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Vacuum configurations for renormalisablenoncommutative scalar field theory
Axel de Goursac
LPT Orsay
in collaboration with:- Adrian Tanasa
- Jean-Christophe Wallet
arXiv:0709.3950 [hep-th],to appear in Eur. Phys. J. C
Axel de Goursac Vienna, November 29, 2007
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Plan
Scalar field theory with harmonic term on the Moyal space
Equation of motion
Vacuum configurations
Linear sigma model
Conclusions and perspectives
Axel de Goursac Vienna, November 29, 2007
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Plan
Scalar field theory with harmonic term on the Moyal space
Equation of motion
Vacuum configurations
Linear sigma model
Conclusions and perspectives
Axel de Goursac Vienna, November 29, 2007
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Plan
Scalar field theory with harmonic term on the Moyal space
Equation of motion
Vacuum configurations
Linear sigma model
Conclusions and perspectives
Axel de Goursac Vienna, November 29, 2007
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Plan
Scalar field theory with harmonic term on the Moyal space
Equation of motion
Vacuum configurations
Linear sigma model
Conclusions and perspectives
Axel de Goursac Vienna, November 29, 2007
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Plan
Scalar field theory with harmonic term on the Moyal space
Equation of motion
Vacuum configurations
Linear sigma model
Conclusions and perspectives
Axel de Goursac Vienna, November 29, 2007
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Moyal space
Deformation of Euclidean RD : Mθ = (S(RD), ?θ)
(f ?θ g)(x) =1
πD | det Θ|
∫dDydDz f (x + y)g(x + z)e−2iyΘ−1z
Θ =
0 −θθ 0
0
. . .
00 −θθ 0
Properties: [xµ, xν ]? = iΘµν (xµ = 2Θ−1
µν xν)∫dDx (f ? g)(x) =
∫dDx f (x)g(x)
∂µφ = − i
2[xµ, φ]? xµφ =
1
2xµ, φ?
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Moyal space
Deformation of Euclidean RD : Mθ = (S(RD), ?θ)
(f ?θ g)(x) =1
πD | det Θ|
∫dDydDz f (x + y)g(x + z)e−2iyΘ−1z
Θ =
0 −θθ 0
0
. . .
00 −θθ 0
Properties: [xµ, xν ]? = iΘµν (xµ = 2Θ−1
µν xν)∫dDx (f ? g)(x) =
∫dDx f (x)g(x)
∂µφ = − i
2[xµ, φ]? xµφ =
1
2xµ, φ?
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Moyal space
Deformation of Euclidean RD : Mθ = (S(RD), ?θ)
(f ?θ g)(x) =1
πD | det Θ|
∫dDydDz f (x + y)g(x + z)e−2iyΘ−1z
Θ =
0 −θθ 0
0
. . .
00 −θθ 0
Properties: [xµ, xν ]? = iΘµν (xµ = 2Θ−1
µν xν)∫dDx (f ? g)(x) =
∫dDx f (x)g(x)
∂µφ = − i
2[xµ, φ]? xµφ =
1
2xµ, φ?
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Matrix base (2 dimensions)
(fmn(x))m,n∈N, eigenfunctions of the 2D harmonic oscillator:base of Mθ
fmn(x) = 2(−1)m√
m!
n!e i(n−m)ϕ
(2r2
θ
) n−m2
Ln−mm
(2r2
θ
)e−
r2
θ
Properties: f †mn(x) = fnm(x)
(fmn ? fkl)(x) = δnk fml(x)
if g(x) =∑∞
m,n=0 gmnfmn(x), then
gmn =1
2πθ
∫d2x g(x)fnm(x)
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Matrix base (2 dimensions)
(fmn(x))m,n∈N, eigenfunctions of the 2D harmonic oscillator:base of Mθ
fmn(x) = 2(−1)m√
m!
n!e i(n−m)ϕ
(2r2
θ
) n−m2
Ln−mm
(2r2
θ
)e−
r2
θ
Properties: f †mn(x) = fnm(x)
(fmn ? fkl)(x) = δnk fml(x)
if g(x) =∑∞
m,n=0 gmnfmn(x), then
gmn =1
2πθ
∫d2x g(x)fnm(x)
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Matrix base (2 dimensions)
(fmn(x))m,n∈N, eigenfunctions of the 2D harmonic oscillator:base of Mθ
fmn(x) = 2(−1)m√
m!
n!e i(n−m)ϕ
(2r2
θ
) n−m2
Ln−mm
(2r2
θ
)e−
r2
θ
Properties: f †mn(x) = fnm(x)
(fmn ? fkl)(x) = δnk fml(x)
if g(x) =∑∞
m,n=0 gmnfmn(x), then
gmn =1
2πθ
∫d2x g(x)fnm(x)
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Matrix base (2 dimensions)
(fmn(x))m,n∈N, eigenfunctions of the 2D harmonic oscillator:base of Mθ
fmn(x) = 2(−1)m√
m!
n!e i(n−m)ϕ
(2r2
θ
) n−m2
Ln−mm
(2r2
θ
)e−
r2
θ
Properties: f †mn(x) = fnm(x)
(fmn ? fkl)(x) = δnk fml(x)
if g(x) =∑∞
m,n=0 gmnfmn(x), then
gmn =1
2πθ
∫d2x g(x)fnm(x)
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Scalar field theory with harmonic term
Renormalisable theory (for Ω 6= 0,hep-th/0401128):
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
Ω2
2(xµφ) ? (xµφ)
+M2
2φ ? φ + λ φ ? φ ? φ ? φ
)Spontaneous symmetry breaking: M2 → −µ2
Special value Ω = 1:
invariant under Langmann-Szabo duality (hep-th/0202039)
(pµ ↔ xµ)
stable under renormalisation group (hep-th/0612251)
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Scalar field theory with harmonic term
Renormalisable theory (for Ω 6= 0,hep-th/0401128):
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
Ω2
2(xµφ) ? (xµφ)
+M2
2φ ? φ + λ φ ? φ ? φ ? φ
)Spontaneous symmetry breaking: M2 → −µ2
Special value Ω = 1:
invariant under Langmann-Szabo duality (hep-th/0202039)
(pµ ↔ xµ)
stable under renormalisation group (hep-th/0612251)
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Scalar field theory with harmonic term
Renormalisable theory (for Ω 6= 0,hep-th/0401128):
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
Ω2
2(xµφ) ? (xµφ)
+M2
2φ ? φ + λ φ ? φ ? φ ? φ
)Spontaneous symmetry breaking: M2 → −µ2
Special value Ω = 1:
invariant under Langmann-Szabo duality (hep-th/0202039)
(pµ ↔ xµ)
stable under renormalisation group (hep-th/0612251)
Axel de Goursac Vienna, November 29, 2007
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Scalar field theory with harmonic term
Renormalisable theory (for Ω 6= 0,hep-th/0401128):
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
Ω2
2(xµφ) ? (xµφ)
+M2
2φ ? φ + λ φ ? φ ? φ ? φ
)Spontaneous symmetry breaking: M2 → −µ2
Special value Ω = 1:
invariant under Langmann-Szabo duality (hep-th/0202039)
(pµ ↔ xµ)
stable under renormalisation group (hep-th/0612251)
Axel de Goursac Vienna, November 29, 2007
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Scalar field theory with harmonic term
Renormalisable theory (for Ω 6= 0,hep-th/0401128):
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
Ω2
2(xµφ) ? (xµφ)
+M2
2φ ? φ + λ φ ? φ ? φ ? φ
)Spontaneous symmetry breaking: M2 → −µ2
Special value Ω = 1:
invariant under Langmann-Szabo duality (hep-th/0202039)
(pµ ↔ xµ)
stable under renormalisation group (hep-th/0612251)
Axel de Goursac Vienna, November 29, 2007
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Equation of motion
For Ω = 1, the equation of motion:
−∂2v(x) + x2v(x)−µ2v(x) + 4λ (v ? v ? v)(x) = 0
1
2x2, v? − µ2v + 4λ v ? v ? v = 0
if v(x) =∑∞
m,n=0 vmnfmn(x), then ((x2)mn = 4θ (2m + 1)δmn)
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
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Equation of motion
For Ω = 1, the equation of motion:
−∂2v(x) + x2v(x)−µ2v(x) + 4λ (v ? v ? v)(x) = 0
1
2x2, v? − µ2v + 4λ v ? v ? v = 0
if v(x) =∑∞
m,n=0 vmnfmn(x), then ((x2)mn = 4θ (2m + 1)δmn)
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
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Equation of motion
For Ω = 1, the equation of motion:
−∂2v(x) + x2v(x)−µ2v(x) + 4λ (v ? v ? v)(x) = 0
1
2x2, v? − µ2v + 4λ v ? v ? v = 0
if v(x) =∑∞
m,n=0 vmnfmn(x), then ((x2)mn = 4θ (2m + 1)δmn)
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Symmetry of the solution
Vacuum invariant under rotations: v(x) = g(x2)
vmn =1
2πθ
∫d2x g(x2)fnm(x)
vmn =1
2πθ
∫rdrdϕ g(r2)2(−1)n
√n!
m!e i(m−n)ϕ
(2r2
θ
)m−n2
Lm−nn
(2r2
θ
)e−
r2
θ
vmn =2(−1)m
θ
∫rdr g(r2)L0
m
(2r2
θ
)e−
r2
θ δmn
The vacuum is of the form:
vmn = amδmn, am ∈ R
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Solutions
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
vmn= amδmn
∀m ∈ N,(4
θ(2m + 1)− µ2 + 4λ a2
m
)am = 0
Solutions: am = 0 or a2m = 2
λθ
(µ2θ8 − 1
2 −m)≥ 0
p = bµ2θ
8− 1
2c → 2p solutions
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Solutions
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
vmn= amδmn
∀m ∈ N,(4
θ(2m + 1)− µ2 + 4λ a2
m
)am = 0
Solutions: am = 0 or a2m = 2
λθ
(µ2θ8 − 1
2 −m)≥ 0
p = bµ2θ
8− 1
2c → 2p solutions
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Solutions
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
vmn= amδmn
∀m ∈ N,(4
θ(2m + 1)− µ2 + 4λ a2
m
)am = 0
Solutions: am = 0 or a2m = 2
λθ
(µ2θ8 − 1
2 −m)≥ 0
p = bµ2θ
8− 1
2c → 2p solutions
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Solutions
(4
θ(m + n + 1)− µ2
)vmn + 4λ
∞∑k,l=0
vmkvklvln = 0
vmn= amδmn
∀m ∈ N,(4
θ(2m + 1)− µ2 + 4λ a2
m
)am = 0
Solutions: am = 0 or a2m = 2
λθ
(µ2θ8 − 1
2 −m)≥ 0
p = bµ2θ
8− 1
2c → 2p solutions
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Propagator
Spontaneous symmetry breaking: φ(x) → v(x) + φ(x)
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
1
2(xµφ) ? (xµφ) +
m2
2φ ? φ
+4λ v ? v ? φ ? φ + 2λ v ? φ ? v ? φ
+ 4λ v ? φ ? φ ? φ + λ φ ? φ ? φ ? φ)
The propagator is given by
Squadr = 2πθ
∞∑m,n=0
(2
θ(m + n + 1)− µ2
2+2λ
p∑k=0
a2k(δmk + δnk)
+2λ
p∑k,l=0
akalδmkδnl
)φmnφnm
Squadr = C−1m,n;k,lφmnφkl
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Propagator
Spontaneous symmetry breaking: φ(x) → v(x) + φ(x)
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
1
2(xµφ) ? (xµφ) +
m2
2φ ? φ
+4λ v ? v ? φ ? φ + 2λ v ? φ ? v ? φ
+ 4λ v ? φ ? φ ? φ + λ φ ? φ ? φ ? φ)
The propagator is given by
Squadr = 2πθ
∞∑m,n=0
(2
θ(m + n + 1)− µ2
2+2λ
p∑k=0
a2k(δmk + δnk)
+2λ
p∑k,l=0
akalδmkδnl
)φmnφnm
Squadr = C−1m,n;k,lφmnφkl
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Propagator
Spontaneous symmetry breaking: φ(x) → v(x) + φ(x)
S [φ] =
∫dDx
(1
2∂µφ ? ∂µφ +
1
2(xµφ) ? (xµφ) +
m2
2φ ? φ
+4λ v ? v ? φ ? φ + 2λ v ? φ ? v ? φ
+ 4λ v ? φ ? φ ? φ + λ φ ? φ ? φ ? φ)
The propagator is given by
Squadr = 2πθ
∞∑m,n=0
(2
θ(m + n + 1)− µ2
2+2λ
p∑k=0
a2k(δmk + δnk)
+2λ
p∑k,l=0
akalδmkδnl
)φmnφnm
Squadr = C−1m,n;k,lφmnφkl
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Spontaneous symmetry breaking
Vacuum condition: ∀m ∈ 0, ..p, a2m > 0 (p = bµ2θ
8 − 12c)
−µ2 > −4θ (p < 0):
v(x) = 0: global minimum of the action
µ2θ8 − 1
2 ∈ N:Local zero mode in the propagator
−µ2 < −4θ (p ≥ 0) and µ2θ
8 − 12 /∈N:
v(x) =
p∑k=0
√2
λθ
(µ2θ
8− 1
2−m
)fmm(x)
is a local minimum of the action
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Spontaneous symmetry breaking
Vacuum condition: ∀m ∈ 0, ..p, a2m > 0 (p = bµ2θ
8 − 12c)
−µ2 > −4θ (p < 0):
v(x) = 0: global minimum of the action
µ2θ8 − 1
2 ∈ N:Local zero mode in the propagator
−µ2 < −4θ (p ≥ 0) and µ2θ
8 − 12 /∈N:
v(x) =
p∑k=0
√2
λθ
(µ2θ
8− 1
2−m
)fmm(x)
is a local minimum of the action
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Spontaneous symmetry breaking
Vacuum condition: ∀m ∈ 0, ..p, a2m > 0 (p = bµ2θ
8 − 12c)
−µ2 > −4θ (p < 0):
v(x) = 0: global minimum of the action
µ2θ8 − 1
2 ∈ N:Local zero mode in the propagator
−µ2 < −4θ (p ≥ 0) and µ2θ
8 − 12 /∈N:
v(x) =
p∑k=0
√2
λθ
(µ2θ
8− 1
2−m
)fmm(x)
is a local minimum of the action
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Spontaneous symmetry breaking
Vacuum condition: ∀m ∈ 0, ..p, a2m > 0 (p = bµ2θ
8 − 12c)
−µ2 > −4θ (p < 0):
v(x) = 0: global minimum of the action
µ2θ8 − 1
2 ∈ N:Local zero mode in the propagator
−µ2 < −4θ (p ≥ 0) and µ2θ
8 − 12 /∈N:
v(x) =
p∑k=0
√2
λθ
(µ2θ
8− 1
2−m
)fmm(x)
is a local minimum of the action
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Linear sigma model(1)
S =
∫d4x
(1
2(∂µΦi ) ? (∂µΦi ) +
1
2x2ΦiΦi
− µ2
2ΦiΦi + λΦi ? Φi ? Φj ? Φj
)Vacuum: < Φ >= (0, . . . , 0, v(x))fluctuation: δΦ = (π1, . . . , πN−1, σ(x))Spontaneous symmetry breaking: Φ →< Φ > +δΦ
S =
∫d4x
(1
2(∂µπi ) ? (∂µπi ) +
1
2x2πiπi −
µ2
2πiπi
+2λv ? v ? πi ? πi + λπi ? πi ? πj ? πj + ...
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Linear sigma model(1)
S =
∫d4x
(1
2(∂µΦi ) ? (∂µΦi ) +
1
2x2ΦiΦi
− µ2
2ΦiΦi + λΦi ? Φi ? Φj ? Φj
)Vacuum: < Φ >= (0, . . . , 0, v(x))fluctuation: δΦ = (π1, . . . , πN−1, σ(x))Spontaneous symmetry breaking: Φ →< Φ > +δΦ
S =
∫d4x
(1
2(∂µπi ) ? (∂µπi ) +
1
2x2πiπi −
µ2
2πiπi
+2λv ? v ? πi ? πi + λπi ? πi ? πj ? πj + ...
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Linear sigma model(1)
S =
∫d4x
(1
2(∂µΦi ) ? (∂µΦi ) +
1
2x2ΦiΦi
− µ2
2ΦiΦi + λΦi ? Φi ? Φj ? Φj
)Vacuum: < Φ >= (0, . . . , 0, v(x))fluctuation: δΦ = (π1, . . . , πN−1, σ(x))Spontaneous symmetry breaking: Φ →< Φ > +δΦ
S =
∫d4x
(1
2(∂µπi ) ? (∂µπi ) +
1
2x2πiπi −
µ2
2πiπi
+2λv ? v ? πi ? πi + λπi ? πi ? πj ? πj + ...
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Linear sigma model(2)
Condition for vanishing π-mass term:∫d4x
(1
2x2πiπi −
µ2
2πiπi + 2λv ? v ? πi ? πi
)=
∫d4x
(Ω′2
2x2πiπi
)+ kinetic terms...
=⇒ v ? v =Ω′2 − 1
4λx2 +
µ2
4λ
This condition is incompatible with the equation of motion
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Linear sigma model(2)
Condition for vanishing π-mass term:∫d4x
(1
2x2πiπi −
µ2
2πiπi + 2λv ? v ? πi ? πi
)=
∫d4x
(Ω′2
2x2πiπi
)+ kinetic terms...
=⇒ v ? v =Ω′2 − 1
4λx2 +
µ2
4λ
This condition is incompatible with the equation of motion
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Linear sigma model(2)
Condition for vanishing π-mass term:∫d4x
(1
2x2πiπi −
µ2
2πiπi + 2λv ? v ? πi ? πi
)=
∫d4x
(Ω′2
2x2πiπi
)+ kinetic terms...
=⇒ v ? v =Ω′2 − 1
4λx2 +
µ2
4λ
This condition is incompatible with the equation of motion
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Conclusions
New sector of the noncommutative scalar theory withharmonic term
Changes in the linear sigma model
Case Ω 6= 1: flows of Ω and µ2
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Conclusions
New sector of the noncommutative scalar theory withharmonic term
Changes in the linear sigma model
Case Ω 6= 1: flows of Ω and µ2
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Conclusions
New sector of the noncommutative scalar theory withharmonic term
Changes in the linear sigma model
Case Ω 6= 1: flows of Ω and µ2
Axel de Goursac Vienna, November 29, 2007