E ective potential for relativistic...
Transcript of E ective potential for relativistic...
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Effective potential for relativistic scattering
Janos Balog
Wigner Research Centre, Budapest and CAS IMP, Lanzhou
in collaboration with
Pengming Zhang Mahmut Elbistan
CAS IMP, Lanzhou CAS IMP, Lanzhou
• Quantum Inverse Scattering
• Nucleon potential from lattice QCD
• Sine-Gordon model
• Sine-Gordon effective potential from QIS
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Quantum Inverse Scattering
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Inverse scattering problems
Inverse scattering: find potential from scattering data
All physics is inverse scattering:
Newton’s law
Rutherford’s experiment
Watson & Crick double helix
Direct scattering: given potential (forces) find scattering data
Inverse scattering: given scattering data find forces
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1-dimensional quantum mechanics on the half-line
Schrodinger operator: `u = −u′′ + qu
self-adjoint Hamiltonian: (~, m etc. scaled out)
`u = −u′′ + qu and boundary condition u(0) = 0
q(x) potential x ≥ 0
q(x) ∼ p(p−1)x2 x→ 0 p > 1
u(x) ∼ xp regular solution
u(x) ∼ x1−p singular solution
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1 2 3 4 5
5
10
15
20
Figure 1: The singular potential q(x)
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Three solutions of the `u = k2u Schrodinger equation:
φ(x, k) ∼ xp x→ 0 physical solution
φ(x, k) ∼ x1−p x→ 0 singular solution
f(x, k) ∼ eikx x→∞ Jost solution
f∗(x, k) = f(x,−k)
Jost function f(k):
f(x, k) = f(k)φ(x, k) + f(k)φ(x, k)
φ(x, k) = 2p−12ik {f(−k)f(x, k)− f(k)f(x,−k)}
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2 4 6 8 10
2
4
6
Figure 2: The singular potential q(x) and the regular wave function φ(x, k) at k = 1.5
The constant total energy is also shown.
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phase shift δ(k):
f(k) = |f(k)|e−iδ(k)
S-“matrix”:
S(k) = f(−k)f(k) = e2iδ(k)
asymptotics of the physical solution:
φ(x, k) ∼ −2p−12ik f(k)
{e−ikx − S(k)eikx
}
∼ sin [kx+ δ(k)]
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(Solvable) example
q(x) = p(p−1)
sinh2(x)
change of variables:
u(x) = eikxF (z) z = 11−e−2x
hypergeometric equation:
z(1− z)F ′′(z) + [c− (a+ b+ 1)z]F′(z)− abF (z) = 0
a = p b = 1− p c = 1 + ik
hypergeometric function: 2F1(a, b, c; z)
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physical solution:
φ(x, k) = 12p
(1− e−2x
)peikx2F1
(p, p− ik, 2p; 1− e−2x
)
Jost solution:
f(x, k) =(1− e−2x
)peikx2F1
(p, p− ik, 1− ik; e−2x
)
Jost function:
f(k) = 12p−1
Γ(1−ik)Γ(2p−1)Γ(p)Γ(p−ik)
S-matrix:
S(k) = Γ(1+ik)Γ(p−ik)Γ(1−ik)Γ(p+ik)
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high energy asymptotics:
δ(k) = π2(1− p)− d1
k + . . . δ(∞) = π2(1− p)
p = 2:
f(k) = 11−ik S(k) = 1−ik
1+ik
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Inverse scattering in three steps
• step 1: scattering data
F (x) = 12πix
∫ ∞
−∞dk eikx S′(k)
• step 2: Marchenko equation
F (x+ y) +A(x, y) +
∫ ∞
x
dsA(x, s)F (s+ y) = 0
• step 3: potential
q(x) = −2 ddxA(x, x)
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p = 2 example
• step 1: scattering data
F (x) = −2 e−x
• step 2: Marchenko equation
A(x, y) = e−y
sinh(x)
• step 3: potential
A(x, x) = coth(x)− 1 q(x) = −2 ddxA(x, x) = 2
sinh2(x)
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Nucleon potential from first principles
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Modern nucleon-nucleon potential
Origin of the N-N Repulsive Core
The Most Fundamental Problem in Nuclear Physics
!!!!!!
r
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(Taketani)
IIIIII
Long range part
one pion exchange potential(OPEP)
I
II Medium range part
σ, ρ, ω exchange2π exchange
III Short range part
repulsive core (RC)
quark ?Bonn: Machleidt, Phys.Rev. C63(Ô01)024001Reid93: Stoks et al., Phys. Rev. C49(Ô94)2950.AV18: Wiringa et al., Phys.Rev. C51(Ô95) 38.
R. Jastrow(1951)
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Our strategy in lattice QCD
Consider “elastic scattering”
NN → NN NN → NN + others (NN → NN + π,NN + NN, · · ·)
Elastic threshold
• S-matrix below inelastic threshold. Unitarity gives
Quantum Field Theoretical consideration
S = e2iδ
Full details: Aoki, Hatsuda & Ishii, PTP123(2010)89.
energy Wk = 2√k2 +m2
N < Wth = 2mN +mπ
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Step 1
define (Equal-time) Nambu-Bethe-Salpeter (NBS) Wave function
ϕk(r) = 〈0|N(x+ r, 0)N(x, 0)|NN,Wk〉QCD eigen-state
N(x) = εabcqa(x)qb(x)qc(x): local operator
Spin model: Balog et al., 1999/2001
“scheme”
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partial wave
r = |r| → ∞
scattering phase shift (phase of the S-matrix by unitarity) in QCD !
ϕlk → Al
sin(kr − lπ/2 + δl(k))
kr
Asymptotic behavior of NBS wave function Lin et al., 2001; CP-PACS, 2004/2005
cf. Luescher’s finite volume method
no interaction
interaction range
L
allowed k at L δl(kn)
NBS wave function scattering wave function in quantum mechanics
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Step 2
• Define a potential through application of a Schrodinger operator:
Vk(r) =
[Ek+
1µ∇
2]ϕk(r)
ϕk(r)
where Ek = k2
µ is the kinetic energy. Note the energy dependence of potential!
Step 3
• Solve the Schrodinger equation with this (zero energy) potential in infinite volume to find phase
shifts and possible bound states below the inelastic threshold.
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Qualitative features of NN potential reproduced!
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1st paper: Ishii-Aoki-Hatsuda, PRL 90(2007)0022001
has been selected as one of 21 papers in Nature Research Highlights 2007. (One from Physics, Two from Japan, the other is on “iPS” by Sinya Yamanaka et al. )
“The achievement is both a computational tour de force and a triumph for theory.”
RVFODIFE
(courtesy of Sinya Aoki)
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1S0
aexp0 (1S0) = 23.7 fm
a0(1S0) = 1.6(1.1) fm
(courtesy of Sinya Aoki)
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Problems with the NBS approach
• Dependence of potential on choice of nucleon operator
• Energy dependence of the NBS potential
Possible solutions:
define nonlocal, but energy-independent potential
define the zero-momentum potential
Uo(r) = limk→0
V NBSk (r)
correctly reproduces scattering lengths, but effective range different
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Sine-Gordon model
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Sine-Gordon model
Very well known: RFT (quantum/classical)
RM (quantum/classical) [alias RS model]
Integrable (solvable) =⇒ (almost) everything calculable
spectrum (solitons, anti-solitons, breathers)
S-matrix
Form-factors, correlators, free energy, ...
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Lagrangian (~ = c = 1 RQFT conventions)
L = 12φ
2 − 12φ′2 + µ2
β2 cos(βφ)
SG coupling β: (equivalence to Thirring model)
0 < β <√
8π β = 2√π FF point
equations of motion: (Kl)Sine-Gordon equation
2ϕ+ µ2 sinϕ = 0 (ϕ = βφ)
parameters:
p = 4πβ2 ν = 1
2p−1
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soliton mass:
m = 2p−1π µ
bound states (breathers):
mk = 2m sin(π2νk
)k = 1, 2, · · · < 2p− 1
Ruijsenaars-Schneider RQM description: zero-momentum potential
qo(x) = 4sinh2(πνx)
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Sine-Gordon S-matrix
Rapidity parametrization: θ = θ1 − θ2
pi = mc sinh(θi) Ei = mc2 cosh(θi) Ei =√
(mc2)2 + (pic)2
soliton-soliton S-matrix (no bound states):
Σ(θ) = exp
{i
∫ ∞
0
dωω sin
(2πθω
) sinh((ν−1)ω)cosh(ω) sinh(νω)
}
phase shift:
Σ(θ) = e2iδ(θ) δ(∞) = π2(1− p)
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for integer p explicit:
Σ(θ) =
p−1∏
m=1
sm−i sinh(θ)sm+i sinh(θ) sm = sin(mνπ)
2-particle scattering
• −→ p1 p2 ←− •x1 x2
initially: p1 > p2 x2 > x1 all times
asymptotic wave function (x2 − x1)→∞:
Φ(x1, x2) ≈ ei(k1x1+k2x2) + S(p1, p2)ei(k2x1+k1x2)
ki = pi~ i = 1, 2 wave numbers
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Relativistic:
SR(p1, p2) = −Σ(θ1 − θ2) θi = arcsinh(pimc
)
NR Schrodinger equation:
HΦ = EΦ H = − ~2
2m∂2
∂x21− ~2
2m∂2
∂x22
+ U(x2 − x1)
separating COM and relative motion:
Φ(x1, x2) = eiK(x1+x2) Ψ(x2 − x1)
effective 1-particle Schrodinger equation:
− ~2
m Ψ′′(x) + U(x)Ψ(x) = ~2
mκ2Ψ(x)
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total energy:
E = ~2
2m(k21 + k2
2) = ~2
m (K2 + κ2)
k1 = K + κ k2 = K − κasymptotic wave function:
Ψ(x) ≈ −A(κ)eiκx + e−iκx x→∞
NR S-matrix:
SNR(p1, p2) = −A(p1−p2
2~)
= −S(p1−p2mc
)
rescaling between physics ←→ maths conventions:
A(κ) = S(2κL) L = ~mc Compton length
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Identification?
SNR(p1, p2) ∼ SR(p1, p2)
S(p1−p2mc
)∼ Σ
(arcsinh
(p1mc
)− arcsinh
(p2mc
))
2 special cases of interest:
I (fixed target) p2 = 0 p1 = kmc
SI(k) = Σ (arcsinh(k))
II (centre of mass) p1 + p2 = 0 p1 − p2 = kmc
SII(k) = Σ (2 arcsinh(k/2))
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Effective Sine-Gordon potential
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effective Sine-Gordon potentials
Sine-Gordon NR S-matrix, I detemination:
SI(k) =
p−1∏
m=1
sm−iksm+ik sm = sin(mνπ)
Sine-Gordon NR S-matrix, II detemination:
SII(k) =
p−1∏
m=1
sm−ik√
1+k2/4
sm+ik√
1+k2/4sm = sin(mνπ)
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simplest case: I; p = 2
SI(k) = s1−iks1+ik = 1−ik/s1
1+ik/s1s1 = sin
(π3
)=√
32
rescaling of basic p = 2 case:
qI(x) = 32
1
sinh2(√
32 x
)
zero-momentum potential:
qo(x) = 4
sinh2(π3x)
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Inverse scattering case I, general p
step1: Fourier transformation
F (x) = −p−1∑
m=1
Rm e−smx
residues:
Rm = 2sm∏
n 6=m
sn+smsn−sm
step2: Marchenko equation
Ansatz:
A(x, y) =
p−1∑
m=1
Rm bm(x) e−sm(x+y)
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Marchenko equation reduced to system of algebraic equations:
bm = 1 +
p−1∑
n=1
znbnsm+sn
zm(x) = Rm e−2smx
step3: calculation of potential
A(x, x) =
p−1∑
m=1
bm(x)zm(x)
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p = 3 solution
A(x, x) = −s1 − s2 +(s2
1−s22) sinh(s1x) sinh(s2x)
D(x)
determinant:
D(x) = s2 cosh(s2x) sinh(s1x)− s1 cosh(s1x) sinh(s2x)
short distance:
A(x, x) ≈ 3x
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Results
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1 2 3 4 5 6
5
10
15
20
Figure 3: Comparison of the integrated effective potential A(x, x) (solid) and the corresponding
zero-momentum Ao(x, x) (dashed) for p = 3.
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1 2 3 4 5 6
10
20
30
40
Figure 4: Comparison of the integrated effective potential A(x, x) (solid) and the corresponding
zero-momentum Ao(x, x) (dashed) for p = 4.
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0.5 1 1.5 2
2
4
6
8
10
12
Figure 5: The integrated effective potential in the COM frame for p = 2 (dots). For comparison
the analytically obtained LAB frame integrated effective potential A(x, x) (solid) is also shown.
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0.5 1 1.5 2
10
20
30
40
Figure 6: The integrated effective potential in the COM frame for p = 3 (dots). For comparison
the analytically obtained LAB frame integrated effective potential A(x, x) (solid) is also shown.
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Figure 7: Comparison of integrated SG effective potentials for p = 3. The solid (red) line, the
(blue) dots and the dashed (black) line are the LAB frame, the COM frame and the zero-momentum
potential, respectively.
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Work in progress
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SO FAR
• SI(k) =⇒ qI(x) p = 2, 3, 4, 5
• SII(k) =⇒ qII(x) [numerically] p = 2, 3, 4
• compare qI(x), qII(x), qo(x)
TO DO
• qI(x) general p?
• QIS with incomplete data?
• method can be used in nuclear physics?
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Thank you!
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