# An extension of Venkatesh’s converse theorem to the Selberg class

Andrew R. Booker  ,  Michael Farmer  and  Min Lee  School of Mathematics,
University of Bristol, Woodland Road, Bristol, BS8 1UG

###### Abstract.

We extend Venkatesh’s proof of the converse theorem for classical holomorphic
modular forms to arbitrary level and character. The method of proof, via the
Petersson trace formula, allows us to treat arbitrary degree $2$ gamma factors
of Selberg class type.

M. L. was supported by Royal Society University Research Fellowship
“Automorphic forms, L-functions and trace formulas”.

##  1\. Introduction

Venkatesh, in his thesis [Ven02], gave a new proof of the classical converse
theorem for modular forms of level $1$, in the context of Langlands’ “Beyond
Endoscopy”. The key analytic input is Voronoi summation, or equivalently the
functional equation for additive twists. More precisely, given a modular form
$f(z)=\sum_{n=1}^{\infty}f_{n}n^{\frac{k-1}{2}}e(nz)$ of weight $k$, level
$N$, and nebentypus character $\chi$, we define the complete additive twist

$\Lambda_{f}(s,\alpha)=\Gamma_{\mathbb{C}}\bigl{(}s+\tfrac{k-1}{2}\bigr{)}\sum_{n=1}^{\infty}\frac{f_{n}e(n\alpha)}{n^{s}}\quad\text{
for }\alpha\in\mathbb{Q},$

where $e(z)=e^{2\pi iz}$ and $\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$.
Then, for any $q\in N\mathbb{Z}_{>0}$ and any $a,\bar{a}\in\mathbb{Z}$ with
$a\bar{a}\equiv 1\pmod{q}$, $\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$ and
$\Lambda_{f}\bigl{(}s,-\tfrac{\bar{a}}{q})$ continue to entire functions and
satisfy the functional equation [KMV02, (A.10)]

$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}=i^{k}\chi(\bar{a})q^{1-2s}\Lambda_{f}\bigl{(}1-s,-\tfrac{\bar{a}}{q}\bigr{)}.$

In this paper we generalize Venkatesh’s proof to forms of arbitrary level and
character. Our precise result is the following:

######  Theorem 1.1.

Consider $\\{f_{n}\\}_{n\geq 1}$, $N$, $\chi$, $\omega$, and $\gamma(s)$ with
the following properties:

  1. (1)

$\\{f_{n}\\}_{n\geq 1}$ is a sequence of complex numbers such that
$\sum_{n=1}^{\infty}f_{n}n^{-s}$ converges absolutely for $\Re(s)>1$;

  2. (2)

$N$ is a natural number and $\chi$ is a Dirichlet character modulo $N$;

  3. (3)

$\omega$ is a nonzero complex number;

  4. (4)

$\gamma(s)=Q^{s}\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})$ for some numbers
$Q,\lambda_{j}\in\mathbb{R}_{>0}$ and $\mu_{j}\in\mathbb{C}$ with
$\Re(\mu_{j})>-\frac{1}{2}\lambda_{j}$ and $\sum_{j=1}^{r}\lambda_{j}=1$.

Given any $\alpha\in\mathbb{Q}$, define the complete twisted $L$-function

$\Lambda_{f}(s,\alpha)=\gamma(s)\sum_{n=1}^{\infty}\frac{f_{n}e(n\alpha)}{n^{s}}.$

Suppose that for every $q\in N\mathbb{Z}_{>0}$ and every pair
$a,\bar{a}\in\mathbb{Z}$ with $a\bar{a}\equiv 1\pmod{q}$,
$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$ and
$\Lambda_{f}\bigl{(}s,-\tfrac{\bar{a}}{q}\bigr{)}$ continue to entire
functions of finite order and satisfy the functional equation

(1.1)
${}\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}=\omega\chi(\bar{a})q^{1-2s}\Lambda_{f}\bigl{(}1-s,-\tfrac{\bar{a}}{q}\bigr{)}.$

Then there exists $k\in\mathbb{Z}_{>0}$ such that
$f(z):=\sum_{n=1}^{\infty}f_{n}n^{\frac{k-1}{2}}e(nz)$ is a modular form of
weight $k$, level $N$, and nebentypus character $\chi$.

######  Remarks 1.2.

  1. (1)

One feature of the argument is that it admits a generalization to gamma
factors of Selberg class type. In this way, our result can be viewed as a
converse theorem for degree $2$ elements of the Selberg class, albeit with
infinitely many functional equations. Recently, Kaczorowski and Perelli [KP20]
have classified the elements of the Selberg class of conductor $1$ without the
need for any twists. Very little is known for higher conductor, however, and
our result is the first that we are aware of to consider both arbitrary level
and degree $2$ gamma factor.

  2. (2)

For $k>1$, it is enough to assume that the finite $L$-functions
$L_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}=\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}/\gamma(s)$
have analytic continuation to $\mathbb{C}$, and in this case we can also
conclude that $f$ is cuspidal. When $k=1$, we need to know that
$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$ is analytic at $s=0$, and there
are noncuspidal examples satisfying all of the hypotheses of Theorem 1.1.

  3. (3)

If we suppose that $L_{f}(s,1)$ lies in the Selberg class then we can combine
the transformation formula in [KP15, Theorem 2] with the Vandermonde argument
in [BK14, Lemma 2.4] to constrain the possible poles of
$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$. In this way, it is likely
possible to prove a result that allows the twisted $L$-functions
$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$ to have arbitrary poles inside the
critical strip, but we have not pursued this.

  4. (4)

Using the Bruggeman–Kuznetsov trace formula and the method of [HLN21], it is
likely possible to prove a similar converse theorem for Maass forms.

##  2\. Lemmas

We begin with some preparatory lemmas.

######  Lemma 2.1.

Let $\gamma(s)=\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})$, where
$\lambda_{j}\in\mathbb{R}_{>0}$, $\mu_{j}\in\mathbb{C}$. If $\gamma(s)$ has
poles at all but finitely many negative integers then
$\sum_{j=1}^{r}\lambda_{j}\geq 1$.

###### Proof.

Since the poles of $\Gamma(\lambda_{j}s+\mu_{j})$ are spaced
$\lambda_{j}^{-1}$ apart, the number of poles of $\gamma(s)$ in
$\mathbb{Z}\cap[-T,0)$ for large $T>0$ is at most
$T\sum_{j=1}^{r}\lambda_{j}+O(1)$. If all but finitely many negative integers
are poles of $\gamma(s)$ then this count is at least $T+O(1)$. The conclusion
follows on taking $T\to\infty$. ∎

######  Lemma 2.2.

Let $\gamma(s)$ be as in the statement of Theorem 1.1, and suppose that
$\gamma(s)$ has poles at all but finitely many negative integers. Then
$\gamma(s)$ is of the form $cP(s)H^{s}\Gamma_{\mathbb{C}}(s)$, where
$c,H\in\mathbb{R}_{>0}$ and $P$ is a monic polynomial whose roots are distinct
non-positive integers.

###### Proof.

Recall that $\gamma(s)=Q^{s}\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})$. By
Lemma 2.1 each factor in the product must contribute infinitely many poles in
the negative integers; in particular $\lambda_{j},\mu_{j}\in\mathbb{Q}$ for
each $j$. Let $\lambda_{j}=a_{j}/q_{j}$ in lowest terms. If $a_{j}>1$ then we
can consider $\tilde{\gamma}(s)=\Gamma(\frac{s-m_{j}}{q_{j}})\prod_{i\neq
j}\Gamma(\lambda_{i}s+\mu_{i})$, where $m_{j}$ is the negative integral pole
of $\Gamma(\lambda_{j}s+\mu_{j})$ of smallest absolute value. Then
$\tilde{\gamma}(s)$ also has poles at all but finitely many negative integers,
contradicting Lemma 2.1.

Hence we have $a_{j}=1$, so
$\gamma(s)=Q^{s}\prod_{j=1}^{r}\Gamma(\frac{s+\nu_{j}}{q_{j}})$, where
$\nu_{j}=\mu_{j}q_{j}\in\mathbb{Z}_{\geq 0}$. Let
$q=\operatorname{lcm}\\{q_{1},\ldots,q_{j}\\}$ and write
$\Gamma(\frac{s+\nu_{j}}{q_{j}})=\Gamma(\frac{q}{q_{j}}\cdot\frac{s+\nu_{j}}{q})$.
By the Gauss multiplication formula, we have

$\gamma(s)=\bigl{(}\sqrt{2\pi}\bigr{)}^{r-n}Q^{s}\prod_{j=1}^{r}\left(\frac{q}{q_{j}}\right)^{\frac{s+\nu_{j}}{q_{j}}-\frac{1}{2}}\prod_{i=0}^{q/q_{j}-1}\Gamma\\!\left(\frac{s+\nu_{j}+iq_{j}}{q}\right),$

which we can rewrite in the form
$cH^{s}\prod_{j=1}^{q}\Gamma\\!\left(\frac{s+\nu_{j}^{\prime}}{q}\right),$ for
some $c,H\in\mathbb{R}_{>0}$ and $\nu_{j}^{\prime}\in\mathbb{Z}_{\geq 0}$.
Moreover, the $\nu_{j}^{\prime}$ must run through a complete set of
representatives for the residue classes mod $q$. Replacing each
$\nu_{j}^{\prime}$ by its mod $q$ reduction
$\nu_{j}^{\prime\prime}\in\\{0,\ldots,q-1\\}$ and using the recurrence formula
to relate $\Gamma(\frac{s+\nu_{j}^{\prime}}{q})$ and
$\Gamma(\frac{s+\nu_{j}^{\prime\prime}}{q})$, we get
$c^{\prime}P(s)H^{s}\prod_{j=1}^{q}\Gamma\\!\left(\frac{s+\nu_{j}^{\prime\prime}}{q}\right),$
with $P$ as described in the statement of the lemma. Finally, applying the
Gauss multiplication formula once more and inserting a factor of
$2(2\pi)^{-s}$, we arrive at the claimed form. ∎

The final lemma that we need is a special case of [HLN21, Lemma 4.1].

######  Lemma 2.3.

Let $r_{q}(n)=\sum_{\begin{subarray}{c}a\bmod q\\\
(a,q)=1\end{subarray}}e(na/q)$ be the Ramanujan sum. Then, for
$n,N\geq\mathbb{Z}_{>0}$ and $\Re(s)>1$, we have

$\sum_{\begin{subarray}{c}q\geq 1\\\ N\mid
q\end{subarray}}\frac{r_{q}(n)}{q^{2s}}=\begin{cases}\frac{\sigma_{1-2s}(n;N)}{\zeta^{(N)}(2s)}&\text{
when }n\neq 0,\\\ N^{1-2s}\prod_{p\mid
N}(1-p^{-1})\frac{\zeta(2s-1)}{\zeta^{(N)}(2s)}&\text{ when }n=0.\end{cases}$

Here, when $\frac{N}{\prod_{p\mid N}p}\mid n$,

(2.1) $\sigma_{s}(n;N)=\prod_{\begin{subarray}{c}p\mid n\\\ p\nmid
N\end{subarray}}\frac{p^{(\operatorname{ord}_{p}(n)+1)s}-1}{p^{s}-1}\cdot\prod_{p\mid
N}\left(\frac{(1-p^{s-1})p^{(\operatorname{ord}_{p}(n)+1)s}-(1-p^{-1})p^{\operatorname{ord}_{p}(N)s}}{p^{s}-1}\right).$

Otherwise $\sigma_{s}(n;N)=0$.

##  3\. Proof of Theorem 1.1

Let $S_{k}(N,\chi)$ denote the space of holomorphic modular forms of weight
$k$, level $N$ and nebentypus character $\chi$, and let $H_{k}(N,\chi)$ be an
orthonormal basis for $S_{k}(N,\chi)$. Each $g\in H_{k}(N,\chi)$ has a Fourier
expansion of the form

$g(z)=\sum_{n=1}^{\infty}\rho_{g}(n)n^{\frac{k-1}{2}}e(nz)$

for some coefficients $\rho_{g}(n)\in\mathbb{C}$. An application of the
Petersson formula gives the following formula found in [IK04, Corollary
14.23]: for positive integers $n$ and $m$, we have

(3.1) $\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\sum_{g\in
H_{k}(N,\chi)}\rho_{g}(n)\overline{\rho_{g}(m)}=\delta_{n=m}+2\pi
i^{-k}\sum_{\begin{subarray}{c}q\geq 1\\\ N\mid
q\end{subarray}}\frac{S_{\chi}(m,n;q)}{q}J_{k-1}\\!\left(\frac{4\pi\sqrt{mn}}{q}\right),$

where

$S_{\chi}(m,n;q)=\sum_{\begin{subarray}{c}a\bmod{q}\\\
\gcd(a,q)=1\end{subarray}}\chi(a)e\\!\left(\frac{ma+n\bar{a}}{q}\right)$

is the twisted Kloosterman sum and $J_{k-1}(y)$ is the classical $J$-Bessel
function.

Fix a choice of data $\\{f_{n}\\}_{n\geq 1}$, $N$, $\chi$, $\omega$, and
$\gamma(s)$ satisfying the hypotheses of Theorem 1.1. Let
$\epsilon\in\\{0,1\\}$ be such that $\chi(-1)=(-1)^{\epsilon}$. For $k\geq 4$,
$\Re(s)\in(\frac{1}{2},\frac{k-1}{2})$, $x>0$, and
$\sigma_{1}\in(\frac{1-k}{2},-\Re(s))$, define

(3.2) ${}F_{k}(s,x)=\frac{1}{2\pi
i}\int_{\Re(u)=\sigma_{1}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})\gamma(1-s-u)}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})\gamma(s+u)}x^{u}\,du.$

By Stirling’s formula, the integral converges absolutely since
$\Re(s)>\frac{1}{2}$. Note also that our choice of $\sigma_{1}$ ensures that
the contour $\Re(u)=\sigma_{1}$ separates the poles of
$\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})$ and $\gamma(1-s-u)$.

The proof will be split into two cases depending on whether $F_{k}(1,1)$ is
nonzero for some $k$ or not.

######  Proposition 3.1.

Suppose that $F_{k}(1,1)\neq 0$ for some $k\geq 4$ with
$k\equiv\epsilon\pmod{2}$. Then $f$ as defined in Theorem 1.1 is in
$S_{k}(N,\chi)$.

###### Proof.

Fix $k$ as in the hypotheses. For $n\in\mathbb{Z}_{>0}$ and $\Re(s)>1$, define

(3.3)
$K_{n}(s,f,\chi)=\zeta^{(N)}(2s)\sum_{m=1}^{\infty}\frac{f_{m}}{m^{s}}\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\sum_{g\in
H_{k}(N,\chi)}\rho_{g}(n)\overline{\rho_{g}(m)}=\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\sum_{g\in
H_{k}(N,\chi)}\rho_{g}(n)L(s,f\times\bar{g}),$

where

$L(s,f\times\bar{g})=\zeta^{(N)}(2s)\sum_{m=1}^{\infty}\frac{f_{m}\overline{\rho_{g}(m)}}{m^{s}}\quad\text{and}\quad\zeta^{(N)}(s)=\zeta(s)\prod_{p\mid
N}\bigl{(}1-p^{-s}\bigr{)}.$

Note that the series for $L(s,f\times\bar{g})$ converges absolutely for
$\Re(s)>1$ thanks to the Ramanujan bound
$\rho_{g}(m)\ll_{\varepsilon}m^{\varepsilon}$.

Now consider $s\in\mathbb{C}$ with $\Re(s)\in(\frac{5}{4},\frac{k-1}{2})$.
Applying (3.1) to (3.3), we get

$\displaystyle K_{n}(s,f,\chi)$
$\displaystyle=\zeta^{(N)}(2s)\sum_{m=1}^{\infty}\frac{f_{m}}{m^{s}}\bigg{\\{}\delta_{n=m}+2\pi
i^{-k}\sum_{\begin{subarray}{c}q\geq 1\\\ N\mid
q\end{subarray}}\frac{S_{\chi}(m,n;q)}{q}J_{k-1}\\!\left(\frac{4\pi\sqrt{mn}}{q}\right)\bigg{\\}}$
$\displaystyle=\zeta^{(N)}(2s)\frac{f_{n}}{n^{s}}+2\pi
i^{-k}\zeta^{(N)}(2s)\sum_{\begin{subarray}{c}q\geq 1\\\ N\mid
q\end{subarray}}\frac{1}{q}\sum_{m=1}^{\infty}\frac{f_{m}S_{\chi}(m,n;q)}{m^{s}}J_{k-1}\\!\left(\frac{4\pi\sqrt{mn}}{q}\right).$

Here the interchange of the sums is justified for $\Re(s)>\frac{5}{4}$ by the
estimates

$J_{k-1}(y)\ll\min\bigl{\\{}y^{k-1},y^{-1/2}\bigr{\\}}\quad\text{and}\quad
S_{\chi}(m,n;q)\ll_{n,\varepsilon}q^{1/2+\varepsilon}.$

Let

$K_{0,n}(s,f,\chi)=K_{n}(s,f,\chi)-f_{n}n^{-s}\zeta^{(N)}(2s).$

By the Mellin–Barnes type integral representation [GR15, (6.422)], we have

$2\pi J_{k-1}(4\pi y)=\frac{1}{2\pi
i}\int_{\Re(u)=\sigma_{0}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}y^{-2u}\,du,$

for any choice of $\sigma_{0}\in(\frac{1-k}{2},0)$. Applying this with
$\sigma_{0}\in(1-\Re(s),0)$, we may change the order of sum and integral, to
obtain

$K_{0,n}(s,f,\chi)=i^{-k}\zeta^{(N)}(2s)\sum_{\begin{subarray}{c}q\geq 1\\\
N\mid q\end{subarray}}\frac{1}{2\pi
i}\int_{\Re(u)=\sigma_{0}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}q^{2u-1}\sum_{m=1}^{\infty}\frac{f_{m}S_{\chi}(m,n;q)}{m^{s}}(mn)^{-u}\,du.$

Opening up the Kloosterman sum, this becomes

$\sum_{\begin{subarray}{c}q\geq 1\\\ N\mid q\end{subarray}}\frac{1}{2\pi
i}\int_{\Re(u)=\sigma_{0}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}q^{2u-1}n^{-u}\sum_{\begin{subarray}{c}a\bmod{q}\\\
a\bar{a}\equiv
1\bmod{q}\end{subarray}}\chi(a)e(n\bar{a}/q)L_{f}\bigl{(}s+u,\tfrac{a}{q}\bigr{)}\,du.$

Next we shift the contour to $\Re(u)=\sigma_{1}\in(\frac{1-k}{2},-\Re(s))$, so
that $\Re(1-s-u)>1$, and apply the functional equation

$\displaystyle\sum_{\begin{subarray}{c}a\bmod{q}\\\ a\bar{a}\equiv
1\bmod{q}\end{subarray}}\chi(a)e(n\bar{a}/q)L_{f}\bigl{(}s+u,\tfrac{a}{q}\bigr{)}=\omega
q^{1-2s-2u}\frac{\gamma(1-s-u)}{\gamma(s+u)}\sum_{\begin{subarray}{c}a\bmod{q}\\\
a\bar{a}\equiv
1\bmod{q}\end{subarray}}e(n\bar{a}/q)L_{f}\bigl{(}1-s-u,-\tfrac{\bar{a}}{q}\bigr{)},$

obtaining

$\displaystyle
K_{0,n}(s,f,\chi)=i^{-k}\omega\zeta^{(N)}(2s)\sum_{\begin{subarray}{c}q\geq
1\\\ N\mid q\end{subarray}}\frac{1}{q^{2s}}\frac{1}{2\pi i}$
$\displaystyle\int_{\Re(u)=\sigma_{1}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}\frac{\gamma(1-s-u)}{\gamma(s+u)}n^{-u}$
$\displaystyle\cdot\sum_{\begin{subarray}{c}a\bmod{q}\\\ a\bar{a}\equiv
1\bmod{q}\end{subarray}}e(n\bar{a}/q)L_{f}\bigl{(}1-s-u,-\tfrac{\bar{a}}{q}\bigr{)}\,du.$

Note that the contour shift is justified by the fact that
$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}$ has finite order and the estimates

$L_{f}\bigl{(}1-s-u,-\tfrac{\bar{a}}{q}\bigr{)}\ll
1\quad\text{and}\quad\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}\frac{\gamma(1-s-u)}{\gamma(s+u)}\ll|u|^{-2\Re(s)}\quad\text{for
}\Re(u)=\sigma_{1}.$

The same estimates show that we may swap the order of sum and integral. We
also expand $L_{f}\bigl{(}1-s-u,-\tfrac{\bar{a}}{q}\bigr{)}$ as a Dirichlet
series, obtaining

$\sum_{\begin{subarray}{c}a\bmod{q}\\\ a\bar{a}\equiv
1\bmod{q}\end{subarray}}e(n\bar{a}/q)L_{f}\bigl{(}1-s-u,-\tfrac{\bar{a}}{q}\bigr{)}=\sum_{\begin{subarray}{c}a\bmod{q}\\\
a\bar{a}\equiv
1\bmod{q}\end{subarray}}e(n\bar{a}/q)\sum_{m=1}^{\infty}\frac{f_{m}e(-m\bar{a}/q)}{m^{1-s-u}}=\sum_{m=1}^{\infty}\frac{f_{m}r_{q}(n-m)}{m^{1-s-u}},$

where $r_{q}$ is the Ramanujan sum. An application of Lemma 2.3 leads to the
following expression.

(3.4) $\displaystyle K_{n}(s,f$
$\displaystyle,\chi)=f_{n}n^{-s}\zeta^{(N)}(2s)$ $\displaystyle+i^{-k}\omega
f_{n}n^{s-1}\zeta(2s-1)N^{1-2s}\prod_{p\mid
N}(1-p^{-1})\cdot\int_{\Re(u)=\sigma_{1}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}\frac{\gamma(1-s-u)}{\gamma(s+u)}\,du$
$\displaystyle+i^{-k}\omega\sum_{\begin{subarray}{c}m\geq 1\\\ m\neq
n\end{subarray}}\frac{f_{m}\sigma_{1-2s}(n-m;N)}{m^{1-s}}\int_{\Re(u)=\sigma_{1}}\frac{\Gamma_{\mathbb{C}}(u+\frac{k-1}{2})}{\Gamma_{\mathbb{C}}(-u+\frac{k+1}{2})}\frac{\gamma(1-s-u)}{\gamma(s+u)}\left(\frac{m}{n}\right)^{u}\,du.$

It is straightforward to see that
$\sigma_{1-2s}(r;N)\ll_{N,\varepsilon}|r|^{\varepsilon}$, uniformly for $r\neq
1$ and $\Re(s)\geq\frac{1}{2}$. Thus, for a fixed $\sigma_{1}$, both integrals
and the sum over $m$ converge absolutely for $\frac{1}{2}<\Re(s)<-\sigma_{1}$.
This establishes the meromorphic continuation of $K_{n}(s,f,\chi)$ to that
region, and hence also of $\sum_{g\in
H_{k}(N,\chi)}\rho_{g}(n)L(s,f\times\bar{g})$, in view of (3.3).

Since the $g$ form a basis for $S_{k}(N,\chi)$, we can choose a finite set
$\\{n_{i}:i=1,\ldots,d\\}$, where $d=\dim S_{k}(N,\chi)$, such that the
vectors $(\rho_{g}(n_{1}),\ldots,\rho_{g}(n_{d}))$ are linearly independent.
Taking a suitable linear combination of (3.4) for $n=n_{i}$, we deduce the
meromorphic continuation of $L(s,f\times\bar{g})$ to $\Re(s)>\frac{1}{2}$ for
each individual $g$. The only possible pole is at $s=1$, and taking residues
we see that

$\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\sum_{g\in
H_{k}(N,\chi)}\rho_{g}(n)\operatorname{Res}_{s=1}L(s,f\times\bar{g})=\tfrac{1}{2}i^{-k}\omega
N^{-1}\prod_{p\mid N}(1-p^{-1})\cdot F_{k}(1,1)f_{n}.$

Since $F_{k}(1,1)\neq 0$, we see that there exist $x_{g}\in\mathbb{C}$ such
that $f_{n}=\sum_{g\in H_{k}(N,\chi)}x_{g}\rho_{g}(n)$ for any positive
integer $n$. Since $S_{k}(N,\chi)$ is a vector space, we have proved the
claim. ∎

We have taken care of the case where the integral (3.2) is nonzero at $s=1$,
$x=1$ for some $k\geq 4$. If this is not the case, we use the following
proposition.

######  Proposition 3.2.

If $F_{k}(1,1)=0$ for all $k\geq 4$ with $k\equiv\epsilon\pmod{2}$, then
$\gamma(s)$ is of the form $cH^{s}\Gamma_{\mathbb{C}}(s+\frac{\ell-1}{2})$,
where $c,H\in\mathbb{R}_{>0}$, and $\ell\in\\{1,2,3\\}$ with
$\ell\equiv\epsilon\pmod{2}$.

###### Proof.

We replace $u$ by $u/2$ in the definition of $F_{k}(1,1)$ and shift the
contour to $\Re(u)=-\frac{5}{2}$, which is permissible for all $k\geq 4$. Our
hypothesis then implies that

(3.5) $\frac{1}{2\pi
i}\int_{\Re(u)=-\frac{5}{2}}\frac{\Gamma_{\mathbb{C}}(\frac{k-1+u}{2})\gamma(-u/2)}{2\Gamma_{\mathbb{C}}(\frac{k+1-u}{2})\gamma(1+u/2)}\,du=0\quad\text{for
all }k\geq 4\text{ with }k\equiv\epsilon\;(\text{mod }2).$

For $n\geq 0$, define

$f_{n}(y)=\frac{\mathbf{1}_{(0,1)}(y)}{\sqrt{1-y^{2}}}\begin{cases}\cos(n\arcsin{y})&\text{if
}2\mid n,\\\ \sin(n\arcsin{y})&\text{if }2\nmid n.\end{cases}$

Using the formulas in [GR15, §3.631], we see that $f_{n}$ has Mellin transform

$\widetilde{f}_{n}(s)=\int_{0}^{\infty}f_{n}(y)y^{s-1}\,dy=\frac{(-1)^{\lfloor{n/2}\rfloor}\Gamma_{\mathbb{C}}(s)}{2^{s}\Gamma_{\mathbb{C}}(\frac{s+n+1}{2})\Gamma_{\mathbb{C}}(\frac{s-n+1}{2})}\quad\text{for
}\Re(s)>0.$

For $k\equiv\epsilon\pmod{2}$ we have

$\displaystyle\frac{\Gamma_{\mathbb{C}}(\frac{k-1+u}{2})\gamma(-u/2)}{2\Gamma_{\mathbb{C}}(\frac{k+1-u}{2})\gamma(1+u/2)}$
$\displaystyle=\frac{\Gamma_{\mathbb{C}}(1-u)}{2^{1-u}\Gamma_{\mathbb{C}}(\frac{k+1-u}{2})\Gamma_{\mathbb{C}}(\frac{3-k-u}{2})}\cdot\frac{2^{-u}\Gamma_{\mathbb{C}}(\frac{3-k-u}{2})\Gamma_{\mathbb{C}}(\frac{k-1+u}{2})}{\Gamma_{\mathbb{C}}(1-u)}\frac{\gamma(-u/2)}{\gamma(1+u/2)}$
$\displaystyle=\frac{(-1)^{\lfloor{(k-1)/2}\rfloor}\widetilde{f}_{k-1}(1-u)}{2^{u-1}\Gamma_{\mathbb{C}}(1-u)\sin(\frac{\pi}{2}(u+k-1))}\frac{\gamma(-u/2)}{\gamma(1+u/2)}$
$\displaystyle=\frac{\widetilde{f}_{k-1}(1-u)}{2^{u-1}\Gamma_{\mathbb{C}}(1-u)\sin(\frac{\pi}{2}(u+1-\epsilon))}\frac{\gamma(-u/2)}{\gamma(1+u/2)}$
$\displaystyle=\widetilde{f}_{k-1}(1-u)\frac{\sqrt{2}\Gamma_{\mathbb{C}}(\frac{1-\epsilon+u}{2})}{\Gamma_{\mathbb{C}}(\frac{2-\epsilon-u}{2})}\frac{\gamma(-u/2)}{\gamma(1+u/2)}.$

Thus, (3.5) can be written as

$\frac{1}{2\pi
i}\int_{\Re(u)=-\frac{5}{2}}\widetilde{f}_{k-1}(1-u)\widetilde{g}(u)\,du=0,\quad\text{where}\quad\widetilde{g}(u)=\frac{\sqrt{2}\Gamma_{\mathbb{C}}(\frac{1-\epsilon+u}{2})}{\Gamma_{\mathbb{C}}(\frac{2-\epsilon-u}{2})}\frac{\gamma(-u/2)}{\gamma(1+u/2)}.$

Applying the inverse Mellin transform, we define

$g(y)=\frac{1}{2\pi
i}\int_{\Re(u)=-\frac{5}{2}}\widetilde{g}(u)y^{-u}\,du\quad\text{for }y>0.$

By Stirling’s formula we have $\widetilde{g}(u)\ll|u|^{-3/2}$ for
$\Re(u)=-\frac{5}{2}$, and it follows that $g(y)$ extends continuously to
$[0,\infty)$.

An application of Fubini’s theorem shows that for any measurable functions
$f,g$ on $(0,\infty)$ satisfying
$\int_{0}^{\infty}|f(y)|y^{-\sigma}\,dy<\infty$ and
$\int_{\mathbb{R}}|\widetilde{g}(\sigma+it)|\,dt<\infty$, we have

$\frac{1}{2\pi
i}\int_{\Re(u)=\sigma}\widetilde{f}(1-u)\widetilde{g}(u)\,du=\int_{0}^{\infty}f(y)g(y)\,dy.$

It is easy to see this is satisfied by the functions $f_{n}$ and $g$ above
with $\sigma=-\frac{5}{2}$. Hence, (3.5) becomes

$\int_{0}^{1}f_{k-1}(y)g(y)\,dy=0\quad\text{for all }k\geq 4\text{ with
}k\equiv\epsilon\;(\text{mod }2).$

Suppose $\epsilon=1$. Then we have

$\int_{-\pi}^{\pi}\cos(\tfrac{k-1}{2}\theta)g(|\sin(\theta/2)|)\,d\theta=0\quad\text{for
every odd }k\geq 5.$

Thus, the function $h(\theta)=g(|\sin(\theta/2)|)$ has Fourier series of the
form $a+b\cos\theta$. Since $h$ is continuous, we have
$h(\theta)=a+b\cos\theta$ for all $\theta$, and thus $g(y)=a+b-2by^{2}$ for
$y\in[0,1]$.

Since $g(y)=O(y^{5/2})$, we must have $a=b=0$. Computing the Mellin transform
again, we see that $\widetilde{g}(u)=\int_{1}^{\infty}g(y)\,y^{u-1}\,dy$, so
$\widetilde{g}(u)$ is analytic for $\Re(u)<-\frac{5}{2}$. Since
$\frac{\gamma(-u/2)}{\Gamma_{\mathbb{C}}(\frac{1-u}{2})}$ is analytic and
nonvanishing for $\Re(u)<-\frac{5}{2}$, it follows that
$\frac{\Gamma_{\mathbb{C}}(\frac{u}{2})}{\gamma(1+u/2)}$ is analytic for
$\Re(u)<-\frac{5}{2}$. This means that $\gamma(s)$ has a pole at each negative
integer.

Applying Lemma 2.2, we have $\gamma(s)=cP(s)H^{s}\Gamma_{\mathbb{C}}(s)$,
where $P$ is a monic polynomial whose roots are distinct non-positive
integers. Since $\gamma(s)$ has poles at all negative integers, either
$P(s)=1$ or $P(s)=s$. Thus, $\gamma(s)$ is of the form
$c^{\prime}H^{s}\Gamma_{\mathbb{C}}(s+\frac{\ell-1}{2})$ for some
$\ell\in\\{1,3\\}$, as required.

Now suppose that $\epsilon=0$. Then

(3.6)
$\int_{-\pi}^{\pi}\frac{\sin(\frac{k-1}{2}\theta)}{\sin(\theta/2)}|\sin(\theta/2)|g(|\sin(\theta/2)|)\,d\theta=0\quad\text{for
every even }k\geq 4.$

Note that
$\frac{\sin(\frac{k-1}{2}\theta)}{\sin(\theta/2)}=W_{\frac{k-2}{2}}(\cos\theta),$
where $W_{n}$ is a polynomial of degree $n$ (the ‘Chebyshev polynomial of the
fourth kind’, see [Mas93]). Writing $v=\cos\theta=1-2\sin^{2}(\theta/2)$,
(3.6) becomes

$0=\int_{-1}^{1}W_{\frac{k-2}{2}}(v)h(v)\sqrt{\frac{1-v}{1+v}}\,dv,\quad\text{where
}h(v)=\frac{g\\!\left(\sqrt{\frac{1-v}{2}}\right)}{\sqrt{1-v}}\text{ for
}v\in(-1,1).$

Since the $W_{n}$ are an orthogonal family with respect to the measure
$\sqrt{\frac{1-v}{1+v}}\,dv$, there exists a constant $a$ such that
$(h(v)-a)\sqrt{\frac{1-v}{1+v}}$ is continuous and absolutely integrable on
$(-1,1)$, and orthogonal to all polynomials. It follows that $h(v)=a$ for all
$v\in(-1,1)$, and thus $g(y)=a\sqrt{2}y$ for $y\in(0,1)$. Since
$g(y)=O(y^{5/2})$, we must have $a=0$, and thus
$\widetilde{g}(u)=\int_{1}^{\infty}g(y)\,y^{u-1}\,dy$.

As before we conclude that
$\frac{\Gamma_{\mathbb{C}}(\frac{1+u}{2})}{\gamma(1+u/2)}$ is analytic for
$\Re(u)<-\frac{5}{2}$. Defining $\tilde{\gamma}(s)=\gamma(s+\frac{1}{2})$ we
see that $\tilde{\gamma}(s)$ satisfies the hypotheses imposed on $\gamma(s)$
in Theorem 1.1 and has a pole at every negative integer. Appealing again to
Lemma 2.2, we find that $\tilde{\gamma}(s)$ is of the form
$cP(s)H^{s}\Gamma_{\mathbb{C}}(s)$, where either $P(s)=1$ or $P(s)=s$. Thus
$\gamma(s)=c^{\prime}H^{s}P(s-\frac{1}{2})\Gamma_{\mathbb{C}}(s-\frac{1}{2})$.
Thanks to the hypothesis $\Re(\mu_{j})>-\frac{1}{2}\lambda_{j}$ in Theorem
1.1, $\gamma(s)$ cannnot have a pole at $\frac{1}{2}$, and therefore $P(s)=s$.
Hence $\gamma(s)=c^{\prime\prime}H^{s}\Gamma_{\mathbb{C}}(s+\frac{1}{2})$, as
required. ∎

Now we can complete the proof of Theorem 1.1. In view of Propositions 3.1 and
3.2, we may assume that
$\gamma(s)=cH^{s}\Gamma_{\mathbb{C}}(s+\frac{\ell-1}{2})$, where
$c,H\in\mathbb{R}_{>0}$ and $\ell\in\\{1,2,3\\}$ with
$\ell\equiv\epsilon\pmod{2}$. In this case we fall back on a more traditional
proof of the converse theorem as in [Raz77], but we must first address the
fact that our gamma factor differs from the expected one by the exponential
factor $H^{s}$.

Suppose first that $H>1$. Equation (3.5) becomes

$\frac{1}{2\pi
i}\int_{\Re(u)=-\frac{5}{2}}H^{-u}\frac{\Gamma_{\mathbb{C}}(\frac{k-1+u}{2})\Gamma_{\mathbb{C}}(\frac{-u}{2}+\frac{\ell-1}{2})}{\Gamma_{\mathbb{C}}(\frac{k+1-u}{2})\Gamma_{\mathbb{C}}(1+\frac{u}{2}+\frac{\ell-1}{2})}\,du=0\quad\text{for
all }k\geq 4\text{ with }k\equiv\epsilon\;(\text{mod }2).$

Since $H>1$, we can shift the contour to the right, as the integrand vanishes
in the limit as $\Re(u)\rightarrow\infty$.

Suppose $\ell=2$. Then when $k=4$, we pick up poles at $u=1,3$ and derive that
$0=H^{-1}-H^{-3}$. Thus $H=1$, giving a contradiction. Similarly, when
$\ell=3$ we take $k=5$; we have poles at $u=2,4$, getting $0=H^{-2}-H^{-4}$,
which results in the same contradiction. When $\ell=1$, looking at $k=5$, we
have poles at $u=0,2,4$; thus $0=1-4H^{-2}+3H^{-4}$, which implies that
$H^{-2}=\frac{1}{3}$. Now looking at $k=7$, we get an extra pole at $u=6$, so
that $0=\frac{2}{3}-6H^{-2}+12H^{-4}-\frac{20}{3}H^{-6}$, which is not
satisfied for $H^{-2}=\frac{1}{3}$.

Hence $H\leq 1$. Let $f(z)=\sum_{n=1}^{\infty}f_{n}n^{\frac{\ell-1}{2}}e(nz)$.
Applying Hecke’s argument [Miy06, Theorem 4.3.5] to our Voronoi formulas
(1.1), we get the modularity relation

(3.7)
${}f\\!\left(\frac{\frac{-1}{H^{2}z}+a}{q}\right)=\omega\chi(\bar{a})(-iHz)^{\ell}f\\!\left(\frac{z-\bar{a}}{q}\right)$

for all $q\in N\mathbb{Z}_{>0}$ and $a,\bar{a}\in\mathbb{Z}$ with
$a\bar{a}\equiv 1\pmod{q}$.

The $\ell$-slash operator is defined for matrices $M=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ of positive determinant by

$(f|M)(z)=(\det M)^{\ell/2}(cz+d)^{-\ell}f\\!\left(\frac{az+b}{cz+d}\right).$

In this notation, (3.7) becomes

$f\bigg{|}\begin{pmatrix}aH^{2}&-1\\\ qH^{2}&0\\\
\end{pmatrix}=i^{-\ell}\omega\chi(\bar{a})f\bigg{|}\begin{pmatrix}1&-\bar{a}\\\
0&q\\\ \end{pmatrix},$

or equivalently

(3.8) ${}f\bigg{|}\begin{pmatrix}aH&\frac{a\bar{a}H^{2}-1}{qH}\\\
qH&\bar{a}H\\\ \end{pmatrix}=i^{-\ell}\omega\chi(\bar{a})f.$

We may assume that $f$ is not identically $0$, since the conclusion is trivial
otherwise. Applying (1.1) twice, we have

$\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)}=\omega\chi(\bar{a})q^{1-2s}\Lambda_{f}\bigl{(}1-s,-\tfrac{\bar{a}}{q}\bigr{)}=\omega^{2}\chi(-1)\Lambda_{f}\bigl{(}s,\tfrac{a}{q}\bigr{)},$

and thus $(i^{-\ell}\omega)^{2}=(-1)^{\ell}\chi(-1)=1$.

Suppose $H<1$. Taking $a=\bar{a}=1$, the matrix above has absolute trace
$2H<2$, so is elliptic. Unless $2H=\zeta+\zeta^{-1}$ for some root of unity
$\zeta$, this matrix has infinite order, and then a generalization of Weil’s
lemma [BBB+18, Lemma 4.2] implies that $f=0$.

Hence $2H$ must be an algebraic integer. For prime $p$ and $i=1,2$, let
$M_{i,p}$ be the matrix $\begin{pmatrix}H&\frac{H-H^{-1}}{q_{i}}\\\
q_{i}H&H\end{pmatrix}$, where $q_{1}=pN$, $q_{2}=(p+1)N$. We compute that

$\operatorname{tr}(M_{1,p}M_{2,p})=(2H)^{2}-2-\bigl{(}4-(2H)^{2}\bigr{)}\bigl{(}4p(p+1)\bigr{)}^{-1}.$

Since $\lim_{p\rightarrow\infty}\operatorname{tr}(M_{1,p}M_{2,p})=(2H)^{2}-2$,
we have $|\operatorname{tr}(M_{1,p}M_{2,p})|<2$ for all sufficiently large
primes $p$, and again by Weil’s lemma, it follows that
$\operatorname{tr}(M_{1,p}M_{2,p})$ is an algebraic integer. Let $K$ be a
number field containing $2H$. Clearly $\operatorname{tr}(M_{1,p}M_{2,p})\in K$
for each $p$. Taking norms,

$N_{K/\mathbb{Q}}\bigl{(}\operatorname{tr}(M_{1,p}M_{2,p})+2-(2H)^{2}\bigr{)}=\bigl{(}-4p(p+1)\bigr{)}^{-[K:\mathbb{Q}]}N_{K/\mathbb{Q}}\bigl{(}4-(2H)^{2}\bigr{)}.$

Taking a sufficiently large prime $p$, this is not an integer, giving a
contradiction.

Hence $H=1$. By (3.8), it follows that

$f|M=i^{-\ell}\omega\chi(M)f\quad\text{for }M=\begin{pmatrix}a&b\\\
q&\bar{a}\end{pmatrix}\in\Gamma_{0}(N)\text{ with }q>0,$

where we define $\chi(M)=\chi(\bar{a})$. Taking $M_{1},M_{2}\in\Gamma_{0}(N)$
such that $M_{1}$, $M_{2}$ and $M_{1}M_{2}$ are all of this form, we have

$i^{-\ell}\omega\chi(M_{1}M_{2})f=f|M_{1}M_{2}=(f|M_{1})|M_{2}=(i^{-\ell}\omega)^{2}\chi(M_{1})\chi(M_{2})f.$

Since $\chi(M_{1}M_{2})=\chi(M_{1})\chi(M_{2})$, we have must have
$\omega=i^{\ell}$. Finally, since the $M$ of the above form, together with
$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ and $\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}$, generate $\Gamma_{0}(N)$, we conclude that $f\in
M_{\ell}(N,\chi)$, as required.

## References

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# Renormalized contact interaction in degenerate unitary Bose gases

Yijue Ding<EMAIL_ADDRESS>Chris H. Greene<EMAIL_ADDRESS>Department
of Physics and Astronomy, Purdue University, West Lafayette, Indiana, USA

###### Abstract

We renormalize the two-body contact interaction based on the exact solution of
two interacting particles in a harmonic trap. This renormalization extends the
validity of the contact interaction to large scattering lengths. We apply this
renormalized interaction to a degenerate unitary Bose gas to study its
stationary properties and elementary excitations using the mean-field theory
and the hyperspherical method. Since the scattering length is no longer a
relevant length scale at unitarity, universal properties are obtained that
depend only on the average particle density. Our treatment shows that the
universal relations for the total energy and for the two-body contact are
$E/N=12.67\hbar^{2}\langle n^{2/3}\rangle/2m$ and $C_{2}/N=11.8\langle
n^{1/3}\rangle$ respectively.

unitary Bose gas, renormalization, mean-field theory, hyperspherical
coordinates

###### pacs:

##  I Introduction

Strongly correlated systems near quantum degeneracy exhibit a wide range of
intriguing phenomena. Paradigmatic examples include helium superfluidity and
the fractional quantum Hall effect. In the atomic physics realm, the ultracold
quantum gas, due to its simplicity, purity and high controllability, is an
excellent candidate to be used to study strongly correlated systems. The
interaction between cold atoms, which is typically characterized by the
$s$-wave scattering length, can be readily controlled through Feshbach
resonancesChin  _et al._ (2010).

The Bose Einstein condensate (BEC) is a highly degenerate quantum system in
which the interparticle interaction can also be tuned via a magnetic or other
types of Feshbach resonanceInouye  _et al._ (1998). When the scattering length
in a BEC is much larger than any length scale of the system, the gas has
reached the so-called unitary regimeCastin and Werner (2012). However,
creating a BEC in the strongly interacting regime or even all the way to
unitarity is extremely difficult. The major reason is that the three-body
recombination rate at zero temperature in dilute gases is proportional to
$a^{4}$Esry  _et al._ (1999); Nielsen and Macek (1999); Bedaque  _et al._
(2000); Braaten and Hammer (2006), which results in a very short lifetime of
the strongly interacting Bose gas. This phenomenon contrasts sharply with the
strongly interacting Fermi gas, for which the three-body recombination is
suppressed by the Pauli exclusion principlePetrov  _et al._ (2004). Because of
the prohibitively high atom loss rate, it has been considered nearly
impossible to access the unitary Bose gas adiabatically. However, a
nonadiabatic approach to unitarity has been developed by the JILA groupMakotyn
_et al._ (2014). In their experimental work, they studied the nonequilibrium
dynamics of a degenerate unitary Bose gas and observed important universal
properties of the system.

Although a few theories have been proposed to treat the degenerate unitary
Bose gasCowell  _et al._ (2002); Song and Zhou (2009); Diederix  _et al._
(2011); Rossi  _et al._ (2014); Zhou and Mashayekhi (2013); Werner and Castin
(2006), no existing theories so far are capable of completely describing this
system. Some theories involve complex derivations such as the renormlization
group theoryZhou and Mashayekhi (2013) or extensive computations like Monte
Carlo simulationRossi  _et al._ (2014). Shortly after the JILA experiment on
the unitary Bose gas, various theoretical descriptions were proposed in an
effort to explain the experimental results, especially the momentum
distributionSmith  _et al._ (2014); Sykes  _et al._ (2014); Laurent  _et al._
(2014); Kira (2015); Comparin and Krauth (2016).

In this article, we introduce a renormalized contact potential similar to that
in Ref. von Stecher and Greene (2007), to extend the validity of the zero-
range potential to the strongly interacting regime. Then we employ this
renormalized potential in company with traditional many-body theories to study
the degenerate unitary Bose gas at zero temperature. The structure of this
article is as follows: In section II, we elaborate the renormalization
procedure and the physical ideas behind it. After that, we apply this
renormalized potential to a few many-body theories and show how they are
modified with the inclusion of the renormalization. In section III, we discuss
the stationary properties and elementary excitations of a degenerate unitary
Bose gas using our renormalization theory, and compare our results with other
theoretical predictions. We particularly focus on a few important physical
observables of the system. Finally, in section IV, we summarize our work and
the most significant findings.

##  II Theory of renormalization

Our renormalization is similar to that in a two-component Fermi gasvon Stecher
and Greene (2007). Here we summarize the procedure and the physical origin of
this renormalization. For a uniform gas system, when the range of the two-body
interaction is much smaller than both the scattering length $a$ and the
average interparticle distance determined by the particle density $n$, the
behavior of the system is characterized by the dimensionless parameter
$na^{3}$. This is also equivalent to the dimensionless parameter $k_{F}a$,
where $k_{F}=(6\pi^{2}n)^{1/3}$ is defined for the Bosonic system in a manner
akin to the Fermi momentum. Our idea is to design an effective scattering
length $a_{eff}$ that can replace the bare scattering length $a$ to describe
the properties of the system. In this case, the dimensionless parameter
becomes $k_{F}a_{eff}$. Therefore, there must be a correspondence between
$k_{F}a_{eff}$ and $k_{F}a$ characterized by a renormalization function
$k_{F}a_{eff}=\zeta(k_{F}a)$. The effective scattering length is designed
specifically for a system of two interacting particles in a harmonic trap such
that it can exactly describe the atomic ground state energy.

For two particles in a harmonic trap with a circular frequency $\omega_{ho}$
interacting with a regularized pseudo potential

$V(\textbf{r})=\frac{4\pi\hbar^{2}a}{m}\delta(\textbf{r})\frac{\partial}{\partial
r}r,$ (1)

the Hamiltonian is given by

$H_{2b}=-\frac{\hbar^{2}}{2m}(\nabla_{\textbf{r}_{1}}^{2}+\nabla_{\textbf{r}_{2}}^{2})+\frac{1}{2}m\omega_{ho}^{2}(\textbf{r}_{1}^{2}+\textbf{r}_{2}^{2})+V(\textbf{r}_{12}),$
(2)

where $\textbf{r}_{12}=\textbf{r}_{1}-\textbf{r}_{2}$ is the relative
coordinate between two particles. The wave function is separable in the center
of mass motion and the relative motion, that is,
$\Psi=\psi_{cm}(\textbf{R}_{cm})\psi_{rel}(\textbf{r}_{12})$, where
$\textbf{R}_{cm}=(\textbf{r}_{1}+\textbf{r}_{2})/2$ is the center of mass
coordinate. The exact solution to the corresponding Schrödinger equation
$H_{2b}\Psi=E_{exact}(a)\Psi$ has been discussed in Ref. Busch  _et al._
(1998). The eigen-energy can be written as $E_{exact}(a)=E_{cm}+E_{rel}(a)$,
where $E_{cm}=(n_{cm}+3/2)\hbar\omega_{ho}$ corresponds to the center of mass
motion in the harmonic trap. The eigen-energy for the relative motion
satisfies the following condition:

$\sqrt{2}\frac{\Gamma\left(-E_{rel}(a)/2\hbar\omega_{ho}+3/4\right)}{\Gamma\left(-E_{rel}(a)/2\hbar\omega_{ho}+1/4\right)}=\frac{l_{ho}}{a},$
(3)

where $l_{ho}=\sqrt{\hbar/m\omega_{ho}}$ is the harmonic trap length.

On the other hand, when the two particles interact with the renormalized
contact potential given by

$\tilde{V}(\textbf{r})=\frac{4\pi\hbar^{2}a_{eff}}{m}\delta(\textbf{r})=\frac{4\pi\hbar^{2}\zeta(k_{F}a)}{mk_{F}}\delta(\textbf{r}),$
(4)

we assume the total wave function has a Hartree-Fock (HF) expression
$\tilde{\Psi}=\psi(\textbf{r}_{1})\psi(\textbf{r}_{2})$. Consequently, the
energy expectation value of the system is given by

$\displaystyle\xi\\{\psi\\}$ $\displaystyle=$
$\displaystyle\int\left[2\psi\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}\right)\psi\right.$
(5)
$\displaystyle\left.+\frac{4\pi\hbar^{2}a_{eff}}{m}\psi^{4}\right]d^{3}\textbf{r}.$

Since there is no Fermi momentum in few body systems, it is natural to replace
$k_{F}$ in Eq. (4) by its average value $\langle
k_{F}\rangle=\int[6\pi^{2}2\psi(\textbf{r})^{2}]^{1/3}\psi(\textbf{r})^{2}d^{3}\textbf{r}$
in this two body case. Minimizing $\xi\\{\psi\\}$ with the normalization
constraint $\langle\psi|\psi\rangle=1$ yields the ground state energy that
depends on the effective scattering length:

$E_{HF}(a_{eff})=\xi\\{\psi\\}|_{\delta\xi/\delta\psi=0}.$ (6)

In order to make the effective scattering length and the bare scattering
length equivalent for this trapped two-particle system, we match the HF energy
to the exact energy:

$E_{HF}(a_{eff})=E_{exact}(a).$ (7)

Before matching these two energies, we should note that the exact energy has
many branches including a molecular branch for $a>0$. Because the HF
approximation describes the lowest atomic gas state, which corresponds to the
branch with $n_{cm}=0$ and
$(1/2)\hbar\omega_{ho}<E_{rel}<(5/2)\hbar\omega_{ho}$, we match the HF energy
to the exact energy in this particular branch. Since Eq. (7) yields a
pointwise correspondence between $\langle k_{F}\rangle a_{eff}$ and $\langle
k_{F}\rangle a$, we can numerically interpolate the renormalization function
$\langle k_{F}\rangle a_{eff}=\zeta(\langle k_{F}a\rangle a)$. This
interpolation can be excellently fitted by an analytical expression

$\zeta(x)=0.395-1.138\arctan(0.362-0.994x),$ (8)

which satisfies the asymptotic conditions $\zeta(+\infty)=2.182$,
$\zeta(-\infty)=-1.392$ and $\zeta(k_{F}a)\rightarrow k_{F}a$ for $|k_{F}a|\ll
1$.

From the renormalization procedure above, we can see that the renormalized
contact potential Eq. (4) reproduces the exact energy solution for the system
of two interacting particles in a trap. The next step is to apply such a
renormalized contact potential to many body systems. Since the many body
Hamiltonian cannot be diagonalized exactly due to the huge number of degrees
of freedom, we must make some aggressive but reasonable approximations, as we
will discuss in the following subsections.

###  II.1 Mean-field approach

One natural and intuitive idea is to generalize the HF approximation employed
above along with the renormalized interaction to many body systems. Such an
approximation for bosons is also called the mean-field approximation.

With the inclusion of renormalized interactions, the $N$-body Hamiltonian now
becomes

$H=\sum_{i=1}^{N}\left(-\frac{\hbar^{2}}{2m}\nabla_{i}^{2}+\frac{1}{2}m\omega_{ho}^{2}r_{i}^{2}\right)+\sum_{i<j}^{N}\frac{4\pi\hbar^{2}\zeta(k_{F}a)}{mk_{F}}\delta(\textbf{r}_{ij}).$
(9)

The mean-field theory assumes the $N$-body ground state wave function to be
$\Psi=\prod_{i=1}^{N}\psi(\textbf{r}_{i})$. By taking the variation $\delta
H/\delta\psi=0$ under the normalization condition $\langle\psi|\psi\rangle=1$,
we can obtain a renormalized $N$-body Gross-Pitaevskii (GP) equation:

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}+\frac{4\pi(N-1)\hbar^{2}}{3m}\right.$
$\displaystyle\left.\times\left(\zeta^{\prime}(k_{F}a)a+2\frac{\zeta(k_{F}a)}{k_{F}}\right)|\psi|^{2}\right]\psi=\epsilon\psi.$
(10)

where $\epsilon$ is the Lagrange multiplier enforcing normalization in the
variation procedure, and is also identified as the orbital energy.
$\zeta^{\prime}(x)$ means the derivative of $\zeta$ with respect to the
variable $x$. $k_{F}$ is the local Fermi momentum. With such a mean-field
approximation and in the framework of the local density approximation (LDA),
the local Fermi momentum is given by $k_{F}=(6\pi^{2}N|\psi|^{2})^{1/3}$.
After solving Eq. (10), we can evaluate the total energy of the system as

$\displaystyle
E=\int\left[N\psi\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}\right)\psi\right.$
$\displaystyle\left.+\frac{N(N-1)}{2}\frac{4\pi\hbar^{2}\zeta(k_{F}a)}{mk_{F}}\psi^{4}\right]d^{3}\textbf{r}.$
(11)

At very large particle numbers, the kinetic energy composes a very small
portion of the total energy of the system. In the meanwhile, the particle
density varies slowly in the trap except for the edge of the cloud, which is
an evidence of the validity of the LDA. In this condition, we can solve Eq.
(10) using the Thomas-Fermi approximation, which neglects the kinetic energy
term and thereby converts Eq. (10) to a regular algebraic equation. In the
unitary regime, if we neglect edge effects, which means assuming
$k_{F}a\rightarrow+\infty$ at any position of the cloud, we can obtain an
analytical expression for the wave function, which is given by

$\psi_{TF}(\textbf{r})=\left[\frac{3(6\pi^{2})^{1/3}(R_{TF}^{2}-r^{2})}{16\pi
N^{2/3}\zeta(+\infty)l_{ho}^{4}}\right]^{3/4},$ (12)

where $R_{TF}$ is the Thomas-Fermi radius indicating the size of the cloud. It
is given by

$R_{TF}=N^{1/6}\left(\frac{256\sqrt{2}}{9}\right)^{1/6}\left(\frac{\zeta(+\infty)}{\pi}\right)^{1/4}l_{ho}.$
(13)

Consequently, the orbital energy is given by

$\epsilon=\frac{1}{2}m\omega_{ho}^{2}R_{TF}^{2}.$ (14)

The fact that $\psi_{TF}$ does not depend on $a$ is another signature that the
scattering length is no longer a relevant length scale in the unitary regime.

In order to calculate the dynamics of a degenerate Bose gas, it is natural to
convert Eq. (10) to a time-dependent GP equation:

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}+\frac{4\pi(N-1)\hbar^{2}}{3m}\right.$
$\displaystyle\left.\times\left(\zeta^{\prime}(k_{F}a)a+2\frac{\zeta(k_{F}a)}{k_{F}}\right)|\tilde{\psi}|^{2}\right]\tilde{\psi}=i\hbar\frac{\partial}{\partial
t}\tilde{\psi}.$ (15)

We should notice that now the local Fermi momentum $k_{F}$ also becomes time-
dependent. Although such a non-linear and time-dependent Shrödinger equation
can be solved directly using a brute force time-evolution method, we can
obtain a clearer physical picture if appropriate approximations are made to
Eq. (15). One typical method is the Bogoliubov approximation, which is
commonly used to predict elementary excitations of a BEC. The Bogoliubov
approximation assumes the time-dependent wave function to be

$\tilde{\psi}_{bog}(\textbf{r},t)=e^{-i\epsilon
t/\hbar}\left(\psi(\textbf{r})+u(\textbf{r})e^{-i\omega
t}+v^{*}(\textbf{r})e^{i\omega t}\right).$ (16)

Inserting this wave function into Eq. (15) and linearizing the equation to the
first order in $u(\textbf{r})$ and $v(\textbf{r})$, we can obtain a pair of
coupled differential equations:

$\displaystyle\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}+f(N,a,\psi)-\epsilon\right)u$
$\displaystyle+g(N,a,\psi)v=\hbar\omega u,$ (17)
$\displaystyle\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{ho}^{2}r^{2}+f(N,a,\psi)-\epsilon\right)v$
$\displaystyle+g(N,a,\psi)u=-\hbar\omega v,$ (18)

where $\omega$ corresponds to the eigen mode frequency. $f$ and $g$ are in
general complicated functions of $N$, $a$, and $\psi$, but they have simple
forms in the asymptotic limits $a\rightarrow 0$ and $a\rightarrow\infty$:

$f\rightarrow\frac{8\pi N\hbar^{2}a}{m}\psi^{2},g\rightarrow-\frac{4\pi
N\hbar^{2}a}{m}\psi^{2}\quad(a\rightarrow 0),$ (19)

and

$\displaystyle f\rightarrow\frac{40\pi
N^{2/3}\hbar^{2}\zeta(+\infty)}{9(6\pi^{2})^{1/3}m}\psi^{4/3},$ $\displaystyle
g\rightarrow-\frac{16\pi
N^{2/3}\hbar^{2}\zeta(+\infty)}{9(6\pi^{2})^{1/3}m}\psi^{4/3}$
$\displaystyle\quad(a\rightarrow\infty).$ (20)

The lowest and most significant eigen-mode is called the breathing mode, which
corresponds to the oscillation of the overall size of the cloud with a fixed
geometry. We will discuss this mode in later sections.

###  II.2 Hyperspherical description

The hyperspherical coordinate system is a powerful toolkit to treat few-body
problemsvon Stecher  _et al._ (2009); Wang  _et al._ (2012a, b); Rittenhouse
_et al._ (2011). To generalize this toolkit to many body systems, aggressive
approximations must be made to significantly reduce the dimension of the
problem. The hyperspherical description of a single component weakly-
interacting BEC and a degenerate Fermi gas have been studied with the bare
Fermi pseudo potential, and important physics has been predicted even with
crude approximationsBohn  _et al._ (1998); Rittenhouse  _et al._ (2006).
Similar to the procedure in Ref. Bohn  _et al._ (1998), we formulate the
hyperspherical theory in a degenerate unitary Bose gas with the renormalized
interaction.

For $N$ particles in a 3D space, which contains $3N$ degrees of freedom, the
hyperspherical coordinates are constructed as follows: The hyperradius $R$,
which is a collective coordinate and indicates the overall size of the cloud,
is given by

$R=\sqrt{\frac{1}{N}\sum_{i=1}^{N}r_{i}^{2}},$ (21)

The remaining $3N-1$ coordinates are called hyperangles. $2N$ of them are the
defined as the regular spherical angles of the $N$ particles, that is,
$\\{\theta_{1},\phi_{1},\theta_{2},\phi_{2},\cdots,\theta_{N},\phi_{N}\\}$.
The remaining $N-1$ hyperangles can be defined as

$\tan\alpha_{i}=\frac{\sqrt{\sum_{j=1}^{i}r_{j}^{2}}}{r_{i+1}},(i=1,\cdots,N-1)$
(22)

With this set of hyperspherical coordinates, the Hamiltonian in Eq.(9) can be
rewritten as

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2M}\frac{1}{R^{3N-1}}\frac{\partial}{\partial
R}R^{3N-1}\frac{\partial}{\partial R}+\frac{\Lambda^{2}}{2MR^{2}}$ (23)
$\displaystyle+\frac{1}{2}M\omega_{ho}^{2}R^{2}+V_{int}(R,\Omega),$

where $M=Nm$ is the total mass of the $N$ particles. $V_{int}(R,\Omega)$
denotes the renormalized interactions written in hyperspherical coordinates,
that is,

$V_{int}(R,\Omega)=\sum_{i<j}^{N}\frac{4\pi\hbar^{2}\zeta(k_{F}a)}{mk_{F}}\delta(\textbf{r}_{ij}).$
(24)

$\Lambda$ is called the grand angular momentum operator. The eigen-functions
of the operator $\Lambda^{2}$, denoted by $\Phi_{\lambda}$, are called
hyperspherical harmonicsAvery (1989). They satisfy the equation

$\Lambda^{2}\Phi_{\lambda}(\Omega)=\lambda(\lambda+3N-2)\Phi_{\lambda}(\Omega),$
(25)

where $\Omega$ represents all hyperangles. For a given $\lambda$,
$\Phi_{\lambda}$ usually has huge degeneracy especially for large $\lambda$.
The eigen-functions $\Phi_{\lambda}$ form a basis in the hyperangular Hilbert
space. An aggressive approximation we make here is to only retain one
hyperangular momentum eigen-state out of this huge basis set. This is also
known as the $K$-harmonics approximation in nuclear theoriesSmirnov and
Shitikova (1977) and it becomes exact in the unitary limit and in the non-
interacting limitWerner and Castin (2006); Blume  _et al._ (2008). The natural
choice of this eigen state for bosons would be the lowest eigen state of
$\Lambda^{2}$, denoted by $\Phi_{0}$, which in fact is a constant. Such a
choice of hyperangular wave function also freezes the geometry of the atomic
cloud into that of the non-interacting case, while the interparticle
interactions modify the overall size of the cloud, which is reflected in the
hyperradial wave function. With this $K$-harmonics approximation, the total
$N$-body wave function can be separated as
$\Psi(R,\Omega)=F(R)\Phi_{0}(\Omega)$. Inserting this expression into the
time-independent Schrödinger equation and integrating over all hyperangles
yields a hyperradial Schrödinger equation:

$\left(-\frac{\hbar^{2}}{2M}\frac{d^{2}}{dR^{2}}+V_{eff}(R)-E\right)R^{(3N-1)/2}F(R)=0,$
(26)

where $V_{eff}(R)$ is an effective hyperradial potential written as

$\displaystyle
V_{eff}(R)=\frac{(3N-1)(3N-3)\hbar^{2}}{8MR^{2}}+\frac{1}{2}M\omega_{ho}^{2}R^{2}$
$\displaystyle+\langle\Phi_{0}(\Omega)|V_{int}(R,\Omega)|\Phi_{0}(\Omega)\rangle_{\Omega},$
(27)

where $\langle\cdots\rangle_{\Omega}$ denotes integration over all
hyperangles. The evaluation of the last term in $V_{eff}$ has been elaborated
in Ref. Bohn  _et al._ (1998). In the large $N$ limit which we are interested
in, $\langle\Phi_{0}|V_{int}(R,\Omega)|\Phi_{0}\rangle_{\Omega}$ can be
readily calculated numerically. We can even obtain analytical expressions in
the unitary limit and the weakly interacting limit: For $a\rightarrow\infty$,

$\langle\Phi_{0}|V_{int}(R,\Omega)|\Phi_{0}\rangle_{\Omega}=\frac{N^{8/3}(3/5)^{3/2}\zeta(+\infty)(4\pi/3)^{1/3}\hbar^{2}}{\pi
MR^{2}},$ (28)

and for $a\rightarrow 0$,

$\langle\Phi_{0}|V_{int}(R,\Omega)|\Phi_{0}\rangle_{\Omega}=\frac{N^{3}(2/\pi)^{1/2}(3/2)^{3/2}\hbar^{2}a}{2MR^{3}}.$
(29)

Figure 1: Effective hyperradial potential curves for $N=5000$ particles with
different scattering lengths $a/l_{ho}=$0.01(solid), 0.1(dashed), 1(dotdashed)
and 10 (dotted) respectively.

Fig. 1 shows the effective hyperradial potential curves at different
scattering lengths. At large hyperradius, $V_{eff}(R)$ is always dominated by
the $R^{2}$ term representing the confinement of the harmonic trap. At a small
hyperradius, the system feels two repulsive forces originating from the
quantum pressure and the two-body interaction. It is interesting that the two-
body interaction term transitions from $R^{-3}$ to $R^{-2}$ as the scattering
length increases to infinity.

Eq. (26) is equivalent to the Schrödinger equation of a particle moving in a
1D potential. Moreover, in the large $N$ limit, the mass of this “particle” is
so huge that it can be treated classically. Consequently, the minimum of
$V_{eff}$ corresponds to the total energy of the system at equilibrium, that
is,

$E=V_{eff}(R_{0}),$ (30)

where $R_{0}$ denotes the equilibrium position. Furthermore, as the
hyperangular wave function is kept frozen, the oscillation of this massive
“particle” in the hyperradial potential indicates the oscillation of the
overall size of the system, which corresponds to the breathing mode. The
breathing mode frequency is associated with the coefficient of the second
order expansion of $V_{eff}$ at $R_{0}$, that is,

$\omega=\omega_{ho}\sqrt{\frac{1}{M}\left.\frac{d^{2}V_{eff}}{dR^{2}}\right|_{R=R_{0}}}$
(31)

##  III Results and discussions

At first, we discuss the total energy of a degenerate Bose gas. Fig. 2 shows
the average energy per particle for $N=10^{4}$ particles as a function of the
scattering length. In the weakly interacting regime where $\langle n\rangle
a^{3}\ll 1$, the mean-field energy obtained by solving the GP equation with
renormalization agrees excellently with that without renormalization. These
two results start to separate near $a/l_{ho}=0.2$, which corresponds to
$\langle n\rangle a^{3}=0.25$. The mean-field energy diverges at unitarity
without renormalization. The energy obtained using the Thomas-Fermi
approximation and the mean-field energy differ in the weakly interacting
regime where the kinetic energy is a significant contribution to the total
energy. However, at large scattering lengths, the total energy of the system
is dominated by the strong interactions between particles and thereby the
Thomas-Fermi approximation agrees excellently with the mean-field result. The
energy obtained using the hyperspherical method agrees qualitatively with the
mean-field result, though it is slightly smaller. Overall, the total energy of
the system saturates at large scattering lengths with the inclusion of the
interaction renormalization, which indicates that the scattering length is no
longer a relevant length scale of the system near unitarity.

Figure 2: The average energy per particle as a function of the scattering
length in the ground state of a degenerate Bose gas for $N=10^{4}$ particles.
The results are obtained with the renormalized interaction using mean-field
approach(blue solid), the Thomas-Fermi approximation(red dashed), and
hyperspherical method(black dotted) respectively. The dotdashed curve shows
the mean-field result with the bare unrenormalized contact interaction for
comparison.

We now discuss the energy of a unitary Bose gas using the Thomas-Fermi
approximation since it is very accurate in the strongly interacting regime.
One advantage of the Thomas-Fermi approximation is that we can obtain
analytical expressions for many physical quantities, which offers us a clear
picture of the unitary Bose gas. With the Thomas-Fermi approximation, the
ground state energy is given by

$\frac{E}{N}=\frac{27}{64}\left(\frac{256\sqrt{2}}{9}\right)^{1/3}\left(\frac{\zeta(+\infty)}{\pi}\right)^{1/2}N^{1/3}\hbar\omega_{ho}$
(32)

For a unitary gas in a uniform space, the only relevant length scale of the
system is the average interparticle distance determined by $n^{-1/3}$, where
$n$ is the particle density. This also defines the only energy scale of the
system $\hbar^{2}n^{2/3}/2m$. When the gas is inhomogeneous while the density
varies slowly in space, the local density approximation can be applied to the
system and thereby $n^{2/3}$ is replaced by its average value $\langle
n^{2/3}\rangle$. For a unitary Bose gas in a harmonic trap, the average value
is given by

$\langle n^{2/3}\rangle=\frac{5\times
3^{2/3}N^{1/3}}{8(2\pi)^{5/6}\zeta(+\infty)^{1/2}l_{ho}^{2}}.$ (33)

Therefore, from Eq. (32) and Eq. (33), we can obtain the universal relation of
the energy of a unitary Bose gas, which is given by

$\frac{E}{N}=\frac{6^{5/3}\pi^{1/3}\zeta(+\infty)}{5}\frac{\hbar^{2}\langle
n^{2/3}\rangle}{2m}\approx 12.67\frac{\hbar^{2}\langle n^{2/3}\rangle}{2m}$
(34)

This universal relation is close to the value $E/N=13.33\hbar^{2}n^{2/3}/2m$
reported in Ref. Cowell  _et al._ (2002).

Figure 3: The average energy per particle, in units of $\hbar^{2}\langle
n^{2/3}\rangle/2m$, as a function of the average particle density for
$a/l_{ho}=0.1$(blue dotted), $1$(red dotdashed) and $10$(green dashed)
respectively. The black solid line shows the universal relation of energy from
Eq. (34).

We show in Fig. 3 the average energy per particle, in units of
$\hbar^{2}\langle n^{2/3}\rangle/2m$, as a function of the average density for
different scattering lengths to verify this universal relation. The results
are obtained by solving the renormalized GP equation Eq. (10). At small
scattering length $a/l_{ho}=0.1$, the value of $2mE/N\hbar^{2}\langle
n^{2/3}\rangle$ varies significantly with the average density. As the
scattering length increases, $2mE/N\hbar^{2}\langle n^{2/3}\rangle$ has weaker
dependence on the average density and the value approaches the universal
constant in Eq. (34). At $a/l_{ho}=10$, where gas has reached the unitary
regime, the result agrees excellently with the universal relation Eq. (34) for
$\langle n\rangle l_{ho}^{3}>5$, The small deviation from the universality at
small densities may be due to the inaccuracy of LDA when the interparticle
distance is comparable to the trapping length.

Besides the total energy, there are many interesting physical quantities
worthy of investigation in unitary Bose gases. An important quantity that
bridges the two-body correlations and the thermodynamics of a many-body system
is called the two-body contact or Tan’s contact. It was first introduced by
Shina Tan to study the universal properties of a two-component Fermi gas with
$s$-wave contact interactionsTan (2008a, b). Universal relations determined by
the two-body contact have also been identified in systems consisting of
identical bosonsBraaten  _et al._ (2011). The two-body contact in bosons has
been measured using rf spectroscopyWild  _et al._ (2012). The two-body contact
is determined by the derivative of the total energy of the system with respect
to the scattering length:

$C_{2}=\frac{8\pi ma^{2}}{\hbar^{2}}\frac{dE}{da}.$ (35)

It is an extensive thermodynamic quantity of the system. Another intrinsic
quantity, which is commonly used in homogeneous systems, is the contact
density $\mathcal{C}_{2}$, which can be obtained from the limit of the high
momentum tail: $\mathcal{C}_{2}=\lim_{k\rightarrow\infty}k^{4}n_{\textbf{k}}$,
where $n_{\textbf{k}}$ is the number of particles in the k momentum state.
Since the interparticle distance is the only relevant length scale of a
homogeneous system at unitarity, the two-body contact density must scale as

$\mathcal{C}_{2}=\alpha n^{4/3},$ (36)

where $\alpha$ is an universal dimensionless coefficient. Such a universal
relation can be generalized to a trapped system under LDA, which is given by

$C_{2}=\alpha N\langle n^{1/3}\rangle.$ (37)

Figure 4: The average two-body contact per particle, in units of $\langle
n^{1/3}\rangle$, as a function of the scattering length for $N=10^{3}$(blue
solid), $10^{4}$(red dashed),$10^{5}$(green dotdashed) and $10^{6}$(black
dotted) particles.

Fig. 4 shows the average two-body contact per particle, in units of $\langle
n^{1/3}\rangle$, as a function of the scattering length. The results are
obtained by directly solving the renormalized GP equation Eq. (10). The value
of $C_{2}/(N\langle n^{1/3}\rangle)$ has similar behavior for different
numbers of particles: It increases drastically in the weakly interacting
regime as the scattering length grows; it then attains a maximum value before
saturating in the unitary regime. For different numbers of particles
$C_{2}/(N\langle n^{1/3}\rangle)$ saturates at the same value, demonstrating
the universality of $\alpha$.

Figure 5: The average two-body contact per particle, in units of $\langle
n^{1/3}\rangle$, as a function of the average particle density for
$a/l_{ho}=0.1$(blue dotted), $1$(red dotdashed) and $10$(green dashed)
respectively. The black solid line shows the universal relation of two-body
contact from Eq. (37).

From our renormalized mean-field model, we can derive the universal relation
Eq. (37) and determine the universal coefficient $\alpha$ analytically with
appropriate approximations. To calculate the two-body contact, we take the
derivative of the renormalized mean-field energy Eq. (11) with respect to the
scattering length. We should note that at unitarity the scattering length
dependence of the wave function $\psi$ is much weaker than the renormalization
function $\zeta(k_{F}a)$. Thus, by approximating $\partial\psi/\partial
a|_{a\rightarrow\infty}\approx 0$ and neglecting the edge effect, we can
readily obtain Eq. (37) and the coefficient $\alpha$ is given by

$\alpha=\frac{1.138(4\pi)^{2}}{0.994(6\pi^{2})^{2/3}}\approx 11.8.$ (38)

Other theoretical works have reported the the universal coefficient to be
$\alpha=10.3$Diederix  _et al._ (2011), $9.04$Rossi  _et al._ (2014) and
$12$Sykes  _et al._ (2014), which agree qualitatively with our prediction from
the renormalized mean-field approach.

To further verify our prediction of the universal relation Eq. (37) and the
value of the coefficient $\alpha$, we show in Fig. 5 the average two-body
contact per particle, in units of $\langle n^{1/3}\rangle$, as a function of
the average particle density for different scattering lengths. At small
scattering length $a/l_{ho}=0.1$, $C_{2}/(N\langle n^{1/3}\rangle)$ increases
significantly with the density. The value of $C_{2}/(N\langle n^{1/3}\rangle)$
at $a/l_{ho}=1$ is larger than its values for both $a/l_{ho}=0.1$ case and
$a/l_{ho}=10$ case because it attains a maximum with the increment of the
scattering length, as shown in Fig. 4. In the unitary regime, $C_{2}/(N\langle
n^{1/3}\rangle)$ converges to the universal coefficient $\alpha$, which
corresponds to $a/l_{ho}=10$ case. The value of $\alpha$ from exact
calculation is slightly smaller than the the analytical approximation
$\alpha\approx 11.8$, which might be due to the edge effect and the small
scattering length dependence of the wave function.

Figure 6: The breathing mode frequency as a function of the scattering length
for $N=10^{4}$ particles. The results are obtained using the hyperspherical
method(blue solid), Bogoliubov approximation(red dashed) and by direct time
evolution(black dots).

Finally, we discuss the elementary excitations of the degenerate Bose gas in a
harmonic trap. Specifically, we focus on the lowest radial excitation, which
corresponds to the breathing mode. The breathing mode frequency can be
determined from the hyperspherical method, the Bogoliubov approximation, or by
directly solving the time-dependent GP equation, as discussed in the theory of
renormalization section above. Fig. 6 shows the breathing mode frequency as a
function of the scattering length for $N=10^{4}$ particles. These three
different methods show overall consistency with each other. The time-evolution
results are obtained by solving Eq. (15) with an interaction quench. At
vanishing scattering length, the breathing mode frequency is apparently
$\omega=2\omega_{ho}$, which corresponds to the beating between two adjacent
harmonic levels with zero angular momentum. The breathing mode frequency
increases gradually with the scattering length before it reaches a maximum
value. It is interesting that the breathing mode frequency regresses to the
non-interacting value $\omega=2\omega_{ho}$ at unitarity. Similar unitary
behavior has also been predicted in Ref. Werner and Castin (2006).

##  IV Conclusion

In summary, we introduced a renormalized contact potential to study degenerate
Bose gases with large scattering lengths. Such a renormalized interaction is
designed by matching the Hartree-Fock energy to the exact energy of two
interacting particles in a trap. We employ this renormalized contact potential
in company with the mean-field theory and the hyperspherical theory to study
the stationary properties and elementary excitations of a degenerate Bose gas,
especially in the unitary regime. In the framework of the local density
approximation, the only relevant length scale of a degenerate unitary Bose gas
is the interparticle spacing $n^{-1/3}$, where $n$ is the particle density.
This length scale also determines the only energy scale $\hbar^{2}n^{2/3}/2m$
and the only two-body contact scale $n^{1/3}$ of the system. Our
renormalization theory offers us a much more clear and convenient approach to
obtain the universal relations for energy and two-body contact at unitarity,
which are given by $E/N=12.67\hbar^{2}\langle n^{2/3}\rangle/2m$ and
$C_{2}/N=11.8\langle n^{1/3}\rangle$ respectively. Our results are in
consistent with other theoretical predictions. Moreover, we studied the lowest
radial excitation of a degenerate Bose gas, which can be induced by an
interaction quench. This excitation is also known as the breathing mode. Our
theory shows an interesting phenomenon that the breathing mode frequency at
unitarity returns to the value of a non-interacting gas.

###### Acknowledgements.

The authors thank Q. Zhou and F. Robicheaux for helpful discussions, and M.
Eiles for careful reading of the manuscript. This work is supported by
National Science Foundation and US-Israel Binational Science Foundation.

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#  Quasiparticles in nonperturbative vacuum

Vladimir Dzhunushaliev<EMAIL_ADDRESS>Department of Theoretical
and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Academician J. Jeenbaev Institute of Physics of the NAS of the
Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan  Vladimir
Folomeev<EMAIL_ADDRESS>Academician J. Jeenbaev Institute of Physics of
the NAS of the Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan

###### Abstract

The model of nonperturbative vacuum in SU(2) Yang-Mills theory coupled to a
nonlinear spinor field is suggested. By analogy with Abelian magnetic monopole
dominance in quantum chromodynamics, it is assumed that the dominant
contribution to such vacuum is coming from quasiparticles described by
dipolelike solutions existing in this theory. Using an assumption of the
behavior of the number density of quasiparticles whose energy approaches
infinity, we derive an approximate expression for the energy density of such
nonperturbative vacuum, which turns out to be finite, unlike an infinite
energy density of perturbative vacuum. Using characteristic values of the
parameters appearing in the expression for the nonperturbative energy density,
it is shown that this density may be of the order of the energy density
associated with Einstein’s cosmological constant. The physical interpretation
of the spinor field self-coupling constant as a characteristic distance
between quasiparticles is suggested. The questions of experimental
verification of the nonperturbative vacuum model under consideration and of
determining its pressure are briefly discussed.

nonperturbative vacuum, quasiparticles, dipole solutions, energy density

###### pacs:

11.15.Tk

##  I Introduction

Vacuum is a well-defined concept in quantum theories with free or weakly
interacting fields. It consists of virtual particles which are created and
annihilated in accordance with the Heisenberg uncertainty relation. One of
basic problems in considering such perturbative vacuum is the presence of an
infinite vacuum or zero-point energy, that in the case of theories which do
not involve gravity is artificially removed by subtraction.

In quantum chromodynamics (QCD), when considering a quark-gluon plasma, one
may introduce quasiparticles filling the plasma [1]. The quark-gluon plasma is
an essentially nonperturbative object where quasiparticles play a quite
important role. Consistent with this, one may assume that in nonperturbative
vacuum quasiparticles might also play a quite essential role. In this
connection, the questions can be asked: How such objects may appear in
nonperturbative vacuum? How one can describe them?

An attempt to answer these questions will be made in the present paper. We
will consider here a hypothesis according to which, among all possible
fluctuations of a quantum field in vacuum, there are such profiles that
satisfy field equations, and they yield a considerable contribution to the
corresponding path integral in quantizing such field. Physically, this means
that the properties of such nonperturbative vacuum would be basically
described by the aforementioned virtual quasiparticles. Such quasiparticles
appear in accordance with the Heisenberg uncertainty principle and exist over
the time $\Delta t\approx\hbar/\Delta E$ determined by this principle. This
idea is analogous to the long-familiar concept that in QCD there exists the
so-called Abelian magnetic monopole dominance. The latter implies that
magnetic monopoles, as topologically nontrivial configurations, yield a
considerable contribution to the functional integral. The Abelian and monopole
dominance [2, 3] has been confirmed by lattice calculations in QCD (see, e.g.,
Refs. [4, 5]).

In the present paper we suggest a model of nonperturbative vacuum where
monopolelike objects found in Refs. [6, 7] are regarded as virtual
quasiparticles in nonperturbative vacuum of SU(2) Yang-Mills theory which
includes a nonlinear spinor field. As pointed out above, such virtual
quasiparticles are created according to the Heisenberg uncertainty principle
and their lifetimes are determined by this principle. Then, if one regards
such quasiparticles as making the main contribution to the nonperturbative
vacuum, it is possible to determine such properties of the nonperturbative
vacuum like, for example, its energy density.

##  II Dipolelike solutions in SU(2) Yang-Mills theory

In this section we briefly describe dipolelike solutions in SU(2) Yang-Mills
theory with a source of gauge field in the form of a nonlinear spinor field
described by the nonlinear Dirac equation. The Lagrangian of such theory is
(here we follow Ref. [7])

$\begin{split}\mathcal{L}=&-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+i\hbar
c\bar{\psi}\gamma^{\mu}D_{\mu}\psi-
m_{f}c^{2}\bar{\psi}\psi+\frac{l_{0}^{2}}{2}\hbar
c\left(\bar{\psi}\psi\right)^{2}.\end{split}$ (1)

Here $m_{f}$ is the mass of the spinor field;
$D_{\mu}=\partial_{\mu}-i\frac{g}{2}\sigma^{a}A^{a}_{\mu}$ is the gauge-
covariant derivative, where $g$ is the coupling constant and $\sigma^{a}$ are
the SU(2) generators (the Pauli matrices);
$F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+g\epsilon_{abc}A^{b}_{\mu}A^{c}_{\nu}$
is the field strength tensor for the SU(2) field, where $\epsilon_{abc}$ (the
completely antisymmetric Levi-Civita symbol) are the SU(2) structure
constants; $l_{0}$ is a constant; $\gamma^{\mu}$ are the Dirac matrices in the
standard representation; $a,b,c=1,2,3$ are color indices and $\mu,\nu=0,1,2,3$
are spacetime indices.

Unlike Refs. [6, 7, 8], we will use below the term “dipole” for the
configurations under consideration. This is because the radial component of
the color magnetic field behaves asymptotically as $H^{a}_{r}\sim D/r^{3}$, as
it takes place in Maxwell’s magnetostatic, and the constant $D$ may be called
the color magnetic moment of the dipole. But, unlike usual dipole solutions,
the solution obtained in Refs. [6, 7, 8] also has a nonvanishing component of
the color magnetic field $H^{a}_{\varphi}$; for this reason, we have enclosed
the word “dipole” in quotation marks. Below we discuss a vector field created
by the color current $j^{a\mu}$.

Using the Lagrangian (1), one can find the corresponding field equations,

$\displaystyle D_{\nu}F^{a\mu\nu}$ $\displaystyle=$ $\displaystyle
j^{a\mu}=\frac{g\hbar c}{2}\bar{\psi}\gamma^{\mu}\sigma^{a}\psi,$ (2)
$\displaystyle i\hbar\gamma^{\mu}D_{\mu}\psi-
m_{f}c\psi+l_{0}^{2}\hbar\psi\left(\bar{\psi}\psi\right)$ $\displaystyle=$
$\displaystyle 0.$ (3)

Here $j^{a\mu}$ is a color current created by the spinor field $\psi$.
“Dipole” solutions are sought in the form of the standard Ansatz used in
describing the ’t Hooft-Polyakov monopole,

$A^{a}_{t}=0,\quad
A^{a}_{i}=\frac{1}{g}\left[1-f(r)\right]\begin{pmatrix}0&\phantom{-}\sin\varphi&\sin\theta\cos\theta\cos\varphi\\\
0&-\cos\varphi&\sin\theta\cos\theta\sin\varphi\\\
0&0&-\sin^{2}\theta\end{pmatrix},\quad i=r,\theta,\varphi\text{ (in polar
coordinates)},$ (4)

and the Ansatz for the spinor field is taken to be [9, 10]

$\psi^{T}=\frac{e^{-i\frac{Et}{\hbar}}}{gr\sqrt{2}}\begin{Bmatrix}\begin{pmatrix}0\\\
-u\\\ \end{pmatrix},\begin{pmatrix}u\\\ 0\\\
\end{pmatrix},\begin{pmatrix}iv\sin\theta e^{-i\varphi}\\\ -iv\cos\theta\\\
\end{pmatrix},\begin{pmatrix}-iv\cos\theta\\\ -iv\sin\theta e^{i\varphi}\\\
\end{pmatrix}\end{Bmatrix},$ (5)

where $E/\hbar$ is the spinor frequency and the functions $u$ and $v$ depend
on the radial coordinate $r$ only.

Substituting the expressions (4) and (5) in the field equations (2) and (3),
one can obtain equations for the unknown functions $f,u$, and $v$:

$\displaystyle-f^{\prime\prime}+\frac{f\left(f^{2}-1\right)}{x^{2}}+\tilde{g}^{2}_{\Lambda}\frac{\tilde{u}\tilde{v}}{x}$
$\displaystyle=$ $\displaystyle 0,$ (6)
$\displaystyle\tilde{v}^{\prime}+\frac{f\tilde{v}}{x}$ $\displaystyle=$
$\displaystyle\tilde{u}\left(-1+\tilde{E}+\frac{\tilde{u}^{2}-\tilde{v}^{2}}{x^{2}}\right),$
(7) $\displaystyle\tilde{u}^{\prime}-\frac{f\tilde{u}}{x}$ $\displaystyle=$
$\displaystyle\tilde{v}\left(-1-\tilde{E}+\frac{\tilde{u}^{2}-\tilde{v}^{2}}{x^{2}}\right),$
(8)

written in terms of the following dimensionless variables: $x=r/\lambda_{c}$,
$\tilde{u}=u\sqrt{l_{0}^{2}/\lambda_{c}g^{2}},\tilde{v}=v\sqrt{l_{0}^{2}/\lambda_{c}g^{2}},\tilde{E}=\lambda_{c}E/(\hbar
c),\tilde{g}^{2}_{\Lambda}=\left(g^{\prime}\lambda_{c}/l_{0}\right)^{2}$,
where $\lambda_{c}=\hbar/(m_{f}c)$ is the Compton wavelength and
${g^{\prime}}^{2}=g^{2}\hbar c$ is a dimensionless coupling constant. The
prime denotes differentiation with respect to $x$.

The color current $j^{a\mu}$ for the spinor (5) has the following components:

$\vec{j}^{1}=\frac{\hbar\,c}{g}\frac{uv}{r^{3}}\left\\{0,-\sin\varphi,-\cot\theta\cos\varphi\right\\},\quad\vec{j}^{2}=\frac{\hbar\,c}{g}\frac{uv}{r^{3}}\left\\{0,\cos\varphi,-\cot\theta\sin\varphi\right\\},\quad\vec{j}^{3}=\frac{\hbar\,c}{g}\frac{uv}{r^{3}}\left\\{0,0,1\right\\},$

where the superscripts $1,2,3$ denote color indices. The current $\vec{j}^{3}$
has only the $\varphi$-component; this means that this is a circular current
creating a dipole with the color index 3. The currents $\vec{j}^{1,2}$ are
shown in Fig. 1. It is seen that these currents also form dipoles; this
suggests that the solution under consideration can be called a “dipole”
solution, enclosed in quotation marks because it also contains a nonvanishing
component of the color magnetic field $H^{a}_{\varphi}$.

Figure 1: Sketches of the force lines of the components of the color current
$g\vec{j}^{1}r^{3}/(\hbar\,c\,uv)$ (left panel) and
$g\vec{j}^{2}r^{3}/(\hbar\,c\,uv)$ (right panel) on the surface
$r=\text{const}$. The plots are given in Cartesian coordinates $\\{x,y,z\\}$.

The total energy density of the “dipole” under consideration is

$\tilde{\epsilon}=\tilde{\epsilon}_{m}+\tilde{\epsilon}_{s}=\frac{1}{\tilde{g}^{2}_{\Lambda}}\left[\frac{{f^{\prime}}^{2}}{x^{2}}+\frac{\left(f^{2}-1\right)^{2}}{2x^{4}}\right]+\left[\tilde{E}\frac{\tilde{u}^{2}+\tilde{v}^{2}}{x^{2}}+\frac{\left(\tilde{u}^{2}-\tilde{v}^{2}\right)^{2}}{2x^{4}}\right].$
(9)

Here the expressions in the square brackets correspond to the dimensionless
energy densities of the non-Abelian gauge fields,
$\tilde{\epsilon}_{m}\equiv\frac{\left(l_{0}\lambda_{c}\right)^{2}}{\hbar
c}\epsilon_{m}$, and of the spinor field,
$\tilde{\epsilon}_{s}\equiv\frac{\left(l_{0}\lambda_{c}\right)^{2}}{\hbar
c}\epsilon_{s}$.

Correspondingly, the total energy of the “dipole” is calculated as

$\tilde{W}_{t}\equiv\frac{m_{f}l_{0}^{2}}{\hbar^{2}}W_{t}=4\pi\int\limits_{0}^{\infty}x^{2}\tilde{\epsilon}dx=\left(\tilde{W}_{t}\right)_{m}+\left(\tilde{W}_{t}\right)_{s},$
(10)

where the energy density $\tilde{\epsilon}$ is taken from Eq. (9). In the
right-hand side of Eq. (10), the first term is the energy of the magnetic
field and the second term corresponds to the energy of the nonlinear spinor
field.

The typical behavior of $\tilde{W}_{t}$ as a function of the spinor frequency
$\tilde{E}$ is sketched in Fig. 2. It is seen that there exists an absolute
minimum of the dimensionless energy (mass gap); call it
$\tilde{\Delta}(g^{2}_{\Lambda})$. Moreover, as demonstrated in Refs. [6, 7],
for $\tilde{E}\to 0,1$ the energy of the “dipole” goes to infinity.

The asymptotic behavior of the functions $f,\tilde{v}$, and $\tilde{u}$ is as
follows:

$\displaystyle f(x)\approx$ $\displaystyle 1-\frac{f_{\infty}}{x},$ (11)
$\displaystyle\tilde{u}(x)\approx$
$\displaystyle\tilde{u}_{\infty}e^{-x\sqrt{1-\tilde{E}^{2}}},\quad\tilde{v}(x)\approx\tilde{v}_{\infty}e^{-x\sqrt{1-\tilde{E}^{2}}},$
(12)

where $f_{\infty},\tilde{u}_{\infty}$, and $\tilde{v}_{\infty}$ are
integration constants.

Figure 2: A sketch of the energy spectrum of the total energy from Eq. (10) as
a function of the spinor frequency $\tilde{E}$ (for details see Ref. [6]).

Consider the question of a linear size of the “dipole” under investigation. It
consists of two fields – the non-Abelian field $A^{a}_{\mu}$ and the spinor
field $\psi$, whose asymptotic behavior is fundamentally different. As follows
from Eq. (11), the color field decreases according to the power law,
$H^{a}_{r,\theta,\varphi}\sim r^{-3}$ [6], whereas the spinor field decreases
exponentially with distance according to Eq. (12). In this connection, it is
difficult to introduce the notion of the size of the “dipole”. However, one
may introduce the notion of the core of the “dipole”, which represents the
region where the spinor field (which is a source of the color Yang-Mills
field) is concentrated. Then, in accordance with Eq. (12), one may define a
characteristic size of the core of the “dipole” as

$l_{cr}\approx\frac{\lambda_{c}}{\sqrt{1-\tilde{E}^{2}}}.$ (13)

This implies that for $\tilde{E}\rightarrow 1$ the size of the core
$l_{cr}\rightarrow\infty$. In turn, for $\tilde{E}\rightarrow 0$, it is
estimated as $l_{cr}\approx\lambda_{c}$, i.e., it remains unchanged.
Unfortunately, the power-law asymptotic behavior of the magnetic field (11)
does not permit one to define a quantity which adequately represents the size
of the “dipole” itself, and not just the size of the core.

Taking into account that the linear size of the core is estimated according to
the expression (13), a maximal number of quasiparticles (“dipoles”) per unit
volume can be estimated as

$\left[n_{qp}\left(\tilde{E}\right)\right]_{\text{max}}\approx\left[l_{cr}\left(\tilde{E}\right)\right]^{-3}.$

Perhaps such concentration of the quasiparticles is not attainable in vacuum,
but it may be reached in a quark-gluon plasma at high temperatures.

##  III “Dipoles” as quasiparticles in the nonperturbative vacuum

In this section we describe a scenario within which “dipoles” found in Refs.
[6, 7] may be regarded as quasiparticles in the nonperturbative vacuum of
SU(2) Yang-Mills theory coupled to a nonlinear spinor field.

The main idea is the same as that used in Abelian magnetic monopole dominance:
among all possible fluctuations of a gauge field there are some that satisfy
Eqs. (6)-(8). Thus, we suppose that, within the nonperturbative vacuum model
under consideration, such configurations give the dominant contribution to the
path integral in the theory with the Lagrangian (1). Since we consider vacuum,
the mean vacuum values of all color potentials and of the spinor field are
zero,

$\left\langle\hat{A}^{a}_{\mu}\right\rangle=0,\quad\left\langle\hat{\psi}\right\rangle=0,$

whereas the dispersions of these fields are nonvanishing,

$\left\langle\left(\hat{A}^{a}_{\mu}\right)^{2}\right\rangle\neq
0,\quad\left\langle\left(\hat{\psi}\right)^{2}\right\rangle\neq 0.$

In the nonperturbative vacuum model under consideration, the dominant
contribution to the nonvanishing magnitudes of the dispersions, as well as to
higher order Green’s functions, is given by quasiparticles whose averaged
spatial distribution is sketched in Fig. 3.

The system under consideration contains two parameters $\lambda_{c}$ and
$l_{0}$ having the dimensions of length. First of them determines the size of
the core of the “dipole” created by the spinor field, and the second one – the
spinor field self-coupling constant, which characterizes the distance between
quasiparticles (see Fig. 3). Thus, the theory contains two constants with the
dimensions of length which determine the properties of the nonperturbative
vacuum: the size of the core of quasiparticles and the distance between them.

Figure 3: A sketch of the averaged spatial distribution of quasiparticles in
the nonperturbative vacuum; their location in space is randomly changed, and
this leads to nonzero values of the dispersions and of higher order Green’s
functions. The size of the core $l_{cr}$ is characterized by the Compton
wavelength of a fermion $\lambda_{c}$ with the mass $m_{f}$ [cf. Eq. (13)].
The average distance between quasiparticles is determined by the spinor field
self-coupling constant $l_{0}$.

Let $n_{qp}\left(\tilde{E}\right)$ is the average density of quasiparticles
per unit volume, which depends on the spinor frequency $\tilde{E}$; then the
average energy density of the nonperturbative vacuum can be defined as

$\varepsilon_{0}\approx\frac{\hbar^{2}}{m_{f}l_{0}^{2}}\left\langle
n_{qp}\left(\tilde{E}\right)\tilde{W}_{t}\left(\tilde{E}\right)\right\rangle,$
(14)

where the dimensionless energy $\tilde{W}_{t}$ is given by the expression (10)
and $\left\langle\cdots\right\rangle$ denotes the quantum averaging over all
possible values of $\tilde{E}$.

The lifetime of a virtual quasiparticle with the energy
$W_{t}\left(\tilde{E}\right)$ is determined according to the Heisenberg
uncertainty principle,

$\Delta
t\approx\frac{\hbar}{W_{t}(\tilde{E})}=\frac{m_{f}l_{0}^{2}}{\hbar}\frac{1}{\tilde{W}_{t}(\tilde{E})}.$
(15)

This expression implies that the maximum lifetime is for a quasiparticle with
the energy equal to the energy of the mass gap
$\tilde{W}_{t}\left(\tilde{E}\right)=\tilde{\Delta}\left(g^{2}_{\Lambda}\right)$.

Consider next the behavior of the concentration of quasiparticles
$n_{qp}\left(\tilde{E}\right)$ for $\tilde{E}\rightarrow 0,1$. For
$\tilde{E}\rightarrow 1$, according to Eq. (13), the linear size of the region
occupied by the core of the “dipole” created by the spinor field approaches
infinity, $l_{cr}\rightarrow\infty$. This actually means that the size of a
quasiparticle approaches infinity as well; that is, one quasiparticle occupies
all the space. This in turn means that

$n_{qp}\left(\tilde{E}\rightarrow 1\right)\rightarrow 0.$ (16)

As we pointed out above, for $\tilde{E}\rightarrow 0$, the size of the core is
estimated as $l_{cr}\approx\lambda_{c}$, whereas the value of the non-Abelian
magnetic field goes to infinity and the field fills all the space. As in the
case with $\tilde{E}\rightarrow 1$, this means that

$n_{qp}\left(\tilde{E}\rightarrow 0\right)\rightarrow 0.$ (17)

Both conditions (16) and (17) can be written as

$n_{qp}\left(\tilde{E}\right)\rightarrow 0\text{ when }\tilde{E}\rightarrow
0,1\text{ or, equivalently, when
}\tilde{W}_{t}\left(\tilde{E}\right)\rightarrow\infty.$ (18)

Thus, the function $n_{qp}\left(\tilde{E}\right)$ is equal to zero at the
boundaries of the interval $0\leqslant\tilde{E}\leqslant 1$; this suggests
that for some $\tilde{E}_{0}$ lying in this interval the concentration
$n_{qp}\left(\tilde{E}_{0}\right)$ has a maximum.

The expression for the energy density (14) is the product of two functions:
the density of quasiparticles and the dimensionless energy of a quasiparticle.
For $\tilde{E}\rightarrow 0,1$, the first function, according to Eq. (18),
goes to zero, whereas the second one – to infinity. Consequently, their
product in Eq. (14) goes either to zero, or to a constant, or to infinity. In
order that the energy density would be finite, it is necessary to choose the
first option. This permits one to estimate the quantity (14) as the product of
the characteristic value of the density of quasiparticles
$n_{qp}\left(\tilde{E}_{0}\right)$ and the characteristic value of the energy
– the mass gap $\tilde{\Delta}(g^{2}_{\Lambda})$:

$\varepsilon_{0}\approx\frac{\hbar^{2}}{m_{f}l_{0}^{2}}n_{qp}\left(\tilde{E}_{0}\right)\tilde{\Delta}(g^{2}_{\Lambda}).$
(19)

This expression indicates that in the model under consideration the energy
density of the nonperturbative vacuum $\varepsilon_{0}$ is a finite quantity,
in contrast to the energy of perturbative vacuum which is infinite. Notice
that in these calculations the presence of a mass gap plays a crucial role. In
the absence of the mass gap, the computation procedure is changed considerably
and the expression (19) would not already be correct.

One of undetermined quantities in the expression (19) is the concentration of
quasiparticles $n_{qp}\left(\tilde{E}_{0}\right)$ with dimensions
$\text{(length)}^{-3}$. In the initial equations (2) and (3), there are two
quantities having the dimensions of length; this is the Compton wavelength
$\lambda_{c}$ and the spinor field coupling constant $l_{0}$. Therefore, one
can assume that, in order of magnitude, the concentration might be estimated
as

$n_{qp}\left(\tilde{E}_{0}\right)\approx\frac{1}{l_{0}^{3-\alpha}\lambda_{c}^{\alpha}}.$
(20)

The quantity $\left(l_{0}^{3-\alpha}\lambda_{c}^{\alpha}\right)^{1/3}$
characterizes the distance between quasiparticles. The simplest case of
$\alpha=0$ is sketched in Fig. 3.

It is of interest to estimate characteristic quantities of the nonperturbative
vacuum under consideration. To do so, we start from the assumption that its
energy density (19) is of the order of the energy density associated with
Einstein’s cosmological constant $\Lambda_{E}$, i.e.,
$\varepsilon_{0}=\varepsilon_{\Lambda_{E}}\approx 5.4\times 10^{-9}\text{erg
cm}^{-3}$. As an example, take the following set of parameters: (i) let
$\alpha=0$ in Eq. (20); (ii) choose $m_{f}$ equal to, say, the mass of the
strange quark $m_{s}\approx 95\,\text{MeV}$; (iii) take the magnitude of the
dimensionless mass gap to be
$\tilde{\Delta}\left(g^{2}_{\Lambda}\right)\approx 50$ [7]; (iv) let
${g^{\prime}}^{2}\approx 10$ (typical value in QCD). Then we have the
following estimates:

$\begin{split}&\tilde{g}^{2}_{\Lambda}\approx 2.09\times
10^{-17},\;\Delta\left(g^{2}_{\Lambda}\right)\approx 1.59\times
10^{-20}\mathrm{erg},\;m_{\Delta}\equiv\Delta\left(g^{2}_{\Lambda}\right)/c^{2}\approx
1.77\times 10^{-41}\mathrm{g},\\\ &\lambda_{c}\approx 2.08\times
10^{-13}\text{cm},\;l_{0}\approx 1.44\times 10^{-4}\text{cm},\;n_{qp}\approx
3.38\times 10^{11}\text{cm}^{-3}.\end{split}$ (21)

For some reasons this does not allow us to regard the energy density of the
nonperturbative vacuum as that associated with Einstein’s cosmological
constant. One of these reasons is that the equation of state corresponding to
the cosmological constant is $p_{\Lambda_{E}}=-\varepsilon_{\Lambda_{E}}.$ The
determination of the pressure of the nonperturbative vacuum (or its equation
of state) is a challenging problem, and we will not address it here. The point
is that the virtual quasiparticles under consideration are extended objects,
and it is in particular unclear how they interact with a reflecting wall.

##  IV Discussion and conclusions

In the present paper we have suggested a speculative model of nonperturbative
vacuum in SU(2) Yang-Mills theory coupled to a nonlinear spinor field. Within
this model, by analogy with Abelian magnetic monopole dominance, it is assumed
that the main contribution to the energy of the nonperturbative vacuum comes
from quasiparticles, which are here represented by “dipoles” constructed in
Refs. [6, 7]. The remarkable feature of such nonperturbative vacuum is that
its energy density is finite.

Summarizing the results,

  * •

The model of nonperturbative vacuum in SU(2) Yang-Mills theory coupled to a
nonlinear spinor field is suggested, assuming that the main contribution to
its characteristics comes from quasiparticles which are described by the
corresponding “dipole” solutions.

  * •

We present arguments to claim that the number density of quasiparticles whose
energy approaches infinity goes to zero.

  * •

We have derived the expression for the energy density of the nonperturbative
vacuum which, being a sum of energies of quasiparticles, is finite.

  * •

Starting from the assumption that the magnitude of the nonperturbative vacuum
energy density is of the order of the energy density associated with
Einstein’s cosmological constant, we have estimated the characteristic
quantities of the nonperturbative vacuum under consideration.

  * •

We suggest the physical interpretation of the spinor field self-coupling
constant, having the dimensions of length, as a quantity characterizing the
distance between quasiparticles in the nonperturbative vacuum.

We have pointed out that, although the energy density of the nonperturbative
vacuum may be of the order of that associated with Einstein’s cosmological
constant, for some reasons it is difficult to identify these two quantities.
For example, for the cosmological constant, it is necessary to have a strictly
defined relationship between its energy density and pressure. In further
studies on this subject it is necessary to get an expression for the pressure
of the nonperturbative vacuum which amounts to obtaining an equation of state
for such vacuum. The problem of principle one faces here is that the
quasiparticles are (i) quantum and virtual; and (ii) extended objects.

Notice that the SU(2) Yang-Mills theory coupled to a nonlinear spinor field
under consideration is non-QCD theory. The difference of principle is that, in
QCD, a spinor field is described by the linear Dirac equation, while here we
deal with the nonlinear Dirac equation. At first glance, the linear and
nonlinear Dirac equations are in no way related to each other. However, it is
worth mentioning here that there is a possibility of obtaining the nonlinear
Dirac equation as a consequence of nonpertubative quantization in QCD, as
discussed in Refs. [6, 7, 8].

In conclusion, we wish to note that the model of nonperturbative vacuum under
consideration is an experimentally testable model, at least in principle. The
lifetime of a virtual quasiparticle is determined by Eq. (15); then, e.g., for
the parameters used in obtaining the estimates (21), one has $\Delta t\approx
6.6\times 10^{-8}\mathrm{sec}$. In turn, one cubic centimeter contains only
$\sim 10^{11}$ virtual quasiparticles. Since this concentration and the
lifetimes are quite large quantities, one may expect that an experimental
verification of the existence of such virtual quasiparticles might be
possible.

## Acknowledgments

This research has been funded by the Science Committee of the Ministry of
Education and Science of the Republic of Kazakhstan. We are also grateful to
the Research Group Linkage Programme of the Alexander von Humboldt Foundation
for the support of this research.

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