---

Magnetic Monopoles As a New Solution to Strong CP

arXiv:hep-ph/9206263v2 21 Jul 1992June 22, 1992LBL-32491 (Revised)Magnetic Monopoles As a New Solution to Strong CPProblem ∗Huazhong ZhangTheoretical Physics GroupLawrence Berkeley LaboratoryMS 50A-3115, 1 Cyclotron RoadBerkeley, California 94720AbstractA non-perturbative solution to strong CP problem is proposed.Itis shown that the gauge orbit space with gauge potentials and gaugetranformations restricted on the space boundary in non-abelian gaugetheories with a θ term has a magnetic monopole structure if there isa magnetic monopole in the ordinary space. The Dirac’s quantizationcondition in the corresponding quantum theories ensures that the vacuumangle θ in the gauge theories must be quantized.

The quantization ruleis derived as θ = 2π/n (n ̸= 0) with n being the topological chargeof the magnetic monopole. Therefore, we conclude that the strong CPproblem is automatically solved non-perturbatively with the existence ofa magnetic monopole of charge ±1 with θ = ±2π.

This is also true whenthe total magnetic charge of monopoles are very large (|n| ≥1092π) if it isconsistent with the abundance of magnetic monopoles. This implies thatthe fact that the strong CP violation can be only so small or vanishingmay be a signal for the existence of magnetic monopoles.∗This work was supported by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy underContract DE-AC03-76SF00098.

Since the discovery of Yang-Mills theories1, the non-perturbative effects ofgauge theories have played one of the most important roles in particle physics.It is known that, in non-abelian gauge theories a Pontryagin or θ term,Lθ =θ32π2ǫµνλσF aµνF aλσ,(1)can be added to the Lagrangian density of the system due to instanton2 effectsin gauge theories. The θ term can induce CP violations.

An interesting fact isthat the θ angle in QCD can be only very small (θ ≤10−9) or vanishing3. Wherein our discussions of QCD, θ is used to denote θ + arg(detM) effectively withM being the quark mass matrix, when the effects of electroweak interactions areincluded.

One of the most interesting understanding of the strong CP problemhas been the assumption of an additional Peccei-Quinn U(1)P Q symmetry4, butthe observation has not given3 evidence for the axions5 needed in this approach.Thus the other possible solutions to this problem are of fundamental interest.In this paper, we will extend the method of Wu and Zee6 for the discus-sions of the effects of the Pontryagin term in pure Yang-Mills theories in thegauge orbit spaces in the Schrodinger formulation.This formalism has alsobeen used with different methods to derive the mass parameter quantizationin three-dimensional Yang-Mills theory with Chern-Simons term6−7. Wu andZee showed6 that the Pontryagin term induces an abelian background field oran abelian structure in the gauge configuration space of the Yang-Mills the-ory.

In our discussions, we will consider the case with the existence of a mag-netic monopole. We will show that magnetic monopoles8−9 in space will inducean abelian gauge field with non-vanishing field strength in gauge configurationspace, and magnetic flux through a two-dimensional sphere in the induced gaugeorbit space is non-vanishing.Then, Dirac condition8−9 in the correspondingquantum theories leads to the result that the relevant vacuum angle θ must bequantized as θ = 2π/n with n being the topological charge of the monopole1

to be generally defined. Therefore, the strong CP problem can be solved withthe existence of magnetic monopoles.

To the knowledge of the author, such aninteresting result has never been given before in the literature.We will now consider the Yang-Mills theory with the existence of a mag-netic monopole at the origin. As we will see that an interesting feature in ourderivation is that we will use the Dirac quantization condition both in the or-dinary space and restricted gauge orbit space to be defined.

The Lagrangian ofthe system is given byL =Zd4x{−14F aµνF aµν +θ32π2ǫµνλσF aµνF aλσ}. (2)We will use the Schrodinger formulation and the Weyl gauge A0 = 0.Theconjugate momentum corresponding to Aai is given byπai = δLδ ˙Aai= ˙Aai +θ8π2ǫijkF ajk.

(3)In the Schrodinger formulation, the system is similar to the quantum systemof a particle with the coordinate qi moving in a gauge field Ai(q) with thecorrespondence6−7qi(t) →Aai (x, t),(4)Ai(q) →Aai (A(x)),(5)whereAai (A(x)) =θ8π2ǫijkF ajk. (6)Thus there is a gauge structure with gauge potential A in this formalism withina gauge theory with the θ term included.

Note that in our discussion with thepresence of a magnetic monopole, the gauge potential A outside the monopolegenerally need to be understood as well defined in each local coordinate region.2

In the overlapping regions, the separate gauge potentials can only differ by a well-defined gauge transformation9. In fact, single-valuedness of the gauge functioncorresponds to the Dirac quantization condition9.

For a given r, we can choosetwo extended semi-spheres around the monopole, with θ ∈[π/2−δ, π/2+δ](0 <δ < π/2) in the overlapping region, where the θ denotes the θ angle in thespherical polar coordinates. For convenience, we will use differential forms10 inour discussions, where A = Aidxi, F = 12Fjkdxjdxk, with F = dA + A2 locally.For our purpose to discuss about the effects of the abelian gauge structure onthe quantization of the vacuum angle, we will now briefly clarify the relevanttopological results needed, then we will realize the topological results explicitly.With the constraint of Gauss’ law, the quantum theory in this formalism isdescribed in the gauge orbit space U/G which is a quotient space of the gaugeconfiguration space U with gauge potentials connected by gauge transforma-tions in each local coordinate region regarded as equivalent.

Where G denotesthe space of continuous gauge transformations, and each gauge potential as anelement in U may be defined up to a gauge transformation in the overlappingregions. Now consider the following exact homotopy sequence11:ΠN(U)P∗−→ΠN(U/G)∆∗−→ΠN−1(G)i∗−→ΠN−1(U) (N ≥1).

(7)Note that homotopy theory has also been used to study the global gauge anoma-lies 12−19, especially by using extensively the exact homotopy sequences and interms of James numbers of Stiefel manifolds14−19. One can easily see that U istopologically trivial, thus ΠN(U) = 0 for any N. Since the interpolation betweenany two gauge potentials A1 and A2At = tA1 + (1 −t)A2(8)for any real t is in U (Theorem 7 in Ref.9, and Ref.6).

since At is transformedas a gauge potential in each local coordinate region, and in an overlapping3

region, both A1 and A2 are gauge potentials may be defined up to a gaugetransformation, then At is a gauge potential which may be defined up to agauge transformation, namely, At ∈U. Thus, we have0P∗−→ΠN(U/G)∆∗−→ΠN−1(G)i∗−→0 (N ≥1).

(9)This implies thatΠN(U/G) ∼= ΠN−1(G) (N ≥1). (10)As we will show that in the presence of a magnetic monopole, the topolog-ical properties of the system are drastically different.

This will give importantconsequences in the quantum theory. Actually, it is interesting to note moregenerally that the topological results in Eq.

(9-10) are true if U and G are thecorresponding induced spaces with A and g restricted to certain region of theordinary space, especially the 2-sphere S2 as the space boundary since the re-stricted gauge configuration space U is topologically trivial. This is in fact thethe relevant case in our discussion, since only the integrals on the space bound-ary S2 are relevant in the quantization equation as we will see.

We will call theinduced spaces of U, G and U/G when A and g are restricted on the space bound-ary S2 as restricted gauge configuration space, restricted gauge orbit space andrestricted gauge transformation space respectively, and restricted spaces collec-tively. Now for the restricted spaces, the main topological result we will use isgiven byΠ2(U/G) ∼= Π1(G),(11)The condition Π2(U/G) ̸= 0 corresponds to the existence of a magnetic monopolein the restricted gauge orbit space.In the usual unrestricted case based onthe whole compactified space M as that for pure Yang-Mills theory, there cannot be monopole structure constructed.

We will first show that in this caseF ̸= 0, and then demonstrate explicitly that the magnetic fluxRS2 ˆF ̸= 0 can be4

nonvanishing in the restricted gauge orbit space, where ˆF denotes the projectionof F into the restricted gauge orbit space.Denote the differentiation with respect to space variable x by d, and the dif-ferentiation with respect to parameters {ti | i = 1, 2...} which A(x) may dependon in the gauge configuration space by δ, and assume dδ + δd=0. Then, similarto A = Aµdxµ with µ replaced by a, i, x, A = Aai Ladxi, F = 12F ajkLadxjdxk andtr(LaLb) = −12δab for a basis {La | a = 1, 2, ..., rank(G)} of the Lie algebra ofthe gauge group G, the gauge potential in the gauge configuration space is givenbyA =Zd3xAai (A(x))δAai (x).

(12)Using Eq. (6), this givesA =θ8π2Zd3xǫijkF ajk(x)δAai (x) = −θ2π2ZM tr(δAF),(13)with M being the space manifold.

With δF = −DA(δA) = −{d(δA) + AδA −δAA}, we have topologicallyF = δA =θ2π2ZM tr[δADA(δA)] =θ4π2ZM dtr(δAδA) =θ4π2Z∂M tr(δAδA). (14)Usually, one may assume A →0 faster than 1/r as x →0 , then6 this would giveF = 0.

However, this is not the case in the presence of a magnetic monopole.Asymptotically, a monopole may generally give a field strength of the form8−9,20Fij =14πr2ǫijk(ˆr)kG(ˆr),(15)with ˆr being the unit vector for r, and this gives A →O(1/r) as x →0.Thus, one can see easily that a magnetic monopole can give a nonvanishing fieldstrength F in the gauge configuration space. To evaluate the F, one needs tospecify the space boundary ∂M in the presence of a magnetic monopole.

we nowconsider the case that the magnetic monopole does not generate a singularity5

in the space. In fact, this is so when monopoles appear as a smooth solution ofa spontaneously broken gauge theory similar to ’t Hooft Polyakov monopole8.For example, it is known that21 there are monopole solutions in the minimalSU(5) model.

Then, the space boundary may be regarded as a large 2-sphereS2 at spatial infinity. For our purpose, we actually only need to evaluate theprojection of F into the gauge orbit space.In the gauge orbit space, a gauge potential can be written in the form ofA = g−1ag + g−1dg,(16)for an element a ∈U/G and a gauge function g ∈G.

Then the projection ofa form into the gauge orbit space contains only terms proportional to (δa)n forintegers n. We can now writeδA = g−1[δa −Da(δgg−1)]g.(17)Then we obtainA = −θ2π2ZM tr(fδa) +θ2π2ZM tr[fDa(δgg−1)],(18)where f = da + a2. With some calculations, this can be simplified asA = ˆA +θ2π2ZS2 tr[fδgg−1],(19)whereˆA = −θ2π2ZM tr(fδa),(20)is the projection of A into the gauge orbit space.

Similarly, we haveF =θ4π2ZS2 tr{[δa −Da(δgg−1)][δa −Da(δgg−1)]}(21)orF = ˆF −θ4π2ZS2 tr{δaDa(δgg−1)+Da(δgg−1)δa−Da(δgg−1)Da(δgg−1)}, (22)6

whereˆF =θ4π2ZS2 tr(δaδa). (23)Now all our discussions will be based on the restricted spaces.

To see that theflux of ˆF through a closed surface in the restricted gauge orbit space U/G canbe nonzero, we will construct a 2-sphere in it. Consider an element a ∈U/G,and a loop in G. The set of all the gauge potentials obtained by all the gaugetransformations on a with gauge functions on the loop then forms a loop C1 inthe gauge configurations space U.

Obviously, the a is the projection of the loopC1 into U/G. Now since Π1(U) = 0 is trivial, the loop C1 can be continuouslyextented to a two-dimensional disc D2 in the U with ∂D2 = C1, then obviously,the projection of the D2 into the gauge orbit space is topologically a 2-sphereS2 ⊂U/G.

With the Stokes’ theorem in the gauge configuration space, We nowhaveZD2 F =ZD2 δA =ZC1 A. (24)Using Eqs.

(19) and (24) with δa = 0 on C1, this givesZC1 A =θ2π2trZS2ZC1[fδgg−1]. (25)Thus, the projection of the Eq.

(26) to the gauge orbit space givesZS2 ˆF =θ2π2trZS2{fZC1 δgg−1},(26)where note that in the two S2 are in the gauge orbit space and the ordinaryspace respectively. We have also obtained this by verifying thatZD2 trZS2 tr{δaDa(δgg−1) + Da(δgg−1)δa −Da(δgg−1)Da(δgg−1)} = 0,(27)or the projection ofRD2 F givesRS2 ˆF.In quantum theory, Eq.

(26) corresponds to the topological result Π2(U/G) ∼=Π1(G) on the restricted spaces. The discussion about the Hamiltonian equation7

in the schrodinger formulation will be similar to that in Refs.6 and 7 includ-ing the discussions for the three-dimensional Yang-Mills theories with a Chern-Simons term. We only need the Dirac quantization condition here for our pur-pose.

In the gauge orbit space, the Dirac quantization condition givesZS2 ˆF = 2πk,(28)with k being integers. Now let f be the field strength 2-form for the magneticmonopole.

The quantization condition is now given by20exp{ZS2 f} = exp{G0} = exp{4πrXi=1βiHi} ∈Z. (29)Where G0 is the magnetic charge up to a conjugate transformation by a groupelement, Hi (i=1, 2,...,r=rank(G)) form a basis for the Cartan subalgebra ofthe gauge group with simple roots αi (i=1,2,...,r).

We need non-zero topologicalvalue to obtain quantization condition for θ. One way to obtain non-zero valuefor Eq.

(26) is to consider g(t) in the following U(1) subgroup on the C1g(t) = exp{4πmtXi,j(αi)jHj< αi, αi >},(30)with m being integers and t ∈[0, 1]. In fact, m should be identical to k accordingto our topological result Π2(U/G) ∼= Π1(G).

The k and m are the topologicalnumbers on each side. Thus, we obtainθ = 2πn (n ̸= 0).

(31)Where we define generally the topological charge of the magnetic monopole asn = −2 < δ, β >= −2Xi< λi, β >,(32)which must be an integer20. Whereδ =Xi2αi< αi, αi > =Xiλi,(33)8

with the λi being the fundamental weights of the Lie algebra, the minus sign isdue to our normalization convention for Lie algebra generators.Therefore, we conclude that in the presence of magnetic monopoles withtopological charge ±1, the vacuum angle of non-abelian gauge theories must be±2π, the existence of such magnetic monopoles gives a solution to the strongCP problem. But CP cannot be exactly conserved in this case since θ = ±2πcorrespond to two different physical sysytems.

The existence of many monopolescan ensure θ →0, and the strong CP problem may also be solved. In this possiblesolution to the strong CP problem with θ ≤10−9, the total magnetic chargespresent are |n| ≥2π109.

This may possiblely be within the abundance allowed bythe ratio of monopoles to the entropy22, but with the possible existence of bothmonopoles and anti-monopoles, the total number of magnetic monopoles maybe larger than the total magnetic charges. Generally, one needs to ensure thatthe total number is consistent with the experimental results on the abundanceof monopoles.

The n = ±2 may also possibilely solve CP if it is consistent withthe experimental observation.In the above discussions, we consider the case that magnetic monopolegenerates no singularity in the space. If we consider the magnetic monopole as asingularity, then with the space outside the monopole the two opposite boundarycontributions are cancelled in the relevant integrations since each inward smallsphere arround the monopole for removing the singularity effectively gives acontribution of the opposite topological charge.

Therefore, only non-singularmagnetic monopoles may provide the solution to the strong CP problem.Moreover, note that our conclusions are also true if we add an additional θterm in QED with the θ angle the same as the effective θ in QCD if there existDirac monopoles as color singlets, or a non-abelian monopoles with magneticcharges both in the color SU(3) and electromagnetic U(1). Then a explana-9

tion of such a QED θ term is needed.The effect of a term proportional toǫµνλσFµνFλσ in the presence of magnetic charges was first considered23 relevantto chiral symmetry. The effect of a similar U(1) θ term was discussed for thepurpose of considering the induced electric charges24 as quantum excitations ofdyons associated with the ’t Hooft Polyakov monopole and generalized magneticmonopoles20,24.

For our purpose, We expect that if a QED θ term is included, itmay possiblely be an indication of the unification. A θ term needs to be includedin the unification gauge theory for Π3(G) = Z for the unification group G, mag-netic monopoles with charges involving the QED U(1) symmetry are generatedthrough the spontaneous gauge symmetry breaking.

Generally, such an inducedθ term in QED may not be discarded in the presence of magnetic monopoles.As a remark, our quantization rule can also be obtained by using constraintsof Gauss’ law. This will be given elsewhere.I thank Y. S. Wu and A. Zee for valuable discussions.10

References[1] C. N. Yang and R. L. Mills, Phys. Rev.

96, 191 (1954). [2] A.

A. Belavin, A. M. Polyakov, A. S. Schwarz and Yu. Tyupkin, Phys.

Lett.59B, 85 (1975); R. Jackiw and C. Rebbi, Phys. Rev.

Lett. 37, 172 (1976); C.Callan, R. Dashen and D. Gross, Phys.

Lett. 63B, 334 (1976); G. ’t Hooft,Phys.

Rev. Lett.

37, 8 (1976) and Phys. Rev.

D14, 334(1976). [3] R. Peccei, in ”CP Violations” (ed.

C. Jarlskog, World Scientific , Singapore,1989). [4] R. D. Peccei and H. R. Quinn, Phys.

Rev. Lett.

38, 1440 (1977); Phys. Rev.D16, 1791 (1977).

[5] S. Weinberg, Phys. Rev.

Lett. 40, 223 (1978); F. Wilczek, Phys.

Rev. Lett.40, 271 (1978).

[6] Y. S. Wu and A. Zee, Nucl. Phys.

B258, 157 (1985). [7] R. Jackiw, in ”Relativity, Groups and Topology II”, (Les Houches 1983, ed.B.

S. DeWitt and R. Stora, Noth-Holland, Amsterdam, 1984). [8] P. A. M. Dirac, Proc.

Roy. Soc.

A133, 60 (1931); Phys. Rev.

74, 817 (1948);’t Hooft, Nucl. Phys.

79, 276 (1974); A. M. Polyakov, JETP Lett. 20, 194(1974).

For a review, see S. Coleman, in ”The Unity of the FundamentalInteractions” and references theirin. [9] T. T. Wu and C. N. Yang, Phys.

Rev. D12, 3845 (1975).

[10] For a brief review, see for example, B. Zumino, in ”Relativity, Groups andTopology II”, (Les Houches 1983, ed. B. S. DeWitt and R. Stora, Noth-Holland, Amsterdam, 1984)11

[11] See for example, S. T. Hu, ”Homotopy Theory”, (Academic Press, NewYork, 1956); N. Steenrod, ”The topology of Fiber Bundles”, (PrincetonUniv. Press, Princeton, NJ 1951).

[12] E. Witten, Phys. Lett.

B117, 324 (1982). [13] S. Elitzur and V. P. Nair, Nucl.

Phys. B243, 205 (1984).

[14] S. Okubo, H. Zhang, Y. Tosa, and R. E. Marshak, Phys. Rev.

D37, 1655(1988). [15] H. Zhang, S. Okubo, and Y. Tosa, Phys.

Rev. D37, 2946 (1988).

[16] H. Zhang and S. Okubo, Phys. Rev.

D38, 1800 (1988). [17] S. Okubo and H. Zhang, in ”Perspectives on Particle Physics (ed.

S. Mat-suda, T. Muta, and R. Sakaki, World Scientific, Singapore, 1989). [18] A. T. Lundell and Y. Tosa, J.

Math. Phys.

29, 1795 (1988). [19] S. Okubo and Y. Tosa, Phys.

Rev. D40, 1925 (1989).

[20] See P. Goddard and D. Olive, Rep. Prog. Phys.

41, 1375 (1978); P. Goddard,J. Nuyts, D. Olive, Nucl.

Phys. B125, 1 (1977).

[21] C. Dokos and T. Tomaras, Phys. Rev.

D21, 2940 (1980). [22] J. Preskill, Phys.

Rev. Lett.

43 1365 (1979); G. Giacomelli, in ”Monopolesin Quantum Field Theory”, (World Scientific, Singapore, 1981). [23] H. Pagels, Phys.

Rev. D13, 343(1976); W. Marciano and H. Pagels, Phys.Rev.

D14, 531(1976). [24] E. Witten, Phys.

Lett. B86, 283 (1979); E. Tomboulis and G. Woo , Nucl.Phys.

B107, 221 (1976); H. Zhang, Phys. Rev.

D36, 1868 (1987).12

---

출처: arXiv:9206.263 • 원문 보기