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QCD Sum Rules for Nucleon-Nucleon Interactions

The QCD sum rules for spin-dependent nucleon-nucleon interactions are formulated and their physical implications are studied. The basic object of the study is the correlation function of the nucleon interpolating field, where the matrix element is taken with respect to the one-nucleon state. The dispersion integral of the correlation function around the nucleon threshold is investigated in detail. It turns out that the integral can be identified as a measure of the nucleon-nucleon interaction strength, which is proportional to the scattering length in the small scattering length limit and to one half of the effective range in the large scattering length limit. New operators must be taken into account in the OPE of the correlation function. There behavior operators do not vanish when the matrix element is taken with respect to the spin-nonaveraged one-nucleon state. The Wilson coefficients of such operators are calculated. The sum rules obtained in this manner relate the spin-dependent nucleon-nucleon interaction strengths with the spin-dependent nucleon matrix elements of the quark-gluon composite operators. The sum rules imply that the interaction is stronger in the spin-triplet channel than in the spin-singlet channel, but that the spin-dependence of the nucleon-nucleon interactions is rather small. In the spin-singlet channel the calculated strength is in qualitative agreement with the empirical strength, which is estimated by the empirical low energy scattering observables.

Therebehavior operators do not vanish when the matrix element is taken with respect to the spin-nonaveraged one-nucleon state. The Wilson coefficients of such operators are calculated. The sum rules obtained in this manner relate the spin-dependent nucleon-nucleon interaction strengths with the spin-dependent nucleon matrix elements of the quark-gluon composite operators. The sum rules imply that the interaction is stronger in the spin-triplet channel than in the spin-singlet channel, but that the spin-dependence of the nucleon-nucleon interactions is rather small. In the spin-singlet channel the calculated strength is in qualitative agreement with the empirical strength, which is estimated by the empirical low energy scattering observables.

I. INTRODUCTION
One of the most important goals of investigating strong interaction physics is to understand the of hadrons and hadronic interactions on the basis of quantum chromodynamics (QCD). Shifman, Vainshtein and Zakharov proposed the method of the QCD sum rule, which provides us with a framework to investigate the properties of hadrons in a model-independent way. [1] This method has been successfully applied to the study of the masses, decay constants, magnetic moments and other properties of various hadrons. [2] Recently, the present authors extended the QCD sum rule to the investigation of hadronic interactions. [3] In Ref. 3) a nucleon-nucleon system was studied as a typical case. The correlation function of the nucleon interpolating field, whose matrix element is taken with respect to the one-nucleon state and averaged over the nucleon spin, was considered. It was noted that the correlation function has a second-order pole at the nucleon on-shell energy as a function of the energy associated with the interpolating field and that its coefficient is the T-matrix for the nucleon-nucleon (N N ) scattering. As-suming that the dispersion integral is dominated by the pole term, sum rules were derived which relate the spin-averaged N N scattering lengths with the spin-averaged matrix elements of the quark-gluon operators with respect to the one-nucleon state. The obtained N N scattering lengths are of the order of several fm, which is rather large in the strong interaction scale, but are rather smaller than the experimental values. The formalism was further applied to other hadron-nucleon systems. [4] [5] The following point, however, remained unclarified in those works. In the analysis of the QCD sum rule for the hadron in the vacuum, the correlation function of the hadron interpolating field, whose matrix element is taken with respect to the vacuum, is considered. The imaginary part of the correlation function consists of a pole term corresponding to the ground state and a continuum term corresponding to the excited states. Under the Borel transformed dispersion integral, the continuum contribution is exponentially suppressed compared to the ground state contribution due to the energy difference between the ground state and the continuum threshold. For this reason the sum rule analysis is expected to be insensitive to the detailed form of the continuum, so that the continuum is usually parametrized in a very simple form. When one deals with the hadron correlation function, whose matrix element is taken with respect to the one-nucleon state, the situation is different. The energy of the continuum threshold is not higher than the pole energy. Therefore, it is not clear if the Borel transformed dispersion integral is really dominated by the pole term or not.
A related question is the following. It is known that there is a loosely bound state in the spin-triplet nucleon-nucleon channel and an almost bound state in the spin-singlet nucleon-nucleon channel. If there is a zero-energy bound state, the scattering length diverges. Therefore, the N N scattering lengths are expected to be very sensitive to the N N interaction strength.
On the other hand, it is hard to believe that the nucleon matrix elements of the quark-gluon operators would be very sensitive to the N N interaction strength. It seems strange that the sum rule relates these two quantities of very different natures.
Another point is that only the spin-averaged sum rules are obtained from the spin-averaged correlation function. The N N channel is special in the sense that selecting the isospin channel automatically selects the spin state.
In Ref. [3], the sum rules for the spin-triplet and singlet scattering lengths are obtained by combining isospin states. As far as the pole term is concerned, the above selection rule is correct, but it does not hold for the continuum.
The sum rule in Ref. 3) is valid only if the pole term is dominant. Therefore, it is more desirable to construct the spin-dependent sum rules.
In this paper we consider the spin-dependent correlation function of the nucleon interpolating field, where the matrix element is taken with respect to the spin-nonaveraged one-nucleon state. The purpose of this paper is twofold. First, we extend the procedure of the sum rule to the case of the spinnonaveraged correlation function. Second, we show that the dispersion integral of the correlation function around the nucleon threshold can be regarded as a measure of the nucleon-nucleon interaction strength. As a result, we derive sum rules which relate the spin-dependent N N interaction strengths with the spin-dependent nucleon matrix elements of the quark-gluon operators.

II. FORMULATION
A. Physical content of the correlation function and Borel sum rules Consider the spin-dependent correlation function, Π(qps), where |ps is the one-nucleon state with momentump and spin s (p 2 = M 2 , s 2 = −1 andps = 0, where M is the nucleon mass) normalized as ps|p ′ s ′ = (2π) 3 δ 3 (p−p ′ )δ ss ′ and ψ is the normalized nucleon field operator 0|ψ(0)|ps = u(ps), where u(ps) is a positive energy solution of the free Dirac equation for the nucleon. In this paper, momentum withˆrepresents the nucleon on-shell momentum. Later, the normalized nucleon field, ψ, is replaced by the unnormalized nucleon interpolating field (quark-gluon composite field), η. The following discussion, however, holds as it is for the interpolating field, except for the normalization. Naively, the dispersion relation for the correlation function, Π(qps), is written as where q ′ = (q ′ 0 , q). Throughout this paper, whenever we take the imaginary part of a quantity, we approach the real energy axis from above in the complex energy plane. Therefore, strictly speaking, ImΠ is the imaginary part of the retarded correlation function. The QCD sum rules are obtained by evaluating the left-hand side of Eq. (1) by the operator product expansion (OPE) and expressing the right-hand side in terms of physical quantities.
Let us consider the singularities of Π(qps) as functions of q 0 . In the complex q 0 plane, Π(qps) has a branch cut from the lowest B = 2 continuum threshold to the right and another branch cut starting from the lowest B = 0 continuum threshold to the left. In addition, Π(qps) has second-order poles at q 0 = ± q 2 + M 2 ≡ ±E q whose coefficients are the N N and NN T-matrices T + and T − , respectively: Hereq = (q 0 , −q),r = (r 0 , −r) and v(qr) is the negative energy solution of the free Dirac equation for the nucleon.
In order to take out the pole contribution from ImΠ(qps) it is convenient to define off-shell N N and NN T-matrices by Note that Eq. (2) is just a definition of the T-matrix off the mass shell, but the LSZ reduction formula shows rigorously that it is the T-matrix on the mass shell.
In order to separate the contribution from the poles at q 0 = E q and q 0 = −E q , we introduce the projection operators Λ + and Λ − by which have the properties The projected correlation functions are related to the off-shell T-matrices as Π ± (qrps) = T ± (qrps) where T ± (qrps) ≡ T ± (qrps; qrps). Clearly, Π ± (qrps) has a second-order pole at q 0 = ±E q but not at q 0 = ∓E q .
Naively, the dispersion relation for the projected correlation function, Π + , is given by Formally, the imaginary part of the correlation function is written as ImΠ + (qrps) = Im 1 However, as it will turn out, when q = 0 and p = 0, the integral of the second term is divergent because u is divergent, and the integral of the third term is Instead of Eq. (3), we consider the dispersion relation for / q−M q 0 Π(qrps), or for q 0 −E q q 0 Π + (qrps) in terms of the projected correlation function, Symmetrizing Eq. (4) we obtain where Now Eq. (5) is well-defined when q = 0 and p = 0 because u does not appear and the second term behaves as Applying the Borel transformation, to both sides of Eq. (5), we obtain where M B is the Borel mass. In order to derive the Borel sum rules we must evaluate the left-hand side by the OPE and parametrize the right-hand side in terms of physical quantities. Now, the question is how to parametrize the integrand of the right-hand side of Eq. (7).
Let us recall the QCD sum rule for the nucleon in the vacuum, where the imaginary part of the correlation function has the form The first term is the contribution from the nucleon pole term, and the second term is due to the excited states. Borel transformation on the dispersion integral of ImΠ + (q) gives where the second term, the contribution from the excited states, starts at the continuum threshold, ω (ω > E q ), and is exponentially suppressed compared to the first term. For this reason, it is possible to use the rough model of the hadron continuum, where Π OP E + is the asymptotic form of the correlation function in the OPE, ω 0 is the effective continuum threshold, and the normalization constant λ is explicitly included ( 0|η(0)|ps = λu(ps)).
Let us turn to the problem at hand. As an extension of Eq. (17) one might by approximating the second term of the right-hand side of Eq. (9) by its asymptotic form. However, the second term starts at , and it is not exponentially suppressed compared to the first term. Therefore, one cannot justify Eq. (9). One has to know the behavior of the second term around the threshold. For this purpose it is important to note that the off-shell optical theorem holds for T . When the center-of-mass energy is above the threshold of the N N channel and below the threshold of the next channel, only the N N states contribute in the intermediate states, and the off-shell optical theorem is simplified as In order to simplify the notation we introduce the scattering amplitude f by . Moreover, we go to the center-of-mass frame (q + p = q ′ + p ′ = 0) and restrict ourselves to the s-wave. We define three scattering amplitudes, f 0 , f 1 and f 2 as It is well known that the on-shell scattering amplitude f 0 has the form where a is the scattering length and r the effective range. Similarly, the off-shell scattering amplitude f 2 has the form which can be shown as follows. First, since the discontinuity of the T-matrix along the real energy axis is proportional to the imaginary part of the Tmatrix, the Taylor expansion of the real and imaginary parts of the scattering amplitude includes only even and odd powers of k, respectively. Second, f 0 , Therefore, we have , Third, from Eq. (10) the following relations hold, Therefore, we have Equation (11) follows from Eqs. (12) and (13).
It should be noted thatr is different from the effective range r, butr coincides with r in the limit a → ∞: This is shown as follows. Equations (12) and (13) indicate that both O(1) and O(k) terms of 1/f 0 (k) and 1/f 2 (k), which are real and imaginary respectively, coincide with each other. Therefore, we have Equation (14) is independent of a and therefore holds also in the limit a → ∞ due to the continuity. Since 1/f 0 (k) and 1/f 2 (k) do not have the O(1) terms in this limit, they must coincide with each other up to the O(k 2 ) terms, i.e.
Having understood the structure of ImT + , we proceed to the integral of the right-hand side of Eq. (7), I, in the vicinity of q 0 = M , The integral, I, can be decomposed as In Eq. (15), the first term, I t , is the threshold contribution, given by The second term, I c , is the continuum contribution, given by where Imf cut The last term, I b , is the bound-state contribution, which has to be taken into account if there is a bound state, given by where Imf pole and iκ 0 is the pole momentum, 1/f 2 (iκ 0 ) = 0.
By performing the integral, the continuum contribution becomes which is simplified in two limits of a as Similarly, the bound-state contribution becomes where ω ′ = 2 M 2 − κ 2 0 − M . In the limit a → ∞, and the bound-state contribution is simplified as Let us suppose that one can freely change the interaction strength of two nucleons and examine how the integral I should change as a function the interaction strength. When the interaction is weak, the scattering length is also small, and the integral I is dominated by I t : As the interaction becomes stronger, the scattering length increases and the integral, I, also increases. As the interaction strength increases further, the scattering length eventually diverges when the bound state is just formed.
Just before the bound state is formed, the integral I becomes and just after the bound state is formed, it becomes This shows that before and after the bound state is formed the integral is continuous, though the scattering length diverges with opposite signs. This observation leads us to conjecture that the integral around the threshold is a measure of the N N interaction strength. Based on this conjecture we define the N N interaction strength, α, by In the dispersion integral, the imaginary part of the correlation function, ImΠ + , contains the contribution from all possible intermediate states such as those of the N N , N N π channels and so on. However, only the N N channel contributes around the threshold. We assume that the contribution from the N N state is taken into account by the form of the right-hand side of Eq. (16) and that the rest is approximated by the asymptotic form of the correlation function starting from an (effective) threshold, ω + , for the B = 2 channels other than the N N channel and ω − for the B = 0 channels: where the normalization constant λ is explicitly included. This is possible now because the contribution from states other than those of the N N channel is exponentially suppressed compared to the N N contribution.

B. OPE of the correlation function and results
Let us turn to the OPE. We take the interpolating field of the neutron as [6] η where C denotes the charge conjugation operator and a, b and c are color indices. We take into account all the operators of dimension less than or equal to four. We also include four-quark operators, they are of dimension six. The operators which involve the quark mass are ignored. In the OPE, the neutron correlation function, where the matrix elements are taken with respect to the one-nucleon state, is given as into the right-hand side of Eq. (7), respectively, and splitting α into spinindependent and spin-dependent parts, α = α indep +α dep , we obtain sum rules for the spin independent and dependent interaction strengths, respectively, as and where and Φ is the error function.
From the QCD sum rule for the nucleon in the vacuum in Ref. [6], the normalization constant λ 2 is related with the vacuum expectation values of the operators according to The sum rule for the spin-independent part of the interaction strength, α indep , is nothing but the sum rule for the N N scattering length in Ref. [3], which was obtained starting from the spin-averaged correlation function. The difference, however, is that the scattering length in Ref. [3] is replaced by the interaction strength in Eq. (19).
Equation (20)  We now discuss the nucleon matrix elements of the quark-gluon operators.
The nucleon matrix elements, qq N and qγ µ q N , are spin-independent and have already been discussed in Ref. 7), while qγ µ γ 5 q N and qγ 5 σ µν q N are spin-dependent and are written in terms of the axial charge, ∆q, and the tensor charge, δq, as qγ µ γ 5 q N = ∆qs µ , The axial charge and the tensor charge are related to the structure func- where g 1 is the longitudinal quark-spin distribution in the nucleon,ḡ 1 is the antiquark-spin distribution, h 1 is the quark-transversity distribution, andh 1 is the antiquark-transversity distribution. h 1 , together with the unpolarized quark distributions f 1 and g 1 , forms a complete set of twist-2 structure functions.
Recently ∆q has received much attention in connection with the spin content of the nucleon. While a naive quark model picture suggests that the nucleon spin is carried by quarks, i.e. ∆u + ∆d = 1, it was experimentally found that ∆u + ∆d << 1. On the other hand, up to now no experimental information is available for δq, since h 1 andh 1 cannot be measured by deep inelastic scattering.
These quantities can, however, be measured by Drell-Yang processes, which are planned in a future RHIC experiment. However, δq for the proton was calculated on the lattice [12] with the result δu = 0.839 ± 0.060, δd = −0.231 ± 0.055 at Q 2 = 2 GeV 2 . In this paper we use the lattice results both for ∆q and δq and ignore the Q 2 evolution of these matrix elements between Q 2 = 1 GeV 2 and 2 GeV 2 .
By neglecting the operator including the quark mass, a 1 can be related to the first moment of the longitudinal quark-spin distribution, g 2 , and it is given at tree-level by [15] a 1 = −2 In addition to the above dimension-three and dimension-four operators we take into account the dimension-six four-quark operators,qqqq,qqqγ µ q, qqqγ µ γ 5 q andqqqγ 5 σ µν q, since four-quark operators are known to give the largest contribution among higher order operators in the QCD sum rule for the nucleon in the vacuum. [ Thus we assume qqqq N = 2 qq 0 qq N , qqqγ µ q N = qq 0 qγ µ q N , qqqγ µ γ 5 q N = qq 0 qγ µ γ 5 q N and qqqγ 5 σ µν q N = qq 0 qγ 5 σ µν q N .
For completeness we list here the values which were used in the calculation.
The spin-independent proton matrix elements of the operators are Borel stability is much more sensitive to ω − than to ω + . This is because the ω − dependence of the coefficients of the dimension-three and dimension-four operators is much stronger than the ω + dependence. We see that the spin-dependent part is rather smaller than the spinindependent part. This is due to the small matrix elements of the spin- In the spin-singlet channel, the scattering length is so large that the strength, α 1 pn , is expected to be approximated well by r 1 pn /2, α 1 pn ≈ r 1 pn /2 = 1.37 fm, which is in rather good agreement with the calculated result.
In the spin-triplet channel, the scattering length is not small. However, it is not as large as in the spin-singlet channel. Therefore, it is not easy to estimate the strength α 3 pn from experimental observables. However, there is a loosely bound state, deuteron, in the spin-triplet channel, while there is an almost bound state in the spin-singlet channel. This implies that the interaction in the spin-triplet channel is stronger (but not very stronger) than that in the spin-singlet channel. This tendency is consistent with the calculated results.

III. SUMMARY AND OUTLOOK
In this paper we have studied spin-dependent nucleon-nucleon (N N ) interactions in the QCD sum rules. The basic object of our study is the spindependent nucleon correlation function, whose matrix element is taken with respect to the one-nucleon state.
The dispersion integral of the correlation function around the nucleon threshold has been investigated in detail. The integral is given by the sum of the threshold contribution, which is due to the second-order pole term proportional to the scattering length, the continuum contribution, and the bound state contribution. When the interaction is weak, the integral is dominated by the threshold contribution and is proportional to the scattering length. As the interaction becomes stronger, the continuum contribution also becomes important. When the interaction is just so strong as to form a bound state, both the threshold contribution and the continuum contribution diverge, but their sum is finite. Thus the sum of these two contributions is given by the same form as in the case of the weak interaction. However, in this case the scattering length is replaced by one half of the effective range. Based on this observation we have defined the N N interaction strength through the dispersion integral around the nucleon threshold.
In the OPE of the correlation function, new operators, such asqγ µ γ 5 q, qγ 5 σ µν q, have to be taken into account. These operators do not vanish when the matrix element is taken with respect to the spin-nonaveraged one-nucleon state. We have calculated the Wilson coefficients of such operators.
The obtained sum rules relate the spin-dependent N N interaction strengths with the spin-dependent nucleon matrix elements of the quarkgluon composite operators. The spin-dependent nucleon matrix elements such as qγ µ γ 5 q N and qγ 5 σ µν q N are related with the spin-dependent structure functions of the nucleon, g 1 , g 2 and h 1 . We found that the interaction strength in the spin-singlet channel is weaker than in the spin-triplet channel, but that the spin-dependent part of the interaction strength is considerably smaller than the spin-independent part. Experimentally, it has been found that there is a loosely bound state, deuteron, in the spin-triplet channel, while there is an almost bound state in the spin-singlet channel, which implies that the interaction is slightly stronger in the spin-triplet channel than in the spinsinglet channel. This seems to be consistent with the sum rule result. In the spin-singlet channel, the scattering length is so large that the interaction strength can be estimated by using the observed effective range, while in the spin-triplet channel the absolute value of the interaction strength is difficult to obtain from observables. The empirical interaction strength thus obtained in the spin-singlet channel agrees rather well with the sum rule calculation.
The method used in the present paper can be extended to other hadronnucleon channels. In particular, it is straightforward to apply it to the hyperon-nucleon channels. The obtained sum rules would relate the hyperonnucleon interaction strengths with the nucleon matrix elements of the quarkgluon operators which include strange quark operators in addition to up and down quark operators. Also, the present sum rules can be used in the oppo-site way. Namely, if one knows detailed information on various spin-dependent hadron-nucleon interaction strengths, one can obtain spin-dependent matrix elements of quark-gluon operators with respect to the one-nucleon state. This provides us with information such as the spin content of the nucleon.