Nucleon-nucleon potential in holographic QCD

Youngman Kima, Sangmin Leeb and Piljin Yic

aAsia Pacific Center for Theoretical Physics and Department of Physics, PohangUniversity of Science and Technology, Pohang, Gyeongbuk 790-784, Korea

bDepartment of Physics, Seoul National University, Seoul 151-747, Korea

cSchool of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

We derive the nucleon-nucleon potential in the D4-D8 model of holographic QCD. Thenucleon-nucleon potential emerges from one-boson exchange picture with massless pseudo-scalars and an infinite tower of spin one mesons. We observe in particular that the tensorcoupling of the ρ meson with nucleon comes out naturally larger than the vector coupling,whereas for ω mesons and axial mesons, such tensor couplings are completely absent. Thepotential is universally repulsive ∼ 1/r2 at short distance, and has the usual long-distanceattractive behavior ∼ −1/r3 along a isospin singlet and spin triplet channel. Both thelarge Nc form and the finite Nc form are given. In the former, a shallow classical minimumof depth ∼ 0.1MKKNc/λ forms at about rMKK � 5.5. We discuss shallow bound statesof nucleons with the holographic potential.

1. INTRODUCTION

The basic but key problem in nuclear physics is to understand the nature of the nuclearforce. Since Yukawa [1], many attempts have made to construct the nucleon-nucleon(NN) potential based on a meson exchange model. Although this approach has made greatachievements with high precision in phenomenology, it remains incomplete in the followingsense. Parameters in the meson exchange approach such as coupling constants and massesare all fitted to NN elastic data. In addition, chiral symmetry of quantum chromodynamics(QCD) is hardly implemented, and it is very difficult to estimate theoretical errors in theapproach.The main goal of this paper is to extract the interaction between a pair of nucleons

in string theoretical framework of holography, and consider possible bound states. Wewill exclusively work with the D4-D8 model [2], which involves a large number of colorsNc, large ’t Hooft coupling λ, and quenching of fermions. Given many approximations,the result should be approached with much caution, yet we have seen often that suchholographic approaches generating realistic numbers. The D4-D8 model has been par-ticularly successful in encoding the spin 1 meson sector coupled with pseudo-Goldstonebosons, and equally successful in describing baryons and the interaction between thesetwo sectors [2–5].

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In this work we derive the NN potential following the meson exchange picture. Fortu-nately, the basic framework for the relevant meson-nucleon interaction has been workedout in great detail [3], where all nucleon-meson couplings can be derived from the D4-D8holographic QCD very precisely.

2. NUCLEON-NUCLEON POTENTIAL FROM HOLOGRAPHY

We first summarize the nucleon-meson interactions derived from the D4-D8 model.After performing integration over the extra dimension, we obtain the four dimensionalnucleon-meson action [3,6]∫dx4 L4 =

∫dx4

(−iN̄γμ∂μN − imN N̄N + Lvector + Laxial

). (1)

The leading large Nc and large λ scaling is such that, for pseudo-scalars (ϕ = π, η′)

gπNN2mN

MKK =gtripletA

2fπMKK � 2 · 3 · π√

5×

√Nc

λ,

gη′NN2mN

MKK =Ncg

singletA

2fπMKK �

√39

2π2 × 1

λNc

√Nc

λ. (2)

For the scaling properties of other mesons, we refer to [6].

2.1. Holographic nucleon-nucleon potentialWe now borrow results on one-boson exchange potentials from Ref. [7] for various

mesons, and truncate to the leading contributions in 1/Nc and in 1/λ, which we callholographic NN potential. The leading contributions arise from the following four classesof couplings

gπNNMKK

2mN∼ gω(k)NN ∼ g̃ρ(k)NNMKK

2mN∼ ga(k)NN ∼

√Nc

λ, (3)

whereas gη′NN is further suppressed by 1/λNc and gρ(k)NN by 1/Nc.The one pion exchange potential is given by

V holographicπ =

1

4π

(gπNNMKK

2mN

)2 1

M2KKr

3S12 τ1 · τ2 , (4)

where S12 = 3(σ · r̂)(σ2 · r̂) − σ1 · σ2. For the isospin singlet vector meson, namely ω(k)-mesons, the derivative coupling is absent and the leading large λNc contribution is verysimple,

V holographic

ω(k) =1

4π(gω(k)NN )2 mω(k) y0(mω(k)r) , (5)

where y0(x) = e−x

x. For ρ(k) which are the isospin triplet vector mesons, the derivative

coupling is dominant over the minimal coupling. The contribution to Vρ(k) due to theminimal coupling gρNN are suppressed by additional 1/Nc.The potential from exchange of isospin singlet axial vectors f (k) is suppressed by addi-

tional 1/(λNc)2 while triplet axial-vector mesons a(k) contributes. In Fig. 1(a), we display

the shape of the large Nc potential with p = 10 for the iso-singlet sector with total angularmomentum one and total spin one [6], where p is the Kaluza-Klein mode number.

Y. Kim et al. / Nuclear Physics A 844 (2010) 224c–228c 225c

(a) (b)

Figure 1. (a) A plot of large Nc nucleon-nucleon potential, truncated at p = 10, along itsmost attractive channel, namely isospin singlet, σ-spin triplet, and even spatial angularmomentum with S12 = 2. The horizontal axis is for the distance, rMKK , while thepotential energy along the vertical axis is in unit ofMKKNc/4πλ (b) A plot of the nucleon-nucleon potential, with Ncλ = 50 and Nc = 3. The horizontal axis is rMKK , while thevertical potential energy is in unit of MKK/4π. .

2.2. Nucleon-nucleon potential for realistic λ and Nc

If one wishes to understand real QCD with Nc = 3, one must consider a different regime.For instance, we neglected Vη′ on account of the small ratio

gη′NNgπNN

∼√37 · 5 · π2

231

λNc

� 1 when λNc � 1 (6)

in the holographic limit. Yet, if we consider Nc = 3 and λ � 17 (determined by measuredvalues of fπ ), we find the ratio to be about 2 and is hardly ignorable. This is in fact ageneric problem in going to the realistic limit.In computing Feynman diagrams and extracting nonrelativistic potential, another small

parameter is p/mN where p is the spatial momentum of the meson being exchanged.However, when translated to real space, this ratio can show up either as m/mN or as1/rmN , which is problematic when the meson mass m exceeds the nucleon mass. Thus,contribution from exchange of heavy mesons cannot be included reliably, forcing us to cutdown to pions, η′, ρ, and ω. Thanks to the universal suppression ∼ e−mr for heavy mesonprocesses, this is a good approximation as long as we are interested in distances strictlylarger that 1/MKK .The relevant NN potential is then

V = Vπ + Vη′ + Vρ(1) + Vω(1) , (7)

where individual term must be computed as a series expansion of m/mN . We plot thispotential in Fig. 1(b) [6].

Y. Kim et al. / Nuclear Physics A 844 (2010) 224c–228c226c

3. HOLOGRAPHIC DEUTERON

In this final section, we explore some basic properties of deuteron with the NN potential.

3.1. Large Nc

The large Nc potential has a simple spin and flavor structure as

V holographic = VC + (V σT σ1 · σ2 + V S

T S12)τ1 · τ2. (8)

The massless pion exchange, the dominant contribution in long distance, contributes onlyto V S

T and positively, so an attractive channel requires S12τ1 · τ2 < 0. Using the fact thatS12 acting on σ-spin singlet vanishes identically, and that the nucleons are fermions, thisforces the isospin singlet (τ1 · τ2 = −3) and the σ−triplet (σ1 · σ2 = 1) channel with evenspatial angular momentum. The Fig. 1(a) is the plot of the potential in this sector, whoseclassical minimum occurs at

rmin � 5.53

MKK

, V (rmin) � −0.0944MKKNc

λ. (9)

Note that the binding potential is very shallow. The stringy picture of the baryon saysthat the individual baryon can be viewed as a D4 brane wrapped on the compact S4 of thedual geometry [8,2]. What we computed here is essentially the potential between two suchobjects separated along the noncompact R3. The binding energy is positive but suppressedrelative to the individual rest mass by 1/λ2, indicating very weak interactions [6].

3.2. Finite Nc

With the NN potential in Fig. 1(b), we calculate a few properties of deuteron. To dothis, we introduce a standard meson-nucleon vertex function,

Fα[(q′ − q)2] =

(Λ2

a −m2α

Λ2α + (q′ − q)2

), (10)

where mα is the meson mass, Λα is the cutoff, and nα is an exponent. The results are sum-marized in the table below, where we used two nucleon masses, mN = (938, 1130) MeV.

Deuteron hQCD (938) hQCD (1130) BonnB

Binding Energy (MeV) 2.3 2.2241 2.2246D− state probability (%) 8.79 10 4.99

Quadrupole moment (fm2) 0.44 0.474 0.278

Since we used Λα and nα as fitting parameters, our results on the deuteron property,hQCD(938) and hQCD(113), are not predictions.

4. SUMMARY

In this work, we computed the nucleon-nucleon potential in the D4-D8 holographicQCD, which is generated by exchange of five-dimensional flavor gauge field. In fourdimensional picture, this amounts to exchange of massless pseudo-scalars and an infinitetower of spin one mesons. In the large λNc limit, it is sensible to sum up to first ∼

Y. Kim et al. / Nuclear Physics A 844 (2010) 224c–228c 227c

√λ/10 vector and axial vector meson pairs, although one may choose to cut it shorter

according to the shortest distance scale interested. This prescription also gives wheretoglue the repulsive short distance regime to the more complicated intermediate and longdistance regime. With the NN potentials derived in this work, we could understand somedeuteron properties. We also tried to study nuclear matter with the potential in theDirac-Brueckner approach. We found that it is very hard to have stable nuclear matterout of the potentials, which may be due to lack of intermediate attractions from scalar(or two pion) exchange. It will be very useful to invent a consistent way to have suchcontributions in our NN potential in the D4-D8 holographic QCD.

ACKNOWLEDGES

P. Y. is grateful to Lenny Susskind for a comment that motivated this work, and alsoDeog-Ki Hong, Shamit Kachru, Mannque Rho, and Ho-Ung Yee for discussions. He alsothanks SITP of Stanford University for hospitality and generous support. Y. K. thanksHyun-Chul Kim for useful comments. Y. K. acknowledges the Max Planck Society(MPG)and the Korea Ministry of Education, Science and Technology(MEST) for the supportof the Independent Junior Research Group at the Asia Pacific Center for TheoreticalPhysics (APCTP). S. L. is supported in part by the KOSEF Grant R01-2006-000-10965-0and the Korea Research Foundation Grant KRF-2007-331-C00073. P. Y. is supported inpart by the Science Research Center Program of KOSEF (CQUeST, R11-2005-021), theKorea Research Foundation (KRF-2007-314-C00052), and by the Stanford Institute forTheoretical Physics (SITP Quantum Gravity visitor fund).

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