Short distance repulsion in 3 nucleon forces from perturbative QCD
We investigate the short distance behavior of 3 nucleon forces (3NF) defined through Nambu–Bethe–Salpeter wave functions, using the operator product expansion(OPE) and calculating anomalous dimensions of 9–quark operators in perturbative QCD. As is the case of NN forces previously considered, we show that 3NF have repulsions at short distance at 1–loop, which becomes exact in the short distance limit thanks to the asymptotic freedom of QCD. Moreover these behaviors are universal in the sense that they do not depend on the energy of the NBS wave function for 3 nucleons.
Realistic nuclear potentials between two nucleons (2N), determined precisely from 2N scattering data together with the deuteron binding energy, have often been used to study nuclear many-body problems. These two-nucleon forces (2NF), however, generally underestimate the experimental binding energies of light nuclei [1, 2] and this fact indicates the necessity of taking into account three-nucleon forces (3NF). In addition, a clear indication of 3NF is observed in high precision deuteron-proton elastic scattering data at intermediate energies .
The 3NF may also play an important role for various phenomena in nuclear physics and astrophysics, which include (i) the backward scattering cross sections in nucleus-nucleus elastic scattering , (ii) the anomaly in the oxygen isotopes near the neutron drip-line , and (iii) the nuclear equation of state at high density relevant to the physics of neutron stars . Universal short distance repulsion for three baryons (nucleons and hyperons) is also suggested to explain the observed maximum mass of neutron stars .
Despite of its phenomenological importance, a microscopic understanding of 3NF is still limited, due to difficulties in studying 3NF experimentally. Pioneered by Fujita and Miyazawa , the long range part of 3NF has been modeled by two-pion exchange , which is known to be attractive at long distance. In addition a repulsive component of 3NF at short distance is introduced in a purely phenomenological way .
To go beyond phenomenology, it is most desirable to determine 3NF directly from the fundamental degrees of freedom , the quarks and the gluons, on the basis of QCD. Recently the first investigation of this kind has been attempted using lattice QCD simulations, where 3NF have been extracted from the Nambu-Bethe-Salpeter (NBS) wave function for a specific alignment of 3 nucleons [11, 12, 13]. The method used there had been previously employed to extract nucleon-nucleon potentials (i.e. 2NF)[14, 15, 16, 17] as follows. The NBS wave function for 2 nucleons is defined by
where is a QCD eigenstate with energy with being the nucleon mass, represents the non-relativistic kinetic energy, and is a nucleon interpolating operator made of 3 quarks such as . The non-local but energy independent potential (more precisely the half off-shell -matrix) is extracted from this NBS wave function as
where . The non-local potential can be expanded in terms of the velocity (derivative) with local function as
at the lowest few orders, where , represents the Pauli-matrices acting on the spin index of the -th nucleon, is the total spin, is the angular momentum, and
is the tensor operator. This method has been shown to work well. The central potentials at the leading order in the expansion have qualitatively reproduced common features of phenomenological 2N potentials: the force is attractive at medium to long distance while it has a characteristic repulsive core at short distance. See also refs. [18, 19] for a summary of results and recent developments.
The present authors have investigated short distance behaviors of the 2NF defined in the framework mentioned above, using the operator product expansion(OPE) and perturbation theory thanks to asymptotic freedom of QCD [20, 21, 22] . (See also a similar attempt for the solvable models in 2 dimensions .) The behavior of the NBS wave function at short distance () is encoded in the operator product expansion (OPE) of the two nucleon operators:
where is a set of local color singlet 6-quark operators with two-nucleon quantum numbers. Asymptotically the -dependence and energy dependence of the wave function is factorized into
Standard renormalization group (RG) analysis gives  the leading short distance behavior of the OPE coefficient function as
where is related to the 1-loop coefficient of the anomalous dimension of the operator , is the tree-level contribution of and finally is some typical non-perturbative QCD scale. Assuming its matrix element does not vanish, the operator with largest RG power dominates the wave function (7) at short distances. We denote the largest power by and the second largest one by .
If is non-zero, this leads to the leading asymptotics of the s-wave potential of the form
which is attractive for and repulsive for .
If , the situation is more complicated. The relative sign of the ratio between the leading and the subleading contributions becomes important and we find:
If is positive, the potential is repulsive, while it is attractive for negative . A system of two nucleons corresponds to this degenerate case. Unfortunately in this case depends on the energy . In  it is argued that in the relevant energy range the relative coefficient is positive, so that the short distance limit of the nucleon potential is repulsive.
In this paper, we extend the above OPE analysis to the 3NF. The corresponding equal time NBS wave function for 3 nucleons is given by
where and denote the energy and the 3N state. We introduce Jacobi coordinates , , . From this wave function, the three nucleon potential is defined by
where with denotes 2NF between -pair, the 3NF, the reduced masses.
In sect. 2, we start with renormalization group considerations and OPE, which are relevant for 3NF. The anomalous dimensions of 9–quark operators are computed in sect. 3. Finally we discuss the short distance behavior of 3NF in sect. 4. For the convenience of the reader we give a brief summary of our results here. The OPE analysis shows that the 3N central potential at short distance behaves as
as , where is given by
where is the number of dynamical quarks. Unlike the 2NF where the situation was not completely determined by PT alone, it is shown that the 3N potential always has a repulsive core. Furthermore it is universal in the sense that it does not depend on details of the 3N state used to define the NBS wave function such as the energy of the state.
2 Renormalization group analysis and operator product expansion for 3NF
2.1 Renormalization group equation for composite operators
In QCD, using dimensional regularization in dimensions, bare local composite operators are renormalized according to
Summation of repeated indices is assumed throughout this paper unless indicated otherwise. The meaning of the above formula is that we obtain finite results if we insert the right hand side into any correlation function of the fundamental gluon and quark fields, provided we also renormalize the bare QCD coupling and the quark and gluon fields. For example, in the case of an –quark correlation function with operator insertion, which we denote by (suppressing the dependence on the quark momenta and other quantum numbers) we have
We recall from renormalization theory that for the analogous –quark correlation function (without any operator insertion) we have
The coupling renormalization is given by
The renormalization constant in the minimal subtraction (MS) scheme we are using has pure pole terms only:
Similarly for the fermion field renormalization constant, we have
where with and is the covariant gauge parameter. The gluon field renormalization constant is also similar, but we do not need it here. Finally the matrix of operator renormalization constants is of the form
The renormalization group (RG) expresses the simple fact that bare quantities are independent of the renormalization scale . Introducing the RG differential operator
the RG equation for –quark correlation functions can be written as
Here the RG beta function is
where is the beta function in dimensions and the RG gamma function (for quark fields) is
It is useful to introduce the RG invariant lambda-parameter by taking the ansatz
and requiring . The solution is the lambda-parameter in the MS scheme () if the arbitrary integration constant is fixed by requiring that for small coupling
Finally the RG equations for –quark correlation functions with operator insertion are of the form
2.2 OPE and RG equations
Let us recall the operator product expansion for :
We will need it in the special case where the operators on the left hand side are nucleon operators and the set of operators on the right hand side are local 9-quark operators of engineering dimension 27/2 and higher. All operators in (31) are renormalized ones, but from now on we suppress the labels . As we know, the nucleon operators are renormalized diagonally as
and we can define the corresponding RG gamma functions by
For the nucleon operator,
Next we write down the bare version of (31) (in terms of bare operators and bare coefficient functions):
and using the -independence of the bare coefficient functions we can derive the RG equations satisfied by the renormalized ones:
where the effective gamma function matrix is defined as
2.3 Perturbative solution of the RG equation and factorization of OPE
We want to solve the vector partial differential equation (37) and for this purpose it is useful to introduce , the solution of the matrix ordinary differential equation
and its matrix inverse . We will assume that the coefficient has the perturbative expansion
where and with , , and are solid angles of the vectors and , respectively. Here is the dimension of the coefficient function. Note that in the massless theory operators of different dimension do not mix. In the full theory quark mass terms are also present, but they correspond to higher powers in and , and therefore can be neglected.
We will also assume that the basis of operators has been chosen such that the 1-loop mixing matrix is diagonal:
In such a basis the solution of (39) in perturbation theory takes the form
where , with possible multiplicative factors, depending on the details of the spectrum of 1-loop eigenvalues .
Here the vector is RG-invariant. Introducing the running coupling as the solution of the equation
can be rewritten as
Since, because of asymptotic freedom (AF), for also as
Putting everything together, we find that the right hand side of the operator product expansion (31) can be rewritten:
There is a factorization of the operator product into perturbative and non-perturbative quantities: is perturbative and calculable (for ) thanks to AF, whereas the matrix elements of are non-perturbative (but -independent). Note that .
2.4 3NF from OPE
Using results in the previous subsection, the NBS wave function for 3N can be written at short distance as
Since a term produces angular momenta and , we can write
up to less singular terms at short distances. Then the sum of 2N and 3NF potentials is extracted as
Since terms with dominate in at short distance, contributions from terms to 2N+3NF potentials are suppressed by an factor, so that they do not contribute at short distance. Therefore we consider terms with () hereafter and do not write dependence in coefficients. We then have
In terms of and we write
so that the dependent part gives a contribution unless , where either or . We assume and in our calculation. Since an dependence appears only at 1-loop or higher orders in , we can neglect it unless an operator which appears at loop has large anomalous dimension such that is larger than other corresponding to operators appearing at tree level. As we will see later such operators are absent; it is then enough to consider the tree level contribution in , so that dependent terms in can be neglected. The largest eigenvalue among operators appearing at tree level is thus denoted by , which corresponds to discussed in the introduction for 2N forces.
We then obtain
The NBS wave function is dominated by the term with largest . If we assume is non-zero, we finally obtain
3 Anomalous dimensions for three nucleons at 1-loop
3.1 OPE for 3N operators at tree level
The general form of a gauge invariant local 3–quark operator is given by
where are spinor, are flavor, are color indices of the (renormalized) quark field . The color index runs from 1 to , the spinor index from 1 to 4, and the flavor index from 1 to . Note that is symmetric under any interchange of pairs of indices (e.g. ) because the quark fields anticommute. For simplicity we sometimes use the notation such as and as indicated in (59).
The usual nucleon operator which is employed in lattice simulations is constructed from the above operators as
where is the projection to the large spinor component, is the charge conjugation matrix, and is the Pauli matrix in the flavor space (for ) given by . Both and are anti-symmetric under the interchange of two indices, so that the nucleon operator has spin and isospin . Although an explicit form of the matrices is unnecessary in principle, we find it convenient to use a (chiral) convention given by
As discussed in the previous section, the OPE at the tree level (generically) dominates at short distance. The OPE of 3N operators given above at tree level becomes
where denote higher dimensional operators, which do not contribute at short distance. The leading operators couple only to the states with (the -state).
If we construct local 3N operators at from for nucleons (which only involve 2 different flavors), there is only one such operator, which has and , due to the Pauli statistics of nucleons. Explicitly it is given by
where , and , , , for the explicit form of the matrices. Above no summation is taken for and .
3.2 General formula for the divergent part at 1-loop
As shown in ref., the gauge invariant part of the divergence from diagrams involving exchange of a gluon between any pair of quark fields is given by
for either (right-handed) or (left-handed), and it vanishes for other combinations.
The operator in eq. (63) can be written as a linear combination of simple operators . According to this 1-loop formula for divergences, such a simple operator mixes only with operators which preserve the set of flavor and Dirac indices in the chiral basis as
Note however that such operators are not all linearly independent. In the case of a 2N operator, we have the following constraint
which comes from the general identity
Here means a simultaneous exchange between the -th indices of and the -th indices of . This identity can be generalized to
from which we have
where the -th index of , the -th index of and the -th index of are cyclically interchanged in . For example,