Renormalization of One Meson Exchange Potentials and Their Currents
aa r X i v : . [ nu c l - t h ] S e p Renormalization of One Meson Exchange Potentialsand Their Currents
A. Calle Cordón and E. Ruiz Arriola Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain.
Abstract.
The Nucleon-Nucleon One Meson Exchange Potential, its wave functions and related Meson Exchange Currents areanalyzed for point-like nucleons. The leading N c contributions generate a local and energy independent potential whichpresents 1 / r singularities, requiring renormalization. We show how invoking suitable boundary conditions, neutron-protonphase shifts and deuteron properties become largely insensitive to the nucleon substructure and to the vector mesons. Actually,reasonable agreement with low energy data for realistic values of the coupling constants (e.g. SU(3) values) is found. Theanalysis along similar lines for the Meson Exchange Currents suggests that this renormalization scheme implies tremendoussimplifications while complying with exact gauge invariance at any stage of the calculation. Keywords:
NN interaction, One Boson Exchange, Renormalization, Strong form factors, Large Nc , Chiral symmetry, Gauge invariance.
PACS:
INTRODUCTION
The original idea of Yukawa that NN interaction at long distances is due to One Pion Exchange (OPE) was verifiedquantitatively by the Nijmegen benchmarking partial wave analysis for NN scattering in the elastic region with c / DOF ∼
1; a partial wave and energy dependent square well potential was considered for distances below 1.4-1.8 fm [1] while the neutral and charged pion masses could be determined from the fit to their currently acceptedPDG values assuming OPE above such distances. The verification of other meson exchanges is less straightforwardfrom NN elastic scattering since the shortest resolution distance probed at the pion production threshold is about l = ¯ h / √ M N m p ∼ . . after renormalization finite nucleon effects parameterizedas strong form factors are indeed marginal. We suggest using renormalization ideas for potentials, wave functions andcurrents computed consistently. In the present contribution we review our findings [7] and apply them to analyze theradiative neutron capture, n + p → d + g , a nuclear reaction where the MEC are known to be essential [5]. Speaker at 12th International Conference On Meson-Nucleon Physics And The Structure Of The Nucleon (MENU 2010) 31 May - 4 Jun 2010,Williamsburg, Virginia
ABLE 1.
Deuteron properties and low energy parameters in the S − D channel for OBE potentials including p , s , r , w mesons [7] as well as axial-vector meson a . We use the same numbers and notation as in Ref. [7]. Here AVMD means taking m a = √ m r ≃ m a = g ( fm − ) h A S ( fm − / ) r m ( fm ) Q d ( fm ) P D h r − i a ( fm ) a ( fm ) a ( fm ) r ( fm ) psrw a (AVMD) Input 0.02557 0.8946 1.9866 0.2792 6.53% 0.639 5.467 1.723 6.621 1.722 psrw a (PDG) Input 0.02552 0.8937 1.9846 0.2780 6.58% 0.671 5.463 1.714 6.607 1.712 psrw ∗ a (AVMD) Input 0.02544 0.8966 1.9909 0.2788 5.90% 0.5087 5.477 1.720 6.604 1.734 psrw ∗ a (PDG) Input 0.02540 0.8951 1.9876 0.2773 6.01% 0.557 5.470 1.708 6.588 1.724NijmII Input 0.02521 0.8845(8) 1.9675 0.2707 5.635% 0.4502 5.418 1.647 6.505 1.753Reid93 Input 0.02514 0.8845(8) 1.9686 0.2703 5.699% 0.4515 5.422 1.645 6.453 1.755Exp. 0.231605 0.0256(4) 0.8846(9) 1.9754(9) 0.2859(3) 5.67(4) − − − MESON EXCHANGE POTENTIALS
A useful and simplifying assumption arises from our observation that the symmetry pattern of the sum rules forthe old nuclear Wigner and Serber symmetries discussed in Refs. [8, 9] largely complies to the large N c and QCDbased contracted SU ( ) C symmetry [10]. In the large N c limit with a s N c fixed, nucleons are heavy, M N ∼ N c , and thedefinition of the NN potential ∼ N c makes sense. The tensorial spin-flavour structure was found to be [10] V ( r ) = V C ( r ) + t · t [ s · s W S ( r ) + S W T ( r )] ∼ N c . (1)Other operators such as spin-orbit or relativistic corrections are O ( N − c ) and hence suppressed by a relative 1 / N c factor. While these counting rules are directly obtained from quark-gluon dynamics, quark-hadron duality and con-finement requires that above the confinement scale one can saturate Eq. (1) with multiple exchanges of mesons whichhave a finite mass for N c ≫ p , s , r and w and a mesons . Thecorresponding potential reads V C ( r ) = − g s NN p e − m s r r + g w NN p e − m w r r , (2) W S ( r ) = g p NN p m p L N e − m p r r + f r NN p m r L N e − m r r r − g a NN p e − m a r r , (3) W T ( r ) = g p NN p m p L N e − m p r r (cid:20) + m p r + ( m p r ) (cid:21) − f r NN p m r L N e − m r r r (cid:20) + m r r + ( m r r ) (cid:21) + g a NN p e − m a r r (cid:20) + m a r + ( m a r ) (cid:21) , (4)where L N = M p / N c and g s NN , g p NN , f r NN , g w NN , g a NN ∼ √ N c and m p , m s , m r , m w , m a ∼ N c . To leading and sub-leading order in N c one may neglect spin orbit, meson widths and relativity. The tensor force W T is singular at shortdistances ∼ / r and requires renormalization. The renormalization is carried out in coordinate space using a boundarycondition at a short distance cut-off r c (see [7] for details) which makes the Hamiltonian self-adjoint for r > r c .Besides being much simpler and efficient, this method allows to deal with cut-off independent potentials. In practiceconvergence is achieved for r c ∼ . h ) .Overall, the agreement is good for realistic couplings and masses as expected from other sources (see Ref. [7]for a short compilation) including a natural SU(3) value for g w NN coupling. The deuteron properties and low energyparameters are shown in table 1. The S phase shift is reproduced for m s ∼ S − D phaseshifts are plotted in Fig. 1. Space-like electromagnetic form factors in the impulse approximation [15] for G pE ( − q ) = Other mesons such as h are sub-leading. Due to the U A ( ) anomaly the h ′ meson appears to be heavy, but in the large N c limit one it becomesdegenerate with the pion m h ′ = m p + O ( / N c ) . So, one might think that this meson would be as important as the pion itself. Actually this is not sosince being an iso-scalar state it generates terms in the potential V S s · s and V T S which are O ( / N c ) and hence do not contribute to the leadingpotential in Eq. (1). In our previous work we left out the a meson. We use the chiral Lagrangian [12] and take g a NN = ( m a / m p ) f p NN = . Imposing a cut-off in momentum space generates an apparent delayed convergence due to the long distances distortion of the potential, so thatunexpectedly large momentum cut-offs are needed. Renormalized results, however, agree in both r − and p − spaces [13]. P h a s e S h i f t s [ d e g ] p cm [MeV] S Channel ppspswr pswr * pswr *a1 pswr a1Nijm data 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 250 300 350 400 P h a s e S h i f t s [ d e g ] p cm [MeV] E Channel ppspswr pswr * pswr *a1 pswr a1Nijm data -30-25-20-15-10-5 0 0 50 100 150 200 250 300 350 400 P h a s e S h i f t s [ d e g ] p cm [MeV] D Channel ppspswr pswr * pswr *a1 pswr a1Nijm data FIGURE 1. np spin triplet eigen phase shifts for the total angular momentum j = a represents the error of changing m a from the AVMD value to the PDG value. G C q [MeV] ppspsrw psrw * pswr *a1 pswr a1 0.001 0.01 0.1 1 0 200 400 600 800 1000 M N G M / M d q [MeV] ppspsrwpsrw * pswr *a1 pswr a1 0.001 0.01 0.1 1 0 200 400 600 800 1000 G Q q [MeV] ppspsrwpsrw * pswr *a1 pswr a1 FIGURE 2.
Deuteron charge (left), magnetic (middle) and quadrupole (right) form factors as a function of transfer q (in MeV).See Table 1 for notation. Data can be traced from Ref. [15] (see also references in [13]). / ( + q / m r ) and without MEC are plotted in Fig. 2 (see [13] for the p case). As we see, including shorter rangemesons 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 . MESON EXCHANGE CURRENTS
Gauge invariance is easily preserved within our coordinate space approach by keeping the same boundary conditionas for the potential in the large N c setup without need of new parameters (see also Ref. [18]). The simplest (purelytransverse) MEC correction to the deuteron magnetic moment in the Impulse Approximation (IA) m IA d = ( m p + m n ) + (cid:0) m p + m n + (cid:1) P D is shown in Fig. 3 (middle panel) as a function of r c . Likewise, we show (right panel) the neutroncapture cross section. The (longitudinal) MEC contribution yields a constant shift at relatively large distances. Thedifferent short distance behaviour between transverse and longitudinal MEC’s will be elaborated elsewhere. We include only the OBE part of the leading N c potential but multiple meson exchanges could also be added as well as D degrees of freedom tocomply with large N c counting rules [11]. Eq. (1) yields V c at leading order in N c . It is worth reminding that for chiral potentials V c = O ( g A / ( f p M N )) is Next-to-next-to-leading order (NNLO). Actually, as noted in Refs. [9, 16, 7] the expected large N c behaviour [11] does not hold for the (Two PionExchange) chiral potentials even after inclusion of D [17]. Likewise, the Wigner symmetry pattern is not fulfilled for those chiral potentials [8, 9]. h r c [fm] h (exp) = 0.0256(4) h ( p ) h ( p + s ) h ( p + s + w + r ) 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 m d [ m N ] r c [fm] p - IA p - MEC psrw - IA psrw - MEC psrw * - IA psrw * - MEC m d = 0.85744 m N
200 220 240 260 280 300 320 340 0 0.1 0.2 0.3 0.4 0.5 0.6 s ( np fi d g ) [ m b ] r c [fm] p -IA p -IA + p -MEC pswr -IA pswr -IA + pr -MEC pswr -IA* pswr -IA* + pr -MEC FIGURE 3.
Short distance Cut-off, r c (in fm), dependence of several observables including when necessary MEC in addition toImpulse Approximation (IA) compared with experimental bands. Asymptotic D/S ratio h = . ( ) (left). Deuteron magneticmoment m d = . m N (middle). Neutron capture cross section and s ( np → d g ) = . ( ) mb. (right) CONCLUSIONS
Self-adjointness of the two-body Hamiltonian and current conservation are simply intertwined within the renormal-ization with boundary conditions approach above a certain cut-off distance. Current conservation is guaranteed at anyvalue of the cut-off as long as matrix elements are consistently evaluated with NN wave functions constructed from themeson exchange Hamiltonian. The conditions for finiteness for both deuteron, scattering and electromagnetic matrixelements of longitudinal MEC’s coincide.
ACKNOWLEDGMENTS
This work is supported by the Spanish DGI and FEDER funds with grant FIS2008-01143/FIS, Junta de Andalucíagrant FQM225-05, and EU Integrated Infrastructure Initiative Hadron Physics Project contract RII3-CT-2004-506078.
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