We consider the scattering amplitudes of five and six gravitons at tree level in superstring theory. Their power series expansions in the Regge slope

Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory

Stephan Stieberger

Max-Planck-Institut fu¨r Physik, Werner-Heisenberg-Institut, 80805 Mu¨nchen, Germany

(Received 1 January 2011; published 16 March 2011)

We consider the scattering amplitudes of five and six gravitons at tree level in superstring theory. Their power series expansions in the Regge slope

⁰

are analyzed through the order

⁰⁸

showing some interesting constraints on higher order gravitational couplings in the effective superstring action such as the absence of

R⁵

terms. Furthermore, some transcendentality constraints on the coefficients of the nonvanishing couplings are observed: the absence of zeta values of even weight through the order

⁰⁸

like the absence of

ð2Þð3ÞR⁶

terms. Our analysis is valid for any superstring background in any space-time dimension, which allows for a conformal field theory description.

DOI:10.1103/PhysRevLett.106.111601 PACS numbers: 04.60.Cf, 04.50.Gh, 11.25.w

Superstring theories contain a massless spin-two state identified as a graviton. Its interactions are studied by graviton scattering amplitudes. Because of the extended nature of strings the latter are generically nontrivial func- tions on the string tension

⁰

. In the effective field theory description this

⁰

dependence gives rise to a series of infinite many higher order gravitational couplings gov- erned by positive integer powers in

⁰

. The classical Einstein-Hilbert term is reproduced in the zero-slope limit

⁰

! 0 . The modification of the Einstein equations through the order

⁰³

has been derived by studying tree-level four graviton scattering amplitudes [1–3] or alternatively by computing four-loop functions of the underlying model [4]. Up to this order the effective (tree- level) superstring action focusing on pure gravitational bosonic terms reads in D space-time dimensions

L

_tree

¼ 1

2

²

R þ

⁰³

2

⁹

4!

²

ð 3 Þt

₈

t

₈

R

⁴

; (1) with the gravitational coupling constant in D dimen- sions, the Riemann scalar R, the Riemann tensor R

and the tensor t

₈

defined in Eq. (4.A.21) of [5]:

t

₈

t

₈

R

⁴

t

₈^12...8

t

₈^12...8

R

₁₂₁₂

R

₃₄₃₄

R

₅₅₅₆

R

₇₈₇₈

: (2) If the indices are restricted to D ¼ 4 the combination t

₈

t

₈

R

⁴

becomes the Bel-Robinson tensor. The absence of R

²

and R

³

terms in superstring theory is shown in [6].

In D ¼ 4 the result simply follows by expanding the (only independent and nonvanishing) four-graviton amplitude

M ð 1

2

3

^þ

4

^þ

Þ ¼ 2

₂

h 12 i

⁸

½ 12 Nð 4 Þh 34 i

Bðs

₁₂

; s

₁₄

Þ Bðs

₁₂

; s

₁₄

Þ (3) through the order

⁰³

. The superscripts denote the helicities of the corresponding gravitons. Above, we have introduced the kinematic invariants s

_ij

¼ 2

⁰

k

_i

k

_j

involving the external (on-shell) momenta k

_i

and the

Euler Beta function B encoding the

⁰

dependence of the full string amplitude. Furthermore, hiji, ½ij are spinor products (see, e.g., [7,8]), s

_ij

¼

⁰

fi; jg :¼¼

⁰

hiji½ji, and NðnÞ ¼ Q

_n1

i¼1

Q

_n

j¼iþ1

hiji [9].

In Eq. (1) further

⁰

corrections L

⁰_tree

arise from terms with higher powers in the Riemann tensor R

supple- mented by covariant derivatives D. In the following, these terms are collectively denoted by t

_m;n

D

^m

R

ⁿ

, with some tensor t

_m;n

contracting D and the Riemann tensor R.

Generically, the set of these additional interactions may be summarized in the series

L

⁰_tree

¼

²

X

¹

n4

X

¹

m¼0

^0n1þm

X

0

ir2N;i1>1 i1þ...þid¼n1þm

ði

₁

; . . . ; i

_d

Þ c

_m;n;i

t

ⁱ_m;n

D

^2m

R

ⁿ

; (4) with multizeta values (MZVs)

ði

₁

; . . . ; i

_d

Þ ¼ X

n₁>...>nd>0

Y

^d

r¼1

n

ⁱr ^r

; i

_r

2 N ; i

₁

> 1 of transcendentality degree P

_d

r¼1

i

_r

¼ n 1 þ m and depth d supplemented by some rational coefficients c

_m;n;i

. The prime at the sum (4) means that the latter runs only over a basis of independent MZVs of weight n 1 þ m [10–13]. For a recent account on MZVs, see Ref. [14].

The terms in the sum (4) are probed by computing scattering amplitudes of n gravitons and analyzing their power series in

⁰

[15]. For n ¼ 4 it is straightforward to extract the relevant information, since in this case each term in the

⁰

expansion of (3) directly translates into a term in the effective action (4). Moreover, the MZV co- efficients of the latter are simply products of Riemann zeta functions ði

₁

Þ of odd degree i

₁

as a consequence of

Bðs

₁₂

; s

₁₄

Þ

Bðs

₁₂

; s

₁₄

Þ ¼ e

²

P

₁

n¼1ð2nþ1Þ

2nþ1ðs^2nþ1₁₂ þs^2nþ1₁₃ þs^2nþ1₁₄ Þ

: (5) In this Letter we discuss the consequences of (on-shell) superstring scattering of five and six gravitons to the PRL 106, 111601 (2011) P H Y S I C A L R E V I E W L E T T E R S

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18 MARCH 2011

0031-9007=11=106(11)=111601(4) 111601-1 Ó 2011 American Physical Society

correction terms (4). We find that some of the coefficients c

_m;n;i

are vanishing and the MZVs of the nonvanishing terms follow some specific pattern.

The string world sheet describing the tree-level string S matrix of N gravitons is described by a complex sphere with N (integrated) insertions z

_i

of graviton vertex opera- tors V

_G

ð ; z

_i

; z

_i

Þ:

M ð 1 ; . . . ; NÞ ¼ V

_CKG¹

Y

^N

j¼1

Z

C

d

²

z

_j

hV

_G

ð

₁

; z

₁

; z

₁

Þ . . . V

_G

ð

_N

; z

_N

; z

_N

Þi: (6) Here, the factor V

_CKG

accounts for the volume of the con- formal Killing group. One of the key properties of graviton amplitudes in string theory is that at tree level they can be expressed as sum over squares of (color ordered) gauge amplitudes in the left—and right—moving sectors. This map, known as Kawai-Lewellen-Tye (KLT) relations [16], gives a relation between a closed string tree-level amplitude on the sphere and a sum of squares of (partial ordered) open string disk amplitudes. On the string world sheet of the sphere describing the N-graviton amplitude (6) the KLT relations are a consequence of decoupling holomorphic and antiholomorphic sectors by splitting the complex sphere integration over the coordinates z, z 2 C into two real ones , 2 R describing products of two open string disk amplitudes. At the level of degrees of freedom this is anticipated from the fact, that the graviton vertex operator V

_G

splits into a product of noninteracting open string states describing the vertex operator of massless vectors V

_g

, i.e., V

_G

ð ; z; zÞ ’ V

_g

ð ; ÞV

_g

ð ; Þ, subject to the decomposi- tion of polarization tensors

¼

. In the graviton amplitude (6) the latter comprises into the linearized Riemann tensor R

¼ k

½

k

½

.

For the two cases N ¼ 5 , 6, which we shall consider in the sequel, we have the following relations [with M ð 1 ; . . . ; NÞ ¼ ð

₂

Þ

^N2

Mð 1 . . . NÞ] [16]:

Mð 12 345 Þ ¼ ð 2

⁰

Þ

²

sin ðs

₁₂

Þ

sin ðs

₃₄

Þ Að 12 345 ÞAð 21 435 Þ þ ð 23 Þ; (7) Mð 123 456 Þ ¼ ð 2

⁰

Þ

³

sin ðs

₁₂

Þ sin ðs

₄₅

Þ Að 123 456 Þ

f sin ðs

₃₅

ÞAð 215 346 Þ þ sin ½ðs

₃₄

þ s

₃₅

Þ Að 215 436 Þg þ ð 234 Þ: (8) The KLT relations (7) and (8), hold for any superstring background and are insensitive to the compactification details or the amount of supersymmetries. The gluon am- plitudes Að 1 . . . NÞ are the color-ordered subamplitudes of the underlying gauge theory. Hence, in superstring theory the tree-level computation of graviton amplitudes boils down to considering squares of tree-level gauge amplitudes A. For this sector explicit computations have been per- formed and results are accessible for the cases N ¼ 4 [5,17], N ¼ 5 [18–21], N ¼ 6 [19–22], and N ¼ 7 [23].

Moreover, through the order

⁰²

the full N-gluon MHV amplitude is given in [20,21], while the order

⁰³

has been constructed in [24]. Based on all these results and the relations (7) and (8), the five- and six-graviton amplitudes can be derived.

The result for the five-gluon subamplitude can be given for any space-time dimension D in the form [25]

Að 12 345 Þ ¼ C

₁

A

_YM

ð 12 345 Þ þ C

₂

A

_F4

ð 12 345 Þ; (9) with the kinematical factors encoding the pure Yang-Mills (YM) part A

_YM

ð 12 345 Þ and the genuine string part A

_F4

ð 12 345 Þ

A

_F4

ð 12 345 Þ ¼ ð 2

⁰

Þ

²

fKð

₁

;

₂

;

₃

; k

₃

;

₄

; k

₄

;

₅

; k

₅

Þ þ ðk

₁

k

₂

Þ

¹

½ð

_{1 2}

ÞKðk

₁

; k

₂

;

₃

; k

₃

;

₄

; k

₄

;

₅

; k

₅

Þ þ ð

₁

k

₂

ÞKð

₂

; k

₁

þ k

₂

;

₃

; k

₃

;

₄

; k

₄

;

₅

; k

₅

Þ ð

₂

k

₁

ÞKð

₁

; k

₁

þ k

₂

;

₃

; k

₃

;

₄

; k

₄

;

₅

; k

₅

Þ

þ cyclic permutationsg; with : Kð

₁

; k

₁

;

₂

; k

₂

;

₃

; k

₃

;

₄

; k

₄

Þ ¼ t

₈^1...8 ₁¹

k

₁²

. . .

₄⁷

k

₄⁸

: (10) Furthermore, we have the two (basis) functions

C

₁

¼ s

₂

s

₅

f

₁

þ ðs

₂

s

₃

þ s

₄

s

₅

Þf

₂

; C

₂

¼ f

₂

; (11) defined by the two hypergeometric functions

₃

F

₂

:

f

₁

¼ Z

1 0

dx Z

1

0

dy I ðx; yÞðxyÞ

¹

; f

₂

¼ Z

1

0

dx Z

1

0

dy I ðx; yÞð 1 xyÞ

¹

;

(12)

with I ðx; yÞ ¼ x

^s2

y

^s5

ð 1 xÞ

^s3

ð 1 yÞ

^s4

ð 1 xyÞ

^s35

. In D ¼ 4 spinor notation the expressions in (9) boil down to the maximally helicity-violating YM five-point partial ampli- tude [26,27]

A

_YM

ð 1

2

3

^þ

4

^þ

5

^þ

Þ ¼ i h 12 i

⁴

h 12 ih 23 ih 34 ih 45 ih 51 i ; (13)

and to the term

A

_F4

ð 1

2

3

^þ

4

^þ

5

^þ

Þ ¼

⁰²

ðh 12 i½ 23 h 34 i½ 41 f 3 ; 4 gf 4 ; 5 g f 1 ; 2 gf 1 ; 5 gÞ

A

_YM

ð 1

2

3

^þ

4

^þ

5

^þ

Þ (14) describing the leading string correction to the YM ampli- tude (13). In the effective action (14) gives rise to the contact term t

₈

F

⁴

t

₈^11...44

F

₁₁

F

₂₂

F

₃₃

F

₄₄

with the field strength F

.

Using in (7) the expression (9) yields the full five- graviton amplitude. Its

⁰

expansion is determined by expanding the basis (12), which is achieved with [28]. In the D ¼ 4 field theory limit, with f

₁

!

^{0!0 1}_s

2s₅

and f

₂

!

^0!0

0 , the only independent five graviton amplitude reduces to [9]

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111601-2

Mð 1

2

3

^þ

4

^þ

5

^þ

Þj

00

¼ 4 i h 12 i

⁸

Nð 5 Þ ð 1 ; 2 ; 3 ; 4 Þ; (15) with 4 i ð 1 ; 2 ; 3 ; 4 Þ ¼ ½ 12 h 23 i½ 34 h 41 i h 12 i½ 23 h 34 i

½ 41 . Through the order

⁰⁸

we find for any dimension D Mð 12 345 Þj

02

¼ 0 ; Mð 12 345 Þj

04

¼ 0 ; Mð 12 345 Þj

⁰ⁿ

¼ ðnÞmð 12 345 Þj

_ðnÞ

þ ð 23 Þ; n ¼ 3 ; 5 ; 7 ; Mð 12 345 Þj

06

¼ ð 3 Þ

²

mð 12 345 Þj

_ð3Þ2

þ ð 3 Þ

²

s ^

₁₂

s ^

₃₄

ð C

₁

A

_YM

þ C

₂

A

_F4

Þj

_ð3Þ

ðC

₁

A

_YM

þ C

₂

A

_F4

Þj

_ð3Þ

þ ð 23 Þ;

Mð 12 345 Þj

08

¼ ð 3 Þð 5 Þmð 12 345 Þj

ð3Þð5Þ

þ ð 3 Þð 5 Þ s ^

₁₂

s ^

₃₄

½ð C

₁

A

_YM

þ C

₂

A

_F4

Þj

ð3Þ

ðC

₁

A

_YM

þ C

₂

A

_F4

Þj

ð5Þ

þ ð C

₁

A

_YM

þ C

₂

A

_F4

Þj

_ð5Þ

ðC

₁

A

_YM

þ C

₂

A

_F4

Þj

_ð3Þ

þ ð 23 Þ: (16) Above we have introduced [ s ^

_ij

¼ ð 2

⁰

Þ

¹

s

_ij

]

mð 12 345 Þ ¼ s ^

₁₂

s ^

₃₄

½ð C

₁

A

_YM

þ C

₂

A

_F4

ÞA

_YM

þ A

_YM

ðC

₁

A

_YM

þ C

₂

A

_F4

Þ: (17) Let us now discuss the implications of the results (16). In [29–31] the four-graviton amplitude has been analyzed, resulting in a series of higher order terms t

₈

t

₈

D

^2m

R

⁴

, which enter Eq. (4) with zeta functions shown in the first column of Table I. The (on-shell) four-graviton amplitude does not contain the momentum terms corresponding to D

²

R

⁴

[32].

The only possible Feynman diagrams contributing at the order

^03þm

, m 0 to the five-graviton amplitude are displayed in Fig. 1. For m ¼ 0 the above diagrams simply reproduce the

⁰³

order of the five-graviton amplitude involving the R

⁴

term. Any D

²

R

⁴

term can always be rewritten as a sum of R

⁵

terms due to the relation D

²

R ’ R

²

. Hence, for m ¼ 1 only the five vertex [right diagram in Fig. 1] stemming from an R

⁵

term may contribute at the order

⁰⁴

to the five-graviton amplitude. Since the latter vanishes at this order, cf. (16), we conclude that an R

⁵

term does not exist. At higher orders

^03þm

; m 2 both D

^2m2

R

⁵

and D

^2m

R

⁴

terms may contribute in Fig. 1. The

⁰

expansion of the five-graviton amplitude does not show ð 2 Þ ð 3 Þ terms at

⁰⁵

, ð 6 Þ terms at

⁰⁶

, ð 3 Þð 4 Þ nor ð 2 Þ ð 5 Þ terms at

⁰⁷

, and not any ð 8 Þ, ð 2 Þ ð 3 Þ

²

, and

ð 5 ; 3 Þ terms at

⁰⁸

, cf. Eq. (16). Since those terms cannot be generated by any reducible diagram with five external gravitons, we conclude the absence of contact interactions with coefficients

⁰⁵

ð 2 Þ ð 3 Þ,

⁰⁶

ð 6 Þ,

⁰⁷

ð 3 Þ ð 4 Þ,

⁰⁷

ð 2 Þð 5 Þ, and

⁰⁸

ð 8 Þ,

⁰⁸

ð 2 Þð 3 Þ

²

,

⁰⁸

ð 5 ; 3 Þ, respec- tively, cf. second column of Table I. On the other hand, the non-vanishing terms (16), which are proportional to

⁰⁵

ð 5 Þ;

⁰⁶

ð 3 Þ

²

,

⁰⁷

ð 7 Þ, and

⁰⁸

ð 3 Þ ð 5 Þ, respectively, account for the two diagrams in Fig. 1. After subtracting the contribution of the left diagram, the remaining terms give rise to combinations of the interactions D

⁴

R

⁴

, D

²

R

⁵

at

⁰⁵

, combinations of D

⁶

R

⁴

, D

⁴

R

⁵

at

⁰⁶

, D

⁸

R

⁴

, D

⁶

R

⁵

at

⁰⁷

, and D

¹⁰

R

⁴

, D

⁸

R

⁵

terms at

⁰⁸

, respectively.

Next, we consider the scattering of six gravitons. The result for the six-gluon subamplitude is given for any space-time dimension D in [19], while in D ¼ 4 spinor notation in [20–22]. Using these expressions in (8) yields the six-graviton amplitude. The

⁰

expansion of this am- plitude gives vanishing results at the orders

⁰²

and

⁰⁴

Mð 123 456 Þj

02

¼ 0; Mð 123 456 Þj

04

¼ 0: (18) The order

⁰³

is proportional to ð 3 Þ and describes dia- grams involving vertices from the R

⁴

coupling. Moreover, through the order

⁰⁸

we find the following properties:

Mð 123 456 Þj

_ð2Þð3Þ05

¼ 0 ; Mð 123 456 Þj

_ð6Þ06

¼ 0 ; Mð 123 456 Þj

_ð2Þð5Þ07

¼ 0 ; Mð 123 456 Þj

_ð3Þð4Þ07

¼ 0 ; Mð 123 456 Þj

_ð8Þ08

¼ 0; Mð 123 456 Þj

_ð2Þð3Þ2⁰⁸

¼ 0;

Mð 123 456 Þj

_ð5;3Þ08

¼ 0 : (19) Together with the previous results the findings (19) restrict the contact interactions at

⁰⁵

to be of the form ð 5 Þ fD

⁴

R

⁴

; D

²

R

⁵

; R

⁶

g, but forbids any ð 2 Þ ð 3 Þ contact terms at this order. Similarly, contact interaction at

⁰⁶

may assume the form ð 3 Þ

²

fD

⁶

R

⁴

; D

⁴

R

⁵

; D

²

R

⁶

g, but no interactions with ð 6 Þ factors are possible. At this order also a reducible TABLE I. Tree-level higher order gravitational couplings and

their corresponding zeta value coefficients probed by the

N-graviton superstring amplitude. Vanishing terms are crossed

out. Those terms, which have not yet been probed by the relevant

N-graviton amplitude, are marked by a question mark.

N¼4 N¼5 N¼6 N¼7 N¼8 ⁰³ð3Þ R⁴

⁰⁴ð4Þ D²R⁴ R⁵

⁰⁵ð5Þ D⁴R⁴ D²R⁵ R^{6 05}ð2Þð3Þ D⁴R⁴ D²R⁵ R⁶

⁰⁶ð3Þ² D⁶R⁴ D⁴R⁵ D²R⁶ R⁷? ⁰⁶ð6Þ D⁶R⁴ D⁴R⁵ D²R⁶ R⁷?

⁰⁷ð7Þ D⁸R⁴ D⁶R⁵ D⁴R⁶ D²R⁷? R⁸? ⁰⁷ð3Þð4Þ D⁸R⁴ D⁶R⁵ D⁴R⁶ D²R⁷? R⁸? ⁰⁷ð2Þð5Þ D⁸R⁴ D⁶R⁵ D⁴R⁶ D²R⁷? R⁸? ⁰⁸ð3Þð5Þ D¹⁰R⁴ D⁸R⁵ D⁶R⁶ D⁴R⁷? D²R⁸? ⁰⁸ð8Þ D¹⁰R⁴ D⁸R⁵ D⁶R⁶ D⁴R⁷? D²R⁸? ⁰⁸ð2Þð3Þ² D¹⁰R⁴ D⁸R⁵ D⁶R⁶ D⁴R⁷? D²R⁸?

⁰⁸ð5;3Þ D¹⁰R⁴ D⁸R⁵ D⁶R⁶ D⁴R⁷? D²R⁸?

FIG. 1. Diagrams contributing at the order

^03þm

to the five- graviton amplitude.

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111601-3

diagram describing the exchange of a graviton between two four vertices of ð 3 ÞR

⁴

contributes. At the order

⁰⁷

only contact terms of the form ð 7 ÞfD

⁸

R

⁴

; D

⁶

R

⁵

; D

⁴

R

⁶

g may appear. However, no contact interactions with ð 3 Þ ð 4 Þ nor ð 2 Þ ð 5 Þ factors exist at this order in

⁰

. Eventually, at

⁰⁸

only contact terms of the form ð 3 Þ ð 5 Þ fD

¹⁰

R

⁴

; D

⁸

R

⁵

; D

⁶

R

⁶

g may appear. However, no contact in- teractions with ð 8 Þ; ð 2 Þ ð 3 Þ

²

nor ð 5 ; 3 Þ factors exist at

⁰⁸

. What remains to be checked is how for a given order in

⁰

the set of contact interactions belonging to one row of Table I can be expressed by a minimal basis of terms. The latter may allow us to reduce the number of Riemann tensors R by converting them into derivatives D

²

, cf. the comment [10].

The results presented here and summarized in Table I hold for any type I or II superstring compactification in D space-time dimensions with eight or more supercharges and suggest that higher order gravitational couplings (4) obey some refined transcendentality properties: at each order in

⁰

only Riemann zeta functions of odd weight or products thereof appear. While for the first column this is obvious to all orders in

⁰

as a result of the relation (5), for the second and third column we have checked this state- ment up to the order

⁰⁸

. Hence, for n 6 and up to order

⁰⁸

the sum (4) runs only over basis elements comprised by MZVs of odd weights [13]. The absence of the MZV ð 5 ; 3 Þ at the order

⁰⁸

fits into this criterion, since it may be written as ð 5 ; 3 Þ ¼

⁵₂

ð 6 ; 2 Þ

²¹₂₅

ð 2 Þ

⁴

þ 5 ð 3 Þð 5 Þ.

For D ¼ 10 type IIB superstring theory, our findings together with the one-loop results [33] restrict the ring of possible modular forms describing the perturbative and nonperturbative completion of the higher order terms.

Our amplitude results, summarized in Table I, have an impact on the recently discussed finiteness of N ¼ 8 supergravity in D ¼ 4 . Counterterms invariant under N ¼ 8 supergravity have an unique kinematic structure and the tree-level string amplitudes provide candidates for them, which are compatible with supersymmetry Ward identities and locality. The absence or restriction on higher order gravitational terms at the order

^0l

together with their symmetries constrain the appearance of possible counter- terms available at l loop; see [34] for a review and refer- ences therein.

[1] D. J. Gross and E. Witten, Nucl. Phys.

B277, 1 (1986).

[2] D. J. Gross and J. H. Sloan, Nucl. Phys.

B291, 41 (1987).

[3] Y. Kikuchi, C. Marzban, and Y. J. Ng, Phys. Lett. B

176,

57 (1986); Y. Cai and C. A. Nunez, Nucl. Phys.

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D²R’R²

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[12] The set of integral linear combinations of MZVs is a ring, since the product of any two values can be expressed by a (positive) integer linear combination of the other MZVs [11], e.g.:

ðmÞðnÞ ¼ðm; nÞ þðn; mÞ þðmþnÞ

(quasishuffle or stuffle relation).

[13] There are many relations over

Q

among MZVs, e.g.,

ð4;1Þ ¼2ð5Þ ð2Þð3Þ. For a given weight w2N

the dimension

d_w

of the space spanned by MZVs of weight

w

is given by

d_w¼dw2þdw3

(d

₁¼0,d₂¼ d₃¼d₄¼1

,

d₅¼2

,

d₆¼2

,

d₇¼3

,

d₈¼4

,

d₉¼5

,

d₁₀¼7;. . .

) [11] and can be constructed by the following basis: for

w¼2

by

ð2Þ, forw¼3

by

ð3Þ, forw¼4

by

ð2Þ²

, for

w¼5

by

ð5Þ,ð2Þð3Þ, for w¼6

by

ð2Þ³

,

ð3Þ²

, for

w¼7

by

ð7Þ,ð2Þð5Þ,ð3Þð2Þ²

, for

w¼8

by

ð2Þ⁴

,

ð2Þð3Þ²

,

ð3Þð5Þ,ð5;3Þ, forw¼9

by

ð9Þ, ð7Þð2Þ, ð5Þð2Þ²

,

ð3Þ³

,

ð3Þð2Þ³

, for

w¼10

by

ð7;3Þ, ð5;3Þð2Þ, ð7Þð3Þ, ð5Þ²

,

ð5Þð3Þð2Þ, ð3Þ²ð2Þ²

,

ð2Þ⁵

, etc.

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^0l1

the alternative

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requires an

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PRL 106, 111601 (2011) P H Y S I C A L R E V I E W L E T T E R S

week ending 18 MARCH 2011

111601-4