Relativistic instant-form approach to the structure of two-body composite systems

26 Pages·0.46 MB·English

by  A. F. Krutov

RELATIVISTIC INSTANT–FORM APPROACH TO THE STRUCTURE OF
TWO-BODY COMPOSITE SYSTEMS
A.F.Krutov∗
Samara State University, 443011, Samara, Russia
V.E.Troitsky†
Nuclear Physics Institute, Moscow State University, 119899, Moscow, Russia
(January, 2001)
1
0
0 needs to take into account relativistic effects. Relativis-
A new approach to the electroweak properties of two–
2 ticeffectsareimportantalsointhetreatmentofcompos-
particle composite systems is developed. The approach is
ite systems of light quarks. In this case the relativistic
n based on the use of the instant form of relativistic Hamilto-
effects are significant even at low energy. However, the
a
nian dynamics. The main novel feature of this approach is
J relativistic treatment of hadron composite systems is a
thenew method of construction of thematrix element of the
9 rather complicated problem. To solve it one needs, in
electroweakcurrentoperator. Theelectroweakcurrentmatrix
2 fact, to solve a many–particle relativistic problem with,
element satisfies the relativistic covariance conditions and in
inaddition, notpreciselyknowninteraction. Letus note
the case of the electromagnetic current also the conservation
1 that the use of the methods of the field theory in this
lawautomatically. Thepropertiesofthesystemaswellasthe
v
case encounters serious difficulties. For example, it is
7 approximations are formulated in terms of form factors. The
well known that the perturbative QCD can not be used
2 approach makes it possible to formulate relativistic impulse
3 approximation in such a way that the Lorentz–covariance of in the case of quark bound states (see, e.g., [1,2]).
1 thecurrentisensured. Intheelectromagneticcasethecurrent In the present paper we will use the relativistic con-
0 conservation law is ensured, too. The results of the calcula- stituent model which describes the hadron properties
1 tions are unambiguous: they do not depend on the choice of at quark level in terms of degrees of freedom of con-
0 thecoordinateframeandonthechoiceof”good”components stituent quarks. The constituent quarks are consid-
/
h of thecurrentas it takesplacein thestandard form oflight– ered as extended objects , the internal characteristics of
p frontdynamics. Ourapproachgivesgoodresultsforthepion which (MSR, anomalous magnetic moments, form fac-
- electromagneticformfactorinthewholerangeofmomentum tors) are parameters of model. As relativistic variant of
p
transfers availableforexperimentsat presenttime,aswell as constituent model we choose the method of relativistic
e
h for lepton decay constant of pion. Hamiltonian dynamics (RHD) (see, e.g., [3–6] and the
: references therein).
v
Our aim is to construct a relativistic invariant ap-
i
X I. INTRODUCTION. proach to electroweak structure of two–particle compos-
r ite systems. The main problem here is the construction
a
All atoms, nuclei, and the main part of the so called of the current operators [7–11].
elementaryparticlesarecompositesystems. Thatiswhy It seems to us that RHD is the method the most ad-
the constructing of correct quantitative methods of cal- equate for our purpose. The use of RHD enables one to
culationforcomposite–particlestructureisanimportant separatethemaindegreesoffreedomandsotoconstruct
line of investigations in particle physics. In nonrelativis- convenient models.
tic dynamics there exist different correct methods which We use one of the forms of RHD, namely the version
use model or phenomenological interaction potentials. of the instant form (IF).
However,inthe caseofhighenergyoneneeds todevelop Ourapproachhasanumberoffeaturesthatdistinguish
relativisticmethods. Itisworthtonotethatnowtheex- it fromother formsof dynamics andother approachesin
periments on accelerators,in particular at JLab are per- the frames of IF.
formed with such an accuracy that the treatment of tra-
The electroweak current matrix element satisfies
ditionally ”nonrelativistic” systems (e.g. the deuteron)
•
the relativistic covariance conditions and in the
case of the electromagnetic current also the con-
servation law automatically.
∗Electronic address: [email protected] We propose a modified impulse approximation
†Electronic address: [email protected] • (MIA). It is constructed in relativistic invariant
1
way. ThismeansthatourMIAdoesnotdependon In the case of interacting systems the situation is dif-
the choice of the coordinate frame, and this con- ferent and the construction of the generators in terms
trasts principally with the ”frame–dependent” im- of dynamical variables encounters some difficulties. To
pulse approximation usually used in instant form make clear the sense of these difficulties let us compare
(IF) of dynamics. 1 RHD with the nonrelativistic quantum mechanics. In
nonrelativistic case it is the Galilean group that is the
Our approach provides with correct and natural
group of invariance. In the case of nonrelativistic inter-
•
nonrelativistic limit (”the correspondence princi-
acting system the interaction operator enters (in addi-
ple” is fulfilled).
tive way)onlythe generatoroftime translations(energy
operator). The interaction operator satisfies the usual
For composite systems (including the spin 1 case)
• conditionsofinvarianceunder translationsandrotations
the approach guarantees the uniqueness of the so-
and of independence on the choice of inertial coordinate
lutionforformfactorsanditdoesnotusesuchcon-
frame. Under these conditions the algebraicrelationsfor
cepts as ”good” and ”bad” current components.
thegeneratorsoftheGalileigroupwrittenintermsofdy-
The approach describes correctly the spin Wigner namicalvariablesremaintobevalidaftertheinteraction
•
rotation and this fact makes it possible to obtain including. ThisisthemeaningoftheGalileaninvariance
the correct (QCD) asymptotic. of the theory. So, in nonrelativistic theory of interacting
particles it is only one generator of Galilei group — the
The RHD method as the relativistic theory of com- Hamiltonian—thatcontainstheinteraction. Othergen-
posite systems is based on the direct realization of the eratorshavethesameformasinthecaseoffreeparticles.
Poincar´e algebra on the set of dynamical observables on Thiswayofinteractionincludingintheobservablesalge-
the Hilbert space (see, e.g., [3–6] and the references brais unique innonrelativisticcaseandgivesthe unique
therein). RHD theory of particles lie between local field nonrelativistic dynamics — the dynamics governed by
theoreticmodelsandnonrelativisticquantummechanical the Schr¨odinger equation.
models. The situation in the relativistic case is quite different.
Let us describe briefly the main statements of RHD. The structure of the Poincar´e algebra is of such kind
As it is known (see, e.g., [12]), the relativistic invari- thattheinteractionincludinginthetotalenergyoperator
ance of a theory means that there exists (on the Hilbert alone results in the breaking of algebraic structure, i.e.
space of states) the unitary representation of the inho- in the relativistic invariance breaking. This is the conse-
mogeneous group SL(2,C), which is the universal cov- quence of the fact that the Lorentz transformations mix
ering of the Poincar´e group. The relativistic invariance space and time. To preserve the algebra it is necessary
means the validity of the Poincar´e algebra commutation to include the interaction in the other generators, too.
relations for the generators of space–time translations Now the generators of Poincar´e algebra can be divided
Pˆµ and rotations Mˆµν. To construct the representation into two classes: containing interaction (interacting gen-
SL(2,C) means to obtain these generators in terms of erators) – hamiltonians, and without interaction (non-
dynamical variables of the system. In the case of free interacting generators). The latter present the so called
particle system this program is not difficult to be re- kinematical subgroup. This division, that is the interac-
alized and the generators Pˆµ , Mˆµν have clear phys- tion inclusion in the Poincar´e group, is not unique. The
ical meaning: Pˆ0 Hˆ — is the total energy opera- different ways of such a division preserving the Poincar´e
tor, Pˆ~ = (Pˆ1,Pˆ2,P≡ˆ3) — is the operator of the total algebra result in different types of relativistic dynamics
3–momentum, Jˆ~ = (Mˆ23,Mˆ31,Mˆ12) — the operator of (see, e.g., [4]).
The idea of this approach — RHD — is originated by
the total angular momentum, Nˆ~ = (Mˆ01,Mˆ02,Mˆ03) — Dirac. In [13] he considered different ways of descrip-
the generators of Lorentz boosts. tion of the evolution of classical relativistic systems —
differentformsofdynamics. Diracseparatedtheconcept
of time as a coordinate and that of time as a parameter
defining the system evolution. Consequently, he defined
1 It is known that correct impulse approximation (IA) real- threemainformsofdynamicswithdifferentevolutionpa-
ization in theframe of traditional version ofIFdynamicsen- rameters: point (PF), instant (IF) and light–front (FF)
countersdifficulties: thestandardIAdependsonthechoiceof dynamics. Each of these forms has its relative advan-
thecoordinate frame. We show below that IA can beformu- tages and disadvantages. The total number of possible
lated in an invariant way, the composite system form factors dynamics is actually five [3] so that the unique non-
being defined bytheone–particle currentsalone. relativistic Hamilton description is changed in relativis-
2
tic case for, generally, five possibilities. Each of these degrees of freedom.
possibledynamicsisconnectedwithathree–dimensional RHD is widely used in the theory of electromag-
hypersurface in the four dimensional space. The initial netic propertiesofcomposite quark andnucleonsystems
conditions are givenon these hypersurfaces and the evo- [6], [8], [11], [15–28]. It was shown that RHD not only
lution of the hypersurfaces is described. The kinemati- presentsaninteresting relativistic modelbut is a fruitful
cal subgroups are the invariance groups of the hypersur- toolandcancompetewithotherapproachesindescribing
faces. Inparticular,thesehypersurfacesare: hyperboloid the existing experimental data ( especially at small and
xµx =a2, t > 0 (PF), hyperplane t=0 (IF), light– moderatemomentum transfers). At presenttime the FF
µ
cone surface x0+x3 =0 (FF). dynamics is the most developed and used for composite
It is worth to notice that RHD and the field theory systems[8,11,15,16,18,19]. Infact,thelightfrontdynam-
are quite different approaches. The establishment of the ics has obvious advantages: a) the minimal number of
connection between RHD and field theory is a difficult operators containing the interaction (three); b) the sim-
and as yet unresolvedproblem. Contraryto field theory, ple relativistic invariant separation of the variables into
RHD is dealing with finite number of degrees of freedom classesof”internal”and”external”variables(while con-
from the very beginning. This is certainly a kind of a sidering the approach as Hamiltonian theory with fixed
model approach. The preserving of the Poincar´e alge- particlenumber); c) the simple vacuum structurefor the
bra ensures the relativistic invariance(see for details the light–front perturbative field theory. However, there are
Sec.II).So,thecovarianceofthedescriptionintheframe somedifficulties inthe FFRHDapproach. Inparticular,
of RHD is due to the existence of the unique unitary itwasshown [15,29]thatthe calculatedelectromagnetic
representation of the inhomogeneous group SL(2,C) on form factors for the systems with the total angular mo-
the Hilbert space of composite system states with finite mentumJ =1(thedeuteron,theρ–meson)varysignif-
number of degrees of freedom. The success of composite icantly with the rotation of the coordinate frame. This
models shows the validity of approximate relativistic in- ambiguityiscausedbythebreakingofthesocalledangle
variant description with fixed number of particles and a condition [15,29], that is really by the breaking of the
finite number of degrees of freedom. The similar situa- rotationinvarianceofthe theory. Some ofthe difficulties
tion (as was pointed out in [3]) takes place in the solid of FF dynamics are discussed in [30]. A possible way
statetheory: manyofsolidstatepropertiesareconnected to solve the problem by adding some new (nonphysical)
directly with its symmetry group and the actual form of formfactorstotheelectromagneticcurrentwasproposed
the particle interaction plays less important role. RHD (see [31] and references therein).
is based on the simultaneous action of two fundamental A different approach to the problem was proposed re-
principles: relativistic invariance and Hamiltonian prin- cently in Ref. [11], where a new method of construction
ciple—andpresentsthe toolthe mostadequatetotreat of electromagnetic current operators in the frame of FF
the systems with finite number of degrees of freedom. dynamics was given. The method of [11] gives unam-
It is worth to notice that the mathematics of RHD is biguous deuteron form factors. However, as the authors
similartothatofnonrelativisticquantummechanicsand of [11] note themselves, their current operator and the
permits to assimilate the sophisticated methods of phe- one used in Ref. [8] are different, since both of them
nomenological potentials and can be generalized to de- are obtained from the free one, but in different reference
scribe three or more particles. frames, related by an interaction dependent rotation.
Now the problem of the choice of the actual form of All these facts naturally cause authors to consider
RHD arises. other forms of relativistic Dirac dynamics.
Some time ago it was proved that S–matrices are Recently the PF dynamics was considered in the pa-
equivalentinthedifferentdynamicsforms [14]. Thisfact pers [6,23,26,27]. The authors used PF dynamics to
is interesting but it does not mean the absolute equiva- calculate the processes under investigations in JLab ex-
lence of the forms. First, there are problems which can periments. It is worth to notice that these experiments
notbe reducedto S–matrix,e.g., the calculationofform enlargetheinteresttothe RHDapproach–therelativis-
factors. Second, one has to keep in mind that any con- tic theory which can be used in the region of soft pro-
crete calculation uses some approximations; the approx- cesses.
imations usually used in different forms of dynamics are Now we present a relativistic treatment of the prob-
nonequivalent. lem of soft electroweak structure in the framework of
Our point of view is the following. One must choose another form – IF of RHD. IF of relativistic dynamics,
theformofdynamicsadequatetotheprobleminquestion althoughnotwidelyused,hassomeadvantages. Thecal-
and to the approximations to be done. It seems us that culations can be performed in a natural straightforward
thisisinthespiritofRHD–thechoosingoftheadequate way without special coordinates. IF is particularly con-
3
venient to discuss the nonrelativistic limit of relativistic The use of canonical parameterization permits to de-
results. This approach is obviously rotational invariant, scribe the electroweak properties of composite systems
so IF is the most suitable for spin problems. satisfyingtheLorentz–covarianceconditiononeachstage
We describe the dynamics of composite systems (the of calculation and to satisfy the electromagnetic cur-
constituentinteraction)inthe frame ofgeneralRHD ax- rent conservation law when describing the electromag-
iomatics. However, our approach differs from the tra- netic properties. In our formalism it is necessary to for-
ditional RHD by the way of constructing of matrix ele- mulate the composite model features in slightly unusual
ments of local operators. In particular, our method of way in terms of matrix elements which are generalized
description of the electromagnetic structure of compos- functions.
ite systems permits the construction of current matrix In particular, the relativistic impulse approximation
elementssatisfyingtheLorentz–covarianceconditionand (IA) has to be reformulated.
the current conservation law. Let us remind the physics of IA. In IA a test particle
To construct the current operator in the frame of IF interacts mainly with each component separately, that
RHD we use the general method of relativistic invariant is the electromagnetic current of the composite system
parameterization of matrix elements of local operators can be described in terms of one–particle currents. In
proposed as long ago as in 1963 by Cheshkov and Shi- fact, the composite–system current is approximated by
rokov [32]. the corresponding free–system current. This means that
The method of [32] gives matrix elements of the op- exchangecurrents are neglected, or,in other words,that
erators of arbitrary tensor dimension (Lorentz–scalar, there is no three–particle forces in the interaction of a
Lorentz-vector,Lorentz–tensor)intermsofafinite num- test particle with constituents. It is well known that
berofrelativisticinvariantfunctions –formfactors. The the traditional IA breaks the Lorentz–covariance of the
formfactorscontainallthedynamicalinformationonthe composite–system current and the conservation law for
transitionsdefinedbytheoperator. Thatiswhyasystem the electromagneticcurrent(see,e.g., [4]for details). As
can be described in terms of form factors. we show in the Sec.III(C) one can overcome these diffi-
The method of parameterization is similar in spirit to culties if one formulates IA in terms of form factors.
the method of presentation of matrix elements of irre- It is worth to notice that all known approaches (in-
ducibletensoroperatorsontherotationgroupintermsof cluding the perturbative quantum field theory (QFT))
reducedmatrixelements. Thismethodextractsfromthe encounter difficulties while constructing a composite–
matrix element ofa tensor operatora partdefining sym- system current operator satisfying Lorentz–covariance
metry properties and selection rules following the well and conservation conditions. This problem is now dis-
known Wigner–Eckarttheorem. cussedwidelyinthe literature[7–11]. Tosatisfythecon-
In the review [4] two possible variants of such kind of servation law in the frame of Bethe–Salpeter equation
representation of matrix elements in terms of form fac- andquasipotentialequations,forexample,itisnecessary
tors are presented – the elementary–particleparameteri- togobeyondIA:onehastoaddthesocalledtwo–particle
zation and the multipole parameterization. The variant currents to the current operator. In the case of nucleon
of parameterization given in [32] is an alternative one. composite systems these currents are interpreted as me-
In [32] the authors propose the construction of matrix son exchange currents [9]. In the case of deuteron this
elements in canonical basis so it can be called canonical means the simultaneous interaction of virtual γ– quanta
parameterization. This method was developed for the with proton and neutron. However, in Ref. [35] it is
case of composite systems in [33,34]. The composite– shown that the current conservation law can be satisfied
system form factors in this approach are in general case without such processes, although they contribute to the
thedistributions(generalizedfunctions),theyaredefined deuteron form factor. It seems that at the present time
by continuous linear functionals on a space of test func- there is an intention to formulate IA with transformed
tions. Thus, for example, the current matrix elements conservation properties without dynamical contribution
forcompositesystemsarefunctionals,generatedbysome of exchange currents [11,26,31].
Lorentz–covariantdistributions,andtheformfactorsare Our formalism also gives, in fact, the description of
functionals generated by regular Lorentz–invariant gen- the covariance properties of the operators in terms of
eralizedfunctions. We demonstrate these facts below, in many–particleaswellasone–particlecurrents. However,
Sec.III, using a simple model as an example. the important feature of our formalism is the fact that
It is worth to notice that the statement that the form form factors or reduced matrix elements describing the
factorsofacompositesystemaregeneralizedfunctionsis dynamicsoftransitionscontainonlythe contributionsof
not something exotic. This fact takes place in the stan- one–particle currents.
dardnonrelativisticpotentialtheory,too(seeSec.III(F)). So, our approach to the construction of the current
4
operator includes the following main points: II. RELATIVISTIC HAMILTONIAN DYNAMICS.
1. We extract from the current matrix element of com-
positesystemthereducedmatrixelements(formfactors) In this Section some basic equations of RHD and
containing the dynamicalinformation onthe process. In some relations from relativistic spin theory are briefly
general these form factors are generalized functions. reviewed.
2. Along with form factors we extract from the matrix The relativistic invariance of a theory means that the
element a partwhich defines the symmetry properties of unitary representation of the Poincar´e group is realized
thecurrent: thetransformationpropertiesunderLorentz on the Hilbert space of system states (see, e.g., [12]).
transformation, discrete symmetries, conservation laws In this case the structure of the Poincar´ealgebra can be
etc. defined on the set of observables (Mˆµν,Pˆσ)
3. The physicalapproximationswhichare used to calcu-
late the currentareformulatednotin terms ofoperators [Mˆµν,Pˆσ]= i(gµσPˆν gνσPˆµ),
but in terms of form factors. − −
In this paper we present the main points of our ap-
proach. To make it transparent we consider here only [Mˆµν,Mˆσρ]= i(gµσMˆνρ gνσMˆµρ) (σ ρ),
− − − ↔
simplesystemswithzerototalangularmomenta,sothat
technical details do not mask the essence of the method. [Pˆµ,Pˆν]=0. (1)
Wedemonstratetheeffectivenessoftheapproachbycal-
In (1) gµν is the metric tensor in Minkowski space.
culatingthepionelectroweakproperties. Inthiscasethe
As we have mentioned, in the case of the particles with
canonicalparameterizationis very simple and can be re-
interactionthe realizationofthe Poincar´ealgebraonthe
alizedwithout difficulties. The case ofmore complicated
setofobservablesismorecomplicatedthaninthecaseof
systems needs using a rather sophisticated mathematics
the free particles. Let us consider one key commutator
for canonical parameterization of local operator matrix
from the set of commutators (1) (see, e.g., [2]):
elements and will be considered elsewhere.
The paper is organized as follows. In Sect. II we re- [PˆjNˆk]=iδjkHˆ (2)
mindbrieflythebasicstatementsofRHD,especiallyofIF
RHD. Two bases in the state space of composite system (we use notations given in the Introduction; j,k =
are considered: the basis with individual spins and mo- 1,2,3).
menta and the basis with separated center–of–mass mo- Since Hˆ is interaction dependent for non-trivial sys-
tion. TheClebsh–GordandecompositionforthePoincar´e tems, either Pˆ~ , Nˆ~, or some combination of Pˆ~ and Nˆ~
group which connects these bases is given. The IF wave
also must be interacting. To preserve the commutation
functions of composite systems are defined. In Sect. III
relations(1)onehastomakeothergeneratorsdepending
our approach to relativistic theory of two–particle com-
on the interaction, too. So, the generators occur to fall
posite systems and their electroweak properties is pre-
intotwogroups: thegeneratorswhichareindependentof
sented. Asimplemodelisconsideredindetails: twospin-
the interaction and form the so called kinematical sub-
lessparticlesin the S-stateofrelativemotion, oneofthe
group, and the generators depending on the interaction
particlesbeinguncharged. Theelectromagneticformfac-
– Hamiltonians. This division is not unique. Different
torofthe systemisderived. Thestandardconditionsfor
ways to obtain kinematical subgroups result in different
thecurrentoperatorarediscussed. Themodifiedimpulse
forms of dynamics.
approximation(MIA) isproposed. TheresultsofIAand
Inthispaperweusethesocalledinstantformdynam-
MIA are compared. The nonrelativistic limit is consid-
ics (IF). In this form the kinematical subgroup contains
ered. TheconnectionbetweenthepresentedversionofIF
the generatorsof the group of rotations and translations
RHD and the dispersion relations in mass is established.
in the three–dimensional Euclidean space:
In Sect.IV the developed formalism is used in the case
of the system of two particles with spins 1/2. The pion
electromagnetic form factor and the lepton decay con- Jˆ~, Pˆ~ . (3)
stant are derived. The model parameters are discussed
The remaining generators are Hamiltonians (interaction
and the comparisonof the results with the experimental
depending):
data is given. The results of calculations in IA and MIA
are compared and are shown to differ significantly. In
Sect.V the conclusion is given. Pˆ0, Nˆ~ . (4)
The additive including of interaction into the mass
squareoperator(Bakamjian–Thomasprocedure[36],see,
5
e.g., [4]for details) presents one of the possible technical P~, √s, J, l, S, mJ P~ ′, √s′, J′, l′, S′, mJ′
h | i
ways to include interaction in the algebra (1):
Mˆ02 →MˆI2 =Mˆ02+Uˆ . (5) =NCGδ(3)(P~ −P~ ′)δ(√s−√s′)δJJ′δll′δSS′δmJmJ′ ,
Here Mˆ0 is the operator of invariant mass for the free N = (2P0)2 , k= 1 s 4M2 . (9)
system and MˆI – for the system with interaction. The CG 8k√s 2p −
interaction operator Uˆ has to satisfy the following com-
mutation relations: Here Pµ = (p1 + p2)µ, Pµ2 = s, √s is the invariant
mass of the two-particle system, l — the orbital angu-
Pˆ~, Uˆ = Jˆ~, Uˆ = ~ , Uˆ =0. (6) lar momentum in the center–of–mass frame (C.M.S.),
P
h i h i h▽ i S~2 = (S~ +S~ )2 = S(S +1) , S — the total spin in
1 2
These constraints (6) ensure that the algebraic relations C.M.S., J — the total angular momentum with the pro-
(1) are fulfilled for interacting system. The constraints jection m .
J
(6)arenottoostrong. Forinstance,a largeclassofnon- The basis (9) is connected with the basis (8) through
relativisticpotentialsatisfies(6). Therelations(6)mean theClebsh–Gordan(CG)decompositionforthePoincar´e
thatthe interactionpotentialdoesnotdependonthe to- group. The decomposition of the direct product (8)
talmomentumofthesystem. Thisfactiswellestablished of two irreducible representations of the Poincar´e group
for a class of potential, for example,for separablepoten- intoirreduciblerepresentations(9)hasthefollowingform
tials [37]. Nevertheless, the conditions (5) and (6) can [34]:
be considered as the model ones. There exists another
approach[38]whereapotentialdependsonthetotalmo- p~ ,m ; ~p ,m = P~, √s, J, l, S, m
1 1 2 2 J
| i | i
mentum but that approachis out of scope of this paper. X
InRHDthe wavefunctionofthe systemofinteracting
particles is the eigenfunction of a complete set of com- ×hJmJ|SlmsmliYl∗ml(ϑ,ϕ)hSmS|1/21/2m˜1m˜2i
muting operators. In IF this set is:
m˜ D1/2(P,p ) m m˜ D1/2(P,p ) m . (10)
1 1 1 2 2 2
Mˆ2, Jˆ2, Jˆ , Pˆ~ . (7) ×h | | ih | | i
I 3 Here the sum is over the variables m˜ , m˜ , m , m , l,
1 2 l S
Jˆ2 is the operator of the square of the total angular mo- S, J, mJ. p~ = (p~1 p~2)/2, p = p~, ϑ,ϕ are the spheri-
− | |
mentum. In IF the operators Jˆ2 , Jˆ , Pˆ~ coincide with cal angles of the vector p~ in the C.M.S., Ylml - a spher-
3 ical harmonics (star means the complex conjugation),
those for the free system. So, in (7) only the operator
Sm 1/21/2m˜ m˜ and Jm Slm m aretheCG
MˆI2 depends on the interaction. hcoefficSie|ntsforthe1gro2uipSUh(2),J|m˜ D1S/2(Pl,ip) m -the
To find the eigenfunctions for the system (7) one has h | | i
three–dimensional spin rotation matrix to be used for
firsttoconstructthe adequatebasisinthe statespaceof
correct relativistic invariant spin addition.
compositesystem. Inthecaseoftwo-particlesystem(for
Letusdiscussbrieflytherelativisticpropertiesofspins.
example, quark-antiquark system qq¯) the Hilbert space
It is known that the Lorentz transformation for spins is
in RHD is the direct product of two one-particle Hilbert
momentum depending (see, e.g., [12]). So, to perform
spaces: .
qq¯ q q¯ Lorentz invariant spin addition for particles with differ-
H ≡H ⊗H
As a basis in qq¯ one can choose the following set of ent momenta ~p and ~p′ one has to ”shift” the spins to
H
two-particle state vectors:
the frame where the momenta are equal to one another.
The spin transforms following the so called small group
p~ ,m ; ~p ,m = p~ m p~ m ,
| 1 1 2 2i | 1 1i⊗| 1 2i which is isomorphic to rotation group and thus, the op-
p~,m ~p′m′ =2p0δ(p~ p~′)δmm′ . (8) eratorofsucha”shift”isa3-dimensionalrotationmatrix
h | i − D(α,β,γ) . The Euler angles α,β,γ can be written in
Herep~ , p~ are3-momentaofparticles,m , m —spin terms of the components of the vectors p~ and ~p′. In this
1 2 1 2
projections on the axis z, p = p~2+M2 , M is the way the ”transplantation” of spins on one and the same
0
constituent mass. p momentum is realized. To understand what means this
One canchoose another basis where the motion of the ”transplantation” let us consider an example: one par-
two-particle center of mass is separated and where three ticle has the momentum ~p1, mass M1, spin j and spin
operators of the set (7) are diagonal: projection m, while another particle with momentum p~2
and mass M has no spin. In the case of free particles
2
P~, √s, J, l, S, m , the vector state of this system is
J
| i
6
|p~1,M1,j,m; ~p2,M2i=|p~1,M1,j,mi⊗ |p~2,M2i. Nˆ~j =ip ∂ [~jp~] , Nˆ~ =ip ∂ . (15)
0 0
(11) ∂~p − p0+M ∂~p
The two–particle vector state in the case when it is the If we perform the transformation (14), the D-matrix
first particle that has no spin is transforms the generators in the following way:
|p~1,M1; p~2,M2,j,mi=|p~1,M1i⊗|p~2,M2,j,mi. Dj(p ,p )(Nˆ~j +Nˆ~ )[Dj(p ,p )]−1 =Nˆ~ +Nˆ~j . (16)
2 1 1 2 2 1 1 2
(12)
It is just the equations (14) and (16) (and only these
In nonrelativistic angular momentum theory the states
equations) that present the exact statement that D-
(11) and (12) are identical. They describe the two–
function shifts spinfrom one momentum to another one.
particle system with momenta p~ and ~p and total spin
1 2 Similar equations (however,more complicated) are valid
j with the projection m. In both cases the total spin
in the case of two non-zerospins. In the case of spin 1/2
can be obtained in simple way and is equal to the spin j
the D – function [34] has the form
of the first particle for (11) or of the second particle for
(12). ω ω [~p ~p ]
In relativistic theory the states (11)and (12)differ es- D1/2(p1,p2)=cos 2i(~k~j) sin , ~k = 1 2 ,
(cid:16)2(cid:17)− (cid:16)2(cid:17) [~p1~p2]
sentially. The difference is caused by the fact that the | |
states transform from one inertial coordinate system to [p~ ~p ]
1 2
ω =2arctan | | . (17)
another in different ways. As was mentioned before the (p +M )(p +M ) (p~ p~ )
10 1 20 2 1 2
Lorentz transformation for spin depends on the particle −
momentum and the spins in (11) and (12) correspond We will use it below.
to particles with different momenta. In relativistic case Letusmakearemarkconcerningthe invarianceofthe
thestates(11)and(12)coincideonlyiftheparticlesmo- decomposition (10). The total spin S and the total or-
menta are equal. In general case, to connect the state bitalangularmomentuml in (10)playthe roleofinvari-
vectors (11) and (12) one has to ”shift” the spin, for antparametersofdegeneracy. However,thesquareofthe
example, to shift the spin in (12) into the frame where total spin S~ = (S~ +S~ ) is not invariant. But one can
1 2
the second particle has the momentum p~ . This shift- define the total spin square in invariant way as follows:
1
ing transformation is realized by the matrix Dj(p ,p )
2 1
which belongs to the small group. Let us consider the [DS1(p ,P)]−1S~ [DS1(p ,P)]
1 1 1
state vector n
2
|p~1,M1i⊗Dj(p2,p1)|p~2,M2,j,mi. (13) + [DS2(p2,P)]−1S~2[DS2(p2,P)] =S(S+1). (18)
o
Let us remind that a transformation belonging to the
Here P is the center-of-mass momentum. One can see
small group does not act on momenta. It is easy to
thatin C.M.S. the definition (18)coincideswith the def-
see that the resulting vector describes the same state as
inition (9). Similarly, one can define the orbital moment
(11) and transforms from one frame to another in the
l in invariant way.
same way as (11). To show this let us use the equation
The described spin rotation effect in (10) is a purely
D(p′,p′′)D(p′′,p) = D(p′,p). Now the following covari-
relativisticeffect. Ifonetakesitintoaccount,oneobtains
ant equality is valid:
interesting observable effects [24].
p~ ,M ,j,m; p~ ,M To obtain the basis vectors (9) in terms of vectors (8)
1 1 2 2
| i
one has to inverse (10). The final equationhas the form:
= p~ ,M ; p~ ,M ,j,m′ m′ Dj(p ,p ) m . (14)
1 1 2 2 2 1
Xm′ | ih | | i P~, √s, J, l, S, mJ
| i
As one can see from (14) the spin has ”changed” the
momentum. As this operation is very important for the dp~ dp~
1 2
= p~ ,m ; p~ ,m
understanding of the parameterization below let us for- 1 1 2 2
Z 2p 2p | i
mulate it in other words, namely, in terms of the gener- mX1m2 10 20
ators of the Lorentz transformations.
p~ ,m ; ~p ,m P~, √s, J, l, S, m . (19)
The Lorentz–transformation generator for the state ×h 1 1 2 2| Ji
(11) is of the form:
Here
Nˆ~ =Nˆ~j +Nˆ~ , p~ ,m ; ~p ,m P~, √s, J, l, S, m
1 2 h 1 1 2 2| Ji
7
=√2s[λ(s, M2, M2)]−1/22P0δ(P p1 p2) u2(k)k2dk =1. (23)
− − Z l
Xl
m1 D1/2(p1P) m˜1 m2 D1/2(p2P) m˜2 Letus note that forcomposite quarksystems one uses
× h | | ih | | i
m˜X1m˜2 sometimes instead of equation (23) the following one:
n u2(k)k2dk =1. (24)
×mXlmSh1/21/2m˜1m˜2|SmSiYlml(ϑ,ϕ) c Xl Z l
Here n – is the number of colours. The wave func-
c
Slm m Jm . tion (22) coincides with that obtained by ”minimal rela-
s l J
×h | i tivization” in [39]. The normalization factors in (22) in
Here λ(a,b,c)=a2+b2+c2 2(ab+bc+ac). To obtain thiscasecorrespondtotherelativizationobtainedbythe
−
(19) the decomposition in terms of spherical harmonics transformation to relativistic density of states
andthesummationofallofthemomentatogivethetotal
momentumJ wereperformedintheC.M.S.andthenthe k2dk k2dk . (25)
obtainedresultwasshiftedtoarbitrarycoordinateframe → 2 (k2+M2)
by use of D-functions. p
It is on the vectors (9), (19) that the Poincar´e–group It is worth to notice that wave functions in RHD (for
representationisrealizedinthe vectorstate spaceoftwo example, the wave function (21) defined as the eigen-
freeparticles. Thevectorinrepresentationisdetermined function of the operators set (7)) in general are not the
by the eigenvalues of the complete commuting set of op- same as relativistic covariant wave functions defined as
erators: solutions of wave equations or as the matrix elements of
local Heisenberg field.
Mˆ02 =Pˆ2, Jˆ2, Jˆ3 . (20) The formalism of this Section is used in the next one
topresentthemethodofcalculationofelectroweakprop-
TheparametersS andl (aswasmentioned)playtherole
erties of composite systems. Particularly, the method
of invariant parameters of degeneracy.
of construction of electroweak current operators is de-
As inthe basis(9)the operatorsJˆ2 , Jˆ , Pˆ~ in(7)are scribed.
3
diagonal,one needs to diagonalize only the operator Mˆ2
I
in order to obtain the system wave function.
TheeigenvalueproblemfortheoperatorMˆ2 intheba- III. THE NEW RELATIVISTIC INSTANT–FORM
I
sis (9) has the form of nonrelativistic Schr¨odinger equa- APPROACH TO THE ELECTROWEAK
tion (see, e.g., [4]). STRUCTURE OF TWO BODY COMPOSITE
The corresponding composite–particle wave function SYSTEMS.
has the form
In this Section we present our approach to elec-
hP~ ′, √s′, J′, l′, S′, m′J|pci= troweak properties of relativistic two–particle systems.
To demonstrate how one describes the electromagnetic
=NCδ(P~ ′−p~c)δJJ′δmJm′J ϕJl′′S′(k′), (21) properties of composite systems in our version of the
RHDinstantformwefirstusethefollowingsimplemodel.
WeconsiderthesystemoftwospinlessparticlesintheS–
NCG state of relative motion, one particle having no charge.
N = 2p ,
C c0r 4k′ Letusnotethatasimilarmodelwasusedin[4]wherethe
p
authorsgavethedescriptionofconstituentinteractionin
p is aneigenvectorofthe set(7); J(J+1)andm are
c J IF of RHD and obtained the mass spectrum. The appli-
| i
the eigenvalues of Jˆ2, Jˆ3, respectively (Eqs. (7), (20)). cation of our method in general case follows the scheme
The two–particle wave function of relative motion for of this Section. The case of π - meson is investigated in
equalmassesandtotalangularmomentumandtotalspin Sec.IV and the S =1 case in [40].
fixed is: Electromagnetic properties of the system are deter-
minedbythe currentoperatormatrixelement. Thisma-
ϕJ (k(s))= √4su (k)k , (22)
lS l trix element is connected with the charge form factor
F (Q2) as follows:
and the normalization condition has the form: c
8
p j (0) p′ =(p +p′) F (Q2), (26) reflections are:
h c| µ | ci c c µ c
iwnhiteiraelapn′cd, fipncaalrset4a–tems,oQm2en=taotf,thqe2c=om(pposipte′)s2y=stetm, qin2 UˆP (cid:16)ˆj0(x0,~x),~ˆj(x0,~x)(cid:17)UˆP−1
− c− c
is the momentum–transfer square. The form (26) is de-
fined by the Lorentz covariance and by the conservation = ˆj0(x0, ~x), ~ˆj(x0, ~x) ,
law only and does not depend on the model for the in- (cid:16) − − − (cid:17)
ternal structure of the system.
Uˆ ˆjµ(x)Uˆ−1 =ˆjµ( x). (31)
The Eq. (26) presents the simplest example of the R R −
extraction of a reduced matrix element. The 4–vector In (31)Uˆ is the unitary operatorfor the representation
(p +p′) describes symmetry and transformationprop- P
c c µ ofspacereflectionsandUˆ isthe antiunitaryoperatorof
ertiesofthematrixelement. Thereducedmatrixelement R
the representation of space-time reflections R=P T.
(the form factor) contains all the dynamical information
(v).Cluster separability condition: If the interaction is
on the process described by the current. Usually, one
switched off then the current operator becomes equal to
does not fix the dependence of form factor on a scalar
the sum of the operators of one–particle currents.
mass of the composite system p 2 =p′2 =M 2, because
c c c (vi).The charge is not renormalized by the interaction
itisdiagonalwithrespecttothisvariable. Therepresen-
including: The electric charge of the system with in-
tation of a matrix element in terms of form factors often
teraction is equal to the sum of the constituent electric
is referred to as the parameterizationof matrix element.
charges.
The scattering crosssection for elastic scattering of elec-
Inthispapertheexplicitequationsfortheformfactors
trons by a composite system can be expressed in terms
areobtainedtakingintoaccountallthelistedconditions.
of chargeform factor F (Q2). So, form factor can be ob-
c
tained from experiment and it is interesting to calculate
it in a theoretical approach.
A. Electromagnetic properties of the system of free
InthisSectionwecalculatetheformfactorofoursim-
particles
ple composite systemusing the versionofRHD IFbased
on the approach of the Section II.
Letusconsiderfirstthesimpletwo–particlesystemde-
Now let us list the conditions for the operator of the
scribed in the beginning of Section III. The elastic scat-
conserved electromagnetic current to be fulfilled in rela-
teringofatestparticle,e.g.,ofanelectron,bythesystem
tivistic case (see, e.g., [10]).
are defined by the operator of electromagnetic current
(i).Lorentz–covariance:
j(0)(0) of the two–particle free system. This operator
µ
Uˆ−1(Λ)ˆjµ(x)Uˆ(Λ)=Λµˆjν(Λ−1x). (27) can be calculated in the representation given by the ba-
ν
sis (8) or in the representationgivenby the basis (9). In
Here Λ is the Lorentz–transformation matrix, Uˆ(Λ) – the first case the operator has the form jµ(0) = j1µ I2.
⊗
the operatorofthe unitary representationofthe Lorentz Here j1µ is the electromagnetic current of the charged
group. particle andI2 is the unity operatorin the Hilbert space
(ii).Invariance under translation: of states of the uncharged particle.
Uˆ−1(a)ˆjµ(x)Uˆ(a)=ˆjµ(x a). (28) hp~1;p~2|jµ(0)(0)|p~′1;p~′2i
−
= p~ p~′ p~ j (0)p~′ . (32)
Here Uˆ(a) is the operator of the unitary representation h 2| 2ih 1| 1µ | 1i
of the translation group. Thematrixelementoftheonespinlessparticlecurrentin
(iii).Current conservation law: the free case contains only one form factor – the charge
form factor of the charged particle f (Q2):
[Pˆ ˆjν(0)]=0. (29) 1
ν
p~ j (0)p~′ =(p +p′) f (Q2). (33)
Intermsofmatrixelements ˆjµ(0) theconservationlaw h 1| 1µ | 1i 1 1 µ 1
h i So,theelectromagneticpropertiesofthesystemoftwo
can be written in the form:
free particles (26) are defined by the form factor f (Q2),
1
q ˆjµ(0) =0. (30) containing all the dynamical information on elastic pro-
µ
h i cesses describedby the matrix element (32)[4]. Particu-
Here q is 4-vector of the momentum transfer. larly, the charge of the system is defined by the value of
µ
(iv).Current–operator transformations under space–time this form factor at Q2 0:
→
9
lim f (Q2)=f (0)=e . (34) of these descriptions are, certainly, equivalent from the
1 1 c
Q2→0 physicalpoint ofview. Let us consider the difference be-
tweenthesedescriptions. Aswewillshowbelowbydirect
e is the system charge.
c calculation the free two–particle form factor g (s,Q2,s′)
Now let us write the electromagnetic–current matrix 0
is not an ordinary function but has to be considered in
element for the two–particle free system in the basis
the sense of distributions in variables s , s′, generated
where the center–of–mass motion is separated (9):
byalocallyintegrablefunction. So,g (s,Q2,s′)isa reg-
0
P~,√s, j(0)(0) P~′,√s′ . (35) ular generalized function. Let us remind that regular
h | µ | i generalized function is that defined through an integral
Here the variables which take zero values are omitted: in the space of test functions. So, all the properties of
J =S =l =0. Onecanconsiderthematrixelement(35) g0(s,Q2,s′) have to be considered as the properties of a
as a matrix element of an irreducible tensor operator on functional given by the integral over the variables s , s′
the Lorentzgroupandone canapplythe Wigner–Eckart ofthefunctiong0(s,Q2,s′)multipliedbyatestfunction.
theorem. Under the conditionofthe theoremthe matrix As test functions it is sufficient to take a large class of
elementofanirreducibletensoroperatoristheproductof smooth functions that give the uniconvergence of the in-
two factors: the invariantpart(reduced matrix element) tegral. Inparticular,thelimit(34)givingthetotalcharge
and the CG coefficient which defines the transformation ofthesystemthroughtwo–particleformfactorisnowthe
properties of the matrix element. Thus, one can write weak limit:
(35) in the form
lim g (s,Q2,s′),φ(s,s′) . (38)
0
P~,√s, j(0)(0) P~′,√s′ = Q2→0h i
h | µ | i
Here φ(s,s′) is a function from the space of test func-
=Aµ(s,Q2,s′)g0(s,Q2,s′). (36) tions. The precise definition of the functional will be
given below.
The motivation for the parameterization (36) is easy to As the invariantvariables s, s′ containthe energies of
be understood for our simple system. The 4–vector A
µ the relativemotionof particlesin initial andfinal states,
describesthetransformationpropertiesofthematrixele-
one can consider the integral in (38) as an integral over
mentandtheinvariantfunctiong (s,Q2,s′)containsthe
0 these energies.
dynamical information on the process. We will refer to
At the first glance it seems that the description of
g (s,Q2,s′)astofreetwo–particleformfactor. Formore
0 the two–particle free system in terms of the form factor
complicatedsystemstheparameterizationcorresponding g (s,Q2,s′) is too complicated. However, so is the real-
0
totheWigner–EckarttheoremfortheLorentzgroupcan
ity, as we will see later in the Subsection III.F. In fact,
beperformedusingaspecialmathematicaltechniquesas
this kind of description is used implicitly for a long time
described in the papers [32], [34].
in nonrelativistic theory of composite systems, without
The vector A (s,Q2,s′) which describes the matrix–
µ calling things by their proper names. It is this kind of
element transformation properties is defined by the 4–
description that makes it possible to construct the elec-
momenta of initial and final states only: we have no
tromagnetic current operator with correct transforma-
other vectors to our disposal. So A (s,Q2,s′) is a lin-
µ tion properties for interacting systems.
ear combination of 4–momenta of initial and final states
and is defined by the current transformation properties
(the Lorentz–covarianceand the conservation law):
B. The form factor of the system of two free particle.
1
Aµ = Q2[(s−s′+Q2)Pµ+(s′−s+Q2)P ′µ]. (37) Thelocallyintegrablefunctiong0(s,Q2,s′)canbeeas-
ily obtained by use of CG decomposition (19) for the
Thus,inthebasis(9)theelectromagneticpropertiesof Poincar´egroup. Using (19) we obtain for (36):
the free two–particlesystemaredefined by the free two–
particle form factor g (s,Q2,s′). P~,√s, j(0)(0) P~′,√s′
0 h | µ | i
So,inbothrepresentations(definedbythebasis(8)as
well as by the basis (9)) we pass from the description of
the system in terms of matrix elements to that in terms = dp~1 dp~2 dp~1′ dp~2′ P~,√s, ~p ;~p
of Lorentz–invariantform factors. Z 2p102p202p′102p′20 h | 1 2i
One can see that (32) and (36) describe electromag-
netic properties in terms of only one form factor. Both p~ ;~p j(0)(0)p~′;~p′ p~′;~p′ P~′,√s′ . (39)
×h 1 2| µ | 1 2ih 1 2| i
10