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Transverse photon effect in mirror nuclei
ANNALS
OF PHYSICS
101,
433-450 (1976)
Transverse
Photon
Effect in Mirror
Nuclei*
SATISH D. JOGLEKAR Institute
for AdLlanced Study, Princeton, New Jersey 08540 Received April 21, 1976
We discuss the effect of the interactions of transverse photons with nucleons in a nucleus on the energy difference between mirror pairs of nuclei. The inclusion of this interaction energy reduces the coulomb energy anomaly by about 10% in the mirror pairs of nuclei considered. (A = 13 - 33).
I.
INTRODUCTION
Systematic studies [l] of differences in the energy levels of mirror nuclei have revealed that the experimental level differences cannot be completely explained. The main source of energy level difference is, of course, the coulomb energy of the last nucleon. Estimates of various corrections to coulomb energy and other possible sources of energy differences have been made with the conclusion that the theoretical values of the level differences are systematically below the experimental results by 5510%. Experimental results on the level differences in isospin analog states in heavier nuclei also lead to the similar conclusion that the r.m.s. radius of the excess neutron distributions are systematically below what the theoretical estimates using reliable potential models require. If interpreted in terms of the n-n and the p-p potentials, both of these results imply that the n-n potential is more attractive than the p-p potential even after coulomb effects are taken out. It is also suggested that in order to explain the results for heavy nuclei, the difference in the potentials must have a long range. The purpose of the calculations in this paper is to see if a part of the coulomb energy anomaly can be explained by the interaction of nuclei with the photon (i.e., with transverse electromagnetic degrees of freedom). In Sections II, III, and IV, we consider the self energy problem for a particle in a potential well, for a two particle bound state and for the case of a nucleus successively. The nuclei we shall be dealing with (which are tabulated in Table I) all have a simple structure of a closed core and a single nucleon in the next shell. Of * Research sponsored by the Energy Research and Development No. E(l l-1)-2220. Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
433
Administration,
Grant
434
S. D. JOGLEKAR
course the net electromagnetic self energy is not simply the total interaction energy due to interactions with photons but there are “renormalization counterterms” to take into account the fact that we are already taking the experimental mass for the nucleon (to compute its kinetic energy, for example) which include the electroelectromagnetic contribution. The problem in Section II is similar to Lamb shift and the net self energy for a nucleon in a potential well similar to that of a nucleus is small, of the order of 1 keV. The main contribution to the self energy comes from the exchange of a photon between them in the case of two charged particles in a bound state. We estimate this contribution for several pairs of mirror nuclei in Section III, using harmonic oscillator wavefunctions of an appropriate potential well and in Section IV, we evaluate the corrections due to the extended size of the core. The conclusion is that the electromagnetic self energy contribution is roughly independent of Z and it reduces the coulomb energy anomaly by about 10 % in the case of light nuclei (A = 13-33). This correction is analagous to the correction to the hydrogenic Lamb shift [2] due to the Breit potential. It is the analog of the collective Lamb shift [3]l in atomic phenomena.
II. CHARGED PARTICLE IN A POTENTIAL
WELL
In this section we shall consider the problem of electromagnetic self energy of a point particle in a bound state of a potential well V(r). The analysis of this problem will then be used in the next section when we consider the electromagnetic selfenergy of a bound system of two charged point particles. We shall quantize the electromagnetic field in the coulomb gauge in the “restframe” of the potential well. The interaction Hamiltonian in the nonrelativistic approximation for the interaction of the charged spin-4 particle of charge e and mass m with photons is (in units fi = c = 1) H’ = 1 d3r (j(r) * A(r) + p(r) - B(r)}
(2.1)
where, j(r) = -(WW~+(r> 44 = Wm)
V+(r) - V++(r) 1clW>
#+@I 4(r)
(2.2)
(2.3)
IcrW = y(r) x and is a two component wavefunction of the particle, (x is a spinor). 1 The author wishes to thank Dr. J. Manassah for pointing out to him the identity of the physical process involved in this phenomenon to the process we are considering.
TRANSVERSE
PHOTON
435
EFFECT
A(r) is the transverse electromagnetic field (V * A = 0) and B = V magnetic field. The quantized electromagnetic field has an expansion {q)(k)
eik’rak,A+ h.c.)
k=lkl
x
A is the
(2.4)
with creation and destruction operators which satisfy [ak,A, aL#,nn]= S3(k - k’) S,,,,,
and polarization
vectors ~&k); w(k)
h = 1, 2 which satisfy
* k = 0,
45 - %‘) = SAA,; (2.5)
1 EC’;&) = Sgj - (kik,/k2). A
The quantized SchrGdinger field $(r) has an expansion Icr(r) = C bjs9j(r) xs j.s
(2.6)
with {bj, , bir,,} = SjjfS,,*
and xs are two orthogonal wavefunctions satisfying
normalized
(-(V2/2m)
spinors and yj(r) are the Schrijdinger
+ V(r) - Ej) vj(r) = 0 (2.7)
s d3r pj*@> p&) = h (j runs over the bound and the possible continuum states.) We shall compute the electromagnetic self energy (i.e., the interaction energy of the particle due to its interaction with the transverse photon) in the ground state. It is of second order in e and is given by Es=-1
IU I H’ I iA2 f E, - Ei -I- k
(2.8)
provided (i]i>
= 1,
7 IfXfl
= 1
where 1i) and 1f} refer to the ground state and an intermediate
(2.9)
state of the
436
S. D.
JOGLEKAR
particle and one photon. Ei and Ef refer to the ground state and the intermediate state energy levels of the particle, k is the photon energy in the intermediate state
If>* Defining, H = H,, + V(r) = -(V2/2m) + V(r) which acting on the states of the particle, we can express
where, &A’(k) is conveniently expressed in the momentum (2k(27~)~)l/~ SA’(k)
is the Hamiltonian
space as
= (e/m) P . en(k) e-k’vP + (ie/2m) Q . Ed x ke-k’vP
(2.11)
where P is the momentum operator. If the particle had not been in the potential well but free, it also would have interacted with photons and gained some electromagnetic self-energy. This selfenergy goes into (a part of) the change in the kinetic energy of the particle that must accompany, for Gallelian covariance, the change in its rest mass on account of the electrostatic self-energy. Since we are already using the physical mass of the particle (i.e., have renormalized the mass) we must subtract this contribution from the electromagnetic self-energy Es computed above. Clearly, the quantity to be subtracted is the electromagnetic self-energy of the particle, were it free, and had for its wavefunction the wavepacket &(r) which can be expressed in terms of the eigenfunctions for the free particle Hamiltonian (plane waves) y&(r) = (1/(25~)~/7 j d3p a(p) eip”Xs .
Then, if C (p) denotes the electromagnetic self-energy of a free particle of momentum p, we shall be subtracting Circe , the weighted average of C (p) with the momentum distribution function / a(p) viz, Cfree = j” d3p C (P) I AIL. The treatment of such a problem is well known (at least the (j * A)2 term in the dipole approximation), from Bethe’s work [4] on the Lamb shift. Further, as it will turn out, this is not the main source of the self-energy difference in mirror nuclei. For these reasons, we shall leave the technical details to Appendix A and simply quote the result. As shown in the Appendix A, the main contribution to the self-energy difference comes from the (j . A)2 term in Eq. (2.10). For a potential, such as say the simple harmonic well which is V(r) = (40r2/R2) MeV
TRANSVERSE PHOTON EFFECT
431
where R is the nuclear radius, this difference is about one keV. Since it will turn out to be much smaller than the final result for the self-energy difference in a pair of mirror nuclei, details about the potential and wavefunctions are irrelevant. In the next section we will only need the facts that the (j . A)2 term dominates and the Lamb-shift-like self-energy contribution is small.
III.
Two CHARGED PARTICLES IN A BOUND STATE
In this section we shall consider the problem of electromagnetic self-energy of a system of a pair of nonrelativistic particles with charges e, and e2 , masses m, and m2. We shall as sume that the particle 1 has spin 4 and the second particle is spinless. (Since this is the analog of the case we shall deal with in nuclei with closed cores.) Let them be bound by a potential V(r). As usual, we define r1c2)= coordinate of particle l(2). r = rl - r, ; M = nz, f
R = (mlrl + m,r,)/M
m3
(3.1)
p = m,m,lM.
The wavefunction of the bound state has the structure $j(r) = yj(r)C(R) xs’ where qi(r) are solutions of the Schrodinger equation (-P2/2p)
+ V(r) - Ej) &r> = 0.
(3.2)
C(R) is a free particle wave packet (with mass = M) for the C.M. coordinate field is quantized in the coulomb gauge in the C.M. frame of the system. We are going to compute the contribution to the rest mass of the system due to its interaction with photons. In order to compute it, it is essential to choose C(R) = eioR = 1 as the C.M. wavefunction. [One cannot choose any wavepacket C(R) = J eiPeRc(P) d3P with (P) = 0, because the calculation of the electromagnetic self-energy of such a system will yield the sum of corrections to the rest mass and the kinetic energy (since (P2)/2M # O).] As in the previous section the correction to the rest mass consists of the electromagnetic self-energy of the system minus the appropriate “renormalization counter terms” to take into account the fact that a part of this self-energy is already included in the kinetic energy of the particle. The photon interactions of the system consist (in the lowest order in e”) of the Feynman diagrams shown in Eig. 1. Figure l(a) and l(b) stand for processes in which a photon is emitted and absorbed by the same particle (1 or 2). These will be grouped with the corresponding “renormalization counter-terms” and using the conclusions of the last section we shall show in Appendix B that in each case this difference is small. Thus remains the diagram of Fig. l(c) which has no counterterm. R and xs is the spinor of the first particle. The electromagnetic
438
S. D. JOGLEKAR
4 1
2
P
I
(la)
2 (lb)
(id FIG. 1. Feynman diagrams for the three kinds of processes involved idthe two particle system.
self-energy of a
This will be the main contributor and therefore we shall deal with it in text in detail. From the experience of the last section and this section, we shall anticipate and prove in Appendix B that the contribution for the process of Fig. l(c) is approximately equal to the corresponding contribution were the particles free and had the same wavepacket for their wavefunction (which is a superposition of free particle eigenfunctions). The latter quantity is much easier to compute. The interaction amplitude for the (j * A) term is
- e,, * VJ,&+(~, R) I/J&, R)] e-ik’rl + (1 -+ 2).
(3.3)
The contribution from the process (Fig. l(c)) arises from the interference terms in ] A I2 while computing the self-energy. (Here V, is the gradient with respect to rI .) We note from Eq. (3.1) that r2 = R - (ml/M) r, rl = R + (m2/M) r, V, = (m2/M) V, - V; V, = (ml/M) VR + V, V = gradient w.r.t. r d%, d3r2 = d3r d3R.
(3.4)
TRANSVERSE
PHOTON
439
EFFECT
We write & = &r) x8, $j(r, R) = I&) eiP.RxS and neglect terms proportional to P . E anticipating P = -k (k . E = 0). We can express A as Afi =
-z(~+$(~P
+ k) j Qri(r) . Ed 1%
ei(m?lM)k.r
e2 e-iWM)kj
d3r
I?12
1
with
Qdr) = v*(r) Vdr> - VTf*(r) Vqdr). From the interference contribution contribution,
(3.5)
proof given in Appendix B it is clear that the contribution from the term to the self-energy is approximately equal to the corresponding were the particles free as explained before. To compute this “free” we take the intermediate states &(r, R) = (]/(2~)~)
eiql.rl+iqz’rzXs
(3.6)
and replace Cr + jisId3q2 . We further define +i(r) = q$(rl - rJ = (1/(27~)~) 1 eipl”l+“pz”za(pl , pz> d3p, d”p,
(3.7)
where a(p, , p2) = (1/(21~)3) f e-“pl”l-“pz%$i(r)
d3rl d3r,
= a3(p, + p2) 1 e”p1”q5i(r) d3r = 43(P1 + P2x27T)3'2 QP,)
(3.8)
where we have made use of Eq. (3.4) in performing the integrations. Then, noting that the initial state is normalized to (2.rr)3S3(0),we can write down the contribution to the self energy from the process of Fig. l(c):
x s d3i’1PI * w’(P,) 6’(q, + PI) a’(q, - ~1 + k) x I d3p,’ PI’ - +*(P,)
S3(q, - ~1’) a’(q, + ~1’ - k) + C.C.
440
S. D.
JOGLEKAR
In general there will be an additional contribution to the self energy coming from the Q . B interaction for the particle 1 and (j . A) interaction for the particle 2. (Note that the particle 2 is assumed to be spinless.) However, when averaged over the initial spin of particle 1, (since the two independent spin states are degenerate) this contribution vanishes. Next, we shall make some estimates for the self-energy of Eq. (3.9). For a(p) we shall choose the normalized simple harmonic oscillator wavefunction. We shall only take those wavefunctions which correspond to the states of the last particle in one of the tabulated (Table I) mirror nuclei and adjust the parameter in the wavefunction such that the Fourier transform of U(P) is the wavefunction for the last nucleon in the appropriate potential well corresponding to that nucleus. TABLE
I
Transverse photon contribution (KeV)
Anomaly [5] (KeV)
FlL017
20 21
210 210
p=Lg=
16
200
CPP
14
240
Mirror
pair
W”p
For a radial quantum number n, angular momentum number m,
I and azimuthal
quantum
where 8 and C$refer to the momentum vector p, Lf$$‘&) are Laguerre polynomials (as defined by Morse and Feshbach), Anln is a normalization factor. We shall compute Ep’ for fixed n, I and averaged over m. We note that
=
A;z,p'
/ p _
k 12 e(-"2+(~--k)2)/202[~Z+(1/2)
~1,2~(n-1)(P2b2)
G:l%?-1~[(P- ~)“/~“I1
441
TRANSVERSE PHOTON EFFECT
where ((7, v’) refer to (p - k). This can be summed using addition spherical harmonics yielding, -!L-
2
+
C 1 m=-z
~nzn,(d
1 x t 4n(21+
4%~
-
formula for
k)
112
1) 1
YZYQJ)
(3.12)
where w is defined by cos w = (p . (p - k))/(p / p - k I). The results are tabulated in Table I. Clearly, the result has no explicit dependence on 2. The variation in Table I is due to the variation of the integral in Eq. (3.9) for different wavefunctions.
IV. SELF-ENERGY DIFFERENCE FOR A PAIR OF MIRROR NUCLEI In this section we shall consider the realistic case of self-energies of a pair of mirror nuclei. In particular, we shall concentrate on the mirror pairs of nuclei which are tabulated in Table I, which have a particular structure of a closed shell and one extra nucleon in the next open shell. It is clear that the electromagnetic self-energy of the cores are equal for both nuclei. Electromagnetic self-energy of the last nucleon while it is in the field of the core was shown in a simplified potential model to be substantially equal to the electromagnetic self-energy when it is free, which is already counted in the total energy of the nucleon. In a realistic case, in computing the self-energy of the last nucleons, the intermediate states will be much more complicated than those in a simple picture of a particle in the potential of a bound core. However, it is evident that presence of all such intermediate states will only bring the self-energy closer to its free value. The main contribution to the difference of self-energies will come from the difference in the electromagnetic interaction of the last nucleon with the core. Since contribution to this is limited to photon momenta of the order of the internal momentum of the last nucleon (-200 MeV), for this calculation at least the model of a bound closed core interacting with the last nucleon is adequate. What we considered in Section III was an oversimplified version of this model in which one considered a spin -4 point particle bound with a spinless point particle. We found that the contribution to E, came from the (j * A)2 term only. That conclusion remains valid here also [even in the presence of spin orbit coupling when a state is specified by ] jmi) rather than ] lmsm,)].
442
S. D.
JOGLEKAR
We had in Section III replaced the nuclear core by a point particle of total charge Ee and mass A’m. Now we shall consider how to improve on the result of Section III by taking into account the finite size of the core. (Taking a point core may seem like a bad approximation to start with, however from Eq. (3.13) it is clear that what matters most is the momentum distribution of the state which is, to a good degree, a function of the size of the system and its angular momentum.) Let r denote now the vector from the last nucleon (still treated as a point particle) to the center of mass of the core. Let pi denote the position vectors of the nucleons measured from the C. M. of the core. Then the amplitude we are interested in will be proportional to C j fl d3pj e,vj”(p) 11 j
. eAeik’(‘+Pn)
(4.1)
where mP>
= (-~P~>(~r*vn$i
- vn$b*?b
v, = WP,)
(4.2)
and #(p) is the core wavefunction. It is clear that what we have done in letting the core be a point particle is to replace the phase factor of each nucleon eik.(r+Pd)-+ e*k’r. Further, 1 j n dpj e,v:;‘(p) * E = -(Ze/A) 11 j
V’
(4.3)
where V’ = (-i/2m)
1 (&*(r’)
V’& - V/4,‘&)
d%’
(4.4)
r’ being the coordinate of the last nucleon and #’ its wavefunctions. This follows from the conservation of momentum and a further assumption that (4.5) To improve on our approximation, -(Ze/A)
e”“V’
we write the expression in (4.1) as
* ~~ j fl(dfJ i
C eik’P~en~*(f) #Cd 12
+ C j fl (&+I 1 e,J(vk) + WI4 #*(P>IcI(dI * wik”+-. ?% i R The second term can be shown to vanish. Let k be in the 2 direction. And eh is in the 5 direction. Consider the operation x, z + x, z and y -+ --y. Under this operation the states of the core are unchanged while E * V changes sign. Since
TRANSVERSE
PHOTON
443
EFFECT
the above operation is a symmetry of the dynamics (consisting of parity and rotation) this term must vanish. The first term can be evaluated easily if we assume a uniform proton distribution in the core to yield, -@e/A) V . qeik’r{jo(kR) +j,(kR)) where j,, and j, are spherical Bessel functions and R is the core radius. Since + j,(kR)} do not depend on anything except the photon momentum, the effect of taking into account the finite size of the core amounts to introducing a form factor {j&R) + j&R)} in the final k integration in the expression of Ep’. This, in general, tends to suppress contribution from photon momenta k 2 l/R; and hence, the final result. We have computed the effective suppression factors for the states of our interest and they are tabulated below. {j&R)
state suppression factor
1P
lf
Id
2s
0.8 1
0.86
0.96
0.72
We have also done the calculation using the experimentally observed [5] form factors where it is possible to evaluate the result analytically and found no substantial change (< 5 %) in the suppression factor. We have also considered the contribution from similar process to He3 - H3 energy difference. The method of calculation is basically different because this is a three-body problem and there are photon interactions with magnetic moments of nucleons which do not vanish. The main contribution comes from a photon exchange between two protons with opposite spins in He3 versus two neutrons in H3. The nonvanishing contributions are (j . A)2; (I* * B)2 direct and exchange contributions. We have evaluated these contributions in a simple harmonic oscillator model. They are small basically because the kinetic energy of nucleons within these nuclei are much smaller (6-7 MeV) as compared with the kinetic energy of the last particle in the mirror nuclei considered (25-30 MeV). It is, however, known [6] that the kinetic energies obtained in more realistic model potentials are larger by a factor of 2 to 3 than the simple harmonic model. If we rescale our results using a dimensional arguments (they should be resealed by 23/2 or 33/2, respectively), the net contribution to He3 -H3 energy difference will be at most -20 keV; (i.e., will increase anomaly by at most 20 keV). APPENDIX
A
In this Appendix, we shall work out the mathematical details needed to reach the conclusions in Section II for the case of a particle in a potential well.
444
S. D. JOGLEKAR
We may expand Eq. (2.10) using the expression for HA’ of Eq. (2.11). There are three kinds of terms that one encounters. We shall call these (j * @, (p * B)” and (j * A)& * B) terms in obvious notations. The (j * A)2 contribution to E, is - 4 I-$&-
c,*“(k) (i / P,,eik’VP H _ L. + k e-k*vpP,,, 1i). t
(A-1)
Here, we need to evaluate an object D = e+k*Vp
-k*Vp E ek.Vp _1 e - Wfp.
1
64.2)
x
H-EEi+ke
D can be evaluated noting
[
1 ek-VpY---X
x]-i 1=- +kk*b, = -- J [e k*Vp,Ho] $, ; =--
i
[ Ho=%]
P2
[up+k - up] ekaVPf
(A-3)
where wp+k = (P + k)2/2m, here an operator. This gives 1
1
D = 7 - x [‘++k - UPI D.
(A.41
This can be further simplified to yield 1 D
=
x
+
(oP+k
(A.3 -
UP)
’
[We note that we have thus eliminated the factor e-k.vp N eik’rfrom our expression, thus taking into account all multipoles via Eq. (A.5)]. To evaluate the expression (A-1) we need 1 p~Dp~ = pn x + wp+k - wp p7lb 1 =
x+
=
x+
@P+k
-
wP
-
wP
pnpm + [P”
1p7n
x + ai+, - cop
1 -
wP+k
1
1 x+
[pn, wP+k
-
UP
x
+
wP+k
-
wPl
x
+
wp+k
_
op
pm
*
TRANSVERSE
The commutator
PHOTON
EFFECT
445
in the second term is equal to [P, , V(r)] = -iV,V(r)
where r in momentum
space is understood as +iV,
pnDp7n = Nothing
1 x + wp+k
64.6)
. And thus
P,P,,, + iDV,VDP,,, -
.
UP
that X [ i) = (H - Ei + k) / i} = k [ i),
but keeping in mind that w~+~ and op are operators, we can after some calculation write 1 w3) 6 1D = ci 1 y _ D(k . VVlrn) where Y = k + CO~+~- wp . We note that k of the order of the average momentum (magnitude) only will give a substantial contribution. The term D(k * Vv/m) can be neglected since D k - vv < m-m-
-
-
k -
y
980 MeV
i.e., D(k * VV/m) < Y. The suppression factor being N (depth of nuclear well)/ (nucleon mass) N l/20. Thus (i 1DP,P,
N (i I (l/Y) P,P,.
(A.9
We shall show that the contribution coming from (i I DP,P, ) i} as given by (A.9) is exactly cancelled by the self-energy counter term. As explained in the text of Section II, this counter term is the self-energy of a free particle with wavefunction &(x), i.e., same as the ground state wavefunction of the particle in the well. The (j . A)2 contribution to this is
(A.10)
446
S. D. JOGLEKAR
which precisely cancels the contribution from first term in Eq. (A.7). The second term in the right-hand side of Eq. (A.7) gives a contribution to Es given by ES = -i : (2cjtk Approximate
* -$
. c:meAn (i I DV,VDP,,
[ i).
(A.ll)
estimate of Es in Eq. (A. 11) can be made letting ; %“Ean = La - (kmwk2) + (213) a,, D + l/k.
Then -iS,,(i
1V,VP,
1i) = (i / V2V 1i) = 240 MeVIRN2
where we have chosen V = 40r2/RN2 MeV
RN = nuclear radius.
Thus E,N--. e2 1 7r 3m2
240R~2ev)
Taking mRN = 20, E, w k (In 2)
keV
where Kmax and Kmin are momentum cutoffs, the former is, for a point particle, a relativistic cutoff, while in the case of a nucleon it will be provided by its form factor and the latter due to the fact that when k - 0, D - l/(k + Km& Kmin being closely related to the energy gap between the ground and the next excited states. With the help of Eq. (AS) (and the discussion following it) it is straightforward matter to show that the (p * B)2 and the (j * A)& * B) contributions to E, are cancelled by the corresponding self-energy counterterms (to the lowest order in V/m). There are no analogs of the second term in Eq. (A.7) arising in these cases.
APPENDIX
B
In this Appendix we shall prove the results assumed in Section III for the case of a system of two particles in a bound state. They are (i) Contribution to the self-energy from processes l(a) and l(b) are approximately cancelled by the corresponding self-energy counter terms.
TRANSVERSE
PHOTON
EFFECT
447
(ii) Contribution Er’ from the process l(c) is approximately equal to the corresponding quantity were the particles free and had the same wavepacket as their wavefunction. The methods used are a straightforward Appendix A.
gneralization
of the methods used in
(i) Contribution to the self-energy from process l(a) is obtained from the square of the 1st term in Afi of Eq. (3.5). It is proportional to (this is only the (j * A)2 term)
1 I P(P + k)12 ericr) e-ik*hlM) Ef - Et + k f (k2/2M) IS a369 - c 1$ a
&
2
T EI _ E, + ; + (k2,2M) 1 1 Q,dr> e-ik’r(mz’M) d3r 1’ z
N ; f d3k (i / Z:*(k)
1 H” _ Ei + k + (k2/2M) s’(k)
(B.1)
’ i)’
where
&y(k) = 2%
l
k.Vp.(m,/M)
ml ((2~)~ 2k)1/2 ”
’ W-
and
Here P’ is the canonically conjugate operator to r (i.e., -iv). But the above expression in Eq. (B.l) has identical form as the expression (A. 1) for self-energy of a particle in a potential will except -& + v(r) + k - Ei + $
+ v@) + k --E,Sm.
k2
It is clear that methods outlined in the Appendix will go through with the same qualitative conclusions. The same will be true for (j * A)@ * B) and (p * B)2 terms. (ii) The contribution Ep’ from the process (lc) is proportional to 1 H” _ ,ITi + k +
(k2/2M)
s%)
1i, + “‘*,
03.3)
448
S. D.
JOGLEKAR
where &‘i defined in Eq. (B.2) and -k*Vphnl/M)
.
(B-4)
The integrand in (B.3) is not quite of the form we dealt with in the Appendix A because of the different exponentials in SF’ and Z”“. If we consider D’ = ek.VphP/M)
1 H” - Ei + k + (k2/2M)
e-k.Vp(m,lM)
.
Then, as in Appendix A, 1
ek.Vp((m,-m,)lM)
D’ = H” - Ei + k + (k2/2M) +
H”
_
Ei
+
1 k
+
@2,2&f)
bUia’+kh,/M))
-
w;‘l
and thus D’ zzz
1 ek.Vp(ha-?nJ/M) x” + W;p’+k(m2/M)) - w;’
with x” = H” - Ei + k + (k2/2M). With this form for D’ all of the arguments of Appendix A will go through leading to the result that expression (B.3) is approximately equal to the corresponding free expression.
APPENDIX
C
In this Appendix we shall give some details of the calculation Nothing that
of Er’.
exp{--p2 - (p - k)2}/2u2 = exp(-k2/402) exp(-c2/02) where C = p - $k, we make this change of variables in the right-hand side of Eq. (3.12). One finds that it is an even function of < * k, when expressed in terms of c and k. The energy denominator is I ‘+-k
-
o+, + k = (k2/2M) - (b * ?YM) + k ’
TRANSVERSE PHOTON EFFECT
Thus in the < integration contribute which is
only the k * < even part of this energy denominator
1 2 (WM)
1 - ((k * Q/M)
449 will
+ (k . ?i -+ --k * C)
+ k
(k + W2M)) = (k + (k2/2M))2
- ((k * t;)/M)2
but loo 2 (k * ?J2< -k212 N -k202 N 938 k’ << k2. M2
M2
M2
(
)
Thus we may drop the (k . <)2/M2 term in the denominator simplifying integration. One may take into account the effect of the form factor [j,(kR) + j,(kR)], performing the k-integration. We note @kRt
=
-f.
(9”
J’nWW
LdW
+ 1)
Cl) the < while
(C.2)
SOthat using tFIPn(t) P,(t) dt = 6,,(2(2n + l)), we learn that j,,(kR)
+ j,(kR)
= 4 1’ dt [P,(t) - P,(t)] eikRt -1
=gs
’ dt (1 -
t2) eikRt.
-1
(C.3)
Thus dk e-k”/40”f(k2)[jO(kR) =Q
The k-integration the t integration function.
’ dt(1 s-1
+ j,(kR)] t”)
L”
&
e-k2/402+iRktf(k2).
(C.4)
can be done by completing the square in the exponent and then can be done by using the tables for the gaussian probability
ACKNOWLEDGMENT I would like to thank Professor S. Adler for suggesting this problem and Professor S. Treiman for raising this question originally. I would also like to thank Professors S. Adler and J. Manassah for discussions and a reading of this manuscript, and the Institute for Advanced Study, for its hospitality, where the work was done.
450
S. D. JOGLEKAR REFERENCES
1. For a review of the subject and a long list of references to previous works, see J. W. NEGELE, “Coulomb Energy Differences,” in “International Conference on Nuclear Structure and Spectroscopy, Amsterdam,” CTP Publication No. 431, MIT, Sept., 1974. 2. For a review of Lambshift, see S. BRODSKY,in Brandeis University Summer Institute in Theoretical Physics, 1969. 3. See, for example, R. FRIEDBERG,S. R. HARTMAN, AND J. T. MANASSAH, Phys. Rep. 7C (1973), 101. 4. H. A. BETHE, Phys. Rev. 72 (1947), 339. 5. R. HOFSTATDER, “Electron Scattering and Nuclear and Nucleon Structure,” Benjamin, New York. 6. Y. AKAISHI, et al., to appear. 7. These figures are taken from NOLEN AND SCHIFFER,Ann. Rev. Nucl. Sci. 19 (1969), 471.
OF PHYSICS
101,
433-450 (1976)
Transverse
Photon
Effect in Mirror
Nuclei*
SATISH D. JOGLEKAR Institute
for AdLlanced Study, Princeton, New Jersey 08540 Received April 21, 1976
We discuss the effect of the interactions of transverse photons with nucleons in a nucleus on the energy difference between mirror pairs of nuclei. The inclusion of this interaction energy reduces the coulomb energy anomaly by about 10% in the mirror pairs of nuclei considered. (A = 13 - 33).
I.
INTRODUCTION
Systematic studies [l] of differences in the energy levels of mirror nuclei have revealed that the experimental level differences cannot be completely explained. The main source of energy level difference is, of course, the coulomb energy of the last nucleon. Estimates of various corrections to coulomb energy and other possible sources of energy differences have been made with the conclusion that the theoretical values of the level differences are systematically below the experimental results by 5510%. Experimental results on the level differences in isospin analog states in heavier nuclei also lead to the similar conclusion that the r.m.s. radius of the excess neutron distributions are systematically below what the theoretical estimates using reliable potential models require. If interpreted in terms of the n-n and the p-p potentials, both of these results imply that the n-n potential is more attractive than the p-p potential even after coulomb effects are taken out. It is also suggested that in order to explain the results for heavy nuclei, the difference in the potentials must have a long range. The purpose of the calculations in this paper is to see if a part of the coulomb energy anomaly can be explained by the interaction of nuclei with the photon (i.e., with transverse electromagnetic degrees of freedom). In Sections II, III, and IV, we consider the self energy problem for a particle in a potential well, for a two particle bound state and for the case of a nucleus successively. The nuclei we shall be dealing with (which are tabulated in Table I) all have a simple structure of a closed core and a single nucleon in the next shell. Of * Research sponsored by the Energy Research and Development No. E(l l-1)-2220. Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
433
Administration,
Grant
434
S. D. JOGLEKAR
course the net electromagnetic self energy is not simply the total interaction energy due to interactions with photons but there are “renormalization counterterms” to take into account the fact that we are already taking the experimental mass for the nucleon (to compute its kinetic energy, for example) which include the electroelectromagnetic contribution. The problem in Section II is similar to Lamb shift and the net self energy for a nucleon in a potential well similar to that of a nucleus is small, of the order of 1 keV. The main contribution to the self energy comes from the exchange of a photon between them in the case of two charged particles in a bound state. We estimate this contribution for several pairs of mirror nuclei in Section III, using harmonic oscillator wavefunctions of an appropriate potential well and in Section IV, we evaluate the corrections due to the extended size of the core. The conclusion is that the electromagnetic self energy contribution is roughly independent of Z and it reduces the coulomb energy anomaly by about 10 % in the case of light nuclei (A = 13-33). This correction is analagous to the correction to the hydrogenic Lamb shift [2] due to the Breit potential. It is the analog of the collective Lamb shift [3]l in atomic phenomena.
II. CHARGED PARTICLE IN A POTENTIAL
WELL
In this section we shall consider the problem of electromagnetic self energy of a point particle in a bound state of a potential well V(r). The analysis of this problem will then be used in the next section when we consider the electromagnetic selfenergy of a bound system of two charged point particles. We shall quantize the electromagnetic field in the coulomb gauge in the “restframe” of the potential well. The interaction Hamiltonian in the nonrelativistic approximation for the interaction of the charged spin-4 particle of charge e and mass m with photons is (in units fi = c = 1) H’ = 1 d3r (j(r) * A(r) + p(r) - B(r)}
(2.1)
where, j(r) = -(WW~+(r> 44 = Wm)
V+(r) - V++(r) 1clW>
#+@I 4(r)
(2.2)
(2.3)
IcrW = y(r) x and is a two component wavefunction of the particle, (x is a spinor). 1 The author wishes to thank Dr. J. Manassah for pointing out to him the identity of the physical process involved in this phenomenon to the process we are considering.
TRANSVERSE
PHOTON
435
EFFECT
A(r) is the transverse electromagnetic field (V * A = 0) and B = V magnetic field. The quantized electromagnetic field has an expansion {q)(k)
eik’rak,A+ h.c.)
k=lkl
x
A is the
(2.4)
with creation and destruction operators which satisfy [ak,A, aL#,nn]= S3(k - k’) S,,,,,
and polarization
vectors ~&k); w(k)
h = 1, 2 which satisfy
* k = 0,
45 - %‘) = SAA,; (2.5)
1 EC’;&) = Sgj - (kik,/k2). A
The quantized SchrGdinger field $(r) has an expansion Icr(r) = C bjs9j(r) xs j.s
(2.6)
with {bj, , bir,,} = SjjfS,,*
and xs are two orthogonal wavefunctions satisfying
normalized
(-(V2/2m)
spinors and yj(r) are the Schrijdinger
+ V(r) - Ej) vj(r) = 0 (2.7)
s d3r pj*@> p&) = h (j runs over the bound and the possible continuum states.) We shall compute the electromagnetic self energy (i.e., the interaction energy of the particle due to its interaction with the transverse photon) in the ground state. It is of second order in e and is given by Es=-1
IU I H’ I iA2 f E, - Ei -I- k
(2.8)
provided (i]i>
= 1,
7 IfXfl
= 1
where 1i) and 1f} refer to the ground state and an intermediate
(2.9)
state of the
436
S. D.
JOGLEKAR
particle and one photon. Ei and Ef refer to the ground state and the intermediate state energy levels of the particle, k is the photon energy in the intermediate state
If>* Defining, H = H,, + V(r) = -(V2/2m) + V(r) which acting on the states of the particle, we can express
where, &A’(k) is conveniently expressed in the momentum (2k(27~)~)l/~ SA’(k)
is the Hamiltonian
space as
= (e/m) P . en(k) e-k’vP + (ie/2m) Q . Ed x ke-k’vP
(2.11)
where P is the momentum operator. If the particle had not been in the potential well but free, it also would have interacted with photons and gained some electromagnetic self-energy. This selfenergy goes into (a part of) the change in the kinetic energy of the particle that must accompany, for Gallelian covariance, the change in its rest mass on account of the electrostatic self-energy. Since we are already using the physical mass of the particle (i.e., have renormalized the mass) we must subtract this contribution from the electromagnetic self-energy Es computed above. Clearly, the quantity to be subtracted is the electromagnetic self-energy of the particle, were it free, and had for its wavefunction the wavepacket &(r) which can be expressed in terms of the eigenfunctions for the free particle Hamiltonian (plane waves) y&(r) = (1/(25~)~/7 j d3p a(p) eip”Xs .
Then, if C (p) denotes the electromagnetic self-energy of a free particle of momentum p, we shall be subtracting Circe , the weighted average of C (p) with the momentum distribution function / a(p) viz, Cfree = j” d3p C (P) I AIL. The treatment of such a problem is well known (at least the (j * A)2 term in the dipole approximation), from Bethe’s work [4] on the Lamb shift. Further, as it will turn out, this is not the main source of the self-energy difference in mirror nuclei. For these reasons, we shall leave the technical details to Appendix A and simply quote the result. As shown in the Appendix A, the main contribution to the self-energy difference comes from the (j . A)2 term in Eq. (2.10). For a potential, such as say the simple harmonic well which is V(r) = (40r2/R2) MeV
TRANSVERSE PHOTON EFFECT
431
where R is the nuclear radius, this difference is about one keV. Since it will turn out to be much smaller than the final result for the self-energy difference in a pair of mirror nuclei, details about the potential and wavefunctions are irrelevant. In the next section we will only need the facts that the (j . A)2 term dominates and the Lamb-shift-like self-energy contribution is small.
III.
Two CHARGED PARTICLES IN A BOUND STATE
In this section we shall consider the problem of electromagnetic self-energy of a system of a pair of nonrelativistic particles with charges e, and e2 , masses m, and m2. We shall as sume that the particle 1 has spin 4 and the second particle is spinless. (Since this is the analog of the case we shall deal with in nuclei with closed cores.) Let them be bound by a potential V(r). As usual, we define r1c2)= coordinate of particle l(2). r = rl - r, ; M = nz, f
R = (mlrl + m,r,)/M
m3
(3.1)
p = m,m,lM.
The wavefunction of the bound state has the structure $j(r) = yj(r)C(R) xs’ where qi(r) are solutions of the Schrodinger equation (-P2/2p)
+ V(r) - Ej) &r> = 0.
(3.2)
C(R) is a free particle wave packet (with mass = M) for the C.M. coordinate field is quantized in the coulomb gauge in the C.M. frame of the system. We are going to compute the contribution to the rest mass of the system due to its interaction with photons. In order to compute it, it is essential to choose C(R) = eioR = 1 as the C.M. wavefunction. [One cannot choose any wavepacket C(R) = J eiPeRc(P) d3P with (P) = 0, because the calculation of the electromagnetic self-energy of such a system will yield the sum of corrections to the rest mass and the kinetic energy (since (P2)/2M # O).] As in the previous section the correction to the rest mass consists of the electromagnetic self-energy of the system minus the appropriate “renormalization counter terms” to take into account the fact that a part of this self-energy is already included in the kinetic energy of the particle. The photon interactions of the system consist (in the lowest order in e”) of the Feynman diagrams shown in Eig. 1. Figure l(a) and l(b) stand for processes in which a photon is emitted and absorbed by the same particle (1 or 2). These will be grouped with the corresponding “renormalization counter-terms” and using the conclusions of the last section we shall show in Appendix B that in each case this difference is small. Thus remains the diagram of Fig. l(c) which has no counterterm. R and xs is the spinor of the first particle. The electromagnetic
438
S. D. JOGLEKAR
4 1
2
P
I
(la)
2 (lb)
(id FIG. 1. Feynman diagrams for the three kinds of processes involved idthe two particle system.
self-energy of a
This will be the main contributor and therefore we shall deal with it in text in detail. From the experience of the last section and this section, we shall anticipate and prove in Appendix B that the contribution for the process of Fig. l(c) is approximately equal to the corresponding contribution were the particles free and had the same wavepacket for their wavefunction (which is a superposition of free particle eigenfunctions). The latter quantity is much easier to compute. The interaction amplitude for the (j * A) term is
- e,, * VJ,&+(~, R) I/J&, R)] e-ik’rl + (1 -+ 2).
(3.3)
The contribution from the process (Fig. l(c)) arises from the interference terms in ] A I2 while computing the self-energy. (Here V, is the gradient with respect to rI .) We note from Eq. (3.1) that r2 = R - (ml/M) r, rl = R + (m2/M) r, V, = (m2/M) V, - V; V, = (ml/M) VR + V, V = gradient w.r.t. r d%, d3r2 = d3r d3R.
(3.4)
TRANSVERSE
PHOTON
439
EFFECT
We write & = &r) x8, $j(r, R) = I&) eiP.RxS and neglect terms proportional to P . E anticipating P = -k (k . E = 0). We can express A as Afi =
-z(~+$(~P
+ k) j Qri(r) . Ed 1%
ei(m?lM)k.r
e2 e-iWM)kj
d3r
I?12
1
with
Qdr) = v*(r) Vdr> - VTf*(r) Vqdr). From the interference contribution contribution,
(3.5)
proof given in Appendix B it is clear that the contribution from the term to the self-energy is approximately equal to the corresponding were the particles free as explained before. To compute this “free” we take the intermediate states &(r, R) = (]/(2~)~)
eiql.rl+iqz’rzXs
(3.6)
and replace Cr + jisId3q2 . We further define +i(r) = q$(rl - rJ = (1/(27~)~) 1 eipl”l+“pz”za(pl , pz> d3p, d”p,
(3.7)
where a(p, , p2) = (1/(21~)3) f e-“pl”l-“pz%$i(r)
d3rl d3r,
= a3(p, + p2) 1 e”p1”q5i(r) d3r = 43(P1 + P2x27T)3'2 QP,)
(3.8)
where we have made use of Eq. (3.4) in performing the integrations. Then, noting that the initial state is normalized to (2.rr)3S3(0),we can write down the contribution to the self energy from the process of Fig. l(c):
x s d3i’1PI * w’(P,) 6’(q, + PI) a’(q, - ~1 + k) x I d3p,’ PI’ - +*(P,)
S3(q, - ~1’) a’(q, + ~1’ - k) + C.C.
440
S. D.
JOGLEKAR
In general there will be an additional contribution to the self energy coming from the Q . B interaction for the particle 1 and (j . A) interaction for the particle 2. (Note that the particle 2 is assumed to be spinless.) However, when averaged over the initial spin of particle 1, (since the two independent spin states are degenerate) this contribution vanishes. Next, we shall make some estimates for the self-energy of Eq. (3.9). For a(p) we shall choose the normalized simple harmonic oscillator wavefunction. We shall only take those wavefunctions which correspond to the states of the last particle in one of the tabulated (Table I) mirror nuclei and adjust the parameter in the wavefunction such that the Fourier transform of U(P) is the wavefunction for the last nucleon in the appropriate potential well corresponding to that nucleus. TABLE
I
Transverse photon contribution (KeV)
Anomaly [5] (KeV)
FlL017
20 21
210 210
p=Lg=
16
200
CPP
14
240
Mirror
pair
W”p
For a radial quantum number n, angular momentum number m,
I and azimuthal
quantum
where 8 and C$refer to the momentum vector p, Lf$$‘&) are Laguerre polynomials (as defined by Morse and Feshbach), Anln is a normalization factor. We shall compute Ep’ for fixed n, I and averaged over m. We note that
=
A;z,p'
/ p _
k 12 e(-"2+(~--k)2)/202[~Z+(1/2)
~1,2~(n-1)(P2b2)
G:l%?-1~[(P- ~)“/~“I1
441
TRANSVERSE PHOTON EFFECT
where ((7, v’) refer to (p - k). This can be summed using addition spherical harmonics yielding, -!L-
2
+
C 1 m=-z
~nzn,(d
1 x t 4n(21+
4%~
-
formula for
k)
112
1) 1
YZYQJ)
(3.12)
where w is defined by cos w = (p . (p - k))/(p / p - k I). The results are tabulated in Table I. Clearly, the result has no explicit dependence on 2. The variation in Table I is due to the variation of the integral in Eq. (3.9) for different wavefunctions.
IV. SELF-ENERGY DIFFERENCE FOR A PAIR OF MIRROR NUCLEI In this section we shall consider the realistic case of self-energies of a pair of mirror nuclei. In particular, we shall concentrate on the mirror pairs of nuclei which are tabulated in Table I, which have a particular structure of a closed shell and one extra nucleon in the next open shell. It is clear that the electromagnetic self-energy of the cores are equal for both nuclei. Electromagnetic self-energy of the last nucleon while it is in the field of the core was shown in a simplified potential model to be substantially equal to the electromagnetic self-energy when it is free, which is already counted in the total energy of the nucleon. In a realistic case, in computing the self-energy of the last nucleons, the intermediate states will be much more complicated than those in a simple picture of a particle in the potential of a bound core. However, it is evident that presence of all such intermediate states will only bring the self-energy closer to its free value. The main contribution to the difference of self-energies will come from the difference in the electromagnetic interaction of the last nucleon with the core. Since contribution to this is limited to photon momenta of the order of the internal momentum of the last nucleon (-200 MeV), for this calculation at least the model of a bound closed core interacting with the last nucleon is adequate. What we considered in Section III was an oversimplified version of this model in which one considered a spin -4 point particle bound with a spinless point particle. We found that the contribution to E, came from the (j * A)2 term only. That conclusion remains valid here also [even in the presence of spin orbit coupling when a state is specified by ] jmi) rather than ] lmsm,)].
442
S. D.
JOGLEKAR
We had in Section III replaced the nuclear core by a point particle of total charge Ee and mass A’m. Now we shall consider how to improve on the result of Section III by taking into account the finite size of the core. (Taking a point core may seem like a bad approximation to start with, however from Eq. (3.13) it is clear that what matters most is the momentum distribution of the state which is, to a good degree, a function of the size of the system and its angular momentum.) Let r denote now the vector from the last nucleon (still treated as a point particle) to the center of mass of the core. Let pi denote the position vectors of the nucleons measured from the C. M. of the core. Then the amplitude we are interested in will be proportional to C j fl d3pj e,vj”(p) 11 j
. eAeik’(‘+Pn)
(4.1)
where mP>
= (-~P~>(~r*vn$i
- vn$b*?b
v, = WP,)
(4.2)
and #(p) is the core wavefunction. It is clear that what we have done in letting the core be a point particle is to replace the phase factor of each nucleon eik.(r+Pd)-+ e*k’r. Further, 1 j n dpj e,v:;‘(p) * E = -(Ze/A) 11 j
V’
(4.3)
where V’ = (-i/2m)
1 (&*(r’)
V’& - V/4,‘&)
d%’
(4.4)
r’ being the coordinate of the last nucleon and #’ its wavefunctions. This follows from the conservation of momentum and a further assumption that (4.5) To improve on our approximation, -(Ze/A)
e”“V’
we write the expression in (4.1) as
* ~~ j fl(dfJ i
C eik’P~en~*(f) #Cd 12
+ C j fl (&+I 1 e,J(vk) + WI4 #*(P>IcI(dI * wik”+-. ?% i R The second term can be shown to vanish. Let k be in the 2 direction. And eh is in the 5 direction. Consider the operation x, z + x, z and y -+ --y. Under this operation the states of the core are unchanged while E * V changes sign. Since
TRANSVERSE
PHOTON
443
EFFECT
the above operation is a symmetry of the dynamics (consisting of parity and rotation) this term must vanish. The first term can be evaluated easily if we assume a uniform proton distribution in the core to yield, -@e/A) V . qeik’r{jo(kR) +j,(kR)) where j,, and j, are spherical Bessel functions and R is the core radius. Since + j,(kR)} do not depend on anything except the photon momentum, the effect of taking into account the finite size of the core amounts to introducing a form factor {j&R) + j&R)} in the final k integration in the expression of Ep’. This, in general, tends to suppress contribution from photon momenta k 2 l/R; and hence, the final result. We have computed the effective suppression factors for the states of our interest and they are tabulated below. {j&R)
state suppression factor
1P
lf
Id
2s
0.8 1
0.86
0.96
0.72
We have also done the calculation using the experimentally observed [5] form factors where it is possible to evaluate the result analytically and found no substantial change (< 5 %) in the suppression factor. We have also considered the contribution from similar process to He3 - H3 energy difference. The method of calculation is basically different because this is a three-body problem and there are photon interactions with magnetic moments of nucleons which do not vanish. The main contribution comes from a photon exchange between two protons with opposite spins in He3 versus two neutrons in H3. The nonvanishing contributions are (j . A)2; (I* * B)2 direct and exchange contributions. We have evaluated these contributions in a simple harmonic oscillator model. They are small basically because the kinetic energy of nucleons within these nuclei are much smaller (6-7 MeV) as compared with the kinetic energy of the last particle in the mirror nuclei considered (25-30 MeV). It is, however, known [6] that the kinetic energies obtained in more realistic model potentials are larger by a factor of 2 to 3 than the simple harmonic model. If we rescale our results using a dimensional arguments (they should be resealed by 23/2 or 33/2, respectively), the net contribution to He3 -H3 energy difference will be at most -20 keV; (i.e., will increase anomaly by at most 20 keV). APPENDIX
A
In this Appendix, we shall work out the mathematical details needed to reach the conclusions in Section II for the case of a particle in a potential well.
444
S. D. JOGLEKAR
We may expand Eq. (2.10) using the expression for HA’ of Eq. (2.11). There are three kinds of terms that one encounters. We shall call these (j * @, (p * B)” and (j * A)& * B) terms in obvious notations. The (j * A)2 contribution to E, is - 4 I-$&-
c,*“(k) (i / P,,eik’VP H _ L. + k e-k*vpP,,, 1i). t
(A-1)
Here, we need to evaluate an object D = e+k*Vp
-k*Vp E ek.Vp _1 e - Wfp.
1
64.2)
x
H-EEi+ke
D can be evaluated noting
[
1 ek-VpY---X
x]-i 1=- +kk*b, = -- J [e k*Vp,Ho] $, ; =--
i
[ Ho=%]
P2
[up+k - up] ekaVPf
(A-3)
where wp+k = (P + k)2/2m, here an operator. This gives 1
1
D = 7 - x [‘++k - UPI D.
(A.41
This can be further simplified to yield 1 D
=
x
+
(oP+k
(A.3 -
UP)
’
[We note that we have thus eliminated the factor e-k.vp N eik’rfrom our expression, thus taking into account all multipoles via Eq. (A.5)]. To evaluate the expression (A-1) we need 1 p~Dp~ = pn x + wp+k - wp p7lb 1 =
x+
=
x+
@P+k
-
wP
-
wP
pnpm + [P”
1p7n
x + ai+, - cop
1 -
wP+k
1
1 x+
[pn, wP+k
-
UP
x
+
wP+k
-
wPl
x
+
wp+k
_
op
pm
*
TRANSVERSE
The commutator
PHOTON
EFFECT
445
in the second term is equal to [P, , V(r)] = -iV,V(r)
where r in momentum
space is understood as +iV,
pnDp7n = Nothing
1 x + wp+k
64.6)
. And thus
P,P,,, + iDV,VDP,,, -
.
UP
that X [ i) = (H - Ei + k) / i} = k [ i),
but keeping in mind that w~+~ and op are operators, we can after some calculation write 1 w3) 6 1D = ci 1 y _ D(k . VVlrn) where Y = k + CO~+~- wp . We note that k of the order of the average momentum (magnitude) only will give a substantial contribution. The term D(k * Vv/m) can be neglected since D k - vv <
-
k -
y
980 MeV
i.e., D(k * VV/m) < Y. The suppression factor being N (depth of nuclear well)/ (nucleon mass) N l/20. Thus (i 1DP,P,
N (i I (l/Y) P,P,.
(A.9
We shall show that the contribution coming from (i I DP,P, ) i} as given by (A.9) is exactly cancelled by the self-energy counter term. As explained in the text of Section II, this counter term is the self-energy of a free particle with wavefunction &(x), i.e., same as the ground state wavefunction of the particle in the well. The (j . A)2 contribution to this is
(A.10)
446
S. D. JOGLEKAR
which precisely cancels the contribution from first term in Eq. (A.7). The second term in the right-hand side of Eq. (A.7) gives a contribution to Es given by ES = -i : (2cjtk Approximate
* -$
. c:meAn (i I DV,VDP,,
[ i).
(A.ll)
estimate of Es in Eq. (A. 11) can be made letting ; %“Ean = La - (kmwk2) + (213) a,, D + l/k.
Then -iS,,(i
1V,VP,
1i) = (i / V2V 1i) = 240 MeVIRN2
where we have chosen V = 40r2/RN2 MeV
RN = nuclear radius.
Thus E,N--. e2 1 7r 3m2
240R~2ev)
Taking mRN = 20, E, w k (In 2)
keV
where Kmax and Kmin are momentum cutoffs, the former is, for a point particle, a relativistic cutoff, while in the case of a nucleon it will be provided by its form factor and the latter due to the fact that when k - 0, D - l/(k + Km& Kmin being closely related to the energy gap between the ground and the next excited states. With the help of Eq. (AS) (and the discussion following it) it is straightforward matter to show that the (p * B)2 and the (j * A)& * B) contributions to E, are cancelled by the corresponding self-energy counterterms (to the lowest order in V/m). There are no analogs of the second term in Eq. (A.7) arising in these cases.
APPENDIX
B
In this Appendix we shall prove the results assumed in Section III for the case of a system of two particles in a bound state. They are (i) Contribution to the self-energy from processes l(a) and l(b) are approximately cancelled by the corresponding self-energy counter terms.
TRANSVERSE
PHOTON
EFFECT
447
(ii) Contribution Er’ from the process l(c) is approximately equal to the corresponding quantity were the particles free and had the same wavepacket as their wavefunction. The methods used are a straightforward Appendix A.
gneralization
of the methods used in
(i) Contribution to the self-energy from process l(a) is obtained from the square of the 1st term in Afi of Eq. (3.5). It is proportional to (this is only the (j * A)2 term)
1 I P(P + k)12 ericr) e-ik*hlM) Ef - Et + k f (k2/2M) IS a369 - c 1$ a
&
2
T EI _ E, + ; + (k2,2M) 1 1 Q,dr> e-ik’r(mz’M) d3r 1’ z
N ; f d3k (i / Z:*(k)
1 H” _ Ei + k + (k2/2M) s’(k)
(B.1)
’ i)’
where
&y(k) = 2%
l
k.Vp.(m,/M)
ml ((2~)~ 2k)1/2 ”
’ W-
and
Here P’ is the canonically conjugate operator to r (i.e., -iv). But the above expression in Eq. (B.l) has identical form as the expression (A. 1) for self-energy of a particle in a potential will except -& + v(r) + k - Ei + $
+ v@) + k --E,Sm.
k2
It is clear that methods outlined in the Appendix will go through with the same qualitative conclusions. The same will be true for (j * A)@ * B) and (p * B)2 terms. (ii) The contribution Ep’ from the process (lc) is proportional to 1 H” _ ,ITi + k +
(k2/2M)
s%)
1i, + “‘*,
03.3)
448
S. D.
JOGLEKAR
where &‘i defined in Eq. (B.2) and -k*Vphnl/M)
.
(B-4)
The integrand in (B.3) is not quite of the form we dealt with in the Appendix A because of the different exponentials in SF’ and Z”“. If we consider D’ = ek.VphP/M)
1 H” - Ei + k + (k2/2M)
e-k.Vp(m,lM)
.
Then, as in Appendix A, 1
ek.Vp((m,-m,)lM)
D’ = H” - Ei + k + (k2/2M) +
H”
_
Ei
+
1 k
+
@2,2&f)
bUia’+kh,/M))
-
w;‘l
and thus D’ zzz
1 ek.Vp(ha-?nJ/M) x” + W;p’+k(m2/M)) - w;’
with x” = H” - Ei + k + (k2/2M). With this form for D’ all of the arguments of Appendix A will go through leading to the result that expression (B.3) is approximately equal to the corresponding free expression.
APPENDIX
C
In this Appendix we shall give some details of the calculation Nothing that
of Er’.
exp{--p2 - (p - k)2}/2u2 = exp(-k2/402) exp(-c2/02) where C = p - $k, we make this change of variables in the right-hand side of Eq. (3.12). One finds that it is an even function of < * k, when expressed in terms of c and k. The energy denominator is I ‘+-k
-
o+, + k = (k2/2M) - (b * ?YM) + k ’
TRANSVERSE PHOTON EFFECT
Thus in the < integration contribute which is
only the k * < even part of this energy denominator
1 2 (WM)
1 - ((k * Q/M)
449 will
+ (k . ?i -+ --k * C)
+ k
(k + W2M)) = (k + (k2/2M))2
- ((k * t;)/M)2
but loo 2 (k * ?J2< -k212 N -k202 N 938 k’ << k2. M2
M2
M2
(
)
Thus we may drop the (k . <)2/M2 term in the denominator simplifying integration. One may take into account the effect of the form factor [j,(kR) + j,(kR)], performing the k-integration. We note @kRt
=
-f.
(9”
J’nWW
LdW
+ 1)
Cl) the < while
(C.2)
SOthat using tFIPn(t) P,(t) dt = 6,,(2(2n + l)), we learn that j,,(kR)
+ j,(kR)
= 4 1’ dt [P,(t) - P,(t)] eikRt -1
=gs
’ dt (1 -
t2) eikRt.
-1
(C.3)
Thus dk e-k”/40”f(k2)[jO(kR) =Q
The k-integration the t integration function.
’ dt(1 s-1
+ j,(kR)] t”)
L”
&
e-k2/402+iRktf(k2).
(C.4)
can be done by completing the square in the exponent and then can be done by using the tables for the gaussian probability
ACKNOWLEDGMENT I would like to thank Professor S. Adler for suggesting this problem and Professor S. Treiman for raising this question originally. I would also like to thank Professors S. Adler and J. Manassah for discussions and a reading of this manuscript, and the Institute for Advanced Study, for its hospitality, where the work was done.
450
S. D. JOGLEKAR REFERENCES
1. For a review of the subject and a long list of references to previous works, see J. W. NEGELE, “Coulomb Energy Differences,” in “International Conference on Nuclear Structure and Spectroscopy, Amsterdam,” CTP Publication No. 431, MIT, Sept., 1974. 2. For a review of Lambshift, see S. BRODSKY,in Brandeis University Summer Institute in Theoretical Physics, 1969. 3. See, for example, R. FRIEDBERG,S. R. HARTMAN, AND J. T. MANASSAH, Phys. Rep. 7C (1973), 101. 4. H. A. BETHE, Phys. Rev. 72 (1947), 339. 5. R. HOFSTATDER, “Electron Scattering and Nuclear and Nucleon Structure,” Benjamin, New York. 6. Y. AKAISHI, et al., to appear. 7. These figures are taken from NOLEN AND SCHIFFER,Ann. Rev. Nucl. Sci. 19 (1969), 471.