Dynamical mass generation

PROCEEDINGS OF THE I N T E R N A T I O N A L C O N F E R E N C E

SELECTED TOPICS IN QUANTUM FIELD THEORY AND MATHEMATICAL PHYSICS

PART III

D Y N A M I C A L MASS GENERATION*)

J. Ho~ek

Institute of Nuclear Physics, Czechosl. Acad. Sci., 250 68 l~e~, Czechoslovakia

We investigate the possibility of calculating the fermion and the gauge-boson masses within an electroweak SU(2) L • U(1)y gauge-invariant model without the Higgs fields. Instead of the whole Higgs sector we introduce one "Abelian" vector boson C with mass M renormalizably interacting with leptons and quarks of chiralities L and R with a strength h. An interplay of all interactions which contribute to the fermion masses results in the fermion mass formula mf = M exp [8g2/(3Y(fL) Y(fR) h2)], where Y(fL,R) are the C hypercharges, and Y(fL) �9 Y(fR) % 0. The intermediate-boson masses m w and m z are expressed in terms of the fermion masses via sum rules.

1. I N T R O D U C T I O N

In the standard SU(3)c • SU(2)L • U(1)y model [1], the masses of leptons, quarks, and of intermediate bosons are generated by the Higgs mechanism [2].

Although field-theoretically unobjectionable, this mechanism of the mass generation must be regarded phenomenological by definition: Each mass is determined by its own coupling constant, and its independent renormalization simulates the ordinary mass renormalization.

A hope for the calculable mass spectrum of leptons, quarks, and intermediate bosons is provided by theories with a dynamical symmetry breakdown [3]. There, the only parameters in the Lagrangian which undergo the renormalization, are the gauge coupling constants. Hence, most of the mass ratios, if dynamically generated,

must be calculable numbers. A theory of this sort should provide a microscopic foundation of the successful phenomenological Higgs approach.

In principle, the strategy is simple. (i) The fermion mass term is a bridge between

a left-handed and a right-handed fermion field. In the Standard model, such a bridge is built up easily from the Yukawa coupling by an assumption of nonzero vacuum expectation value of the Higgs field. Without the Higgs field such a bridge

can be built up provided there exist the gauge bosons interacting both with the left-

*) Invited talk presented at the Internat ional Conference "Selected Topics in Quan tum Field Theory and Mathematical Physics", Bechyn~, Czechoslovakia, June 23--27, 1986.

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J. Ho~ek: Dynamical mass generation

handed and the right-handed fermion fields. Technically this amounts to finding a finite solution of the Schwinger-Dyson (SD) equation for the fermion proper self- -energy part Z.

(ii) The 9auge-boson mass squared is given by the residue at the massless pole of the polarization tensor H,# of the gauge field. In the Standard model, the massless pole in the polarization tensor is due to the "would be" Nambu-Goldstone (NG) boson described by the Higgs field, and the residue is related to a nonzero vacuum expectation value of this field. Without the Higgs field, the massless pole in the polarization tensor can only be due to a dynamical "would be" N G boson built up from some fermion fields. The residue is related to a nonzero vacuum expectation value of some fermion-field bilinear combination.

In practice, the situation is sad. A microscopic theory underlying the Higgs one should, as one always demands on microscopic theories, (1) reproduce good features of the phenomenological theory more economically, (2) be able to calculate the parameters of the phenomenological theory, (3) provide new predictions lying outside the range of validity of the phenomenological theory, and (4) lead to the phenomenological theory by a controllable sequence of approximations. A scheme which would fulfil these requirements simply does not exist.

Here we present a model and its solution which, bona fide, can be regarded as an attempt in approaching this difficult task.

2. THE MODEL

Perturbatively, the model is defined by its Lagrangian:

(1) s = ~rL i7~(0~ -- ig �89 + ig' �89 - ih �89 +

+ Vra i?~(0~ -- ih �89 vfa + Ira i?~(g, - ig'B~ - ih �89 Ira +

+ qrL iY~(O~ - ig �89 - i9' ~B~ - ih �89 qfe +

+ UfR i7r - ig ~B~ - ih �89 + afR i7~(0r + ig' �89 -- ih �89 - -

-- �88 F'P(C) + �89 ~ - �88 F~P(B) - �88 F~P(A) + QCD .

It is the SU(3)c x SU(2)L x U(1)y gange-invariant model without the Higgs sector supplemented with an interaction of a massive vector boson C with fermions of both chiralities. We take the liberty of ascribing the different C hypercharges YfH to the different fermions. For the sake of brevity we shall use the term heaviness for these hypercharges.

In the standard notation, fis the family index. We note that a possibility of employ- ing the Abelian vector boson with an explicit mass term [4] for the dynamical mass generation was ah'eady mentioned in the pioneering ref. 5.

To be defined, the model has to be renormalizable. The conditions on the heaviness following from the anomaly freedom can be found in the literature. For an illustration

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J. Hogek: Dynamical mass 9eneration

we explicitly elaborate the case wi th o u t new f e r m i o n s which, .however, can be made only SU(Z)L x U(1)v x U(1)n anomaly-free [6].

If we define YrI - (Y(qL), y(UR), y(dg); Y(~'L), y(vg), Y(IR)) then the model (1) is SU(2)L x U(1)y x U(1)H anomaly-free for the heaviness which for each fermion family f can be parametrized as [6]

(2) Yfn = ~ 1) + flrY(n 2) ,

where Yn (1) = (�89 ~, - ]; - 1, 0, - 2), YH (2) = (0, 1, - 1; 0, 1, - 1), or [6]

(3) Ym = 7,u (1) + 6,Y(n a) ,

where I,-(3) = (0, 5, 1; 0, - (35) 1/3, -7 ) . In (2) and (3), el, fir, 7f, and 6f are arbitrary real parameters.

A complete analysis of the mass generation with the SU(3)c x SU(2)L X U(1)y x x U(1)H anomaly freedom including a gravity condition requires the existence

of new fermions [7], and it deserves a separate study. The model (1) has to be renormalized at a nonzero p2 = _/~2 due to the infrared

divergences (all particles except the C boson are perturbatively massless). A nice feature of the Lagrangian (1) is that there is no genuine ultraviolet renormalization of the mass parameter M [4, 8]. M is renormalized by the same infinite renormalization constant as the coupling constant h: M = Z~/2(ho, A/I t ) Mo. A finite renormalization is necessary to convert the renormalized mass parameter into a physical mass.

Finally we note that the numbers Ym can in principle be completely fixed by embedding properly the model (1) (or another)into a GUT group [9]. The mass M can also viewed as a remnant of the grand-unification. We, however, do not make any attempt in this direction, and appreciate that the model (1) is perturbatively well defined in isolation.

3. FERMION MASSES

The gauge SU(2)L • U(1)v symmetry of the Lagrangian (1) guarantees massless- ness of fermions, and W and Z bosons in every order in the perturbation theory. A nonzero fermion mass m = S(p 2 = m 2) can nevertheless exist provided a chiral- symmetry breaking proper self-energy part 2;(p2) is found as a finite solution of the SD equation for the inverse fermion propagator S- l (p) = puyU _ 2;(p2) of a chirally invariant Lagrangian.

A. C h a r g e d l e p t o n s

For Dirac fermions, only those vector bosons in (1) which interact both with the left- and the right-handed fermion fields contribute to 2;. In the case of charged leptons, these are the massless U(1)y boson B and the new massive boson C. The corresponding SD equation for 27~(p2), i = e, g, z . . . . (we neglect here and in the following the fermion mixing for simplicity), with the Wick's rotation already

Czech. j. ehy,, B 37 [1987] 1 0 0 3

d. Hogek: Dynamical mass generation

performed, is [101

(4) Zi(p 2) = 3 ~ d4k ff �9 2g'2(#) Y(~/iL) Y_(21iR) h2(]/) 1 ZI(k2) J(2rc) 4 L(P - k) ~ + ~ _ z k TM J k + z~(k2)"

The fermion propagator in (4) is full, the vertices and the boson propagators (in the Landau gauge) are taken bare. This (ladder) approximation is justified for small coupling constants and it is the first step in a systematic iteration procedure [111. 22 is the only quantity which must be found nonperturbatively.

In contrast to the models without the bare masses where the mass scale is given by the renormalization point #, the mass scale in our model is provided by the mass M of the C boson. Since one arbitrary mass scale is enough, it seems natural to fix # in (4). Physically preferred renormalization points #i are those at which the two independently running coupling constants 2g'Z(p) and Y(@iL)Y(liR)hZ(P) meet, provided that Y(~IL) y(liR) < O:

(5) 29'2(/0 + Y(~/L) y(liR) h2(/~) = 0.

This condition is reminiscent of the condition imposed on two independent coupling constants e and 2 in the model of the dynamical mass generation of Coleman and Weinberg [121.

With (5) taken into account the eq. (4) becomes [10, l l 1

221(p2) = _ _ 3Y(O/L) y(liR) h2(/*i) Mz(#,) •

f d4k 1 Zi(k 2) x (2n) 4 (P _ k) 2 [(P -k) 2 + MZ(pi)] (k 2 + X~(k2)),

i.e., its kernel is Fredholm. Although the exact solution of this equation is not known, the approximate value of the fermion mass ml ~- 2;i(0) defined at Pi is, fortunately, determined with a good accuracy [10, 111:

(6) mi(1) = M(pl) exp [Sn2/(3y(~iL) Y(/,R)hZ(Pi))l = M(pi) exp [-n/(3c((#,))] .

An important point is that the conclusion of refs. 10, 11 about an inconsistency of the solution (6) for a small coupling constant does not apply here, since M is not a dynamically generated mass.

Due to the relation g' = e/cos Ow valid in the present model, we express the different lepton masses m~(1) in terms of the fine structure constant c~ = �88 and the Weinberg angle tg O w = g'/g, both determined at the different scales Pi-

B. N e u t r i n o s

In general, neutrinos, being the only electrically neutral fermions, can dynamically acquire both the Dirac and the Majorana masses. However, the Fredholm kernel cannot be constructed either for the Dirac mass term 'TLZDV• or for the Majorana

- M C mass term VRZ R(vR) of the right-handed neutrino field. The reason is that the weak

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J. Hogek: Dynamical mass generation

hypercharge of Vg is zero and, consequently, only the C boson contributes to Z D and S M in these cases. Fredholm kernel cannot be constructed either for the Majorana mass term ~LZM(vL) c of the left-handed neutrino field, although for a different reason: all contributions to S M (i.e. A 3, B, C) have necessarily the same sign:

g2(it)/( p _ k)Z + g,Z(#)/(p _ k)2 + yZ(~iL ) hZ(#i)/[(p _ k)2 + M z] =t= O.

Hence, the neutrinos stay massless in our approach (without mixing).

C. Q u a r k s

At small distances, where the strong-interaction coupling constant gs(P) is small due to the asymptotic freedom, the QCD is treated perturbatively. This means that the current quarks, and the gluons at small distances behave like the ordinary particles, i.e. like leptons, and the electroweak gauge bosons. Consequently, we are obliged to include into the dynamical generation of the current quark masses, be- sides the B and C exchanges, also the attractive exchange of the massless gluons.

The conditions analogous to (5) that fix the renormalization points #i are then

(7) 4 2 xgs(/A) + ~-g'2(P/) + Y(qiL) y(UlR) h2(pl) = 0

for i ---- u, e, t . . . . quarks, and

,gs(#i) ~g,2(/,i) + Y(qiL) y(dia) h2(pl) = 0 ( s ) , 2 _

for i = d, s, b . . . . quarks. The quark mass formulas are

(9) mi(u) = M(p/)exp [8r~2/(3y(qiL) Y(UIR) hZ(Pi))] =

= M(#i)exp [--r~/(2as(/~,) + }a'(#,))]

for i = u, c, t . . . . quarks, and

(10) m,(d) = M(m) exp [8~2/(3y(q,L) y(d,R) h2(/~,))] =

= M(p,) exp [-=l(2~s(/*,) -- �89 .

It seems natural that the quark current masses are expressed not only in terms of the electric charge and the Weinberg angle like the leptons, but also in terms of the

QCD charge. Fermion mixing means that Z become the nondiagonal matrices in the space

of the fermions with equal electric charges. Straightforward consequences of this are that (i) the unitary Kobayashi-Maskawa mixing matrix appears in the charged quark weak current, and (ii) the current coupled to the C boson becomes flavour- changing. This latter fact severely restricts the value of the mass [13 3 M:

(11) M > 10 3 T e V .

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3". Hogek: Dynamical mass generation

4. GAUGE BOSON MASSES

The dynamically generated fermion masses 2; break spontaneously the gauge SU(2)L x U(1)y symmetry down to U(1)e m. Consequently, the W and Z bosons acquire the masses dynamically. To show this, we must calculate the residue at the massless pole of the polarization tensor of the W and Z gauge fields. The massless poles correspond to the "would be" NG bosons. They are seen in the proper vertex functions F~v and /'~ as necessary consequences of the SU(2)L x U(1)y Ward- Takahashi identities [14] which we assume renormalizably maintained:

g r~v(P + q, P) = ~ o ~ 2 7~(1 - 7.,) -

g q "

2x/2 q2 [(1 - 75) 2;u(p + q ) - (1 + 75) 27D(p)] ,

g {t37~( 1 - 7) -- Q7 ~ sin20w - r i ( p + q, p) = ~-,o 2 cos Ow

} q2 ta[Z(P + q) + S(p)] 75 �9

From the pole terms of F~v and F) we extract the effective vertices between the fermions and the dynamical "would be" NG bosons [3, 5]:

e • (1 -T- 75) mu - (1 _+ 75) mD 27st3m = m Z l r ] 1 / 2 , Po = ~ 2 I ~ 1 / 2 "

(m~Iu ;D + " 'D 'D;W ( m 2 I u ; u + ' "D 'D;D,

The indices U and D stand for the upper (U) and the down (D) fermion in a SU(2) doublet. The dimensionless quantities IF;F, will be defined in the following.

In accordance with the definition of P• the pole term of F~v can be decomposed [3, 5] as shown in fig. 1. The only quantity to be calculated is the loop integral

g f d C k (12) J~v(q) = ~ 3 (2rQ 4 Tr P_ Su(k) 7~(1 - 7) SD( k -- q) .

We evaluate it with the fermion propagators approximated by SF(p)= (puTU + + SF(pZ))/(p = -- rag). The most divergent part of the integral identically vanishes due to the trace, while the rest is finite due to the fact [10, 11] that Sv(p 2) ,-, p -4 for large p. I f we define 2;F(p z) = mvav(p2), and

f d ' k (k -- q)~ au(k 2) _ (_iq~)iu;o(qZ) Iu ;D(q) = 8G0(2•)4 (k z _ 7 ~ - [ ~ - ~ ~)2 ~ mDZ]

(n~ = 1 for leptons, ne = 3 for quarks) we can rewrite the integral (12) as

2 2 2 2 g mulu;D(q ) + mDlD;u(q ) J~v(q) = ~ 2 (- iq~)

(mg IU;D(O) + m 2 ID;u(0)) l/z

] 006 Czech. ;. Phys. B a7 ['~9871


J~v(q) represents the direct coupling of the charged "would be" NG boson with the W boson (see fig. 1). Its knowledge is crucial for the determination of rn~ as a residue at the massless pole of the longitudinal part of the polarization tensor of the W boson (see fig. 2). The rest of H~p follows automatically [3, 5] from the property of transversality. Since both lepton and quark doublets operate in this mechanism

K-q

~W, po|e = =

P U

U• W . . . . -ju.

Fig. 1. The effective coupling of charged NG bosons with fermions and W boson.

W l/q 2 W

~' 1 Ju,D(q) q q JU;D(q

Fig. 2. The pole term in the vacuum polarization tensor of the W bosom

incoherently, we arrive at the sum rule for m~:

(13) rn~ = �88 ) + rnglD;u(0)).

The evaluation of m~ proceeds quite analogously, and we state merely the result:

(14) m~ = �88 + g,2) X(m~jfu;u(O ) + mglD;o(0)) �9

This completes our program of the dynamical calculation of the elementary- particle masses. To one's satisfaction the knowledge of the function Xi(p 2) is, however, clearly missing.

5. CONCLUSIONS

While the Higgs mechanism describes the fermion masses by introducing the independently renormalized Yukawa coupling constants which differ by many orders of magnitude, we calculate the fermion masses in terms of a new hypercharge called heaviness of the same order of magnitude for all fermions, and not undergoing any renormalization. While the generation of the W and Z boson masses is completely independent of the generation of the fermion masses in the Higgs

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mechanism (there is no theoretical reason at all for using just the Higgs doublets), the W and Z boson mass generation is a necessary and unique consequence of the fermion mass generation in our approach.

A. G a u g e b o s o n masses

The sum rules (13) and (14) imply that:

(i) The canonical tree-level ratio of the standard model

(15) 2 2 mw/m zoos 2 0 w = 1

is valid only provided the fermion masses in the weak doublets are degenerate. In reality this is not the case. Hence, we predict a definite (hopefully small) departure from the relation (15).

(ii) There is no fundamental weak-interaction scale in the present model. The value (x/2 GF)- 1/2 ~_ 250 GeV is a remnant of the heavy fermions. The only fundamental mass scale is fixed by the mass of the C boson, and it is much higher, see (11).

(iii) We can determine an absolute upper bound on the heaviest fermion in the world, and to study the saturation of the sum rules (13) and (14) with presently known fermions.

For a numerical illustration of these points we use a model [15] which satisfies the correct behaviour of Si(p 2) both at p2 = O, and p2 __, o0:

(16) Si(p 2) = m, M4(p 2 - M2) -2 = m i a ( p 2 ) .

With (16) the integrals IF;V, (0) are easily calculated. For M >> mu, mo which is 2 2 a limit we are really interested in, we get the sum rules for m w and m z expressed

explicitly in terms of the experimentally accessible quantities:

{E -' J "~ +

(17) m 2 = �88 2 2; (2~) z m 2

I M 2 - ( 1 +A(r + m 2 l n m ~

2 n c m 2 I n - - - 23- + m 2 I n - - - ~ . (18) mz = �88 + 9 , z ) S ( ~

Here ~ 2 2 = mu/mo, and A(~) = ~(~2 _ 1)-x In ~.

In accordance with the general formulas the relation (15)is exact with (17) and (18) at ~ = 1(A(1) = �89 Deviation from this relation cannot be large even at the extreme limit ~ ~ ~ , since A(~) monotonously decreases from �89 to zero at ~ = o% and the symmetric logarithmic terms always dominate for M ~> mu, roD.

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An absolute upper bound on the heaviest fermion in the world is obtained in the approximation (16) assuming that the sum rules (17) and (18) are saturated by just

one quark:

2 3 2 M2 3 m 2 In __-5- -- ~ " m w = 192(2rQ a m u I n - 1 , rnz z =�88 + g , 2 ) ~

For M = 106 GeV we get from the first equation mum,x - 230 GeV. With this value 2 2 = mw/rn z 1.032 cos 2 0 w. Similarly, with M = 10 s GeV we get mOmax "~ 275 GeV, and 2 2 mw/m z = 1.048 cos = O w. It is good that the dependence of mma x upon M is rather weak. Even for M = 1 0 1 5 GeV we get a reasonable ram, x ~- 120 GeV.

2 Finally, we check how the presently known fermions saturate the sum rule for m z. With m u = 5 . 1 0 -a GeV, m d = 9 . 1 0 -3 GeV, m e = 5 . 1 0 -4 GeV, me = 1"35 GeV,

ms = 0 ' 1 7 5 G e V , m r = 0 . 1 0 5 G e V , and r o t = 50GeV, rn b = 5"6GeV, ms = = 1.785 GeV we get (with M = 106 GeV) mz 2 - (�88 + 9 '2) (60 GeV) z instead of m z2 _ �88 (92 + 9 '2) (250 GeV) 2. Hence, new heavy families are necessary.

B. F e r m i o n m a s s e s

First, we should answer the question of how many parameters we deal with in the Lagrangian (1). We distinguish between those which undergo an infinite renormalization, and between the pure finite numbers.

To the former category belong the coupling constants 9, 9', 9s, h, and the mass of the C boson M. These must be fixed once for ever f rom an experiment at some point Po, say #o~ = row. Three of these parameters are already known [16] (9, 9' , 9s), although 9s not with a satisfactory accuracy.

To the latter category belong the hypercharges, and the number of the fermion families F. We should notice the trivial fact that the group s t r u c t u r e SU(2)L • x U(1)v • "U(1)H" admits arbitrary Y and H hypercharge assignments. The weak

hypercharges are fixed by agreement with the experimentally observed fermion electric charges, and the heaviness will be fixed by agreement with the experimentally observed fermion masses. Thus the concept of the fermion mass is put on the same footing with the fermion electric charge. A general observation is that more than three fermion families are necessary. With F = 3, and ct'(p0) = 0.0099, ~'(#) remains too small up to the Planck mass, and to get the electron mass f rom eq. (6) would require M -~ 1 0 2 4 GeV, With F = 6, 7, 8 :r grows faster, and the situation looks rea-

sonable. We have performed a crude numerical analysis of the fermion mass formulas (6),

(9), (10) using the one-loop renormalization group formulas for 9' , gs, h neglecting the fermion mixing. The only acceptable heaviness assignment is (3) assuming the standard fermion content. We should repeat that the heaviness assignment (3) is not SU(3)c x SU(Z)L X U(1)y x U(1)H anomaly-free. By fitting any two fermion masses in the family to their experimental value we "predict" the third. The predictions do not agree with reality. Thus, we are inclined to conclude that newfermions

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a re n e c e s s a r y al though other explanations are possible (fermion mixing, two-loop effects, ...).

We believe that the mechanism of the fermion, and the gauge-boson mass genera- t ion invented above represents a real computa t ional framework. It should, however,

be improved in several respects. The main point is to find a t rustworthy solution of the SD equat ion for S(pZ).

Shortly, all good features of the s tandard electroweak low-energy phenomenology are reproduced by the model presented here. A more detailed compar ison of the

present, hopefully microscopic approach with the canonical but phenomenological Higgs one is taste-dependent to some extent, and we leave it to the reader.

Received 16. 3. 1987.

References

[1] Glashow S. L.: Nucl. Phys. 22 (1961) 579. Weinberg S.: Phys. Rev. Lett. 19 (1967) 1264. Salam A.: in Proc. 8-th Nobel Symp. (ed. N. Svartholm). Almquist and Wiksell, Stockholm, 1968.

[2] Higgs P. W.: Phys. Lett. 12 (1964) 132; Engl6rt F., Brout R.: Phys. Rev. Lett. 13 (1964) 321. Guralnik G. S., Hagen C. R., Kibble T. W. B.: Phys. Rev. Lett. 13 (1964) 585.

[3] Farhi E., Jackiw R.: in Dynamical gauge symmetry breaking. World-Scientific, 1982 (a collec- tion of papers relevant to the subject).

[4] Itzykson C., Zuber J .- B.: Quantum field theory. Mc Graw Hill Inc., New York, 1980. [5] Jackiw R., Johnson K.: Phys. Rev. D 8 (1973) 2386. [6] Chanowitz M. S., Ellis J., Gaillard M. K.: Nucl. Phys. B 128 (1977) 506. [7] Barr S. M., Bednarz B., Benesh C.: Phys. Rev. D 34 (1986) 235. [8] Weinberg S.: Phys. Rev. D 8 (1973) 3497, see footnote 10. [9] Georgi H., Glashow S. L.: Phys. Rev. Lett. 32 (1974) 433.

[10] Pagels H.: Phys. Rev. D 21 (1980) 2336. [11] Engl6rt F.: in Lectures at Cargese Summer Institute, 1975. [12] Coleman S., Weinberg E.: Phys. Rev. D 7 (1973) 1888. [13] Cahn R. N., Harari H.: Nucl. Phys. B 176 (1980) 135;

Dimopoulos S., Ellis J.: Nucl. Phys. B 182 (1981) 505. [14] Huang K., Mendel R.: Preprint MIT-CTP 876, 1980. [15] Pagels H., Stokar S.: Phys. Rev. D 20 (1979) 2949;

Cornwall J. M.: Phys. Rev. D 22 (1980) 1452. [16] Maiani L., Petronzio R.: Preprint CERN-TH 4415/86, 1986.

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