Isospin-breaking two-nucleon force with explicit Delta-excitations

0.21 MB·English

by  E. Epelbaum

FZJ-IKP-TH-2007-36
HISKP-TH-07/29
Isospin-breaking two-nucleon force with explicit ∆-excitations
E. Epelbaum,1,2,∗ H. Krebs,2,† and Ulf-G. Meißner2,1,‡
1Forschungszentrum Ju¨lich, Institut fu¨r Kernphysik (Theorie), D-52425 Ju¨lich, Germany
2Universit¨at Bonn, Helmholtz-Institut fu¨r Strahlen- und Kernphysik (Theorie), D-53115 Bonn, Germany
(Dated: February 2, 2008)
We study the leading isospin-breaking contributions to the two-nucleon two-pion exchange po-
tential due to explicit ∆ degrees of freedom in chiral effective field theory. In particular, we find
importantcontributionsduetothedeltamasssplittingstothechargesymmetrybreakingpotential
that act opposite tothe effects induced bythenucleon mass splitting.
8
PACSnumbers: 13.75.Cs,21.30.-x
0
0
2

n
a
I. INTRODUCTION
J

8

] Isospin-violating(IV)two-(2NF)andthree-nucleonforces(3NF)haveattractedalotofinterestintherecentyearsin
h
thecontextofchiraleffectivefieldtheory(EFT),see[1]andreferencestherein. Inthetwo-nucleonsector,IVone-pion
t
- [2, 3, 4, 5, 6], two-pion [4, 6, 7, 8, 9], one-pion-photon [10, 11] and two-pion-photon exchange [11, 12, 13] potentials
l
c as well as short-range contact interactions [4, 6] have been studied in this framework up to rather high orders in the
u EFT expansion, as also reviewed in [1]. In addition, the leading and subleading IV 3NF have been worked out in
n Refs. [14, 15]. As found in Ref. [6], the charge-symmetry breaking (CSB) two-pion exchange potential (TPEP) in
[
EFT withoutexplicit ∆degreesoffreedomexhibits anunnaturalconvergencepatternwith the (formally)subleading

1 contribution yielding numerically the dominant effect. This situation is very similar to the isospin-conserving TPEP,
v where the unnaturally strong contribution at next-to-next-to-leading order in the chiral expansion can be attributed
9 to the large values of the low-energy constants (LECs) c accompanying the subleading ππNN vertices. The large
3.4
9
values of these LECs are well understood in terms of resonance saturation [16]. In particular, the ∆-isobar provides
2
the dominant (significant) contribution to c (c ). Given its low excitation energy, ∆ m m = 293 MeV,
1 3 4 ≡ ∆ − N
and strong coupling to the πN system, one expects that the explicit inclusion of the ∆ in EFT utilizing e.g. the
.
1 so-called small scale expansion (SSE) [17]. Such a scheme allows to resum a certain class of important contributions
0
and improves the convergence as compared to the delta-less theory. For the isospin-invariant TPEP, the improved
8
convergence in the delta-full theory has indeed been verified via explicit calculations [18, 19]. It is natural to expect
0
thatincludingthe∆-isobarasanexplicitdegreeoffreedomwillalsoimprovetheconvergencefortheIVTPEP.Inour
:
v recent work [20] we already made an important step in this direction and analyzed the delta quartet mass splittings
i
X in chiral EFT. In addition, we worked out the leading ∆-contribution to IV 3NF. As expected based on resonance
saturation, a significant part of the (subleading) CSB 3NF proportional to LECs c in the delta-less theory was
r 3,4
a demonstrated to be shifted to the leading order in the delta-full theory. We also found important effects which go
beyond resonance saturation of LECs c .
3,4
In this paper we derive the leading ∆-contributions to the IV two-pion exchange 2NF and compare the results with
the calculations based on the delta-less theory. Our manuscript is organized as follows: in sec. II, we briefly discuss
our formalism and present expressions for the IV TPEP in momentum space. We Fourier transform the potential
into coordinate space and compare the obtained results at leading order in the delta-full theory with the ones found
at subleading order in the delta-less theory [6] in sec. III. Our work is summarized in sec. IV.
∗Email: [email protected]
†Email: [email protected]
‡Email: [email protected];URL:www.itkp.uni-bonn.de/∼meissner/
2
FIG. 1: Leading isospin-breaking contributions to the 2π-exchange NN potential due to intermediate ∆-excitation. Solid,
dashed and double lines represent nucleons, pions and deltas, respectively. Solid dots and filled circles refer to the leading
isospin-invariant and isospin-breaking vertices, in order. Diagrams resulting by the interchange of the nucleon lines are not
shown.
II. ∆-CONTRIBUTIONS TO THE LEADING ISOSPIN-BREAKING TWO-PION EXCHANGE
POTENTIAL
Toobtaintheleadingisospin-violating∆-contributionstotheTPEPonehastoevaluatethetriangle,boxandcrossed-
box diagrams with one insertion of isospin-breaking pion, nucleon and delta mass shifts δM , δm , δm1 and δm2 ,
π N ∆ ∆
respectively. In addition, one needs to consider triangle diagrams involving the leading strong IV ππNN vertex.
Following Ref. [1], we also include the leading electromagnetic ππNN vertices. Notice that these IV ππNN vertices
do not contribute to the 3NF with the intermediate ∆-excitation and were not considered in [20]. We follow the
same strategyas in Refs. [5, 20] and eliminate the neutron-protonmass difference term fromthe effective Lagrangian
in favor of new isospin-violating vertices proportional to δm . This allows one to directly use the Feynman graph
N
technique to derive the corresponding NN potential and thus considerably simplifies the calculations. The leading
∆-contribution to the IV TPEP arises from Feynman diagrams shown in Fig. 1. The Feynman rules for the relevant
isospin-invariantverticescanbe found e.g. inRef. [21], see also[19]. To the orderwe are working,the Feynman rules
for IV vertces after eliminating the neutron-proton mass shift have the following form.
Two pions, no nucleons, no ∆:
•
iδM2δ δ +δm ǫ (q q ) v. (2.1)
π a3 b3 N 3ab 2− 1 ·
Here, v is the baryon four-velocity, q and q denote the incoming pion momenta with the isospin quantum
µ 1 2
numbers a and b, respectively, δM2 M2 M2 is the difference of the squared charged and neutral pion
π ≡ π± − π0
masses, and δm m m is the neutron-proton mass difference. Here and in what follows, we express,
N p n
≡ −
whenever possible, the LECs accompanying isospin-violating vertices in terms of the pion, nucleon and delta
mass shifts.
3
No pions, no nucleons, one ∆:
•
τ3 3
i δm1 3δm δ +δm2 δ δ g , (2.2)
h(cid:0) ∆− N(cid:1) 2 ij ∆4 i3 j3i µν
where i, j and µ, ν refer to the isospin and Lorentz indices of the Rarita-Schwinger field and τi denotes the
Pauli isospin matrix with the isospin index i. Further, δm1 and δm2 are the two isospin-violating delta mass
∆ ∆
shifts introduced and analyzed in Ref. [20].
Two pions, one nucleon, no ∆:
•
δmstr
i N τaδ +τbδ τ3δ +i2e2f δ δ , (2.3)
2F2 b3 a3− ab 1 a3 b3
π (cid:0) (cid:1)
where F is the pion decay constant and f is a LEC accompanying one of the leading-order electromagnetic
π 1
operators, see Ref. [22, 23] for more details. Notice that the last term in the above expression leads to the IV
3NF, see [15], but does not generate a IV two-pion exchange 2NF at the order considered here, see also [6] for
the same conclusion made using the delta-less EFT.
The leading IV NN potential in the center-of-mass system (CMS) can be conveniently written in the form:
V =τ3τ3 VII+VII~σ ~σ +VII~σ ~q ~σ ~q +(τ3+τ3) VIII+VIII~σ ~σ +VIII~σ ~q ~σ ~q , (2.4)
1 2 h C S 1· 2 T 1· 2· i 1 2 h C S 1· 2 T 1· 2· i
where ~p and p~′ are the initial and final CMS momenta, ~σ (τ ) refers to the spin (isospin) matrices of nucleon i and
i i
~q p~′ ~p, ~k 1(p~′ +p~). Further, the superscripts C, S, T of the scalar functions V , V and V denote the
≡ − ≡ 2 C S T
central, spin-spin and tensor components while the superscripts refer to the class-II and class-III isospin-violating
2NFs in the notation of Ref. [24]. The class-II interactions conserve charge symmetry and are often referred to as
charge-independence breaking while the class-II 2NF are charge-symmetry breaking, see also Ref. [15]. Notice that
at this order there are no class-IV contributions to the TPEP which lead to isospin-mixing. Evaluating the diagrams
shown in Fig. 1 and utilizing the spectral function regularization framework [25] we obtain the following expressions
for the scalar functions in Eq. (2.4):
∆-excitation in the triangle graphs:
•
h2
VII = A 2∆2Σ(4∆(3∆δm2 +δM2) 3δm2Σ)DΛ˜(q)+2 M2+∆2 2∆(9∆δm2 +δM2)
C −144Fπ4π2∆Σ(cid:16) ∆ π − ∆ (cid:0)− π (cid:1)(cid:0) ∆ π
3δm2ω2 HΛ˜(q)+6Σ 8∆2δm2 +∆δM2+δm2 Σ ω2 LΛ˜(q) ,
− ∆ (cid:1) (cid:0) ∆ π ∆(cid:0) − (cid:1)(cid:1) (cid:17)
h2∆
VIII = A 6∆2(7δm 2δmstr 5δm1)+( 15δm +6δmstr+5δm1) 2∆2+Σ DΛ˜(q)
C −216F4π2 N − N − ∆ − N N ∆
π (cid:0) (cid:0) (cid:1)(cid:1)
h2
A M2+∆2 6∆2(11δm 4δmstr 5δm1)+ ( 15δm +6δmstr+5δm1 )ω2 HΛ˜(q)
−216F4π2∆Σ − π N − N − ∆ − N N ∆
π (cid:0) (cid:1)(cid:0) (cid:1)
h2
A 3∆2(9δm 3δmstr 5δm1)+ M2( 15δm +6δmstr+5δm1) LΛ˜(q),
−108F4π2∆ N − N − ∆ π − N N ∆
π (cid:0) (cid:1)
VII = VII =VIII =VIII =0 (2.5)
S T S T
Single ∆-excitation in the box and crossed-box graphs:
•
g2h2
VII = A A ∆2Σ2(8∆(3∆δm2 +δM2)+3δm2Σ)ω2DΛ˜(q)
C 144Fπ4π2∆3Σω2 (cid:16)− ∆ π ∆
M2 ∆2 ω2 72∆4δm2 +40∆3δM2 3δm2ω4 6∆2δm2 Σ+ω2 2∆δM2 Σ+5ω2 HΛ˜(q)
− π − ∆ π − ∆ − ∆ − π
+Σ(cid:0) 2∆δM2(cid:1) 2∆(cid:0)2+Σ 2+2∆ 42∆3δm2 +23∆2δM2+6∆δm(cid:0)2Σ+4δ(cid:1)M2Σ ω2 (cid:0) (cid:1)(cid:1)
π ∆ π ∆ π
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1)
+ ( 2∆(3∆δm2 +5δM2)+3δm2 Σ)ω4 3δm2ω6 LΛ˜(q) ,
− ∆ π ∆ − ∆ (cid:1) (cid:17)
4
g2h2
VII = q2VII = A A q2 8∆δM2+3δm2ω2 AΛ˜(q),
S − T 576F4π∆2 π ∆
π (cid:0) (cid:1)
VIII = gA2h2A 10π∆(3δm δm1 )Σ 2∆2+Σ 2ω2AΛ˜(q)+2∆2Σ2 8∆2(3δm 5δm1)
C −864Fπ4π2∆3Σω2 (cid:16) N − ∆ (cid:0) (cid:1) (cid:0) N − ∆
+5(3δm δm1 )Σ ω2DΛ˜(q)+2 M2+∆2 ω2 120∆4(δm +δm1)+5(3δm δm1)ω4
N − ∆ − π N ∆ N − ∆
+2∆2 3δm Σ 1(cid:1)5ω2 5δm1 (cid:0)Σ+ω2 H(cid:1)Λ˜(q(cid:0))+2Σ 24∆2δm 2∆2+Σ 2
N − − ∆ (cid:16) N
(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:1)
+4∆2 ∆2(33δm +35δm1)+(9δm +5δm1 )Σ ω2 5 2∆2(9δm +δm1 )
N ∆ N ∆ − N ∆
+(3δm(cid:0) δm1 )Σ ω4+5(3δm δm1)ω6 LΛ˜((cid:1)q) , (cid:0)
N − ∆ (cid:1) N − ∆ (cid:1) (cid:17)
g2h2
VIII = q2VIII = A A q2 10π∆(3δm δm1 )Σω2AΛ˜(q)+2∆2Σ 4∆2(3δm 5δm1) (2.6)
S − T 3456Fπ4π2∆3Σ (cid:16) N − ∆ (cid:0) N − ∆
+5(3δm δm1 )ω2 DΛ˜(q) 5(3δm δm1) 2 M2+∆2 4∆2 ω2 HΛ˜(q) 4M2ΣLΛ˜(q) .
N − ∆ (cid:1) − N − ∆ (cid:16) (cid:0)− π (cid:1)(cid:0) − (cid:1) − π (cid:17)(cid:17)
Double ∆-excitation in the box and crossed-box graphs:
•
h4
VII = A 16∆3 M2+∆2 (3∆δm2 +δM2)Σ
C −1296F4π2∆3Σ(4∆2 ω2) − − π ∆ π
π − (cid:0) (cid:0) (cid:1)
+2∆2Σ 32∆3(3∆δm2 +δM2) 8∆(6∆δm2 +δM2)Σ 3δm2Σ2
∆ π − ∆ π − ∆
4∆2 (cid:0) ω2 DΛ˜(q)+ 8∆3 8∆4(69∆δm2 +17δM2)+2∆2(63∆δm(cid:1) 2 +17δM2)Σ+δM2Σ2
× − ∆ π ∆ π π
2(cid:0)∆ 16∆4(8(cid:1)7∆δm2 +(cid:0)22δM(cid:0)2)+2∆2(87∆δm2 +25δM2)Σ (6∆δm2 +δM2)Σ2 ω2 (cid:1)
− ∆ π ∆ π − ∆ π
+4∆(cid:0)129∆3δm2 +37∆2δM2+3∆δm2Σ+2δM2Σ ω4+( 2∆(6∆δm2 +5δM2)(cid:1)+3δm2Σ)ω6
∆ π ∆ π − ∆ π ∆
3δm(cid:0)2ω8 HΛ˜(q)+2Σ 1296∆6δm2 +320∆5δM2+(cid:1)12∆3δM2 3Σ 10ω2 +156∆4δm2 Σ 3ω2
− ∆ ∆ π π − ∆ −
+3δm2ω4(cid:1) Σ+ω2 (cid:0)2∆δM2 Σ2+4Σω2 5ω4 6∆2δm2 (cid:0)Σ2+4Σω2(cid:1) 4ω4 LΛ˜(q)(cid:0) , (cid:1)
∆ (cid:0)− (cid:1)− π(cid:0) − (cid:1)− ∆(cid:0) − (cid:1)(cid:1) (cid:17)
h4
VII = q2VII = A q2 2∆2Σ 12∆2δm2 8∆δM2 3δm2ω2 DΛ˜(q)
S − T −10368Fπ4π2∆3Σ (cid:16) (cid:0) ∆− π − ∆ (cid:1)
+ 4∆2( 4∆δM2+3δm2Σ) 3δm2 8∆2+Σ ω2+3δm2ω4 HΛ˜(q) 12M2δm2 ΣLΛ˜(q) ,
(cid:0) − π ∆ − ∆(cid:0) (cid:1) ∆ (cid:1) − π ∆ (cid:17)
h4
VIII = A 48∆4 M2+∆2 (7δm 5δm1)Σ
C 3888F4π2∆3Σ(4∆2 ω2) − π N − ∆
π − (cid:0) (cid:0) (cid:1)
2∆2Σ 96∆4(7δm 5δm1)+16∆2( 3δm +5δm1)Σ+5(3δm δm1)Σ2
− N − ∆ − N ∆ N − ∆
4∆2 (cid:0) ω2 DΛ˜(q)+ 192∆8(89δm 75δm1)+5(3δm δm1 )ω6 Σ ω2(cid:1)
× − − N − ∆ N − ∆ −
1(cid:0)6∆6 297δ(cid:1)m Σ 23(cid:0)5δm1Σ 714δm ω2+590δm1ω2 +4∆4 8( 9δ(cid:0)m +5(cid:1)δm1)Σ2
− N − ∆ − N ∆ − N ∆
+5(93δm(cid:0) 71δm1)Σω2+3( 229δm +175δm1)ω4 +(cid:1)4∆2ω2(cid:0) 3δm Σ2+19Σω2 25ω4
N − ∆ − N ∆ − N −
+5δm1 Σ2+7Σω2 9ω4 HΛ˜(q) 2Σ 80∆6(63δm(cid:1) 53δm1(cid:0))+5(3δm(cid:0) δm1)ω4 Σ ω(cid:1)2
∆ − − N − ∆ N − ∆ −
+4∆4 1(cid:0)29δm Σ 115δm1(cid:1)(cid:1)Σ(cid:1) 495δm ω2(cid:0)+405δm1ω2 + 2∆2 3( 7δm +5δm1)Σ2 (cid:0) (cid:1)
N − ∆ − N ∆ − N ∆
+12( (cid:0)7δm +5δm1 )Σω2+40(3δm 2δm1 )ω4 LΛ˜((cid:1)q) , (cid:0)
− N ∆ N − ∆ (cid:1)(cid:1) (cid:17)
h4
VIII = q2VIII = A q2 2∆2Σ 4∆2(57δm 35δm1)+5(3δm δm1)ω2 DΛ˜(q)
S − T −31104Fπ4π2∆3Σ (cid:16) (cid:0) N − ∆ N − ∆ (cid:1)
+ 64∆4( 9δm +5δm1) 20∆2(3δm δm1 ) Σ 2ω2 +5(3δm δm1 )ω2 Σ ω2 HΛ˜(q)
− N ∆ − N − ∆ − N − ∆ −
+(cid:0)20M2(3δm δm1)ΣLΛ˜(q) . (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (2.7)
π N − ∆ (cid:17)
Here,g ,h andδmstr denotethenucleon,thedelta-nucleonaxial-vectorcouplingandthestrongcontributiontothe
A A N
neutron-proton mass splitting, in order. The quantities Σ, LΛ˜, AΛ˜, DΛ˜ and HΛ˜ in the above expressions are defined
as follows:
Σ = 2M2+q2 2∆2,
π −
5
ω Λ˜2ω2+q2s2+2Λ˜qωs
LΛ˜(q) = θ(Λ˜ 2M ) ln , ω = q2+4M2 , s= Λ˜2 4M2,
− π 2q 4M2(Λ˜2+q2) π q − π
π p
1 q(Λ˜ 2M )
AΛ˜(q) = θ(Λ˜ 2M ) arctan − π ,
− π 2q q2+2Λ˜M
π
1 Λ˜ dµ µ2 4M2
DΛ˜(q) = arctan − π ,
∆ Z µ2+q2 p 2∆
2Mπ
2Σ
HΛ˜(q) = LΛ˜(q) LΛ˜(2 ∆2 M2) . (2.8)
ω2 4∆2(cid:20) − − π (cid:21)
− p
Notice that the spectral function cutoff Λ˜ can be set to in order to obtain the expressions corresponding to
dimensional regularization. We further emphasize that the∞factors of Σ−1 in Eqs. (2.5-2.7) always show up in the
combination Σ−1HΛ˜(q) and thus do not lead to singularities.
It is instructive to verify the consistency between the results obtained in EFT with andwithout explicit ∆ degreesof
freedom as done in Ref. [19] for the isospin-conserving TPEP.Since both formulations differ from each other only by
the different counting of the delta-to-nucleon mass splitting, ∆ M Λ versus M ∆ Λ for the delta-full
π χ π χ
∼ ≪ ≪ ∼
and the delta-less theory, respectively. Thus expanding the various terms in Eqs. (2.5-2.7) in powers of 1/∆ and
counting ∆ Λ should yield either terms polynomial in momenta (i.e. contact interactions) or non-polynomial
χ
∼
contributions absorbable into a redefinition of the LECs in the delta-less theory (in harmony with the decoupling
theorem). ExpandingEqs.(2.5-2.7) inpowersof1/∆andkeepingthe 1/∆terms yields the followingnon-polynomial
contributions:
g2h2δM2
VII = q2VII = A A πq2AΛ˜(q)+... ,
S − T 72πF4∆
π
g2h2δm
VIII = q2VIII = A A Nq2LΛ˜(q)+... ,
S − T 72π2F4∆
π
g2h2 128M4+112M2q2+23q4 δm LΛ˜(q) h2 8M2+5q2 (2δm δmstr)LΛ˜(q)
VIII = A A π π N A π N − +... , (2.9)
C − (cid:0) 216F4π2(4M2+q2)∆(cid:1) − (cid:0) 4(cid:1)32F4π2∆
π π π
where the ellipses refer to higher-order terms. These expressions agree with the subleading IV contributions to the
TPEP in the delta-less theory given in Ref. [6] if one uses the values for the LECs c resulting from ∆ saturation:
i
4h2
c =0, c = c =2c = A . (2.10)
1 2 3 4
− 9∆
Notice that there is a factor 1/2 missing in Eq. (3.52) of Ref. [6]. The correct expression for the central component
of the subleading CSB TPEP in the delta-less theory W(5) reads
C
LΛ˜(q) 48M4(2c +c )
W(5) = g2δm π 1 3
C −96π2F4 (cid:26)− A N 4M2+q2
π π
1
+4M2 g2δm (18c +2c 3c )+ (2δm δmstr)(6c c 3c )
π(cid:20) A N 1 2− 3 2 N − N 1− 2− 3 (cid:21)
1
+q2 g2δm (5c 18c ) (2δm δmstr)(c +6c ) . (2.11)
(cid:20) A N 2− 3 − 2 N − N 2 3 (cid:21)(cid:27)
III. RESULTS FOR THE POTENTIAL IN CONFIGURATION SPACE
We are now in the position to discuss the numerical strength of the obtained IV TPEP and to compare the results
with the ones arising in the delta-less theory. The coordinate space representations of the various components of the
TPEP up to NNLO are defined according to
V˜(r) = τ3τ3 V˜II(r)+V˜II(r)~σ ~σ +V˜II(r)(3~σ rˆ~σ rˆ ~σ ~σ ) +(τ3+τ3) V˜III(r)+V˜III(r)~σ ~σ
1 2h C S 1· 2 T 1· 2· − 1· 2 i 1 2 h C S 1· 2
+V˜III(r)(3~σ rˆ~σ rˆ ~σ ~σ ) . (3.1)
T 1· 2· − 1· 2 i
6
150 150
8 8
]
V
100 100
e
k
[

) 4 4
r 50 50
(
II C
~V
0 0 0 0
] 100 4 100 4
V
e
k
[

r) 50 2 50 2
(
II T
~V
0 0 0 0
0 0
0 0
]
V
-1 -1
e -20 -20
k
[

) -40 -2 -40 -2
r
(
II S
~V -60 -3 -60 -3
-80 -4 -80 -4
1 1.2 1.4 1.6 1.8 22 2.2 2.4 2.6 2.8 3 1 1.2 1.4 1.6 1.8 22 2.2 2.4 2.6 2.8 3
r [fm] r [fm]
FIG. 2: Class-II two-pion exchange potential. The left (right) panel shows the results obtained at leading order in chiral
EFT with explicit ∆ resonances (at subleading order in chiral EFT without explicit ∆ degrees of freedom). The dashed and
dashed-dotted lines depict the contributions due to the delta and squared pion mass differences δm2∆ and δMπ2, respectively,
while thesolid lines give thetotal result. In all cases, the spectral function cutoff Λ˜ =700 MeV is used.
ThefunctionsV˜II (r)andV˜III (r)canbedeterminedforanygivenr >0usingthespectralfunctionrepresentation
C,S,T C,S,T
as described in [18, 19]. We use the following values for the various LECs which appear in the TPEP: g = 1.27,
A
h = 3g /(2√2) = 1.34 from SU(4) (or large N ), F = 92.4 MeV, δM2 = 1260 MeV2 and δm = 1.29 MeV.
FoAr the sAtrong nucleon mass shift, we adopt the vcalueπfrom Ref. [26] δmstπr = 2.05 MeV, see alsoN[27]−for a recent
determination from lattice QCD. The IV delta mass shifts δm1 and δmN2 ha−ve been determined in [20] from the
∆ ∆
physical values of m , m and either the average delta mass m¯ =1233 MeV leading to
∆++ ∆0 ∆
δm1 = 5.3 2.0 MeV, δm2 = 1.7 2.7 MeV (3.2)
∆ − ± ∆ − ±
or the quark mass relation m m =m m leading to
∆+ ∆0 p n
− −
δm1 = 3.9 MeV, δm2 =0.3 0.3 MeV. (3.3)
∆ − ∆ ±
Let us first discuss the charge-independence-breakingcontributions to the TPEP.In Fig. 2, we compare the strength
of the corresponding central, spin-spin and tensor components V˜II (r) obtained at leading order in the delta-full
C,S,T
theory with the ones resulting at subleading order in the EFT without explicit delta. In the former case, we add
the leading-order contributions given in Eq. (3.40) of Ref. [6], see also [7] for an earlier calculation, to the leading
7
300 300
10 10
] 200 200
V
e 5 5
k 100 100
[

)
r
( 0 0 0 0
III C
~V
-100 -100
-5 -5
-200 -200
60 60
1.5 1.5
] 40 40
V
e 1 1
k
[ 20 20
0.5 0.5
)
r
(
III T 0 0 0 0
~V
-20 -0.5 -20 -0.5
20 1 20 1
]
V
e 0 0 0 0
k
[

)
r
(
III S-20 -1 -20 -1
~V
-40 -2 -40 -2
1 1.2 1.4 1.6 1.8 22 2.2 2.4 2.6 2.8 3 1 1.2 1.4 1.6 1.8 22 2.2 2.4 2.6 2.8 3
r [fm] r [fm]
FIG. 3: Class-III two-pion exchange potential. The left (right) panel shows the results obtained at leading order in chiral
EFTwithexplicit∆resonances(atsubleadingorderinchiralEFTwithoutexplicit∆degreesoffreedomandassumingcharge
independenceoftheπN couplingconstant,β =0). Thedashedanddashed-double-dottedlinesdepictthecontributionsdueto
the delta and nucleon mass differences δm1∆ and δmNstr,em, respectively, while the solid lines give the total result. In all cases,
thespectral function cutoff Λ˜ =700 MeV is used.
∆-contributions in Eq. (2.5-2.7). In the latter case, we adopt the expressions given in Ref. [6] and use the central
values of the LECs c found in Ref. [19], namely:
i
c = 0.57, c =2.84, c = 3.87, c =2.89, (3.4)
1 2 3 4
− −
inunitsofGeV−1. Noticethatwhileinthe delta-lesstheory,the leadingandsubleadingclass-IITPEParisesentirely
from the pion mass difference δM2, in the delta-full theory one also finds contributions proportional to δm2 . The
π ∆
results shown in Fig. 2 for the contributions δM2 are consistent with the observations made in Ref. [19] for
∝ π
the isospin-invariant TPEP, namely that the next-to-leading order (NLO) isovector central (spin-spin and tensor)
components in the delta-full theory are overestimated (underestimated) as compared to the next-to-next-to-leading
order(NNLO)calculationinthedelta-lesstheory. Weremindthereaderthatthecharge-independence-breakingTPEP
duetothepionmassdifferencecanbeexpressedintermsofthecorrespondingisospin-invariantTPEPasdemonstrated
in Ref. [7]. The contributions proportional to δm2 are numerically smaller than the ones proportional to δM2 if
one adopts the central value δm2 = 1.7 MeV∝and∆lead to a slight enhancement of the δM2-contributions. Notπice
that the δm2 -terms provide a cl∆ear m−anifestation of effects which go beyond the subleadingπorder in the delta-less
∆
theory, see Ref. [20] for a related discussion.
Let us now regardthe charge-symmetry-breakingTPEP.Again,we comparein Fig.3 the leading-orderresults in the
8
delta-full theorywith the subleading calculationsin the EFT without explicit ∆ using the results of [6]. The class-III
TPEP is generated in the delta-less theory by the strong and electromagnetic nucleon mass shifts and the charge
dependent pion-nucleon coupling constant β [6] whose value is not known at present. In our numerical estimations,
we set β =0. In EFT with explicit ∆ degrees of freedom, the class-II TPEP also receives contributions proportional
to the delta massshiftδm1 . The δm -partsofVIII turnouttobe verysimilarinboth caseswhile there aresizeable
∆ N S,T
deviations for VIII. Notice that although the subleading contributions in the delta-full theory have not yet been
C
worked out and thus the convergence of the EFT expansion cannot yet be tested, the obtained results imply that
the significant part of the unnaturally big subleading contribution for the class-III TPEP in the delta-less theory
is now shifted to the lower order leading to a more natural convergence pattern. The improved convergence of the
delta-full theory was also demonstrated for the isospin-invariantTPEP [19]. In addition to the CSB terms generated
bythenucleonmassshift,therearealsocontributionsproportionaltoδm1. Forourcentralvalue,δm1 = 5.3MeV,
these contributions are numerically large and tend to cancel the ones pr∆oportional to δm and δm∆str le−ading to a
N N
significantly weaker resulting class-III TPEP as compared to the ones at subleading order in the delta-less theory.
Similar cancellations were observed recently for the IV 3NF [20]. This can be viewed as an indication that certain
higher-order IV contributions still missing at subleading order in the delta-less theory are unnaturally large in the
theorywithoutexplicit deltadegreesoffreedom. We wouldfurther liketo emphasize thatthereis a largeuncertainty
in the obtained results for the IV TPEP due to the uncertainty in the values of δm1 and δm2. This is visualized in
∆ ∆
Fig. 4 where the bands refer to the variation in the values δm1,2 according to Eq. (3.2).
∆
IV. SUMMARY AND CONCLUSIONS
In this paper we have studied the leading IV contributions to the TPEP due to explicit ∆ degrees of freedom. The
pertinent results can be summarized as follows:
i) We have calculated the triangle, box and crossed box NN diagrams with single and double delta excitations
whichgiverisetothe leadingIVTPEP,seeFig.1. Tofacilitate the calculations,weusedthe formulationbased
on the effective Lagrangianwith the neutron-proton mass difference being eliminated.
ii) We have verified the consistency of our results with the previous calculations based on the delta-less theory by
expanding the non-polynomial contributions in powers of 1/∆ and using resonance saturation for the LECs c .
i
iii) Wefoundimportantcontributionstothe IVTPEPdueto themasssplittings withinthedelta quartetwhichgo
beyond the subleading order of the delta-less theory. In particular, the strong CSB potential found in Ref. [6]
is significantly reduced by the contributions proportional to δm1 .
∆
In the future, it would be interesting to derive the subleading ∆-contributions to the IV TPEP in order to test the
convergence of the chiral expansion in the delta-full theory. The explicit expressions for the IV TPEP worked out in
this paper can (and should be) incorporated in the future partial wave analysis of nucleon-nucleon scattering.
Acknowledgments
The work of E.E. and H.K. was supported in parts by funds provided from the Helmholtz Association to the young
investigatorgroup“Few-NucleonSystems inChiralEffective FieldTheory”(grantVH-NG-222)andofU.M. through
the virtual institute “Spin and strong QCD” (grant VH-VI-231). This work was further supported by the DFG
(SFB/TR 16 “Subnuclear Structure of Matter”) and by the EU Integrated Infrastructure Initiative Hadron Physics
Project under contract number RII3-CT-2004-506078.
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9
250 8
~ ~
II 150 III
V (r) [keV] V (r) [keV]
200 C 10 C 6
100
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4
100 5 50
2
50
0
0 0 0
60 2
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II 4 III
V (r) [keV] V (r) [keV]
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