Physics Letters B 765 (2017) 109–112 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Skyrme model study of proton and neutron properties in a strong magnetic field Bing-Ran He Department of Physics, Nanjing Normal University, Nanjing 210023, PR China a r t i c l e i n f o Article history: Received 29 September 2016 Received in revised form 27 November 2016 Accepted December 2016 Available online 12 December 2016 Editor: J.-P Blaizot a b s t r a c t The proton and neutron properties in a uniform magnetic field are investigated The Gell–Mann–Nishijima formula is shown to be satisfied for baryon states It is found that with increasing magnetic field strength, the proton mass first decreases and then increases, while the neutron mass always increases The ratio between magnetic moment of proton and neutron increases with the increase of the magnetic field strength With increasing magnetic field strength, the size of proton first increases and then decreases, while the size of neutron always decreases The present analysis implies that in the core part of the magnetar, the equation of state depend on the magnetic field, which modifies the mass limit of the magnetar © 2016 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 Introduction Recently, experiments have observed that there exists a strong magnetic field when baryons collide with each other, and astrophysicists have observed that the strong magnetic field exists in magnetars [1–3] The baryon states have electric charge distribution, thus the interaction between baryons and magnetic fields modifies the properties of baryons The Skyrme model [4], which identifies the soliton solution from mesons theories as the baryon, has been widely accepted, and also have lots of applications to hadron physics, astrophysics and also condensed matter physics The study of Skyrmion in a uniform magnetic field shows that, in the leading order of large N C , i.e., O ( N C ), the mass and shape of Skyrmion depend on the strength of magnetic field [5] In this letter, the O ( N C−1 ) effects are introduced in the semiclassical quantization approach [6], and then the physical baryon states, i.e., proton and neutron, in a uniform magnetic field are studied The Gell–Mann–Nishijima formula for baryon states is shown to be satisfied The semi-classical quantization of Skyrmion introduces time dependence to (e B ) terms of the model Because the wave functions for baryon states are different, the magnetic response of baryon states is different It is found that with the increase of the magnetic field strength, the effective proton mass first decreases and then increases, consequently, the proton size first increases and then decreases On the other hand, the effective neutron mass always increases, and consequently, the neutron size E-mail address: hebingran@njnu.edu.cn always decreases Furthermore, the ratio between magnetic moment of proton and neutron increases with the increase of the magnetic field strength Finally, since both the mass and size of proton and neutron depend on the strength of the magnetic field, the equation of state for magnetar is modified The model The action of the model contains two parts: = d4 xL + (1) WZW , where L is expressed as f L = π Tr( D μ U † D μ U ) + 16 32g m2 f + π π Tr(U + U † − 2) 16 Tr([U † D μ U , U † D ν U ]2 ) (2) Here f π is the pion decay constant, mπ is the pion mass, and g is a dimensionless coupling constant The covariant derivative for U is expressed as D μ U = ∂μ U − i Lμ U + iU Rμ , where L and R are the external fields expressed as Lμ = Rμ = e Q B V B μ + e Q E H μ for the present purpose Here e is the unit electric charge, Q B = 1 is the baryon number charge matrix, Q E = 16 + 12 τ3 is the electric charge matrix, is the rank unit matrix, τ3 is the third Pauli matrix, and V B μ is the external gauge field of the U (1)V baryon number In the symmetric gauge, the magnetic field H μ is http://dx.doi.org/10.1016/j.physletb.2016.12.019 0370-2693/© 2016 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 110 B.-R He / Physics Letters B 765 (2017) 109–112 expressed as H μ = − 12 B y ημ1 + 12 Bxημ2 , where η is the geometry with diag(+1, −1, −1, −1) The Wess–Zumino–Witten (WZW) action WZW ≡ d4 xLWZW , represents the chiral anomaly effects, which is given in Refs [7,8] The semi-classical quantization Following Ref [9], the x, y, and z in the elliptic coordinate system are expressed as Considering the external fields have a fluctuation as Lμ = a Rμ = e Q B V B μ + e Q E H μ − δ(Vμa ) τ2 , the corresponding iso-vector dV j V sin(2F ) e Bc ρ2 r D 33 =− y = c ρ r sin(θ) sin(ϕ ) , = I3 (3) where c ρ and c z are positive dimensionless parameters, r ≡ x2 cρ + y2 cρ + z2 , c 2z and θ and ϕ are polar angles with θ ∈ [0, π ] and ϕ ∈ [0, 2π ] The U is decomposed in the Cartesian coordinate system as U = cos( F (r ))1 + i sin( F (r )) τ1 r cρ x+ τ2 cρ y+ τ3 cz z (4) The ansatz equations (3) and (4) are the solution of the statical case of single Skyrmion, the physical baryon states are obtained by semi-classical quantization of the Skyrme model The quantization of single Skyrme model is proposed by Ref [6], and then the discussion is extended to many baryon states [10,11] Following [10], the time dependence of U is expressed as Uˆ = A (U ( R )) A † , (5) where A is the rotation matrix of isospin space, and R is the rotation matrix of spatial space in x − y plane for the present purpose The rotation matrix A and R are expressed as ˙= A −1 A i ωa τa , ( R −1 R˙ )i j = − i j3 (6) 3, where a = 1, 2, and i , j = 1, Insert (5) in action (1) we obtain ˆ = d4 x(Lˆ + LˆWZW ) = d xLˆtotal The canonical conjugate momenta of the isospin and spin are obtained by taking a functional derivative of the action with ωa and , respectively, as ∂ Lˆtotal Ia = ∂ ωa , V B μ →0 ∂ Lˆtotal J3 = ∂ (7) V B μ →0 The Gell–Mann–Nishijima formula The baryon number current of the model is obtained by takμ ing a functional derivative of the WZW term with V B μ , i.e., j B = ∂ LˆWZW ∂(e V B μ ) |V B μ →0 The baryon number N B is obtained as dV j 0B NB = = sin(2F ) e Bc ρ2 r D 33 + − 12F F (∞)=0 12π F (0)=π = 1, (8) where dV = c ρ c z r sin(θ)drdθ dϕ and D 33 = Tr[ A τ3 A τ3 ] In the present calculation, the boundary conditions F (0) = π and F (∞) = are imposed 2 † ) 3,0 N V 3,0 = x = c ρ r sin(θ) cos(ϕ ) , z = c z r cos(θ) , ∂(Lˆ a ,μ current is obtained as j V = ∂(δ(Vtotal |V B μ →0,δ(Vμa )→0 The cona μ )) served charge corresponding to the third component of SU (2) isovector current is obtained as 24π F (∞)=0 F (0)=π + I3 (9) The Gell–Mann–Nishijima formula for electric charge of baryon is given as NE = j 0B dV + j 3V,0 = NB + I3 (10) Here the electric charges of baryon states are shown to be always conserved in a uniform magnetic field, which is consistent with the fact that both U (1)V and the third component of SU (2)isospin symmetries are conserved, respectively Numerical results The parameters c ρ and c z in Eqs (3), (4) have two effects: deform the ansatz and scale the volume Since the scale effect of c ρ and c z can be absorbed by performing the scale transform of r, with no loss of generality, the restriction between c ρ and c z is im√ posed as c ρ ≡ 1/ c z In N B = sector, following Ref [6], after properly choosing the Skyrme units, a standard set of parameters is considered: mπ = 138 [MeV], f π = 108 [MeV], and g = 4.84 [12] The equation of motion for proton and neutron are obtained from |ˆ| at O ( N C ) order, respectively Here | expresses the wave functions for | p ↑ , | p ↓ , |n ↑ and |n ↓ which are given in Ref [6] The N C counting for the parameters of the present model 1/ −1/2 are f π ∼ O ( N C ), g ∼ O ( N C ), mπ ∼ O( N C0 ), e B ∼ O( N C0 ), ωa ∼ O( N C−1 ) and ∼ O( N C−1 ) The Hamiltonian up to O ( N C−1 ) is obtained as (ωa I a ) + H= a=1,2,3 J3 − Lˆtotal V B μ →0 (11) The nucleon mass and the nucleon magnetic moment are de∂M fined as M ≡ | dV H| and μ ≡ − ∂( , respectively It e B) is easy to check that M p ≡ M p ↑ = M p ↓ , M n ≡ M n↑ = M n↓ , μ p ≡ μ p↑ = −μ p↓ and μn ≡ μn↑ = −μn↓ The parameter c z is fixed to minimize the proton and neutron mass for a given |e B |, respectively The |e B | dependence of c z for proton and neutron are shown in Fig Fig shows that when the strength of the magnetic field increases, c z increases, which implies that the shape of proton and neutron are stretched The |e B | dependence of proton and neutron mass are shown in Fig Fig shows that, with increasing magnetic field strength, the proton mass first decreases then increases This is because the Hamiltonian of proton contains linear term of (e B ) and higher order terms of (e B ), the linear term of (e B ) has a different sign with higher order terms of (e B ) Therefore, for a weak |e B |, the linear term of (e B ) takes a dominant role which causes the proton mass to decrease; for a strong |e B |, with the increase of |e B |, the dominant role is shifted to higher order terms of (e B ), which causes the proton mass to increase Fig also shows that, the neutron mass B.-R He / Physics Letters B 765 (2017) 109–112 Fig |e B | dependence of c z for proton and neutron Fig |e B | dependence of M p and Mn always increases when the magnetic field strength |e B | increases, since in the Hamiltonian of neutron, the linear term of (e B ) has a same sign with higher order terms of (e B ), which causes the neutron mass to increase The |e B | dependence of μ p and μn are shown in Fig Fig shows that with increasing magnetic field strength, the magnitude of magnetic moment for proton first decreases and then increases, while the magnitude of magnetic moment for neutron first increases and then decreases Notice the magnetic moment of proton flips the sign when |e B | 0.062 [GeV2 ], which is consistent with Fig that when |e B | 0.062 [GeV2 ], the proton mass increases with the increase of |e B | The |e B | dependence of μ p /μn is shown in Fig Fig shows that with the increase of the magnetic field strength, μ p /μn increases Theoretical analysis of the present model shows that μ p /μn → when |e B | → ∞, which agrees with the tendency shown in Fig Notice that in Gell–Mann–Nishijima formula (10), the induced charge from U (1) baryon sector just cancel with the induced charge from the third component of SU (2) iso-vector sector, thus the electric charge density for nucleon state is defined as ρ E = 12 ρ I =0 + | I | ρ I =1 , where ρ I =0 ≡ j 0B |e B →0 , ρ I =1 ≡ a=1,2,3 a | dV a| and a ≡ ∂ Lˆ ∂ ωa2 The proton root mean square (RMS) electric charge radius and neutron mean square (MS) electric charge radius are defined 1/ as R 2p E ≡ p | dV R ρ E | p 1/2 and R n2 E ≡ n| dV R ρ E |n , respectively Here R represents x2 + y + z2 , x and z, respectively 111 Fig |e B | dependence of μ p and μn Fig |e B | dependence of μ p /μn Fig |e B | dependence of the proton RMS electric charge radius R 2p 1/2 E The |e B | dependence of the proton RMS electric charge radii are shown in Fig Fig shows that for proton state: (i) the magnitude of the RMS electric charge radii first increases and then decreases, this tendency is understandable from that: for weak |e B |, the proton mass decreases, which causes the proton size to increase; for strong |e B | the freedom of charged meson π +,− is restricted in the x − y plane, which causes the proton size to de1/ 1/ crease; (ii) the magnitude of z2 E is slightly larger than x2 E , which is because the freedom of charged meson π +,− is restricted 112 B.-R He / Physics Letters B 765 (2017) 109–112 Fig |e B | dependence of the neutron MS electric charge radius R n2 E in the x − y plane, while the neutral meson π is free to move along z axis, thus, the shape of proton is stretched along z axis The |e B | dependence of the neutron MS electric charge radii are shown in Fig Fig shows that (i) the neutron MS electric charge radii have a minus sign, this is because the distribution of ρ I =1 is more apart from the centre point of the soliton than that of ρ I =0 ; (ii) the magnitude of neutron MS electric charge radii decrease with the increase of |e B |, this fact can be understood from that: for all range of |e B |, the neutron mass always increases, which causes the neutron size to decrease, i.e., the magnitude of MS electric charge radii will decrease; (iii) the magnitude of x2 E is slightly larger than z2 E , which is because the ρ I =0 part is more sensitive with c z than that of ρ I =1 part Conclusions and discussion In this letter, the properties of proton and neutron in a uniform magnetic field were studied The nucleon states are separated by introducing the O ( N C−1 ) effects in the semi-classical quantization approach It was first shown that the baryon number and the charge corresponding to the third component of SU (2) iso-vector current are conserved in a uniform magnetic field, respectively It was found that the induced charge from U (1) baryon sector cancel with the induced charge from the third component of SU (2) iso-vector sector in the Gell–Mann–Nishijima formula Next, the |e B | dependence of proton and neutron mass were studied It was found that with the increase of the magnetic field strength, the proton mass first decreases and then increases, while the neutron mass always increases When |e B | ∼ 2.4m2π , the proton mass has a minimal point, which decreases about 23 [MeV] compared to that in vacuum After that, the proton and neutron magnetic moment were investigated It was found that the magnitude of proton magnetic moment first decreases and then increases, while the magnitude of neutron magnetic moment first increases and then decreases For an extreme weak magnetic field |e B | ∼ 0, the magnetic moment of proton and neutron are 1.94 [μ N ] and −1.21 [μ N ], respectively, which are consistent with Ref [13] The ratio of μ p /μn is about −1.60 when |e B | ∼ 0, and 0.61 when |e B | ∼ 28m2π Theoretical analysis of the present model implies μ p /μn → when |e B | → ∞, which is consistent with the tendency of μ p /μn The RMS electric charge radii of proton and MS electric charge radii of neutron were also investigated It was found that, the pro- ton RMS electric charge radii first increase and then decrease with the increase of the magnetic field While the magnitude of neutron MS electric charge radii always decrease For an extreme weak magnetic field |e B | ∼ 0, the proton RMS electric charge radius 1/ R 2p E is about 0.865 [fm], which agrees with the experimental result 0.84 ∼ 0.87 [fm]; while the neutron MS electric charge radius R n2 E is about −0.278 [fm2 ], which is against the experimental result −0.116 [fm2 ] The present analysis shows that in the core part of the magnetar (|e B | ∼ 10−2 [GeV2 ]), the proton density decreases about 3.4% and the neutron density increases about 15.3% compared to that in vacuum, respectively Thus, the equation of state in the core part of magnetar is modified In the vacuum of the present analysis, i.e., |e B | ∼ 0, the magnitude of proton and neutron magnetic moment are about 30% and 36% smaller than the experimental results, respectively The magnitude of neutron MS electric charge radius is about 1.39 times larger than the experimental result There are several possibilities to cure these problems, e.g., (i) inclusion of vector mesons and also scalar mesons [14]; (ii) inclusion of the sextic term, which moves the model towards a BPS model [15]; (iii) inclusion of new potential part [16] In the present analysis, the dynamical reaction of magnetic field is neglected, which could change the magnitudes of the results [17] The set of equations (3), (4) is one particular choice of the axially symmetric ansatz, for a generalized axially symmetric ansatz, these behaviours of the present model need to be checked These perspectives will be reported elsewhere Acknowledgements The author thanks Masayasu Harada, Sven Bjarke Gudnason and Muneto Nitta for discussions References [1] D.E Kharzeev, K Landsteiner, A Schmitt, H.U Yee, Lect Notes Phys 871 (2013) [2] V.A Miransky, I.A Shovkovy, Phys Rep 576 (2015) [3] V Skokov, A.Y Illarionov, V Toneev, Int J Mod Phys A 24 (2009) 5925 [4] T.H.R Skyrme, Nucl Phys 31 (1962) 556 [5] B.R He, Phys Rev D 92 (2015) 111503(R) [6] G.S Adkins, C.R Nappi, E Witten, Nucl Phys B 228 (1983) 552 [7] J Wess, B Zumino, Phys Lett B 37 (1971) 95 [8] E Witten, Nucl Phys B 223 (1983) 422 [9] G Holzwarth, B Schwesinger, Rep Prog Phys 49 (1986) 825 [10] E Braaten, L Carson, Phys Rev D 38 (1988) 3525 [11] S Krusch, Ann Phys 304 (2003) 103; C.J Halcrow, Nucl Phys B 904 (2016) 106; C.J Halcrow, C King, N.S Manton, arXiv:1608.05048 [nucl-th] [12] The alternative set of parameters might change the magnitudes of the results There are 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The ansatz equations (3) and (4) are the solution of the statical case of single Skyrmion, the physical baryon states are obtained by semi-classical quantization of the Skyrme model The quantization... where A is the rotation matrix of isospin space, and R is the rotation matrix of spatial space in x − y plane for the present purpose The rotation matrix A and R are expressed as ˙= A −1 A i ? ?a ? ?a. .. the neutron mass always increases When |e B | ∼ 2.4m2π , the proton mass has a minimal point, which decreases about 23 [MeV] compared to that in vacuum After that, the proton and neutron magnetic