Strong coupling structure of $\mathcal{N}=4$ SYM observables with matrix Bessel kernel

Prep ared for submissio n to JHEP Strong coupling structure of N = 4 SYM observables with matrix Bessel k ernel Bercel Boldis a,b a Dep artment of The or etic al Physics, Institute of Physics, Budap est University of T e chnolo gy and Ec onomics M˝ ue gyetem rkp. 3., 1111 B udap est, Hungary b HUN-REN Wigner R ese ar ch Centr e for Physics, Konkoly-The ge Miklos ut 29-33, 1121 Budap est, Hungary Abstra ct: In this pap er I con tin ue the program of studying the strong coupling expansion of certain observ ables in N = 4 sup ersymmetric Y ang–Mills theory , which are giv en by a determinan t with a matrix Bessel kernel. I sho w that, by reorganizing the transseries of the determinant at large v alues of the ’t Ho oft coupling, a simple underlying structure emerges, in whic h each exp onen tially suppressed correction is related to the p erturbativ e series in a simple w a y . This new approac h pro vides an eﬃcien t metho d to generate the full transseries for N = 4 SYM observ ables, suc h as the cusp anomalous dimension, multi-gluon scattering amplitudes, and the octagon form factor. Using high-precision n umerical analysis, I verify the results and pro vide a complete description of the resurgence structure of the strong coupling expansion. Con ten ts 1 In tro duction and summary 1 2 Strong coupling expansion 5 3 The transseries structure from a diﬀerent angle 10 3.1 1 /g expansion 13 3.2 Stok es constants 16 3.3 A practical example 19 4 Resurgence relations 23 4.1 Stok es automorphism 24 4.2 Asymptotic analysis 26 4.3 Median resummation 29 A Bridge equations 30 1 In tro duction and summary Computing observ ables in four-dimensional sup erconformal Y ang–Mills theories at ﬁnite ’t Ho oft cou- pling λ = g 2 YM N is cen tral to the study of gauge/gravit y dualit y . Within the AdS/CFT corresp ondence [ 1 ], observ ables in the planar limit of the gauge theory admit diﬀeren t descriptions dep ending on the v alue of λ . A t w eak coupling, they can b e computed directly in the gauge theory using standard p erturbativ e techniques. In this case, the expansion in the coupling constant is well con trolled, and calculations can b e systematically organized in terms of F eynman diagrams. How ever, for larger v alues of λ , the series typically b ecome div ergen t and the weak coupling expansion loses its v alidit y . In contrast, at strong ’t Ho oft coupling, the same observ ables are describ ed b y a dual string theory . In this regime, the leading contribution is typically captured by a classical or semiclassical string conﬁguration, whereas subleading terms arise from quantum ﬂuctuations of the string world- sheet. Determining these corrections from direct string theory calculations is extremely diﬃcult; th us, obtaining a ﬁnite λ answ er for a given observ able from the strong coupling expansion is a challenging task. Since accessing observ ables at ﬁnite ’t Hooft coupling lies b ey ond the reach of both p erturbativ e gauge theory and semiclassical string theory , therefore, a complete description of observ ables requires alternativ e metho ds, suc h as those based on integrabilit y or lo calization [ 1 – 3 ]. In recent years, signiﬁ- can t progress has b een made in these directions to determine certain observ ables for arbitrary ’t Ho oft coupling. In this pap er, I inv estigate a sp ecial class of these observ ables. In the planar limit, at arbitrary v alues of the coupling constant, they admit the representation as a determinant: Z ℓ ( g ) = det  δ nm + K nm ( α )     1+ ℓ ≤ n,m< ∞ . (1.1) – 1 – These observ ables depend on the eﬀectiv e coupling constan t g = √ λ/ (4 π ), which en ters the observ able via the elements of the semi-inﬁnite matrix K ( α ). This matrix is casted in the blo ck form: K ( α ) = 2 cos α " cos α K oo sin α K oe − sin α K eo cos α K ee # . (1.2) Here α is a real parameter. Eac h blo c k in K ( α ) is itself a semi-inﬁnite matrix. Their en tries are expressed as the in tegrals: K nm = √ nm Z ∞ 0 dt t χ  √ t 2 g  J n ( √ t ) J m ( √ t ) , (1.3) where J n ( x ) is the Bessel function. K nm is the so-called truncated Bessel kernel [ 4 ]. The subscripts of the blo cks K oo , K eo , etc. in ( 1.2 ) denote the parity of the indices in K n,m that contribute to that sp eciﬁc block; namely , their elemen ts are giv en b y: [ K oo ] n,m = K 2 n − 1 , 2 m − 1 , [ K oe ] n,m = K 2 n − 1 , 2 m , [ K eo ] n,m = K 2 n, 2 m − 1 , [ K ee ] n,m = K 2 n, 2 m . (1.4) The parameter α can b e considered as a mixing angle betw een the subspaces of K nm deﬁned b y in tegrating o v er ev en or o dd Bessel functions. The coupling constan t en ters the determinant via the function χ ( x ), typically referred to as the symbol of the Bessel matrix. Determinan ts similar to ( 1.1 ) hav e previously app eared in sev eral contexts in the study of super- symmetric gauge theories [ 5 – 20 ]. Hence the form ( 1.1 ) is quite generic. Beside the ’t Ho oft coupling, Z ℓ ( g ) dep ends on the t w o real parameters α and ℓ , and the explicit form of the sym b ol χ ( x ). F or diﬀeren t v alues of the parameters and diﬀerent functions χ ( x ), the determinant describ es v arious observ ables of sup ersymmetric gauge theories: 1. F or α = 0, only the oﬀ-diagonal blo cks in ( 1.2 ) v anish, and Z ( g ) can b e written as a pro duct of the determinants of the diagonal blo cks. F or diﬀerent forms of the symbol, and v alues of ℓ , these determinants describ e several observ ables in planar N = 4 SYM, for instance, the ﬂux tub e correlators [ 5 , 11 , 13 ] and correlation functions of inﬁnitely heavy half-BPS op erators [ 6 – 12 ]. F or other c hoices of the symbol, they also give the leading non-planar correction to the partition function of N = 2 SYM [ 16 – 20 ], and in another case coincide with the T racy-Widom distribution that describ es the eigen v alue distribution in the Laguerre ensemble near the hard edge [ 21 ]. 2. F or α = π / 4 and with sym bol: χ ( x ) = 2 e x − 1 , (1.5) the matrix ( 1.2 ) is known as the BES kernel [ 5 ], and the determinant ( 1.1 ) go v erns the cusp anomalous dimension Γ cusp of N = 4 SYM. Γ cusp is given b y the ratio of the determinants ev aluated at ℓ = 1 and ℓ = 0 [ 19 , 22 ], namely: Γ cusp ( g ) = 4 g 2 Z ℓ =1 ( g ) Z ℓ =0 ( g ) . (1.6) – 2 – 3. F or the same symbol as in ( 1.5 ), but for diﬀeren t v alues of α = π r , with r b eing a rational n um ber, Z ℓ ( g ) also appears in the study of m ulti-gluon scattering amplitudes and form factors in planar N = 4 SYM [ 13 , 23 – 25 ]. As discussed in [ 22 ], the determinan t ( 1.1 ) can b e reform ulated in the form of a F redholm deter- minan t of an in tegral op erator whose k ernel can b e written as a tw o-b y-tw o matrix of Bessel k ernels (for this reason, I simply refer to ( 1.1 ) as a determinant with a matrix Bessel k ernel). In recent y ears a h uge developmen t w as made in the study of F redholm determinants with Bessel kernel. Based on pre- vious results from the mathematical literature (see, for instance, [ 21 , 26 – 34 ]), it was shown that these determinan ts and the resolven ts of the Bessel kernel satisfy a system of coupled integro-diﬀeren tial equations and they pro vide a systematic w a y to expand them b oth for weak and strong coupling in an eﬃcien t wa y [ 14 , 15 , 22 , 35 – 37 ]. By high order numerical computations, it was sho wn that these series are asymptotic and a resurgence analysis [ 38 – 40 ] is essential to obtain a ﬁnite ph ysical answer. The resurgence prop erties for the w eak and strong coupling regimes w ere studied in [ 22 , 36 , 37 , 41 – 45 ]. Relations for the determinan t ( 1.1 ) for diﬀerent v alues of the parameter ℓ were also found in [ 46 ]. In this pap er, I con tin ue the program of [ 22 , 36 , 37 , 45 ] and in v estigate further prop erties of the strong-coupling expansion of ( 1.1 ) for arbitrary v alues of α and ℓ , with the sp eciﬁc sym bol ( 1.5 ). In [ 22 ] w e ha v e already presen ted a metho d to compute the large- g expansion for Z ℓ ( g ) in this case up to arbi- trary orders in the transseries parameters and inv estigated its resurgence prop erties. Our metho d was based on ﬁrst computing the logarithm of the determinan t F ℓ ( g ) through a set of integro-diﬀeren tial equations and then exp onen tiating the resulting series to obtain the strong coupling expansion of Z ℓ ( g ) = e F ℓ ( g ) . In this w a y , w e found that at large g , the determinant Z ℓ ( α ) can b e expanded as the transseries: Z ℓ ( g ) = A ℓ ( g ) X n,m ≥ 0 Λ 2 n − Λ 2 m + Z ( n,m ) ( g ) , (1.7) where a = α/π . Each co eﬃcien t function Z ( n,m ) ( g ) is giv en by series in 1 /g . The exponentially small corrections – which corresp onds to ( n, m )  = (0 , 0) – are go v erned b y the parameters Λ 2 − = g 2 a e − 4 π g (1 − 2 a ) , Λ 2 + = g − 2 a e − 4 π g (1+2 a ) . (1.8) The deriv ation of the large- g expansion in ( 1.7 ) heavily relied on the analytical prop erties of the sym b ol in ( 1.5 ), or more precisely , on the com bination: χ α ( x ) = e iα + χ ( x ) cos α = cosh( x/ 2 + iα ) sinh( x/ 2) . (1.9) The exp onential scales in ( 1.8 ) and their pro ducts deﬁning the large- g transseries are related to the zeros of this function, lo cated a w a y from the real axis at: x = 2 π i  l + 1 2 − a  ≡ 2 π ix + l , and x = − 2 π i  j + 1 2 + a  ≡ − 2 π ix − j , (1.10) with l, j ∈ N 0 . The prefactor A ( g ) in ( 1.7 ) contains an o v erall dependence on g and the Widom-Dyson constant B ℓ ( α ) [ 22 ]: A ℓ ( g ) = e π (1 − 4 a 2 ) g (8 π g ) 1 / 4+ ℓ + a 2 e B ℓ ( a ) . (1.11) – 3 – By numerically ev aluating the determinan t for diﬀerent v alues of the angle α = π a and analytically in v estigating its b ehavior around a = 0 and a = 1 / 2, w e previously found a ﬁtting expression for Widom-Dyson constan t for arbitrary a and ℓ (see equation (5.18) of [ 22 ]). Although w e w ere able to obtain sev eral analytic expressions for the strong coupling coeﬃcients of Z ℓ ( g ), we observed cancellations of certain non-p erturbative contributions, and non-trivial resurgence relations w ere found b et w een the remaining corrections. The gov erning principle behind this structure remained unkno wn. In our latest pap er [ 45 ], w e ha v e shown that in case of α = 0 (which we studied previously in [ 36 ] and [ 37 ]), there is a simple underlying structure in the large g expansion of the determinant itself. W e found that the non-p erturbative contributions app ear only at ﬁrst orders in certain exp onential scales, which are related to the zeros of the function 1 − χ ( x ). F urthermore, these contributions can be easily obtained from the p erturbative sector by applying simple transformations on the co eﬃcients to the p erturbative 1 /g series. W e also found that in this new structure the resurgence b et w een diﬀerent non-p erturbativ e sectors app ears in a natural wa y . The question then arises whether the same principle can b e generalized to observ ables given b y the matrix Bessel k ernel deﬁned in ( 1.2 ) with sym b ol ( 1.5 ) and arbitrary v alues of α . In this paper, I demonstrate that, b y rearranging the strong coupling expansion found in [ 22 ], the results of [ 45 ] can b e extended to the observ ables discussed in p oints 2. and 3. ab o v e, making it extremely eﬃcient to generate the complete transseries for these ph ysical quan tities as well. This new form of the transseries also provides the opportunity to give a complete description to the resurgence structure of the strong coupling expansion of these observ ables. Summary and conclusions The pap er is organized as follo ws: In Section 2, I summarize the results obtained in [ 22 ] for the strong coupling expansion ( 1.7 ). I sho w some examples of the transseries co eﬃcien ts and I discuss that they ha v e an explicit dep endence on tw o diﬀerent ingredien ts: the parameter a = α/π and a set of in tegrals I n related to the moments of ∂ x log χ α ( x ). I also motiv ate that there is an underlying structure that connects the exp onentially suppressed corrections in the transseries to the p erturbativ e sector. In the ﬁrst part of Section 3, I rearrange the transseries and show that the strong coupling expansion in ( 1.7 ) can b e written in the form: Z ℓ ( g ) = A ( g ) X δ + ,δ − (8 π g ) − ∆(∆ − 2 a ) e − 8 π g ( P l ∈ δ + x + l + P j ∈ δ − x − j ) e iπ a ∆ S ( δ + ,δ − ) D ( δ + ,δ − ) ( g ) , (1.12) The exp onen tial w eigh ts x + l and x − j are the zeros of the function χ α ( x ) giv en in ( 1.10 ). The summation runs o v er all possible pairs of ﬁnite sets δ + and δ − , both con taining only non-negativ e in tegers. This ensures that all terms in the transseries are at most ﬁrst order in each distinct exp onential factor e − 8 π g x + l and e − 8 π g x − j . F or a ﬁxed pair of δ + and δ − , ∆ denotes the diﬀerence b et w een the n um ber of elemen ts in δ + and δ − , that is, it is equal to ∆ = | δ + |−| δ − | . The functions D ( δ + ,δ − ) ( g ) are given b y expansions o ver 1 /g and S ( δ + ,δ − ) are the corresp onding Stok es constants. In Section 3.1, I sho w that it is conv enient to introduce the notation for the p erturbativ e ( δ + , δ − ) = ( {} , {} ) sector: D ( {} , {} ) ( g ) = D [ I n ] ( g ) , (1.13) to mak e the dep endence on the moments I n explicit. With the system of in tegro-diﬀeren tial equations studied in [ 22 ], the 1 /g expansion of D [ I n ] ( g ) can be eﬃcien tly generated both analytically and – 4 – n umerically . Then applying the same ideas as in [ 45 ], I sho w that the exp onentially small corrections can be easily generated from the p erturbativ e functions with the simple rule: D ( δ + ,δ − ) ( g ) = D h I ( δ + ,δ − ) n i ( g )    a → a − ∆ . (1.14) This expression means that the non-p erturbative corrections D ( δ + ,δ − ) ( g ) are obtained from the p er- turbativ e sector by shifting the explicit a dependence by − ∆ and the momen ts should b e replaced by suitably modiﬁed in tegrals denoted b y I ( δ + ,δ − ) n (for their explicit form see equation ( 3.25 )) In Section 3.2, I use the same idea to show that there exist tw o recurrence re lations, that generate all the corresp onding Stokes constants S ( δ + ,δ − ) as w ell. These relations are given in ( 3.43 ) and ( 3.44 ). The rule ( 1.14 ) for the 1 /g -expansions together with the recurrence relations for the Stokes constan ts completely generates the full strong coupling expansion ( 1.12 ) of the determinant with matrix Bessel k ernel in an eﬀectiv e w a y . In Section 3.3, as a practical example, I sp ecify the parameter α = π / 4. Using the relation b et w een the exp onentially small corrections and the perturbative sector, I present several analytic results for the strong coupling corrections for the determinant ev aluated at the ab ov e-men tioned v alue of α and ℓ = 0 , 1. Then, b y ( 1.6 ), I also giv e some explicit non-p erturbativ e corrections for the cusp anomalous dimension. In Section 4, I inv estigate the resurgence prop erties of the transseries ( 1.12 ). Through the Bridge equations (discussed in more detail in App endix A) I derive the Alien algebra for ( 1.12 ) and I sho w that this transseries is more natural from the p oin t of view of resurgence than the expansion given in ( 1.7 ). With high precision numerical analysis of certain exp onen tial corrections, I also explain some resurgence relations that w ere found in [ 22 ]. The adv an tage of writing the transseries as ( 3.9 ) instead of ( 1.7 ), is that the exp onen tial scales are in one-to-one corresp ondence with the zeros of the relev an t sym b ol ( 1.9 ). Moreov er, the exp onentially small corrections are at most ﬁrst order in ev ery scale e − 8 π g x + l and e − 8 π g x − j , and the corresp onding 1 /g expansions can b e easily obtained from the p erturbativ e sector with the simple rule in ( 1.14 ). This is the same structure as was studied in [ 45 ]. Therefore, ( 3.9 ) puts the observ ables in p oints 2. and 3. deﬁned through the matrix Bessel kernel with diﬀeren t v alues of the mixing angle α in the univ ersal strong coupling structure of [ 45 ]. It also provides a more natural framework for describing the resurgence prop erties of the ab ov e-men tioned observ ables at large v alues of the ’t Ho oft coupling. According to the results of this pap er together with [ 45 ], there is a simple underlying structure in the strong coupling expansion for certain observ ables of N = 4 SYM, at least for those given by the determinan t ( 1.1 ). It is an op en question whether the same structure holds for other observ ables in N = 4 SYM as w ell, or physical quantities in diﬀerent but related mo dels, suc h as the O (6) mo del [ 47 – 49 ]. It w ould also be in teresting to relate the obtained transseries structure to the results of [ 50 , 51 ] where a complete resurgence analysis was carried out for a family of generalized energy densities in the O ( N ) models. 2 Strong coupling expansion In [ 22 ] w e presented a technique to determine the strong coupling expansion of the determinant ( 1.1 ) for arbitrary v alues of a and ℓ . Our metho d w as based on a complicated system of in tegro- diﬀeren tial equations for the logarithm of the determinant F ℓ ( g ) = log Z ℓ ( g ). With this, w e w ere able to – 5 – analytically compute the perturbative part Z (0 , 0) ( g ) in ( 1.7 ) and man y non-p erturbativ e con tributions Z ( n,m ) , with ( n, m )  = (0 , 0) up to several orders in 1 /g . In the follo wing, I highligh t some of the analytic results for the perturbative and non-perturbative sectors of ( 1.7 ). P erturbativ e part First, I consider the p erturbative sector giv en b y Z (0 , 0) ( g ). The 1 /g expansion of Z (0 , 0) ( g ) can parametrized in the follo wing wa y: Z (0 , 0) ( g ) = 1 + ( a 2 − ℓ 2 ) X k ≥ 1 ( − 1) k k ! f (0 , 0) k (8 π g ) k , (2.1) with the ﬁrst few functions f (0 , 0) k b eing: f (0 , 0) 1 = I 2 , f (0 , 0) 2 = ( a 2 − ℓ 2 + 1) I 2 2 + 2 a I 3 , f (0 , 0) 3 =  a 2 − ℓ 2 + 1   a 2 − ℓ 2 + 2  I 3 2 + 6 a  a 2 − ℓ 2 + 2  I 2 I 3 + 2(5 a 2 − ℓ 2 + 1) I 4 , f (0 , 0) 4 =  a 2 − ℓ 2 + 1   a 2 − ℓ 2 + 2   a 2 − ℓ 2 + 3  I 4 2 + 12 a  a 2 − ℓ 2 + 2   a 2 − ℓ 2 + 3  I 2 2 I 3 + + 6  2 a 4 − 2 a 2 ℓ 2 + 9 a 2 − ℓ 2 + 1  I 2 3 + 8  a 2 − ℓ 2 + 3   5 a 2 − ℓ 2 + 1  I 2 I 4 + + 12 a  7 a 2 − 3 ℓ 2 + 5  I 5 . (2.2) The modiﬁed sym bol ( 1.9 ) enters the p erturbativ e co eﬃcients f (0 , 0) n through the moments 1 : I n = 1 π 2 Re  Z ∞ 0 dz  2 π i z  n z ∂ z log χ α ( z )  , (2.3) Eac h function f (0 , 0) n is a m ultilinear com bination of I n . F or n ≥ 2, these integrals div erge at the origin; ho w ev er, with analytical regularization, they can b e ev aluated, giving: I n = 1 ( n − 2)!  ψ ( n − 2)  1 2 − a  − ψ ( n − 2) (1) + ( − 1) n  ψ ( n − 2)  1 2 + a  − ψ ( n − 2) (1)  , (2.4) where ψ ( n ) ( x ) is the p olygamma function of order n . Notice that the functions I n ha v e the symmetry under a → − a : I n = ( − 1) n I n | a →− a (2.5) F ollowing the method of [ 22 ], the perturbative part can b e eﬀectiv ely determined up to arbitrary order in 1 /g . Leading (1 , 0) non-p erturbativ e correction The determinant ( 1.1 ) is an ev en function of α and p erio dic in α → α + π therefore, it is enough to consider 0 < α < π / 2. In this case, for large v alues of g , Λ − ≫ Λ + , whic h giv es an ordering 1 F or conv enience, compared to the notations of [ 22 ], I used a diﬀeren t normalization for the integrals of the sym b ol χ α ( x ). The in tegrals I n deﬁned in [ 22 ] are related to I n via I n = (2 π ) n − 1 I n . – 6 – for the exp onentially suppressed terms. Hence the leading non-p erturbativ e sector corresp onds to ( n, m ) = (1 , 0). Its 1 /g expansion lo oks as: Z (1 , 0) ( g ) = e iπ a P (1 , 0) (8 π ) 2 a 8 π g   1 + (( a − 1) 2 − ℓ 2 ) X k ≥ 1 ( − 1) k k ! f (1 , 0) k (8 π g ) k   (2.6) where the prefactor (8 π ) 2 a w as introduced to matc h the pow er of g included in Λ 2 − . The series starts at O (1 /g ), and the functions f (1 , 0) n are again multilinear combinations of the in tegrals I n . Up to 1 /g 3 they are given b y: f (1 , 0) 1 = I 2 + 2(1 / 2 − a ) − 1 , f (1 , 0) 2 = (( a − 1) 2 − ℓ 2 + 1)  I 2 + 2(1 / 2 − a ) − 1  2 + 2( a − 1)  I 3 − 2(1 / 2 − a ) − 2  , f (1 , 0) 3 =  ( a − 1) 2 − ℓ 2 + 1   ( a − 1) 2 − ℓ 2 + 2   I 2 + 2(1 / 2 − a ) − 1  3 + + 6( a − 1)  ( a − 1) 2 − ℓ 2 + 2   I 2 + 2(1 / 2 − a ) − 1   I 3 − 2(1 / 2 − a ) − 2  + + 2(5( a − 1) 2 − ℓ 2 + 1)  I 4 + 2(1 / 2 − a ) − 3  , (2.7) and the prefactor P (1 , 0) is: P (1 , 0) = − ( − 1) ℓ  1 2 − a  2 a − 1 Γ(1 + ℓ − a ) Γ( ℓ + a ) . (2.8) It can b e seen that the series Z (1 , 0) is complex v alued, as it contains an ov erall factor of e iπ a . The rest of the expansion is real. Notice that functions Z (1 , 0) and f (1 , 0) n are related to f (0 , 0) n corresp onds to the p erturbativ e part and is shown in ( 2.1 ) and ( 2.2 ). f (1 , 0) n , as w ell as their prefactors in the series ( 2.6 ) can b e obtained from ( 2.1 ) and ( 2.2 ) b y simply shifting the explicit a dep endence by a → a − 1 and c hanging the in tegrals I n as: I 2 → I 2 + 2  1 2 − a  , I 3 → I 3 − 2  1 2 − a  2 , I 4 → I 4 + 2  1 2 − a  3 , . . . , (2.9) or in general: I n → I n − ( − 1) n − 1 2  1 2 − a  n − 1 . (2.10) (0 , 1) correction The next subleading exp onentially small contribution is Λ 2 − Z (0 , 1) ( g ). Its 1 /g expansion can b e parametrized as: Z (0 , 1) ( g ) = P (0 , 1) (8 π ) − 2 a 8 π g   1 + (( a + 1) 2 − ℓ 2 ) X k ≥ 1 ( − 1) k k ! f (0 , 1) k (8 π g ) k   (2.11) F rom the deﬁnition of K ( α ) it follo ws that the determinant is symmetric under the exc hange α → − α , therefore the strong-coupling expansion should reﬂect this symmetry as w ell. This leads to a relation b et w een the t w o non-p erturbative sectors: Z (0 , 1) ( g ) = Z (1 , 0) ( g )    a →− a , (2.12) – 7 – and the same property for the corresp onding exp onential scales Λ 2 − and Λ 2 + . Therefore the prefactor P (0 , 1) and the co eﬃcients f (0 , 1) k can be easily obtained from P (1 , 0) and f (1 , 0) k b y changing a → − a . The series Z (0 , 1) is again complex v alued, since it con tains an o v erall factor of e − iπ a . It is imp ortant to note, that up to an ov erall prefactor, the series ( 2.11 ) can b e easily obtained also from the p erturbativ e series ( 2.1 ), by shifting the explicit a dependence with a → a + 1 and transforming the integrals I n as: I 2 → I 2 + 2  1 2 + a  , I 3 → I 3 + 2  1 2 + a  2 , I 4 → I 4 + 2  1 2 + a  3 , . . . , (2.13) or: I n → I n + 2  1 2 + a  n − 1 . (2.14) The series again starts with a O (1 /g ) term. (1 , 1) correction The next subleading con tribution app ears at order Λ 2 − Λ 2 + . At this order we ﬁnd the 1 /g expansion: Z (1 , 1) ( g ) = P (1 , 1)   1 + ( a 2 − ℓ 2 ) X k ≥ 1 ( − 1) k k ! f (1 , 1) k (8 π g ) k   . (2.15) The functions f (1 , 1) n are similar to ( 2.16 ): f (1 , 1) 1 = I 2 + 2(1 / 2 − a ) − 1 + 2(1 / 2 + a ) − 1 , f (1 , 1) 2 = ( a 2 − ℓ 2 + 1)  I 2 + 2(1 / 2 − a ) − 1 + 2(1 / 2 + a ) − 1  2 + + 2 a  I 3 − 2(1 / 2 − a ) − 2 + 2(1 / 2 + a ) − 2  , f (1 , 1) 3 =  a 2 − ℓ 2 + 1   a 2 − ℓ 2 + 2   I 2 + 2(1 / 2 − a ) − 1 + 2(1 / 2 + a ) − 1  3 + + 6 a  a 2 − ℓ 2 + 2   I 2 + 2(1 / 2 − a ) − 1 + 2(1 / 2 + a ) − 1  × ×  I 3 − 2(1 / 2 − a ) − 2 + 2(1 / 2 + a ) − 2  + + 2(5 a 2 − ℓ 2 + 1)  I 4 + 2(1 / 2 − a ) − 3 + 2(1 / 2 + a ) − 3  . (2.16) The o verall prefactor P (1 , 1) is giv en b y: P (1 , 1) = −  1 2 − a  1+2 a  1 2 + a  1 − 2 a (2.17) In this case Z (1 , 1) ( g ) is purely real. It can b e easily seen, that f (1 , 1) n can b e directly obtained from the perturbative co eﬃcients f (0 , 0) n b y ﬁrst applying the shift ( 2.9 ) then ( 2.13 ) subsequently on the integrals I n , or vica v ersa. General observ ations F rom the deﬁnition of K ( α ) it follo ws that the determinant is symmetric under the exc hange α → − α , therefore the strong-coupling expansion should reﬂect this symmetry as w ell. This leads to relation b et w een the non-p erturbativ e sectors: Z ( n,m ) ( g ) = Z ( m,n ) ( g )    a →− a . (2.18) – 8 – Figure 1 : The horizontal axis denotes the pow ers of Λ 2 − , while the v ertical one represen ts the p o w ers of Λ 2 + . The colored squares corresp ond to corrections that satisfy ( 2.19 ), hence giv e non-zero con tribution to the strong coupling expansion. Diﬀerent colors represent how many diﬀeren t con tributions they con tain from the ( δ + , δ − ) represen tation. The exponential scales in ( 1.8 ) also resp ect this symmetry . By generating several large- g con tributions b oth analytically and numerically for the logarithm of the determinant F ( g ) and taking its exp onen tial to compute the determinant, we observ ed the cancellation of sev eral non-perturbative corrections in Z ℓ ( g ). W e found that Z ( n,m ) is only non-zero, if ( n, m ) satisﬁes the following constrain t: 1 2 p ( p − 1) ≤ n < 1 2 p ( p + 1) , 0 ≤ m − n ≤ p , (2.19) with p b eing an in teger such that p ≥ 0. In Figure 1, I show at which orders of Λ 2 − and Λ 2 + there is a non-zero series Z ( n,m ) . F or the non-zero con tributions w e also found that at order Λ 2 n − Λ 2 m + , the leading term in 1 /g is of order: Z ( n,m ) ( g ) = O (1 /g ( n − m ) 2 ) . (2.20) F urthermore, if n  = m , the functions Z ( n,m ) ( g ) are complex v alued. Eac h of these con tributions gain an o v erall prefactor of e iπ ( n − m ) a . As it can b e seen, the 1 /g p ow ers of the leading order contribution in Z ( n,m ) ( g ) dep ends on the com bination n − m . Also the scale Λ 2 n − Λ 2 m + giv es an additional factor of 1 /g 2 a ( n − m ) . Later we will see, that the diﬀerence n − m app ears in several other wa ys in describing the large- g expansion of the determinan t, therefore it is con v enien t to in tro duce the diﬀerence: ∆ = n − m , (2.21) that is diﬀeren t for diﬀerent non-p erturbative sectors. Collecting the leading p ow er of g in eac h non-p erturbativ e contribution, we ﬁnd: Λ 2 n − Λ 2 m + Z ( n,m ) ( g ) ∼ O  g − ∆(∆ − 2 a )  . (2.22) – 9 – As men tioned previously , in addition to the α → − α symmetry of the determinan t, it is also p erio dic in α → α + π . Therefore, it is enough to restrict the v alues of α to the interv al 0 < α < π / 2. Ho w ev er, for further reasons, it is w orth commenting on the results if α is outside of this branc h. The abov e results are, strictly sp eaking, only v alid in the branch − π / 2 ≤ α ≤ π / 2. Outside this regime a more careful analysis is necessary . First, consider the mo diﬁed symbol deﬁned in ( 1.9 ). This function has the follo wing prop erties: χ α ( x ) = χ ∗ − α ( x ) = − χ α + π ( x ) = χ ∗ π − α ( x ) , (2.23) where ” ∗ ” means complex conjugation. The momen ts ( 2.3 ) hav e to reﬂect these symmetries, namely I n has to satisfy: I n = ( − 1) n I n | − a = I n | a +1 = ( − 1) n I n | 1 − a . (2.24) Although the ﬁrst prop ert y is automatically satisﬁed b y the ev aluated form ( 2.4 ) of I n , the function do es not admit the rest of the prop erties. Since ( 2.4 ) has singularities at a = ± 1 / 2, this problem can b e solv ed b y deﬁning I n outside the branch − 1 / 2 < a < 1 / 2 b y: I n | a → a + m ≡ I n , (2.25) with some integer m . This ensures that the p erio dicity in ( 2.24 ) holds. Since the strong coupling e xpansion of Z ℓ ( g ) do es not dep end on p erio dic functions of a , the remaining explicit dep endence on a in the transseries should b e shifted according to the branc h in whic h a lies. This, together with the p erio dicity ( 2.25 ) can b e form ulated as follows: suppose that for − 1 / 2 < a < 1 / 2, the strong coupling expansion of Z ℓ ( g ) is a function of the in tegrals I n : Z ℓ ( g ) ≡ Z 0 ℓ [ I n ]( g ) , (2.26) and it also has an explicit dep endence on the parameter a . Then for m − 1 / 2 < a < m + 1 / 2, the strong coupling expansion is given by: Z ℓ ( g ) = Z 0 ℓ [ I n ]   a → a − m ( g ) . (2.27) That is, the large- g expansion is the same as in the principal branc h with the same momen ts given in ( 2.4 ), but the explicit dep endence on a is shifted according to the branch of a . Later w e will see, that the same prop ert y app ears in the relation connecting the perturbative series with the non-perturbative corrections. One ﬁnal remark is that although it is formally p ossible to determine the transseries ( 1.7 ) up to arbitrary order in 1 /g and Λ 2 ± with our previous metho d established in [ 22 ], ho w ev er, due to the scaling ( 2.20 ), b eyond a certain exponential order, it becomes diﬃcult to gain analytical results for the strong coupling co eﬃcien ts. F ollo wing the idea of [ 45 ], in the next section I will sho w, that ( 1.7 ) can be recast in to a new form, in which a natural connection b etw een the p erturbative and non-p erturbativ e con tributions emerges, making the computation of non-p erturbative corrections extremely eﬃcient. 3 The transseries structure from a diﬀeren t angle The reason for the cancelation of terms that do not satisfy the condition in ( 2.19 ) remained unknown, as it do es not follo w trivially from the metho d presented in [ 22 ]. It can also b e seen that, by apply- ing certain transformations to the parameter a and the in tegrals I n , the non-p erturbative functions – 10 – Z ( n,m ) ( g ) with ( n, m )  = (0 , 0) can easily b e obtained from the p erturbativ e part Z (0 , 0) ( g ). In the fol- lo wing I will sho w that the pro cedure of [ 45 ] can b e generalized to the matrix Bessel kernel ( 1.2 ) with sym b ol ( 1.5 ), and these prop erties follow automatically from the analytic structure of the mo diﬁed sym b ol ( 1.9 ) and its successive redeﬁnition at each exp onen tial level. In [ 19 ] it w as shown that for α = 0 and arbitrary sym bol function χ ( x ), the non-p erturbative co eﬃcien ts for ( 1.1 ) app ear at exp onential orders e − 8 π g x n , where x n denote the zeros of the function 1 − χ ( x ). As follows from our metho d in [ 22 ], in case of the matrix Bessel kernel with mixing angle α , this function is replaced by the mo diﬁed symbol ( 1.9 ). The solution of the underlying in tegro- diﬀeren tial equations in v olv es the Wiener-Hopf decomp osition of χ α ( x ), whic h can be written as: χ α ( x ) = 2 cos π a x Φ + ( x )Φ − ( − x ) , (3.1) with Φ ± ( x ) are given b y: Φ + ( x ) = Γ  1 2 − a  Γ  1 + ix 2 π  Γ  1 2 − a + ix 2 π  = Y j ≥ 0 1 + ix 2 π x + j 1 + ix 2 π y j , Φ − ( x ) = Γ  1 2 + a  Γ  1 + ix 2 π  Γ  1 2 + a + ix 2 π  = Y j ≥ 0 1 + ix 2 π x − j 1 + ix 2 π y j . (3.2) These functions are both analytic in the lo w er half-plane and hav e inﬁnitely man y zeros and p oles in the upper half-plane. They v anish at x = 2 π ix + j and x = 2 π ix − j resp ectiv ely , with x ± j giv en by: x ± j = j + 1 2 ∓ a, j ∈ N 0 , (3.3) while b oth of the functions hav e p oles at x = 2 π iy j , where y j = j + 1 and j is a non-negative in teger. The functions Φ ± ( x ) and the zeros x ± j dep end on the parameter a and Φ + and x + n are related to Φ − and x − n via the exc hange a → − a . Ho w ev er, for reasons explained later, I suppress their a dep endence and treat these quantities as if they w ere a independent. F rom the metho d of [ 22 ] and the structure studied in [ 45 ], we expect the app earance of non- p erturbativ e contributions at orders of e − 8 π g x ± j . If w e lo ok at the exp onen tial scales Λ 2 n − Λ 2 m + m ulti- plying Z ( n,m ) ( g ) in the transseries ( 1.7 ), w e ﬁnd that the exp onential prefactors are: Λ 2 n − Λ 2 m + ∼ exp  − 8 π g  n  1 2 − a  + m  1 2 + a  . (3.4) F or diﬀerent v alues of n and m , the exp onent contains b oth + a and − a terms and can be written as ﬁnite linear combinations of the zeros x ± j , with each x ± j app earing with a multiplicit y of one. Therefore, together with the ﬁndings of [ 45 ], this suggests that in the case of matrix Bessel k ernel, non- p erturbativ e con tributions contain exp onential factors that are at most ﬁrst order in eac h parameter e − 8 π g x + j and e − 8 π g x − j . Therefore, the exp onential orders can b e parametrized as: e − 8 π g P l ∈ δ + x + l e − 8 π g P j ∈ δ − x − j , (3.5) where δ + and δ − are ﬁnite sets of non-negative integers, b oth con taining an y integer at most once. T o b e more precise, b y equating: X l ∈ δ + x + l + X j ∈ δ − x − j = n (1 / 2 − a ) + m (1 / 2 + a ) , (3.6) – 11 – and using ( 3.3 ), it can b e shown that to a sp eciﬁc combination of the sets δ + and δ − , a pair of integers ( n, m ) can b e uniquely assigned, so that ( 3.6 ) is satisﬁed and ( 3.5 ) con tains the same exp onential factor as Λ 2 n − Λ 2 m + . F or example: δ + = { 0 } , δ − = {} → n = 1 , m = 0 , δ + = { 0 } , δ − = { 0 } → n = 1 , m = 1 , δ + = { 1 } , δ − = {} → n = 2 , m = 1 , δ + = { 0 } , δ − = { 1 } → n = 2 , m = 2 , δ + = { 1 } , δ − = { 0 } → n = 2 , m = 2 , δ + = { 0 , 1 } , δ − = {} → n = 3 , m = 1 , δ + = { 0 , 1 } , δ − = { 0 } → n = 3 , m = 2 , δ + = { 2 } , δ − = {} → n = 3 , m = 2 , δ + = { 1 } , δ − = { 1 } → n = 3 , m = 3 . (3.7) By inv estigating further combinations of δ + and δ − and assigning pairs of integers ( n, m ) to them via ( 3.6 ), w e ﬁnd that not every pair of ( n, m ) occurs (eg. ( n, m )  = (0 , 2)). In fact, we only ﬁnd a con tribution at order Λ 2 n − , Λ 2 m + , if n and m satisfy the condition ( 2.19 ), so at the same orders as indicated in Figure 1. This conﬁrms that instead of ( 1.7 ), it is more conv enient to parametrize the strong coupling expansion of the determinan t in terms of ( 3.5 ). F rom the examples in ( 3.7 ), it can also b e seen that diﬀerent combinations of δ + and δ − could pro duce the same pair of ( n, m ). Indeed, by in v estigating higher order con tributions, the num b er of exp onen tial corrections in ( 3.5 ) corresp ond to the same order Λ 2 n − Λ 2 m + rapidly increases. In Figure 1 I indicated the order of this degeneracy using diﬀerent colors. It is easy to see that for diﬀerent sets ( δ + , δ − ) that corresp ond to the same pair of ( n, m ), the n um ber ∆ deﬁned in ( 2.21 ) is in v arian t and giv en by: ∆ = n − m = | δ + |−| δ − | . (3.8) Here, | δ | denotes the num ber of elements in the set δ . Therefore, the v alue of ∆ represen ts the diﬀerence betw een the n um b er of exp onential corrections of t yp e e − 8 π g x + l and e − 8 π g x − j . Finally , collecting all the prop erties discussed ab ov e and in Section 2 , I conjecture that the strong coupling expansion of Z ℓ ( g ) is given by: Z ℓ ( g ) = A ℓ ( g ) X δ + ,δ − (8 π g ) − ∆(∆ − 2 a ) e − 8 π g ( P l ∈ δ + x + l + P j ∈ δ − x − j ) e iπ a ∆ S ( δ + ,δ − ) D ( δ + ,δ − ) ( g ) . (3.9) This series is an alternative form of the strong coupling expansion in ( 1.7 ). Here, the exponential w eigh ts are precisely giv en b y the zeros of the relev ant symbol χ α ( x ) and each exp onential correction is at most ﬁrst order in every e − 8 π g x + l and e − 8 π g x − j factor. The functions D ( δ + ,δ − ) ( g ) are given by series in 1 /g : D ( δ + ,δ − ) ( g ) = X k ≥ 0 d ( δ + ,δ − ) k (8 π g ) k , (3.10) – 12 – and eac h term in the series ( 3.9 ) is normalized in such a w a y that d ( δ + ,δ − ) 0 = 1 and S ( {} , {} ) = 1. S ( δ + ,δ − ) are the Stokes constants of the transseries. Notice that in each term, I hav e pulled out the complex factors e iπ a ∆ , so that S ( δ + ,δ − ) and D ( δ + ,δ − ) ( g ) are real. The a → − a symmetry of the determinant is reﬂected in the Stokes constants and the non- p erturbativ e functions as: S ( δ + ,δ − ) = S ( δ − ,δ + ) | a →− a , (3.11) D ( δ + ,δ − ) ( g ) = D ( δ − ,δ + ) ( g ) | a →− a . (3.12) In the perturbative sector δ ± = {} the coeﬃcients d ( {} , {} ) k with k ≥ 1 are related to the functions f (0 , 0) k giv en in ( 2.2 ) via: d ( {} , {} ) k = ( a 2 − ℓ 2 ) ( − 1) k k ! f (0 , 0) k . (3.13) F or the leading order non-perturbative contribution δ + = { 0 } and δ − = {} w e ha v e: d ( { 0 } , {} ) k = (( a − 1) 2 − ℓ 2 ) ( − 1) k k ! f (1 , 0) k , (3.14) with the ﬁrst few f (1 , 0) k are given in ( 2.7 ). A t this exp onen tial level the Stok es constant coincides with P (1 , 0) and b y ( 2.8 ) its explicity v alue is: S ( { 0 } , {} ) = − ( − 1) ℓ  1 2 − a  2 a − 1 Γ(1 + ℓ − a ) Γ( ℓ + a ) . (3.15) The next subleading co eﬃcients d ( {} , { 0 } ) k and the corresp onding Stokes constant S ( {} , { 0 } ) can b e ob- tained from the ( { 0 } , {} ) sector by changing a → − a . F or subleading corrections, the situation becomes more complicated. As can b e seen in ( 3.7 ), for example, the Λ 4 − Λ 4 + term receives a con tribution from t w o diﬀerent sources in the sum ( 3.9 ): from the corrections with δ + = { 0 } , δ − = { 1 } and δ + = { 1 } , δ − = { 0 } . Therefore, at this order Z (2 , 2) ( g ) is exp ected to b e written as a sum: Z (2 , 2) ( g ) = S ( { 1 } , { 0 } ) D ( { 1 } , { 0 } ) ( g ) + S ( { 0 } , { 1 } ) D ( { 0 } , { 1 } ) ( g ) . (3.16) In the follo wing I presen t a metho d to compute the non-p erturbativ e functions D ( δ + ,δ − ) ( g ) from the perturbative one D ( {} , {} ) . By comparing n umerically the results with the series ( 1.7 ) obtained in [ 22 ], I verify the degeneracy of higher order contributions Z ( n,m ) ( g ) in the δ ± prescription. I also sho w a recursive w a y to eﬀectiv ely generate the Stokes constants S ( δ + ,δ − ) . This fully describ es the strong coupling expansion of the determinant ( 1.1 ). 3.1 1 /g expansion First, I describ e the structure of the series D ( δ + ,δ − ) ( g ). As w as discussed in Section 2 , the 1 /g expansion of these functions is gov erned by the integrals I n . Their v alues are given in ( 2.3 ). Using the series represen tation of the p olygamma function, these in tegrals can b e directly expressed in terms of the zeros and p oles of the the symbol χ α ( x ) (or Φ ± ( x )): I n = ∞ X k =0  ( − 1) n − 1  1 ( x + k ) n − 1 − 1 ( y k ) n − 1  −  1 ( x − k ) n − 1 − 1 ( y k ) n − 1  (3.17) – 13 – the locations of the zeros x ± k are giv en in ( 3.3 ). In Section 2 , I ha v e shown, that while the perturbative series D ( {} , {} ) is given in terms of ( 3.17 ), the 1 /g expansions of the leading order non-p erturbative sectors Z (1 , 0) ( g ), Z (0 , 1) ( g ) and Z (1 , 1) ( g ) – therefore D ( { 0 } , {} ) ( g ), D ( {} , { 0 } ) ( g ) and D ( { 0 } , { 0 } ) ( g ) – can b e easily obtained from the p erturbativ e series by applying the transformations ( 2.10 ) and ( 2.14 ) and shifting the explicit dep endence on a b y a → a − 1 and a → a + 1 and a → a resp ectively . In terms of the zeros of Φ ± ( x ), for example, the transformation ( 2.10 ) is simply written as: I n → I ( { 0 } , {} ) n ≡ I n − ( − 1) n − 1 2  x + 0  n − 1 (3.18) F rom the p oint of view of expansion ( 3.17 ), this rule corresponds to changi ng the sign before the terms con taining x + 0 to contribute to I n in the same wa y as the lo cations of the p oles y k do. In other words, the ”new” in tegral I ( { 0 } , {} ) n en tering the non-p erturbative series Z (1 , 0) ( g ) is giv en by the same in tegral as in ( 2.3 ), but with a new sym bol, χ ( { 0 } , {} ) α ( x ): χ ( { 0 } , {} ) α ( x ) = 2 cos π a x Φ { 0 } + ( x )Φ − ( − x ) , (3.19) where the function Φ { 0 } + ( x ) is: Φ { 0 } + ( x ) = Φ + ( x )  1 + ix 2 π x + 0  2 . (3.20) This function has zeros at x = 2 iπ x + j with j ≥ 1 and p oles at x = 2 iπ x + 0 and x = 2 iπ y j with j ≥ 0. W e can interpret the transformation ( 2.14 ) in the same w a y , but with another symbol: χ ( {} , { 0 } ) α ( x ) = 2 cos π a x Φ + ( x )Φ { 0 } − ( − x ) , (3.21) with the function Φ { 0 } − ( x ) being: Φ { 0 } − ( x ) = Φ − ( x )  1 + ix 2 π x − 0  2 . (3.22) No w, this function has zeros at x = 2 iπ x − j with j ≥ 1 and p oles at x = 2 iπ x − 0 and x = 2 iπ y j with j ≥ 0. These transformation rules are analogous to what we hav e found in [ 45 ]. This suggests that all the subleading non-p erturbative corrections can b e obtained in a similar wa y , b y promoting the corresp onding zeros to b e p oles of the functions χ α ( x ), namely at every exp onen tial lev el introducing a new symbol function χ ( δ + ,δ − ) : χ ( δ + ,δ − ) α ( x ) = 2 cos π a x Φ δ + + ( x )Φ δ − − ( − x ) , (3.23) with: Φ δ + + ( x ) = Φ + ( x ) Q j ∈ δ +  1 + ix 2 π x + j  2 , Φ δ − − ( x ) = Φ − ( x ) Q j ∈ δ −  1 + ix 2 π x − j  2 . (3.24) – 14 – This new sym b ol enters the 1 /g expansion of D ( δ + ,δ − ) through the in tegrals of the form ( 2.3 ) but with χ α ( x ) replaced by χ ( δ + ,δ − ) α ( x ). This results in the mo diﬁed momen ts: I ( δ + ,δ − ) n = I n − ( − 1) n − 1 X l ∈ δ + 2 ( x + l ) n − 1 + X j ∈ δ − 2 ( x − j ) n − 1 = = I n − ( − 1) n − 1 X l ∈ δ + 2  l + 1 2 − a  n − 1 + X j ∈ δ − 2 ( j + 1 2 + a ) n − 1 (3.25) where in the second relation, I restated the exact form of x ± l . T aking in to account all the proper- ties discussed ab o v e, the function D ( δ + ,δ − ) ( g ) is giv en b y the p erturbative function by replacing the in tegrals I n b y ( 3.25 ) and shifting the explicit a dep endence b y a → a − ∆, with ∆ deﬁned in ( 3.8 ) In other words, if w e indicate the functional dependence of the p erturbativ e part on I n : D ( {} , {} ) ( g ) ≡ D [ I n ] ( g ) , (3.26) then the non-p erturbativ e con tribution D ( δ + ,δ − ) ( ˜ g ) is giv en by: D ( δ + ,δ − ) ( g ) = D h I ( δ + ,δ − ) n i ( g )    a → a − ∆ . (3.27) T o put this in an explicit form, using ( 2.1 ) an ( 3.13 ), w e can parameterize any non-p erturbative function D ( δ + ,δ − ) ( g ) as: D ( δ + ,δ − ) ( g ) = 1 +  ( a − ∆) 2 − ℓ 2  X k ≥ 1 ( − 1) k k ! f ( δ + ,δ − ) k (8 π g ) k , (3.28) with the co eﬃcients f ( δ + ,δ − ) k giv en by: f ( δ + ,δ − ) 1 = I ( δ + ,δ − ) 2 , f ( δ + ,δ − ) 2 = (( a − ∆) 2 − ℓ 2 + 1) h I ( δ + ,δ − ) 2 i 2 + 2( a − ∆) I ( δ + ,δ − ) 3 , f ( δ + ,δ − ) 3 =  ( a − ∆) 2 − ℓ 2 + 1  h I ( δ + ,δ − ) 2 i 3 + + 6( a − ∆)  ( a − ∆) 2 − ℓ 2 + 2  I ( δ + ,δ − ) 2 I ( δ + ,δ − ) 3 + + 2(5( a − ∆) 2 − ℓ 2 + 1) I ( δ + ,δ − ) 4 , (3.29) etc. With this simple rule, up to the o v erall factor S ( δ + ,δ − ) , it is p ossible to generate an y non- p erturbativ e contributions to the determinan t ( 1.1 ), by computing the perturbative series of a ”new” determinan t with a suitably mo diﬁed sym bol function. A t the end of Section 2, I discussed the eﬀect of the p erio dicity of Z ℓ ( g ) in α → α + π on the strong coupling expansion. According to ( 2.26 ) and ( 2.27 ), the righ t-hand side of ( 3.27 ) is the large- g p erturbativ e series of the determinan t ev aluated in the branc h ∆ − 1 / 2 < a < ∆ + 1 / 2 with the original sym b ol χ α ( x ) replaced by χ ( δ + ,δ − ) α ( x ). It is imp ortan t to emphasize that although the zeros x ± j and hence the integrals I n are a dep enden t, the shift a → a − ∆ in ( 3.27 ) only stands for the explicit a dep endence in the coeﬃcients f (0 , 0) k . This is the reason wh y I ha v e suppressed the a dependence in the notation x ± j and I n . – 15 – F or the parameter v alues of ( a, ℓ ) = (1 / 4 , 0), ( a, ℓ ) = (1 / 4 , 1), and ( a, ℓ ) = (1 / (2 √ 2) , 2), I n umer- ically generated 50 1 /g terms in the expansions of D ( { 0 } , {} ) ( g ), D ( {} , { 0 } ) ( g ) and D ( { 0 } , { 0 } ) ( g ) with 50 digit precision. By normalizing the functions Z ( n,m ) ( g ) with their leading 1 /g co eﬃcients, I was able to verify ( 3.27 ) for these non-degenerate con tributions. F or the next few degenerate non-p erturbative sectors, from the analytic v alues of their leading 1 /g coeﬃcients, I chec k ed that there exist certain o v erall constant factors S ( δ + ,δ − ) , such that relations like ( 3.16 ) hold, with D ( δ + ,δ − ) ( g ) determined via ( 3.27 ). 3.2 Stok es constants The only remaining task to completely describ e the large- g expansion of the determinant is to give a closed formula for the Stokes constan ts S ( δ + ,δ − ) . In the follo wing I show that there is a consistent prescription to determine these constan ts with the argument giv en for D ( δ + ,δ − ) ( g ) ab ov e, whic h allo ws to deriv e a pair of recurrence relations for the Stok es constants. I start b y rewriting the leading order constan ts S ( { 0 } , {} ) , S ( {} , { 0 } ) and S ( { 0 } , { 0 } ) – whic h, according to ( 3.7 ), are essen tially the same as P (1 , 0) , P (0 , 1) and P (1 , 1) – in terms of the functions Φ ± ( x ) and the lo cations of their zeros and p oles x ± j and y j . F rom the metho d presented in [ 22 ] it follows that the factors S ( { 0 } , {} ) and S ( {} , { 0 } ) can be written as: S ( { 0 } , {} ) = ( − 1) ℓ 2 π i Γ(1 + ℓ − a ) Γ( ℓ + a ) Φ − ( − 2 iπ x + 0 ) ∂ x Φ + (2 iπ x + 0 ) ( x + 0 ) 2 a − 2 S ( {} , { 0 } ) = ( − 1) ℓ 2 π i Γ(1 + ℓ + a ) Γ( ℓ − a ) Φ + ( − 2 iπ x − 0 ) ∂ x Φ − (2 iπ x − 0 ) ( x − 0 ) − 2 a − 2 (3.30) By inv estigating higher order contributions, it turns out that similar combinations app ear in the Stok es constan ts at each exp onential le v el, but with x ± 0 replaced b y x ± j . Therefore, it is con v enien t to in tro duce the functions: S { j } + = ( − 1) ℓ 2 π i Γ( ℓ − a ) Γ(1 + ℓ + a ) Φ − ( − 2 iπ x + j ) ∂ x Φ + (2 iπ x + j ) ( x + j ) 2 a S { j } − = ( − 1) ℓ 2 π i Γ( ℓ + a ) Γ(1 + ℓ − a ) Φ + ( − 2 iπ x − j ) ∂ x Φ − (2 iπ x − j ) ( x − j ) − 2 a (3.31) Then the leading order co eﬃcients are giv en by: S ( { 0 } , {} ) = S { 0 } +    a → a − 1 , S ( {} , { 0 } ) = S { 0 } −    a → a +1 . (3.32) Notice that the shift in a is again understo o d only for the explicit dependence on a and the functions Φ ± ( x ) and their zeros x ± j are treated as indep endent ob jects. S { j } ± w as chosen in such a wa y that this shift is consistent with a → a − ∆ appearing in ( 3.27 ). By generating exact analytical expressions for subleading corrections up to Λ 6 − Λ 6 + , I found that at orders O  e − 8 π g x ± j  , similar relations to ( 3.32 ) still hold, namely at orders ( δ + , δ − ) = ( { j } , {} ), the Stok es constants are: S ( { j } , {} ) = S { j } +    a → a − 1 , (3.33) – 16 – whereas if δ + = {} and δ − = { j } w e ha v e: S ( {} , { j } ) = S { j } −    a → a +1 . (3.34) The problem arises for mixed terms, that is, when δ + and δ − together contain more than one elemen t. By generating subleading corrections D ( δ + ,δ − ) ( g ) with ( 3.27 ) and, with their prop er degener- acy , comparing their linear com binations with Z ( δ + ,δ − ) , for the ﬁrst few subleading Stok es constants, I found: S ( { 0 } , { 0 } ) = −  1 2 − a  1+2 a  1 2 + a  1 − 2 a S ( { 0 , 1 } , {} ) = −  1 2 − a  − 1+2 a  3 2 − a  − 3+2 a Γ( ℓ − a + 1)Γ( ℓ − a + 2) Γ( ℓ + a )Γ( ℓ + a − 1) S ( { 0 , 1 } , { 0 } ) = ( − 1) ℓ 4  1 2 + a  3 − 2 a  1 2 − a  1+2 a  3 2 − a  − 1+2 a Γ( ℓ − a + 1) Γ( ℓ + a ) S ( { 0 , 1 , 2 } , {} ) =  1 2 − a  − 1+2 a  3 2 − a  − 3+2 a  5 2 − a  − 5+2 a Γ( ℓ − a + 1)Γ( ℓ − a + 2)Γ( ℓ − a + 3) Γ( ℓ + a )Γ( ℓ + a − 1)Γ( ℓ + a − 2) S ( { 0 , 1 } , { 0 , 1 } ) = 1 144  1 2 − a  3+2 a  3 2 − a  1+2 a  1 2 + a  3 − 2 a  3 2 + a  1 − 2 a (3.35) F urther can b e obtained b y using the symmetry property in ( 3.11 ). The ﬁrst tw o co eﬃcients ab ov e (together with their a → − a coun terparts) can also b e expressed in terms of the functions in ( 3.31 ): S ( { 0 , 1 } , {} ) = −  1 − x + 1 x + 0  2  S { 0 } +    a → a − 1   S { 1 } +    a → a − 2  , S ( {} , { 0 , 1 } ) = −  1 − x − 1 x − 0  2  S { 0 } −    a → a +1   S { 1 } −    a → a +2  , S ( { 0 } , { 0 } ) = −  1 + x − 0 x + 0  − 2  S { 0 } +    a → a − 1  S { 0 } − . (3.36) Therefore, in addition to the pro ducts of S { j } ± with certain shifts in parameter a , we ha ve additional factors containing the zeros x ± j . Moreo v er, for the mixed terms, there is more than one wa y to express them in terms of ( 3.31 ). F or example, b eside ( 3.36 ), w e can also write: S ( { 0 , 1 } , {} ) = −  1 − x + 0 x + 1  2  S { 1 } +    a → a − 1   S { 0 } +    a → a − 2  , S ( {} , { 0 , 1 } ) = −  1 − x − 0 x − 1  2  S { 1 } −    a → a +1   S { 0 } −    a → a +2  , S ( { 0 } , { 0 } ) = −  1 + x + 0 x − 0  − 2  S { 0 } −    a → a +1  S { 0 } + . (3.37) In this case, there are diﬀerent shifts in a and diﬀeren t prefactors, but surprisingly the results are the same as ( 3.37 ). F ollowing the same argumen t that I presented for the functions D ( δ + ,δ − ) ( g ), these prop erties can b e understo o d in a simple w a y . Assume that w e start with the exponential correction ( δ + , δ − ) = – 17 – ( { j } , {} ) whose Stok es constant is given in ( 3.33 ). Now app end an elemen t k  = j to δ + , so w e ha v e the exp onen tial correction at level ( δ ′ + , δ − ) = ( { j, k } , {} ). With this step, ∆ increased and b ecame ∆ ′ = | δ ′ + |−| δ − | = 2. Then motiv ated b y the structure of D ( δ + ,δ − ) ( g ), it is reasonable to guess that the new Stokes constant is ( 3.33 ) m ultiplied by S { k } + with its explicit a dep endence shifted by a → a − ∆ ′ = a − 2. Ho w ev er, in addition to the shift, w e saw that at every exp onential lev el, to generate the functions D ( δ + ,δ − ) ( g ), w e ha v e to remov e the corresponding zeros of Φ ± ( x ) and promote them to b e p oles. Therefore, w e ha v e to use the new functions in ( 3.24 ) to describ e the 1 /g expansion. Since at the level ( δ + , δ − ) = ( { j } , {} ) we already ”remov ed” the x + j zero, this suggests that to obtain S ( { j,k } , {} ) from S ( { j } , {} ) , w e ha v e to m ultiply ( 3.33 ) with: ( − 1) ℓ 2 π i Γ( ℓ − a ) Γ(1 + ℓ + a ) Φ − ( − 2 iπ x + k ) ∂ x Φ { j } + (2 iπ x + k ) ( x + k ) 2 a      a → a − 2 = 1 − x + k x − j ! 2  S { k } +    a → a − 2  . (3.38) In case of j = 0, k = 1 and j = 1, k = 0, up to an ov erall min us sign, this reproduces the ﬁrst lines of ( 3.36 ) and ( 3.37 ). Similarly if w e start with ( δ + , δ − ) = ( { j } , {} ) and go to lev el ( δ + , δ ′ − ) = ( { j } , { k } ) then we get the same num ber of ”+” type con tributions as ” − ” ones, therefore at this lev el we ha v e ∆ ′ = | δ + |−| δ ′ − | = 0. In the ﬁrst step x + j w as remo v ed from the symbol. Therefore, it suggests that the new Stok es constan t is giv en b y the pro duct of ( 3.34 ) and: ( − 1) ℓ 2 π i Γ( ℓ + a ) Γ(1 + ℓ − a ) Φ { j } + ( − 2 iπ x − k ) ∂ x Φ − (2 iπ x − k ) ( x − k ) − 2 a = 1 + x − k x + j ! − 2 S { k } − . (3.39) With j = 0 and k = 0, up to an ov erall min us sign, this repro duces the result in the last line of ( 3.36 ). The similar pro cedure can b e done b y starting from ( δ + , δ − ) = ( {} , { j } ) to repro duce the rest of ( 3.36 ) and ( 3.37 ). The simultaneous shifts of a and remo v al of zeros guarantees that the tw o prescriptions giv e the same answer indep enden tly of the path of adding elements to δ ± . This pro cedure can b e generalized to arbitrary exp onentially small corrections. By app ending more and more elements to δ + and δ − , at each step w e ha ve to m ultiply the previous Stokes constant with ( 3.31 ), but with Φ ± ( x ) replaced b y Φ δ ± ± ( x ) and the explicit a dep endence shifted b y the new ∆. The pro cedure can b e summarized in the following wa y: the Stok es constan t for the p erturbative part is: S ( {} , {} ) = 1 . (3.40) No w assume that we know the v alue of S ( δ + ,δ − ) at a speciﬁc lev el ( δ + , δ − ). Then the Stok es constan t at level ( δ ′ + , δ − ) = ( δ + ∪ { k } , δ − ), which is obtained from ( δ + , δ − ) by adding the integer k / ∈ δ + to the set δ + is giv en b y: S ( δ ′ + ,δ − ) = ( − 1) ∆ Q j ∈ δ +  1 − x + k x + j  2 Q j ∈ δ −  1 + x + k x − j  2  S { k } +    a → a − ∆ ′  S ( δ + ,δ − ) , (3.41) where ∆ = | δ + |−| δ − | and ∆ ′ = | δ ′ + |−| δ − | = ∆ + 1. – 18 – Similarly , the Stokes constan t at level ( δ + , δ ′ − ) = ( δ + , δ − ∪ { k } ), which diﬀers from the original ( δ + , δ − ) b y adding the element k / ∈ δ − to δ − , is given b y: S ( δ + ,δ ′ − ) = ( − 1) ∆ Q j ∈ δ −  1 − x − k x − j  2 Q j ∈ δ +  1 + x − k x + j  2  S { k } −    a → a − ∆ ′  S ( δ + ,δ − ) , (3.42) where in this case ∆ ′ = | δ + |−| δ ′ − | = ∆ − 1. Using these recurrence relations, the Stok es constants can b e determined up to arbitrary non-perturbative orders. The prefactors ( − 1) ∆ in ( 3.42 ) and ( 3.41 ) w ere in troduced by in v estigating n umerical results with diﬀeren t v alues of a and ℓ up to Λ 5 − Λ 5 + . The origin of these factors can b e understo o d by absorbing the e iπ a ∆ factors from ( 3.9 ) as e ± iπ a phases in ( 3.31 ). Then the Stokes constan ts will no longer b e real, but the a → a − ∆ shifts in the recurrence relations ( 3.41 ) and ( 3.42 ) will pro duce exactly the same factors ( − 1) ∆ as in ( 3.41 ) and ( 3.42 ) and will produce the same complex phases e iπ a ∆ as in ( 3.9 ). W e will see later that the prescription that I used abov e is more conv enien t regarding the resurgence analysis of the transseries ( 3.9 ). T o put the recurrence relations ( 3.41 ) and ( 3.42 ) into a more practical form, I recov er the explicit forms of the functions Φ ± ( x ) and the zeros x ± l . With this, the ﬁrst recurrence relation b ecomes S ( δ + ∪{ k } ,δ − ) = − ( − 1) ℓ +∆ Γ ( ℓ − a + ∆ + 1) Γ ( ℓ + a − ∆) Γ 2  k + 1 2 − a  Γ 2  1 2 − a  Γ 2 ( k + 1) × × Q j ∈ δ +  j − k j + 1 2 − a  2 Q j ∈ δ −  j + k +1 j + 1 2 + a  2  k + 1 2 − a  2 a − 1 − 2∆ S ( δ + ,δ − ) , (3.43) while the second one is: S ( δ + ,δ − ∪{ k } ) = − ( − 1) ℓ − ∆ Γ ( ℓ + a − ∆ + 1) Γ ( ℓ − a + ∆) Γ 2  k + 1 2 + a  Γ 2  1 2 + a  Γ 2 ( k + 1) × × Q j ∈ δ − j  j − k j + 1 2 + a  2 Q j ∈ δ + j  j + k +1 j + 1 2 − a  2  k + 1 2 + a  − 2 a − 1+2∆ S ( δ + ,δ − ) . (3.44) These relations repro duce the examples giv en in ( 3.35 ). The recurrence relations ( 3.41 ) and ( 3.42 ) together with ( 3.27 ) completely describ e the large- g expansion of the determinant Z ( g ) and allo w us to eﬀectiv ely determine the physical observ ables dis- cussed in Section 1 up to arbitrary non-p erturbativ e and 1 /g orders b y only knowing the p erturbative con tributions of the determinan t. The relations ( 3.27 ), ( 3.43 ) and ( 3.44 ) w ere numerically veriﬁed for the parameter v alues of ( a, ℓ ) = (1 / 4 , 0), ( a, ℓ ) = (1 / 4 , 1) and ( a, ℓ ) = (1 / (2 √ 2) , 2) by comparing diﬀeren t corrections in ( 3.9 ) with the corresp onding terms ( 1.7 ) up to order Λ 8 − Λ 8 + . 3.3 A practical example In this subsection, I sp ecify the parameter a as a = 1 / 4 and generate exact strong-coupling results for the determinant. As was discussed in the In tro duction, this is a relev ant example for the N = 4 SYM, – 19 – since the ratio of the determinants ev aluated at ℓ = 1 and ℓ = 0 giv es the cusp anomalous dimension (see equation ( 1.6 )). F or a = 1 / 4, the zeros of Φ ± ( x ) are lo cated at: x + j = j + 1 4 , x − j = j + 3 4 , (3.45) with j ≥ 0 an integer. Therefore, in this case, the exp onen tial corrections scale as p ow ers of e − 2 π g and w e can parametrize the strong coupling expansion for a = 1 / 4 as: Z ℓ ( g ) = A ℓ ( g ) X n ≥ 0 e − 2 π ng C ( n ) ℓ ( g ) . (3.46) Using ( 3.27 ), the ﬁrst few co eﬃcient functions C ( n ) ℓ ( g ) are: C (0) ℓ ( g ) = D ( {} , {} ) ( g )    a =1 / 4 , C (1) ℓ ( g ) = e iπ 4 (8 π g ) − 1 2 S ( { 0 } , {} ) D ( { 0 } , {} ) ( g )    a =1 / 4 , C (2) ℓ ( g ) = 0 , C (3) ℓ ( g ) = e − iπ 4 (8 π g ) − 3 2 S ( {} , { 0 } ) D ( {} , { 0 } ) ( g )    a =1 / 4 , C (4) ℓ ( g ) = S ( { 0 } , { 0 } ) D ( { 0 } , { 0 } ) ( g )    a =1 / 4 , C (5) ℓ ( g ) = e iπ 4 (8 π g ) − 1 2 S ( { 1 } , {} ) D ( { 1 } , {} ) ( g )    a =1 / 4 (3.47) In the examples ab o v e, for a given C ( n ) ℓ ( g ), only one non-p erturbativ e function D ( δ + ,δ − ) ( g ) con- tributes. How ever, for a = 1 / 4, at levels with n ≥ 8, certain com binations of ( δ + , δ − ) pro duce an exp onen tial contribution to ( 3.46 ) at the same lev el, therefore, in general C ( n ) ( g ) is a linear combina- tion of m ultiple D ( δ + ,δ − ) ( g ). At higher orders, to compute C ( n ) ( g ), one should sum up all the ( δ + , δ − ) corrections in ( 3.9 ), for which: X l ∈ δ + (4 l + 1) + X j ∈ δ − (4 j + 3) = n (3.48) Since I n are indep enden t of the parameter ℓ , b efore giving explicit expressions for the diﬀerent co eﬃcien ts in ( 3.46 ) with ℓ = 0 , 1, it is con v enien t to rewrite the momen ts in a more familiar form. By substituting x ± k with a = 1 / 4 in to ( 3.17 ), the moments I n with odd n can b e written as: I 2 k +1 = 2 2 k β (2 k ) , (3.49) while for even n : I 2 k = − (2 4 k − 2 − 2 2 k − 1 − 2) ζ (2 k − 1) , (3.50) with ζ ( x ) is the Riemann zeta function and β ( x ) is the Diric hlet b eta function. The same sp ecial functions already app eared in the analysis of the strong coupling expansion of the cusp anomalous dimension. F or example, in equations (28) and (29) of [ 52 ] the p erturbative part of the cusp anomalous dimension were given in terms of some c oeﬃcients c n prop ortional to ζ ( x ) and β ( x ) with odd and ev en int eger arguments resp ectively . Hence these co eﬃcients are in one-to-one corresp ondence with momen ts I n , namely: I 2 = − 8 π c 1 , I 3 = 256 π 2 c 2 , I 4 = − 4096 π 3 c 3 , I 5 = 262144 21 π 4 c 4 , I 6 = − 4194304 87 π 5 c 5 , I 7 = 134217728 1605 π 6 c 6 , (3.51) – 20 – etc. T o a v oid cumbersome expressions, in the follo wing I will express ev erything in terms of I n . According to ( 3.25 ) the momen ts I ( δ + ,δ − ) n en tering the higher order con tributions are obtained from ( 3.49 ) and ( 3.50 ) via: I ( δ + ,δ − ) n = I n − ( − 1) n − 1 X l ∈ δ + 2  l + 1 4  n − 1 + X j ∈ δ − 2 ( j + 3 4 ) n − 1 (3.52) Then using rule ( 3.27 ), all subleading D ( δ + ,δ − ) ( g ) ev aluated at a = 1 / 4 can b e easily obtained from the perturbative part: D ( δ + ,δ − ) ( g )    a =1 / 4 = D h I ( δ + ,δ − ) n i ( g )    a =1 / 4 − ∆ , (3.53) where D [ I n ] ( g ) coincides with the p erturbative function in ( 2.1 ), and the in tegrals I ( δ + ,δ − ) n are given b y ( 3.52 ), with I n expressed in ( 3.49 ) and ( 3.50 ). Substituting these in tegrals in to the p erturbative function ( 2.1 ), ev aluating them at a = 1 / 4 − ∆ with the appropriate shift, and summing them up according to ( 3.48 ), arbitrary non-p erturbative correction can b e determined to ( 3.46 ). No w I turn to the special cases of ℓ = 0 and ℓ = 1. ℓ = 0 F or a = 1 / 4 and ℓ = 0, ﬁrst w e hav e to determine the prefactor A ( g ). By ( 1.11 ) and equation (5.18) of [ 22 ], in this sp ecial case its v alue is given as: A ℓ =0 ( g ) = e 3 πg 4 g 5 16 Γ  1 4  1 4 Γ  3 4  − 1 4 e 1 8 2 37 48 π 5 16 A 3 2 , (3.54) Where A is the Glaisher–Kinkelin constant. In this case, by ( 3.29 ), ( 3.47 ) and ( 3.52 ), the ﬁrst few non-trivial exp onentially small con tributions to Z ℓ =0 , up to the ﬁrst three leading order terms in 1 /g , are: C (0) ℓ =0 = 1 − I 2 128 π g + 17 I 2 2 + 8 I 3 32768 π 2 g 2 − 187 I 3 2 + 264 I 2 I 3 + 224 I 4 4194304 π 3 g 3 + O  1 g 4  , C (1) ℓ =0 = − e iπ 4 (8 π g ) 1 2 2Γ  3 4  Γ  1 4   1 − 9( I 2 + 8) 128 π g + 225( I 2 + 8) 2 − 216( I 3 − 32) 32768 π 2 g 2 − − 3075( I 2 + 8) 3 − 8856( I 2 + 8)( I 3 − 32) + 5856( I 3 + 128) 4194304 π 3 g 3 + O  1 g 4  , C (3) ℓ =0 = e − iπ 4 (8 π g ) 3 2 2Γ  5 4  3 √ 3Γ  3 4   1 − 25(3 I 2 + 8) 384 π g + 1025(3 I 2 + 8) 2 + 1000(9 I 3 + 32) 294912 π 2 g 2 − − 19475(3 I 2 + 8) 3 + 57000(3 I 2 + 8)(9 I 3 + 32) + 37600(27 I 4 + 128) 113246208 π 3 g 3 + O  1 g 4  , (3.55) while C (2) ℓ =0 = 0. By generating higher order corrections in 1 /g for the p erturbativ e function D ( {} , {} ) ( g ), the pro cedure of computing exponentially suppressed corrections to ( 3.46 ) can b e done up to arbitrary order in e − 2 π ng . – 21 – ℓ = 1 F or a = 1 / 4 and ℓ = 1, one ﬁnds for the v alue of the o v erall prefactor A ( g ) : A ℓ =1 ( g ) = e 3 πg 4 g 21 16 Γ  1 4  5 4 Γ  3 4  3 4 e 1 8 2 109 48 π 21 16 A 3 2 (3.56) Rep eating the same argumen t as for the ℓ = 0 case, the ﬁrst few non-p erturbative functions C ( n ) ( g ) are: C (0) ℓ =1 = 1 + 15 I 2 128 π g − 15 I 2 2 + 120 I 3 32768 π 2 g 2 + 85 I 3 2 + 2040 I 3 I 2 + 800 I 4 4194304 π 3 g 3 + O  1 g 4  , C (1) ℓ =1 = e iπ 4 (8 π g ) 1 2 2Γ  7 4  Γ  5 4   1 + 7( I 2 + 8) 128 π g − 63( I 2 + 8) 2 − 168( I 3 − 32) 32768 π 2 g 2 + + 1575( I 2 + 8) 3 − 12600( I 2 + 8)( I 3 − 32) + 10080( I 4 + 128) 12582912 π 3 g 3 + O  1 g 4  , C (3) ℓ =1 = e − iπ 4 (8 π g ) 3 2 2Γ  9 4  √ 3Γ  7 4   1 − 9 I 2 + 24 128 π g + 25(9 I 2 + 24) 2 + 360(9 I 3 + 32) 294912 π 2 g 2 − − 1025(9 I 2 + 24) 3 + 44280(9 I 2 + 24)(9 I 3 + 32) + 108000(27 I 4 + 128) 1019215872 π 3 g 3 + O  1 g 4  (3.57) While C (2) ℓ =1 ( g ) is again equal to zero. Cusp anomalous dimension Finally , using ( 1.6 ), I p esent some analytic results for the strong coupling expansion of the cusp anomalous dimension. If we put the expansion ( 3.46 ) into ( 1.6 ), w e ﬁnd that the strong coupling expansion of Γ cusp is go v erned by the pow ers of e − 2 π g . Hence Γ cusp ( g ) can b e parametrized as: Γ cusp ( g ) = X n ≥ 0 e − 2 π ng γ ( n ) ( g ) , (3.58) in agreemen t with [ 41 ]. Substituting ( 3.55 ) and ( 3.57 ) and the v alues of A ( g ) in to Z ℓ =0 ( g ) and Z ℓ =1 ( g ) and expanding – 22 – the ratio ( 1.6 ) in p ow ers of e − 2 π g and 1 /g , for the ﬁrst few γ ( n ) ( g ) corrections one obtains: γ (0) ( g ) = 2 g  1 + I 2 8 π g − I 3 256 π 2 g 2 + 2 I 2 I 3 + I 4 4096 π 3 g 3 + O  1 g 4  , γ (1) ( g ) = e iπ 4 2 3 2 3 1 8 g 1 2 π 1 2 k 1 4  1 + 3 + I 2 16 π g − 54 + 12 I 2 + 2 I 2 2 − I 3 1024 π 2 g 2 + + 2 I 3 2 + 18 I 2 2 − 3 I 3 I 2 + 162 I 2 − 21 I 3 + 6 I 4 + 714 16384 π 3 g 3 + O  1 g 4  , γ (2) ( g ) = e iπ 2 3 1 4 π k 1 2  1 − 3 8 π g + 24 I 2 − 3 I 3 + 252 512 π 2 g 2 − − 24 I 2 2 − 6 I 3 I 2 + 504 I 2 − 57 I 3 + 4 I 4 + 3684 4096 π 3 g 3 + O  1 g 4  , γ (3) ( g ) = e − iπ 4 ( k − 9) 2 3 2 3 13 8 π 3 2 k 3 4 g 1 2  1 − 3( k − 9) I 2 + 5( k − 81) 48 π g ( k − 9) − + 2(125 k − 56133) + 180( k − 81) I 2 + 54( k − 9) I 2 2 + 9(7 k + 117) I 3 9216 π 2 g 2 ( k − 9) + O  1 g 3  , (3.59) with the moments I n giv en in ( 3.49 ) and ( 3.50 ). F or conv enience, I ha v e introduced the notation: k = √ 3Γ  1 4  8 64 π 4 . (3.60) F or a = 1 / 4, all the Stokes constants S ( δ + ,δ − ) and the prefactors A ( g ) con tain combinations of Γ( n/ 4) with | n | b eing an o dd in teger. Hence, using w ell-kno wn relations for the Gamma functions, every function γ ( n ) ( g ) can b e completely expressed in terms of the constant k and the momen ts I n . The p erturbativ e part in ( 3.59 ) repro duces the expansion presen ted in [ 52 ], while the ﬁrst non- p erturbativ e correction coincides with the results of [ 41 ]. With the metho d discussed ab o v e, an y higher order non-p erturbative contribution to the cusp anomalous dimension can b e easily obtained in an extremely eﬃcien t w a y . 4 Resurgence relations W e see that ( 3.27 ) together with the recurrence relations ( 3.43 ) and ( 3.44 ) is extremely eﬃcient to gain information ab out the large- g expansion of the determinan t with matrix Bessel kernel, hence to compute the strong coupling expansion of several ph ysical observ ables in sup ersymmetric gauge theories. Although the method presen ted ab ov e is pow erful to obtain analytic results, b y c ho osing sp eciﬁc v alues of a and ℓ , higher order con tributions with h undreds of 1 /g terms can b e achiev ed n umerically up to arbitrary exp onential orders. Using relation ( 3.27 ) and n umerically ev aluating 1 /g co eﬃcients for the functions D ( δ + ,δ − ) ( g ) with sp eciﬁc v alues of a and ℓ , one ﬁnds that these co eﬃcien ts are asymptotic or, in other w ords, d ( δ + ,δ − ) k gro w factorially for large v alues of k . This means that the transseries in ( 3.9 ) is only formal and a resurgence analysis [ 38 – 40 ] is essential to b e able to resum the transeries and obtain a reasonable ph ysical answer. In [ 22 ] w e ha v e already in vestigated the resurgence prop erties of the determinan t in represen tation ( 1.7 ). There we ha v e studied the asymptotic b ehavior of the leading non-p erturbativ e contributions – 23 – with high precision n umerical analysis. Although w e found that certain con tributions are connected via direct resurgence relations, it turned out that not all of them are related, and the reason of the absence of certain relations remained unknown. The structure of direct connections is illustrated in Figure 3. of [ 22 ]. In this section I will follo w the ideas discussed in App endix A to derive some basic resurgence prop erties for the restructured strong coupling expansion ( 3.9 ). I sho w that the asymptotic behavior of the non-perturbative functions and the analytic structure of their Borel transforms immediately follo w from the transseries structure and that the new form of the expansion shed light to the resurgence structure found in [ 22 ]. With high precision numerical analysis, I v erify these resurgence relations and b y that, I giv e another strong evidence on the v alidity of ( 3.9 ). Since in the transseries ( 3.9 ), the coupling constant g only app ears together with a factor of 8 π , to a void lengthy expressions, in the follo wing I will use the notation: ˜ g = 8 π g = 2 √ λ . (4.1) In terms of the rescaled coupling constan t ˜ g , with a slight abuse of notation, the strong coupling expansion tak es the form: Z ℓ ( g ) = A ℓ ( g ) X δ + ,δ − ˜ g − ∆(∆ − 2 a ) e − ˜ g ( P l ∈ δ + x + l + P j ∈ δ − x − j ) e iπ a ∆ S ( δ + ,δ − ) D ( δ + ,δ − ) ( ˜ g ) . (4.2) D ( δ + ,δ − ) ( g ) giv en by the series: D ( δ + ,δ − ) ( ˜ g ) = X k ≥ 0 d ( δ + ,δ − ) k ˜ g k , (4.3) with the same expansion co eﬃcients d ( δ + ,δ − ) k as before. 4.1 Stok es automorphism The asymptotic b ehavior of the p erturbative function D ( {} , {} ) is captured by the analytic prop erties of its Borel transform: B h D ( {} , {} ) i ( s ) = X k ≥ 0 d ( {} , {} ) k s k Γ( k + 1) . (4.4) F or a con v ergen t series of the form ( 4.3 ), the function B  D ( {} , {} )  ( s ) can b e resummed, and its in verse, giv en by the in tegral ( A.4 ) o v er the positive real line, gives back the original function. Ho w ev er, since the co eﬃcien ts d ( {} , {} ) k gro w factorially with k , the function B  D ( {} , {} )  ( s ) pro- duces singularities and cuts on the complex s plane, called the Borel plane. These singularities should b e a v oided while taking the inv ers transformation either b elow, or ab o v e the real axis. Therefore, one has to deﬁne the lateral Borel resummations: S ± h D ( {} , {} ) i ( ˜ g ) = ˜ g Z ∞ e iϵ 0 dse − ˜ g s B h D ( {} , {} ) i ( s ) . (4.5) This means that there is an ambiguit y in the resummation of D ( {} , {} ) ( ˜ g ). This am biguit y is the reason of the appearance of non-p erturbativ e con tributions in the transseries ( 3.9 ). The same argumen t holds for the higher order, exp onentially small corrections as w ell. The t wo lateral Borel resummations S + and S − are related by the Stok es automorphism S : S + = S − ◦ S . (4.6) – 24 – Its logarithm is deﬁned by the so called Alien deriv ativ es ∆ ω : ln S = X ω e − ˜ g ω ∆ ω (4.7) where ω denotes the singular p oints of ( 4.4 ) on the real line. The Alien deriv ativ es describ e the discon tinouities of the Borel transform across the singular p oints ω . F or a giv en transseries, suc h as ( 3.9 ), obtained from some system of integro-diﬀeren tial equations, the Alien deriv ativ es can be computed via the Bridge equations [ 38 – 40 ]. If we assign a parameter σ ± l to ev ery distinct exponential weigh t x ± l , w e can deﬁne: Z ℓ  g , { σ ± }  = A ℓ ( ˜ g ) X δ + ,δ −   Y l ∈ δ + σ + l e − ˜ g x + l     Y j ∈ δ − σ − j e − ˜ g x − j   ˜ g − ∆(∆ − 2 a ) e iπ a ∆ S ( δ + ,δ − ) D ( δ + ,δ − ) ( ˜ g ) (4.8) Then the Bridge equations connect the Alien deriv ativ es of Z ℓ ( g , { σ ± } ) to ordinary diﬀeren tials with resp ect to the parameters σ ± l . In App endix A, using the Bridge equations, I deriv e the Alien deriv atives for a similar, but somewhat less complicated transseries than ( 3.9 ). It is simpler in the sense, that it contains only one set of exp onential w eigh ts { x l } l ≥ 0 instead of having t w o sectors { x ± l } l ≥ 0 , and it do es not contain the complex phases e iπ a ∆ . Ho w ev er, for the relev ant v alues 0 < a < 1 / 2, the ph ysical w eigh ts can b e ordered as x + 0 < x − 0 < x + 1 < x − 1 < . . . , therefore, we can assign x 2 l → x + l , x 2 l +1 → x − l , and redeﬁne D ( δ + ,δ − ) ( ˜ g ) b y collecting its prefactors, so w e arriv e at the same transseries as in ( A.1 ). The pow er λ ( δ ) of the leading 1 /g terms is given by − ∆(∆ − 2 a ) in this case. Rep eating the same argument as in Appendix A, one ﬁnds that the singular p oints of B h D ( δ + ,δ − ) i ( s ) app ear on the positive real line at x ± j with j / ∈ δ + , δ − and the corresp onding Alien deriv atives are: ∆ + j D ( δ + ,δ − ) ( ˜ g ) = ˜ g − 1 − 2∆+2 a A + j S ( δ + ∪{ j } ,δ − ) S ( δ + ,δ − ) D ( δ + ∪{ j } ,δ − ) ( ˜ g ) , ∆ − j D ( δ + ,δ − ) ( ˜ g ) = ˜ g − 1+2∆ − 2 a A − j S ( δ + ,δ − ∪{ j } ) S ( δ + ,δ − ) D ( δ + ,δ − ∪{ j } ) ( ˜ g ) . (4.9) The factors A ± j are y et undetermined n um bers, and for simplicity , I ha v e in troduced the notation ∆ ± j ≡ ∆ x ± j . F or j ∈ δ + or j ∈ δ − the Alien deriv atives give zero. F rom the Bridge equations it also follo ws that ( 4.9 ) is supplemen ted with the relations:  ∆ ± j  2 = 0 , h ∆ + l , ∆ + j i = h ∆ − l , ∆ − j i = h ∆ + l , ∆ − j i = 0 , (4.10) for all j, l ≥ 0 in tegers. Equations ( 4.9 ) and ( 4.10 ) mean that D ( δ + ,δ − ) ( ˜ g ) resurges only with the subleading corrections D ( δ + ∪{ j } ,δ − ) ( ˜ g ) and D ( δ + ,δ − ∪{ j } ) ( ˜ g ) (so that their exp onential orders diﬀer only b y a factor of e − ˜ g x ± j ). This is in accordance with the ﬁndings of [ 22 ]. There w e found a direct resurgence relation, for example, b et w een the ( n, m ) = (0 , 0) and ( n, m ) = (2 , 1) terms of ( 1.7 ), but not b etw een the ( n, m ) = (0 , 0) and ( n, m ) = (2 , 2) con tributions. The ( n, m ) = (0 , 0) p erturbativ e part b elongs to ( δ + , δ − ) = ( {} , {} ) in the ( 3.9 ) prescription. Due to ( 4.9 ) it has a direct resurgence relation with ( δ + , δ − ) = ( { 1 } , {} ), which b y ( 3.7 ) corresponds to ( n, m ) = (2 , 1). In contrast, the ( δ + , δ − ) = ( { 1 } , { 0 } ) term is not related to the p erturbativ e part, since they diﬀer in more than t w o diﬀeren t exp onential factors of e − ˜ g x ± j . – 25 – Therefore, by ( 3.7 ), ( n, m ) = (0 , 0) should not b e in direct resurgence relation with ( n, m ) = (2 , 2). This is in complete agreement with Figure 3. of [ 22 ]. The Alien algebra in ( 4.9 ) and ( 4.10 ) is useful to determine the asymptotic b eha vior of the co eﬃcien ts d ( δ + ,δ − ) k . Using the algebra ( 4.10 ) we ﬁnd from deﬁnition ( 4.7 ) that S is: S = Y l ≥ 0  1 + e − x + l ˜ g ∆ + l  Y j ≥ 0  1 + e − x − j ˜ g ∆ − j  . (4.11) As it is discussed in App endix A, b y acting with S on the non-p erturbativ e functions D ( δ + ,δ − ) ( g ), one can ﬁnd that the diﬀerence b et w een the lateral resummations, hence the discon tinouities of their Borel transforms can be easily related to the subleading strong coupling corrections. 4.2 Asymptotic analysis In the follo wing I illustrate how the Stok es automorphism ( 4.11 ) giv es the opportunity to determine the asymptotic b ehavior of the non-p erturbativ e corrections. F or simplicit y , I presen t the metho d only for a sp ecial set of corrections that correspond to labels of the form ( δ + , δ − ) = ( { 0 , 1 , . . . , l } , { 0 , 1 , . . . , l − 1 } ) and ( δ + , δ − ) = ( { 0 , 1 , . . . , l } , { 0 , 1 , . . . , l } ), where the elemen ts of δ + and δ − go along all non-negative in tegers up to l and l − 1. These corrections are sp ecial, since b y n umerically inv estigating the asymptotic b ehavior of their 1 /g expansions, the co eﬃcients A ± l can b e determined and the results for the corresp onding Stok es constants S ( δ + ,δ − ) can be tested. Since the exp onen tial w eigh ts are ordered as x + 0 < x − 0 < x + 1 < x − 1 < . . . , by acting with S on the ﬁrst few such corrections and using ( 4.9 ) w e ﬁnd: S + h D ( {} , {} ) i ( ˜ g ) − S − h D ( {} , {} ) i ( ˜ g ) = ˜ g − 1+2 a e − ˜ g x + 0 A + 0 S ( { 0 } , {} ) D ( { 0 } , {} ) ( ˜ g ) + . . . , S + h D ( { 0 } , {} ) i ( ˜ g ) − S − h D ( { 0 } , {} ) i ( ˜ g ) = ˜ g 1 − 2 a e − ˜ g x − 0 A − 0 S ( { 0 } , { 0 } ) S ( { 0 } , {} ) D ( { 0 } , { 0 } ) ( ˜ g ) + . . . , S + h D ( { 0 } , { 0 } ) i ( ˜ g ) − S − h D ( { 0 } , { 0 } ) i ( ˜ g ) = ˜ g − 1+2 a e − ˜ g x + 1 A + 1 S ( { 0 , 1 } , { 0 } ) S ( { 0 } , { 0 } ) D ( { 0 , 1 } , { 0 } ) ( ˜ g ) + . . . , S + h D ( { 0 , 1 } , { 0 } ) i ( ˜ g ) − S − h D ( { 0 , 1 } , { 0 } ) i ( ˜ g ) = ˜ g 1 − 2 a e − ˜ g x − 1 A − 1 S ( { 0 , 1 } , { 0 , 1 } ) S ( { 0 , 1 } , { 0 } ) D ( { 0 , 1 } , { 0 , 1 } ) ( ˜ g ) + . . . , (4.12) and so on. The dots denote exp onen tially suppressed terms. Then the leading asymptotic expansion of the 1 /g -co eﬃcients can be parametrized as: d ( δ + ,δ − ) k = 1 π X j ≥ 0 c j Γ( k + λ − j ) A k + λ − j + . . . , (4.13) where the parameters A , λ and c j can be ﬁxed b y ( 4.12 ). By taking the Borel transform ( 4.4 ) of D ( {} , {} ) ( ˜ g ), D ( { 0 } , {} ) ( ˜ g ), etc. parametrized as ( 4.13 ), and ev aluating their lateral Borel resummations S ± , their diﬀerence should b e compared with ( 4.12 ). F rom the jumps across the leading cuts, one – 26 – ﬁnds: d ( {} , {} ) k = A + 0 2 π i S ( { 0 } , {} ) X j ≥ 0 d ( { 0 } , {} ) k Γ( k − j − 1 + 2 a ) ( x + 0 ) k − j − 1+2 a + . . . , d ( { 0 } , {} ) k = A − 0 2 π i S ( { 0 } , { 0 } ) S ( { 0 } , {} ) X j ≥ 0 d ( { 0 } , { 0 } ) k Γ( k − j + 1 − 2 a ) ( x − 0 ) k − j +1 − 2 a + . . . , d ( { 0 } , { 0 } ) k = A + 1 2 π i S ( { 0 , 1 } , { 0 } ) S ( { 0 } , { 0 } ) X j ≥ 0 d ( { 0 , 1 } , { 0 } ) k Γ( k − j − 1 + 2 a ) ( x + 1 ) k − j − 1+2 a + . . . , d ( { 0 , 1 } , { 0 } ) k = A − 1 2 π i S ( { 0 , 1 } , { 0 , 1 } ) S ( { 0 , 1 } , { 0 } ) X j ≥ 0 d ( { 0 , 1 } , { 0 , 1 } ) k Γ( k − j + 1 − 2 a ) ( x − 1 ) k − j +1 − 2 a + . . . , (4.14) where the dots denote corrections that corresp ond to subleading cuts. The leading asymptotic b e- ha vior of higher order con tributions with ( δ + , δ − ) = ( { 0 , 1 , . . . , l } , { 0 , 1 , . . . , l − 1 } ) and ( δ + , δ − ) = ( { 0 , 1 , . . . , l } , { 0 , 1 , . . . , l } ) contain the ov erall prefactors A − l and A + l resp ectiv ely . Since with formula ( 3.27 ) the co eﬃcien ts d ( δ + ,δ − ) k can b e determined numerically for arbitrary δ ± eﬃcien tly , the asymp- totic behaviors in ( 4.14 ) could be v eriﬁed and the coeﬃcients A ± l could be extracted. F or the parameters ( a, ℓ ) = (1 / 4 , 0), ( a, ℓ ) = (1 / 4 , 1) and ( a, ℓ ) = (1 / (2 √ 2) , 2) I hav e generated d ( δ + ,δ − ) k for ( δ + , δ − ) = ( {} , {} ) , ( { 0 } , {} ) , ( { 0 } , { 0 } ) , ( { 0 , 1 } , { 0 } ) , ( { 0 , 1 } , { 0 , 1 } ) and ( { 0 , 1 , 2 } , { 0 , 1 } ) up to k = 200 with 500 digit precision. The v alues ( a, ℓ ) = (1 / 4 , 0), ( a, ℓ ) = (1 / 4 , 1) for the parameters a and ℓ were motiv ated by ph ysical relev ance (see the discussions ab ov e ab out the cusp anomalous dimension), while the v alue a = 1 / (2 √ 2) was chosen to b e an irrational n um ber in the range 0 < a < 1 / 2 to hav e a clear separation betw een the non-perturbative contributions. First, I computed the Borel transforms of the ab o v e-men tioned non-p erturbative series and to ok their diagonal P ad ´ e approximations. With a suitable conformal map (see Appendix G of [ 22 ]) I separated their diﬀeren t cuts in the Borel plane. The analytical structures of the Borel transforms of D ( { 0 } , {} ) ( ˜ g ), D ( { 0 } , { 0 } ) ( ˜ g ) and D ( { 0 , 1 } , { 0 } ) ( ˜ g ) are depicted in Figure 2. The cut structure is in agreemen t with the argument giv en in Section 3. The Borel transform of the p erturbative function D ( {} , {} ) ( ˜ g ) has cuts on the negative real line of the Borel plane starting from − y i and their linear com binations, while on the p ositiv e real line starting from the com binations of x ± l . They corresp ond to the p oles and zeros of Φ ± ( x ), resp ectiv ely . T o obtain the non-p erturbative con tribution D ( { 0 } , {} ) ( ˜ g ) from the p erturbativ e one, one has to promote the ﬁrst zero x + 0 of Φ + ( x ) to b e a p ole instead. This means that the cuts on the p ositiv e real line starting from v alues, which contain x + 0 , are remo v ed and new cuts app ear on the negativ e side starting from − x + 0 and its linear com binations with − y i . In case of going from D ( { 0 } , {} ) ( ˜ g ) to D ( { 0 } , { 0 } ) ( ˜ g ), the same thing happ ens with the cut starting from x − 0 , and so on. This phenomenon is reﬂected in the cut structures illustrated in Figure 2. As a next step, with the metho d discussed in App endix G of [ 22 ], I was able to capture the asymptotic b ehaviors of d ( { 0 } , {} ) k , d ( δ + ,δ − ) k with ( δ + , δ − ) = ( {} , {} ) , ( { 0 } , {} ) , ( { 0 } , { 0 } ) , ( { 0 , 1 } , { 0 } ) and ( { 0 , 1 } , { 0 , 1 } ) around the leading cuts. I was able to verify that with some suitable v alues of A ± j , the asymptotic expansions in ( 4.14 ) hold. F or each v alue of ( a, ℓ ) mentioned ab ov e, with a 10 − 6 relativ e error, I found that the ﬁrst few A ± l – 27 – − y 0 x + 0 x − 0 x + 0 + x − 0 x + 1 x − 1 − y 0 − x + 0 − y 0 − x + 0 x − 0 x + 1 x − 1 − y 0 − x − 0 − y 0 − x + 0 − y 0 − x + 0 − x − 0 − x − 0 − x + 0 x + 1 x − 1 Figure 2 : The ﬁgures from top to b ottom illustrate the cut structures of B h D ( δ + ,δ − ) i ( s ) on the Borel plane with ( δ + , δ − ) = ( {} , {} ) , ( { 0 } , {} ) and ( { 0 } , { 0 } ). The black lines represent cuts starting from the branc h p oints on the real line and ending at inﬁnit y . The green lines represen t whic h cuts has to b e remo v ed from the p ositive line to go from one correction to the next one. The red lines appearing on the negativ e real line denote the additional cuts compared to the previous correction. are the following: A + 0 = − 2 i sin ( π a ) , A − 0 = 2 i sin ( π a ) , A + 1 = − 2 i sin ( π a ) , A − 1 = 2 i sin ( π a ) , A + 2 = − 2 i sin ( π a ) . (4.15) Notice that the leading asymptotic behavior of d ( {} , {} ) , d ( { 0 } , {} ) and d ( {} , { 0 } ) with the coeﬃcients giv en in ( 4.15 ) is compatible with results found in [ 22 ] for the functions Z (0 , 0) ( g ), Z (1 , 0) ( g ) and Z (0 , 1) ( g ), ho w ev er there are sign diﬀerences in the co eﬃcients A + 0 and A − 0 . As a veriﬁcation of ( 4.15 ), I also p erformed a Richardson extrap olation on the coeﬃcients of the p erturbative function. Its asymptotic b eha vior agreed with ( 4.14 ) with the same co eﬃcient A + 0 as in ( 4.15 ). F urthermore, with the me thod discussed in App endix G of [ 45 ], I was also able to separate the ﬁrst subleading cut of B  D ( {} , {} )  ( s ). I found that around this cut, its asymptotic b ehavior is giv en by: d ( {} , {} ) k ∼ A − 0 2 π i S ( {} , { 0 } ) X j ≥ 0 d ( {} , { 0 } ) k Γ( k − j − 1 + 2 a ) ( x − 0 ) k − j − 1+2 a , (4.16) with A − 0 b eing the same as in ( 4.15 ) also in agreement with ( 4.9 ). The v alues found in ( 4.15 ) suggest that the general form of the co eﬃcien ts A ± l is giv en b y: A ± l = ∓ 2 i sin ( π a ) (4.17) Notice that, as exp ected, A + l is related to A − l via a → − a . With this the Alien deriv atives in ( 4.9 ) b ecome: ∆ + j D ( δ + ,δ − ) ( ˜ g ) = − 2 i sin ( π a ) ˜ g − 1 − 2∆+2 a S ( δ + ∪{ j } ,δ − ) S ( δ + ,δ − ) D ( δ + ∪{ j } ,δ − ) ( ˜ g ) , ∆ − j D ( δ + ,δ − ) ( ˜ g ) = 2 i sin ( π a ) ˜ g − 1+2∆ − 2 a S ( δ + ,δ − ∪{ j } ) S ( δ + ,δ − ) D ( δ + ,δ − ∪{ j } ) ( ˜ g ) (4.18) – 28 – 4.3 Median resummation The app earance of trigonometric factors sin( π a ) in the Alien deriv ativ es is due to the complex phases e iπ a ∆ in the transseries. As discussed in [ 22 ], strictly sp eaking, the strong coupling expansion in ( 1.7 ) is only v alid for Im g < 0 and corresp onds to the lateral resummation S + . The same holds for the transseries ( 3.9 ), namely the resummation of Z ℓ ( g ) is understo o d as: Z ℓ ( g ) = A ℓ ( g ) X δ + ,δ − ˜ g − ∆(∆ − 2 a ) e − ˜ g ( P l ∈ δ + x + l + P j ∈ δ − x − j ) e iπ a ∆ S ( δ + ,δ − ) S + h D ( δ + ,δ − ) i ( ˜ g ) . (4.19) If w e use S − instead, the complex conjugate of ( 4.19 ) should be tak en: Z ℓ ( g ) = A ℓ ( g ) X δ + ,δ − ˜ g − ∆(∆ − 2 a ) e − ˜ g ( P l ∈ δ + x + l + P j ∈ δ − x − j ) e − iπ a ∆ S ( δ + ,δ − ) S − h D ( δ + ,δ − ) i ( ˜ g ) , (4.20) whic h is v alid for Im g > 0. If w e consider, for instance, the p erturbative part, the ambiguit y in c hoosing either S +  D ( {} , {} )  ( ˜ g ) or S −  D ( {} , {} )  ( ˜ g ) should cancel the ( { 0 } , {} ) term. This means that: S + h D ( {} , {} ) i ( ˜ g ) − S − h D ( {} , {} ) i ( ˜ g ) ∼ − ( e iπ a − e − iπ a ) ˜ g − 1+2 a e − ˜ g x + 0 S ( { 0 } , {} ) D ( { 0 } , {} ) ( ˜ g ) (4.21) The prefactor on the righ t-hand side is exactly the constant A + 0 in ( 4.15 ). Similarly , the discontin uit y around the ﬁrst subleading cut cancels against: − ( e − iπ a − e iπ a ) ˜ g − 1 − 2 a e − ˜ g x + 0 S ( {} , { 0 } ) D ( {} , { 0 } ) ( ˜ g ) (4.22) Whic h giv es A − 0 = 2 i sin ( π a ). The same argumen t holds for higher order con tributions in agreement with ( 4.17 ). As w e see, neither ( 4.19 ) nor ( 4.20 ) are real functions of the coupling constan t; therefore, they do not giv e a prop er physical answer for the strong coupling expansion of the observ ables giv en b y the determinan t ( 1.1 ). Instead, the physical result is obtained via the median resummation. The median resummation is deﬁned as: S med = S + ◦ S − 1 / 2 = S − ◦ S 1 / 2 (4.23) The square ro ot of the Stok es automorphism can be easily expressed in terms of Alien deriv ativ es by expanding the square ro ot of ( 4.11 ) and using the algebraic prop erties in ( 4.9 ). Then by ( 4.18 ), the eﬀect of S ± 1 / 2 on the non-p erturbativ e functions D ( δ + ,δ − ) ( ˜ g ) can b e easily determined. After substituting S + = S med ◦ S 1 / 2 in to ( 4.19 ), and, for conv enience, redeﬁning the non- p erturbativ e functions as: ¯ D ( δ + ,δ − ) ( ˜ g ) ≡ ˜ g − ∆(∆ − 2 a ) S ( δ + ,δ − )  Q l ∈ δ + S { l } +   Q j ∈ δ − S { j } −  D ( δ + ,δ − ) ( ˜ g ) , (4.24) with S { l } ± giv en in ( 3.31 ), we ﬁnd that the median resummation of the determinan t Z ℓ ( g ) is giv en by: Z ℓ ( g ) = A ℓ ( g ) X δ + ,δ −   Y l ∈ δ + σ + l e − ˜ g x + l     Y j ∈ δ − σ − j e − ˜ g x − j   S med ¯ D ( δ + ,δ − ) ( ˜ g ) , (4.25) – 29 – with the parameters σ ± l b eing: σ ± l = cos ( π a ) S { l } ± . (4.26) Notice that ( 4.25 ) is now a real function of the coupling constant ˜ g . Therefore, the median resummation in ( 4.25 ) pro vides the ph ysical answer for the strong coupling expansion of the determinant with matrix Bessel k ernel. This completes the full resurgence structure of the strong coupling expansion of ( 1.1 ) and provides strong evidence on the v alidit y of the transseries structure ( 3.9 ). It would b e interesting to inv estigate ho w the resurgence structure discussed abov e manifests itself in the strong coupling expansion of the cusp anomalous dimension giv en by ( 1.6 ). Ac kno wledgemen ts I w ould lik e to thank Zoltan Ba jnok, Gregory P . Korc hemsky and Dennis Le Plat for useful discussions and helpful commen ts on the draft of this paper. The research w as supp orted b y the Do ctoral Excel- lence F ello wship Programme funded b y the National Researc h Developmen t and Innov ation F und of the Ministry of Culture and Innov ation and the Budap est Universit y of T ec hnology and Economics, under a grant agreement with the National Research, Developmen t and Inno v ation Oﬃce (NKFIH). It w as also supported b y the gran t NKKP Adv anced 152467. A Bridge equations In this app endix, based on [ 38 – 40 ] I deriv e the Alien algebra for a transseries of the form: F ( g ) = X δ e − g P l ∈ δ x l g λ ( δ ) Φ ( δ ) ( g ) , (A.1) with an inﬁnite set of distinct exp onential w eigh ts x 0 < x 1 < . . . . The summation go es ov er all p ossible unordered sets δ of distinct non-negative in tegers. This means that the sum contains terms that are at most ﬁrst order in eac h factor of e − g x l . The functions Φ ( δ ) are giv en b y the expansions: Φ ( δ ) ( g ) = X k ≥ 0 ϕ ( δ ) k g k , (A.2) and λ ( δ ) is a function of δ that go v erns the leading 1 /g order at exp onential level δ . F or simplicit y in this App endix I considered only one sec tor of exp onential weigh ts. Although the large- g expansion of the determinant Z ( g ) in ( 3.9 ) contains tw o sets of weigh ts x + j and x − j , the argumen t b elow remains the same for Z ( g ) as well. The function F ( g ) with λ ( δ ) = 0 and g → 8 π g coincides with the strong coupling expansions inv estigated in [ 45 ]. The non-p erturbativ e structure is deeply connected to the large order b ehavior of the co eﬃcients ϕ ( δ ) k . This property is captured by the Borel transformation: B h Φ ( δ ) i ( s ) = X k ≥ 0 ϕ ( δ ) k Γ( k + 1) s k (A.3) If the series ( A.2 ) is con v ergen t, then its inv erse is simply a Laplace transform: S h Φ ( δ ) i ( g ) = g Z ∞ 0 dse − g s B h Φ ( δ ) i ( s ) . (A.4) – 30 – and it gives bac k the original function Φ ( δ ) ( g ). The problem arises if the coeﬃcients ϕ ( δ ) k gro w factorially , for example if they ha v e the asymptotic form: ϕ ( δ ) k = 1 π X j ≥ 0 c j Γ( k + λ − j ) A k + λ − j , (A.5) where A , λ and c j (with j ≥ 0) are diﬀerent parameters of the expansion. In general, the expansion of ϕ ( δ ) k is a linear com bination of terms similar to ( A.5 ) with v arious sets of parameters. In this case, the series in ( A.2 ) cannot b e resummed directly , only through its Borel transform. After the resummation of ( A.3 ), its inv erse ( A.4 ) should b e taken to compute the function Φ ( δ ) . Ho w ev er, for an asymptotic series, the function ( A.3 ) produces p oles and cuts on the Borel plane, and the in tegration contour in ( A.4 ) crosses these singularities. T o av oid them, one has to shift the integration con tour in ( A.4 ) slightly b elo w or ab ov e the real axis and deﬁne the lateral Borel summations: S ± h Φ ( δ ) i ( g ) = g Z ∞ e iϵ 0 dse − g s B h Φ ( δ ) i ( s ) . (A.6) F or an asymptotic series, it pro duces an am biguit y in calculating the resummation of ( A.2 ) and hence in computing the function F ( g ). Due to this ambiguit y , diﬀeren t terms in the expansion dep end on the c hoice of regularization, and one has to choose either from: F + ( g ) = X δ e − g P l ∈ δ x l g λ ( δ ) S + h Φ ( δ ) i ( g ) , (A.7) or: F − ( g ) = X δ e − g P l ∈ δ x l g λ ( δ ) S − h Φ ( δ ) i ( g ) (A.8) to compute F ( g ). W e will see that due to the presence of exp onentially small corrections, the am bigu- ities kill eac h other, and the resummation of F ( g ) b ecomes indep endent of the regularization, namely: F + ( g ) = F − ( g ). The diﬀerence b etw een the t w o prescriptions in ( A.6 ) contains all the information about the full discon tin uit y of B  Φ ( δ )  ( s ) along the real axis: S + − S − = S − ◦ Disc . (A.9) Therefore, it is conv enien t to in tro duce the so called Stokes automorphism S , which connects the t w o lateral resummations, and tells us ho w B  Φ ( δ )  ( s ) jumps across the real line: S + = S − ◦ ( I − Disc) ≡ S − ◦ S . (A.10) F or the simplest example, one can take the Borel transform ( A.3 ) of Φ ( δ ) , with its large g co ef- ﬁcien ts giv en in ( A.5 ). Then it is straigh tforw ard to sho w that the function exhibits a singularity at s = A . The discon tinouit y across this singularity is: lim ϵ → 0  B [Φ ( δ ) ]( s + iϵ ) − B [Φ ( δ ) ]( s − iϵ )  = 2 iθ ( s − A ) X j ≥ 0 c j ( s − A ) j − λ Γ(1 + j − λ ) . (A.11) Therefore, the diﬀerence betw een the lateral Borel resummations is: S + [Φ ( δ ) ]( g ) − S − [Φ ( δ ) ]( g ) = 2 ig λ e − g A X j ≥ 0 c j g j . (A.12) – 31 – F rom this simple example, w e immediately see that for an asymptotic series, the exp onen tial corrections are related to the ambiguit y in the resummation. Since the discon tinouit y is directly related to the Stok es automorphism, if one can compute S for a transseries of the form ( A.1 ), then the asymptotic expansions (namely the parameters A , λ and c j ) of the coeﬃcients ϕ ( δ ) k can be determined. As w as men tioned previously , the asymptotic series in ( A.5 ) could con tain several similar terms, pro ducing diﬀeren t singularities and cuts on the Borel plane. T o separate the jumps along diﬀerent cuts on the real line, it is useful to write the Stokes automorphism in terms of the Alien deriv ativ es. By deﬁnition, they are related to S via: S = exp X ω e − ω g ∆ ω ! (A.13) where the summation runs along all the singular p oints ω of the Borel transform. ∆ ω called a deriv a- tiv e, since it satisﬁes the Leibnitz rule. The calculation of the Alien deriv atives can b e done via the so called Bridge equations. The ﬁrst step is to assign a parameter σ j to eac h exp onen tial weigh t x j in the transseries ( A.1 ) and deﬁne: F ( g , { σ } ) = X δ Y l ∈ δ σ l ! e − g P l ∈ δ x l g λ ( δ ) Φ ( δ ) ( g ) , (A.14) so every exponential factor e − g x l comes with a factor of σ l . Now it can b e shown (see [ 38 – 40 ]), that the com bination: ˙ ∆ ω = e − ω g ∆ ω (A.15) comm utes with diﬀeren tiation with resp ect to the v ariable g . The same holds for diﬀerentiation with resp ect to the parameters σ j , so that:  ∂ g , ∂ σ j  = h ∂ g , ˙ ∆ ω i = 0 . (A.16) If we supp ose that F ( g ) is a solution of any non-linear problem, it follows that ∂ σ j F ( g , { σ } ) (with ev ery parameter σ j ) and ˙ ∆ ω F ( g , { σ } ) solve the same linear, homogeneous equation whose exact form dep ends on the original problem. As a consequence, ˙ ∆ ω can b e written as a linear combination of diﬀeren tiations with resp ect to the parameters σ j : ˙ ∆ ω = X l ≥ 0 A ω ,l ∂ σ l , (A.17) where A ω ,l are related to the Stoke constan ts. This is called the Bridge equation and it relates the partial deriv ativ es with respect to the resurgence parameters σ j and the Alien deriv ativ es ∆ j . Acting on F ( g , { σ } ) with ( A.15 ) and the right-hand side of ( A.17 ), and comparing the exp onen tial factors and diﬀerent pow ers of the parameters σ j , one ﬁnds that ω should b e one of the exp onential w eigh ts x l . F urthermore, the diﬀeren t non-p erturbativ e sectors are related as: ∆ j Φ ( δ ) = g λ ( δ ∪{ j } ) − λ ( δ ) A j Φ ( δ ∪{ j } ) , (A.18) where j / ∈ δ , and for simplicit y , I introduced the notation ∆ x j ≡ ∆ j and relab eled the co eﬃcien ts A ± x ± j ,j ≡ A ± j . It also follows from the Bridge equations that: (∆ j ) 2 = 0 , [∆ j , ∆ l ] = 0 (A.19) – 32 – for ev ery in teger j, l ≥ 0. With the help of the algebra in ( A.19 ), the exp onentiation in ( A.13 ) can easily b e ev aluated to get a more reasonable expression for the Stok es automorphism. F rom ( A.19 ) it immediately follows that S is formally equal to: S = Y l ≥ 0  1 + e − x l g ∆ l  = 1 + X j ≥ 1 X 0 ≤ n 1 ≤ ...n j e − g ( x n 1 + x n 2 + ··· + x n j ) ∆ n 1 ∆ n 2 . . . ∆ n j . (A.20) Acting on the function Φ ( δ ) with ( A.20 ) and using the deﬁnition of S from ( A.10 ), one ﬁnds for the lateral Borel resummations S ±  Φ ( δ )  ( g ) that: S + Φ ( δ ) = S − Φ ( δ ) + X j ≥ 1 X 0 ≤ n 1 ≤ ...n j ∀ l ≤ j, n l / ∈ δ g λ ( δ ∪{ n 1 ,...,n j } ) − λ ( δ ) e − g ( x n 1 + x n 2 + ··· + x n j ) A n 1 A n 2 . . . A n j S − Φ ( δ ∪{ n 1 ,...,n j } ) . (A.21) This relation means, that the Borel transform of Φ ( δ ) con tains cuts starting from every p oint x n 1 + x n 2 + · · · + x n j on the real line, with each x n l / ∈ δ . According to ( A.18 ), Φ ( δ ) has direct resurgence relations only with the exp onential corrections Φ ( δ ∪{ j } ) with j / ∈ δ . The rest of the terms in ( A.21 ) (and hence in the asymptotic expansion of Φ ( δ ) ) come from the asymptotic b eha vior of higher order corrections Φ ( δ ∪{ j } ) . By substituting ( A.21 ) into F + ( g ) in ( A.7 ), higher order terms in ( A.21 ) kill eac h other and the series b ecomes equal to F − ( g ) in ( A.8 ). This means that the ambiguities cancel and F ( g ) b ecomes indep enden t of the regularization: F ( g ) = F + ( g ) = F − ( g ). A t ﬁrst, the expression in ( A.21 ) looks tedious; how ev er, taking sp eciﬁc v alues of δ clariﬁes that ( A.21 ) is enough to determine the asymptotic behavior of the functions Φ ( δ ) ( g ) and a p ossible w a y to compute the constants A j that appear in the Alien deriv atives in ( A.18 ). Consider, for example, the set δ = { 0 , 1 , . . . , l } that con tains all integers from 0 to l . F or the corresp onding functions Φ ( { 0 , 1 ,...,l } ) ( g ), the expression in ( A.21 ) lo oks as: S + Φ ( { 0 ,...,l } ) − S − Φ ( { 0 ,...,l } ) = g λ ( { 0 ,...,l +1 } ) − λ ( { 0 ,...,l } ) e − g x l +1 A l +1 S − Φ ( { 0 ,...,l +1 } ) + . . . , (A.22) where the dots denote exp onen tially suppressed terms. Now supp ose that Φ ( { 0 ,...,l } ) ( g ) is given by a series in 1 /g as in ( A.2 ) and its co eﬃcients ϕ ( { 0 ,...,l } ) k can be parametrized as: ϕ ( { 0 ,...,l } ) k = 1 π X j ≥ 0 c j Γ( k + λ − j ) A k + λ − j + . . . , (A.23) with the dots standing for higher order cuts. Then b y ( A.12 ) and ( A.22 ) it follo ws that the parameters A , λ and c j in the asymptotic series are giv en as: A = x l , λ = λ ( { 0 , . . . , l + 1 } ) − λ ( { 0 , . . . , l } ) , c j = A l +1 2 i ϕ ( { 0 ,...,l +1 } ) k , (A.24) where ϕ ( { 0 ,...,l +1 } ) k are the 1 /g coeﬃcients of Φ ( { 0 ,...,l +1 } ) ( g ). Therefore, the leading asymptotic expan- sion of ϕ ( { 0 ,...,l } ) k is: ϕ ( { 0 ,...,l } ) k = A l +1 2 π i X j ≥ 0 ϕ (0 ,...,l +1) j Γ( k + λ − j ) ( x l ) k + λ − j + . . . , (A.25) – 33 – with λ giv en in ( A.24 ). 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Strong coupling structure of $\mathcal{N}=4$ SYM observables with matrix Bessel kernel

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