NUCLEAR POTENTIALS IN QCD AND THEIR EXTENSIONS Sinya Aoki, Yukawa - - PowerPoint PPT Presentation
NUCLEAR POTENTIALS IN QCD AND THEIR EXTENSIONS Sinya Aoki, Yukawa Institute for Theoretical Physics, Kyoto University Miniworkshop on Lattice QCD Rm 716, CCMS/Physics Building, National Taiwan University, October 18-19, 2013 1. INTRODUCTION
Sinya Aoki, Yukawa Institute for Theoretical Physics, Kyoto University
Miniworkshop on Lattice QCD Rm 716, CCMS/Physics Building, National Taiwan University, October 18-19, 2013
POTENTIALS IN QUANTUM FIELD THEORIES?
matching between QFT and quantum mechanics ? scattering matrix element
QFT
+ · · · +
Born approximation
V 1 H0 − E V +
More precisely, Scattering in QFT Scattering via “potential” in QM “Inverse scattering problem” A sumilar example
mid-range attraction short-range repulsion virtual state
NN scatering phase shifts(experiment) potentials which reproduce phase shift in QM
calculate NN phase shift in QCD “Nuclear potential” which reproduces the QCD phase shift Questions Is this “nuclear potential calculated in QCD” ? ambiguity of potentials ? (previous figure) Are potentials observables ? Potential can be defined in classical mechanics
V (x) V (x)
input in QM same in CM (no “quantum corrections”) cannot define potential in QM alone ? nuclear potential : no correspondence in CM VNN(x)
?
“output” from QCD In this talk, I present our proposal for a definition of nuclear potentials in QCD and some results based on it. I also discuss extentions of our definition to more general cases.
Plan
Aoki, Hatsuda & Ishii, PTP123(2010)89.
conditions satisfied for nuclear potentials reproduce NN scattering phase shift via calculations in QM important for applications to many body problems in nuclear physics calculable in QCD (without using QM calculations) Assumption 1 consider elastic scattering only
NN → NN
NN → NN + others (NN → NN + π, NN + ¯ NN, · · ·)
Energy
Wk = 2
N < Wth = 2mN + mπ
inelastic threshold Assumption 2 QCD interactions are short-ranged.（no long-ranged force like Coulomb force.）
Basic quantity: (Equal time) Nambu-Bethe-Salpeter (NBS) wave functions
ϕk(r) = 0|N(x + r, 0)N(x, 0)|NN, Wk
QCD eigenstate
N(x) = εabcqa(x)qb(x)qc(x): local operator
QCD vacuum local quarks “a scheme” possible to use smeared quarks but unnecessary possible to use unequal time, but complicated and no new informations properties of NBS wave function. In particular, asymptotic behaviors in large |r|. It is very difficult or almost impossible to calculate NBS wave function “analytically”. However, it is possible to calculate it “numerically” in lattice QCD. Nucleon operator
Lippman-Schwinger equation in QFT
0β|T|α0 = 2πδ(Eα − Eβ)Tαβ.
(H0 + V )|αin = Eα|αin,
H0|α0 = Eα|α0.
in-state(full) free
energy conservation
H0 + V
QCD Hamiltonian in terms of quarks and gluons (with gauge fixing).
H0
Free Hamiltonian. No explicit form but “operation” is known.
|αin = |α0 +
|β0 Tβα Eα Eβ + iε
Tβα = 0β|V |αin
V
“potential” our target ? energy conservation T-matrix (physical observables)
NBS wave function For simplicity, consider scalars.
ZkΨk(r) = eik·r +
Zp eip·r T(p; k) Ek − Ep + iε
plain wave energy
00|ϕ1(x + r, 0)ϕ2(x, 0)|k0
center of mass frame
Ek = 2
partial wave
Ψk(r) = in0|ϕ1(x + r, 0)ϕ2(x, 0)|kin
spehrical Bessel spherical harmonics
eik·r = 4π
iljl(kr)Ylm(Ωr)Ylm(Ωk)
solid angle
Ψl(r, k) = jl(kr) + ∞ p2dp 2π Zk Zp jl(pr)Tl(p, k) Ek − Ep + iε
no pole on real axes(assumption)
Tl(p, k)
denominator contributes
r → ∞
ZkΨk(r) = 4π
ilΨl(r, k)Ylm(Ωr)Ylm(Ωk)
T(p; k) =
Tl(p, k)Ylm(Ωp)Ylm(Ωk)
Complicated for nucleons.
Unitarity of S-matrix + elastic condition Ek < Eth
Tl(k, k) = − 4 kEk eiδl(k) sin δl(k)
final result Ψl(r, k) eiδl(k) [jl(kr) cos δl(k) + nl(kr) sin δl(k)]
eiδl(k) kr sin(kr lπ/2 + δl(k))
NBS wave function =scattering wave in QM at large |r|. “phase shift”= phase of S-matrix due to its unitarity phase of S-matrix =phase shift (in QFT) potentials from this NBS wave function ?
Lin et al., 2001; CP-PACS, 2004/2005; Ishizuka, 2009 Aoki, Hatsuda & Ishii, 2010
∞ p2dp 2π Zk Zp jl(pr)Tl(p, k) Ek Ep + iε kEk 4 Tl(k, k) [nl(kr) + ijl(hr)]
spherical Neumann
no interaction interaction range
L
|r| → ∞
no interaction (under our assumption on short-ranged force)
eiδl(k) kr sin(kr lπ/2 + δl(k))
2 particle energy in finite box with length L.
E = 2
k is discrete due to PBC
k = 2π L n (n ∈ Z3)
free (no interaction) Due to interactions, k is different from free case Luescher’s finite volume method
k cot δ0(k) = 2 √πLZ00(1; q2)
Z00(s; q2) = 1 √ 4π
(n2 − q2)−s
q = kL 2π
k = |k|
integer for free case
E = 2
generalized zeta function
Our proposal: extract informations of interactions directly at short distance (?) informations of interactions
Our proposal
Step 1 define non-local but energy-independent potential as
[k − H0] k(x) =
k = k2 2µ
non-relativistic kinetic energy
µ = mN/2
reduced mass
H0 = −∇2 2µ
non-relativistic free Hamiltonian
“proof of existence” Using NBS wave functions below inelastic threshold, we can construct U(x, y) =
Wk,Wk≤Wth
[k − H0] k(x)−1
k,k† k(y)
ηk,k = (ϕk, ϕk)
inner product inverse of
∀Wp ≤ Wth
Indeed, it is easy to see
elastic elastic
[k − H0]k(x)−1
k,kk,p = [p − H0]p(x)
If we solve Schroedinger equation with this potential, we obtain NBS wave function as a solution, which gives correct the “phase shift” by construction. Finite size effect to the potential is small if the box is large enough. non-local potential which satisfies the Schroedinger equation is NOT unique. there exist many potentials different only for inelastic scatterings. No non-relativistic approximation is used. Equal NBS WF has no time. time dependence can be derived from Lorentz transformation, but no new information on scattering. Schroedinger equation is convenient for nuclear physics calculations. In practice, it is almost impossible to calculate non-local potential from the above construction. We will introduce some “approximation”.
Step 2 velocity(derivative) expansion
U(x, y) = V (x, r)δ3(x y)
V (x, ∇) = V0(r) + Vσ(r)(σ1 · σ2) + VT (r)S12 + VLS(r)L · S + O(∇2)
LO LO LO NLO NNLO tensor operator S12 = 3
r2 (σ1 · x)(σ2 · x) − (σ1 · σ2)
spin
To calcilate non-local potentil in terms of this expansion is a part of “our scheme”. Ex：Leading order
VLO(x) = [k − H0]k(x) k(x)
Step 3
equation in infinite volume with the potential obtained with this expansion. Physical observables have some errors due to the truncation of the expansion. It is possible to estimate these errors.
Comments “potentials” are NOT physical observables, and therefore depend on their definition (scheme). Phase shifts and binding energies do NOT depend on the scheme. analogy:running coupling in QCD. Scheme dependent. Cross section is an observable. choice of nucleon operator、definition of non-local potential, derivative expansion “good” scheme ? fast convergence of the derivative expansion（convenient）。 analogy: fast convergence of perturbative expansion (“good” running coupling). it is best if local potential is exact. energy-independent “local” potential？（”inverse scattering problem”） Potential is useful to understand “physics”, though it is not observable. analogy:asymptotic freedom. attractions at long and intermediate distances, repulsive core Our proposal: give a method to extract observables in QCD via potential.
Extraction of NBS wave function in lattice QCD NBS wave function Potential 4-pt Correlation function
F(r, t − t0) = 0|T{N(x + r, t)N(x, t)}J (t0)|0
source for NN
F(r, t − t0) = 0|T{N(x + r, t)N(x, t)}
|2N, Wn, s1, s22N, Wn, s1, s2|J (t0)|0 =
An,s1,s2ϕWn(r)e−Wn(t−t0), An,s1,s2 = 2N, Wn, s1, s2|J (0)|0.
complete set for NN
− → ∞ lim
(t−t0)→∞ F(r, t − t0) = A0ϕW0(r)e−W0(t−t0) + O(e−Wn=0(t−t0))
NBS wave function This is a standard method in lattice QCD and was employed for our first calculation. ground state saturation at large t
ϕk(r) = 0|N(x + r, 0)N(x, 0)|NN, Wk
[k − H0]k(x) =
+ · · ·
Improved method normalized 4-pt function
R(r, t) ≡ F(r, t)/(e−mNt)2 =
AnϕWn(r)e−∆Wnt
∆Wn = Wn − 2mN = k2
n
mN − (∆Wn)2 4mN
− ∂ ∂tR(r, t) =
1 4mN ∂2 ∂t2
potential Leading Order
10 20 30 40 0.5 1 1.5 2 2.5 VC(r) [MeV] r [fm] total 1st term 2nd term 3rd term
∂t + 1 4mN ∂2 ∂t2
1st 2nd 3rd total 3rd term(relativistic correction) is negligible. Ground state saturation is no more required. (advantage over finite volume method.)
Ishii et al. (HALQCD), PLB712(2012) 437
energy-independent
2+1 flavor QCD, spin-singlet potential
10 20 30 40 0.5 1 1.5 2 2.5 VC(r) [MeV] r [fm]
a=0.09fm, L=2.9fm
mπ ≃ 700 MeV
10 20 30 40 50 60 50 100 150 200 250 300 350 [deg] Elab [MeV] exp lattice
phase shift
1S0
a0(1S0) = 1.6(1.1) fm
aexp (1S0) = 23.7 fm
NN potential
Qualitative features of NN potential are reproduced. (1)attractions at medium and long distances (2)repulsion at short distance(repulsive core) It has a reasonable shape. The strength is weaker due to the heavier quark mass. Need calculations at physical quark mass( 140 MeV pion).
Convergence of velocity expansion: estimate１
mπ ≃ 0.53 GeV
a=0.137fm, L=4.0 fm
PTP 125 (2011)1225.
If the higher order terms are large, LO potentials determined from NBS wave functions at different energy become different.(cf. LOC of ChPT). Numerical check in quenched QCD Higher order terms turn out to be very small at low energy in our scheme.
Potential vs Luescher (I=2 pi-pi scattering. Quenched QCD)
[°] ECM[MeV]
V=(1.84 fm)3 V=(2.76 fm)3 V=(3.7 fm)3 V=(5.5 fm)3
50 100 150 200 250 300 350 400
mπ = 940 MeV
a = 0.115 fm
Kurth, Ishii, Doi, Aoki & Hatsuda, arXiv:1305.4462[hep-lat]
both methods agree very well. This establishes a validity of the potential method and shows a good convergence of the velocity expansion. Convergence of velocity expansion: estimate 2 phase shifts
Limitations of ”potential” method
NN → NN
W < Wth
Key Property１
In order to remove these limitations and extend the potential method to inelastic and/or multi-particle scatterings, we have to show Asymptotic behaviors of NBS wave functions for more than 2 particles
Key Property 2
Existence of energy independent potentials above inelastic thresholds
NBS wave functions for multi-particles
Sinya Aoki, et al., PRD88(2013)014036.
Key Property１ Unitarity
T † − T = iT †T.
0[pA]n|T|[pB]n0 ≡ δ(EA − EB)δ(3)(P A − P B)T([qA]n, [qB]n)
T-matrix
T([qA]n, [qB]n) ≡ T(QA, QB) =
T[L][K](QA, QB)Y[L](ΩQA)Y[K](ΩQB)
re QX = (qX
1, qX 2, · · · , qX n−1) f
momentum in D=3(n-1)dim. Jacobi momenta
ˆ L2Y[L](Ωs) = L(L + D − 2)Y[L](Ωs)
solution to the Unitarity constraint
T[L][K](Q, Q) =
U[L][N](Q)T[N](Q)U †
[N][K](Q),
T[L](Q) = − 2n3/2 mQ3n−5eiδ[L](Q) sin δ[L](Q),
“phase shift” δ[L](Q)
1 2 3
r1 r2
Jacobi coordiante
For simplicity, (1) we consider scalar particles with “flavors” (2) we assume no bound state exists. hyper-spherical harmonic function with non-relativistic approximation
As before, Lippmann-Schwinger equation gives NBS wave function
Ψn
α([x]) = in0|ϕn([x], 0)|αin,
ϕn([x], t) = T{
n
ϕi(xi, t)},
Ψn
α([x]) =
1 Zα
00|ϕn([x], 0)|α0 +
Zβ
00|ϕn([x], 0)|β0Tβα
Eα − Eβ + iε .
00|ϕn([x], 0)|[k]n0 =
1
n n
1
eikixi
plain wave
in terms of coordinates in D-dim.
Ψn(R, QA) = C
2m 2πn3/2
eiQ·R Q2
A − Q2 + iεT(Q, QA)
eiQ·R = (D − 2)!! 2πD/2 Γ(D/2)
iL jD
L (QR) Y[L](ΩR) Y[L](ΩQ),
Ψn(R, QA) =
Ψn
[L],[K](R, QA)Y[L](ΩR)Y[K](ΩQA),
n-scalar fields with different flavors
hyper-spherical Bessel function
Expansion in terms of hyper-spherical harmonic function
Ψn
[L],[K](R, QA) ≃ CiL (2π)D/2
(QAR)
D−1 2
U[L][N](QA)eiδ[N](QA)U †
[N][K](QA)
×
π sin
∆L = 2LD − 1 4 π.
R → ∞
At large R, perform Q-integral as before. Non-relativistic approximation is needed. Even for n-body, NBS wave function =scattering wave in QM, at large R. phase shift in n-body scattering = phase of S-matrix phase of S-matrix in QFT =phase shift (n-body generalization). a use of hyper-spherical harmonic function in D=3(n-1) dim is a key. However, non-relativistic approximation, unnecessary for 2-body case, is required for n-body cases. Asymptotic behavior of NBS wave functions scattering wave with “phase shift” !
Energy-independent potential above inelastic thresholds
Key Property２
Sinya Aoki, et al., Phys. Rev. D87(2013)34512
NN → NN, NNπ
W 0
th = 2mN
W 1
th = 2mN + mπ
W 2
th = 2mN + 2mπ
∆0 = [W 0
th, W 1 th)
∆1 = [W 1
th, W 2 th)
W ∈ ∆1
ZNϕ00
W,c0(x0) = 0|T{N(x, 0)N(x + x0, 0)}|NN, W, c0in,
ZNZ1/2
π ϕ10 W,c0(x0, x1) = 0|T{N(x, 0)N(x + x0, 0)π(x + x1, 0)}|NN, W, c0in,
ZNϕ01
W,c1(x0) = 0|T{N(x, 0)N(x + x0, 0)}|NN + π, W, c1in,
ZNZ1/2
π ϕ11 W,c1(x0, x1) = 0|T{N(x, 0)N(x + x0, 0)π(x + x1, 0)}|NN + π, W, c1in,
２ operator ２states
N(x)N(y) N(x)N(y)π(z)
|NN, W, c0 |NN + π, W, c1
energy
×
４NBS wave functions
ϕij
W,cj([x]i)
i(j): number of π’s in the operator(state)
e [x]0 = x0
nd [x]1 = x0, x1.
scattering Ex.
(Ek
W − Hk 0 )ϕki W,ci =
d3yn U kl([x]k, [y]l)ϕli
W,ci([y]l),
k, i ∈ (0, 1),
En
W =
p2
1
2mN + p2
2
2mN +
n
k2
i
2mπ , W
, W =
N + p2 1 +
N + p2 2 + n
π + k2 i ,
re p1+p2+n
i=1 ki = 0.
total energy We can show an existence of non-local but energy-independent potential matrix satisfying the above equation. non-local potential matrix
NN + nπ → NN + kπ
ΛΛ → ΛΛ, NΞ, ΣΣ
with non-relativistic approximation
momentum conservation
coupled channel equation kinetic energy non-relativistic
The construction of U can easily be generalized to
Baryon-Baryon potentials in the flavor SU(3) limit
Inoue et al. (HAL QCD Coll.), Progress of Theoretical Physics 124(2010)591
mu = md = ms
8 ⊗ 8 = 27 ⊕ 8s ⊕ 1 ⊕ 10 ⊕ 10 ⊕ 8a
200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 V(r) [MeV] r [fm]
V
(1)
(a)
0.0 0.5 1.0 1.5 2.0 L=4 [fm] L=3 [fm] L=2 [fm] Fit
flavor singlet potential
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Bound state energy E0 [MeV] Root-mean-square distance r2 [fm]
H-dibaryon
MPS = 1171 [MeV] MPS = 1015 [MeV] MPS = 837 [MeV] MPS = 672 [MeV] MPS = 469 [MeV]
H-dibaryon real world with SU(3) breaking ?
H-dibaryon with the flavor SU(3) breaking
size binding energy
mu = md = ms
SU(3) limit
ΛΛ − NΞ − ΣΣ
H
25-50 MeV real world 2386 MeV
ΣΣ
NΞ
ΛΛ
2257 MeV 2232 MeV 25 MeV 129 MeV
H ? H ?
u d s U d s
gauge ensembles
u,d quark masses lighter
π 701±1 570±2 411±2 K 789±1 713±2 635±2 mπ/mΚ 0.89 0.80 0.65 N 1585±5 1411±12 1215±12 Λ 1644±5 1504±10 1351± 8 Σ 1660±4 1531±11 1400±10 Ξ 1710±5 1610± 9 1503± 7 Esb 1 Esb 1 Esb 2 Esb 2 Esb 3 Esb 3
In unit
Esb 1 Esb 1 Esb 2 Esb 2 Esb 3 Esb 3 ΛΛ : 3288MeV ΝΞ : 3295MeV ΣΣ : 3320MeV 3008MeV 3021MeV 3062MeV
2702MeV 2718MeV 2800MeV
SU(3) breaking effects becomes larger
All channels have repulsive core
π Diagonal elements Off-diagonal elements
shallow attractive pocket Deeper attractive pocket Strongly repulsive
3x3 coupled channel potentials
Preliminary ! Bound H-dibaryon couplded to NΞ H as ΛΛ resonance H as bound NΞ H as ΛΛ resonance H as bound NΞ This suggests that H-dibaryon becomes resonance at physical point. Below or above NΞ ? Need simulation at physical point.
Esb3 : mπ= 411 MeV Esb3 : mπ= 411 MeV Esb1 : mπ= 701 MeV Esb1 : mπ= 701 MeV Esb2 : mπ= 570 MeV Esb2 : mπ= 570 MeV
ΛΛ
NΞ
Physically, it is essential that H-dibaryon is a bound state in the flavor SU(3) limit.
ΛΛ and NΞ phase shift
Linear setup
r r
1 2 3
Triton(I = 1/2, JP = 1/2+)
S-wave only (1,2) pair
1S0, 3S1, 3D1
1e-39 2e-39 3e-39 4e-39 5e-39 6e-39 7e-39 8e-39 9e-39 0.5 1 NBS wave function r [fm]
10 20 30 40 50 0.5 1 VTNF [MeV] r [fm] ), ϕM ≡
1 √ 2(+ψ1S0 + ψ3S1),
simulations. ψ3D1 wave
1 √ 2(−ψ1S0 + ψ3S1),
Doi et al. (HAL QCD), PTP 127 (2012) 723
Three nucleon force (TNF)
scalar/isoscalar TNF is observed at short distance. Analysis by OPE (operator product expansion) in QCD predicts universal short distance repulsions in TNF.
Aoki, Balog and Weisz, NJP14(2012)043046
give a definition of 2-body potential in QCD Elastic region: NBS wave function =scattering wave. Phase shift = phase of S-matrix. can easily apply the method to meson-baryon & meson-meson systems. Extension to multi-particles and inelastic scatterings, with non-relativistic approx. QCD as a generalized quantum mechanics QCD at given energy W QM with coupled channel non-local potential matrix among stable particles
N, ¯ N, π, · · ·
resonance
∆, ρ, · · ·
Nπ, ππ, · · ·
bound ? deuteron, H,...
NN, ΛΛ, · · ·
D, H, · · ·
stable
N, ¯ N, π, · · ·