Gauge-Higgs unification and related topics

Precision tests and CP violation
in gauge-Higgs unification scenario
@ 「余剰次元物理」研究会
(Jan. 20, ’10)
C.S. Lim （林青司）
(Kobe University)
I. Gauge-Higgs unification (GHU)
unification of gravity (s=2) & elemag (s=1)
Kaluza-Klein theory
unified theory of gauge (s=1) & Higgs (s=0) interactions
“Gauge-Higgs unification”
: realized in higher dimensional gauge theory
extra dimension
5D gauge field
Higgs
4D space-time
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the idea of gauge-Higgs unification itself is not new:
・N.S. Manton, Nucl. Phys. 58(’79)141.
・Y. Hosotani, Phys. Lett. B126 (‘83) 309 : ``Hosotani mechanism”
The scenario was revived:
・H. Hatanaka , T. Inami and C.S.L., Mod. Phys. Lett. A13(’98)2601
( the main point )
・ The quantum correction to mH is finite because of the higher
dimensional gauge symmetry → A new avenue to solve the
hierarchy problem without invoking SUSY
(N.B.) The scenario may also shed some light on the arbitrariness
problem in the interactions of Higgs.
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II. Finite observables and the precision tests of GHU
scenario
To see whether the scenario is viable, it will be of crucial
importance to address the questions,
(1)Does the scenario have characteristic (generic) predictions on
the observables subject to the precision tests ?
(2) The problem of the arbitrariness of Higgs interactions may be
solved. On the other hand, how is the variety of Yukawa
couplings explained and how CP violation is realized ?
(flavor physics → Maru’s talk)
Concerning (1), it will be desirable to find out finite (UVinsensitive) and calculable observables subject to the precision
tests , although the theory is non-renormalizable and very UV
sensitive in general.
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Are there such calculable observables other than the Higgs mass,
protected by the higher dimensional gauge symmetry ?
Yes !
Calculable one-loop contributions to S and T parameters in the
Gauge-Higgs Unification w./ N. Maru (Phys. Rev. D75(’07)115011)
It will be natural to suspect that such observables are hidden in the
Gauge-Higgs sector, just as the case of the Higgs mass.
We thus consider the S, T parameters due to gauge boson selfenergies as the typical candidate.
We find (@ one-loop) :
・ For 6D, we find S – (4 cosθW) T is calculable
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(operator analysis)
In 4D, S and T are described by gauge invariant operators
including Higgs doublet with higher mass dimension 6:
However, in the Gauge-Higgs unification scenario, Higgs is a
gauge field, and these operators should be described by AM alone.
Thus we expect that the operators are no longer independent
(operators are also ``unified” ).
In fact, we find that the gauge invariant operator responsible
for S and T with mass dimension 6 (from 4D point of view)
is unique (under the Bianchi identity):
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The local operator yield a UV-divergence for a specific linear
combination of S and T, and the orthogonal combination ,
should be finite !
This expectation relying on an operator analysis has been
confirmed by explicit calculations of Feynman diagrams.
Finite anomalous magnetic moment in the GHU scenario
(w./ Y. Adachi and N. Maru)
We consider another observable including fermions, which has
been subject to the precision test: anomalous magnetic moment
of fermions, a= (g-2)/2.
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・ U(1) theory (D+1 dimensional) (Y. Adachi, C.S.L. and N. Maru,
Phys.Rev. D76(‘07)075009)
The result is striking ! We find the anomalous moment is finite
(calculable) in any space-time dimensions (in the simplified model)
in G-H unification scenario.
Simple operator analysis shows it is the case.
In 4D, an operator relevant for g-2 is given as
In GHU, the Higgs should be replaced by Ay and the higher
dimensional gauge symmetry implies Ay appears through
covariant derivative Dy.
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Thus, the relevant local gauge invariant operator should read as
Actually to get g-2, DA should be replaced by <DA>, where <AM> =
δMy <Ay>. On the other hand, the on-shell condition for the fermion
reads as
Thus we realize that the local operator relevant for g-2 just
disappears !
Hence, we expect g-2 is observable in any space-time
dimension, just as in the case of Higgs mass.
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The 1-loop diagram contributing to a = (g-2)/2.
(
)
+
The UV divergent and finite parts are nicely separated by use of
Poisson resummation.
We find the divergent part cancels out for arbitrary dimension
D!
・ The anomalous moment in a “realistic” model: SU(3) on
MD x (S1/Z2) (Y. Adachi, C.S.L. and N. Maru, Phys. Rev.
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D79(‘09)075018)
III. CP violation in higher dimensional theories
In spite of the great success of Kobayashi-Maskawa model,
the origin of CP violation still seems to be not conclusive .
(N.B.) The observed baryon asymmetry in the universe cannot
be attributed to the SM. → some new mechanism of CP
violation (?)
We may have some new mechanism to break CP, once space-time
is extended.
In fact, CP violation due to compactification of extra space was
discussed: C.S. Lim, Phys. Lett. B256(’91)233 (A.Strominger
and E. Witten, Commun. Math Phys. 101(’85)231).
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(How to define C, P transf.s corresponding to ordinary 4D ones ? )
We can easily fin matrix C, for instance, in higher dim. space-time,
so that it satisfies
.
However, we find (C.S. L., 1991):
・Such defined higher dimensional C, P transf.s do not
correspond to the 4-dimensional ones, in general, and some
modification is necessary.
・Interestingly, the modified CP transformation act on the extra
space coordinates non-trivially: it acts as a complex conjugation
of the complex homogeneous coordinates for the extra space.
・If the compactfied space has “complex structure”, the breaking
of CP can be realized.
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Take D=6 case for the illustrative purpose.
In the basis, where 6D spinor decomposes into two 4-D spinors,
gamma matrices are given as
The C matrix satisfying
,
found not to reduce to ordinary 4-D transf., because of
.
We thus modify C and P such that they correspond to ordinary 4D ones,
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Accordingly the transformation properties of a vector
is uniquely determined and we find:
(N.B.) The reason of the peculiar transf. under C was that
extra dimensional gamma matrices are half symmetric and
half anti-symmetric.
Thus introducing, a complex coordinate as
CP transf. is nothing but a complex conjugation:
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This peculiar property persists for higher (even) dimensions:
For instance, in 10D
Consider Type-I superstring theory with 6-dimensional Calabi-Yau
manifold defined by a quintic polynomial for the coordinates of
CP4 ,
“4 generation model”
CP is broken only when the coefficient C is complex, since
otherwise the above defining equation is invariant under
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・In fact, resultant Yukawa couplings is known to have a CP
violating phase for complex C (M. Matsuda, T. Matsuoka, H.
Mino, D. Suematsu and Y. Yamada, Prog. Theor. Phys.
79(’88)174).
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IV. CP violation in the GHU
How to break CP symmetry is a challenging issue in the
scenario of GHU, where the Higgs interactions are originally
gauge interactions with real couplings, in contrast to the case of
UED.
(N.B.) The low-energy limit of the open string sector of
superstring theory is a sort of gauge-Higgs unification model,
such as 10-dimensional (SUSY) Yang-Mills theory.
So the same problem should be addressed also in string theory.
As far as the original theory is CP invariant, possible way to
break CP would be, say, ``spontaneous CP violation".
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CP violation due to the orbifold compactification
(w./ N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-ph] )
One possibility to break CP symmetry is to invoke the manner of
compactificaion, which determines the vacuum state of the
theory.
(N.B.)
Another possibility: CP violation through Hosotani mechanism,
due to the VEV of the Higgs (Wilson-line phase).
→ Y. Adachi ‘s talk
In our paper we consider much simpler compactification than the
C.-Y. compactification. Namely, we discuss the CP violation in
the 6-dimensional U(1) gauge theory due to the compactification
on the orbifold
We easily know that CP transfomration is not compatible with
the condition of orbifolding.
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In terms of a complex coordinate
the orbifold condition is written as
After the CP transf.,
the condition reads as
: “orientation-changing operator”
(Strominger and Witten)
Thus, CP tranf. is not compatible with
orbifolding condition , and CP symmetry
is broken.
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Even the interaction vertices for non-zero K-K photons generally
have CP violating phases:
・We have shown the phases survive even after the re-phasing.
・We have identified re-phasing invariant quantity a la Jarlskog
parameter .
The EDM of electron, as a typical CP violating observable,
however, is found to vanish at 1-loop level. Unfortunately, we
anticipate that we cannot get a non-vanishing contributions even
at higher loops.
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This is because the P transformation acts as
Thus P symmetry is not violated by the compactification, as is
naively expected in QED.
Since, EDM necessitates both of P and CP violations, we anticipate
EDM vanishes in our model, although we expect EDM will get
contributions in a realistic theory including the SM, since P should
be violated anyway in such a realistic theory.
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