Carlos R. Mafra (work with Oliver Schlotterer) · Carlos R. Mafra (work with Oliver Schlotterer) Department of Applied Mathematics and Theoretical Physics University of Cambridge

Multiparticle superfields and superstring amplitudes

Carlos R. Mafra

(work with Oliver Schlotterer)

Department of Applied Mathematics and Theoretical PhysicsUniversity of Cambridge

C.R. Mafra (DAMTP) Non-linear SYM 25.03.2015 1 / 26

Breaking news

At the Strings 2014 I talked about the calculation of the superstring3-loop amplitude with the pure spinor formalism

And emphasized one point: we determined the overall normalizationfrom first principles

However, there was a factor of 3 mismatch w.r.t a S-dualityprediction by Green and Vanhove from 2005

Last week we found the mistake in our calculation

And now the two results agree!


Breaking news

At the Strings 2014 I talked about the calculation of the superstring3-loop amplitude with the pure spinor formalism

And emphasized one point: we determined the overall normalizationfrom first principles

However, there was a factor of 3 mismatch w.r.t a S-dualityprediction by Green and Vanhove from 2005

Last week we found the mistake in our calculation

And now the two results agree!


Motivation

The long-term goal is to compute the N-point superstring amplitudefor any number of loops

This is a very difficult problem, it hasn’t been solved yet

Major issues: the form of the pure spinor b-ghost and complicatednature of Berkovits–Nekrasov regularization prescription, Schottkyproblem, non-vanishing OPEs between b-ghost and vertices etc

In this talk:1 Discuss another type of difficulty: the complicated nature of

higher-loop PS kinematic factors2 Present a solution at 3-loops (comment briefly at 4-loops)


Motivation

As I will show in a moment, the massless vertex operators in the purespinor formalism are written in terms of 10D SYM superfields,

Aα(x , θ), Am(x , θ), W α(x , θ), Fmn(x , θ)

The problem is that their mass dimension is not high enough

Mass dimension of the field-strength is +1; there is only one derivative

Describing operators of higher-mass dimensions is not “natural”; needa covariant way to increase powers of momenta

In the non-minimal pure spinor formalism this task is done bycovariant superspace derivatives Dα acting on the superfields

Lead to cumbersome expressions which are difficult to analyse


Motivation

One of the simplifying features of the pure spinor formalism is theso-called pure spinor superspace

Compact representation for supersymmetric operators coming fromamplitude calculations

1-loop (dimension f 4mn) (Berkovits ‘04)

〈(λA)(λγmW )(λγnW )Fmn〉

2-loop (dimension k2f 4mn) (Berkovits ‘05)

〈(λγmnpqrλ)(λγsW )FmnFpqFrs〉

That’s it. Cannot write down higher-dimension operators using onlythe standard SYM superfields. Need covariant derivatives to increasethe dimension

How to describe a 3-loop operator of dimension k4f 4mn?


Motivation

3-loop (dimension k4f 4mn) (CM, Gomez ‘13)

T12,3,4 = 〈(λγabcD)(λγdefD)(λγghiD)(λγmnpD)(λγqrsD)(λγtuvD)

× (λγadefmλ)(λγbghitλ)(λγuqrsnλ)(λγcW12)(λγpW3)(λγvW4)〉

This is definitely not trivial as the examples before

The non-minimal pure spinor superspace (λα, λα, θα) is harder to

work with than the minimal superspace (λα, θα)

Finding its component expansion (in terms of gluons and gluinos) iscomplicated

In the end the calculations can be done, but there must be a simplerway


Motivation

In this talk I will describe how this problem has been solved usingsuperfields of higher-mass dimensions (CM, Schlotterer ‘15)

New 3-loop representation is very simple, using only the minimal purespinor superspace

T12,3,4 = 〈(λγmW n12)(λγnW

p[3)(λγpW

m4] )〉

BRST variation trivial to compute

Component expansion (gluons, gluinos) trivial to obtain

To understand how this was done, need to make a detour to tree-levelstring amplitudes with the pure spinor formalism (Berkovits ‘00, ‘05)

As we will see, the string loop amplitudes benefit from our quest forsimplicity at tree-level


String tree-level amplitude

N-point prescription (Berkovits ‘00):

Atree = 〈V1V2V3

∫U4. . .

∫Un〉

where V and U are massless vertex operators

V = λαAα(x , θ), QV = 0

U = ∂θαAα + AmΠm + dαWα +

1

2NmnFmn, QU = ∂V

λα is the pure spinor satisfying (λγmλ) = 0

[Aα,Am,Wα,Fmn] are 10D superfields describing gluon and gluino

Suitable for a cohomology analysis with BRST charge Q = λαDα

General N-point amplitude explicitly worked out (CM, Schlotterer,Stieberger ‘11)


SYM superfields in 10D

There is a covariant description of SYM theory in D=10 using thesuperfields (Witten ‘86)

Aα(x , θ), Am(x , θ), W α(x , θ), Fmn(x , θ)

After defining the covariant derivatives

∇α ≡ Dα − Aα(x , θ), ∇m ≡ ∂m − Am(x , θ)

Non-linear equations of motion{∇α,∇β

}= γmαβ∇m,

[∇α,∇m

]= −(γmW )α{

∇α,W β}

= 14(γmn)α

βFmn,[∇α,Fmn

]=[∇[m, (γn]W )α

]were shown to reproduce the standard component equations of motion

∂mfmn + [am, fmn] = 0, γmαβ(∂mχ

β + [am, χβ])

= 0


SYM superfields in 10D

For amplitudes, it’s enough to use the linearized version of EOM

DαAβ + DβAα = γmαβAm, DαAm = (γmW )α + ∂mAα

DαWβ =

1

4(γmn) β

α Fmn, DαFmn = ∂[m(γn]W )α

The superfields have well-known θα expansions:

Aα(x , θ) =1

2am(γmθ)α −

1

3(χγmθ)(γmθ)α + · · ·

Am(x , θ) = am − (χγmθ) + · · ·

W α(x , θ) = χα − 1

4(γmnθ)αfmn + · · ·

Fmn(x , θ) = fmn − 2(∂[mχγn]θ) + · · ·

These are the some of the key ingredients of pure spinor superspace


Pure Spinor Superspace

The CFT computation of the tree-level correlator is usual:

〈V1V2V3

∫U4. . .

∫Un〉

Use OPEs to integrate out non-zero modes of ∂θα, dα,Πm and Nmn

contained in the vertex operator U(z), e.g.

dα(z)dβ(w)→ −γmαβΠm

z − w

Surviving zero-modes of the pure spinor λα and θα give rise toexpressions written in pure spinor superspace (λα, θα)

PS bracket 〈. . .〉 non-vanishing only for terms with 3 λ’s and 5 θ’s(Berkovits ‘00)

〈(λγmθ)(λγnθ)(λγpθ)(θγmnpθ)〉 = 1


Pure Spinor Superspace

PS superspace leads to compact expressions for kinematic factors

Straightforward to analyse the cohomology properties of theamplitudes

Component expansions can be computed using Berkovits’ measure,e.g.

〈V1V2V3〉 = (e1 · e2)(k2 · e3) + (e1 · e3)(k1 · e2) + (e2 · e3)(k3 · e1)

+[e1m(χ2γ

mχ3) + cyc(1, 2, 3)]

Computations are easy for low number of points, higher-pointamplitudes require many OPE contractions and become potentiallycomplicated

We need an organizing principle to simplify the calculations


Multiparticle SYM superfields (CM, Schlotterer ‘14, ‘15)

Multiparticle superfields defined by the simple poles in the OPE ofintegrated vertices

U1(z1)U2(z2) ∼ 1

z1 − z2

[∂θαA12

α + A12m Πm + dαW

α12 +

1

2NmnF 12

mn

]where, e.g.

W α12 =

1

4(γmnW 2)αF 1

mn + W α2 (k2 · A1)− (1↔ 2)

The multiparticle superfields obey the same linearized SYM equationsof motion with the addition of contact terms, e.g.

DαWβ12 =

1

4(γmn)α

βF 12mn + (k1 · k2)(A1

αWβ2 − A2

αWβ1 )


Multiparticle SYM superfields

Recursive construction of KB ∈ {ABα , A

mB , W

αB , Fmn

B } withmultiparticle label B = 123 . . .

For example,

W α123 = −(k12 · A3)W α

12 +1

4(γrsW 3)αF 12

rs − (12↔ 3)

+1

2(k1 · k2)

[W α

2 (A1 · A3)− (1↔ 2)]

satisfies the EOM

DαWβ123 =

1

4(γmn)α

βF 123mn

+ (k1 · k2)[A1αW

β23 + A13

α W β2 − (1↔ 2)

]+ (k12 · k3)

[A12α W β

3 − (12↔ 3)],

For a given multiparticle label, the different label orderings are not allindependent


Multiparticle symmetries

The superfields KB ∈ {ABα , A

mB , W

αB , Fmn

B } satisfy generalized Liesymmetries

0 = K12 + K21,

0 = K123 + K231 + K312, (Jacobi identity)

0 = K1234 − K1243 + K3412 − K3421

0 = general formula known

These are the same symmetries obeyed by nested commutators

Can make symmetries manifest in the notation

K12 → K[1,2], K123 → K[[1,2],3], K1234 → K[[[1,2],3],4] . . .


Multiparticle symmetries and BCJ numerators

This notation also suggests a mapping between multiparticlesuperfields and trees with cubic vertices (planar binary trees)

1 2 1 2 3 1 2 3 4 1 2 3 4 5

K[1,2] K[[1,2],3] K[[[1,2],3],4] K[[[1,2],3],[4,5]]

This is related to BCJ numerators, but the connection is a bit indirect:

Local N-pt numerators at tree-level (CM, Schlotterer, Stieberger ‘11)

Local 5-pt numerators at 1-loop (CM, Schlotterer ‘14)

Local 5-pt numerators at 2-loops (CM, Schlotterer, to appear soon)

Local 4-pt numerators at 3-loops (CM, Schlotterer, to appear later)


Berends–Giele currents

Define new superfields KB ∈ {ABα ,Am

B ,WαB ,Fmn

B } with inverseMandelstam variables by summing over all binary trees (Catalannumber), e.g

They satisfy much simpler equations of motion, e.g for KB →WαB ,{

Dα,WβB

}= 1

4(γmn)αβFB

mn +∑

XY=B

(AXαW

βY −A

YαW

βX

)∑

XY=B means the deconcatenation of the multiparticle label, forexample when B = 123∑

XY=B

AXαW

βY = A1

αWβ23 +A12

α Wβ3


Lie series

The Berends–Giele currents KB satisfy the following symmetry

KA�B = 0, ∀A,B 6= ∅

For example,K1�2 = K12 +K21 = 0,

K12�34 = K1234 +K1324 +K1342 +K3412 +K3124 +K3142 = 0

This symmetry allows the definition of a Lie series (Ree ‘57)

K ≡∑i

Ki ti +∑i ,j

Kij ti t j +

∑i ,j ,k

Kijkti t j tk + . . .

=∑i

Ki ti + 1

2

∑i ,j

Kij [ti , t j ] + 1

3

∑i ,j ,k

Kijk [[t i , t j ], tk ] + · · ·

t i denote generators in the Lie algebra


Equations of motion of Lie series

We have just defined four Lie series K ∈ {Aα,Am,Wα,Fmn}Using the equations of motion of their Berends–Giele constituents,can show{

D(α,Aβ)}

= γmαβAm + {Aα,Aβ}[Dα,Am

]=[∂m,Aα

]+ (γmW)α + [Aα,Am]{

Dα,Wβ}

= 14(γmn)α

βFmn + {Aα,Wβ}[Dα,Fmn

]=[∂[m, (γn]W)α

]+ [Aα,Fmn]− [A[m, (γn]W)α]

What’s the structure of these equations?


Non-linear SYM

Defining the covariant derivatives using the Lie series Aα(x , θ) andAm(x , θ) instead of the standard single-particle superfields

∇α ≡ Dα − Aα(x , θ), ∇m ≡ ∂m − Am(x , θ)

the above equations become the non-linear equations of motion ofD=10 SYM: {

∇α,∇β}

= γmαβ∇m[∇α,∇m

]= −(γmW)α{

∇α,Wβ}

= 14(γmn)α

βFmn[∇α,Fmn

]=[∇[m, (γn]W)α

].

So the series {Aα,Am,Wα,Fmn} solve the non-linear EOM!


Higher mass dimension superfields

Can define higher-mass dimension superfields using Lie series(similarly for Fm1...mk |pq)

Wm1...mkα ≡[∇m1 ,Wm2...mkα

]Translates to

Wm1...mkαB = km1

B Wm2...mkαB

+∑

XY=B

(Wm2...mkα

X Am1Y −W

m2...mkαY Am1

X

),

Single-particle case is trivial: Wm1...mkαi = km1

i Wm2...mkαi

but non-trivial structure for multiparticle label


Higher mass dimension superfields

For simplicity, consider the simplest example Wmα

Non-linear equation of motion{∇α,Wmβ

}= 1

4(γpq)αβFm|pq −

{(Wγm)α,Wβ

},

or after expanding the series

DαWmβB = 1

4(γpq)αβFm|pq

B +∑

XY=B

(AXαW

mβY −AY

αWmβX )

−∑

XY=B

[(WXγ

m)αWβY − (WY γ

m)αWβX

]The deconcatenation terms play an essential role in the newrepresentation of the (scalar) 3-loop kinematic factor


Simpler representation for the 3-loop k4F 4mn

Having superfields with higher mass dimensions allows much simplerrepresentations for operators appearing in higher-loop amplitudes

Recall dimensions of [λα] = −12 , [θα] = −1

2 ; therefore removal of thePS measure λ3θ5 accounts for dim +4

Recall dimensions of the standard superfields

[Aα] = −1

2, [Am] = 0, [W α] = +

1

2, [Fmn] = +1

For example, can check that 〈(λA1)(λγmW2)(λγnW3)F 4mn〉 has

dimension of [F 4mn] = +4

To construct a 3-loop operator T12,3,4 of dimension [k4F 4mn] = +8 we

need〈total dimension +4〉


Simpler representation for the 3-loop k4f 4mn

One additional requirement for T12,3,4: one superfield must havemultiparticle label 12

Simplest solution is straightforward to guess:

T12,3,4 = 〈(λγmW n12)(λγnW

p[3)(λγpW

m4] )〉

Dimensions: [Wmα12 ] = +5/2, [W p

3 ] = +3/2, [W p3 ] = +3/2 and

(λα)3 = −3/2, total = +4

What is good about this particular choice?


Simpler representation for the 3-loop k4f 4mn

The BRST variation reproduces the expectations from the 3-loopcomputation:

QT12,3,4 = s12(V1T2,3,4 − V2T1,3,4)

Which means that the following is BRST invariant

Aα′→04 =

〈T12,3,4 + T34,1,2〉s12

+〈T23,4,1 + T41,2,3〉

s23

This was identified as the 3-loop open string kinematic factor in theα′ → 0 limit (CM, Gomez ‘13)

Finally, one can also construct a vector kinematic factor Tm1,2,3,4

Tm1,2,3,4 ≡ −

1

3km1 〈V1(λγqW n

[2)(λγnW p3 )F qp

4] + (1↔ 2)

such that

|〈T12,3,4〉|2

s12+ 〈Tm

1,2,3,4〉〈T̃m1,2,3,4〉+ perm = D6R4


Conclusions

Originally designed to simplify tree-level computations, multiparticlesuperfields and their higher-mass dimension partners also simplifyloop calculations

The 4-point closed-string 3-loop amplitude can now be roughlyderived with zero-mode counting arguments in a few lines (up tooverall coefficient)

Similar reasoning leads to 4-loop k6f 4mn operator

T12,34 = 〈(λγmnpqrλ)(λγsWmnpq12 )F rs

34〉+ 12↔ 34

How to incorporate the new superfields in the PS formalism itself?New types of multiparticle vertex operators containing higher-masssuperfields? E.g.

UeffA = · · ·+ Πm(dWm

A ) + · · ·

Many interesting possibilities ahead!


Documents

Carlos R. Mafra (work with Oliver Schlotterer) · Carlos R. Mafra (work with Oliver Schlotterer) Department of Applied Mathematics and Theoretical Physics University of Cambridge