Precise top-quark mass from the diphoton mass spectrum

We calculate the $gg\to\gamma\gamma$ amplitude by including the $t\bar t$ bound-state effects near their mass threshold. In terms of the non-relativistic expansion of the amplitude, the LO contribution is an energy-independent term in the one-loop amplitude. We include the NLO contribution described by the non-relativistic Green function and part of the NNLO contribution. Despite a missing NLO piece which can be accomplished with the two-loop-level amplitude via massive quarks, the shape of the diphoton mass spectrum is predicted with a good accuracy. Thanks to the simple and clean nature of the observable, its experimental measurement can be a direct method to determine the short-distance mass of the top quark at hadron colliders.

The threshold structure of the massive-quark-loop amplitude deserves particular interests [21,26,27] where the massive quark is regarded as the top quark or a hypothetical particle beyond the SM. Beyond the one-loop level, the amplitude receives large QCD corrections due to the Coulomb-gluon exchanges between the nearly on-shell and low-velocity heavy quarks in s-channel. Thus, the description of the amplitude requires an elaborate treatment based on the non-relativistic QCD formalism. For gg → γγ process, such a study cannot be found in the literature. The aim of this paper is to compile the present knowledge of the non-relativistic QCD theory for the description of the bound-state effects in the massive-quark-loop amplitude, and to present a dedicated and quantitative study on the diphoton mass spectrum near the tt threshold. Our framework follows the preceding studies on h → γγ [28,29], and some of our numerical results overlap with that in Ref. [26].
We discuss further to utilize the predicted mass spectrum for a precise determination of the top-quark mass, which is one of the fundamental parameters in the SM. Although the top-quark mass has been measured with an error of sub-GeV level [30], its interpretation in terms of well-defined mass parameters is not settled yet in perturbative QCD. It is known that the well-defined mass parameters can be determined by using the threshold scan method at future e + e − colliders [31][32][33]. We show that the diphoton mass spectrum measurement can be a considerable alternative to it at hadron colliders. The application of the formula for physics beyond the SM will be reported elsewhere.

II. SCATTERING AMPLITUDE IN THE THRESHOLD LIMIT
We start the main content of the paper by introducing the scattering amplitude for gg → γγ at the one-loop level with the top quark, and provide an easy-to-use expression for its threshold behavior. By using the all-outgoing convention for the momenta (p i ) and where M q is the contribution from the massless-quark loop with five flavors (n f = 5), and M t from the top-quark loop with the top-quark pole-mass, m t . The amplitude for the top-quark loop near the threshold is expressed as i is the tt Green function in S-wave without QCD effects. The first term which is energyindependent, represents the contribution from the hard-momentum integral. The second term which is O(v), represents the contribution from the soft-momentum loop where the top-quarks can be on-shell. At the one-loop level, all the imaginary part of the amplitude originates from G (0) above the threshold, E ≥ 0. A t depends on the scattering angle θ, while B t is independent of θ because only the spin-singlet tt state contributes at this order. For For the other combinations of the helicity, B t as well as A t can be written in terms of them.
For a description of A t , we make use of the partial-wave decomposition with numerical coefficients. The A t term is expanded as where µ = −λ 1 + λ 2 and µ = λ 3 − λ 4 . Because B t is constant, B t has only the J = 0 In Table I, we list the numerical values of A J t for J up to 4. The Wigner d-functions d J µµ can be found in the literature. We find that the expansion up to J = 4 gives a sufficiently good approximation.

III. THRESHOLD EFFECTS
We incorporate the tt threshold effects into the Green function by evaluating it with the QCD potential [34,35]. The amplitude with the threshold effects is expressed as [28] where we define A J>0 t and E = E + iΓ t with the top-quark decay width, Γ t . The Green function is defined by the following Schrödinger equation: where V (r) is the QCD potential. For the tt system, we can utilize the perturbativelycalculated potential. The real part of the Green function at r = 0 is known to be divergent, thus has to be renormalized. We adopt the MS renormalization scheme in dimensional regularization [36][37][38]. An artificial scale µ is introduced to the renormalized Green function.
By matching with the one-loop amplitude, the amplitude is finally expressed as Before moving to the numerical evaluation, we discuss the order of the corrections in the non-relativistic QCD formalism. Taking v and α s as the expansion parameters, the leading-order contribution is the A J=0 t term which is constant, and the B t G( 0; E) term is at the next-to-leading order (NLO). There is another NLO term in the two-loop amplitude, which is an O(α s ) correction to A t . However, this has not been calculated yet for the massive-quark contribution. Indeed, this term is required for the consistent calculation of the threshold corrections up to NLO in order that the scale dependence of the real part of the Green function is canceled with the O(α s ) term of A t [29]. In our calculation, we do not include the O(α s ) A t -term, thus the scale dependence remains in the threshold amplitude.
We treat it as an uncertainty of our calculation.
Since the leading contribution to the squared amplitude is the absolute square of the sum of M q and the A t term where both are independent of energy, the uncertainty of A t term mainly affects the overall normalization of the diphoton mass spectrum. On the other hand, some of the NNLO corrections improve the description of the tt resonances.
Therefore, for the sake of a precise and stable prediction of the resonance structure, it is worthwhile to include the available NNLO corrections even though we cannot reach the full NLO accuracy. The known corrections are (1) the NLO correction to the Green function, (3) the O(α s ) correction to Γ t . First, the NLO correction to the Green function is incorporated by solving the Schrödinger equation with the NLO QCD potential [39,40] given by where β 0 = 11/3 · C A − 2/3 · n f and a 1 = 31/9 · C A − 10/9 · n f with C F = 4/3 and C A = 3.
We will show later that evaluating the Green function beyond LO is crucial for the reliable prediction. Second, the O(α s ) correction to B t can be derived from the O(α s ) hard-vertex corrections to the gg → tt and tt → γγ processes. The hard-vertex factor to the gg → tt cross-section in the color-singlet channel reads 1 + (α s /π)h 1 with [41][42][43] where µ R is the renormalization scale of α s . The corresponding factor for tt → γγ reads only the first term of Eq. (8). By using them, B t with the O(α s ) correction is given as Finally, the O(α s ) correction to Γ t has been calculated in Refs. [44][45][46]. However, we treat Γ t as an input parameter in our study. Identification and derivation of the remaining NNLO corrections are beyond the scope of this paper.

IV. NUMERICAL RESULTS
We present numerical studies for the gg → γγ amplitude as well as the cross sections at the LHC. In Fig. 1, we plot dσ/dm γγ via gg → γγ for the LHC 13 TeV with kinematical cuts of |η γ | < 2.5 and p γ T > 40 GeV [8]. Both the massive-and massless-quark loops are included. We use the CT14NLO gluon distribution function [47], and take the renormalization and factorization scales as µ R = µ F = m γγ . The Green function is evaluated by numerically solving Eq. (5) with the LO or NLO QCD potential following the method described in Ref. [48]. The scale of α s in the QCD potential is taken as the same as the renormalization scale µ of the Green function, which we vary from 20 GeV to 160 GeV. The result with the one-loop amplitude is also plotted for comparison. In the plots, we observe that the distributions show a characteristic structure near m γγ 2m t = 346 GeV; it shows a dip and then a small bump below the threshold [26]. We find that, if we employ the LO Green function, the shape of the distribution changes by the scale choice. In contrast, by with the one-loop-level M J=0 t,++++ is drawn in the black line. Below the threshold, the two amplitudes, M q and M t , are pure real, and their relative sign is negative. Therefore, there is a destructive interference, and the total amplitude goes toward the origin by increasing m γγ until the threshold. Above the threshold, the amplitude gains an imaginary part and the real part tends to increase along with m γγ . At the high-energy limit, where the top quark can be assumed to be massless, the imaginary part goes to zero and the total amplitude   [53,54] in non-relativistic QCD, and it is well-known that the prediction becomes significantly accurate when it is expressed in terms of the short-distance mass to cancel the renormalon ambiguity. By using the O(α 5 s m t ) formula for the spin-singlet case [53,54] and the MS mass with m t = 163 GeV, we also plot the 1S energy-level, M (n) 1S , at N n LO up to n = 3 in Fig. 3. It can be seen that the convergency is good, and the scale uncertainty is reduced to around 100 MeV or below. 2 By combining these arguments, the dip and bump positions can be accurately predicted by including higher-order corrections with the short-distance mass. More detailed studies will be presented in a future publication. any m t . The setup for the gluon distribution function and acceptance cuts is same as that for Fig. 1. An additional cut p γ T > 0.4m γγ is applied for the FCC case which enhances the selection efficiency of the J = 0 partial-wave contribution. One can clearly see in Fig. 4 that the bump position shifts in proportion to m t . Consequently, we can extract m t from the diphoton mass spectrum. Since a photon is a clean object and not directly affected by finalstate QCD interactions, this measurement would be quite transparent experimentally and theoretically. Especially, systematic errors of photon momentum reconstruction are much smaller than those of jet momentum which are the major source of the systematic error in the current m t measurement. These virtues are shared with leptonic-observable methods proposed in Refs. [59][60][61].
In order to estimate the sensitivity of the method, we perform pseudo-experiments assuming the LHC 13 TeV with 3 ab −1 data, and the FCC 100 TeV with 1 ab −1 and 10 ab −1 data.
We prepare event samples of the signal gg → γγ events and the background events by other The background events are generated by Diphox [15] at LO with qq → γγ, one-direct-onefragmentation, and two-fragmentation contributions. The total number of events is fixed by using the observed data to take into account detector efficiency and a K-factor from higher-order corrections. We read off a corresponding correction factor of C 1.2 from the LHC 13 TeV diphoton analysis by the ATLAS Collaboration [10]. For simplicity we apply the same C for the FCC case. The signal-to-background ratio, which is crucial to the mass sensitivity, is subject to theoretical uncertainties of the cross-section calculations, such as the choice of scales µ R , µ F and µ, uncalculated higher-order corrections, and also a definition of isolated photons [62]. Based on the LO calculations for both the signal and background processes, the ratio is estimated to be 10% at the LHC 13 TeV and 30% at the FCC 100 TeV. On the other hand, the ratio is estimated to be 5% at the LHC where the QCD NNLO corrections are included in the background calculation [12,14], while 10% at the FCC where the QCD NLO corrections are included in the background calculation [58].
We note that a recent study in Ref. [14] indicates that the ratios become closer to the LO estimates when the NLO corrections are included additionally to the signal process. Considering these estimations, we take the ratio to be 5% to 10% at the LHC, while 10% to 30% at the FCC in this study.
The sample m γγ distributions are fitted with the sum of the signal prediction which depends on m t plus an analytic smooth function for the background, taking the signal-tobackground ratio as a fitted parameter. The background function is taken as where x = m γγ / √ s and a is a parameter to be fitted. Notice that our fitting procedure does not rely on the value of the signal-to-background ratio nor the accurate prediction of the background shape. We perform least-squares fits to the binned m γγ distribution in the interval [300, 400] GeV with the bin width of 1 GeV. By repeating the pseudo-experiment, we obtain the expected statistical error ∆m t from the distribution of the fitted m t . For the LHC 3 ab −1 , the obtained m t distribution is not Gaussian, while it has a peak at m t = m true t .
We approximate the distribution as Gaussian and obtain ∆m t 2 GeV to 3 GeV for the signal ratio 10% to 5%. For the FCC, by assuming the signal-to-background ratio to be 30%, the distribution behaves as Gaussian and we obtain ∆m t = 0.2 GeV (0.06 GeV) for 1 ab −1 (10 ab −1 ). When the ratio is assumed to be 10%, we obtain ∆m t = 0.6 GeV (0.2 GeV) for 1 ab −1 (10 ab −1 ). We find that the correlation between two fitted parameters, m t and the signal-to-background ratio, is weak.
Before closing, we present several comments. The systematic error of photon energy scale is about 0.5% [63] in the ATLAS detector and about 0.3% [64] in the CMS detector. Thus we naively expect the systematic error of δm sys. well-defined masses, the top-quark MS mass can be extracted directly from the diphoton mass spectrum. Measuring the short-distance mass from the resonance structure is conceptually equivalent with the threshold scan method in e + e − → tt. In the e + e − case, the threshold production cross-section is established up to N 3 LO in non-relativistic QCD [65,66]. In the diphoton case at hadron colliders, only the one-loop gg → γγ amplitude has been known, and thus the NLO calculation has not been completed yet. To complete, one requires the two-loop gg → γγ, one-loop gg → γγg and gq → γγq amplitudes. In the one-gluon emission processes, corrections via an initial-state gluon emission, color-octet tt effects, and an ultrasoft gluon emission from the on-shell tt state appear. These corrections can be sizable because of the large partonic luminosity of the color-octet gluons. Investigations of these effects are left for future works. However, we expect that these would not severely spoil the characteristic shape of the spectrum in the resonance region, because the initial-state radiation does not affect the bound-state formation and the color-octet tt Green function is known to have a smooth slope in the resonance region. Finally, it might be possible to determine Γ t simultaneously with m t at the FCC.

VI. CONCLUSIONS
To conclude, we have studied the gg → γγ amplitude with the tt bound-state effects near their mass threshold by collecting the available higher-order corrections in non-relativistic QCD. We have predicted a characteristic structure in the diphoton mass spectrum near the threshold whose shape is stable under the scale uncertainty, while the overall normalization has an uncertainty of 10% level due to the lack of the two-loop amplitude. We have proposed a new method to determine m t from the diphoton mass spectrum at the LHC and the FCC.
We have shown that the estimated statistical errors are fairly small at the FCC, which deserves further realistic experimental studies and also motivates to calculate higher-order corrections in theory.