Flavor Asymmetry of the Sea Quarks in the Baryon Octet

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1 Flavor Asymmetry of the Sea Quarks in the Baryon Octet Susumu Koretune Department of Physics, Shimane Medical University, Izumo,Shimane 6,Japan We show that the chiral SU(n) SU(n) flavor symmetry on the null-plane severely restricts the sea quarks in the baryon octet. It predicts large asymmetry for the light sea quarks (u, d, s), and universality and abundance for the heavy sea quarks. Further it is shown that eistence of the heavy sea quarks constrained by the same symmetry reduces the theoretical value of the Ellis-Jaffe sum rule substantially. Introduction Many years ago, based on the current anti-commutation relation on the null-plane, the Gottfried sum rule was re-derived. Since the re-derived sum rule had a slightly different physical meaning from the original one, I called it as the modified Gottfried sum rule. Several years ago, this sum rule was found to take the following form 4,5 : d ep m Km N (, Q ) F en(, Q )} = ( 4f K dν ν ν (m K m N ) {σ K+n (ν) σ K+p (ν)}), () where σ K+N (ν) withn=por n is the total cross section of the K + N scatterings and f K is the kaon decay constant. This gave us a new way to investigate the vacuum properties of the hadron based on the chiral SU(n) SU(n) flavor symmetry on the null-plane. Here I briefly eplain the fact and show that it severely restricts the sea quarks in the 8 baryon. For details see Ref. 6. A physical meaning of the modified Gottfried sum rule The Gottfried sum multiplied by / takes the following form: d ep (, Q ) F en(, Q )} = d{ u v d v} + d{ λ u λ d} d{ λ ū + λ d}. () On the other hand Adler sum rule takes the following form: 4 d νp (, Q ) F νp (, Q )} = + d{ λ u λ d} + π d{ u v d v} d{ λ ū + λ d }. () Here the subscript v means the valence quark and λ i means the i type sea quark. The fundamental difference between the Gottfried sum and the Adler sum rule is the sign in front of the antiquark distribution function. Thus, under the physically reasonable assumption dλ i(, Q ) = dλī (, Q ), we can say that the Adler sum rule measures the mean I of the {[quark] + [antiquark]} and hence the one of the valence quarks being equal to the I of the proton, while the modified Gottfried sum rule measures the mean I of the {[quark] - [antiquark]}in the proton. Now the current commutation relation on the null-plane is given as [J a + (),J+ b ()] + = =[Ja 5+ (),J5+ b ()] + = = if abc δ( )δ ( )J c + (4) (). and the current anti-commutation relation on the nullplane is given as <p {J a + (),J+ b ()} p > c + = =< p {Ja 5+ (),J5+ b ()} p > c + = = π P ( )δ ( )[d abc A c (p, (5) =) +f abc S c (p, =)]p +. Both-hand sides of the modified Gottfried sum rule are equal to π P α A (α, ), (6) where as far as we are discussing the moment at n = A a (α, )can be related to the bilocal currents on the nullplane as <p i λa λa [: q()γµ q() : : q()γµ q() :] p> c =p µ A a (p, )+ µ Ā a (p, (7) ). We decompose the quark field on the null-plane into the particle mode and the anti-particle mode as q() = a n φ (+) n ()+ b nφ ( ) n (). (8) n n The normal ordered product on the null-plane is : q()γ +λ q() := n,m a na φ(+) m n ()γ + λ φ(+) m () () n,m b mb φ( ) n n ()γ + λ φ( ) m (). By setting α = p + y,we define a part contributing to the quark distribution function as f() = πp + ep[ iα] <p n,m a n a (+) m φ n (y )γ + λ φ(+) m () p > c, ()

2 and a part contributing to the antiquark distribution function as g() = πp + ep[ iα] <p n,m b nb φ( ) m m ()γ + λ n (y ) p > c. φ( ) () By using the integral representation of the sign function we obtain ɛ() = iπ P da a ep[ia], () iπp P + α <p n,m a na φ(+) m n (y )γ + λ φ(+) m () p > c = dɛ()f() = df(), () iπp P + α <p n,m b mb φ( ) n n (y )γ + λ φ( ) m () p > c = dɛ( )g() = dg(). (4) The integral of the type P α originates from the factor P in Eq. (5). In case of the commutation relation we obtain the integral of the type δ(α) which originates from the factor δ( )in Eq. (4), hence we have no sign chage as in Eq. (4). Thus the current anti-commutation relation on the null-plane naturally eplains the physical difference between the Gottfried sum and the Adler sum rule, and at the same time, we can see that the physical importance of this relation lies in the fact that it gives us information of the vacuum property of the hadron. The regularization of the divergent sum rule Now the sum rule from the current anti-commutation relation can be epected to diverge in general. Here we eplain the method to regularize such divergence by taking the following sum rule. d F ep = 8π P α { 6A (α, ) + A (α, ) + A 8 (α, )}. (5) The left-hand side of this sum rule definitely diverges because of the pomeron. We assume this component as the flavor singlet. Then we can identify the divergent piece on the right-hand side as that coming from A (α, ). If the trajectory of the pomeron takes the value α P (t) < at some t near zero, we can regularize the sum rule by the analytical continuation from the non-forward direction, since in this case we can derive the finite sum rule. The soft pomeron by Donnachi and Landshoff 7 is one eample which makes it possible to carry out the program easily. Now the assumption α P (t) < forsome small t can not be satisfied by the hard pomeron based on the fied-coupling constant 8. However, there are great efforts to improve the defect of this pomeron. The netto-leading corrections seems to suggest a substantial reduction of the value of the intercept. The multiple scatterings of the pomeron gives us important unitary corrections at low. Thuseveninsuchaperturbative approach there is a hope to satisfy the assumption. Though we use the soft pomeron to eplain the regularization, in view of the situation, we clarify the quantities which do not depend on the assumed high energy behavior in the following. Now we continue to discuss by the effective method which gives the same results as those in the non-forward direction. We take the leading high energy behavior of the F ep isgivenbythepomeronas ( Q ) αp() β ep (Q, α P ())(ν) αp (), and assume it to be the flavor singlet. It should be noted that what we αp () assume here is only the high energy behavior (ν) and no assumption is made about the Q dependence, since all the unknown Q dependence is absorbed in β ep. This also applies to the scale factor in ν. We rewrite the left-hand side of the sum rule as d F ep d ep = + β ep (Q, α P ()) αp () } dβ ep(q, α P ()) αp (), and the right-hand side of it as 6 π P = 6 π P α A (α, ) α {A (α, ) f(α)} + 6 π P (6) α f(α). (7) By setting α P () = +b ɛ, weepandβ ep as βep(q ) (ɛ b)βep (Q )+O((ɛ b) ). The pole term as ɛ b should be canceled out from both-hand sides of Eq. (5) since the sum rules are convergent for the arbitrary finite positive (ɛ b) which corresponds to the small negative t in the non-forward case, hence there must eists f(α) such that the quantity 6 f(α) =( 6 π P α f(α) β ep ɛ b )becomes finite in the limit ɛ b, whereβ ep is Q independent since Eq. (5) holds at any Q. After taking out the singular piece we take the limit ɛ andobtain d ep = 8π P β ep b } α { 6S (α,,q )+A p (α, ) + A p 8 where S (α,,q )and S are defined as 6 S = 6 [ f(α)+ = 6 π P π P α S (α,,q ). α {A (α, ) f(α)}]+β ep(q ) (α, )}, (8) () Here the superscript in S(α,,Q )and S means the singlet in the SU(). We find that the regularization of the sum rule simply results in the Q dependence in

3 the singlet component and that all the relation from the symmetry is inherited. Thus if we get the relation which do not include the singlet component, we can have regularization independent relation. The modified Gottfried sum rule belongs to this type of the sum rules and we give such sum rule in the following. 4 The symmetry constraint on the light sea quark distributions in the baryon octet A a (α, ) is governed by the chiral SU() SU() flavor symmetry. If we take the state as the 8 baryon, the matri element becomes α, p i [: q()γµ λ aq() q()γ µ λ aq() :] β,p c = p µ (A a (p, )) αβ + µ (Ā a (p, )) αβ, () where the α, β are the symmetry inde specifying each member of the 8 baryon. Since the matri element can be classified by the flavor singlet in the product 8 8 8, (A a (α, )) αβ is decomposed as (A a (α, )) αβ = if αaβ F (α, ) + d αaβ D(α, ) () for a. Using the value of the modified Gottfried sum rule, we obtain 4 π P α Ap = π (α, ) α (α, ) + D(α, )} =.6. () The mean hypercharge sum rule gives us π P = π α Ap 8 α (α, ) (α, ) D(α, )} =.. () Here we use the notation A B a (α, ) with B = (p, n, Σ ±, Σ, Λ, Ξ, Ξ ) to specify each member of the 8 baryon. Thus we obtain F π D π F (α, ) =.8, (4) α D(α, ) =.5. (5) α Now we define the sea quark distribution of the i type one in the 8 baryon as λ B i, and regularizes its mean number as λ B i = d{λ B i a αp () }, (6) where α P () here is α P () = +b and is taken as.88, and a is defined as lim () αp λ B i = a. (7) The value a is proportional to βep and was determined through the sum rule as a =.. 5 Though this relation can be lost ecept the soft pomeron case, we still have the fact that the singlet component contributes universally to all the sea quarks. Thus similar regularized sea quark number as Eq. (6) eists in the general case. The constraints on the sea quark distributions are obtained as follows. Let us take an eample of the proton matri element with K = λa = diag( ) = 6 6 λ + λ + 6 λ 8. Since α = (4 + i5),β = (4 i5), by taking < λ B i >=< λ B >, weobtain<u p ī v >+ < λ p u >= 6 S + F + D. Thus we have many relations for the sea quarks in the 8 baryon. Then by using the fact that each valence part is merely the number of the valence quark, we get many sum rules from these relations. Among them the sum rules for the mean quantum numbers of the light sea quarks are fundamental since they do not depend on the singlet component. In other words they do not depend on the regularization. Here the light sea quarks mean the u, d, s type sea quarks. We summarize them in the Table. Note that λ B i is replaced by λ B i because the divergent part is canceled out in each epression. The perturbative QCD corrections to these relations begin from the loops and they enter the same way as the one in the modified Gottfried sum rule.,4 Therefore they are negligibly small compared with the non-perturbative values listedinthetable. 5 The symmetry constraint on the heavy sea quark distributions in the baryon octet Here we etend the symmetry from the SU() SU() to the SU(n) SU(n) withn 4. Let us now discuss the heavy sea quarks in the 8 baryon. For concreteness we take chiral SU(4) SU(4) flavor symmetry. In this case the 8 baryon belongs to the M, and the currents to the 5. The matri element in this case can be classified by the singlet component in the product M M 5. Since the adjoint representation 5 appears twice in the product as M M = , and since only these two 5 can make the singlet with the remaining 5, we have only two different terms in the matri element. Further these two 5 can be represented by the 4 4matriwhose sub-matri agrees with the matriinthesu(). Thus the two different terms are F (α, ) and D(α, ) in the SU(). However, in this generalization, the singlet in the SU() is not the singlet in the SU(4). To see this fact more concretely, we take the matri K = diag( ), and decomposes it as K = 4 λ4 + λ4 + 6 λ λ4 5. Here the λ 4 k is the Gell-Mann matri generalized to the SU(4). The SU() singlet part in this decomposition

4 Table : The mean quantum number of the light sea quarks. I Y Q B { λb u λb d } { λb u + λb d λb s } { λb u λb d λb s } p ( F + D) 4 ( F D) ( F + D) =.55 =.56 =. n ( F + D)+ 4 ( F D) D =.55 =.56 =.4 Σ + F D 6 ( F + D) =.54 =.4 =. Σ D D = =.4 =.7 Σ F + D 6 ( F + D)+ =.54 =.4 =. Ξ ( F + D)+ 4 ( F + D)+ ( F D)+ =.45 =. =.56 Ξ ( F D) 4 ( F + D)+ D =.45 =. =.4 Λ D D = =.4 =.7 is 4 λ4 + 6 λ4 5 = diag( ). Since the sub-matri diag( ) is epressed as 6 λ in the SU(), the coefficient of the singlet part in the SU() is different from the one in the SU(4). On the other hand, sub-matri in the part λ4 + 6 λ4 8 has the same epression in these two cases. Thus we find one relation between the singlet contribution in the SU() and the one in the SU(4). If we denote the SU(4) singlet contribution as S 4 corresponding to the S in the SU(), we obtain 6 S = S 4 + D. Epressed in the parton model, this generalization from the SU() to the SU(4) corresponds to the addition of the charm sea quark with the symmetric condition without changing anything in the light sea quarks. Thus we see all the charm sea quark distributions in the 8 baryon corresponding to the matri element of diag( ) are the same. Eplicitly we obtain < λ c >= 6 S 4 D for all members in the 8 baryon. For the proton, we obtain <λ c λ p d >=.5 and <λ c λ p s >=.4. This means that the charm sea quark in the proton is abundant especially in the small region. 5,5 It should be noted that the gluon fusion like term is in general included in our definition of the charm quark distribution function, since the virtual photon couples to the gluon through the quark and since the classification of the sea quark in our case is done by this coupling with the virtual photon. Hence we reach the conclusion that the charm sea quark is universal and abundant in the 8 baryon. The same kind of the discussions can be repeated in the SU(5) or SU(6), and we get < λ b >= < λ t >= 6 S 4 D for the bottom and the top sea quarks. 6 Flavor asymmetry of the spin-dependent sea quark distribution It is interesting to note that similar discussion to etend the symmetry from the SU() to the SU(4) can be applied to the matri element <p,s,α Ja 5µ () p, s, β >. We define p, s, α Ja 5µ () p, s, β = sµ A αβ a, (8) where s µ is the spin vector, and A αβ a = if αaβ F + d αaβ D, () for a. The Ellis-Jaffe sum 6 for the 8 baryon is I B f = dg B (, Q ), () where the subscript f specifies the flavor group. If B is proportional to d αaβ and in case of the proton it is well known to take the form I p = 6 [4 Qp + Qp + Qp 8 ], () where Q p = up + d p + s p, Q p = up d p, and Q p 8 = up + d p s p,and q p is the fraction of the spin of the proton carried by the spin of quarks of flavor q. Here q p includes the contribution from the 4

5 antiquark as usual. We obtain u p = S+ D+F, dp = S D, sp = S+ D F, () where Q p = S. It is straightforward to get the spin fraction of the quarks in other baryons. Now in the SU(4), Q B 5 can be defined as Q B 5 = 6A αβ 5 = ub + d B + s B c B. () Applying similar discussion in the previous section to this quantity we obtain Q B 5 =Dfor all members in the 8 baryon. Since Q B a for a 8isthesameasinthe SU(), we obtain c = S D, (4) for all members in the 8 baryon. Note that we use the same S as in the SU(). Thus we get I p 4 = [ 4 up + dp + sp + 4 cp ] = 5 7 S + 6 F 5 54 D. (5) The generalization to the SU(5) or the SU(6) is straightforward, and we obtain b = t = S D. (6) Using eperimental value of F =.46 ±. and D =.7±. 7, we see that for a reasonable value of the S, the theoretical value of the Ellis-Jaffe sum rule is reduced substantially by the charm quark. It is usually considered that the light sea quark gets contribution from the gluon anomaly because of the small-ness of the quark mass compared with the infra-red cutoff 8. The magnitude of this gluon contribution is determined by input information. Then, to make the Ellis-Jaffe sum rule consistent with the eperiment by this gluon polarization, it must be taken very large. In our case, such large gluon polarization is not necessary. The heavy quarks such as the charm and the bottom ones are suffice to make the Ellis-Jaffe sum rule consistent with the eperiment. References. S.Koretune, Phys. Rev. D, 8 (8).. K.Gottfried, Phys. Rev. Lett., 74 (67).. S.Koretune, P rog. T heor. P hys. 7, 8 (84). 4. S.Koretune, Phys. Rev. D 47, 6 (). 5. S.Koretune, Phys. Rev. D 5, 44 (5). 6. S.Koretune, hep-ph8448, to be published in the Nuclear Physics B. 7. A.Donnachie and P.V.Landshoff, Phys. Lett. B 6, 7 (). 8. E.A.Kuraev, L.N.Lipatov, and V.S.Fadin, Sov. Phys. JETP 45, (77); Ya.Ya.Balitsky and L.N.Lipatov, Sov. J. Nucl. P hys. 8, 8 (78).. J.R.Forshaw and D.A.Ross, Quantum Chromodynamics and the Pomeron (Cambridge Univ. Press,Cambridge,UK,7), pp.-8 and pp M.Ciafaloni and G.Camici, hep-ph88. V.S.Fadin and L.N.Lipatov, hep-ph8.. A.H.Mueller, hep-ph75.. S.Koretune, P rog. T heor. P hys. 8, 74 (7).. D.A.Ross and C.T.Sachrajda, Nucl. Phys. B 4, 47 (78), 4. G.Curci, W.Furmanski, and R.Petronzio, Nucl. Phys. B 75, 7 (8). 5. S.Koretune, P rog. T heor. P hys. 88, 6 (). 6. J.Ellis and R.L.Jaffe, Phys. Rev. D, 444 (74);ED 66(74). 7. S.Y.Hsueh et al., Phys. Rev. D 8, 56 (88). 8. Mankiewicz and Schäfer., Phys. Lett. B 4, 455 (); and references cited therein. 7 Conclusions We show that the chiral SU(n) SU(n) flavor symmetry on the null-plane combined with the fied-mass sum rule derived from the current anti-commutation relation on the null-plane severely restricts the sea quark in the 8 baryon. It predicts a large asymmetry for the light sea quarks, and universality and abundance for the heavy sea quarks. Further we show that the same symmetry restricts the fraction of the spin of the 8 baryon carried by the quark. Especially we show that this effect is outstanding for the intrinsic charm sea quark in the nucleon and that it plays the role to reduce the theoretical value of the Ellis-Jaffe sum rule substantially. 5