RU98-04-B NYU-TH/98/03/01 hep-th/9803073

SUPERSYMMETRIC SUM RULES FOR ELECTROMAGNETIC

MULTIPOLES

Ioannis Giannakis,
James T. Liu and Massimo Porrati ^{†}^{†} e-mail:
, ,

Physics Department, The Rockefeller University

1230 York Avenue, New York, NY 10021-6399

Department of Physics, New York University

4 Washington Pl., New York, NY 10003

Abstract

We derive model independent, non-perturbative supersymmetric sum rules for the magnetic and electric multipole moments of any theory with supersymmetry. We find that in any irreducible supermultiplet the diagonal matrix elements of the -multipole moments are completely fixed in terms of their off-diagonal matrix elements and the diagonal -multipole moments.

1. Introduction.

Supersymmetry imposes constraints on the magnetic moments of the particle states [1], [2]. These constraints are model independent, valid for any massive and supermultiplet. They are also in agreement with the results of Ferrara and Remiddi [3], who showed that to all orders in perturbation theory for any chiral multiplet and those of Bilchak, Gastmans and Van Proeyen [4] who demonstrated that when spin- fields are present supersymmetry does not necessarily demand , but nevertheless leads to a relation between the -factors of the spin-1/2 and spin-1 particles of the superspin 1/2 multiplet.

The model independent magnetic dipole moment sum rules were derived in [1] by noting that supersymmetry relates the matrix elements of the conserved electromagnetic current within the various states of a general massive supermultiplet. By selecting the magnetic dipole term in the multipole expansion of the electromagnetic current, the authors of [1] found, for the gyromagnetic ratios, the following sum rule:

Note that is the superspin labeling the massive supermultiplet which contains states of spins states have identical gyromagnetic ratios, we see that all the -factors are determined in terms of a single real number, , corresponding to an off-diagonal magnetic dipole matrix element between the and states of the supermultiplet. In the special cases , the sum rules read . Since both spin-

Notice that chiral multiplets () have a fixed gyromagnetic ratio .

In this letter, we generalize the above gyromagnetic ratio sum rule to encompass higher multipole moments (both electric and magnetic). This is easily done by working to all orders in the momentum transfer in the appropriate electromagnetic matrix elements. The resulting multipole sum rules have a similar structure as (1), and take the form

where the electric/magnetic -pole generalization of is
denoted by and is defined in Eqn. (21) below^{†}^{†} More precisely these sum rules hold for the generic case
, where denotes the superspin. Note that
is meaningless whenever , as may be
infered from Eqn. (21)..
These sum rules indicate the general
structure imposed by supersymmetry that the electric (magnetic) -pole
moments are completely determined solely in terms of a single magnetic
(electric) -pole moment and the real quantity
parameterizing an off-diagonal transition between the spin states
of the multiplet.
Note that the upper and lower signs in (3) and subsequent
equations correspond to the first and second entries in e.g. .
This difference in sign between the electric and magnetic sum rules may be
understood intuitively from electromagnetic duality which exchanges
electric and magnetic fields,
and .

When , the magnetic part of the sum rule (3) reduces to the result of Ferrara and Porrati, (1), since is defined as a ratio: and . Furthermore, just as for the magnetic dipole moments, we note that setting yields the “preferred” value for the -pole moments

generalizing the notion of as the preferred value of the gyromagnetic ratio.

2. Derivation of the Sum Rules.

Derivation of the sum rules (3) follows the method of [1], and involves the transformation properties of a conserved current that commutes with the supersymmetry algebra. The main complication in obtaining the present results is the requirement of working to all orders in the multipole expansion (and as a result having to keep track of higher-order in momentum transfer terms in the matrix elements).

Recall that the algebra has the form , where is the Majorana conjugate and is the charge conjugation matrix obeying and . For a massive single particle state, we may work in the rest frame . Defining chiralities

and helicities

the supersymmetry algebra can be recast as follows:

while the remaining anticommutators vanish^{†}^{†} To fix our phase conventions, we work in the Dirac
representation for the -matrices and take
and
. The spinors then decompose as

Since the supercharges are operators of spin , this leads to a shorthand notation for labeling the states of a massive multiplet in the following manner: the spin Clifford vacuum is denoted by , acting on this state with the normalized supercharges or then results in the spin ‘up’ or ‘down’ states or respectively. The action of two ’s on the Clifford vacuum is denoted by .

For supersymmetry, any conserved current commuting with the supersymmetry generators must belong to a real linear multiplet. The components of a real linear multiplet multiplet are , where is a real scalar and a Majorana spinor. As a result of current conservation, , the multiplet consists of fermionic and bosonic degrees of freedom. The transformation properties of the components under a supersymmetry variation are given by

It follows that two successive supersymmetry transformations on the conserved current gives

The matrix elements of this equation between single particle states which belong to the same multiplet give rise to sum rules for the electromagnetic multipoles of the particle states.

To obtain the connection between the matrix elements of and the terms in the multipole expansion, we first recall the standard definitions (see e.g. [5]) for the electric -pole moments

and the magnetic -pole moments

While ordinarily defined in terms of spherical tensors (see e.g. [6]), the above multipole moments, expressed as cartesian tensors, are more naturally related to the expansions for the matrix elements of ,

In particular, the traceless components of and correspond exactly to and respectively. Note that the matrix elements of are completely determined by current conservation, .

The multipole moment sum rules are derived by taking the double supersymmetry variation of the conserved current ,

and evaluating it between single particle states and
. Since the supercharge generates superpartners
(), this expression relates
matrix elements of between different states of a supermultiplet
in terms of , which is given by (9).
The electromagnetic -pole sum rules then follow by using (12) to
expand the matrix elements in terms of multipoles and then by collecting
terms of order . We note that an important simplification occurs since
we are only interested in sum rules on the static multipole moments. This
means in practice that all terms depending explicitly on the contracted
momentum may be ignored, as they do not contribute to the static
-pole moments (and instead correspond to the trace terms in
)^{†}^{†} In principle supersymmetry would give complete relations between
electromagnetic form factors of superpartners.
However in this case it appears the moments of the “auxiliary field”
enter in a non-trivial manner..

The general double supersymmetry variation procedure is simplified in practice by choosing the global supersymmetry transformation parameters and in such a way that several terms on the right hand side of (13) act as annihilation operators on the initial or final states and hence may be dropped. In particular, by choosing , we find

where denotes the Lorentz boost of , namely , and .

By further choosing , and noting from (9) that , we easily obtain the “vanishing” sum rule,

This demonstrates that all matrix elements of the electromagnetic current vanish between states and , and hence that there are no off-diagonal moments between the two spin- states of the supermultiplet.

If instead we choose and make use of the fact that transforms as a spinor,

we obtain from (14) the expression

Equation (17) can be simplified significantly if we ignore (i.e. trace) terms which do not contribute to the electromagnetic multipole sum rules. After some manipulation, the time and space components of Eqn. (17) can be written as follows:

where we have omitted terms explicitly proportional to . Note in particular that matrix elements of do not enter.

We now use the explicit multipole expansion of the matrix elements, (12), and equate terms of the same order in . Because of the explicit factor of in (18), we see that multipole terms of order and are explicitly related; heuristically Eqn. (18) states that the -pole moment of a superpartner is given by the same -pole moment of the original state plus a correction based on the opposite (electric/magnetic) -pole. Explicitly, at order , we find

where the indices are to be explicitly symmetrized, and all tensor quantities are assumed traceless. Note that the last term in (19) is responsible for subtracting out the trace from the spin-1 spin-() combination .

Because of rotational invariance, each -pole moment may be completely characterized by a single quantity—essentially a reduced matrix element according to the Wigner-Eckart theorem. In particular, for a single particle state of spin and -component , we define the reduced -pole moment by

The sum rules may now be established by examining the components of (19). Furthermore, the spin- angular momenta manipulations are simplified by picking the particular state in the matrix elements, in which case (20) may be reexpressed as

With the same motivation we define the multipole transition moments as

Recall that in (19) both and denote the spin- Clifford vacuum state, , which may be abbreviated as . By choosing the spinor parameters and appropriately, we then relate the electromagnetic multipoles of the different members of the massive multiplet. With a total of two and two parameters, we find

where in the indices denote the combinations , and the states and are implicitly understood in terms of the Clebsch-Gordon combination of spin- spin-. This is the main result of our paper. The matrix elements of the -electric (magnetic) multipole moment between different members of the supermultiplet are given in terms of the matrix elements of the -electric (magnetic) multipole moment and the -magnetic (electric) multipole moment between the Clifford vacuum.

Finally by carrying out the addition of the superspin to the supersymmetry generated spin we find the following sum rules:

which may be written in a completely equivalent form as presented in Eqn. (3). Note that both spin- states carry identical -pole moments, as may be established using the same argument as in [1].

3. Discussion.

While the sum rules were derived for generic superspin , it is important to realize that angular momentum selection rules forbid both diagonal () and non-diagonal () -pole electromagnetic moments whenever . For (dipole moment), the magnetic sum rule reduces to that of Ref. [1], while the electric sum rule gives rise to the relation between EDM’s:

where .

The special cases and are noteworthy. For only dipole moments are allowed (for the spin- particle), in which case the gyromagnetic ratio of the spin-1/2 particle in the supermultiplet is , as shown by Ferrara and Remiddi [3]. For (massive vector multiplet) only dipole and quadrupole moments are allowed. Robinett [7] and Bilchak, Gastmans and Van Proeyen [4] showed that the electric quadrupole of the spin-1 particle is completely determined in terms of its anomalous magnetic dipole moment. Our sum rule reproduces this result. Indeed, by setting and in Eqn. (3) we find the following relation between electric quadrupole and magnetic dipole:

Since the conventional quantum definition of the electric quadrupole moment is given by , and is related to by , the above relation may in fact be rewritten as (cfr. [4]):

(where the -factor sum rule (1) was also used). This result can be understood in the following way. The action of a massive, charged vector multiplet coupled to a real, massless vector multiplet can be written in superfields as [1]:

Here is the mass of and is an arbitrary constant; and are defined as in [1]. The term proportional to is the only superfield expression that contributes to the magnetic dipole. Expanding in components, indeed, one finds a term proportional to

The magnetic-dipole contribution comes by setting (). On the other hand, by setting , (), one finds a contribution to the electric quadrupole, since on shell and at low momenta :

No other quadrupole term can be written in superfields; therefore, the electric quadrupole is completely determined by the magnetic dipole, as explicitly found in [4] and implied by our sum rules. Acknowledgments. We would like to thank V. P. Nair and H. C. Ren for useful discussions. This work was supported in part by the Department of Energy under Contract Number DE-FG02-91ER40651-TASK B, and by NSF under grant PHY-9722083.

References.

[1]S. Ferrara and M. Porrati, Phys. Lett. B288 (1992) 85. [2]I. Giannakis and J. T. Liu, Rockefeller University preprint, RU-97-08-B,

hep-th/9711173. [3]S. Ferrara and E. Remiddi, Phys. Lett. 53B (1974) 347. [4]C. L. Bilchak, R. Gastmans and A. Van Proeyen, Nucl. Phys. B273 (1986) 46. [5]J. D. Jackson, Classical Electrodynamics, pp. 755–758, Wiley, New York (1975). [6]V. Rahal and H. C. Ren, Phys. Rev. D41(1989) 1989. [7]R. W. Robinett, Phys. Rev. D31 (1985) 1657.