Perturbative anomalies in quantum mechanics

PER TURBA TIVE ANOMALIES IN QUANTUM MECHANICS MAXIM GRITSKO V, ANDREY LOSEV, AND SA VELIY TIMCHENKO Abstract. In this work, we propose a cohomological approach to studying perturbative anomalies in quantum mechanics. The Hamiltonian ˆ H together with the symmetry generator ˆ S forms a unitary representation of the t wo- dimensional Ab elian Lie algebra g ∼ = R 2 on the Hilb ert space V . W e show that perturbations of suc h a system are related to the first Chev alley-Eilen b erg cohomology group H 1 CE ( R 2 , u( V )) . In turn, the perturbative anomalies of the symmetry ˆ S are related to the second cohomology group H 2 CE ( R 2 , u( V )) . Contents 1. In tro duction: Lie algebra cohomological program 1 2. P erturbative anomalies as the cohomological obstructions 2 2.1. Motiv ation 2 2.2. Deformation of the representation and the CE complex 2 2.3. Application to quantum mechanics 3 3. Graded Lie algebra structure on H • ( R 2 , u( V )) 6 4. Conclusion 7 Ac knowledgemen ts 7 References 7 1. Introduction: Lie algebra cohomological pr ogram In this article, we present our initial arguments in fa vor of the idea that sym- metry anomalies in physics are essentially deformation obstructions. One of the classic questions in quan tum field theory is what happ ens to symmetry when the theory is deformed. It turns out that when studying the question of p erturbativ e symmetry breaking, it is sufficient to study the represen tation of the symmetry al- gebra on the state space of the initial theory . In this case, the appropriate language for describing deformations of a theory with symmetry is the language of homolog- ical algebra. Namely , let us consider a theory with Hamiltonian H and symmetry algebra generated b y S α suc h that [ S α , S β ] = f γ αβ S γ . Then we define the so-called Cheval ley-Elienb er g differential for the quan tum system ( H , S α ) [ LS23 ]: (1) d CE = c H H + c α S α − 1 2 f γ αβ c α c β ∂ ∂c γ . The space of infinitesimal deformations of theories with symmetries is the first cohomology group H 1 CE of the Chev alley-Eilenberg algebra of this symmetry . Ho wev er, not every infinitesimal deformation defines a curv e in the space of theories passing through a giv en p oint. The first obstruction is the map (2) µ 2 : H 1 CE ⊗ H 1 CE → H 2 CE In eac h order of p erturbation theory , obstructions µ n : ( H 1 CE ) ⊗ n → H 2 CE arise. The maps µ n form a structure of the L ∞ -algebra on H • CE and, in particular, satisfy the quadratic relations [ KS00 ]. In particular, if S satisfies the dilation of space, then 1 2 MAXIM GRITSK OV, ANDREY LOSEV, AND SA VELIY TIMCHENKO µ n can b e interpreted as co efficients of the b eta function and therefore m ust satisfy quadratic relations coming from the L ∞ -structure [ LMZ06 ; GLS24 ]. In this text, w e implemen t the program described ab ov e using a minimal example whic h we analyze in detail. 2. Per turba tive anomalies as the cohomological obstructions 2.1. Motiv ation. Consider a quantum mechanical system with state space V , Hamiltonian ˆ H , and symmetry ˆ S . By symmetry ˆ S , w e mean a self-adjoint op erator that commutes with the Hamiltonian. F or simplicit y , we will assume that ˆ H and ˆ S ha ve discrete sp ectra, i.e., that V can b e decomp osed into a direct sum (p ossibly infinite) of subspaces that are eigenspaces for b oth ˆ H and ˆ S simultaneously . No w we will p erturb the Hamiltonian, adding a new term ˆ H → ˆ H + t δ ( 1 ) ˆ H . Generally sp eaking, now [ ˆ S , ˆ H + t δ ( 1 ) ˆ H ]  = 0 and it would seem that the symmetry is broken. But is it p ossible to deform the symmetry generator ˆ S → ˆ S + t δ ( 1 ) ˆ S so that it b ecomes the symmetry of the p erturbed problem in first order in t ? The corresp onding correction δ ( 1 ) ˆ S to the generator ˆ S satisfies the equation: (3) [ ˆ H , δ ( 1 ) ˆ S ] = [ ˆ S , δ ( 1 ) ˆ H ] . Supp ose that w e hav e managed to restore symmetry in the first order of per- turbation theory by selecting the appropriate correction δ ( 1 ) ˆ S . But what ab out the next orders of p erturbation theory? Is it p ossible to deform the system ( ˆ H , ˆ S ) along the direction ( δ ( 1 ) ˆ H , δ ( 1 ) ˆ S ) in all orders of p erturbation theory? F ormally , the problem can b e stated as finding a curve ( ˆ H ( t ) , ˆ S ( t )) in the space of systems passing at time t = 0 through a given p oin t ( ˆ H , ˆ S ) with velocity ( δ ( 1 ) ˆ H , δ ( 1 ) ˆ S ) . The p erturbative solution to the problem consists in finding tw o formal series ˆ H ( t ) = ˆ H + t δ ( 1 ) ˆ H + X n ⩾ 2 t n δ ( n ) ˆ H , ˆ S ( t ) = ˆ S + t δ ( 1 ) ˆ S + X n ⩾ 2 t n δ ( n ) ˆ S (4) suc h that [ ˆ H ( t ) , ˆ S ( t )] = 0. In the second order, condition [ ˆ H ( t ) , ˆ S ( t )] = 0 requires: (5) [ ˆ H , δ ( 2 ) ˆ S ] + [ δ ( 2 ) ˆ H , ˆ S ] + [ δ ( 1 ) ˆ H , δ ( 1 ) ˆ S ] = 0. It turns out that this equation is not alwa ys solv able. Moreo ver, it turns out that if the equation (5) is solv able, then the equations in all higher orders of p erturbation theory will also be solv able. T o demonstrate this, we will consider this problem from the p ersp ectiv e of a c ohomological approach to deformations. F or this approac h, it is more natural to study not the deformation of the tw o Hermitian op erators system, but the deformation of the tw o anti-Hermitian op er- ators system, since the space of anti-Hermitian op erators is closed with resp ect to the commutator. Thus, we define the an ti-Hermitian op erators H = i ˆ H and S = i ˆ S , whic h realize the unitary r epr esentation of the ab elian Lie algebra g ∼ = R 2 in the Hilb ert space V . The deformations of such a system are essentially deformations of this representation, which is describ ed by the Chev alley-Elienberg complex [ CE48 ]. 2.2. Deformation of the representation and the CE complex. Let g b e a Lie algebra with representation ρ : g → End ( V ) in the complex vector space V . Definition 2.1. Consider the basis e i in g , dual basis c i , and the structure con- stan ts f k ij . Then the CE complex of g with co efficien ts in V ⊗ V ∗ is given by (6) CE • (g , V ⊗ V ∗ ) = S • g ∗ [− 1 ] ⊗ V ⊗ V ∗ , PER TURBA TIVE ANOMALIES IN QUANTUM MECHANICS 3 with the differential (7) d = c i ad ρ ( e i ) − 1 2 f k ij c i c j ∂ ∂c k . Hereafter, the summation conv en tion for rep eated indices is assumed. Prop osition 2.2. The first cohomology group H 1 (g , V ⊗ V ∗ ) corresp onds to the non trivial infinitesimal deformations of a Lie algebra represen tation ρ . Prop osition 2.3. The second cohomology group H 2 (g , V ⊗ V ∗ ) corresp onds to the obstructions to infinitesimal deformations of a Lie algebra represen tation ρ . Example 2.4. Consider the Lie algebra g = sl( 2, C ) with the standard basis ele- men ts e , h , f satisfying the following commutation relations: (8) [ h , e ] = 2 e ; [ h , f ] = − 2 f , [ e , f ] = h . Let V = C [ x ] b e the space of p olynomials in a v ariable x . W e consider the represen- tation ρ : g → End ( V ) corresp onding to the V erma mo dule with the highest weigh t λ ∈ C . The action of the basis elements is given b y the differential operators: (9) ρ ( e ) = ∂ x ; ρ ( h ) = − 2 x∂ x + λ ; ρ ( f ) = − x 2 ∂ x + λx . F or this represen tation, the Chev alley-Eilen b erg differen tial d can b e written as (10) d = c h ad ρ ( h ) + c e ad ρ ( e ) + c f ad ρ ( f ) − 2 c h c e ∂ ∂c e + 2 c h c f ∂ ∂c f − c e c f ∂ ∂c h . W e analyze an infinitesimal deformation ρ t = ρ + t δρ of this represen tation: (11) δρ ( e ) = 0; δρ ( h ) = 1; δρ ( f ) = x . According to Prop osition 2.2, for δρ to define an infinitesimal deformation, it m ust corresp ond to a 1-co cycle c h δρ ( h ) + c e δρ ( e ) + c f δρ ( f ) in the CE complex: (12) d ( c h δρ ( h ) + c e δρ ( e ) + c f δρ ( f )) = 0. W e verify this condition by p erforming an explicit calculation: (13) d ( c h + c f x ) = − c e c f − 2 c h c f [ x∂ x , x ] + c e c f [ ∂ x , x ] + 2 c h c f x = 0. Since this deformation corresp onds to the shift of the highest w eight b y t , it is not obstructed and defined for an y finite t , not just infinitesimally . 2.3. Application to quantum mec hanics. Let g ∼ = R 2 b e the tw o-dimensional ab elian Lie algebra with basis e 1 , e 2 . Let ρ : g → u( V ) b e a unitary represen tation on a vector space V . W e in tro duce the notation H = ρ ( e 1 ) and S = ρ ( e 2 ) . Since the representation ρ is unitary , H and S are commuting an ti-Hermitian op erators. W e construct the Chev alley-Eilen b erg complex with co efficients in u( V ) . Let c H , c S b e the dual basis elements generating the exterior algebra S • g ∗ [− 1 ] . Then (14) 0 → u( V ) d − − → ( c H ⊕ c S ) ⊗ u( V ) d − − → c H c S ⊗ u( V ) → 0, where the differential d is the CE differential (7) for the ab elian Lie algebra: (15) d = c H · ad H + c S · ad S . Consider the sp ectral decomp osition of V (16) V = M ( a , α ) V ( a , α ) , where for v ∈ V ( a , α ) , Hv = i λ a v and Sv = i µ α v with λ a , µ α ∈ R . The sub- spaces V ( a , α ) are pairwise orthogonal since they are eigenspaces of an ti-Hermitian op erators. W e denote the orthogonal pro jector onto space V ( a , α ) b y Π V ( a , α ) . 4 MAXIM GRITSK OV, ANDREY LOSEV, AND SA VELIY TIMCHENKO Lemma 2.5. The Lie algebra u( V ) decomp oses into a direct sum of vector spaces (17) u( V ) = M ( a , α ) , ( b , β ) B ( a , α ) , ( b , β ) where B ( a , α ) , ( b , β ) is the image of the linear op erator π ( b , β ) ( a , α ) : (18) π ( b , β ) ( a , α ) ( x ) = 1 2 · Π V ( a , α ) x Π V ( b , β ) + 1 2 · Π V ( b , β ) x Π V ( a , α ) . Then, the complex (14) decomp oses in to a direct sum of the complexes B • ( a , α ) , ( b , β ) : (19) 0 → B ( a , α ) , ( b , β ) d − − → ( c H ⊕ c S ) ⊗ B ( a , α ) , ( b , β ) d − − → c H c S ⊗ B ( a , α ) , ( b , β ) → 0. Pr o of. Consider the action of the differential (15) on the general element of (19): d ( w + c H x + c S y + c H c S z ) = [ c H H + c S S , w ] + { c H H + c S S , c H x + c S y } = = c H · [ H , w ] + c S · [ S , w ] + c H c S · [ H , y ] − c H c S · [ S , x ] . (20) Then it is sufficient to show that for an y element ω ∈ B ( a , α ) , ( b , β ) , it is true that (21) [ H , ω ] ∈ B ( a , α ) , ( b , β ) , [ S , ω ] ∈ B ( a , α ) , ( b , β ) . W e will pro ve this for [ H , ω ] . Let there exist such an ˜ ω that ω = π ( b , β ) ( a , α ) ( ˜ ω ) , then (22) [ H , ω ] = i λ ab 2 · Π V ( a , α ) ˜ ω Π V ( b , β ) − i λ ab 2 · Π V ( b , β ) ˜ ω Π V ( a , α ) ∈ u( V ) , where λ ab = λ a − λ b . But then it is easy to find its π ( b , β ) ( a , α ) -preimage: (23) [ H , ω ] = π ( b , β ) ( a , α ) ( i λ ab · Π V ( a , α ) ˜ ω Π V ( b , β ) − i λ ab · Π V ( b , β ) ˜ ω Π V ( a , α ) ) . Then it follows that [ H , ω ] ∈ B ( a , α ) , ( b , β ) . F or [ S , ω ] , the pro of is similar. □ Lemma 2.6. Consider L ( a , α )  =( b , β ) B • ( a , α ) , ( b , β ) . This sub complex is acyclic. Pr o of. It suffices to sho w this comp onen twise. Consider 0-cycle ω in B • ( a , α ) , ( b , β ) : dω = c H ·  i λ ab 2 · Π V ( a , α ) ˜ ω Π V ( b , β ) − i λ ab 2 · Π V ( b , β ) ˜ ω Π V ( a , α )  + + c S ·  i µ αβ 2 · Π V ( a , α ) ˜ ω Π V ( b , β ) − i µ αβ 2 · Π V ( b , β ) ˜ ω Π V ( a , α )  = 0. (24) Ho wev er, λ ab and µ αβ are not equal to zero sim ultaneously . Suppose that λ ab  = 0: (25) Π V ( a , α ) ˜ ω Π V ( b , β ) − Π V ( b , β ) ˜ ω Π V ( a , α ) = 0. It follows that ω = 0 and, therefore, (26) H 0 ( B • ( aα ) , ( b , β ) , d ) = 0. No w, let us consider the 1-co cycle ω = c H ω H + c S ω S : (27) dω = c H c S [ H , ω S ] − c H c S [ S , ω H ] = 0. Note that again λ ab and µ αβ cannot b oth b e zero at the same time. Therefore i λ ab 2 · Π V ( a , α ) ˜ ω S Π V ( b , β ) − i λ ab 2 · Π V ( b , β ) ˜ ω S Π V ( a , α ) = = i µ αβ 2 · Π V ( a , α ) ˜ ω H Π V ( b , β ) − i µ αβ 2 · Π V ( b , β ) ˜ ω H Π V ( a , α ) . (28) Supp ose that λ ab  = 0. Then it follows from this equation that (29) ω s = µ αβ λ ab · ω H . PER TURBA TIVE ANOMALIES IN QUANTUM MECHANICS 5 Then we should find suc h a 0-chain Ω that dΩ = ω = c H ω H + µ αβ λ ab · c S ω H = = c H ·  1 2 · Π V ( a , α ) ˜ ω H Π V ( b , β ) + 1 2 · Π V ( b , β ) ˜ ω H Π V ( a , α )  + + µ αβ λ ab · c S ·  1 2 · Π V ( a , α ) ˜ ω H Π V ( b , β ) + 1 2 · Π V ( b , β ) ˜ ω H Π V ( a , α )  . (30) Then Ω can b e chosen as follows: (31) Ω = 1 2i λ ab · Π V ( a , α ) ˜ ω H Π V ( b , β ) − 1 2i λ ab · Π V ( b , β ) ˜ ω H Π V ( a , α ) . The fact that dΩ = ω can b e v erified by direct calculation. The fact that the co c hain Ω ∈ B ( a , α ) , ( b , β ) w as shown in (23). Then we obtained that (32) H 1 ( B • ( aα ) , ( b , β ) , d ) = 0. Finally , it remains to show that ev ery 2-co c hain ω has a preimage. T o do this, w e will again use the fact that λ ab and µ αβ are not equal to zero at the same time. Let λ ab  = 0 again, then as a 1-co chain, w e will consider (33) Ω = c S Ω S = c S ·  1 2i λ ab · Π V ( a , α ) ˜ ω Π V ( b , β ) − 1 2i λ ab · Π V ( b , β ) ˜ ω Π V ( a , α )  . Then dΩ = c H c S [ H , Ω S ] and using the fact that Ω S ∈ B ( a , α ) , ( b , β ) w e obtain (34) dΩ = c H c S ·  1 2 · Π V ( a , α ) ˜ ω Π V ( b , β ) + 1 2 · Π V ( b , β ) ˜ ω Π V ( a , α )  = ω . It finally follows from this that H 2 ( B • ( aα ) , ( b , β ) , d ) = 0. □ Lemma 2.7. Let Z • = L ( a , α ) B • ( aα ) , ( aα ) . Then d | Z • = 0. Pr o of. This immediately follows from the fact that for every B ( a , α ) , ( a α ) (35) λ aa = µ αα = 0. Therefore, the complex b ecomes a sequence of v ector spaces with zero maps: (36) 0 → Z 0 − − → ( c H ⊕ c S ) ⊗ Z 0 − − → ( c H c S ) ⊗ Z → 0. The cohomology groups are simply the vector spaces themselv es. □ Theorem 2.8. The Chev alley-Eilenberg cohomology of the tw o-dimensional ab elian Lie algebra g ∼ = R 2 with co efficients in the mo dule u( V ) is given by: H 0 ( R 2 , u( V )) ∼ = Z ; H 1 ( R 2 , u( V )) ∼ = Z ⊕ Z ; H 2 ( R 2 , u( V )) ∼ = Z , (37) where Z = { A ∈ u( V ) | [ H , A ] = [ S , A ] = 0 } is the comm utant of the represen tation. Pr o of. This immediately follows from the lemmas 2.5, 2.6 and 2.7. □ Example 2.9. Let V = C 3 with the standard pairing ⟨· | ·⟩ . Consider the op erators: (38) ˆ H =   1 0 0 0 1 0 0 0 0   , ˆ S =   0 0 0 0 0 0 0 0 1   . 6 MAXIM GRITSK OV, ANDREY LOSEV, AND SA VELIY TIMCHENKO They satisfy [ ˆ H , ˆ S ] = 0. Consider the first-order p erturbation of the Hamiltonian: (39) δ ( 1 ) ˆ H =   0 1 1 1 0 0 1 0 0   . Note that [ ˆ S , δ ( 1 ) ˆ H ]  = 0, so the symmetry requires a correction to remain conserv ed: (40) δ ( 1 ) ˆ S =   0 0 − 1 0 1 0 − 1 0 0   . It is easy to verify that [ ˆ H + tδ ( 1 ) ˆ H , ˆ S + tδ ( 1 ) ˆ S ] = 0 in the first order with resp ect to the coupling constant t . Ho wev er, to satisfy the second order condition (5), the generators must b e corrected in the second order in t . Substituting matrices ˆ H , ˆ S , δ ( 1 ) ˆ H , δ ( 1 ) ˆ S into equation (5) and considering ma- trices δ ( 2 ) ˆ H , δ ( 2 ) ˆ S in the most general form, we obtain the following expression:   0 0 ( δ ( 2 ) ˆ H + δ ( 2 ) ˆ S ) 13 0 0 ( δ ( 2 ) ˆ H + δ ( 2 ) ˆ S ) 23 −( δ ( 2 ) ˆ H + δ ( 2 ) ˆ S ) 31 −( δ ( 2 ) ˆ H + δ ( 2 ) ˆ S ) 32 0   = =   0 − 1 0 1 0 1 0 − 1 0   . (41) It is a pro of that there are no matrices δ ( 2 ) ˆ H , δ ( 2 ) ˆ S that satisfy condition (5). Because this equation has no solution, [ δ ( 1 ) ˆ H , δ ( 1 ) ˆ S ] is not the element of the CE differen tial image. Then obstruction is the second cohomology class. 3. Graded Lie algebra str ucture on H • ( R 2 , u( V )) Prop osition 3.1. There are no higher orders obstructions. The L ∞ -algebra struc- ture on the cohomology H • ( R 2 , u( V )) reduces to a differential graded Lie algebra (dgLa) structure with a v anishing differential, meaning that all higher obstruction maps µ n v anish for n ⩾ 3. Pr o of. According to Theorem 2.8, the Chev alley-Eilenberg cohomology groups are isomorphic to the sub complex Z • = L ( a , α ) B • ( a , α ) , ( a , α ) . As established in Lemma 2.7, the differential d v anishes iden tically on this subcomplex ( d | Z • = 0) b ecause the corresp onding eigenv alues satisfy λ aa = µ αα = 0. In the context of deformation theory , the L ∞ -structure on cohomology is in- duced from the structure of the initial complex via the homotop y transfer theorem [ Arv+22 ]. Since the complex decomp oses into a direct sum of an acyclic part and a part with zero differential Z • , we can choose a homotopy transfer where the only non-trivial higher brack et is the binary commutator inherited from u( V ) . Sp ecifically , the obstruction maps µ n for n ⩾ 3 in volv e higher l n brac kets which v anish when the differential on the representativ e subcomplex is zero and no further corrections to the generators are required b eyond the second order. Th us, the only p ossible obstruction to the deformation ( ˆ H ( t ) , ˆ S ( t )) arises in the second order. □ Remark 3.2. The Prop osition 3.1 has an interesting physical consequence: if the original Hamiltonian ˆ H or the symmetry generator ˆ S had no degenerate eigen v alues, the anomaly up on p erturbation of such a system would b e zero. Prop osition 3.3. The only p erturbativ e anomaly in quantum mechanics is realized in the second order of p erturbation theory and satisfies a quadratic equation. REFERENCES 7 Pr o of. As sho wn earlier, the deformation space is the comm utant of the represen- tation of the original algebra. It is isomorphic to the direct sum of the A-type Lie algebras L ( a , α ) u( n ( a , α ) ) , where n ( a , α ) = dim V ( a , α ) . So w e can decomp ose δ ( 1 ) H and δ ( 1 ) S in to a basis e i of the commutan t Z : δ ( 1 ) H = h i e i , δ ( 2 ) S = s i e i . W e will denote by f k ij the structure constants of the commutator Z in the basis e i . Then, deforming along the basis directions in the first order, we obtain the follo wing equation in the second order of p erturbation theory: (42) [ δ ( 2 ) ˆ S , ˆ H ] + [ ˆ S , δ ( 2 ) ˆ H ] = f k ij e k . The obstruction is prop ortional to f k ij . It has a quadratic equation that comes from the dgLa structure on H • ( R 2 , u( V )) , which is essentially the Jacobi identit y: (43) f l ij f m lk + f l jk f m li + f l ki f m lj = 0. Therefore, the anomaly in this example is limited to the s econd order of p erturba- tion theory and, according to general ideology , satisfies a quadratic equation. □ 4. Conclusion In this pap er, we examine the simplest example of an anomaly that arises as a cohomological obstruction. Next, we plan to implemen t the cohomological program in a crucial example: conformal symmetry in t w o-dimensional quan tum field theory . Another interesting sp ecial case is where the anomaly arises due to deformation along the Planc k constant h . Then we can consider a quan tum mec hanical example in whic h deformation o ccurs simultaneously in tw o directions: the Planck constant h is deformed and the coupling constant t is deformed. The deformation of the coupling constant t at h = 0 corresp onds to a transition to an action that is classically inv arian t with resp ect to the deformed symmetry . On the other hand, the deformation of the Planck constant h at t = 0 corresp onds to the existence of quantum symmetry in the initial theory . Then, simultaneous deformation along these tw o directions can lead to a mixe d-typ e anomaly (with parameter t h ), which w e interpret as a one-lo op anomaly . W e in tend to construct an example of this phenomenon in one of our next pap ers. A cknowledgements W e are grateful to Vy achesla v Lysov for helpful discussions. W e are grateful to the Shanghai Institute for Mathematics and In terdisciplinary Sciences for pro viding us a w orkspace. The first author is supported b y the Ministry of Science and Higher Education of the Russian F ederation (Agreement No. 075-15-2025-013). References [Arv+22] Alex S. Arv anitakis et al. “Homotopy T ransfer and Effective Field The- ory I: T ree-lev el”. In: F ortschritte der Physik 70.2-3 (2022), p. 2200003. doi : 10.1002/prop.202200003 . arXiv: 2007.07942 [hep-th] . 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Sulimov. “New Ob jects in Scattering The- ory with Symmetries”. In: JETP L ett. 117.7 (2023), pp. 487–491. doi : 10.1134/S0021364023600428 . arXiv: 2302.09464 [math-ph] . Skolk ovo Institute of Science and Technology, 121205, Moscow, Russia Saint Petersburg St a te University, Universitetska y a nab. 7/9, 199034 St. Peters- burg, Russia Email address : m.gritskov@spbu.ru Shanghai Institute for Ma thema tics and Interdisciplinar y Sciences, Building 3, 62 Weicheng Road, Y angpu District, 200433, Shanghai, China Email address : aslosev2@yandex.ru F acul ty of Physics, Na tional Research University Higher School of Economics, St ara y a Basmanna y a ul. 21/4s1, 105066, Moscow, Russia Email address : sgtimchenko@edu.hse.ru