Old nuclear symmetries and large N_{c} as long distance symmetries in the two nucleon system

Old nuclear symmetries and large Nc as long distance symmetries in the two nucleon system

E. Ruiz Arriola
Speaker.Supported by the Spanish DGI and FEDER funds with grant FIS2008-01143/FIS, Junta de Andalucía grant FQM225-05, and EU Integrated Infrastructure Initiative Hadron Physics Project contract RII3-CT-2004-506078.
A. Calle Cordón

E-mail: alvarocalle@ugr.es
Abstract

Wigner and Serber symmetries for the two-nucleon system provide unique examples of long distance symmetries in Nuclear Physics, i.e. symmetries of the meson exchange forces broken only at arbitrarily small distances. We analyze the large picture as a key ingredient to understand these, so far accidental, symmetries from a more fundamental viewpoint. A set of sum rules for NN phase-shifts, NN potentials and coarse grained NN potentials can be derived showing Wigner SU(4) and Serber symmetries not to be fully compatible everywhere. The symmetry breaking pattern found from the partial wave analysis data, high quality potentials in coordinate space at long distances and their relatives is analyzed on the light of large contracted symmetry. Our results suggest using large potentials as long distance ones for the two-nucleon system where the meson exchange potential picture is justified and known to be consistent with large counting rules. We also show that potentials based on chiral expansions do not embody the Wigner and Serber symmetries nor do they scale properly with . We implement the One Boson Exchange potential realization saturated with their leading contributions due to and mesons. The short distance singularities stemming from the tensor force can be handled by renormalization of the Schrödinger equation. A good description of deuteron properties and deuteron electromagnetic form factors in the impulse approximation for realistic values of the meson-nucleon couplings is achieved.

Old nuclear symmetries and large as long distance symmetries in the two nucleon system

A. Calle Cordón

E-mail: alvarocalle@ugr.es

\abstract@cs

International Workshop on Effective Field Theories: from the pion to the upsilon 2-6 February 2009 Valencia, Spain

1 Introduction

The standard point of view in Particle Physics has often been that increasing the energy implies a higher degree of symmetry. In QCD, for instance, scale invariance roughly sets in for momenta much higher than the quark masses. In Nuclear Physics the situation may be exactly the opposite; some symmetries such as those introduced by Wigner [1] and Serber 111There is no reference. According to R. Serber [2] the name "Serber force" was coined by E. Wigner around 1947. are unveiled at low energies where the wavelength becomes larger than a certain scale. For obvious reasons we call them Long Distance Symmetries (LDS) [3, 4]. In the meson exchange picture this implies the presence of arbitrarily large symmetry breaking counterterms. We analyze these, so far accidental, LDS in the two-nucleon system below pion production threshold corresponding to CM momenta .

2 Wigner symmetry

The Wigner SU(4) spin-flavour symmetry corresponds to the algebra of isospin , spin and Gamow-Teller generators in terms of the one particle spin and isospin Pauli matrices,

 Ta=12∑AτaA,Si = 12∑AσiA,Gia=12∑AσiAτaA. (2.0)

The two-body Casimir operator is . The one-nucleon irreducible representations is a quartet made of a spin and isospin doublet

 4=(p↑,p↓,n↑,n↓)=(S=1/2,T=1/2).

Two nucleon states with relative angular momentum and total spin and isospin fulfilling due to Fermi statistics correspond to an antisymmetric sextet and a symmetric decuplet which, in terms of representations of the subgroup, are

 6A = (1,0)⊕(1,0)L=0,2,…→(1S0,3S1),(1D2,3D1,2,3),(1G2,3G1,2,3),… (2.0) 10S = (0,0)⊕(1,1)L=1,3,…→(1S0,3S1),(1P1,3P0,1,2),(1F1,3F0,1,2),… (2.0)

In particular, one obtains which seems verified for (see Fig. 1, left) for high quality potentials [5], i.e. having for 6000 data !. However, one might think that since a symmetry of the potential implies a symmetry of the S-matrix one should also have at low energies, in total contradiction to the data in Fig. 1. (see Sect. 4).

3 Serber symmetry

A vivid demonstration of Serber symmetry is demonstrated in Fig. 2 (left) where the pn differential cross section at low CM momenta, , fulfills to a good approximation

 dσpndΩ=|fpn(π−θ)|2=|fpn(θ)|2, (3.0)

suggesting no interaction in odd L-waves as , a fact verified by NN potentials in the spin-triplet states for , see Fig. 2 (middle) for the P-wave case. This assumption can also be tested by looking at Deuteron photodisintegration, , dominated above threshold by the transition . Neglecting tensor force and meson exchange currents (MEC) the cross section for a normalized deuteron state with binding energy reads [6]

 σE1(γd→pn)=απ3p(p2+2μpnBd)∣∣∫∞0drud(r)ru3P(r)∣∣2 (3.0)

with . For a free spherical P-wave , the agreement is good using from effective range (ER) theory [6] or from a potential [3] (POT), see Fig. 2 (right).

A further hint for Serber symmetry comes from the late 50’s Skyrme proposal [7] to introduce a pseudopotential representing the NN effective interaction in nuclei in the form

 Veffective(p′,p) = t0(1+x0Pσ)+t1(1+x1Pσ)(p′2+p2)+t2(1+x2Pσ)p′⋅p+… (3.0)

with the spin exchange operator. for spin singlet and for spin triplet states. Serber symmetry corresponds to take in the P-wave term, . Mean field theory calculations fitting single nucleon states yield  [8].

4 Renormalization and Long Distance Symmetry

In the meson exchange picture [9] the NN interaction can be decomposed as the sum

 V(x)=Vshort(x)+Vlong(x) (4.0)

where the short range and scheme dependent piece is given by distributional contact terms

 Vshort(r) = C0δ(→x)+C2{∇2,δ(→x)}+…, (4.0)

whereas the long distance piece is scheme independent and usually produces power divergences at short distances. We introduce a short distance cut-off,, which will be removed in the end 222The constants , etc. are scale dependent. The equivalence with momentum space renormalization is shown in Ref. [10] where the limit implies the irrelevance of in the presence of a singular chiral potential.. LDS means that even if for any one has . We analyze the implications by looking at finite energy wave scattering states

 up(r)=up,c(r)+pcotδ0(p)up,s(r)→cos(pr)+cotδ0(p)sin(pr), (4.0)

where is CM momentum. For then and zero energy states are

 u0(r)=u0,c(r)−u0,s(r)/α0→1−r/α0, (4.0)

Here , , and depend on only. Orthogonality in requires

 0=∫∞rcdr[u0,c(r)−1α0u0,s(r)][up,c(r)+pcotδ0(p)up,s(r)]. (4.0)

Note that the potential and the scattering length are independent variables. Thus we assume Wigner symmetry for the potential but experimentally different scattering lengths and , yielding from Eq. (4.0) the structure for ,

 pcotδ1S0(p)=α1S0\@fontswitchA(p)+\@fontswitchB(p)α1S0\@fontswitchC(p)+\@fontswitchD(p),pcotδ3S1(p)=α3S1\@fontswitchA(p)+\@fontswitchB(p)α3S1\@fontswitchC(p)+\@fontswitchD(p), (4.0)

showing that a symmetry of the potential for any , , is not necessarily a symmetry of the S-matrix. The result for exchange, while not exact, works rather well (see Fig. 1).

5 Sum rules

Based on the LDS idea we have recently derived the sum rules for phase shifts [3, 4]

 (5.0)

where we have defined the multiplet center . From data Fig. 3 shows that one has Wigner for even L and Serber for triplet odd L. The LDS character accommodates the symmetry for increasing and ; what matters is the impact parameter, .

The previous sum rules have a parallel long distance potential analog , and are also well verified for  [4]. This suggests that a coarse graining of the interaction using e.g. the potentials [11] works and justifies per se the symmetry obtained phenomenologically by fitting single particle states [8] for the Skyrme effective force, Eq. (3.0), [4]. We find that for and for .

6 Large Nc nucleon-nucleon potentials

As it is well known, in the large limit with fixed, nucleons are heavy,  [12], and the NN potential becomes meaningful. The amazing aspect is that the symmetry pattern of the sum rules for the old nuclear Wigner and Serber symmetries largely complies to the large and QCD based contracted symmetry [13, 14] where the tensorial spin-flavour structure is

 V(r)=VC(r)+τ1⋅τ2[σ1⋅σ2WS(r)+S12WT(r)]∼Nc (6.0)

Other operators are and hence suppressed by a relative factor. One has the sum rules

 V1L(r)=V3L(r) = VC(r)−3WS(r)+\@fontswitchO(N−1c),(−1)L=+1 (6.0) V1L(r) = VC(r)+9WS(r)+\@fontswitchO(N−1c),(−1)L=−1 (6.0) V3L(r) = VC(r)+WS(r)+\@fontswitchO(N−1c),(−1)L=−1 (6.0)

Thus, large implies Wigner symmetry only in even-L channels, exactly as observed in Fig. 3. Serber symmetry is possible but less evident (see [4]). This suggests to use large itself and its contracted spin-flavour group as a long distance symmetry. Actually, the energy independent potential may be obtained in a multi-meson exchange picture consistently with large counting rules [15] 333The LDS character implies relaxing the contact interaction piece not to be of the same form as the long distance potentials, i.e. avoiding the extra symmetry,  [16].. Retaining one boson exchange (OBE) with ,, and mesons one has

 VC(r) = −g2σNN4πe−mσrr+g2ωNN4πe−mωrr, (6.0) WS(r) = g2πNN48πm2πΛ2Ne−mπrr+f2ρNN24πm2ρΛ2Ne−mρrr, (6.0) WT(r) = (6.0)

where and and . To leading and subleading order in one may neglect spin orbit, meson widths and relativity. The tensor force is singular at short distances and requires renormalization (see [17] for the case). Deuteron properties are shown in Table 1 for parameters always reproducing the phase shift, Fig. 1 (middle). Space-like electromagnetic form factors in the impulse approximation [20] for and without MEC are plotted in Fig. 4 (see [21] for the case). Overall, the agreement is good for realistic couplings 444The Goldberger-Treiman relation gives for pions and for scalars for and . Sakurai’s universality and KSFR yield . From we have using OZI rule, . meson dominance yields with with and . Adding states yields and thus .. The inclusion of shorter range mesons induces moderate changes, due to the expected short distance insensitivity embodied by renormalization, despite the short distance singularity and without introducing strong meson-nucleon-nucleon vertex functions. In practice convergence is achieved for . Our calculation includes only the OBE part of the leading potential but multiple meson exchanges could also be added [15].

For large , the central potential is leading, Eq. (6.0). Energy independent potentials using power counting within Chiral Perturbation Theory (ChPT) [22] yield a central force only to i.e. NLO and ChPT potentials do not scale properly with since , and there are terms scaling as and not as , even after inclusion of  [23]. Moreover, Wigner and Serber symmetries are violated at long distances since

 VChPT2π(r)=(1+2τ1⋅τ2)e−2mπrr3g4Am5π1024f4πMNπ2+… (6.0)

These features might perhaps explain why renormalizing ChPT potentials in different schemes a mismatch of at for the phase shift is persistently obtained [24, 25, 10, 26].

7 Conclusions

Wigner and Serber symmetries in the NN system are realized as long distance ones and are largely compatible with the large picture. When large NN-potentials are saturated by ,, and exchange and subsequently renormalized, we obtain satisfactory results for the deuteron and central partial waves. This suggests that large potentials might eventually provide a workable scheme, less directly related to ChPT but closer in spirit to the common wisdom of Nuclear Physics.

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