Table Of ContentExploring pseudo-Nambu–Goldstone bosons by stimulated photon colliders in the
mass range 0.1 eV to 10 keV
Kensuke Homma1,2 and Yuichi Toyota1
1Graduate School of Science, Hiroshima University,
Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
2International Center for Zetta-Exawatt Science and Technology,
Ecole Polytechnique, Route de Saclay, F-91128 Palaiseau Cedex, France
(Dated: January 17, 2017)
Searchingforpseudo-Nambu–Goldstonebosons(pNGBs)inweak-couplingdomainsiscrucial for
understanding the dark components in the universe. We propose searching for pNGBs coupled to
two photons in the mass range from 0.1 eV to 10 keV. This could provide opportunities to test
7 string-theory-based pNGBs beyond the GUT scale M ∼ 1016 GeV included in the weak coupling
1 proportional to M−1. We provide formulae that are applicable to photon–photon scattering via
0
a pNGB resonance exchange with a stimulation process in an asymmetric head-on photon–photon
2
colliderbymixingthreelaserpulsesinlaboratory experiments. Wediscussthequantumelectrody-
n namic effects on the pNGB exchange in the same mass–coupling domain as a background process
a from the standard model. We find that a large unexplored mass–coupling domain is accessible by
J combining existing laser facilities, including free-electron lasers.
6
1
I. INTRODUCTION single galaxy and galaxy clusters [8, 9], and the argu-
] ments are still actively ongoing [10]. The possible inter-
h
p Spontaneous symmetry breaking could be one of the pretationofapNGBdecayingintotwophotonshasbeen
- most robust guiding principles for general discussions of discussed [11]. Indeed, string theories predict pNGBs to
p dark components in the universe. Whenever a global be homogeneously distributed on a log scale in the mass
he symmetry is broken, a massless boson may appear as a rangepossiblyupto108 eV[3]. Thissituationmotivates
[ Nambu–Goldstoneboson(NGB). Innature,however,an us to try to extend the same method up to 10 keV in
NGB emerges as a pseudo-NGB (pNGB) with a finite general. Given the effective Lagrangian , we evaluate
1 L
mass. Even if a pNGB is close to being massless, its de- the sensitivity to γγ φ γγ scattering by means of
v → →
2 cayintolighterparticlessuchasphotonsiskinematically stimulation using coherent X-ray sources. However, we
8 allowed. TheneutralpionissuchanexampleofapNGB emphasize that the formulae we provide here are never-
2 via chiral symmetry breaking. The effective interaction theless applicable to a wide mass range from 0.1 eV to
4 Lagrangianis defined as 10keVbycombiningdifferenttypesofcoherentandinco-
0 herentlightsources. Thismassrangeisyettobe probed
1. =gM−11FµνFµνφ, (1) intensively, and thus the proposed approach will open
0 −L 4 a window through which to explore string-theory-based
7 where we assume a scalar-type field φ for simplicity, and pNGBs beyond the GUT scale M 1016 GeV.
∼
1 M hasthedimensionofenergywhereasg isadimension-
:
v less constant.
i There are several theoretical models that predict low- II. FORMULAE FOR STIMULATED
X ASYMMETRIC PHOTON–PHOTON
mass pNGBs such as axions [1], dilatons [2], and string-
r theory-based axion-like particles [3]. However, pinning SCATTERING VIA PNGBS
a
down the physical mass of a pNGB by means of such
models is commonly non-trivial. Therefore, experiments To distinguish clearly between the photons in the col-
are indispensable for investigating the physical mass as liding beams and the signal photons whose frequencies
widely as possible in the lower mass range. have been shifted by the scattering process, we consider
We have previously advocated a novel method [4] for an asymmetric collision system (ACS) as illustrated in
stimulating γγ φ γγ scattering in a quasi-parallel Fig. 1. The ACS is obtained by boosting a center-of-
collision system→(QPS→) via an s-channel resonant pNGB mass system (CMS) along the head-on collision axis (z
exchange by utilizing the coherent nature of laser fields. axis)withaspeedβc,wherecisthespeedoflight(hence,
Kinematically, this is analogous to four-wave mixing in the Lorentz factor is γ =1/ 1 β2). Four-momenta in
−
matter[5]ifthenonlinearatomicprocessisreplacedwith the ACS are defined as
p
the pNGB exchange. A QPS is intended for searching
for pNGBs in the sub-eV mass range. Indeed, we have p1 =ω1(1,0,0,1),p2=ω2(1,0,0, 1), (2)
−
performed two previous laboratory searches [6, 7] based p =ω (1,sinθ ,0,cosθ ),p =ω (1,sinθ ,0, cosθ ).
3 3 3 3 4 4 4 4
−
on this approach.
Recently, an unidentified emission line, ω 3.5 keV, For later convenience, with an incident photon energy ω
∼
has been reported in the photon energy spectra from a intheCMS,wedefinetheincidentenergiesintheACSas
2
where
m2/2 mΓ/2
ω2 = , a= . (8)
r 1 cos2ϑ 1 cos2ϑ
− −
BecausethescatteringamplitudeisLorentzinvariant,we
have only to consider ϑ=π/2 in the above formulation,
corresponding to the CMS, in order to apply it to the
ACS with the arbitrary parameter u and u 1 associated
−
with a Lorentz boost along the head-on collision axis in
the CMS. In the CMS, Eq. (8) is thus simplified as
FIG. 1: Asymmetric collision system (ACS) realized in the
m 2 m
laboratoryframe,interpretedastheresultofaLorentzboost ω2 = , a= Γ. (9)
r 2 4
byarelativevelocityβofthelaboratoryframealongthehead-
(cid:16) (cid:17)
on collision (z) axis in thecenter-of-mass system (CMS). All From Eqs. (6) and (8), a is also expressed as
four photons are confined within a common reaction plane,
thex–z plane. a= ωr2 gm 2, (10)
16π M
(cid:16) (cid:17)
ω1 uω and ω2 u−1ω via the following relationships: which shows explicitly the proportionality to M−2. The
u ≡ (1+β)/(1≡ β) = γ + γ2 1. With u+ u+ squared amplitude is then expressed as
u−≡1 and u− u −u−1, energy–mom−entum conser≡vation a2
resulpts in the≡foll−owing relationpships: 2 (4π)2 . (11)
|M| ≈ χ2+a2
0-axis: u+ω =ω +ω , (3)
3 4 In the off-resonance case χ a, equivalent to Eq. (4),
z-axis: u ω =ω cosθ ω cosθ , 2 is largely suppressed b≫ecause of the factor a2
− 3 3 4 4
x-axis: ω3sinθ3 =ω4sin−θ4. M|M−|42 for(la4rπg)e2 Mis.exIpnecctoendtrfarsotm, inEqth.e(1l1im).itInofpωrin→cipωl∝er,,
We have formulated a QPSwith a symmetric incident t|Mhis|is→independent of how small a M 2 is.
−
angleϑ,whichisthehalfanglebetweentwoincidentpho- Physically, however, it is difficult∝to satisfy χ = 0 ex-
tonswhenalaserbeamwiththephotonenergyω 1eV actly in the case of an extremely small a because Γ be-
∼
isfocusedbyalenselement. This formulationis alsoap- comessonarrow. ACMSenergyuncertaintyiscausedby
plicable to the CMS. In the coplanar condition whereby momentum (angle) and energy uncertainties originating
the plane determined by p1(1) and p2(1) coincides with from the spatial and temporal localizations of the laser
that determined by p3(1) and p4(1) (where the numbers fieldsbythefocusingandshorteningpulses,respectively.
in parenthesesspecify the linearpolarizationstates), the This uncertainty due to the wavy aspect of a photon is
Lorentz-invariantscatteringamplitude ofthe scalar-field unavoidable,evenatthelevelofapairofphotonsinprin-
exchange is expressed as ciple. For a given uncertainty χ ηa with η 1, we
still can expect an enhancemen±t≡fro±m the aver≫aged ef-
ω4(cos2ϑ 1)2
MS =−(gM−1)22ω2(cos2ϑ −1)+m2, (4) fect [12] over the uncertainty as follows:
− 1 χ+
wherethe subscriptS denotescombinationsoflinearpo- 2 = 2dχ (12)
|M| χ χ |M|
larization states of four photons in the initial and final +− − Zχ−
states. The nonzero amplitudes are = = (4π)2
1122 = 2211 where the linearMly1p11o1larizMed2s2t2a2tes = 2ηa 2atan−1(η)=(4π)2η−1tan−1(η)
−M −M
(1) and (2) are orthogonal to each other. Since all the π a
(4π)2η 1 =8π3 ,
squared amplitudes are common, we omit the polariza- ≈ − 2 χ
tionsubscriptS below. Inthefollowing,thedenominator | ±|
where the approximationis applied to the situation η
of Eq.(4) is denoted by . We then introduce the imag- ≫
D 1. By introducinga beam-energyuncertainty ∆ω result-
inary part due to the resonance state by the following
ing in a bandwidth ω m/2 ∆ω in the CMS with
replacement: respect to the resonan±ce≡conditi±on ω = m/2, the uncer-
m2 (m iΓ/2)2 m2 imΓ, (5) tainty in χ in the CMS is expressed as
→ − ≈ −
where the decay rate Γ is expressed as χ =ω2 (m/2)2 =∆ω2 m∆ω m∆ω, (13)
± ±− ± ≈±
Γ=(16π) 1 gM 1 2m3 (6) where ∆ω2 m∆ω is assumedinthe aboveapproxima-
− − ≪
tion. Therefore, with Eqs.(6) and (9), the averageof the
foragivenmassm[4]. Subst(cid:0)ituting(cid:1)thisintothedenom- squared scattering amplitude is approximated as
inator in Eq. (4) and expanding around m, we obtain
a π2 g 2
2(1 cos2ϑ)(χ+ia) withχ=ω2 ω2, (7) 2 8π3 = m3. (14)
D≈− − − r |M| ≈ m∆ω 8∆ω M
(cid:16) (cid:17)
3
Compared to the off-resonant case 2 a2, 2 a
stillhasagainofa 1 M2. Thisis|Mon|eo∝fthe|tMwo|m∝ost
−
∝
promising features of our approach. We note that the
uncertainty of the CMS energy is related to the beam-
energyuncertaintiesδω andδω intheACS(laboratory
1 2
frame) as follows:
2 2
δω δω mR
1 2
∆ω =√ω ω + , (15)
1 2
s(cid:18) ω1 (cid:19) (cid:18) ω2 (cid:19) ≡ √2
where in the last step we assume that an experiment
tunesthebeamenergiessothattheysatisfy2√ω1ω2 =m
with a common relative energy uncertainty R δω /ω
i i
≡
for i = 1,2. By substituting Eq. (15) into Eq. (14),
weeventuallyexpressthe massandcouplingdependence
FIG. 2: Collision geometry between three pulsed photon
of the squared scattering amplitude for a given relative-
beams. Creation photon pulses p1 and p2 collide head-on at
beam-energy uncertainty R as theoriginofthexyz-coordinatesandacoherentlaserpulsep4
tostimulatethescattering issimultaneously focused intothe
√2π2 g 2 origin. Asaresult of thestimulated γγ →γγ scattering, sig-
2 m2. (16)
|M| ≈ 8R M nalphotonsp3 areproducedonthesamereactionplane. The
(cid:16) (cid:17) solidangleof~p3 enhancedbytheinducingbeamp4 isrelated
We then parametrize the differential cross section to the angular spread of p~4 via energy-momentum conserva-
dσ [6] for a pNGB exchange in the ACS (laboratory tion. This relationship gives the rotation angle ∆φ3 of the
ngb
reaction plane around thez-axis.
frame) as
1
dσngb = 2dLips, (17) now express the partially integrated cross section σ˜ngb
K(ϑ)2ω12ω2|M| over dΩ3 in the ACS as follows:
where 2 φ3 θ3 ω 2
3
σ˜ngb = (8|Mπω|)2 dφ3 2ω sinθ3dθ3 (22)
dL =(2π)4δ(p +p p p ) d3p3 d3p4 (18) Zφ3 Zθ3 (cid:16) (cid:17)
ips 3 4− 1− 2 2ω3(2π)32ω4(2π)3 = √82Rπ2 Mg 2m2 ∆φ3(cosθ3−cosθ3)
(8πω)2 (u+ u cosθ )(u+ u cosθ )
and the relative velocity K is defined as [13] (cid:0) (cid:1) − − 3 − − 3
√2 gm 2 √2 g 2
= = ,
(v~ v~)2 2(16ω)2R M I 128R M I
K(ϑ)≡r(v~1−v~2)2− 1×c2 2 =2csin2ϑ=2 (19) where Eqs. (21) and(cid:16)(16(cid:17)) are substitu(cid:16)ted,(cid:17)and we intro-
duce
for ϑ= π/2 and ¯h = c = 1. With d3p = p~ ω dω dΩ ,
3 3 3 3 3
| |
we can express the differential cross section as a func- ∆φ3(cosθ3 cosθ3)
− . (23)
tfoiollnowofs:the solid angle dΩ3 of signalphoton energy ω3 as I ≡ (u+−u−cosθ3)(u+−u−cosθ3)
As shown in the last step of Eq. (22), σ˜ eventually
ngb
dσ 1 2 ω 2 becomes independent ofmassbecause the beamenergies
ngb = 3 2 (20)
dΩ3 (cid:18)8πω(cid:19) (cid:16)2ω(cid:17) |M| icnortrheespAoCndSinagreCtMunSe.d so that ω =m/2 is satisfied in the
with The second important feature of our approach is to
induce a specific two-photon final state by supplying an
2ω additional coherent field with a different wavelength ω
ω , (21) 4
3 ≡ u+−u−cosθ3 from any of ω1, ω2, or ω3. Simultaneously, the momen-
tum of the induction field p must coincide with that of
4
which is a result of energy–momentum conservation. one of the two photons in the final state. Therefore, the
In what follows, for simplicity, we assume perfect en- stimulatedrangeofthesignalphotonp isdeterminedby
3
ergyresolutionofsignalphotonω3 sothatwecandiscuss the momentum range of the inducing coherent photons
the momentum range of p3 based only on the angular p4 after all. For simplicity, we assume that the momen-
spread of ~p3 from θ3 to θ3 on the reaction plane and tum spread of p4 is dominated by the angular spread of
∆φ φ φ in the perpendicular direction with re- ~p due to the short focal length of a lens element by ne-
3 3 3 4
≡ −
spect to the reaction plane in the ACS (see Fig. 2). We glecting the intrinsic energy spread of ω . In this case,
4
4
wecanexpresstheangularrangeθ viathethirdrelation time, and the beam radius w(ct) = w 1+(ct/Z )2 is
3 0 R
in Eq. (3) as given for wavelength λ, beam waist w = 2Fλ/π, and
0
Rayleigh length Z = πw2/λ. The volpume for the nor-
R 0
θ3 =sin−1 u+v v sin(θ4+∆θ4) (24) malization is thus obtained as
θ3 =sin−1 (cid:18)u+v−v sin(θ4−∆θ4)(cid:19), V =Z−∞∞ IE(x02i)dxi =(π/2)3/2w02cτ. (29)
(cid:18) − (cid:19)
Based on the three-beam arrangement in the identical
whereω4 ≡vω and∆θ4 ∼1/(2F4)by assumingthatthe reaction plane in Fig .2, the normalized densities ρk =
p4 beamisfocusedwiththef-numberF4 asillustratedin Ik/Vk are defined as follows:
Fig.2. Figure2depictsalltheopticalbeamaxesasbeing
olinkeafcoocmusmedonbreeaamctiboansepdlanone,Ganadusps4iaisnporpotviicdse.dTahsearecfoonree-, ρ1(xi)=N1w(22/(cπt))3c/τ2 e−2xw212+(cyt2)e−2(cid:16)zc−τ1ct(cid:17)2, (30)
1 1
daestleirgmhtindeedvbiaytitohneoafng~pu3lafrrosmpretahde ocfopp~l4a.nTarhicsoanzdiimtiuotnhaisl ρ2(xi)=N2w(22/(cπt))3c/τ2 e−2xw222+(cyt2)e−2(cid:16)zc+τ2ct(cid:17)2,
angular spread, δφ , of p~ from the reaction plane gives 2 2
4 4
athxeis.roHtaetnicoen, wanegcleasnoefxpthreessre∆acφt3ioansplanes around the z P4(xi)=V4ρ4(xi)=N4ww42(02c4t)e−2Xw242+(cyt2)e−2(cid:16)Zc−τ4ct(cid:17)2,
∆φ3 =2sin−1 sin2θ4cossin2δδφφ44+sin2δφ4!, (25) woθ4fh)texhr−eescXionh(≡πer+seiθnn4t()πizn+adruθec4it)nixotnr+ofidceuolcsd(eπdw+fiothrθt4rh)ezespraoenctdattZteod≡tihncecoidhs(eeπnacd+e-
p on collision axis of the ACS.
where δφ = ∆θ holds because of the axial symmetry
4 4
The geometrical overlap factor integrated over the
around the optical axis of the p beam. By substitut-
4
longest pulse duration τ among the three pulses whose
ing these parameters into Eq. (23), we can evaluate an L
central positions are all at the origin at t = 0 is then
induciblepartiallyintegratedcrosssectionfromEq.(22).
expressed as
With the parametrization for signal yield, , devel-
Y
oped in Appendix A of Ref. [6], we express the expected τL
numberofsignalswithω3perpulsecrossingforthespon- G=(2/π)3/2(w04/(cτ))2 dt(w1w2w4)−2× (31)
taneous scattering process as Z−τL
A(1/w2+1/w2+1/w2)B 1/2e 2Dt2,
{ 1 2 4 }− −
= dt dxiρ (t,xi)ρ (t,xi)K 1/L2 σ˜ L2 ,(26)
s 1 2 ngb where for simplicity we assume that experiments intro-
Y
Z Z duce τ =τ =τ with
(cid:2) (cid:3) (cid:2) (cid:3) 1 2
whereirunsfrom1to3,andthe firstfactorcorresponds
A 1/w2+1/w2+(cosθ /w )2+(sinθ /(cτ ))2,
totimeintegratedbeamluminositywiththedimensionof ≡ 1 2 4 4 4 4
1/L2 with length unit L. We then express the induction B (sinθ /w )2+(cosθ /(cτ ))2+2/(cτ)2
4 4 4 4
≡ −
effectbyconvolutingtheprobabilitydistributionfunction [sinθ cosθ 1/w2 1/(cτ )2 ]2/A,
of an induction field P as follows: 4 4{ 4− 4 }
4 D 2/τ2+1/τ2 sin2θ /A+
≡ 4 − 4
i = dt dxiρ1(t,xi)ρ2(t,xi)Kσ˜ngbP4(t,xi)(27) cos2θ4{1+(1/(cid:2)w42−1/(cτ4)2)sin2θ4}2/B ]/(cτ42)2.
Y
Z Z Because the total number Y of detected signal photons
Kσ˜ N N N G,
ngb 1 2 4
≡ is described as Y = fTǫ with collision repetition rate
i
Y
whereρ indicatesanormalizeddensitydistributionover f [Hz], data accumulation time T [s], and detection effi-
k
an infinite space-time range for creation beams k = 1,2 ciencyǫ,areachablecouplingstrengthisfinallyexpressed
andtheinductionbeamk =4byassumingpulsedGaus- as
sian beams with the number of photons contained in in-
g RY
dividual pulses, N1, N2, and N4, as defined explicitly in =21/48 (32)
the following. The factorGisthe integratedgeometrical M s fTǫKGN1N2N4
I
overlap factor. At a space point xi for a given common
time t in the ACS, the squared field strength of a Gaus- from Eqs. (22) and (27). We note that g/M has m−1
sian laser pulse can be parametrized as [14] dependence because G ω2 m2.
∝ ∝
I(x,y,z =ct)=E02w2w(02ct)e−2xw22+(cyt2)e−2(z−cτct)2, (28) III. SENSITIVITY AND QED BACKGROUND
whereE istheelectricfieldandE2isproportionaltothe The three dashedlines in Fig.3show g/M [1/GeV]as
0 0
number of photons N in a pulse, τ is the pulse duration a function of m [eV] for N = 1010,1015, and 1020 for
k
5
model. Probing QED-based photon–photon scattering
via four-wave mixing has been proposed in Ref. [17].
These calculations are based on the Euler–Heisenberg
effective Lagrangian and the signal yield is evaluated
within a classicalwavepicture. The formulationwe pro-
videhereisbasedontheparticlepictureofcollidingpho-
tonsandthus allowsus tosimply plug the partiallyinte-
gratedcrosssection, σ˜ ,forthe QEDboxdiagraminto
qed
Eq. (27), which has been applied to σ˜ on the same
ngb
footing. The differential crosssectionof the elastic QED
scattering process [15] with respect to scattering angles
in the ACS is expressed as
(αr )2139 γ2(cosθ β)2 2
dσ = 0 ω6 3+ 3− (33)
qed 4π2 902 γ2(cosθ β)2+sin2θ ×
(cid:18) 3− 3(cid:19)
160 ω2sin2θ
3
1+
1394γ2(cosθ β)2+3sin2θ ×
(cid:18) 3− 3(cid:19)
sin2θ γ(1 βcosθ )
3 3
sγ2(cosθ3 β)2+sin2θ3γ2(cosθ3− β)2+sin2θ3dθ3dφ3
− −
with α = 1/137 and r = 2.8 10 13 cm. This cor-
0 −
×
FIG. 3: Sensitivity to the mass–coupling domains. Exper- responds to the unpolarized case that is comparable to
imentally excluded and theoretically predicted domains are the s-channel scalar-pNGB exchange with S =1111 dis-
all imported from Ref. [16], where all details are explained. cussed here, because the S = 1111 cross section even-
Anaturalconstraint thatthelifetime ofdarkmatterisequal tually coincides with that of the unpolarized photon–
to the age of the universe at the shortest is expressed as the photon scattering. Since the solid angle and the ef-
dash-dottedline. Thethreedashedlinesshowaccessiblecou- fect of the induction laser field are common to both
plinglimitswithN =1010,1015,1020 commonfork=1,2,4,
k the pNGB exchange and the QED process, we have
respectivelybasedonEq.(32)forthecommon parameterset
only to plug the partially integrated QED cross section
in Table I. This figure shows the domain in which the QED
into Eq. (27) in order to get the QED-based scattering
process dominates thecross section of a pNGB exchange ob-
yield. The boundary line of the gray translucent area
served by the same method. The boundary line of the gray
translucent area shows m–g/M, at which σ˜ngb =σ˜qed is sat- in Fig.3 shows m–g/M, at which σ˜ngb = σ˜qed is satis-
isfied. fied. This equi-cross-sectionline intersects with the con-
straint from a natural requirement that the life time of
a pNGB is equal to the age of the universe [3] at around
k = 1,2,4, respectively based on Eq. (32) in the range (m,g/M) = (2 keV,10−11 GeV−1). Although the QED
0.1 eV to 10 keV. The experimentally excluded and the- cross section exceeds the pNGB cross section consistent
oretically expected domains of the mass–coupling rela- with a natural dark matter candidate (dash-dotted line)
tionship are imported from Ref. [16], where all details at m=7 keV, the search window of m < 2 keV is still
areexplained. Forthisfigure,thesetofparameterssum- wide-open with relatively low QED backgroundlevels.
marized in Table I is assumed, where the parameters in
the upper rows can be commonly used for any ω =m/2.
We note here that a long f-number F is assumed delib- IV. LIGHT SOURCES
erately, thereby guaranteeing that the CMS energy does
not fluctuate because of the uncertainty in the incident There are already several short-pulsed coher-
angles but rather because of the energy spread of the ent/incoherent light sources that cover the range
creation beams. This allows the use of nearly parallel 0.1 eV to 10 keV. In the 1–10 keV range, free-electron
incoherent creation pulses, for instance, those generated lasers are already available. For instance, an X-ray
by laser-electronCompton scattering, as we discussed in free-electron laser (XFEL), SACLA [19], can provide
all-optical-based γ–γ scattering [18]. As an example of N 1011 at f = 60 Hz in that energy range with
k
∼
ω, we show concrete energies and scattering angles in an undulator length of 90 m. Although introducing
the second set of rows if we aim at m = 7 keV in order three long XFEL lines would likely be difficult from
to test the possibility of pNGB exchange to explain the a practical point of view, incoherent photon–photon
excess [8]. collisions for only the creation part combined with
Above the keV range, we must take photon–photon an XFEL for the stimulation part would be the least
scattering based on quantum electrodynamics (QED) time-consuming approach. This is because incoherent
into account as a background source from the standard light sources can be attainable using relatively compact
6
Lorentzfactor to boost CMS energies γ =1.5 ric head-on collision system that would be applicable to
themassrangeof0.1eVto10keV.Abovem=2 keV,we
Scattering angle in CMS θ=π/4
foundthattheQEDphoton–photonscatteringcrosssec-
Incidentangle of induction beam θ4 =1.65 rad
tiondominatesthatofapNGBexchangewhoselifetimeis
Scattering angle of signal photons θ3 =0.31 rad
consistentwith the ageofthe universe. Especiallyinthe
Common f-numberof creation beams F =100 0.1-100eV range,the domainwith g/M <10 12 GeV 1
− −
Inductionbeam f-number F4 =10 has not been tested against any observations to date.
Common duration time for creation beams τ =ZR2/c Therefore, the proposed method could provide a wide
Integration time from longest Rayleigh length τL =ZR2/c search window onto an unexplored valley in the sensi-
Common energy uncertaintyof creation beams R=5% tivity curve with relatively suppressed QED background
Collision repetition rate f =1 Hz levels. Thesensitivitycouldeventuallyreachthedomain
Data accumulation time T =106 s beyond the GUT scale M 1016 GeV, hence, our pro-
∼
posal could provide opportunities to test string-theory-
Total numberof signal photons Y = 100
based models [23].
Detectorefficiency to signal photons ǫ=100%
Creation beam energy in CMS ω=3.50 keV Itwouldbevaluableforfutureexperimentstotunethe
Highercreation beam energy in ACS ω1 =9.16 keV beamenergy andintensity so that the expected sensitiv-
ity could reach the equi-cross-section line. Confirmation
Lower creation beam energy in ACS ω2 =1.34 keV
oftheQEDprocessisfirstlyimportanttoguaranteethat
Inductionbeam energy in ACS ω4 =2.48 keV
the experiment is indeed performing properly. Once the
Signal photon energy in ACS ω3 =8.02 keV
QEDscatteringhasbeenconfirmed,itwouldbeinterest-
ingtotestwhetherinterferenceexistsbetweenthepNGB
exchange and the QED process, for example, by look-
TABLE I: Three-beam parameters in CMS and ACS (labo-
ratory frame). The second set of rows gives example values ing at the angular distribution of the signal photons by
when tunedat m=2ω=7 keV. changing the incident angle of the induction laser beam
andalsothedependenceonthecombinationoflinearpo-
larization states. By repeating this test over four orders
all-optical laser systems [18]. In addition, such a long- of magnitude in the CMS energies, if we were to see a
scale undulator might be drastically shortened by future statistically significant deviation from the QED predic-
developments in compact coherent X-ray sources, such tioninaparticularmassrange,wecouldreducethelaser
as a graphene-based undulator [21]. In the 10 eV–1 keV intensity at which the QED effect becomes insignificant.
range, the generation of higher harmonics by shooting If we still see the scattering phenomenon, we would be
high-intensity laser pulses into material targets could be able to claim that something dark is exchanged in the
useful [20]. In the 0.1–10 eV range, variable-wavelength photon–photon scattering process.
lasers based on optical parametric amplification are
available as commercial products. State-of-the-art laser
facilities such as the Extreme Light Infrastructure [22]
can reachnumbers of optical coherentphotons per pulse
beyond N 1020 with τ 10 fs. These facilities can
k L
∼ ∼
generate multi-wavelength laser fields up to the keV
Acknowledgments
range with high intensity by combining several of the
methods mentioned above.
We thank Y. Fujii for helpful discussions. K. Homma
acknowledges the support of the Collaborative Research
V. CONCLUSION Program of the Institute for Chemical Research, Kyoto
University(GrantsNos. 2016–68)andtheGrants-in-Aid
We formulateda stimulatedphoton–photonscattering for Scientific Research Nos. 15K13487 and 16H01100
processviaans-channelpNGBexchangeinanasymmet- from MEXT of Japan.
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