Renormalization group: new relations between the parameters of the Standard Model
S. Rebeca Juárez W., Piotr Kielanowski, Gerardo Mora, Arno Bohm
RRenormalization group: new relations betweenthe parameters of the Standard Model
S. Rebeca Ju´arez W.
Departamento de F´ısica, Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnicoNacional, U.P. “Adolfo L´opez Mateos”. C.P. 07738 Ciudad de M´exico, Mexico
Piotr Kielanowski
Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados, C.P. 07000Ciudad de M´exico, Mexico
Gerardo Mora
Divisi´on Acad´emica de Ciencias B´asicas, Universidad “Ju´arez” Aut´onoma de Tabasco,Mexico
Arno Bohm
Department of Physics, University of Texas at Austin
Abstract
We analyze the renormalization group equations for the Standard Model atthe one and two loops levels. At one loop level we find an exact constant ofevolution built from the product of the quark masses and the gauge couplings g and g of the U (1) and SU (3) groups. For leptons at one loop level wefind that the ratio of the charged lepton mass and the power of g varies (cid:39) × − in the whole energy range. At the two loop level we have found tworelations between the quark masses and the gauge couplings that vary (cid:39)
4% and (cid:39) g and g that varies (cid:39) . Keywords: renormalization group, Standard Model
PACS:
Email addresses: [email protected] (S. Rebeca Ju´arez W.), [email protected] (Piotr Kielanowski), [email protected] (Gerardo Mora), [email protected] (Arno Bohm)
Preprint submitted to Elsevier November 14, 2018 a r X i v : . [ h e p - ph ] M a r . Introduction In particle physics the renormalization group is used for the study of theasymptotic properties of the theory [1, 2]. The renormalization group equations(RGE) for the Standard Model [3–14] is a set of coupled nonlinear differentialequations, derived perturbatively, for the parameters of the theory (couplingsand masses). The full set of RGE for the Standard Model is known up to twoloops [15] and some partial results are known at higher orders (see [16] andreferences therein). There are no known exact solutions of the full set of RGEfor the Standard Model and only some partial results were obtained. At oneloop, equations for the gauge couplings decouple and are solved exactly, but attwo loops this is not the case. Another approach, is to use the hierarchy of theparameters of the Standard model, keeping only certain powers of the quark andlepton masses and of λ CKM ≈ .
21 of the Cabibbo-Kobayashi-Maskawa (CKM)matrix [17]. In such a way one obtains the exact solutions of the approximateequations [18]. The most precise analysis of the renormalization group evo-lution of the Standard Model is done by numerical methods, which give veryprecise predictions for the evolution of the couplings, masses and CKM matrixparameters.The aim of this paper paper is to find relations between the parameters ofthe Standard Model that remain constant (or are slowly varying) during therenormalization group evolution. We start with one loop equation and find an exact constant for the quark masses and gauge couplings. Next we consider alepton sector and find that with great accuracy ( ∼ × − ) the charged leptonmasses flow proportionally to the g − / . For the two loop case we find twogeneralizations of the one loop constant for quarks and one for leptons.The study of the renormalization group invariants in the Standard Model andits extensions has been done before, see e.g., Ref [19] (and references therein),where such invariants were studied for the minimal supersymmetric extension ofthe Standard Model. The invariants in the lepton sector were analyzed in [20].In Ref. [21] an approximation of two flavors was used to simplify the problem ofthe analysis of complicated non linear equations. Recently, such invariants wereanalyzed within the powerful scheme of the flavor invariants in the minimalflavor violating extension of the Standard Model [22, 23]. One should noticethat all these attempts have been limited to the one loop renormalization groupequations. Our approach is different: we directly analyze the structure of therenormalization group equations, first at the one loop level and then at twoloops. The invariant relations are the result of such analysis in the quark andlepton sectors.In Sec. 2 we briefly recall the renormalization group equations for the Stan-dard Model. Next in Sec. 3 we derive and discuss an exact constant of evolutionfor the quark masses and gauge couplings. In Sec. 4 we consider the two loopcase for the quark section and derive two expressions with very slow flow. InSec. 5 we analyze slowly varying expressions in the lepton sector for one and twoloop equations. We discuss our results in Sec. 6. We also include an Appendix,where we derive an equation needed in our analysis.2 . General Considerations The Standard Model has the following set of parameters g , g , g – gaugecouplings, y u , y d , y l – Yukawa couplings of the up and down quarks and ofthe leptons, λ , m – Higgs quartic coupling and mass parameter. Frequentlythe Higgs field vacuum expectation value v is used instead of m . All theseparameters are functions of the renormalization point energy and fulfill therenormalization group equations, which have the following generic form for aparameter g dgdt = β g = 1(4 π ) β (1) g + 1(4 π ) β (2) g + · · · (1)Here t = ln µ and µ is the renormalization point energy. The β g is the β -function,which describes the evolution of the parameter g . The β g function dependson all the parameters of the theory, but does not depend explicitly on therenormalization point energy t . The β -functions are calculated perturbativelyand β ( i ) g is its contribution at the i -loop level.The β -functions of the gauge and Yukawa couplings and of the Higgs fieldvacuum expectation value v have the property that they can be factorized inthe following way β ( i ) g i = g i ˜ β ( i ) g i , β ( i ) y u,d,l = y u,d,l ˜ β ( i ) y u,d,l , β ( i ) v = v ˜ β ( i ) v . (2)This means that the renormalization group equation for the Standard Modelhave the following form d ln g i dt = 1(4 π ) ˜ β (1) g i + 1(4 π ) ˜ β (2) g i + · · · , (3a) dy u,d,l dt = y u,d,l (cid:18) π ) ˜ β (1) u,d,l + 1(4 π ) ˜ β (2) u,d,l + · · · (cid:19) , (3b) d ln vdt = 1(4 π ) ˜ β (1) v + 1(4 π ) ˜ β (2) v + · · · . (3c)Yukawa couplings y u,d,l are complex matrices 3 ×
3, so Eq. (3b) is a matrixdifferential equation. Yukawa couplings y u,d couple to the left- and right-handedquarks. Let us observe that the diagonalization of the matrix y u,d by a biunitarytransformation requires also the knowledge of the right diagonalizing matrix,which is not related to any observable of the Standard Model. For this reason itis more convenient to use the matrices H u,d = y † u,d y u,d which are hermitian andare diagonalized by the left diagonalizing unitary matrices. which are relatedto the Cabibbo-Kobayashi-Maskawa matrix. It follows from Eq. (3) that the3atrices H u,d = y † u,d y u,d fulfill the following differential equation dH u,d dt = H u,d (cid:18) π ) ˜ β (1) u,d + 1(4 π ) ˜ β (2) u,d + · · · (cid:19) + (cid:18) π ) ˜ β (1) u,d + 1(4 π ) ˜ β (2) u,d + · · · (cid:19) † H u,d = 1(4 π ) (cid:16) H u,d ˜ β (1) u,d +( ˜ β (1) u,d ) † H u,d (cid:17) + 1(4 π ) (cid:16) H u,d ˜ β (2) u,d +( ˜ β (2) u,d ) † H u,d (cid:17) + · · · , (4)and the symbol † means here the hermitian conjugate matrix.The one loop approximation consists in keeping only the terms ˜ β (1) on theright hand side of Eqs. (3) and (4), the two loop approximation consists inkeeping the terms ˜ β (1) and ˜ β (2) on the right hand side of Eqs. (3) and (4), etc.
3. A constant for the one loop evolution
The explicit form of the one loop β (1) functions for the gauge and Yukawacouplings and for the Higgs field vacuum expectation value v is the following˜ β (1) g i = − b i g i , { b , b , b } = {− , , } , (5a)˜ β (1) y u = 32 ( y † u y u − y † d y d ) + Y ( S ) − (cid:18) g + 94 g + 8 g (cid:19) , (5b)˜ β (1) y d = 32 ( y † d y d − y † u y u ) + Y ( S ) − (cid:18) g + 94 g + 8 g (cid:19) , (5c)˜ β (1) y l = 32 y † l y l + Y ( S ) −
94 ( g + g ) , (5d)˜ β (1) v = 94 (cid:18) g + g (cid:19) − Y ( S ) , (5e) Y ( S ) = Tr(3 y † u y u + 3 y † d y d + y † l y l ) . Now let us analyze the RGE for the H u,d matrices couplings. From Eq. (4)we know that they are the first order ordinary differential matrix equations ofvery specific form discussed in Appendix A. From this discussion it follows that4ne can derive equations for the determinant of the H u and H d matrices: d det H u dt = 1(4 π ) det H u (Tr( ˜ β (1) u + ( ˜ β (1) u ) † ) ⇒ d ln(det H u ) dt = 1(4 π ) Tr( ˜ β (1) u + ( ˜ β (1) u ) † ) , (6a) d det H d dt = 1(4 π ) det H d (Tr( ˜ β (1) d + ( ˜ β (1) d ) † ) ⇒ d ln(det H d ) dt = 1(4 π ) Tr( ˜ β (1) d + ( ˜ β (1) d ) † ) . (6b)Thus, taking into account Eqs. (3), (5) and (6) we obtain at the one loop level d ln det H u dt + d ln det H u dt + 12 d ln vdt = 1(4 π ) (cid:18) − g − g (cid:19) . (7)Now, from Eqs. (3) and (5a) we have1(4 π ) g i = − b i d ln g i dt (8)and Eq. (7) can be rewritten as ddt ln det H u det H d v g b g b = d ln det H u dt + d ln det H u dt + 12 d ln vdt − b d ln g dt − b d ln g dt = 1(4 π ) (cid:16) Tr( ˜ β (1) u + ( ˜ β (1) u ) † + ˜ β (1) d + ( ˜ β (1) d ) † + 12 ˜ β (1) v − b ˜ β (1) g − b ˜ β (1) g (cid:17) = 0 . (9)This means that the following function of the parameters of the Standard Modelis constant upon the renormalization group evolutiondet H u det H d v g b g b = const. (10)Let us now express the constant in Eq. (10) in terms of observables. Theeigenvalues of the hermitian matrices H u,d = y † u,d y u,d are the squares of theeigenvalues of the Yukawa coupling matrices, corresponding to the up- anddown- quarks { Y t , Y c , Y u } and { Y b , Y s , Y d } . The determinants of H u and H d are thus equaldet H u = ( Y t Y c Y u ) and det H d = ( Y b Y s Y d ) . (11)5he quark masses are equal m i = Y i v √ . (12)Taking into account Eqs. (11) and (12) the evolution constant from Eq. (10)can be rewritten in terms of the quark masses and gauge couplings64 m t m c m u m b m s m d g b g b = const. (13)Thus K defined below is the one loop constant of the renormalization groupevolution K = m t m c m u m b m s m d g b g b = m t m c m u m b m s m d g − g = const. (14) K We will now display the evolution of the constant K from Eq. (14), usingthe evolution of the parameters m i and g i obtained from the numerical solutionof the one and two loop renormalization group equations. The results are shownin Fig. 1, where we draw the constant K normalized to 1 at t = 0 (by dividing itby its value at t = 0). The one loop evolution of K produces a perfect straightline with a constant value, which demonstrates that the numerical analysis iscompatible with the analytical one. The K relation at the two loop solutionof the renormalization group equations is not constant, what mathematically isexpected, because K was derived from the one loop equations. However it israther surprising that the variation is so large, more that 15%. We will discussit later after the analysis of the two loop equations.
4. Discussion of the two loop evolution
The evolution of the constant K , at one loop level, given in Fig. 1 showsthat K is not constant for the two loop solution of the renormalization groupequations. Unfortunately the two loop equations are too complicated to deriveanalytically another quantity that might be constant. However, we will analyzethe situation and we will introduce some improvements.6 ne loop solutiontwo loop solution t K ( t ) / K ( ) c on s t an t Figure 1: Plot of the K relation (which is constant at the one loop level) normalized to 1 at t = 0. The line labeled “one loop solution” has been drawn using the numerical solution of therenormalization group equations at the one loop level. The line labeled “two loop solution”has been drawn using the numerical solution of the renormalization group equations at the two loop level. From Eq. (9) generalized to two loops it follows that ddt ln det H u det H d v g b g b = 1(4 π ) (cid:16) Tr( ˜ β (2) u + ( ˜ β (2) u ) † + ˜ β (2) d + ( ˜ β (2) d ) † ) + 12 ˜ β (2) v − b ˜ β (2) g − b ˜ β (2) g (cid:17) = 1(4 π ) (cid:32) k g + k g + k g + k g g + k g g + k g g + 172 Tr( H u + H d ) − H u H d ) + (cid:16) g + 18 g + 1927 g − λ − Y ( S ) (cid:17) Tr H u + (cid:16) g + 18 g + 1927 g − λ − Y ( S ) (cid:17) Tr H d − g Tr H l (cid:33) (cid:54) = 0 , (15) k = − , k = 2618 , k = − , k = − , k = 107481435 , k = 3247 . Here λ is the Higgs quartic coupling. The right hand side of Eq. (15) cannot beanalytically expressed as a derivative and thus it is not possible to find a twoloop analogue of constant K . However the terms of the type g i and g i g j can7e expressed as derivatives using the one loop renormalization group equations1(4 π ) g i = − π ) b i d g i dt = ddt (cid:32) ln exp (cid:18) − b i π ) g i (cid:19)(cid:33) , (16a)1(4 π ) g i g j = 1(4 π ) g i g j b i g i − b j g j ( b i g i − b j g j )= − π ) g i g j b i g i − b j g j (cid:32) d ln g i dt − d ln g j dt (cid:33) = ddt (cid:32) ln (cid:18) g i g j (cid:19) − π )2 g i g j big i − bjg j (cid:33) . (16b)Here g i is the value of the g i coupling at t = 0.Now, inserting Eqs. (16) into Eq. (15) and moving all derivatives to the lefthand side we obtain ddt ln det H u det H d v g b + r g r g b + r exp( − π ) ( k b g + k b g + k b g )) = 1(4 π ) (cid:32)
172 Tr( H u + H d ) − H u H d )+ (cid:16) g + 18 g + 1927 g − λ − Y ( S ) (cid:17) Tr H u + (cid:16) g + 18 g + 1927 g − λ − Y ( S ) (cid:17) Tr H d − g Tr H l (cid:33) , (17) r = − π ) (cid:18) k g g b g − b g + k g g b g − b g (cid:19) ,r = 1(4 π ) (cid:18) k g g b g − b g − k g g b g − b g (cid:19) ,r = 1(4 π ) (cid:18) k g g b g − b g + k g g b g − b g (cid:19) . Using Eq. (17) we define K = m t m c m u m b m s m d g b + r g r g b + r exp (cid:0) − π ) ( k b g + k b g + k b g ) (cid:1) . (18)Let us also define the quantity K , which is a slight modification of K K = m t m c m u m b m s m d g b + b r g b r g b + b r exp (cid:0) − π ) ( k b g + k b g + k b g ) (cid:1) . (19)The two loop evolution of the quantities K and K normalized to 1 at t = 0 isshown in Fig. 2 8 ( t ) K ( ) K ( t ) K ( ) t Figure 2: Plot of the evolution of the two quantities K and K normalized to 1 at t = 0.This figure should be compared with Fig. 1, where the constant K for the two loop evolutionshows a significant variation. K changes only 4% in the whole range of the energy and K is more stable than K and it changes only 1%.
5. Lepton case
The matrix of the Yukawa couplings for leptons is diagonal y l = Diag(Y e , Y µ , Y τ ) and the renormalization group equations for the leptonsector decouple, so one does not have to consider the equation for the determi-nant. If one takes the sum of Eqs. (5d) and (5e) and uses Eq. (8) one obtains d ln Y i dt + d ln vdt = 1(4 π ) (cid:16) Y i − g (cid:17) ⇒ ddt ln m i g b = 32(4 π ) Y i , i = { e, µ, τ } (20)and we define the quantity K , which is approximate constant for leptons K = m i g b = m i g − = exp (cid:18) π ) (cid:90) tt Y i dt (cid:19) ≈ const. , i = { e, µ, τ } . (21)In Fig. 3 we plot the evolution of K . In Fig. 3 a) we use the one loop evolutionand one sees that for electron and muon one cannot notice any variation and forlepton τ the evolution is linear but very small. In Fig. 3 b) we plot K and usethe two loop solution of the renormalization group equations and one cannotnotice any difference between electron, muon and τ .9 , μτ t a ) e, μ , τ t b ) Figure 3: Plot of the approximate one loop constant for leptons. In a) we draw the evolutionof K using the one loop solution of the renormalization group equations and in b) we drawthe evolution of K for the two loop solution. The two loop analogue of Eq. (20) has the following form (cid:18) d ln Y i dt + d ln vdt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) two loop = 1(4 π ) (cid:16) k g + k g + k g g + 32 Y i + (cid:0) g + 13516 g − Y ( S ) − λ (cid:1) Y i (cid:17) , (22) k = 4191800 , k = 8732 , k = 8180 . and the two loop approximate lepton relation K is equal K = m i g b + r g − r exp (cid:0) − π ) ( k b g + k b g ) (cid:1) = exp (cid:18) π ) (cid:90) tt Y i dt + 1(4 π ) (cid:90) tt (cid:16) Y i + (cid:0) g + 13616 g − Y ( S ) − λ (cid:1) Y i (cid:17) dt (cid:19) ≈ const. , i = { e, µ, τ } , (23) r = − π ) k g g b g − b g . In Fig. 4 we plot the relation K given in Eq. (23) for the two loop solution ofthe renormalization group equations. One can see that the variation of K inthe whole range of energy is approximately 0.1%, so with great accuracy onecan say that K behaves like a constant.10 , μτ t K Figure 4: The plot of the relation K given in Eq. (23) for electron, µ and τ . There is novisible difference between the evolution of K for electron and µ . One can see the significantimprovement in comparison with Fig. 3b.
6. Discussion of the results
We have analyzed the renormalization group equations for the StandardModel and we have derived various expressions built from the parameters of thetheory that are constant or almost constant during the evolution. The form ofthe expressions depends whether we use the one or two loops equations.In case of the one loop equations the evolution constant K is given inEq. (14). The numerator of K is the product of the quark masses and thedenominator is equal to the product g g ( g and g are the gauge couplingsof U (1) and SU (3), respectively). This means that the dependence on g of theproduct of quark masses cancels out. This cancellation is interesting, because itshows that the product of masses of all quarks depends only on electromagneticand strong interactions.For leptons there is no exact constant of one loop evolution, but the relation K , for each charged lepton, given in Eq. (21) has very small variation in thewhole energy range, ∼ × − for the τ lepton and much smaller for electronand µ . From Eq. (21) we also see that at the one loop level the lepton massesdepend only on electromagnetic interactions through the coupling g .The relations K and K plotted with the two loop solutions are no moreconstant: see Figs. 1 and 3b. For this reason we extended our analysis to thetwo loop renormalization group equations. At this level the equations becomemore complicated and it is not possible to analytically derive exact evolutionconstants. However, using the one loop equations we were able to find severalrelations, which have very small variation during the evolution. For quarks, theevolution of these constants, K and K , is shown in Fig. 2 and the variationof K is of the order of 4% and that of K is of the order of 1%. In the case oftwo loops the product the quark masses depends on three gauge couplings.11e were also able to obtain a two loop expression for leptons. This gen-eralized expression K describes the evolution of the charged lepton masses asfunctions of g and g . The variation of K is of the order of 0.1% and is verysimilar for all leptons (see Fig. 4).Summarizing, we have found several relations between the quark and leptonmasses and gauge couplings, which remain (almost) constant upon the renor-malization group evolution. Remarkably, all our relations are between the quarkand lepton masses and the gauge couplings, but do not contain explicitly theHiggs quartic coupling λ or the Higgs field vacuum expectation value v . It isalso interesting to note that our relations contain the product of the two flavorinvariants: determinants of the quark Yukawa coupling matrices [22, 23]. Ouranalysis clarifies the picture of the renormalization group flow of the StandardModel, which is governed by a set of coupled non-linear differential equations. Acknowledgments
Supported in part by
Proyecto SIP:20161034 y SIP:20170819, Secretar´ıade Investigaci´on y Posgrado, Beca EDI y Comisi´on de Operaci´on y Fomentode Actividades Acad´emicas (COFAA) del Instituto Polit´ecnico Nacional (IPN),Mexico . P.K would also like to thank Professor Duane Dicus for kind hospitalityat the Department of Physics, University of Texas at Austin, where part of thework on the paper has been done.
Appendix A. Differential equation for determinant
Let us suppose that A and T are square matrices of the same dimensionand A fulfills the following differential equation dAdt = A · T. (A.1)We will derive from Eq. (A.1) the differential equation for det A , the determinantof the matrix A .The Jacobi formula for derivative of a determinant reads d det Adt = Tr (cid:18) adj( A ) dAdt (cid:19) , (A.2)where adj( A ) is adjugate of the matrix A with the following property A · adj( A ) = adj( A ) · A = det A · I, (A.3)and I is the identity matrix. If we insert Eq. (A.1) into Eq. (A.2) and useEq. (A.3) then we immediately obtain the differential equation for the determi-nant of the matrix A d det
Adt = Tr( T ) det A. (A.4)12uppose now that the matrix A is not hermitian and let us consider thehermitian matrix H = A † A . From Eq. (A.1) it is easy to show that matrix H fulfills the following differential equation dHdt = H · T + T † · H. (A.5)Using again the Jacobi formula (A.2) for the derivative of the determinant weobtain the differential equation for det Hd det Hdt = Tr( T + T † ) det H. (A.6) References [1] E.C.G. Stueckelberg and Andr´e Petermann, Helv. Phys. Acta , 499(1953).[2] M. Gell-Mann and F.E. Low, Phys. Rev. , 1300–1312 (1954).[3] M. E. Machacek and M. T. Vaughn, Nucl. Phys. B222 , 83 (1983);
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