- Email: [email protected]
Local current algebra sum rules and electron-positron annihilation into baryons
PHYSICS
Volume 31B, number 6
LOCAL
CURRENT
LETTERS
ALGEBRA SUM ANNIHILATION
RULES INTO
16 March 1970
AND ELECTRON-POSITRON BARYONS
M. S. CHANOWITZ * Laboratory
of Nuclear
Studies,
Cornell
Received
University,
Ithaca,
New York
14850,
USA
30 January 1970
We derive a set of local current algebra sum rules which can be studied in e+e-colliding beam experiments. With satur_ation assumptions , _we calculate the nucleon isovector radius and the slope at threshold for o(e+e --+NN)I=l and o(e+e- *NA)fzl.
Global current algebra commutators, involving charges or charges and currents, have been investigated and confirmed by a variety of sum rules and soft pion calculations. However, we still know very little about the local commutators of currents with currents. Our ignorance is most perfect in the case of commutators which contain one or two spatial components, largely because the infinite momentum (P-infinity limit) technique probably cannot be applied to such commutators [l]. Another method for studying commutators of spatial components, the Bjorken limit as used by Callan and Gross [2], is not consistent with some model perturbation theory calculations [3]; whether the fault lies with the limit or an inappropriate use of perturbation theory remains to be seen. In any case, we observe that such elegant methods as the P-infinity and Bjorken limits do have the disadvantage of interposing additional assumptions between the posited commutator and the conclusions obtained. We are therefore motivated to re-examine what is perhaps the most straightforward method for studying local current algebra- - the fixed three - momentum, finite moment sum rules. Such sum rules were first used to show that results previously obtained from SU6 symmetry could also be obtained from current algebra [4]. More recently, the method has been used by Bietti and by Sidhu and Dresden [5] to investigate the consequences of quark model and field algebra current commutation relations. The principal obstacle to a wider application of these sum rules has been the practical impossibility of For instance, to measure the quantities which appear achieving a detailed comparison with experiment. in Bietti’s sum rules, one would have to scatter timelike photons from proton and neutron targets ( or detect the inverse decays). We have found, however, that the situation is actually not quite so bad and that the sum rules contain a previously neglected class of terms, which make an important contribution to the sum rules and which have the virtue of being accessible to experimental study. These are the “pair state” state contributions, which can be measured in electron-positron colliding beam experiments. In particular, we have derived five sum rules which when saturated with the leading resonances yield i) a determination of the isovector charge radius of the nucleon, which is within 25% of the known experimental value and ii) a prediction of the slope of the cross sections at threshold for electron-positron annihilation into isovector nucleon-antinucleon pairs and isovector nucleon-anti ~(1236) pairs. We will sketch the derivation of one of the five sum rules. Rather than use the multipole operators directly, as previous workers have done, we will use a related technique which has the advantage of being free of ambiguity. Elsewhere we will show that commutators of high order multipole operators do not always yield unique sum rules, but that different sum rules may result, depending on how the the usual way of determining possible interimplicit derivatives and limits are defined. Furthermore, mediate states by inspection of the angular momentum and parity of the multipole operators can lead to incorrect conclusions [6]. For the sake of definiteness, we consider free quark model and field algebra commutation relations
171.
* National Science
374
Foundation Predoctoral
Fellow.
Volume 31B, number 6
PHYSICS
LETTERS
16 March 1970
(d)
(b)
(0)
Fig. 1. Types of contributions: (a) connected, (b) pairs, (c) semiconnected and (d) disconnected. (the wavy lines denote the currents. ) Then an equal time commutator [e(x7,
=
v3_(0)]
where the gradient
of SU2 vector
26(x) v3,(%) + i &
term C is a c-number.
[sd3x exp(i+ x) I&?),
=
v!!(O)]
currents
* is
6(.?)C Taking a three-dimensional
2$(O)
Fourier
we find
f43 C
(2)
We bracket (2) by protons polarized along the positive z-axis and of opposite complete set of intermediate states. With the choice c =p, we obtain ~(N+(P)(VOcIn(~))(n(~)jV3/N+(-~))
transform,
- {V~++V~}
=
momenta 55,
and insert
P3(2~)36(2$)C
a
(3)
where the sum is over a complete set of quantum numbers except for the total three-momentum, which we have already integrated to obtain eq. (3). Eq. (3) appears to be a bizarre representation for the gradient term C. In fact, eq. (3) contains the usual spectral representation for C [8], plus a sum rule which has nothing to do with C. To understand this, we must rememeber that there are disconnected contributions to the matrix elements which appear in eq. (3). If the state n contains a proton and other particles, /n(G)) = / N,(g)n(-z)), then the corresponding matrix element will have a connected and a disconnected part,
(N&j) / Vgl n(o)
>
=
(Nt( Vgl n(o)),
+ (2n)3Q5G-fi)(0
(4)
\V$ln(-k))
where the subscript “c” denotes the connected part. When n contains several protons, eq. (4) must be modified to include several disconnected terms. Inserting expressions like eq. (4) into eq. (3), we find four classes of contributions to the sum rules, which are known as connected, pairs, semi-connected and disconnected ( see fig. 1). The disconnected terms contain a factor b(2$) which matches the singular coefficient on the right-hand side of eq. (3). Thus the disconnected terms provide the representation of C, while the other terms must cancel among themselves. Eq. (3) is true for all values of F. We may extract fi$te moment s>rn rules from (3) in an unambiguous way by calculating its derivatives with respect top at the point P = 0. Here we consider the first this is equivalent to considering the comderivative with respect to p,; in the language of multipoles, Current conservation (we work to zeroth order mutator of the electric dipole with the spatial charge**. in weak and electro-magnetic interactions) allows us to write (N+($)IV~In(~)), Finally we substitute tain *
=
1
pO_,;=l
the equations
.cPi(N+(p)
/ VJj n(o)),
like (4) and (5) into (3) and evaluate the derivative
(5) a/ap3 133
to ob-
denotes the vector current with SU2 index “anI understood to be at time t=O. Vt is understood to be at (Z,t) = (6,O). SU2 raising and lowering currents are defined in the usual way, Vg = VI/J* i V2@. Proton states are normalized by(Ns@)INsr(P’)) = (2Q36($-p)ass,. ** It turns out that the commutator of the electric dipole with the spatial charge does lead to a unique sum rule, which is just our eq. (7). But, for instance, the commutator of the electric dipole with the magnetic quadrupole , which is considered in ref. 5, leads to different sum rules depending on exactly how the implicit derivatives are chosen. This will be discussed elsewhere. V*(2)
375
PHYSICS
Volume 31B, number G
LETTERS
o=& - 2 /Gv(4m2) I2+$&p&l(N+(@jV~ IN*@)),I2- &I(0 =+F
$
n
Re[@ iv: jn@))(n(@N+(fi)
IV! lN+(G))c]
16 March 1970
jv!l~+(O)~*(b))/~
+ {V:++V!)
In eq. (7) we have explicitly displayed the nucleon and antinucleon contributions - m is the nucleon mass, m * is the mass corresponding to N* and GV(4m2) is the Sachs isovector nucleon form factor [9] (at 4m2 the electric and magnetic Sachs form factors are equal). The connected terms all contribute to eq. (‘7) with a positive sign, so that the pair and semiconnected terms are of equal importance with the connected terms in satisfying the sum rule. For the two sum rules given below which are derived from the commutator of two spatial components, we then find (by using current conservation as in eq. (5) ) that the pair and semiconnected contributions are more important than the connected contributions. It is straightforward to verify that eq. (6) is the usual spectral representation for the c-number part of the gradient term [8], displayed noncovariantly by choosing the rest frame of the intermediate states. It can be shown that if there were an isospinsymmetric operator gradient in the commutator (l), it would contribute to the left-hand side of eq. (‘7) [6]. We have also made the amusing discovery that the analogue of eq. (7) derived for pions instead of nucleons is, in the soft pion limit, precisely the first spectral function sum rule of Weinberg [6,10]. In eq. (‘7), the intermediate states are kinematically restricted as follows: a) Connected contributions N* have angular momentum and pa;ity, JP = $-, f- . b) Pair contributions N* are the charge;on_jugates of states N which have JP = L+ 2 9 I+. 2 c) Semiconnected contributions n have J - 1 . Eventually we will saturate (7) with the leading resonances. But at first sight there appears to be-a catastrophe: The rho meson pole contribution to the rho meson semiconnected term (N(O)p(O)/ V:] N(O)), The difficulty is resolved when we notice that the infinity is is infinite in the zero width limit, I?,-0. cancelled, via unitarity, by rho pole contributions to the connected terms. A very similar cancellation has been pointed out by Weisberger [ll] 1. For our present purpose, the important point is that assuming rho pole dominance of the matrix element (oN / VjN)c and neglecting the rho meson width (i.e. neglecting terms of order Fo /mp), we find that the rho meson semi-connected term does not contribute to the sum rule. We shall adopt these approximations and neglect the rho. Together with the assumption of resonance saturation, this means that we will neglect the semiconnected terms altogether. Finally we write eq. (7) in a more compact form, without the semiconnected terms. Defining the “inelastic mean square radii”, R,,,~J(N*)
-
(8)
+JJJ J(N*) where
I
(m’l_m)2 /(N+(fi)/V~IN*(@)/2
=
and Jdenote
&
IGv(4m2)12
= the isospin -r
8m
=
and angular $?..
momentum
(m*-m)R
of N*, we rewrite
21,2J-(“*+“)‘2I,2J
eq. (7) as (1’)
‘&,f
Similarly, @X),
from the commutators V!(O) ] = 26 (3 V$)
[v~(X),V~(O)]
= 26(rlVj(Z)+i$--_6(MC
(9) (10)
We obtain a sum rule without singularities $ Our proof of the cancellation is somewhat different from Weisberger’s. by choosing the protons in eq. (3) with opposite but slightly unequal momenta. The parts of the rho pole contributions which become singular as the proton momenta become equal and opposite are then seen to cancel using unitarity.
376
Volume
31B, number 6
PHYSICS
26(X) Vi%)
[vp),v”(o)l=1 o [v’,(%v:(o)]
8m2 4
+(,,2, __ 3
l/2 0
+ ($f A,$?) + (f)’ A$(?)
m
=
=
G*2R11+2R13-R31-
(12)
R33+ {R--w)
x (m* - m)( -4R11 +2R13 +2R31 -R33) N’
2 / GV(4m2) / 2+ $m*
=
Quark
sum rules 1s :
- 1GV(4m2) I2 22
1GV(4m2) I2
Pv-2
(11)
Field alg.
we obtain the corresponding 1
Quark Field alg.
46(X) I’$@
=
16 March 19’70
LETTERS
- m)2 (2Rll+2R13
(9’)
- ( R--R,
- R31 - R33) + (R-+-R,
m --m >
n--‘-m
(10’)
>
(11’)
I
(piAi
f + (N+(a) 1
+ (if Ai /N+(O))
I
=
0
= 41~‘(4m~)/~
+$m*-m)2(4R~l+R~3+R3~-2R33)
+(R--,
m--m
1
(12’)
These are all the independent sum rules which can be derived for the “inelastic radii” defined in eq. (8). yv and pv are the nucleon isovector charge radius and magnetic moment, both known experimentally. The currentsAp are a U3 nonet of axial currents. For the quar % model eq. (12’) is no help to us since it is not even known whether the posited neutral axial currents occur in nature. We therefore attempt to saturate the four remaining sum rules. Our strategy is to determine the unknown parameters from three of the sum rules and to use the fourth to obtain a consistency relation. In particular, we saturate the connected terms with the nucleon and N*(1518) and the pair terms with the antinucleon and 2 (1236). From eqs. (l’),(ll’) and (12’) we calculate, using quark model (field algebra) commutators, that IGV(4m2)i2 = 0.43(0.29), R13(1518) = = 0.87(0.98) m-2 and R 3(1236) = O.lg(O.25) m-2 (we have used the experimental value, 1-1~ = =$(/JP-/L~) =2.35). I?rom eq. (10’) we then calculate the isovector charge radius to be rV = 0.46(0.48) fm, which may be compared with the experimental value, yy = 0.62rtO.01 fm [9]. These predictions for / GV(4m2) 12, while large, are not incompatible with the experimental bound j~G(5.8 m2))<0.2 [12], considering the interval A s = 1.8 m2 between the two points and the fact that the bound is for the proton and not for the isovector combination. The simple rho dominance model for the Pauli form factors, which satisfies GEV(4m2) = G V(4m2), predicts 1GV(4m2)12 = 0.22, which is in amusing (though most certainly spurious) agreemen r with the result we have obtained from field algebra commutators. The dipole fit to the Pauli form factors for spacelike momenta predicts a value which is smaller by one order of magnitude, lGv(4m2)l 2 = 0.022. The pair state inelastic mean square radius RIJ (N*) determines the slope of the cross section at threshold for e+e--N? in the I = 1 channel. In particular, for the creation of isovector nucleon-antinucleon pairs, we have
g
(e+e--m)
p=.
=$I GV(4m2)1
2
$$ In eqs. (9’)-(12’), (m* f m) is factored for the sake of brevity; degeneracy in the mass of the intermediate states.
(13)
of course
we do not mean to imply a necessary
Volume
31B, number 6
PHYSICS
LETTERS
where P is the magnitude of the nucleon three-momentum numerical results into eq. (13), we find
16 March 1970
in the center-of-mass
frame.
Substituting our
(14) Assuming there is little variation in the nucleon form factors from s = 4m2 to s = 4.16m2, we may use eq. (14) to estimate the magnitude of the cross section at s = 4.16m2 (p = 0.2 m). We find oV(4.16m2) - 3.2(2.2) x 1O-33 cm2 from the quark model (field algebra) results. (With the expected luminosity of the Frascati colliding beams, this would correspond to a rate of two or three events per hour. ) As in eq. (13), we may calculate for the production of Z = 1 nucleon-anti ~(1236) pairs that Quark X 10-32m-1cm2
Field alg.
(15)
The fact that we have been able to obtain a plausible value for the isovector nucleon radius suggests that our predictions for e+e--m may be taken seriously at least as indication of order of magnitude. The rather large value of our prediction is a clear consequence of the structure of the sum rules, together with the resonance saturation assumption and the assumption that the semiconnected terms are less important than the connected and pair state terms. Whether such sum rules can eventually be used to probe the local commutators on a more refined scale will depend heavily on the nature of the results obtained in the future experimental study of electron-positron annihilation into hadrons. I am grateful to P. Carruthers, and helpful discussions.
K. Gottfried,
J. Pestieau,
P. Roy and K. Wilson for many interesting
References 1. S. L. Adler and R. F. Dashen, Current Algebras (W. A. Benjamin, Inc., New York, 2. C. G. Callan and D. J. Gross, Phys. Rev. Letters 22 (1969) 156. 3. R. Jackiw and G. Preparata, Phys, Rev. Letters 22 (1969) 975; S. Adler and W. Tung, Phys. Rev. Letters 22 (1969) 978. 4. B. W. Lee, Phys. Rev. Letters 14 (1965) 676; R. F. Dashen and M. Gell-Mann, Phys. Letters 17 (1965) 145. 5, A. Bietti, Phys. Rev. 142 (1966) 1258; D. Sidhu and M. Dresden, Phys. Rev. Letters 23 (1969) 447. 6, Details will be presented in a paper to be submitted to Phys. Rev. 7, R. Dashen and M. Gell-Mann, Phys. Letters 17 (1965) 142; T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Letters 18 (1967) 1029. 8. T. Gato and T. Imamura, Progr. Theor. Phys. 14 (1955) 396. 9. L. Hand, D. Miller and R. Wilson, Rev. Mod. Phys. 35 (1963) 335. 10. S. Weinberg, Phys. Rev. Letters 18 (1967) 507. 11. W. Weisberger, Phys. Rev. Letters 14 (1965) 1047. 12. D. Hartill et al., Phys. Rev. 184 (1969) 1415.
378
1968),
Ch. 4 and 5.
Volume 31B, number 6
LOCAL
CURRENT
LETTERS
ALGEBRA SUM ANNIHILATION
RULES INTO
16 March 1970
AND ELECTRON-POSITRON BARYONS
M. S. CHANOWITZ * Laboratory
of Nuclear
Studies,
Cornell
Received
University,
Ithaca,
New York
14850,
USA
30 January 1970
We derive a set of local current algebra sum rules which can be studied in e+e-colliding beam experiments. With satur_ation assumptions , _we calculate the nucleon isovector radius and the slope at threshold for o(e+e --+NN)I=l and o(e+e- *NA)fzl.
Global current algebra commutators, involving charges or charges and currents, have been investigated and confirmed by a variety of sum rules and soft pion calculations. However, we still know very little about the local commutators of currents with currents. Our ignorance is most perfect in the case of commutators which contain one or two spatial components, largely because the infinite momentum (P-infinity limit) technique probably cannot be applied to such commutators [l]. Another method for studying commutators of spatial components, the Bjorken limit as used by Callan and Gross [2], is not consistent with some model perturbation theory calculations [3]; whether the fault lies with the limit or an inappropriate use of perturbation theory remains to be seen. In any case, we observe that such elegant methods as the P-infinity and Bjorken limits do have the disadvantage of interposing additional assumptions between the posited commutator and the conclusions obtained. We are therefore motivated to re-examine what is perhaps the most straightforward method for studying local current algebra- - the fixed three - momentum, finite moment sum rules. Such sum rules were first used to show that results previously obtained from SU6 symmetry could also be obtained from current algebra [4]. More recently, the method has been used by Bietti and by Sidhu and Dresden [5] to investigate the consequences of quark model and field algebra current commutation relations. The principal obstacle to a wider application of these sum rules has been the practical impossibility of For instance, to measure the quantities which appear achieving a detailed comparison with experiment. in Bietti’s sum rules, one would have to scatter timelike photons from proton and neutron targets ( or detect the inverse decays). We have found, however, that the situation is actually not quite so bad and that the sum rules contain a previously neglected class of terms, which make an important contribution to the sum rules and which have the virtue of being accessible to experimental study. These are the “pair state” state contributions, which can be measured in electron-positron colliding beam experiments. In particular, we have derived five sum rules which when saturated with the leading resonances yield i) a determination of the isovector charge radius of the nucleon, which is within 25% of the known experimental value and ii) a prediction of the slope of the cross sections at threshold for electron-positron annihilation into isovector nucleon-antinucleon pairs and isovector nucleon-anti ~(1236) pairs. We will sketch the derivation of one of the five sum rules. Rather than use the multipole operators directly, as previous workers have done, we will use a related technique which has the advantage of being free of ambiguity. Elsewhere we will show that commutators of high order multipole operators do not always yield unique sum rules, but that different sum rules may result, depending on how the the usual way of determining possible interimplicit derivatives and limits are defined. Furthermore, mediate states by inspection of the angular momentum and parity of the multipole operators can lead to incorrect conclusions [6]. For the sake of definiteness, we consider free quark model and field algebra commutation relations
171.
* National Science
374
Foundation Predoctoral
Fellow.
Volume 31B, number 6
PHYSICS
LETTERS
16 March 1970
(d)
(b)
(0)
Fig. 1. Types of contributions: (a) connected, (b) pairs, (c) semiconnected and (d) disconnected. (the wavy lines denote the currents. ) Then an equal time commutator [e(x7,
=
v3_(0)]
where the gradient
of SU2 vector
26(x) v3,(%) + i &
term C is a c-number.
[sd3x exp(i+ x) I&?),
=
v!!(O)]
currents
* is
6(.?)C Taking a three-dimensional
2$(O)
Fourier
we find
f43 C
(2)
We bracket (2) by protons polarized along the positive z-axis and of opposite complete set of intermediate states. With the choice c =p, we obtain ~(N+(P)(VOcIn(~))(n(~)jV3/N+(-~))
transform,
- {V~++V~}
=
momenta 55,
and insert
P3(2~)36(2$)C
a
(3)
where the sum is over a complete set of quantum numbers except for the total three-momentum, which we have already integrated to obtain eq. (3). Eq. (3) appears to be a bizarre representation for the gradient term C. In fact, eq. (3) contains the usual spectral representation for C [8], plus a sum rule which has nothing to do with C. To understand this, we must rememeber that there are disconnected contributions to the matrix elements which appear in eq. (3). If the state n contains a proton and other particles, /n(G)) = / N,(g)n(-z)), then the corresponding matrix element will have a connected and a disconnected part,
(N&j) / Vgl n(o)
>
=
(Nt( Vgl n(o)),
+ (2n)3Q5G-fi)(0
(4)
\V$ln(-k))
where the subscript “c” denotes the connected part. When n contains several protons, eq. (4) must be modified to include several disconnected terms. Inserting expressions like eq. (4) into eq. (3), we find four classes of contributions to the sum rules, which are known as connected, pairs, semi-connected and disconnected ( see fig. 1). The disconnected terms contain a factor b(2$) which matches the singular coefficient on the right-hand side of eq. (3). Thus the disconnected terms provide the representation of C, while the other terms must cancel among themselves. Eq. (3) is true for all values of F. We may extract fi$te moment s>rn rules from (3) in an unambiguous way by calculating its derivatives with respect top at the point P = 0. Here we consider the first this is equivalent to considering the comderivative with respect to p,; in the language of multipoles, Current conservation (we work to zeroth order mutator of the electric dipole with the spatial charge**. in weak and electro-magnetic interactions) allows us to write (N+($)IV~In(~)), Finally we substitute tain *
=
1
pO_,;=l
the equations
.cPi(N+(p)
/ VJj n(o)),
like (4) and (5) into (3) and evaluate the derivative
(5) a/ap3 133
to ob-
denotes the vector current with SU2 index “anI understood to be at time t=O. Vt is understood to be at (Z,t) = (6,O). SU2 raising and lowering currents are defined in the usual way, Vg = VI/J* i V2@. Proton states are normalized by(Ns@)INsr(P’)) = (2Q36($-p)ass,. ** It turns out that the commutator of the electric dipole with the spatial charge does lead to a unique sum rule, which is just our eq. (7). But, for instance, the commutator of the electric dipole with the magnetic quadrupole , which is considered in ref. 5, leads to different sum rules depending on exactly how the implicit derivatives are chosen. This will be discussed elsewhere. V*(2)
375
PHYSICS
Volume 31B, number G
LETTERS
o=& - 2 /Gv(4m2) I2+$&p&l(N+(@jV~ IN*@)),I2- &I(0 =+F
$
n
Re[@ iv: jn@))(n(@N+(fi)
IV! lN+(G))c]
16 March 1970
jv!l~+(O)~*(b))/~
+ {V:++V!)
In eq. (7) we have explicitly displayed the nucleon and antinucleon contributions - m is the nucleon mass, m * is the mass corresponding to N* and GV(4m2) is the Sachs isovector nucleon form factor [9] (at 4m2 the electric and magnetic Sachs form factors are equal). The connected terms all contribute to eq. (‘7) with a positive sign, so that the pair and semiconnected terms are of equal importance with the connected terms in satisfying the sum rule. For the two sum rules given below which are derived from the commutator of two spatial components, we then find (by using current conservation as in eq. (5) ) that the pair and semiconnected contributions are more important than the connected contributions. It is straightforward to verify that eq. (6) is the usual spectral representation for the c-number part of the gradient term [8], displayed noncovariantly by choosing the rest frame of the intermediate states. It can be shown that if there were an isospinsymmetric operator gradient in the commutator (l), it would contribute to the left-hand side of eq. (‘7) [6]. We have also made the amusing discovery that the analogue of eq. (7) derived for pions instead of nucleons is, in the soft pion limit, precisely the first spectral function sum rule of Weinberg [6,10]. In eq. (‘7), the intermediate states are kinematically restricted as follows: a) Connected contributions N* have angular momentum and pa;ity, JP = $-, f- . b) Pair contributions N* are the charge;on_jugates of states N which have JP = L+ 2 9 I+. 2 c) Semiconnected contributions n have J - 1 . Eventually we will saturate (7) with the leading resonances. But at first sight there appears to be-a catastrophe: The rho meson pole contribution to the rho meson semiconnected term (N(O)p(O)/ V:] N(O)), The difficulty is resolved when we notice that the infinity is is infinite in the zero width limit, I?,-0. cancelled, via unitarity, by rho pole contributions to the connected terms. A very similar cancellation has been pointed out by Weisberger [ll] 1. For our present purpose, the important point is that assuming rho pole dominance of the matrix element (oN / VjN)c and neglecting the rho meson width (i.e. neglecting terms of order Fo /mp), we find that the rho meson semi-connected term does not contribute to the sum rule. We shall adopt these approximations and neglect the rho. Together with the assumption of resonance saturation, this means that we will neglect the semiconnected terms altogether. Finally we write eq. (7) in a more compact form, without the semiconnected terms. Defining the “inelastic mean square radii”, R,,,~J(N*)
-
(8)
+JJJ J(N*) where
I
(m’l_m)2 /(N+(fi)/V~IN*(@)/2
=
and Jdenote
&
IGv(4m2)12
= the isospin -r
8m
=
and angular $?..
momentum
(m*-m)R
of N*, we rewrite
21,2J-(“*+“)‘2I,2J
eq. (7) as (1’)
‘&,f
Similarly, @X),
from the commutators V!(O) ] = 26 (3 V$)
[v~(X),V~(O)]
= 26(rlVj(Z)+i$--_6(MC
(9) (10)
We obtain a sum rule without singularities $ Our proof of the cancellation is somewhat different from Weisberger’s. by choosing the protons in eq. (3) with opposite but slightly unequal momenta. The parts of the rho pole contributions which become singular as the proton momenta become equal and opposite are then seen to cancel using unitarity.
376
Volume
31B, number 6
PHYSICS
26(X) Vi%)
[vp),v”(o)l=1 o [v’,(%v:(o)]
8m2 4
+(,,2, __ 3
l/2 0
+ ($f A,$?) + (f)’ A$(?)
m
=
=
G*2R11+2R13-R31-
(12)
R33+ {R--w)
x (m* - m)( -4R11 +2R13 +2R31 -R33) N’
2 / GV(4m2) / 2+ $m*
=
Quark
sum rules 1s :
- 1GV(4m2) I2 22
1GV(4m2) I2
Pv-2
(11)
Field alg.
we obtain the corresponding 1
Quark Field alg.
46(X) I’$@
=
16 March 19’70
LETTERS
- m)2 (2Rll+2R13
(9’)
- ( R--R,
- R31 - R33) + (R-+-R,
m --m >
n--‘-m
(10’)
>
(11’)
I
(piAi
f + (N+(a) 1
+ (if Ai /N+(O))
I
=
0
= 41~‘(4m~)/~
+$m*-m)2(4R~l+R~3+R3~-2R33)
+(R--,
m--m
1
(12’)
These are all the independent sum rules which can be derived for the “inelastic radii” defined in eq. (8). yv and pv are the nucleon isovector charge radius and magnetic moment, both known experimentally. The currentsAp are a U3 nonet of axial currents. For the quar % model eq. (12’) is no help to us since it is not even known whether the posited neutral axial currents occur in nature. We therefore attempt to saturate the four remaining sum rules. Our strategy is to determine the unknown parameters from three of the sum rules and to use the fourth to obtain a consistency relation. In particular, we saturate the connected terms with the nucleon and N*(1518) and the pair terms with the antinucleon and 2 (1236). From eqs. (l’),(ll’) and (12’) we calculate, using quark model (field algebra) commutators, that IGV(4m2)i2 = 0.43(0.29), R13(1518) = = 0.87(0.98) m-2 and R 3(1236) = O.lg(O.25) m-2 (we have used the experimental value, 1-1~ = =$(/JP-/L~) =2.35). I?rom eq. (10’) we then calculate the isovector charge radius to be rV = 0.46(0.48) fm, which may be compared with the experimental value, yy = 0.62rtO.01 fm [9]. These predictions for / GV(4m2) 12, while large, are not incompatible with the experimental bound j~G(5.8 m2))<0.2 [12], considering the interval A s = 1.8 m2 between the two points and the fact that the bound is for the proton and not for the isovector combination. The simple rho dominance model for the Pauli form factors, which satisfies GEV(4m2) = G V(4m2), predicts 1GV(4m2)12 = 0.22, which is in amusing (though most certainly spurious) agreemen r with the result we have obtained from field algebra commutators. The dipole fit to the Pauli form factors for spacelike momenta predicts a value which is smaller by one order of magnitude, lGv(4m2)l 2 = 0.022. The pair state inelastic mean square radius RIJ (N*) determines the slope of the cross section at threshold for e+e--N? in the I = 1 channel. In particular, for the creation of isovector nucleon-antinucleon pairs, we have
g
(e+e--m)
p=.
=$I GV(4m2)1
2
$$ In eqs. (9’)-(12’), (m* f m) is factored for the sake of brevity; degeneracy in the mass of the intermediate states.
(13)
of course
we do not mean to imply a necessary
Volume
31B, number 6
PHYSICS
LETTERS
where P is the magnitude of the nucleon three-momentum numerical results into eq. (13), we find
16 March 1970
in the center-of-mass
frame.
Substituting our
(14) Assuming there is little variation in the nucleon form factors from s = 4m2 to s = 4.16m2, we may use eq. (14) to estimate the magnitude of the cross section at s = 4.16m2 (p = 0.2 m). We find oV(4.16m2) - 3.2(2.2) x 1O-33 cm2 from the quark model (field algebra) results. (With the expected luminosity of the Frascati colliding beams, this would correspond to a rate of two or three events per hour. ) As in eq. (13), we may calculate for the production of Z = 1 nucleon-anti ~(1236) pairs that Quark X 10-32m-1cm2
Field alg.
(15)
The fact that we have been able to obtain a plausible value for the isovector nucleon radius suggests that our predictions for e+e--m may be taken seriously at least as indication of order of magnitude. The rather large value of our prediction is a clear consequence of the structure of the sum rules, together with the resonance saturation assumption and the assumption that the semiconnected terms are less important than the connected and pair state terms. Whether such sum rules can eventually be used to probe the local commutators on a more refined scale will depend heavily on the nature of the results obtained in the future experimental study of electron-positron annihilation into hadrons. I am grateful to P. Carruthers, and helpful discussions.
K. Gottfried,
J. Pestieau,
P. Roy and K. Wilson for many interesting
References 1. S. L. Adler and R. F. Dashen, Current Algebras (W. A. Benjamin, Inc., New York, 2. C. G. Callan and D. J. Gross, Phys. Rev. Letters 22 (1969) 156. 3. R. Jackiw and G. Preparata, Phys, Rev. Letters 22 (1969) 975; S. Adler and W. Tung, Phys. Rev. Letters 22 (1969) 978. 4. B. W. Lee, Phys. Rev. Letters 14 (1965) 676; R. F. Dashen and M. Gell-Mann, Phys. Letters 17 (1965) 145. 5, A. Bietti, Phys. Rev. 142 (1966) 1258; D. Sidhu and M. Dresden, Phys. Rev. Letters 23 (1969) 447. 6, Details will be presented in a paper to be submitted to Phys. Rev. 7, R. Dashen and M. Gell-Mann, Phys. Letters 17 (1965) 142; T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Letters 18 (1967) 1029. 8. T. Gato and T. Imamura, Progr. Theor. Phys. 14 (1955) 396. 9. L. Hand, D. Miller and R. Wilson, Rev. Mod. Phys. 35 (1963) 335. 10. S. Weinberg, Phys. Rev. Letters 18 (1967) 507. 11. W. Weisberger, Phys. Rev. Letters 14 (1965) 1047. 12. D. Hartill et al., Phys. Rev. 184 (1969) 1415.
378
1968),
Ch. 4 and 5.