Relativistic constraints on 3N contact interactions

In this paper we analyze the relativistic corrections to the leading order three-nucleon (3N) contact interactions. These boost corrections are derived first from the nonrelativistic reduction of covariant Lagrangians and later from the Poincar\'e algebra constraints on nonrelativistic theories. We show that in order to describe the 3N potential in reference frames other than the center-of-mass frame, the inclusion of five additional terms with fixed coefficients is required. These terms will be relevant in systems with mass number A>3. How they will affect EFT calculations of binding energies and scattering observables in these systems should be investigated.


Introduction
Nowadays, effective field theories (EFTs) are recognized as the standard framework for dealing with the nuclear interaction [1][2][3][4][5][6][7].The starting points are the identification of the most general effective Lagrangian preserving all the low energy symmetries of the fundamental theory and a power counting to organize the infinite tower of permitted interactions.This leads to the emergence of a predictive setting in which the interactions are expressed at each order of the low-energy expansion in terms of a finite number of low-energy constants (LECs), which can be treated as fitting parameters and extracted from phenomenology.Of particular interest among these fitting parameters are the LECs related to contact interactions between nucleons.They are strongly constrained by discrete symmetries and also by Poincaré symmetry, although the typical setting of nuclear physics is a nonrelativistic quantum-mechanical context.
Relativistic effects in nuclear interaction vertices can be determined by the nonrelativistic reduction of a relativistic quantum-field theoretical Lagrangian and evaluated order by order in the low-energy expansion, since they scale with the soft nucleon momenta [8,9].An alternative approach derives from the Poincaré algebra constraints in a purely quantum mechanical setting [10,11].Indeed, at sufficiently low energy scales, when the effects of creation and annihilation of particles can be ignored, the system can be considered as constituted by a fixed number of particles, and interactions can be described as direct (i.e., they explicitly depend on the physical variables associated to the constituents of the system) rather than mediated by fields.
The analysis of relativistic corrections on twonucleon contact forces up to order 1/m 2 , m being the nucleon mass, has already been discussed in Refs.[12,13] both via the nonrelativistic reduction of covariant Lagrangians and from the point of view of the constraints imposed by the Poincaré algebra.Furthermore, the analysis of these constraints up to 1/m 4 in Refs.[14,15] led to the to the identification of two free LECs that parameterize a nucleon-nucleon (NN) interaction dependent on the overall momentum of the pair.
In this work we extend the above results to the three-body contact forces.
The first contribution to the contact 3N force is represented by a single operator O 0 , accompanied by the LEC E 0 , whose matrix elements are constant in momentum space and take the form of the identity operator in spin-flavor space.The subleading terms involve two powers of momenta and were classified in Ref. [16] as consisting of 13 independent operators in the 3N center of mass frame.In a general frame the 3N contact potential reads where k i = p i − p ′ i and Q i = p i + p ′ i are related to the initial and final momenta of the i-th nucleon, respectively p i and p ′ i , and we indicate with O * i=1−5 the operators depending on the overall momentum The study of this three-body contact force proves particularly interesting in the context of many unsolved problems in nuclear physics [17,18].As shown in Ref. [14,19], some terms of the 3N contact potential are related via a unitary transformation to the P -dependent two-nucleon potential and seem to be crucial to solve the p − d A y puzzle [20].Similarly, the study of the threebody force and its relativistic corrections could have an impact on the study of systems with A > 3, where still there exist large and unexplained discrepancies between theory and experiment [21][22][23][24].
The paper is structured as follows.In Section 2 the relativistic corrections are calculated by nonrelativistic reduction of covariant Lagrangians.In Section 3 we show how the same corrective terms can be derived from Poincaré algebra constraints.Finally, in Section 4 we present the conclusions of this work.

Boost corrections from a covariant 3N contact Lagrangian
The objective of this section is to determine the P -dependent relativistic correction to the leading order 3N contact potential in any given frame by applying the non-relativistic reduction to the covariant Lagrangian.We begin by establishing a complete nonminimal set of relativistically invariant 3N contact operators Õi that contribute to the nonrelativistic expansion starting from order Q 0 , while satisfying the requirements of hermiticity and CPT invariance, following general principles outlined in Refs.[15,[25][26][27].Formally, the operators Õi are structured as the composition of fermion bilinears [13,28] Here, ψ represents the relativistic nucleon field, which is a doublet in isospin space.The symbol The symbols Γ A,B,C denote generic elements of the Clifford algebra, expanded in the basis 1, γ 5 , γ µ , γ µ γ 5 , Table 1 Transformation proprieties of the different elements of the Clifford algebra, metric tensor, Levi-Civita tensor and derivative operators under parity (P), charge conjugation (C) and hermitian conjugation (h.c.) σ µν , as well as the metric tensor or the Levi-Civita tensor ǫ µνρσ (with the convention ǫ 0123 = −1).
The Lorentz indices on the partial derivatives must be contracted among themselves and/or with those in the Γ A,B,C in order to preserve Lorentz invariance.
Regarding the isospin degrees of freedom, the allowed isospin-invariant flavor structures are In Table 2 are displayed the transformation properties under parity, charge conjugation, and Hermitian conjugation of the fermion bilinears with the aforementioned different elements of the Clifford and flavour algebra.
If charge conjugation and parity symmetries are satisfied, time reversal symmetry is automatically fulfilled, according by the CPT theorem.
We now outline power counting criteria needed to establish which operators contribute to the nonrelativistic expansion starting from order Q 0 [13,15].
Derivatives ∂ acting on the whole bilinears are of order Q, while derivatives ← → ∂ µ acting inside a bilinear are of order Q 0 due to the presence of the heavy fermion mass scale.Therefore, we can restrict ourselves to retain only operators containing the latter kind of derivatives.Nevertheless, a generic operator contributing at order Q 0 may in principle include arbitrary powers of space-time derivatives of the fields.However, it is possible to restrict ourselves to consider only a finite number of structure, as detailed in what follows.Whenever ← → ∂ µ is contracted with an element of the Clifford algebra inside of the same bilinear, the fields' equations of motion can be used to remove it [29,30].The same is true when As a result, by the equation of motions, no two Lorentz indices can be contracted among themselves inside the same bilinear, exept for the Levi-Civita tensors and for the suppressed ∂ 2 .
For what concerns pairwise contracted ( ← → ∂ A • ← → ∂ B ) between two different bilinears, we observe that these structures too generate redundant contributions.For instance, can be neglected in the non relativistic expansion, as it starts from Q 2 .(We see from Eq. (A1) that the contribution of Õ1 starts from order Q 0 ).
The Dirac matrix γ 5 can be thought of as of order O(Q) since it mixes the large and small components of the Dirac spinor.This also applies to the spatial components of γ µ and to the temporal component of γ µ γ 5 , as well as to σ 0i .For instance, operators associated to the structure γ γ σ, such as ( ψγ α ψ) 1 ( ψγ β ψ) 2 ( ψσ αβ ψ) 3 , do not contribute at order Q 0 due to this mixing.
The antisymmetry properties of ǫ µνρσ and σ µν restrict the maximum number of their possible contractions with a derivative ← → ∂ operating within a bilinear to one.Any additional contraction with a derivative would lead to a contribution at a higher order than O(Q 0 ).
On the basis of these properties we derive a complete (but non minimal) set of 92 different relativistic operators, displayed in Table 2, that contribute to the non-relativistic expansion starting from order Q 0 .Obviously, none of the 92 operators presents derivatives ∂ acting on the entire bilinear, as they are of order Q.
The relativistic nucleon field ψ can be expanded in a non-relativistic manner up to second order in Q by utilizing the non-relativistic field, We obtain 92 resulting non-relativistic operators Õi as combinations of the 146 subleading 3N contact operators o 1,...,146 compatible with the which starts at O(Q 0 ) and contain P-dependent drift corrections.Formally the non-relativistic expansions of the relativistic operators Õi , up to O(Q 2 ), take the form where α i , β k i are coefficients which can be read from the explicit expressions.
Thus, starting from the relativistic 3N contact Lagrangian written in terms of 92 (redundant) LECs, Ẽi , the potential can be written as and comparing with Eq. ( 1), while considering Eq. ( 5), we obtain the following identification and specific constraints on the LECs E * i=1,...,5 , Finally, we can identify as the P -dependent component of the 3N contact potential in an arbitrary frame of Eq. ( 1), i.e. the boost correction up to order Q 2 to the 3N leading order contact potential in the rest frame of the system, 3 The 3N contact interaction boost corrections from Poincaré algebra As an alternative to the procedure discussed in the previous Section, we review the calculation of the boost correction to the leading order 3N contact interaction up to Q 2 order from the Poincaré algebra constraints.Relativistic many-body descriptions of systems consisting of a fixed number of interacting particles can be achieved using relativistic Hamiltonians.These Hamiltonians are defined as the sum of relativistic one-body kinetic energies, two-and many-body interactions and, importantly, their corresponding boost corrections.For a generic system of interacting particles with momenta p ν and masses m ν , such Hamiltonians may be expressed as follows [31], where P νµ = p ν + p µ is the total momentum of particles ν and µ, and P νµρ = p ν + p µ + p ρ is the total momentum of particles ν, µ and ρ.The term v νµ corresponds to the two-body potential in the rest frame of the sub-system constituted by particles of indices ν,µ.Analogously, V νµρ is the three-body potential in the rest frame of particles ν,µ,ρ.Terms δv νµ (P νµ ) and δV νµρ (P νµρ ) are referred to as "boost interactions".Clearly, these quantities vanish in the rest frame of their corresponding sub-system (i.e., δv νµ (0) = 0 if P νµ = 0, and δV νµρ (0) = 0 if P νµρ = 0).However, it is essential to take them into account to attain accurate descriptions in reference frames where P = 0.Both v µν and V µνρ are determined by the fields and by the internal structure of the interacting particles.Realistic models of v µν and v µνρ are obtained by choosing their theoretical model and by fitting them to experimental data; as a consequence, they may contain some form of model-dependent relativistic effects.
Starting from v µν (respectively, v µνρ ) it is possible to obtain δv µν (P µν ) ( respectively, δV µνρ (P µνρ )) without any further model dependence, through relations fixed by the general principle of relativistic covariance.
For our present purposes, we are considering a system of three particles, each with spin s and mass m.The dynamical variables for the νth particle (ν = 1, 2, 3) are spin σ ν , isospin τ ν , momentum p ν , and position r ν .The momenta and positions are canonically conjugate operators, as are the center-of mass variables R = r1+r2+r3 3 and P = p 1 + p 2 + p 3 .The spin and isospin operators satisfy the well-known angular momentum commutation relations: In the instant form of relativistic dynamics [32], interactions affect the Hamiltonian H and, necessarily, the boost generators K.We write where V , W are the interaction terms, and the subscripts 0 indicate the corresponding operators in the absence of interactions, where ν is the single-particle energy of the νth particle.
The generators in Eq. ( 15) must satisfy the commutation relations of the Poincaré group: [H, where i, j, k ∈ {1, 2, 3}, ǫ ijk is the Levi-Civita tensor, δ ij is the Kronecker delta tensor, and summation convention on repeated indices is in force.Units are such that = 1 and c is the speed of light in vacuum.
As it is well known, the relations of Eq. ( 17) are satisfied by the free generators in Eq. ( 16).The problem of describing an interacting system of relativistic particles consists of determining functions V and W such that the commutation relations (17) are still satisfied [10].
It is assumed that we are considering an interacting system for which H and K can be expanded in powers of 1 m 2 as where M = i ν m ν and the superscripts refer to the order in powers of 1 m 2 .Under this assumption, interactions can be introduced at each order as additional terms V (n) , W (n) to the noninteracting components at the order n, respectively with both V (n) and W (n) depending on the dynamical variables of the system, and not explicitly on time [10].
The commutation relations in Eq. ( 17) can be consequently expanded in powers of 1 m 2 and, in principle, they could be solved at each order by mean of a "direct integration".It is conjectured that the obtained solutions represent the most general case for systems of the depicted kind for which the expansion exists.
We assume, following [10], that the representation for a relativistic system is chosen so that in the nonrelativistic limit it holds W (0) = 0 and K = M R − tP , and J and P are those given in Eqs.(15) and (16).
In particular, we focus on the P -dependent component of such correction, i.e. the boost correction δV (P ), which provides the relationship between descriptions of the system in different reference frames.
An expression for δV up to order 1 m 2 beyond the non-relativistic limit has been derived by Friar [33].
It is convenient to write it in terms of normalized canonical Jacobi coordinates of momentum, π a,b and position ρ a,b , which are related to the physical variables p 1,2,3 , r 1,2,3 through the change of coordinates Thus, for a system of three particles, δV reads where We identify V (0) in Eq. ( 22) with the leading order 3N contact interaction parameterized by the LEC E 0 in Eq. (1).
The vector w in Eq. ( 23) is a translationally invariant function of P and ρ a,b , π a,b .It satisfies ∇ P × w = 0, making the integral in (23) independent of the path. .As a minimal choice, we set w = 0, so that χ v (P ) = 0, as in Refs.[11,12,34,35]; this corresponds to assuming the existence of an appropriate unitary transformation absorbing w [14].
We evaluate δV between 3N states Ψ and Ψ ′ as Ψ ′ | δV (P ) |Ψ .The details of the calculation can be found in Appendix (C).
The result, expressed in terms of variables When written in the basis of the 146 3N subleading contact operators o i , i = 1, . . ., 146, Eq. ( 25) reads However, when we use the basis of the 18 independent 3N subleading contact operators O 1−13 , O * 1−5 , the same result can be written as Since it is completely characterized by the low energy constant E 0 , we recognize δV (P ) as the boost correction of our interest, in perfect agreement with Eq. ( 12).

Conclusions
In this paper we derived the relativistic corrections of the leading order 3N contact potential in an arbitrary frame with two different approaches.The first approach, developed within the framework of field theory, involves identifying the operators that contribute to the covariant Lagrangian at the zeroth-order in the low-energy expansion and afterwards performing relativistic reduction; the result is represented by Eqs. ( 5) and (12) or, equivalently, by constraints on the LECs E * i parametrizing P-dependent interactions, Eqs.(9,10,11).The second approach, formulated within the context of relativistic quantum mechanics, is directly based on fundamental principles of covariance and the constraints that arise from them through Poincaré algebra; the result is Eq.(27).In both cases it is evident that the P -dependent boost correction is entirely determined by the interaction at the leading order.The expression for the boost correction is given by indicating overall agreement within the two approaches.
The consistency between the two approaches supports the validity of the minimal ansatz w = 0 made in Eq. ( 22).Hence, it is possible to assume the existence of an appropriate unitary transformation absorbing w [14].A similar assumption has been made in the 2N case in Refs.[31] and [13]; nevertheless, the validity of its extension for systems composed of a N > 2 nucleons is not straightforward [11].
Furthermore, the results obtained in Eqs (B94)-(B185) provide a non-trivial check of the Fierz's relations.In fact, if there had been an error in the Fierz's identities, as in Ref. [16] where only 14 nonrelativistic operators were found, the relativistic corrections would have had different expressions for each operator thus precluding the unambiguous determination of the boost correction.
It is conjectured that the obtained solutions are the most general for systems of the kind depicted for which the expansion exists.Any relativistic description of a system of interacting particles of finite mass and spin, whether exact or approximate, should fall in this framework.

Appendix A Non-relativistic expansions
Here we give the non-relativistic expansions of the operators Õi defined in Table 2 in terms of the 146 subleading 3N contact operators o i and of the six leading order operators O (0) i listed in Ref. [16].

Appendix B Reduction to the minimal basis
Here we use the Fierz's relations among the subleading 3N contact operators derived in Ref. [16] to rewrite the non-relativistic expansions of the 92 relativistic operators in terms of the minimal basis of operators appearing in Eq. ( 1).Notice that, as a further consequence of Fierz' identities, the six leading order operators O (0) i=1,...,6 are all proportional to the operator O 0 , which is the identity operator in spin and isospin space, To proceed with the calculation of Ψ ′ | δV (P ) |Ψ , we write χ 0 ≡ χ (1) + χ (2) + χ (3) , with where s a ≡ s 1 − s 2 and s b ≡ s 3 − s1+s2 2 .We find the following intermediate results, leading to the final result in Jacobi coordinates, In matrix elements such as those appearing in the above Eqs.(C191), (C192), (C193), (C194), exchange terms will always be present, analogously as in Eqs.(D207), (D208), (D209), due to the symmetry of the operators under the exchange of particles (see Appendix D).Such exchange terms will be implicitly understood during calculations.
In this appendix, we undertake the calculations leading to Eq. (C192); Eq. (C193) and Eq.(C194) are obtained with a similar procedure.

Appendix D Orthogonality and closure relations
We define physical states in terms of the variables p ν , r ν , σ ν , τ ν as follows.We indicate a generic permutation of indices 1, 2, 3 with α i = (α 1 i , α 2 i , α 3 i ) and its sign with ǫ α i : In view of the above, Jacobi variables of momentum and position satisfy the following orbital closure and orthogonality relations, where permutation terms are understood: dP dq a dq b |P, q a , q b P, q a , q b | = 1 (P,qa,q b ) ; (D210) P, q a , q b |P ′ , q ′ a , q ′ b = δ(P − P ′ )δ(q a − q ′ a )δ(q b − q ′ b ); (D211)

1 )
= ρ a • P π a • P + π a • P ρ a • P + ρ b • P π b • P + π b • P ρ b • P .Developing the calculation only for the component involving Jacobi variables ρ a , π a , we get

Table 2
List of three-nucleon relativistic contact operators contributing to the leading order.