Topquark mass from the diphoton mass spectrum
Eur. Phys. J. C
Topquark mass from the diphoton mass spectrum
Sayaka Kawabata 1
Hiroshi Yokoya
0 Quantum Universe Center , KIAS, Seoul 02455 , Korea
1 Institute of Convergence Fundamental Studies, Seoul National University of Science and Technology , Seoul 01811 , Korea
We calculate the gg → γ γ amplitude by including the t t¯ boundstate effects near their mass threshold. In terms of the nonrelativistic expansion of the amplitude, the LO contribution is an energyindependent term in the oneloop amplitude. We include the NLO contribution described by the nonrelativistic Green function and part of the NNLO contribution. Despite a missing NLO piece which can be accomplished with the twolooplevel 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 shortdistance mass of the top quark at hadron colliders. At the LHC, a diphoton mass spectrum dσ/dmγ γ has attracted broad attention for observations of the properties of the Higgs boson in the standard model (SM) [15] and searches for new phenomena beyond the SM [611]. At hadron colliders, pairs of high pT photon are produced by qq¯ annihilation and gluonfusion mechanisms [1214], and processes which involve fragmentationphoton contributions [15]. The gg → γ γ process, which is the main focus of this letter, is described by loop diagrams with quarks in the SM. The analytic expression of the oneloop amplitude has been known for a long time for both the massless and the massivequark loops [1622]. The twoloop amplitude has been calculated only for the masslessquark loops [2325]. The threshold structure of the massivequarkloop 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 oneloop level, the amplitude receives large QCD corrections due to the Coulombgluon exchanges between the nearly onshell and lowvelocity heavy quarks in schannel. Thus, the description of the amplitude requires an elaborate treatment based on the nonrelativistic QCD formalism. For gg → γ γ process,

such a study cannot be found in the literature. The aim of this
letter is to compile present knowledge of the nonrelativistic
QCD theory for the description of the boundstate effects in
the massivequarkloop amplitude, and to present a dedicated
and quantitative study on the diphoton mass spectrum near
the t t¯ threshold. Our framework follows the preceding
studies on h → γ γ [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 topquark mass, which is
one of the fundamental parameters in the SM. Although the
topquark mass has been measured with an error of subGeV
level [30], its interpretation in terms of welldefined mass
parameters is not settled yet in perturbative QCD. It is known
that the welldefined mass parameters can be determined by
using the threshold scan method at future e+e− colliders [31–
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.
We start by introducing the scattering amplitude for
gg → γ γ at the oneloop level with the top quark, and
provide an easytouse expression for its threshold
behavior. By using the alloutgoing convention for the momenta
( pi ) and helicities (λi ), ga1 (− p1, −λ1) + ga2 (− p2, −λ2) →
γ ( p3, λ3) + γ ( p4, λ4), the oneloop amplitude is written as
Mgg→γ γ ({ pi }; {λi }; a1, a2) = 4ααs δa1a2
j=1
where Mq is the contribution from the masslessquark loop
with five flavors (n f = 5), and Mt from the topquark loop
with the topquark polemass, mt . The amplitude for the
topquark loop near the threshold is expressed as
Mt,{λi } = At,{λi }(θ ) + Bt,{λi }G(0)(0; E ) + O(v2),
mt v2 and v =
G(0)(0; E ) ≡ −mt2/(4π )√−E /mt − i is the t t¯ Green
function in Swave without QCD effects. The first term which
is energyindependent, represents the contribution from the
hardmomentum integral. The second term which is O(v),
represents the contribution from the softmomentum loop
where the topquarks can be onshell. At the oneloop level,
all the imaginary part of the amplitude originates from G(0)
above the threshold, E ≥ 0. At depends on the
scattering angle θ , while Bt is independent of θ because only
the spinsinglet t t¯ state contributes at this order. For {λi }
= λ1λ2λ3λ4, we find Bt,++++ = −Bt,−−++ = −4π 2/mt2,
while Bt,−+++ = Bt,−+−+ = 0. For the other
combinations of the helicity, Bt as well as At can be written in
terms of them. For a description of At , we make use of the
partialwave decomposition with numerical coefficients. The
At term is expanded as
J =0
where μ = −λ1 + λ2 and μ = λ3 − λ4. Because Bt is
constant, Bt has only the J = 0 component, Bt = BtJ =0. In
Table 1, we list the numerical values of AtJ for J up to 4.
The Wigner dfunctions dμJμ can be found in the literature.
We find that the expansion up to J = 4 gives a sufficiently
good approximation.
We incorporate the t t¯ 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]
Mtt,h{rλi } = At,{λi } + Bt,{λi }G(0; E ) + At,{λi }
J =0 J >0 (θ ),
where we define AtJ >0(θ ) = At (θ )−AtJ =0 and E = E +i t
with the topquark decay width, t . The Green function is
defined by the following Schrödinger equation:
− ∇mt + V (r )
− E
where V (r ) is the QCD potential. For the t t¯ system, we can
utilize the perturbatively calculated potential. The real part
where α¯ s = αs (μB ), β0 = 11/3 · C A − 2/3 · n f and a1 =
31/9 · C A − 10/9 · n f with CF = 4/3 and C A = 3. We
AtJ=0
AtJ=1
AtJ=2
AtJ=3
−0.00060737
AtJ=4
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 [36–38]. An artificial
scale μ is introduced to the renormalized Green function.
By matching with the oneloop amplitude, the amplitude is
finally expressed as
Mtm,{aλtic}h = Mt,{λi } + Bt,{λi }[G(0; E ) − G(0)(0; E )].
Before moving to the numerical evaluation, we discuss
the order of the corrections in the nonrelativistic QCD
formalism. Taking v and αs as the expansion parameters, the
leadingorder contribution is the AtJ =0 term which is
constant, and the Bt G(0; E ) term is at the nexttoleading order
(NLO). There is another NLO term in the twoloop
amplitude, which is an O(αs ) correction to At . However, this has
not been calculated yet for the massivequark 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(αs ) term of At [29]. In our calculation, we do
not include the O(αs ) At 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 Mq and the At term where
both are independent of energy, the uncertainty of At 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 t t¯ 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, (2) the O(αs ) correction to Bt , and (3)
the O(αs ) correction to 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
will show later that evaluating the Green function beyond
LO is crucial for a reliable prediction. Second, the O(αs )
correction to Bt can be derived from the O(αs ) hardvertex
corrections to the gg → t t¯ and t t¯ → γ γ processes. The
hardvertex factor to the gg → t t¯ cross section in the
colorsinglet channel reads 1 + (αs /π )h1 with [41–43]
h1 = C F
−5 + 4
where μR is the renormalization scale of αs . The
corresponding factor for t t¯ → γ γ reads only the first term of Eq. (8).
By using them, Bt with the O(αs ) correction is given as
Bt = Bt(0)[1 + (αs /π )b1] with
b1 = C F
−5 + 4
Finally, the O(αs ) correction to t has been calculated in
Refs. [44–46]. However, we treat t as an input parameter
in our study. Identification and derivation of the remaining
NNLO corrections are beyond the scope of this letter.
We present numerical studies for the gg → γ γ amplitude
as well as the cross sections at the LHC. In Fig. 1, we plot
the gg → γ γ contribution to the hadronic differential cross
section, dσ/dmγ γ , for the LHC 13 TeV with kinematical cuts
of ηγ  < 2.5 and pTγ > 40 GeV [8]. Both the massive and
the masslessquark loops are included. We use the CT14NLO
gluon distribution function [47], and take the renormalization
and factorization scales as μR = μF = mγ γ . 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 αs in the QCD potential is taken as the
same as the renormalization scale μ of the Green function,
which we vary from 20 to 160 GeV. The result with the
oneloop amplitude is also plotted for comparison. In the plots, we
observe that the distributions show a characteristic structure
near mγ γ 2mt = 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 using the NLO Green
function the shape of the distribution is quite stable apart
from the overall normalization. The positions of the dip and
the bump are shifted by the choice of μ by around 0.6 GeV. A
relatively large uncertainty appears as the overall size of the
cross section, which amounts to about 10%. This uncertainty
originates mainly from the lack of the O(αs ) correction in
the At term. We note that there exists another source of the
uncertainty for the overall normalization, which is the scale
choice of μR and μF . For the LHC 13 TeV, changing these
]
eV 0.24
G
/
f[b 0.22
γ
γ
m 0.2
d
/
dσ 0.18
]
eV 0.24
G
/
[fb 0.22
γ
γ
/dm 0.2
dσ 0.18
355360
Re[MJ+=+0++]
Fig. 2 gg → γ γ amplitudes for {λi } = + + ++ in the J = 0
channel for mγ γ = 330–360 GeV with points in a 5GeV step. For the
illustrative convenience, we set mt = 172.5 GeV in this plot
scales from mγ γ /2 to 2mγ γ varies the cross section by about
20%.
For a better understanding of the behavior of the cross
section, we plot in Fig. 2 the gg → γ γ amplitudes in a complex
plane for {λi } = + + ++ in the J = 0 channel, by varying
Fig. 1 gg → γ γ contribution to the diphoton invariantmass
distribution near the t t¯ threshold at the LHC 13 TeV. Top panel is for the LO
Green function, and bottom panel is for the NLO Green function
M = 11/9 Mq + 4/9 Mt
mγ γ from 330 to 360 GeV. The masslessquarkloop
amplitude gives a constant contribution, 11/9 · MqJ,=+0+++ = 11/9.
The total amplitude M J =0
++++ = 11/9 · MqJ,=+0+++ + 4/9 ·
MtJ,+=+0++ with the onelooplevel MtJ,+=+0++ is drawn in the
black line. Below the threshold, the two amplitudes, Mq
and Mt , 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 mγ γ until
the threshold. Above the threshold, the amplitude gains an
imaginary part and the real part tends to increase along with
mγ γ . At the highenergy limit, where the top quark can be
assumed to be massless, the imaginary part goes to zero and
the total amplitude arrives at M J =0
++++ = 15/9.1 The
amplitude with the threshold corrections calculated with the NLO
(LO) Green functions are plotted in colored solid (dotted)
lines for μ = 40, 80 and 160 GeV. The imaginary part of
the amplitude is nonzero even below the threshold, which
comes from the finiteness of t . The size of the imaginary
part increases rapidly above mγ γ = 340 GeV with showing
a resonancelike curve just below the threshold. The scale
dependence of the Green function originates from the two
sources, one in the QCD potential and the other from the
realpart renormalization. For the NLO Green function, the former
is well suppressed and the latter affects only the real part of
the amplitude by a constant for any mγ γ . For the LO Green
function, both effects are large and the amplitude shows a
complicated scale dependence. Especially, there remains a
scale dependence in the imaginary part of the amplitude.
This explains the reason that the shape of the invariantmass
distribution is stable by using the NLO Green function in
contrast to the LO Green function. Although the uncertainty
in the real part of the amplitude is significant, it leads only the
10% level uncertainty to the cross section, due to the
presence of the large imaginary part and the lightquarkloop
contribution.
In Fig. 3, we show the scale dependence of the dip and
bump positions, Mdip and Mbump, respectively, in the
diphoton mass spectrum at the LHC evaluated with the NLO
Green function. In addition, we plot the 1S energy level
of the t t¯ boundstate (toponium) at NLO [O(αs3mt )], M1(1S),
which is in good approximation the resonance peak
position in the NLO Green function. We find the scale
variation of the Green function affects the difference of the two
mass scales, Mdip and M1S (and also, Mbump and M1S ), by
only around 20 MeV (40 MeV). This indicates that the
connection of the dip (bump) position and the 1S resonance
mass is sufficiently solid under uncertainties of the Green
function. The toponium energy levels have been calculated
up to O(αs5mt ) [53, 54] in nonrelativistic QCD, and it is
well known that the prediction becomes significantly
accu]
eV 344
G
[
M
343.5
Fig. 3 Scale dependence of the dip and bump positions in the diphoton
mass spectrum at the LHC evaluated with the NLO Green function,
and the energy level of the 1S toponium evaluated in the polemass
scheme at NLO as well as those in the MSmass scheme up to N3LO.
mtpole = 173 GeV or mt = 163 GeV is used
rate when it is expressed in terms of the shortdistance mass
to cancel the renormalon ambiguity. By using the O(αs5mt )
formula for the spinsinglet case [53, 54] and the MS mass
with mt = 163 GeV, we also plot the 1S energy level, M (1nS),
at Nn 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
higherorder corrections with the shortdistance mass. More
detailed studies will be presented in a future publication.
We propose to use the diphoton mass spectrum near the t t¯
threshold for a precise determination of the topquark mass
in hadroncollider experiments. Figure 4 shows the diphoton
mass spectra via gg → γ γ with different values of mt for
the LHC at √s = 13 TeV (top panel) and the proposed future
circular collider (FCC) at √s = 100 TeV [56–58] (bottom
panel). We utilize the NLO Green function with μ = 40 GeV.
t = 1.498 GeV is fixed for any mt . The setup for the gluon
distribution function and acceptance cuts is the same as that
for Fig. 1. An additional cut pTγ > 0.4mγ γ is applied for
the FCC case which enhances the selection efficiency of the
J = 0 partialwave contribution. One can clearly see in Fig. 4
that the bump position shifts in proportion to mt .
Consequently, we can extract mt 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 mt
mea
1 Interference effects with schannel resonant diagrams have been stud
ied in Refs. [13,21,49–52].
2 In Ref. [55], the scale variation is examined for the range from 80 to
320 GeV, and the uncertainty is claimed to be about 40 MeV.
0
300 310 320 330 340 350 360 370 380 390 400
mγγ [GeV]
Fig. 4 gg → γ γ contribution to the diphoton invariantmass
distribution for different mt . Top panel is for the LHC 13 TeV and bottom
panel is for the FCC 100 TeV
surement. These virtues are shared with leptonicobservable
methods proposed in Refs. [59–61].
In order to estimate the sensitivity of the method, we
perform pseudoexperiments 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 → γ γ
events and the background events by other sources for the
range mγ γ = [300, 400] GeV with applying the above
acceptance cuts. The signal events are generated based on
the predicted distribution assuming mttrue = 173 GeV. The
background events are generated by Diphox [15] at LO
with qq¯ → γ γ , onedirect–onefragmentation, and
twofragmentation contributions. The total number of events is
fixed by using the observed data to take into account detector
efficiency and a K factor from higherorder 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 signaltobackground ratio, which is crucial to the
mass sensitivity, is subject to theoretical uncertainties of the
crosssection calculations, such as the choice of scales μR ,
μF and μ, noncalculated higherorder corrections, and also
a definition of isolated photons [62]. Based on the LO
calculations for both the signal and the 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 5–10% at the LHC, while 10–30% at the
FCC in this study.
The sample mγ γ distributions are fitted with the sum of
the signal prediction which depends on mt plus an analytic
smooth function for the background, taking the
signaltobackground ratio as a fitted parameter. The background
function is taken as (1 − x 1/3)a where x = mγ γ /√s and a is a
parameter to be fitted. Notice that our fitting procedure does
not rely on the value of the signaltobackground ratio nor
the accurate prediction of the background shape. We
perform leastsquares fits to the binned mγ γ distribution in the
interval [300, 400] GeV with the bin width of 1 GeV. By
repeating the pseudoexperiment, we obtain the expected
statistical error mt from the distribution of the fitted mt . For
the LHC 3 ab−1, the obtained mt distribution is not
Gaussian, while it has a peak at mt = mtrue. We approximate the
t
distribution as Gaussian and obtain mt 2 to 3 GeV for
the signal ratio 10–5%. For the FCC, by assuming the
signaltobackground ratio to be 30%, the distribution behaves as
Gaussian and we obtain mt = 0.2 GeV (0.06 GeV) for
1 ab−1 (10 ab−1). When the ratio is assumed to be 10%, we
obtain mt = 0.6 GeV (0.2 GeV) for 1 ab−1 (10 ab−1). We
find that the correlation between two fitted parameters, mt
and the signaltobackground 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
δmtsys. 1 GeV at the future LHC measurement. For more
realistic estimation at the LHC as well as at the FCC,
simulation studies with detailed detector performance are required.
Beyond the oneloop level, the mass renormalization scheme
becomes explicit. With the signal distribution expressed in
terms of theoretically welldefined masses, the topquark
MS mass can be extracted directly from the diphoton mass
spectrum. Measuring the shortdistance mass from the
resonance structure is conceptually equivalent with the threshold
scan method in e+e− → t t¯. In the e+e− case, the
threshold production cross section is established up to N3LO in
nonrelativistic QCD [65,66]. In the diphoton case at hadron
colliders, only the oneloop gg → γ γ amplitude has been
known, and thus the NLO calculation has not been
completed yet. To complete, one requires the twoloop gg → γ γ ,
oneloop gg → γ γ g and gq → γ γ q amplitudes. In the
onegluon emission processes, corrections via an initialstate
gluon emission, coloroctet t t¯ effects, and an ultrasoft gluon
emission from the onshell t t¯ state appear. These corrections
can be sizable because of the large partonic luminosity of
the coloroctet gluons. Investigations of these effects are left
for future work. However, we expect that these would not
severely spoil the characteristic shape of the spectrum in the
resonance region, because the initialstate radiation does not
affect the boundstate formation and the coloroctet t t¯ Green
function is known to have a smooth slope in the resonance
region. Finally, it might be possible to determine t
simultaneously with mt at the FCC.
To conclude, we have studied the gg → γ γ amplitude
with the t t¯ boundstate effects near their mass threshold
by collecting the available higherorder corrections in
nonrelativistic 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 twoloop amplitude. We have proposed a new
method to determine mt 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 one
to calculate higherorder corrections in theory.
Acknowledgements We are grateful to Yukinari Sumino and Yuichiro
Kiyo for valuable discussions and encouragements. We also thank
Hyung Do Kim and Michihisa Takeuchi for useful discussions. The
research of S.K. was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by
the Ministry of Science, ICT and Future Planning (Grant No.
NRF2014R1A2A1A11052687).
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