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Symmetry breaking by Wilson-loop elements in string theories and rescaling of gauge coupling constants
Volume 176, number 1,2
PHYSICS LETTERS B
21 August 1986
S Y M M E T R Y BREAKING BY W I L S O N - L O O P E L E M E N T S IN S T R I N G T H E O R I E S AND RESCALING OF GAUGE COUPLING CONSTANTS
Kiwoon C H O I and Jihn E. K I M Department of Physics. Seoul National University, Seoul 151, Korea
Received 15 May 1986 It is found that there exists nontrivial rescaling of gauge coupling constants at the compactification scale by the Wilson-loop elements in the heterotic string theory. This phenomenon appears at the string loop level, and the result modifies the previous phenomenological analysis of sin20w.
Recent discoveries of consistent superstring theories with gauge groups SO(32) or E~ x E s render a new insight in constructing a theory of q u a n t u m gravity [1]. Furthermore, these theories, in particular the E s x E s theory, imply various interesting phenomenological applications [2]. For physical applications, we consider the spontaneous compactification of M~0 with the vacuum configuration defined o n M 4 x K where M 4 iS the four-dimensional Minkowski space and K is some c o m p a c t six-manifold [2,3]. In many compactification schemes, the gauge symmetry breaking by means of the Wilson-loop elements is crucial in obtaining realistic low energy gauge groups [2,4]. The Wilson-loop vacuum configuration is the Yang Mills gauge field configuration which gives vanishing field strength tensor but cannot be gauged away. It is allowed only when ¢rl(K ) is nontrivial. The gauge invariant expression is given by a h o m o m o r p h i s m exp(i2z-cb) : z'l(K ) --* H 0, where exp(i2~-tb) = P exp(i ~ A,,, d y " ' ) and H o is a discrete subgroup of the ten-dimensional gauge group G. Then the unbroken D = 4 gauge group G should c o m m u t e with H 0. In the compactification schemes of the E s × E~ heterotic string theory on the C a l a b i - Y a u manifold, the Wilson-loop vacuum configuration has been used to break E 6 without breaking a D = 4 supersymmetry [2]. In this case, q) behaves like a
D = 4 Higgs field transforming as 78 under E s. But q) has a novel property due to its topological origin. If ~I(K) is finite, the eigenvalues of q) are quantized, and hence we can naturally give superheavy masses to the color triplet partners of Higgs doublets with a suitable choice of q) [5,6]. Another bizarre property of ¢b is the presence of magnetic monopoles with unconventional magnetic charges [7]. In this paper, we discuss the effect of the Wilson-loop vacuum configuration on the D = 4 gauge coupling constants. We will also include some general discussions on the physics of Wilson-loop vacuum configuration. Our main conclusion is that couplings of SU(3) c, SU(2) and U(1) of the standard model can differ at the compactification scale. For the sake of simplicity, let us consider that H 0 is a discrete group Z~,, H 0 = Z,~, ={exp(i2~kq)); k=0,1,2
..... N-l},
(1)
where the eigenvalues NO A of Nq) are integers. Then H 0 is invariant under the transformations 0 A~-O
A,
0 A~O A+integers,
(2,3)
from which we conclude that all low energy parameters depending on q) should be even and periodic functions of q). 103
Volume 176, number 1,2 After account the pure effective "~cff
PHYSICS LETTERS B
the compactification which takes into the Wilson-loop vacuum configuration, Y a n g - M i l l s part of the D = 4 low energy lagrangian can be written as
I t : F" = - a,.,, .. F
hI'".
(4)
where F~'. corresponds to the field strength of the unbroken D = 4 gauge field A~, and the information of compactification is carried by f~e. The most general form of f~h which respects the unbroken D = 4 gauge s y m m e t r y [8] is
f.:,=
Tr F[2¢rCb]T"T h,
(5)
where F[2Trq)] is an even periodic function of q~ which can be expanded by F[2~r~] = ~
B k cos(2rrkq~).
(6)
k=0
For the quantized q~, i.e. exp(i2¢rNq~) = 1, we can write
21 August 1986
action pull-backed on 2; is given by [10]
S = ½ ~ d r dO[gMx ( X ) O,~XM 3"X N + B.~( X) 3oX ~ aexNy, o~
a+ +
3+xM)x '
q-vltl(isij a + AMI J 3_xM)q* *J
+ ½F,,.,:X'X/,t,'¢~],
(9)
where 8+ = 3, + 3°, 3_ = 3~ - 8o and wM,: denotes the SO(1,9) spin connection and ~:tJ is the gauge field strength tensor. Here the )t' are 32 left-handed fermions. For the SO(32) theory, the q't form the fundamental representation of SO(32); in the E x × Estheory, they are split into two sets of 16 each that correspond to the 16 of the SO(16) subgroup of Eg. Hereafter, we pay attention on one SO(16) subgroup of Es. Consider the gauge field configuration A M such that
[ N/2I
F[Zvrq~] =
Y'~ C k cos(2rrkqJ), k=0
(7)
where all other details of compactification are included in the constant coefficients C a . Up to now, all we have used are the invariance properties of the Wilson-loop vacuum configuration whose gauge invariant informations are specified by H o. Any K a l u z a - K l e i n theories with the Wilson-loop vacuum configuration H o can give rise to the effective lagrangian (4) for which f,,h is given by eq. (7). But in the heterotic string theory, eq. (7) gets a clear meaning, namely it is viewed as the result of the string dynamics under the Wilson-loop vacuum configuration. To see it, let us discuss the string dynamics under the Wilson-loop vacuum configuration. The propagation of a closed string o n M 4 )< K is defined by a m a p p i n g
S M : ~ ~ M 4 X K,
(8)
where N is a closed oriented two-surface with genus n [9]. If one considers the propagation X M of the heterotic string under the background fields ( D = 10 metric gMN, antisymmetric tensor BMX and gauge field AM), the corresponding o-model 104
A,,=A~(y)~, A =A~(x)T",
F,,, = 0,
F~v=F~(x)T",
(10)
where ~, u, O (m, n, p ) denote the tangent indices of M 4 ( K ) and x ( y ) is the coordinate of M 4 (K). Here A~ corresponds to the Wilson-loop a vacuum configuration and A~, denotes the fluctuation of unbroken D = 4 gauge field (i.e. [q), T"] = 0). For % ( K ) which is defined by %(K) =Zs.=
(yz: l = 0 , 1, 2 . . . . . N - l } ,
(11)
y:S1 = [0, ~r ] --+ K, the condition ~ A., d y ' = 2~r gives an isomorphism between % ( K ) and H 0. If we assume that 4) and T" are in the SU(8) subalgebra of SO(16), the interesting Y a n g - M i l l s interaction of the fermionic oscillators can be written as
Sy M=l.fd'~do'I'*A[ia~a_+A~.,(X) a X ' ~ + A"~( X) ~_X"T; B] g'~,
(12)
where q'A (q,.A) behaves like 8 (8) of SU(8). Hereafter we work in the basis in which 4) is
diagonal. For X M= xM+ ~M where x M denotes the background string propagation and fM is its quantum fluctuation, Sy_ M is given by [11]
Sy_M = '2f d ¢ do'l"*A[iS~ 3 + A ~ ( X ) 3 xmq)AB +A~(2) 3 2"T;8 + Ffl,(2)~" 3 2~G 'B + ...]%.
(13)
As can be seen from eq. (13). all the effects of the Wilson-loop vacuum configuration appear throug_h the fermion propagator S + - ( i I 3_ + A ,~, 3 X " q~) 1. To see the properties of S+, let us define the class of string propagations X~ as
Xl=(--M
= o, o ) - - v ' }.
--m
(14)
Then X(t --g) denotes the propagation of a string configuration which is homotopic in K to the loop element ~,~ in %(K). We will therefore call the class Xt as the l-soliton sector. For X(t --M), the corresponding Dirac operator can be written by
iD~ }= iI 3 + A.,* 3_L7,
~/~mdY m¢19
= i / 3 - --~rl
=ii3
21 August 1986
PHYSICS LETTERSB
Volume 176, number 1,2
-2l*,
(15)
which clearly shows the spectral flow of the fermion oscillators in the l-soliton sector [12]. All physical implications of the Wilson-loop vacuum configuration are due to this spectral flow. The fermion propagator in the l-soliton sector is
invariant under
X,,I---, - X , l ,
X,,I ~ X,I + 2NCb.
(19,20)
Notice that exp(2ioNeb)q,,(z) is a well-defined wave function because N~b has integer eigenvalues. (We normalize the closed string parameter o as 0 ~
sT"(z-
z':
s T ' ( z - z': =
(21)
S ~ ' ( z - z'" • + ( N / I ) ~ ) = exp[2i(o - o')NCb] S~'(z - z'" rb),
(22)
which enables us to see clearly the physical implications of q~. Now let us discuss about our main quantity f,h in superstring compactification. One relevant fact for the nontrivial f,h is the gauge charge Qe of the ground state of the l-soliton sector. It was noticed that the ground state of the l-soliton sector takes non-vanishing Q~ which may take fractional values [7]. Then the charge spectrum of the existing states in the /-soliton sector reveals the symmetry breaking due to the non-vanishing Qe of the ground state. This non-trivial gauge charge structure of the l-soliton sector (l 4= 0) implies the nontrivial renormalization of the gauge coupling constants. Following Callan et al. [11], we observe that the kinetic terms of the ga_uge fields_ are generated by the operator F~"~(X)~" 3 X"q '*a × TA~Bq'B in the action (13). Then we have /,,~ = C(fd2z d 2 z ' [ ~ ' ( z ) 3 X " ( z )
S~]'(z-z':q~)=(zl(iD '°) 'lz') 1
= E 2%1- 2lq)q~"(z)eO*(z')'
×~,(z') 3 X,(z- ')J+(z)Jb+(z')),"
n
(16) where z = (r, o), z ' = ( r ' , o ' ) and X ~ . ( z ) . From the following relations:
i3_q,.(z)=
il 3 q , . ( - z ) = - X . l q , . ( - z ) ,
where C is a constant and J+ = '/'*ATA"B'/"B. Let f ~ l be the contribution to f,h from the l-soliton sector. After integrating the quantum degrees, we get
(17)
f,~/,'( O) = A"' f d2z d2z'[Tr{ T"S~'(z - z'; q~)
i I 3 [exp(2ioN~)q~.(z)]
= (X,,I + 2Nrb)[exp(2ioNCb)q~.(z)],
(23)
(18)
we can see that the spectral distribution {X. } is
• ×3
.g~)(z) 3_Xl~)(z')],
(24) 105
Volume 176, number 1,2
PHYSICS LETTERS B
T ° 0-X (p)
Tb
Fig. 1. The diagram for the amplitude (24). The solid lines represent the fermion oscillator '/'A and the dotted line represents the propagator of the string fluctuation ,~,.
where D ~ is the propagator of ~,. The corresponding diagram is given in fig. 1. Using eqs. (21) and (22), we obtain
f (th' ) / \~ )
= rJ(al b) l -~
~)) = f ~ ) ( ~
+
(U/l)~)
(25)
which implies that f,~)((b) = ~ C ~ l) Tr T~T h cos(2~rk/(b). k
21 August 1986
appear only at string loop level which can generate an /-soliton sector ( / 4 0). One important result from these string loop effects is the non-trivial structure of f,t, at the compactification scale. Finally let us briefly discuss about the phenomenological implication of our result. The eigenvalues of f~t, corresponds to 1/g~, where the g,, are the gauge coupling constants. If we embed the standard SU(5) [13] in SU(8) as
(su(s)) =
=(1/N)diag.(1,1,1,3,3, - 3 , - 3 , -3), (29) then we get
1/g.~ : l/g2, : 1/g~l ......p.c,mc~u~........ i,~ (26)
f,,,,(~) = E f , g'(~,)
a=
2 ~rk
[ N/21
l
~
Gcos~-,
/,=1
[N/2I
where C a depends on the details of the string dynamics under the compactification. This form of f,,e is the same as expression (7) which is obtained from the invariance of H 0. In general, the C~ are non-vanishing. Summarizing our analysis, all the physical implications of the Wilson-loop vacuum configuration appear through the fermion propagator S([)(l =g 0). The string propagation in X0 (i.e. the l = 0 sector) cannot detect the Wilson-loop vacuum configuration because S~ ~) does not depend on ~b. One trivial corollary of this result is that the Wilson-loop vacuum configuration gives no contribution at the string tree level because any string propagation X M : S z --, M 4 × K corresponds to the zero-soliton sector X o. This was already noticed by Witten in his discussion about E6-gauge coupling [5]. But our analysis clearly shows that the effects of the Wilson-loop vacuum configuration 106
[ N/2]
6 ~rk
,',= ~, C k cos N k=l
(31,32)
(27)
0
(30)
= a . b. ~(2,, + 3h), where
k
(28) SU(3)
and choose
Finally, we obtain
= E Ck Vr{cos(2rrk~)T"U'},
sv(5)
which should be positive for the proper definition /5 of F~2, terms. Here g~, gw and g l ( = ~'~ gY) denote the gauge coupling constants of SU(3)~ × SU(2),, × U(1)y with the normalization Tr T"T h= ~. Clearly, eq. (30) shows the difference between various gauge coupling constants which should be considered in the G e o r g i - Q u i n n - W e i n b e r g analysis [14]. We thank C. Lee for useful discussions. This work is supported in part by the Korean Science and Engineering Foundation and by the Research institute of Basic Sciences, Seoul National University.
References [1] M.B. Green and J.H. Schwarz, Phys. Lett. B149 (1984) 117; D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B256 (1985) 253: and Princeton preprint (1985).
Volume 176, number 1,2
PHYSICS LETTERS B
[2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys, B258 (1985) 246. [3] J.-P. Derendinger, L.E. Ib~.ez and H.P. Nilles, Nucl. Phys. B267 (1986) 365; M. Dine, V. Kaplunovsky, C. Nappi, M. Mangano and N. Seiberg, Nucl. Phys. B259 (1985) 549. [4] Y. Hosotani, Phys. Lett. B129 (1983) 193. [5] E. Witten, Nucl. Phys. B258 (1985) 75. [6] A. Sen, Phys. Rev. Lett. 55 (1985) 33; J. Breit, B. Ovrut and G. Segr& Phys. Left. B158 (1985) 33. [7] X.-G. Wen and E. Witten, Nucl. Phys. B261 (1985) 651.
21 August 1986
[8] Q. Shaft and C. Wetterich, Phys. Rev. Lett. 52 (1984) 875; C.T. Hill, Phys. Lett. B135 (1984) 47. [9] J.H. Schwarz, Phys. Rep. 89 (1982) 223. [10] C.M. Hull and E. Witten, Phys. Lett. B160 (1985) 398. [11] C.G. Callan, D, Friedan, E. Martinec and M.J. Perry, Nucl. Phys. B262 (1985) 593. [12] N.S. Manton, preprint NSF-ITP-84-15. [13] H. Georgi and S.L. Glashow. Phys. Rev. Lett. 32 (1974) 438. [14] H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. kett, 33 (1974) 451.
107
PHYSICS LETTERS B
21 August 1986
S Y M M E T R Y BREAKING BY W I L S O N - L O O P E L E M E N T S IN S T R I N G T H E O R I E S AND RESCALING OF GAUGE COUPLING CONSTANTS
Kiwoon C H O I and Jihn E. K I M Department of Physics. Seoul National University, Seoul 151, Korea
Received 15 May 1986 It is found that there exists nontrivial rescaling of gauge coupling constants at the compactification scale by the Wilson-loop elements in the heterotic string theory. This phenomenon appears at the string loop level, and the result modifies the previous phenomenological analysis of sin20w.
Recent discoveries of consistent superstring theories with gauge groups SO(32) or E~ x E s render a new insight in constructing a theory of q u a n t u m gravity [1]. Furthermore, these theories, in particular the E s x E s theory, imply various interesting phenomenological applications [2]. For physical applications, we consider the spontaneous compactification of M~0 with the vacuum configuration defined o n M 4 x K where M 4 iS the four-dimensional Minkowski space and K is some c o m p a c t six-manifold [2,3]. In many compactification schemes, the gauge symmetry breaking by means of the Wilson-loop elements is crucial in obtaining realistic low energy gauge groups [2,4]. The Wilson-loop vacuum configuration is the Yang Mills gauge field configuration which gives vanishing field strength tensor but cannot be gauged away. It is allowed only when ¢rl(K ) is nontrivial. The gauge invariant expression is given by a h o m o m o r p h i s m exp(i2z-cb) : z'l(K ) --* H 0, where exp(i2~-tb) = P exp(i ~ A,,, d y " ' ) and H o is a discrete subgroup of the ten-dimensional gauge group G. Then the unbroken D = 4 gauge group G should c o m m u t e with H 0. In the compactification schemes of the E s × E~ heterotic string theory on the C a l a b i - Y a u manifold, the Wilson-loop vacuum configuration has been used to break E 6 without breaking a D = 4 supersymmetry [2]. In this case, q) behaves like a
D = 4 Higgs field transforming as 78 under E s. But q) has a novel property due to its topological origin. If ~I(K) is finite, the eigenvalues of q) are quantized, and hence we can naturally give superheavy masses to the color triplet partners of Higgs doublets with a suitable choice of q) [5,6]. Another bizarre property of ¢b is the presence of magnetic monopoles with unconventional magnetic charges [7]. In this paper, we discuss the effect of the Wilson-loop vacuum configuration on the D = 4 gauge coupling constants. We will also include some general discussions on the physics of Wilson-loop vacuum configuration. Our main conclusion is that couplings of SU(3) c, SU(2) and U(1) of the standard model can differ at the compactification scale. For the sake of simplicity, let us consider that H 0 is a discrete group Z~,, H 0 = Z,~, ={exp(i2~kq)); k=0,1,2
..... N-l},
(1)
where the eigenvalues NO A of Nq) are integers. Then H 0 is invariant under the transformations 0 A~-O
A,
0 A~O A+integers,
(2,3)
from which we conclude that all low energy parameters depending on q) should be even and periodic functions of q). 103
Volume 176, number 1,2 After account the pure effective "~cff
PHYSICS LETTERS B
the compactification which takes into the Wilson-loop vacuum configuration, Y a n g - M i l l s part of the D = 4 low energy lagrangian can be written as
I t : F" = - a,.,, .. F
hI'".
(4)
where F~'. corresponds to the field strength of the unbroken D = 4 gauge field A~, and the information of compactification is carried by f~e. The most general form of f~h which respects the unbroken D = 4 gauge s y m m e t r y [8] is
f.:,=
Tr F[2¢rCb]T"T h,
(5)
where F[2Trq)] is an even periodic function of q~ which can be expanded by F[2~r~] = ~
B k cos(2rrkq~).
(6)
k=0
For the quantized q~, i.e. exp(i2¢rNq~) = 1, we can write
21 August 1986
action pull-backed on 2; is given by [10]
S = ½ ~ d r dO[gMx ( X ) O,~XM 3"X N + B.~( X) 3oX ~ aexNy, o~
a+ +
3+xM)x '
q-vltl(isij a + AMI J 3_xM)q* *J
+ ½F,,.,:X'X/,t,'¢~],
(9)
where 8+ = 3, + 3°, 3_ = 3~ - 8o and wM,: denotes the SO(1,9) spin connection and ~:tJ is the gauge field strength tensor. Here the )t' are 32 left-handed fermions. For the SO(32) theory, the q't form the fundamental representation of SO(32); in the E x × Estheory, they are split into two sets of 16 each that correspond to the 16 of the SO(16) subgroup of Eg. Hereafter, we pay attention on one SO(16) subgroup of Es. Consider the gauge field configuration A M such that
[ N/2I
F[Zvrq~] =
Y'~ C k cos(2rrkqJ), k=0
(7)
where all other details of compactification are included in the constant coefficients C a . Up to now, all we have used are the invariance properties of the Wilson-loop vacuum configuration whose gauge invariant informations are specified by H o. Any K a l u z a - K l e i n theories with the Wilson-loop vacuum configuration H o can give rise to the effective lagrangian (4) for which f,,h is given by eq. (7). But in the heterotic string theory, eq. (7) gets a clear meaning, namely it is viewed as the result of the string dynamics under the Wilson-loop vacuum configuration. To see it, let us discuss the string dynamics under the Wilson-loop vacuum configuration. The propagation of a closed string o n M 4 )< K is defined by a m a p p i n g
S M : ~ ~ M 4 X K,
(8)
where N is a closed oriented two-surface with genus n [9]. If one considers the propagation X M of the heterotic string under the background fields ( D = 10 metric gMN, antisymmetric tensor BMX and gauge field AM), the corresponding o-model 104
A,,=A~(y)~, A =A~(x)T",
F,,, = 0,
F~v=F~(x)T",
(10)
where ~, u, O (m, n, p ) denote the tangent indices of M 4 ( K ) and x ( y ) is the coordinate of M 4 (K). Here A~ corresponds to the Wilson-loop a vacuum configuration and A~, denotes the fluctuation of unbroken D = 4 gauge field (i.e. [q), T"] = 0). For % ( K ) which is defined by %(K) =Zs.=
(yz: l = 0 , 1, 2 . . . . . N - l } ,
(11)
y:S1 = [0, ~r ] --+ K, the condition ~ A., d y ' = 2~r gives an isomorphism between % ( K ) and H 0. If we assume that 4) and T" are in the SU(8) subalgebra of SO(16), the interesting Y a n g - M i l l s interaction of the fermionic oscillators can be written as
Sy M=l.fd'~do'I'*A[ia~a_+A~.,(X) a X ' ~ + A"~( X) ~_X"T; B] g'~,
(12)
where q'A (q,.A) behaves like 8 (8) of SU(8). Hereafter we work in the basis in which 4) is
diagonal. For X M= xM+ ~M where x M denotes the background string propagation and fM is its quantum fluctuation, Sy_ M is given by [11]
Sy_M = '2f d ¢ do'l"*A[iS~ 3 + A ~ ( X ) 3 xmq)AB +A~(2) 3 2"T;8 + Ffl,(2)~" 3 2~G 'B + ...]%.
(13)
As can be seen from eq. (13). all the effects of the Wilson-loop vacuum configuration appear throug_h the fermion propagator S + - ( i I 3_ + A ,~, 3 X " q~) 1. To see the properties of S+, let us define the class of string propagations X~ as
Xl=(--M
= o, o ) - - v ' }.
--m
(14)
Then X(t --g) denotes the propagation of a string configuration which is homotopic in K to the loop element ~,~ in %(K). We will therefore call the class Xt as the l-soliton sector. For X(t --M), the corresponding Dirac operator can be written by
iD~ }= iI 3 + A.,* 3_L7,
~/~mdY m¢19
= i / 3 - --~rl
=ii3
21 August 1986
PHYSICS LETTERSB
Volume 176, number 1,2
-2l*,
(15)
which clearly shows the spectral flow of the fermion oscillators in the l-soliton sector [12]. All physical implications of the Wilson-loop vacuum configuration are due to this spectral flow. The fermion propagator in the l-soliton sector is
invariant under
X,,I---, - X , l ,
X,,I ~ X,I + 2NCb.
(19,20)
Notice that exp(2ioNeb)q,,(z) is a well-defined wave function because N~b has integer eigenvalues. (We normalize the closed string parameter o as 0 ~
sT"(z-
z':
s T ' ( z - z': =
(21)
S ~ ' ( z - z'" • + ( N / I ) ~ ) = exp[2i(o - o')NCb] S~'(z - z'" rb),
(22)
which enables us to see clearly the physical implications of q~. Now let us discuss about our main quantity f,h in superstring compactification. One relevant fact for the nontrivial f,h is the gauge charge Qe of the ground state of the l-soliton sector. It was noticed that the ground state of the l-soliton sector takes non-vanishing Q~ which may take fractional values [7]. Then the charge spectrum of the existing states in the /-soliton sector reveals the symmetry breaking due to the non-vanishing Qe of the ground state. This non-trivial gauge charge structure of the l-soliton sector (l 4= 0) implies the nontrivial renormalization of the gauge coupling constants. Following Callan et al. [11], we observe that the kinetic terms of the ga_uge fields_ are generated by the operator F~"~(X)~" 3 X"q '*a × TA~Bq'B in the action (13). Then we have /,,~ = C(fd2z d 2 z ' [ ~ ' ( z ) 3 X " ( z )
S~]'(z-z':q~)=(zl(iD '°) 'lz') 1
= E 2%1- 2lq)q~"(z)eO*(z')'
×~,(z') 3 X,(z- ')J+(z)Jb+(z')),"
n
(16) where z = (r, o), z ' = ( r ' , o ' ) and X ~ . ( z ) . From the following relations:
i3_q,.(z)=
il 3 q , . ( - z ) = - X . l q , . ( - z ) ,
where C is a constant and J+ = '/'*ATA"B'/"B. Let f ~ l be the contribution to f,h from the l-soliton sector. After integrating the quantum degrees, we get
(17)
f,~/,'( O) = A"' f d2z d2z'[Tr{ T"S~'(z - z'; q~)
i I 3 [exp(2ioN~)q~.(z)]
= (X,,I + 2Nrb)[exp(2ioNCb)q~.(z)],
(23)
(18)
we can see that the spectral distribution {X. } is
• ×3
.g~)(z) 3_Xl~)(z')],
(24) 105
Volume 176, number 1,2
PHYSICS LETTERS B
T ° 0-X (p)
Tb
Fig. 1. The diagram for the amplitude (24). The solid lines represent the fermion oscillator '/'A and the dotted line represents the propagator of the string fluctuation ,~,.
where D ~ is the propagator of ~,. The corresponding diagram is given in fig. 1. Using eqs. (21) and (22), we obtain
f (th' ) / \~ )
= rJ(al b) l -~
~)) = f ~ ) ( ~
+
(U/l)~)
(25)
which implies that f,~)((b) = ~ C ~ l) Tr T~T h cos(2~rk/(b). k
21 August 1986
appear only at string loop level which can generate an /-soliton sector ( / 4 0). One important result from these string loop effects is the non-trivial structure of f,t, at the compactification scale. Finally let us briefly discuss about the phenomenological implication of our result. The eigenvalues of f~t, corresponds to 1/g~, where the g,, are the gauge coupling constants. If we embed the standard SU(5) [13] in SU(8) as
(su(s)) =
=(1/N)diag.(1,1,1,3,3, - 3 , - 3 , -3), (29) then we get
1/g.~ : l/g2, : 1/g~l ......p.c,mc~u~........ i,~ (26)
f,,,,(~) = E f , g'(~,)
a=
2 ~rk
[ N/21
l
~
Gcos~-,
/,=1
[N/2I
where C a depends on the details of the string dynamics under the compactification. This form of f,,e is the same as expression (7) which is obtained from the invariance of H 0. In general, the C~ are non-vanishing. Summarizing our analysis, all the physical implications of the Wilson-loop vacuum configuration appear through the fermion propagator S([)(l =g 0). The string propagation in X0 (i.e. the l = 0 sector) cannot detect the Wilson-loop vacuum configuration because S~ ~) does not depend on ~b. One trivial corollary of this result is that the Wilson-loop vacuum configuration gives no contribution at the string tree level because any string propagation X M : S z --, M 4 × K corresponds to the zero-soliton sector X o. This was already noticed by Witten in his discussion about E6-gauge coupling [5]. But our analysis clearly shows that the effects of the Wilson-loop vacuum configuration 106
[ N/2]
6 ~rk
,',= ~, C k cos N k=l
(31,32)
(27)
0
(30)
= a . b. ~(2,, + 3h), where
k
(28) SU(3)
and choose
Finally, we obtain
= E Ck Vr{cos(2rrk~)T"U'},
sv(5)
which should be positive for the proper definition /5 of F~2, terms. Here g~, gw and g l ( = ~'~ gY) denote the gauge coupling constants of SU(3)~ × SU(2),, × U(1)y with the normalization Tr T"T h= ~. Clearly, eq. (30) shows the difference between various gauge coupling constants which should be considered in the G e o r g i - Q u i n n - W e i n b e r g analysis [14]. We thank C. Lee for useful discussions. This work is supported in part by the Korean Science and Engineering Foundation and by the Research institute of Basic Sciences, Seoul National University.
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Volume 176, number 1,2
PHYSICS LETTERS B
[2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys, B258 (1985) 246. [3] J.-P. Derendinger, L.E. Ib~.ez and H.P. Nilles, Nucl. Phys. B267 (1986) 365; M. Dine, V. Kaplunovsky, C. Nappi, M. Mangano and N. Seiberg, Nucl. Phys. B259 (1985) 549. [4] Y. Hosotani, Phys. Lett. B129 (1983) 193. [5] E. Witten, Nucl. Phys. B258 (1985) 75. [6] A. Sen, Phys. Rev. Lett. 55 (1985) 33; J. Breit, B. Ovrut and G. Segr& Phys. Left. B158 (1985) 33. [7] X.-G. Wen and E. Witten, Nucl. Phys. B261 (1985) 651.
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