880 Brazilian Journal of Physics, vol. 35, no. 3B, September, 2005

Subtractive Renormalization of One-Pion-Exchange
and Contact Interactions

V. S. Timóteo1, T. Frederico2, A. Delfino3, and L. Tomio4
1Centro Superior de Educação Tecnoĺogica, Universidade Estadual de Campinas, 13484-370, Limeira, SP, Brazil

2Departamento de F́ısica, Instituto Tecnológico de Aerońautica, CTA, 12228-900, São Jośe dos Campos, SP, Brazil
3Instituto de F́ısica, Universidade Federal Fluminense, 24210-900 Niterói, RJ, Brazil and

4Instituto de F́ısica Téorica, Universidade Estadual Paulista, 01405-900, São Paulo, SP, Brazil

Received on 10 August, 2005

A recursive subtractive renormalization of the scattering equation is applied to the nucleon-nucleon1S0 chan-
nel with one-pion-exchange plus derivative contact interactions. This method can be easily extended to any
derivative order of the singular interaction. Although we limit this work to the singlet partial wave, the method
can be used as well in higher waves and coupled channels. The1S0 renormalization parameters are fitted to the
data.

I. INTRODUCTION

The inspiring work of Weinberg [1] provided the basis for
the effective field theory (EFT) of nuclear forces starting from
the expansion of an effective chiral Lagrangian. In leading
order, it gives the one-pion-exchange potential (OPEP) plus a
Dirac-δ contact interaction. Many works where EFT methods
were applied to the NN system have been developed by many
groups with important results [2–11].

Recent works [12, 13] handle the OPEP plus derivative
Dirac-δ interactions making use of dimensional and boun-
dary conditions regularizations. The leading order interaction,
renormalized with subtracted scattering equations, dominates
the coupled3S1−3 D1 channel while the singlet1S0 channel
requires high order terms in the effective interaction [11].

The description of the1S0 singlet wave up topLab ∼
300MeV/c demands an effective NN interaction with second
order derivatives of the Dirac-δ, which in the relative momen-
tum space reads

〈~p′|VEFT|~p〉= 〈~p′|Vs
π |~p〉 +

1

∑
i, j=0

λi j p
′2i p2 j , (1)

where theλ’s are unregulated strengths and the matrix element
of the one-pion-exchange potential is〈~p′|Vs

π |~p〉. The effective
bare potential of Eq. (1) generates integrals that diverge as
much asp5 in the scattering equation. Therefore, it is neces-
sary at least three subtractions in the kernel of the Lippman-
Schwinger (LS) equation, since each subtraction introduces a
factor ofp−2. Differently from the recent works [12] and [13],
we implement the alternative method of subtracted scattering
equations [15] to handle the divergencies.

The one-subtraction scheme used in our previous work [11]
was generalized in [15] to allow multiple subtractions, which
makes possible to treat derivatives of the contact interaction
in the effective two-body potential. The driving term of the
n-subtracted LS equation is constructed recursively, so that
the model is renormalized at each subtraction order, being the
approach renormalization group invariant [15].

In this work we obtain the1S0 NN amplitude from the effec-
tive interaction Eq. (1), using a scattering equation with three

subtractions. Within this framework, we perform a systematic
analysis of the physical contribution coming from each order
term in the recursive order-by-order renormalization method.

The presentation of this paper is as follows. In section II,
we revise the recursive subtraction method to treat the scat-
tering equation of ultraviolet singular potentials, and we des-
cribe in detail the renormalized scattering equation, also we
briefly discuss the invariance of the scattering amplitude un-
der the dislocation of the subtraction point. In section III, we
present the results for the singlet nucleon-nucleon phase-shift
and in section IV, we conclude.

II. SUBTRACTED T-MATRIX EQUATIONS

The bare effective potential of Eq. (1) is ultraviolet di-
vergent, making the Lippman-Schwinger equation singular
which requires regularization and renormalization to allow a
sensible scattering amplitude. The use of the subtracted scat-
tering equations which by construction are regularized and re-
normalized where the Green’s functions appear subtracted at
certain scales, which are convenient for introducing the phy-
sical inputs.

In the present case we consider a four-term singular inte-
raction of Eq. (1), which after partial-wave decomposition to
the singlet s-wave state, gives the bare potential in the form:

VEFT,s(p′, p) = Vreg
π,s (p′, p)+λ00︸ ︷︷ ︸

Vπ+δ

+ λ01p′2 +λ10p2 +λ11p′2p2

︸ ︷︷ ︸
Vδ′

, (2)

and we introduceVπ+δ which corresponds to the regular part
of OPEP plus a Dirac-delta interactionVδ. For the1S0 state,the
regular part is

Vreg
π,s (p′, p) =− g2

a

32π f 2
π

Z 1

−1
dx

m2
π

p2 + p′2−2pp′x+m2
π
, (3)

which has a finite scattering matrix, solution of the partial-



V. S. Tiḿoteo et al. 881

wave projected LS equation:

Tπ(p′, p;k2) =

Vreg
π,s (p′, p)+

2
π

Z ∞

0
dqq2 Vreg

π,s (p′,q)
k2−q2 + iε

Tπ(q, p;k2). (4)

In our normalization conventions, the scattering amplitude
in the angular momentum basis isT(k,k, ;k2) = − 1

kcotδ−i k ,
and reminding that the low-energy parameters are defined by
the effective range expansionkcotδ =−1

a + 1
2r0k2 + · · · , with

a the scattering length andr0 the effective range.

A. Renormalized T-matrix for Vπ+δ

The T-matrix of the one-pion-exchange plus the Dirac-delta
potential [11] is the input of the subtracted scattering equati-
ons for the complete singular potential of Eq. (2). Here, it is
obtained using Distorted Wave Theory [16] (see Ref. ([12]):

Tπ+δ(E) = Tπ(E)+ [Ω−(E)]†Tδ(E)Ω+(E) , (5)

whereΩ+(E) = 1+ G(+)
0 (E)Tπ(E). The singular part of the

T-matrix is:

Tδ(E) = Vδ +VδG(+)
π (E)Tδ(E) , (6)

where

G(+)
π (E) = G(+)

0 (E)+G(+)
0 (E)Tπ(E)G(+)

0 (E) , (7)

is the Green’s function obtained from the regular part of the
pion-exchange interaction.

Eq. (6) is ill-defined due to ultraviolet divergences in the
momentum integration. In our method, this scattering equa-
tion is reformulated in a subtracted form, which allows to get
a finite scattering amplitude with only one subtraction [11].
Therefore, we apply the method of subtracted equations to
Eq. (6) subtracting the interacting Green’s function, Eq. (7),
at a certain energy scale−µ2 and the driving termTδ(−µ2) of
the subtracted equation has now a finite value. Then, we have:

Tδ(E) =

Tδ(−µ2)+Tδ(−µ2)
[
G(+)

π (E)−Gπ(−µ2)
]

Tδ(E). (8)

We observe that Eq. (8) has the same operator form as the ori-
ginal one-fold subtracted T-matrix equation [11] with the only
difference being that the interacting Green’s function appears
in the place of the free one.

The renormalized strength of the interaction is the value of
Tδ(E) at the subtraction point,Tδ(−µ2) = λR 00, which allows
to solve Eq. (8) resulting in a finite scattering amplitude for
the OPEP plus a Dirac-delta:

Tπ+δ(p′, p;−µ2) =

Tπ(p′, p;−µ2)+ω(p′;−µ2) λR 00 ω(p;−µ2), (9)

where

ω(p;−µ2) = 1+
2
π

Z ∞

0
dq q2 Tπ(p,q;−µ2)

−µ2−q2 , (10)

with Eq. (4) giving the T-matrix for the regular part of the
one-pion-exchange potential in the1S0 channel.

B. Recursively subtracted T-matrix equations

The singular potential of Eq. (2) requires a three-fold sub-
tracted T-matrix equation to allow a finite scattering amplitude
from its solution. The n-fold subtracted equation in operator
form is written as [15]:

T(E) = V(n)(−µ2;E)+

V(n)(−µ2;E)G(+)
n (E;−µ2)T(E), (11)

where

V(n)(−µ2;E)≡
[
1− (−µ2−E)n−1V(n−1)(−µ2;E)G3

0(−µ2)
]−1

×
V(n−1)(−µ2;E) , (12)

with the n-fold subtracted Green’s function defined by

G(+)
n (E;−µ2) ≡

[
(−µ2−E)G0(−µ2)

]n
G(+)

0 (E), (13)

and G(+)
0 (E) = [E−H0 + iε]−1 with H0 = p2 in the two-

nucleon rest-frame.
The driving termV(3)(−µ2) is constructed recursively star-

ting fromV(1)(−µ2) = Tπ+δ(−µ2) given by Eq. (9). Then, the
higher order singular terms of the potential Eq.(2) are introdu-
ced in the driving term of the three-fold subtracted equation as
we are going to show in detail below.

The driving term of the three-fold subtracted scattering
equation is obtained numerically by solving recursively the in-

tegral equations for the matrix elementsV(n)
π+δ(p′, p ; −µ2;k2)

which are explicitly given by:

V(n)
π+δ(p′, p;−µ2;k2) = V(n−1)

π+δ (p′, p;−µ2;k2)

+
2
π

Z ∞

0
dqq2

(
µ2 +k2

µ2 +q2

)n−1 V(n−1)
π+δ (p′,q;−µ2;k2)

−µ2−q2 ×

×V(n)
π+δ(q, p;−µ2;k2), (14)

with k=
√

E. Then, the higher singular terms of the full effec-
tive interaction of Eq. (2) are introduced directly in the matrix

element ofV(3)
π+δ+δ′(−µ2) in the form:

V(3)
π+δ+δ′(p′, p;−µ2;k2) = V(3)

π+δ(p′, p;−µ2;k2)+

λR 10(p′2 + p2)+λR 11p′2p2 . (15)

The driving term of the three-fold subtracted scattering equa-
tion gets contribution from the derivatives of the Dirac-delta
interaction, that haveλR i j as the renormalized strengths of the
corresponding singular part of the potential.

Finally, within our method we solve numerically the three-



882 Brazilian Journal of Physics, vol. 35, no. 3B, September, 2005

fold subtracted LS equation given by:

T(p′, p;k2) = V(3)
π+δ+δ′(p′, p;−µ2;k2)

+
2
π

Z ∞

0
dqq2

(
µ2 +k2

µ2 +q2

)3

V(3)
π+δ+δ′(p′,q;−µ2;k2)

k2−q2 + iε
T(q, p;k2) , (16)

where in all steps to get and solve the above equation the ultra-
violet divergences were removed in favor of the renormalized
strengths of the singular interaction at the subtraction energy
−µ2. Indeed, the arbitrary subtraction point can be moved
without affecting the physics at the expense of changing the
driving term of the subtracted LS-equation.

C. Renormalization group invariance

The physics of the nucleon-nucleon scattering should not
depend on the arbitrary subtraction energy scale,−µ2. One
could work with any convenient value ofµ2. However, the
detailed form of the driving term in Eq. (15) which defines
the scattering amplitude depends on the prescription used to
define the renormalized theory. The renormalization group
method guide us how to modify this prescription while kee-
ping unchanged the predictions of the theory [17].

The invariance of the theory under the changes of the re-
normalization prescriptions defines a rule to modify consis-
tentlyV(n) in Eq.(11) to keep unchanged the T-matrix. Using
Eq. (11) and∂T(E)

∂µ2 = 0, one can derive the non-relativistic
Callan-Symanzik (NRCS) differential equation [15, 18]:

∂V(n)(−µ2;E)
∂µ2 =

−V(n)(−µ2;E)
∂G(+)

n (E;−µ2)
∂µ2 V(n)(−µ2;E) . (17)

The solution of the above equation gives the driving term at
any subtraction point with the boundary condition given by
Eq. (15). The NRCS equation concretizes the invariance of
the renormalized T-matrix under dislocation of the subtraction
point. From that one immediately realizes that the dependence
on the subtraction point appearing in the driven term of the
subtracted scattering equation is highly nontrivial, although
the physical results of the model are kept unchanged.

III. NUMERICAL RESULTS

The free parameters of our calculation are the renorma-
lized strengthsλR 00, λR 10, λR 11 and the subtraction point
µ. To simplify the fitting procedure we fixλR 10, λR 11 and
µ, and adjustλR 00 to reproduce the singlet scattering length
as = −23.739 fm. The three parameters left are adjusted to
reproduce the Nijgemen data [14] up to the center of mass mo-
mentum ofk =300 MeV/c. From the fitting procedure we got

0 100 200 300
p [ MeV/c ]

0

10

20

30

40

50

60

70

P
h

as
e 

S
h

if
t 

[ 
D

eg
re

es
 ]

Nijmegen

 δ’

 π + δ

 π + δ + δ’

FIG. 1: 1S0 phase shift as a function of the c.m. momentum calcula-
ted forVπ+δ, Vδ′ andVπ+δ+δ′ . The dots are the Nijmegen data [14].

µλR 00 = −0.1465, µ3λR 01 = 4.7124 and µ5λR 11 = 5.0265,
with µ = 214 MeV. The resulting effective range isr0,s =
2.73 fm compared with the value of 2.68 fm from Ref. [14].

In the figure, we show our study for the different contributi-
ons of the effective potential as well as the full calculation. We
performed calculations for the1S0 phase shifts obtained with
Vπ+δ, Vδ′ andVπ+δ+δ′ , with the parameters fixed at the values
we found by the fitting procedure. Belowp∼ 20 MeV/c the
calculation with onlyVπ+δ underestimates the data, while for
Vδ′ the phase shifts are better described. The dominant contri-
bution in this channel comes fromVδ′ , while the pion appears
at low energies providing the long range part of the NN inte-
raction.

It is interesting to study how the observables depend on the
scaleµ. The dependence of the singlet1S0 phase shifts on the
subtraction pointµ was studied in Ref. [19].

IV. CONCLUDING REMARKS

We show how to apply the method of subtracted scattering
equations [15] to calculate the NN singlet phase-shift, when
the matrix elements of the effective interaction diverges in the
ultraviolet region asp2. In this case three subtractions are
required to renormalize the model. Then, the integrand of
the original scattering equation is automatically regularized
by the subtractions of the propagator at some energy scale.
The Born term in our calculation is the T-matrix at the sub-
traction energy scale differently from the usual potential term
of the standard LS equation. For regular potentials the sub-
tracted equations is fully equivalent to the LS formalism. If
one desires the subtraction scale can be moved without mo-
difying the calculated observables as long as new driven term
of the subtracted scattering equation comes from the solution
of the Non-Relativistic Callan-Symanzik (CS) equation. In
our calculation, the boundary condition of the CS equation
is determined by the renormalized coupling constants and by
the T-matrix of the OPE and Dirac-δ potentials at the speci-
fied value of the subtraction point. And finally, we would like
to add that the method of subtracted scattering equation is in
principle suitable to treat also higher order singular terms of



V. S. Tiḿoteo et al. 883

the effective interaction.
Acknowledgements.This work was partially supported by

FAPESP, CNPq, FAPERJ, and FAEPEX/UNICAMP.

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