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Quark selection principle. II. Pseudoscalar meson form factor
ANNALS
OF PHYSICS:
Quark
63, 248-260
Selection
(1971)
Principle.
II. Pseudoscalar
Meson
Form
Factor*+
L. I. SCHIFF Institute of Theoretical Stanford University,
Physics, Department Stanford, California
of Physics, 94305
Received July 29, 1970
The quark selection principle proposed earlier is applied to the calculation of the elastic electromagnetic form factor of a pseudoscalar meson. The Bethe-Salpeter equation is inapplicable, and it is assumed instead that the quark-antiquark wavefunction satisfies a free Dirac equation for each particle. A spatial boundary condition is applied instantaneously in the rest system, and the dependence on relative time is inferred by adapting an argument given by Wick. Initial and final states are then obtained by applying appropriate Lorentz boosts to the resulting rest-system wavefunction. It is found that the form factor falls off for large momentum transfer like l/q2.
1. INTRODUCTION The assumption of a quark selection principle that has a range built into it was proposed several years ago as a means for reconciling the successesof the naive quark model of hadronic structure with the apparent experimental unobservability of free quarks. This principle may be expressedas follows. Quarks and antiquarks can exist in any combinations provided that clusters of noninteger baryon number or electric charge cannot separate from other clusters by more than some characteristic distance R. As a special example of this principle, a single quark cannot exist at a distance greater than R from other hadronic matter. In a recent paper [I] the principle was formulated as a boundary condition which causesthe manyparticle wavefunction to vanish whenever the selection principle is violated. It may then be thought of as analogous to the exclusion principle, which causesall fermion wavefunctions that are not fully antisymmetric to vanish. It must of course be recognized that should the existence of fractionally charged hadrons be established experimentally [2], the work reported here would have at the very least to be greatly modified. * Dedicated to the memory of Amos de-Shalit, physicist, public servant, and friend. + Research supported in part by the Air Force Office of Scientific Research under AFOSR Contract F44620-68C-0075.
248
QUARK
SELECTION
PRINCIPLE,
II
249
The earlier work [l] was based on the nonrelativistic Schrodinger equation. This is a good approximation for the internal motion, since quarks were found to have a rest energy of about 336 MeV and much smaller kinetic energy [3]. However, the calculation of the electromagnetic form factor for large momentum transfer cannot be carried through convincingly as suggested earlier, since there is no unique way of applying a Lorentz boost to a nonrelativistic wavefunction. The present paper, therefore, makes use of relativistic particle theory. The wavefunction for the simplest hadronic system, the pseudoscalar meson composed of a quark and an antiquark, is found in its rest system on the basis of certain assumptions concerning the relativistic wave equation and boundary condition. Fully covariant boosts are then applied to provide initial and final wave-functions from which the electromagnetic current matrix element is calculated for arbitrarily large momentum transfer. As before, since the interactions between quarks are assumed to be only moderately strong, they are neglected entirely and binding is provided by the boundary condition, It is then inappropriate to start with the Bethe-Salpeter equation [4], since this assumes that binding is caused by an interaction. If Oa is the operator that annihilates the wavefunction for a single free particle a, the Bethe-Salpeter equation can be written schematically as
where G is an operator that represents the interaction. Then if G if set equal to zero, Eq. (1) clearly has solutions that correspond to one or the other of the particles being free, but not both; these must be regarded as physically unacceptable. We therefore assume that the correct substitute for Eq. (1) is the pair of equations
Oa$= 0,
o,* = 0
It is not difficult to solve these equations in the rest system (total three momentum equal to zero); a spatial boundary condition can then be applied instantaneously, i.e., when ta = tb . An additional assumption must be made in order to find the dependence of the wavefunction on the relative time ta, - tb . The solution thus obtained can then be boosted to any other Lorentz frame. Because of the complications associated with nonzero spin, it is instructive to solve the problem first for the unphysical case of spin 0 quarks; this is done in Section 2. Section 3 then develops the solution for spin one-half quarks. The electromagnetic current matrix element, or elastic form factor, for a pseudoscalar meson is calculated in Section 4.
250
SCHIFF 2. SPIN ZERO QUARKS
We assign masses m, and mt, to the quark and antiquark, and space-time coordinates xa and Xb [5]. It is convenient to make use of relative and center-ofmass coordinates x = & - xb ,
x
=
Tda
+
mb),
+
Tbxb
,
(3)
7a.b = ma,b/h
m
=
T&%3
p
=
ma
-
mb
.
These imply that the relative and total four-momentum, defined in terms of gradient operators on the relative and center-of-mass coordinates, are related by p
=
Tbpa
-
Tapb
,
(4)
P=pafpb.
We wish to find solutions to Eqs. (2) when OaJ., are Klein-Gordon (b2
f
ma2)
$(Xa
, xb)
=
0,
cob2
f
mb”>
$(&,
xb)
=
operators, 0.
(5)
In the rest system, the total three-momentum is zero. Thus # has no dependence on the space components of X, and has the form $h(&,Xb)
=
u(x)eXp(-ihfT)
(6)
where T is the time component of X. The quantity M is then the rest mass of the quark-antiquark system. Since binding is provided by the spatial boundary condition to be applied below, M is greater than ma + mb . Direct substitution of (6) into (5) shows that U(X) has the following dependence on the relative space-time coordinates t, x, y, z: u(x) = 24(x,y, 2) exp(--iwt), w = --(hi2
- 4m2)/4Mm,
(V2 + a”) 4% y, 4 = 0, 01~= (Al2 - p2)(M2 - 4m2)/4M2.
(7)
The relative time dependence of Eq. (7) seems surprising at first, since experience with the Bethe-Salpeter equation suggests that $ should obey Feynman boundary conditions in the relative time; that is, $ should contain only positive frequences for t > 0 and only negative frequencies for t < 0. We shall see, however, how the result (7) for w arises in our situation from the same argument that was used by Wick [6] to establish Feynman boundary conditions in the usual situation. Repeating Wick’s argument, we assume, as is usually done, that the two-particle wavefunction is defined as the matrix element of the time-ordered product of Heisenberg field operators between the bound state and the vacuum [7],
QUARK
SELECTION
PRINCIPLE,
251
11
For positive relative time, ta > tb , the time-ordering operator can be omitted, and a complete set of states inserted between the field operators, so that Eq. (8) becomes 9% 9 xb)
=
c
n
(0
! p&a)
1 n>(n
1 yb(xb)
(9)
1 B).
In the case considered by Wick, the only intermediate states that contribute are those with the quantum numbers of a single particle of type a, and the lowest possible mass of such a state is m, ; thus E, 3 ma. Also, the mass of the bound state is M, and of the vacuum state is zero. Equation (9) then gives for the time dependence of the n term in the series for #,
exp(-iE&J
. exp[-i(M
- E&b]
= exp(-iMT)
* exp[-i(E,
- ~&f)t].
(10)
It follows that for t > 0, # contains only relative frequencies that are greater than or equal to m, - T&M, which is greater than zero since stability requires that M < ma + mb . In similar fashion, it is easily seen that only negative frequencies appear for t < 0. In our case, there is an inconsistency inherent in the expansion (9) since it assumes that there can be single-particle states n. This conflicts with the quark selection principle, which states that isolated fractionally charged particles do not exist. Nevertheless, we assume that the t-dependence of I,Lis the same as would be obtained with particles that do not obey this principle, and that the principle manifests itself only in the spatial boundary condition, which is to be applied below. Since the particles described by Eqs. (5) are noninteracting, the only states E, that appear in the expansion (9) are those of a single particle with mass ma ; from Eq. (4), this particle has three-momentum of magnitude a. On substituting this En into (lo), we find that the only relative frequency that appears is (ma2 + ,2)1/2 - TJM, which is equal to the w given in Eq. (7). For negative relative time, the only frequency that appears is -(mbO + a2)lj2 + TbM, which again is equal to w. Thus Wick’s argument accounts for the appearance of a single frequency, which may be positive, negative, or zero, for all values of the relative time. We now wish to impose a spatial boundary condition which is a generalization of that used in Ref. [l]. We assume that it is instantaneous in the rest system, in analogy with the instantaneous interaction often assumed in solving the BetheSalpeter equation. This means that we set t = 0 in (7) and solve the equation for u(x, y, z). It is plausible to require u to vanish for r 3 R, where r is the magnitude of the relative space coordinate in the rest system. The unnormalized solution with smallest M or 01is then
u = $ar),
r < R; aR = rr,
u = 0,
r > R, (11)
252
SCHIFF
where j,(p) = (sin p)/p is a spherical Bessel function. This agrees with the corresponding result obtained in Ref. [I], in the nonrelativistic limit. There remains the probIem of finding the dependence of Z/Jon t for t # 0. We
assume that this is determined by Wick’s argument, when account is taken of the fact that, because of the boundary condition, the u given by Eq. (11) contains spatial Fourier components with three-momenta of all magnitudes, not just OLas in the completely free case discussed above. The Fourier transform of (11) is
s
d3r exp(--ik
* r) u(x, y, z) = -4~Rj,(kR)/(k~
-
12).
(12)
Thus the bound state B can be regarded as a superposition of free particles of relative momenta k. In the same way, the intermediate states n that appear in the expansion (9) represent single free particles of type a and momentum k, since in accordance with (4), pa is equal to p if P is zero. Wick’s argument then gives for the relative frequency as a function of k, t > 0: w+(k)
= Es(k)
- T&I, &b(k)
t < 0: w-(k)
= -Et,(k)
+
TbM,
(13)
E (m2,J + k2)?
It should be noted that since A4 > ma + mb in our case, w+ extends from a negative lower limit to + co, and w- extends from - 00 to a positive upper limit, so that there are no Feynman boundary conditions on the relative time. This is not surprising, since in our case binding is the result of the spatial boundary condition rather than of an attractive interaction. We then construct U(X) in Eq. (6) by multiplying its Fourier transform (12) by exp(ik . I) . exp[--iw*(k)t] and integrating d3k/(2.rr)3. The angle integrations are easily performed, and yield u(t > 0, x, y, z) = -(2R/n-)
s, k2 dk exp[-
b+(k)
The corresponding expression for t -X 0 has w+(k) dimensional Fourier transform of (14) is
t] j,(kr)
j,,(kR)/(k2
replaced by w-(k).
- a”).
(14)
The four-
The four-vector k” has time component k,, and spatial magnitude k; E is a real positive infinitesimal, so that (15) is unchanged if the E’S are omitted and the usual prescription of giving ma and mb infinitesimal negative imaginary parts is adopted. The wavefunction adjoint to U(X) or x(k) is needed for the calculation of matrix
QUARK
SELECTION
PRINCIPLE,
253
11
elements. Mandelstam [8] has shown that U(x) is obtained from the expression for u(x) by replacing time-ordering in Eq. (8) by anti-time-ordering, and taking the complex conjugate. This means that LI(t > 0, x, y, z) is given by (14) with --iw+(k)t replaced by +iw$c)t in the exponent of the integrand. It then follows that x(k) is equal to x(k).
3.
SPIN
We now wish to find solutions
ONE-HALF
QUARKS
to Eqs. (2) when Oa,b are Dirac operators,
Here Pa = yauV,,, , Gsp = a/a~,~, and similarly for F’1,. The subscript a denotes derivatives with respect to the coordinate xa and Dirac y matrices that operate on the four spinor components of the quark a; similarly, the subscript b refers to the antiquark b. The wavefunction $I therefore has 16 components, and is most readily handled by writing it as a 4 x 4 matrix [9, lo]; it can then be expressedin terms of y matrices, which have well-known algebraic properties. We shall make use of some of the results obtained by Smith [l 11,who has adapted this formalism to the solution of the Bethe-Salpeter equation for pseudoscalar and vector mesonsthat are composed of a quark and an antiquark. We expect that the matrix wavefunction $Jshould transform like the product of a positive-energy spinor #(a) for a quark a and the transposeof a positive-energy spinor #r.(b)T for antiquark b, where the superscript T denotes the transpose. The latter can be expressedin terms of the corresponding negative-energy quark spinor #(b) by means of the operator C = iy2y0, through $,(b) = C$(b)T, where the adjoint spinor 4(b) = #(b)+yO,and the dagger denotes the hermitian conjugate [12]. Then making use of the relation CTyaT = -yuCT, we find that w%(a> %@Y = rw%(4 %c(b>T= v%(a) $@I CT, Yb”%(a) %@Y = %(a) %,(blT yuT, = #(a) $(b) CryUT = -#(a) $(b) yuCr. It is thus convenient to redefine the # of Eqs. (16) by multiplying it from the right by (F-l, so that the new 16transforms Iike #(a) q(b) and satisfiesthe relations YavJ = vi4
yb”‘$
=
(17)
-47”.
Equations (16) then become, in terms of the new $J, (iy ’ o’, - ?I?,) $(xa , xb) = 0,
$‘(xa
, xb)(iy
* o’b
+
Mb)
=
0,
(18)
254
WHIFF
where the arrow over each gradient operator indicates the direction in which it acts. As in Section 2, we write # in the form (6) for the rest system, and make use of Eqs. (3) to substitute for the gradient operators Vb = TbVX - v, .
v, = TaV, + v, , With P = (t, x, y, z), we have that
y - v, = yoajat+ y . v, where V is the usual spatial gradient operator on the relative coordinates, and y = (yl, y2, y3). Equations (18) then become
[yyTsM+ iqst) + iy * ii - ma]u(x)= 0, (19) U(X)[yO(T,,M - i&
- iy * ti + mb] = 0,
where U(X) is a 4 x 4 matrix function of the relative space-time coordinates. The four-dimensional Fourier transform of U(X), defined as in (15), satisfies equations analogous to (19), [yO(cM
+ ko) -
Y
* k - ma1 x(k) = 0, (20)
X(k)[y’(TbM
-
k,)
+
Y * k
+
mb]
=
0,
where the four-vector k” = (k. , k). We now follow Smith [l I], and make use of the result that the most general pseudoscalar form for x(k) is a linear combination of y5
, y&
- W,
nP,
y5VP
- PJQ,
with coefficients that are even functions of k * P if the bound state is to be even under charge conjugation. In our case, the total four-momentum vector P is equal to (M, 0, 0, 0), so that k . P = Mko and P = My”. We may then put x(k)
=
y5[A
+ By0 + C(y . k) y” + Dko(y . k)].
(21)
Substitution of (21) into (20) yields eight simultaneous algebraic equations in the four coefficients A, B, C, D, with k, and k = 1k 1 as parameters. These equations can be solved if and only if k. = w and k = (Y,where w and 01are given in Eqs. (7). The coefficients are then related by B = -(2m/M)A,
C = -[2M/(M2
- p2)]A,
k,D = -[2p/(M2
- p2)]A.
(22)
QUARK
SELECTION
PRINCIPLE,
255
II
The pseudoscalar wavefunction, obtained by substituting thus be written in terms of Pauli spin matrices,
&?0,h)=const[l”2i
The corresponding
(22) into (21)
l~+~~~(~~-~)~(,~,-~).
configuration
space wavefunction
2i(o . 0) zr(t, r) = const i
can
(23)
is
I+% 2i(o . V) MS- p- i
exp( - iwt) j,(w).
(24)
The solutions (23) and (24) refer to completely free particles, before the spatial boundary condition is applied, and are analogous to Eq. (7) for spin zero quarks. We must now find the analog of Eq. (11). It is apparent, however, that we cannot simply set CLR = n without further justification, for while this makes the offdiagonal components of U(X) continuous at r = R, the diagonal components are not. We therefore look back at the conclusion, in the spin zero case, that u(x, y, z) = 0 for r > R when r = 0. This would have been obtained from Eqs. (7) by assuming that the instantaneous spatial boundary condition corresponds to letting m + + cc for r > R. We therefore attempt to apply the same condition to the solution of Eqs. (19). In accordance with Eq. (21), we put for u(x), u(r,r)=~5[A+Bdl-i(Y.VC)4+:((Y.VD)],
(25)
where A,..., D are now space-time functions. Substitution of (25) into (19) yields eight simultaneous differential equations. These are satisfied for r < R provided that A,..., D are each proportional to f(r) exp(-iwt), where (Cz + G)f = 0 and w and 01are given by (7). Further, the proportionality constants must be related by Eqs. (22) with k,D replaced by wD. Thusf = jo(o(r) for r < R. For r > R, we assume that m is replaced by ??I’, which is large and positive and will later approach f co. We also assume that p is unchanged. Then w and A,..., D are all replaced by primed quantities, and 01is replaced by ifl, with (7) and (22) still valid. A’,..., D’ are now proportional to g(r) exp(-iw’t), where (Vz - p2)g = 0 so that g = I#‘($r) = -(l/fir) exp(-/h). We would expect A, B, WC, and wVD to be continuous at r = R when t = 0. It is apparent from (22) that if A is continuous, then C and WD are also, but B
256
SCHIFF
is not. Further, A4 can be chosen so that A and VA can both be made continuous at r = R, in which case VC and wVD are also continuous; the requirement is that aR cot olR = -j3R. Thus as m’ and /3 approach + co, aR approaches rr in such a way that sin olR is approximately equal to n-//3R. Then VC and wVD are finite at r = R, and drop infinitely rapidly to zero for r > R as /3 -+ + co. In this limit, A and B approach zero as r + R from below, and A remains zero for r > R. For large finite /3, however, B is discontinuous at r = R; it jumps from a small value just below R to a large but finite value just above R and then falls off rapidly as r increases further. We thus find that A, VC, and wVD are all continuous for large finite m’, although the latter two are not in the limit m’ + + co. On the other hand, B is not continuous so long as m’ is finite, but is in the limiting case. This is the most favorable way to apply the instantaneous spatial boundary condition to (24); it leads to the value of M determined by the condition aR = n, as in the spin zero case, and also to agreement with Ref. [l] in the nonrelativistic limit. We thus obtain as the analog of (1 l),
40,r>= ~5 [l
- $
y” + J’M,,
(Y
* V) y” + M2 y pL2(Y . V)] jo(ar)
(26)
for I < R, and ~(0, r) = 0 for r > R, with aR = 7r. The argument that led from (11) to (15) is unchanged, and takes (26) into
(27)
where k = 1k 1 and F(k, , k) is the right side of Eq. (15). As in the spin zero case, the adjoint wavefunction f(k) has the same coefficient F(ko , k) as appears in x(k). We must however take the Hermitian conjugate of the matrix part of x(k), y5[.*.], and multiply it on the right by yaOybO. As remarked above Eq. (21), this bound state is even under charge conjugation if F(ko , k) is an even function of k . P, which in the rest system is equal to Mk, . It is easily seen from (15) that this is true if and only if w-(k) = -w+(k), and it follows from (13) that this occurs only when ma = mb . In the remainder of this paper, we shall assume that this is the case, and accordingly drop the subscripts from ma, mb , l&(k), &(k), and w+(k), and set w-(k) = -w(k), 7a = 5-b = t, and p = 0. The right side of (15) then becomes Ftko,
k)
=
4TiWf
(k2 -
- 2@ jotW a2)(ko2 - co”)
where it is understood that m has an infinitesimal
negative imaginary part.
(28)
257
QUARK SELECTION PRINCIPLE, II 4. ELASTIC
ELECTROMAGNETIC
FORM FACTOR
The elastic form factor is proportional to the matrix element of the electromagnetic current operator between the initial and final states that are obtained from (27) and (28) by appropriate boosts. Since this part of the calculation is fully covariant, any Lorentz frame can be employed. It is most convenient to use the Breit frame, in which the initial and final states have equal and opposite threemomenta. The initial and final total four-momenta are then Pi” = (My, 0, 0, -Mvy),
A”
=
(My,
(40,
(29)
Mvy),
where y = (1 - Do)-* and D is the speed of the meson in the Breit frame. The square of the four-momentum transfer is 9’ = (Pf - Pi)” = -4MW’y2
= -4M2(y2
- 1)
(30)
and is of course an invariant quantity. The boosts that generate the initial and final momenta (29) from the rest-system momentum P,,= = (M, 0, 0,O) are
(6.f):
=
cash z 0 10 0 i i sinh z 0 0
0
where tanh z = v and cash z = y; that is, PLf = (ai,r)“, Pow.The corresponding transformations for each four-component spinor are [13] S(ui,f) = cash &Z F y”y3 sinh 4~.
(31)
The initial state obtained from the rest-systemfunction *+a. >Xb)
=
u(X)
eXp(-Z?o
’ x)
is $i(xa
, xb)
=
&(Ui)
Sb(ai)
U(U;lX)
eXp[--iP,
(32)
* (u;‘x)],
where the subscript on each S refers to the spinor on which the y’s of (31) operate. In the exponent of (32), it is easily seen that PO. (a;%) = (PO@) . X = Pi . X. We can expressu in Eq. (32) in terms of its Fourier transform, to give *i(Xa
595/63/I-18
*
Xb) = exP(--PI
* 1)(27~)-~ j” d4k exp[--i(ka;l)
- X] S&)
sb(&) x(k).
(33)
258
SCHIFF
The matrix character of the adjoint of the final state wavefunction is the same as that of uf(x)+
ya”?/bo
=
U(&x)+
&(&)+
sb(@)+
=
f&‘x)+
~a”~b”&&f)-’
3/a”ybo s&f)-’
since y”Styo = S-l. We thus have $(xa
, Xb)
=
exp(iPr * X)(~?F)-~ 1 d4k exp[i(ka;l)
* X] x(k) &(a&’
sb(af)-I.
(34)
We now assume that the operator for the current density at the space-time point x0 , caused for example by the quark a with charge e, , is that originally given by Mandelstam [8, 141, = -ieas4(xo - xa) yau(iyb ’ vb - mb).
jam
We must insert this operator between the initial and final states (33) and (34), integrate over X%and Xb , and sum over spinor indices. Once the Dirac y matrices with subscripts a and b are properly placed with respect to the wavefunctions, in accordance with (17), the sum over spinor indices is accomplished by calculating the trace of the product 4 x 4 matrices [lo]. The eightfold coordinate integration is most easily performed in terms of relative and center-of-mass coordinates, and leads to Ul jdxo>~ I 0 = - -ci”n”,4 exp(iq * x0) / d4k g($P, - kar) S&f)-l * ya”(yb
* k
-
m)
&&a)
sb(@)
x(Bpo
-
&(a$1
hi).
Here q = Pf - Pi is the four-momentum transfer, and we have set ms = mb , The calculation of the trace is quite tedious, and is better done by multiplying the matrices out than by using trace theorems. The final result is that the integral vanishes by symmetry for the three space components of j,(x,); then since q” = (0, 0, 0,2Mvy), the current is conserved, as of course it must be. The time component of the current is
0-l jtdxo)o I i> = 7
exp(iq . x0) / d4k
(M - 2En)(M - 2Ei) j&R) jo(ktR) (k,2 - a2)(k,2 - G)(k;,, - w,2)(kfo - wi”)
+qko3+4ko[1 + q k;.t
=
+f$(E2+
mzv2)
(k12 + k22)] - 91, k12
+
h2
E?.f = m2 -I- kt,r ,
+
y2(k,
f
(35) vko12,
ui.t = El.! - BM,
k io.io = iM
-
y(ko
f vk,),
E2 = m2 + k12 + k22 + k32.
QUARK SELECTION PRINCIPLE, II
259
The integral in (35) is actually three-dimensional, since k1 and k, enter the integrand only as the sum of their squares. The normalization integral, obtained from (35) with y = 1 or u = 0, can be reduced to one dimension and is equal to 32ea m (A4 + 2E) sin2 kR dk (k2 - a2)2 ’ 4M 0
(36)
where E2 = m2 i- k2. It would seem at first that the electromagnetic form factor is simply the ratio of the current matrix element (39, with exp(iq * x0) omitted, to the normalization integral (36). Actually, however, we must divide this ratio by the corresponding quantity that would be obtained with a point meson. For a point meson at rest, the unnormalized wavefunction is exp(-NT); boosting it to initial or final four-momentum P, as we did in obtaining (33) and (34), leads to the wavefunction exp(--iP * X), without the usual multiplicative factor (2P”)-‘l”. The matrix element of the point meson current operator in the Breit frame is then proportional to (Pi + Pi)” = (2My, 0, 0,O). Thus in order to calculate the form factor, we must divide (35) not only by (36), but also by y. 5. CONCLUDING
REMARKS
The integral in (35) converges for all real y > 1, i.e., for q2given by (30) on the negative real axis. Its analytic properties for general complex values of q2 seem to be very difficult to decipher. However, there is no trace of the anomalous threshold that one would expect to find from the vertex diagram in perturbation theory, which is at q2 = -MM2(M2 - 4m2)/m2.This is not surprising when it is remembered that there is no field theory underlying our calculation which shows how it can be cast in terms of perturbation theory diagrams. The convergence of (35) is quite slow for large values of the variables of integration, except for y = 1 where the normalization (36) is rapidly convergent. The rate of convergence of (35), and therefore also the feasibility of numerical computation, would probably be improved if the sharp spatial boundary condition represented by (26) were to be smoothed over. The asymptotic behavior of the form factor for large negative q2 is not difficult to find. In this case y > 1 and we can set u = 1 in calculating the leading term. The change of variables from k, , k, to x, y = y(k, f k3) separates the parts of the integrand involving ki and kio from the parts involving kf and kf, . With the substitution w = k12+ kz2, the leading term of (35) for large y can be written z
exp(iq * x0) 1: dw (m” + w) Io(w)[211(~) - Ml,(w)], ~“(44 - 2E) j,(kR) In(W) = I”“, dx (k2 _ a2)(E - x)(&f - E - x + ig) ’
(37)
260
SCHIFF
where k2 = x2 + w and E2 = x2 + w + m2. The form factor should be real; this is evidently the case when y = 1 [Eq. (36)], but is not apparent in general. It follows from (37) and the comment at the end of the preceding section that the electromagnetic form factor falls off like l/y2 or l/q2 for large momentum transfer. It is therefore of the “pole” form rather than the “dipole” form which gives a better fit to the experiments on high-energy elastic electron-proton scattering. There is of course no reason why the meson and proton form factors must have the same asymptotic form, and there is no experimental evidence with respect to mesons. Finally, it should be noted that the asymptotic behavior calculated here is independent of the functional form of F(k,, , k) in (28), and hence of the choice of the spatial boundary condition and the relative time dependence. This is because the change of variables from k,, , k, to x, y used in deriving (37) effectively eliminates the contribution of F(k, , k) to the y dependence of the form factor. ACKNOWLEDGMENTS It is a pleasure to thank M. Jacob and L. Van Hove for stimulating conversations. The hospitality of the Theoretical Physics Group at Imperial College, London, is gratefully acknowledged.
REFERENCES 1. L. I. SCHIFF,Phys. Lett. B 31 (1970), 79; this paper contains references to related earlier work. 2. The most recent papers that bear on the experimental observability of quarks are W. T. CHU, Y. S. KIM, W. J. BEAM, AND N. KWAK, Phys. Rev. Len. 24 (1970), 917; H. FAISSNER, M. HOLDER, K. KRISOR, G. MASON, Z. SAWAF, AND H. UMBACH, Phys. Rev. Lett. 24 (1970), 1357. These papers contain references to earlier work. 3. This mass refers to nonstrange quarks, and was obtained from the proton and neutron magnetic moments. In the same way, the mass of the strange quark, obtained from the d magnetic moment, is found to be about 430 MeV/?. 4. E. E. SALPETER AND H. A. BETHE, Phys. Rev. 84 (1951), 1232; E. E. SALPETER, Phys. Rev. 87 (1952), 328. 5. We use the now standard notation of J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Mechanics,” Appendix A, McGraw-Hill Book Co., New York, 1964. 6. G. C. WICK, Phys. Rev. 96 (1954), 1124. 7. M. GELL-MANN AND F. Low, Phys. Rev. 84 (1951), 350. 8. S. MANDELSTAM, Proc. Roy. Sot. London, Ser. A 233 (1955), 248. 9. L. DE BROGLIE, “ThCorie G&ale des Particules g Spin (Mtthode de Fusion),” Gauthier Villars, Paris, 1943. 10. J. S. GOLDSTEIN, Phys. Rev. 91 (1953), 1516. 11. C. H. L. SMITH, Ann. Phys. (New York) 53 (1969), 521. 12. Reference [5], p. 67. 13. Reference [5], Chapter 2. 14. S. J. BRODSKY AND J. R. PRIMACK, Ann. Phys. (New York) 52 (1969), 315.
OF PHYSICS:
Quark
63, 248-260
Selection
(1971)
Principle.
II. Pseudoscalar
Meson
Form
Factor*+
L. I. SCHIFF Institute of Theoretical Stanford University,
Physics, Department Stanford, California
of Physics, 94305
Received July 29, 1970
The quark selection principle proposed earlier is applied to the calculation of the elastic electromagnetic form factor of a pseudoscalar meson. The Bethe-Salpeter equation is inapplicable, and it is assumed instead that the quark-antiquark wavefunction satisfies a free Dirac equation for each particle. A spatial boundary condition is applied instantaneously in the rest system, and the dependence on relative time is inferred by adapting an argument given by Wick. Initial and final states are then obtained by applying appropriate Lorentz boosts to the resulting rest-system wavefunction. It is found that the form factor falls off for large momentum transfer like l/q2.
1. INTRODUCTION The assumption of a quark selection principle that has a range built into it was proposed several years ago as a means for reconciling the successesof the naive quark model of hadronic structure with the apparent experimental unobservability of free quarks. This principle may be expressedas follows. Quarks and antiquarks can exist in any combinations provided that clusters of noninteger baryon number or electric charge cannot separate from other clusters by more than some characteristic distance R. As a special example of this principle, a single quark cannot exist at a distance greater than R from other hadronic matter. In a recent paper [I] the principle was formulated as a boundary condition which causesthe manyparticle wavefunction to vanish whenever the selection principle is violated. It may then be thought of as analogous to the exclusion principle, which causesall fermion wavefunctions that are not fully antisymmetric to vanish. It must of course be recognized that should the existence of fractionally charged hadrons be established experimentally [2], the work reported here would have at the very least to be greatly modified. * Dedicated to the memory of Amos de-Shalit, physicist, public servant, and friend. + Research supported in part by the Air Force Office of Scientific Research under AFOSR Contract F44620-68C-0075.
248
QUARK
SELECTION
PRINCIPLE,
II
249
The earlier work [l] was based on the nonrelativistic Schrodinger equation. This is a good approximation for the internal motion, since quarks were found to have a rest energy of about 336 MeV and much smaller kinetic energy [3]. However, the calculation of the electromagnetic form factor for large momentum transfer cannot be carried through convincingly as suggested earlier, since there is no unique way of applying a Lorentz boost to a nonrelativistic wavefunction. The present paper, therefore, makes use of relativistic particle theory. The wavefunction for the simplest hadronic system, the pseudoscalar meson composed of a quark and an antiquark, is found in its rest system on the basis of certain assumptions concerning the relativistic wave equation and boundary condition. Fully covariant boosts are then applied to provide initial and final wave-functions from which the electromagnetic current matrix element is calculated for arbitrarily large momentum transfer. As before, since the interactions between quarks are assumed to be only moderately strong, they are neglected entirely and binding is provided by the boundary condition, It is then inappropriate to start with the Bethe-Salpeter equation [4], since this assumes that binding is caused by an interaction. If Oa is the operator that annihilates the wavefunction for a single free particle a, the Bethe-Salpeter equation can be written schematically as
where G is an operator that represents the interaction. Then if G if set equal to zero, Eq. (1) clearly has solutions that correspond to one or the other of the particles being free, but not both; these must be regarded as physically unacceptable. We therefore assume that the correct substitute for Eq. (1) is the pair of equations
Oa$= 0,
o,* = 0
It is not difficult to solve these equations in the rest system (total three momentum equal to zero); a spatial boundary condition can then be applied instantaneously, i.e., when ta = tb . An additional assumption must be made in order to find the dependence of the wavefunction on the relative time ta, - tb . The solution thus obtained can then be boosted to any other Lorentz frame. Because of the complications associated with nonzero spin, it is instructive to solve the problem first for the unphysical case of spin 0 quarks; this is done in Section 2. Section 3 then develops the solution for spin one-half quarks. The electromagnetic current matrix element, or elastic form factor, for a pseudoscalar meson is calculated in Section 4.
250
SCHIFF 2. SPIN ZERO QUARKS
We assign masses m, and mt, to the quark and antiquark, and space-time coordinates xa and Xb [5]. It is convenient to make use of relative and center-ofmass coordinates x = & - xb ,
x
=
Tda
+
mb),
+
Tbxb
,
(3)
7a.b = ma,b/h
m
=
T&%3
p
=
ma
-
mb
.
These imply that the relative and total four-momentum, defined in terms of gradient operators on the relative and center-of-mass coordinates, are related by p
=
Tbpa
-
Tapb
,
(4)
P=pafpb.
We wish to find solutions to Eqs. (2) when OaJ., are Klein-Gordon (b2
f
ma2)
$(Xa
, xb)
=
0,
cob2
f
mb”>
$(&,
xb)
=
operators, 0.
(5)
In the rest system, the total three-momentum is zero. Thus # has no dependence on the space components of X, and has the form $h(&,Xb)
=
u(x)eXp(-ihfT)
(6)
where T is the time component of X. The quantity M is then the rest mass of the quark-antiquark system. Since binding is provided by the spatial boundary condition to be applied below, M is greater than ma + mb . Direct substitution of (6) into (5) shows that U(X) has the following dependence on the relative space-time coordinates t, x, y, z: u(x) = 24(x,y, 2) exp(--iwt), w = --(hi2
- 4m2)/4Mm,
(V2 + a”) 4% y, 4 = 0, 01~= (Al2 - p2)(M2 - 4m2)/4M2.
(7)
The relative time dependence of Eq. (7) seems surprising at first, since experience with the Bethe-Salpeter equation suggests that $ should obey Feynman boundary conditions in the relative time; that is, $ should contain only positive frequences for t > 0 and only negative frequencies for t < 0. We shall see, however, how the result (7) for w arises in our situation from the same argument that was used by Wick [6] to establish Feynman boundary conditions in the usual situation. Repeating Wick’s argument, we assume, as is usually done, that the two-particle wavefunction is defined as the matrix element of the time-ordered product of Heisenberg field operators between the bound state and the vacuum [7],
QUARK
SELECTION
PRINCIPLE,
251
11
For positive relative time, ta > tb , the time-ordering operator can be omitted, and a complete set of states inserted between the field operators, so that Eq. (8) becomes 9% 9 xb)
=
c
n
(0
! p&a)
1 n>(n
1 yb(xb)
(9)
1 B).
In the case considered by Wick, the only intermediate states that contribute are those with the quantum numbers of a single particle of type a, and the lowest possible mass of such a state is m, ; thus E, 3 ma. Also, the mass of the bound state is M, and of the vacuum state is zero. Equation (9) then gives for the time dependence of the n term in the series for #,
exp(-iE&J
. exp[-i(M
- E&b]
= exp(-iMT)
* exp[-i(E,
- ~&f)t].
(10)
It follows that for t > 0, # contains only relative frequencies that are greater than or equal to m, - T&M, which is greater than zero since stability requires that M < ma + mb . In similar fashion, it is easily seen that only negative frequencies appear for t < 0. In our case, there is an inconsistency inherent in the expansion (9) since it assumes that there can be single-particle states n. This conflicts with the quark selection principle, which states that isolated fractionally charged particles do not exist. Nevertheless, we assume that the t-dependence of I,Lis the same as would be obtained with particles that do not obey this principle, and that the principle manifests itself only in the spatial boundary condition, which is to be applied below. Since the particles described by Eqs. (5) are noninteracting, the only states E, that appear in the expansion (9) are those of a single particle with mass ma ; from Eq. (4), this particle has three-momentum of magnitude a. On substituting this En into (lo), we find that the only relative frequency that appears is (ma2 + ,2)1/2 - TJM, which is equal to the w given in Eq. (7). For negative relative time, the only frequency that appears is -(mbO + a2)lj2 + TbM, which again is equal to w. Thus Wick’s argument accounts for the appearance of a single frequency, which may be positive, negative, or zero, for all values of the relative time. We now wish to impose a spatial boundary condition which is a generalization of that used in Ref. [l]. We assume that it is instantaneous in the rest system, in analogy with the instantaneous interaction often assumed in solving the BetheSalpeter equation. This means that we set t = 0 in (7) and solve the equation for u(x, y, z). It is plausible to require u to vanish for r 3 R, where r is the magnitude of the relative space coordinate in the rest system. The unnormalized solution with smallest M or 01is then
u = $ar),
r < R; aR = rr,
u = 0,
r > R, (11)
252
SCHIFF
where j,(p) = (sin p)/p is a spherical Bessel function. This agrees with the corresponding result obtained in Ref. [I], in the nonrelativistic limit. There remains the probIem of finding the dependence of Z/Jon t for t # 0. We
assume that this is determined by Wick’s argument, when account is taken of the fact that, because of the boundary condition, the u given by Eq. (11) contains spatial Fourier components with three-momenta of all magnitudes, not just OLas in the completely free case discussed above. The Fourier transform of (11) is
s
d3r exp(--ik
* r) u(x, y, z) = -4~Rj,(kR)/(k~
-
12).
(12)
Thus the bound state B can be regarded as a superposition of free particles of relative momenta k. In the same way, the intermediate states n that appear in the expansion (9) represent single free particles of type a and momentum k, since in accordance with (4), pa is equal to p if P is zero. Wick’s argument then gives for the relative frequency as a function of k, t > 0: w+(k)
= Es(k)
- T&I, &b(k)
t < 0: w-(k)
= -Et,(k)
+
TbM,
(13)
E (m2,J + k2)?
It should be noted that since A4 > ma + mb in our case, w+ extends from a negative lower limit to + co, and w- extends from - 00 to a positive upper limit, so that there are no Feynman boundary conditions on the relative time. This is not surprising, since in our case binding is the result of the spatial boundary condition rather than of an attractive interaction. We then construct U(X) in Eq. (6) by multiplying its Fourier transform (12) by exp(ik . I) . exp[--iw*(k)t] and integrating d3k/(2.rr)3. The angle integrations are easily performed, and yield u(t > 0, x, y, z) = -(2R/n-)
s, k2 dk exp[-
b+(k)
The corresponding expression for t -X 0 has w+(k) dimensional Fourier transform of (14) is
t] j,(kr)
j,,(kR)/(k2
replaced by w-(k).
- a”).
(14)
The four-
The four-vector k” has time component k,, and spatial magnitude k; E is a real positive infinitesimal, so that (15) is unchanged if the E’S are omitted and the usual prescription of giving ma and mb infinitesimal negative imaginary parts is adopted. The wavefunction adjoint to U(X) or x(k) is needed for the calculation of matrix
QUARK
SELECTION
PRINCIPLE,
253
11
elements. Mandelstam [8] has shown that U(x) is obtained from the expression for u(x) by replacing time-ordering in Eq. (8) by anti-time-ordering, and taking the complex conjugate. This means that LI(t > 0, x, y, z) is given by (14) with --iw+(k)t replaced by +iw$c)t in the exponent of the integrand. It then follows that x(k) is equal to x(k).
3.
SPIN
We now wish to find solutions
ONE-HALF
QUARKS
to Eqs. (2) when Oa,b are Dirac operators,
Here Pa = yauV,,, , Gsp = a/a~,~, and similarly for F’1,. The subscript a denotes derivatives with respect to the coordinate xa and Dirac y matrices that operate on the four spinor components of the quark a; similarly, the subscript b refers to the antiquark b. The wavefunction $I therefore has 16 components, and is most readily handled by writing it as a 4 x 4 matrix [9, lo]; it can then be expressedin terms of y matrices, which have well-known algebraic properties. We shall make use of some of the results obtained by Smith [l 11,who has adapted this formalism to the solution of the Bethe-Salpeter equation for pseudoscalar and vector mesonsthat are composed of a quark and an antiquark. We expect that the matrix wavefunction $Jshould transform like the product of a positive-energy spinor #(a) for a quark a and the transposeof a positive-energy spinor #r.(b)T for antiquark b, where the superscript T denotes the transpose. The latter can be expressedin terms of the corresponding negative-energy quark spinor #(b) by means of the operator C = iy2y0, through $,(b) = C$(b)T, where the adjoint spinor 4(b) = #(b)+yO,and the dagger denotes the hermitian conjugate [12]. Then making use of the relation CTyaT = -yuCT, we find that w%(a> %@Y = rw%(4 %c(b>T= v%(a) $@I CT, Yb”%(a) %@Y = %(a) %,(blT yuT, = #(a) $(b) CryUT = -#(a) $(b) yuCr. It is thus convenient to redefine the # of Eqs. (16) by multiplying it from the right by (F-l, so that the new 16transforms Iike #(a) q(b) and satisfiesthe relations YavJ = vi4
yb”‘$
=
(17)
-47”.
Equations (16) then become, in terms of the new $J, (iy ’ o’, - ?I?,) $(xa , xb) = 0,
$‘(xa
, xb)(iy
* o’b
+
Mb)
=
0,
(18)
254
WHIFF
where the arrow over each gradient operator indicates the direction in which it acts. As in Section 2, we write # in the form (6) for the rest system, and make use of Eqs. (3) to substitute for the gradient operators Vb = TbVX - v, .
v, = TaV, + v, , With P = (t, x, y, z), we have that
y - v, = yoajat+ y . v, where V is the usual spatial gradient operator on the relative coordinates, and y = (yl, y2, y3). Equations (18) then become
[yyTsM+ iqst) + iy * ii - ma]u(x)= 0, (19) U(X)[yO(T,,M - i&
- iy * ti + mb] = 0,
where U(X) is a 4 x 4 matrix function of the relative space-time coordinates. The four-dimensional Fourier transform of U(X), defined as in (15), satisfies equations analogous to (19), [yO(cM
+ ko) -
Y
* k - ma1 x(k) = 0, (20)
X(k)[y’(TbM
-
k,)
+
Y * k
+
mb]
=
0,
where the four-vector k” = (k. , k). We now follow Smith [l I], and make use of the result that the most general pseudoscalar form for x(k) is a linear combination of y5
, y&
- W,
nP,
y5VP
- PJQ,
with coefficients that are even functions of k * P if the bound state is to be even under charge conjugation. In our case, the total four-momentum vector P is equal to (M, 0, 0, 0), so that k . P = Mko and P = My”. We may then put x(k)
=
y5[A
+ By0 + C(y . k) y” + Dko(y . k)].
(21)
Substitution of (21) into (20) yields eight simultaneous algebraic equations in the four coefficients A, B, C, D, with k, and k = 1k 1 as parameters. These equations can be solved if and only if k. = w and k = (Y,where w and 01are given in Eqs. (7). The coefficients are then related by B = -(2m/M)A,
C = -[2M/(M2
- p2)]A,
k,D = -[2p/(M2
- p2)]A.
(22)
QUARK
SELECTION
PRINCIPLE,
255
II
The pseudoscalar wavefunction, obtained by substituting thus be written in terms of Pauli spin matrices,
&?0,h)=const[l”2i
The corresponding
(22) into (21)
l~+~~~(~~-~)~(,~,-~).
configuration
space wavefunction
2i(o . 0) zr(t, r) = const i
can
(23)
is
I+% 2i(o . V) MS- p- i
exp( - iwt) j,(w).
(24)
The solutions (23) and (24) refer to completely free particles, before the spatial boundary condition is applied, and are analogous to Eq. (7) for spin zero quarks. We must now find the analog of Eq. (11). It is apparent, however, that we cannot simply set CLR = n without further justification, for while this makes the offdiagonal components of U(X) continuous at r = R, the diagonal components are not. We therefore look back at the conclusion, in the spin zero case, that u(x, y, z) = 0 for r > R when r = 0. This would have been obtained from Eqs. (7) by assuming that the instantaneous spatial boundary condition corresponds to letting m + + cc for r > R. We therefore attempt to apply the same condition to the solution of Eqs. (19). In accordance with Eq. (21), we put for u(x), u(r,r)=~5[A+Bdl-i(Y.VC)4+:((Y.VD)],
(25)
where A,..., D are now space-time functions. Substitution of (25) into (19) yields eight simultaneous differential equations. These are satisfied for r < R provided that A,..., D are each proportional to f(r) exp(-iwt), where (Cz + G)f = 0 and w and 01are given by (7). Further, the proportionality constants must be related by Eqs. (22) with k,D replaced by wD. Thusf = jo(o(r) for r < R. For r > R, we assume that m is replaced by ??I’, which is large and positive and will later approach f co. We also assume that p is unchanged. Then w and A,..., D are all replaced by primed quantities, and 01is replaced by ifl, with (7) and (22) still valid. A’,..., D’ are now proportional to g(r) exp(-iw’t), where (Vz - p2)g = 0 so that g = I#‘($r) = -(l/fir) exp(-/h). We would expect A, B, WC, and wVD to be continuous at r = R when t = 0. It is apparent from (22) that if A is continuous, then C and WD are also, but B
256
SCHIFF
is not. Further, A4 can be chosen so that A and VA can both be made continuous at r = R, in which case VC and wVD are also continuous; the requirement is that aR cot olR = -j3R. Thus as m’ and /3 approach + co, aR approaches rr in such a way that sin olR is approximately equal to n-//3R. Then VC and wVD are finite at r = R, and drop infinitely rapidly to zero for r > R as /3 -+ + co. In this limit, A and B approach zero as r + R from below, and A remains zero for r > R. For large finite /3, however, B is discontinuous at r = R; it jumps from a small value just below R to a large but finite value just above R and then falls off rapidly as r increases further. We thus find that A, VC, and wVD are all continuous for large finite m’, although the latter two are not in the limit m’ + + co. On the other hand, B is not continuous so long as m’ is finite, but is in the limiting case. This is the most favorable way to apply the instantaneous spatial boundary condition to (24); it leads to the value of M determined by the condition aR = n, as in the spin zero case, and also to agreement with Ref. [l] in the nonrelativistic limit. We thus obtain as the analog of (1 l),
40,r>= ~5 [l
- $
y” + J’M,,
(Y
* V) y” + M2 y pL2(Y . V)] jo(ar)
(26)
for I < R, and ~(0, r) = 0 for r > R, with aR = 7r. The argument that led from (11) to (15) is unchanged, and takes (26) into
(27)
where k = 1k 1 and F(k, , k) is the right side of Eq. (15). As in the spin zero case, the adjoint wavefunction f(k) has the same coefficient F(ko , k) as appears in x(k). We must however take the Hermitian conjugate of the matrix part of x(k), y5[.*.], and multiply it on the right by yaOybO. As remarked above Eq. (21), this bound state is even under charge conjugation if F(ko , k) is an even function of k . P, which in the rest system is equal to Mk, . It is easily seen from (15) that this is true if and only if w-(k) = -w+(k), and it follows from (13) that this occurs only when ma = mb . In the remainder of this paper, we shall assume that this is the case, and accordingly drop the subscripts from ma, mb , l&(k), &(k), and w+(k), and set w-(k) = -w(k), 7a = 5-b = t, and p = 0. The right side of (15) then becomes Ftko,
k)
=
4TiWf
(k2 -
- 2@ jotW a2)(ko2 - co”)
where it is understood that m has an infinitesimal
negative imaginary part.
(28)
257
QUARK SELECTION PRINCIPLE, II 4. ELASTIC
ELECTROMAGNETIC
FORM FACTOR
The elastic form factor is proportional to the matrix element of the electromagnetic current operator between the initial and final states that are obtained from (27) and (28) by appropriate boosts. Since this part of the calculation is fully covariant, any Lorentz frame can be employed. It is most convenient to use the Breit frame, in which the initial and final states have equal and opposite threemomenta. The initial and final total four-momenta are then Pi” = (My, 0, 0, -Mvy),
A”
=
(My,
(40,
(29)
Mvy),
where y = (1 - Do)-* and D is the speed of the meson in the Breit frame. The square of the four-momentum transfer is 9’ = (Pf - Pi)” = -4MW’y2
= -4M2(y2
- 1)
(30)
and is of course an invariant quantity. The boosts that generate the initial and final momenta (29) from the rest-system momentum P,,= = (M, 0, 0,O) are
(6.f):
=
cash z 0 10 0 i i sinh z 0 0
0
where tanh z = v and cash z = y; that is, PLf = (ai,r)“, Pow.The corresponding transformations for each four-component spinor are [13] S(ui,f) = cash &Z F y”y3 sinh 4~.
(31)
The initial state obtained from the rest-systemfunction *+a. >Xb)
=
u(X)
eXp(-Z?o
’ x)
is $i(xa
, xb)
=
&(Ui)
Sb(ai)
U(U;lX)
eXp[--iP,
(32)
* (u;‘x)],
where the subscript on each S refers to the spinor on which the y’s of (31) operate. In the exponent of (32), it is easily seen that PO. (a;%) = (PO@) . X = Pi . X. We can expressu in Eq. (32) in terms of its Fourier transform, to give *i(Xa
595/63/I-18
*
Xb) = exP(--PI
* 1)(27~)-~ j” d4k exp[--i(ka;l)
- X] S&)
sb(&) x(k).
(33)
258
SCHIFF
The matrix character of the adjoint of the final state wavefunction is the same as that of uf(x)+
ya”?/bo
=
U(&x)+
&(&)+
sb(@)+
=
f&‘x)+
~a”~b”&&f)-’
3/a”ybo s&f)-’
since y”Styo = S-l. We thus have $(xa
, Xb)
=
exp(iPr * X)(~?F)-~ 1 d4k exp[i(ka;l)
* X] x(k) &(a&’
sb(af)-I.
(34)
We now assume that the operator for the current density at the space-time point x0 , caused for example by the quark a with charge e, , is that originally given by Mandelstam [8, 141, = -ieas4(xo - xa) yau(iyb ’ vb - mb).
jam
We must insert this operator between the initial and final states (33) and (34), integrate over X%and Xb , and sum over spinor indices. Once the Dirac y matrices with subscripts a and b are properly placed with respect to the wavefunctions, in accordance with (17), the sum over spinor indices is accomplished by calculating the trace of the product 4 x 4 matrices [lo]. The eightfold coordinate integration is most easily performed in terms of relative and center-of-mass coordinates, and leads to Ul jdxo>~ I 0 = - -ci”n”,4 exp(iq * x0) / d4k g($P, - kar) S&f)-l * ya”(yb
* k
-
m)
&&a)
sb(@)
x(Bpo
-
&(a$1
hi).
Here q = Pf - Pi is the four-momentum transfer, and we have set ms = mb , The calculation of the trace is quite tedious, and is better done by multiplying the matrices out than by using trace theorems. The final result is that the integral vanishes by symmetry for the three space components of j,(x,); then since q” = (0, 0, 0,2Mvy), the current is conserved, as of course it must be. The time component of the current is
0-l jtdxo)o I i> = 7
exp(iq . x0) / d4k
(M - 2En)(M - 2Ei) j&R) jo(ktR) (k,2 - a2)(k,2 - G)(k;,, - w,2)(kfo - wi”)
+qko3+4ko[1 + q k;.t
=
+f$(E2+
mzv2)
(k12 + k22)] - 91, k12
+
h2
E?.f = m2 -I- kt,r ,
+
y2(k,
f
(35) vko12,
ui.t = El.! - BM,
k io.io = iM
-
y(ko
f vk,),
E2 = m2 + k12 + k22 + k32.
QUARK SELECTION PRINCIPLE, II
259
The integral in (35) is actually three-dimensional, since k1 and k, enter the integrand only as the sum of their squares. The normalization integral, obtained from (35) with y = 1 or u = 0, can be reduced to one dimension and is equal to 32ea m (A4 + 2E) sin2 kR dk (k2 - a2)2 ’ 4M 0
(36)
where E2 = m2 i- k2. It would seem at first that the electromagnetic form factor is simply the ratio of the current matrix element (39, with exp(iq * x0) omitted, to the normalization integral (36). Actually, however, we must divide this ratio by the corresponding quantity that would be obtained with a point meson. For a point meson at rest, the unnormalized wavefunction is exp(-NT); boosting it to initial or final four-momentum P, as we did in obtaining (33) and (34), leads to the wavefunction exp(--iP * X), without the usual multiplicative factor (2P”)-‘l”. The matrix element of the point meson current operator in the Breit frame is then proportional to (Pi + Pi)” = (2My, 0, 0,O). Thus in order to calculate the form factor, we must divide (35) not only by (36), but also by y. 5. CONCLUDING
REMARKS
The integral in (35) converges for all real y > 1, i.e., for q2given by (30) on the negative real axis. Its analytic properties for general complex values of q2 seem to be very difficult to decipher. However, there is no trace of the anomalous threshold that one would expect to find from the vertex diagram in perturbation theory, which is at q2 = -MM2(M2 - 4m2)/m2.This is not surprising when it is remembered that there is no field theory underlying our calculation which shows how it can be cast in terms of perturbation theory diagrams. The convergence of (35) is quite slow for large values of the variables of integration, except for y = 1 where the normalization (36) is rapidly convergent. The rate of convergence of (35), and therefore also the feasibility of numerical computation, would probably be improved if the sharp spatial boundary condition represented by (26) were to be smoothed over. The asymptotic behavior of the form factor for large negative q2 is not difficult to find. In this case y > 1 and we can set u = 1 in calculating the leading term. The change of variables from k, , k, to x, y = y(k, f k3) separates the parts of the integrand involving ki and kio from the parts involving kf and kf, . With the substitution w = k12+ kz2, the leading term of (35) for large y can be written z
exp(iq * x0) 1: dw (m” + w) Io(w)[211(~) - Ml,(w)], ~“(44 - 2E) j,(kR) In(W) = I”“, dx (k2 _ a2)(E - x)(&f - E - x + ig) ’
(37)
260
SCHIFF
where k2 = x2 + w and E2 = x2 + w + m2. The form factor should be real; this is evidently the case when y = 1 [Eq. (36)], but is not apparent in general. It follows from (37) and the comment at the end of the preceding section that the electromagnetic form factor falls off like l/y2 or l/q2 for large momentum transfer. It is therefore of the “pole” form rather than the “dipole” form which gives a better fit to the experiments on high-energy elastic electron-proton scattering. There is of course no reason why the meson and proton form factors must have the same asymptotic form, and there is no experimental evidence with respect to mesons. Finally, it should be noted that the asymptotic behavior calculated here is independent of the functional form of F(k,, , k) in (28), and hence of the choice of the spatial boundary condition and the relative time dependence. This is because the change of variables from k,, , k, to x, y used in deriving (37) effectively eliminates the contribution of F(k, , k) to the y dependence of the form factor. ACKNOWLEDGMENTS It is a pleasure to thank M. Jacob and L. Van Hove for stimulating conversations. The hospitality of the Theoretical Physics Group at Imperial College, London, is gratefully acknowledged.
REFERENCES 1. L. I. SCHIFF,Phys. Lett. B 31 (1970), 79; this paper contains references to related earlier work. 2. The most recent papers that bear on the experimental observability of quarks are W. T. CHU, Y. S. KIM, W. J. BEAM, AND N. KWAK, Phys. Rev. Len. 24 (1970), 917; H. FAISSNER, M. HOLDER, K. KRISOR, G. MASON, Z. SAWAF, AND H. UMBACH, Phys. Rev. Lett. 24 (1970), 1357. These papers contain references to earlier work. 3. This mass refers to nonstrange quarks, and was obtained from the proton and neutron magnetic moments. In the same way, the mass of the strange quark, obtained from the d magnetic moment, is found to be about 430 MeV/?. 4. E. E. SALPETER AND H. A. BETHE, Phys. Rev. 84 (1951), 1232; E. E. SALPETER, Phys. Rev. 87 (1952), 328. 5. We use the now standard notation of J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Mechanics,” Appendix A, McGraw-Hill Book Co., New York, 1964. 6. G. C. WICK, Phys. Rev. 96 (1954), 1124. 7. M. GELL-MANN AND F. Low, Phys. Rev. 84 (1951), 350. 8. S. MANDELSTAM, Proc. Roy. Sot. London, Ser. A 233 (1955), 248. 9. L. DE BROGLIE, “ThCorie G&ale des Particules g Spin (Mtthode de Fusion),” Gauthier Villars, Paris, 1943. 10. J. S. GOLDSTEIN, Phys. Rev. 91 (1953), 1516. 11. C. H. L. SMITH, Ann. Phys. (New York) 53 (1969), 521. 12. Reference [5], p. 67. 13. Reference [5], Chapter 2. 14. S. J. BRODSKY AND J. R. PRIMACK, Ann. Phys. (New York) 52 (1969), 315.