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Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Effective Field Theory of Gravity:Leading Quantum Gravitational Corrections to
Newtons and Colulombs Law
Sven Faller
Theoretische Physik 1Universität Siegen
Theorieseminar Universität Köln – 02.02.2009
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
table of contents
1 Introduction
2 Quantum Gravity
3 EFT of Gravity
4 Potential
5 Summary
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Motivation
all known field theories: quantum field theoriesgravity quantization – Feynman (1962)
Is it possible that gravity is not quantized and all the rest ofthe world is? – Answer: No!illustration: Feynman’s two-slit diffraction Gedankenexperiment→ quantum nature of a field could not be destroyed
gravity must be a quantum field theoryproblem: consistent quantization method unknownQuantum Gravity: De Witt, Feynman, ’t Hooft and Veltman
present energies: quantum gravity non-renormalizablebut: low-energy predictions independent of high-energyinfluence
possible solution:Effective Field Theory of Gravity (Donoghue 1994)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Motivation
simple dimensional analysis: lowest-order corrections toNewtonian potential
V (r) = −Gm1m2
r
(1+α
G(m1 + m2)
rc2︸ ︷︷ ︸relativistic correction
+ βG~r2c3︸ ︷︷ ︸
quantum correction
+ . . .
)
lot of papers with different results for coefficients α and β, e.g.Paper α β
Donoghue (1994) -1 − 12730π2
Akhundov et al. (1997) +1 − 10730π2
Bjerrum-Bohr et al. (2003) +3 4110π
correct numbers:sufficiently interesting from theoretical point of view
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Newton’s Gravity (1687)
gravity force
~F12(~r) = mi ~a = −G mG MG~r1 −~r2
|~r1 −~r2|3
three different masses: inertial mass mi ,passive gravitational mass mG and activegravitational mass MG
third law - “actio est reactio„- inert and activegravitational mass equal→ problem: no explanation for equality
– seems to be coincidental↪→ experimental measurements:
verification of equality, basic for↪→ Einstein’s Principle of Equivalence
Torsionfaden
Spiegel
Kugel derMasse m
Kugel derMasse M
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Einstein’s Special Relativity
Newton: Galilei transformations between ISEinstein (1905):Newton’s Theory must be specialized by universality of thevelocity of light in all frames
x 7−→ x ′ = ΛΛΛx + a (Lorentz transformation)
Postulategeneral transformation for the line element must satisfy
ds2 = ηαβ dxµ dxν = c2 dt2 − d~x2
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
General Relativity
Einstein (1916):Die Grundlagen der allgemeinen Relativitätstheorie.Ann. d. Physik, 322(10):891-921Newton: space R3 and parameter time Rt
Einstein: new relations between space-time and mass⇒ curved space-time mannifoldcurvature of space = measure for mass:„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Some Definitions
units: ~ = c = 1flat space: ηµν = diag(1,−1,−1,−1)
metric tensor gµν and g = detgµνRieman Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12 gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Background Field Method
introduced by ’t Hooft and Veltman (1974)gravitational field expanded about smooth backgroundmetric gµν
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
classical equations of motion: gµν satifies
Rµν − 12
gµν R = −κ2
4Tµν (Einstein’s equation)
quantum field hµν : all dynamical information
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν +O(R3) + Lm
}upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Scalar QED
general: scalar (charged or neutral) matterLm = 1
2
(gµν ∂µφ ∂νφ∗ −m2 |φ|2)
generally covariant Lagrangian for scalar QED
LSQED =√−g
[−1
4(gαµgβνFαµFβν
)+gµν DµφDνφ
∗−m2|φ|2]
photon field strength Fµν = DµAν − DνAµQED covariant derivative Dµ = ∂µ − ieqAµ(x)
Feynman Rules at 1 loop order→ expansion LSQED tosecond order in hphoton: Lorenz gauge LC = −1
2 (∂µAµ)2
this talk: only scalar theory discussed
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Quantization Problems
nonlinear nature of the theorydimensionful coupling constant κ =
√32πG
↪→ divergences appear which can not absorbed by introducedparameters
coupling grows with energy↪→ strong energy coupling at very high energies E > MPl
possible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Effective Quantum Gravity
quantization process: first order curvaturerenormalization higher order needed→ value of c1, c2 shifted↪→ renormalization effects absorbed by c(r)
1 and c(r)2
↪→ one-loop Feynman rules: no contribution from higher order
gravitational action at one loop
S = Svac + Sm + Sgf + Sghost
=
∫d4x
√−g{
2R
κ2 + Lgf + Lghost︸ ︷︷ ︸= Lgr
+Lm
}
quantum degrees of freedom:gravitation field hµν and ghostfields ηµ, η∗µ
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Effective Lagrangian
low-energy d.o.f.: hµν + ghost fields + matter fields
Z[J] =
∫[dφ][dhµν ]eiSeff(φ,g,h,J)
Seff =∫
d4x√−g Leff, Leff = Lgr + Lm
effective Lagrangian = expansion in powers of hµν
Lgr = L(0)gr + L(2)
gr + L(4)gr + . . .
Lm = L(0)m + L(2)
m + . . .
→ Feynman rules from Leff
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
S-Matrix
two particle scattering process: momentum transfer qFeynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-spacepotential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r
higher order effects: O(v2/c2), O(Gm/rc2)
pure gravity interaction general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis:loop diagrams→ extra power of κ2 ∼ G, factor ~
pure gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
One Particle Irreducible Diagrams - Scalar Theory
k1 k2
k′1 k′
2q
= + +q
+set
of+ +
+
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
One Particle Irreducible Diagrams - SQED
(m1, e1) (m2, e2) =
+set
of
{+ +
}
+set
of+
set
of
{+
}
+ +vacuum polarization
diagrams
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Vacuum Polarization Diagrams
scalar theory:
self-energy ghost massless scalar particle photon
k1 k2
k′1 k′
2
′t Hooft and Veltman (1974) Capper et al. (1974) Capper et al. (1974)
Hamber, Liu (1995)
SQED: only one diagram: Bjerrum-Bohr (2002)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Result: Scalar Theory
Scattering Potential
V (r) = −G m1 m2
r
[1 + 3
G(m1 + m2)
r c2 +131 + 6Nν
30πG ~r2c3
]
new result for massless scalar loopphoton loop calculation:in agreement with Capper et al. (1974)results from previous publications could be verified, e.g.Bjerrum-Bohr (2003)non-relativistic potential:~Gc3 ∼ 10−70m2 corrections should be unmeasureable
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Result: SQED
Scattering Potential
VSQED(r) =VGR(r)
+αe1e2
r
(1 + 3
G (m1 + m2)
c2r+
6π
G~c3r2
)+
12
(m1e22 + m2e
21) Gα
c2r2 − 43π
(m2
2e21 + m2
1e22
m1m2
)Gα~c3r3
rescaled by α = ~c/137charges e1, e2 normalized in units of elementary chargesagreement with previous publication: Bjerrum-Bohr (2002)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictionsEFT only valid at energies below Planck scale ∼ 1019 GeVand long distances
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravityfurther interesting papers:
M. S. Butt: Leading quantum gravitational corrections toQED, [arXiv:gr-qc/0605137] (2006)B. R. Holstein and A. Ross: Long Distance Effects in MixedElectromagnetic-Gravitational Scatterin,[arxiv:0802.0717] (2008)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Feynmans Gedankenexperiment (Feynman Lectures on Gravitation 1962-1963)
postulate of quantum mechanical behavior: there is anamplitude ψ for different processes↪→ particle described by ψ cannot have interaction described
by a probabilitytwo-slit diffraction experiment with an electron
classical gravity detector: information about which slit theelectron has passed
Intensity
gravity
detector
wavegravitational
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Feynmans Gedankenexperiment (Feynman Lectures on Gravitation 1962-1963)
no signal: electron position described by an
Amplitude =12ψ(upper slit) +
12ψ(lower slit)
if gravitational interaction transmitted by a field→ gravity field must have an amplitude:
12 of the amplitude corresponds to gravity field of anelectron which went through either slit
↪→ precisely characteristic of a quantum field
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Recall: Lorentz Invariance
global coordinate change: xµ 7−→ x ′µ = Λµν xν
Minkowski metric ηµν invariantfields transform as scalars, vectors, etc.
φ(x) 7−→ φ′(x ′) = φ(x)
Aµ(x) 7−→ A′µ(x) = Λµν(x) Aν(x)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
General relativity as a gauge theory (Sabbata 1985)
Poincaré group is non abeliancf. Yang-Mills theory
Lgauge = −14
F aµν F aµν = −1
2trF 2
gravity: introduction of vierbein- or tetrad fields e λµ
Lgauge = − e2g
eµλeνσR λσ
µν (ωωω) ≡ 2κ2
√−g R
with g = det[gµν ] and κ2 = 32πG.
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion - Affine Connection
background field method: metric expansion
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
affine connection: Γλµν = Γλµν +−Γλµν +
=Γλµν
with
Γλµν =12
gλσ(∂µgσν + ∂ν gσµ − ∂σgµν
)(O(h0)) ,
−Γλµν =
κ
2gλσ(Dµhσν + Dνhσµ − Dσhµν
)(O(h1)) ,
=Γλµν = −κ
2
2hλγ(Dµhγν + Dνhµγ − Dγhµν
)(O(h2)) .
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion: Curvature
Riemann curvature tensor:
Rβαµν = DµΓβαν − DνΓβαµ + ΓλανΓβλµ − ΓλαµΓβλν ≡ Rβαµν +
−Rβαµν +
=Rβαµν
Ricci scalar:
R = gαµ=Rαµ − κhαµ
−Rαµ + κ2hαγ hγµRαµ
= κ2−
12
Dµ`hβγDµhγβ
´+
12
Dβˆhβν`2Dµhνµ − Dνhµµ
´˜+
14
`Dµhνβ + Dβhνµ − Dνhµβ
´`Dµhβν + Dνhβµ − Dβhµν
´−
14
`2Dµhνµ − Dνhµµ
´Dνhββ −
12
hαµDµDαhββ
+12
hµαDβ`Dαhβµ + Dµhβα − Dβhαµ
´+ κ2hβµhαβ Rµ
α
ff
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν − 12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cn
two different typs of quantum field theoriesasympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Evaluation Vertex Factors
momentum space vertex factors
Vµ1ν1,...,µmνn = +iZ
d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)
· δ
δJ1(x1)· . . . · δ
δJn(xn)· δ
δHν1µ11 (y1)
· . . . · δ
δHµmνmm (ym)
· Lint`φ1, . . . , φn,H1, . . .Hm
´(x)
sources of gravity: J1, . . . , Jn
external and internal gravity field: Hµ1ν11 , . . . ,Hµmνm
m
incoming p1, . . . ,pn, outgoing q1, . . .qm momentum
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Graviton Progpagator
second order Lagrangian Lgr
harmonic gauge→ gauge fixing Lagrangian Lgf
quantum field hµν bilinear Lagrangian Lfreegr = − 1
2 hαβ ∆−1αβγδ hγδ
graviton propagator in harmonic gauge
qαβ µν =
12
iq2 + iε
(ηαµηβν + ηανηβµ − ηαβηµν)
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Vertex Factors
vertex factors at one-loop order
−→q
p
p′
ℓ′ ր p
ℓտ p′
−→k
ց q
ր ℓ
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Scalar-Graviton-Vertex
vertex factor
τµν = iZ
d4x d4x1 d4x2 d4x3 ei(px1−p′x2+qx3) · ∂
∂φ(x1)
∂
∂φ(x2)
∂
∂hµν(x3)
·−κ
2hαβ ·
»∂αφ(x)∂βφ(x)− 1
ηαβ`∂γφ(x)∂γφ(x)−m2φ(x)2´–ff
scalar-graviton-vertex
−→q
p
p′
µν = − iκ2
{pµp′ν + pνp′µ − ηµν
[(p · p′)−m2]}
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Graviton-Graviton-Scalar-Vertex
Lagrangian O(h2)
L(2)m = κ2
„12
hµνhνλ−14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ−
12
hh«ˆ∂µφ∂
µφ−m2φ2˜vertex factor
Vηλρσ = +iZ
d4x d4x1 d4x2 d4x3 d4x4 ei(px1−p′x2+kx3−kx4)
· ∂
∂φ(x1)· ∂
∂φ(x2)· ∂
∂hηλ(x3)· ∂
∂hρσ(x4)
· κ2
2hηλ»
1ηλαδ1δ
ρσβ −14`ηηλ1ρσαβ + ηρσ1ηλαβ
´–∂αφ(x)∂βφ(x)
− 14
„1ηλρσ −
12ηηλ −
12ηηληρσ
«ˆ∂γφ(x)∂γφ(x)−m2φ(x)2˜ffhρσ
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Graviton-Graviton-Scalar-Vertex
ℓրρσ
p
ℓ′ տ p′ηλ
Vηλρσ =iκ2
2
»1ηλαδ1
δρσβ −
14
`ηηλ1ρσαβ + ηρσ1ηλαβ
´–`pαp′β + pβp′α
´−
12
»1ηλρσ −
12ηηληρσ
–`(p · p′)−m2´ff
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Three-Graviton-Vertex
τµναβγδ(k , q) = − iκ2
Pαβγδ
»kµkν + (k − q)µ(k − q)ν + qµqν +
32ηµνq2
–+ 2qλqσ
ˆ1
σλαβ 1
µνγδ + 1
σλγδ 1
µναβ − 1 µσ
αβ 1νλ
γδ − 1 µσγδ 1
νλαβ
˜+ˆqλqµ
`ηαβ1
νλγδ + ηγδ1
νλαβ
´+ qλqν
`ηαβ1
µλγδ + ηγδ1
µλαβ
´− q2`ηαβ1 µν
γδ + ηγδ1µν
αβ
´− ηµνqλqσ
`ηαβ1
σλγδ + ηγδ1
σλαβ
´˜+ˆ2qλ
˘1
λσαβ 1
νγδσ (k − q)µ + 1
λσαβ 1
µγδσ (k − q)ν
− 1 λσγδ 1
ναβσ kµ − 1 λσ
γδ 1µ
αβσ kν¯
+ q2`1
µαβσ 1
νσγδ + 1
νσαβ 1
µγδσ
´+ ηµνqσqλ
`1λραβ1
σγδρ + 1
λργδ 1
σαβρ
´˜+
`k2 + (k − q)
´»1
µσαβ 1
νγδσ + 1
νσγδ 1
µαβσ −
12ηµνPαβγδ
–−`1
µνγδ ηαβk2 − 1 µν
αβ ηγδ(k − q)2´ffffffff
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)
nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Back
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributions
li : one-loop contributions
l1, l4 : divergent high enery contributions
l2, l3, l5, l6 : finite non-analytic low energy contributions
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Renormalization
combination l1, l4 and di → renormalized values
d (r)1 (µ2) = d1 + κ2l1
d (r)2 (µ2) + d (r)
3 (µ2) = d2 + d3 − κ2 l44
experiments: measure renormalized values
d (r)i (µ2)→ measured values depend on µ2 choice in
logarithms but all physics independent of µ2
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion: Vacuum Lagrangian
metric expansion:
p−g =
p−g
1−κ
2hαα −
κ2
4hαβhβα +
κ2
8
`hαα´2
+O(h3)
ffLagrangian expansion
2κ2
p−gR =
p−g»
2κ2
R + L(1)gr + L(2)
gr + . . .
–,
L(1)gr =
1κ
hµνˆgµνR − 2Rµν
˜,
L(2)gr =
12
Dαhµν Dαhµν −12
Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα
+ R
„14
h2 −12
hµνhµν«
+ Rµν`2hλµhνλ − h hµν
´.
Introduction Quantum Gravity EFT of Gravity Potential Summary Backup
Expansion: Matter Lagrangian
e.g. scalar particle: Lm =√−g
[12gµν∂µφ∂νφ− 1
2m2φ2]Lagrangian expansion:
Lm =p−g˘L(0)
m + L(1)m + L(2)
m + . . .¯
L(0)m =
12
`∂µφ∂
µφ−m2φ2´L(1)
m = −κ
2hµνTµν
Tµν ≡ ∂µφ∂µφ−12
gµν`∂λφ∂
λφ−m2φ2´ (energy-momentum-tensor)
L(2)m = κ2
„12
hµλhνλ −14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ −
12
hh«ˆ∂µφ∂
µφ−m2φ2˜