Colored and electrically charged gauge bosons and their related quarks

Transcription

1 Colored and electrcally charged gauge bosons and ther related quarks Eu Heung Jeong We propose a model of baryon and lepton number conservng nteractons n whch the two states of a quark, a colored and electrcally charged state and a colorless and electrcally neutral state, can transform nto each other through the emsson or absorpton of a colored and electrcally charged gauge boson. A novel feature of the model s that the colorless and electrcally neutral quarks carry away the mssng energy n decay processes as do neutrnos. euhjeong@hanmal.net; ehjeong@kda.or.kr

2 2 I. INTRODUCTION The known gauge bosons can be classfed accordng to ther color charge and electrc charge nto four types: 1. Colorless and electrcally neutral: photon and Z Colorless and electrcally charged: W ±. 3. Colored and electrcally neutral: gluons. 4. Colored and electrcally charged: X and Y bosons. All the known gauge bosons of the frst, second and thrd types medate the baryon and lepton number conservng nteractons, whereas the X and Y bosons of the fourth type,.e., the colored and electrcally charged gauge bosons, medate the baryon and lepton number volatng nteractons [1, 2]. The queston arses as to whether there exst the colored and electrcally charged gauge bosons whch medate the baryon and lepton number conservng nteractons lke the known gauge bosons of the frst, second and thrd types. In ths paper, the possblty s explored of exstence of such colored and electrcally charged gauge bosons. In secton II, we consder the varous cases of transtons of a quark from one state to another wth the emsson or absorpton of a colored and electrcally charged gauge boson. For the descrpton of the transtons wth whch we are to deal, we postulate the exstence of the colorless and electrcally neutral quarks. In secton III, we dscuss the propertes of the colorless and electrcally neutral quarks. In secton IV, we construct a model of gauge nvarant Lagrangan whch nvolves the newly ntroduced quark and gauge boson felds. In secton V, we show the applcaton of the model to the problem of the measured K + π + + mssng energy branchng rato. We shall hereafter denote by p c c the state of a partcle p, of whch the color charge and electrc charge are c and c e respectvely, and denote by C 0 colorless. For any color c, the relaton c+ c = C 0, eng the ant-c, holds. II. QUARKS AND GAUGE BOSONS Let us consder a transton of a quark from a colored and electrcally charged state to another state q e 2 wth the emsson of a colored and electrcally charged gauge boson b e b, q e 2 + b e b, and ts reverse transton wth the absorpton of the gauge boson, q e 2 + b e b : q e 2 + b e b, (1)

3 3 where C 0, e 1 0, ( = r, g, b, e 1 = 1 3, + 2 ), (2) 3 C 0, e b 0. (3) It can easly be shown that n the nteractons of the quark currents q e 2 medated by the colored and electrcally charged gauge boson b e b, the baryon and lepton numbers are strctly conserved. We shall now determne the color and electrc charges of the gauge boson b e b, and those of the quark q e 2. The transtons (1) must satsfy the law of conservaton of color and electrc charges: = +, e 1 = e 2 + e b, (4) whch gves = +, e b = e 1 e 2. (5) Substtutng for and e b n (1) ther values, we have q e 2 + b e 1 e 2 +. (6) Snce we are consderng the case C 0, e b 0, (.e.,, e 1 e 2 ), and C 0, e 1 0, we may consder four cases of equalty and nequalty between the and C 0, and e 2 and 0: Case I. When = C 0 and e 1 e 2 = 0, Case II. When = C 0 and e 1 e 2 0, q 0 C 0 + b e 1. (7) Case III. When C 0 and e 1 e 2 = 0, q e 2 C 0 + b e 1 e 2. (8) Case IV. When C 0 and e 1 e 2 0, q 0 + b e 1 +. (9) q e 2 + b e 1 e 2 +. (10) Each case wll requre consderable dscusson and lead to many theores. In ths paper, we shall restrct ourselves to the frst case, and construct a model of nteractons based on the frst case: The transton of a quark from a colored and electrcally charged state

4 4 to a colorless and electrcally neutral state q 0 C 0 wth the emsson of a colored and electrcally charged gauge boson b e 1, and ts reverse transton wth the absorpton of the gauge boson: q 0 C 0 + b e 1, (11) whch also mean that the colored and electrcally charged quark and the colorless and electrcally neutral quark q 0 C 0 transform nto each other through the emsson or absorpton of the colored and electrcally charged gauge boson b e 1. Of course, n our case (11), the exstence s postulated of colorless and electrcally neutral quarks. nvolvng ant-partcles may be made as follows: The extenson of the transtons (11) to the cases of q e 1 q C b e 1, + b e 1 qc 0 0, b e 1 + q C 0 0, qe 1 + b e 1 q C 0 0, (12) etc., where Ā denotes the ant-partcle of A. It should be noted that the colored and electrcally charged gauge boson b e 1 n (11) has the same color and electrc charges as the colored and electrcally charged quark,.e., b e b = b e 1, t thus has the color charges r, g, b and the electrc charges 1 3 e, e. The transtons (11) can be rewrtten n the form q Q/e κ q + W Q/e, (13) where we have put κ q = q 0 C 0, W = b, =, Q/e = e 1. The (13) takes the form when q = u, c or t, (Q/e = ), and when q = d, s or b, (Q/e = 1 3 ), q κ q + W + 2 3, (14) q 1 3 κ q + W 1 3. (15) III. COLORLESS AND ELECTRICALLY NEUTRAL QUARKS We may see from (14) and (15) that there can be sx colorless and electrcally neutral quarks, whch we shall call cen-quarks, κ q (q = u, c, t, d, s, b) : κ u, κ c, κ t, κ d, κ s, κ b, (16) and sx pars of (q, κ q ): (u, κ u ), (c, κ c ), (t, κ t ), (d, κ d ), (s, κ s ), (b, κ b ). Cen-quarks are colorless and electrcally neutral quarks, whereas neutrnos are colorless and electrcally neutral leptons. Snce cen-quarks have nether color charge nor electrc charge, they

5 5 partcpate nether n strong nteractons nor n electromagnetc nteractons. cen-quarks can carry away the mssng energy as do neutrnos. Let us consder two knds of transtons Ths means that q Q/e κ q + W Q/e, (17) l ν l + W, (18) where Q/e = (q = u, c, t), Q/e = 1 3 (q = d, s, b), l = e, µ, τ. We may see that the transtons (17) take the same form as the transtons (18): A colored and electrcally charged quark loses ts color and electrc charges completely through the emsson of a colored and electrcally charged gauge boson, just as an electrcally charged lepton loses ts electrc charge completely through the emsson of an electrcally charged gauge boson. Ths suggests that the descrpton of the transtons of a quark (17) can be made n the same way as the transtons of a lepton (18), that s, n the form of electroweak theory of leptons. The cen-quarks must have spn 1 2 and baryon number 1 3. To descrbe wthn the framework of SU(2) U(1) model, the left-handed cen-quarks have sospn 1 2 : Each left-handed colored and electrcally charged quark and ts left-handed cen-quark have the same magntude of sospn charge but opposte n sgn. Thus from the sospn charge T q, L for q = u, c, t, T 3 q, L = (19) 1 2 q, L for q = d, s, b, and Q e = T Y, the sospn charge T 3 and hypercharge Y of each left-handed cen-quark are 1 2 T 3 κ q, L = κ q, L for q = u, c, t, κ q, L for q = d, s, b, (20) and +1 κ q, L for q = u, c, t, Y κ q, L = 1 κ q, L for q = d, s, b. (21) Accordngly, quarks can be classfed nto four types and three generatons: u type : u, c, t κ d type : κ d, κ s, κ b κ u type : κ u, κ c, κ t d type : d, s, b (22)

6 6 and u c t κ G I = d κ, G II = s κ, G III = b. (23) κ u κ c κ t d s b We may see that cen-quarks resemble neutrnos n many respects. We utlze the resemblance between them by determnng the masses of cen-quarks from the mass condtons of neutrnos: We assume that the mass of each cen-quark s ether zero or very small n comparson to the mass of the correspondng colored and electrcally charged quark,.e., m κq = 0 or m κq m q. IV. MODEL Let us consder ten column vectors Ψ L 1 = u L c L t L d L s L, Ψ L 2 = u L c L t L κ L u κ L c, Ψ L 3 = κ L d κ L s κ L b d L s L, Ψ L 4 = κ L d κ L s κ L b κ L u κ L c, (24) b L κ L t b L κ L t where = r, g, b, q L = P L q, κ L q = P L κ q, (q = u, c, t, d, s, b), P L = 1 γ5 2. By ntroducng the 6 6 sospn matrces T jα (j = 1, 2, 3, α = 1, 2, 3, 4), T 1α = 1 0 U α, T 2α = 1 0 U α, T 3α = 1 I 0, (25) 2 U α 0 2 U α I the I beng the 3 3 unt matrx, the U α s 3 3 untary matrces, whch satsfy the commutaton relatons [T α, T jα ] = ɛ jk T kα, (α : unsummed), (26) and the 6 6 dagonal hypercharge matrces Y α Y 1 = 1 3 I 0, Y 2 = 1 3 I 0, I 0 +I Y 3 = I 0, Y 4 = I 0, (27) I 0 +I

7 7 we may construct the Lagrangan, whch s nvarant under T jα and Y α gauge transformatons, beng of the form L = 3 j=1 =r,g,b + Ψ L j γ µd µ Ψ L j + Ψ L 4 γ µd µ Ψ L 4 q=u,c,t,d,s,b =r,g,b q R γ µd µ q R + q=u,c,t,d,s,b κ R q γ µ D µ κ R q + L, (28) where q R = P R q, κ R q = P R κ q, (q = u, c, t, d, s, b), P R = 1+γ5 2, the L nvolves the terms of free gauge felds and Hggs felds, and the covarant dervatves are defned as The terms D µ Ψ L 1 = ( µ + D µ Ψ L 2 = ( µ + D µ Ψ L 3 = ( µ + D µ Ψ L 4 = ( µ + 3 gt j1 W µ j1 + 1 g 2 Y 1B µ )Ψ L 1, (29) j=1 2 gt j2 W µ j2 + gt 32W µ g 2 Y 2B µ )Ψ L 2, (30) j=1 2 gt j3 W µ j3 + gt 33W µ g 2 Y 3B µ )Ψ L 3, (31) j=1 3 gt j4 W µ j4 + 1 g 2 Y 4B µ )Ψ L 4, (32) j=1 D µ q R = ( µ + g 2 3 Bµ )q R, (q = u, c, t), (33) D µ q R = ( µ g 1 3 Bµ )q R, (q = d, s, b), (34) D µ κ R q = µ κ R q. (35) =r,g,b Ψ L 1 γ µd µ Ψ L 1 + are well-known. Newly ntroduced terms are 3 j=2 =r,g,b q=u,c,t,d,s,b =r,g,b Ψ L j γ µd µ Ψ L j + Ψ L 4 γ µd µ Ψ L 4 + q=u,c,t,d,s,b q R γ µd µ q R (36) κ R q γ µ D µ κ R q + L. (37) Snce the charged currents constructed out of Ψ L 1 do not carry color charges, they are coupled not to colored and electrcally charged gauge bosons, but to colorless and electrcally charged gauge bosons lke W ±. Whereas, the charged currents constructed out of Ψ L 2 or ΨL 3 electrc charges, and are coupled to colored and electrcally charged gauge bosons. The frst term of (37) can be wrtten n the form 3 j=2 =r,g,b Ψ L j γ µd µ Ψ L j = 3 j=2 =r,g,b carry color and Ψ L j γ µ µ Ψ L j + L IC + L IN, (38)

8 8 where The L IC L IC = g L IN = 3 2 j=2 k=1 =r,g,b 3 j=2 =r,g,b Ψ L j γ µt kj W µ kj ΨL j, (39) Ψ L j γ µ(gt 3j W µ 3j + g 1 2 Y jb µ )Ψ L j. (40) descrbes the nteractons n whch each current nvolvng a cen-quark and a colored and electrcally charged quark s coupled to a colored and electrcally charged gauge boson: L IC = g 2 2 =r,g,b where W µ 2 = 1 2 (W µ 12 W µ 22 ), W µ 3 = 1 2 (W µ 13 + W µ 23 ), (W µ 2 J 2µ + W µ 2 J 2µ + W µ 3 J 3µ + W µ 3 J 3µ ), (41) J jµ = 2 Ψ L j γ µh j Ψ L j, H j = 0 U j 0 0. (42) We shall denote the quanta of the feld W µ 2 and those of the feld W µ 3 by W (or W ) and W 1 3 (or W 1 3 ) respectvely. The L IC where = r, g, b. The L IN descrbes the processes such as u κ u + W + 2 3, d κ d + W 1 3, (43) descrbes the nteractons n whch neutral currents of quarks are coupled to colorless and electrcally neutral gauge bosons: L IN = 3 j=2 =r,g,b Ψ L j γ µ(gt 3 W µ 3j + g 1 2 Y jb µ )Ψ L j, (44) where T 3 T 3j. By ntroducng Hermtan felds Z µ j and Aµ substtutng for W µ 3j and Bµ n (44) ther values, we obtan L IN = = 3 W µ 3j = cos θ jz µ j + sn θ ja µ, (45) B µ = sn θ j Z µ j + cos θ ja µ, (46) j=2 =r,g,b Ψ L j γ µ[gt 3 (cos θ j Z µ j + sn θ ja µ ) +g 1 2 Y j( sn θ j Z µ j + cos θ ja µ )]Ψ L j 3 j=2 =r,g,b Ψ L j γ µ[(g cos θ j T 3 g 1 sn θ j 2 Y j)z µ j +(g sn θ j T 3 + g cos θ j 1 2 Y j)a µ ]Ψ L j. (47)

9 9 Snce Q j e = T Y j and g sn θ j T 3 + g cos θ j 1 2 Y j = Q j, they agree f we take g sn θ j = g cos θ j = e. Thus t becomes L IN = = 3 j=2 =r,g,b 3 g [ (J (T 3) cos θ j j=2 Ψ L j γ g µ[ (T 3 sn 2 Q j θ j cos θ j e )Zµ j + Q ja µ ]Ψ L j µj sn 2 θ J (Qj) µj j e where J (T 3) µj = =r,g,b Ψ L j γ µt 3 Ψ L j and J (Q j) µj = =r,g,b Ψ L j γ µq j Ψ L j. From (41) and (48), we have L IC + L IN = g 2 2 =r,g,b 3 g [ (J (T 3) cos θ j j=2 )Z µ j + J (Q j) µj A µ ], (48) (W µ 2 J 2µ + W µ 2 J 2µ + W µ 3 J 3µ + W µ 3 J 3µ ) µj sn 2 θ J (Qj) µj j e )Z µ j + J (Q j) µj A µ ]. (49) V. APPLICATION It should be noted that cen-quarks can be produced n non-leptonc decays, and they can carry away mssng energy as do neutrnos. Thus we must take nto account the processes nvolvng the cen-quarks n non-leptonc weak nteractons where mssng energes occur. For example, for the descrpton of the rare kaon decay K + π + + mssng energy, we must take nto account the followng quark level processes: () s d ν l ν l (l = e, µ, τ), (50) () s d κ q κ q (q = u, c, t, d, s, b), (51) () s d κ d κ s, (52) where = r, g, b, and κ d and κ s are mxed states of κ d, κ s and κ b. We may see that the process () s smlar to the muon decay µ ν µ e ν e n many respects. The nteractons of the well-known processes () are medated by the gauge bosons W ± and Z 0. Whereas the nteractons of () and () are medated by the gauge bosons W 1 3, W 1 3 and Z3 0. We may nfer from extremely short range of the nteractons of () and () that the mass of the gauge bosons W 1 3, W 1 3 and Z3 0 must be very massve. The decay rate of K + π + + mssng energy,.e., Γ(K + π + + Nothng), can be wrtten from (50), (51) and (52) as Γ(K + π + + Nothng) = Γ () + Γ () + Γ (), (53)

10 10 where Γ () = Γ(K + π + ν ν) ν=νe,ν µ,ν τ, (54) Γ () = Γ(K + π + κ κ ) κ =κ u,κ c,κ t,κ d,κ s,κ b = Γ( s d κ κ ) κ =κ u,κ c,κ t,κ d,κ s,κ b, (55) Γ () = Γ(K + π + κ d κ s) = Γ( s d κ d κ s). (56) The terms among the Lagrangan terms n (49) responsble for the processes () and () are where L I3 = g 2 2 =r,g,b g [ (J (T 3) cos θ 3 (W µ 3 J 3µ + W µ 3 J 3µ ) µ3 sn 2 θ J (Q3) µ3 3 e )Z µ 3 + J (Q 3) µ3 A µ ], (57) J 3µ = 2Ψ L κ γ µ U 3 Ψ L q = 2Ψ L κ γ µ Ψ L q, (58) where Ψ L κ = U 3 ΨL κ, and Ψ L κ = κ L d κ L s, ΨL q = d L s L. (59) κ L b b L In the lmt m W 1 3, the W 1 3 propagator reduces to g µν m 2. (60) W 1 3 Smlarly, n the lmt m Z 0 3, the Z 0 3 propagator reduces to g µν m 2. (61) Z3 0 In the lowest order, the process () has one nternal W 1 3 boson lne, whereas each of the processes () has two nternal W 1 3 boson lnes for the box dagram, or at least one nternal W 1 3 boson lne plus one nternal Z3 0 boson lne for the Z0 3-pengun dagrams. Thus from g2 W m 2 W and 8g 2 W cos 2 θ 3 m 2 Z 0 3 1, we have Γ () Γ (), (62) and Γ(K + π + + Nothng) Γ () + Γ (). (63)

11 11 From (57), we may construct the nvarant ampltude of the lowest order for the process () whch takes n the form M 4g2 W m 2 s L γµ κ L s κ L d γ µd L, (64) W 1 3 where g W = g 2 2 and κ L s = U 3(s L,κ) κ, κl d = U κ. (65) 3(d L,κ) κ=κ L d,κl s,κl b κ=κ L d,κl s,κl b Assumng that U 3 I, m κd 0, m κs 0, m d m s, and that the process () s unaffected by strong and electromagnetc nteractons except some neglgble hgher order correctons, we have from (64) Γ(K + π + κ d κ s) g4 W m5 s 96π 3 m 4 W 1 3, (66) where g 2 W 1 2 Gm 2 W ± well-known n electro-weak theory. The (63) suggests that the branchng rato B(K + π + κ d κ s) can be determned by the dscrepancy between the measured K + π + + mssng energy branchng rato and the predcted K + π + ν ν branchng rato. Thus f the dscrepancy s determned, from B(K + π + κ κ d s) = τ K +Γ(K + π + κ κ d s) τ K +G2 m 4 W m 5 ± s 192π 3 m 4, (67) W 1 3 we may calculate the mass of W 1 3 m W 1 3 ( τ K +G 2 m 4 W ± m 5 s 192π 3 B(K + π + κ d κ s) ) 1 4, (68) where G GeV 2, m W ± GeV, m s 100MeV, τ K s. However, at the present stage, f we compare the measured K + π + + mssng energy branchng rato [3] wth the predcted K + π + ν ν branchng rato [4], B(K + π + + Nothng) Exp = ( ) 10 10, (69) B(K + π + ν ν) = ( ± 0.029) 10 10, (70) the range of the expermental uncertanty s so wde that we cannot know the exact value of B(K + π + κ d κ s),.e., the dscrepancy between the (69) and (70). In ths stuaton, f we postulate that B(K + between the representatve values of (69) and (70),.e., π + κ d κ s) s approxmately the dscrepancy B(K + π + κ d κ s) , (71)

12 12 the value for the m W 1 3 becomes from (68) m W TeV. (72) To confrm our speculaton, we should look for colored and electrcally charged bosons wth spn-1 consstent wth the propertes descrbed n ths paper. [1] H. Georg, and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974). [2] A. J. Buras, J. Ells, M. K. Gallard, and D. V. Nanopoulos, Nucl. Phys. B 135, 66 (1978). [3] A. V. Artamonov, et al. (E949 Collab.) Phys. Rev. D 79, (2009). [4] J. Brod, M. Gorbahn, and E. Stamou, Phys. Rev. D 83, (2011).