We have investigated the present renormalization prescriptions of
Cabibbo-Kobayashi-Maskawa (CKM) matrix, and found there is still not an
integrated prescription to all loop levels. In this paper We propose a new
prescription which is designed for all loop levels in the present perturbative
theory. This new prescription will keep the unitarity of the bare CKM matrix
and make the amplitude of an arbitrary physical process involving quark mixing
convergent and gauge independent.Comment: 4 pages, has been revise

Zhou, Yong

Physics Letters B

English

arXiv

AbstractWe have investigated the present renormalization prescriptions of Cabibbo–Kobayashi–Maskawa (CKM) matrix, and found there is still not an integrated prescription to all loop levels in the on-shell renormalization scheme. In this Letter we attempt proposing a new prescription designed for all loop levels in the present perturbative theory. This new prescription will keep the unitarity of the CKM matrix and make the amplitude of an arbitrary physical process involving quark mixing convergent and gauge independent

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Physics Letters B 577 (2003) 67–70
www.elsevier.com/locate/physletb
Renormalization of the Cabibbo–Kobayashi–Maskawa matrix
in standard model
Yong Zhou
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
Received 3 August 2003; received in revised form 23 September 2003; accepted 4 October 2003
Editor: T. Yanagida
Abstract
We have investigated the present renormalization prescriptions of Cabibbo–Kobayashi–Maskawa (CKM) matrix, and found
there is still not an integrated prescription to all loop levels in the on-shell renormalization scheme. In this Letter we attempt
proposing a new prescription designed for all loop levels in the present perturbative theory. This new prescription will keep the
unitarity of the CKM matrix and make the amplitude of an arbitrary physical process involving quark mixing convergent and
gauge independent.
 2003 Published by Elsevier B.V.
PACS: 11.10.Gh; 12.15.Lk; 12.15.Hh
Open access under CC BY license.As an important part of standard model (SM) [1],
the renormalization of Cabibbo–Kobayashi–Maskawa
(CKM) quark mixing matrix is a matter of great ac-
count in theory. At present, along with the devel-
opment of exact determination of CKM matrix ele-
ments [2], the importance of renormalization of CKM
matrix becomes more and more apparent. This was re-
alized for the Cabibbo angle with two fermion gener-
ations by Marciano and Sirlin [3] and for the CKM
matrix of the three-generation SM by Denner and
Sack [4] more than a decade ago. Though Denner and
Sack’s prescription is very delicate and simple, it re-
duces the physical amplitude involving quark mixing
E-mail address: zhouy@itp.ac.cn (Y. Zhou).0370-2693 2003 Published by Elsevier B.V.
doi:10.1016/j.physletb.2003.10.013
Open access under CC BY licengauge dependent [5]. Recently many authors have dis-
cussed this problem [5,6], but because of its complex-
ity all of them are limited to one-loop level and an in-
tegrated prescription beyond one-loop level in the on-
shell renormalization scheme has been not obtained.
So we want to propose a new prescription to solve this
problem.
As we know a CKM matrix renormalization pre-
scription must satisfy the three conditions [6]:
(1) In order to keep the transition amplitude of any
physical process involving quark mixing ultra-
violet finite, the CKM counterterm must cancel
out the ultraviolet divergence left in the loop-
corrected amplitude.
(2) It must guarantee such transition amplitude gauge
parameter independent [7].se.
68 Y. Zhou / Physics Letters B 577 (2003) 67–70(3) SM requires the bare CKM matrix V 0 is unitary,
(1)
∑
k
V 0ikV
0∗
jk = δij ,
with i , j , k the generation index and δij the
unit matrix element. If we split the bare CKM
matrix element into the renormalized one and its
counterterm
(2)V 0ij = Vij + δVij
and keep the unitarity of the renormalized CKM
matrix, Eq. (1) requires
(3)
∑
k
(
δVikV
∗
jk + VikδV ∗jk + δVikδV ∗jk
)= 0.
In order to satisfy these conditions we will renormal-
ize the CKM matrix through two steps. First we intro-
duce a CKM counterterm which makes the physical
amplitude of W+ → uid¯j convergent and gauge in-
dependent below certain loop levels. Next we mend
it to satisfy the unitary condition of Eq. (3) below the
loop levels, and simultaneously keep the divergent and
gauge-dependent (if it has) part of it unchanged. Then
by recursion we construct the CKM counterterms till
infinite loop levels.
In order to elaborate our idea clearly we firstly
introduce the n-loop (n  1) decay amplitude of
W+ → uid¯j as follows (here all of the contributions
of the counterterms lower than n-loop level have been
included in the formfactors):
Tnij =AL
[
FLnij + Vij
(
δgn
g
+ 1
2
δZWn
)
+ 1
2
δZ¯uLnikVkj +
1
2
VikδZ
dL
nkj + δVnij
]
(4)+ARFRnij +BLGLnij +BRGRnij ,
with g and δg the SU(2) coupling constant and
its counterterm, δZW the W boson wave-function
renormalization constant (WRC), δZ¯uL and δZdL the
left-handed up-type and down-type quark’s WRC [8].
The added denotation n represents the n-loop result,
and
AL = g√
2
u¯i(p1)/εγLνj (q −p1),
(5)BL = g√
2
u¯i(p1)
ε · p1
MW
γLνj (q − p1),with εµ the W boson polarization vector, γL and γR
the left-handed and right-handed chiral operators, and
MW the W boson mass. Similarly, replacing γL with
γR in Eq. (5) we get AR and BR , respectively. FL,R
and GL,R are four formfactors. Here we will only care
about the coefficient of AL which contains the n-loop
CKM counterterm. The simplest method to make the
amplitude Tn convergent and gauge independent is to
make the coefficient of AL equal to zero, i.e.,
δVnij =−FLnij − Vij
(
δgn
g
+ 1
2
δZWn
)
(6)− 1
2
δZ¯uLnikVkj −
1
2
VikδZ
dL
nkj .
Obviously such CKM counterterm cannot be guar-
anteed to satisfy the unitary condition of Eq. (3),
so needs to be mended. Here we introduce a new
set of denotation: δV¯n, to denote the amended CKM
counterterm which satisfies the unitary condition. Our
method is to construct δV¯n through δVn, δV¯n−1, . . . ,
δV¯1. Here we state that δVn is obtained by using
δV¯n−1, . . . , δV¯1 as the lower-loop CKM counterterms
in Eq. (6). Now the unitary condition of Eq. (3) be-
comes
δV¯1V
† + V δV¯ †1 = 0,
δV¯2V
† + V δV¯ †2 =−δV¯1δV¯ †1 ,
δV¯3V
† + V δV¯ †3 =−δV¯1δV¯ †2 − δV¯2δV¯ †1 ,
...
δV¯nV
† + V δV¯ †n =−δV¯1δV¯ †n−1 − δV¯2δV¯ †n−2 · · ·
− δV¯n−2δV¯ †2 − δV¯n−1δV¯ †1 ,
(7)...
In order to solve these equations, we introduce a set of
symbols Bn
B1 = 0,
(8)Bn =
n−1∑
i=1
−δV¯iδV¯ †n−i .
Obviously Bn satisfies
(9)Bn = B†n.
Assuming the CKM counterterms δV¯1, δV¯2, . . . , δV¯n−1
and δVn have been obtained, the n-loop amended
Y. Zhou / Physics Letters B 577 (2003) 67–70 69CKM counterterm δV¯n can be determined as follows:
(10)δV¯n = 12
(
δVn − V δV †n V +BnV
)
.
This method takes example by Diener and Kniehl’s
prescription [6]. It is easy to check that such CKM
counterterm satisfies Eqs. (7) at n-loop level. Since the
counterterms δg and δZW are both real in the on-shell
renormalization scheme, we can obtain the following
result from Eq. (10) and Eq. (6)
δV¯nij = 12
(∑
kl
VikF
∗
LnlkVlj − FLnij
)
+ 1
4
∑
k
(
δZ¯uL∗nki − δZ¯uLnik
)
Vkj
+ 1
4
∑
k
Vik
(
δZdL∗njk − δZdLnkj
)
(11)+ 1
2
∑
k
BnikVkj .
The remaining problem is to test whether the
amended CKM counterterm δV¯n has the same diver-
gent and gauge-dependent part as δVn, which is the re-
quirement of making the physical amplitude involving
quark mixing finite and gauge independent. Based on
the renormalizability and predictability of SM, we can
predict that the divergent and gauge-dependent part of
δVn (if it has) must satisfy the unitary condition of
Eq. (3) at n-loop level
(12)δV DGn V † + V δV DG†n = BDGn ,
where the superscript DG denotes the divergent or
gauge-dependent part of the quantity. This is because
if not so the unitary condition of Eq. (3) will require
the divergent or gauge-dependent part of the right
CKM counterterm different from δVn, thus will reduce
the physical amplitude of W+ → ui d¯j divergent or
gauge dependent (see Eq. (4)). In fact Eq. (12) is
satisfied at one-loop level [4,6]. From Eqs. (10) and
(12), it is easy to obtain(
δV¯ DGn − δV DGn
)
V †
(13)= 1
2
(
BDGn − δV DGn V † − V δV DG†n
)= 0.
Times CKM matrix V at the right-hand side of Eq.
(13), we have
(14)δV¯ DGn = δV DGn .Now we have obtained the proper CKM countert-
erm at n-loop level. We can construct CKM coun-
terterms till infinite loop levels by recursion, which
will satisfy the unitary condition of Eq. (3) and make
the physical amplitude involving quark mixing con-
vergent and gauge independent. Since the renormal-
ization of CKM matrix is a very complex problem
(one can see it from the fact that at present an in-
tegrated prescription applicable to all loop levels has
not been obtained in the on-shell renormalization
scheme), our solution is quite simple and practical
(see Eq. (11)). On the other hand, we suppose our
prescription will not break the present symmetries of
SM, e.g., Ward–Takahashi identity, because it only
changes the values of CKM matrix elements from V 0ij
to Vij + δV¯ij .
Lastly we want to point out that the problem
of infrared divergence is unclear in our renormal-
ization prescription. As we know, the correction of
Eq. (4) to the amplitude of W+ → uid¯j has the
infrared divergence coming from the Feynman di-
agrams including photons. This divergence should
be cancelled in the inclusive decay width W+ →
(ui d¯j , ui d¯j γ , ui d¯jγ γ, . . .) by the corresponding di-
vergence of the real photon emission processes. How-
ever, there are no a priori reasons for the cancellation
of this divergence in the proposed CKM matrix coun-
terterm, Eqs. (10), (11). Since this problem looks very
difficult to be solved, we want to leave it for the next
work.
Acknowledgements
The author thanks professor Xiao-Yuan Li for his
useful guidance and the referee for pointing out the
problem of infrared divergence in our manuscript. The
author also thanks Dr. Hu Qingyuan for his sincerely
help (in my life).
Appendix A
In this appendix we give the explicit result of δV¯1.
From Eqs. (11) and (8), we obtain
70 Y. Zhou / Physics Letters B 577 (2003) 67–70δV¯1ij = 12
(∑
kl
VikF
∗
L1lkVlj − FL1ij
)
+ 1
4
∑
k
(
δZ¯uL∗1ki − δZ¯uL1ik
)
Vkj
(A.1)+ 1
4
∑
k
Vik
(
δZdL∗1jk − δZdL1kj
)
,
which is gauge independent since δV1ij is gauge
independent in Eq. (6) [5] and Eq. (10). Eq. (A.1)
is similar as Eq. (12) of Ref. [6]. The ultraviolet
divergence of δV¯1ij is
δV¯1ij
∣∣
UV
= 3α∆
64πM2Ws2W
×
[
−2
∑
k,l 
=j md,jm2u,kVilV ∗klVkj
md,l −md,j
+ 2
∑
k,l md,jm
2
u,kVilV
∗
klVkj
md,l +md,j
− 2
∑
k 
=i,l mu,im2d,lVilV ∗klVkj
mu,k −mu,i
+ 2
∑
k,l mu,im
2
d,lVilV
∗
klVkj
mu,k +mu,i
+ Vij
(∑
k
VikV
∗
ikm
2
d,k
(A.2)
+
∑
k
VkjV
∗
kjm
2
u,k − 2m2d,j − 2m2u,i
)]
,with α the fine structure constant, sW the sine of
the weak mixing angle θW , and ∆ = 2/(D − 4) +
γE − ln(4π) + ln(M2W/µ2) (D is the space–time
dimensionality, γE is the Euler’s constant, and µ is an
arbitrary energy scale). mu,i and md,j , etc. are up-type
and down-type quark’s masses. The Rξ -gauge and the
dimensional regularization have been used. This result
agrees with the results of Refs. [4,5] and [6].
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Renormalization of the Cabibbo-Kobayashi-Maskawa Matrix in Standard Model

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