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Figure 10 – uploaded by Piyush Kumar

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Here we see immediately that in the limit w — —oo for the solution above en — 0. Therefore we conclude that the de Sitter extremum cannot be obtained from (72) which correlates with the previous observation that for A = 1 (72) has only one solution - the SUSY extremum. Nevertheless, as we will see in the next subsection the real solutions plotted in Figure 8 are a very good approximation to the exact numerical solutions corresponding to the AdS vacua with spontaneously broken supersymmetry. — Figure 8 Here we see immediately that in the limit w — —oo for the solution above en — 0. Therefore we conclude that the de Sitter extremum cannot be obtained from (72) which correlates with the previous observation that for A = 1 (72) has only one solution - the SUSY extremum. Nevertheless, as we will see in the next subsection the real solutions plotted in Figure 8 are a very good approximation to the exact numerical solutions corresponding to the AdS vacua with spontaneously broken supersymmetry.

Related Figures (30)

In the next section we will go on to analyze the vacua of this potential with unbroken supersymmetry. The vacua in which supersymmetry is spontaneously broken are described in sections V and VI. It turns out that in order for the potential (9) to have minima, the axions must take on the values such that cos[(by N! — byN?)-¢] =—1 for Aj, Ay > 0. Otherwise the potential has a runaway behavior. After choosing the minus sign, the potential takes the form

FIG. 1: Positive values of s; plotted as a function of a for a case with two condensates and three bulk moduli for the following choice of constants 6; = am be = 27 Ni = {1,2,2},N? = {2,3,5},a; = {1,1/7, 25/21}. The qualitative feature of this plot remains the same for different choices of constants as well as for different 7. The vertical line is the locus fot 2 bo N3 ‘ : a = +51, where the denominator of (20) vanishes. 14V5 to be very important as we will see shortly. From (21), we know that there are two branches for allowed values of a. Here we consider branch a) for concreteness, branch b) can be analyzed similarly. Figure (2) shows plots for Aj/A, and s; as functions of a for a case with two condensates and three bulk moduli. The plots are for a given choice of the constants {b,, bo, N}, N?,a;, i = 1,2,3}. The qualitative feature of the plots remains the same even if one has a different value for the constants.

FIG. 2: Left - A2/Ai plotted as a function of a for a case with two condensates and three bulk moduli. The function 2 diverges as it approaches the loci of singularities of (20), viz. a = oat Right - Positive s;, 1 = 1, 2,3 for the same case plotted as functions of a. s1 is represented by the solid curve, s2 by the long dashed curve and s3 by the short dashed curve.The vertical lines again represent the loci of singularities of (20) which the respective moduli s; asymptote to. The horizontal solid (red) line shows the value unity for the moduli, below which the supergravity approximation is not valid. Both plots are for bi = 27, be = 32, N} = {1, 2,2}, N? = {2,3, 5}, ai = {1, 1/7, 25/21}.

FIG. 3: Plots of positive s;,i = 1,2,3 as functions of A2/A1. Top Left: Same choice of constants as in Figure(2), i.e. bi = 24, bo = 2%, Ni = {1,2,2}, N? = {2,3,5}, ai = {1,1/7, 25/21}. 30° 29° Top Right: We increase the ranks of the gauge groups but keep them close (keeping everything else same) - b1 = os , bg = =# Bottom Left: We introduce a large difference in the ranks of the gauge groups (with everything else same) - b} = 34, be = Bottom Right: We keep the ranks of the gauge groups as in Top Left but change the integer coefficients to Nj = {1, 2, 2}, Ni = {3, 3, 4}.

FIG. 4: Plots of positive s;,i = 1,2,3 as functions of A2/Ai. The constants are bi 2m bo 2t Nj {1,0,1},.N? = {1, 1,1}, a; = {1/10, 1,37/30}. s1 is represented by the solid curve, s2 by the long dashed curve and s3 by the short dashed curve. The red curve represents the volume of the internal manifold as a function of A2/A1. Right - the same plot with the vertical plot range decreased. B. Special Case A very interesting special case arises when the gauge kinetic functions f; and fo in (5) are equal. Since in this case N} = N?, the moduli vevs are larger in the supersymmetric vacuum; hence this case is representative of the vacua to be found within the supergravity approximation. special case, in section IV, we will describe explicit examples of Gz manifolds in w In the special case. we have Even though this is a hich N} = N?.

FIG. 5: Plot of v as a function of A2/A1 for the choice 6; = =n bo = on The solid (red) curve represents the exac numerical solution whereas the black dashed curve is the leading order approximation given by (29). Therefore, the hierarchy between the moduli sizes is completely determined by the ratios a;/N; for different values of 7. In addition, from Figure 5 it can be seen that v keeps increasing indefinitely if we keep increasing A/A, (though theoretically there may be a reasonable upper limit for A2:/Aj), which is not possible for the general case as there are N v;’s. This implies that it is possible to have a wide range of the constants which yield a solution in the supergravity regime.

In addition, each solution in (52) is a function of the auxiliary variable a defined in (36). By substituting (52) into (36) we obtain two equations for a) and a®) By solving (53) numerically and keeping only those solutions that result in real positive values for the modulus s in (52) we get

FIG. 6: Potential multiplied by 10°? plotted as a function of one modulus s. For our particular choice of constants in (54), the modulus is stabilized at the supersymmetric AdS minimum s) = 26.101. The maximum is de Sitter, given by (2) _ ss? = 27.185. Let us now consider a particularly simple example with two moduli. Here we will choose both moduli to appear on an equal footing in the Kahler potential (1) by choosing

FIG. 7: Potential multiplied by 10°? plotted as a function of two moduli s; and sz for the values in (67). The SUSY AdS extremum given by (68) is a saddle point, located between the non-supersymmetric AdS minima given by (70).

Thus, we have the following two solutions for the moduli for the AdS vacua with broken SUSY To verify the approximation we can check it for the previously considered special case with two moduli when a; = a2 = 7/6, i.e. the case when A = 0. By solving (72) we obtain:

By combining (92) with (79) and (23) it is easy to obtain the corresponding expressions for the moduli that include the first subleading order correction:

FIG. 9: Potential in units of M3 2M, along the slice s1 = s2 for a manifold with two moduli with the meson field equal to its value at the minimum of the potential (134). The microscopic constants are as in (136). Although hard to see from the graph, the value of the potential at the minimum (i.e. the cosmological constant) is 0.194 m3 jgMp- where the volume of the associative three cycle Q is again

0 corresponding to the minimum value 6 = 27/4 we see that neither of the two solutions satisfy the bove requirement since both short-dashed curves corresponding to L A,— for the two solutions are always ositive for the entire range 0.877781 < A < 1, whereas both L A,+ and B 4 remain negative. Therefore or 6 = 27/4 and A < 1 there are no solutions for which all the moduli are stabilized at positive values. loreover, as parameter 6 is further increased, the range of possible values of A for which the system as two real solutions gets smaller and more importantly, the values of L A,— remain positive and only ncrease while both L A,+ and Ba remain negative, which can be seen from the right plot in Figure 10, vhere 6 = 8. This trend continues as we increase 6. Thus, we can make the following general claim: If the solution for A = 1 has a positive vacuum energy, Again, since the volume 7, is always positive, from (145) we see that for all moduli to be stabilized in the positive range, all three quantities L A,+) L A,— and Ba must have the same sign. For 5 = 27/4, the system (149) has two real solutions when 0.877781 < A < 1. However, from the left plot in Figure

FIG. 11: log; (ms3/2) as a function of P for case 1) - top plots and case 2) - bottom plots. The light grey area represents possible values of logy, (m3/2) in the range where perit < p < 10 consistent with the SUGRA approximation. The dark area indicates the region of interest where —16 < log, 9(mg3,2) < —14 such that 240 GeV < m3 2 < 24TeV. The dark solid curve corresponds to log;)(m3/2) when p = perit. The lower boundary of the light grey area represents the log, (73/2) curve when p = 10. The dashed curve corresponds to p = 1.01. Top left: Case 1) when P— Q = 1. Top right: Case 1) when P—Q = 3. Bottom left: Case 2) when P — Q = 1. Bottom right: Case 2) when P — @ = 2.

FIG. 12: log, o(ms3/2) as a function of P for case 1) - top plots and case 2) - bottom plots. The light grey area represents possible values of log9(™m3/2) in the range where ferit < P < 100 consistent with the SUGRA approximation. The dark area indicates the region of interest where —16 < log,)(m3/2) < —14 such that 240 GeV < mg3/2 < 24TeV. The dark solid curve corresponds to logi9(m3/2) when f = ferit. The lower boundary of the light grey area represents the log,9(m3/2) curve when # = 100. The dashed curve corresponds to 6 = 1.01. Top left: Case 1) when Q — P = 3. Top right: Case 1) when Q — P = 6. Bottom left: Case 2) when Q — P = 3. Bottom right: Case 2) when Q — P = 4. For the de Sitter vacua, we use a less restrictive upper bound p = a g < 100. For a manifold witl N = 100 moduli we obtain: ar

The gravitino mass (195) depends on three integers: the two gauge group dual coxeter numbers JN, -Q and the number of moduli N.’ This will give us an idea about the distribution of the gravitinc mass (which sets the superpartner masses) obtained after imposing the constraint (191) that the vacuurr energy is acceptable. Asides from only considering the vacua within the supergravity approximation (ie s; > 1) we expect an upper bound on the dual coxeter numbers of the hidden sector gauge groups P anc (). Based on duality with the heterotic string, it seems reasonable to assume that they can be at least as large as 30 - the dual coxeter number of Eg. Of course, values of P,Q larger than 30 cannot be rulec out, and here we assume an upper bound P < 100. Notice that from the constraint in (191), the ratic (Ai Q)/(A2P) can get exponentially large when Q — P = 3 and the values of P are small. Here we are voing to completely relax the requirement on the upper bound of the ratio because as we will see, fot generic manifolds with a large numer of moduli, P has to be large for the supergravity approximatior to hold. The distribution can be constructed as follows. The three integers: P, Q — P and WN are variec subject to (135) and the supergravity constraint s; > 1. For each point in the resulting two dimensiona subspace, logyg(™m3/2) can be computed and rounded off to the closest integer value. One can ther count how many times each integer value is encountered in the entire scan and plot the corresponding distribution. FIG. 13: The gravitino mass distribution with the x-axis denoting the logarithm of the gravitino mass (to base 10). Left: Distribution corresponding to scan one in (200). Middle: Distribution corresponding to scan two in (201) for which manifolds with the number of moduli N < 50 were excluded from the scan. Right: Distribution corresponding to scan three in (202) for which manifolds with the number of moduli N < 100 were excluded from the scan.

FIG. 15: The gravitino mass distribution with the x-axis denoting the logarithm of the gravitino mass (to base 10). Scans for the smallest possible choice (Q — P)min = 3. Left: Distribution corresponding to the scan with Pnaz = 200. Middle: Distribution corresponding to the scan with Pnaz = 100. Right: Distribution corresponding to the scan with Pmaz = 30. In all three plots in Figure 15 we see the same peak at m3/2 ~ O(1 — 100) TeV, which narrows around M3 2 ~ O(100) TeV as Prax is decreased.

FIG. 16: Plot of log,)(m3/2) as a function of P for Q — P = 3. Short-dashed curve corresponds to N = 50. Long-dashed curve corresponds to N = 500.

FIG. 17: Plots of the potential at the barrier in units of MM3 19 as a function of P. Solid line corresponds to 422 = 100 AiQ while the dashed line corresponds to a5 = 10. Left: Plot for P= Q+3. Right: Plot for P=Q+1.

FIG. 18: Potential in units of M3 joM> plotted as a function of the meson field and the moduli along the slice s1 = s2, for a manifold with two moduli and the microscopic constants as in (136).

FIG. 19: Left: Potential in units of M3 )oMz along the slice s1 = s2 for a manifold with two moduli with the meson field equal to its value at the minimum of the potential (as in (137)). Right: Potential in units of M3)2M> along the slice s1 = s2 for a manifold with two moduli with the meson field ¢ = 0.102 (this is such that the height of the potential barrier is at its minimum). The microscopic constants for both cases are the same as in (136).

FIG. 20: Absolute values of (M12) left and (My 2) 2) -right in units of gravitino mass as functions of g. As parameter A varies over 0 < A < 1/7 - on the left and 0 < A <1 - on the right and the whole light grey region is covered. Left plot: A = 0- long dashed line, A = 1/9 - solid line, A = 5/36 - short dashed line. Right plot: A = 0- long dashed line, A = 0.5 - solid line, A = 0.95 - short dashed line. Note that the general formula (255) which relates the gravitino and gaugino masses is completely independent of the number of moduli.

hierarchy between Mj/2 and m3/ in the Type IB flux vacua [11], where the gaugino mass is generically suppressed by In (ms /2): For the special case A = 0, system (72) yields two solutions with positive moduli. Therefore, there will be two different values for the gaugino mass corresponding to these solutions. After some algebra we obtain Again, this relation is valid for any AdS vacuum with broken SUSY with A = 0 with an arbitrary number of moduli. hierarchy between Mj /2 and m3 in the Type IIB flux vacua [11], where the gaugino mass is generically suppressed by In (m3 72).

We first compute the F-term contributions where 0% implicitly depends on the moduli. However, it is much simpler to keep the dependence as a function of 6%, as is explained in the Appendix. Depending on the values of the Casimir invariants Ca and C* for the three gauge groups, the anomaly mediated contribution can either add to or cancel the tree level contributions. Here we also took into account that m3/2 = eK/? || Ape” but & < 0, since Q—-—P > 3. Using the expression for Li in (129) along with the definitions of %, y and Z in (114) in terms of @ in (133) and assuming that Q — P ~ O(1), in the limit when P is large we obtain where C, and C® are the casimir invariants of the a‘” gauge group and a runs over the number of fields charged under the a‘” gauge group. For a given spectrum such as that of the MSSM, C, and C® are known.

where, in the large vy approximation, K,, Ko, K3, K4 are given by

Next, we normalize the above vectors and construct an orthogonal (N +1) x (N +1) matrix We can diagonalize the matrix (314) in two steps. We first construct a set of orthogonal (unnormalized) N +1 vectors given by

Hence, the first N — 1 eigenvalues are given by Diagonalizing the 2 x 2 block is then trivial and boils down to solving a simple quadratic equation. Sinc for typical values of Q and P we have

With these identifications, and using (A6), we can write the individual Gz moduli in terms of the IIA moduli: which is the same as (A10). So, for the particular case, this suggests the following generalization:

For more general G2 manifolds with many moduli, the precise map of the individual moduli is not completely clear. However, it seems plausible that the complex structure moduli appearing in the Kahler metric in IIA map to a subset of the Gy moduli in a similar way, as in the particular case.

FIG. 21: Left: F as a function of 0%. Middle: G as a function of 6%. Right: H as a function of 0% (for 1=1). From [28], On € [0, 1). The dependence of the soft parameters in section VIIIB on 6% is extremely simple. Instead of re- expressing the dependence on 9° in terms of the moduli, it is much more convenient to retain the dependence on 0¢, as 0° € [0,1) [28] and the variation of relevant functions with 0° in the allowed range can be plotted easily. In particular, the function F(0%) = +{wW® sin(270¢)} appears in the ex- pression for the anomaly mediated gaugino masses (eqn. (279)) and the trilinears (eqn. (287)), the function G(@¢) = In (4 “rant) appears in the expression for the trilinears (eqn. (287)) and the function H(02) = F{P? ve, sin?(270%) + 2? pe sin(47%) — 21% sin(2702)} appears in the expression for the scalars (eqn. (291)). The variation of these functions with 6¢ is quite mild as seen from Figure A 2: Since the functions F, G and H vary very mildly with 0¢ and are O(1) in the whole range, it is reasonable to replace the above functions by O(1) numbers, as is done in section VIIIB. This is justified

Related topics:

Mathematical Physics Physics Theoretical Physics Quantum Physics Particle Physics Supersymmetric Models De Sitter Effective Potential

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