The necessity for a Melosh-type transformation

Nuclear Physics Bl17 (1976) 356-364 © North-Holland Publishing Company

THE NECESSITY FOR A MELOSH-TYPE TRANSFORMATION Jos6 A. de A Z C A R R A G A * Department of Theoretical Physics, Oxford University A n t h o n y J.G. HEY Department of Physics, Southampton University Received 17 June 1976

The identification of null-plane U(6)W, currents with U(6)W' constituents is shown to exclude the existence of the 56 (L z = 0) baryons. The subgroups SU(2)W, currents and U(3) × U(3)current s also lead to similar theoretical contradictions. These results, obtained by investigating the implications of these symmetries on the NN% ~A3, and AN7 vertices, lend additional support to the need of a Melosh-type transformation relating the constituent and current charges. 1. Introduction In order to avoid Coleman's theorems for spacelike charges [1], the search for a relativistic U(6) symmetry has focused on the U(6)w algebra of null-plane charges [2] - the U(6)w, currents -- and subgroups such as null-plane chiral U(3) ® U(3) [ 3 ] However, attempts to identify this algebra of currents with the symmetry group of physical hadronic states - i.e., to identify the generators of U(6)w, currents with those of U(6)w, constituents -- lead to experimental and theoretical contradictions. Dashen and Gell-Mann [3] showed long ago that such a symmetry predicts zero anomalous magnetic moment for the nucleons, which is clearly not a good phenomenological approximation to the physical world. Following the suggestion of Gell-Mann [4], Melosh [ 5 - 7 ] * * and others [8] have constructed a non-trivial unitary transformation Vrelating the two sets of charges***. Melosh [6] argued that an algebra of current charges symmetry is not theoretically possible. However, since his argument involved the use of free quark and antiquark wave packets to simulate hadronic bound states, there was a possible loophole On leave of absence from Salamanca University. Partially supported by a British Council Scholarship. Present address: Fi'sicaMatem~tica, Facultad de Ciencias, Salamanca, Spain. ** Ref. [5] contains the "first" Melosh transformation. A review of the motivations and difficulties of the two Melosh transformations is given in ref. [7]. *~'~ Mixing schemes having their origin in the failure of SU(6)W charge symmetry were earlier proposed in ref. [9] and others. 356

J.A. de Azcdrraga, A.J.G. Hey / Melosh-type transformation

357

This loophole was closed by the demonstration that a hypothetical current charges symmetry (i.e., V =/) would exclude [10] the existence of the simplest composite system, the 35 (Lz = 0) mesons. In this paper, we extend these considerations by a study of the complete set of constraints for the ground state 56 (Lz = 0) baryon multiplet. Our conclusion is that the consideration of the baryon sector alone shows that the U(6)w, currents symmetry and its subgroups SU(2)w and U(3) ® U(3) are untenable when all electromagnetic transitions within the 56 (Lz = 0) are taken into account. Thus, any attempt to relate the U(6)w algebra of currents to the U(6)w classification algebra of hadronic states must involve some non-trivial Melosh-type transformation V. These results are obtained by exploiting the conditions that the transformation properties of the "good" components of the electromagnetic current --

1

J~e.m. = X/~(J~e.rn. +Je3.m.) ,

(1.1)

under the U(6)w ' currents generated by

x/2 f d4xa(x+)q+(x)t a i q+(x) , A i = l x i ® ( 1 , ~ O x , ~ { 3 ~ y , lOz),

(1.2) i = 0 , 1 ..... 8 ,

(~=O,x,y,z ,

(1.3)

impose on the structure of the general electromagnetic vertex, whose form is in turn dictated by Lorentz invariance and current conservation. Since ordinary boosts do not commute with null-plane charges, states of arbitrary momentum must be defined from those at rest by means of the special boost B [ 11 ] which belongs to the stability group of null-plane charges. This boost differs from the hermitian boost by a rotation, and it is given by

B(p) = exp(-i V±E±) exp(-iM<3) ,

(1.4)

where

Vl=p±/~,

e~ =x/~rl/m,

rl = (oo + p3)/x/2,

(1.5)

and E± are the "Galilean boosts" defined by E1K

1 +j2

E 2 _ K 2 _ j1 '

'

(1.6)

Y and K being the usual Lorentz generators. Arbitrary momentum states in the B basis (null-plane or infinite momentum helicity basis) are obtained from those at rest by means of (1.4) and labelled Ir/, p£, X) -= I11,X) where X is the eigenvalue of Fz - Wz (and of 4 for L z = 0 states). Thus, for calculations with the NNT, ANy and AA7 vertices we require the corresponding Dirac and Rarita-Schwinger [12] null-plane helicity spinors. These are readily ob*ained by means of (1.4) and their full expressions are given in the appendix.

J.A. de A zchrraga,A.J.G. Hey /Melosh-typetransformation

358

The plan of the paper is as follows. In sect. 2 we examine the consequences of the SU(2)w subgroup of the U(6)w ' currents symmetry for the NN% AA3, and AN'), vertices and give a physical argument concerning the minimal analyticity assumption required to continue the results to zero m o m e n t u m transfer. Sect. 3 is devoted to the "larger" symmetries U(3) ® U(3) and U(6). Sect. 4 collects our conclusions and the appendix contains the form of the null-plane helicity spinors required for the calculations. 2. Electromagnetic form factors of the (56, Lz -- 0) and SU(2)w, currents symmetry +

Under the W-spin [131 subgroup A = (½(3Ox,12(3Oy,½Oz)of U(6)w, currents Je.m. is a singlet operator. Thus, the SU(2)w, currents symmetry prediction takes the form (11', W', X'lJ~e.m.I~, W, X) cc ~iww, 6x~, ,

(2.1)

where W(W') is the W-spin of the initial (final) state. Let us consider first the NN'), vertex. This is described by the well-known expression (N0f), X' IJ~(0)LN01 ), X)

E

=u01,X)

u ( F l ( q )+FN2(q2)) - ~ F N 2 ( q 2 u(rl, X)

(2.2)

with q = p' - p and P = p' + p. The requirement (N(q'), 12IJ+ W(~), _ l ) = 0 ,

(2.3)

evaluated with null-plane helicity spinors leads to the Dashen and Gell-Mann result

FN(q 2) = 0 ,

(2.4)

for q2 4= 0 and, assuming a minimal amount of analyticity for the form factor, also at zero m o m e n t u m transfer. No constraint is, however, obtained on FU(q2). The general expression of the AA7 vertex is given in terms of 2. ~ + 1 form factors and, for the purpose of calculations, is best written in a form (involving a minimum number of 7%) which parallels (2.2),

~/x(ff), x' G(o) L/x(n), x> = --U'o(TI', X') {7u [(F~ + F~)g °° + (F'~ + F~)q°q°/2M2 l k

e.

[F~g°° +F~q°qa/2M2 ]1 UoO1,X) J

(2.5)

where F/zx = F~(q2). The structure of (2.5) is obtained [14] from general invariance

J.A. de A zchrraga, A.J.G. He), / Melosh-type transformation

359

principles and the fact that the spinors Up satisfy the usual Rarita-Schwinger conditions. At zero momentum transfer, F ~ gives the electric charge; the values of the other form factors are given in terms of the electric charge, the quadrupole electric moment and the dipole and octupole magnetic moments [14]. Using the expression of the null-plane helicity Rarita-Schwinger spinors (appendix) we obtain from

= o,

(2.6)

Fa~(q2) = 0 ,

(2.7)

that

for q2 4= 0. Now, from (A(rl'), --~ iJ+ lAx(n), 3) = 0 ,

(2.8)

we get (with F4a = 0) F3~(q2 ) = r/+r/'- F2~(q2), r?

(2.9)

and since q2 is not determined by the ratio (r? + 7')/77 alone, this requires (also for q2 @ 0) that *

F3~(q 2) = F2~(q 2) = 0 .

(2.10)

Finally, from (A01'), ~ IJ + fAx(rl), ~) = 0 ,

(2.1 1)

we obtain with F~(q 2) = 0, i = 2, 3, 4, the key result

F~(q 2) = 0 .

(2.12)

Again, minimal analyticity for the form factors gives that they also vanish at q2 = 0. This would imply, in particular, Fla(0) = 0 in direct contradiction with the charge normalization condition (for AX+)F~(0) = 1. As for the AN,), vertex, its general structure [1 5]** is given, for instance, in the form (A01 '), X' IJu(0 ) IN(n), X} = ~o(~ ,, X,)3,s [C7O + Dp,O ] (qoguo - qogua)u(R, X). • Eq. (2.9) is also incompatible with the condition one obtains from

(A(n'),-~-~J+,A(W,~)=O,

F~(qb=(~+'/) F2~(qb. r~

• * General forms of vertices are given by Barbour and Malone [15 ].

(2.13)

360

ZA. de A zcdrraga, A.J.G. Hey /Melosh-type transformation

The W-spin constraints for this vertex immediately require C(q 2) and D(q 2) to vanish, with the result that the A cannot be photoproduced! It is clear therefore that the consideration of all the W-spin e.m. transitions within the 5 6 Lz -- 0 multiplet produces serious theoretical contradictions for this hypothetical current charges symmetry. These are due mainly to the seemingly innocent continuation from arbitrarily small momentum transfer to zero momentum transfer - a continuation that any bound state model with reasonable dynamics should clearly allow. However, our results reflect exactly the behaviour to be expected for baryon " b o u n d " states made from non-interacting quarks. At zero momentum transfer one merely measures the charge, but at any non-zero q2 momentum will be transferred to one of the quarks. Since there are no forces binding them, the "baryon" breaks up and the elastic form factor vanishes (this free quark model "bound state" clearly does not satisfy our minimal analyticity assumption). In fact, our results lead us to suspect, remembering some of the old SU(6) S-matrix "no go" theorems [16], that the SU(2)w, currents symmetry -- which here appears "combined" with the Lorentz symmetry through the structure of the vertices - is too strong to allow any interaction.

3. Consequences of the SU(6)w and U(3) ® U(3) current charges symmetries Since SU(2)w ' currents is contained in SU(6)w ' currents, this larger symmetry also gives zero form factors for the AA7 and AN3, vertices. However, for the NN7 vertex the additional result FIN(q2) = 0 is obtained. To see this, it is sufficient to note that f~e.m, transforms as an element of a 35 with zero W-spin. Thus, we can use the WignerEckart theorem to relate the matrix elements of (56, W = ~ IJ+(35, W = 0)156, W = ½) (known from the NN7 vertex general expression) and (56, W = 2a 1J+(35, W = 0)[56, W = ~) (determined by the AA 7 vertex expression). Since 56 ® 35 contains the 56 only once, both matrix elements will depend on only one common reduced matrix element. For instance * for the nucleon we shall have (56, W = 2I 1J+(35, W = 0)156, W = 1) cc

{unitary SU 6 /fW sPin /f //SU 2 / (5611351156). scalar J[CG coeff,) / is°sCalarl factor

_factor

) [ C G coeff.J

(3.1)

From a similar expression for the A we find (5611351156) -- 0 and then from (3.1) the

SU(6)W, c u r r e n t s symmetry prediction F~(q 2) = O. Similar results are obtained by using the null-plane chiral charges [U(3) ® U(3)] ® 0(2)L z as a symmetry. In fact, under this subgroup of U(6)w ® O(2)Lz, the various helicity states of the 5_6(Lz = 0 ) [10, X = ~], [10 • 8, X = ~], [8 . 10, X = - ½ ] , [10, X = - 3 ] transform [181 as [(10, 1)3/z, 01, [(6, 3)1/2, 01, [(3, 6)_~/a, 01, [(1, • SU(6) Clebsch-Gordan coefficients are given in ref. [ 17 ] but they are not necessary here.

J.A. de A zehrraga, A.J. G. Hey / Melosh-type transformation

361

10)_3/2,0] ;J+ transforms as (8, 1)0 + (1,8)0. Thus, since Wz is a good quantum number for U(3) ® U(3), Wz conservation immediately gives, as in sect. 2, the predictions F2N = Fff = Clvza = D Nzx = 0. To obtain in addition F f ( q 2) = 0 it is sufficient, for instance, to consider the U-spin subgroup [U(2)u ~ U(2)u ]. Since J+ is a singlet under this subgroup, we can relate by means of the appropriate group element the matrix elements (AIJ + IA),

(NIJ + IN),

and since the former is zero, the result FN(q 2) = 0 follows from the second also being zero.

4~ Concluding remarks The contradictions we have found amongst the electromagnetic couplings of the ground state (~_6_,Lz = 0) multiplet of baryons under the assumption of a current charge symmetry suggest that this symmetry may be consistent only with the "dynamics" of free quarks. The crucial element of our argument is the fact that the nullplane symmetry generators, and in particular Wz, commute with the generators of Kz and Euclidean E± boosts - which belong to the stability group of the null plane. Thus, although the U(6)w symmetry singularizes the z direction, it can nevertheless be used to obtain predictions for non-collinear situations. In order to avoid such contradictions, the presence of some non-trivial Melosh type transformation which does not respect at least the E± invariance of the null-plane charges seem to be necessary Melosh's explicit free quark examples [5,6] did not commute with the E~ boosts and thus only allowed consideration of collinear situations, avoiding these inconsistencies. Nevertheless, the construction of a transformation for an interacting situation seems to be required for a theoretical understanding of the successes of the U(6)w scheme - its absence certainly leads to inconsistencies and perhaps to free quark dynamics. However, our analysis does not illuminate the specific form of the transformation. In the same way, this general approach does not show why the present examples of the Melosh transformation can be understood in terms of Wigner rotations [19] nor clarifies other intuitive arguments [20] which interpret such transformations as relating certain reference frames. Clearly, one possible way for progress is to investigate all the implications of a Melosh transformation both with respect to general model independent properties and in the context of a specific interacting quark model * These questions are presently being considered. J.A.A. Wishes to thank the Department of Theoretical Physics, Oxford, for its hospitality, and A.J.G.H. thanks Professor K.J. Barnes and Dr. H. Osborn for helpful * Work in this line has been done, e.g, by the authors ofref. [21].

362

J.A. de A zc&raga, A.J.G. Hey

/Melosh-type transformation

discussions. A.J.G.H. also wishes to acknowledge a valuable discussion between members of the CERN theory division, and in particular remarks made by Drs. Close and Amati.

Appendix Rarita-Schwinger infinite momentum helicity spinors The required null-plane spinors may be obtained by boosting the corresponding rest spinors by means o r B [(1.4)]. Substituting into (1.4) the appropriate form of the infinitesimal generators we find that the boost B takes the form (to avoid too many ~/2 factors we have used here a = X/2r7 = co + p3)

I a + m 2 + p2 BO/2, WZ)(p) =

2mpl

2mp2

a 2 _ rn 2 + pat

2ap i

2am

0

2ap 2

2ap 2

0

2am

2ap z

_2mpl

_2mp2

a 2 + rrt2 _ pat

La 2 _ m 2 _ p2

(A.1)

in the vectorial representation, and the form [P+ -= 21(1 -+ a3)]

B(1/2,0 * O, 1/2)(/9) -

P+a + P _ m + P_7°"/ip± N/~

(A.2)

in the Dirac spinorial representation. By using (A.1) and (A.2) one gets

c~(,1, ~) = e°(,1, 1)u(~, ~),

(A.3a)

u°(~, ½) -_ / vr ~ e o,m, Ou(~, ~) + x/~3 c°(q, o).(n., t)

(A.3b)

UO(q, _ l ) = ~

(A.3c)

eO(q, - 1)u(q, "0 + ~

vo(n, _3) __ eO(n, -1)u(n,

e°(rl, O)u(q, $),

~) ,

(A.3d)

where [p+ =__-pl + ip2],

-1

e°(q, 1) = 7 ~ 2 (p+,a, , a , - p + ) ,

(A.4a)

[e°(rest, 1) : -X/~-(0, l, i, 0)] ,

e°(n, o) -_ ~ -i m (a 2 _ m 2 + p 2 , 2 a p l , 2ap2, a 2 + m 2 _ pat) ,

(A.4b)

J.A. de Azc6rraga, A.J. G. Hey/Melosh-type transformation

363

[eO(rest, 0) = (0, 0, 0, 1)] ,

e°OI, -- 1) = +

(19_, a, --ia, --p_ ) ,

(A.4c)

[c°(rest, - 1 ) = X/}-21-(0, 1, - i , 0)] , and the Dirac spinors (with normalisation u+u = 2co or K3`+u = 2a) are given b y

,

=

uOI, t $ )

o3~bt,t = +q~t+ ,

(A.5)

~ rno3 - ~ P i

in the realization o f the 3' matrices where c~3 is diagonal,

0~3 =

(, o) ,

0

--1

3`0 =

,~ =

(o o o) \

0

-03~r I

,),s = 03

0

--o 3

3`3 = '

(o f) 03

'

)

It is simple to check that the Uo's satisfy (~ - m)U o = O, "T°Uo = 0 and are normalised to --U~aU° = 2co (we use timelike metric + - - - ) .

References [1] S. Coleman, Phys. Letters 19 (1965) 144; J. Math. Phys. 7 (1966) 787; S. Okubo, Nuovo Cimento 42A (1966) 1029; 44A (1966) 1015; C.A. Orzalesi, Rev. Mod. Phys. 42 (1970) 381. [2] J. Jersfik and J. Stern, Nucl. Phys. B7 (1968) 413; H. Leutwyler, Springer Tracts in Mod. Phys. 50 (1969) 29. [3] M. Gell-Mann, Strong and weak interactions, ed. A. Zichichi (Academic Press, NY, 1966); S. Fubini, G. Segr6 and J.D. Walecka, Ann. of Phys. 39 (1966) 381; R. Dashen and M. Gell-Mann, Symmetry principles at high energy, 1966 Coral Gables Conf. eds. A. Perhnutter et al. (Freeman, 1966); It. Harari, Spectroscopic and group theoretical methods in physics, eds. F. Bloch el al. (North-Holland, 1968). [4] M. Gell-Mann, Acta Phys. Austr. Suppl. 9 (1972) 733. [5] lt.J. Melosh, Ph.D. Thesis, Cal. Inst. of Technology (26-11-1973), unpublished. [6] H.J. Melosh, Phys. Rev. D9 (1974) 1095. [7] J.S. Bell, Acta Phys. Austr. Suppl. 13 (1974) 395. [8] S.P. De Alwis, Nucl. Phys. B55 (1973) 427; E. Eichten, F. Feinberg and J.F. Willemsen, Phys. Rev. D8 (1973) 1204; S.P. De Alwis and J. Stern, Nucl. Phys. B77 (1974) 509; H. Osborn, Nucl. Phys. B80 (1974) 90.

364

J.A. de A zc6rraga, A.J.G. Hey / Melosh-type transformation

[9] F. BuceUa, H. Kleinert, C.A. Savoy, E. Celeghini and E. Sorace, Nuovo Cimento 69 (1970) 133. [10] J.S. Bell and A.J.G. Hey, Phys. Letters 51B (1974) 365. [11] K. Bardacki and G. Segr~, Phys. Rev. 159 (1967) 1263; H. Leutwyler, ref. [2] ; J.M. Kogut and D.E. Soper, Phys. Rev. D1 (1970) 2901; D.E. Soper, Phys. Rev. D5 (1972) 1956. [12] Yu. V. Novozhilov, Introduction to elementary particle theory (Pergamin, 1975) p. 110; P.R. Auvil and J.]. Brehm, Phys. Rev. 145 (1966) 1152. [13] H.J. Lipkin and S. Meshkov, Phys. Rev. Letters 14 (1966) 670; Phys. Rev. 143 (1966) 1269; K.J. Barnes, P. Carruthers and F. von Hippel, Phys. Rev. Letters 14 (1965) 82. [14] M. Gourdin, Nuovo Cimento 36 (1965) 129; M. Gourdin and J. Micheli, Nuovo Cimento 40A (1965) 225. [15] M. Gourdin and Ph. Salin, Nuovo Cimento 27 (1963) 309; 1.M. Barbour and W. Malone, Nucl. Phys. B82 (1974) 477. [16] T.F. Jordan, Phys. Rev. 139 (1965) B149; 140 (1965) B766; S. Coleman and J. Mandula, Phys. Rev. 159 (1967) 1251. [17] C.L. Cook and G. Murtaza, Nuovo Cimento 39 (1965) 531; J.G. Carter, J.J. Coync and S. Meshkov, Phys. Rev. Letters 14 (1965) 523. [18] H. Harari, ref. [3]; J. Weyers, Particle interactions at very high energies, eds. F. Halzen et al. (Plenum Press, 1974. [19] E. Eichten et al., ref. [9] ; F. Bucella, C.A. Savoy and P. Sorba, Nuovo Cimento Letters 25A (1974) 331; V. Aldaya and J.A. de Azc~trraga, Pbys. Rev. DI4 (1976) 1049. [20] H.J. Lipkin, New directions in hadron spectroscopy, ANL-HEP-75-58; eds. S.L. Kramer and E.L. Berger (Argonne National Lab., 1975). [21] J.S. Bell and H. Ruegg, Nucl. Phys. B93 (1975) 12; B104 (1976) 245; R. Carlitz and Wu-Ki Tung, Phys. Rev. D13 (1976) 3446.