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# Gauge theory bootstrap

Yifei He, Martin Kruczenski

► To cite this version:

Yifei He, Martin Kruczenski. Gauge theory bootstrap. Physical Review Letters, 2024, 133 (19), pp.191601. 10.1103/PhysRevLett.133.191601. hal-04235631

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# Gauge Theory Bootstrap

Yifei He*

Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, F-75005 Paris, France

Martin Kruczenski†

Department of Physics and Astronomy and PQSEI, Purdue University, West Lafayette, Indiana 47907, USA

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(Received 29 February 2024; revised 6 June 2024; accepted 17 June 2024; published 4 November 2024)

We propose the gauge theory bootstrap, a method to compute the pion $S$ matrix that describes the low-energy physics of the strong interaction and other similar gauge theories. The phase shifts of the $S0$, $P1$, and $S2$ waves obtained are in good agreement with experimental results. The only numerical inputs are the quark mass $m_q$, the QCD scale $\Lambda_{QCD}$, the pion mass $m_\pi$, and the pion decay constant $f_\pi$ without any other experimental data. We make use of the form-factor bootstrap recently proposed by Karateev, Kuhn, and Penedones together with a finite energy version of the Shifman-Veinshtein-Zakharov sum rules.

DOI: 10.1103/PhysRevLett.133.191601

**Introduction**—Finding the low-energy description of a system described in terms of its microscopic variables is an important physical problem. In the case of the strong interactions, the microscopic description is in terms of an asymptotically free gauge theory: quantum chromodynamics (QCD) with light quarks that undergoes confinement and chiral symmetry breaking at low energy. The low-energy theory is a scalar field theory of pions—the pseudo-Goldstone bosons of chiral symmetry breaking. However, there is a large range of intermediate energies where both the pion description and the gauge theory description are strongly coupled. In this Letter, we show how to use the $S$-matrix bootstrap method to fill this gap and compute pion scattering in this energy regime. The method looks for the most general pion scattering matrix that matches the gauge theory at high energy and the low-energy theory of pions given by chiral perturbation theory [1–4]. We call this method the *gauge theory bootstrap*, and it is defined by the following steps. (i) *Pure $S$-matrix bootstrap*.—In the first step, the bootstrap method maps out the space of low-energy $S$ matrices allowed by unitarity, crossing, analyticity, and the global $SU(2)_V$ flavor symmetry. (ii) *Chiral symmetry breaking*.—In the second step, we add the constraints of chiral symmetry breaking that determine an approximate $S$ matrix in the very-low-energy region in terms of $m_\pi$ and $f_\pi$. (iii) *QCD, form factors, and Shifman-Veinshtein-Zakharov (SVZ) sum rules*.—Finally,

we pick a scale $s_0 \simeq 1.2$ GeV where the strong coupling constant is $\alpha_s \simeq 0.4$, an adequate value to match perturbative QCD at high energy. Gauge theory information is introduced into the bootstrap via form factors together with a finite energy version of the SVZ sum rules. As shown below, this procedure for two flavors and gauge group SU(3) is in good agreement with experiment.

Historically, the $S$-matrix bootstrap [5,6] was proposed in the 1960s as a fundamental formulation of the strong interactions, before it was found to be an asymptotically free gauge theory. Recently, the $S$-matrix bootstrap was revived in [7–9] (see also [10]) as a way to study the allowed space of $S$ matrices of generic theories. In particular, pion scattering was studied from a pure low-energy perspective in [11–13] by including low-energy information such as resonance masses and scattering lengths. The question still remained of how to incorporate high-energy data into the bootstrap in order to *identify the low-energy theory corresponding to a given high-energy theory*. An important development is [18], where form factors are used to connect the low-energy bootstrap with the UV theory (see also [19–21]). This leads to one of the main points of our Letter, namely, that to identify the correct low-energy theory we have to introduce information from the microscopic or high-energy theory, which we do using the famous SVZ sum rules [22–27].

The naive physical intuition that motivated the gauge theory bootstrap and helps understand why the method works is the following. The tree-level effective field theory gives the dynamics below threshold. Above threshold, a simple unitarization provides reasonable predictions for the $S0$ and $S2$ waves [28]. The $P1$ wave containing the $\rho$ resonance does not follow this way [29]. In fact, the $\rho$ meson can be considered the signature of QCD and requires

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP³.</footnote>
specifying the correct underlying UV theory. It is known that the SVZ sum rules use gauge theory information to compute the $\rho$ mass assuming that the spectral density is dominated by a narrow resonance. Putting both things together provided a good initial step to identify the scattering phase shifts corresponding to the given gauge theory.

**Pion form factor bootstrap with finite energy sum rules**—Low-energy QCD [4] is described in terms of three pions with $2 \to 2$ scattering ($a, b, c, d = 1, 2, 3$)

$$ \pi_a(p_1) + \pi_b(p_2) \to \pi_c(p_3) + \pi_d(p_4). \quad (1) $$

Defining $s = (p_1 + p_2)^2$, $t = (p_1 - p_3)^2$, and $u = (p_1 - p_4)^2$, the scattering amplitude can be written as

$$ T_{ab,cd} = A(s,t,u)\delta_{ab}\delta_{cd} + A(t,s,u)\delta_{ac}\delta_{bd} + A(u,t,s)\delta_{ad}\delta_{bc} \quad (2) $$

with $A(s, t, u) = A(s, u, t)$ an analytic function of the Mandelstam variables $s$, $t$, and $u$. Setting $m_\pi = 1$, the analyticity and crossing symmetry can be encoded into the Mandelstam representation:

$$ A(s, t, u) = T_0 + \frac{1}{\pi} \int_4^\infty dx \frac{\sigma_1(x)}{x-s} + \frac{\sigma_2(x)}{x-t} + \frac{\sigma_2(x)}{x-u} \\ + \int_4^\infty \frac{dxdy}{\pi^2} \frac{\rho_1(x,y)}{(x-s)(y-t)} + \frac{\rho_1(x,y)}{(x-s)(y-u)} \\ + \frac{\rho_2(x,y)}{(x-t)(y-u)} \quad (3) $$

with $\rho_2(x, y) = \rho_2(y, x)$. Projecting onto channels of given isospin $I$ and angular momentum $\ell$, we get the $S$ matrices ($S_\ell^I$), partial waves ($f_\ell^I$), and phase shifts ($\delta_\ell^I$):

$$ S_{\ell}^{I}(s) = \eta_{\ell}^{I}(s) e^{2i\delta_{\ell}^{I}(s)} = 1 + i\pi \sqrt{\frac{s-4}{s}} f_{\ell}^{I}(s). \quad (4) $$

Unitarity implies $\eta_\ell^I(s) = |S_\ell^I(s)| \le 1$, $s > 4$. Also, $\ell$ is even for $I = 0, 2$ and odd for $I = 1$.

From chiral symmetry breaking [1–4], it follows that

$$ f_0^0(s) \approx \frac{2s-1}{16\pi^2 f_\pi^2}, \quad f_1^1(s) \approx \frac{s-4}{48\pi^2 f_\pi^2}, \quad f_0^2(s) \approx \frac{2-s}{16\pi^2 f_\pi^2} \quad (5) $$

in the unphysical region $0 < s < 4$.

Consider now a set of gauge theory currents $j_\ell^I(x)$ with isospin $I$ and angular momentum $\ell$. In this Letter, using the standard $u, d$ quark notation, we take

$$ j_0^0(x) = m_q(\bar{u}u + \bar{d}d), \quad (6a) $$

$$ j_{1,\mu}^{1}(x) = \frac{1}{2}(\bar{u}\gamma_{\mu}u - \bar{d}\gamma_{\mu}d). \quad (6b) $$

Using conservation laws and Lorentz symmetry, the corresponding pion form factors have the form

$$ \begin{aligned} \langle \pi^+(p_2) | j_0^0(x) | \pi^+(p_1) \rangle &= e^{i(p_1 - p_2) \cdot x} F_0(t), \\ \langle \pi^+(p_2) | j_{1,\mu}^1(x) | \pi^+(p_1) \rangle &= e^{i(p_1 - p_2) \cdot x} (p_{1\mu} + p_{2\mu}) F_1(t), \end{aligned} \quad (7) $$

where $t = (p_1 - p_2)^2$. Using crossing symmetry and states of fixed angular momentum and isospin, we find

$$ \langle I = \ell = 0, I_3 = \sigma = 0; P | j_0^0(0) | 0 \rangle = \mathcal{F}_0^0(s), \quad (8a) $$

$$ \langle I = \ell = 1, I_3 = \sigma = 0; P | j_{1,3}^1(0) | 0 \rangle = \mathcal{F}_1^1(s) \quad (8b) $$

with $P = p_1 + p_2$, $s = P^2$ and

$$ \mathcal{F}_0^0(s) = \frac{\sqrt{6\pi}}{16\pi^3} \frac{1}{s^{1/4}} \left(\frac{s-4}{4}\right)^{1/4} F_0(s), \quad (9a) $$

$$ \mathcal{F}_1^1(s) = \sqrt{\frac{4\pi}{3}} \frac{1}{8\pi^3} \frac{1}{s^{1/4}} \left(\frac{s-4}{4}\right)^{3/4} F_1(s). \quad (9b) $$

Now, using that $\int d^3x j_{1,0}^I(x) = \frac{1}{2}(n_u - n_d)$, we obtain

$$ F_1(0) = 1 \quad (10) $$

for $\pi^+$ with quark content $u\bar{d}$. For the scalar current [31,32],

$$ F_0(0) = m_q \frac{\partial m_{\pi}^2}{\partial m_q} \approx m_{\pi}^2 = 1. \quad (11) $$

Thus, given that the form factors are analytic functions of $s$ with a cut on the real axis for $s \ge 4$, we can write a subtracted dispersion relation

$$ F_{\ell}(s) = 1 + \frac{1}{\pi} \int_{4}^{\infty} dx \left( \frac{1}{x-s} - \frac{1}{x} \right) \mathrm{Im} F_{\ell}(x). \quad (12) $$

**Current correlators and bootstrap**—Finally, we introduce the current correlators ($s = P^2$)

$$ \Pi_0^0(s) = i \int \frac{d^4 x}{(2\pi)^4} e^{iPx} \langle 0 | \hat{T} \{ j_0^0(x) j_0^0(0) \} | 0 \rangle, \quad (13a) $$

$$ \Pi_1^1(s) = i \int \frac{d^4 x}{(2\pi)^4} e^{iPx} \langle 0 | \hat{T} \{ j_{1,-}^1(x) j_{1,+}^1(0) \} | 0 \rangle, \quad (13b) $$

analytic in $s$ with a cut for $s \in \mathbb{R}_{\ge 4}$ and discontinuity:

$$ \rho_{\ell}^{I}(s) = 2 \mathrm{Im} \Pi_{\ell}^{I}(x + i\epsilon) = \int \frac{d^4 x}{(2\pi)^4} e^{iPx} \langle 0 | j_{\ell}^{I\dagger}(x) j_{\ell}^{I}(0) | 0 \rangle. \quad (14) $$
In (13b), we define the components $j_{\pm} = (1/\sqrt{2})(j_1 \pm ij_2)$. Defining an operator

$$ \mathcal{O}_{P,I,\ell} = \int \frac{d^4 x}{(2\pi)^4} e^{-iPx} j_{\ell}(x), \qquad (15) $$

we find positive semidefinite matrix of overlaps

$$ \begin{array}{ccc} |\text{out}\rangle_{P,I,\ell} & |\text{in}\rangle_{P,I,\ell} & \mathcal{O}_{P,I,\ell}|0\rangle \\ \langle\text{out}|_{P',I,\ell} & \left( \begin{array}{ccc} 1 & S_{\ell}^{I}(s) & \mathcal{F}_{\ell}^{I} \\ S_{\ell}^{I*}(s) & 1 & \mathcal{F}_{\ell}^{I*} \\ \mathcal{F}_{\ell}^{I*} & \mathcal{F}_{\ell}^{I} & \rho_{\ell}^{I}(s) \end{array} \right) & \ge 0, \\ \langle\text{in}|_{P',I,\ell} & & \\ \langle 0|\mathcal{O}_{P',I,\ell}^{\dagger} & & \end{array} \qquad (16) $$

where we removed an overall factor $\delta^4(P - P')$. This positivity condition defines the form factor bootstrap [18]. The spectral density $\rho_{\ell}^I(s)$ appearing in the matrix is the same as the discontinuity defined in (14) and, for $s \gg \Lambda_{QCD}$, follows from the QCD correlators [22,33]

$$ \begin{aligned} \Pi_0^0(s) &\approx \frac{m_u^2 + m_d^2}{(2\pi)^4} \left\{ -\frac{3}{8\pi^2} \left( 1 + \frac{13}{3} \frac{\alpha_s}{\pi} \right) s \ln \left( -\frac{s}{\mu^2} \right) \right\}, \\ \Pi_1^1(s) &\approx \frac{1}{2} \frac{1}{(2\pi)^4} \left\{ -\frac{1}{4\pi^2} \left( 1 + \frac{\alpha_s}{\pi} \right) s \ln \left( -\frac{s}{\mu^2} \right) \right\}, \end{aligned} \qquad (17) $$

where we ignored the small contributions from the quark and gluon condensates. The finite energy SVZ sum rules [34] impose that the integrals of the analytic functions $s^n \Pi_{\ell}^I(s)$ along a contour that goes around the cut $s > 4$ and closes on a large circle of given radius $s_0$ vanish:

$$ \begin{aligned} \int_4^{s_0} \rho_0^0(x) \frac{x^n}{s_0^{n+2}} dx &= \frac{3(m_u^2 + m_d^2)}{2^6 \pi^5 (n+2)} \left(1 + \frac{13 \alpha_s}{3 \pi}\right), & n \ge 0, \\ \int_4^{s_0} \rho_1^1(x) \frac{x^n}{s_0^{n+2}} dx &= \frac{1}{2^6 \pi^5 (n+2)} \left(1 + \frac{\alpha_s}{\pi}\right), & n \ge -1, \end{aligned} \qquad (18) $$

where the left-hand side is the contribution from the cut and the right-hand side from the circle. Also, $n \ge -1$ on the second line of (18), since $\Pi_1^1(s)$ has a zero at $s = 0$.

Lastly, we need the high-energy behavior of the form factor from QCD computed, for example, in [35]:

$$ F_1(s) \approx -\frac{16\pi\alpha_s(s)f_{\pi}^2}{s}. \qquad (19) $$

The scalar form factor $F_0(s)$ should behave similarly with the ratio between form factors given by the respective charges ($|F_0|/|F_0| \sim 2m_q$).

**Results**—The numerical inputs for the calculation are $\alpha_s(s_0 = 1.2 \text{ GeV}) \approx 0.4$, the pion mass $m_\pi \approx 140 \text{ MeV}$, the pion decay constant $f_\pi \approx 92 \text{ MeV}$, and the quark masses $m_u \approx 4 \text{ MeV}$ and $m_d \approx 7.3 \text{ MeV}$ using data from, e.g., [34].

<figure><img src="image_4.png" /><figcaption>FIG. 1. Allowed space of S matrices restricted by the chiral constraints (blue) dots and the sum rules (red). The phase shifts corresponding to the highlighted dots are depicted in Fig. 2.</figcaption></figure>

We parametrize the space of allowed S matrices by the values of the partial waves $f_1^1(s = 3)$ and $f_0^0(s = 3)$ at the unphysical point $s = 3$ where they are real. This plane is depicted in Fig. 1. The line is the approximation (5) with arbitrary $f_\pi$, with the black dot corresponding to $f_\pi \approx 92 \text{ MeV}$.

Now we impose the constraints of analyticity, unitarity, and crossing and require that $f_1^1(s_j)$ and $f_0^0(s_j)$ agree with (5) at the four points $s_j = 1/2, 1, 3/2, 2$ within a certain tolerance. See the companion paper [36] for details of the numerical implementation. The allowed values of the partial waves at $s = 3$ are contained in the region bounded by the blue points in Fig. 1. Each of those points corresponds to a particular extremal amplitude, and, remarkably, for points close to the black dot, such as the highlighted blue point, the phase shifts of the S0 ($I = \ell = 0$) and S2 ($I = 2, \ell = 0$) waves agree quite well with experiment as seen in Fig. 2 (blue dots). Notice that the experimental S0 wave has a bump at $\sim 1 \text{ GeV}$ due to kaon production that we do not have since we do not include the s quark. However, the P1 ($I = \ell = 1$) wave does not display the $\rho$-meson resonance. Now we impose the sum rules (18) and asymptotic form of the form factors (19), thus incorporating gauge theory information. This shrinks the allowed region in Fig. 1 to the one bounded by the red dots. Remarkably, the extremal amplitudes corresponding to the highlighted red points, close to the black dot, now display the $\rho$ meson resonance in the P1 wave; in fact, all three phase shifts agree quite well with experiment. These are depicted again in Fig. 2, where the phase shifts corresponding to three different points on the boundary of the region are displayed.

Thus, in the gauge theory bootstrap, including gauge theory information is crucial in order to identify the correct low-energy theory and, in particular, to find the $\rho$ resonance. This agrees with phenomenological ideas [42] that indicate the $\rho$ is a consequence of QCD and not just chiral dynamics.
<figure><img src="image_5.png" /><figcaption>FIG. 2. The phase shifts as a function of energy: experiment [37–40] (gray dots), phenomenological fit [41] (gray line), and bootstrap result (see the text).</figcaption></figure>

In summary, we presented a method to determine the low-energy dynamics of QCD, including resonances. It borrows from old (SVZ sum rules) and new (form factor bootstrap) techniques and uses only a few gauge theory parameters ($m_q$, $\Lambda_{QCD}$) and the low-energy parameters ($m_\pi$, $f_\pi$) to determine the strongly coupled pion dynamics.

The phase shifts of pion scattering thus obtained in the S0, P1, and S2 channels are in good agreement with experiment. Other asymptotically free gauge theories with different number of flavors and gauge groups can be treated with the same procedure.

*Acknowledgments*—M. K. thanks LPENS and Shanghai Tech, and we both thank Swissmap Les Diablerets Research Station for hospitality. We are grateful to J. Penedones, B. Van Rees, P. Vieira, and especially to A. Guerrieri for various discussions and L. Cordova, K. Häring, M. Paulos, S. Rychkov, A. Shapere, and A. Zhiboedov for comments. This work was supported in part by DOE through Grant No. DE-SC0007884 and the QuantiSED Fermilab consortium.

<footnote>*Contact author: yifei.he@ens.fr</footnote>

<footnote>†Contact author: markru@purdue.edu</footnote>

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