*Article* **Causality and Renormalization in Finite-Time-Path Out-of-Equilibrium** *φ***<sup>3</sup> QFT**

**Ivan Dadi´c <sup>1</sup> and Dubravko Klabuˇcar 2,\***


Received: 30 November 2018; Accepted: 9 January 2019; Published: 18 January 2019

**Abstract:** Our aim is to contribute to quantum field theory (QFT) formalisms useful for descriptions of short time phenomena, dominant especially in heavy ion collisions. We formulate out-of-equilibrium QFT within the finite-time-path formalism (FTP) and renormalization theory (RT). The potential conflict of FTP and RT is investigated in *gφ*<sup>3</sup> QFT, by using the retarded/advanced (*R*/*A*) basis of Green functions and dimensional renormalization (DR). For example, vertices immediately after (in time) divergent self-energy loops do not conserve energy, as integrals diverge. We "repair" them, while keeping *<sup>d</sup>* < 4, to obtain energy conservation at those vertices. Already in the S-matrix theory, the renormalized, finite part of Feynman self-energy Σ*F*(*p*0) does not vanish when |*p*0| → ∞ and cannot be split to retarded and advanced parts. In the Glaser–Epstein approach, the causality is repaired in the composite object *GF*(*p*0)Σ*F*(*p*0). In the FTP approach, after repairing the vertices, the corresponding composite objects are *GR*(*p*0)Σ*R*(*p*0) and Σ*A*(*p*0)*GA*(*p*0). In the limit *d* → 4, one obtains causal QFT. The tadpole contribution splits into diverging and finite parts. The diverging, constant component is eliminated by the renormalization condition 0|*φ*|0 = 0 of the S-matrix theory. The finite, oscillating energy-nonconserving tadpole contributions vanish in the limit *t* → ∞.

**Keywords:** out-of-equilibrium quantum field theory; dimensional renormalization; finite-time-path formalism

#### **1. Introduction and Survey**

In many regions of physics, the interacting processes are embedded in a medium and require a short-time description. To respond to such demands, neither vacuum S-matrix field theory [1–5], nor equilibrium QFT [6–16] with the Keldysh-time-path [17–28] suffice. The features, a short time after the beginning of evolution, where uncertainty relations do not keep energy conserved, are to be treated with the finite-time-path method. Such an approach includes many specific features that are not yet completely understood. A particular problem, almost untreated, is handling of UV divergences of the QFT as seen at finite time. The present paper is devoted to this problem. We consider it in the simplest form of *λφ*<sup>3</sup> QFT, but many of the discussed features will find their analogs in more advanced QED and QCD.

Starting with perturbation expansion in the coordinate space, one performs the Wigner transform and uses the Wick theorem. The propagators, originally appearing in matrix representation, are linearly connected to the Keldysh base with *R, A*, and *K* components. For a finite-time-path, the lowest order propagators and one-loop self-energies taken at *t* = ∞ correspond to Keldysh-time-path propagators and one-loop self-energies. For simplicity, the label "∞" is systematically omitted throughout the paper, except in the Appendix with technical details.

To analyze the vertices, one further separates *K*-component [27,28] into its retarded (*K,R*) and advanced (*K,A*) parts:

$$G\_{R}(p) = G\_{A}(-p) = \frac{-i}{p^{2} - m^{2} + 2ip\_{0}\epsilon},$$

$$G\_{K}(p) = 2\pi\delta(p^{2} - m^{2})[1 + 2f(\omega\_{p})]$$

$$\omega\_{p} = G\_{K,R}(p) - G\_{K,A}(p),$$

$$G\_{K,R}(p) = -G\_{K,A}(-p) = h(p\_{0}, \omega\_{p})G\_{R}(p),$$

$$\omega\_{p} = \sqrt{\vec{p}^{2} + m^{2}}, \quad h(p\_{0}, \omega\_{p}) = -\frac{p\_{0}}{\omega\_{p}}\left[1 + 2f(\omega\_{p})\right].\tag{1}$$

Matrix propagators are (*i* and *j* take the values 1, 2):

$$\mathcal{G}\_{\vec{l}\vec{\jmath}}(p) = \frac{1}{2} [\mathcal{G}\_K(p) + (-1)^{\vec{l}} \mathcal{G}\_R(p) + (-1)^{\vec{l}} \mathcal{G}\_A(p)]. \tag{2}$$

Specifically:

$$\mathcal{G}\_{\mathbb{F}}(p) = \mathcal{G}\_{11}(p)\_{f(\omega\_p) = 0} = \frac{-i}{p^2 - m^2 + 2i\epsilon'} \, \mathcal{G}\_{\mathbb{F}}(p) = -\mathcal{G}\_{\mathbb{F}}^\*(p). \tag{3}$$

#### **2. Results**

#### *2.1. Conservation and Non-Conservation of Energy at Vertices*

Having done all this, one obtains the vertex function (for simplicity, all the four-momenta are arranged to be incoming to the vertex). For the simplicity of discussion, all the times corresponding to the external vertices (*j*) of the whole diagram are assumed equal (*x*0,*j*,*ext* = *t*, all *j*; otherwise, some factors, oscillating with time, but inessential for our discussion, would appear), so that the vertex function becomes:

$$\frac{i}{2\pi} \frac{e^{-it\sum\_{i} p\_{0i}}}{\sum\_{i} p\_{0i} + i\epsilon}. \tag{4}$$

This expression [27–29] integrated over some *dpo*,*<sup>k</sup>* by closing the time-path from below gives the expected energy conserving *δ*(∑*<sup>i</sup> p*0*i*), with the oscillating factor reduced to one. If the integration path catches additional singularity, say the propagator's *D*(*pk*) pole at *p*¯0*k*, for this contribution, conservation of energy is "spoiled" by a finite amount Δ*E* = ∑ *<sup>i</sup> p*0*<sup>i</sup>* + *p*¯0*k*, and there is an oscillating vertex function (*i*/2*π*)*e*−*it*Δ*E*/(Δ*E* + *i* ). Note: the fact that some time is lower or higher than another, i.e., *<sup>t</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>2</sup> or *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>t</sup>*2, survives Wigner transform in the character of ordering (retarded or advanced) of the two-point function.

In general, we have the following possibilities:

• If the vertex time is lower than the other times of all incoming propagators, there are additional contributions, and energy is not conserved at this vertex. The oscillations are just what we would expect from the Heisenberg uncertainty relations. It is how the time dependence emerges in the finite-time-path out-of-equilibrium QFT. The ill-defined pinching singularities—products of retarded and advanced propagators with the same (*p*0,*p*), only partially eliminated for the Keldysh time-path [30]—do not appear here as the propagator energies *p*<sup>0</sup> and *p* <sup>0</sup> are different variables, so that the singularities do not coincide except at the point *p*<sup>0</sup> = *p* <sup>0</sup>. Thus, the pertinent mathematical expressions are well defined.


In the *λφ*<sup>3</sup> QFT, there are two divergent subdiagrams: the tadpole diagram and self-energy diagram, considered separately in the following subsections.

#### *2.2. UV Divergence at the Tadpole Subdiagram*

In the perturbation expansion, the tadpole diagram (Figure 1) appears as a propagator with both ends attached to the same vertex, which is the (lower-time) end-point vertex of the second propagator.

The tadpole subdiagram without a leg is simple. Of the three components, the loop integral vanishes for the *R* and *A* components and diverges for the *K*, *R* and *K*, *A* ones. At finite *κ* = 4 − *d*, these integrals are real constants related to the *F* and *F*¯ components. In the limit *d* = 4, the renormalization performed on *F* and *F*¯ makes them finite.

$$\begin{split} &ig\boldsymbol{\mu}^{\kappa/2} \frac{\int d^d p}{(2\pi)^d} \mathcal{G}\_R(p) = ig\boldsymbol{\mu}^{\kappa/2} \frac{\int d^d p}{(2\pi)^d} \mathcal{G}\_A(p) = 0, \\ &\mathcal{G}\_{\text{Tad}} \equiv -ig \frac{\int d^d p}{(2\pi)^d} \mathcal{G}\_{K,A}(p) = -ig\boldsymbol{\mu}^{\kappa/2} \frac{\int d^d p}{(2\pi)^d} \frac{p\_0}{p^2 - m^2 - 2ip\_0\varepsilon} = ig\boldsymbol{\mu}^{\kappa/2} \frac{\int d^d p}{(2\pi)^d} \mathcal{G}\_{K,R}(p), \\ &\Longrightarrow -\frac{1}{2} \mathcal{G}\_{\text{Tad}} = -\frac{ig\boldsymbol{m}^2}{8\pi^2\kappa} - \frac{ig\boldsymbol{m}^2}{16\pi^2} [1 - \gamma\_\varepsilon + \ln(\frac{4\pi\mu^2}{m^2})] + \mathcal{O}(\kappa) + ig \frac{\int d^3 p}{(2\pi)^3} 2f(\omega\_p) \\ &= -\frac{ig\boldsymbol{m}^2}{8\pi^2(\kappa)} + finite \text{vacuum term} + finite \, f(\omega\_p) \text{ term.} \end{split} \tag{5}$$

(Above, and throughout the paper, *γ<sup>E</sup>* denotes the Euler-Mascheroni constant, *γ<sup>E</sup>* ≈ 0.5772.)

For a tadpole subdiagram with a leg (see Figure 1), we have two vertices; higher in time (*t*2), which is the connection to the rest of the diagram, and lower in time (*t*1, *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>t</sup>*2) with the tadpole loop. The lower vertex does not conserve energy.

One has to add contributions from vertices of Type 1 and Type 2. We write it symbolically with the help of the Wigner transform, the connection between the Keldysh-time-path propagators and the finite-time-path propagators at the time *t* = ∞ and transition to the *R*/*A* basis. The derivation given in the Appendix A shows that:

$$\mathcal{G}\_{\text{tail},\text{j}}(\mathbf{x}\_2) = -\mathcal{G}\_A(0,0)\mathcal{G}\_{\text{Tad}} + \int \frac{dp\_{02}}{2\pi} \frac{i e^{ip\_{02}\mathbf{x}\_{02}}}{p\_{02} - i\varepsilon} [\mathcal{G}\_A(p\_{02}, 0) - \mathcal{G}\_A(0,0)] \mathcal{G}\_{\text{Tad}}\,. \tag{6}$$

The contribution is split into the first, energy-conserving term, and the second term, oscillating with time, in which energy is not conserved at the vertex 1 [31].

The tadpole counterterm follows the same pattern:

$$\mathcal{G}\_{\text{count},\text{j}}^{\text{tadpole}}(\mathbf{x}\_{2}) = -G\_{A}(0,0) + \int \frac{dp\_{02}}{2\pi} \frac{i e^{ip\_{02}\mathbf{x}\_{02}}}{p\_{02} - i\epsilon} \left[ G\_{A}(p\_{02}, 0) - G\_{A}(0, 0) \right]. \tag{7}$$

Notice the similarity of the expressions (6) and (7).

An important point here is that the tadpole contribution splits into two: (1) the energy-conserving part and (2) the energy nonconserving part.

In the energy conserving part, the constant multiplying the counterterm may be adjusted to satisfy the renormalization condition 0|*φ*|0 = 0 of the S-matrix theory, by which the tadpoles are completely eliminated from perturbation expansion. Nevertheless, the terms proportional to *f* survive. The energy nonconserving terms oscillate with time, with the frequency depending on the energy increment. In the competition with the contributions of subdiagram without tadpoles, they fade with time, thus giving the same *t* → ∞ limit as expected from S-matrix theory.

The *g*<sup>3</sup> order tadpoles and tadpoles with the resummed loop propagator (obtainable after renormalizing the self-energy; see further in the text) do not change our conclusions.

**Figure 1.** The tadpole diagram with a leg.

#### *2.3. UV Divergence at the Self-Energy Subdiagram*

While in the S-matrix theory, there is only Feynman (Σ<sup>1</sup> *<sup>F</sup>*(*p*0,*p*)) one-loop self energy, which does not depend on the frame, in out-of-equilibrium FT, we have self-energies Σ<sup>1</sup> *<sup>R</sup>*(*p*0,*p*), <sup>Σ</sup><sup>1</sup> *<sup>A</sup>*(*p*0,*p*), and Σ<sup>1</sup> *<sup>K</sup>*(*p*0,*p*), which is frame dependent through *<sup>f</sup>*(*ωp*) (notice here that we distinguish the "true" retarded and advanced functions from those that carry index *<sup>R</sup>* (*A*), but do not vanish for *<sup>t</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>1</sup> (*t*<sup>2</sup> <sup>&</sup>lt; *<sup>t</sup>*1), except at *<sup>d</sup>* <sup>&</sup>lt; 4).

$$\Sigma\_R^1(p\_0, \vec{p}) = -i\mathfrak{g}^2 \mu^\kappa \int \frac{d^d q}{2(2\pi)^d} [G\_R(p\_0 - q\_0, \vec{p} - \vec{q}) G\_{K,R}(q\_0, \vec{q})]$$

$$\Xi\_{K,R}(p\_0 - q\_0, \vec{p} - \vec{q}) G\_R(q\_0, \vec{q})] = \Sigma\_A^{1,\*}(p\_0, \vec{p}),$$

$$\Sigma\_K^1(p\_0, \vec{p}) = -\Sigma\_{K,R}^1(p\_0, \vec{p}) + \Sigma\_{K,A}^1(p\_0, \vec{p})\tag{8}$$

$$\Sigma\_{\mathcal{K},\mathcal{R}}^{1}(p\_{0\prime}\vec{p}) = -i\,g^{2}\,\mu^{\mathrm{x}}\int \frac{d^{d}q}{2(2\pi)^{d}} \left[\mathcal{G}\_{\mathcal{K},\mathcal{R}}(p\_{0} - q\_{0\prime}\vec{p} - \vec{q})\mathcal{G}\_{\mathcal{K},\mathcal{R}}(q\_{0\prime}\vec{q})\right]$$

$$+\mathcal{G}\_{\mathcal{R}}(p\_{0} - q\_{0\prime}\vec{p} - \vec{q})q\_{\prime}\mathcal{G}\_{\mathcal{R}}(q\_{0\prime}\vec{q})\right] = -\Sigma\_{\mathcal{K},\mathcal{A}}^{1,\*}(p\_{0\prime}\vec{p}).\tag{9}$$

Now, all the integrals containing *f*(*ωp*) are UV finite owing to the assumed UV cut-off in the definition of *f* . Vacuum contributions to Σ<sup>1</sup> *<sup>K</sup>*,*<sup>R</sup>* are finite separately at *d* → 4; at *d* → 6, this is no longer the case, but their sum is finite.

For retarded and advanced self-energies, imaginary parts and parts proportional to *f*(*ωp*) are UV finite and could be calculated directly from (8). Real, vacuum parts of Σ<sup>1</sup> *<sup>R</sup>* are connected to <sup>Σ</sup><sup>1</sup> *<sup>F</sup>*, and we use the results already available from S-matrix renormalization. The connection is:

$$
\Sigma\_{j,k}^1 = \frac{1}{2} \left[ -\Sigma\_{K,R}^1 + \Sigma\_{K,A}^1 - (-1)^k \Sigma\_R^1 - (-1)^j \Sigma\_A^1 \right],
$$

$$
Rc\Sigma\_{R,f=0}^1 = Rc\Sigma\_{11}^1 + \Sigma\_{K,R,f=0}^1 = \Sigma\_F^1 + \Sigma\_{K,R,f=0}^1. \tag{10}
$$

The regularization procedure (either by making *<sup>d</sup>* < 4 or by introducing fictive massive particles as in Pauli–Villars regularization) is usually considered artificial. Nevertheless, there are efforts to generate necessary massive particles (virtual wormholes) dynamically [32].

For Σ<sup>1</sup> *<sup>F</sup>*(*p*), we find in the literature [33]:

$$\Sigma\_{F}^{1}(p) = \frac{1}{2} 2^{2} g^{2} \frac{\int d^{4}q\_{1} d^{4}q\_{2}}{(2\pi)^{8}} G\_{F}(q\_{1})G\_{F}(q\_{2})(2\pi)^{4} \delta^{(4)}(q\_{1} - q\_{2} - p),$$

$$= \frac{1}{2} g^{2} \frac{\int d^{4}q\_{1} d^{4}q\_{2}}{(2\pi)^{8}} \frac{(2\pi)^{4} \delta^{(4)}(q\_{1} - q\_{2} - p)}{(q\_{1}^{2} - m^{2} + i\epsilon)(q\_{2}^{2} - m^{2} + i\epsilon)},$$

$$\implies \frac{1}{2} g^{2}(\mu)^{\kappa} \int\_{0}^{1} dz \int \frac{d^{4}q'}{(2\pi)^{d}} \frac{1}{[q'^{2} - m^{2} + p^{2}z(1 - z) + i\epsilon]^{2}},$$

$$= \frac{ig^{2}}{32\pi^{2}} (\mu^{2})^{\kappa/2} \Gamma(\kappa/2) \int\_{0}^{1} dz [\frac{p^{2}z(1 - z) - m^{2} + i\epsilon}{4\pi\mu^{2}}]^{-\kappa/2}.\tag{11}$$

The last relation above is still causal. It is UV finite, and it allows the separation into the sum of the retarded and advanced term. However, the expansion of [*p*2*z*(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*) <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* /4*πμ*2] <sup>−</sup>*κ*/2 in power series of <sup>|</sup>*κ*<sup>|</sup> is allowed only when *<sup>κ</sup>* ln[*p*2/(4*πμ*)] << 1; thus, it is a "low energy" expansion, and in spite of the fact that *κ* may be taken arbitrarily small, the limit |*p*0| → ∞ is never allowed.

$$\Sigma\_F^1(p) \approx \frac{ig^2\mu^\kappa}{16\pi^2(\kappa)} - \frac{ig^2\mu^\kappa}{32\pi^2} [\gamma\_\varepsilon + \int\_0^1 dz \ln[\frac{p^2z(1-z) - m^2 + i\varepsilon}{4\pi\mu^2}] ]$$

$$= \frac{ig^2\mu^\kappa}{16\pi^2(\kappa)} + finite. \tag{12}$$

This expression is no longer causal; it is valid only if *<sup>κ</sup>* ln[*p*2/(4*πμ*)] << 1. One needs the vanishing of self-energy for |*p*0| → ∞, i.e., the region where the opposite condition *<sup>κ</sup>* ln[*p*2/(4*πμ*)] >> 1 is fulfilled. Then, <sup>|</sup>Σ<sup>1</sup> <sup>∞</sup>,*F*(*p*)| → 0 as |*p*0| → ∞ as far as *κ* = 0.

The integration over *z* gives:

$$\Sigma\_{\mathbb{F}}^{1}(p) = -\frac{g^2}{16\pi^2} \left\{ \frac{1}{\kappa} - \frac{\gamma\_\varepsilon}{2} + 1 + \frac{1}{2} \ln(4\pi \frac{\mu^2}{m^2}) - \frac{1}{2} \sqrt{1 - \frac{4m^2}{p^2 + i\epsilon}} \ln \left[ \frac{\sqrt{1 - \frac{4m^2}{p^2 + i\epsilon}} + 1}{\sqrt{1 - \frac{4m^2}{p^2 + i\epsilon}} - 1} \right] \right\} \tag{13}$$

with a high *p*<sup>0</sup> limit:

$$\Sigma\_F(p^2, m^2)\_{p^2 \to \infty} \approx -\frac{g^2}{16\pi^2} \left\{ \frac{1}{\kappa} - \frac{\gamma\_E}{2} + 1 + \frac{1}{2} \ln(4\pi \frac{\mu^2}{m^2}) - \frac{1}{2} \ln\left[ -\frac{m^2}{p^2} \right] \right\}.\tag{14}$$

To verify the causality of the two-point function, one may try to project out the retarded part of the finite (subtracted) part of Σ<sup>1</sup> *<sup>F</sup>*(*p*), namely −*i dp* 0 <sup>2</sup>*<sup>π</sup>* <sup>Σ</sup><sup>1</sup> *<sup>F</sup>*, *finite*(*p*)/(*p*<sup>0</sup> − *p* <sup>0</sup> <sup>−</sup> *<sup>i</sup>* ), by integration *dp*<sup>0</sup> over a large semicircle. However, the contribution over a very large semicircle does not vanish, and the integral is ill defined.

Indeed, we have started from the expressions for *GF* (Σ*F*) containing only retarded and advanced functions, and in the absence of divergence, we expect this to be the truth at the end of calculation. Instead, the function in the last two lines of Expression (12) is not a combination of the *R* and *A* functions, otherwise it should vanish when |*p*0| → ∞ and *κ* are chosen as arbitrarily small; such a behavior can be shifted to an arbitrarily high scale. However, the limit *κ* → 0 remains always out of reach. To preserve causality, we should keep the whole *p*<sup>0</sup> complex plane. Specifically, we need the region with large |*p*0|, to be able to integrate over a large semicircle in the complex *p*<sup>0</sup> plane, at least to get *dp*0Σ<sup>1</sup> *<sup>R</sup>*(*p*)*GK*,*A*(*p*0) = 0. Thus, we have obtained a result correct at *κ* = 0 and problematic at *d* = 4.

Fortunately enough, there is a way to "repair" causality: the composite object *GF*(*p*)Σ<sup>1</sup> *<sup>F</sup>*(*p*) is vanishing when |*p*0| → ∞; it can be split into its retarded and advanced parts; thus, it is causal. This sort of reparation of causality is possible in other QFT in which logarithmic UV divergence appears. It is similar to the Glaser–Epstein [34–36] approach, where not just Σ, but *G*Σ are the subjects of expansion.

In this spirit, we agree with the conclusion of [37–39]: "Our amplitudes are manifestly causal, by which we mean that the source and detector are always linked by a connected chain of retarded propagators."

Similar is the problem we can see by considering *λφ*<sup>4</sup> theory. In this theory, the loop of Figure 2 is a vertex diagram, and the above Glaser–Epstein philosophy does not apply. Nevertheless, the propagator attached to the vertex depends on *p*<sup>0</sup> and "improves" the convergence of *dp*<sup>0</sup> integration.

**Figure 2.** The vertex diagram.

#### *2.4. Self-Energy Diagram with Legs*

To be able to introduce composite objects with Σ*R*(*A*), we need one of Σ*R*(*A*)'s vertices to conserve energy. The lower in time vertex may be the minimal time vertex, so it does not help in all cases. However, the higher in time vertex would do it, if both the integrals *dq*<sup>0</sup> and *dp*<sup>0</sup> converge.

The Σ*ij* self-energy contributions with legs (see Figure 2) are:

$$
\begin{array}{cccc}
\ G\_R \Sigma\_{K,R}^1 \ast \ G\_{A\prime} & \mathcal{G}\_R \ast \Sigma\_{K,A}^1 \mathcal{G}\_{A\prime} & \mathcal{G}\_R \Sigma\_R^1 \ast \mathcal{G}\_{K,A\prime} & \mathcal{G}\_{K,R} \ast \Sigma\_A^1 \mathcal{G}\_{A\prime} \\\\ \mathcal{G}\_R \Sigma\_R^1 \ast \mathcal{G}\_{A\prime} & \mathcal{G}\_A \Sigma\_A^1 \mathcal{G}\_{A\prime} & \mathcal{G}\_R \Sigma\_R^1 \mathcal{G}\_{K,R\prime} & \mathcal{G}\_{K,A} \Sigma\_A^1 \mathcal{G}\_A \\\\ \end{array}
\tag{15}
$$

In the above expression, Σs are introduced in Equation (8). "∗" indicates the convolution product, which includes the energy nonconserving vertex. Terms containing Σ<sup>1</sup> *<sup>K</sup>*,*<sup>R</sup>* and <sup>Σ</sup><sup>1</sup> *<sup>K</sup>*,*<sup>A</sup>* are UV-finite, creating no problems. The other terms, containing Σ<sup>1</sup> *<sup>R</sup>* and <sup>Σ</sup><sup>1</sup> *<sup>A</sup>*, are finite as long as *<sup>d</sup>* < 4, and we may obtain their real part through (11).

Two features seem potentially suspicious: (1) UV divergence in the loop defining Σ<sup>1</sup> *R*(*A*) , (2) the ill-defined vertex function between *GR* and Σ<sup>1</sup> *<sup>R</sup>* and between <sup>Σ</sup><sup>1</sup> *<sup>A</sup>* and *GA*.

Nevertheless, both problems are resolved at *<sup>d</sup>* < 4: "to be" UV divergence is subtracted and energy conservation is recovered in the above-mentioned vertices. The composite objects *GR*(*p*)Σ<sup>1</sup> *<sup>R</sup>*(*p*) and Σ<sup>1</sup> *<sup>A</sup>*(*p*)*GFA*(*p*) are now well defined.

#### **3. Discussion and Conclusions**

We examined renormalization prescriptions for the finite-time-path out-of-equilibrium *λφ*<sup>3</sup> QFT in the basis of *GR*, *GA*, *GK*,*R*, and *GK*,*<sup>A</sup>* propagators.

As expected, the number of counterterms did not change, and the formalism enables term by term finite perturbation calculation.

There are some interesting features:


The procedure is therefore generalized for application to more realistic theories (QED and QCD, electro-weak QFT, etc.) by the following:

(A) regularize; (B) do energy-conserving integrals; (C) subtract "to be" UV infinities; (D) deregularize (do limit *d* → 4).

Again, the above described Features (1) and (2) will emerge.

This work contains many of the features [40] arising in the more realistic theories like QED or QCD. Such finite-time-path renormalization is a necessary prerequisite for the calculation of damping rates, and other transition coefficients under the more realistic conditions truly away from equilibrium as opposed to the results obtained within the linear response approximation.

Our plan is to extend the exposed methods to the case of QED. Specifically, we resolve the controversy of the UV diverging number of direct photons in the lowest order of quark QED, as calculated by Boyanovsky and collaborators [41,42] and criticized by [43]. We find that, at the considered one-loop order of perturbation, it is only the vacuum-polarization diagram contributing. The renormalization leaves only finite contributions to the photon production [44].

**Author Contributions:** Conceptualization, I.D. and D.K.; Formal analysis, I.D.; Investigation, I.D. and D.K.; Methodology, I.D. and D.K.; Validation, I.D. and D.K.; Visualization, D.K.; Writing—original draft, I.D.; Writing—review & editing, I.D. and D.K.

**Funding:** This work was supported in part by the Croatian Science Foundation under Project Number 8799 and by STSMgrants from COST Actions CA15213 THORand CA16214 PHAROS.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix**

This Appendix provides the derivation of Equation (6).

The tadpole diagram, Figure 1, appears as a propagator with both ends attached to the same vertex. We start in coordinate representation. To sum contributions from the vertices of Types 1 and 2, we write the propagators with the help of the Wigner transform. Keldysh-time-path propagators and the finite-time propagators become identical in the limit *t* → ∞. To translate to the *R*/*A* basis, we use *Gi*,*<sup>j</sup>* = <sup>1</sup> <sup>2</sup> [*GK* + (−1)*<sup>j</sup> GR* + (−1)*<sup>i</sup> GA*].

*Gtad*,*j*(*x*2) = *igμκ*/2 *ddx*<sup>1</sup> ×[*G*1,1(*x*1, *x*1)*G*1,*j*(*x*1, *x*2) − *G*2,2(*x*1, *x*1)*G*2,*j*(*x*1, *x*2)], <sup>=</sup> *igμκ*/2 *dd*−1*x*<sup>1</sup> ∞ 0 *dx*01*e* <sup>−</sup>*ip*2(*x*1−*x*2) *<sup>d</sup><sup>d</sup> <sup>p</sup>*<sup>1</sup> (2*π*)*<sup>d</sup> d<sup>d</sup> p*<sup>2</sup> (2*π*)*<sup>d</sup>* <sup>×</sup>[*G*1,1,*x*<sup>01</sup> (*p*1)*G*1,*j*,*t*(*p*2) <sup>−</sup> *<sup>G</sup>*2,2,*x*<sup>01</sup> (*p*1)*G*2,*j*,*t*(*p*2)], *<sup>t</sup>* <sup>=</sup> *<sup>x</sup>*<sup>01</sup> <sup>+</sup> *<sup>x</sup>*<sup>02</sup> <sup>2</sup> , <sup>=</sup> *igμκ*/2 *dd*−1*x*<sup>1</sup> ∞ 0 *dx*<sup>01</sup> *d<sup>d</sup> p*<sup>1</sup> (2*π*)*<sup>d</sup> d<sup>d</sup> p*<sup>2</sup> (2*π*)*<sup>d</sup>* ×*e* −*ip*2(*x*1−*x*2) *dp* 01*dp* <sup>02</sup>*Px*<sup>01</sup> (*p*01, *p* <sup>01</sup>)*Pt*(*p*02, *p* 02) ×[*G*1,1,∞(*p* 1)*G*1,*j*,∞(*p* <sup>2</sup>) − *G*2,2,∞(*p* 1)*G*2,*j*,∞(*p* <sup>2</sup>)], *p* <sup>1</sup> = (*p* 01,*p*1), *<sup>p</sup>* <sup>2</sup> = (*p* 02,*p*2), (A1)

where we have used the projection operator *P* connecting time-dependent lowest order propagators with time-independent lowest order propagators [27,28]:

$$\begin{split} G\_{t}(p\_{0},\vec{p}) &= \int\_{-\infty}^{\infty} dp'\_{0} P\_{t}(p\_{0},p'\_{0}) \mathbb{G}\_{\infty}(p'\_{0},\vec{p}),\\ P\_{t}(p\_{0},p'\_{0}) &= \frac{\Theta(t)}{2\pi} \int\_{-2t}^{2t} ds\_{0} e^{i s\_{0}(p\_{0}-p'\_{0})} = \frac{\Theta(t)}{\pi} \frac{\sin 2(p\_{0}-p'\_{0})t}{(p\_{0}-p'\_{0})},\\ \lim\_{t \to \infty} \Pr\_{t \to \infty} \Pr(p\_{0},p'\_{0}) &= \delta(p\_{0}-p'\_{0}),\\ \int\_{-\infty}^{\infty} dp\_{0} e^{-i s\_{0}p\_{0}} P\_{t}(p\_{0},p'\_{0}) &= e^{-i s\_{0}p'\_{0}} \Theta(t) \Theta(2t-s\_{0}) \Theta(2t+s\_{0}). \end{split} \tag{A2}$$

Here, *G* is a bare propagator (matrix propagator or *R*, *A*, or *K* propagator.) A similar relation holds for lowest order self-energies:

$$
\Sigma\_t^1(p\_{0\prime}\vec{p}) = \int\_{-\infty}^{\infty} dp\_0' P\_t(p\_{0\prime}, p\_0') \Sigma\_\infty^1(p\_{0\prime}'\vec{p}),\tag{A.3}
$$

where Σ<sup>1</sup> *<sup>t</sup>* is the retarded, advanced, or Keldysh self-energy.

By using the above relations, we obtain:

*Gtad*,*j*(*x*2) = *igμκ*/2 *dd*−1*x*<sup>1</sup> ∞ 0 *dx*01*e* −*ip* <sup>2</sup>(*x*1−*x*2) *<sup>d</sup><sup>d</sup> <sup>p</sup>* 1 (2*π*)*<sup>d</sup> d<sup>d</sup> p* 2 (2*π*)*<sup>d</sup>* ×[*G*1,1,∞(*p* 1)*G*1,*j*,∞(*p* <sup>2</sup>) − *G*2,2,∞(*p* 1)*G*2,*j*,∞(*p* <sup>2</sup>)], = *igμκ*/2(2*π*)−<sup>1</sup> −*i p* <sup>02</sup> − *i δ*(*d*−1) (*p* 2)*e ip* <sup>02</sup>*x*<sup>02</sup> *<sup>d</sup><sup>d</sup> <sup>p</sup>* 1 (2*π*)*<sup>d</sup> <sup>d</sup><sup>d</sup> <sup>p</sup>* 2 ×[*G*1,1,∞(*p* 1)*G*1,*j*,∞(*p* <sup>2</sup>) − *G*2,2,∞(*p* 1)*G*2,*j*,∞(*p* <sup>2</sup>)], = *igμκ*/2(2*π*)−<sup>1</sup> −*i p* <sup>02</sup> − *i δ*(*d*−1) (*p* 2)*e ip* <sup>02</sup>*x*<sup>02</sup> *<sup>d</sup><sup>d</sup> <sup>p</sup>* 1 (2*π*)*<sup>d</sup> <sup>d</sup><sup>d</sup> <sup>p</sup>* 2 ×1 2 [−*GK*,∞(*p* 1)*GA*,∞(*p* <sup>2</sup>) − *GR*,∞(*p* 1)*GK*,∞(*p* <sup>2</sup>) − *GA*,∞(*p* 1)*GK*,∞(*p* 2) +(−1)*<sup>j</sup> GR*,∞(*p* 1)*GR*,∞(*p* 2)+(−1)*<sup>j</sup> GA*,∞(*p* 1)*GR*,∞(*p* <sup>2</sup>)], (A4)

By taking the fact that tadpoles with *GR* and *GA* vanish, we obtain:

$$\begin{split} G\_{\mathrm{tad},j}(\mathbf{x}\_{2}) &= i g \mu^{\mathbf{x}/2} \frac{(2\pi)^{-1}}{2} \int \frac{i}{p\_{02}^{\prime} - i\epsilon} \delta^{(d-1)}(\vec{p}\_{2}^{\prime}) e^{i p\_{02}^{\prime} \mathbf{x} \mathbf{q} \mathbf{z}} \\ &\times \frac{d^{d} p\_{1}^{\prime}}{(2\pi)^{d}} d^{d} p\_{2}^{\prime} G\_{\mathrm{K},\infty}(p\_{1}^{\prime}) G\_{\mathrm{A},\infty}(p\_{2}^{\prime}), \\ &= (2\pi)^{-1} \int \frac{i}{p\_{02}^{\prime} - i\epsilon} e^{i p\_{02}^{\prime} \mathbf{x}\_{02}} G\_{\mathrm{A},\infty}(p\_{02}^{\prime}, 0) d p\_{02}^{\prime} G\_{\mathrm{Tad}} \\ &G\_{\mathrm{Tad}} = \frac{i g \mu^{\mathbf{x}/2}}{2} \int G\_{\mathrm{K},\infty}(p\_{1}^{\prime}) \frac{d^{d} p\_{1}^{\prime}}{(2\pi)^{d}}. \end{split} \tag{45}$$

Thus,

$$\mathcal{G}\_{\rm tad,j}(\mathbf{x}\_2) = -\mathcal{G}\_{A,\infty}(0,0)\mathcal{G}\_{\rm Tab} + \int \frac{dp'\_{02}}{2\pi} \frac{i\varepsilon^{ip'\_{02}\,\mathbf{x}\alpha}}{p'\_{02} - i\varepsilon} [\mathcal{G}\_{A,\infty}(p'\_{02}, 0) - \mathcal{G}\_{A,\infty}(0,0)]\mathcal{G}\_{\rm Tab}.\tag{A6}$$

The contribution is split into the first, energy-conserving term, and the second term, oscillating with time, in which energy is not conserved at the vertex 1.

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **NA61/SHINE Experiment—Program beyond 2020**

#### **Ludwik Turko for the NA61/SHINE Collaboration**

Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50-205 Wroclaw, Poland; ludwik.turko@ift.uni.wroc.pl; Tel.: +48-71-375-9355

Received: 29 October 2018; Accepted: 26 November 2018; Published: 30 November 2018

**Abstract:** The fixed-target NA61/SHINE experiment (SPS CERN) looks for the critical point (CP) of strongly interacting matter and the properties of the onset of deconfinement. It is a scan of measurements of particle spectra and fluctuations in proton–proton, proton–nucleus, and nucleus– nucleus interactions as a function of collision energy and system size. This gives unique possibilities to researching critical properties of the dense hot hadronic matter created in the collision process. New measurements and their objectives, related to the third stage of the experiment after 2020, are presented and discussed here.

**Keywords:** QCD matter; phase transition; critical point

#### **1. Introduction**

The NA61/SHINE, **S**uper Proton Synchrotron (SPS) **H**eavy **I**on and **N**eutrino **E**xperiment, is the continuation and extension of the NA49 [1,2] measurements of hadron and nuclear fragment production properties in fixed-target reactions induced by hadron and ion beams. It has used a similar experimental fixed-target setup as NA49 (Figure 1), but with an extended research program. Beyond an enhanced strong interactions program, there are the measurements of hadron production for neutrino and cosmic ray experiments realized. NA61/SHINE is a collaboration, with about 150 physicists, 33 institutions, and 14 countries being involved.

The strong interaction program of NA61/SHINE is devoted to studying the onset of deconfinement and search for the critical point (CP) of hadronic matter, related to the phase transition between hadron gas (HG) and quark-gluon plasma (QGP). The first order phase boundary between the HG and QGP phase is expected to end at the CP, as seen in Figure 2. At the CP, the sharp first-order phase transition turns into a rapid crossover, resulting in the appearance of large fluctuations of various observables, which are sensitive to the vicinity of the CP. The CP has long been predicted for thermal quantum chromodynamics (QCD) at finite *μB*/*T* [3–5]. Lattice QCD calculations, which are becoming more and more accurate, have led to the present conclusions that the cross-over region occurs at *Tc*(*μ<sup>B</sup>* = 0) = 154 ± 9 MeV [6] and the location of the CP is not expected for *μB*/*T* 2 and *<sup>T</sup>*/*Tc*(*μ<sup>B</sup>* <sup>=</sup> <sup>0</sup>) <sup>&</sup>gt; 0.9 [7]. A more detailed exploration of the QCD phase diagram would need both new experimental data with extended detection capabilities and improved theoretical models [8].

The NA49 experiment studied hadron production in Pb + Pb interactions, while the NA61/SHINE collects data varying collision energy (13A–158A GeV) and the size of the colliding systems, as shown in Figure 3. This is, in a sense, equivalent to the two-dimensional scan of the NA61/SHINE piece of the hadronic phase diagram in the (*T*, *μB*) plane, as depicted in Figure 2. Changes in the collision energy lead to different values of the net baryon number chemical potential *μ<sup>B</sup>* and temperature *T*. Different sizes of colliding systems allow to identify the minimum hadronic volume, which can be excited to the state where statistical physics concepts of HG/QGP phase transition are still meaningful. The research program was initiated in 2009, with thep+p collisions used later on as reference measurements for heavy-ion collisions.

**Figure 1.** The present NA61/SHINE detector consists of a large acceptance hadron spectrometer, followed by a set of six Time Projection Chambers (TPCs), as well as time-of-flight (ToF) detectors. The high resolution forward calorimeter, the projectile spectator detector (PSD), measures energy flow around the beam direction. For hadron–nucleus interactions, the collision volume is determined by counting the low momentum particles emitted from the nuclear target with the low momentum particle detector (a small TPC) surrounding the target. An array of beam detectors identifies beam particles, secondary hadrons, and nuclei, as well as primary nuclei, and measures precisely their trajectories.

**Figure 2.** Phase structure of hadronic matter covered by NA61/SHINE (green), compared to present and future heavy ion experiments.

**Figure 3.** For the program on strong interactions NA61/SHINE scans in the system size and beam momentum. In the plot, the recorded data are indicated in green and the approved future data in red.

Hadron production measurements for neutrino experiments are just reference measurements of p+C interactions for the T2K experiment, since they are necessary for computing initial neutrino fluxes at J-PARC. These measurements have been extended to the production of charged pions and kaons in interactions with thin carbon targets and replicas of the T2K target, to test accelerator neutrino beams [9]. The collection of data began in 2007.

Collected p + C data also allow to better understand nuclear cascades in the cosmic air showers—necessary in the Pierre Auger and KASCADE experiments [10,11]. These are reference measurements of p + C, p + p, *π* + C, and K + C interactions for cosmic ray physics. The cosmic ray collisions with the Earth's atmosphere produce air shower secondary radiation. Some of the particles produced in such collisions subsequently decay into muons, which are able to reach the surface of the Earth. Cosmic ray induced muon production would allow to reproduce primary cosmic ray composition if the related hadronic interactions are known [12].

As seen in Figure 2, the phase structure of hadronic matter is involved. Progress in the theoretical understanding of the subject and collecting more experimental data will allow to delve into the subject. While the highest energies achieved at LHC and RHIC colliders provide data related to the crossover HG/QGP regions, the SPS fixed-target NA61/SHINE experiment is particularly suited to explore the phase transition line HG/QGP with the CP included.

#### *Results of Initial NA61/SHINE Research*

The production properties of light and medium mass hadrons, in particular pions and kaons, have been measured [13] according to the NA61/SHINE proposal [1]. The Be + Be results are close to p+p independently of collision energy. Moreover, the data show a jump between light (p + p, Be + Be) and intermediate, heavy (Ar + Sc, Pb + Pb) systems [14]. The observed rapid change of hadron production properties that starts when moving from Be + Be to Ar + Sc collisions can be interpreted as the beginning of the creation of large clusters of strongly interacting matter—the onset of fireball. One notes that non-equilibrium clusters produced in p + p and Be + Be collisions seem to have similar properties at all beam momenta studied here.

The *K*+/*π*<sup>+</sup> ratio inp+p interactions is below the predictions of statistical models. However, the ratio in central Pb + Pb collisions is close to statistical model predictions for large volume systems [15].

In p + p interactions, and thus also in Be + Be collisions, multiplicity fluctuations are larger than predicted by statistical models. However, they are close to statistical model predictions for large volume systems in central Ar + Sc and Pb + Pb collisions [16].

The two-dimensional scan conducted by NA61/SHINE by varying collision energy and nuclear mass number of colliding nuclei indicates four domains of hadron production separated by two thresholds: The onset of deconfinement and the onset of fireball [17]. The sketch presented in Figure 4 illustrates this preliminary conclusion. Collected Ar + Sc and Xe + La data are being analyzed to provide further information.

**Figure 4.** The onset of deconfinement and the onset of fireball. The onset of deconfinement is well established in central Pb + Pb (Au + Au) collisions. Its presence in collisions of low mass nuclei, inelastic p + p interactions in particular, is questionable.

Total production cross-sections and total inelastic cross-sections for reactions *π*++C,Al and *K*++C,Al at 60 GeV/c and *π*++C,Al at 31 GeV/c were measured. These measurements are a key ingredient for neutrino flux prediction from the reinteractions of secondary hadrons in current and future accelerator-based long-baseline neutrino experiments [18].

#### **2. New Measurements Requested**

The third stage of the experiment, starting after the Long Shutdown 2 (LS-2) of the CERN accelerator system, would include:


The proposed measurements and analysis are requested by heavy ion, cosmic ray, and neutrino communities. A careful analysis of fluctuations and intermittency phenomena in NA61/SHINE data collected so far is necessary to look for the CP [19].

The objective of **charm hadron production measurements** in Pb + Pb collisions is to obtain the first data on the mean number of *cc*¯ pairs produced in the full phase space in heavy ion collisions. Moreover, further new results on the collision energy and system size dependence will be provided. This will help to answer the questions about the mechanism of open charm production, about the relation between the onset of deconfinement and open charm production, and about the behavior of *J*/*ψ* in quark-gluon plasma.

The objective of **nuclear fragmentation cross-section measurements** is to provide high-precision data needed for the interpretation of results from current-generation cosmic ray experiments. The proposed measurements are of crucial importance to extract the characteristics of the diffuse propagation of cosmic rays in the Galaxy.

The objectives of **new hadron production measurements for neutrino physics** are to further improve the precision of hadron production measurements for the currently used T2K replica target, to perform measurements for a new target material, both for T2K-II and Hyper-Kamiokande experiments, and to study the possibility of measurements at low incoming beam momenta (below 12 GeV/c), relevant for improved predictions of both atmospheric and accelerator neutrino fluxes.

NA61/SHINE is the only experiment which will conduct such measurements in the near future. Together with other HIC experiments, it creates a full-tone physical picture of QCD in dense medium. Especially concerning the strong interaction heavy-ion program, the NA61/SHINE has unique capabilities in comparison with the other experiments (see Figure 2):

The limitations of other experiments are related to: (i) Limited acceptance, (ii) measurement of open charm not considered in the current program, or (iii) very low cross-section at SIS-100.

Concerning other experiments' capabilities shown at Figure 5:


**Figure 5.** Recent (red) and future (green) heavy ion facilities in the phase diagram of strongly interacting matter.

The beam momentum range provided to NA61/SHINE by the SPS and the H2 beam line is highly important for the heavy ion, neutrino, and cosmic ray communities. It covers:


#### *Specific Research Goals*

The NA61/SHINE charm program addresses questions about the validity and the limits of statistical and dynamical models of high energy collisions in the new domain of quark mass, *mc* ≈ 1300 MeV *TC* ≈ 150 MeV [33]. To answer these questions, knowledge is needed on the mean number of charm–anticharm quark pairs *cc*¯ produced in the full phase space of heavy ion collisions.

Such data do not exist yet and NA61/SHINE aims to provide them within the coming years. The related preparations have started already. In 2015 and 2016, a Small Acceptance Vertex Detector (SAVD) was constructed and first measurements of open charm production started in 2016—Figure 6. Vertex resolution has appeared precisely enough (30 μm) to distinguish *D*<sup>0</sup> decay. That was possible due to the fixed-target experiment-specific property, where the Lorenz factor *βγ* ≈ 10 makes short-living *D*<sup>0</sup> an observable particle, even in such a small acceptance vertex detector.

**Figure 6.** Present NA61/SHINE setup with the SAVD included.

Successful performance of the SAVD in 2016 led to the decision to also use it during the collection of Xe + La data in 2017. About 5 <sup>∗</sup> 106 events of central Xe + La collisions at 150A GeV/c were collected. The Xe + La data are currently under analysis and are expected to lead to physics results in the coming months. One expects to reconstruct several hundreds of *D*<sup>0</sup> and *D*¯0 decays. Beyond this, about 4000 *D*<sup>0</sup> and *D*¯0 decays can be expected to be reconstructed from the collection of Pb + Pb data in 2018. Further data collection on Pb + Pb collisions and the reconstruction of decays of various open charm mesons are planned by NA61/SHINE for the years 2022–2024. This would be combined with the required detector upgrades, including a full scale, large acceptance vertex detector—now under construction.

Another domain of NA61/SHINE activity will be to measure fragmentation cross-sections relevant for the production of Li, Be, B, C, and N nuclei. These elements are of particular importance for the physics of cosmic rays in the Galaxy. The NA61/SHINE facility has already successfully taken data with light ion beams [34] and can be used with practically no modifications to perform the

needed cross-section measurements at isotope level. The ability to separate different isotopes from fragmentation interactions for a given charge was validated with simulations [35].

**Funding:** The author acknowledges support by the Polish National Science Center under contract No. UMO-2014/15/B/ST2/03752, the COST Actions CA15213 (THOR), CA16117 (ChETEC) and CA16214 (PHAROS) for supporting their networking activities, and the Bogolubov-Infeld Program for supporting the author's stay at JINR Dubna.

**Conflicts of Interest:** The author declare no conflicts of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


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