Dark Photon Search at A Circular $e^+e^-$ Collider

0.5 MB·

by  Min He

NCTS-PH1706
Dark Photon Search at A Circular e+e− Collider
Min He1∗, Xiao-Gang He2,3,1†, Cheng-Kai Huang2‡
1INPAC,Department of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai.
2Department of Physics, National Taiwan University, Taipei.
3Physics Division, National Center for
Theoretical Sciences, Hsinchu, Taiwan 30013.
(Dated: March 24, 2017)
Abstract
7
1 One of the interesting portals linking a dark sector and the standard model (SM) is the
20 kineticmixingbetweentheSMU(1)Y fieldwithanewdarkphotonA(cid:48) fromaU(1)A(cid:48) gauge
interaction. Stringentlimitshavebeenobtainedforthekineticmixingparameter(cid:15)through
r
a various processes. In this work, we study the possibility of searching for a dark photon
M
interaction at a circular e+e− collider through the process e+e− → γA(cid:48)∗ → γµ+µ−. We
3 findthattheconstrainton(cid:15)2 fordarkphotonmassinthefewtensofGeVrange, assuming
2 that the µ+µ− invariant mass can be measured to an accuracy of 0.5%mA(cid:48), can be better
] than 3×10−6 for the proposed CEPC with a ten-year running at 3σ (statistic) level, and
h √
better than 2×10−6 for FCC-ee with even just one-year running at s = 240 GeV, better
p
- thantheLHCandotherfacilitiescandoinasimilardarkphotonmassrange. ForFCC-ee,
p √
e running at s = 160 GeV, the constraint can be even better.
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Electronic address: hemin [email protected]
∗
Electronic address: [email protected]
†
Electronic address: [email protected]
‡
1
Introduction
There may be a dark sector which only interacts indirectly with the standard
model (SM) particles through some sort of portal. One of the interesting portal pos-
sibilities is the dark photon A(cid:48) of an additional U(1) gauge symmetry interacting
A(cid:48)
withSMparticlesresultedfromkineticmixingwiththeSMU(1) gaugeboson[1,2].
Y
There are many interesting consequences if such a dark photon exists[3, 4]. Great
efforts have been made to search for a dark photon through various processes and
stringent limits have been obtained for the kinetic mixing parameter for a given
dark photon mass m [5–7]. Most of the constraints on dark photon kinetic mixing
A(cid:48)
are for dark photon with a low mass (less than 10 GeV or so) from various low en-
ergy facilities and rare decays of known particles. There is no convincing theoretical
argument that the dark photon must have a low mass. It would be better to have
experimental data to tell whether larger dark photon mass is allowed. There are
less studies of constraints on dark photon with a larger mass. LHC may provide
some important information[6]. LHCb can provide stringent constraint on the ki-
netic mixing for dark photon mass larger than 10 GeV. It has been shown that the
ATLAS and CMS may provide even better constraint at dark photon mass around
40 to 50 GeV by studying pp → Xµ+µ−. The LHC can provide higher energy to
probe larger dark photon mass. However, if analysis can be carried out at a high
energy e+e− colliders , such as the CEPC and FCC-ee, the background and signal
may be easier to separate and provide better information. A possible process is
e+e− → γµ+µ−. The final states γ and µ+µ− in this case can be more easily studied
compared with the Xµ+µ− final states in the pp collision case. Better constraint
may be possible. In this work, we study the possibility to search for dark photon
effects at a circular e+e− collider through the process e+e− → γA(cid:48)∗ → γµ+µ−.
A dark photon A(cid:48) from an extra U(1) gauge interaction, which does not couple
A(cid:48)
to SM fields directly, can indirectly interact through a renormalizable kinetic mixing
term F(cid:48) Bµν with SM particles. Here F(cid:48) = ∂ A(cid:48) −∂ A(cid:48) and B = ∂ B −∂ B ,
µν µν µ ν ν µ µν µ ν ν µ
and A(cid:48) and B are the U(1) and U(1) gauge fields, respectively. With kinetic
A(cid:48) Y
mixing, the renormalizable terms involving these two U(1) gauge fields are given
by[1]
1 1 1
L = − BµνB − σF(cid:48) Bµν − F(cid:48) F(cid:48)µν . (1)
kinetic 4 0 0,µν 2 0,µν 0 4 0,µν 0
The U(1) gauge field B is a linear combination of the photon A and the Z
Y 0 0 0
boson fields, B = c A − s Z with c = cosθ and s = sinθ . θ is the
0 W 0 W 0 W W W W W
weak interaction Weinberg angle. To have the above Lagrangian in the canonical
form, that is, there are no crossing terms, one needs to redefine the fields. Letting
the redefined fields to be A, Z and A(cid:48), we have[1]
A  1 0 −√cWσ  A 
0 1−σ2
 Z0  = 0 1 √s1W−σσ2  Z  . (2)
A(cid:48) 0 0 √ 1 A(cid:48)
0 1−σ2
2
The interaction of A, Z and A(cid:48) with SM currents, to the first order in σ is given
by
Jµ (A −c σA(cid:48) )+Jµ(Z +s σA(cid:48) )+JµA(cid:48) , (3)
em µ W µ Z µ W µ D µ
where Jµ , Jµ are the SM electromagnetic and Z boson interaction currents, respec-
em Z
tively. Jµ is the dark current in the dark sector.
D
After the electroweak symmetry breaking, the Z boson obtains a non-zero mass
0
m . Depending on how the U(1) symmetry is broken, A(cid:48) boson can receive a
Z A(cid:48) 0
non-zero mass which may or may not cause mixing with Z . If one introduces a SM
0
singlet S with a non-trivial U(1) quantum number s to break the symmetry, A(cid:48)
A(cid:48) √ A(cid:48) 0
boson will receive a mass m = g s v / 2. Here g is the U(1) gauge coupling
√ A(cid:48) A(cid:48) A(cid:48) s A(cid:48) A(cid:48)
constant and v / 2 is the vacuum expectation value (cid:104)S(cid:105) of S field, In the Z and A(cid:48)
s
basis, they mix with each other with the mixing matrix given by
(cid:32) (cid:33)
m2 √σsW m2
Z 1−σ2 Z (4)
√σsW m2 1 m2 + s2Wσ2m2
1−σ2 Z 1−σ2 A(cid:48) 1−σ2 Z
For a small σ, in the mass eigenstate bases, A(cid:48) only interacts with Jµ at the second
Z
order in σ.
In general if the vacuum expectation value of the scalar which breaks U(1) also
A(cid:48)
carries U(1) charge[2], there is an additional mixing parameter between Z and A(cid:48)
Y
which can lead to an interaction of A(cid:48) with J . But, the interaction of A(cid:48) with J
Z em
remainsthesame[2]. Inourlaterdiscussions,wewilltakethesimplecasethatS does
notcarryanyU(1) chargeandconcentrateonthestudyofthesearchofdarkphoton
Y
A(cid:48) at a circular e+e− collider through e+e− → γµ+µ−. To have a similar notation
compared with that used by many in the literature, we use the notation −c σ = (cid:15).
W
Besides A(cid:48) contribution to e+e− → γµ+µ− through intermediate A(cid:48), there are also
SM contributions from intermediate A and Z interactions. The effective Lagrangian
concerning photon and dark photon interaction with SM currents to be used is in
the following form
L = Jµ A +JµZ +(cid:15)Jµ A(cid:48) . (5)
int em µ Z µ em µ
At low energy, way below Z boson mass, the contribution from intermediate
Z boson is small. When the collision energy is large compared with the Z
boson mass, the interaction term JµZ may also be important. For searching
Z µ
for dark photon effects through e+e− → γA(cid:48)∗ → γµ+µ−, the SM contributions,
e+e− → γ(γ∗,Z∗) → γµ+µ− become the background. We find that the constraint
on (cid:15)2 for dark photon mass in the few tens of GeV range, assuming that the
µ+µ− invariant mass can be measured to an accuracy of 0.5%m , can be better
A(cid:48)
than 3 × 10−6 for the proposed CEPC with a ten-year running at 3σ (statistic)
level, and better than 2 × 10−6 for FCC-ee with even just one-year running at
√
s = 240GeV,betterthantheLHCandotherfacilitiescandoinasimilardarkpho-
√
ton mass. For FCC-ee, running at s = 160 GeV, the constraint can be even better.
3
The e+e− → γ(γ∗,Z∗,A(cid:48)∗) → γµ+µ− processes
The Feynman diagram for e+e− → γ(γ∗,Z∗,A(cid:48)∗) → γµ+µ− are shown in Fig.
1. The contributions from intermediate A(cid:48) state are suppressed by a factor of (cid:15)4
which can naively be thought to be negligible compared with intermediate γ,Z
contributions[8]. However, since the final invariant mass of the µ+µ− pair is not
fixed, the value can hit the A(cid:48) mass pole, the cross section σmµµ(e+e− → γA(cid:48)∗ →
γA(cid:48)
γµ+µ−) at the A(cid:48) mass pole can be large, even larger than the SM cross section
σmµµ (e+e− → γ(γ∗,Z∗) → γµ+µ−),withtheµ+µ− invariantmasss = (k +k )2 =
γ(γ,Z) 3 1 2
m2 close to m2 , that is, m in the range of m −σ ∼ m +σ with σ to
µµ A(cid:48) µµ A(cid:48) µµ A(cid:48) µµ µµ
be much smaller than m . The subscript (γ,Z) indicates contributions from both
A(cid:48)
intermediate γ and Z. Measurement at that region with small enough σ can
µµ
provide information about the dark photon interaction. In the following we explain
how this can be done.
To this end we defined below two measurable quantities,
σmµµ = (cid:90) (mA(cid:48)+σµµ)2(dσ /ds )ds
γA(cid:48) γA(cid:48) 3 3
(mA(cid:48)−σµµ)2
σmµµ = (cid:90) (mA(cid:48)+σµµ)2(dσ /ds )ds . (6)
γ(γ,Z) γ(γ,Z) 3 3
(mA(cid:48)−σµµ)2
e− γ µ− e− µ−
k
1
p1 k3
γ,Z,A γ,Z,A
′ ′
k
2
p
2
e+ µ+ e+ γ µ+
e− µ− e− µ−
γ,Z,A γ,Z,A γ
′ ′
γ
e+ µ+ e+ µ+
FIG. 1: Feynman diagrams for e−e+ → γµ−µ+ process.
Evaluating the Feynman diagrams with intermediate γ, Z and A(cid:48) shown in Fig.
4
1, we obtain
dσ 4α3 (s2 +s2)
γ(γ,Z) = em 3 (cid:0)s(ln(s/m2)−1)+s (ln(s /m2)−1)(cid:1)
ds 3s3s (s−s ) e 3 3 µ
3 3 3
α3 (8sin4θ −4sin2θ +1)2 s2 +s2
+ em W W 3
48sin4θ cos4θ s2(s−s )
W W 3
(cid:18)s (ln(s/m2)−1) s(ln(s /m2)−1)(cid:19)
× 3 e + 3 µ
(s −m2)2 (s−m2)2
3 Z Z
α3 (1−4sin2θ )2 s+s ss
− em W 3 3
16sin4θ cos4θ s2(s−s )(s−m2)(s −m2)
W W 3 Z 3 Z
α3 (1−4sin2θ )2 s2 +s2 (cid:18)ln(s/m2)−1 ln(s /m2)−1(cid:19)
+ em W 3 e + 3 µ
6sin2θ cos2θ s2(s−s ) s −m2 s−m2
W W 3 3 Z Z
α3 s+s (cid:18) s s (cid:19)
− em 3 3 + ;
4sin2θ cos2θ s2(s−s ) s −m2 s−m2
W W 3 3 Z Z
(7)
dσ 4α3 (cid:15)4s (s2 +s2)
γA(cid:48) ≈ em 3 3 (cid:0)ln(s/m2)−1(cid:1)
ds 3s2(s−s )((s −m2 )2 +Γ2 m2 ) e
3 3 3 A(cid:48) A(cid:48) A(cid:48)
4α3 (cid:15)4s (s2 +s2) π
≈ em 3 3 δ(s −m2 )(cid:0)ln(s/m2)−1(cid:1) ,
3s2(s−s ) Γ m 3 A(cid:48) e
3 A(cid:48) A(cid:48)
where α is the fine structure constant, and s = (p +p )2.
em 1 2
For dσ /ds , we have kept the leading contribution which is, in the narrow
γA(cid:48) 3
width approximation, proportional to δ(s − m2 ). Γ is the decay width of the
3 A(cid:48) A(cid:48)
dark photon
(cid:115)
Γ = (cid:88)Γ(A(cid:48) → f¯f) , Γ(A(cid:48) → f¯f) = (cid:15)2Q2fαemmA(cid:48)(1+ 2m2f) 1− 4m2f . (8)
A(cid:48) 3 m2 m2
f A(cid:48) A(cid:48)
ThesummationaboveistosumoverfermionpairsintheSMwithmassm < m /2.
f A(cid:48)
We find that in the range of a few tens of GeV up to 60 GeV or so (significantly
below the Z pole), the process may be able to provide stringent constraints on (cid:15)2,
we will work in this region. In this case the summation of f will need to sum over
u, d, s, c, b, e, µ and τ. With lower A(cid:48) mass, one should be careful to only sum over
states which are below the threshold.
Multiplying the integrated luminosity I = time × luminosity, one obtains the
event numbers for the SM contributions N and dark photon contribution N
γ(γ,Z) γA(cid:48)
respectively
N = σmµµ I , N = σmµµI . (9)
γ(γ,Z) γ(γ,Z) γA(cid:48) γA(cid:48)
With information from event numbers which can be obtained, one can analyze
the sensitivity for a given experiment and then obtain the constraints on (cid:15)2 as a
function of the dark photon mass m .
A(cid:48)
5
Sensitivities for CEPC and FCC-ee circular colliders
The sensitivity on the mixing parameter (cid:15) at an e+e− collider depends on how
well one can control over the µ+µ− invariant mass measurement, that is how small
one can accurately control σ . We will take a similar accuracy as what can be done
µµ
at the LHC[7], assuming σ = 0.5%m . In the study of searching for dark pho-
µµ A(cid:48)
ton in Ref.[7], for SM contribution, only intermediate γ contribution was considered.
WhenincludeintermediateZ contribution, thedetailedvaluesforthesensitivitywill
be changed. The statistic sensitivity depends on how one can measure the back-
(cid:112)
ground events from SM contribution. We take N as the statistic sensitivity
γ(γ,Z)
(cid:112)
for SM background. We then obtain the signal S to background error N
γ(γ,Z)
(cid:112)
ratio χ = N / N as the indicator how well one can obtain constraints on
γA(cid:48) γ(γ,Z)
(cid:15)2 and m . As long as statistic errors are concerned, the value of χ corresponds
A(cid:48)
to the number of σ. There may be other background coming from other processes
and also systematic errors which required a more involved analysis with full knowl-
edge of the detectors. Here we will only consider statistic error discussed above.
We will obtain the event numbers using the benchmark luminosities planed for the
√
CEPC[9]: 2×1034cm−2s−1 at s = 240 GeV, and FCC-ee[10]: 1.5×1036cm−2s−1,
√
3.5 × 1035cm−2s−1, and 8.4 × 1034cm−2s−1 at s equal to 160 GeV, 240 GeV and
350 GeV, respectively. The results on the cross section, event numbers per year and
the sensitivity for the mixing parameter (cid:15) are shown in Figs. 2, 3, 4 and 5.
The cross sections for various energies relevant to CEPC and FCC-ee are shown
√
in Fig. 2. The cross sections for SM contributions are above 5×10−40cm2 for s
in the whole range covered by CEPC and FCC-ee. The SM background have two
contributions, the intermediate γ and Z contributions. In the parameter space we
areconsideringtheintermediateγ contributiondominates. Numericallywefindthat
the intermediate Z contribution is less than 10% of the intermediate γ contribution
in the parameter spaces discussed above. If the machine is running at the Z pole,
the intermediate Z contribution become important. Also when the dark photon
mass is close to the Z mass, the dark photon contribution also become important.
For these reasons, we have limited the dark photon mass to be significantly away
from the the Z mass and also have chosen machine energies to be away from the Z
boson mass.
Multiplying the luminosity of each machine at different energies on the cross
section shown in Fig. 2, we obtain the one-year running event numbers shown in
Fig. 3. We see that even for the lowest case, the CEPC case, the event number
per year for SM contribution can be more than 1000. This provides a large enough
number for analysis with some accuracy.
The sensitivity of (cid:15)2 as a function of dark photon mass m for a given χ are
A(cid:48)
shown in Figs. 4 and 5. We plot (cid:15)2 as a function of m for various values χ for
A(cid:48)
different machines and running times.
For CEPC, in the range of 10 GeV to 60 GeV for m , the one-year running
A(cid:48)
sensitivity for (cid:15)2 can reach 9×10−6 at 3σ level. If one just wants an 1σ limit, the
sensitivity for (cid:15)2 can reach 3 × 10−6. With ten-year running time, the sensitivity
6
√
can lower a factor of 10. These limits are better than ATLAS and CMS at the
LHC can reach[7]. In the range of 1 GeV to 10 GeV for m , the sensitivity for (cid:15)2 is
A(cid:48)
better because the width Γ is smaller. However, other processes, such as Belle II
A(cid:48)
experiment, can give better constraints[7]. At 10 GeV to 20 GeV m mass range,
A(cid:48)
LHCb may give a slightly better constraint[7].
√
For FCC-ee, since the luminosity at s = 240 GeV is higher than that for
CEPC, the sensitivity can be much better. The one-year running sensitivity for
√
(cid:15)2 at s = 160 GeV, can reach 0.7 × 10−6 at 3σ level. The 1σ limit can reach
√
2.4 × 10−7. At s = 240 GeV, the one-year running sensitivity for (cid:15)2 can reach
√
2.2×10−6 at 3σ level. The 1σ limit can reach 0.72×10−7. At s = 350 GeV, the
one-year running sensitivity for (cid:15)2 can reach 6.5 × 10−6 at 3σ level. The 1σ limit
can reach 2×10−6.
5⨯10-38
mA′=60GeV 5⨯10-34 mA′=60GeV
2cm()σZ,()γγ 2512⨯⨯⨯⨯11110000----33339988 mmmAAA′′′===42100GGGeeeVVV 22cm/()σϵA′γ 2512⨯⨯⨯⨯11110000----33335544 mmmAAA′′′===42100GGGeeeVVV
1⨯10-39 1⨯10-35
5⨯10-40 5⨯10-36
100 150 200 250 300 350 400 100 150 200 250 300 350 400
s(GeV) s(GeV)
√
FIG. 2: Cross sections σγm(µγµ,Z) and σγmAµ(cid:48)µ as functions of s. mA(cid:48) = 1,20,40 and 60 GeV.
The mA(cid:48) dependence of σγm(µγµ,Z) is due to integration ranges depend on mA(cid:48).
106 1010
FCC-ee, s=160GeV FCC-ee, s=160GeV
105 109
FCC-ee, s=240GeV FCC-ee, s=240GeV
NZ,()γγ 104 FCC-ee, s=350GeV 2N/ϵA′γ 108 FCC-ee, s=350GeV
103 107
CEPC, s=240GeV CEPC, s=240GeV
102 106
10 20 30 40 50 60 10 20 30 40 50 60
mA′(GeV) mA′(GeV)
√
F√IG. 3: Nγ(γ,Z) and NγA(cid:48)/(cid:15)2 as functions of mA(cid:48) for CEPC ( s = 240 GeV) and FCC-ee
( s = 160, 240, 350 GeV) for one-year running.
Conclusions
Insummary, wehavestudiedthepossibilityofsearchingfordarkphotoninterac-
tion at a circular e+e− collider through the process e+e− → γA(cid:48)∗ → γµ+µ−. There
7
1⨯10-5
2⨯10-5 χ=5 5⨯10-6 χ=5
1⨯10-5 χ=3 χ=3
χ=2
2ϵ 5⨯10-6 2ϵ 2⨯10-6 χ=2
χ=1
2⨯10-6 1⨯10-6 χ=1
1⨯10-6 5⨯10-7
10 20 30 40 50 60 10 20 30 40 50 60
mA′(GeV) mA′(GeV)
FIG. 4: Sensitivity on (cid:15)2 as functions of dark photon mass mA(cid:48) for a given χ for CEPC.
The left figure is for one-year running and the right figure is for ten-year running.
2⨯10-6 5⨯10-6 2⨯10-5
χ=5 χ=5 χ=5
1⨯10-6 χ=3 2⨯10-6 χ=3 1⨯10-5 χ=3
2ϵ 5⨯10-7 χ=2 2ϵ χ=2 2ϵ 5⨯10-6 χ=2
2⨯10-7 χ=1 1⨯10-6 χ=1 2⨯10-6 χ=1
5⨯10-7
1⨯10-7 1⨯10-6
10 20 30 40 50 60 10 20 30 40 50 60 10 20 30 40 50 60
mA′(GeV) mA′(GeV) mA′(GeV)
FIG. 5: Sensitivity on (cid:15)2 as functions√of dark photon mass mA(cid:48) for a given χ for FCC-ee.
The figures from left to right are for s = 160 GeV, 240 GeV and 350 GeV for one-year
running, respectively.
are two contributions in the SM, the intermediate γ and intermediate Z. When the
dark photon mass and also the center-of-mass frame energy are significantly away
from the Z boson mass, the dominate contribution to the SM background is from
intermediate γ interaction. The CEPC and FCC-ee e+e− can provide sensitive con-
straints on the dark photon mixing parameter and dark photon mass. We find that
the constraints on (cid:15)2 for dark photon mass in the few tens of GeV range, assuming
that the µ+µ− invariant mass can be measured to an accuracy of 0.5%m , can be
A(cid:48)
better than 3×10−6 for the proposed CEPC with a ten-year running at 3σ (statis-
tic) level, and better than 2×10−6 for FCC-ee with even just one-year running at
√
s = 240 GeV, better than the LHC and other facilities can do in a similar dark
√
photon mass. For FCC-ee, running at s = 160 GeV, the constraint can be even
better. In the range of 20 GeV to 60 GeV for m , the smallest σ is 100 MeV
A(cid:48) µµ
which is reachable at the CEPC and FCC-ee. With a smaller σ , the sensitivity
µµ
can be improved.
Acknowledgments
XGH was supported in part by MOE Academic Excellent Program (Grant
No. 105R891505) and MOST of ROC (Grant No. MOST104-2112-M-002-015-MY3),
8
and in part by NSFC (Grant No. 11575111) of PRC. This work was also supported
by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry
of Education, and Shanghai Key Laboratory for Particle Physics and Cosmology
(SKLPPC) (Grant No. 11DZ2260700).
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