Particle Interferometry in a Moat Regime

As illustrated in Fig. 1, in situations like heavy-ion collisions, a fixed temperature, for example, is typically found on a three-dimensional hypersurface $\Sigma$ of 4$d$ Minkowski spacetime which differs from flat $\mathbb{R}^{3}$. Thus, instead of describing these particles in Minkowski space, it is advantageous to describe them in terms of an appropriate foliation of spacetime. To this end, we use that any hypersurface $\Sigma$ can be defined by parametric equations of the form

$\displaystyle x^{\mu}=x^{\mu}(w^{i})\,,$

(4)

where $w^{i}(x)$ with $i=1,2,3$ are coordinates intrinsic to $\Sigma$ [36]. We define tangent vectors pointing into ‘spatial’ directions,

$\displaystyle e_{i}^{\mu}=\frac{\partial x^{\mu}}{\partial w^{i}}\,,$

(5)

which give rise to the induced metric (first fundamental form) on $\Sigma$,

$\displaystyle G_{ij}=-g_{\mu\nu}\,e_{i}^{\mu}e_{j}^{\nu}\,,$

(6)

where $g^{\mu\nu}$ is the spacetime metric, i.e., $g^{\mu\nu}={\rm diag}(1,-1,-1,-1)$ in our case. Given the coordinates $w^{i}$, we can express the ‘temporal’ direction via the normal vector

$\displaystyle\hat{v}^{\mu}=\frac{e_{0}^{\mu}}{\sqrt{e_{0}^{\mu}e_{0\mu}}}\,,$

(7)

with

$\displaystyle e_{0}^{\mu}=\bar{\epsilon}^{\mu\alpha\beta\gamma}\,e_{1\alpha}e_{2\beta}e_{3\gamma}\,,$

(8)

where $\sqrt{|{\mathrm{det}}\,g|}\,\bar{\epsilon}^{\mu\alpha\beta\gamma}={\rm sign}({\mathrm{det}}\,g)\,\epsilon^{\mu\alpha\beta\gamma}$ is the covariant Levi-Civita tensor. We consider only spacelike hypersurfaces, defined by having a timelike normal vector, $\hat{v}^{\mu}\hat{v}_{\mu}=+1$.

Using these definitions, we can decompose the ambient metric as

$\displaystyle g^{\mu\nu}=\hat{v}^{\mu}\hat{v}^{\nu}-\Delta^{\mu\nu}\,,$

(9)

where $\Delta^{\mu\nu}=G^{ij}e_{i}^{\mu}e_{j}^{\nu}$ projects onto the hypersurface. This allows us define the temporal and spatial coordinates

$\displaystyle x_{\parallel}=\hat{v}^{\mu}x_{\mu}\,,\qquad\mathbb{x}_{\perp}=\mathbb{e}^{\mu}x_{\mu}\,.$

(10)

Thus, ‘time’ and ‘space’ are defined as projections of the ambient-space coordinates normal and tangential to $\Sigma$. This construction defines a foliation of spacetime, where instead of Minkowski space $\{x^{0}\}\times\mathbb{R}^{3}$, we describe spacetime as $\{x_{\parallel}\}\times\Sigma$.

Since the spatial coordinates are the components of tangent vectors $x_{\perp}^{i}e_{i}^{\mu}$ on $\Sigma$, the corresponding derivatives are covariant derivatives with the induced connection,

$\displaystyle\Gamma_{ijk}=\frac{1}{2}\Bigg{(}\frac{\partial G_{ij}}{\partial w^{k}}+\frac{\partial G_{ik}}{\partial w^{j}}-\frac{\partial G_{jk}}{\partial w^{i}}\Bigg{)}\,.$

(11)

This defines the intrinsic covariant derivative, which for a vector field on $\Sigma$, $a_{\perp\,i}=a_{\mu}e^{\mu}_{i}$, is

$\displaystyle\partial_{\perp\,j}\,a_{\perp\,i}=\frac{\partial a_{\perp\,i}}{\partial w^{j}}-\Gamma^{k}_{ij}\,a_{\perp\,k}\,.$

(12)

This can also be expressed in terms of the covariant derivative of the ambient space. Since this is ordinary Minkowski space, this is just the partial derivative, and

$\displaystyle\partial_{\perp\,j}\,a_{\perp\,i}=\partial_{\nu}\,a_{\mu}\,e^{\mu}_{i}e^{\nu}_{j}.$

(13)

Thus, for a hypersurface embedded into flat space, we can define the intrinsic covariant derivative through the projection

$\displaystyle\partial_{\perp\,i}=e_{i}^{\mu}\partial_{\mu}\,.$

(14)

In analogy the time derivative is

$\displaystyle\partial_{\parallel}=\hat{v}^{\nu}\partial_{\nu}\,.$

(15)

Note that contractions of spatial coordinates always involve the induced metric, e.g., $\mathbb{a}_{\perp}\!\!\cdot\mathbb{b}_{\perp}=G_{ij}\,a_{\perp}^{i}b_{\perp}^{j}$. The product of four-vectors becomes $a\!\cdot\!b=a_{\parallel}b_{\parallel}-\mathbb{a}_{\perp}\!\!\cdot\mathbb{b}_{\perp}$.

Spacetime integrals translate into

$\displaystyle d^{4}x=dx_{\parallel}\,d^{3}w\,\bigg{|}e_{0}^{\mu}\frac{\partial x_{\mu}}{\partial x_{\parallel}}\bigg{|}=dx_{\parallel}\,d^{3}w\,e_{0}^{\mu}\hat{v}_{\mu}\equiv dx_{\parallel}\,d\Sigma\,,$

(16)

and we define

$\displaystyle d\Sigma^{\mu}=d\Sigma\,\hat{v}^{\mu}\,.$

(17)

With the induced metric defined in Eq. (6), we can also write $d\Sigma=\sqrt{|{\mathrm{det}}G|}\,d^{3}w$.

We assume that the fields we are interested in can be described as quasiparticles on $\Sigma$. The corresponding effective action of a scalar field can be written as

$\displaystyle\begin{split}S=\int\!\!\!dx_{\parallel}\!\int\!\!\!d\Sigma\bigg{\{}&\frac{1}{2}\phi(x)\Big{[}\!-\partial_{\parallel}^{2}+Z\big{(}-\partial_{\perp}^{2}\big{)}\,\partial_{\perp}^{2}-m^{2}\Big{]}\phi(x)\\ &+J(x)\phi(x)\bigg{\}}\,.\end{split}$

(18)

We introduced a source $J(x)$ which encodes interactions with external particles or fields. We allow for a general spatial ‘wave-function renormalization’ $Z\big{(}-\partial_{\perp}^{2}\big{)}$, which can itself be an arbitrary function of the spatial derivative operator $\partial_{\perp}^{2}$. In a free Lorentz-invariant theory $Z=1$. However, if boost invariance is broken, $Z\neq 1$ and higher-order spatial derivative terms can be induced, e.g., by spatial modulations, as in a moat regime. For example, in a low-momentum expansion up to order $\partial_{\perp}^{4}$, one recovers the effective action used in Refs. [17, 19]. Since the specific conditions where particles are in a given regime are in general met only on nontrivial hypersurfaces of, e.g., the expanding fireball of a heavy-ion collision, it is most sensible to formulate the resulting effective field theory also on this surface, instead of in flat space.

Due to the lack of Lorentz invariance, the effective action is frame-dependent. We assume Eq. (18) is the action in the local frame of the particles on $\Sigma$, i.e., in the rest frame of the medium. Because of higher-order spatial derivative terms in the effective action, boosting into a different frame would introduce higher-order time derivatives and the resulting theory will be plagued by Ostrogradsky instabilities. We restrict ourselves to theories with two time derivatives and use ordinary canonical quantization in the following.

Given a specific foliation, we can use the well-known techniques of quantum field theory in curved spacetime to canonically quantize the theory defined by Eq. (18). To this end, we define the symplectic form

$\displaystyle(\chi_{1},\chi_{2})=i\int\!d\Sigma^{\mu}\big{[}\chi_{1}^{*}(\partial_{\mu}\chi_{2})-(\partial_{\mu}\chi_{1}^{*})\chi_{2}\big{]}\,,$

(19)

where $\chi_{1,2}(x)$ are solutions of the equations of motion (EoM) for the action (18). We emphasize that if Lorentz invariance is broken, the symplectic form depends on the choice of the hypersurface $\Sigma$. The free EoM,

$\displaystyle\Big{[}\partial_{\parallel}^{2}-Z\big{(}-\partial_{\perp}^{2}\big{)}\,\partial_{\perp}^{2}+m^{2}\Big{]}\phi_{0}(x)=0\,,$

(20)

are solved by plane waves

$\displaystyle u_{\mathbb{p}_{\perp}}=\frac{1}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}e^{-i\bar{p}\cdot x}\,.$

(21)

We use a notation where an overline, $\,\overline{\phantom{p}}\,$, indicates that the momentum is on the mass-shell of the particle. Thus $\bar{p}^{\mu}$ is the on-shell momentum and

$\displaystyle\omega_{\mathbb{p}_{\perp}}\equiv\hat{v}^{\mu}\bar{p}_{\mu}=\sqrt{Z\big{(}\mathbb{p}_{\perp}^{2}\big{)}\,\mathbb{p}_{\perp}^{2}+m^{2}}$

(22)

is the energy of the particle on $\Sigma$. The waves are normalized such that

$\displaystyle(u_{\mathbb{p}_{\perp}},u_{\mathbb{q}_{\perp}})=(2\pi)^{3}\delta^{(3)}(\mathbb{p}_{\perp}-\mathbb{q}_{\perp})=\int\!d\Sigma\,e^{-i(\mathbb{p}_{\perp}-\mathbb{q}_{\perp})\cdot\mathbb{x}_{\perp}}\,.$

(23)

The solution $\phi_{0}(x)$ of Eq. (20) can now be expanded in terms of these waves, with the annihilation and creation operators $\tilde{a}_{\mathbb{p}_{\perp}}$ and $\tilde{a}_{\mathbb{p}_{\perp}}^{\dagger}$ as coefficients,

$\displaystyle\phi_{0}(x)=\int\!\frac{d^{3}\mathbb{p}_{\perp}}{(2\pi)^{3}}\,\big{(}\tilde{a}_{\mathbb{p}_{\perp}}u_{\mathbb{p}_{\perp}}+\tilde{a}_{\mathbb{p}_{\perp}}^{\dagger}u_{\mathbb{p}_{\perp}}^{*}\big{)}\,.$

(24)

Using this relation, or through the symplectic form in Eq. (19), the ladder operators at time $x_{\parallel}$ can be expressed as

$\displaystyle\begin{split}\tilde{a}_{\mathbb{p}_{\perp}}&=\big{(}u_{\mathbb{p}_{\perp}},\phi_{0}(x)\big{)}\\ &=i\int\!d\Sigma^{\mu}\,e^{i\bar{p}\cdot x}\frac{1}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\big{(}\partial_{\mu}-i\bar{p}_{\mu}\big{)}\phi_{0}(x)\,,\\ \tilde{a}_{\mathbb{p}_{\perp}}^{\dagger}&=-\big{(}u_{\mathbb{p}_{\perp}}^{*},\phi_{0}(x)\big{)}\\ &=-i\int\!d\Sigma^{\mu}\,e^{-i\bar{p}\cdot x}\frac{1}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\big{(}\partial_{\mu}+i\bar{p}_{\mu}\big{)}\phi_{0}(x)\,.\end{split}$

(25)

This completes the quantization of the homogeneous case defined by the free EoM in Eq. (20). We are, however, interested in the more general case where particles can emerge from external fields or interactions. The corresponding EoM which follow from Eq. (18) are then

$\displaystyle\Big{[}\partial_{\parallel}^{2}-Z\big{(}-\partial_{\perp}^{2}\big{)}\,\partial_{\perp}^{2}+m^{2}\Big{]}\phi(x)=J(x)\,.$

(26)

Under the assumption that the source $J$ is only turned on for $x_{\parallel}>0$, and by using the retarded propagator on $\Sigma$,

$\displaystyle D_{R}(p)=\frac{1}{-(p_{\parallel}+i\epsilon)^{2}+\omega_{\mathbb{p}_{\perp}}^{2}}\,,$

(27)

Eq. (26) is solved by

$\displaystyle\phi(x)=\phi_{0}(x)+\int\!d^{4}x^{\prime}\,D_{R}(x-x^{\prime})\,J(x^{\prime})\,.$

(28)

Consequently, the shifted ladder operators,

$\displaystyle\begin{split}a_{\mathbb{p}_{\perp}}&=\tilde{a}_{\mathbb{p}_{\perp}}+\frac{i}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\int\!d^{4}x\,e^{i\bar{p}\cdot x}J(x)\,,\\ a_{\mathbb{p}_{\perp}}^{\dagger}&=\tilde{a}_{\mathbb{p}_{\perp}}^{\dagger}-\frac{i}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\int\!d^{4}x\,e^{-i\bar{p}\cdot x}J^{*}(x)\,,\end{split}$

(29)

are given by

$\displaystyle\begin{split}a_{\mathbb{p}_{\perp}}&=\big{(}u_{\mathbb{p}_{\perp}},\phi(x)\big{)}\\ &=i\int\!d\Sigma^{\mu}\,e^{i\bar{p}\cdot x}\frac{1}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\big{(}\partial_{\mu}-i\bar{p}_{\mu}\big{)}\phi(x)\,,\\ a_{\mathbb{p}_{\perp}}^{\dagger}&=-\big{(}u_{\mathbb{p}_{\perp}}^{*},\phi(x)\big{)}\\ &=-i\int\!d\Sigma^{\mu}\,e^{-i\bar{p}\cdot x}\frac{1}{\sqrt{2\omega_{\mathbb{p}_{\perp}}}}\big{(}\partial_{\mu}+i\bar{p}_{\mu}\big{)}\phi(x)\,.\end{split}$

(30)

We can use these expressions to compute any particle spectrum, such as the ones shown in Eq. (3), on an arbitrary spacelike hypersurface $\Sigma$. Furthermore, this equation is valid for arbitrary dispersion relations $\omega_{\mathbb{p}_{\perp}}$.

For an ordinary relativistic particle, the relations in Eq. (30) are typically the starting point of the derivation of the LSZ reduction formula. There, $x_{0}\rightarrow\pm\infty$ in order to define in- and out-states. In contrast, here, $x_{\parallel}$ can be any time where a quasiparticle picture applies, i.e., where interactions can be considered as classical sources. In the following, we will show that this allows us to express spectra of these particles in terms of their real-time correlation functions. For applications of related ideas to the Schwinger effect and scattering amplitudes, see Refs. [37, 38].

With the help of Eq. (30) we can express particle spectra on $\Sigma$ in terms of correlation functions of fields. Given the generating functional

$\displaystyle Z[j]=\int\!\mathcal{D}\phi e^{iS[\phi]+i\int\!dx_{\parallel}\int\!d\Sigma\,j(x)\phi(x)}\,,$

(31)

we can write in general

$\displaystyle\begin{split}&\big{\langle}a_{\mathbb{p}_{1}}^{\dagger}\cdots a_{\mathbb{p}_{n}}^{\dagger}a_{\mathbb{q}_{1}}\cdots a_{\mathbb{q}_{n}}\big{\rangle}\\ &=\lim_{\begin{subarray}{c}x_{\parallel,1},\dots,x_{\parallel,n}\rightarrow x_{\parallel}\\ y_{\parallel,1},\dots,y_{\parallel,n}\rightarrow x_{\parallel}\end{subarray}}\,\Bigg{[}\prod_{k=1}^{n}\int\!d\Sigma_{x_{k}}^{\mu}\,d\Sigma_{y_{k}}^{\nu}\,\frac{1}{2\omega_{\mathbb{p}_{k}}}\,e^{-i\bar{p}_{k}\cdot x_{k}}e^{i\bar{q}_{k}\cdot y_{k}}\\ &\quad\times\big{(}\partial_{\mu}^{x_{k}}+i\bar{p}_{k,\mu}\big{)}\big{(}\partial_{\mu}^{y_{k}}-i\bar{q}_{k,\mu}\big{)}\Bigg{]}\\ &\quad\times\frac{(-i)^{2n}}{Z[0]}\Bigg{(}\prod_{k=n}^{1}\frac{\delta}{\delta j(y_{k}))}\Bigg{)}\Bigg{(}\prod_{k=n}^{1}\frac{\delta}{\delta j(x_{k}))}\Bigg{)}\,Z[j]\Bigg{|}_{j=0}\,,\end{split}$

(32)

Note that the path integral automatically generates time-ordered correlation functions. The above expression only describes spectra on $\Sigma$ after the equal-time limit is taken. This is indicated by the limit in Eq. (32), where all times $x_{\parallel,k}$ and $y_{\parallel,k}$ are set to the same time $x_{\parallel}$. In the following, we explicitly work out the expressions for single-particle spectra. Since this corresponds to two-point functions of the fields, this is sufficient to describe all $n$-particle spectra if the generating functional is Gaussian. All correlations/spectra can then be expressed in terms of two-point functions/single-particle spectra by virtue of Wick’s theorem.

For illustration, and since it is most relevant to the application below, see Sec. IV.1, we compute the mixed single-particle spectrum,

$\displaystyle n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})=\sqrt{\omega_{\mathbb{p}_{\perp}}\omega_{\mathbb{q}_{\perp}}}\,\big{\langle}a_{\mathbb{p}_{\perp}}^{\dagger}a_{\mathbb{q}_{\perp}}\big{\rangle}\,.$

(33)

For $\mathbb{p}_{\perp}=\mathbb{q}_{\perp}$ this reduces to the ordinary single-particle spectrum in Eq. (3). Using Eq. (30) we get

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})&=\frac{1}{2}\lim_{y_{\parallel}\rightarrow x_{\parallel}}\int\!d\Sigma_{x}^{\mu}\,d\Sigma_{y}^{\nu}\,e^{-i\bar{p}\cdot x}e^{i\bar{q}\cdot y}\\ &\quad\times\big{(}\partial^{x}_{\mu}+i\bar{p}_{\mu}\big{)}\big{(}\partial^{y}_{\nu}-i\bar{q}_{\nu}\big{)}\langle\phi(x)\phi(y)\rangle\,.\end{split}$

(34)

Since the particle spectrum is defined at equal time, so is the correlation function on the right-hand side. In order to express this as something resembling a phase-space distribution, we define the average and relative positions

$\displaystyle X=\frac{1}{2}(x+y)\,,\qquad\Delta X=x-y\,.$

(35)

By also introducing the average and relative momenta,

$\displaystyle\begin{split}P&=\frac{1}{2}(p+q)\,,\qquad\Delta P=p-q\,,\\ \overline{P}&=\frac{1}{2}(\bar{p}+\bar{q})\,,\qquad\overline{\Delta P}=\bar{p}-\bar{q}\,,\end{split}$

(36)

the mixed spectrum becomes

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})&=\frac{1}{2}\lim_{\Delta X_{\parallel}\rightarrow 0}\int\!d\Sigma_{X}^{\mu}\,d\Sigma_{\Delta X}^{\nu}\,e^{-i\overline{\Delta P}\cdot X}e^{-i\overline{P}\cdot\Delta X}\\ &\quad\times\widetilde{\Box}_{\mu\nu}\,\bigg{\langle}\phi\bigg{(}X+\frac{1}{2}\Delta X\bigg{)}\phi\bigg{(}X-\frac{1}{2}\Delta X\bigg{)}\bigg{\rangle}\,.\end{split}$

(37)

Here, $d\Sigma_{X}^{\mu}=d\Sigma_{X}\hat{v}^{\mu}$ and $d\Sigma_{\Delta X}^{\nu}=d\Sigma_{\Delta X}\hat{v}^{\nu}$ are the integration measures for the average and relative hypersurfaces at times $X_{\parallel}=\hat{v}^{\mu}X_{\mu}$ and $\Delta X_{\parallel}=\hat{v}^{\mu}\Delta X_{\mu}$. The operator $\widetilde{\Box}_{\mu\nu}$ is

$\displaystyle\begin{split}\widetilde{\Box}_{\mu\nu}&=\frac{1}{4}\partial_{\mu}^{X}\partial_{\nu}^{X}-\frac{1}{2}\partial_{\mu}^{X}\partial_{\nu}^{\Delta X}+\frac{1}{2}\partial_{\mu}^{\Delta X}\partial_{\nu}^{X}-\partial_{\mu}^{\Delta X}\partial_{\nu}^{\Delta X}\\ &\quad-\frac{i}{2}\bigg{(}\overline{P}_{\nu}-\frac{1}{2}\overline{\Delta P}_{\nu}\bigg{)}\big{(}\partial_{\mu}^{X}+2\partial_{\mu}^{\Delta X}\big{)}\\ &\quad+\frac{i}{2}\bigg{(}\overline{P}_{\mu}+\frac{1}{2}\overline{\Delta P}_{\mu}\bigg{)}\big{(}\partial_{\nu}^{X}-2\partial_{\nu}^{\Delta X}\big{)}\\ &\quad+\bigg{(}\overline{P}_{\mu}+\frac{1}{2}\overline{\Delta P}_{\mu}\bigg{)}\bigg{(}\overline{P}_{\nu}-\frac{1}{2}\overline{\Delta P}_{\nu}\bigg{)}\,.\end{split}$

(38)

Through the integrations over $d\Sigma_{X}^{\mu}$ and $d\Sigma_{\Delta X}^{\mu}$ this operator is projected normal to these hypersurfaces,

$\displaystyle\begin{split}\hat{v}^{\mu}\hat{v}^{\nu}\widetilde{\Box}_{\mu\nu}&=\frac{1}{4}\big{(}\partial_{\parallel}^{X}\big{)}^{2}-\big{(}\partial_{\parallel}^{\Delta X}\big{)}^{2}+\frac{i}{2}\overline{\Delta P}_{\parallel}\partial_{\parallel}^{X}-2i\overline{P}_{\parallel}\partial_{\parallel}^{\Delta X}\\ &\quad+\overline{P}_{\parallel}^{2}-\frac{1}{4}\overline{\Delta P}_{\parallel}^{2}\,,\end{split}$

(39)

where $\overline{P}_{\parallel}=\frac{1}{2}(\omega_{\mathbb{p}_{\perp}}+\omega_{\mathbb{q}_{\perp}})$ and $\overline{\Delta P}_{\parallel}=\omega_{\mathbb{p}_{\perp}}-\omega_{\mathbb{q}_{\perp}}$. The integration over $\Delta X$ in Eq. (37) is almost a Wigner transformation, which replaces the relative position $\Delta X$ by the average momentum $\overline{P}$. What is missing is the $\Delta X_{\parallel}$ integration. It can be included by introducing and then cancelling the corresponding Fourier transformation (note that this is a transformation with negative momentum $-P$),

$\displaystyle h(\Delta X_{\parallel})=\int\!\frac{dP_{\parallel}}{2\pi}\,e^{iP_{\parallel}\Delta X_{\parallel}}\int\!d\Delta X_{\parallel}^{\prime}\,e^{-iP_{\parallel}\Delta X_{\parallel}^{\prime}}\,h(\Delta X_{\parallel}^{\prime})\,,$

(40)

where $h$ is an arbitrary function. Applying this to Eq. (37), we can carry out the full Fourier transform with respect to $\Delta X$. We define the Wigner-transformed spectral and statistical functions from the commutators and the anti-commutators of the fields,

$\displaystyle\begin{split}\rho(X,P)&=\int\!d\Delta X_{\parallel}\int\!d\Sigma_{\Delta X}\,e^{iP\cdot\Delta X}\\ &\quad\times\bigg{\langle}\bigg{[}\phi\bigg{(}X+\frac{1}{2}\Delta X\bigg{)},\phi\bigg{(}X-\frac{1}{2}\Delta X\bigg{)}\bigg{]}\bigg{\rangle}\,,\\ F(X,P)&=\frac{1}{2}\int\!d\Delta X_{\parallel}\int\!d\Sigma_{\Delta X}\,e^{iP\cdot\Delta X}\\ &\quad\times\bigg{\langle}\bigg{\{}\phi\bigg{(}X+\frac{1}{2}\Delta X\bigg{)},\phi\bigg{(}X-\frac{1}{2}\Delta X\bigg{)}\bigg{\}}\bigg{\rangle}\,.\end{split}$

(41)

With this, and taking the limit $\Delta X_{\parallel}\rightarrow 0$, we finally arrive at

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})&=\frac{1}{2}\int\!d\Sigma_{X}\,e^{-i\overline{\Delta P}\cdot X}\int\!\frac{dP_{\parallel}}{2\pi}\\ &\quad\times\bigg{[}\frac{1}{4}\big{(}\partial_{\parallel}^{X}\big{)}^{2}+\frac{i}{2}\overline{\Delta P}_{\parallel}\partial_{\parallel}^{X}+\big{(}P_{\parallel}+\overline{P}_{\parallel}\big{)}^{2}-\frac{1}{4}\overline{\Delta P}_{\parallel}^{2}\bigg{]}\\ &\quad\times\bigg{[}F(X,P)-\frac{1}{2}\rho(X,P)\bigg{]}\,.\end{split}$

(42)

Note that it was important not to go to equal times from the beginning. Otherwise we would not have been able to do the Wigner transformation.

Equation (42) is the most general expression we can derive for the mixed particle spectrum. We see that the relative momentum $\overline{\Delta P}$ is correlated with the average location $X$. The Wigner-transformed correlation functions are functions of the average pair momentum $P$.

To get the single-particle spectrum, we set $p=q$, so that $\overline{\Delta P}=0$ and $P=p$, and

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp})&=\frac{1}{2}\int\!d\Sigma_{X}\,\int\!\frac{dp_{\parallel}}{2\pi}\,\bigg{[}\frac{1}{4}\big{(}\partial_{\parallel}^{X}\big{)}^{2}+\big{(}p_{\parallel}+\omega_{\mathbb{p}_{\perp}}\big{)}^{2}\bigg{]}\\ &\quad\times\bigg{[}F(X,p)-\frac{1}{2}\rho(X,p)\bigg{]}\,,\end{split}$

(43)

since the on-shell longitudinal momentum is $\bar{p}_{\parallel}=\omega_{\mathbb{p}_{\perp}}$. Although these expressions have been derived assuming a quasiparticle picture, they can in principle be used to define the spectra non-perturbatively and out of equilibrium since they depend on the general spectral and statistical functions of the theory.

We can simplify these expressions by making a few assumptions which are reasonable if one has, as we do, the hydrodynamic regime of a heavy-ion collision in mind. First, we assume that the system is in local thermodynamic equilibrium. This means that $F$ and $\rho$ are related through a generalized fluctuation-dissipation relation (sometimes called the Kadanoff-Baym ansatz) [39],

$\displaystyle F(X,P)=\bigg{[}\frac{1}{2}+f(X,P)\bigg{]}\rho(X,P)\,,$

(44)

where $f(X,P)$ is the single-particle distribution. Using this relation, the Boltzmann equation for $f(X,P)$ can be derived from the Kadanoff-Baym equations [40].

Furthermore, we will neglect any derivatives in $X$. This is certainly true for isotropic systems in thermodynamic equilibrium, where only relative positions matter. In general, such gradients are accompanied by additional powers of $\hbar$ and can be neglected in a semi-classical approximation.

Using these approximations, we finally arrive at

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})&=\frac{1}{2}\int\!d\Sigma_{X}\,e^{-i\overline{\Delta P}\cdot X}\int\!\frac{dP_{\parallel}}{2\pi}\bigg{[}\big{(}P_{\parallel}+\overline{P}_{\parallel}\big{)}^{2}-\frac{1}{4}\overline{\Delta P}_{\parallel}^{2}\bigg{]}\\ &\quad\times f(X,P)\,\rho(X,P)\,,\\ n_{1}(\mathbb{p}_{\perp})&=\frac{1}{2}\int\!d\Sigma_{X}\int\!\frac{dp_{\parallel}}{2\pi}\,\big{(}p_{\parallel}+\omega_{\mathbb{p}_{\perp}}\big{)}^{2}\,f(X,p)\,\rho(X,p)\,.\end{split}$

(45)

These equations can be viewed as generalized Cooper-Frye formulas [32]. In fact, if we use a covariant Bose-Einstein distribution for $f$ and the spectral function of a Lorentz-invariant free particle, $n_{1}(\mathbb{p}_{\perp})$ reduces to the original expression of Cooper and Frye, see Sec. IV.2. Thus, with our formalism we have derived the phenomenological Cooper-Frye formula from the underlying microscopic quantum field theory. Equation (45) also explicitly accounts for off-shell effects. The advantage of our formalism is its generality. The original versions of the Cooper-Frye formalism and HBT, which have been derived in a similar manner as in the present work in Ref. [33], are limited to free particles on the mass-shell. We are not restricted to these limitations here.

With the procedure from above any $n$-particle spectrum can be expressed in terms of $2n$-point functions of fields. For simplicity, we assume that the generating functional in Eq. (31) is Gaussian. As mentioned above, in the Gaussian approximation $n$-particle spectra can be expressed solely in terms of single-particle ones. This turns out to be sufficient for the theoretical description of HBT correlations. Here we explicitly work out the two-particle spectrum $n_{2}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})$, defined in Eq. (3). In the Gaussian approximation the 4-point function is

$\displaystyle\begin{split}&\big{\langle}\phi(x_{1})\phi(x_{2})\phi(y_{1})\phi(y_{2})\big{\rangle}\\ &\qquad=\big{\langle}\phi(x_{1})\phi(y_{1})\big{\rangle}\big{\langle}\phi(x_{2})\phi(y_{2})\big{\rangle}\\ &\qquad\quad+\big{\langle}\phi(x_{1})\phi(y_{2})\big{\rangle}\big{\langle}\phi(x_{2})\phi(y_{1})\big{\rangle}\\ &\qquad\quad+\big{\langle}\phi(x_{1})\phi(x_{2})\big{\rangle}\big{\langle}\phi(y_{1})\phi(y_{2})\big{\rangle}\,.\end{split}$

(46)

By comparison with Eq. (32), we can identify the different contributions:

$\displaystyle\begin{split}&{\rm(a)}\;\;\big{\langle}\phi(x_{1})\phi(y_{1})\big{\rangle}\big{\langle}\phi(x_{2})\phi(y_{2})\big{\rangle}\\ &\qquad\longrightarrow\,n_{1}(\mathbb{p}_{\perp})\,n_{1}(\mathbb{q}_{\perp})\,,\\[5.0pt] &{\rm(b)}\;\;\big{\langle}\phi(x_{1})\phi(y_{2})\big{\rangle}\big{\langle}\phi(x_{2})\phi(y_{1})\big{\rangle}\\ &\qquad\longrightarrow\,n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\,n_{1}(\mathbb{q}_{\perp},\mathbb{p}_{\perp})=\big{|}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\big{|}^{2}\,,\\[5.0pt] &{\rm(c)}\;\;\big{\langle}\phi(x_{1})\phi(x_{2})\big{\rangle}\big{\langle}\phi(y_{1})\phi(y_{2})\big{\rangle}\\ &\qquad\longrightarrow\omega_{\mathbb{p}_{\perp}}\omega_{\mathbb{q}_{\perp}}\,\big{\langle}a_{\mathbb{p}_{\perp}}^{\dagger}a_{\mathbb{q}_{\perp}}^{\dagger}\big{\rangle}\big{\langle}a_{\mathbb{p}_{\perp}}a_{\mathbb{q}_{\perp}}\big{\rangle}\equiv\big{|}\bar{n}_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\big{|}^{2}\,.\end{split}$

(47)

(a) and (b) we already computed. (a) is just a product of two single-particle distributions. (b) stems from particle-particle interference. This is the actual HBT effect. It describes the correlation of identical particles generated by their Bose-Einstein statistics. (c) can be interpreted as particle-antiparticle interference. For particles that are their own antiparticles, like the neutral scalar fields considered here, we cannot distinguish this. Otherwise, one of the ladder operators in each expectation value would correspond to the creation/annihilation operator of an antiparticle. This is what is called the ‘surprising effect’ in Ref. [27].

The resulting two-particle spectrum is

$\displaystyle\begin{split}n_{2}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})&=n_{1}(\mathbb{p}_{\perp})\,n_{1}(\mathbb{q}_{\perp})+\big{|}n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\big{|}^{2}\\ &\quad+\big{|}\bar{n}_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\big{|}^{2}\,.\end{split}$

(48)

The first two terms are usually considered when studying HBT correlations. Once the spectral function, the single-particle distribution function, and the hypersurface are specified, we can compute this using Eq. (45). The missing particle-antiparticle interference term is computed in App. A. In the following, we set up appropriate models for the required quantities in order to illustrate the relevant qualitative effects.

But first, we note that HBT is often expressed in terms of the emission function (or source function) $S(x,\mathbb{P}_{\perp})$, where $x$ is a spacetime coordinate. It describes the phase-space distribution, i.e., the distribution of spacetime positions and momenta of the particles emitted from a phase-space element $d^{3}x\,d^{3}P_{\perp}$. The particle-particle interference term is then given by the (on-shell) Fourier transformation of the emission function with respect to the relative momentum [24, 26],

$\displaystyle n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})\approx\int\!d^{4}x\,e^{-i\overline{\Delta P}\cdot x}\,S(x,\mathbb{P}_{\perp})\,.$

(49)

In this form, the connection between the behavior of the correlation function and the spacetime structure of the emitting source is most transparent: the range of the correlation in relative momentum $\mathbb{\Delta P}_{\perp}$ is related to the inverse size of the source. More precisely, it relates to the region of homogeneity inside the source for a given average momentum $\mathbb{P}_{\perp}$ [41].

An underlying assumption in Eq. (49) is that the dependence of the correlation on the relative momentum $\mathbb{\Delta P}_{\perp}$ arises solely from the Fourier transformation of the location $x$. As is evident from Eq. (42), this is not the case here. Instead, we can define the emission function plus a correction, $S(x,\mathbb{P}_{\perp})+\Delta S(x,\mathbb{P}_{\perp},\mathbb{\Delta P}_{\perp})$, with the emission function

$\displaystyle\begin{split}&S(x,\mathbb{P}_{\perp})\\ &\quad=\frac{1}{2}\int\!d\Sigma_{X}\,\delta(X-x)\int\!\frac{dP_{\parallel}}{2\pi}\,\bigg{[}\frac{1}{4}\big{(}\partial^{X}_{\parallel}\big{)}^{2}+\big{(}P_{\parallel}+\overline{P}_{\parallel}\big{)}^{2}\bigg{]}\\ &\quad\quad\times\bigg{[}F(X,P)-\frac{1}{2}\rho(X,P)\bigg{]}\\ &\quad\approx\frac{1}{2}\int\!d\Sigma_{X}\,\delta(X-x)\int\!\frac{dP_{\parallel}}{2\pi}\,\big{(}P_{\parallel}+\overline{P}_{\parallel}\big{)}^{2}f(X,P)\,\rho(X,P)\,,\end{split}$

(50)

and the correction

$\displaystyle\begin{split}&\Delta S(x,\mathbb{P}_{\perp},\mathbb{\Delta P}_{\perp})\\ &\quad=\frac{1}{4}\int\!d\Sigma_{X}\,\delta(X-x)\int\!\frac{dP_{\parallel}}{2\pi}\,\bigg{(}i\,\overline{\Delta P}_{\parallel}\partial^{X}_{\parallel}-\frac{1}{2}\overline{\Delta P}_{\parallel}^{2}\bigg{)}\\ &\quad\quad\times\bigg{[}F(X,P)-\frac{1}{2}\rho(X,P)\bigg{]}\\ &\quad\approx-\frac{1}{8}\,\overline{\Delta P}_{\parallel}^{2}\int\!d\Sigma_{X}\,\delta(X-x)\int\!\frac{dP_{\parallel}}{2\pi}\,f(X,P)\,\rho(X,P)\,.\end{split}$

(51)

In the last lines of both equations we again assumed local thermodynamic equilibrium and neglected gradients in $X$. With this we have

$\displaystyle n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})=\int\!d^{4}x\,e^{-i\overline{\Delta P}\cdot x}\,\big{[}S(x,\mathbb{P}_{\perp})+\Delta S(x,\mathbb{P}_{\perp},\mathbb{\Delta P}_{\perp})\big{]}\,.$

(52)

Thus, the assumption underlying Eq. (49) that the HBT correlation can be described as the Fourier transformation of a phase-space distribution is only warranted if the difference of the energies of the on-shell particles on the hypersurface, $\overline{\Delta P}_{\parallel}$, can be neglected. This is obviously not true in general. We will discuss the validity of neglecting this contribution in Sec. IV.4.

For the spectral function that enters Eq. (45) we use a similar approximation to the one used in Ref. [19], where thermodynamic fluctuations in a moat regime have been studied. For consistency with our derivation in Sec. III, we assume a quasiparticle as described by the action in Eq. (18). The retarded propagator on $\Sigma$ is then given by Eq. (27), which results in the spectral function

$\displaystyle\begin{split}\rho(p)&=2\,{\rm Im}D_{R}(p)\\ &=\frac{\pi}{\omega_{\mathbb{p}_{\perp}}}\,\big{[}\delta(p_{\parallel}-\omega_{\mathbb{p}_{\perp}})-\delta(p_{\parallel}+\omega_{\mathbb{p}_{\perp}})\big{]}\,.\end{split}$

(53)

In general, any spectral function with a sufficiently sharp peak is suitable for our formalism. We choose a quasiparticle with infinite lifetime for simplicity.

For a quasiparticle with energy $p_{\parallel}$ on $\Sigma$, the relativistic Boltzmann equation in local thermodynamic equilibrium and in absence of external forces yields the single-particle distribution function

$\displaystyle f(X,p)=\theta(p_{\parallel})\,n_{B}(p_{\parallel})=\frac{\theta(p_{\parallel})}{e^{p_{\parallel}/T}-1}\,.$

(54)

$f$ is a number density in phase space and as such has to be non-negative. This is ensured by the theta function.

The spectral and distribution functions depend on the location of the particle in spacetime through the projection of the momentum onto the hypersurface. Plugging Eqs. (53) and (54) into Eqs. (45) and (72) yields for the single-particle spectrum,

$\displaystyle\begin{split}n_{1}(\mathbb{p}_{\perp})=\int\!d\Sigma_{X}\,\omega_{\mathbb{p}_{\perp}}\,n_{B}(\omega_{\mathbb{p}_{\perp}})\,,\end{split}$

(55)

for the particle-particle interference contribution,

$\displaystyle\begin{split}&n_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})=\\ &\quad\frac{1}{4}\int\!d\Sigma_{X}\Bigg{\{}e^{-i\overline{\Delta P}\cdot X}\frac{1}{\omega_{\mathbb{P}_{\perp}}}\bigg{[}\big{(}\omega_{\mathbb{P}_{\perp}}+\overline{P}_{\parallel}\big{)}^{2}-\frac{1}{4}\overline{\Delta P}_{\parallel}^{2}\bigg{]}\\ &\quad\times n_{B}(\omega_{\mathbb{P}_{\perp}})\Bigg{\}}\,,\end{split}$

(56)

and for the particle-antiparticle interference term,

$\displaystyle\begin{split}&\bar{n}_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})=\\ &\quad-\frac{1}{8}\int\!d\Sigma_{X}\Bigg{\{}e^{-2i\overline{P}\cdot X}\frac{1}{\omega_{\mathbb{\Delta P}_{\perp}/2}}\bigg{[}\big{(}2\omega_{\mathbb{\Delta P}_{\perp}/2}+\overline{\Delta P}_{\parallel}\big{)}^{2}\\ &\quad-4\overline{P}_{\parallel}^{2}\bigg{]}\,n_{B}(\omega_{\mathbb{\Delta P}_{\perp}/2})\Bigg{\}}\,.\end{split}$

(57)

We see that with the present approximations, we recover the original Cooper-Frye formula for the single-particle spectrum $n_{1}(\mathbb{p}_{\perp})$, but with a more general dispersion relation. We have therefore provided a microscopic derivation for the expression proposed in Ref. [19].

The interference terms (56) and (57) receive contributions from both on- and off-shell momenta of the individual particles. In the quasiparticle picture also the on-shell average and relative energies $\omega_{\mathbb{P}_{\perp}}$ and $\omega_{\mathbb{\Delta P}_{\perp}/2}$ contribute. They are manifestly different from the average and relative energies of the on-shell particles, $\overline{P}_{\parallel}$ and $\overline{\Delta P}_{\parallel}$. This reflects the generality of our formalism, as it is able to capture off-shell effects by taking into account the full spectrum of the particles involved. In previous work it was necessary to resort to on-shell approximations to describe interference [24].

Furthermore, our microscopic derivation reveals a different momentum dependence of the particle-particle interference contribution (56) as compared to the literature, cf., e.g., Refs. [42, 24]. In previous work where correlations on a sharp hypersurface have been considered (as is customary in hydrodynamical applications), it has always been assumed that interference can be described by the ordinary Cooper-Frye formula for average momenta, and an additional Fourier transformation with respect to the average position of the particles ¹¹1See, e.g., Eq. (85) in [24]. Our results show that this is not correct, as the particles on the hypersurface cannot be described by their average momentum alone.

To be specific, in addition to the assumptions that lead from the most general Eqs. (42), (43), and (71) to the above equations, our results reduce to the ones in the literature [24, 26] using the following assumptions: First, the contribution from particle-antiparticle interference, $\bar{n}_{1}(\mathbb{p}_{\perp},\mathbb{q}_{\perp})$ needs to be neglected. Second, the difference between the on-shell average energy $\omega_{\mathbb{P}_{\perp}}$ and the average of the on-shell energies $\overline{P}_{\parallel}$ has to be neglected, $\overline{P}_{\parallel}=\omega_{\mathbb{P}_{\perp}}$. Furthermore, the relative on-shell energy needs to be neglected as well, $\overline{\Delta P}_{\parallel}=0$. Evidently, these assumptions cannot be true in general, so the known equations in the literature have to be considered approximations of the more general ones derived here. Since a quantitative analysis is not in the scope of this work, we defer an analysis of the validity of these approximations to future work.