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UCSD/PTH 94-01 hep-ph/9406209
BARYONS IN THE LARGE N LIMIT*
arXiv:hep-ph/9406209v1 1 Jun 1994
ANEESH V. MANOHAR Physics Department, University of California, 9500 Gilman Drive La Jolla, CA 92093, USA
Abstract The properties of baryons can be computed in a systematic expansion in 1/Nc , where Nc is the number of colors. Recent results on the axial couplings and masses of baryons (for the case of three flavors) are presented. The results give insight into the structure of flavor SU (3) breaking for baryons.
* Talks presented at the Symposium on Internal Spin Structure of the Nucleon (Yale University, Jan 94) and at the Workshop on Continuous Advances in QCD (Theoretical Physics Institute, University of Minnesota, Feb 94).
1. Introduction and Summary of Results In this talk, I will present some recent results on baryons in the 1/Nc expansion obtained in collaboration with Roger Dashen and Elizabeth Jenkins., ,  This work shows that one can use the 1/Nc expansion to obtain predictions for baryon properties that are in good agreement with the experimental results for Nc = 3. The results are derived directly from QCD, and do not make any model assumptions about the structure of baryons. The success of the predictions indicates that 1/Nc corrections are under control. Other recent work on baryons in the 1/Nc expansion can be found in papers by Carone, Georgi, and Osofsky, Luty and March-Russell, and by Broniowski. A good introduction to the 1/Nc expansion for QCD are the original papers of ’t Hooft and Witten, and the lecture notes of Coleman. The 1/Nc expansion can be used to obtain quantitative information about baryons. The basic approach is to study baryon-meson scattering amplitudes. One finds that these scattering amplitudes violate unitarity unless certain cancellation conditions are satisfied. The cancellation conditions lead to an additional symmetry of baryons in the Nc = ∞ limit, which is discussed in the next section. This symmetry allows one to obtain results for the baryons in terms of the irreducible representations of the symmetry algebra. The 1/Nc corrections can also be classified in terms of the symmetry algebra. Many of the results for the case of three light flavors can be obtained without any assumptions about the mass of the s-quark. These results provide important constraints on the structure of SU (3) breaking in the baryons. The 1/Nc expansion can also be used for baryons containing a heavy quark. Some of the results obtained so far include, ,  • The baryon sector of QCD has a (contracted) SU (6) spin-flavor symmetry in the large Nc limit. This symmetry can be used to compute the ratio of pion-baryon couplings, such as gNNπ /gN∆π . Similar results hold for the pion couplings of baryons containing a heavy quark. • The F/D ratio for the baryon axial currents is determined to be 2/3 + O 1/Nc2 , in good agreement with the experimental value of 0.58 ± 0.04. • The F/D ratio for the baryon magnetic moments is determined to be 2/3 + O 1/Nc2 , in good agreement with the experimental value of 0.72. The difference between the F/D ratios for the axial currents and magnetic moments is an indication of the size of 1/Nc2 corrections. • The ratios of all pion-baryon couplings are determined up to corrections of order 1/Nc2 , and the ratios of all kaon-baryon couplings are determined to leading order. These results are independent of the mass of the s-quark. • The SU (3) breaking in the pion couplings must be linear in strangeness at order 1/Nc . This leads to an equal spacing rule for the pion couplings, which agrees well with the data. The SU (3) breaking in the decuplet-octet transition axial currents is related to the SU (3) breaking in the octet axial currents. 2
• The baryon mass relations
Σ∗ − Σ = Ξ∗ − Ξ 1 3
(Σ + 2Σ∗ ) − Λ =
3 4Λ 1 2
+ 41 Σ −
(∆ − N )
(N + Ξ) = − 14 (Ω − Ξ∗ − Σ∗ + ∆)
(Σ∗ − ∆) − (Ξ∗ − Σ∗ ) +
Σ∗Q − ΣQ = Ξ∗Q − Ξ′Q ∗ 1 3 2ΣQ + ΣQ − ΛQ =
(Ω − Ξ∗ ) = 0
(∆ − N )
are valid up to corrections of order 1/Nc2 without assuming SU (3) symmetry. Some of these relations are also valid using broken SU (3), with octet symmetry breaking. Relations which can be proved using either large Nc spin-flavor symmetry or broken flavor SU (3) work extremely well, because effects which violate these relations must break both symmetries. • The chiral loop correction to the baryon axial currents cancels to two orders in the 1/Nc expansion, and is of order 1/Nc instead of order Nc . • The order Nc non-analytic correction to the baryon masses is pure SU (3) singlet, and the order one contribution is pure SU (3) octet. Thus violations of the Gell-Mann– Okubo formula are at most order 1/Nc . The baryon masses can be strongly non-linear functions of the strange quark mass, and still satisfy the Gell-Mann–Okubo formula. This helps resolve the σ-term puzzle. I do not have time to discuss all of these results here. I will concentrate on the pion-baryon couplings and the baryon masses in this talk. The analysis in Secs. 2 and 3 is for the case of two light flavors (u and d). The extension to three flavors is discussed in Sec. 4. 2. Pion-Baryon Couplings Baryons in the 1/Nc expansion were studied by Witten. He was able to discuss qualitative features of the baryons, and to show that (at least for heavy quarks) large Nc baryons could be described by a Hartree picture. Witten derived 1/Nc counting rules for meson-baryon scattering by studying the Nc dependence of Feynman diagrams. For example, he showed that the meson-baryon scattering amplitude is order one, and the baryon-baryon scattering amplitude is order Nc . One can use the qualitative 1/Nc counting rules to obtain consistency conditions on the baryons. These consistency conditions can be solved to obtain relations for baryons that are in good agreement with the experimental data at Nc = 3. The basic large Nc scaling results I will use are that the pion decay √ coupling of constant fπ is of order Nc , the baryon mass is of order Nc , and the axial √ the baryon, gA , is of order Nc . The pion-baryon coupling is of order gA /fπ ∼ Nc , if the FIGURE 1. interaction is written using a gradient coupling. Graphs contributing to pion-baryon scattering at leading order in 1/Nc .
The leading contributions to pion-baryon scattering in the large Nc limit are given in Fig. 1. The axial current matrix element in the nucleon can be written as hN | ψγ i γ5 τ a ψ |N i = gNc hN | X ia |N i ,
where hN | X ia |N i and g are of order one. The coupling constant g has been factored out so that the normalization of X ia can be chosen conveniently. X ia is an operator (or 4 × 4 matrix) defined on nucleon states p ↑, p ↓, n ↑, n ↓, which has a finite large Nc limit. The pion-nucleon scattering amplitude for π a (q) + N (k) → π b (q ′ ) + N (k ′ ) is Nc2 g 2 1 jb ia 1 ia jb −i q q , X X − ′0 X X fπ2 q0 q i ′j
where the amplitude is written in the form of an operator acting on nucleon states. Both initial and final nucleons are on-shell, so q 0 = q ′ 0 . The product of the X’s in Eq. (2.2)√then sums over the possible spins and isospins of the intermediate nucleon. Since fπ ∼ N c , the overall amplitude is of order Nc , which violates unitarity, and also contradicts the large Nc counting rules of Witten. Thus large Nc QCD with a I = J = 1/2 nucleon multiplet interacting with a pion is an inconsistent field theory. There must be other states that cancel the order Nc amplitude in Eq. (2.2) so that the total amplitude is order one, and consistent with unitarity. One can then generalize X ia to be an operator on this degenerate set of baryon states, with matrix elements equal to the corresponding axial current matrix elements. With this generalization, the form of Eq. (2.2) is unchanged. Thus we obtain the first consistency condition for baryons,
X ia , X jb = 0,
so that the axial currents are represented by a set of operators X ia that commute in the large Nc limit. This consistency condition was also derived by this method by Gervais and Sakita. There are additional commutation relations,
J i , X jb = i ǫijk X kb ,
J i , J j = i ǫijk J k ,
a jb I ,X = i ǫabc X jc ,
I a , I b = i ǫabc I c ,
a i I , J = 0,
since X ia has spin one and isospin one. The algebra in Eqs. (2.3) and (2.4) is a contracted SU(4) algebra. Consider the embedding SU (4) → SU (2) ⊗ SU (2) where 4 → (2, 2). If the generators of SU (2) ⊗ SU (2) in the defining representation are I a , and J i , the SU (4) generators in the defining representation are J i ⊗ 1, 1 ⊗ I a and J i ⊗ I a , which we will call I a , J i√and Gia respectively. √ √ (The properly normalized SU (4) generators are I a / 2, J i / 2 and 2 Gia .) The algebra for large Nc baryons in QCD is given by taking the limit Gia , Nc →∞ Nc
X ia = lim 4
(up to an overall normalization of the X ia ). The commutation relations of SU (4), i j a b J , J = i ǫijk J k , I , I = i ǫabc I c , a ia i jb I , G = i ǫabc Gjc , J , G = i ǫijk Gkb , a i ia jb I , J = 0, G , G = 4i ǫijk δab J k + 4i ǫabc δij I c ,
turn into the commutation relations eqs. (2.3)–(2.4) in the large Nc limit. The large Nc limit of QCD has a contracted SU (4) symmetry in the baryon sector. This explains the success of the non-relativistic SU (4) symmetry of the quark model. The irreducible representations of the contracted Lie algebra can be obtained using the theory of induced representations. The irreducible representation can be labelled by a quantum number K = 0, 1/2, 1, . . .. The K = 0 irreducible representation contains baryons with no strange quarks, such as the nucleon N , the delta ∆, and other baryon states which do not exist for Nc = 3. The K = 1/2 tower has the quantum numbers of baryons with one strange quark, etc. The contracted Lie algebra is sufficient to determine the ratio of all the pion-baryon couplings. The normalization of the couplings is arbitrary, since all the commutation relations are homogeneous in X ia . The explicit form for the pion-baryon couplings (the matrix elements of X ia ) can be written using 6j-symbols. It is easy to show that the large Nc QCD predictions for the pion-baryon coupling ratios are the same as those obtained in the non-relativistic quark model,  or in the Skyrme model, in the Nc → ∞ limit, because both these models also have a contracted SU (4) symmetry in this limit. (The contracted SU (4) symmetry is sufficient to determine the ratios of all the pion-baryon couplings.) In the Skyrme model, the axial current in the Nc → ∞ limit is proportional to X ia ∝ Tr Aτ i A−1 τ a . The X’s commute, since A is a coordinate, and the expression for X does not contain any factors of the conjugate momentum. In the quark model, X is proportional to 1 X † i a X ia ∝ q σ τ qα , Nc α α where the sum on α is over the color index. The commutator of two X’s is i ia jb 1 Xh † i a qα σ τ qα , qβ† σ j τ b qβ , X ,X ∝ 2 Nc αβ h i X 1 = 2 qα† σ i τ a qα , qβ† σ j τ b qβ → 0, Nc α=β
where the last equality follows since quarks of different colors commute. The sum is at most of order Nc , and so vanishes as Nc → ∞ because of the overall factor of 1/Nc2 . 3. 1/Nc Corrections What makes the 1/Nc expansion for baryons interesting is that it is possible to compute the 1/Nc corrections. This allows one to compute results for the physical case Nc = 3, rather than for the strict Nc = ∞ limit, which is only of formal interest. 5
FIGURE 2. Diagrams contributing to π+N→π+π+N .
The 1/Nc corrections to the axial couplings X ia are determined by considering the scattering process π a + N → π b + π c + N at low energies. The nucleon pole graphs that contribute in the large Nc limit are shown in Fig. 2. Define the matrix element of the axial current to order 1/Nc by hN | ψγ i γ5 τ a ψ |N i = g0 Nc hN | X ia |N i ,
where g0 is a constant independent of Nc . X ia then can be expanded in a series in 1/Nc , X ia = X0ia +
1 ia X +... Nc 1
(The X used in Sec. 2 is now denoted byX0 .) The amplitude for pion-nucleon scattering from the diagrams in Fig. 2 is proportional to Nc3/2 X ia , X jb , X kc , −3/2
and violates unitarity unless the double commutator vanishes at least as fast as Nc , so that the amplitude is at most of order one. (In fact, one expects that the double commutator is of order 1/Nc2 since the corrections should only involve integer powers of 1/Nc . This result also follows from √ the large Nc counting rules which imply that each additional pion has a factor of 1/ Nc in the amplitude.) Substituting eq. (3.2) into the constraint gives ii ii h h h h (3.3) X0ia , X1jb , X0kc + X0ia , X0jb , X1kc = 0, h i using X0ia , X0jb = 0 from Eq. (2.3). The only solution to the consistency equation (3.3) is that X1ia is proportional to X0ia . This can be verified by an explicit computation of X1ia using reduced matrix elements, or by using group theoretic methods discussed in ref. . Thus we find that 1 c ia ia X0 + O , (3.4) X = 1+ Nc Nc2
where c is an unknown constant. The first correction to X ia is proportional to the lowest order value X0ia , so the 1/Nc correction to the axial coupling constant ratios vanishes. The overall normalization factor (1 + c/Nc ) can be reabsorbed into a redefinition of the unknown axial coupling g0 by the rescaling g0 → g = g0 (1 + c/Nc ), X ia → X0ia , so there are no new parameters at order 1/Nc in the axial current sector. The 1/Nc rescaling of g depends, in general, on the particular irreducible representation K, i.e. on the strangeness of the baryon. Thus in the case of three flavors, there is a 1/Nc term in the pion-baryon couplings from g(K), that depends on the strangeness of the baryons. This 1/Nc term can be shown to be linear in K, and has a coefficient is calculable in the SU (3) limit. ia which Eq. (3.4) implies that the commutator X , X jb is of order 1/Nc2 , since there are no 1/Nc terms in the expansion of X, and the leading term vanishes by Eq. (2.3). This yields 6
a pion-baryon scattering amplitude in Eq. (2.2) of order 1/Nc , when one includes the 1/Nc corrections for the pion-baryon vertices only. The full pion-baryon scattering amplitude is expected to be of order one, not of order 1/Nc . The order one contribution comes from the 1/Nc correction to the intermediate baryon propagator, that arises from the baryon mass splittings. At order 1/Nc , the baryons in an irreducible representation of the contracted SU (4) Lie algebra with a given value of K are no longer degenerate, but are split by an order 1/Nc mass term ∆M . The intermediate baryon propagator in Eq. (2.2) should be replaced by 1/(E − ∆M ). The energy E of the pion is order one, whereas ∆M is of order 1/Nc , so the propagator can be expanded to order 1/Nc as 1 ∆M 1 = + 2 +... E − ∆M E E
The order one contribution to the pion-baryon scattering amplitude arises from the second ia jb term in the expansion of the propagator, and is proportional to X , X , ∆M . Including the 1/Nc corrections to the propagator does not affect the derivation of Eq. (2.3), as the two terms in Eq. (3.5) have different energy dependences. The first term leads to the consistency condition Eq. (2.3) and the second gives the consistency condition on the baryon masses,,  ia jb kc X , X , X , ∆M = 0. (3.6) This constraint can be used to obtain the 1/Nc corrections to the baryon masses. The constraint Eq. (3.6) is equivalent to a simpler constraint obtained by Jenkins using chiral perturbation theory ia ia X , X , ∆M = constant. (3.7)
The solution of Eq. (3.6) or (3.7) is that the baryon mass splitting ∆M must be proportional to J 2 /Nc = j(j + 1)/Nc , where j is the spin of the baryon. 4. The Extension to Three Flavors The analysis so far has concentrated on the two flavor case. The 1/Nc expansion for baryons for three flavors is more complicated than for two flavors, because the flavor SU (3) representation of the baryons changes with the number of colors. The SU (3) weight diagram of the spin-1/2 baryons for Nc colors is shown in Fig. 3. For Nc = 3, this reduces FIGURE 3. to the familiar weight diagram for the baryon octet. The SU(3) weight diagram for the spin-1/2 baryons for Nc colors. The long edge of the triangle has Nc −1 states.
The method used in Ref.  for three flavors was to work with the SU (2) of isospin, and use the results of the previous sections for the different strangeness (i.e. K) sectors. The different strangeness sectors can then be related using K-meson–baryon scattering, in essentially the same manner as the analysis of Jenkins for baryons containing heavy quarks. The disadvantage of this method is that it does not use a manifestly SU (3) 7
invariant formalism. However, the big advantage is that this method can be used without assuming SU (3) symmetry, which allows one to analyze the structure of SU (3) breaking in the baryons in the 1/Nc expansion. The constraints on SU (3) breaking from the 1/Nc expansion are extremely interesting, and help resolve some long-standing puzzles about why SU (3) symmetry works so well in the baryon sector. We will see that for some quantities, SU (3) breaking is suppressed by powers of 1/Nc or 1/Nc2 . 4.1. F/D The K = 1/2 baryon tower contains the S = −1 baryons, and includes two spin-1/2 baryons, the Λ and Σ. The ratio of the pion couplings within a given tower is known to order 1/Nc2 . The explicit results from Ref.  give Σ+ → Σ0 π + =1+O Σ+ → Λ0 π +
√ This result√can be compared with the result for Nc = 3 in the SU (3) limit, 3F/D, to give F/D = 1/ 3 = 0.58, which is in excellent agreement with the experimental value. One can show that the F/D ratio of 2/3 obtained using the non-relativistic quark model is also consistent with all the pion-baryon couplings up to corrections of order 1/Nc2 . Thus one could equally well use 2/3 = 0.67 for the predicted value of F/D. There is an ambiguity in the value of F/D at order 1/Nc2 , which is connected with the fact that large Nc flavor representation of the baryons depends on Nc for three or more flavors. We will use the quark model value of 2/3 in what follows. This value is in good agreement with the experimental F/D ratio of 0.58 ± 0.04. The 1/Nc results for the baryon magnetic moments are very similar to those for the pion-baryon couplings. The F/D ratio is again predicted to be 2/3 plus corrections of order 1/Nc2 , and is in good agreement with the experimental value of 0.72. The difference between the experimental values of F/D for the baryon magnetic moments and pion couplings is an indication of the size of 1/Nc2 effects in QCD—they appear to be small, so that the 1/Nc expansion in the baryon sector appears to be under control. 4.2. Masses The baryon mass constraints can be analyzed for three flavors. Only the results will be presented here. It can be shown that the baryon masses must have the form,  1 M = Nc a + b K + Nc
c I +d J +e K
where a–e are constants which have an expansion in powers of 1/Nc . This equation is valid without assuming SU (3) symmetry, and holds for arbitrary values of the s-quark mass. It provides some interesting information on SU (3) breaking in the baryon masses. 8
which are valid in the 1/Nc expansion irrespective of the value of ms . One can also derive mass relations using SU (3) perturbation theory in ms , without using the 1/Nc expansion. This leads to the well-known relations valid including terms of first order in ms , 3Λ + Σ N + Ξ − = GMO = 0, 4 2 (Ω − Ξ∗ ) − (Σ∗ − ∆) = ESRI = 0,
1 1 ∗ (Σ − ∆) + (Ω − Ξ∗ ) − Ξ∗ − Σ∗ = ESRII = 0, 2 2 which are the Gell-Mann–Okubo formula, and the two equal spacing rules. Comparing Eqs. (4.3) with Eqs. (4.4) we see that ESRII is true in either the 1/Nc or the ms expansions. It is only violated at higher order than m2s /Nc2 , and so works extremely well (LHS=146 MeV, RHS=149 MeV).1 Similarly, the difference between GMO and 3/2 ESRI is only violated at order ms /Nc2 , and also works extremely well (LHS=7 MeV, RHS=3 MeV). The violation of ESRII and GMO − ESRI, which is of second order in both ms and 1/Nc , is comparable in size to isospin violating effects. The GMO relation is respected by the order Nc and order one terms, and is violated in the 1/Nc expansion at order 1/Nc rather than 1/Nc2 . It works about as well as the difference between GMO and ESRI. This means that a particular linear combination of the coefficients in Eq. (4.2) is about a third, instead of being one. The 1/Nc expansion partly explains the success of this relation; the other part is an accidental cancellation. The relation Σ∗ − Σ = Ξ∗ − Ξ is valid only in the 1/Nc expansion, but not in the ms expansion, and does not work as well (LHS=191 MeV, RHS=215 MeV) as the relations that are valid in both expansions. A systematic study of the mass relations obtained at different orders in the 1/Nc and ms expansions shows that SU (3) breaking effects are of order 25%, and 1/Nc effects are of order 1/3. Some relations, such as the GMO relation, work slightly better than this naive estimate. However, none of the relations has an unexpectedly large 1/Nc correction, which is a signal for the breakdown of the 1/Nc expansion. 1
In principle, the ms expansion has non-analytic terms of order ms . This contribution
vanishes for ESRII.
5. Conclusions The 1/Nc expansion provides a systematic method of computing the properties of the baryons. The results are in good agreement with the experimental values at Nc = 3 when the 1/Nc corrections are included. Whether this success holds for other quantities remains to be seen. The 1/Nc expansion also helps explain the nature of SU (3) breaking in the baryons. There are some interesting results on non-linear SU (3) breaking effects, and the connection with chiral perturbation theory, which I do not have time to discuss here. At present, the 1/Nc expansion can be used to find the operator form of 1/Nc corrections, but not the absolute normalization. The connection between SU (3) symmetry and the 1/Nc corrections is subtle, and needs to be explored further. Other approaches to the 1/Nc expansion for baryons,,  than the one presented here give additional insight. An analysis of the Adler-Weisberger sum rule in large Nc can be found in Ref. . Hadron scattering in the Nc → ∞ limit has been studied in Ref. , which derives an It = Jt selection rule for meson-baryon scattering.
6. Acknowledgements The work presented in this talk was done in collaboration with R. Dashen and E. Jenkins. This work was supported in part by the Department of Energy under grant number DOE-FG03-90ER40546 and by the National Science Foundation under PYI award PHY8958081.
R. Dashen and A.V. Manohar, Phys. Lett. B315 (1993) 425, 438. E. Jenkins, Phys. Lett. B315 (1993) 431, 441, 447. R. Dashen, E. Jenkins, and A.V. Manohar, Phys. Rev. D49 (1994) 4713. C. Carone, H. Georgi, and S. Osofsky, Phys. Lett. B322 (1994) 227. M. Luty and J. March-Russell, LBL-34778 [hep-ph/9310369]. W. Broniowski, TPR-93-39 [hep-ph/9402206]. G. ’t Hooft, Nucl. Phys. B72 (1974) 461, Nucl. Phys. B75 (1974) 461. E. Witten, Nucl. Phys. B160 (1979) 57. S. Coleman, Aspects of Symmetry (Cambridge, 1985). J.-L. Gervais and B. Sakita, Phys. Rev. D30 (1984) 1795. A.V. Manohar, Nucl. Phys. B248 (1984) 19. G. Karl and J.E. Paton, Phys. Rev. D30 (1984) 238; M. Cvetic, and J. Trampetic, Phys. Rev. D33 (1986) 1437. G.S Adkins, C.R. Nappi, and E. Witten, Nucl. Phys. B228 (1983) 552; N. Dorey, J. Hughes, and M.P. Mattis, [hep-ph/9404274]. R.L. Jaffe and A.V. Manohar, Nucl. Phys. B337 (1990) 509. E. Jenkins, Nucl. Phys. B368 (1992) 190. M.P. Mattis and M. Mukerjee, Phys. Rev. Lett. 61 (1988) 1344; M.P. Mattis and E. Braaten, Phys. Rev. D39 (1989) 2737.
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