V Belavin

Modular Integrals in Minimal Super Liouville Gravity

come

Modular Integrals in Minimal Super Liouville Gravity

Introduction

The continuum formulation of the noncritical string theory is equivalent to 2D quantum gravity coupled to some critical matter, i.e., the matter described by a conformal field theory M c . Simple reaction of conformal theories to the scaling of the metric leads to the universal form of the effective action of the generated gravity, which is called the Liouville gravity (LG) [START_REF] Polyakov | Quantum geometry of fermionic strings[END_REF]. Because of the peculiarities of two-dimensional metric geometry, many technical simplifications immediately come into play. Thus, LG is perhaps the simplest example of quantum gravity, but it nevertheless shares the same basic questions of interpretation and can hence be considered useful and worth studying. The problem of choosing observables correctly and the problem of calculating the corresponding correlation functions are of primary importance in any quantum theory. The field of LG has experienced considerable progress in recent years. Recently discovered higher equations of motion (HEM) [START_REF] Al | Higher equations of motion in Liouville field theory[END_REF] in LFT have allowed reaching the four-point level in calculating the correlation functions in LG [START_REF] Belavin | Integrals over moduli spaces, ground ring, and fourpoint function in minimal Liouville gravity[END_REF][START_REF] Belavin | Moduli integrals and ground ring in minimal Liouville gravity[END_REF]. The results were tested against the calculation in the framework of the relatively independent approach to 2D quantum gravity usually called the matrix models (see, e.g., [START_REF] Ginsparg | Lectures on 2-D gravity and 2-D string theory[END_REF] and the references therein). Moreover, a deeper understanding of the correspondence between these two approaches was achieved based on these results [START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF], although the complete picture of the relations between these techniques is still missing.

In the context of string theory applications, the construction of 2D quantum gravity in superspace is one of the most interesting questions. The first possible generalization is N =1 supersymmetry. Here, essential progress was also achieved recently. The study was motivated by a series of works concerned with super LFT in which the profound understanding of the properties of the conformal blocks (which are the basic elements of any CFT [START_REF] Belavin | Infinite conformal symmetry in twodimensional quantum field theory[END_REF]) was 1 achieved. This allowed "taking off" the necessity of treating the correlations involving only special degenerate [START_REF] Kac | Infinite-dimensional Lie algebras[END_REF] excitations. Another important result was the discovery of the HEM in super LFT [START_REF] Belavin | Higher equations of motion in Super Liouville field theory[END_REF]. All these results have served as a good starting point for a more profound study of super Liouville gravity (SLG). In [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF], the structure of the physical fields in the Neveu-Schwarz (NS) sector of SLG was clarified, and the general expression for the npoint correlation numbers on the sphere in terms of integrals over moduli space was written explicitly. Then the super HEM in the N =1 super LFT and the analysis of the structure of the super ground ring (and its logarithmic counterpart) allowed deriving the explicit analytic expressions for the four-point correlators. Because little is now known for the supersymmetric matrix model, there are no independent results analogous to those obtained in [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF]. In this situation, more checks of the validity of these results would seem desirable. This paper is devoted to directly calculating the four-point correlation numbers in SMLG.

The paper is organized as follows. To make the presentation self-contained, we collect all necessary information related to the subject in Sec. 2 and Sec. 3. The remaining part of the paper deals with evaluating the four-point integrals numerically. In Sec. 4, we consider two examples of the four-point integral. We reduce the expressions to a number of integrals over the fundamental region of the modular group, and the integrands represent the products of the various correlation functions both in the matter and in the Liouville sectors. The elliptic transformation is the subject of Sec. 5. The numerical results are presented in Sec. 6. Some useful details omitted from the main text are given in Appendix A.

Minimal Super Liouville Gravity

In the framework of the so-called DDK approach [START_REF] David | Conformal Field theories coupled to 2-D gravity in the conformal gauge[END_REF][START_REF] Distler | Conformal Field theory and 2-D quantum gravity or who's afraid of Joseph Liouville?[END_REF][START_REF] Distler | Superliouville Theory as a Two-Dimensional, Superconformal Supergravity Theory[END_REF], SLG is represented as a tensor product of superconformal matter (SM), super Liouville, and super ghost systems

A SLG = A SM + A SL + A SG (1)
with the interaction via the relation for the central charge parameters

c SM + c SL + c SG = 0 (2)
and also due to the construction of the physical fields. The superconformal algebra is

[L n , L m ] = (n -m)L n+m + ĉ 8 (n 3 -n)δ n,-m , {G r , G s } = 2L r+s + ĉ 2 r 2 - 1 4 δ r,-s , [L n , G r ] = 1 2 n -r G n+r , (3)
where r, s ∈ Z + 1 2 for the NS sector, r, s ∈ Z for the R sector.

The SLFT central charge is

c SL = 1 + 2Q 2 , (4)
where the "background charge" parameter Q is related to the SLFT basic quantum param-

eter b Q = b -1 + b (5)
The fields belong to the highest-weight representations of the superconformal algebra. The basic fields of interest in this paper belong to the primary supermultiplet (V a , Y a , Ȳa , W a ) with the bottom component V a having the conformal dimension

∆ L a = a(Q -a) 2 , ( 6
)
where a is a continuous (complex) parameter, and the other components of the primary supermultiplet being

Y a = G L -1/2 V a , Ȳa = ḠL -1/2 V a , W a = ḠL -1/2 G L -1/2 V a . (7)
Here and hereafter, we use the superscripts M, L, and G to specify the matter, Liouville, and ghost sectors of the superconformal generators. The generators without sector superscripts are related to the total super Virasoro algebra.

At certain special values of the parameter a = a m,n , one singular vector appears at the level mn/2 in the Verma module over V am,n = V m,n [START_REF] Kac | Infinite-dimensional Lie algebras[END_REF]. Here,

a m,n = Q/2 -λ m,n , (8)
where (m, n) is a pair of positive integers (m -n ∈ 2Z) and

λ m,n = mb -1 + nb 2 . (9)
The basic super Liouville operator product expansion [START_REF] Belavin | Bootstrap in Supersymmetric Liouville Field Theory. I. NS[END_REF] (for the sake of brevity we write ∆ = ∆ Q/2+iP and ∆

i = ∆ a i ) V a 1 (x)V a 2 (0) = (10) ′ dP 4π (xx) ∆-∆ 1 -∆ 2 C Q/2+iP a 1 ,a 2 V Q/2+iP (0) ee + CQ/2+iP a 1 ,a 2 V Q/2+iP (0) oo
This OPE is continuous and involves integration over the "momentum" P . In [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF], [V p ] ee,oo denotes the contribution of the primary field V p and its "even" and "odd" superconformal descendants to the operator product expansion. As usual, the prime on the integral indicates possible discrete terms; in this study, we consider only the region b where such extra terms do not appear and the integral can be understood literally. All other OPEs of two arbitrary local fields in SLFT can be derived from [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF]. The basic structure constants

C Q/2+iP a 1 a 2 and 3 CQ/2+iP a 1 ,a 2
in (10) were evaluated using the bootstrap technique in [START_REF] Poghossian | Structure Constants in the N=1 Super-Liouville Field Theory[END_REF][START_REF] Rashkov | Three point correlation functions in N=1 super Liouville theory[END_REF][START_REF] Fukuda | Super-Liouville theory with boundary[END_REF] and have the explicit form (here a denotes a 1 + a 2 + a 3 )

C Q-a 3 a 1 a 2 = πµγ Qb 2 b 1-b 2 (Q-a)/b Υ R (b)Υ NS (2a 1 )Υ NS (2a 2 )Υ NS (2a 3 ) 2Υ NS (a -Q)Υ NS (a 1+2-3 )Υ NS (a 2+3-1 )Υ NS (a 3+1-2 ) , CQ-a 3 a 1 a 2 = -πµγ Qb 2 b 1-b 2 (Q-a)/b iΥ R (b)Υ NS (2a 1 )Υ NS (2a 2 )Υ NS (2a 3 ) Υ R (a -Q)Υ R (a 1+2-3 )Υ R (a 2+3-1 )Υ R (a 3+1-2 ) , (11)
where we use the convenient notation in [START_REF] Fukuda | Super-Liouville theory with boundary[END_REF] for the special functions

Υ NS (x) = Υ b x 2 Υ b x + Q 2 , Υ R (x) = Υ b x + b 2 Υ b x + b -1 2 (12)
expressed in terms of the "upsilon" function Υ b , which is the standard element in the Liouville field theory (see [START_REF] Dorn | On correlation functions for non-critical strings with c < 1 but d > 1[END_REF][START_REF] Zamolodchikov | Conformal bootstrap in Liouville field theory[END_REF]).

Because of central charge balance condition (2), the central charge of the matter sector is given in terms of the same basic parameter b:

c SM = 1 -2q 2 , (13)
where q = b -1 -b. We let (Φ α , χ α , χα , Ψ α ) denote the primary multiplet in the matter sector with the dimension of the bottom component Φ a being given by

∆ M α = α(q -α) 2 . ( 14
)
The super ghost system (see, e.g., [START_REF] Polchinski | String theory. Vol. 2: Superstring theory and beyond[END_REF][START_REF] Verlinde | Lectures On String Perturbation Theory[END_REF][START_REF] Friedan | A Tentative theory of large distance physics[END_REF]) is described by the free super conformal field theory with the central charge c SG = -10. The fermionic part of the SG system involves two anticommuting fields (b, c) of spins (2, -1), and the bosonic part involves two bosonic fields (β, γ) of spins (3/2, -1/2). The formal fields (see [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF]) of the form δ(γ(0)) of dimension 1/2 are essential in constructing the gravitational amplitudes.

Physical Fields and the Correlation Numbers

The physical fields form a space of cohomology classes with respect to the nilpotent BRST charge Q,

Q = m : L M+L m + 1 2 L g m c -m : + r : G M+L r + 1 2 G g r γ -r : - 1 4 c 0 . (15)
In this work, we deal with the correlators of physical fields of the two types

W a (z, z) = U a (z, z) • c(z)c(z) • δ(γ(z))δ(γ(z)), (16)
and

Wa (z, z) = ḠM+L -1/2 + 1 2 Ḡg -1/2 G M+L -1/2 + 1 2 G g -1/2 U a (z, z) • c(z)c(z), (17)
where

U a (z, z) = Φ a-b (z, z)V a (z, z) (18)
Here the parameter a can take generic values. The general form of the n-point correlation numbers on the sphere for these observables [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF] is

I n (a 1 , • • • , a n ) = n i=4 d 2 z i Ḡ-1/2 G -1/2 U a i (z i ) Wa 3 (z 3 ) W a 2 (z 2 ) W a 1 (z 1 ) . ( 19
)
An additional "discrete" physical state arises when the representation in the matter sector is degenerate,

O m,n (z, z) = Hm,n H m,n Φ m,n (z, z)V m,n (z, z). ( 20
)
The operators H m,n are composed of the super Virasoro generators and are defined uniquely modulo exact terms. Moreover, if we introduce the logarithmic counterparts of the discrete states

O m,n , O ′ m,n = Hm,n H m,n Φ m,n V ′ m,n , (21)
then we have the important relations

[10] QQO ′ m,n = B m,n Wm,-n (22)
and

Ḡ-1/2 G -1/2 U m,-n = B -1 m,n ∂∂O ′ m,n mod Q, ( 23
)
where B m,n are the coefficients arising in the higher equations of motion of SLFT [START_REF] Belavin | Higher equations of motion in Super Liouville field theory[END_REF]. For four points, relation [START_REF] Al | Gravitational Yang-Lee model: Four point function[END_REF] allows reducing the moduli integral in general expression [START_REF] Zamolodchikov | Conformal bootstrap in Liouville field theory[END_REF] to boundary integrals if one of the fields is degenerate, i.e., a i = a m,-n . The explicit result is (see [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF])

I 4 (a m,-n , a 1 , a 2 , a 3 ) = κN (a m,-n ) 3 i=1 r,s∈(m,n) q (m,n) r,s (a i ) + 2mnλ m,n 3 i=1 N (a i ), (24)
where

q (m,n) r,s (a) = |a -λ r,s -Q/2| Re -λ m,n (25)
and the fusion set is (m

, n) = {1 -m : 2 : m -1, 1 -n : 2 : n -1}. The coefficient is κ = -2µ -1 b -2 πµγ 1 2 + b 2 2 2+b -2 γ b -2 2 - 1 2 , ( 26
)
and the "leg" factors are

N (a) = πµγ 1 2 + b 2 2 -a/b γ ab - b 2 2 + 1 2 γ a b - b -2 2 + 1 2 1/2 . ( 27
)
4 Direct Calculation

Here, we verify analytic result [START_REF] Hadasz | Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks[END_REF]. The space of parameters (a 1 , a 2 , a 3 , and also b) is rather big to present a comprehensive analysis. In what follows, we focus on the two examples where one of the fields is either W b or W 2b . Moreover, we restrict ourself to considering the most symmetric situation of four identical fields I 4 (a) = I 4 (a, a, a, a). In the four-point case [START_REF] Zamolodchikov | Conformal bootstrap in Liouville field theory[END_REF] reduces to

I 4 (a) = d 2 z Ḡ-1/2 G -1/2 U a (z)W a (0) Wa (1) W a (∞) . (28)
The analytic results following from expression [START_REF] Hadasz | Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks[END_REF] are

I 4 (b) = κ b N 4 (b)Σ (1,1) (b), (29)
where

Σ (1,1) (b) = |2b 2 -1|. ( 30
)
For the second integral, we have

I 4 (2b) = κ b N 4 (2b)Σ (1,3) (b) (31)
and

Σ (1,3) (b) = 3 2 |5b 2 -1| + |3b 2 -1| + |b 2 -1| -3b 2 -1 . ( 32
)
Let us consider the integral I 4 (b). Taking into account that we deal with the unit operators in the matter sector in this case, we have

Ḡ-1/2 G -1/2 U b = W b , W b = V b cc δ(γ)δ(γ), (33)
Wb = ḠM+L -1/2 + 1 2 Ḡg -1/2 G M+L -1/2 + 1 2 G g -1/2 V b cc.
Taking the explicit form of the correlation functions in the ghost sector into account,

C(0)C(1) = 0, C(0)C(1)C(∞) = 1, (34) δ(γ(0))δ(γ(1)) = 1,
we conclude that the only nonzero contribution comes from the term in Wb , which is proportional to cc,

I 4 (b) = d 2 z W b (z)V b (0)W b (1)V b (∞) . (35)
In the same way, we have

Ḡ-1/2 G -1/2 U 2b = ḠM+L -1/2 G M+L -1/2 Φ b V 2b , W 2b = Φ b V 2b cc δ(γ)δ(γ), (36)
W2b = ḠM+L -1/2 + 1 2 Ḡg -1/2 G M+L -1/2 + 1 2 G g -1/2 Φ b V 2b cc,
for the second integral, and taking (34) into account, we obtain

I 4 (2b) = d 2 z Ψ b (z)Φ b (0)Ψ b (1)Φ b (∞) V 2b (z)V 2b (0)V 2b (1)V 2b (∞) , + χ b (z)Φ b (0)χ b (1)Φ b (∞) Ȳ2b (z)V 2b (0) Ȳ2b (1)V 2b (∞) , (37)
+ χb (z)Φ b (0) χb (1)Φ b (∞) Y 2b (z)V 2b (0)Y 2b (1)V 2b (∞) , + Φ b (z)Φ b (0)Φ b (1)Φ b (∞) W 2b (z)V 2b (0)W 2b (1)V 2b (∞) .
We now use the symmetry of the integrals under modular transformations to reduce the integration from the whole complex plane to the fundamental domain. The modular subgroup of projective transformations divides the complex plane into six regions. The fundamental region is defined as G ={Re x < 1/2; |1 -x| < 1}. The other five regions are mapped to the fundamental one using one of the transformations A, B, AB, BA, ABA, where A: z → 1/z and B: z → 1 -z. Combining the projective transformations of the fields and the corresponding change of variables in the integrals, we reduce the integration to the fundamental region. We note that the Jacobian of the transformation exactly cancels the transformation of the fields because their total conformal dimension is 1. Then,

I 4 (b) = 2 G d 2 z W b (z)V b (0)W b (1)V b (∞) + W b (z)V b (0)V b (1)W b (∞) + + W b (z)W b (0)V b (1)V b (∞) , ( 38
)
where the factor 2 in front of the integral takes the equivalent projective images into account.

The expression for the second integral is rather bulky:

I 4 (2b) = 2 G d 2 z Ψ b (z)Φ b (0)Ψ b (1)Φ b (∞) V 2b (z)V 2b (0)V 2b (1)V 2b (∞) + Ψ b (z)Ψ b (0)Φ b (1)Φ b (∞) V 2b (z)V 2b (0)V 2b (1)V 2b (∞) + Ψ b (z)Φ b (0)Φ b (1)Ψ b (∞) V 2b (z)V 2b (0)V 2b (1)V 2b (∞) + χ b (z)Φ b (0)χ b (1)Φ b (∞) Ȳ2b (z)V 2b (0) Ȳ2b (1)V 2b (∞) + χ b (z)χ b (0)Φ b (1)Φ b (∞) Ȳ2b (z) Ȳ2b (0)V 2b (1)V 2b (∞) + χ b (z)Φ b (0)Φ b (1)χ b (∞) Ȳ2b (z)V 2b (0)V 2b (1) Ȳ2b (∞) (39) + χb (z)Φ b (0) χb (1)Φ b (∞) Y 2b (z)V 2b (0)Y 2b (1)V 2b (∞) + χb (z) χb (0)Φ b (1)Φ b (∞) Y 2b (z)Y 2b (0)V 2b (1)V 2b (∞) + χb (z)Φ b (0)Φ b (1) χb (∞) Y 2b (z)V 2b (0)V 2b (1)Y 2b (∞) + Φ b (z)Φ b (0)Φ b (1)Φ b (∞) W 2b (z)V 2b (0)W 2b (1)V 2b (∞) + Φ b (z)Φ b (0)Φ b (1)Φ b (∞) W 2b (z)W 2b (0)V 2b (1)V 2b (∞) + Φ b (z)Φ b (0)Φ b (1)Φ b (∞) W 2b (z)V 2b (0)V 2b (1)W 2b (∞) .
We now use the conformal block decomposition of the correlation functions. It is useful to introduce a compact notation. For a while, we omit some arguments that are easily reconstructed in the final expressions. In the matter sector,

Φ(z)Φ(0)Φ(1)Φ(∞) = c k |A (0) k (z)| 2 , Ψ(z)Φ(0)Ψ(1)Φ(∞) = c k |A (1) k (z)| 2 , Ψ(z)Ψ(0)Φ(1)Φ(∞) = c k |A (2) k (z)| 2 , ( 40
) Ψ(z)Φ(0)Φ(1)Ψ(∞) = c k |A (3) k (z)| 2 .
Here, the index k = +, 0, -corresponds to the three channels in the degenerate OPE of the field Φ b (and also of its super partners), and we assume summation with respect to k. The coefficients c k are related to the basic structure constants (see [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF]):

c + = C 2 + (b) = γ(1/2 + b 2 /2)γ(-1/2 + 5b 2 /2) γ(-1/2 + 3b 2 /2)γ(1/2 + 3b 2 /2) , (41)
c 0 = -C2 0 (b) = -1, (42)
c -= C 2 -(b) = - γ(2b 2 )γ(-1/2 + b 2 /2)γ 2 (1/2 + b 2 /2) b 4 γ 3 (b 2 )γ(-1 + b 2 )γ(-1/2 + 3b 2 /2) . (43)
In (40), A

denotes the conformal blocks appearing in the k channel for the given correlation function. Here and hereafter, the normalization is chosen such that all but the basic combinations c k are absorbed inside the conformal blocks. In Appendix A, we recapitulate some details and explicit constructions concerning conformal blocks. In the Liouville sector,

V (z)V (0)V (1)V (∞) = R dP 4π r l (P )|B (0) l (P, z)| 2 , W (z)V (0)W (1)V (∞) = R dP 4π r l (P )|B (1)
l (P, z)| 2 , W (z)W (0)V (1)V (∞) = R dP 4π r l (P )|B (2)
l (P, z)| 2 , (44)
W (z)V (0)V (1)W (∞) = R dP 4π r l (P )|B (3)
l (P, z)| 2 .
Because the correlation functions do not contain the degenerate fields in the Liouville sector, the index l here assumes summation of the "even" and of the "odd" conformal blocks in accordance with general OPE [START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF]. Again, the normalization leaves only two basic combinations outside the conformal blocks,

Rr 0 (P ) = C Q/2+iP a,a C Q/2-iP a,a , Rr 1 (P ) = CQ/2+iP a,a CQ/2-iP a,a , (45)
where we separate the factor R for convenience (it is independent of P ) and the parameter a is related to the external conformal dimension (i. e. either b or 2b). All other correlation functions in (39) are already not independent. They are expressed in terms of the same ingredients as in ( 40) and (44). For example,

χ(z)Φ(0)χ(1)Φ(∞) = c k A (1)
k (z)A (0) k (z), χ(z)χ(0)Φ(1)Φ(∞) = c k A (2) k (z)A (0) k (z), (46)
χ(z)Φ(0)Φ(1)χ(∞) = c k A (3) k (z)A (0) k (z),
and

r 0 (P ) = P 2 exp ∞ 0 dt t - (5 -18b 2 + 37b 4 )e -t 2b 2 + (66) 2 cos( P t 2 )(cosh( (1-7b 2 )t 4b ) + 2 cosh( (1+b 2 )t 4b ) + cosh( (3-5b 2 )t 4b
)) -cos(P t)(cosh( (1-b 2 )t 2b ) + 1) -6 sinh(t/(2b)) sinh(bt/2) ,

r 1 (P ) = 4P 2 exp ∞ 0 dt t - (5 -26b 2 + 37b 4 )e -t 2b 2 + (67) 2 cos( P t 2 )(cosh( (3-7b 2 )t 4b ) + 2 cosh( (1-b 2 )t 4b ) + cosh( (1-5b 2 )t 4b
)) -cos(P t)(cosh( (1-b 2 )t 2b ) + 1) -6 sinh(t/(2b)) sinh(bt/2) .

The conformal blocks can be evaluated effectively using a numerical algorithm based on the recurrence relations developed in [START_REF] Belavin | Bootstrap in Supersymmetric Liouville Field Theory. I. NS[END_REF][START_REF] Hadasz | Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks[END_REF][START_REF] Hadasz | Recursion representation of the Neveu-Schwarz superconformal block[END_REF][START_REF] Belavin | N=1 SUSY Conformal Block Recursive Relations[END_REF][START_REF] Belavin | On the N=1 super Liouville four-point functions[END_REF][START_REF] Chorazkiewicz | Braiding and fusion properties of the Neveu-Schwarz superconformal blocks[END_REF]. We do not use the elliptic recursion to construct the necessary conformal blocks here. It turns out that to attain a convincing accuracy of the results, we need to know a very few first terms in the x-(as well as q-) expansion of the conformal blocks. This information can be obtained directly starting from the very definition of the conformal blocks in terms of the chain vectors (see Appendix A). Nevertheless, the elliptic representation and especially the form of the prefactor (i.e., the ∆ asymptotic of the conformal blocks; see [START_REF] Hadasz | Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks[END_REF]) is very useful. It allows verifying the explicit expressions for the correlation functions by checking the crossing symmetry requirement, and in particular, fixing up all the signs, which is difficult to do starting from general principles.

Numerics

With (55) and (57), calculating reduces to numerically integrating several integrals of the general form

F |z(1 -z)θ 4 3 (q)F P (z)| 2 d 2 τ, (68)
where F P (z) is some Liouville conformal block like in (55) or some more complicated composite expression like in (57). The integrand can be developed as a double power series in q and q in accordance with the general expansion

z(1 -z)θ 4 3 (q)F P (z) = (16q) α ∞ r=0 b r (P )q r , (69)
where α and the coefficients b r are defined by the concrete choice of the function F P (z). In each term, we can integrate in τ 2 = Im τ explicitly with the result in terms of the function

Φ(A, r, l) = F d 2 τ |16q| 2A q r ql = (16) 2A π(2A + r + l) 1/2 -1/2 cos(π(r -l)x)e -π √ 1-x 2 (2A+r+l) dx. (70)
For (55), we have the sum of six integrals of form (68), and we obtain the series

I 4 (b) = πR ∞ L=0 A (1,e) L + A (1,o) L + A (2,e) L + A (2,o) L + A (3,e) L + A (3,o) L , (71)
where in the last sum

A (1,e) L = ∞ 0 r 2 (P )dP L k=0 b (1,e) k (P )b (1,e) L-k (P )Φ P 2 /2 + Q 2 /8 -1/2, k, L -k , A (1,o) L = ∞ 0 r 1 (P )dP L k=0 b (1,o) k (P )b (1,o) L-k (P )Φ P 2 /2 + Q 2 /8, k, L -k , A (2,e) L = ∞ 0 r 2 (P )dP L k=0 b (2,e) k (P )b (2,e) L-k (P )Φ P 2 /2 + Q 2 /8 -1/2, k, L -k , A (2,o) L = ∞ 0 r 1 (P )dP L k=0 b (2,o) k (P )b (2,o) L-k (P )Φ P 2 /2 + Q 2 /8, k, L -k , (72)
A (3,e) L = ∞ 0 r 1 (P )dP L k=0 b (3,e) k (P )b (3,e)
L-k (P )Φ P

2 /2 + Q 2 /8 -1, k, L -k , A (3,o)
L = ∞ 0 r 2 (P )dP L k=0 b (3,o) k (P )b (3,o) L-k (P )Φ P 2 /2 + Q 2 /8 -1/2, k, L -k .
Each term in (72) is suppressed by a factor max F |q| 2L , and the series in L converges very rapidly in practice. We found that it suffices to sum up to L = 4 to reach the three-to four-digit precision (see Table 1). In the case (1, 3), we numerically integrate in the same way although we now have 18 integrals of form (68). In this case summing up to L = 4 we were able to reach the two-to three-digit precision (see Table 2). In Figs. 1 and2, the results of numerically evaluating integrals (29) and (31) are shown as circles while the lines correspond to the exact result.

b

Σ (1,1) (b) num. Σ (1,
In the main text, we use the brief notation F 00 (∆, z) = F e a a a a ∆ z , F 01 (∆, z) = F 0 a a a a ∆ z , (81)

F 10 (∆, z) = F e â â a a ∆ z , F 11 (∆, z) = F 0 â â a a ∆ z , (82)

( 1 , 3 )Figure 1 :

 131 Figure 1: Direct numerical evaluation of reduced integral (29) (circles) versus the exact formula (continuous line).

2 Σ ( 1 , 3 )

 213 (b)

Figure 2 :F e,o a 1 a 3 a 2 a 4 ∆ 2 N ∈Z,Z/ 2 z N 12 NF e,o â1 â3 a 2 a 4 ∆ 2 NF e,o â1 a 3 â2 a 4 ∆ 1 NF e,o â1 a 3 a 2 2 N

 242212424122 Figure 2: Direct numerical evaluation of reduced integral (31) (circles) versus the exact formula (continuous line).

F 20 (F

 20 ∆, z) = F e â a â a ∆ z , F 21 (∆, z) = F 0 30 (∆, z) = F e â a a â ∆ z , F 31 (∆, z) = F 0 â a a â ∆ z .(84)

Table 1 :

 1 Numerical data for Σ(1,1) (b) at µ = 1.

	1) (b) exact

Table 2 :

 2 Numerical data for Σ

Acknowledgments The author thanks the LPTA of University Montpellier II, for the warm hospitality and stimulating scientific air. Special gratitude is extended to V. Fateev and A. Neveu for encouraging the interest in this work. He is grateful to A. Belavin for the useful discussions. This work was supported by the Russian Foundation for Basic Research (Grant No. 08-01-00720) and also by the RBRF-CNRS project PICS-09-02-91064. Part of the calculations were performed while visiting the Mathematical Department of Kyoto University in January 2008. He acknowledges the hospitality of this division and personally T. Miwa.

The remaining six correlation functions not written explicitly are obtained from ( 46) and (47) by complex conjugation. Using the introduced notation, we can rewrite the integrals under consideration in the compact forms

and

Bulky expression (39) has a remarkably compact and clear structure in terms of the conformal blocks. We again note that in (48) and (49), we respectively assume the different values of external conformal dimensions in the Liouville sector ∆ b and ∆ 2b .

The Modular Integral

It turns out efficient [START_REF] Al | Gravitational Yang-Lee model: Four point function[END_REF] to use elliptic transformations in the integration. We use the standard map

where the complete elliptic integral of the first kind is

and

where q = e iπτ (53) and

Integral (48) becomes

where

Similarly, for (49), we have

We now define the conformal blocks more explicitly,

Here, the first argument of the symmetric conformal blocks defines the internal conformal dimension. The first lower index corresponds to one of the four basic types of conformal blocks we consider with respect to the set of external fields (see Appendix A); the second index is 0 if the corresponding block with the given internal conformal dimension is "even" and 1 if it is "odd". In the Liouville sector,

1 (z) = F L 30 (P, z).

11

For the first integral, the complicated expressions for the Liouville structure constants give

The special function Υ b (x) is the standard element of the LFT (see [START_REF] Zamolodchikov | Conformal bootstrap in Liouville field theory[END_REF] for the definition and properties). Explicitly,

where we use "shift" relations (see [START_REF] Zamolodchikov | Conformal bootstrap in Liouville field theory[END_REF]) for the last upsilon function to extract the additional factor (γ( bQ 2 )) 4 . This allows improving the accuracy of the comparison with the analytic result, which contains the same factor with a singularity at b = 1. The P dependent parts are

where we again use the shift relations to move the arguments of all upsilon functions inside the strip [0, Q] where the standard integral representation is applicable and we use the notation

.

For the second integral, we analogously find the explicit expressions

Appendix A. Conformal Blocks and the Chain Vectors For definiteness, we use the notation for the Liouville sector, though the results concerning the conformal block are universal (i.e., are independent of the sector). Schematically, the contribution of the given conformal family in the four basic OPE can be written as

where the so-called chain vectors

for k > 0. The normalization of the chain vectors chosen in this text is determined by the requirements

Relations (74) and (75) are equivalent to the linear problem for the coefficients determining the chain vectors in terms of the Virasoro basis vectors of the same level. These systems can be solved numerically up to a rather high level.


A El Kaabouchi

Q A Wang

C J Ou

J C Chen

G Z Su

A Le Méhauté

A counterexample against the Lesche stability of a generic entropy functional

We provide a counterexample to show that the generic form of entropy

p g p S ) ( ) (
is not always stable against small variation of probability distribution (Lesche stability) even if g is concave function on [0,1] and analytic on ]0,1]. Our conclusion is that the stability of such a generic functional needs more hypotheses on the property of the function g, or in other words, the stability of entropy cannot be discussed at this formal stage.

If a physical quantity observable is continuous function of the characteristic variables of motion such as time, configuration, velocity, energy, probability distribution etc., this quantity, and of course its mathematical definition, should undergo smooth variation for the system in smooth motion. Such a condition can be referred to as experimental robustness or observability and can be used to examine the validity of mathematical. An example of such quantity is the entropy which is characteristic of probabilistic uncertainty in stochastic dynamics and considered as continuous function of probability distribution. From this consideration, the mathematical definition of Shannon entropy, Renyi entropy and Tsallis entropy has been reviewed in [1] and [2], in which the robustness was called stability against small perturbation of probability or subsequently Lesche stability after Lesche who initialized the discussion by defining a restrictive uniform continuity criterion [1]. That stability criterion was afterwards used to examine many other quantities including the kappa-entropy [3], the stretched exponential entropy [3], the quantum group entropy [3], the incomplete entropy [START_REF] Ou | Lesche stability of incomplete statistics[END_REF][5] and the escort expectation [START_REF] Abe | Instability of q-averages in nonextensive statistical mechanics[END_REF], and has also been extended to a generic forms of entropy

∑ = i i p g p S ) ( ) (
in [7] where the authors advocated for the stability of such a formal definition if g is analytic and concave function of probability distribution p i . This conclusion would be important and very useful if it was well founded. However, we found that a key argument, the calculation from Eq.(2.10) to Eq. (2.15) of [7], would contain some mistakes.

As a matter of fact, keeping the Taylor expansion of entropy only to first order in these equations leads to vanishing entropy variation hence invalids the pretended proof of stable S(p). Although this conclusion can be saved by keeping the term of second order in the expansion, the omission of higher order terms cannot be justified. In this letter, we provide a counterexample in order to show that the general conclusion drawn in [7] does not hold. The stability of entropy in that generic form may needs more restrictions on the entropic functional expression. We suppose that:

i) f is of class 2 C on ] , ] b a , increasing on ] , [ b a , concave on ] , [ c a and convex on ] , [ b c ; ii) g is of class 2 C on )] ( ), ( ] b f a f , concave on )] ( ), ( [ b f a f increasing on )] ( ), ( [ c f a f , decreasing on )] ( ), ( [ b f c f . Then f g o is concave on ] , [ b a . Proof a) Let ] , [ , c a y x ∈ and ] 1 , 0 [ ∈ α . We have (
)

) ( ) 1 ( ) ( ) 1 ( y f x f y x f α α α α - + ≥ - + because f is concave on ] , [ c a , and
( ) ( ) (
)

) ( ) 1 ( ) ( ) 1 ( y f x f g y x f g α α α α - + ≥ - + because g is increasing on )] ( ), ( [ c f a f .
On the other hand,

( ) ( ) ( ) ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( y f g x f g y f x f g α α α α - + ≥ - + because g is concave on )] ( ), ( [ b f a f It follows that, ( ) ( ) ( ) ( ) ) ( ) 1 ( ) ( ) 1 ( y f g x f g y x f g α α α α - + ≥ - + .
We deduced thus,

( )( ) ( )( ) ( )( ) y f g x f g y x f g o o o ) 1 ( ) 1 ( α α α α - + ≥ - + Consequently, f g o is concave on ] , [ c a or equivalently ( ) 0 ' ' ≤ f g o on ] , ] c a b) Let ] , [ , b c y x ∈ and ] 1 , 0 [ ∈ α . We have ( ) ) ( ) 1 ( ) ( ) 1 ( y f x f y x f α α α α - + ≤ - + because f is convex on ] , [ b c , and
( ) ( ) (
)

y x f g y f x f g ) 1 ( ) ( ) 1 ( ) ( α α α α - + ≤ - + because g is decreasing on )] ( ), ( [ b f c f , On the other hand, ( ) ( ) ( ) ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( y f g x f g y f x f g α α α α - + ≥ - + because g is concave on )] ( ), ( [ b f a f , It follows that, ( ) ( ) ( ) ( ) ) ( ) 1 ( ) ( ) 1 ( y f g x f g y x f g α α α α - + ≥ - + We deduce thus ( )( ) ( )( ) ( )( ) y f g x f g y x f g o o o ) 1 ( ) 1 ( α α α α - + ≥ - + Consequently, f g o is concave on ] , [ b c or equivalently ( ) 0 ' ' ≤ f g o on ] , [ b c . c) By a) ( ) 0 ' ' ≤ f g o on ] , ] c a , and b) ( ) 0 ' ' ≤ f g o on ] , [ b c , we conclude that ( ) 0 ' ' ≤ f g o on ] , ] b a . Thus f g o is concave on ] , [ b a .

Lemma 2. Let h be the function defined on

] 1 , 0 [ by 0 ) 0 ( = h and       - = x x h ln 1 sin ) ( π if 0 ≠ x .
Then h is concave and verifies 2

1 1 1 lim 2 =             +∞ → x h x h x . Proof Let f be the function defined on ] 1 , 0 [ by 0 ) 0 ( = f and x x f ln 1 ) ( - = π if 0 ≠ x ,
and g be the function defined on ] , 0

[ π by x x g sin ) ( = .
We have:

f is continuous on ] 1 , 0 [ , f is of class 2 C on ] 1 , 0 ] . For all ] 1 , 0 ] ∈ x , ( ) 0 ln 1 ) ( ' 2 > - = x x x f π , thus f is increasing on ] 1 , 0 [
and we conclude that ( )

] , 0 [ )] 1 ( ), 0 ( [ ] 1 , 0 [ π = = f f f . For all ] 1 , 0 ] ∈ x , ( ) 3 2 ln 1 ln 1 ) ( ' ' x x x x f - + = π , thus ' ' f is negative on       e 1 , 0
and ' ' f is positive on

      1 , e 1 , consequently, f is concave on       e 1 , 0
and

f is convex on       1 , e 1 .
On the other hand, g is of class 2

C on ] , 0 [ π , g is concave on ] , 0 [ π , increasing on             =       e 1 ), 0 ( 2 , 0 f f π and decreasing on             =       ) 1 ( , e 1 , 2 f f π π . From lemma 1, f g h o = is concave on ] 1 , 0 [ . Finally, 2 1 1 1 lim 2 =             +∞ → x h x h x , because we have ( ) π π π π π π x x x x x x x h x h ln 1 ln 2 1 ln 1 sin ln 2 1 sin ) / 1 ln( 1 sin / 1 ln 1 sin 1 1 2 2 + + ≈       +       + =         -         - =             ∞ .
Thus lemma 2 is proved.

Lemma 3. Let g be a function continuous on

] 1 , 0 [ with 0 ) 1 ( ) 0 ( = = g g and g is of class 2 C on ] 1 , 0 ] with 0 ' ' < g on ] 1 , 0 ] .
The entropy associated with g is defined on ( )

U * ∈ = N n n f E [0,1] by ( ) , , , 1 f n E p p p ∈ = ∀ L ∑ = = n i i p g p S 1 ) ( ) ( . i) For all * ∈ N n ,       = ∈ ∑ = 1 , [0,1] ), (
( ) n n x x x [0,1] , , 1 ∈ = L , ∑ = = n i i x g x S 1 ) ( ) ( is continuous on n [0,1] and       = ∈ ∑ = 1 , [0,1] 1 n i i n x x is a compact of n [0,1]
n n p p p [0,1] , , 1 ∈ = ∀ L ,       - - = Φ ∑ ∑ = = 1 ) ( ) (
{ } n i , , 2 , 1 L ∈ , ( )
0 ) 0 ( = ∂ Φ ∂ p x i n .
On the other hand, for all

{ } n i , , 2 , 1 L ∈ , ( ) α - = ∂ Φ ∂ ) ( ' i i n p g p x .
Because ' g is strictly decreasing on ] 1 , 0

] , let x be the unique element of ]

= ) ( ' x g , we have { } n i , , 2 , 1 L ∈ ∀ , ( 1 , 0 ] such that α
x p p x i i i n = ⇔ = ∂ Φ ∂ ) 0 ( ) 0 ( 0 Thus, )
= ∑ = n i i p implies that n x 1 = . Consequently, ( ) ( ) ( )       = = = = ∑ = n ng p ng p g p S S n i i n n 1 ) 0 ( 1 1 ) 0 (
. Thus lemma 3 is proved.

Definition 4.

A function S defined on

U * 1 , [0,1] 1 N ∈ =       = ∈ = ∑ n n i i n p p A is stable (Lesche stability) if and only if 0 > ∀ε , 0 > ∃δ , * N ∈ ∀N ,       = ∈ ∈ ∀ ∑ = 1 , [0,1] ' , 1 N i i N p p p p , ε δ < - ⇒ < - max , 1 ) ' ( ) ( ' N S p S p S p p .

Proposition 5.

Let h be the function defined on ] 1 , 0 [ by

0 ) 0 ( = h and       - = x x h ln 1 sin ) ( π if 0 ≠ x . Then h is concave, h is of class ∞ C on ] 1 , 0 ] , continuous on 0 and 0 ) 1 ( ) 0 ( = = h h .
We consider the function S defined on

U * 1 , [0,1] 1 N ∈ =       = ∈ = ∑ n n i i n p p A by * N ∈ ∀n , ( ) n n p p p [0,1] , , 1 ∈ = ∀ L with 1 1 = ∑ = n i i p , ∑ = = n i i p g p S 1 ) ( ) ( .
Then S is not Lesche stable.

Proof

By lemma 2, we can find * 0

N ∈ N such that for all 0 N n ≥ , ( ) 4
1 ) / 1 ( / 1 1 2 > - n h n h n n . For 0 8 1 > = ε , for all 0 > δ , there is * 0 , 1 2 max N ∈         +       = N E N δ , let ' , p p defined by ( ) N i i p p , , 1 L = = , i i p 1 δ = ( ) N i i p p , , 1 ' ' L = = , 2 ' 1 1 1 1 N N p + - = , 2 ' 1 N p i = if 1 > i .
We have

1 1 ' 1 = = ∑ ∑ = = W i i N i i p p , δ < ≤       - = - +       + - - = - ∑ = N N N N N N p p N i 2 1 1 2 1 0 1 1 1 1 ' 2 2 2 2 1
, and

ε > ≥             - ≥             - +       + - =             - -       + - - =                     - +       + - - = - ∑ = 4 1 1 1 ) 1 ( 1 1 ) 1 ( 1 1 1 1 1 ) 1 ( 1 1 1 1 1 ) 0 ( 1 1 1 ) 1 ( ) ' ( ) ( 2 2 2 2 2 2 2 2 max , N Nh N h N N Nh N h N N N h N Nh N h N N N h N Nh N h h N N h h S p S p S N i N Consequently, S is not Lesche stable.
In summary, a counterexample is given to show that the entropy

∑ = i i p g p S ) ( ) (
is not always stable or robust even if g is concave function on [0,1] and analytic on ] 1 , 0 ] . To our opinion, the stability of such a generic form of entropy needs more hypotheses on the property of the function g, or the stability of entropy cannot be discussed at this formal stage.

We would like to indicate in passing that the present counterexample can also be used to review the conclusion of [3] about the stability of a generic definition of entropy, given by Eq.( 6) of [3] based on maximum entropy principle (maxent). In fact, if one replaces the or f(t) by g' -1 (t) which satisfies the properties of f(t), then the function B(•) defined in Eq.( 18) of [3] has a limit value of 1 instead of zero as obtained in [3]. Note however that the eventual failure of establishing stability for this formal entropy has nothing to do with the validity of that definition which we esteem on the contrary a logical and useful proposal based on maxent. The stability problem can be addressed after an entropy is derived from this definition, as has been done in [3] for several functional. This same statement also applies to the definition, proposed by mimicking the second law entropy for reversible process, of a variational entropy (called varentropy) dx x d dS -= for any single random variable x [START_REF] Wang | Probability distribution and entropy as a measure of uncertainty[END_REF]. Varentropy has been shown to be maximized by the distributions used to derive it's functional [START_REF] Wang | Probability distribution and entropy as a measure of uncertainty[END_REF]. For this purpose, it coincides with the entropy of Eq.( 6) in [3].

Lemma 1 .

 1 Let f be a function continuous on an interval ]

  multiplier method, we consider the function n Φ defined on n [0,1] by :

Acknowledgements

We thank Professor Abe for valuable discussions. Two of the authors (Chen and Ou) thank the Région des Pays de la Loire of France for the grants N° 2007-6088 and N°2007-11953, Ou also thanks Huaqiao University, People's Republic of China for the Science Research Fund (No. 07BS105). Su thanks ISMANS for the support of his participation in the research works around this subject.


Espen Gaarder Haug
email: espenhaug@mac.com.orcid

Can Rest-Mass Acceleration Explain Galaxy Rotation Without Dark Matter?

Keywords: galaxy rotation, minimum acceleration, rest-mass acceleration, Planck mass acceleration

Standard Newton theory, as well as general relativity theory, is not able to predict galaxy rotation without the hypothesis of dark matter. Even after considerable e↵ort, no direct observations of dark matter have been made. This could simply mean dark matter cannot be detected directly or that it is extremely hard to detect due to very weak interactions, but it could also mean there exist other explanations for the observed galaxy rotation curves currently not accounted for in the ⇤-CDM model. Other alternative theories should therefore also be investigated until we have the "final" answer. Here we demonstrate that the matter definition given by collision space-time leads to a minimum acceleration somewhat similar to that of modified Newton dynamics (MOND), but di↵erent in both size and mathematical function. Further, MOND does not have a deeper theory for why there should be a minimum acceleration. We introduce rest-mass acceleration and explain why this is naturally to be expected in matter, and how it is also likely linked to Planck acceleration. We are testing the modified acceleration predicted by this model against 2,793 data points from 153 galaxies from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database and it seems to give very accurate predictions without the need for dark matter.

Introduction

In 1983, Milgrom [START_REF] Milgrom | A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis[END_REF] published a theory he called modified Newton dynamics (MOND) that basically states:

G Mm R 2 = m ⇣ v 2 r ⌘ 2 a0 (1)
where a0 basically is an "arbitrary" constant that gives an extra degree of freedom to calibrate the model to observational data. The ao parameter is seen as a minimum acceleration, and has been calibrated to be a0 = 1.2 ⇥ 10 10 m/s 2 . What is of particular interest here is that this simple adjustment of the Newton gravity formula makes it fit galaxy rotations very well without any need for dark matter. The Milgrom theory has been criticized for not being a very deep theory and to be more of a curve-fitting model. It does not have any solid foundation or explanation for why there should be such a minimum acceleration. Still the Milgrom model could be important, as it was the first model giving us a hint to where the answer to the missing dark matter can be found, it could be linked to minimum acceleration. If that is the case, the dark matter needed in the standard ⇤-CDM model could simply be a fudge factor because, for some unknown reason, it is missing a minimum acceleration.

Several di↵erent minimum acceleration models have been suggested in recent years that, unlike the Milgrom model, also give possible explanations for why there should minimum acceleration. In 2021, Tonin [START_REF] Tonin | MOUND: An alternative to Milgrom's MOND[END_REF], as well as Haug [START_REF] Haug | Can the standard model predict a minimum acceleration that gets rid of dark matter?[END_REF], proposed a slightly di↵erent model that suggests the mass of the observable universe should also be taken into account. Also a model suggested in 2006 by Brownstein and Mo↵at [START_REF] Brownstein | Galaxy rotation curves without non-baryonic dark matter[END_REF] gives a minimum acceleration linked to the cosmological scale. Yet another minimum acceleration model is the so-called quantized inertia that McCulloch [START_REF] Mcculloch | Galaxy rotations from quantised inertia and visible matter only[END_REF] tested against galaxy rotation curves in 2017, and this model also predicts rotation curves very accurately without the need for dark matter. Quantized inertia modifies the standard inertial mass in a way that links the adjustment to the scale of the diameter of the universe. So, these four models all link the minimum acceleration to the cosmic scale. These are all interesting minimum acceleration models that should be investigated further. However, in this paper, the focus is on a new minimum acceleration model that indicates that matter also could have rest-mass acceleration. That is, this model claims such a minimum acceleration comes from the quantum scale internally in matter, rather than from interactions with cosmological scales; something we will look at in the next sections.

Collision-Space Time Minimum Acceleration from Rest Mass Acceleration

In collision space-time [6,[START_REF] Haug | Quantum Gravity Hidden In Newton Gravity And How To Unify It With Quantum Mechanics[END_REF] everything consists of only indivisible particles and empty space; this is very similar to Newton's [START_REF] Newton | Philosophiae Naturalis Principia Mathematica[END_REF][START_REF]I Newton[END_REF] corpuscular theory. These indivisible particles move at the speed of light when not colliding and are massless, and are then the building blocks of photons. The wavelength of the photon is then simply the distance between such indivisible particles traveling after each other so also in this theory, they have a wave-particle duality. This a very di↵erent interpretation than the one found in standard physics.

When a photon collides with another indivisible particle, this collision itself is defined as mass: the most essential o↵ all masses, as it makes up all observable masses. Calibration to observable gravitational phenomena, as is described in the two references above, shows that such a collision remarkably lasts for the Planck time and also that the mass of this collision corresponds to the Planck mass and, further, the diameter of the indivisible particle is the Planck length. So collision space-time is a quantum theory directly linked to the Planck scale, and we have recently published a quantum gravity field equation, rooted in the Planck scale, and that gives a series of predictions of observable gravity phenomena identical to standard gravitational theory, but there are also a few di↵erences, such as the radius where the escape velocity is c is half that of the Schwarzschild radius [START_REF] Haug | A new full relativistic escape velocity and a new Hubble related equation for the universe[END_REF][START_REF] Haug | Unified quantum gravity field equation describing the universe from the smallest to the cosmological scales[END_REF]. This is not, however, easily testable. The main focus in this paper is on whether our theory can also predict galaxy rotation, something that is testable.

The Planck mass is much larger than any known particle or even any known atom, but still, it makes up all known particles, even the electron, in this theory. This is possible since the Planck mass, which is the collision between two indivisible particles, only lasts the Planck time before again dissolving into non-mass (internal energy). Inside all matter, our theory suggests, collisions between indivisible particles happen at the reduced Compton frequency f = c ¯ , where ¯ is the reduced Compton [START_REF] Compton | A quantum theory of the scattering of x-rays by light elements[END_REF] wavelength, ¯ = h mc . This is more than pure speculation, as recent research indicates that matter can be seen as Compton clocks, see [6,[START_REF] Lan | A clock directly linking time to a particle's mass[END_REF][START_REF] Dolce | On the Compton clock and the undulatory nature of particle mass in graphene systems[END_REF]. The question is then what exactly happens at the Compton periodicity. We claim a collision between indivisible particles making up the mass is happening at this interval, but that the collision itself only lasts the Planck time. To give a practical example, in an electron there is an internal collision between two indivisible particles the following number of times per second:

fe = c ¯ e ⇡ 7.76 ⇥ 10 20 (2)
That is, the electron is a Planck mass 7.76 ⇥ 10 20 times per second, but since each of these Planck masses only last the Planck time, this gives the correct electron mass (and also correct output dimensions):

me = femptp = c ¯ e mptp ⇡ 9.11 ⇥ 10 31 kg (3)
What is interesting here in relation to galaxy rotations is that, in each of these events, the indivisible particles making up the electron stand still for the Planck time in a collision but then, after the Planck time, move away from each other at the speed of light. This means that an acceleration from zero to the speed of light is happening every time such a collision takes place. This is, interestingly, identical to the Planck acceleration over the Planck time, that indeed leads to an object at rest going to speed c. We will soon get back to the Planck acceleration, but what is important at this stage is to understand this acceleration as something that happens internally in matter due to the building blocks making up the mass. One can think of two indivisible particles each traveling over the reduced Compton wavelength at the speed of light but, at each reduced Compton time periodicity, they are colliding and standing still for the Planck time. After this Planck time, they again move away from each other at the speed of light. Since this all happens internally inside matter in this model, we can coin it rest-mass acceleration.

In standard theory, there is no such thing as rest-mass acceleration, but we are all familiar with rest-mass energy, so that there can also be a rest-mass acceleration should not be excluded before being investigated properly. It is easy to be overly negative towards totally new concepts, but we ask the reader to carefully consider this before rejecting it.

As planets, stars, and galaxies in this model consist of baryonic matter, and since protons (and neutrons) make up almost all of baryonic matter, then it is the rest-mass acceleration of the proton that seems relevant, and it is given by:

ar = c 1s lp c ¯ P c = c 1s lp ¯ P ⇡ 2.3 ⇥ 10 11 m/s 2 (4)
where

lp ¯ P
is simply the percentage of total time we have rest-mass acceleration of c internally in the proton. The division by one second can be mistaken for arbitrary, but it is simply because acceleration is normally expressed as meters per second squared. So to divided by 1 second is only relevant when working with SI units, that is the speed of light c is also expressed in meters per second. In the natural unit system where one set G = h = c = 1, then the minimum acceleration would simply be lp ¯ , that is dimensionless. However, it is most normal and perhaps practical to work in SI units here, so we will stay to that in the rest of the paper. That is, even if the acceleration is enormous and happens an enormous number of times per second, the fraction of the total time inside, for example, a second that such enormous acceleration happens in, is very small. The calculated minimum rest-mass acceleration can be seen almost as an average between most of the time, when there is no acceleration, and the many Planck time periods when there is enormous acceleration internally in matter.

Photon-photon collisions happen at the Compton frequency inside the proton. That photon-photon collisions can create mass has been predicted even by standard theory for many years; see, for example, [START_REF] Pike | A photon-photon collider in a vacuum hohlraum[END_REF]. What is di↵erent in this model is that this is how matter is created and maintained: by photon-photon collisions internally in matter at the Compton periodicity.

This acceleration of approximately 2.3 ⇥ 10 11 m/s 2 can be seen as the minimum acceleration in a proton, and since almost all matter is built from protons or neutrons (and we assume approximately the same mass and Compton wavelength as the proton), then this is also the minimum acceleration for large masses, such as planets and stars and even galaxies. This means we must likely have the following relation:

m(a ar) = G Mm R 2 a = G M R 2 + ar a = G M R 2 + c 1s lp ¯ (5)
That is, the total observed acceleration is adjusted to allow for the fact that there likely is a rest-mass acceleration. A good starting point is to try to find out if this makes any sense by testing the predictions from this simple model and comparing them with actual observations. This is the topic of section 4, but before that we will shortly look at Planck mass acceleration and how it seems linked to the minimum acceleration in our theory.

3 Is Planck mass acceleration linked to the missing rest-mass acceleration?

Max Planck [START_REF] Planck | Natuerliche Masseinheiten[END_REF][START_REF] Planck | Vorlesungen über die Theorie der Wärmestrahlung[END_REF] assumed there were three important universal constants: the Newton gravitational constant, the Planck constant, and the speed of light. Based on dimensional analysis, he came up with a unique length:

lp = q Gh c 3 , time; tp = q Gh c 5 , mass; mp = q hc G , temperature; Tp = q hc 5
k b G , today known as the Planck units. Scarpetta [START_REF] Scarpetta | [END_REF] introduced what today is known as the Planck acceleration (see also [START_REF] Caianiello | Maximal acceleration and sakharov's limiting temperature[END_REF]):

ap = r c 7 Gh (6)
which corresponds to a = c 2 lp . Also, Fala and Landsberg [START_REF] Falla | Black holes and limits on some physical quantities[END_REF] in 1994 thought the Planck acceleration was the maximum acceleration. As nothing can move faster than the speed of light, then the Planck acceleration cannot last longer than the Planck time, as we have:

aptp = r c 7 Gh tp = c 2 lp lp c = c (7)
However, nothing with mass can move at the speed of light as it would take infinite energy to bring a mass to speed c as can be easily seen from Einstein's relativistic mass formula, E = mc 2 = mc 2 / p 1 v 2 /c 2 , that when setting v = c, needs infinite energy. Electromagnetic radiation (photons) is the only thing that always moves at the speed of light so, in standard theory, it would be meaningless to talk about Planck acceleration for light. However, in our theory, when the Planck mass is indeed created or defined as the collision between two light particles, (indivisible particles) that collide for one Planck time and then dissolve into light again, the Planck mass acceleration fits very well. The Planck mass acceleration is therefore related to the internal acceleration that happens inside matter; it is a rest-mass acceleration, but it only lasts a very small fraction of time, as it only happens for the Planck time, and this again happens at the reduced Compton frequency.

Testing our theory for galaxy rotation

To test our theory, we use 2,793 data points from 153 galaxies from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database. SPARC is likely the largest galaxy database to date with rotation curves for galaxies. In Figure 1, we can see that our model, only using baryonic matter, fits these observations very well. Our model is the red line, and the observations are the black dots. We see the standard model, the green line, is far o↵, which is why the standard model needs to introduce the hypotheses of dark matter to fit the data. The standard model needs both dark matter and a distribution of this dark matter to make it fit the observation.

The figure also shows the MOND model as the yellow line. McGaugh et al [START_REF] Mcgaugh | The radial acceleration relation in rotationally supported galaxies[END_REF] have shown that the following function fits the observational data very well: The MOND model gives a better fit to observation than our suggested model, but we must bear in mind that MOND has a free parameter that is optimized to observed data. Our model has no free parameters here, but it is blind and simply rooted in an assumption about matter and energy at the depth of reality. We have not taken into account any relativistic e↵ects, which could be an additional adjustment to look into. One would then have to take into account the relativistic Compton wavelength in the minimum acceleration calculation; see [START_REF] Haug | Derivation of a relativistic Compton wave[END_REF]. Still, even this very simple model seems to give a very good fit. So we think this approach should be interesting enough to encourage further research to discern whether there could be something to the idea of rest-mass acceleration, which is basically an internal acceleration in matter.

Conclusion

We have shown that the Planck acceleration could be linked to what we will call rest-mass acceleration. The mass definition in collision-space-time theory leads to such a rest-mass acceleration, as all matter here consists of indivisible particles (building blocks of photons) moving back and forth over the reduced Compton wavelength of the particle in question, to then collide for the duration of the Planck time. This rest-mass acceleration leads to a minimum acceleration in all matter and seems to lead to predictions that fit quite accurately with observations from galaxy rotation curves without the need to introduce dark matter. This is, therefore, yet another minimum acceleration theory. However, compared to MOND that also predicts galaxy rotation curves very well without dark matter, this theory provides a deeper explanation for why this is the case. Naturally, the model should not be taken for granted, but the idea of rest-mass acceleration should be worth further investigation by more researchers.

  g bar is the Newton (and GR) acceleration predicted from baryonic matter only and gmin is an ad-hoc minimum acceleration. This minimum acceleration corresponds to the MOND minimum acceleration of approximately 1.2 ⇥ 10 10 m/s 2 .

Figure 1 :

 1 Figure 1: Galactic accelerations in Collision-Space-Time compared to data and other models. Exactly 2,793 individual data points for 153 SPARC galaxies are shown in black. Predictions by general relativity theory, when not taking into account dark matter, are shown in green. The yellow line is MOND and the red line is collision-space-time. The red line takes into account rest-mass acceleration as discussed in this paper.