Can negative bare couplings make sense? The $\vec{\phi}^{4}$ theory at large $N$

The thermal partition function of the positive-coupling theory is formally given by

$$Z_{+}\propto\int\mathcal{D}^{N}\vec{\phi}\,e^{-\int S_{+}[\vec{\phi}]}$$

(2)

where the path integral is over real-valued fields $\vec{\phi}(x)\in\mathbb{R}^{N}$ with periodic boundary conditions in Euclidean time $\tau$: $\vec{\phi}(\tau+\beta,\mathbf{x})=\vec{\phi}(\tau,\mathbf{x})$. The Euclidean action $S_{+}[\vec{\phi}]$ is

$$S_{+}[\vec{\phi}]=\int_{\beta,V}\textrm{d}^{4}x\,\bigg{(}\frac{1}{2}(\partial_{\mu}\vec{\phi})^{2}+\frac{\lambda}{N}\vec{\phi}^{4}\bigg{)},$$

where $\int_{\beta,V}\textrm{d}^{4}x=\int_{0}^{\beta}\textrm{d}\tau\int_{V}\textrm{d}^{3}\mathbf{x}$. In this work I only consider the massless case, but it is easy to generalize to the case with a mass term $M\vec{\phi}^{2}$ as done in [4, 5].

For this theory, the expression for $p(\beta)$ in (1) has already been calculated and can be found for example in [4]. In dimensional regularization in (3–$2\epsilon$)+1D, the expression is

$$\begin{split}p(\beta)=&\frac{m^{4}}{16\lambda}+\frac{m^{4}}{64\pi^{2}}\bigg{(}\frac{1}{\epsilon}+\ln\bigg{(}\frac{\bar{\mu}^{2}}{m^{2}}\bigg{)}+\frac{3}{2}\bigg{)}\\ &+\frac{m^{2}}{2\pi^{2}\beta^{2}}\sum_{n=1}^{\infty}\frac{K_{2}(n\beta m)}{n^{2}}.\end{split}$$

(3)

Here, $\bar{\mu}$ is the $\overline{\textrm{MS}}$ scale and $K_{2}(x)$ is a modified Bessel function of the second kind. $m$ is a gap parameter that is fixed by solving the gap equation $\partial p(\beta)/\partial m^{2}=0$. Only the solution to the gap equation which gives the larger pressure contributes in the large-volume limit.

The expression (3) for $p(\beta)$ has a $1/\epsilon$ divergence from the dimensional regularization which can be absorbed into the renormalized coupling $\lambda_{\textsc{r}}(\bar{\mu})$ by defining

$$\frac{1}{\lambda_{\textsc{r}}(\bar{\mu})}=\frac{1}{\lambda(\bar{\mu})}+\frac{1}{4\pi^{2}\epsilon}.$$

(4)

The renormalized coupling depends on $\bar{\mu}$ such that the pressure per component $p(\beta)$ does not depend on $\bar{\mu}$. This requirement, $\partial p(\beta)/\partial\bar{\mu}=0$, which gives the beta function, fixes the running coupling to be

$$\lambda_{\textsc{r}}(\bar{\mu})=\frac{4\pi^{2}}{\ln(\Lambda_{\textsc{lp}}^{2}/\bar{\mu}^{2})},$$

(5)

with $\Lambda_{\textsc{lp}}$ a scale that appears from the constant in the integration of the beta function. $\Lambda_{\textsc{lp}}$ is determined for example by fixing the value $\lambda_{0}$ of the coupling $\lambda_{\textsc{r}}(\Lambda_{0})=\lambda_{0}$ at some UV scale $\Lambda_{0}$. The running of the coupling is visualized in figure 1.

One immediate objection can be made. If the coupling has some non-zero, positive value at some finite scale $\bar{\mu}$, then the scale $\Lambda_{\textsc{lp}}>\bar{\mu}$ will also be finite. Above the scale $\Lambda_{\textsc{lp}}$, the coupling in (5) becomes negative. Therefore, there is no way to take the UV scale $\Lambda_{0}$ to infinity and keep the UV coupling $\lambda_{0}$ positive. Since a positive coupling was assumed in order to make the calculation (3) of $p(\beta)$ and of the partition function $Z_{+}$, this is problematic⁵⁵5However, I am being a little heuristic. It is really the coupling $\lambda$, and not $\lambda_{\textsc{r}}$, that was assumed to be positive here. In dimensional regularization in (3–2$\epsilon$)+1D this means that $\lambda(\bar{\mu})$ becomes negative at a scale $\bar{\mu}=\Lambda_{\textsc{lp}}\,e^{-1/2\epsilon}$. The fact remains that the coupling diverges and becomes negative at some scale. In cutoff regularization with cutoff $\Lambda$, there is a similar scale $\Lambda=\Lambda_{\textsc{lp}}$ above which the coupling $\lambda(\Lambda)$ becomes negative [5], and $\lambda(\Lambda)$ is identical to $\lambda_{\textsc{r}}(\bar{\mu})$ in (5) if $\bar{\mu}$ is identified with $\Lambda$.. The interacting theory appears to have no continuum limit $\Lambda_{0}\to\infty$, and if one tries to fix this problem by taking $\Lambda_{\textsc{lp}}$ to infinity, the coupling at any finite scale $\bar{\mu}<\infty$ will be zero, such that the theory is trivial (non-interacting). This is a feature of scalar $\phi^{4}$ field theory, which is known to be quantum trivial in 3+1D for $N=1,2$ components [21, 22].

As an additional note, the coupling $\lambda_{\textsc{r}}(\bar{\mu})$ will diverge at the scale $\bar{\mu}=\Lambda_{\textsc{lp}}$. Therefore the $\vec{\phi}^{4}$ theory at large $N$ has a Landau pole even non-perturbatively (as opposed to only perturbatively).

Despite these potential issues, assuming that $\bar{\mu}<\Lambda_{\textsc{lp}}$ and that the $\vec{\phi}^{4}$ theory is simply an effective (cut-off) theory, the calculation of the pressure $p(\beta)$ in (3) can proceed, and has been done by Romatschke, and the result plotted, in [2, 4]. The renormalized pressure is given by

$$\begin{split}p(\beta)=&\frac{m^{4}}{64\pi^{2}}\bigg{(}\ln\bigg{(}\frac{\Lambda_{\textsc{lp}}^{2}}{m^{2}}\bigg{)}+\frac{3}{2}\bigg{)}\\ &+\frac{m^{2}}{2\pi^{2}\beta^{2}}\sum_{n=1}^{\infty}\frac{K_{2}(n\beta m)}{n^{2}}.\end{split}$$

(6)

At low temperatures $T\leq T_{\textsc{c}}\approx 0.616\,\Lambda_{\textsc{lp}}$, one finds two solutions $m$ to the gap equation $\partial p(\beta)/\partial m^{2}=0$. The dominant solution has $m^{2}=e\Lambda_{\textsc{lp}}^{2}$ at $T=0$ and determines the pressure $p(\beta)$ via (6). At higher temperatures $T>T_{\textsc{c}}$ there are also two solutions to the gap equation, but now they are a complex-conjugate pair of solutions $m_{+},m_{-}=m_{+}^{*}$ corresponding to complex-conjugate pressures $p_{+}(\beta),p_{-}(\beta)=p_{+}^{*}(\beta)$. Putting the partition function $Z_{+}$ into the large-$N$ form of (1) leaves one with

$$\begin{split}Z_{+}&\propto e^{N\beta V\operatorname{Re}p_{+}(\beta)+\ln\cos\big{(}N\beta V\operatorname{Im}p_{+}(\beta)\big{)}}\\ &\approx e^{N\beta V\operatorname{Re}p_{+}(\beta)},\,\,\,\,\beta<\beta_{\textsc{c}},\end{split}$$

(7)

which follows formally at large $N$ from only keeping the part of $\ln Z_{+}$ that scales like $N$ ⁶⁶6Note that the argument for setting $p(\beta)=\operatorname{Re}p_{+}(\beta)$ in [2, 4] was based on the unproven conjecture in [20] for the $\mathcal{P}\mathcal{T}$-symmetric $-g\phi^{4}$ theory. Here instead I’ve alternatively based it on the formal expansion of $\ln Z$ in large $N$. Later, in subsection III.2, I discuss the consequences of not neglecting the imaginary parts of both saddles, and how it suggests $\mathcal{P}\mathcal{T}$ symmetry breaking at high $T$.. One finds that $p(\beta<\beta_{\textsc{c}})=\operatorname{Re}p_{+}(\beta)$ is physically well-behaved (e.g. increases with temperature) and $p(\beta)$ has a continuous first derivative at $\beta=\beta_{\textsc{c}}$.

Interestingly enough, the pressure per component $p(\beta)$ of the $\lambda\vec{\phi}^{4}/N$ theory, calculated in this way, asymptotes toward the Stefan–Boltzmann limit for a single free boson at high temperatures $\beta\to 0$ [2], suggesting that the $\vec{\phi}^{4}$ theory at large $N$ is asymptotically free. This should not be true based on the argument that the Landau pole in (5) prevented the positive-coupling theory from having an interacting continuum limit. Moreover, the beta function $\partial\lambda_{\textsc{r}}/\partial\ln\bar{\mu}$ has the same sign as the coupling assuming $\bar{\mu}<\Lambda_{\textsc{lp}}$. Asymptotically free theories, on the other hand, are interacting, do have a continuum limit, and have a beta function whose sign is opposite that of the coupling as $\bar{\mu}\to\infty$.

The solution to this this apparent contradiction, and to the problem of triviality, is to allow the bare coupling to be negative. Non-triviality and a negative bare coupling were argued in [5]. But if the bare coupling is negative, the calculation of $p(\beta)$ must be done in some other way that assumes negative coupling from the start.

That is the main new result and next part of this work.

The thermal partition function for the negative-coupling theory is given by

$$Z_{-}\propto\int_{\mathcal{C}_{-}}\mathcal{D}^{N}\vec{\phi}\,e^{-\int S_{-}[\vec{\phi}]}$$

(8)

where the fields still have periodic boundary conditions in Euclidean time but now the path integral is over complexified fields that live on the domain $\mathcal{C}_{-}$ (which will be specified shortly). The Euclidean action $S_{-}[\vec{\phi}]$ is

$$S_{-}[\vec{\phi}]=\int_{\beta,V}\textrm{d}^{4}x\,\bigg{(}\frac{1}{2}(\partial_{\mu}\vec{\phi})^{2}-\frac{g}{N}\vec{\phi}^{4}\bigg{)},$$

(9)

where $-g<0$ is now the coupling.

A choice for the domain $\mathcal{C}_{-}$ that one might consider for the $\mathcal{P}\mathcal{T}$-symmetric $-g\vec{\phi}^{4}/N$ theory is the domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ parametrized by $N$ real-valued fields $\chi(x)\in\mathbb{R}$, $\vec{\eta}(x)\in\mathbb{R}^{N\textrm{--}1}$ as

$$\begin{split}\vec{\phi}(x)&\cdot\hat{e}=\chi(x)f\big{(}\chi(x)\big{)}\\ \vec{\phi}(x)-&\big{(}\vec{\phi}(x)\cdot\hat{e}\big{)}\hat{e}=\vec{\eta}(x)f\big{(}\chi(x)\big{)},\end{split}$$

(10)

where $\hat{e}$ is some unit vector in $\mathbb{R}^{N}$ and the function $f$ is defined as

$$f(\chi)\equiv\theta\big{(}\chi)e^{-\textrm{i}\pi/4}+\theta(\textrm{--}\chi)e^{\textrm{i}\pi/4},$$

(11)

where $\theta$ is the Heaviside step function. This forces one component $\vec{\phi}(x)\cdot\hat{e}$ of the field $\vec{\phi}$ to live on the union of two half-lines at angles $-\pi/4$, $-3\pi/4$ in the lower half of the complex plane at every point $x$ in Euclidean spacetime. If every component lived independently on the same union of half-lines, the path integral would be unbounded, as was pointed out in [23]. This domain respects the $\mathcal{P}\mathcal{T}$ symmetry $\vec{\phi}\to-\vec{\phi}$, $\textrm{i}\to-\textrm{i}$, which is also a symmetry of the action $S_{-}[\vec{\phi}]$, and any deformation of this domain with the same Lefschetz thimble decomposition (e.g. terminating within the associated Stoke’s wedges) gives the same partition function $Z_{-}=Z_{\mathcal{P}\mathcal{T}}$.

The path integral on the domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ in (10), however, is difficult to solve analytically using standard large-$N$ techniques, because there are an infinite number of sharp kinks in the domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$, with one kink at $\chi(x)=0$ for every point $x$ in spacetime. These kinks can be managed straightforwardly in a lattice calculation (and the $N=1$ lattice calculation in 3+1D was done in [5] and in 1D in [23]), but on the lattice the path integral (8) with $\mathcal{C}_{-}=\mathcal{C}_{\mathcal{P}\mathcal{T}}$ from (10) has a sign problem, so the calculation is difficult to scale to $N$ components.

One could consider a domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ without kinks by generalizing the hyperbola $\phi(x)=-2\textrm{i}\sqrt{1+\textrm{i}\chi(x)}$ for $\chi(x)\in\mathbb{R}$ in [9, 24] to $N$-component scalar fields, but it is unclear how to do this. One such generalization to $N$ components for the 1D theory was done in [25] by letting the radial coordinate live on this complex hyperbola, i.e. $\vec{\phi}^{\,2}(x)=-4\big{(}1+\textrm{i}\chi(x)\big{)}$ for $\chi(x)\in\mathbb{R}$, but this generalization gives an unbounded Hamiltonian spectrum and thus an unbounded partition function in 1D once angular momentum is considered⁷⁷7The interested reader can check this by including the angular momentum quantum number $\ell$ in the radial Schrödinger equation and finding the Hermitian Hamiltonian which is equivalent to the $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian Hamiltonian along the lines of [24, 9, 25].; it likewise gives an unbounded path integral in 3+1D. Another parametrization in the literature, $\phi_{i}(x)=-2\textrm{i}\sqrt{c_{i}+\textrm{i}\chi_{i}(x)}$ [26], also gives an unbounded path integral upon closer inspection because there is a flat direction $\chi_{i}-\chi_{j}$ for $i\neq j$ in the potential.

There is an alternative choice for the domain $\mathcal{C}_{-}$ that gets rid of most of these kinks by only forcing one component of the Fourier zero mode of the scalar field to live on the half-lines at $-\pi/4$, $-3\pi/4$ in the lower half of the complex plane. Let $\vec{\phi}(x)=\vec{\phi}_{0}+\vec{\phi}^{\prime}(x)$, where $\vec{\phi}_{0}$ is the zero mode of the scalar field and $\vec{\phi}^{\prime}$ contains all the non-zero modes. Then the domain $\mathcal{C}_{-}$ can be parametrized in terms of the real-valued fields $\chi(x)=\chi_{0}+\chi^{\prime}(x)\in\mathbb{R}$ and $\vec{\eta}(x)\in\mathbb{R}^{N\textrm{--}1}$ as

$$\begin{split}\vec{\phi}_{0}\cdot\hat{e}&=\chi_{0}f(\chi_{0}),\\ \vec{\phi}^{\prime}(x)\cdot&\hat{e}=\chi^{\prime}(x)f(\chi_{0}),\\ \vec{\phi}(x)-&\big{(}\vec{\phi}(x)\cdot\hat{e}\big{)}\hat{e}=\vec{\eta}(x)f(\chi_{0}),\end{split}$$

(12)

where $\chi_{0}$ and $\chi^{\prime}$ denote the zero mode and non-zero modes of the field $\chi$, respectively, and where $f$ is the same function defined in (11). See figure 2 for a visualization of this domain and the domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ in $\eqref{eq:domain-PT}$. This domain still gives a bounded path integral and respects the $\mathcal{P}\mathcal{T}$ symmetry $\vec{\phi}\to-\vec{\phi}$, $\textrm{i}\to-\textrm{i}$, but now it makes the path integral amenable to large-$N$ techniques, as we will see. Moreover, it gives a physical partition function. However, this domain does not (as far as I can tell) correspond to a Hamiltonian picture where quantum states and an inner product like the $\mathcal{C}\mathcal{P}\mathcal{T}$ inner product can be defined. The domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ in equation (10) however will have a Hamiltonian, inner product, states, and an associated notion of unitary. For ease of calculation, the domain $\mathcal{C}_{-}$ specified by (12) is the choice that will be considered in this paper for the negative-coupling theory.

One more remark should be made before continuing on to the calculation of the partition function $Z_{-}$. It is perhaps unsatisfactory that the domain $\mathcal{C}_{-}$ in (12), and also the domain $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ in (10), breaks the usual $\textrm{O}(N)$ symmetry of the theory (which is only a symmetry of the action) down to $\textrm{O}(N-1)$. However, it is not apparent how to keep the $\textrm{O}(N)$ symmetry and the $\mathcal{P}\mathcal{T}$ symmetry while keeping the path integral bounded. As was already mentioned, letting the radial coordinate live on the $\mathcal{P}\mathcal{T}$-symmetric hyperbola $\vec{\phi}^{\,2}(x)=-4\big{(}1+\textrm{i}\chi(x)\big{)}$, as was done for the 1D $N$-component theory in [25], gives an unbounded path integral, even though it keeps the $\textrm{O}(N)$ symmetry.

It should be pointed out that the calculation of the $\mathcal{P}\mathcal{T}$-symmetric theory’s partition function using the constraint method (also applied in this paper in subsection II.3) was done by Ogilvie and Meisinger in [26] without explicitly breaking the $\textrm{O}(N)$ symmetry. These authors acquired a result equivalent to (23) in this work and also to the Hermitian theory. However in using the constraint calculation they do not specify a domain of integration for the field $\vec{\phi}$ or for the auxiliary fields, nor do they renormalize the theory or do the Lefschetz thimble analysis that is done later in this work. The only domain they specify, earlier in the work, gives an unbounded path integral and also breaks the $\textrm{O}(N)$ symmetry, and is not used explicitly in the calculation, unlike what will be done with the domain $\mathcal{C}_{-}$ here.

Alternatively, there are composite $\mathcal{P}\mathcal{T}$-symmetric domains that keep the $\textrm{O}(N)$ symmetry, for example by extending the half-lines in the lower half of the complex plane in figure 2 to full lines, with the sacrifice that the domain of integration intersects itself and is no longer parametrized simply by $N$ real fields. I note that if this is done for the domain $\mathcal{C}_{-}$, the $\textrm{O}(N)$ symmetry can be maintained, and one still gets the same result as in this work. One can also restore the $\textrm{O}(N)$ symmetry, allowing it only to be spontaneously broken, by integrating the unit vector $\hat{e}$ in (10) and (12) over the sphere $S^{N-1}$, and one will again get the same result as this work.

Here, I will stick to the domain $\mathcal{C}_{-}$ in (12) with the intent of demonstrating there is at least some domain $\mathcal{C}_{-}$ for which the negative-coupling theory “renormalizes into” the positive-coupling theory.

The partition function $Z_{-}$ will now be calculated. With the choice in (12) for the domain $\mathcal{C}_{-}$, and in terms of the real fields $\chi$, $\vec{\eta}$, the path integral (8) for the negative-coupling theory becomes

$$\begin{split}Z_{-}\propto\sum_{+,-}\pm&e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\bigg{(}\int_{0}^{\pm\infty}\textrm{d}\chi_{0}\\ &\int\mathcal{D}\chi^{\prime}\,\mathcal{D}^{N\textrm{--}1}\vec{\eta}\,\,e^{-S_{-}^{\pm}[\chi,\vec{\eta}]}\bigg{)}.\end{split}$$

(13)

Here, the symbol $\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}$ is shorthand for $\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}\equiv\sum_{n}\int\textrm{d}^{3}\mathbf{k}/(2\pi)^{3}$ (a sum over Matsubara frequencies $\omega_{n}$ and a integral over 3-momenta $\mathbf{k}$), and $S_{-}^{\pm}[\chi,\vec{\eta}]$ is given by

$$\begin{split}S_{-}^{\pm}[\chi,\vec{\eta}]=\int_{\beta,V}\textrm{d}^{4}x\,\bigg{(}\pm\frac{1}{2\textrm{i}}&\big{(}(\partial_{\mu}\chi)^{2}+(\partial_{\mu}\vec{\eta})^{2}\big{)}\\ &+\frac{g}{N}(\chi^{2}+\vec{\eta}^{\,2})^{2}\bigg{)}.\end{split}$$

(14)

The fields $\chi$, $\vec{\eta}$ have periodic boundary conditions in Euclidean time. We see the path integral becomes a sum of two pieces, each corresponding to one of the half-lines at angles $-\pi/4$, $-3\pi/4$ on which the zero mode $\vec{\phi}_{0}\cdot\hat{e}$ lives.

In order to solve this integral at large $N$, it is convenient to perform a Hubbard–Stratonovich transformation by introducing the auxiliary fields $\zeta$ and $\sigma=\chi^{2}+\vec{\eta}^{\,2}$, as follows:

$$\begin{split}Z_{-}\propto\sum_{+,-}&\pm e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\bigg{(}\int_{0}^{\pm\infty}\textrm{d}\chi_{0}\\ &\int\mathcal{D}\zeta\,\mathcal{D}\sigma\,\mathcal{D}\chi^{\prime}\,\mathcal{D}^{N\textrm{--}1}\vec{\eta}\,\,e^{-S_{-}^{\pm}[\chi,\vec{\eta},\zeta,\sigma]}\bigg{)},\end{split}$$

(15)

where $S_{-}^{\pm}[\chi,\vec{\eta},\zeta,\sigma]$ is given by

$$\begin{split}S_{-}^{\pm}[\chi,\vec{\eta},\zeta,\sigma]=\int_{\beta,V}&\textrm{d}^{4}x\,\bigg{(}\pm\frac{1}{2\textrm{i}}\big{(}(\partial_{\mu}\chi)^{2}+(\partial_{\mu}\vec{\eta})^{2}\big{)}\\ &+\frac{g}{N}\sigma^{2}+\frac{\textrm{i}}{2}\zeta(\chi^{2}+\vec{\eta}^{\,2}-\sigma)\bigg{)}.\end{split}$$

(16)

This amounts to the insertion of a delta function into the path integral that enforces $\sigma=\chi^{2}+\vec{\eta}^{\,2}$. The action (16) is quadratic in the auxiliary field $\sigma$, so $\sigma$ can be integrated out to give

$$\begin{split}Z_{-}\propto\sum_{+,-}&\pm e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\bigg{(}\int_{0}^{\pm\infty}\textrm{d}\chi_{0}\\ &\int\mathcal{D}\zeta\,\mathcal{D}\chi^{\prime}\,\mathcal{D}^{N\textrm{--}1}\vec{\eta}\,\,e^{-S_{-}^{\pm}[\chi,\vec{\eta},\zeta]}\bigg{)},\end{split}$$

(17)

where $S_{-}^{\pm}[\chi,\vec{\eta},\zeta]$ is now given by

$$\begin{split}S_{-}^{\pm}[\chi,\vec{\eta},\zeta]=\int_{\beta,V}&\textrm{d}^{4}x\,\bigg{(}\pm\frac{1}{2\textrm{i}}\big{(}(\partial_{\mu}\chi)^{2}+(\partial_{\mu}\vec{\eta})^{2}\big{)}\\ &+\frac{\textrm{i}}{2}\zeta(\chi^{2}+\vec{\eta}^{\,2})+\frac{N\zeta^{2}}{16g}\bigg{)}.\end{split}$$

(18)

The auxiliary field $\zeta$ can be split into a zero mode $\zeta_{0}$ and non-zero modes $\zeta^{\prime}$ as $\zeta(x)=\zeta_{0}+\zeta^{\prime}(x)$. At leading order in large $N$, only the zero mode contributes to the pressure⁸⁸8Including the non-zero modes $\zeta^{\prime}$ amounts to including $1/N$ corrections, and can be done via R$n$ resummation methods [27, 28]., so the integral over the non-zero modes can be neglected [29] and the partition function becomes

$$\begin{split}Z_{-}\propto\sum_{+,-}&\pm e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\bigg{(}\int_{0}^{\pm\infty}\textrm{d}\chi_{0}\\ &\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\int\mathcal{D}\chi^{\prime}\mathcal{D}^{N\textrm{--}1}\vec{\eta}\,\,e^{-S_{-}^{\pm}[\chi,\vec{\eta},\zeta_{0}]}\bigg{)},\end{split}$$

(19)

with $\zeta$ in expression (18) now replaced by only the zero mode $\zeta_{0}$.

There is a term $\textrm{i}\zeta_{0}(\chi^{2}+\eta^{2})/2$ in the action $S_{-}^{\pm}[\chi,\vec{\eta},\zeta_{0}]$ in (19) which contains a cross-term $\textrm{i}\zeta_{0}\chi_{0}\chi^{\prime}$ between the zero and non-zero modes of $\chi$. Because $\zeta_{0}\chi_{0}$ does not vary with $x$ and $\chi^{\prime}$ contains only non-zero modes, this cross-term vanishes under the integral $\int_{\beta,V}\textrm{d}x^{4}$, leaving one with a term $\textrm{i}\zeta_{0}(\chi_{0}^{2}+\chi^{\prime 2}+\eta^{2})/2$ in the action. The action $S_{-}^{\pm}[\chi,\vec{\eta},\zeta_{0}]$ is thus an even function of $\chi_{0}$, and so the bounds of the integral over $\chi_{0}$ can be extended from $[0,\pm\infty)$ to $(-\infty,\infty)$ without consequence, which gives

$$\begin{split}Z_{-}\propto\sum_{+,-}&e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\int\mathcal{D}^{N}\vec{\chi}\,\,e^{-S_{-}^{\pm}[\vec{\chi},\zeta_{0}]},\end{split}$$

(20)

where $\vec{\chi}(x)=\big{(}\chi(x),\vec{\eta}(x)\big{)}\in\mathbb{R}^{N}$ has been introduced simply to group together the fields $\chi(x)$ and $\vec{\eta}(x)$, and $S_{-}^{\pm}[\vec{\chi},\zeta_{0}]$ is

$$\begin{split}S_{-}^{\pm}[\vec{\chi},\zeta_{0}]=&\beta V\frac{N\zeta_{0}^{2}}{16g}\\ &+\int_{\beta,V}\textrm{d}x^{4}\bigg{(}\pm\frac{1}{2\textrm{i}}(\partial_{\mu}\vec{\chi})^{2}+\frac{\textrm{i}}{2}\zeta_{0}\vec{\chi}^{\,2}\bigg{)}.\end{split}$$

(21)

The action (21) is quadratic in $\vec{\chi}$, and so $\vec{\chi}$ can be integrated out, yielding

$$\begin{split}Z_{-}\propto\,\,&\sum_{+,-}e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}\bigg{(}\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\\ &\,\,\,\,\,\,\,\,e^{-N\ln\det(\pm\textrm{i}\partial_{\mu}^{2}+\textrm{i}\zeta_{0})/2-\beta VN\zeta_{0}^{2}/16g}\bigg{)}\\ \propto\,\,&\sum_{+,-}\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{-N\textrm{tr}\ln(-\partial_{\mu}^{2}\mp\zeta_{0})/2-\beta VN\zeta_{0}^{2}/16g}\\ &=\,\,\sum_{+,-}\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{N\beta Vp(\beta,\mp\zeta_{0})}.\end{split}$$

(22)

Note that the complex Jacobian term $e^{\mp\textrm{i}N\pi V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}/4}$ has been canceled by pulling out a $e^{-N\textrm{tr}\ln(\mp\textrm{i})/2}$ from inside the path integral, where the trace of the operator on a single field component brings out a phase space factor $V\mathord{\mathchoice{\ooalign{\hfil$\displaystyle\Sigma$\hfil\cr\hfil$\displaystyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\textstyle\Sigma$\hfil\cr\hfil$\textstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptstyle\Sigma$\hfil\cr\hfil$\scriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}{\ooalign{\hfil$\scriptscriptstyle\Sigma$\hfil\cr\hfil$\scriptscriptstyle\textrm{\hskip 1.0pt\scalebox{1.2}{$\int$}}$\hfil\cr}}}$. Here, also, I’ve introduced the pressures per component $p(\beta,\mp\zeta_{0})$, which are given more explicitly by

$$\begin{split}p(\beta,\mp\zeta_{0})=&-\frac{\zeta_{0}^{2}}{16g}\\ -\frac{1}{2\beta}\sum_{n=-\infty}^{\infty}&\int\mu^{2\epsilon}\frac{\textrm{d}^{3-2\epsilon}\mathbf{k}}{(2\pi)^{3}}\ln(\omega_{n}^{2}+\mathbf{k}^{2}\mp\zeta_{0}),\end{split}$$

(23)

where $\omega_{n}=2\pi n/\beta$ is the $n$th Matsubara frequency and the integral over 3-momenta $\mathbf{k}$ can be done using dimensional regularization in $3-2\epsilon$ dimensions. $\mu$ is the scale of dimensional regularization, related to the $\overline{\textrm{MS}}$ scale $\bar{\mu}$.

The thermal sum-integral in (23) has been done, for example, in [30], and the resulting pressures per component are

$$\begin{split}p(\beta,\mp\zeta_{0})=&-\frac{\zeta_{0}^{2}}{16g}+\frac{\zeta_{0}^{2}}{64\pi^{2}}\bigg{(}\frac{1}{\epsilon}+\ln\bigg{(}\frac{\bar{\mu}^{2}}{\mp\zeta_{0}}\bigg{)}+\frac{3}{2}\bigg{)}\\ &\mp\frac{\zeta_{0}}{2\pi^{2}\beta}\sum_{n=1}^{\infty}\frac{K_{2}(n\beta\sqrt{\mp\zeta_{0}})}{n^{2}}.\end{split}$$

(24)

This is the same expression as (3) from the positive-coupling theory, with $m^{2}$ replaced by $m^{2}=\mp\zeta_{0}$ and $\lambda$ replaced by $-g$.

As with the positive-coupling theory, there is a $1/\epsilon$ divergence from the dimensional regularization that can be absorbed by defining the renormalized coupling $g_{\textsc{r}}(\bar{\mu})$ as

$$\frac{1}{g_{\textsc{r}}(\bar{\mu})}=\frac{1}{g(\bar{\mu})}-\frac{1}{4\pi^{2}\epsilon}.$$

(25)

Then, requiring the pressures to be independent of the $\overline{\textrm{MS}}$ scale $\bar{\mu}$, such that $\partial p(\beta,\mp\zeta_{0})/\partial\bar{\mu}=0$, yields the beta function and fixes the running coupling to be

$$g_{\textsc{r}}(\bar{\mu})=\frac{4\pi^{2}}{\ln(\bar{\mu}^{2}/\Lambda_{\textsc{lp}}^{2})},$$

(26)

where $\Lambda_{\textsc{lp}}$ is again a scale introduced from the integration of the beta function. The renormalized pressures $p(\beta,\mp\zeta_{0})$ are then given by

$$\begin{split}p(\beta,\mp\zeta_{0})=&\frac{\zeta_{0}^{2}}{64\pi^{2}}\bigg{(}\ln\bigg{(}\frac{\Lambda_{\textsc{lp}}^{2}}{\mp\zeta_{0}}\bigg{)}+\frac{3}{2}\bigg{)}\\ &\mp\frac{\zeta_{0}}{2\pi^{2}\beta^{2}}\sum_{n=1}^{\infty}\frac{K_{2}(n\beta\sqrt{\mp\zeta_{0}})}{n^{2}}.\end{split}$$

(27)

Note that the running coupling in (26) is exactly the same expression as the running coupling in (5) for the positive-coupling theory up to a minus sign. That is, $g_{\textsc{r}}(\bar{\mu})=-\lambda_{\textsc{r}}(\bar{\mu})$ if the scale $\Lambda_{\textsc{lp}}$ in both theories is identified as the same scale.

At large $N$, the sum of integrals in (22) can be replaced by

$$Z_{-}\approx\sum_{+,-}n_{j}^{\pm}\sum_{j}e^{N\beta Vp(\beta,\zeta_{0j})},$$

(28)

where $\zeta_{0j}$ denotes the $j$th saddle of $p(\beta,\zeta_{0})$ and $n^{\pm}_{j}$ denotes the intersection number of the contour of integration with the Lefschetz anti-thimble passing through the $j$th saddle. For an introduction to Lefschetz thimbles, see [31]. Only the relevant saddles with $n_{j}\neq 0$ contribute to the partition function, and only the saddle with the largest pressure dominates in the large-volume limit. In order to determine the saddles, one has to apply the saddle condition (or gap equation) $\partial p(\beta,\mp\zeta_{0})/\partial\zeta_{0}=0$ to fix the gap $m=\sqrt{\mp\zeta_{0}}$.

However, the function for the pressures in (27) has a branch point at $\zeta_{0}=0$ and is multi-sheeted. See figure 3 for an illustration of this feature. A branch cut can be made on the real axis for $\mp\zeta_{0}<0$, defining the principle sheet, and the integrals of $e^{N\beta Vp(\beta,\mp\zeta_{0})}$ in (22) can be done slightly above or below the real axis in order to avoid the branch point. Here there is an apparent ambiguity whether the integration should occur below or above the branch point, but the requirement of $\mathcal{P}\mathcal{T}$ symmetry settles the ambiguity. Since $\mathcal{P}\mathcal{T}$ symmetry interchanges $\zeta_{0}\to-\zeta_{0}$ and $\textrm{i}\to-\textrm{i}$ ⁹⁹9This is because $\mathcal{P}\mathcal{T}$ exchanges the two half lines at angles $-\pi/4$, $-3\pi/4$ in figure 2, or equivalently switches the two integrals over $\chi_{0}>0$ and $\chi_{0}<0$ in (13), which in (22) amounts to swapping $p(\beta,-\zeta_{0})$ with $p(\beta,\zeta_{0})$, i.e. $\zeta_{0}\to-\zeta_{0}$., the integrals in (22) should be

$$\begin{split}Z_{-}\propto&\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{N\beta Vp(\beta,-\zeta_{0}+\textrm{i}0^{+})}\\ &+\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{N\beta Vp(\beta,\zeta_{0}-\textrm{i}0^{+})}\\ =&\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{N\beta Vp(\beta,\zeta_{0}+\textrm{i}0^{+})}\\ &+\int_{-\infty}^{\infty}\textrm{d}\zeta_{0}\,e^{N\beta Vp(\beta,\zeta_{0}-\textrm{i}0^{+})}\end{split}$$

(29)

where here I have used the fact that $\int_{-a}^{a}\textrm{d}x\,f(-x)=\int_{-a}^{a}\textrm{d}x\,f(x)$ for any function $f$.

The relevant saddles of $p(\beta,\zeta_{0})$ in (29) can be determined from the downward flow of the contours $\zeta_{0}\pm\textrm{i}0^{+}\in\mathbb{R}$ onto the Lefschetz thimbles (contours of steepest descent) passing through the saddles. The downward flow is defined as

$$\frac{\partial\zeta_{0}}{\partial s}=-\bigg{(}\frac{\partial p(\beta,\zeta_{0})}{\partial\zeta_{0}}\bigg{)}^{*},$$

(30)

where $s$ is “flow time” and the the lines following the downward flow toward the saddle are the Lefschetz anti-thimbles. A visualization of this downward flow at one selected temperature is in figure 4.

For $T<T_{\textsc{c}}\approx 0.616\,\Lambda_{\textsc{lp}}$ there are two real saddles which are relevant for both contours $\zeta_{0}\pm\textrm{i}0^{+}\in\mathbb{R}$, and the saddle with the larger value of $\zeta_{0}>0$ gives the dominant pressure per component $p(\beta)$ in (1). At zero temperature this saddle has $\zeta_{0}=e\Lambda_{\textsc{lp}}^{2}$, which corresponds to a non-vanishing mass gap $m=\sqrt{\zeta_{0}}$ for the scalar field $\vec{\phi}$ in the vacuum. The saddle of the non-preferred phase is trivial at $T=0$. When the scale $\Lambda_{\textsc{lp}}$ is equated to the scale $\Lambda_{\textsc{lp}}$ in the positive-coupling theory, these saddles correspond exactly to those for the positive-coupling theory and those found by Romatschke in [2], and the pressures of these saddles are the same in both theories. Above the critical temperature $T>T_{\textsc{c}}$, there are, just as in the positive-coupling theory, a complex conjugate pair of saddles. The second integral in (29), over the contour slightly below the branch cut (shown in figures 3 and 4), picks up a contribution only from the saddle with $\operatorname{Im}(\zeta_{0})<0$. Meanwhile, the first integral, on the contour above the branch cut in figure 4, picks up a contribution from the saddle with $\operatorname{Im}(\zeta_{0})>0$. The saddles have the same pressures $p_{+}(\beta)$, $p_{-}(\beta)=p^{*}_{+}(\beta)$ as the complex conjugate pair of saddles in the positive-coupling theory. Just as in (7), keeping only the part of $\ln Z_{-}$ that scales like $N$ gives a pressure per component $p(\beta)$ in (1) that is equal to $\operatorname{Re}p_{+}(\beta)$.

The pressure per component $p(\beta)$ as a function of temperature is plotted in figure 5. I discuss results in the next section.

A notable feature of the pressure in figure 5 is that it approaches the Stefan–Boltzmann limit $p(\beta)=\pi^{2}T^{4}/90$ for a free scalar boson at high temperatures. This is a characteristic feature of asymptotically free theories. This makes perfect sense from having a negative bare coupling $\lambda_{0}=-g_{0}<0$ as $\Lambda_{0}\to\infty$ as seen in figure 1; the sign of the beta function $\partial g_{\textsc{r}}/\partial\ln(\bar{\mu})$ is opposite the sign of the coupling as $\bar{\mu}\to\infty$ and the coupling (26) approaches zero in the UV, so that the theory is almost free at high energies.

In doing the calculation of the partition function $Z_{-}$ with negative coupling, the only trouble we ran into was that the scale $\bar{\mu}$ could not be taken below the scale $\Lambda_{\textsc{lp}}$ or else the assumption of negative coupling would be violated¹⁰¹⁰10If the coupling were taken to be positive, as happens for $\bar{\mu}<\Lambda_{\textsc{lp}}$, the path integral (8) on the domain $\mathcal{C}_{-}$ would be unbounded.. Thus the negative-coupling theory has a Landau pole, albeit a Landau pole¹¹¹¹11Here, by Landau pole, I mean any place where the coupling diverges and becomes infinite. in the IR and not in the UV. A Landau pole in the IR, unlike in the positive-coupling case, does not prevent there from being an interacting continuum limit of the theory. Indeed, the negative-coupling $\vec{\phi}^{4}$ theory at large $N$ is not quantum trivial.