## February 4, 2005

### Multiloop Amplitudes

Surprising to say, at this late date, but there’s been considerable recent progress in multiloop string perturbation theory.

D’Hoker and Phong have a pair of new papers, looking at genus-2 scattering amplitudes. I’ve written about their previous work in some detail. The current papers extend their story to N-point functions at genus-2.

Meanwhile, prodded by Luboš who, in his weblog post and privately, has been championing Nathan Berkovits’s pure-spinor approach to the covariant Green-Schwarz superstring, I decided to take a closer look.

The action, in Nathan’s theory looks deceptively simple:

(1)$S = \int d^2z[-\textstyle{\frac{1}{2}} \partial x^m\overline{\partial} x_m - p_\alpha\overline{\partial}\theta^\alpha - \tilde{p}_\alpha\partial\tilde{\theta}^\alpha + w_\alpha\overline{\partial}\lambda^\alpha + \tilde{w}_\alpha\partial\tilde{\lambda}^\alpha ]$

The only wrinkle is that the commuting ghost fields $\lambda^\alpha$, $\tilde{\lambda}^\alpha$ obey a pure-spinor condition

(2)$\lambda^t \gamma^\mu \lambda = 0,\qquad \tilde{\lambda}^t \gamma^\mu \tilde{\lambda} = 0$

so, despite appearances, this is not a free field theory. For compatibility with the pure-spinor constraint, the antighosts, $w_\alpha$, have a gauge-invariance

(3)$w_\alpha \to w_\alpha + \Lambda^m(\gamma_m\lambda)_\alpha$

The “BRST operator” is

(4)\array{\arrayopts{\colalign{right left}} Q & =\oint \lambda^\alpha d_\alpha\\ d_\alpha & = p_\alpha -\frac{1}{2}\gamma^m_{\alpha\beta} \theta^\beta \partial x_m - \frac{1}{8}\gamma^m_{\alpha\beta}\gamma_{m\gamma\delta} \theta^\beta\theta^\gamma\partial\theta^\delta }

Because of the gauge-invariance, however, $w_\alpha$ can only appear in gauge-invariant combinations like

(5)$N_{m n} = \frac{1}{2} w_\alpha (\gamma_{mn})^\alpha_\beta \lambda^\beta,\qquad J= w_\alpha \lambda^\alpha$

and correlation functions involving these objects (and the $\lambda$s), says Nathan, can be computed using free fields. Unfortunately, there’s no candidate (composite) local operator, $b$, which satisfies $\{Q,b\}= T$. Instead, Nathan has a rather strange prescription to contruct a bilocal operator, of ghost number zero, which satisfies

(6)$\{Q,\hat{b}(y,z)\} = T(y)Z_B(z)$

where

(7)$Z_B = \frac{1}{2} B_{mn} \lambda\gamma^{mn}d \delta (B N)$

for some constant antisymmetric tensor $B$. Aside from the strange bilocality, we by construction break the Lorentz-invariance in our definition of the $\hat{b}$s.

Similarly, the dimension-(1,1) (integrated) vertex operators are not built from the dimension-0 BRST cohomology, $V$, by acting with $b$. Instead, they’re constructed in an ad-hoc way as ghost-number zero fields satisfying $[Q,U]= \partial V$. And, in order to define the amplitudes, one needs a plethora of further insertions of non-Lorentz-invariant “Picture-changing Operators” (of which $Z_B$ above was an example).

All of these various sources of non-Lorentz-invariance, says Nathan, only change the integrand by surface terms. And, if you use a certain prescription for integrating over the zero modes of the $\lambda$s (remember, it’s a nonlinear space), all will be OK.

As you can tell, I have many, many questions about this — very interesting — proposal. But I’ll close with four:

1. Is it true that

(8)$\partial_z \hat{b}(y,z) = [Q,\cdot]$

(as surely is required for a sensible amplitude)? The expression for $\hat{b}(y,z)$ is deucedly complicated, and I can’t see why this is true.

2. The usual relation that $b_{-1}\overline{b}_{-1}$ acting on the dimension-0 (fixed-location) vertex operator gives you the dimension-(1,1) (integrated) vertex operator is crucial to the proof of unitarity of multiloop amplitudes (so crucial, that we rarely think about it). What replaces that here?
3. The unphysical poles that one encounters in multiloop NSR amplitudes when one naïvely uses the picture-changing formalism is a consequence of the index theorem applied to the bosonic ghosts (not, as implied in footnote 12, of bosonizing those ghosts). One might worry that similar poles arise here.
4. Is it really true that the only legacy of the nonlinear nature of the pure-spinor constraint is in the zero-mode integration?

#### Update:

Let me explain what the deal with question 3) is about. Let us review where the unphysical poles arise in the usual RNS story.

Consider a spin-$\lambda$ $bc$ ghost system with the usual free first-order action. The partition function

(1)$\Omega(m_1,\dots,m_{3g-3};z_1,\dots,z_n) = \int [Db Dc] e^{-S} b(z_1)\dots b(z_n)$

where $n=(2\lambda-1)(g-1)$, is a section of a certain line bundle $L\to \mathcal{M}_{g,n}$. Now consider the corresponding spin-$\lambda$ $\beta\gamma$ system,

(2)$℧(m_1,\dots,m_{3g-3};z_1,\dots,z_n) = \int [D\beta D\gamma] e^{-S} \delta(\beta(z_1))\dots \delta(\beta(z_n))$

The well-known facts of the matter are that $℧$ is a section of the dual bundle, $L^*$ and, moreover, $℧=\Omega^{-1}$. Wherever $\Omega$ has a zero, $℧$ has a pole.

Now, why might $\Omega$ have a zero for some values of the moduli? Well, it could happen that — while the net number of zero modes is $n$ — the actual number of $b$ and $c$ zero modes could jump for some values of the moduli. The presence of these extra zero modes makes the fermionic path integral vanish. And it makes the bosonic path integral diverge. That actually happens for $\lambda=3/2$, the case of relevance for the RNS string, which is why the picture-changing formalism (with its delta-function supported gravitini) is bad at higher genus.

What about $\lambda=1$, the case vaguely relevant to Berkovits? For free fields, there are always $n+1$ zero modes of $\beta$ (called $w$ in his paper) and 1 zero mode of $\gamma$ (called $\lambda$). Moreover, these numbers do not jump as you move about in the interior of the moduli space. So ignoring the effects of the pure-spinor constraint (the nonlinearities, the funky form of the picture-changing operators, etc.) we would not find any unwanted poles in the interior of the moduli space.

#### Update (2/6/2005):

The discuss below of question 1) now seems pretty devastating to me. Recall that “ordinary” string theories require $3g-3$ insertions of a conformal primary field, $b(y)$, which satisfies $\{Q,b(y)\}=T(y)$. When folded into Beltrami differentials, $\mu(y,\overline{y})$, these insertions generate the desired measure on the moduli space of Riemann surfaces of genus $g$. Since, in his theory, there are no gauge-invariant operators of ghost-number -1, Nathan replaces this by $3g-3$ insertions of a bilocal operator $\hat{b}(y,z)$. Folded into $3g-3$ Beltrami differentials, as before, this generates a measure on the moduli space, but one that depends on the choice of $3g-3$ arbitrary point, $z_i$. If it were true that $\partial_z \hat{b}(y,z)=[Q,\cdot]$, then changing $z$ would change the measure by an exact form and the integrated amplitude (modulo the usual worries about surface terms) would be independent of the locations of the $z_i$.

Unfortunately, as Luboš reminded me, $\partial_z \hat{b}(y,z)$ cannot be $Q$-exact, because it’s not even $Q$-closed. So the integrated amplitude depends (continuously!) on the locations of these arbitrary $3g-3$ point. Which means it can’t be correct.

Posted by distler at February 4, 2005 12:30 PM

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Read the post Two-Loop Superstring Amplitudes
Weblog: Not Even Wrong
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Tracked: February 4, 2005 1:00 PM
Read the post Distler on Multi-loop Amplitudes
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Tracked: February 4, 2005 4:57 PM

### Re: Multiloop Amplitudes

Hi Jacques!

Berkovits is spelled differently than in your text.

I think that you worry too much. For example,

1. the z-derivative of b-hat is “Q of something” simply because Q of this z-derivative of b-hat vanishes by nilpotency and the defining relation for b, {Q,bhat}=T.Z. Therefore this z-derivative is a Q-closed operator, and consequently it is also Q-exact because there is no cohomology at the same ghost number. If there were some homologies in this ghost number sector, you could define “b” directly as a composite field, I think.

2. As Nathan says privately, the most efficient way to prove unitarity is probably through the equivalence with the light cone gauge where a hermitean Hamiltonian exists. I agree that the unitarity is probably not proved yet, on the other hand, I see no really good reason why it should be wrong if the building blocks of the amplitudes seem to match those in other formalisms.

3. I don’t have anything to say here.

4. My feeling is that Nathan developed this lambda path-integral in depth, although I don’t understand it well enough.

Moreover, I absolutely sympathize with Hirosi and Nathan (and others) in the claim that the formalism(s) with manifest spacetime SUSY are more suitable and natural to prove the finiteness, and d’Hoker Phong are just doing things a hard way.

Best
Lubos

Posted by: Lubos Motl on February 4, 2005 8:04 PM | Permalink | Reply to this

### Re: Multiloop Amplitudes

1. the z-derivative of b-hat is “Q of something” simply because Q of this z-derivative of b-hat vanishes by nilpotency and the defining relation for b, {Q,bhat}=T.Z. Therefore this z-derivative is a Q-closed operator, and consequently it is also Q-exact because there is no cohomology at the same ghost number. If there were some homologies in this ghost number sector, you could define “b” directly as a composite field, I think.

Try thinking some more, then.

How does

(1)$\{Q,\partial_z\hat{b}(y,z)\}={0}$

follow from $\{Q, \hat{b}(y,z)\}=T(y)Z(z)$? If I take the $z$-derivative of the latter equation, I get

(2)$\{Q,\partial_z\hat{b}(y,z)\}= T(y) \partial_z Z$

which sure ain’t zero-looking to me. (It’s $Q$-exact, but we knew that already.)

To the contrary, this looks almost to be a proof that the path integral measure Nathan defines does depend (and not just up to $Q$-trivial pieces) on the arbitrary points $z_i$.

I see no really good reason why it should be wrong if the building blocks of the amplitudes seem to match those in other formalisms.

Maybe, in an appropriate gauge, the “building blocks” match those of lightcone gauge. But they certainly do not match those of any Lorentz-invariant formulation (like covariant-gauge NSR strings). The string integrand, in this formulation, is manifestly non-Lorentz-invariant.

(Showing that the integrated amplitudes agree is, obviously, much, much harder than showing that the integrand factorizes correctly, which is the way standard proofs of higher-loop perturbative unitarity proceed.)

Moreover, I absolutely sympathize with Hirosi and Nathan (and others) in the claim that the formalism(s) with manifest spacetime SUSY are more suitable and natural to prove the finiteness, and d’Hoker Phong are just doing things a hard way.

I strongly suspect that a Lorentz-invariant formulation is, and may well remain, a more convenient formulation for doing actual calculations.

Posted by: Jacques Distler on February 4, 2005 10:01 PM | Permalink | PGP Sig | Reply to this

### Re: Multiloop Amplitudes

Hi Jacques:

Is there a similar calculation for one-loop case using supermoduli space and picture changing operators? It would be great if you can provide the reference paper(s)!

Posted by: QGravity on March 5, 2016 10:56 AM | Permalink | Reply to this

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