Isospin symmetry breaking in hadrons and nuclei

NUCLEAR PHYSICS A

Nuclear Physms A577 (1994) 361c-368c North-Holland, Amsterdam

Isospln Symmetry

Breaking

m Hadrons

and Nuclei

Tetsuo H a t s u d a ~ ~Instltute of Physics, University of Tsukuba, Tsukuba, Ibarakl 305, Japan A b n e f review of the lsospln symmetry breaking in hadrons and nuclei is given with emphasis on the u - d quark mass difference The off-shell p0 _ w r u l i n g is studmd as a typical example of the symmetry breaking, and its relevance to the nuclear force and nuclei is discussed 1. I n t r o d u c t i o n In hadron and nuclear physics, the lsospin symmetry has been known to hold fairly well In the quark level, the symmetry is defined as a global vector rotation in u - d lsospace It is also useful to define the charge symmetry as a 180 ° rotation around y axis in the lsospln space By this operation R, the u and d mass eigenstates transform as R ] u) -- - I d),

R i d ) --[ u)

(1)

Whenever charge symmetry breaking (CSB) occurs, so does the lsospin symmetry breaking Hereafter, we will focus on CSB for simplicity CSB has two detectable origins (i) Electromagnetic (EM) effect Since the electric charges of the u quark and the d quark are different (e~ -- 2/3 e, ed = --1/3 e), QED induces CSB of O ( a ) (n) Quark mass difference Since the current quark masses of the u quark and the d quark are different (rn~(1GeV) _~ 4MeV, md(1GeV) -~ 9MeV [1]), the quark-mass term H ~ CO in the QCD hamlltonian induces CSB This is ease to see from 1 m ~ - rnd)(fzu HQCD : ~1 ( m ~ + . ~ ) ( ~ -u + dd) + :~(

dd)

(2)

The second term proportional to m~, - m d gives O ( ( m d -- m ~ ) / h Q c D ) breaking of the charge symmetry Since CSB is generally small (1% level or less), it is a good approximation to expand arbitrary physical quantity A up to first order m a and md -- m~, A ( a , md -- m~,) ~-- A(0, 0) + b a + c

md -- m u AQCD

(3)

1.1. C S B i n h a d r o n s Among various examples of CSB phenomena in hadron physics, the following three are well known and relatively important 7r± - 7r° mass difference, proton-neutron mass difference and the p0 _ w mLx~ng (See Table 1) The first one is a typical example of 0375-9474/94/$07 00 © 1994 - Elsevier Scrence B V All nghts reserved SSDI 0375-9474(94)00397-1

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T Hatsuda / lsospm symmetry breakmg m hadrons and nuclet

the EM dominated CSB Since 7r± and 7r° have equal number of u and d quarks on the average, O(md -- mu) effect vanishes in the mass difference This is the reason why the naive expectation "charged particles should have larger inertia mass than neutral partMes due to the photon cloud" holds m this case TheoretmM estmlate of the EM effect for the pmn self-energy is conmstent with this picture On the other hand, the neutron is heavmr t h a n the proton m the real world, which indicates that there must be a sizable CSB from the quark mass difference with opposite sign to EM effect The recent measurement of the e+e - --~ 7r+Tr- shows an unainblguous determination of the p0 _ w inlxlng with negative sign [2,3] The EM contribution due to p0 __, 7 ~ w is pomtlve and small, thus a relatively large and negative c o n t n b u t m n from the quark mass difference is again necessary The numbers in the 4th column of Table 1 are the expected O(rnd--mu) contributions obtained by subtracting the theoretical EM effects (3rd column) from the experimental numbers (2nd column) The QCD sum rules as well as other effective models have been used to estimate the O(md -- mu) effect and the results are conmstent with those in Table 1 (See e g [4,5] ) Thus we are rather in a good shape to understand the origin of CSB m hadrons, although the first principle calculatmns on the lattice are stall mmsmg

A m.± - m~0 (MeV) m p - rn,~ (MeV) p°-w mixing (MeV 2)

experiments 46 - 1 29 - 4 5 2 0 • 600

O(a)

O(md - m~)

4 6 4- 0 1 (theory) 0 76 -t- 0 3 (theory) .-~ 610 (theory)

0 ~ - 2 05 ~ -5130

Table 1 CSB examples in hadron masses and the mixing pO _ a~ mixing is defined by the covanant matrix element (pO I Hcsa I w) at the pO _ aJ mass shell with Hcsn being the second term m eq (2) See [1] for the details of the O(a) estimate One should note here that mpo ¢ m~o even without CSB

1.2. C S B m n u c l e i W h a t about CSB in nucleon-nucleon interactions and in nuclei~ There are, in fact, accumulated evidences that EM effect is not enough to explain the observed CSB (In the following, we will put " bar" to express CSB with the EM effect subtracted ) For example, (1) the CSB in N-N scattering length such as ~,,,, - ~pp = - 1 5 4- 0 5 fm [3,6], (n) binding energy &fferenees in mirror nuclei such a s / ) ( 3 H ) - [~(3He) = 80 =t=24keV [3] and the Okamoto-Nolen-Schlffer anoInaly [7,8], and (nl) the asymmetry m the polarized n - p scattering at zero-crossing angle AA(00) [9] See ref [6] for a more complete list of the nuclear CSB As far as the two-body N-N interaction dominates in nuclei, all the above quantities are related to the EM subtracted CSB in the elementary N-N interaction fztcN,

G,~ ¢ ~ # ~

(4)

Among various contributions to fTyN, the exchange of the p0 _ w complex (the one in Fig l(b)) is recently claimed to play a crucial role m the above examples (~)-(ln) In the following sectmns, I will discuss whether thin idea is plausible or not on the basis of the recent work by Henley, Melssner, Kreln and myself [10]

T Hatsuda / lsospm symmetry breakmg tn hadrons and nuclei

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2. T h e o f f - s h e l l pO _ w m i x i n g Let's start with the following hadromc correlation function H.~(q 2) - z / d*x(Tp°(x)w~(0))o = - ( g . . -

q.q~' )__ 1 q2 q2 - mp2 0 ( q 2 )

1

q2 - m,~2

(5)

The mLxmg m a t r i x O is m general q2 dependent It can be determined at least at q2 ..~ m~ through the interference of e+e - ~ pO ~ ~r+~_- w~th e+e - ~ w ~ pO ___.~r+Tr- (F~g l ( a ) ) m the e+e - ~ 7r+Ir - experiment [2,3] The process m F~g l ( a ) is a s-channel exchange of the pO _ o~ complex for q2 > 0 On the other hand, m the elastic N-N scattering, the pO _ o~ complex is exchanged in the t-channel (F~g l(b)), thus the relevant q2 ~s inevitably off-shell and space hke (q2 < 0) All the prewous eshmates of CSB m N-N force and in nuclei have however been done with an assumptmn that O(q 2) is q2 independent m the wide range of the m o m e n t u m -m2,., < q2 < m~,~ Is thin a vahd assumptmn or not ? Thin is the questmn first raised by Goldman, Henderson and Thomas [11] Subsequently, Melssner, Henley, Krem and myself have proved that O must be q2 dependent in a model independent way, and have extracted the dependence using the QCD sum rules [10] 71.+

7r-

ip

e÷

e-

(a)

(b)

Figure 1 (a) s-channel exchange of the p0 _ w complex m e+e - --* ~r+Tr- experiment (b) t-channel exchange in the N-N interaction

2.1. W h y O m u s t b e q2 d e p e n d e n t ? T h e heunstlc argument goes as follows Let us concentrate on the non-EM part of O and start with an u n s u b t r a c t e d dispersion relatmn for H(q 2) - - I I . , ( q 2 ) / 3 ReH(q2) = PTr/

I m H ( s ) ds s - q-----Y

(6)

Instead of II(q 2) here, one can make the same argument starting with the QCD correlation HQCD~v "~ (WJ~(x)J~(O))o with J~,~ = (~7.u T dTd)/2 In that case, one can prove t h a t the s u b t r a c h o n is not necessary for the non-EM part Also one can take into account the higher resonances (p' and w') See [10] for detmls

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T Hatsuda / Isosptn symmetry breaking m hadrons and nuclet

S a t u r a t i n g the imaginary part m (6) by the p0 and w resonances as ~lmH(s) -= FpS(s rn2v) - F,og(s _ m,o) , 2 one immediately arrives at -Fp F~ _ 1 ReII(q2) - q2 _ rn~ + q2 _ rn~ q2 _ rn~

1 q2 _ rn2 '

O(q2)

(7)

with O being the non-EM part of the mL,ong matrix O(q 2) q~ ® ( m 2 ) - - 1 + ) ~ ( ~ - 1), where

=

+

and

=

(Y. -

(8)

+

+

2

2

-

One

should note the following points in the above formula (1) Fp and F,~ are O ( m d -- m~,) quantities However, there is no reason to believe t h a t Fp -- F~, since they are the residues at different pole positions Also note that mp ~ raw even if there is no CSB (n) ;~, which dictates the q2 dependence of the p0 _ ~; mixing, is an O(1) quantity This suggests a large variation of (9 from the on-shell point to the space like points However, in the previous applications of the p0 _ w mLxmg to the nuclear CSB, ,~ is assumed to be zero with no specific reason (nl) It is unlikely t h a t the relatively small EM effect (see the 4th row of Table 1) substantially changes the above conclusion at q2 < 0 2 2. O(q 2) m Q C D s u m r u l e s In ref [10], we have evaluated A with the use of the QCD sum rules A mare advantage of this m e t h o d over other estimates is that one can relate A to the basic QCD parameters such as md/m~, and (~U)o/(dd)o For these parameters, we take the results of the chlral p e r t u r b a t i o n theory and QCD sum rules [1,10] _

md _- m~ _ 0 28 -4- 0 03 -( -u>0 = 1 - (2 ~ 10) 10 -3 md+ m~ ' (dd)o After the Borel + finite energy analyses of the QCD sum rules, one obtains =143~185

(9)

(10)

The corresponding ®(q2) is shown in Fig 2 1

q

T

~

-08

-06

-04

1

F

1

r

06

O8

05 0

o(q2)-o -15 -2 25 -3 -1

02

02

04

Figure 2 q2 dependence of the non-EM mtx_mg matrix ® with A = [1 43, 1 85] [10]

T Hatsuda I Isospm symmetry breaktng m hadrons and nuclet

365c

Our result is insensitive to the second input in eq (9) which has a large uncertainty Since A is positive and larger t h a n 1, O(q 2) m the space hke region, whmh is relevant to N-N force, changes sign from its on-shell value Thus we estabhsh the mappllcablhty of the assumption O(q 2) = O(rn 2) It is in order here to mention other model calculations of O(q 2) the constituent-quark loop model (p0 _~ qq ~ w) [11,12] and the nucleon-loop model (p0 ~ /YN ~ w) [13] Although they also give a large q2 variation of O, there is a crucial assumption with no theoretical justification m these models In fact, CSB other than the EM effect is taken into account only as a constituent-quark mass difference or the n-p mass difference, while the current-quark mass difference m d - rn~ is m principle hidden in e g the coupling constants of the vector mesons with the constituent quarks or the nucleons In the QCD sum rules, one does not suffer from this deficiency, since all the CSB effects are automatically taken Into account m the operator product expansion of the current correlator (For the 7r° - ~7mixing, the chlral perturbation theory is an alternative method to calculate the q2 dependence in a consistent way [14] ) To see the physical effect of O(q2), let us make a Fourier transform of the mixed p0 _ w propagator eq (7) and extract a static and central part of the CSB N-N potential, which results in

l)n.

o ( m 2)

-

l;"m

~

~(1

2~.

-

~r)e-

m~

(11)

A = 0 corresponds to the potential by Coon and Barrett [3], while )~ = 1 78 is ours They are compared In Fig 3 Although the nn interaction m more attractive than pp when A = 0, the use of A larger t h a n 1 completely kills the attraction m the relevant range of the CSB nuclear force This indicates that the diagram such as Fig l(b) alone is not effective to explain the scattering length difference ~.. - alv and other nuclear CSB contrary to the previous claims

off shell on shell - -

0 02

001 v

(r) 0 -0 01 -0 02

8

I

I

I

I

1

12

14

16

r Cfm] Fxgure 3 CSB N-N potential V ~ - V~,~-I)'pv in an arb,trary umt [10] The solid (dotted) hne corresponds to A = 0 ()~ = 1 78)

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T Hatsuda / lsosptn symmetry breakmg m hadrons and nuclet

3. O u t l o o k - - t o w a r d r e l i a b l e c a l c u l a t m n s - We have here analysed a specific CSB effect "the p0 _ w mLxang" and found that the effect is substantially changed going from the tlme-hke region of q2 to the space-hke region The existence of the q2 dependence can be shown in a model independent way using the u n s u b t r a c t e d dispersion relation We could also calculate the quantitative q2 dependence using the QCD sum rules The resultant CSB potential from the p0 _ w mlxang is much weaker t h a n the C o o n - B a r r e t t ' s potential and does not seem to be a dominant contribution to the various CSB in N-N force and m nuclei contrary to the prevmus claims My aim here was not to solve the nuclear CSB puzzles but to spotlight a serious problem of the previous approaches by taking the pO _ w mixing as an example One of the m a j o r difficulties to s t u d y the CSB in many body systems lies In the fact that CSB is hidden in all over the place in hadromc Interactions As an example, let's take a N-N elastic scattering in the one-boson exchange model (Fig 4)

)(

---~---v,s,p

(a)

V,S,p

V,S,p

(b)

(c)

Figure 4 T h e N-N s c a t t e n n g with CSB m the one-boson exchange model (a) CSB m the mLvang matrix, (b) CSB in external hne and (c) CSB m the meson-nucleon vertex The quark mass difference and the EM effect denoted by the "cross" are hidden not only m the meson mL~mgs (Fig 4(a)) and the external hnes (Fig 4(,t)) but also m the meson-nucleon vertmes (Fig 4(c)) They are all the same order and it is inconsistent to take only a part of them Further dlfficultms arise from the following facts Firstly, one should consider all possible CSB m scalar (s), vector (v), pseudoscalar (p) channels in Fig 2(a,c) q2 dependence should be also taken into account hele Secondly, it is inconsistent to estmlate the meson mLxangs and the CSB meson-nucleon vertmes by the loop calculahons m hadromc models In fact, there is always CSB effects m the tree level of the hadromc effechve lagranglans The strength of them cannot be determined in the effechve theory and must be calculated in QCD or must be fixed by experunents These two remarks lead us to the two ways to attach the CSB problem m m a n y - b o d y systems (1) Write down an effective lagranglan with all possible CSB terms Calculate the coefficmnts of these terms m QCD (e g by the lattice QCD or the QCD sum rules) Then use the lagranglan to evaluate the relevant hadromc processes (11) Calculate the

T Hatsuda / Isospm symmetry breakmg m hadrons and nuclet

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physical CSB processes directly in QCD The second approach is formidable for manybody systems but may be apphcable for N-N force In fact, the studies of the N-N scattering lengths on the lathce have been already started [15] and one may generalize the method for the md ~ m . case in the future Finally, I should briefly remark on the fundamental importance of CSB in hadrons and nuclei The stability of our world is actually intimately related to the u-d quark mass difference If it were m~ > m d just like the quarks in other generahons (note that mc > me and mt > mb), the proton becomes heavier than the neutron and the hydrogens are unstable by the/3+ decays It will be a fun to see what sort of nucleosyntheses takes place in the early universe m such a sltuatmn It may be also interesting to note that CSB by the EM lnteractmn and that by the quark-mass difference are roughly the same order in magnitude although they are dictated by independent parameters in the standard model (a = e2/4zr versus (rod -- m , , ) / A Q c D = (gd -- g,~)(¢)/AQcD with g, being the Higgs-quark couphng and (¢) being the H~ggs expectatmn value) The two effects may turn out to have the same ongm m some unffied theory of the strong, weak and EM interactions Acknowledgement I would like to thank the organizing committee for giving me an opportunity to talk at thin conference In particular I thank Prof Y Mizuno and Prof Toru Suzuki This talk is based on a work in collaboration with Th Melssner, E M Henley and G Kreln to whom I am also grateful

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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