Hadron masses and the sigma commutator in light of chiral perturbation theory

ANNALS OF PHYSICS 136, 62-l 12 (1981)

Hadron Masses and the Sigma Commutator in Light of Chiral Perturbation Theory*

J. GASSER

Instituteeor Theoretical Physics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland

Received March 23, 1981

We analyze carefully the impact of non-analytic chiral corrections to the mass spectrum of

the pseudoscalar meson octet Jp = O- and the baryon octet Jp = f’. We find that the quark

mass ratios must lie in the range 21 < m,/A < 32 and 1.6 ,< m&n, ,< 2.2. We also calculate the analogous corrections to the pion-nucleon sigma commutator u,~. It turns out that the

value onN = 60 MeV is not compatible with the structure of the meson and baryon spectrum,

unless the nucleon mass is smaller than 600 MeV in the chiral limit m, = m,, = m, = 0.

1. INTRODUCTION

In a previous paper [ 1 ] (referred to as [I ] below) we developed a technique to do perturabation theory in the quark mass parameter ‘9JI around the chiral symmetric limit which is described by the QCD Hamiltonian H,,,(%Jl = 0). The present article deals with a specific application of this method: We evaluate the leading non-analytic corrections (LNAC) to the mass spectrum of the pseudoscalar mesons (Jp = (7) and baryons (Jp = $ ’ ).

The main motivation for this calculation is the fact that it allows us to give bounds on the quark mass ratios 2m,/(m, + md) = m,/rii and md/mu. These bounds follow from rather weak assumptions. As a consequence, any specific field theory model which allows the evaluation of these quark mass ratios but disagrees with our bounds is also in conflict with the current understanding of QCD and its spontaneous chiral symmetry breakdown.

It turns out that LNAC to the meson mass spectrum are reasonably small. We shall see, however, that LNAC to baryon masses are in general large. In fact we must abondon here the idea of improving first order results by evaluating non-analytic corrections. We shall instead use a model (which goes beyond chiral perturbation theory) for the cloud of virtual particles surrounding mesons and baryons. This model allows to estimate the effects of non-analytic corrections. It reproduces the LNAC found before, and it gives corrections to the meson masses which are in agreement with those found from LNAC. These features give us the confidence to

* Supported in part by Schweizerischer Nationalfonds.

62 0003.4916/81/110062-51$05.00/O

Copyright % 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.

HADRON MASSES AND SIGMACOMMUTATOR 63

apply the model to the baryon masses in order to estimate the uncertainties in the evaluation of the ratio

R = (rii - m,)/(m, - m,)

(We shall always use fi = (m, + m,)/2 in the following.) Part of the results which concern meson masses have been obtained before by

Langacker and Pagels [ 2, 31. All the results which are presented here could, in prin- ciple, be obtained with the method developed by these authors, As we already pointed out in [I], we do think that our method is much simpler than theirs. Furthermore we disagree on the interpretation of the results. This will become clearer in Sections 4 and 6-8.

The paper is organized as follows. In Section 2 we briefly recapitulate our technique to calculate LNAC. Sections 3-5 deal with the meson spectrum and the resulting values of the quark mass ratios. The barvon octet and the evaluation of R from P-N,Z’-Z-and3°-Z~ is discussed in Sections 6-8. Section 9 deals with LNAC to the pion-nucleon sigma term o,~. In Section 10 we do simultaneous fits to the meson and baryon masses. from which we derive bounds on the quark mass ratios. The results of this section are applied in Section 11 to the evaluation of hadronic wave functions (p 1 qq / p) and used in Section 12 to give bounds on possible values of ‘T,,.

2. CHIRAL PERTURBATION THEORY REVISITED

We restrict our considerations to QCD with the three flavors up. down and strange. The Hamiltonian is written as

H Qco=hloQc-+m [!i5r(~)d~~.

where

sj,(x> = I,, + &@d)R + t-s(Ss)~: (2.1)

ti are real numbers such that m, = m<, , md = m& and m, = mt,. We denote renormalized operators by ( )R. The dependence of the mass parameters m on the mass scale p, which is introduced by renormalization is suppressed. We do not discuss in this article the dynamical origin of the quark mass parameters m,, md and m,.

It is a general feature of quantum field theory that matrix elements (oi0(n(x)l/?) (‘L‘ is any operator) are non-analytic functions of the mass parameters which occur in the theory. This is also true for QCD. In this case we can show more than just non- analyticity: The coefficient of the leading non-analytic term is fixed by the chiral symmetric theory itself.

The basic idea to actually calculate these LNAC is the following (for details see II]). Since the matrix element under consideration is non-analytic at 1132 = 0. its

64 J. GASSER

derivative (of sufficiently high order) with respect to !UI must be singular at ‘38 = 0. Once the coefficient of this singularity is found, one obtains LNAC by a quadrature. To be specific, consider LNAC to the hadron mass spectrum. It turns out that a*M*/aYJl* is not defined at ‘iUl = 0 (M = hadron mass). Therefore we need a general expression for a*M*/aW as %I + 0. Since e,(x) in Eq. (2.1) is independent of m, we find from the Feynman-Hellmann theorem [4] that

aM* am= (Pl8mP)

(We use covariantly normalized states (p’ ]p) = 2(27r)3 p”a3(p’ - p). Discrete quantum numbers which specify the state are suppressed). The second derivative is most easily obtained from the (Fourier transformed) retarded amplitude

R,(P,q)=j j-d 4x ew[Wl (P21 W”)[~&/2), sj,(-x/2)llP,),,,,,,,,, y

where

A’ = (~2 --P,)‘? P’ = gp, +p*y. (2.2)

“Connected” means that we must disregard all vacuum bubbles. From R, we obtain the desired derivative,

a2M2

amz = - lim lim lim lim R,(P, q)

qO+~+ qL0 ~4 Au+0 (2.3)

(For a discussion of this and related formulae in the case of free fields see the Appendix of [I]). The dispersion relation for R,(P, q) reads

where

and

q’w = G&Y 9)

(2.4)

VA@, p, = f 1 d 4x expk4 (PA Kf&W, bI(--x/2)ll~l)connected. (2.5)

The nice feature of this representation is the fact that the asymptotic freedom property of QCD guarantees the convergence of the dispersion integral in Eq. (2.4), so we need not worry about subtractions. Furthermore only a very limited set of intermediate states (to be inserted in the right-hand side of Eq. (2.5)) contributes to LNAC, provided the following statements hold:

HADRON MASSES AND SIGMA COMMUTATOR 65

FIG. I. Intermediate states in the expansion of V,(q. P) (Eq. (2.5): q” > 0) which contribute to the LNAC of tnesons. Solid line: Meson under consideration. Dashed line: intermediate meson. Wavy line:

B,(O).

(1) Chiral symmetry is spontaneously broken. In the symmetry limit, we have an octet of massless pseudoscalar Goldstone bosons. No other massless particles are present (e.g. M,, f 0).

(2) We consider all particles to be stable.

(3) The current algebra low-energy theorems for on-shell (Goldstone-) meson-hadron scattering are valid.

(4) Let

0) = I’d4.~exp[Wl (al IBr(x)q sj,(O)l la>.

where la) is any one-particle eigenstate of the total Hamiltonian Hoco. Then

exists.

The graphs which are relevant for LNAC are shown in Fig. 1 (Fig. 2) for mesons (baryons).

We add one comment about q-q’ mixing. Since we assume that M,, # 0 in the chiral limit, the q’ particle may be considered like any other massive pseudoscalar

y- -A.$ 1

FIG. 2. Intermediate states in the expansion of V,(q, P) (Eq. (2.5): q” > 0) which contribute to the LNAC of baryons. Solid line: Baryon under consideration. Dashed line: Intermediate meson. Dashed&dotted line: Intermediate baryon.

66 J.GASSER

particle with no special relation to the massless pseudoscalar octet. Specifically, one does not have any problem with q-q’ mixing in chiral perturbation theory: We can just ignore the presence of q’ in all calculations. This is not true for perturbations around the SU(3)-symmetric world, where v and n’ do mix.

3. THE PSEUDOSCALAR OCTETJ~=O-

In this section we evaluate LNAC to Mi, Mi and Mi. The result allows us to calculate the corrections to the first order values of G/m,, md/mu and (fii - m,)/(m, - mJ. Since it is known that chiral perturbations around SU(3) x SU(3) can be substantial [2], we consider in the first paragraph the closely related problem of expanding the matrix elements of the “current” operators Uu, dd and Ss. Although these matrix elements are not measurable, we will get a feeling of how chiral expanions work for such matrix elements. We shall find that LNAC to the meson matrix elements are reasonably small, while they are huge in the case of baryons (considered in Section 6). Once LNAC for these matrix elements are calculated, we obtain immediately the expansion of the mass spectrum via

m$$= m,(pl (fiuh IP) + dpl (@rx IP) + 4~1 (is), 1~).

3A. Chiral Expansion of Current Matrix Elements

Let j(x) denote any of the operators (UU)~ (x), (dd)R (x) or (Ss), (x). We obtain a representation for (a/am) (p/j(O) 1 p) a ain g via Feynman’s trick, compare the case j(x) = a,(x) considered in Section 2. Define

R{(P,q)=i \d 4x exdivl (p2 I @(~“N.W2)~ 8,(-x/2)1 ~~~~~~~~~~~~~ 5

where

P’ = %P, + PZK Au = (~2 -P?.

Then

&(olj(O)lo)=- 1’ ,/z+ h& litit limo Ri(P, q). (3.1)

The details of the calculation can be found in Appendix A. We present the results below.

We shall use here and in the following the notation

z = o(n’ I (UU>R 17r+ >o; 61 = (m, + mJ2;

o,=(rS+m,)Z; u, = 2AZ; 0, = f(4a, - 0,);

A = 1/167r2pX2.


1 ),, is an eigenstate of HocD in the chiral symmetry limit !JJI = 0, and with F”, we denote the pion decay constant F,(-93 MeV) in this same limit. We set m, = md and have for the

Pion.

(7r”~(iiu)R~7ro)=Z 1 +A 1 L

u,ln-$--&(o,+3u,)ln$ II

,

(no / (ad), j 71’) = (no ( (6~)~ (R”)

(n”~(~s),~no)=-Z.4~u,ln~.

Kaon.

(K”~(tiu),~Ko)=Z~ jfuyln$-u,ln [““2~“y2]2~,

(K”I(;id),IKo)=C !I+$([u,+fu,]ln$

+u,ln [“‘2;uY2]2) 1,

(K”I(Ss),jKo)=C /1+$[u,+fo,]ln~/.

Eta.

(rll(Uu),/9)=$t. 1 +A 1 L

(3u,+o,)ln$

++(u,-5u,)ln+-3u,ln-$ , II

(~l(ss),l?)=+ 11 +A [ 3uv:o, In-$-+$(u”--8u,)ln$- . II

(3.2)

Let us discuss the usefulness of these expansions. For sufficiently small values of the quark mass parameters, the LNAC evaluated above are a small correction to the chiral symmetric values of the matrix elements and hence the expansion makes sense. To check this statement for the actual values of the quark masses, we may proceed as Follows.

68 J. GASSER

Consider the expansion of (K’I (Ss), II(“). Define the quantity E by

(K” I@), I K”) = o(K” I (Ss), I K’), (1 + ~1,

.5-=+ [o,+fo,] In%.

E depends on the three parameters lit& m,C and the cutoff /1. To the order considered, we may replace rFiZ and m,Z by Mfg and Mi,

Mf, = 2riiZ,

M~=(fi+mm,)L,

and we get

e=0.15 Iris A 2 ’ A >M,.

From this result we see that 1 E I ( 0.2 for M, < A < 1 GeV, and therefore LNAC to the chiral symmetric value of (K’I (Ss), IK’) are less than 20%. We have checked that the same is true for the other matrix elements listed in Eq. (3.2), provided they do not vanish for ‘iIJI = 0. Consider now the chiral corrections to the two matrix elements (no I(&), Ix”) and (K’J (Uu)a II(“). As we can see from Eq. (3.2), their values off the chiral limit is of the same order of magnitude (in units of C) as the correction E discussed above.

In conclusion, we find that LNAC to the meson matrix elements of the currents Uu, dd and Ss are of the order of 20% or less for n 2 1 GeV. Therefore we expect that LNAC to meson masses are reasonably small. This is confirmed in the next paragraph.

3B. Chit-al Expansion of Meson Masses

In this paragraph we list LNAC to the pseudoscalar meson masses. no-q mixing which occurs for m, # md does not pose any technical problem in our formalism, since we always work with matrix elements of qmq between physical states, see Eqs. (2.2~(2.5). As a result, intermediate states Irr’q) do not contribute to LNAC, since the matrix element (rr” I B,(O) I q) vanishes like YJI In !Ul for YJI + 0.

The final result is presented in Appendix B, since the expressions are lengthy. The formulae simplify considerably if we drop terms of second order in isospin breaking. Let ( )’ denote the photonic contributions to the mass. Then

M;,=o,+~~;~~ Z+Ok+O,ln$-+(Sm:,)Y

+(m,-m,)Z$o,ln~+o,++ I o,ln”-o,lns

A2 A2 I


In the case m, = ntd, things simplify further:

(3.4)

To compare these results with earlier calculations, we give all the logarithms in Eqs. (3.3) and (3.4) the same value. Then the results in Eq. (3.4) agree with the expressions given by Langacker and Pagels 121, whereas Eq. (3.3) agrees with the expressions obtained by the same authors in [3]. (For the comparison we replaced 6, in Eq. (2.18) of Ref. [3] by the exact expression 2Mk 6,/(2M~ - Mi)).

Finally consider the mass splitting Mi+-M$. Keeping terms of second order in isospin breaking, we find

MZ,, - Mio = & (m, - $2) t; + (dmf, +)‘- (6m;o)’

+-$(3 2

rn: - 2riim, - 5A*) X2 In 3

-&(6 Am, M2

- 8ri12) C2 In -;ii’-. (3.5)

The term (m, - $r) Z/(4R2), which is due to no-~ mixing 15 1, does not contain In !DI

70 J. GASSER

singularities. It contributes -0.13 MeV to M,+-M,,. LNAC tend to cancel this contribution. In fact one finds from Eq. (3.5)

0.12 MeV, /i = 600 MeV M,, - M,, = 0.05 MeV, /i = 800 MeV

0.005 MeV, /i = 1 Gev I + photonic corrections. (3.6)

4. QUARK MASS RATIOS FROM THEJ~=O- OCTET

As is well known, quark mass ratios are renormalization point independent as long as one ignores effects due to QED [6]. We start the discussion with the ratio G/m,. Consequently, we ignore the dependence of the renormalized quark mass on the mass scale p. Effects of QED are discussed below.

4A. The Ratio &/m,

The first order chiral expansion reads

h4;=+,

A4f+Jn,

Mfi = i(4aK - a,).

We may obtain a.value for film, from pion/kaon or from pion/eta mass values:

Cz/m, = 0.038 (pion, kaon),

tqm, = 0.04 1 (pion, eta), (4.1)

where we have used Mi = + (Mt + + Mi,) and MS, = M&. We consider these two different values as a measure of the uncertainties introduced

by higher order corrections. These are of two kinds: LNAC (Eq. (3.4)) and O(mm’) terms. We are not able to calculate these latter corrections. In order to at least guess their sign and order of magnitude, we write

M’K = (#I + m,) .z + (fi + m,y = UK( 1 + X;“(3)),

M2, = 2AZ + 4rit* s o,(l + PXU’3)),

Mf, = j(2fi + 4m,) z + j(2AZ + 4mi) s a,( 1 + PqU(3)).

Numerical values of O(!JJI*) terms are calculated with [7]

(4.2a)

tSz = 5.5 MeV, m, = 130 MeV. (4.2b)

These formulae should be interpreted as follows. Consider the kaon. Since binding energies are negligible for 9Jl large, we expect that Mk N (fi + m,)* in this limit.


Therefore the (kinematic) model for Mi in Eq. (4.2) is a simple interpolating formula between small and large values of the quark mass parameters. The same applies for Mi.The case of Mi is more subtle. The probability for the eta to be in a Ss state is +. This contribution to Mi is therefore i(m, + m,)* = 8mi/3, and similarly for 2. This consideration does not take into account the mixing q--q’ 151, and therefore the expression for Mi so obtained is even less reliable than those for Mi and Mi. Lacking anything better, we shall nevertheless use in what follows Eq. (4.2) as an educated guess of the sign and order of magnitude of 0(9JI*) corrections, and we shall refer to these estimates as kinematic corrections. We will use similar O(YJI*) corrections for isospin breaking (Eq. (4.8)) and for O(YJI*) terms in baryon mass formulae (see Eqs. (6.5) and (8.3)). One finds

‘Ys,“‘” = 0.07, ‘q”‘3) ‘v 0.005, J-y‘” v 0.15.

These corrections tend to increase the values for G/m, obtained above:

film, = 0.042 (pion, kaon),

film, = 0.049 (pion, eta).

Next consider LNAC and their influence on A/m,. We write

Mt, = UK( 1 + XkNAC + XF’3)),

M5, = a,( 1 + X;NAc + ,7”‘J’),

M; = a,( 1 t X,LNAC + F0U’3’).

(4.3)

We find from Eq. (3.4) that IXkNACI < l.2~Ku’3’, IX,“““‘/ < 4p,“‘3’, IX\NACj < 1 “(‘) for /i < 1 GeV. Therefore these exactly calculable corrections to the mass 4-v values are of the same order of magnitude as SU(3) corrections, except for the pion, where both are small. Furthermore, the signs of LNAC are such as to decrease S/m,. See Fig. 3, where we have evaluated G/m, by calculating r?rC and m,C from either (Mi. Mi) or from (MS, Mi), also see Eqs. (3.4) and (4.2). Then g/m, = A,?T/(m,Z).

From this discussion we derive the following value of film, from the spectrum of the pseudoscalar octet:

film, = 0.042 f 0.008

or

m,/riz = 25 f 5. (4.4)

Finally, we comment on the slightly different value fi/ms = 0.03 1 obtained by Langacker and Pagels from a very similar calculation (31. These authors evaluate rA/m, from pion/kaon and they do not include kinematic corrections, se we should compare their result with line (2) in Fig. 3. The cutoff used in Ref. 131 is n = I GeV.

72 .I. GASSER

5.0

4.5

8 x 4.0

4 <E

3.5

3.0

-\‘\ ‘\:; 8

8 I I 500 1000

Cutoff A [MeV]

20

22

24

cE \

26 E”

28

30

32

34

FIG. 3. The ratio film, from Eq. (3.4). (l), (2): GC and m,C are obtained from the solution of the equations for Mi, M:. Then one has S/m5 = rfiZ/(m,Z). (3), (4): G/m, from M:, M:, same procedure as for Mk, Mi. (I) and (3) include the kinematic corrections in Eq. (4.2), (2) and (4) result from LNAC alone. (P,): g/m, determined as in (2), but all the logarithms are given the same value In 0.17/A*, see Ref. 131. (P,): A/m, from Eq. (2.18) in Ref. [3] with common logarithms In 0.17/n’.

from which we find film, = 0.034. If we give all the logarithms in Eq. (3.4) the same value In 0.17/A’ (this is what has been done in [3]) and calculate Ritz and m,Z from (Mi, Mz), we obtain S/m, = 0.030 (line (P,) in Fig. 3). Langacker and Pagels use instead in their Eq. (2.18) an explicit expression for A/m, which agrees with Eq. (3.4) up to terms O(!LR*) and which gives A/m, = 0.031 (see line (Pz) in Fig. 3. This line shows #z/m, calculated i la Langacker and Pagels with a common value In 0.17/A* for the logarithms).

4B. The Ratio md/mu

The inclusion of isospin breaking effects requires a new treatment of the renormalization problem. We do not intend to repeat here the relevant points [6,8 J. The upshot of the whole discussion is as follows: We can extract the ratio md/m, from the physical masses if we collect the electromagnetic correction to the mass shift in an educated guess for the contribution of the Born term and resonance region in the Cottingham formula.

We evaluate md/mu from Mi+ - M& and IV:, as follows. Thefirst order formulae are

A4: + = (6 + m,) 2 + j(m, - md) Z + (Jrni +)?

M& = (A + m,) C - f(m, - md) Z + (srniJy,

IV:+ = 2riiC + (Smi+)y, (4.5)


where ( )’ means contribution from the photon cloud. For the pion we take

(~Irnz)~= (~3rni+)~ - (6mf$)Y = 1.16 . 10e3 GeV*. (4.6)

For the kaons we consider three alternatives:

(4.7a)

which is Dashen’s sum rule 191. The other two possibilities are

(Ami)y= 2 . low3 GeV’. (4.7b)

This value corresponds to the Born term contribution in the Cottingham formula, and

(~Irni)~= 2.97 x IO-’ GeV’ (4.7c)

is Socolows estimate [ 101 used recently by Langacker and Pagels 13 I. The resulting values for md/mu obtained from Eqs. (4.5t(4.7) are shown in the

first column of Table I. (We have set, in addition, (GmI$)Y= 0). The next step consists in the extension of the (kinetic) estimates of O(9JI’) terms to

include isospin breaking effects:

M:+ = (m, + m,) C + (m, + ms)* + (am;-)‘:

Mt,,, = (md + m,) C + (m, + ms)* + (Sm’,,,)Y.

Ma+ = (m, + md) Z + (m, + mdJ2 + (drn3 ,)Y (48a)

Numerical values of O(!IB’) terms are calculated with md/mu - 1.8 and therefore 17 1

m, = 4 MeV, md = 7 MeV, m, = 130 MeV. (4.8b)

TABLE I

The Ratio m,,/m, (R = (& - m,)/(m, - wzd)) from the First Order Formula for

Mi,, M&and MS,

mJm,, (fi - m,)lW, - md)

(3tn;)Y a b a’ b’ -~- ~-~~_____

Dashen 1.8 1.6 44 48 Born term 2.0 1.8 38 40 Sokolow 2.2 2.0 33 34

Note. Column a (a’): md/mu (R) from Eq. (4.5) for three different values of (Am:): see Eq. (4.7). Column b (b’): m,/m, (R) from Eq. (4.5). but including kinematic corrections according to Eq. (4.8). (We use (d/n:)‘= 1.16 X 10.‘GeV* and (&:,)Y=t).)

74

We find from Eq. (4.8)

J. GASSER

A4;+ - A4:o = (m, - rnd) Z( 1 + 2m,/.z) + (dmt,)Y

= 1.2(??2, - rnd) c + (.4&)Y. (4.9)

These kinematic corrections tend to decrease the ratio md/mu, see column b of Table I.

Now include LNAC. From Eq. (3.3)

M;, - A4;0 = (m, - rnd) Z( 1 + x;:;: + Xp$) + (d&)Y,

where

P,“c’,‘! = 2nl,/z = 0.2

from Eq. (4.9) and

p;!T,“s ( < x;:c,“i for li < 1 GeV.

Next we calculate m,Z, m,C and m,X from Eqs. (Bl)-(B3) for Mi,, Mio and Mi,. Then md/mu = m,Z/(m,X). The result is shown in Fig. 4 as a function of the cutoff A. Obviously, LNAC tend to increase m4/m,. We conclude from this figure that

md/mu = 2.0 f 0.5. (4. IO)

Finally we comment on the value md/mu = 3.6 obtained by Langacker and Pagels

I I 0 500 1000

Cutoff A [MeV]

FIG. 4. The ratio md/mu from Eqs. (Blt(B3). We set (Gm$)Y= 0 and (6mi+)Y= 1.16 x lO-3 GeV’. Then we evaluate m,Z, m,C and m,Z from the equations for mi,, Mi, and Mi,, where mdm, = m,Z/(m,C). (1): (Am',)" = (drni)Y (Dashen sum rule). (2): (Amf#= 2 X 10-l GeV* (Born term). (3): (Am:)" = 2.97 x lo-’ GeV* (Socolow [lo]). (l)-(3) are calculated with the kinematic corrections in Eq. (4.8). (4): (Amk)y = 2 x 10e3 GeV’, without kinematic corrections. (P): md/mu from Eq. (2.18) in Ref. 13) with a common logarithm In 0.17/A2.


13 1 from a similar calculation. Their large value for md/mu is due to several reasons. Firstly they use exclusively Socolow’s estimate of (dmi)Y. Secondly the fact that they use a common value for the logarithm for all LNAC tends to increase md/rnu considerably, as is seen in Fig. 4. There we plot m,/m, using their formula (2.18). In fact the use of a common value for the logarithms in Eqs. (B 1 )-(B3) makes the whole expansion useless since the corrections become too large compared to the first order terms.

4c. The Ratio R

We may easily evaluate R from the quantities m,C, m,Z and m,C determined above. One finds

R= $2-m, .

= 40 f 10. mu -md

(4.11)

This value includes LNAC and kinematic corrections. See also Table I.

4D. Corrections to the Gell-Mann-Okubo Relation

Finally we comment on LNAC to the Gell-Mann-Okubo relation

dGM0 = ;(4Mt, -M;) -M; = 0 (exp. 0.02 GeV’).

LNAC contribute a small correction which has the correct sign,

0 < QGMO),,,, < 0.015 GeV’ M,<A< 1 GeV.

The kinematic corrections from Eq. (4.2) contribute with the opposite sign,

(dGMO),,i, = - +rn: z - 0.02 GeV2.

Therefore

-0.02 GeV’ < (AGMO),,, + (dGMO),,,,c < -0.005 GeV’. (4.12)

These corrections are small on the scale of Mi and Mt, and therefore we do not put any heavy weight on the wrong sign in Eq (4.12): Let us note that the missing 0.025 - 0.04 GeV* in Eq. (4.12) amounts to a correction of less than 10% in the first order formula for M’, and Mi.

5. A MODEL FOR THE CLOUD OF VIRTUAL MESONS AROUND MESONS

We have found in the previous section that LNAC are reasonably small for mesons. This will turn out not to be the case for baryons, and we must rely on a model for the cloud of virtual mesons around baryons. The same simple model can also be applied to the virtual mesons surrounding mesons. We want to show here that

76 J.GASSER

the results for the ratios G/m, and mdmu derived with this model agree with what we found above. The reader may skip this section and return to it later, after having met the difficulties with chiral corrections in the baryon spectrum.

The model for virtual mesons is based on the observation that LNAC are due to tadpole graphs calculated with a non-linear Lagrangian describing low-energy meson-meson scattering (I). In fact the contribution of the tadpole to the mass shift is given by

*M2 = KLA,(O; m’) ei

= 167r2Fz2 M2 iA’-m’ln ($+ I)/,

where the bare mass of the external (virtual) particle is M(m). Obviously (l/i) d,(O; m’) is a non-analytic function of m2, and the leading non-analytic contributions from these tadpole graphs are exactly those found in Section 3. Our model announced above consists in the replacement of the terms Am2 ln(m2//i2) in Eqs. (3.3) and (3.4) by J(m*) = (l/F,2)(l/i) d,(O; m’).

To be definite, the mass formula for kaons and pions reads in the case m, = md

M;=o.(l +;J(Mz,)],

h4; = a,{ 1 + f[J(@) - fJ(M;)]}. (5.1)

This result goes over into Eq. (3.4) for M,, M, + 0 (as far as LNAC are concerned). The first two terms in the expansion of J(m’),

J(m2) - (167r2~)-’ {A’ + m2 ln(m’//i’)), (5.2)

reproduce J(m’) rather well up to values m - 550 MeV for A - 1 GeV (discrepancy at m - 550 MeV < 15%). If we now evaluate HI/m, from Eq. (5. l), we find fi/ms - 0.036(0.037) for li = 770 MeV and h/m, - 0.034(0.034) for ,4 = 1 GeV. The values in brackets are those obtained from LNAC alone. The same game may be played in the case m, # md. We find again values for md/mu which are consistent with those found from LNAC.

In conclusion we see that the quark mass ratios evaluated in Section 4 can also be obtained from a model where LNAC are replaced by tadpole graphs with a cutoff of the order of 0.7-l GeV, inserting physical mass values for the virtual particles.

6. THE BARYON OCTET JP=ft

We evaluate in this section LNAC to the baryon octet. We wish to check whether the quark mass ratio R obtained from the K+ - K” mass difference (Eq. (4.11)) is consistent with the value obtained from the baryons.


First we consider the expansion of the current matrix elements (B 1 (qq)R 1 B). The results show that chiral perturbation theory is not applicable in this case. The same is true for the expansion of the baryon masses. We shall consider in the next section a model for higher order quark mass corrections which does not have this disease.

6A. Chid Expansion of Current Matrix Elements

Let

where M, is the nucleon mass in the chiral limit. g, 2: 1.26 is the axial vector coupling constant. Furthermore we shall use the notation

c,p = (PI @J)R lP>? 2, = o(Pl (fiu)a IP)o*

where 1~)~ denotes the proton state in the chiral limit, and Ip) is the proton eigenstate

of HQCDI !JII # 0. The analogous notation will be used for the matrix elements of the operators (dd)R and (Ss),. Then one has

Z;=sf,+JC 3M,+f(8a2-12a+6)M,+ I

(4a - 3)2 M 9 0

3-4a! + JZ(hf, + M,) --j---,

C,‘=&+JZ 3M,+2(2a-1)2M,+ 9 i

(4a - 3)2 M n

i

3 - 4a - J-Wf, + Mv’-,

+(20n2-24a+9)M,+ 4(4a - 3)’ M

9 (6.1)

a is related to the F/D ratio of the axial vector coupiing in the usual way (1 I),

a = & GAzOn~Gppzo I 0.62. (6.2)

We do not list the expansion for the other members of the octet for the following reason:

Consider the D-coupling of the currents (qq),,

(27uP+~gP-2~dP)=(~u+~.s-2~.d)(l +X,+X,+X,),

18

where

J. GASSER

x = 2 M%G

n TM;-M:,

J*-Aa.M,,

x =lMi,-M: K 3 M2-M2 J.-$@-3)‘+M,,

z A

x =-T_JK-M:, 7 3 M2-MM2 J.j(4a-3)(2a-3).M,.

z A (6.3)

Now

2 Mf,-M; 0.5 ~M;-M;~~M~

for M,, N 0.8 GeV. Inserting this into Eq. (6.3), we find that chiral perturbation is not applicable since LNAC are huge. The same is true for the matrix elements of the other members of the octet.

One expects the same thing to happen for the expansion of the baryon masses themselves, unless cancellations between large terms occur. In fact these cancellations do not occur, and we have to abandon the idea of improving first order results with the evaluation of LNAC. We want to stress, however, that this does not signal a breakdown of SU(3) symmetry, but shows the impotence of chiral perturbation in this case. (A breakdown of chiral perturbation theory was noted long ago in the context of meson electromagnetic mass shifts [ 121 and of baryon magnetic moments [ 131. See also the review article by Pagels, Ref. [2]).

6B. Chiral Expansion of Baryon Masses

We list all LNAC for the case m, = md.

(M;)LNAC = 3M:+f(ZOu’-24a+9)Mk+ (4a - 3)2 M3 3 I tl ’

(M;)“““” = C f(7a2-12a+6)M:+4(2a2-2a+I)Mi+$Mf, , I

(@JLNAc = 4a’M~+~(10a2- 18a+9)Mi+ TM; , 1 (M;)LNAC = 3( 1 - 2a)‘Mi + f (8a2 - 12a + 9) Mi + Pa - 3)* M3

3 v 1 ’

where

gf, MO C=--- 167r pr2’

(6.4)


These corrections are huge compared to first order terms and to O(!UI*) corrections which we take to be, in analogy with the kinematic estimates made for mesons (see Eq. (4.2) and comments thereafter)

MZ, = Iv,** + +I - m,)(C, + 2zd - 2‘q) + 9?Fl*,

M; = My + +<ti? - m,)(f, + ‘c’, - 2&) + (2ri + mJ2,

Mf, = Iv,** - f(fi - rn,@” + c, - 2&) + (2fi + rnsJ2,

M2, = kf,** + ;@-i - m,)(C, + gj - 22”) + (r?l + 2m,)?

A4z2 = lw; + 3(2rit + rn,)@” + icd + 2,). (6.5a)

Numerical values of O(W*) terms are calculated with 17 ]

A = 5.5 MeV, m, = 130 MeV. (6Sb)

Finally we note that the correction to the Cell-Mann-Okubo relation takes the particularly simple form

Mk + Mi - + (3Mfi + MC) = T (2a* - 6a + 3)(Ma - 4Mi + 3Mi)

which is a nice result, but useless in light of the above remark. We conclude this section with the observation that we cannot straightforwardly

apply chiral perturbation theory in this case. We have to work harder to get an estimate for the influence of higher order corrections to the quark mass ratio R.

7. A MODEL FOR THE CLOUD OF VIRTUAL MESONS AROUND BARYONS

All LNAC for the baryon octet can be obtained from the one-loop contribution to the mass shift induced by the phenomenological Lagrangian

pint(X) = Gp(x) Y”Y5 y(X) a,cO(X). (7.1)

In fact we find for the contribution from a single meson to the nucleon mass

(fimN)LNAC = -G2M3/87r

(M = meson mass). If we include the whole octet of mesons and baryons in the usual way [ 141 in the interaction in Eq. (7.1), one finds the result (6.4) for the LNAC, provided we introduce physical values for the meson masses in the loop integral. This shows again explicitly that LNAC are due to the cloud of soft mesons surrounding the baryons.

The model which we have in mind consists in the replacement of LNAC in Eq. (6.4) by the one-loop integrals evaluated with the Lagrangian in Eq. (7.1). including

80 J. GASSER

the whole baryon and meson octet. The cutoff which regularizes the divergent integral is provided by the (square of the) axial vector form factor which enters the expression for the propagator.

To be more specific, consider the modified expansion for the nucleon (m, = md). It reads

where

AJ,(M,,M,,M)=J,(M,,M,,M)-J,(M,,M,,O)

and

As before, M, is the nucleon mass in the chiral limit. This model reproduces the correct LNAC. We expect that we can obtain in this

way an estimate of the corrections induced by non-analytic quark mass terms. The reader should go back at this time to Section 5, where we analyzed an analogous model for the mesons and where we found that the quark mass ratios calculated with it agreed with those obtained from LNAC within the uncertainties which we had to tolerate anyway.

It is instructive to compare the exact value of Am, = ,dJ,,(M,, MN, M)/(8M~) with its chiral expansion up to and including the terms O(M3) (keeping M, fixed),

(7.3)

In Fig. 5 is shown the mass shift Am, as a function of the fictitious meson mass M, together with the first three terms of its chiral expansion as given in Eq. (7.3). The


-20

-40

T I3

i- 60 a

-80

-100

FIG. 5. Comparison between the exact value of Am, = AJ,,(M,, M,, M)/(8Mk) (see Eq. (7.2)) and its chiral expansion up to and including the terms O(M)). Solid line: Exact result. Dashed line: Chiral expansion according to Eq. (7.3). Cutoff’ A = 700 MeV.

disaster is obvious: For M > 150 MeV, chiral expansion (up to and including the terms 0(M3)) does not reproduce within reasonable erros the original function dm, .

Next we discuss the choice of the cutoff A. One natural choice is II w 1 GeV, which corresponds to the axial vector form factor fit given by Sehgal [ 151. This value of /1 is somewhat exotic, since then the mass shift of the nucleons due to (zero mass) pions is -200 MeV (-90 MeV for /i N 700 MeV). We will allow in our estimates a variation of /i in the range 0.5 GeV < ,4 < 1 GeV. The uncertainty in the choice of the cutoff is the main reason for the uncertainty in the value of (fi - m,)/(m, - m,).

8. QUARK MASS RATIO R FROM THE Jp = $ ‘OCTET

We discuss first the corrections to the Gell-Mann-Okubo formula implied by our model calculation. The ratio R is considered afterwards.

8A. Correclions to the Gell-Mann-Okubo Relation

The corrections to the Gell-Mann-Okubo relation,

dGM0 = M:, + M; - f(3M’n + M;) = 0 (exp 0.04 GeV’),

are obtained from the expansion in Eq. (7.2) (and its analog for M:, AI: and M:). One finds

-0.01 GeV2 < dGM0 < 0.03 GeV’

82

for

J. GASSER

0.4 Q a < 0.8; 0.5 GeV <A < 1 GeV.

(We allowed a 30% variation of a around its best value a = 0.62.) Hence the experimental value dGM0 N 0.04 GeV* is consistent with our model for cloud effects, provided we include O(!III’) corrections according to the kinematic estimates in Eq. (6.5) which increase dGM0 by ~0.03 GeV’.

8B. The Ratio R

As is explained in Section 4B, we take the Born terms as a measure for the photon contribution to isospin breaking. Renormalization point dependence of the quark masses are negligible and hence ignored in what follows [3, 6, 8, 271.

For the Born terms we use SU(3) symmetry for the electromagnetic form factors [16, 271 and take physial masses in the kinematic part of the Cottingham formula.

Then

(Mp - Z~YI,)~ = 0.7 f 0.3 MeV,

(M, + - MI -)’ = -0.2 + 0.1 MeV,

(M,,-MM,-)Y= -1.0 f 0.5 MeV,

(M,, + M,- - 2M,Jy= 1.73 f 0.3 MeV. (8.1)

The errors are an educated guess of resonance contributions, A, --f-trajectories and the like [16, 271.

The first order expansion reads

(8.2)

where ( )tad means that electromagnetic corrections are subtracted. We obtain for R’ the values listed in column a, Table II. Higher order corrections are again of two kinds: O(W*) and LNAC. As (kinematic) O(9JI”) corrections we use Eq. (6.5) and

(@ - Mh)tad = (mu - md)(fu - 2.d) + 3(m, - md)(mu + md)?

(Mi. - A4-)‘ad = (m, - m,)@, - 2,) + 4(m, - md)(mu + md + m,), (8.3a)i

(M$ - Mi-)tad = (m, - md)(Zd - 2,) + (m, - md)(m, + md + 4mJ.


TABLE II

The Ratio R = (fi - m,)/(m, - md) Calculated from

M2p-MZNrM:+-M:-andMZ,-M4,

(rit - m,)l(m,, - m,)

a b

M; ~ ML 79 71 M;, - M$- 46 47

M&M;- 40 42

Note. Column a: First order formula, Eq. (8.2). Column b: same calculation, but including kinematic corrections according to Eqs.

(6.5) and (8.3). Elecrtromagnetic corrections are listed in Eq. (8.1).

Numerical values of these 0(9X’) terms are again alculated with

m, = 4 MeV, md = 7 MeV, m, = 130 MeV, (8.3b)

see Eq. (4.8b). From Eqs. (8.2), (6.5) and (8.3) we obtain for R the values quoted in Table II,

column b. We see that O(m2) corrections act in the right direction. Consider now meson cloud effects. According to the model discussed in Section 7,

we evaluate all one-loop contributions to the self-mass of the baryons, including

““r-----l -7 - - - - -__

- - - - - - - , ‘ - . w

____ - .A‘ ; - . - -

/____.--~ T

““L 0.4 0.5 0.6 0.7 0.8

a

FIG. 6. The ratio R from (Mi - Mk) (solid lines), (Mi+ - Mj -) (dashed lines), (Mi o - Mi J (dashed dotted lines). Upper curves correspond to A = 500 MeV, lower ones to .4 = 700 MeV. R > 30 also for ,4 = 1 GeV. Corrections according to Eqs. (6.5) and (8.3) are included. a is related to the F/D

-atio of the axial vector current as usual, see Eq. (6.2). We allow for a 30% variation of a around its ?est value a = 0.62 in order to show that the result depends only weakly on this quantity.

84 J.GASSER

isospin breaking effects (and hence A-,?? mixing). The result of this analysis is shown in Fig. 6, where we plot R as a function of a for 2 different values of the cutoff/i. We see that meson cloud effects are small and do not invalidate the corrections expected from the estimates in Eq. (8.3). Furthermore it is clear from Fig. 6 that the value of R is rather insensitive to the precise choice of a.

In conclusion we find that the value of R found from the meson spectrum, R = 40 f 10, is perfectly consistent with the baryon spectrum.

9. LNAC TO THE PION-NUCLEON SIGMATERM

Next we discuss higher order corrections to the pion-nucleon sigma term [ 171

From Eq. (6.1) we find

1 u = - F&f’, + &)

nN 2A4,

g: x -- 64~ FE I 3M, + f (20a* - 24a + 9) M, + + (4a - 3)*M,

I - (9.1)

LNAC are w-35 MeV. This shows that chiral perturbation theory is also not applicable in this case. We may proceed, instead, similarly to the expansion of baryon masses in Section 7 and replace the O(W”*) corrections in Eq. (9.1) by the contribution from the one-loop graph. This is done as follows.

According to the Feynman-Hellmann theorem [4],

Therefore, from Eq. (7.2),

where Sm’, denotes the contribution to the nucleon mass from the one-loop graph. The derivatives &(a/%) can approximately be replaced by derivatives with respect to the meson masses,

This approximation amounts to keep only those graphs in fi(a/afi) ami, which lead


to the same (leading) non-analytic terms as the formal chiral expansion of &$a/%?) Smi. Finally, we obtain

(9.3)

6mg(dmi) denotes one-loop corrections to Mi(Mi). For a given value of 6/m, and M,, we may now evaluate u,, from Eq. (9.2). We

postpone the discussion of the value for M, to the last section and set here M, = 750 MeV. In Fig. 7 we have plotted the resulting values for rrzN with three different choices of the cutoff/i, including the somewhat exotic value /i = 1 GeV (see Section 7). Furthermore we have chosen 6/m, = 0.04. Obviously the result depends only weakly on a.

The crucial question: Is it conceivable to have [ II] onN = 60 MeV? is answered in the last section in light of the baryon and meson spectrum.

Let us mention that LNAC to u,, were also calculated by Pagels and Pardee I18 1. These authors calculate, however, LNAC to the quantity C,,(2Mi) defined as

G(P’ I (fiu)R + (ad), Ip) = ti(p’) Z,,(t) u( pk t = (p’ -p)T

Therefore their result has no direct relation to our Eq. (9.3) above.

(9.4)

~=700MeV -

A = 500 t&V

FIG. 7. The sigma term evaluated according to Eq. (9.3) for M, = 750 MeV, A/m, = 0.04. We calculated u,,~ for three different values of the ultraviolet cutoff in Eq. (7.2): A = 500 MeV, 700 MeV. I GeV. Obviously unN is rather insensitive to the value of the F/D ratio of the axial vector current. la is defined in Eq. (6.2).)

86 J.GASSER

10. MESON-BARYON CONSTRAINTS ON QUARK MASS RATIOS

In this section we evaluate bounds on the quark mass ratios by simultaneously fitting the meson and the baryon mass spectrum with the six free parameters m,[ A, (m, - &)I (m, - md), S, F, D and d, see below. This calculation serves also as a welcome check of the previous evaluation of mass ratios. As a byproduct, we shall obtain information about the hadronic wave functions (pi (qq)a Ip), as is discussed in detail in Section 11.

10A. Setting up the Fitting Procedure

We express the 13 baryon and meson masses through the six parameters

S = Mi + &(2’, + f’,) + rnsZd,

F = f@ - m&f’, - f,),

D = f(lii - m,)@‘, + f’, - 2fd),

d=ritC,

R = (fi - m,)/(m, - m&

r = m,/rit.

One finds for the:

Baryons.

SU(3)-splittings:

M;=S+F-D+AN,

M;=S+A,,

M:,=S-$D+A,,,

M;=S-F-D+AE.

SU(2)-splittings:

M&I4;=(1/R)(F+D)+A,-Ad.,

I@+-A4-=(2/R)F+A,+-A,-,

M5.++M~--22M5o=A,++A,_-24,,,

M&,-M;-=(l/R)(F--)+A,-,,-A:-.

(10.1)


Mesons.

SU(3)-splittings:

SU(2)-splittings:

Mk=(l +r)d+d,,

Mf,=2d+A,,

M:=+(l +2r)d+A,.

M:+-MZ,o=(l/R)(l-r)d+A.,-A,,,

M~+-h4~,=A~+-Ad,,. (10.2)

Ai denote the contributions from the photon cloud, from LNAC and from kinematic corrections. The electromagnetic piece is chosen according to Eq. (8.1) for the baryons. For the mesons, we set (MK+ - M,O)Y = 1.5 MeV and (M,+ - M,+)Y=

TABLE 111

Individual Contributions to the Corrections d, Defined in Eq. (10.2)

QED LNAC kinematic correction --. ~--~~--. ---__

M: (ckV2) -1.6 x 10 L 1.8 x 10 ? MZ hlff

1.3 x lomJ I x10 -2.2 x 10 ? 4.5 x lo- 1

M; 18 x 10.’ 2 x lo-?

M, -M, (MeV) 21 0

M, ~ M, 29 31 ML ~ M, 32 9

M,, - M,” 1.50 0.80 -0.82 M,. ~- ,%,I+ 4.30 0.04 -0.03

M, -- hl, 0.70 0.61 -0.05 M, + ~ hl, -0.20 0.24 -0.71

M,, - M, -1.0 -0.48 -0.60

ML. + M,. - 2M,, 1.73 4.03 0.0 I

iVole. LNAC for mesons are listed in Appendix B. LNAC for baryons correspond to the one-loop correction. see Section 7. (Cutoff A = 850 MeV both for mesons and baryons.) The contribution

(m, - ~2) C,‘(8R’M,) to M,, -M,, which is due to no-v mixing and which is free from In W singularities is also included in the LNAC piece of M,, - Mqo. The corresponding correction from Z”-,1 mixing to M,, + M,- - 2M,, is w-0.02 MeV and was neglected. Kinematic corrections are defined in Eqs. (4.2). (4.8). (6.5). (8.3) and were evaluated with m, = 4 MeV. vzd = 7 MeV. m* = 130 MeV. di was calculated with M, = 900 MeV.

88 J. GASSER

4.3 MeV. LNAC for mesons are listed in Appendix B. Nonanalytic chiral corrections for the baryons are replaced by the one-loop graph, as we discussed in detail in Section 7. The cutoff was chosen to be A = 850 MeV (both for baryons and mesons) and

a = t/3/4 Gnro,o/Gpp,o = 0.62.

Kinematic corrections were defined in Eqs. ((4,2), (4.8), (6.5), (8.3)). We list in Table III these different contributions to di individually.

Remarks. (i) We included in the LNAC-piece of d,,, the contribution -(m, - ti)Z/(4R*), h’ h * d t w ic IS ue o no-q mixing (and which does not contain In !UI singularities). Recall that this term is nearly cancelled by LNAC, as we showed in Section 3, Eq. (3.6). The analogous contribution to M,, + M,_ - 2iI4,, from Z”-/i mixing (free from fl singularities) is -D/(4R2Mz) - -0.02 MeV in leading order. It was neglected in our calculations.

(ii) We see from Eq. (10.2) that (rit - m,)/(m, - md) is the only variable which occurs in both baryon and meson splittings.

Next we define erros in order to set up the fitting procedure. Let

(10.3)

and use analogous definitions for

The parameter S appears only in Mi and is therefore fixed by S = il4: -A,. Hence er = 0 (whatever definition for ez we use). The two remaining SU(2)-splittings Mi+ - M$ and MS. + + Mg _ - 2M& do not contain terms which are of first order in (m, - md). Hence we put

M:+ - M2,o = A,+ -A,, + e,+,o(Mf,+ - M:o)

and

M;++M;--2M&

=A,++A,~-2A,,+e,+,-,,(M;++M;~-2M~,). (10.4)

The reason for the error definitions given above is the following. Consider the splitting M’p - M’, and define errors ebN, egN by

Mi-MM;:=(l/R)(J’+D)(I +~lpN)+Ap-AN

and

M; -M; = (I/R)(F + 0) + A, -A, + &FN(M; -ML).


Obviously

E GN PN = 1 +&&

&;N

= 1 -(A,-A,)(MZ,-M;)-l’

The error ~6~ (and hence E,, = ekN + O[ (~6~)’ ]) . is a measure of the contributions from higher order terms (in 93131) as compared to the first order term (l/R)(F + D). We used spN (and the analogous quantities eEN,..., sk+k(,) instead of &(E;~,..., c’kAKt,) for calculational reasons. We discard siN since sSN # F& also in lowest order in t.ip%.

TheJts which we present below were done by minimizing

( 10.5 )

We do not include E,+,~ and sZtz .y0 in A&. In order to disentangle the influence of electromagnetic, LNAC and kinematic corrections, we used three different choices for the corrections Ai (see also Table III):

Fit F,: Ai = (QED corrections only),

Fit F,: Ai = (QED + LNAC),

Fit F,: Ai = (QED + LNAC + kinematic corrections). (10.6)

IOB. Results

We list in Table IV the result of six different fits to the spectrum. In the lines a, p and y (which correspond to the fits F, , F, and F, respectively, see Eq. (10.6)), we exclude Mz - ML from the fit (hence N= 10 in Eq. (10.5)). The ratio Fp/Dp is defined as

(10.7)

see also Eq. (11.11). We see that the isospin splitting M’, - M’, plays a special role: Its error is much larger than all other errors in the two octets. We checked that this is also true if we replace the electromagnetic part of the baryon isospin splittings by the following prescription [20]: Up to a multiplicative constant C (which is the same for all baryons) it is taken to be the sum of quark-quark Coulomb interactions, e.g..

(MN)Y= [2. ($. -4)+$. 41. c.

Obviously LNAC and kinematic corrections act in the right direction; including both of them reduces the error in M’, -M’, by more than a factor of 2.

The lines a,, /I, and yr contain the fits obtained by excluding Mi (again N = 10 in Eq. (10.5)). The three lines correspond to the fits F,, F, and F,, respectively.

90 J. GASSER

Obviously y1 represents the best lit. But it is also seen quite clearly that the kinematic corrections to Mi (Eq. (4.2)) do not properly take into account higher order terms (in

W In Fig. 8 (Table V) we present fits to all 1 I states. Columns (lines) A, B and C

correspond to the fits F,, F,and F,. In addition, we made fits by minimizing the maximal value of the 11 errors,

One finds min(smax) = 18.5% (Fit F,),

min(smax) = 14.5% (Fit Fd,

min(smax) = 9.3% (Fit F,). (10.9)

Columns (lines) A,, B, and C, contain the fits obtained by minimizing AC in Eq. (10.5) under the constraints

E max = 19% (Fit F, , column A,),

& max = 15% (Fit F, , column B ,),

& max = 10% (Fit F,, column C ,). (10.10)

Obviously the mean square deviation does not change much if we include these additional constraints.

Column (line) A, finally corresponds to a lit F, with R as the only free variable. The five remaining parameters were fixed by the “natural choice”

S=Mi,

F = (M2, - M2,)/2,

D = 3(M; - Mf,)/4,

d = M;/2,

s 2Mf,-M5, m A- MS, ’

(10.11)

It is quite clear that this fit is inferior to the one presented in columns (lines) A and 4.

To summarize these results, we show in Fig. 9 the values of m,/& and (ti - m,)/ (m, - md) which correspond to the columns A, B and C in Fig. 8. In addition, we plotted contours in the (m,/rfi, (A - m,)/(m, - mJ} plane. Points within each contour result in nearly equally good fits, i.e., As < 12% (Fit F,, dashed dotted line), As < 10% (Fit F,, dashed line) and AE < 7.5% (Fit F,, solid line). For the benefit of

HADRON MASSES AND SIGMA COMMUTATOR

TABL

E V

Num

erica

l Re

sults

fo

r th

e Fi

ts

Show

n in

Fi

g.

8

SU(3

)

Erro

rs

(%)

SUP)

M

ean

squa

re

Quar

k m

ass

ratio

s an

d FP

/DP

Fit

K x

r] C-

A 2-

N C-

N C

K+-K

” I[+

--no

P-N

z+-z-

z”-

.-8-

C++

,,--2

,3

mu-

md

mu

---

4 Fi

rst

orde

r P

A 3

0 -3

-2

0 22

8

0 -1

I

-17

10

13

4 11

.7

25

40

1.86

2.

I

z

A,

3 0

-5

-18

19

7 0

3 I

-19

12

16

4 12

.0

25.6

42

.3

1.82

2.

2 EJ

A,

0

0 -6

0

0 -4

0

26

I -4

0 18

30

4

18.0

25

.9

56.3

1.

57

3.2

LNAC

B 2

1 -4

-1

3 15

5

0 3

6 -1

9 10

12

6

9.6

21.4

39

.2

2.02

2.

2 B,

00

-5

-1

2 14

5

0 5

6 -1

5 13

15

6

9.8

21.7

40

.6

1.98

2.

2

LNAC

+

kinem

atic

C 5

0 -8

-8

9

4 0

3 6

-13

7 5

6 6.

8 24

.3

40.2

1.

82

2.2

C,40

-9

-9

8 30

6

6 -1

0 9

8 6

1 24

.5

41.6

1.

79

2.2

Norc

. FP

/DP

= (zz

-

SF)/@

: +

Zr

- 2Z

:),

wher

e CL

, ZI

an

d Zi

de

note

m

atrix

el

emen

ts

of

(tiu)

a,

(ad)

, an

d (is

), be

twee

n ph

ysica

l pr

oton

sta

tes


mu= md#m,

-

1300 r-?

--

1200 1 ___ -- --

m, = md # m5

600.

-- ‘I _-- --

550 -

100 t t t tt tt A A, A, B B, c Cl

FIG. 8. Fit to I I meson and baryon masses, obtained by minimizing the mean square error As. see Eqs. (10.2t(lO.5). Mi+ - MiO and Mi+ + M:-- 2Mfo are excluded from the tit. Columns (A, B. C)

correspond to (F,, F,, F,), see Eq. (10.6). In columns A,(F,), B,(F,) and C,(F,) we present the tits which were obtained by minimizing Aa under the additional constraints listed in Eq. (10.10). Column A,

finally corresponds to a tit F, with R as the only variable. The five remaining parameters are taken from Eq. ( 10.1 I ). We list numerical values in Tables V and VI.

94 J. GASSER

30 ,.---.

,’ *\

(E I’

s; B ,-.

I’ /6 c

E : ,’ , “?

25 \‘J’ A,,’ ! , ‘-/,ooc

CI; ‘L.--H

.A’

FIG. 9. Each line encircles those points in the (mdrfi, (fi - m,)/(m, -M,,)) plane which result in equally good fits to the 11 meson and baryon masses (M:, - M& and MC + + MS - 2Mi 0 are excluded from the fit). Dashed-dotted line: Fit F, (Eq. (10.6)), LIE < 12% (Eqs. (10.2k(10.5)). Dashed line: Fit F,, AC < 10%. Solid line: Fit F,, de < 7.5%. The points A, B and C in the figure correspond to the fits shown in Fig. 8, columns A, B and C, respectively.

those who prefer to use the parameter md/mu instead of ($I - m,)/(m, - m,), we mapped these regions into the {m,/A, mdm,} plane, see Fig. 10. We conclude that the following values of quark mass ratios result in equally good tits to the meson and baryon spectrum:

9= 24.5 zt 3 (24.3),

liz - m, =4oi4

mu --%I (40.2),

md -= 1.8 f 0.2 (1.82). mu

(The values in brakets correspond to column C in Fig. 8, see line C in Table V.) Figures 11, 12 show those values of m,/rii, (fi - m,)/(m, - md) and md/mu for which

----. 30 - I* \ I

I’ : <E ‘;; E

25 -

2oc. md/m”

FIG. 10. The regions in Fig. 9 mapped into the (m,/rit, nq,/m, 1 plane.


35

I 30

tE ‘;;

! E

25 1 I I

20 A FIG. I I. The line encircles those points in the (m,/&. (A - m,)/(m, - m,)t plane for which we can

obtain a fit to the spectrum such that max(le,i, IE,~, l&K+KO /) < 10% and max{l&L,~.i~rx~.~g~~~.lEr~.

ICpl*:lr /Ez’l- 1. 1 Earn+ I} < 15%. In fact Ed = 0. see comment after Eq. (10.3).

one can obtain a fit to the spectrum (including LNAC and kinematic corrections) such that

max{lE,,l, IeENI, I~zNl,l~~l~ Ib.l~ lkztx I9 I&+ It < 15%. (10.13)

The errors in the individual mass splittings are quite generous, compare the best lit in column C of Fig. 8 and line C, Table V. Note that we did not include M”, in the fit. As a result the allowed regions for m,/fi and (ti - m,)/(m, - md) are enlarged. We conclude that the quark mass ratios must lie in the range

21< m,/S < 32,

34 < (fi - m,)/(m, - md) < 5 1,

1.6 < md/mu ,< 2.2 (10.14)

if a somehow reasonable fit to the meson and baryon spectrum should emerge.

Remark. Comparing Fig. 9 and Fig. II, we see that the region with “equally good fits of type Fz” (dashed line in Fig. 9) does not lie entirely within the region shown in Fig. 11. Since this might be due to an erroneous guess of higher order corrections (i.e., our “kinematic corrections”), we made the following check. Instead of adding kinematic corrections to the baryon mass splittings, we evaluated the fits with linear mass formulae for the baryons. This amounts to an independent guess of higher order corrections. The corresponding “region of equally good fits” is then moved totally within the region shown in Fig. 11. The best lit was obtained for m,/& = 24.5, (SI - m,)/( m, - md) = 41, in close agreement with the values obtained by including kinematic corrections.

595/136/l 7

96 J. GASSER

30 -

<E ‘;;

E 25y, 20

‘0, 1

1.6 1.8 2.0 2.2

md/mu

FIG. 12. The region in Fig. 11 mapped into the {m,/ti, m,/m,} plane.

TABLE VI

The Value of the Six Parameters in Eq. (10.1) as Determined by the Various Fits Described in the Text

s F D dx lo2 m, G-m - I Fit (GeV2) (GeV*) (GeV2) (GeV’) H-l mu-mm,

a 1.422 -0.4106 0.1356 0.910 25.3 41.1 aI 1.422 -0.3380 0.1585 0.911 25.6 40.8

i, 1.242 1.242 -0.3819 -0.3417 0.0987 0.1101 0.904 0.903 27.1 28.3 40.0 38.3

Y 1.222 -0.3524 0.0985 0.904 24.2 39.5 YI 1.222 -0.3342 0.1059 0.904 25.9 41.6

A 1.422 -0.3363 0.1612 0.911 25.0 40.0 A, 1.422 -0.3466 0.1587 0.906 25.6 42.3 A, 1.422 -0.4279 0.1341 0.911 25.9 56.3

B 1.242 -0.3376 0.1112 0.903 27.4 39.2 B, 1.242 -0.3380 0.1107 0.902 27.7 40.6

C 1.222 -0.3270 0.1067 0.903 24.3 40.2 C, 1.222 -0.3301 0.1069 0.907 24.5 41.6

Note. S was calculated with M, = 900 MeV. (Quark mass ratios are independent of S.)

Finally we list in Table VI the vlues of the six parameters in Eq. (10.1) as they were determined by the various fits described above.

HADRONMASSESAND SIGMACOMMUTATOR 97

11. MESON AND BARYON WAVE FUNCTIONS (~l(ijq)~ Ip)

The formalism developed so far gives u also information about the hadronic wave functions (pi (qq)R / p). They depend on the mass scale ,u introduced by renormalization and on the absolute value of, e.g., the strange quark mass m,@) as may be seen from

Mk,, = (m, + m,) o(Ko 1 (Ss), 1 K’), + higher order terms.

As an example, we chose m, = 130 MeV in the following for the evaluation of absolute values of the wave functions. (For rn: # 130 MeV, multiply the values cited by 130/m: (MeV)). The ratio of two wave functions is independent of ,B and m,(p). Consider first the absolute value of the six meson wave functions

{i$,,TC~,Z~} = (i/ ((UU)~, (dd),, (is),} Ii): i = K”, no. (11.1)

Ii) denotes an eigenstate of the full Hamiltonian HQcD including mass terms. LNAC to Cj were already evaluated in Section 3; kinematic corrections can be obtained from

(11.2)

(Of course we could also calculate LNAC from this equation and from the expressions for &I; as given in the Appendix.)

Now write

z;” = C( 1 + E;),

c;” = C( 1 + Ed”),

c;” = c . E;.

c;“=& Ef,

zy = Z( 1 + Et),

‘ry = Z( 1 + &,K). (11.3)

The quantities sj: include both LNAC and kinematic corrections. We set m, = md and obtain from Section 3 (cutoff n = 850 MeV)

(E:, Ed”+ E:),,,c = (-0.016, -0.0 16, 0.003),

(E,K, E,K, E;),,,,, = (0.009, -0.108, -0.134). (11.4)

Kinematic corrections give the contributions

(E:, E:, E&,N = (0.011, 0.011, o),

(E,K, $9 E:)KIN = (0, 0.149, 0.149). (11.5)

98 J. GASSER

Finally we evalmate the variation of .?Y by varying m,/A and m4/m, in the region which is shown in Fig. 12, demanding that the constraints listed in Eq. (10.13) are fulfilled. One finds, using Eqs. (11.4), (11.5),

(E,““, zf, c;“) = ~(0.995,0*995,0.003),

(go, go, Zf”) = Z(O.009, 1.041, 1.015),

1.61 GeV < 2 < 1.97 GeV.

(11.6)

Next consider the baryon wave functions

@, , &, Ci I= (iI {@u>~, @d), , (is), } Ii); i=P,Z+,A,E’. (11.7)

LNAC and kinematic corrections were calculated with the help of Eq. (11.2). We write

2Y;=#??“+Bup, g+=g+3~+ (11.8)

and define analogous errors S{ for the remaining wave functions. One finds (for mu = md>

<a 6 7 m.,,, = (1.93, 1.77,0.098) GeV,

(6: +, 6: +, 8; +)LNAC = (1.24,0.94,0.24) GeV,

(sy, go, 6f”),,,, = (0.45,0.22,0.58) GeV,

cc a wm*c = (0.8,0.8,0.52) GeV,

and

<s,‘, a:, K),,, = (0.06,0.03,0) GeV,

(di’, 6:+, 6f’),,, = (0.56,0,0.28) GeV,

(cyO, go, df”),,, = (0.53,0, 1.06) GeV,

@:‘, &j, mm = (0.28,0.28,0.28) GeV, (11.9)

With the help of these results, we may calculate the wave functions in Eq. (11.7) as a function of either M,, the common mass of the Jp = f ’ members in the chiral limit !JJI = 0, or of the ratio of wave functions. From Eq. (10. l),

ms2u = g-(S-M;)+&F+zLD, l--r 2+r

rnsZd = -&S-M3 - 2

-rD, l-r 2+r

msZs = $--(s-M:)--&F+-~D. l-r 2+r

(11.10)


As an example, we plot in Fig. 13 the proton wave functions and ratios thereof as a function of M,. The values of the six parameters in Eq. (10.1) were taken from line C, Table VI. In the same figure we also plot these quantities as a function of the ratio of wave functions by using the relations

m,C,P = & [ y(FP + 0’) - 4FP],

m,C,P = & [2(DP - FP) -y(FP + D’JL

m,.Zi = c [D’ - 3FP],

where

1 r

FP = + (A - m,)(c,’ - CSp),

Dp = i (fi - m,)(ZL + C,P - 2.X:). (11.11)

In Fig. 14 we plot error bands on Z:,Zi and CF. They were obtained as follows.

FIG. 13. Proton wave functions Z,P, Cz. ZF and ratios thereof. We use m, = 130 MeV (7) and evaluate the wave functions from the six parameters in Eq. (10.1). determined by the best fit which is

shown in Fig. 8. column C. The value of the parameters is listed in Table VI. LNAC and kinematic corrections are included according to Eq. (I 1.9).

100 J. GASSER

6

T Le.

"Li 4

600 800 1000

b '4, [MeVl

FIG. 14. Error bands for the proton wave functions E:. .Zi and ZF. We use ms = 130 MeV [ 7 1 and vary the six parameters in Eq. (10.1) such that the constraints in Eq. (10.13) ((Eq. 11.16)) are fulfilled in (a) ((b)). For (b) we use the error s2 defined in Eq. (11.14). LNAC and kinematic corrections are included according to Eq. (11.9).

Consider first Fig. 14a, where we plot everything as a function of the parameter 2Zt/(Z[ + C,‘). We varied m,/ti and md/mu in the region shown in Fig. 12, insisting that the constraints in Eq. (10.13) are fulfilled. This leads to the minimal/maximal values of the various quantities as shown in Fig. 14a.

Now consider Fig. 14b. Since in this case the wave functions also depend on S = Mi + fi@, + i’,) + ms.f,, (see Eq. (1 l.lO)), we must also allow for a variation in this parameter. What is the variation which we should allow for? We considered two possibilities. Note that S tixes the center of mass of the baryon octet,

(M~)~5(M~+M:+M~+M~)=S-dD+~(dN+dr+dn+dz)

= (1.15 GeV)‘. (11.12)

Since we have to choose Mi for the evaluation of L{, we might set

(M~)=M~+ [Ij2(.?Z,+2’,)+ms2’d-~D](1 +E~)

+&l,+d,+Ll*+&) (11.13)

and demand that ] er ] < 15%, a value which is chosen in correspondence to Eq. (10.13). As the second possibility, we set

(Mu) = S - :D + :(A, + A, + A,, + A,-) + E* . [0.44 GeV’] (11.14)


measuring in this way that the deviation of the center of mass in units of the SU(3) splittings

(Mu) - ML = ,0.44 GeV’.

(Mu) - Ati = -0.42 GeV*. (11.15)

We have checked that the following constraints on the value of the six parameters in Eq. (10.1) lead to nearly identical results for the error bands in the wave functions:

max(ls,l+ l&,1, IcKIKOIJ < 10%

compare the constraints in Eq. (10.13) which lead to the bounds on m,/A and md/mu as shown in Fig. 12. (The bands in Fig. 14b were calculated with error E,. Eq. (11.14)).

12. WHAT CAN THE VALUE OFTHE PION-NUCLEON SIGMA TERM POSSIBLY BE?

Discussion about the value of the pion-nucleon sigma term, (T,~, are invariantly intertwined with four problems:

(i) Theoretical estimate of its magnitude.

(ii) (Theoretical) connection between uXN and pion-nucleon scattering amplitudes.

(iii) Evaluation of these scattering amplitudes from the data and therefore. by (ii), determination of (T,~.

(iv) Discrepancy between (i) and (iii).

We first comment briefly on these points. Then we show how our previous results narrow the possible values of B,,.

Problem (i). Knowledge of the six parameters in Eq. (10.1) does not sufftce to fix the value of oRN. As an additional input one needs, e.g., M,, the nucleon mass in the chiral limit. In fact,

1 2sl u

“N=~2A+m, I y+pLp+A ms 2 m,

i +I’-F’) -hl:( 1

where

Sp = Mi + ti(C~ + C,‘) + m,C,P. (12.1)

102 J. GASSER

(The relation between (S’, FP, Dp) and (S, F, D) can be found from Eqs. ((lO.l), (11.7)-(11.9) and (11.11)). Often one prefers to use, instead, as an additional input the parameter y = 2zi/(,?Yi + JY,‘). Then one has

u zN=&-&(Dp-3Fp)&. (12.2)

Therefore we are faced with the crucial question: What is the value of MO (or y)? We mention two extreme attitudes: Assuming the validity of the Okubo-Zweig-Iizuka rule [2 I] for the matrix element (p] (Ss), 1 p) one has z,’ = 0. Since the Gell-Mann- Okubo mass formla works better for linear baryon masses, one is inclined to write, to lowest order in ?Ul,

u 7tN =---&(&+M,-2M,) s

and therefore (m,/rit = 26 from Eq. (4.1))

u nN N 25 MeV. (12.3)

On the other hand, we know that (O] (Ss), IO) - -(250 MeV)3, see Refs. [22,7]. This indicates [23] that Ci might be substantial, thus increasing the value of unN from its magnitude at y = 0.

Problem (ii). It can be shown that u,, is equal to a particular combination of on-shell pion-nucleon scattering amplitudes (at an unphysical point) plus correction terms. It has been argued that these corrections contain terms O(Mz) which are exactly calculable and which introduce corrections to unN of the order of 15 MeV, see Ref. [ 181. The existence of these non-analytic terms hinges on the fact that according to Ref. [18] the sigma term must be evaluated at (p’ -p)’ = 2Mi (p and p’ are nucleon momenta), see Eq. (9.4). This was recently contested by Dominguez and Langacker [ 191.

Problem (iii). The difficulty of extracting u,~ from the data is illustrated by the scattering of published values for u,, [ll, 17, 24, 251. It has even been argued recently [ 191 that the experimental value of the sigma term is uncertain by as much as lOO%, with values of 30-70 MeV giving equally acceptable fits to the data.

Problem (iv). The evaluation of unN from the data seems to merge in recent years to a value around 60 MeV [ 1 I]. This differes by more than a factor of 2 from the estimate in Eq. (12.3), hence problem (iv) is manifest. (There is the school which likes m,/rii - 5 and which seems to be happy with present experimental values of unN [26]. In our language m,/fi N 5 corresponds to (K 1 Ss 1 K) = O(W) as W -+ 0, and thus ML,,,, = O(!JJI’). We are concerned in this paper with the problem of a consistent picture in a world where ,?Y # 0; thus we do not discuss m$fi - 5 in what follows.)


Next we discuss the implications of our previous results on the value of unN. We do not intend to argue about possible values of M, or y. Instead we determined those values of unN which are compatible with the meson and baryon spectrum, leaving open the actual value of M, or y. Consider first Fig. 15. There we plot error bands (solid lines) to uzN which was calculated from Eq. (12.2). The parameters m$& D and F were varied under the constrains listed in Eq. (10.13). We also show in this figure those values of u,~ which correspond to the fits A, B and C shown in Fig. 8. We have, for y = 0,

U rrN = 26 MeV (Fig. 8, column A),

U nN = 32 MeV (Fig. 8, column B), (12.4)

U nN = 35 MeV (Fig. 8, column C).

We see that both LNAC and kinematic corrections increase the value of urrN. For y = 0 the total increase is

AuzN N 9 MeV.

This is consistent with Eq. (11.9) from which

AUXN - 11 MeV.

We also varied 111,/S, md/mu (and consequently the four remaining parameters in Eq.

60 t

T 50 - Lz

E 40- tJ

I- 30-

ii .!z 20 - m

FIG. 15. The values of onN which are compatible with the constraints listed in Eq. (10.13) lie between the two solid lines. LNAC and kinematic corrections are included according to Eq. (1 1.9). Dashed lines B and C: (J,~ calculated with the parameters found from the fits B and C in Fig. 8. see Table VI. The lower bound on uWN coincides with onN calculated from the tit A, Fig. 8.

104 J. GASSER

(10.1)) in the regions of “equally good tits” shown in Fig. 10 with the result (y = 0)

23.5 MeV < anN < 29 MeV (Fit F,, AE < 12%),

28 MeV ( cnN < 36 MeV (Fit F,, As < lo%),

31 MeV<u,,(40MeV (Fit F,, As < 7.5%). (12.5)

Next we discuss Fig. 16 where we plot error bands on cnN evaluated as a function of M,, according to Eq. (12.1). Upper and lower limits were obtained by varying the parameters in Eq. (10.1) under the constraints listed in Eq. (11.16). The solid (dashed) lines show the bounds obtained by the use of the error e2 (cl) defined in Eq. (11.14) (Eq. (11.13)). Obviously the value unN - 60 MeV is not compatible with the meson and baryon spectrum, unless

13. SUMMARY AND CONCLUSION

(1) We apply chiral perturbation theory (developed in I) to the calculation of LNAC to the masses of mesons (Jp = O-) and baryons (Jp = 4’).

(2) The complete LNAC for the meson octet are listed in Appendix B. In Eq. (3.3) we present the expansion for k$+, i& and Mi+ as it reads up to terms of

FIG. 16. The values of (I.~ which are compatible with the constraints listed in Eq. (11.16) lie between the two solid (dashed) lines. LNAC and kinematic corrections are included according to Eq. (11.9). Solid (dashed) lines: Error .s2 (E,) is used in Eq. (11.16), see Eq. (11.14) (Eq. (11.13)).


order O((m, - m,,)‘), and in Eq. (3.4) we have set m, = md. LNAC for the barpns can be found in Eq. (6.4) for m, = md.

(3) In addition to LNAC, we estimated O(m”) terms, e.g..

kf;+ = (m, + m,) c + (m, + m$

see Eqs. (4.2), (4.8) for mesons and Eqs. (6.5), (8.3) for baryons. It turns out that these kinematic corrections to the mass shifts are of the same order of magnitude as LNAC if we rely on the Leutwyler quark mass values rfr = 5.5 MeV, m, = 130 MeV [ 7 ] and choose md/mu Y 1.8.

(4) We did not consider chiral perturbations for the electromagnetic piece of the mass shift, since chiral perturbation theory breaks down in this case, as was noted long ago by Langacker and Pagels [ 121, see also the review article by Pagels ]2 1. (These authors considered the mesonic case. We expect that a similar phenomenon occurs for baryons, see also point (6) below). Isospin breaking effects from QED were, instead [ 3, 8, 271, taken into account with an educated guess of the contribution from the Born term and from the resonance region, see Eqs. (4.6), (4.7) and (8.1).

(5) From the mesons alone we found the following values of the quark mass ratios:

m,/A = 25 + 5,

mJm, = 2.0 + 0.5.

R= ri -

112, =40 f 10. (13.1) mu - md

The bulk part of the uncertainty in m,/fi is due to the unknown value of the cutoff n in Eq. (3.4). The errors in md/mu and (G’t - mJ(m, - md) are mainly due to the uncertaintes in the electromagnetic contribution to the K’ - K” mass difference, see Fig. 4.

(6) LNAC are not reliable for baryons, see Fig. 5. Therefore we base the estimate for R on the one-loop correction to the mass shift calculated with the phenomenological Lagrangian (7.1). This correction reproduces the LNAC found in Eq. (6.4) and tames the disastrous result from chiral perturbation theory. (This calculation is similar in spirit to the one done for electromagnetic mass shifts, where meaningless chiral perturbation theory [ 121 is replaced 13, 8, 271 by Born term and resonance contributions retaining physical mass values in the Cottingham formula.) We found that the value of R obtained from the meson spectrum is perfectly consistent with baryon isospin splittings, see Eq. (13.1) and Fig. 6.

(7) In Section 10 we made various fits to the baryon and meson spectrum. We

106 J. GASSER

found that the following values of the quark mass ratios lead to equally good tits:

m,/rii = 24.5 f 3,

md/mu = 1.8 l 0.2,

$--mm, =40*4.

mu -%

In addition, these ratios must lie in the range

21< m,/& < 32,

1.6 < md/m,, < 2.2,

34< lit-ms <51, mu-m.3

in order that a reasonable tit to the meson and baryon spectrum is possible. A series of figures (Figs. 8-12) illustrates these results.

(8) The bounds in Eq. (13.2) are compatible with the values of quark mass ratios found earlier in Refs. [3, 27, 281. We wish to comment at this time on the value R = 85 f 15 found in Ref. [27]. As we emphasized in that publication (see especially summary and conclusion, point (x)), the above value of R was based on afirst order calculation. Furthermore only the proton-neutron mass difference was used as the input. In fact we can easily repeat this calculation, including LNAC and kinematic correction: Set (M, - MJ = 0.7 MeV and

This expression was also used in Ref. [27]. Then we find

R-855 (First order),

R ~65 (LNAC alone),

R N 55 (LNAC + kinematic corrections).

We obtain an independent guess of the kinematic corrections if we use linear mass formulae instead. Then

R=45 (LNAC, linear mass formulae).

We conclude that in this case second order symmetry breaking corrections are substantial, a result which was aniticipated in Ref. [27]. In view of this, the result, R = 85 f 15, found in this reference does not contradict the bounds in Eq. (13.2).

(9) Our bounds are inconsistenf with those estimates which obtain m, = 0 [29].


(10) The formalism developed in this article gives also some information about the hadronic wave functions (p] (qq), / p), as is illustrated in Figs. 13, 14 and discussed in Section 11.

(11) Finally we comment on the pion-nucleon sigma term

(3 -TN = &- <c:: + c:, = & (PI (fiu)R + (ad), ip), N

which is related to the chiral expansion of the nucleon mass in the usual way. We found that LNAC and kinematic corrections increase unN by -10 MeV (at .Zi = 0). Furthermore we calculated error bands on cnN, see Figs. 15 and 16. It is seen that

- 60 MeV [ 111 is not compatible with the meson and baryon spectrum, unless >y< 600 MeV (M,, is the nucleon mass in the chiral limit) or 2Cr/(Z,’ + Z,‘) > 0.3.

In conclusion we find that chiral perturbation theory leads to a consistent picture as far as the meson and baryon spectrum is concerned. However, it remains to be seen whether the values of M, (or of 2Cr/(C,P + CT;)) needed for (T,, - 60 MeV can be understood in the framework of QCD.

APPENDIX A: LNAC FOR (plj(0) jp)

In this appendix we derive the result Eq. (3.2). The dispersion relation for the retarded amplitude Rh(P, q) (Eq. (3.1)) reads

Vi (q, P) = ; 1 d4 -&74 (~~1 ]j(x/2), 8,(-x/2)] IPJ. (Al)

Convergence of the dispersion relation is guaranteed by the asymptotic freedom property of QCD, see [I]. Completely analogous to the case where j(x) = b,(x) (considered in [I]) one proves that only the intermediate states which are shown in Fig. 17 contribute to the LNAC. From this we conclude that states which contain r’ need not be considered. This solves the problem with q--7’ mixing.

First we discuss the contributions from the direct (D) and Z graphs in Fig. 17. Let

Jdf) = (~wMo)l M(P)),

HMO) = @WI b,(O) IWP)), t = (p’ - py.

]M(p)) is a one-meson eigenstate (mass r(lJ of the total Hamiltonian HQ,,. Then

aJ,K’> am fl+z

= $T- [J,(O) 443 - W J,d+&) f-C!X+&)l I. M

108 J. GASSER

El E2

FIG. 17. Intermediate states in the expansion of Vi(q, P) (Eq. (Al); q” > 0) which contribute to the LNAC of ‘$1 (qq)R Ip). Solid line: Meson under consideration. Dashed line: Intermediate meson states. Wavy line: b,(O). Duble line: (qq),.

Next we write a once-subtracted dispersion relation for the form factors JM(r) and H&

ImJ,(s)=-+C(M(p’),G((p)in((S- l)lain)(OIbr(0)lai~)* a

and analogous for H&). Only two-particle intermediate states contribute to LNAC (see [I]). To proceed further, we need the low-energy theorem for elastic meson-meson scattering. The expansion (in s, t) for the scattering amplitudes in question reads

(m(p,), l(p,) in [(S - l)] i(p,), k(p,) in) = $ (Sfi’rfkmr + ,fik’f’“‘);

s = (PI + P*)*, f = (PI -P312.

The indices i, k, 1 and m denote octet components (e.g., ] rr+) = -(l/\/z)(I 1) + i I2))), f ik’ are the SU(3) structure constants. Inserting this expression into the dispersion relation for J,(s) and H,(s) leads to the desired result.

Next consider the contributions from the graphs E, and E, (Fig. 17). A subtlety occurs in the evaluation of M(p,,p,,p;p’) = (M,(P,), M,(p,), M(p)1 8,(O) IWP’)), in contrast to the case of perturbation theory around SU(2) x SU(2) considered in [I]. Due to the one particle poles floating around in this matrix element, we must be careful about the order in which we set the four momenta equal to zero, since


in general. We state without proof that the target particles (i.e., M(p), M(p’)) must be made soft first. (This guarantees by the way that the pion mass vanishes in the SU(2) x SU(2) symmetry limit, as can be seen from the mass formula in Eq. (3.4)). The rest of the calculation is straightforward.

APPENDIX B: LNAC FOR MESON MASSES

This appendix contains the complete LNAC to the (Jp = O-) meson mass spectrum. We mentioned already in the text that no chiral expansion of the photonic part of the mass shifts is done, since the chiral expansion breaks down in this case (Langacker and Pagels, Ref. [12].)

Let (&+z~)~ be the phoronic contribution to the meson M, including quark mass effects. From Eqs. (2.1)-(2.5) and Fig. 1 one finds the following expansions (valid up to terms which vanish faster than 9JI* In 9.B as far as the non photonic corrections are concerned):

Kaons.

Mi+=h,++h,++ sinaCSP”‘h,,ln$ I

- cos a SCM”* h, In s !

+ (&n2, +)y,

Mio = h,, + hKo $ I -sin a CSM”2h,Jn 2

+ cos a SCP”’ h, In -$- I

+ (Gmi,)?

Pions.

Mi+ = h,+ + h,+ $ I CSP”’ CSM”’ h,, In $

+ SCP”* SCMV2 h, In -$ 1

+ (Gmi,)Y

M$,== h.,+A{h.+D,,,++h,,D,o,,} I,-$

+ A I hg+DnoR+ + + hqoDnoTo

I

031)

tB2)

110

Eta.

J. GASSER

We made use of the following notations:

SCP = (fi sin 0 + cos 0)‘; SCM = (fisin 0 - cos O)2;

CSP = (ficos 0 + sin O)*; CSM = (ficos 0 - sin O)2;

.

hi = m iii (iI 8,(O) 1 i); HQcD Ii) = ~@Tm;

2Y= lii(“‘I (iiu)R I7r’)

D nOK+ = fhno SCP - f{m,Z CSP + 4m,C sin’ a},

D noKo = ihno SCM - {{m,Z CSM + 4m,Z sin2 a},

D +lnO = - i{m,Z CSP2 + m,Z CSM* + 16m,Z sin4 a},

D nOn+ = 2h,o~~~2 a - ${m,Z CSP + m,Z CSM},

D 1% = - ${m,Z CSP . SCM + m,C CSM . SCP

+ 16m,Z sin2 a cos2 a},

D tlK+ = )h, CSM - j{m,Z SCM + 4m,Z cos’ a},

D ,,KO = fh, CSP - f{m,Z SCP + 4m,X cos2 a},

D ?lnO = - j(m,Z SCM . CSP + m,Z SCP . CSM

+ 16m,Z sin2 a cos’ a},

D ,,,+=2h,,sin*a -~{m,ZSCM+m,CSCP},

Dm = - $(m,Z SCM’ + m,Z SCP* + 16m,C cos4 a},

A= 1

1 67c2Fz2 ; F”, = 93 MeV: Pion decay constant in the chiral limit.

(B3)


ACKNOWLEDGMENTS

I thank H. Leutwyler, P. Minkowski and P. Schwab for informative discussions.

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Hadron masses and the sigma commutator in light of chiral perturbation theory