he Oscillating Neutrino The Oscillating Neutrino

exception of neutrino oscillations. If oscillations are confirmed, the mixing Family Mixing and the Origin of Mass angles measured in the neutrino experiments will become part of a CKM mixing The difference between weak eigenstates and mass eigenstates matrix for the leptons. This sidebar derives the form of the CKM matrix and shows how it reflects Stuart Raby the difference between the rotation matrices for the up-type quarks (Q ϭϩ2/3) and those for their weak partners, the down-type quarks (Q ϭϪ1/3). This he Standard Model of elementary particle physics contains two disjoint difference causes the family mixing in weak-interaction processes and is an sectors. The gauge sector describes the interactions of quarks and leptons example of the way in which the Higgs sector breaks the weak symmetry. We T(fermions, or spin-1/2 particles) with the spin-1 gauge bosons that mediate will also show that, because the neutrino masses are assumed to be degenerate the strong, weak, and electromagnetic forces. This sector has great aesthetic appeal (namely, zero), in the Standard Model, the rotation matrices for the neutrinos can because the interactions are derived from local gauge symmetries. Also, the three be defined as identical to those for their weak partners, and therefore the CKM families of quarks and leptons transform identically under those local symmetries matrix for the leptons is the identity matrix. Thus, in the minimal Standard Model, and thus have the same basic strong, weak, and electromagnetic interactions. in which neutrinos are massless, no family mixing can occur among the leptons, The Higgs sector describes the interactions of the quarks and leptons with the and individual-lepton-family number is conserved. spin-0 Higgs bosons h+ and h0. This sector is somewhat ad hoc and contains many This discussion attributes the origin of mixing to the mismatch between weak free parameters. The Higgs bosons were originally introduced to break the weak eigenstates and mass eigenstates caused by the Higgs sector. A more fundamental isospin gauge symmetry of the weak interactions by giving mass to the weak understanding of mixing would require understanding the origin of fermion masses gauge bosons, the W and the Z0. The W and the Z0 must be very heavy to explain and the reason for certain symmetries, or approximate symmetries, to hold in why the weak force is so weak. But in the Standard Model, interactions with those nature. For example, a fundamental theory of fermion masses would have to Higgs bosons are also responsible for giving nonzero masses to the three families explain why muon-family number is conserved, or only approximately conserved. 0 0 of quarks and leptons. Those interactions must yield different masses for the parti- It would also have to explain why the K – Kෆ mixing amplitude is on the order 2 cles from different families and must cause the quarks from different families to of GF and not larger. The small amount of family mixing observed in nature mix, as observed in experiment. But neither the nine masses for the quarks and puts severe constraints on any theory of fermion masses. Developing such a theory charged leptons nor the four parameters that specify the mixing of quarks across is an outstanding problem in particle physics, but it may require a significant families are determined by any fundamental principle contained in the Standard extension of the Standard Model. Model. Instead, those thirteen parameters are determined from low-energy experi- To discuss mixing as it appears in the Standard Model, it is necessary to explic- ments and are matched to the free parameters in the Standard Model Lagrangian. itly write down the parts of the Standard Model Lagrangian that contain the By definition, weak eigenstates are the members of the weak isospin doublets Yukawa interactions between the fermions and the Higgs bosons (responsible for that transform into each other through interaction with the W boson (see Figure 5 fermion masses) and the weak gauge interaction between the fermions and the W on page 38). Mass eigenstates are states of definite mass created by the interaction boson (responsible for charge-changing processes such as beta decay). But first, with Higgs bosons. Those states describe freely propagating particles that are iden- we must define some notation. As in the sidebar “Neutrino Masses” on page 64, tified in detectors by their electric charge, mass, and spin quantum numbers. Since we describe the fermion states by two-component left-handed Weyl spinors. c c c the Higgs interactions cause the quark weak eigenstates to mix with each other, Specifically, we have the fields ui, di, ui , di , ei, ␯i , and ei , where the family the resulting mass eigenstates are not identical to the weak eigenstates. index i runs from one to three. The ui are the fields for the three up-type quarks u, Each set of eigenstates provides a description of the three families of quarks, c, and t with electric charge Q ϭϩ2/3, the di are the fields for the three down- and the two descriptions are related to each other by a set of unitary rotations. type quarks d, s, and b with Q ϭϪ1/3, the ei stand for the three charged leptons Most experimentalists are accustomed to seeing the Standard Model written in e, ␮, and ␶ with Q ϭϪ1, and the ␯i stand for the three neutrinos ␯e, ␯␮, and ␯␶ c the mass eigenstate basis because the quarks of definite mass are the ingredients with Q ϭ 0. The fields ui and ui , for example, are defined as follows: of protons, neutrons, and other metastable particles that the experimentalists measure. In the mass eigenstate basis, the Higgs interactions are diagonal, and ui annihilates the left-handed up-type quark uL and creates the right-handed

the mixing across families appears in the gauge sector. In other words, the unitary up-type antiquark ෆuR in family i, and rotations connecting the mass eigenstate basis to the weak eigenstate basis appear c in the gauge interactions. Those rotation matrices could, in principle, appear in all ui annihilates the left-handed up-type antiquark uෆL and creates the right-handed the gauge interactions of quarks and leptons; but they do not. The Standard Model up-type quark uR in family i. symmetries cause the rotation matrices to appear only in the quark charge- † c† changing currents that couple to the W boson. To describe the Hermitian conjugate fields ui and ui , interchange the words c The specific product of rotation matrices that appears in the weak charge- annihilate and create used above. Thus ui, ui , and their Hermitian conjugates changing currents is just what we call the CKM matrix, the unitary 3 ϫ 3 mixing describe the creation and annihilation of all the states of the up-type quarks. matrix deduced by Cabibbo, Kobayashi, and Maskawa. The elements in the The down-type quark fields and the charged lepton fields are similarly defined.

CKM matrix have been determined by measuring, for example, the strengths For the neutrinos, only the fields ␯i containing the states ␯L and ෆ␯R are observed; c of the strangeness-changing processes, in which a strange quark from the second the fields ␯i are not included in the Standard Model. In other words, the Standard family of mass states transforms into an up quark from the first family. So far, Model includes right-handed charged leptons, but it has no right-handed neutrinos family mixing has not been observed among the leptons, with the possible (or left-handed antineutrinos).

2 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

0 The Weak Eigenstate Basis. We begin by defining the theory in terms of the density, whereas the spatial component is the flux. Similarly, J3 is the weak iso- weak eigenstates denoted by the subscript 0 and the color red. Specifically, the topic charge density. It contains terms of the form f † f , which are number opera- 0 0 – weak gauge coupling to the W is given by tors Nf that count the number of f particles minus the number of f antiparticles g present. When this density is integrated over all space, it yields the weak isotopic ϩ ␮ Ϫ ␮† w ᏸweak ϭ ϩᎏ (W␮ J ϩ W␮ J ) , (1) charge I3 : ͙2ෆ ␮ 0 3 w where the charge-raising weak current J is defined as ͵J3 (x)d x ϭ I3 .

␮ † ␮ † ␮ J ϭ Α u0i ෆ␴ d0i ϩ ␯0i ෆ␴ e0i , (2) Now, let us consider the Higgs sector. The fermion fields interact with the Higgs i weak isospin doublet (h+, h0) through the Yukawa interactions given by The Pauli Matrices for Spin-1/2 and the charge-lowering current J␮† is defined as Particles c 0 + c + † 0 † c + † 0 † ␮† † ␮ † ␮ ᏸYukawa ϭ Αu 0i (Yup )ij [u0j h Ϫ d0j h ] ϩ d 0i (Ydown)ij [u0j (h ) + d0j (h ) ] ϩ e 0i (Ylepton)ij [␯0j (h ) + e0j (h ) ] , The Pauli spin matrices generate all J ϭ Α d0i ෆ␴ u0i ϩ e0i ෆ␴ ␯0i . (3) i,j rotations of spin-1/2 particles. i Spin-1/2 particles have only two The constant g in Equation (1) specifies the strength of the weak interactions, and where Yup, Ydown, and Ylepton are the complex 3 ϫ 3 Yukawa matrices that give ␮ j j possible spin projections along, say the ෆ␴ is a four-component space-time vector given by (1, Ϫ␴ ), where the ␴ are the strengths of the interactions between the fermions and the Higgs bosons. ϩ1/2, and the standard Pauli spin matrices for spin-1/2 particles with j ϭ x, y, z, the spatial Because the Higgs fields form a weak isospin doublet, each expression in brackets ؍ the 3-axis: spin up, or s3 spin down, or s3 ϭ Ϫ1/2. ⌻he step-up directions. These 2 ϫ 2 matrices act on the spin components of the spin-1/2 fields is an inner product of two weak doublets, making an isospin singlet. Thus, each operator ␴ϩ raises spin down to spin and are totally independent of the family index i. Each term in the charge-raising term in the Lagrangian is invariant under the local weak isospin symmetry since The charge-raising weak Ϫ c up, the step-down operator ␴ and charge-lowering currents connects states from the same family, which means the conjugate fields (for example, u 0) are weak singlets. The lepton terms in interaction in the first family lowers spin up to spin down, and ␴ the weak interactions in Equation (1) are diagonal in the weak eigenstate basis. In Equation (5) are introduced in the sidebar “Neutrino Masses” (page 64), where ϩ ␮ ϩ † ␮ 3 (W␮ J )first family ϭ W␮ u0 ␴ෆ d0 . gives the value of the spin projection fact, those interactions define the weak eigenstates. masses are shown to arise directly from the Yukawa interactions because h0 has a † ␮ 0 d along the 3-axis. The basis set for To understand the action of the currents, consider the first term, u0 ෆ␴ d0, in nonzero vacuum expectation value = v/͙ෆ2 that causes each type of fermion the spin quantized along the 3-axis the charge-raising current J␮. It annihilates a left-handed down quark and creates a to feel an everpresent interaction. These interactions yield mass terms given by → u is given by left-handed up quark (d0L u0L) and, thereby, raises the electric charge by one ϩ Ϫ → c 0 c 0† c 0† 1 0 unit. Electric charge is conserved because the W field creates a W (see top dia- ᏸYukawa ᏸmass ϭ u 0i (Yup )ij u0j ϩ d 0i (Ydown)ij d0j ϩ e 0i (Ylepton)ij e0j . (6) ␮ ΂ 0΃ and ΂ 1΃ , gram at right). The first term in the charge-lowering current J † does the reverse: † ␮ c d0 ෆ␴ u0 annihilates a left-handed up quark and creates a left-handed down quark Notice that each term in ᏸmass contains a product of two fermion fields f0 f0, W → and the matrices are given by (u0L d0L) and, thereby, lowers the electric charge by one unit; at the same time, which, by definition, annihilates a left-handed fermion and creates a right-handed Ϫ ϩ 0 1 0 Ϫi the W field creates a W (see bottom diagram at right). Thus, the members of fermion. Thus, these Yukawa interactions flip the handedness of fermions, a pre- ␴1 ϭ ␴2 ϭ ΂ 1 0΃ ΂ i 0 ΃ each pair u0i and d0i transform into each other under the action of the charge- requisite for giving nonzero masses to the fermions. These terms resemble the A down quark changes to an up raising and charge-lowering weak currents and therefore are, by definition, a weak Dirac mass terms introduced in the sidebar “Neutrino Masses,” except that the quark with the emission of a WϪ. 1 0 isospin doublet. The quark doublets are (u0, d0), (c0, s0), and (t0, b0), and the lep- matrices Yup, Ydown, and Ylepton are not diagonal. Thus, in the weak eigenstate ␴3 ϭ ΂ 0 Ϫ1 ΃ . ton doublets are (␯e0, e0), (␯␮0, ␮0), and (␯␶0, ␶0). The first member of the doublet basis, the masses and the mixing across families occur in the Higgs sector. w w has weak isotopic charge I3 ϭ ϩ1/2, and the second member has I3 ϭ Ϫ1/2. Defining the matrices ␴Ϯ as Finally, note that J␮ and J␮† are left-handed currents. They contain only the The Mass Eigenstate Basis and the Higgs Sector. Let us examine the theory in c fermion fields f0 and not the fermion fields f0 , which means that they create and the mass eigenstate basis. We find this basis by diagonalizing the Yukawa Ϯ 1 1 2 ␴ ϭ ᎏᎏ΂␴ Ϯ i␴ ΃ , annihilate only left-handed fermions f (and right-handed antifermions fෆ . The matrices in the mass terms of Equation (6). In general, each Yukawa matrix is The charge-lowering weak 2 0L 0R) right-handed fermions f0R (and left-handed antifermions fෆ0L) are simply impervi- diagonalized by two unitary 3 ϫ 3 transformation matrices. For example, interaction in the first family c ^ one arrives at the following ous to the charge-changing weak interactions, and therefore, the f are weak the diagonal Yukawa matrix for the up quarks Y is given by Ϫ ␮† Ϫ † ␮ 0 up (W␮ J )first family ϭ W␮ d0 ␴ෆ u0 . commutation relations: isotopic singlets. They are invariant under the weak isospin transformations. ^ R L† (7) u d Weak isospin symmetry, like strong isospin symmetry from nuclear physics and Yup ϭ Vu YupVu , [␴3, ␴ Ϯ] ϭ Ϯ 2␴ Ϯ , and the symmetry of rotations, is an SU(2) symmetry, which means that there are three ϩ Ϫ 3 R c [␴ , ␴ ] ϭ ␴ . generators of the group of weak isospin symmetry transformations. Those genera- where matrix Vu acts on the right-handed up-type quarks in the fields u 0, and L tors have the same commutation relations as the Pauli spin matrices. (The Pauli matrix Vu acts on the left-handed up-type quarks in u0. The diagonal elements ␮ ^ + matrices, shown at left, generate all the rotations of spin-1/2 particles.) The J and of Yup are (␭u, ␭c, ␭t ), the Yukawa interaction strengths for all the up-type quarks: ␮† ϩ ^ ^ J are the raising and lowering generators of weak isospin analogous to ␴ and the up, charm, and top, respectively. Matrices Ydown and Ylepton are similarly Ϫ ␮ c W ␴ . The generator analogous to 1/2␴3 is J3 given by diagonalized. If u0 and u 0 are the fields in the weak eigenstate, the fields in the mass eigenstate, uc and u, are defined by the unitary transformations An up quark changes to a down ␮ † ␮ † ␮ † ␮ † ␮ ϩ J3 ϭ1/2 Α u0i ෆ␴ u0i Ϫ d0i ෆ␴ d0i ϩ ␯0i ␴ෆ ␯0i Ϫ e0i ␴ෆ e0i , (4) quark with the emission of a W . i c c R L† u 0 ϭ u Vu and u0 ϭ Vu u . (8) and the time components of these three currents obey the commutation relations 0 0† 0 † † [J , J ] ϭ 2J3 . In general, the time component of a current is the charge Since the Vs are unitary transformations, V V ϭ VV ϭ ⌱, we also have

4 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

c c R † L L u = u 0 Vu and u = Vu u0 . difference between the rotation matrices for the up-type quarks Vu and those for L the down-type quarks Vd . It is that difference that determines the amount of In this new mass basis, ᏸmass in Equation (6) takes the form family mixing in weak-interaction processes. For that reason, all the mixing can be placed in either the up-type or down-type quarks, and by convention, the CKM c ^i 0 c ^i 0 c ^i 0 ᏸmass ϭ Α u i Yup ui ϩ di Ydown di ϩ e i Ylepton ei matrix places all the mixing in the down-type quarks. The weak eigenstates for the i down-type quarks are often defined as d′: c ^ i c ^ i c ^ i ϭ Α u i Mup ui ϩ di Mdown di ϩ e i Mlepton ei , (9) i d′ = V d ϭ VL V L† d ϭ VL d , (14) ^ i ^ i CKM u d u 0 where the matrices M ϭ Y vր͙ෆ2 are diagonal, and the diagonal elements are just the masses of the fermions. In particular, we can write out the three terms for the in which case, the up-type weak partners to d′ become u′: up-type quarks u, c, and t: ′ L u ϭ Vu u0 ϵ u . c ^ i c 0 c 0 c 0 Α u i M up ui ϭ ␭u u u ϩ ␭c c c ϩ ␭t t t i When all the mixing is placed in the down-type quarks, the weak eigenstates for c c c ϭ ␭u v/͙ෆ2 u u ϩ ␭c v/͙ෆ2 c c ϩ ␭t v/͙ෆ2 t t the up-type quarks are the same as the mass eigenstates. (We could just as easily place the mixing in the up-type quarks by defining a set of fields u′ given in c c c ϭ mu u u ϩ mc c c ϩ mt t t , (10) terms of the mass eigenstates u and VCKM.) Independent of any convention, the weak currents J␮ couple quark mass eigenstates from different families. The form with the masses of the up, charm, and top quarks given by of the CKM matrix shows that, from the Higgs perspective, the up-type and down-type quarks look different. It is this mismatch that causes the mixing across mu ϭ ␭uvր͙ෆ2 , mc ϭ ␭cvր͙ෆ2 , and mt ϭ ␭tvր͙ෆ2 . quark families. If the rotation matrices for the up-type and down-type left-handed L L quarks were the same, that is, if Vu ϭ Vd , the CKM matrix would be the Thus, the Higgs sector defines the mass eigenstate basis, and the diagonal elements identity matrix, and there would be no family mixing in weak-interaction of the mass matrices are the particle masses. processes. The existence of the CKM matrix is thus another example of the way in which the mass sector (through the Higgs mechanism) breaks the weak isospin Mixing in the Mass Eigenstate Basis. Now, let us write the weak gauge interac- symmetry. It also breaks nuclear isospin symmetry (the symmetry between tion with the W in the mass eigenstate. Recall that up-type and down-type quarks), which acts symmetrically on left-handed and g right-handed quarks. ϩ ␮ Ϫ ␮† R ᏸweak ϭ ϩ ᎏ (W␮ J ϩ W␮ J ) , Note that the mixing matrices V associated with the right-handed fermions 2ෆ ͙ do not enter into the Standard Model. They do, however, become relevant in but to write the charge-raising weak current J␮ in the mass eigenstate, we extensions of the Standard Model, such as supersymmetric or left-right-symmetric substitute Equation (8) into Equation (2), models, and they can add to family-number violating processes. Finally, we note that, because the neutrinos are assumed to be massless in the ␮ † ␮ † ␮ J ϭ Α u0i ෆ␴ d0i ϩ ␯0i ෆ␴ e0i Standard Model, there is no mixing matrix for the leptons. In general, the leptonic i analog to the CKM matrix has the form † L ␮ L† † ␮ ϭ Α ui (Vu )ikෆ␴ (Vd )kjdj ϩ ␯i ෆ␴ ei i,k,j L L† Vlepton ϭ V␯ Ve . † ␮ † ␮ ϭ Αui ෆ␴ (VCKM)ijdj ϩ ␯i ෆ␴ ei , (11) i,j But we are free to choose any basis for the neutrinos because they all have the and to rewrite the charge-lowering current J␮†, we substitute Equation (8) into same mass. By choosing the rotation matrix for the neutrinos to be the same as L L Equation (3): that for the charged leptons V␯ ϭ Ve , we have

␮† † ␮ † ␮ L† L† J ϭ Α d0i ෆ␴ u0i ϩ e0i ෆ␴ ␯0i ␯0 ϭ Ve ␯ and e0 ϭ Ve e . i † L ␮ L† † ␮ ϭ Α di (Vd )ikෆ␴ (Vu )kjuj ϩ ei ෆ␴ ␯i The leptonic part of, for example, the charge-raising current is i,k,j † ␮ † † ␮ † ␮ † L ␮ L† † ␮ ϭ Α di ෆ␴ (VCKM) ij uj ϩ ei ෆ␴ ei , (12) Α␯0i ෆ␴ e0i ϭ Α ␯i (Ve )ijෆ␴ (Ve )jkek ϭ Α␯i ෆ␴ ei , i,j i i,k,j i and the leptonic analog of the CKM matrix is the identity matrix. This choice L L† (13) where VCKM ϭ Vu Vd . of eigenstate would not be possible, however, if neutrinos have different masses. On the contrary, the neutrinos would have a well-defined mass eigenstate and there Thus, the charge-raising and charge-lowering quark currents are not diagonal would likely be a leptonic CKM matrix different from the identity matrix. It is this in the mass eigenstate basis. Instead, they contain the complex 3 ϫ 3 mixing leptonic mixing matrix that would be responsible for neutrino oscillations as well → ■ matrix VCKM. This matrix would be the identity matrix were it not for the as for family-number violating processes such as ␮ e ϩ ␥.

6 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science