We establish the potential existence of natural relations between the Cabibbo
angle and the quark mass ratios, in a Standard Model with one Higgs doublet and
two quark generations. The argument is based on the calculation of the
divergent one-loop radiative corrections to the quark mass matrices.Comment: 10 pages, 1 figur

Buysse, M.

English

arXiv

We establish the potential existence of natural relations between the Cabibbo
angle and the quark mass ratios, in a Standard Model with one Higgs doublet and
two quark generations. The argument is based on the calculation of the
divergent one-loop radiative corrections to the quark mass matrices.
 Comment: 10 pages, 1 figur

Buysse, Martin

DIAL UCLouvain

arXiv:hep-ph/0104244v1  24 Apr 2001UCL-IPT-01-06Natural Relations in the Standard ModelM. Buysse∗Institut de Physique The´oriqueUniversite´ Catholique de Louvain2, Chemin du CyclotronB-1348 Louvain-La-Neuve, BelgiumAbstractWe establish the potential existence of natural relations betweenthe Cabibbo angle and the quark mass ratios, in a Standard Modelwith one Higgs doublet and two quark generations. The argument isbased on the calculation of the divergent one-loop radiative correctionsto the quark mass matrices.One of the outstanding questions raised by the Standard Model (SM) ofelementary particles concerns the set of free parameters it seems to involve.Many of those parameters find their origin in the Yukawa sector of thetheory: six quark masses, three mixing angles and one phase, not to mentionthe leptons. Furthermore, their empirical values show a strongly hierarchicalpattern. Since the early times of its discovery, one has tried to completethe SM in order to reduce the size of its set of free parameters and hencepropose an explanation for this observed hierarchy. The simplest way toreach this goal is to look for potential relations between apparently freeparameters. But one has to make sure that those relations are preserved bythe renormalization; they must be natural [1, 2]. Until now, most attemptsconsisted in enlarging the symmetry group of the SM by adding a horizontalcomponent to it [3, 4, 5, 6]. The horizontal symmetry imposes constraintson the structure of the Yukawa couplings. After spontaneous breakdown ofthe symmetry, the fermion mass matrices that are generated still bear thestamp of those constraints and through bidiagonalization, they give rise torelations between mass ratios and mixing angles. Such an implementationguarantees that the relations survive to renormalization; they are callednatural. However one soon realized [7, 8, 9, 10] that it could not be achieved∗E-mail: buysse@fyma.ucl.ac.be1without extending the particle content – by considering models with morethan one Higgs doublet, thereby increasing the number of couplings...We consider the SM in its minimal realization, namely built up with onesingle Higgs doublet. We ignore the leptons and concentrate on the quarksector. The approach we put forward here is based on the calculation of thedivergent one-loop radiative corrections to the quark mass matrices, whichexclusively involves self-energy and tadpole diagrams. More precisely, sincewe are interested in natural relations between up-type quark mass ratiosand down-type quark mass ratios on the one hand, and mixing angles onthe other hand, we will solely compute the divergent one-loop radiativecorrections to those specific parameters. This considerably simplifies ourtask. One indeed notices that neither QED nor QCD, which are flavour-blind, will bring in divergent contributions that would affect the mixing angleor the mass ratios. The same argument holds for the diagrams involving thetransverse polarizations of the Z0 and of the W± vector bosons, as well asthe tadpoles. To persuade oneself, it is worth checking that the contributionof the latter diagrams to the renormalization of the quark mass ratios readsmumc7−→(1 + Cγ + CG + CZ0 + CW± + CT )mu(1 + Cγ + CG +CZ0 + CW± + CT )mc=mumcwhere Cγ , CG, CZ0 and CW± respectively originate from the interventionsof the photon, the gluons, the transverse Z0 and the transverse W± in theup-type quark self-energies, while CT originates from the tadpole diagrams.Those C’s are identical for the mass renormalization of any up-type quark– as far as the divergent part is concerned. The reasoning can be applied tothe down-type quarks. For the mixing angles, the proof is even more directsince none of those diagrams leads to divergent non-diagonal correction tothe tree-level diagonal mass matrices. In other words, the only diagramsone has to consider are the quark self-energies due to the exchange of thescalars (Higgs and would-be-Goldstone bosons) – for the complete list ofthe relevant divergent diagrams, see figure 1. This is not astonishing sincethe scalars are the only fields that know about the difference between thefermion families. We are looking for a special structure inside the Yukawacouplings (or the mass matrices), hence the fact that the Yukawa sector isthe only one responsible for the naturalness of this structure will come asno surprise.We look into a two-quark-generation SM. Once we have the expressionof the divergent one-loop radiative corrections to the Cabibbo angle andto the two mass ratios, we suppose the existence of a tree-level naturalrelation between them. Since it is natural, this putative relation must hold2at the one-loop level up to some finite corrections. This gives us a severecriterion to constrain the shape of any natural relation. We show thatthere are indeed potential one-loop natural relations in a SM with one Higgsdoublet and two quark generations, contrary to what has been thought sofar. We conclude that those relations cannot originate from any additionnalhorizontal symmetry. We finally state our a result in a different way andclaim there is one apparently free parameter, in a SM with one Higgs doubletand two quark generations, to which the one-loop radiative corrections arefinite. In other words, one apparently free parameter whose presence inthe SM seems not to be necessary to render it renormalizable; or else, oneapparently free parameter whose one-loop β-function vanishes. We call it astable parameter.Let us briefly recall the input we need. We start from a two-fermion-generation SM; the scalar sector includes one single Higgs doublet. Thehadronic Yukawa sector, which we are interested in, contains three useful freeparameters: the Cabibbo angle θ [11] and the two mass ratios ru = mu/mcand rd = md/ms. The approach is the following: we assume the existenceof a natural relation between the Cabibbo angle and the mass ratios; wethen make use of the definition of a natural relation to establish its generalform. Namely, assuming the existence of a natural relation automaticallydetermines that relation.A general natural relation between the Cabibbo angle and the mass ratiosreads:F (θ) = G(ru, rd) (1)so thatF (θ + δ θ) = G(ru + δ ru, rd + δ rd) (2)where δ θ, δ ru and δ rd are the divergent parts of the radiative corrections toθ, ru and rd respectively. This last expression is the mathematical transla-tion of naturalness. To be natural, (1) cannot be broken by infinite radiativecorrections. That is, (2) must be verified. ThusFpθ δ θ = Gpru δ ru +Gprd δ rdwhere fpx denotes the (partial) derivative of f with respect to x. Computingthe one-loop radiative corrections to the Yukawa couplings (see Appendix),3one obtains:δ θ = ǫ[1 + r2u1− r2u(m2d −m2s) +1 + r2d1− r2d(m2u −m2c)]sin θ cos θδ ru = ǫ ru [(m2d −m2s) cos 2θ − (m2u −m2c)] (3)δ rd = ǫ rd [(m2u −m2c) cos 2θ − (m2d −m2s)]with ǫ = 342v2(14π2ln Λ2µ2), Λ being a cut-off, µ an arbitrary energy scale andv the vacuum expectation value of the scalar field. Let us bear in mind thatwe do not care about the finite part of the corrections. Introducing thoseexpressions into equation (2) splits it into two independent equations (sincewe are not interested in considering relations involving mass ratios differentfrom ru and rd):Fpθ sin θ cos θ1 + r2u1− r2u= Gpru ru cos 2θ −Gprd rd (4)Fpθ sin θ cos θ1 + r2d1− r2d= Gprd rd cos 2θ −Gpru ru (5)which turn out to be compatible if and only ifcos 2θ =1−r2u1+r2urdGprd −1−r2d1+r2druGpru1−r2u1+r2uruGpru −1−r2d1+r2drdGprd(6)Now thismust be the natural relation (1) whose existence has been supposed,i.e.1F (θ) = cos 2θ (7)andG(ru, rd) =1−r2u1+r2urdGprd −1−r2d1+r2druGpru1−r2u1+r2uruGpru −1−r2d1+r2drdGprd(8)1One checks that the arbitrary character of those identifications will not show itself inthe expected solution. To prove it, we imagine (7) would rather read f(F (θ)) = cos 2θ.One should then replace G by f(G) in the left-hand side of (8). But the right-hand sideof it is invariant under G 7→ f(G). Namely, one can solve (8) with respect to the variablef(G) which we finally identify to f(F (θ)) = cos 2θ.4Exploiting the remaining information in (4) and (5), and using (6) and (7),yields1+r2u1−r2uG− ruGpru+1+r2d1−r2d= 01+r2d1−r2dG− rdGprd +1+r2u1−r2u= 0This system – from which one obviously recovers equation (8) – is integrable,and the general solution readsG(ru, rd) =−(1 + r2u)(1 + r2d) + 2λ rurd(1− r2u)(1− r2d)where λ is the integration constant. One concludes that, if a relation of thekind suggested in (1) exists, it necessarily belongs to the following class:cos 2θ =−(m2u +m2c)(m2d +m2s) + 2λ mumcmdms(m2u −m2c)(m2s −m2d)(9)We have found an infinite number of potential one-loop natural relationsinside the Yukawa sector. This set is parametrized by a dimensionless con-stant λ. One has no further theoretical argument to constrain the value ofλ; and one cannot evade the difficulty by asking one of the quark masses tovanish, since it would lead to a cosine smaller than minus one2. Now, onehas to select the unique viable natural relation by making λ fit the data, i.e.λ =(m2u +m2c)(m2d +m2s) + (m2u −m2c)(m2d −m2s) cos 2θ2mumcmdms(10)This last step gives the impression one goes round in a circle; it does of coursenot increase the predictive power of the reasoning, but it allows us to shed anew light on the result. One might indeed think of λ as a physical quantity(since it is built from physical ones) which is not renormalized, i.e. whichis not useful for absorbing divergences. Or else, the one-loop λ differs fromthe tree-level one by a finite quantity. In other words, the Sandard Modelwith two quark generations and one Higgs doublet seems to be vast enoughto allow one of its apparently free parameters not to absorb any divergent2Consequently we may already conclude that it is not possible to naturally set themass of one single quark to zero, together with the requirement of the existence of anatural relation between the Cabibbo angle and the quark mass ratios. In this context, byexcludingmu = 0 our result rules out the natural vanishing of the QCD vacuum parameterθS .5one-loop radiative correction (we show in the Appendix that δ λ = 0 asexpected by construction,). This means that the one-loop β-function of λvanishes, i.e. at the one-loop level, λ does not run; λ is stable.Stating that several natural relations potentially exist in a one-Higgs-doublet model apparently contradicts previous results obtained in the con-text of family symmetries [7, 8, 9, 10]. But since we do not appeal to suchkind of symmetries, we do not expect our result to respect the conclusionsderived in their context. Namely, the potential natural relations (9) cannotbe associated with the presence of any extra horizontal symmetry3. Oneshould examine the validity of the results at the n-loop level; then look forthe reason of the existence of a stable parameter inside the SM; and finallylook for some possible “determination principle” of it outside or beyond theSM.The analysis we have conducted here can be applied to the leptons,provided that the neutrinos are massive, their mass being of the Dirac typeexclusively. The result, since it depends on the sole structure of the Yukawasector, is strictly identical.The extension of the present calculation to a three-quark-generation SMseems to be doomed to failure because of the complexity of the one-loopradiative corrections to the Cabibbo-Kobayashi-Maskawa parameters [11,12]. Those corrections are indeed too cumbersome to be manipulated andintroduced into a solvable partial differential equations system. We willhowever expound, in a forthcoming paper, an alternative approach to derivesimilar results in a n-quark-generation SM.AcknowledgmentsThis work is supported by the Fonds pour la Formation a` la Recherchedans l’Industrie et dans l’Agriculture (FRIA). We would like to thank J.-M.Ge´rard and J. Weyers for a critical reading of the manuscript.Appendix3According to [8], in a one-Higgs-doublet model, the Cabibbo angle which is determinedby a family symmetry can only take the values 0 or pi/2, yielding cos 2θ = 1 or cos 2θ = −1respectively. In our approach, (10) then reads λ = (m2um2d + m2cm2s)/(mumcmdms) orλ = (m2cm2d + m2um2s)/(mumcmdms), and there still is an opportunity for relating themass ratios among each other. However, again, this potential natural relation is not to beregarded as the consequence of the presence of any family symmetry.6A Self-energies in the Yukawa sectorAfter spontaneous breakdown of the symmetry, the Yukawa sector Lagrangianfor quarks reads:LY = u¯LΓddRφ+ + d¯LΓddRφ0 + d¯RΓ†duLφ− + d¯RΓ†ddLφ0 ⋆+ u¯LΓuuRφ0 ⋆ − d¯LΓuuRφ− + u¯RΓ†duLφ0 − u¯RΓ†udLφ++ d¯LMddR + u¯LMuuR + d¯RM†ddL + u¯RM†duLThe Lagrangian fields and parameters are renormalized:uL,R 7−→ u′L,R = (ZuL,R)− 12uL,RdL,R 7−→ d′L,R = (ZdL,R)− 12dL,RMu 7−→ M′u = (ZuL)12 (ZMu)−1Mu(ZuR)12Mu 7−→ M′d = (ZdL)12 (ZMd)−1Md(ZdR)12The one-loop calculation of the fermion self-energies leads to (the first termcorresponding to the neutral current intervention ; the second, to the chargedcurrent intervention – see figure 1):ZuL = 1−ǫ2[ΓuΓ†u + ΓdΓ†d] ZdL = 1−ǫ2[ΓdΓ†d + ΓuΓ†u]ZuR = 1−ǫ2[ΓuΓ†u + ΓuΓ†u] ZdR = 1−ǫ2[ΓdΓ†d + ΓdΓ†d]ZMu = 1− ǫ[0 + ΓdΓ†d] ZMd = 1− ǫ[0 + ΓuΓ†u](11)with ǫ =(14π2ln Λ2µ2)where Λ is a cut-off and µ an arbitrary energy scale(one checks that ZuL = ZdL as expected). Those results are true only up to aterm proportional to the identity in the flavour space, which would take intoaccount the electromagnetic, weak transversal and strong contributions, aswell as the tadpole ones. But since this term would factor out in the finalresult, which is supposed to involve exclusively mixing parameters and up-or down-type mass ratios, we chose not to write it down. The finite parts ofthe diagrams are omitted.From (11), one derives the mass matrices corrections in the weak baseM ′u =Mu + ǫ [MdM†dMu −MuM†uMu]M ′d =Md + ǫ [MuM†uMd −MdM†dMd]and in the physical baseU †LM′uUR = Du + ǫ [KD2dK†Du −D3u]V †LM′dVR = Dd + ǫ [K†D2uKDd −D3d]7where we have absorbed a 342v2factor in ǫ, v being the scalar VEV, andwhere Du and Dd are the tree-level diagonal mass matrices while K is thetree-level Cabibbo-Kobayashi-Maskawa matrix. Let us repeat that thoseexpressions do not include the finite parts of the radiative correction, andthat they account for the sole (neutral and charged) scalar exchanges in thefermion self-energies.One can rewrite those last expressions as followsM ′′u = Du + ǫuM ′′d = Dd + ǫdand proceed to the diagonalization of M ′′u and M′′d , i.e.U ′†LM′′uU′R = D′uV ′†L M′′d V′R = D′dwhereD′u =(mu + ǫu 11mc + ǫu 22)U ′L =(1 θu−θu 1)D′d =(md + ǫd 11ms + ǫd 22)V ′L =(1 θd−θd 1)withθu =muǫu 21 +mcǫu 12m2c −m2uand θd =mdǫu 21 +msǫu 12m2s −m2dThe one-loop mixing matrix is defined byK ′ = U ′†LKV′Lso that the one-loop mixing angle readsθ′ = θ − θu + θdInserting the value of ǫu and ǫd in θ′, D′u and D′d, leads to the equations (3).8B Divergent radiative corrections to λδλ = λpθδθ + λpruδru + λprdδrd= −2 sin 2θ(1 − r2u)(1− r2d)2rurdδθ+−rd(1− r2u)(1 + r2d)− rd(1 + r2u)(1− r2d) cos 2θ2r2ur2dδru+−ru(1 + r2u)(1− r2d)− ru(1− r2u)(1 + r2d) cos 2θ2r2ur2dδrd= −ǫ sin2 2θ[(1 + r2u)(1− r2d)2rurd(m2d −m2s) +(1− r2u)(1 + r2d)2rurd(m2u −m2c)]+ ǫ sin2 2θ(1 + r2u)(1− r2d)2rurd(m2d −m2s)+ ǫ sin2 2θ(1 − r2u)(1 + r2d)2rurd(m2u −m2c)= 0References[1] S. Weinberg. Rev. Mod. Phys., 46:255–277, 1974.[2] S. Weinberg. Trans. New York Acad. Sci., 38:185–201, 1977.[3] H. Fritzsch. Phys. Lett., B70:436, 1977.[4] H. Harari, H. Haut, and J. Weyers. Phys. Lett., B78:459, 1978.[5] S. Pakvasa and H. Sugawara. Phys. Lett., B73:61, 1978.[6] H. Fritzsch and J. Plankl. Phys. Lett., B237:451, 1990.[7] D. Wyler. Phys. Rev., D19:330, 1979.[8] R. Barbieri, R. Gatto, and F. Strocchi. Phys. Lett., B74:344–346, 1978.[9] R. Gatto, G. Morchio, and F. Strocchi. Phys. Lett., B83:348–350, 1979.[10] R. Gatto, G. Morchio, and F. Strocchi. Phys. Lett., B80:265, 1979.9[11] N. Cabibbo. Phys. Rev. Lett., 10:531–532, 1963.[12] M. Kobayashi and T. Maskawa. Prog. Theor. Phys., 49:652, 1973.                                            /                      /                                        /                      / =−1Mi (Z     −  1) M d dφd u u dLRLR−+0Lu u+φLR Rd d =−1Mi (Z     −  1) M uu +0φR Ld d dR0*R L Rd u dφ − =i (Z     − 1) pRd −1+φu udR L R+φuLuu0R R=i (Z     − 1) pu −1R +φL L0d d dRφL LRd u d− =Li (Z     − 1) pd −1+φL Lu uuR0* φL Lu udR+ =Lu −1i (Z     − 1) p +Fig. 1: relevant divergent diagrams involved in the calculation of the renor-malization constants.10

Natural Relations in the Standard Model

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