4

5 10 15 20"2

"1

0

1

2

t!bm#f #4$%% 3"L!2

.5fm#

n ! 5

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 6

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 7

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 8

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 9

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 10

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 11

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n ! 12

5 10 15 20"2

"1

0

1

2

t!b

m#f #4$%% 3"L!2

.5fm#

n=7