Non-Conservation of Baryon Number at High Energy in the Standard Model Citation Espinosa, Olivier R. (1990) Non-Conservation of Baryon Number at High Energy in the Standard Model. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/74cf-xv77. https://resolver.caltech.edu/CaltechETD:etd-06062007-160632 Abstract We calculate the coupling constant and energy dependence of the scattering amplitudes for baryon- and lepton-number violating processes in the context of the standard model, in the semiclassical approximation. It is found that, to leading order in this expansion, the spin-averaged total cross sections for these processes grow as a power of the CM-energy and thus violate the bound imposed by unitarity. This result has a twofold implication: first, perturbation theory in the instanton sector of the electroweak theory must break down at high energies and, second, it strongly suggests that baryon and lepton number non-conservation might be observed experimentally at energies accesible in the near future. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Preskill, John P. Group: Caltech Theory Thesis Committee: Preskill, John P. (chair) Frautschi, Steven C. McKeown, Robert D. Zachariasen, Fredrik Defense Date: 25 May 1990 Record Number: CaltechETD:etd-06062007-160632 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-06062007-160632 DOI: 10.7907/74cf-xv77 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2482 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 13 Jun 2007 Last Modified: 11 Feb 2022 17:55 Thesis Files PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

NON-CONSERVATION OF BARYON NUMBER AT HIGH ENERGY IN THE STANDARD MODEL Thesis by Olivier R. Espinosa In· Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1990 (Submitted May 25, 1990) 11 ACKNOWLEDGEMENTS I would like to thank my advisor, John Preskill, for his encouragement and guidance and, in particular, for having interested me in the subject of this thesis. I would also like to thank my fellow graduate students, particularly Arun Gupta, Tim Yen, and most especially Sandip Trivedi for having listened to me in times of confusion and for enjoyable conversations, some of them even on physics. I was very fortunate to have Sandip as a classmate and friend. Many thanks go to him, among many other things, for his friendly encouragement in times when my research was not going well and for providing an example to me of love for and perseverance in physics. My warmest gratitude goes to my father, whose support has been invaluable to me. I wish I could have thanked my mother, for all her love. Above all, I would like to thank my wife, Nina, for her continuing support and love throughout the years of graduate school and for all the sacrifices she made to enable me to pursue my studies. I dedicate this thesis to her, and to the memory of my mother. lll ABSTRACT We calculate the coupling constant and energy dependence of the scattering amplitudes for baryon- and lepton-number violating processes in the context of the standard model, in the semiclassical approximation. It is found that, to leading order in this expansion, the spin-averaged total cross sections for these processes grow as a power of the CM-energy and thus violate the bound imposed by unitarity. This result has a twofold implication: first, perturbation theory in the instanton sector of the electroweak theory must break down at high energies and, second, it strongly suggests that baryon and lepton number non-conservation might be observed experimentally at energies accesible in the near future. IV TABLE OF CONTENTS Acknowledgements 11 Abstract Ill Table of contents IV Chapter I. Introduction 1 Chapter Baryon and lepton number non-conserving Green functions 2.1 Anomalous currents in a simplified standard model 2.2 Anomalous Green functions in Euclidean space 2.3 Index theorem in the presence of Yukawa couplings Chapter III. Saddle point expansion of the bosonic functional integral 3.1 Constrained instantons 3.2 Collective coordinates 3.3 Momentum space Green functions in the Gaussian approximation Chapter IV. Fermionic zero mode in the constrained instant on background 4.1 The fermionic zero mode Chapter V. Cross sections for anomalous processes 5 .1 Fourier transforms of the fermionic zero mode and the constrained instanton 5.2 The scattering amplitude for anomalous processes 5.3 High energy behavior of anomalous cross sections and violation of unitarity to leading order 5 .4 Conclusions 5 9 12 16 22 23 27 31 33 34 36 Appendices Appendix A Appendix B Appendix C 42 References 44 38 39 1 Chapter Introduction It has been known for some time that baryon and lepton numbers are not exactly conserved in the standard model. Due to the chiral- nature of the fermionic representations the baryonic and leptonic currents suffer from [SU(2)]2 and [U(1)]2 anomalies [1]. However, any nonvanishing scattering amplitude that violates baryon and lepton number conservation must involve (in Euclidean space) topologically nontrivial large gauge field configurations with Euclidean actions greater or equal than 81r 2 / g 2 , where g is the SU(2) coupling constant. These instanton configurations [2] describe tunnel- ing events of the gauge field under the potential barrier that separates topologically distinct vacua. Thus the rate for any process mediated by tunneling of the gauge field is (for g small) suppressed by the exponentially small factor exp(-l61r 2 / g2). Taking this potential barrier picture seriously, several authors [3-5] suggested that these anomalous processes can be unsuppressed (enhanced considerably) if high temperatures or high energies are involved: instead of tunneling through the barrier the gauge ( and Higgs) field can pass over the barrier. The discovery [5,6] of a static and unstable solution of the classical equations of motion for the SU(2) Yang-Mills-Higgs field theory which interpolates between topologically distinct vacua and therefore corresponds to the configuration sitting at the top of the barrier (the so-called sphaleron), made possible a semi-quantitative estimate of the magnitude of the rate of transitions over the barrier at finite temperatures [3, 7]. These authors further computed the corresponding dissipation rate of the baryonic asymmetry of the universe and concluded that the effect is big enough to wipe out an initial baryonic asymmetry generated at an earlier epoch (such as GUT baryogenesis). The connection between transitions over the barrier and baryon and lepton production is rather ad-hoc in their calculations, however. (In Ref. [8] it is claimed that although fluctuations in FF can be unsuppressed at high temperatures, 2 the net rate of baryon and lepton number generation is negligible due to the absence of occupation number factors in the rates.) In an attempt to reconcile their sphaleron-based calculation of the baryonic asymmetry dissipation with a more conventional one based on individual anomalous scattering events, Arnold and McLerran suggest in Ref. [9] that although the scattering amplitude for a purely fermionic process like q + q _,. (3n 1 - 2)q + 3n 1 l (n 1 is the number of families) is exponentially suppressed, an anomalous scattering event containing many gauge bosons and/ or Higgs particles in the final state might not be suppressed. The motivation to consider the latter kind of processes comes from the suggestion that a classical object like the sphaleron with energy Esp rv mw / aw and size rv m; 1 has typical Fourier components k rv mw and thus is likely to decay into rv 1/aw W's with typical momenta mw, They claim that a Green function of the type ( ( qqqlr 1 An) although proportional to the small factor exp( -81r 2 / g 2 ) is also proportional to (1/gr, the 1/g's coming from the instanton configuration. So naively, for large n the latter factor can compensate for the smallness of the former, avoiding the exponential suppression. This observation cannot be taken too seriously however, since perturbation theory around the instanton breaks down for n too large. There is a peculiar point in the previous reasoning however: for a given small value of the coupling constant g it seems that the scattering amplitude should be largest for largest n. This is not what one would expect for the emission of weakly coupled gauge bosons. Furthermore, it would be hard to understand how the unitarity equation for a process like 3n 1 q+n 1 l-,. 3n 1 q+n 1 l that can produce real intermediate states containing T¥-bosons could work, for in the equation A Ji - Aif = L i(21r) 8C \pi - Pm)AJmAin, 4 4 (1.1) m where i = 3n 1 q+n 1 l and f = 3n 1 q+n 1l, the LHS carries no factors of 1/g (apart from the usual (g- 8 ) 2 coming from the collective coordinate Jacobian) whereas a term in the RHS containing an intermediate state with n gauge bosons would carry a factor of (1/g) 2n. 3 The purpose of this thesis is to study the coupling constant and energy dependence of the baryon and lepton number violating amplitudes involving in general also some number of gauge and Higgs bosons, in leading order in the semiclassical expansion (g 2 ~ 0, keeping mH and mw fixed). It is essential for the feasibility of the calculation to introduce a Higgs field that spontaneously breaks the SU(2) gauge symmetry completely, since otherwise the S-matrix is not defined due to the severe infrared divergences of the theory (reflected in an instanton calculation in the divergent integration over the instanton size p). We also investigate the effects of small Yukawa couplings (i.e., fermion masses) on the anomalous scattering amplitudes. The main results of our leading order calculation are the following: i) for fixed gauge and Higgs boson masses, the amplitude for a scattering process involving n 2 gauge bosons and m Higgs bosons is proportional to gn+me- 81r f g2. Thus, the emission of an extra gauge or Higgs boson is suppressed by an extra power of g. ii) surprisingly enough, amplitudes grow with external momenta as pNtl 2 qn, where NJ, n are respectively the number of fermions and gauge bosons participating in the scattering process, and p, q represent respectively typical fermion and gauge boson momenta. We find thus that the unitarity bound is not respected by the leading term in the semiclassical expansion. Work substantially overlapping with the work presented in this thesis has been published in a recent paper [16]. The author of Ref. [16] has performed a calculation very similar to ours, reaching basically the same qualitative conclusions as we do. However, he neglects to include the effect of the global isospin orientation of the constrained instanton (see chapter 3), which does induce a nontrivial dependence of the anomalous scattering amplitudes on momentum transfers. This nontrivial dependence of the amplitudes on external momenta makes the corresponding cross sections grow faster with energy, which would make the anomalous processes experimentally detectable at fairly lower energies than estimated in Ref. [16] (provided, of course, higher order corrections, which should be responsible for the unitarization the cross sections at high energies, do not make the amplitudes start falling off when they are still too small.) 4 The rest of this thesis is organized as follows: after setting up the problem in chapter 2, our main calculational tool, the method of constrained instantons developed by Affleck [10], is described in some length in chapter 3, and used to obtain a simple expression for the relevant momentum space Green functions. In chapter 4 we study the fermionic zero modes in the field of the constrained instanton for nonzero Yukawa couplings. The result for the total spin-averaged cross section for the process q + q --t (3n 1 - 2)q + 3n 1 l + nvV + nH and its high energy behavior, to leading order in the semiclassical approximation, are presented in chapter 5. Our conclusions are discussed at the end of that chapter. Technical details are relegated to three appendices. 5 Chapter Baryon and lepton number non-conserving Green functions 2.1 Anomalous currents in a simplified standard model Our aim is to compute a semiclassical approximation to the baryon- and leptonnumber-nonconserving scattering amplitudes in the standard model with n 1 families of quarks and leptons. Right from the outset we make several simplifications: a) We neglect interfamily mixings, that is, we set the (generalized) KobayashiMaskawa matrix equal to the identity. b) We neglect strong interaction corrections, that is, we set g 8 = 0. c) We set the U(l) coupling constant g 1 equal to zero. We think this is a reasonable approximation since on the one hand the Weinberg angle is small (sin2 0w c::::'. 0.22) and on the other, although the anomalous divergence equation for the baryon and lepton currents involves g 1 in addition to the SU(2) coupling constant g, (F F)u(I) fluctuates around zero in the semiclassical expansion. The model reduces therefore to a SU(2) chiral gauge theory with the following matter content: one scalar Higgs doublet H and n 1 identical families of twocomponent fermions each containing four left-handed doublets lL and qLa and seven right-handed singlets eR and uRa, clRa, where a= 1, 2, 3, is the SU(3) color index. As we shall soon see, the multiplicity of families as well as of quark colors is trivially taken into account. Thus, we only need to consider the role of one lepton field lL, eR and one quark field qL, uR, clR in the scattering process. 6 Componentwise, The Lagrangian (in Minkowski space) is: £ = £YM + £H + n1 3 (Jam) a=l L (£z + I: £qa), (2.1) where l pµv , (2.2a) CH= (DµH)t(DI' H) - ¾cHtH - v 2 ) 2 , (2.2b) £1 = illaµDµlL + ielaµ8µeR - (gzl1HeR + h.c.), (2.2c) £YM = - 2tr Fµv £q = iqlaµDµqL + iuk<iµ8µuR + idk<iµ8µdR - (guqlcH*uR + gaq1H dR + h.c. ), (2.2d) and (2.3) a Dµ = 8µ - igA: ~ . (2.4) The hypercharge assignments are those of the standard model. In (2c, d) aµ = (l, if), cfµ = (l, -if), E = i7 2 = (_~ ~) and the phases of the fermionic fields have been adjusted so that gz, 9u and ga are positive. 7 Although lepton and baryon numbers are global symmetries of£, at the quantum level the corresponding currents are not conserved [1]: (2.5) where (2.6a) (2.6b) and (2.7) We shall calculate the "anomalous" scattering amplitudes by the LSZ procedure: we first compute the corresponding multiparticle Green function in Euclidean space and after continuing it back to Minkowski space we amputate the propagators for the external particles and put the latter on mass shell. One must exercise some care when writing down the fermionic piece of the Euclidean space Lagrangian if one wants to obtain an SO( 4) invariant object. The subtlety here is that complex conjugation does not interchange the two spinor representations of SO( 4). Defining A- and B-type spinors in SO( 4) (the analogs of Rand L-spinors of the Lorentz group, see appendix A), the correct transition is made as follows: if 1/JR and 1/JL are two-component Weyl spinors appearing in (Minkowski space) £, then one has to substitute (2.8) and -µ -0- --+ -0-µ ( -a-, ➔ ') i . (2.9) 8 In this way the Euclidean counterparts of eqns. (2.2a-d) are: ~ £YM = 1 (2.10a) 2tr FµvFµv, l,H = (D,,H)t(D,,H) + i(HtH - v 2 )2, (2.10b) (2.10c) ~ -1-,q a a . t D µqB - 'l,UBO'µ . t - µUA + 'I,'dtBO'µ - µd A - guqAE t H* UA iqAaµ - 9dq1HdA + 9uu1HTEqB - 9dd1HtqB. (2.10d) Defining Q=GJ, G:), (2.11) (2.12) L= and iJQ = iaµDµ -guEH* -gdH 9uHTE -iciµ8µ 0 -g'JHt 0 -icfµ8µ DL = ( icr,,D,, -91H) -gzHt -ici,t8µ (2.13) (2.14) ' we can write (2.15a) and (2.15b) 9 2.2 Anomalous Green functions in Euclidean space We are interested in the Fourier transform into momentum space of Green functions of the type (Ol1f'(x1) · · · 1P(xp)A(y1) · · · A(yn)r,(z1) · · · r,(zm)IO) = j 1Jµ(,P, A, H) e-SE ,P(x1) · · · ,P(xp)A(y1) · · · A(yn)1J(z1) · · · 1J(zm) j 1Jµ(,P, A, H) (2.16) e-SE where we have suppressed all (Euclidean) spacetime and isospin indices and 1/', A, 77 denote any fermionic field, the gauge field, and the (shifted) Higgs field respectively. 'Dµ( 11', A, H) denotes the path integral measure including the appropriate gauge fixing and ghost terms. The fermionic piece of the path integrals in (2.16) can be computed exactly in terms of the backgroud bosonic fields A, H since the fermionic fields appear quadratically in SE. More generally, we consider the generating functional (2.17) where w(x) and wt(x) are, in general, independent multi-component objects, their entries being two-component Grassmann spinors. (( x) and (( x) are the external sources, and iJ is, in general, a nonhermitian operator (with respect to the standard inner product (1/', x) = J d4x 1f' t x). The result for Z is: N Z[(,(] = N' IT vi(() IT ttj(()[Det'(iJtiJ)] i=l j=l 112 exp J d4xd4y((x)S'(x,y)((y), (2.18) 10 where i = 1, ... ,N', (2.19a) i = 1, ... , N, (2.19b) N, N' being the number of normalizable zero modes 'l/Joi, <poi of iJ and !Jt respectively: o, i = 1, ... ,N, (2.20a) !Jt <poi = 0, i = 1, ... ,N', (2.20b) D'l/Joi = chosen such that (2.20c) The meaning of the primed objects in (2.18) is explained in appendix B, where a rough derivation of formula (2.18) is given as well. Green functions are obtained by differentiating Z[(, (] with respect to ( and/or ( and then putting ( = ( = 0. It is clear that from the sector with no zero modes (N = N' = 0, and hence no Ui or Vi factors) the only nonvanishing Green functions are those containing an equal number of \J! and wt fields which therefore conserve any fermionic number. Only the sector of the sum over histories over the bosonic fields A, H that leads to the existence of normalizable zero modes of either of the operators iJ, !Jt contributes to an anomalous Green function. It is possible to prove (see Section 3.3) that the index of the operators DQ and DL (recall that index A = dim ker A - dim ker At) is independent of the Higgs field configuration and depends only on the Pontryagin index of the SU(2) gauge field as in the pure Yang-Mills case. (One might argue that this is a trivial result since the index of an operator is unchanged by continuous deformations of the operator. This is not 11 so, however, because finiteness of the Euclidean action implies a nontrivial topology for the Higgs field as well [11]. Moreover, the configuration H = 0 has infinite action and hence cannot be taken as the "undeformed" configuration.) The result above is a reflection of the fact that the anomalous divergence equation (2.5) is independent of the scalar Higgs field [12]. We shall consider the simplest case: iJ has exactly one (normalizable) zero mode and iJt none. Then the simplest nonvanishing Green function is J 1)\lf 1)whl!(x) exp{ - J d4x(wtiJw)} = __J:_z[(, (] I 8((x) (=(=O = ¢o(x )Det '(iJtiJ). (2.21) In our theory (2.1) we have 4n 1 independent fermionic fields to integrate over. So the simplest anomalous Green function must contain p = 4n 1 different fermionic fields and violates L-number and R-number by n 1 units. It is proportional to (2.22) where (2.23) and the subscript v = 1 indicates that we are to integrate over the winding number one bosonic configurations. To very good approximation, widely separated instantonanti-instanton configurations will not contribute to the integration since they imply too many zero modes. In the following chapter we shall perform a semiclassical evaluation of (2.22). 12 2.3 Index theoren.1 in the presence of Yukawa couplings We want to compute the index of the operators DQ and DL defined in (2.13) and (2.14). Using the equality of the nonzero eigenvalues of the operators iJtiJ and iJiJt (see (B.1-3)), the index of the operator D, index iJ = dim ker iJ - dim ker iJ, (2.24) can be written as AtA A At indexD = Tr e-tD D -Tr e-tDD , A (t > 0). (2.25) To get a manageable form for (2.25) we write £=n,Oi where a, /3 are spin indices and a, b are isospin indices. Using the completeness of the eigenfunctions of the hermitian operator fJtjJ and a momentum representation for the delta function inside brackets in the last formula, we can write (2.26) and thus the index of the operator iJ can be computed as (2.27) where the trace is both over spin and isospin indices. 13 vVe will first consider (2.28) where we have absorbed the coupling constants g and gz into the definition of the fields. Using formulas ( C.6) one gets (2.29) (2.30) It is seen that !JliJL and !JLiJ1 can be obtained from each other by interchanging O" µv +-+ cf µv and aµ +-+ cfµ• We will compute (2.27) in the limit t -+ 0, where several simplifications occur. First we split -tbl DL = tA + tB and -t!JL!J1 = tA + tB', where (2.31) and B' can be obtained from B replacing O"µv -+ cfµv, and interchanging O"µ +-+ cfµ. Using then the operator identity limet(A+B) = limetAetB = limetBetA, t-o t-o t-o (2.32) we can write (2.33) Since the index of the Dirac operator DL is a gauge invariant quantity, without loss of generality we can require the gauge field to satisfy the gauge condition BµAµ = 0, 14 so that the following simple formula holds: e tD 2 ik-x ik-x -t(kµ,-Aµ,) 2 e =e e . (2.34) Hence, from (2.27, 2.33, 2.34) we have (2.35) Rescaling kµ = k~/ 0, the k-integral becomes which in the limit t -+ 0 has the value (16?r 2 t 2 )- 1 and is proportional to the identity matrix in isospin space. Thus, etB and etB' in (2.35) need to be expanded up to order t 2 only: t2 tr (etB - etB) = t tr (B - B') + -tr (B 2 - B' 2 ) + O(t3). I 2 B - B' is easily seen to have vanishing trace in isospin space, whereas for the term of order t 2 one finds tr (B 2 - B' 2 ) = tr (aµvO'af3 - aµvaa13)tr FµvFa/3 + tr (aµav - aµav )tr (DµH)(DvH)t + 2tr (aµav - aµav )(DµH/ (DvH) (2.36) where we have used equations (C.6) and (C.7), and the tracelessness of the matrices O' µv and cfµv• 15 We have proven therefore that 1 ; 4 ~ mdex DL = - -2 d x tr FµvFµv, l61r o A (2.37) where (2.38) completely independent of the Higgs field H. An analogous calculation for iJQ shows that index !JQ = index DL. (2.39) 16 Chapter Saddle point expansion of the bosonic functional integral 3.1 Constrained instantons Our strategy is to perform a semiclassical evaluation of the integral (2.22). From now on we will drop the prime in S~ so that SE will stand for the right-hand-side of equation (2.23). It is convenient to work in terms of the rescaled fields 1 Aµ--+ -Aµ, g 1 H--+ y'AH, (3.1) so that where we have defined (3.3) and (3.4) and now Fµv = OµAv - OvAµ - i[Aµ, Av], (3.5) Dµ = 8µ-iAµ- (3.6) From (3.2) it is seen that the semiclassical approximation, n --+ 0, corresponds to taking g 2 --+ 0, keeping K, and (ef>) fixed. (This is, mH = v'Av = (ef>) and mw = gv/./2 = #(ef>) should stay fixed as g2 --+ 0.) 17 The following analysis has been given by Affleck [10]. We want to expand SE around some stationary field configuration and approximate (for g small) the full integral by its Gaussian approximation. When (</>) -::j; 0, SE has no nontrivial stationary points, however. This is seen from a simple scaling argument: for a given finite action configuration Aµ(x), H(x) we consider the rescaling Aµ(x)-+ aAµ(ax), H(x)-+ H(ax) (which preserves the finite action boundary conditions), under which SE scales as Thus, given any field configuration we can always rescale it to get a smaller action (except in the trivial case Aµ== 0, H == constant). From the point of view of the Euler-Lagrange equations of motion, (DµFµv Ta -H- K, t - iKHt 2 8vH - 2A~HtH == 0, (3.7) (D H)a - ½(HtH - (¢,) )Ha = 0, 2 2 the situation is peculiar, however. We know that for (</>) == 0 a finite action solution with unit winding number of (3. 7) exists, namely the instanton Aµ= 2xv 2 X + p2Tµv, H = 0. (3.8) (For the definition and properties of the matrices Tµv and r µv, see Appendix C.) For (</>) small enough ( (</>) ~ 1/ p) we should expect to find a solution of (3.7) in perturbation theory in p( </>) that reduces to (3.8) (actually, a gauge transformation of it, as explained below) when (</>) -+ 0. The finite action boundary conditions (Fµv -+ 0, DµH -+ 0, HtH -+ (</>) 2 faster than 1/ x 2 as x -+ oo) take different forms in different gauges. In the particularly 18 convenient gauge where H -+ (cp)h_ at infinity, where h.. is a constant isospinor with h..th.. = 1, finiteness of the action requires Aµ -+ 0 faster than 1/x 2 at infinity. The solution (3.8) does not have this asymptotic behavior; it actually approaches a pure gauge at infinity: (3.9) where (3.10) To obtain the desired asymptotic behavior of the instanton, we have to recast it in the so-called "singular gauge" (obtained from (3.8) by the gauge transformation singular at the origin n- 1 ): 2p2 Aµ = X 2 ( X 2 +p 2 ) XvAµ 11 , (3.11) where Aµv = T µ 11 for the instanton (T µ 11 and T µv get interchanged when one goes from the regular gauge to the singular gauge and vice versa) but in general Aµv will stand for Urµ 11 Ut or UrµvUt, where the SU(2) matrix U represents the orientation of the instanton. Aµ 11 satisfies the commutation relations of the so( 4) Lie algebra. If we neglect terms of order (p(cp) ) 2 then equations (3.7) are approximately solved by (3.11) and . H= ( (In fact, (3.11,3.12) satisfy DµFµv 2+p2) x2 X 1/2 (cp)h_. (3.12) = 0, D 2 H = 0 and render the source terms negligible.) A perturbative solution of (3.7) around the zeroth-order solution (3.11,3.12) does not exist because the appropiate finite action boundary conditions for the higher order terms cannot be enforced. The operators that act on higher order terms possess zero modes that determine a priori the behavior of these terms at infinity (which is not the required one). 19 A way of getting around this difficulty is to extremize SE subject to a constraint. We follow Affleck [10] who chooses to constrain the spacetime integral of the sum of two gauge invariant local operators of dimension greater than four and vanishing rapidly enough at infinity, one depending only on the gauge field and the other only on the Higgs field. If we insert the identity (3.13) in the functional integral (2.22), then the relevant stationary configuration will be a solution of the constrained Euler-Lagrange equations (3.14) Given OA and OH, the Lagrange multipliers aA and aH are to be determined order by order in perturbation theory in p( cp) in order to enforce the correct boundary conditions for the higher order terms. The "constrained instanton" A, fl is the unique solution of (3.14) obtained by this procedure that reduces to (3.11,3.12) when p(cp) - t 0. It is easy to see how the constrained inst anton behaves at large distances (x ~ p). Since the extra terms added to (3. 7) decay more rapidly than the linear mass terms as H - t (c/J)h. and Aµ - t 0, at infinity A, fl - (c/J)h. become proportional to the solutions of 8µ(8µAv - 8vAµ) - K, 2 2 (c/J) Av= 0, oµoµiI - (c/J) iI = o. 2 (3.15) In terms of the function (3.16) 20 (where K1 is a modified Bessel function) which satisfies (3.17) the solutions to (3.15) can be written as (3.18) where mw =#,(cf>)= gv/0, and mH =(cf>)= v'Xv are the gauge boson and Higgs masses respectively. Gm (x) decays exponentially at infinity and has the series expansion 1 1 Gm(x) = x 2 + m--lnmx + (const.)m ? 2 2 + ••• 1 1 2 2 1 = x 2 + m lnmp + m (const. + lnx/ p) + .... 2 (3.19) 2 From (3.19) one can determine the constants of proportionality in (3.18) and the asymptotic behavior of the terms in the correct perturbative expansion of the constrained inst anton (which contains logarithms of m). One finds Aµ (x) = Aoµ (x; p) + (p(cf>)) 2 A 1 µ(x; p) + •••, fl ( x) = Ho (x; p) + (p (cf>)) ln p(cf>) H1 ( x; p) + · · · , 3 (3.20) where Aoµ(x; p) and Ho(x; p) are given by (3.11,3.12). The higher order terms in these expansions generally diverge at infinity and therefore (3.20) is useful only at small distances ( x /f> p). 21 For large distances ( x ~ ( </> )- 1 ) one finds expansions of the type Aµ(x) = p2 [A~(x; (</>)) + (p(</>) ) 2 At(x; (</>)) + •••], 2 0 fl ( x) = p [ H ( x; (</>)) + (p( </>) )2 H ( x; (</>)) + •••], 1 (3.21) where (3.22) All the terms in these expansions vanish exponentially at infinity but blow up as x~o. It turns out that the ansatz Aµ(x) = XvAµvA(x 2 ), (3.23) H(x) = cp(x 2 )h., where Ji is a constant isospinor and A, </> are real functions of x 2 , reduces the constrained Euler-Lagrange equations (3.14) to two coupled real nonlinear equations for A and </> (at least for the particular choice DAA) = i tr F 3 = i tr FµvFv>.F>,.µ and OH ( H) = (ntH - (</> )2 )3, but this should be true in general). The perturbative expansions for A and</> are analogous to (3.20) and (3.21) with 2p2 Ao= X 2 (x 2 + p2 ) ' 112 (x2: p2) ('P), (3.24) q,o = ('P) - lp2mH('P) K1(:Hx). (3.25) 'Po = The constrained instanton resembles the instanton (3.11,3.12) at short distances but decays exponentially at large distances (recall that I<n(z) ~ e-z / ~ as z ~ oo). 22 The action of the constrained instanton can be calculated using expansions (3.20) and (3.21 ). One finds j d4x F (A) = 16Jr2 + 0 ((p(<f,) ) tr 4 2 J IDµH12 = d4x ], 2Jr2 p2(<f,)2 + 0 [(p(<f,) )4 ln p(ql)], j d4x (HtH - (<f,) 2) = 0 [(p(<p) ) In p( <f,)], 2 4 (3.26a) (3.26b) (3.26c) so that (3.27) For ,\ (and g) small, only those values of p such that p(</>) ~ ~ ~ 1 will contribute appreciably to the p-integral present in (2.22) after we use (3.13). In what follows we shall work to lowest order in p(cp). To this order the operators OA and OH play a role in the integration over the constrained bosonic fluctuations (see (3.34)), but otherwise they drop out of the calculation altogether. 3.2 Collective coordinates It has been noted that A and ¢> satisfy a set of equations that is independent of the matrix U appearing in Aµv and of the isospinor /J.. This is because the action SE(Aµ = xvAµvA(x 2 ), H = ¢>(x 2 )!J.) is invariant under independent global rotations of Aµ and H. In fact, the possibly noninvariant Higgs kinetic term takes the form independently of U and fl., where the prime denotes d/ dx 2 • Thus we see that in the constrained instanton the gauge field and Higgs field orientations are not correlated. The latter, however, is completely fixed once we choose our spontaneously-broken vacuum state. Conventionally we take h_ = (~). 23 The orientation of Aµ remains a symmetry of the constrained instanton action and therefore has to be treated as a collective coordinate along with the position xo of the constrained instanton (of course we are assuming that the gauge choice is invariant under independent global rotations of the gauge field). The size of the instanton is no longer a collective coordinate, but it acts like one since it parametrizes a family of local minima of the action when the constraint is imposed. 3.3 Momentum space Green functions in the Gaussian approximation Schematically we have (,P(x1) · · · 1J(zm)) ~ J JJ d!xo dU dp !':-,.(p, (¢,) )J(p, (¢,)) j 'Dµj_(A, H)8(C(A, H)) [Dee .z)tb(A, H)] 2n1 e-SE(A,H)'l/Jo(x1 - xo) · · · rJ(Zm - xo), (3.28) where J(p, (cp)) is the Jacobian factor coming from the transition to the collective coordinates xo, U, the measure 'DµJ_ does not include fluctuations along the zero modes associated with the collective coordinates, and C(A, H) denotes the constraint in (3.13). The Gaussian approximation gives (,P(x1) · · · 1J(zm)) = j d!xo j dU j dp !':-,.(p, (¢,) )J(p, (¢,) )e-SE(A,H) F(p, (¢>); µ)'l/Jo(x1 - xo) · · · 77(zm - xo), (3.29) where F(p(ef>); µ) includes the (regularized) contributions coming from the fermionic determinants and the quadratic fluctuations about the constrained instanton, µ being the renormalization point. 24 When we go to momentum space, the integration over the position of the constrained instanton xo (which ensures translational invariance) produces a total momentum conserving delta function. Defining (21r) 4 84 (p1 + · · · + q1 + · · · + k1 + · · ·) G~ ( {p}, {q}, {k} ) = j d x1 · · · d!yl : · · d4z1 · · · ein-x, · · · iq,-y, · · · eik,-z, · · · (if,(x1) · · · A(y1) · · · 17(z1) · · ·), 4 we can write for G: G( {p}, {q}, { k}) = j dU j dp ii(p, (¢,) )J(p, (¢,) )e- 8E(A,H) F(p, (</>); µ)'l/Jo(p1) · · · A(q1) · · · 'fJ(k1) · · ·, (3.30) where 'l/Jo(p), Aµ(q) and 'fJ(k) are the Fourier transforms of the zero mode and the constrained instanton respectively: (3.31) etc. Dimensional analysis requires f),.(p, (</>) )J(p, (</>) )F(p, (</>); µ) ~ P2n 1 -s f(p(</>), pµ) ~ p2n,-5 [fo(pµ) + (p(</>))2 fi(pµ) + .. ·]' (3.32) where f, Jo, Ji, etc., are dimensionless functions of the indicated variables. The coupling constant dependence of the factors in (3.30) is obtained as follows: i) from the Jacobian factor J we pick up a factor g- 7 [13], g- 4 coming from the zero 25 modes associated with translational invariance, and g- 3 coming from the zero modes associated with global isospin invariance. ii) ~(p, (c/>)) defined in (3.13) is given by (3.33) Thus ~ carries no factors of g; iii) The integration over constrained bosonic fluctuations in (3.28) is of a form analogous to the finite dimensional integral (3.34) (only linear fluctuations need to be kept inside the delta function in the Gaussian approximation) where A is proportional to g- 2 . The powers of g 2 coming from the determinant get cancelled out against powers of g present in the (rescaled) measure, but a factor g- 1 coming from A- 1 survives (this factor corresponds to the one coming from the collective coordinate pin the case (c/>) = O); iv) finally, we get n factors g- 1 and m factors ,\- 112 = y!K,g- 1 from the external gauge and Higgs fields. In fact, in (3.30) A(q) and ry(k) are the Fourier transform of the gauge and Higgs components of the constrained instanton respectively. The rescaled fields appearing in (3.2) do not have the correct normalization to create relativistically normalized one particle states. To get these fields we have to undo the rescaling (3.1), which has the effect of multiplying the expressions for A and cf> in (3.24), (3.25) by the factors g- 1 and ,\- 1 / 2 respectively. Putting everything together we can write, using (3.27) and (3.32) (3.35) where now g and 11, are understood to be running coupling constants at the scale µ. 26 The function fo(pµ) was computed by 't Hooft [14]: fo(pµ)=c(pµ) 43-8nf 6 , (3.36) ensuring the RC-invariance of the scattering amplitudes (to lowest order). The constant c is given by (3.37) ( c '.'.:'.: '.'. 1.03 x 10 19 for n 1 = 3). In the following chapter we give the explicit forms of the zero modes of DQ and DL in the field of the constrained instanton. Their Fourier transforms, as well as the Fourier transform of the constrained instanton, are given in chapter 5. 27 Chapter IV. Fermionic zero n-iode in the constrained instanton background 4.1 The fermionic zero mode We shall consider the slightly more general case of DQ first ( the result for DL can then be obtained by letting 9d --1' gz and gu --1' 0). We want to solve the following system of simultaneous differential equations: (4.la) (4.lb) (4.lc) in perturbation theory in p(cp). Here ?Ju,d = A- 112 gu,d and Aµ, Hare given by (3.20). We assume that 9u,d are small ( at least of order A1 / 2 ) so that ?Ju,d are of order one. The ansatz (4.2) where <.p carries spin and isospin degrees of freedom, reduces equations (4.1) to the simpler Du<.p(u) + i?Juc/J(u)uA(u)c!l/ + igdcp(u)dA(u)h_ = 0, (4.3a) ?Juc/J(u)h_Tc<.p(u) - 2iu:(u) = 0, (4.3b) -gdcp(u)h_tc.p(u) - 2id~(u) = 0, (4.3c) where (4.4) u = x 2 = xµxµ and the prime denotes d/du. Upon substituting A= Ao and¢= ¢a 28 (see (3.24)) it is seen that to lowest order in p(c/>) (4.3a-c) are solved by <p = <po, (4.5) ( 4.6a) (4.6b) Here xµcf µ<po is the (normalized) zero mode of the Dirac operator iJ = O"µDµ in the field of the pure Yang-Mills instanton (in the singular gauge) Aµ = AoxvUTµvUi". The explicit form of <po is (4.7) where <p 8 is the singlet in the coupled spin-isospin space, satisfying (cf• f)<ps = -3<p 8 and <p!<ps = 1, and the matrix U acts on isospin indices. From (4.6a, b) we find ( 4.8a) (4.8b) where ut is now acting on the two-component spinors indicated. As with the constrained instanton, the correct forms of the perturbative series for <p, uA and dA are obtained analyzing the behavior of the solutions of (4.3a-c) at infinity. In this limit uA-* 0 and cf>-* (cf>) so that (4.3a-c) reduce to (4.9a) (4.9b) (4.9c) 29 From (4.9a) we can solve for uA and dA (at infinity): (4.10a) (4.10b) Substituting these expressions in (4.9b, c) respectively, we find equations for the behavior of h/e.<p and h.t <p at infinity: u(b/E<p)" + 3(b/E<p) 1 - ~m~ (!J/E<p) = 0, 1 (4.11) u(h.t <p)" + 3(h.t <p), - 4m~(h.t <p) = 0, where we have identified mu,d = 9u,d(cp). The solutions of these equations that vanish at infinity and match the leading x ~ p term of h_Te.<po and h.t <po respectively, when (cp) -+ 0 can be written as (4.12a) and (4.12b) and consequently (4.13a) and (4.13b) We note that uA, dA are correctly suppressed by mu, md respectively. 30 We have shown, therefore, that the zero mode of Dq in the field of the constrained instanton, as in the case of the latter, can be expressed perturbatively in p(c/>) by means of two series expansions analogous to (3.20) and (3.21 ). The respective leading terms in these series are given by (4.8a) and (4.13a) for uA, and (4.8b) and (4.13b) for dA. For uB = -l{EqB and dB = h_tqB we have, putting together (4.2),(4.7) and (4.12a, b): (4.14a) (4.14b) The expressions for the components eB, eA, vB of the zero mode of DL can readily be obtained from (4.14b), (4.8b,4.13b) and (4.9a) respectively. For the massless neutrino the large distance expansion becomes: (4.14c) with a power law falloff. 31 Chapter V. Cross section for anomalous processes 5.1 Fourier transforms of the fermionic zero mode and the constrained instant on The Fourier transforms we need to compute are of the generic forms (5.la) (5.lb) where p = (PµPit) 112 • For high energies, IPµI ~(¢),we expect the integrals (5.1) to be sensitive to the form off for small values of r. As we shall see, however, the pole term of these integrals turns out to depend only on the long range tail off. The functions f are integrable at the origin and decay exponentially at infinity. We cannot use the perturbative expansion f (x 2 ) = f o( x 2 ; p) + (p (cp)) 2 ln p(cp) Ji (x 2 ; p) + (p(cp)) 2 h (x 2 ; p) + ·· · (5. 2) in the whole integration range since the functions Ji, h, ... do not decay to zero at infinity (in fact, all but a finite number of them diverge there). The correct thing to do is to arbitrarily split the integration range into the two intervals (0, R) and (R, oo) and use (5.2) in the first, but the large distance expansion of f in the second. It is clear that the first integral will have no singularities in the finite complex p-plane. The only singularities in p may come from the integration over the second interval. It is easy to see that if f decays at infinity as ra e-M r, then the second integral will exist for all p inside the circle of radius M in the complex p-plane but will diverge 32 for p = ±iM, since Jn(±iz) = (±it In(z), n integer, and In(z) grows as ez / vz as z -+ oo. This singularity is actually a pole (as expected): in general, in (n == 1, 2), to low~st order in p(</>), f will have the form (see (3.25,4.13,4.14)) and the relevant integral will be the second integral being again an entire function of p 2 (I<n(z) rv z-n, Jn(z) rv zn as z -+ 0). Ref. [15] gives us (5.3) After amputating the propagators for the external particles and putting the latter on mass shell, only the residue of the pole piece of the Fourier transforms survive. These are given by: [Aµ(P)Lmp = 47f" 2 i p2pvUTµ11ut, (5.4a) [11(p) Lmp == - 27[" 2 p2 (<1>), (5.4b) [7PA(P)Lmp = -27f"i pm fut X, (5.4c) [7PB(P)Lmp = 27f"i ppµaµut X, (5.4d) where [f(p)]amp - p2_,._m2 lim (p 2 + m 2 )f(p) and X = (~1) for 7P = u, v, X = (~) for 7,b = d, e. 33 5.2 The scattering amplitude for anomalous processes The p-integral appearing in (3.35) can now be explicitly calculated. Collecting all the factors of p we have: (5.5) where 19 · 7 t = 12 + n + m + 3n1. A note on the applicability of equation (5.5) is in order. Although (5.5) is true for any values of the particle multiplicities n, m, the approximation of neglecting the terms of higher order in p(</>) in the action of the constrained instanton (3.27) will be justified only if the dominant contribution to the p-integral in (5.5) comes from the region where p(</>) ~ 1. Now, the function = 21r 2 Kj g 2 = 21r 2 / A, N = 1l +2n +2m + \4 n 1) appearing in the p-integration (5.5) 112 is peaked at p( </>) = (fcJ . Hence, if we want to treat the constrained inst anton in (a perturbation theory in p( </>), we must require that 13 14 41r 2 K N = -6 + 2n + 2m + -n 3 f ~ -g2- . (5.6) The amplitude for a scattering process that violates lepton and baryon numbers by n I units, involving the least possible number of fermions (4n 1 ) in addition to n gauge 34 bosons and m Higgs particles (n, m satisfying (5.6)), will therefore be equal to (5.7) where 19 7 c1 = 1.26 X 10 6 e 6 ·516 nif(- + n + m + -n 1 ) 12 3 ' (5.8) and all the information about spins, polarizations and momenta is encoded in the function h: Here x~ is a two-component spinor representing the spin of the i-th fermion, the xi's were defined in (5.4), and Pj is the 2 x 2 matrix that projects out the charge state of the j-th gauge boson (with polarization c(j)). 5.3 High energy behavior of anomalous cross sections and violation of unitarity to leading order After analytically continuing back to Minkowski space by means of p4 = -ipo, po = E > 0, and performing the U-integration (which enforces charge conservation), one can perform the average and summation over initial and final spins and polarizations respectively of the quantity If dU h(U)l 2 to obtain (5.10) where the ci's are numerical constants and each 7ri is a relativistically invariant function of the momenta Pl, ... , P4n 1 , q1, ... , qn, consisting in a product of factors ki • kj 35 and/or €µvo:J3kf kJkf k!/n (k = p or q), such that each fermion momentum appears once and each gauge boson momentum appears twice. For a scattering process of the type (5.11) the phase space integral for the total unpolarized cross section will be where S is the symmetry factor for identical particles in the final state. We can extract the dependence of D on the total energy in the extreme relativistic case (E = I.Pl). In terms of s = (p1 + p2)2, we can rescale Pi = VSXi, etc., to obtain D _ D(n n m) 8 6n1+2n+m-4 ' ' ,, (5.13) where To calculate the total cross section for the process (5.11), we need to give a value to the renormalization pointµ. Normally this is chosen in order to minimize the error introduced by neglecting higher order corrections of the type g~ ln sf µ 2 , which vanish if we set µ = VS· In the case at hand, however, we have two energy scales, vs and (<p), and we expect to get possibly large additional logarithmic corrections of the type g~ ln(</>)2 / µ 2 . We choose to set µ = (¢) = mH. Therefore, large logarithms will be unimportant only up to moderately large s, i.e., if number of order one. g;,m ln s/miJ stays less than some 36 Our result for the total unpolarized cross section for the process (5.11) in the extreme relativistic case is then (5.15) where c II (nt, n, m ) 7 )] 2 D~ ( nJ, n, m ) , = 7.94 x 10 11 e13 ·On f [r ( -19 + n + m + -nf 12 3 (5.16) and we have rewritten (</>) as mH and substituted "-mH = g;m/ AmH = 2m!/ m'fr. From (5.15) we see that the emission of an extra gauge boson or Higgs particle in an anomalous scattering process costs an extra power of g 2 in the cross section, and not an extra factor of g- 2 as conjectured in Ref. [9]. Therefore, as far as the semiclassical expansion is concerned, the unitarity equation (1.1) for anomalous processes has a chance to work. However, it is apparent from (5.15) that for s high enough a can be big even to the point of violating the boundedness requirement imposed by unitarity. Can this unphysical result be attributed to the breakdown of our approximations when s is very large? Only a detailed analysis of higher order corrections can tell us the answer. These should include corrections to the stationary point approximation of the path integral (2.22) in the one instanton sector, as well as contributions from multiinstanton-anti-instanton configurations, treated beyond the dilute gas approximation. 5.4 Conclusions We have calculated in the semiclassical approximation the energy and coupling constant dependence of cross sections for baryon and lepton number violating processes involving the least possible number of fermions and an arbitrary number of gauge and Higgs bosons. 37 The calculated cross section displays the unphysical behavior of growing like a power of the total CM-energy squared s, in contradiction with the unitarity-bound. We must conclude therefore that perturbation theory in the instanton sector breaks down for large energies, although we have no idea as to what the mechanism responsible for this might be. (That perturbation theory may be completely unreliable when high energies are involved was suggested in Ref. [6], in an analogy with the quantum pendulum. If this were so, the most we could expect is that our results are valid only when vs is much less than the height of the potential barrier between topologically distinct vacua, that is, the sphaleron energy Esp f"-.1 (1.5-2.7)41rv'2v / g = (1.5-2.7)81rmw/g 2 ). This breakdown of perturbation theory has nothing to do with the high multiplicity of particles that would be expected in a high energy ( B + L )non-conserving process: the unitarity bound is also violated in a purely fermionic anomalous process (n, m = 0). Our approach, however, is limited by the condition (5.6) to small multiplicities of gauge and Higgs bosons, and cannot be expected to apply to processes involving f"-.1 1/ aw W's or H's, claimed to be the relevant ones at high energy [9], although it strongly suggests that these processes may be, in fact, unsuppressed in this regime. A different method to calculate amplitudes at high energy and high multiplicity seems to be needed if one wants to draw any reasonable conclusions about the many particle processes that possibly play a role in baryon number violation at high energies. We would also like to point out, disagreeing with Ref. [8], that our calculation shows that the fermions involved in an anomalous scattering event can be in arbitrary momentum states ( consistent with four-momentum conservation), and therefore should carry the corresponding occupation factors that arise due to the Pauli exclusion principle in a given physical situation. 38 APPENDIX A For a rotation R in S0(4) we define SU(2) matrices A(R) and B(R) such that a four-vector in R 4 , Vµ, transforms as R --j, (A.l) where aµ is defined in (2.9). ((A.l) is an explicit realization of the group homomorphism S0(4) ~ SU(2) x SU(2).) A-type and B-type spinors transform then as (A.2) One can verify that the fermion bilinears 1/-Jl/J A, 'l/Jl'l/JB are SO( 4) invariants, whereas 'l/Jla µ"PA, 'l/J !a-µ'1/)B transform as SO( 4)-vectors. 39 APPENDIX We want to sketch the derivation of formula (2.18) for (2.17) In general, \J! is a "multi-spinor" like Q or L in (2.11, 2.12), containing both Atype and B-type spinors. The operators iJ, iJt do not have well-defined eigenvalue problems since they take A-spinors into B-spinors and vice versa. The hermitian operators !Jt!J and fJ !Jt, on the other hand, preserve the spinor type and thus have the well-defined eigenvalue problems, (B.1) (B.2) their eigenfunctions forming complete orthonormal basis sets for the objects \J! and (wt)t respectively. One can adjust the relative phase of 7Pn and c.pn such that for An -=I=- 0 the latter is given by 1 c.pn = An D'l/Jn• A (B.3) The zero modes of fJ and fJtiJ are the same ( and therefore this also holds for !Jt and iJiJt): iJtiJ'l/Joi = 0 D'l/Joi = o, i = l, ... ,N, (B.4) D iJt c.poi = 0 iJt c.poi = 0, i = 1, ... ,N'. (B.5) We expand N \J!(x) = L ai'lpoi(x) + L bn'l/Jn(x), i=l An#O (B.6) 40 N' wt(x) = Laicphi(x) + L bncpl(x), (B.7) II dai II daj II dbndbn. (B.8) i=l An#O and define the fermionic measure as '.D\J!'.Dwt = N N' i=l j=l n Upon substituting (B.6,7) into (2.17), the integration over nonzero modes results in ,., t,., ] 1/2 exp [ Det'(D D) J d4.xdLy -((x)S'(x, y)((y), (B.9) where "t" Det I (DD)= II ,\n, (B.10) 2 An#O and S'(x,y) = L 'Pn(xl<pl(y) An;ifO (B.11) n satisfies DxS'(x, y) = 5( 4 )(x - y)l Po(x, y), (B.12) N' Po(x, y) = L <poi(x)cpt(y) (B.13) i=l being the projector onto the subspace spanned by the zero modes of !Jt. The integrations over the zero modes are (B.14) 41 Jg N' ( N' da;exp ~?iu;(() ) = g N' u;((), (B.15) i == 1, ... , N', (B.16) i == 1, ... , N. (B.17) where j d4xcpbi(x)((x) V;(() = j d4x((x),f,o;(x) u;(() = Equation (2.18) is now obtained putting together results (B.9,14,15). 42 APPENDIX C Throughout this thesis the 2x2 matrices denoted by the letters a and T represent the same numerical matrices, but they act on spin and isospin spaces respectively. \Ve define (C.l) (C.2) where (C.3) Here cf are the Pauli matrices and the indices µ, v run from 1 to 4. a µ 11 and cfµ 11 are antisymmetric in µ, v. Explicitly they are: 1 O'ij = ciij = -r=,ijkO'k, 1 O'i4 (C.4) = -O'i4 = 20'i, for i,j = 1,2,3, which also shows that they are traceless. 0'µ 11 is selfdual and cfµ 11 antiselfdual: 1 0'µ11 = 2Eµ110:f30'0:f3, cfµ11 = - 1 (C.5) 2Eµ110:t3cf o:t3• There are a number of useful properties: cfµ0' 11 = 8µ 11 + 2iaµ 11 , (J' µ cf11 = 8 µ11 + 2i cf µ11' (C.6) 43 tr(crµvCia,B) = tr (cfµvcfa,a) = 1 (8µaDv,a- 8µ,aDva) + 1 2 2cµva,8, 1 1 (C.7) 2(8µaDv,B - Dµ,BDva:) - 2sµva:,8· The matrices er µv and 7Jµv satisfy the commutation relations of the so( 4) Lie algebra, e.g., (C.8) The 't Hooft symbols 'T/aµv [14] relate the S0(4) generators CTµv , aµv to the SU(2) generators era /2: era 2, era CTµv = 'T/aµv 2• CTµv = 'T/aµv (Recall that S0(4)=SU(2)xSU(2).) (C.9) 44 References [1] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tyupkin, Phys. Lett. 59B (1975) 85. [3] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. 155B (1985) 36. [4] V.A. Rubakov, Nucl. Phys. B256 (1985) 509. [5] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D30 (1984) 2212. [6] N.S. Manton, Phys. Rev. D28 (1983) 2019. [7] P. Arnold and L. McLerran, Phys. Rev. D36 (1987) 581. [8] A.G. Cohen, M. Dugan and A.V. Manohar, Phys. Lett. B222 (1989) 91. [9] P. Arnold and L. McLerran, Phys. Rev. D37 (1988) 1020. [10] I. Affleck, Nucl. Phys. B191 (1981) 429. [11] V.N. Romanov and V.A. Fateev, Theor. Math. Phys. 33 (1977) 1025. [12] K. Fujikawa, Phys. Rev. D29 (1984) 285, and references therein. [13] L. Gervais and B. Sakita, Phys. Rev. Dll (1975) 2943. [14] G. 't Hooft, Phys. Rev. D14 (1976) 3432, ibid.D18 (1978) 2199 (Erratum). [15] LS. Gradshteyn and I.M. Ryzhik, Table of integrals) series and products (Academic Press, 1980) p. 672. [16] A. Ringwald, Nucl. Phys. B330 (1990) 1.