Photodisintegration of $^3H$ in a three dimensional Faddeev approach

0.06 MB·English

by  S. Bayegan

Photodisintegration of 3H in a three dimensional faddeev approach
S.Bayegan1,a,M.A.Shalchi1,andM.R.Hadizadeh2
1 DepartmentofPhysics,UniversityofTehran,P.O.Box14395-547,Tehran,Iran
2 Instituto de F´ısica Teo´rica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bl. II, Barra Funda,
01140-070,Sa˜oPaulo,Brazil
0
Abstract. Aninteractionofaphotonwith3HisinvstigatedbasedonathreedimensionalFaddeevapproach.
1
0 Inthisapproachthethree-nucleonFaddeevequationswithtwo-nucleoninteractionsareformulatedwithconsid-
2 eration of the magnitude of the vector Jacobi momenta and the angle between them with the inclusion of the
spin-isospin quantum numbers, without employing a partial wave decomposition. In this formulation the two
n
bodyt-matricesandtritonwavefunctionarecalculatedinthethreedimensionalapproachusingAV18potential.
a
Inthefirststepweusethestandardsinglenucleoncurrentinthisarticle.
J

3

1 Introduction position are two-body off-shell t-matrices, which depend
]
h onthemagnitudesoftheinitialandfinalJacobimomenta
t and the angle between them. Fachruddin et al. have cal-
- Sincetheearlydaysofthestudyofthenuclearphysicsso
cl manyeffortshavebeenperformedon3Nsystemsconsider- culatedtheNNboundandscatteringstatesina3Drepre-
u ingrealorvirtualphotoninteractions[1]-[2].Alsoseveral sentationusingtheBonn-BandtheAV18potentials[15]-
n studiesonthebehaviorof3Nboundstatesinrealorvirtual [16].Recentlytherehasbeeneffortstodothesamecalcu-
[ lation using chiral potential [23]. our aim in this work is
photon absorbtion have been reported[3]-[4]. Before six-

to formulate photodisintegration of 3H in a three dimen-
1 ties variational approach was used for these calculations
sional Faddeev approach. In the firststep we ignore three
v and works using this approach are still continuing. After
2 introducing Faddeev formulation for three body systems bodyforcesandwejustusethesinglenucleoncurrent.We
6 [5]-[6],neweffortsusingthisschemewerestarted.Asan willuseAV18potentialandtritonwavefunctionwhichhas
3 example one can point out the early calculations of elec- beencalculatedinourpreviouswork[20].
0 trodisintegration [7] and photodisintegration[8] with 3He Thismanuscripthasbeenorganizedasfollow:insec-
1. and3H.Animprovementinthephotodisintegrationcalcu- tion 2 we explain our basic states and we evaluate all of
0 lation of the bound and 3N continuum with the same 3N the matrix elements in these basis. In section 3 we intro-
0 hamiltonianhavebeenperformed[9].Therearealsoother duceoursingularityproblemanditssolution.Wefinishin
1 approaches to calculate electromagnetic interactions with section4withasummaryandoutlook.
:
v light nuclei such as Green-function-Monte-Carlo method
i [10],hypersphericalharmonicexpansions[11],andLorentz
X
integraltransform(LIT)method[12].Thereisaverygood 2 Integral equation of nuclear matrix
r
review of Faddeev calculations on the interaction of real
a elements without partial wave
or virtual photon with 3He [13]. In this work like previ-
decomposition
ous calculations the partial wave decomposition has been
used. In PW approach one should sum all PW to max-
imum angular momentum where the calculation is con- Tocalculatethephotodisintegrationobservablewefirstneed
verged. The problem is that in higher energies this max- tocalculatenuclearmatrixelementsintheFaddeevscheme.
imum angular momentum increases and we should solve FormoredetailsseeRef.[13].
morecomplicatedequations.Toavoidthiscomplexityone
should use vector momentum as basis states [14]. To this N = 1 φ (1+tG )PU (1)
0 0
aim in the past decade the main steps have been taken 2h | | i
byOhio-Bochumcollaboration(Elster,Glo¨ckleetal.)and
Bayeganetal.toimplementthe3Dapproachinfew-body U =(1+P)Jψ +tG PU (2)
0
boundandscatteringcalculations(seeforexamplesRefs. | i | i | i
[15]-[22]).Itshouldbeclearthatthebuildingblockstothe InaboveequationstisNNt-operatorwhichobeysLipmann-
few-bodycalculationswithoutangularmomentumdecom- Schwingerequations,G isfreepropagator,Pispermuta-
0
tion operator, ψ is three body bound state and U is an
| i | i
a e-mail:[email protected] axillarystate.Threebodyforceshavebeenignored. φ is
0
| i
EPJWebofConferences
asubsectionofthefullyantisymmetricfreestate, Φ ,in Where:
0
| i
whichnucleons2and3areinsubsystem. 1 1
π =q+ q π = q+q (10)
1 2 ′ 2 2 ′
Φ =(1+P)φ (3)
| 0i | 0i Nowwithrespecttoaboverelationandsymmetryconsid-
φ0 isalsoourbasicstatetosolvetheintegralequation erationswecanevaluateequations(7)and(8)asfollow:
| i
(2)andisantisymmetricunderpermutationofnucleons2
1 1 3 1
and3. N = p q, p qm m m ν ν ν U
2{h−2 − 4 − 2 2 3 1 2 3 1| i
φ0 pqm1m2m3ν1ν2ν3 a (4) 1p+ 3q, p 1qm m m ν ν ν U
| i≡| i h−2 4 − − 2 3 1 2 3 1 2| i}
In equation (4) p and q are jacobi momenta and m’s
1
andν’sarethespinandisospinoftheindividualnucleons + d3qa pm m ν ν t q+q,m m ν ν a
respectively. X Z h 2 3 2 3||2 ′ ′2 ′3 2′ 3′i
m m
Orthonormalityandcompletenessrelationsoftheseba- ′2 ′3
ν ν
sicstatescanbeconsideredasbellow: 2′ 3′
1 1
q q, q m m m ν ν ν U (11)
q2+q2+qq h−2 ′− ′ ′2 ′3 1 2′ 3′ 1| i
ahpqm1m2m3ν1ν2ν3|p′q′m′1m′2m′3ν1′ν2′ν3′ia E− ′m · ′
1
= δ(p p)δ δ δ δ pq, m m m ν ν ν U
2{ − ′ m2m′2 m3m′3 ν2ν2′ ν3ν3′ h 1 2 3 1 2 3| i
δ(p+p)δ δ δ δ δ(q q)δ δ (5) = pq, m1m2m3ν1ν2ν3(1+P)Jψ
− ′ m2m′3 m3m′2 ν2ν3′ ν3ν2′} − ′ m1m′1 ν1ν1′ h | |1i
+ d3qa pm m ν ν t q+q,m m ν ν a
X Z h 2 3 2 3||2 ′ ′2 ′3 2′ 3′i
m m
′2 ′3
X Z d3pd3q|pqm1m2m3ν1ν2ν3ia ν2′ ν3′
m m m 1 1
1 2 3 q q, q m m m ν ν ν U (12)
ν1 ν2 ν3 E q2+q′2+q·q′h−2 ′− ′ ′2 ′3 1 2′ 3′ 1| i
a pqm m m ν ν ν 1 (6) − m
× h 1 2 3 1 2 3|≡ Thefirstermintheequation(12)canbeevaluatedas:
Considering these properties we can rewrite the integral a pq, m m m ν ν ν (1+P)Jψ
1 2 3 1 2 3
equations(1)and(2)inourbasicstats. h | | i
= d3pd3q
N = 1a pqm m m ν ν ν (1+tG )PU mX,ν Z ′ ′
2 h 1 2 3 1 2 3| 0 | i a pq′,′m m m ν ν ν (1+P)Jpq, m m m ν ν ν a
1 h 1 2 3 1 2 3| | ′ ′ ′1 ′2 ′3 1′ 2′ 3′i
= 2ahpqm1m2m3ν1ν2ν3|P|Ui ×ahp′q′, m′1m′2m′3ν1′ν2′ν3′|ψi (13)
1 Nowweconcentrateontheelementsoftheseequationsi.e.
+ a pqm m m ν ν ν tG PU (7)
2 h 1 2 3 1 2 3| 0 | i current,two-bodyt-matrixandtritonwavefunction,more
precisely.
a pqm m m ν ν ν U
1 2 3 1 2 3
h | i
=a pqm m m ν ν ν (1+P)Jψ 2.1 current
1 2 3 1 2 3
h | | i
+a pqm m m ν ν ν tG PU (8)
1 2 3 1 2 3 0 Consideringthesymmetrypropertieswehave:
h | | i
The effect of permutation operator on our basic states a pq, m m m ν ν ν (1+P)JSNψ
1 2 3 1 2 3
h | | i
canbeconsideredasfollow: =3a pq, m m m ν ν ν (1+P)JSN(1)ψ
1 2 3 1 2 3
h | | i
pqm m m ν ν ν Ppqm m m ν ν ν (14)
h 1 2 3 1 2 3| | ′ ′ ′1 ′2 ′3 1′ 2′ 3′i
1 3 1 Matrixelementsofsinglenucleoncurrentcanbeevaluated
=δ(p+ p + q)δ(q p + q)
2 ′ 4 ′ − ′ 2 ′ asfollow:
δ δ δ δ δ δ
× m1m′2 1m2m′3 m33m′1 ν1ν2′ ν2ν3′ ν3ν1′1 ahpq, m1m2m3ν1ν2ν3|J(1)|p′q′, m′1m′2m′3ν1′ν2′ν3′i
+δ(p+ 2p′− 4q′)δ(q+p′+ 2q′) =δ(q′ q+ 2Q)
− 3
δ δ δ δ δ δ
× m1m′3 m2m′1 m3m′2 ν1ν3′ ν2ν1′ ν3ν2′ 1
=δ(p+π2)δ(p′ π1) ×2[δ(p−p′)δm2m′2δm3m′3δν2ν2′δν3ν3′
−
δ δ δ δ δ δ δ(p+p)δ δ δ δ ] J (Q,q)
× m1m′2 m2m′3 m3m′1 ν1ν2′ ν2ν3′ ν3ν1′ − ′ m2m′3 m3m′2 ν2ν3′ ν3ν2′ × m1 m′1
+δ(p π )δ(p +π ) ν ν
− 2 ′ 1 1 1′
δ δ δ δ δ δ (9) (15)
× m1m′3 m2m′1 m3m′2 ν1ν3′ ν2ν1′ ν3ν2′
19th InternationalIUPAPConferenceonFew-BodyProblemsinPhysics
InaboveequationQisthemomentumofphoton.Weneed
to rewrite the single nucleon current operator in a form
which is suitable for our basic states. The current opera- tπSt p,p,cosθ =VπSt p,p,θ
ΛΛ ′ ΛΛ ′
torwhichwewilluseis: ′ ′
(cid:0) 1(cid:1) (cid:0) (cid:1)
1
J =GE(Q)k12m+Nk′1 + 2miNGM(Q)σ×(k1−k′1) (16) + 2Z dp′′p′′2Z1 d(cid:0)cosθ′′(cid:1)vΛπS′1t,Λ(cid:0)p′,p′′,θ′,θ′′(cid:1)G0(cid:0)p′′(cid:1)
−
tπSt p ,p,θ
Which is summation of convection current and spin cur- 1Λ ′′ ′′
rent.G (Q)andG (Q)areelectricandmagneticformfac- (cid:0) (cid:1) 1
E M 1
torsrespectively.Fortheconvectionpartwehave: + dp p 2 d cosθ vπSt,Λ p,p ,θ ,θ G p
2Z ′′ ′′ Z ′′ Λ′0 ′ ′′ ′ ′′ 0 ′′
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
k1+k′1 =2q+Q+ 32K (17) t0πΛSt p′′,p,θ′′−1 (22)
(cid:0) (cid:1)
As we will show we have to choose coordinate system in Where
which the z axis is along the Q vector and we also need
tensor component of current so the second and the third 2π
tTehrmussfoofrtthheecroignhvtechtaionndcsuidrreenotfweequhaatvioen:(17) will vanish. vΛπS′Λt,′Λ′(p′,p′′,θ′,θ′′)=Z dφ′′e−iΛ(φ′−φ′′)VΛπS′Λt′′ p′,p′′
(cid:0) (cid:1)
0
q
Jco1nvec =GE(Q)m±1 (18) (23)
± N
Andthetensorcomponentofspinpartcanalsobeeval-
uatedas: 2.3 tritonwavefunction
√2Q For evaluating the Triton wave function we need to make
J±S1pin = −2mN GM(Q)S± (19) athreeolanteiownhbiecthwheaesnbteheinswcaalvceulfautendctiinonthienfooullrobwaisnigcbstaastiess[2t0o]:
1 1
2.2 two-bodyt-matrix pq(s )Sm (t )Tm ψ = pq,X ,αψ (24)
S T pq
h 2 2 | i h | i
Two body t-matrices can be related to the one which cal-
culatedinhelicitybasis: Wecanrelatethesestatestoourfreespinandisospinstates
withClebsch-Gordancoefficients.
1
ahpm1m2nν1ν2|t|p′m1m2ν1′ν2′ia = 4δ(ν1+ν2),(ν1′+ν2′) gγα = γα (25)
h | i
11 11
ei(Λ0φp−Λ′0φ′p)X(1−ηπ)C(22t,ν1ν2)C(22t,ν1′ν2′) Where
πst
11 11
C(22S,m1m2Λ0)C(22S,m′1m′2Λ′0) |αi=|(s21)SmS,(t21)TmTi
XΛΛ′ dΛS0Λ(θp)dΛS′0Λ′(θ′p)PeiN(φp−dφ′pΛS)′dΛNS(θΛp(θp′p))dNSΛ′(θ′p) It is very import|aγnit=to|mm1emn2timon3νt1hνa2tνt3hie spin of the nu(c2l6e-)
tπSt(p,p,cosθ ,z) (20)
ΛΛ′ ′ pp′ onsisquantizedinthedirectionofthezaxiswhichinthe
calculationofwavefunctionithasbeenchosentobeinthe
Intheaboverelationz = E 3q2 istheenergyofsubsys- directionofq.Butwehavetoconsiderthezaxisalongthe
− 4m
tem. As we know two-body function has a singularity in direction of incident photon Q. So we should first rotate
theenergyofdeuteron,z = Ed.Toremovethissingularity the spin of the nucleons in our basis to be settled in the
weshouldconsidert-operatorasfollow: direction of q axis. Then we should use Clebsch-Gordan
coefficients to obtain the wave function in the calculated
basismentionedintheequation(24):
a pm m nν ν tpm m ν ν a
h 1 2 1 2|| ′ 1 2 1′ 2′i
a pm m nν ν tˆpm m ν ν a pqm m m ν ν ν ψ
= h 1 2 1 2|| ′ 1 2 1′ 2′i (21) h 1 2 3 1 2 3| i
z−Ed = X XDm1m′1(θq,φq)Dm2m′2(θq,φq)Dm3m′3(θq,φq)gγα
Twobodyt-matrixinhelicitybasishasbeencalculated m′1m′2m′3 α
before[15]. pq,Xpq,αψ (27)
×h | i
EPJWebofConferences
3 Singularity problem q/2+q′′
dp pG¯(q,q ,p) dqˆ δ(x x )
Inordertoconsiderthesingularityproblemwecanrewrit- Z ′ ′ ′′ ′ Z ′′ ′′− 0
ten the equation (11) an (12) in a unified form ignoring |q/2−q′′|
isospindependentwhichissimilartospindependent. Um′2m′3m1(p′′πˆ2,q′′,Q)tˆma2m3m′3m′3(p,p′πˆ1,z) (31)
U (p,q,Q)=U (p,q,Q)
m1m2m3 m′1m2m3 IntheaboveequationG¯whichisalwayspositiveisdefined
+ d3q Um′2m′3m1(π2,q′′,Q)tˆma2m3m′3m′3(p,π1,z) as:
mX′2m′3Z ′′ E− q2+q′′2m−q·q′′ E+iǫ−Ed− 34qm2 G¯(q,q′′,p′)= 1 (32)
(28) E 3q′′2 + 1(p2+ 3q2)
− d− 4m m ′ 4
Tosolvethisintegralequationweshouldevaluatesingular- x = cosθ indicatestheanglebetweenqandq and
′′ ′′ ′′
ity in the denominator of the propagator which is a func- x isintroducedasfollow:
0
tion of q and angle between q and q. So instead of sin-
′′ ′′
gularpointwehavearegionofsingularityinq q plane. 1 1 1 1
There is a solution to this moving singularity i−n R′′ef.[24]. x0 = qq (p′2− 4q2−q′′2)= qq (p′′2− 4q′′2−q2)
′′ ′′
For using this method we have to put z axis along the q.
(33)
Butbecauseofsimplificationincurrentoperatorandfinal
cross section we should choose the z axis in the direction
ofthemomentumofthephoton,Q.Soinordertoevaluate
4 Summary and outlook
thesingularityweshoulduseanothermethodwhichisin-
troducedinRef.[25].Thereforeoneshouldseparateangle
partofdeltafunctionsasfollow: InthispaperwehaveformulatedtheFaddeevintegralequa-
tionsforcalculatingthephotodisintegrationobservableof
tritoninathreedimensionalapproach.Tothisaimwein-
δ(p π )δ(p π )
δ(p′+π1)δ(p′′−π1)= ′p−′2 1 ′p′′−′2 1 tirnodvueccteodrofourrmbsasaics wstaetlelsaswihnidchivicdounatlaisnpsinjaacnodbiismoospminenotaf
δ(pˆ +πˆ)δ(pˆ πˆ) each nucleon. So we have avoided to decompose angle
′ 1 ′′ 1
−
δ(p π )δ(p π ) states in terms of angular momentum states (partial wave
δ(p′−π1)δ(p′′+π1)= ′p−2 1 ′p′−2 1 approach)whichistraditionallyusedtosolvethesekindof
′ ′′ equations.Thefinalintegralequationsarelesscomplicated
δ(pˆ πˆ)δ(pˆ +πˆ) (29)
′− 1 ′′ 1 than the PW ones and are unique in number of the equa-
Andthentheintegralequationcanberewriteasfollow: tions in all energies. We have also explained about over-
comingofthemovingsingularityinourwork.
ThecalculationofthisobservableusingtheAV18po-
Um1m2m3(p,q,Q)=Um′1m2m3(p,q,Q) tentialisunderwayandtheresultswillbepublishedsoon.
Adding two and three body currents as well as three
+ d3q dpdp
Z ′′ ′ ′′ bodyforcesinourcalculationsareotherfuturemajorworks.
X
m′2m′3 The same calculation for radiative capture is also under
Um′2m′3m1(π2,q′′,Q) tˆma2m3m′3m′3(p,π1,z) consideration.
E− m1(p′′2+ 34q′′2)E+iǫ−Ed− 34qm2
(30) Acknowledgments
Aftersomesimplificationtheintegralequationtransforms
Thisworkwassupportedbycenterofexcellenceonstruc-
tothisequation:
tureofmatter,DepartmentofPhysics,UniversityofTehran.
U (p,q,Q)=U (p,q,Q)
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