# On the convex Pfaff-Darboux Theorem
of Ekeland and Nirenberg

Robert L. Bryant  Duke University Mathematics Department
P.O. Box 90320
Durham, NC 27708-0320<EMAIL_ADDRESS>http://www.math.duke.edu/~bryant

(Date: December 22, 2015)

###### Abstract.

The classical Pfaff-Darboux Theorem, which provides local ‘normal forms’ for
$1$-forms on manifolds, has applications in the theory of certain economic
models [3]. However, the normal forms needed in these models come with an
additional requirement of _convexity_ , which is not provided by the classical
proofs of the Pfaff-Darboux Theorem. (The appropriate notion of ‘convexity’ is
a feature of the economic model. In the simplest case, when the economic model
is formulated in a domain in $\mathbb{R}^{n}$, convexity has its usual
meaning.) In [4], Ekeland and Nirenberg were able to characterize necessary
and sufficient conditions for a given $1$-form $\omega$ to admit a convex
local normal form (and to show that some earlier attempts [2, 5] at this
characterization had been unsuccessful).

In this article, after providing some necessary background, I prove a
strengthened and generalized convex Pfaff-Darboux Theorem, one that covers the
case of a Legendrian foliation in which the notion of convexity is defined in
terms of a torsion-free affine connection on the underlying manifold. (The
main result in [4] concerns the case in which the affine connection is flat.)

###### Key words and phrases:

Pfaff-Darboux theorem, convexity, utility theory

###### 1991 Mathematics Subject Classification:

58A15, 91B16

Thanks to Duke University for its support via a research grant and to the
National Science Foundation for its support via DMS-9870164 (during which most
of the research for this article was done) and DMS-1359583 (during which this
article was written). This is Draft Version $1.0$.

###### Contents

  1. 1 Introduction
  2. 2 Classical Pfaff-Darboux theorems
  3. 3 Convexity and affine manifolds

##  1\. Introduction

The Pfaff-Darboux Theorem provides a local ‘normal form’ for $1$-forms on
manifolds, assuming that certain constant rank conditions are met. A common
version111There are many variants. See [1, Chapter II, §3]. of this classical
theorem is the following: Let $\omega$ be a smooth $1$-form on an $n$-manifold
$M$ and suppose that there is an integer $k>0$ such that

$\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k}$
vanishes identically on $M$

while

$\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k-1}$
is nowhere vanishing on $M$.

Then each $x\in M$ has an open neighborhood $U\subset M$ on which there exist
(smooth) functions $y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$ and a nonvanishing
function $a$ such that222Throughout this article, I adopt the convention that,
when $L\subset M$ is a submanifold and $\psi$ is a differential form on $M$,
then $L^{*}\psi$ denotes the _pullback_ of $\psi$ to $L$.

(1.1)
$U^{*}\omega=a\,(\mathrm{d}y^{1}+p_{2}\,\mathrm{d}y^{2}+\cdots+p_{k}\,\mathrm{d}y^{k}).$

Since

$U^{*}\bigl{(}\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k-1}\bigr{)}=(-1)^{k(k-1)/2}(k{-}1)!\,a^{k}\,\mathrm{d}y^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\cdots{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}y^{k}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}p_{2}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\cdots{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}p_{k},$

the functions $y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$ in this representation
must be independent on $U$.

The normal form (1.1) is often written more symmetrically as

(1.2)
$U^{*}\omega=a_{1}\,\mathrm{d}u^{1}+a_{2}\,\mathrm{d}u^{2}+\cdots+a_{k}\,\mathrm{d}u^{k},$

where the $a_{i}$ do not simultaneously vanish in $U$. In this representation,
the independence of the functions $y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$
translates into the condition that the mapping

(1.3)
$\bigl{(}u,[a]\bigr{)}:U\to{\mathbb{R}}^{k}\times{\mathbb{R}}{\mathbb{P}}^{k-1}={\mathbb{P}}\bigl{(}T^{*}{\mathbb{R}}^{k}\bigr{)}$

be a submersion.

In fact, the representation (1.2) is more common in treatises on mathematical
economics, where the Pfaff-Darboux Theorem plays an important role [2].
However, the normal forms needed in these models sometimes come with an
additional requirement of _convexity_ , i.e., the underlying manifold is
$M={\mathbb{R}}^{n}$ (or an open domain in ${\mathbb{R}}^{n}$), and one would
like to have the functions $a_{i}$ be _positive_ while the functions $u^{i}$
are _strictly convex_ , i.e., they have positive definite
Hessians.333Sometimes one only requires _weak convexity_ , i.e., that the
Hessian of $u^{i}$ be positive definite on each of its level sets.A useful
reference for the role of convexity in economic models is the book [3].

Now, it turns out that constructing such a _convex_ Pfaff-Darboux
representation is not always possible, which raises the question of how to
determine when one exists. In [4], Ekeland and Nirenberg were able to provide
necessary and sufficient conditions for a given $1$-form
$\omega\in\Omega^{1}({\mathbb{R}}^{n})$ to admit a local convex Pfaff-Darboux
normal form. They also constructed examples that showed that some earlier
attempts [2, 5] to find such conditions had been unsuccessful.

In this note, after providing some necessary background, I prove a
generalization of the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg.
This treatment has some notable features that make it of interest for the
general problem. First, the proof of Ekeland and Nirenberg does not rely on
the classical Pfaff-Darboux Theorem; instead, it constructs the required
convex representation directly using PDE techniques, thereby reproving the
Pfaff-Darboux Theorem in this special case. The proof below assumes the
classical Pfaff-Darboux Theorem, and so the argument can more directly focus
on choosing a Pfaff-Darboux representation that satisfies the convexity
requirements. This results in a shorter proof, one that also brings the nature
of the convexity requirements more sharply into focus. Second, the notion of
_strict convexity_ turns out to be meaningful on any manifold endowed with a
torsion-free affine connection, and the proof below covers this more general
situation with no extra work. Third, the proof yields a stronger result, in
that it produces a local convex Pfaff-Darboux representation of $\omega$
adapted to any $\omega$-Legendrian foliation that satisfies a certain
geometrically natural positivity condition, one that is equivalent to the
condition of Ekeland and Nirenberg.

##  2\. Classical Pfaff-Darboux theorems

Let $\omega$ be a smooth $1$-form on an $n$-manifold $M$ that, for some
integer $k>0$, satisfies

(2.1)
$\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k}$
vanishes identically on $M$

while

(2.2)
$\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k-1}$
is nowhere vanishing on $M$.

The integer $k{-}1$ is known as the _Pfaff rank_ of $\omega$ [1, Chapter II,
§3]. Note that $k\leq\tfrac{1}{2}(n{+}1)$; when $k=\tfrac{1}{2}(n{+}1)$,
$\omega$ is said to be a _contact form_ on $M$.

When $\omega$ satisfies (2.1) and (2.2), so does $\tilde{\omega}=f\,\omega$
for any nonvanishing function $f$ on $M$, since
$\tilde{\omega}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\tilde{\omega})^{r-1}=f^{r}\,\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{r-1}$
for all integers $r>0$.

###  2.1. Canonical subbundles

An $\omega$ satisfying (2.1) and (2.2) defines a _kernel subbundle_
$K=\omega^{-1}(0)\subset TM$ of corank $1$ and a subbundle $A\subset K$ of
corank $2(k{-}1)$ in $K$ by the rule that, for each $x\in M$,

(2.3) $A_{x}=\left\\{v\in K_{x}\ \vrule\ \mathrm{d}\omega(v,w)=0,\ \forall\
w\in K_{x}\ \right\\}.$

Replacing $\omega$ by $\tilde{\omega}=f\,\omega$ for any nonvanishing function
$f$ does not change $K$ or $A$.

When $k>1$, then $K\subset TM$ is not an integrable plane field, but the
subbundle $A\subset TM$ is always integrable, since it is the Cauchy
characteristic plane field of the differential ideal ${\mathcal{I}}$ generated
by $\omega$ (see [1, Chapter II, Prop. 2.1]). In the contact case, i.e., when
$n=2k{-}1$ (which is, in some sense, generic), one has $A=(0)$.

There is a nondegenerate, skew-symmetric bilinear pairing
$B_{\omega}:K{/}A\times K{/}A\to{\mathbb{R}}$ defined by

$B_{\omega}\bigl{(}v{+}A_{x},w{+}A_{x}\bigr{)}=\mathrm{d}\omega(v,w),$

when $v,w\in K_{x}$. It satisfies $B_{f\omega}=fB_{\omega}$ for any
nonvanishing function $f$ on $M$.

Note that any subspace $W\subset T_{x}$ on which both $\omega$ and
$\mathrm{d}\omega$ vanish, must, first of all, satisfy $W\subset K_{x}$ (since
$\omega$ vanishes on $W$), and second, must have codimension at least $k{-}1$
in $K_{x}$, since $\mathrm{d}\omega$, as a skew-symmetric form on $K_{x}$, has
Pfaff rank $k{-}1$. Moreover, if $W$ does have codimension $k-1$ in $K_{x}$,
then it must contain $A_{x}$, so that $W/A_{x}$ is a null subspace of
$B_{\omega}$.

###  2.2. Legendrian submanifolds and Grassmannians

Any submanifold $L\subset M$ that satisfies $L^{*}\omega=0$, i.e., an
_integral manifold of $\omega$_, must also satisfy $L^{*}\mathrm{d}\omega=0$
and hence, by the above linear algebra discussion, must have codimension at
least $k$ in $M$. When $L\subset M$ is an integral manifold of $\omega$ of
codimension $k$, it is said to be an _$\omega$ -Legendrian submanifold_.

In particular, if $L\subset M$ is $\omega$-Legendrian, then, for each $x\in
L$, the tangent space $T_{x}L$ satisfies $A_{x}\subset T_{x}L\subset K_{x}$
while $B_{\omega}$ vanishes identically on $T_{x}L/A_{x}\subset K_{x}/A_{x}$.

This motivates defining the _Legendrian Grassmannian_
$\operatorname{Leg}_{x}(\omega)\subset\operatorname{Gr}^{k}(T_{x}M)$ to be the
set of subspaces $W\subset K_{x}$ that have codimension $k$ in $T_{x}M$ and on
which both $\omega$ and $\mathrm{d}\omega$ vanish. By the above remarks, it
follows that $\operatorname{Leg}_{x}(\omega)$ can be canonically identified
with the _Lagrangian Grassmannian_
$\operatorname{Lag}(K_{x}/A_{x})\subset\operatorname{Gr}^{k-1}(K_{x}/A_{x})$
consisting of the $(k{-}1)$-dimensional subspaces of $K_{x}/A_{x}$ on which
$B_{\omega}$ vanishes. Hence, $\operatorname{Leg}_{x}(\omega)$ is naturally a
smooth manifold of dimension $\tfrac{1}{2}k(k{-}1)$. Moreover, the (disjoint)
union

$\operatorname{Leg}(\omega)=\bigcup_{x\in
M}\operatorname{Leg}_{x}(\omega)\subset\operatorname{Gr}^{k}(TM)$

is a smooth subbundle, and
$\operatorname{Leg}(f\omega)=\operatorname{Leg}(\omega)$ for all nonvanishing
$f$.

###  2.3. A local normal form

One version of the _Pfaff-Darboux Theorem_ [1, Chapter II, Theorem 3.1] states
that, when $\omega\in\Omega^{1}(M)$ satisfies (2.1) and (2.2), each $x\in M$
has an open neighborhood $U\subset M$ on which there exist smooth functions
$u^{1},\ldots,u^{k}$ and $a_{1},\ldots,a_{k}$ (with not all $a_{i}$
simultaneously vanishing) so that

(2.4) $U^{*}\omega=a_{1}\,\mathrm{d}u^{1}+\cdots+a_{k}\,\mathrm{d}u^{k}\,.$

Moreover, the mapping
$\bigl{(}u,[a]\bigr{)}:U\to{\mathbb{R}}^{k}\times\mathbb{RP}^{k-1}$ is a
submersion. (In fact, the kernel subbundle of the differential of this mapping
is the restriction of $A$ to $U$.)

Conversely, the existence of functions $u^{i}$ and $a_{i}$ for $1\leq i\leq k$
on an open set $U\subset M$ satisfying (2.4) with the $a_{i}$ not all
simultaneously vanishing and having the property that
$\bigl{(}u,[a]\bigr{)}:U\to{\mathbb{R}}^{k}\times\mathbb{RP}^{k-1}$ be a
submersion implies that both (2.1) and (2.2) hold on $U$.

###  2.4. Geometry of the normal form

It will be useful to have a geometric interpretation of the Pfaff-Darboux
Theorem. Now, in the representation (2.4), the functions $u^{i}$ have
independent differentials, i.e.,
$\mathrm{d}u^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\cdots{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}u^{k}$
does not vanish on $U$. Consequently, the simultaneous level sets of the
functions $u^{i}$ define a foliation ${\mathcal{L}}$ of $U\subset M$ by
$\omega$-Legendrian submanifolds, i.e., an $\omega$-Legendrian foliation.

Conversely, given an $\omega$-Legendrian foliation ${\mathcal{L}}$ on an open
subset $V\subset M$, each point $x\in V$ will have an open neighborhood
$U\subset V$ in which the leaves of ${\mathcal{L}}$ are the fibers of a
submersion $u=(u^{i}):U\to{\mathbb{R}}^{k}$. Since $\omega$ vanishes when
pulled back any fiber of $u$, it follows that there exists a mapping
$a=(a_{i}):U\to{\mathbb{R}}^{k}$ such that
$U^{*}\omega=a_{1}\,\mathrm{d}u^{1}+\cdots+a_{k}\,\mathrm{d}u^{k}$.

Thus, a geometric interpretation of the Pfaff-Darboux Theorem is the statement
that, when $\omega\in\Omega^{1}(M)$ satisfies (2.1) and (2.2), each point
$x\in M$ has an open neighborhood $U\subset M$ on which there exists an
$\omega$-Legendrian foliation.

###  2.5. Variants and extensions

There are a number of variants and extensions of the classical Pfaff-Darboux
Theorem that can all be seen to be equivalent to the above versions by
elementary arguments [1, Chapter II, §3]. In this article, two such variants
will be important. For convenience of reference, they will be stated as
propositions.

######  Proposition 1.

Suppose that $\omega\in\Omega^{1}(M)$ satisfies (2.1) and (2.2). Then for each
$x\in M$ and $W\in\operatorname{Leg}_{x}(\omega)$, there exists a
$\omega$-Legendrian submanifold $L\subset M$ such that $x\in L$ and
$W=T_{x}L$.

######  Proposition 2.

Suppose that $\omega\in\Omega^{1}(M)$ satisfies (2.1) and (2.2) and that
$L\subset M$ is an embedded $\omega$-Legendrian submanifold. Then each $x\in
L$ has an open neighborhood $U\subset M$ on which there exists an
$\omega$-Legendrian foliation ${\mathcal{L}}$ with the property that $L\cap U$
is a leaf of ${\mathcal{L}}$.

##  3\. Convexity and affine manifolds

###  3.1. Classical convexity

When $M={\mathbb{R}}^{n}$, there is a notion of _strict convexity_ of a
function $u$, which is the condition that the Hessian quadratic form $H(u)$ be
positive definite, where

(3.1) $H(u)={{\partial^{2}u}\over{\partial x^{i}\partial
x^{j}}}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$

and where $x^{1},\ldots,x^{n}$ are the usual affine linear coordinates in
${\mathbb{R}}^{n}$. Note that strict convexity is an affine-invariant notion
on ${\mathbb{R}}^{n}$.

Motivated by applications in economics, Ekeland and Nirenberg [4] asked what
further conditions one must impose on an
$\omega\in\Omega^{1}({\mathbb{R}}^{n})$ satisfying (2.1) and (2.2) in order to
know that one can choose the functions $u^{j}$ and $a_{j}$ in the
representation (2.4) so that the $u^{j}$ be strictly convex and the $a_{j}$ be
positive. It is not hard to show, by example, that _some_ further condition on
$\omega$ is necessary to guarantee the existence of such a convex
representation. (See the discussion at the beginning of §3.3.)

They showed that two earlier articles [2, 5] claiming to provide such
necessary and sufficient conditions were flawed (indeed, they exhibited
counterexamples to the claims of these articles) and then produced their own
condition, which they showed to be necessary and sufficient.

In this note, I will show that their main result, properly formulated, holds
good on an $n$-manifold $M$ endowed with a torsion-free affine connection, not
just on ${\mathbb{R}}^{n}$ endowed with the (flat) affine connection it
inherits as a vector space.

###  3.2. Affine connections and convexity

Let $\nabla$ be a torsion-free affine connection on an $n$-manifold $M^{n}$,
i.e., $\nabla$ is a first-order, linear differential operator

(3.2) $\nabla:\Omega^{1}(M)\to\Omega^{1}(M)\otimes\Omega^{1}(M)$

that obeys the Leibnitz rule

(3.3) $\nabla(f\eta)=\mathrm{d}f\otimes\eta+f\,\nabla(\eta)$

for all smooth functions $f$ on $M$ and smooth $1$-forms $\eta$ on $M$. The
assumption that $\nabla$ be torsion-free is the condition that the associated
(second-order) _Hessian operator_ $H(u)=\nabla(\mathrm{d}u)$ be a symmetric
$(0,2)$-tensor for each smooth function $u$ on $M$.

A smooth function $u$ on $M$ is said to be strictly $\nabla$-convex if, as a
quadratic form, $H(u)$ is positive definite at every point of $M$.

When $M={\mathbb{R}}^{n}$ and $\nabla$ is the standard (flat) connection,
satisfying $\nabla(\mathrm{d}x^{i})=0$ for all of the coordinate functions
$x^{i}$, then $H(u)$ is the usual Hessian tensor (3.1), and this notion of
convexity is simply the classical one.

In the more general case, when $x=(x^{i}):U\to{\mathbb{R}}^{n}$ is a local
coordinate chart, one has

(3.4)
$H(x^{k})=\nabla(\mathrm{d}x^{k})=\Gamma^{k}_{ij}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$

where $\Gamma^{k}_{ij}=\Gamma^{k}_{ji}\in C^{\infty}(U)$ are the coefficients
of the connection $\nabla$ relative to the coordinate chart $x=(x^{i})$. The
general coordinate formula for $H$ then becomes

(3.5) $H(u)=\left({{\partial^{2}u}\over{\partial x^{i}\partial
x^{j}}}+\Gamma^{k}_{ij}\,{{\partial u}\over{\partial
x^{k}}}\right)\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}.$

Thus, $\nabla$-convexity of $u$ is expressible in terms of a condition on the
$2$-jet of $u$, slightly more general than the condition for classical
convexity.

Adopting the usual conventions

(3.6)
$\displaystyle\alpha{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\beta$
$\displaystyle={\textstyle{1\over 2}}(\alpha\otimes\beta-\beta\otimes\alpha)$
$\displaystyle\alpha\circ\beta$ $\displaystyle={\textstyle{1\over
2}}(\alpha\otimes\beta+\beta\otimes\alpha),$

one sees that, for a $1$-form $\omega$ of the form

(3.7) $\omega=a_{1}\,\mathrm{d}u^{1}+\cdots+a_{k}\,\mathrm{d}u^{k},$

one has (using the summation convention)

(3.8) $\displaystyle\nabla\omega$
$\displaystyle=\mathrm{d}a_{i}\otimes\mathrm{d}u^{i}+a_{i}\,H(u^{i})$
$\displaystyle=\mathrm{d}a_{i}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}u^{i}+\mathrm{d}a_{i}\circ\mathrm{d}u^{i}+a_{i}\,H(u^{i})$
$\displaystyle=\mathrm{d}\omega+\mathsf{S}\omega,$

where I have introduced the notation $\mathsf{S}\omega$ to denote the
symmetrization of $\nabla(\omega)$. Thus,
$\mathsf{S}\omega=\nabla\omega-\mathrm{d}\omega$ is a well-defined quadratic
form on $M$. (Of course, the linear, first-order differential operator
$\mathsf{S}$ depends on $\nabla$.)

###  3.3. A positivity condition

If, in a local representation (3.7), the $a_{i}$ are positive and the
functions $u^{j}$ are strictly $\nabla$-convex, then the quadratic form

$\mathsf{S}\omega=\mathrm{d}a_{i}\circ\mathrm{d}u^{i}+a_{i}\,H(u^{i})$

is strictly positive definite on the plane field $N\subset TM$ defined by
$\mathrm{d}u^{1}=\mathrm{d}u^{2}=\cdots=\mathrm{d}u^{k}=0$ since, on $N$, the
terms $\mathrm{d}a_{i}\circ\mathrm{d}u^{i}$ in $\mathsf{S}\omega$ vanish and
one is left with the positive definite expression $a_{i}\,H(u^{i})$. (Of
course, both $\omega$ and $\mathrm{d}\omega$ vanish when pulled back to the
plane field $N$.)

Thus, one sees that the $\omega$-Legendrian foliation ${\mathcal{L}}$ defined
in $U$ by $\mathrm{d}u^{1}=\mathrm{d}u^{2}=\cdots=\mathrm{d}u^{k}=0$ has the
property that the quadratic form $\mathsf{S}\omega$ is positive definite on
each of the leaves of ${\mathcal{L}}$. It turns out that this necessary
condition for a local ‘convex’ Pfaff-Darboux representation compatible with
the $\omega$-Legendrian foliation ${\mathcal{L}}$ is also sufficient.444It is
worth pointing out that the same conclusion about the positive definiteness of
$\mathsf{S}\omega$ on the leaves of ${\mathcal{L}}$ would have followed if one
had merely assumed that each $u^{i}$ be only ‘strictly $\nabla$-_quasi_
-convex’, i.e., that $\mathrm{d}u^{i}$ be nonvanishing and $H(u^{i})$ be
positive definite when restricted to the hyperplane field $\mathrm{d}u^{i}=0$.
Compare [4, Lemma 1], and the preceding discussion about their Problem 2.

######  Theorem 1.

Suppose $\nabla$ be a torsion-free affine connection on $M$, that
$\omega\in\Omega^{1}(M)$ satisfy (2.1) and (2.2) for some $k>0$, and that
${\mathcal{L}}$ be an $\omega$-Legendrian foliation on $M$ with the property
that $\mathsf{S}\omega$ pulls back to each leaf of ${\mathcal{L}}$ to be
positive definite. Then each $x\in M$ has an open neighborhood $U\subset M$ on
which there exist strictly $\nabla$-convex functions $u^{1},\ldots,u^{k}$ that
are constant on the leaves of ${\mathcal{L}}$ in $U$ and positive functions
$a_{1},\ldots,a_{k}$ such that

(3.9) $U^{*}\omega=a_{1}\,\mathrm{d}u^{1}+\cdots+a_{k}\,\mathrm{d}u^{k}.$

Before giving the proof of Theorem 1, I will state one of its corollaries, so
that it can be compared with the main result of Ekeland and Nirenberg [4,
Theorem 1].

First, some useful terminology. As always, assume that $\omega$ satisfies
(2.1) and (2.2) for some $k>0$.

######  Definition 1.

An $\omega$-Legendrian subspace $W\subset T_{x}M$ is _$\nabla$ -positive for
$\omega$_ if the restriction of the quadratic form $\mathsf{S}\omega$ to $W$
is positive definite.

Let $\operatorname{Leg}^{+}(\omega,\nabla)\subseteq\operatorname{Leg}(\omega)$
denote the set of $\omega$-Legendrian subspaces that are $\nabla$-positive for
$\omega$. Then $\operatorname{Leg}^{+}(\omega,\nabla)$ is a (possibly empty)
open subset of $\operatorname{Leg}(\omega)$. Consequently, the set of points
$x\in M$ for which there exists a $\nabla$-positive, $\omega$-Legendrian
subspace $W\subset T_{x}M$ is an open subset of $M$. Also, note that, since
such a $W$ contains $A_{x}$, it follows that $\mathsf{S}\omega$ must be
positive definite on $A_{x}$.

######  Corollary 1.

Suppose that $\nabla$ be a torsion-free affine connection on $M$, that
$\omega\in\Omega^{1}(M)$ satisfy (2.1) and (2.2) for some $k>0$, and that
there exist a $W\in\operatorname{Leg}^{+}(\omega,\nabla)$ with $W\subset
T_{x}M$. Then $x\in M$ has an open neighborhood $U\subset M$ on which there
exist strictly $\nabla$-convex functions $u^{1},\ldots,u^{k}$ and positive
functions $a_{1},\ldots,a_{k}$ such that

(3.10) $U^{*}\omega=a_{1}\,\mathrm{d}u^{1}+\cdots+a_{k}\,\mathrm{d}u^{k}.$

The proof of Corollary 1 follows by applying Propositions 1 and 2 to produce
an $\omega$-Legendrian foliation ${\mathcal{L}}$ on an open neighborhood $V$
of $x$ whose leaf through $x$ has $W$ as a tangent space. Since
$\mathsf{S}\omega$ is positive definite on $W$, it follows that it is positive
definite on all the tangent spaces to the leaves of ${\mathcal{L}}$ in some
(possibly) smaller $x$-neighborhood $V^{\prime}\subset V$. Now apply Theorem 1
to ${\mathcal{L}}$ on $V^{\prime}$.

######  Remark 1.

In the special case in which $M={\mathbb{R}}^{n}$ and $\nabla$ is the flat
connection satisfying $\nabla(\mathrm{d}x^{i})=0$ for $x^{i}$ the standard
coordinates on ${\mathbb{R}}^{n}$, Corollary 1 simply becomes Theorem 1 of
Ekeland and Nirenberg [4], since their Condition 3 turns out to be equivalent
to the existence of a $W\in\operatorname{Leg}^{+}(\omega,\nabla)$ with
$W\subset T_{x}M$ in this case.

###### Proof of Theorem 1..

There exists an $x$-neighborhood $V_{0}\subset M$ on which there exist smooth
functions $y^{1},\ldots,y^{k}$ vanishing at $x$ so that the leaves of
$\mathrm{d}y^{1}=\cdots=\mathrm{d}y^{k}=0$ are intersections of the leaves of
${\mathcal{L}}$ with $V_{0}$ as well as functions $p_{2},\ldots,p_{k}$, also
vanishing at $x$, and a nonvanishing function $a$ so that

$V_{0}^{*}\omega=a\bigl{(}\mathrm{d}y^{1}+p_{2}\,\mathrm{d}y^{2}+\cdots+p_{k}\,\mathrm{d}y^{k}\bigr{)}$

By reversing the signs of $a$ and the $y^{i}$, if necessary, one can assume
that $a(x)>0$. Let $W\subset T_{x}M$ be the tangent to the leaf of
${\mathcal{L}}$ that passes through $x$, so that $W$ is the common kernel of
the $\mathrm{d}y^{i}$ evaluated at $x$.

Set $\bar{\omega}=a^{-1}\omega$ and note that, since
$\mathrm{d}\bar{\omega}\equiv a^{-1}\,\mathrm{d}\omega\bmod\omega$, it follows
that ${\mathcal{L}}$ is also $\bar{\omega}$-Legendrian. Moreover, since

$\mathsf{S}\bar{\omega}=\mathrm{d}(a^{-1})\circ\omega+a^{-1}\,\mathsf{S}\omega,$

it follows that the tangent spaces of ${\mathcal{L}}$ (which, of course,
satisfy $\omega=0$) are also $\nabla$-positive for $\bar{\omega}$. Since
$\omega=a\,\bar{\omega}$ and $a>0$, finding the desired convex representation
for $\bar{\omega}$ will also yield one for $\omega$. Thus, it suffices to
prove the theorem with $\bar{\omega}$ in the place of $\omega$, i.e., to
assume that $a=1$, so I will do that from now on. Thus,

$V_{0}^{*}\omega=\mathrm{d}y^{1}+p_{2}\,\mathrm{d}y^{2}+\cdots+p_{k}\,\mathrm{d}y^{k}.$

Since
$\omega{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}(\mathrm{d}\omega)^{k-1}\not=0$,
the functions $y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$ have linearly
independent differentials at $x$.

Restricting to $V_{0}$, i.e., setting $M=V_{0}$, one has

$\mathsf{S}\omega=H(y^{1})+\mathrm{d}p_{2}\circ\mathrm{d}y^{2}+\cdots+\mathrm{d}p_{k}\circ\mathrm{d}y^{k}+p_{2}\,H(y^{2})+\cdots+p_{k}\,H(y^{k}).$

Since the $p_{j}$ vanish at $x$, it follows that, when restricted to $W\subset
T_{x}M$, the two quadratic forms $H(y^{1})$ and $\mathsf{S}\omega$ are equal.
Thus $H(y^{1})$ is positive definite on $W$, and so there is a constant $c>0$
so that $H(y^{1})+c\,(\mathrm{d}y^{2})^{2}+\cdots+c\,(\mathrm{d}y^{k})^{2}$ is
positive definite on $K_{x}=\\{v\in T_{x}M\ \vrule\ \mathrm{d}y^{1}(v)=0\\}$.
Writing

$\omega=\mathrm{d}\bigl{(}y^{1}+\tfrac{1}{2}c\,(y^{2})^{2}+\cdots+\tfrac{1}{2}c\,(y^{k})^{2})+(p_{2}-c\,y^{2})\,\mathrm{d}y^{2}+\cdots+(p_{k}-c\,y^{k})\,\mathrm{d}y^{k}$

and observing that

$\displaystyle
H(y^{1}+\tfrac{1}{2}c\,(y^{2})^{2}+\cdots+\tfrac{1}{2}c\,(y^{k})^{2})$
$\displaystyle=H(y^{1})+c\,(\mathrm{d}y^{2})^{2}+\cdots+c\,(\mathrm{d}y^{k})^{2}$
$\displaystyle\qquad+c\,y^{2}\,H(y^{2})+\cdots+c\,y^{k}\,H(y^{k})$

shows that, setting

$\bar{y}^{1}=y^{1}+\tfrac{1}{2}c\,(y^{2})^{2}+\cdots+\tfrac{1}{2}c\,(y^{k})^{2},\quad\bar{y}^{i}=y^{i},\quad\text{and}\qquad\bar{p}_{i}=p_{i}-c\,y^{i},$

one has
$\omega=\mathrm{d}\bar{y}^{1}+\bar{p}_{2}\,\mathrm{d}\bar{y}^{2}+\cdots+\bar{p}_{k}\,\mathrm{d}\bar{y}^{k}$.

Thus, one could have chosen the functions
$y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$ with $H(y^{1})$ being positive
definite on the hyperplane $K_{x}$. Assume now that this was done.

It still needs to be shown that one can choose the functions
$y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}$ with $H(y^{1})$ being positive
definite on all of $T_{x}M$, not just on $K_{x}$, which is the kernel of
$\mathrm{d}y^{1}$ at $x$. To do this, note that, if $\phi$ is any smooth
function on a neighborhood of the origin in ${\mathbb{R}}$, then

$H\bigl{(}\phi(y^{1})\bigr{)}=\phi^{\prime}(y^{1})\,H(y^{1})+\phi^{\prime\prime}(y^{1})\,(\mathrm{d}y^{1})^{2}.$

Hence, by choosing a $\phi$ with $\phi(0)=0$, $\phi^{\prime}(0)=1$ and
$\phi^{\prime\prime}(0)>0$ sufficiently large, I can arrange that
$\phi(y^{1})$ be strictly $\nabla$-convex at $x$. Since

$\omega={1\over{\phi^{\prime}(y_{1})}}\,\left(\mathrm{d}\bigl{(}\phi(y^{1})\bigr{)}+\phi^{\prime}(y_{1})p_{2}\,\mathrm{d}y^{2}+\cdots+\phi^{\prime}(y_{1})p_{k}\,\mathrm{d}y^{k}\right),$

one sees that the functions
$\bar{y}^{1},\ldots,\bar{y}^{k},\bar{p}_{2},\ldots,\bar{p}_{k}$, where

$\bar{y}^{1}=\phi(y^{1}),\quad\text{and}\qquad\bar{y}^{i}=y^{i},\quad\bar{p}_{i}=\phi^{\prime}(y_{1})p_{i},\quad
2\leq i\leq k,$

(with $a=1/\phi^{\prime}(y_{1})>0$), give a Pfaff-Darboux representation for
$\omega$ that is compatible with the foliation ${\mathcal{L}}$ and for which
$\bar{y}^{1}$ is strictly $\nabla$-convex.555This is the same idea that
Ekeland and Nirenberg used in their generalization of their Theorem 1 to cover
the quasi-convex case.

Thus, one can assume henceforth that, on an open $x$-neighborhood
$V_{1}\subset M$, one has a representation of the form

$\omega=\mathrm{d}y^{1}+p_{2}\,\mathrm{d}y^{2}+\cdots+p_{k}\,\mathrm{d}y^{k},$

where the functions $y^{1},\ldots,y^{k},p_{2},\ldots,p_{k}\in
C^{\infty}(V_{1})$ all vanish at $x$, the equations $\mathrm{d}y^{i}=0$ define
the tangents to the leaves of ${\mathcal{L}}$ in $V_{1}$, and $y^{1}$ is
strictly $\nabla$-convex.

Under these assumptions, there is a constant $b>0$ sufficiently large so that
$H(y^{i}+by^{1})=H(y^{i})+b\,H(y^{1})$ is positive definite at $x$ for $2\leq
i\leq k$. Thus, writing

$\omega=\bigl{(}1-b(p_{2}{+}\cdots{+}p_{k})\bigr{)}\,\mathrm{d}y^{1}+p_{2}\,\mathrm{d}(y^{2}+b\,y^{1})+\cdots+p_{k}\,\mathrm{d}(y^{k}+b\,y^{1}),$

it follows that I can, after restricting to an $x$-neighborhood $V_{2}\subset
V_{1}$ on which the function $a=\bigl{(}1-b(p_{2}{+}\cdots{+}p_{k})\bigr{)}$
is positive, dividing by $a>0$, and replacing $y^{j}$ by $y^{j}+by^{1}$ and
$p_{j}$ by $p_{j}/a$ for $2\leq j\leq k$, assume that I have a representation

$\omega=\mathrm{d}y^{1}+p_{2}\,\mathrm{d}y^{2}+\cdots+p_{k}\,\mathrm{d}y^{k},$

in which all of the $H(y^{j})$ are positive definite at $x$, i.e., the $y^{j}$
are strictly $\nabla$-convex on some neighborhood of $x$ and the $p_{i}$ all
vanish at $x$.

Finally, for $\varepsilon>0$ and sufficiently small, write

$\omega=\mathrm{d}\bigl{(}y^{1}-\varepsilon(y^{2}{+}\cdots{+}y^{k})\bigr{)}+(p_{2}+\varepsilon)\,\mathrm{d}y^{2}+\cdots+(p_{k}+\varepsilon)\,\mathrm{d}y^{k}.$

Then, setting $u^{1}=y^{1}-\varepsilon(y^{2}+\cdots+y^{k})$ and $u^{j}=y^{j}$
for $j>1$ and setting $a_{1}=1$ and $a_{j}=\varepsilon+p_{j}$ for $j>1$, one
achieves the desired convex Pfaff-Darboux representation on an open
$x$-neighborhood $U\subset V_{2}$. ∎

######  Remark 2 (Global considerations).

While Theorem 1 gives necessary and sufficient conditions for the existence of
local $\nabla$-convex Pfaff-Darboux representations, for applications one
would like to know something about how large an open set in the model $M$ one
can cover with such a representation, and this seems to be a subtle problem.

Even in the simplest case of a $3$-manifold $M$ endowed with a contact
$1$-form $\omega$ and a torsion-free affine connection $\nabla$ for which
$\mathsf{S}\omega$ is positive definite on the $2$-plane bundle $K\subset TM$,
it is not clear how to characterize the domains $U\subset M$ that support a
$\nabla$-convex Pfaff-Darboux representation. It is clear that such a $U$ must
be $\omega$-tight, but this does not appear to be sufficient.

## References

  * [1] R. Bryant, S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, _Exterior Differential Systems_ , Springer-Verlag, 1991. MR1083148
  * [2] P.A. Chiappori and I. Ekeland, _A convex Darboux theorem_ , Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) 25 (1997), 287–297. MR1655519
  * [3] P.A. Chiappori and I. Ekeland, _The Economics and Mathematics of Aggregation: Formal Models of Efficient Group Behavior_ , Foundations and Trends in Microeconomics 5 (2009) Nos. 1-2, 1–151.
  * [4] I. Ekeland and L. Nirenberg, _A convex Darboux theorem_ , Methods Appl. Anal. 9 (2002), 329–344. MR2023129
  * [5] V. Zakalyukin, _Concave Darboux theorem_ , C. R. Acad. Sci. Paris Sér. I, 327 (1998), 633–638. MR1652709


Polarized $Z$ cross sections in Higgsstrahlung for the determination of
anomalous $ZZH$ couplings

Kumar Raoa, Saurabh D. Rindanib, Priyanka Sarmaha, Balbeer Singhc

aDepartment of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India

bTheoretical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad 380009, India

cDepartment of Theoretical Physics,
Tata Institute of Fundamental Research, Dr. Homi Bhabha Road,
Colaba, Mumbai 400005, India

Abstract

The production of a Higgs boson in association with a $Z$ at an electron-
positron collider is one of the cleanest methods for the measurement of the
couplings of the Higgs boson. In view of the large production cross section at
energies a little above the threshold, it seems feasible to make a more
detailed study of the process by measuring the cross sections for polarized
$Z$ in order to measure possible anomalous $ZZH$ couplings. We show that
certain combinations of cross sections in $e^{+}e^{-}\to ZH$ with different
$Z$ polarizations help to enhance or isolate the effect of one of the two
kinds of anomalous $ZZH$ couplings possible on general grounds of CP and
Lorentz invariance. These combinations can be useful to get information on the
$ZZH$ coupling in the specific contexts of an effective field theory, two-
Higgs-doublet models, and composite Higgs models, in a relatively model-
independent fashion. We find in particular that the longitudinal helicity
fraction of the $Z$ is expected to be insensitive to anomalous couplings, and
would be close to its value in the standard model in the scenarios we
consider. We also discuss the sensitivity of the proposed measurements to the
anomalous couplings, including longitudinal beam polarizations, which suppress
backgrounds, and can improve the sensitivity if appropriately chosen.

Experiments at the Large Hadron Collider (LHC) have studied in great detail
the properties of the Higgs boson, especially its couplings to fermions and
gauge bosons, with increasing precision. The results seem to be in agreement
with the predictions of the standard model (SM) to a good degree of accuracy.
However, there is still a possibility that the Higgs couplings are not
precisely those predicted by the SM, and more data from future experiments at
the LHC should be able to improve the accuracy of the comparison.

Another prospect for acquiring more information on the Higgs being considered
is the construction of an electron-positron collider. At such a collider, the
process of associated $Z$ and Higgs production would be an attractive way to
study Higgs properties because of a clean environment and a reasonably large
cross section at energies not far above threshold. In addition, the Higgs
energy-momentum can be reconstructed from the $Z$ decay products regardless of
the Higgs decay final state.

In the context of the process $e^{+}e^{-}\to ZH$ alluded to above, deviations
from the SM predictions, if seen, would most likely indicate $ZZH$ couplings
differing from those in the SM, and perhaps also $\gamma ZH$ couplings. We
concentrate here on so-called anomalous $ZZH$ couplings and their effect on
the $ZH$ production process. Our aim is to investigate how $Z$ polarization
can be used to determine the coefficients of different Lorentz tensors in a
general $ZZH$ vertex written in a model-independent way.

In [1] we considered angular asymmetries of charged leptons arising in $Z$
decay which characterize the $Z$ spin density matrix and could be used to
determine $ZZH$ couplings. These included polar and azimuthal angle
asymmetries of leptons, the azimuthal angular dependence arising from the off-
diagonal elements of the density matrix, based on the formalism of [2, 3].
Here we concentrate on the simpler diagonal elements of the density matrix,
which are the degrees of polarization of the $Z$, and which can be probed
without a detailed study of the azimuthal distribution of leptons. A
significant result is that certain combinations of the polarized $Z$ cross
sections enable isolation of specific anomalous couplings.

Our study is in the context of a few specific scenarios for $ZZH$ couplings
differing from those in the SM. Ref. [1] dealt with a completely model
independent set of form factors, only constrained by Lorentz invariance. While
one could measure several asymmetries and use them simultaneously to determine
or limit the form factors, the process would be fairly complicated, and would
be lacking in accuracy because of the many variables involved. However, in the
special scenarios considered in this paper, using combinations of polarized
cross sections, a certain conceptual and practical simplification results, and
the dependence of the observables is only on one, or at most two parameters,
leading to better accuracy. Though these scenarios vary in the assumptions
that are made in them, they are reasonably model-independent, within those set
of assumptions. The different combinations of polarized cross sections that we
suggest have a varied advantage in each scenario.

In all cases, we assume that longitudinally polarized beams would be
available, whose signs can be chosen to suppress background and improve the
sensitivity of the observables.

Other studies involving $Z$ polarization are [4], in the context of the
process $e^{+}e^{-}\to ZH$ and [5, 6] in the context of determination of
anomalous $ZZH$ couplings at the LHC.

There have been several studies in the past for the measurement of anomalous
$ZZH$ couplings using final state asymmetries in $e^{+}e^{-}\to ZH$ without
reference to $Z$ polarization, see [7, 8] and references therein. These do not
have the advantage of the kind of observables we are suggesting here.

We consider the process $e^{+}e^{-}\rightarrow ZH$, where the vertex
$Z^{\ast}_{\mu}(k_{1})\rightarrow Z_{\nu}(k_{2})H$ has the Lorentz structure

$\Gamma^{V}_{\mu\nu}=\frac{g}{\cos\theta_{W}}m_{Z}\left[a_{Z}g_{\mu\nu}+\frac{b_{Z}}{m_{Z}^{2}}\left(k_{1\nu}k_{2\mu}-g_{\mu\nu}k_{1}.k_{2}\right)+\frac{\tilde{b}_{Z}}{m_{Z}^{2}}\epsilon_{\mu\nu\alpha\beta}k_{1}^{\alpha}k_{2}^{\beta}\right],$
(1)

where $g$ is the $SU(2)_{L}$ coupling and $\theta_{W}$ is the weak mixing
angle. The couplings $a_{Z}$, $b_{Z}$ and $\tilde{b}_{Z}$ are Lorentz scalars,
and depending on the framework employed, are either real constants, or
complex, momentum-dependent, form factors. The $a_{Z}$ and $b_{Z}$ terms are
invariant under CP, while the $\tilde{b}_{Z}$ term corresponds to CP
violation. In the SM, at tree level, the coupling $a_{Z}=1$, whereas the other
two couplings $b_{Z}$ and $\tilde{b}_{Z}$ vanish.

We consider here some possibilities for the couplings $a_{Z}$, $b_{Z}$ and
$\tilde{b}_{Z}$ in various scenarios.

(a) In an effective field theory (EFT) description of new physics [9, 10] (for
applications to $ZH$ production, see the works of Hagiwara and Stong in [8]
and of Beneke et al., in [7]), where the SM is the low-energy limit of an
extended theory, $a_{Z}$ would be normalized in the SM to the value of 1 and
would get a contribution $\delta a_{Z}$ of order $1/\Lambda^{2}$ from
dimension-six operators, so that $a_{Z}=1+\delta a_{Z}$. $b_{Z}$ and
$\tilde{b}_{Z}$ would get contributions of order $1/\Lambda^{2}$ from
dimension-six operators, and would be suppressed. They would, however, be
real, from Hermiticity.

The EFT Lagrangian, including terms up to dimension 6 takes the form

${\cal L}_{\rm eff}={\cal L}_{\rm
SM}^{(4)}+\frac{1}{\Lambda^{2}}\sum^{59}_{k=1}\alpha_{k}{\cal O}_{k},$ (2)

with a sum of 59 independent terms of dimension 6, of which 11 are relevant
for our process. Of these, we can identify $\delta
a_{Z}=\hat{\alpha}^{(1)}_{ZZ}$ and $b_{Z}=\hat{\alpha}_{ZZ}$, where
$\hat{\alpha}^{(1)}_{ZZ}$ and $\hat{\alpha}_{ZZ}$ in the notation of [11] are
combinations of coefficients of dimension-6 operators with a weak coupling
factor $m_{Z}^{2}(\sqrt{2}G_{F})^{1/2}$ pulled out.

(b) In two-Higgs-doublet models (for a review, see [12]) at tree level,
$a_{Z}=\sin(\alpha-\beta)$, where $\tan\beta=v_{2}/v_{1}$, $v_{1,2}$ being the
vacuum expectation values of the two neutral Higgs fields, and $\alpha$ is the
mixing angle between the physical scalar eigenstates and Lagrangian scalar
fields. $\alpha$ can have a real value different from 1, whereas $b_{Z}$ and
$\tilde{b}_{Z}$ are zero. In these models, we will consider the coupling of
the lighter of the two CP-even Higgs particles. In the case of other
extensions of the sector with more doublets, singlets or triplets, the
situation would be similar.

(c) In composite Higgs models [13, 14, 15] the coupling $a_{Z}$ is different
from unity, modified by a model-dependent reduction factor. This factor in the
so-called minimal composite Higgs models [14, 16] is described by one
parameter $\xi$, and is given by $\sqrt{1-\xi}$, where $\xi=v^{2}/f^{2}$, $v$
being the scalar vacuum expectation value characterizing the electroweak
breaking scale, and $f$ being the scale of compositeness. For $f$ of order
TeV, $v/f\ll 1$, and $a_{Z}=\sqrt{1-\xi}\approx 1-\frac{1}{2}\xi$. $b_{Z}$ in
these models is expected to be small, of order $m_{Z}^{2}/f^{2}$ [15].

As mentioned earlier, we aim to constrain the above anomalous couplings with
specific combinations of polarized cross sections. A hermitian, traceless,
$3\times 3$ density matrix gives a complete description of the polarization
states of spin one, characterized by 8 polarization parameters [17]. The
diagonal elements of the density matrix correspond to pure polarization
states. With the final-state phase space appropriately put in, these diagonal
elements would give us production cross sections with definite $Z$
polarization. We are interested in constructing combinations of these
polarized $Z$ cross sections, such that each combination would be dominantly
sensitive to one of the two couplings.

On the experimental side, polarization of weak gauge bosons has been measured
at the LHC in $W+\rm jet$ production [18, 19], $Z+\rm jet$ production [20,
21], $W$ produced in the decay of top quarks [22] and more recently in $WZ$
production [23] and same-sign $WW$ production [24]. The gauge-boson
polarizations and helicity fractions are inferred from the angular
distributions of the fermions to which the gauge bosons decay [25]. It should
be possible to determine the $Z$ polarization in the $e^{+}e^{-}\to ZH$ in the
same way. See also [26] for a recent suggestion for measurement of $W$
polarization.

For the following, we assume for simplicity of some of our expressions that
there is no CP violation, $\tilde{b}_{Z}=0$. However, there is no loss of
generality because a nonzero $\tilde{b}_{Z}$ would not contribute to the
observables we consider here. The implications on including loop-level
contributions from, say, triple-Higgs couplings can be different and
important.

The density matrix elements and the helicity amplitudes from which they are
constructed were obtained in [1]. The relevant diagonal elements of the
density matrix integrated over phase space are given by

$\sigma(\pm,\pm)=\frac{F(P_{L},\bar{P}_{L})g^{4}m^{2}_{Z}|\vec{k}_{Z}|}{96\pi\sqrt{s}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left[|a_{Z}|^{2}-2{\rm
Re}(a_{Z}b_{Z}^{*})\frac{E_{Z}\sqrt{s}}{m_{Z}^{2}}+|b_{Z}|^{2}\frac{E_{Z}^{2}s}{m_{Z}^{4}}\right],$
(3)
$\sigma(0,0)=\frac{F(P_{L},\bar{P}_{L})g^{4}E^{2}_{Z}|\vec{k}_{Z}|}{96\pi\sqrt{s}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left[|a_{Z}|^{2}-2{\rm
Re}(a_{Z}b_{Z}^{*})\frac{\sqrt{s}}{E_{Z}}+|b_{Z}|^{2}\frac{s}{E_{Z}^{2}}\right].$
(4)

Here, $E_{Z}=(s-m_{H}^{2}+m_{Z}^{2})/(2\sqrt{s})$ is the $Z$ energy in the
c.m. frame, and $|\vec{k}_{Z}|=\sqrt{E_{Z}^{2}-m_{Z}^{2}}$ is the magnitude of
the $Z$ three-momentum. $c_{V}$ and $c_{A}$ are respectively the vector and
axial-vector couplings of the $Z$ to the electron, given by
$c_{V}=\frac{1}{2}(-1+4\sin^{2}\theta_{W})$ and $c_{A}=-\frac{1}{2}$. The beam
polarization dependence enters through the combination

$F(P_{L},\bar{P}_{L})=(1-P_{L}\bar{P}_{L})(c_{V}^{2}+c_{A}^{2}-2P_{L}^{\rm
eff}c_{V}c_{A}),$ (5)

where $P_{L}$ and $\bar{P}_{L}$ are respectively the degrees of electron and
positron beam longitudinal polarization, and $P_{L}^{\rm
eff}=(P_{L}-\bar{P}_{L})/(1-P_{L}\bar{P}_{L})$. The total cross section is
then $\sigma=\sigma(+,+)+\sigma(0,0)+\sigma(-,-)$.

We now take up three combinations constructed out of the polarized cross
sections, and discuss their features and advantages.

1. The quantity $\sigma_{T}-2\sigma_{L}$, where $\sigma_{T}\equiv\sigma(+,+)+\sigma(-,-)$ and $\sigma_{L}\equiv\sigma(0,0)$ are respectively the cross sections for the production of transverse and longitudinally polarized $Z$, is independent of $b_{Z}$ to first order, and depends only quadratically on $b_{Z}$.

In [1] we showed that a decay-lepton angular asymmetry
$A_{zz}=\frac{3}{16}\;(\sigma_{T}-2\sigma_{L})/\sigma$ can be used to put a
limit on Re $b_{Z}$. However, the numerator of this asymmetry, which is
proportional to

$\Delta\sigma_{1}\equiv\sigma_{T}-2\sigma_{L}=\frac{F(P_{L},\bar{P}_{L})g^{4}|\vec{k}_{Z}|^{3}}{48\pi\sqrt{s}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left[-|a_{Z}|^{2}+\frac{s}{m_{Z}^{2}}|b_{Z}|^{2}\right],$
(6)

is actually independent of $b_{Z}$, to first order in $b_{Z}$. The limit on Re
$b_{Z}$ from the asymmetry used in [1] actually comes from the $b_{Z}$
dependence of the denominator, which is the cross section. Thus, in models
where $b_{Z}$ arises only as a small effect, possibly from loops, the quantity
$\sigma_{T}-2\sigma_{L}$ can be used to determine $a_{Z}$ (assumed non-SM)
independently of $b_{Z}$.

(a) In EFT, with $a_{Z}=1+\delta a_{Z}$, and keeping only first order in
higher-dimensional couplings,

$\Delta\sigma_{1}=\frac{F(P_{L},\bar{P}_{L})g^{4}|\vec{k}_{Z}|^{3}}{48\pi\sqrt{s}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left[-1-2{\rm}\delta
a_{Z}\right],$ (7)

and can be used to determine $\delta a_{Z}$.

(b) In 2HDM like models, there is no tree-level higher derivative $b_{Z}$-type
coupling (which is non-renormalizable). $b_{Z}$ may arise only at loop level
and is therefore small. Hence the quadratic term in $b_{Z}$ can be neglected.
So $\Delta\sigma_{1}$ can be used to determine $a_{Z}$, which is
$\sin(\alpha-\beta)$, as mentioned earlier.

(c) Again, in the case of composite models, $b_{Z}$ is expected to be small,
and hence the quadratic term can be neglected. Hence $\Delta\sigma_{1}$ can
again be used to determine $a_{Z}$, and hence the parameter $\xi$.

2. Another useful combination of polarized cross sections is $\Delta\sigma_{2}\equiv\sigma_{T}-(2m_{Z}^{2}/E_{Z}^{2})\sigma_{L}$, which evaluates to

$\Delta\sigma_{2}=\frac{F(P_{L},\bar{P}_{L})g^{4}|\vec{k}_{Z}|^{3}}{48\pi\sqrt{s}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left[-2{\rm
Re}(a_{Z}b_{Z}^{*})\frac{\sqrt{s}}{E_{Z}}+\displaystyle|b_{Z}|^{2}\frac{(E_{Z}^{2}+m_{Z}^{2})s}{E_{Z}^{2}m_{Z}^{2}}\right].$
(8)

This is independent of $|a_{Z}|^{2}$, and hence proportional to $b_{Z}$.

(a) To linear order in the EFT couplings,

$\Delta\sigma_{2}=\frac{F(P_{L},\bar{P}_{L})g^{4}|\vec{k}_{Z}|^{3}}{48\pi
E_{Z}\cos^{4}\theta_{W}(s-m_{Z}^{2})^{2}}\left(-2{\rm Re}~{}b_{Z}\right).$ (9)

$\Delta\sigma_{2}$ can thus be used to determine $b_{Z}$, thus giving
information complementary to that obtained from $\Delta\sigma_{1}$.

(b) In SM and models like 2HDM, $\Delta\sigma_{2}$ will be zero. Of course, at
the loop level, the answer will be non-zero, and this would get dominant
contribution from say triple-Higgs couplings, since tree-level contributions
are eliminated.

(c) In composite models $\Delta\sigma_{2}$ will also be zero at tree level, or
in some models, suppressed by $m_{Z}^{2}/f^{2}$ [15].

3. The longitudinal helicity fraction $F_{0}\equiv\sigma_{L}/\sigma$ is given by

$\frac{\sigma_{L}}{\sigma}=\frac{E_{Z}^{2}|a_{Z}|^{2}-2{\rm
Re}(a_{Z}b_{Z}^{*})E_{Z}\sqrt{s}+s|b_{Z}|^{2}}{(2m_{Z}^{2}+E_{Z}^{2})|a_{Z}|^{2}-6{\rm
Re}(a_{Z}b_{Z}^{*})E_{Z}\sqrt{s}+s|b_{Z}|^{2}(1+2E_{Z}^{2}/m_{Z}^{2})}.$ (10)

The value of the helicity fraction is independent of beam polarization. To
first order in $b_{Z}/a_{Z}$,

$F_{0}=\frac{E_{Z}^{2}}{2m_{Z}^{2}+E_{Z}^{2}}\left(1+\frac{{\rm
Re}(a_{Z}b_{Z}^{*})}{|a_{Z}|^{2}}\frac{4\sqrt{s}|\vec{k}_{Z}|^{2}}{E_{Z}(2m_{Z}^{2}+E_{Z}^{2})}\right).$
(11)

(a) To first order in dimension-six EFT couplings,

$F_{0}\approx\frac{E_{Z}^{2}}{2m_{Z}^{2}+E_{Z}^{2}}\left(1+{\rm
Re}(b_{Z})\frac{4\sqrt{s}|\vec{k}_{Z}|^{2}}{E_{Z}(2m_{Z}^{2}+E_{Z}^{2})}\right).$
(12)

We see that the SM contribution dominates. There is a correction from the
$b_{Z}$ coupling, but no dependence on the $\delta a_{Z}$ type of couplings. A
measurement of the helicity fraction would thus be mainly model independent,
with only a mild dependence on the dimension-six couplings entering $b_{Z}$.

(b) In models like SM and 2HDM, where $b_{Z}=0$, the helicity fraction is
independent of $a_{Z}$ and therefore a model-independent quantity,
$E_{Z}^{2}/(2m_{Z}^{2}+E_{Z}^{2})$, which approaches 1 at high energies [27].
This high-energy behaviour is as expected because the longitudinally polarized
$Z$ bosons give a dominant contribution, which can be seen in eq. (4). The
model-independence of $F_{0}$ seems interesting to verify experimentally.
Corrections from loops would be interesting to check.

(c) In composite Higgs models, since $b_{Z}$ is expected to be small, the
value of $F_{0}$ is approximately the same as in the SM, or as that in the
2HDM.

We now discuss how sensitive the measurements of these three quantities would
be in the determination of the relevant couplings.

The statistical sensitivity is determined by comparing the expected number of
new-physics events $\Delta N$ with the statistical fluctuation $\sqrt{N_{\rm
SM}}$ in the number of events in the SM. Thus, the 1-$\sigma$ limit on a
coupling can be obtained by equating these two quantities. To linear order,

$\Delta N_{i}={\cal L}(\Delta\sigma_{i}-\Delta\sigma_{i}^{\rm SM})={\cal
L}\frac{\partial\Delta\sigma_{i}}{\partial c_{j}}c_{j}=\sqrt{{\cal
L}\sigma_{i}^{\rm SM}},$ (13)

where $c_{j}$ ($j=1,2$) is the anomalous coupling ($\delta a_{Z}$ or $b_{Z}$)
and ${\cal L}$ is the integrated luminosity. Then, the 1-$\sigma$ limit on
$c_{j}$ from a measurement of $\Delta_{i}$ is given by

$c_{j}^{\rm lim}=\frac{\sqrt{\sigma_{i}^{\rm SM}}}{\sqrt{\cal
L}|\partial\Delta\sigma_{i}/\partial c_{j}|}.$ (14)

At the LHC, a measurement of $W$ polarization in top decay with c.m. energy of
8 TeV and an integrated luminosity of 20 fb-1, an accuracy of about 2% for
$F_{0}$ and about 3% for the left-handed helicity fraction $F_{L}$ [22]. This
implies an overall accuracy of 2-3% for our observable $F_{0}$, though it is
likely to be better for $Z$ polarization at an $e^{+}e^{-}$ collider. However,
for the measurement of our first two observables, which are combinations of
polarized cross sections and not ratios, the efficiency would be expected to
be lower. The efficiency would also depend on the actual final states into
which $Z$ and $H$ decay. The corresponding backgrounds and the kinematic cuts
needed to suppress the backgrounds would determine the efficiency. A
discussion of backgrounds for ealier proposals for linear colliders may be
found in [28] and references therein, and for more recently planned colliders
can be found in [29] for the CEPC and [30] for the FCC-ee and the CEPC. Here,
we have not incorporated effects of cuts and assumed an ideal efficiency. We
reiterate that our main motive is to show the usefulness of certain
polarization combinations which have not been considered before.

Figure 1: The SM cross section $\sigma_{\rm SM}$ as a function of $\sqrt{s}$
for different electron and positron polarization combinations.

To get an idea of the event rates, we have first plotted in Fig. 1 the SM
cross section as a function of the c.m. energy for various polarization
combinations.

We estimate the 1-$\sigma$ limits possible on $\delta a_{Z}$ using
$\Delta\sigma_{1}$ and on $b_{Z}$ using $\Delta\sigma_{2}$ for various
configurations of $e^{+}e^{-}$ colliders, for both polarized and unpolarized
beams. We observe that there is improvement in the limits in the presence of
negative electron polarization and/or positive positron polarization as
compared to the unpolarized case. Furthermore, inclusion of (negative)
positron polarization results in an improvement as compared to the case when
only the electron beam is polarized.

Figure 2:  1-$\sigma$ limits on $\delta a_{Z}$ (left panel) and on $b_{Z}$
(right panel) which may be obtained respectively from $\Delta\sigma_{1}$ and
$\Delta\sigma_{2}$ for ${\cal L}=2\,{\rm ab}^{-1}$ plotted as functions of
$\sqrt{s}$ for different electron and positron polarization combinations.

We have plotted in Fig. 2, for the case of ${\cal L}=2\,{\rm ab}^{-1}$, the
limits on $\delta a_{Z}$ and $b_{Z}$ as functions of the c.m. energy for
various combinations of beam polarization. It is clear from these plots that
the optimal limits in either case are obtained for a c.m. energy in the region
of 350 GeV. We also see that longitudinal beam polarization we have chosen
helps to improve the sensitivity to a great extent. In addition to the results
shown in the plots we find that further increase in positron polarization, if
feasible, would improve the sensitivity further. As for example, for
$\sqrt{s}=500$ GeV, in going from $P_{e^{-}}=-0.8$ and $P_{e^{+}}=+0.3$ to
$P_{e^{-}}=-0.8$ and $P_{e^{+}}=+0.6$, the limit on $\delta a_{Z}$ improves
from $9.1\times 10^{-4}$ to $8.2\times 10^{-4}$ and on $b_{Z}$ from $4.4\times
10^{-4}$ to $4.0\times 10^{-4}$.

We may compare the limits we consider possible here using polarized cross
section combinations with the limits estimated in [1] with the use of
asymmetries. The limits on $b_{Z}$ we find here are better by an order of
magnitude, even though in [1] $a_{Z}$ was assumed to be exactly equal to 1,
whereas here we only make a linear approximation. It should of course be borne
in mind that in either case, in practice, the limits would be somewhat lower
and dependent on a more detailed analysis including the effects of cuts and
detector efficiency, which we have not attempted here.

To conclude, we have studied how $Z$ polarization in $e^{+}e^{-}\to HZ$ can be
used to measure general Lorentz-invariant $HZZ$ interaction described by three
couplings, $a_{Z}$, $b_{Z}$ and $\tilde{b}_{Z}$, of which the first two are CP
conserving. Instead of the full $Z$ spin density matrix which would require
complicated distributions or asymmetries of the $Z$ decay products, we suggest
looking at only the diagonal elements of this matrix. These correspond to
cross sections for the production of $Z$ with definite polarizations, which
are more easily accessible. We find that certain combinations of $Z$
production cross sections with definite $Z$ polarization can help to enhance
or isolate the effect of one of the two CP-even couplings.

The specific combinations of longitudinally polarized ($\sigma_{L}$) and
transversely polarized ($\sigma_{T}$) cross sections we consider are
$\Delta\sigma_{1}\equiv\sigma_{T}-2\sigma_{L}$,
$\Delta\sigma_{2}\equiv\sigma_{T}-(2m_{Z}^{2}/E_{Z}^{2})\sigma_{L}$, and the
longitudinal helicity fraction
$F_{0}\equiv\sigma_{L}/(\sigma_{L}+\sigma_{T})$. We show that
$\Delta\sigma_{1}$ is independent of $b_{Z}$ to first order, and can therefore
be used to determine $a_{Z}$. On the other hand, $\Delta\sigma_{2}$ is
proportional to $b_{Z}$ and has no leading $|a_{Z}|^{2}$ dependence, so that
it can be used to determine $b_{Z}$. To leading order in $b_{Z}/a_{Z}$,
$F_{0}$ takes its SM value, viz., $E_{Z}^{2}/(2m_{Z}^{2}+E_{Z}^{2})$,
regardless of the value of $a_{Z}$.

We consider in a relatively model-independent fashion three scenarios, viz.,
EFT, 2HDM and composite Higgs, and analyze what role the measurement of the
three cross section combinations $\Delta\sigma_{1}$, $\Delta\sigma_{2}$ and
$F_{0}$ can play in these scenarios. We also estimate the 1-$\sigma$ limits on
the couplings that could be placed using $\Delta\sigma_{1}$ and
$\Delta\sigma_{2}$.

We have also considered the effect of longitudinal beam polarization. While
the limits on the couplings from $\Delta\sigma_{1}$ and $\Delta\sigma_{2}$ are
improved by a choice of electron and positron beam polarizations $-0.8$ and
$+0.3$, $F_{0}$ is independent of beam polarization.

We find that limits of the order of a few times $10^{-4}$ to $10^{-3}$ can be
obtained for ranges of $\sqrt{s}$ from 250 to 500 GeV and ${\cal L}$ values
from 2 to 30 ab-1 envisaged at various electron-positron colliders.

The cross section combinations we have described here would be interesting to
measure at future colliders. A more detailed analysis using event generators
and detector simulation would be necessary to check how well the estimates
made here survive after realistic kinematic cuts and detection efficiencies.
Further, it would be interesting to investigate if such cross section
combinations yield useful results in a $pp$ environment, like at the LHC.

Acknowledgement The work of SDR was supported by the Senior Scientist
programme of the Indian National Science Academy, New Delhi.

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