Hadrons and Nuclei : Scattering

Lattice Summer SchoolLattice Summer School

Martin Savage

Summer 2007

University of Washington

Why Scattering with Lattice QCD ?

Reproducing what is known is important check of lattice QCD, but intrinsically not interesting----not new physics e.g. NN scattering at physical pion mass

Calculating quantities that cannot be determined (well) any other way is the underlying motivation e.g. YN scattering , nnn , weak-YN,

d¾d (mq)

Present Nuclear Theory fails to Reproduce (a few!) Precise Expts

Courtesy of W. Tornow,data from TUNL,calculations by

Present Nuclear Theory fails to Reproduce Precise expts (2)

Worlds YN DataPolinder et al

p → KProduction)

p → p Scattering)

n → KProduction)

p → p Scattering)

Hyperon-Nucleon Scattering Experiments

Kozi Nakai (KEK)(talk given at Hypernuclear 2006 (Mainz))

Beam

SCITIC0

Production p Scattering

Production p Scattering

Hyperon-Nucleon Scattering Experiments Kozi Nakai (KEK)

susuu

Weak decay

Strong scatter

Neutron Stars (1)Why are we interested in scattering of

strange baryons (and mesons) ?

Supernova Remnant ? neutron stars or black holes ,or black holes ,

…….. kaon condensation , strange baryons ? ….... kaon condensation , strange baryons ? …..

Neutron Stars (2)

AtmosphereEnvelopeCrustOuter CoreInner Core

HomogeneousMatter

LasagnaNuclei +

Neutron superfluid

n-superfluid and p superconductor

Reddy and Pageastro-ph/0608360

Neutron Stars (3) : Hyperons in Neutron Matter

n

n

YN interactions shift the mass of Y in a neutron background

H » ®§ y§ nyn + M (0)§ § y§ + ::::

!³M (0)

§ + ®½n

´§ y§ + :::: Mean¡ Field

Neutron Stars (4)

Vacuum - Masses

With Interactions -- meson exchange model

WithOUT Interactions

We Need QCD Calculations to Improve upon Model Calcs

Reddy and Pageastro-ph/0608360

Low-Energy Scattering, Phase-Shifts and Scattering

Parameters (1)

Ã+ = ei kz + f (µ)ei kr

r

¾ =

Zd

d¾d =

Zd jf (µ)j2

f (µ) =1k

X

l

(2l +1) ei±l sin±l Pl(cosµ)

!1k

ei±0 sin±0 + ::: =1

kcot±0 ¡ ik

Analytic function of com energy

Review :

Low-Energy Scattering, Phase-Shifts and Scattering

Parameters (2)

kcot± = ¡ 1a

+ 12rk2 + :::

a = Scattering Length r = Effective Range

Can take any value Size dictated by range of interaction

Low-Energy Scattering, Phase-Shifts and Scattering

Parameters (3)

k = 0k ==0

a

±Ã = 1¡ r

a

Maiani-Testa no-go Theorem (1)

(s) ?

S-matrix elements cannot be extracted from infinite-volume Euclidean-space correlation functions except at kinematic thresholds.

Maiani-Testa no-go Theorem (2)j ppiout = Sy j ppi in

outhpp j ppi in = inhpp jSj ppi in

= ei2± (below inelastic thresholds)

Consider the Euclidean-space correlation function associated with a source J(x) that couples to two protons

GE (t1; t2;q) = h0j©q(t1)©¡ q(t2)J (0)j0i

J(0)

(s-wave to s-wave)

©q(t)

©¡ q(t)Interpolating fields

for proton

Maiani-Testa no-go Theorem (3)

This dominates at long times unless 2 Eq is equal to the minimum value of En

Away from Kinematic threshold…..cannot isolate S-matrix elements from Euclidean space correlators

h0jÁ(0;0)jpi =p

Z

©q =R

d3x e¡ iq¢x Á(x;t)

GE (t1;t2;q) = h0j©q(t1)©¡ q(t2)J (0)j0i

=Z

(2Eq)2e¡ E q (t1+t2 ) [ R (J ) + 2Eq Pq(J ;t2) ]

R (J ) =12

( outhppjJ (0)j0i + inhppjJ (0)j0i )

Pq(J ;t2) = PX

n

hpqj©¡ q(0)jniconnout outhnjJ (0)j0ie¡ (E n ¡ 2E q )t2

Maiani-Testa no-go Theorem (4)

R(J ) =12

( outhppjJ (0)j0i + inhppjJ (0)j0i )

= jouthppjJ (0)j0i j cos±

GM (t1; t2; q) =Z

(2Eq)2e¡ iE q (t1+t2 )

outhppjJ (0)j0i

=Z

(2Eq)2e¡ iE q (t1+t2 ) jouthppjJ (0)j0i jei±

In Minkowski space 2EqPq(J ;t2) !

12

( outhppjJ (0)j0i ¡ inhppjJ (0)j0i )

Luscher (1)

Compute something else !!!! Work in finite-volume and look at energy-levels

Non-Relativistic QM analysis = Lee + YangE (j )

0 = h0(0)j V j0(j ¡ 1) i ;

J-th order g.s. energy-shift

Unperturbed g.s. wavefunction

J-th order contribution to g.s. wavefunction

Lee + Yang (2)

k =2¼L

n

n = (nx;ny;nz)

hrjki =1

L3=2ei k¢r

hr1;r2jk;¡ ki =1L3

eik¢(r1¡ r2)

Periodic B.C. give r ! r + mL

Lee + Yang (3) : Energies

V = ´ ±3(r1 ¡ r2)

¢ E0 = E (1)0 + E (2)

0 + :::

=´L3

2

4 1 ¡´M4¼2L

X

n6=0

1jnj2

+ :::

3

5

Lee + Yang (4) : Threshold Scattering

V = ´ ±3(r1 ¡ r2)

f = ¡¹2¼

Zd3r V(r) ¡

¹ 2

¼

Zd3r1

Zd3r2 V(r1) G+(r1;r2)V(r2) + :::

= ¡¹ ´2¼

+¹ 2´2

¼

Zd3p

(2¼)3

1jpj2 + i²

+ :::: = a

´ = ¡4¼aM

·1 ¡ 4¼a

Zd3p

(2¼)3

1jpj2 + i²

+ :::¸

Lee + Yang (4) : Combining

V = ´ ±3(r1 ¡ r2)

¢ E0 = ¡4¼aM L3

2

4 1 +³ a

¼L

´0

@¤ jX

n6=0

1jnj2

¡ 4¼¤ j

1

A + :::

3

5

Luscher (5) : True in QFT !!Below Inelastic thresholds and lattices L >> R

UV regulator

Measure on lattice

S(x) = lim¤ j ! 1

¤ jX

j

1jj j2 ¡ x2

¡ 4¼¤j

pcot ±(E ) =1

¼LS

µpL2¼

¶

Luscher (6) : Methodology?

(x,t)

(y,t)

source

G(0;0)2 (t)

hG(0)

1 (t)i 2 ! e¡ (E 2¡ 2M )t

G(p1=0;p2=0)2 (t) =

Zd3x

Zd3y G2(y;t;x;t;0;0) ! A2 e¡ E 2t

G(p=0)1 (t) =

Zd3x G1(x;t;0;0) ! A1 e¡ M t

Luscher (7) : Methodology?

G(0;0)2 (t)

hG(0)

1 (t)i 2 ! e¡ (E 2¡ 2M )t

T = E2 ¡ 2M = 2p

q2 + M 2 ¡ 2M

pcot ±(T) =1

¼LS

µqL2¼

¶

Many-Bosons (1)

3-boson interaction

E0(n;L) =

4¼aM L3

nC2

(

1¡³ a

¼L

´I +

³ a¼L

´2 £I 2 +(2n ¡ 5)J

¤

+³ a

¼L

´3 h¡ I 3 ¡ (2n ¡ 7)I J ¡

¡5n2 ¡ 41n +63

¢K ¡ 8(n ¡ 2)(2Q +R)

i)

+nC364¼a4

M L6(3

p3¡ 4¼) log(¹ L) +n C2

8¼2a3

M L6r +n C3

´3(¹ )L6

+O¡L ¡ 7

¢:

numbers

3-Bosons (1)

E0(3;L) =12¼aM L3

(

1¡³ a

¼L

´I +

³ a¼L

´2 £I 2 + J

¤

+³ a

¼L

´3 h¡ I 3 +I J + 15K ¡ 8(2Q +R)

i)

+64¼a4

M L6(3

p3¡ 4¼) log(¹ L) +

24¼2a3

M L6r

+1L6

´3(¹ ) + O¡L ¡ 7

¢

I=2 Simplest hadronic scattering process

Physics Wise Computationally

C(t) =G(0;0)

2 (t)hG(0)

1 (t)i 2 !

A2

A21

e¡ (E 2¡ 2M )t

log·

C(t)C(t + 1)

¸

! E2(t) ¡ 2M = T(t) = 2p

q2(t) + M 2 ¡ 2M

pcot±(T(t)) =1

¼LS

µq(t)L2¼

¶

I=2

[ m¼ a¼+ ¼+ ](t) m¼ » 350 MeV

e.g.

Beane et al, arXiv:0706.3026

I=2 Beane et al, arXiv:0706.3026

m¼ aI =2¼¼ = ¡

m2¼

8¼f 2¼

(

1+m2

¼

16¼2f 2¼

"

3logµ

m2¼

¹ 2

¶¡ 1 ¡ lI =2

¼¼ (¹ )

#)

(2007)

I=2 Beane et al, arXiv:0706.3026

à m¼; f ¼

Lattice m/f is the way to go

m¼aI =2¼¼ = ¡

m2uu

8¼f 2

(

1+m2

uu

(4¼f )2

"

4lnµ

m2uu

¹ 2

¶

+ 4~m2

j u

m2uu

ln

Ã~m2

j u

¹ 2

!

¡ 1+ 0¼¼(¹ )

#

¡m2

uu

(4¼f )2

"~¢ 4

j u

6m4uu

+~¢ 2

j u

m2uu

·ln

µm2

uu

¹ 2

¶+1

¸ #

+~¢ 2

j u

(4¼f )20P Q (¹ ) +

a2

(4¼f )20a2 (¹ )

)

:

Two-flavor mixed-action Chen, O’Connell, Walker-Loud

m¼aI =2¼¼ = ¡

m2¼

8¼f 2¼

(

1 +m2

¼

(4¼f ¼)2

·3ln

µm2

¼

¹ 2

¶¡ 1 ¡ lI =2

¼¼ (¹ ) ¡~¢ 4

j u

6m4¼

¸)

I=0 ?

Computationally expensiveNeed to compute N ~ Volume propagators

I = 2

I = 0

u

u

d

d

u

u

I=0

Point to allpropagators

all to allpropagators

CP-PACS : Phase-shift (extrapolation)

CP-PACS

123 X 24163 X 32 243 X 48

L = 2.5 fm

nf = 2

(2002)

– Scattering phase-shift

CP-PACS

Kand KK Scattering

Lattice QCD + Chiral Symmetry

b = 0.125 fm

(S. Beane, P. Bedaque, T. Luu, K. Orginos, E. Pallante, A.Parreno, mjs)

K+ + K+ K+

Baryon Potentials from LQCD

Why bother with the scattering amplitude…just calculate the potential,and use that in the Schrodinger equation !!!

h0jO1(x;t0)i®O1(y;t0)

j¯ jÃ0i = Z(S;I )

N N (jrj) h0jN (x;t0)i®N (y;t0)

j¯ jÃ0i + :::

O1(x;t)i® = ²abc qi ;c

®

¡qa;T C°5¿2qb

¢(x;t)

GN N (x;y;t) = h0jO1(x;t)i®O1(y;t)j

¯ J (0)j0i

=X

n

h0jO1(x;0)i®O1(y;0)j

¯ jÃn ihÃn jJ (0)j0ie¡ E n t

2En

Baryon Potentials from LQCD (2)

• Potential between two infinitely massive mesons is well-defined• e.g. B-mesons in the HQ limit – E = V(R)• r is a constant of the motion

UE (r) = E +12¹

r 2GN N

GN N

12¹

r 2GN N + UE (r) GN N = E GN N

trivially

1 ] Energy-dependent potential…i.e. each different energy requires a different potential….not as useful as it first sounds !!!

2 ] NOT unique … only constrained to reproduce ONE quantity …. ( E0 )

Q Q Q QDirect

Exchange

BB t-channel Potentials ... Insight into NN ?

B B

B B

¼ ¼¼

B-meson has I = 12 ; sl = 1

2

BB (2)(W. Detmold, K.Orginos, mjs , 2007)

Quenched : a = 0.1 fm

Periodic Boundary Conditions and Images

BB (3) (W. Detmold, K.Orginos, mjs )

Tensor Force between BB ?

Deuteron

Quenched Potential : Hairpins

No strong anomaly´0 is also a pseudo-Goldstone boson

G´´ (q2) =i

(q2 ¡ m2¼+ i²)

+i(M 2

0 ¡ ®©q2)(q2 ¡ m2

¼+ i²)2

V (Q)(r) =1

8¼f 2¾1 ¢r ¾2 ¢r

µg2

A¿1 ¢¿2

r+ g2

01¡ ®©

r¡ g2

0M 2

0 ¡ ®©m2¼

2m¼

¶e¡ m¼r

Dominates at long-distances

Nucleons on the Lattice

~

¾x =

phG2i ¡ hGi2

hGi ! e(M N ¡ 32 m¼)t

3 ¼

Mesons are easy, Nucleons are hardand two nucleons are even harder !!!

G.P. Lepage, Tasi 1989

Large Scattering Lengths are OK !

Require : L >> r0

but ANY a

M = 350 MeV 2.5 fm lattices... YESTERDAY !!

M = 350 MeV 2.5 fm lattices... TODAY !!

NN on the Lattice

~E L2

~E L2

Deuteron1st continuum

1st continuum2nd continuum

(S. Beane, P. Bedaque, A.Parreno, mjs)

Effective Field Theory Calculation at Finite-Volume

NN Correlators

1S0

3S1 - 3D1

G

Fully-Dynamical QCDDomain-Wall Valence on rooted-Staggered Sea

Cn Exp[-En t]nC0 Exp[-E0 t]

NN Scattering (S. Beane, P. Bedaque, K. Orginos, mjs ; PRL97, 012001 (2006))

1S0 : pp , pn , nn 3S1-3D1 : pn : deuteron

a ~ 1/m

Scale-Invariance

NN is Fine-Tuneda1S0

np = ¡ 23:710§ 0:030fm ; r1S0np = +2:73§ 0:03

a3S1np = +5:432§ 0:005fm ; r3S1

np = +1:73§ 0:02

a >> r

Toy-Model r = 0 a = +1

~E L2

Hyperon-N Interactions(S.Beane, P.Bedaque, T.Luu, K.Orginos, E.Pallante, A.Parreno, mjs , 2006)

|k| = 261 MeV |k| = 179 MeV

|k| = 255 MeV |k| = 169 MeV

Interactions --- I=0, J=0, s=2

-ve Energy Shift ?

M = 590 MeV

Luscher Relation -- revisited

pcot ±(E ) =1

¼LS

µpL2¼

¶

V = 0 ! a = r = :::: = 0

S(´) = 1

Non-interacting particlesk =

2¼L

n

n = (nx;ny;nz)

T < 0 T > 0 Bound-state or Scattering state ?

Bound states are also described !!

° + pcot ±jp2=¡ ° 2 = 0

E ¡ 1 = ¡°2

M

·1 +

12°L

11¡ 2°(pcot ±)0

e¡ ° L + :::¸

pcot±(E ) =1

¼LS

µpL2¼

¶(S. Beane, P. Bedaque, A.Parreno, mjs)

Bound States vs Scattering States (1)

L = 200 b 1/r = M = 350 MeVa = 10, 5, 1, -1, -5, -10 fm

( )P L 2

2

p cot

´ =

b = 0.125 fm

Bound States

Bound States vs Scattering States (3)

L = 20 b 1/r = M = 350 MeVa = 10, 5, 1, -1, -5, -10 fm

( )P L 2

2

p cot

´ << ´0 ! most probably a bound state

´0´ =

NN Resource Requirementswith Current AlgorithmsNN Scattering Length fixed at 2 fm

for demonstrative purposes

Domain-Wall Propagator Generation ONLY !!

Does not include time for lattice generation

Contractions Usually

Lattice generation > propagator generation > contractions

Not so for nuclear physics

Need high statistics .. Many propagators per lattice

Large number of quarks in initial and final states

contractions » u! d! s!

= (A +Z)! (2A ¡ Z)! S!

P roton : N cont: = 223592 U : N cont: = 101494

nnn

Important for neutron rich nuclei Lattice calc. is not as easy as

Cannot have all at rest as they are fermions

Closing Remarks on Scattering

Two hadron scattering can be studied with lattice QCD by studying the energy eigenstates at finite-volume

Simplest system well-understood

Baryon-baryon systems still very primitive ..

a lot of room for improvement and contribution

The END