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Fermion mass hierarchy and the strong CP problem
Volume 219, number 2.3
PHYSICS LETTERS B
16 March 1989
FERMION MASS HIERARCHY AND THE STRONG CP PROBLEM K.S.BABU 1 Department of Physics and Astronomy, Universityof Rochester, Rochester, NY 14627, USA and X i a o - G a n g HE Research Centerfor High Energy Physics, School of Physics, Universityof Melbourne, Parkville, Victoria 3052, Australia Received 16 November 1988; revised manuscript received 3 January 1989
We present a model which resolves the strong CP problem by the Peccei-Quinn (PQ) mechanism and at the same time provides a natural explanation to the fermion mass hierarchy. Only the third generation quarks acquire masses at the tree level due to the PQ symmetry present in the model; the second and first generation quark masses arise as one-loop and two-loop radiative corrections, respectively. The coupling of the axion in the model to the second and third generation quarks is three to four orders of magnitude larger than in conventional invisible axion models.
The smallness of the CP violating 0 p a r a m e t e r o f the Q C D lagrangian ( 0 < 10-9) is a well-known problem of the standard SU ( 3 ) c X SU ( 2 ) LX U ( 1 ) y m o d e l (the strong CP problem ) ~~. The P e c c e i - Q u i n n mechanism [ 2 ] of resolving the puzzle results in a neutral pseudoscalar boson - the axion - whose coupling to quarks and leptons should be rather weak in order to be consistent with laboratory experiments [ 3 ]. This will be the case if the PQ chiral s y m m e t r y is broken at an energy scale much larger than the electroweak s y m m e t r y breaking scale. There are two consistent ways of i m p l e m e n t i n g this idea. One way is to introduce extra isodoublet and isosinglet Higgs scalars to the theory as discussed by Zhitnitskii and Dine, Fishler and Srednicki ( Z D I S ) [4]. An alternative way is proposed by Kim, and Shifman, Vainshtein and Z a k h a r o v ( K S V Z ) [5] in which vector-like isosinglet quarks and an isosinglet Higgs scalar are introduced. In both cases, the PQ s y m m e t r y is broken by the v a c u u m expectation value ( V E V ) o f the isosinglet scalar at a high energy scale. The c o m b i n e d cosmological and astrophysical constraints require the PQ s y m m e t r y breaking scale to be between l0 s and 10 ~2 GeV [ 1 ]. A second puzzling feature of the s t a n d a r d model which begs for an explanation is the observed hierarchical pattern in the fermion masses. The down, strange and b o t t o m quark masses, for example, obey m,l: m , : m / , ~ 1:20:600. A similar hierarchy exists in the up quark and the charged lepton sectors as well [6]. Such a geometric hierarchy in the masses may be suggestive that only the third generation quarks have tree-level masses, the second generation quarks acquire masses at the one-loop level, whereas the first generation quark masses are generated by two-loop radiative corrections. Recently, a model i m p l e m e n t i n g this idea has been proposed based on the left-right symmetric gauge group SU (3) c X SU (2) LX SU (2) R X U ( 1 ) ~_ L [ 7 ]. The model requires the existence of vector-like isosinglet quarks (from now on we refer to them as exotic quarks) for its success. Models which utilize such exotic quarks in order to explain the smallness of the fermion masses as c o m p a r e d to the W-boson mass in terms of a generalized see-saw m e c h a n i s m are also known [ 8 ]. We first note that both the KSVZ solution to the strong CP problem and the attempts to explain the fermion Present address: Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA. ~ See ref. [ 1] for recent reviews. 342
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
V o l u m e 219, n u m b e r 2.3
PHYSICS LETTERS B
16 M a r c h 1989
mass hierarchy require the existence of singlet quarks. It is tempting then to speculate that the observed fermion mass hierarchy is perhaps related to the PQ symmetry. In this letter we present a model which realizes such a connection. The model contains one generation of exotic isosinglet quarks (P of electric charge 2e3, N of electric charge ~e) and one isosinglet complex Higgs scalar a. Their gauge group transformation properties and the PQ charges along with those of the ordinary quarks g-~,L= (Ui. d,)L, Urn, d,R (where i is the generation index, L and R indicate left- and right-handedness, respectively) and the isodoublet Higgs 0 are as follows: -
g",e:(3,2,~)_l,
um:(3,1,4)~,
4
PR" (3, 1, 3 ) - ' ,
dm:(3,1,_])t
NR: (3, 1,---~) 1, PL: (3, 1, 4),,
2 NL: (3, 1, -- ~)~,
0: (1,2,--1 )0,
O: (1, 1,0)2.
(1)
The numbers in the parentheses indicate the transformation under SU (3) c × SU ( 2 ) , X U ( 1 ) y and the subscript is the PQ charge. The particle spectrum is similar to the KSVZ model except that in our case the ordinary quarks also transform under PQ chiral symmetry ~2. With this quantum number assignment, as we will show below, the first two generation quark masses will be zero. Additional ingredients have to be added in order to give radiative masses to the first two generation quarks. For this purpose, we introduce two color triplet, isosinglet Higgs scalars t/t and t/2. Their transformation properties are t/,: (3,
1,--~)2,
?72: (3, 1, - ~2) - 2 -
(2)
The Yukawa coupling invariant under the gauge symmetry as well as the PQ symmetry is Lv = -
~
i-- 1,2,3
[hi(~Ti, d~)cONR +h~(ffi, a~)Lir20*PR] - (hs*6LPR +h's?TeNR)a
- Z ( Hq ~ii_ C17~2t/I ~L +HI~ u ~, Ct/2 dRj ) -- (fpT Ct/2 NL + f 'P~ Ct/~ N R ) "k h.c.
(3)
0
Here C is the charge conjugation operator, H,j is a symmetric matrix. In addition, bare mass terms connecting the exotic and ordinary quarks are also allowed:
L=-
Y,
i-- 1,2,3
(miPcum+m;NLdRi).
(4)
The most general Higgs potential is v= -~
Ial2-~t~ 1012+21 [a14+221014+).121<21012
+ Y" (-,u',21t/i12-1-Ti]t/i14-1-71,la121t/i12-1-72ilO121~.hl 2) i= 1,2
+0~,2 It/, 12l t/212+0{'121 t/~t/212+
['~-(t/~t/2)o'2"Fh.c. ].
(5)
When the scalar fields ~ and a develop VEVs denoted by ( # ) = Vdand ( a ) = v~, the quarks acquire masses given by the matrix (for the up quark)
- -
(uctP)L
(0o00 0
1
0
0
h'2Vdl
0
0
h'3Vdl
m2
m3
h~vs / \ P / R
'
(6)
~-" It is p e r h a p s w o r t h n o t i n g t h a t the P Q s y m m e t r y acts u n i f o r m l y o n all g e n e r a t i o n s o f o r d i n a r y q u a r k s - t h u s it is not a h o r i z o n t a l symmetry.
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PHYSICS LETTERS B
16 March 1989
and similarly for the down quarks. It is clear from eq. (6) that at this stage, two o f the quarks have zero masses; the two non-zero masses are m,,,~ hsvs, and m,'~ mhvd/h,.vs with m = x / m 2 + m s~+ m 32 and h=x/hl ,2 + h ~ + h S 2. Here we assumed hsvs >> ms >> h', yd. Clearly, m, cannot be too small c o m p a r e d to hsvs or else the top mass will be too small. In order to generate non-zero masses for the first two generation quarks, we have to consider loop corrections. The one-loop diagrams which contribute finite corrections to m v (where i a n d j run from 1 to 3) are shown in fig. 1. There are also one-loop corrections to the rni~, and mix entries. However, these corrections are proportional to the tree-level entries. They do not change the form o f the mass matrix. Evaluating the diagram in fig. 1, we obtain the one-loop corrected up-quark mass matrix
\ S m i lP/)
x~(1)!'
(7)
vHt'l'l" /
where 8m(, ) _2aibj2VdV~mx i dt 0 = ~ 167r (t+m~,O(t+m2)(t+m~2) .
(8)
o
Here m x = h;v~ and m,,, are the tree-level masses o f N and qi, respectively, a, = ~lHitht, and b~= E,H},.m ~.. Notice that 6m ~ ) is factorizable which implies that the d e t e r m i n a n t o f the o n e q o o p corrected mass matrix is zero. The second generation has acquired mass at this stage while the first generation quark mass is still zero. Now we have to show that if we go to higher loop corrections the first generation acquires a non-zero mass. This is not a difficult job. One o f the two-loop diagrams, which guarantees non-zero mass for the first generation upquark is shown in fig. 2. F o r this diagram, we have
8m~ "-)~C Z
H, kHak' hk,' m,,H
(9)
k,k',r.r'
We have not displayed the c o m p l i c a t e d two-loop integral I. We simply note that 6m o is no longer factorizable. W i t h p r o p e r choice of the Yukawa couplings the correct quark mixings can be reproduced [ 7 ]. It is interesting to note that given the pattern m , > mr,, the model predicts m,.> m, and r n , < mj. We now turn to discuss the a x i o n - q u a r k interaction. The axion field in our model is exactly the same as in the KSVZ model, that is A ( a x i o n ) = x ~ Ima. At the classical level A is massless but acquires a finite mass through the color anomaly: m~ = rn,~f~x~/( 1 + z)f~, with f~ = , , / 2 Vs [ 1 ], and z = m,,/md= 0.56 [6 ]. The interaction o f the axion in our model with the ordinary quarks is different from that o f the KSVZ model since in our model the exotic quarks mix with the ordinary ones. There are several sources which generate the coupling o f the axion to the ordinary quarks. The color anomaly generates the familiar coupling o f the axion A to the light quarks [11:
L ( , ) = _ 1 m,,m<~i(UTsu+d75d) A" f, m, + md
(10) av~
2
~.Vs
T~1 t
1 1 ~ ~-- K ' - - - ~ . l.V-2 r~1,,4/
\ ~2
i! /
,
h, Vd mXk =- ~HkjdLi N dRk URj
UL i
x /
Hik~ 2kn/ dL k
" x "2 2
\ \
h'nVd UL n
p
m~
\
kHm, ~HIj URm dR l dR j
Fig. 1. The one-loop diagram that contributes to the up-quark
Fig. 2. The two-loop diagram which generates the first generation
mass matrix,
quark masses.
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PHYSICS LETTERS B
The coupling of the axion to two photons induces the
L~2)-3°~mQ~mJ( [~12 -2nf~ 6 ~C' eQ~i In -fo -
mj
followingfysfAcoupling
-- -2 4 + z In m~) iJ~ysfjA. 3 l+z
16 March 1989
[9]: ( 11 )
where CbQ is the PQ charge of the ith f e r m i o n f of electric charge Qi and mass mi and the summation is over all fermions. In our model there is an additional coupling of A to the ordinary quarks through their mixing with the exotic quarks. At the tree level, the axion only couples to the exotic quarks which is given by (ihsP75P+ih'sNysN)A/ , , ~ in the weak interaction basis. After diagonalizing the tree-level mass matrix, we find that the coupling of the axion to the super heavy and the third generation mass eigenstates is (for the up quark)
~PLtR--
L'3'~-i~
{LPr~+PLPRA +h.c.
(12)
In obtaining eq. ( 12 ), we have assumed, for simplicity, that hsvs>> m, hVd.Clearly this coupling of A to the third generation is about four orders of magnitude larger than that of eq. ( 11 ). To determine the additional coupling of the axion to the first two generation quarks, we have to consider the loop corrections. At the one-loop level, such a coupling will be generated through a diagram similar to fig. 1 with the tht/2 vertex replaced by ~h*t/2A. The couplings can be obtained by simply substituting the thq2 mixing coefficient 2v~ by 22vd~2 in eq. (8). There are similar diagrams which generate axion-quark-exotic-quark couplings. We can write the one-loop corrected axion-quark coupling as . 1+75 _ _ L~= - i ._1_1 ,~ ; ChM,j---~ qjA*h.c.,
(13)
where L J = 1, 2, 3, P and
1(26m],"
Mn~ --
6LJ~,)
•
(14)
Here 8A/vsdenotes the one-loop correction to the axion coupling to the exotic quark. Letting VL and VR be the unitary matrices which diagonalize the one-loop mass matrix and writing L~ in terms of the mass eigenstate basis, we find the axion coupling to the first generation up quark to be Lfi,-~ ~ i x/~m' VLI 3 VR3 l / ~ 5 / ~ . Us
( 15 )
The coupling to the charm quark is
(m~
L . . . . . d ~ --ix/2 ~
mtVL23VR32)C,sC" V~ /
(16)
The mixing matrix elements VL,jand VROare expected to be of the same order of magnitude as the corresponding mixing matrix of the charged current UKM, then, VLI3VR31 ~ 0 ( U,b) 2 ~ 10 -5 and g L 2 3 V R 3 2 ~ O ( U 2 c b ) ~ l O - 3 . U s ing these numbers, we find that the dominant contribution to the coupling of A to the u quark is from eq. (10) and to the c quark is from the second term in eq. (16). Similar results hold for the down quarks as well. On the other hand, if VLI3VR31 is of order (sin 0~) 2, the first generation coupling to the axion [eq. (15) ] is a factor of 100 larger than eq. (10). Astrophysical considerations will then require that the PQ symmetry breaking scale be greater than 10 ~° GeV. There is an additional contribution to the coupling of A to the first generation coming from two-loop graphs. If we replace the ~ht/2 mixing vertex in fig. 2 by A/v~,we have (rn~/v~)a75uA,of the same order as eq. (10). The mechanism for explaining the quark mass hierarchy can be easily generalized to the leptonic sector. How345
Volume 219, number 2.3
PHYSICS LETTERS B
16 March 1989
ever, lacking any i n f o r m a t i o n on the n e u t r i n o mass s p e c t r u m , we do not pursue it here. I f we assign t e n t a t i v e l y zero P Q charge to the leptons, the charged l e p t o n s will a c q u i r e masses t h r o u g h t h e i r Y u k a w a couplings to the i s o d o u b l e t 0 as in the c o n v e n t i o n a l S U ( 3 ) c × S U ( 2 ) c × U ( 1 ) y m o d e l a n d n e u t r i n o s will be exactly massless. I f n e u t r i n o s h a v e finite but small masses, m o r e particles h a v e to be i n t r o d u c e d . M o d e l s w h i c h c o n n e c t small neut r i n o masses with the strong CP p r o b l e m h a v e b e e n s t u d i e d in ref. [ 10 ]. X . G . H . w o u l d like to t h a n k the T h e o r y group at F e r m i l a b for h o s p i t a l i t y while this w o r k was c o m p l e t e d . T h e w o r k o f K.S.B. is s u p p o r t e d by the U S D e p a r t m e n t o f Energy u n d e r c o n t r a c t No. D E - A C 0 2 - 7 6 E R 1 3 0 6 5 . X . G . H . is s u p p o r t e d by the A u s t r a l i a R e s e a r c h G r a n t C o m m i t t e e .
References [ 1] J.E. Kim, Phys. Rep. 150 (1987) 1; H.Y. Cheng, Phys. Rep 158 (1988) 1. [ 2 ] R. Peccei and H. Quinn, Phys. Rev. Lett. 38 ( 1977 ) 1440; Phys. Rev. D 16 ( 1977 ) 1791. [3] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [4] A. Zhilnitskii, Sov. J. Nucl. Phys. 31 (1980) 260; M. Dine, W. Fishler and M. Srednicki, Phys. Lett. B 104 ( 1981 ) 199. [5] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103; M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B 166 (1980) 493. [6 ] H. Gasser and H. Leutwyler, Phys. Rep. 85 (1982) 77. [7] B.S. Balakrishna, Phys. Rev. Lett. 60 (1988) 1602; B.S. Balakrishna, A.L. Kagan and R.N. Mohapatra, Phys. Lett. B 205 ( 1988 ) 345. [8] S. Rajpoot, Phys. Rev. D 36 (1987) 1479; A. Davidson and K. Wali, Phys. Rev. Lett. 60 (1988) 1813. [9] M. Srednicki, Nucl. Phys. B 260 (1985) 689. [ 10] P. Langacker, R. Peccei and T. Yanagida, Mod. Phys. Lett. A 1 (1986) 541; M. Shin, Phys. Rev. Len. 59 (1988) 2515.
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FERMION MASS HIERARCHY AND THE STRONG CP PROBLEM K.S.BABU 1 Department of Physics and Astronomy, Universityof Rochester, Rochester, NY 14627, USA and X i a o - G a n g HE Research Centerfor High Energy Physics, School of Physics, Universityof Melbourne, Parkville, Victoria 3052, Australia Received 16 November 1988; revised manuscript received 3 January 1989
We present a model which resolves the strong CP problem by the Peccei-Quinn (PQ) mechanism and at the same time provides a natural explanation to the fermion mass hierarchy. Only the third generation quarks acquire masses at the tree level due to the PQ symmetry present in the model; the second and first generation quark masses arise as one-loop and two-loop radiative corrections, respectively. The coupling of the axion in the model to the second and third generation quarks is three to four orders of magnitude larger than in conventional invisible axion models.
The smallness of the CP violating 0 p a r a m e t e r o f the Q C D lagrangian ( 0 < 10-9) is a well-known problem of the standard SU ( 3 ) c X SU ( 2 ) LX U ( 1 ) y m o d e l (the strong CP problem ) ~~. The P e c c e i - Q u i n n mechanism [ 2 ] of resolving the puzzle results in a neutral pseudoscalar boson - the axion - whose coupling to quarks and leptons should be rather weak in order to be consistent with laboratory experiments [ 3 ]. This will be the case if the PQ chiral s y m m e t r y is broken at an energy scale much larger than the electroweak s y m m e t r y breaking scale. There are two consistent ways of i m p l e m e n t i n g this idea. One way is to introduce extra isodoublet and isosinglet Higgs scalars to the theory as discussed by Zhitnitskii and Dine, Fishler and Srednicki ( Z D I S ) [4]. An alternative way is proposed by Kim, and Shifman, Vainshtein and Z a k h a r o v ( K S V Z ) [5] in which vector-like isosinglet quarks and an isosinglet Higgs scalar are introduced. In both cases, the PQ s y m m e t r y is broken by the v a c u u m expectation value ( V E V ) o f the isosinglet scalar at a high energy scale. The c o m b i n e d cosmological and astrophysical constraints require the PQ s y m m e t r y breaking scale to be between l0 s and 10 ~2 GeV [ 1 ]. A second puzzling feature of the s t a n d a r d model which begs for an explanation is the observed hierarchical pattern in the fermion masses. The down, strange and b o t t o m quark masses, for example, obey m,l: m , : m / , ~ 1:20:600. A similar hierarchy exists in the up quark and the charged lepton sectors as well [6]. Such a geometric hierarchy in the masses may be suggestive that only the third generation quarks have tree-level masses, the second generation quarks acquire masses at the one-loop level, whereas the first generation quark masses are generated by two-loop radiative corrections. Recently, a model i m p l e m e n t i n g this idea has been proposed based on the left-right symmetric gauge group SU (3) c X SU (2) LX SU (2) R X U ( 1 ) ~_ L [ 7 ]. The model requires the existence of vector-like isosinglet quarks (from now on we refer to them as exotic quarks) for its success. Models which utilize such exotic quarks in order to explain the smallness of the fermion masses as c o m p a r e d to the W-boson mass in terms of a generalized see-saw m e c h a n i s m are also known [ 8 ]. We first note that both the KSVZ solution to the strong CP problem and the attempts to explain the fermion Present address: Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA. ~ See ref. [ 1] for recent reviews. 342
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
V o l u m e 219, n u m b e r 2.3
PHYSICS LETTERS B
16 M a r c h 1989
mass hierarchy require the existence of singlet quarks. It is tempting then to speculate that the observed fermion mass hierarchy is perhaps related to the PQ symmetry. In this letter we present a model which realizes such a connection. The model contains one generation of exotic isosinglet quarks (P of electric charge 2e3, N of electric charge ~e) and one isosinglet complex Higgs scalar a. Their gauge group transformation properties and the PQ charges along with those of the ordinary quarks g-~,L= (Ui. d,)L, Urn, d,R (where i is the generation index, L and R indicate left- and right-handedness, respectively) and the isodoublet Higgs 0 are as follows: -
g",e:(3,2,~)_l,
um:(3,1,4)~,
4
PR" (3, 1, 3 ) - ' ,
dm:(3,1,_])t
NR: (3, 1,---~) 1, PL: (3, 1, 4),,
2 NL: (3, 1, -- ~)~,
0: (1,2,--1 )0,
O: (1, 1,0)2.
(1)
The numbers in the parentheses indicate the transformation under SU (3) c × SU ( 2 ) , X U ( 1 ) y and the subscript is the PQ charge. The particle spectrum is similar to the KSVZ model except that in our case the ordinary quarks also transform under PQ chiral symmetry ~2. With this quantum number assignment, as we will show below, the first two generation quark masses will be zero. Additional ingredients have to be added in order to give radiative masses to the first two generation quarks. For this purpose, we introduce two color triplet, isosinglet Higgs scalars t/t and t/2. Their transformation properties are t/,: (3,
1,--~)2,
?72: (3, 1, - ~2) - 2 -
(2)
The Yukawa coupling invariant under the gauge symmetry as well as the PQ symmetry is Lv = -
~
i-- 1,2,3
[hi(~Ti, d~)cONR +h~(ffi, a~)Lir20*PR] - (hs*6LPR +h's?TeNR)a
- Z ( Hq ~ii_ C17~2t/I ~L +HI~ u ~, Ct/2 dRj ) -- (fpT Ct/2 NL + f 'P~ Ct/~ N R ) "k h.c.
(3)
0
Here C is the charge conjugation operator, H,j is a symmetric matrix. In addition, bare mass terms connecting the exotic and ordinary quarks are also allowed:
L=-
Y,
i-- 1,2,3
(miPcum+m;NLdRi).
(4)
The most general Higgs potential is v= -~
Ial2-~t~ 1012+21 [a14+221014+).121<21012
+ Y" (-,u',21t/i12-1-Ti]t/i14-1-71,la121t/i12-1-72ilO121~.hl 2) i= 1,2
+0~,2 It/, 12l t/212+0{'121 t/~t/212+
['~-(t/~t/2)o'2"Fh.c. ].
(5)
When the scalar fields ~ and a develop VEVs denoted by ( # ) = Vdand ( a ) = v~, the quarks acquire masses given by the matrix (for the up quark)
- -
(uctP)L
(0o00 0
1
0
0
h'2Vdl
0
0
h'3Vdl
m2
m3
h~vs / \ P / R
'
(6)
~-" It is p e r h a p s w o r t h n o t i n g t h a t the P Q s y m m e t r y acts u n i f o r m l y o n all g e n e r a t i o n s o f o r d i n a r y q u a r k s - t h u s it is not a h o r i z o n t a l symmetry.
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and similarly for the down quarks. It is clear from eq. (6) that at this stage, two o f the quarks have zero masses; the two non-zero masses are m,,,~ hsvs, and m,'~ mhvd/h,.vs with m = x / m 2 + m s~+ m 32 and h=x/hl ,2 + h ~ + h S 2. Here we assumed hsvs >> ms >> h', yd. Clearly, m, cannot be too small c o m p a r e d to hsvs or else the top mass will be too small. In order to generate non-zero masses for the first two generation quarks, we have to consider loop corrections. The one-loop diagrams which contribute finite corrections to m v (where i a n d j run from 1 to 3) are shown in fig. 1. There are also one-loop corrections to the rni~, and mix entries. However, these corrections are proportional to the tree-level entries. They do not change the form o f the mass matrix. Evaluating the diagram in fig. 1, we obtain the one-loop corrected up-quark mass matrix
\ S m i lP/)
x~(1)!'
(7)
vHt'l'l" /
where 8m(, ) _2aibj2VdV~mx i dt 0 = ~ 167r (t+m~,O(t+m2)(t+m~2) .
(8)
o
Here m x = h;v~ and m,,, are the tree-level masses o f N and qi, respectively, a, = ~lHitht, and b~= E,H},.m ~.. Notice that 6m ~ ) is factorizable which implies that the d e t e r m i n a n t o f the o n e q o o p corrected mass matrix is zero. The second generation has acquired mass at this stage while the first generation quark mass is still zero. Now we have to show that if we go to higher loop corrections the first generation acquires a non-zero mass. This is not a difficult job. One o f the two-loop diagrams, which guarantees non-zero mass for the first generation upquark is shown in fig. 2. F o r this diagram, we have
8m~ "-)~C Z
H, kHak' hk,' m,,H
(9)
k,k',r.r'
We have not displayed the c o m p l i c a t e d two-loop integral I. We simply note that 6m o is no longer factorizable. W i t h p r o p e r choice of the Yukawa couplings the correct quark mixings can be reproduced [ 7 ]. It is interesting to note that given the pattern m , > mr,, the model predicts m,.> m, and r n , < mj. We now turn to discuss the a x i o n - q u a r k interaction. The axion field in our model is exactly the same as in the KSVZ model, that is A ( a x i o n ) = x ~ Ima. At the classical level A is massless but acquires a finite mass through the color anomaly: m~ = rn,~f~x~/( 1 + z)f~, with f~ = , , / 2 Vs [ 1 ], and z = m,,/md= 0.56 [6 ]. The interaction o f the axion in our model with the ordinary quarks is different from that o f the KSVZ model since in our model the exotic quarks mix with the ordinary ones. There are several sources which generate the coupling o f the axion to the ordinary quarks. The color anomaly generates the familiar coupling o f the axion A to the light quarks [11:
L ( , ) = _ 1 m,,m<~i(UTsu+d75d) A" f, m, + md
(10) av~
2
~.Vs
T~1 t
1 1 ~ ~-- K ' - - - ~ . l.V-2 r~1,,4/
\ ~2
i! /
,
h, Vd mXk =- ~HkjdLi N dRk URj
UL i
x /
Hik~ 2kn/ dL k
" x "2 2
\ \
h'nVd UL n
p
m~
\
kHm, ~HIj URm dR l dR j
Fig. 1. The one-loop diagram that contributes to the up-quark
Fig. 2. The two-loop diagram which generates the first generation
mass matrix,
quark masses.
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Volume 219, number 2.3
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The coupling of the axion to two photons induces the
L~2)-3°~mQ~mJ( [~12 -2nf~ 6 ~C' eQ~i In -fo -
mj
followingfysfAcoupling
-- -2 4 + z In m~) iJ~ysfjA. 3 l+z
16 March 1989
[9]: ( 11 )
where CbQ is the PQ charge of the ith f e r m i o n f of electric charge Qi and mass mi and the summation is over all fermions. In our model there is an additional coupling of A to the ordinary quarks through their mixing with the exotic quarks. At the tree level, the axion only couples to the exotic quarks which is given by (ihsP75P+ih'sNysN)A/ , , ~ in the weak interaction basis. After diagonalizing the tree-level mass matrix, we find that the coupling of the axion to the super heavy and the third generation mass eigenstates is (for the up quark)
~PLtR--
L'3'~-i~
{LPr~+PLPRA +h.c.
(12)
In obtaining eq. ( 12 ), we have assumed, for simplicity, that hsvs>> m, hVd.Clearly this coupling of A to the third generation is about four orders of magnitude larger than that of eq. ( 11 ). To determine the additional coupling of the axion to the first two generation quarks, we have to consider the loop corrections. At the one-loop level, such a coupling will be generated through a diagram similar to fig. 1 with the tht/2 vertex replaced by ~h*t/2A. The couplings can be obtained by simply substituting the thq2 mixing coefficient 2v~ by 22vd~2 in eq. (8). There are similar diagrams which generate axion-quark-exotic-quark couplings. We can write the one-loop corrected axion-quark coupling as . 1+75 _ _ L~= - i ._1_1 ,~ ; ChM,j---~ qjA*h.c.,
(13)
where L J = 1, 2, 3, P and
1(26m],"
Mn~ --
6LJ~,)
•
(14)
Here 8A/vsdenotes the one-loop correction to the axion coupling to the exotic quark. Letting VL and VR be the unitary matrices which diagonalize the one-loop mass matrix and writing L~ in terms of the mass eigenstate basis, we find the axion coupling to the first generation up quark to be Lfi,-~ ~ i x/~m' VLI 3 VR3 l / ~ 5 / ~ . Us
( 15 )
The coupling to the charm quark is
(m~
L . . . . . d ~ --ix/2 ~
mtVL23VR32)C,sC" V~ /
(16)
The mixing matrix elements VL,jand VROare expected to be of the same order of magnitude as the corresponding mixing matrix of the charged current UKM, then, VLI3VR31 ~ 0 ( U,b) 2 ~ 10 -5 and g L 2 3 V R 3 2 ~ O ( U 2 c b ) ~ l O - 3 . U s ing these numbers, we find that the dominant contribution to the coupling of A to the u quark is from eq. (10) and to the c quark is from the second term in eq. (16). Similar results hold for the down quarks as well. On the other hand, if VLI3VR31 is of order (sin 0~) 2, the first generation coupling to the axion [eq. (15) ] is a factor of 100 larger than eq. (10). Astrophysical considerations will then require that the PQ symmetry breaking scale be greater than 10 ~° GeV. There is an additional contribution to the coupling of A to the first generation coming from two-loop graphs. If we replace the ~ht/2 mixing vertex in fig. 2 by A/v~,we have (rn~/v~)a75uA,of the same order as eq. (10). The mechanism for explaining the quark mass hierarchy can be easily generalized to the leptonic sector. How345
Volume 219, number 2.3
PHYSICS LETTERS B
16 March 1989
ever, lacking any i n f o r m a t i o n on the n e u t r i n o mass s p e c t r u m , we do not pursue it here. I f we assign t e n t a t i v e l y zero P Q charge to the leptons, the charged l e p t o n s will a c q u i r e masses t h r o u g h t h e i r Y u k a w a couplings to the i s o d o u b l e t 0 as in the c o n v e n t i o n a l S U ( 3 ) c × S U ( 2 ) c × U ( 1 ) y m o d e l a n d n e u t r i n o s will be exactly massless. I f n e u t r i n o s h a v e finite but small masses, m o r e particles h a v e to be i n t r o d u c e d . M o d e l s w h i c h c o n n e c t small neut r i n o masses with the strong CP p r o b l e m h a v e b e e n s t u d i e d in ref. [ 10 ]. X . G . H . w o u l d like to t h a n k the T h e o r y group at F e r m i l a b for h o s p i t a l i t y while this w o r k was c o m p l e t e d . T h e w o r k o f K.S.B. is s u p p o r t e d by the U S D e p a r t m e n t o f Energy u n d e r c o n t r a c t No. D E - A C 0 2 - 7 6 E R 1 3 0 6 5 . X . G . H . is s u p p o r t e d by the A u s t r a l i a R e s e a r c h G r a n t C o m m i t t e e .
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