Sep 13, 2000 - âTalk presented at the IX Mexican School on Particles and Fields, August 9â19, Metepec, Pue., Mexico. To be published in the AIP Conference ...
Energy dependence of the quark masses and mixings∗ S.R. Ju´arez W.† , S.F. Herrera H.† , P. Kielanowski‡ and G. Mora H.‡ †
Departamento de F´ısica, Escuela Superior de F´ısica y Matem´ aticas, IPN, M´exico Departamento de F´ısica, Centro de Investigaci´ on y Estudios Avanzados, M´exico ‡ Institute of Theoretical Physics, University of Bialystok, Poland
arXiv:hep-ph/0009148v1 13 Sep 2000
‡
The one loop Renormalization Group Equations for the Yukawa couplings of quarks are solved. From the solution we find the explicit energy dependence on t = ln E/µ of the evolution of the down quark masses q = d, s, b from the grand unification scale down to the top quark mass mt . These results together with the earlier published evolution of the up quark masses completes the pattern of the evolution of the quark masses. We also find the energy dependence of the absolute values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix |Vij |. The interesting property of the evolution of the CKM matrix and the ratios of the quark masses: mu,c /mt and md,s /mb is that they all depend on t through only one function of energy h(t). 12.15.Ff, 12.15.Hh, 11.30.Hv
In a recent paper Ref. [1], a systematical investigation of the evolution of the CKM matrix and the quark Yukawa couplings yu (t) and yd (t) was performed. Exact solutions of the one loop Renormalization Group Equation (RGE) and some general properties of the RGE evolution for the quark masses mq , q = u, c, t, d, s, b, and CKM matrix, compatible with the observed hierarchy Ref. [2], were obtained. The objective of this talk is to present some complementary results to the previous ones. We show here the explicit solutions for the masses of the quarks of the down sector and how they are derived. The quark masses are the eigenvalues of the Yukawa couplings obtained after its diagonalization by biunitary transformations with the help of the unitary matrices (Uu,d )L,R Diag(mu , mc , mt ) = (Uu )L yu (Uu )†R , Diag(md , ms , mb ) = (Ud )L yd (Ud )†R . From the diagonalizing matrices we obtain the flavor mixing in the charged current described by the CKM matrix VCKM = (Uu )L (Ud )†L . The Yukawa couplings are scale dependent. Very frequently one makes various assumptions about their properties at the Grand Unification (GU) scale and then one has to compare the predictions with the measured values at low energies Ref. [3]. The RGE are an important tool for the search of the properties of the quark masses and the CKM matrix at different energy scales. The RGE have been worked out by various authors Ref. [4–7]. The structure of the one loop RGE for the gauge coupling constants gk and the Yukawa couplings yu,d,e,ν is the following dgk dyu,d,e,ν 1 1 (1) 3 bk gk , β yu,d,e,ν . = = dt (4π)2 dt (4π)2 u,d,e,ν (1)
Here t ≡ ln(E/µ) is the energy scale parameter, the coefficients bk are defined in Table 1 and the functions βu,d,e,ν are defined for various models in the Appendix. The approximate form of the equations for the quark Yukawa couplings, neglecting all the terms of λ4 and higher (λ = 0.22) have the following form dyu 1 [αu (t) + αu2 yu yu† + αu3 Tr(yu yu† )]yu , = dt (4π)2 1 1 dyd [αd (t) + αd2 yu yu† + αd3 Tr(yu yu† )]yd . = dt (4π)2 1 The explicit solutions for mq (t) with q = u, c, t and yd (t) previously obtained in Ref. [1] are: p p (b+2) mu,c (t) = mu,c (t0 ) rg (t)(h(t))b/c , mt (t) = mt (t0 ) rg (t) (h(t)) c , q yd (t) = rg′ (t)(h(t))2a/c (Uu )†L Z(t)(Uu )L yd (t0 ),
(1)
(2)
∗ Talk presented at the IX Mexican School on Particles and Fields, August 9–19, Metepec, Pue., Mexico. To be published in the AIP Conference Proceedings.
1
where the (a, b, c) are equal to (0, 2, 2/3), (0, 1, 1/3), (1, 1, −1) in the MSSM, DHS and SM, respectively, Z 1 3c t 2 m (τ )dτ ) h(t) = exp( (4π)2 2 t0 t c 2(b+2) 1 = . Rt 3(b+2) 2 1 − (4π)2 mt (t0 ) t0 rg (τ )dτ
(3)
and
[Z(t)]ij = δij + (h(t) − 1)δi3 δj3 . These solutions Eqs. (2) do depend on the energy through the overall factors rg (t), rg′ (t) (see Appendix) and the matrix Z(t). The procedure Ref. [8], to obtain the energy dependence of the masses for the down sector is the following: Step 1.- Differentiate with respect to t the following equation d
d
(Uu )L yd (t)yd (t)† (Uu )†L = rg (t)(h(t))(2α3 /α2 ) Z(t)(Uu )L yd (t0 )yd (t0 )† (Uu )†L Z(t), ′
which can be written in this way † (t) (Uu )L yd (t)yd (t)† (Uu )†L = VCKM (t)Md2 (t)VCKM
where Md2 (t) is the diagonal matrix with the squares of the physical down quarks Yukawa couplings on the diagonal, that become the squares of the down quark masses after the spontaneous symmetry breaking. Next we obtain the following matrix differential equation dVCKM dVCKM 2 † = (Md2 )−1 VCKM (t) Md dt dt dMd2 d † . −(Md2 )−1 VCKM (t) ((Uu )L yd (t)yd (t)† (Uu )†L )VCKM (t) + (Md2 )−1 dt dt
† VCKM (t)
(4)
Eq. (4) becomes simpler after using the following relation (see Eq. (2)) d ((Uu )L yd (t)yd (t)† (Uu )†L )VCKM (t) dt ′ d d ′ d ln(rg (t)(h(t))(2α3 /α2 ) ) h † 2 −1 † 2 I = (R R + (Md ) R RMd ) + h dt † (Md2 )−1 VCKM (t)
where the vector R = (Vtd , Vts , Vtb ). Step 2.- Extract the differential equations for the diagonal matrix elements of Eq. (4): i h ′ 4a/c d 1 d ln rg (t)(h(t)) d ln md (t) = + ln h(t) |Vtd (t)|2 , dt 2 dt dt i h ′ 4a/c d ln r (t)(h(t)) g 1 d d ln ms (t) = + ln h(t) |Vts (t)|2 , dt 2 dt dt i h ′ 4a/c d ln r (t)(h(t)) g 1 d d ln mb (t) = + ln h(t) |Vtb (t)|2 . dt 2 dt dt
(5)
Step 3.- From the off diagonal matrix elements of Eq. (4) we deduce equations for the squares of the CKM matrix elements |Vij |2 . We solve equations for the |Vtd |2 , |Vts |2 and |Vtb |2 matrix elements that are needed in Eq. (5): d d 2 |Vtd (t)| = −2 ln h(t) |Vtd (t)|2 (1 − |Vtd (t)|2 ), dt dt d d 2 |Vtb (t)| = 2 ln h(t) |Vtb (t)|2 (1 − |Vtb (t)|2 ), dt dt i h d d 2 (6) |Vts (t)| = −2 ln h(t) |Vts (t)|2 |Vtb (t)|2 − |Vtd (t)|2 . dt dt 2
Obtaining |Vtd (t)|2 = |Vts (t)|2 =
|Vtd0 |2 , h2 (t) + (1 − h2 (t))|Vtd0 |2
|Vtb (t) |2 =
|Vts0 |2 |Vtb0 |2 [1 − |Vtd0 |2 ] h 2 h (t) +
|Vtb0 |2 h2 (t) , 1 + |Vtb0 |2 (h2 (t) − 1)
h2 (t) i 0 |2 1−|Vtb h2 (t) + 0 |V |2 tb
0 |2 |Vtd
[1−|Vtd0 |2 ]
,
(7)
where Vij0 = Vij (t0 ). Step 4.- Finally, Eqs. (7) are used to solve in an analytical way the differential equations (5). The formulas which give the explicit energy dependence of the quark masses for the down sector are q (h(t))(c+2a)/c md (t) = rg′ (t) p , md (t0 ) h2 (t) + |Vtd0 |2 (1 − h2 (t)) p q h2 (t) + |Vtd0 |2 (1 − h2 (t)) ms (t) ′ = rg (t) p (h(t))2a/c , ms (t0 ) 1 + |Vtb0 |2 (h2 (t) − 1) q q mb (t) = rg′ (t) 1 + |Vtb0 |2 (h2 (t) − 1) (h(t))2a/c . mb (t0 )
(8)
u,c t d,s b
1.6
1.5
m q(t) /m q(t0)
1.4
1.3
1.2
1.1
1.0
0.9 0
5
10
15
20
25
30
t=ln(E/m t)
FIG. 1. Evolution of the quark masses in the Standard Model
The results presented in this paper together with those of Ref. [1] form the complete set of the renormalization group evolution predictions for the observables derived from the quark Yukawa couplings. The consistent approximation scheme based on the hierarchy of the quark masses and the CKM matrix is strictly observed in all the derivations and it is shown that the final results for the ratios of the quark masses and the CKM matrix depend only on one function of energy in agreement with the theorem presented in Ref. [1]. As an illustration we show in Fig. 1 the evolution of the quark masses for the energy range from E = mt to E = 1014 GeV for the Standard Model. The explicit form of the evolution given here can be very useful for the phenomenological analysis of the models that specify the properties of the Yukawa interactions at the GU scale. 3
We acknowledge the financial support from CONACYT (M´exico) projects 3512P-E9608 and 26247E. S.R.J.W. also thanks to “Comisi´ on de Operaci´ on y Fomento de Actividades Acad´emicas” (COFAA) from Instituto Polit´ecnico Nacional. APPENDIX (1)
βl
(1)
(1)
= αl1 (t) + αl2 Hu(1) + αl3 Tr(Hu(1) ) + αl4 Hd + αl5 Tr(Hd ) (1)
+ αl6 He(1) + αl7 Tr(He(1) ) + αl8 Hν(1) + αl9 Tr(Hν(1) ), Hl
= yl yl† .
(A1)
l = u, d, e, ν. The αl1 (t) and the αli , i = 2, ..., 9 are given in Tables 2 and 3 2 rg (t) = exp( (4π)2
Z
t
2 rg′ (t) = exp( (4π)2
Z
t
t0
Πk=3 k=1
gk2 (ti ) gk2 (t)
cbk
αd1 (τ )dτ ) = Πk=3 k=1
gk2 (ti ) gk2 (t)
cbk
αu1 (τ )dτ )
=
k
,
′
t0
k
,
(A2)
where gk (t) = q 1−
gk (t0 ) 2 (t )(t−t ) 2bk gk 0 0 (4π)2
.
The coefficients bk , ck and c′k are defined in Table 1. TABLE I. The parameters for the various models. Model
b1
b2
b3
c1
c2
c3
c′1
c′2
c′3
MSSM
33 5
1
−3
13 15
3
16 3
7 15
3
16 3
DHM
21 5
−3
−7
17 20
9 4
8
1 4
9 4
8
SM
41 10
−19 6
−7
17 20
9 4
8
1 4
9 4
8
TABLE II. The coefficients αl1 for various models. SM and DHM
MSSM
αu1 (t) =
−( 17 g 2 + 49 g22 + 8g32 ) 20 1
−( 13 g 2 + 3g22 + 15 1
16 2 g ) 3 3
αd1 (t) =
−( 14 g12 + 49 g22 + 8g32 )
7 2 −( 15 g1 + 3g22 +
16 2 g ) 3 3
αe1 (t) =
9 2 −( 20 g1 + 49 g22 )
−( 95 g12 + 3g22 )
αν1 (t) =
9 2 −( 20 g1 + 49 g22 )
−( 35 g12 + 3g22 )
4
TABLE III. The coefficients αlk for various models; the constants (a, b, c) are equal to (0, 2, 2/3), (0, 1, 1/3), (1, 1, −1) in the MSSM, DHS and SM. l
αl2
αl3
αl4
αl5
αl6
αl7
αl8
αl9
u
3 b 2
3
3 c 2
3a
0
a
0
1
d
3 c 2
3a
3 b 2
3
0
1
0
a
e
0
3a
0
3
3 b 2
1
3 c 2
a
ν
0
3
0
3a
3 c 2
a
3 b 2
1
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