On Wess–Zumino terms of brane–antibrane systems

Nuclear Physics B 800 [PM] (2008) 502–516 www.elsevier.com/locate/nuclphysb

On Wess–Zumino terms of brane–antibrane systems Mohammad R. Garousi a,b,∗ , Ehsan Hatefi a a Department of Physics, Ferdowsi University, PO Box 1436, Mashhad, Iran b Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran, Iran

Received 10 November 2007; accepted 22 January 2008 Available online 3 February 2008

Abstract We calculate the disk level S-matrix element of one closed string RR field, two open string tachyons and one gauge field in type II superstring theory. An expansion for the S-matrix element has been found that its four leading order terms are reproduced exactly by the symmetric trace tachyon DBI and the Wess– Zumino actions of D-brane–anti-D-brane systems. Using this consistency, we have also found the first higher derivative correction to the some of the WZ terms. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Study of unstable objects in string theory might shed new light in understanding properties of string theory in time-dependent backgrounds [1–6]. Generally speaking, source of instability in these processes is appearance of some tachyonic modes in the spectrum of these objects. It then makes sense to study them in a field theory which includes those modes. In this regard, it has been shown by A. Sen that an effective action of Born–Infeld type proposed in [7–10] can capture many properties of the decay of non-BPS Dp -branes in string theory [2,3]. Recently, unstable objects have been used to study spontaneous chiral symmetry breaking in holographic model of QCD [11–13]. In these studies, flavor branes introduced by placing a set of parallel branes and antibranes on a background dual to a confining color theory [14]. Detailed study of brane–antibrane system reveals when brane separation is smaller than the string length scale, spectrum of this system has two tachyonic modes [15]. The effective action should then include these modes because they are the most important ones which rule the dynamics of the system. * Corresponding author at: Department of Physics, Ferdowsi University, PO Box 1436, Mashhad, Iran.

E-mail address: [email protected] (M.R. Garousi). 0550-3213/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2008.01.024

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The effective action of a Dp D¯ p -brane in type IIA(B) theory should be given by some extension of the DBI action and the WZ terms which include the tachyon fields. The DBI part may be given by the projection of the effective action of two non-BPS Dp -branes in type IIB(A) theory with (−1)FL [16]. We are interested in this paper in the appearance of tachyon, gauge field and the RR field in these actions. These fields appear in the DBI part as the following [17]:     SDBI = − d p+1 σ Tr V (T ) −det(ηab + 2πα  Fab + 2πα  Da T Db T ) . (1) The trace in the above action should be completely symmetric between all matrices of the form Fab , Da T , and individual T of the tachyon potential. These matrices are  (1)      Fab 0 0 Da T 0 T Fab = (2) Da T = , T = (2) , 0 T∗ 0 (Da T )∗ 0 Fab (i)

(i)

(i)

(1)

(2)

where Fab = ∂a Ab − ∂b Aa and Da T = ∂a T − i(Aa − Aa )T . If one uses ordinary trace, instead, the above action reduces to the action proposed by Sen [18] after making the kinetic term symmetric and performing the trace. This latter action is not consistent with S-matrix calculation. The tachyon potential which is consistent with S-matrix element calculations has the following expansion: V (|T |) = 1 + πα  m2 |T |2 +

1   2 2 2 πα m |T | + · · · , 2

where Tp is the p-brane tension, m2 is the mass squared of tachyon, i.e., m2 = −1/(2α  ). The  2 2 above expansion is consistent with the potential V (|T |) = eπα m |T | which is the tachyon potential of BSFT [19]. The terms of the above action which has contribution to the S-matrix element of one gauge field and two tachyons in which we are interested in this paper are the following [17]:    πα   (1) (1)  2 2 ∗ (2) (2) F ·F +F ·F LDBI = −Tp (2πα ) m |T | + DT · (DT ) − 2    2 + Tp (πα  )3 DT · (DT )∗ F (1) · F (1) + F (1) · F (2) + F (2) · F (2) 3  2m2 2  (1) (1) |τ | F · F + F (1) · F (2) + F (2) · F (2) + 3   4  μ ∗ μα (1) μα (2) μα (2)  D T Dβ T + D μ T (Dβ T )∗ F (1) Fαβ + F (1) Fαβ + F (2) Fαβ . − 3 (3) Note that if one uses the on-shell value for the tachyon mass, i.e., m2 = −1/(2α  ), the above terms would not be ordered in terms of power of α  . The WZ term describing the coupling of RR field to gauge field of brane–antibrane is given by [20,21]    (1)  (2)  C ∧ ei2πα F − ei2πα F , S = μp (4) Σ(p+1)

where Σ(p+1) is the world volume and μp is the RR charge of the branes. In above equation,  p−m+1 C is a formal sum of the RR potentials C = m=p+1 (−i) 2 Cm . Note that the factors of

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i disappear in each term of (4). The inclusion of the tachyon fields into this action has been proposed in [22–24] using the superconnection of noncommutative geometry [25–27]   SWZ = μp (5) C ∧ STr ei2πα F Σ(p+1)

where the curvature of the superconnection is defined as: F = dA − iA ∧ A,

(6)

the superconnection is  (1)  iA βT ∗ iA = , βT iA(2) √ where β is a normalization constant with dimension 1/ α  which we shall find it later, and a “supertrace” is defined by   A B STr = Tr A − Tr D. C D Using the multiplication rule of the supermatrices [23]          A B A B AA + (−)c BC  AB  + (−)d BD    · = C D C  D DC  + (−)a CA DD  + (−)b CB 

(7)

where x  is 0 if X is an even form or 1 if X is an odd form, one finds that the curvature is   (1) iF − β 2 |T |2 β(DT )∗ , iF = βDT iF (2) − β 2 |T |2 (i)

(1)

(2)

where F (i) = 12 Fab dx a ∧ dx b and DT = [∂a T − i(Aa − Aa )T ] dx a . The WZ action (5) has the following terms:   C ∧ STr iF = Cp−1 ∧ F (1) − F (2) ,

C ∧ STr iF ∧ iF = Cp−3 ∧ F (1) ∧ F (1) − F (2) ∧ F (2)  

+ Cp−1 ∧ −2β 2 |T |2 F (1) − F (2) + 2iβ 2 DT ∧ (DT )∗ , C ∧ STr iF ∧ iF ∧ iF

= Cp−5 ∧ F (1) ∧ F (1) ∧ F (1) − F (2) ∧ F (2) ∧ F (2)   + Cp−3 −3β 2 |T |2 F (1) ∧ F (1) − F (2) ∧ F (2)  

+ 3iβ 2 F (1) + F (2) ∧ DT ∧ (DT )∗  

+ Cp−1 3β 4 |T |4 ∧ F (1) − F (2) − 6iβ 4 |T |2 DT ∧ (DT )∗ .

(8)

The appearance of Cp−1 ∧ dT ∧ dT ∗ has been checked in [22] by studying the disk level Smatrix element of one RR field and two tachyons. In the present paper we will check, among other things, the appearance of Cp−1 ∧ DT ∧ (DT )∗ and Cp−1 ∧ |T |2 (F (1) − F (2) ) terms and fix their coefficients using the S-matrix element of one RR field, two tachyons and one gauge field. The coupling of one RR field, two tachyons and one gauge field in the above terms can be

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combined into the following form:       

  C(p−1) ∧ d A(1) − A(2) T T ∗ − A(1) − A(2) d T T ∗ μp (2πα  )2 −β 2 Σ(p+1)

  = μp (2πα  )2 −β 2



  H(p) ∧ A(1) − A(2) T T ∗ .

(9)

Σp+1

This combination actually appears naturally in the S-matrix element in the string theory side. An outline of the rest of paper is as follows. In Section 2, we review the calculation of the S-matrix element of one RR and two tachyons [22]. The low energy expansion of this amplitude produces one massless pole and infinite number of contact terms. The massless pole is the one reproduce by the above field theories, and its first contact term by Cp−1 ∧ dT ∧ dT ∗ . This fixes the normalization of tachyon in WZ action with respect to the tachyon in the DBI part. In Section 3, we calculate the S-matrix element of one RR, two tachyons and one gauge field. We find an expansion for the amplitude whose leading order terms are fully consistent with the above field theories. 2. The T –T –C amplitude The three-point amplitude between one RR field and two tachyons has been studied in [22] where a non-zero coupling C ∧ dT ∧ dT ∗ has been found. To fix the coefficient of this term we reexamine this amplitude in this section. The tachyon vertex operator corresponds to the real components of the complex tachyon, i.e., 1 T = √ (T1 + iT2 ). (10) 2 The three point amplitude between one RR and two tachyons is given by the following correlation functions:   (−1) (w, w) ¯ , AT ,T ,RR ∼ dx dy d 2 w VT(0) (x)VT(−1) (y)VRR (11) where the vertex operators are (0)

VT (x) = 2ik·ψ(x)e2ik·X(x) , (−1)

VT



(y) = e−φ(y) e2ik ·X(y) ,

¯ ¯ VRR (w, w) ¯ = (P−H / (n) Mp )αβ e−φ(w)/2 Sα (w)eip·X(w) e−φ(w)/2 Sβ (w)e ¯ ip·D·X(w) . (−1)

(12)

The tachyon vertex operator corresponds to either T1 or T2 . The projector in the RR vertex operator is P− = 12 (1 − γ 11 ) and an Hμ ···μ γ μ1 · · · γ μn , n! 1 n where n = 2, 4 for type IIA and n = 1, 3, 5 for type IIB. an = i for IIA and an = 1 for IIB the/ (n) )αβ = ory. The spinorial indices are raised with the charge conjugation matrix, e.g., (P−H αδ β / (n) )δ (further conventions and notations for spinors can be found in Appendix B C (P−H of [28]). The RR bosons are massless so p 2 = 0 and for the tachyons k 2 = k  2 = 1/4. In the string theory calculation we always set α  = 2. The world-sheet fields has been extended to the H / (n) =

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entire complex plane. That is, we have replaced X˜ μ (w) ¯ → Dνμ X ν (w), ¯ ψ˜ μ (w) ¯ → Dνμ ψ ν (w), ¯ β ˜ w) ¯ → Mα Sβ (w), ¯ φ( ¯ → φ(w), ¯ and S˜α (w) where  D=

1p+1 0

0 −19−p

 ,

and Mp =

±i a0 a1 (p+1)! γ γ ±1 a0 a1 (p+1)! γ γ

· · · γ ap a0 ···ap

for p even,

· · · γ ap γ11 a0 ···ap

for p odd.

Using these replacements, one finds the standard propagators for the world-sheet fields X μ , φ, i.e.,  μ X (z)X ν (w) = −ημν log(z − w),  φ(z)φ(w) = − log(z − w). (13) One also needs the correlation function between two spin operators and one ψ. The correlation function involving an arbitrary number of ψ ’s and two S’s is obtained using the following Wicklike rule [29]:  μ ψ 1 (y1 ) · · · ψ μn (yn )Sα (z)Sβ (¯z) =

(z − z¯ )n/2−5/4  μn ···μ1 −1  C αβ Γ |y1 − z| · · · |yn − z|    + ψ μ1 (y1 )ψ μ2 (y2 ) Γ μn ···μ3 C −1 αβ ± perms     + ψ μ1 (y1 )ψ μ2 (y2 ) ψ μ3 (y3 )ψ μ4 (y4 ) Γ μn ···μ5 C −1 αβ  ± perms + · · · , 1

2n/2

(14) Γ μn ···μ1

is the totally where dots means sum over all possible contractions. In above equation, antisymmetric combination of the gamma matrices and the Wick-like contraction, for real yi , is given by 

Re[(y1 − z)(y2 − z¯ )] ψ μ (y1 )ψ ν (y2 ) = 2ημν . (y1 − y2 )(z − z¯ )

The number of ψ in the correlators (11) is one. Using the above formula for one ψ and performing the other correlators using (13), one finds that the integrand is invariant under SL(2, R) transformation. Gauge fixing this symmetry by fixing the position of vertex operators at (x, y, w, w) ¯ = (x, −x, i, −i), one finds [22] A



1   (1 + x 2 )2 2 +u 2 ∼ dx Tr P−H / (n) Mp γ a ka , 2 2 16x 1+x −∞     iμp Γ [−2u] Tr P−H / (n) Mp γ a ka , 2π 1 = 2 4 Γ [ 2 − u] ∞

T ,T ,RR

(15)

where u = −(k + k  )2 and conservation of momentum along the world volume of brane, i.e., k a + k  a + p a = 0, has been used. We have also normalized the amplitude by iμp /4. The trace

M.R. Garousi, E. Hatefi / Nuclear Physics B 800 [PM] (2008) 502–516

(a)

507

(b)

Fig. 1. The Feynman diagrams corresponding to the amplitudes (16) and (19).

is zero for p = n, and for n = p it is   32 / (n) Mp γ a = ± Ha0 ···ap−1  a0 ···ap−1 a . Tr H p! We are going to compare string theory S-matrix elements with field theory S-matrix element including their coefficients, however, we are not interested in fixing the overall sign of the amplitudes. Hence, in above and in the rest of equations in this paper, we have payed no attention to the sign of equations. The trace in (15) containing the factor of γ 11 ensures the following results also hold for p > 3 with H / (n) ≡ ∗H(10−n) for n  5. If one replaces ka in (15) with −ka − pa and uses using the conservation of momentum, one will find that the pa term vanishes using the totally antisymmetric property of  a0 ···ap−1 a . Hence the amplitude (15) is antisymmetric under interchanging 1 ↔ 2. This indicates that the three point amplitude between one RR and two T1 or two T2 is zero. The effective Lagrangian of massless and tachyonic particles, (3) and (8), produces the following massless pole: A = Va (Cp−1 , A)Gab (A)Vb (A, T1 , T2 ), where the gauge field in the off-shell line is iδab ,  (2πα )2 Tp (−pa2 )   Vb A(1) , T1 , T2 = iTp (2πα  )(kb

A(1)

(16) and

A(2) .

The propagator and vertexes are

Gab (A) = 



− kb ),

Vb A(2) , T1 , T2 = −iTp (2πα  )(kb − kb ),   1 Va Cp−1 , A(1) = iμp (2πα  ) a0 ···ap−1 a H a0 ···ap−1 , p!   1 (2) Va Cp−1 , A = −iμp (2πα  ) a0 ···ap−1 a H a0 ···ap−1 . p!

(17)

Replacing them in (16), one finds A = 4iμp

1 a ···a a H a0 ···ap−1 k a . p!u 0 p−1

(18)

The field theory (8) has also the following contact term: Ac = iμp (2πα  )2 β 2

1 a ···a a H a0 ···ap−1 k a . p! 0 p−1

The Feynman diagrams corresponding to the amplitudes (16) and (19) appear in Fig. 1.

(19)

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The massless pole of field theory (18) can be obtained from the S-matrix element (15) by expanding it at low energy (u = −pa2 → 0). The prefactor of (15) has the expansion  2    Γ [−2u] −1 π 2π 1 (20) = + 4 ln(2) + − 8 ln(2)2 u + O u2 . 2 u 6 Γ [ 2 − u] The first term is exactly the field theory massless pole (18). The second term should be the contact term in (19). This fixes the normalization constant β to be  1 2 ln(2) . β= (21) π α Note that in the string theory calculations we have set α  = 2, so to compare the string theory amplitudes with the corresponding amplitudes in field theory one has to set α  = 2 in the field theory side too. Since the expansion (20) is in terms of the powers of pa2 , the other terms in (20) correspond to the higher derivative corrections of the WZ action. For example, it is easy to check that the following higher derivative term reproduces the third term in (20):  2    π  2 2 − 8 ln(2) Cp−1 ∧ D a Da DT ∧ DT ∗ . i(α ) μp (22) 6 On the other hand, one may write u = −pa2 = −1/2 − 2k · k  using the on-shell condition k 2 = k  2 = 1/4. Then one may conclude that because of the −1/2 term, the last term in (20) does not correspond to the higher derivative of the tachyons. As we mentioned, it is obvious that the O(u) terms in the expansion (20) correspond to the higher derivative of the tachyons, however, for other S-matrix elements, e.g., the S-matrix element of four tachyons, it is hard to prove it. It is speculated though that the non-leading terms of the expansion of any S-matrix element of tachyons correspond to the higher derivative of the tachyon field [30]. It is interesting to compare the above higher derivative term with the higher derivative term of one RR and two gauge fields. The string theory S-matrix element of one RR and two gauge fields is given by [31,32] A∼2

Γ [−2u] K, Γ [1 − u]2

where K is the kinematic factor. Expansion of the prefactor at low energy is  2   −1 π Γ [−2u] = − u + O u2 . 2 2 u 6 Γ [1 − u]

(23)

(24)

In this case actually there is no massless pole at u = 0 as the kinematic factor provides a compensating factor of u. The amplitude has the following expansion:    2   (4π)2 μp a0 a1  a2 a3 a4 ···ap π A=i (25) f ε a0 ···ap 1 + f u2 + O u3 δp,n+2 , 4(p − 3)! 6 where fab = i(ka ξb − kb ξa ), f  ab = i(ka ξb − kb ξa ) and ε is the polarization of the RR potential. The first term is reproduced by the couplings in the ZW terms (8), and the second term by the following higher derivative term   (πα  )2 μp (2πα  )2 Cp−3 ∧ ∂ a ∂ b F (1) ∧ ∂a ∂b F (1) − ∂ a ∂ b F (2) ∧ ∂a ∂b F (2) . 6 2!

(26)

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509

While the leading higher derivative term in (22) has two extra derivatives with respect to the corresponding coupling in the WZ terms, in above coupling there are four extra derivatives with respect to the corresponding coupling in (8). 3. The T –T –A–C amplitude The S-matrix element of one RR field, two tachyons and one gauge field is given by the following correlation function:   (−1) (0) (0) (−1) AAT T C ∼ dx1 dx2 dx3 dz d z¯ VA (x1 )VT (x2 )VT (x3 )VRR (z, z¯ ) , (27) where we have chosen the vertex operators according to the rule that the total superghost number must be −2. The tachyon and RR vertex operators are given in (12) and the gauge field vertex operator in (−1)-picture is (−1)

VA

= ξ . ψ(x1 )e−φ(x1 ) e2ik1 .X(x1 ) ,

(28)

where ξa is the gauge field polarization. Introducing x4 ≡ z = x + iy and x5 ≡ z¯ = x − iy, the scattering amplitude reduces to the following correlators:   ∗ / (n) Mp )αβ ξa :e−1/2φ(x4 ) :e−1/2φ(x5 ) :e−φ(x1 ) : AAT T C ∼ dx1 · · · dx5 (P−H  × :e2ik1 .X(x1 ) :e2ik2 .X(x2 ) :e2ik3 .X(x3 ) :eip.X(x4 ) :eip.D.X(x5 ) :  × :Sα (x4 ):Sβ (x5 ):ψ a (x1 ):2ik2 . ψ(x2 ):2ik3 . ψ(x3 ): . The correlators in the first and the second lines can be calculated using the propagators in (13), and the correlator in the last line can be read from (14). The result is √  AAT T C ∼ 2 dx1 · · · dx5 k2b k3c ξa (x34 x35 )−1/2+2k3 .p (x14 x15 )−1+2k1 .p 4k1 .k2 × (x24 x25 )−1/2+2k2 .p x45 x23 2 3 x12 x13 1 3 (P−H / (n) Mp )αβ    Re[x14 x25 ]  c −1  Γ C αβ × Γ cba C −1 αβ + 2ηab x12 x45  Re[x14 x35 ]  b −1  Re[x24 x35 ]  a −1  Γ C αβ + 2ηbc Γ C αβ , − 2ηac x13 x45 x23 x45 p.D.p 4k .k

4k .k

where xij ≡ xi − xj . One can show that the integrand is invariant under SL(2, R) transformation. Gauge fix this symmetry by fixing x1 = 0, One finds A

AT T C

x2 = 1,

x3 → ∞,

dx1 dx2 dx3 → x32 .

√  −2(t+s+u)−1 ∼ 2 dx4 dx5 k2b k3c ξa x45 |x4 |2t+2s−1 |1 − x4 |2u+2t−1/2    −x + x4 x5  c −1  x  b −1  Γ C αβ − 2ηac Γ C αβ × Γ cba C −1 αβ − 2ηab x45 x45  x − 1  a −1  Γ C αβ (P−H + 2ηbc / (n) Mp )αβ , x45

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where we have also introduced the Mandelstam variables s = −(k1 + k3 )2 ,

t = −(k1 + k2 )2 ,

u = −(k2 + k3 )2

and used the conservation of momentum along the brane, i.e., k1a + k2a + k3a + p a = 0. Using the fact that Mp , H / (n) , and Γ cba are totally antisymmetric combinations of the Gamma matrices, one realizes that the first term is non-zero only for p = n + 2, and the last three terms are non-zero only for p = n. The integral in the above equation can be written in terms of the Gamma functions, using the following identity [33]:  d 2 z |1 − z|a |z|b (z − z¯ )c (z + z¯ )d = (2i)c 2d π

b+c a+c a+b+c 1+c 2 )(1 + 2 )(−1 − 2 )( 2 ) (− a2 )(− b2 )(2 + c + d + a+b 2 )

(1 + d +

for d = 0, 1 and arbitrary a, b, c. The region of integration is the upper half complex plane, as in our case. Using this integral, one finds  iμp   Tr (P−H / (n) Mp )(k3 . γ )(k2 . γ )(ξ . γ ) I δp,n+2 AAT T C = √ 2 2π   + Tr (P−H / (n) Mp )γ a J δp,n k2a (t + 1/4)(2ξ . k3 ) + k3a (s + 1/4)(2ξ . k2 )

 − ξa (s + 1/4)(t + 1/4) , (29) √ where we have normalized the amplitude by iμp /2 2π . In above equation, I, J are: (−u)(−s + 1/4)(−t + 1/4)(−t − s − u) , (−u − t + 1/4)(−t − s + 1/2)(−s − u + 1/4) (−u + 1/2)(−s − 1/4)(−t − 1/4)(−t − s − u − 1/2) J = 21/2 (2)−2(t+s+u+1) π . (−u − t + 1/4)(−t − s + 1/2)(−s − u + 1/4) The traces are:   32 Tr H / (n) Mp (k3 . γ )(k2 . γ )(ξ . γ ) δp,n+2 = ±  a0 ···ap Ha0 ···ap−3 k3ap−2 k2ap−1 ξap δp,n+2 , n!   32 a0 ···ap−1 a a Tr H (30) / (n) Mp γ δp,n = ±  Ha0 ···ap−1 δp,n . n! Examining the poles of the Gamma functions, one realizes that for the case that p = n + 2, the amplitude has massless pole and infinite tower of massive poles. Whereas for p = n case, there are tachyon, massless, and infinite tower of massive poles. Now the nontrivial question is how to expand this amplitude such that its leading terms correspond to the effective actions (3) and (8), and its non-leading terms correspond to the higher derivative terms? Let us study each case separately. I = 21/2 (2)−2(t+s+u)−1 π

3.1. p = n + 2 case For p = n + 2, the amplitude is antisymmetric under interchanging 2 ↔ 3, hence the fourpoint function between one RR, one gauge field and two T1 or two T2 is zero. The electric part of the amplitude for one RR, one gauge field, one T1 and one T2 is given by  a ···a  8iμp  0 p Ha0 ···ap−3 k3ap−2 k2ap−1 ξap I. AAT1 T2 C = ± √ 2π (p − 2)!

(31)

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(a)

511

(b)

Fig. 2. The Feynman diagrams corresponding to the amplitudes (32) and (35).

Note that the amplitude satisfies the Ward identity, i.e., the amplitude vanishes under replacement ξ a → k1a . The effective Lagrangian of the massless field and tachyon, (3) and (8), produces the following massless pole for p = n + 2:       A = Va Cp−3 , A(1) , A(1) Gab A(1) Vb A(1) , T1 , T2 , (32) where   Gab A(1) =

iδab , (2πα  )2 Tp (u)   Vb A(1) , T1 , T2 = Tp (2πα  )(k2 − k3 )b ,   1 a a ···a a H a0 ···ap−3 k1 p−2 ξ ap−1 . Va Cp−3 , A(1) , A(1) = μp (2πα  )2 (p − 2)! 0 p−1

(33)

The amplitude (32) becomes A = μp (2πα  )

2i a a a ···a a H a0 ···ap−3 k2 p−2 k3 p−1 ξ a . (p − 2)!u 0 p−1

(34)

The WZ action (8) has also the following contact term: Ac = μp β 2 (2πα  )3

i a a a ···a a H a0 ···ap−3 k2 p−2 k3 p−1 ξ a . 2!(p − 2)! 0 p−1

(35)

The Feynman diagrams corresponding to the amplitudes (32) and (35) appear in Fig. 2. The massless pole (34) and the contact term (35) can be exactly reproduced by the string theory amplitude (29) if one expands it around t → −1/4,

s → −1/4,

u → 0.

In fact expansion of I around this point is   2   √ −1 π π 2 (s + t + 1/2)2 2 I = π 2π + 4 ln(2) + − 8 ln(2) u − + ··· , u 6 6 u

(36)

(37)

replacing it in (31), one finds that the first term is reproduced by (34) and the second term by (35). Now what about the other terms of (37)? A natural extension of the higher derivative term (22) to Cp−3 which is similar to the extension of Cp−1 ∧ (DT ∧ DT ∗ ) to Cp−3 in (8) is the following:  2      i π   2 2 (38) (2πα )(α ) μp − 8 ln(2) Cp−3 ∧ F (1) + F (2) ∧ D a Da DT ∧ DT ∗ . 2 6

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This higher derivative term in field theory reproduces exactly the contact term in the second line of (37). The last term in (37), on the other hand, is given by the Feynman amplitude (32) where the vertex Va (Cp−3 , A(1) , A(1) ) should be derived from the higher derivative term (26), that is   (2π)2 1 a μp (2πα  )2 a ···a a H a0 ···ap−3 k1 p−2 ξ ap−1 (k1 · p)2 . Va Cp−3 , A(1) , A(1) = 6 (p − 2)! 0 p−1 (39) Note that the vertex Vb (A(1) , T1 , T2 ) has no higher derivative correction as it arises from the kinetic term of the tachyon. The tachyon pole of string amplitude indicates that the kinetic term has no higher derivative correction. Now replacing Vb (A(1) , T1 , T2 ) and the above vertex in (32) one again finds exact agreement with the string theory amplitude. Hence the expansion of the string amplitude (31) around (36) produces an expansion whose leading order terms are reproduced by the Feynman amplitudes resulting from DBI and WZ actions and the next terms are related to the higher derivative corrections to the WZ action. In particular, its massless poles gives information about the higher derivative corrections of Cp−3 ∧ F ∧ F and its contact terms about Cp−3 ∧ F ∧ DT ∧ DT ∗ . 3.2. p = n case Now we consider n = p case. The string theory amplitude in this case is symmetric under interchanging 2 ↔ 3. On the other hand, there is no Feynman amplitude in field theory corresponding to four-point function of one RR, one gauge field, one T1 and one T2 . Hence, for p = n the string theory amplitude (29) is the S-matrix element of one RR, one gauge field and two T1 or two T2 . Its electric part is,  8iμp  a0 ···ap−1 a  Ha0 ···ap−1 J AAT1 T1 C = ± √ 2π p!

 × k2a (t + 1/4)(2ξ . k3 ) + k3a (s + 1/4)(2ξ . k2 ) − ξa (s + 1/4)(t + 1/4) . (40) Note that the amplitude satisfies the Ward identity, i.e., the amplitude vanishes under replacement ξ a → k1a . The effective field theory, (3) and (8), has the following massless pole for p = n:   A = Va (Cp−1 , A)Gab (A)Vb A, T1 , T1 , A(1) , (41) where A should be A(1) and A(2) . The propagator and vertexes Va (Cp−1 , A) and Vb (A, T1 , T1 , A(1) ) are iδab , (2πα  )2 Tp (u + t + s + 1/2)   1 Va Cp−1 , A(1) = iμp (2πα  ) a0 ···ap−1 a H a0 ···ap−1 , p!   1 Va Cp−1 , A(2) = −iμp (2πα  ) a0 ···ap−1 a H a0 ···ap−1 , p!  (1)  (1) = −2iTp (2πα  )ξb , Vb A , T1 , T1 , A  (2)  Vb A , T1 , T1 , A(1) = 2iTp (2πα  )ξb . Gab (A) =

(42)

M.R. Garousi, E. Hatefi / Nuclear Physics B 800 [PM] (2008) 502–516

(a)

513

(b)

Fig. 3. The Feynman diagrams corresponding to the amplitudes in (41) and (44).

The amplitude (41) then becomes A=

4iμp  a0 ···ap−1 a Ha0 ···ap−1 ξa . p!(u + s + t + 1/2)

(43)

The couplings in (9) has also the following contact term: Ac = iμp ln(2)

16 a0 ···ap−1 a Ha0 ···ap−1 ξa .  p!

(44)

The Feynman diagrams corresponding to the amplitudes (41) and (44) appear in Fig. 3. Now in string theory side, expansion of (s + 1/4)(t + 1/4)J in (40) around (36) is (s + 1/4)(t + 1/4)J √   2  2π −1 π 2 = + 4 ln(2) + − 8 ln(2) (s + t + u + 1/2) 2 (t + s + u + 1/2) 6  π 2 (t + 1/4)(s + 1/4) + ··· . − 3 (t + s + u + 1/2)

(45)

Replacing it in (40), one finds that the first term above is reproduced by the field theory massless pole (43) and the second term by (44). To find the higher derivative term corresponding to the first term in the second line above, we note that the WZ terms in the third line of (8) indicates that the following higher derivative term should accompany the term in (22):  2     π −(α  )2 μp (46) − 8 ln(2)2 Cp−1 ∧ D a Da F (1) − F (2) |T |2 . 6 Combining the above with the coupling of one RR, two tachyons and one gauge field of (22), one finds the following coupling:  2     π  2 2 −(α ) μp (47) − 8 ln(2) Hp ∧ ∂ a ∂a A(1) − A(2) T T ∗ 6 which reproduce exactly the first term in the second line of (45). The effective field theory has also the following poles for p = n:   A = Va (Cp−1 , A)Gab (A)Vb (A, T1 , T2 )G(T2 )V T2 , T1 , A(1)   + V (Cp−1 , T1 , T2 )G(T2 )V T2 , T1 , A(1) where the off-shell gauge field A should be A(1) and A(2) , and

(48)

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(a)

(b)

Fig. 4. The Feynman diagrams corresponding to the amplitudes (48).

1 V (Cp−1 , T1 , T2 ) = β 2 μp (2πα  )  a0 ···ap−1 a Ha0 ···ap−1 k2a , p!   V T2 , T1 , A(1) = Tp (2πα  )(k3 − k) · ξ, Vb (A, T1 , T2 ) = Tp (2πα  )(k2a + ka ), i G(T2 ) = ,  (2πα )Tp (s + 1/4)

(49)

where ka is momentum of the off-shell tachyon. Replacing them in (48), one finds   4 ln(2) 1 a0 ···ap−1 a −1 Ha0 ···ap−1 k2a (2k3 · ξ ). (50) +  4iμp (s + 1/4)(t + s + u + 1/2) (s + 1/4) p! Similar result appear for the Feynmann diagram in which k2 ↔ k3 . The Feynman diagram corresponding to the amplitude (48) appear in Fig. 4. Now in string theory side, expansion of (t + 1/4)J around (36) is  1√ −1 4 ln(2) (t + 1/4)J = 2π + 2 (s + 1/4)(t + s + u + 1/2) (s + 1/4)    2 π2 (t + 1/4) π 2 (s + t + u + 1/2) − 8 ln(2) − + ··· . + 6 (s + 1/4) 3 (t + s + u + 1/2) (51) Replacing it in the string theory amplitude (40), one finds exact agreement between the first two terms and the field theory results (50). The first term in the second line should be reproduced by the following Feynman amplitude in field theory:   A = V (Cp−1 , T1 , T2 )G(T2 )V T2 , T1 , A(1) , (52) where the propagator and the vertex V (T2 , T1 , A(1) ) is the same as in (49), however, the vertex V (Cp−1 , T1 , T2 ) should be derived from the higher derivative term (22), that is  2  π 1 a0 ···ap−1 a V (Cp−1 , T1 , T2 ) = (α  )2 μp Ha0 ···ap−1 k2a (s + t + u + 1/2). − 8 ln(2)2  6 p! Replacing them in (52), one finds exact agreement with the string theory result. Finally, the sum of last term in (51) and the corresponding term when k2 ↔ k3 , and the last term in (45) is   a0 ···ap−1 a Ha0 ···ap−1 4i − π 2 μp k2a (t + 1/4)(2ξ . k3 ) 3 p!(s + t + u + 1/2)  1 − ξa (s + 1/4)(t + 1/4) + (3 ↔ 2) . 2

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To find the corresponding term in field theory, consider the following Feynman amplitude:   A = Va (Cp−1 , A)Gab (A)Vb A, A(1) , T1 , T1 (53) where the vertex Va (Cp−1 , A) is the same as the one appears in (42) and Vb (A, A(1) , T1 , T1 ) is derived from the second line of (3), that is   4i   Vb A(1) , A(1) , T1 , T1 = Tp (πα  )3 kb (s + 1/4)(2k2 · ξ ) + (t + 1/4)(2k3 · ξ ) 3  4i + Tp (πα  )3 k2b (t + 1/4)(2ξ . k3 ) + k3b (s + 1/4)(2ξ . k2 ) 3  − ξb (s + 1/4)(t + 1/4) ,   2i   Vb A(2) , A(1) , T1 , T1 = Tp (πα  )3 kb (s + 1/4)(2k2 · ξ ) + (t + 1/4)(2k3 · ξ ) 3  2i + Tp (πα  )3 × k2b (t + 1/4)(2ξ . k3 ) + k3b (s + 1/4)(2ξ . k2 ) 3  − ξb (s + 1/4)(t + 1/4) , where k a is the momentum of the off-shell gauge field. Replacing them in (53), one finds exact agreement with the string theory result. Note that the F (1) · F (2) term in the tachyon DBI action (3) is necessary for the above consistency. The above consistency indicates that the coupling Cp−1 ∧ F has no higher derivative correction. Now let us speculate on the other terms of the string theory expansion. Since there is no higher derivative corrections to Cp−1 ∧ F and to the kinetic term, the expansion should not have the double poles other than the one appears in the leading term. For the simple tachyonic poles, since the kinetic term has no correction, the non-leading poles of tachyon should gives information about the higher derivative corrections to Cp−1 ∧DT ∧DT ∗ . For the simple massless poles, since there is no correction to Cp−1 ∧ F , the non-leading massless poles should give information about the higher derivative corrections to the coupling of two tachyons and two gauge fields. Finally, the contact terms of the string theory amplitude gives information about the higher derivative correction to Cp−1 ∧ DT ∧ DT ∗ and to Cp−1 ∧ F T T ∗ . It would be interesting to study these terms in details and find higher derivative corrections to tachyon DBI and to WZ actions. Acknowledgement We would like to thank A. Ghodsi for comments. References [1] M. Gutperle, A. Strominger, JHEP 0204 (2002) 018, hep-th/0202210; A. Sen, JHEP 0204 (2002) 048, hep-th/0203211. [2] A. Sen, JHEP 0207 (2002) 065, hep-th/0203265. [3] A. Sen, Mod. Phys. Lett. A 17 (2002) 1797, hep-th/0204143. [4] A. Sen, JHEP 0210 (2002) 003, hep-th/0207105; P. Mukhopadhyay, A. Sen, JHEP 0211 (2002) 047, hep-th/0208142; A. Strominger, hep-th/0209090; F. Larsen, A. Naqvi, S. Terashima, JHEP 0302 (2003) 039, hep-th/0212248; M. Gutperle, A. Strominger, Phys. Rev. D 67 (2003) 126002, hep-th/0301038; A. Maloney, A. Strominger, X. Yin, JHEP 0310 (2003) 048, hep-th/0302146; T. Okuda, S. Sugimoto, Nucl. Phys. B 647 (2002) 101, hep-th/0208196;

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