The 1/N_c expansion in hadron effective field theory
aa r X i v : . [ h e p - ph ] D ec The /N c expansion in hadron effective field theory Guo-Ying Chen
Department of Physics and Astronomy, Hubei University of Education, Wuhan 430205, China ∗ (Dated: December 4, 2018)We study the N c scalings of pion-nucleon and nucleon-nucleon scatterings in hadron effectivefield theory. By assuming Witten’s counting rules are applied to matrix elements or scatteringamplitudes which use the relativistic normalization for the nucleons, we find that the nucleon axialcoupling g A is of order N c , and a consistent large N c counting can be established for the pion-nucleon and nucleon-nucleon scatterings. We also justify the nonperturbative treatment of the lowenergy nucleon-nucleon interaction with the large N c analysis and find that the deuteron bindingenergy is of order 1 /N c . PACS numbers:
In the seminal paper Ref. [1], ’t Hooft showed that QCD has a hidden expansion parameter 1 /N c . Withthe 1 /N c expansion the QCD coupling constant g is of order 1 / √ N c , and gluons are denoted by double lines.By counting the number of interaction vertexes and closed color loops, one can figure out the N c order ofa specific Feynman diagram. It is then found that in the large N c limit, the leading contribution comesfrom planar diagrams with minimal quark loops. The idea to expand QCD in 1 /N c is very attractive, asit explains hadron phenomenology successfully, for instance the OZI suppression rule. The extension of the1 /N c expansion to baryons was first carried out by Witten [2]. The 1 /N c expansion of baryons, which arebound states of N c valence quarks and have masses of order N c , is more complicated than that of mesons.Using quarks and gluons as degrees of freedom, Witten showed that meson-baryon coupling is of order √ N c , meson-baryon scattering amplitude is of order N c , and baryon-baryon scattering amplitude is of order N c (these results are called the large N c counting rules of Witten in this manuscript). It is interesting tostudy whether a realistic hadron effective field theory using baryons and mesons as degrees of freedom canreproduce the large N c counting rules of Witten. We will see in the following that this is not a clearly solvedproblem. (a) (b) (c) FIG. 1: Pion-nucleon scattering diagrams in leading chiral expansion.
First of all, we consider the pion-nucleon scattering. The leading pion-nucleon scattering diagrams inchiral expansion are shown in Fig. 1. In chiral theory, the pion-nucleon vertex is proportional to the factor g A f π , where f π is the pion decay constant with order √ N c , and g A is the nucleon axial coupling constantwhich is assumed to be of order N c in both the nonrelativistic quark model[3] and the skyrme model [4].The nucleon propagator is of order N c [5]. Therefore, both amplitudes of Fig. 1(a) and (b) are proportionalto the factor ( g A f π ) and are of order N c , and Fig. 1(c) is proportional to 1 /f π and of order 1 /N c . Clearlythese results contradict the large N c counting rules of Witten. An solution to this puzzle has been proposedin Ref. [6–8], and we give a short review here for the convenience of further discussions. The axial vector ∗ Electronic address: [email protected] urrent matrix element in the nucleon is of order N c and can be written as [8] h N | ¯ ψγ i γ τ a ψ | N i = N c g h N | X ia | N i , (1)where the N c dependence has been explicitly factored out, so h N | X ia | N i and g are of order one. The sumof the scattering amplitudes for the pole diagrams of π a ( q ) + N ( P ) → π b ( q ′ ) + N ( P ′ ) reads − iq i q ′ j N c g f π (cid:20) q X jb X ia − q ′ X ia X jb (cid:21) , (2)where the amplitude is written in the form of an operator acting on nucleon states, and q = q ′ as bothinitial and final nucleons are on-shell. One can see explicitly that the above amplitude is of order N c andviolates the unitarity. The solution to this puzzle is the commutation relation in the large N c limit[ X ia , X jb ] = 0 , (3)which is usually called the large N c consistency condition. The solution of the large N c consistency conditionrequires to consider all the possible baryon states degenerate with the nucleon in the intermediate statesof the scattering. For instance, besides the nucleon, ∆ should also be included as an intermediate state inthe pion-nucleon scattering, and the coupling g πN ∆ can be determined in terms of g πNN using the large N c consistency condition. The large N c consistency condition can be derived from the spin-flavor algebra.Consider the two-flavor case, the SU (4) generators J i , I a and G ia satisfy the spin-flavor algebra (cid:2) J i , J j (cid:3) = iǫ ijk J k , (cid:2) I a , I b (cid:3) = iǫ abc I c , (cid:2) I a , G ib (cid:3) = iǫ abc G ic , (cid:2) J i , G ja (cid:3) = iǫ ijk G ka , (cid:2) J i , I a (cid:3) = 0 , (cid:2) G ia , G jb (cid:3) = i δ ab ǫ ijk J k + i δ ij ǫ abc I c . (4)Rescaling the generator G ia by a factor of 1 /N c and taking the large N c limit, X ia = lim N c →∞ G ia N c , (5)we can find that the commutation relation for [ G ia , G jb ] turns into Eq. (3) in the large N c limit, as the orderof baryon matrix elements of J and I are at most of N c . A careful study find that the commutator [ X ia , X jb ]is of order 1 /N c [9], hence Eq. (2) is of order 1 /N c , which is the same as the amplitude of Fig. 1(c). SinceWitten’s counting rules suggest that the leading contribution should come from order N c diagrams in the1 /N c expansion, one may wonder why the pion-nucleon scattering amplitude is dominated by order 1 /N c diagrams instead of order N c diagrams. NN FIG. 2: A typical diagram of order N c which contributes to the nucleon-nucleon interaction in large N c QCD.
Secondly, there is difficulty in matching the N c counting of the nucleon-nucleon potential calculated atthe quark-gluon level to that calculated at the hadronic level. The dominant nucleon-nucleon interaction isgenerically of order N c with the analysis at the quark-gluon level [2]. This can be understood from the one-2luon exchange diagrams in Fig. 2, where each gluon coupling has a factor 1 / √ N c , and there are N c ways tochoose a pair of quarks. In studying the N c counting of the nucleon-nucleon potential, Ref. [10, 11] considerthe kinematic region p ∼ N c for the nucleon and assume that quark-line connected diagrams such as Fig. 2are of order N c leads to a potential of order N c . It should be note that the kinematic region is different fromthat was considered in Witten’s paper, where p ∼ N c [2]. The reason is that, if p ∼ N c which is the sameorder as the nucleon mass, the nucleon is a relativistic particle, and this kinematic region is beyond the scopeof the low energy hadron effective field theory [12]. At the hadronic level, it seems straightforward to see thatthe one-meson exchange potential is of order N c , for example the one-pion exchange potential is proportionalto the factor ( g A f π ) and of order N c . At the two-meson exchange level, diagrams(for instance, the crossed-box diagram) with four nucleon-meson vertexes are of order N c and will contribute to the potential, butRef. [12] shows that there is a cancelation of the retardation effect of the box graph against the contributionof the crossed-box diagram, therefore the dominant nucleon-nucleon potential remains to be of order N c .Nevertheless, it is difficult to show that whether such cancelations will happen in all multi-meson exchangelevels. Actually, it is found that at the three- and higher-meson exchange level the potential derived fromthe hadronic theory can be larger than that of the quark-gluon theory [13]. This problem is called the large N c nuclear potential puzzle.Two possible resolutions to the large N c nuclear potential puzzle are suggested in Ref. [13]. One possibilityis that necessary cancelations might happen if the hadronic-level calculation is reorganized in some other way.A tentative study on this direction is done in Ref. [14], where the hadronic-level calculation is reorganizedin the way that the potential is energy-independent. The other possibility is that the N c scaling rule of thenucleon-nucleon potential proposed in Ref. [10, 11] may be invalid. This can be plausible by noting thatsome nuclear phenomena may be difficult to understand if the nucleon potential is of order N c , for example,the deuteron binding energy is a small number in reality, i.e., B = 2 . N c = 3 [15, 16], it will stillbe interesting to find an alternative solution which does not rely on the fact that N c = 3 in the real world.In addition, since the nucleon kinetic energy is order N − c , if the potential is order N c nucleon matter formsa crystal in the large N c limit. However, nuclear matter appears to be in liquid state rather than in crystalstate in reality. Generally, nuclear phenomena seem to prefer a small potential value, and this observationmotivates Ref. [17, 18] to proposed a refined quark model which gives g A ∼ N c , thus the one-pion exchangepotential is of order N − c .From the above discussions, we can see that it is not clear how to reproduce the large N c counting rules ofWitten consistently in a realistic hadron effective field theory. In this paper we try to propose a simple wayto resolve the above difficulties. We find that we can have a consistent large N c counting in hadron effectivefield theory if we assume Witten’s counting rules are applied to matrix elements or scattering amplitudeswhich use the relativistic normalization for the nucleons. Our work is organized as following: we first discussthe difference of N c scalings between the relativistic and nonrelativistic normalizations. With the relativisticnormalization, we then discuss the N c scalings of pion-nucleon and nucleon-nucleon scatterings. We finallyextend our study to the meson-meson scatterings and find that loosely-bound meson-meson molecular statesmay not exist in the large N c limit.The leading non-relativistic chiral Lagrangian for the pion-nucleon coupling reads L πN = g A f π ψ † ~σ · ( ~∂π ) ψ, (6)where ψ is the nucleon doublet and π = τ i π i . If g A is of order N c , the pion-nucleon coupling is proportionalto g A f π and of order √ N c , which is just the result of Witten. But if g A ∼ N c as suggested in Ref. [17, 18], thepion-nucleon coupling is of order N − / c , which seems to contradict the large N c counting rules of Witten.One should note that in the above N c counting Feynman rules for the external nucleon lines are assumed tobe independent of N c , or explicitly one use the nonrelativistic normalization for the nucleon h p ( ~k, s ) | p ( ~p, s ) i = δ ( ~k − ~p ) δ s ,s . (7)However from the Dirac theory, we know that the Feynman rule for the external nucleon line is u s ( p ), whichreduces to √ m N χ s ( χ s is the two-component spinor) in the nonrelativistic limit and obviously depends on3 c . Thus the relativistic normalization for the nucleon reads h p ( ~k, s ) | p ( ~p, s ) i = 2 m N δ ( ~k − ~p ) δ s ,s , (8)where we have taken the nonrelativistic limit for the Dirac spinor u s ( p ) to simplify the N c counting in thelow energy effective field theory. Scattering amplitudes which use Eq. (8) as the normalization conditioncan be understood as the nonrelativistic reduction of the relativistic form. In this work, we will use therelativistic normalization for the scattering amplitudes, thus the relation between our defined scatteringamplitude M ( p , p → p f ) and the S -matrix reads S = 1 + i T , (9)where < p f | i T | p , p > = i (2 π ) δ (4) ( p + p − X p f ) M ( p , p → p f ) . (10)To obtain the scattering amplitudes with the nonrelativistic normalization M NR , one takes the limit | ~p | → √ m N which come from external nucleon spinor function u s ( p ). For example, for thenonrelativistic nucleon-nucleon elastic scattering, we can have M NR = M / (2 m N ) . One should note thatwe only discuss the nonrelativistic scattering in this work as the relativistic normalization reduce to the formin Eq.(8). We will find that Witten’s large N c counting rules can be consistent, if one assume that theserules are applied to matrix elements or scattering amplitudes which use the relativistic normalization for thenucleon.We now return back to the discussion of the pion-nucleon coupling. With the normalization in Eq.(8),there is a factor of √ m N for each external nucleon line, and the pion-nucleon coupling is proportional to m N g A f π . Our matching scheme is that we assume Witten’s counting rules are applied to matrix elements orscattering amplitudes which use the relativistic normalization for the nucleon, we then find that g A ∼ N c as the pion-nucleon coupling is order √ N c from Witten’s rules. Thus we have shown that if g A ∼ N c assuggested in Ref. [17, 18], the N c counting of the pion-nucleon coupling can be consistent with Witten’sresult. In the following we will count g A as order N c , since it is the consequence of our assumption.Now we come to discuss the N c scaling of the pion-nucleon scattering. The scattering amplitude for thepion-nucleon scattering given in Eq. (2) is written with the nonrelativistic normalization. But noting that,while factors of √ m N which come from the external nucleon lines are dropped in Eq. (2), such factor iskept for the baryon propagator. This can be obviously since the baryon propagator iq in Eq. (2) is obtainedfrom the Dirac propagator by taking the nonrelativistic limit i ( /P + /q + m B )( P + q ) − m B → iq · v ( 1 + /v , (11)where m B is the mass of intermediate baryon, v µ is the four-velocity of the initial nucleon, i.e., P µ = m N v µ ,and it is taken to be v µ = (1 ,~
0) in the nonrelativistic limit. With the the relativistic normalization, factorsof √ m N should be restored for external nucleon lines in the scattering amplitudes, then the scatteringamplitudes for Fig. 1(a,b) are proportional to m N ( g A fπ ) , and the amplitude for Fig. 1(c) is proportional to m N /f π , all of which are of order N c and consistent with the large N c counting rules of Witten. Actuallywhat we have illustrated in the above is that if the assumption that Witten’s rules are applied to matrixelements or scattering amplitudes which use the relativist normalization for the nucleon is adopted for thepion-nucleon coupling, the N c scaling of the pion-nucleon scattering will be consistent with this assumption.To have a consistent N c counting with the nonrelativistic normalization, one can redefine the nucleon fieldby dividing a factor of √ m N , then factors of √ m N are not needed in the Feynman rules for the externalnucleon lines. Meanwhile, the nucleon propagator should be i m N q instead of iq as the new defined nucleonfield has the dimension one, and g A should absorb a factor of 2 M N which is dimension one and of order N c . With this convention the scattering amplitudes for Fig. 1(a,b) are proportional to ( g A fπ ) m N q , whichis also of order N c as g A is of the order N c . Therefore, we can see that physical results do not depend onthe normalization conventions as long as they are used consistently. The advantage of using the relativistic4ormalization is that g A is a dimensionless constant as usually defined in the text book.It is worth to mention that, although with the relativistic normalization g A is of order N c , the commutationrelation in Eq. (3) still holds in the large N c limit. This can be shown by rewriting the axial current matrixelement in Eq. (1) with the normalization in Eq. (8) h N | ¯ ψγ i γ τ a ψ | N i = 2 m N g h N | X ia | N i = N c m N N c g h N | X ia | N i , (12)where h N | X ia | N i and g are of order one, and the product g h N | X ia | N i is different from that in Eq. (1) bya N c independent factor m N N c . Since the matrix element of X ia defined in Eq. (12) is different from that inEq. (1) by a N c independent factor, operators X ia defined in Eq. (12) still satisfy the commutation relationin Eq. (3), i.e., the large N c consistency condition. The commutation relation Eq. (3) arises because theaxial vector current matrix element grows with N c . With our normalization, g A is of order N c , but the axialvector current matrix element is still of order N c , thus the commutation relation Eq. (3) still holds. Takingthis contracted spin-flavor symmetry into account, we should also include ∆ particle in the intermediatestate, then the contributions from pole diagrams Fig. 1(a) and (b) vanish in the large N c limit. + + + · · · FIG. 3: Feynman diagrams generated by the contact interaction.
We then come to the nucleon-nucleon scatterings. As in the above, we assume Witten’s counting rules areapplied to the nucleon-nucleon scattering amplitudes which use the relativistic normalization. In this way,nucleon-nucleon scattering amplitudes with the relativistic normalization are of order N c . The order N c scattering amplitude leads to a potential of order N − c , as factors of √ m N which come from the externalnucleon lines should be dropped in the scattering amplitude to obtain the non-relativistic potential. Thereforeour resolution to the nuclear potential puzzle corresponds to the second possibility suggested in Ref. [13], i.e.,the assumption that the dominant nucleon-nucleon potential is of order N c is somewhat heuristic and may beinvalid. We will take the pion-exchange as an example to show explicitly how the large N c nuclear potentialpuzzle can be resolved. The nucleon-nucleon scattering amplitude for the one-pion exchange diagram isproportional to the factor ( m N g A f π ) , which is of order N c and again consistent with our assumption. Theamplitude of two-pion exchange crossed-box diagram is proportional to m N ( g A f π ) , which is of order N c and thus is suppressed by 1 /N c . It is obvious that more-pion exchange diagrams which are two-nucleoninreducible, will be suppressed by more powers of 1 /N c . Thus one can see that the nucleon-nucleon potentialis at most of order N − c , and this large N c counting rule will not be violated at the multi-pion exchange level.It is straightforward to see that similar conclusion can be obtained for other meson-exchange diagrams, inparticularly the sigma-exchange potential is also of order N − c .So far, we have shown that a consistent large N c counting in hadron effective field theory can be established,if we assume Witten’s counting rules are applied to matrix elements or scattering amplitudes which usethe relativistic normalization for the nucleons. Nevertheless there are still two points in nucleon-nucleonscatterings need to be further investigated. The first point is that one may wonder whether the order N c scattering amplitude violates the unitarity. We now show that this does not happen in the nucleon-nucleonscattering. To be specific, let’s consider the single channel S -wave nucleon-nucleon scattering. The S -matrixfor the S -wave nucleon-nucleon scattering can be written as S = 1 + i p πm N M , where p is the nucleonmomentum and M is the scattering amplitude which use the relativistic normalization. Unitary condition,i.e. SS † = 1, requires that Im M = p πm N |M| . (13)5e consider the kinematic region p ∼ N c as in Ref. [11] and denote the N c scaling of the amplitude asRe M ∼ N n c , Im M ∼ N n c . The unitary condition in the large N c limit can then be written as n = 2max( n , n ) − . (14)The solution to the above condition readsIf n > n , then n n = 1 , M ∼ O ( N c );If n < n , then n < n < , M < O ( N c ) . (15)Thus one can see that the unitary condition can be satisfied if the scattering amplitude is of order N c .The second point is that in hadron effective field theory deuteron corresponds to a bound state polewhich emerges from the non-perturbative summation of the nucleon-nucleon scattering amplitudes, henceit is interesting to investigate whether such non-perturbative summation is justified in the large N c limit.Following, we will study this point in detail with a low energy pionless effective field theory. NN FIG. 4: A typical quark-disconnected diagram.
Let’s now consider the low energy pionless effective field theory in the S channel [19–21]. The leadingorder operator in the chiral expansion for nucleon-nucleon interactions reads L NN = − C ( ψ † ψ ) . (16)Feynman diagrams generated by this interaction vertex are shown in Fig. 3. We can identify the order N c quark-connected diagrams, such as Fig. 2, as the tree diagram in Fig. 3, and identify the quark-disconnecteddiagrams, such as Fig. 4, as the one-loop diagram in Fig. 3. The amplitude for the tree diagram in Fig. 3reads i M tree = − i m N C , (17)where the factor 4 m N comes from four external nucleon lines. M tree is of order N c , according to ourassumption that Witten’s counting rules are applied to the scattering amplitudes which use the relativisticnormalization. We can then find that C is of order N − c . Amplitudes for the loop diagrams can be obtainedafter treating the loop integral. In the pionless effective field theory, the loop integral in can be done in the6onrelativistic approximation. Using the minimal subtraction scheme, the loop integral reads [20] I = Z d D ℓ (2 π ) D iℓ − ~ℓ / (2 m N ) + iǫ · iE k − ℓ − ~ℓ / (2 m N ) + iǫ , = Z d D − ℓ (2 π ) D − iE k − ~ℓ / ( m N ) + iǫ , = i m N π ( − m N E k − iǫ ) / , = m N p π , (18)where E k = ~p /m N is the total kinetic energy of the two-nucleon in the center of mass frame. In the chiralexpansion the three-momentum of the nucleon is treated as small scale and has the same order as m π , thuswe assume p to be independent of N c as in Ref. [11]. It is worth mentioning that the nucleon propagatorused here is order N c and different from that in Eq. (2), as the term ~ℓ / m N should also be included in thepropagator to avoid the infrared divergence in the two-nucleon reducible diagrams [19]. With the result forloop integral, one can obtain the one-loop amplitude i M loop = − p πm N M tree = − π m N C p. (19)We can find that the one-loop amplitude is of order N c , which is the same as that of the tree diagram.This result is somewhat surprising at the first sight, as quark-disconnected diagram Fig. 4 seems to be oforder N c . Actually to count the N c order of Fig. 4 one should note that such a diagram contains nucleonpropagators and loop, thus its N c order can be the same as that obtained in the hadron effective field theory,i.e., O ( N c ). It is straightforward to count the N c order of all the other diagrams in Fig. 3. One can find thatall the diagrams in Fig. 3 are of order N c , hence the amplitude at the leading order in the 1 /N c expansionshould come from the non-perturbative summation of all the diagrams in Fig. 3. The re-summed amplitudereads i M sum = − i m N C i C m N π p , (20)which is obviously of order N c . The partial wave S -matrix can then be written as S = 1 + i p πm N M sum = 1 − i m N p π C i C m N π p . (21)One can find that, although M sum is of order N c , the S-matrix is independent of N c and satisfies the unitarycondition, i.e. SS † = 1. Actually it has already been shown in Ref. [20] that all the diagrams in Fig. 3need to be summed, because they all are at the leading chiral order O ( p − ). Here we have shown that thissummation is also justified in the large N c expansion. The deuteron corresponds to a bound state pole at E k = − B in Eq.(20), where B is the binding energy, B = 16 π C m N , (22)which is of order N − c . Similar analysis has been done in Ref. [22], but the conclusion is different. Because C is taken to be O ( N c ), B is found to be O ( N − c ) in Ref. [22]. However, if C is order N c and B is order N − c , it will then be a puzzle why the binding energy is so small, while the nonrelativistic potential is large.We can also study the N c scaling of the deuteron binding energy in the pionfull effective field theory withthe method used in Ref. [22]. In the chiral limit, the Schr¨odinger equation in coordinate space can be simply7ritten as [22] ( ∇ m N − α π r ) | Ψ > = − B | Ψ >, (23)where α π = g A πf π . (24)Because the three-momentum of the nucleon and g A are of order N c , both kinetic energy and tensor potentialenergy in the left hand side of Eq.(23) are of order N − c , and the binding energy B is of order N − c which is thesame as that in the pionless effective field theory. This conclusion remains unchanged even if an additionalterm C δ (3) ( r ) is included in the potential. In contrast, by treating g A as O ( N c ), Ref. [22] assumes that thecoordinate scales as N c , or equivalently the three-momentum carried by the nucleon scales as 1 /N c , then B scales as N − c . However, as mentioned in Ref. [22], if the three-momentum carried by the nucleon scales as1 /N c , the effective field theory will be constrained to threshold.It is interesting to extend the above analysis to discuss the existence of S -wave meson-meson molecularstates. We will find that the N c analysis of the deuteron cannot be straightforwardly extended to the meson-meson case as the meson-meson scattering amplitudes and meson masses have different N c countings. Ineffective field theory approach, an S -wave meson-meson molecular state corresponds to a bound state pole inthe elastic meson-meson scattering amplitude coming from the summation of all the diagrams in Fig. 3 [23].Similar to Eq. (22), the binding energy ˜ B for the meson-meson molecular state reads˜ B = 2 π ˜ C µ , (25)where ˜ C is the coefficient of the contact four-meson operator, and µ is the reduced mass of the two-mesonsystem. ˜ C is of order N − c , as the tree level amplitude for the meson-meson scattering scales as N − c . µ is of order N c , as the meson mass is independent of N c . Therefore, we can find that the binding energyof the S -wave meson-meson molecular state scales as N c , and the binding momentum is of order N c . Thisindicates that if the meson-meson molecular state exists, it should be deeply bounded. However, mesonswith the momenta of order N c are extremely-relativistic particles, and their scatterings cannot be treatedin the nonrelativistic effective field theory. If the momentum of the meson is taken to be independent of N c as that of the meson mass, we will find that the diagrams in Fig. 3 have different N c scalings. The treediagram in Fig. 3 scales as N − c , and the n -loop diagram are suppressed which scales as N − ( n +1) c . Theseresults are different from the S -wave nucleon-nucleon scattering, due to the fact that the meson mass isof order N c , while the nucleon mass is of order N c . Thus for meson-meson scatterings, only the tree leveldiagram contributes to the leading order amplitude in the 1 /N c expansion, which is just the conclusion inthe standard large N c analysis for meson-meson interactions [2], and the summation of all the diagrams inFig. 3 is unnecessary. We then conclude that, there is no loosely-bound meson-meson molecular state inthe large N c limit, as the meson-meson interaction is weak, and similar conclusion has also been given inRef. [24]. Finally, we would like to mention that we only discuss the existence of the meson-meson molecularstate in the large N c limit, for other configurations such as tetraquark and polyquark states one can refer toRef. [25] and references therein.In summary, we have tried to propose a possible solution to overcome the difficulties which are encounteredin reproducing the large N c counting rules of Witten in hadron effective field theory. We find that aconsistent large N c counting can be established if we assume Witten’s counting rules are applied to matrixelements or scattering amplitudes which use the relativistic normalization for the nucleons. We also findthat at the leading order in the 1 /N c expansion, the S -wave nucleon-nucleon scattering should be treatednonperturbatively, and the deuteron binding energy is of order 1 /N c which is consistent with the nuclearphenomena. In contrast, the S -wave meson-meson interaction is weak, and loosely-bound meson-mesonmolecular states may not exist in the large N c limit. 8 CKNOWLEDGMENTS
I would like to thank Yu Jia for helpful discussions and J.Ruiz de Elvira for valuable comments. Partof this work was done during my visit to Institute of Theoretical Physics, Chinese Academy of Science inBeijing. This work is supported, in part, by National Natural Science Foundation of China (Grant Nos.11305137). [1] G. ’t Hooft, Nucl. Phys. B (1974) 461 .[2] E. Witten, Nucl. Phys. B (1979) 57.[3] A. V. Manohar, hep-ph/9802419.[4] G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B (1983) 552.[5] E. E. Jenkins, Ann. Rev. Nucl. Part. Sci. (1998) 81.[6] J. L. Gervais and B. Sakita, Phys. Rev. Lett. (1984) 87.[7] J. L. Gervais and B. Sakita, Phys. Rev. D (1984) 1795.[8] R. F. Dashen and A. V. Manohar, Phys. Lett. B (1993) 425.[9] R. F. Dashen and A. V. Manohar, Phys. Lett. B (1993) 438.[10] D. B. Kaplan and M. J. Savage, Phys. Lett. B (1996) 244.[11] D. B. Kaplan and A. V. Manohar, Phys. Rev. C (1997) 76.[12] M. K. Banerjee, T. D. Cohen and B. A. Gelman, Phys. Rev. C (2002) 034011.[13] A. V. Belitsky and T. D. Cohen, Phys. Rev. C (2002) 064008.[14] T. D. Cohen, Phys. Rev. C (2002) 064003.[15] F. Gross, T. D. Cohen, E. Epelbaum and R. Machleidt, Few Body Syst. (2011) 31[16] T. D. Cohen and V. Krejˇciˇr´ık, Phys. Rev. C (2013) no.5, 054003.[17] Y. Hidaka, T. Kojo, L. McLerran and R. D. Pisarski, Nucl. Phys. A (2011) 155.[18] T. Kojo, Nucl. Phys. A (2013) 76.[19] S. Weinberg, Phys. Lett. B251 (1990) 288, Nucl. Phys. B363 (1991) 3.[20] D.B. Kaplan, M.J. Savage and M.B. Wise, Phys. Lett. B (1998),390; Nucl.Phys. B (1998),329.[21] U. van Kolck, Prog. Part. Nucl. Phys. (1999) 337.[22] S. R. Beane, hep-ph/0204107.[23] M. T. AlFiky, F. Gabbiani and A. A. Petrov, Phys. Lett. B (2006) 238.[24] C. Liu, Eur. Phys. J. C (2008) 413.[25] T. Cohen, F. J. Llanes-Estrada, J. R. Pelaez and J. Ruiz de Elvira, Phys. Rev. D (2014) no.3, 036003.(2014) no.3, 036003.