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The neutron electric dipole moment in the cloudy bag model
Volume 179, number 4
PHYSICS LETTERS B
30 October 1986
T H E N E U T R O N E L E C T R I C D I P O L E M O M E N T IN T H E C L O U D Y BAG M O D E L Michael A. M O R G A N Department of Phystc~, Seattle Unwerstt~, Seattle, WA 98122, USA
and Gerald A M I L L E R lnstttute for Nuclear Theory, Department of Physics, FM-15, l]mverstty of Washington, Seattle, WA 98195, USA Received 14 July 1986 An evaluation of the neutron electric dipole moment (NEDM), using the cloudy bag model (CBM) shows that two CPviolating effects (a quark mass term and a p i o n - q u a r k interaction) have contributions that are about equal in magnitude, but opposite in sign. This cancellation allows the upper limit on the 0 parameter to increase by about an order of magnitude.
The experimental upper limit 5 × I0 -25 e c m [1] on the neutron electric dipole moment (along with the assumption of invariance under CPT) places strong limitations on theories of CP violation [2]. Many possible sources o f such CP violation have been suggested [2,3]. Here we restrict out attention to the "strong" CP violation present m the fundamental QCD lagrangian describing the strong interactions between quarks within the neutron [4]. It has become a standard textbook exercise to show that the term
-0
g2 F ' F * ,
(1)
32n 2 present in the QCD lagrangian because of the topological properties of the vacuum, can be transferred to the quark sector through an appropriate global chiral rotation of the quark fields. The result is an additional interaction between quarks
5 ~CP = iOmCt75q
(2)
which violates b o t h P and T. Here ~ -= mumdms/ (mum d + mum s + mdm s) --~ 5.3 MeV if we take the MIT bag value o f m s = 300 MeV and use the ratios md/m u = 1.8 and ms/m d = 20 as determined by chiral perturbation theory [5]. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A calculation o f the NEDM (Dn)within the framework of the MIT bag has previously been done using (2) as a perturbation and estimating the contributions of the N*(1535) and N*(1700) intermediate states [6]. The result D n = 2.7 X 10-160 e cm
(3)
was obtained. Since the bag radms R is the only length scale xn the MIT bag model, simple &menslonal analysis leads to the result that D n is proportional to ~ R 2. This means that the numerical value o f D n is quite sensitive to the bag radius A value o f 1.4 fm was used in obtaining eq. (3). However, there is a bigger problem. We do not believe that the calculation leading to eq. (3) is vahd. This is because the interaction of eq. (2) does not connect two quark orbitals with positive energy, as we show below. Thus the simple, N*(qqq) excited states used should not contribute to D n. Nevertheless, we recompute this term below and find that the magnitude of this term is in agreement with eq. (3). Another calculation of the NEDM has been based on an effectwe CP-violating 7rN interaction derwed from eq. (2) [7]. These authors use current algebra to show that D n = g . N N ~ N N ln(mN/mr)/4~r2m N
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m the limit that m~r becomes zero. Here glrNN = 0.0270 is the CP-violating coupling constant computed from the SU(3) mass splittings between the E and Z. The authors obtain the estxmate D n ~ 3.6 × 10 -16 0 e cm. This result is due solely to 7rN intermediate states which are the only ones used in an expression (eq. (14) of ref. [7] ) in which a sum over all states appears. There is also a large uncertainty as to what mass to insert in the numerator of the logarithm and therefore this approach can only give an order of magnitude estimate for D n. These two calculations may be only parts of a more complete calculation since both pionlc and quark terms should be included. To do this one needs to calculate the NEDM in a model which simultaneously incorporates the effects of the neutron's quark core and pion cloud. In the last few years chiral bag models have been successfully employed in the computation of static nucleon properties. In this letter we report on the computation o f D n using the cloudy bag model (CBM) [8]. The previous calculations [6,9] o f D n make it clear that both effects, considered alone, can be sigmficant. Further, if cancellation between these two contributions did occur [7], then a dimmution of D n for a fixed value of 0 would result. Alternatively, one could raise the value of 0 needed to allow a calculated NEDM to be consistent with the enormously small experimental upper bound on D nFirst we evaluate the contribution to D n from the quark core. In the CBM the quark fields are given by their mode sums
30 October 1986
places. Here we shall only need the lowest positiveenergy mode (1S1/2), (xl 1 , - I , ~, m) = ~ 1,-1,1/2,m (X)
-
N [ ]0 (wr/R) ~ (i]l(cOr/R)~. f] Xm
X exp(-iEt) O(R - r)
with energy E = co/R and eigenfrequency 6o = 2.04 determined by the boundary condition in eq. (7). The normalization constant is N = w/2R3(w - 1)]2(6o). To lowest order in the perturbation of the neutron wave function we have D n = ( n [ i . d i n ) = 2 ~ (NI~ "dlX)(XI6LcpIN) X EX _ EN
qa(x)
= ~l [~l(X)bal + "~t(x)d~l],
where c~labels the type of quark and l - n , k , j , m the bag mode; b destroys quarks in positwe-energy states ~l and d ÷ creates anti-quarks in negative-energy states qJl" The bag states satisfy the massless Dlrac equation reside the bag of radius R,
oz.pll)=Elll)
(6)
6Lcp = i~O f
d3x ~(x)3"sq(x):
- i t "~ It) = II)
(7)
at the surface o f the bag. Here ¥ are the Dlrac gamma matrices (Blorken and Drell convention) and at = 3'0¥. The complete set of wave functions is given in many 380
(10)
given by eq. (2). The dipole operator d = fxp(x) d3x receives contributions only from the quark core charge density (11)
where Q is the quark charge matrLx. Substitution of eq. (5) into eqs. (10) and (11) results in the expression D -- 2ir~0e ~
n
Qc~
X Ex~E N
× (OIx31J)(m[FIn)(NJb+~ib(~ib;mb~n{N) + (mlFl'ff)(TIx311) (Ic~ibcqbomdcnlN) N d + + )
and also obey the boundary condition
(9)
for the component of the NEDM along the spin of the neutron. IN) is the MIT state for the ground-state neutron with spin up, IX) is an arbitrary intermediate state of odd parity and 6Lcp is the CP-violating interaction
p(x) = e Ft(x)3"°Qq(x) :
(5)
(8)
(12)
where repeated indices are summed and the tilde indicates a state of negative energy. Here Qa --- Qaa, F 703'5 and the single-particle matrix elements are defined as usual by (110 Ij) - f d3x ~i~ (x)O (x) ~j(x). From eq. (6) and the fact that ~ and F anti-commute,
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PHYSICS LETTERS B
it follows that if In) is a bag state of (positive) energy E n then Pl n) is a bag state of (negative) energy - E n ,
with the boundary condition for both given by eq. (7). Because of this Fin) is orthogonal to In), and to any positive-energy state Ira). Therefore, the first term in eq. (12) vanishes and intermediate states with all quarks in positive-energy states do not contribute to • 1t h i s piece o f D n. In particular, the odd parity : excited states of the nucleon are not mixed into the wavefunction by 6 L c p . However, eq. (12) does allow intermediate q~ fluctuations which result in a non-zero contribution to D n. Using I'fi')= Pl n) and keeping only the connected diagrams one finds the second term in eq. (12) to give
30 October 1986
It is interesting to note that the magnitude of this result is in rough agreement with the calculation of Bahim [6] when his value is scaled down to R = 1 fm, but our sign is opposite to his. Our result in eq. (15) is exact in lowest-order perturbation theory if one ignores center-of-mass motion of the bag. Next we compute the contribution to D n from the pion cloud. The CBM is used to account for the strong interaction between pion and quarks. An effective CPviolating pion-quark interaction will be induced by eq. (2). The only possible P-violatlng interaction terms linear in the plon field are [10] Z? 1 - ~Trq,
~22 - r:l(~,u0uTr)q,
~o3 ~ ~(ouv0uOvTr)q , Dn = "u'~Oe 3E ((~lx31 f ) _ (~tx31¢))
(13)
corresponding to an intermediate state with an extra quark and anti-quark in the 1S1/2 and 1S1/2 states respectively (see fig. 1). The energy denominator is E X - E N = 2E = 2 w / R . The arrows refer to the quark spin. The matrix elements are (+1 ix3l+l ) = ~_iRft°
(14)
where f w = [360(60 -- 1) sin2w]-1 1
X [~co(1 + ~ cos 260) - ~ sin 2co1. Therefore the quark core contribution is D(quark)n = -- 2 (fco/w)~R 20e - 1.8 × 10-160 e cm
(15)
at a bag radius o f R = 1.0 fm.
~4 - 7q(~ Xz)3q.
(16)
Of these, only Z? 1 is acceptable. Integration by parts shows .122 to vanish because of conservation of quark probability current. "~3 = 0 because ouv is antisymmetric. "~4 does not vanish but is even under T and therefore cannot give a NEDM. Since QCD is isospin invariant we shall assume (17) qq~r for our effective lagrangian. The coupling constant g x can be computed by comparison withg~rNN in eq. (4). With the inclusion of pions we must consider contributlons to D n from both the quark and plon charge densities. For now let us only consider the contributions due to intermediate states with all quarks remainmg in the ground state 1S1/2. We expect this to be the dominant contribution since large energy denominators will tend to suppress contributions from excited quark states. Furthermore, effects due to the finite size of the plon have been shown to reduce the contributions from these intermediate excited states in calculations of other static nucleon observables [ 11 ]. Our interaction hamlltonian at the nucleon level is .12 CP = g x ~ t ' c ' ~ q
HI = HCBM + Hx =
-f(~c~M + ~
CP
qqrr) d3x
(18)
where "~CBM = q 7 5 ~ " n @ (r - R )/21f7 r is the standard CBM interaction [8], and £? qqn C? is given above. Projection of eq. (18) into the subspace of colorless baryonic bags gives Fig. 1. Quark contribution to the NEDM. Only the 1S1/2 negative-energy intermediate quark state contributes• 381
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PHYSICS LETTERS B
3
i=~ fd
k A,B
* (19) ~j %}
30 October 1986
Proceeding with the calculation we use perturbation theory. One may write D n = (nl£ "din>
where ak/destroys a pion with momentum k and isospin component ], A destroys a baryon bag [A), and the sum runs over all baryons m the octet and decuplet. The vertex functmn contains a strong p~ece and a CPviolating piece,
The CP-violating vertex is
v~) AI = _gx TAB
(20)
Ux(kR)
X 4(2rr)32CokSSASB6SAsB(TBtBlnlTAtA)(e])2 where S(s) and T(t) stand for the spin and xsospin of the baryon and n stands for the nth spherical component. T AB is a reduced matrix element of the baryon isospin operator and 1
.2 -J~)(c°X)Jo(kRx) dx Ux(kR) = N2R3 fJ x 2(Jo
(21)
0 is the CP-violatlng form factor. The coupling constant
gx is determined by comparison with the effective 7rNN interaction of Crewther et al. [7] at zero momentum transfer. The result is
gx =gTrNN/Ux(0) =gnNN/0"480 = --0.0560.
(22)
= (N[d3GoHIGoHIIN) + (N IHIGod3GoHIIN) + (N IHIGoHIGod3IN)
with H I given by eqs. (19), (20) and (23), and G O = (m N - H 0 ) - I is the free static-nucleon propagator. Now the dipole operator receives contributions from both charged quarks and charged plons. We need not consider the contribution from the quarks since by assumption they remain in the ground state and cannot produce a non-zero expectation of a P-odd operator. The charge density of the pion field is
O(x) = ie [c~t(x)~ (x) - c~(x)4)~ (x)]
oAB ki
u(kR )
mn
4(2~)326o k
d3k Ux(kR)
egx ( Dn =_~_~ 4fNN lim
n
×
q~O d(2n)J~
cok
d [ k;u(k'm
-
X (SBSB lmlSaSA)(TBtBln [TAtA)(k)m(e.l) * " n* (23) and the coupling constantsf AB have been given many places [8]. The CBM form factor is u(kR) = 3/'1 (kR)/ kR. Now since the CBM vertex conserves strangeness and the CP-violating vertex only connects baryons of the same spin, our intermediate states will be restricted to the nucleon. Therefore the only reduced matrix elements needed are T NN = (1 ~1 101~ ~2) -1 a n d f NN = ~fNN~r, where fNN~r = 0.28 lS the physical lrN coupling constant. 382
(25)
where ~(x) = [nl(X ) + in2(×)]/x/~is the field associated with the destruction of a negatwely charged plon as in ref. [12]. Insertion of eq. (19) into eq. (24) results in twelve terms, only half of which are non-zero. These are the interference terms with the strong coupling at one vertex and the CP-violatlng coupling at the other as shown in fig. 2. The non-interference terms are obviously zero since d 3 is odd under P. Using the expressions for the two vertex functions in eqs. (20) and (23) one finds that the total contribution from all six diagrams in fig. 2 is
For completeness the CBM vertex is
if AB
(24)
where q -= k' - k is the momentum transferred by the photon. Taking the limit and using the expression for u(kR) one finds
D n - e g xfNN y(Zll(Z2~_aZ " 1 2.12 (z) m~r 7r2 0
(z2 + o~2)3/2
lZ3f l (Z)
where a = rn~.R and I 1 and J2 are spherical lessel functions. The function u x is the CP-violating form factor
Volume 179, number 4
PHYSICS LETTERS B
30 October 1986
-¢ J" [
n ///
3O
/
P n
IIITI'-
x// E
25
20 X\
/
\
i
x
I
\\
0
/
d [O
/ /
05
/ /
l''
1
I
O4
06
_
I
08
I
1.0
I
I 2
4
R (fro) Fig. 2. Piomc contributions to the NEDM (quarks restricted to the ground state of the bag). A cross indicates the wolating interactmn.
CP-
given in eq. (21). A sphne fit to u x was made to simplify the numerical integral in (27). The results for mTr = 140 MeV a n d f NN = 3V'4-~fNN~r = 2.98 are given in fig. 3 as a function of the bag radius. At R = 1.0 fm D (pl°n) ~ 1.4 X 10 -16 0 e cm. n Notice the positive sign o f this result agrees with the current algebra results o f Crewther et al. [7] and that it is opposite to the contribution from the quark core in eq. (15). Also, we see that, whereas the quark contributlon IS proportional to R 2, the ploncloud contribution is a slowly varying function o f R . Both contributions are plotted as a function of the bag radius in fig 3. Here we see that at some radius just under 1 fm the two contrlbunons are equal so that D n is exactly zero, lrregardless o f the value o f 0. It is interesting to note that this value o f R <~ 1.0 fm lies within previous phenomenologlcal determinations. Thus the cancellation o f the two terms is a physically reasonable possibihty within the cloudy bag model. Since the quark-core contribution [eq. (15)] is entirely due to intermediate states with qV= 1 pairs one may wonder about the size o f contributions due to antiquarks in intermediate states of plon-cloud terms. We have computed these contributions [13] and found
Fig 3. The plonic (sohd) and the negative of the quark (dashed) contributions to the NEDM as a function o f bag radlu s, R.
that they only increase the results of fig. 3 by about 10% If the pion is treated as a point-like particle. Actually, contributions to the plon-cloud terms with Intermediate 1P1/2 quarks should be expected to be as large as these anti-quark contributions since they have comparable energy denominators. Our calculations show that these increase the results o f fig. 3 by about 17% [13]. These higher-order corrections to the pion-cloud contribution to D n are small and probably should be ignored since proper account of the pion structure will make them even less important. At most, these corrections slightly increase the value o f R for which D n is zero. It is tempting to use fig. 3 to conclude that by choosing an appropriate value of R , one can set the computed value o f the NEDM to zero, thereby allowing 0 to be as large as one likes. However, to specify the correct value o f R , one would have to know it to about SEXsignificant figures. This is clearly impossible since R is a model-dependent quantity, defined in a static cavity approximation. Indeed, one expects a variety of effects such as center-of-mass motion and various quantum fluctuations to provide corrections on the order o f ten percent. Thus we say that the bag radius is defined to within only one significant figure and any cancellation is Incomplete. One can only rea383
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sonably expect to gain (at most) a factor of ten m the upper limit of 0 allowed by the experimental upper limit on D n. One therefore finds that 0 is of the order of 1(3-8 instead of 10 -9 . This change may be relevant in allowing certain theories of CP violation (e.g. some of those m ref. [3]) to survwe [14]. This work is supported in part by the US Department of Energy, and is included in a 1984 University of Washington Ph.D. thesis [ 13]. We thank G. Veneziano and R. Peccei for useful discussions.
References [ 1] [2] [3] [4] [5]
J.M. Pendlebury et al., Phys. Lett. B 136 (1984) 327. N.F. Ramsey, Annu. Rev. Nucl. Part. Sci. 32 (1982) 211. S.M. Barr and A. Zee, Phys. Rev. Lett. 55 (1985) 2253. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. S. Wemberg, in. A festschrift for I.I. Rabi (ed. L. Motz, New York Academy of Science, New York, 1977) [6] V. Baluni, Phys. Rev. D 19 (1979) 2227.
384
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[7] R.J. Crewther, P DiVecchm, G. Veneziano and E. Witten, Phys. Lett. B 88 (1979) 123,B 91 (1980) 487(E); G. Veneziano, private communication. [8] S. Thdberge, A.W Thomas and G.A. Miller, Phys. Rev. D 22 (1980) 2838,D 23 (1981) 2106(E); A.W. Thomas, S. Th~berge and G.A. Miller, Phys. Rev. D 24 (1981) 216; S. Thdberge, G.A. Miller and A.W. Thomas, Can. J. Phys. 60 (1982) 59; S. Th~berge, Ph.D. thems, University of British Columbia (1982); A.W. Thomas, Adv. Nucl. Phys. 13 (1983) 1; G.A. Miller, in- Internationalreview of nuclear physics, ed. W. Weise (World Scientific, Singapore, 1984). [9] M.M. Musakhanov and Z.Z Israilov, Phys Lett. B 137 (1984) 419. [ 10] E.M. Henley, private commumcation. [ 11] G.A. Crawford and G.A. Miller, Phys. Lett. B 132 (1983) 173. [ 12] M.A. Morgan and G.A. Miller, Phys. Rev. D 33 (1986) 817. [ 13 ] M.A. Morgan, Ph.D. thesis, Unwersity of Washington (1984). [ 14] R. Peccei, private communication.
PHYSICS LETTERS B
30 October 1986
T H E N E U T R O N E L E C T R I C D I P O L E M O M E N T IN T H E C L O U D Y BAG M O D E L Michael A. M O R G A N Department of Phystc~, Seattle Unwerstt~, Seattle, WA 98122, USA
and Gerald A M I L L E R lnstttute for Nuclear Theory, Department of Physics, FM-15, l]mverstty of Washington, Seattle, WA 98195, USA Received 14 July 1986 An evaluation of the neutron electric dipole moment (NEDM), using the cloudy bag model (CBM) shows that two CPviolating effects (a quark mass term and a p i o n - q u a r k interaction) have contributions that are about equal in magnitude, but opposite in sign. This cancellation allows the upper limit on the 0 parameter to increase by about an order of magnitude.
The experimental upper limit 5 × I0 -25 e c m [1] on the neutron electric dipole moment (along with the assumption of invariance under CPT) places strong limitations on theories of CP violation [2]. Many possible sources o f such CP violation have been suggested [2,3]. Here we restrict out attention to the "strong" CP violation present m the fundamental QCD lagrangian describing the strong interactions between quarks within the neutron [4]. It has become a standard textbook exercise to show that the term
-0
g2 F ' F * ,
(1)
32n 2 present in the QCD lagrangian because of the topological properties of the vacuum, can be transferred to the quark sector through an appropriate global chiral rotation of the quark fields. The result is an additional interaction between quarks
5 ~CP = iOmCt75q
(2)
which violates b o t h P and T. Here ~ -= mumdms/ (mum d + mum s + mdm s) --~ 5.3 MeV if we take the MIT bag value o f m s = 300 MeV and use the ratios md/m u = 1.8 and ms/m d = 20 as determined by chiral perturbation theory [5]. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A calculation o f the NEDM (Dn)within the framework of the MIT bag has previously been done using (2) as a perturbation and estimating the contributions of the N*(1535) and N*(1700) intermediate states [6]. The result D n = 2.7 X 10-160 e cm
(3)
was obtained. Since the bag radms R is the only length scale xn the MIT bag model, simple &menslonal analysis leads to the result that D n is proportional to ~ R 2. This means that the numerical value o f D n is quite sensitive to the bag radius A value o f 1.4 fm was used in obtaining eq. (3). However, there is a bigger problem. We do not believe that the calculation leading to eq. (3) is vahd. This is because the interaction of eq. (2) does not connect two quark orbitals with positive energy, as we show below. Thus the simple, N*(qqq) excited states used should not contribute to D n. Nevertheless, we recompute this term below and find that the magnitude of this term is in agreement with eq. (3). Another calculation of the NEDM has been based on an effectwe CP-violating 7rN interaction derwed from eq. (2) [7]. These authors use current algebra to show that D n = g . N N ~ N N ln(mN/mr)/4~r2m N
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Volume 179, number 4
PHYSICS LETTERS B
m the limit that m~r becomes zero. Here glrNN = 0.0270 is the CP-violating coupling constant computed from the SU(3) mass splittings between the E and Z. The authors obtain the estxmate D n ~ 3.6 × 10 -16 0 e cm. This result is due solely to 7rN intermediate states which are the only ones used in an expression (eq. (14) of ref. [7] ) in which a sum over all states appears. There is also a large uncertainty as to what mass to insert in the numerator of the logarithm and therefore this approach can only give an order of magnitude estimate for D n. These two calculations may be only parts of a more complete calculation since both pionlc and quark terms should be included. To do this one needs to calculate the NEDM in a model which simultaneously incorporates the effects of the neutron's quark core and pion cloud. In the last few years chiral bag models have been successfully employed in the computation of static nucleon properties. In this letter we report on the computation o f D n using the cloudy bag model (CBM) [8]. The previous calculations [6,9] o f D n make it clear that both effects, considered alone, can be sigmficant. Further, if cancellation between these two contributions did occur [7], then a dimmution of D n for a fixed value of 0 would result. Alternatively, one could raise the value of 0 needed to allow a calculated NEDM to be consistent with the enormously small experimental upper bound on D nFirst we evaluate the contribution to D n from the quark core. In the CBM the quark fields are given by their mode sums
30 October 1986
places. Here we shall only need the lowest positiveenergy mode (1S1/2), (xl 1 , - I , ~, m) = ~ 1,-1,1/2,m (X)
-
N [ ]0 (wr/R) ~ (i]l(cOr/R)~. f] Xm
X exp(-iEt) O(R - r)
with energy E = co/R and eigenfrequency 6o = 2.04 determined by the boundary condition in eq. (7). The normalization constant is N = w/2R3(w - 1)]2(6o). To lowest order in the perturbation of the neutron wave function we have D n = ( n [ i . d i n ) = 2 ~ (NI~ "dlX)(XI6LcpIN) X EX _ EN
qa(x)
= ~l [~l(X)bal + "~t(x)d~l],
where c~labels the type of quark and l - n , k , j , m the bag mode; b destroys quarks in positwe-energy states ~l and d ÷ creates anti-quarks in negative-energy states qJl" The bag states satisfy the massless Dlrac equation reside the bag of radius R,
oz.pll)=Elll)
(6)
6Lcp = i~O f
d3x ~(x)3"sq(x):
- i t "~ It) = II)
(7)
at the surface o f the bag. Here ¥ are the Dlrac gamma matrices (Blorken and Drell convention) and at = 3'0¥. The complete set of wave functions is given in many 380
(10)
given by eq. (2). The dipole operator d = fxp(x) d3x receives contributions only from the quark core charge density (11)
where Q is the quark charge matrLx. Substitution of eq. (5) into eqs. (10) and (11) results in the expression D -- 2ir~0e ~
n
Qc~
X Ex~E N
× (OIx31J)(m[FIn)(NJb+~ib(~ib;mb~n{N) + (mlFl'ff)(TIx311) (Ic~ibcqbomdcnlN) N d + + )
and also obey the boundary condition
(9)
for the component of the NEDM along the spin of the neutron. IN) is the MIT state for the ground-state neutron with spin up, IX) is an arbitrary intermediate state of odd parity and 6Lcp is the CP-violating interaction
p(x) = e Ft(x)3"°Qq(x) :
(5)
(8)
(12)
where repeated indices are summed and the tilde indicates a state of negative energy. Here Qa --- Qaa, F 703'5 and the single-particle matrix elements are defined as usual by (110 Ij) - f d3x ~i~ (x)O (x) ~j(x). From eq. (6) and the fact that ~ and F anti-commute,
Volume 179, number 4
PHYSICS LETTERS B
it follows that if In) is a bag state of (positive) energy E n then Pl n) is a bag state of (negative) energy - E n ,
with the boundary condition for both given by eq. (7). Because of this Fin) is orthogonal to In), and to any positive-energy state Ira). Therefore, the first term in eq. (12) vanishes and intermediate states with all quarks in positive-energy states do not contribute to • 1t h i s piece o f D n. In particular, the odd parity : excited states of the nucleon are not mixed into the wavefunction by 6 L c p . However, eq. (12) does allow intermediate q~ fluctuations which result in a non-zero contribution to D n. Using I'fi')= Pl n) and keeping only the connected diagrams one finds the second term in eq. (12) to give
30 October 1986
It is interesting to note that the magnitude of this result is in rough agreement with the calculation of Bahim [6] when his value is scaled down to R = 1 fm, but our sign is opposite to his. Our result in eq. (15) is exact in lowest-order perturbation theory if one ignores center-of-mass motion of the bag. Next we compute the contribution to D n from the pion cloud. The CBM is used to account for the strong interaction between pion and quarks. An effective CPviolating pion-quark interaction will be induced by eq. (2). The only possible P-violatlng interaction terms linear in the plon field are [10] Z? 1 - ~Trq,
~22 - r:l(~,u0uTr)q,
~o3 ~ ~(ouv0uOvTr)q , Dn = "u'~Oe 3E ((~lx31 f ) _ (~tx31¢))
(13)
corresponding to an intermediate state with an extra quark and anti-quark in the 1S1/2 and 1S1/2 states respectively (see fig. 1). The energy denominator is E X - E N = 2E = 2 w / R . The arrows refer to the quark spin. The matrix elements are (+1 ix3l+l ) = ~_iRft°
(14)
where f w = [360(60 -- 1) sin2w]-1 1
X [~co(1 + ~ cos 260) - ~ sin 2co1. Therefore the quark core contribution is D(quark)n = -- 2 (fco/w)~R 20e - 1.8 × 10-160 e cm
(15)
at a bag radius o f R = 1.0 fm.
~4 - 7q(~ Xz)3q.
(16)
Of these, only Z? 1 is acceptable. Integration by parts shows .122 to vanish because of conservation of quark probability current. "~3 = 0 because ouv is antisymmetric. "~4 does not vanish but is even under T and therefore cannot give a NEDM. Since QCD is isospin invariant we shall assume (17) qq~r for our effective lagrangian. The coupling constant g x can be computed by comparison withg~rNN in eq. (4). With the inclusion of pions we must consider contributlons to D n from both the quark and plon charge densities. For now let us only consider the contributions due to intermediate states with all quarks remainmg in the ground state 1S1/2. We expect this to be the dominant contribution since large energy denominators will tend to suppress contributions from excited quark states. Furthermore, effects due to the finite size of the plon have been shown to reduce the contributions from these intermediate excited states in calculations of other static nucleon observables [ 11 ]. Our interaction hamlltonian at the nucleon level is .12 CP = g x ~ t ' c ' ~ q
HI = HCBM + Hx =
-f(~c~M + ~
CP
qqrr) d3x
(18)
where "~CBM = q 7 5 ~ " n @ (r - R )/21f7 r is the standard CBM interaction [8], and £? qqn C? is given above. Projection of eq. (18) into the subspace of colorless baryonic bags gives Fig. 1. Quark contribution to the NEDM. Only the 1S1/2 negative-energy intermediate quark state contributes• 381
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3
i=~ fd
k A,B
* (19) ~j %}
30 October 1986
Proceeding with the calculation we use perturbation theory. One may write D n = (nl£ "din>
where ak/destroys a pion with momentum k and isospin component ], A destroys a baryon bag [A), and the sum runs over all baryons m the octet and decuplet. The vertex functmn contains a strong p~ece and a CPviolating piece,
The CP-violating vertex is
v~) AI = _gx TAB
(20)
Ux(kR)
X 4(2rr)32CokSSASB6SAsB(TBtBlnlTAtA)(e])2 where S(s) and T(t) stand for the spin and xsospin of the baryon and n stands for the nth spherical component. T AB is a reduced matrix element of the baryon isospin operator and 1
.2 -J~)(c°X)Jo(kRx) dx Ux(kR) = N2R3 fJ x 2(Jo
(21)
0 is the CP-violatlng form factor. The coupling constant
gx is determined by comparison with the effective 7rNN interaction of Crewther et al. [7] at zero momentum transfer. The result is
gx =gTrNN/Ux(0) =gnNN/0"480 = --0.0560.
(22)
= (N[d3GoHIGoHIIN) + (N IHIGod3GoHIIN) + (N IHIGoHIGod3IN)
with H I given by eqs. (19), (20) and (23), and G O = (m N - H 0 ) - I is the free static-nucleon propagator. Now the dipole operator receives contributions from both charged quarks and charged plons. We need not consider the contribution from the quarks since by assumption they remain in the ground state and cannot produce a non-zero expectation of a P-odd operator. The charge density of the pion field is
O(x) = ie [c~t(x)~ (x) - c~(x)4)~ (x)]
oAB ki
u(kR )
mn
4(2~)326o k
d3k Ux(kR)
egx ( Dn =_~_~ 4fNN lim
n
×
q~O d(2n)J~
cok
d [ k;u(k'm
-
X (SBSB lmlSaSA)(TBtBln [TAtA)(k)m(e.l) * " n* (23) and the coupling constantsf AB have been given many places [8]. The CBM form factor is u(kR) = 3/'1 (kR)/ kR. Now since the CBM vertex conserves strangeness and the CP-violating vertex only connects baryons of the same spin, our intermediate states will be restricted to the nucleon. Therefore the only reduced matrix elements needed are T NN = (1 ~1 101~ ~2) -1 a n d f NN = ~fNN~r, where fNN~r = 0.28 lS the physical lrN coupling constant. 382
(25)
where ~(x) = [nl(X ) + in2(×)]/x/~is the field associated with the destruction of a negatwely charged plon as in ref. [12]. Insertion of eq. (19) into eq. (24) results in twelve terms, only half of which are non-zero. These are the interference terms with the strong coupling at one vertex and the CP-violatlng coupling at the other as shown in fig. 2. The non-interference terms are obviously zero since d 3 is odd under P. Using the expressions for the two vertex functions in eqs. (20) and (23) one finds that the total contribution from all six diagrams in fig. 2 is
For completeness the CBM vertex is
if AB
(24)
where q -= k' - k is the momentum transferred by the photon. Taking the limit and using the expression for u(kR) one finds
D n - e g xfNN y(Zll(Z2~_aZ " 1 2.12 (z) m~r 7r2 0
(z2 + o~2)3/2
lZ3f l (Z)
where a = rn~.R and I 1 and J2 are spherical lessel functions. The function u x is the CP-violating form factor
Volume 179, number 4
PHYSICS LETTERS B
30 October 1986
-¢ J" [
n ///
3O
/
P n
IIITI'-
x// E
25
20 X\
/
\
i
x
I
\\
0
/
d [O
/ /
05
/ /
l''
1
I
O4
06
_
I
08
I
1.0
I
I 2
4
R (fro) Fig. 2. Piomc contributions to the NEDM (quarks restricted to the ground state of the bag). A cross indicates the wolating interactmn.
CP-
given in eq. (21). A sphne fit to u x was made to simplify the numerical integral in (27). The results for mTr = 140 MeV a n d f NN = 3V'4-~fNN~r = 2.98 are given in fig. 3 as a function of the bag radius. At R = 1.0 fm D (pl°n) ~ 1.4 X 10 -16 0 e cm. n Notice the positive sign o f this result agrees with the current algebra results o f Crewther et al. [7] and that it is opposite to the contribution from the quark core in eq. (15). Also, we see that, whereas the quark contributlon IS proportional to R 2, the ploncloud contribution is a slowly varying function o f R . Both contributions are plotted as a function of the bag radius in fig 3. Here we see that at some radius just under 1 fm the two contrlbunons are equal so that D n is exactly zero, lrregardless o f the value o f 0. It is interesting to note that this value o f R <~ 1.0 fm lies within previous phenomenologlcal determinations. Thus the cancellation o f the two terms is a physically reasonable possibihty within the cloudy bag model. Since the quark-core contribution [eq. (15)] is entirely due to intermediate states with qV= 1 pairs one may wonder about the size o f contributions due to antiquarks in intermediate states of plon-cloud terms. We have computed these contributions [13] and found
Fig 3. The plonic (sohd) and the negative of the quark (dashed) contributions to the NEDM as a function o f bag radlu s, R.
that they only increase the results of fig. 3 by about 10% If the pion is treated as a point-like particle. Actually, contributions to the plon-cloud terms with Intermediate 1P1/2 quarks should be expected to be as large as these anti-quark contributions since they have comparable energy denominators. Our calculations show that these increase the results o f fig. 3 by about 17% [13]. These higher-order corrections to the pion-cloud contribution to D n are small and probably should be ignored since proper account of the pion structure will make them even less important. At most, these corrections slightly increase the value o f R for which D n is zero. It is tempting to use fig. 3 to conclude that by choosing an appropriate value of R , one can set the computed value o f the NEDM to zero, thereby allowing 0 to be as large as one likes. However, to specify the correct value o f R , one would have to know it to about SEXsignificant figures. This is clearly impossible since R is a model-dependent quantity, defined in a static cavity approximation. Indeed, one expects a variety of effects such as center-of-mass motion and various quantum fluctuations to provide corrections on the order o f ten percent. Thus we say that the bag radius is defined to within only one significant figure and any cancellation is Incomplete. One can only rea383
Volume 179, number 4
PHYSICS LETTERS B
sonably expect to gain (at most) a factor of ten m the upper limit of 0 allowed by the experimental upper limit on D n. One therefore finds that 0 is of the order of 1(3-8 instead of 10 -9 . This change may be relevant in allowing certain theories of CP violation (e.g. some of those m ref. [3]) to survwe [14]. This work is supported in part by the US Department of Energy, and is included in a 1984 University of Washington Ph.D. thesis [ 13]. We thank G. Veneziano and R. Peccei for useful discussions.
References [ 1] [2] [3] [4] [5]
J.M. Pendlebury et al., Phys. Lett. B 136 (1984) 327. N.F. Ramsey, Annu. Rev. Nucl. Part. Sci. 32 (1982) 211. S.M. Barr and A. Zee, Phys. Rev. Lett. 55 (1985) 2253. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. S. Wemberg, in. A festschrift for I.I. Rabi (ed. L. Motz, New York Academy of Science, New York, 1977) [6] V. Baluni, Phys. Rev. D 19 (1979) 2227.
384
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[7] R.J. Crewther, P DiVecchm, G. Veneziano and E. Witten, Phys. Lett. B 88 (1979) 123,B 91 (1980) 487(E); G. Veneziano, private communication. [8] S. Thdberge, A.W Thomas and G.A. Miller, Phys. Rev. D 22 (1980) 2838,D 23 (1981) 2106(E); A.W. Thomas, S. Th~berge and G.A. Miller, Phys. Rev. D 24 (1981) 216; S. Thdberge, G.A. Miller and A.W. Thomas, Can. J. Phys. 60 (1982) 59; S. Th~berge, Ph.D. thems, University of British Columbia (1982); A.W. Thomas, Adv. Nucl. Phys. 13 (1983) 1; G.A. Miller, in- Internationalreview of nuclear physics, ed. W. Weise (World Scientific, Singapore, 1984). [9] M.M. Musakhanov and Z.Z Israilov, Phys Lett. B 137 (1984) 419. [ 10] E.M. Henley, private commumcation. [ 11] G.A. Crawford and G.A. Miller, Phys. Lett. B 132 (1983) 173. [ 12] M.A. Morgan and G.A. Miller, Phys. Rev. D 33 (1986) 817. [ 13 ] M.A. Morgan, Ph.D. thesis, Unwersity of Washington (1984). [ 14] R. Peccei, private communication.