Relativity constraints on the two-nucleon contact interaction

0.19 MB·English

by  L. Girlanda

Relativity constraints on the two-nucleon contact interaction
L. Girlandaa,b, S. Pastorec, R. Schiavillac,d, and M. Vivianib
aDepartment of Physics, University of Pisa, 56127 Pisa, Italy
bINFN-Pisa, 56127 Pisa, Italy
cDepartment of Physics, Old Dominion University, Norfolk, VA 23529, USA
dJefferson Lab, Newport News, VA 23606, USA
0 (Dated: January 20, 2010)
1
Abstract
0
2 We construct the most general, relativistically invariant, contact Lagrangian at order Q2 in

n the power counting, Q denoting the low momentum scale. A complete, but non-minimal, set of
a
J (contact) interaction terms is identified, which upon non-relativistic reduction generate 2 leading
0 independent operator combinations of order Q0 and 7 sub-leading ones of order Q2—a result
2
derived previously in the heavy-baryon formulation of effective field theories (EFT’s). We show

] that Poincar´e covariance of the theory requires that additional terms with fixed coefficients be
h
included, in order to describe the two-nucleon potential in reference frames other than the center-
t
-
l of-mass frame. These terms will contribute in systems with mass number A > 2, and their impact
c
u on EFT calculations of binding energies and scattering observables in these systems should be
n studied.
[

1 PACS numbers: 12.39.Fe, 21.30.-x,11.30.Cp, 13.75.Cs
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I. INTRODUCTION AND CONCLUSIONS
Chiral effective field theory (χEFT), pioneered by Weinberg in a series of seminal pa-
pers almost two decades ago [1], has led to a novel understanding of strong interactions in
nuclei by providing a direct link between these interactions and the symmetries of quan-
tum chromodynamics, including chiral symmetry with its explicit and dynamical breaking
mechanisms (see review papers in Refs. [2]). In its original form, χEFT is formulated in
terms of pions and (non-relativistic) nucleons, whose interactions, strongly constrained by
chiral symmetry, are organized as an expansion in powers of small momenta Q. All heavier
degrees of freedom are “integrated out”, and their effects are implicitly subsumed in the
coupling constants accompanying local vertices. At sufficiently low energy, even the pions
can be integrated out, and the nucleons only interact through contact vertices. In either
case, two-nucleon (NN) contact interactions are an important aspect of EFT descriptions.
In the present paper we examine the constraints that relativistic covariance imposes on the
resulting NN potential up to order Q2 (or next-to-next leading order, N2LO).
At LO (Q0) in the low energy expansion there are only 2 independent contact interac-
tions [1]
1 1
(0)
L = − C O − C O , (1.1)
I 2 S S 2 T T
where C and C denote low-energy constants (LEC’s), and the operators O and O are
S T S T
defined in terms of the non-relativistic nucleon field N(x),
dp
N(x) = b (p)χ e−ip·x , (1.2)
(2π)3 s s
Z
and its adjoint in Table I. Here b (p) and b†(p) are annihilation and creation oper-
s s
ators for a nucleon in spin state s, satisfying standard anticommutation relations, i.e.
bs(p), b†s′(p′) = (2π)3δ(p − p′)δss′. A sum over the repeated index s = ±1/2 is im-
+
hplied, and it isiunderstood that field operator products are normal-ordered in O and O
S T
(as well as in the O ’s defined below). We have suppressed isospin indices, since they will
i
not enter in the discussion to follow (see Sec. IIA). The corresponding NN potential reads
vCT0 = C +C σ ·σ . (1.3)
S T 1 2
At the next non-vanishing order, N2LO, the contact Lagrangian involving two derivatives
of the nucleon fields has been written in Ref. [3] as consisting of 14 operators
14
L(2) = − C′O , (1.4)
I i i
i=1
X
where the O ’s are listed in Table I and the C′ are LEC’s. In fact, we showed in Ref. [4]
i i
that, after partial integrations, only 12 out of the above 14 operators are independent, since
O +2O = O +2O , O +O = O . (1.5)
7 10 8 11 4 5 6
In a general frame in which the NN pair has total momentum P and initial and final
relative momenta, respectively, p and p′, the potential derived from the Lagrangian L(2) is
I
2
O (N†N)(N†N)
S
O (N†σN)·(N†σN)
T
−→
O (N†∇N)2+h.c.
1
−→ ←−
O (N†∇N)·(N†∇N)
2
−→
O (N†N)(N†∇2N)+h.c.
3
−→ ←−
O i(N†∇N)·(N†∇×σN)+h.c.
4
←− −→
O i(N†N)(N†∇·σ×∇N)
5
←− −→
O i(N†σN)·(N†∇×∇N)
6
−→ −→
O (N†σ·∇N)(N†σ·∇N)+h.c.
7
−→ −→
O (N†σj∇kN)(N†σk∇jN)+h.c.
8
−→ −→
O (N†σj∇kN)(N†σj∇kN)+h.c.
9
−→ ←−
O (N†σ·∇N)(N†∇·σN)
10
−→ ←−
O (N†σj∇kN)(N†∇jσkN)
11
−→ ←−
O (N†σj∇kN)(N†∇kσjN)
12
←− −→
O (N†∇·σ∇jN)(N†σjN)+h.c.
13
←− −→
O 2(N†∇σj ·∇N)(N†σjN)
14
TABLE I:OperatorsenteringtheLO(Q0)andN2LO(Q2)contact interactions [3]. Theleft(right)
arrow on ∇ indicates that the gradient acts on the left (right) field. Normal-ordering of the field
operator products is understood.
conveniently separated into a term, vCT2, independent of P [3, 5] and one, vCT2, dependent
P
on it [4]:
σ +σ
vCT2(k,K) = C k2 +C K2 +(C k2 +C K2)σ ·σ +iC 1 2 ·K×k
1 2 3 4 1 2 5
2
+ C σ ·k σ ·k+C σ ·K σ ·K , (1.6)
6 1 2 7 1 2
σ −σ
vCT2(k,K) = iC∗ 1 2 ·P×k+C∗(σ ·P σ ·K−σ ·K σ ·P)
P 1 2 2 1 2 1 2
+ (C∗ +C∗σ ·σ )P2+C∗σ ·P σ ·P , (1.7)
3 4 1 2 5 1 2
where the momenta k and K are defined as k = p′ −p and K = (p′ +p)/2, and the C ’s
i
(i = 1,...,7) and C∗ (i = 1,...,5) are in a one-to-one correspondence with the LEC’s C′’s
i i
multiplying the 12 independent operators (see Refs. [3–5]).
We argued in Ref. [4] that the P-dependent terms represent boost corrections to the LO
potential vCT0, and that the C∗, rather than being independent LEC’s, are in fact related
i
to C and C as
S T
C −C C C C
C∗ = S T , C∗ = T , C∗ = − S , C∗ = − T , C∗ = 0 , (1.8)
1 4m2 2 2m2 3 4m2 4 4m2 5
where m is the nucleon mass. This result is derived in relativistic quantum mechanics—its
instant-form formulation [6]—by requiring that the commutation relations of the Poincar´e
group generators are satisfied, which, to order P2/m2, leads to the elegant relation [7, 8]
P2 i i
vCT2 = − vCT0 + P·r P·p, vCT0 + (σ −σ )×P·p, vCT0 , (1.9)
P 8m2 8m2 8m2 1 2
(cid:2) (cid:3) (cid:2) (cid:3)
3
where r and p are, respectively, the relative position and momentum operators. The po-
tential vCT2(k,K) then follows by evaluating the commutators in momentum space, and by
P
retaining only contributions of order Q2 (we assume here P ∼ k ∼ K ∼ Q). That there
are dynamical corrections to the NN interaction, when it is boosted to an arbitrary frame,
is not surprising, as in instant-form relativistic quantum mechanics interactions enter both
the Hamiltonian and boost generators.
Inthepresentpaper, wejustifytheclaimmadeaboveinaEFTsetting. Weproceedintwo
steps. First, we construct, up to order Q2 included, the most general hermitian Lagrangian
density allowedby invariance under transformationsoftheLorentzgroupandby thediscrete
symmetries of the strong interaction. After performing its non-relativistic reduction, we find
that there are 2 independent operator combinations of order Q0, accompanied by specific Q2
corrections, and 7 independent operator combinations of order Q2—a result also obtained [9]
(2)
in the heavy baryon formulation [10] of L by requiring that it be re-parametrization
I
invariant [11].
Second, we show that this same picture emerges within the non-relativistic theory in the
context of a systematic power counting, by enforcing that the commutation relations among
the Poincar´e group generators are satisfied order by order in the low energy expansion (for
a similar approach, in a different context, see Ref. [12]). The above correspondence between
the C∗’s and C , C is recovered, showing that the commutator relation in Eq. (1.9) remains
i S T
valid in a EFT framework. Thus, in order to determine the boost corrections of order Q2
to the complete LO chiral potential, which also includes the one-pion-exchange term, one
can either use Eq. (1.9) or compute the potential in an arbitrary frame starting from the
Lagrangian of the covariant theory.
These boost corrections should be taken into account in χEFT (and EFT) calculations
of nuclei with mass number A > 2. So far, they have been evaluated, for the case of
realistic potentials, in the A=3 and 4 binding energies, where they have been found to give,
respectively, about 400 keV and 1.9 MeV repulsive contributions [8], as well as in three-
nucleon scattering observables [13], where, in particular, they have led to an increase of the
discrepancy between theory and experiment in the nd vector analyzing power.
II. NON-RELATIVISTIC REDUCTION
To begin with, we observe that, while the relativistic theory is written in terms of fermion
fields ψ = ψ(+)+ψ(−) containing both positive- and negative-energy components, the latter
play no role in the NN contact potential at order Q2 of interest here. This is because
antinucleon degrees of freedom only enter via loop corrections, and each loop is suppressed
by a factor Q3 in time ordered perturbation theory (examples are shown in Fig. 1). Hence,
the two-derivative contact Lagrangian (of order Q2) can be derived, without any loss of
generality, startingfromtherelativistictheoryandignoringthenegativeenergycomponents.
A. Generalities and strategy
ThebuildingblocksoftherelativisticcontactLagrangianareproductsoffermionbilinears
with space-time structures
1 ←→ ←→ ←→ ←→
(ψi∂ αi∂ β···Γ ψ)∂λ∂µ···(ψi∂ σi∂ τ ···Γ ψ) , (2.1)
A B
(2m)Nd
4
FIG. 1: Time ordered diagrams contributing to the NN scattering amplitude and involving nucle-
ons (solid lines) and antinucleons (dashed lines) interacting through the contact vertices at order
Q0 and Q2 (solid circle). Note that at order Q2 the diagrams with antinucleons do not contribute
(see text for explanation).
←→ −→ ←−
where ∂ α = ∂ α− ∂ α and theΓ’s denote generic elements of the Cliffordalgebra, expanded
in the usual basis 1, γ , γµ, γµγ , σµν, or the Levi-Civita tensor ǫµνρσ (with the convention
5 5
←→
ǫ0123 = −1). Inthe above equation, N stands forthe number of four-gradients(both ∂ and
d
∂λ) entering the formula, and the factor 1/(2m)Nd has been introduced so that all contact
terms will have the same dimension. The Lorentz indices α,...,τ on the partial derivatives
are contracted among themselves and/or with those in the Γ (for ease of presentation,
A,B
these indices, unless necessary, will be suppressed hereafter). In order to have flavor singlets,
the isospin structure of the two bilinears must be either 1⊗1 or τa⊗τa. However, the latter
needs not be considered, as it can be eliminated by Fierz rearrangement.
A few remarks are now in order. First, the Lagrangian density should be hermitian and
invariant under charge conjugation(C)andparity(P). Welist thetransformationproperties
of the fermion bilinears in Table II. While the hermiticity condition does not impose any
←→
1 γ γ γ γ σ ǫ ∂ ∂
5 µ µ 5 µν µνρσ µ µ
P + – + – + – + +
C + + – + – + – +
h.c. + – + + + + – +
TABLE II: Transformation properties of the fermion bilinears (with the different elements of the
Cliffordalgebra), Levi-Civitatensor, andderivative operatorsunderparity(P), chargeconjugation
(C),andhermitianconjugation(h.c.). Thesymbol∂ inthelastcolumnstandsforthefour-gradient
µ
acting on the whole fermion bilinear.
constraint, since one can always multiply the individual bilinears by appropriate factors of
i, the C and P symmetries must be enforced.
Second, we observe that derivatives ∂ acting on the whole bilinear are of order Q, while
←→
derivatives ∂ acting inside a bilinear are of order Q0 due to the presence of the nucleon
mass. Therefore, at each order in the power counting, only a finite number of ∂ appears,
←→
while it is possible to have, in principle, any number of ∂ . The situation is not so hopeless,
←→ ←→
however. For instance, the contracted product ∂ ∂ µ inside a bilinear yields a squared
µ
massterm(without derivatives) plusa∂ ∂µ actingonthewholebilinear, whichissuppressed
µ
←→ ←→
by Q2. Similarly, a term like ∂/ ≡ ∂ γµ, resulting from the contraction, in a bilinear, of
µ
←→
∂ with one of the elements of the Clifford algebra, can be replaced by a term without
µ
derivatives by making use of the equations of motion, i.e. i∂/ψ = mψ and its adjoint; for
5
example,
←→ −→ ←−
ψi∂ σµν ψ = ψ γν ∂/ + ∂/γν ψ −∂ν ψψ = −∂ν ψψ . (2.2)
µ
(cid:16) (cid:17)
Hence, in general, no two Lorentz indices in a fermion(cid:0)bilin(cid:1)ear can b(cid:0)e co(cid:1)ntracted with one
another, except for the Levi-Civita tensor and for the (suppressed) ∂2 acting on the whole
bilinear.
Some of the most problematic terms are of the type
1 ←→ ←→ ←→ ←→ ←→ ←→
O(n) = (ψi∂ µ1 i∂ µ2 ···i∂ µn Γα ψ)(ψi∂ i∂ ···i∂ Γ ψ) , (2.3)
ΓAΓB (2m)2n A µ1 µ2 µn Bα
e
since, as stated above, n can be any integer. In fact, a little thought shows that terms with
n > 1 donot introduceanynew operatorstructure uptoO(Q2) included. This ismost easily
seen by considering the matrix elements of such terms between initial and final two-nucleon
states with momenta, respectively, p ,p and p ,p . These matrix elements consist of the
1 2 3 4
product of two factors: one given (in a schematic notation) by (u Γα u ) (u Γ u )—the
3 A 1 4 Bα 2
u denote Dirac spinors—and another involving the particles’ four-momenta,
i
[(p +p )·(p +p )]n
1 3 2 4
, (2.4)
(2m)2n
which to O(Q2) can be approximated as
n
1+ p2 +p2 +p2 +p2 −(p +p )·(p +p ) . (2.5)
4m2 1 2 3 4 1 3 2 4
(cid:20) (cid:21)
Therefore, as discussed in more detail in the next section, one only needs, in practice, to
account for terms of type (2.3) with n = 0,1.
B. Lagrangian to order Q2
Following the criteria laid out in the previous section, a complete but non-minimal set
consisting of 36 P- and C-conserving operators, denoted as O , is obtained. They are listed
i
in Table III. Note that some operator structures are missing, since they do not contribute
at order Q2. For instance, operators having the 1⊗γ structeure are at least of order Q4: P
5
symmetry requires the presence of an ǫµναβ whose indices (three of which space-like) must
be contracted with partial derivatives, and an additional factor Q comes from the presence
of γ , which mixes large and small components of the Dirac spinors.
5
The non-relativistic reduction of the O up to terms of order Q2 included is tedious but
i
straightforward. Tothisend, therelativisticfield(specifically, itspositive-energycomponent,
where the (+) superscript has been dropeped for simplicity)
dp m
ψ(x) = b (p)u(s)(p)e−ip·x, (2.6)
(2π)3E s
p
Z
e
with normalizations
E
bs(p), bs†′(p′) + = mp (2π)3δ(p−p′)δss′ , u(s)(p)u(s′)(p) = δss′ , (2.7)
h i
e e
6
1⊗1 O (ψψ)(ψψ)
1
←→ ←→
O 1 (ψi ∂ µψ)(ψi ∂ ψ)
2 4m2 µ
Oe 1 (ψψ)∂2(ψψ)
1⊗γ Oe3 14(mψ2i←→∂ µψ)(ψγ ψ)
e4 2m←→ ←→ µ ←→
O 1 (ψi ∂ µi ∂ νψ)(ψγ i ∂ ψ)
e5 8m3 ←→ µ ν
O 1 (ψi ∂ ψ)∂2(ψγµψ)
e6 8m3 ←→ µ ←→
1⊗γγ O 1 ǫ (ψi ∂ µψ)∂ν(ψi ∂ αγβγ ψ)
5 7 8m3 µναβ 5
e
γ ⊗γ O (ψiγ ψ)(ψiγ ψ)
5 5 8 5 5
e ←→
γ ⊗σ O 1 ǫ (ψiγ ψ)∂µ(ψi ∂ νσαβψ)
5 e9 4m2 µναβ 5 ←→
O 1 ǫ (ψiγ i ∂ µψ)∂ν(ψσαβψ)
10 4m2 µναβ 5
e
γ⊗γ O (ψγµψ)(ψγ ψ)
11 µ
Oe 1 (ψγµi←→∂ νψ)(ψγ i←→∂ ψ)
12 4m2 µ ν
Oe 1 (ψγµψ)∂2(ψγ ψ)
13 4m2 ←→ µ←→
Oe 1 (ψγµi ∂ νψ)(ψγ i ∂ ψ)
Oe14 1 (4ψmγ2µi←→∂ νi←→∂ αψ)(ψγν i←→∂µi←→∂ ψ)
e15 16m4 ←→ν µ α
γ ⊗γγ O 1 ǫ (ψγµψ)∂ν(ψi ∂ αγβγ ψ)
5 e16 4m2 µναβ ←→ 5
O 1 ǫ (ψγµi ∂ νψ)∂α(ψγβγ ψ)
Oe17 1 ǫ4m2 µ(ψναγβγi←→∂ µψ)∂ν(ψi←→∂ i←→∂5αγβγ ψ)
18 16m4 µναβ γ 5
e
γγ ⊗γγ O (ψγµγ ψ)(ψγ γ ψ)
5 5 19 5 µ 5
Oe 1 (ψγµγ i←→∂ νψ)(ψγ γ i←→∂ ψ)
20 4m2 5 µ 5 ν
Oe 1 (ψγµγ ψ)∂2(ψγ γ ψ)
21 4m2 ←→5 µ 5←→
Oe 1 (ψγµγ i ∂ νψ)(ψγ γ i ∂ ψ)
e22 4m2 5 ←ν→5 µ
γγ ⊗σ O 1 ǫ (ψγµγ ψ)(ψi ∂ νσαβψ)
5 e23 4m µναβ ←→5 ←→ ←→
O 1 ǫ (ψγµγ i ∂ γψ)(ψi ∂ νi ∂ σαβψ)
Oe24 16m3 µ1ναǫβ (ψ5γµγ i←→∂ νψ)(ψσαβψγ)
e25 4m µναβ ←→5 ←→ ←→
O 1 ǫ (ψγµγ i ∂ νi ∂ γψ)(ψσαβi ∂ ψ)
Oe26 16m3 1µναǫβ (ψγ5µγ ψ)∂2(ψi←→∂ νσαβψ)γ
e27 16m3 µναβ 5 ←→
O 1 ǫ (ψγµγ i ∂ νψ)∂2(ψσαβψ)
Oe28 1 16ǫm3 µ(νψαβγγγ i←→∂5µψ)(ψi←→∂ i←→∂ νσαβψ)
Oe29 161m3ǫµναβ(ψγµγ5i←→∂ νi←→∂ ψ)(ψγi←→∂ ασβγψ)
30 16m3 µναβ 5 γ
σ⊗σ Oe 1(ψσµνψ)(ψσ ψ)
31 2 ←→ µν ←→
Oe 1 (ψσµνi ∂ αψ)(ψσ i ∂ ψ)
32 8m2 µν α
Oe 1 (ψσµνψ)∂2(ψσ ψ)
33 8m2 ←→ ←→µν
Oe 1 (ψσµαi ∂ νψ)(ψi ∂ σ ψ)
e34 8m2 ←→ ←→ α µν ←→ ←→
O 1 ǫ ǫ (ψσµνi ∂ γi ∂ ρψ)(ψσαβi ∂ δi ∂ σψ)
Oe35 32m41µνǫγδ αβǫρσ (ψσµνi←→∂ γi←→∂ ρψ)∂δ∂σ(ψσαβψ)
36 32m4 µνγδ αβρσ
e
TABLE III: A complete, buet non-minimal, set of relativistic contact interactions. Note that the
field operator products are understood to be normal-ordered.
7
is expanded in terms of the non-relativistic field N(x), defined in Eq. (1.2), as
1 i 0 1 ∇2
ψ(x) = − + N(x)+O(Q3) . (2.8)
0 2m σ ·∇ 8m2 0
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Note that the relativistic (b) and non-relativistic (b) versions of the annihilation operator are
related to each other by b (p) = m/E b (p). Partial integrations and use of the fields’
s p s
equations of motion to elimeinate time derivatives,
p
e
i ←→ i ←− −→
ψ ∂ 0ψ = ψ† 1− γ · ∇+∇ ψ
2m 2m
(cid:20) (cid:21)
(cid:16) (cid:17)
1 ←− −→ 2 ←− −→
= N†N − N† ∇+∇ +2iσ·∇×∇ N +O(Q3) , (2.9)
8m2
(cid:20) (cid:21)
(cid:16) (cid:17)
lead to the operators ONR of Table IV. They are given there as linear combinations of the
i
operator basis O , defined previously (see Table I).
i
Returning briefly toethe discussion of the terms of type O(n) in Eq. (2.3), it is useful to
ΓAΓB
separatesuchtermsintothreeclasses, dependingonwhetherthenon-relativisticexpansionof
thespinormatrixelement(u Γαu )(u Γ u )isi)1+O(Qe2)(classI),orii)±σ ·σ +O(Q2)
3 A 1 4 Bα 2 1 2
(class II), or iii) O(Q2) (class III). Making use of the relations (2.8) and (2.9), we find that
terms in class I reduce to
n
O(n) = O(n=0) + (O −2O −2O )+O(Q4) , (2.10)
ΓAΓB ΓAΓB 4m2 1 2 3
while those in classeII reduceeto
n
O(n) = O(n=0) ± (3O +2O +2O )+O(Q4) , (2.11)
ΓAΓB ΓAΓB 4m2 9 12 14
and lastly the terems in clases III are simply given, for any n, by O(n=0) up to corrections
ΓAΓB
O(Q4). A quick glance at Tables III and IV shows that the relations above are verified:
consider, for example, O and O in class I, O and O in class II,eand O in class III, and
1 2 20 21 8
their corresponding non-relativistic reductions.
Inspection of TableeIV showes that a set eof linearley independent opereator combinations
can be defined as
O +(O +O +O +O )/(4m2)
S 1 3 5 6
O −(O +O −O +O +2O +O )/(4m2)
T 5 6 7 8 12 14
O +2O
1 2
2O +O
2 3
O +2O (2.12)
9 12
O +O
9 14
O −O
5 6
O +2O
7 10
O +O +2O
7 8 13
consisting of 2 leading (of order Q0) and 7 sub-leading (Q2) ones, in agreement with the
results of an analysis based on the heavy-baryon formulation of EFT [9, 14], so that the
8
ONR O + 1 (O +2O +2O +2O )
1 S 4m2 1 2 3 5
ONR O + 1 (2O +2O )
2 S 4m2 1 5
OeNR 1 (O +2O )
3 4m2 1 2
OeNR O + 1 (O −2O +2O )
4 S 4m2 1 2 6
OeNR O + 1 (2O −4O −2O +2O )
5 S 4m2 1 2 3 6
OeNR 1 (O +2O )
6 4m2 1 2
OeNR 1 (−2O +2O )
7 4m2 5 6
OeNR 1 (O +2O )
8 4m2 7 10
OeNR 1 (2O +4O )
9 4m2 7 10
OeNR 1 (−2O −4O )
10 4m2 7 10
OeNR O + 1 (−4O −2O +4O +O −O +2O −2O )
11 S 4m2 2 5 6 7 9 10 12
OeNR O + 1 (O −6O −2O −2O +4O +O −O +2O −2O )
12 S 4m2 1 2 3 5 6 7 9 10 12
OeNR 1 (O +2O )
13 4m2 1 2
OeNR O + 1 (O −6O −2O −2O +4O )
14 S 4m2 1 2 3 5 6
OeNR O + 1 (2O −8O −4O −2O +4O )
15 S 4m2 1 2 3 5 6
OeNR 1 (−2O +2O +O −O +2O −2O )
16 4m2 5 6 7 9 10 12
OeNR 1 (−O +O −2O +2O )
17 4m2 7 9 10 12
OeNR 1 (−2O +2O )
18 4m2 5 6
OeNR −O − 1 (−2O +O −O −2O −2O +2O −2O )
19 T 4m2 6 7 9 10 12 13 14
OeNR −O − 1 (−2O +O +2O −2O +2O )
20 T 4m2 6 7 9 10 13
OeNR 1 (−O −2O )
21 4m2 9 12
OeNR 1 (−2O −2O −4O )
22 4m2 7 8 13
OeNR −O − 1 (−2O +2O −O −2O +2O −2O )
23 T 4m2 6 7 9 12 13 14
OeNR −O − 1 (−2O +2O +2O +2O )
24 T 4m2 6 7 9 13
OeNR −O − 1 (−2O −2O +O −2O −2O )
25 T 4m2 5 8 9 12 13
OeNR −O − 1 (−2O −2O +4O −2O +2O )
26 T 4m2 5 8 9 13 14
OeNR 1 (−O −2O )
27 4m2 9 12
OeNR 1 (−O −2O )
28 4m2 9 12
OeNR 1 (−2O −2O −4O )
29 4m2 7 8 13
OeNR 1 (−O +O −O −O +2O −2O +2O )
30 4m2 5 6 7 8 9 13 14
OeNR O + 1 (−O −2O −4O +2O +O −2O +2O −4O −2O )
31 T 4m2 1 2 5 6 7 8 10 12 13
OeNR O + 1 (−O −2O −4O +2O +O −2O +3O +2O −2O −2O +2O )
32 T 4m2 1 2 5 6 7 8 9 10 12 13 14
OeNR 1 (O +2O )
33 4m2 9 12
OeNR 1 −1O −O −2O +2O −O −O +2O −2O +2O
34 4m2 2 1 2 5 6 7 8 9 13 14
OeNR 1 (4O +4O +8O )
35 (cid:0) 4m2 7 8 13 (cid:1)
OeNR 1 (2O +4O )
36 4m2 7 10
e
TAeBLE IV:Thenon-relativistic expressions,uptoorderQ2 included,correspondingtothecontact
interactions of Table III.
9
effective Lagrangian can be written as
1 1 1 1
L = − C O + (O +O +O +O ) − C O − O +O −O +O
2 S S 4m2 1 3 5 6 2 T T 4m2 5 6 7 8
" # "
1 1 1
+2O +O − C (O +2O )+ C (2O +O )− C (O +2O )
12 14 1 1 2 2 2 3 3 9 12
2 8 2
!#
1 1 1 1
− C (O +O )+ C (O −O )− C (O +2O )− C (O +O +2O ).
4 9 14 5 6 5 6 7 10 7 7 8 13
8 4 2 16
(2.13)
Evaluation of the matrix elements of the operators O between initial and final two-nucleon
i
states with momenta P/2+p,P/2−p and P/2+p′,P/2−p′, i.e.
O (p′,p;P) = dxhP/2+p′,P/2−p′ | O | P/2+p,P/2−pi , (2.14)
i i
Z
shows that the 7 sub-leading combinations above give vanishing P-dependent contributions,
and in fact lead, in the center-of-mass frame, to the 7 k- and K-dependent terms occur-
ring in vCT2(k,K), Eq. (1.6). Similarly, the 2 leading combinations and associated 1/m2
corrections—first 2 lines of Eq. (2.12)—give rise, respectively, to the P-dependent terms
P2 i
− + (σ −σ )·P×k (2.15)
2m2 4m2 1 2
and
P2 i 1
− σ ·σ − (σ −σ )·P×k+ (σ ·P σ ·K−σ ·K σ ·P) , (2.16)
2m2 1 2 4m2 1 2 m2 1 2 1 2
which, after multiplication by C /2 and C /2, are precisely the terms entering the potential
S T
vCT2(k,K) in Eq. (1.7).
P
III. POINCARE´ ALGEBRA CONSTRAINTS
As an alternative to the procedure discussed in the previous section, one can impose the
Poincar´e algebra constraints on the Hamiltonians derived from the Lagrangians in Eqs. (1.1)
(n) (n)
and (1.4), H = −L . In the instant form of relativistic dynamics, the interactions affect
I I
not only the Hamiltonian H but also the boost generators K. We write
H = H +H , K = K +K , P = P , J = J , (3.1)
0 I 0 I 0 0
to distinguish between the operators in the absence (with subscript 0) and in the presence
(without subscript) of interactions, and impose the following commutation relations:
[Ji, Jj] = iǫijkJk , [Ki, Kj] = −iǫijkJk , [Ji, Kj] = iǫijkKk , [Pµ, Pν] = 0 ,
(3.2)
[Ki, Pj] = iδijH , [Ji, Pj] = iǫijkPk , [Ki, H] = iPi, [Ji, H] = 0 .
10