- Email: [email protected]
Baryon masses and pion-nucleon σ-term to second order in the quark masses
4 January 1996 PHYSICS
LETTERS
B
Physics Letters B 365 (1996) 285-291
EISVIER
Baryon masses and pion-nucleon a-term to second order in the _j, quark masses ‘. B. Borasoy Institutfir
Theoretische
Kernphysik.
‘,
Universittit
Ulf-G. MeiBner 2 Bonn, NuJallee
14-16,
D-531 15 Bonn, Germany
Received 28 August 1995; revised manuscript received 25 September 1995 Editor: R. Gatto
Abstract We analyze the octet baryon masses and the pion-nucleon c-term in the framework of heavy baryon chiral perturbation theory. In contrast to previous investigations, we include all terms up-to-and-including quadratic order in the light quark masses. The pertinent low-energy constants are fixed from resonance exchange. This leaves as the only free parameter the baryon mass in the chiral limit, A. We find A= 749 f 125 MeV together with a,~(0) = 48 f 10 MeV. We discuss various implications
of these results.
eral, the quark mass expansion of the octet baryon masses takes the form
1. Introduction
The analysis of the octet baryon masses in the framework of chiral perturbation theory already has a long history, see e.g. [ l-l 21. In this paper, we present the results of a first calculation including allterms of order O( mi), where m, is a generic symbol for any one of the light quark masses mu,&. We work in the isospin limit m, = md and neglect the electromagnetic corrections. Previous investigations only considered mostly the so-called computable corrections of order rnior included some of the finite terms at this order. This, however, contradicts the spirit of chiral perturbation theory (CHPT) in that all terms at a given order have to be retained, see e.g. [ 13-l 51. In gen-
m-ii+ c 4
B,m,+~Cqm~'2+~D,m~+... Y
v
(1) modulo logs. Here, i is the mass in the chiral limit of vanishing quark masses and the coefficients B,,C,,D, arestate-dependent. Furthermore, they include contributions proprotional to some low-energy constants which appear beyond leading order in the effective Lagrangian. In this letter, we evaluate these coefficients for the octet baryons N, A, Z and E. In addition, we also calculate the pion-nucleon o-term which is intimately related to the quark mass expansion of the baryon masses [ 3,7,16]. For some comprehensive reviews, see e.g. [ 17-201.
*Work supported in part by the Deutsche Forschungsgemeinschaft. ’ E-mail: [email protected]. * E-mail: [email protected]. 0370-2693/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01286-9
286
B. Borasoy, U.-G. Meipner/Physics
in via loops. There are seven terms contributing at dimension four,
2. Effective Lagrangian To perform the calculations, we make use of the effective meson-baryon Lagrangian. Our notation is identical to the one used in [ 71 and we discuss here only the new terms. Denoting by 4 the pseudoscalar Goldstone fields (v, K, 77) and by B the baryon octet, the effective Lagrangian takes the form
where the chiral dimension (i) counts the number of derivatives and/or meson mass insertions. The baryons are treated in the extreme non-relativistic limit [21,22], i.e. they are characterized by a fourvelocity ucL.In this approach, there is a one-to-one correspondence between the expansion in small momenta and quark masses and the expansion in Goldstone boson loops, i.e. a consistent power counting scheme emerges. The form of the lowest order mesonbaryon Lagrangian is standard, see e.g. [ 71, and the meson Lagrangian is given in [ 151. The dimension two meson-baryon Lagrangian can be written as (we only enumerate the terms which contribute)
(3) i=l
with the 012’ monomials in the fields of chiral dimension two. Typical forms are Tr( B [ uP , [up, B] ] ), Tr(B[u . 1.4,[u. u,BJ]) or BBTr(u,u”), with ucL= iutdpUut, u = I/?? and U = exp(i4/Fp) collects the pseudoscalars. The form of f$~cPnd) is [ 71, $ttand)
= bDTr(B{X+,B})
+ bo Tr(BB) Tr(x+) ,
Letters B 365 (1996) 285-291
where typical forms of the 0i4) are BBTr( x:) or Tr(B[x+, [,y+,B]]). At this stage, we take m, = md # m,y. The explicit expressions for the various terms in Eqs. (3), (5) can be found in [ 241. We point out that there are 20 a priori unkown constants (to be precise, only some linear combinations of these are relevant for the calculation discussed here). In addition, there are the F and D coupling constants (subject to the constraint F + D = gA = 1.25) from the lowest order Lagrangian Coda).What we have to calculate are all one-loop graphs with insertions from CFi2’ and tree graphs from L(p2d4’.We stress that we do not include the spin-312 decuplet in the effective field theory [ 61, but rather use these fields to estimate the pertinent low-energy constants (resonance saturation principle). We therefore strictly expand in small quark masses and external momenta (collectively denoted by ‘q ‘) with no recourse to large NCarguments.
3. Baryon masses and pion-nucleon
a-term
The form of the terms N my and N m:‘2 for the baryon masses and c?rN(0) is standard, we use here the same notation as Ref. [ 71. The q4 contribution to any octet baryon mass mg takes the form
+ bFTr(B[X+,B])
(6) (4)
i.e. it contains three low-energy constants and x+ = utxut + u,ytu is proportional to the quark mass matrix M = diag(m,, md, m,) since x = 2BM. Here, B = -(Ol~~lO>lF;z and Fp is the pseudoscalar decay constant. All low-energy constants in ‘CL: are finite. A subset of the bi has been estimated in [ 231 by analyzing kaon-nucleon scattering data. The splitting of the dimension two meson-baryon Lagrangian in Fq. (3) is motivated by the fact that while the first three terms appear in tree and loop graphs, the latter ten only come
with P, Q = V, K, v and A the scale of dimensional regularization. The explicit form of the state-dependent prefactors Ei,Bcan be found in Ref. [ 241. Notice the appearance of mixed terms N MS Mi which were not considered in most of the existing investigations. The fourth order contribution to the baryon masses contains divergences proportional to (using dimensional regularization) L _ -_ /P4
__
1
16~2 { d-4
- i[ln(47r)
+ 1 -yEI},
(7)
B. Borasoy, U.-G. Meipner/Physics
287
Letters B 36s (1996) 285-291
with yE - 0.5772215. These are renormalized by an appropriate choice of the low-energy constants di, d/-d,‘(h)
+IiL.
/
/
(8)
In fact, all seven di are divergent. The appearance of these divergences is in marked contrast with the q3 calculation which is completely finite (in the heavy fermion approach). In what follows, we set A = 1 GeV. A part of the M; corrections considered here was already discussed by Lebed and Luty [ 81. For a direct comparison to their work, we refer to [ 241. Similarly, the fourth order contribution to the pion-nucleon gterm can be written as
with the ei,c given in [ 241. Here, the renormalization is somewhat more tricky. It can most efficiently be performed in a basis of a linearly independent subset of the operators dOj4)/dm,, q = (u, d, s), as detailed in Ref. [ 241. A good check on the rather lengthy expressions for the nucleon mass and a,~(0) is given by the Feynman-Hellmann theorem, riz(&z~/&?r) = U,N( 0), with iit the average light quark mass, riz = (m, + md) /2. We remark here that in contrast to the order q3 calculation, the shift to the Cheng-Dashen point, grrN(2Mi) - a,N(O), is no longer finite, i.e. there appear r-dependent divergences. We therefore do not consider this a-term shift in what follows. We will also not discuss in detail the two kaon-nucleon a-terms, c&!;*)(t) , for similar reasons in this letter.
Clearly, we are not able to fix all the low-energy (2,4) from data, even if we constants appearing in L#, would resort to large N, arguments. We will therefore use the principle of resonance saturation to estimate these constants. This works very accurately in the meson sector [ 25-271. In the baryon case, one has to account for excitations of meson (R) and baryon (N* ) resonances. One writes down the effective Lagrangian with these resonances chirally coupled to the Goldstones and the baryon octet, calculates the
\
/’
\
\
\
/
-i
/
(b)
(4
Fig. 1. Resonance saturation for the masses. In (a), a baryon resonance (from the decuplet) is excited, whereas (b) represents the r-channel meson (scalar or vector) excitation. Solid and dashed lines denote the octet baryons and Goldstones, respectively.
Feynman diagrams pertinent to the process under consideration and, finally, lets the resonance masses go to infinity (with fixed ratios of coupling constants to masses). This generates higher order terms in the effective meson-baryon Lagrangian with coefficients expressed in terms of a few known resonance parameters. Symbolically, we can write
t,dUB,R,N*l
-)&du,Bl.
(10)
Here, there are two relevant contributions. One comes from the excitation of the spin-3 /2 decuplet states and the second from t-channel scalar and vector meson excitations, cf. Fig. 1. It is important to stress that for the resonance contribution to the baryon masses, one has to involve Goldstone boson loops. This is different from the normal situation like e.g. in form factors or scattering processes. Consider first the decuplet contribution. We treat these field relativistically and only at the last stage let the mass become very large. The pertinent interaction Lagrangian between the spin-3/2 fields (denoted by A), the baryon octet and the Goldstones reads (suppressing SU( 3) indices)
c 4. Resonance saturation
C--_
/-\
c - -A”TO,,(Z) *” - fiFp
BaP++h.c.,
(11)
with T the standard 4 -+ 5 isospin transition operator, Fp = 100 MeV the (average) pseudeoscalar decay constant and C = 1.5 determined from the decays A -+ Bv. The Dirac matrix operator O,,(Z) is given by @P,(Z) =&Lu -(z+;)Y,Yv.
(12)
For the off-shell parameter Z, we use Z = -0.3 from the determination of the A contribution to the TN scattering volume ~33 [ 281. The tadpole graphs shown in Fig. lb with an intermediate vector meson only contribute at order $. This is evident in the conventional
288
B. Borasoy, U.-G. Meipner/Physics
vector field formulation. In the tensor field formulation, the vertex Tr( Vpvu~uy) seems to lead to a contribution at order q4. However, in a tadpole graph one needs the index contraction ,u = Y and thus no terms of order q4 can be obtained. Matters are different for the scalars. Denoting by S and Si the scalar octet and singlet with Ms N MS, N 1 GeV, respectively, the lowest order coupling to the baryon octet reads ,C$’ = DsTr(B{S,B})
+ F..Tr(B[S,B])
+ Ds,SITr(BB)
(13)
where the coupling constants Ds, FS and Ds, have chiral dimension zero. 3 In fact, there are no empirical data to really pin them down. From the decay pattern of the low-lying baryons one can, however, estimate these numbers to be small. We will generously vary them between zero and one. For the couplings of the scalars to the Goldstones, we use the notation of [ 251 and the parameters determined therein. Putting pieces together, all low-energy constants are expressed via resonance parameters and the baryon masses take the form
Letters B 365 (1996) 285-291
graphs with insertions proportional to the low-energy constants be,n,~ and bi, respectively. Note, however, that in the resonance exchange approximation not all of the ten bi are contributing. The 6 terms subsume the contributions from ,!Zc, these are proportional to the low-energy constants d;. Finally, the terms of the type 0; are the scalar meson contributions to the mass. They amount to a constant, state-dependent shift and consist of terms of the types N M& NM:lnMiand N M;M$ In M$, see [ 241. To be specific, we give the coefficients Dp.‘*’ and A for the nucleon (the constants di of Eq. (8) are expressed in terms of resonance parameters), 1 -$(D 8,rr2F*P 1
AN = -
-
D; = Df’*’
-3F)*M~ln$
+ Df4’
c* 1tin-* F; {
+pD;+cYD;+cD;+D;,
(-4r*F;)(M:,
+ 4M;)
M*
M2
+~M$ln-$+~M~ln$+$M~ln-$ + -;M; [
E=~(222+z-l). A
(14)
with mf’ the contribution of o(m:‘*) and mA = 1.455 GeV is the average decuplet mass. Notice that it is convenient to lump the c3( my) and the 0( rni) corrections togther as it was done in Eq. ( 14) (in the constants DE and 0;). We have kept explict the baryon mass in the chiral limit. Of course, in the fourth order terms it could be substituted by the corresponding physical values. At second order, however, we would get a state-dependent shift, see e.g. [ 291, and we thus prefer to work with & Alternative representations for the baryon masses are given in [ 241. Let us briefly explain the origin of the various terms in Eq. ( 14). The A contributions are tadpoles with 1/m insertions from C’*’ 68’ Similarly, the p and E terms stem from tadpole ’ In what follows, we neglect the singlet field.
z
-k(D2-2DF+3F2)M:ln~},
=-
rnB =& +mi3’ + Ag (ii) -’
&(D
+ F)*M:ln
M2
+ z4M;
+i(D-3F)*(T+Mi)]M:ln$ + [$D
+ F)*(f$-
+ Mi)]Miln$
+ -%(D - F)2~n:, [
+&(223D*-318DF+351F2)M:]M~ln~},
The corresponding coefficients for the A, Z and z and also the Di can be found in [ 241. The tree con-
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B. Bomoy, U.-G. Meipner/ Physics Letters B 365 (19%) 285-291
Table 1 Numerical values of the state-dependent coefficients in E$. ( 14). The 0: are dimensionless. For the scalar coupliugs, we use the results of 1251 and MS - 1 GeV. B
AB
N
0.0049
A z E
0.0179 0.0144
t~v*l
0;
4
-39.875 -68.044 -132.90
-126.46
0.0216
0;
[GeV*]
0;
[GeV*]
-0.017 -0.047 +0.051 -0.037
0.0321 0.0617 0.1422 0.1324
-2.3617 1.0831 -11.273 -5.4799
[GeV] Ds Ds Ds Ds
+ + -
0.061 Fs O.OIOFs O.OlOFs
0.082 Fs
Table 2 Some low-energy
(*) in GeV-' . In the first row, only the decuplet contribution is given. In the second row, scalar meson constants from L1, exchange is added.
bl
b2
Reso. ( DS - Fs - 0)
0.084
-0.144
Reso. (Ds-Fs-1) Ref. [23]
0.100 0.044
-0.111 -0.145
tribution
from L$S’and) is subsumed
hR
b3 0.108 0.125 -0.054
in the D$. The
numerical values of the AB, D$, Di, 05 and 0; are given in Table 1 (using F = 0.5, D = 0.75 and qrn from [ 25 I) . We see that the dominant terms at
U( nz’,) are indeed the tadpole graphs with an insertion from f$iscand) (this holds for the masses but not for fl&O)). It is ak0 instructive to compare the values we find from resonance exchange with the ones previously determined from KN scattering data. We have transformed the results of Ref. [ 231 into our notation. As can be seen from Table 2, most (but not all) coefficients agree in sign and magnitude. We remark that the procedure used in [ 23 I involves the summation of arbitrary high orders via a Lippmann-Schwinger equation and is thus afflicted with some uncertainty not controled within CHPT.
5. Results and discussion The only free parameter in the formula for the baryon masses, Eq. ( 14), is the mass of the baryons in the chiral limit, &, since all low-energy constants are fixed in terms of resonance parameters. In particular, in contrast to Ref. [7], the parameters 60, bD and by are no longer free. Also, at quadratic order in the quark masses the ambiguity between & and ba is resolved, it is not necessary to involve any one of
ho
bF
t-U
-0.216
-0.738
-0.264
0.317
-0.239 -0.165
-0.733 -0.493
-0.208 -0.213
0.345 0.066
the a-terms in the fitting procedure [5,7]. In fact, one can not find one single value of h to fit all four octet masses, mN = 0.9389 GeV, rnh = 1.1156 GeV, mx = 1.1931 GeV and ma = 1.3181 GeV exactly. We therefore fit these masses individually and average the corresponding values for & The contribution from scalar meson exchange is included in the uncertainties of the numbers given. This is justified since the numerical values for the couplings FS and DS are supposedly small. A more thorough discussion on this point can be found in [ 241. We have &= 711 MeV, 0 m,,==679 MeV, ?%I- 877 MeV, & 728 MeV, with the following average: &=749f
125 MeV.
(16)
This number is compatible with the one found in the analysis of the pion-nucleon a-term, where approximately 130 MeV to the nucleon mass were attributed to the strange matrix element m&($s)p) (with a sizeable uncertainty) [ 301. The spread of the various values is a good measure of the uncertainties related to this complete q” calculation. Let us take a closer look at the quark mass expansion of the nucleon mass, in the notation of Eq. ( 1) , mN=711+202-272+29s = 939 MeV .
MeV (17)
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B. Borasoy, U.-G. Meipner/ Physics Letters B 365 (1996) 28S-291
This looks similar for the other octet baryons. We conclude that the quark mass corrections of order my, my312 and rni are all of the same size. 4 In Ref. [ 161, it was argued that only the leading non-analytic corrections (LNAC) N m:'2arelarge and that further terms like the ones w rnzaremoderately small, of the order of 100 MeV. This would amount to an expansion in M$/(47rFp)2 w l/6 with a large leading term. This expectation is not borne out by our results, the next corrections are as large as the LNACs. These findings agree with the meson cloud model calculation of Gasser [ 31. A last remark about the baryon masses concerns the deviation from the Gell-Mann-Okubo relation,Ao~o = (3mA+mx-2mN-2mz)/4 whichempirically is about 6.5 MeV. We find Ao~o = 31 MeV, which is larger in magnitude than the value found in [ $61. We remark that in our case the decuplet contribution is contained in the rni contributions and not in the nil2asin [ 51. Therefore, in our case, Ao~o is dominated by the rni piece. The sizeable uncertainty in the chiral limit masses, Eq. ( 16)) does not allow for a very accurate statement about this very small quantity. It is also very sensitive to the scalar couplings. To get a better handle on this issue, one either has to be able to fix all pertinent low-energy constants at order rnifrom data or improve upon the resonance saturation estimate used here by including e.g. the mass splitting within the decuplet and the SU( 3) breaking of the decupletoctet-meson couplings. A better understanding of this topic is, of course, at the heart of the determination of the quark mass ratio R = (md - m,)/(m, - 62) from the baryon masses (once the electromagnetic corrections have been included). The pion-nucleon a-term is completely fixed. Using for & its average, we find (no scalar resonance contribution) a,+,(O) = 48 MeV,
(18)
with an uncertainty of about f10 MeV due to the spread in &. This number compares favourably with the one found in Ref. 1301, USN(O) = 44 & 9 MeV. We stress that this result reflects a very non-trivial con4Note that for certain values of the scalar couplings, the fourth order term m$’ can be significantly smaller. However, the nucleon mass is much more sensitive to the scalar contribution than the other octet masses.
sistency for the complete calculation to quadratic order in the quark masses using the resonance saturation principle in the scalar sector. The additional contribution from the scalar meson exchange is accounted for in the f10 MeV uncertainty. To be specific, we have u,,v( 0) = (48 + Ds .0.5- Fs .2.0)MeV. It is furthermore instructive to disentangle the various contributions to ffnN (0)of order q2, q3 and q4,respectively, URN(0) = 54 - 33 + 27 MeV = 48 MeV ,
(19)
which shows a moderate convergence, i.e. the terms of increasing order become successively smaller. Still, the q4 contribution is important. Also, at this order no ~TT~T rescattering effects are included. We notice that using the values for be,~~ and b; as determined in Ref. [ 231 leads to a much increased fourth order contribution.
6. Summary and outlook In this paper, we have used heavy baryon chiral perturbation theory to calculate the octet baryon masses to quadratic order in the quark masses, including 20 local operators with unknown coeffcients. These lowenergy constants were fixed by resonance exchange. The dominant contribution comes indeed from the excitation of the spin-3/2decuplet fields. Tadpole graphs with scalar meson exchange only lead to small corrections. This left us with one free parameter, the baryon mass in the chiral limit, which could be determined within 18% accuracy, and is compatible with the value inferred for the strange matrix-element m,(&slp) in Ref. [ 301. Furthermore, the pion-nucleon a-term comes out surprisingly close to its empirical value, (+lrN(0) = 48 MeV, with an uncertainty of about f 10 MeV. This first exploratory q4 study of the three flavor scalar sector of baryon CHPT points towards a significant improvement compared to previous investigations which were mostly confined to so-called “computable” corrections and/or fitted a few of the pertinent low-energy constants. However, the calculation is not yet accurate enough to determine the quark mass ratio R reliably from the octet masses. Topics not addressed here are the kaon-nucleon @-terms, the corresponding shifts to the pertinent Cheng-Dashen points together with the strangeness content of the proton, see Ref. [ 241.
B. Borasoy, U.-G. Me@ner/
Physics Letters B 365 (19%) 28.5-291
Acknowledgements We are grateful to Daniel Wyler for a very useful remark.
References [II P Langacker and H. Pagels, Phys. Rev. D 8 ( 1971) 4595. [21 H. Pagels, Phys. Rep. 16 (1975) 219. [31 [41 [51 [61
t71 [81 191 [lOI
[Ill [I21
J. Gasser, Ann. Phys. 136 ( 1981) 62. J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. E. Jenkins, Nucl. Phys. B 368 ( 1992) 190. E. Jenkins and A.V. Manohar, in: Effective Field Theories of the Standard Model, ed. U.-G. MeiDner (World Scientific, Singapore, 1992). V. Bernard, N. Kaiser and U.-G. MeiBner, Z. Pbys. C 60 (1993) 111. RF. L&red and M.A. Luty, Phys. Lett. B 329 (1994) 479. S. Mallik, Z. Phys. C 68 ( 1995) 149. M.K. Banerjee and J. Milana, Baryon Mass Splittings in Chiral Perturbation Theory, preprint UMPP 95-058, 1994 [ hep_ph/9410398]. S. Dentin, Thesis, Universite d’Aix-Marseille II, 1990 (unpublished). U. Btirgi, Diploma Thesis, University of Bern, 1994 (unpublished).
291
[ 131 S. Weinberg, Physica A 96 ( 1979) 327. [ 141 J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142. [ 151 J. Gasser and H. Leutwyler, Nucl. Pbys. B 250 ( 1985) 465. [ 161 E. Jenkins and A.V. Manohar, Phys. Len. B 281 ( 1992) 336. [ 171 U.-G. MeiBner, Rep. Prog. Phys. 56 (1993) 903. [ 181 G. Ecker, Prog. Nucl. Part. Phys. 35 ( 1995) 1. [ 191 V. Bernard, N. Kaiser and U.-G. Meithrer, Int. J. Mod. Pbys. E 4 (1995) 193. 1201 A. Pith, Rep. Prog. Phys. 58 (1995) 563. [21] E. Jenkins and A.V. Manohar, Phys. Lett. B 255 ( 1991) 558. [22] V. Bernard, N. Kaiser, J. Kambor and U.-G. MeiRner, Nucl. Phys. B 388 (1992) 315. 1231 PB. Siegel, N. Kaiser and W. Weise, Nucl. Phys. A ( 1995), to appear. [24] B. Borasoy and U.-G. Meigner, in preparation. [25] G. Ecker, J. Gasser, A. Pith and E. de Rafael, Nucl. Phys. B 321 (1989) 311. [ 261 G. Ecker, J. Gasser, H. Leutwyler, A. Pith and E. de Rafael, Phys. Len. B 223 (1989) 425. [27] J.E Donoghue, C. Ramirez and G. Valencia, Pbys. Rev. D 39 (1989) 1947. [28] A. Schmidt, Thesis Technische Universitiit Mtinchen, 1994 (unpublished). 1291 V. Bernard, N. Kaiser, U.-G. MeiSner and A. Schmidt, Z. Phys. A 348 (1994) 317. [30] J. Gasser, H. Leutwyler and M.E. Sainio, Phys. Len. B 253 (1991) 252.
LETTERS
B
Physics Letters B 365 (1996) 285-291
EISVIER
Baryon masses and pion-nucleon a-term to second order in the _j, quark masses ‘. B. Borasoy Institutfir
Theoretische
Kernphysik.
‘,
Universittit
Ulf-G. MeiBner 2 Bonn, NuJallee
14-16,
D-531 15 Bonn, Germany
Received 28 August 1995; revised manuscript received 25 September 1995 Editor: R. Gatto
Abstract We analyze the octet baryon masses and the pion-nucleon c-term in the framework of heavy baryon chiral perturbation theory. In contrast to previous investigations, we include all terms up-to-and-including quadratic order in the light quark masses. The pertinent low-energy constants are fixed from resonance exchange. This leaves as the only free parameter the baryon mass in the chiral limit, A. We find A= 749 f 125 MeV together with a,~(0) = 48 f 10 MeV. We discuss various implications
of these results.
eral, the quark mass expansion of the octet baryon masses takes the form
1. Introduction
The analysis of the octet baryon masses in the framework of chiral perturbation theory already has a long history, see e.g. [ l-l 21. In this paper, we present the results of a first calculation including allterms of order O( mi), where m, is a generic symbol for any one of the light quark masses mu,&. We work in the isospin limit m, = md and neglect the electromagnetic corrections. Previous investigations only considered mostly the so-called computable corrections of order rnior included some of the finite terms at this order. This, however, contradicts the spirit of chiral perturbation theory (CHPT) in that all terms at a given order have to be retained, see e.g. [ 13-l 51. In gen-
m-ii+ c 4
B,m,+~Cqm~'2+~D,m~+... Y
v
(1) modulo logs. Here, i is the mass in the chiral limit of vanishing quark masses and the coefficients B,,C,,D, arestate-dependent. Furthermore, they include contributions proprotional to some low-energy constants which appear beyond leading order in the effective Lagrangian. In this letter, we evaluate these coefficients for the octet baryons N, A, Z and E. In addition, we also calculate the pion-nucleon o-term which is intimately related to the quark mass expansion of the baryon masses [ 3,7,16]. For some comprehensive reviews, see e.g. [ 17-201.
*Work supported in part by the Deutsche Forschungsgemeinschaft. ’ E-mail: [email protected]. * E-mail: [email protected]. 0370-2693/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01286-9
286
B. Borasoy, U.-G. Meipner/Physics
in via loops. There are seven terms contributing at dimension four,
2. Effective Lagrangian To perform the calculations, we make use of the effective meson-baryon Lagrangian. Our notation is identical to the one used in [ 71 and we discuss here only the new terms. Denoting by 4 the pseudoscalar Goldstone fields (v, K, 77) and by B the baryon octet, the effective Lagrangian takes the form
where the chiral dimension (i) counts the number of derivatives and/or meson mass insertions. The baryons are treated in the extreme non-relativistic limit [21,22], i.e. they are characterized by a fourvelocity ucL.In this approach, there is a one-to-one correspondence between the expansion in small momenta and quark masses and the expansion in Goldstone boson loops, i.e. a consistent power counting scheme emerges. The form of the lowest order mesonbaryon Lagrangian is standard, see e.g. [ 71, and the meson Lagrangian is given in [ 151. The dimension two meson-baryon Lagrangian can be written as (we only enumerate the terms which contribute)
(3) i=l
with the 012’ monomials in the fields of chiral dimension two. Typical forms are Tr( B [ uP , [up, B] ] ), Tr(B[u . 1.4,[u. u,BJ]) or BBTr(u,u”), with ucL= iutdpUut, u = I/?? and U = exp(i4/Fp) collects the pseudoscalars. The form of f$~cPnd) is [ 71, $ttand)
= bDTr(B{X+,B})
+ bo Tr(BB) Tr(x+) ,
Letters B 365 (1996) 285-291
where typical forms of the 0i4) are BBTr( x:) or Tr(B[x+, [,y+,B]]). At this stage, we take m, = md # m,y. The explicit expressions for the various terms in Eqs. (3), (5) can be found in [ 241. We point out that there are 20 a priori unkown constants (to be precise, only some linear combinations of these are relevant for the calculation discussed here). In addition, there are the F and D coupling constants (subject to the constraint F + D = gA = 1.25) from the lowest order Lagrangian Coda).What we have to calculate are all one-loop graphs with insertions from CFi2’ and tree graphs from L(p2d4’.We stress that we do not include the spin-312 decuplet in the effective field theory [ 61, but rather use these fields to estimate the pertinent low-energy constants (resonance saturation principle). We therefore strictly expand in small quark masses and external momenta (collectively denoted by ‘q ‘) with no recourse to large NCarguments.
3. Baryon masses and pion-nucleon
a-term
The form of the terms N my and N m:‘2 for the baryon masses and c?rN(0) is standard, we use here the same notation as Ref. [ 71. The q4 contribution to any octet baryon mass mg takes the form
+ bFTr(B[X+,B])
(6) (4)
i.e. it contains three low-energy constants and x+ = utxut + u,ytu is proportional to the quark mass matrix M = diag(m,, md, m,) since x = 2BM. Here, B = -(Ol~~lO>lF;z and Fp is the pseudoscalar decay constant. All low-energy constants in ‘CL: are finite. A subset of the bi has been estimated in [ 231 by analyzing kaon-nucleon scattering data. The splitting of the dimension two meson-baryon Lagrangian in Fq. (3) is motivated by the fact that while the first three terms appear in tree and loop graphs, the latter ten only come
with P, Q = V, K, v and A the scale of dimensional regularization. The explicit form of the state-dependent prefactors Ei,Bcan be found in Ref. [ 241. Notice the appearance of mixed terms N MS Mi which were not considered in most of the existing investigations. The fourth order contribution to the baryon masses contains divergences proportional to (using dimensional regularization) L _ -_ /P4
__
1
16~2 { d-4
- i[ln(47r)
+ 1 -yEI},
(7)
B. Borasoy, U.-G. Meipner/Physics
287
Letters B 36s (1996) 285-291
with yE - 0.5772215. These are renormalized by an appropriate choice of the low-energy constants di, d/-d,‘(h)
+IiL.
/
/
(8)
In fact, all seven di are divergent. The appearance of these divergences is in marked contrast with the q3 calculation which is completely finite (in the heavy fermion approach). In what follows, we set A = 1 GeV. A part of the M; corrections considered here was already discussed by Lebed and Luty [ 81. For a direct comparison to their work, we refer to [ 241. Similarly, the fourth order contribution to the pion-nucleon gterm can be written as
with the ei,c given in [ 241. Here, the renormalization is somewhat more tricky. It can most efficiently be performed in a basis of a linearly independent subset of the operators dOj4)/dm,, q = (u, d, s), as detailed in Ref. [ 241. A good check on the rather lengthy expressions for the nucleon mass and a,~(0) is given by the Feynman-Hellmann theorem, riz(&z~/&?r) = U,N( 0), with iit the average light quark mass, riz = (m, + md) /2. We remark here that in contrast to the order q3 calculation, the shift to the Cheng-Dashen point, grrN(2Mi) - a,N(O), is no longer finite, i.e. there appear r-dependent divergences. We therefore do not consider this a-term shift in what follows. We will also not discuss in detail the two kaon-nucleon a-terms, c&!;*)(t) , for similar reasons in this letter.
Clearly, we are not able to fix all the low-energy (2,4) from data, even if we constants appearing in L#, would resort to large N, arguments. We will therefore use the principle of resonance saturation to estimate these constants. This works very accurately in the meson sector [ 25-271. In the baryon case, one has to account for excitations of meson (R) and baryon (N* ) resonances. One writes down the effective Lagrangian with these resonances chirally coupled to the Goldstones and the baryon octet, calculates the
\
/’
\
\
\
/
-i
/
(b)
(4
Fig. 1. Resonance saturation for the masses. In (a), a baryon resonance (from the decuplet) is excited, whereas (b) represents the r-channel meson (scalar or vector) excitation. Solid and dashed lines denote the octet baryons and Goldstones, respectively.
Feynman diagrams pertinent to the process under consideration and, finally, lets the resonance masses go to infinity (with fixed ratios of coupling constants to masses). This generates higher order terms in the effective meson-baryon Lagrangian with coefficients expressed in terms of a few known resonance parameters. Symbolically, we can write
t,dUB,R,N*l
-)&du,Bl.
(10)
Here, there are two relevant contributions. One comes from the excitation of the spin-3 /2 decuplet states and the second from t-channel scalar and vector meson excitations, cf. Fig. 1. It is important to stress that for the resonance contribution to the baryon masses, one has to involve Goldstone boson loops. This is different from the normal situation like e.g. in form factors or scattering processes. Consider first the decuplet contribution. We treat these field relativistically and only at the last stage let the mass become very large. The pertinent interaction Lagrangian between the spin-3/2 fields (denoted by A), the baryon octet and the Goldstones reads (suppressing SU( 3) indices)
c 4. Resonance saturation
C--_
/-\
c - -A”TO,,(Z) *” - fiFp
BaP++h.c.,
(11)
with T the standard 4 -+ 5 isospin transition operator, Fp = 100 MeV the (average) pseudeoscalar decay constant and C = 1.5 determined from the decays A -+ Bv. The Dirac matrix operator O,,(Z) is given by @P,(Z) =&Lu -(z+;)Y,Yv.
(12)
For the off-shell parameter Z, we use Z = -0.3 from the determination of the A contribution to the TN scattering volume ~33 [ 281. The tadpole graphs shown in Fig. lb with an intermediate vector meson only contribute at order $. This is evident in the conventional
288
B. Borasoy, U.-G. Meipner/Physics
vector field formulation. In the tensor field formulation, the vertex Tr( Vpvu~uy) seems to lead to a contribution at order q4. However, in a tadpole graph one needs the index contraction ,u = Y and thus no terms of order q4 can be obtained. Matters are different for the scalars. Denoting by S and Si the scalar octet and singlet with Ms N MS, N 1 GeV, respectively, the lowest order coupling to the baryon octet reads ,C$’ = DsTr(B{S,B})
+ F..Tr(B[S,B])
+ Ds,SITr(BB)
(13)
where the coupling constants Ds, FS and Ds, have chiral dimension zero. 3 In fact, there are no empirical data to really pin them down. From the decay pattern of the low-lying baryons one can, however, estimate these numbers to be small. We will generously vary them between zero and one. For the couplings of the scalars to the Goldstones, we use the notation of [ 251 and the parameters determined therein. Putting pieces together, all low-energy constants are expressed via resonance parameters and the baryon masses take the form
Letters B 365 (1996) 285-291
graphs with insertions proportional to the low-energy constants be,n,~ and bi, respectively. Note, however, that in the resonance exchange approximation not all of the ten bi are contributing. The 6 terms subsume the contributions from ,!Zc, these are proportional to the low-energy constants d;. Finally, the terms of the type 0; are the scalar meson contributions to the mass. They amount to a constant, state-dependent shift and consist of terms of the types N M& NM:lnMiand N M;M$ In M$, see [ 241. To be specific, we give the coefficients Dp.‘*’ and A for the nucleon (the constants di of Eq. (8) are expressed in terms of resonance parameters), 1 -$(D 8,rr2F*P 1
AN = -
-
D; = Df’*’
-3F)*M~ln$
+ Df4’
c* 1tin-* F; {
+pD;+cYD;+cD;+D;,
(-4r*F;)(M:,
+ 4M;)
M*
M2
+~M$ln-$+~M~ln$+$M~ln-$ + -;M; [
E=~(222+z-l). A
(14)
with mf’ the contribution of o(m:‘*) and mA = 1.455 GeV is the average decuplet mass. Notice that it is convenient to lump the c3( my) and the 0( rni) corrections togther as it was done in Eq. ( 14) (in the constants DE and 0;). We have kept explict the baryon mass in the chiral limit. Of course, in the fourth order terms it could be substituted by the corresponding physical values. At second order, however, we would get a state-dependent shift, see e.g. [ 291, and we thus prefer to work with & Alternative representations for the baryon masses are given in [ 241. Let us briefly explain the origin of the various terms in Eq. ( 14). The A contributions are tadpoles with 1/m insertions from C’*’ 68’ Similarly, the p and E terms stem from tadpole ’ In what follows, we neglect the singlet field.
z
-k(D2-2DF+3F2)M:ln~},
=-
rnB =& +mi3’ + Ag (ii) -’
&(D
+ F)*M:ln
M2
+ z4M;
+i(D-3F)*(T+Mi)]M:ln$ + [$D
+ F)*(f$-
+ Mi)]Miln$
+ -%(D - F)2~n:, [
+&(223D*-318DF+351F2)M:]M~ln~},
The corresponding coefficients for the A, Z and z and also the Di can be found in [ 241. The tree con-
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B. Bomoy, U.-G. Meipner/ Physics Letters B 365 (19%) 285-291
Table 1 Numerical values of the state-dependent coefficients in E$. ( 14). The 0: are dimensionless. For the scalar coupliugs, we use the results of 1251 and MS - 1 GeV. B
AB
N
0.0049
A z E
0.0179 0.0144
t~v*l
0;
4
-39.875 -68.044 -132.90
-126.46
0.0216
0;
[GeV*]
0;
[GeV*]
-0.017 -0.047 +0.051 -0.037
0.0321 0.0617 0.1422 0.1324
-2.3617 1.0831 -11.273 -5.4799
[GeV] Ds Ds Ds Ds
+ + -
0.061 Fs O.OIOFs O.OlOFs
0.082 Fs
Table 2 Some low-energy
(*) in GeV-' . In the first row, only the decuplet contribution is given. In the second row, scalar meson constants from L1, exchange is added.
bl
b2
Reso. ( DS - Fs - 0)
0.084
-0.144
Reso. (Ds-Fs-1) Ref. [23]
0.100 0.044
-0.111 -0.145
tribution
from L$S’and) is subsumed
hR
b3 0.108 0.125 -0.054
in the D$. The
numerical values of the AB, D$, Di, 05 and 0; are given in Table 1 (using F = 0.5, D = 0.75 and qrn from [ 25 I) . We see that the dominant terms at
U( nz’,) are indeed the tadpole graphs with an insertion from f$iscand) (this holds for the masses but not for fl&O)). It is ak0 instructive to compare the values we find from resonance exchange with the ones previously determined from KN scattering data. We have transformed the results of Ref. [ 231 into our notation. As can be seen from Table 2, most (but not all) coefficients agree in sign and magnitude. We remark that the procedure used in [ 23 I involves the summation of arbitrary high orders via a Lippmann-Schwinger equation and is thus afflicted with some uncertainty not controled within CHPT.
5. Results and discussion The only free parameter in the formula for the baryon masses, Eq. ( 14), is the mass of the baryons in the chiral limit, &, since all low-energy constants are fixed in terms of resonance parameters. In particular, in contrast to Ref. [7], the parameters 60, bD and by are no longer free. Also, at quadratic order in the quark masses the ambiguity between & and ba is resolved, it is not necessary to involve any one of
ho
bF
t-U
-0.216
-0.738
-0.264
0.317
-0.239 -0.165
-0.733 -0.493
-0.208 -0.213
0.345 0.066
the a-terms in the fitting procedure [5,7]. In fact, one can not find one single value of h to fit all four octet masses, mN = 0.9389 GeV, rnh = 1.1156 GeV, mx = 1.1931 GeV and ma = 1.3181 GeV exactly. We therefore fit these masses individually and average the corresponding values for & The contribution from scalar meson exchange is included in the uncertainties of the numbers given. This is justified since the numerical values for the couplings FS and DS are supposedly small. A more thorough discussion on this point can be found in [ 241. We have &= 711 MeV, 0 m,,==679 MeV, ?%I- 877 MeV, & 728 MeV, with the following average: &=749f
125 MeV.
(16)
This number is compatible with the one found in the analysis of the pion-nucleon a-term, where approximately 130 MeV to the nucleon mass were attributed to the strange matrix element m&($s)p) (with a sizeable uncertainty) [ 301. The spread of the various values is a good measure of the uncertainties related to this complete q” calculation. Let us take a closer look at the quark mass expansion of the nucleon mass, in the notation of Eq. ( 1) , mN=711+202-272+29s = 939 MeV .
MeV (17)
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B. Borasoy, U.-G. Meipner/ Physics Letters B 365 (1996) 28S-291
This looks similar for the other octet baryons. We conclude that the quark mass corrections of order my, my312 and rni are all of the same size. 4 In Ref. [ 161, it was argued that only the leading non-analytic corrections (LNAC) N m:'2arelarge and that further terms like the ones w rnzaremoderately small, of the order of 100 MeV. This would amount to an expansion in M$/(47rFp)2 w l/6 with a large leading term. This expectation is not borne out by our results, the next corrections are as large as the LNACs. These findings agree with the meson cloud model calculation of Gasser [ 31. A last remark about the baryon masses concerns the deviation from the Gell-Mann-Okubo relation,Ao~o = (3mA+mx-2mN-2mz)/4 whichempirically is about 6.5 MeV. We find Ao~o = 31 MeV, which is larger in magnitude than the value found in [ $61. We remark that in our case the decuplet contribution is contained in the rni contributions and not in the nil2asin [ 51. Therefore, in our case, Ao~o is dominated by the rni piece. The sizeable uncertainty in the chiral limit masses, Eq. ( 16)) does not allow for a very accurate statement about this very small quantity. It is also very sensitive to the scalar couplings. To get a better handle on this issue, one either has to be able to fix all pertinent low-energy constants at order rnifrom data or improve upon the resonance saturation estimate used here by including e.g. the mass splitting within the decuplet and the SU( 3) breaking of the decupletoctet-meson couplings. A better understanding of this topic is, of course, at the heart of the determination of the quark mass ratio R = (md - m,)/(m, - 62) from the baryon masses (once the electromagnetic corrections have been included). The pion-nucleon a-term is completely fixed. Using for & its average, we find (no scalar resonance contribution) a,+,(O) = 48 MeV,
(18)
with an uncertainty of about f10 MeV due to the spread in &. This number compares favourably with the one found in Ref. 1301, USN(O) = 44 & 9 MeV. We stress that this result reflects a very non-trivial con4Note that for certain values of the scalar couplings, the fourth order term m$’ can be significantly smaller. However, the nucleon mass is much more sensitive to the scalar contribution than the other octet masses.
sistency for the complete calculation to quadratic order in the quark masses using the resonance saturation principle in the scalar sector. The additional contribution from the scalar meson exchange is accounted for in the f10 MeV uncertainty. To be specific, we have u,,v( 0) = (48 + Ds .0.5- Fs .2.0)MeV. It is furthermore instructive to disentangle the various contributions to ffnN (0)of order q2, q3 and q4,respectively, URN(0) = 54 - 33 + 27 MeV = 48 MeV ,
(19)
which shows a moderate convergence, i.e. the terms of increasing order become successively smaller. Still, the q4 contribution is important. Also, at this order no ~TT~T rescattering effects are included. We notice that using the values for be,~~ and b; as determined in Ref. [ 231 leads to a much increased fourth order contribution.
6. Summary and outlook In this paper, we have used heavy baryon chiral perturbation theory to calculate the octet baryon masses to quadratic order in the quark masses, including 20 local operators with unknown coeffcients. These lowenergy constants were fixed by resonance exchange. The dominant contribution comes indeed from the excitation of the spin-3/2decuplet fields. Tadpole graphs with scalar meson exchange only lead to small corrections. This left us with one free parameter, the baryon mass in the chiral limit, which could be determined within 18% accuracy, and is compatible with the value inferred for the strange matrix-element m,(&slp) in Ref. [ 301. Furthermore, the pion-nucleon a-term comes out surprisingly close to its empirical value, (+lrN(0) = 48 MeV, with an uncertainty of about f 10 MeV. This first exploratory q4 study of the three flavor scalar sector of baryon CHPT points towards a significant improvement compared to previous investigations which were mostly confined to so-called “computable” corrections and/or fitted a few of the pertinent low-energy constants. However, the calculation is not yet accurate enough to determine the quark mass ratio R reliably from the octet masses. Topics not addressed here are the kaon-nucleon @-terms, the corresponding shifts to the pertinent Cheng-Dashen points together with the strangeness content of the proton, see Ref. [ 241.
B. Borasoy, U.-G. Me@ner/
Physics Letters B 365 (19%) 28.5-291
Acknowledgements We are grateful to Daniel Wyler for a very useful remark.
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