SLAC-PUB-8184 SU-ITP-99/32 hep-th/ June 30, 1999 One-loop corrections to the D3 brane action. Marina Shmakova Stanford Linear Accelerator Cente

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1 SLA-PU-88 SU-ITP-99/3 hep-th/99639 June 3, 999 One-loop corrections to the D3 brane action. Marina Shmakova Stanford Linear Accelerator enter, Stanford University, Stanford, A 939; University of Tennessee, Knoxville, TN ASTRAT We calculate one-loop corrections to the eective Lagrangian for the D3 brane. We perform the gauge-xing of the -symmetric orn-infeld D3 brane action in the at background using Killing gauge. The linearized supersymmetry of the gauge- xed action coincides with that of the N Yang-Mills theory. We use the helicity amplitude and unitarity technique to calculate the one-loop amplitudes at order. The counterterms and the nite -loop corrections are of the form (@F) and their supersymmetric generalization. This is to be contrasted with the orn-infeld action which contains (F ) and other terms which do not depend on derivatives of the vector eld strength. This research was partly supported by the Department of Energy under contract DE-A3-76SF55. Submitted for publication in Phys. Rev. D. shmakova@slac.stanford.edu.

2 Introduction The action for supersymmetric D3-branes in at and curved type II supergravity backgrounds has been studied extensively during the last few years [,, 3, ]. In addition to extended supersymmetry these actions have a local -symmetry and half of the 3 fermionic elds can be \gauged away" by xing the -gauge. Moreover, of the 3 supersymmetries of type II supergravity, 6 are realized linearly and 6 non-linearly. The interpretation of all these supersymmetries is closely related to the problem of the correspondence between supersymmetric N U(N)-symmetric Yang-Mills theories and D3-brane orn-infeld-type U()-symmetric actions []. The N Yang-Mills theory in d is not only renormalizable but even nite [5]. The D3 brane action has been considered so far only as an eective action as it is not renormalizable by power counting in d. Still, there are 6+6 linear and non-linear global symmetries in the gauge-xed action (or equivalently a local -symmetry of the classical action). Is it possible that these symmetries are strong enough to forbid the counterterms in the D3 brane action which have a higher number of derivatives than the classical orn-infeld action? One has to take into account that the D3 brane action has terms of the form ((F ) n + :::) however, terms of the form (@F) n or (@ F ) n are not present. As a rst step in resolving these issues, in particular to clarify the relation between the eective action of the open string theory, Yang-Mills theory and eective action of the D3-brane theory, we will treat the -symmetric action of the D3-brane on the same footing as supergravity in d or supersymmetric Yang-Mills theory in d>. A priori these theories are not renormalizable. However a careful analysis of supersymmetric higher derivative counterterms was performed in the past, as well as some actual calculations [6]. In this paper we will present one-loop corrections to the D3-brane Abelian orn- Infeld action in a at background, and we will show that there are ultraviolet divergences, in contrast to the nite d N Super-Yang-Mills [5], and that the form of the couterterm is (s + t + u )F (@F). To x the -symmetry we will employ the so-called Killing gauge used in [7] for gauge-xing of the GS type II string in

3 the AdS 5 S 5 background. In this gauge we will be able to directly identify the linear supersymmetries of the gauge-xed D3 brane action with the YM supersymmetries. We will compare this to the symmetries of the D3 brane action in gauge, proposed in [3]. The calculation of the tree amplitudes and one-loop corrections gives the same results in the other gauge, of course. However, in the Killing gauge the lowest order Lagrangian contains only quartic interaction terms while in gauge there are cubic as well as quartic interactions. Therefore it is easier to perform one-loop calculations in the Killing gauge. It is interesting that the terms with the (@F) structure were found in the eective action for the open superstrings in the paper by Andreev and Tseytlin [8] from the string S-matrix term (s + t + u )F : This kind of terms were also found in the more recent papers by Hashimoto and Klebanov [9] in which they computed the scattering amplitude for massless vectors on a D-brane in string theory. In the low energy limit their result also contain the term proportional to (s + t + u )F : The appearance of this structure in our calculations and in the string eective action could mean that the supersymmetry xes this structure in the unique way. This would also mean that the local -symmetry after gauge xing corresponds to the global N supersymmetry at least in this approximation. To perform one-loop calculations we will use the helicity amplitude method that has been developed extensively in the last years (for reviews see [, ]). The main ideas of this method that we will employ here are the spinor helicity representation for polarization vectors [], supersymmetry identities [3] and unitarity cuts []. This method has been used for loop calculations in Yang-Mills theory and recently also in gravity [5]. It turns out that it is extremely useful for the D3 brane calculations. In Sec. wewill discuss the D3-brane action and nd the SUSY transformations in the Killing gauge. In Sec. 3 the N supersymmetric Lagrangian of the orn- Infeld theory will be presented up to the quartic order in the elds. In Sec. we will apply the helicity amplitude formalism to the D3-brane action. We will present tree and one-loop amplitudes for the quartic Lagrangian in the Killing gauge and give the expressions for the one-loop corrections and counterterms in this theory. In Sec. 5

4 we will discuss the form of the Lagrangian up to the quartic order in gauge and Sec. 6 is a conclusion. The D3 brane action in a at background In this paper we will closely follow the notations of []. The supersymmetric D3- brane action is a sum of the orn-infeld (I) and Wess-Zumino (WZ) terms [, 3, ] that depend on superspace coordinates X M (x^a ; I ) and an Abelian world-volume gauge eld strength da S S I + S WZ ; () where the orn-infeld part is S I T 3 ZM d q det (G ij + F ij ) () where T 3 ( ) G ij is the D3-brane tension [6] and is the inverse string tension. is the pullback to the d world-volume metric of the d Minkowski metric ^a^b ( +:::+) with i; j :::3; ^a;^b :::9 : G ij L^a i ^a^b L^b i ; L^a (X()) d i L^a i (3) and in the at background the supervielbeins are L I d I ; I ; L^a dx^a i I ^a d I : () The eld strength F F ijd i ^ d j is the supersymmetric extension of da: The WZ part of the action is Z S WZ it 3 Z M F da +i Z dt L^a t ^ ab ^b KL t : (5) dt 6 b L t ^ bl t ^ bl t E L t + b L t^f t^jl t +T3 Z M (bos) ; (6) 3

5 where b L L^a ^a and L^a t (x; ) L^a (x; t); L t (x; ) L(x; t): The independent part of the -form in the at background is (bos) d : The ^a are the d Dirac matrices and the matrices E ; J ; K in (6) and (5) act on the SO() indices of type II spinors I : A ; A ; A : (7) The supersymmetry transformations with global parameter " and -symmetry transformations with local parameter () for this action are [, ] " I " I (8) x^a " i( ^a ") (9) " A idx^a ^a K" 6 (d ^a )( ^a K") 6 (d ^a K)( ^a ") () I I () x^a i( ^a ) () A idx^a ^a K + (d ^a )( ^a K)+ (d ^a K)( ^a ); (3) where the SUSY parameter " I is a constant type II Majorana-Weyl spinor. Half of the components of the -transformation parameter are projected out by the condition (+ ); ; () A i :::i! i :::i E + i i F i3 i J + ( ) F 8 i i F i3 i E q det(gij + F ij ) (5) where i :::i n b L [i ::: b L in]: Following [7, 7] we will choose the \Killing gauge" for the -symmetry. In [7] ; were choose to be the Killing spinors of the AdS 5 background, for the at background we can simply introduce new variables U ; + A () () A (6)

6 where () 3 with p t p ; p ;:::;3 5 t ; t;:::;9 (7) Here p, and 5 are standard Dirac matrices with n p ; po pp and t are 8 8 six-dimensional Dirac matrices with n t ; to tt : The supersymmetry and -symmetry transformations () and (3) with the new parameters " U " I, U I +" + +" "+ Here the matrices A and are " A A A + + A: (8) A () + () ; () () (9) and explicit expressions for and are given in (5). We gauge away half of the fermionic coordinates by imposing :This condition also requires that +" :Gauge xing -symmetry by the condition + weget: +" + A + "+ () +" ( + ) + " : () Eq. () gives a relation between and " ; Finally we get ( + ) " : () +" + () + () " + "+ : (3) Using this transformation on A p and x^a x p ; y t (p ;;3; t ;;9): +" x p i + p (" " + ) () +" y t i + t " (5) +" A p i + p " p + a + + a " i@p y t + t (" " + ) (6) 5

7 where () : (7) () + The additional general coordinate reparametrization is necessary to satisfy the static gauge (x p p ) condition x p p (8) x^a p x^a (9) then from +"+ x p i x p p i it follows that i i + i (" " + ): Finally the SUSY-transformations for remaining nonzero fermions, scalars and vectors are: +"+ + (" + " + )+ i + (3) +"+ y t i + t " + i y t (3) +"+ A i i + i " i + a + + a " (3) i@ i y t + t (" " + )+( j A i +@ i j A j ) (33) It is easy to see that the transformations for y t are linear and do not contain an " + part. From here on we will drop the \+" in the spinor notation i.e. + ) : In the new variables the supervierbeins () are: L d ; L p dx p i p d ; L t dy t : (3) 3 The Quartic N Lagrangian The D3-brane I action () simplies considerably in the Killing gauge. In this case G ij + F ij ij i j i i i j i y i y t + (F ij i@ i y t j + i@ j y t i ) (35) where F i A j A i and the WZ-term is L WZ T 3 (i 6@ l jkl t y k y t + (( 6@) i ) 6

8 k () t t y j y t k t i y j y t + i 6 ijkl abcl j k + ijkl abcd j k l ( ) l ijkl F ij y t ) (36) Using a series expansion for q det (G ij + F ij ) and a standard redenition of the elds we get the eective Lagrangian up to the fourth order in elds (and coupling constant ): L I+WZ i 6@ + F ijf iy i y t + (( 6@) i ) + 8 (F ijf jk F kl F li (F ijf ji ) )+ (@ iy j y i y y t (@ iy i y t ) ) (@ iy j y t F ik F iy i y t F jk F kj ) i (( j )(@ i y j y t ) ( 6@)(@ i y i y t k i y t j y t t ) + i ( j F jk F ki 6@F ij F ji ) + i i y t j F i y t t l F jk ijkl ) (37) To establish the analogy between this theory and N super-yang-mills theory it is helpful to use the chiral representation for t (see Appendix) where the SU() symmetry is manifest. Using this representation we will introduce new notations for the scalar elds s IJ ( t ) IJ y t with SU() indices I; J ;;3;: The 6- component spinor will be represented as four d Majorana spinors, I _I A ; (38) where the extra factor was introduced for convenience and the d chiral projections are + I ( + 5 ) (I) ; I ( 5 ) (I) : The quartic N Lagrangian in terms of the new -dim YM variables (g; I ;s IJ ) with g i A i is L I+WZ i I 6@ I + F ijf is i s IJ 7

9 + ( 6 I 6@ I ) I I j J J i + Fij F jk F kl F li 8 (F ijf ji ) )+ (@ is j s i s I s I J (@ is i s IJ i s j s IJ F ik F is i s IJ F jk F kj i ( I j I )(@ i s j s IJ ) ( I 6@ I )(@ i s i s IJ k I i s j s K JK + i I j I F jk F ki I 6@ I F ij F ji + + i + is IJ j + J F ij i s IJ j J F ij + I i s IJ l + J F jk I i s IJ l J F jk : (39) where i i k k! [i i k] : The parameter from Eq.(7) is, up to linear order in the elds: IJ 6@yt t IJ ij F ij IJ : () The supersymmetry transformation now looks like: susy (I) susy g i i i ij F ij IJ 6@ 5 s IJ 6@ 5 s IJ ij F ij IJ susy s IJ i" I J i" J I + i" K L IJKL : " J " J A Finally, using a chiral representation for i we get susy I ij F ij " I i s IJ " J () susy g i i" I i I + i I i (" ) I () susy s IJ i" I J i" J I + i" K L IJKL (3) which is exactly the N transformation given in [8]. 8

10 Helicity amplitudes. To calculate one loop corrections to the D3-brane action we use the helicity amplitude technique (see for example the review paper [] and references therein). In the spinor helicity formalism, positive- and negative-helicity massless spinors are represented as jpi (p) ( 5 ) (p) () with antisymmetric spinor products hp i jp j +i hp i p j ihiji (5) hp i + jp j i [p i p j ] [ij] (6) hiji[ji] p i p j s ij : (7) Negative- and positive-helicity polarization vectors also can be represented through spinors jpi; " + (p; k) hk j jp i p hkpi ; " (p; k) hk+j p jp+i ; (8) [kp] where k (k ) is a \reference" momentum which corresponds to the particular choice of gauge for the external legs. Due to the gauge invariance of the amplitudes, the vector k drops out of the nal expressions. The supersymmetry transformations (),() and (3) can also be rewritten in terms of bosonic states with denite helicity g (p) " (p)g and spinors (p) jpi as [3] [Q I (p); J (k)] (k; p)g (k) IJ (k; p)s IJ ; (9) [Q I (p); g (k)] i (k; p) I ; (5) [Q I (p); s JK(k)] i IJ K i IK J i L IJKL ; (5) where + (k; p) [pk]; (k; p) hpki; and isanumerical Grassmann parameter. Using the Lagrangian (39) it is easy to nd the tree-level helicity amplitudes. For the vector bosons we have:!(f ij F jk F kl F li (F ijf ji ) ) (t 8 ) 3 3 F F F 3 3 F 6(t 8 ) k k k 3 3 k 3 3 (5) 9

11 where (t 8 ) is given in eq.(9.a.9) in [9] (t 8 ) 3 3 ( )( ) + ( antisym: of [ i i ]) + (3) + (3) permutations (53) and k i and i are the momentum and polarization vectors for an external vector boson leg. In the YM theory with non-abelian gauge group SU(N c ) the tree amplitudes are usually represented according to color decomposition [, ] as a sum of partial subamplitudes with xed cyclic ordering of external legs multiplied by the color factors: X n (; ; n) Tr(T a() T a () T a (n) ) n ((); (); (n)) (5) Sn Zn where Sn Z n is the set of all non-cyclic permutations and T a (n) are matrices of the fundamental representation of the SU(N c ) color group. In the D3-brane case the gauge group is an Abelian U() and color factors will be simply reduced to. All four-particle tree amplitudes are proportional to (39) and to simplify the notations we will let :Due to the supersymmetric Ward identities (SWI) [3] the only nonzero four vector-boson tree amplitude in this case is (g g g + 3 g+ ) (with two and two + helicity bosons)! ( ; ; 3 + ; + ) i 8 h3; j(f ijf jk F kl F li (F ijf ji ) )j; i i (t 8) 3 3 k k k 3 3 k 3 3 i sthi hih3ih3ihi (55) where s s ; t s ; u s 3 are Mandelstam variables. This amplitude is related to the standard N SYM four-gluon color-ordered sub-amplitude [] by: ( ; ; 3 + ; + ) statree;y M (; ; 3; ): (56)

12 It is important to notice that the RHS of (56) is totally symmetric under external leg permutations [3]. The nonzero vector-scalar tree amplitudes are: g g + ( g ; s IJ ;s + IJ ;g+ ) i (t 8) t t 3 3 k k k 3 3 k 3 3 i hih3i[3][] s IJ s IJ sthi h3i hi ;Y M (; ; 3; ): (57) There are two kinds of four-scalar amplitudes: s IJ s IJ s KL(IJ) s KL(IJ) ( s IJ ; s KL ;s + IJ ;s+ KL) i st (58) sthih3ih3ihi hi ;Y M (; ; 3; ) ( s IJ ; s IJ ;s + IJ ;s+ IJ) i s (59) sthi h3i hi ;Y M (; ; 3; ): The four fermion tree amplitudes include two nonzero ones: I + I J(I) + J(I) ( I ; J ; + J ; + I ) i [3][]hi hi h3ihi hi (; ; 3; ) (6) ( I ; I ; + I ; + I ) i [][3]hi sthi 3 h3i hi ;Y M (; ; 3; ): (6) The next set of diagrams contains a mixture of two fermions and two bosons. There are ve nonzero tree amplitudes:

13 g g + ( g ; I ; + I ;g+ ) (i) i [3][]hi (6) (i) sthi 3 h3i hi ;Y M (; ; 3; ) (63) I + I K + K s IJ s IJ + ( I ; s JK ;s + JK ; + I ) (i)i [3][]hih3i (i) sthih3i hi hi ;Y M (; ; 3; ) (6) ( I ; s JK ;s + IJ ; + K) (i) i [][3]hih3i (i) sthih3ihih3i hi ;Y M (; ; 3; ) (65) K g K s IJ ( I ; J ;s + IJ ;g+ ) (66) sthi h3ih3i i hi ;Y M (; ; 3; ) ( g ; s IJ ; + I ; + J ) (67) sthi h3ihi i hi ;Y M (; ; 3; ) This set of tree amplitudes has exactly the same relative factors between them as for N SUSY Yang-Mills theory given in [5] (extra i's come from dierence between metrics signature which are g (+ )in [5] and g ( + ++) in this paper ). The general expression connecting SYM amplitudes and the D3 brane

14 amplitudes is: (l ;l ;l 3 ;l ) st Atree;Y M (l ;l ;l 3 ;l ) (68) where l ;l ;l 3 ;l are the momenta of the particles from N SUSY multiplet. The main dierence between the standard N SYM and the D3-brane eective theory (39) is that the rst one is nite in d. The following calculation shows that the D3-brane action is not one-loop nite in the at background. Avery useful relation for one-loop calculations in the standard N SYM theory is [5, ]: X S ;S (N ) ;Y M ( l S ; ; ; l S ) ;Y M ( l S ; 3; ; l S ) ist;y M (; ; 3; ) (l k ) (l k 3 ) (69) where the sum is over all particles in the N multiplet and ; ; 3; stands for the momenta of the external particles. This statement is true for any dimension and its proof can be found in [, ]. For our case (with relation (68)) it becomes: X S ;S (N ) (D3) ( l S ; ; ; l S ) (D3) ( l S ; 3; ; l S ) (7) is statree;y M (; ; 3; ) is Atree(D3) (; ; 3; ) here (D3) (; ; 3; ) is the four vector-boson D3-brane amplitude (56). The N supersymmetry of the D3-brane action allows us to use a unitaritybased construction of the one loop amplitudes, as it has done in N Super- Yang-Mills theory (see review [] and references therein). In this technique, one obtains the imaginary part of the one-loop amplitude from the product of the tree amplitudes, and then reconstructs the real part up to a possible polynomial function. The good ultraviolet behavior of N super-yang-mills theory makes it possible to reconstruct the whole amplitude without the additive polynomial ambiguity. The one-loop amplitudes that can be reconstructed that way (\cut-constructible") must 3

15 satisfy a certain loop-momentum \power-counting" criterion given, for example, in []. One-loop amplitudes in the standard N SYM theory obtained by using the cut-reconstruction formalism can be expressed as: loop (; ; 3; ) ig s s 3 (; ; 3; )( 3 I loop A N ; (s ;s 3 ) + 3 I loop (s ;s 3 )+ 3 I loop (s 3 ;s 3 )) (7) where 3 is a color factor for the non-abelian gauge group and I loop is a one-loop four-point (box) integral, I loop (s s 3 ) Z d D p () D p (p k ) (p k k ) (p + k ) : (7) The expression (69) was used to simplify the direct product of the tree amplitudes. The expressions (69)and (7) for N amplitudes are correct in all dimensions and can be used [5, 3] to reconstruct complete massless loop amplitudes. The D3- brane eective Lagrangian (39) in the Killing gauge does not contain three-particle vertices, hence only bubble diagrams will contribute to the one-loop corrections to the four-particle amplitudes. considerably simplies and can be written as: A D3; loop (; ; 3; ) In this case the expression for one-loop corrections i s s 3 ;Y M (; ; 3; )(s I loop (s )+s 3 I loop (s 3 )+s I loop (s )) i Atree(D3) (; ; 3; )(s I loop (s)+t I loop (t)+u I loop (u)): (73) Here we have taken into account that s s 3 ;Y M (; ; 3; ) from (56) is symmetric under momenta permutations and color factors are gone because gauge eld in (5) is an Abelian one. Instead of the box integrals (7) expression (73) contains only two-point one-loop integrals: I loop (s ) Z d p () p (p k k ) : (7) This integral is ultra-violet divergent and in dimensional reduction it has the form: ( + ) I loop ( ) (s) i() ( ) ( ) ( s )

16 i () [ + ( log s + log + )] (75) as! : It is easy to nd counterterms for eective D3-brane Lagrangian (39). structure at the quartic level will be Their (s + t + u )(t 8 )F! (t 8 ) 3 F 3 F ; (76) or, in the notations of [5] and [6], the counterterms are: where T D3 (F ij F jk F kl F li (F ijf ji ) ); (77) T D3 () (s + t + u ) (78) 6 Using the relations (55)-(67) between the amplitudes with dierent particle content it is easy to nd the corrections and the counterterms for all other four-particle interactions of N multiplets in this theory. 5 Dierent -gauges The Killing gauge is obviously not the only choice for xing the -gauge. In [3] it was proposed to use the gauge where and. Then the supersymmetry transformations will be: +( ) X m m ( ) m X m : (79) Awas dened in (5). The requirement and the static gauge condition X p determine the connection between the and parameters, so that the nal supersymmetry 5

17 transformations are [3]: +"+ + ; +"+ y t ( ) t y t ; +"+ A ( )( + y t ) +( 3 ) m A A : (8) The index m is a d index, which includes both (p) and t values (t is a sixdimensional index for scalar elds). The induced world-volume metric in this case is: G mn L m Ln ; X m i (8) and F F i ( + y t )@ +i ( + y t )@ : (8) The choice of -gauge aects only the part of the Lagrangian that contains fermions leaving the purely bosonic terms unchanged. In the gauge the fermionic part of the Lagrangian up to the quartic order in the elds is: L () ferm i 6@ + i y + i )F + ( ( 6@) ) i (@ y y + i )F F 6@F F y y t 6@) + i F ( y y ): (83) In this gauge the Lagrangian contains the three-particles vertices along with the quartic terms. Four-particle tree amplitudes with fermions will contain a set of tree diagrams where line denotes both gauge A and scalar y t bosons and stands for the fermions : + + 6

18 + + alculations of these tree amplitudes show that they are the same for both cases of the -gauges. oth theories have the same one loop corrections and nontrivial counterterms in d. 6 onclusion The orn-infeld type actions have received a lot of attention in the last few years. This type of action plays a crucial role for the D3 brane theory. The rich structure of interaction terms in this theory makes it especially interesting to study them. The other interesting question that arises in this theory is a question about the form of supersymmetry transformations after the xing of the gauge. We have presented the supersymmetry transformations, Feynman rules and oneloop corrections to the -gauge xed D3 brane orn-infeld theory up to the quartic order. We would like to stress here that the Feynman rules following from the gauge- xed supersymmetric orn-infeld action turned out to be rather simple at least for the -point one loop calculations, which has allowed us to perform the quantum calculation in this very unusual theory. Our calculations show that D3 brane theory in the at II background has a nontrivial counterterm in d. We have found that the structure of the counterterm as well as nite corrections includes terms like (@F) : We hope that our calculations may help to shed some light on the relations between the properties of the fundamental theory including the D3 branes and the quantum N supersymmetric YM gauge theory. We also consider these calculations as a preparation for the studies of the more complicated theories, e.g. the D3 brane action in AdS 5 S 5 background or, possibly, for a few interacting D3 branes with non-abelian gauge eld in the action. We have 7

19 found that the use of the Killing gauge for -symmetry combined with the helicity amplitudes technique has simplied our calculations signicantly. Hopefully these methods will be useful for calculations in some other class of interesting problems. Acknowledgments I am grateful to R. Kallosh for the help in the statement of the problem and for support and encouragement. I am thankful to L. Dixon for very useful discussions that helped me in my calculations. I am also thankful to J. Rahmfeld and M. Perelstein with whom we have discussed many issues in the D-brane theory and helicity amplitude formalism. Iwould also like to thank Michael Peskin and Arkady Tseytlin for the important comments on this paper. This research was supported by the Department of Energy under grant DOE- FG5-9ER67 and partly under contract DE-A3-76SF55. This work was also supported in part by the NSF grant PHY Appendix We use chiral representation for d6 dimensional (6) matrices following M. Sohnius notations [8]. In this representation (6) is written as ( t ) IJ ( t ) IJ where matrices ( t ) IJ satises the conditions: tr t t tt ( t ) IJ A (8) IJKL ( t ) KL (85) 8

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