Unification and fermion mass structure.
Graham G. Ross ∗and Mario Serna †
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP
November 18, 2021
Abstract
Grand Unified Theories predict relationships between the GUT-scale quark and lepton masses. Using
new data in the context of the MSSM, we update the values and uncertainties of the masses and mixing
angles for the three generations at the GUT scale. We also update fits to hierarchical patterns in the
GUT-scale Yukawa matrices. The new data shows not all the classic GUT-scale mass relationships remain
in quantitative agreement at small to moderate tan β. However, at large tan β, these discrepancies can be
eliminated by finite, tan β-enhanced, radiative, threshold corrections if the gluino mass has the opposite
sign to the wino mass.
Explaining the origin of fermion masses and mixings remains one of the most important goals in our
attempts to go beyond the Standard Model. In this, one very promising possibility is that there is an
underlying stage of unification relating the couplings responsible for the fermion masses. However we are
hindered by the fact that the measured masses and mixings do not directly give the structure of the underlying
Lagrangian both because the data is insufficient unambiguously to reconstruct the full fermion mass matrices
and because radiative corrections can obscure the underlying structure. In this letter we will address both
these points in the context of the MSSM.
We first present an analysis of the measured mass and mixing angles continued to the GUT scale. The
analysis updates previous work, using the precise measurements of fermion masses and mixing angles from
the b-factories and the updated top-quark mass from CDF and D0. The resulting data at the GUT scale
allows us to look for underlying patterns which may suggest a unified origin. We also explore the sensitivity
of these patterns to tanβ-enhanced, radiative threshold corrections.
We next proceed to extract the underlying Yukawa coupling matrices for the quarks and leptons. There
are two difficulties in this. The first is that the data cannot, without some assumptions, determine all elements
of these matrices. The second is that the Yukawa coupling matrices are basis dependent. We choose to work
in a basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small relative
to the appropriate combinations of on-diagonal matrix elements. This is the basis we think is most likely to
display the structure of the underlying theory, for example that of a spontaneously broken family symmetry
in which the hierarchical structure is ordered by the (small) order parameter breaking the symmetry. With
this structure to leading order the observed masses and mixing angles determine the mass matrix elements
on and above the diagonal, and our analysis determines these entries, again allowing for significant tanβ
enhanced radiative corrections. The resulting form of the mass matrices provides the “data” for developing
models of fermion masses such as those based on a broken family symmetry.
The data set used is summarized in Table 1. Since the fit of reference [4] (RRRV) to the Yukawa texture
was done, the measurement of the Standard-Model parameters has improved considerably. We highlight
a few of the changes in the data since 2000: The top-quark mass has gone from Mt = 174.3 ± 5 GeV to
Mt = 170.9 ± 1.9 GeV. In 2000 the Particle Data Book reported mb(mb) = 4.2 ± 0.2 GeV [5] which has
improved to mb(mb) = 4.2 ± 0.07 GeV today. In addition each higher order QCD correction pushes down
the value of mb(MZ) at the scale of the Z bosons mass. In 1998 mb(MZ) = 3.0 ± 0.2 GeV [6] and today
it is mb(MZ) = 2.87 ± 0.06 GeV [7]. The most significant shift in the data relevant to the RRRV fit is a
downward revision to the strange-quark mass at the scale µL = 2 GeV from ms(µL) ≈ 120± 50 MeV [5] to
∗g.ross@physics.ox.ac.uk
†serna@physics.ox.ac.uk
http://arxiv.org/abs/0704.1248v2
Low-Energy Parameter Value(Uncertainty in last digit(s)) Notes and Reference
mu(µL)/md(µL) 0.45(15) PDB Estimation [1]
ms(µL)/md(µL) 19.5(1.5) PDB Estimation [1]
mu(µL) +md(µL) [8.8(3.0), 7.6(1.6)] MeV PDB, Quark Masses, pg 15
[1]. ( Non-lattice, Lattice )
−(md+mu)2/4
22.8(4) Martemyanov and Sopov [2]
ms(µL) [103(20) , 95(20)] MeV PDB, Quark Masses, pg 15
[1]. [Non-lattice, lattice]
mu(µL) 3(1) MeV PDB, Quark Masses, pg 15
[1]. Non-lattice.
md(µL) 6.0(1.5) MeV PDB, Quark Masses, pg 15
[1]. Non-lattice.
mc(mc) 1.24(09) GeV PDB, Quark Masses, pg 16
[1]. Non-lattice.
mb(mb) 4.20(07) GeV PDB, Quark Masses, pg 16,19
[1]. Non-lattice.
Mt 170.9 (1.9)GeV CDF & D0 [3] Pole Mass
(Me,Mµ,Mτ ) (0.511(15), 105.6(3.1), 1777(53) ) MeV 3% uncertainty from neglect-
ing Y e thresholds.
A Wolfenstein parameter 0.818(17) PDB Ch 11 Eq. 11.25 [1]
ρ Wolfenstein parameter 0.221(64) PDB Ch 11 Eq. 11.25 [1]
λ Wolfenstein parameter 0.2272(10) PDB Ch 11 Eq. 11.25 [1]
η Wolfenstein parameter 0.340(45) PDB Ch 11 Eq. 11.25 [1]
|VCKM |
0.97383(24) 0.2272(10) 0.00396(09)
0.2271(10) 0.97296(24) 0.04221(80)
0.00814(64) 0.04161(78) 0.999100(34)
 PDB Ch 11 Eq. 11.26 [1]
sin 2β from CKM 0.687(32) PDB Ch 11 Eq. 11.19 [1]
Jarlskog Invariant 3.08(18)× 10−5 PDB Ch 11 Eq. 11.26 [1]
vHiggs(MZ) 246.221(20) GeV Uncertainty expanded. [1]
( α−1EM (MZ), αs(MZ), sin
2 θW (MZ) ) ( 127.904(19), 0.1216(17), 0.23122(15)) PDB Sec 10.6 [1]
Table 1: Low-energy observables. Masses in lower-case m are MS running masses. Capital M indicates pole
mass. The light quark’s (u,d,s) mass are specified at a scale µL = 2 GeV. VCKM are the Standard Model’s
best fit values.
today’s value ms(µL) = 103± 20 MeV. We also know the CKM unitarity triangle parameters better today
than six years ago. For example, in 2000 the Particle Data book reported sin 2β = 0.79 ± 0.4 [5] which is
improved to sin 2β = 0.69± 0.032 in 2006 [1]. The sin 2β value is about 1.2 σ off from a global fit to all the
CKM data [8], our fits generally lock onto the global-fit data and exhibit a 1 σ tension for sin 2β. Together,
the improved CKM matrix observations add stronger constraints to the textures compared to data from
several years ago.
We first consider the determination of the fundamental mass parameters at the GUT scale in order simply
to compare to GUT predictions. The starting point for the light-quark masses at low scale is given by the
χ2 fit to the data of Table 1
mu(µL) = 2.7± 0.5 MeV md(µL) = 5.3± 0.5 MeV ms(µL) = 103± 12 MeV. (1)
Using these as input we determine the values of the mass parameters at the GUT scale for various choices of
tanβ but not including possible tanβ enhanced threshold corrections. We do this using numerical solutions
to the RG equations. The one-loop and two-loop RG equations for the gauge couplings and the Yukawa
couplings in the Standard Model and in the MSSM that we use in this study come from a number of sources
[6] [9][10] [11]. The results are given in the first five columns of Table 2. These can readily be compared to
expectations in various Grand Unified models. The classic prediction of SU(5) with third generation down-
quark and charged-lepton masses given by the coupling B 5f .10f .5H
1 is mb(MX)/mτ (MX) = 1 [12]. This
ratio is given in Table 2 where it may be seen that the value agrees at a special low tanβ value but for large
tanβ it is some 25% smaller than the GUT prediction2. A similar relation between the strange quark and
the muon is untenable and to describe the masses consistently in SU(5) Georgi and Jarlskog [14] proposed
that the second generation masses should come instead from the coupling C 5f .10f .45H leading instead to
the relation 3ms(MX)/mµ(MX) = 1. As may be seen from Table 2 in all cases this ratio is approximately
0.69(8). The prediction of Georgi and Jarlskog for the lightest generation masses follows from the relation
Det(Md)/Det(M l) = 1. This results from the form of their mass matrix which is given by3
 , M l =
A −3C
 (2)
in which there is a (1, 1) texture zero4 and the determinant is given by the product of the (3, 3), (1, 2) and
(2, 1) elements. If the (1, 2) and (2, 1) elements are also given by 5f .10f .5H couplings they will be the same
in the down-quark and charged-lepton mass matrices giving rise to the equality of the determinants. The
form of eq(2) may be arranged by imposing additional continuous or discrete symmetries. One may see from
Table 2 that the actual value of the ratio of the determinants is quite far from unity disagreeing with the
Georgi Jarlskog relation.
In summary the latest data on fermion masses, while qualitatively in agreement with the simple GUT
relations, has significant quantitative discrepancies. However the analysis has not, so far, included the SUSY
threshold corrections which substantially affect the GUT mass relations at large tanβ [15]. A catalog of the
full SUSY threshold corrections is given in [16]. The particular finite SUSY thresholds discussed in this letter
do not decouple as the super partners become massive. We follow the approximation described in Blazek,
Raby, and Pokorski (BRP) for threshold corrections to the CKM elements and down-like mass eigenstates
[17]. The finite threshold corrections to Y e and Y u and are generally about 3% or smaller
δY u, δY d . 0.03 (3)
and will be neglected in our study. The logarithmic threshold corrections are approximated by using the
Standard-Model RG equations from MZ to an effective SUSY scale MS.
The finite, tanβ-enhanced Y d SUSY threshold corrections are dominated by the a sbottom-gluino loop,
a stop-higgsino loop, and a stop-chargino loop. Integrating out the SUSY particles at a scale MS leaves the
matching condition at that scale for the Standard-Model Yukawa couplings:
δmsch Y
uSM = sinβ Y u (4)
δmsch Y
d SM = cosβ U
1 + Γd + V
CKM Γ
u VCKM
Y ddiag U
R (5)
Y e SM = cosβ Y e. (6)
All the parameters on the right-hand side take on their MSSM values in the DR scheme. The factor δmsch
converts the quark running masses from MS to DR scheme. The β corresponds to the ratio of the two
Higgs VEVs vu/vd = tanβ. The U matrices decompose the MSSM Yukawa couplings at the scale MS :
Y u = U
diagU
R and Y
d = U
diagU
R. The matrices Y
diag and Y
diag are diagonal and correspond to
the mass eigenstates divided by the appropriate VEV at the scale MS . The CKM matrix is given by
VCKM = U
L . The left-hand side involves the Standard-Model Yukawa couplings. The matrices Γ
u and
Γd encode the SUSY threshold corrections.
If the squarks are diagonalized in flavor space by the same rotations that diagonalize the quarks, the
matrices Γu and Γd are diagonal: Γd = diag(γd, γd, γb), Γ
u = diag(γu, γu, γt). In general the squarks are
15f , 10f refer to the SU(5) representations making up a family of quarks and leptons while 5H is a five dimensional
representation of Higgs scalars.
2We’d like to thank Ilja Dorsner for pointing out that the tan β dependence of mb/mτ (MX) is more flat than in previous
studies (e.g. ref. [13]). This change is mostly due to the higher effective SUSY scale MS , the higher value of αs(MZ) found in
global standard model fits, and smaller top-quark mass Mt.
3The remaining mass matrix elements may be non-zero provided they do not contribute significantly to the deteminant
4Below we discuss an independent reason for having a (1, 1) texture zero.
Parameters Input SUSY Parameters
tanβ 1.3 10 38 50 38 38
γb 0 0 0 0 −0.22 +0.22
γd 0 0 0 0 −0.21 +0.21
γt 0 0 0 0 0 −0.44
Parameters Corresponding GUT-Scale Parameters with Propagated Uncertainty
yt(MX) 6
−5 0.48(2) 0.49(2) 0.51(3) 0.51(2) 0.51(2)
yb(MX) 0.0113
+0.0002
−0.01 0.051(2) 0.23(1) 0.37(2) 0.34(3) 0.34(3)
yτ (MX) 0.0114(3) 0.070(3) 0.32(2) 0.51(4) 0.34(2) 0.34(2)
(mu/mc)(MX) 0.0027(6) 0.0027(6) 0.0027(6) 0.0027(6) 0.0026(6) 0.0026(6)
(md/ms)(MX) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7)
(me/mµ)(MX) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2)
(mc/mt)(MX) 0.0009
+0.001
−0.00006 0.0025(2) 0.0024(2) 0.0023(2) 0.0023(2) 0.0023(2)
(ms/mb)(MX) 0.014(4) 0.019(2) 0.017(2) 0.016(2) 0.018(2) 0.010(2)
(mµ/mτ )(MX) 0.059(2) 0.059(2) 0.054(2) 0.050(2) 0.054(2) 0.054(2)
A(MX) 0.56
+0.34
−0.01 0.77(2) 0.75(2) 0.72(2) 0.73(3) 0.46(3)
λ(MX) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1)
ρ̄(MX) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6)
η̄(MX) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4)
J(MX) × 10
−5 1.4+2.2−0.2 2.6(4) 2.5(4) 2.3(4) 2.3(4) 1.0(2)
Parameters Comparison with GUT Mass Ratios
(mb/mτ )(MX) 1.00
+0.04
−0.4 0.73(3) 0.73(3) 0.73(4) 1.00(4) 1.00(4)
(3ms/mµ)(MX) 0.70
−0.05 0.69(8) 0.69(8) 0.69(8) 0.9(1) 0.6(1)
(md/3me)(MX) 0.82(7) 0.83(7) 0.83(7) 0.83(7) 1.05(8) 0.68(6)
(detY
detY e
)(MX) 0.57
+0.08
−0.26 0.42(7) 0.42(7) 0.42(7) 0.92(14) 0.39(7)
Table 2: The mass parameters continued to the GUT-scale MX for various values of tanβ and threshold
corrections γt,b,d. These are calculated with the 2-loop gauge coupling and 2-loop Yukawa coupling RG
equations assuming an effective SUSY scale MS = 500 GeV.
not diagonalized by the same rotations as the quarks but provided the relative mixing angles are reasonably
small the corrections to flavour conserving masses, which are our primary concern here, will be second order
in these mixing angles. We will assume Γu and Γd are diagonal in what follows.
Approximations for Γu and Γd based on the mass insertion approximation are found in [18][19][20]:
γt ≈ y
t tanβ
, µ2) ∼ y2t
γu ≈ −g
2 M2 µ
,m2χ2 ,m
ũ) ∼ 0 (8)
M3 µ I3(m
M3 µ I3(m
where I3 is given by
2, b2, c2) =
a2b2 log a
+ b2c2 log b
+ c2a2 log c
(a2 − b2)(b2 − c2)(a2 − c2)
. (11)
In these expressions q̃ refers to superpartner of q. χj indicate chargino mass eigenstates. µ is the coefficient
to the Hu Hd interaction in the superpotential. M1,M2,M3 are the gaugino soft breaking terms. A
t refers to
the soft top-quark trilinear coupling. The mass insertion approximation breaks down if there is large mixing
between the mass eigenstates of the stop or the sbottom. The right-most expressions in eqs(7,9,10) assume
the relevant squark mass eigenstates are nearly degenerate and heavier than M3 and µ. These expressions (
eqs 7 - 10) provide an approximate mapping from a supersymmetric spectra to the γi parameters through
which we parameterize the threshold corrections; however, with the exception of Column A of Table 4, we
do not specify a SUSY spectra but directly parameterize the thresholds corrections through γi.
The separation between γb and γd is set by the lack of degeneracy of the down-like squarks. If the
squark masses for the first two generations are not degenerate, then there will be a corresponding separation
between the (1,1) and (2,2) entries of Γd and Γu. If the sparticle spectra is designed to have a large At and
a light stop, γt can be enhanced and dominate over γb. Because the charm Yukawa coupling is so small, the
scharm-higgsino loop is negligible, and γu follows from a chargino squark loop and is also generally small
with values around 0.02 because of the smaller g2 coupling. In our work, we approximate Γ
22 ∼ Γ
11 ∼ 0.
The only substantial correction to the first and second generations is given by γd [15].
As described in BRP, the threshold corrections leave |Vus| and |Vub/Vcb| unchanged to a good approxi-
mation. Threshold corrections in Γu do affect the Vub and Vcb at the scale MS giving
V SMub − V
V MSSMub
V SMcb − V
V MSSMcb
⋍ − (γt − γu) . (12)
The threshold corrections for the down-quark masses are given approximately by
md ⋍ m
d (1 + γd + γu)
ms ⋍ m
s (1 + γd + γu)
mb ⋍ m
b (1 + γb + γt)
where the superscript 0 denotes the mass without threshold corrections. Not shown are the nonlinear effects
which arise through the RG equations when the bottom Yukawa coupling is changed by threshold effects.
These are properly included in our final results obtained by numerically solving the RG equations.
Due to our assumption that the squark masses for the first two generations are degenerate, the combina-
tion of the GUT relations given by
detM l/ detMd
(3ms/mµ)
(mb/mτ ) = 1 is unaffected up to nonlinear
effects. Thus we cannot simultaneously fit all three GUT relations through the threshold corrections. A best
fit requires the threshold effects given by
γb + γt ≈ −0.22± 0.02 (13)
γd + γu ≈ −0.21± 0.02. (14)
giving the results shown in the penultimate column of Table 2, just consistent with the GUT predictions. The
question is whether these threshold effects are of a reasonable magnitude and, if so, what are the implications
for the SUSY spectra which determine the γi? From eqs(9,10), at tanβ = 38 we have
∼ −0.5,
∼ 1.0
The current observation of the muon’s (g − 2)µ is 3.4 σ [21] away from the Standard-Model prediction.
If SUSY is to explain the observed deviation, one needs tanβ > 8 [22] and µM2 > 0 [23]. With this sign
we must have µM3 negative and the d̃, s̃ squarks only lightly split from the b̃ squarks. M3 negative is
characteristic of anomaly mediated SUSY breaking[24] and is discussed in [25][26][20][27]. Although we have
deduced M3 < 0 from the approximate eqs(9,10), the the correlation persists in the near exact expression
found in eq(23) of ref [17]. Adjusting to different squark splitting can occur in various schemes[28]. However
the squark splitting can readily be adjusted without spoiling the fit because, up to nonlinear effects, the
solution only requires the constraints implied by eq(13), so we may make γb > γd and hence make m
by allowing for a small positive value for γt. In this case A
t must be positive.
It is of interest also to consider the threshold effects in the case that µM3 is positive. This is illustrated
in the last column of Table 2 in which we have reversed the sign of γd, consistent with positive µM3 , and
chosen γb ≃ γd as is expected for similar down squark masses. The value of γt is chosen to keep the equality
between mb and mτ . One may see that the other GUT relations are not satisfied, being driven further away
by the threshold corrections. Reducing the magnitude of γb and γd reduces the discrepancy somewhat but
still limited by the deviation found in the no-threshold case (the fourth column of Table 2).
Parameter 2001 RRRV Fit A0 Fit B0 Fit A1 Fit B1 Fit A2 Fit B2
tanβ Small 1.3 1.3 38 38 38 38
a′ O(1) 0 0 0 0 −2.0 −2.0
ǫu 0.05 0.030(1) 0.030(1) 0.0491(16) 0.0491(15) 0.0493(16) 0.0493(14)
ǫd 0.15(1) 0.117(4) 0.117(4) 0.134(7) 0.134(7) 0.132(7) 0.132(7)
|b′| 1.0 1.75(20) 1.75(21) 1.05(12) 1.05(13) 1.04(12) 1.04(13)
arg(b′) 90o +93(16)o − 93(13)o +91(16)o − 91(13)o +93(16)o − 93(13)o
a 1.31(14) 2.05(14) 2.05(14) 2.16(23) 2.16(24) 1.92(21) 1.92(22)
b 1.50(10) 1.92(14) 1.92(15) 1.66(13) 1.66(13) 1.70(13) 1.70(13)
|c| 0.40(2) 0.85(13) 2.30(20) 0.78(15) 2.12(36) 0.83(17) 2.19(38)
arg(c) − 24(3)o − 39(18)o − 61(14)o − 43(14)o − 59(13)o − 37(25)o − 60(13)o
Table 3: Results of a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold corrections.
We set a′ as indicated and set c′ = d′ = d = 0 and f = f ′ = 1 at fixed values.
At tanβ near 50 the non-linear effects are large and b − τ unification requires γb + γt ∼ −0.1 to −0.15.
In this case it is possible to have t − b − τ unification of the Yukawa couplings. For µ > 0,M3 > 0, the
“Just-so” Split-Higgs solution of references [29, 30, 31, 32] can achieve this while satisfying both b → s γ and
(g − 2)µ constraints but only with large γb and γt and a large cancellation in γb + γt. In this case, as in the
example given above, the threshold corrections drive the masses further from the mass relations for the first
and second generations because µM3 > 0. It is possble to have t−b−τ unification with µM3 < 0, satisfying
the b → s γ and (g − 2)µ constraints in which the GUT predictions for the first and second generation of
quarks is acceptable. Examples include Non-Universal Gaugino Mediation [33] and AMSB; both have some
very heavy sparticle masses ( & 4 TeV) [20]. Minimal AMSB with a light sparticle spectra( . 1 TeV), while
satisfying (g − 2)µ and b → s γ constraints, requires tanβ less than about 30 [23].
We turn now to the second part of our study in which we update previous fits to the Yukawa matrices
responsible for quark and lepton masses. As discussed above we choose to work in a basis in which the
mass matrices are hierarchical with the off-diagonal elements small relative to the appropriate combinations
of on-diagonal matrix elements. This is the basis we think is most likely to display the structure of the
underlying theory, for example that of a spontaneously broken family symmetry, in which the hierarchical
structure is ordered by the (small) order parameter breaking the symmetry. With this structure to leading
order in the ratio of light to heavy quarks the observed masses and mixing angles determine the mass matrix
elements on and above the diagonal provided the elements below the diagonal are not anomalously large.
This is the case for matrices that are nearly symmetrical or for nearly Hermitian as is the case in models
based on an SO(10) GUT.
For convenience we fit to symmetric Yukawa coupling matrices but, as stressed above, this is not a critical
assumption as the data is insensitive to the off-diagonal elements below the diagonal and the quality of the
fit is not changed if, for example, we use Hermitian forms. We parameterize a set of general, symmetric
Yukawa matrices as:
Y u(MX) = y
d′ǫ4u b
′ ǫ3u c
′ ǫ3u
b′ ǫ3u f
′ ǫ2u a
′ ǫ2u
c′ ǫ3u a
′ ǫ2u 1
 , (15)
Y d(MX) = y
d ǫ4d b ǫ
d c ǫ
b ǫ3d f ǫ
d a ǫ
c ǫ3d a ǫ
 . (16)
Although not shown, we always choose lepton Yukawa couplings at MX consistent with the low-energy lepton
masses. Notice that the f coefficient and ǫd are redundant (likewise in Y
u). We include f to be able to
discuss the phase of the (2,2) term. We write all the entries in terms of ǫ so that our coefficients will be
O(1). We will always select our best ǫ parameters such that |f | = 1.
RRRV noted that all solutions, to leading order in the small expansion parameters, only depend on two
Parameter A B C B2 C2
tanβ 30 38 38 38 38
γb 0.20 −0.22 +0.22 −0.22 +0.22
γt −0.03 0 −0.44 0 −0.44
γd 0.20 −0.21 +0.21 −0.21 +0.21
a′ 0 0 0 −2 −2
ǫu 0.0495(17) 0.0483(16) 0.0483(18) 0.0485(17) 0.0485(18)
ǫd 0.131(7) 0.128(7) 0.102(9) 0.127(7) 0.101(9)
|b′| 1.04(12) 1.07(12) 1.07(11) 1.05(12) 1.06(10)
arg(b′) 90(12)o 91(12)o 93(12)o 95(12)o 95(12)o
a 2.17(24) 2.27(26) 2.30(42) 2.03(24) 1.89(35)
b 1.69(13) 1.73(13) 2.21(18) 1.74(10) 2.26(20)
|c| 0.80(16) 0.86(17) 1.09(33) 0.81(17) 1.10(35)
arg(c) − 41(18)o − 42(19)o − 41(14)o − 53(10)o − 41(12)o
Y u33 0.48(2) 0.51(2) 0.51(2) 0.51(2) 0.51(2)
Y d33 0.15(1) 0.34(3) 0.34(3) 0.34(3) 0.34(3)
Y e33 0.23(1) 0.34(2) 0.34(2) 0.34(2) 0.34(2)
(mb/mτ )(MX) 0.67(4) 1.00(4) 1.00(4) 1.00(4) 1.00(4)
(3ms/mµ)(MX) 0.60(3) 0.9(1) 0.6(1) 0.9(1) 0.6(1)
(md/3me)(MX) 0.71(7) 1.04(8) 0.68(6) 1.04(8) 0.68(6)∣∣∣detY
d(MX)
detY e(MX )
∣∣∣ 0.3(1) 0.92(14) 0.4(1) 0.92(14) 0.4(1)
Table 4: A χ2 fit of eqs(15,16) including the SUSY threshold effects parameterized by the specified γi.
phases φ1 and φ2 given by
φ1 = (φ
b − φ
f )− (φb − φf ) (17)
φ2 = (φc − φa)− (φb − φf ). (18)
where φx is the phase of parameter x. For this reason it is sufficient to consider only b
′ and c as complex
with all other parameters real.
As mentioned above the data favours a texture zero in the (1, 1) position. With a symmetric form for the
mass matrix for the first two families, this leads to the phenomenologically successful Gatto Sartori Tonin
[34] relation
Vus(MX) ≈
∣∣bǫd − |b′|ei φb′ ǫu
∣∣∣∣ . (19)
This relation gives an excellent fit to Vus with φ1 ≈ ± 90
o, and to preserve it we take d, d′ to be zero in our
fits. As discussed above, in SU(5) this texture zero leads to the GUT relation Det(Md)/Det(M l) = 1 which,
with threshold corrections, is in good agreement with experiment. In the case that c is small it was shown
in RRRV that φ1 is to a good approximation the CP violating phase δ in the Wolfenstein parameterization.
A non-zero c is necessary to avoid the relation Vub/Vcb =
mu/mc and with the improvement in the data,
it is now necessary to have c larger than was found in RRRV 5. As a result the contribution to CP violation
coming from φ2 is at least 30%. The sign ambiguity in φ1 gives rise to an ambiguity in c with the positive
sign corresponding to the larger value of c seen in Tables 3 and 4.
Table 3 shows results from a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold
corrections. The error, indicated by the term in brackets, represent the widest axis of the 1σ error ellipse
in parameter space. The fits labeled ‘A’ have phases such that we have the smaller magnitude solution of
|c|, and fits labeled ‘B’ have phases such that we have the larger magnitude solution of |c|. As discussed
above, it is not possible unambiguously to determine the relative contributions of the off-diagonal elements
of the up and down Yukawa matrices to the mixing angles. In the fit A2 and B2 we illustrate the uncertainty
5As shown in ref. [35], it is possible, in a basis with large off-diagonal entries, to have an Hermitian pattern with the (1,1)
and (1,3) zero provided one carefully orchestrates cancelations among Y u and Y d parameters. We find this approach requires
a strange-quark mass near its upper limit.
associated with this ambiguity, allowing for O(1) coefficients a′. In all the examples in Table 3, the mass
ratios, and Wolfenstein parameters are essentially the same as in Table 2.
The effects of the large tanβ threshold corrections are shown in Table 4. The threshold corrections
depend on the details of the SUSY spectrum, and we have displayed the effects corresponding to a variety
of choices for this spectrum. Column A corresponds to a “standard” SUGRA fit - the benchmark Snowmass
Points and Slopes (SPS) spectra 1b of ref([36]). Because the spectra SPS 1b has large stop and sbottom
squark mixing angles, the approximations given in eqns(7-10) break down, and the value for the correction
γi in Column A need to be calculated with the more complete expressions in BRP [17]. In the column A
fit and the next two fits in columns B and C, we set a′ and c′ to zero. Column B corresponds to the fit
given in the penultimate column of Table 2 which agrees very well with the simple GUT predictions. It is
characterized by the “anomaly-like” spectrum with M3 negative. Column C examines the M3 positive case
while maintaining the GUT prediction for the third generation mb = mτ . It corresponds to the “Just-so”
Split-Higgs solution. In the fits A, B and C the value of the parameter a is significantly larger than that
found in RRRV. This causes problems for models based on non-Abelian family symmetries, and it is of
interest to try to reduce a by allowing a′, b′ and c′ to vary while remaining O(1) parameters. Doing this for
the fits B and C leads to the fits B2 and C2 given in Table 4 where it may be seen that the extent to which
a can be reduced is quite limited. Adjusting to this is a challenge for the broken family-symmetry models.
Although we have included the finite corrections to match the MSSM theory onto the standard model
at an effective SUSY scale MS = 500 GeV, we have not included finite corrections from matching onto a
specific GUT model. Precise threshold corrections cannot be rigorously calculated without a specific GUT
model. Here we only estimate the order of magnitude of corrections to the mass relations in Table 2 from
matching the MSSM values onto a GUT model at the GUT scale. The tanβ enhanced corrections in eq(7-10)
arise from soft SUSY breaking interactions and are suppressed by factors of MSUSY /MGUT in the high-scale
matching. Allowing for O(1) splitting of the mass ratios of the heavy states, one obtains corrections to yb/yτ
(likewise for the lighter generations) of O( g
(4π)2
) from the X and Y gauge bosons and O(
(4π)2
) from colored
Higgs states. Because we have a different Higgs representations for different generations, these threshold
correction will be different for correcting the 3ms/mµ relation than the mb/mτ relation. These factors can
be enhanced in the case there are multiple Higgs representation. For an SU(5) SUSY GUT these corrections
are of the order of 2%. Plank scale suppressed operators can also induce corrections to both the unification
scale [37] and may have significant effects on the masses of the lighter generations [38]. In the case that the
Yukawa texture is given by a broken family symmetry in terms of an expansion parameter ǫ, one expects
model dependent corrections of order ǫ which may be significant.
In summary, in the light of the significant improvement in the measurement of fermion mass parameters,
we have analyzed the possibility that the fermion mass structure results from an underlying supersymmetric
GUT at a very high-scale mirroring the unification found for the gauge couplings. Use of the RG equations
to continue the mass parameters to the GUT scale shows that, although qualitatively in agreement with
the GUT predictions coming from simple Higgs structures, there is a small quantitative discrepancy. We
have shown that these discrepancies may be eliminated by finite radiative threshold corrections involving
the supersymmetric partners of the Standard-Model states. The required magnitude of these corrections is
what is expected at large tanβ, and the form needed corresponds to a supersymmetric spectrum in which
the gluino mass is negative with the opposite sign to the Wino mass. We have also performed a fit to the
recent data to extract the underlying Yukawa coupling matrices for the quarks and leptons. This is done
in the basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small
relative to the appropriate combinations of on-diagonal matrix elements, the basis most likely to be relevant
if the fermion mass structure is due to a spontaneously broken family symmetry. We have explored the
effect of SUSY threshold corrections for a variety of SUSY spectra. The resulting structure has significant
differences from previous fits, and we hope will provide the “data” for developing models of fermion masses
such as those based on a broken family symmetry.
M.S. acknowledges support from the United States Air Force Institute of Technology. The views expressed
in this letter are those of the authors and do not reflect the official policy or position of the United States
Air Force, Department of Defense, or the US Government.
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  Grand Unified Theories predict relationships between the GUT-scale quark and
lepton masses. Using new data in the context of the MSSM, we update the values
and uncertainties of the masses and mixing angles for the three generations at
the GUT scale. We also update fits to hierarchical patterns in the GUT-scale
Yukawa matrices. The new data shows not all the classic GUT-scale mass
relationships remain in quantitative agreement at small to moderate tan beta.
However, at large tan beta, these discrepancies can be eliminated by finite,
tan beta-enhanced, radiative, threshold corrections if the gluino mass has the
opposite sign to the wino mass.

Unification and fermion mass structure.
Graham G. Ross ∗and Mario Serna †
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP
November 18, 2021
Abstract
Grand Unified Theories predict relationships between the GUT-scale quark and lepton masses. Using
new data in the context of the MSSM, we update the values and uncertainties of the masses and mixing
angles for the three generations at the GUT scale. We also update fits to hierarchical patterns in the
GUT-scale Yukawa matrices. The new data shows not all the classic GUT-scale mass relationships remain
in quantitative agreement at small to moderate tan β. However, at large tan β, these discrepancies can be
eliminated by finite, tan β-enhanced, radiative, threshold corrections if the gluino mass has the opposite
sign to the wino mass.
Explaining the origin of fermion masses and mixings remains one of the most important goals in our
attempts to go beyond the Standard Model. In this, one very promising possibility is that there is an
underlying stage of unification relating the couplings responsible for the fermion masses. However we are
hindered by the fact that the measured masses and mixings do not directly give the structure of the underlying
Lagrangian both because the data is insufficient unambiguously to reconstruct the full fermion mass matrices
and because radiative corrections can obscure the underlying structure. In this letter we will address both
these points in the context of the MSSM.
We first present an analysis of the measured mass and mixing angles continued to the GUT scale. The
analysis updates previous work, using the precise measurements of fermion masses and mixing angles from
the b-factories and the updated top-quark mass from CDF and D0. The resulting data at the GUT scale
allows us to look for underlying patterns which may suggest a unified origin. We also explore the sensitivity
of these patterns to tanβ-enhanced, radiative threshold corrections.
We next proceed to extract the underlying Yukawa coupling matrices for the quarks and leptons. There
are two difficulties in this. The first is that the data cannot, without some assumptions, determine all elements
of these matrices. The second is that the Yukawa coupling matrices are basis dependent. We choose to work
in a basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small relative
to the appropriate combinations of on-diagonal matrix elements. This is the basis we think is most likely to
display the structure of the underlying theory, for example that of a spontaneously broken family symmetry
in which the hierarchical structure is ordered by the (small) order parameter breaking the symmetry. With
this structure to leading order the observed masses and mixing angles determine the mass matrix elements
on and above the diagonal, and our analysis determines these entries, again allowing for significant tanβ
enhanced radiative corrections. The resulting form of the mass matrices provides the “data” for developing
models of fermion masses such as those based on a broken family symmetry.
The data set used is summarized in Table 1. Since the fit of reference [4] (RRRV) to the Yukawa texture
was done, the measurement of the Standard-Model parameters has improved considerably. We highlight
a few of the changes in the data since 2000: The top-quark mass has gone from Mt = 174.3 ± 5 GeV to
Mt = 170.9 ± 1.9 GeV. In 2000 the Particle Data Book reported mb(mb) = 4.2 ± 0.2 GeV [5] which has
improved to mb(mb) = 4.2 ± 0.07 GeV today. In addition each higher order QCD correction pushes down
the value of mb(MZ) at the scale of the Z bosons mass. In 1998 mb(MZ) = 3.0 ± 0.2 GeV [6] and today
it is mb(MZ) = 2.87 ± 0.06 GeV [7]. The most significant shift in the data relevant to the RRRV fit is a
downward revision to the strange-quark mass at the scale µL = 2 GeV from ms(µL) ≈ 120± 50 MeV [5] to
∗g.ross@physics.ox.ac.uk
†serna@physics.ox.ac.uk
http://arxiv.org/abs/0704.1248v2
Low-Energy Parameter Value(Uncertainty in last digit(s)) Notes and Reference
mu(µL)/md(µL) 0.45(15) PDB Estimation [1]
ms(µL)/md(µL) 19.5(1.5) PDB Estimation [1]
mu(µL) +md(µL) [8.8(3.0), 7.6(1.6)] MeV PDB, Quark Masses, pg 15
[1]. ( Non-lattice, Lattice )
−(md+mu)2/4
22.8(4) Martemyanov and Sopov [2]
ms(µL) [103(20) , 95(20)] MeV PDB, Quark Masses, pg 15
[1]. [Non-lattice, lattice]
mu(µL) 3(1) MeV PDB, Quark Masses, pg 15
[1]. Non-lattice.
md(µL) 6.0(1.5) MeV PDB, Quark Masses, pg 15
[1]. Non-lattice.
mc(mc) 1.24(09) GeV PDB, Quark Masses, pg 16
[1]. Non-lattice.
mb(mb) 4.20(07) GeV PDB, Quark Masses, pg 16,19
[1]. Non-lattice.
Mt 170.9 (1.9)GeV CDF & D0 [3] Pole Mass
(Me,Mµ,Mτ ) (0.511(15), 105.6(3.1), 1777(53) ) MeV 3% uncertainty from neglect-
ing Y e thresholds.
A Wolfenstein parameter 0.818(17) PDB Ch 11 Eq. 11.25 [1]
ρ Wolfenstein parameter 0.221(64) PDB Ch 11 Eq. 11.25 [1]
λ Wolfenstein parameter 0.2272(10) PDB Ch 11 Eq. 11.25 [1]
η Wolfenstein parameter 0.340(45) PDB Ch 11 Eq. 11.25 [1]
|VCKM |
0.97383(24) 0.2272(10) 0.00396(09)
0.2271(10) 0.97296(24) 0.04221(80)
0.00814(64) 0.04161(78) 0.999100(34)
 PDB Ch 11 Eq. 11.26 [1]
sin 2β from CKM 0.687(32) PDB Ch 11 Eq. 11.19 [1]
Jarlskog Invariant 3.08(18)× 10−5 PDB Ch 11 Eq. 11.26 [1]
vHiggs(MZ) 246.221(20) GeV Uncertainty expanded. [1]
( α−1EM (MZ), αs(MZ), sin
2 θW (MZ) ) ( 127.904(19), 0.1216(17), 0.23122(15)) PDB Sec 10.6 [1]
Table 1: Low-energy observables. Masses in lower-case m are MS running masses. Capital M indicates pole
mass. The light quark’s (u,d,s) mass are specified at a scale µL = 2 GeV. VCKM are the Standard Model’s
best fit values.
today’s value ms(µL) = 103± 20 MeV. We also know the CKM unitarity triangle parameters better today
than six years ago. For example, in 2000 the Particle Data book reported sin 2β = 0.79 ± 0.4 [5] which is
improved to sin 2β = 0.69± 0.032 in 2006 [1]. The sin 2β value is about 1.2 σ off from a global fit to all the
CKM data [8], our fits generally lock onto the global-fit data and exhibit a 1 σ tension for sin 2β. Together,
the improved CKM matrix observations add stronger constraints to the textures compared to data from
several years ago.
We first consider the determination of the fundamental mass parameters at the GUT scale in order simply
to compare to GUT predictions. The starting point for the light-quark masses at low scale is given by the
χ2 fit to the data of Table 1
mu(µL) = 2.7± 0.5 MeV md(µL) = 5.3± 0.5 MeV ms(µL) = 103± 12 MeV. (1)
Using these as input we determine the values of the mass parameters at the GUT scale for various choices of
tanβ but not including possible tanβ enhanced threshold corrections. We do this using numerical solutions
to the RG equations. The one-loop and two-loop RG equations for the gauge couplings and the Yukawa
couplings in the Standard Model and in the MSSM that we use in this study come from a number of sources
[6] [9][10] [11]. The results are given in the first five columns of Table 2. These can readily be compared to
expectations in various Grand Unified models. The classic prediction of SU(5) with third generation down-
quark and charged-lepton masses given by the coupling B 5f .10f .5H
1 is mb(MX)/mτ (MX) = 1 [12]. This
ratio is given in Table 2 where it may be seen that the value agrees at a special low tanβ value but for large
tanβ it is some 25% smaller than the GUT prediction2. A similar relation between the strange quark and
the muon is untenable and to describe the masses consistently in SU(5) Georgi and Jarlskog [14] proposed
that the second generation masses should come instead from the coupling C 5f .10f .45H leading instead to
the relation 3ms(MX)/mµ(MX) = 1. As may be seen from Table 2 in all cases this ratio is approximately
0.69(8). The prediction of Georgi and Jarlskog for the lightest generation masses follows from the relation
Det(Md)/Det(M l) = 1. This results from the form of their mass matrix which is given by3
 , M l =
A −3C
 (2)
in which there is a (1, 1) texture zero4 and the determinant is given by the product of the (3, 3), (1, 2) and
(2, 1) elements. If the (1, 2) and (2, 1) elements are also given by 5f .10f .5H couplings they will be the same
in the down-quark and charged-lepton mass matrices giving rise to the equality of the determinants. The
form of eq(2) may be arranged by imposing additional continuous or discrete symmetries. One may see from
Table 2 that the actual value of the ratio of the determinants is quite far from unity disagreeing with the
Georgi Jarlskog relation.
In summary the latest data on fermion masses, while qualitatively in agreement with the simple GUT
relations, has significant quantitative discrepancies. However the analysis has not, so far, included the SUSY
threshold corrections which substantially affect the GUT mass relations at large tanβ [15]. A catalog of the
full SUSY threshold corrections is given in [16]. The particular finite SUSY thresholds discussed in this letter
do not decouple as the super partners become massive. We follow the approximation described in Blazek,
Raby, and Pokorski (BRP) for threshold corrections to the CKM elements and down-like mass eigenstates
[17]. The finite threshold corrections to Y e and Y u and are generally about 3% or smaller
δY u, δY d . 0.03 (3)
and will be neglected in our study. The logarithmic threshold corrections are approximated by using the
Standard-Model RG equations from MZ to an effective SUSY scale MS.
The finite, tanβ-enhanced Y d SUSY threshold corrections are dominated by the a sbottom-gluino loop,
a stop-higgsino loop, and a stop-chargino loop. Integrating out the SUSY particles at a scale MS leaves the
matching condition at that scale for the Standard-Model Yukawa couplings:
δmsch Y
uSM = sinβ Y u (4)
δmsch Y
d SM = cosβ U
1 + Γd + V
CKM Γ
u VCKM
Y ddiag U
R (5)
Y e SM = cosβ Y e. (6)
All the parameters on the right-hand side take on their MSSM values in the DR scheme. The factor δmsch
converts the quark running masses from MS to DR scheme. The β corresponds to the ratio of the two
Higgs VEVs vu/vd = tanβ. The U matrices decompose the MSSM Yukawa couplings at the scale MS :
Y u = U
diagU
R and Y
d = U
diagU
R. The matrices Y
diag and Y
diag are diagonal and correspond to
the mass eigenstates divided by the appropriate VEV at the scale MS . The CKM matrix is given by
VCKM = U
L . The left-hand side involves the Standard-Model Yukawa couplings. The matrices Γ
u and
Γd encode the SUSY threshold corrections.
If the squarks are diagonalized in flavor space by the same rotations that diagonalize the quarks, the
matrices Γu and Γd are diagonal: Γd = diag(γd, γd, γb), Γ
u = diag(γu, γu, γt). In general the squarks are
15f , 10f refer to the SU(5) representations making up a family of quarks and leptons while 5H is a five dimensional
representation of Higgs scalars.
2We’d like to thank Ilja Dorsner for pointing out that the tan β dependence of mb/mτ (MX) is more flat than in previous
studies (e.g. ref. [13]). This change is mostly due to the higher effective SUSY scale MS , the higher value of αs(MZ) found in
global standard model fits, and smaller top-quark mass Mt.
3The remaining mass matrix elements may be non-zero provided they do not contribute significantly to the deteminant
4Below we discuss an independent reason for having a (1, 1) texture zero.
Parameters Input SUSY Parameters
tanβ 1.3 10 38 50 38 38
γb 0 0 0 0 −0.22 +0.22
γd 0 0 0 0 −0.21 +0.21
γt 0 0 0 0 0 −0.44
Parameters Corresponding GUT-Scale Parameters with Propagated Uncertainty
yt(MX) 6
−5 0.48(2) 0.49(2) 0.51(3) 0.51(2) 0.51(2)
yb(MX) 0.0113
+0.0002
−0.01 0.051(2) 0.23(1) 0.37(2) 0.34(3) 0.34(3)
yτ (MX) 0.0114(3) 0.070(3) 0.32(2) 0.51(4) 0.34(2) 0.34(2)
(mu/mc)(MX) 0.0027(6) 0.0027(6) 0.0027(6) 0.0027(6) 0.0026(6) 0.0026(6)
(md/ms)(MX) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7)
(me/mµ)(MX) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2)
(mc/mt)(MX) 0.0009
+0.001
−0.00006 0.0025(2) 0.0024(2) 0.0023(2) 0.0023(2) 0.0023(2)
(ms/mb)(MX) 0.014(4) 0.019(2) 0.017(2) 0.016(2) 0.018(2) 0.010(2)
(mµ/mτ )(MX) 0.059(2) 0.059(2) 0.054(2) 0.050(2) 0.054(2) 0.054(2)
A(MX) 0.56
+0.34
−0.01 0.77(2) 0.75(2) 0.72(2) 0.73(3) 0.46(3)
λ(MX) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1)
ρ̄(MX) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6)
η̄(MX) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4)
J(MX) × 10
−5 1.4+2.2−0.2 2.6(4) 2.5(4) 2.3(4) 2.3(4) 1.0(2)
Parameters Comparison with GUT Mass Ratios
(mb/mτ )(MX) 1.00
+0.04
−0.4 0.73(3) 0.73(3) 0.73(4) 1.00(4) 1.00(4)
(3ms/mµ)(MX) 0.70
−0.05 0.69(8) 0.69(8) 0.69(8) 0.9(1) 0.6(1)
(md/3me)(MX) 0.82(7) 0.83(7) 0.83(7) 0.83(7) 1.05(8) 0.68(6)
(detY
detY e
)(MX) 0.57
+0.08
−0.26 0.42(7) 0.42(7) 0.42(7) 0.92(14) 0.39(7)
Table 2: The mass parameters continued to the GUT-scale MX for various values of tanβ and threshold
corrections γt,b,d. These are calculated with the 2-loop gauge coupling and 2-loop Yukawa coupling RG
equations assuming an effective SUSY scale MS = 500 GeV.
not diagonalized by the same rotations as the quarks but provided the relative mixing angles are reasonably
small the corrections to flavour conserving masses, which are our primary concern here, will be second order
in these mixing angles. We will assume Γu and Γd are diagonal in what follows.
Approximations for Γu and Γd based on the mass insertion approximation are found in [18][19][20]:
γt ≈ y
t tanβ
, µ2) ∼ y2t
γu ≈ −g
2 M2 µ
,m2χ2 ,m
ũ) ∼ 0 (8)
M3 µ I3(m
M3 µ I3(m
where I3 is given by
2, b2, c2) =
a2b2 log a
+ b2c2 log b
+ c2a2 log c
(a2 − b2)(b2 − c2)(a2 − c2)
. (11)
In these expressions q̃ refers to superpartner of q. χj indicate chargino mass eigenstates. µ is the coefficient
to the Hu Hd interaction in the superpotential. M1,M2,M3 are the gaugino soft breaking terms. A
t refers to
the soft top-quark trilinear coupling. The mass insertion approximation breaks down if there is large mixing
between the mass eigenstates of the stop or the sbottom. The right-most expressions in eqs(7,9,10) assume
the relevant squark mass eigenstates are nearly degenerate and heavier than M3 and µ. These expressions (
eqs 7 - 10) provide an approximate mapping from a supersymmetric spectra to the γi parameters through
which we parameterize the threshold corrections; however, with the exception of Column A of Table 4, we
do not specify a SUSY spectra but directly parameterize the thresholds corrections through γi.
The separation between γb and γd is set by the lack of degeneracy of the down-like squarks. If the
squark masses for the first two generations are not degenerate, then there will be a corresponding separation
between the (1,1) and (2,2) entries of Γd and Γu. If the sparticle spectra is designed to have a large At and
a light stop, γt can be enhanced and dominate over γb. Because the charm Yukawa coupling is so small, the
scharm-higgsino loop is negligible, and γu follows from a chargino squark loop and is also generally small
with values around 0.02 because of the smaller g2 coupling. In our work, we approximate Γ
22 ∼ Γ
11 ∼ 0.
The only substantial correction to the first and second generations is given by γd [15].
As described in BRP, the threshold corrections leave |Vus| and |Vub/Vcb| unchanged to a good approxi-
mation. Threshold corrections in Γu do affect the Vub and Vcb at the scale MS giving
V SMub − V
V MSSMub
V SMcb − V
V MSSMcb
⋍ − (γt − γu) . (12)
The threshold corrections for the down-quark masses are given approximately by
md ⋍ m
d (1 + γd + γu)
ms ⋍ m
s (1 + γd + γu)
mb ⋍ m
b (1 + γb + γt)
where the superscript 0 denotes the mass without threshold corrections. Not shown are the nonlinear effects
which arise through the RG equations when the bottom Yukawa coupling is changed by threshold effects.
These are properly included in our final results obtained by numerically solving the RG equations.
Due to our assumption that the squark masses for the first two generations are degenerate, the combina-
tion of the GUT relations given by
detM l/ detMd
(3ms/mµ)
(mb/mτ ) = 1 is unaffected up to nonlinear
effects. Thus we cannot simultaneously fit all three GUT relations through the threshold corrections. A best
fit requires the threshold effects given by
γb + γt ≈ −0.22± 0.02 (13)
γd + γu ≈ −0.21± 0.02. (14)
giving the results shown in the penultimate column of Table 2, just consistent with the GUT predictions. The
question is whether these threshold effects are of a reasonable magnitude and, if so, what are the implications
for the SUSY spectra which determine the γi? From eqs(9,10), at tanβ = 38 we have
∼ −0.5,
∼ 1.0
The current observation of the muon’s (g − 2)µ is 3.4 σ [21] away from the Standard-Model prediction.
If SUSY is to explain the observed deviation, one needs tanβ > 8 [22] and µM2 > 0 [23]. With this sign
we must have µM3 negative and the d̃, s̃ squarks only lightly split from the b̃ squarks. M3 negative is
characteristic of anomaly mediated SUSY breaking[24] and is discussed in [25][26][20][27]. Although we have
deduced M3 < 0 from the approximate eqs(9,10), the the correlation persists in the near exact expression
found in eq(23) of ref [17]. Adjusting to different squark splitting can occur in various schemes[28]. However
the squark splitting can readily be adjusted without spoiling the fit because, up to nonlinear effects, the
solution only requires the constraints implied by eq(13), so we may make γb > γd and hence make m
by allowing for a small positive value for γt. In this case A
t must be positive.
It is of interest also to consider the threshold effects in the case that µM3 is positive. This is illustrated
in the last column of Table 2 in which we have reversed the sign of γd, consistent with positive µM3 , and
chosen γb ≃ γd as is expected for similar down squark masses. The value of γt is chosen to keep the equality
between mb and mτ . One may see that the other GUT relations are not satisfied, being driven further away
by the threshold corrections. Reducing the magnitude of γb and γd reduces the discrepancy somewhat but
still limited by the deviation found in the no-threshold case (the fourth column of Table 2).
Parameter 2001 RRRV Fit A0 Fit B0 Fit A1 Fit B1 Fit A2 Fit B2
tanβ Small 1.3 1.3 38 38 38 38
a′ O(1) 0 0 0 0 −2.0 −2.0
ǫu 0.05 0.030(1) 0.030(1) 0.0491(16) 0.0491(15) 0.0493(16) 0.0493(14)
ǫd 0.15(1) 0.117(4) 0.117(4) 0.134(7) 0.134(7) 0.132(7) 0.132(7)
|b′| 1.0 1.75(20) 1.75(21) 1.05(12) 1.05(13) 1.04(12) 1.04(13)
arg(b′) 90o +93(16)o − 93(13)o +91(16)o − 91(13)o +93(16)o − 93(13)o
a 1.31(14) 2.05(14) 2.05(14) 2.16(23) 2.16(24) 1.92(21) 1.92(22)
b 1.50(10) 1.92(14) 1.92(15) 1.66(13) 1.66(13) 1.70(13) 1.70(13)
|c| 0.40(2) 0.85(13) 2.30(20) 0.78(15) 2.12(36) 0.83(17) 2.19(38)
arg(c) − 24(3)o − 39(18)o − 61(14)o − 43(14)o − 59(13)o − 37(25)o − 60(13)o
Table 3: Results of a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold corrections.
We set a′ as indicated and set c′ = d′ = d = 0 and f = f ′ = 1 at fixed values.
At tanβ near 50 the non-linear effects are large and b − τ unification requires γb + γt ∼ −0.1 to −0.15.
In this case it is possible to have t − b − τ unification of the Yukawa couplings. For µ > 0,M3 > 0, the
“Just-so” Split-Higgs solution of references [29, 30, 31, 32] can achieve this while satisfying both b → s γ and
(g − 2)µ constraints but only with large γb and γt and a large cancellation in γb + γt. In this case, as in the
example given above, the threshold corrections drive the masses further from the mass relations for the first
and second generations because µM3 > 0. It is possble to have t−b−τ unification with µM3 < 0, satisfying
the b → s γ and (g − 2)µ constraints in which the GUT predictions for the first and second generation of
quarks is acceptable. Examples include Non-Universal Gaugino Mediation [33] and AMSB; both have some
very heavy sparticle masses ( & 4 TeV) [20]. Minimal AMSB with a light sparticle spectra( . 1 TeV), while
satisfying (g − 2)µ and b → s γ constraints, requires tanβ less than about 30 [23].
We turn now to the second part of our study in which we update previous fits to the Yukawa matrices
responsible for quark and lepton masses. As discussed above we choose to work in a basis in which the
mass matrices are hierarchical with the off-diagonal elements small relative to the appropriate combinations
of on-diagonal matrix elements. This is the basis we think is most likely to display the structure of the
underlying theory, for example that of a spontaneously broken family symmetry, in which the hierarchical
structure is ordered by the (small) order parameter breaking the symmetry. With this structure to leading
order in the ratio of light to heavy quarks the observed masses and mixing angles determine the mass matrix
elements on and above the diagonal provided the elements below the diagonal are not anomalously large.
This is the case for matrices that are nearly symmetrical or for nearly Hermitian as is the case in models
based on an SO(10) GUT.
For convenience we fit to symmetric Yukawa coupling matrices but, as stressed above, this is not a critical
assumption as the data is insensitive to the off-diagonal elements below the diagonal and the quality of the
fit is not changed if, for example, we use Hermitian forms. We parameterize a set of general, symmetric
Yukawa matrices as:
Y u(MX) = y
d′ǫ4u b
′ ǫ3u c
′ ǫ3u
b′ ǫ3u f
′ ǫ2u a
′ ǫ2u
c′ ǫ3u a
′ ǫ2u 1
 , (15)
Y d(MX) = y
d ǫ4d b ǫ
d c ǫ
b ǫ3d f ǫ
d a ǫ
c ǫ3d a ǫ
 . (16)
Although not shown, we always choose lepton Yukawa couplings at MX consistent with the low-energy lepton
masses. Notice that the f coefficient and ǫd are redundant (likewise in Y
u). We include f to be able to
discuss the phase of the (2,2) term. We write all the entries in terms of ǫ so that our coefficients will be
O(1). We will always select our best ǫ parameters such that |f | = 1.
RRRV noted that all solutions, to leading order in the small expansion parameters, only depend on two
Parameter A B C B2 C2
tanβ 30 38 38 38 38
γb 0.20 −0.22 +0.22 −0.22 +0.22
γt −0.03 0 −0.44 0 −0.44
γd 0.20 −0.21 +0.21 −0.21 +0.21
a′ 0 0 0 −2 −2
ǫu 0.0495(17) 0.0483(16) 0.0483(18) 0.0485(17) 0.0485(18)
ǫd 0.131(7) 0.128(7) 0.102(9) 0.127(7) 0.101(9)
|b′| 1.04(12) 1.07(12) 1.07(11) 1.05(12) 1.06(10)
arg(b′) 90(12)o 91(12)o 93(12)o 95(12)o 95(12)o
a 2.17(24) 2.27(26) 2.30(42) 2.03(24) 1.89(35)
b 1.69(13) 1.73(13) 2.21(18) 1.74(10) 2.26(20)
|c| 0.80(16) 0.86(17) 1.09(33) 0.81(17) 1.10(35)
arg(c) − 41(18)o − 42(19)o − 41(14)o − 53(10)o − 41(12)o
Y u33 0.48(2) 0.51(2) 0.51(2) 0.51(2) 0.51(2)
Y d33 0.15(1) 0.34(3) 0.34(3) 0.34(3) 0.34(3)
Y e33 0.23(1) 0.34(2) 0.34(2) 0.34(2) 0.34(2)
(mb/mτ )(MX) 0.67(4) 1.00(4) 1.00(4) 1.00(4) 1.00(4)
(3ms/mµ)(MX) 0.60(3) 0.9(1) 0.6(1) 0.9(1) 0.6(1)
(md/3me)(MX) 0.71(7) 1.04(8) 0.68(6) 1.04(8) 0.68(6)∣∣∣detY
d(MX)
detY e(MX )
∣∣∣ 0.3(1) 0.92(14) 0.4(1) 0.92(14) 0.4(1)
Table 4: A χ2 fit of eqs(15,16) including the SUSY threshold effects parameterized by the specified γi.
phases φ1 and φ2 given by
φ1 = (φ
b − φ
f )− (φb − φf ) (17)
φ2 = (φc − φa)− (φb − φf ). (18)
where φx is the phase of parameter x. For this reason it is sufficient to consider only b
′ and c as complex
with all other parameters real.
As mentioned above the data favours a texture zero in the (1, 1) position. With a symmetric form for the
mass matrix for the first two families, this leads to the phenomenologically successful Gatto Sartori Tonin
[34] relation
Vus(MX) ≈
∣∣bǫd − |b′|ei φb′ ǫu
∣∣∣∣ . (19)
This relation gives an excellent fit to Vus with φ1 ≈ ± 90
o, and to preserve it we take d, d′ to be zero in our
fits. As discussed above, in SU(5) this texture zero leads to the GUT relation Det(Md)/Det(M l) = 1 which,
with threshold corrections, is in good agreement with experiment. In the case that c is small it was shown
in RRRV that φ1 is to a good approximation the CP violating phase δ in the Wolfenstein parameterization.
A non-zero c is necessary to avoid the relation Vub/Vcb =
mu/mc and with the improvement in the data,
it is now necessary to have c larger than was found in RRRV 5. As a result the contribution to CP violation
coming from φ2 is at least 30%. The sign ambiguity in φ1 gives rise to an ambiguity in c with the positive
sign corresponding to the larger value of c seen in Tables 3 and 4.
Table 3 shows results from a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold
corrections. The error, indicated by the term in brackets, represent the widest axis of the 1σ error ellipse
in parameter space. The fits labeled ‘A’ have phases such that we have the smaller magnitude solution of
|c|, and fits labeled ‘B’ have phases such that we have the larger magnitude solution of |c|. As discussed
above, it is not possible unambiguously to determine the relative contributions of the off-diagonal elements
of the up and down Yukawa matrices to the mixing angles. In the fit A2 and B2 we illustrate the uncertainty
5As shown in ref. [35], it is possible, in a basis with large off-diagonal entries, to have an Hermitian pattern with the (1,1)
and (1,3) zero provided one carefully orchestrates cancelations among Y u and Y d parameters. We find this approach requires
a strange-quark mass near its upper limit.
associated with this ambiguity, allowing for O(1) coefficients a′. In all the examples in Table 3, the mass
ratios, and Wolfenstein parameters are essentially the same as in Table 2.
The effects of the large tanβ threshold corrections are shown in Table 4. The threshold corrections
depend on the details of the SUSY spectrum, and we have displayed the effects corresponding to a variety
of choices for this spectrum. Column A corresponds to a “standard” SUGRA fit - the benchmark Snowmass
Points and Slopes (SPS) spectra 1b of ref([36]). Because the spectra SPS 1b has large stop and sbottom
squark mixing angles, the approximations given in eqns(7-10) break down, and the value for the correction
γi in Column A need to be calculated with the more complete expressions in BRP [17]. In the column A
fit and the next two fits in columns B and C, we set a′ and c′ to zero. Column B corresponds to the fit
given in the penultimate column of Table 2 which agrees very well with the simple GUT predictions. It is
characterized by the “anomaly-like” spectrum with M3 negative. Column C examines the M3 positive case
while maintaining the GUT prediction for the third generation mb = mτ . It corresponds to the “Just-so”
Split-Higgs solution. In the fits A, B and C the value of the parameter a is significantly larger than that
found in RRRV. This causes problems for models based on non-Abelian family symmetries, and it is of
interest to try to reduce a by allowing a′, b′ and c′ to vary while remaining O(1) parameters. Doing this for
the fits B and C leads to the fits B2 and C2 given in Table 4 where it may be seen that the extent to which
a can be reduced is quite limited. Adjusting to this is a challenge for the broken family-symmetry models.
Although we have included the finite corrections to match the MSSM theory onto the standard model
at an effective SUSY scale MS = 500 GeV, we have not included finite corrections from matching onto a
specific GUT model. Precise threshold corrections cannot be rigorously calculated without a specific GUT
model. Here we only estimate the order of magnitude of corrections to the mass relations in Table 2 from
matching the MSSM values onto a GUT model at the GUT scale. The tanβ enhanced corrections in eq(7-10)
arise from soft SUSY breaking interactions and are suppressed by factors of MSUSY /MGUT in the high-scale
matching. Allowing for O(1) splitting of the mass ratios of the heavy states, one obtains corrections to yb/yτ
(likewise for the lighter generations) of O( g
(4π)2
) from the X and Y gauge bosons and O(
(4π)2
) from colored
Higgs states. Because we have a different Higgs representations for different generations, these threshold
correction will be different for correcting the 3ms/mµ relation than the mb/mτ relation. These factors can
be enhanced in the case there are multiple Higgs representation. For an SU(5) SUSY GUT these corrections
are of the order of 2%. Plank scale suppressed operators can also induce corrections to both the unification
scale [37] and may have significant effects on the masses of the lighter generations [38]. In the case that the
Yukawa texture is given by a broken family symmetry in terms of an expansion parameter ǫ, one expects
model dependent corrections of order ǫ which may be significant.
In summary, in the light of the significant improvement in the measurement of fermion mass parameters,
we have analyzed the possibility that the fermion mass structure results from an underlying supersymmetric
GUT at a very high-scale mirroring the unification found for the gauge couplings. Use of the RG equations
to continue the mass parameters to the GUT scale shows that, although qualitatively in agreement with
the GUT predictions coming from simple Higgs structures, there is a small quantitative discrepancy. We
have shown that these discrepancies may be eliminated by finite radiative threshold corrections involving
the supersymmetric partners of the Standard-Model states. The required magnitude of these corrections is
what is expected at large tanβ, and the form needed corresponds to a supersymmetric spectrum in which
the gluino mass is negative with the opposite sign to the Wino mass. We have also performed a fit to the
recent data to extract the underlying Yukawa coupling matrices for the quarks and leptons. This is done
in the basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small
relative to the appropriate combinations of on-diagonal matrix elements, the basis most likely to be relevant
if the fermion mass structure is due to a spontaneously broken family symmetry. We have explored the
effect of SUSY threshold corrections for a variety of SUSY spectra. The resulting structure has significant
differences from previous fits, and we hope will provide the “data” for developing models of fermion masses
such as those based on a broken family symmetry.
M.S. acknowledges support from the United States Air Force Institute of Technology. The views expressed
in this letter are those of the authors and do not reflect the official policy or position of the United States
Air Force, Department of Defense, or the US Government.
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