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 )

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• 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?

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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

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, 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

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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

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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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