#### Universality in string interactions

Received: April
Universality in string interactions
Yu-tin Huang 0 1 2 5
Oliver Schlotterer 0 1 2 3
Congkao Wen 0 1 2 4
Congkao.Wen@roma 0 1 2
.infn.it 0 1 2
Open Access 0 1 2
c The Authors. 0 1 2
0 Via della Ricerca Scientifica , 00133 Roma , Italy
1 14476 Potsdam , Germany
2 Taipei 10617 , Taiwan, R.O.C
3 Max-Planck-Institut fu ̈r Gravitationsphysik, Albert-Einstein-Institut
4 Dipartimento di Fisica, Universita` di Roma “Tor Vergata”
5 Department of Physics and Astronomy, National Taiwan University
In this note, we provide evidence for universality in the low-energy expansion of tree-level string interactions. More precisely, in the α′-expansion of tree-level scattering amplitudes, we conjecture that the leading transcendental coefficient at each order in α′ is universal for all perturbative string theories. We have checked this universality up to seven points and trace its origin to the ability to restructure the disk integrals of open bosonic string into those of the superstring. The accompanying kinematic functions have the same low-energy limit and do not introduce any transcendental numbers in their α′-corrections. Universality in the closed-string sector then follows from KLT-relations.
Bosonic Strings; Scattering Amplitudes
1 Introduction 2 Open-string amplitudes 2.1
The bosonic open string
Universality and BCJ identities
Closed-string amplitudes
Universality for closed string theories
Universality of UV completion
One of the formidable challenges for a theory of quantum gravity is the construction of
a gravitational S-matrix which respects unitarity at high energies. Perturbative string
theories provide candidate solutions, as its four-point graviton S-matrix is exponentially
suppressed in the high-energy limit for fixed-angle scattering [1, 2]. In fact, assuming
tree-level causality [3] and unitarity [4] imposes stringent constraints, under which string
theories provide the only known analytic solutions so far.
Different string theories are understood to be equivalent through a web of strong-weak
dualities which relate different orders in the perturbative expansion [5, 6]. At tree level,
however, the low-energy description in the form of an effective action with expansion in
curvature tensors and covariant derivatives is largely unconstrained by string dualities.
More precisely, the coefficients of these higher-dimensional operators are expected to be
distinct for different string theories. Thus, if some of these coefficients turn out to be
universal, it is then conceivable that such a phenomenon reflects a deeper principle in the
theory of quantum gravity beyond the known dualities.
At low energies, closed-string theories yield an effective action that augments the
Einstein-Hilbert term SEH with higher-dimensional operators. At tree level, type-II
superstring theories exhibit the following expansion in the inverse string tension (or cut-off
contractions of covariant derivatives and Riemann tensors. The tensor structure of each
operator as well as its coefficient furnished by multiple zeta values (MZVs)
turally categorized according to their transcendental weight n1 +n2 +. . .+nr and constitute
a fruitful domain of common interest between high-energy physics and number theory. In
fact, for type-II theories, the transcendental weight for each coefficient matches the order
for open strings in the type-I theory. The type-I effective action is now an expansion in
non-abelian field-strength operators tr(DnF m). In this light, uniform transcendentality for
closed strings is inherited from open strings through the Kawai, Lewellen and Tye (KLT)
In this letter, we conjecture that the leading transcendental coefficient at each order
closed-string theories. We have explicitly verified this up to the seven-point level, and the
conjectural all-multiplicity extension is further investigated in a companion paper [8]. This
that correspond to the UV completion of tree-level Einstein-Hilbert graviton amplitudes.
This is given by the type-II theories. For Heterotic and Bosonic closed string theories,
this is augmented by separate terms that correspond to UV completions of amplitudes
dilaton or the Tachyon. This remarkable property can be best understood by inspecting
the world-sheet correlator of the open-string amplitudes.
It was shown in [9] that the n-point tree amplitude of the open superstring can be cast
of different color-orderings. These basis integrals exhibit uniform transcendentality upon
be cast into the very same basis where — in contrast to the superstring — the augmented
Yang-Mills tree amplitude, additional rational functions that contain Tachyon poles. Thus
involve rational numbers upon Taylor-expansion, i.e. they do not carry any transcendental
have the same leading transcendental pieces as found for the superstring.
The same property can be extended to closed strings by utilizing the KLT-relations [7],
which assemble closed-string tree amplitudes from products of two open-string trees. The
scendentality of the type-II theory. Different double-copies of open bosonic strings and
superstrings give rise to three different closed-string theories — bosonic, heterotic and type-II
integrals of uniform transcendentality inherited from the open-string constituents. Only
specific to open bosonic strings do not introduce any transcendental weight and thereby
do not affect the leading-transcendental piece. This completes the argument for
univerweight-n coefficient is universal for all perturbative closed-string theories. More over, this
representation also allows us to separate a piece of the closed string graviton amplitude
that is universal and is given by the type II theories.
This paper is organized as follows: in the next section, we give a brief review of the
organization of open superstring amplitude as a matrix of disk integrals multiplied by
the basis of Yang-Mills amplitude. We will show that through a series of Integration By
Parts (IBP) identity, the bosonic string amplitude can be cast into the very same same
relations. Finally, we show that the closed string graviton amplitude for all perturbative
string theories can be naturally separated into a piece that can be identified as the UV
completion of pure tree-level Einstein-Hilbert graviton amplitudes, which is given by that
of type II theories.
Open-string amplitudes
The tree-level amplitude for n gluon-multiplet states in open superstring theory can be
conveniently written as [9]
where AS and AYM indicate color-ordered amplitudes of the superstring and super
YangMills field theory, respectively. The ordering of the two amplitudes does not have to be
( n−2 k−1
k=2 m=1 zkm
monodromy relations [11, 12].
weight matches the degree of the accompanying polynomials in sij . Since AYM do not
Once undoing the above choice of SL(2) frame, the functions (2.2) can be identified as
respectively, and the measure is given by
The integral reductions performed in [9] rely on partial-fraction manipulations and
integraHowever, as already exploited in a superstring context [9, 19], IBP additionally allows to
techniques to gluon amplitudes of the bosonic string yields our main result to be reported
in the following.
The bosonic open string
The tree-amplitude prescription for n-gluon scattering in the bosonic string introduces
significantly more rational functions of zij of suitable SL(2) weight than captured by the
single cycles in (2.4). In particular, one obtains more terms involving higher order poles as
well as multicycle denominators. Still, repeated use of IBP is expected to reduce all of them
to the single-cycle form and thereby to the same integral basis as seen in (2.1) and (2.2).
For example, at four-points, all double poles can be reduced by using the following identity:
1 − s23
The denominator on the right-hand side signals tachyon exchange specific to the bosonic
open superstring, while such double cycle denominators are also present, the OPE among
supersymmetric vertex operators guarantees that tachyon poles as in (2.6) are suppressed
by numerators 1 − sij . For explicit examples see e.g. [9, 19].
Extending the integral reduction along the lines of (2.6) to arbitrary multiplicity leads
us to conjecture the following structure for the n-gluon tree in bosonic string theory:
In comparison to the superstring result (2.1), the kinematic factors AYM(. . .) are replaced
of sij and multilinear in the polarizations ej entering via (ei · ej ) and (ei · kj ), and crucially
Yang-Mills tree amplitudes, and is therefore identical to that of the superstring, i.e.
B(1,. . ., n; α′) = AYM(1,. . ., n) + X(2α′)kBk(1,. . ., n) .
k=1
The simplest instances of the subleading terms occur at the three-point level and signal
the F 3 interaction specific to the bosonic string,
B1(1,2,3) = (e1 ·k2)(e2 ·k3)(e3 ·k1) , Bk≥2(1,2,3) = 0 .
The higher-point case requires integral reductions as in (2.6), and the resulting geometric
series yield non-zero Bk(. . .) for any value of k.
A. Four-points. In the case of n=4, we find
s212(1 − s12)
the Bk(1, 2, 3, 4)’s from the second line of (2.10).
At five points, after partial fractional manipulations, we require
following two identities to reduce all the integrals to a single-cycle basis (2.4):
s23 − 1
s13Z(1, 3, 2, 4, 5)
1 − s51
1 − s51
s23 − 1
notebook containing the full expression is attached.
To arrive at (2.7) at six points, we find that after partial fractions, besides
the single-cycle basis we encounter integrals of following forms,
(z23z34z42) (z15z56z61)
2 ,
We have checked that indeed all above integrals can be reduced to single-cycle integral via
IBP, for instance,
(z23z34z42) (z15z56z61)
Note that all identities of (2.6), (2.11) and (2.13) can alternatively be derived by imposing
linearized gauge invariance under ej → kj , and the same is believed to hold for the integral
reduction at arbitrary multiplicity.
The analogous seven-point checks to arrive at (2.7) have been performed as well.
Universality and BCJ identities
must agrees with the superstring amplitude. This leads to the conclusion that the
leadingare universal in open-string theories.
Although the kinematic factors Bk(. . .) in (2.8) differ from AYM(. . .) in tensor structure
and mass dimension, we will now argue that they obey the same KK- and BCJ-relations [15–
17]. The universal monodromy relations [11, 12] among bosonic-string subamplitudes have
inserting (2.7) into the lowest-transcendentality pieces of the monodromy relations and
identifying B0(. . .) ≡ AYM(. . .) yields
0 = Bk(1, 2, . . . , n) + Bk(2, 1, 3, . . . , n) + Bk(2, 3, 1, . . . , n)
+ . . . + Bk(2, 3, . . . , n − 1, 1, n)
0 = s12Bk(2, 1, 3, . . . , n) + (s12 +s13)Bk(2, 3, 1, 4, . . . , n)
+ . . . + (s12 +s13 +. . .+s1,n−1)Bk(2, 3, . . . , n − 1, 1, n)
ploited in [20] to derive BCJ-relations for subamplitudes of the F 3 operators as well as
the supersymmetrized D2F 4 + F 5. Moreover, an all-order argument for single-trace gluon
amplitudes of the heterotic string has been given in [21]. By the same reasoning, (2.14)
(Mj1 Mj2 . . . Mjp )στ Bk(1, 2τ , . . . , (n−2)τ , n−1, n) .
in (2.15) are inevitable to verify permutation invariance of the world-sheet integrand for
supergravity trees [29].
Closed-string amplitudes
Closed-string amplitudes at tree level can be obtained from squares of open-string
ampliable basis w.r.t. Q [22, 24]. The selection rules were identified in [25] with the single-valued
projection of MZVs [26, 27]. A representation of the massless closed-superstring tree MnS
which manifests the effect of these cancellations has been firstly given in [22]:
X A˜YM(1, 2σ, . . . , (n−2)σ, n, n−1)(S0)σρ
The polarizations of the type-II supergravity multiplets stem from tensor products of
the gauge-multiplet polarizations in A˜YM and AYM. The matrix S0 has entries of order
F (α′) = 1 + ζ2P2 + ζ3M3 + ζ22P4 + ζ5M5 + ζ2ζ3P2M3 + ζ23P6 +
sij with rational coefficients. Once the MZVs are expressed in terms of their conjectural
basis over Q, only one independent matrix Pw or Mw occurs at each weight w along with
seen in (3.2) recycling information from lower weights. The rational prefactors 51 , 265 , . . .
the terms in (3.2) [22],
along with the matrix commutators [M5, M3], . . . can be understood from the coproduct
of MZVs which can only be rigorously defined for their motivic versions. A description
of motivic MZVs which lead to a rewriting of (3.2) with unit rational coefficients can be
G(α′) = 1 + 2ζ3M3 + 2ζ5M5 + 2ζ32M3M3 + 2ζ7M7 + 2ζ3ζ5{M3, M5}
spirations in the KLT-formula. Moreover, the properties of the Mw matrices additionally
These selection rules have been identified in [25] with the single-valued projection of MZVs,
see [26, 27] for further mathematical background.
Together with the polarization-dependence from A˜YM, AYM, they encode the tensor
contractions of the DnRm operators in the tree-level effective action to the orders seen
in (1.1). Given the ubiquitous matrix products with summations over permutations in
Universality for closed string theories
II closed-string amplitudes are a property of the sphere integrals involving two copies
of the integrands in (2.2). Accordingly, the results on the integrals can be exploited in
further contexts such as gravitational tree amplitudes MnH or MnB in the heterotic or
the closed bosonic string which rest on one or two copies of the bosonic-string integrand
in (2.7). The only modification as compared to the superstring (3.1) is an exchange of
Clearly, the Einstein-Hilbert interaction can be recovered from (3.6) and (3.7) at
leadstructure of (3.6) is expected to capture multitrace interactions in the gauge sector of the
In complete analogy to the open bosonic string, it is natural to organize (3.6) and (3.7)
While the kinematic
factherefore identical in (3.5), (3.6) and (3.7). Hence, we have shown that, at leading
transcendentality, gravitational tree-level interactions are universal to the bosonic, heterotic and
type-II closed-string theories.
Universality of UV completion
It is instructive to consider the four-dimensional spinor helicity form of B(1, 2, 3, 4). For
different helicity choices, it takes on the following simple form:
B(1−, 2−, 3−, 4−) = −u
B(1−, 2−, 3−, 4+) = u
B(1−, 2−, 3+, 4+) = u
− 1 .
− 1
For MHV amplitudes we will only need the last form. Thus the MHV amplitude in closed
string theories can be written as
M4S(1−, 2−, 3+, 4+) = h12i4[34]4 f (s, t, u)
with the function f (s, t, u) given as:
Super f (s, t, u) =
Heterotic f (s, t, u) =
Bosonic f (s, t, u) =
tree-level Einstein-Hilbert graviton amplitudes. For the heterotic string the extra term
Thus we see that the UV completion of the the tree-level Einstein-Hilbert graviton
amplitude is in fact unique in all perturbative string theories. The low energy amplitudes
that stem from the presence of three-point higher-dimensional operators are completed
separately. Note that the uniqueness of the UV completion for Einstein-Hilbert, follows
directly from our conjectural form of the bosonic tree function in eq. (2.7), which has been
proven up to seven-points. It will be extremely interesting to see whether the additional
terms for heterotic and bosonic string graviton amplitudes at higher points can also be
written as separate completion for tree-amplitudes of higher dimensional operators. We
leave this for future study.
In this letter, tree-level amplitudes in all perturbative open- and closed-string theories are
versality can be achieved by casting the world-sheet correlators of the bosonic open string
matic factors. We have explicitly shown that such a reorganization can be achieved up to
seven points, and the conjectural all-multiplicity extension is relegated to future work [8].
Generalizations to closed-string interactions in bosonic, heterotic and type-II theories
directly follow from the KLT-relations. These universality results have greatly facilitated the
construction of matrix elements for counterterms in half-maximal supergravity [30].
It would be interesting to apply the same organizing principles to massive-state
scattering. We expect the same basis of disk integrals to capture tree amplitudes among any
combination of massive open-string resonances. Moreover, the structure of (3.5) is believed
to apply to closed-string trees among massive resonances upon appropriate replacements
of AYM and A˜YM.
Finally the conjectural form of eq. (2.7) leads to the interesting conclusion that the
UV completion of pure Einstein-Hilbert gravity is in fact unique amongst perturbative
string theory. The difference lies in the presence of separate terms in the amplitude that
R3. This structure is straight forward at four-points, and it will be interesting to see if the
latter persists to higher points.
We are grateful to Johannes Bro¨del, Paolo Di Vecchia, Michael Green and Henrik Johansson
for inspiring discussions and valuable comments on a draft of the manuscript.
Massimo Bianchi and Andrea Guerrieri are thanked for enlightening discussions.
would like to thank Nima Arkani-Hamed for the interesting observation of uniqueness
in UV completion of Einstein-Hilbert gravity. Y-t.H. is supported by MOST under the
grant No. 103-2112-M-002-025-MY3, and O.S. is grateful to the Universit`a di Roma Tor
Vergata for kind hospitality during finalization of this work.
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any medium, provided the original author(s) and source are credited.
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