s-matrix solution of the lippmann-schwinger equation for regular … · 2016. 9. 18. · s-matrix...
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S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
S-matrix solution of the Lippmann-Schwinger
equation for regular and singular potentials
J. A. Oller
Departamento de FısicaUniversidad de Murcia 1
in collaboration with D. R. Entem
[I] arXiv:1609, next month
[II] Long version, in preparation
TIDS16, Granada, August 23rd, 2016
1Partially funded by MINECO (Spain) and EU, project FPA2013-40483-P
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Overview
1 Lippmann-Schwinger equation
2 New exact equation in NR scattering theory
3 N/D method with non-perturbative ∆(A)
4 Regular interactions
5 Singular Interactions
6 T (A) in the complex plane
7 T (A) in the complex plane
8 Conclusions
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Lippmann-Schwinger equation (LS)
Scattering T -matrix T (z) , Im(z) 6= 0
T (z) =V − VR0(z)T (z)
R0(z) = [H0 − z ]−1
H0 =− 1
2µ∇2
H =H0 + V
• Resolvent of H, R(z):
R(z) =[H − z ]−1
R(z) =R0(z)− R0(z)T (z)R0(z)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
• Spectrum of H: H|ψp〉 = Ep|ψp〉Continuous spectrum: Povzner’s result
|ψp〉 =|p〉 − limǫ→0+
R0(Ep + iǫ)T (Ep + iǫ)|p〉
Bound States: Poles in T (p,p′; z) for z ∈ R−
• LS in momentum spaceFor definiteness we consider uncoupled spinless case by now :
T (p′,p, z) = V (p′,p)−∫
d3q
(2π)3V (p′,q)
1q2
2µ − zT (q,p, z)
POTENTIAL
+ . . .
p
p′
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
• LS in partial waves
Tℓ(p′, p, z) =
1
2
∫ +1
−1dcos θ Pℓ(cos θ)T (p′,p, z)
cos θ =p′ · p
T (p′,p, z) =∞∑
ℓ=0
(2ℓ+ 1)Pℓ(cos θ)Tℓ(p′, p, z)
Tℓ(p′, p, z) =Vℓ(p
′, p) +µ
π2
∫ ∞
0dqq2
Vℓ(p′, q)Tℓ(q, p, z)
q2 − 2µz
Convention: V (p′,p) → −V (p′,p) , T (p′,p, z) → −T (p′,p, z)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
• On-shell unitarity (extensively used later)Propagation of real two-body states
p′ =p
Ep =p2
2µ
ImTℓ(p) =µp
2π|Tℓ(p)|2 , p > 0
Im1
Tℓ(p)=− µp
2π
Unitarity cut for p2 > 0
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Criterion for Singular Potentials
Vℓ(r) −−−→r→0
αr−γ
α =α+ ℓ(ℓ+ 1)
Potential Ordinary Singular
γ < 2 > 2
γ = 2 1 + 4α > 1 1 + 4α ≤ 1
Ordinary/Regular Potentials:
Standard quantum mechanical treatmentBoundary condition: u(0) = 0 and behavior at ∞No extra free parameters
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
The One-Pion-Exchange (OPE) potential for the singlet NNinteraction (r > 0):
Yukawa potential V (r) =− τ1 · τ2(
gAmπ
2fπ
)2e−mπr
4πr
Exchange of a pion between two nucleons
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
In many instances one has singular potentials
Multipole expansion
Φ(x) =
∫
ρ(x′)
|x− x′|d3x ′ = 4π
∑
ℓ,m
qℓm
2ℓ+ 1
Yℓm(θ, φ)
r ℓ+1
qℓm =
∫
Y ∗ℓm(θ
′, φ′)r ′ℓρ(x′)d3x ′
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Van der Waals Force (molecular physics)
V (r) = −3
2
αAαB
r6IAIB
IA + IB
In QFT/EFT we treatcomposite objects aspoint like
Physical meaning for r → 0?:
De Broglie length1p>> rA ∼ 2A
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Nuclear physicsOPE is singular attractive for the Deuteron in NN scattering
2S+1LJ :3P0 NN partial wave
V (r) =m2
π
12π
(
gA
2fπ
)2
[−4T (r) + Y (r)]
Y (r) =e−mπr
r
T (r) =e−mπr
r
[
1 +3
mπr+
3
(mπr)2
]
Triplet part of the OPE potential V (r) −−−→r→0
g2A
4πf 2πr−3
Quark Models, pNRQCD, pNRQED, QCD’s EFTs, etc
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Math: Taking r → 0 for a singular potential
• Full range r ∈]0,∞[: Case, PR60,797(1950)
r
E
• Singular Attractive Potential
Near the origin the solution is thesuperposition of two oscillatorywave functionsOne has to fix a relative phase,ϕ(p) How to do it?
Which are the appropriateboundary conditions?(self-adjoint extensions of theHamiltonian → dϕ(p)/dp = 0)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
The potential does not determine uniquely the scattering problemPlesset, PR41,278(1932), Case, PR60,797(1950)
r
E
• Singular Repulsive Potential
There is only one finite(vanishing) reduced wavefunction at r = 0
The solution is fixed
Typically, the phenomenology is not accurate
E.g. this scheme a la Case does not fit well NN phase shifts.
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Lippmann-Schwinger equation
Low-energy EFT paradigm:Contact terms are necessary to reproduce short-distance physics
They are allowed by symmetryThey are required to make loops finite. NonrenormalizableQFT/EFTThey are expected to be relatively important because ofpower-counting
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
New exact equation in NR scattering theory
Yukaway potential,
V (q) =2g
q2 +m2π
Singularity for q2 = −m2π
1S0 potential: (2S+1LJ)
V (p) =g
2p2log(4p2/m2
π + 1)
Left-hand cut (LHC) discontinuity
p2 <−m2π/4 = L
Born approximation
∆1π(p2) =
V (p2 + i0+)− V (p2 − i0+)
2i= ImV (p2 + i0+) =
gπ
2p2
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
Full LHC Discontinuity , p2 = −k2 < L
2i∆(p2) =T (p2 + i0+)− T (p2 − i0+)
∆(p2) =ImT (p2 + i0+)
.
.
.
.
The LS generates contributionswith any number of pions to∆(p2), p2 < L
∆nπ(p2) for p2 < −(nm2
π/2)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
How to calculate ∆(A)?
G.E. Brown, A.D. Jackson “The Nucleon-Nucleon interaction”,North-Holland, 1976. Page 86: “In practice, of course, we do not
know the exact form of ∆(p2) for a given potential . . . ”
p =ik ± ε , ε = 0+ , p2 = −k2 < L
T (ik ± ε, ik ± ε) =V (ik ± ε, ik ± ε)
+µ
2π2
∫ ∞
0dqq2
V (ik ± ε, q)T (q, ik ± ε)
q2 + k2
T (ik ± ε, ik ± ε), so calculated, IS PURELY REAL!!
You can try to calculate numerically just the first iterated OPE
µ
2π2
∫ ∞
0dqq2
V (ik ± ε, q)V (q, ik ± ε)
q2 + k2∈ R
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
Notation: A = p2
This is an example of:Not all what you can calculate with a computer is the rightanswer!!• 1st particular method: Algebraic calculationsOPE 1S0 is simple enough
∆1π(p2) =
gπ
2p2θ(L− A)
∆2π(A) =θ(4L− A)
(
g2Am
2π
16f 2π
)2MN
A√−A
log
(
2√−A
mπ
− 1
)
∆3π(A) =θ(9L− A)
(
g2Am
2π
4f 2π
)3 (MN
4π
)2π
4A
∫ 2√−A−mπ
2mπ
dµ11
µ1(2√−A− µ1)
θ(µ1 − 2mπ)
∫ µ1−mπ
mπ
dµ21
µ2(2√−A− µ2)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
∆4π(A) =θ(16L− A)
(
g2Am
2π
4f 2π
)4 (MN
4π
)3π
4A
∫ 2√−A−mπ
3mπ
dµ11
µ1(2√−A− µ1)
×θ(µ1 − 3mπ)
∫ µ1−mπ
2mπ
dµ21
µ2(2√−A− µ2)
×θ(µ2 − 2mπ)
∫ µ2−mπ
mπ
dµ31
µ3(2√−A− µ3)
.
This can be generalize for a diagram with n pions to
∆nπ(A) =θ(n2L− A)
(
g2Am
2π
4f 2π
)n (MN
4π
)n−1 π
4A
×n−1∏
j=1
θ(µj−1 − (n + 1− j)mπ)
∫ µj−1−mπ
(n−j)mπ
dµj1
µj(2√−A− µj)
with µ0 = 2√−A
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
Formal solution of the integral equation (IE):
∆(A, µ) =∆1π(A) +
(
MNA
π2
)
θ(µ− 2mπ)
∫ µ−mπ
mπ
dµ∆1π(A)∆(A, µ)
µ(2√−A− µ)
∆(A) =∆(A, 2√−A)
• Asymptotic solution for√−A ≫ mπ
∆(A) =λπ2
MNAe
2λ√−A
arctanh
(
1− mπ√−A
)
λ =gMN
2π
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
• 2nd GENERAL method:Analytic extrapolation of the LS from its integral expression
f(ν) = ∆v(ν, k)− θ(k − 2mπ − ν)m
2π2
∫ k−mπ
mπ+ν
dν1k2 − ν21
∆v(ν, ν1)f(ν1)
∆(A) = − f(−k)
2A, k =
√−A , v(p1, p2) = p1p2V (p1, p2)
IE : − k +mπ < ν < k −mπ
The limits in the IE ARE FINITE
The denominator never vanishes , |ν1| < k in the IE
NO FREE PARAMETERS
Reason: Contact interactions (monomials) do not contribute tothe discontinuity of T (A)Short-distance physics is not resolved→ Contact interactions
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
f(ν) = ∆v(ν, k)− θ(k − 2mπ − ν)m
2π2
∫ k−mπ
mπ+ν
dν1k2 − ν21
∆v(ν, ν1)f(ν1)
∆(A) = − f(−k)
2A, k =
√−A , v(p1, p2) = p1p2V (p1, p2)
It can be applied to:
Any local potential (spectral decomposition:)
V (p′,p) =2
π
∫ ∞
m0
dmmη(m2)
q2 +m2, q = p′ − p
Higher partial waves, ℓ ≥ 0
Coupled Channels
Nonlocal potentials due to relativistic corrections
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
New exact equation in NR scattering theory
log-log plot for 1S0 ∆(A); gA = 6.80
1e-05
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
1 10 100 1000 10000 100000 1e+06
|∆(A
)| (|L
|-1)
|A| (|L|)
∆1π, ∆2π, ∆3π, ∆4π, Asymptotic sol. (dots) |A| ≫ m2π
Full solution ∆(A)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
N/D method with non-perturbative ∆(A)
N/D method with non-perturbative ∆(A)
Once we now the exact ∆(A) for a given potential we can useS-matrix theory to solve the LS: N/D method with the full ∆(A)
TJℓS(A) =NJℓS(A)
DJℓS(A)
NJℓS(A) has Only LHC
DJℓS(A) has Only RHC
RHC
ǫ → 0
R → ∞
CI
ǫ → 0
R → ∞CII
−m2π4
LHC
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
N/D method with non-perturbative ∆(A)
Uncoupled Partial Waves
Exact knowledge of discontinuities
Tℓ(A) =Nℓ(A)
Dℓ(A)
Im1
Tℓ(A)= −ρ(A) ≡ µ
√A
2πA > 0 (RHC)
ImDℓ(A) = −Nℓ(A)ρ(A) A > 0 (RHC)
ImNℓ(A) = Dℓ(A)∆(A) A < L (LHC)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
N/D method with non-perturbative ∆(A)
(m1,m2) N/D equations for D(A) and N(A)
N/Dm1 m2
N(A) =
m1∑
i=1
νi (A− C )m1−i +(A− C )m1
π
∫ L
−∞
dk2 ∆(k2)D(k2)
(k2 − A)(k2 − C )m1
D(A) =
m2∑
i=1
δi (A− C )m2−i − (A− C )m2
π
∫ ∞
0
dq2ρ(q2)N(q2)
(q2 − A)(q2 − C )m2
N(A) is substituted in D(A)
Linear IE for D(A) arises
D(0) = 1. To fix a floating constant in the ratioT (A) = N(A)/D(A)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
Regular interactions
• N/D01: Regular solution for an ordinary potential
Scattering is completely fixed by the potential
N(A) =1
π
∫ L
−∞
dωLD(ωL)∆(ωL)
(ωL − A)
D(A) = 1− A
π
∫ ∞
0dωR
ρ(ωR)N(ωR)
(ωR − A)ωR
= 1− iµ√A
2π2
∫ L
−∞
dωL∆(ωL)D(ωL)
√ωL
(√ωL +
√A)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
• N/D11: Additional subtraction in N(A) is fixed in terms ofscattering length
D(A) = 1 + ia√A+ i
MN
4π2
∫ L
−∞
dωLD(ωL)∆(ωL)
ωL
A√A+
√ωL
N(A) = −4πa
MN
+A
π
∫ L
−∞
dωLD(ωL)∆(ωL)
(ωL − A)ωL
Effective Range Expansion (ERE)
kcotδ(k) = −1
a+
1
2rk2 +
∑
i=2
vik2i
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
• N/D12: Additional subtraction in D(A), r is fixed
D(A) = 1 + ia√A− ar
2A− i
MNA
4π2
∫ L
−∞
dωLD(ωL)∆(ωL)
ωL
×[
√A
(√ωL +
√A)
√ωL
− i
aωL
]
N(A) = −4πa
MN
+A
π
∫ L
−∞
dωLD(ωL)∆(ωL)
(ωL − A)ωL
The results are just dependent on ∆(A) (input potential) andexperimental ERE parameters
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
• N/D22: Additional subtraction in N(A), v2 is fixed
D(A) = (1− 2v2
rA)(1 + ia
√A)− ar
2A
+iMN
4π2A
∫ L
−∞
dωLD(ωL)∆(ωL)
ω2L
×[
A√A+
√ωL
+ i2
ra2ωL
(1 + ia√ωL)(1 + ia
√A)
]
N(A) = −4πa
MN
+ A8πav2
MN r+
A
π
∫ L
−∞
dωLD(ωL)∆(ωL)
ω2L
×[
A
(ωL − A)+
2
raωL
(1 + ia√ωL)
]
The more subtractions are included the more perturbative N/D iswith respect to ∆(A). ∆nπ(A) contributes for A < n2L
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
Example: 1S0 projected Yukawa potential
gA = 1.26
0
2
4
6
8
10
12
14
0 50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
(a)
gA = 6.80 → a = −23.75 fm
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300 350 400δ1
S 0
k (MeV)
(c)
LS (black dots), N/D01 (red line)1π, 2π, 3π,4π
Non-perturbative problemOnly Full ∆(A) gives agreementwith LS
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Regular interactions
ERE parameters from N/D01 and N/D11
gA = 6.80
as r v2 v3 v4 v5 v6
Regular (fm) (fm) (fm3) (fm5) (fm7) (fm9) (fm11)N/D01
∆1π 1.66 0.714 -0.168 0.847 -5.35 35.5 -241∆2π 3.53 2.03 -5.70 10−2 3.38 -27.4 234 -1.99 103
∆3π 1.80 1.15 -8.71 10−2 0.924 -5.69 36.9 -247∆4π -6.89 13.7 47.5 356 2.63 103 1.97 104 1.46 105
Full -23.75 8.90 18.7 89.8 411 2.00 103 8.98 103
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Singular Interactions
Analytical properties determine the solutions for singularpotentials
Singular Interactions. E.g. NLO and NNLO 1S0 potentials
Potentials are calculated within Chiral Perturbation TheoryLow-energy EFT of QCD
∆(A) is now divergent
limA→−∞∆(A)NLOBorn/k2 < 0 , ∆(A)NNLOBorn /k3 < 0
Attractive singular interaction
N/D11 and N/D22 are convergent
N/D01 (at least one parameter is needed) and N/D12 are notconvergent
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Singular Interactions
We compare with
LS renormalized with one contact term:
V (p1, p2) → V (p1, p2) + C0
V (p1, p2) → V (p1, p2) + C0 + C1(p21 + p22) + . . .
LS is insensitive to C1, etc.
Repulsive Singular Potential: LS is insensitive to all Ci
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Singular Interactions
NLO NNLO
-10
0
10
20
30
40
50
60
70
50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
-30
-20
-10
0
10
20
30
40
50
60
70
50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
N/D11 , N/D22
Subtractive-renormalized LS: pointsOne counterterm It cannot reproduce r
Yang,Elster,Phillips, PRC80,044002(’09)
a = −23.75 , r = 2.655 fm , v2 = −0.6265 fm3
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
T (A) in the complex plane
• As a bonus the non-perturbative-∆ N/D method allows tocalculate T (A) for A ∈ C in the 1st/2nd Riemann sheet
This is not trivial with LS
Look for bound states, resonances and virtual states
Bound State A = (ik)2
Binding energy of near threshold bound state, gA = 7.45One does not need to solve Schrodinger equation
Poles of T (A) ↔ zeros of D(A)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
• As a bonus the non-perturbative-∆ N/D method allows tocalculate T (A) for A ∈ C in the 1st/2nd Riemann sheet
This is not trivial with LS
Binding energy of near threshold bound state, gA = 7.45One does not need to solve Schrodinger equation
Poles of T (A) ↔ zeros of D(A)
A = (ik)2 N/D01 N/D11 Schrodinger
∆1π 2.02∆2π 2.18∆3π 2.21∆4π 0.89 2.22
Non-perturbative 2.22 2.22 2.22
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
Anti-bound (virtual) state for 1S0
T−1II (A) = T−1
I (A) + 2iρ(A)
=DI + NI 2iρ(A)
NI
, Im√A ≥ 0
Look for zero of DII (A) . E = A/MN =
N/D11:−0.070 (LO) , −0.067 (NLO,NNLO) MeV
For the other N/Dm1m2 : −0.066 MeV always
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
Back to Yukawa 1S0gA = 1.26
0
2
4
6
8
10
12
14
0 50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
(b)
gA = 6.80 → a = −23.75 fm
-140
-120
-100
-80
-60
-40
-20
0
20
40
60
0 50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
(d)
LS (black dots), N/D11 (red line) 1π, 2π, 3π,4π
• Perturbative-∆ N/D + subtractions JAO et al PRC93,024002(2016);
PRC89,014002(’14); PRC86,034005(’12);PRC84,054009(’11)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
Two-nucleon reducible diagrams [II]; Guo,Rıos,JAO,PRC89,014002(’14);
Similar size to the other NLOirreducible diagrams
.
.
.
.
All pion lines must be put on-shell −→ A ≤ −n2M2π/4.
As n increases their physical contribution fades away.
This only occurs for the imaginary part!
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
ERE parameters from N/D01 and N/D11
gA = 6.80
as r v2 v3 v4 v5 v6
Regular (fm) (fm) (fm3) (fm5) (fm7) (fm9) (fm11)
∆1π 1.66 0.714 -0.168 0.847 -5.35 35.5 -241∆2π 3.53 2.03 -5.70 10−2 3.38 -27.4 234 -1.99 103
∆3π 1.80 1.15 -8.71 10−2 0.924 -5.69 36.9 -247∆4π -6.89 13.7 47.5 356 2.63 103 1.97 104 1.46 105
Full -23.75 8.90 18.7 89.8 411 2.00 103 8.98 103
N/D11
∆1π -23.75 8.56 15.3 60.5 221 906 3.08 103
∆2π -23.75 8.80 17.7 80.4 346 1.60 103 6.71 103
∆3π -23.75 8.88 18.4 87.5 395 1.90 103 8.40 103
∆4π -23.75 8.90 18.6 89.3 408 1.98 103 8.87 103
Full -23.75 8.90 18.7 89.8 411 2.00 103 8.98 103
The more subtractions are included the more perturbative N/D iswith respect to ∆(A).
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
LS renormalized with a counterterm
We add to the Yukawa potential Local Terms:
Monopole Form Factor:
V1(p′, p) = C 1
0
Λ2
2p′pQ0(zΛ)
Q0(zΛ) =1
2log
(
zΛ + 1
zΛ − 1
)
, zΛ =p′2 + p2 + Λ2
2p′p
Gaussian type form factors
V2(p′, p) = C 2
0 e−
(
p′
Λ
)4−( p
Λ)4
V3(p′, p) = C 3
0 e−
(
p′
Λ
)6−( p
Λ)6
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
gA = 1.26 , a = −23.70 fm , Λ = 100 GeV
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
LS (black dots; Λ = 100 GeV)N/D11 (red line) , 1π, 2π, 3π,4π
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
75.0
75.5
76.0
76.5
77.0
77.5
78.0
78.5
0 100 2 10-5 4 10-5 6 10-5 8 10-5 1 10-4
δ1S 0
(k=
400
MeV
)
1/Λ (MeV-1)
V1, V2, V3. Λ → ∞ by a linear fit.Subtractive renormalized LS (solid black line)N/D11 by dashed lines, 1π, 2π, 3π,4π, Full ∆(A)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
Yukawa 1S0
-10
0
10
20
30
40
50
60
70
80
50 100 150 200 250 300 350 400
δ1S 0
k (MeV)
N/D11, N/D12, N/D33
Phase shifts: Granada analysis Navarro et al. PRC88,064002(’13)
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
T (A) in the complex plane
G.E. Brown, A.D. Jackson “The Nucleon-Nucleon interaction”,North-Holland, 1976. Page 86: “In practice, of course, we do not
know the exact form of ∆(p2) for a given potential and the N/Dequations do not represent a practical alternative to the exact
solution of the LS equation for potential scattering. . . ”
Now (2016), this statement is superseded
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Conclusions
Conclusions
A new non-singular IE allows to calculate the exact ∆(A) inpotential scattering for a given potential
One can calculate the scattering amplitude for regular/singularpotentials from its analytical/unitarity properties
Any proper solution for singular potentials can be found withthis method
We reproduce the LS outcome with/without one counterterm
S-matrix solution of the Lippmann-Schwinger equation for regular and singular potentials
Conclusions
It can be straightforwardly used in the whole complex plane(bound states, resonances, virtual states)
It can reproduce accurately 1S0 phase shifts