Renormalization of chiral two-pion exchange NN interactions. Momentum vs. coordinate space

0.62 MB·English

by  D. R. Entem

Renormalization of chiral two-pion exchange NN interactions.
Momentum vs. coordinate space.
D. R. Entem,1,∗ E. Ruiz Arriola,2,† M. Pav´on Valderrama,3,‡ and R. Machleidt4
1Grupo de F´ısica Nuclear, IUFFyM, Universidad de Salamanca, E-37008 Salamanca,Spain
2Departamento de F´ısica At´omica, Molecular y Nuclear,
Universidad de Granada, E-18071 Granada, Spain.
3Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
4Department of Physics, University of Idaho, Moscow, Idaho 83844§
(Dated: February 1, 2008)
Therenormalizationofthechiralnpinteractioninthe1S0channeltoN3LOinWeinbergcounting
for the long distance potential with one single momentum and energy independent counterterm
is carried out. This renormalization scheme yields finite and unique results and is free of short
8 distance off-shell ambiguities. We observe good convergence in the entire elastic range below pion
0
productionthresholdandfindthattherearesomesmallphysicaleffectsmissinginthepurelypionic
0
chiral NN potential with or without inclusion of explicit ∆ degrees of freedom. We also study
2
the renormalizability of the standard Weinberg counting at NLO and N2LO when a momentum

n dependentpolynomial counterterm is included. Ournumerical results suggest that theinclusion of
a thiscounterterm does not yield a convergent amplitude (at NLO and N2LO).
J

2 PACSnumbers: 03.65.Nk,11.10.Gh,13.75.Cs,21.30.Fe,21.45.+v
Keywords: NNinteraction, OneandTwoPionExchange, Renormalization.

]
h
t I. INTRODUCTION cutoff dependent [27, 35] and hence to be incompatible
-
l with renormalizability 1. Thus, some acceptable com-
c
u promise must be made. Actually, the chosen approach
n The modern Effective Field Theory (EFT) analysis of betweenthisdichotomydependsstronglyonthepursued
[ theNNinteractionusingchiralsymmetryasaconstraint goals and it is fair to say that any choice has both ad-

has a recent but prolific history [1, 2] (for comprehen- vantages and disadvantages. In any case, the reason for
2
v sive reviews see e.g. Ref. [3, 4, 5]). Most theoretical bothfailurescanbetracedbacktothenatureofthelong-
0 setupsareinvariablybasedonaperturbativedetermina- distancechiralpotentials;whilepion-exchangepotentials
7 tion of the chiral potential [6, 7, 8, 9, 10, 11, 12, 13, 14] falloffexponentiallyatlongdistancestheyincludestrong
7 and the subsequent solution of the scattering prob- power-law singularities at short distances. Those sin-
2 lem [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Actu- gularities become significant already at distances com-
.
9 ally, the theory encounters many problems in the low parable with the smallest de Broglie wavelength probed
0 partialwavesandinparticularin thes−waves(see how- in NN scattering below pion production threshold. Ob-
7 ever Ref. [26] for a more optimistic view). Indeed, there viously, any development of NN interactions based on
0 is at present an ongoing debate on how an EFT pro- chiral dynamics will presumably require a deeper under-
:
v gramshouldbesensiblyimplementedwithintheNNcon- standingandproperinterpretationofthe peculiaritiesof
i text and so far no consensus has been achieved (see e.g. these highly singular chiral potentials. Although singu-
X
[27, 28, 29, 30]). The discussion is concerned with the lar potentials where first analyzed many years ago [36]
ar issue ofrenormalizationvs.finite cutoffs, a priori(power (for an early review see e.g. [37] and for a more updated
counting) vs. a posteriori error estimates or the appli- view within an EFT context see [38]), their short dis-
cability of perturbation theory both on a purely short tance singular character within the NN interaction has
distance theory or around some non-perturbative dis- seriously been faced more recently within a renormal-
torted waves. At the moment, it seems fairly clear that ization context for the one-pion exchange (OPE) poten-
an EFT scheme with a cutoff-independent and system- tial[27,29,39,40,41,42,43,44,45,46]andthetwo-pion
atic perturbative power counting for the S-matrix (the exchange (TPE) potential [35, 47, 48].
so-called KSW counting) fails [31, 32, 33, 34]. On the
other hand, the original EFT inspired scheme [1, 2] (the In the np scattering problem the 1S0 channel is very
so-called Weinberg counting) has recently been shown special since the scattering length is unnaturally large
to produce many results which turn out to be strongly as compared to the range of the strong interaction,
α = −23.74(2)fm ≫ 1/m = 1.4 fm. In fact, even
0 π
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] 1 Inthispaperwerefertorenormalizabilityinthesenseofparam-
‡Electronicaddress: [email protected] eterizingtheshortdistancephysicsbyapotential whichmatrix
§Electronicaddress: [email protected] elementsinmomentumspaceareapolynomialinthemomenta.
2
at zero energy the wave function probes relatively short justing the physical scattering length. In the derivation
distancecomponentsofthechiralpotential[35]2. Higher ofthisresultthemathematicalrequirementsofcomplete-
energiesbecomeevenmoresensitivetoshortdistancein- ness and self-adjointness for the renormalized quantum
teractions. Consequently,this channellookslikeanideal mechanicalproblemforalocalchiralpotentialplayade-
place to learn about the size of the most relevant short cisive role. This surprising result is in contrast to the
range corrections to the NN force in the elastic scatter- standard Weinberg counting where an additional coun-
ing region. Actually, in the 1S channel, most EFT in- terterm C is included already at NLO. This C coun-
0 2 2
spired schemes yield at leading order (LO) (which con- terterm could, in fact, be determined by fitting the ex-
sists of OPE plus a nonderivative counterterm) an al- perimental value of the effective range; the physics of
most constant phase of about 75o around k =250MeV, C is to provide a short distance contribution to the ef-
2
and an effective range of rLO = 1.44fm. On the other fective range in addition to the contribution from the
0
hand, most determinations from Partial Wave Analy- known long distance chiral potential. Within this con-
sis[49,50]andhighqualitypotentialmodels[51]yieldan text it is remarkable that according to Ref. [35], where
almost vanishing phase at this center-of-mass (CM) mo- such a short distance contribution vanishes (or equiva-
mentum while the experimental effective range is about lently C = 0 when the cut-off is removed), rather ac-
2
twice the OPE value, rexp = 2.77(5)fm. 3 It is quite curate values are predicted yielding rNLO = 2.29fm and
0 0
unbelievable that such large changes can be reliably ac- rN2LO =2.86fm after renormalization. This latter value
0
commodated by perturbation theory starting from this is less than 3% larger than the experimentally accepted
LO result despite previous unsuccessful attempts treat- value and it suggests that most of the effective range
ing OPE and TPE perturbatively [31, 32, 33, 34]. Ac- is saturated by N2LO TPE contributions and calls for
tually, for the singlet channel case, short distance com- pinning down the remaining discrepancy. This trend to
ponents are enhanced due to the large value of the scat- convergenceandagreementisalsosharedbyhigherorder
tering length and the weakness of the OPE interaction slope parameters in the effective range expansion with-
in this channel. This is why TPE contributions have outstrongneedofspecificcountertermsalthoughthereis
been treated with more success in a non-perturbative stillroomfor improvement. Motivatedby this encourag-
fashion [2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. ingresult,onegoalofthe presentpaperisto analyzethe
Despite phenomenological agreement with the data, the size of the next-to-next-to-next-to-leadingorder (N3LO)
inclusion of finite cutoffs suggests that there might be corrections to the results found in [35].
some regulator dependence in those calculations.
The calculations in Ref. [35, 44, 48, 52] exploit explic-
In a series of recent papers [35, 44, 48], two of us itlythelocalcharacterofthechiralpotentialbyconduct-
(M.P.V. and E.R.A.) have proposed not only to iterate ing the calculationsincoordinatespacewhichmakesthe
but also to renormalize to all orders the NN chiral po- analysis more transparent. Many results, in particular
tentialwithinalongdistanceexpansion. Byallowingthe the conditions under which a renormalized limit exists,
minimal number of counterterms to yield a finite result, can be established a priori analytically. Moreover, the
long-distance regulator-independent correlations are es- highly oscillatory character of wave functions at short
tablished. In practice, the potential must be computed distances is treated numerically using efficient adaptive-
withinsomepowercountingscheme. Whilethepotential stepdifferentialequationstechniques. Thissituationcon-
is used within Weinberg’s power counting to LO, next- trastswithmomentumspacecalculationswhere,withthe
to-leading order (NLO), and next-to-next-to-leading or- exception of the pion-less theory, there is a paucity of
der (N2LO), we only allow for those counterterms which analytical results, and one must mostly rely on numeri-
yieldafiniteanduniquescatteringamplitude. Inthe1S0 cal methods. Moreover, the existence of a renormalized
np channel a single energy and momentum independent limit is not obvious a priori and one may have to resort
countertermC0 is consideredwhichisdeterminedbyad- to some trial and error to search for counterterms. Fi-
nally, renormalization conditions are most naturally for-
mulated at zero energy for which the momentum space
treatment may be challenging, at times. Of course, be-
2 Thiscanbebestseenbymeansoftheeffectiverangeformula sides these technical issues, there is no fundamental dif-
r0=2 ∞dr u0(r)2− 1− r 2 fsepraecnec,epbaerttwiceuelnarplyrocaefteedrinrgeninormmoamlizeanttiuomn, oprrocvoiodreddintahtee
Z0 " „ α0« # same renormalization conditions are specified, since dis-
whereu0(r)isthezeroenergywavefunction,fulfillingtheasymp- parate regulators stemming from either space are effec-
toticconditionu0(r)→1−r/α0. Mostoftheintegrandislocated tively removed. Indeed, we will check agreement for the
intheregionaroundr=1fmwhichisinbetweenOPEandTPE
phase-shiftsdeterminedindifferentspaceswheneversuch
ranges. Moreover,the lowenergy theorem ofRef.[35] allowsto
writer0=A+B/α0+C/α20whichintheextremelimitα0→∞ acomparisonbecomespossible. Thisequivalenceisinit-
yields r0 → A. Numerically it is found that A is far more de- self a good motivation for renormalization.
pendent than B and C when being evaluated at LO, NLO and
However,atN3LOsomeunavoidablenon-localitiesap-
NNLO.
3 Infact,thesehighqualitypotentialmodelsyieldslightlysmaller pearinthechirallongdistancepotential. Althoughthey
values,r0≈2.67fm. could be treated in configuration space, we adopt here a
3
momentum space treatment. This will also allow us to momentumandcoordinatespaceatNLOandN2LOand
answer anintriguing question which was left open in the shown to potentially have some problems. Finally, in
coordinate space analysis of previous works [35, 44, 48], Sec. VI, we summarize our main points.
namely, the role played by the conventionalmomentum-
polynomialrepresentationsof the shortdistance interac-
tion used in most calculations [15, 16, 18, 19, 20, 21, 22, II. THE RENORMALIZATION PROBLEM
23, 24, 25]in the renormalizationproblem. More specifi-
cally,Ref.[35]showedthattakingC =0wasconsistent, A. General overview and main results
2
and a regularization scheme exists where a fixed C was
2
irrelevant, but could not discriminate whether C 6= 0
2 Let us define the scope and goals of the present work.
was inconsistent as far as it was readjusted to the ef-
The standardnon-perturbativeformulationofthe renor-
fective range parameter for any cut-off value. 4 In this
malization problem starts with an effective Lagrangian
regard, the present paper yields a definite answer mak-
or Hamiltonian (see e.g. [1, 2] and [3, 4] and references
ing the surprising agreementof the effective range found
therein),fromwhichacertainsetofirreducibleFeynman
in Ref. [35] an inevitable mathematical consequence of
diagrams (usually up to a certain order) is calculated.
renormalization.
Theseirreduciblediagramsaredefinedtorepresentapo-
tentialV. Thepotentialistheninsertedintoascattering
The paper is organized as follows. The problem is
equation where it is iterated infinitely many times or, in
stated in Sec. II where a general overview of the renor-
other words, re-summed non-perturbatively. In the CM
malization problem is given both in momentum as well
frame,wherethenpkineticenergyisgivenbyE =p2/M,
as coordinate space. In Sec. IIB, we particularize the
with M = 2µ = 2M M /(M +M ), the scattering
momentum space formulation of the scattering problem np n p p n
processisgovernedbytheLippmann-Schwingerequation
with counterterms for the 1S channel within a sharp
0
three-momentum cutoff scheme. Likewise, in Sec. IIC T =V +VG T , (1)
0
we proceed similarly in the coordinate space formula-
tion within a boundary condition regularization with a with V the potential operatorand G =(E−H )−1 the
0 0
short distance cutoff r . In Sec. III, we discuss some resolvent of the free Hamiltonian. The outgoing bound-
c
features of both coordinate and momentum space for- ary condition corresponds to E → E +i0+. Using the
mulations in the pion-less theory and try to connect the normalization h~x|~ki=ei~k·~x/(2π)3/2 one has
high momentum cutoff Λ with the short distance radial
cutoff rc. This allows a one-to-one mapping of coun- h~k′|T(E)|~ki=h~k′|V|~ki+ Λd3qh~k′|V|~qih~q|T(E)|~ki.
terterms in both spaces which will prove useful later on Z E−(q2/2µ)
in the pion-full theory. The identification between the
(2)
sharp momentum cutoff and the short distance radius
found in the pion-less theory is discussed further in Ap-
HereΛmeansagenericregulatorandrepresentsthescale
pendix A in the presence of a long distance potential in
below which all physical effects are taken into account
the lightofthe Nyquist theorem. In Sec.IV, we come to
explicitly. The degrees of freedom which are above Λ
the central discussion on N3LO corrections to the phase
are taken into account implicitly by including a suitable
shiftswhenthescatteringamplitudeisrenormalizedwith
cutoff dependence inthe potential. The precise equation
only one short distance counterterm. A wider perspec-
governingthiscutoffdependencewasdescribedandstud-
tiveisachievedbyfurtherconsideringthe roleofexplicit
ied in some detail in Ref. [53] with particular emphasis
∆-excitations in intermediate states and the subsequent
on infrared fixed points. We will analyze the cutoff de-
one-countertermrenormalizationofthescatteringampli-
pendencebelowfocusingontheultravioletaspectsofthe
tude. We also discuss the role of three pion exchange as
interaction.
well as how the results depend on the renormalization
Motivated by the low energy nature of the effective
scheme used to compute the potential based on cut-off
theory, the potential is usually separated into short and
independent counterterms. The mathematical justifica-
long distance components in an additive form
tion for using just one counterterm is providedin Sec. V
where the standard Weinberg scheme is pursued both in h~k′|V|~ki=V (~k′,~k)+V (~k′,~k), (3)
S L
where the long distance contribution is usually given by
successive pion exchanges
4 We mean of course the case when the cut-off is being removed.
Theessential issueiswhether or notone can fixbyashortdis- V (~k′,~k)=V (~k′,~k)+V (~k′,~k)+... , (4)
L 1π 2π
tancepotential whichisapolynomialinthemomentatheeffec-
tiverangeindependentlyonthepotentialandremovethecut-off and the short distance component is characterized by a
at the same time. Of course, the very definition of the poten-
power series expansion in momentum
tialisambiguousandrequiresaspecificchoiceonthepolynomial
Cpa2r,tbs.ecToemchensiicrarlelyle,vwaentfinindtthheatliamniytfiΛx→ed,∞cut(-soeffeibnedloenwd)e.pendent VS(k′,k)=C0+C1~k·~k′+C2(~k2+k~′2)+... , (5)
4
where for simplicity we assume a spin singlet channel 5. on the momentum transfer 7. In such a case, if we for-
NotethatC andC contributetos-waveswhileC con- mally take a Fourier transformation of the long distance
0 2 1
tributes to p-waves, and so on. One should face the fact potential
that,althoughthedecompositiongivenbyEq.(3)isper-
turbatively motivated and seems quite natural, the ad- d3p
V (~x)= V (p~)eip~·~x, (6)
ditivity between short and long range forces is actually L Z (2π)3 L
an assumption which has important consequences, as we
will see. 6 and take the limit Λ → ∞ one has the standard
Schr¨odinger equation in coordinate space,
In principle, for a given regularization scheme charac-
terizedbyacutoffΛ,thecountertermsC0,C1,C2 andso 1
on are determined by fixing some observables. One nat- − ∇2Ψk(~x)+V(~x)Ψk(~x)=EΨk(~x), (7)
M
urally expects that the number of renormalization con-
ditions coincides with the number of counterterms in a where the coordinate space potential is
way that all renormalization conditions are fully uncor-
related. The statement of UV-renormalizability is that V(~x) = V (~x)+C δ(3)(~x)+C ∇~δ(3)(~x)∇~
L 0 1
such a procedure becomes alwayspossible when the cut-
+ C ∇2δ(3)(~x)+δ(3)(~x)∇2 +... . (8)
off Λ is removed by taking the limit Λ → ∞. This may 2
h i
notbethecaseastheremayappearredundantcontribu-
tions (see the discussion below in Sec. III) meaning that The whole discussion, which has been carried on for
one counterterm or counterterm combination can take years now, concerns the precise meaning of these delta
anyvalue. Anotherpossiblesituationisjusttheopposite; and derivatives of delta interactions, particularly when
onemaywanttoimposemorerenormalizationconditions a long distance potential is added to the short distance
than possible. In this case some counterterms or coun- one. Acrucialfindingofthepresentpaperbasedonadi-
terterm combinations are forbidden. Rather than being rectanalysisinmomentumspaceisthatnon-perturbative
intricate mathematical pastimes, these features have al- renormalization imposes restrictions on the number of
ready been investigatedrecently [27, 35] for largecutoffs terms and form of the short distance potential which
castingsomedoubtonthe regulatorindependence ofthe depend also on the particular long distance potential.
original proposal [1, 2]. Some of these restrictions were discussed in previous
works [27, 35, 44]. Remarkably these new renormaliz-
EvenifoneadmitsgenericallyEq.(3)aswellasEq.(4)
ability restrictions were conjectured in coordinate space
and Eq.(5), it is not obvioushow many terms shouldbe in Ref. [28, 35, 44] for the 1S channel based on self-
0
consideredandwhether there isa clearwayofdefining a
adjointness and completeness of states and apply to the
convergence criterium or identifying a convergence pat-
TPE potential; a single C 6= 0 counterterm is allowed
0
tern. It is fairly clear that, on physical grounds, one
while two counterterms, C 6= 0 and C 6= 0, are forbid-
0 2
shouldconsider anexpansionofthe potential thatstarts den 8.
at long distance and decreases in range as the number
of exchanged particles increases. Note, however, that,
while OPE is well-defined, the general form of the TPE
B. Momentum space formulation
potential is not uniquely determined. Within this con-
text, one of the main attractive features of the EFT
In the 1S channel the scattering process is governed
approach has been the definition of a power counting 0
by the Lippmann-Schwinger equation
scheme which provides a hierarchy and an a priori cor-
relation between long and short range physics. In the
Λ q2M
present paper, we will assume Weinberg power count- T(k′,k)=V(k′,k)+ dqV(k′,q) T(q,k),
ing [6, 7, 8, 9, 10, 11, 12, 13, 14] (see below for a more Z p2−q2+i0+
0
precise definition), where the long distance potential is (9)
determined in a dimensional power expansion.
where T(k′,k) and V(k′,k) are the scattering amplitude
The long distance component V is obtained by par-
L and the potential matrix elements, respectively, between
ticle exchanges and, in some simple cases, depends only
off-shell momentum states k and k′ in that channel and
the sharp three-momentum cutoff Λ represents the scale
below which all physical effects are taken into account
5 Moredetailedexpressionsincludingspintripletchannels canbe
lookedupe.g.inRef.[4].
6 Specifically, Eq. (3) does not foresee for instance terms of the
form V(p′,p)C(Λ), i.e. terms which are not polynomial but in- 7 This assumption will be relaxed immediately below when dis-
fluence the renormalization process. Of course, once additivity cussingtheWeinbergcounting inthe1S0 channel.
isrelaxedthere aremanypossiblerepresentations, inparticular 8 Again, we mean cut-off dependent counterterms designed to fit
fortheshortdistancecomponents. physicalobservables.
5
explicitly. From the on-shell scattering amplitude the Note that this counting involves both unknown short-
phase shift can be readily obtained distancephysicsandchirallong-distancephysicsinaun-
correlated way. Notealsothatthereisnofirstordercon-
2
T(p,p)=− eiδsinδ(p). (10) tributionandthatthereisnothirdordercontributionto
πMp
the short distance potential.
The short range character of the nuclear force implies Regarding Eq. (12) one should stress that the separa-
thatatlowenergiesonehastheeffectiverangeexpansion tion between long and short range contributions to the
(ERE) potentialisnotunique. Infact,thereisapolynomialam-
biguity in the long range part which can freely be trans-
2 1 ferred to the short distance contribution. However, the
pcotδ(p) = − Re
Mπ (cid:20)T(p,p)(cid:21) non-polynomial part is unambiguous as it is directly re-
1 1 lated to the left hand cut of the partial wave amplitude
= − + r p2+v p4+v p6+...(11)
0 2 3 which for nπ exchange is located at p = inm /2 but
α 2 π
0 presumably becomes incomplete for |p|>m /2. 9
ρ
where α is the scattering length, r the effective range
0 0
and v , v etc. are slope parameters.
2 3
Inthe 1S0 channel,the potentialis decomposedasthe C. Coordinate space formulation
sum of short and long range pieces
V(k′,k)=V (k′,k)+V (k′,k). (12) In coordinate space, the problem in the 1S0 channel
S L
is formulated as follows [47, 52]. Assuming a local long
In the standard Weinberg counting, the short distance distancepotentialV (r)onehastosolvetheSchr¨odinger
L
contribution is written as follows equation
V (k′,k) = C (Λ)+(k2+k′2)C (Λ) −u′′(r)+U (r)u (r)=p2u (r), r >r , (17)
S 0 2 p L p p c
+ C′(Λ)k2k′2+C (Λ)(k4+k′4)+...(13)
4 4 where UL(r) = 2µnpVL(r) is the reduced potential (in
fact, the Fourier transformationofV (q) )andu (r) the
wherethe countingis relatedtothe orderofthe momen- L p
reduced wave function for an s-wave state. Here r is
tum which appears explicitly. The long distance compo- c
the short distance cutoff and the reduced wave function
nent of the potential is taken to be the sum of explicit
is subject to the boundary condition at r = r and the
pion exchanges c
standard long distance free particle behaviour
V =V +V +V +... (14)
L 1π 2π 3π u′(r )
p c
= pcotδ (p), (18)
where [1] u (r ) S
p c
V1π = V1(π0)+V1(π2)+V1(π3)+V1(π4)+... up(r) → sin(pr+δ(p)). (19)
sinδ(p)
V = V(2)+V(3)+V(4)+...
2π 2π 2π 2π
whereδ (p)istheshortdistancephase-shiftencodingthe
V = V(4)+... S
3π 3π physics for r <r . In the case of a vanishing long range
c
(15) potential U (r)=0 the phase shift is given by δ (p,r ).
L S c
Ontheotherhand,ifwetakeδ (p)=0wegetastandard
using dimensional power counting. Ideally, one should S
problemwithahardcoreboundarycondition,u (r )=0
determinethephysicallyrelevantlongrangeregulatorin- p c
which for r → 0 becomes the standard regular solution
dependent correlations,i.e., long distance effects of simi- c
at the origin. At low energies both the full phase-shift
larrange. Thiswouldamounttoconsiderallnπexchange
δ(p) and the short distance phase-shift δ (p) can be de-
effects on the same footing, since they yield a long dis- S
scribed by some low energy approximation, like e.g., an
tance suppression ∼ e−nmπr modulo power corrections.
effective range expansion,
At present, the only way how these long distance poten-
tials can be systematically computed is by dimensional 1 1
pcotδ (p) = − + r p2+... (20)
power counting in perturbation theory, as represented S α 2 0,S
0,S
schematically in Eqs. (14) and (15).
1 1
In the standard Weinberg counting one has pcotδ(p) = − + r0p2+... (21)
α 2
0
V = V(0)+V(0)
LO S 1π
V = V +V(2)+V(2)+V(2)
NLO LO S 1π 2π
(3) (3) 9 Thebestwaytorecognizetheambiguityisintermsofthespec-
V = V +V +V
N2LO NLO 1π 2π tral function representation of the potential [7], where the sub-
V = V +V(4)+V(4)+V(4)+V(4) traction constants can be fixed arbitrarily. In coordinate space
N3LO N2LO S 1π 2π 3π the non-ambiguous part corresponds to the potential V(r) for
(16) anynon-vanishingradius(seee.g. thediscussioninRef.[35,48].)
6
where α is the short range scattering length, r the 62, 63, 64, 65, 66] although without much considera-
0,S 0,S
shortrangeeffectiverange,andα andr thefullones10. tion on how this problem might be embedded into the
0 0
If we also make an expansion at low energies of the re- wider and certainly more realistic situation where the
duced wave function finite range and short distance singular chiral NN po-
tentials are present. In fact, much of the understanding
up(r)=u0(r)+p2u2(r)+... (22) of non-perturbative renormalization within the modern
NN context has been tailored after those and further
we get the hierarchy of equations studies based on the non-singular OPE singlet 1S po-
0
tential [63, 67, 68, 69] plus the standard perturbative
−u′′(r)+U(r)u (r) = 0, (23)
0 0 experience. In previous [35, 43, 44, 47, 48] and in the
α0,Su′0(rc)+u0(rc) = 0, present work, we pursue exactly the opposite goal: we
r will only consider renormalization procedures which can
u (r) → 1− ,
0 α directly be implemented in the presence of long distance
0
potentials since, after all, contact NN interactions are
and always assumed to approximate truly finite range inter-
actionsinthelongwavelengthlimit. Thus,itisusefulto
−u′′(r)+U(r)u (r) = u (r), (24)
2 2 0 review here those developments with an eye on the new
α u′(r )+u (r ) = 1r α u (r ), ingredients which appear in the non-perturbative renor-
0,S 2 c 2 c 2 0,S 0,S 0 c malization of singular pion exchange potentials as ana-
r
u (r) → r2−3α r+3α r , lyzed in later sections. In addition, the deduced running
2 0 0 0
6α0 (cid:0) (cid:1) of the counterterms in the contacttheory in the infrared
domain serves as a useful starting point when the long
and so on. The standard way to proceed would be to
distance pion exchangepotentialis switchedon. Finally,
integratetheequationsforu (r),u (r),etc. frominfinity
0 2 we will also discuss the size of finite cutoff correctionsto
downwards, with a known value of α , using Eq. (23) to
0 therenormalizedresultdependingbothontheparticular
obtain α and then one can use Eq.(17) together with
0,S regularizationaswellasthecorrespondingrepresentation
Eq.(19)andEq.(20)tocomputeδ(k)foranyenergywith
of the short distance physics.
a given truncated boundary condition. This procedure
provides by construction the low energy parameters we
started with and takes into account that the long range
A. Momentum space
potential determines the form of the wave function at
long distances. The only parameter in the procedure is
Although the previously described momentum space
theshortdistanceradiusr ,whichiseventuallyremoved
c framework has extensively been used in the past to de-
by taking the limit r →0.
c scribesuccessfullythedata[15,16,18,19,21,22,23,24,
One should mention at this point that the coordinate
25] with a finite cutoff Λ it is worth emphasizing some
space is particularly suited for the case of local long dis-
puzzling features regarding the off-shell ambiguities of
tance potentials, but the renormalization with an arbi-
theshortdistancepotentialwhenfiniterangecorrections,
trary number of countertersm requires an energy depen-
encoded in the C , C etc. counterterms, are included.
dentbutrealboundaryconditionatshortdistanceswhich 2 4
In momentum space, the pion-less theory corresponds
eventually violates self-adjointness. On the other hand,
to taking V (k′,k) = 0. In such a case the Lippmann-
the momentum space formulation allows the discussion L
Schwinger equation reduces to a simple algebraic equa-
ofnonlocallongdistancepotentialsandtherenormaliza-
tion[58,64]. AtverysmallvaluesofthecutoffΛ<m /2,
tion is done in terms of a momentum dependent short π
the long range part of the potential may be neglected
distancepolynomialpotential. Althoughthislookslikea
since they scale with powers of momentum and a sim-
self-adjoint problem, we will see that in this formulation
ple contact theory of the form of Eq. (13) may be used.
the counterterms may in fact become complex.
For instance, when V (k′,k) = C (Λ), the Lippmann-
S 0
Schwingerequation(LSE)maybe directly solved. Using
the basic integral 11
III. THE RENORMALIZATION PROBLEM FOR
THE PION-LESS THEORY
Λ dqq2 πp p Λ+p
J = =−Λ−i + log ,(25)
0 Z p2−q2+i0+ 2 2 Λ−p
The renormalization of the pion-less theory, i.e., a set 0
of pure contactinteractions,has been treated with great
detail in the literature [54, 55, 56, 57, 58, 59, 60, 61,
11 The result for a different momentum cutoff scheme such as
V(k′,k)→g(k′,Λ)V(k,k′)g(k,Λ)correspondstomakingthere-
placement Λdq → dqg(q,Λ)2. In dimensional regularization
0
10 This is not the only possible short distance representation [52]. (minimal subtraction scheme) the integral is just the unitarity
SeeSec.IIIBforafurtherdiscussiononthis. piece,J0=R−iπ2p. R
7
for a sharp momentum cutoff Λ and 0 ≤ p ≤ Λ, one The first equation allows to eliminate uniquely C in
0
obtains for the phase shifts favourofα andC ,butasweseetherearetwobranches
0 2
for the solutions. However, we choose the branch for
2 2Λ p Λ+p
pcotδ(p) = − − + log . (26) which C2 decouples in the infrared domain, i.e. fulfills
MπC0 π π Λ−p C2 →0 for Λ→0. In fact, at small cutoffs, one gets for
this branch
Atzeroenergy,T(0,0)=2α /Mπand,thus,therunning
0
of C0 is given by 2α Λ 2α Λ 2 2 2α Λ 3
0 0 0
MC (Λ)Λ = + + +...
α 0
0 π (cid:18) π (cid:19) 3(cid:18) π (cid:19)
MΛC (Λ)=− . (27)
0 α − π
0 2Λ 1 2α Λ 2
MC (Λ)Λ3 = − 0 +...
2
In this case, the phase shift is given by 2(cid:18) π (cid:19)
(32)
1 p Λ+p
pcotδ(p) = − + log
α0 π Λ−p Thefactor2/3appearinginthesmallcutoffexpansionfor
1 1
= − +O( ), (28) C0 differsalreadyfromthecoefficientinthecaseC2 =0.
α0 Λ Eliminating C0 and C2 in favourof α0 and r0, the phase
shift becomes
which corresponds to an ERE with r =v = ···= 0 in
0 2
thelimitΛ→∞. Notethatfinitecutoffcorrectionsscale
2Λ (π−2Λα )2
as 1/Λ. This indicates a relatively slow convergence to- pcotδ(p) = − 0
πα 2Λ(π−2Λα )+α p2(r πΛ−4)
wardstheinfinitecutofflimitandhencethatfinitecutoff 0 0 0 0
effects are quantitatively important and might even be- 2Λ p Λ+p
− + log
come a parameter of the theory. Actually, one might π π Λ−p
determineΛbyfixingtheeffectiverangefromthefirstof 1 1 1
= − + r p2+O( ). (33)
Eq.(28),r0(Λ)=4/(πΛ)=2.77fm,yieldingtheaccurate α0 2 0 Λ
numerical value Λ = 90.7MeV. In this case, this par-
ticular three-momentum regularization method becomes Notethatthe finitecutoffcorrectionsare,afterfixingr ,
0
itselfamodel,sincewehavenocontrolontheremainder. again O(Λ−1). So, fixing more low energy constants in
In any case, it is straightforward to check that for any the contact theory does not necessarily imply a stronger
finite cutoff there is no off-shellness: T(k′,k)=T(p,p). short distance insensitivity, as one might have naturally
The running given by Eq. (27) must be used for any expected 12. In other words, the inclusion of a higher
cutoff Λ if we want to renormalize in the end. However, dimensional operator such as C does not improve the
2
thinking of the more generalcase where finite range cor- ultraviolet limit, at least in the polynomial representa-
rections are relevant such a running is only reliable for tiongivenbyEq.(13). InSec.IVwewillshow,however,
verysmallcutoffsΛ≪π/2α . IfweconsideralsoaC (Λ) that with just one counterterm C the inclusion of pion
0 2 0
coefficientinthepotential,thecorrespondingLSEcanbe exchange long distance contributions generates a much
solved with the ansatz faster convergencetowardsthe renormalizedlimit as an-
ticipated in Refs. [35, 44, 48] (see also Ref. [52] for a
T(k′,k)=T0(p)+T2(p)(k2+k′2)+T4(p)k2k′2, (29) quantitative estimate). In Sec. V we will also show that
whenaC countertermisaddedthisscalingbehaviouris
which yields a set of three linear equations for T (p), 2
0 notonlybrokenbut alsothe phaseshift fails to converge
T (p) andT (p). After some algebraicmanipulation, the
2 4 in the limit Λ→∞.
final resultfor the phase shift canthen be written in the
Thus, we see that one canestablisha one-to-onemap-
form
ping betweenthe countertermsC , C andthe threshold
0 2
10(C MΛ3+3)2/(Mπ) parameters α and r . Nevertheless, this is done at the
2 0 0
pcotδ(p) =
9(C22MΛ5−5C0)−15C2(C2MΛ3+6)p2 expense of operator mixing, i.e., both C0 and C2 are in-
tertwined to determine both the scattering length and
2Λ p Λ+p
− + log . (30) theeffectiverange. Inotherwordsthecutoffdependence
π π Λ−p
ofC isdifferentdependingonthepresenceofC . Aswe
0 2
Matching at low energies to the ERE, Eq. (11), we get haveseenthis is nota problemsince for smallcutoffs we
the running of C0 and C2 expect the running of C0 to be fully independent of C2
1 10(C MΛ3−3)2 2Λ
2
− = −
α 9Mπ(−C2MΛ5+5C ) π
0 2 0
1 50C2 3+C2MΛ3 2 6+C2MΛ3 2 12 Wzereoheanveergiynamndinddedriivsapteirvseisonthreerleaotfioonfsthwehdeirsepearnsyivesupbatrrtaicmtiponrovaet
r = + .
2 0 (cid:0)27π(−5C +(cid:1)C(cid:0)2Λ5M)2 (cid:1) πΛ the high energy behaviour and become more insensitive in the
0 2 ultraviolet. Asweseethisisnotthecaseinthecontactpion-less
(31) theory.
8
and hence on r . However, unlike the one counterterm subtractionschemeyieldeddifferent renormalizedampli-
0
case,whereC =0,thesolutionsofEq.(32)maybecome tudes for a truncated potential. This non-uniqueness in
2
complex when theresultduetoadifferentregularizationhappenswhen
a non-vanishing C counterterm is considered. In any
α2r πΛ3−16α2Λ2+12α πΛ−3π2 ≤0. (34) 2
0 0 0 0 case, the dimensional regularization scheme has never
For the physical 1S threshold parameters this happens beenextendedtoinclude thelongrangepartoftheTPE
0
already for Λ > Λ = 382MeV (the other two roots are potential which usually appear in the present NN con-
c
complex). Abovethiscriticalvaluethepotentialviolates text. Thus, for the momentum space cutoffs which have
self-adjointness. For r → 0 one has Λ → 16/(πr ) → been implemented in practice the short distance repre-
0 c 0
∞. Thus, the cutoff can only be fully removed with a sentation is somewhat inconsistent at least for a finite
self-adjoint short distance potential if r = 0. This is value of the cutoff Λ.
0
consistentwiththeviolationoftheWignercausalitycon- Alternatively, one may choose an energy dependent
ditionreportedin[56,57,58,59]. Notethattheviolation representation of the short distance physics as
of self-adjointness is very peculiar since once C and C
0 2
havebeeneliminatedthephase-shift(33)remainsreal13. V =C +2p2C +p4(2C +C′)+... (36)
S 0 2 4 4
One feature in the theory with two counterterms C
0
and C is that the off-shell T−matrix becomes on-shell
2 In this case the correspondence between counterterms
only in the infinite cutoff limit,
and threshold parameters α ,r , v , etc. is exactly one-
0 0 2
T(k′,k)=T(p,p)+O Λ−1 . (35) to-one, and the parameter redundancy is manifest, since
(cid:0) (cid:1) the on shell T-matrix depends only on the on-shell po-
This isunlike the theorywithone countertermC where
0 tential. Actually, under dimensional regularization the
there is no off-shellness at any cutoff. Thus, finite cutoff
representations of the potential Eq. (13) and Eq. (36)
effects are also a measure of the off-shellness in this par-
yield the same scattering amplitude. Although this on-
ticular problem. This will have important consequences
shell equivalence is certainly desirable it is also unnatu-
inSec.Vwhenattemptingtoextendthetheorywithtwo
ral, if the long distance potential is energy independent.
counterterms in the presence of the long distance pion
We willnevertheless analyzesucha situationin the next
exchange potentials since the off-shellness of the short
subsection in coordinate space.
distance contribution of the potential becomes an issue
Thepreviousdiscussionhighlightsthekindofundesir-
in the limit Λ→∞.
able but inherent off-shell ambiguities which arise when
The situation changes qualitatively when the fourth
finiterangecorrectionsareincludedintheshortdistance
ordercorrectionsdepending ontwocountertermsC and
C′ are considered. Obviously, we cannot fix bot4h C potential15. Inourviewtheseareunphysicalambiguities
an4d C′ simultaneously by fixing the slope parameter v4 whichhavenothingtodowiththe unambiguousoff-shell
4 2 dependence of the long distance potential. Of course,
of the effective range expansion, Eq. (11). Clearly, one
expects some parameter redundancy between C and C′ one way to get rid of the ambiguities is to take the limit
4 4 Λ→∞whichcorrespondstothe casewhereatrulyzero
or else an inconsistency would arise since a sixth order
range theory is approached. However, even for a finite
parameter in the effective range expansion v should be
3
cutoff there is a case where one is free from the ambi-
fixed. The situation worsens if higher orders in the mo-
guities, namely when the short distance potential is both
mentum expansion are considered due to a rapid pro-
energyandmomentumindependentfors-wavescattering
liferation of counterterms while there is only one more
threshold parameter for each additional order in the ex-
pansion. Thisrequiredparameterredundancyisactually VS(k′,k)=C0(Λ). (37)
a necessaryconditionforconsistencywhichis manifestly
fulfilled within dimensional regularizationbut not in the Thekeypointisthatweallowonlythiscountertermtobe
three-momentum cutoff method 14. Moreover,it was re- cutoffdependentandreal,asrequiredbyself-adjointness.
alized some time ago [57, 58] that the finite cutoff regu- Of course, the discussion above for the contact theory
larizationand dimensional regularizarionin the minimal suggests the benefits of using just one C counterterm
0
but does not exactly provide a proof that one must take
further counterterms such as C to zero. The extension
2
ofthisanalysistothe caseofsingularchiralpotentialsin
13 Nonetheless,off-shellunitaritydeducedfromsandwichingthere- Sec. V will yield the definite conclusion that renormaliz-
lationT−T†=−2πiT†δ(E−H0)T betweenoff-shellmomentum ability is indeed equivalent to take C2 =0.
states, is violated, since the Schwartz’s reflection principle fails
T(E+i0+) †6=T(E−i0+). Thiswouldalsohavefarreaching
consequences forthethree bodyproblem, sincethreebody uni-
ˆ ˜
tarity rests on two-body off-shell unitarity and self-adjointness
ofthreebodyforces. 15 This fact becomes more puzzling if the potential V = C2(k2+
14 This operator redundancy has also been discussed on a La- k′2−2p2)isconsidered. Itvanishesonthemassshellk=k′=p
grangeanlevel[70]basedonequations ofmotionandintheab- but nonetheless generates non trivial on shell scattering for the
senceoflongdistanceinteractions (seealso[71]). three-momentum cutoff.
9
B. Coordinate space yielding
1
The previous renormalization scheme is the momen-
=pcot(pr +δ(p)), (44)
c
tum spaceversioncorrespondingto the coordinatespace rc−α0
renormalizationadoptedin a previous workby two ofus
(MPV and ERA) [35, 44, 48]. Actually, in the pure con- and thus
tact theory, we can relate the renormalization constant
1−p(α −r )tan(pr )
with the momentum space wave function explicitly. At 0 c c
pcotδ(p) = −p
large values of the short distance cutoff rc, the zero en- p(α0−rc)+tan(prc)
ergy wave function reads, 1
= − +O(r ). (45)
c
α
r 0
c
u (r )=1− . (38)
0 c
α
0 This is in qualitative agreement with the momentum
space result when the cutoff is being removed, Eq. (28)
Thus, the following relation holds
and, as we can see, the approach to the renormalized
α0 =1−rcu′0(rc). (39) vNaoltueefiusrstihmerilathraifttshineciedethnetifibcoautniodnarrycc=onπd/i(t2ioΛn)iissemneardgey.
α −r u (r )
0 c 0 c
independenttheproblemisself-adjointandhenceorthog-
onality between different energy states is guaranteed.
Comparing with Eq. (27), we get
The theory with two counterterms where both α and
0
MΛC (Λ)=r u′0(rc) −1, (40) r0 are fixed to their experimental values opens up a new
0 c possibility, already envisaged in Ref. [52], related to the
u0(rc)(cid:12)rc=π/2Λ
(cid:12) non-uniqueness of the result both for a finite cutoff as
(cid:12)
where the momentum cutoff Λ and the short distance well as for the renormalized phase-shift. As pointed out
cutoff r are related by the equation above, this non-uniqueness was noted first in momen-
c
tum space Ref. [57, 58] when using a finite three dimen-
π
Λr = (41) sionalcutoffordimensionalregularizarion(minimalsub-
c
2 traction). Remarkably, within the boundary condition
regularization we will be able to identify both cases as
whichisnothingbutanuncertaintyprinciplerelationbe-
different short distance representations.
tweencutoffs16. Notethatforthestandardregularsolu-
Actually, when fixing α and r we are led to
tion u (r) ∼ r one has a vanishing counterterm C = 0. 0 0
0 0
Incontrast,C 6=0fortheirregularsolution. Inthecase
of the singula0r attractive potentials the solution is reg- u′p(rc) =d (r )= u′0(rc)+p2u′2(rc) +O(p4). (46)
ular but highly oscillatory and the C takes all possible u (r ) p c u (r )+p2u (r )
0 p c 0 c 2 c
values for r → 0. A more detailed discussion on these
c
issuescanbeseeninRefs.[43,47,72]. Ofcourse,strictly where u and u are defined in Sec. IIC. Note that now
0 2
speaking both Eq. (40) and Eq. (41) are based on a zero self-adjointnessisviolatedfromthe beginningdue tothe
energystate,andinthefiniteenergycasewewillassume energydependenceoftheboudarycondition. Withinthe
these relations having the limit r → 0 or Λ → ∞ in second order approximation in the energy the neglected
c
mind. termsareO(p4),soanyrepresentationcompatibletothis
Let us now deal with finite energy scattering states. ordermightinprinciplebeconsideredasequallysuitable.
Sincethereisnopotential,U (r)=0,forr >r wehave The close similarity to a Pad´e approximant suggests to
L c
the free wave solution compare the following three possibilities for illustration
purposes
sin(pr+δ(p))
u (r)= . (42)
p
sinδ(p) u′(r )+p2u′(r )
dI = 0 c 2 c ,
p u (r )+p2u (r )
Inthetheorywithonecounterterm,wefixthescattering 0 c 2 c
length α0 by using the zero energy wave function and dII = u′0(rc) +p2 u′2(rc) − u′0(rc)u2(rc) ,
matching at r=r so we get p u (r ) (cid:20)u (r ) u (r )2 (cid:21)
c 0 c 0 c 0 c
u′(r )2
uu′0((rrc)) = uu′p((rrc)) =pcot(prc+δ(p)), (43) dIpII = u0(rc)u′0(rc)+p2[u2(0rc)cu′0(rc)−u0(rc)u′2(rc)],
0 c p c (47)
and study what happens as the cutoff is removed, r →
c
16 Thisrelationwillbeshowntoholdalsointhepresenceofalocal 0. Note that all three cases possess by construction the
potential,seeAppendixA. same scattering length α0 and effective range r0 and no
10
potential for r>r . Straightforwardcalculation yields
c
40
1 1
pcotδ(p) = − + r p2+O(r2) (I)
α0 2 0 c 20
1 1
pcotδ(p) = − + r p2+O(r ) (II)
α0 2 0 c 2) 0
1 1 -V
pcotδ(p) = −α 1− 1α r p2 +O(rc) (III) Ge -20
0 2 0 0 ) (
(48) Λ -40
(
C
As we see, all three representations provide the same -60 OPE
threshold parameters, but do not yield identical renor- NLO
N2LO
malizedamplitudeforfiniteenergy. Actually,casesIand -80
N3LO
II coincide with the three-dimensional cutoff regulariza- contact
tionmethod(seeSec.IIIA),whereascaseIIIcorresponds -100
0 100 200 300 400 500 600
to dimensional regularization (MS). Moreover, the finite
cutoff corrections to the renormalized result are, gener- Λ (MeV)
ally,O(r )whiletherationalrepresentationyieldscorrec-
c
tions O(r2). These observations survive at higher orders
c
when v , v , etc. threshold parameters are further taken
2 3
into account. This indicates that not all short distance FIG. 1: LO, NLO, N2LO and N3LO running of the coun-
representations are equally “soft” in the UV-cutoff. The terterm (in GeV−2 as a function of the cutoff Λ in the 1S0
generalizationoftheseresultstothecaseofsingularTPE channel for small cutoffs Λ≤ 600MeV. The renormalization
condition is determined by fixing the scattering length to its
chiralpotentials was studied in Ref. [52] andwill be also
re-analyzedin Sec. V while discussing the consistency of experimental value α0 = −23.74fm. We use the parameters
of Ref. [24] for thepion exchangepotential V .
the standard Weinberg’s power counting. L
In any case, when finite range corrections are consid-
eredwithin the boundarycondition regularization,there
in the long range description.
aretwopossiblerenormalizedsolutionsdependingonthe
In this context there is of course the question of con-
particular parameterization of short distance physics. A
vergence or cutoff insensitivity of the phase shift, when
nicefeatureofthisregularizationisthattheycanbeiden-
Λ → ∞, provided we keep at any rate the scattering
tifiedwithsimilarresultsfoundalreadyinthemomentum
length α to its physical value by suitably adjusting the
space analysis of Ref. [57, 58] when confronting three- 0
unique counterterm C (Λ). In coordinate space this is
momentum cut-off and dimensional regularization. We 0
a rather trivial matter if the long distance potential is
note alsohere that no ambiguity ariseswhen the bound-
local [35, 44, 48], as it happens in the LO, NLO, and
ary condition is assumed to be energy independent, in
N2LO Weinberg counting. The analysis in momentum
which case self-adjointness is guaranteed.
space involves detailed large momenta behaviour of the
Lippmann-Schwinger equation and one must resort to
trial and error. Indeed, the N3LO case analyzed below
IV. RENORMALIZATION OF PION
EXCHANGES WITH ONE COUNTERTERM includes nonlocalities and it turns out to provideconver-
gent results.
For numerical calculations, we take the values for the
The study of the contact theory in Sec. III provides
c andd parametersappearingin the pionexchangepo-
suggestive arguments why it is highly desirable to carry i i
tential V used in Ref. [24], which do a good job for pe-
outaregularizationwith asinglecountertermin the1S π
0
ripheralwaves,wherere-scatteringeffectsaresuppressed
channel by adjusting it to the physical scattering length
and where one is, thus, rather insensitive to cutoff ef-
for any cutoff value. In this section we want to extend
fects. We will only consider TPE contributions to the
that study when the long distance chiral potential orga-
N3LO potential.
nized according to Weinberg power counting enters the
game and the cutoff is removed. By taking the cutoff to
infinity,weareactuallyassumingthatalldegreesoffree-
dom not included in the present calculation become in- A. Renormalized N3LO-TPE
finitelyheavy. Thiswayweexpecttolearnaboutmissing
physics in a model and regularization independent fash- To determine the running of the counterterm we start
ion. The traditional strategy of adjusting an increasing from low cutoffs Λ ≪ m since the long range part of
π
number of counterterms may obscure the analysis. In the potential is suppressed and Eq. (27) may be used.
other words, by using this minimal number of countert- Actually, the analytical result is well reproduced by the
erms, we try not to mock up what might be still missing numericalmethodusedtosolvetheLippmann-Schwinger