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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=d 2z[12x m¯x mp α¯θ αp˜ αθ˜ α+w α¯λ α+w˜ αλ˜ α] 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 λ α\lambda^\alpha, λ˜ α\tilde{\lambda}^\alpha obey a pure-spinor condition

(2)λ tγ μλ=0,λ˜ tγ μλ˜=0 \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 αw_\alpha, have a gauge-invariance

(3)w αw α+Λ m(γ mλ) α w_\alpha \to w_\alpha + \Lambda^m(\gamma_m\lambda)_\alpha

The “BRST operator” is

(4)Q =λ αd α d α =p α12γ αβ mθ βx m18γ αβ mγ mγδθ βθ γθ δ \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 αw_\alpha can only appear in gauge-invariant combinations like

(5)N mn=12w α(γ mn) β αλ β,J=w αλ α 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 λ\lambdas), says Nathan, can be computed using free fields. Unfortunately, there’s no candidate (composite) local operator, bb, which satisfies {Q,b}=T\{Q,b\}= T. Instead, Nathan has a rather strange prescription to contruct a bilocal operator, of ghost number zero, which satisfies

(6){Q,b^(y,z)}=T(y)Z B(z) \{Q,\hat{b}(y,z)\} = T(y)Z_B(z)

where

(7)Z B=12B mnλγ mndδ(BN) Z_B = \frac{1}{2} B_{mn} \lambda\gamma^{mn}d \delta (B N)

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

Similarly, the dimension-(1,1) (integrated) vertex operators are not built from the dimension-0 BRST cohomology, VV, by acting with bb. Instead, they’re constructed in an ad-hoc way as ghost-number zero fields satisfying [Q,U]=V[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 BZ_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 λ\lambdas (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) zb^(y,z)=[Q,] \partial_z \hat{b}(y,z) = [Q,\cdot]

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

  2. The usual relation that b 1b¯ 1b_{-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-λ\lambda bcbc ghost system with the usual free first-order action. The partition function

(1)Ω(m 1,,m 3g3;z 1,,z n)=[DbDc]e Sb(z 1)b(z n) \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=(2λ1)(g1)n=(2\lambda-1)(g-1), is a section of a certain line bundle L g,nL\to \mathcal{M}_{g,n}. Now consider the corresponding spin-λ\lambda βγ\beta\gamma system,

(2)(m 1,,m 3g3;z 1,,z n)=[DβDγ]e Sδ(β(z 1))δ(β(z n)) ℧(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 *L^* and, moreover, =Ω 1℧=\Omega^{-1}. Wherever Ω\Omega has a zero, ℧ has a pole.

Now, why might Ω\Omega have a zero for some values of the moduli? Well, it could happen that — while the net number of zero modes is nn — the actual number of bb and cc 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 λ=3/2\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 λ=1\lambda=1, the case vaguely relevant to Berkovits? For free fields, there are always n+1n+1 zero modes of β\beta (called ww in his paper) and 1 zero mode of γ\gamma (called λ\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 3g33g-3 insertions of a conformal primary field, b(y)b(y), which satisfies {Q,b(y)}=T(y)\{Q,b(y)\}=T(y). When folded into Beltrami differentials, μ(y,y¯)\mu(y,\overline{y}), these insertions generate the desired measure on the moduli space of Riemann surfaces of genus gg. Since, in his theory, there are no gauge-invariant operators of ghost-number -1, Nathan replaces this by 3g33g-3 insertions of a bilocal operator b^(y,z)\hat{b}(y,z). Folded into 3g33g-3 Beltrami differentials, as before, this generates a measure on the moduli space, but one that depends on the choice of 3g33g-3 arbitrary point, z iz_i. If it were true that zb^(y,z)=[Q,]\partial_z \hat{b}(y,z)=[Q,\cdot], then changing zz 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 iz_i.

Unfortunately, as Luboš reminded me, zb^(y,z)\partial_z \hat{b}(y,z) cannot be QQ-exact, because it’s not even QQ-closed. So the integrated amplitude depends (continuously!) on the locations of these arbitrary 3g33g-3 point. Which means it can’t be correct.

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

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2 Comments & 2 Trackbacks

Read the post Two-Loop Superstring Amplitudes
Weblog: Not Even Wrong
Excerpt: Eric D'Hoker and D.H. Phong this past week finally posted two crucial papers with results from their work on two-loop superstring amplitudes. The first one shows gauge slice independence of the two-loop N-point function, the second shows that, for N...
Tracked: February 4, 2005 1:00 PM
Read the post Distler on Multi-loop Amplitudes
Weblog: Not Even Wrong
Excerpt: Jacques Distler has a new posting about multi-loop string amplitudes. It's mainly devoted to the Berkovits superstring formalism, and explains in some detail the possible problems with this formalism that one might worry about. I'd alluded to some of t...
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, zb^(y,z)}=0 \{Q,\partial_z\hat{b}(y,z)\}={0}

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

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

which sure ain’t zero-looking to me. (It’s QQ-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 QQ-trivial pieces) on the arbitrary points z iz_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

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