Pions are neither perturbative nor nonperturbative: Wilsonian renormalization group analysis of nuclear effective field theory including pions

0.37 MB·English

by  Koji Harada

KYUSHU-HET-129
Pions are neither perturbative nor nonperturbative:
Wilsonian renormalization group analysis of nuclear effective field theory
including pions
Koji Harada∗
Department of Physics, Kyushu University
Fukuoka 810-8560 Japan
Hirofumi Kubo†
Synchrotron Light Application Center, Saga University
1 Honjo, Saga 840-8502, Japan
Yuki Yamamoto‡
Department of Community Service and Science,
Tohoku University of Community Services and Science
Iimoriyama 3-5-1, Sakata 998-8580 Japan
(Dated: January 24, 2011)
1
Pionful nuclear effective field theory (NEFT) in the two-nucleon sector is examined from the
1
Wilsonian renormalization group point of view. The pion exchange is cut off at the floating cutoff
0
scale,Λ,withtheshort-distancepartbeingrepresentedascontactinteractionsinaccordancewiththe
2
generalprincipleofrenormalization. Wederivethenonperturbativerenormalizationgroupequations
n intheleadingorderofthenonrelativisticapproximationintheoperatorspaceincludinguptoO(p2)
a andfindthenontrivialfixedpointsinthe1S0 and3S1–3D1 channelswhichareidentifiedwiththose
J in thepionless NEFT. Thescaling dimensions, which determinethepower counting,of thecontact

1 interactions at the nontrivial fixed points are also identified with those in the pionless NEFT. We
2 emphasize the importance of the separation of the pion exchange into the short-distance and the
long-distance parts, since a part of the former is nonperturbativewhile thelatter is perturbative.

]
h
t
- I. INTRODUCTION Wise [10] pointed out that the Weinberg power counting
l
c schemeis inconsistent,in the sense thatthe higher order
u Nucleareffectivefieldtheory(NEFT)isthelow-energy counterterms are required to cancel the divergences at
n the given order. They (and, independently, van Kolck)
effective field theory[1]ofnucleonsbasedonsymmetries
[
of QCD, and is expected to give a model-independent proposedan alternativepowercounting scheme in which

3 description of nuclear phenomena at low energies. Since onlythenon-derivativecontactinteraction,the C0-term,
v theseminalpapersofWeinberg[2–4],alotofapplications istreatednonperturbativelysothatthepionsaretreated
6 perturbatively [11–13]. This scheme (widely known as
have been done with success for more than two decades.
1
See Refs [5–9] for reviews. It still has a wide range of KSWscheme)isfreefromtheinconsistency,butFleming,
7
2 phenomena to be explored. Mehen, and Stewart [14] showed that the KSW scheme
doesnotleadtothe convergingresultsinthechannelsin
. In spite of these successes, however, there are unset-
2 which the singular tensor part of the pion exchange con-
tled issues at the fundamental level even in the simplest
1
tributes. Beane, Bedaque, Savage, and van Kolck [15]
0 two-nucleon sector; whether the pions should be treated
proposed a remedy, known as the hybrid approach, in
1 perturbativelyornotandhow to treatthe contactinter-
which the amplitudes are expanded around the chiral
: actions. Because the understanding of the two-nucleon
v limit so that the 1/r3 part (which survives in the limit)
scatteringisessentialalsoforotherphenomena,thisissue
i
X is of central importance in NEFT. is treated nonperturbatively.
r The original power counting due to Weinberg [2–4] The treatment of the tensor part of the one-pion ex-
a for the “effective potential” is nothing but the naive di- change (OPE) is the source of controversy. See also
mensional analysis. The “effective potential” is plugged Refs. [16–21] for more details.
into the Lippmann-Schwinger equation and the scatter- Recently, Beane, Kaplan, and Vuorinen (BKV) [22]
ing amplitude is calculated, so that the pion exchange is considered a Pauli-Villars type regularization for the
iteratedinfinitetimes. InthisWeinbergschemethepions OPE in order to separate the short-distance part of
arethus treatednonperturbatively. Kaplan,Savage,and the tensor interaction from the long-distance part. The
short-distancepartisrepresentedbycontactinteractions.
They employed the power divergence subtraction (PDS)
renormalizationand obtained the convergentresults.
∗ [email protected] Note thatnotallthe 1/r3 partofthe OPEissingular.
† [email protected] It is the short-distance part that is singular and spoils
‡ [email protected] the convergenceofthe amplitude asfound inRef [14]. It
2
means that all the 1/r3 part of the OPE does not need Note that, although the contact interactions are at-
to be iterated. tributed to the effects of the heavy particles in the usual
Theideaofseparatingthesingularshort-distancepart EFT lore, this is not precise. They arise also from the
ofthe tensorinteractionfromthe long-distancepartand short-distance physics of light particles, such as pion
representing the former by contact interactions (BKV exchanges. Note also that the representation of short-
prescription)isessentiallytheWilsonianrenormalization distance physics by local interactions is a general treat-
group (RG) idea of effective interactions [23, 24]. The mentandhasnothingtodowithhowsingulartheS-OPE
separation scale in Ref. [22] may be viewed as an analog is.
of the floating cutoff in the Wilsonian RG. We find nontrivial fixed points in the 1S and 3S –
0 1
The Wilsonian RG is a useful tool to investigate the 3D channels,whichareresponsibleforthelargescatter-
1
effects of the short-distance physics on the long-distance ing lengths. Importantly these fixedpoints areidentified
physics. When the quantum fluctuations with momenta withthose foundinthe WilsonianRGanalysisinthe pi-
higherthanthefloatingcutoffΛareintegratedout,their onless NEFT. Thus the scaling dimensions at the fixed
effects aresimulatedby aseriesoflocaloperators,which points are also the same. There is one relevant operator
serve as low-momentum effective interactions. The cou- in each channel. That is, the pion interactions do not
pling constants thus depend on the cutoff Λ, while the alter the scaling dimensions.
physicalquantities suchasscattering amplitudes do not. We emphasize that the question of whether the pions
TheWilsonianRGhasbeenappliedtoNEFTinorder are perturbative or nonperturbative is not well posed.
to examine the power counting issue [19, 25–30]. (See Since the S-OPE is represented as contact interactions
also Ref. [31–33] for other use of the Wilsonian RG in in the Wilsonian analysis, only the long-distance part
NEFT.) The existence of the nontrivial fixed points of of the OPE (L-OPE) is the proper interactions due to
the RG equations (RGEs) in the S-waves explains un- OPE. On the other hand, the S-OPE cannot be distin-
naturally large scattering lengths. The power counting guished from other contributions, such as heavier meson
can be determined by the scaling dimensions of opera- (ρ, ω, etc.) exchanges. (Because of the cutoff, we do
tors around the fixed points. In particular, it is relevant not have enough resolution.) The RGEs tell us that, at
operators that should be resummed to all orders. the nontrivial fixed point, the L-OPE should be treated
Birse [19] examines the pionful NEFT by using the perturbatively, while there is a relevant operator (a part
so-called “distorted-wave RG [26]” and claims that the of which is the S-OPE) that should be resummed to all
scaling dimensions are shifted from those of the pionless orders, i.e., nonperturbatively.
theory due to the singularity of the tensor force in the Theexistenceoftherelevantoperatoratthenontrivial
spin-triplet channel. Also, his results imply that the ef- fixed point sharpens the distinction between the S-OPE
fectsofpionsdonotdecoupleevenatverylowmomenta. and the L-OPE. If there were no relevant operator, the
It sounds strange however that pions do not decouple separation would not make much difference.
at the momenta where the pionless NEFT is valid. The OurWilsonianRGpermitsthesmoothtransitionfrom
effectsshouldbeeventuallyrepresentedbycontactinter- the pionful NEFT to the pionless NEFT. The nontrivial
actionsatverylowmomenta. Oneshouldexpectthatthe fixed points do not change nor the scaling dimensions.
transition from the pionful NEFT to the pionless NEFT The L-OPE transmutes into contact interactions which
is smooth. A formulation of the Wilsonian RG which represent the S-OPE as the cutoff is lowered. At the
permits the smooth transition is desired. value of Λ lower than the pion mass, the most of the
Inthispaper,weperformtheWilsonianRGanalysisof L-OPE has been changed into contact interactions, thus
the pionful NEFT in the two-nucleon sector, and exam- the system is well described by the pionless NEFT.
ine the effects of pions. In the Wilsonian RG approach, The structure of the paper is the following. In Sec II
thereisasinglescaleΛwhichseparatesthelong-distance werecapitulatethemainpointsofthepreviouspapersin
physics from the short-distance physics so that one does order to introduce the notations and the main concepts
not need to implement an additional regularization for in our analysis. In Sec. III we discuss the nonrelativistic
the short-distance part of the OPE, as does in the BKV approximation. Starting with the nonrelativistic nucle-
prescription. ons and the relativistic pions, we estimate the order of
The key idea here is the separation of the short- magnitude of a various kind of diagrams contributing to
distance part from the long-distance part. NEFT, as a the RGEs and determine the leading terms. In Sec. IV
low-energyeffective theory, has a finite range of applica- we present the RGEs and examine the structure of the
bility, specified by the physical cutoff Λ . It is a general flows in the 1S and 3S –3D channels. The nontrivial
0 0 1 1
principle of EFT that the physics with the momentum fixed points are found to be identified with those of the
scale larger than Λ is represented as local operators. pionless theory. Sec V is devoted to the summary and
0
Theshort-distancepartoftheOPE(S-OPE)shouldalso the comments on the related works. In Appendix A the
be represented as local operators (contact interactions). RGEsforthecaseofΛ<m arepresentedandcompared
π
(IntheWilsonianRGanalysis,thefloatingcutoffΛplays withthepionlesscase. InAppendixBwediscussthesim-
theroleofΛ ,afterintegratingthemodeswithmomenta ilarity andthe difference betweenthe pionful NEFT and
0
between Λ and Λ .) QED.
0
3
II. WILSONIAN RG ANALYSIS OF PIONLESS nucleon system, the spin-singlet channel is considered
NEFT IN TWO-NUCLEON SECTOR to be in the weak-coupling phase, while the spin-triplet
channelis consideredto be inthe strong-couplingphase,
In the previous papers [28–30], we have explained the because of the (non)existence of a bound state in these
basic concepts of the Wilsonian RG and its relevance to channels.
thepowercountingintheNEFT.Here,webrieflyrecapit- To summarize: the two-nucleon system with large S-
ulate it, give some remarks, and introduce the notations wave scattering lengths is governed by the existence of
used in later sections. nontrivial fixed points, and the Wilsonian RG is a sys-
tematic tool to study the scaling dimensions on which
thepowercountingshouldbe based,aroundthenontriv-
A. What is the use of the Wilsonian RG in NEFT? ial fixed points.
The most basic idea behind the power counting is the
B. Remarks on Wilsonian RGEs with Galilean
order of magnitude estimate based on dimensional anal-
invariance
ysis. In an EFT with the physical cutoff Λ , the dimen-
0
sionalanalysisisusuallybasedonthisscale. Attheclas-
There are several formulations for the Wilsonian RG
sicallevel,the(canonical)massdimensionofanoperator
analysis [34–39], which are however essentailly equiva-
isdeterminedwithrespecttothekineticterm. Forexam-
ple, a Dirac field ψ with a kinetic term =ψ¯i∂/ψ has lent. ThemostpopularoneisthefunctionalRGmethod.
Lkin (SeeRefs.[40,41]forreviews.) Inthisformulation,acut-
mass dimension three halves, [ψ]=3/2. The dimensions
offfunctionisintroducedforeachpropagatortosuppress
of other operators are determined accordingly. The op-
erator (ψ¯ψ)2 has dimension six, so that it enters in the the low-frequency fluctuations. The effective averaged
Lagrangian as (c/Λ2)(ψ¯ψ)2, where c is a dimensionless actionΓΛ[Φ],whichinterpolatesthe bareclassicalaction
0 S[Φ] (Λ = Λ ) and the effective action Γ[Φ] (Λ = 0),
constant. The coupling constant c/Λ2 associated with 0
the operator (ψ¯ψ)2 has dimension 2,0which counts the depends on the floating cutoff scale, Λ, as
−
power of Λ0. dΓΛ 1 dRΛ −1
Quantumfluctuationsmay,ingeneral,changetheclas- dΛ = 2Tr dΛ Γ(2)+RΛ , (2.1)
sical dimensional analysis. The quantum counterpart of (cid:20) (cid:21)
(cid:0) (cid:1)
the(canonical)dimensioniscalledthescalingdimension, where Φ is the classical field, Γ stands for the sec-
(2)
whichcanbeobtainedbytheRGanalysis. WilsonianRG ond derivative of the averagedaction Γ with respect to
Λ
is a nonperturbative tool to handle the quantum fluctu- Φ, and R is the cutoff function which suppresses the
Λ
ations. fluctuations with p . Λ. Note that, although it is an
An operator whose coupling has a negative (scaling) “one-loop” equation, it contains all the nonperturbative
dimension is called irrelevant because it becomes less information.
important at lower energies. An operator whose cou- A straightforward application of this formulation to a
pling hasa positive(scaling)dimensionis calledrelevant nonrelativisticsystemhoweverencountersdifficulties. In
because it becomes more important at lower energies. the usual formulation for a relativistic system, one con-
An operator whose coupling is dimensionless is called siders the theory in Euclidean space. The cutoff is im-
marginal. The scaling dimension of an operator is the posed on the magnitude of four-momentum of the prop-
measureofhow importantitis. Itis thereforenaturalto agator. On the other hand, in a nonrelativistic system,
consider the power counting on the basis of the scaling spaceandtimeshouldbetreateddifferently,andthusthe
dimensions. Euclideanformulationcannotbeused. Onemightrather
IntheS-wavescatteringoftwonucleons,thescattering liketoconsideracutoffimposedonthethree-momentum
lengths are known to be much larger than the “natural” in the propagator. But such a cutoff necessarily breaks
size, 1/Λ . (In the case of the pionless NEFT, the phys- Galilean invariance of the nonrelativistic system. There
0
ical cutoff is of order of the pion mass, Λ (m ).) is no obvious way to impose a Galilean invariant cutoff
0 π
∼ O
From the RG point of view, the “fine-tuning” is related at the averaged action level. This is a general feature
to the existence ofa nontrivialfixedpoint(andacritical independent of the choice of the cutoff function. Fur-
surface) of the RG flow. thermore, if the cutoff function is not smooth enough,
Around the nontrivial fixed points the scaling dimen- non-analytic terms in momenta arise1. See Ref. [27] for
sions are drastically different from the canonical dimen- an example in a similar context.
sions. It has been shown [25, 29, 30] that the coupling
which corresponds to the scattering length becomes rel-
evant, although it is irrelevant at the classical level.
1 Ifoneconsidersonlyafewleadingordertermsinthederivative
There are values of coupling constants with which the
expansion, typically in the local potential approximation, non-
scattering length is infinite. This set of coupling con- smooth cutoff does notcause the problem. Itisthe reasonwhy
stants forms a critical surface. It separates the weak- the problem of non-analyticity is not revealed in many applica-
coupling and the strong-coupling phases. For the two- tions.
4
In a system of two nonrelativistic particles, there is a nonrelativistic nucleons, which interact with themselves
simple and physically sensible way out of this problem: only through contact interactions. In the two-nucleon
it is to impose a cutoff on the relative three-momentum sector,they arefour-nucleonoperatorswithanarbitrary
of the two particles, which is Galilean invariant. In ad- numberofderivatives. Anoperatorwithmorederivatives
dition, the results are very insensitive to the choice of has higher canonical dimensions than the ones with less
the cutoff function. See Appendices of Refs [29, 30]. In derivatives.
particular, the results with a sharp cutoff are the same
as those with a smooth one. It is a technical advantage
that a sharp cutoff can be used because it simplifies the
calculations considerably. Since there are infinitely many operators involved in
the flowequation,oneneeds tointroduceatruncationof
the space of operators in the averagedaction in order to
C. Pionless NEFT up to O(p2) in the 1S0 and solveit. Weretainonlytheoperatorswithderivativesup
3S1–3D1 channels to a certain order. We simply count the number of spa-
tial derivatives ( p) and a time derivative is counted
In Ref. [29], we consider the pionless NEFT without as two spatial d∇eri∼vatives (∂ p2). We consider the
t
∼
isospin breaking. The relevant degrees of freedom are following ansatz for the averagedaction up to (p2),
O
Γ(π/) = d4x N† i∂ + ∇2 N
Λ t 2M
Z (cid:20) (cid:18) (cid:19)
C(S) (S)+C(S) (S)+2B(S) (SB) , (1S channel)
− 0 O0 2 O2 O2 0
 (cid:21) (2.2)
 −C0(T)O0(T)+C2(T)O2(T)+2B(T)O2(TB)+C2(SD)O2(SD) , (3S1–3D1 channel)
(cid:21)
where the operators in the 1S are given by
0
†
(S) = NTP(S)N NTP(S)N , (2.3a)
O0 a a
(cid:16) (cid:17) †(cid:16) (cid:17)
(S) = NTP(S)N NTP(S)←→2N +h.c. , (2.3b)
O2 a a ∇
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
2 †
(SB) = NTP(S) i∂ + ∇ N NTP(S)N +h.c. , (2.3c)
O2 "(cid:26) a (cid:18) t 2M(cid:19) (cid:27) (cid:16) a (cid:17) #
and in the 3S –3D channel,
1 1
†
(T) = NTP(T)N NTP(T)N , (2.4a)
O0 i i
(cid:16) (cid:17) †(cid:16) (cid:17)
(T) = NTP(T)N NTP(T)←→2N +h.c. , (2.4b)
O2 i i ∇
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
† 1
(SD) = NTP(T)N NT ←→ ←→ δ ←→2 P(T)N +h.c. , (2.4c)
O2 i ∇i∇j − 3 ij∇ j
(cid:20)(cid:16) (cid:17) (cid:26) (cid:18) (cid:19) (cid:27) (cid:21)
2 †
(TB) = NTP(T) i∂ + ∇ N NTP(T)N +h.c. , (2.4d)
O2 "(cid:26) i (cid:18) t 2M(cid:19) (cid:27) (cid:16) i (cid:17) #
(2.4e)
wherewehaveintroducedthenotation←→2 ←−2+−→2 triplet) channel respectively. The nucleon field N(x)
2←− −→ and the projection operators[14∇], ≡∇ ∇ − with mass M is an isospin doublet nonrelativistic two-
∇· ∇ component spinor. Pauli matrices σi and τa act on spin
1 1
P(S) σ2τ2τa, P(T) σ2σkτ2, (2.5) indices and isospin indices respectively. The two chan-
a ≡ √8 k ≡ √8 nels are completely decoupled, and thus we can consider
each channel separately.
for the 1S (spin singlet) channel and the 3S (spin
0 1
5
Note that the possible forms of the operators are re- for the spin-triplet channel, with
stricted by Galilean invariance. Note also that we have
p2
included “redundant operators,” (SB) and (TB), be- S =p0 i , p =p p , r =(p p )2. (2.12)
O2 O2 i i − 2M ij i− j ij i− j
cause they are necessary in a consistent Wilsonian RG
analysis [28]. p and p are the incoming momenta of the nucleons to
1 2
Weintroducethefollowingdimensionlesscouplingcon- thevertex,andp andp aretheoutgoingmomentafrom
3 4
stants, the vertex. The notation stands for the operation of
C
|
taking the coefficient of F (p ,p ) in the sum.
MΛ MΛ3 Λ3 C i f
x C(S), y 4C(S), z B(S), (2.6) ByusingtheformulaEq.(2.8)theRGEsareobtained.
≡ 2π2 0 ≡ 2π2 2 ≡ 2π2 The explicit expressions are not presented here, but can
bereadeasilyfromtheRGEs(Eqs.(4.6)andEqs.(4.12))
for the spin-singlet channel and
for the pionful theory discussed in Sec. IV.
x′ MΛC(T), y′ MΛ34C(T), z′ Λ3 B(T), Thereisanontrivialfixedpointineachchannel,which
≡ 2π2 0 ≡ 2π2 2 ≡ 2π2 is relevant to the physical two-nucleon system:
MΛ34
w′ C(SD), (2.7) 1 1
≡ 2π2 3 2 (x⋆,y⋆,z⋆)= 1, , , (2.13)
− −2 2
forthespin-tripletchannel. Withthesharpcutoffonthe (cid:18) (cid:19)
relative momenta, the flow equations that determine the in the spin-singlet channel, and
dependence on t = ln(Λ /Λ) of the coupling constants
0 1 1
can be written as [30] x′⋆,y′⋆,z′⋆,w′⋆ = 1, , ,0 , (2.14)
− −2 2
(cid:18) (cid:19)
(cid:0) (cid:1)
dxC MΛFA(pi,Λ)FB(Λ,pf) in the spin-triplet channel. Atthe nontrivialfixed point,
+d x = x x ,
dt C C A B 2π2 1 A˜(p ) (cid:12) the operators get large anomalous dimensions and there
AX,B − i (cid:12)(cid:12)C isoneoperatorthatbecomesrelevant withthescalingdi-
(cid:12)(2.8)
(cid:12) mensionofthecouplingconstantbeingoneineachchan-
where x stands for one of the dimensionless coup(cid:12)ling
C nel.
constants, and d is the power of Λ in the definition of
C
thedimensionlesscouplingconstant. Inthefollowing,we
call dC thecanonicaldimension2ofthecoupling. A˜(P) III. NONRELATIVISTIC APPROXIMATION, IR
−
is defined as ENHANCEMENT AND THE LEADING ORDER
IN Λ/M
P2
A(P)=P0 , A˜(P)=MA(P)/Λ2, (2.9)
− 4M
In this section, we consider the inclusion of pions as
whereP =(P0,P)isthetotalmomentumofthesystem. dynamical degrees of freedom. It extends the range of
F (p ,p )isthe momentum-dependentfactorassociated applicability of NEFT to higher momenta beyond the
A i f
with the coupling x : pionmass scale. The contactinteractions in the pionless
A
theory resolve into the effects by pion propagation and
2π2 2π2
the rest. The physical cutoff Λ is now larger than m ,
F = − , F = − (r +r ), 0 π
x MΛ y 4MΛ3 12 34 and we suppose that Λ is of order 400 MeV.
0
2π2 4
F = S , (2.10)
z Λ3 i
i=1 A. Chiral symmetry and nonrelativistic nucleons
X
for the spin-singlet channel, and
The most important feature of the pionful theory is
2π2 2π2 chiralsymmetry,SU(2) SU(2) spontaneouslybroken
Fx′ = −MΛ , Fy′ = 4−MΛ3 (r12+r34), to SU(2)V. It is convLen×ient toRintroduce the field Σ,
2π2 4 which transforms linearly as
Fz′ = Λ3 Si, Σ(x) LΣ(x)R†, (3.1)
i=1 →
X
Fij = 3 −2π2 pi pj +pi pj 1δij(r +r ) , where L and R are the elements of SU(2)L and SU(2)R
w′ 4 MΛ3 12 12 34 34− 3 12 34 respectively. Thepionfieldπa(x)maybedefinedthrough
(cid:18) (cid:19)(cid:20) (cid:21)
(2.11)
Σ(x)=exp(iπa(x)τa/f), (3.2)
where f is the pion decay constant in the chiral limit.
The nucleon field transforms as
2 This definition of the canonical dimension reflects the nonrela-
tivisticscalingproperty. SeeRef.[42]. N(x) U(x)N(x), (3.3)
→
6
where U(x) is a function of L, R, and Σ(x) and defined notseemtoexistamanifestlychiralinvariantcutofffunc-
through tion which controls all the fluctuations, because of the
nonlinearity of the transformation [43]. Furthermore, it
ξ(x) Lξ(x)U(x)† =U(x)ξ(x)R†, (3.4) is known that perturbation theory generates apparently
→
noninvariantterms(ANTs)evenwiththelatticeandthe
where ξ2(x)=Σ(x), i.e., ξ(x)=exp(iπa(x)τa/2f).
dimensional regularization, which preserve chiral sym-
The chiralinvariant Lagrangianfor the nonrelativistic
metry [44, 45]. ANTs are also expected to appear in the
nucleon interacting with pions is given as
WilsonianRGanalysis,butitisnotobvioushowtotreat
(σ D)2 them.
=N† iD + · N +g N†σ AN In addition, the inclusion ofpions makes the notionof
NR 0 A
L " 2M # · relativemomentumobscure. Furthermore,becausepions
C(c) (c)+C(c) (c)+2B(c) (cB) are relativistic, Galilean invariance does not make good
− 0 O0 2 O2 O2 sense as a constraint.
m2
D(c) πTr(Σ+Σ†) (c) Fortunately,however,itturns outthatthese problems
− 2 2 O0 donotinterferewiththeleadingordercalculationsinthe
+ , (3.5) nonrelativisticapproximation. Theargumentisbasedon
···
the order of magnitude estimate of the contribution of
where the chiral covariantderivative D is defined as
µ each diagram to the Wilsonian RGEs.
We are going to obtain the RGEs for the two-nucleon
D N =(∂ +V )N, (3.6)
µ µ µ
sector in the next-to-leading order in momentum expan-
and V and A are defined as sion(to (p2))oftheaveragedactioninthenonrelativis-
µ µ O
tic formulation. Let us first remind the following things:
1
V ξ†∂ ξ+ξ∂ ξ† , (3.7)
µ ≡ 2 µ µ The contributions to the Wilsonian RGEs only
•
i (cid:0) (cid:1) come from one-loop diagrams.
A ξ†∂ ξ ξ∂ ξ† . (3.8)
µ µ µ
≡ 2 −
Because of the nonrelativistic feature, diagrams
(cid:0) (cid:1) •
The superscript (c) denotes the channel to be speci- with anti-nucleon lines are absent. Thus, the di-
fied, and the summation is also implied, i.e., the term agrams are divided into sectors, each of them is
C(c) (c) for the spin-triplet channel contains C(T) (T) specified by the nucleon number. The n-nucleon
an2dOC2(SD) (SD). Note that the derivatives in th2e fOou2r- sector has n nucleons at each time slice.
2 O2
nucleon operators should be replaced by the covariant
The contributionstothe two-nucleonsectordonot
derivatives,though we do not explicitly showthem here. •
come from n-nucleon operators with n 6.
The ellipsis in Eq. (3.5) denotes other terms, e.g., the ≥
higher order operators, including six-nucleon operators, Our primary concern is the renormalization of the
•
four-nucleon operators with more than two derivatives, four-nucleon operators with no pion emission and
etc. It also contains the counterterms of the form N†N absorption. Chiral symmetry constrains the renor-
and N† 2N, which play an important role in the renor- malizationofthe operatorsinwhichthe pionfields
∇
malization of the nucleon mass. Chiral symmetry does appear through covariant derivatives.
not prevent the appearance of the operators that con-
tain more than one A, such as N†A2N. But it is not Supplementarily, one needs to consider the self-
•
generated by nonrelativistic reduction from the simple energy diagram of the nucleon and the diagrams
(chiral invariant) Dirac action. It implies that the oper- for the nucleon-pion vertex that contribute to the
ators have smaller coefficients than 1/M. Furthermore, RGEs for the four-nucleon operators.
wewillseethatiteitherdoesnotcontributetotheRGEs
Note also that there is no contributions to the pion
for the four-nucleon operators, or gives only suppressed
self-energy to this order.
contributions to the order we are working.
The Lagrangianfor the pions is given by
f2 1. Four-nucleon operators
= Tr ∂ Σ†∂µΣ +m2Tr Σ†+Σ + . (3.9)
Lπ 4 µ π ···
Accordingtotheabove-mentionedremarks,weneedto
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
considerthediagramsgiveninFig.1forthefour-nucleon
B. Order of magnitude estimation of the operators.
contributions to the Wilsonian RGEs In order to obtain the RGEs, we need to evaluate the
contributions from the so-called “shell-mode,” the loop-
In order to perform the Wilsonian RG analysis, one contributions with the magnitude of the loop (relative)
usually needs to use a cutoff function that preserves all three-momentum k = k is between Λ dΛ and Λ. The
| | −
the symmetries of the theory. Unfortunately, there does argument is similar to the “power counting” with the
7
that the nucleon pole gives a dominant contributions,
d3k k2
−
Zshell(2π)3[−p0+(p+k)2/2M]2−ωk2
1
×(p0+p′0) (p+k)2/2M (p′ k)2/2M
(a) (b) (c) (d) − − −
d3k (k (p p′)/2)2
− − −
∼Zshell (2π)3[−p0+(k+P/2)2/2M]2−ωk2−(p−p′)/2
1
(3.11)
×E P2/4M k2/M
− −
where ω k+m2, and E p0+p′0 andP p+p′
k ≡ π ≡ ≡
are the total energy and the total three-momentum of
p
(e) (f) (g) (h) the two nucleons, respectively. In going to the second
line, we have made a shift of the integration momentum
so that k is now the relative momentum. Since M
≫
k =Λ P , p0, and E P2/4M Λ2/M, one may
| | ≫| | − ≪
estimate the loop integral as
1
− MdΛ. (3.12)
∼ 2π2
(i) (j) (k) (l) The pion pole gives ΛdΛ/4π2, which is smaller than
∼
the dominant contribution by a factor of Λ/M.
FIG. 1. Contributions to the four-nucleon operators. The On the other hand, the diagram (e) in Fig. 1 contains
four-nucleon vertices represent contact interactions collec-
tively. The dotted lines represent the pion propagators. For d4k ik2 i
thediagrams(e),(f),(g),(h),and(l),therearealsomirrored (2π)4k2 m2 +iǫ(p0 k0) (p k)2/2M +iǫ
diagrams with theleft and theright interchanged. Zshell − π − − −
i
, (3.13)
×(p′0 k0) (p′ k)2/2M +iǫ
− − −
which is similar to Eq. (3.10), but there is a crucial
scale Q found in the literature, but here, (i) we do not difference. No nucleon pole appears in the lower half
considertheamplitudebutthecontributiontotheRGEs, plane. Thus the contribution can be evaluated by the
and (ii) the magnitude of the momentum in the loop is residue of the pion pole in the lowerhalf plane andgives
actually Λ, while in the “Q-counting”it is assumed that ΛdΛ/4π2. This is suppressed by a factor of Λ/M
∼ −
the dominant contributions come from the loop momen- compared with the diagram (b) in Fig. 1.
tum of order Q. It is a general feature that the two-nucleon reducible
(2NR)diagramacquirestheso-called“IRenhancement,”
To see how the dominant contributions to the RGEs
first noted by Weinberg [3], where the residue of the nu-
emerge in certain diagrams, let us consider the diagram
cleon pole gets an enhancement factor M/Λ. Note that
(b) in Fig. 1 as an illustrative example. At the moment,
theIRenhancementisaconsequenceofthe nonrelativis-
we assume that the floating cutoff Λ is larger than m
π tic kinematics. In addition, with the residue of the nu-
for simplicity. The diagram contains the loop integral
cleon pole, the pion propagator becomes
i
. (3.14)
[ p0+(p+k)2/2M]2 ω2
d4k ik2 i − − k
(2π)4k2 m2 +iǫ(p0+k0) (p+k)2/2M +iǫ Noting p0 p2/2M Λ2/M Λ, it is approximated
Zshell − π − − ≪ ≪
i by
, (3.10)
×(p′0 k0) (p′ k)2/2M +iǫ
− − − i
− (3.15)
∼ k2+m2
π
where denotes the shell mode integral with the for k Λ. That is, the effects of the pion can be repre-
shell ∼
restriction on the magnitude of the relative three- sented as the instantaneous potential.
momenRtum. There are four poles in the complex k0 Although we have shown that the mechanism of “IR
plane, one nucleon and one pion poles in both the upper enhancement”worksinthecaseofm <Λ,itisactually
π
and the lower half planes. One can evaluate the integral independent of this assumption, and it also works in the
bytheresiduesofthepolesineitherhalfplane,andfinds case of Λ<m .
π
8
(Thecontributionsfromthenucleonwavefunctionrenor-
malization is not considered here.) Note that the last
term has an explicit Λ-dependence. In Eq. (4.7) the last
term has been neglected because Λ/M is much smaller
than one.
Theself-energydiagramΣ(p)showninFig3givescon-
(a) (b) tributions
dΣ(p)
Λ =A+Bp0+Cp2+ (Λ2/M2), (3.18)
dΛ O
whereA, B,andC arethe constantsthatdependonthe
couplings and the cutoff Λ. The constants A and C are
canceledbythe counterterms,leavingthepole ofthe nu-
cleon propagator intact. After doing so, the propagator
with this shell-mode contribution becomes
(c) (d)
−1
dΛ i
1 B , (3.19)
FIG. 2. Contributions to thegA-vertex. (cid:18) − Λ (cid:19) p0− 2pM2 +iǫ
so that the contributions to the wavefunction renormal-
izationconstantforthenucleonfield,Z ,canbewritten
N
as
dZ
N
=B. (3.20)
dt
FIG. 3. Contributions tothe self-energy of nucleon.
The order of magnitude estimate of B gives
Λ
2. Vertex corrections and the nucleon self-energy B γ, (3.21)
∼ M
(cid:18) (cid:19)
We also need to consider the renormalization of the so that the inclusion of the effect of wavefunction renor-
g -term. The relevant diagrams are depicted in Fig. 2. malizationdoes not alter the results of the leading order
A
Therearecontributionstotheself-energyofthenucleon, calculations.
shown in Fig. 3. These are potentially important to the
renormalizationof the four-nucleon operators.
C. Averaged action with L-OPE
The examples in the previous subsection imply that
the nucleonself-energydiagramandthe contributionsto
thenucleon-pionvertexdonothaveIRenhancement. We We have seen that the leading contributions to the
willseeshortlythatthesecontributionscanbe neglected RGEsconsistoftheone-loop2NRdiagramswithcontact
in our leading order calculations. The key point is that interactions and/or the instantaneous pion exchanges.
the appropriate dimensionless coupling constant for the Actuallythesecontributionscanbegeneratedbyamuch
nucleon-pion coupling is, as we will explain in the next simpleractionthanthatinEq.(3.5). Thuswestartwith
section (Sec. IVA), given by γ, the following ansatz for the averagedaction ΓΛ,
MΛ gA 2 ΓΛ =Γ(Λπ/)+ d4x −D2(c)m2πO0(c)
γ ≡ 2π2 (cid:18)2f(cid:19) . (3.16) + gA2 Zdt nd3xd3y (S)(xo,y) 2
4f2 O ∇x
The contributions from the nucleon-pion vertex to γ Z Z (cid:20)
should be written in terms of γ. + (T)(x,y) 2
There are four diagrams contributing to the nucleon- Oii ∇x
1
pionvertexshowninSec.IIIB. Thetadpolediagram(b) 6 (T)(x,y) ∂x∂x δ 2 Y(x y ),
− Oij i j − 3 ij∇x | − |
can be absorbed in the redefinition of the coupling con- (cid:18) (cid:19)(cid:21)
stant g . Each of the diagrams (c) and (d) has an extra (3.22)
A
1/M factor because the two-pion vertex comes from the where we have introduced
kinetic term D2/2M so that they are suppressed com-
†
pared to the diagram (a). The diagram (a) gives rise to (S)(x,y)= NT(x)P(S)N(y) NT(y)P(S)N(x) ,
O a a
the second term of the right-hand side of the following
(cid:16) (cid:17)(cid:16) ((cid:17)3.23)
RGE expressed in terms of γ,
†
(T)(x,y)= NT(x)P(T)N(y) NT(y)P(T)N(x) ,
dγ Λ Oij i j
= γ 3 γ2. (3.17)
dt − − M (cid:16) (cid:17)(cid:16) (3(cid:17).24)
(cid:18) (cid:19)
9
and F (p ,Λ)F (Λ,p ) in Eq. (2.8) is replaced by
A i B f
F (p ,Λ)F (Λ,p ) , where is defined as
1 e−mπr h A i B f i h···i
Y(r)= . (3.25)
4π r 1
= dΩ ( ), (4.5)
h···i 4π kˆ ···
Note that all the derivatives are now the usual ones, not Z
the covariant derivatives. It means that chiral symme-
whereΩ standsfortheangularvariablesofk. See
try is broken explicitly, but the breaking is of higher or- kˆ
the original derivation of the formula for the de-
der in Λ/M, as is seen from the derivation. Note also
tails [30].
that the operator corresponds to D(c) is the same as
2
that to C(c), but the former is a part of the operator In the following, we assume that mπ <Λ and expand
0
that emits/absorbs pions and is of higher order in the in powers of mπ/Λ. The case of Λ < mπ is discussed in
p-expansion than the latter. Appendix A.
The effects of pions are represented as one-pion ex-
change interactions. It is the L-OPE because the av-
eraged action is defined with the cutoff Λ, so that the A. Spin-singlet channel
S-OPE (with the momenta larger than Λ) is included in
the contact interactions. In the spin-singlet channel, we have the following
The last term in Eq. (3.22) represents the tensor force RGEs:
of L-OPE in the spin-triplet channel.
dx
= x x2+2xy+y2+2xz+2yz+z2
dt − −" #
IV. WILSONIAN RGES FOR PIONFUL NEFT
O(p2) IN THE 1S0 AND 3S1-3D1 CHANNELS −2(x+y+z)γ−γ2, (4.6a)
dy 1 3 1
= 3y x2+2xy+ y2+yz z2
We are now ready to calculate the RGEs for the two- dt − −"2 2 − 2 #
nucleon system in the S-waves. The formula Eq. (2.8)
1
can be used with slight modification. (x+2y)γ γ2, (4.6b)
− − 2
We introduce the dimensionlesscoupling constants
dz 1 1 3
• γ definedinEq.(3.16)forL-OPE,anduandu′ for = 3z+ x2+xy+ y2 xz yz z2
D(c), defined by dt − "2 2 − − − 2 #
2 1
+(x+y z)γ+ γ2, (4.6c)
MΛ3 MΛ3 − 2
u D(S), u′ D(T). (4.1)
≡ 2π2 2 ≡ 2π2 2 du = 3u 2(x+y+z)(u γ) 2uγ+2γ2,(4.6d)
dt − − − −
Note that g and f appear in Eq. (3.22) only
A
through the combination of (g /f)2. and for γ,
A
dγ
The momentum-dependent factors associated with
= γ. (4.7)
• γ, u and u′ are given by dt −
The first lines of Eqs (4.6a) – (4.6c) are the same as
2π2 1 r r
F (p ,p )= − 13 + 14 , those in the pionless calculations obtained in Ref. [29].
γS f i MΛ 2 r +m2 r +m2
(cid:18) (cid:19) (cid:20) 13 π 14 π(cid:21) The terms in the second lines express how the L-OPE
(4.2) is rearrangedinto the S-OPE when the floating cutoff is
6π2 1 δijr 2pi pj lowered.
FγijT(pf,pi)= −MΛ 2 r13−+m123 13 We emphasize the choice made here of the dimension-
(cid:18) (cid:19) (cid:20) 13 π lesscouplingconstantγ. Thereareseveralwaystomake
+ δijr14−2pi14pj14 , (4.3) a dimensionless quantity from the combination (gA/f)2,
r +m2 M, andΛ. Our choice is the one for which the RGEs for
14 π (cid:21)
2π2m2 the coupling constantsof the contactinteractionsdo not
Fu =Fu′ = −MΛ3π. (4.4) haveexplicitΛ-dependence. IftheexplicitΛ-dependence
were present in the RGEs, the iterative property (self-
similarity) would be lost and the concept of fixed points
Since the F and Fij have a bit more compli- wouldbecome obscure. Since the unnaturally largescat-
• γS γT
cated momentum dependence than those in the teringlengths inthe S-wavescatteringarebelievedtobe
pionless theory, the formula contains nontrivial related to the nontrivial fixed points, one needs to use
integrations over angular variables. The part such dimensionless variables that allow fixed points.
10
The nontrivial fixed point of the RGEs (4.6) and vectors. It is easily seen that the eigenvalue 1 is triply
−
Eq.(4.7)relevanttotherealtwo-nucleonsystemisfound degenerate. Twoofthe(nonzero)eigenvectorsareu and
2
to be u , but the third one does not exist. This is not a math-
4
ematical inconsistency, however. Although we do not
1 1
(x⋆,y⋆,z⋆,u⋆,γ⋆)= 1, , ,0,0 , (4.8) understand very well the reason why the third eigenvec-
− −2 2 tor is missing, it clearly has something to do with the γ-
(cid:18) (cid:19)
direction,becausethevectorintheγ-directioncannotbe
whichisidentifiedwiththatfoundinthepionlessNEFT,
expressed as a linear combination of u ’s (i=1, ,4).
i
given in Eq. (2.13). ···
Eq. (4.7) shows that the L-OPE is irrelevant. It im-
We linearize the RGEs around the fixed point, and
pliesthattheL-OPEshouldbetreatedasaperturbation.
find the following eigenvalues (scaling dimensions) and
Note that there is a typical scale in the pionful NEFT,
the corresponding eigenvectors:
Λ ,
NN
1 0
1 1 4π 2f 2
ν1 =+1: u1 =−01, ν2 =−1: u2 = −10 , ΛNN = M (cid:18)gA(cid:19) , (4.10)
   
 0   0  and our γ is related to it as
   
2 0
1 0 2 Λ
ν3 =−2: u3 =−−2, ν4 =−1: u4 = 0. γ(Λ)= π (cid:18)ΛNN(cid:19). (4.11)
0 1
   
 0   0 Kaplan, Savage, and Wise [11, 12] regard p/ΛNN as an
   (4.9) expansionparameter. Ourfindingisconsistentwiththeir
approach.
The eigenvalues ν , ν , and ν and corresponding eigen-
1 2 3
vectorsu , u , andu canbe identified withthose found
1 2 3
in the pionless theory. Therefore, the power counting is
B. Spin-triplet channel
not modified by the inclusion of the pions; only the one
relevantoperator(u ) shouldbe resummedto allorders.
1
Note that the eigenvalue problem is five dimensional In the spin-triplet channel, we have the following
and one thus expects five pairs of eigenvalues and eigen- RGEs:
dx′
= x′ x′2+2x′y′+y′2+2x′z′+2y′z′+z′2+2w′2
dt − −" #
2(x′+y′+z′ 4w′)γ 9γ2, (4.12a)
− − −
dy′ 1 3 1
= 3y′ x′2+2x′y′+ y′2+y′z′ z′2+w′2
dt − −"2 2 − 2 #
7
(x′+2y′)γ+ γ2, (4.12b)
− 2
dz′ 1 1 3
= 3z′+ x′2+x′y′+ y′2 x′z′ y′z′ z′2+w′2
dt − "2 2 − − − 2 #
9
+(x′+y′ z′ 4w′)γ+ γ2, (4.12c)
− − 2
dw′
= 3w′ x′w′+y′w′+z′w′
dt − −" #
1
+ (2x′+2y′+2z′ 9w′)γ+2γ2, (4.12d)
5 −
du′
= 3u′ 2(x′+y′+z′)u′+2(x′+y′+z′ 4w′)γ 2u′γ+18γ2. (4.12e)
dt − − − −
Here, again, the first lines of Eqs (4.12a) – (4.12d) are in Ref. [29], while the second lines are the contributions
the same as those in the pionless calculations, obtained