Download Full-Text

JHEP04(2020)072 ). in terms 7 Springer ( O March 4, 2020 April 14, 2020 March 20, 2020 : : : dimensions. Using Received Published − Accepted = 4 d -expansion of the Wilson-Fisher (1) symmetric model to O Published for SISSA by https://doi.org/10.1007/JHEP04(2020)072 model -expansion in ) in terms of the beta function coeﬃcients. The 1 ( -expansion, Lagrange inversion and . 3 1910.12631 The Authors. c Conformal Field Theory, Renormalization Group

We discuss properties of the , [email protected] -Origins, University of Southern Denmark, 3 We finally use our general results to compute the values for the Wilson-fisher fixed Campusvej 55, Odense M 5230, Denmark E-mail: CP -expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at Open Access Article funded by SCOAP Keywords: ArXiv ePrint: by the facial structurevalue of of anomalous associahedra. dimensions at We theof then Wilson-Fisher the fixed write coefficients point down order of by an the order beta exact in function expression and for anomalouspoint the dimensions. coupling and critical exponents for the scalar Lagrange inversion we writefixed down an point exact coupling expression order for by the order value of in each the order Wilson-Fisher depends onproperties the of beta Lagrange function inversionfixed we coefficients point then via coupling argue equally Bell well that polynomials. can the be viewed Using as certain a geometric expansion which is controlled Abstract: Thomas A. Ryttov Properties of the associahedra and the O JHEP04(2020)072 ]. ]. 13 14 . In N and using N ] long stood as 9 – ) in the attached 4 5 ] for general ( 12 5 = 1 has appeared [ O N . At a fixed point these dimensions. But recently the dimensions is extremely rich in − 9 − = 4 d = 4 9 d ) 7 in ( 2 – 1 – ) O i φ ] were brought to the end in [ i 3 φ 11 , ] as well as the five loop calculation [ 3 10 , 2 ]. It suffices to say that further understanding of the theory dimensions to higher and higher order in the coupling. The 1 the one loop beta function possesses a non-trivial zero [ = 3 it describes the critical behavior of isotropic ferromagnets − N ]. When combining these results with resummation techniques they = 4 12 d 1 ). For ) symmetric theory ( ) symmetric model and its close relatives for which the reader can find a 12 = 2 the theory describes the superfluid transition of liquid helium close to N N ( (1) symmetric model to XY ( N O O O -expansion as a geometric expansion: associahedra = 0 it coincides with the critical behavior of polymers (self-avoiding walks) while for -point ( At sufficiently small For these reasons much effort has been put into computing various renormalization λ N = 1 it lies in the same universality class as liquid-vapor transitions and uniaxial magnets This is the Wilson-Fisher fixedassociated point. critical At exponents this are fixedcritical point exponents expressed anomalous are as dimensions scheme power andthe independent their series six physical in loop quantities. calculationsMathematica For file they in general can [ be found in analytical form to six loop calculations initiated inaddition an [ impressive seven loop calculationThese in the now special case constitute of functions the which highest include loop theanomalous order beta dimension. calculations function, of the the field anomalous renormalization dimension group and the mass is of great importance. group functions in three and four loop calculationthe [ state-of-the-art of higher loop calculations in (Ising). For the (Heisenberg). There are manythe more examples scalar of critical phenomenamore that complete are discussion described in by [ The scalar the number of physicalfor phenomena that it canN accommodate and describe. For instance 6 Discussion 1 Introduction 3 The 4 Anomalous dimensions and critical5 exponents Scalar Contents 1 Introduction 2 The Wilson-Fisher fixed point JHEP04(2020)072 , i 2 1 = x L i c 2 /c , . As d 1 =1 , L ∞ i 1 − = 1 and show P 1 /c d . The zero = 4 i x for three sets d = 1 ) = i c are each others x 1 x ( d g f =1 in ∞ i i while the Wilkinson to any desired order P x and i 1 i d c + L f a =1 5 i − -expansion it will be more P where ]. + ) = i y x a we press on and discuss how the 37 i ( – d 3 f . The Lagrange inversion theorem 17 y =1 + one might ask whether it is possible to ∞ i . ) of the beta function in 2 i a we derive the value of the Wilson-Fisher P x L i 2 c )) = y ≡ − ) = ( =1 y ∞ i g . Of course one again finds ) i ( ( ]. We also note several recent complimentary x – 2 – g a . The closed form all orders expressions are then f ( P i ˜ f d 16 , ) = =1 and . ∞ i x 15 ( , x P f , etc. There is also a general closed form expression for 5 1 12 = ) where , /c x 7 ) ( x )) = 3 -expansion. ˜ , etc. as above. It is more natural for us to consider Lagrange f is perhaps best illustrated by the fact that in the Earth-Sun c x 1 5 1 ( 3 c f /c ( L ) − g 3 in the power series c 2 2 i 1 c c and d . One can in principle compute the coefficients i 2 − a = (2 L 2 2 i c d , 3 1 d L =1 , ∞ i 3 1 = (2 3 P /c ) and be written as d 2 x c . Although it is impossible to find general solutions in terms of radicals of a quintic , = ) we are looking for should then be given as a power series in the first coefficient ( 3 3 1 ]. ˜ − x f x L The paper is organized as follows. In section The use of Lagrange inversion of power series in physics has a beautiful history in In our case where we study critical phenomena via the Given the formal power series In this work we do not attempt to extend any of the higher loop calculations. We ( /c 38 ˜ = f 2 presented in both a combinatoric and geometric language within the physical setting of at any order which is given in terms of Bell polynomials. This can be found for instance of as c i 2 i -expansion of the Wilson-Fisher fixed point coupling also can be understood as a geometric Microwave Anisotropy Probe (WMAP) occupies fixed point coupling inthat terms it of is the given beta in terms function of coefficients Bell to polynomials. all In orders section in and equation it is possible toa find a single realusing solution the which is Lagrange an inversionlocations expansion theorem. in of the The coefficient successsystem the of Solar the and Lagrange Heliospheric method Observatory to (SOHO) find occupies the critical phenomena and the celestial mechanics andproblem. originates in Here the oneof has study coefficients to of in solve the order a restricted to quintic Newtonian equation find 0 the three-body = location of the three collinear Lagrange points of a − inversion in this way.d Further details will follow below where we will see the coefficients in [ convenient for us tobut phrase equivalent the way. question Insteadwe of of will power asking ask series for for the inversion the in compositional zero a inverse of of slightly different find the coefficients compositional inverses provides a procedure ford doing to thisd and for the first few orders one finds functions of all theneed coefficients to of be the beta calculated).it function The is and main an anomalous feature expansion dimension ofwe in (which the will the still Wilson-Fisher see first fixed coefficient this pointof (which allows is is the us the beta to fact function use that to the all Lagrange orders inversion in theorem to formally find the zero studies of scalar field theories in various dimensions [ will instead entertain ourselvesfixed by point providing value explicit of the andWilson-Fisher coupling fixed closed and point form any to expressions anomalous all orders dimension for in or the critical exponent at the agree well with experiments [ JHEP04(2020)072 k in ˜ so 1 i ... i 0 − g (2.2) (2.3) (2.4) (2.1) b b + k i g ˜ i = c + ) for the i 7 + b ˜ ( ...... O + + to all orders in and 2 3 dimensions it is ˜ ˜ we need to know ∗ 1 0 i b g be until the term g c ) and then expand 2 b ˜ − k + 3 ... 0 = 2.2 g ˜ 0 + ... + g 1 3 = 4 ). If we first look at the b ˜ + g i 1 d ... 1 3 ∗ g b (˜ g 0 + 2 g O + b ˜ 3 ˜ 2 +2 + 2 ’th coefficient g 2 i g 0 g 2 ∗ b g ) and in + 1 + + b ˜ 2 dimensions. Clearly the trivial fixed 2 ) back into eq. ( ˜ ˜ g, 1 ). How large can + (˜ g ( i g ∗ we discuss anomalous dimensions while ∗ 1 β − − g (˜ b g ˜ + 4 2 0 O = g ˜ as + 0 ) = 0 we want to solve as + ]. ˜ = 4 g g 1 +1 . As already explained this is encoded in the , i − g 38 ) amounts to determining the coefficients – 3 – d ∗ = g g )? Once we take the finite number of terms to 1 (˜ = i i ( + ∗ ˜ − i ∗ β i g 1 (˜ . b g − 1) ˜ 1 O i 6 g − − =1 ∞ i 0 i X b ˜ g =1 ∞ i + X =1 ∞ = ( i X g j + ) = − ! l ˜ ˜ (˜ is unity. This is purely a matter of convenience. We want to find 1 ∗ − then clearly only a finite number of terms g − ∗ ) = l l ˜ g g 1 are the standard beta function coefficients. We are interested in the 1 0 = g, − =1 ∞ l ( ) = 0 of the theory in l X g β , ,...

∗ 1 1 , can eventually contribute at =1 g , ∞ l i − ( j . This is the Wilson-Fisher fixed point. ) which is a power series in ˜ b β ˜ P 0 ≤ we compute the fixed point coupling and critical exponents to b = 0 (˜ to arrive at ∗ =1 ∞ i 5 k j X g = = 0 always exists. At one loop the beta function also possesses a non-trivial fixed , + i will no longer contribute at ∗ ∗ ˜ b g g k − ˜ We will be interested in the physics of the Wilson-Fisher fixed point and how to derive An alternative method can again be found in [ 1 1 − (1) model. We conclude in section k 0 = . First write the fixed point equation what terms in theinfinite composite sum set of sumsfor will some contribute at g In order to write an explicit closed form expression for the One way to do thisagain is in to ˜ first plug the ansatz for where for convenience wethat have the chosen coefficient to of normalizethe everything zero as ˜ Lagrange inversion theorem. Finding the power series fixed points point point a closed form exact expression for the fixed point value of the coupling generally given as a formal power series expansion where in section O 2 The Wilson-Fisher fixedWe point write the beta function of the coupling expansion controlled by associaheda. In section JHEP04(2020)072 i . 3 is +1 = 1 ). = j +1 l j , the (2.5) (2.6) (2.7) − i +1 ]. so we − 0 j i +1 j g j j − 39 i − allows to +1) i j = 2 subsets j = 3 subsets. 3 − l i , which can j k +( +1 +1) + 1)! j , . . . , x ... j j + 1. Using the 1 − + i − x 1 j i x j − ( . ’s should of course 1 i i ˜ = 1 elements and − i,j ( g +( ) k i x B ( . +1 ... j = ) j − + +1) j ··· i − j allows to solve for 1 j i k 1 = 1 elements and − and is lowest order in ˜ − j i j ’s and g i i k g ( b ˜ k 1 ˜ ··· 1! 1 x 1 j 0 − g + 1)! k = 10 different ways. All this is precisely , . . . , h . ) should hold for varying ˜ ! ) and the coefficient of ˜ ) g 3 j 0 x ! m x g ( non-empty, non-overlapping subsets. +1 2 1 2.4 = 15 different ways. The partitioning into − j 00 x +1 j that we want to partition into i = 2 subsets contain j − i , h ( m l ! − } ) j i i ! x ) then clearly j + 10 j ( i 0 . Precisely for 1 2 2 in terms of the Bell polynomials ··· h (˜ x ··· ! ,..., i +1 1 in the Bell polynomial ! where 1 0 = 1 subsets contain c O ’th derivative of a composite function. Now j 1 x j g i l l k j i,j − i i – 4 – }} . The coefficient of ˜ a, b, c, d, e j B e = { (1! such subsets and this specific partitioning can be = { )) mx , l +1 x j ) = 15 ). This can be done order by order to any order. In +1 } i,j j ( j where 3 1 d = − h − B i { g ( , . . . , j i are combinatorial in the (already knowing j , x 1 ) , j +1 }} 1 j 2 X } i j 1 ( e − j + c j − { g f , x +1) i ... b , ˜ j j 1 and ! } + X i x =1 j + − 1 0 ( a, b, c j i X j 3 = 5 elements g ... , c, d i {{ 5 i +( =1 { + ’th derivative of a composite function. However the above = ) and we can therefore read off the desired coefficient as j X i , 1 i B ... j j } ! )) = 1 + i (˜ 1 x j ( a, b O +1 1 + 1. If there are h = j ( {{ i j − f i c i ˜ i − ) = j d dx − Also for a discussion on Bell polynomials we refer the reader to [ i +1 g j allows to solve for 2 = 3 elements. This partitioning can be done in − + . So if this is to be of order 2 i elements that we want to partition into k k , . . . , i ˜ i , the term that involves a single factor of ... (already knowing 1 j ways then this is encoded as + 2 − 2 = 1 k g = 2 elements. This partitioning can be done in ˜ m g 1 , . . . , x k 1 k g 1 − x + j ( The fact that the coefficients Bell polynomials are well known in combinatorics. They give a way to encode, in a ˜ ) An example: take a set with The Faàdi Bruno formula is 0 ˜ 3 2 i,j g 0 g B First this can becontain done as three subsets can alsosubsets be contain done as encoded in the Bell polynomial solve for not come as ait surprise is important since to theycan realize are equate that the coefficients the order expansioncoefficient by of in order ˜ eq. in ˜ ( polynomial, the partitioning of ahave a set set into of non-empty, non-overlapping subsets.Each subset Assume we in thebe partition any will contain somedone number in of original elements For completeness we notedi that it Bruno is formula alsoderivation for possible is the to quite find straightforwardBruno the and formula. coefficient perhaps by not using everyone the is Faà familiar with the Faàdi We have here chosen to write the coefficient where the sum is over all sequences we pick up the term of ( multinomial formula we can therefore write the power JHEP04(2020)072 2. . = 1 and − (2.9) j i (2.14) (2.13) (2.10) (2.11) (2.12) = 1 0 ] we can 11 , g ] for an intro- 48 j − so one can always i b 1 0 )! − b i j g and hence the value of ] (see [ j − i 1 loop beta function coef- − 47 ( i − i – b i ˜ since the Bell polynomials g )! 1 j 45 , − ,..., i 4 − 1 g which comes about in the b ˜ i 0 b 41 ( , 1 b − 1! − i 40 3 g b ˜ ! and is linear in 1 i ,j ,..., b ˜ 1 1 1 b ˜ − − i i as a convex polytope of dimension ) = 1! + 6 depends on the 1 i iB 2 2 5. Upon insertion of the six loop beta func- ˜ 1 − j b ˜ i ,j 3 − K g i 1 b ˜ 1) . . . , g ! ] that Lagrange inversion also has a geometric g − – 5 – 0 + 3 ]. This is a check of our formal manipulations. i − in terms of the beta function coefficients − ,..., ). It is linear in 42 g ( 2 2 – 2 1 11 b ˜ iB ˜ b ˜ − =1 2 1 j − i 1 i 40 j = 0 X , . . . , i b ˜ b ˜ 0 1) ! i 1 i g 21 2 , − 5 we find complete agreement with the reported results i b ˜ ( + 5 g − = (1! 1 3 1 , . . . , g 1 − 4 1 =1 − 1 b 1 ˜ 1 i, i j X b ˜ 2 1 b 5 g ˜ − b (2.8) depends on i ˜ ! ,..., B 1). This gives us every coefficient 1 i − − the coefficient i − as a combinatorial exercise we can see it as being controlled i . . . -expansion of the Wilson-Fisher coupling order by order is some = 0 = = 1 = = 2 = = 14 j , g of ˜ i 1 0 1 2 3 4 i i + , g g g g g − c i i i . Order by order we then find g ( g i 0 only and does not receive corrections from higher orders. So for the c b ]. 1 ··· − 1 i 44 − , i + 1) g 43 i ) symmetric model where the beta function is known to six loops [ -expansion as a geometric expansion: associahedra ( i =1 ∞ , . . . , b i X N 0 ( = b O i ˜ ) = There are many ways to realize an associahedron [ Note that for a given ( ∗ g by a specific polytopedron known also as has found an itsamplitudes associahedron. way [ into We other note branches that of recently theoretical physics the including associahe- duction). scattering We will define the associahedron It was expected that the combinatorial function of the betabecome function clear coefficients. to However the itinterpretation. mathematicians has [ only In very recently factcients instead at of each viewing order the in arrangement of the beta function coeffi- calculate the first six coefficients tion coefficients into in the Mathematica file accompanying [ 3 The This is a very conciseis and combinatoric compact in formula the for sense the that fixed it point is value givenficients to in all terms orders of in thescalar Bell polynomials. the fixed point coupling to all orders in where solve for it (alreadysatisfy knowing the following identity term in the sum in general the coefficient JHEP04(2020)072 , 4 3 e 5. ) K K ))). d = 4 = 2 , 3 )) de i i ( ,..., ) K c bc e ( ( , )) b 2 a ( cd = 1 ( a K ) b e i e (( , ) = 5 elements ))) = 3 elements a )) d e ) i )), (( ) cd i ( bc cd d b ( ( de ) ((( ) a b a ( c )) ( d )( ) ) e ) a bc cd bc ab (( (( ) e (( )) b ))) e . ( a ) )) ) a 5 a de de cd cd ( )( , ((( cd K c )( ), ( e )(( . So there are five vertices bc is a point. )( )) b ab (( d ( ab (pentagon). a ( (( de a 2 de ab ))) ( )) ( 4 c and d K )( ))( cd bc K )) 4 ( ( ab ) ( e bc b K ) a bc ( ( ( d . de d , ) ) (b) 5 a a a c ))( 3 c ( ) ) K ) bc K ( ab a de ab , ( ((( )), ( )), ( )( 2 (( c e ) and ( ) is the pentagon. ) K de d ab ( ) 4 (( c cd bc K )( (( b (( ( ab a – 6 – ) a e )) ), ( ), )), e ) cd ( ) de cd e b )) e ( bc ) (( ))) ( is defined to be the empty set. )( b d a ) a c cd ( e cd 1 ( ) ( (line). a )) bc (c) Front and back of b b d ( K ) 3 ((( ab a ), (( a ( . The associahedra bc K a (( cd a ), (( ), )( e )) ) e e )) ) ) de elements then each vertex corresponds to inserting parentheses ab ab cd d )( ( )( ) cd i ,( bc Figure 1 ab (( d bc a (( )(( ) )( c ). So there are two vertices and the one dimensional associahedron (point) and d ((( ) ab )) ). So the zero dimensional associahedron bc a 2 ) e ( ab ) bc ,( , ( ab de K d a c e e ) a ) we show the zero, one, two and three dimensional associahedra c ))( (( ) )) )) ) ab bc 1c d (a) ( ( ab de ) cd a and – ( ((( )( = 2 there is only one way to put a set of parentheses in the string of = 4 there are five different ways to put the parentheses in the string of c = 3 there are two ways to put the parentheses in the string of )( c bc = 1 the associahedron ) = 5 there are fourteen ways to put the parentheses in the string of ) 1a i i i i i (( ab ab . (( a ab 5 elements (( and the two dimensional associahedron If (( ( If ( is a line. If If If elements ( K • • • • • and If we have ain string this of string andparentheses each a edge single corresponds time.In to We figure using now the construct associativity the first rule few for associahedra replacing with the JHEP04(2020)072 , m 2. of i 41 1 of 1 − K , (3.1) (3.2) − ] i − i 40 i g is com- 42 ≤ – = 5 j ] -expansion 40 K is known as m 49 × · · · × 1 − 2 i 1 with is [ − m − i K i j i i 2 . We will present ) of the associahe- i and is [ K + = i 1 K i i, j ( which are isomorphic F ... K T 5 + . 0) = 1 K i i, ( 2 2 T − − with F j b ˜ j m F i − − i 2 which is the associahedron itself. K i 2 dim . This also includes the faces which − − i i K +1 -dimensional faces 2 i j 3 of the associahedron = ×· · ·× 2 2 have faces of dimension 1) − j 1 − i j − − − i ( K i i i − -expansion of the coupling at the Wilson-Fisher ≤ K – 7 – i = j is the empty set and for this, by definition, we F 1 X 5 5 1 — — 2 1 — — — 1 — — — — 0 1 2 3 4 1 42 84 56 14 1 14 21 9 1 — ————— face of − K 1 j F is controlled by the associahedron j we show the number of faces of different dimensions for − 6 5 4 3 2 1 \ = 1 i i 1 of dimension 1 − i i − g i g and then motivate it by looking at a number of examples. The ) = K for each face i 1 i, j ( − m T i . The sum is over all faces of the associahedron. In this way the facial i b ˜ K ··· -dimensional faces with 1 = 1. j − 0 1 g i b ˜ . = 1 the associahedron 3 = K 1)’th Catalan number and is the leftmost column in the number triangle. Finally i . Number triangle showing the number of F . i b ˜ × − If choose So there are fourteenposed vertices of and six the pentagons and three three dimensional squares. associahedron ]. The associahedron K i 3 Having introduced the associahedra we now turn to how they control the A few facts about the associahedron are relevant for us. Details can be found in [ • 48 K – structure of the associahedra controlsfixed the point. Consider nowworks. the first few examples so we can obtain a better feel for how it where the associahedron to of the value ofthe the Wilson-Fisher coupling fixed at point coupling thethe we Wilson-Fisher general associate result fixed the for point. associahedron any calculation To of each the coefficient coefficient the ( we need that Cartesianare products isomorphic of to lower dimensional theare associahedra faces not themselves of an the associahedron associahedron such as the three squares in the first few associahedra. Specifically the number of vertices The number of In the number triangle in table 45 There is of course only one face of dimension Table 1 dron JHEP04(2020)072 2 3 3 4 = 1. 1 K K K K 1 − 2 − 4 (3.6) (3.7) (3.3) (3.4) (3.5) − × F i × = b ˜ 5 = which 2 3 b ˜ 2 3 F 2 5 2 ). Note 1 K F F K F K − which are × dim m = 1 2 2 i 2.12 × dim − add to b − 2 2 − 2 K K 3 b 1 ˜ +1 F b )–( ˜ i K − +1 × ··· 1 × i i and is a point of − 2 1) 1 − g 2 2.8 × 2 1 2 2 1) − − K b b ˜ ˜ 2 1 K i 1 − K b K = − + ( 2 = F = 2 = . It is dictated by the 1 b + ( ˜ 1 = = 2 and one face shaped 1 2 1 F 1 − F = 2, six faces shaped like F − 1 dim F 4 3 − 3 b ˜ 3 4 F − F b ˜ = 1, three faces shaped like 1 b ˜ 2 F dim 1 1 − +1 2 F i b − 2 ˜ − F b ˜ 2 − 1) 2 b +1 ˜ dim i F 4 1. − = − F ( 1) 1 − dim +1 m 0 − 1 i i − b dim − 2 = 1 and one face shaped like i − b 1) ˜ is uniquely related to its facial structure = 2 +1 = 3. The associated coefficient is found K = 0, twenty one faces shaped like F i − F +1 = 1. The associated coefficient is i 5 + 21( m = 0, five faces shaped like 1 1) 2 F 1 dim F 1) + ( 1 − − F X − − 4 = 0, and a single face shaped like F 1 − 2 face of b ˜ m b − ˜ +1 – 8 – i 1 2 1 i F − + 5( b ˜ F . − + 1 + 6( 1) 3 2 1 = m b b − ˜ ˜ i 1 − − 2 1 1 1 has two faces shaped like 2 − ... b b ˜ ˜ has five faces shaped like K − b ˜ − 3 has a single face shaped like 1 i 3 2 has fourteen faces shaped like 1 b ˜ 4 F = ( + b ˜ 2 1 − 5 K 1 + 6 1 1 2 K − i K − b ˜ g 3 2 2 K dim 2 1 b b = 2. The associated coefficient is ˜ ˜ b ˜ − 3 − 1 3 3 , g 2 × · · · × F F b ˜ b +1 ˜ F + 3 i 1 i 1 i 2 F − dim 1) b dim ˜ K 0 2 − 2 1 − b − b ˜ = 0. The associated coefficient is b ˜ dim = 1 +1 i b +1 ˜ − which are lines and of dimension dim which are squares and of dimension dim F 21 i 1 which are pentagons and of dimension dim F which is of dimension dim 1) 3 3 = 2( +1 − 1) 4 − i i + 5 5 − 2 K g K 4 1 − K 3 1 g 1) b ˜ K b ˜ the sum of the indices of the beta function coefficients in × × − × =1 5 ∞ +3( i i 2 X 3 = 2 − K 5 K K = 14( = 14 F = 3 the associahedron = 5( = = 4 the associahedron ) = = 2 the associahedron = 5 the associahedron 4 × = = i g i i 3 i ( 2 3 4 g ∗ K F F like by the rule If which are points and of dimension dim If are points and eachwhich of are dimension lines dim and eachand of of dimension dimension dim dim points and each ofwhich is dimension a dim line and is of dimension dim dimension dim If If g What we have arrived at is a simple and stunningly beautiful closed form expression • • • • geometry of the associahedra and at each order in for each of its faces In all cases isthat there for each complete agreementThis with is our a combinatorial result eq.’s of the ( condition for the coupling at the Wilson-fisher fixed point to all orders in JHEP04(2020)072 ) )) N ( ( . In ∗ (4.1) (4.2) ) and ... r O g ( + φ through 3 2.6 γ = 2. This ν 0 γ b ]. The same + 11 ) = 2 ). For instance and ) 3 g β ( ( η γ η γ + loop beta function + 2 2 ). We find complete γ i 6 α 1 ) or by the geometric g ] we can compute the ( for some integer ) O j 11 +2 r and 2.13 ). To our knowledge these ) 1 − i i 7 γ ... related to g γ 2 φ ( i g δ + φ + O 3 +( +1)! β g 2 j 3 and γ = − 0 i ( b γ critical exponents + is scheme dependent, anomalous di- , βδ 2 ∗ ) β , g loop anomalous dimension. It does not , the anomalous dimension of the mass ) computed in [ g 2 2 7 , α φ ,..., i γ ] so we can convert theses computations γ 2 0 α only depends on the g + − ( m 13 (1) model to 1 + i γ γ k O (1! g O 1 1 g = 2 i,j by the same method that led us to eq. ( γ – 9 – ν i B +( ) j = k 0 γ b ) and the two i ! 1 j g − , i γ ) γ i =1 = j X ( ∗ ! j =1 ∞ 1 ), (4 i i g X ! ( η 0 k the coefficient at the Wilson-Fisher fixed point to = β − i i 1 ) = 0 − b g (2 ’th coefficient ( i ν ) = ))] γ 1 ( ( = − , k ∗ ω k γ g i g ), and the coefficients of the ( 1 2 ). This constitutes an exact closed form expression for the anomalous =1 − 0 ∞ i m X k b g γ 3.2

(1) symmetric model to i j γ k O =1 =1 ∞ ∞ i j X X ’s are given either by the combinatoric expression eq. ( ) = [2 + ). Our results in this section require only little effort to arrive at and has only been = 1 7 − ( ) = i ( ν Note that at any order We write the anomalous dimension of some operator in the theory as g ( and six loop mass anomalous dimension γ O φ point coupling and critical exponentsresults for have the still not appearedsions in have the been literature. calculatedinto The to beta a seven function prediction loops and of anomalousto in the dimen- [ Wilson-Fisher fixed point coupling and all the critical exponents the scaling relations is a final confirmation of our formal results. 5 Scalar We now use our general results to explicitly provide the value of the Wilson-Fisher fixed γ correction to scaling and agreement with the results reportedholds in true the for Mathematica file the accompanying additional [ critical exponents dimension of any operator at the Wilson-Fisher fixed point. coefficients (via receive corrections from higher orderssymmetric and is model exact to using this order. the Again six for loop the scalar beta function, six loop field anomalous dimension where we have found the the expression eq. ( At the Wilson-Fisher fixed point we find to all orders in or the anomalous dimensiongeneral the of anomalous some dimension composite is written operator in ( terms of the formal power series Although the Wilson-Fisher fixedmensions point and coupling critical exponentsphysical at quantities the charaterizing fixed the point system. are not. They areit scheme could independent be the anomalous dimension of the field 4 Anomalous dimensions and critical exponents JHEP04(2020)072 (4) ζ 4 (5.1) π (5) 5 (5) 2 ζ (4) ζ 6 ζ (4) π (3) 5 ζ (3) (3) 1889568 ζ 34951705 316009 ζ ζ (7) 8 8 25194240 2 ζ π 5) 7 4 π 12349 11664 , (8) 5)+ 6227 ζ 49 27 (5) (7) ] and seven loops , (3 3) 229635 19243 ζ ζ 314928 4 , 2187 ζ (7)+ − (8) 116 (3 5)+ 221281 − 385046 11 ζ 6075 ζ 1180980 − , (7 ζ 9 (5) (3) ζ (6) 196 − (11) ζ − 567 5) (3 (3) 88181 18225 ζ 5030 (3) ζ ζ , (3) ζ ζ ζ (5) 20425591 − (5)+ 40 81 ζ 6 (3) 459270000 (3) 25 54 (3 1215 ζ ζ 27 940 88181 ζ π ζ 112 5103 ζ 6 (6)+ − 6 3 9 (7) 49 (5)+ 405 40 6 ζ π 224 π − 5776 − 810 ζ 1458 ζ π (4)+ 9 − 651319 (5)+ 5) − 50 ζ (7)+ 5776 6075 (7) , ζ 3 (4) 1010 ζ − 5)+ ζ − 2 ζ (4) 413343 27 , + (3 7946 3645 4 (7)+ 2020 ζ 12454 25515 11 45927 (3) 3 ζ , ζ 7383787 π 373 729 81 + − (3 2 3 (5) 95659380 − ζ 7 961 6 − 640 729 ζ ζ 129631 P (3) − 1837080 π − (3) (6) ζ − (5)+ (3) ζ 10 ζ − 243 (4)+ 8 ζ 6227 1280 (3)+ ζ 4 (6) π 729 18225 4 ζ 4 2324 (5) ζ 161678 164025 ζ 569957 393660 2187 π 256 729 − π ζ π 106 729 10935 2 508228 2 − − + 1117 2187 − 2 158 243 443 + 129631 (5)+ 5832 (3) (4)+ (3) (3) 5405 1458 2 (6)+ (9) 33067440 ζ − 1226 3645 ζ ζ ζ ζ ζ ζ 4 − (3) 4453 2 21870 8 2 45106 2 ζ (5)+ π (3) 316009 π 1399680 π ζ – 10 – 79 ζ − (3) (4) 162 286446699 (5)+ 10909 243 (3) ζ 944784 − ζ (5)+ 1735 ζ 2187 ζ − + 16904 ζ (3) 81 613 2 178417 314928 58 − ζ 6227 22429 32805 32805 16 − (3) 11 81 400 243 + 243 54675 6561 − ) they are 64312 ζ 3 63481 (3) − (5) 590490 − + ζ (3) 2 ζ − 8 ζ 4.2 4 9 2 + (9)+ (3)+ π 198223 314928 (3)+ (6) 3 (4)+ 243 (3) ζ ζ (3) 4960 − ζ ζ ζ ζ (5)+ + (3) ζ (3) 4 2 − ζ ζ ζ (3) 4 π (3) 480656027 102036672 99202757785 ζ 4 ζ 132239526912 1661059517 π 27 (4) (5) (3) (3) 991796451840 ζ ζ ζ 6886777 2834352 ζ 11221 18707 − 16 104976 423301 118098 177147 19683 8 9 32 1347170 7085880 2187 163879 − + 243 16 27 6 6561 1156 − − + 20425591 8748 ) and eq. ( + 448 729 (11)+ − 12349 − − 6 8266860000 − ζ 2 − 3) 709 95713 39336 + , + 17496 (8)+ 10 (3)+ (5) 3 (4) 3 ζ 2.14 − ζ (3) (4) π ζ , 1603 2916 (4)+ ζ (9) 4 ζ ζ ζ (9) ζ (5 (3) + π (4) ζ 2 (3) ζ ζ 321451 ζ + (3) ζ 2 853678429 1224440064 408146688 ζ 32174329 17 81 (3) 20734249 34012224 9183300480 − − 90212 2187 492075 ζ 78732 4453 17 27 6561 729 1215 38 + 230560 11992616 46112 31827411 1737593 4907 2 3 19683 1599413 1417176 10590889 85030560 12454 393660 243 19696 − − − + 1 3 + + − − + + + + + + − + + + ]. Using eq. ( ) = ) = 13 ( ( ∗ ω g in [ made possible due to the extraordinary computations to six loops in [ JHEP04(2020)072 5 (5.5) (5.4) (5.2) (5.3) (7) 7 ζ 8 5 4 4 π (4) (5) π 7 (5) ζ ζ ζ (9) (4) ζ (3) (7) 40 (5) (5) (8) ζ 729 (3) ζ ζ 11 6227 ζ ζ , ζ (6) 72900 ζ 7 ζ 6 5) (3) 10 81 P , 22 49 π (4) 729 108 ζ 7085207 ζ (4)+ 46112 59049 3) (3 3664579 6323 − (7)+ ζ 68024448 , 2 ζ 16337 11664 11022480000 279936 ζ 1 79 2187 1 + (4)+ k 1 (7 200 729 84 (4)+ (6) − + − 2 ζ ζ 91854 n (8)+ ζ 6 ζ − 157464 − − 5) ζ 1 π 10053541 (7) 36 1 (4) , 13633 ζ ··· 1951 25 ζ 1978411 (5) 131220 (3) 567 162 − (3 i 1880 157464 324 (3)+ 204073344 ζ ζ 6227 2 k i ζ 3)+ − ζ 2 81 7 − , 3 4 − n (3) 11969 88181 3 8 ζ π 164025 3779136 4255 61 1 (3) (5)+ , 11664 243 π (7)+ (6) (3) 329 ζ ζ (9)+ ≥ 463493 − ζ ζ ζ 17496 (5 (4)+ 11 1 − + ζ , ζ 396809280 ζ 3 191 7 5)+ 35 − 5832 74 (7) (3) , (3) 729 P (6) + 11664 972 ζ ζ ζ 1735 (3) X − (3) 4 ζ (3 ζ 3991 ζ π ζ 19683 − >...>n 135 2063 16 243 i 24908 27 (4)+ n + 725 (3)+ 64 2654189 2834352 1421 1458 (5) 229635000 ζ 2 3707 729 − ) ζ 959 17496 1615 ζ 244 8748 2 321511 11664 1 2187 − − + 92969 68024448 − (9)+ (3) 6 − 2 7558272 + ζ (3) ζ (5) 191 π (8)+ (7)+ (6) 3 7776 2 4 ζ ζ 312061 (6) ζ ζ ζ (4) (3) + π (5)+ π ζ 4 − + 4 100776960 ζ ζ ζ – 11 – (3) 2 π 3 , . . . , k ζ i + 8 71 9 k (3) (3) 53182423 23827 19683 ( 1701 3645 ζ ζ 2448880128 12454 226820 1180980 2807 256 (3) − 26244 42397 + 2025 6561 314928 2 3800527 3779136 (3) 12454 − ζ − 248687 839808 2 − 81 7873200 − 52036931 ζ 184 729 + 4 − (5) − − 2 4 π − (9) 1 ζ (3) (5) 27 243 3) (3)+ (11) 6 ζ 7) ζ ζ , ζ (5)+ (7) , 3) ζ ζ , (3)+ (3) π − ζ − , (5)+ 3 ζ k ζ ζ , 1 ζ (4) (3 n (7 3466530079 ζ ζ (3) (5 ζ ζ 4247 3173748645888 ζ =1 1783 3240 59049 6227 81 69984 7217 244339 X 640 n 47 58565 25194240 651319 4459444 6323 3779136 918540 26244 629856 50 10027 755965 5103 2519424 235 729 209952 157464 + 5103 − − − 132893 6 6227 2430 − − + 1889568 − − − 10 2 ) = + (5)+ + 5) (8)+ (3)+ (5) 3 π k (6) 2 ζ , (5)+ ζ (3)+ ζ ζ ( (7) 2 ζ ζ 8 4 ζ 7 ζ ζ (3 (4)+ (9) 4 162 (5) π π ζ ζ ζ (3) (3) ζ π 109 + (3) ζ ζ 5832 ζ (3) (3) 4349263 + ζ ζ 22553 29 1 1 46425175 816293376 2 12 65712521 164025 59917 1393 27 81 35 6561 1 972 11528 47970464 1889568 26008 15309 477411165 109350 28 27 262440 27 21870 3470 1300 29386561536 264479053824 + 1 − + 2 + 1 54 + − + + + − + + − + + ) = ) = ( ( η ν where JHEP04(2020)072 0 6 < (5.6) (5.7) (5.8) (5.9) 7 0 b dimen- 829313 − -expansion . 6 314749 . 7 +0 = 4 5 +0 d 6 742 . We also evaluate 15437 . . 0). One could also ... +3 217018 5 . > +458 0 6 − 0812726 . 4 4, ( 0 0). In this way we envision 357566 . − 1113 843685 5 (1) model. We also gave the . . d < 0 O < 93 − − 4 = 200 0708838 5 . 4, ( 11 , ) symmetric model in 7 +0 n 0256565 -expansion in terms of the coefficients of . 3 ( P d > 7498 . O ) for the 268653 +0 . 7 4 ( +20 +0 4 O 3 – 12 – 0190434 . 0 − 23514 2 00832877 . . 137559 5 . 0 0 − − 3 3 − 2 ]. Numerically it is 50 0432099 . 61822 01869 . . +0 209877 . +1 +0 . 2 2 7 7 +0 083333 . 4831 62963 . . 57525 . 0 13 0185185 5+0 3 333333 . . . − was calculated in [ − − 11 , 7 0 so that the Wilson-Fisher fixed point appears for ) = 0 ) = 0 ) = 0 ) = P Finally we used our general results to compute the Wilson-Fisher fixed point coupling One of our original hopes was that in deriving a closed form expression order by Although we explicitly discussed the scalar ( ( ( ( > ∗ η ν ω g 0 numerical values. Acknowledgments We thank G. Dunne, E. Mølgaard andsupported R. by Shrock for the important Danish discussions. National TAR is Research partially Foundation under the grant DNRF:90. this hope however hasresults unfortunately are not useful been since one fulfilled.coefficients at enter least Nevertheless at has we a an believe given understanding order that of for our how example. theand various beta the function associated critical exponents to as given here order by order. order for the Wilson-Fisherasymptotic nature fixed of point the couplingterms expansion. we of Bell would Whether polynomials gain we or view further as the insight geometric expansion into in as the terms combinatoric of in the facial structure of associahedra imagine extending our analysisone to could multiple perform couplings. similarmany Also studies new for projects as theories worth above fordent investigating. for of which We whether should the mentionof various that different convergence our renormalization or analysis group are is functions asymptotic. indepen- have a This finite is radius irrelevant to the Lagrange inversion procedure trolled by the associahedra. sions in the introductionfact our in analysis the is above certainlyb derivations not it restricted could to be this any set theory of with theories. a In single coupling and for which 6 Discussion In this work we have usedfixed Lagrange point inversion coupling to and derive critical thethe exponents exact appropriate in form renormalization the of group the functions. Wilson-Fisher of We have the also argued Wilson-Fisher that fixed the point coupling can be viewed in a geometric sense and is con- the fixed point coupling and critical exponents numerically. They are and JHEP04(2020)072 , in 46 77 28 (1998) Nucl. Model , Phys. 4 , (2018) A 31 gφ ]. (1983) 351 model (1981) 457] 4 J. Phys. Lett. D 97 Five loop ϕ SPIRE (1993) 545] J. Phys. B 132 IN theory in the , Phys. Rev. Lett. B 101 4 ][ , φ -expansions of critical (2017) 036016 Zh. Eksp. Teor. Fiz. B 319 Phys. Rev. Five Loop Renormalization -symmetric , ) D 96 n Phys. Lett. ( , O dimensions Six loop analytical calculation of the ]. theory and Erratum ibid. ]. (1979) 521 [ in 99 4 . [ ]. η 3 ϕ ]. 50 Calculation of Critical Exponents by Phys. Rev. arXiv:1606.09210 SPIRE Erratum ibid. Five Loop Calculations in the , SPIRE [ IN Perturbation Theory at Large Order. 1. The IN [ SPIRE SPIRE ][ IN , Elsevier (2012), pp. 554–558 [ IN [ (1981) 147 (2016) [ ][ symmetric – 13 – ) (1991) 39 n ]. ( B 99 Accurate critical exponents from the epsilon expansion ]. O (1977) 1544 Minimally subtracted six loop renormalization of Sov. Phys. JETP (1983) 406 , Renormalization group functions of B 272 SPIRE in Four-dimensions Theory Critical exponents in ]. ), which permits any use, distribution and reproduction in Critical exponents of the N vector model IN 4 SPIRE D 15 Critical phenomena and renormalization group theory ]. ][ IN gφ arXiv:1601.01960 Phys. Lett. B 133 ][ PoS(LL2016)038 [ , SPIRE , η cond-mat/0012164 ]. IN SPIRE [ ]. theory and critical exponents ][ Phys. Lett. IN , 4 CC-BY 4.0 ][ 5 Phys. Rev. φ , SPIRE This article is distributed under the terms of the Creative Commons The Method of Uniqueness, a New Powerful Technique for Multiloop SPIRE IN (2016) 147 Phys. Lett. , Numbers and Functions in Quantum Field Theory IN (2002) 549 [ ]. ]. ]. arXiv:1606.08598 [ B 906 368 cond-mat/9803240 Interaction -symmetric [ ) n SPIRE SPIRE N ( 2 arXiv:1705.06483 hep-th/9503230 IN IN [ 085018 (1972) 240 Phys. MS-scheme to six loops O [ 8103 field anomalous dimension and the critical exponent Large-order behaviour of perturbation theory (1985) 137 renormalization group functions of exponents up to and the Critical Index [ Calculations Quantum Field Theory Methods (1979) 1035] [ Group Calculations in the [ Rept. ϕ R. Guida and J. Zinn-Justin, J.C. Le Guillou and J. Zinn-Justin, H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin and S.A. Larin, K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, D.I. Kazakov, S.G. Gorishnii, S.A. Larin, F.V. Tkachov and K.G. Chetyrkin, A. Pelissetto and E. Vicari, E. Brézin,J.C. Le Guillou and J. Zinn-Justin, D.I. Kazakov, O.V. Tarasov and A.A. Vladimirov, O. Schnetz, K.G. Wilson and M.E. Fisher, M. Kompaniets and E. Panzer, M.V. Kompaniets and E. Panzer, D.V. Batkovich, K.G. Chetyrkin and M.V. Kompaniets, [9] [7] [8] [5] [6] [4] [1] [2] [3] [13] [14] [11] [12] [10] Attribution License ( any medium, provided the original author(s) and source are credited. References Open Access. JHEP04(2020)072 ) ) N N B , ]. ( ( ]. O O JHEP B 21 , , Solving Solving ε SPIRE SPIRE − IN A 48 A 49 IN Phys. Rev. ][ , ][ = 4 Archipelago ]. Nucl. Phys. d ) , Phys. Rev. N ( , ]. J. Phys. J. Phys. O SPIRE , , ]. IN (2012) 025022 dimensions [ (2019) 201 arXiv:1909.01337 SPIRE , IN 10 SPIRE − (2017) 056 ][ 6 IN D 86 arXiv:1411.1099 04 ]. ][ [ ]. arXiv:1512.00013 (1977) 95 [ ]. The large charge limit of scalar JHEP On the CFT Operator Spectrum at 39 -expansion Meets Semiclassics -Expansion SPIRE JHEP Three loop analysis of the critical Bootstrapping the SPIRE Precision Islands in the Ising and The analytic structure of conformal = 0, models in , IN IN ]. SPIRE Bounding scalar operator dimensions in Phys. Rev. ) ][ ]. ][ IN , The N ( Unitarity violation at the Wilson-Fisher fixed ][ (2015) 045011 ]. O ]. SPIRE ]. ]. (2016) 125025 arXiv:1505.01537 IN SPIRE arXiv:1403.4545 [ IN ][ [ – 14 – D 91 SPIRE ][ SPIRE Phys. Rev. Lett. SPIRE SPIRE D 93 IN Critical IN , IN IN Critical Exponents for the N Vector Model in Critical Exponents from Field Theory ][ ][ ][ ][ -expansion from conformal field theory arXiv:0807.0004 (2015) 071 [ (2014) 869 arXiv:1603.04436 ]. ]. General Properties of Multiscalar RG Flows in [ ]. Phys. Rev. 12 arXiv:1810.10541 The , [ Phys. Rev. 157 , SPIRE SPIRE SPIRE arXiv:1504.07997 On critical exponents without Feynman diagrams IN IN arXiv:1404.1094 JHEP ]. [ IN (2008) 031 [ , ][ ][ ][ (2016) 036 12 arXiv:1701.04830 (2019) 008 arXiv:1505.00963 dimensions arXiv:1510.07770 arXiv:1909.01269 [ 08 Accessing Large Global Charge via the SPIRE [ 6 dimensions ε [ [ IN − [ J. Stat. Phys. Revisiting the dilatation operator of the Wilson-Fisher fixed point (2015) 106 6 − , JHEP ]. , 4 JHEP Ising Model with the Conformal Bootstrap 11 Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical , (2014) 025018 D d (2017) 368 CFT 3 3 (2019) 110 SPIRE D arXiv:1702.03938 IN arXiv:1908.11347 arXiv:1203.6064 blocks and the generalized Wilson-Fisher[ fixed points 920 [ 11 (2016) 445401 Large Global Charge field theories, and the[ Wilson-Fisher fixed point at JHEP Models the [ the Exponents D 90 models in 4 point in (2015) 29FT01 Three-Dimensions from Field Theory (1980) 3976 SciPost Phys. F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, P. Liendo, G. Badel, G. Cuomo, A. Monin and R. Rattazzi, K. Sen and A. Sinha, S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, G. Arias-Tamargo, D. Rodriguez-Gomez and J.G. Russo, M. Watanabe, F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, M. Hogervorst, S. Rychkov and B.C. van Rees, S. Rychkov and Z.M. Tan, L. Fei, S. Giombi and I.R. Klebanov, J.C. Le Guillou and J. Zinn-Justin, S. Rychkov and A. Stergiou, J.C. Le Guillou and J. Zinn-Justin, [32] [33] [30] [31] [27] [28] [29] [25] [26] [23] [24] [21] [22] [18] [19] [20] [16] [17] [15] JHEP04(2020)072 . ]. . . D B JHEP , SPIRE (2017) IN (1963) (1963) 275 (1954) 53 ][ 08 (2019) . 108 Phys. Rev. 82 108 , Nucl. Phys. 123 -expansion , JHEP , arXiv:1709.07504 , (1997) 149 , McGraw-Hill, New 18 Humboldt University of Conformal dimensions in the arXiv:1711.09102 ]. [ (1974). Phys. Rev. Lett. , Bull. Soc. Math. Fr. , Trans. Am. Math. Soc. SPIRE Trans. Am. Math. Soc. , , Clay Mathematics Institute IN , [ , Ph.D. Thesis, (2018) 096 Springer Adv. Appl. Math. ]. , , Grundlehren der Mathematischen 05 ]. Scattering Forms and the Positive Geometry of ]. models in the complex field plane ) SPIRE Global Wilson-Fisher fixed points – 15 – N IN JHEP SPIRE ( SPIRE , ][ IN ]. O IN , first edition, ][ , Graduate Texts in Mathematics, volume 152, ][ ]. Wilson-Fisher fixed point arXiv:1506.07243 SPIRE Methods of Theoretical Physics. Part I IN Critical Algebraic Operads (4) SPIRE ][ O Hopf monoids and generalized permutahedra IN ][ Towards a Bootstrap approach to higher orders of arXiv:1607.02030 [ arXiv:1701.05168 arXiv:1711.01173 [ Homotopy Associativity of H-Spaces. II Homotopy Associativity of H-Spaces. I Lectures on polytopes [ The multiple facets of the associahedron ooıepódn´stchaˆınesde Malcev Mono¨ıdespréordonnéset Advanced Combinatorics Convex Polytopes and Enumeration Feynman integrals and hyperlogarithms Combinatorics and Topology of Kawai-Lewellen-Tye Relations arXiv:1902.09542 [ , Berlin Germany (2015), arXiv:1706.08527 (2017) 769 . [ (2018) 153 (2017) 025026 Springer-Verlag (1995). Berlin 293 097 Kinematics, Color and the Worldsheet Publication (2005). Wissenschaftem, volume 346, Springer (2012). 921 York U.S.A. (1953), pp. 411–413. 02 large charge sectors at the 051603 95 R. Simion, E. Panzer, D. Tamari, J.D. Stasheff, J.D. Stasheff, G.M. Ziegler, M. Aguiar and F. Ardila, S. Mizera, N. Arkani-Hamed, Y. Bai, S. He and G. Yan, L. Comtet, J.-L. Loday, J.-L. Loday and B. Vallette, A. Jüttner,D.F. Litim and E. Marchais, P.M. Morse and H. Feshbach, D. Banerjee, S. Chandrasekharan, D. Orlando and S. Reffert, D.F. Litim and E. Marchais, P. Dey and A. Kaviraj, [49] [50] [45] [46] [47] [48] [42] [43] [44] [39] [40] [41] [37] [38] [35] [36] [34]