text:
Disclosure Debate Reheats In Wake Of FTC Report
In the wake of the mammoth study of mortgage disclosures released in mid-June by the Federal Trade Commission (FTC), the real estate industry is actively analyzing the new forms and the FTC's conclusions about what needs changing. But the opinions on the study and what it could mean for RESPA reform are unsurprisingly varied. Paul Schieber of Blank Rome LLP perhaps said it best when he observed, "To be honest, any new form is probably going to be an improvement, but no form will satisfy everyone."
Brokers pleased
Marc Savitt, vice president and president-elect of the National Association of Mortgage Brokers (NAMB), was very pleased with the FTC's findings.
"I can't tell you how happy I am to see this report the way it came out, because it validates everything NAMB has been saying for the past few years — we need to simplify these disclosures to make them easier for consumers to understand. Whoever originates a loan should disclose on the exact same forms in the exact same manner. As the report shows, the way it is done now confuses the consumer, and when it confuses the consumer, they make mistakes, and they are costly mistakes."
Savitt noted, "One of the most surprising things from this is that you'd think more disclosure would come out as a result of this report or that they'd be recommending more disclosure — not just clearer disclosure. But actually what they're saying is that there is too much disclosure."
He added that even the savviest consumers are often confused by the forms, saying, "I've had people that are lawyers and/or involved in the mortgage business [applying for loans] and you can see in their eyes that they don't understand [the forms]. We always tell them that if you don't understand, don't be afraid to ask what you may think is a stupid question, because the only stupid question is the one you don't ask."
Breaking down the findings
Howard Lax of Lipson Neilson Cole Seltzer Garin PC took a very analytical look at the report.
"I cannot fault the empirical findings of the study," he said. "However, there is still a lot more work to be done before new disclosure models are proposed. This is clearly evident from the conclusions of the study."
Lax went through each conclusion one-by-one, offering comments on each.
(1) Current mortgage cost disclosures failed to convey key mortgage costs to many consumers.
"We know this already," Lax said.
(2) Prototype disclosures developed for the study significantly improved consumer recognition of mortgage costs, demonstrating that better disclosures are feasible.
"Yes, but they are up against a very low standard," Lax said. "The issue is not whether better disclosures are feasible - that is a given. The issue is whether better disclosures will be effective. This is just a first stab at providing effective disclosures. More work needs to be done."
Lax emphasized that "significant improvement is needed in the next prototype disclosure about prepayment and balloon loans, and explaining how the APR differs from the interest rate (and why this is important)."
Lax also pointed out the following quote from page ES-9 of the study:
"Although the prototype form provided significant improvements in consumer understanding, some consumers still failed to recognize key costs, and, in some cases, represented substantial proportions of prototype form respondents. Forty-one percent of prototype form respondents, for example, could not identify the amount of prepayment penalties (though this was a substantial improvement over the 95 percent who could not do so with the current forms), and 30 percent did not recognize that the loan included a large balloon payment, an identical percentage as in the current forms group. Further development of the disclosures may provide additional improvements that better convey these costs."
Regarding this, Lax said, "It is significant that borrowers who thought they were happy with their loan were not, and borrowers who thought they understood the terms of their loan did not — even those who considered themselves sophisticated consumers. Further study may reveal that there is no good way to explain certain terms, such as what the APR is and the benefit this disclosure provides. If it is not possible to make this disclosure effective (especially since almost all homeowners prepay their loan), the Federal Reserve Board and Congress should think about abandoning this disclosure or significantly revising it in favor of a more effective yardstick (perhaps by showing the cost of credit over the first seven years of a loan rather than over the life of the loan)."
(3) Both prime and subprime borrowers failed to understand key loan terms, and both benefited from the improved disclosures.
"Was anyone able to identify how they could have benefited from the improved understanding?" Lax asked. "Were there any 'Gee, I could have had a V-8!' moments? Did anyone go back to their lender and demand a modification of their loan?"
(4) Improved disclosures provided the greatest benefit for more complex loans, where both prime and subprime borrowers had the most difficulty understanding loan terms.
"I would like to see the FTC repeat the study with people who never had a mortgage loan, across a broader range of the population," Lax said. "The study produced minor differences between the understanding of loan terms by prime and subprime borrowers. We hear anecdotal stories of loan officers taking advantage of subprime borrowers because they are uneducated or inexperienced. Perhaps subprime borrowers are 'educated' by the origination process so that the study results are similar for prime and subprime borrowers. Perhaps the anecdotal stories were not supported by the study because the sample population for the study was from Montgomery County, Md., an area where education and income levels are substantially higher than elsewhere in the U.S.," he noted.
"Testing individuals who fall into the prime and subprime credit categories in various financially homogeneous regions (e.g. the City of Detroit, the City of San Francisco, and two rural communities) but who have not purchased a home may show that there are greater differences in understanding disclosures among persons of different financial circumstances. This issue was not part of this study," Lax said.
He added, "I suspect that creditworthiness is strongly related to the level of parental training and other informal education concerning financial services received by teenagers and young adults. Disclosures need to be developed for the inexperienced (first time) home buyer. Perhaps we need an additional set of disclosures for the experienced consumer."
Regarding this point, James Lacko of the FTC's Bureau of Economics and author of the study, disagreed about the study's scope.
Lacko told RESPAnews, "Although [Lax] is correct that we did not focus the study on first-time homebuyers, we did examine mortgage customers in many areas of the country," Lacko said. "The Montgomery County, Md. sample was used for the 36 in-depth interviews conducted in the study. The study also conducted consumer testing with over 800 mortgage customers in 12 locations across the country, including Boston, Westchester County (New York), Akron, Chicago, Nashville, Atlanta, Denver, Dallas, Phoenix, Las Vegas, Portland and Seattle."
Making it mean something
Lax also had some observations about other portions of the study.
He pointed out pages 31-34 of the report (Section 3.4.2) which explain how the study found that applicants really do not understand the various services that a broker or lender performs that are listed on the GFE.
"These statements underscore the need for mandatory financial education in public schools," Lax said.
Further, Lax said, "Page 61 of the report states that the enhanced GFE disclosure used in the study provided a total closing cost estimate rather than an itemized cost list. Many brokers and lenders have been providing a total of the estimated closing costs for a long time without any apparent benefit to consumers. This is evident from the high number of subjects that failed to identify the cash due at closing when the closing costs are financed (pages 99-100 of the report). Anecdotal stories indicate that the average uninformed consumer only cares how much his or her payment is going to be. Hence, there are only two significant items of information. First, how much money do I need to bring to closing (the report correctly identifies this as the borrower's first payment), and how much is my monthly payment thereafter.
"If the total closing costs is going to be a prime disclosure on the GFE, it has to mean something," he continued. "I believe that consumers are confusing the total closing costs and the amount needed to close. Both are important, but they are different concepts and they should be disclosed in different manners to recognize the differences. The total loan closing costs should be identified as a hidden cost of credit — much like taxes and title fees when buying a car —that may be built into the amount that is financed to buy the car. The amount that needs to be brought to closing should be identified as the first loan payment, as alluded to in the report."
A second disclosure?
Lax further said that the disclosure did not address some of the more complex decisions that a borrower should consider in structuring the loan.
"I think that the FTC should consider adding a second disclosure as an addendum that addresses subtle issues beyond the 'what is my payment' level," he said. "For example, the concept that the consumer can lower his closing costs and monthly payment by paying the closing costs rather than financing them should be disclosed in terms of the borrower's loan — perhaps in a short table attached as an addendum at the end of the disclosure."
The HUD factor
After the study came out last week, HUD's Brian Sullivan said the department was looking at the report and prototype disclosure and felt that the agencies were both "singing the same general song here, and that is better disclosure is better."
Sullivan said HUD certainly will be considering the FTC's suggestions as it moves forward with drafting the new rule and noted that HUD has continuously been in contact with its federal partners throughout the whole reform process.
For his part, Savitt said he hopes HUD limits itself to "RESPA reform lite" as Phil Schulman of K&L Gates put it in 2005, commenting on a reform proposal that would include only a revamped GFE.
Savitt added, "We hope HUD implements these changes and takes the recommendations seriously. It would be a tremendous benefit to the industry and consumers."
RESPA reform and policy implications
With regard to HUD's RESPA reform effort, Rich Andreano Jr. of Weiner Brodsky Sidman Kider PC felt that "the FTC's finding that consumer comprehension of mortgage transactions can be materially enhanced by simply improving the loan disclosures is significant."
Andreano noted, "While the FTC prototype disclosure presents a lump sum amount for settlement charges as a package, the FTC makes clear that it is not proposing the packaging concept that HUD included in its 2002 reform effort. Specifically, the FTC states that 'although the prototype form uses the phrase 'package' when referring to the settlement services charge, it would not necessarily require the types of packages outlined by HUD in its 2002 proposal. All that is necessary is that the cost of the various settlement services be disclosed as a single price rather than itemized as in the current GFE. Whether this is accomplished through a HUD-type package or simply an aggregation of the individual costs is not material to the disclosure or its intended use by consumers.' It will be interesting to see if HUD hears the FTC and proposes only improvements in the GFE and related changes, or attempts to adopt more wide-sweeping changes to RESPA as it did in 2002.
"HUD may be successful if it takes the former approach, and it likely will not be successful if it takes the latter approach," Andreano said.
Giving teeth to the GFE
Lax went a bit further in his policy speculation, stating that there is another issue Congress needs to address.
"State regulators recently began criticizing brokers for providing a GFE that varies significantly from the figures on the final HUD Settlement Statement. In many cases, the variance is due to differences between the processing fees, lender paid broker fees, and origination fees for (a) the loan amount and loan program used to provide the GFE, and (b) the loan program offered to the borrower after underwriting the borrower's credit," he said.
"In Michigan, state examiners have been recommending that brokers and lenders redisclose the GFE three days before closing to 'correct' the estimates made at the time of application," Lax continued. "There is no basis in state or federal law to require the redisclosure of the GFE. Some brokers, to avoid a charge by state regulators that the GFE is not accurate, are providing ranges of fees (e.g. the dollar equivalent of 0 percent to 5 percent of the loan amount) rather than fixed amounts. If only a total range of fees is provided, the consumer will not be able to use this disclosure for its intended purposes — to shop for credit.
"Either we must drop the pretense that consumers can use the GFE to shop for credit, or we need to make this disclosure binding on the broker for a period of time to permit the borrower to shop it around," he concluded.
RESPAnews will continue to follow this story as it progresses.
Robin Wardzala is the editor of http://www.respanews.com the first and only online publication dedicated solely to covering issues and news related to the Real Estate Settlement Procedures Act (RESPA). RESPAnews.com provides its 32,000 readers with exclusive reporting and analysis on HUD settlements, RESPA court cases and class actions, ongoing compliance updates and multi-sided perspectives on regulatory reform debates and proposals.
RESPAnews.com also offers a Q&A service with questions answered by expert attorneys. RESPAnews.com is a publication of October Research Corporation, the nation's premier provider of real estate industry news and analysis.
Article Source: http://EzineArticles.com/?expert=Robin_Wardzala


The new additions to Disney Springs so far have been terrific. Earlier on this list we have discussed how we were blown away by The BOATHOUSE, but Jock Lindsey's Hangar Bar may be the biggest surprise. Disney's recent attempts to make food and drink experiences from the Star Wars brand have been, at best, unique. A bar based around a character from the Indiana Jones films that no one remembers could have been a complete disaster, but instead it's actually kinda good. Actually, it's really good. While not as elaborately themed as Trader Sam's Grog Grotto, Jock Lindsey's has a much better food menu and generally has much less of a wait to get into.
Our recommendation for the Hangar Bar is the Cool-headed Monkey. One of the most memorable moments from Indiana Jones and the Temple of Doom is a dinner scene in which our heroes were served a meal consisting of snakes, live bugs, and monkey brains. The brains were served out of what appeared to be the head of a monkey, causing one character to faint. This is where we should point out that Temple of Doom is by far the darkest of all the Indiana Jones films and is usually the one people like or dislike the most. In a great touch, Jock Lindsey's Hangar Bar has a drink and souvenir mug to commemorate this classic film moment with the Cool-headed Monkey.
Jock Lindsey's Hangar Bar is surprisingly intimate inside, but has a good amount of seating outdoors as well. There are nods to Indiana Jones films throughout the restaurant, although thankfully none featuring Shia LaBeouf.
This set-up is also the exact same one Jim Hill uses up in New Hampshire to record The Unofficial Guid's Disney Dish podcasts.
I know that people "drink around the world", is there a name for when someone drinks all 25 Christmas drinks, like Mickey's Very Merry Liver Transplant or something?
When you order the Cool-Headed Monkey, do you get to keep that awesome cup?
Annoying Twitter tweets show up blocking written content on my iPad. Please fix.


"Will Barnet and Bob Blackburn: An Artistic Friendship in Relief"
Wesley and Missy Cochran, two contemporary print collectors from LaGrange, Georgia, and Patricia Phagan, curator of prints and drawings at the Georgia Museum of Art, have organized this exhibition of 51 lithographs, woodcuts, etchings and screenprints drawn from the Cochrans collection.
The two printmakers met first as teacher and student, the older Barnet having been instrumental in obtaining a scholarship for Blackburn to the Art Students League in the 1930s. Barnets simple, uncluttered, classical aesthetic at first seems opposed to Blackburns improvisational, abstract works, yet both emphasize the two-dimensional nature of their medium, and one can trace the progression of their careers through the variety of styles both have adopted and then discarded.


Oh my goodness, what to say about these two cute little people in my family. I am the luckiest Auntie! When I get to spend time with this brother and sister duo, all bets are off. It's time for lollipops, chocolate, juice with heart straws, chicken nuggets, and whatever toy they want to play with. My job is to spoil these two souls and always be a friend to them. Sloane will be eighteen months, going on...teenage...with her baby talk and..."don't you understand what I'm saying?" look. Finn will be three soon. He is my sweet little Disney loving' pal. We can find all toys Disney related and play for hours...including Jake and the Neverland Pirates, Mickey Mouse Clubhouse, and we have to give thumbs up to Bubble Guppies from Nickelodeon! Our photo session was an afternoon, that turned into dinner and after play. What a lucky girl I am to have two little ones that can sit on my lap while we watch Disney movies on my IMAC. Thank you Finn and Sloane for an awesome Saturday night date!


text:
\section{Introduction}

\noindent
Theories with extra spatial dimensions have received great attention during the past decade,
offering plausible solutions to the gauge hierarchy problem \cite{RS}, and inspiring
 many attempts to understand the fermion mass generation mechanism.

\noindent
Large flat extra dimensions \cite{ADD} can provide a suppression mechanism of fermion
masses via a reduction of Yukawa couplings by a volume factor $1/\sqrt{V}$, where $V$ is
the volume of the compact extra dimensional space \cite{ADD}. Many attempts have been
considered in this context to generate small masses for the Standard Model neutrinos,
see e.g. \cite{ADDR, DS, MP}.

\noindent
 Interesting alternatives arise when a non-factorizable geometry is considered
\cite{RS, Gob, RS2} .In this case the background metric entangles the four dimensional
space to
 the extra dimensions via ``warp'' factors $\sim e^{-k|y|}$ induced by the (constant)
curvature $k$ of the extra spatial dimensions $y$.
The role of the volume of large flat extra dimensions is now played by the warp factor,
 without implying stringent constraints on the size of the extra dimension.
A particularly successful construction has been proposed in \cite{RS} where an
$\mbox{AdS}_5$ space has been considered with the fifth
dimension $y$ circularly compactified on a  radius $R$ and orbifolded to
$S^1/{Z}_2$. The orbifold fixed points $y=0$ and $y=\pi R$ can be identified with 3-branes
with positive and negative tension: the Planck (or invisible) brane at $y=0$ and the TeV
 (or visible) brane at $y=\pi R$. Any fundamental mass parameter $m_0$ in the original
theory will be
 converted to an effective mass parameter $m \sim m_0 e^{-k\pi R}$ on the visible brane,
via the warping action of the slice of $\mbox{AdS}_5$ between the two branes.
For $kR\sim 11$ all masses of particles confined to the visible brane are naturally of
order the weak scale.

\noindent
One of the long standing questions in particle physics remains the one of finding a
 mechanism that naturally explains the apparent hierarchy of fermion masses in the Standard
 Model and why neutrino masses are much smaller than all charged lepton masses.
The addition of extra dimensions offers many new possibilities, along with new challenges:
the suppression of additional operators might become a far more complicated task in an extra
 dimensional framework. Any plausible mechanism of generation of observed phenomena, must
also guarantee a suppression of processes like proton decay, flavour changing neutral
 currents and lepton flavour violation.

\noindent
Many interesting attempts have been made along the years in order to explain the
smallness of neutrino masses with mechanisms induced by the presence of extra dimensions.
 This has been
 done by adding fermion fields to the bulk and maintaining their Dirac nature \cite{GN}, or
by adding a Majorana mass term \cite{HuberShafi} or considering higher dimensional Majorana
couplings \cite{HuberShafi_2}, or adding extra scalar fields to induce a small Majorana mass
 on the visible brane \cite{Chen}. The dynamics of bulk fermion, scalar and vector fields
has been
extensively considered in a supersymmetric context in e.g. \cite{GP,GP_2}.

\noindent
In this paper we consider a particular mechanism to generate a Majorana spinor on the
 visible brane by imposing a $Z_2$ transformation property of the bulk fermion fields
which mixes the
fields and their charge conjugates, analogously to the Majorana condition in four
dimensions. This boundary condition is tailored for neutrinos, and it is the generalization
 to a warped geometry of the boundary conditions first proposed in \cite{flat} in the case
 of flat extra dimensions. We show how these
{\em pseudo-Majorana} boundary conditions far more constrain the Kaluza-Klein spectrum
with respect to the ordinary $Z_2$ boundary conditions when reducing from five to four
 dimensions.

\noindent
The paper is organized as follows. In Section 2 we describe the framework,  impose the
 {\em pseudo-Majorana} boundary conditions and derive the Kaluza-Klein spectrum of the
 theory. Here
 we build the connection between the 5D and the 4D charge conjugation operators and we show
 that
the boundary conditions generate one Majorana spinor with peculiar properties induced by
the warped geometry. In Section 3 we attempt some more phenomenological considerations,
 identifying the bulk Majorana spinor with a sterile neutrino or with a Majorana Standard
Model neutrino.
In Section 4 we summarize alternative attempts to generate small Majorana masses and we
reconsider the derivation of ref.~\cite{Chen}. Our conclusions are in Section 5.

\section{Bulk fermions with pseudo-Majorana boundary conditions}

\noindent
We consider a scenario with one extra dimension as first introduced in ref.~\cite{RS}.
The fifth dimension $y$ is circularly compactified on a radius $R$ and orbifolded to
$S^1/{Z}_2$ on the
interval $-\pi R \le y \le \pi R$. The background metric is non-factorizable and describing
a slice of $\mbox{AdS}_5$. The orbifold fixed points are $y=0$ and $\pi R$ which can
be naturally
intepreted as the location of two 3-branes, where the stress-energy tensor solution of
the Einstein's equation is non zero.
The Planck scale brane is located at $y=0$ and the Tev or visible brane is located
at $y=\pi R$.
This thin-brane case can also be understood as the limiting behaviour of a thick
domain-wall solution.

\noindent
The action for a spin-1/2 Dirac fermion $\psi$ living in the bulk is the Dirac action
generalized to curved five-dimensional space with constant curvature and can be written as
follows
\begin{equation}
\label{eq:action_5d}
S= \int\, d^4x \int\, dy\, \sqrt{-g}\, \left ( i\bar{\psi} g^{MN}\gamma_M\,D_M \psi -
m \bar{\psi}\psi \right )\, ,
\end{equation}
where $g=det(g_{MN})= -e^{-8\sigma}$ is the determinant of the metric tensor
with $\sigma = k|y|$ and $k$ the constant curvature along the fifth dimension.
We use capital indices $M,N\ldots$ for objects defined in curved space and lower-case greek
indices $\mu ,\nu,\ldots \alpha ,\beta$ for objects defined in the
tangent frame: in our notation $\mu = 0,1,2,3$ and $M = (\mu , 5)$.

\noindent
A line-element solution of the Einstein's equations in five-dimensions, which respects
four-dimensional Poincare invariance, is given by \cite{RS}
\begin{equation}
\label{eq:ds2}
ds^2 = e^{-2\sigma}\eta_{\mu\nu}dx^\mu dx^\nu +dy^2\, ,
\end{equation}
 with flat metric tensor $\eta_{\mu\nu} = (-1, 1, 1 , 1)$ and $\mu ,\nu = 0,1,2,3$.
The warp factor $e^{-2\sigma}$
entangles the four dimensional space-time to the fifth dimension $y$
rendering the geometry non-factorizable. This is a special case of an $\mbox{AdS}_5$
 geometry.
This solution can only be trusted if $k<M$, i.e. the bulk curvature is small compared with
the fundamental Planck scale.
The gamma matrices $\gamma^M$ are the curved space Dirac matrices, $\gamma^M =
(\gamma^\mu , \gamma_5)$\footnote{We use the Dirac representation of the gamma
matrices unless differently stated}. They are related  to
the flat space ones through the {\emph {vielbein}} $e^M_\alpha$ ,
$\gamma^M = e^M_\alpha\gamma^\alpha$ with $e^M_\alpha = (e^\sigma ,  e^\sigma ,
 e^\sigma , e^\sigma , 1)$.

\noindent
We have introduced the covariant derivative acting on the fermion field in curved space
 $D_M = \partial_M + \Gamma_M$, where $\Gamma_M$ is the spin connection\footnote{This is
in agreement with ref.~\cite{GP} and one can easily check this to be equivalent to the
prescription derived in ref.~\cite{GN}.}
\begin{equation}
\Gamma_\mu = \frac{1}{2} \gamma_5\gamma_\mu {d\sigma\over dy}\qquad\Gamma_5 =0\, .
\end{equation}
For the sake of simplicity, we do not consider a bulk gauge field coupled to the bulk
fermion in this paper. An investigation of this case for flat extra dimensions can be
 found in ref.~\cite{flat}. We have introduced a
bulk mass term with mass $m$ for the fermion field.
The mass $m$ can be typically generated by scalar fields acquiring a vacuum expectation
value in the higher dimensional theory.
It is in general complex valued $m= |m|e^{i\alpha}$, however its phase is irrelevant at
this level, since we can always rotate the Wyel spinors by a phase which
 cancels the phase in the mass term and leaves the action invariant. Hence, we can set
$m=|m|$ without loss of generality.

\noindent
The invariance of the bulk mass term under the ${Z}_2$ parity transformation
$y \to -y$ can be
guaranteed in two ways: the mass $m$ and the scalar fermion bilinear $\bar{\psi}\psi$
are both odd or both even under a ${Z}_2$-parity transformation. A $Z_2$ odd mass parameter
 can be explained as the vev of a scalar field with an odd ``kink'' profile. The vacuum
 configuration of the scalar field resembles in this case the background 3-form field in
 the dimensional reduction of the Horava-Witten theory~\cite{HoravaWitten}.  The $Z_2$-even
 case appears when pseudo-Majorana boundary conditions are imposed, as it will be discussed
 in the following.
\vspace{1truecm}

\noindent {\bf{The odd case}}

\noindent
A $Z_2$ odd mass parameter is required when ordinary boundary conditions for the bulk
fermion fields are imposed. These have been extensively considered in the literature~
\cite{GN,GP}.
The bulk Dirac field transforms under ${Z}_2$ as
\begin{equation}
\label{ordinaryBC}
\psi (x, -y)=\pm \gamma_5 \psi (x,y)\, ,
\end{equation}
 where the
sign can only be determined by the fermion interactions. This transformation property
implies that the fermion scalar bilinear $\bar{\psi}\psi$ is odd under ${Z}_2$,
$\bar\psi (-y)\psi(-y) = - \bar\psi (y)\psi (y)$. The invariance of the action under $Z_2$
 thus requires a ${Z}_2$ odd mass of the type $m(y) = m\,\epsilon (y)$ with
$\epsilon (y)$ the sign function.
Given that the left- and right-handed components of the Dirac field satisfy
$\psi_{L,R} =\mp \gamma_5\psi_{L,R}$, ordinary boundary conditions will imply that the
L-handed field is ${Z}_2$-odd and the R-handed field even, or viceversa.
The odd component will vanish at the fixed points of the orbifolding
$y=0$ and $y=\pi R$ - where $\pi R$ is indentified with -$\pi R$\footnote{The orbifolding
condition that identifies fields at $y=\pi R$ and $y= -\pi R$ is equivalent to periodic
boundary conditions imposed on universal flat dimensions of size $L$, i.e.
$\psi (y) = \psi (y+L)$.}.

\noindent The odd bulk mass term vanishes at the boundaries $y=0, \pm \pi R$.
Thus we conclude that the most general
mass term allowed by ordinary boundary conditions contains a bulk contribution and no
boundary terms.

\vspace{1truecm}
\noindent {\bf The even case}

\noindent The ${Z}_2$-even case has instead an even mass paramater $m$ and an even fermion
bilinear which
does not vanish at the orbifold fixed points, so that both a bulk mass contribution and a
boundary contribution are in general allowed.
We consider an especially suitable type of unordinary boundary conditions, first introduced
in ref.~\cite{flat} in the context of universal flat extra dimensions.
We call them {\emph{pseudo-Majorana}} boundary conditions, since they relate the bulk
fermion field to its charge-conjugate field in the five-dimensional theory and
they are able to generate the Majorana condition $\psi = \psi^c$ back into four dimensions.

\noindent The pseudo-Majorana boundary conditions are imposed on the Dirac fields living
in the 5D bulk as follows
\begin{equation}
\label{eq:tbc}
\psi (x, -y) = e^{-i\delta} \gamma_5\psi^c (x,y)\, ,
\end{equation}
where a phase factor is allowed in the most general case, and $\psi^c =C \left
 ( \bar{\psi}\right )^T$ is the charge conjugate field with
$C (\equiv C_5)$ the charge conjugation matrix of the five dimensional theory.
At this point we only need to know the following properties of $C$:
\begin{equation}
\label{eq:c5}
C^\dagger = C^T = C^{-1} = -C\qquad C\;\Gamma_a \;C^{-1} = \Gamma_a^T\, ,\qquad
\Gamma_a = (\gamma_\mu, \gamma_5)\, ,
\end{equation}
while in section~(\ref{sec:C5D4D}) we shall state the relation between the 5D charge
conjugation
operator and the charge conjugation operator in four dimensions. A valid matrix
representation of the 5D charge conjugation operator in Dirac or Wyel basis is
$C=\gamma_1\gamma_3$. It is clear that any boundary condition which induces a mixing of a
fermion field and its charge conjugate will only be allowed for electrically-neutral fields,
and this is the case of neutrinos. Any additional $U(1)$ symmetry, under which the fermion
 is charged, will be broken by the boundary conditions in eq.~(\ref{eq:tbc}).

\noindent In order to make connection to the charge conjugate field, notice that the action
in eq.~({\ref{eq:action_5d}}) can equivalently be written in terms of the field $\psi$
and its charge conjugate in a compact form, also extensively used in ref.~\cite{flat}
\begin{equation}
\label{eq:action_bi}
S= \int\, d^4x \int\, dy\, \sqrt{-g}\, \left ( i\bar{\Psi} g^{MN}\gamma_M\,D_N \Psi -
\bar{\Psi}{\cal{M}}\Psi \right )\, ,
\end{equation}
where we have defined the following vector of fields and mass matrix
\begin{equation}
\Psi\, =\, \pmatrix{\psi^c \cr \psi }\qquad {\cal{M}}\,=\,\pmatrix{-m &0\cr 0 & m } .
\end{equation}
To derive eq.~(\ref{eq:action_bi}) one observes that the action of the 5D charge
conjugation operator implies $(\psi^c)^c = -\psi$, hence $\bar{\psi}\psi = -\bar{\psi^c} \psi^c$ and
$\bar{\psi}\gamma^M\,D_M\psi = \bar{\psi^c}\gamma^M\,D_M\psi^c$. These identities
 lead straightforwardly to the form of eq.~(\ref{eq:action_bi}).
The pseudo-Majorana boundary condition on the vector of fields $\Psi$ reads as follows
\begin{equation}
\label{eq:tbc_compact}
\Psi (x, -y) = \sigma_1\, e^{-i\sigma_3\delta}\,\gamma_5\,\Psi (x,y)\,=
-\sigma_3\, e^{i\sigma_3\delta}\,\gamma_5\,\Psi^c (x,y) .
\end{equation}
The vector of fields $\Psi (x,y)$ is also subject to periodic boundary conditions, so
that $\Psi (x, \pi R) = \Psi (x, -\pi R)$. This is a very natural condition to impose.
However, this type of constraint can also be generalized, see for example~\cite{flat}.

\noindent We perform a
Kaluza-Klein decomposition of the 5D fields in order to be able to derive
an action for fermions in four-dimensions. We have the choice of working in terms of
 the fields $\psi$ and $\psi^c$, or in terms of L- and R-handed fields $\psi_{L,R}$,
 or in terms of the
vector of fields $\Psi$.
We choose to decompose the 5D fields in terms of L- and R-handed components to keep our
derivation easily comparable with the Dirac case of ordinary boundary conditions
extensively analyzed in the literature~\cite{GN,GP}. We write
\begin{equation}
\label{eq:KK}
\psi_{L,R} (x, y) = \frac{1}{\sqrt{2\pi R}} \sum_{n=0}^\infty\, \psi_{n,L,R}(x)\,
e^{2\sigma}\,\hat{f}_{n,L,R}(y)\, ,
\end{equation}
where in the most general case the $y$-dependent functions $\hat{f}_{n,L,R}(y)$ are
complex valued, and subject to the specific boundary conditions. The same decomposition
for the charge conjugate field
$\psi^c(x,y)= C{\bar{\psi}}^T(x,y) = C\gamma_0\psi^\ast (x,y)$ reads
\begin{equation}
\label{eq:KKC}
\left (\psi_{L,R}\right )^c (x, y) = \frac{1}{\sqrt{2\pi R}} \sum_{n=0}^\infty\,
\left (\psi_{n,L,R}\right )^c (x)\,
e^{2\sigma}\,\hat{f}_{n,L,R}^\ast (y)
\end{equation}
and the identity $(\psi^c)_{L,R} = (\psi_{R,L})^c$ holds under the action of the 5D charge
 conjugation operator.

\subsection{ Imposing Pseudo-Majorana boundary conditions}

\noindent
The boundary conditions in eq.~(\ref{eq:tbc}) constrain the functions $\hat{f}_{n,L,R} (y)$
 through the Kaluza-Klein decomposition of eqs.~(\ref{eq:KK}) and (\ref{eq:KKC}),
so that one
only independent function remains, associated to one 4D Weyl spinor analogously to what
happens when imposing the Majorana condition in four dimensions.
Writing eq.~(\ref{eq:tbc}) in terms of L- and R-handed fields
we obtain
\begin{eqnarray}
e^{i\delta}\psi_L (x,-y) &=& -(\psi^c)_{L}(x,y) =  -(\psi_{R})^c(x,y) \nonumber\\
e^{i\delta}\psi_R (x,-y) &=& +(\psi^c)_{R}(x,y) =  +(\psi_{L})^c(x,y)\, .
\end{eqnarray}
When deriving the boundary conditions on the KK fields we have the choice to absorb the
phase factor $e^{i\delta}$ into a) the constraints on the 4D fields $\psi_n(x)$ or into b)
 the constraints on the $y$-dependent functions $\hat{f}_n(y)$. Clearly, both choices are
 equivalent. In the first case one has
\begin{equation}
\psi_n(x)=e^{-i\delta}\gamma_5 \psi_n^c(x) \nonumber
\end{equation}
and
\begin{eqnarray}
\label{eq:4D_BC}
e^{i\delta}\psi_{n,L}(x) &=& - (\psi_n^c)_L(x)= -(\psi_{n,R})^c(x)\nonumber\\
e^{i\delta}\psi_{n,R}(x) &=&  (\psi_n^c)_R(x)= (\psi_{n,L})^c(x)\, ,
\end{eqnarray}
while the functions $\hat{f}_{n,L,R} (y)$ must satisfy the relation
\begin{equation}
\label{eq:BC_fLR}
\hat{f}_{n,L,R} (-y) = \hat{f}^\ast_{n,R,L} (y)\, ,
\end{equation}
therefore leaving one independent function, as expected.
In the second case, one has $\psi_n(x)=\gamma_5 \psi_n^c(x)$ and $\hat{f}_{n,L,R} (-y) =
e^{-i\delta}\hat{f}^\ast_{n,R,L} (y)$.
When instead ordinary boundary conditions are imposed, the Dirac nature of
the 5D fermion fields is preserved and the two $y$-dependent functions $\hat{f}_{n,L,R}$ are
subject to the constraints $\hat{f}_{n,L,R} (-y) = \mp \hat{f}_{n,L,R} (y)$, or viceversa.

\noindent In both cases, and after Kaluza-Klein decomposition, the periodic boundary
conditions $\psi_{L,R}(x,\pi R) = \psi_{L,R}(x,-\pi R)$ lead to $\hat{f}_{n,L,R}(\pi R) =
\hat{f}_{n,L,R}(-\pi R)$, implying that odd functions of the fifth coordinate
 will vanish at the boundaries $\pm \pi R$\footnote{Notice that in the most general case
 a phase factor is allowed, so that $\psi_{L,R}(x,\pi R) = e^{i\alpha}\psi_{L,R}(x,-\pi R)$.
 Antiperiodic boundary conditions have been considered in ref.~\cite{GP_2} in a
supersymmetric scenario.}.

\subsection{ Boundary conditions in terms of Weyl spinors.}

\noindent
We can decompose the 5D Dirac spinor $\psi = \psi_L+\psi_R$ in terms of a L-handed and a
 R-handed Weyl spinor as follows
\begin{equation}
\psi_L = \pmatrix{\phi \cr 0}\qquad  \psi_R = \pmatrix{0 \cr \epsilon \xi^\ast
\equiv \chi}\, ,
\end{equation}
where $\epsilon\equiv i\sigma_2$. Charge conjugation in 5D is realized by
\begin{equation}
C = \gamma_1\gamma_3 = i\sigma_2 \otimes {\mathbbm{1}} =
\pmatrix{\epsilon &0 \cr 0 &\epsilon} .
\end{equation}
Working in the Weyl basis with $\gamma^0 = \pmatrix{0 & \mathbbm{1} \cr \mathbbm{1} &0}$ and
$\gamma_5 = \pmatrix{-\mathbbm{1} & 0 \cr 0 & \mathbbm{1}}$ we obtain
\begin{equation}
\psi^c (x,y) = C\gamma^0\psi^\ast = \pmatrix{\epsilon\chi^\ast \cr \epsilon\phi^\ast}
\end{equation}
and the pseudo-Majorana boundary conditions thus imply (in the simplified case $\delta =0$)
\begin{eqnarray}
\phi (x,-y) &=& -\epsilon \chi^\ast (x,y) \nonumber\\
\chi (x,-y) &=& \epsilon \phi^\ast (x,y)\, .
\end{eqnarray}
The boundary condition $\psi (x) =\gamma_5 \psi^c (x)$ on the 4D fields implies
\begin{eqnarray}
\phi (x) &=& -\epsilon \chi^\ast (x) \nonumber\\
\chi (x) &=& \epsilon \phi^\ast (x)\, .
\end{eqnarray}
The 4D spinor can hence be written in terms of one Weyl spinor
\begin{equation}
\psi (x) = \pmatrix{ \phi (x) \cr \epsilon \phi^\ast (x)} =  \pmatrix{ -\epsilon
\chi^\ast (x)\cr \chi (x)}
\end{equation}
under the 5D pseudo-Majorana conditions.

\subsection{The action in four dimensions}

\noindent We determine the spectrum of the KK modes, their masses and wave functions,
by solving an eigenvalue problem which can be obtained once an ansatz for
the 4D fermion action is imposed to the 5D action in eq.~(\ref{eq:action_5d}).
The problem is: find the eigenfunctions $\hat{f}_{n,L,R}$ and the associated
eigenvalues of a $y$-dependent differential operator, subject to the given boundary
conditions.
 After rewriting the 5D action in terms of L- and R-handed fields
\begin{eqnarray}
\label{eq:action_LR}
S&=& \int\, d^4x \int\, dy\, \sqrt{-g}\, \bar{\psi} (x,y) \left ( i e^\sigma \partial\hspace{-0.2truecm}/
+2\gamma_5 \sigma^\prime -\gamma_5 \partial_5 - m\right ) \psi (x,y)\nonumber\\
&=&\int\, d^4x \int\, dy\, \sqrt{-g}\,\left \{
\bar{\psi}_L (x,y)\, i e^\sigma \partial\hspace{-0.2truecm}/ \psi_L (x,y) +
\bar{\psi}_R (x,y)\, i e^\sigma \partial\hspace{-0.2truecm}/ \psi_R (x,y) + \right .\nonumber\\
&+&\left . \bar{\psi}_R (x,y) (-2\sigma^\prime +\partial_5 - m)\psi_L (x,y) +
\bar{\psi}_L (x,y) (  2\sigma^\prime -\partial_5 - m   )\psi_R (x,y)
\right\} ,
\end{eqnarray}
and performing the Kaluza-Klein decomposition according to eq.~(\ref{eq:KK})
\begin{eqnarray}
\label{eq:action_KK}
S&=& \int\, d^4x \int\, dy\, \frac{1}{2\pi R}\sum_{n,m}\left \{
 \bar{\psi}_{n,L} (x) i \partial\hspace{-0.2truecm}/  \psi_{m,L} (x)\hat{f}_{n,L}^\ast e^\sigma\hat{f}_{m,L}
+ \bar{\psi}_{n,R} (x) i \partial\hspace{-0.2truecm}/  \psi_{m,R} (x)\hat{f}_{n,R}^\ast e^\sigma\hat{f}_{m,R}
+ \right .\nonumber\\
&+&\left .
\bar{\psi}_{n,R} (x)\psi_{m,L} (x)\hat{f}_{n,R}^\ast (\partial_5 - m)\hat{f}_{m,L} +
\bar{\psi}_{n,L} (x)\psi_{m,R} (x)\hat{f}_{n,L}^\ast  ( -\partial_5 - m   ) \hat{f}_{m,R}
\right\}\,
\end{eqnarray}
we require that the effective 4D action is given by
\begin{equation}
\label{eq:action_4D}
S_{4D} = \int\,d^4x\,\sum_n\, \left [\bar{\psi}_n \, i\partial\hspace{-0.2truecm}/ \psi_n -
{\cal L}(m_n)\right ]
\, ,
\end{equation}
with a Majorana mass term ${\cal L}(m_n)$. This is guaranteed by the presence of the
pseudo-Majorana boundary conditions.
The form of the 4D kinetic term in eq.~(\ref{eq:action_4D}) and the use of the
 pseudo-Majorana boundary conditions imply the following
normalization and orthonormality condition on the $y$-dependent functions
\begin{equation}
\frac{1}{2\pi R} \int_{-\pi R}^{\pi R}\, dy\, e^\sigma \hat{f}_{n,L,R}^\ast (y)
 \hat{f}_{m,L,R}(y) = \frac{1}{2}\delta_{n,m}\, .
\end{equation}
The most general 4D Majorana mass term involving one Majorana spinor can be
conveniently expressed as a function of
L- or R-handed fields and in terms of the 4D charge conjugate fields. This reads
\begin{equation}
{\cal L}(m_n) =  m_n ( \bar{\psi^c_L} \psi_L + \bar{\psi}_L \psi^c_L )_n \qquad{\mbox{or}}\qquad
 m_n ( \bar{\psi^c_R} \psi_R + \bar{\psi}_R \psi^c_R )_n\, ,
\end{equation}
where we defined $\psi^c_{L,R} \equiv (\psi_{L,R})^c$ and the charge conjugate
fields are now
$\psi^c = C_4(\bar{\psi})^T$, where $C_4$ is the charge conjugation matrix in four
dimensions.

\subsection{Charge conjugation from five to four dimensions}
\label{sec:C5D4D}

\noindent Once the Clifford algebra is built for a given number of spacetime
 dimensions $d=t+s$, with t timelike and s spacelike dimensions, there always exists a
 charge conjugation matrix $C$, such that
\cite{Tools}
\begin{equation}
C^T = -\epsilon\, C\qquad \Gamma_a^T = -\tilde{\eta}\, C\Gamma_a C^{-1}
\end{equation}
for $\epsilon =\pm 1$ and $\tilde{\eta} = \pm 1$, and $\{\Gamma_a, \Gamma_b\} =
2\eta_{ab}$ defines
the Clifford algebra in the given spacetime. A formal proof and further details
can be found in \cite{Tools,Scherk4,Kugo5}.
The unitarity of the charge conjugation matrix $C^\dagger = C^{-1}$ is true in any
representation, while the charge conjugation matrix not necessarily squares to $\mathbbm{1}$.

\noindent
What it is relevant to observe is that for odd dimensions only one of the two choices,
$\tilde{\eta} = +1$ or $\tilde{\eta} = -1$, can be used \cite{Tools}. While in $d=4$ the
two solutions
$\epsilon = +1, \tilde{\eta} = +1$ and  $\epsilon = +1, \tilde{\eta} = -1$ are allowed,
in $d=5$ only the
solution $\epsilon = +1, \tilde{\eta} = -1$ can be realized.

\noindent
An often used realization of the charge conjugation in $d=4$ is $C=i\gamma^2\gamma^0$, which
satisfies
\begin{eqnarray}
&&C^\dagger = C^T = C^{-1} = -C\qquad C^2 = -\mathbbm{1} \nonumber\\
&&C\gamma^\mu C^{-1} = -{(\gamma^\mu )}^T\, .
\end{eqnarray}
This choice gives the relation $\gamma_5 C_5 = -C_4$.
In the most general case we can choose a charge conjugation matrix in four dimensions with
a phase factor, such that $C_4 = \eta\gamma^2\gamma^0$, and $|\eta |^2 = 1$.
The matrix $C_4$ satisfies the following relations
\begin{eqnarray}
&&C^\dagger  = C^{-1} = (\eta^\ast )^2 C\qquad C^2 = \eta^2 \mathbbm{1} \nonumber\\
&& C^T = -C  \nonumber\\
&&C\gamma^\mu C^{-1} = -{(\gamma^\mu )}^T\, ,
\end{eqnarray}
and it is immediate to verify that $\gamma_5 C_5 = -i\eta^\ast C_4$.
In the case in which the phase factor $e^{-i\delta}$ of eq.~(\ref{eq:tbc}) is absorbed
 into the boundary conditions for the 4D fields, they will satisfy the following relation
\begin{equation}
\label{eq:C5C4}
\psi (x)= e^{-i\delta}\gamma_5 C_5\bar{\psi}^T(x) = -i\eta^\ast e^{-i\delta}C_4
\bar{\psi}^T(x)=-i\eta^\ast e^{-i\delta}
 \psi^c (x)\, ,
\end{equation}
implying the usual Majorana condition in four dimensions, generalized with a phase
 factor coming from the freedom in the choice of 5D boundary conditions and the
freedom in the construction of the charge conjugation matrix in four dimensions.
Eq.~(\ref{eq:C5C4}) leads to the identitites
\begin{eqnarray}
\bar{\psi}_R\psi_L &=& -i\eta^\ast e^{-i\delta}\bar{\psi}_R\psi^c_R  =
i\eta e^{i\delta}\bar{\psi^c_L}\psi_L \nonumber\\
\bar{\psi}_L\psi_R &=& -i\eta^\ast e^{-i\delta} \bar{\psi}_L\psi^c_L  = i\eta e^{i\delta}
\bar{\psi^c_R}\psi_R \, ,
\end{eqnarray}
thus implying that the 5D action in eq.~(\ref{eq:action_KK}) generates a 4D Majorana mass
term if subject to pseudo-Majorana boundary conditions.
Equivalently, the phase factor $e^{-i\delta}$ can instead be absorbed into the boundary
conditions for $\hat{f}_{L,R}(y)$.

\subsection{ Solutions to the eigenvalue problem}

\noindent In order to obtain the 4D action in eq.~(\ref{eq:action_4D}) the $y$-dependent
functions $\hat{f}_{n,L,R}$ must be solutions of the equations
\begin{eqnarray}
\label{eq:EIGEN}
(-\partial_5 +m) \hat{f}_{n,L}(y) &=& -i\eta^\ast e^{-i\delta}    m_n e^\sigma
\hat{f}_{n,R}(y)\nonumber\\
(\partial_5 +m) \hat{f}^\ast_{n,R}(y) &=& -i\eta^\ast e^{-i\delta} m_n e^\sigma
\hat{f}^\ast_{n,L}(y)\, ,
\end{eqnarray}
on the interval $[-\pi R, \pi R]$, with real masses $m, m_n$ and  $\hat{f}_{n,L,R}$
complex functions, differently from the ordinary Dirac case. Notice that the $Z_2$
 invariance of the 5D action implies that we can restrict all $y$-dependent
integrations from $0$ to $\pi R$ and the behaviour of the solutions for negative
$y$ should be fully determined by their $Z_2$ transformation properties.
The solutions must also satisfy the pseudo-Majorana boundary conditions
 $\hat{f}_{n,L,R} (-y) = \hat{f}^\ast_{n,R,L} (y)$ and periodic boundary conditions
$\hat{f}_{n,L,R} (-\pi R) = \hat{f}^\ast_{n,L,R} (\pi R)$.
As anticipated, we can equivalently choose to remove the phase factor
$e^{-i\delta}$ from equations (\ref{eq:EIGEN}), while imposing the boundary conditions
$\hat{f}_{n,L,R} (-y) = e^{-i\delta} \hat{f}^\ast_{n,R,L} (y)$.
We choose to illustrate this second case, in which we also set $ -i\eta^\ast = 1$,
so that $\psi_n (x) = \psi_n^c(x)$ is satisfied. With these substitutions,
equations (\ref{eq:EIGEN}) should then be
solved separately for the real and imaginary parts, giving
\begin{eqnarray}
\label{eq:EIGEN_Real}
(-\partial_5 +m) Re \hat{f}_{n,L}(y) &=&  m_n e^\sigma Re \hat{f}_{n,R}(y)\nonumber\\
(\partial_5 +m) Re \hat{f}_{n,R}(y) &=& m_n e^\sigma Re \hat{f}_{n,L}(y)\, ,
\end{eqnarray}
and the equations for the imaginary parts are obtained by replacing
$ Re f_{L,R} \to Im f_{L,R}$.
Notice that eqs.~(\ref{eq:EIGEN_Real}) together with periodic boundary conditions
tell us that we should seek for solutions with definite and opposite $Z_2$ parity,
i.e. $Re \hat{f}_{n,L}(y)$ even and $Re \hat{f}_{n,R}(y)$ odd or viceversa.
This constrains the phase $\delta$ of the boundary conditions and the shape of the bulk
mass parameter $m$.

\noindent For real $e^{-i\delta}$ the pseudo-Majorana conditions imply that $Re f_L$ and
$Re f_R$ always carry the same parity (the same being true for the imaginary parts).
This condition only allows for the zero mode solution of eqs.~(\ref{eq:EIGEN_Real}) to
survive, thus $m_n =0$, the mass term $m=0$ and $\hat{f}_{0,L}(y)=\hat{f}_{0,R}(y)\sim
const$.
For imaginary $e^{-i\delta}$
the entire Kaluza-Klain tower is allowed; the pseudo-Majorana boundary condition implies
in this case  that $Re \hat{f}_L$ and $Im \hat{f}_R$ have the same parity, and opposite to
$Re \hat{f}_R$ and
$Im \hat{f}_L$, a condition always compatible with eqs.~(\ref{eq:EIGEN_Real}).

\noindent
We expect that the complete solutions $\hat{f}_{n,L,R}(y)$ will be plane waves - the
solutions of the flat case~\cite{flat} - deformed by the warped geometry. Some more
subtleties arise from the comparison with the ordinary Dirac case as it is shown below.
\vspace{1truecm}

\noindent {\bf The solutions}

\noindent
Solutions to eqs.~(\ref{eq:EIGEN_Real}) can conveniently be found  by deriving a
second order partial differential equation which decouples the L- and R-handed modes,
in complete analogy with what has been done in refs.~\cite{GP,GN}.
The coupled equations (\ref{eq:EIGEN}) are equivalent to the following second order
partial differential equation on the interval $[-\pi R, \pi R]$
\begin{equation}
\label{eq:EIGEN_II}
\left ( -e^\sigma\partial_5 e^{-\sigma}\partial_5 + M^2\right ) Re \hat{f}_{n,L,R}(y) =
m_n^2 e^{2\sigma} Re \hat{f}_{n,L,R}(y)\, ,
\end{equation}
with the mass-like term $M^2$ given by
\begin{equation}
\label{eq:M2}
M^2= \mp m\sigma^\prime \pm m^\prime +m^2\, ,
\end{equation}
where $\sigma^\prime = k \epsilon (y)$ and the upper (lower) sign for the $L$ ($R$) modes.
The form of $M^2$ depends on the shape of the bulk mass parameter $m$.
In the ordinary Dirac case~\cite{GP,GN} the bulk mass must be $Z_2$ odd, thus
usually defined as $m=c\sigma^\prime$ proportional to the derivative of the warp factor.
In the pseudo-Majorana case the bulk mass is $Z_2$ even, and we can in general allow for
a bulk and a boundary contribution. However, no further calculation is needed while
noticing that for $m$ even the mass-like term $M^2$ will carry mixed parity. Hence, for
any choice of $m$ even, a definite $Z_2$-parity solution to ~(\ref{eq:EIGEN_Real})
only exists for $m =0$. This seems to be a genuine consequence of the warped $\mbox{AdS}_5$
geometry with the extra dimension compactified on a $S^1/Z_2$ orbifold. We expect that
some of these constraints might be released for more general choices of the geometry
of extra dimensions,
for example in the flat case \cite{flat}.

\noindent It is convenient to rewrite eq.~(\ref{eq:EIGEN_II}) in terms of the new variable
$x= e^\sigma m_n/k$, and maintain for a few steps more the explicit dependence upon
the mass-like parameter $M^2$. We obtain
\begin{equation}
\label{eq:EIGEN_x}
x^2\frac{\partial^2 f_n}{\partial x^2} +\frac{\sigma^{\prime\prime}}{k^2} x
\frac{\partial f_n}{\partial x} -\left ( \frac{M^2}{k^2} -|x|^2\right ) f_n(x) =0
\, .
\end{equation}
The boundary terms, proportional to $\sigma^{\prime\prime} =  2 k (\delta (y) - \delta
(y-\pi R))$, should be treated separately and depend on the form of $M^2$.
Thus we seek for the general solution of
\begin{equation}
\label{eq:AlmostBessel}
 x^2\frac{\partial^2 f_n}{\partial x^2}  -\left ( \frac{\hat{M}^2}{k^2} -|x|^2\right )
 f_n(x) =0
\, ,
\end{equation}
where the reduced mass-like term $\hat{M}^2$ is obtained by subtracting the boundary terms.
As noticed in ref.~\cite{GP} a further redefinition $f(x) = e^{\sigma/2} \tilde{f}(x)$
 reduces equation~(\ref{eq:AlmostBessel}) to a Bessel equation for $\tilde{f}_n(x)$
\begin{equation}
\label{eq:Bessel}
 x^2\frac{\partial^2 \tilde{f}_n}{\partial x^2} + x \frac{\partial \tilde{f}_n}{\partial x}
  -\left ( \frac{\hat{M}^2}{k^2} +\frac{1}{4} -
|x|^2\right ) \tilde{f}_n(x) =0\,
\end{equation}
the most general solution of which is a linear combination of Bessel functions
$J_\alpha (x)$ and $Y_\alpha (x)$ of order $\alpha = \sqrt{ {\hat{M}^2}/{k^2} +{1}/{4} }$,
 for $x$ real~\cite{GP,GN}. The complete solutions $f_n(y)$ are thus given by
\begin{equation}
\label{eq:solution}
f_n(y) =\frac{e^{\sigma /2}}{N_n}\left [ J_\alpha \left (\frac{m_n}{k}e^\sigma\right )
+ b_\alpha (m_n)  Y_\alpha \left (\frac{m_n}{k}e^\sigma\right )          \right ]\, ,
\end{equation}
with normalization factor
\begin{equation}
N_n^2 = \frac{2}{\pi R}\int_{-\pi R}^{\pi R}\, dy\, e^{2\sigma} \left ( J_\alpha \left
 (\frac{m_n}{k}e^\sigma\right ) + b_\alpha (m_n)  Y_\alpha
\left (\frac{m_n}{k}e^\sigma\right )      \right )^2\, ,
\end{equation}
where the  coefficients $b_\alpha (m_n)$  and the mass spectrum eigenvalues $m_n$ are
 determined by the specific boundary conditions.

\noindent  When $\hat{M}^2$ has mixed parity, we must solve eq.~(\ref{eq:Bessel})
separately on the two branches $y>0$ and $y<0$, thus obtaining Bessel functions of
different order on the two branches for the same mode $n$. As already concluded before,
 the existence of a solution implies that $M^2 =0$.
The case $m=0$, $M^2=0$ is immediately solved, the solutions being Bessel functions of
 order $\alpha =1/2$. They are trigonometric functions
$J_{1/2}(x)=\sqrt{2/\pi x}\, \sin{x}$ and $Y_{1/2}= -J_{-1/2}(x)= -\sqrt{2/\pi x}\,
 \cos{x}$.
The coefficients $b_{1/2}(m_n)$ are determined by the boundary conditions at $y=0, \pi R$
(the term proportional to $\sigma^{\prime\prime}$ in eq.~(\ref{eq:EIGEN_x}))
\begin{equation}
\label{eq:WALL}
{\frac{df_n}{dy}}\vline_{\vspace{0.1cm}y=0, \pi R} = 0
\end{equation}
for $Z_2$ even solutions and
\begin{equation}
f_n(y)\vert_{y=0, \pi R} = 0
\end{equation}
for $Z_2$ odd solutions. In this case $f_n(y) = \sigma^\prime /k f_n(|y|)$ and the condition
$f_n(0)=f_n(\pi R) =0$ directly implies (\ref{eq:WALL}). We obtain the simple result
\begin{eqnarray}
\label{eq:bcoeff}
b_{1/2}(m_n) &=& - \frac{ \frac{1}{2} J_{1/2}(\frac{m_n}{k}) + \frac{m_n}{k}
J_{1/2}^\prime(\frac{m_n}{k})  } { \frac{1}{2} Y_{1/2}(\frac{m_n}{k}) + \frac{m_n}{k}
 Y_{1/2}^\prime(\frac{m_n}{k})  }  = - \cot \left (\frac{m_n}{k}\right )\nonumber\\
b_{1/2}(m_n) &=& b_{1/2}(m_n e^{\pi kR})
\end{eqnarray}
for the even solutions and
\begin{eqnarray}
\label{eq:bcoeff_odd}
b_{1/2}(m_n) &=& - \frac{ J_{1/2}(\frac{m_n}{k}) }{  Y_{1/2}(\frac{m_n}{k}) }  =
\tan\left (\frac{m_n}{k}\right ) \nonumber\\
b_{1/2}(m_n) &=& b_{1/2}(m_n e^{\pi kR})
\end{eqnarray}
for the odd solutions respectively, with $n=1,2\ldots$.
Thus the even parity solutions are
\begin{equation}
f_n(y) = \sqrt{\frac{k\pi R}{4(e^{k\pi R}-1)}}\, \cos{\left ( \frac{m_n}{k}
\left (e^\sigma -1\right ) \right )}
\end{equation}
and the odd parity solutions are
\begin{equation}
f_n(y) = \frac{\sigma^\prime}{k}\sqrt{\frac{k\pi R}{4(e^{k\pi R}-1)}}\,
\sin{\left ( \frac{m_n}{k}\left (e^\sigma -1\right ) \right )}\, .
\end{equation}
The same conditions, (\ref{eq:bcoeff}) or equivalently (\ref{eq:bcoeff_odd}), give the
exact spectrum of eigenvalues $m_n$:
\begin{equation}
m_n =n\pi k \frac{e^{-k\pi R} }{ 1- e^{-k\pi R}}\, .
\end{equation}
This can be compared with the Dirac case with ordinary boundary conditions, where one gets
an estimate in the limit $kR\gg 1$ and $m_n\ll k$ given by~\cite{GP}
 $m_n \approx (n+\alpha/2 -3/4) \pi k e^{- k\pi R}$ for even solutions and
$m_n \approx (n+\alpha/2 -1/4) \pi k e^{- k\pi R}$ for odd solutions,
and $\alpha =1/2$ in our case.

\noindent Summarizing, the pseudo-Majorana boundary conditions imposed on fermions living on
a slice of $\mbox{AdS}_5$ generate a Kaluza-Klein tower of Majorana spinors in the four
dimensional action. No Dirac type bulk mass term seems to be allowed in this case by the
warped geometry with orbifolding. The phase
factor $e^{i\delta}$ in the pseudo-Majorana boundary conditions must be chosen along with
the choice of the 4D charge conjugation matrix. We have derived the
explicit solutions for pseudo-Majorana boundary conditions given by (imaginary phase)
$\psi (x, -y) = \pm i\gamma_5 \psi^c (x, y)$, with $C_4= i\gamma^0\gamma^2$. This induces
 the Majorana condition $\psi = \psi^c$ in four dimensions.
A massless zero mode exists and all Kaluza-Klein modes, including the zero mode, are
 localized on the visible ($y=\pi R$) brane. In the same context, we have noticed that
the opposite
choice (real phase) $\psi (x, -y) = \pm \gamma_5 \psi^c (x, y)$ removes all excited states
 and only allows for a zero mode. For the moment we leave open the question of what
mechanism might be in place that favours one choice and not the other.

The $y$-dependent wave functions of the Majorana Kaluza-Klein tower are deformed
plane waves, as expected, and given by
\begin{eqnarray}
\hat{f}_n(y) &=& \sqrt{\frac{k\pi R}{4(e^{k\pi R}-1)}}\, \left ( \cos{
\left (\frac{m_n}{k}\left (e^\sigma -1\right )\right )  } +i \frac{\sigma^\prime}{k}\,
\sin{\left (\frac{m_n}{k}\left (e^\sigma -1\right )\right )  }  \right )
\qquad n=1,2\ldots \nonumber\\
\hat{f}_0(y) &=& \sqrt{\frac{k\pi R}{4(e^{k\pi R}-1)}}\, .
\end{eqnarray}
These results show that the case of bulk fermion fields with pseudo-Majorana boundary
conditions is far more constrained than the ordinary Dirac case.
A bulk Majorana mass term can still be added to our case, in analogy to what has been done
in the ordinary case in ref.~\cite{HuberShafi}. Again, the mass parameter will acquire
opposite $Z_2$ parity with respect to the ordinary case if pseudo-Majorana boundary
conditions are imposed.
Notice finally that in the ordinary Dirac case the localization of the zero modes is
monitored by the bulk Dirac mass, which accelerates or reduces the exponential decay of the
zero mode $y$-dependent wave function. By tuning the bulk mass parameter one can achieve
localization of the zero modes on the Planck brane, delocalization, or localization on the
TeV brane. The pseudo-Majorana case produces a zero mode Majorana spinor which is localized
on the TeV brane.

\section{Neutrino phenomenology, Dirac or Majorana}

The main purpose of this paper is the one of solving the exercise of
generating Majorana spinors in four dimensions, starting from a five dimensional scenario
with a warped geometry and bulk Dirac fermions. We did this by imposing non-standard
pseudo-Majorana boundary conditions in 5D, instead of the ordinary boundary conditions
which preserve the Dirac nature of bulk fermions.
This exercise motivates the more phenomenological question of whether the found solution
 can at all give rise to a plausible mechanism to generate small masses for neutrinos.
This is part of the many attempts in the literature at finding a convincing
explanation of why the observed masses of fermions, quarks and leptons and in particular
neutrinos, are the way they are, respecting a hierarchy among families and with neutrinos
much lighter then all their Standard Model partners.

The pseudo-Majorana boundary conditions do not leave much space for tuning. With this
mechanism we can generate a zero mode Majorana spinor which is localized on our brane,
 with or without a Kaluza-Klein tower of states.
We can either identify the bulk fermion field with a ``sterile'' Majorana neutrino,
 i.e. a neutrino which is not charged under the SM gauge group, or with the Standard
 Model neutrino.
In the first case a well known successful mechanism in four dimensions is the see-saw
mechanism, where a sterile R-handed heavy neutrino is added to the particle content of
the Standard Model.
 We might identify the first Kaluza-Klein mode (or the tower of KK modes) of the Majorana
spinor in the bulk with the ``heavy'' sterile neutrino and induce a see-saw mechanism,
while all SM particles are localized on the TeV brane.
The mass of the Kaluza-Klein
 modes is naturally of the TeV order, and might induce a too small suppression of the SM
neutrino
masses. However, a complete description of this mechanism can only be achieved with the full
understanding of how Kaluza-Klein higher modes are entangled to the four dimensional
effective theory. This might imply deviations from the assumptions implicit in the
Kaluza-Klein decomposition.

\noindent A complete mass term in 4D might be of the type
\begin{equation}
S_M = \int~d^4 x \sum_{n\geq 1} m_n \bar{\psi}^{c (n)}_R\psi^{(n)}_R + \sum_{n\geq 0} Y_5
f^{(n)}_R(y=\pi R)\bar{L}(x)<\tilde{H}> \psi^{(n)}_R(x)\, .
\end{equation}
The usual see-saw mechanism in four dimensions predicts $m_{light} \sim v^2/M$ and
 $m_{heavy} \sim M$, with $v$ the weak-scale vev of the neutral component of the
Higgs doublet and $M$ a GUT scale. In the 5D case here discussed, and considering
the tower of Kaluza-Klein excitations we can generate Dirac masses in the four
dimensional theory through Yukawa couplings
proportional to $Y_5^{(n)} f^{(n)}_R(y=\pi R) v$ and
Majorana masses $m_n$ which are naturally of order the TeV. One needs to know the
behaviour of $Y_5^{(n)} f^{(n)}_R(y=\pi R)$ for the excitations in the KK tower in
 order to predict a
hierarchy of scales. In the simplest scenario in which all couplings and masses are
 of order the weak scale, no hierarchy will be generated. However, a suppression might
 be induced by the suppression of the couplings of KK excitations to the visible brane.

\noindent
As a side note we add that in the case in which Majorana masses and Dirac masses are of the
same order of magnitude and sufficiently small, the see-saw mass matrix would give rise to
large mixing angles, inducing neutrino-antineutrino oscillations of the type
$\nu_L \to (\nu_R)^c$. This would mean
oscillations in which the SM neutrinos turn into ``sterile'' particles. This alternative
idea has been explored long ago in \cite{Barger1980,ChengLi1980}.

The depicted scenario can be compared with the Dirac case discussed in ref.~\cite{GN},
where the R-handed bulk fermion zero mode is identified with a sterile neutrino.
In this case the bulk mass term can be used as a free parameter
to tune the amount of localization of the zero mode.
Above a critical value of the bulk mass (c=1/2) the $Z_2$ even (R-handed) zero mode is
localized on the Planck brane, so that its wave function at the TeV brane is
strongly suppressed. By introducing a ``new Yukawa'' interaction
$Y_5 \bar{L}(x)\tilde{H}(x)\psi^{(0)}(x, y=\pi R)$, the four dimensional Yukawa coupling
will be given by $Y_5 f_{0,R}(y=\pi R)$, with $Y_5=O(1)$ and the SM neutrino mass
induced via the Higgs mechanism $m_\nu \sim Y_5 vf_{0,R}(y=\pi R)$ receives an
exponential suppression given by $ f_{0,R}(y=\pi R)$ the value of the $y$-dependent
 function at the TeV brane.
This beautiful mechanism however requires a fine-tuning of the bulk mass term.
Nevertheless it has been
shown~\cite{GN} to be very effective in producing plausible ranges of neutrino masses and
oscillation parameters.

\noindent The second possibility is to identify bulk fermion fields with SM particles.
 This implies that gauge fields should also be
allowed in the bulk. We do not consider here this
possibility, rather we observe that a scenario with all SM fields in the bulk offers many
more appealing features if supersymmetry is imposed. In the simplest case, and disregarding
gauge interactions, the zero mode of the bulk Majorana field can be identified with the
SM L-handed neutrino. Given its Majorana nature, no Yukawa coupling to the Higgs doublet and
therefore Dirac mass term can be generated, while a mass term can be induced by the
coupling with a Higgs triplet. The smallness of neutrino masses would in this case be
explained by the smallness of the Higgs triplet vev, a well known mechanism in the four
dimensional theory. The fitted value of the $\rho$ parameter of the Standard Model
 restricts the ratio of the Higgs triplet to the doublet vevs to be approximately
$v_T/v_H < 0.17$, thus naturally implying a suppression of neutrino masses with respect to
charged SM fermions. In this scenario, the question of the smallness of neutrino masses is
moved into the question of the smallness of a Higgs triplet vev and a hierarchy of
fermion masses would be due to a hierarchy of scalar boson masses.

\noindent Alternatively, a mechanism in which higher order radiative corrections give rise
to a small mass for the Majorana zero mode remains a viable solution.

\section{Other mechanisms to generate small Majorana masses in a warped geometry}

\noindent
It is now part of textbooks the list of recipes to follow in order to generate a small
neutrino mass in our four-dimensional world. Three possibilities are foreseen
i) enlarging the SM lepton sector (e.g. with the addition of an $SU(2)$ singlet
 R-handed neutrino),
ii) enlarging the SM Higgs sector (e.g. with the addition of a Higgs triplet and
extra doublets) and iii) enlarging both.
Fermions in the bulk of (compactified) extra dimensions offer an additional mechanism of
enhancement or suppression of interactions and generation of mass terms.

\noindent
However, no clear mechanism has been isolated that can explain why neutrino masses are much
smaller than charged lepton masses, and most of the proposed mechanisms do rely on a fine
tuning procedure. Experimentally, no evidence in favour or against the Majorana
nature of neutrinos has been reported, apart from a long debated claim of the Gran Sasso
neutrinoless double $\beta$-decay experiment. For an up to date analysis of neutrino
data see e.g.~\cite{Fogli}.
Results collected along the years seem to indicate that some relevant
ingredient is missing, perhaps a fundamental symmetry principle, a group theory
consideration~\cite{Volkov, Davidson} that might explain the recurrence of the three-family
structure in the fermion spectrum of the standard model without the need for a fine tuning.
Might a random process have a role in explaining the observed hierachies? It should
possibly occur jointly with an enhancement mechanism acting on an initial tiny symmetry
 breaking effect.An intriguing use of randomness has been made in ref.~\cite{Hall} in
 order to describe the apparent hierarchy of athmospheric and solar neutrino oscillation
 parameters.

\noindent
In the context of warped extra dimensions, the main attempts to generate a small Majorana
mass for neutrinos on the visible brane, can be summarized as follows:
i) Higher dimensional operators such as the operator $\psi^TC H^2\psi$~\cite{HuberShafi_2}
can induce a Majorana mass term on the visible brane. The suppression of the neutrino
mass w.r.t.
the charged lepton mass is obtained when the 5D wave function of the zero
mode is localized on the Planck brane, similarly to what originally proposed in
ref.~\cite{GN}. ii) The addition of a bulk Majorana mass on the Planck brane~
\cite{HuberShafi} in the case of bulk fermion fields with ordinary boundary conditions
can induce an uplifting of the mass of the bulk fermion zero mode, which can thus
be identified with the R-handed heavy neutrino of a see-saw mechanism. iii) Finally
the role of additional scalar fields in the bulk has been explored in ref.~\cite{Chen}.
Assuming a L-handed neutrino on the visible brane, we can generate a Majorana mass term
via the interaction with an $SU(2)$ Higgs triplet: $\lambda_{ij} L_i^T C^{-1}(i\tau_2 T)
 L_j$, with obvious notation.
The vev of the neutral component, call it $\xi_0$, of the Higgs triplet $T$ will thus induce
a Majorana mass term $\sim \lambda_{ii} \xi_0$. Therefore, a suppression of the triplet
vev w.r.t. the doublet vev would account for a suppression of the Majorana neutrino mass
w.r.t. the charged fermion masses.
It was concluded in ref.~\cite{Chen} that the introduction of bulk scalar fields would
suffice to guarantee a suppressed triplet vev on the visible brane, thus offering a viable
mechanism for the generation of a suppressed Majorana mass for SM neutrinos.
However, after reconsidering the proposed model, we find~\cite{Tesi} that no suppression
mechanism can actually
be obtained\footnote{A rescaling of a quartic interaction in going from the 5D to
the 4D action of the singlet scalar field seems to be missing in ref.~\cite{Chen}}.
We briefly summarize the most relevant steps of the derivation and provide some physics
considerations.
The model contains one bulk scalar singlet field $S(x,y)$, a Higgs triplet $T(x)$ and a
Higgs doublet $H(x)$ on the visible brane. Denoting with $\xi_0$ and $v$ the vevs of the
 neutral triplet and doublet respectively, the desired hierachy of vevs should be
$\xi_0 \ll v$, while the vev of the bulk scalar singlet, call it $S_0$,
should induce such hierarchy.

\noindent
The action for a bulk scalar field has also been derived in ref.~\cite{GP}, where the 5D
equations of motion for bulk scalar, vector and fermion fields are represented in a useful
 compact form. Through power counting, we can write the most general potential for the
 singlet field and its interactions with the triplet and doublet on the brane, up to and
including quartic couplings. The potential we find does agree with that of ref.~\cite{Chen}.
The bulk singlet field with a generic mass term $ak^2 +b\sigma^{\prime\prime}$ -- a bulk
 and a boundary contributions are allowed by $Z_2$ invariance --
has a zero mode if $b=2-\alpha$, with $\alpha = \sqrt{4+a}$. This relation between the
 bulk and boundary terms can be justified in the supersymmetric limit~\cite{GP}.
As a consequence, the localization of the singlet zero mode is driven by the
parameter $\alpha$: it will be localized on the Planck brane
for $\alpha > 1$, on the TeV brane for $\alpha <1$ and delocalized for $\alpha =1$.
The author of ref.~\cite{Chen} points out that the Planck localized case can provide
the required hierarchy.
However, we find a different rescaling of the coupling of the quartic self-interaction of
the singlet field $\lambda_S$ when reducing from five to four dimensions, with crucial
 consequences for the physics conclusion.
The coupling $\lambda_S$ rescales as follows
\begin{equation}
\tilde{\lambda}_S = \frac{\lambda_S}{M_P^2}\frac{1}{4\pi^2 R^2}\frac{1}{N_0^4}
e^{4(1-\alpha )k\pi R}
\end{equation}
where
\begin{equation}
N_0^{-1} = \sqrt {\frac{2(1-\alpha )k\pi R  }{e^{2(1-\alpha )k\pi R} -1  }    }
\end{equation}
and, for $\alpha >1$, $kR\gg 1$
\begin{equation}
N_0^2 (\alpha >1, kR \gg 1) \simeq \frac{1}{2(\alpha -1)k\pi R}\, .
\end{equation}
For $\alpha >1$ all couplings of the 5D potential rescale as follows
\begin{eqnarray}
&& \tilde{\lambda}_H = \lambda_H\qquad \tilde{\lambda}_T = \lambda_T\qquad
 \tilde{\lambda}_S = \frac{\lambda_S}{M_P^2}k^2 (\alpha -1)^2 e^{-4(\alpha -1)k\pi R}
 \nonumber\\
&&\tilde{\eta} =\eta\qquad \tilde{\xi} = \xi \sqrt{\alpha -1} \sqrt{\frac{k}{M_P}}
 e^{-(\alpha -1)k\pi R}
 \nonumber\\
&&\tilde{\chi}_i =\chi_i ({\alpha -1}) \frac{k}{M_P} e^{-2(\alpha -1)k\pi R}\,\,\, i=1,2
\qquad
\tilde{\mu}_{H,T}^2 =\mu_{H,T}^2k^2 e^{-2k\pi R}\, ,
\end{eqnarray}
where the first line contains all quartic self-interactions, and the remaining two
lines contain the couplings of the allowed mixed interactions of singlet, triplet and
doublet, $\eta ,
\xi ,\chi_{1,2}$ and the doublet and triplet masses $\mu_{H,T}$.
For $k\simeq M_P$ the warped geometry is providing a suppression of the couplings.
Denoting with $A$ the suppression factor
\begin{eqnarray}
A &=& \sqrt{2(\alpha - 1)} \sqrt{\frac{k}{M_P}} e^{(1-\alpha )k\pi R} \qquad \alpha >1
\nonumber\\
A &=& \sqrt{2(1-\alpha )} \sqrt{\frac{k}{M_P}} \qquad \alpha <1 \nonumber\\
A &=& \frac{1}{\sqrt{kR M_P} } \qquad \alpha =1\, ,
\end{eqnarray}
we obtain $ \tilde{\lambda}_S =A^4 {\lambda}_S$,
$ \tilde{\chi}_{1,2} =A^2 {\chi}_{1,2}$ and $\tilde{\xi} = A\xi$, thus providing the
hierarchy of couplings $ \tilde{\lambda}_S \ll  \tilde{\chi}_{1,2} \ll  \tilde{\xi}$.
Once the hierarchy of couplings is obtained, the solutions of the equations which minimize
the complete potential will provide the hierarchy of the associated vevs.
The set of coupled equations
\begin{eqnarray}
&&2\tilde{\lambda}_H v^3 + v (\tilde{\mu}_H^2 +\tilde{\eta}\xi_0^2 +\tilde{\xi} S_0\xi_0
+\tilde{\chi}_2 S_0^2) = 0 \nonumber\\
&&4\tilde{\lambda}_T \xi_0^3 + 2\xi_0 (\tilde{\mu}_T^2 +\tilde{\eta}v^2 +\tilde{\chi}_1
S_0^2)
+\tilde{\xi} S_0 v^2 = 0 \nonumber\\
&&4\tilde{\lambda}_S S_0^3 + 2S_0 (\tilde{\chi}_1 \xi_0^2 + \tilde{\chi}_2 v^2)
+\tilde{\xi} \xi_0 v^2 = 0
\end{eqnarray}
can be easily analyzed without the need for an exact solution.
From the third equation, by assuming $\xi_0\ll v$ we get $\xi_0 \ll AS_0$ and $S_0\sim v/A$,
 thus enhanced! The second equation, using $S_0\sim v/A$, implies that $\xi_0\sim v$ is the
only allowed non trivial solution.
Therefore, the produced hierarchy is $S_0\sim ve^{(\alpha -1) k\pi R} \gg v$ and
$\xi_0 \sim v$.
This conclusion is completely in agreement with the fact that once the singlet field is
localized on the Planck brane, it has to acquire a vev of order the Planck scale.
Unfortunately, this also means that no mechanism is left to provide a suppressed triplet
vev on the visible brane via the addition of a singlet bulk scalar field with
self-interacting
potential.
The cases of a delocalized and TeV-localized singlet field are also not relevant to our
purpose.
For $\alpha =1$, a power-like enhancement of the singlet field  $S_0\sim \sqrt{kR} v$ is
generated, while again $\xi_0 \sim v$. For $\alpha <1$ no hierarchy is produced, with
$S_0\sim  \xi_0 \sim v$.

\section{Conclusions}

In the context of warped extra dimensions, we have explored the possibility of imposing
boundary conditions on the bulk fermion fields which mix fields and their charge conjugate
 under a $Z_2$ parity transformation involving the warped extra dimension.
We have called these unordinary boundary conditions {\em pseudo-Majorana} conditions,
since they induce the usual Majorana constraint on spinors in four dimensions. They
can be written as
\begin{equation}
\psi (x, -y) = e^{-i\delta} \gamma_5\psi^c (x,y)\, ,
\end{equation}
where a phase factor is allowed in the most general case.
This is a particular case of a general analysis proposed in \cite{flat} in the case of
 universal flat extra dimensions. The warped geometry with orbifolding induces peculiar
behaviours of the allowed spectrum of fermion fields obtained in  the reduction from five
to four dimensions via the usual Kaluza Klein decomposition.

\noindent In particular, we have shown that the pseudo-Majorana boundary conditions generate
 a Kaluza-Klein tower of Majorana spinors in the four
dimensional action. No Dirac type bulk mass term seems to be allowed by these boundary
conditions embedded in the
warped geometry with orbifolding. The phase
factor $e^{i\delta}$ in the pseudo-Majorana boundary conditions must be chosen along with
the choice of the phase in the 4D charge conjugation matrix.
A massless zero mode exists and all Kaluza-Klein modes, including the zero mode, are
 localized on the visible ($y=\pi R$) brane. We have also noticed that a particular
choice of the phase factor  $e^{i\delta}$ forbids all excited states and only allows
for a zero mode solution.
The $y$-dependent wave functions of the Majorana Kaluza-Klein tower are
plane waves deformed by the warped geometry.

\noindent
These results show that the case of bulk fermion fields with pseudo-Majorana boundary
conditions is far more constrained than the ordinary Dirac case. No tuning of a
localization mechanism via the bulk mass term is allowed in the Majorana case.
It remains interesting the possibility to generalize these boundary conditions allowing
 for a superposition of $\psi$ and $\psi^c$ after a $Z_2$ transformation.
There is also the option of adding a bulk Majorana mass term, analogously to
ref.~\cite{HuberShafi}. But again, the mass parameter will acquire
opposite $Z_2$ parity with respect to the ordinary case if pseudo-Majorana boundary
conditions are imposed. Bulk Majorana spinors can be naturally identified with sterile
neutrinos with masses of order the weak scale or with a massless
Majorana Standard Model neutrino.

\noindent
Finally, after reconsidering a derivation proposed in \cite{Chen}, we have shown that
an additional bulk singlet scalar field cannot induce a suppression of a Higgs triplet
vev w.r.t. a Higgs doublet vev both confined on the visible brane.
Unfortunately, our conclusion seems to rule out the possibility that a singlet bulk scalar
field could induce a small mass for a Standard Model
Majorana neutrino via the Yukawa interaction with a Higgs triplet. Although we suspect that
this result extends to more general choices of scalar fields and their interactions,
an interesting task is still the one of generating small Majorana masses through a
hierarchy of
scalar vevs, possibly induced by the geometry of extra dimensions.

\subsection{Acknowledgments}
We aknowledge interesting discussions with Eric Bergshoeff, Mees de Roo, Avihay Kadosh,
Jos Postma, Duurt Johan van der Hoek, and correpondence with M-C. Chen.

redpajama_set_name: RedPajamaArXiv


text:
Visible G-string Friday
Filed under: General — Downunder @ 10:45 am Thu 27th November 2014
This story starts back here with Roger Sutton – another scalp for feminism followed by Andrew Litte feminazis' new top white-knight and A big THANK YOU to Roger Sutton
following Sutton's resignation from CERA (Canterbury Earthquake Recovery Authority) after a complaint of sexual harassment.
Amongst Roger Sutton's impropriety is his 'Visible G-string Friday' comment, for which he's taking a beating.
Of course sexual references and innuendo are about as rare as wide-mouth frogs, should never happen, and apparently women need to be protected from this.
Likewise, how many times have any of us been the butt of a joke (shit, am I allowed to say that) by other people deliberately taking an innocent comment the wrong way.
Life is full of it.
Sitcoms are full of it, and it's nothing new.
Shakespeare's The Taming of the Shrew …
'tongue in your tail'
… is at the more risque end of his lively references.
Today, it's commonplace in movies and songs, especially in children's stories, where innuendo is used to attract adult attention.
SHREK, like many other children's movies is a good example, and it starts right from the beginning and appears in all the sequells.
Lord Farquaad being shown Snowwhite in the mirror.
"Although she lives with seven other men, she's not easy."
Let's not forget the regular use of sexual innuendo in marketing, that's thrown at us daily, over radio and TV.
I've seen John Key making a lot of noise about this but have a little look back here: around the time of his mincing down the cat-walk at the Rugby World Cup.
However he (Key) has not resiled from remarks he made on his radio spot with broadcaster Tony Veitch just over a week ago when he listed a number of female celebrities he would have on his "wish-list", and called British actress Liz Hurley "hot".
And this …
International media labelled the remarks sexist, but Mr Key did not step back from the comments when asked about them this week.
Questions were also raised about the appropriateness of Mr Key appearing on a radio show with Veitch, who was convicted in 2009 of the violent assault against his former partner Kristin Dunne-Powell.
Well, I certainly hope Kibblewhite threw that back in Key's face after being accused of associating with a feminist declared felon.
Key is happy to tag along with the 'sail into Roger Sutton club' and the 'sail into Andrew Kibblewhite club' when he has happily defended himself for exactly the same thing.
One Response to "Visible G-string Friday"
Thu 27th November 2014 at 4:25 pm
Let's look at the story Stuffed News ran about John Key's sexism:
Prime Minister John Key has brushed off claims that he's a sexist for calling British actress and model Liz Hurley "hot", claiming she'd be flattered with his endorsement.
Key's comments were picked up by the British tabloid the Daily Mail after he listed glamour-girl Hurley among his celebrity crushes during an interview on Radio Sport last week.
He revealed his list of favourite females to host Tony Veitch after being asked if he'd like to be cricketer Shane Warne.
"Yeah, well, given, his current liaisons with Liz Hurley," Key said.
"I like Liz Hurley, actually. I reckon she is hot."
He also listed Jessica Alba on his 'hot' list, along with Angelina Jolie.
Meanwhile, Labour leader Phil Goff said Liz Hurley "was pretty close" to being on his top five hot women list.
Today is Goff's wedding anniversary and he put his wife Mary in the number one spot.
He said Elle Macpherson and Julia Roberts would also feature in his top five, "all being totally hypothetical of course".
The Mail today ran a story titled Liz Hurley is hot: New Zealand PM slammed over 'sexist' remarks, saying Key's comments had sparked a "sexist row".
"Mr Key, who has two teenagers, Stephie and Max, said that in his dream date top three – and there were 'a range' – Miss Hurley was 'a definite', especially as she's only 'slightly older'," the article read.
Despite criticisms from women's rights campaigner Sue Kedgley – who the Mail said went after Key with "talons sharpened" – the Prime Minister was unfazed by the implications he'd said something wrong.
When quizzed today, Key said: "I think [Liz Hurley] will be thrilled by the endorsement."
He was quite comfortable appearing on Veitch's show and would do so weekly.
Kedgley said she thought the comments were a bit "1960s".
"It is a bit stereotypical and rather odd, coming from a Prime Minister," she said.
PR expert Claudia Macdonald, managing director of Mango Communications, agreed with Key's approach, saying his comments were "refreshing" to hear from a Prime Minister.
"The days of tightly buttoned-up politicians with carefully managed facades are waning," she said.
"His popularity in New Zealand is high, partly because of this persona as it appeals to a wide range of people."
Macdonald said with the Rugby World Cup and election approaching, she would counsel Key to moderate the number of times he revealed his blokey side, "so that he is seen more as a capable leader; then he can let it loose again once re-elected".
The article is not the first the Mail has run on Key. Last June it ran a story about his vasectomy.
« A real HERO Are protection orders more harm than good? »

redpajama_set_name: RedPajamaCommonCrawl


text:
Arrests for all offences proposed
community socialist | 12.08.2004 21:31 | Repression | Social Struggles | Liverpool
All offences could prompt arrest under Home Office proposals
Police in England and Wales could be given powers to arrest people for minor offences such as graffiti or litter.
Police can currently arrest people only for crimes with sentences of at least five years, meaning breach of the peace is often used as a "catch all" offence.
A Home Office consultation paper also proposes to allow drug tests of people when arrested, and to make it easier to search suspects and their property.
Home Office minister Hazel Blears says the aim is to modernise police powers.
But shadow home secretary David Davis dismissed the proposals as "another headline grabbing initiative from a gimmick crazy home secretary".
All offences to be arrestable
Search warrants to cover more than one property, one search and to stay in force longer
Community Support Officers to get range of new powers, including to search detainees
Drug tests allowed on arrest rather than only when a suspect is charged
Fixed penalty fines to pay for more use of automatic number plate recognition technology
New offence of having an incorrectly registered vehicle
New power to stop and search under-18s suspected of carrying fireworks
Allow police to take DNA, fingerprints, shoeprints covertly
Key points of police plans
The measures are now out for consultation, with comments invited by the Home Office until 8 October.
Under the new plans, all offences including minor "anti-social behaviour" ones could in theory lead to an arrest, but there would be strict guidelines to determine when an arrest was appropriate.
Officials described their consultation document as a "tidying-up exercise" which would resolve confusion over the law, rather than result in many more arrests.
"What we are saying here is that in future all offences will be arrestable but that does not necessarily mean that everybody will be arrested because there will be a necessity test," Ms Blears told BBC Radio 4's Today Programme.
"A constable could come upon an offence and he might not be sure whether it is arrestable or not. In future that would be very clear.
I have yet to hear any serious voice in policing in this country say that the problem the police faces is that they don't have powers to arrest people for dropping litter
Shami Chakrabarti
Head to head: Wider arrest powers
"The range of offences on the statute book, some are serious arrestable offences, some are prosecutable by a summons - that's the way our legal system is .
"We are saying [we could make] every offence arrestable but what we are also saying is that you have to show that it was necessary in these circumstances," she added.
'Arbitrary'
But Shami Chakrabarti, director of civil rights group Liberty, said it was "disproportionate" to make offences which could not carry a prison sentence arrestable.
What is the point? I've not seen a policeman on the streets in ages.
Gary Davey, Colchester
The broader the powers the police were given, the more they were used in an arbitrary and racially discriminatory manner, she claimed.
"I have yet to hear any serious voice in policing in this country say that the problem the police faces is that they don't have powers to arrest people for dropping litter," she told Today.
Keith Taylor, Principal Speaker of the Green Party, said: "These proposals are the latest in a series of attacks on civil liberties by a government that already monitors its citizens more closely than any in Britain's history."
Civilian wardens
The government is also considering allowing police to test anyone they have arrested for drugs, regardless of whether they have been charged with an offence.
Police believe that testing suspects at an earlier stage would help isolate drug users more efficiently.
The paper also proposed giving more powers to community support officers.
They could get greater powers to direct traffic, tackle beggars, search people for weapons and enforce bylaws, so that police officers were free to concentrate on front-line duties, the consultation paper said.
We hope that by having a full range of powers available to us, we can deal effectively with antisocial behaviour, be one step ahead of the criminal
Chief Constable Denis O'Connor
Police could also have the power to fingerprint drivers at the roadside, as well as at police stations.
The Home Office suggests changes the way search warrants work so they would be granted for any address connected to a suspect, not just a single address.
The Association of Chief Police Officers (Acpo) said it had not asked for all offences to be made arrestable, although there was a need to rationalise some powers and strengthen others.
A spokesman said making minor offences arrestable could act as a deterrent as it sent a strong message to perpetrators.
Acpo vice president Chief Constable Denis O'Connor welcomed the consultation on the power to take fingerprints by the roadside and take footwear impressions without consent.
But he sounded a "note of caution" on plans to extend the powers of community safety officers.
"We hope that by having a full range of powers available to us, we can deal effectively with antisocial behaviour, be one step ahead of the criminal and give people the confidence to live safely," he added.
WELCOME TO THE FASCIST POLICE STATE!
community socialist
600 new offences since this government first came to power — dh
appalling — Moderate
George Orwell - notorious anti-communist and agent of the state! — Kai Andersen
NEW INVENTION — Mr. Blunkett
It's a fair cop? — beyond cheese butties


Q&A with Matthew Y.C. Lin, M.D.
We are pleased to announce that Emily Finlayson, M.D., M.S., FACS, professor of Surgery, Medicine (Geriatrics), and Health Policy at the University of California San Francisco has been named the new Vice Chair for Faculty Affairs and Development in the Department of Surgery. Dr. Finlayson received her MD at Harvard Medical School and MS from the Center for Evaluative Clinical Sciences at Dartmouth Medical School. She completed her general surgery residency at UCSF and her fellowship in colorectal surgery at the Mayo Clinic. Her NIH-funded work includes population-based analyses, clinical trials, and qualitative research on surgical outcomes focused on frail older adults.
Dr. Finlayson is founder and Program Director of the UCSF Center for Surgery in Older Adults and is a nationally recognized thought leader in the field. Most recently, she was senior author of a seminal paper in JAMA Surgery demonstrating that breast cancer surgery in female nursing home residents was an overused surgical modality generally conferring no clinical benefit.
In her role as the inaugural Vice Chair for Faculty Affairs and Development, Dr. Finlayson will lead efforts to ensure faculty success through development, promotion, diversity, recruitment, and retention. She will collaborate with the School of Medicine and University leadership to advance these objectives. Dr. Finlayson will also oversee the Department's Merit & Promotions process and serve as a member of the Chair's Leadership Council, which provides strategic advice and policy recommendations to the Chair, and partner with the Department's to be named Vice Chair of Diversity, Equity & Inclusion to foster an organizational culture grounded on these important principles.
Dr. Finlayson has demonstrated a strong commitment to the career development of academic surgeons, mentoring students, residents, fellows, and junior faculty who have gone on to launch successful academic careers. As Director of the Department of Surgery's Faculty Mentoring Program, Dr. Finlayson has been instrumental in the development and implementation of robust career plans for incoming faculty and as lead mentor for the Clinical Discovery & Health Services Research track on the Resident Research Committee.


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Cruise news /	Cruise passengers in best place to witness the Solar Eclipse
Cruise passengers in best place to witness the Solar Eclipse
Passengers on no fewer than nine cruise ships were in prime position to witness the Solar Eclipse. All had made their way to waters off the Faroe Islands where the area of totality could be seen at its best. This was the first total eclipse to be seen from Northern Europe since 1999; the next one is
Passengers on no fewer than nine cruise ships were in prime position to witness the Solar Eclipse.
All had made their way to waters off the Faroe Islands where the area of totality could be seen at its best. This was the first total eclipse to be seen from Northern Europe since 1999; the next one is not expected to happen until August 2026.
Fred. Olsen Cruise Lines' had Boudicca, Braemar and Black Watch in the area. Jostling for position were Saga Sapphire, Voyages of Discovery's Voyager, P&O Cruises' Oriana. and three ships from Cruise & Maritime Voyages' fleet: Magellan, Marco Polo and Azores.
A selection of magnificent photos capturing this event from on-board Fred Olsen's vessels is shown below.
Braemar with the eclipse overhead, taking from on-board Black Watch
The eclipse at its best, seen from Black Watch
The eclipse, seen from on-board Black Watch
The eclipse, seen from on-board Braemar
The trio of Fred. Olsen cruise ships (Braemar, Black Watch and Boudicca) off the Faroe Islands
Follow this link to our video footage of the eclipse as seen from Marco Polo.
(Photo credits to Fred. Olsen Cruise Lines)


Offshore drilling contractor Seadrill has sold its remaining stake of approximately 490 million shares in SapuraKencana, an oilfield services company based in Malaysia.
Seadrill said on Wednesday it has sold its 8.2 percent stake in the Malaysian company for approximately $195 million.
The offshore drilling contractor noted that the proceeds from the share sale would provide additional liquidity and would be used for general corporate purposes.
To remind, in April 2013, Seadrill increased its ownership stake in SapuraKencana to 12% as a result of the integration of its tender rig business into SapuraKencana aiming to develop a player in the Far East integrated service market.
In April 2014, Seadrill sold 230 million shares of SapuraKencana, realizing total economic gain of approximately $165 million, and leaving Seadrill with 490 million shares it sold on Wednesday.
Posted on April 28, 2016 with tags SapuraKencana, Seadrill.