Renormalization-group constraints on Yukawa alignment in multi-Higgs-doublet models

0.1 MB·English

by  P. M. Ferreira

CFTP/10-001
Renormalization-group constraints on Yukawa
alignment in multi-Higgs-doublet models
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1
P.M. Ferreira,(1,2)∗ L. Lavoura,(3)† and Jo˜ao P. Silva(1,3)‡
0
2
n (1) Instituto Superior de Engenharia de Lisboa
a
J 1959-007 Lisboa, Portugal

4
1 (2) Centro de F´ısica Te´orica e Computacional, Universidade de Lisboa

1649-003 Lisboa, Portugal
]
h
p (3) Centro de F´ısica Te´orica de Part´ıculas, Instituto Superior T´ecnico
-
p
1049-001 Lisboa, Portugal
e
h
[ 14 January 2010

1
v
1
6
5 Abstract
2
.
1 We write down the renormalization-group equations for the Yukawa-coupling
0
matrices inageneral multi-Higgs-doublet model. We then assumethat thematrices
0
of the Yukawa couplings of the various Higgs doublets to right-handed fermions of
1
: fixed quantum numbers are all proportional to each other. We demonstrate that,
v
i in the case of the two-Higgs-doublet model, this proportionality is preserved by the
X
renormalization-group running only in the cases of the standard type-I, II, X, and
r
a Y models. We furthermore show that a similar result holds even when there are
more than two Higgs doublets: the Yukawa-coupling matrices to fermions of a given
electric charge remain proportional under the renormalization-group running if and
only if there is a basis for the Higgs doublets in which all the fermions of a given
electric charge couple to only one Higgs doublet.
∗E-mail: [email protected]
†E-mail: [email protected]
‡E-mail: [email protected]
1
The standard model of the electroweak interactions has an extension with two Higgs
doublets, called the two-Higgs-doublet model (THDM). In that extension the Yukawa
couplings are given by
L = −Q¯ (Γ φ +Γ φ )n + ∆ φ˜ +∆ φ˜ p −L¯ (Π φ +Π φ )ℓ +H.c., (1)
Y L 1 1 2 2 R 1 1 2 2 R L 1 1 2 2 R
h (cid:16) (cid:17) i
where the φ (k = 1,2) are the Higgs doublets, φ˜ ≡ iτ φ∗, and Q , L , n , p , and
k k 2 k L L R R
ℓ are 3-vectors in flavour space; Γ , ∆ , and Π are the 3×3 complex matrices of the
R k k k
Yukawa couplings to the right-handed down-type quarks, up-type quarks, and charged
leptons, respectively.
Recently [1], it has been proposed that the Yukawa-coupling matrices may be propor-
tional to each other:
Γ = dΓ , (2a)
2 1
∆ = u∆ , (2b)
2 1
Π = eΠ , (2c)
2 1
where d, u, and e are complex numbers. This ‘Yukawa alignment’ might be the result of
some symmetry and, as such, be stable under renormalization. The purpose of this paper
is to investigate that possibility by making use of the renormalization-group equations
(RGE).
The one-loop RGE for a model with n Higgs doublets have been presented in [2] for
the special case in which L only includes the matrices Π . It is easy to extend them to
Y k
the most general case. Let µ be the renormalization scale and D ≡ 16π2µ(d/dµ). Then,
DΓ = a Γ (3a)
k Γ k
n
† † †
+ 3tr Γ Γ +∆ ∆ +tr Π Π Γ (3b)
k l k l k l l
Xh (cid:16) (cid:17) (cid:16) (cid:17)i
l=1
n
1 1
+ −2∆ ∆†Γ +à Æà + ∆ ∆†Γ + à Æà , (3c)
(cid:18) l k l k l l 2 l l k 2 l l k(cid:19)
X
l=1
D∆ = a ∆ (4a)
k ∆ k
n
† † †
+ 3tr ∆ ∆ +Γ Γ +tr Π Π ∆ (4b)
k l k l k l l
Xh (cid:16) (cid:17) (cid:16) (cid:17)i
l=1
n
1 1
+ −2Γ Γ†∆ +∆ ∆†∆ + Γ Γ†∆ + ∆ ∆†∆ , (4c)
(cid:18) l k l k l l 2 l l k 2 l l k(cid:19)
X
l=1
DΠ = a Π (5a)
k Π k
n
+ 3tr Γ Γ† +∆†∆ +tr Π Π† Π (5b)
k l k l k l l
Xh (cid:16) (cid:17) (cid:16) (cid:17)i
l=1
n
1
+ Π Π†Π + Π Π†Π , (5c)
(cid:18) k l l 2 l l k(cid:19)
X
l=1
2
where tr denotes the trace and the factors 3 in (3b), (4b), and (5b) are from colour.
Furthermore,
9 5
a = −8g2 − g2 − g′2, (6a)
Γ s 4 12
9 17
a = −8g2 − g2 − g′2, (6b)
∆ s 4 12
9 15
a = − g2 − g′2, (6c)
Π
4 4
where g , g, and g′ are the gauge coupling constants of SU(3) , SU(2), and U(1),
s colour
respectively.1 We emphasize that equations (3)–(6) are valid for a model with an arbitrary
number of Higgs doublets and for any number of fermion generations.
Letusexaminethecontribution(3b)intheTHDMunderthealignmenthypothesis(2).
One obtains
DΓ = 3+3|d|2 tr Γ Γ† +(3+3ud)tr ∆†∆
1 1 1 1 1
h (cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1)
+(1+e∗d)tr Π Π† Γ +··· , (7a)
1 1 1
(cid:16) (cid:17)i
DΓ = 3|d|2d+3d tr Γ Γ† + 3|u|2d+3u∗ tr ∆†∆
2 1 1 1 1
h (cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
+ |e|2d+e tr Π Π† Γ +··· . (7b)
1 1 1
(cid:16) (cid:17)i
(cid:0) (cid:1)
We now require that the hypothesis (2) be stable under the RGE, i.e. we require
DΓ = d(DΓ ), (8a)
2 1
D∆ = u(D∆ ), (8b)
2 1
DΠ = e(DΠ ). (8c)
2 1
Using (7), one finds that (8a) leads to
(u∗ −d)(1+ud) = 0, (9a)
(e−d)(1+e∗d) = 0. (9b)
It iseasy tocheck thatthecondition (9a) ensures that (8a) alsoholds fortheterms in(3c).
One may furthremore check that (8b) and (8c) do not yield any further conditions on d,
u, and e beyond (9). We have thus found that the alignment hypothesis is (one-loop)
renormalization-group invariant only in one of the four cases
d = e = u∗, (10a)
1
d = e = − , (10b)
u
1
d∗ = u = − , (10c)
e
1
u = e∗ = − . (10d)
d
1The normalization of g′ is such that the φk have hypercharge 1/2.
3
Instead of using the basis (φ ,φ )T for the two Higgs doublets, one may use any other
1 2
basis, related to that one through an SU(2) transformation. In particular, one may use
any of the following three bases:
φd1 = 1+|d|2 −1/2 1 d φ1 , (11a)
(cid:18) φ (cid:19) (cid:18) −d∗ 1 (cid:19)(cid:18) φ (cid:19)
d2 2
(cid:0) (cid:1)
φu1 = 1+|u|2 −1/2 1 u∗ φ1 , (11b)
(cid:18) φ (cid:19) (cid:18) −u 1 (cid:19)(cid:18) φ (cid:19)
u2 2
(cid:0) (cid:1)
φe1 = 1+|e|2 −1/2 1 e φ1 . (11c)
(cid:18) φ (cid:19) (cid:18) −e∗ 1 (cid:19)(cid:18) φ (cid:19)
e2 2
(cid:0) (cid:1)
If the alignment conditions (2) hold, then the Yukawa Lagrangian (1) may be rewritten
L = −Q¯ φ 1+|d|2Γ n +φ˜ 1+|u|2∆ p −L¯ φ 1+|e|2Π ℓ +H.c.,
Y L d1 1 R u1 1 R L e1 1 R
(cid:18) q q (cid:19) q
(12)
where φ˜ ≡ iτ φ∗ . The various cases (10) may then be interpreted in the following way:
u1 2 u1
• In the case (10a), φ = φ = φ , hence
d1 u1 e1
L
Y = −Q¯ φ Γ n +φ˜ ∆ p −L¯ φ Π ℓ +H.c.,
L d1 1 R d1 1 R L d1 1 R
1+|d|2 (cid:16) (cid:17)
q
which means that, in the basis (φ ,φ ), only φ has Yukawa couplings. This is
d1 d2 d1
called the type-I THDM [3].
• In the case (10b),
L ∆
Y = −Q¯ φ Γ n −φ˜ 1 p −L¯ φ Π ℓ +H.c.,
L d1 1 R d2 R L d1 1 R
(cid:18) d (cid:19)
1+|d|2
q
which means that only φ has Yukawa couplings to the down-type quarks and to
d1
the charged leptons, while only φ has Yukawa couplings to the up-type quarks.
d2
This is called the type-II THDM [4].
• In the case (10c),
L Π
Y = −Q¯ φ Γ n +φ˜ ∆ p +L¯ φ 1 ℓ +H.c.,
L d1 1 R d1 1 R L d2 d∗ R
1+|d|2 (cid:16) (cid:17)
q
which means that only φ has Yukawa couplings to the quarks while only φ has
d1 d2
Yukawa couplings to the charged leptons. This is called the type-X THDM [5].
• In the case (10d)
L ∆ Π
Y = −Q¯ φ Γ n −φ˜ 1 p +L¯ φ 1 ℓ +H.c.,
L(cid:18) d1 1 R d2 d R(cid:19) L d2 d∗ R
1+|d|2
q
4
which means that only φ has Yukawa couplings to the down-type quarks while
d1
only φ has Yukawa couplings to the charged leptons and to the up-type quarks.
d2
This is called the type-Y THDM [5].
Itiswell knownthatallthesecasesmaybeobtainedthroughtheimpositionofappropriate
symmetries on the Yukawa Lagrangian (1).
2
Z
We proceed to analyze the case of an arbitrary number n of Higgs doublets, where
n n
L = −Q¯ Γ φ n +∆ φ˜ p −L¯ Π φ ℓ +H.c., (13)
Y L k k R k k R L k k R
X(cid:16) (cid:17) X
k=1 k=1
with a generalized alignment hypothesis
Γ = d Γ , (14a)
k k 1
∆ = u ∆ , (14b)
k k 1
Π = e Π , (14c)
k k 1
where d = u = e = 1. Inserting the conditions (14) in equation (3) leads to
1 1 1
DΓ = a d Γ
k Γ k 1
n
+ 3d |d |2tr Γ Γ† +3u∗u d tr ∆†∆ +e e∗d tr Π Π† Γ
k l 1 1 k l l 1 1 k l l 1 1 1
Xh (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)i
l=1
n
1 3
+ −2u u∗d + |u |2d ∆ ∆†Γ + |d |2d à Æà . (15)
(cid:20)(cid:18) l k l 2 l k(cid:19) 1 1 1 2 l k 1 1 1(cid:21)
X
l=1
It follows that
n
DΓ −d DΓ = 3(u∗ −d )u d tr ∆†∆ +(e −d )e∗d tr Π Π† Γ
k k 1 k k l l 1 1 k k l l 1 1 1
Xh (cid:16) (cid:17) (cid:16) (cid:17)i
l=1
n
−2(u∗ −d ) u d ∆ ∆†Γ . (16)
k k l l 1 1 1
X
l=1
We require the quantity (16) to vanish for all k. A similar requirement is imposed on
D∆ −u D∆ and on DΠ −e DΠ . We find that the generalized alignment hypothe-
k k 1 k k 1
sis (14) is preserved by the renormalization-group evolution if and only if
n
(u∗ −d ) u d = 0, (17a)
k k l l
X
l=1
n
(e −d ) e∗d = 0, (17b)
k k l l
X
l=1
n
(u∗ −e ) u e = 0 (17c)
k k l l
X
l=1
5
aresatisfiedforallk. Now, ifthegeneralizedalignment hypothesis holds, thentheYukawa
Lagrangian (13) may be rewritten as
L = −Q¯ N φ Γ n +N φ˜ ∆ p −L¯ N φ Π ℓ +H.c., (18)
Y L d d1 1 R u u1 1 R L e e1 1 R
(cid:16) (cid:17)
where
n n
N φ = d φ , N = v |d |2, (19a)
d d1 k k d u k
Xk=1 uXk=1
t
n n
N φ = u∗φ , N = v |u |2, (19b)
u u1 k k u u k
Xk=1 uXk=1
t
n n
N φ = e φ , N = v |e |2. (19c)
e e1 k k e u k
Xk=1 uXk=1
t
Thus, condition (17a) means that either φ = φ or else φ and φ are orthogonal;
u1 d1 u1 d1
the conditions (17b) and (17c) have analogous interpretations. This finally leaves us with
only five possibilities:
1. φ = φ = φ ;
d1 u1 e1
2. φ is orthogonal to φ = φ ;
u1 d1 e1
3. φ is orthogonal to φ = φ ;
e1 d1 u1
4. φ is orthogonal to φ = φ ;
d1 e1 u1
5. φ , φ , and φ are all orthogonal to each other.
d1 u1 e1
The possibilities 1–4 reproduce the four cases (10a)–(10d) already present in the THDM.
The possibility 5 is new: if there are three or more Higgs doublets, it is possible that each
charged-fermion sector couples to a separate Higgs doublet—this is of course impossible
in the THDM.
Itiseasytoshowthatallfivecases1–5maybeenforcedthroughsimple symmetries.
2
Z
To summarize, in this paper we have shown that Yukawa alignment in the two-Higgs-
doublet model, as proposed in [1], can only be stable under the renormalization group
in the well-known cases in which only one Higgs doublet has Yukawa couplings to the
right-handed fermions of each electric charge. All other cases of Yukawa alignment are
not consistent with the renormalization group, which means that, either they cannot be
enforcedbyanysymmetries, ortheycanbeenforcedbysymmetries onlyinthecontextofa
modellargerthantheTHDM. Wehavegeneralized thenotionofalignment tomodelswith
an arbitrary number of Higgs doublets and have shown that, also in that generalization,
alignment is only stable under the RGE when the right-handed fermions of each electric
charge all couple to only one Higgs doublet.
6
Acknowledgements: We thank A. Pich and P. Tuzo´n for reading and commenting on
the manuscript. The work of P.M.F. is supported in part by the Portuguese Fundac¸˜ao
para a Ciˆencia e a Tecnologia (FCT) under contract PTDC/FIS/70156/2006. The work
of L.L. and of J.P.S. is funded by FCT through the project U777–Plurianual.
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