The sum rule for the structure functions of the deuteron from the current algebra on the null-plane
aa r X i v : . [ h e p - ph ] S e p The sum rule for the structure functions of the deuteron from the currentalgebra on the null-planeSusumu Koretune
Department of Physics, Shimane University, Matsue,Shimane,690-8504,JapanThe fixed-mass sum rules for the deuteron target have been derived by using the connectedmatrix element of the current anti-commutation relation on the null-plane. From thesesum rules we obtain the relation between the pseudo-scalar meson deuteron total crosssections and the structure functions of the deuteron. We show that the nuclear effecton the mean hypercharge of the sea quark of the proton can be studied by this relation.Further we obtain the relation among the Born term, the resonances and the non-resonantbackground in the small Q region, and as a new aspect of the spin 1 target, explicitrelation of the tensor structure function b at small or moderate Q to that at large Q . The current algebra based on the canonical quantization at equal-time gives us very gen-eral constraints. These constraints are essential ingredient in QCD. [1] The sum rules inthe current-hadron reaction in this formalism are called as the fixed-mass sum rules be-cause the mass of the current takes the fixed space-like or null value. Adler sum rule andAdler-Weisberger sum rule are typical examples of these sum rules. In the former case,the momentum transfer of the weak boson which couple to the hadronic weak current isfixed at the space-like value, and in the latter case the square of the momentum of thepseudo-scalar meson which is related to the divergence of the axial-vector current is fixedat the off-shell value 0. This algebra had been extended to the one based on the canonicalquantization at equal null-plane time. The superior points of the null-plane formalismover the equal-time one are the followings.(1) We need not take the infinite momentumframe. (2)Some sum rules in the equal-time formalism get corrections from the bilocalcurrents, which without them were considered to be peculiar. (3)A technical aspect indealing with some graphs contributing to the intermediate state. These were explained inRef.[2]. Apart from these facts, null-plane formalism involved a further extension to thecurrent anti-commutation relation.[3, 4] We briefly explain the fact in the following andtechnical aspects are summarized in the Appendix A.In the late 60’s, through the experimental finding of the parton at SLAC, the light-conecurrent algebra was proposed. This algebra was abstracted from the leading light-conesingularity of the current commutation relation in the free quark model [5]. Further, sincethe leading singularity was mass-independent, it was suggested that the reasoning to reachthis algebra could be extended to the current product if we sacrificed the causal nature ofthe current commutation relation. Though this type of the relation had been suggestedbut had not been considered further in Ref.[5]. On the other hand, since the assumption toextract the light-cone singularity was too restricted, another method which was based onthe canonical quantization on the null-plane was considered. [2] This algebra was a directgeneralization of the equal-time formalism. The canonical quantization on the null-planeoriginated from Dirac[6] and was unrelated to the light-cone current algebra. However,similar bilocal quantities appeared in the both methods. The bilocal quantities in the1ight-cone current algebra were regular operators where all singularities in the light-conelimit were taken out and hence were different from those in the null-plane formalism wheresuch manipulation had not been imposed. However because of the similarity they wereoften identified on the null-plane as a heuristic method to obtain some physical insight.Among them the works in Ref.[4] would be a first attempt to obtain some relations atthe wrong signature point in this sense. Now through the finding of the scaling violationswhich led to the QCD, it was recognized that the method of taking the leading light-conesingularity should be refined. In fact, the short distance expansion was taken first, andwith use of the dispersion relation, this expansion was analytically continued to the re-gion near the light-cone. This light-cone expansion utilized the causality of the currentcommutation relation, hence, the moment sum rules obtained in this expansion were atalternate integers. Further, each moment corresponded to the matrix element of the localoperator obtained by the expansion of the bilocal operator and it was found that becauseof the anomalous dimension we could not take out the light-cone singularity uniformlyfrom each moment. Thus the expansion by the singular coefficient function multiplied bythe regular bilocal current in the light-cone current algebra had broken down. The rela-tion at the missing integers was later shown to be obtained by the cut vertex formalism[7]which suggested that these quantities were related to the non-local quantities. A physicalapplication of the light-cone expansion were restricted to the deep-inelastic region.Now through the study of the fixed-mass sum rules in the semi-inclusive lepton-hadronscatterings where the one soft pion was observed, we encountered the current anti-commutation relation on the null-plane. Since at that time we knew the simple methodto take out the leading light-cone singularity was wrong and the bilocal operator in thismethod should not be taken literally, we needed some methods to abstract the currentanti-commutation relation on the null-plane. It was in this point where Deser, Gilbertand Sudarshan (hereafter called as DGS) representation[8] played an important role.[3]Through this method it became possible to consider the fixed-mass sum rules at thewrong signature point with use of the connected matrix element of the current anti-commutation relation between the stable hadron. The application of this method hasbeen so far restricted to the hadrons. However, as far as the s channel and the u channelare disconnected and that the target particle is stable, this method can be used. Nowthe sum rules from the current anti-commutation relation gave us information of the seaquarks in the hadrons. A typical example of this fact can be seen in the modified Got-tfried sum rule.[9, 10, 11, 12] Compared with the Adler sum rule which is obtained bythe current commutation relation, we have the extra factor ( −
1) from the contribution ofthe anti-quark distributions.[12] Hence the contribution from the sea quark distributionremains in the sum rule. Thus the study of the sum rules can give us information of thehadronic vacuum. In other words, we can say that the sum rule controls how the quark-antiquark pair is produced or annihilated in the hadrons. From this point of view, it isinteresting to extend the method to the nuclear target since an nuclear effect is differentfrom that of the hadron. In this paper, as a first step of the application to the nucleartargets, we apply the method to the deuteron. In section 2, the kinematics of the spin 1deuteron target is given, and in section 3, the sum rules from the good-good componentare derived. In section 4, the sum rules are transformed to various forms and physicalmeanings are explained. Summary is given in Section 5.2
Kinematics
The imaginary part of the forward reaction ”current(q) + deuteron(p) → current(q) +deuteron(p)” is proportional to the total cross section of the inclusive reaction ”current(q)+ deuteron(p) → anythings(X)”, where q is the momentum of the current and p is that ofthe deuteron with its mass m d . This part is called as the hadronic tensor and is expressedby assuming the completeness of the sum over X as W µνab ( p, q, E, E ∗ ) = 14 π Z d x e iq · x h p, E | [ J µa ( x ) , J νb (0)] | p, E i c , (1)where E is the polarization vector of the deuteron and the suffix c on the right-handside of the equation means to take the connected part. Since the current is the inducedhadronic current in the inclusive reaction ”lepton + deuteron → lepton + anythings”,the momentum q is the difference of the momentum of the initial lepton and that of thefinal lepton and hence it takes the space-like value. We first discuss the conserved vectorcurrent J µa ( x ) where the suffice a denotes the flavor index. The generalization to the non-conserved and the parity violating case is given later in this section. Since the hadroniccurrent is color singlet we ignore a color suffix in the quark field. Now, by requiring parityand time reversal invariance, we obtain[13] W µνab ( p, q, E, E ∗ ) = − F ab G µν + F ab P µ P ν ν − b ab r µν + 16 b ab ( s µν + t µν + u µν )+ 12 b ab ( s µν − u µν ) + 12 b ab ( s µν − t µν ) + ig ab ν ǫ µνλσ q λ s σ + ig ab ν ǫ µνλσ q λ ( νs σ − s · qp σ ) , (2)where ν = p · q , κ = 1 − m d q /ν , P µ = p µ − ( ν/q ) q µ , G µν = g µν − (1 /q ) q µ q ν ,s µ = − ( i/m d ) ǫ µαβγ E ∗ α E β p γ , E · E ∗ = − m d , p · E = p · E ∗ = 0 , and r µν = 1 ν ( q · E ∗ q · E − ν κ ) G µν , s µν = 2 ν ( q · E ∗ q · E − ν κ ) P µ P ν ν ,t µν = 12 ν ( q · E ∗ P µ e E ν + q · E ∗ P ν e E µ + q · EP µ e E ∗ ν + q · EP ν e E ∗ µ − ν P µ P ν ) ,u µν = 1 ν ( e E ∗ µ e E ν + e E ∗ ν e E µ + 2 m d G µν − P µ P ν ) , (3)with e E µ = E µ − ( q · E/q ) q µ and e E ∗ µ = E ∗ µ − ( q · E ∗ /q ) q µ .Similar hadronic tensor f W µνab ( p, q, E, E ∗ ) can be defined by the current anti-commutationrelation as f W µνab ( p, q, E, E ∗ ) = 14 π Z d x e iq · x h p, E |{ J µa ( x ) , J νb (0) }| p, E i c = − e F ab G µν + e F ab P µ P ν ν − e b ab r µν + 16 e b ab ( s µν + t µν + u µν ) + 12 e b ab ( s µν − u µν )+ 12 e b ab ( s µν − t µν ) + i e g ab ν ǫ µνλσ q λ s σ + i e g ab ν ǫ µνλσ q λ ( νs σ − s · qp σ ) . (4)The structure functions defined by the current commutation relation and those of thecurrent anti-commutation relation are the same quantity in the s channel but oppositein sign in the u channel. The crossing relation under µ ↔ ν , a ↔ b and q → − q , are3 ab ( − x, Q ) = − F ba ( x, Q ) , F ab ( − x, Q ) = F ba ( x, Q ) , b ab ( − x, Q ) = − b ba ( x, Q ), b ab ( − x, Q ) = b ba ( x, Q ), b ab ( − x, Q ) = b ba ( x, Q ) , b ab ( − x, Q ) = b ba ( x, Q ) , g ab ( − x, Q ) = g ba ( x, Q ) , g ab ( − x, Q ) = g ba ( x, Q ), while the structure functions defined by the anti-commutation relation are opposite in sign, where x = Q / ν with q = − Q .Now we take the current as J µa ( x ) =: ¯ q ( x ) γ µ λ a q ( x ) : in the chiral SU ( N ) × SU ( N ) model.On the null-plane x + = 0, the quark field is decomposed as q ( ± ) ( x ) = Λ ± q ( x ) where theprojection operator is define as Λ ± = (1 ± γ γ ) with x ± = √ ( x ± x ) and the suffixesof internal symmetry are discarded since inclusion of them do not affect the followingdiscussion. Through the equation of motion, q ( − ) ( x ) is related to the q (+) ( x ). Hence the q ( − ) ( x ) depends on the q (+) ( x ), and the independent field on the null-plane is q (+) ( x ),hence the canonical quantization is given as { q (+) † ( x ) , q (+) (0) }| x + =0 = √ + δ ( ~x ⊥ ) δ ( x − ).Since J + a ( x ) =: ¯ q ( x ) γ + λ a q ( x ) := √ q (+) † ( x ) λ a q (+) ( x ) :, the current for µ = + dependsonly on the q (+) ( x ), and does not depend on the equation of motion. In this sense thecurrent commutation relation on the null-plane for µ = ν = + is called as the good-good component. The current J ia ( x ) depends on one q ( − ) ( x ) and then called as a badcomponent. Thus the good-good component is < p, E | [ J + a ( x ) , J + b (0)] | p, E > c | x + =0 = iδ ( x − ) δ ( ~x ⊥ ) (cid:2) d abc < p, E | A + c ( x | | p, E > c + f abc < p, E | S + c ( x | | p, E > c (cid:3) , (5)where S µa ( x |
0) = 12 [: ¯ q ( x ) γ µ λ a q (0) : + : ¯ q (0) γ µ λ a q ( x ) :] ,A µa ( x |
0) = 12 i [: ¯ q ( x ) γ µ λ a q (0) : − : ¯ q (0) γ µ λ a q ( x ) :] ,< p, E | S µa ( x | | p, E > c = p µ S a ( p · x, x ) + x µ ¯ S a ( p · x, x )+ p µ { ( E ∗ · x )( E · x ) −
13 (( x · p ) − m d x ) } S Pa ( p · x, x )+ x µ { ( E ∗ · x )( E · x ) −
13 (( x · p ) − m d x ) } ¯ S Pa ( p · x, x )+ { E µ ( E ∗ · x ) + E ∗ µ ( E · x ) −
23 (( x · p ) p µ − m d x µ ) } e S Pa ( p · x, x ) ,< p, E | A µa ( x | | p, E > c = p µ A a ( p · x, x ) + x µ ¯ A a ( p · x, x )+ p µ { ( E ∗ · x )( E · x ) −
13 (( x · p ) − m d x ) } A Pa ( p · x, x )+ x µ { ( E ∗ · x )( E · x ) −
13 (( x · p ) − m d x ) } ¯ A Pa ( p · x, x )+ { E µ ( E ∗ · x ) + E ∗ µ ( E · x ) −
23 (( x · p ) p µ − m d x µ ) } e A Pa ( p · x, x ) . (6)The target polarization dependent parts are defined so that their contributions vanishwhen the target polarizations are averaged. The right-hand side of Eq.(5) is equal to iδ ( x − ) δ ( ~x ⊥ ) f abc < p, E | J + c (0) | p, E > c because of the delta function constraint, however,we write the expression before this constraint is applied, since what the DGS representa-tion gives us is that the term corresponding to this term remains in the anti-commutationrelation and that the other terms are zero on the null-plane.[3, 9] Then the corresponding4elation for the current anti-commutation relation is < p, E |{ J + a ( x ) , J + b (0) }| p, E > c | x + =0 = π P ( x − ) δ ( ~x ⊥ ) [ d abc < p, E | A + c ( x | | p, E > c + f abc < p, E | S + c ( x | | p, E > c ] , (7)where P means to take the principal value. Before going to a detailed derivation of the sumrule we explain the hadronic tensor for the non-conserved and parity violating currentsincluding the cases for the weak boson mediated reactions. In such a general case, wehave the 36 independent helicity amplitudes since we have two types of the helicity 0 statefor the non-conserved current.[14] Then by the time reversal invariance the independentamplitudes are reduced to 24 and by the parity invariance they are further reduced to 14.Among the 14 amplitudes, 6 amplitudes enter due to the non-conservation of the current.The tensors corresponding to these amplitudes are p µ q ν + p ν q µ , ( q · E ∗ q · E − ν κ )( p µ q ν + p ν q µ ) , ( q · E ∗ q · E − ν κ ) q µ q ν , q µ q ν , n q · E ∗ q µ e E ν + q · E ∗ q ν e E µ + q · Eq µ e E ∗ ν + q · Eq ν e E ∗ µ − ν ( p µ q ν + p ν q µ ) + ν q q µ q ν o ,iǫ µναβ p α s β . (8) Now, with use of Eqs.(5)-(7), the sum rule can be obtained by integrating W ++ ab and f W ++ ab over q − and assuming the interchange of setting q + = 0 and ν integration. For thepolarization averaged quantities, we obtain from the current commutation relation Z dxx F [ ab ]2 ( x, Q ) = 14 f abc Γ c , (9)where F [ ab ]2 = ( F ab − F ba ) / i and < p, E | J µa (0) | p, E > = p µ Γ a , and from the currentanti-commutation relation Z dxx F ( ab )2 ( x, Q ) = 14 π d abc P Z ∞−∞ dαα A c ( α, , (10)where F ( ab )2 = ( F ab + F ba ) / E h , and take p = ( p , , , p ) , q =( q , q , q , q ) , √ E ± = m d (0 , ∓ , − i, , E = ( p , , , p ). Then, by taking the polariza-tion vector E ± , we obtain from the commutation relation Z dxx (cid:16) ( κ − b [ ab ]2 ( x, Q ) + 3( κ − b [ ab ]3 ( x, Q ) + 3( κ − b [ ab ]4 ( x, Q ) (cid:17) = 0 , (11)and from the anti-commutation relation Z dxx (cid:16) ( κ − b ( ab )2 ( x, Q ) + 3( κ − b ( ab )3 ( x, Q ) + 3( κ − b ( ab )4 ( x, Q ) (cid:17) = − π d abc Z ∞−∞ dα { αA Pc ( α,
0) + 2 e A Pc ( α, } , (12)5here symmetric and antisymmetric combination of the spin dependent structure func-tions are defined similarly as the structure function F ab . From the case E , we obtainno new sum rules. Now as the transverse polarization vector, we take the combination E = m d (0 , , ,
0) and E = m d (0 , , , p = ( p , , , p ) , q = ( q , q , , q ). Then,since E · q = E ∗ · q = 0, we obtain from the commutation relation Z dxx (cid:16) ( κ + 2) b [ ab ]2 ( x, Q ) + 3( κ − b [ ab ]( x,Q )3 + 3( κ − b [ ab ]4 ( x, Q ) (cid:17) = 0 , (13)and from the anti-commutation relation Z dxx (cid:16) ( κ + 2) b ( ab )2 ( x, Q ) + 3( κ − b ( ab )3 ( x, Q ) + 3( κ − b ( ab )4 ( x, Q ) (cid:17) = 34 π d abc Z ∞−∞ dα { αA Pc ( α,
0) + 2 e A Pc ( α, } . (14)Since κ − x m d /Q , from Eq.(11) and Eq.(13) we obtain Z dxx b [ ab ]2 ( x, Q ) = 0 , (15)and Z dxx (cid:16) b [ ab ]2 ( x, Q ) + 3 b [ ab ]3 ( x, Q ) + 3 b [ ab ]4 ( x, Q ) (cid:17) = 0 . (16)Similarly, from Eq.(12) and Eq.(14) we obtain Z dxx b ( ab )2 ( x, Q ) = 14 π d abc Z ∞−∞ dα { αA Pc ( α,
0) + 2 e A Pc ( α, } , (17)and Z dxx (cid:16) b ( ab )2 ( x, Q ) + 3 b ( ab )3 ( x, Q ) + 3 b ( ab )4 ( x, Q ) (cid:17) = 0 . (18)The sum rules (10) ,(17) and (18) are the ones for the symmetric combination, hence theycan be applied to the electromagnetic current.Now, since < p, E | [ J a ( x ) , J b (0)] | p, E > c | x + =0 = < p, E | [ J + a ( x ) , J + b (0)] | p, E > c | x + =0 ,we obtain the relation < p, E |{ J a ( x ) , J b (0) }| p, E > c | x + =0 = < p, E |{ J + a ( x ) , J + b (0) }| p, E > c | x + =0 with use ofthe DGS representation[3, 9]. The hadronic tensor for the axial-vector current is non-conserved one. Then the tensors given in Eq.(8) are necessary. When we derive the sumrule we set q + = 0. Since all tensors in Eq.(8) are proportional to q + , we see that theydo not affect the derivation of the sum rule in the above discussion. Thus the sum rule(10) also holds in this case. Then by using the PCAC relation, we can transform the sumrule (10) to the ones for the pseudo-scalar deuteron total cross section as in the nucleoncase[3, 9]. 6 Application
Now, the sum rule (10) is the equality of the possible divergent quantity which definitelybreaks the condition necessary to derive the sum rule. Including such case, importance ofthe regularization of possible divergent sum rules was explained in Ref.[15]. Here we followthe method in Ref.[16]. We first derive the sum rule in the non-forward direction. Thenwe see that the right-hand side of the sum rule given by the integral of the non-forwardmatrix element of the bilocal current is Q independent. We assume the high-energybehavior is controlled by the moving Regge pole or cut and the divergence comes fromthe flavor singlet part corresponding to the Pomeron. Then we take sufficiently large | t | such that the sum rule is convergent. Next we change | t | to smaller value, and subtract thepole singularity from both-hand sides of the sum rule. From this we obtain the conditionthat the residue of the pole is Q independent. After that, we can take still smaller | t | in the subtracted sum rule, and we finally obtain the relation at t = 0. A net result ofthis manipulation can be mimicked in the forward sum rule by changing the intercept ofthe slope parameter appropriately. [9, 10, 11, 12] Now we take the chiral SU (3) × SU (3)flavor symmetry and obtain B π + 2 f π π Z ∞ ν π dνν { σ π + d ( ν )+ σ π − d ( ν ) } = 12 π P Z ∞−∞ dαα { √ A ( α, √ A ( α, } , (19) B K + 2 f K π Z ∞ ν K dνν { σ K + d ( ν )+ σ K − d ( ν ) } = 12 π P Z ∞−∞ dαα { √ A ( α, A ( α, − √ A ( α, } , (20)where σ means the off-shell q = 0 total cross section of the reaction specified by itsupper suffix and can be assumed to be smoothly continued to the on-shell one, and ν π = m π m d and ν K = m K m d . The f π and the f K are the pion and the kaon decay constantrespectively. B π and B K is the contribution from the Born terms and the unphysical regionbelow the threshold of the continuum contribution. In the neutrino reactions, we obtain Z dx x { F ¯ νd ( x, Q ) + F νd ( x, Q ) } = 12 π P Z ∞−∞ dαα { √ A ( α,
0) + 2 √ A ( α, } , (21)and in the electroproduction we obtain Z dxx F ed ( x, Q ) = 118 π P Z ∞−∞ dαα { √ A ( α,
0) + 3 A ( α,
0) + √ A ( α, } . (22)In the left-hand side of the sum rules (21) and (22), the contribution from the Born termis included but can be neglected in the deep-inelastic region. We regularize the sum rules(19) - (22) by the method explained just before Eq.(19)[9, 10, 11, 12]. In addition herewe assume P Z dαα A ( α,
0) = 0 since the deuteron is iso-singlet. Note that this quantitycorresponds to the difference of the mean I of the quark and the anti-quark in the protonand that in the neutron in the deuteron, and hence it is zero under the isospin symmetry.In this way, we obtain the relation Z dxx (cid:26)(cid:18) F ¯ νd ( x, Q ) + F νd ( x, Q )2 (cid:19) − F ed ( x, Q ) (cid:27) = I dπ − I dK , (23)7here, by assuming the smooth extrapolation to the on-shell quantity, I π and I K aredefined as I dπ = B π + 2 f π π Z ∞ ν π dνν h ( ν − m π m d ) / { σ π + d ( ν ) + σ π − d ( ν ) } − νs bπ β πd i + 2 f π β πN π ln (cid:20) ν π (cid:21) , (24) I dK = B K + 2 f K π Z ∞ ν K dνν h ( ν − m K m d ) / { σ K + d ( ν ) + σ K − d ( ν ) } − νs bK β Kd i + 2 f K β KN π ln (cid:20) ν K (cid:21) , (25)with the leading high energy behavior being given by the soft Pomeron [17]as n σ π + d ( ν ) + σ π − d ( ν ) o ∼ β πd s α P (0) − π , n σ K + d ( ν ) + σ K − d ( ν ) o ∼ β Kd s α P (0) − K , (26)where α P (0) = 1 + b with b = 0 . s π = m π + m d + 2 ν and s K = m K + m d + 2 ν , andas a result of the assumption that the divergence in the forward direction comes from thesinglet, we obtain f π β πd = f K β Kd , and the relation between the residue of the Pomeronin the pion deuteron cross section and that of the structure function in the lepton-hadronscatterings.[9, 10, 11, 12] In terms of the sea quark distribution function λ i ( x, Q )of theproton in the deuteron where i = u, d, s specifys the quark, the sum rule (23) can betransformed as13 Z dx (cid:8) λ u ( x, Q ) + λ d ( x, Q ) − λ s ( x, Q ) (cid:9) = 12 (cid:18) − I dπ − I dK (cid:19) . (27)In Eq.(27), we have assumed the isospin symmetry of the quark distribution function andexpressed the quark distribution function of the neutron in the deuteron by the one ofthe proton in the deuteron. Further, though we take λ ¯ i = λ i for simplicity, what is reallyrequired in our formalism is the equality of the integrated quantity. Hence they can takedifferent value locally. The left-hand side of Eq.(27) is the mean hypercharge of the seaquark of the proton in the deuteron. If we neglect the nuclear effect, the deuteron crosssection is the sum of that of the proton and the neutron. In this case we obtain therelations I dπ ≈ I π and I dK ≈ I pK + I nK . Using these relations on the right-hand side ofEq.(27), we find that the left-hand side of it exactly agrees with the mean hypercharge ofthe sea quark of the proton given in Refs.[11, 12]. Since these relations break down due tothe nuclear effects, we see the sum rule (23) or (27) gives us information of the hadronicvacuum under the nuclear environment. Now, in a phenomenological analysis, we mayneed modification of the sum rules (23) and (27). These sum rules are derived by theassumption where the Pomeron is flavor singlet. The condition f π β πd = f K β Kd obtainedby this assumption is violated phenomenologically. One way to account this effect isexplained in Ref.[10]. However, it should be noted that the sum rules (23) and (27)correspond to the quantity related to the hypercharge and hence have a clear physicalmeaning. They show that the large symmetry restoration of the strange sea quark isnecessary in the small x region. Since the strange sea quark distribution is suppressedabove x = 0 .
01 greatly, this symmetry restoration itself is an interesting phenomena.Hence, these relation should be studied first by neglecting the symmetry breaking effectand taking the symmetry limit of sea quark distributions.[18] We explain the possiblesymmetry breaking effects together with the symmetry relation in the Appendix B.8nother application of Eq.(22) is to consider the relation for arbitrary two different Q and Q by separating out the Born term from F ed . Z x c ( Q ) dxx F ed ( x, Q ) − Z x c ( Q ) dxx F ed ( x, Q ) = B ( Q , Q ) + K ed ( Q , Q ) , (28)where the contribution from the Born term is given as B ( Q , Q ) = (cid:20) G C ( Q ) + 89 η G Q ( Q ) + 23 η G M ( Q ) (cid:21) − (cid:20) G C ( Q ) + 89 η G Q ( Q ) + 23 η G M ( Q ) (cid:21) , (29)with η i = Q i / m d . G C , G M and G Q are charge, magnetic and quadrupole moment of thedeuteron defined as < n, E ′ | J µem (0) | p, E > = − m d (cid:16)(cid:8) G ( Q )( E ′∗ · E ) − G ( Q ) ( E ′∗ · q )( E · q )2 m d (cid:9) ( p + n ) µ + G M ( Q ) (cid:8) E µ ( E ′∗ · q ) − E ′∗ µ ( E · q ) (cid:9)(cid:17) (30)with q = n − p for the electromagnetic current J µem (0), and G and G are related to G C , G M and G Q as G C = G + ηG Q , G Q = G − G M + (1 + η ) G with η = Q / m d . Thederivation of the Born term is straightforward but tedious hence we give its sketch in theAppendix C. K ed ( Q , Q ) is defined as K ed ( Q , Q ) = − Z x c ( Q )0 dxx F ed ( x, Q ) + Z x c ( Q )0 dxx F ed ( x, Q ) , (31)where x c ( Q ) = Q / ν c ( Q ) with ν c ( Q ) = ( W c − m d + Q ) / W =( p + q ) and W c is the cutoff invariant mass W . In Eq.(31), the integral over x should betaken after subtracting the small x behavior of F ed ( x, Q ) and F ed ( x, Q ) by obtainingthe condition that the residue of the pole is Q independent. It should be noted that , inthis regularization, we need not consider the symmetry breaking effect of the Pomeron. Inthese sum rules, we take Q fixed and can investigate the Q dependence of the sum rule.Further, if we take Q and Q small value such that K ed ( Q , Q ) is negligibly small, wehave the relation which express an intimate relation among the Born term, the resonancesand the non-resonant background.The regularization of the sum rule (17) can be done similarly, and we can transformit to Z x c ( Q ) dxx b ed ( x, Q ) − Z x c ( Q ) dxx b ed ( x, Q ) = B b ( Q , Q ) + K edb (( Q , Q ) , (32)where B b ( Q , Q ) = 4 η (cid:20) η η { G C ( Q ) + η G Q ( Q ) − G M ( Q ) } G Q ( Q ) + 14 G M ( Q ) (cid:21) − η (cid:20) η η { G C ( Q ) + η G Q ( Q ) − G M ( Q ) } G Q ( Q ) + 14 G M ( Q ) (cid:21) , (33)9nd K edb ( Q , Q ) = − Z x c ( Q )0 dxx b ed ( x, Q ) + Z x c ( Q )0 dxx b ed ( x, Q ) . (34)In Eq.(34), the integral over x should be taken after subtracting the small x behavior of b ed ( x, Q ) and b ed ( x, Q ). Now, we take Q large by keeping Q small or moderate value.Then,since the Born term at large Q is negligible, we can neglect it in Eq.(33). Whenthe integral is convergent, we can take x c ( Q ) = x c ( Q ) = 0, hence K edb ( Q , Q ) = 0.Then the sum rule (32) relates the tensor polarization and the elastic form factors atsmall or moderate Q to the tensor polarization at large Q . Especially, if Callan Grosslike relation b ed = 2 xb ed [13] with the vanishing tensor polarization of the sea quark atlarge Q holds, the second term on the left-hand side of Eq.(32) is zero [19]. In this casethe sum rule becomes the one at small or moderate Q . Now the recent experiment atHERMES shows[20, 21] Z . . dxx b ed ( x, Q = 5GeV ) > . (35)Though there are unmeasured region, HERMES result possibly suggest the non-zero ten-sor polarization at Q = 5GeV . Since the Born term at Q = 5GeV is negligibly smallin Eq.(33), we can set B ( Q , Q ) = 0. Then the sum rule (32) shows that the non-zeropolarization persist in the large Q region. When the integral over b ed ( x, Q ) /x diverges,since the main contribution comes from the small x region and that ,at large Q , b ed ( x, Q )behaves similarly as F ed ( x, Q )[13], we can expect K edb ( Q , Q ) >
0. Thus HERMES re-sult does not contradict with the zero tensor polarization of the sea quark at large Q inthe regularized sense. We have derived the sum rules for the structure functions of the deuteron from the currentanti-commutation relation on the null-plane. The sum rules correspond to the ones at thewrong signature point. As explained in the Introduction they give us information of thevacuum of the deuteron.From the spin independent part, we obtain the sum rule for the mean hypercharge of thesea quark of the proton in the deuteron. Further, in the small Q region, we obtain therelation among the Born term, the resonances and the non-resonant background. Fromthe spin dependent part, we obtain the relation between the tensor polarization at smallor moderate Q and that at large Q .Now, the application of these sum rules in other forms such as the ones in the photopro-duction are possible as in the nucleon target case. Further, though only the sum rulesfrom the good-good component are discussed, the same method can be applied to thegood-bad component. In this case, we obtain the sum rules for the spin dependent struc-ture functions g ed and g ed . These sum rules take the same form as in the nucleon targetcase.[22] 10 σ β pq + q σ=−β m Figure 1: The support property of the spectral function h ab ( λ , β ) in the ( β, σ ) plane. Itis not zero only in the shaded region and, below the parabola σ = − β m , it is zero bythe causality. A Current anti-commutation relation on the null-plane through DGS representation
Let us consider DGS representation of the connected matrix element of the current com-mutation relation on the null-plane between the stable hadron.[8] W ab ( p · q, q ) = Z d x exp( iq · x ) < p | [ J a ( x ) , J b (0)] | p > c = Z d x exp( iq · x ) Z ∞ dλ Z − dβ exp ( iβp · x ) h ab ( λ , β ) i ∆( x, λ )= (2 π ) Z ∞ dλ Z − dβδ (( q + βp ) − λ ) ǫ ( q + βp ) h ab ( λ , β ) , (36)where W ab ( p · q, q ) can be expressed as W ab ( p · q, q ) = X n (2 π ) δ ( p + q − n ) h p | J a (0) | n ih n | J b (0) | p i− X n (2 π ) δ ( p − q − n ) h p | J b (0) | n ih n | J a (0) | p i . (37)We denote the lowest mass in the s channel continuum as M s and that in the u channelas M u . At the rest frame, p = ( m, ~ m + q = n , q satisfys q ≧ M s − m . Similarly, since the second term is restricted as m − q = n , q satisfys q ≦ m − M u . Hence the first and the second terms in Eq.(37)are disconnected as far as m < ( M s + M u ) / h ab ( λ , β ) is not zero only in the shaded region inFig.1. The integration path is σ = 2 βp · q + q , where σ = λ − β m . At the rest frame,we see that the point in the integration path where the sign changes through the factor ǫ ( p · q + βm ) lies always in the region σ < − β m . In the s channel, since p · q >
0, slopeof the integration path is positive. Thus only the region ǫ ( p · q + βm ) = 1 contributes,and11ence, in the s channel,we obtain(2 π ) Z ∞ dλ Z − dβδ (( q + βp ) − λ ) h ab ( λ , β ) θ ( q + βp )= X n (2 π ) δ ( p + q − n ) h p | J a (0) | n ih n | J b (0) | p i . (38)Similarly, in the u channel, we obtain(2 π ) Z ∞ dλ Z − dβδ (( q + βp ) − λ ) h ab ( λ , β ) θ ( − ( q + βp ))= X n (2 π ) δ ( p − q − n ) h p | J b (0) | n ih n | J a (0) | p i . (39)By combining these two relations we obtain DGS representation of the current anti-commutation relation as f W ab ( p · q, q ) = Z d x exp( iq · x ) < p |{ J a ( x ) , J b (0) }| p > c = Z d x exp( iq · x ) Z ∞ dλ Z − dβ exp ( iβp · x ) h ab ( λ , β )∆ (1) ( x, λ )= (2 π ) Z ∞ dλ Z − dβδ (( q + βp ) − λ ) h ab ( λ , β ) . (40)The null-plane restriction of the current commutation relation or the anti-commutationrelation can be obtained by the integration of W ab and f W ab with respect to q − .Now we take the scalar current J a ( x ) =: φ † ( x ) τ a φ ( x ) : where[ φ † ( x ) , φ (0)] | x + =0 = i ∆( x ) (41)with ∆( x ) = − ǫ ( x − ) δ ( ~x ⊥ ) / x + = 0. Using this relation, the current commutationrelation at x + = 0 becomes < p | [ J a ( x ) , J b (0)] | p > c | x + =0 = i ∆( x ) < p | : φ † ( x ) τ a τ b φ (0) : + : φ † (0) τ b τ a φ ( x ) : | p > c . (42)From Eq.(36) restricted at the null-plane, we have < p | [ J a ( x ) , J b (0)] | p > c | x + =0 = Z ∞ dλ Z − dβ exp ( iβp · x ) h ab ( λ , β ) i ∆( x, λ ) . (43)Since ∆( x, λ ) = ∆( x ) = − ǫ ( x − ) δ ( ~x ⊥ ) / x + = 0, we obtain the relation < p | : φ † ( x ) τ a τ b φ (0) : + : φ † (0) τ b τ a φ ( x ) : | p > c = Z ∞ dλ Z − dβ exp ( iβp · x ) h ab ( λ , β ) . (44)By using this relation, the equation restricted at the null-plane obtained from Eq.(40)becomes < p |{ J a ( x ) , J b (0) }| p > c | x + =0 = Z ∞ dλ Z − dβ exp ( iβp · x ) h ab ( λ , β )∆ (1) ( x, λ )= ∆ (1) ( x ) < p | : φ † ( x ) τ a τ b φ (0) : + : φ † (0) τ b τ a φ ( x ) : | p > c , (45)12here we use the fact that ∆ (1) ( x, λ )at x + = 0 is also independent on the mass λ andgiven as ∆ (1) ( x, λ ) = ∆ (1) ( x ) = − ln | x − | δ ( ~x ⊥ ) / π . A net result is that the current anti-commutation relation on the null-plane can be obtained from the current commutationrelation on the null-plane simply by changing i ∆( x, λ ) to ∆ (1) ( x, λ ) at x + = 0. Morerigorous reasoning can be done by taking the Fourier transform and by considering theconditions necessary to restrict them to the null-plane such aslim Λ →∞ Z ∞−∞ dq − exp ( − ( q − ) / Λ ) H ab ( p · q, q ) , (46)where H ab is W ab or f W ab . This results in the condition to the h ab and is known to beequivalent to the superconvergence relation which is required to get the fixed-mass sumrule, and in the null-plane formalism is equivalent to the interchange to set q + = 0 and ν = p · q integration. It is in this point where the difference between the connected matrixelement of the stable hadron of the current commutation relation and that of the currentanti-commutation relation appears. B SU(3) symmetry breaking effect on the symmetryrelation
Here we consider the sum rules for SU (3). The regularization of the sum rules (19) ∼ (22) can be done as explained in the paragraph before Eq.(19). The detailed method isgiven for example in Ref.[12]. We first summarize the result. We assume the leading highenergy behavior is given by the soft Pomeron as { F ¯ νd + F νd } ∼ (cid:16) Q /Q (cid:17) α P (0) − β νd ( Q , − α P (0))(2 ν ) α P (0) − , (47)where Q = 1GeV and F ed ∼ (cid:16) Q /Q (cid:17) α P (0) − β ed ( Q , − α P (0))(2 ν ) α P (0) − . (48)We expand β ld with l = e or ν as β ld − ( ǫ − b ) β ld + O (( ǫ − b ) ) . (49)where the intercept of the Pomeron is set as α P (0) = 1 + b − ǫ and ǫ approaches b from the above. The parameter ǫ goes to 0 finally after taking out the pole terms fromboth-hand side of the sum rule. This change of the parameter mimics the − t in thenon-forward sum rules. Then by assuming that the Pomeron is flavor singlet and that itcomes from the term A ( α,
0) being flavor singlet, we obtain the relations, β νd = 6 β ed and β νd = 6 β ed from the sum rules (21) and (22). Further from the sum rules (19) and (21) weobtain the condition πβ νd = 4 f π β πd and from the sum rules (19) and (20) the condition f π β πd = f K β Kd . The regularization dependent terms in the sum rules are related by theserelations and we obtain the relations independent of the regularization. In this way, weobtain the relation (23) and C d = 19 (2 I π + I K ) , (50)13here C d = Z dxx { F ed − β ed x − b } − β ed . (51)Now the condition f π β πd = f K β Kd is violated about 20% phenomenologically. Hence thesum rules (23) and (27) still diverge if we use the phenomenological value. To remedythis we consider the mixing of the singlet and the octet as[10] f A ( α,
0) = A ( α,
0) cos θ + A ( α,
0) sin θ, f A ( α,
0) = − A ( α,
0) sin θ + A ( α,
0) cos θ, (52)and assume the contribution of the Pomeron is given by the term f A ( α, ∼ (22) with use of Eq.(52) and by regularizing them, we obtain the relations f π β πd √ θ + sin θ ) = f K β Kd √ θ − sin θ , (53)and β iνd = 12( √ θ + sin θ )2 √ θ + sin θ · β ied , (54)where i = 0 ,
1. The relation between β νd and β πd is the same as the one before the mixing.In case of the nucleon target, a similar relation as Eq.(53) was derived, and was foundthat the relation is satisfied phenomenologically about at θ ∼ − ◦ . In the deuteroncase, the relation (53) seems to be satisfied well phenomenologically about at the sameangle because the large constant term at high energy in the cross section formula by theParticle data group[23] satisfys the relation (53) with this angle. Now by this mixing, wefind that the relation (50) also holds and the sum rule (23) can be rewritten as Z dxx ((cid:18) F ¯ νd ( x, Q ) + F νd ( x, Q )2 (cid:19) − √ θ + sin θ )2 √ θ + sin θ F ed ( x, Q ) ) = I dπ − I dK − θ √ θ + sin θ · C d . (55)Then the sum rule (27) becomes(2 √ θ − sin θ ) I π − √ θ + sin θ ) I K = 6 √ θ − √ θ )+ Z dx n √ θ − √ θ )( λ u + λ d ) − √ θ + sin θ ) λ s o . (56)Since the strange sea quark is suppressed above x = 0 .
01 greatly, the large symmetryrestoration of the sea quark exists in the small x region[18], and that to satisfy Eq.(56)the small x limit of the strange sea quark distribution must be larger than that of the u or d type sea quark. 14 The Born term contributions
The Born term contribution to Eq.(1) is given as12 δ (2 ν − Q ) B µν , (57)with B µν = < p, E | J µem (0) | n, E ′ >< n, E ′ | J νem (0) | p, E > = 1 m d X λ (cid:16)(cid:8) G ( Q )( E ∗ · E ′ ) − G ( Q ) ( E ∗ · q )( E ′ · q )2 m d (cid:9) ( p + n ) µ + G M ( Q ) (cid:8) − E ′ µ ( E ∗ · q ) + E ∗ µ ( E ′ · q ) (cid:9)(cid:17) × (cid:16)(cid:8) G ( Q )( E ′∗ · E ) − G ( Q ) ( E ′∗ · q )( E · q )2 m d (cid:9) ( p + n ) ν + G M ( Q ) (cid:8) E ν ( E ′∗ · q ) − E ′∗ ν ( E · q ) (cid:9)(cid:17) , (58)where n = p + q and λ is the polarization of E ′ . Here we denote it as E ′ ( n, λ ). Thenusing the relation X λ E ′∗ µ ( n, λ ) E ′ ν ( n, λ ) = n µ n ν − m d g µν (59)with E ′ · n = E ∗′ · n = 0 and E ′ · E ∗′ = − m d , we take the product on the right-hand side ofEq.(58). Then we classify the product into the symmetric terms and the antisymmetricones under the interchange of the E and E ∗ . We first calculate the symmetric ones andtake the polarization average of the initial deuteron and obtain the Born term contributionto F and F as F = δ (2 ν − Q ) Q η + 1) G M , (60)and F = δ (2 ν − Q ) Q (cid:16) G C + 89 η G Q + 23 ηG M (cid:17) . (61)The rest of the symmetric terms contribute to b ∼ b . By noting that the polarizationaveraged parts are subtracted, and that g µν is only in the tensor G µν we find the con-tribution to the sum of the b and the b − b with an appropriate coefficient. Furthersince E µ E ∗ ν + E ∗ µ E ν is only in the tensor u µν , we obtain the contribution to the b − b .Hence we can separate the contribution to the b and the b − b . Similar considerationcan be done to the coefficient of t µν which gives the contribution to the b − b and p µ p ν which gives the contribution to the b + 3 b + 3 b . Thus we obtain b = δ (2 ν − Q ) Q ηG M , (62) b = δ (2 ν − Q )4 Q η (cid:16)
11 + η (cid:0) G C + η G Q − G M (cid:1) G Q + 14 η G M (cid:17) , (63) b = δ (2 ν − Q )4 Q η (cid:16) η ) (cid:0) G C + η G Q − G M (cid:1) G Q − η + 212 η G M (cid:17) , (64)15 = δ (2 ν − Q )4 Q η (cid:16) η ) (cid:0) G C + η G Q − G M (cid:1) G Q + 1 + 6 η η G M + G Q G M (cid:17) . Now the antisymmetric parts under the interchange of the E and E ∗ give the contributionto the g and g . In this case, we first note the identity a µ ǫ νραβ q ρ s α p β − a ν ǫ µραβ q ρ s α p β = − ( a · q ) ǫ µναβ s α p β − ( a · s ) ǫ µναβ p α q β + ( a · p ) ǫ µναβ s α q β . (65)Since s µ = − ( i/m d ) ǫ µαβγ E ∗ α E β p γ , we have ǫ νραβ q ρ s α p β = i ( E ∗ ν ( q · E ) − E ν ( q · E ∗ )) . (66)Thus we obtain the relation i (cid:16) a µ (cid:0) E ∗ ν ( q · E ) − E ν ( q · E ∗ ) (cid:1) − a ν (cid:0) E ∗ µ ( q · E ) − E µ ( q · E ∗ ) (cid:1)(cid:17) = − ( a · q ) ǫ µναβ s α p β − ( a · s ) ǫ µναβ p α q β + ( a · p ) ǫ µναβ s α q β . (67)In case of a = 2 p + q , a · q = 2 p · q + q = 0 for the Born term. Another useful relation is ǫ µναβ s α p β = i ( E ∗ µ E ν − E ∗ ν E µ ) . (68)In this way we obtain g = δ (2 ν − Q ) Q G M ( G C + η G Q + η G M ) , (69) g = δ (2 ν − Q ) Q η G M ( G C + η G Q − G M ) . (70) References [1] S.L.Adler and R.F.Dashen,
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