IC/83'126 INTEHNAL REPORT In a recoct paper ; it was claimed that the monopole catalysis of proton (Limited distribution) 'f-i,.,;;1' ' ig absent for SU(5) tnonopoles , because of the non-existence of the ?,f 1.0-oner.^y fermicn-monopole bound state in the theory for ( Higga induced)
it. International Atomic Energy Agency :r:5.Rsive form ions. Cue to the t ar-reaching implications of this claim we reanalyze the problem for the SU(5) fundamental monopole by paying due atten-
United Nations Educational Scientific and Cultural Organization - r ,0 the geometrical structure of the monopole. Our results disagree with : r,x -it reference 1.
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS It was shown by Jackiw and Rebbi that if the fenaion mass ie induced by rhe Higgs field that created the SU(2) monopole, there exist nonnaliiable aero anergy a-nrave bound states. Then one may wonder why it is necessary to reconsider the problem for the Stl(5) monopole again, given that their gauge
DOES THE SU(5) MONOPOLE CATALYZE PROTOH DECAY' * field strucures are the same. However, this is a well—posed problem; because although the gauge field structures are the same, the Higgs field structure oi" the 5Ct(5) is a lot richer. For instance, it is the ^-Higgs which Faheem Hussain * and Mamik K. Pak ** fjenerates the monopole, whereas the fermion mass generation is induced by International Centre for Theoretical Physics, Trieste, Italy. the ^-Higgs. Therefore, there is no a priori reason why the problem of Zero- fiiicgy hound states should be the same for the 311(5) monopole. However, we shall prove that the zero-energy bound states still exist for the SU(5) monopole.
ABSTRACT We start by briefly reviewing the S0(5) monopoles. In the standard
scenario, the 3U(5) symmetry is broken down to SU(3)0 X SU(2)L X U(l)T, at The role of Higgs Induced mass for the fermions in the an energy scale of 10 ' Qev, by a superheavy Higgs multiplet f in the SU(5) monopole catalysis of the borj/on derny problem is investigated. adjoint representation At the Salam-Weinberg stage the symmetry is fur- We find that the inclusion of such a mass does not rule out the ther broken down to 30(3) X U(lV by another Higgs,H, in the fundamental Rubakov effect but it does suppress the catalysis eroES-section. c e .m. reprasentation ^, A monopole is a oonfiguration carrying non-trivial topology in a U(l) subgroup generated by the Abelian diagonal generators. The simplest realisat- ion 'can be obtained by a. transposition of the 't Hooft - Polyakov monopole MIRAMARE - TRIESTE to ar. appropriate 5U(2) subgroup of SU(5). The stable (least massive) August 1983 corresponds to the 1+1+2+1 imbedding of a Stl(2) into SU(5) given by
T = -L (« ,° > (1)
+ To be submitted for publication. * Permanent address: Department of Physics, Quaid-I-Azam University, Islamabad, Pakistan. •• Address after September 1983: CERN, Geneva, Switzerland.
-2- where X are the 2x2 Pauli matrices. The general form of this taonopole is a ';i\re of radius m in vhich all the SU{5) degrees of freedom are excited. 1'he Salam-Weinberg phase is located in the shell m < r < E .In this 1 W - (T» T) K^ - (2a) 'xgion, y has attained its v.e.v. but H is still zero. It attains its v.e.v. i i__ A* about m ~ - 0(10 ) GeV. Outside m " radius, only colour and ordinary W - 0 (2b) rra^netic fields survive. These long range fields are superpositions of two ordinary single Dirac-unit monopoles. The quarks and leptons have both ordinary charge and colour hypercharge. (2c > I'ecause $ breaks SU(2) also, SU{2) —»• U(1)Q, outside the main core, quarks and leptona move as particles of total charge Q - Q + T in the field of a simple Abelian Birac monopole.
The standard choice of fermions is a right — handed ^j, denoted by and a left - handed 10, denoted by If. Q, « 0 fermions cannot interact with monopoles. The others with charge - monopole strength Qg - +^ form the '2d) doublets -u2 (4)
under the SU(2) . The fermion-monopole interaction ia usually treated by taking the monopole The analog of the Julia - Zee dyon is obtained as well by the sane method : as an external field, i.e. monapole fields are assumed not to be perturbed by the fertoions. We study the zero energy solutions of the Dirac equation in the \ monopole field as Rubakov has shown that it is these sero energy states vhich (3) control the asymptotic behaviour of the Ferns ion Green's function. The Hubakov effect depends on tha existence of these zero energy states. We have to also stress that the anomaly plays a oentral role in the Rubakov effect. Thus in order to have an understanding of the zero energy bound state problem in this context, we have to consider configurations which can supply the anomaly effect as a topologically non - trivial background. 6) This configuration will be useful later. The functions K(r), J±(r) and Dyona are thus the right objecys. In any case, quantum monopoles are dyons. are raal as reauired by the Hermiticity of Vy, and £ . We can also take h(r) I'ta, we otall carry our1 analysis for the dyon configuration depicted in to be real as this minimizes the energy. o'-iuatLons (?,3). Quantum mechanically, the fernion condensate which forms abound tho dyon can be taken as the "zero-energy" bound state. Thus the The boundary conditions at infinity are defined by requiring the finite- "Size" of this bound state really determines the "size" of the mono—fermion ness of the energy, which in turn requires the fields to approach the Higgs hairon. We ghall see shortly that for the fundamental nonopole of SU(5) the vacuums away from the origin. In the phenomenologioally interesting two-stage 5 uir.e is of the order 0(m ~ ), rather than 0(»i ~ ) as is commonly believed. breaking case, the $-Higgs vacuum is attained at 0(10 ) GeV, whereas the H-Higga vaouum is attained at 0(10 ) GeV. Therefore, geometrically, the Now let ua consider the Dirac equation for one of the doublets in Kq.4, fundamental aonopole of SU(5) has the following nested structure : There is -h-
-3-
-- - " say Ui t*), in the background of a SIJ(5) fundamental dyon and look for zero Indeed, h(r) does not change sign ; it increases exponentially from lero to energy solutions. its vacuum value around r - m ~ . Thus the lower limit of the integral starts around m ~ . Therefore, all that eq.(lO) is telling us is that 4f, H, vanish identically for r J,ro ~ , but can be nonzero for r < m ~ , a huge region (5) surrounding the monopole core (extends over twelve orders of magnitude). Thus the equations for the aero energy s—wave are the same as those of Jackiw where G ia the standard Higgs coupling strength of the ^ and 10_ fermions (ci U^ and Rebbi (without their Higgs term) inBide the Salam-Weinberg domain, Introducing the 2x2 matrices ft1 ( + and - stand for the upper and lower oomponanta of the Dirac apinor), Although the dyon equations are more complicated, the WQ being like a second Higgs in the adjoint representation (which does not participate in "M-^ - On*w T-™ i,B,n « 1,2. {6) the mass generation), we expect the massive fermion case for the dyon to we can reduce the Dirac equation, for S = 0, into the following form follow the same pattern as in the monopole ca.se. 1 T*. V 'Vl±- i i . f iwW We have just demonstrated that the Higgs induced mass for the fernions = o does not effect the results on the existence of zero modes which were obtained for the massless case, although Hubakov himself was worried that in the case where m(r) - X Gh(r) and w(r) - i(k^-l). of massive fermions higher order corrections could destroy hia boundary Next we decompose the matrices "f¥l~ as conditions, thus invalidating his analysis. Our results have a very simple explanation if we recall the meaning of spontaneously generated maBB for the -TfTl- = <^i(^+ T^T, a - 1,2,3. (8) fermions. Kass for fermions is a well-defined concept only in the ^Higgs and go through the same manipulations a3 in reference(i) to get vacuum far away from the monopole. But the monopole"s neighbourhood is a region where different symmetry phases appear in a nested fashion. So a "massive" feroion approaching a monopole will experience changes in this attribute as it moves closer, by crossing different phase boundaries. As it crosses the Salam — Weinberg boundary, at i " , it quickly loses its naes and from then on, in a huge region, moves as a massless partiole. Our analysis shows, however, that the fermion mass is not entirely without effects on In contrast to ref .(l), we definitely cannot conclude from this physics in the neighbourhood of a monopole. Although the condensate was expression that the fermion wave function vanishes identically, because of attending out to infinity like r~ in the maesless case , now it is confined the extra terms due to the dyons. Thus the existence of the zero energy states inside r < i " , So a monofermion system is not a hadron with size m ~ , is not ruled out by this argument. However, let ua see whether vre get the but rather the size is - m ~ . Therefore, the cross section for the fermion result of ref.(l) by restricting ourselves to a static monopole configuration. number violating process will be 0(m "" ), agreeing with a previous result of In this case, eq..(9) reduces to eq.(3.8) of ref.(l), due to the pure imaginary WilcsselJ; nature of the second term in eq.(9) i The implications of a catalysis cross section of the order of m ~ are important in the context of recent attempts to measure monopole induced proton decay i If we take our arguments as correct, then the geometrical (10) cross section for catalysed proton decay is
-5- -6- 9) REFERENCES Combining this with the Parker bound for the monopole flu* 1. T.F. Walsh, P. Veisz, and T.T. Vu, DBS! preprint, 83 - 022, March 1983. < 6 X 1O"16 cm.'2 sr"1 a"1 (12) 2. V.A. Rubakov, JETP Lett. 33., 644 (1981). we get V.A. Eubakov, Nucl Phye. B2O3.311 (1982). X lO-*7 sr"1 a"1 Fm* (13) C-G. Callan Jr., Phys. Bev. D2£, 2141 (1982). which is orders of magnitude lower than the recently reported 1MB CO. Callan Jr., Phys. Rev. H26, 2058 (1982). result I 3. C.P. DokOB and J.N. Tomaras, Phys. Rev. D21, 2940 (i960). F X 37 Tc > 3.5 X 1O yrs. (15) A. Polyakov, JSTP Lett. 20, 430 (1974). Such a long lifetime would seem to be beyond the capabilities of current 6. E. Witton, Phye. Lett. 86B, "&*, (1.979). experiments. N. Christ and H. Jackiw, Phys. Lett. 91B, ??8 (i960). ».K. Pak, Prog, of Theor.Phys. 64., 2187 (1980). Here we would like to comment on the D"B conclusion that the flux is H.K. Pak and R. Percacci, Nucl. Phys. Bl88, 355 (1981). < 7.2 X 10~15 cm~2 sr"1 s"1. (16) 7. F. Hilczek, Phya. Rev. Lett. 48, 1146 (1982). We feel that thiB upper limit on the flux cannot be taken seriously, as 8. S. Errede, et. al. Phya. Bev. Lett. £1, 245 (1983). what is actually measured in the experiment is P^ I ^ ; and F^ is deduced by assuming a strong interaction value for <^ . If we use our 9. K. Turner, E. Parker and T.J. Bogdan, Phys. Sev. D26, 1296 (1982). value for ^ , we find that the 1KB experiment would yield a much less 10. E. Piorini, Talk at the European Physical Society Conference, Brighton stringent limit on the flux July, 1983. 5.4 X 10"8 cm^sr"1 a"1. (17) We Bee that this bound 1B just about Cabrerra'a present limit l0>of cm"2 sr"1 a"1 (18) Both these limits are far above the Parker bound. However, we feel tint the calculation of the flux from a measurement of F X -T- -8-