A central concept in understanding the Standard Model of particle physics, and relating experimental results to the 19 fundamental parameters in the Standard Model Lagrangian, is the running of the parameters as a function of energy: the β functions. Understanding the behavior of β functions is also essential in any attempt to go beyond the Standard Model. The running of the gauge couplings and the discovery of asymptotic freedom [1,2] was a crucial step in the development of the Standard Model and the running of the Higgs quartic coupling, and the top quark Yukawa coupling is important for the stability of the Higgs sector at high energies [3,4].

The running of the parameters in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [5] is not so well known perhaps for two reasons: the measured magnitude of the quark Yukawa couplings is too small for the running of the CKM parameters to have any physical relevance, and, even at one-loop, the β functions are not simple. The running of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [6,7] is more relevant beyond the Standard Model, and the mixing matrices in both the quark and the lepton sectors can be treated in parallel with the same techniques: for the Standard Model with the addition of three gauge singlet, Weyl, right-handed neutrinos the analysis is identical to that of the quark sector.

The β functions for the CKM parameters were investigated in [8,9], in the limit where the top Yukawa coupling yt dominates, and it was observed there is an infrared fixed point. The full one-loop β functions for the CKM matrix were calculated in [10]. The one-loop β functions for mixing matrices were further analyzed in [11–17] but the full expressions are somewhat complicated, and they are significantly simpler in the small angle approximation: approximate one-loop β functions, with one dominant Yukawa coupling and/or at least one small mixing angle, were analyzed in [18,19] (for higher loops, see Refs. [20–25]). However these approximations obscure an analytic pattern in the β functions which shall be explored in the present paper.

We shall therefore study the analytic form of the full one-loop massless renormalization group (RG) equations for the mixing matrix: fixed points are determined and the matrix of anomalous dimensions is calculated at each of the fixed points (full two-loop β functions are given in [26,27], but we leave this for later analysis). The β functions are of course scheme dependent, but the existence of fixed points, and the eigenvalues of the matrix of anomalous dimensions at the fixed points, are scheme independent. There are six fixed points¹

After this work was complete we became aware of Ref. [25] where these six fixed points were also found.

The parameters in the mixing matrices consist of three angles and one phase, parametrizing a space that is topologically the double coset U(1)×U(1)\SU(3)/U(1)×U(1),where U(1)×U(1)≈T is the Cartan torus of SU(3) (generated by λ3 and λ8 in the Gell-Mann representation). The right-coset SU(3)/U(1)×U(1) is the flag manifold F3, a compact complex manifold that admits a metric with isometry group SU(3). The left action of T on F3 has six fixed points. and these are precisely the six fixed points of the one-loop RG running.

If the RG running is lifted to F3, which can be done in a well-defined manner, and it is assumed that the left action of T on F3 commutes with this running, then a proof is given²

I thank Charles Nash for pointing out the significance of the fact that these actions commute for the fixed points.

The one-loop running is reviewed in Sec. II and a technique for extracting the running of the mixing parameters from the running of the Yukawa matrices is presented in Sec. III. As a simple example the special case of two generations, the Cabibbo angle, is presented in detail in Sec. IV before the three generation one-loop β functions for the mixing matrices are presented in Sec. V. The eigenvalues of the matrix of anomalous dimensions at each of the fixed points are determined in Sec. VI, and presented in detail in Appendix C. A proof that fixed points of the left action of T on F3 must be fixed points of the RG is given in Sec. VII. The significance of the results and possible future developments are discussed in Sec. VIII. Full expression for the one-loop β functions are given in Eqs. (21), (22), and Appendix B.

The Yukawa couplings in the Standard Model are LYukawa=-∑a¯,b=13(Ya¯bΨ¯La¯ΦcΨRb+Ya¯b′Ψ¯La¯ΦΨR′b)-H.c.,(1)where Φ=(ϕ+ϕ0) is the Higgs doublet; Φc=-iσ2Φ* the conjugate Higgs; ΨLa are left-handed SU(2) doublets; ΨRa and ΨR′a right-handed singlets. The Yukawa couplings Ya¯b and Ya¯b′ are 3×3 complex matrices, with a,b=1,2,3 labeling generations.

In the quark sector ΨL=(uLcLtLdLsLbL),ΨR=(uR,cR,tR)andΨR′=(dR,sR,bR),but the ensuing analysis is equally applicable to the Standard Model with the leptonic sector extended by three right-handed gauge-singlet Weyl neutrinos, in which case³

Majorana neutrinos are not considered here: if the neutrinos are Majorana the analysis would be more complicated as there are more mixing parameters and extra terms involving Majorana masses [29].

There are 18 parameters in Ya¯b and Ya¯b′ but, as is well known, these are not all physical. The Weyl fermions ΨL and ΨR can be rotated by two different U(3) transformations in generation space, to bring Ya¯b to real diagonal form, with just three Yukawa couplings; a further eight parameters can be removed from Ya¯b′ with a further SU(3) transformation on ΨR′, leaving 10 parameters in Ya¯b′. But we are still free to perform individual U(1) phases transformations on the three generations in ΨL (compensating with phases transformation on ΨR to keep Ya¯b real) to remove another three parameters from Ya¯b′, leaving seven parameters in Ya¯b′: three real Yukawa couplings and the four mixing parameters of the CKM (or PMNS) matrix.

The one-loop running of the Yukawa matrices Y and Y′ can be determined by standard techniques; the Feynman diagrams that contribute are shown in Appendix A. Using dimensional regularization, with t=lnμ, the massless running is [10], dYdt=116π2[32(YY†-Y′Y′†)+4π(Π-α)1]Y,(2)dY′dt=116π2[32(Y′Y′†-YY†)+4π(Π-α′)1]Y′,(3)where Π arises from summing over fermion loops in the Higgs leg of the Yukawa vertex and α and α′ are quadratic combinations of the gauge couplings α3, α2, and α1 (their explicit form will not be relevant for the following analysis).⁴

Explicitly, Π=14π(3Tr(YqYq†+Yq′Yq′†)+Tr(Yl′Yl′†)), where q stands for quarks and l for leptons (for νSM there is a fourth term: 14πTr(YlYl†)). For quarks α=8α3+94α2+1712α1 and α′=8α3+94α2+512α1, while for leptons α=94α2+34α1 and α′=94α2+154α1, αi=gi24π being the gauge couplings (α1 is hypercharge).

It is convenient to express the running in terms of Hermitian matrices Z=14πYY†andZ′=14πY′Y′†,in terms of which dZdt=14π{3Z2-32(ZZ′+Z′Z)+2(Π-α)Z},(4)dZ′dt=14π{3Z′2-32(ZZ′+Z′Z)+2(Π-α′)Z′}.(5)

The Hermitian matrices Z and Z′ can be diagonalized with U and U′∈SU(3): Λ=U†ZU,Λ′=U′†Z′U′,(6)where the diagonal components of Λ and Λ′ are related to the usual Yukawa couplings by Λa=ya24π,Λa′=ya′24π.

If Z and Z′ do not commute, then U≠U′ and V=U†U′is the mixing matrix, the CKM matrix for quarks and the PMNS matrix for leptons.

If the right-handed neutrinos are Weyl, and not Majorana, the U(1) phases of the leptons, as well as the quarks, are unobservable and, for both quarks and leptons, the space of physical parameters in the mixing matrix V is the double coset U(1)×U(1)\F3. This is a four-dimensional space so there are four physical parameters in V.⁵

In [30] a proof is given that T\F3 is topologically S4, though it is not everywhere differentiable—in the same way as the surface of a cube, while not a differentiable manifold, is topologically S2. I thank Charles Nash for bringing my attention to this reference.

The physics is invariant under independent left and right actions of U(1)×U(1), V→hVh′†,with h acting on ΨR and h′ acting on ΨR′. Such transformations will be referred to as phase transformations, it being understood that h and h′ are global, but they may depend on the RG scale μ. We shall also refer to ya and ya′ as Yukawa couplings and (θ1,θ2,θ3,δ) as mixing angles and a phase.

Allowing for possible scale dependence t=lnμ in U(μ), U′(μ), the RG evolution of Eq. (6) gives dZdt=U(dΛdt-i[A,Λ])U†,(9)dZ′dt=U′(dΛ′dt-i[A′,Λ′])U′†,(10)where A=iU†dUdt,A′=iU′†dU′dt.(11)In terms of A and A′ dVdt=i(AV-VA′),(12)or dVdt=iBVwith B=-idVdtV†=A-VA′V†.(13)

The off-diagonal components of the two matrices A and A′ can be obtained from Eqs. (4) and (5), expressed in terms of the diagonal matrices Λ and Λ′: dΛdt=14π{3Λ2-32(ΛVΛ′V†+VΛ′V†Λ)+2(Π-α)Λ}+i[A,Λ],(14)dΛ′dt=14π{3Λ′2-32(Λ′V†ΛV+V†ΛVΛ′)+2(Π-α′)Λ′}+i[A′,Λ′],(15)by demanding that the right-hand side of each these equations is diagonal. Since A and A′ are Hermitian matrices in the Lie algebra of SU(3) they can both be expanded in terms of Gell-Mann matrices, A=AIλI, A′=AI′λI. However Eqs. (14) and (15) do not involve the diagonal components A3, A8, A3′, or A8′; (14) and (15) are sufficient to fix A1, A2, A4, A5, A6, A7 and A1′, A2′, A4′, A5′, A6′, A7′ in terms of the off-diagonal components of the anticommutators, {Λ,VΛ′V†}and{Λ′,V†ΛV},but A3, A8, A3′, and A8′ remain undetermined. A natural choice is to choose fermion phases which preserve the parametrization (8) of V at all energies. This requires that the four components dV11dt, dV12dt, dV23dt, and dV33dt of dVdt must be real and Eq. (12) then gives linear equations for A3, A8, A3′, and A8′, which can be solved to determine them uniquely as functions of the parameters in V and the off-diagonal components of A and A′, which have already been calculated.

An important consequence of this is that A and A′ do not depend on Π, α, or α′; they are completely determined by demanding that the off-diagonal components of -38π{Λ,VΛ′V†}+i[A,Λ]and-38π{V†ΛV,Λ′}+i[A′,Λ′]vanish, together with the phase convention that V˙11, V˙12, V˙23, and V˙33 be real. In this way A and A′ can be expressed uniquely as functions of the four parameters in V and the six Yukawa couplings in Λ and Λ′. Once A and A′ are known the β functions for the parameters in V are obtained from (12).

More generally one can use (7), with the weaker condition that only V˙11 and V˙33 are real. This is sufficient to fix the differences A3-A3′ and A8-A8′, but leaves the sums A3+=A3+A3′ and A8+=A8+A8′ arbitrary. Only δ=ϕ1+ϕ2+ϕ3 is physical; one is free to choose arbitrary energy-dependent phases ϕ1(t) and ϕ3(t) in order to determine A3+ and A8+, but all physical quantities are independent of A3+ and A8+ and only the combination δ remains. It is useful to perform all calculations in a general such “gauge,” as a check on our results: ϕ1, ϕ3, A3+, and A8+ will be left arbitrary, but they must drop out in the calculation of any physically quantity, only δ=ϕ1+ϕ2+ϕ3 should remain. This is the strategy that will be adopted in Sec. V, but we first examine the case of two generations to warm up.

Consider first the quark sector in the case of two generations, (ud) and (cs), when the parameters of the CKM matrix reduce to the Cabibbo angle θC, with θC∈[0,π2], and V reduces to the 2×2 matrix V=(cosθCsinθC-sinθCcosθC).Equations (14) and (15) give the off-diagonal components of i[A,Λ] and i[A,Λ′] from the off-diagonal components of the anticommutators 32{VΛ′VT,Λ}and32{VTΛV,Λ′}(16)respectively, with Λ=116π2(yu200yc2)andΛ′=116π2(yd200ys2).The anticommutators are purely real and A=(A3-3i64π2((yc2+yu2)(ys2-yd2)yc2-yu2)sin2θC3i64π2((yc2+yu2)(ys2-yd2)yc2-yu2)sin2θC-A3),A′=(A3′-3i64π2((yd2+ys2)(yc2-yu2)yd2-ys2)sin2θC3i64π2((yd2+ys2)(yc2-yu2)yd2-ys2)sin2θC-A3′),with A3 and A3′ yet to be determined. Demanding that dVdt is real in the two-generations version of (12), dVdt=i(AV-VA′),then requires A3=A3′=0, so A=364π2((yc2+yu2)(ys2-yd2)sin2θCyc2-yu2)σ2,A′=364π2((yd2+ys2)(yc2-yu2)sin2θCyd2-ys2)σ2.These expressions for A and A′ then give the Cabibbo angle β function for the two-generation version of (13) as B=A-VA′VT-A=A-A′=βCσ2with [8] βC=3sin2θC64π2[(yc2+yu2yc2-yu2)(ys2-yd2)+(ys2+yd2ys2-yd2)(yc2-yu2)].

It proves convenient to define the variables (similar variables will be used extensively in the next section) z=yc2-yu2yc2+yu2,z′=ys2-yd2ys2+yd2,which lie in the range [-1,1]: observationally these are both close to 1. In terms of z and z′ dθCdt=βC=3sin2θC32π2[yc2zz′(1+z)+ys2z′z(1+z′)].(17)There are fixed points of θC at θC=0 and π2.

Equation (17) is easily solved, with θC=θ0 at t=lnμμ0=0, to give tanθC=tanθ0e∫bdt≈tanθ0(μμ0)b,(18)where b=316π2[yc2zz′(1+z)+ys2z′z(1+z′)].Thus θC→0 or π2 for μ→∞ (depending on the sign of b).⁷

This analysis is only for illustrative purposes: Eq. (18) is not valid for energies of the order of, or less than, the Higgs mass, since only massless RG evolution is considered here: the running will freeze for energies close to, and below, the Higgs mass.

Note that the poles in βC at z=0 and z′=0 are not a pathology [14]. For example, if the RG trajectories of ys and yd cross at some energy t, so ys2=yd2 and z′=0 there, then Λ′ is proportional to the identity matrix at that energy. With yc2>yu2 and ys2-yd2=ε>0, dθCdt→∞ as ε→0 and the RG evolution drives θC infinitely quickly to π2 as ε→0. This is physically perfectly reasonable.

Geometrically the Cabibbo angle parametrizes the double coset U(1)\SU(2)/U(1). Performing the right action first SU(2)/U(1)≈S2, however, the left action of U(1) on S2 has fixed points at the N and S poles. Using (ϑ,φ) as standard polar coordinates on S2, ϑ=2θC and the left action of U(1) is the Killing vector ∂dφ. In terms of sinθC, U(1)\SU(2)/U(1) is just the unit line interval [0, 1] (see Fig. 1).

With three generations, including three right-handed Weyl, gauge singlet, neutrinos, the CKM matrix, and the PMNS matrix, can be treated with the same formalism. Let V denote either the CKM matrix in the quark sector or the PMNS matrix in the leptonic sector, so (y1,y2,y3)=(yu,yc,yt), (y1′,y2′,y3′)=(yd,ys,yb) in the quark sector and (y1′,y2′,y3′)=(ye,yμ,yτ), (y1,y2,y3)=(yν1,yν2,yν3) in the leptonic sector. To exhibit the explicit form of the β functions it is convenient to define the ratios z1=y32-y22y32+y22,z2=y32-y12y32+y12,z3=y22-y12y22+y12,(19)z1′=y3′2-y2′2y3′2+y2′2,z2′=y1′2-y3′2y1′2+y3′2,z3′=y2′2-y1′2y2′2+y1′2.(20)These all lie in the range [-1,1]. They are not all independent, of course: for example z3=z2-z11-z1z2.

In these variables the β functions are βi=θ˙i=332π2∑j,k=13[y32Pijkzkzj′(1+zk)+y3′2Qijkzk′zj(1+zk′)],(21)βδ=δ˙=3sinδ32π2∑j,k=13[y32Pδjkzkzj′(1+zk)+y3′2Qδjkzk′zj(1+zk′)],(22)with the Ps and Qs polynomials in the trigonometric functions in V. Explicit expressions for all the Ps and Qs are given in Appendix B.

There are poles when any of the zj or zj′′ are zero: as for the Cabibbo angle above, near a pole the RG flow will push V very quickly to a fixed point of the left action of U(1)×U(1) on the flag manifold SU(3)/U(1)×U(1).

Using the Ps and Qs in Appendix B, the β functions (21) and (22) vanish when (θ1*,θ2*,θ3*)=(0,0,0),(0,0,π2),(π2,0,0),(π2,0,π2),⁢(0,π2,0),(π2,π2,0),(0,π2,π2),(π2,π2,π2),with δ=0 or π. For θ2=0 there are four such points V*=(100010001),(010−100001),(1000010−10),(010001100),choosing δ=0 or π does not give different points, δ only appears in the mixing matrix V in the combination sinθ2e±iδ.

For θ2=π2 the situation is more subtle. For δ=0 the four points give V*=(001010100),(0010−10100),(001−1000−10),(001−1000−10),for δ=π they give V*=(00−1010100),(00−1010100),(00−11000−10),(00−1−100010).Up to phase transformations only two of these eight are distinct physical points. Furthermore note that, when θ2=π2, the mixing matrix V in (8) takes the form V=(001-sinθ+cosθ+0-cosθ+-sinθ+0)for  δ=0,(23)with θ+=θ3+θ1∈[0,π] and V=(00-1sinθ-cosθ-0cosθ--sinθ-0)for  δ=π,(24)with θ-=θ3-θ1∈[-π2,π2], but Eqs. (23) and (24) are physically the same up to phase transformation. For δ=0 the range 0≤θ+≤πdouble counts physical values, as the sign of cosθ+ in (23) can be changed by a phase transformation. All distinct physical values of V with θ2=π2 are achieved with the smaller range 0≤θ+≤π2.

The upshot of this is that θ2=π2, with δ either 0 or π, is a one-dimension line in physical V space, given by (23) with 0≤θ+≤π2, and there are only two physically distinct fixed points where the β functions vanish, (21) and (22), θ+=0 and θ+=π2.⁸

This line is a RG invariant space in (U(1)×U(1))\SU(3)/(U(1)×U(1)); it is a double coset U(1)\SU(2)/U(1).

Although there are a total of ten parameters in the Yukawa sector, it makes physical sense to focus on the fixed points of the mixing parameters alone. Label the six Yukawa couplings by yα=(ya,ya′), α=1,…,6, and the four mixing parameters by ϑμ=(θ1,θ2,θ3,δ), μ=1,…,4. A fixed point of the full Yukawa sector requires βα=dyαdt=0 and βμ=dϑμdt=0. But the six fixed points of βμ, determined by (21) and (22), require all the P’s and Q’s in Appendix B to vanish, regardless of the prefactors, so if ϑμ are chosen so that βμ(ϑ)=0 for some yα, then βμ(ϑ)=0 for all yα, independently of the values of the βα. It is shown in Appendix C that the 10×10 matrix of anomalous dimensions is block diagonal when (21) and (22) vanish, so it makes physical sense to view the above six points where βμ=0 to be fixed points of the mixing matrix flow, independently of the Yukawa coupling β-functions βα.

In summary, there are six fixed points of the one-loop RG flow of the mixing matrix, and V at these points is (up to phase transformations) V1*=(100010001),(θ1*,θ2*,θ3*)=(0,0,0);V2*=(010100001),(θ1*,θ2*,θ3*)=(0,0,π2);V3*=(010001100),(θ1*,θ2*,θ3*)=(π2,0,π2);V4*=(100001010),(θ1*,θ2*,θ3*)=(π2,0,0);V5*=(001100010),(θ1*,θ2*,θ3*)={(0,π2,π2),(π2,π2,0);V6*=(001010100),(θ1*,θ2*,θ3*)={(0,π2,0),(π2,π2,π2).(25)These are labeled 1 to 6 in the above order: with this labeling of the fixed points the situation can be visualised as shown in Fig. 2.

The six matrices V1*-V6* actually form a group; they furnish a unitary representation of the symmetric group acting on three objects. The appearance of this group is related to the fact that S3 is the Weyl group of SU(3) and these six points are also fixed points of the left action of the Cartan torus of SU(3), T≈U(1)×U(1), on the flag manifold F3.⁹

The fact that there are six fixed points is related to fixed point theorems: the Euler characteristic of F3 is 6. I am grateful to Charles Nash for bringing my attention to fixed point theorems and the significance of the Weyl group of SU(3) in this context.

Note that the Jarlskog invariant [28], J=i4det[Z′,Z]=(y32-y22)(y22-y12)(y12-y32)(y3′2-y2′2)(y2′2-y1′2)(y1′2-y3′2)J,withJ=18sin2θ1sin2θ2sin2θ3cosθ2sin(ϕ1+ϕ2+ϕ3)=18sin2θ1sin2θ2sin2θ3cosθ2sinδ,vanishes at all of the fixed points.