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Quark Mass from Tachyon
Shigenori Seki
This talk is based on O. Bergman, S.S., J. Sonnenschein, JHEP 0712 (2007) 037; S.S., JHEP 1007 (2010) 091.
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13 September 2010 Crete Conference on Gauge Theories and the Structure of Spacetime
Introduction : Many models of holographic QCD
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• Holographic models of QCD in string theory
• D4-branes/D6-brane model[Kruczenski-Mateos-Myers-Winter (2004)]
• D4-branes/D8-anti-D8-branes modelSakai-Sugimoto model[Sakai-Sugimoto (2005)]
• D4-branes model[Van Raamsdonk-Whyte (2010)]
• and many others
• AdS/QCD model5-dimensional gauge theory in anti-de Sitter space[Erlich-Katz-Son-Stephanov (2005)]
Sakai-Sugimoto model
• D4-branes
D8-branes and anti-D8-branes
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Nf
Nc background
probe D8-branes
D4-branes
U(Nf )
Nclarge
chiral symmetry
D8-branesanti-D8-branes
open string
U-shaped D8-branes
U(Nf )R × U(Nf )L
x4 U0 1 2 3 6 7 8 9
D4 x x x x x
D8 x x x x x x x x x
• background [Witten (1998)]
• period of
• In this background, the probe D8-branes are analysed in terms of Dirac-Born-Infeld action.
U
x4
ds2 =
U
R
32
(ηµνdxµdxν + f(U)(dx4)2) +
R
U
32
dU2
f(U)+ U2dΩ2
4
f(U) = 1− U3KK
U3
F4 = dC3 =2πNc
V44
eφ = gs
U
R
34
R3 = πgsNcl3s
δx4 =4π
3R
32
U12KK
MKK =2π
δx4
x4
4
• chiral symmetry is manifest
• good agreement with experimental data
• holographic dual of massless QCDD4-D8 open strings :massless quark massless pion
• large probe analysis is valid in
Advantages
Disadvantages
Nc
Nf Nc
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Mass?
• The quark mass term in QCD
: a fundamental of : a fundamental of
The mass and the quark bi-linear should be identified with the bi-fundamental field in the bulk.
(cf. in AdS/QCD [Casero, Kiritsis, Paredes (2007)])
• Pion massGell-Mann-Oakes-Renner relation [Gell-Mann-Oakes-Renner (1968)]
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m2π ∝ mqqq
mqqq = mqq†RqL + q†LqR
U(Nf )R
U(Nf )L
qR
qL
the open string stretched between D8 and anti-D8-branes
Quark mass from tachyon
• Non-compact limit ( ) of Sakai-Sugimoto model as a toy model
• We are now interested in the contribution of the bi-fundamental field. In this background, one can consider parallel D8-branes and anti-D8-branes.
Following the usual holographic dictionary,
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UKK → 0
ds2 =
U
R
32
(ηµνdxµdxν + (dx4)2) +
R
U
32
dU2 + U2dΩ24
expectation value of quark bi-linear
quark mass
normalizable mode
non-normalizable mode
bi-fundamental field In the dual gauge theory
[Bergman, S.S., Sonnenschein, JHEP 0712 (2007) 037]
Dirac-Born-Infeld action with tachyon
• DBI action of Dp/anti-Dp-branes [Garousi (2005)]
• D8-brane and anti-D8-brane ( )
: separation of D8 and anti-D8 : bi-fundamental “tachyon” field
• tachyon potential [Buchel-Langfelder-Walcher (2002), Leblond-Peet (2003), Lambert-Liu-Maldacena (2007)]
Nf = 1
S[T,L] = −2N
d4xdu V (T )u4D[T,L]
u ≡ U/R N ≡ T8V4R5/gs
D =1u3
+1
4R2L2 +
2πα
R2u3/2T 2 +
12παu3/2
L2T 2
u
x4
L(u)
anti-D8 D8
L(u)T (u)
V (T ) =1
cosh(√
πT )
8
• mass of “tachyon”
The tachyon is localized around .
m2T = − 1
2α +L2
proper
(2πα)2Lproper = u3/4L
u = 0
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• Let us remember tachyon condensation.
At the top of the potential, open string theory Dp and anti-Dp
At the bottom of the potential, Dp and anti-Dp annihilate closed string theory
Tachyon condensation
[Sen (1998)]
V (T )
T
tachyon condense
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• Tachyon is localized around .
Why U-shape?
T
V (T ) uu: large
u: small
u = 0
T =∞
T = 0
condense
not condense
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Analyses of D8/anti-D8 DBI action
• The trivial solution of the equations of motion:
describes the parallel D8-brane and the anti-D8-brane.
• It is difficult to solve the full equations of motion of tachyonic DBI action. So we shall find the asymptotic solutions in the small u region (IR) and the large u region (UV).
• u: small
U-shape solution and
L(u) = (u− u0)p[l0 + l1(u− u0) + · · · ]T (u) = (u− u0)q[t0 + t1(u− u0) + · · · ]
q = −2, 0 < p < 1, t0 =√
π
2R2pu3/2
0
T (u0) =∞
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L(u) = constant, T (u) = 0
• u: largeWe consider the perturbation from the parallel D8 and anti-D8.
The solution is
Note that we assumed .
Then we guess the following interpretations:
normalizable mode
non-normalizable mode
L(u) = L∞ + L(u) , T (u) = 0 + T (u)
L(u) = CLu−9/2 ,
T (u) = u−2(CT e−RL∞u + C T e+RL∞u)
C T e−RL∞u∞
C T
CT qq
mq
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• Assume that the current quark mass is given by
• quark condensate(cf. )LQCD = · · · + mq qq + · · ·
qq =δE(CT )δmq
mq=0
=2NL∞
ΛRCT
mq = ΛC T
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Pion
• Consider the fluctuations of gauge fields
A±(xµ, u) :=12(AD8 ±AD8)
• gauge fix
• vector meson
• scalar meson and pseudo-scalar meson (pion)
• In terms of mode functions, the pion mass and decay constant can be evaluated.
A+u = 0
GOR relation up to numerical factorM20 ∼ 4mqqq
f2π
A+µ
A−u , A−µ
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How is the quark mass generated in other HQCD models?
16
Van Raamsdonk-Whyte’s D4-branes model
• D4-branes ----- background D4-branes ----- probe
Advantages The description of baryon is simpler.
Disadvantages The chiral symmetry is not manifest. The current quark mass = 0....
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Nf
Nc
x4 r θ0 1 2 3 7 8 9
D4 x x x x x
D4 x x x x x
[Van Raamsdonk-Whyte (2009)]
Intersecting D4-branes model
• D4-branes intersecting D4 and anti-D4-branes
When , this model is reduced to the model by Van Raamsdonk-Whyte.
Nf
Nc
T = 0
tachyonic DBI action+
Chern-Simons terms
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[S.S. JHEP 1007 (2010) 091]
• The background metric
• The action
ds2 =
ρ
R
32
ηµνdxµdxν + dx24
+
R
ρ
32
dr2 + r2dθ2 + dx2T
eφ = gs
ρ
R
34
, ρ2 = r2 + x2T
S = −2T4
gs
d4xdr V (T )
r
R
32
1 +r2
4
dΘdr
2
+ 2πα
r
R
32
dT
dr
2
+r2
2πα
R
r
32
Θ2T 2
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SCS =
D4P (1)[C5]−
D4P (2)[C5]
C5 =Nc
16π3αR6ρ3dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4
• Trivial solution
• Non-trivial solutionIR
UV
Θ(r) = Θ∞(constant) , T (r) = 0
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Θ(r) =θ0R3a
(2πα)2a(r − r0)a +O
(r − r0)a+1
, 0 < a < 1
T (r) =(2πα)
√πar3/2
0
2R3/2(r − r0)−2 +O
(r − r0)−1
.
Θ(r) = Θ∞ +2παCθ
R3/2
1√r
,
T (r) =(2πα)2
R3Θ2∞
1r
CnnI2
Θ∞R3/2
πα√
r
+ CnK2
Θ∞R3/2
πα√
r
.
• quark mass and condensate
qq =δE
δmq
mq=0
= ΛδE
δCnn
Cnn=0
=(2πα)5T4
2gsR9
ΛΘ4∞
Cn
Cnn = Λmq
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Gauge fields
• gauge fields
• mode expansion
U(Nf = 1)
A−µ = A⊥µ + Aµ (∂µA⊥µ = 0)
(gauge )
vector meson
axial vector meson
pseudo-scalar meson
A(+)µ (xµ, r) =
n
a(+)nµ (xµ)ψn(r)
Aµ(xµ, r) =
n
∂µωn(xµ)ξn(r)
A(−)r (xµ, r) =
n
ωn(xµ)ζn(r)
A(+)r = 0
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A(±)i =
12(A(1)
i ±A(2)i )
A⊥µ (xµ, r) =
n
a(−)nµ (xµ)ξ⊥n (r)
• Anti-symmetric sector
S[A(−)] = −
d4xdr
C1
F (−)µν
2 + C2
F (−)µr
2 + C3
A(−)µ
2 + C4
A(−)r
2 + C5F(−)µr A(−)µ
D = 1 +r2
4Θ2 + 2πα
r
R
32
T 2 +r2
2πα
R
r
32
Θ2T 2 ,
Q = 1 +1
2πα R3/2r1/2Θ2T 2
C1 =(2πα)2T4
2gsV (T )
R
r
32√D , C2 =
(2πα)2T4
gsV (T )
r
R
32 Q√
D
C3 =8παT4
gsV (T )
√DQ T 2 +
T4
gsV (T )
r4
Q√D
R
r
32
Θ2Θ2T 4 ,
C4 =8παT4
gsV (T )
r
R
3 1√D
T 2 , C5 =4παT4
gsV (T )
r2
√D
ΘΘT 2 .
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• eigen equations and normalisation conditions
−12
d
dr
C2ξ
⊥n +
12C3ξ
⊥n +
14
dC5
drξ⊥n =
m(−)
n
2C1ξ⊥n ,
C4ζn = M2n
C2
ζn − ξ
n +
12C5ξ
n
,
∂r
C2
ζn − ξ
n +
12C5ξ
n
+ C3ξ
n +
12C5
ζn − ξ
n = 0 ,
S[A(−)] = −
d4x
n
14f (−)
nµν
2 +12m(−)
n
2a(−)nµ
2 +12∂µωn
2 +12M2
nω2n
24
14δmn =
dr C1ξ
⊥mξ⊥n ,
12δmn =
dr
C2
ζm − ξ
mζn − ξ
n + C3ξ
mξ
n
+12C5
ζm − ξ
mξ
n + ξm
ζn − ξ
n
.
• We assume that the quark mass is small. ( )
• Pion decay constant:In QCD, it appears in the correlator of axial vector currents for .
We evaluate the pion decay constant by differentiating the action twice with respect to and imposing .
If we set the boundary condition ,
Pion
a−µ
fπ
Seff =
d4p
· · · +
n
p2 (f (−)
n )2
p2 + (m−n )2
+ f2π
a−(n)
µ a−(n)µ
a−(0)µ p2 = 0
ξ0 = α +β√r
ξ⊥0 (∞) = c
12f2
π = (C2ξ⊥0 ξ⊥0
)r=∞
f2π =
(2πα)4T 24 c2β2
g2sR3
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Cnn(∼ mq) 1
• Pion mass
• Gell-Mann-Oakes-Renner relation is satisfied up to a numerical factor.
M2π = −8c2 mqqq
f2π
M20 = 2
dr C4ζ
20
M2π(= M2
0 ) ≈ −8παgsCnCnn
R6T4Θ4∞β2
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Conclusion and Future directions
• intersecting D4-branes model
• (tachyonic) bi-fundamental fields provide current quark mass in various models of holographic QCD.
• The solution interpolating between UV and IR?
• baryon?
• back reaction?
• more flavours?
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