Scholarly article on topic '(Non)-supersymmetric marginal deformations from twistor string theory'

(Non)-supersymmetric marginal deformations from twistor string theory Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — Peng Gao, Jun-Bao Wu

Abstract The tree-level amplitudes in the β-deformed supersymmetric gauge theory are derived from twistor string theory. We first show that a simple generalization of [M. Kulaxizi, K. Zoubos, Marginal deformations of N = 4 SYM from open/closed twistor strings, Nucl. Phys. B 738 (2006) 317, hep-th/0410122] gives us the desired results for all the tree-level amplitudes up to first order in the deformation parameter β. Then we provide a new proposal which matches field theory to all orders of β. With our deformed twistor string theory, it can be shown that integration over connected instantons and disconnected instantons remain equivalent. The tree-level amplitudes in the non-supersymmetric γ-deformed theory can be obtained using a similar deformation of the twistor string. Finally, we find improved twistor string results for the tree-level purely gluonic amplitudes in theories with more general marginal deformations.

Academic research paper on topic "(Non)-supersymmetric marginal deformations from twistor string theory"

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Nuclear Physics B 798 (2008) 184-197

www. elsevier. com/locate/nuclphysb

(Non)-supersymmetric marginal deformations from twistor string theory

Peng Gaoa, Jun-Bao Wub*

a Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA b International School for Advanced Studies (SISSA), via Beirut 2-4, I-34014 Trieste, Italy

Received 9 January 2008; accepted 29 January 2008

Available online 5 February 2008

Abstract

The tree-level amplitudes in the f-deformed supersymmetric gauge theory are derived from twistor string theory. We first show that a simple generalization of [M. Kulaxizi, K. Zoubos, Marginal deformations of N = 4 SYM from open/closed twistor strings, Nucl. Phys. B 738 (2006) 317, hep-th/0410122] gives us the desired results for all the tree-level amplitudes up to first order in the deformation parameter f . Then we provide a new proposal which matches field theory to all orders of f . With our deformed twistor string theory, it can be shown that integration over connected instantons and disconnected instantons remain equivalent. The tree-level amplitudes in the non-supersymmetric y-deformed theory can be obtained using a similar deformation of the twistor string. Finally, we find improved twistor string results for the tree-level purely gluonic amplitudes in theories with more general marginal deformations. © 2008 Elsevier B.V. All rights reserved.

PACS: 11.15.Bt; 11.30.Pb

Keywords: Perturbative gauge theory; Twistor string theory

1. Introduction

One of the important issues in AdS/CFT correspondence [1-3] is to study the closed string dual of the gauge theories with less supersymmetries. In [4], the gravity dual of some gauge theories with less supersymmetries was studied using an elegant solution generating transformation.

* Corresponding author. Tel.: +39 040 3787502; fax: +39 040 3787528. E-mail addresses: gaopeng@physics.upenn.edu (P. Gao), wujunbao@sissa.it (J.-B. Wu).

0550-3213/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2008.01.027

One of the gauge theories studied in [4] is the 3-deformed theory which has N = 1 super-symmetry. Later this discussion was generalized to y-deformed theory which has no supersym-metry [5-7].1 These deformed SYM can be obtained from a certain kind of star product among the superfields. Using this star product, it is easy to see that the planar amplitudes in such theories are identical to those of the N = 4 theory up to an overall phase factor [4,7,17].

On the other hand, Witten found recently a new relation between perturbative gauge theory and a topological string theory whose target space is the supertwistor space [18]. While the proposed AdS/CFT correspondence is a strong-weak duality, the new correspondence found by Witten is a perturbative one. The gauge theory amplitudes can then be computed using localization to either the connected instantons in twistor space [18-21], or the completely disconnected instantons [22]. It has been shown that indeed these two prescriptions are equivalent [23] and a family of intermediate prescriptions can also be used [23,24]. Following Witten's original paper [18], there appeared a lot of works on computing tree-level and/or one-loop amplitudes in four-dimensional gauge theories, some of which are [25-41].

The twistor string theory corresponding to the gauge theories with less supersymmetries obtained by the Leigh-Strassler deformation [42] was explored in [43], motivated by some earlier work on deformation operators in the open-closed topological B model string [44,45]. There it was suggested that the effect of the marginal deformations in field theory can be captured in twistor string theory by introducing a non-anticommutative star product among the fermionic coordinates of the supertwistor space. This star product can be shown heuristically to partially arise from a particular fermionic deformation in the closed B model string background. A prescription for calculating the tree-level amplitudes in the deformed theory using the connected instantons was also proposed in that paper and shown to produce desired field theory results to the first order of the deformation parameter. In this prescription, a non-anti-commutative star product among the wavefunctions of the external particles plays an important role. The authors of [43] computed examples of amplitudes corresponding to degree one curves in twistor space (these amplitudes are dubbed 'analytic amplitudes' in [46,47]) and found that the results coincide with field theory results. Later it was claimed one can generalize the prescription to amplitudes localized on higher degree curves [48]. The precise origin of this prescription is not completely understood. In particular, a prescription valid to all orders of the deformation parameter is absent. It is therefore interesting to make some progress in this direction.

Since the planar amplitudes in 3-deformed and y -deformed theories are rather simple, one may wonder whether they can be reproduced using the twistor string theory. At present it is not clear how to calculate the amplitudes beyond tree-level from pure twistor string technique, we will from now on focus on tree-level amplitudes. We first show that the tree-level amplitudes in 3-deformed theory can be obtained from the prescription in [43] to first order of the deformation parameter. Notice our result is not restricted to the degree one instanton case. In fact, we produce all of the tree-level amplitudes exploiting a rather simple generalization of [43] based on previous work for the N = 4 theory [19-21]. We then propose an all-order exact prescription for the non-anti-commutative star product and show that this new prescription gives the right complete field theory results. The MHV diagrams and the equivalence between the prescriptions using connected instantons and completely disconnected instantons are also discussed in the case of the 3-deformed theory. The y -deformed theory is no longer supersymmetric, and does not belong to

1 The deformations in gauge theory were also studied in [8-15] and some generalizations of the work in [5] including a generalization of y-deformation can be found in [16].

the class of field theories studied in [43]. In this case, we also give an prescription to all orders of y to reproduce tree-level field theory amplitudes from the deformed twistor string theory.

Encouraged by these success, we proceed to find some further evidence for a possible all-degree closed form answer for theories with generic Leigh-Strassler deformations. We consider tree-level scattering amplitudes involving only particles in the N = 1 vector supermultiplet. One can easily see that the particles in the N = 1 chiral supermultiplets cannot appear in the contributing tree-level Feynman diagrams. So these amplitudes in the theory with generic Leigh-Strassler deformations are the same as the ones in the undeformed N = 4 theory. We improve existing results in [43] by showing that this holds true for all of the tree-level amplitudes involving only particles in N = 1 vector supermultiplet.

This paper is organized as follows. In Section 2 we briefly review the method of computing field theory tree-level amplitudes from connected instantons in twistor string theory. In Section 3, we study the amplitudes in j-deformed gauge theory from the twistor string theory. An exact prescription for the tree level amplitudes in y -deformed theory is given in Section 4. In Section 5, we discuss the tree-level amplitudes with only particles in the N = 1 vector supermultiplet in the case of generic Leigh-Strassler deformations. The final section is devoted to conclusions and directions for further studies.

2. Review of tree-level Yang-Mills amplitudes

2.1. Amplitudes of the N = 4 theory from twistor strings

We denote the tree-level partial amplitude of N = 4 super-Yang-Mills theory, which is the coefficient of

Tr(Ta1 ■■■Tan), (1)

AN=4{ni, ni,hi,ti;...; nn,7tn,hn,tn), (2)

or simply by AN=4({ni, ni,hi,ti}), here we decompose the momenta of the ith massless external particles into two Weyl spinors, i.e., we define the 'bi-spinor' piaa as piaa = and choose a decomposition of this 'bi-spinor' as piaa = nia7tia (for more details, see, for example [18,49]), and we use hi to denote the helicity of this external particle and ti its SU(4)R quantum numbers.

This amplitude can be computed in twistor string theory from localization on the connected instantons [18-21] (some aspects of this approach were summarized in [49]). The main formula is:

AN=4({ni,ni,hi,ti}) = j dMdlf № ■■■ J Jn#\ (3)

\C C '

in the following we will explain the ingredients in this formula. First we note that in the su-pertwistor space CP3'4 with coordinates denoted by ZI = (ka,^a), fA,(a, a = 1,2,I,A = 1,..., 4), we have the following expansion [18]:

a = A + f 4X4 + fIXi + f V^i + 1 tItJ€uK'PK + 1 f 4tItJciJKX K

+ 3 fIfJfK^iJKX 4 + 3 f 4fIfJfK€UKG. (4)

As in [43], we split the four fermionic coordinates in CP3'4, tyA(A = 1,...,4), into tyI (I = 1,2, 3) and ty4. For latter convenience, we write the above expansion as

A = J2 gh ty(5)

A= —1,-1/2,0,1/2,1

where @ha = A, x4,XI,$I,<t>1, XK, X4,G respectively. ^i(ni, 7ti,hi,ti) in Eq. (3) is the wave-function of the i th particle in the supertwistor space, which is given by

/ — \2h—1

Vi(ni,iïi,hi,ti) = 8( — )){ —j exp(i [Hi,ri(ni/V)ga(* A,fI), (6)

where the definition of the delta function 1(f) is

~S(f) = dfS2(f), (7)

following [22]. In Eq. (3), Ji is a holomorphic current made of free fermions in the wouldvolume theory of the ,D5-branes (the details can be found in [18,49]). Since what we actually compute is the partial amplitude which is the coefficient of Tr(Ta1 ■ ■ ■ Tan), in Eq. (3) we only pick out the following coefficient of Tr(Ta1 ■ ■ ■ Tan) in the correlation function (J1(u1) ■ ■ ■ Jn(un)):

Ui (ui, dui) (8)

Elk (uk,ut+1)'

In the prescription using the connected D -instantons which are D-strings wrapped on algebraic curves, these amplitudes only receive contributions from the curves with genus zero and degree satisfying d = "=1(1 - hi) - 1.

If we choose homogeneous coordinates (u, v) on an abstract CP1, then the genus 0, degree d curve C, which is a map from CP1 to CP3'4, can be parametrized as

Z1 = PI(u,v) = Pluavd-a, fA = QA(u, v) = J2 QAuavd-a. (9)

a=0 a=0

Then the measure on the moduli space of the genus 0, degree d curves can be written as

dMd = nq=, nA=1 n4=1dpI dQA. (,0)

d GL(2, C)

Now we may write Eq. (3) as

f f (k(u-)X 2hi-1 A=4/r_. _ /^A /(.TT i /ui dui )l(lk(ui),n^1 1

an -( to.*^ }=f dm„n/<ui-dui)H (—(ui)-ni)

x exp(i [p(ui), ni](ni/—(ui)))ghi. (tyi)--. (11)

i Hk (uk,uk+1)

2.2. Amplitudes in deformed SYMfrom twistor strings

In [43], a proposal was made to calculate the amplitudes in deformed SYM from twistor string theory to the linear order of the deformation parameters.

The superpotential in the deformed theory considered in [43] can be written as2

W = Wn=4 + 3hIJK Tr($I$j$K), (12)

to linear order of hIJK, where hIJK is totally symmetric. Here

Wn=4 = i Tr(01 [02,03]), (13)

is the superpotential of the N = 4 SYM and 0i, i = 1,2, 3 are three chiral superfields in the adjoint representation of the gauge group. We note that this deformed theory only has N = 1 supersymmetry.

The authors of [43] conjectured a star product among the f -dependent part of the wavefunc-tions as follows 3

f(f1) * g(f2) = f(f1)g(f2) - ^Kifif^fyKfL^dfJJgf) ), (14)

where the definition of VKJL is

vkjl = hijq€qkl + eijqh qkl, (15)

then the amplitudes is given by

a deformed

Anei0imed{{nl,ni,hl,tl}) = j dMdlj Jx^x * J J2&2 *■■■*!

\ r" f r"

ni, n i,hi,ti ^ = J dMAj J1&1 * I J2&2 *■■■* I Jn^n ), (16)

again to first order of the deformation parameters.4

3. fi-deformed gauge theory from twistor strings

3.1. All tree-level amplitudes to linear order in j3

The superpotential in the ¡i-deformed SYM theory [4,17] is

Wp-deformed = i Tr(eini 010203 - e-ini 0103 02). (17)

Here ¡ is a real deformation parameter.

Now we compute the tree-level amplitudes in this theory using the all-degree generalized prescription we give above. First, we expand Wp-deformed to first order of i:

Wp -deformed = WN=4 - np Tr( 01{02,03}) + O(p2). (18)

Then one can easily read off in this case the hIJK's in Eq. (12) are simply -2np\eIJK\, so we find

VKL = hIJQ(QKL + eIJQh qkl = -4npSIKS[€UQaQ, (19)

2 As in [43], we pull the gauge coupling constant out of the superpotential. For ease of comparison, we also use the same normalization as [43].

3 We note that the superscript of f denotes its SU(4) R-symmetry quantum number (the SU(4)r symmetry of the N = 4 SYM has been broken by the deformation) and the subscript of f numbers the corresponding external particle.

4 As we have mentioned in the introduction, in [43] only the analytic amplitudes were found, the formula here is in fact valid for all tree-level amplitudes.

where the definition of aj's is simply a1 = a2 = a3 = 1.

Using this, it can be easily found that for an arbitary pair of wavefunctions f and g we have

f(fi) * gf)

= f(fi)g(f2) + i*P№i)( J2 eJJKzdTfI1aK^-d7)gW2) + O(p2). (20)

dfj dfj'

Because the wavefunctions are all monomials in fermionic directions, we further have

9 r \f(fi), if I e f,

f(fi)-vf! = (21)

dfI [ 0, otherwise,

where by I e f, we simply meant fI is a factor of f(fi). In other words,

ff )^rrjfT = Njf(fi), (22)

where Nj is the number of fj in f(f\) (Nf < 1 since fj is fermionic). Similarly,

fg^TTjgf) = Nggf). (23)

2 dfg g

So Eq. (20) can also be re-written as

f(fi) * gf) = (l + inp £ eUKNIfNJgaAf(fi)g(f2) + O (p2)

= (l + inp £ eijKaAfWiMfc) + O(p2). (24)

^ Ief,Jeg '

We can slightly rephrase the above results by considering the following U(1) x U(1) symmetry acting on the fermionic part of the supertwistor space CP3'4, under which (fi,f2,f3,f4) are assigned the following charges5:

from which we have £ijk®k — Q f Qf - Q f Q f . Using the identity

Q(ffl ...fIn) = Qff +...+ QfIn, (26)

we get

^i * ^ = fi + i*pJ2( (Q1Q 2 - Q 2Q1)) ^ n + O (p2). (27)

f1 f2 f3 f

Q i 0 -1 1 0

Q 2 1 -1 0 0

5 In [50], similar star product was introduced in N = 4 light-cone superspace in stead of supertwistor space. Recently, a possible new relation between p-deformation and noncommutative field theories was discussed in [5i].

Here Q1 and Q2 are the charges of the ith wavefunction. We notice that there is the following relation between the Qi defined here and the Qi defined in [4,17],

Qi($ha) + Q i g f 4,fI )) = 0. (28)

Replacing the charges using this identification, we find

ai-deformed = (1 + inp £(q1q2 - Q2Q1 )) aN=4 + O (p2). (29)

Here Q1 and Q2 are the charges of the wavefunction of i th external particle. This result coincides with the results in [17] to first order of p.

3.2. The star product to all orders of p

Based on the calculation in the previous subsection, we propose the following star product among the wavefunctions

( I j ~lT\

f(f 1) * g(f 2) = f(f1) exp inp > eIJK—jfiaKf2—rJ )g(f2)

^ i?K dfI dfJ'

= E^ E ••• E f(f1)(j fl1f

n=0 h J1K1 InJnKn V °f2

( d J J d \

"■■i'^ffwj-j™. (30)

will reproduce all tree-level field theory amplitudes to all orders of p.

Since -^-f^ commutes with fJm-dr-, we have

df1 1 2 dfjm

(inp)n y—^. y—^.

n ■ ■ ■ eIJ1Kr ■ ■ eInJnKnaKr ■ ■ aKn

n=0 n' I1 J1K1 InJnKn

d I, d I Jt d j d

X f(f\)-T-f{----^fifJ -r- ' ' f2n-rgf)

dfI1 1 dfIn 1 2 dfJ 2 dfJn

^ (inp)n ^ ^

= ¿^-j— ¿^ ■ ■ ehJ1Kr ■ ■ ^InJnKnaK1■ ■ ■ aKn

n=0 n' I1 J1K1 InJnKn

x Nf1 ■ ■ ■ NfNJ1 ■ ■ ■ NJnf(f1)g(f2)

= £ ^n-i £ (IJKttKNfNJ^j f(f1)g(f2) n=0 U' IJK '

~ (inp)n ( ^ \n

= -j—( (IJKaKj f(f1)g(f2)

n=0 n' ^IefJeg '

= expi inp J^ €IJKaK\f(f\)g(f2). (31)

^ IsfJeg '

As in the previous subsection, we can now re-write

f (f i) * g(f2) = exp(inp(Qf Qf - Qf Qg))f(fi)g(f2). (32)

From Eq. (28), we know that the star product among the wavefunctions defined here is isomorphic to the one among the fields defined in [4,i7], and hence the former star product is also associative due to the associativity of the latter one. This can be verified explicitly as follows:

(f * g) * h = exp(inp(Qf Q2 - Qf Qg))(fg) * h

= exp(inp(QfQ2 - QfQi) + inp(QfgQh - QfgQ\))fgh = exp (inp(Q fQ gh - Q (Qgh) + inp(Q gQ h - Q gQ h))fgh

= exp(inp(QgQh - QgQh))f * (gh)

= f * (g * h), (33)

where Qfg = Qf + Qg(i = i, 2) is used.

Similar to what we have done in the previous subsection, one can easily verify that

^p-deformed = exp(mp £(QiQ2 - Q2Qi)) AN=4. (34)

This result is indeed the same as the one obtained in field theory computations [4,i7]. 3.3. The MHV diagrams and the equivalence between different twistor space prescriptions

In this subsection we will discuss the MHV diagrams and the equivalence between the prescription using connected instantons and the prescription using completely disconnected instantons for the p -deformed theory.6

The analytic amplitude in p -deformed theory is independent of X, since this amplitude equals the product of the corresponding analytic amplitude in N = 4 theory and a phase factor,7 both of which are independent of X. Then in the p-deformed theory, we can use the same off-shell continuation as in [22,47] to define the analytic vertex. We can compute all of the tree-level amplitudes in p-deformed theory from the MHV diagrams obtained by connecting the analytic vertices with propagators as in [47]. Similar to the proof using Feynman rules in [i7], we can prove that the amplitudes from the MHV diagrams are exactly the same as expected.

As in the N = 4 theory [22], the extended-CSW rules in the p-deformed theory mentioned above can be obtained from the prescription for calculating the tree-level amplitudes using completely disconnected instantons in the twistor string theory. Consider an expression in the completely disconnected prescription corresponding to a given MHV diagram. Notice that the twistor space propagator D proportional to S4(f - f') and the latter is just (f - f ')4 [23,49] (f and f' are fermionic coordinates of the two ends of the twistor space propagator). In a given MHV diagram, each end of a propagator has a fixed helicity and a fixed SU(4)R quantum number. According to this helicity and quantum number, only one term in the expansion of (f - f ')4 will be picked out. This term is a product of two factors, one factor is a product of f 's the other of

6 In this subsection, we will only give some brief discussions, the details are omitted since they are similar to the discussions in the N = 4 case.

7 Notice that the phase factor also depends on the internal particles.

f''s. So for every analytic vertex and a propagator connected to this vertex, there is an end of the propagator corresponding to the vertex. Then there is a corresponding product of the fermionic coordinates. In the expression corresponding to the MHV diagram, for every vertex, these factors from the propagators connected to it and the wavefunctions of the external particles connected to it should be multiplied using the star product in supertwistor space to give the amplitudes in j -deformed theory. Then, similar to the proof using Feynman rules in [17], we will finally get a star product among all of the external wavefunctions multiplied by ordinary multiplication with the ordinary product of the twistor space propagators. The associativity of the all-order star product defined in the last subsection is essential for the these arguments.

Following the proof in [23] for the N = 4 case, one can prove that the prescription using the connected instantons and the prescription using the completely disconnected instantons will give the same result. One need only notice that in the proof in [23], the poles of the integrands which contribute to the residues never came from the wave-functions themselves, nor from the case when two points where the wavefunctions are inserted come close to each other.

4. y -deformed gauge theory from twistor strings

A generalization of the j-deformation is the y -deformation [5-7]. It is obtained by introducing the following star product

f+g = exp(-inQf Q8:€ijkYk)fg

among the component fields in the Lagrangian of N = 4 theory. Here Yi,i = 1,2, 3 are three real deformation parameters and Qi, i = 1, 2, 3 are the charge of three U(1) symmetries of the theory. The charges of the component fields are the following [7]:

The y-deformed theory is non-supersymmetric and the component Lagrangian of this theory can be found for example in [7]. To reproduce the tree-level amplitudes in the y -deformed theory [5-7] from connected instantons in twistor string theory, we consider the following global ^(1)3 symmetry acting on the fermionic coordinates of the supertwistor space, under which

A,, 01 02 03 /1 /2 /3 X4

Qi 0 1 0 0 1/2 -1/2 -1/2 1/2

Q2 0 0 1 0 -1/2 1/2 -1/2 1/2

Q3 0 0 0 1 -1/2 -1/2 1/2 1/2

and for every component field $, we have Qi (0t) = - Qi ($)

(f 1,f2,f3,f4) have the following charge assignment

f1 f2 f3 f4

Q1 -1/2 1/2 1/2 -1/2

Q 2 1/2 -1/2 1/2 -1/2

Q 3 1/2 1/2 -1/2 -1/2

and we define the star product among the wave functions as

f(fi) * g(f2 ) = f(fi) expf-inV] -^jfAQ fjYkQ f*2TTB)g(*2),

\ ABdf* j dfB)

which is equivalent to

f(fi) * gf) = expf-in £ Q fAQ feijkn^WiMfc). (39)

^ Aef,Beg '

This star product is also associative and it gives rise to the following tree-level amplitudes:

.y-deformed ( . ST^ Aa „ 1 /|N=4

An = expl iQ jdjkYk An

^ a<b '

= exp(-in £ Qa Qbj-^j AN=4,

J ^iJk r k I" <-n 7 (40)

again coinciding with field theory results [7] to all orders of y • Similarly to what we have done in the previous section, we can show that the prescription using the connected instantons and the one using the completely disconnected instantons are equivalent which relies crucially on the associativity of the current star product defined above.

5. Pure gluonic amplitudes in the general deformed theory

The tree-level purely gluonic amplitudes in theories with generic Leigh-Strassler deformations are the same as in the N = 4 theory [43]. This is a trivial result in the field theory side since if the external particles are all gluons, the internal particles in the tree-level Feynman diagrams can only be gluons too. In the twistor string theory side, this was proved for the analytic amplitudes to first order of the deformation parameters in [43]. Here we will show that it is also true for all of the tree-level amplitudes obtained from twistor string theory. We take this as another evidence for the existence of a generalized prescription to produce field theory amplitudes in general marginally deformed theories. We only consider the amplitudes An(1-, 2-, 3-, 4+ ,...,n+) as an example. The demonstration of this result for other purely gluonic amplitudes is identical in spirit.

The star product one needs to compute is

(e/1/2/3 fi fi f'i3) * (e/1/2/3f2Jf/2 f2 ) * f3K ff2 ff3)

= e/1/2/3 /2/3 efl K2K3I f 1 f{2 f'l3 f'2 f{2 fJ f3Kl ff2 ff3

- - (3V!1 f2 f 13f/1 f/2 / f?1 f 3?2 f 3?3 v73/1

4\ r1 r 1 r 1 ' 2 ' 2 ' 2 r3 r3 r3 l^J-y

J3K1 /3K1

+ 32f 11 f 12 f 13 f/1 f J2 f/3 f?1 f?2 f?3 V/3K1

+ 32 fl1 fl2f3f/1 f/f/3f?1 f?2fK3V,73?1 ) I, (41)

to the linear order of deformation.

The first term in the right-hand side of the above equation gives the amplitudes in N = 4 theory [18-21,23]. Now we will show that the second term gives no contributions. Since there are 3 external gluons with negative helicity, the contributing algebraic curves in supertwistor space should be the ones with genus zero and degree 2. Such curves can be parametrized as 2

Z7 = £ Pluav2-a, (42)

fA = £ QAuav2-a. (43)

Now we define

fW^W^3 / j / j d QiVh ^h fh / / . (44)

J I = A a=0 /

It is not hard to see that [19]

FI1I2I3/1/2/3K1K2K3 a çhhhç/1/2/3ÇK1K2K3 (45)

here the proportional coefficient is independent of the indices Ii ,Ki (we will not need the concrete value of this coefficient in the following). From this result, we can prove that

eW3 /1/2 /3 fK1K2K3 F^3 /1/2/3K1K2K3 VI/1 = 0. (46)

Using Eq. (45), we know that the left-hand side of the above equation is proportional to

Q1I2I3 €/1/2 /3 ^K1K2K3 e^3 /1/2/3 e™3 {tI3/1Qh Q/ + hI3/1 QeQhi1 )

a 51/ (eI3/1QhQI3/1 + hi3/1q€q/)

= e I3/1Qh q I3/1 + hI3 /1Q€Q I3/1

= 0. (47)

So the second term in Eq. (41) vanishes after integral over the Grassmann odd coordinates of the moduli space of these curves. By the same calculation we find that neither the third term nor the forth term gives contributions. This completes our proof.

Using the same method, we can show that all amplitudes with only gluons and/or gluinos in the N = 1 vector supermultiplet (i.e., A.4) in the deformed theory are the same as in the N = 4 theory to first order of deformations considered here.

6. Conclusion and discussions

In this paper, we studied the particle scattering amplitudes in four-dimensional gauge theories with marginal deformations using the recently proposed twistor string theory, especially in the supersymmetric j -deformed theory and the non-supersymmetric y -deformed theory.

In the j-deformed theory, we first re-investigate the amplitudes to linear order in j using a generalization of results of [43]. We find that the corresponding star product among the wavefuc-tions in this special case is drastically simplified compared to theories with general deformations. The special combinations j^t^i and f/ J/ that appear in our expressions make it easy to convert our star-product into the form found in [4,17]. These special operators merely count the number of f [ and f/ and this fact lead us to make a conjecture for the all-order star product and show that this conjecture is indeed correct. In other words, this identifies the exact star product we need to use to multiply two general functions f(f1) and g(f2) in the corresponding twistor string theory. In sharp contrast to our success in the real j case, it turned out much harder to find the exact star product in other more general cases at this moment. We hope to return to this problem in the near future.

In the case of real y deformed theory, we were also able to find an all-order description which leads correctly to the field theory amplitudes. This is done essentially by guesswork and we know even less about aspects of the B-model description in this case. It is an interesting open problem to find the correct closed string background which would give the self-dual part of the Lagrangian of this string theory.

Finally we showed that the amplitudes involving only the fields in the N = 1 vector supermultiplet obtained from the prescription is the same as the one expected from the field theory. Due to the difficulty alluded to above in generalizing our results in the special cases, we do this to linear order in the deformation parameters. This general result may be taken as yet another evidence of the possibility of completely understanding the closed B-model string background for the general marginal deformation of four-dimensional gauge theory.

Despite the relative ease with which we were able to reproduce correct field theory amplitudes, we still lack a proper conceptual understanding of the precise origin of the twistor string theory deformations found here and in [43]. Notice the difficulty here is closely related to the fact that the scattering amplitudes involve products among different wavefunctions while the holomorphic Chern-Simons theory concerns only the wavefunction of the five-brane. This means necessarily that modifying the product in the holomorphic Chern-Simons theory would not give rise to the full star product required to reproduce four-dimensional field theory results. We hope that our exact results in the above special cases may shed some more light on this issue.

Another interesting problem is to prove that the tree-level amplitudes obtained from twistor string theory satisfy the constraints from the parity invariance of the gauge theory. Since in twistor theory, the gluons with opposite helicities are treated in different manners, the parity invariance is not manifest at all. It has been proven that the amplitudes in N = 4 super-Yang-Mills theory obtained from twistor string theory satisfy the constraints from parity invariance [21,52]. It is quite interesting to generalize this proof in the theories with deformations, possibly first at the linear order of the marginal deformation parameters.

Acknowledgements

We are thankful to the CMS of Zhejiang University where the current work was initiated, Edna Cheung for organizing a great workshop, and to Konstantinos Zoubos for very helpful discussions. P.G. thanks Wu-yen Chuang for useful discussions and acknowledges AEI Potsdam, ICTP Trieste and the 4th Simons Workshop in Stony Brook for hospitality during the course of this work. J.W. would also like to thank Bin Chen, Bo Feng, Bo-Yu Hou, Miao Li, Xing-Chang Song, Xiang Tang, Peng Zhang and Chuan-Jie Zhu for useful discussions, and Peking University, Northwest University for hospitality. The work of P.G. was supported in part by the DOE grant DE-FG02-95ER40893. The work of J.W. is supported in part by the European Community's Human Potential Programme under contract MRTN-CT-2004-005104 'Constituents, fundamental forces and symmetries of the universe' as a postdoc of the node of Padova.

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