Quark Mass from Tachyon

Transcription

1 Quark Mass from Tachyon Shigenori Seki This talk is based on O. Bergman, S.S., J. Sonnenschein, JHEP 071 (007) 07; S.S., JHEP 1007 (010) September 010 Crete Conference on Gauge Theories and the Structure of Spacetime 1

2 Introduction : Many models of holographic QCD Holographic models of QCD in string theory D4-branes/D6-brane model [Kruczenski-Mateos-Myers-Winter (004)] D4-branes/D8-anti-D8-branes model Sakai-Sugimoto model [Sakai-Sugimoto (005)] D4-branes model [Van Raamsdonk-Whyte (010)] and many others AdS/QCD model 5-dimensional gauge theory in anti-de Sitter space [Erlich-Katz-Son-Stephanov (005)]

3 Sakai-Sugimoto model N c D4-branes background N f D8-branes and anti-d8-branes probe D8-branes x U D4 x x x x x D8 x x x x x x x x x chiral symmetry U(N f ) R U(N f ) L anti-d8-branes D8-branes U(N f ) U-shaped D8-branes D4-branes open string large N c

4 background [Witten (1998)] ds = U R (ηµν dx µ dx ν + f(u)(dx 4 ) )+ du R U f(u) + U dω 4 f(u) =1 U KK U e φ = g s U R 4 x 4 period of F 4 = dc = πn c V 4 4 x 4 R = πg s N c l s U δx 4 = 4π R U 1 KK M KK = π δx 4 In this background, the probe D8-branes are analysed in terms of Dirac-Born- Infeld action. 4

5 Advantages chiral symmetry is manifest good agreement with experimental data Disadvantages holographic dual of massless QCD D4-D8 open strings : massless quark massless pion N c large probe analysis is valid in N f N c 5

6 Mass? The quark mass term in QCD m q qq = m q q R q L + q L q R q R q L : a fundamental of : a fundamental of U(N f ) R U(N f ) L The mass and the quark bi-linear should be identified with the bi-fundamental field in the bulk. the open string stretched between D8 and anti-d8-branes (cf. in AdS/QCD [Casero, Kiritsis, Paredes (007)]) Pion mass Gell-Mann-Oakes-Renner relation [Gell-Mann-Oakes-Renner (1968)] m π m q qq 6

7 Quark mass from tachyon Non-compact limit ( U KK 0) of Sakai-Sugimoto model as a toy model U ds R = (ηµν dx µ dx ν +(dx 4 ) )+ du + U dω 4 R U 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, [Bergman, S.S., Sonnenschein, JHEP 071 (007) 07] bi-fundamental field normalizable mode non-normalizable mode In the dual gauge theory expectation value of quark bi-linear quark mass 7

8 Dirac-Born-Infeld action with tachyon DBI action of Dp/anti-Dp-branes [Garousi (005)] anti-d8 u D8 D8-brane and anti-d8-brane ( N f =1) L(u) : separation of D8 and anti-d8 T (u) : bi-fundamental tachyon field S[T,L]= N d 4 xdu V (T )u 4 D[T,L] L(u) D = 1 u + 1 4R L + πα R u / T + 1 πα u / L T x 4 u U/R N T 8 V 4 R 5 /g s tachyon potential [Buchel-Langfelder-Walcher (00), Leblond-Peet (00), 1 Lambert-Liu-Maldacena (007)] V (T )= cosh( πt) 8

9 mass of tachyon m T = 1 α + L proper (πα ) L proper = u /4 L The tachyon is localized around u =0. 9

10 Tachyon condensation Let us remember tachyon condensation. [Sen (1998)] At the top of the potential, open string theory Dp and anti-dp V (T ) tachyon condense At the bottom of the potential, Dp and anti-dp annihilate closed string theory T 10

11 Why U-shape? Tachyon is localized around u =0. V (T ) T =0 not condense u: large u condense u: small T T = 11

12 Analyses of D8/anti-D8 DBI action The trivial solution of the equations of motion: L(u) = constant, T(u) = 0 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 L(u) = (u u 0 ) p [l 0 + l 1 (u u 0 )+ ] T (u) = (u u 0 ) q [t 0 + t 1 (u u 0 )+ ] π q =, 0 <p<1, t 0 = R pu/ 0 U-shape solution and T (u 0 )= 1

13 u: large We consider the perturbation from the parallel D8 and anti-d8. L(u) =L + L(u), T(u) =0+ T (u) The solution is L(u) = C L u 9/, T (u) = u (C T e RL u + CT e +RL u ) Note that we assumed CT e RL u. Then we guess the following interpretations: normalizable mode C T qq non-normalizable mode C T m q 1

14 Assume that the current quark mass is given by m q =ΛC T quark condensate (cf. L QCD = + m q qq + ) qq = δe(c T ) δm q mq =0 = N L ΛR C T 14

15 Pion Consider the fluctuations of gauge fields A ± (x µ,u):= 1 (AD8 ± A D8 ) gauge fix A + u =0 vector meson A + µ scalar meson and pseudo-scalar meson (pion) A u,a µ In terms of mode functions, the pion mass and decay constant can be evaluated. GOR relation M 0 4m q qq f π up to numerical factor 15

16 How is the quark mass generated in other HQCD models? 16

17 Van Raamsdonk-Whyte s D4-branes model [Van Raamsdonk-Whyte (009)] N c D4-branes background N f D4-branes probe 0 1 x 4 r θ D4 x x x x x D4 x x x x x Advantages The description of baryon is simpler. Disadvantages The chiral symmetry is not manifest. The current quark mass =

18 Intersecting D4-branes model [S.S. JHEP 1007 (010) 091] N c N f D4-branes intersecting D4 and anti-d4-branes tachyonic DBI action + Chern-Simons terms When T =0, this model is reduced to the model by Van Raamsdonk-Whyte. 18

19 The background metric ds = ρ R ηµν dx µ dx ν + dx R 4 + ρ e φ = g s ρ R 4, ρ = r + x T dr + r dθ + dx T The action S = T 4 g s r d 4 xdr V (T ) 1+ r dθ r +πα R 4 dr dt R dr + r πα R r Θ T S CS = D4 P (1) [C 5 ] D4 P () [C 5 ] C 5 = N c 16π α R 6 ρ dx 0 dx 1 dx dx dx 4 19

20 Trivial solution Θ(r) =Θ (constant), T(r) =0 Non-trivial solution IR Θ(r) = θ 0 R a (πα ) a (r r 0) a + O (r r 0 ) a+1, T (r) = (πα ) πar / 0 (r r R / 0 ) + O (r r 0 ) 1. 0 <a<1 UV Θ(r) = Θ + πα C θ 1 r, R / T (r) = (πα ) 1 Θ R C / R Θ nn I r πα Θ R / r + C n K r πα. 0

21 quark mass and condensate C nn =Λm q q q = δe δm q mq =0 =Λ δe δc nn Cnn =0 = (πα ) 5 T 4 g s R 9 Λ Θ 4 C n 1

22 Gauge fields U(N f = 1) gauge fields A (±) i = 1 (A(1) i ± A () i ) mode expansion vector meson A (+) µ (x µ,r)= n axial vector meson a (+) nµ (x µ )ψ n (r) A µ = A µ + A µ ( µ A µ = 0) (gauge ) A (+) r =0 A µ (x µ,r)= n a ( ) nµ (x µ )ξ n (r) A µ(x µ,r)= n µ ω n (x µ )ξ n(r) pseudo-scalar meson A ( ) r (x µ,r)= n ω n (x µ )ζ n (r)

23 Anti-symmetric sector S[A ( ) ]= d 4 xdr C 1 F ( ) µν + C F ( ) µr + C A ( ) µ + C 4 A ( ) r + C 5 F µr ( ) A ( )µ C 1 = (πα ) T 4 R V (T ) D, C = (πα ) T 4 r Q V (T ) D g s r g s R C = 8πα T 4 D V (T ) g s Q T + T 4 V (T ) g s r 4 Q D R r Θ Θ T 4, C 4 = 8πα T 4 r 1 V (T ) T, C 5 = 4πα T 4 V (T ) r ΘΘ T. g s R D g s D D =1+ r r 4 Θ +πα T + r R Θ R πα T, r Q =1+ 1 πα R/ r 1/ Θ T

24 eigen equations and normalisation conditions 1 d C ξ n + 1 dr C ξn C 4 ζ n = M n C ζn ξ n + 1 C 5ξ n dc 5 dr ξ n = m ( ) C1 n ξn,, r C ζn ξ n + 1 C 5ξ n + C ξ n + 1 C 5 ζn ξ n =0, 1 4 δ mn = 1 δ mn = dr C 1 ξ mξ n, dr C ζm ξm ζ n ξn + C ξmξ n + 1 C 5 ζm ξ m ξ n + ξ m ζn ξ n. S[A ( ) ]= d 4 x n 1 f ( ) nµν m ( ) n a ( ) nµ + 1 µ ω n 1 + M nω n 4

25 Pion We assume that the quark mass is small. ( ) C nn ( m q ) 1 Pion decay constant: f π In QCD, it appears in the correlator of axial vector currents for. S eff = d 4 p + n p (f ( ) n ) p +(m n ) + f π We evaluate the pion decay constant by differentiating the action twice with respect to a (0) and imposing. µ p =0 1 f π =(C ξ 0 ξ 0 ) r= a µ a (n) µ a (n)µ If we set the boundary condition ξ0 ( ) =c, f π = (πα ) 4 T 4 c β g sr ξ 0 = α + β r 5

26 Pion mass M0 = dr C 4 ζ 0 M π(= M 0 ) 8πα g s C n C nn R 6 T 4 Θ 4 β Gell-Mann-Oakes-Renner relation is satisfied up to a numerical factor. M π = 8c m qq q f π 6

27 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? 7