Nuclear Science and Technology, Vol.4, No. 4 p 0 1 # , ^ . ' 3 8 - 4 4 .
Multipole expansion for the electron-nu'ete^s Vcsit^iHng at high energies in the unified electroweak theory
Luong Zuyen Phu "', Nguyen Huu Ha ^' (I) Vanlang Universily, Ho ChiMinh City
(2) University of Dalat, Lamdong
(Received 03 November 2014, accepted 20 March 2015)
"^i
Abstract: Tbe article presents the multipole e>g)ansion for the electron-nucleus scattering cross^' * section at hig^ energies within fiamework ofthe unified electroweak theory. The electroweak currents of the nucleus is expanded into simple components with definite angular momentum, whicb are called the multipole form &ctors. The multipole expansion ofthe cross-section is a consequence ofthe abov?
expansion. Besides the &miUar electromagnetic form factors F^ , there are new form fectors ) and A,^ related to weak interactions, coiresponding to the vector and axial weak currents. 1 obtaiMd general expressions are applied to the nucleus \Li, where the partial form ftfctorS^^J
n the multiparticie shell model.
L INTRODUCTION The method of studying the nuclear structure by electron scattering, or more generally, lepton scattering, is highly effective because it provides most detailed resiUts about tiie inner structure of the nuclei, especially when the attained electron energy becomes more and more higher.
In order to relate the stmcture of the nucleus with tiie scattered electrons, the best way is to expand flie scattering amplitude into multipole components, each term corresponds to a definite angular momentum Lm (of the interaction carrier). Weigert and Rose [1] were the first to perform a con^lete expansion when the interaction is purely electromagnetic, and the expansion was improved afterwards by Donnelly and Raskm [2]. Owing to ttus expansion, every partial amplihide corresponding to each multipole can be calculated in details for nuclei, and they clarify many properties ofthe nuclei.
At high energies tiiat tiie electton accelerators reached at present, of order GeV,
the electron-nucleus scattering must fe^
described by the unified electroweak theory..
We shall extent the Weigert-Rose metiiod te^
expand the scattering amplitude in this,(
For simplicity, we consider oitiy the soatil in which the nuclei are unoriented and 1 electrons are unpolarized.
n . THE ELECTRON-NUCLEUS SCATTERING AT HIGH ENERGIES; I
We shall consider the scattering electrons at high energies, of order GeV,oai\
nuclei. In order to apply the B(M»«
approximation, the target nuclei are supposes to be Ught or medium, i.e. they have the C1MII|^
number Z which is fairly less tiian Ma = islto At this energy scale, in the scattering M l electron exchanges a photon / and anj intermediate boson Z" witii the nucleus, lSs^
scattering amplitude is of flie fimn:
^. - ^[s'?'.«^{0+'iirr„(^,-^g,y,X(e)]O)
©2014 Vietnam Atomic Energy Society and Vietnam •*
-—-«
LUONOZUKB^PHU, wheic u is tbe (spinor) ^ate atapUiak ofthe electron, Q is die 4-nianiailiim transfo of election (to nudeas), Q ^K~K;= (,iD,q),K==
(« k) and i:'= («•, kO are Hie 4-inonKnlaof the electtDn before and after scatterjng lespectively, y ; ( g ) jj die electromagnetic euneot and J J ( g ) is tbe weak current of tbe nucleus.
X = .
J is die weak interaction constant, fl, is die Weinberg angle, bi die Weinberg-Salam model gv=-ll2 + 2xw, gi = -1/2, x r - sin'6V.
Tbe scattering cross section of die process is defined by;
(2)
where die summation 2 = S 2 " ^ ' y
>f trtt) itifi) die aveiage over die initial spin projection states and die summation over die ttoal spin projection states, which are performed for bodi lepton (i) and nucleus (fl), and we call briefly die summation. After die summation we obtain die results for die expression after die second equality m (2) as follows [3,4]:
« = RF+Ra + l (3)
* « " Y, JF'J'. (44) NGUYEN HUU HA
i>,=X'li^.Hf^XjfJ^. (4c)
srcn
Tbe quantities i ^ , l}^, i ^ are caUed die lepton tensois, and Hf, H^, Hf are nuclear tensors, which are also called die hadion tensors when considering die hadrons instead of nuclei. We see diat die quantities Rr, RFZ, RZ are die contiaction products of a lepton tensor widi a nuclear tensor. After performing the summation over lepton states V , we
Ef(£)
obtain:
il. - ^[r•{'*r^){'^.*t)r,{g.*s^,)[^^r,s)(m.*K)]
= SrL.^+g.,L;^, (54) ' i - Sp[r.(>+r/)(»..+«')r,(r,+s)(,^+ie)], (5c)
r.(^*r,s-](m,*k-)r,(g,^g^y,)
= ( « p + ? ' ) i , « + 2 g „ g . , i ; , . (5rf) Thus diere remains die summation over nuclear states 2|,whicb we sbaU perform m die
tf(H)
following.
m . MULTIPOLE EXPANSION FOR THE SCATTERING CROSS SECTION
Our multipole expansion for Rp essentiaUy concides widi die results of Weigert and Rose [1], but diere are some changes for being compatible with high energies [4]. At energies of order GeV, die contiibution of * ^ is of several percents 6om Rp, and die
MULTIPOLE EXPANSION FOR THE ELECTRON-NUCLEUS SCATTERING ...
contribution fixim liz is of the same order compared to Rfz, so Rz csn be neglected.
We employ i Cartesian coordinate system with the following unit vectors:
C; - a , e„ = kxk'. e , = fkxk')xq, where a = aflal,
i.e. the OZ axis is along the direction of the momentum transfer q, the OX axis lies on the scattering plane, and the OY axis is perpendicular to this plane. The axis along q is called longitudinal, two other axes are transverse. The next step is to change to the cyclic coordinate system
?. = e,, ;., _1_
"V2 (±e:,+ /ej.).
and we shall expand the electromagnetic and weak currents of the nucleus into multipoles m this cyclic system
^ q ) = Z V M 2 i + i ) r i ( ? ) ,
Lm
J ( q ) = £ V ' ' T ( 2 i + l)7Z,(?)?;,(p=0.±l),(6)
The coefficient T^^ is the Coulomb component of the expansion, three other coefficients r i ( p = 0, ±1) are composed of
Tt., ^'so denoted by T'^, which is called the longitudinal component, and two coefficients T'c corresponding to the transverse components, which can be written as 'rS.=~(.Tl±Tl^)l^, where T^ is the electric component and T^ is the magnetic ones. The inveree expressions of (6) are
TLii) = ' ' J p ( r ) S f . ( 9 , r W ' r ,
? ! ( ? ) = ' • " J J ( r ) . B L ( ? , r ) r f V ,
T^(.<!) = i'jJ(.r)Bl(g,r)d'r. (7)^'
where fif^, B'].„,Bf^ and B J ^ are the basfe4 multipole fields (of a vector field) [5], of Q^i Coulomb, longitudinal, electric and magielicl
types respectively, | We rewrite here the general form of tii^v
stnicture of the electromagnetic and weafe currents in the unified theory [6]:
j",= r'+A^ V = ^<X+#;>f^^i
where V° is the vector weak current. A" is fl»
pseudovector weak current, which is also '•
called the axial (weak) current, two subscripts at V° \k A" in (8a) and (86) express the isospia (0 and 1) together with their principal projections. In the Weinberg-Salam model we have p'^^= - Ixw, Pf= 1 - •h:w, Pf= 0.
jff^''^ ' Thus the expansions (6) and (7) correspond to tiuee currents: when J" = Jp then T^=F^, when r J^ flien
T^=ZtiZisVoTA).
Note that in quantum theory, the currents are operators, so tiie expressions (6) and (7) in fact are operators. For computing the scattering cross section, w e need to compute fust the transition amplitude of the current (6), or equivalentiy, of their multipole componente (7), between the initial |i) = \J^f) and final ^m'$
\J'M') states, i.e. tiie expressions ^•
{J'M'\J'']JM) o r ( / ' M ' | f £ ^ | y M ) . O n t i i e ;.
other hand, / ) r ^ is a spherical tensor, or IT a-M
J
LUONG ZUYEN PHU, NGUYEN HUU HA
multipole tensor, of dK Older in,, so according Y H " - ^ ! • * « U> die Wigner-Eckart dieorem in quantiuf " " ^ ^^cPrP.-u.JtJ', mechanics, die angular momentum projection i I • „ ^ \ „ 2 ,.„ . , ,, quantum nmnbeis M. tn, M- are g L r e d i " < ' ( ' ' ' ^ ' ^ - ^ ' ' ' ) - ^ 2 . V j ; j i + 2 ( o ; ) ( k . , togedier into a Clebsch-Goidan coefBcient, and
(1 the remaming ftctor will not contain diem; " (« + 0 [ / ' ; ( k . J ^ ) - l - ( k . J ^ " ) p ^ ]
{j'M-\f-\jM} = cz.{j'i\fniJ). (9) -(*,H')[-/r(kJi)+(kj^-)i]} •"""
'I The ftctor ( y Iff/II J ) is called die reduced , ^ r
(multipole) matrix elemem and will be denoted '-"" ' l,*'"'-W''^'-'-'''H*-'*) O ^ ) ] " ^ ' ^ ' y ' f = ( - f ' l | J f IU>,widi7-=F, C ^ a n d X rutting AT = 4/r(2/'+1)/(Z/-H 1) and usmg (6)
= C. II, £, M. The quantities F"' F ^ /(•' are """ ^ " ^ ° " " ^ ' " ' ' ' " ^ ^ "'"'='' = " P"='=°' ™ '• ' ^ ' i (14a 6)*
I also called electiomagnetic, vector and axial '
multipole form factors respectively. 2 Rf^PrPz = JfT/^'^f"^, iftH}
dp mni
^•i-divj = 0. (10) at
'/(ff)
Smce die cuirents y ; and (" are ^F^ ,„ ^, „
conserved, so diey satisfy die continuity nUi '^Z.'^L<'L.
equation ^^=P L. ^^(P'FA+J^PZ) = KY (F,^y;+F'p
i f ( » | ' ^ \ C L L
From it we derive diat die longinidinal
component is expressed m terms of die scaftr Z •'<'('<J?)(kJi) = -Kk^y(F'y'+F"y (Coulomb) component '"'" 2 • ^ > ' «
1 . = -FL, yl = - P i , . (II) - | ( * - ' 1 ' H J - . ' < J . ) = ' ^ ; Z ( « * « ) ( 1 5 )
The axial current is not conserved and ^ ^ " " ^ " ' " S ""ree terms give die does not satisfy die equation (10), so diere is " " P " " ' " " ^'^ "> ==«>•
not die relation Uke (11) for it. The results of calculation are Now we compute « „ . First of aU it is "fz = SMgvBi + g^B2) (16) easy to obtain [4];
C = <grX^^gJ^), (12) "' = '^Z^ff-Z-.f.*; 4 v ( « . ^ f ' , ' )
2 ^ « , y , = A r I [ « ' ^ K / + „ , ( F / K / + ^«K«)],(17») Kfi = '^afi^K''K"' (13 J
S — Ku' ^ {F^A'^ + F'^A^\
Next, we compute die contiaction product ~ r ^ ' ' ' ' • ' ' *''"''
le fi^vj-usDific-o ' MULTIPOLE EXPANSION FOR THE ELECTRON-NUCLEUS SCATTERING . where the coefficients uc, tfj, Wcj, "T, ^ "r
are called tiie kinematic coefficients, they
£^)peared when conqniting the lepton tensors L^„, /.L,, and have the form:
i = ^K &
= 2££'(l-A
2 ^ £ ^ ' ( 1 - X ' ) ,
= - 2 ( ^ ; + £r't) = -4-e£-'(l-x==),
& 2
: £-A:j[-f'^i -(£+f')e£-'x^.
j « , = J—'j;*(I-.t )>UoJ
where the coefficient W^ is present due to the application of (11).
In order to obtain the expression for Rp we use (4o), and get R^ =AX^Hf, after tiiat by using tiie computation (15) or tiie computed results (17), where we must substitute J^ —>J°, then
^Z
«c(^f)^+"„(F«)^-f«^,F,-f'll= ^l:{mF,y•^u,[iFff^F-f•
.(\9) This expression comcides witii tiie results of [1], but tiie ways of computing are different [7 8].The expressions (19), (16) and (17) computing RF and Rpj. achieve tiie expansion of tile scattering cross section into tiie quadric
forms of tiie mul%ole form foctors.
multipole form ^ t o r is a transiticm an^Iitu of the transition current con^onent definite angular momentum Lm, wheve I^M form factor in feet dqjends only on L, mrt c m. In principle the sums in above formulas a infinite, but the selection rules derived &om fl symmetries limit the quantity of remaining' terms, which becomes not only finite but also c rather few. ' f.
IV. MULTIPOLE FORM FACTORS , | The multipole form fectors of flifr ' nucleus are computed on the basis of somt ^ model on the nuclear structure. By putting tlie, ' obtained form factors into tiie scattering cross t- section formula, the comparison with experiments will give us a verification of th§ : model. Up to present, there have been many such verifications, but tiie verifications at h i ^ •'
energies in fact are few. ;•
In the foUowing we present an exan^ile on the computation ofthe form factors, relatel • to the nucleus jLi, and the scattering ^^' supposed to be elastic. This nucleus has spin i
= 1, so L = 0,1, 2. The selection rules allow fbs . existence of 8 following multipole fonn, factors: F^, F / , F^", V^, Vf, V,", 4,^\
AJ . Since tiie scattering is elastic so J'=J, iw have K = An. The contraction products in tfe cross section become
AIAK :
.{F,^)'].u,(Fr)\
BJAK = U^(F^V^+F;V^)+U,F,"V,''
BJAa = 4F,"^', (20) We see diat ,4," does not contiibute to dl6|
scattering cross section.
LUONG ZUYEN PHU, NQUYEN HUU HA On die ofter hand, die nucleus J i ; has
Z = JV, so die isospin projection m, = 0, which leads to >;; = 4« = 0 , because diese terms are proportional to ntr. Therefore we get
i.e. die weak current is proportional to the eleOromagnetic current, diis is a common characteristic of the nuclei having Z = N.
Furdiermore iu die Weinberg-Salam model we have ff = 0, so die nuclei with Z = # do not have die axial current A". This is a doubtfiil issue, diefe are evidences diat ;8™ * 0 and die above mentioned data become not taie. We hope to talk about diis topic in a report in die future.
WiUey [9] was die fnst to present die way of computing die multipole form factors using die many-particle dieory. The author started ftom die expression of current density which is die sum of die currents created by each nucleon, and averaged die interactions between diem. After diat he computed die sum hy considering die distribution of nucleons in die nucleus according to die shell model. The influence of individual nucleons on observable quantities is taken into account by introducing additional parameteis related to its charge and magnetic moment as follows:
y. = | ( i + O r , + i ( i - r . 3 ) y . . At low energies we have ^ = 1 , ^ = 0,;'
= 2,79, r. = -1,91. At high energies, we adjust the expression by substihiting die form factors
ofthe nucleon for above quantities; s, -> Gi^,
«, -> Ga, ;(, -> G^ y -> Gu,, i.e.
"•. ^ * . = | ( l + r 3 J G ^ - ^ i ( l - r • 3 J G ^ , n ^ y . = | ( I - ^ ^ , J G ^ - ^ i ( l ^ r 3 , ) G ^ . The nucleon form factors which were fitted in experiments are [10]
GEP = Go = ~,
GE„ = - 1,91 ;;Go/a +5,67), 7 = -&Aml, G^ = 2,19Go, GM, = -1,91GO,
nty = 0,71 GeV^, m/f-. nucleon mass. (22) The next step is to compute tiie multipole form fectors according to such a distribution, we obtain the following
<- r. —^ ^I-(L + l)MgrJYl (r J . o J
T S . ^/^A-, («<•.) V^-'(r.)
A/Vm., I ^•[+Vmj„(,^,)Y'"(r.)
HL+l) rV?fTA-,(?';)Y^-'(r.) 2 ''°\-^h.M)Yl'\r,)r° .(23)
By computing die tiansition matiix element and using the Wigner-Eckart rule of reduction, we get
MULTIPOLE EXPANSION FOR THE ELECTRON-NUCLEUS SCATTERING ...
f /
JA- (24)
J„ - (1 — X ) e " , X= (flAp= 26,589',
^ r = G ^ = (Gs.+Gay2, fG^+G^V2.
The weak form factors in this case are proportional to the electromagnetic form factors, as mentioned above, so there is no need to write them down.
V. CONCLUSION We performed the multipole expansion for the transition currents in the nucleus mto the multipole form factors and expressed the cross section in terms of them. These fectors are tiie reduced matrix elements of the multipole components which appear when expanding the transition currents. The multipole form factors are tiie simplest components of the transition currents, which can be computed directiy from the nuclear stiuchire raodels. Therefore flie method of multipole expansion allows us to obtain the more detailed information about nuclear Sttucttire, first of all tiie information which manifests only in the high energy processes. It also opens new perspectives in checking tiie unified electtoweak interaction model. The computatton of tiie multipole form factors for nuclei as well as the evaluation of the parameters of the electtoweak theory will become an extensive area of the nuclear structure theory.
One of tiie authors (L.Z.P.) gratefiilly acknowledges Professor K.A. Gridnev (Saint Petersburg Stale University, Russia) for helpfiil
discussions calculations.
while performing
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