Charge symmetry breaking and the neutron-proton mass difference

Nuclear Physics A562 (1993) 644-658

North-Holland

NUCLEAR PHYSICS A

Charge symmetry breaking and the neutron-proton mass difference *

Thomas Schafer ‘, Volker Koch, Gerald E. Brown

Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA

Received 15 December 1992

(Revised 26 February 1993)

Abstract We study the behaviour of the neutron-proton mass difference in the nuclear medium. The

main contribution to this quantity from charge symmetry breaking in the meson sector is given by

po mixing. In order to include this effect in the QCD sum rules, we extend the sum rules to

finite density and discuss the importance of additional sources of isospin breaking which appear

at finite density.

1. Introduction

The Nolen-Schiffer anomaly [l] is a long standing problem in nuclear physics related to the discrepancy between experimental and theoretical determinations of the binding energy difference of mirror nuclei. The recent work of Henley and Krein [2], indicating that the anomaly might be resolved by the rapid density dependence of the neutron-proton mass difference AMnp = M,, - Mp, triggered a number of papers [3-51 which studied the problem in the QCD sum-rule approach. The result of the QCD sum-rule calculations can be schematically written in the form

A Mnp = 5.2 MeV - 1.5(m,-m,),

where (44) is the vacuum quark condensate, ((?4)* the condensate at finite density, and md and m, are the current quark masses. From eq. (1) one can see

Correspondence to: Prof. T. Schafer, Department of Physics, Nuclear Theory Group, State University

of New York at Stony Brook, Stony Brook, NY 11784-3800, USA.

l Supported by the U.S. Dept. of Energy Grant No. DE-FG02-88ER40388. ’ Supported in part by the Alexander von Humboldt Foundation.

03759474/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

T. Setifer et al. / Charge ante breaking 645

why the drop in AMnp is so rapid with increasing density. At zero density AMnp = 1.29 MeV results from a small difference of the two larger terms in eq. (1). At finite density, chiral symmetry is restored and the quark condensate decreases. Since the second term in eq. (1) has a different sign than the first one and does not change with density, even a small change in the quark condensate is sufficient to result in a rather large modification of AM,,P.

Basically, the structure of eq. (1) arises because, although the proton is com- posed of two up quarks and one down quark, in the leading log approximation the proton mass arises completely from the don-quark condensate. Of course, the neutron mass is determined in the same way by the up-quark condensate. Since the down quark is heavier than the up quark, one expects the down-quark condensate to be smaller than the up-quark condensate. This is corraborated by studies of charge symmetry breaking in the framework of the QCD sum rules which give [7]

Gu> y=l- (&> - = -0.009 + 0.003. (2)

Similar values for the isospin-breaking parameter y have been obtained from Nambu-Jona-Lasinio type models [3,5,6].

Note that the mechanism discussed in the last two paragraphs is very counter intuitive from the point of view of the constituent quark model. In particular, one would expect a different sign for the current quark-mass contribution in eq. (1). It is therefore not at all clear why QCD sum rules lead to results that are qualita- tively similar to the conclusions of Henley and Krein [2], which are based on a constituent quark model of the nucleon.

Although, over two decades, the Nolen-Schiffer effect seemed to defy theoretical explanation, the work of refs. [2,4,5] gave a straightforward explanation in terms of the rapid density dependence of AM,,p. However, Blunden and Iqbal[8] had obtained N 60% of the Nolen-Schiffer effect through the mechanism of pw mixing (an additional 15% is explained by other charge s~et~-brea~ng forces). In this work we want to address the question of how this mechanism, based on the classical meson-exchange picture of the nucleon-nucleon interaction, is related to the density dependence of AM,,p calculated within the QCD sum rules.

In the next section we will argue that the Blunden-Iqbal mechanism can be understood as a density-dependent neutron-proton mass difference arising from the pw mixing contribution to the nucleon self-energy in the nuclear medium. The quantitative result of Blunden and Iqbal however, is probably too large by a factor w 2. In sect. 3 we will discuss how pw mixing arises in the context of the QCD sum rules. In sect. 4 we shall try to incorporate pw mixing in the description of the nucleon within the QCD sum rules. For this purpose we have to extend the sum rules to finite density and introduce isospin breaking into the continuum part of the spectral function.

646 T. Schiifer et al. / Charge symmetry breaking

2. Charge symmetry breaking

Recently, Christiansen et al. calculated AMnp from po mixing at zero density [lo]. The parameters used in this calculation are l

A = -4520 MeV’,

where A is the po mixing parameter defined by

h=2mp(p01Hlo). (4)

In a relativistic description of the pwN system pw mixing can be incorporated by adding an interaction lagrangian of the type _Y= hpiw*. Note that the sign of this term is fixed by the fact that the physical components of the p- and w-fields are spacelike. The po mixing term in the lagrangian gives a contribution to the nucleon self-energy which violates charge symmetry. Using a form factor

A$-m2, bb2) = A2 _42

V (5)

with A, = 2m,, one finds from the diagram shown in fig. la

AMPO = - 1.1 MeV nP 3 (6)

where we have corrected a sign error contained in the work of Christiansen et al. Like the contribution from the electromagnetic self-energy, which was estimated to be of the order AM,,p em = -0.7 MeV in ref. [12], this term has the wrong sign to explain the observed mass difference. It does, however, give an important contribution to the density dependence of the neutron-proton mass difference.

At finite density, po mixing also contributes to the tadpole term in the vector self-energy of the nucleon. In isospin-symmetric nuclear matter, only the graph shown in fig. lb is present. This term gives a density-dependent contribution to the neutron-proton mass difference,

AM,,p = gpgo A

- m2m2 p, P 0

(7)

l We use the nuclear-physics definition of these coupling constants, which differs by a factor of 4 from that of Christiansen et al. Also, our definition of A differs by a minus sign from the definition

employed by these authors.

T. Schafer et al. / Charge symmetry breaking 647

b)

Fig. 1. Charge s~et~-brewing ~n~ibutions to the nucleon self-energy arising from po mixing. Figs. (a), (b), (c) show the Fock term, the Hartree term and the finite-density correction to the Fock term. Nucleon propagators are shown as solid lines, p-propagators as dashed lines and w-propagators

as wavy lines.

where p is the density of the nuclear medium. There is another important contribution which arises from the Pauli blocking of the intermediate nucleon in the Fock term l (fig. 1~). At small densities, this cuts down the rest& from the Hartree term by a factor f. Terms of higher order in the density are not very important around nuclear matter density. Using the parameters given above, one finds

Ah&, =

which leads to a decrease of the neutron-proton mass difference in medium and therefore helps to resolve the Nolen-Schiffer anomaly. Using the Skyrme II calculation of Sato 191, the theoretical discrepancy is roughIy 650 keV (j?/p,), where p’ is the average density seen by the valence nucleon. This would leave - 70% of the anomaly to be explained by other effects.

Blunden and Iqbal [8] performed a calculation of the binding energy difference of mirror nuclei using a very general isospin-breaking nucleon-nucleon potential. They find that by far the largest contribution (- 80%) comes from the pw mixing potential. Numericahy, their resuh can be parametrized by A&&, = - 328 keV

l This term was recently considered in ref. [ll]. These authors, however, do not include the contribution of the Hartree term.


(p/pa) where we have used average nuclear densities quoted in Cohen et al. [14]. This density dependence is significantly stronger than our result quoted above, essentially due to the use of much larger coupling constants for the p- and o-mesons l :

gp’ - = 0.95, 4r

g = 19.32, 4r

h = -3850 MeV2.

We believe the gi/4a = 0.6 used by Christiansen et al. to correctly describe the (universal) coupling of the elementary p-meson. This coupling constant correctly reproduces the p-meson width in p + 27r decay and is used to extract the mixing parameter from e+e-+ (na> data. The gi/47r = 0.95 used by Blunden and Iqbal, describes in the Bonn potential not only the elementary p, but also effects of the 2~ continuum. This is made clear in the recent paper by Hippchen ef al. [15] in which gi/47r is lowered from 0.92 to 0.74 by separately taking into account effects of the 2~r continuum. We believe that gi/4r should be lowered even more, to the value in eq. (3). This would, of course, involve readjustment of other parameters, because Hippchen et al. fit the observed NN-scattering phase shifts.

The value of gz/4r = 5 used by Christiansen et al. is close to the SU(3) value of 9 times gz/4r. Blunden and Iqbal employ the value gi/4r = 19.32 from Machleidt [16]. This number has changed to 25 in the later nonrelativistic configu- ration space OBEP [17]. It is becoming clear, however, that most of the repulsion in the NN system does not result from w-meson exchange, because nothing like the attraction that would result from changing the sign of the w-exchange potential, as dictated by G-parity, is seen in the NN system [181. Corraboration of this conclusion is seen in comparison of the KN and m interaction [19].

In conclusion we have shown that the result of Blunden and Iqbal [81 can be interpreted as a density-dependent neutron-proton mass difference arising due to the po mixing contribution to the nucleon self-energy in the nuclear medium. The combination gpg,h used by these authors is a factor of 2.5 larger than the one of Christiansen et d., which we favor, and this explain the largeness of the po mixing contribution in the work of the former authors.

3. Rho-omega mixing

If charge symmetry holds, the p” is a member of an isotriplet of p-mesons and the o is an isospin singlet. Due to charge symmetry breaking, the p” and o”

l Also taking into account the different cutoffs used, one would expect the result of Blunden and Iqbal to be a factor of 1.9 larger than our estimate.

i? Schiifer et al. / Charge qwnmetry breaking 649

physical states are mixtures of the bare states. As mentioned in the last section, the

mixing matrix element has been determined to be [13]

A= 2f7~,{p~ I El,,, IO> = -4250 MeV2. (10)

The electromagnetic contribution to this matrix element comes from the annihila- tion process p” -+ y + o and gives h,, = 660 MeV2, which has the opposite sign and is an order of magnitude too small.

Using the ~nstituent quark modeI, one can get a simple estimate of the strong contribution to the mixing matrix element. Writing the unperturbed basis states as

1 PO) = fi( I uu> - I a), (11)

Iw>=~(liiu>+ G-d)), (12)

and taking the charge symmetry-breaking hamiitonian to be

(13)

where M,, and lMd denote the constituent up and down quark masses, one finds

h =2m,(M,-A4,). (14)

For the generally accepted value SM = &f, - Md = -3 MeV, this nicely reproduces the measured value of X.

A more comprehensive analysis of pw mixing can be given using the method of the QCD sum rules. Shifman, Vainshtein and Zakharov [20] have derived the zero-density sum rule

12

----I (

ff exp -s/M2) Im ~(~~)(~) ds = - -

2~~~~u} - 2m,(i%d)

PM2 167r3 M;:

224~ lys( @)*

+81---- % (15)

where ~(~~)(~) denotes the nondiagonal correlation function

(q,q, - q2g,,).@pw)(q2) = i/d4.x e’q’“(O I T( jz( x)j,“(O)) IO). (16)

Here, a, and cy are the strong and electromagnetic fine-structure constants and y as defined in eq. (2) above measures the isospin breaking in the light quark

6.50 T. Schiifer et al. / Charge symmetry breaking

condensates. The Bore1 mass M, is usually taken to be around the vector-meson mass.

In writing down the sum rule we have neglected a small perturbative contribution proportional to m2, - rnz. The leading term which drives pw mixing involves the expectation value m,( EL) - md( dd). In order to determine the mixing parameter A from the sum rule eq. (15) one has to make a model for the spectral function (l/r) Im II (PO)(s) associated with po mixing. Shifman et al. use a very simple model which only contains zero width p- and w-meson contribution as well as correction from the first radial excitation. Taking into account only the ground- state mesons, the leading isospin breaking term (m, - m,>((Eu> + (ld>> gives

g&o mu - md f,?mt 2e A=- - = -5870 MeV*,

3 m,+md mi 1-p (17)

where p = 3 is a parameter which describes the relative strength of the p- and w-residues in the spectrum function [20]. Higher-order contributions soften the dependence on the Bore1 mass, but do not very much affect the result at Mg = rni.

Also, one can use a more realistic spectral function containing the 25~ and 37~ intermediate states. This improvement can be incorporated into the result (17) by readjusting the parameter p.

4. QCD sum rules and charge symmetry breaking

The neutron-proton mass difference in the framework of the QCD sum rules has been investigated in refs. [3-51. The basic mechanism discussed in these works has already been outlined in the Introduction. At finite density, AMnp is determined by the competition between the charge symmetry breaking in the condensates, which adds to AM,,, and the difference in the current quark masses, which subtracts from this quantity.

In this work we are interested in the question of whether the Blunden-Iqbal mechanism can be recovered in the context of the QCD sum rules. In sect. 2 we have shown that this mechanism can be understood as the pw mixing contribution to the vector self-energy of the nucleon in the nuclear medium. Therefore, we have to study the effects of isospin breaking in the QCD sum rules at finite density.

QCD sum rules for the nucleon at finite density have been studied in refs. [22,23]. The method is based on the correlation function

(18)

T. Schiifer et al. / Charge symmetry breaking 651

where q = •abc(uaCyCLub)y~ysdc and 77 = E abc(d”Cypdb)ypy,uc are the Ioffe cur-

rents [21] with the appropriate quantum numbers for the proton (neutron) and I To) is the ground state of nuclear matter. Following the procedure adopted in

[4,23] the correlation function can be decomposed

where uP is the four velocity of the nuclear medium. In order to extend the sum rules to finite density, one has to include the effects of nonscalar operators like the vector condensate ( qtq>. The corresponding Wilson coefficients are most easily calculated by using the background field method [23]. For our purposes, we need to include all terms that are linear in the isospin breaking parameters m,, -m, and (UU) - (dd). Note that by comparing the np and ZB systems one can argue [5] that (&u) - (dd) is roughly linear in m,, - m,. We will consistently neglect all terms that are of higher order in the quark masses. In particular, for our estimates we will expand m,(Eu> - m,(zd) 1: (m, -m,)(G) and (UU)~ - (zd)2 = 2((L) - (Jd))(iiu).

In the rest frame of the nuclear matter, in which q. u + qo, we find the following contributions to the invariant functions for the proton:

1 4(q2, qo) = - &( -q2)2 ln( -q2)m, - G( -s2) ln( -q2)(t&

1 ln( -q2)m,(u+u) -

1 - + za2q0 sqo ln( -q2)(ADod)

- $n)(u+u) - $(iiu)(m,(iiu) + $m,(dd)), (20)

17,(q2, qo) = - &( -q2)2 ln( -q2) - &ln( -q2)md(dd)

1 +- 6_2 q. ln( -q’)( (U+u> + (d+d>)

+ $ln(-g’)( (1- T)(dtDod)+ (4- $)(utDou))

2p)Lz!$ 4

(m,Gu)(d+d) + $m,(dd)(u+u)), (21)

652 T. Sctifer et al. / Charge symmetry breaking

n”(C?“, 40) = 1

gq2 ln( -q2)(7(U+U) + Cd+&)

- -J&q0 In( -q2)(( d+iD,d)+ 4( u+iD,u))

- &nJEu)(d+d) +m,(dd>(u+u)). 4

(22)

The result for the neutron can be obtained simply by making the replacement u t) d. Here we have neglected contributions that are of higher order in the density, since these are still small around nuclear matter density. For each of the Lorentz structures fli = flS,s,U we assume a dispersion relation

(23)

where we have omitted subtraction constants. We saturate the phenomenological part of the sum rule with a nucleon propagating in the nuclear medium

At4 n(q)= - (q-$,(q, u))*~+(hf+&(qT 4)’

where A, denotes the coupling of the Ioffe current to the physical nucleon state. The vector and scalar self-energies &, are momentum as well as density dependent. Taking the imaginary part of eq. (24) we get a nucleon pole contribution shifted by the real part of the self-energy and a continuum determined by the imaginary part of the self-energy. The strategy is to determine the real part of the self-energy at the position of the nucleon pole from the sum rules and to use a model for the continuum part. Since we are interested only in the effect of isospin breaking, we use the imaginary part of the po self-energy as a model for the continuum. Note that only the Fock term (fig. la,c) develops an imaginary part above a certain threshold whereas the Hartree term (fig. lb) is real and momentum independent.

At zero density, one usually performs a Bore1 transformation with respect to Q2 = -q2 in order to improve the overlap between the theoretical and phenomenological description of the correlation function. At finite density the dispersion relation depends on Q2 and q. u independently. In order to facilitate the comparison of our treatment with the zero density results we perform a Bore1 transform in Q2 but keep q. u fixed. The leading terms in the sum rules for the


three invariant structures then give [23]

AiM* exp( -p’/Mz) = - -&Mi(zd) + *. *,

(27)

Here M” = M + 2Ts is the effective mass of the nucleon, $ = M *2 - 2,” + 2q,& the on-shell value of the invariant mass and Ma the Bore1 mass. Note that the sum rules depend on two parameters, q0 and MB. The dependence on q0 enters through p2 and the momentum dependence of the self-energies on the left-hand side and via finite-density corrections on the right-hand side. For qa = M *(qo,

q=O)+Ls,(q,, q=Ol th e quasiparticle is at rest with respect to the nuclear medium. Different choices of q0 correspond to different ~nematical situations. We will study the dependence on @a in our discussion of the numerical results.

Since the compIete expression for the correlator, eqs. (20)~(22), is very lengthy, we want to discuss the most important contributions to the neutron-proton mass difference separately. The energy of the nucleon in the nuclear medium is determined by taking the sum of the first eq. (25) and the third sum rule eq. (27) and dividing by the second one eq. (26). Keeping only terms that involve the quark condensate alone, the effective mass of the proton is given by

M*= --th2(~d>M;:

M; + +“{ Eu)~ * (28)

As discussed in the Introduction, this formula leads to the conclusions that the proton mass is essentially determined by the down-quark condensate. Expanding eq. (28) around the isospin symmetric case (Iiiu) = (zd), the contribution of the quark condensate to the neutron-proton mass difference is

129)

At zero density this term gives AM,,p = 9.3 MeV, where we have used y = -0.006. At finite density, chiral symmetry is restored and in the dilute nucleon-gas appro~mation the quark condensate decreases with density at a rate

(30)


where ZrN = (45 k 7) MeV is the pion-nucleon sigma term [24]. Note that the factor ,Z~r,&z, + md> = 4 is rather large, resulting in a very strong density dependence of the condensate. In addition to the condensate, the isospin breaking parameter y is also expected to be density dependent [3,51, decreasing at roughly the same rate as the third root of the quark condensate,

y"_ (a4)* l/3

Y- i I (44) * (31)

Using these estimates, the change of the neutron-proton mass difference in the medium is given by

Ai&, 2: 9.3 MeV - 1.4 MeV (32)

In the same way, we can collect all the terms which involve either the quark masses alone or the masses together with the quark condensate. Using the fact that the Bore1 mass scales roughly like the nucleon effective mass, the mass terms con- tribute

AM,,P= -:(m,-m,) = -8 MeV, (33)

where we have used md - WI, = 3 MeV. Together with eq. (29) this term reproduces the neutron-proton mass difference in vacuum. Furthermore, since the contribution from eq. (29) decreases with density whereas eq. (33) remains constant, the neutron-proton mass difference changes sign at around nuclear matter density.

So far we have discussed only those terms which are already present in the sum rules at zero density. These contributions give a straightforward explanation of the reduction of the neutron-proton mass difference in terms of the competition between the density-dependent term eq. (29) and the constant piece eq. (33). From a theoretical point of view, however, corrections due to the vector condensate (q+q> may not be small since these terms are also linear in the density and the isospin breaking parameter md - m,.

As one can see from the sum rule for the II, structure, eq. (271, the most important effect of the vector condensate is to produce a repulse vector self-energy 2, = (32r2/M& for the nucleon. Here we are considering isospin symmetric matter with (u+u> = (dtd) = $p. In this case the leading term in the vector self-energy does not distinguish between neutrons and protons.

The leading charge symmetry-breaking piece in the vector self-energy involves


the density times the expectation value of rn,(E.~) - m,(dd), giving l

A~%I&,~ = (34)

Note that this is precisely the same matrix element that is responsible for the pw mixing in the QCD sum rules. Therefore, we expect this term to reproduce the pw mixing contribution to the neutron-proton mass difference. Numerically, we get

AMnP= -160keV

which in fact nicely reproduces the result eq. (8) we obtained in sect. 2. The Fock term (fig. la) which appeared in the meson-exchange calculation in sect. 2 is explicitly included in the continuum part of the spectral function. Note that this is consistent with the QCD sum-rule method in which the real part of the correlation is calculated from quark diagrams and the imaginary part is parametrized in terms of the contributions from physical intermediate states. After Bore1 transformation, however, the high-energy part of the spectrum is suppressed so that the effect of isospin breaking in the continuum is very small.

Additional density-dependent charge symmetry-breaking terms appear in the scalar self-energy, eq. (20). All of these contributions vanish at q. = 0. However, if q. is chosen to be equal to the energy of a nucleon at rest, q. = M * + Z,, they become quite significant. The m,(q+q) term is easy to evaluate and gives

AM,,P = 0.75 MeV (36)

To a large extent,. this term, is cancelled by the (qq)(q+q) term which contributes

AA& = -0.55 MeV (37)

An additional source of charge symmetry breaking at finite density is given by nonlocal corrections to the quark condensate (@D,q) and the vector condensate (qtiDoq). Using the equations of motion, we get (@Dpq) =m,(qy,q) for the nonlocal correction to the quark condensate and

(38)

l This estimate is based on setting q0 equal to the energy of a nucleon at rest, q,, = kf * + 2,.

T. Schiifer et al. / Charge symmetry breaking

Fig. 2. Neutron-proton mass difference as a function of density calculated from the QCD sum rules. The upper and lower curve correspond to the two cases q. = 0 and q. = M* + 2,.

for the vector condensate. Additional information is needed in order to determine the expectation value of the twist 2 operator (q+iD,q). To linear order in the density, one can evaluate this matrix element in a dilute nucleon gas, giving (q+iD,q) = (N 1 qtiD,q I N) * p, where (N 1 . I N) denotes the matrix element between spin- and isospin-averaged nucleon states at zero momentum. This matrix element can be determined in deep inelastic scattering [25]. Using the quark distribution functions at a low scale p2 = 1 GeV2 one finds (q?iD,q) = 150 MeV . p. Since isospin breaking is already included in the trace part of eq. (38), we will neglect the isospin dependence of the nucleon matrix element (N I qtiD,q I N). In this case, applying the sum rules at q0 = M * + Z,, leads to AM,,, = -0.7

MeV(p/p,), which is quite large as compared to the leading order result, eq. (29). In fig. 2, we show our numerical results for the density dependence of the

neutron-proton mass difference. We have considered the two scenarios q0 = 0 and q,, = M * + z’,. In each case we determine the Bore1 mass by minimizing the sensitivity to this parameter.

If one does not take the corrections from nonlocality into account, the value of AM,,p from the sum rules is quite insensitive to the choice of qO. The numerical result corresponds roughly to

AMnP = 1.4 MeV - 1.1 MeV (39)

This density dependence is pretty much in line with what is needed to resolve the Nolen-Schiffer anomaly. However, if one includes the effects of isospin breaking

T. Schiifer et al. / Charge Emmett breaking 657

in the nonlocal corrections to the quark condensate the result depends on the parameter q,,. This might reflect a rather strong momentum dependence of the charge symmetry-breaking part of the nucleon self-energy. Applying the sum rules at qa=M* +C, we find A~~~ = 1.4 MeV - 1.8 MeV(p/p,), which is too large in comparison with the empirical magnitude of the anomaly.

5. Conclusions

We have studied the density dependence of the neutron-proton mass difference, The most important contribution arising from charge symmetry breaking in the meson sector is given by the po mixing self-energy of the nucleon. At finite density, this terms reduces the neutron-proton mass difference. Our result is in agreement with the findings of Blunden and Iqbal IS]. We have argued, however, that the quantitative result of Blunden and Iqbal is too large by a factor 2-3, leaving roughly 70% of the Nolen-Schiffer anomaly to be explained by other effects.

In order to understand the role of pw mixing we have extended the treatment of the neutron-proton mass difference in the QCD sum rules to finite density. In this case charge symmetry breaking in the nucleon self-energy is determined by isospin symmetry breaking in the density-dependent part of the operators con- tributing to the nucleon correlator in the medium. The density dependence of the nucleon vector self-energy calculated in this approach does in fact reproduce the result from pw mixing. We conclude that the effect discussed by Blunden and Iqbal is correctly included in the QCD sum rules.

In the QCD sum rules, the largest part of the density dependence of the proton-neutron mass difference comes from the scalar self-energy. The magnitude of this term is hard to determine since it depends on the value and density dependence of the isospin breaking parameter y_ Depending on the choice of the sum rules, we find A~~~ = 1.4 MeV - u(~/~*) with a = (1.1-1.7) MeV.

References

[l] J.A. Nolen and J.P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 471 [Z] EM. Henley and G. Krein, Phys. Rev. Lett. 62 (1989) 2586 [3] T. Hatsuda, H. Hiagaasen and M. Prakash, Phys. Rev. C42 (1990) 2212 141 T. Hatsuda, H. Hogaasen and M. Prakash, Phys. Rev. Lett. 66 (1991) 2851 [Sl C. Adami and G.E. Brown, Z. Phys. A340 (1991) 93 [61 M. Lutz, H.-K. Lee and W. Weise, Z. Phys. A340 (1991) 393 [7] S. Narison, Rev. Nuovo Cimento 10.2 (1987) 1 I81 P.G. Blunden and M.J. Iqbal, Phys. Lett. B198 (1987) 126 [9] H. Sato, Nucl. Phys. Al65 (1976) 378

1101 H.R. Christiansen, L.N. Epele, H. Franchiotti and C.A. Garcia Canal, Phys. Lett. B267 (1991) 164


[ll] G. Krein, D.P. Menezes and M. Nielsen, Phys. L&t. B294 (1992) 7 [12] J. Gasser and H. Leutwyler, Phys. Reports 87 (1982) 77 [13] G.A. Miller, B.M.K. Netlcens and I. Slaus, Phys. Reports 194 (1990) 1 [14] T.D. Cohen, R.J. Furnstahl and M.K. Banejee, Phys. Rev. C43 (1991) 357 [15] T. Hippchen, J. Speth and M.B. Johnson, Phys. Rev. C40 (1989) 1316 [16] R. Machleidt, Relativistic dynamics and quark nuclear physics, ed. M.B. Johnson and A. Pick-

lesheimer (Wiley, New York, 1986) [17] R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189, p. 352, table A.3 [18] J. Speth, Nucl. Phys. A527 (1991) 411~ [19] K. Holinde, AIP Conf. Proc. on particles and fields series, vol. 43, Proc. LAMPF Workshop on

(r, K) physics, ed. B.F. Gibson,. W.R. Gibbs and M.B. Johnson [20] M.A. Shifman, AI. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (19791 519 [21] B.L. Ioffe, Nucl. Phys. B188 (19811 317 [22] E.G. Drukarev and E.M. Levin, Nucl. Phys. A511 (1990) 679 [23] T.D. Cohen, R.J. Fumstahl and D.K. Griegel, Phys. Rev. Lett. 67 (1991) 961;

R.J. Furnstahl, D.K. Griegel and T.D. Cohen, Phys. Rev. C46 (1992) 1507; X. Jin, T.D. Cohen, R.J. Fumstahl and D.K. Griegel, preprint, University of Maryland UMP#92- 242

[24] J. Gasser, H. Leutwyler and M.E. Sainio, Phys. Lett. B253 (1991) 252 [25] T. Hatsuda and S.H. Lee, Phys. Rev. C46 (1992) R34

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Charge symmetry breaking and the neutron-proton mass difference