A Concentration of Measure Phenomenon in the Principal Chiral Model

A Concen tration of Measure Phenomenon in the Principal Chiral Mo del T. Tlas Abstract W e utilize the concen tration of measure phenomenon to study the large N limit of the O ( N ) principal c hiral mo del. The partition function in this limit is demonstrated to be that of a free massive theory . In tro duction The principal chiral mo del is widely considered to b e a simpler v ersion of the Y ang-Mills theory which nonetheless has some of its nonp erturbativ e features. The mo del has been solved [1, 2] around 40 y ears ago using metho ds whic h, alas, do not generalize to the more physically relev ant case of Y ang-Mills. The problem of re-deriving those classical results in an alternative fashion, b egin- ning with the path integral, has remained op en. The reason for doing so is not purely academic for it has recently b een demonstrated that Y ang-Mills theory can b e considered to b e a principal chiral model in loop space [3]. The aim of this manuscript is to tak e a step in this direction. Notably , we will obtain the partition function in the large N limit, which turns out to be that of (an infinite n umber of copies of ) a free, massiv e scalar field. W e will also obtain the mass gap of this limiting free theory explicitly . A t this p oin t w e need to clarify a subtle issue regarding the v arious limits w e will b e taking. If we lab el the ultraviolet regulator by Λ, then the regularized partition function is Z Λ ,N [ J ]. What we shall do b elow is obtain the asymptotic form of this function as N → ∞ , holding the Λ fixed, and tak e the contin uum limit Λ → ∞ afterwards. In other w ords, w e will consider the limit lim Λ →∞ lim N →∞ Z Λ ,N [ J ] . This is a similar approac h to that in [4, 5] and is also in line with the constructiv e quan tum field theory work [6]. One could also ask what is the result of inter- c hanging the order of the t w o limits. There is evidence [7] that in this case the limiting theory is no longer free, ev en though it do es still ha ve a trivial S-matrix. Since a free theory is clearly a more con venien t p oin t to use as a starting p oint for approximating a theory with finite N , we shall not discuss this alternative order of limits further. Let us briefly outline the main ideas of the work b elow. The general approac h could b e though t of as a realization of the strategy sk etched in [8]. One starts b y swapping the in tegral ov er the original field with that ov er the Lagrange m ultiplier. Contrary to what happ ens with the O ( N ) vector model, one cannot simply apply Laplace’s metho d to the resulting integral. The reason for this is well understo od and lies in the fact that, in addition to the action, there is a comparable contribution coming from the entrop y [8, 9], as the num ber of comp onen ts of the Lagrange multiplier is large. It has been suggested in [8] to diagonalize the Lagrange multiplier and then perform the integral o ver the di- agonalizing matrices. Unfortunately , this integral is intractable. W e bypass this problem below using the idea of concentration of measure; it shows that in the large N limit, the effect of the entrop y can b e mo deled by a Gaussian, similarly to how it was shown recen tly in lattice Y ang-Mills [5]. Using this ansatz, we will b e able to rewrite the partition function in a w ay amenable to the standard asymptotic analysis. Curiously , it turns out that, due to the sp ecific details of the w ay concentration of measure manifests itself, the entropic fluctuations for the diagonalized field are suppressed in the con tinuum limit. This is differen t from the w ay the same phenomenon app ears in lattice Y ang-Mills [5] and will p erhaps be the most technically in volv ed asp ect of the text b elo w. This sup- pression easily allo ws us to find the mass gap of the principal chiral mo del and explain wh y it coincides with the result obtained b y the, a priori incorrect, naive asymptotic analysis. The outline of the pap er is as follows: in the next section we giv e the necessary bac kground and describ e our setting and conv entions. In the subsequent sec- tion, w e perform the preliminary analysis of the problem so that it’s amenable to treatmen t using the concentration of measure approac h. This treatment is started in the section that follows. The necessary calculations of the mean and the v ariance o ccup y the section after that. W e finish with the asymptotic anal- ysis obtaining the main results at the v ery end. Bac kground and Notation Let us describ e our setting. W e will b e studying the principal chiral mo del in the Euclidean setting. W e will only deal with the O ( N ) case, but essentially ev- erything b elo w can be straightforw ardly generalized to other groups of in terest. The theory is regularized by putting it on a toroidal lattice, with the deriv atives replaced b y the finite differences. The num b er of lattice sites will b e denoted b y Λ 2 . The ph ysical v olume of the lattice will b e denoted by V , and th us eac h step of the lattice has length ∆ = q V Λ 2 . The elemen ts of the lattice will usually b e denoted by x (or sometimes y ). The elements of the (F ourier) dual lattice will b e denoted by p (or sometimes q ). W e take the step of the dual lattice to b e 2 π √ V . With this c hoice, the F ourier transform relations for any function f ( x ) ↔ b f ( p ) b ecome b f ( p ) = X x e − ipx f ( x )∆ 2 ⇝ ˆ e − ipx f ( x ) d 2 x f ( x ) = 1 (2 π ) 2 X p e ipx b f ( p ) (2 π ) 2 V ⇝ 1 (2 π ) 2 ˆ e ipx b f ( p ) d 2 p, where the equations on the right are the corresp onding contin uum limit equa- tions. Note that the contin uum limit is achiev ed by taking Λ → ∞ , V → ∞ and ∆ = V Λ 2 → 0. Since w e are in terested even tually in the contin uum limit of the theory , w e will usually use the con tinuum limit form for most expressions b elo w, as they are often more transparen t. W e will of course revert bac k to the explicit discrete notation when issues of regularization b ecome relev an t. Note that since p ranges ov er a square 1 of side length 2 π Λ √ V ≡ ˜ Λ, we shall consider our momen tum ultra violet cut-off to be ˜ Λ. Note that as a consequence of our con v entions, if b f ( p ) is rotationally in v ariant 2 as a function of p , we hav e the follo wing imp ortant formula, whic h is used rep eatedly b elo w X p b f ( p ) ⇝ V (2 π ) 2 ˆ b f ( p ) d 2 p ∼ V 2 π ˆ ˜ Λ 2 0 b f ( p ) | p | d | p | , where ∼ stands for “asymptotic to”. The justification for the right hand side is that we’v e replaced the in tegral ov er the square of side length ˜ Λ with that ov er the inscrib ed disc of radius ˜ Λ 2 . 3 Preliminary Analysis The action of the principal c hiral mo del model is N 2 λ ˆ ∂ µ ϕ ab ∂ µ ϕ ab , where λ is the ’t Hooft coupling and the fields ϕ satisfy the orthogonality con- strain t ϕ ba ϕ bc = δ ac . As mentioned ab o ve, w e’re giving it in the contin uum form. Of course, if we wan t to b e fastidious, w e should replace the integral ab o v e with a sum o ver x weigh ted by a factor of ∆ 2 , and the deriv ativ es with finite differences. It is undoubtedly clear to the reader that the con tin uum ex- pression is significantly less cluttered than the discrete one. W e will con tinue to use the con tinuum expressions freely and will not b elab or this issue an y more. The partition function Z [ J ] of the theory is giv en b y 4 ˆ D ϕ D M exp  − N 2 λ ˆ ∂ µ ϕ ab ∂ µ ϕ ab + i ˆ M ac  ϕ ba ϕ bc − δ ac  + i r N λ ˆ J ab ϕ ab  , where the factor q N λ is inserted in the source term in order to obtain a non triv- ial large N limit. Note that the Lagrange m ultiplier field satisfies M ac = M ca 1 W e shall tak e this square to be centered at (0 , 0). There is no loss of generalit y in doing so due to p eriodi cit y of the F ourier transform. 2 More precisely , if it is a restriction of a rotationally inv ariant function to the lattice. 3 Needless to say , for this to work, one should assume sufficient decay of b f ( p ). The functions we’ll be dealing with will satisfy the necessary decay so w e will not dw ell on this p oin t further. 4 This is of course meant to be regularized, and th us if we are awfully nitpic ky , should b e written as Z Λ ,N [ J ]. There will b e no loss of clarit y in simply ignoring the subscripts in what follows. since the orthogonalit y constrain t is symmetric. Rescaling the fields q N λ ϕ → ϕ and 2 λ N M → M , we see that Z [ J ] is, up to an irrelev ant ov erall constan t, equal to ˆ D ϕ D M exp  − 1 2 ˆ ∂ µ ϕ ab ∂ µ ϕ ab + i 2 ˆ M ac  ϕ ba ϕ bc − N λ δ ac  + i ˆ J ab ϕ ab  = ˆ D ϕ D M exp  − 1 2 ˆ ϕ ab K ab,a ′ b ′ ϕ a ′ b ′ + i N 2 λ ˆ M aa + i ˆ J ab ϕ ab  , where K ab,a ′ b ′ = − ∂ 2 δ aa ′ δ bb ′ − iM aa ′ δ bb ′ = ( − ∂ 2 δ aa ′ − iM aa ′ ) ⊗ δ bb ′ = K aa ′ ⊗ δ bb ′ . W e can no w perform the integral ov er the ϕ ’s and get 5 Z [ J ] = ˆ D M exp  − N 2 T r  ln( K aa ′ )  − i N 2 λ ˆ M aa − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  , No w, using the fact that M is a symmetric matrix, we change v ariables M = O t ˆ M O , where ˆ M is diagonal and O is the diagonalizing orthogonal matrix. The Jacobian of this transformation is w ell-kno wn [10] and is equal to Q a  = b | ˆ M a − ˆ M b | , where { ˆ M a } a =1 ,N is the set of eigenv alues of ˆ M . W e th us get that Z [ J ] = ˆ D ˆ M D O  Y x,a  = b ln | ˆ M a − ˆ M b |  exp  − N 2 T r  ln( K aa ′ )  − i N 2 λ ˆ X a ˆ M a − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  , where the product ov er x in the Jacobian go es ov er all the points of space. A t this stage, motiv ated by the form of the propagator K aa ′ , we p erform a change of v ariables ˆ M → ˆ M + iµ where µ > 0. 6 With this change, we get that Z [ J ] is equal to ˆ D ˆ M D O  Y x,a  = b ln | ˆ M a − ˆ M b |  exp  − N 2 T r  ln( K aa ′ )  − i N 2 λ ˆ X a ˆ M a + N 2 V µ 2 λ − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  , 5 Strictly speaking, the form ula b elo w is incorrect. This is b ecause there is a factor of ∆ 2 in the quadratic term in the ϕ ’s (it comes from the in tegral) which filters down to a rescaling of the K . How ever, this only amoun ts to an additive constan t in the logarithm term, whic h contributes an irrelev ant o v erall constant to Z [ J ] and thus, will be ignored below. Note that the same factor does not cause an issue in the J term. This is because the factor of 1 ∆ 2 coming from the K − 1 aa ′ term cancels with (∆ 2 ) 2 coming from the J ’s, leaving a ∆ 2 hidden in the expression of the in tegral abov e. 6 This change of variables can truly b e justified only a p osteriori once w e deduce the large N limit and find that µ is in fact the mass 2 of the theory . where V ab o ve is the volume of space. No w, as is customary [11], w e in tro duce the probabilit y densit y ρ ( ˆ M ) = 1 N X a δ ( ˆ M − ˆ M a ) . This allows us to rewrite Z [ J ] as ˆ D ρ D O exp  N 2  X x ˆ d ˆ M d ˆ M ′ ρ ( ˆ M ) ρ ( ˆ M ′ ) ln | ˆ M − ˆ M ′ | − 1 2 N T r  ln( K aa ′ )  + V µ 2 λ − i 2 λ ˆ ˆ d ˆ M ρ ( ˆ M ) ˆ M  − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  . Note that in the penulti mate term, one of the in tegrals is ov er ˆ M while the other is ov er space. Also, observe that D ρ , which stands for a functional integral o ver ρ , is constrained to go o v er probabilit y measures. Concen tration of Measure W e would like to obtain the large N asymptotic of Z [ J ]. T o this end, we utilize the concentration of measure phenomenon [12] along the lines demonstrated in [5]. As a first step, w e need the following simple Lemma. The function O → exp  − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  is Lipschitz with r esp e ct to the metric d ( O , O ′ ) = ´ || O − O ′ || H S wher e || · || H S stands for the Hilb ert-Schmidt norm. Pr o of. Since e − z is Lipschitz as a function of z if ℜ z is bounded from b e- lo w, it is enough to show that ´ J ab ( K aa ′ ) − 1 J a ′ b is Lipsc hitz as a function of O , and that its real part is bounded from below. T o this end, consider K aa ′ = ( − ∂ 2 + µ ) δ aa ′ − iM aa ′ . Note that the real part of the its eigenv al- ues is b ounded from b elo w b y µ . It follows that the real part of the singu- lar v alues of K − 1 aa ′ is contained in [0 , µ ]. F rom this, tw o facts follow at once: ℜ ( ´ J ab ( K aa ′ ) − 1 J a ′ b ) ≥ 0 and || K − 1 aa ′ || ≤ µ where || · || is the usual sup norm. W e thus hav e that | ˆ J ab ( K aa ′ ( O )) − 1 J a ′ b − ˆ J ab ( K aa ′ ( O ′ )) − 1 J a ′ b | ≤ ˆ | J ab ( K − 1 aa ′ ( O ) − K − 1 aa ′ ( O ′ )) J a ′ b | ≤ ˆ || J || 2 H S || K − 1 ( O ) − K − 1 ( O ′ ) || ≤ ˆ || J || 2 H S || K − 1 ( O ) || || K − 1 ( O ′ ) || || K ( O ) − K ( O ′ ) || ≤ ˆ || J || 2 H S µ 2 || O t ˆ M O − O ′ t ˆ M O ′ || ≤ ˆ || J || 2 H S µ 2  || O t ˆ M O − O t ˆ M O ′ || + || O t ˆ M O ′ − O ′ t ˆ M O ′ ||  ≤ ˆ 2 || J || 2 H S || ˆ M || 2 µ 2 || O − O ′ || ≤ 2 max x ( || J || 2 H S || ˆ M || 2 ) µ 2 ˆ || O − O ′ || H S , whic h concludes the pro of. Before w e proceed, note that w e are mainly interested in v arious deriv atives of Z [ J ] ev aluated at 0. The modifications of the pro of ab o ve needed to cov er this case are utterly straigh tforw ard and will be left to the reader. W e now use the fact that Lipschitz functions on O ( N ) concentrate as N → ∞ . 7 This means that suc h a function conv erges in probabilit y to its mean, or more in tuitively , that the function is essentially constan t on its domain. This allows us to replace the term exp  − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  in Z [ J ] with ˆ D O exp  − 1 2 ˆ J ab ( K aa ′ ) − 1 J a ′ b  , whic h in turn, by applying the same argumen t to the exp onen t whic h has been sho w abov e to b e Lipschitz as w ell, can b e replaced with exp  − 1 2 ˆ D O ˆ J ab ( K aa ′ ) − 1 J a ′ b  . Before we pro ceed and calculate this expression, let us note that if the reader is not entirely con vinced by the somewhat abstract argumen t abov e, we shall sk etch b elo w a different, more direct, justification of this manipulation. In view of the discussion ab o ve, we see that in the large N limit w e get the follo wing asymptotic relation 7 See for example Theorem 5.17 in [13]. Z [ J ] ∼ ˆ D ρ exp  − 1 2 ˆ D O ˆ J ab ( K aa ′ ) − 1 J a ′ b  ! (1) × exp  N 2  X x ˆ d ˆ M d ˆ M ′ ρ ( ˆ M ) ρ ( ˆ M ′ ) ln | ˆ M − ˆ M ′ | + V µ 2 λ − i 2 λ ˆ ˆ d ˆ M ρ ( ˆ M ) ˆ M  ! ˆ D O exp  N 2  − 1 2 N T r  ln( K aa ′ )   ! . W e w ould now lik e to handle the last brack ets in the expression ab ov e. Note that in this case, contrary to what was done in the first term, it is not justified to simply replace the exp onen t with its a v erage. The reason for this is the presence of the N 2 factor in the exp onen t. Thus, one needs a more refined metho d to handle this term. W e shall proceed b y a method similar to that used in [5] to deal with lattice Y ange-Mills. Notably , we would like to pushforward the D O measure with the function O → t ( O ) = 1 2 N V T r  ln( K aa ′ )  , where V is the volume of space. How ev er, b efore we do this, we rotate the con tour of in tegration o ver the ˆ M , or equiv alently , that ov er the ρ , so that it is along the imaginary axis. This has the effect of replacing ˆ M everywhere with i ˆ M , whic h would guaran tee that the integrand in Z [ J ] abov e is real. The rea- son for this contour change is that the reality of the resulting expressions would p ermit us to use the simple Laplace’s metho d to obtain the asymptotics instead of the saddle point metho d. Before w e proceed, a couple of quic k remarks: first we’v e done this rotation at this stage and not on the original expression since otherwise, we w ould not b e able to guaran tee Lipsc hitzness of K aa ′ in the Lemma ab o ve. Therefore, we’v e only rotated the contour after the e − ´ J K − 1 J term is safely taken outside of the O in tegral. Second, note that the fact that the contour of integration can b e rotated with impunity needs to b e justified as the function integrated is not analytic (the culprit b eing the | M a − M b | = e ln | M a − M b | term). It is how ev er a very simple argument to show that this rotation is v alid, for the integral w e ha ve is a m ulti-v ariable generalization of ´ C f ( z ) | z | dz , where C is along the real axis and f is assumed to b e analytic and v anishing sufficiently fast at infinity . W e can now decomp ose C as C + + C − where C + /C − is the positive/negativ e part of the real axis. W e th us ha ve that ˆ C f ( z ) | z | dz = ˆ C + f ( z ) z dz − ˆ C − f ( z ) z dz = i 2 ˆ C + f ( it ) tdt − i 2 ˆ C − f ( it ) tdt = i 2 ˆ ∞ −∞ f ( it ) | t | dt. Note that the second equalit y is obtained by rotating the contours C + /C − coun ter-clo c kwise b y π 2 . This is justified as the t wo in tegrands are analytic in the first and third quadrants. So, having rotated the contour, let us consider the pushforward of D O . Denote the pushforw ard measure b y dν N [ t ]. What can w e sa y about this measure? If w e mak e the reasonable assumption that this measure is asymptotic to e − N 2 f ( t ) dt , with f ( t ) b eing sufficiently regular, then in fact, using Laplace’s metho d, w e can take f ( t ) = A 2 ( t − t 0 ) 2 . Moreov er, this form of the measure do vetails nicely with the concentration phenomenon. Finally , the co efficien ts A, t 0 , and the fact that the o verall factor in front of f is N 2 (as opp osed to some other sequence going to infinity) can b e read off from the large N asymptotics of the first tw o momen ts of dν N [ t ] dt . 8 W e are thus faced with computing the asymptotics of the mean and the v ari- ance of dν N [ t ] dt . Since we will b e computing v arious in tegrals o ver O ( N ), let us adapt the graphical notation used in the literature [14] to p erform such inte- grals. The Kroneck er’s delta, δ aa ′ will b e denoted b y an undecorated line, not necessarily straight. Note that it is irrelev ant which index is attac hed to which end of the line. An entry of an element of O ( N ), O aa ′ will b e denoted by a v ertical line decorated b y a disk. The upp er end of the line represents the index a while the lo wer represents the index a ′ . Other matrices, e.g. ˆ M aa ′ will b e denoted by other (non-circular) shap es with tw o ‘legs’, the left one representing the index a and the right one representing the index a ′ . Connecting tw o shap es represen ts iden tifying the t wo relev ant indices and summing, that is, a product of the relev ant matrices. Figure 1 shows a summary of the notation ab o v e. δ aa ′ , a a ′ O aa ′ , a a ′ ˆ M aa ′ , O t aa ′ ˆ M a ′ a ′′ O a ′′ a ′′′ a a ′′′ Figure 1: Summary of the gr aphic al notation use d b elow. Note that we’ve only chosen a p articular r epr esentation of the Kr one cker’s delta. We c ould have dr awn any other line. Also note that the last diagr am is inde e d e qual to the displaye d mathematic al expr ession sinc e the tr ansp ose inter changes the indic es. W e need to compute integrals of p olynomials ov er O ( N ). This is a mature sub ject (see [15] for a bird’s eye ov erview) where there are explicit formulas for the expression b elo w. How ev er, since our interest is in the large N limit, w e only need the following simple form ula (first obtained in [16]), giving the dominan t asymptotic of a product of an ev en n um b er of O ’s: ˆ dO O a 1 b 1 . . . O a 2 k b 2 k = 1 N k X δ a α 1 a β 1 δ b α 1 b β 1 . . . δ a α k a β k δ b α k b β k + o ( 1 N k ) , 8 Note that one could try to obtain further justification of the asymptotic form abov e by attempting to compute al l the moments of dν N [ t ] dt along the lines that were done in [5]. How ev er, in this case it is a significantly more inv olved combinatorial problem and will b e left to p ossible future work. where the sum goes o v er all possible splittings of the collection { 1 , 2 , . . . , 2 k } into pairs { ( α 1 , β 1 ) , ( α 2 , β 2 ) , . . . , ( α k , β k ) } . Figure 2 gives a graphical represen tation of the form ula ab o v e for the k = 1 and k = 2 cases. ´ dO = 1 N ´ dO = 1 N 2 ( + + ) + o ( 1 N 2 ) Figure 2: The gr aphic al r epr esentation of the c ases k = 1 and k = 2 of the asymptotic inte gr ation formula ab ove. Note that in the k = 1 c ase, the dominant asymptotic is the exact answer. The Mean and the V ariance W e are now ready to compute the asymptotic mean and v ariance of the measure dν N [ t ]. Let us begin with the mean t 0 : t 0 = ˆ tdν N [ t ] = ˆ D Ot ( O ) = 1 2 N V ˆ D O T r  ln( K aa ′  = 1 2 N V ˆ D O T r  ln(( − ∂ 2 + µ ) δ aa ′ + O t ˆ M aa ′ O )  = 1 2 V T r ln( − ∂ 2 + µ ) + 1 2 N V ˆ D O T r ln  δ aa ′ + ( − ∂ 2 + µ ) − 1 O t ˆ M aa ′ O )  . No w, b efore pro ceeding an y further, we shall mak e the simplifying ansatz of taking the field M aa ′ to b e indep endent of space. The justification for this is that we anticipate, similarly to what happ ens in the large N vector mo del, that the configuration of this field that gives the dominant asymptotic of the parti- tion function is translation inv arian t. Keeping this simplification in mind, w e get 1 2 N V ˆ D O T r ln  δ aa ′ + ( − ∂ 2 + µ ) − 1 O t ˆ M O  = 1 2 N V ∞ X n =1 ˆ D O T r ( − 1) n n  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n . F o cusing now on a the particular term in the sum ov er n and reverting to the discrete notation for clarit y , w e get that ( − 1) n 2 nN V ˆ D O T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n = ( − 1) n 2 nN V X x 1 ,...,x n X a 1 ,b 1 ,c 1 ,...,a n ,b n ,c n ˆ D O ( − ∂ 2 + µ ) − 1 x 1 ,x 2 O t a 1 b 1 ( x 2 ) ˆ M b 1 c 1 O c 1 a 2 ( x 2 ) . . . . . . ( − ∂ 2 + µ ) − 1 x n ,x 1 O t a n b n ( x 1 ) ˆ M b n c n O c n a 1 ( x 1 ) . The sum o ver the x ’s can b e split into the follo wing classes: the class where all the x ’s are different, the class where all are differen t except tw o, the class where all different except three and so on. W e shall consider only the first t w o classes since, as will be apparen t below, the rest are suppressed. Let us now perform the D O integration in these tw o cases. In the case where all the x ’s are different, w e get n rep etitions of the following graphical form ula ´ dO = 1 N Figure 3: The dashe d lines r epr esent indic es which ar e c ontr acte d with other terms. The triangle stands for the matrix ˆ M aa ′ . P erforming all the in tegrals, w e will b e left with the follo wing expression 1 N ! n =  1 N P a ˆ M a  n  P b δ bb  = N  ´ d ˆ M ρ ( ˆ M ) ˆ M  n ≡ N  ˆ M  n Figure 4: The r esult of p erforming the O inte gr al in the c ase of non-c oincident p oints. The b ar denotes aver aging with r esp e ct to the ρ me asur e. In the case when all the x ’s are different except tw o, the O integrations o ver those O ’s lo cated at the non-coinciden t points proceeds as ab o ve. The graphical calculation of in tegration o ver the remaining O ’s is sho wn in figure 5. In order to proceed, w e use the follo wing, easily v erified form ula ( − ∂ 2 + µ ) − 1 x,y = 1 Λ 2 X p e ip ( x − y ) p 2 + µ . Putting it all together, we ha ve that 1 N ! n − 2 ´ dO = 1 N ! n − 2 ´ dO = 1 N ! n − 2 =  1 N P a ˆ M a  n − 2  P b ˆ M 2 b  = N  ´ d ˆ M ρ ( ˆ M ) ˆ M  n − 2  ´ d ˆ M ρ ( ˆ M ) ˆ M 2  ≡ N  ˆ M  n − 2  ˆ M 2  Figure 5: The r esult of doing the O inte gr als when exactly two x ’s ar e c o- incident. As ab ove, the b ar denotes aver aging with r esp e ct to the ρ me asur e. The starting expr ession is that obtaine d after inte gr ating over the O ’s at non- c oincident p oints. The first and se c ond e qualities ab ove fol low fr om the fact that O t O = I and that the dO is a pr ob ability me asur e. ( − 1) n 2 nN V X x 1 ,...,x n X a 1 ,b 1 ,c 1 ,...,a n ,b n ,c n ˆ D O ( − ∂ 2 + µ ) x 1 ,x 2 O t a 1 b 1 ( x 2 ) ˆ M b 1 c 1 O c 1 a 2 ( x 2 ) . . . . . . ( − ∂ 2 + µ ) x n ,x 1 O t a n b n ( x 1 ) ˆ M b n c n O c n a 1 ( x 1 ) = ( − 1) n  ˆ M  n 2 nV X x 1 ,...,x n ( − ∂ 2 + µ ) − 1 x 1 ,x 2 . . . ( − ∂ 2 + µ ) − 1 x n ,x 1 +  n 2  ˆ M 2 ˆ M 2 − 1  X x 1 ,...,x n − 1 ( − ∂ 2 + µ ) − 1 x 1 ,x 2 . . . ( − ∂ 2 + µ ) − 1 x n − 1 ,x 1 + . . . = 1 (2 π ) 2 " ( − 1) n  ˆ M  n 2 n + ( − 1) n ( n − 1)  ˆ M  n 4Λ 2  ˆ M 2 ˆ M 2 − 1  + o  1 Λ 2  # ˆ d 2 p ( p 2 + µ ) n . Therefore, t 0 = 1 2 V T r ln( − ∂ 2 + µ ) + 1 2 ˆ d 2 p ln  1 + ˆ M p 2 + µ  + O (1) = 1 2(2 π ) 2 ˆ d 2 p ln( p 2 + µ + ˆ M ) + O (1) . Note that the correction is indeed of order 1 since the second term in the square brac kets ab ov e when summed ov er n giv es an integrand of order 1. The latter, when in tegrated, giv es a term of order Λ 2 whic h in turn cancels with the iden- tical factor in the denominator. Before w e compute the v ariance, let us compute ´ D O ´ J ab ( K aa ′ ) − 1 J a ′ b , since the relev ant calculation is almost iden tical with the one done ab ov e to obtain t 0 . In fact, w e hav e ˆ D O ˆ J ab ( K aa ′ ) − 1 J a ′ b = ˆ ˆ D OJ ab  − ∂ 2 + µ + O t ˆ M O  − 1 aa ′ J a ′ b = ˆ ˆ D OJ ab ( − ∂ 2 + µ ) − 1  I + ( − ∂ 2 + µ ) − 1 O t ˆ M O  − 1 aa ′ J a ′ b = ∞ X n =0 ( − 1) n ˆ ˆ D OJ ab ( − ∂ 2 + µ ) − 1  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n aa ′ J a ′ b . The calculation then pro ceeds by integrating ov er the O ’s as shown in figure 3. Th us, ˆ D O ˆ J ab ( K aa ′ ) − 1 J a ′ b = ˆ J ab  − ∂ 2 + µ + ˆ M  − 1 J ab . Mo ving on to computing the v ariance of dν N [ t ], we need to compute A − 1 = ˆ ( t 2 − t 2 0 ) dν N [ t ] = ˆ D Ot 2 ( O ) −  ˆ D Ot ( O )  2 = 1 4 N 2 V 2  ˆ D O  T r ln  δ aa ′ + ( − ∂ 2 + µ ) − 1 O t ˆ M aa ′ O )   2 −  ˆ D O T r ln( δ aa ′ + ( − ∂ 2 + µ ) − 1 O t ˆ M aa ′ O ))  2  = 1 4 N 2 V 2 X n,m ( − 1) n + m nm  ˆ D O T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n × T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  m −  ˆ D O T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n × ˆ D O ′ T r  ( − ∂ 2 + µ ) − 1 O ′ t ˆ M O ′  m  . Consider the term in the square brack ets ˆ D O T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  m − − ˆ D O T r  ( − ∂ 2 + µ ) − 1 O t ˆ M O  n ˆ D O ′ T r  ( − ∂ 2 + µ ) − 1 O ′ t ˆ M O ′  m = X x 1 ,...,x n ; y 1 ,...,y m ( − ∂ 2 + µ ) − 1 x 1 ,x 2 . . . ( − ∂ 2 + µ ) − 1 x n ,x 1 ( − ∂ 2 + µ ) − 1 y 1 ,y 2 . . . . . . ( − ∂ 2 + µ ) − 1 y m ,y 1  ˆ D O tr  O t ( x 1 ) ˆ M O ( x 1 ) . . . O t ( x n ) ˆ M O ( x n )  × tr  O t ( y 1 ) ˆ M O ( y 1 ) . . . O t ( y m ) ˆ M O ( y m )  − ˆ D O tr  O t ( x 1 ) ˆ M O ( x 1 ) . . . O t ( x n ) ˆ M O ( x n )  × ˆ D O ′ tr  O ′ t ( y 1 ) ˆ M O ′ ( y 1 ) . . . O ′ t ( y m ) ˆ M O ′ ( y m )   . The D O and D O ′ in tegrals will depend, as before, on the fact whether there are any coincidences betw een the points at whic h the relev ant O ’s and O ′ ’s are lo cated. It is ob vious that unless there are coincidences b et ween the x ’s and the y ’s, the terms in the square brack ets cancel eac h other. Therefore, we only need to consider the situation with coincidences b et w een the x ’s and the y ’s. Of course, each extra coincidence “costs” a factor of 1 Λ 2 , and thus, w e need only lo ok at the minimal possible n um b er of coincidences whic h give nonzero answers. It is easy to see that if there is a single x coinciding with a single y , then the tw o terms still cancel out. The reason for this is that after doing the in tegral ov er all the other O ’s and O ′ ’s, the dependence on the remaining orthogonal matrix will disapp ear (similarly to ho w it happened in the calculation in figure 5). W e th us need to consider the case when there are precisely t w o pairs of coinciden t p oin ts. Performing the in tegrals o ver all the other p oin ts, w e see that the term in brack ets is calculated graphically as sho wn in figure 6. Th us, w e ha ve that, up to subdominant terms, A − 1 = 1 4 N 2 V 2 X n,m ( − 1) n + m nm  4  ˆ M  2 ˆ M 2 +  ˆ M 2  2   ˆ M  n + m − 4 × X x 1 ,...,x n ; y 1 ,...,y m ; x i = y i ′ ,x j = y j ′ ( − ∂ 2 + µ ) − 1 x 1 ,x 2 . . . ( − ∂ 2 + µ ) − 1 x n ,x 1 × . . . · · · × ( − ∂ 2 + µ ) − 1 y 1 ,y 2 . . . ( − ∂ 2 + µ ) − 1 y m ,y 1 = 4 ˆ M 2 ( ˆ M ) 2 + ( ˆ M 2 ) 2 ( ˆ M ) 4 4 N 2 V 2 Λ 8 X n,m ( − 1) n + m ( ˆ M ) n + m X x,y n − 1 X α =1 m − 1 X β =1 X p,p ′ ,q ,q ′ e i ( p + p ′ + q + q ′ )( x − y ) × 1 ( p 2 + µ ) α 1 ( p ′ 2 + µ ) n − α 1 ( q 2 + µ ) β 1 ( q ′ 2 + µ ) m − β ´ d d − N 2  ˆ M  4 !  ˆ M  n + m − 4 = 1 N 4 ( 2 × + 2 × + + o ( N 4 ) )  ˆ M  n + m − 4 =  4  ˆ M  2 ˆ M 2 +  ˆ M 2  2 + o (1)   ˆ M  n + m − 4 Figure 6: The starting expr ession ab ove is that obtaine d after inte gr ation over al l the O ’s at non-c oincident p oints. The o ( N 4 ) terms in the curly br ac es c ome fr om the r emaining two ways of r e c onne cting the e dges. = 4 ˆ M 2 ( ˆ M ) 2 + ( ˆ M 2 ) 2 ( ˆ M ) 4 4 N 2 V 2 Λ 8 X n,m ( − 1) n + m ( ˆ M ) n + m X x,y X p,p ′ ,q ,q ′ e i ( p + p ′ + q + q ′ )( x − y ) ×  1 p 2 + µ 1 p ′ 2 + µ ( 1 p 2 + µ ) n − 1 − ( 1 p ′ 2 + µ ) n − 1 1 p 2 + µ − 1 p ′ 2 + µ  ×  1 q 2 + µ 1 q ′ 2 + µ ( 1 q 2 + µ ) n − 1 − ( 1 q ′ 2 + µ ) n − 1 1 q 2 + µ − 1 q ′ 2 + µ  = 4 ˆ M 2 ( ˆ M ) 4 + ( ˆ M 2 ) 2 4 N 2 V 2 Λ 8 " X x,y X p,p ′ ,q ,q ′ e i ( p + p ′ + q + q ′ )( x − y ) ×  1 p 2 + µ 1 p ′ 2 + µ 1 q 2 + µ 1 q ′ 2 + µ  p 2 + µ p 2 + µ + ˆ M − p ′ 2 + µ p ′ 2 + µ + ˆ M  ×  q 2 + µ q 2 + µ + ˆ M − q ′ 2 + µ q ′ 2 + µ + ˆ M  # . W e no w obtain the con tinuum limit asymptotics of the term in the square brac k- ets abov e. In this limit, note that the terms in the curly brack ets are asymptotic to 1. W e th us consider X x,y X p,p ′ ,q ,q ′ e i ( p + p ′ + q + q ′ )( x − y )  1 p 2 + µ 1 p ′ 2 + µ 1 q 2 + µ 1 q ′ 2 + µ  ∼ V 4 (2 π ) 8 X x,y  ˆ e ip ( x − y ) p 2 + µ dp  4 ∼ Λ 4 V 2 (2 π ) 8 ˆ dxdy  ˆ e ip ( x − y ) p 2 + µ dp  4 ≃ Λ 4 V 3 , where ≃ stands for “asymptotic up to a constan t”. Putting ev erything together, w e see that A − 1 ≃ 1 N 2  4 ˆ M 2 ( ˆ M ) 4 + ( ˆ M 2 ) 2  V Λ 4 . Large N Asymptotics W e are no w ready to use ev erything we’v e learned and plug it in (2). Then, we get Z [ J ] ∼ ˆ D ρdt exp  − 1 2 ˆ J ab  − ∂ 2 + µ + ˆ M  − 1 J ab  × exp  N 2  Λ 2 ˆ d ˆ M d ˆ M ′ ρ ( ˆ M ) ρ ( ˆ M ′ ) ln | ˆ M − ˆ M ′ | + V µ 2 λ + ˆ M 2 λ  ! × e − N 2 ( V t + A 2 ( t − t 0 ) 2 ) ! . (2) This expression is now in the appropriate form for the large N limit analysis using the usual Laplace’s metho d. Before w e do so, let us briefly sk etch an alter- nativ e wa y , alluded to after the lemma ab ov e, to arrive at the expression we hav e. Instead of the argument given ab o v e, using the Lipschitzness of the J term whic h allow ed us to take the aforementioned term outside the D O in tegral, we could hav e proceeded as ab o ve, pushing forw ard the measure directly , with the c hange that ˜ t ( O ) = 1 2 N V T r  ln( K aa ′ )  + 1 2 N 2 J ab ( K aa ′ ) − 1 J a ′ b . In other words, we could hav e incorp orated the J term into the pushforw ard measure dν N [ t ]. This would ha ve shifted t 0 obtained ab o ve by δ N 2 where, up to sub dominan t terms, δ = 1 2 N 2 ˆ J ab  − ∂ 2 + µ + ˆ M  − 1 J ab . In this case, shifting in (2) the in tegral ov er t by δ , we w ould hav e reobtained the expression ab o v e. Going back to the asymptotic analysis, we hav e three v ariables, t , µ and ρ . Pro ceeding with the standard Laplace’s metho d we set the v ariation of the exp onen t with resp ect to the first t wo to zero giv es the follo wing set of equations V + A ( t − t 0 ) = 0 − V 2 λ + A ′ 2 ( t − t 0 ) 2 − A ( t − t 0 ) t ′ 0 = 0 . Of course, strictly sp eaking, w e still need to v ary with resp ect to ρ . Ho w ever, this will turn out to be unnecessary for our goals as we’ll see below. Therefore, obtaining ( t − t 0 ) from the first equation and plugging it into the second, we get − V 2 λ + V 2 A ′ 2 A 2 + t ′ 0 = 0 = ⇒ − V 2 λ − V 2 ( A − 1 ) ′ + V t ′ 0 = 0 . (3) Next, let us compute t ′ 0 = 1 4 π ˆ ˜ Λ 2 0 | p | | p | 2 + µ + ˆ M d | p | ∼ 1 8 π ln  ˜ Λ 2 µ + ˆ M  . No w, let us compare the three terms in (3). Note that the first and third terms are of the same order in Λ, while the second is greatly sub dominan t (the culprit b eing the 1 Λ 4 in A − 1 ). W e shall drop this term from the equation 9 obtaining 1 8 π ln  ˜ Λ 2 µ + ˆ M  = 1 2 λ = ⇒ µ + ˆ M = Λ 2 e − 4 π λ = µ 0 . Plugging this bac k in to Z [ J ], we see that Z [ J ] ≃ exp  − 1 2 ˆ J ab  − ∂ 2 + µ 0  − 1 J ab  . W e ha ve th us obtained the claimed fact that the principal chiral mo del in the large N limit is a an infinite collection (since the a and b indices hav e an infinite range) of free theories of massiv e particles, all with mass µ 0 . Ac knowledgmen ts: The author w ould lik e to thank J. Merhej for reading a preliminary v ersion of this paper and for the n umerous comments which hav e greatly improv ed its readability . 9 Of course, we can’t simply ignore a term in an equation. The rigorous w ay to proceed would be to solve the equation with this term present, and then verify that the result matches asymptotically with the one obtained by simply dropping this term. References [1] A. P olyak ov, P . B. Wiegmann, “Theory of nonabelian goldstone b osons in t wo dimensions”, Ph ysics Letters B, 131, 1–3, (1983), 121–126. [2] L.D F addeev, N. Y u. Reshetikhin, “Integrabilit y of the principal c hiral field mo del in 1+1 dimension”, Annals of Ph ysics, 167, 2, (1986), 227–256. [3] T. Tlas, “Y ang-Mills as a constrained Gaussian”, Adv. Theor. Math. Ph ys. 27, no. 2, (2023), 563–581. [4] D. Gross, E. Witten, “Possible third-order phase transition in large- N lat- tice gauge theory”, Ph ys. Rev. D, 21, 2, (1980), 446–453. [5] T. Tlas, “A Concen tration of Measure Phenomenon in Lattice Y ang-Mills”, arXiv: 2603.25236 [math-ph]. [6] C. Kopp er, “Mass Generation in the Large N -Nonlinear σ -Model”, Comm. Math. Phys. 202, (1999), 89–126. [7] A. Cub ero, P . Orland, “Correlation functions of the S U ( ∞ ) principal chiral mo del”, Phys. Rev. D, 88, 2, (2013), 025044. [8] A. Poly ak ov, “Gauge fields and strings”, Contemp. Concepts Phys., 3, Har- w o o d Academic Publishers, (1987). [9] Y u. Mak eenko, “Metho ds of con temp orary gauge theory”, Cambridge Monogr. Math. Ph ys., Cam bridge Univ ersity Press, Cam bridge, (2002). [10] M. Meh ta, “Random matrices”, third edition, Pure and Applied Mathe- matics (Ams terdam), 142, Elsevier/Academic Press, Amsterdam, 2004 [11] S. Coleman, “Aspe cts of Symmetry: Selected Erice Lectures”. Cambridge Univ ersity Press; 1985. [12] M. Ledoux, “The concen tration of measure phenomenon”, Math. Surveys Monogr., 89, American Mathematical So ciet y , Providence, 2001. [13] E. Mec kes, “The Random Matrix Theory of the Classical Compact Groups”, Cambridge Universit y Press, 2019. [14] M. Creutz, “Quarks, Gluons and Lattices”, Cambridge Univ ersity Press, 2023. [15] B. Collins, S. Matsumoto, J. Nov ak, “The W eingarten Calculus”, Notices Amer. Math. So c. 69 (2022), no. 5, 734–745. [16] D. W eingarten, “Asymptotic b eha vior of group integrals in the limit of infinite rank”, J. Mathematical Phys. 19 (1978), no. 5, 999-1001. Department of Mathematics, American University of Beirut, Beirut, Lebanon. Email address : tamer.tlas@aub.edu.lb