LARGE N c QCD AND THE SKYRME MODEL

LARGE N c QCD AND THE SKYRME MODEL Norberto Scoccola (CNEA Buenos Aires). Introduction: Hadrons. QCD as the theory of strong interactions. Asymptotic freedom and confinement. Non-perturbative QCD 1/N c expansion of QCD: 1/N c counting rules for QCD Feynman diagrams. Application to meson phenomenology. Baryons in the 1/N c expansion of QCD. Extension to the study of excited baryons and their properties. Skyrme Model: Solitons. The SU(2) Skyrme model. The hedgehog skyrmion and its quantization. Extension to N f > 2 flavors. Multibaryon configurations.

Introduction Bibliography F. Close, Introduction to Quarks and Partons (Academic Press, 1979). F. Halzen and A.D. Martin Quarks and leptons (Wiley, 1984). D. Griffiths, Introduction To Elementary Particles (Wiley, 1987). Large N c QCD A. Manohar, Large N QCD. Proc. of Probing the Standard Model of Particle Interactions. F. David and R. Gupta editors (hep-ph/9802419). E. Jenkins, Large N(c) baryons, Ann. Rev. Nucl. and Part. Sci. 48 (1998) 81 (hep-ph/9803349). R. Lebed, Phenomenology of large N(c) QCD. Czech.J.Phys.49 (1999)1273 (nucl-th/9810080).

Skyrme Model R. Rajaraman, Solitons and Instantons (North-Holland, 1982). I. Zahed and G. Brown, Phys.Rept.142:1-102,1986. G. Holzwarth, B. Schwesinger, Rept.Prog.Phys.49:825,1986. H. Weigel, Int.J.Mod.Phys.A11:2419-2544,1996; Lect. Notes Phys.743:1-274,2008. G.E. Brown and M. Rho eds. The Multifaceted Skyrmion (World Scientific, to appear). Most contributions are already in the arxiv s

INTRODUCTION The first indication of the existence of the strong interactions appeared immediately after the discovery of the atomic nucleus (Rutherford, 1911). Nuclei contain protons (first known baryon), that repel each other due to their electric charge: it must exist an stronger attractive interaction. In 1932 the other baryon that composes the atomic nucleus is discovered: the neutron. En 1935, Yukawa, based on the ideas that explained the electromagnetic interactions in terms of photon exchanges, proposed that the strong interaction should be mediated by the exchange of a boson: he introduces the first meson, the pion (experimentally discovered in1948) During the 50 s other baryons (Δ, Λ, Σ, etc) and mesons (K, ρ, etc.) are discovered. There starts to be a proliferation of elementary particles. At the beginning of the 60 s the quark model (QM) is proposed (Gell-Mann, Zweig). According to the QM, hadrons (baryons and mesons) are composed by quarks. Strong interactions should be formulated in terms of quark interactions. This idea, together with the development of non-abelian gauge theories (Yang- Mills theories), led to the formulation of QCD

Hadrons and their properties Meson and baryon spectra according to Particle Data Group, C. Amsler et al., Physics Letters B667, 1 (2008). [ http://pdg.lbl.gov/ ]

Of course, many other hadronic properties are listed in the PDG. A (very) short list of them is The aim is to develop methods that could allow us to understand and predict these and other properties (i.e. such form factors, etc)

Flavor symmetries Low-lying baryon mass spectrum Degeneracies Symmetries

If we assume the breaking of the degeneracy is small, we see that these states can be accommodated in a mutliplet of degeneracy 8 (spin ½ octet) and another one of degeneracy 10 (spin 3/2 decuplet). These correspond to the dimension of some low dimensional SU(3) irreducible representations (irrep s) Similarly, for low-lying mesons Spin 0 meson octet Spin 1 meson octet

But these are not the lowest dimensional irreps of SU(3). They can be obtained by combining the fundamental irrep s 3 and 3

This is at the basis of the flavor S(3) quark model. Hadrons are form by combinations of quarks transforming in the fundamental irrep of SU(3). For example, explicitely, the flavor wave functions of the decuplet baryons are As we see, for example, the flavor wf of the Ω - is completely symmetric. Since this particle has spin 3/2 its spin wf is also symmetric. We have a problem here: Ω - is a fermion which implies that the full wf should be antisymmetric! A new quantum number has to be added COLOR. The idea is the wf is antisymmetric in color. We need three colors N c =3

QCD as the theory of strong interactions The QCD action has a SU(3) gauge symmetry. It reads 4 1 a μν cf f cf μν a ( μ γ ) μ S = d x q i m q G G 4 + μ a a g qcfγ λ qcf Gμν 1 cc 12 2 where c=r,g,b; a=1,..,8; f=u,d,s,..t; μ,ν = 0,..3 and the field tensor is G = G G + ig f G G a a a b c μν μ ν ν μ abc μ ν λ a are Gell-mann matrices and f abc current quark masses are the SU(3) structure constants. The quark u d s c b t ~3 MeV ~6 MeV ~170 MeV ~1.3 GeV ~4.3 GeV ~175 GeV

Schematically, the interactions are quark-gluon interaction 3-gluon and 4-gluon interactions g μ a a g qcfγ λ qcf Gμ 1 cc 12 1 Terms of 3rd and 4th order in G a m included in g g 2 1 4 G a G μν μν a The big difference with QED comes from the existence of the 3-gluon and 4-gluon interactions. Those interactions are due to the non-abelian nature of the gauge group. Let us see how this affects the behavior of the coupling constant.

Let us start by the case of QED. Quantum loop corrections lead to the renormalization of the electric charge. Namely, when one takes into account diagrams of higher order in α (=e 2 /4π) of the form Analytically, one gets that the running coupling constant is (for large values of Q 2 ) α( Q ) = 2 0 2 α 0 Q 1 log 2 3π α Γ where Γ is the cutoff needed to regularize the loop divergences Γ can be eliminated by defining α at a certain scale μ α 2 ( Q ) = αμ 2 ( ) 2 2 αμ ( ) Q 1 β1 log 2 2π μ where β 1 =2/3 As Q 2 increases, so does the running coupling constant and viceversa. That is, Screening for large distances (small Q 2 ), Strong coupling for short distances (large Q 2 )

In the case of QCD, there are extra diagrams due to the gluon-gluon interactions Following the same steps as before one gets α Q 2 s ( ) = 2 αs ( μ ) 2 2 αs ( μ ) Q 1 2 1 β log 2π μ but now β = 1 2N 11N f 6 c For N c = 3 and N f < 16 one has β 1 < 0. This implies 2 ( 2 Q ) lim α = 0 Q s Asymptotic freedom (Gross, Wilczek, Politzer, 73)

This figure shows the behavior of α s as a function of Q as obtained from more precise theoretical calculations, and compared to some empirical data. For Q ~ 1 GeV the coupling constant α s is of the order of 1. We have to use nonperturbative methods to treat QCD in that regime.

Another important observation (which has not been yet proven directly from QCD) is that quarks are confined. That is, they cannot be observed as isolated objects. The idea is that only color singlet quark configurations are allowed by QCD. 1 mesons: ( rr + gg + bb) 3 c1 c2 c baryons: ε 3 c1, c2, c where c, c, c= r, g, 3 q q q b 1 2 3 Anti-symmetric under color exchange

LARGE N c QCD As we have seen, for low momenta, the situation in QCD is very different from the one in QED in the sense that the strong coupling constant α s cannot be used as an expansion parameter. With no obvious expansion parameter, we must find an expansion parameter that is not obvious. It was pointed out by t Hooft that there is in QCD a not-so-obvious hidden candidate for a possible expansion parameter. t Hooft proposed a generalization of QCD in which the color group is extended from SU(3) to SU(N c ), where N c is an arbitrary large number. The idea behind this suggestion being that it could be possible to solve the theory in the large N c limit, and that the N c =3 theory may be qualitatively and quantitatively close to the large N c limit. As we will see QCD simplifies as N c becomes larger, and there is a systematic expansion in powers of 1/N c. Moreover, we will also see that there are indications that this expansion may be good approximation for 1/N c = 1/3.

Counting rules To analyze the N c counting rules for QCD, one needs a simple way to count the powers of N c in a given Feynman diagram. This can be done with the help of a trick due originally to t Hooft. The quark propagator is < q a ( x) q b ( y) >= δ ab S( x y) where a, b are indices in the fundamental representation a, b =1,.. N c. This is represented diagrammatically by a single line, and the color at the beginning of the line is the same as at the end of the line, because of the δ ab.

The gluon propagator is A B AB < G ( x) G ( y) >= δ D ( x y) μ ν μν where A, B are indices in the adjoint representation, A,B=1,.. N c2-1. Instead of treating a gluon as a field with a single adjoint index, it is convenient to treat it as an N c x N c matrix with two indices in the fundamental N c and N c representations, G m a b = G ma (T A ) a b. The gluon propagator can be rewritten as where the SU(N c ) identity a c 1 a c 1 a c < Gμ b( x) Gν d( y) >= δ dδ b δ bδ d Dμν( x y) 2 2Nc A a A ( T ) ( ) b c 1 a c 1 a c T = δ d dδ b δ b δ d 2 2Nc has been used. Note that for U(N c ) the corresponding identity does not have the last term. Thus, for large values of N c the difference between SU(N c ) and U(N c ) becomes irrelevant, and for the N c counting we can make the replacement A ( Gμ ) ( Gμ ) A = N ab= N 2 1,.., c 1, 1,.., a b c or graphically Strictly speaking we have to include a ghost field to cancel the extra U(1) gluon field. However, its effect is 1/N c 2 suppressed.

Similar replacements can be done for the interactions vertices g g g g g 2 g 2 Let us use these rules to consider the one gluon loop correction to the gluon propagator N c 2 g N c g g In order to have a consistent theory, the large N c limit has to be taken in such a way that 2 lim g Nc N c = constant t Hooft limit Namely, the QCD coupling constant g has to scale as g N 2 1/ c This analysis can be generalized to higher order diagrams

Using the double line notation one can easily determine the order in N c of any QCD diagram in the t Hooft limit. For example N g ϑ( N ) 2 6 1 c c

Using these counting rules we can consider different contributions to the gluon propagator planar diagrams ϑ(1) Quark loop diagrams ϑ(1/ N c ) Non-planar diagrams ϑ 2 (1/ N c )

This behaviour can be shown for any other diagram. That is, in general, For large N c planar diagrams dominate the dynamics. Non-planar diagrams are suppressed with respect to the planar ones by a factor 1/N c2 for each non-planar gluon Diagram containing quark loops are suppressed by a factor 1/N c for each quark loop. It should be noticed that even to perform the summation of the dominant diagrams (i.e. the planar ones) is a very formidable task. In fact, we do not know how to evaluate complicated planar diagrams with many gluon lines. However, as we will see, it is possible to obtain a clear qualitative picture of the behavior of the theory in the large N c limit from the knowledge that planar diagrams dominate, together with an additional hypothesis: we will assume the sum of planar diagrams leads to a confining theory.

Mesons in large N c QCD For arbitrary N c the normalized wave function of a meson reads Let us consider a typical correction to a meson propagator given Creation of a meson Annihilation of a meson 1 Nc g 1/ Nc Nc 1 ϑ 0 ( N c ) N c g 1/ Nc N c Therefore, the meson mass m behaves as m = ϑ(1)

In general, one can show that the n-meson amplitude is of order ( 1 ϑ N n /2 ) c Meson decay amplitude into the vacuum: f π f π = ϑ ( ) N c Meson decay amplitude into two mesons ( ) 1/ N c ϑ Meson strong decays suppressed Meson-meson scattering amplitude ( ) 1/ N c ϑ Mesons interact weakly

qq, qγ μ q We consider now the quark bilinear operators J (i.e., etc). We will establish that, in the large N c limit, J acting on the vacuum creates only one-meson states. Namely, it has vanishing matrix elements to create a two meson state, or meson and a glue particle, etc. This is equivalent to claim that any two point function < J(k) J(-k) > can be written as 2 fn < Jk ( ) J( k) >= k m n 2 2 n where f n = < 0 J n > is the amplitude to a create a meson from the vacuum and n runs over meson states Let us consider an arbitrary leading contribution to this two point function gggqq We have to show that the intermediate state forms a single singlet color state corresponding to a pertubative approximation to a meson. gqq The alternative would be that, for example, are coupled to a color singlet while the other two gluons are also coupled to a singlet, corresponding to perturbative approximations to a meson and glueball, respectively.

l k j i ql A k A j A i q Intermediate state forms a single color singlet state On the other hand the expansion in terms of one meson intermediate states should hold for arbitrary momentum k. For large k we can use the perturbative QCD result 2 ( 2 2) < J( k) J( k) > ln k / μ k To obtain this logarithmic behavior out of a sum over intermediate meson states the number of these meson states should be infinite (for each channel). For large N c there are infinite mesons states, which are narrow and weakly coupled between each other. (t Hooft 74)

Other result for mesons Using the corresponding currents something similar can be shown for glueballs. The glueball-meson mixing is suppressed since it is proportional to 1/ Nc qqqq Multiquark meson states are suppressed by 1/N c The mesonic processes that require the annihilation of pairs are suppressed (OZI rule). For example, φ(1020) decays mainly into two kaons although nonstrange modes (ρπ, etc) have larger phase space. qq Predictions for Chiral Perturbation Theory coefficients

Baryons in large N c QCD The situation for baryons is qualitatively different than that for mesons since a Large N c Baryon is formed by N c quarks in a completely anti-symmetric color combination ( color singlet ) ε ii i qq q i1 i i 2... Nc 12,.. N c To appreciate the problem let us consider a few low order perturbative corrections to the free quark propagation of N c quarks N c N c 2 N c 3 With each diagram diverging more severely than the one before the situation seems to be hopeless!!

To understand what is going on let us assume that quarks are heavy. In such a case we can use a non-relativistic picture and write H = N m+ N t+ V c c Current quark mass Kinetic energy of each quark Interaction potential energy Considering only linked contributions V = +... 2 2 1 g Nc ϑ( N c ) 6 4 1 g Nc ϑ( N c ) g 6 4 1 Nc ϑ ( N c ) Combinatorial 4 3 1 g Nc ϑ( N c ) 4 3 1 g Nc ϑ( N c )

Therefore 1 c c c c V = ϑ( N ) = N v H = N ( m + t + v) = N h and we expect to have M N N 1 B = cε = ϑ( c) Once this has been understood it is not difficult to understand the apparently divergent behavior of the perturbative series found before. Such a series corresponds to an expansion of the amplitude for a free baryon propagation 1 i 2 3! 2 2 3 3 exp(- im Bt) = 1 i M Bt M Bt M B t +... The apparent bad behavior of the series does not imply that large N c baryons do not exist but that the baryon mass is order N c.

The problem can be conveniently treated using the methods of many body theory. In particular, a Hartree picture arises. The logic behind this approximation is as follows. For large N c. the interaction between each pair of quarks (e.g. one gluon exchange) is negligible (i.e. of order 1/ N c. ). But the total potential experienced by any one quark is of order one, since any one quark interacts with N c. -1 other quarks. To find the ground state baryon, each quark should be placed in the ground state of the average potential it experiences. We can do that because the color part of the wave function is already antisymmetric. Thus, the rest of the full many-body wavefunction can be written as Nc S a1... a x N 1 xn = x c c a... a i 1 N c i= 1 ψ (,..., ) χ φ( ) Spin-flavor symmetric wave function N c Ground state baryon

φ(x i ) is solution of the equation hφ( x) = ( m+ t+ v) φ( x) = εφ( x) For example, if we consider that the quark-quark interaction is given by a simple Coulomb potential, we have that the total baryon potential energy is V = g N c 2 1 x x i< j i j and v= g d y * 2 3 φ ( y) φ( y) x y Of course, in general we do not know the detailed form of V and, thus, of the functional v. However, the important conclusion that emerges from this analysis is that the wavefunction is order N c 0 Therefore, for large N c a Hartree picture of the baryons arises: the mass of the baryon is of order N c, while its size and shape are of order N c 0 (Witten 79)

We can now consider the large N c behavior of other baryon properties Combinatorial Baryonbaryon interaction g ϑ( N ) 2 2 1 N c c Baryon-meson interaction 1 N N c c ϑ 1/2 ( N c )

Baryon-meson scattering amplitude ( a) ( b) 2 1 N c N c 2 1 2 2 g Nc Nc ϑ 0 ( Nc )

Spin-flavor symmetry for baryons. Consistency relations Let us start by considering baryon meson scattering not in terms of quarks but rather hadrons. Suppose that the meson is a pion which, as NG boson of chiral a a symmetry, couples derivatively through the axial vector current A μ = qγ μ γτ where τ a 5 q is a generator of the flavor group. The scattering process consists of two insertions of A ma on the baryon line which can occur in two orderings (a) (b) Note that from the considerations given before the sum of these two contribution should be of order N c 0

To calculate them we need the explicit form of the interaction vertices and the baryon propagator. For the case of the absorption of the incoming meson (diagram (a)) the momentum of the intermediate baryon is P μ = M v m = M v Bm + k m where v Bm is the four-velocity of the initial baryon and v m that of the intermediate one. Note that v B.v B = v.v =1. Clearly, No recoil v m = v B m + ϑ(1/n c ) Then, baryon propagator is S 1 i( P a ( P μ ) = 2 2 + M) i 1+ v P M k. v 2 B B Using that in the rest frame of the baryon k.v B =k 0 = E π S i ( P ) = ϑ( N ) 1 μ 0 a c Eπ Using similar arguments for diagram (b) S b 1 μ ( P ) i E π we get

We consider now the πbb vertex which, as we have already seen, is. At low energies we can use the usual derivative coupling iπ f π a μ f π π a ( μa A ) ' a π a A = < B' q γγ τ q B> BB ' f ia i i ( ) 5 π BB ϑ 1/2 ( N c ) 1/2 μa Since fπ ϑ ( N c ) we see that ( A ) ϑ( Nc). In the large N c limit the BB ' baryon is infinitely heavy as compared to the pion, so the πbb coupling reduces to the static one Where i=1,2,3. Thus, (A ia ) B B carries spin and isospin 1. To keep the N c- dependence explicit we write ( ia ) ia ' ' c A = gn < B X B> BB g and < B X ia B > are both of order N c0 and, thus, have well defined large N c limit.

With all these ingredients the contribution of the relevant diagrams to the πb scattering amplitude is = gk k' N jb ia i < B' [ X, X ] B > 2 i j 2 c 2 Eπ f π Note that the product of X-matrices sums over all possible spin and flavor quantum numbers of the intermediate states which have the same mass as B and B. Naively, it appears that is of order N c which is in contradiction with our previous finding. The only way to avoid this contradiction is to have many degenerate states in such a way that cancellations appear and the matrix elements of the conmutator are of order ϑ(1/ N c ) even though the matrix elements of each X are individually of order ϑ(n c 0 ). Thus we must have jb ia 1 [ X, X ] = O( N c ) Consistency relations Gervais-Sakita 84 Dashen-Manohar 93 Writing X X ia = X + +, X0 = 0 ia ia 1 jb 0.. this implies 0 N X c

Since X ia transforms as a vector under spin and flavor transformations we have that in the large N c limit J i, T a, X ia form the algebra J, J = iε J ; J, X = iε X i j k i ja ka ijk 0 ijk 0 T, T = i f T ; T, X = i f X a b c a ib ic abc 0 abc 0 ia jb X0, X0 = 0 which is the contracted algebra SU(2 N f ) c. Therefore large N c baryons should fall into irrep of the corresponding Lie group. Note that this algebra can be obtained from the usual SU(2 N f ) i j k i ja ka J, J = iεijk J ; J, G = iεijkg a b c a ib ic T, T = i fabc T ; T, G = i fabcg ia jb i c δab k 1 ck G, G = δij fabc T + iεijk J + dabc G 4 2N f 2 by considering X 0 ia = G ia /N c and taking the limit N c