Duality and Electric Dipole Moment of Magnetic Monopole

Transcription

1 Progress of Theoretical Physics Supplement No. 167, Duality and Electric Dipole Moment of Magnetic Monopole Makoto Kobayashi High Energy Accelerator Research Organization (KEK), Tsukuba 35-81, Japan and International Institute for Advanced Studies, Kizugawa , Japan After a few personal recollections on Professor Shoichi Sakata and the theory group of Nagoya Univiersity, the electric dipole moment of magnetic monopoles is discussed. In the N = 2 supersymmetric gauge model, the explicit calculation shows that the fraction of the fermion contribution to the moment is given by a curious number. 1. Personal recollections I studied at Nagoya University for 9 years, 4 years as an undergraduate and 5 years as a graduate student. I entered the undergraduate course in There were about 8 students majoring in physics in that year. Professor Shoichi Sakata gave us a course of lectures on Quantum mechanics. At the time, I thought that his lectures were quite orthodox, but now I feel they were very unique. For instance, one day he wrote down just one line of equation, the Schrödinger equation, on the blackboard and kept on talking for more than half an hour. I don t remember what he said then but I remember the scene clearly. He spent a lot of time explaining principles and interpretation. In 1967, I entered to the graduate course and came to attend seminars of the theory group. Sakata-sensei was there occasionally, although not every time, because he was busy as the dean of the faculty. Anyhow it was valuable experience for me to hear what he said on such occasions. Unfortunately, this did not continue for long. He suffered from a fatal disease and passed away when I was a graduate student. It was a pity that I could not have much time to learn from Prof. Sakata directly. I enjoyed, however, the atmosphere of the theory group of Nagoya, which he fostered. I think that the quality of daily discussion was very high. Fundamental principles and strategy were discussed frequently beside daily topics. I particularly benefited from the discussion with Ohnuki-san. Meanwhile I started collaboration with Maskawa-san and others. We focused on the dynamical aspects of chiral symmetry in the quark model. Maskawa-san was a research associate at that time. But he moved to Kyoto in 197. In 1972, I also moved to Kyoto, after obtaining my Ph.D. Soon after, we wrote the paper on CP violation. The paper was published in the following year. 1) Anyhow, what I learned at Nagoya has been the most valuable asset for me.

2 96 M. Kobayashi 2. Electric dipole moment of magnetic monopoles The rest of this article is devoted to the study of the electric dipole moment of magnetic monopoles from the viewpoint of duality. This work started with a very simple question: Does a monopole with spin have an electric dipole moment? We expect the existence of the electric dipole moment as an electro-magnetic dual to the fact that an electrically charged particle with spin has a magnetic moment. 2) If this is the case, we can define a coefficient similar to the gyro-magnetic ratio as follows: where d = g m Q m 2M J, Q m = n 4π e, according to the Dirac quatization condition, and J is the spin angular momentum of the monopole. We call the coefficient g m as a gyro-electric ratio or simply as a g-factor in the following discussion. The minus sign is attached following the generic rule of the duality transformation. From the duality, we conjecture that, for a spin one-half monopole, the g-factor is 2 with possible radiative corrections. As we will see, we can show that this is the case in a special model. Furthermore we will be able to clarify how the electric dipole moment is generated in the model. Here we note that the existence of edm does not necessarily imply parity violation. Since the parity operation transforms a monopole state to an anti-monopole state with opposite magnetic charge, the usual argument forbidding the electric dipole moment cannot be applied for the monopole system. Parity transformation in the magnetic sector plays a similar role as CP transformation in the electric sector. We can say that parity and CP transformation are mutually dual under the electro-magnetic duality. It is well-known that monopoles are given by topological soliton solutions for certain types of the gauge theory. The simplest model is a system described by the following Lagrangian, in which the gauge group is SU(2) and the scalar field belongs to the triplet representation: L = FµνF a aµν a=1 3 D µ φ a D µ φ a V (φ), a=1 with V (φ) = 1 4 λ ( 3 a=1 φ a φ a v 2 ) 2. The potential energy V (φ) constrains the asymptotic value of the scalar field on a sphere of a radius v: 3 φ a 2 = v 2, as r. a=1

3 Duality and Electric Dipole Moment of Magnetic Monopole 97 In this model, there is a classical solution of the following form: with the boundary conditions φ a =ˆr a H(evr), er A a i = ɛ a ij ˆr j 1 K(evr) er K(evr) 1, H(evr), as r, K(evr), H(evr)/(evr) 1, as r., The asymptotic value of the scalar field is a topologically non-trivial mapping from a sphere to a sphere. The magnetic field is the projection of the field strength onto the direction of the scalar field, B i = 1 φa ɛijk 2 φ F jk a, and from the boundary condition, the asymptotic behavior is given by B i ˆr i 1 er 2. This implies that the magnetic charge of the solution is Q m = 4π e. It is known that when the potential V =, the energy is bounded from below E vg. The equality holds when the following BPS equation is satisfied B a = Dφ a. Therefore we can find a monopole solution by solving the BPS equation instead of the field equations. It is rather easy to solve the BPS equation under the previous ansatz and the solution is given by H(x) =x coth x 1, K(x) = x sinh x. We note that the monopole mass is given by gv, and this should be compared with the gauge boson mass after the spontaneous break down: M = gv m W = ev. Another point we note is that, since the potential energy is zero, there is a massless scalar particle in this case.

4 98 M. Kobayashi The monopole solution we obtained is not invariant under the rotation J nor the global gauge SU(2) transformation T. But it is invariant under J + T. Therefore, we can regard the monopole as the spin singlet state. The spin of the monopole can be introduced by considering an additional fermion field. In the case of the triplet fermion, the Lagrangian for the fermion fields is and the Dirac equation is given by L f = i ψ a γ µ (D µ ψ) a if ψ a φ b ɛ abc ψ c, µ ψ a + eɛ abc A b µψ c fɛ abc φ b ψ c =. This equation has two zero energy solutions for the fermion. Then, we consider the excitation of the zero energy mode fermion in the background monopole state: with a + + Ω, a+ Ω, ψ = a + ψ + + a ψ +. These states can be interpreted as the spinning monopole states. Since they are doublet, they are spin 1/2 monopole states. With these things in mind, we proceed to a specific model, in which we can calculate the electric dipole moment explicitly. We consider the N = 2 supersymmetric gauge model with the gauge group SU(2). The Lagrangian is given by L = Tr ( 14 F µνf µν + 12 Dµ SD µ S + 12 Dµ PD µ P + e2 [S, P ]2 ) 2 +i ψγ µ D µ ψ e ψ[s, ψ]+ie ψγ 5 [P, ψ], where S and P are the scalar and pseudoscalar fields, respectively. All the fields belong to the triplet representation of SU(2) and we have adopted the matrix notation. The fields appearing in this Lagrangian, except P, can be identified with the fields we considered previously. (The scalar field is denoted here as S instead of φ.) Therefore, it is obvious that this system has monopole solutions. Since the potential term vanishes when P =, the monopole should be of the BPS type. As we have seen, a spinning monopole can be obtained by creating a zero mode fermion. In the present model, however, a spinning state can be obtained by applying a supersymmetry transformation to the spinless monopole state also. Using this fact, we can easily obtain the zero energy solutions for the fermion. The supersymmetry transformation of ψ for the general field configulation is given by ( ) 1 δψ = 2 σµν F µν DS a + i DPγ 5 e[p, S]γ 5 α. Substituting the BPS monopole solution into this expression, we have δψ = 2γ i D i SP + α.

5 Duality and Electric Dipole Moment of Magnetic Monopole 99 This is nothing but the zero energy solution for the Dirac equation. Here, α is the supersymmetry transformation parameter and P + is the projection operator defined by P + = 1 ( 1+Γ 5), Γ 5 = iγ γ 5. 2 In the following, we will adopt the representation of the gamma matrices such that Γ 5 is diagonal. In order to obtain the spinning monopole we consider the creation of zero energy mode fermion. After some consideration, the following trick is useful for this purpose. We consider α as a creation-annihilation operator, instead of the supersymmetry transformation parameter, which satisfies α = ( χ ), {χ a,χ b } = 1 4M δa b. We note that, since the number of independent zero energy modes is two, α can be expressed in terms of a two component spinor. 3. Calculation of the electric dipole moment We know the configuration of the zero mode fermion explicitly, so that we can calculate the charge distribution of the created fermion in the spinning monopole, which is a possible origin of the electric dipole moment of the monopole. The charge density is given by the projection of the time-component of the SU(2) current onto the direction of the scalar field: ρ ψ = ieɛ abc Ŝ a ψb γ ψ c = 4 em H2 K 2 4 4ˆr i J i. The second line is obtained by inserting the zero energy solution and J is the angular momentum of the monopole given by the fermion spin J k =2iM(α γ k α)=2m(χ σ k χ). Now the dipole moment of the above charge distribution can be written as d i ψ = dv r i ρ ψ = 2 4π dr2r 1 H 2 K 2 J i. em 3 Extracting the g-factor from this expression, we have g ψ = 4 3 = 8 3 dr2r 1 H 2 K 2 y(y cosh y sinh y)2 dy sinh 4 y

6 1 M. Kobayashi = 4 3 ζ(3) 4 9 = We note that the fermion contribution to the g-factor is given by a curious number, whichshouldbecomparedwiththediracmoment,g = 2. Since we have calculated only the fermion contribution so far, the discrepancy could be attributed to the bosonic field contributions to the electric dipole moment. The bosonic contributions can also be calculated with the help of the supersymmetry transformation. For this purpose, however, we need the expression for the second order in α. It is not difficult to obtain the finite transformation for the monopole configuration: ψ = 2γ k D k SP + α, à = P = 2iD k S(α γ k P + α), à i = A i, S = S, where the transformed fields are indicated by tildes. It should be noted that this is an exact finite transformation. Actually we can show that the transformed fields satisfy the field equations exactly. According to the above mentioned trick, α in these expressions can be interpreted as the operators. Then, if we sandwich the transformed fields with the spinning monopole states, we can obtain exact classical field configurations of the bosonic fields for the spinning monopole. The value of the electric dipole moment is easily obtained by looking at the asymptotic behavior of the electric field at long distance. The electric field is given by projecting the field strength onto the direction of the scalar field and the longdistance behavior is obtained from the explicit form determined by the finite supersymmetry transformation: ˆr a Ẽ ai = ˆr a D i à a 1 emr 3 (3ˆriˆr k δ ik )J k. From the coefficient of this dipole field, we can conclude that g = 2. This argument was first made by Kastor and Na several years ago. 3) In the present model, however, we can go further because we know the exact form of the field configuration. In the following we calculate the bosonic field contribution to the g-factor and clarify how thedipolefieldiscreated. 4) For this purpose, first we define the electric charge density. We start with the field equation for E i = F i, D i E i = ie[s, D S]+ie[P, D P ]+e[ ψ, γ ψ]. This is still a matrix equation. Projecting this onto the direction of the scalar field, we have the Gauss-law equation for the electric field i (Ŝa E ai )=ρ A1 + ρ A2 + ρ S + ρ P + ρ ψ,

7 Duality and Electric Dipole Moment of Magnetic Monopole 11 where ρ A1 =( i Ŝ a )r 1 E ai, ρ A2 = eɛ abc Ŝ a A b ie ci, ρ S = eɛ abc Ŝ a S b (D S) c, ρ P = eɛ abc Ŝ a P b (D P ) c, ρ ψ = ieɛ abc Ŝ a ψb γ ψ c. We can regard the right hand side of the Gauss-law equation as the charge density, which consists of terms expressing the contributions of each field. The gauge field contribution is divided to two pieces. The term ρ A1 originates from the exchange of the derivative and Ŝ. We note that the above definition of the charge density is a generic one, not only for the monopole configuration. To obtain the charge distribution of the spinning monopole state, we insert the field configuration into these expressions. ρ A1 = 2 em K(H + K2 1)r 4ˆr k J k, ρ A2 = 2 em ( K + K2 )(H + K 2 1)r 4ˆr k J k, ρ ψ = 4 em H2 K 2 r 4ˆr k J k, ρ S = ρ P =. Since we have calculated the fermion contribution already, here we consider the gauge boson contributions to the electric dipole moment. The calculation is straightforward and we obtain the following results in terms of the g-factor. g A1 = 4 3 = 7 3 ζ(3) + 1 3, = , dy (y2 + y cosh y sinh y 2sinh 2 y) sinh 3 y g A2 = 4 dy (y sinh y)(y2 + y cosh y sinh y 2sinh 2 y) 3 sinh 4 y = ζ(3) 14 9, = ,, Now we find that the sum of three contribution to the g-factor is g ρ = g ψ + g A1 + g A2 = 4 3. The ζ-functions cancel out in the sum, but it is still 2/3 of the value determined by the long-distance behavior.

8 12 M. Kobayashi To understand this point, let us consider the following simple mathematics. We start with the Gauss-law relation, ρ = E. Multiplying r on the both sides and integrating them, we have dv r ρ= dv r E = lim dω rr 2 E r dv E r = 2 d 3 dv E, where we have assumed the following dipole behavior for the electric field as r E i (3ˆriˆr j δ ij ) 4πr 3 d j. We note that the surface integral gives only 2/3 of the coefficient of the dipole field, d. Therefore if the second term on the right hand side vanishes, the dipole moment of the charge distribution should be 2/3 of d. Actually this is the case for the present system. We can easily show that the angular dependence of the electric field is proportional to (3ˆr iˆr j δ ij ) in the entire space, so that the second term vanishes. Where does the remaining 1/3 of the g-factor come from? Obviously, if a circulating magnetic current exists, it will generate an electric dipole moment. In order to find the magnetic current, we consider the Bianchi identity, D B i + ɛijkd j E k =. By extracting the rotation of the electric field from the Bianchi identity, we can define the magnetic current as follows, where ɛ ijk j (Ŝa E ak )= j i 1 j i 2 j i 3, j i 1 = ɛ ijk ( j Ŝ a )E ak, j i 2 = eɛ ijk ɛ abc Ŝ a A b je ck, j i 3 = eɛ abc Ŝ a A b B ci. We note that the magnetic current consists of the gauge field only. moment due to the magnetic current can be expressed as d i = 1 dv ɛ ijk r j jm k. 2 The dipole

9 Duality and Electric Dipole Moment of Magnetic Monopole 13 The remaining calculation is straightforward. Inserting the solutions into the expression of the current, we have g j1 = 2 drr 1 K(1 K 2 H H 2 ) 3 = 2 3 = 4 3, g j2 = 2 3 = 2 3 dy y2 sinh 2 y y cosh y sinh y + y 2 cosh 2 y sinh 3 y drr 1 (K K 2 )(1 K 2 H H 2 ) = 1 3 ζ(3) 5 9 = , g j3 = 2 drr 1 H 2 K 2 3 = 2 3 = 1 3 ζ(3) 1 9 = dy (y sinh y)(y2 sinh 2 y y cosh y sinh y + y 2 cosh 2 y) sinh 4 y y(sinh y y cosh y)2 dy sinh 4 y The result contains the ζ-function again. However the sum of these three contributions is given by which is what we expected. g j1 + g j2 + g j3 = 2 3, 4. Discussion As we have seen, in the N = 2 supersymmetric gauge model, the electric dipole moment of the magnetic monopole with spin 1/2 can be calculated explicitly. The results are summarized as follows: The gyroelectric ratio is 2, the value inferred from the Dirac fermion, but it consists of various contributions. We found that the contribution of the charge distribution of the fermion field to the electric dipole moment is given by a curious number and that 1/3 comes from the magnetic current. The above results are derived based on the classical solutions. The values of the g-factor, however, would not be modified by radiative corrections, because the present model is exactly supersymmetric. It is interesting, therefore, to determine the effects of supersymmetry breaking. This is under investigation. According to the duality argument, there will be a dual description, in which magnetic monopoles are described as local fields. In such a description, the decomposition of the electric dipole moment into the various components we have discussed

10 14 M. Kobayashi here would not be seen explicitly. This in turn suggests that there would be a structure in the magnetic moment of an ordinary charged particles, which is hidden in the ordinary description. For the moment, I do not know the implication for the observation of this possible structure, but it may be worth for further investigation. References 1) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973), ) C. Montonen and D. Olive, Phys. Lett. B 72 (1977), ) D. Kastor and E.S. Na, Phys. Rev. D 6 (1999), 252. See also H. Osborn, Phys. Lett. B 115 (1982), ) M. Kobayashi, Prog. Theor. Phys. 117 (27), 479; hep-th/