Gradient-flowed thermal correlators: how much flow is too much?

Gradient flow has been proposed in the lattice community as a tool to reduce the sensitivity of operator correlation functions to noisy UV fluctuations. We test perturbatively under what conditions doing so may contaminate the results. To do so, we compute gradient-flowed electric field two-point correlators and stress tensor one- and two-point correlators at finite temperature in QCD. Gradient flow has almost no influence on the value of correlators until a (temperature- and separation-dependent) level of flow is reached, after which the correlator is rapidly compromised. We provide a prescription for how much flow is"safe."

the determination of thermodynamical information. One could evaluate the energy density at temperature by evaluating the difference T 00 β − T 00 vac , the thermal-to-vacuum difference in the 00 component of the stress tensor. The configuration-by-configuration squared fluctuations in this quantity are set by the 2-point function lim x→0 T 00 (x)T 00 (0) . Based on operator dimension, we see that this quantity diverges at small x as x −8 . Of course on the lattice this divergence is cut off by the lattice spacing and is O(a −8 ). This squared fluctuation must be compared to β −8 , the square of the size of the energy density difference; the number of spacetime points times configurations must compensate this large ratio to obtain a statistically significant measurement. On the other hand, under gradient flow to a depth τ F , we expect the overlapping 2-point function to be O(τ −4 F ) (τ F has dimensions of length 2 , not length). Therefore the UV fluctuations which inhibit a statistically significant evaluation are ameliorated and the number of configurations we must evaluate to obtain good statistics is reduced by a factor 1 of (a 2 /τ F ) 2 .
A little gradient flow is certainly a good thing, improving statistics, fixing some operator renormalization issues [3], and making the lattice more continuum-like. However, too much gradient flow is definitely bad, as eventually we erase the fluctuations responsible for the physics we want to study. In particular, we want to know, for the study of thermal one-operator and multi-operator correlators, exactly how much gradient flow one may apply before one changes the physics of interest. In this note we will study this problem perturbatively. To our knowledge this is the first perturbative study of thermal correlation functions, and of correlators of spacetime-separated operators, under gradient flow. Therefore we will content ourselves for the moment with a leading-order perturbative evaluation. It is possible that interactions reveal some new physics which makes the situation worse than what we find here, so it would be valuable to extend these calculations to the loop level. However we will leave this for future work.
Here we will consider three types of correlation functions. First and simplest, we consider the stress tensor one-point function at finite temperature. As discussed above, this can be used to measure rather directly the energy density as a function of temperature (if the operator renormalization issues can be resolved; so far the renormalization of a gradientflowed stress tensor has only been studied perturbatively [5,8], while a nonperturbative treatment is probably necessary). Second, we will consider the correlator of two electric field operators, embedded along a Polyakov line: x 2 ) and E i is the electric field. This operator was introduced in [20], who show that its analytical continuation to Minkowski frequency determines the (momentum-space and coordinate-space) diffusion of a heavy quark, m q T in a thermal bath. Recently there has been a vigorous effort to measure this correlation function on the lattice [21], but so far only quenched results are available and the issue of the E-field renormalization has not been resolved.
Gradient flow would fix the renormalization issue and will hopefully improve statistical power such that the correlation function can be reliably measured at the nonperturbative level. Finally, we will consider the correlation function of two stress tensors at vanishing spatial momentum (equivalently, integrated over spatial separation) as a function of the Euclidean time separation τ . For those T µν components which couple to hydrodynamical modes, such as T 00 T 00 and T 0i T 0i , the correlator should be τ -independent and should reproduce thermodynamical information (the heat conductivity and enthalpy density respectively). For the = 2 space component, e.g. T xy T xy , the analytical continuation of the correlator holds information about the shear viscosity [22][23][24][25].
In the next section we will develop perturbative tools for gradient flow at finite temperature in coordinate space, which turns out to be the most convenient for the problems we study here. Next, Section 3 contains the specific details of the leading-order calculations of each correlator mentioned above. In every case we find that there is a (τ -dependent) range of flow times τ F for which the correlator feels exponentially suppressed corrections; but for more flow it quickly goes wrong. We end with a discussion which presents our recommendations for the amount of flow which can be applied "safely," given our lowest-order perturbative results.

Gradient flow at temperature in coordinate space
We write the unflowed gauge field as A a µ (x) and will generally suppress the color index a. The flowed gauge field B µ (x, τ F ) is defined at nonnegative flow time τ F through the τ F = 0 boundary condition and the flow equation where G νµ (x, τ F ) is the field strength tensor written using B µ (x, τ F ) rather than A µ (x). The second term in the flow equation constitutes as τ F -dependent gauge choice, which is convenient to make in the context of perturbative calculations [3]. Choosing α 0 = 1 and working to linearized order, the flow equation simplifies to which is the heat equation.
In vacuum, the Feynman-gauge momentum-space propagator after flow is and Fourier transforming leads to the zero temperature flowed propagator in coordinate space .

(2.5)
This result was found by Lüscher in [3]. But we could have reached this result faster by noting that the coordinate-space propagator before flow is the solution to the Poisson . At tree level and in Feynman gauge, flow is the application of the heat equation to this propagator, which is the same as convolving it with a Gaussian envelope, .
Alternatively, one may take the right-hand expression as an Ansatz and verify that it satisfies the τ F = 0 boundary conditions and the heat equation.
To introduce finite temperature, we restrict the Euclidean time to lie in x 0 ∈ [0, β] with periodic boundary conditions. The fast way to find the coordinate-space propagator is to note that the Poisson equation is now solved using the method of images; and that flow again corresponds to evolving this propagator under the heat equation or convolving with a Gaussian: . (2.8) We could also arrive at this result the "hard way" by Fourier transforming the finitetemperature, flowed momentum-space propagator by use of Poisson's summation formula [26] T m∈Z to rewrite the summation over p 0 as a sum over coordinate-space copies -essentially, the same images as above. At this point each dp 0 term represents a vacuum contribution with a different x 0 position, shifted into one of the image copies. This leads rather directly back to Eq. (2.8). In the following we will only work at finite temperature so we will suppress the subscript β.
When we take correlation functions, we will have to include a sum over images for each propagator which appears. We present a cartoon of this procedure in Figure 1.

Calculations
We will now use the coordinate-space propagator to compute the desired correlation functions. While the last section has introduced propagators between fields with different amounts of flow, here we will only consider correlators where all operators are evaluated after the same amount of flow τ F ; so the previous formulae should be modified by writing

Energy-Momentum Tensor One-Point Function
The tree-level energy-momentum tensor in Yang-Mills theory is 2 We evaluate the correlator of two field strengths by splitting the field strengths to reside at points x, y, write which we found in Eq. (2.8), take derivatives, and then set x = y. Introducing a dimensionless rescaled flow timeτ F ≡ 8τ F /β 2 , we evaluate the two field strength correlators which we need, where each n 2 arises as x 2 n /β 2 . Then we combine them to find a closed expression for the stress-tensor one-point function after flow, Here d A = N 2 c − 1 = 8 is the dimension of the group, which counts gluon colors. The sum over n is a sum over images; the vacuum result is the n = 0 term, which is defined as the n → 0 limit and which actually vanishes. In the τ F → 0 limit the exponential terms vanish and we have only the 1/n 4 term, confirming as expected that the standard Stefan-Boltzmann result.
In the opposite limit,τ F 1, many terms contribute to the sum and we may approximate it with an integral, giving rise to This result corresponds to the contribution arising from the zero Matsubara frequency, as all other Matsubara frequencies are damped away by the flow. For finiteτ F we evaluate the sum numerically and display the result in Figure 2. The plot shows that, for smallτ F , the corrections to Stefan-Boltzmann are exponentially small, physically representing the exponentially small amplitude for the "smearing" due to flow to stretch all the way around the periodic direction. However the stability of the result then rather abruptly breaks down aboveτ F ∼ 0.12, and for largeτ F values the thermal contribution is almost completely lost. If we require that the flow change the determined energy density by at most 1%, then we can constrain the allowed flow depth to be 8τ F /β 2 ≤ 0.12. On the lattice with N t lattice points around the temporal direction, that corresponds to τ F /a 2 ≤ 0.015N 2 t with a the lattice spacing.

Electric-Field Correlation Function at Finite Temperature
In Eq. derivatives of the gauge-field propagator are involved. The leading-order contribution reads .
(3.8) Differentiating and introducing the dimensionless scaled coordinatex n = x n /β and the ratio of squared coordinate to flow timeξ 2 n =x 2 n /τ F , we find (3.9) In this expression we have allowed the electric fields to be at different spatial coordinates, but the correlator relevant for heavy quark transport involves x = 0, which we will set from now on. Our result then simplifies to This is the main result of this section.
To explore this result further, we consider first the limit of small flow time,τ F → 0 or ξ → ∞. In this limit ξ 4 +ξ 2 + 1 e −ξ 2 0. The sum can be performed analytically and   Left: plot of the free-theory electric-field electric-field correlation, Right:the same, normalized to the unflowed behavior the result is

Scaled E-Field E-Field Correlatorτ
The correlation function is negative, as expected; the electric field is odd under the time- and so its correlation function should be negative. Note however that the time-integrated EE correlator could still be positive due to contact terms when the operators overlap. We can also explore the opposite limit of large flow time,τ F 1, which allows us to approximate the sum over n with an integral, which is the same result we would get by considering only the contribution of the zero Matsubara frequency. In contrast to the smallτ F limit, this result is positive. There is no contradiction with fundamental theorems, because the operator after flow is no longer local, so Θ-odd behavior does not ensure negative correlations. But this indicates that the result at large flow times has been thoroughly contaminated with contact-term type contributions. Once a correlator which is expected to be negative becomes positive due to flow, the character of the correlation function has been fundamentally altered. The sum in Eq. (3.10) can be evaluated numerically. In Figure 3, the behavior of the correlator is shown for different values ofτ F . The black curve is the analytic result for   Figure 3 shows that as we increase the amount of flow, the x 0 range for which the correlator remains almost unchanged gets narrower; for the larger flow times shown, the two never coincide. Therefore the amount of flow which we can "get away with" is x 0 dependent, which should not be too surprising. It is also instructive to explore the correlator as a function of flow time at fixed separation. Figure 4 shows a plot of this function. The behavior of the function is exactly as expected. For small flow times, the correlator shows a plateau of the unflowed value and the amount of flow which damages the correlator depends on the separation. If we use more flow, the correlator changes sign. For enough flow it becomes small as all fluctuations are damped away.
In Table 3.2 we show the maximum amount of flow before the correlator changes by 1% as a function of x 0 . We believe that this can be used as a criterion for how much flow one can "get away with" in measuring the EE correlator at a given x 0 value.

Stress Tensor Two-Point Functions
The calculation of the stress tensor two-point correlator is similar to the electric field correlator, except that each stress tensor contains two field strengths. Since there are two gauge field propagators, there is now a double sum over images. The connected stress Table 1. List of the flow times needed to change the EE correlator by 1% relative to the zero-flow value, for different x 0 separations.
(3.15) The derivatives can be applied for each sum separately, with the dimensionless scalar functions defined as Because at leading order T 00 = −T ii , there are three independent stress-tensor correlators (at vanishing spatial momentum) for which we can apply these formulae, which evaluate to These are the main analytic results of this section. We are interested in the p = 0 channel and thus need to integrate over d 3 x. At this point we resort to a numerical evaluation. In Figure 5 the results for the energy density-energy density component T 00 T 00 are shown. Energy conservation implies that for vanishing flow time the correlator should be a flat line at a value set by the heat capacity, which in the free theory is 4π 2 d A 15β 5 . Such consequences of stress conservation hold up to exponentially small corrections so long asξ 2 1. However, for larger flow extents, operators effectively overlap, and contact terms contaminate consequences of stress conservation. Therefore, when the flow depth approaches the squared separation, the constancy  of the correlator will be lost. This is indeed what we observe. For large values of flow timẽ τ F 1, the correlator becomes flat again, as it is dominated by the time-independent zero Matsubara frequency contribution.
The momentum-momentum component T 0i T 0i is related to momentum fluctuations in the medium. Without flow, it should be constant and negative, with value set by the enthalpy density times temperature, 4π 2 d A 15β 5 . The behavior under flow is shown in Figure 6. The flow time dependence is similar to that for the T 00 T 00 correlator, for the same physical reasons.
The stress-stress component T ij − 1 3 δ ij T ll T ij − 1 3 δ ij T kk is physically interesting because its continuation to a spectral function determines the shear viscosity [22][23][24][25]. Because it is not constrained by conservation laws, no short-distance cancellations occur and it shows strong short-distance divergent behavior; the unflowed behavior is dominated by the vacuum contribution which diverges at the origin. If we use gradient flow, the correlator is finite at the origin and for intermediate flow times 0.01 <τ F < 0.1 we find a non-trivial behavior. The numerical results are presented in Figure 7. For large flow times the zero Matsubara frequency again dominates the correlator, which is nearly x 0 independent.
The main result of this numerical evaluation is that if we are using flow to suppress fluctuations in our correlators, then the T 00 T 00 and T 0i T 0i correlators are best evaluated at x 0 = β/2 and with at most 8τ F /β 2 < 0.027. For T ij T ij one should use the same τ F values as for the electric field correlator with the same x 0 value.

Discussion and conclusions
Gradient flow successfully reduces short-distance fluctuations, which is a boon for reducing statistical fluctuations in the lattice determination of local operator correlation functions. Therefore there is an interest in applying it to lattice measurements of correlation functions.
Here we made a first exploration of how reliable this approach may be at finite temperature, for the evaluation of the energy density T 00 and of electric field and stress tensor two-point functions. At lowest order in perturbation theory, we found that the energy density of the thermal bath is obtained reliably provided that the flow depth obeys τ F < 0.015β 2 (or τ F /a 2 = 0.015N 2