G. Orlandini Saclay, May 19 2009
The Lorentz Integral Transform (LIT) The Lorentz Integral Transform (LIT) method and its connections with other method and its connections with other
approachesapproaches
First proposed inV. D. Efros, W. Leidemann and G. Orlandini, Phys. Lett. B338, 130 (1994)
Recent Topical Review:V. D. Efros, W. Leidemann, G. Orlandini and N. Barnea “The Lorentz Integral Transform (LIT) method and its applications to perturbation induced reactions” arXiv:0708.2803
J. Phys. G: Nucl. Part. Phys. 34 (2007) R459-R528
G. Orlandini Saclay, May 19 2009
The LIT methodThe LIT method
it is an it is an ab initioab initio method for method for calculationscalculations of reactions involving of reactions involving states states in the (far) continuumin the (far) continuum
it is general enough to be applied to strong as well as electroit is general enough to be applied to strong as well as electroweak (perturbative) weak (perturbative) reactionsreactions of inclusive as well as exclusive of inclusive as well as exclusive naturenature
the applications so far have been to the applications so far have been to electromagnetic and weakelectromagnetic and weak reactions on light nuclei.reactions on light nuclei.
G. Orlandini Saclay, May 19 2009
There are many examples in physics where one uses “integral transform approaches”
T = K F
Integral transform approaches
object of interestaccessible object
There are many classes of problems that are difficult to solve in their original representations. An integral transform "maps" an equation from its original "domain" into another domain. Manipulating and solving the equation in the target domain is sometimes much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.
G. Orlandini Saclay, May 19 2009
T = K F
The instrument “integrates” F with the form of its “window” K and gives T
In experimental physics this is common in that one wantsto extract information on the observable from the datawhich is obtained by means of the instrumentation.
object of interestaccessible object
G. Orlandini Saclay, May 19 2009
Laplace Kernel
Φ ∫†xd3x∫ e – S() d
In theoretical physics:
In Nuclear Physics:
= Charge or current density operatorS() = R() “Response” Function (to external perturbative probe)Φ is obtained with Monte Carlo Methods
In Condensed Matter Physics:
= Density OperatorS() = Dynamical Structure FunctionΦis obtained with Monte Carlo Methods
In QCD
= quark or gluon creation operatorS() = Hadronic Spectral FunctionΦ is obtained by OPE QCD sum rules or Lattice
= it
G. Orlandini Saclay, May 19 2009
One is able to calculate (or measure) Φ but wants S(), which is the quantity of direct physical meaning.Problem:The “inversion” of Φ may be an “ill posed problem”
Φ = ∫ dω K(ω,σ) S(ω )
G. Orlandini Saclay, May 19 2009
Definition of “well-posed” problems in making mathematical models of physical phenomena ( by Jacques Hadamard 1865-1963): 1. Equations have solutions (existence) 2. The solution is unique 3. The solution depends continuously on the inputs in some reasonable topology .
Continuum problems must often be discretized in order to obtain a numerical solution (==> Ax=b ). While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision i.e. a small error in the input (b) can result in much larger errors in the output (x) i.e. they become “ill-posed” problems.
They need to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution (x). This process is known as regularization. The regularization can reduce the condition number (a measure of the degree of instability) to acceptable levels.
G. Orlandini Saclay, May 19 2009
It is well known that the numerical inversion of the Laplace Transform is a terribly ill-posed problem
G. Orlandini Saclay, May 19 2009
It is well known that the numerical inversion of the Laplace Transform is a terribly ill-posed problem
ΦS
G. Orlandini Saclay, May 19 2009
It is well known that the numerical inversion of the Laplace Transform is a terribly ill-posed problem
ΦS
???
G. Orlandini Saclay, May 19 2009
a “good” Kernel has to satisfy two requirements
1) one must be able to calculate the integral transform
2) one must be able to invert the transform, minimizing instabilities
G. Orlandini Saclay, May 19 2009
What is the perfect Kernel?
G. Orlandini Saclay, May 19 2009
What is the perfect Kernel?
the delta-function!
G. Orlandini Saclay, May 19 2009
What would be the “perfect” Kernel?
the delta-function!
in fact
Φ S ∫ S d
G. Orlandini Saclay, May 19 2009
the LIT method is based on the idea to use one of the so-called “representations of the delta-function”:
it turns out that a very good Kernel is the Lorentzian function
Γ
0
Φ ∫ [ S d
G. Orlandini Saclay, May 19 2009
The Lorentz Kernel satisfies the two requirements !
N.1. one can calculate the integral transform
N.2 one is able to invert the transform, minimizing instabilities
G. Orlandini Saclay, May 19 2009
Illustration of requirement Illustration of requirement N.1:N.1: one can calculate the integral one can calculate the integral transformtransform
G. Orlandini Saclay, May 19 2009
Suppose we want an R() defined as (for example for perturbation induced inclusive reactions)
�
G. Orlandini Saclay, May 19 2009
choosing the 2parameter kernel L(
) a theorem based on closure
states that the integral transform
Φ(ω0,Γ) is given by:
ω0
Γ
where
Φ(ω0,Γ) =
G. Orlandini Saclay, May 19 2009
Closure = 1
Proof of the theorem:Φ (ω
0,Γ) =
G. Orlandini Saclay, May 19 2009
Closure = 1
Φ (ω0,Γ) =
Proof of the theorem:
G. Orlandini Saclay, May 19 2009
The LIT in practice:The LIT in practice:
is found solving for fixed Γ and many ω0
1.
G. Orlandini Saclay, May 19 2009
the overlap is calculated
3.
2.
the transform is inverted
G. Orlandini Saclay, May 19 2009
main point of the LIT :main point of the LIT :
Schrödinger-like equation with a source
S =
G. Orlandini Saclay, May 19 2009
main point of the LIT :main point of the LIT :Schrödinger-like equation with a source
The solution is unique and has bound state asymptotic behavior
Theorem:
G. Orlandini Saclay, May 19 2009
main point of the LIT :main point of the LIT :Schrödinger-like equation with a source
one can apply bound state methods
The solution is unique and has bound state asymptotic behavior
Theorem:
G. Orlandini Saclay, May 19 2009
The LIT methodThe LIT method reduces the reduces the continuumcontinuum problem to a problem to a bound state bound state
problemproblem needs needs onlyonly a “good” method for a “good” method for bound statebound state
calculations (FY, HH, NCSM, ...???)calculations (FY, HH, NCSM, ...???) applies both to applies both to inclusiveinclusive reactions (straightforward!) reactions (straightforward!)
and to and to exclusiveexclusive ones ones has beenhas been benchmarked benchmarked in “directly solvable” systems in “directly solvable” systems
(A=2,3) (A=2,3)
G. Orlandini Saclay, May 19 2009
A very good method to solve A very good method to solve bound states:bound states:
similar idea as for the No Core Shell Model method i.e. usesimilar idea as for the No Core Shell Model method i.e. use of Effective Interactionof Effective Interaction applied to H.H. instead of H.O.applied to H.H. instead of H.O.
avoids the avoids the ΩΩH.O.H.O.
parameter dependenceparameter dependence
fast convergencefast convergence can be applied to A>3 can be applied to A>3
N.Barnea, W.Leidemann, G.O. PRC61(2000)054001
the Effective Interaction in Hyperspherical Harmonics method (EIHH)
G. Orlandini Saclay, May 19 2009
For A>3 no other benchmark is possible! NO viable solution of the scattering problem beyond the 3-body break up threshold for A=4 and larger.
The LIT approach is at present the only viable one!
Benchmarks:
with R() calculated with traditional differential equation algorithm in A=2 [ V.D.Efros et al.Phys. Lett. B338, 130 (1994) ]
with R() calculated with Faddeev solutions in the continuum in A=3 [J.Golak et al. [Nucl. Phys. A707 (2002) 365]
G. Orlandini Saclay, May 19 2009
Practical calculation of Φ
1. Eigenvalue method:
can be expanded on localized functions !!!
n n n
H is represented in this basis -----> Hmn
Hmn
is diagonalized -----> |
G. Orlandini Saclay, May 19 2009
Practical calculation of Φ
1. Eigenvalue method: can be expanded on localized functions |
G. Orlandini Saclay, May 19 2009
Practical calculation of Φ
1. Eigenvalue method:
sum of Lorentzians around ευN
can be expanded on localized functions |
G. Orlandini Saclay, May 19 2009
Practical calculation of Φ
2. Lanczos method
G. Orlandini Saclay, May 19 2009
Illustration of requirement Illustration of requirement N.2: N.2: one can invert the integral one can invert the integral transform minimizing instabilitiestransform minimizing instabilities
G. Orlandini Saclay, May 19 2009
Inversion of the LIT: the regularization method
Works well with bell shaped kernels
G. Orlandini Saclay, May 19 2009
Phys Lett. B338 (1994) 130
RL (
)
[MeV
-1]
MeV
test on the Deuteron:
R() is the longitudinal (e,e') response function
q =2.3 fm-1
G. Orlandini Saclay, May 19 2009
66--Body total photodisintegrationBody total photodisintegration
EIHH
MT6Li
6He
classical GT mode
soft mode
Bacca et al. PRL89(2002)052502Bacca et al. PRL89(2002)052502
G. Orlandini Saclay, May 19 2009
77-Body total -Body total photodisintegrationphotodisintegration
'75
S.Bacca et al. PLB 603(2004) 159
G. Orlandini Saclay, May 19 2009
Realistic interactions?Realistic interactions?
NN=AV18 + NNN=UIX NN=AV18 + NNN=UIX
44HeHe
G. Orlandini Saclay, May 19 2009
dotted: PWIA
full: AV18+UIX
Role of FSI:
S.Bacca et al., PRL 102 (2009) 162501
Inclusive electron scatteringInclusive electron scattering44HeHe
G. Orlandini Saclay, May 19 2009
Role of 3-body force
Inclusive electron scatteringInclusive electron scattering
S.Bacca et al., PRL 102 (2009) 162501
dotted: AV18
full: AV18+UIX
44HeHe
G. Orlandini Saclay, May 19 2009
SURPRISE:LARGE EFFECT OF
3-BODY FORCE AT LOW Q
NO MEASUREMENTSAT LOW q !!!
44HeHe
S.Bacca et al., PRL 102 (2009) 162501
G. Orlandini Saclay, May 19 2009
Effect of different 3-body forces
G. Orlandini Saclay, May 19 2009
Importance of the regularization:
G. Orlandini Saclay, May 19 2009
Remember the practical calculation of Φ
G. Orlandini Saclay, May 19 2009
1. Eigenvalue method:
sum of Lorentzians around ευN
can be expanded on localized functions
Remember the practical calculation of Φ
G. Orlandini Saclay, May 19 2009
G. Orlandini Saclay, May 19 2009
However, here comes the problem of the Continuum!
A regularization is needed!
G. Orlandini Saclay, May 19 2009
test on deuteron photodisintegration
h.o. basis:fixed MeV
“true”
Nho
=2400
Nho
=150
MeV
G. Orlandini Saclay, May 19 2009
test on deuteron photodisintegration
h.o. basis:fixed
“true”
MeV
MeV
MeV
MeV
MeV
G. Orlandini Saclay, May 19 2009
Γ=10 MeV + inversion / regularization
Nho
=150 is enough for accuracies at the % level!!
MeV
ΦN
ho=2400
Nho
=150
G. Orlandini Saclay, May 19 2009
ConclusionsConclusions
it allows to calculate reactions to the “far” continuum it allows to calculate reactions to the “far” continuum where the many-body scattering problem (all channels!) where the many-body scattering problem (all channels!) is not solvableis not solvable
only only bound statebound state technique is needed technique is needed
the LIT represents an accurate viable method on the way from
ab initio NUCLEAR STRUCTURE
ab initio NUCLEAR REACTIONS
also for A>3
G. Orlandini Saclay, May 19 2009
ConclusionsConclusions
since the LIT is calculated numerically a since the LIT is calculated numerically a regularizationregularization procedure is demanded to solve procedure is demanded to solve the integral equation (inversion of the LIT)the integral equation (inversion of the LIT)
there is no discretization of the continuum: the LIT equation is bound-state like
the bell shaped kernel makes the the bell shaped kernel makes the regularizationregularization procedure “inexpensive”, and allows to control procedure “inexpensive”, and allows to control instabilities. instabilities.
G. Orlandini Saclay, May 19 2009
(A < 8)
ab initio, small A
understand some “physics”
translate into languages we are used to in
many-body theory
=
( )
understand the only input
( nuclear interaction )
Future workFuture work
Apply LIT to larger systems with other bound state methods (CC, ??)
G. Orlandini Saclay, May 19 2009
Future workFuture work
Apply LIT to larger systems with other bound state methods (CC, ??)