Knockout-reaction with RIB
A3F-CNSSS20, August 17th-21st, 2020
My aim is:– to explain very basic (but important) things on proton-“induced” knockout reactions,– to instruct how to perform DWIA calculations with a code pikoe (with thought!),– not to demonstrate our recent activities or to go into the very details of theories.

Kazuyuki Ogata
RCNP, Osaka University

Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, NP 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIA

Plan of this talk (2/2)
4) Some theoretical achievements (for future)
K. Yoshida, M. Goméz-Ramos, KO, and A. M. Moro, PRC 97, 024608 (2018).

4-1. Microscopic optical potential
4-2. Benchmark study on 15C(p,pn) with DWIA, TC, and Faddeev,-AGS.

5) Divergence of the TDX in inverse kinematics
KO+, in preparation.

5-1. Two-value feature of the kinematics and divergence of PV
5-2. When occurs?

6) Some recent/ongoing KO reaction studies around RCNP/RIBF
6-1. 2n correlation study via (p,pn)
6-2. a KO reactions
6-3. deuteron KO reactions…

7) Summary

Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIA

What is the knockout reaction?
Qualitatively speaking, by quasi-free scattering a process is meant in which a highenergy (100–1000 MeV) particle knocks a nucleon out of a nucleus and no further
violent interaction occurs between the nucleus and the incident or the two outgoingparticles.
G. Jacob and Th. A. J. Maris, Rev. Mod. Phys. 38, 121 (1966).

In essence, a proton induced knockout reaction is a nuclear reaction in which an
incident proton interacts with either a nucleon or a nuclear cluster in a target nucleus
and knocks this entity out of the nucleus, generating a one-hole or a clusterhole state.
This process, in particular the one nucleon knockout reaction, is the most dominant
reaction at intermediate (200–1000 MeV) energies.
T. Wakasa, KO, T. Noro, Prog. Part. Nucl. Phys. 96, 32 (2017).

Knockout or removal?
RemovalKnockOut

Elastic Breakupp(A,B+p)p

T(A,B+p)T

STRipping
T(A,B+x)B

B

pA(= B+p)ppplarge energy-momentum transfer

A(= B+p)T

B
pT*(or T+p)

small energy-momentum transfer

NOTE: At Michigan State University (MSU), nucleon removal processes haveintensively been measured and studied. Some people call them “knockout” processes
but those reaction mechanism is quite different from that of the knockout process.

What we can study via KO reaction?
K1
K0

K2
A

MB, KBWe have 9+1 d.o.f. in the final state. Because of the energy-momentum conservation, 6 out of 10 are independent.
If K1 and K2 are specified, all the kinematics are determined as well as MB (internalenergy of the residue B). This is called kinematically complete measurement.
In what follows, I assume that MB has been specified. Then, there are 5 d.o.f. left.
In the picture of KO reactions, B behaves as a spectator. This indicates that before
the KO, the nucleon had a momentum KN = –KB in the nucleus A.

What we can study via KO reaction? (Con’t)
K1
K0
KN

j
KN = –KB = K1 + K2 – K0

K2
AMB, KBIf B is in a bound state, it has a specific Jp. Then, thes.p. orbit j of the nucleon inside A can be specified.
(In some cases, j is not unique but the contribution of each j isincoherent so we can consider j one by one.)

Momentum conservation and the spectator assumption
for B allows a specification of KN.

By KO reactions, one may take a “snapshot” of a s.p. momentum of a nucleus.

Plane-Wave Impulse Approxn (PWIA) for p2p (1/2)
s0 or 1

R0 R
R1

2

K1

R2
A

B

K0

K2
A

MB, KB
proton bound state W.Fn.

just for simplicity(in this school)

Plane-Wave Impulse Approxn (PWIA) for p2p (2/2)=

Snapshot!

Momentum transfer q (usually large):
Missing momentum Q (usually intended to be small):
NOTE: Q = 0 corresponds to the recoilless condition.

An example of PWIA calculation
E1 = 251 MeV
q1 = 32.5 deg.
f1 = 0 deg.

2B

q2 : varied
f2 = 180 deg.

d3s / dE1dW1dW2(mb sr-2MeV-1)

1

250

A key observable:
Triple-differential cross section (TDX)

12C(p,2p)11B @392 MeV
g.s.

200
150
100
50
0
20

40

60
q2 (deg)

At q2 = 50 deg., KB ~ 10.7 MeV/c, which corresponds to the recoilless condition(RLC). E1 and q2 were chosen so that the RLC is achieved at a value of q2.

80

Input for pikoe (read the manual!)

sample1.cnt

You need to put FLtbl_rede.dat at the directory specified here. In this example, it must be put on the directory
where pikoe1.exe (or a.out etc.) exits. You can change the path accordingly to your directory structure.
atomic # and mass #of 12Cj = 3/2, l = 1, theS-factor is set to 1,
the # of nodes is 0.

incident energy

WS parameter usedin 12C(e,e’p) analysis

proton separationenergy of 12C

q2 is varied from 0
deg. to 180 deg. withthe step of0.5 deg.

not use the BohrMottelson s.p. pot.
T1, q1, f1, and f2 are
fixed at 251 MeV,
32.5 deg., 0 deg.,and 180 deg.

Distorting pots. for particles 0, 1, and 2 are switched off.

200

40Ca(p,2p)39K @392 MeV
g.s.

150
100
50
0
20

0d3/2

40

60
q2 (deg)80

d3s / dE1dW1dW2 (mb sr-2MeV-1)

d3s / dE1dW1dW2 (mb sr-2MeV-1)

PWIA results for 40Ca(p,2p)
10

4

10

3

10

2

10

1

10

0

10

40Ca(p,2p)39K
1/2+,2.5MeV@392 MeV

1s1/2

-1

20

40

60

80

q2 (deg)

The orbital angular momentum l can be determined by the shape of the TDX.
HW: By preparing the input files, reproduce the results shown above. The kinematicalcondition for T1, q1, f1, and f2 is the same as for the 12C target case.

Profile of KB
0.5

12C
40Ca_gs
40Ca_ex

KB (fm-1)

0.4
0.3
0.2
0.1
0
40

45

50
q2 (deg)55

60

From PWIA to Distorted-Wave IA (DWIA)

The asymptotic momentum approximation (AMA) on the propagation of the DW
for a short distance:
Factorization approxn to the T matrix

NOTE: Some people call this the zero-range approxn but it will be misleading.
does contain an integration over s. The zero-range approxn is not adopted for tpp.

Why DWIA, not DWBA?
Answer: Because the transition interaction is the NN effective interaction in which
all the ladder diagrams regarding vNN are taken into account.

vNN
1 step+ … ++2 step…

+ … = tNN

N stepcf. Lippmann-Schwinger Eq.

We need an NN effective interaction in the many-body system but it is often
replaced with tNN. This is the essence of the impulse approximation, which will bevalid at intermediate energies.

NN t-matrix effective interaction (in free space)・An effective interaction has no repulsive core andis easily handled.
・The wave function (of a many-body system)
to be operated does not need to be very accurate.
An example of the comparison between v and t
M. Yahiro, K. Minomo, KO, M. Kawai, PTP 120, 767 (2008).

vNN or tNN (MeV)

Transition by a bare interaction with infinite order processes is expressed bya single step transition by an effective interaction.

vNN (s)Re tNN (s)Im tNN (s)s (fm)

200

12C(p,2p)11B @392 MeV
g.s.

pikoe
Noro 2020

Sp2p = 1.77

150

(See’p = 1.72)

100

0p3/2

50
0
20

40

60
q2 (deg)80

d3s / dE1dW1dW2 (mb sr-2MeV-1)

d3s / dE1dW1dW2 (mb sr-2MeV-1)

Spectroscopic study via p2p with DWIA
100

40Ca(p,2p)39K
g.s. @392 MeV

Sp2p = 2.47 (See’p = 2.58)
0d3/2

50

0
20

40

60

80

q2 (deg)

The spectroscopic factor can be determined by the magnitude of the TDX.
NOTE: In the PPNP review and PTEP paper, the finite solid-angle and momentumbite (W-DP) corrections have been applied to the DWIA calculations.

200

12C(p,2p)11B @392 MeV
g.s.

Sp2p = 1.77

150

(See’p = 1.72)

100

0p3/2

50
finite W-DP corr.

0
20

40

60
q2 (deg)80

d3s / dE1dW1dW2 (mb sr-2MeV-1)

d3s / dE1dW1dW2 (mb sr-2MeV-1)

Spectroscopic study via p2p with DWIA
100

40Ca(p,2p)39K
g.s. @392 MeV

Sp2p = 2.47 (See’p = 2.58)
0d3/2

50

0
20

40

60

80

q2 (deg)

The spectroscopic factor can be determined by the magnitude of the TDX.
NOTE: In the PPNP review and PTEP paper, the finite solid-angle and momentumbite (W-DP) corrections have been applied to the DWIA calculations.

Input for 12C(p,2p) with DWIA

sample2.cnt

The optical potential file for particle 0 (unit # is set to 12) and that for particles 1 and 2 (unit # 13). Because
the KD potential is not applicable, you need to prepare optical potentials as ext. files.
S-factor. In an actual
study, it is determined
to reproduce exp. data.
Distorted waves arecalculated with orbitalang. mom. L up to 60.

Distorting pots. aregiven in the files ofunit #s = 12, 13, and13 for particles 0, 1,
and 2.

This value is negative.
So the nonlocality
correction function isread from the ext.
files (the absolutevalue has no meaning
in this case).
If you use KD (when
it is applicable), 0.85
is recommended forthese values.

d3s / dE1dW1dW2 (mb sr-2MeV-1)

T. Noro+, PTEP 2020 (in press).

Spectroscopic study via p2p with DWIA 210340Ca(p,2p)39K @392 MeV
ex

sum

10

10

2

1

1s1/2
0f7/2

10

0

20

40

60
q2 (deg)

When the states are not well separated, themultipole decomposition analysis (MDA)
can be used (if l are different).

1/2+: Sp2p = 1.03 (See’p = 1.03)
7/2-: Sp2p = 0.42 (See’p = 0.38)

HW: Draw this figure by yourself.

80

Consistency between Sp2p and See’p
Based on data at 392 MeV taken at RCNP

T. Noro+, PTEP 2020 (in press).

They are consistent within uncertainties of 15–20 % except for 208Pb.
ppN can be applied to n KO and KO for unstable nuclei.

Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIA

Keys to “establish” p2p as a spectroscopic tool
Bound state W.Fn. used in the analysis of (e,e’p).
If not available, the s.p. model of Bohr-Mottelson can be used (IBMC=IBMS=1).
Well-constrained optical potential Uopt
The Koning-Delaroche (KD) global Uopt are adopted when IPOT=1. It is limited
for the target mass 24 ≤ A ≤ 209 and the incident energy 1 keV ≤ E ≤ 200 MeV.
The Dirac phenomenology (Dirac PH) was employed in the PPNP review article.
The EDAD1 set is applicable to nuclei from 12C to 208Pb for 21 MeV ≤ E ≤ 1040
MeV. To use the Dirac PH, you need to prepare an external file for the potential.
Nonlocality corrections to the bound-state W.Fn. and distorted waves (next page).
Moller factor: the Jacobian for the NN t-matrix amplitude from the NN c.m. frame tothe p + A c.m. frame. pikoe always takes this into account.
Fermi motion of the nucleon in A (for Ein = 392 MeV, the NN scattering energy
varies from 95 MeV to 550 MeV). pikoe always takes this into account.

Nonlocality corrections (NLC)
Local phenomenological N-A potentials are not constrained so as to generate a proper
W.Fn in the nuclear interior region. (Its asymptotic form is OK.)d3s / dE1dW1dW2 (mb sr-2MeV-1)

N-A potentials are nonlocal in general. (cf. the projection operator formalism by Feshbach)As a phenomenological prescription, theW.Fn. is multiplied by the following Pereyfactor to include the nonlocality effect.

12C(p,2p)11B @392 MeV (Sg.s.
p2p = 1.77)

200
150

local DW
all local

default
local b.s. W.Fn.

range of nonlocality (0.85 fm)

100
renormalization factor for b.s. W.Fn.

50
0
20

40

60
q2 (deg)

80

A similar correction can be made by usinga Darwin factor in Dirac PH, though its
correspondence with the Perey factor has
not yet been proved.

Feshbach’s projection operator formalism
The entire space is divided into the P-space (to be described explicitly) and the
Q-space (complement).

The potential to describe the P-space is complex, energy-dependent, and nonlocal.

Input for checking the NLC effectsample2.cnt

You can switch-off the NLC for the b.s. W.Fn. by setting this value to 0.

HW:
Draw a figure forseeing the NLC effecton 40Ca(p,2p)39Kex at
392 MeV that issimilar to the figureon the previous slide.

You can switch-off
the NLC for the DWs
by setting these values
to 0. You can also
investigate the NLC
effect on each particle.

The Moller factor
It is just a Jacobian but its (trivial) importance has not been recognized well in somestudies (in my observation).

d3s / dE1dW1dW2 (mb sr-2MeV-1)

Neglect of the Moller factor results in an overshooting of the TDX at high energies.
12C(p,2p)11B @392 MeV
g.s.

200

w/ Moller
w/o Moller

Sp2p = 1.77

150
100
50
0
20

40

60
q2 (deg)80Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIAMDA for two states having the same lT. Wakasa, KO, and T. Noro, PPNP 96, 32 (2017).

MDA fails to differentiate the twostates.

The (vector) analyzing power Ay has
a strong j dependence because of theMaris polarization (Maris effects).

MDA for two states having the same lT. Wakasa, KO, and T. Noro, PPNP 96, 32 (2017).

Ay (for > 0) represents to what
extent a spin-up projectile contributes the process considered.
The
Maris
is useful for
MDA
fails effectto differentiatethethetwo
p specification in general.
jstates.

The (vector) analyzing power Ay has
a strong j dependence because of theMaris polarization (Maris effects).

The Maris effect (1/2)
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.
q1= 30 deg. and q2 = -30 deg.

quantization

j>

axis
up

up

T1 << T2

KN

up

1. For NN collision, s↑↑ and s↓↓
dominate s↑↓ and s↓↑.
2. The mean free path for a lowenergy nucleon is short.

KN
T1 >> T2

l // sKN is fixed so as to satisfy
the energy and momentum
conservation.

Assumptions

T1 << T2

When spin-up incident proton
hits a nucleon in a j> orbit, (p,pN)
events are observed only when
T1 >> T2.
(This is also the case when spin-down
incident proton hits a j< orbit nucleon.)

The Maris effect (2/2)
T1 >> T2
q1= 30 deg. and q2 = -30 deg.

KNup

T1 << T2

up

KN

up

l // -sj<

When spin-up incident proton
hits a nucleon in a j< orbit, (p,pN)
events are observed only when
T2 >> T1.
(This is also the case when spin-down
incident proton hits a j> orbit nucleon.)quantization
axis

Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

T1 >> T2

Ay is positive for a j> (j<) orbit
with T1 >> T2 (T1 << T2).

MDA for two states having the same lT. Wakasa, KO, and T. Noro, PPNP 96, 32 (2017).

MDA fails to differentiate the twostates.

The (vector) analyzing power Ay has
a strong j dependence because of theMaris polarization (Maris effects).

Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIANOTE: An integral with no specificrange means an integration overentire regions.

The pp scatteringqdefined by reaction probability

defined by # of counts at a detector
pp cross section at 500 MeV

Counting rule(classical mechanics)

Antisymmetrization of
the wave function(quantum mechanics)

ds/dW (mb/sr)

10

FL
SP07
5

0

0

30

60

90

q (deg)

120

150

180

The (p,2p) process: case 1
defined by # of counts incoincident at two detector

12C(p,2p)11B
g.s. at 392 MeV (RCNP data)

12Bq2 (deg)defined byreaction probability

When pikoe outputs an integrated TDX, regardless of the integration region, the value
is divided by 2.

The (p,2p) process: case 2
defined by # of counts at a detector
AND reaction probability

12C(p,2p)11B
g.s. at 400 MeV/u (GSI data)

V. Panin+, PLB753, 204 (2016).

1

2
B

Whether the factor 1/2 is needed
depends on the definition of theobservables. For the integrated(total) p2p cross sections, thedivision by 2 is necessary.

Plan of this talk (1/2)
1) Overview of (p,2p) studies (on stable nuclei)
T. Wakasa, KO, T. Noro, PPNP 96, 32 (2017); T. Noro+, PTEP 2020 (in press).

1-1. “Definition” of the KO reaction
1-2. What we can learn from PWIA analysis of KO reactions.
1-3. What we can learn from DWIA analysis of KO reactions.

2) A bit more advanced aspects of (p,2p) studies
Th. A. J. Maris, Nucl. Phys. 9 (1958–1959) 577.

2-1. Key ingredients for spectroscopic studies
2-2. The Maris effect
2-3. Treatment of the identical particles in (p,2p)3) Momentum distribution in inverse kinematics
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

3-1. Peak shift and asymmetric shape3-2. Phase volume (PV) and attractive distortion effects
3-3. SEASTAR, SHARAQ, and GSI data analysis with DWIA

Question:
Why do you expect a symmetricshape for the TDX?

d3s / dE1dW1dW2 (mb sr-2MeV-1)

T. Noro+, PTEP2020 (in press).

Spectroscopic study via p2p with DWIA 210340Ca(p,2p)39K @392 MeV
ex

sum

10

10

2

1

1s1/2
0f7/2

10

0

20

40

60
q2 (deg)

When the states are not well separated, themultipole decomposition analysis (MDA)
can be used (if l are different).

1/2+: Sp2p = 1.03 (See’p = 1.03)
7/2-: Sp2p = 0.42 (See’p = 0.38)

HW: Draw this figure by yourself.

80Symmetric or asymmetric?
A

2

A-rest frame

=

B

2

Lab. framec.m. of A

A-rest frame
in final channelBc.m. of A≠

Kinematics are asymmetric so theobservables. Symmetric shape isobtained only when the effect of
the asymmetric kinematics can beneglected.

2Lab. frameBc.m. of A changes its motion!

Momentum distribution (MD) of the residue B
K1
K0

K2
MB, KBA=
ds/dKB, with specifying MB, will be an ideal observable for the s.p. structure of A.
It will be rather easy to measure in inverse kinematics experiments.

MD of 11B of 12C(p,2p)11B in inverse kinematics
A-frame (A is at rest)

KB

K1

z

K0

A
5

5

0
0

200
0

-400

-200

0

200

ds / dKBz [mb / (MeV/c)]

×10-4

ds / dKBb [mb / (MeV/c)]

K2
Longitudinal (LG)
40

20

0
-400

-200

0

200

Transverse (TR)

60
40
20

400

KBz (MeV/c)

The 2-dim. MD directly reflects the proton s.p. structure in 12C.

0

0

200

KBb (MeV/c)

400Input for MD calc. of 12C(p,2p) in A-frame & inv. kin.
sample3.cnt

A-frame (V-frame forinv. kin.)inverse kinematics
MD calc.
KBz in the A-frame isvaried from -2.0 to
2.0 with the step sizeof 0.05 (unit is fm-1).
KBb is varied from0 to 2.0 with the step
size of 0.05.

output files for the
1-dimensional MDs

The unit of 1dimMD is mb/(MeV/c)
These lines haveno meaning when
IVAR=9You must makethese #s finite (15is enough in usualcases).

MD of 11B of 12C(p,2p)11Bg.s.@100 MeV/u in inv. kin.
A-frame (A is at rest)

KB

K1
A
×10-4

5

0
0

200
0

-400

-200

0

200

ds / dKBb [mb / (MeV/c)]

K2
ds / dKBz [mb / (MeV/c)]

5

z

K0
Longitudinal (LG)
40

20

0
-400

-200

0

200

KBz (MeV/c)

400

Transverse (TR)

40

20

0

0

200

KBb (MeV/c)

The asymmetric LG-MD is due to the phase-volume & attractive distortion effects.

400What is the phase volume (PV)?
Answer: The size (volume) of the phase space (here, momentum space) that cansatisfy the energy and momentum conservation. It depends on the choice ofindependent variables.
Infinitesimal cross section defined in the p-A c.m. frame (the starting point)

Lorentz invariance of the 4-dim. delta function and dK/E for each particle
Superscript A means the A-rest frame

Let us decide to chooseandas independent variables. Our aim is to calculateCalculation of the PV
We perform an integration over wh. By the mom. cons. it is fixed at
momentum transfer to B

Infinitesimal cross section for which the mom. cons. is satisfied (in the A-rest frame)PVCalculation of the PV (Con’t)
One can perform the integration overWe implicitly assume thatby using:

satisfying the energy cons. is unique and is written as
Experimental condition
Independent variables
Fixed quantities

for simplicity

Calculation of MD in the A-rest framePhase Volume
×10-4

5

5

392 MeV

×10-4

5

0

5

100 MeV

0

0

200
0

-400

-200

0

0

200

200
0

-400

-200

0

200

Role of the PV
A-frame (A is at rest)
Positive KB//

KB

K1

Ep + En : must be large

K2

A

K1 + K2 = K0 - KB

K0

KB

K1
Negative KB//

Ep + En = E0 - EB

Ep + En : can be small

K2
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

PV effect on MD at 100 MeV (PWIA)
2

1.5
2

1

1.5

0.5

1

0

0.5

200

0

1
1

0

0

200
0

-400

-200

0

200

0p3/2 proton MD in 12C

×10-4

0

20
15

20

10

15

5

10

0

NOTE:
Energy dependence of the NN t-matrix isdisregarded for a transparent interpretation(w/ IELM=0).

5
0

200
0

-400

-200

0

200

-400

-200

0

Phase Volume200

PV effect on LG-MD at 100 MeV (PWIA)
w/ PV
ds / dKBz [mb / (MeV/c)]

w/o PVds / dKBz [a.u.]

20

10

0
-400

-200

0

200

400

150
100
50
0
-400

KBz (MeV/c)

-200

0

200

400

KBz (MeV/c)

The PV effect gives a cut on the high-mom side.
This effect becomes large at low energies and/or for deeply bound nucleons.
KO, K. Yoshida, K. Minomo, PRC 92, 034615 (2015).

Attractive distortion effectA-frame (A is at rest)

K1

K0

30

PW
PW+V

100

50

0
-400

-200

0

200

KBz (MeV/c)

400

ds / dKBz [mb / (MeV/c)]ds / dKBz [mb / (MeV/c)]Longitudinal (LG)
150

B

K2
Longitudinal (LG)
PW+W
DW
20

10

0
-400

-200

0

200

KBz (MeV/c)

Transverse (TR)

50

400

ds / dKBb [mb / (MeV/c)]

A

KO, K. Yoshida, K. Minomo,
PRC 92, 034615 (2015).

PW+W
DW
25

0

0

200

KBb (MeV/c)

400Input for checking the attractive distortion effectsample4.cnt

Incident energy is 100
MeV (/u)Distorted waves arecalculated with orbitalang. mom. L up to 30.

A very simplified NN
cross section is usedby setting IELM=0

Now you need a p12C potential at 100
MeV, which can be
downloaded fromthe web.

Change these values
if you want to control
real and imaginaryparts of opticalpotentials

Attractive distortion effect
HW: Draw the figures below. Note that the DWIA calculations on the MD with thecurrent setting take time; on RCNP HPCI (miho), it takes roughly two hours.

PW
V only

100

50

0
-400

-200

0

200

KBz (MeV/c)

400

Longitudinal (LG)W only
DW

20

10

0
-400

-200

0

200

KBz (MeV/c)

400

Transverse (TR)50

ds / dKBb [mb / (MeV/c)]

150

30

ds / dKBz [mb / (MeV/c)]ds / dKBz [mb / (MeV/c)]Longitudinal (LG)W only
DW

25

0

0

200

KBb (MeV/c)

400

MD of EB in the Glauber model
K. Hencken, Bertsch, and Esbensen, PRC 54, 3043 (1996).

c
n

K0

k
c

K
n

-k

(AD + forward scattering
approximation)

・Taking a Jacobi representation results in the simplest form of the PV.
・small w-q is assumed, with neglecting the E-conservation.
Integration over K⊥ in the whole region

・MD in the A-frame if the mom. of the c-n c.m. is 0.

Application of DWIA to SEASTAR data analysis
Y. Sun+, PLB 802.

54Ca(p,2p)53K

80Zn(p,2p)79Cu: L. Olivier+, PRL 119.
77Cu(p,2p)76Ni: Z. Elekes+, PRC 99.
79Cu(p,2p)78Ni: R. Taniuchi+, Nature 569.
54Ca(p,pn)53Ca: S. Chen+, PRL 123.
54,52Ca(p,2p)53,51K: Y. Sun+, PLB 802.
63V(p,2p)62Ti: M. L. Cortés+, PLB 800.
70,72,74Ni(p,2p)69,71,73Co: T. Lokotko+. PRC 101.

and more …

1s1/2 (837 keV)

DWIA analysis of RIBF/RCNP data
S. Kawase+, PTEP 2018, 021D01 (2018).

with T > 30 MeV, 20 deg. < q < 65 deg., and |f| < 15 deg.

A puzzle
12C(p,2p)11B
g.s. at 392 MeV

q2 (deg)

DWIA triple-differential cross sections (TDX),
are integrated over E1, W1, and W2, and comparedwith the GSI data.
V. Panin+, PLB 753, 204 (2016).

Pikoe for the GSI data
local + w/o
Moller

2.94

w/oMoller
2.68

0.34

0.31

0.32

0.28

DWIA

local

3/2- g.s.

3.36

1/2- 2.13 MeV
3/2- 5.02 MeV

2.33

PLB757(GSI)
2.11

PPNP96(RCNP)
1.82(3)(e,e’p)1.72(11)

0.27

0.25

0.26

0.30(2)0.26(2)

0.25

0.21

0.21

0.23(3)0.20(2)

A puzzle

HW: Try to get these numbers.

DWIA triple-differential cross sections (TDX),
are integrated over E1, W1, and W2, and comparedwith the GSI data.
V. Panin+, PLB 753, 204 (2016).

Pikoe for the GSI data
local + w/o
Moller

2.94

w/oMoller
2.68

0.34

0.31

0.32

0.28

DWIA

local

3/2- g.s.

3.36

1/2- 2.13 MeV
3/2- 5.02 MeV

2.33

PLB757(GSI)
2.11

PPNP96(RCNP)
1.82(3)(e,e’p)1.72(11)

0.27

0.25

0.26

0.30(2)0.26(2)

0.25

0.21

0.21

0.23(3)0.20(2)

Plan of this talk (2/2)
4) Some theoretical achievements (for future)
K. Yoshida, M. Goméz-Ramos, KO, and A. M. Moro, PRC 97, 024608 (2018).

4-1. Microscopic optical potential
4-2. Benchmark study on 15C(p,pn) with DWIA, TC, and Faddeev,-AGS.

5) Divergence of the TDX in inverse kinematics
KO+, in preparation.

5-1. Two-value feature of the kinematics and divergence of PV
5-2. When occurs?

6) Some recent/ongoing KO reaction studies around RCNP/RIBF
6-1. 2n correlation study via (p,pn)
6-2. a KO reactions
6-3. deuteron KO reactions…

7) Summary

From phenomenology to microscopic theory

N

vi

A

N
i th

+

i th

…
i th

+ … =

A

A
1 step+ … +N

2 stepN stepA

Multiple scattering theory (MST)
N

vi

A

N
i th

+

+ … +

i th

A
1 step

N

…
i th

+ … =

A
2 stepN stepResummation(for all boundary conditions)(for a specific b. c.)

L. L. Foldy, Phys. Rev. 67, 107 (1945); K. M. Watson, Phys. Rev. 89, 115 (1953).
A. K. Kerman, H. McManus, and R. M. Thaler, Ann. Phys. (NY) 8, 551 (1959).
M. Yahiro, K. Minomo, KO, and M. Kawai, PTP 120, 767 (2008).
Extension to nucleus-nucleus scattering

The folding model potential based on the MST
An “expectation value” of a nucleon-nucleon (NN) effective interaction
NN t-matrix (or g-matrix)

Nucleonr

OT

rT

one-body density

TargetMicroscopic description of nucleon-nucleus scattering
proton at 65 MeV
in-medium veff
free veff

neutron at 65 MeV
in-medium veff
free veff(x106)(x104)(x10-2)(x102)(x10-5)

No free parameter (“prediction”)

cf. K. Amos+, Adv. Nucl. Phys. 25, 275 (2000).
T. Furumoto+, PRC 78, 044610 (2008).
M. Toyokawa+, PRC 92, 024618 (2015).

Benchmark with Transfer to the Continuum model
K. Yoshida, M. Goméz-Ramos, KO, and A. M. Moro, PRC97, 024608 (2018).
15C(p,pn)14C
g.s. at 420 MeV/u

15C(p,pn)14C
g.s. at 420 MeV/u

TC justifies the impulse approximation (use of tNN, with choosing NN kinematics
according to the two asymptotic nucleon momenta and including the Moller factor)
DWIA justifies fixing the optical potentials of outgoing nucleons at one energy.

DWIA vs. Faddeev-AGS
R. Crespo+, PRC77, 024601 (2008); PRC90, 044606 (2014).

Plan of this talk (2/2)
4) Some theoretical achievements (for future)
K. Yoshida, M. Goméz-Ramos, KO, and A. M. Moro, PRC 97, 024608 (2018).

4-1. Microscopic optical potential
4-2. Benchmark study on 15C(p,pn) with DWIA, TC, and Faddeev,-AGS.

5) Divergence of the TDX in inverse kinematics
KO+, in preparation.

5-1. Two-value feature of the kinematics and divergence of PV
5-2. When occurs?

6) Some recent/ongoing KO reaction studies around RCNP/RIBF
6-1. 2n correlation study via (p,pn)
6-2. a KO reactions
6-3. deuteron KO reactions…

7) Summary

12C(p,2p)11B

g.s. at 392 MeV/u

L-frame1
0

2
A

V-frameE1 = 251 MeV
q1 = 32.5 deg.
f1 = 0 deg.

B

q2 : varied
f2 = 180 deg.

E1 = 129 MeV
q1 = 50.7 deg.
f1 = 0 deg.

q2 : varied
f2 = 180 deg.

200

E1 = 251 MeV
q1 = 32.5 deg.
f1 = 0 deg.

1Very peculiar behavior of TDX
100

10

0
20

d3s / dE1dW1dW2(mb sr-2MeV-1)

g.s. at 392 MeV/u (PWIA)

40

80 q2 (deg)

60

10
5
0

5

10

15

20

25

30

35 q2 (deg)d3s / dE1dW1dW2(mb sr-2MeV-1)

d3s / dE1dW1dW2(mb sr-2MeV-1)

12C(p,2p)11B

1

10 B
0

10

-1

10

-2

10

-3

20

2

q2 : varied
f2 = 180 deg.

E1 = 129 MeV
q1 = 50.7 deg.
f = 0 deg.
25 130
q2 (deg)
q2 : varied
f2 = 180 deg.

35

Input for TDX calc. of 12C(p,2p) in inv. kin.

sample5.cnt

output in the V-frame

The direction of thez-axis is inverted.

When you investigate
the correspondence
between the forward
and inverse kinematics,
we recommend to use
ICTREIN=1.
Usually it is better
to use a smaller
step size of q in
inverse kinematics.
HW: You can directly control the kinetic variables in inverse kinematics measurement by putting IKIN=1,
IFRM=0, and IMIR=0. Reproduce the result in the lower panel on the previous slide in that way.

12C(p,2p)11B

E2 (MeV)

130g.s. at 392 MeV/u
1

125

120

115

2
B
0

20

40

60

80

q2 (deg)

100

600

E2 (MeV)

E1 = 251 MeV
q1 = 32.5 deg.
f1 = 0 deg.

divergence point400

q2 : varied
f2 = 180 deg.

E1 = 129 MeV
q1 = 50.7 deg.
f1 = 0 deg.

200

0

0

5

10

15

20

q2 (deg)

25

30

35

q2 : varied
f2 = 180 deg.

Plan of this talk (2/2)
4) Some theoretical achievements (for future)
K. Yoshida, M. Goméz-Ramos, KO, and A. M. Moro, PRC 97, 024608 (2018).

4-1. Microscopic optical potential
4-2. Benchmark study on 15C(p,pn) with DWIA, TC, and Faddeev,-AGS.

5) Divergence of the TDX in inverse kinematics
KO+, in preparation.

5-1. Two-value feature of the kinematics and divergence of PV
5-2. When occurs?

6) Some recent/ongoing KO reaction studies around RCNP/RIBF
6-1. 2n correlation study via (p,pn)
6-2. a KO reactions
6-3. deuteron KO reactions…

7) Summary

6He(p,pn)

Proving 2n in 6He via (p,pn) (in inverse kinematics)
Y. Kikuchi, KO, Y. Kubota, M. Sasano, and T. Uesaka, PTEP 2016, 103D03 (2016).

250 A MeV

6Hep

(p,pa)20Ne(p,pa) at 101.5 MeV
K. Yoshida, Y. Chiba, M. Kimura, Y. Taniguchi, Y. Kanada-En’yo, and KO, PRC 100, 044601 (2019).

RWA obtained by AMD (20-body calculations)

DWIA

Other systems:
120Sn(p,pa): K. Yoshida, K. Minomo, and KO, PRC 94, 044604 (2016).

10Be(p,pa): M. Lyu, K. Yoshida, Y. Kanada-En'yo, and KO, PRC 97, 044612 (2018).

12Be(p,pa): M. Lyu, K. Yoshida, Y. Kanada-En'yo, and KO, PRC 99, 064601 (2019).

(p,pd)Experimental fact
C. Samanta+, RRC 26, 1379 (1982).
16O(p,pd)14N
1+(3.95) at 101.3 MeV

C. Samanta+, RRC 34, 1610 (1986).

16O(p,2p)15N
gnd at 101.3 MeV

Remarks on pn knockout
deuteron breakup/formation during
the emission described by CDCC
pn pair distribution calculated with EDF
in collaboration w/ K. Yoshida (Kyoto)

The elementary process can be either
knockout “d” or breakup the pn pair.
A pickup type of (p,pd) can also be considered (NP1912-SAMURAI53)

Pairing strength vs. TDX
Y. Chazono, K. Yoshida, K. Yoshida, and KO, arXiv:2007.06771

・The peak height of the TDX clearly reflects the pn pairing strength.
・The deuteron breakup is neglected.
・The elementary process is assumed to be the pd elastic scattering.

Breakup effect of the emitted deuteronY. Chazono, K. Yoshida, and KO, in preparation.

・The deuteron breakup effect is very large.
・A naïve pn single-particle wave functionis adopted.
・The elementary process is assumed to be
the pd elastic scattering.

4n and 7H

4n and 7HApproved as an RCNP COREnet programwith Hiyama-san (Kyushu U / RIKEN )5-body CDCC

8He

8He(p,pa): NP1406-SAMURAI 194n6-body CDCC

8He
7H (= t + 4n)8He(p,2p): NP1512-SAMURAI 34

Summary1) (p,pN) is a powerful tool for investigating proton/neutron s.p. structure of stableand unstable nuclei. Determination of the S-factor is, however, not so trivial evenfor stable nuclei via kinematically complete measurement in forward kinematics.
2) Momentum distribution (MD) is a key observable in inverse kinematics. Its shapeis asymmetric in general because of the asymmetry in the kinematics. The phasevolume and attractive distortion effects are responsible for the asymmetric MD.
3) The triple differential cross section (TDX) diverges in some kinematical conditionsin inverse kinematics. It happens when the solution to the energy conservation is adouble root. Although an integrated cross section becomes finite, the TDX issignificantly enhanced around the divergence point, which is nothing to do with thes.p. structure of nuclei.
4) 2n correlation, a clustering, eff. polarization of residue, deuteron-like pn pair (andpn tensor correlation), 4n, and 7H are under investigation via knockout reactions.