Interpolating the Coulomb phase of little string theory
JHE
Interpolating the Coulomb phase of little string theory
YingHsuan Lin 0 1 3
ShuHeng Shao 0 1 3
Yifan Wang 0 1 2
Xi Yin 0 1 3
0 Cambridge , MA 02139 , U.S.A
1 Cambridge , MA 02138 , U.S.A
2 Center for Theoretical Physics, Massachusetts Institute of Technology
3 Je erson Physical Laboratory, Harvard University
We study up to 8derivative terms in the Coulomb branch e ective action of (1; 1) little string theory, by collecting results of 4gluon scattering amplitudes from both perturbative 6D superYangMills theory up to 4loop order, and treelevel double scaled little string theory (DSLST). In previous work we have matched the 6derivative term from the 6D gauge theory to DSLST, indicating that this term is protected on the entire Coulomb branch. The 8derivative term, on the other hand, is unprotected. In this paper we compute the 8derivative term by interpolating from the two limits, near the origin and near the in nity on the Coulomb branch, numerically from SU(k) SYM and DSLST respectively, for k = 2; 3; 4; 5. We discuss the implication of this result on the UV completion of 6D SYM as well as the strong coupling completion of DSLST. We also comment on analogous interpolating functions in the Coulomb phase of circlecompacti ed (2; 0) little string theory. gauge theory, Superstrings and Heterotic Strings

HJEP12(05)
1 Introduction
2 The Coulomb branch e ective action
3 Perturbative 6D SYM in the Coulomb phase
4 The 0 expansion of little string amplitude
4.1 An interpolating function from weak to strong coupling
5 Discussion
6 Comments on (2; 0) LST and 5D SYM
A 6D SYM loop amplitudes contributing to D4F 4
A.1 Oneloop
A.2 Twoloop
A.3 Threeloop
A.4 Fourloop
B Evaluation of the little string amplitudes
the other hand, may be regarded as an expansion near the origin of the Coulomb branch,
and describes the strong coupling limit of DSLST. The goal of this paper is to exploit this
correspondence, by connecting the two limits of the Coulomb phase of (1; 1) LST.
We will inspect the derivative expansion of the Coulomb branch e ective action,
focusing on terms of the structure fn(r)D2nF 4, n = 0; 1; 2; etc. Here r stands for the distance
from the origin of the Coulomb branch, as measured by the scalar expectation values,
and F the eld strength of the U(
1
)k 1 vector multiplets in the Cartan of the SU(k) gauge
group. The most convenient way to organize the supersymmetric completion of these higher
derivative terms in the e ective action is through the massless superamplitudes they
generate [9]. For our purpose, it su ces to focus on the 4point superamplitudes, which take
{ 1 {
the form1 8(Q)F (s; t; u), where Q is the total supermomentum and s; t; u the Mandelstam
variables [10{12]. F (s; t; u) will depend on the color assignment of the Cartan gluons, and
depend on r through the W boson masses.
The 4point superamplitude can be computed in the large r regime by the perturbative
double scaled LST [13, 14]. In previous work we have formulated the tree amplitude in the
DSLST in terms of an explicit double integration over the cross ratio of four points on the
Riemann sphere and over a continuous family of conformal blocks, which is then evaluated
numerically. In this paper we will present some higher order terms in the 0expansion of
the DSLST tree amplitude, giving the leading 1=r2 term of the fn(r)D2nF 4 coupling on
the Coulomb branch, at large r.
In the small r regime, on the other hand, we will perform a perturbative computation
in 6D SU(k) SYM. The 4point amplitude is reduced to 8(Q) times a set of scalar box
type integrals, which can be evaluated straightforwardly up to 3loops. We will present
some numerical results for k = 2; 3; 4; 5. Starting at 4loop order, the 4point amplitude
of Cartan gluons su ers from logarithmic UV divergences. This divergence structure is a
bit intricate, as the nonabelian 4point amplitude already diverges at 3loop and a 3loop
counterterm of the form D2trF 4 is needed [15, 16]. While this counterterm vanishes when
restricted to the Cartan, it gives a nontrivial contribution to the 4loop amplitude, which
has been studied in [16]. In the end, after taking into account suitable 4loop counterterms,
of the form D4trF 4 and D4tr2F 4, one obtains a 4loop contribution to f2(r) that involves
logarithmic dependence on r, of the form (ln r)2 and ln r. While the
nite shifts of the
3loop and 4loop counterterms are not a priori determined in SYM perturbation theory
(but should be ultimately xed in the LST), the coe cients of the leading logarithms are
unambiguously determined. The results of [16] on the 4loop divergence of double trace
terms then allows for determining certain leading log coe cients, which when combined
with 1; 2; 3loop results produce the rst few terms in the small r expansion of fn(r).
The agreement of the r 2F 4 term between a 1loop computation of 6D SYM and low
energy limit of DSLST found in [13], was expected as a consequence of the supersymmetry
constraints on the F 4 coupling in the Coulomb branch e ective action [2, 17, 18]. The
agreement of r 2D2F 4 term between a 2loop computation of 6D SYM, the next order
0expansion of the DSLST amplitude was found in our previous work [14], numerically
for k = 2; 3; 4; 5. One anticipates that this agreement should follow from supersymmetry
constraints on D2F 4 coupling, namely the function f1(r) should be xed to be the form
C1=r2, and the coe cient C1 can then be computed from either small r (SYM) or large r
(DSLST). Indeed, the agreement we found in the SU(3) case can be understood in terms of
the (sixteensupercharge) nonrenormalization theorem of [18].2 Although the result of [18]
1For comparison, the colorordered treelevel superamplitude is given by Atree =
i
2For SU(2) gauge theory, the D2F 4 term in the Lagrangian is proven to be twoloop exact by [19, 20].
The focus of this paper is the f2(r)D4F 4 term. This is the lowest order in the derivative
expansion of the Coulomb branch e ective action where we anticipate a nontrivial
interpolating function f2(r) from small r (SYM) to large r (DSLST). Indeed, f2(r) receives all
loop perturbative contributions. Collecting numerical results on both sides, we will be able
to estimate the interpolating function on the entire Coulomb branch. We will nd that,
while the small and large r limits are obviously di erent expansions, when naively
extrapolated to the intermediate regime they are not far from one another. In the next section,
we describe the general structure of the Coulomb branch e ective action and its relation to
1) massless scalars in 6 dimensions [3]. We denote these massless scalar by
i, i = 1; 2; 3; 4, which take values in the U(
1
)k 1 Cartan of the SU(k) gauge group, in
the 6D SYM description (which is a priori valid near the origin of the Coulomb branch).
We will focus on a Zkinvariant 1dimensional subspace of the Coulomb moduli space,
corresponding to
Z
1 + i 2 = r diag(1; e2 i=k;
; e2 i(k 1)=k);
3 = 4 = 0:
The large r regime along this 1dimensional subspace is then described by the perturbative
double scaled little string theory [7, 8], with the worldsheet CFT given by
R1;5
(SL(2)k=U(
1
))
(SU(2)k=U(
1
))
Zk
:
The string coupling at the tip of the cigar (target space of SL(2)=U(
1
) coset CFT) is
identi ed with 1=r.
The massless degrees of freedom in the Coulomb phase, consisting of k
1 Abelian
vector multiplets of the 6D (1; 1) supersymmetry, are governed by a quantum e ective
action, that is the U(
1
)k 1 supersymmetric gauge theory action together with an in nite
series of higher derivative couplings.
We will focus on couplings of the schematic form
f ( )D2nF 4 +
. Such higher derivative deformations of the Abelian (1; 1) gauge theory
{ 3 {
(2.1)
(2.2)
are constrained by supersymmetry, though the constraints become weaker with increasing
number of derivatives. An illuminating way to organize the higher derivative couplings
is through the corresponding supervertex, namely, a set of (super)amplitudes that obey
supersymmetry Ward identities with no poles [9]. If we
x the scalar vev (say of the
form (2.1)), and consider terms of the form D2nF 4 +
, then a supersymmetric completion
of such a coupling corresponds to a 4point supervertex of the form
where Q is the total supermomentum, de ned by [10{12]
8(Q)F (s; t; u);
Q =
4
X qi;
i=1
qi = (qiA; qeiB);
q
iA = i
Aa
ia; qeiB = eiBb_ ei :
_
b
(2.3)
(2.4)
indices, a and b_ on the other hand are SU(2)
SU(2) little group indices. i
Aa and eiBb_ are
6 dimensional spinor helicity variables, with the null momentum of the ith particle related
a_ b_ = 12 ABCDpiCD. ia and ei_ are a set of 4
Grassby piAB =
i
Aa Bb
i
ab, piAB = eiAa_ eiBb_
mannian variables that generate the 24 = 16 states in the supermultiplet of the ith particle.
Corresponding to D2nF 4 coupling, F (s; t; u) would be a function of Mandelstam
variables s; t; u of total degree n.
For instance, if we
x the color structure (choice
of Cartan generators), there is a unique supersymmetric completion of the F 4 term,
corresponding to the constant term in F (s; t; u). In the SU(2) gauge theory, the massless
elds on the Coulomb branch are in a single U(
1
) gauge multiplet, and thus F (s; t; u)
must be symmetric in s; t; u. From this we immediately learn that there is no independent
D2F 4 vertex, since s + t + u = 0. This result is also an immediate consequence of the
nonrenormalization theorem of Paban, Sethi, and Stern [19] which is later extended
to the SU(3) case by [18]. In the more general SU(k) theory with k > 3, to the best
of our knowledge, there isn't a nonrenormalization theorem that determines the D2F 4
completely in terms of the F 4 coupling on the Coulomb branch. In fact, since di erent
Cartan generators can be assigned to the 4 external lines of the superamplitude, one can
construct nontrivial superamplitudes with F (s; t; u) a linear function of s; t; u. These are
the terms computed in [14], from both the SYM at 2loop and from DSLST. It is likely
that by consideration of higher point superamplitudes, and consistency with unitarity, one
can derive the supersymmetry constraint on the rdependence of the f1(r)D2F 4 coupling
as in the work of Sethi, but we not will pursue this topic in the current paper.
The consideration of superamplitudes allows for an easy classi cation of D2nF 4
couplings for all n. In below we will mostly think in terms of the superamplitudes rather
than the terms in the e ective Lagrangian. Now to be precise we will introduce a color
label ai 2 Zk for each external line, corresponding to a Cartan gluon in the U(
1
)k 1 that
transforms under the Zk cyclic permutation of k NS5branes by the phase e2 iai=k. The
4point superamplitude is of course subject to the constraint Pi4=1 ai = 0 (mod k), and
{ 4 {
(2.5)
(2.6)
(2.7)
takes the form
where our convention, s = s12 =
(p1 + p2)2, t = s14, u = s13 =
s
t. We also have the
following identi cation between the 6D gauge coupling gY M and the little string scale,
as seen by matching the tension of the instanton string with the fundamental string of
DSLST, and also veri ed in [14]. In this paper we work in units of 0, and so gY2 M = 32 3.
Our convention for the Coulomb branch radius parameter r is such that the W boson
corresponding to the D1brane stretched between the ith and jth NS5 brane has mass
8(Q)Fa1a2a3a4 (s; t; u; r);
2
1
0
=
In the next two sections, we will study the expansion of the function Fa1a2a3a4 (s; t; u; r) in
detail, from perturbative SYM and from DSLST.
3
Perturbative 6D SYM in the Coulomb phase
Near the origin of the Coulomb branch, the W bosons are light compared to the scale set by
gY M , and we can compute the 4point amplitude of Cartan gluons in SYM perturbation
theory. A priori, one may expect such a computation to run into two di culties: the
loop expansion of the massless scattering amplitude su ers from UV divergence at 4loop
order [16] (while the mixed Cartan gluon and W boson amplitude diverges at 3loop [15]),
and there may be higher dimensional operators that deform the SYM Lagrangian [21].
The consistency of DSLST [13] combined with nonrenormalization theorems of Sethi et
al. implies that the SYM Lagrangian at the origin of the Coulomb branch is not deformed
by trF 4 terms. The result of [14] further indicates that the 1=4 BPS operator of the form
D2tr2F 4 is absent at the origin of the Coulomb branch as well. On the other hand, the
3loop divergence in the nonAbelian sector means that the nonBPS dimension 10 operator
D2trF 4 is needed as a counterterm [15]. Likewise, at 4loop order we will need
counterterms of the form D4trF 4 and D4tr2F 4 [16]. It appears that one can proceed with the SYM
perturbation theory, and add the appropriate counterterms whenever a new divergence
is encountered at a certain loop order. Of course, the perturbative SYM does not give
a prescription for determining the
nite part of these counterterms. Such ambiguities
however do not a ect the leading logarithmic dependence on r, and so these leading logs
can be computed unambiguously in the framework of SYM perturbation theory at small r.
On the other hand, the nite shifts of the counterterms that cannot be determined by SYM
perturbation theory are in principle determined in the full little string theory, and one could
hope for extracting such information from the opposite regime, namely the large r limit.
Let us begin with the F 4 term in the Coulomb e ective action, or more precisely, its
supersymmetric completion, along the 1dimensional subspace as speci ed in (2.1). The
{ 5 {
k
3
4
5
`
corresponding superamplitude takes the form
As was shown in [13], the coe cient C0 is given by
where c0 is a constant that is independent of the color assignment.
Next consider the D2F 4 term, which can be written as
The result of [14] indicates that the corresponding superamplitude takes the form
8(Q) r2 C1;a1a2a3a4 s12 + (1 $ 3) + (2 $ 3) ;
i
and is twoloop exact. By symmetry of permutation on external lines, C1;a1a2a3a4 is
invariant under the permutations (12), (34), as well as (13)(24). Note that there is no 1loop
contribution to f1(r)D2F 4, of order r 4, simply because a 1loop contribution would come
with a C1;a1 a4 factor that is completely symmetric under permutation of a1;
; a4, and
thus must be proportional to s+t+u, which is zero. Therefore f1(r) takes the simple form3
with higher numerical precision.
We can write the D4F 4 couplings as
In [14], the C1 coe cients were computed for k = 2; 3; 4; 5 and color assignment
2
`
0 . The results are listed here in table 1
Now let us consider the D4F 4 term, which receives contributions from all loop orders.
fS;a1a2a3a4 (r)(s2 + t2 + u2)Fa1
Fa4 + fA;a1a2a3a4 (r)s2Fa1
Fa4
(3.6)
3When there is no potential confusion, we will often omit the color indices a1a2a3a4 if a1 = a2 = a3 =
a4 = ` + 1. For example, C1 = C1; `+1; `+1; (`+1); (`+1).
where S and A stand for symmetric and asymmetric in the Mandelstam variables. fS(r)
and fA(r) each admits a small r expansion4
The coe cients CS1=A, CS2=A, CS3=A (which depend on the color factors) are computed from
1, 2, 3loop amplitudes. The coe cients BS=A and BS0=A come from the 4loop amplitudes,
after canceling the log divergences by 3loop and 4loop counterterms. Note that the
appearance of the double log terms is due to nested divergences at 4loop order. In the UV
completed theory, namely the full LST, the divergence of SYM at 4loop order and higher
is re ected as a branch cut in the analytic structure of the function f2(r). The detailed
computation and numerical results for the 1, 2, and 3loop contributions are given in
appendix A, for k = 2; 3; 4; 5 and color assignment a1 = a2 =
a4 = `+1 with k 2
`
0. For 3loop, we need to sum up the scalar integrals represented by the nine diagrams
in gure 4. Each of diagrams (e) (f) (g) (i) is in fact UV divergent by itself at linear order in
s, t, u, and would potentially contribute to D2F 4. However, these divergences cancel after
we sum up these diagrams and the permutations of the external legs, and the remaining
parts are quadratic or higher in s, t, u and give nite contributions to D2nF 4 for n
2.
The 4loop divergence can be computed at the origin of the Coulomb branch, as in [16].
After moving away from the origin on the Coulomb branch, in the expansion in external
momenta, the logarithmic divergences appear in the form ln( =r), and in the case of nested
divergences, (ln( =r))2. After canceling the logarithmic divergences with counterterms,
we are left with logarithmic dependence on r, and the coe cient of the leading log (or
double log) is independent of nite shifts of the counterterm.
The logarithmic divergence at the origin of the Coulomb branch involves three possible
terms, of the form (s2 + t2 + u2)trF 4, (s2 + t2 + u2)(trF 2)2, and s2(trF 2)2 + (2 more). The
terms proportional to (s2 + t2 + u2) also contain double pole divergences (in dimensional
regularization). To cancel the divergences we need a 3loop counterterm D2trF 4 (it
vanishes when restricted to the Cartan, but is now needed to cancel subdivergences in the
4loop amplitude) and 4loop counterterms of the form D4trF 4 as well as D4(trF 2)2. In
the end, one obtain unambiguously the coe cient of
and the coe cient of
In principle, one can also determine unambiguously the (ln r)2 coe cient of the single
trace term proportional to (s2 + t2 + u2)trF 4, but this double pole coe cient has not been
evaluated explicitly in [16].
4The colorordered oneloop superamplitude is permutation invariant, hence the full amplitude is
completely symmetric in s, t, and u, and so CA1 = 0. The log2 divergence is also completely symmetric, as can
be seen from (A.50), and hence BA0 = 0.
ln r s2(trF 2)2 + (2 more) ;
(ln r)2(s2 + t2 + u2)(trF 2)2:
(3.8)
(3.9)
{ 7 {
The CS1=A, CS2=A, CS3=A and BS=A coe cients for k = 2; 3; 4; 5 and color assignment
in the (R,R) sector, of the form [13, 14],
Vab_;` = e '2 '2e eip X
2
a eBb_ SASeBV `sl;( 1
A 2
; `+22;; 12`)+2 V 2`s;u;(2` ; 21 ;2` 12 );
2
(4.1)
the form
with ` = 0; 1;
; k
2 labeling the color index of the U(1)k 1 gluons according to their
eigenvalues e2 i(`+1)=k with respect to the Zk cyclic permutation of the NS5 branes.
spin
and eBb_ are the 6D spinor helicity variables as before, and SA; SeB are the left and right
elds of the R1;5 part of the worldsheet CFT. There is also an identi cation Vab_;`
+
Vab_;k 2 ` [13, 14, 22]. It was shown in [14] that the sphere 4point superamplitude takes
A
a
ADSLST (1`+1; 2`+1; 3 ` 1; 4 ` 1) = 8(Q) Nk;`
Z
C
d2zjzj
(`+1)2 s 12 1
k
j
j
z ` (`+1)2 u+ 12
k
; ;
; P ; z)j2 : (4.2)
3; 4; 2
Q
{ 8 {
; P ; z) is the Liouville 4point conformal block. See [14]
for the precise identi cation of the parameters i
, i etc.
The evaluation of the conformal block integral and the integration over the cross ratio
z are performed numerically, order by order in the 0 expansion.5 For k = 2; 3; 4; 5, the two
leading terms in the expansion were given in [14]. We carry out this computation to 0
order, with the order 0n terms corresponding to D2nF 4 coupling in the Coulomb branch
e ective action. In the following we normalize the amplitudes by their 00 order terms.
3
1 + 2:10359958(s2 + t2 + u2) + 17:42982502stu +
:
(4.3)
1 + 6:1080323(s2 + t2 + u2) + 96:795814stu +
:
1
1
k = 5; ` = 0; 3 :
k = 5; ` = 1; 2 :
2:39679s + 14:9055(s2 + t2 + u2) 14:8295s2 + 302:54stu
The omitted terms are of quartic and higher degrees in s; t; u, corresponding to D8F 4 and
higher derivative couplings in the e ective action.
Note that the DSLST fourpoint amplitude is invariant under ipping the Zk charges of
the vertex operators. In addition when ` + 1 = k=2 (i.e. the vertex operators are identical),
the amplitude is invariant under permutation of the Mandelstam variables.
5Since we set 0 = 1, the 0 expansion is an expansion in the Mandelstam variables.
{ 9 {
×
×
( )
 ( )
=
=
=
=
( )
=
=
 ( )
=
=
HJEP12(05)
line is given by the DSLST tree level superamplitude (valid for large r). The lower green line comes
from 6D SYM one loop, the middle orange line comes from one and two loops combined, and the
upper blue line combines the contributions up to three loops (valid for small r). We interpolate the
two ends by a naive extension beyond their regimes of validity.
4.1
An interpolating function from weak to strong coupling
On one hand, perturbative 6D SYM gives a small r expansion of the coe cient f2(r) of
each higher derivative term in the Coulomb branch e ective action. On the other hand,
perturbative string scattering in DSLST gives an expansion valid at large r. The exact
f2(r) is a function that interpolates the two ends.
Let us rst consider the D2F 4 term. A nonrenormalization theorem by [18] shows
that f1(r) is twoloop exact in SU(3) maximal SYM, which means that (3.5) should hold
for arbitrary r. Indeed, the result of [14] was that the coe cients of s in the treelevel
DSLST superamplitudes exactly match with the C1 obtained from the SYM twoloop
superamplitudes (see table 1), for k = 4; 5 as well as k = 3. It is not inconceivable that
f1(r) is twoloop exact in 6D SYM for all k, which also implies that all higher genus
superamplitudes for the scattering of four Cartan gluons should vanish at 01 order.
Next let us consider D4F 4.
With the color index assignment a1 = a2 =
a3 =
a4 = ` + 1 (labeling the Zk charge), the two independent structures are proportional
to s2 + t2 + u2 and s2. We will compare the large and small r expansions. On the 6D
SYM side, the ra(ln(r= ))b terms after resummation will correct the power of r when one
interpolates the function f2(r) to large r. Here the coe cient of (ln(r= ))2 in the small r
expansion can be determined by the 4loop UV divergence at the origin of the Coulomb
branch. However, as already mentioned, this computation involves the divergence in the
single trace D4trF 4 term, which has not yet been computed in 6D SYM. The scale
has
absorbed the contribution from the counter term, and is expected to be of order gY M1 in
the full LST. Since the actual numerics depends on the precise value of the mass scale ,
we will not include the ln(r= ) terms in the interpolation function.
In each case, the coe cients of 1=r2 are close but not equal between the large and
small r expansions. There is no reason for them to be equal, since the large r expansion
should be corrected by higher genus contributions of order 1=r2(g+1), and the small r
expansion includes oneloop 1=r6 and twoloop 1=r4 terms, and should further be corrected
by higherloop contributions of the form ra(log(r= ))b.
For concreteness, we explicitly make the comparison for k = 5, noting that the other
cases are qualitatively the same.
k = 5; ` = 0 : Large r expansion:
and 3 loops) are plotted in solid lines. We interpolate the two ends by a naive extension
beyond their regimes of validity.
5
Discussion
To summarize our results so far, while the r 2F 4 and r 2D2F 4 terms in the Coulomb
branch e ective action are computed exactly by perturbative SYM at oneloop and
equal) to the result obtained from
captures the large r limit of f2(r).
twoloop orders respectively, and match precisely with the corresponding 0expansion of
the tree level amplitude in DSLST, the f2(r)D4F 4 terms involve a set of nontrivial
interpolation functions f2(r), that receive a priori allloop contribution in SYM perturbation
theory. We have determined f2(r) in its small r expansion up to 3loop orders in 6D SYM.
Interestingly, the 3loop contribution that scales like r 2, is numerically close (but not
02 order terms in the tree amplitude of DSLST, which
Starting at 4loop order in the perturbative SYM description, one encounters UV
divergences and while the leading log coe cients can be determined unambiguously in
perturbation theory, the subleading logs and constant shifts depend on
nite parts of 3
and 4loop counter terms (D2trF 4, D4trF 4, and D4tr2F 4 at the origin of the Coulomb
branch), and are a priori undetermined in 6D SYM perturbation theory.
In principle, the (1; 1) LST provides an unambiguous UV completion of the
perturbative amplitudes of 6D SYM. If one could somehow compute the exact 4gluon amplitude in
DSLST, nonperturbatively in gs, then one should recover all the perturbative SYM loop
amplitudes, and
x the
nite parts of all counter terms. While we do not have the
technology for such exact computations on the string theory side, the interpolation results on
the Coulomb moduli space so far suggests that, despite the nonrenormalizability of the 6D
SYM, the naive perturbative expansion is a valid prescription provided that appropriate
counter terms are included at each loop order.6
The UV divergences that arise at 4loop order and higher in the massless amplitudes of
6D SYM in the Coulomb phase, indicate not a trouble with SYM perturbation theory, but
rather a feature of the amplitudes and the corresponding couplings in the Coulomb branch
e ective action. Namely, the function f2(r), as an analytic function of r on the Coulomb
branch, has a branch cut starting from the origin. Where does this branch cut end, in the
analytic continuation of Coulomb moduli space? A natural expectation is that perhaps the
branch cut goes all the way to r = 1, where the Coulomb phase is described by weakly
coupled DSLST. In fact, we generally expect nonanalyticity in f2(r) at r = 1, due to the
nonconvergence of the string perturbation series, and the need for stringy nonperturbative
contributions (e.g. Dinstanton amplitudes). In fact, due to the identi cation gs
we could speculate that nonperturbative string amplitudes of the form exp( 1=gs)
contributes to the nite counter terms at the origin of the Coulomb moduli space!
1=r,
e r,
Going beyond massless amplitudes, the scattering of gluons with W bosons in 6D SYM
may be compared to Dbrane scattering amplitudes in DSLST.7 We hope to report on these
results in the near future.
6For instance, one could have said that since the 6D SYM theory is expected to be strongly coupled at
the scale gY M1 , a UV cuto should be imposed at the scale
gY M1 , and there would seem to be no reason
to perform the loop integral over momenta above this scale. However, the exact agreement of oneloop
and twoloop contributions to the F 4 and D2F 4 terms with DSLST indicates that the naive loop integrals,
which happen to be free of UV divergences in these cases, give the correct answer.
7At the level of 3point amplitude of gluon emission by a W boson, the agreement with the disc 1point
amplitude in DSLST was known in [23].
Comments on (2; 0) LST and 5D SYM
In this section we discuss the compacti cation of the (2,0) DSLST to ve dimensions, and
constrain the higher derivative terms in the e ective action of the resulting 5D gauge theory
on the Coulomb branch. In particular, we will show that the trF 4 coupling at the origin of
the Coulomb branch of the circlecompacti ed (2,0) superconformal eld theory is absent.
At the perturbative level, or equivalently in the 1=r expansion on the Coulomb branch,
the structure of (2; 0) DSLST is very similar to (1; 1) DSLST, di ering only through GSO
projection. As far as the massless 4point amplitude is concerned, at string tree level the
only di erence between the (2; 0) and (1; 1) case is the interpretation of the supermomentum
delta function 8(Q) in terms of the polarizations of the massless supermultiplets involved.
The scalar function of s; t; u that multiplies 8(Q) is identical. An analogous statement
holds for the genus one 4point amplitude as well. In the NSR formalism, this can be seen by
noting that the contribution from the (P,P) spin structure vanishes,8 and therefore the IIA
and IIB GSO projections yield the same amplitudes, up to reassignment of polarization
tensor structure. It is not inconceivable that the massless 4point amplitudes in (2; 0) and (1; 1)
DSLST involve the same scalar function of s; t; u to all order in perturbation theory, though
we do not have an argument for this. On the other hand, it appears that the Dinstanton
amplitudes of massless string scattering will be quite di erent in the two theories, as the
BPS Dinstanton that is pointlike in the R6 and localized at the tip of the cigar exists only
in the (2; 0) DSLST and not in the (1; 1) theory. Such contributions could alter the e ective
action near the origin of the Coulomb branch signi cantly, and give rise to entirely di erent
low energy dynamics of the (2; 0) and (1; 1) LST at the origin of the Coulomb branch.
Nonetheless, in view of the idea that the (2; 0) SCFT, when compacti ed on a circle, is
described in the low energy limit as 5D maximally supersymmetric YangMills theory [27]
together with an in nite series of higher dimensional operators/counterterms [28{31], one
could ask whether there is a similar interpolation on the Coulomb branch of the (2; 0) LST
compacti ed on a circle. In this case, the W boson comes from Dbranes located at the tip
of the cigar in the Tdual picture [23]. The parameters in the circlecompacti ed DSLST are
the string length `s, the W boson mass mW which is related to the string coupling9 gs by
(6.1)
8It su ces to look at the scattering of the scalars which correspond to (NS,NS) vertex operators. At one
loop in the (P,P) sector (here we are following the convention of [24] although historically this had been also
referred as the (odd,odd) sector [25]), we need to have three (0; 0)picture and one ( 1; 1)picture vertex
operators plus one PCO. Hence in the path integral we have a total of 4 insertions of
to a vanishing contribution to the total amplitude due to the presence of six zero modes for
which leads
(and ~ ).
and ~
One can also reach the slightly stronger statement that the four point amplitudes in (1; 1) and (2; 0)
DSLST agree up to 2loops following a version of Berkovits' argument in section 3.2 in [26].
9In this paper, we use gs to denote the string coupling at the tip of the cigar in the IIB picture , not
to be confused with the asymptotic string coupling gs1 before taking the decoupling limit of NS5branes in
asymptotically at spacetime in the IIA picture. They are related by gs
`sgs1=r [7, 32{34].
mW
R
gs`s2
;
and the compacti cation radius R, which is related to the 5D gauge coupling g5 by
form
then we expect
R =
g
2
From the 5D perspective, the natural mass scale is set by g5 or R, and the two
dimensionless parameters are
mW R (parameterizing distance from the origin on the Coulomb
branch) and R=`s. The 5D gauge theory obtained from compacti cation of (2; 0) SCFT,
in its Coulomb phase, is obtained in the limit R=`s ! 1, while holding R and
This in particular requires sending gs ! 1 at the same time.
If we write the amplitude of massless particles in the compacti ed (2; 0) DSLST in the
and the corresponding amplitude in the UV completion of 5D SYM in the form
gs!1
lim A(2;0) DSLST
E2g2
gs; gsmW
5 ; g52E
= A5D GT(g52mW ; g52E):
The l.h.s. cannot be captured by DSLST perturbation theory in a straightforward manner.
For instance, we can write the D2nF 4 terms in the Coulomb branch quantum e ective
Lagrangian in the schematic form
1
n=0
X fn( )D2nF 4;
where
mW R is the distance parameter on the Coulomb branch, and the subscript n
indicates the \number of derivatives". If we assume that the UV completion of the 5D
SYM perturbation theory is such that higher dimensional operators are added only when
needed as counterterms,11 then the SYM loop expansion of fn( ) has the structure
f0( ) =
f1( ) =
f2( ) =
f (
1
)
Here the coe cient fn(L) comes from the Lloop 4point amplitude. Note that the
1loop contribution f1(
1
)= 5 is absent; this is because the 1loop amplitude involves only
10Note that it is a di erent limit than taking R=`s ! 1 while keeping gs and `s xed, which is the limit
of decompacti ed (2,0) DSLST.
11As we will see shortly, while this is expected for the compacti ed (2; 0) SCFT, this is not true for the
compacti ed (2; 0) LST. We thank C. Cordova and T. Dumistrescu for a key discussion on this point.
(6.2)
xed.10
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
a single color structure that is invariant under permuting the 4 external lines, and the
D2F 4 amplitude would be proportional to s + t + u which vanishes. Note that while the
5D SYM 4point amplitude is known to have UV divergence at 6loop order [35], such a
divergence vanishes when the external gluons are restricted to the Cartan subalgebra. This
is because the counterterm responsible for this divergence is the unique dimension 10
non
BPS operator of the form D2trF 4 +
[21, 36], which in fact vanishes upon Abelianization
(i.e. restricting to the Cartan subalgebra). The 4point amplitude of Cartan gluons in 5D
SYM is expected to diverge rst at 8loop order, with the counterterm being a nonBPS
operator of the form D4tr F 4 +
. In the UV completion that is expected to arise from the
compacti cation of (2; 0) theory, the D4tr F 4 counterterm should cancel the log divergence,
dependence in the Coulomb e ective action, hence the f2(8) ln
term in (6.7).
Let us focus on the f0( )F 4 coupling for the moment. The argument of [19] and [20]
indicates that, at least in the SU(2) case where the Coulomb branch moduli space is just
a single R5, f0( ) is a harmonic function on the R5.12 Assuming SO(5) Rsymmetry, such
a harmonic function must be of the form
f0( ) = c + 03 :
f (
1
)
The constant c, if nonvanishing, would correspond to a trF 4 coupling in the nonAbelian
SYM at the origin of the Coulomb branch moduli space. In writing (6.7) we have assumed
that such coupling is absent in the low energy limit of the compacti ed (2; 0) theory. We
will now justify this assumption.
The Coulomb phase of the circlecompacti ed A1 (2; 0) LST has a moduli space of
vacua R
4
and has size
S1. The S1 coming from the compact scalar in the 6D (2; 0) tensor multiplet,
(R=`s)2 in units of R.13 In the Coulomb phase of the compacti ed (2; 0)
LST, the D2nF 4 couplings come with the coe cients fn(~; R=`s), such that
lim
R=`s!1
fn(~; R=`s) = fn( ):
Here ~ parameterize a point on the R
4
S1 moduli space, and the function fn(~; R=`s) is
invariant under SO(4) Rsymmetry in 6 dimensions, while the SO(5) is only restored in the
R=`s ! 1 limit. Note that, importantly, the limit is taken with
and so taking R=`s ! 1 requires sending gs ! 1 at the same time. From the 5D
perspective, gs of DSLST is related to the vev of a massless scalar eld, whereas R=`s is a rigid
parameter (there is no massless graviton propagating in the R1;5 of the DSLST and hence there
is no massless 5D scalar associated with the compacti cation radius); in particular, the
dependence on gs is constrained by supersymmetry, whereas the dependence on R=`s is not.
= R2=(gs`s2) held
xed,
12This is consistent with the v4= 3 e ective potential between separate D4 branes moving at a relative
velocity [37].
13To see the size of the S1, we can go back to the NS5brane picture in type IIA string theory,
separated in the transverse R4, with the world volume of the NS5branes compacti ed on a circle of radius
R. A W boson coming from D2brane stretched between a pair of the NS5branes and wrapping the
circle has mass mW
Rr=(gs1`s3)
2
R=(gs`s) as before. On the other hand, if we are to separate the
NS5branes along the Mtheory circle, the M2brane stretched between the M5branes and wrapping the
compacti cation circle of radius R has mass
R=`s2.
(6.8)
(6.9)
At nite R=`s, f0(~; R=`s) is an SO(4)invariant harmonic function on the R
We can write ~ = (~ ; y), where ~ parameterizes the R4 and y is the coordinate on the S1.
The harmonic function f0(~; R=`s) is restricted to be of the form
f0(~; R=`s) = c(R=`s) + X
n2Z
f0(
1
)(R=`s)
While c may no longer be a constant, it must be a function of the rigid parameter R=`s
only. In the limit of large j j, f0 can be expanded as
Matching this with the tree level (2; 0) DSLST, we conclude that c(R=`s)
argument we also expect that the corrections to the tree level contribution to F 4 coupling
in the compacti ed DSLST are entirely nonperturbative in gs.
Now, near the origin of Coulomb branch, ( ; y) = (0; 0), f0 can be written as
:
3 is generated from 5D SYM by integrating out W bosons
at 1loop. The second term is nonvanishing at the origin of the Coulomb branch and can be
understood in terms of 6D SYM compacti ed on a circle (as in the Tdual (1; 1) LST), with
massive KaluzaKlein modes integrated out at 1loop. This term vanishes in the R=`s ! 1
limit, and thus the trF 4 coupling is absent in the compacti ed (2; 0) superconformal theory
(at the origin of its Coulomb branch). The third term comes from the 1loop diagram with
6D W bosons in the loop that also carry nonzero KK momenta, expanded to the second
order in the W boson mass parameter, and gives rise to an SO(5)R breaking dimension 10
BPS operator at the origin of the Coulomb branch of the 5D gauge theory.
It should be possible to extend this discussion to higher rank cases as well. A more
detailed investigation of the twoparameter interpolation function in the Coulomb phase of
compacti ed (2; 0) DSLST, and its interplay with the perturbative structure of 5D SYM,
are left to future work.
Acknowledgments
We would like to thank ChiMing Chang for collaboration during the early stage of this
project. We are grateful to Ofer Aharony, Lance Dixon, Daniel Ja eris, Cumrun Vafa for
useful conversations and correspondences, to Zohar Komargodski and Shiraz Minwalla for
key suggestions on the use of nonrenormalization theorems in the Coulomb e ective action,
to Clay Cordova and Thomas Dumitrescu for important discussions on the compacti cation
of (2; 0) theories, and to Travis Max eld and Savdeep Sethi for comments on a
preliminary draft. We would like to thank the 7th Taiwan String Workshop at National Taiwan
University, Weizmann institute, and Kavli IPMU for their support during the course of
this work. The numerical evaluations of loop integrals are performed using FIESTA on the
Harvard Odyssey cluster, whereas the conformal block integrations and DSLST amplitudes
are computed with Mathematica. S.H.S. is supported by the Kao Fellowship at Harvard
University. X.Y. is supported by a Sloan Fellowship and a Simons Investigator Award from
the Simons Foundation. Y.W. is supported in part by the U.S. Department of Energy
under grant Contract Number DESC00012567. This work is also supported by NSF Award
PHY0847457, and by the Fundamental Laws Initiative Fund at Harvard University.
6D SYM loop amplitudes contributing to D4F 4
The term f2(r)D4F 4 receives contribution from all loop orders of the scattering amplitude
of four Carton gluons. At each loop order, we need expand the superamplitude to quadratic
order in the Mandelstam variables. Each loop order is proportional to the colorordered
fourpoint treelevel scattering amplitude
A
tree(1; 2; 3; 4) =
i
The oneloop amplitude of four Cartan gluons can be written as14
Here I41 loop(s12; s14; mij ) is the scalar box integral ( gure 2)15
I
1 loop(s12; s14; mij )
4
=
Z
d6`
1
(2 )6 (`2 +mi2j )((`+p1)2 +mi2j )((`+p1 +p2)2 +mi2j )((` p4)2 +mi2j ) :
mij is the mass of the W boson with gauge indices (ij), and vaj is the polarization vector
for the external Cartan gluons.
14The perturbative expansion of the amplitude of massless Cartan gluons takes the form
A = gY4 M A
1 loop + gY6 M A
2 loop +
+ gY2+M2LAL loop +
:
15In contrast to the more common convention in the scattering amplitude literature (for example ([16])
where the mostly minus signature is used and s = (p1 + p2)2, here we work in the mostly plus signature
and de ne s =
(p1 + p2)2. Hence when comparing the two, the Mandelstam variables are the same, but
we di er in the de nition of the scalar box integrals by factors of i from Wick rotating d`0 and minus signs
from the propagator 1=p2.
(A.1)
(A.4)
(A.5)
(A.2)
1
4
1 loop to s2=r6 order. It is straightforward to show that
where we have made the following replacements in the integrand:
` pi ` pj !
1 `2 pi pj =
6
1
192
1
12
2
` sij ;
(` pi)(` pj )(` pk)(` pm) !
(`2)2(sij skm + siksjm + simsjk):
Summing up with A1324
1 loop and A1243
1 loop, we obtain the order s2=r6 term for the full
oneloop amplitude
r6
A
1 loop(1; 2; 3; 4) s2 =
va`);
A.2
Twoloop
The full twoloop amplitude is given by
A
2 loop(1; 2; 3; 4) =
h
s12(A1234
s12s14A
+ A3421
tree(1; 2; 3; 4)
Let us start with the planar contribution,
(vam
2 loop;P is the planar scalar twoloop integral ( gure 3(a)),
I
2 loop;P (mij; m`i; mj`) =
4
( )
4
1
The order s2=r4 terms in A
can be computed straightforwardly,
I
4
3(`2 + m2 )(`2 + m2 )
ij
2
`i
Z
1
`22 +m2
`i
6 6
d `1 d `2
`
2
1
(`21 +m2 )2
ij
#
:
2 loop(1; 2; 3; 4) correspond to the s=r4 terms in I
`
2
2
+
4`1 `2
(`22 +m2 )2
`i
3(`21 +m2 )(`22 +m2 )
ij `i
Moving on to the nonplanar diagram,
A1234
=
X
is the nonplanar scalar twoloop integral ( gure 3(b)),
I
4
(mij; m`i; mj`) =
(2 )6 (2 )6 (`21 +mi2j)((`1 +p1)2 +mi2j)(`22 +m`2i)((`2 +p4)2 +m`2i)
((`2
p1
p2)2 + m`2i)((`1 + `2
p2)2 + mj2`)((`1 + `2)2 + mj2`)
As in the planar case, we are interested in the s=r4 term in I
computed straightforwardly,
(mij; m`i; mj`)
"
s12
`
2
2
(`22 + m`2i)2
2`1 `2 + 2`22
s
r4
=
1
!
(2 )6 (2 )6 (`21 + mi2j)2(`22 + m`2i)3((`1 + `2)2 + mj2`)2
`22 + m`2i
3(`21 + mi2j)(`22 + m`2i)
3(`21 + mi2j)((`1 + `2)2 + mj2`)
+
+
`21 + `1 `2
`1 `2 +`22
3(`22 +m`2i)((`1 +`2)2 +mj2`)
3(`21 +mi2j)(`22 +m`2i)
3(`22 +m`2i)((`1 +`2)2 +mj2`)
1
+
s14
`1 `2
`1 `2
1
:
1
1
2
b
( )
4
(A.12)
(A.13)
(A.14)
(A.15)
!#
:
1
2
1
2
1
a
( )
d
( )
`
1
g
( )
3
4
4
3
4
1
2
1
2
1
`
1
`
7
b
( )
`
1
`
3
`
6
`
4
e
( )
`
5
h
( )
`
9
`10
`
2
`
8
3
4
3
4
4
1
2
1
2
1
c
( )
`
1
(i)
f
( )
`
5
`
3
`
2
`
1
`
6
The full threeloop amplitude is given by
A
3 loop(1; 2; 3; 4) =
1 (d)
4 A1234 + 2A(1e2)34 + 2A(1f2)34 + 4A(1g2)34 +
1 (h)
2 A1234 + 2A(1i2)34
where we have summed over contributions from individual diagrams in gure 4 and
permutations of external legs. The coe cients in front of A1234 combined with the overall 1=4 are the
symmetry factors. The numerators for the scalar integrals in gure 4 are given in table 4.16
(x)
In below we will listed the contribution from each of the nine graphs, with external lines
restricted to Cartan gluons, and with the appropriate W boson mass assignments in the
16In contrast to the convention in [38] where the external momenta are all outgoing, our external momenta
are all ingoing. Furthermore, as mentioned before, the momentum square p2 di ers by a sign due to di erent
conventions on the signature, while the Mandelstam variables are the same.
Moreover since we consider W bosons propagating through the loops, the loop momenta `i (not all
independent) in the expressions of table 4 are taken to be higher dimensional with their extra components
constrained by the mass of the propagating particle. These will be made explicit in the expressions for the
full scalar integrals below.
`
4
3
4
3
4
4
(A.16)
HJEP12(05)
Integral I(x)
(a)(d)
(e)(g)
(h)
(i)
2
s12
s12(`1
p4)2
s12(`1 + `2)2
s14(`3 + `4)2 + s12`52 + s14`62
W boson mass square m2 term associated to each (` + p)2 factor in the numerator. We later restore
these factors in the explicit expressions for A1234 below.
internal propagators. The scalar loop integral will then be expanded in powers of external
momenta, or in terms of the Mandelstam variables s; t; u. At order s, while some of the loop
integrals are subject to UV divergence, these divergences cancel in the full 3loop amplitude
of Cartan gluons. For the purpose of extracting D4F 4 e ective coupling in the Coulomb
e ective action, we will expand the scalar integrals to s2 order. Below we will also list these
expanded expressions, which can then be evaluated numerically using FIESTA program.
Diagram (a) gives, including color factors,
(a)
Ia(mij ; mi`; mim; mj`; m`m)
X
X
i;j;`;m
i;j
Ia(mij ; mi`; m`m; mj`; mim)
+ 4 X Ia(mij ; 0; mij ; mij ; mij ) Y(vai
vaj)
Y (vai
a=1;2
vaj)
Y (vai
a=3;4
Y (vai
a=1;2
vaj)
Y (vam
a=3;4
vam)
va`)
where the scalar integral is
Ia(mij ; mi`; mim; mj`; m`m)
= s122 Z
1
(`22 +mi2m)((`2 +p4)2 +mi2m)((`2 +p3 +p4)2 +mi2m)(`23 +mi2`)((`3 +p1 +p2)2 +mi2`)
1
((`1
`3)2 + mj2`)((`2 + `3)2 + m`2m)
Before proceeding, let's introduce some shorthand notation,
dL
A1234 =
2
2
X
X
i;j
Ib(mij ; mi`; mj`; m`m; mim)
Ib(mij ; mi`; mj`; mim; m`m)
Ib(mij ; 0; mij ; mij ; mij )
4
Y
a=1
Y
a=1;2
Y
a=1;2
(vai
(vai
(vai
vaj);
where
Ib(mij; mi`; mj`; m`m; mim)
Z
Expanding in external momenta, we have
r2
Ic(mij ; mim; m`m; mj`; mjm; mi`)
Z
dL
2
3
1j1 13j3 2j5 23j4 3j2
2
2
:
Y
a=1;2
(vai
j
va)
Y
a=3;4
(va`
Expanding in external momenta and extracting the order s2 terms, we have
Note that by power counting the loop integral scales like m
vaj)(v4m
v4`)(v3i
v3m)
(A.22)
vam)
4
Y
a=1
(vai
vaj);
(A.23)
(A.24)
(A.25)
(A.26)
Diagram (c) gives
X
where
Ic(mij; mim; m`m; mj`; mjm; mi`)
Z
(`22 +m`2m)((`2 +p4)2 +m`2m)((`2 +p3 +p4)2 +m`2m)(`23 +mj2`)((`1 +p1 +p2 `2 `3)2 +mi2m)
Ic(mij ; mij ; mij ; mij ; 0; 0) + Ic(mij ; 0; mij ; 0; mij ; mij )
1
1
:
Z
1
((`1
`3)2 + mi2`)((`2 + `3)2 + m2 )
jm
Expanding in external momenta,
2
Ic(m1; m2; m3; m4; m5; m6)j s2 = s12
r2
dL
3
3
1j1 2j3 3j4 123j2 13j6 23j5
:
(A.27)
Id(mij ; mjm; m`m; mi`; mim)(v1i
v1)(v2
v2m)(v3`
v3m)(v4i
`
v4)
Id(mim; mij ; m`m; mj`; mjm)(v1m
v1i)(v2i
j
v2)(v3m
v3`)(v4`
j
v )
4
Id(mij ; mij ; mij ; mij ; 0)
(vai
j
va)
4
Y
a=1
Diagram (d) gives
X
X
where
Id(mij; mjm; m`m; mi`; mim)
Z
1
Y
a=1;2
(vai
Expanding in external momenta, and after some simpli cation of the loop integrals,
Ie(m1; m2; m3; m4; m5; m6)j s2 =
r2
s12 Z
3
dL
2
2
2
1j1 2j2 3j3 13j4 12j5 23j6
3s12
j
3s12
12j5
"
(2 )6 (2 )6 (2 )6 (`21 + mi2j)((`1 + p1)2 + mi2j)((`3
`1)2 + mj2m)((`3
`1 + p2)2 + m2 )
jm
1
:
(`22 + mi2`)((`2 + p4)2 + mi2`)((`2 + `3
Expanding in external momenta
2
Id(m1; m2; m3; m4; m5)j s2 = s12
r2
Z
dL
2
2
2
1j1 13j2 2j4 23j3 3j5
2
2
:
Ie(mij ; mi`; mim; mjm; mj`; m`m)
vaj)(v3i
v3`)(v4i
v4m)
Ie(mij ; mij ; mij ; 0; 0; 0)
(vai
vaj);
Diagram (e) gives
X
where
s12
we have
Ie(mij ; mi`; mim; mjm; mj`; m`m)
Z
6 6 6
d `1 d `2 d `3
(`1
1 1
1;23( s12
1j1 23j6
13;12s12
+
+
3 3 2 2
31j4 23j6
Diagram (f ) gives
12;23s12
+
1;31(s14
1 1 2 2
2;31s12
2j2 13j4
2;12s12
2j2 12j5
1 1 3 3
3;12s12
3j3 12j5
+
2 X If (mij ; mij ; mij ; 0; mij ; 0) Y(vai
If (mij ; mj`; mim; mjm; m`m; mi`) Y (vai
vaj)(v3`
v3j)(v4i
v4m)
where
s12
If (mij; mj`; mim; mjm; m`m; mi`)
((`1 + `2)2 + mj2m)(`23 + mi2`) :
Expanding in external momenta, we have
(A.33)
(A.34)
(A.35)
(A.36)
(A.37)
a=1;2
vaj);
(`1
a=1;2
(`1
p4)2 + m21
4
a=1
1
4
a=1
(2 )6 (2 )6 (2 )6 (`21 + m21)((`1 + p1)2 + m21)((`1 + p1 + p2)2 + m21)(`22 + m23)
If (m1; m2; m3; m4; m5; m6)j s2 =
r2
"
3s12
1 1
3s12 +
13j2
1;23(s14
1j1 23j5
Diagram (g) gives
s12 Z
3
2
1 1
j
+
1;1(s14 + 2s12)
1;2(s14
s12)
13;23s12
13j2 23j5
3 13;13s12 #:
2
13j2
j
j
2;13s12
2j3 13j2
+
1;13(3s12
s14)
1j1 13j2
where
2
i;j
X
2 X Ig(mij ; mij ; mij ; 0; mij ; 0) Y(vai
vaj)
Ig(mij ; m`m; mim; mjm; mj`; mi`) Y (vai
vaj)(v3`
v3m)(v4i
v4m);
Expanding in external momenta, we have
Ig(m1; m2; m3; m4; m5; m6)j s2 =
r2
s12 Z
3
3s12 + 1;1(s14 +2s12) + 1;2(s14 s12) +
23j2
+
2;23s12
2j3 23j2
2
2
1j1 2j3 13j5 23j2 3j6 12j4
dL
2
3 13;23s12
13j5 23j2
+
`3)2 + mi2j)((`1
`3 + p1)2 + mi2j)((`2 + `3)2 + mj2m)((`2 + `3 + p4)2 + mj2m)
p2
1
p3)2 + mi2m)(`23 + mj2`) :
Expanding in external momenta, we have
Ih(m1; m2; m3; m4; m5; m6)j s2 =
"
3 +
312;1
1 2
j
312;2
j
r2
13j1
s12s23 Z
3
312;23
23j4
#
2 12;12 :
12j6
dL
2
2
2
2
1j2 2j3 13j1 23j4 12j6 3j5
+
"
3s12
1 1
3s12
13j5
1;23( 3s12 + s14)
1j1 23j2
Diagram (h) gives
i;j
+ 2
X
vaj)
Ih(mij; mi`; m`m; mjm; mj`; mim)
where
where
1 1
2 1;13s12
1j1 13j5
2 13;13s12 #:
2
13j5
2;13s12
2j3 13j5
(A.38)
(A.39)
(A.40)
v4j);
(A.41)
(A.42)
v4m);
(A.43)
Diagram (i) gives
2
X
Ii(mij; mj`; mi`; mim; mjm; m`m)
s12((`1
p4)2 + mi2j) + s14((`1 + `2)2 + mi2`) + 13 (s12
s14)(`22 + mj2`)
(`21 + mi2j)((`1 + p1)2 + mi2j)(`22 + mj2`)((`2 + p2)2 + mj2`)
12;1s12
12j3
!
+
2
2
2
1j1 2j2 12j3 3j4 13j5 123j6
j
1 1 3 4
Expanding in external momenta, we have
Ii(m1; m2; m3; m4; m5; m6)j s2 =
3
1;2(s12 + s14)
r2
2 2
1
3
s12 1j1 +s14 12j3 +
(s12 s14) 2 2
2;12(2s12 + s14)
2;3(s12 + s14)
3 12;12s12
2j2 12j3
2 2 3 4
12j3
Note that the above expressions for the scalar loop integrals expanded in external
momenta to order s2 do not always exhibit symmetries of the graphs in a manifest way. In
the numerical evaluation of the loop integrals, veri cation of these symmetries is a basic
and useful consistency check.
Results for 6D SYM in the Coulomb phase.
To make contact with the consideration
of 6D SYM in section 3, we set the mass of the W boson with gauge indices (ij) to be
3
12j3
+
12;3s12
12j3 3j4
1j1 12j3
: (A.45)
((`1 + `2 + p1 + p2)2 + mi2`)((`1 + `2
p4)2 + mi2`)(`23 + mi2m)((`3 + p4)2 + mi2m)
1
((`1 + `3)2 + mj2m)((`1 + `2 + `3)2 + m`2m)
and the polarization vector for the external Cartan gluons to be
mij = 2r sin
(i
j)
k
;
vaj = !(j 1)na ; j = 1;
; k;
where ! = e2 i=k. For the four Cartan gluon scattering of interest,
n1 = n2 = ` + 1;
n3 = n4 = k
(` + 1)
with values ` = 0; 1;
; k
The partial amplitudes and full amplitudes for each case are listed in the tables below.
The quantity listed is the threeloop contribution to D4F 4 normalized by the oneloop F 4
amplitude
A
1 loop(1; 2; 3; 4) s2
r6
s12s14Atree(1; 2; 3; 4) s122 + s213 + s214
r6
k
184320
L=1
Xk1 sin2 L(k`+1) sin2 L(kk ` 1)
:
(A.49)
sin6 L
k
In the notation of section 3, this quantity is CS3 (s2 + t2 + u2) + CA3s2.
(A.44)
HJEP12(05)
(A.46)
(A.47)
(A.48)
diagram
(a)
(b)
(c)
(d)
total
3:772838(s2 + t2 + u2)
k = 3; ` = 0 :
diagram
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
gY M A3 loop=A1 loop
4
14:39876(s2 + t2 + u2) 10:376120s2
5:976425(s2 + t2 + u2) 4:223506s2
5:1697610(s2 + t2 + u2) 3:7469277s2
3:8144749(s2 + t2 + u2) 1:2321663s2
0:56439858(s2 + t2 + u2) + 0:42112441s2
0:68831287(s2 + t2 + u2) + 0:37394094s2
1:0393916(s2 + t2 + u2) 0:73705051s2
0:17584295(s2 + t2 + u2) 0:12991690s2
0:030527986(s2 + t2 + u2) + 0:091583958s2
symmetry factor
16
4
4
8
2
2
1
8
2
total
7:086485(s2 + t2 + u2) 4:505248s2
diagram
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
k = 4; ` = 1 :
total
11:619831(s2 + t2 + u2) 8:729678s2
diagram
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
gY M A3 loop=A1 loop
4
17:16058(s2 + t2 + u2)
6:703913(s2 + t2 + u2)
6:131683(s2 + t2 + u2)
4:8779369(s2 + t2 + u2)
0:762594(s2 + t2 + u2)
1:252848(s2 + t2 + u2)
1:2121707(s2 + t2 + u2)
0:21142504(s2 + t2 + u2)
0
total
8:521180(s2 + t2 + u2)
symmetry factor
16
4
4
8
2
2
1
8
2
diagram
k = 5; ` = 1 :
diagram
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
9:645323(s2 + t2 + u2) 7:922393s2
6:151710(s2 + t2 + u2) 1:9295194s2
1:538466(s2 + t2 + u2) + 0:7303561s2
1:308312(s2 + t2 + u2) + 0:645134s2
1:879007(s2 + t2 + u2) 1:451108s2
0:32813257(s2 + t2 + u2) 0:26805768s2
0:0512934(s2 + t2 + u2) + 0:1579127s2
total
17:38894(s2 + t2 + u2) 13:955903s2
symmetry factor
total
12:88988(s2 + t2 + u2) 8:901921s2
A.4
Fourloop
dimensions is
The result of [16] for the 4loop 4point amplitude of maximal SU(k) SYM in D = 6
2
A
(
(k2 3 + 25 5) Tr12Tr34s2 + Tr14Tr23t2 + Tr13Tr24u2
+ (single trace):
When restricted to the Cartan gluons, of charge na 2 Zk (a = 1; 2; 3; 4) with respect to the
Zk action, the single trace term is always proportional to (s2 + t2 + u2) P na ( here stands
for Kronecker delta modulo k). The coe cient will involve 1= 2 and 1= divergences. These
have not been computed explicitly.
On the other hand, for the double trace terms, we have
(k; na + nb
0; otherwise:
0
mod k;
For the amplitude of gluons with Zk charge (n; n; n; n) (n = ` + 1 in our notation),
we always have Tr13 = Tr14 = Tr23 = Tr24 = k. Tr12 = Tr34 = 0 for n 6= k=2, and
Tr12 = Tr34 = k for n = k=2. In the case k = 4, by comparing ` = 0 with ` = 1, we can
separate a contribution from double trace terms only,
4 loop
Ak=4;`=1
4 loop
Ak=4;`=0 = (stAtree) (4 )12 4 64 (s2 + t2 + u2)
e 4
(
2 2
16 + 36 3 +
1
16
35
18
+ 4 3 + 9 4 + 20 5
3
(16 3 + 25 5)s2
(A.51)
(A.52)
(A.53)
3
3
)
(8 + 18 3)(8 ln r)2 + A ln r + B
After subtracting o the 4loop counterterms, we expect
4 loop
Ak=4;`=1
4 loop
Ak=4;`=0 =
(
(s(t4A)tr1e2e) 64 (s2 + t2 + u2)
+ s2 3(16 3 + 25 5)(8 ln r + C) :
Here A is a constant that depends on
and B; C are constants that depend on
nite shifts of the 3loop D2trF 4 counterterm,
nite shifts of the 4loop D4trF 4 and D4tr2F 4
counterterms. They cannot be determined from SYM perturbation theory alone.
In the n = k=2 cases, all terms are proportional to s2 + t2 + u2, and we cannot separate
the double trace terms from the single trace terms at all. In the k = 3 and k = 5 cases, as
well as the k = 4; ` = 0 case, since Tr12 = Tr34 = 0, we can determine
(
Ak;`
4 loop =
(s(t4A)tr1e2e) k3 (s2 + t2 + u2)(unknown)
s
2 3(k2 3 + 25 5)(8 ln r + C) : (A.54)
)
B
Evaluation of the little string amplitudes
In this appendix, we discuss some machinery that went into the numerical evaluation of
the double scaled little string theory amplitude (4.2). The conformal block can be written
F ( i; P jz) = (16q)P 2 z Q42
1
2 (1
Q2
z) 4
1
3
3(q)3Q2 4( 1+ 2+ 3+ 4)H( i; P jq);
q(z) = e i (z);
(z) = i
K(1
K(z)
K(z) =
1 Z 1
2 0
where
P = Q42 + P 2, z is the cross ratio
q is the nome of z, de ned by
and 3 is the Jacobi theta function de ned by
z =
z12z34 ;
z14z32
3(p) =
1
X
n= 1
pn2 :
H satis es Zamolodchikov's recurrence formula [39, 40], which allows one to obtain H as
a series expansion in q. Alternatively, we can compute F as a series expansion in z by
computing inner products between Virasoro descendants of the external primary states.
The resulting expression is manifestly a rational function in c,
i, and
P . For this
reason the latter bruteforce method is more advantageous for obtaining simple analytic
expressions, although its computational complexity (with respect to the order of the series
in q) is much higher than the complexity of the recurrence method.
The conformal block written in the form (B.1) converges much faster than a naive
series expansion in z, due to the fact that jq(z)j is much smaller than z (note for example
that 16jq(z)j
jzj and jq(z)j < 1 for all z 2 C). Given an orderN series in z, we can
rewrite it in the form of (B.1) by performing a variable transformation and then truncate
H to order qN . If we want to integrate z over regions far from the origin, it is crucial that
we approximate the conformal block by a truncation of (B.1) instead of a series in z.
The Liouville structure constant C( 1; 2
; 3) is expressed as ratios of the special
function
, which has an integral representation [41, 42]
log (x) =
Z 1 dt "
0
Q
2
x
2
e t
sinh2( Q
2
sinh b2t sinh 2tb
x) 2t #
that is is convergent for 0 < Re x < Q. For x lying outside this region,
can be analytically
continued via the shift formulae
b(x + b) = (bx)b1 2bx
b(x + 1=b) = (x=b)b 2bx 1
where
(1
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
numerically, the oscillatory behavior of the second term at large t must
be taken care of by stripping out an exponential integral function
Z 1 dt
e( Q2 x)t
4t sinh b2t sinh 2tb = E1(xt0) +
4t
sinh b2t sinh 2tb
Z 1 dt (e bt + e b
e Qt)e( Q2 x)t
:
(B.8)
To obtain the Liouville fourpoint function, we then integrate over the Liouville momentum
P of the intermediate state. This integral is performed by a simple Riemann sum.
Finally we are in place to evaluate the integral with respect to the cross ratio z. We
break the integral over the complex plane into six regions. These regions are mapped to
each other under the S3 action generated by z ! 1 z and z ! 1=z. A fundamental region
z
j j
1;
Re z <
1
2
(B.9)
is chosen and the integrals over the other regions are mapped to Region I using crossing
symmetry of the fourpoint functions. In Region I, jzj is bounded by 1, and jqj by 0:066,
thus with the conformal block expressed in the form of (B.1), even if H is truncated to q6
order, we still have at least 10 7 precision for F !
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Lett. B 408 (1997) 105 [hepth/9704089] [INSPIRE].
[2] N. Seiberg, New theories in sixdimensions and matrix description of Mtheory on T 5 and
T 5=Z2, Phys. Lett. B 408 (1997) 98 [hepth/9705221] [INSPIRE].
[3] O. Aharony, M. Berkooz, D. Kutasov and N. Seiberg, Linear dilatons, NS vebranes and
holography, JHEP 10 (1998) 004 [hepth/9808149] [INSPIRE].
[4] A. Giveon, D. Kutasov and O. Pelc, Holography for noncritical superstrings, JHEP 10
(1999) 035 [hepth/9907178] [INSPIRE].
[hepth/9911147] [INSPIRE].
034 [hepth/9909110] [INSPIRE].
023 [hepth/9911039] [INSPIRE].
[5] O. Aharony, A brief review of `little string theories', Class. Quant. Grav. 17 (2000) 929
[6] D. Kutasov, Introduction to little string theory, prepared for ICTP Spring School on
Superstrings and Related Matters, Trieste Italy April 2{10 2001 [INSPIRE].
[7] A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999)
[8] A. Giveon and D. Kutasov, Comments on double scaled little string theory, JHEP 01 (2000)
[9] H. Elvang, D.Z. Freedman and M. Kiermaier, SUSY Ward identities, superamplitudes and
counterterms, J. Phys. A 44 (2011) 454009 [arXiv:1012.3401] [INSPIRE].
[10] C. Cheung and D. O'Connell, Amplitudes and spinorhelicity in six dimensions, JHEP 07
(2009) 075 [arXiv:0902.0981] [INSPIRE].
[11] T. Dennen, Y.T. Huang and W. Siegel, Supertwistor space for 6D maximal super
YangMills, JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].
HJEP12(05)
unreasonable e ectiveness of 6D SYM), JHEP 12 (2014) 176 [arXiv:1407.7511] [INSPIRE].
[15] Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The complete fourloop
fourpoint amplitude in N = 4 superYangMills theory, Phys. Rev. D 82 (2010) 125040
[arXiv:1008.3327] [INSPIRE].
[16] Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying multiloop
integrands and ultraviolet divergences of gauge theory and gravity amplitudes, Phys. Rev. D
85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
[17] S. Paban, S. Sethi and M. Stern, Constraints from extended supersymmetry in quantum
mechanics, Nucl. Phys. B 534 (1998) 137 [hepth/9805018] [INSPIRE].
[18] S. Sethi and M. Stern, Supersymmetry and the YangMills e ective action at nite N , JHEP
06 (1999) 004 [hepth/9903049] [INSPIRE].
[19] S. Paban, S. Sethi and M. Stern, Supersymmetry and higher derivative terms in the e ective
action of YangMills theories, JHEP 06 (1998) 012 [hepth/9806028] [INSPIRE].
[20] T. Max eld and S. Sethi, The conformal anomaly of M 5branes, JHEP 06 (2012) 075
[arXiv:1204.2002] [INSPIRE].
maximally supersymmetric YangMills theories, JHEP 05 (2011) 021 [arXiv:1012.3142]
[22] V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the twodimensional
conformal quantum
eld theory and selfdual critical points in Zn invariant statistical
systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [INSPIRE].
[23] D. Israel, A. Pakman and J. Troost, Dbranes in little string theory, Nucl. Phys. B 722
(2005) 3 [hepth/0502073] [INSPIRE].
[24] J. Polchinski, String theory. Vol. 2: superstring theory and beyond, Cambridge University
Press, Cambridge U.K. (1998).
(1988) 917 [INSPIRE].
[hepth/0609006] [INSPIRE].
[hepth/9705117] [INSPIRE].
[25] E. D'Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60
[26] N. Berkovits, New higherderivative R4 theorems, Phys. Rev. Lett. 98 (2007) 211601
[27] N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. Proc. Suppl. 67 (1998) 158
[28] K.M. Lee and J.H. Park, 5D actions for 6D selfdual tensor eld theory, Phys. Rev. D 64
(2001) 105006 [hepth/0008103] [INSPIRE].
quantum 5D superYangMills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
[arXiv:1012.2880] [INSPIRE].
HJEP12(05)
[hepth/9811167] [INSPIRE].
025018 [arXiv:1210.7709] [INSPIRE].
maximally supersymmetric YangMills theory diverges at six loops, Phys. Rev. D 87 (2013)
part 1: onshell formulation, arXiv:1403.0545 [INSPIRE].
for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
[hepth/0108121] [INSPIRE].
[1] M. Berkooz , M. Rozali and N. Seiberg , Matrix description of Mtheory on T 4 and T 5 , Phys . [12] Z. Bern , J.J. Carrasco , T. Dennen , Y.T. Huang and H. Ita , Generalized unitarity and sixdimensional helicity , Phys. Rev. D 83 ( 2011 ) 085022 [arXiv: 1010 .0494] [INSPIRE]. [13] O. Aharony , B. Fiol , D. Kutasov and D.A. Sahakyan , Little string theory and heterotic/typeII duality, Nucl . Phys. B 679 ( 2004 ) 3 [ hep th/0310197] [INSPIRE]. [14] C.M. Chang , Y.H. Lin , S.H. Shao , Y. Wang and X. Yin , Little string amplitudes (and the [29] N. Lambert , C. Papageorgakis and M. SchmidtSommerfeld , M 5branes, D4branes and [30] M.R. Douglas , On D = 5 super YangMills theory and (2; 0) theory , JHEP 02 ( 2011 ) 011 [31] N. Lambert , C. Papageorgakis and M. SchmidtSommerfeld , Deconstructing (2; 0) proposals , Phys. Rev. D 88 ( 2013 ) 026007 [arXiv: 1212 .3337] [INSPIRE]. [32] H. Ooguri and C. Vafa , Twodimensional black hole and singularities of CY manifolds , Nucl.
Phys. B 463 ( 1996 ) 55 [ hep th/9511164] [INSPIRE]. [33] D. Kutasov , Orbifolds and solitons , Phys. Lett. B 383 ( 1996 ) 48 [ hep th/9512145] [INSPIRE]. [34] K. Sfetsos , Branes for Higgs phases and exact conformal eld theories , JHEP 01 ( 1999 ) 015 [35] Z. Bern , J.J. Carrasco , L.J. Dixon , M.R. Douglas , M. von Hippel and H. Johansson , D = 5 [36] C.M. Chang , Y.H. Lin , Y. Wang and X. Yin , Deformations with maximal supersymmetries [37] M.R. Douglas , D.N. Kabat , P. Pouliot and S.H. Shenker , Dbranes and short distances in