Experimental consequences of massless quarks

Volume 48B, number 5

PHYSICS LETTERS

EXPERIMENTAL

CONSEQUENCES

4 March 1974

OF MASSLESS QUARKS

R . L . HEIMANN

Cavendish Laboratory, Free School Lane, Cambridge, UK Received 15 October 1973 We calculate the Bjorken scaling function for v OL/OT in the light-cone algebra with no order 1/v corrections to scaling of rt¢2. Quarks are then massless, and polarized electroproducfion is suppressed by a power of to at large to. In inelastic electron-proton scattering, Bjorken scaling is observed for vI4/2, with a small value for R = OL/OT [1]. It is this that has suggested that one abstracts the leading light-cone singularity in local current commutators from that in the free (or formal gluon) quark parton model [2, 3]. From this follows the scaling law [4, 5]

lim(v/M)R = r(60),

co =- 2 M v / - q 2.

(1)

Bj It is fully consistent with the available data [ 1], and in more recent analyses may even be preferred over other rules [6]. Here we consider the generally unknown function r(co) in the context o f the light-cone algebra. We show r(co) to be calculable in terms of F2(co)=limBj(V/M)W 2 to within one constant, whenever there are no,order 1Iv corrections to the scaling of vW2: the el-ectromagnetic current commutator has a 6'(x2), but no 6(x2), singularity .1 . In formal field theory, the bare quark 'mass is zero. These conclusions rely upon gauge invariance for the photon coupling to hadrons* 2. we fmd that with massless quarks in the above sense, [7], (A) as 60 --> ~ in the Regge region of F2(60 ) cc coC~(0)-i r(60) -->[2/c~(0)] (1/6o) if a(0) > 0

(2)

sharply contrasting with the behaviour expected otherwise, r(60) ~ co [8] , a . (B) For any co, r(co) X F2(co ) :1:I p([4/1 + [ i)2/q 2 ] W2) will also have no or der 1fi, correctiofis, but I¢1 and vR will. Scaling for vWi s ~ e c o c i o u s " . More precisely, we should use w1-=M/(x/v'-q ~- 9)--,to (l+M/2vto...). ~2 They do not apply to neutrino scattering, unless (as may be the ease), the weak interactions were to be gauge invariant. ,a These powers of to are independent of a(0). Powers between them are forbidden for maximal Regge behaviour. Of course, a(0) ~- 1 in practice (pomeron).

satisfies a first order linear differential equation; the solution has one unknown constant, G(1):

(2/60) [f ~ d60'F2(60')--2 G(1)l r(60) =

F2(co)

(3)

(C) in polarized inelastic electron-nucleon scattering, the scaling function o f u d(u,q2), [e.g. 9 ] , can have no otherwise allowed Regge singularity above / = 0. This follows from the Schwarz inequality [10] relating R and d(v,q2) .4 . Thus

lim ud(u,q 2) ~ const .60

as co ~

Bj instead o f the expected behaviour ~ [co2/(log 60)c] from the exchange of the (allowed) pomeron cut [ 1 1] The uncertainty of the value of G(1) in eq. (3) has no significant effect Otl r(60) for large 60 if, as appears experimentally, a(0) > 0. We have previously [7] used this convenient device of avoiding G(1) to study (A), eq. (2), as a means o f detecting zero mass for partons. Although extensive (and accurate) data on r(60) does not exist beyond 60 -~ 5 - 7 , measurements of pmeson electroproduction [ 12] and parton model considerations both gave no serious indication of (A) being correct, but not conclusively so in view of the limitations of the data and attendant assumptions. To consider (B), we argue that G(1) = 0 in eq. (3), by regarding r(w) near threshold, w ~-I, where G(1) is most important. As 6o --, 1, the Drell-Yan-West relation or BloomGilman duality give F2(60 ) ~ a(60-1) n, where , 4 d --- (G1 + [ v/M] G2) in the notation of ref. [9]. It corre-

sponds to scattering with the nucleon polarized transversely to the virtual photon direction. The Schwarz inequality is WIX/(u/M)R ~ (2/.v/'~-)IMpd L 453

Volume 48B, n u m b e r 5

r(~,) I

l

[Jl't(x),jv(O)] =

(~.o~..o+~o~,o_~,.~oo~

. . . . . . . . L- ~

0 2

I0

I0O

CO

Hg. 1. Graph of the predicted scaling function, (v/M)R~r(to), with G(1) = 0 = r(1). The dashed line is the asymptotic curve 2/to. The data have (r(to)> ~ 3 ~: 3 over to = 1-5. The lower bound on r(to) from the highest _q2 data of ref. [12] is shown.

GM(q2 ) = ( _ q 2 ) - ( n + l ) / 2 as _ q 2 -. oo. The data agree well and give n ~ 3. If these rules were also to apply to R, the nucleon Born term yields r(1): lim _r(,o)= 2

lim

J PUV(x'O) + [other

terms].

r(W)~n~(6o_l)_4

(4)

G(1)

(6)

Singularities stronger than 6'(x 2) are forbidden by the algebra; the other terms of eq. (6) include terms weaker than 6(x2), and those between these two. The latter are suppressed in the Bjorken limit by an extra 1/u 2, owing to the current conservation restriction: the divergence of eq. (6) must be less than 6'(x 2) as x 2 -~ 0. Similarly, the bilocalpu~'(x,O), with 8(x 2) singularity, is related to Bo(x, O) [7] :


(7) (plx°~VB o(X, O)-x.aBV(x, O)-xVa .B(x,O) lp) (plPUV(O,O) lp) = - lgUV (plb .B(0,0) lp)

_q2ookGM(q2)J

(8) - ig ~v
to compare with that from eq. (3):

1

(6°- 1) n .

(5)

Unless G(1) = 0, eqs. (4, 5) require GE(q2 ) to decrease ( _ q 2 ) - 1 / 2 only. SU(6) predicts GE/G M ~ la~ 1 ; the data (extending out to _ q 2 ~ 3(GeV/c)2) suggest the ratio to be less than this and quite possibly falling. We take this to indicate that r(1) is finite and quite possibly zero, so G ( 1 ) =- 0 in eq. (5). Fig. 1 shows r(6o) calculated from eq. (3) with G(1) = 0. Typically, r(co) is of order "- 0.1 - 0.2 except near ~o ~ 1 and 6o- 1 ~ 0. To within ~ 10%, r(co) ~ ¼ ( c o - l ) for w ~< 1.3, and ~ 2/w for u~ >~20 *s . The available data go up to w ~ 7; typically, r(co) "~ 3 -+ 3, so very much greater accuracy is needed 454

r18(x)e(x 2 o )] c aoL 47r jBIx.0.

so that as x -* 0 (locally)

[%(02)]2 to~l ¢.o

4 March 1974

to test B. By contrast, (A) can be ruled out with only moderate accuracy at large co >~ 1 0 - 2 0 , possibly by using polarized electroproduction, (C) [7], or p-meson electro-production [ 12]. The theoretical argument leading to eqs. (1) to (3) involves the role of gauge invariance of the electromagnetic coupling of hadrons in restricting the bilocal operators in the light-cone expansion:

rt

0"75 1

19"1

PHYSICS LETTERS

in the quark model [16]. Eq. (8) def'mes the bare quark mass matrix M, and with no 6(x 2) singularity present, PUV(0,0) - 0; this mass is zero. Then, eq. (7) gives a differential equation upon Fourier transforming [ 171 :

*5 For to ~<10, we have computed the integral over the data

of ref. [ 1 ]. For to >~ 10, we have used the fit of Barger [ 13 ], which gives an excellent fit to the ep and en data, is well motivated by parton and Regge ideas, and predicts v,'~N data in startling agreement with experiment [ 14]. (I thank Prof. Feynman for sending me ref. [14]). For to > 1 0 , the Regge parameters should be taken less seriously, as they are not fixed by fitting in the Regge region. But they are very similar to those in a fit [15] to the smaller _q2 data, with equal amounts of pomeron and f - A 2 exchanges.

Volume 48B, number 5

PHYSICS LETTERS

d [coG(co_l) ] = ½F2(co )

(9)

dw

where G(co-1), F2(co ) are given by the matrix element

(plBo(x,O)lp)=4paFl(X'p) + 2iM2xo G(x'p) (1o)

4 March 1974

where ~pu + ku is the i-th parton m o m e n t u m with probabilityfi(~,k2 ;pz) asPz ~ oo.pu is the hadron m o m e n t u m , ku essentially the transverse parton momentum, with binding corrections. As co -+ ~, [7], r(o0) ~ A r(1) co, A ~ 1 for rough SU(3) symmetry, using simple parton model ideas on the fi at ~ = 0 and 1:

FI (~) = : d(---X2:exp [-i~(x'p)] Fl(X'p) etc., ~=l

¢0

from which in the Bjorken limit v /:2 W2 -+ F2(w ) = 2 ~FI(~) and W1 +~-~ W2 ~ 0, R -~ 0 but it is only with gauge invariance that pR scales,

__, .~n 2~

(~)

2 (m 2 +(k2)u,~_l); i.e. finite or zero. M2 "-

Here, scaling for vR may be avoided [20], but only by having a bizarre Pz'dependence for the first k2-moment, lim

~a~f d2kk2.f,(~,k2;ez)

pz-+~ l

=

- r(w).

With no 6(x 2) singularity present, eq. (9) then yields

r(w) -

r(1)proton_

4 G(to - I ) w F2(w)

(11)

and eqs. (9, 11) derive eqs. (2, 3), and the quark mass is zero. If F2(6o ) is dominated by a Regge pole at j = a(0), eq. (9) forces G(co -1) to be dominated b y j = a ( 0 ) - 2 and not by a(0), and pd(u,q 2) b y j ~<~(0) - 1 from the Schwarz inequality [7]. A sharper result follows only with a stronger assuhaption. Consider the high spin ( < j ) local operators of twist 2 within the bilocal:

oe2....~jI Bo(x'O) = f= 2(L~everl) (Xotlxot2......Xoe./_l)bf(0)~I

even though the zero moment, i.e. 2 F 1 (~), is still to scale. Analogously, the light-cone algebra could have I G ( x ' p ) l ~ Ix.p1-8 asx'p ~ O.Provided 6 < 1 , the bilocal matrix element is still finite, and vl-~R scales. The Wilson expansion would fail, making the algebra implausible. A full account has been presented elsewhere [7]. Scaling of vR is a prerequisite for our considerations. Experiments on R, as well as electro-production with a polarized target and beam, and of p-mesons, will decide this, and whether massless quarks are possible. I thank Profs. R_P. Feynman and R. Jackiw, and Drs. A J . G . Hey and H. Osborn, for relevant discussions; and Profs. B. Zumino, S.D. Drell and M. Gell-Mann for hospitality at CERN, SLAC, and at Caltech where this work was begun, at various times during the summer of 1973.

For massless quarks, eq. (8) yields b2(0)~--Trace b2(0 ) = 0. If aU these operators b/(O)were to form a single Regge trajectory [3,18], then (pl Trace b/(O)Ip) = 0 for all/ and from eq. (10) one again finds r(co) ~ 2/co, but if and only if the quark is massless. In the quark model, b2(0)°a is a U(3) partner of the stress tensor, traceless for massless quarks. Similarly, in the quark parton model [19] : 4

R ~_q2

(M2+k2) so r(~o) = 2 ~ (M2+_k2)l

M 2 1 ~ =t°-I

References [1] D. Miller et al., Phys. Rev. D5 (1972) 528. [2] H. Fritzsch and M. Cell-Mann, Proc. 7th Coral Gables Conf., Gordon and Breach, New York (1971). [3] J.M. Cornwall and R. Jaekiw, Phys. Rev. D4 (1971) 367. [4] D.A. Dicus, R. Jaekiw and V.L. Teplitz, Phys. Rev. D4 (1971) 1733; D. Palmer, Phys. Letters 39B (1972) 517. [5] J.E. Mandula, Phys. Rev. D8 (1973) 328. [6] R. Jackiw, private communication [7] R.L. Heimann; University of Cambridge, Cavendish Laboratory preprint, HEP 73/11, August 1973.

455

Volume 48B, number 5

PHYSICS LETTERS

[8] J.F. Gunion and R.L. Jaffe, MIT preprint 342 (March 1973, unpublished). [9] A.J.G. Hey and J.E. Mandula, Phys. Rev. D5 (1972) 2610. [10] M.G. Doncel and E. de Raphael, Nuovo Cimento 4A (1971) 363. [11] R.L. Heimann, Nucl. Phys. B (to appear). [12] J.J. Sakurai, Phys. Rev. Lett 30 (1973) 245; J.T. Dakin et al., Phys. Rev. D8 (1973) 687. [13] V. Barger, Wisconsin preprint (1973). [14] R.P. Feynman, Talk given at Dansk Ingeni¢rforening, Copenhagen; Caltech preprint (Oetober 1973). [15] Y. Matsumoto et al., Phys. Lett. 39B (1972) 258.

456

4 March 1974

[16] R. Jackiw and H.J. Schnitzer, Phys. Rev. D7 (1973) 3116, Phys. Rev. D5 (1972) 2008. [17] K.T. Mahanthappa and T. Yao, Phys. Lett. 39B (1972) 549. [18] M. Bander, Phys. Rev. D4 (1971) 1237; M. Testa, Phys. Lett. 42B (1972) 267. [19] R.P. Feynman, Photon-hadron interactions (W.A. Benjamin Inc., Reading, Mass. 1972). [20] H. Osborn and G. Woo, Cambridge preprint DAMTP 73/13 (April 1973); R. Jackiw and R.E. Waltz, Phys. Rev. D6 (1972) 702.