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# On the implicit constant fields and key polynomials for valuation algebraic
extensions

Arpan Dutta  Department of Mathematics, IISER Mohali, Knowledge City, Sector
81, Manauli PO, SAS Nagar, Punjab, India, 140306<EMAIL_ADDRESS>
###### Abstract.

This article is a natural continuation of our previous works [References] and
[References]. In this article, we employ similar ideas as in [References] to
provide an estimate of $IC(K(X)|K,v)$ when $(K(X)|K,v)$ is a valuation
algebraic extension. Our central result is an analogue of [References, Theorem
1.3]. We further provide a natural construction of a complete sequence of key
polynomials for $v$ over $K$ in the setting of valuation algebraic extensions.

###### Key words and phrases:

Valuation, minimal pairs, key polynomials, valuation algebraic extensions,
ramification theory, implicit constant fields, extensions of valuation to
rational function fields

###### 2010 Mathematics Subject Classification:

12J20, 13A18, 12J25

This work was supported by the Post-Doctoral Fellowship of the National Board
of Higher Mathematics, India.

##  1\. Introduction

Throughout this article, we assume that $(\overline{K}(X)|K,v)$ is an
extension of valued fields, where $\overline{K}$ is a fixed algebraic closure
of $K$ and $X$ is an indeterminate. The extension $(K(X)|K,v)$ satisfies the
celebrated Abhyankar inequality:

(1) $\mbox{\rm rat rk}\,vK(X)/vK+\mbox{\rm trdeg}\,[K(X)v:Kv]\leq 1,$

where $vK$ and $Kv$ denote respectively the value group and residue field of
$(K,v)$ and $\mbox{\rm rat rk}\,vK(X)/vK$ is the ${\mathbb{Q}}$-dimension of
the divisible hull ${\mathbb{Q}}\otimes_{{\mathbb{Z}}}(vK(X)/vK)$. The above
inequality is a consequence of [References, Chapter VI, §10.3, Theorem 1]. The
extension $(K(X)|K,v)$ is said to be valuation algebraic if we have strict
inequality in (1). Otherwise, it is said to be valuation transcendental.
Moreover, the extension is said to be value transcendental if we have
$\mbox{\rm rat rk}\,vK(X)/vK=1$ and residue transcendental if $\mbox{\rm
trdeg}\,[K(X)v:Kv]=1$.

Several different objects have been introduced to study extensions of
valuations to rational function fields. Minimal pairs of definition (cf.
Section 3) are central to the investigation of valuation transcendental
extensions [cf. References, References, References, References]. The notion of
implicit constant fields (cf. Section 6) was introduced by Kuhlmann in
[References] to construct valuations with prescribed value groups and residue
fields. The connections between these two notions were studied thoroughly in
[References] and [References]. In particular, it has been observed in
[References, Theorem 1.3] that given a minimal pair of definition $(a,\gamma)$
for a valuation transcendental extension $(K(X)|K,v)$ and an extension of $v$
to $\overline{K(X)}$, we have

$(K(a)\cap K^{r})^{h}\subseteq IC(K(X)|K,v)\subseteq K(a)^{h},$

where $K^{r}$ denotes the absolute ramification field of $(K,v)$ and $K^{h}$
denotes the henselization of $K$ (cf. Section 2).

Valuation algebraic extensions have been explored in detail in [References].
Following an idea of MacLane [References], the authors show that any valuation
algebraic extension can be expressed as a limit of residue transcendental
extensions. We employ similar observations to estimate the implicit constant
fields of valuation algebraic extensions. When $(K(X)|K,v)$ is valuation
algebraic, we observe in Section 5 that there exist sequences
$\\{\gamma_{\nu}\\}_{\nu<\lambda}\subseteq v(X-\overline{K})$ and
$\\{a_{\nu}\\}_{\nu<\lambda}\subseteq(\overline{K},v)$, where
$v(X-\overline{K}):=\\{v(X-a)\mid a\in\overline{K}\\}$ and $\lambda$ is some
limit ordinal, satisfying the following conditions for all $\nu<\lambda$:

  * •

$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is a cofinal well-ordered subset of
$v(X-\overline{K})$,

  * •

$v(X-a_{\nu})=\gamma_{\nu}$,

  * •

$(a_{\nu},\gamma_{\nu})$ is a minimal pair of definition for
$v_{a_{\nu},\gamma_{\nu}}$ over $K$,

  * •

$v(X-a_{\nu})\geq v(X-\sigma a_{\nu})$ for all $K$-conjugates $\sigma a_{\nu}$
of $a_{\nu}$.

For each $\nu<\lambda$, we then choose $b_{\nu}\in\overline{K}$ such that
$K(b_{\nu})=K(a_{\nu})\cap K^{r}$. After fixing an extension of $v$ to
$\overline{K(X)}$, we obtain in Theorem 6.2 that

$K(b_{\nu}\mid\nu<\lambda)^{h}\subseteq IC(K(X)|K,v)\subseteq
K(a_{\nu}\mid\nu<\lambda)^{h}.$

This observation is an analogue of Theorem 1.3 of [References], which in turn
is a modification of Theorem 1.1 of [References].

Key polynomials (cf. Section 7) are fundamental objects which play a pivotal
role in the problem of describing extensions of valuations to rational
function fields. They were introduced by MacLane [References] and later
generalized by Vaquié [References]. An alternative form of key polynomials,
also known as abstract key polynomials, was given by Spivakovsky and Novacoski
in [References]. Throughout this article, we will work with abstract key
polynomials and we will simply call them key polynomials for brevity. In
Theorem 7.4 we obtain that when $(K(X)|K,v)$ is a valuation algebraic
extension, then

$\\{Q_{\nu}\\}_{\nu<\lambda}$ forms a complete sequence of key polynomials for
$v$ over $K$,

where $Q_{\nu}$ is the minimal polynomial of $a_{\nu}$ over $K$. A special
case of this result is Theorem 1.2 of [References].

##  2\. Ramification Theory

A valued field $(K,v)$ is said to be henselian if $v$ admits a unique
extension to $\overline{K}$. Every valued field has a minimal separable-
algebraic extension which is henselian. This extension is unique up to
valuation preserving isomorphisms over $K$. We will call this extension the
henselization of $(K,v)$ and denote it by $K^{h}$. Henselization is an
immediate extension, that is, $vK^{h}=vK$ and $K^{h}v=Kv$. An algebraic
extension of henselian valued fields is again henselian. For any algebraic
extension $L$ of $K$, we have that $L^{h}=L.K^{h}$.

We define the distinguished group:

$G^{r}:=\\{\sigma\in\mathrm{Gal}\,(\overline{K}|K)\mid v(\sigma a-a)>va\text{
for all }a\in K^{\mathrm{sep}}\setminus\\{0\\}\\},$

where $K^{\mathrm{sep}}$ denotes the separable-algebraic closure of $K$. The
corresponding fixed field in $K^{\mathrm{sep}}$ will be denoted by $K^{r}$ and
we will call it the absolute ramification field of $(K,v)$. It is well-known
that $K^{h}\subseteq K^{r}$ and hence $K^{r}$ is a henselian field. Further,
the extension $(K^{r}|K^{h},v)$ is defectless, that is, for any tower of field
extensions $K^{h}\subseteq L\subseteq L^{\prime}\subseteq K^{r}$ where
$L^{\prime}|L$ is finite, we have

$[L^{\prime}:L]=(vL^{\prime}:vL)[L^{\prime}v:Lv].$

######  Lemma 2.1.

Assume that $(K(y)|K,v)$ is an immediate extension. Then $v(y-K)$ does not
admit a maximum.

###### Proof.

Suppose that $\alpha:=v(y-a)=\max v(y-K)$ for some $a\in K$. The fact that
$(K(y)|K,v)$ is an immediate extension implies that $v(y-K)\subset vK$. Take
$b\in K$ such that $vb=\alpha$. Then $v\frac{y-a}{b}=0$ and $\frac{y-a}{b}v\in
K(y)v=Kv$. Consequently, there exists $c\in K$ such that
$v(\frac{y-a}{b}-c)>0$, that is, $v(y-a-bc)>vb=\alpha$. The fact that $a+bc\in
K$ then contradicts the maximality of $\alpha$. ∎

##  3\. Minimal Pairs of Definition

Take any $a\in\overline{K}$ and $\gamma$ in some ordered abelian group
containing $v\overline{K}$. Any polynomial $f(X)\in\overline{K}[X]$ has a
unique expression of the form $f(X)=\sum_{i=0}^{n}c_{i}(X-a)^{i}$ where
$c_{i}\in\overline{K}$. Consider the map $v_{a,\gamma}:\overline{K}[X]\to
v\overline{K}+{\mathbb{Z}}\gamma$ given by

$v_{a,\gamma}f:=\min\\{vc_{i}+i\gamma\\},$

and extend $v_{a,\gamma}$ canonically to $\overline{K}(X)$. Then
$v_{a,\gamma}$ is a valuation transcendental extension of $v$ from
$\overline{K}$ to $\overline{K}(X)$ [References, Theorem 3.11]. Further,
$(\overline{K}(X)|\overline{K},v_{a,\gamma})$ is value transcendental whenever
$\gamma\notin v\overline{K}$. Otherwise, the extension is residue
transcendental.

It has further been observed in [References, Theorem 3.11] that $(K(X)|K,v)$
is value (residue) transcendental if and only if
$(\overline{K}(X)|\overline{K},v)$ is also value (residue) transcendental,
which in turn holds if and only if $v=v_{a,\gamma}$ for some
$a\in\overline{K}$ and $\gamma\in v\overline{K}(X)$. Such a pair
$(a,\gamma)\in\overline{K}\times v\overline{K}(X)$ is said to be a pair of
definition for $v$ over $K$. A valuation transcendental extension can have
multiple pairs of definition. It has been observed in [References, Proposition
3] that $(a^{\prime},\gamma)$ is also a pair of definition for $v$ over $K$ if
and only if $v(a-a^{\prime})\geq\gamma$.

######  Definition 3.1.

Assume that $(K(X)|K,v)$ is a valuation transcendental extension. A pair
$(a,\gamma)\in\overline{K}\times v\overline{K}(X)$ is said to be a minimal
pair of definition for $v$ over $K$ if it satisfies the following conditions:

(MP1) $(a,\gamma)$ is a pair of definition for $v$ over $K$,

(MP2) $v(a-a^{\prime})\geq\gamma\Longrightarrow[K(a^{\prime}):K]\geq[K(a):K]$
for all $a^{\prime}\in\overline{K}$.

In other words, $a$ is of minimal degree over $K$ with the property that
$(a,\gamma)$ is a pair of definition for $v$ over $K$. It follows from
[References, Theorem 3.11] that a valuation transcendental extension always
admits a pair of definition. The well-ordering principle then implies that a
valuation transcendental extension always admits a minimal pair of definition.

######  Lemma 3.2.

Assume that $(K(X)|K,v)$ is a valuation transcendental extension. Take a
minimal pair of definition $(a,\gamma)$ and a pair of definition
$(a^{\prime},\gamma)$ for $v$ over $K$. Then,

$vK(a)\subseteq vK(a^{\prime})\text{ and }K(a)v\subseteq K(a^{\prime})v.$

###### Proof.

We first assume that $(K(X)|K,v)$ is a value transcendental extension. It
follows from [References, Remark 3.3] that

$vK(X)=vK(a)\oplus{\mathbb{Z}}vQ,$

where $Q(X)\in K[X]$ is the minimal polynomial of $a$ over $K$. Moreover,
observe that $(a^{\prime},\gamma)$ is a minimal pair of definition for $v$
over $K(a^{\prime})$. As a consequence,

$vK(a^{\prime},X)=vK(a^{\prime})\oplus{\mathbb{Z}}\gamma.$

Take any $\alpha\in vK(X)$. The fact that $vK(X)\subseteq vK(a^{\prime},X)$
then implies that $\alpha=\alpha^{\prime}+n\gamma$ for some
$\alpha^{\prime}\in vK(a^{\prime})$ and $n\in{\mathbb{Z}}$. Since $\gamma$ is
not a torsion element modulo $vK$, it follows that $n=0$ and consequently,
$\alpha=\alpha^{\prime}$. We have thus obtained that

$vK(a)\subseteq vK(a^{\prime}).$

It further follows from [References, Remark 3.3] that

$K(a)v=K(X)v\subseteq K(a^{\prime},X)v=K(a^{\prime})v.$

We now assume that $(K(X)|K,v)$ is a residue transcendental extension. Then
$\gamma\in v\overline{K}$. Take the ordered abelian group
$v\overline{K}\oplus{\mathbb{Z}}$ endowed with the lexicographic order. Embed
$v\overline{K}$ into $(v\overline{K}\oplus{\mathbb{Z}})_{\text{lex}}$ by
setting

$\alpha\mapsto(\alpha,0)\text{ for all }\alpha\in v\overline{K}.$

Set

$\Gamma:=(\gamma,-1).$

Take the extension $w:=v_{a,\Gamma}$ of $v$ from $\overline{K}$ to
$\overline{K}(X)$. Observe that,

$v(a-b)\geq\Gamma\Longleftrightarrow v(a-b)\geq\gamma\text{ for all
}b\in\overline{K}.$

It follows that $(a,\Gamma)$ is a minimal pair of definition and
$(a^{\prime},\Gamma)$ is a pair of definition for $w$ over $K$. The fact that
$\Gamma\notin v\overline{K}$ implies that $(K(X)|K,w)$ is a value
transcendental extension. It now follows from our preceding discussions that

$vK(a)\subseteq vK(a^{\prime})\text{ and }K(a)v\subseteq K(a^{\prime})v.$

∎

##  4\. Pseudo-Cauchy Sequences

A well-ordered set $\\{z_{\nu}\\}_{\nu<\lambda}$ in $(K,v)$, where $\lambda$
is some limit ordinal, is said to form a pseudo-Cauchy sequence (pCs) if it
satisfies the following condition:

$v(z_{\nu_{1}}-z_{\nu_{2}})<v(z_{\nu_{2}}-z_{\nu_{3}})\text{ for all
}\nu_{1}<\nu_{2}<\nu_{3}<\lambda.$

Given a polynomial $f(X)\in K[X]$ and a pCs $\\{z_{\nu}\\}_{\nu<\lambda}$ in
$(K,v)$, the sequence $\\{vf(z_{\nu})\\}_{\nu<\lambda}$ is either ultimately
constant, or ultimately monotonically increasing [References, Lemma 5]. We say
that $\\{z_{\nu}\\}_{\nu<\lambda}$ is a pCs of transcendental type if
$\\{vf(z_{\nu})\\}_{\nu<\lambda}$ is ultimately constant for all $f(X)\in
K[X]$. Otherwise, we say that $\\{z_{\nu}\\}_{\nu<\lambda}$ is a pCs of
algebraic type.

Observe that $v(z_{\nu}-z_{\nu+1})=v(z_{\nu}-z_{\mu})$ for all
$\nu<\mu<\lambda$. Set $\gamma_{\nu}:=v(z_{\nu}-z_{\nu+1})$. An element $\ell$
in some valued field extension $(L,v)$ of $(K,v)$ is said to be a limit of
$\\{z_{\nu}\\}_{\nu<\lambda}$ if

$v(\ell-z_{\nu})=\gamma_{\nu}\text{ for all }\nu<\lambda.$

If $\ell^{\prime}\in L$ is another limit of $\\{z_{\nu}\\}_{\nu<\lambda}$,
then it follows from the triangle inequality that
$v(\ell-\ell^{\prime})\geq\gamma_{\nu}$ for all $\nu<\lambda$. The fact that
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is a monotonically increasing sequence
without a final element then implies the following:

######  Lemma 4.1.

Assume that $\\{z_{\nu}\\}_{\nu<\lambda}$ is a pCs in a valued field $(K,v)$.
Take an extension $(L|K,v)$ and assume that $\ell\in L$ is a limit of
$\\{z_{\nu}\\}_{\nu<\lambda}$. Then $\ell^{\prime}\in L$ is also a limit of
$\\{z_{\nu}\\}_{\nu<\lambda}$ if and only if

$v(\ell-\ell^{\prime})>\gamma_{\nu}\text{ for all }\nu<\lambda.$

######  Definition 4.2.

An extension $(K(X)|K,v)$ is said to be a pure extension if it satisfies one
of the following conditions:

(PE1) $v(X-a)$ is not a torsion element modulo $vK$ for some $a\in K$,

(PE2) $vb(X-a)=0$ and $b(X-a)v$ is transcendental over $Kv$ for some $a,b\in
K$,

(PE3) $X$ is a limit of some pCs of transcendental type in $(K,v)$.

If $(K(X)|K,v)$ is a pure extension satisfying (PE3), then it follows from
[References, Theorem 2] that $(K(X)|K,v)$ is immediate.

##  5\. Valuation Algebraic Extensions

Assume that $(K(X)|K,v)$ is a valuation algebraic extension. Then
$(\overline{K}(X)|\overline{K},v)$ is an immediate extension [References,
Lemma 3.3]. It follows from Lemma 2.1 that $v(X-\overline{K})$ does not admit
a maximum. Take a cofinal well-ordered subset
$\\{\delta_{\nu}\\}_{\nu<\lambda}$ of $v(X-\overline{K})$, where $\lambda$ is
some limit ordinal. Further, for each $\nu<\lambda$, take
$d_{\nu}\in\overline{K}$ such that $v(X-d_{\nu})=\delta_{\nu}$.

Take a minimal pair of definition $(a,\delta_{1})$ of $v_{d_{1},\delta_{1}}$
over $K$. Then $v(a-d_{1})\geq\delta_{1}$ and hence $v(X-a)\geq\delta_{1}$.
Set $\gamma:=v(X-a)$. By definition, $(a,\gamma)$ is a pair of definition for
$v_{a,\gamma}$ over $K$. Take another pair of definition $(a^{\prime},\gamma)$
for $v_{a,\gamma}$ over $K$. Then $v(a-a^{\prime})\geq\gamma\geq\delta_{1}$
and hence $(a^{\prime},\delta_{1})$ is a pair of definition for
$v_{d_{1},\delta_{1}}$ over $K$. The minimality of $(a,\delta_{1})$ then
implies that $[K(a):K]\leq[K(a^{\prime}):K]$. It follows that $(a,\gamma)$ is
a minimal pair of definition for $v_{a,\gamma}$ over $K$. If $a\notin K$,
assume that $\tau a\neq a$ is a conjugate of $a$ over $K$ such that $v(X-\tau
a)\geq v(X-a)$. It follows that $v(a-\tau a)\geq v(X-a)\geq\delta_{1}$ and
hence $(\tau a,\delta_{1})$ is also a minimal pair of definition for
$v_{d_{1},\delta_{1}}$ over $K$. Replacing $a$ by $\tau a$ and employing the
same arguments, we can thus further assume that $v(X-a)\geq v(X-\sigma a)$ for
all $K$-conjugates $\sigma a$ of $a$. Set

$a_{1}:=a\text{ and }\gamma_{1}:=\gamma.$

Assume that we have obtained some $(a_{\nu},\gamma_{\nu})$. The fact that
$\\{\delta_{\nu}\\}_{\nu<\lambda}$ is a well-ordered cofinal subset of
$v(X-\overline{K})$ implies that there exists some minimal $\mu<\lambda$ such
that $\gamma_{\nu}<\delta_{\mu}$. We can then construct a pair
$(a^{\prime},\gamma^{\prime})$ such that
$v(X-a^{\prime})=\gamma^{\prime}\geq\delta_{\mu}$,
$(a^{\prime},\gamma^{\prime})$ is a minimal pair of definition for
$v_{a^{\prime},\gamma^{\prime}}$ over $K$ and $v(X-a^{\prime})\geq v(X-\sigma
a^{\prime})$ for all $K$-conjugates $\sigma a^{\prime}$ of $a^{\prime}$. We
set

$a_{\nu+1}:=a^{\prime}\text{ and }\gamma_{\nu+1}:=\gamma^{\prime}.$

Observe that $\delta_{\nu}\leq\gamma_{\nu}$ for all $\nu<\lambda$.
Consequently, the fact that $\\{\delta_{\nu}\\}_{\nu<\lambda}$ is a cofinal
well-ordered subset of $v(X-\overline{K})$ implies that
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is a cofinal well-ordered subset of
$v(X-\overline{K})$. We have thus constructed sequences
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ and $\\{a_{\nu}\\}_{\nu<\lambda}$
satisfying the following conditions for all $\nu<\lambda$:

  * •

$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is a cofinal well-ordered subset of
$v(X-\overline{K})$,

  * •

$v(X-a_{\nu})=\gamma_{\nu}$,

  * •

$(a_{\nu},\gamma_{\nu})$ is a minimal pair of definition for
$v_{a_{\nu},\gamma_{\nu}}$ over $K$,

  * •

$v(X-a_{\nu})\geq v(X-\sigma a_{\nu})$ for all $K$-conjugates $\sigma a_{\nu}$
of $a_{\nu}$.

We fix this choice of $\\{\gamma_{\nu}\\}_{\nu<\lambda}$ and
$\\{a_{\nu}\\}_{\nu<\lambda}$ for the rest of this article. From [References,
Theorem 2.3], we obtain that

$vK(a_{\nu})\subseteq vK(a_{\mu})\text{ and }K(a_{\nu})v\subseteq
K(a_{\mu})v\text{ for all }\nu<\mu<\lambda.$

Further, it follows from [References, Theorem 5.1] that

$vK(X)=\bigcup\limits_{\nu<\lambda}vK(a_{\nu})\text{ and
}K(X)v=\bigcup\limits_{\nu<\lambda}K(a_{\nu})v.$

By definition, $\\{a_{\nu}\\}_{\nu<\lambda}$ is a pCs in
$(K(a_{\nu}\mid\nu<\lambda),v)$ with $X$ as a limit. Suppose that
$\\{a_{\nu}\\}_{\nu<\lambda}$ is a pCs of algebraic type. In light of
[References, Theorem 3], there exists some $\ell\in\overline{K}$ such that
$\ell$ is also a limit of $\\{a_{\nu}\\}_{\nu<\lambda}$. It then follows from
Lemma 4.1 that

$v(X-\ell)>\gamma_{\nu}\text{ for all }\nu<\lambda.$

However, this contradicts the fact that $\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is
a cofinal subset of $v(X-\overline{K})$. As a consequence,
$\\{a_{\nu}\\}_{\nu<\lambda}$ is a pCs of transcendental type in
$(K(a_{\nu}\mid\nu<\lambda),v)$.

##  6\. Implicit Constant Fields

Fix an extension of $v$ to $\overline{K(X)}$. The implicit constant field of
the extension $(K(X)|K,v)$ is defined as

$IC(K(X)|K,v):=\overline{K}\cap K(X)^{h}.$

Henselization being a separable-algebraic extension implies that
$IC(K(X)|K,v)$ is a separable-algebraic extension of $K$. Moreover, the fact
that $K^{h}\subseteq K(X)^{h}$ implies that $K^{h}\subseteq IC(K(X)|K,v)$.
Consequently, $(IC(K(X)|K,v),v)$ is a henselian valued field.

######  Lemma 6.1.

Assume that $(K(X)|K,v)$ is a valuation transcendental extension. Take a
minimal pair of definition $(a,\gamma)$ and a pair of definition
$(a^{\prime},\gamma)$ for $v$ over $K$. Take $b\in\overline{K}$ such that
$K(b)=K(a)\cap K^{r}$. Then

$vK(b,a^{\prime})=vK(a^{\prime})\text{ and }K(b,a^{\prime})v=K(a^{\prime})v.$

###### Proof.

Fix an extension of $v$ to $\overline{K(X)}$. We have observed in [References,
Theorem 1.3] that $K(b)^{h}\subseteq IC(K(X)|K,v)$. Observe that
$IC(K(X)|K,v)\subseteq IC(K(a^{\prime},X)|K(a^{\prime}),v)$ by definition. It
has been observed in [References, Lemma 5.1] that
$IC(K(a^{\prime},X)|K(a^{\prime}),v)=K(a^{\prime})^{h}$. It then follows that
$K(b)^{h}\subseteq K(a^{\prime})^{h}$. In particular, $b\in K(a^{\prime})^{h}$
and consequently,

$K(b,a^{\prime})^{h}=K(a^{\prime})^{h}.$

The fact that the henselization is an immediate extension now yields the
lemma. ∎

We are now ready to provide an estimate of $IC(K(X)|K,v)$ when the extension
$(K(X)|K,v)$ is valuation algebraic.

######  Theorem 6.2.

Assume that $(K(X)|K,v)$ is a valuation algebraic extension. Fix an extension
of $v$ to $\overline{K(X)}$. Take $\\{\gamma_{\nu}\\}_{\nu<\lambda}$ and
$\\{a_{\nu}\\}_{\nu<\lambda}$ as in Section 5. For each $\nu<\lambda$, take
$b_{\nu}\in\overline{K}$ such that

$K(b_{\nu})=K(a_{\nu})\cap K^{r}.$

Then,

(2) $K(b_{\nu}\mid\nu<\lambda)^{h}\subseteq IC(K(X)|K,v)\subseteq
K(a_{\nu}\mid\nu<\lambda)^{h}.$

###### Proof.

We have observed in Section 5 that $\\{a_{\nu}\\}_{\nu<\lambda}$ is a pCs of
transcendental type in $(L,v)$ with limit $X$, where
$L:=K(a_{\nu}\mid\nu<\lambda)$. Thus $(L(X)|L,v)$ is a pure extension by
definition. It follows from [References, Lemma 3.7] that $IC(L(X)|L,v)=L^{h}$.
Consequently,

$IC(K(X)|K,v)\subseteq IC(L(X)|L,v)=L^{h}=K(a_{\nu}\mid\nu<\lambda)^{h}.$

Take some $\nu_{0}<\lambda$. Observe that $K(b_{\nu_{0}})=K(a_{\nu_{0}})\cap
K^{r}$ and hence $vK(b_{\nu_{0}})\subseteq vK(a_{\nu_{0}})$. It follows from
[References, Theorem 2.3] and [References, Theorem 5.1] that
$vK(X)=\bigcup\limits_{\nu<\lambda}vK(a_{\nu})$. Consequently,

$vK(b_{\nu_{0}})\subseteq vK(a_{\nu_{0}})\subseteq vK(X).$

We now obtain from [References, Theorem 3(1)] that

(3) $vK(b_{\nu_{0}},X)=vK(X).$

Following the same arguments as in Section 5, we construct sequences
$\\{\beta_{\nu}\\}_{\nu<\lambda}$ and $\\{c_{\nu}\\}_{\nu<\lambda}$ satisfying
the following conditions for all $\nu<\lambda$:

  * •

$\gamma_{\nu}\leq\beta_{\nu}=v(X-c_{\nu})$,

  * •

$(c_{\nu},\beta_{\nu})$ is a minimal pair of definition for
$v_{c_{\nu},\beta_{\nu}}$ over $K(b_{\nu_{0}})$.

It follows from [References, Theorem 2.3] and [References, Theorem 5.1] that
$K(b_{\nu_{0}},c_{\nu})v\subseteq K(b_{\nu_{0}},c_{\mu})v$ for all
$\nu<\mu<\lambda$. Further,

$K(b_{\nu_{0}},X)v=\bigcup\limits_{\nu<\lambda}K(b_{\nu_{0}},c_{\nu})v.$

As a consequence,

$K(b_{\nu_{0}},X)v=\bigcup\limits_{\nu_{0}<\nu<\lambda}K(b_{\nu_{0}},c_{\nu})v.$

Take some $\nu>\nu_{0}$. The fact that $\\{\gamma_{\mu}\\}_{\mu<\lambda}$ is
cofinal in $v(X-\overline{K})$ implies that there exists some $\mu<\lambda$
such that

$\beta_{\nu}<\gamma_{\mu}.$

Thus $v(X-c_{\nu})<v(X-a_{\mu})$. It follows from the triangle inequality that
$v(a_{\mu}-c_{\nu})=\beta_{\nu}$ and hence $(a_{\mu},\beta_{\nu})$ is a pair
of definition for $v_{c_{\nu},\beta_{\nu}}$ over $K(b_{\nu_{0}})$. Applying
Lemma 3.2 to the extension
$(K(b_{\nu_{0}},X)|K(b_{\nu_{0}}),v_{c_{\nu},\beta_{\nu}})$, we obtain that

$K(b_{\nu_{0}},c_{\nu})v\subseteq K(b_{\nu_{0}},a_{\mu})v.$

Observe that $\gamma_{\mu}>\beta_{\nu}\geq\gamma_{\nu}$ and hence $\mu>\nu$.
Then the choice $\nu>\nu_{0}$ implies that $\mu>\nu_{0}$. As a consequence,
$v(a_{\mu}-a_{\nu_{0}})=\gamma_{\nu_{0}}$. It follows that
$(a_{\mu},\gamma_{\nu_{0}})$ is a pair of definition for
$v_{a_{\nu_{0}},\gamma_{\nu_{0}}}$ over $K$. Applying Lemma 6.1 to the
extension $(K(X)|K,v_{a_{\nu_{0}},\gamma_{\nu_{0}}})$, we obtain that

$K(b_{\nu_{0}},a_{\mu})v=K(a_{\mu})v.$

We have thus observed that for each $\nu>\nu_{0}$, there exists some $\mu>\nu$
such that

$K(b_{\nu_{0}},c_{\nu})v\subseteq K(a_{\mu})v.$

As a consequence,

$\bigcup\limits_{\nu_{0}<\nu<\lambda}K(b_{\nu_{0}},c_{\nu})v\subseteq\bigcup\limits_{\nu_{0}<\nu<\lambda}K(a_{\nu})v.$

It follows from [References, Theorem 2.3] and [References, Theorem 5.1] that

$K(X)v=\bigcup\limits_{\nu<\lambda}K(a_{\nu})v=\bigcup\limits_{\nu_{0}<\nu<\lambda}K(a_{\nu})v.$

The fact that $K(X)v\subseteq K(b_{\nu_{0}},X)v$ now yields that

(4)
$K(b_{\nu_{0}},X)v=\bigcup\limits_{\nu_{0}<\nu<\lambda}K(b_{\nu_{0}},c_{\nu})v=\bigcup\limits_{\nu_{0}<\nu<\lambda}K(a_{\nu})v=K(X)v.$

It follows from (3) and (4) that $(K(b_{\nu_{0}},X)|K(X),v)$ is an immediate
extension. The fact that the henselization is an immediate extension then
implies that $(K(b_{\nu_{0}},X)^{h}|K(X)^{h},v)$ is also immediate. Now,
$K(b_{\nu_{0}})$ being contained in $K^{r}$ implies that
$K(b_{\nu_{0}},X)\subseteq K(X)^{r}$ [References, Theorem 3(1)]. Consequently,
$(K(b_{\nu_{0}},X)^{h}|K(X)^{h},v)$ is a defectless extension. The fact that
$(K(b_{\nu_{0}},X)^{h}|K(X)^{h},v)$ is immediate now implies that
$K(b_{\nu_{0}},X)^{h}=K(X)^{h}$, that is, $b_{\nu_{0}}\in K(X)^{h}$. As a
consequence,

$b_{\nu_{0}}\in\overline{K}\cap K(X)^{h}=IC(K(X)|K,v).$

Thus $b_{\nu}\in IC(K(X)|K,v)$ for all $\nu<\lambda$. Consequently,
$K(b_{\nu}\mid\nu<\lambda)\subseteq IC(K(X)|K,v)$. The fact that
$IC(K(X)|K,v)$ is a henselian valued field now implies that

$K(b_{\nu}\mid\nu<\lambda)^{h}\subseteq IC(K(X)|K,v).$

∎

A valued field $(K,v)$ is said to be a tame field if $(K,v)$ is henselian and
$\overline{K}=K^{r}$. The following corollary is now immediate:

######  Corollary 6.3.

Assume that $(K,v)$ is a tame valued field and $(K(X)|K,v)$ is a valuation
algebraic extension. Fix an extension of $v$ to $\overline{K(X)}$. Take
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ and $\\{a_{\nu}\\}_{\nu<\lambda}$ as in
Section 5. Then,

$IC(K(X)|K,v)=K(a_{\nu}\mid\nu<\lambda).$

##  7\. Key Polynomials

Take a polynomial $f(X)\in K[X]$. Following [References], we define

$\delta(f):=\max\\{v(X-a)\mid a\in\overline{K}\text{ and }f(a)=0\\}.$

A root $a$ of $f$ such that $\delta(f)=v(X-a)$ is said to be a maximal root of
$f$. A monic polynomial $Q(X)\in K[X]$ is said to be a key polynomial for $v$
over $K$ if for any $f(X)\in K[X]$, we have

$\deg f<\deg Q\Longrightarrow\delta(f)<\delta(Q).$

Take any $f(X)\in K[X]$ and a monic $Q(X)\in K[X]$. Then we have a unique
expansion

$f=f_{0}+f_{1}Q+\dotsc f_{r}Q^{r},$

where $f_{i}(X)\in K[X]$ such that $\deg f_{i}<\deg Q$. Consider the map
$v_{Q}:K[X]\longrightarrow vK(X)$ given by

$v_{Q}f:=\min\\{vf_{i}+ivQ\\},$

and extend it to $K(X)$ by defining $v_{Q}(f/g):=v_{Q}f-v_{Q}g$. A sufficient
condition for $v_{Q}$ to be a valuation on $K(X)$ is that $Q$ be a key
polynomial for $v$ over $K$ [References, Proposition 2.6], but it is not a
necessary condition [References, Proposition 2.3].

######  Lemma 7.1.

Take $a\in\overline{K}$ and $f(X)\in K[X]$. Set $\gamma:=v(X-a)$. Then,

$\displaystyle vf$ $\displaystyle>v_{a,\gamma}f\text{ if and only if
}\delta(f)>\gamma,$ $\displaystyle vf$ $\displaystyle=v_{a,\gamma}f\text{ if
and only if }\delta(f)\leq\gamma.$

###### Proof.

Write $f(X)=z\prod_{i=1}^{n}(X-z_{i})$ where $z,z_{i}\in\overline{K}$. By
definition, $v_{a,\gamma}(X-z_{i})=\min\\{\gamma,v(a-z_{i})\\}$. It then
follows from the triangle inequality that $v_{a,\gamma}(X-z_{i})\leq
v(X-z_{i})$. As a consequence,

$v_{a,\gamma}f\leq vf.$

Further, $v_{a,\gamma}f<vf$ if and only if $v_{a,\gamma}(X-z_{i})<v(X-z_{i})$
for some $i$, which holds if and only if $v(X-z_{i})>\gamma=v(a-z_{i})$. The
lemma now follows. ∎

######  Lemma 7.2.

Take a key polynomial $Q(X)$ for $v$ over $K$ and a maximal root $a$ of $Q$.
Then,

$v_{a,\delta(Q)}|_{K(X)}=v_{Q}.$

###### Proof.

By definition, $(a,\delta(Q))$ is a pair of definition for $v_{a,\delta(Q)}$
over $K$. Take another pair of definition $(a^{\prime},\delta(Q))$. Then
$v(a-a^{\prime})\geq\delta(Q)$ and hence $v(X-a^{\prime})\geq\delta(Q)$. Take
the minimal polynomial $Q^{\prime}(X)$ of $a^{\prime}$ over $K$. The fact that
$v(X-a^{\prime})\geq\delta(Q)$ implies that $\delta(Q^{\prime})\geq\delta(Q)$.
As a consequence of $Q(X)$ being a key polynomial for $v$ over $K$, we
conclude that $\deg Q^{\prime}\geq\deg Q$, that is,
$[K(a^{\prime}):K]\geq[K(a):K]$. It follows that $(a,\delta(Q))$ is a minimal
pair of definition for $v_{a,\delta(Q)}$ over $K$. In light of [References,
Theorem 1.1], we conclude that $v_{a,\delta(Q)}|_{K(X)}=v_{Q}$. ∎

######  Definition 7.3.

A family $\Lambda$ of key polynomials for $v$ over $K$ is said to form a
complete sequence of key polynomials for $v$ over $K$ if it satisfies the
following conditions:

(CSKP1) $\delta(Q)\neq\delta(Q^{\prime})$ for $Q,Q^{\prime}\in\Lambda$ with
$Q\neq Q^{\prime}$,

(CSKP2) $\Lambda$ is well-ordered with respect to the ordering given by
$Q<Q^{\prime}$ if and only if $\delta(Q)<\delta(Q^{\prime})$,

(CSKP3) for any $f(X)\in K[X]$, there exists some $Q\in\Lambda$ such that
$vf=v_{Q}f$.

######  Theorem 7.4.

Assume that $(K(X)|K,v)$ is a valuation algebraic extension. Take
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ and $\\{a_{\nu}\\}_{\nu<\lambda}$ as in
Section 5. For each $\nu<\lambda$, take the minimal polynomial $Q_{\nu}(X)$ of
$a_{\nu}$ over $K$. Then $\\{Q_{\nu}\\}_{\nu<\lambda}$ forms a complete
sequence of key polynomials for $v$ over $K$.

###### Proof.

Observe that by our construction, $a_{\nu}$ is a maximal root of $Q_{\nu}$ for
all $\nu<\lambda$. Consequently,

$\delta(Q_{\nu})=\gamma_{\nu}<\gamma_{\mu}=\delta(Q_{\mu})\text{ for all
}\nu<\mu<\lambda.$

Take some $\nu<\lambda$. Assume that $\delta(g)\geq\delta(Q_{\nu})$ for some
$g(X)\in K[X]$. Then there exists a root $a$ of $g$ such that
$v(X-a)\geq\delta(Q_{\nu})$. Consequently, $v(X-a)\geq v(X-a_{\nu})$ and hence
$v(a-a_{\nu})\geq v(X-a_{\nu})=\gamma_{\nu}$. It follows that
$(a,\gamma_{\nu})$ is a pair of definition for $v_{a_{\nu},\gamma_{\nu}}$ over
$K$. The minimality of $(a_{\nu},\gamma_{\nu})$ then implies that
$[K(a):K]\geq[K(a_{\nu}):K]$, that is, $\deg g\geq\deg Q_{\nu}$. It follows
that $Q_{\nu}(X)$ is a key polynomial for $v$ over $K$ for all $\nu<\lambda$.

We now take some $f(X)\in K[X]$ and a maximal root $a$ of $f$. The fact that
$\\{\gamma_{\nu}\\}_{\nu<\lambda}$ is cofinal in $v(X-\overline{K})$ implies
that there exists some $\nu<\lambda$ such that
$\delta(f)\leq\gamma_{\nu}=v(X-a_{\nu})=\delta(Q_{\nu})$. It follows from
Lemma 7.1 that $vf=v_{a_{\nu},\delta(Q_{\nu})}f$. In light of Lemma 7.2, we
conclude that $vf=v_{Q_{\nu}}f$. ∎

## References

  * [1] V. Alexandru and A. Zaharescu, Sur une classe de prolongements à $K(x)$ d’une valuation sur une corp $K$, Rev. Roumaine Math. Pures Appl., 5 (1988), 393–400.
  * [2] V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extensions of a valuation, J. Math. Kyoto University, 28 (1988), 579–592.
  * [3] V. Alexandru, N. Popescu and A. Zaharescu, Minimal pairs of definition of a residual transcendental extension of a valuation, J. Math. Kyoto University, 30 (1990), 207–225.
  * [4] V. Alexandru, N. Popescu and A. Zaharescu, All valuations on $K(X)$, J. Math. Kyoto University, 30–2 (1990), 281–296.
  * [5] N. Bourbaki, Commutative Algebra, Hermann, Paris (1972).
  * [6] A. Dutta, Minimal pairs, inertia degrees, ramification degrees and implicit constant fields, https://arxiv.org/abs/2111.13641, Submitted.
  * [7] A. Dutta, Minimal pairs, minimal fields and implicit constant fields, Journal of Algebra, 588 (2021), 479–514.
  * [8] A. Dutta and F.-V. Kuhlmann, Eliminating tame ramification: generalizations of Abhyankar’s lemma, Pacific Journal of Mathematics, 307(1) (2020), 121–136.
  * [9] I. Kaplansky, Maximal fields with valuations I, Duke Math. Journal 9 (1942), 303–321,
  * [10] F.-V. Kuhlmann, Value groups, residue fields and bad places of rational function fields, Trans. Amer. Math. Soc., 356 (2004), 4559–4600.
  * [11] S. MacLane, A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc., 40 (1936), 363–395.
  * [12] J. Novacoski, Key polynomials and minimal pairs, Journal of Algebra, 523 (2019), 1–14.
  * [13] J. Novacoski and M. Spivakovsky, Key polynomials and pseudo-convergent sequences, Journal of Algebra, 495 (2018), 199–219.
  * [14] M. Vaquié, Extension d’une valuation, Trans. Amer. Math. Soc., 359 no. 7 (2007), 3439–3481.


# Supersymmetry breaking and 4 dimensional string models

Michael Hewitt
Computing Services, Canterbury Christ Church University College,
North Holmes Road, Canterbury, CT1 1QU, U.K.
email<EMAIL_ADDRESS>tel:+44(0)1227-767700

(28 March 2003)

###### Abstract

A family of superpotentials is constructed which may be relevant to
supersymmetry breaking in 4 dimensional (0,1) heterotic string models. The
scale of supersymmetry breaking, as well as the coupling constant, would be
stable and could not run away to zero.

PACS numbers: 11.25.Mj, 04.65.+e, 11.25.Sq

##  1 Introduction

In this paper we will study a family of superpotentials which may be relevant
to supersymmetry (susy) breaking in 4 dimensional (0,1) heterotic string
models, and which generalise those given in [1,2]. By 4 dimensional string
models we have in mind that the extra string degrees of freedom are subject to
twisted boundary conditions in such a way that no ‘breathing’ modes for extra
dimensions appear in the level zero spectrum [3]. Stability is a feature of
such $D=4$ string models, for which decompactification cannot take place.
There is a fixed, arbitrary gauge coupling, related to string topology through
the genus expansion. There may be a possible restriction of this coupling
constant required for the consistent propagation of knotted and linked
strings, which are features which appear when $D$ is reduced to 4, with
strings producing orbifold-like singularities in space-time. The origin of
susy breaking is often ascribed to the gaugino condensation mechanism [4,5].
This relies on a variation of the gauge coupling with a scalar field, which is
therefore absent in these models. Consequently, there would need to be susy
breaking by some fundamental rather than composite Goldstone supermultiplet.
If this is possible, there would be benefits in avoiding problems with the
stability of the gauge hierarchy and a shallow potential well which could lead
to decompactification during inflation [6]. A vanishing 4 dimensional
cosmological constant also seems more natural in a model without extra
dimensions. For the present, we will consider the scalar fields only for
simplicity. A vanishing cosmological constant may be viewed as a balance
between local and global susy breaking. This can be described geometrically as
a requirement that the field-space vector formed by all auxiliary fields be
null. In terms of the geometry of the chiral scalar fields, this is equivalent
to the superpotential $G$ having a critical slope of $\sqrt{3}$ in field
space. The polarization of positive and negative energy auxiliary fields due
to this local/global susy breaking is much greater than the energy due to non-
cancellation between bosons and fermions from global susy breaking. A small
adjustment of the balance between gravitational and other auxiliaries would
cancel the residue to keep $\Lambda=0$. We will assume that such a balancing
mechanism applies. A small failure of the balance mechanism could give a small
$\Lambda$, as may be the actual case in cosmology, but we will not attempt to
understand this here.

In $N=1$ susy, we may make a decomposition of the chiral fields into the
Goldstino $z$ and other fields $s_{i}$,where $z$,$s_{i}$ are orthogonal at the
vacuum expectation values $z_{0},s_{0}$,

$\frac{\partial^{2}G}{\partial z\partial
s^{*}_{i}}(z_{0},s_{0})=\frac{\partial G}{\partial s_{i}}(z_{0},s_{0})=0$ (1)

We propose that the Goldstino scalar $z$ can be approximated by the spin 0
part of the dilaton supermultiplet, which may be regarded as scalar and
pseudoscalar components of the graviton, as we can form a massive gravitino
multiplet, or extended graviton multiplet by taking the full content of the
spin $(1)_{L}\otimes(1\oplus 1/2)_{R}$ heterotic level zero fields. Our
motivation is that the dilaton multiplet in $D=4$ is naturally present, and
has a flat potential, or critical-slope superpotential (implementing the
local/global balance principle) in the absence of other sources of susy
breaking [7]. The pseudoscalar graviton mass is zero, protected by a gauge
invariance, and therefore the pseudoscalar part of $z$ will be a pseudo-
Goldstone boson with a relatively long range. Mixings between the
gravitational and other scalars will be suppressed by factors of $1/m_{3/2}$,
since the contributions of different superfields to the Goldstone
supermultiplet would be in proportion to their auxiliary components. CP
violation may give a small scalar component to the pseudo-Goldstone boson, but
we will neglect this below. The situation in realistic models where there are
also terms in $V$ from the gauge auxiliary $D$ fields is more complicated, but
providing that the local/global balance principle applies, there will again be
zero cosmological constant with a small mixing $O(1/m_{3/2})$ between $z$ and
other superfields.

##  2 Derivation of the Superpotentials

Decomposing $z$ into scalar and pseudoscalar parts, we have $z=x+\mathrm{i}y$
where

$N(x)dy=*(dB+\omega_{GS})+O(1/m_{3/2})$ (2)

where $N(x)$ is the normalization factor to give a standard kinetic term for
the pseudoscalar $Ny$ and $\omega_{GS}$ is the Green-Schwarz 3-form, so that
under the transformation $y\rightarrow y+\epsilon$, $\delta G=\epsilon
O(G/m_{3/2})$ or

$\frac{\partial G(z)}{\partial y}=O(G/m_{3/2}),\ \frac{\partial G(z)}{\partial
y}(z_{0},s_{0})=0$ (3)

i.e. $G(z)=G(x)+O(G/m_{3/2})$. For any function $h(z)$ which depends only on
$x$ we have

$\frac{\partial h(x)}{\partial z}=\frac{\partial h(x)}{\partial
z^{*}}=1/2\frac{dh}{dx}\equiv\frac{h^{{}^{\prime}}}{2}$ (4)

so that the potential for $x$ becomes (neglecting for now the contribution
from other fields)

$V(x)=\mathrm{e}^{G}(\frac{G^{{}^{\prime}2}}{G^{{}^{\prime\prime}}}-3)=\frac{\mathrm{e}^{G}}{G^{{}^{\prime\prime}}}(G^{{}^{\prime}2}-3G^{{}^{\prime\prime}}).$
(5)

Introducing the notation

$G^{{}^{\prime}}=g(x)$ (6)

we can write

$g^{2}-3g^{{}^{\prime}}=f^{2}.$ (7)

For a vanishing cosmological constant, we require that $f$ should have a zero.
The simplest ansatz with this property is essentially

$f^{2}=\alpha(g-3)^{2}.$ (8)

Here we have used the linear transformation $x\rightarrow ax+b$ to arrange
$x=0,g=3$ at $f=0$ without loss of generality (these values are chosen to
simplify the formulas below). The case $\alpha=0$ gives the flat, no-scale
solution

$G=-3\ln{(x)},\ \ V=0.$ (9)

We will now derive the solution corresponding to eq.(8). Rearranging eq.(7)
gives the first order differential equation

$\frac{dx}{dg}=\frac{1}{(1-\alpha)g^{2}+6\alpha g-9\alpha}$ (10)

with solution

$x=\tan^{-1}{(\frac{1-\alpha}{3\alpha}g+1)}-\tan^{-1}{(\frac{1}{\alpha})}$
(11)

giving

$g=\frac{3\alpha}{1-\alpha}(\tan{(x+c)-1)}$ (12)

where

$c=\tan^{-1}{(\frac{1}{\alpha})}.$ (13)

Now

$G(x)=\int_{u=0}^{x}g(u)du+G_{0}$ (14)

where the constant of integration $G_{0}$ gives the value of
$\ln{(m_{3/2}^{2})}$ at the potential minimum. We have

$G=G_{0}+\frac{3\alpha}{1-\alpha}(\log\sec(x+c)-\log\sec(c)-x)$ (15)

and using

$G^{{}^{\prime\prime}}=g^{{}^{\prime}}=\frac{3\alpha}{1-\alpha}\sec^{2}{(x+c)}$
(16)

gives us, on eliminating $c$ by using elementary identities,

$V=3\mathrm{e}^{G_{0}}\frac{(1+\alpha^{2})}{1-\alpha}[\mathrm{e}^{-x}(\cos{(x)}-\frac{1}{\alpha}\sin{(x)})]^{\frac{3\alpha}{\alpha-1}}\sin^{2}{(x)}.$
(17)

Here $x$ will be confined to the range where $V$ is finite i.e.
$\tan(x)<\alpha$ for $0<\alpha<1$. For the mass of the scalar $x$ we have

$m_{S}^{2}=\frac{V^{{}^{\prime\prime}}}{G^{{}^{\prime\prime}}}|_{x=0}=\frac{\mathrm{e}^{G_{0}}}{G^{{}^{\prime\prime}}}\frac{(f^{{}^{\prime}})^{2}}{G^{{}^{\prime\prime}}}|_{x=0}=\alpha
m_{3/2}^{2}$ (18)

giving the interpretation of the parameter $\alpha$ as
$m_{S}^{2}/m_{3/2}^{2}$. As argued above the mass of the pseudoscalar $y$ will
be

$m_{P}^{2}=O(m_{3/2}^{4}).$ (19)

The solution eq.(17) may be written implicity as a function of the sigma model
parameter

$\phi(x)=\int_{u=0}^{x}\sqrt{2G^{{}^{\prime\prime}}(u)}du=\sqrt{\frac{2\alpha}{3(1-\alpha)}}\ln{(\frac{\sec(x+c)+\tan(x+c)}{\sec(c)+\tan(c)})}$
(20)

with standard kinetic term. Now consider the limit $\alpha\rightarrow 1$
corresponding to the physically interesting case $m_{S}=m_{3/2}$. Going back
to eq.(7) with $\alpha=1$ gives

$x=1/2\ln{(2g/3-1)}.$ (21)

Introducing an alternative parametrization by

$w=\exp{(z)},|w|=\exp{(x)}$ (22)

(so that $y$ and $y+2\pi n$ are identified), which is natural when we recall
the origin of $x$ as the 4 dimensional dilaton, gives the solution

$V=\frac{3\mathrm{e}^{G_{0}}}{4}(|w|^{2}\exp{(|w|^{2}-1)})^{3/4}(|w|-|w|^{-1})^{2}.$
(23)

Here the value of $w$ is kept away from zero by the divergence in $V(0)$. In
this case the sigma model field is $\phi=\sqrt{6}|w|$, so that if we write
$\psi=|w|$ we can write the Lagrangian terms for $\psi$ as

$L_{\psi}=3\sqrt{-g}((\partial\psi)^{2}-\frac{\mathrm{e}^{G_{0}}}{4}(\psi^{2}\exp{(\psi^{2}-1}))^{3/4}(\psi-\psi^{-1})^{2}).$
(24)

(Here $g$ is the usual $D=4$ metric determinant.) An interesting feature of
this model is that the scalar and pseudoscalar fields will be described by a
sigma model with the flat Kahler metric

$d\sigma^{2}=6dwdw^{*}=6(d\psi^{2}+\psi^{2}d\theta^{2}).$ (25)

We will now consider the origin of the integration constant $G_{0}$, which
determines $m_{3/2}$. Suppose that, near $(z_{0},s_{0})$, $G$ takes the form

$G(z,s)=G_{g}(z)+G_{s}(s)+H(z,s)=G_{g}(z)+K(s)+\ln(|W(s)|^{2})+H(z,s)$ (26)

solving eq.(1), where $G_{g}(z)$ takes the form discussed above and for the
remaining chiral fields $s$, $K(s)$ is the Kahler potential (generating the
field kinetic terms) and $W(s)$ represents the chiral scalar superfield
interactions. $H(z,s)$ represents mixing terms of higher order in $1/m_{3/2}$
relative to $G(z,s)$, with $H(z_{0},s_{0})=0$. We expect $K_{0}\sim 0$, giving

$\exp{(G_{0})}\sim|W_{0}|^{2}$ (27)

where $W_{0}$ is the minimum value of $W(s)$ \- since the vacuum represents
the minimum of $G$ w.r.t. $s$, it also represents the minimum of $W(s)$. Now
the leading renormalizable contribution gives $W\sim g_{4}X_{1}X_{2}X_{3}$
where $X_{i}$ may acquire expectation values by a Coleman-Weinberg mechanism
well below the string tension scale. For $m_{3/2}\sim$ 1TeV, the $X_{i0}$ may
come from GUT symmetry breaking, a neutrino mass see-saw mechanism or some
other hidden sector. String diagrams with zero background field give $V(z)=0$.
While this may represent the $\alpha\rightarrow 0$ limit of the family of
models represented by eq.(8), it may alternatively represent the limit
$\exp{(G_{0})}\rightarrow 0$, in particular with $\alpha=1$, allowing stable
supersymmetry breaking in 4 dimensional string models. For $\alpha\leq 1$,
$m_{3/2}$ is prevented from running away to zero, so that it is natural in
this scenario for susy breaking to survive inflation.

References

[1] J.Polonyi, Budapest Preprint KFKI-1997-93.

[2] E.Cremmer, S.Ferrara, C.Kounnas, and D.V.Nanopoulos, Phys. Lett B133
(1983) 61.

[3] H.Kawai, D.C.Lewellen and S.H.H.Tye, Nucl. Phys. B288 (1987) 1.

[4] S.Ferrara, L.Girardello and H.P.Nilles, Phys. Lett B125 (1983) 457.

[5] M.Dine, R.Rohm, N.Seiberg and E.Witten, Phys. Lett B156 (1985) 55.

[6] R.Brustein and P.J.Steinhardt, Phys. Lett B302 (1993) 196.

[7] M.Dine and N.Seiberg, Phys. Rev. Lett57 (1986) 2625.


# Celestial amplitude for 2d theory

Sarthak Duary

###### Abstract

We explore celestial amplitude corresponding to $2d$ bulk
$\mathcal{S}$-matrix. We consider scalar particles with identical mass and
show that the celestial amplitude becomes the fourier transform of the $2d$
$\mathcal{S}$-matrix written in the rapidity variable. We translate the
crossing and unitarity conditions into the conditions on the celestial
amplitude. For the $2d$ Sinh-Gordon model, we calculate the celestial
amplitude perturbatively in coupling constant and check that the crossing and
unitarity conditions are satisfied for the celestial amplitude. Imposing the
crossing and unitarity conditions to the celestial amplitude, we want to find
amplitudes to the higher order in perturbation theory from the lower order
i.e., to provide a “proof of principle” to show we can apply the bootstrap
idea to the celestial amplitude. We find that imposing the crossing and
unitarity conditions is not enough for bootstrapping celestial amplitude,
there is an extra term which can’t be fixed by the crossing and unitarity
conditions. We also study the gravitational dressing condition in $2d$ QFT for
massless particles in celestial space and see that for the gravitationally
dressed celestial amplitude, the poles on the right half-plane get erased for
several ansatzes.

##  1 Introduction

While evaluating Quantum field theory (QFT) scattering amplitudes, we
generally express the asymptotic states in terms of asymptotic plane wave
solutions to the free wave equation which is energy-momentum eigenstates. This
conventional plane wave basis makes spacetime translation invariance manifest
due to energy-momentum conservation but obscures conformal invariance.

The Lorentz group in $\mathbb{R}^{1,d-1}$ is the same as the Euclidean
conformal group in $(d-2)$-dimensions. This indicates that the $d$-dimensional
$\mathcal{S}$-matrix is related to $(d-2)$-dimensional conformal field theory
correlation function.

As opposed to the plane wave basis which is energy-momentum eigenstate, in [1,
2], a new basis called the conformal primary basis which is the boost
eigenstate is constructed for both massive and massless particles. In this
basis, free fields transform as conformal primaries under the Lorentz group
and the $\mathcal{S}$-matrix elements transform manifestly as conformal
correlation functions on the celestial sphere i.e., the sphere at null
infinity. The scattering amplitude computed in this basis are known as
celestial amplitude and we can express the bulk $\mathcal{S}$-matrix in terms
of a boundary correlation function of ‘celestial CFT’ that lives on the sky as
in fig.1.

Figure 1: Bulk $\mathcal{S}$-matrix mapped to Boundary Celestial CFT
correlation function

The motivation for seeking this conformal basis is to understand the
holographic nature of quantum gravity in asymptotically flat spacetimes.
Realization of the holographic duality [4, 5, 6] from the bottom up by finding
the symmetries that both sides of the holographic dual pair obey [14, 15]
gives rise to the enhancement of the Lorentz symmetry to full Virasoro [16]
and the existence of a stress tensor in a $2d$ CFT obeying Ward identity
constructed from subleading soft-graviton mode in the bulk [17]. These
observations along with the equivalence between soft theorems and asymptotic
symmetries i.e., soft theorems recasted as conservation laws associated with
large gauge symmetries lead to the proposal that there exists a holographic
duality between the theory of gravity in four-dimensional asymptotically flat
spacetimes and some sort of exotic CFT living on the two-dimensional celestial
sphere at null infinity.

The flat space holography is initiated by the work of de Bohr and Solodukhin
[19]. From dS and AdS slicing of Minkowski space, they postulate that the flat
space in four dimensions has some description in terms of a theory living on
the boundary of these dS and AdS slices which is identified with the celestial
sphere.

In this celestial holography paradigm, we study the celestial amplitude for
the $2d$ bulk scattering. We construct the map to evaluate celestial amplitude
from bulk $\mathcal{S}$-matrix which is implemented by the change of basis
from energy-momentum eigenstates to boost eigenstates. To summarize, for
massive scalar particles the celestial amplitude is the fourier transform of
the $2d$ $\mathcal{S}$-matrix written in the rapidity variable. We calculate
the perturbative celestial amplitude for the $2d$ Sinh-Gordon model. One
subtle point here is that in $2d$, the perturbative expansion of the Sinh-
Gordon $\mathcal{S}$-matrix contains pole at rapidity $\theta=0$. Therefore
while evaluating the perturbative celestial amplitude, we should implement
$i\epsilon$ prescription and as a consequence of it there are two types of
celestial amplitude which we indicate by the retarded and advanced celestial
amplitude. In the celestial space, we map the crossing and unitarity
conditions to calculate the celestial amplitude to the higher order in
perturbation theory from the lower one by imposing these constraints which we
refer as “celestial bootstrap”. We check the crossing and unitarity conditions
for the Sinh-Gordon model.

In AdS, solving conformal crossing equation, the one-loop correction to the
four point amplitude is evaluated for scalar $\phi^{4}$ theory in [20] from
which one can analytically calculate the anomalous dimensions. To apply the
CFT-bootstrap technique in flat space to find the amplitude, we map the
crossing and unitarity conditions in the celestial space for $2d$ scattering.
For massive scalar case, in higher than $2d$, technically it’s a bit
challenging to evaluate the celestial amplitude even at tree level say for
massive $\phi^{4}$ theory in $4d$, for that as a stepping stone towards
bootstrapping celestial amplitude we restrict to $2d$ scattering.

We also find that imposing the crossing and unitarity conditions to the
celestial amplitude is not enough to determine the higher-order perturbative
celestial amplitude from the lower order. There is an extra term which can’t
be fixed by the crossing and unitarity conditions.

We study the gravitational dressing condition for $2d$ QFT amplitude for
massless particles in celestial space. We see that after the gravitational
dressing in celestial space, the poles on the right half-plane of the
celestial amplitude get erased for the functions having multiple poles on the
right half-plane.

In the conclusion, authors of [27] commented on studying the celestial
amplitude for the bulk $2d$ spacetime, which we initiate in this work.

### Outline:

The paper is organized as follows. In section 2, we review the construction of
massive celestial amplitude. In section 3, we define the celestial $4$-point
amplitude for $2\to 2$ scattering of massive scalar particles in $2d$. The
massive celestial amplitude in $2d$ is given by the fourier transform of the
$\mathcal{S}$-matrix written in the rapidity variable. In section 4, we
evaluate the celestial amplitude in $2d$ Sinh-Gordon model by perturbatively
expanding the $\mathcal{S}$-matrix. In section 5, we translate the crossing
and unitarity conditions in celestial space and check the crossing and
unitarity conditions for the Sinh-Gordon model. In section 6, we discuss about
reconstructing the higher-order correction to the $4$-point celestial
amplitude from the lower-order amplitude using the “crossing and unitarity”
conditions. In section 7, we translate the gravitational dressing condition of
the $2d$ QFT amplitude in celestial space. For massless particles, we see that
the gravitational dressing condition translated to celestial space acts as a
pole eraser from the right half-plane corresponding to the celestial amplitude
for the functions having multiple poles on the right half-plane. In section 8,
we summarize our main results and discuss open problems and future directions.

##  2 Massive Celestial amplitude

In this section, we review the construction of massive celestial amplitude
described in [2].

The massive scalar conformal primary wavefunction
$\phi_{\Delta}^{\pm}(X^{\mu};\vec{w})$ in $\mathbb{R}^{1,d-1}$ admits the
fourier expansion on the plane waves

$\phi_{\Delta}^{\pm}(X^{\mu};\vec{w})=\int_{H_{d-1}}[d\hat{p}]~{}~{}G_{\Delta}(\hat{p};\vec{w})~{}\text{exp}[\pm~{}im~{}\hat{p}.X]~{}~{}~{},$
(2.1)

where the on-shell momenta per mass, a unit timelike vector
$\hat{p}(y,\vec{z})$ satisfying $\hat{p}^{2}=-1$ can be parametrized using the
$H_{d-1}$ coordinates $y$ ($y>0$) and $z\in\mathbb{R}^{d-2}$ as,

$\hat{p}(y,\vec{z})=\Bigg{(}\frac{1+y^{2}+|\vec{z}|^{2}}{2y},\frac{\vec{z}}{y},\frac{1-y^{2}-|\vec{z}|^{2}}{2y}\Bigg{)}~{}~{}~{}.$
(2.2)

Here, $[d\hat{p}]$ is the $SO(1,d-1)$ invariant measure on $H_{d-1}$

$\begin{split}\int_{H_{d-1}}[d\hat{p}]&\equiv\int\frac{d^{d-1}\hat{p}^{i}}{\hat{p}^{0}}\\\
&=\int_{0}^{\infty}\frac{dy}{y^{d-1}}\int
d^{d-2}\vec{z}~{}~{}~{},~{}~{}~{}i=1,\ldots,d-1~{}~{}~{},~{}~{}~{}\hat{p}^{0}=\sqrt{\hat{p}^{i}\hat{p}^{i}+1}~{}~{}~{}.\end{split}$
(2.3)

Here, $G_{\Delta}(\hat{p};\vec{w})$ is the scalar bulk-to-boundary propagator
in $H_{d-1}$ given by [5]

$G_{\Delta}(\hat{p};\vec{w})=\Bigg{(}\frac{y}{y^{2}+|\vec{z}-\vec{w}|^{2}}\Bigg{)}^{\Delta}~{}~{}~{},$
(2.4)

where $\vec{w}\in\mathbb{R}^{d-2}$ lies on the boundary of $H_{d-1}$. The
scalar bulk-to-boundary propagator written in terms of
$\hat{p}^{\mu}(y,\vec{z})$ and null momentum
$q^{\mu}(\vec{w})=(1+|\vec{w}|^{2},2\vec{w},1-|\vec{w}|^{2})$ in
$\mathbb{R}^{1,d-1}$ is given by [3]

$G_{\Delta}(\hat{p};q)=\frac{1}{(-\hat{p}.q)^{\Delta}}~{}~{}~{}.$ (2.5)

Using the mapping from the plane wave to the conformal primary wavefunction
given by eq.(2.1), the $\mathcal{S}$-matrix elements in the conformal prinary
basis is given in terms of an integral transform

$\mathcal{\tilde{A}}(\Delta_{i},\vec{w}_{i})=\prod_{k=1}^{n}\int_{H_{d-1}}[d\hat{p}_{k}]G_{\Delta_{k}}(\hat{p}_{k};\vec{w}_{k})\mathcal{A}(\pm
m_{i}\hat{p}^{\mu}_{i})~{}~{}~{},$ (2.6)

where $\pm m_{i}\hat{p}^{\mu}_{i}$ parametrization depends on whether the
particle is incoming or outgoing.

In the r.h.s., $\mathcal{A}(\pm m_{i}\hat{p}^{\mu}_{i})$ is the momentum space
amplitude along with the momentum conserving delta function.

We define $\mathcal{\tilde{A}}(\Delta_{i},\vec{w}_{i})$ by the massive
celestial amplitude. Under the conformal symmetry action the massive celestial
amplitude transforms covariantly as a $(d-2)$-dimensional CFT $n$-point
function of scalar primaries with dimension $\Delta_{i}$

$\mathcal{\tilde{A}}(\Delta_{i},\vec{w}^{\prime}_{i}(\vec{w}_{i}))=\prod_{k=1}^{n}\Bigg{|}\frac{\partial\vec{w}^{\prime}_{k}}{\partial\vec{w}_{k}}\Bigg{|}^{\frac{-\Delta_{k}}{d-2}}\mathcal{\tilde{A}}(\Delta_{i},\vec{w}_{i})~{}~{}~{}.$
(2.7)

##  3 Massive Celestial amplitude for $2\to 2$ scattering: The Celestial point

In this section, we define the celestial $4$-point amplitude for $2\to 2$
scattering of massive scalar particles in $2d$.

In $2d$, the on-shell momenta of particles of mass $m$ can be written in the
rapidity parametrization $\theta$

$\begin{split}p^{\mu}&=(p^{0}=E,~{}p^{1}=p)\\\
&=m(\cosh\theta,\sinh\theta)~{}~{}~{}.\end{split}$ (3.1)

We consider the elastic scattering process of identical real scalar particles
of mass $m$ with rapidities $\theta_{1}$ and $\theta_{2}$.

Energy and momentum conservation for identical particles give

$\begin{split}&\theta_{1}=\theta_{4}~{}~{},~{}~{}\theta_{2}=\theta_{3}~{}~{}~{}.\\\
\end{split}$ (3.2)

The $2\to 2$ $\mathcal{S}$-matrix is given by

$\begin{split}\mathcal{S}_{2\to
2}=\langle\theta_{4},\theta_{3}\ket{\theta_{1},\theta_{2}}&=4p_{1}^{0}p_{2}^{0}\times(2\pi)^{2}\delta(p_{1}^{1}-p_{4}^{1})\delta(p_{2}^{1}-p_{3}^{1})+i(2\pi)^{2}\delta^{(2)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})~{}\mathcal{T}\\\
&=4(2\pi)^{2}\delta(\theta_{1}-\theta_{4})\delta(\theta_{2}-\theta_{3})\Bigg{(}1+\frac{i~{}\mathcal{T}(\theta_{1}-\theta_{2})\csch(\theta_{1}-\theta_{2})}{4m^{2}}\Bigg{)}~{}~{}~{}.\end{split}$
(3.3)

We define $\mathcal{S}$-matrix which is dependent on the difference of
rapidities as

$S(\theta_{1}-\theta_{2})\equiv
1+\frac{i\mathcal{T}(\theta_{1}-\theta_{2})\csch(\theta_{1}-\theta_{2})}{4m^{2}}~{}~{}~{}.$

The Mandelstem variables are

$s=(p_{1}+p_{2})^{2}=4m^{2}\cosh^{2}\Big{(}\frac{\theta}{2}\Big{)}~{}~{}~{},~{}~{}~{}t=4m^{2}-s~{}~{}~{},~{}~{}~{}u=0~{}~{}~{}.$

$s$ is the cenrer of mass energy squared, $t$-channel gives $\theta\to
i\pi-\theta$.

In $2d$, the celestial $4$-point amplitude of the massive conformal primary
wavefunction is

$\tilde{\mathcal{A}}=\Bigg{(}\prod_{i=1}^{4}\int\frac{d\hat{p}_{i}^{1}}{\hat{p}_{i}^{0}}\Bigg{)}\times\prod_{i=1}^{4}G_{\Delta_{i}}(\hat{p}_{i})~{}\mathcal{S}_{2\to
2}$ (3.4)

where,
$\hat{p}_{i}^{\mu}\equiv\hat{p}^{\mu}(\theta_{i})=\frac{p_{i}^{\mu}}{m}=(\cosh\theta_{i},\sinh\theta_{i})$
and $G_{\Delta_{i}}(\hat{p}_{i})$ is the bulk-to-boundary propagator in
$H_{1}$.

The scalar bulk-to-boundary propagator in $H_{1}$ is

$\begin{split}G_{\Delta}(\hat{p};q)&=\frac{1}{(-\hat{p}.q)^{\Delta}}=\frac{1}{(\cosh\theta-\sinh\theta)^{\Delta}}\\\
&=e^{\Delta\theta}~{}~{}~{}.\end{split}$ (3.5)

Here, $q=(1,1)$ since, conformal boundary of $H_{1}$ are specified by $2$
dimesnional points on the projective null cone

$-(q^{0})^{2}+(q^{1})^{2}=0~{}~{},~{}~{}~{}q\sim\lambda q~{}~{}~{}.$

The massive scalar conformal primary wavefunctions in $d$ dimensions are
delta-function normalizable when $\Delta$ belongs to the principal continuous
series of the irreducible unitary $SO(1,d-1)$ representations,

$\Delta\in\frac{d-2}{2}+i\mathbb{R}~{}~{}.$

Therefore, for the $2d$ scattering, $\Delta$ is pure imaginary. Now, we
redefine $i\Delta$ as the conformal dimension of the conformal primary
wavefunction in $2d$. The $H_{1}$ scalar bulk-to-boundary propagator
$G_{\Delta}(\hat{p})=e^{i\Delta\theta}$.

The normalization condition for the $H_{1}$ scalar bulk-to-boundary propagator
is

$\int_{-\infty}^{\infty}d\theta
e^{i\Delta_{i}\theta}e^{i\Delta_{j}\theta}=2\pi~{}\delta(\Delta_{i}+\Delta_{j})~{}~{}~{}.$
(3.6)

Now, the integral measure is

$\prod_{i=1}^{4}\int\frac{d\hat{p}_{i}^{1}}{\hat{p}_{i}^{0}}=\prod_{i=1}^{4}\int_{-\infty}^{\infty}d\theta_{i}~{}~{}~{}.$

We can express celestial $4$-point amplitude in eq.(3.4) as

$\small\begin{split}\tilde{\mathcal{A}}&=4(2\pi)^{2}\prod_{i=1}^{4}\int_{-\infty}^{\infty}d\theta_{i}~{}~{}e^{i\Delta_{i}\theta_{i}}S(\theta_{1}-\theta_{2})\delta(\theta_{1}-\theta_{4})\delta(\theta_{2}-\theta_{3})\\\
&=4(2\pi)^{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\theta_{4}~{}d\theta_{3}~{}S(\theta_{4}-\theta_{3})~{}e^{i\Delta_{1}\theta_{4}}e^{i\Delta_{2}\theta_{3}}e^{i\Delta_{3}\theta_{3}}e^{i\Delta_{4}\theta_{4}}\\\
&=4(2\pi)^{2}\frac{1}{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\theta_{+}~{}d\theta_{-}~{}S(\theta_{-})e^{\frac{i}{2}(\Delta_{1}+\Delta_{4}+\Delta_{2}+\Delta_{3})\theta_{+}}e^{\frac{i}{2}(\Delta_{1}+\Delta_{4}-\Delta_{2}-\Delta_{3})\theta_{-}}\\\
&=16\pi^{3}~{}\delta(\Delta_{+})~{}\mathcal{A}(\Delta_{-})~{}~{}~{}.\end{split}$
(3.7)

where, we define light-cone coordinates
$\theta_{\pm}=\theta_{4}\pm\theta_{3}$,
$\Delta_{\pm}=\frac{1}{2}(\Delta_{1}+\Delta_{4}\pm\Delta_{2}\pm\Delta_{3})$
and

$\mathcal{A}(\Delta_{-})\equiv\int_{-\infty}^{\infty}d\theta
e^{i\Delta_{-}\theta_{-}}S(\theta_{-})~{}~{}~{}.$

Now, we strip off the overall delta function $\delta(\Delta_{+})$ and name the
conjugate pair of $\theta_{-}$ as
$\omega\equiv\Delta_{-}=\frac{1}{2}(\Delta_{1}+\Delta_{4}-\Delta_{2}-\Delta_{3})~{},$
the $2d$ celestial amplitude is the fourier transform of the $S$-matrix with
respect to rapidity.

$\mathcal{A}(\omega)\equiv\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta)~{}~{}~{}.$ (3.8)

Physically, the rapidity $\theta$ shifts under the boost as
$\theta\to\theta+c$. Therefore, in order to diagonalize the boost which is
achieved by the conformal basis, we need to perform the fourier transform of
the $\mathcal{S}$-matrix $S(\theta)$ with respect to rapidity.

Here, there is no celestial coordinate and the dual theory is zero-dimensional
which we refer as the “celestial point”. The CFT correlation function “dual”
to the bulk $\mathcal{S}$-matrix would be a zero-dimensional CFT correlation
function or said differently, an operator algebra without coordinates.

##  4 Celestial amplitude in 2d Sinh-Gordon model

In this section, we evaluate the celestial amplitude in $2d$ Sinh-Gordon model
by perturbatively expanding the $\mathcal{S}$-matrix of the Sinh-Gordon model.

The largangian of the Sinh-Gordon model [8, 18] is

$\mathcal{L}=\frac{1}{2}(\partial\phi)^{2}+\frac{m^{2}}{b^{2}}(\cosh(b\phi)-1)~{}~{}~{}.$
(4.1)

The $\mathcal{S}$-matrix for Sinh-Gordon model with the parameter $\alpha$
related to the coupling $b$ is

$S(\theta)=\frac{\sinh\theta-i\sin\alpha}{\sinh\theta+i\sin\alpha}~{}~{}~{},~{}~{}~{}\alpha=\frac{\pi
b^{2}}{8\pi+b^{2}}~{}~{}~{}.$ (4.2)

We can verify the $\mathcal{S}$-matrix by expanding the lagrangian
perturbatively in coupling $b$

$\mathcal{L}=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}\phi^{2}+\frac{m^{2}b^{2}}{4!}\phi^{4}+\frac{m^{2}b^{4}}{6!}\phi^{6}+\cdots~{}~{}~{},$
(4.3)

and calculate the perturbative expansion of the $\mathcal{S}$-matrix to first
few orders in $b^{2}$.

The perturbative expansion of the $\mathcal{S}$-matrix with respect to $b^{2}$
is given by

$\small\begin{split}S^{(0)}(\theta)&=1\\\
S^{(1)}(\theta)&=-\frac{1}{4}ib^{2}\text{csch}\theta\\\
S^{(2)}(\theta)&=-\frac{b^{4}\text{csch}\theta(\pi\text{csch}\theta-i)}{32\pi}\\\
S^{(3)}(\theta)&=\frac{ib^{6}\text{csch}\theta\left(6\pi^{2}\text{csch}^{2}\theta-12i\pi\text{csch}\theta+\pi^{2}-6\right)}{1536\pi^{2}}\\\
S^{(4)}(\theta)&=\frac{b^{8}\text{csch}\theta\left(6\pi^{3}\text{csch}^{3}\theta-18i\pi^{2}\text{csch}^{2}\theta+2\pi\left(\pi^{2}-9\right)\text{csch}\theta-3i\left(\pi^{2}-2\right)\right)}{12288\pi^{3}}\\\
S^{(5)}(\theta)&=-\frac{ib^{10}\csch\theta\left(120\pi^{4}\text{csch}^{4}\theta-480i\pi^{3}\text{csch}^{3}\theta+60\pi^{2}(\pi^{2}-12)\text{csch}^{2}\theta-160i\pi(\pi^{2}-3)\text{csch}\theta+\pi^{4}-120\pi^{2}+120\right)}{1966080\pi^{4}}.\end{split}$
(4.4)

Since, the perturbative expansion of the $\mathcal{S}$-matrix $S(\theta)$
contains poles at $\theta=0$, we define the celestial amplitude using
$i\epsilon$ prescription as

$\mathcal{A}^{\pm}(\omega)\equiv\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta\pm i\epsilon)$

where

$\mathcal{A}^{\pm}(\omega)=2\pi\Big{[}\delta(\omega)+b^{2}f_{1}^{\pm}(\omega)+b^{4}f_{2}^{\pm}(\omega)+\cdots\Big{]}~{}~{}~{},$
(4.5)

$\begin{split}f_{n}^{\pm}(\omega)=\frac{1}{2\pi(b^{2})^{n}}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S^{(n)}(\theta\pm i\epsilon)~{}~{}~{}.\end{split}$ (4.6)

Here, we call $\mathcal{A}^{+}(\omega)$ as the retarded celestial amplitude
and $\mathcal{A}^{-}(\omega)$ as the advanced celestial amplitude.

Here, eq.(4.5) can be understood as the perturbative expansion of the retarded
and advanced celestial amplitude and we call $f_{n}^{+}(\omega)$ and
$f_{n}^{-}(\omega)$ as the perturbative retarded and advanced celestial
amplitude.

Now, we evaluate the fourier transform by $+i\epsilon$ prescription. We shift
the pole at $\theta=0$ to $\theta=-i\epsilon$ and enclose the contour in the
upper half-plane.

Evaluating the fourier transform using $+i\epsilon$ prescription, we get the
perturbative retarded celestial amplitude

$\small\begin{split}f_{1}^{+}(\omega)&=-\frac{1}{4(1+e^{\pi\omega})}\\\
f_{2}^{+}(\omega)&=\frac{\left(\pi\omega+e^{\pi\omega}(\pi\omega+1)-1\right)(\coth(\pi\omega)-1)}{64\pi}\\\
f_{3}^{+}(\omega)&=-\frac{e^{\frac{\pi\omega}{2}}(\coth(\pi\omega)-1)\left[\left(\pi^{2}\left(6\omega^{2}+5\right)+6\right)\sinh\left(\frac{\pi\omega}{2}\right)+12\pi\omega\cosh\left(\frac{\pi\omega}{2}\right)\right]}{1536\pi^{2}}\\\
f_{4}^{+}(\omega)&=\frac{e^{\frac{\pi\omega}{2}}(\coth(\pi\omega)-1)\left[3\left(\pi^{2}\left(6\omega^{2}+5\right)+2\right)\sinh\left(\frac{\pi\omega}{2}\right)+\pi\omega\left(\pi^{2}\left(\omega^{2}+2\right)+18\right)\cosh\left(\frac{\pi\omega}{2}\right)\right]}{12288\pi^{3}}\\\
f_{5}^{+}(\omega)&=-\frac{e^{\frac{\pi\omega}{2}}(\coth(\pi\omega)-1)}{1966080\pi^{4}}\times\\\
&\Bigg{[}\left(120\pi^{2}\left(6\omega^{2}+5\right)+\pi^{4}\left(5\omega^{2}\left(\omega^{2}-2\right)-14\right)+120\right)\sinh\left(\frac{\pi\omega}{2}\right)+80\pi\omega\left(\pi^{2}\left(\omega^{2}+2\right)+6\right)\cosh\left(\frac{\pi\omega}{2}\right)\Bigg{]}~{}~{}~{}.\end{split}$
(4.7)

Next, we evaluate the integral by $-i\epsilon$ prescription. We shift the pole
at $\theta=0$ to $\theta=i\epsilon$ and enclose the contour in the lower half-
plane.

Evaluating the fourier transform using $-i\epsilon$ prescription, we get the
perturbative advanced celestial amplitude

$\small\begin{split}f_{1}^{-}(\omega)&=\frac{1}{4(e^{-\pi\omega}+1)}\\\
f_{2}^{-}(\omega)&=\frac{e^{\pi\omega}\left(\pi\omega+e^{\pi\omega}(\pi\omega-1)+1\right)(\coth(\pi\omega)-1)}{64\pi}\\\
f_{3}^{-}(\omega)&=\frac{e^{\pi\omega}(\coth(\pi\omega)-1)\left[e^{\pi\omega}\left(\pi^{2}\left(3\omega^{2}+2\right)-12\pi\omega+6\right)-3\pi\omega(\pi\omega+4)-2\left(3+\pi^{2}\right)\right]}{3072\pi^{2}}\\\
f_{4}^{-}(\omega)&=\frac{e^{\pi\omega}(\coth(\pi\omega)-1)\left[\pi(\pi\omega+3)\left(\pi\left(\omega^{2}+2\right)+6\omega\right)+e^{\pi\omega}\left(\pi(\pi\omega-3)\left(\pi\left(\omega^{2}+2\right)-6\omega\right)-6\right)+6\right]}{24576\pi^{3}}\\\
f_{5}^{-}(\omega)&=\frac{e^{\frac{3\pi\omega}{2}}(\coth(\pi\omega)-1)}{1966080\pi^{4}}\times\\\
&\Bigg{[}\left(120\pi^{2}\left(3\omega^{2}+2\right)+\pi^{4}\left(5\left(\omega^{2}+4\right)\omega^{2}+16\right)+120\right)\sinh\left(\frac{\pi\omega}{2}\right)-80\pi\omega\left(\pi^{2}\left(\omega^{2}+2\right)+6\right)\cosh\left(\frac{\pi\omega}{2}\right)\Bigg{]}.\end{split}$
(4.8)

In appendix A, we give the details of the computation of the perturbative
retarded and advanced celestial amplitude.

##  5 Crossing and unitarity conditions in celestial space

In this section, our aim is to translate the crossing and unitarity conditions
into the conditions on the celestial amplitude.

In terms of rapidity, $\theta$ we express the crossing and the unitarity
conditions as [7, 8, 9, 10]

$\begin{split}&S(\theta)=S(i\pi-\theta)\\\
&|S(\theta)|^{2}=1~{}~{}~{},\end{split}$ (5.1)

where $\theta=\theta_{1}-\theta_{2}$ .

The crossing condition physically implies the symmetry of the
$\mathcal{S}$-matrix under the exchange of the $s$ and $t$ channels. Unitarity
condition physically implies the probablity of getting $2$-particle final
state given initial state should be less than or equal to one, i.e.,
$|S(\theta)|^{2}\leq 1$. Assuming integrability condition, we have
$|S(\theta)|^{2}=1.$

The crossing condition in celestial space is obtained by taking the fourier
transform of both sides of the crossing condition in the rapidity variable.
Since, the perturbative expansion of the $\mathcal{S}$-matrix $S(\theta)$
contains poles at $\theta=0$, therefore, perturbatively if we expand upto a
given order we should take the fourier transform of $S(\theta\pm i\epsilon)$.

The crossing condition in celestial space is obtained by taking the fourier
transform of both sides

$\begin{split}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta+i\epsilon)&=\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(i\pi-\theta+i\epsilon)\\\
&=-\int_{i\pi+\infty}^{i\pi-\infty}d\theta^{\prime}e^{i\omega(i\pi-\theta^{\prime})}S(\theta^{\prime}+i\epsilon)~{}~{}~{}(i\pi-\theta\equiv\theta^{\prime})\\\
&=\int_{i\pi-\infty}^{i\pi+\infty}d\theta^{\prime}e^{i\omega(i\pi-\theta^{\prime})}S(\theta^{\prime}+i\epsilon)\\\
&=e^{-\omega\pi}\int_{-\infty}^{+\infty}d\theta^{\prime}e^{-i\omega\theta^{\prime}}S(\theta^{\prime}+i\epsilon)\\\
&=e^{-\omega\pi}\mathcal{A}^{+}(-\omega)\\\
\implies\mathcal{A}^{+}(\omega)&=e^{-\omega\pi}\mathcal{A}^{+}(-\omega)~{}~{}~{}.\end{split}$
(5.2)

where, in the last step we use

$\int_{i\pi-\infty}^{i\pi+\infty}d\theta^{\prime}e^{i\omega(i\pi-\theta^{\prime})}S(\theta^{\prime})=\int_{-\infty}^{+\infty}d\theta^{\prime}e^{i\omega(i\pi-\theta^{\prime})}S(\theta^{\prime})~{}~{}~{}.$
(5.3)

which is valid when we have no poles in the physical strip
$0<\text{Im}(\theta)<\pi.$

Similarly, the crossing condition while taking the fourier transform of

$\int_{-\infty}^{\infty}d\theta e^{i\omega\theta}S(\theta-i\epsilon)$

becomes

$\begin{split}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta-i\epsilon)&=\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(-i\pi-\theta-i\epsilon)\\\
\implies\mathcal{A}^{-}(\omega)&=e^{\omega\pi}\mathcal{A}^{-}(-\omega)~{}~{}~{}.\end{split}$
(5.4)

The crossing condition relates the retarded (advanced) celestial amplitude of
positive $\omega$ to the retarded (advanced) celestial amplitude of negative
$\omega$ and vice-versa.

Perturbatively expanding $\mathcal{A}^{\pm}(\omega)$ as

$\mathcal{A}^{\pm}(\omega)=2\pi\Big{[}\delta(\omega)+b^{2}f_{1}^{\pm}(\omega)+b^{4}f_{2}^{\pm}(\omega)+b^{6}f_{2}^{\pm}(\omega)+\cdots\Big{]}~{}~{}~{},$

$f_{n}^{\pm}(\omega)$ satisfy the crossing condition

$f_{n}^{\pm}(\omega)=e^{\mp\omega\pi}f_{n}^{\pm}(-\omega)~{}~{}~{}.$ (5.5)

The unitarity condition gives

$\begin{split}&S(\theta+i\epsilon)S(\theta+i\epsilon)^{*}=1\\\
\implies&S(\theta+i\epsilon)S(-\theta-i\epsilon)=1~{}~{}~{}.\end{split}$ (5.6)

Here, the unitarity condition $|S(\theta)|^{2}=1$ becomes
$S(\theta+i\epsilon)S(-\theta-i\epsilon)$. We combine the original unitarity
condition with the real analyticity of the $S$-matrix

$S(\theta+i\epsilon)^{*}=S(-\theta-i\epsilon)~{}~{}~{}.$

.

The unitarity condition in celestial space is obtained by taking the fourier
transform of both sides of

$S(\theta+i\epsilon)S(-\theta-i\epsilon)=1~{}~{}~{}.$

Now, the multiplication of functions $S(\theta+i\epsilon)S(-\theta-i\epsilon)$
gets converted into the convolution under the fourier transform as follows

$\begin{split}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta+i\epsilon)S(-\theta-i\epsilon)&=\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta+i\epsilon)\Bigg{[}\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega^{\prime}e^{i\omega^{\prime}\theta}\mathcal{A}^{-}(\omega^{\prime})\Bigg{]}\\\
&=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\mathcal{A}^{-}(\omega^{\prime})\int_{-\infty}^{\infty}S(\theta+i\epsilon)e^{i(\omega+\omega^{\prime})\theta}d\theta\\\
&=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\mathcal{A}^{-}(\omega^{\prime})\mathcal{A}^{+}(\omega+\omega^{\prime})~{}~{}~{}.\end{split}$
(5.7)

Here,

$\mathcal{A}^{\pm}(\omega)\equiv\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S(\theta\pm i\epsilon)~{}~{}~{}.$

Here, $\mathcal{A}^{+}(\omega)$ is the retarded celestial amplitude which is
the fourier transform using $+i\epsilon$ prescription enclosing the contour in
the upper half-plane in counterclockwise way and $\mathcal{A}^{-}(\omega)$ is
the advanced celestial amplitude which is the fourier transform using
$-i\epsilon$ prescription enclosing the contour in the lower half-plane in
clockwise way.

Therefore, the unitarity condition in celestial space becomes

$\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\mathcal{A}^{+}(\omega+\omega^{\prime})\mathcal{A}^{-}(\omega^{\prime})=2\pi\delta(\omega)~{}~{}~{}.$
(5.8)

Perturbatively expanding $\mathcal{A}^{\pm}(\omega)$ as

$\mathcal{A}^{\pm}(\omega)=2\pi\Big{[}\delta(\omega)+b^{2}f_{1}^{\pm}(\omega)+b^{4}f_{2}^{\pm}(\omega)+b^{6}f_{2}^{\pm}(\omega)+\cdots\Big{]}~{}~{}~{},$

$\begin{split}f_{n}^{\pm}(\omega)=\frac{1}{2\pi(b^{2})^{n}}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}S^{(n)}(\theta\pm i\epsilon)~{}~{}~{},\end{split}$ (5.9)

and then put it in eq.(5.8) we have the unitarity condition order by order in
perturbation theory

$\begin{split}f_{1}^{+}(\omega)+f_{1}^{-}(-\omega)=0~{}~{}~{},\end{split}$
$\begin{split}f_{n}^{+}(\omega)+f_{n}^{-}(-\omega)+\sum_{j=1}^{n-1}\int_{-\infty}^{\infty}d\omega^{\prime}f_{n-j}^{+}(\omega+\omega^{\prime})f_{j}^{-}(\omega^{\prime})=0~{}~{}~{}(n>1)~{}~{}~{}.\end{split}$
(5.10)

One important thing to note is that while translating the crossing and
unitarity conditions in celestial space the retarded and the advanced
celestial amplitudes naturally appears.

Now, since

$S^{(n)}(\theta+i\epsilon)+S^{(n)}(-\theta-i\epsilon)=2~{}\text{Re}~{}S^{(n)}(\theta+i\epsilon)$

$f_{n}^{+}(\omega)+f_{n}^{-}(-\omega)$ is related ro the fourier transform of
the real part of $S^{(n)}(\theta+i\epsilon)$

$f_{n}^{+}(\omega)+f_{n}^{-}(-\omega)=2~{}\frac{1}{2\pi(b^{2})^{n}}\int_{-\infty}^{\infty}d\theta
e^{i\omega\theta}\text{Re}S^{(n)}(\theta+i\epsilon)~{}~{}~{}.$ (5.11)

Therefore, we can say that the unitarity condition in perturbation theory
translated into celestial amplitude relates the fourier transform of the real
part of the perturbative $\mathcal{S}$-matrix at a given order to the
convolution of the retarded and advanced celestial amplitude of lower-orders.
We illustrate the unitarity condition by the diagram 2.

Figure 2: Pictorial representation of the unitarity condition in celestial
space

###  5.1 Checking the crossing $\&$ unitarity conditions in celestial space
for the 2d Sinh-Gordon model

We check the crossing and unitarity conditions for the $2d$ Sinh-model using
celestial amplitudes obtained in section 4.

$f_{n}^{\pm}(\omega)$ satisfy the crossing condition

$f_{n}^{\pm}(\omega)=e^{\mp\omega\pi}f_{n}^{\pm}(-\omega)~{}~{}~{}.$ (5.12)

The unitarity conditions are satisfied

$\begin{split}f_{1}^{+}(\omega)+f_{1}^{-}(-\omega)=0~{}~{}~{},\end{split}$
$\begin{split}f_{2}^{+}(\omega)+f_{2}^{-}(-\omega)+\int_{-\infty}^{\infty}d\omega^{\prime}f_{1}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})=0~{}~{}~{},\end{split}$
(5.13)

where the integral is given by

$\int_{-\infty}^{\infty}d\omega^{\prime}f_{1}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})=\frac{\omega}{16(1-e^{\pi\omega})}~{}~{}~{}.$
(5.14)

In deriving the crossing and unitarity conditions in celestial space we should
be extremely careful about the $i\epsilon$ prescription.

The convolution of the perturbative retarded and retarded celestial amplitude
and the perturbative advanced and advanced celestial amplitude diverges

$\int_{-\infty}^{\infty}d\omega^{\prime}f_{1}^{\pm}(\omega+\omega^{\prime})f_{1}^{\pm}(\omega^{\prime})\rightarrow\infty~{}~{}~{}.$

The proper $i\epsilon$ prespription involving the convolution of the retarded
and advanced celestial amplitude cures this divergence.

##  6 Bootstrapping Celestial amplitude

In this section, we see that using the crossing and unitarity conditions, how
much we can get for the higher order celestial amplitude from the lower order
celestial amplitude.

Crossing condition translated in celestial space gives

$f_{n}^{\pm}(\omega)=e^{\mp\pi\omega}f_{n}^{\pm}(-\omega)~{}~{}~{}.$ (6.1)

Unitarity condition translated in celestial space gives

$f_{n}^{+}(\omega)+f_{n}^{-}(-\omega)+\sum_{j=1}^{n-1}\int_{-\infty}^{\infty}d\omega^{\prime}f_{n-j}^{+}(\omega+\omega^{\prime})f_{j}^{-}(\omega^{\prime})=0~{}~{}~{}.$
(6.2)

Now, the relation between $f_{n}^{+}(\omega)$ and $f_{n}^{-}(\omega)$ is

$f_{n}^{+}(\omega)-f_{n}^{-}(\omega)+\frac{1}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(n)}(\theta)\Big{]}_{\theta=0}=0~{}~{}~{}.$
(6.3)

Figure 3: Contours for perturbative retarded and advanced celestial amplitudes

From the fig.3, we see that the difference between $f_{n}^{+}(\omega)$ and
$f_{n}^{-}(\omega)$ is same as the negative of
$\frac{1}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(n)}(\theta)\Big{]}_{\theta=0}$. This
is because when we enclose the contour in the upper half-plane, the small
semicircle
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\mathcal{C}_{\varepsilon_{1}}}$
is in clockwise sense which adds to the
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\mathcal{C}_{\varepsilon_{2}}}$
semicircle in clockwise sense after taking the difference of
$f_{n}^{+}(\omega)$ and $f_{n}^{-}(\omega)$ and by convention while
calculating the residue at $\theta=0$ we enclose the semicircle in
anticlockwise sense.

As a consistency check we satisfy this equation by evaluating the residue of
$e^{i\omega\theta}S^{(n)}(\theta)$ at $\theta=0$ for $n=1,\ldots,5$.

$\small\begin{split}\frac{1}{2\pi b^{2}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(1)}(\theta)\Big{]}_{\theta=0}&=\frac{1}{4}\\\
\frac{1}{2\pi(b^{2})^{2}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(2)}(\theta)\Big{]}_{\theta=0}&=\frac{\pi\omega-1}{32\pi}\\\
\frac{1}{2\pi(b^{2})^{3}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(3)}(\theta)\Big{]}_{\theta=0}&=\frac{3\pi^{2}\omega^{2}-12\pi\omega+2\pi^{2}+6}{1536\pi^{2}}\\\
\frac{1}{2\pi(b^{2})^{4}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(4)}(\theta)\Big{]}_{\theta=0}&=\frac{\pi^{3}\omega^{3}-9\pi^{2}\omega^{2}+2\pi^{3}\omega+18\pi\omega-6\pi^{2}-6}{12288\pi^{3}}\\\
\frac{1}{2\pi(b^{2})^{5}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(5)}(\theta)\Big{]}_{\theta=0}&=\frac{5\pi^{4}\omega^{4}-80\pi^{3}\omega^{3}+20\pi^{4}\omega^{2}+360\pi^{2}\omega^{2}-160\pi^{3}\omega-480\pi\omega+16\pi^{4}+240\pi^{2}+120}{1966080\pi^{4}}~{}~{}~{}.\end{split}$
(6.4)

Using crossing and unitarity conditions in eq.(6.1) and eq.(6.2) along with
eq.(6.3) we get

$\begin{split}&f_{n}^{+}(\omega)=\frac{1}{(1+e^{-\pi\omega})}\Bigg{[}-\underbrace{\frac{e^{-\pi\omega}}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(n)}(\theta)\Big{]}_{\theta=0}}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{not
fixed by crossing and unitarity
conditions}}}-\underbrace{\sum_{j=1}^{n-1}\int_{-\infty}^{\infty}d\omega^{\prime}f_{n-j}^{+}(\omega+\omega^{\prime})f_{j}^{-}(\omega^{\prime})}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{fixed
by crossing and unitarity conditions}}}\Bigg{]}~{}~{}~{}.\end{split}$ (6.5)

The second term in eq.(6.5) is fixed by the crossing and unitarity conditions
while the first term is not fixed by the crossing and unitarity conditions.

Therefore, we can calculate $f_{2}^{+}$, $f_{3}^{+}$ and so on

$\small\begin{split}&f_{2}^{+}(\omega)=\frac{1}{(1+e^{-\pi\omega})}\Bigg{[}-\frac{e^{-\pi\omega}}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(2)}(\theta)\Big{]}_{\theta=0}-\int_{-\infty}^{\infty}d\omega^{\prime}f_{1}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})\Bigg{]}\\\
&f_{3}^{+}(\omega)=\frac{1}{(1+e^{-\pi\omega})}\Bigg{[}-\frac{e^{-\pi\omega}}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(3)}(\theta)\Big{]}_{\theta=0}-\int_{-\infty}^{\infty}d\omega^{\prime}\Big{[}f_{1}^{+}(\omega+\omega^{\prime})f_{2}^{-}(\omega^{\prime})+f_{2}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})\Big{]}\Bigg{]}\\\
&f_{4}^{+}(\omega)=\frac{1}{(1+e^{-\pi\omega})}\Bigg{[}-\frac{e^{-\pi\omega}}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(4)}(\theta)\Big{]}_{\theta=0}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\int_{-\infty}^{\infty}d\omega^{\prime}\Big{[}f_{1}^{+}(\omega+\omega^{\prime})f_{3}^{-}(\omega^{\prime})+f_{2}^{+}(\omega+\omega^{\prime})f_{2}^{-}(\omega^{\prime})+f_{3}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})\Big{]}\Bigg{]}\\\
&f_{5}^{+}(\omega)=\frac{1}{(1+e^{-\pi\omega})}\Bigg{[}-\frac{e^{-\pi\omega}}{2\pi(b^{2})^{n}}~{}2\pi
i~{}\text{Res}\Big{[}e^{i\omega\theta}S^{(4)}(\theta)\Big{]}_{\theta=0}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\int_{-\infty}^{\infty}d\omega^{\prime}\Big{[}f_{1}^{+}(\omega+\omega^{\prime})f_{4}^{-}(\omega^{\prime})+f_{2}^{+}(\omega+\omega^{\prime})f_{3}^{-}(\omega^{\prime})+f_{3}^{+}(\omega+\omega^{\prime})f_{2}^{-}(\omega^{\prime})+f_{4}^{+}(\omega+\omega^{\prime})f_{1}^{-}(\omega^{\prime})\Big{]}\Bigg{]}~{}~{}~{}.\end{split}$
(6.6)

##  7 Gravitational dressing of the $2d$ QFT amplitude in celestial space

For integrable field theories, in presence of irrelevant deformation
$T\bar{T}$, the $2d$ QFT $\mathcal{S}$-matrix is modified by a pure phase [11]

$S_{ij}^{kl}(\theta)\to
S_{ij}^{kl}(\theta)e^{i\delta_{ij}^{(t)}(\theta)}~{}~{}~{},$ (7.1)

where, the diagonal phase shift $\delta_{ij}^{(t)}(\theta)$ is given by
deformation parameter $t$ and difference of rapidities
$\theta=\theta_{i}-\theta_{j}$

$\delta_{ij}^{(t)}(\theta)=tm_{i}m_{j}\sinh\theta~{}~{}~{}.$ (7.2)

The deformation parameter $t$ is related to the string length $t=2l_{s}^{2}$
in effective string theory. For same mass particles, we have

$S(\theta)\to S(\theta)e^{2il_{s}^{2}m^{2}\sinh\theta}~{}~{}~{}.$ (7.3)

This factor is also discussed in the paper [7] as a solution to the crossing
and unitarity conditions i.e., $S(\theta)=S(i\pi-\theta)$ and
$S(\theta)S(-\theta)=1$ .

In terms of $s$-variable we have

$S(s)\to S(s)e^{il_{s}^{2}\sqrt{s(s-4m^{2})}}~{}~{}~{},$ (7.4)

the dressing factor for massless particle reduces to $e^{il_{s}^{2}s}$.

This dressing factor introduced in [12] in the context of gravitational
scattering of relativistic point particles in trans-Planckian regime and large
impact parameter is referred to as the gravitational dressing factor.

In this section, we study the gravitational dressing of the $2d$ QFT amplitude
in the celestial space restricting to massless particles.

The gravitational dressing of the $2d$ QFT amplitude for massless particles is
given by

$\begin{split}S(s)\to S(s)S_{\text{grav}}(s)~{}~{}~{},\end{split}$ (7.5)

where, the gravitational dressing factor is

$S_{\text{grav}}(s)=e^{il_{s}^{2}s}~{}~{}~{}.$ (7.6)

The $\mathcal{S}$-matrix is a function of mandelstem variable $s$ which is
center-of-mass energy squared. In the celestial space we define the celestial
amplitude $\mathcal{A}(\omega)$ as

$\begin{split}\mathcal{A}(\omega)\equiv&\int_{0}^{\infty}dss^{\omega-1}S(s)~{}~{}~{}\\\
\implies S(s)&=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}d\omega
s^{-\omega}\mathcal{A}(\omega)~{}~{}~{},\end{split}$ (7.7)

where, we trade the mandelstem variable $s$ for a rindler energy or conformal
dimension $\omega$ diagonalizing boosts in the directions of the particles.

Now, Mellin transform of $e^{il_{s}^{2}s}$ is

$\begin{split}\int_{0}^{\infty}dss^{\omega-1}e^{il_{s}^{2}s}=(-il_{s}^{2})^{-\omega}\Gamma(\omega)~{}~{}~{}.\end{split}$
(7.8)

The gravitational dressing gives

$\begin{split}\int_{0}^{\infty}dss^{\omega-1}~{}S(s)\to\int_{0}^{\infty}dss^{\omega-1}~{}S(s)e^{il_{s}^{2}s}\end{split}$
(7.9)

The gravitational dressing condition becomes

$\begin{split}\mathcal{A}(\omega)&\to\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}d\omega^{\prime}(-il_{s}^{2})^{-\omega^{\prime}}\Gamma(\omega^{\prime})\mathcal{A}(\omega-\omega^{\prime})~{}~{}~{}.\\\
\end{split}$ (7.10)

Now, we take several ansatzes for $\mathcal{A}(\omega)$ that has a pole on the
right half-plane of $\omega$ and see what we get after the convolution.

### Ansatz for $\mathcal{A}(\omega)$

### Ansatz 1

We take $\mathcal{A}(\omega)$ as

$\mathcal{A}(\omega)=\csc\pi\omega~{}~{}~{}.$ (7.11)

The function has poles at $\omega=n~{},~{}~{}n\in\mathbb{Z}~{}~{}.$

Performing the convolution we get

$\begin{split}&\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}d\omega^{\prime}(-il_{s}^{2})^{-\omega^{\prime}}\Gamma(\omega^{\prime})\csc(\pi(\omega-\omega^{\prime}))\\\
&=\frac{e^{-il_{s}^{2}}\Gamma(\omega)\Gamma(1-\omega,-il_{s}^{2})}{\pi}~{}~{}~{}.\end{split}$
(7.12)

where we use the Mellin-Barnes integral representation of the upper incomplete
gamma function $\Gamma(a,z)$[Eq.(3.4.11) in [13], p. 113]

$\Gamma(a,z)=-\frac{z^{a-1}e^{-z}}{\Gamma(1-a)}\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}ds~{}\Gamma(s+1-a)\pi z^{-s}\csc\pi
s~{}~{}~{}.$ (7.13)

we put $s=\omega^{\prime}-\omega,~{}a=1-\omega,~{}z=-il_{s}^{2}$ in eq.(7.13)
to prove eq.(7.12).

Now, the upper incomplete gamma function $\Gamma(a,z)$ is an entire function
of $a$ when $z\neq 0$. Therefore, for $l_{s}^{2}\neq 0$,
$\Gamma(1-\omega,-il_{s}^{2})$ is an entire function of $1-\omega$. The
function $\Gamma(\omega)$ has poles at
$\omega=n~{},~{}~{}n=\mathbb{Z}^{-}\displaystyle\cup\\{0\\}$.

$\csc\pi\omega$ has poles at $\omega=n~{},~{}~{}n\in\mathbb{Z}$, after
gravitational dressing the function
$\frac{e^{-il_{s}^{2}}\Gamma(\omega)\Gamma(1-\omega,-il_{s}^{2})}{\pi}$ has
poles at $\omega=n~{},~{}~{}n=\mathbb{Z}^{-}\displaystyle\cup\\{0\\}$.

We can see that after gravitational dressing the poles on the right half-plane
of $\omega$ are absent.

### Ansatz 2

We take $\mathcal{A}(\omega)$ as

$\mathcal{A}(\omega)=\frac{1}{\omega-\mathfrak{C}}~{}~{}~{},$ (7.14)

where, $\mathfrak{C}$ is located on the right half plane, i.e.,
$\text{Re}(\mathfrak{C})>0$.

Performing the convolution we get

$\begin{split}&\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}d\omega^{\prime}(-il_{s}^{2})^{-\omega^{\prime}}\Gamma(\omega^{\prime})\frac{1}{\omega-\omega^{\prime}-\mathfrak{C}}~{}~{}~{}.\\\
&=(-il_{s}^{2})^{\mathfrak{C}-\omega}\gamma(\omega-\mathfrak{C},-il_{s}^{2})~{}~{}~{}.\end{split}$
(7.15)

where we use the Mellin-Barnes integral representation of the lower incomplete
gamma function $\gamma(a,z)$ [Eq.(3.4.10) in [13], p. 113]

$\begin{split}\gamma(a,z)=\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}~{}ds~{}\frac{\Gamma(-s)}{s+a}z^{s+a}ds~{}~{}~{}.\end{split}$
(7.16)

Now, the lower incomplete gamma function
$\gamma(\omega-\mathfrak{C},-il_{s}^{2})$ is meromorphic with simple poles at

$\begin{split}\omega-\mathfrak{C}&=-n~{}~{}~{},~{}~{}~{}n=\mathbb{Z}^{+}\displaystyle\cup\\{0\\}\\\
\implies\omega&=\mathfrak{C}-n~{}~{}~{}.\end{split}$ (7.17)

For $n=0$ , $\text{Re}(\mathfrak{C})>0$ therefore we have a pole at
$\omega=\mathfrak{C}$ which lies on the right half-plane.

Now, if we impose $\text{Re}(\mathfrak{C}-n)<0$ , $n=\mathbb{Z}^{+}$ then we
say that there are no other poles on the right half-plane but still there is a
pole at $\omega=\mathfrak{C}$, $\text{Re}(\mathfrak{C})>0$.

### Ansatz 3

We take $\mathcal{A}(\omega)$ as

$\mathcal{A}(\omega)=\Gamma(-\omega)~{}~{}~{}.$ (7.18)

The function has poles at
$\omega=n~{},~{}~{}n=\mathbb{Z}^{+}\displaystyle\cup\\{0\\}~{}~{}~{}.$

Performing the convolution we get

$\begin{split}&\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}d\omega^{\prime}(-il_{s}^{2})^{-\omega^{\prime}}\Gamma(\omega^{\prime})\Gamma(-(\omega-\omega^{\prime}))\\\
&=2\left(-il_{s}^{2}\right)^{-\frac{\omega}{2}}K_{\omega}\left(2\sqrt{-il_{s}^{2}}\right)~{}~{}~{}.\end{split}$
(7.19)

where we use the Mellin-Barnes integral representation of the modified Bessel
function of the second kind $K_{\nu}(z)$[Eq.(3.4.18) in [13], p. 114]

$K_{\nu}(z)=\frac{\Big{(}\frac{1}{2}z\Big{)}^{\nu}}{4\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}ds~{}\Gamma(s)\Gamma(s-\nu)\Big{(}\frac{z}{2}\Big{)}^{-2t}~{}~{}~{}.$
(7.20)

Now, the modified Bessel function of the second kind $K_{\nu}(z)$ has only one
singular point at $\nu=\infty$ for fixed $z$. Here, for fixed $l_{s}^{2}$,
$K_{\omega}(2\sqrt{-il_{s}^{2}})$ has pole when $\omega\to\infty$.

We can see that after gravitational dressing the poles on the right half-plane
of $\omega$ are absent.

Therefore, from this analysis we see that the poles on the right half-plane
are absent after gravitational dressing if the function contains multiple
poles.

The absence of poles has nice analogy with the absence of the bulk point
singularity for AdS/CFT correlators [21]. For celestial amplitude, in [22] we
see that the exponentially soft high-energy behavior when translated into
celestial amplitude erases the poles on the right half-plane. We can think the
gravitaional dressing condition in $2d$ as a constraint which gives some
constraint on the analytic structure of the celestial amplitude.

##  8 Conclusions and furure directions

In this paper, we study celestial amplitude for $2d$ bulk $\mathcal{S}$-matrix
and show that for massive scalar particles the celestial amplitude is just the
fourier transform of the $\mathcal{S}$-matrix written in the rapidity
variable. For the Sinh-Gordon $\mathcal{S}$-matrix we evaluate the
perturbative celestial amplitude and see that there should be two types of
celestial amplitude, the retarded and the advanced due to the presence of the
pole in the origin of the complex rapidity-plane.

Here, for the Sinh-Gordon model, the exact $\mathcal{S}$-matrix has no pole at
rapidity, $\theta=0$, since $\alpha$ is real. The rapidity, $\theta=0$ pole is
a perturbative manifestation, for this there are these two perturbative
celestial amplitudes corresponding to two $i\epsilon$ prescriptions. The
difference between these two perturbative amplitudes is related to the residue
of the pole at $\theta=0$. One important point to note is that these two
perturbative amplitudes also naturally appear in unitarity equation. After
resumming the perturbative celestial amplitude it’s nice to get the celestial
amplitude corresponding to the exact $\mathcal{S}$-matrix. Since there are two
perturbative amplitudes, which one will resum to be the celestial amplitude
for the exact $\mathcal{S}$-matrix? This is a very interesting question. To
evaluate the celestial amplitude corresponding to the exact
$\mathcal{S}$-matrix by resummation of the perturbative celestial amplitude is
difficult.

In the celestial space, we translate the crossing and unitarity conditions and
check these conditions for the Sinh-Gordon model. From the celestial CFT
perspective, we ask about determining the higher order celestial amplitude
from the lower order i.e., bootstrapping celestial amplitude by imposing the
“crossing and unitarity” conditions. Finally, we analyze the gravitational
dressing condition for the $2d$ QFT amplitude in celestial space restricted to
massless particles. We see that this condition manifests itself as eraser of
poles from the right half-plane in the celestial space. There are several
promising future directions related to this work.

Bootstrapping celestial amplitude in $4d$ for massive scalar theory: It would
be interesting to extend our analysis to the massive $\phi^{4}$ theory in $4d$
with an aim to compute the one-loop correction to the four-point amplitude in
$\phi^{4}$ theory in flat space by imposing the “crossing and unitarity
conditions” to the celestial amplitude.

Gravitational dressing for massive scaterring in $2d$: Another question is to
translate the gravitational dressing for massive scattering in $2d$ into
celestial space and see it’s consequence. In doing this, the exact analytical
result for inverting the gravitational dressing factor for the massive case in
the celestial space can’t be obtained. But, one can try to analyze this using
some approximation.

Space of celestial amplitude in 2d: Another interesting direction in massive
QFTs in $2d$ is to find the allowed dual celestial space of $2\to 2$
$\mathcal{S}$-matrix by imposing the crossing, unitarity, analyticity and
$O(N)$ symmetry into the celestial amplitude i.e., to understand the dual
space of [24] in the celestial space.

Flat limit of massive scalar AdS amplitude and it’s contact with massive
scalar celestial amplitude: For the massive scalar celestial amplitude, it’s
nice to connect the celestial amplitude to the flat limit of AdS amplitude.
For this, we can use the dictionary between the positions of operators at the
boundary of AdS and the momenta of particles in the flat limit of AdS [25].
The celestial amplitude written in conformal basis which is an integral
transform of the flat space amplitude in momentum space [2] can be translated
to an integral transform of positions of operators at the boundary of AdS. We
can then try to see the interpretation of the integral transform in terms of
CFT living on the boundary. The light-ray operators are some class of
operators in CFT defined by an integral, it would be interesting to make
connections with the integral transform with light-ray operators. In this
context, we can try to perform the integral transform by taking some AdS
contact correlators and analyze the result.

Flat limit of the massless scattering in AdS: We can take the bulk-point limit
related to high energy scattering of the massive scalar celestial amplitude
expressed in the boundary of AdS in which the mass of the particles can be
neglected. It’s nice to make a connection with the flat limit of massless
scattering in AdS [26].

Conformal block expansion for the celestial amplitude: In this line of
thinking, it would be useful to understand the analog of conformal block
expansion for the celestial amplitude. In AdS, we can convolute the conformal
block expansion of the boundary four-point function with the integral
transform and see how the block gets modified. We can work entirely in flat
space and using the partial wave expansion of the four point amplitude in a
basis of Gegenbauer polynomials [23] we can perform the integral transform and
the question is after the integral transform whether the transformed object
has a nice structure in terms of conformal block decomposition, it would be
nice to make connection with the conformal block expansion in the Celestial
CFT [28].

## Acknowledgements

I would like to thank Shota Komatsu for his guidance throughout this work and
comments on the earlier version of the draft. I also thank R. Loganayagam for
useful discussions, giving me mental support and comments on the earlier
version of the draft. I also thank Chethan Krishnan and Suvrat Raju for useful
discussions. This work is supported by the Department of Atomic Energy,
Government of India, under project no. RTI4001.

##  Appendix A Perturbative retarded and advanced celestial amplitude
computation in $2d$ Sinh-Gordon model

In this appendix A, we calculate the perturbative retarded and advanced
celestial amplitude in $2d$ Sinh-Gordon model.

In the complex $\theta$-plane, the perturbative $\mathcal{S}$-matrices
$S^{(n)}(\theta)$ for $n=1,\ldots,5$ contain poles at $\theta=n\pi i$,
$n\in\mathbb{Z}$.

While calculating the perturbative retarded celestial amplitude
$f_{n}^{+}(\omega)$, we choose the contour in the upper half-plane as in fig.4
and enclose the poles as $\theta=n\pi i$, $n\in\mathbb{Z}^{+}$.

Figure 4: Contour for evaluating perturbative retarded celestial amplitude

While calculating the perturbative advanced celestial amplitude
$f_{n}^{-}(\omega)$, we choose the contour in the lower half-plane as in fig.5
and enclose the poles as $\theta=n\pi i$, $n\in\mathbb{Z}^{-}$.

Figure 5: Contour for evaluating perturbative advanced celestial amplitude

The residues of the functions appeared in the perturbative expansion of the
$\mathcal{S}$-matrix at the poles as $\theta=n\pi i$, $n\in\mathbb{Z}$ are
given by

$\begin{split}\text{Res}\Big{[}\csch\theta~{}e^{i\omega\theta}\Big{]}_{\theta=n\pi
i}&=(-1)^{n}e^{-n\pi\omega}\\\
\text{Res}\Big{[}\csch^{2}\theta~{}e^{i\omega\theta}\Big{]}_{\theta=n\pi
i}&=i\omega~{}e^{-n\pi\omega}\\\
\text{Res}\Big{[}\csch^{3}\theta~{}e^{i\omega\theta}\Big{]}_{\theta=n\pi
i}&=-\frac{1}{2}(-1)^{n}(\omega^{2}+1)e^{-n\pi\omega}\\\
\text{Res}\Big{[}\csch^{4}\theta~{}e^{i\omega\theta}\Big{]}_{\theta=n\pi
i}&=-\frac{1}{6}i\omega(\omega^{2}+4)e^{-n\pi\omega}\\\
\text{Res}\Big{[}\csch^{5}\theta~{}e^{i\omega\theta}\Big{]}_{\theta=n\pi
i}&=\frac{1}{24}(-1)^{n}(\omega^{2}+1)(\omega^{2}+9)e^{-n\pi\omega}~{}~{}~{}.\end{split}$
(A.1)

## References

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