A Spinor helicity formalism

bibliosimply4.bib

ROM2F/2015/10

Simplifying one-loop amplitudes in superstring theory

Massimo Bianchi, Dario Consoli

Dipartimento di Fisica, Università di Roma “Tor Vergata”

I.N.F.N. Sezione di Roma “Tor Vergata”

Via della Ricerca Scientifica, 00133 Roma, Italy

Abstract

We show that 4-point vector boson one-loop amplitudes, computed in [Bianchi:2006nf] in the RNS formalism, around vacuum configurations with open unoriented strings, preserving at least SUSY in , satisfy the correct supersymmetry Ward identities, in that they vanish for non MHV configurations and . In the MHV case we drastically simplify their expressions. We then study factorisation and the limiting IR and UV behaviours and find some unexpected results. In particular no massless poles are exposed at generic values of the modular parameter. Relying on the supersymmetric properties of our bosonic amplitudes, we extend them to manifestly supersymmetric super-amplitudes and compare our results with those obtained in the hybrid formalism, pointing out difficulties in reconciling the two approaches for contributions from sectors.

## Introduction

There is a revival of interest in superstring loop amplitudes from different perspectives [Bianchi:2006nf, Matone:2005vm, Tourkine:2013rda, Witten:2013tpa, Witten:2013cia, Witten:2012bh, Green:2013bza]. The original one-loop computations by Brink, Green and Schwarz [Green:1982sw] paved the way to remarkable developments both in string theory and in field theory related contexts. Some time ago the BGS amplitude for four vector bosons in Type I superstrings was generalised to in vacuum configurations with open and unoriented strings preserving at least supersymmetry [Bianchi:2006nf]. Although the final result could be expressed in a compact form as a sum over the various sectors, only the contribution of the sector and the ‘irreducible’ contributions of the sectors could be easily seen to be proportional to the tree-level amplitude. Supersymmetry Ward identities imply that the only non-vanishing 4-point amplitudes be Maximally Helicity Violating (MHV) [Brandhuber:2007vm]. Thus loop corrections should reproduce similar structures to tree-level amplitudes in supersymmetric vacuum configurations in .

Aim of the present paper is to simplify the results of [Bianchi:2006nf] using the spinor helicity formalism and to analyse the singularities of the resulting amplitudes. We will also comment on the soft behaviour of the amplitudes and compare our results with the ones obtained in [Berkovits:2001nv] within the ‘hybrid formalism’ [Berkovits:1994wr]1 and find it hard to reconcile the two approaches. The main difference is that despite vector boson vertex operators are compactification independent, i.e. they are proportional to the identity operator of the internal SCFT, only in the sector one has a complete factorisation of the space-time and internal part, encoded in the sum of KK momenta or alike. In sectors the computation does not factorise even for the ‘irreducible’ contributions, encoded in a function of the modular parameter and of the compactification moduli, which does not simply coincide with the internal partition function.

The plan of the paper is as follows. In Section 1, we will briefly review one-loop open superstring amplitudes in order to fix the notation. In Section 2 and 3 we rewrite 2- and 3-point ‘amplitudes’2, which vanish in the case of sector, in the helicity formalism. We show that they satisfy the correct Ward identities, in that, for instance, 3-point amplitudes with 3 positive helicity vector bosons vanish, and identify their divergences. The results of [Bianchi:2006nf] for 4-point amplitudes are reviewed in Section 4 and systematically simplified in Section 5, where we show that only MHV amplitudes are non-vanishing. The case of ‘regular’ branes at orbifold singularities [Kachru:1998ys, Zoubos:2010kh] is discussed in Section 6. In Section 7 we study factorisation and find some unexpected result, i.e. no massless poles appear for generic modular parameter, even in sectors with reduced SUSY. We then discuss the IR and UV behaviours of the independent amplitudes: planar, non-planar ( and ) and unorientable. After extending our bosonic amplitudes to full super-amplitudes, in Section 8 we draw a comparison between our results and the results of the hybrid formalism [Berkovits:2001nv] and we discuss potential sources of disagreement. We will conclude with some speculations about higher points, higher loops, soft limits and KLT relations beyond tree-level in Section 9.

## 1 Superstrings at one loop

In theories with open and unoriented strings scattering amplitudes can be computed inserting the corresponding vertex operators on the boundaries [Kiritsis:2007nutshell]. The vector boson vertex (in the super-ghost pictures ) reads

 V(0)B=aμ(∂Xμ+ik⋅ψψμ)eikX=(a⋅∂X−i2fμνψμψν)eikX (1.1)

where is the linearised field strength.3

The tree-level disk contribution is similar to Veneziano amplitude

 A\textuptree4(1,2,3,4)=g2sF4[B(s,t)stT1234+B(t,u)tuT1423+B(u,s)usT1342] (1.2)

where is the coupling constant for open string. The Veneziano factor and the Chan-Paton factor read

 B(x,y)=Γ(1−α′x)Γ(1−α′y)Γ(1−α′x−α′y),Tabcd=tr(tatbtctd) (1.3)

while the totally symmetric kinematic factor is given by

 F4=[2(f1f2f3f4)−12(f1f2)(f3f4)+cyclic(234)] (1.4)

In , is non-vanishing only in the Maximally Helicity Violating (MHV) case

 F4++++=F4−+++=0,F4−−++=⟨12⟩3⟨23⟩⟨34⟩⟨41⟩st (1.5)

thereby, for a given color ordering, the partial amplitudes read

 A\textuptree4[1−,2−,3+,4+]=A\textuptree\textupYM[1−,2−,3+,4+]B(s,t) (1.6)

while . Generalization to higher points can be found in [Mafra:2011nv, Mafra:2011nw, Barreiro:2012aw, Barreiro:2013dpa].

The one-loop four-point amplitude in was computed long ago by Brink, Green and Schwarz [Green:1982sw]. It receives three contributions: planar, non-planar and un-orientable. Setting the modular parameter of the covering torus to be for the annulus and for the Möbius strip, all contributions can be written in the form

 A1−loop4(1,2,3,4)=g4sTCPα′2F4∫∞0dTT5+1∫RCPd4ziΠ4(zi,ki) (1.7)

where is the Chan-Paton factor, is the integration region, depending on color ordering, and

 Π4(zi,ki)=∏i

is the ubiquitous Koba-Nielsen factor with the scalar propagator (Bargmann kernel) for boundary insertions at one-loop

 GA(z1,z2;τA)=−[logθ1(z1−z2|τ)θ′1(0|τ)−2π[Im(z1−z2)]2Imτ] (1.9)

We will often write instead to .

In the planar case all vertex operators are inserted on the same boundary of an annulus

 Tplan1234=tr(t1t2t3t4)tr(1),Rplan1234={z1>z2>z3>z4=0} (1.10)

plus cyclic permutations of 234. The parametrization of the world-sheet variable on a boundary is with .

In the non-planar case vertex operators are equally distributed among the two boundaries of an annulus

 Tnon−pl12|34=tr(t1t2)tr(t3t4),Rnon−pl12|34={z1>z2;z3>z4=0} (1.11)

plus permutations of 2 with 3 and 4. The parametrization of the world-sheet variable on the other boundary is with .

For gauge groups with (anomalous) factors there is an additional non-planar contribution with 3 vertices inserted on a boundary and the remaining one on the other boundary

 Tanom123|4=tr(t1t2t3)tr(t4),Ranom123|4={z1>z2>z3=0;z4} (1.12)

plus permutations of 4 with 1,2,3.

In the un-orientable case vertex operators are inserted along the single boundary of a Möbius strip with twice the length of the strip itself

 Tun−or1234=2tr(t1t2t3t4)TΩ,Run−or1234={z1>z2>z3>z4=0} (1.13)

plus cyclic permutations of 234, with the tension of the relevant -plane in units of , quantized charge. The parametrization of the world-sheet variable on the unique boundary is with .

At low-energies , and one can trivially integrate over the insertion points producing a factor of . The remaining integral over the modular parameter is IR finite (for ) but UV divergent (for ), due to the dilaton tadpole associated to the empty boundary and the ‘cross-cap’. Yet for , the dilaton tadpole cancels and the Type I theory if free of both UV divergences and chiral anomalies [Green:2012pqa]. Non-planar amplitudes are regulated by momentum flow between the two boundaries.

A subtle issue related to potential anomalies is the role of the odd spin structure in the computation of scattering amplitudes. In order to detect potential anomalies, one of the gauge boson vertex operators should appear with longitudinal polarisation and should decouple thanks to BRST invariance in a consistent theory. The standard procedure requires the insertion of a vertex operator in the super-ghost picture and an additional world-sheet super-current insertion brought down by integration over the super-modulus associated to the world-sheet gravitino zero. In hexagon gauge anomaly could be detected this way [Green:1984qs, Gross:1987pd]. In the case under consideration, 4-point amplitudes are not anomalous and BRST invariance allows to replace the combination of the super-modulus and world-sheet super-current with a picture changing operator [Friedan:1985ey]. The latter can act on the vertex operator in the super-ghost picture and change its picture . As in [Bianchi:2006nf], one can then proceed with all vertex operators in the picture4.

### 1.1 Partition function

In order to generalize BGS formula to (supersymmetric) vacuum configurations for open and unoriented strings in , one has to first recall the structure of the one-loop partition function. As in [Bianchi:2006nf], we will mainly focus here on magnetised or intersecting D-branes at (non-compact) orbifold singularities [Kachru:1998ys, Zoubos:2010kh]. Consistency requires local RR tadpole cancellation [Bianchi:1988fr, Bianchi:1990tb, Bianchi:1990yu, Angelantonj:1996uy, Angelantonj:2002ct].

The partition function depends on the choice of brane configuration, including number and type of intersections or magnetic fluxes thereon, -planes and orbifold group . For simplicity we will focus on , i.e.  with in order to preserve SUSY. We will label the branes of type by , and the orbifold sectors by . We define three combinations that express the twist or shift in the open string spectrum

 uIab=hvIab+iϵIabT2 (1.14)

where , denotes the angles between brane and brane or the shift in the string spectrum induced by the relative magnetic flux and denotes the twist caused by the orbifold. The combinations determine the amount of supersymmetry preserved in each sector.

### 1.2 N=1 sectors

The weakest condition one can impose to have ‘minimal’, i.e.  , supersymmetry is , with . In this case the partition function assumes the form

 ZN=1α=XN=1abθα(0)η33∏I=1θα(uIab)θ1(uIab)withXN=1ab=VXIab2\textupGSO2Ωn\textuporb(α′T)2 (1.15)

where labels the four spin structures, represents the ‘regulated’ volume of space-time (to be replaced by in scattering amplitudes), denotes the number of brane intersections or the degeneracy of Landau levels. We have traced the origin of various integers in the denominator, wherein the factor accounts for integration over loop-momenta in .

### 1.3 N=2 sectors

In sectors with supersymmetry one of the vanishes, let us say . As a consequence . The partition function reads

 ZN=2α=XN=2abθ2α(0)θ2α(uab)η6θ21(uab)withXN=2ab=VXΛ∥abI⊥ab2\textupGSO2Ωn\textuporb(α′T)2 (1.16)

Where , denotes the number of intersections or degeneracy of Landau levels in the ‘twisted/transverse’ directions and denotes the lattice sum in the two (one complex) ‘untwisted/longitudinal’ compact directions.

### 1.4 N=4 sectors

Sectors with maximal supersymmetry correspond to and the partition function is simply given by

 ZN=4α=XN=4abθ4α(0)η12withXN=4ab=VXΛab2\textupGSO2Ωn\textuporb(α′T)2 (1.17)

where denotes the lattice sum in the six internal directions.

Later on, we will compute 2, 3 and 4-point scattering amplitudes. Although the first two formally vanish on-shell due to collinear kinematics, we report their derivation using the spinor helicity formalism since it highlights the meaning of some of the structures that will later appear in the more interesting 4-pt amplitudes. Definitions and notation for elliptic functions and helicity spinors can be found in the appendices.

## 2 Two-point amplitudes

Let us begin with the two-point amplitude without specifying for the time being whether the amplitude is planar, non-planar or un-oriented. We will see that the results are substantially the same up to minor modifications. Although momentum conservation implies and then , we will formally keep . The one-loop amplitude is given by

 Extra close brace or missing open brace (2.1)

where and denote the contributions of the even and odd spin structures. Contractions with zero and one bilinear are zero, we have only the two bilinears contribution. In the even spin structures, the reduced contraction of two bilinears yields

 Extra close brace or missing open brace (2.2)

where is the one-loop fermionic propagator (Szego kernel)5

 Unknown environment '%' (2.3)

where is the scalar propagator in 1.9. We often use the notation instead of . Using the identity , where is Weierstrass function, that does not contribute to the sum over spin structures, and , we have

 E(2)=−α′22(f1f2)ENe−α′k1⋅k2G12 (2.4)
 EN=−∑αcαeα−1ZNα (2.5)

where the function introduced in [Bianchi:2006nf], labelled by the number of preserved SUSY, depends on the world-sheet modular parameter , on the parameters of the brane configuration coded in and the moduli of the compactification. vanishes in sectors due to Riemann identity.

In the odd spin structure, which only contributes in sectors6 we can absorb the four zero modes in a unique way

 Extra close brace or missing open brace (2.6)

For brevity we define the function

 CN=1=−2c\textupGSO1π2XN/T2 (2.7)

Combining the contributions of even and odd spin structures, the two-point amplitude thus reads

 A\textup1−loop2[1,2]=g2sα′2∫∞0dTT[CN=1(~f1f2)+EN(f1f2)]∫dμ(2)e−α′k1⋅k2G12 (2.8)

Fixing the helicities and noting that we have two simple results. Choosing and using we obtain

 A\textup1−loop2[1±,2±]=−g2sα′24(f±1f±2)∫∞0dTT[EN±iCN=1]T2 (2.9)

while choosing the amplitude vanishes, viz.

 A\textup1−loop2[1±,2∓]=0 (2.10)

To obtain planar, non-planar and un-oriented contributions one has to choose the specific modular parameter and the corresponding integration domain. The result is generically divergent for both sectors, wherein it encodes -functions and one-loop threshold corrections to the gauge kinetic functions [Kiritsis:1997hf, Anastasopoulos:2006hn]. As already observed, it vanishes in sectors, which points to the no-bubble conjecture in SYM, i.e. to the absence of one-loop massless amplitudes with two (bunches of) insertion points.

## 3 Three-point amplitudes

We continue our preliminary analysis and compute the three-point one-loop amplitude that reads:

 Extra close brace or missing open brace (3.1)

Momentum conservation for massless vector bosons implies i.e. collinear momenta. In order to proceed one could either relax momentum conservation [Minahan:1987ha, Kiritsis:1997hf, Berg:2011ij, Berg:2014ama] yet with or analytically continue to complex momenta [Elvang:2013cua]. In the spinor helicity formalism, reviewed in appendix A, one has and there are two options: either , with , convenient for MHV or helicity configurations or the other way around. In the even spin structures, we have two types of contributions from two and three bilinears. The term with three bilinears produces

 Extra close brace or missing open brace (3.2)

where

 ω123=S12+S23+S31, (3.3)

is the Koba-Nielsen factor and in the last step we have used

 Sα(z12)Sα(z23)=−ω123Sα(z13)−S′α(z13)2Sα(z12)S′α(z12)=P′12 (3.4)

The terms with two bilinears produce

 Extra close brace or missing open brace (3.5)

where , with .

In the odd spin structure we have similar contributions, from three bilinears we have three terms depending on the choice of the points where the two zero modes are absorbed7

 Extra close brace or missing open brace (3.6)

where the sum on exchanges means summing terms with and exchanged in the same vertex. From two bilinears we have three terms

 Extra close brace or missing open brace (3.7)

Let us consider the two independent helicity configurations. First the case . We begin from even spin structures, we must compute the scalar products :

 Missing or unrecognized delimiter for \left (3.8)

Using momentum conservation and , so that one has and one can thus factor out and get

 E(3)=i2α′3EN(f+1f+2f+3)∑\textupcyclic(S31−S23)Π3(zi,ki)=0 (3.9)

The contribution of the odd spin structure becomes proportional to the contribution of the even spin structure after using , thus the complete amplitude vanishes for this choice of helicities, as well as for ,

 A\textup1−loop3[1+,2+,3+]=0=A\textup1−loop3[1−,2−,3−] (3.10)

as expected from SUSY Ward identities8. Indeed, barring anomalous ’s, the only supersymmetric invariant for 2- and 3-points is giving rise to the standard bosonic term, since , though present in the bosonic string even at tree level, does not admit a SUSY completion.

Let us then consider the case . For this helicity configuration one has a single term. The even spin structures produce

 E(3)=i2α′3ENa−1⋅P1(f+2f+3)Π3(zi,ki) (3.11)

It is convenient to compute factorizing the MHV amplitude. In the factor would give zero due to collinear kinematics. In order to circumvent these subtleties, one can analytically continue to complex momenta and choose . This yields the ‘right’ result:

 a−1⋅P1(f+2f+3)=−1√2⟨12⟩[23][13][23]2S12=−√2[23]3[12][31]k1⋅k2S12 (3.12)

Using partial integration one can replace with to make it look more symmetric and finally find

 E(3)=−i√2α′3EN[23]3[12][31]k2⋅k3S23Π3(zi,ki) (3.13)

In the odd spin structure, only sectors contribute. Following similar steps, one finds a similar result with replaced by . Now we can write the complete three-point amplitude

 A\textup1−loop3[1−,2+,3+]=−i√2g3sα′3[23]3[12][31]∫∞0dTT∫dμ(3)(EN−iCN=1)k2⋅k3S23Π3(zi,ki) (3.14)

To compute planar, non-planar and un-oriented contributions one has to choose the specific modular parameter and the corresponding integration domain [Sagnotti:1987tw, Bianchi:1988fr, Pradisi:1988xd, Bianchi:1988ux, Bianchi:1989du]. The result is generically divergent for both sectors, wherein it is related by gauge invariance to the 2-point amplitude, encoding -functions and one-loop threshold corrections to the gauge kinetic functions. As already observed, it vanishes in sectors, in a way reminiscent of the no-triangle conjecture in SYM, i.e. the absence of one-loop massless amplitudes with three (bunches of) insertion points.

## 4 Four-point amplitudes

We are now ready to compute four-point amplitudes. We start by briefly reviewing and summarising the results of [Bianchi:2006nf] and then analyse them in terms of helicity configurations, color orderings and limiting behaviours.

The starting point is

 Extra close brace or missing open brace (4.1)

where the integration region and the Chan-Paton factor depend on the distributions of insertions on the two boundaries for the annulus (planar , non-planar and ). For the un-orientable case there is no choice, except for the relative ordering of the insertions. Here, we will only summarise the results, the details can be found in [Bianchi:2006nf].

### 4.1 Even spin structures, four bilinears

 Extra close brace or missing open brace (4.2)

The fermionic contribution consists in two types of terms connected and disconnected. The result for connected contractions is

 E(4)\textup4−bil,conn=α′4∑\textupconn(f1f2f3f4)[12EN(P13+ω123ω341+P24+ω234ω412)−FN]Π4(zi,ki) (4.3)

where (vanishing for ) was defined previously in (2.5) and are defined in 3.3, while

 FN=4∑α=2cαe2α−1ZNα (4.4)

depends on the number of preserved SUSY, on the world-sheet modular parameter , on the parameters of the brane configuration coded in and the moduli of the “compactification”. The disconnected contractions yield

 E(4)\textup4−bil,disconn=α′4∑\textupdisconn14(f1f2)(f3f4)[EN(P12+P34)+FN]Π4(zi,ki) (4.5)

### 4.2 Even spin structures, three bilinears

Aside from the bosonic contractions, the fermionic contractions are the same as for three-point amplitudes, discussed previously, thus one finds

 Extra close brace or missing open brace (4.6)

### 4.3 Even spin structures, two bilinears

We already computed the fermionic contractions, thus the contribution to the amplitude is

 Extra close brace or missing open brace (4.7)

Notice that each term is gauge invariant per se up to total derivatives. For instance, replacing (or ) with the momentum and noting that (with for brevity) and , the bosonic contractions in can be rewritten as a total derivative that vanishes upon integration

 (k1⋅a2∂1∂2G12−α′k1⋅P1a2⋅P2)Π4=∂1(a2⋅P2)Π4+∂1Π4a2⋅P2=∂1(a2⋅P2Π4) (4.8)

### 4.4 Odd spin structure, four bilinears

Four fermionic bilinears allow three types of contractions.

First one can absorb the four zero modes at two points (for example and ):

 Extra close brace or missing open brace (4.9)

Second, one can absorb two zero modes in a point and the others in two separate points. There are twelve ways to do this:

 Extra close brace or missing open brace (4.10)

Third, one can absorb the zero modes in four different points:

 Extra close brace or missing open brace (4.11)

### 4.5 Odd spin structure, three bilinears

With three bilinears, one has three ways to absorb zero modes and the contractions yield

 Extra close brace or missing open brace (4.12)

### 4.6 Odd spin structure, two bilinears

With two fermionic bilinears, there are six ways to absorb the four fermionic zero-modes

 Extra close brace or missing open brace (4.13)

## 5 Simplifying 4-pt amplitudes

Let us now simplify the above results and show that non MHV amplitudes vanish. We will also identify the regions of the integration domain that generically expose singularities and later on discuss which (tadpole) conditions the brane configurations must satisfy in order to cancel or mitigate the singular behaviours.

After analyzing the symmetry properties of the integration variables, that allow to manipulate the integrands and reduce the number of independent contributions, we will study the three independent helicity configurations and check that and .

The -term is proportional to thus reproduces the MHV structure, so we will focus on the -term.

### 5.1 A\textup1−loop4[1+,2+,3+,4+]=0

This case is the most laborious because none of the traces over the Lorentz indices of the vanish.

The six terms arising from contractions of two bilinears are separately gauge invariant, thus we can always choose and get and fix so as to make some other product between momenta and polarizations vanish. For example one can compute with the choice and get

 a+1⋅P1a+2⋅P2(f+3f+4)=−12[12]2(S12−S13)(S21−S23)(f+3f+4)=−12[12]2[34]2(S212+Ω123) (5.1)

For brevity we define

 Ω123=S12S23+S23S31+S31S12 (5.2)

Collecting the various Lorentz invariant structures yields

 E(4)\textup2−bil=−14EN∑\textupdisconn(f+1f+2)(f+3f+4)(S212+S234+Ω123+Ω134) (5.3)

Using Schouten’s identity traces with two and four ’s can be related

 (f+1f+2)(f+3f+4)=[12]2[34]2=[12][23][34][41]+[12][24][43][31]=2(f+1f+2f+3f+4)+2(f+1f+2f+4f+3) (5.4)

One can easily obtain two more similar formulae permuting the external legs. These formulae can be inverted to give

 (f+1f+2f+3f+4)=14[(f+1f+2)(f+3f+4)−(f+1f+3)(f+2f+4)+(f+1f+4)(f+2f+3)] (5.5)

We rewrite all the traces in terms of single traces of four ’s and obtain

 E(4)\textup2−bil=−12EN∑\textupconn(f+1f+2f+3f+4)(S212+S234+S214+S223+2Ω123+2Ω134)Π4(zi,ki) (5.6)

The three-bilinear term can be simplified using a cyclic gauge choice, for example , and momentum conservation . With this choice all the kinematic factors become equal . Thus one finds

 E(4)\textup3−bil=−(f+1f+2f+3f+4)EN∑i(Sii+1+Si+3i)ωi+1i+2i+3Π4(zi,ki) (5.7)

Expanding the sum we obtain

 E(4)\textup3−bil=−(f+1f+2f+3f+4)EN⎡⎣4(S12S34+S23S41)+∑\textupcyclic(2S41S12−S12S24−S24S41)⎤⎦Π4(zi,ki) (5.8)

The terms can be rewritten in four ways using partial integrations, for example choosing the tern :

 S12S34+S23S41=−2S41S12−us(S41S12+S12S24)−ut(S41S12+S24S41) (5.9)

To obtain the other three it’s enough to perform a cyclic permutation on the indices . Replacing last equation in one gets

 E(4)\textup3−bil=(f+1f+2f+3f+4)EN∑\textupcyclic[(S12S24+S24S41)+us(S41S12+S12S24)+ut(S41S12+S24S41)]Π4(zi,ki) (5.10)

The ratios of can be used to transform the traces of ’s into one another according to

 u(f+1f+2f+3f+4)=t(f+1f+2f+4f+3)=s(f+1f+3f+2f+4) (5.11)

that can be easily proved using the helicity formalism and momentum conservation. Thus one gets

 (f+1f+2f+3f+4)(S24(S12+S41)+