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Electromagnetic form factors of bound nucleons
Volume 92B, number 3,4
PHYSICS LETTERS
19 May 1980
ELECTROMAGNETIC FORM FACTORS OF BOUND NUCLEONS J.O. EEG Fysisk Institutt, Universitetet, N Oslo 3, Norway
M. MARTINIS Rudjer Boskovi6 Institute, 41 O0 Zagreb, Yugoslavia
H. PILKUHN Universitdt, D 75 Karlsruhe 1, Fed. Rep. Germany
Received 6 February 1980
The isovector form factors of bound nucleons are more pointlike than those of free nucleons, at least in the impulse approximation. Inclusion of the proton-neutron mass difference changes the true isoscalar magnetic moment of nucleons by 5%.
In the theory of nuclear form factors and mass differences, one includes diagram (a) of fig. 1 only implicitly, through the isovector nucleon form factor. It is true that the "Born part'" of this diagram (obtained by replacing the lengthy blob between the bound nucleons b and b' by a single nucleon N) is considerably quenched by the presence of other nucleons, due to the Pauli principle [! ], but this effect is included in the "pion exchange current" contribution, in which the pion by definition is exchanged between two different nucleons. The remaining binding correction, once claimed to be large [2], comes out negligibly small in
TI:
/
(a)
'l
(b)
Fig. 1. The pion current contribution to nuclear form factors (a) in the impulse approximation, (b) including a nuclear potential but excluding the p resonance and other structure of (a).
a nuclear model calculation including the necessary wave function renormalization [3]. In this note we wish to emphasize that the binding effect is not negligible in the impulse approximation, which neglects diagram (b) of fig. 1. The potential model calculation of Nogami [3[ includes that diagram, but keeps only the Born part in (a). The danger of such a procedure is illustrated by the fact that the integral over the Born part diverges, whereas the full dispersion integral of (a) converges, once the p-meson is properly included. Finally, the use of a harmonic oscillator potential may be misleading for 3He and 3H, which are presently the only nuclei for which this discussion is relevant [4]. Since the neutron-proton mass difference cannot be neglected in light nuclei, we first study its influence on the magnetic moments of free nucleons, by the method of ref. [5]. The anomalous magnetic moment is the value of the Pauli form factor F2(t ) at t = 0, for which we use an unsubtracted dispersion relation: dt t¢ =rr -1 4fm --~- I m F 2 ( t ) .
(1)
We assume that the small mass difference, 6 m = 1.3 MeV is negligible in the 31r-cut contribution to Im F 2. 271
Volume 92B, number 3,4
PHYSICS LETTERS
If we denote by Kv the isovector nucleon magnetic tool ment for an average nucleon mass m = ~ (mp + ran), we find Kp =K s+K v - 0 . 0 0 3 ,
Kn = K s - K v - 0 . 0 0 3 ,
(2)
where the isoscalar part Ks has no mr-cut contribution. Consequently, Ks is 0.003 above its old value -0.061 : Ks = -0.061 + 0.003 = - 0 . 0 5 8 .
(3)
The effect is small, but more of the order of 5 m / 2 m , than 5m/2m. Note that since the numerical values of Kp and Kn have not changed, there is no change in the deuteron magnetic moment: The change in Ks is compensated by a contribution from 7rrr-states. Next, we consider a nucleon of cms momentum p in a nucleus of altogether A nucleons. It is below its mass shell at least by an amount [6] m 2 _ p2 = 2mb + p2m A / m A _ 1 -= 2rnf ,
19 May 1980
2 Howremains small until t reaches the region of m o. ever, it has been emphasized by H6hler and Pietarinen [7] that the p3-behaviour is completely obscured by the Born term G B. To find Im G B, we write m 2 _ p 2 = 2pnpN( z _ cos Ot),
PN = (t/4 -- m 2 ) 1/2,
(6) the partial-wave projection of (z - cos Or)- 1 being the Legendre function of the second kind, Q1 (z). The complete expression including nucleon spin is Im G EB -_ - f ~2 IFTr(t){2 p~m~- 2 t - 1 ] 2 X ( t - 2m2)Ql(Z),
f 2 = 0.080,
(7)
where we have approximated PN by im. As z is imaginary in the interesting region, we write Q1 = (1/x)arc t a n x - 1~
(4)
where b is the separation energy of this nucleon. The fact that the nucleons b and b' are virtual affects only the legs of the blob, but not the blob itself. Therefore is treated in the same way as the mass difference 6m above. Forgetting for the time being the 1.3 MeV n - p mass difference, we find that Kv is decreased for = 5.9 MeV (which is the average separation energy of 3He and t) 10 and 20 MeV by the amounts given in table 1 (the decrease is roughly linear in 5). Similar effects occur in the isovector electric form factor G~, for which one must use a subtracted dispersion relation:
x = (1/iz) = 4rnp~(t - 2m 2 - 4m5) - 1 .
(8)
F o r x ,~ 1, we can approximate Q1 ~ - ~ x2, and P~rQ1 in (7) behaves in fact as p3. But due to the factor 4rn, x rises rapidly with t, and around t = 6m 2 we have already x >> 1, Q1 ~ n/2x - 1, and PTr [Ql(X (5)) - Ql(X (0))] ~ 7r6/2.
(9)
The isovector electric radius is defined by r~ 2 = _6(dG~/dq2)q2 =0 0o
(10)
= 6_n4fm~ t - 2 dt Im G~ (t).
G { ( - q 2) = 1 _ 7r-lq2 X
~f -~(t+q2)-lImG~(t). 4m 2
(5)
p2._
s 2
v 2
(r E ) - ( r E ) + ( r E) - 0 . 0 0 3 ,
Since the two pions in 3' -+ n~" are in a P-wave, the integrand starts off as p3 at t = 4m 2, with P~r = (t/4 ~ 2 x l / 2 . One might then expect that the integrand ,,,~) -
With the same conventions as in eq. (2), we obtain
-
(r~)2 _ (r~)2 _ ( ~ ) 2 _ 0.003.
(11)
Insertion of 0.702 and - 0 . 1 1 7 for (r~) 2 and (r~) 2 [8] gives
Table 1 The change of the isoveetor anomalous magnetic moment and (electric radius) 2 as function of the binding energy 6.
(r~) 2 = 0.294 + 0.003 = 0.297.
8[MeV] KV(m) - KV(m- 8) [r~(m)] 2 - [r~(m- 5)] 2 [fm21
272
(12)
This change is still smaller than the experimental error. The binding effects on ( r [ ) 2 are included in table 1. The signs of 6K and 5rE2 are easily understood: The form factors have a triangle singularity at
5.9
10
20
0.0136 0.010
0.023 0.017
0.045 0.032" ttr = 4m 2 - (m 2 + 2m6)2/m 2
(13)
Volume 92B, number 3,4
PHYSICS LETTERS
on an unphysical sheet. For ~ = 0, its distance from the physical threshold tth = 4m 2 (which is on the physical sheet) is only m4/m2. As ~ increases, the singularity recedes from the threshold, and consequently the nucleon becomes more pointlike. In particular, its anomalous magnetic moment must decrease. This is opposite to the (smaller) effect found by Nogami [3]. We wish to thank Dr. S. Coon for bringing the work ofref. [3] to our attention.
19 May 1980
References [1] S.D. Drell and J.D. Walecka, Phys. Rev. 120 (1960) 1069; M. Chemto and M. Rho, Nucl. Phys. A163 (1971) 1. [2] R.V. Gerstenberger and Y. Nogami, Phys. Rev. Lett. 29 (1972) 233. [3] Y. Nogami, Revista Brasileira de Fisica 7 (1977) 19. [41 R.A. Brandenburg, S.A. Coon and P.U. Sauer, Nucl. Phys. A294 (1978) 305. [5] J.O. Eeg and H. Pilkuhn, Z. Phys. A287 (1978) 407. [6] H. Pilkuhn, Relativistic particle physics (Springer 1979). [7] G. HShler and E. Pietarinen, Phys. Lett. 53B (1975) 471; Nucl. Phys. B95 (1975) 210. [8] G. H~Shleret al., Nucl. Phys. Bl14 (1976) 505.
273
PHYSICS LETTERS
19 May 1980
ELECTROMAGNETIC FORM FACTORS OF BOUND NUCLEONS J.O. EEG Fysisk Institutt, Universitetet, N Oslo 3, Norway
M. MARTINIS Rudjer Boskovi6 Institute, 41 O0 Zagreb, Yugoslavia
H. PILKUHN Universitdt, D 75 Karlsruhe 1, Fed. Rep. Germany
Received 6 February 1980
The isovector form factors of bound nucleons are more pointlike than those of free nucleons, at least in the impulse approximation. Inclusion of the proton-neutron mass difference changes the true isoscalar magnetic moment of nucleons by 5%.
In the theory of nuclear form factors and mass differences, one includes diagram (a) of fig. 1 only implicitly, through the isovector nucleon form factor. It is true that the "Born part'" of this diagram (obtained by replacing the lengthy blob between the bound nucleons b and b' by a single nucleon N) is considerably quenched by the presence of other nucleons, due to the Pauli principle [! ], but this effect is included in the "pion exchange current" contribution, in which the pion by definition is exchanged between two different nucleons. The remaining binding correction, once claimed to be large [2], comes out negligibly small in
TI:
/
(a)
'l
(b)
Fig. 1. The pion current contribution to nuclear form factors (a) in the impulse approximation, (b) including a nuclear potential but excluding the p resonance and other structure of (a).
a nuclear model calculation including the necessary wave function renormalization [3]. In this note we wish to emphasize that the binding effect is not negligible in the impulse approximation, which neglects diagram (b) of fig. 1. The potential model calculation of Nogami [3[ includes that diagram, but keeps only the Born part in (a). The danger of such a procedure is illustrated by the fact that the integral over the Born part diverges, whereas the full dispersion integral of (a) converges, once the p-meson is properly included. Finally, the use of a harmonic oscillator potential may be misleading for 3He and 3H, which are presently the only nuclei for which this discussion is relevant [4]. Since the neutron-proton mass difference cannot be neglected in light nuclei, we first study its influence on the magnetic moments of free nucleons, by the method of ref. [5]. The anomalous magnetic moment is the value of the Pauli form factor F2(t ) at t = 0, for which we use an unsubtracted dispersion relation: dt t¢ =rr -1 4fm --~- I m F 2 ( t ) .
(1)
We assume that the small mass difference, 6 m = 1.3 MeV is negligible in the 31r-cut contribution to Im F 2. 271
Volume 92B, number 3,4
PHYSICS LETTERS
If we denote by Kv the isovector nucleon magnetic tool ment for an average nucleon mass m = ~ (mp + ran), we find Kp =K s+K v - 0 . 0 0 3 ,
Kn = K s - K v - 0 . 0 0 3 ,
(2)
where the isoscalar part Ks has no mr-cut contribution. Consequently, Ks is 0.003 above its old value -0.061 : Ks = -0.061 + 0.003 = - 0 . 0 5 8 .
(3)
The effect is small, but more of the order of 5 m / 2 m , than 5m/2m. Note that since the numerical values of Kp and Kn have not changed, there is no change in the deuteron magnetic moment: The change in Ks is compensated by a contribution from 7rrr-states. Next, we consider a nucleon of cms momentum p in a nucleus of altogether A nucleons. It is below its mass shell at least by an amount [6] m 2 _ p2 = 2mb + p2m A / m A _ 1 -= 2rnf ,
19 May 1980
2 Howremains small until t reaches the region of m o. ever, it has been emphasized by H6hler and Pietarinen [7] that the p3-behaviour is completely obscured by the Born term G B. To find Im G B, we write m 2 _ p 2 = 2pnpN( z _ cos Ot),
PN = (t/4 -- m 2 ) 1/2,
(6) the partial-wave projection of (z - cos Or)- 1 being the Legendre function of the second kind, Q1 (z). The complete expression including nucleon spin is Im G EB -_ - f ~2 IFTr(t){2 p~m~- 2 t - 1 ] 2 X ( t - 2m2)Ql(Z),
f 2 = 0.080,
(7)
where we have approximated PN by im. As z is imaginary in the interesting region, we write Q1 = (1/x)arc t a n x - 1~
(4)
where b is the separation energy of this nucleon. The fact that the nucleons b and b' are virtual affects only the legs of the blob, but not the blob itself. Therefore is treated in the same way as the mass difference 6m above. Forgetting for the time being the 1.3 MeV n - p mass difference, we find that Kv is decreased for = 5.9 MeV (which is the average separation energy of 3He and t) 10 and 20 MeV by the amounts given in table 1 (the decrease is roughly linear in 5). Similar effects occur in the isovector electric form factor G~, for which one must use a subtracted dispersion relation:
x = (1/iz) = 4rnp~(t - 2m 2 - 4m5) - 1 .
(8)
F o r x ,~ 1, we can approximate Q1 ~ - ~ x2, and P~rQ1 in (7) behaves in fact as p3. But due to the factor 4rn, x rises rapidly with t, and around t = 6m 2 we have already x >> 1, Q1 ~ n/2x - 1, and PTr [Ql(X (5)) - Ql(X (0))] ~ 7r6/2.
(9)
The isovector electric radius is defined by r~ 2 = _6(dG~/dq2)q2 =0 0o
(10)
= 6_n4fm~ t - 2 dt Im G~ (t).
G { ( - q 2) = 1 _ 7r-lq2 X
~f -~(t+q2)-lImG~(t). 4m 2
(5)
p2._
s 2
v 2
(r E ) - ( r E ) + ( r E) - 0 . 0 0 3 ,
Since the two pions in 3' -+ n~" are in a P-wave, the integrand starts off as p3 at t = 4m 2, with P~r = (t/4 ~ 2 x l / 2 . One might then expect that the integrand ,,,~) -
With the same conventions as in eq. (2), we obtain
-
(r~)2 _ (r~)2 _ ( ~ ) 2 _ 0.003.
(11)
Insertion of 0.702 and - 0 . 1 1 7 for (r~) 2 and (r~) 2 [8] gives
Table 1 The change of the isoveetor anomalous magnetic moment and (electric radius) 2 as function of the binding energy 6.
(r~) 2 = 0.294 + 0.003 = 0.297.
8[MeV] KV(m) - KV(m- 8) [r~(m)] 2 - [r~(m- 5)] 2 [fm21
272
(12)
This change is still smaller than the experimental error. The binding effects on ( r [ ) 2 are included in table 1. The signs of 6K and 5rE2 are easily understood: The form factors have a triangle singularity at
5.9
10
20
0.0136 0.010
0.023 0.017
0.045 0.032" ttr = 4m 2 - (m 2 + 2m6)2/m 2
(13)
Volume 92B, number 3,4
PHYSICS LETTERS
on an unphysical sheet. For ~ = 0, its distance from the physical threshold tth = 4m 2 (which is on the physical sheet) is only m4/m2. As ~ increases, the singularity recedes from the threshold, and consequently the nucleon becomes more pointlike. In particular, its anomalous magnetic moment must decrease. This is opposite to the (smaller) effect found by Nogami [3]. We wish to thank Dr. S. Coon for bringing the work ofref. [3] to our attention.
19 May 1980
References [1] S.D. Drell and J.D. Walecka, Phys. Rev. 120 (1960) 1069; M. Chemto and M. Rho, Nucl. Phys. A163 (1971) 1. [2] R.V. Gerstenberger and Y. Nogami, Phys. Rev. Lett. 29 (1972) 233. [3] Y. Nogami, Revista Brasileira de Fisica 7 (1977) 19. [41 R.A. Brandenburg, S.A. Coon and P.U. Sauer, Nucl. Phys. A294 (1978) 305. [5] J.O. Eeg and H. Pilkuhn, Z. Phys. A287 (1978) 407. [6] H. Pilkuhn, Relativistic particle physics (Springer 1979). [7] G. HShler and E. Pietarinen, Phys. Lett. 53B (1975) 471; Nucl. Phys. B95 (1975) 210. [8] G. H~Shleret al., Nucl. Phys. Bl14 (1976) 505.
273