Determination the Determination the FlipFlip and and Non-FlipNon-Flip parts parts of of npnp -- elastic scattering at 0 degree elastic scattering at 0 degree over the energy region 0.55 – 2.0 GeVover the energy region 0.55 – 2.0 GeV

and two interpretations of this process asand two interpretations of this process as np→np(π – θ) and np→pn(θ)

R.A. Shindin

• History of this matter and History of this matter and LuboshitzLuboshitz remark remark• Mutual transformations between the scattering matrixes Mutual transformations between the scattering matrixes

М М npnp––npnp((ππ – – θθ)) and and М М npnp––pnpn((θθ)) using using WolfensteinWolfenstein amplitudes amplitudes• Transition toTransition to the the Goldberger-WatsonGoldberger-Watson amplitudes and amplitudes and

determination thedetermination the FlipFlip and and Non-FlipNon-Flip parts of parts of npnp -- elastic elastic scattering in these both casesscattering in these both cases

• New “Delta-Sigma”New “Delta-Sigma” experiment data for the rations experiment data for the rations RRdpdp, , rrnf/flnf/fl andand good agreement with the Phase-Shift-Analysis forgood agreement with the Phase-Shift-Analysis for the the Charge-Exchange Charge-Exchange processprocess

RRdpdp

nfl fl1 1 1 3np np

d d dnd p nn F t F tdt dt dt

dp

nfl/f

fl

fl nfl2 2 1 3 3 1

np

np np

dd nd dtdtd np d ddt dt dt

lrR

At the impulse approximationAt the impulse approximation thethe differential cross section differential cross section of of nndd →→ pp((nnnn)) c can be expressedan be expressed as a as a Dean formulaDean formula using using

spin flipspin flip and and spin non-flipspin non-flip contributioncontribution ofofcharge exchange charge exchange nnpp →→ ppnn processprocess::

ForFor RRdpdp - ration we have - ration we have::

fl

023

lim 1 npt

d dF t nd p nndt dt

V.L. Luboshitz remarkV.L. Luboshitz remark

The The Dean Dean formula have been obtainedformula have been obtainedfor small momentum transferfor small momentum transfer

when the scattering angle when the scattering angle θθ closes to 0. closes to 0. And for the calculation the And for the calculation the RRdpdp rationration

we can use the amplitudeswe can use the amplitudesof the Charge Exchange only!of the Charge Exchange only!

--------------------------------------------------------------------------------------------------------------------------------------------------------V.V.Glagolev, V.L.Luboshitz, V.V.Luboshitz, N.M.PiskunovV.V.Glagolev, V.L.Luboshitz, V.V.Luboshitz, N.M.Piskunov

CHARGE-EXCHANGE BREAKUP OF THE DEUTRONCHARGE-EXCHANGE BREAKUP OF THE DEUTRONWITH THE PRODUCTION OF TWO PROTONSWITH THE PRODUCTION OF TWO PROTONSAND SPIN STRUCTURE OF THE AMPLITUDEAND SPIN STRUCTURE OF THE AMPLITUDE

OF THE NUCLEON CHARGE TRANSFER REACTIONOF THE NUCLEON CHARGE TRANSFER REACTION

wrong approachwrong approachwhich used amplitudeswhich used amplitudes

of of nnpp--nnpp(180)(180)

N.W. Dean: Phys. Rev D 5 1661N.W. Dean: Phys. Rev D 5 1661N.W. Dean: Phys. Rev D 5 2832N.W. Dean: Phys. Rev D 5 2832

RRdp dp ration of yieldsration of yields ndnd →→pp((nnnn) and ) and npnp →→pnpnMeasurement of neutron-proton spin obsevables at 0°Measurement of neutron-proton spin obsevables at 0°using highest energy polarized d, n probesusing highest energy polarized d, n probes

------------------------------------------------------------------------------------------------------------------------------L.N. StrunovL.N. Strunov et al.: Czechoslovak Journal of Physics, Vol. 55 et al.: Czechoslovak Journal of Physics, Vol. 55

(2005)(2005)

npnp interaction in the c.m.s. interaction in the c.m.s.

__

__

k

k`n

pn

p

__

__

k

k`

n

pn

p

n

pn

p

n

pn

pElastic backwardElastic backwardCharge Exchange processCharge Exchange process

These both cases These both cases havehave

identical cinematicidentical cinematicand therefore and therefore

can`t be separated can`t be separated using experimentusing experiment

Spin-Singlet scattering – SSpin-Singlet scattering – S == 00

Variant 1.Variant 1.Initial and outgoing neutronsInitial and outgoing neutronshave have parallelparallel spin projection spin projection

Variant 2.Variant 2.Initial and outgoing neutronsInitial and outgoing neutronshave have antiparallelantiparallel spin projection spin projection

INTERPRETATIONINTERPRETATION

Elastic backward Charge ExchangeElastic backward Charge Exchange Non-FlipNon-Flip Spin-FlipSpin-Flip

INTERPRETATIONINTERPRETATION

Elastic backward Charge ExchangeElastic backward Charge Exchange Spin-FlipSpin-Flip Non-FlipNon-Flip

This anti-symmetry (between the definition of the This anti-symmetry (between the definition of the spin-flip spin-flip and and spinspin non-flipnon-flip parts at the S=0 station) parts at the S=0 station)

shows the possible reason for the difference between shows the possible reason for the difference between these interpretations of the np-interaction. Though the these interpretations of the np-interaction. Though the differential cross section should be same, these both differential cross section should be same, these both cases will can unequaled for the cases will can unequaled for the flipflip and and non-flipnon-flip..

NNNN formalism formalism ( (Lapidus, Bilenky, RyndinLapidus, Bilenky, Ryndin))

1 M M M nn nnpp pp

1 01 2

M M M M p p p pn n n n

1 01 2

M M M M n n n np p p p

1 2 1 20 1

1 3, , ,

4 4M k k M k k M k k

1 2

1 2 1 2

1 2 1 2

1 2

1

21

2

,

T

T

T T

T

T

C n n

G m m l lM k k B S T

H m m l l

N n n

n np p

1 2 1 2 1 11 34 4

S T , , k k k k k kn m lk k k k k k

GeneralGeneral view of the view of the NN NN scattering scattering matrixmatrix

If both If both nucleonsnucleons are are identicalidentical then then

For the For the npnp Elastic Elastic scattering we have scattering we have

For theFor the ChargeCharge ExchangeExchange WolfensteinWolfenstein amplitudes amplitudes BB,, CC,, GG,, HH,, NN are the are the complex functions complex functionsof the interacted particles of the interacted particles EnergyEnergy andand the scattering the scattering angle angle θθ..

n

m

l

k

k̀k-k`

k+k`

k

k̀ *

Charge-ExchangeCharge-Exchange np→pn(θ)

1 2

1 2 1 2

1 2 1 2

1 2

1

21

2

, +

CEXT

CEXTCEX

T TCEXT

CEXT

C n n

G m m l lM k k B S T

H m m l l

N n n

n np p

k

k̀k-k̀k+k̀

k k

`*

n

m

l

As a analogy for this process we can writeAs a analogy for this process we can writethe the general general view of the scattering view of the scattering matrixmatrix as follows as follows

However here we have the other amplitudesHowever here we have the other amplitudesBBCEXCEX, , CCCEXCEX, , GGCEXCEX, , HHCEXCEX, , NNCEXCEX

which belong to the which belong to the Charge ExchangeCharge ExchangeThe vectors The vectors ((nn,,mm,,ll)) form the form the newnew coordinates coordinates systemsystem also also

For comparisonFor comparison the the nnpp––nnpp((ππ –– θθ) ) and and nnpp––ppnn((θθ)) interpretations among themselves interpretations among themselves we should obtainwe should obtain

the the matrixmatrix MMnnpp––ppnn((––k`k`,,kk)) from the from the matrix matrix MMnnpp––nnpp((k`k`,,kk) ) directlydirectlySuchSuch representation of the representation of the Charge ExchangeCharge Exchange matrixmatrix

is absolutely is absolutely uselessuseless for our purposes for our purposes

Accordingly to the Accordingly to the antisymmetry antisymmetry ofof two fermions two fermions wave functionwave functionrelative to the total relative to the total permutationpermutation, including permutation of scattering, including permutation of scattering

vector (vector (k`k`→ –→ –k`k` )), permutation of , permutation of spinspin and and isotopic-spinisotopic-spin ((nn↔↔pp) ) we definewe define

1,2 1,2 , ,n pP P k k k k

1,2 1 21

2

1P

1,2 1 21

2

1P

1,2 1,2, ,M k k P P M k k n n n pnp p p

1 2

1 2 1 2

1 2 1 2

1 2

1

21

2

,

T

T

T T

T

T

C n n

G m m l lM k k B S T

H m m l l

N n n

p pn n

d ddt dt

pn np p pn n

Operators Operators PP1,21,2((σσ)) and and PP1,21,2 ( (ττ)) are the are the unitaryunitary and and HermitianHermitian

therefore the therefore the differential cross sectionsdifferential cross sections for the for the npnp-elastic -elastic and for the and for the Charge-ExchangeCharge-Exchange are equalare equal

Goldberger-WatsonGoldberger-Watson amplitudes representation amplitudes representation

(1) (2) (1) (2) (1) (2) (1) (2), n n n n m m l lM k k a b c e f

1

41

4

1

41

4

3

2

2

a B G N

b N B G

c C

e G H B N

f G H B N

1

41

4

1

41

4

3

2

2

CEX

CEX

CEX

CEX

CEX

a B G N

b G B N

c C

e N H B G

f N H B G

2 22 2 2 2d a b c e fdt

Goldberger-WatsonGoldberger-Watson amplitudes representation amplitudes representation

1

21

2

1

21

2

CEX CEX CEX CEX

CEX CEX CEX CEX

CEX

CEX CEX CEX CEX

CEX CEX CEX CEX

a a b e f

b a b e f

c c

e a b e f

f a b e f

1

21

2

1

21

2

CEX

CEX

CEX

CEX

CEX

a a b e f

b a b e f

c c

e a b e f

f a b e f

If scattering angle θ equal 0°, then:

0 CEX CEX CEXc c b f b e

SS

ST

Non-Flip

Flip

0

Ba

CEXa

SS

ST

Non-Flip

Flip

0

Ba

CEXa

CEX a

1 01 2

M M M pn np

T=0T=0

It does not give anything newand will be simple retelling

IfIf suddenly occurred that amplitudes suddenly occurred that amplitudes a and and aCEXCEX are are equalequalthen the coordinates systems would coincidethen the coordinates systems would coincide

and the and the Non-FlipNon-Flip would be would be precisely equal precisely equal the the SSSS amplitude amplitude

CEXCEX

T=0T=0

RRdpdpFor calculation the For calculation the RRdpdp energy dependence energy dependencethe the PSAPSA solutions solutions VZ40VZ40, , FA91FA91, , SP07SP07 from from

SAID DATA BASE was usedSAID DATA BASE was usedsaid@gwdac.phys.gwu.edusaid@lux2.phys.va.gwu.edu

The values of the The values of the Charge ExchangeCharge Exchange amplitudes amplitudesat the at the θθ = 0° have been obtain = 0° have been obtain fromfrom the the

npnp --Elastic backwardElastic backward amplitudes amplitudesusing presented formulasusing presented formulas

The experimental The experimental Delta SigmaDelta Sigma points points of of RRdpdp are the directly relation of yieldsare the directly relation of yieldsof of ndnd→→pp((nnnn) ) and and npnp→→pnpn process process

r r nfl/flnfl/fl

2 1 13

nfl / fl

dpr

R

The ratio The ratio r r nfl/flnfl/fl is defined as follows is defined as follows

non flip flip

np pn np pn

d ddt dt

nfl / flrTeoretical values from Teoretical values from PSAPSA

Experimental pointsExperimental points

CONCLUSIONCONCLUSION

• Using Dean formula and experimental Using Dean formula and experimental RRdpdp points wepoints we separated separated FlipFlip && Non-FlipNon-Flip parts parts for for npnp -- ElasticElastic scattering over the energy scattering over the energy region 0.55 – 2.0 GeVregion 0.55 – 2.0 GeV

• Good agreementGood agreement with with PSAPSA have been obtain have been obtain due to the transition to due to the transition to Charge ExchangeCharge Exchange