Electromagnetic multipole moments of baryons Alfons Buchmann University of Tübingen 1.Introduction 2.Method 3.Observables 4.Results 5.Summary NSTAR 2007, Bonn, 5-8 September 2007

1. Introduction

What can we learn from electromagnetic multipole moments? Electromagnetic multipole moments of baryons are interesting observables. They are directly connected with spatial distributions of charges and currents inside baryons. They provide fundamental information on baryonic structure, size, shape.

Experimental discovery by Frisch and Stern in 1933  p (exp)  2.5  N   p is very different from value predicted by Dirac equation  p (Dirac) = 1.0  N Conclusion: proton has an internal structure. Measurement of proton charge radius by Hofstadter et al. in 1956 r² p (exp)  (0.81 fm)² Example: Proton magnetic moment

Multipole expansion of charge distribution  for a system of point charges q 0 + d 1 + Q 2 +  3 +   ~ octupole quadrupole dipole monopole +q –q–q –q–q –q–q –q–q –q–q –q–q –q–q  +++ J=0 J=1 J=2 J=3

Advantage of multipole expansion multipole operators M J transfer definite angular momentum and parity angular momentum and parity selection rules apply, e.g. only even charge multipoles few multipoles suffice to describe charge and current density Example: N(939) J i =1/2 charge monopole C0 magnetic dipole M1 Baryon B with total angular momentum J i  2 J i + 1 electromagnetic multipoles

Multipole expansion of baryon charge density

Example: N  quadrupole moment Recent electron-proton scattering experiments provide evidence for a nonzero p   (1232) transition quadrupole moment Recent electron-proton scattering experiments provide evidence for a nonzero p   + (1232) transition quadrupole moment Buchmann et al., PRC 55 (1997) 448 data Tiator et al., EPJ A17 (2003) 357 Blanpied et al., PRC 64 (2001) theory neutron charge radius

What can be learned from these results? Definition of intrinsic quadrupole moment both N and  have nonspherical charge distributions to learn more about the geometric shape of both systems, one has to calculate their intrinsic quadrupole moments Q 0  concentrated in equatorial plane r² -term dominates Q 0 < 0  concentrated along z-axis 3z²- term dominates Q 0 > 0 prolate oblate

Intrinsic (Q 0 ) vs. spectroscopic (Q) quadrupole moment body-fixed frame lab frame projection factor J=1/2  Q=0 even if Q 0  0.

Buchmann and Henley ( PRC 63 (2001) ) have calculated Q 0 in three nucleon models. All models agree as to the sign of Q 0. For example, in the quark model they find Neutron charge radius determines sign and size of N and  intrinsic quadrupole moments. Q 0 (N) = -r n 2 > 0 Q 0 (  ) = r n 2 < 0.

Interpretation in pion cloud model A. J. Buchmann and E. M. Henley, Phys. Rev. C63, (2001) prolateoblate Q 0 > 0Q 0 < 0

Summary In the last decade interesting experimental and theoretical results on the charge quadrupole structure of the N and  system have been obtained. Based on model calculations of the intrinsic quadrupole moments of both systems, it has been proposed that the charge distributions of N(939) and  (1232) possess considerable prolate and oblate deformations. Review: V. Pascalutsa, M. Vanderhaeghen, S.N. Yang, Phys. Rep. 437, 125 (2007)

What about higher multipole moments? Presently, practically nothing is known about magnetic octupole moments of decuplet baryons. Information needed to reveal structural details of spatial current distributions in baryons.

2. Method

General parametrization method Basic idea 1) Define for observable at hand a QCD operator Ô and QCD eigenstates  2) Rewrite QCD matrix element  B|Ô|B  in terms of spin-flavor space matrix elements including all spin-flavor operators allowed by Lorentz and inner QCD symmetries (G. Morpurgo, 1989)

... exact QCD state containing q, (q  q), (q  q) 2, g, auxilliary 3 quark QCD state... unitary operator dresses auxilliary state with quark-antiquark pairs and gluons... spin-flavor state... spin-flavor operator QCD matrix element spin-flavor matrix element V...unitary operator  B...auxilliary 3q states

O [i]  all allowed invariants in spin-flavor space one-bodytwo-body three-body General spin-flavor operator O constants A, B, and C  parametrize orbital and color matrix elements. These are determined from experiment. Which spin-flavor operators are allowed? Operator structures determined from symmetry principles.

Strong interaction symmetries Strong interactions are approximately invariant under SU(3) flavor and SU(6) spin-flavor symmetry transformations.

                 np S T3T /2+1/20+1-3/2-1/2+3/2+1/2 J=1/2 J=3/2 SU(3) flavor multiplets octet decuplet

SU(6) spin-flavor symmetry ties together SU(3) multiplets with different spin and flavor into SU(6) spin-flavor supermultiplets

SU(6) spin-flavor supermultiplet S T3T3 ground state baryon supermultiplet

SU(6) spin-flavor is a symmetry of QCD SU(6) symmetry is exact in the large N C limit of QCD. For finite N C, the symmetry is broken. The symmetry breaking operators can be classified according to powers of 1/N C attached to them. This leads to a hierarchy in importance of one-, two-, and three-quark operators, i.e., higher order symmetry breaking operators are suppressed by higher powers of 1/N C.

1/N C expansion of QCD processes two-bodythree-body strong coupling

SU(6) spin-flavor symmetry breaking by spin-flavor dependent two- and three-quark operators These lift the degeneracy between octet and decuplet baryons. k m σkσk

Example: quadrupole moment operator no one-quark contribution two-body three-body

SU(6) spin-flavor symmetry breaking by spin-flavor dependent two- and three-quark operators e.g. electromagnetic current operator e i... quark charge  i ... quark spin m i... quark mass   eiei ekek 3-quark current2-quark current

1-quark operator 2-quark operators (exchange currents) Origin of these operator structures

based on symmetries of QCD quark-gluon dynamics reflected in 1/N c hierarchy employs complete operator basis in SU(6) spin-flavor space Summary Parametrization method

3. Observables

Magnetic octupole moments of decuplet baryons Decuplet baryons have J i =3/2  4 electromagnetic multipoles Example:  + (1232) charge monopole moment q  + =1 charge quadrupole moment Q  +  r n 2 magnetic dipole moment   +   p magnetic octupole moment   + = ?

Definition of magnetic multipole operator special cases: J=1 J=3

Magnetic octupole moment  analogous to charge quadrupole moment Q replace magnetic moment density by charge density

Magnetic octupole moment measures the deviation of the magnetic moment distribution from spherical symmetry  > 0 magnetic moment density is prolate  < 0 magnetic moment density is oblate

Construction of octupole moment operator  in spin-flavor space  ~ M J=3... tensor of rank 3  rank 3 tensor built from quark spin operators  1  ~  i 1   j 1   k 1   involves Pauli spin matrices of three different quarks  is a three-quark operator 

If two Pauli spin operators in  had the same particle index e.g. i=j=1  reduction to a single spin matrix due to the SU(2) spin commutation relations  is neccessarily a three-quark operator

Allowed spin-flavor operators e i...quark charge  i...quark spin two-quark quadrupole operator multiplied by spin of third quark three-quark quadrupole operator multiplied by spin of third quark Do we need both?

Spin-flavor selection rules M  0 only if  R transforms according to representations found in the product 0-body 3-body 2-body 1-body

There is a unique three-quark operator transforming according to 2695 dim. rep. of SU(6).  We can take either one of the two spin-flavor structures. Explicit calculation of both operators give the same results for decuplet baryons.

4. Results

Matrix elements q B baryon charge In the flavor symmetry limit, magnetic octupole moments of decuplet baryons are proportional to the baryon charge

Introduce SU(3) symmetry breaking SU(3) symmetry breaking parameter flavor symmetry limit r=1

Baryon magnetic octupole moments 

Efficient parametrization of baryon octupole moments in terms of just one constant C.

Relations among octupole moments There are 10 diagonal octupole moments. These are expressed in terms of 1 constant C.  There must be 9 relations between them.

Diagonal octupole moment relations These 6 relations hold irrespective of how badly SU(3) is broken.

3 r-dependent relations Typical size of SU(3) symmetry breaking parameter r=m u /m d =0.6

Information on the geometric shape of current distribution in baryons from sign and magnitude of .

Numerical results Determine sign and magnitude of  using various approaches 1) pion cloud model 2) experimental N  N*(1680) transition octupole moment 3) current algebra approach

Estimate of  in pion cloud model spherical  core surrounded by P-wave  cloud ´´ 00 Y01Y01 bare  core  cloud

N      N Electromagnetic current of pion cloud model

Insert spatial current density J(r)  only  N interaction current contributes to 

Matrix element quark model relation pion Compton wave length

Comparison with experimental N  N*(1680) transition octupole moment A p ½ = -15  GeV –1/2 A p 3/2 = 133  GeV – 1/2 Rewrite helicity amplitudes in terms of charge quadrupole and magnetic octupole form factors J i =1/2  J f =5/2 transition between positive parity states only charge quadrupole and magnetic octupole can contribute G M3 (0) =  = 0.16 fm³ G C2 (0) = Q = 0.20 fm² Order of magnitude agrees with estimate in pion cloud model

 (  + ) = 4C = fm³  C=-0.04 fm³ We can now determine the constant C and predict the sign and size of  for all decuplet baryons

diagonal octupole moments [fm³]

5. Summary magnetic octupole moments provide unique opportunity to learn more about three-quark currents SU(6) spin-flavor analysis  relations between baryon octupole moments estimated sign and magnitude of  in pion cloud model  ~ Q  decuplet baryons have negative octupole moments  oblate deformation of magnetic moment distribution experimental determination of   is a challenge

END Thank you for your attention.

n u d d d d u 00 Interpretation in quark model In the  0, all quark pairs have spin 1. Equal distance between down-down and up-down pairs.  planar (oblate) charge distribution  zero charge radius. Two-quark spin-spin operators are repulsive for quark pairs with spin 1. In the neutron, the two down quarks are in a spin 1 state, and are pushed further apart than an up-down pair.  elongated (prolate) charge distribution  negative neutron charge radius. A. J. Buchmann, Can. J. Phys. 83 (2005)

Estimate of degree of nonsphericity a/b=1.1 large! Use r= 1 fm, Q 0 = 0.11 fm², then solve for a and b A. J. Buchmann and E. M. Henley, Phys. Rev. C63, (2001)

Another configuration +q –2q +q –q–q  ++ + –2q +q+q  + q + d 1 + Q 2 +  3 +   (r) = octupole quadrupole dipole monopole

Group algebra relates symmetry breaking within a multiplet (Wigner-Eckart theorem) Y hypercharge S strangeness T 3 isospin symmetry breaking along strangeness direction by hypercharge operator Y Relations between observables M 0, M 1, M 2 from experiment

Gell-Mann & Okubo mass formula Equal spacing rule

Gürsey-Radicati mass formula Relations between octet and decuplet masses SU(6) symmetry breaking part e.g.

Example: charge radius operator e i...quark charge  i...quark spin

Decomposition of 2695 into irred. reps. of SU(3) F and SU(2) S subgroups 2695 = (1,7) + (1,3) + (8,7) + 2 (8,5) + (8,3) + (8,1) +  There is only one (8,7) in the decomposition of 2695  unique three-quark magnetic octupole operator transforming as flavor octet. flavorspin

Relations between N   transition form factors and elastic neutron form factors Buchmann, Phys. Rev. Lett. 93 (2004)

Definition of C2/M1 ratio neutron elastic form factor ratio

data: electro-pionproduction curves: elastic neutron form factors A.J. Buchmann, Phys. Rev. Lett. 93, (2004).