On the impact of kinetic mixing in beta functions at two-loop
SSMU-HEP-16-10
On the impact of kinetic mixing in beta functions at two-loop
F. Lyonnet ∗ Southern Methodist University, Dallas, TX 75275, USA (Dated: November 5, 2018)Kinetic mixing is a fundamental property of models with a gauge symmetry involving severalU(1) group factors. In this paper, we perform a numerical study of the impact of kinetic mixing onbeta functions at two-loop. To do so, we use the recently published PyR@TE 2 software to derivethe complete set of RGEs of the SM B-L model at two-loop including kinetic mixing. We show thatit is important to properly account for kinetic mixing as the evolution of the parameters with theenergy scale can change drastically. In some cases, these modifications can even lead to a differentconclusion regarding the stability of the scalar potential.
PACS numbers: 11.10.Hi,12.60.Cn,12.60.Fr
CONTENTS
I. Intro 1II. Kinetic Mixing in PyR@TE 1III. SM B-L 2A. Lagrangian and symmetry breaking 2B. Parameters and stability 3IV. Effect of kinetic mixing at two-loop 3A. Running of the parameters 3B. Stability of the potential 3C. Impact of ˜ g I. INTRO
Renormalization group equations (RGEs) play a cen-tral role in studying the high-energy behaviour of ex-tensions of the Standard Model (SM). The RGEs for anarbitrary gauge field theory at two-loop have been knownfor a long time [1–6] and their derivation for specific mod-els has been automated in [7, 8]. An extra complicationarises when the gauge structure contains multiple U(1)group factors as kinetic mixing can occur. This intro-duces n ( n −
1) extra dynamical parameters for eachof which RGEs must be calculated. In addition, kineticmixing induces modifications to the RGEs of the otherparameters that also need to be taken into account forconsistency. This work was carried out in [9, 10] for ∗ fl[email protected] the gauge couplings and extended in [11] for the dimen-sionless parameters at two-loop. Recently, an alternativemethod was presented in [12] in which the modificationsto the RGEs for the dimensionful parameters are alsoderived. We implemented the method of [12] in the newversion [13] of the computer code PyR@TE [7] whichallows us to simply derive RGEs for a given model attwo-loop taking into account kinetic mixing.In this note we show that the impact of such kineticmixing can be important for the high-energy behaviourof the theory parameters. To do so, we study the RGEsof the SM B-L model in which the SM has been supple-mented by an additional U(1) B − L group factor. We startin Sec. II by giving some details on how the kinetic mix-ing is implemented in PyR@TE. Sec. III summarizes theprinciple properties of the SM B-L model while Sec. IV isdevoted to the study of the impact of kinetic mixing onthe running of its parameters. Finally, our conclusionsare presented in Sec. V. II. KINETIC MIXING IN PYR@TE
Let’s first consider a gauge field theory with n U(1)group factors
G ×
U(1) × U(1) ×· · ·× U(1) n where G cor-responds to the non Abelian part of the gauge structure.Gauge invariance then allows one to write the followingterm in the Lagrangian: L ⊃ (cid:126)F µν ξ (cid:126)F µν , (2.1)where ξ is an n × n symmetric matrix, and F µν is thevector of the n field strength tensors associated to the n gauge groups U(1) i , i = 1 . . . n . This indeed introduces n ( n −
1) extra dynamical parameters. These additionalparameters can be exchanged by corresponding effectivegauge couplings that populate the off-diagonal entries ofan extended gauge coupling matrix G = ˜ Gξ − / [12].Of course the two approaches are equivalent and one canrecover the results obtained working with ξ by performingsuitable rotations.The advantage of the effective gauge coupling methodis that it does not require the introduction of new pa-rameters ξ and their RGEs but only promotes the gauge a r X i v : . [ h e p - ph ] S e p Field Quantum Numbers Q L ( , , / , / ) u R ( , , / , / ) d R ( , , − / , / ) L L ( , , − / , − ) e R ( , , − , − ) ν R ( , , , − ) H ( , , / , ) χ ( , , , ) Table I: Particle content of the SM B-L model and theirquantum numbers underSU(3) c × SU(2) L × U(1) Y × U(1) B − L .couplings to a non-diagonal matrix. The modificationsdue to kinetic mixing on the beta functions of the otherparameters can be expressed as more or less complex re-placement rules [12]. For instance one has g S ( R ) → G (cid:88) p W Rp ( W Rp ) T , (2.2)with S ( R ) the Dynkin index of representation R . Thesum p runs over all the fermions ( R = F ) or scalars( R = S ). Finally, W Ri ≡ G T Q Ri , with Q Ri the vectorof Abelian charges of the field i in the representation R . All the replacement rules derived in [12] have beenimplemented in the computer code PyR@TE at two-loop. III. SM B-L
Our goal is to assess the size of the modifications due tokinetic mixing on the running of Lagrangian parameters.In order to do so, we will concentrate on the SM extendedby an additional U(1) B − L group factor, i.e. B-L is pro-moted to a gauge symmetry. We will assume that we havean extra singlet scalar χ transforming as ∼ (1 , , ,
2) un-der SU(3) c × SU(2) L × U(1) Y × U(1) B − L . In addition, wewill consider three right-handed neutrinos ∼ (1 , , , − A. Lagrangian and symmetry breaking
Table I list all the particles in the model along withtheir quantum numbers under SU(3) c × SU(2) L × U(1) Y × U(1) B − L gauge symmetry. Note that our setup is iden-tical to [15].With two scalars the most general scalar potential con-tains 5 parameters and reads V ( H, χ ) = µ H H † H + µ χ χ † χ + λ (cid:0) H † H (cid:1) + λ (cid:0) χ † χ (cid:1) + λ (cid:0) H † H (cid:1) (cid:0) χ † χ (cid:1) . (3.1) The two scalars take the following vacuum expectationvalues (VEVs), leading to spontaneous symmetry break-ing (SSB) < H > = 1 √ (cid:32) v (cid:33) , < χ > = v (cid:48) √ . (3.2)The minimization conditions lead to the following expres-sions for v and v (cid:48) v = µ χ λ / − µ H λ λ λ − λ / , (3.3) v (cid:48) = µ H λ / − µ χ λ λ λ − λ / . (3.4)After SSB, the two neutral scalars mix leading to twophysical states of mass m , . Defining the mixing anglebetween the two scalars, θ , one can derive the followingrelations [15] λ = m v (1 + cos 2 θ ) + m v (1 − cos 2 θ ) ,λ = m v (cid:48) (1 − cos 2 θ ) + m v (cid:48) (1 + cos 2 θ ) ,λ = sin 2 θ (cid:18) m − m vv (cid:48) (cid:19) . (3.5)There is an additional mixing angle, α , in the gaugesector which has been tightly constrained by LEP [16], | α | ≤ − , and the masses of Z and Z (cid:48) bosons in thislimit can be approximated by M Z (cid:39) v (cid:113) g + g , M Z (cid:48) (cid:39) v (cid:112) ˜ g + (4 g (cid:48) v (cid:48) /v ) , (3.6)in which g is the gauge coupling of the SU(2) L gaugegroup and (cid:32) g ˜ g g (cid:48) (cid:33) are the Abelian gauge couplings in theupper triangular basis which is linked to G ≡ (cid:32) g g g g (cid:33) via the rotation˜ G = G · (cid:32) cos φ − sin φ sin φ cos φ (cid:33) , (3.7)cos φ = g (cid:112) g + g , sin φ − g (cid:112) g + g . (3.8)Finally, the Yukawa interactions of the model are dic-tated by the following Lagrangian − L Y = Y ijd Q iL Hd jR + Y iju Q iL ˜ Hu jR + Y ije L i He jR + Y ijν L i ˜ Hν jR + Y ijN ( ν iR ) c ν jR χ + h . c . . (3.9)After SSB this leads to the following mass term for theneutrinos − L νY = Y ijν v √ (cid:124) (cid:123)(cid:122) (cid:125) M ijd ( ν iL ) c ν jR + 1 √ Y ijN v (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) M ijm ν iR ν jR + h . c . , (3.10)which requires Y ν ∼ O (10 − ) for light neutrinos while Y N ∼ O (1) for heavy right-handed neutrinos in the TeVrange, see [15]. B. Parameters and stability
In the numerical analysis, we will neglect the Yukawacouplings, Y e , Y d , Y ν , and retain only the top Yukawacoupling, y t . For simplicity, the right-handed neutrinoYukawa will be reduced to Y ijN = δ ij y N .For the numerical analysis, we select the following setof parameters B = { θ, M Z (cid:48) , M m , g (cid:48) , ˜ g, m , m } , (3.11)from which we derive initial values for { v (cid:48) , y N , λ , λ , λ , µ H , µ χ } . Indeed, v (cid:48) can be ex-tracted from Eq. (3.6) and the knowledge of M Z (cid:48) , g (cid:48) and ˜ g ; y N is obtained directly from Eq. (3.10) whereasthe values of the quartic couplings λ , λ , λ result fromEq. (3.5). Finally, the initial values for the Lagrangianmass parameter, µ H and µ χ are fixed via the mini-mization conditions, Eq.(3.3) and Eq.(3.4). The SMHiggs mass is identified with the physical scalar mass m = 126 GeV.It is well known, that the stability of the scalar poten-tial is achieved by the following conditions [15] λ > , λ > , λ λ − λ , (3.12)which we will investigate in the next section. IV. EFFECT OF KINETIC MIXING ATTWO-LOOP
The RGEs for the SM B-L as derived with PyR@TEhave been given recently in [13] which we refer to for thefull expressions and details. Our goal here is not to obtainprecise conclusions regarding the physics of the model athand. It is rather to quantify the impact of kinetic mix-ing one can expect in a physical situation on the runningof the parameters. Therefore, we neglect the electroweakmatching conditions as well as the scalar threshold cor-rections due to the heavy Higgs. These corrections wouldnot affect our conclusions regarding the amplitude of theimpact of the kinetic mixing. The initial values of all theparameters are set at the scale of the Z-boson mass, M Z ,and evolved to 10 GeV using C ++ routines as providedby the PyR@TE package. A. Running of the parameters
To begin, we simplify the situation and select a sce-nario in which there is no mixing between the two scalars, Where y t is third diagonal entry of Y u . i.e. θ = 0. In addition, we set ˜ g = 0 and will investi-gate the effect of ˜ g (cid:54) = 0 later. Note that this does notcorrespond to neglecting kinetic mixing as it is gener-ated automatically by radiative corrections. The massof the new heavy gauge boson M Z (cid:48) is set to 2 . B = { , , , . , , , } , see Eq.(3.11) wherethe dimensionful parameters are given in GeV.Fig. 1 shows the running of some of the parametersof the B-L model when the kinetic mixing is fully takeninto account at two-loop or neglected. More specifically,the gauge couplings g and g , the quartic couplings λ , λ , λ and the Yukawa couplings y t and y N areshown. On Fig. 2 we display the corresponding ratioof the beta functions including kinetic mixing over thebeta functions neglecting kinetic mixing.While the impact on the gauge couplings is somewhatlimited to 1-3 % it reaches 2-6% for the Yukawa couplingsat the GUT scale. For the quartic couplings, the changeis dramatic and deserves some comments.(i) λ actually goes to zero when the kinetic mixing isneglected, see Fig. 1, causing the ratio to blow up;(ii) λ gets large kinetic contributions at one-loop of theform ∼ g λ and ∼ g coming from Λ Sabcd and A abcd in the notation of [3]. Moreover, there is a backreaction coming from the impact of kinetic mixingon λ since β λ ∼ λ . These effects ultimately leadto a ratio of 1.5 at 10 GeV;(iii) the kinetic mixing in λ is such that it turns the signof the beta function around, from negative to posi-tive at a scale of about 10 GeV, ultimately leadingto a positive λ at a scale of 10 GeV.Therefore, as expected the impact of kinetic mixing isgoverned by the evolution of the off-diagonal effectiveAbelian gauge couplings, g , g . In Fig. 3, we show therunning of these two gauge couplings with the energy.It is important to note that the values of these gaugecouplings stay perturbative all the way up to the Planckscale and that the effects seen in the other parameters arenot the result of extreme values of the gauge couplings. B. Stability of the potential
As we have already seen, the impact of kinetic mixingcan be quite large on the running of the different param-eters. In this section, we investigate how this translatesin terms of the stability conditions, Eq. (3.12).Fig. 4, shows the stability condition 4 λ λ − λ withand without kinetic mixing for B . The difference in this g x g kin . no . kin . g kin . no . kin . y t , y N kin . no . kin . kin . no . kin . t = log ( Q /M Z ) g x λ kin . no . kin . t = log ( Q /M Z ) λ kin . no . kin . t = log ( Q /M Z ) λ kin . no . kin . θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 1: Running of the parameters in the B-L model in the case where the kinetic mixing is taken into account(solid lines) or neglected (dashed lines), for the benchmark point B . See text and figure title for the value of theinput parameters. The parameters plotted are g , g , y t , y N , λ , λ , λ . R = g k i n . m i x . x / g n o . k i n . m i x . x g g y t y N t = log ( Q /M Z ) R = g k i n . m i x . x / g n o . k i n . m i x . x λ t = log ( Q /M Z ) λ λ θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 2: Ratio of the couplings taking into account or neglecting the kinetic mixing. The parameters plotted are g , g , y t , y N , λ , λ , λ . The initial values are those of B .case is striking as the scenario develops an instabilityat around 10 GeV when kinetic mixing is accountedfor. This behaviour is actually simple to understand asthe instability is the direct consequence of λ turningnegative at around the same scale, Λ ∼ GeV.One might think that such a drastic change in re-sults is limited to a small region of the parameter space,however that is not the case. Indeed, one can eas- ily find regions where even the opposite situation hap-pens, i.e. where the kinetic mixing rescues the stabil-ity of the potential. Fig. 5 shows such an example, for B = { . , , , . , , , } . This scenario isthe result of several competing effects.(i) θ (cid:54) = 0 leads to λ B ( M Z ) (cid:29) λ B ( M Z ) which remainstrue at all scales, see Eq. 3.5. t = log ( Q /M Z ) R G E s g g θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 3: Running of the Abelian off-diagonal gaugecouplings, g and g . The input parameters are thesame as in Fig. 1. t = log ( Q /M Z ) λ λ − λ with kin . mix . no kin . mix . θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 4: Stability condition for the benchmark point B of the B-L model in the case where the kineticmixing is (solid line) or is not (dashed line) taken intoaccount. The impact of kinetic mixing in this case leadsto different conclusions.(ii) λ ( M Z ) ∼ m / (4 v )(1 − cos 2 θ ), therefore θ (cid:54) = 0greatly increases λ ( M Z ) which leads to λ > λ gets large kinetic corrections sufficientto overcome the increase in λ .One more thing to note is that the mass of the heavyHiggs plays a crucial role here. The initial values of λ , , decrease with m ( M Z ) and for m ( M Z ) ∼
500 GeV orlighter, λ quickly runs negative without kinetic mixingleading to an unstable potential. t = log ( Q /M Z ) λ λ − λ with kin . mix . no kin . mix . θ = 0 . , M Z = 2500 GeV , M m = 1100 GeV , m = 800 GeV , ˜ g = 0 Figure 5: Stability condition for B . In this example,the kinetic mixing rescues the stability of the potential C. Impact of ˜ g Finally, we investigate the impact of the value of ˜ g on the running of the parameters and the stability ofthe potential. To do so, we consider B but with ˜ g ∈{ ., . , . , . } . The results are presented in Figs. 6and 7 in which we show the running of the stability con-dition and the corresponding quartic couplings for dif-ferent values of ˜ g respectively. Values of ˜ g > .
15 willlead to non-perturbative couplings at the highest ener-gies around 10 GeV (not shown). The spread in λ , values, Fig. 7, due to different values of ˜ g is 35% and 15%respectively at the scale 10 GeV while the λ parame-ter is extremely small for ˜ g = 0 resulting in a spread oftwo orders of magnitude at the same scale. t = log ( Q /M Z ) λ λ − λ ˜ g = 0 . g = 0 .
05 ˜ g = 0 . g = 0 . θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV Figure 6: Stability condition in the B benchmark pointfor several values of ˜ g ∈ { , . , . , . } . t = log ( Q /M Z ) R G E s λ ˜ g = 0 . g = 0 .
05 ˜ g = 0 . g = 0 . t = log ( Q /M Z ) λ ˜ g = 0 . g = 0 .
05 ˜ g = 0 . g = 0 . t = log ( Q /M Z ) λ ˜ g = 0 . g = 0 .
05 ˜ g = 0 . g = 0 . θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV Figure 7: Running of the quartic couplings in the B benchmark point for several values of ˜ g ∈ { , . , . , . } . D. Two-loop kinetic mixing contribution t = log ( Q /M Z ) g f u ll k i n . x / g L k i n . x λ λ λ µ H µ χ y t θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 8: Running of some of the couplings of the B-Lmodel for benchmark point B . We show the ratio ofpredictions including the full two-loop correctionscoming from kinetic mixing over those including onlythe corrections up to two-loop order in the gaugecouplings and one-loop elsewhere.In this last part we focus on determining the amplitudeof the kinetic mixing contributions at two-loop versus theone-loop order ones. To do so, we consider the two-loopbeta functions without kinetic mixing to which we addeither the two-loop, or one-loop order corrections comingfrom kinetic mixing. For simplicity, we keep the kineticmixing contribution up to two-loop in the gauge cou-plings but switch on and off the two-loop order contribu-tions in the other parameters. This would correspond toa simplified implementation of the rules of [12] in whichthe more involved replacement rules are ignored, and inparticular the ones involving the scalar generators.Fig. 8 shows ratios of couplings at two-loop as a func-tion of the scale, in which the numerator is the couplingincluding the whole set of two-loop kinetic contributions,while the denominator is the same coupling in which only λ λ − λ
1L kin . full kin . t = log ( Q /M Z ) R θ = 0 . , M Z = 2500 GeV , M m = 1000 GeV , m = 750 GeV , ˜ g = 0 . Figure 9: Stability condition in the B-L model forbenchmark point B . We show the predictions includingthe full two-loop corrections (dashed line) coming fromkinetic mixing and those including only the correctionsup to two-loop order in the gauge couplings andone-loop elsewhere (solid line).the one-loop kinetic mixing contributions have been re-tained . The benchmark point, B , used here is definedby B = { . , , , . , . , , } . The ratiosare below 1% for the parameters µ H , µ χ and y N , whereasfor the quartic terms, they are typically larger than 1% atthe scale 10 GeV and can reach 5% at the GUT scale.The corresponding effect on the stability condition,Eq.(3.12), are shown in Fig. 9. In the upper panel, thedashed line represent the stability condition in the casewhere all the corrections are included while the solid lineis the result of retaining only the one-loop kinetic contri-bution. From the bottom panel, showing the ratio, it iseasy to extract that the difference in the two approachescan be of the order 10%. Note that for the gauge couplings, the two-loop contributions arealso taken into account.
While B is a particular point in the parameter space,it illustrates that neglecting the two-loop contributionscoming from kinetic mixing might lead to an error of theorder a couple of percents. In addition, it is not un-likely, that in some combinations of couplings like thestability condition of Eq.(3.12) these differences are en-hanced leading to larger discrepancies. Furthermore,since contributions from the kinetic mixing terms and thebase beta functions mix together, a coherent perturba-tive treatment at two-loop demands to take into accountcontributions coming from kinetic mixing at the sameorder. V. CONCLUSION
Kinetic mixing is a fundamental property of modelswith an extended Abelian gauge structure like the SMB-L. Taking into account the kinetic mixing in the RGEsat two-loop is a complex procedure. To this end, wehave consistently implemented the kinetic mixing at two-loop in the software PyR@TE. Therefore, these modelsnow benefit from the same level of automation as theircounterparts without kinetic mixing.After reviewing the theoretical setup of the SM B-L,we studied in detail the impact of kinetic mixing on therunning of the parameters of the model. We showed that neglecting the kinetic mixing can lead to erroneous con-clusions regarding the stability of the scalar potential.In addition, we studied the impact of the kinetic mixingcontributions at two-loop and showed that they can besignificant with respect to their one-loop counterparts.Of course, our goal was only to illustrate that the fea-tures of a model can dramatically change when prop-erly accounting for kinetic mixing; a careful treatment ofthe electroweak matching conditions and scalar thresholdcorrections would be required to draw detailed physicalconclusions regarding the B-L model investigated here.We leave this to future work.We believe that the order of magnitude of the effectsexemplified with the SM B-L can be similar in other mod-els and that taking kinetic mixing into account is crucialto obtain meaningful results. This is now a simple taskthanks to PyR@TE.
ACKNOWLEDGMENTS
I would like to thank T. Jezo, A. Kusina, F. Olness,I. Schienbein, and F. Staub for their useful commentsand for reviewing this manuscript. This work was alsopartially supported by the U.S. Department of Energyunder Grant No. DE-SC0010129. [1] M. E. Machacek and M. T. Vaughn, “Two LoopRenormalization Group Equations in a GeneralQuantum Field Theory. 1. Wave FunctionRenormalization,”
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