2016年通用UUT模型：Ms、d与εK的新张力
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本文主要探讨了2016年在通用单位三角（Universal Unitarity Triangle, UUT）和具有约束性最小味素违背（Constrained Minimal Flavour Violation, CMFV）模型框架下的研究进展。费米实验室晶格和MILC合作的最新改进结果对强相互作用引起的B0s-d0-B's-d0混合中的矩阵元素产生了显著影响。研究团队利用这些新数据，精确测量了|Vub|/|Vcb|的比值，即0.0864±0.0025，以及CP不对称参数γ的值，即(62.7±2.1)°。在CMFV模型中，介子混合的新贡献可以通过一个单一的通用变量S(v)来描述。
作者进一步通过将实验观测的ΔMd/ΔMs与CP不对称SψKS关联起来，计算了 CKM矩阵元素|Vts|、|Vtd|、|Vcb|和|Vub|。值得注意的是，他们在2006年的研究中确定的S(v)的下限导致了这些元素的上限，这在当时明显低于实验值，从而揭示了UUT模型与εK（电荷-parity违反参数）之间的张力。使用ΔMs的实验值作为S(v)的函数，避免了依赖于树级近似下的|Vub|和|Vcb|测量所带来的不确定性。
然而，当使用εK作为输入时，发现ΔMs和d的数值与实验数据相比明显偏高。这一结果强调了新晶格数据对CMFV模型内ΔMs、d和εK关系的显著增强，使得这种张力更加显著。这篇论文通过细致的理论分析和实验数据的结合，深入探究了在CMFV模型下，基本粒子物理中的UUT参数与不同物理量之间的微妙平衡及其可能的限制。这对于理解标准模型的局限性和潜在的新物理现象具有重要意义。
Eur. Phys. J. C (2016) 76 :197 Page 3 of 12 197
Both strategies use the determination of the UUT by means
of (7) and allows to determine the whole CKM matrix, in
particular |V
ts
|, |V
td
|, |V
ub
| and |V
cb
| as functions of S(v).
Yet their outcome is very different, which signals the ten-
sion between M
s,d
and ε
K
in this framework. As we will
demonstrate below, this tension, known already from previ-
ous studies [27,28], has been sharpened significantly through
the results in (3) and (4). Using these two strategies sepa-
rately allows one to exhibit this tension transparently. Indeed
we have the following:
• The lower bound in (8) implies in S
1
upper bounds on
|V
ts
|, |V
td
|, |V
ub
|and |V
cb
|which are saturated in the SM,
and in turn it allows to derive an upper bound on ε
K
in
CMFV models that is saturated in the SM but turns out
to be significantly below the data.
• The lower bound in (8) implies in S
2
also upper bounds
on |V
ts
|, |V
td
|, |V
ub
| and |V
cb
| which are saturated in
the SM. However the S(v) dependence of these ele-
ments determined in this manner differs from the one
obtained in S
1
, which in turn allows to derive lower
bounds on M
s,d
in CMFV models that are reached
in the SM but turn out to be significantly above the
data.
It has been known since 2008 that the SM experiences
some tension in the correlation between S
ψ K
S
and ε
K
[29–
33]. It should be emphasised that in CMFV models only the
version of this tension in [30], i. e. NP in ε
K
, is possible
as in these models there are no new CP-violating phases.
Therefore S
ψ K
S
has to be used to determine the sole phase
in these models, the angle β in the UT, or equivalently the
CKM phase, through the unitarity of the CKM matrix. The
resulting low value of ε
K
can be naturally raised in CMFV
models by enhancing the value of S(v) and/or increasing the
value of |V
cb
|. However, as pointed out in [27,28], this spoils
the agreement of the SM with the data on M
s,d
, signalling
the tension between M
s,d
and ε
K
in CMFV models. The
2013 analysis of this tension in [34] found that the situation of
CMFV with respect to F = 2 transitions would improve
if more precise results for F
B
s
ˆ
B
B
s
and F
B
d
ˆ
B
B
d
turned
out to be lower than the values known in the spring of 2013.
The recent results from [3]in(3) show the opposite. Both
F
B
s
ˆ
B
B
s
and F
B
d
ˆ
B
B
d
increased. Moreover the more pre-
cise and significantly smaller value of ξ enlarges the tension
in question.
In view of the new lattice results, in this paper we take
another look at CMFV models. Having more precise values
for F
B
s
ˆ
B
B
s
, F
B
d
ˆ
B
B
d
and ξ than in 2013, our strategy
outlined above differs from the one in [34]. In particular we
take γ to be a derived quantity and not an input as done in the
latter paper. Moreover, we will be able to reach much firmer
conclusions than it was possible in 2013. In particular, in
contrast to [34] and also to [3] at no place in our paper tree-
level determinations of |V
ub
|, |V
cb
|and γ are used. However,
we compare our results with them.
It should be mentioned that Fermilab-MILC identified a
significant tension between their results for the B
0
s,d
−
¯
B
0
s,d
mass differences and the tree-level determination of the CKM
matrix within the SM. Complementary to their findings, we
identify a significant tension within F = 2 processes, that
is between ε
K
and M
s,d
in the whole class of CMFV mod-
els. Moreover, we determine very precisely the UUT, in par-
ticular the angle γ in this triangle and the ratio |V
ub
|/|V
cb
|,
both valid also in the SM.
Our paper is organised as follows. In Sect. 2 we deter-
mine first the UUT as outlined above, that in 2016 is signif-
icantly better known than in 2006 [25] and in particular in
2000, when the UUT was first suggested [23]. Subsequently
we execute the strategies S
1
and S
2
defined above. The val-
ues of |V
ts
|, |V
td
|, |V
cb
| and |V
ub
|, resulting from these two
strategies, differ significantly from each other which is the
consequence of the tension between ε
K
and M
s,d
in ques-
tion. In Sect. 3 we present the implications of these results
for K
L
→ π
0
ν ¯ν, K
+
→ π
+
ν ¯ν and B
s,d
→ μ
+
μ
−
within
the SM, obtaining again rather different results in S
1
and S
2
.
In Sect. 4 we briefly discuss how the U (2)
3
models match
the new lattice data and comment briefly on other models.
We conclude in Sect. 5.
2 Deriving the UUT and the CKM
2.1 Determination of the UUT
We begin with the determination of the UUT. For the mass
differences in the B
0
s,d
−
¯
B
0
s,d
systems we have the very accu-
rate expressions
M
s
= 17.757/ps ·
⎡
⎣
ˆ
B
B
s
F
B
s
276.0MeV
⎤
⎦
2
S(v)
2.322
|V
ts
|
0.0389
2
×
η
B
0.5521
, (9)
M
d
= 0.5055/ps ·
⎡
⎣
ˆ
B
B
d
F
B
d
229.4MeV
⎤
⎦
2
S(v)
2.322
×
|V
td
|
7.95 · 10
−3
2
η
B
0.5521
. (10)
The value 2.322 in the normalisation of S(v) is its SM value
for m
t
(m
t
) = 163.5 GeV obtained from
123
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