The physical origin of the nucleon-nucleon attraction by exchange Nicolae Mandache, Dragos Palade

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The physical origin of the nucleon - nucleon attraction by pion exchange

Nicolae Bogdan Mandache and Dragos Iustin Palade

National Institute for Laser, Plasma and Radiation Physics, Plasma Physics and Nuclear Fusion L aboratory, Bucharest - Magurele, Romania E - mail: [email protected]

Abstrac t A formula is derived for the central nucleon - nucleon potential, based on an analysis of t he physical origin of the nucleon - nuc l eon attraction by pion exchange . T he decrease of the dynamical mass of the interaction fiel d, exchange d pion in this case, is the principal mechanism respons ible for the nuclear attraction in a similar way the decrease of the kinetic energy of the exchange electron in the diatomic molecule is directly responsible for the covalent molecular attra ction. The minimum value of this central nucleon - nucleon potential and the position of the minimum are similar with the values reported in literature for a pote ntial calculated by lattice QCD, which shares the features of the phenomenological nucleon - nucle on potential s . The Schrodinger equation with this potential was solved numerically for different values of the pion mass. It results that t he nucleon - nucleon binding energy increases with the decrease of the pion mass. A very small binding energy ( 0.05 MeV ) was obtained f or the real pion mass. We discuss on the two pion exchange and hard core repulsion . The minimum value of the potential for two pion exchange is comparable with the minimum value of the CD Bonn potential.

Key words : pion exchange, dynamical mass, central nucleon - nuc leon potential,

1 . I ntroduction

The effective degrees of freedom in the nuclear interaction at low energy, in particular in the nuclear bound state , are the nucleons and the . The pion exchange is the basic mechanism of nucleon - nucleon ( N N) attraction at low energy [1 - 6 ]. As it is well known the current masses of quarks (antiquarks) u (ū) and d ( đ) are very small ; the nucleon and pion masses are mainly of dynamical origin (“kinetic” energy) . The confinement of the quarks into nucleon and pion associate an energy , given by the Heisenberg uncer tainty relation , which is just the dynamical mass . The long range structure of the nucleon is given roughly by the pion [7, 8 ] , which also gives the range of nuclear . In particular, the Compton wavelength of the pion     / m  c , which has a

value of 1.4 1 fm for the charged pion , gives the range r N of the nucleon extension. The pion in nucleon

can be represented by a degree of freedom of current mass  0, localized into a region of radius r N    . 2 This localization into the nucleon associates an energy (dynamical mass) E  pc   c /    m  c , given by the uncertainty relation , which is just the mass of the pion . At the for mation of a nuclear bound state this dynamic al mass decrease s , wich means a mass defect and consequently a binding energy for the nucleon - nucleon state [ 9 ] . Indeed , when two nucleons approach each other to form a bound state, in particular the deuteron, they put in common their pion degree s of freedom ( pion exchange). This is equivale nt with a slight de - localization of the pion degree of

freedom from a region of radius r N    to a region of radius r N   ( R )      ( R ) , where  ( R ) is direct proportional to the distance R between the two bound nucle ons and is strongly dependent on the probability of the pion to penetrate the potential barrier between the two nucleons . The dynamical mass gets :

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 c E       ( R )

and is lower than the initial one (that in the free nucleon). To form a boun d state , the decrease of the dynamical mass of the pion degree of freedom :

c c  E     (1)       ( R )

must be larger than the kinetic energy acquired by the system of two nucleons due to their localization at a distance R each other : Th is mechanism of nuclear binding has similarities with the mechanism of molecular binding of diatomic molecules [10 ] . In fact Heisenberg was the first who presented the attractive between  and in analogy to that in the hydrogen molecular ion H 2 , where the electron is the particle exchanged between the two [2 ] .

2 . Fey n man approach to the molecular and nuclear exchange interaction s

+ Let’s start with the physical interpretation of the mechanism of H 2 ion binding presente d by Feynman in [10].  I n the H 2 ion, since there are two protons, there is more space where the electron can have a low potential energy than in the case of hydrogen atom. T he electron spreads out lowering its kinetic energy , in accord with uncertainty relation , without increasing its potential energy . This energy decrease is at the  origin of the binding energy of H 2 ion . T he attractive force comes from the reduced energy of the system due to the possibility of the electron jumping from one proton to the other [10, 11, 12, 13] .  For large distances between the two protons of the H 2 ion the electrostatic potential energy of the electron is nearly zero over most of the space it must go when it make s its jump . In this space, then, the electron moves nearly like a free particle in empty s pace but with a negative energy [10] :

p 2   W H , 2 m e

where W H is the binding energy (+13.6 eV) of the hydrogen atom. This mea ns that p is an imaginary number:

p  i 2 m e W H .

The amplitude A for a particle of definite energy to get from o ne place to another a distance R away is proportional to :

e ( i /  ) pR A ~ R

w here p is the momentum corresponding to the definite ene rgy. Replacing p one obtains that t he amplitude of jumping of electron from one proton to the other , for large separation R between the two protons , will vary as [10] :

2

 R / a e  ( 2 m e W H /  ) R e 0 A ~  R R

w here a 0 is the Bohr radius. If the partic le goes in one direction the amplitude is [10]:

 R / a A ~ e 0

O ne can note that t he exponential factor limits drastically the amplitude of electron exchange for large separation. T he nuclear interaction which takes place between a ne utron and a proton by pion exchange is described by Feyman , for large R, w ith similar arguments in the same chapter [10 ]. Since in the nuclear process the proton and the neutron have almost equal masses, the exchanged pion w ill have zero total energy. But t he relation between the total energy E and the momentum p for a

pion of mass m  is:

2 2 2 2 2 E  p c  m  c .

Since E is practically zero the momentum is again imaginary:

p  im  c This means the amplitude for the pi on to jump from one nucleon to another is for large R :

A ~ e  ( m  c /  ) R  e  R /   (2)

The exponential factor is typical for a Yukawa potential or exponential potential , and again it limits drastically the exchange for large R.

3 . The central nucleon - n ucleon potential due to pion exchange

In fact , the exponential factor is well known from the tunneling of a potential barrier of width R by a particle with an energy much lower than the barrier height . The probability of transmission is proportional to th e square of the amplitude , this means in our case :

P ~ e  2 R /   (3)

The increase  ( R ) of the radius of localization region of the pion degree of freedom , which appears in formula (1), is strongly limited by this exponent ial factor (3), i.e. the probability for the pion to penetrate the potential barrier between the two nucleons . Therefore  ( R ) , which is direct proportional to the distance R between the nucleons, is taken equal to :

 ( R )  Re  2 R /   ( 4 )

By replacing (4) in formula (1), we obtain the decrease of the dynamical mass of the pion degree of freedom , which is at t he origin of nuclear attraction. In fact w ith sign minus this decrease is just the NN potential due to pion exchange :

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c c c Re  2 R /   V ( R )    E        * ( 5)  2 R /    2 R /      Re        Re

2 where  c /    m  c . Th e N N potential V(R) as a function of inter - nucleon distance R is shown in Fig.1 for a charged

pion exchange between the two nucleons (    1 . 41 fm ) . This potential has some similarities with that obtained by lattice QCD [14 ] , which shares the features of the phenomenological NN potentials: an attractive well at intermediate and larger distances and a hard core repulsion at small range , with the maximum depth of the potential in the inter mediate range [ 1 - 5 ]. The minimum value of the potential ( - 22 MeV) and the position of the minimum (0.7 fm) in Fig.1 , are comparable with the values shown in Fig 3 from [ 14 ] : about - 25 MeV and 0.7 fm. The fall of the potential towards zero value in Fig.1 fo r small R ( R <0.6 fm) is compatible with the beginning of the hard core repulsion region [ 1 - 5, 14 ] which gets dominant at short range . The potential in Fig 1 is a little la rger, in particular the fall towards higher values of R is slower than in the c ase of potential obtained in [14 ]. On the other hand t he results in [14 ] were obtained with a pion mass (530 MeV) higher than the real pion mass. For a lower pion mass (360 MeV) the range of the potential gets wider also [14] .

0

-5 )

V -10 e M ( V -15

-20

-25 0 1 2 3 4 5 6 7 R(fm)

Fig. 1 Th e N N potential in the c ase of charged pion exchange.

We solved numerically t he Schrodinger equation for the potential V(R) given in relation (5) , for

di fferent values of the pion mass m  :

2     ( R )  V ( R )  ( R )  E  ( R ) (6) 2 

where  is the reduced mass of the two nucleons . From a numerical point of view, the Schrodinger equation in central potential has been solved using the substitution:  ( R )   ( R ) / R which, in turn, gives a standard Sturm - Liouville eigenvalue problem with a cons tant coefficient for the second order derivative:

4

2 d 2  ( R )    V ( R )  ( R )  E  ( R ) (7) 2  dR 2

We look for the ground state, this means zero centrifugal energy (l=0). The spatial dimension has been truncated as R  [ 0 , 20 ] fm with a standard discretization in equal intervals of  R  10  3 fm . Such large radial extension is needed in order to resolve properly the states lying closely to the continuum ( E  0 ). The eigenvalue problem is solved by means of a finite difference method w ith the boundary values  ( 0 )   ' ( 0 )  0 and imposing an exponential decay at large radius. The resulting Hamiltonian is diagonalized and the eigenvalues (energy) are obtained. In F ig. 2 is given the dependence of t he total energy of the two nucleon s in function of the ratio of 0 0 pion mass m  to the real pion mass m  . For the real pion mass ( m  / m   1 ) a very small binding energy, about 0.05 MeV, was obtained.

0 Fig. 2 The total energy of the tw o nucleons as a function of the ratio of pion mass m  to the real pion mass m 

The total energy of the two nucleons decreases, this means the binding energy increases wit h the decrease of the pion mass, a result o btained in [4] for the binding energy of deuteron in function of pion mass . In the same time the range of the potential gets wider. For masses higher than the real pion mass the total energy gets positive. This highlights the role of the pion as main playe r in the production of nuclear attraction. To underline this conclusion we mention that for a mass of exchanged equal to 770 MeV (the ρ meson mass) the potential energy given by relation (5) has significant values only for R lower than about 0.8 fm. This means a very low contribution to the nuclear potential in the intermediate range attraction region.

4 . The two pion exchange and the hard core repulsion.

T he maximum molecular attraction is realized in the diatomic molecular bound by exchange of tw o

electrons [10] . This is the case of the hydrogen molecule H 2 in which the t wo hydrogen atoms put in common ( exchange ) their electrons.

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Similar the maximum of N N attraction is gi ven by the exchange of two pions [1 - 5 ]. W hen the two nu cleons approach each other to form a bound state, in particular the deuteron, they put in common (exchange) their pion degrees of freedom . The attractive potential in this case is practically two times the potential given by relation (5). The minimum value of the potential gets - 44 MeV, which is comparable with the minimum value ( - 50 MeV) of the CD Bonn potential [2].

50

40

30

20

10 ) V e 0 M ( V -10

-20

-30

-40

-50 0 1 2 3 4 5 6 7 R(fm)

Fig. 3 The N N potential in the case of two pion exchange and hard core repulsion at 0.6 fm

If one adds a hard core repulsion at r 0  0 . 6 fm to this two pion exchange pote ntial 2V(R), where V(R) is given by relation (5), one obtains the NN potential shown in Fig 3. The Schrodinger equation for this two pion e x change potential 2V(R) with hard core repulsion was

solved numerical ly for different values of the hard core radius r 0 . In Fig 4 it is shown the total energy of the two nucleons in function of the hard core radius.

Fig. 4 The total energy of the two nucleons as a function of hard core repulsion ra dius r 0 for the two pion exchange potential

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For a value of the hard core repulsion radius equal to 0.6 fm (Fig.3) , a typical value for phenomenological nucleon - nucleon potentials [1 - 5], the binding energy is 0.7 MeV.

5 . Discussion a nd c onclusions

The pion exchange is at the origin of the nuclear attraction, in a similar way the electron exchange is at the origin of attraction in the molecular covalent bound. The decrease of the kinetic energy (in fact  dynamical mass decrease from a relativistic point of view) of the exchange electron in the H 2 ion , directly responsible for the formation of the molecular bound state, is replaced by the decrease of the dynamical mass of the pion degree of f r e edom in the case of th e nuclear bound by pion exchange . The slight de - localization of the pion degree of freedom , which is at the origin of this dynamical mass decrease, is drastically limited by an exponential factor, which represents the probability for the pion to penetrate the potential barrier between the two nucleons. A similar exponential factor exists in the case of molecular bound. The analytical formula of the central nuclear potential (5) derived for the N N interaction by pion exchange does not contain any unknown par ame ter. The minimum value of the N N potential and the position of the minimum are similar with the values reported in literature for the central N N potential obtained by lattice QCD , which shares the features of the phenomenological NN potentials . A very s mall binding energy (0.05 MeV) was obtained by solving numerically t he Schrodinger equation . The binding energy increases for pion masses lower than the real pion mass . The fall of the potential (5 ) towards zero value for small R (Fig.1) is compatible with the beginning of the well known hard core repulsion region which is dominant at short range . The Yukawa potential, which was derived in analogy with the coulomb attraction which originates in exchange, gets infinite attractive for R  0 due to factor  1 / R [1, 2] .  2 R /   Since the maximum value of Re is 0.26 fm (for R    / 2 ), this means substantially smaller than    1 . 41 fm , relation (5) can be written in a good approximati on as:

2 R  2 R /   V ( R )   m  c e ( 8 )  

The N N nuclear potential is proportional to th e mass of the interaction field . It is also proportional to the ratio R /   , which is directly related to the slight de - localization of the pion degree o f freedom and its (dynamical) mass decrease. This de - localization is drastically limited by the exponential function , which is similar to an exponential potential except the factor 2. In the case of two pion exchange the minimum of the potential gets comp arable with the minimum value of the CD Bonn potential. The potential in this case is in a good aproximation:

2 R  2 R /   V 2  ( R )   2 m  c e ( 9 )  

 x The dependence of the type xe of the nuclear potential , where x  2 R /   is simi lar with the attractive part of the Rydberg potential used to describe the molecular covalent bonding [15]. A hard core repulsion was added to the two pion exchange potential and the Scrodinger equation was solved numerically for different values of hard core repulsion radius. For a radius of 0.6 fm the binding energy is about 0.7 MeV.

Let’s compare the strength of the nuclear potential V 2  ( R ) from relation ( 9 ) with the coulombian q 2 potential V C  . The ratio of the two pote ntials for R    is: R

7

 c  2 1  2 V 2  (   ) / V C (   )  2 2 e  2 e ( 10 ) q 

where  is the e.m , a typical result for th e relative strength of the nucl ear interaction to the e.m. interaction. If we analyze the mechanism of NN interaction at quark level, we could say that by pion exchange between two nucleons some quark degrees of freedom are implicitly exchanged and in consequence are slightly de - localized. The confinement region of a quark slightly increases and accordi ngly its dynamical mass decreases. This suggests to interpret the nuclear interaction as a residual .

Acknowledgments. We thank E. Dudas for helpful discussions and suggestions.

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